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United Nations University – New York City, United States
Date added: March 19, 2018 Application deadline: April 13, 2018 Contract type: fixed-term Hours: full-time Salary: In accordance with the relevant UN salary scale
For the past four decades, the United Nations University (UNU)has been a go-to think tank for impartial research on the pressing global problems of human survival, conflict prevention, development, and welfare. With more than 400 researchers in 13 countries, UNU's work spans the full breadth of the 17 Sustainable Development Goals, generating policy-relevant knowledge to effect positive global change. UNU maintains more than 200 collaborations with UN agencies and leading universities and research institutions across the globe.
About UNU Centre for Policy Research (UNU-CPR)
UNU's Centre for Policy Research in New York is an independent think tank within the United Nations system. We combine research excellence with deep knowledge of the multilateral system to generate innovative solutions to current and future global public policy challenges.
This work has been recognised and directly incorporated into key UN reform processes, including the Review by the High-Level Independent Panel on Peace Operations (HIPPO), the Review by the Advisory Group of Experts on the Peacebuilding Architecture, and the process leading up to the World Humanitarian Summit. UNU-CPR is staffed by recognised experts in key areas related to the peace and security, sustainable development, and human rights pillars of the Organisation, with recognised expertise in anti-slavery, anti-trafficking, global drugs policys and migration. Established in 2014 in Tokyo, in July 2018 UNU-CPR will merge with the UNU Office at the United Nations (UNU-ONY) in New York, and operate from New York.
The Chief Operations Officer (COO) will work closely with the Director of the Centre for Policy Research (CPR), serving as the Director's primary deputy on all finance, administration, human resources, and operational matters and managing personnel responsible for the back-office support functions across the Centre. S/he will also play a representational role.
Key roles and responsibilities
Representational role
The COO will serve as the primary deputy for the Director in representing the University into the UN system. Under the Director's supervision and direction, the COO may:
Serve as a strategic liaison between UNU Institutes and researchers, including the UNU Migration Network, and ongoing intergovernmental and interagency discussions at the UN in New York, notably the negotiation of and implementation of the Global Compact on Migration;
Represent UNU in formal appearances before the Economic and Social Council, Advisory Committee on Administrative and Budgetary Questions, and General Assembly's Fifth Committee;
Working with the Advisor to the Director, oversee the coordination of UNU inputs into cross-UN reporting;
Substitute for the Director in participation at the UNU Academic Council of Directors, UNU Council, or otherwise; and
Represent UNU-CPR at official and informal meetings, events, receptions, conferences, and workshops
Operational role
Working under the supervision and oversight of the Director, the COO will:
Manage personnel responsible for the back-office support functions across the Centre, including finance and administration, procurement, and human resources support; event management; and communications management;
Working closely with UNU Centre's Finance & Administration team, oversee the daily administration of all financial, accounting, and budgetary matters across the Centre's budget, including contract negotiation and management, legal review and sign-off; procurement workflow, including approval of requisitions and purchase orders; cash flow, accounts, and budget monitoring and reporting: and exercising delegated financial authority, if any;
Ensure effective administration of the UNU-CPR office(s), including stocking, inventory, reception, telephony, lease management, and any related matters;
Working closely with UNU Centre's Human Resources team, and with UNDP Human Resources personnel as needed, be responsible for the daily
administration of all HR-related functions at UNU-CPR, including personnel contract management; planning, approval, and recording of leave entitlements;
Administration of other entitlements, including insurance coverage; and performance appraisal processes;
Supervise UNU-CPR's Communications Manager and Events Manager, and ensure effective integration of UNU-CPR communications and events with its research functions and execution of UNU-CPR strategy;
Provide support to the Director in preparing and pitching fundraising proposals; and
Perform any other related operational duties as required for the effective functioning of UNU-CPR.
Experience and skills
Required qualifications and experience include:
An advanced university degree (master's or equivalent) in public policy, political science, or the humanities
At least five (5) years of progressively responsible experience in international relations, political affairs, conflict resolution, or economic development
Knowledge of and strong networks within UN institutions and policymaking, especially as they relate to multilateral negotiation and migration
Proven excellence in writing, editing, and communication
Experience providing close support to senior members of a think tank, UN entity, or research organisation
The ability to interact with colleagues and others of diverse cultural backgrounds
Fluency in English (fluency in at least one other official language of the United Nations is desirable)
Excellent team player with strong interpersonal skills and ability to work in a multicultural environment, with demonstrated sensitivity and respect for diversity and gender
UNU is committed to building a pluralistic and culturally diverse personnel. Applications from suitably qualified women candidates are particularly encouraged.
The position is a P-3 fixed-term appointment and will be remunerated in accordance with the relevant UN salary scale. The post carries the standard set of United Nations benefits and entitlements for international positions in the UN Common Systems, including participation in the United Nations Joint Staff Pension Fund, the possibility of participation in a health insurance programme, education grant for eligible children, removal expenses, and home leave.]
Interested applicants should submit their applications by email to [email protected] and must include the following:
A cover letter setting out how the qualifications and experience match the requirements of the position
A single piece of published writing solely authored by the candidate, demonstrating their analytical capacities on issues pertaining to global development
A completed and signed UNU Personal History (P.11); please avoid using similar forms provided by other United Nations organisations
An updated curriculum vitae
Full contact information of three (3) referees
An indication of the reference number of the vacancy announcement (2018/UNU/CPR/FTA/COO/21)
The United Nations shall place no restrictions on the eligibility of men and women to participate in any capacity and under conditions of equality in its principal and subsidiary organs. (Charter of the United Nations – Chapter 3, article 8)
Apply for this job see all Job Postings
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## GATEWAYS
## ALSO BY F. PAUL WILSON
Repairman Jack Novels:
The Tomb
Legacies
Conspiracies
All the Rage
Hosts
The Haunted Air
Healer
Wheels Within Wheels
An Enemy of the State
Black Wind
Soft & Others
Dydeetown World
The Tery
Sibs
The Select
Implant
Deep as the Marrow
Mirage (with Matthew J. Costello)
Nightkill (with Steven Spruill)
Masque (with Matthew J. Costello)
The Barrens & Others
The Christmas Thingy
Sims
The Adversary Cycle:
The Keep
The Tomb
The Touch
Reborn
Reprisal
Nightworld
Editor:
Freak Show
Diagnosis: Terminal
## GATEWAYS
A Repairman Jack Novel
## F. PAUL WILSON
The author and publisher have provided this e-book to you without Digital Rights Management software (DRM) applied so that you can enjoy reading it on your personal devices. This e-book is for your personal use only. You may not print or post this e-book, or make this e-book publicly available in any way. You may not copy, reproduce or upload this e-book, other than to read it on one of your personal devices.
**Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com/piracy.**
for Daniel and Quinn
## AUTHOR'S NOTE
Thanks to the usual crew for their editorial help with the manuscript: my wife, Mary; my editor, David Hartwell; his assistant, Moshe Feder; Elizabeth Monteleone; Steven Spruill (who also allowed me to tap into his store of knowledge about the Korean War); and my agent, Albert Zuckerman.
Thanks, too, to the many friendly South Florida folk and air-boat pilots who helped me along the way, especially the rangers at the Royal Palm and Shark Valley Visitor Centers in Everglades National Park who introduced me to the amazing diversity of wildlife in the Glades.
Special thanks to Stuart Schiff for being so generous with his fabulous single malts, and to Blake Dollens for his keen eye.
Finally, thanks to NY Joe (Joe Schmidt) and Angel (Janada Oakley) for advice on the weaponry. I did a little improvising along the way, so any errors in that area are my own.
## Contents
TUESDAY
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
WEDNESDAY
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
THURSDAY
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
FRIDAY
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
SATURDAY
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
SUNDAY
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
TUESDAY
Chapter 1
AFTERWORD
## GATEWAYS
## TUESDAY
## 1
Blessed be the blackmailers, Jack thought as he pawed through the filing cabinet.
He had a penlight clamped in his teeth and kept it trained on the labels of the hanging folders while his latex-gloved fingers fanned through them.
What a trove. If someone could be called a professional blackmailer, Richie Cordova fit the bill. Private investigation was his legitimate line, if such a line could be legit. But apparently he dug up lots of additional dirt during the course of his investigations, and put that to work for him. Never against his clients, Jack had learned. Did his blackmailing anonymously. That kept his professional rep clean, kept that stream of referrals from satisfied clients flowing. But Jack had picked him up on a money drop Cordova had set up for his latest fish and took an instant dislike to the fat slob. Nine days of shadowing him hadn't mellowed that initial impression. The guy was a jerk.
Cordova's PI office occupied a second floor space over an Oriental deli on the other side of Bronx Park. But his other line of work, probably the more profitable one, was here on the third floor of his house. Small and stuffy, furnished with the filing cabinet, a computer, a high-end color printer, and a rickety desk, it appeared to be a converted attic.
Where was the letter? Jack was counting on it being in this cabinet. If not—
There...Jank. Could that stand for Jankowski? He pulled out the file and opened it. Yep. This was it. Here was the handwritten letter at the root of Stanley Jankowski's problems. Cordova had found it and was using it to squeeze the banker for all he was worth.
Jack tucked it in his pocket.
Yes, blessed be those blackmailers, he thought as he began emptying the folders from both drawers of the cabinet and dropping their contents—letters, photos, negatives—onto the floor, for they help keepeth me in business.
Blackmail was the reason a fair percentage of Jack's customers came to him. Stood to reason: They were being blackmailed because they had something they wanted kept secret; couldn't go to officialdom because then it would no longer be a secret. So they were left with two options: pay the blackmailer again, and again, and again, or go outside the system and pay Jack once to find the offending photos or documents and either return them or destroy them.
Destroying was better and safer, Jack thought. But untrusting customers feared Jack might simply use the material to start blackmailing them on his own. Jankowski had been burned and wasn't about to trust no one no how no more. He wanted to see the letter before he paid the second half of Jack's fee.
Jack spread the two drawers' worth of photos and documents on the floor. A small, voyeuristic part of him wanted to sit and sift through them, looking for names or faces he recognized, but he resisted. No time. Cordova would be back in an hour.
He pulled a pair of glass Snapple bottles out of his backpack and unwrapped the duct tape from around their tops. He was about to do a big favor for some of the people in that pile. Not all. Cordova had probably scanned all this stuff into a computer and had digital copies stashed away somewhere. But a scan couldn't sub for a handwritten letter. Cordova needed the original, with its ink and fingerprints and all, to have any real leverage. A copy, no matter how close to the original, was not the real deal and could be dismissed as a clever fake.
He looked down at the pile of damning evidence. Some of these folks were about to get a freebie. Not because Jack particularly cared about them—for all he knew, some of them might deserve to be blackmailed—but because if he took just the Jankowski letter, Cordova would know who was behind this little visit. Jack didn't want that. With everything destroyed or damaged beyond repair, Cordova could only guess.
Burning the pile would have been best but the guy lived in a tight little Williamsbridge neighborhood in the upper Bronx. Lots of nice, old, post-war middle-class homes stacked cheek by jowl in a neat grid. If Cordova's place burned, it wouldn't burn alone. So Jack had come up with another way.
He held one of the Snapple bottles at arm's length as he unscrewed the cap. Even then the sharp odor stung his nose. Sulfuric acid. Very carefully—this stuff would burn right through his latex gloves—he began to sprinkle it on the pile, watching the glossy surfaces of the photos smoke and bubble, the papers turn brown and shrivel.
He'd used up most of the first bottle and the room was filling with acrid smoke when he heard the front door slam three floors below.
Cordova?
Checked his watch: about a quarter past midnight. In the past week or so that Jack had been shadowing him, Cordova had hit a neighborhood bar over on White Plains Road three times, and on each night he'd hung till 1 A.M. or later. If that was Cordova downstairs, he was home at least an hour early. Damn him.
Dumped the rest of the acid from the first bottle and sloshed the contents of the second over the pile, then left them atop the filing cabinet. Now to get out of here. Wouldn't be long before Cordova detected the stink.
Opened the window and slid out onto the roof. Looked around. He'd planned on leaving as he'd entered—through the back door. Now he was going to have to improvise.
Jack hated to improvise.
Looked over at the neighboring roof. Pretty close, but close enough to...?
Through the open window behind him he heard Cordova's heavy feet pounding up the stairs. Another glance at the neighboring roof. Guessed it was going to have to be close enough.
Hauling in a deep breath, Jack took three running steps down the shingled slope and leaped. One sneakered foot, then the other, landed on the opposing roof and found traction. Without pausing to congratulate himself, Jack used his forward momentum to keep going, his rubber soles slipping and scraping up the incline toward the peak.
A loud, whiny "Noooooo!" followed by a bellow of rage and dismay echoed from Cordova's house, but Jack didn't turn to look—didn't want Cordova to see his face. Then he heard a shot and almost simultaneously felt the slug zing past his ear.
Cordova had a gun! Jack had figured he'd have one somewhere, but hadn't expected him to shoot up his own neighborhood. Two miscalculations tonight. He hoped he hadn't miscalculated on getting home alive.
Dove over the peak of the roof and slid down toward the gutter, the shingles shredding the palms of his latex gloves and wearing away the front of his nylon windbreaker like an electric sander. Halfway to the gutter he slowed his slide and angled his body ninety degrees. That slowed him a little more. Further angling around allowed him to get his foot in the gutter and stop altogether.
Not home free yet. Still two stories up with Cordova no doubt pelting down his stairs and heading for the street. Plus this house was occupied, probably with two families, since that seemed the rule around here. He could see the glow of lights turning on inside. He was sure the owners were dialing 9-1-1 right now to report the racket on their roof. Probably thought he was a clumsy second-story burglar.
Jack peeked over the gutter and positioned himself over a dark window. Slid off the roof feet first and belly down, easing his weight onto the gutter. It groaned and creaked and sagged as he hung by his fingers. Before it could give way he managed to place his feet on the windowsill and let that take his weight. Eased himself into a crouch to where he could grip the sill with his hands, then dropped again. He clung to the sill only a second or two, poising his feet a mere six feet off the ground, then let go. He twisted in the air and hit the ground running.
His sneakers made no sound as he sprinted along the sidewalk. He bent as low as he could without compromising his speed and waited for a second shot. But none came. Took a left at the first corner and a right at the next and kept running. At least now he was out of the line of fire—if Cordova stayed on foot. But if he got into a car and started cruising...
Plus, cops should be on their way.
What a mess. This was supposed to be a simple in-and-out job with no one the wiser until later.
Kept moving in a crouch, watching the passing cars, on alert for flashing lights. Slipped out of his partially shredded windbreaker—he was wearing a WWE Lance Storm T-shirt beneath—and pulled the Mets cap from the pocket. Jammed the cap on his head and bunched the jacket into a nylon lump the size of a softball. Palmed that and slowed to a speedy walk.
Slowed further when he hit 232nd Street. Stuffed the windbreaker down into a trash receptacle as he walked to the elevated subway station on 233rd. Caught the 2 train and settled down for a long ride back to Manhattan.
He patted the letter folded in his jeans pocket. Another problem fixed. Jankowski would be happy, and Cordova...
Jack smiled. Fat Richie Cordova had to be fuming as much as the sulfuric acid on his photos and papers.
## 2
A man who was something more than a man crouched among the foundation plantings of a two-story house in a quiet Connecticut community. He moved through the world under different guises, using different names, but never his own, never his True Name. And as he traveled, doing what must be done to prepare the way, he searched out places such as this family home.
He sat with his spine and the back of his head pressed against the house's concrete foundation. Someone coming upon him might have thought he was an indigent sleeping off a bender. But he hadn't been sleeping. He required very little rest. He could go for days without closing his eyes.
And even if this had been one of those rare occasions when he needed rest, he would have found sleep impossible while basking in the exhilarating emanations from the basement of this house.
On the other side of the wall...systematic torture, mutilation, and defilement. The victim wasn't the first so abused by this family of three, and would not be the last. Or so the man who was something more than a man hoped.
What the two adults within had done to the ones they'd captured and imprisoned over the years would have been sustenance enough for this man. But the fact that they had debased their own child and made him a willing participant in the systematic defilement of another human being...this was exquisite.
He flattened his back more firmly against the wall, drinking, feasting...
## 3
After stopping at Julio's for a couple, Jack fell into bed when he got home. Jankowski could wait till morning for the good news.
Somewhere around 3 A.M. the ringing of the front-room phone dragged him from slumberland. The answering machine clicked on and out came a voice he hadn't heard in fifteen years.
"Jackie. This is your brother Tom. Long time no see. I assume you're still alive, though it's hard to tell. Well, anyway, Dad was in a car accident earlier tonight. He's in pretty bad shape, in a coma, they tell me. So give me a call, prontissimo. We need to talk."
He rattled off a number with a 215 area code.
Jack had been up and moving at the mention of his father's accident, but didn't reach the receiver in time to pick up. He stood over the phone in the dark.
Dad? In an accident? In a coma? How the hell—?
Unease trickled through his gut. The past he'd cut himself off from was worming its way back into his life. First he runs into his sister Kate last June, and a week later she's dead. Now, three months after that, he hears from big brother Tom that his father's in a coma. Was he detecting a scary symmetry here? A pattern?
Deal with that later, he told himself. First find out what happened to Dad.
Jack replayed the message, writing down the phone number. He used his Tracfone to return the call. That same voice answered.
"Tom? Jack."
"Well, I'll be. The long lost brother. The prodigal son. He lives. He returns a call."
Jack didn't have time for this. "What's the story with Dad?"
Jack had never particularly liked his brother. Hadn't disliked him either. They'd never had any sort of a relationship growing up. Tom—Tom, Jr., officially—was ten years older and seemed to have viewed his little brother as an inconvenient pet, one that belonged to his parents and his sister but had nothing to do with him. He'd always been self-involved to a fault. Kate had said he was on his third wife and hinted that the latest marriage was headed for the same fate as his others. Jack hadn't been surprised.
Tom had been a Philadelphia lawyer for a couple of decades and was now a Philadelphia judge. Which meant he was an officer of the court, a cog in the wheels of officialdom. All the more reason for Jack to keep his distance. Courts gave him the creeps.
"Pretty much what I told you. I got a call from this nurse at the Novaton Community Hospital that Dad was involved in an MVA and—"
"M-V—?"
"Motor vehicle accident—and that he's in bad shape."
"Yeah. A coma, right? Jeez, what do we do?"
"Not we, Jackie. You."
Jack didn't like the sound of this. "I don't get you."
"One of us has to go down there. I can't, and since Kate's not exactly available, that leaves you."
"What do you mean, you can't?"
"I—I'm in the middle of a bunch of legal business...judicial matters that have me tied up."
"You can't get away to see a comatose father?"
"It's complicated, Jackie. Too complicated to go into on the phone at this hour of the morning. Suffice it to say that I can't leave the city now."
Jack sensed a lot more going on here than Tom was telling.
"Are you in some sort of trouble?"
"Me? Christ, why would you ask something like that?"
"Because you sound funny."
Tom's tone took on a sharp edge. "How would you know what I sound like? We haven't spoken in, what, ten years, and you're going to tell me how I sound?"
"It's been fifteen years"—not quite long enough, Jack thought—"and yeah, I'm telling you you sound funny."
"Yeah, well, don't worry about me. Worry about Dad. He gave me your number before he moved to Florida. 'Just in case,' he said. Well, 'just in case' just happened. Tag, you're it."
Jack sighed. "All right. I guess I'll go."
"Don't sound so enthusiastic."
Jack shook his head. First off, he hated to leave New York for any reason, period. Plus, this wasn't a good time for him to be heading for Florida or anywhere else. He had another fix-it in the early stages of development, but he'd have to let it wait. Worse, an emergency trip like this meant that driving and Amtrak were out. He'd have to take a plane. He didn't mind flying itself, but all the extra security since 9-11 made an airport a scary place for a guy with no official identity.
But then, it was his father down there.
Tom said, "In a way you're lucky he's in a coma."
Strange thing to say. "How's that?"
"Because he's pissed at you for not showing up for Kate's funeral. Come to think of it, so am I. Where the hell were you?"
As if he'd tell a judge, even if that judge happened to be his big brother.
Big Brother...judge. How Orwellian.
"Suffice it to say," he said, deciding to give Tom a dose of his own medicine, "that it's too complicated to go into on the phone at this hour of the morning."
"Very funny. I tell you, though, I can't say I was unhappy about him taking a turn on you. All we heard for years from him was how he wanted to reach you and bring you back into the fold. That was how he put it: 'Bring Jack back into the fold.' It became his mantra. He obsessed on it. But he's not obsessing anymore."
Jack felt he should be glad to hear that—he'd had no intention of ever returning to any fold anywhere—but he wasn't. Instead he felt a pang of regret, as if he'd lost something.
A decade and a half ago, when Jack had dropped out of college, out of his family, and out of society in general, his father spent years tracking him down. Somehow he found someone who had Jack's number. He started calling. Eventually he wore Jack down to the point where he agreed to meet him in the city for dinner. After that they got together maybe once a year for a meal or a set of tennis.
A tenuous relationship at best. The get-togethers were always uncomfortable for Jack. Though his father had never said it, Jack knew he was disappointed in his younger son. Thought he was an appliance repairman and was always pushing him to better himself—finish college, get a pension plan, think about the future, retirement will be here before you know it, blah-blah-blah.
Dad didn't have a clue about what his younger son was about, the crimes he'd committed, the people he'd had to kill while earning his living, and Jack never would tell him. The old guy would be devastated.
"Where'd you say he was?"
"Novaton Community Hospital, and don't ask me where that is because I don't know. Someplace in Dade County, I'd imagine. That's where he had his place."
"Where's—?"
"South of Miami. Look, the best thing to do is call the hospital—no, I don't have the number—and ask for directions from Miami International. That's where you'll have to fly into."
"Swell."
"If he wakes up, explain to him that I'd be there if I could."
Sure you would, Jack thought. And then it hit him.
"'If he wakes up'?"
"Yeah. If. They say he's banged up pretty bad."
Jack's chest ached. "I'll leave as soon as I tie up a few loose ends here," he said, suddenly tired.
He hung up. He had nothing more to say to his brother.
## 4
Semelee awoke alone in the dark. She opened her eyes and lay perfect still, listenin. She heard the breathin sounds of her clansmen around her, some soft, some rough. She heard the creak of the old houseboat timbers as it rocked gentle like, the soft lap of the lagoon water against the hull, the croakin of frogs and the chirpin of crickets among the night sounds of the other Everglades critters. She jumped as someone nearby—Luke, most likely—made a coughin sound that turned into a snore.
The thick hot air lay like a damp sheet on the exposed skin of her arms and legs, but she was used to it. This September was provin to be a hot one, but not like August. That had been a hot one, hottest she could remember.
Why was she awake? She usually slept straight through the night. And then she remembered the dream—not the details, for they had vanished into the night like mornin mist before a storm, but the overall feel of movement...movement toward her.
"Someone's comin," she whispered aloud.
She didn't know how she knew, she just did. This weren't the first time she'd had a second sight. Every so often, without warnin, she'd get a sense of somethin about to happen, and then it did, it always did.
Someone was comin her way. A him, a man, was on his way. She didn't know if that was a good thing or a bad thing. Didn't matter. Either way, Semelee would be ready.
## 5
"Such bounty," Abe Grossman said, staring down at the half dozen donuts laid out in the box before him. "I've done what to deserve this?"
Jack said, "Nothing...everything."
Abe's raised eyebrows sent wrinkles like sets of surfing waves up his brow and into the balding bay of his scalp to crash on the receding gray shore of his hairline. "But Krispy Kremes? For me?"
"For us."
Jack dipped into the box and extracted one of the crustier, sour-cream models, heavy with grease and glazed to within an inch of its life. He took a big bite and closed his eyes. Damn, these were good.
Abe made a face. "But they're full of fat, those things." He rubbed his bulging waistline as if he had a belly ache. "Like ladling concrete into the arteries."
"Probably."
"And to me you brought them?"
The two of them flanked the scarred rear counter of Abe's store, the Isher Sports Shop, Jack on the customer side, Abe across from him, perched like Humpty Dumpty on a stool. Jack made a show of looking around at the dusty cans of tennis balls, the racquets, the basketballs and hoops, footballs and Rollerblades along with their attendant padding shoved helter skelter onto sagging shelves lining narrow aisles. Bikes and SCUBA gear hung from the ceiling. If the Collyer brothers had been into sporting goods instead of newspapers, this is what their place might have looked like.
"You see anyone else around?"
"We're not open yet. I should see no one."
"There you go." Jack pointed to the donuts. "Come on. What are you waiting for?"
"This is a trick, right? You're trying to pull one over on your old friend. You brought them for Parabellum."
As if in response to his name, Abe's little blue parakeet peeked out from behind a neon-yellow bicycle safety helmet, spotted the donut box, and hopped across the counter to it.
Jack spoke around a mouthful of donut. "Absolutely not."
Parabellum cocked his head at the donuts, then looked up at Jack.
"Better not deny him," Abe warned. "He's a fierce predator, that Parabellum. A raptor in disguise, even."
"Oh, right." Jack tore off a tiny piece and tossed it to the bird, who leaped on it.
"What happened to the fat-free Entenmann's and the low-fat cream cheese?"
"We're taking a vacation from all that."
Abe rubbed his belly again. "Nu? I shouldn't be worried about my heart? You want I should die before my time?"
"Jesus, Abe. Can we have one breakfast without you complaining? If I bring in low-cal stuff, you bitch. So here I bring the kind of stuff you always say you wish you were eating instead, and you accuse me of trying to kill you."
Abe was past sixty and his weight ran in the eighth-of-a-ton range, which wouldn't have been so bad if he were six-eight; but he missed that by a foot, maybe more. Jack had become concerned last year about his oldest and dearest friend's potential lack of longevity and had been trying to get him to lose weight. His efforts had not engendered an enthusiastic response.
"Such a crank he is this morning."
Abe was right. Maybe he was feeling a little short. Well, he had his reasons.
"Sorry," Jack said. "Look at it this way: Think of them as a going-away present."
"Going? I'm going somewhere?"
"No, I am. To Florida. Don't know how long I'll be there so I figured I'd pre-load you with some calories to tide you over."
"Florida? You want to go to Florida? In September? In the middle of the worst drought they've had in decades?"
"It's not a pleasure trip."
"And the humidity. It seeps into your pores, heads for the brain, makes you meshugge. Water on the brain—it's not healthy."
"Swell." Jack drummed his fingers on the counter. "Eat a damn donut, will you."
"All right," Abe said. "If you insist. A bisel."
He picked one, took a bite, and rolled his eyes. "Things should not be allowed to taste this good."
Jack had a second donut while he told Abe about his brother's call.
"I'm sorry to hear this," Abe said. "This is why you're so cranky? Because you don't want to see him?"
"I don't want to see him like that...in a coma."
Abe shook his head. "First your sister, and now..." He looked up at Jack. "You don't think...?"
"The Otherness? I hope not. But with the way things have been going lately, I wouldn't be surprised."
After hanging up with Tom last night he'd called the hospital and learned that his father was stable but still on the critical list. He got directions from the airport, then tried to watch a movie. He'd started a Val Lewton festival, watching The Cat People Sunday night. He'd been looking forward to seeing I Walked with a Zombie, but after starting it he couldn't get into it. Thoughts about his father in a coma and getting through airport security proved too distracting. He'd shut if off and lain in the dark, trying to sleep, but thoughts about an indefinable something pulling the strings of his life kept him awake.
So this morning he was tired and irritable. The chance that the accident might not have been so accidental put him on edge.
"You have any details on what happened?"
"Car accident is all I know."
"That doesn't sound too sinister. How old is he?"
"Seventy-one. But he's in great shape. Still plays tennis. Or at least he did."
Abe nodded. "I remember when he roped you into a father-son doubles match last summer."
"Right. Just before all hell broke loose up here."
"Another summer like that I don't need." Abe shook himself, as if warding off a chill. "Oh, I may have something for you on that citizenship matter."
"Yeah? What?"
Since he'd found out last month that he was going to be a father, Jack had been looking for a way to sneak up from underground without having to answer the inevitable questions from various agencies of the government as to where he'd been and what he'd been doing for the last fifteen years, and why he'd never applied for a Social Security Number and never filed a 1040 or paid a cent in taxes in all that time.
He'd thought of simply telling them he'd been ill—disoriented, possibly drug addled—wandering the country, depending on the kindness of strangers, and now he was better and ready to become a productive citizen. That would work, but in these suspicious times it meant he'd be put under extra scrutiny. He didn't want to live the rest of his life on the Department of Homeland Security's watch list.
"A contact in Eastern Europe called and said he thought maybe he had a way. Maybe. It's going to take a little more research."
This bit of good news felt like a spotlight through the gloom that had descended since Tom's call.
"Didn't he give you even a hint?"
Abe frowned. "Over an international phone line? From his country? He should be so foolish. When he works out the details—if he can—he will let me know."
Well, maybe it wasn't such good news. But at least it was potentially good news.
Abe was staring at him. "Nu? You're leaving for Florida when?"
"Today. Haven't booked a flight yet though. Want to talk to Gia first, see if I can convince her to come along."
"Think she'll go?"
Jack smiled. "I'm going to make her an offer she can't refuse."
## 6
"Sorry, Jack," Gia said, shaking her head. "It won't work."
They sat in the old-fashioned kitchen of number eight Sutton Square, one of the toniest neighborhoods in the city, he nursing a cup of coffee, she sipping green tea. Gia had been letting her corn-silk-colored hair grow out a little; it wasn't so close to her head anymore, but still short by most standards. She wore low-cut jeans and a white scoop-neck top that clung to her slim torso. Although into her third month of pregnancy, she had yet to show even the slightest bulge.
Gia's discovery last month that she was pregnant had thrown them both for a loop. It had not been on the radar, and they hadn't been prepared for it. It meant changes for both of them, most drastically for Jack, but they were dealing with it.
Jack had told her about his father as soon as he stepped through her door this morning. Gia had never met him but had been upset by the news and urged Jack to hurry down to Florida. Jack didn't share her sense of urgency. All he could do down there was stand next to his unconscious father's bed and feel helpless; he could think of few things in the world he hated more than feeling helpless. And if and when his father awoke, how long before he started in on why Jack had missed Kate's funeral.
So Jack had sprung his plan on Gia and she had shot him down.
He tried to hide his disappointment. He'd thought it was a sure thing. He'd offered to fly her and Vicky down to Orlando and put them up in Disney World. He'd shuttle back and forth between his father and Orlando.
"How can you say no?" he said. "Think of Vicky. She's never been to Disney World."
"Yes, she has. We went with Nellie and Grace when she was five."
Jack saw a cloud pass through her sky-blue eyes at the mention of Vicky's two dead aunts.
"That was three years ago. She needs another trip."
"Did you forget school?"
"Let her play hooky for a week. She's a bright kid. How much of a challenge can third grade be for her?"
Gia shook her head. "Uh-uh. New year, new class, new teacher. She just started two weeks ago. I can't pull her out for a week this early in the year. If it was November, maybe, but then"—she patted her tummy—"I'd be far enough along to where I wouldn't want to fly."
"Swell," Jack said. He took a turn patting her tummy. "How's Little Jack coming along?"
"She's doing just fine."
This had been their tug-of-war since learning she was pregnant. Jack was sure it was a boy—had to be—while Gia insisted it was a girl. So far the fetal doppler had been inconclusive as to sex.
"Hey, I just had an idea. What do you think about hiring Vicky a nanny for a week and..."
Gia's azure stare stopped him. "You're kidding, right?"
He sighed. "Yeah, I guess so."
What had he been thinking? Obviously he hadn't. Gia going off to Disney World without her daughter? Never. It would crush Vicky. And Jack would be as uncomfortable as Gia about leaving her with anyone else for a week.
He leaned back and watched her take tiny sips of her tea. He loved the way she drank tea, loved the way her whole face crinkled up when she laughed. Loved the way she did everything. They'd met a little over two years ago—twenty-six months, to be exact—but it seemed as if he'd known her all his life. All the women before her, and there'd been more than a few, had faded to shadows the first time he saw her smile. No one had a smile like Gia's. They'd hit a few speed bumps along the way—her discovery of how he earned his living had almost derailed them—and still didn't see eye to eye on everything, but the deep regard and trust they'd developed for each other allowed them to live with their differences.
Jack couldn't remember feeling about anyone as he felt about Gia. Every time he saw her he wanted to touch her—had to touch her, even if only for an instant brush of his fingertips against her arm. The only other person who approached Gia in his affections was her daughter Vicky. Jack and Vicks had bonded from the get-go. He couldn't think of too many people or things worth dying for, but two of them lived in this house.
"Aww," Gia said, smiling that smile and patting his knee. "Feeling shot down?"
"In flames. Looks like I'll be going alone. Usually you're the one getting on a plane and leaving." Gia made regular trips back to Iowa to keep Vicky in touch with her grandparents. Those weeks were like holes in his life. This one would be worse. "Now it's me."
"I've got a cure for those hurt feelings." She put her cup down, rose, and took his hand. "Come on."
"Where?"
"Upstairs. It's going to be a week. Let's give you a bon voyage party."
"Do we get to wear dopey hats?"
"No hats allowed. No clothes allowed either."
"My kind of party."
## 7
Jack was feeling a little cross-eyed and weak in the knees when they left Gia's. She had that effect on him.
On their way to his apartment on the West Side—she'd volunteered to help him pack—he stopped at a mailing service and picked up a couple of FedEx overnight boxes, along with some bubble wrap.
"What are those for?"
"Oh...just have to mail a couple of things before I go."
He didn't want to tell her more than that.
When they reached his third-floor apartment in a West-Eighties brown-stone, he opened the windows to let in some air. The breeze carried a tang of carbon monoxide and the throbbing bass of a hip-hop song with the volume turned up to 11.
Gia said, "How are you going to work this?"
"What do you mean?"
"Buying the ticket."
They stood in the cluttered front room filled with Victorian wavy-grained golden oak furniture laden with gingerbread carving.
"How else? Buy a ticket and go."
"Who are you going to be this time?"
"John L. Tyleski."
After careful consideration, Jack had settled on Tyleski as his identity for the trip. Tyleski's Visa card, secured with a dead kid's Social Security Number, was barely six months old, and so far he'd made all his payments on time. Tyleski had a New Jersey driver license with his photo on it, courtesy of Ernie's ID. It was as bogus as everything else Ernie sold, but the quality was Sterling.
"Isn't that risky?" she said. "You get caught buying a ticket under an assumed identity these days and you're in trouble. Big, Federal trouble."
"I know. But the only way I can get caught is if someone checks the number on the driver license with the Jersey state DMV. Then I'm screwed. But they don't do that at airports."
"Not yet."
He looked at her. "You're not making this any easier, Gia."
She dropped into a wing-back chair, looking worried. "I just don't want to turn on the news tonight and hear that they're investigating some man with no identity who tried to board a plane, and see a picture of you."
"Neither do I."
Jack shivered. What a nightmare. The end of his life in the interstices. But even worse would be having his picture in the papers and on TV. He'd made a fair number of people very unhappy during the course of his fix-it career. The only reason he was still alive was because they didn't know who he was or where to find him. A very public arrest would change all that. Might as well paint a bull's-eye on his chest.
While Gia checked the Miami weather on the computer in the second bedroom, Jack seated himself at the claw-foot oak table and took out a spare wallet. He removed all traces of other identities, leaving only the Tyleski license and credit card, then added about a thousand in cash.
Gia returned from the other room. "The three-day forecast for Miami is in the nineties, so I'd better pack you light clothes."
"Fine. Throw in some running shorts while you're at it." He was dressed in jeans, sneakers, and a T-shirt now, but he needed something more for the trip. "While you're in there, pull me out a long-sleeved shirt, will you?"
She made a face. "Long-sleeved? It's hot."
"I have my reasons."
She shrugged and disappeared into his bedroom.
While she was digging through his drawers, Jack swathed his 9mm Glock 19 in bubble wrap, then wrapped that in aluminum foil, and shoved it into the FedEx box; he did the same with his .38 AMT Backup and its ankle holster, then packed in more wrap to keep them from shifting around in the box. That done he wrapped duct tape around the box wherever the FedEx logo appeared.
"How many days should I pack for?" Gia called from the other room.
"Three or four. If I stay longer I'll have them washed."
Gia popped back into the front room holding a lightweight cotton shirt with a tight red-and-blue check.
"You sure you want long sleeves?"
He nodded. "Need them to hide this."
He held up a plastic dagger. It was dark green, almost black, with a three-inch blade and a four-inch handle, all molded from a single piece of super-hard plastic fiber compound that Abe guaranteed would breeze past any metal detector on earth. The blade had no cutting edge to speak of, but the point was sharp enough to pierce plywood.
No one was hijacking his flight.
Gia's eyes widened. "Oh, Jack! You're not really thinking of—"
"I'll have it taped to the inside of my arm. No one will find it."
"This is insane! Do you know what will happen to you if you're caught?"
"I won't be." He held up a roll of adhesive tape. "Help me tape it on?"
"Absolutely not! I'll have no part in this craziness. It's irresponsible. You have a child on the way! Do you want to be in jail when she's born?"
"Of course not. But Gia, you should understand by now, this is the way I am, this is the way I have to do it."
"You're afraid of giving up control is what it is."
"Maybe so. Getting on a plane piloted by someone I don't know puts a crimp in my comfort zone. But I can handle that. What I can't handle is handing some out-to-lunch airline full responsibility for making sure that all the other passengers are going to behave."
"You've got to learn to trust, Jack."
"I do. I trust me, I trust you, I trust Abe, I trust Julio. Beyond that..." He shrugged. "Sorry. It's the way I'm wired." He held up the tape again. "Please?"
She helped, but he could tell her heart wasn't in it.
He blunted the point with a small piece of tape, then held it in place against the inside of his left upper arm, the butt of the handle almost in his armpit, while she secured it with three long strips that encircled his arm. Not the most comfortable arrangement, but he'd remove it in the restroom once they were in the air and transfer the knife to the inside of one of his socks for the rest of the flight.
When she finished taping she stepped back and looked at her work.
"That should hold. I..." She shook her head.
"What?"
"I can't help thinking that if there'd been someone like you on those 9-11 planes, the Trade Towers might still be standing."
"Maybe. Maybe not. I'm not Superman. I can't take on five alone. But along with guys like the ones on Flight 93, who knows?"
He pulled on the shirt, rolled the cuffs halfway up his forearms, and struck a pose.
"How do I look?"
"Suspicious," she said.
"Really?"
She sighed. "No. You look like you always look: Mister Everyday People."
That was what he wanted to hear. "Great. Am I packed?"
"I put it all on the bed. Where's your suitcase?"
"Suitcase? I don't have one. I've never needed one."
"That's right. You don't travel. How about a gym bag or something along that line?"
"Yeah, but it's filled with tools." His kind of tools.
"Well, if it's not too dirty inside, empty it out and we'll see if it'll do the job."
Jack pulled the bag out of a closet and emptied its contents on the kitchen counter: glass cutter, suction cup, rubber mallet, pry bar, slim jim for car doors, lock picks, an assortment of screwdrivers and clamps in various sizes and configurations.
"What is all this?" Gia asked as she watched the growing pile.
"Tools of the trade, m'dear. Tools of the trade."
"If you're a burglar, maybe."
He wiped out the inside of the bag with a damp paper towel and handed it to her. "Will this do?"
It did. His wardrobe down south would consist of shorts, T-shirts, socks, and boxers. They managed to stuff it all into the bag.
"You're going to look wrinkled," she warned.
"I'm going to Florida, remember? Wrinkle City."
"Touché."
He hefted the bag. "Do I check this or will they let me carry it on board?"
"That looks plenty small enough for the overhead."
"Overhead...? Oh, right. I know what you mean."
She looked up at him. "When was the last time you were on a plane?"
Jack had to think about that. The answer was a little embarrassing. "I think it was sophomore year of college. Spring break in Lauderdale."
He barely remembered it. Seemed like a lifetime ago. In a way it was. A different life.
"Not once since?"
He shrugged. "No place I want to go."
She stared at him. "Is that the truth?"
"Of course. Anything I could ever want is right here in this city."
"You don't think the fact that flying is so much of a hassle, a risky hassle for you, has anything to do with it?"
"Maybe some." Where was this going?
Gia slipped her arms around him and squeezed, pressing herself against him.
"Don't you see?" she said. "Don't you see? You've built this anonymous, autonomous life for yourself, but it's become a trap. Sure, no one knows you exist and you don't spend the first four or five months of every year working for the government like the rest of us, and that's great in its way, but it's also a trap. Everywhere you go you've got to pretend to be someone else and run the risk of being found out. I go anywhere I want without a second thought. If I go to an airport and someone scrutinizes my ID, I'm not worried. But you've got the anxiety that someone will spot a flaw."
She released him and fixed him with her blue stare.
"Who's freer, Jack? Really."
She didn't understand. Jack figured she'd never fully understand. But that was okay. It didn't make him love her any less, because he knew where she was coming from. She'd been on her own for years, a single mother trying to make a career for herself and a life for her child. She had responsibilities beyond herself. Her days, spent dealing with the nuts and bolts of everyday life, were hectic and exhausting enough without adding multiple layers of complexity.
"It's not subject to comparison, Gia. I've lived the way I felt I had to live. By my rules, my code. My not paying taxes has nothing to do with money, it has to do with life, and who owns mine, or who owns yours, or Vicky's, or anyone's."
"I understand that, and philosophically I'm with you all the way. But in the practical, workaday world, how does that work for a man with a family? 'Oh, I'm sorry, honey. Daddy's not traveling with us because he's using a false identity and doesn't want us involved if he's picked up. But don't worry, he'll meet us there. I hope.' That's no way to bring up a child."
"We could all have false identities. We could be an under-the-radar family." He quickly held up his hands. "Only kidding."
"I hope so. What a nightmare that would be."
This time he pulled her close. "I'm working on it, Gi. I'll find a way."
She kissed him. "I know you will. You're Repairman Jack. You can fix anything."
"I'm glad you think so."
But coming back from underground with his freedom intact...that was a tall order.
You'd better come through for me, Abe, he thought, because I've hit a wall.
He didn't want the hassle of parking at the airport so he called a cab to take him to LaGuardia. Since Gia lived in the shadow of the Fifty-ninth Street Bridge, a minimal detour would allow him to drop her off at home along the way.
"Be careful," she whispered after a long good-bye kiss. "Come back to me, and don't get into any trouble down there."
"I'm visiting my comatose father. How on earth could I possibly get into any trouble?"
## 8
Jack reached the OmniShuttle Airways counter an hour before the next scheduled flight.
Before dropping Gia off, he'd had the cab take him over to Abe's where he left the package to be overnighted to his father's place. Abe used a small, exclusive, expensive shipping company that didn't ask questions. The cab ride had been uneventful, but it felt so odd to be moving about the city without a gun either tucked into the small of his back or strapped to his ankle. He didn't dare risk trying to sneak one onto the plane, though, even in checked luggage, now that they were x-raying every piece.
The ticket purchase went smoothly: A mocha-skinned woman with an indeterminate accent took the Tyleski Visa card and the Tyleski driver license, punched a lot of keys—an awful lot of keys—then handed them back along with a ticket and a boarding pass. Jack had chosen OmniShuttle because he didn't want any round-trip-ticket hassles. The airline sold one-way tickets without regard to Saturday stayovers or any of that other nonsense: When you want to go, buy a ticket; when you want to come back, buy another.
Jack's kind of company.
He asked for an aisle seat but they were all already taken. But he did manage to snag an exit row, giving him more leg room.
He had some time so he treated himself to a container of coffee with a trendoid name like mocha-latte-java-kaka-kookoo or something like that; it tasted pretty good. He bought some gum and then, steeling himself, headed for the metal detectors with their attendant body inspectors.
He made sure to get on the end of the longest line, to give him a chance to see how they conducted the screening process. He noticed that a much higher percentage of the people who set off the metal alarm were taken aside for more thorough screening than the ones who didn't. Jack wanted to be in the latter category.
This is how a terrorist must feel, he realized. Standing on line, sweating, praying that no one sees through his bogus identity. Except I'm not looking to hurt anyone. I'm just looking to get to Florida.
When it came his time, he placed his bag on the belt and watched as it was swallowed by the maw of the fluoroscope. Then it was his turn to step through the metal detector. He put his watch, change, and keys into a little bowl that was passed around the detector, then stepped through.
His heart skipped a beat and jumped into high gear when a loud beep sounded. Damn!
"Sir, have you emptied your pockets?" said a busty bottle-blonde woman in a white shirt with epaulettes, a gold badge, and a name tag that read "Delores." She was armed with a metal detecting wand. A dozen feet behind her, two security guards stood with carbines slung over their shoulders.
"I thought I did. Let me check again." He patted his pants pockets front and rear but, except for his wallet, they were empty. He pulled out the wallet. "Could this be the culprit?"
She waved her wand past it without a beep. "No, sir. Step over here, please."
"What for?"
"I have to wand you."
When had "wand" become a verb?
"Is something wrong?"
"Probably just your belt buckle or jewelry. Stand here, back to the table. Good. Now spread your legs and raise your arms out from your body."
Jack assumed the position. The moisture deserting his mouth seemed to be migrating to his palms. She waved the wand up and down the inside and outside of his legs, then across his waist where she got a beep from his belt buckle—no problem—and then she started on his arms. Right one first—inside and outside, okay; then the left—outside okay, but a loud beep as the wand approached his armpit.
Oh shit, oh hell, oh Christ. Abe you promised me, you swore to me the knife would pass the detectors. What's happening?
Without moving his head, Jack checked out the two security guards from the corner of his right eye. They looked bored, and certainly weren't paying attention to him. To his left a handful of unarmed security personnel were busy screening—wanding—other travelers. He could barrel past them and dash back out into the terminal, but where to go from there? His chances of escaping were nil, he knew, but he damn well wasn't simply going to stand here and put his hands out for the cuffs. If they wanted him, they were going to have to catch him.
"Sir?"
"Hmmm? What?" Jack could feel the sweat breaking out on his forehead. Had she noticed?
"I said, do you have anything in your breast pocket?"
"My—?"
He jammed his hand into the pocket and came out with his package of Dentyne Ice. Gum in a blister pack...sealed with foil...
She ran her wand over it and was rewarded with a beep. She took the pack, opened it to make sure it was only gum, then dropped it on the table. The rest of the wanding was beepless.
The future that had been telescoping closed at warp-10 now opened wide again. Feeling as giddy as a man with a reprieve from death row, Jack retrieved his watch, keys, and chain, but he left the damn gum. It had put him on a train to heart attack city. Let Delores have it.
As he hefted his gym bag strap onto his shoulder he fought an urge to ask Delores if she wanted to inspect that too. Inspect anything you want! The mad inspectee strikes again!
But he said nothing, contenting himself with a friendly nod as he started toward his gate. He reached it with just enough time to put in a quick to call Gia.
"I made it," he said when she answered. "I board the plane in a couple of minutes."
"Thank God! Now I won't have to figure out how to bake a cake with a file inside."
"Well, there's still the flight home."
"Let's not think about that yet. Call me when you've seen your father, and let me know how he is."
"Will do. Love ya."
"Love you too, Jack. Very much. Just be careful. Don't talk to strangers or go riding in strange cars, or take candy from—"
"Gotta run."
He wound up in a window seat in the left emergency row with the perfect traveling companion: The guy fell asleep before takeoff and didn't wake up until they were on the Miami tarmac. No small talk and Jack got to eat the guy's complimentary bag of peanuts.
The only glitch in the trip was a slight westward alteration of the usual flight path due to tropical storm Elvis. Elvis...when Jack had heard the name announced on TV the other night he'd done a double take that would have put Lou Costello to shame.
He wondered now if there'd ever been a tropical storm named Eliot. If so, had it been designated on the maps as T. S. Eliot?
Elvis was not expected to graduate to hurricane status, but was presently off the coast near Jacksonville, cruising landward and stirring things up, just as its namesake had in the fifties. Though the plane swung westward to avoid the turbulence, Jack could see the storm churning away to the east. From his high perch he looked out over the rugged terrain of cloud tops broken dramatically here and there by fluffy white buttes from violent updrafts. Elvis was entering the building.
## 9
"Don't let her bite me, Semelee!" Corley cried.
Semelee lifted the shells away from her eyes and looked at Corley.
Corley's good eye, the one he could open, rolled in its socket under his bulging forehead as he looked up at her from where he stood waist deep in the lagoon. Normally at that spot in the lagoon the water'd be up to his neck. But with this drought...
Corley was hard on the eyes, that was for sure, but that made him good for beggin. They'd take him to town, sit him in a shady spot on the sidewalk, put a beat-up old hat in front of him, and wait. That hat wouldn't stay empty for long. People'd take one look at that face and empty their pockets of all their spare change, even toss in a few bills now and then.
But Tuesdays weren't no good for beggin—not as bad as Mondays, but bad. So Mondays and Tuesdays became fishin days.
"Tell her not to bite!" Corley wailed.
"Hesh up and hold the net," Luke told him.
Semelee smiled as she watched the two clansmen from the deck of the second, smaller houseboat, the Horse-ship. They stood in the water beside the boat, each holdin a four-foot pole with a net of half-inch nylon mesh stretched between them. Twisted trees with tortured trunks on the bank leaned over the water.
Luke was Corley's half brother, and he was special too. Not in ways you could see so plain like Corley's, and not in ways that was much good on the beggin front. So he mostly just ferried the beggin folk around. But Luke was special in his own way. Maybe too special. He'd tried the beggin thing, takin off his shirt to show the little fins runnin down his spine and all the big scales that covered his back, but he was a flop. Didn't collect a dime. People was heard to say it looked fake, that no one could really have a back that ugly, and wouldn't drop a dime. The cops tried to arrest him for public disgustation or somethin like that, but he run off before they could catch him.
Semelee was glad she wasn't misshapen like Corley or Luke or the other members of the clan. But she was special too. She had a weird look that had been enough to bring her a lot of pain, but not weird enough to bring in loose change. She was special in another way. In her own way. Special on the inside.
"Ain't like this is the first time you ever done this," she told Corley.
"I know, but I hate it. If'n I do it a million times I'm still gonna hate it. That thing could take my leg off with one bite if it got a mind to."
"Not just one leg, Corley," Luke said with a grin. "When you think about it, she could take both off at once—if she got a mind to, that is."
"Or if I got tired of your whining and told her to," Semelee added.
"That ain't funny!" Corley said, dancing in place like a little boy who had to take a wizz.
"Stand still!" Luke said. "We're tryin to catch fish, not scare 'em away! Just be glad it ain't Devil doin the herdin."
Corley's hands shook. "If'n it was Devil, I wouldn't be in the water! Hell, I wouldn't even be on the bank!"
Semelee spotted a dark shape, maybe a foot or two deep, slidin through the water toward them, rippling the surface above as it moved.
Dora was comin, drivin the fish before her.
"Get ready," she told them. "Here we go."
Corley let out a soft, high-pitched moan of fear but held his ground and his end of the net.
The shape glided closer and closer to Luke and Corley, and then suddenly the net bowed backward and the water between them was alive with fish, frothing the surface as they thrashed against the net. The two men pushed their poles together and lifted the net out of the water. A coupla dozen or more good-size mollies and even a few bass wiggled in the mesh.
"Fish fry tonight!" Luke cried.
"She touched me!" Corley said, looking this way and that. If his neck would've allowed it, it'd be swivelin round in circles. "She tried to bite me!"
"That was just her flipper," Luke said.
"I don't care! Let's get these things ashore!"
"Don't forget to leave me some," Semelee said. "Dora'll be very unhappy if you don't."
"Oh, right! Right!" Corley said. He reached into the net and pulled out a wriggling six-inch molly. "The usual?"
"A couple should do."
He flipped one and then another onto the deck, then headed for shore.
Semelee picked up one of the flopping, gasping fish and held it by its slick, slippery tail over the water.
"Dora," she sing-songed. "Dora, dear. Where are you, baby?"
Dora must have been waitin on the bottom because she popped to the surface right away. The snapping turtle's mountainous shell with its algae-and grass-covered peaks and valleys appeared first, runnin a good three-four feet stem to stern. Then her heads broke the surface, all four beady little eyes fixed on her, both hooked jaws open and waitin. Semelee could see the little wormlike growth on each of her tongues that Dora used like fishin lures when she sat on the bottom during the daytime and waited for lunch. Finally the long tail broke the surface and floated behind her like a big fat water moccasin.
Semelee was sure scientists would give anything for a look at Dora, the biggest, damnedest, weirdest-looking alligator snapper anyone had ever seen, but she was Semelee's, and no one else was gettin near her.
She tossed a fish at the left head. The sharp, powerful jaws snapped closed across the center, severing the head and tail. The right head snatched those up as they hit the water. A pair of convulsive swallows and the mouths were open again.
Semelee gave the right head first crack at the second fish, with similar results, then she stretched her hands out over the water. Dora reared up so that her heads came in reach.
"Good girl, Dora," she cooed, stroking the tops of the heads. Dora's long tail thrashed back and forth with pleasure. "Thanks for your help. Better get outta sight now before the dredgers come."
Dora gave her one last look before sinkin from sight.
As Semelee straightened she caught a glimpse of her reflection in the churned-up water and took another peek. She didn't hold much with mirror gazin, but every once in a while she took a look at herself and wondered how different things mighta been for her if she'd had a head of normal hair—black or brown or red or blond, didn't matter, just so long as it wasn't what she'd been born with.
The surface of the water showed someone in her mid-twenties with a face that wasn't no head turner but not ugly neither. If heads did turn, it was cause of her hair, a tangled silver-white mane that trailed after her like a cloud—a very tangled, twisted stormy cloud that no amount of combin or brushin could straighten. No amount at all. She should know. She'd spent enough hours as a kid workin on it.
That hair had been a curse for as long as she could recall. She didn't remember bein born here, right here on the lagoon, and didn't remember her momma leavin the lagoon and takin her to Tallahassee. But she did remember grammar school in Tallahassee. Did she ever.
Her earliest memories there was of kids pointin to her hair and callin her "Old Lady." Nobody wanted Old Lady Semelee on their team no matter what they was playin, so she used to spend recesses and after school mostly alone. Mostly. Being left out would have been bad enough, but the other girls couldn't let it go at that. No, they had to crowd around her and pull off the hat she wore to hide her hair, then they'd yank on that hair and make fun of it. The days she came home from school cryin to her momma were beyond countin. Home was her safe place, the only safe place, and her momma was her only friend.
Semelee remembered how she'd cursed her hair. If not for that hair she wouldn't be teased, she'd be allowed into the other kids' games, she'd have friends—more than anything else in the world little Semelee wanted a friend, just one lousy friend. Was that too much to ask? If not for that hair she'd belong. And little Semelee so wanted to belong.
Since hats wasn't helpin, she decided one day at age seven to cut it all off. She took out her momma's sewin scissors and started choppin. Semelee smiled now at the memory of the mess she'd made of it, but it hadn't been funny then. Her momma'd screamed when she seen it. She was fit to be tied and that scared Semelee, scared her bad. Her only friend was mad.
Momma took the scissors and tried to make somethin outta the chopped-up thatch but she couldn't do much.
And the kids at school only laughed all the harder when they saw it.
But they ain't laughin now, Semelee thought with grim satisfaction as she threaded the holes in the eye-shells through the slim leather thong she wore around her neck. At least some of them ain't. Some of them'll never laugh again.
She watched the ripples and eddies that remained behind on the surface in Dora's wake. Something about their crisscrossing pattern reminded her of her dream last night, the one about someone coming from someplace far away. As she watched the water she had a flash of insight. Suddenly she knew.
"He's here."
## 10
Miami International had been a mob scene, far more hectic and crowded than LaGuardia. Jack wound his way through the horde of arrivees and departees toward the ground transportation area. There he caught a shuttle bus to Rent-a-Car Land. In order to help them out of second place, Jack decided to rent from Avis. He settled on an "intermediate" car and chose the most anonymous looking vehicle they had: a beige Buick Century.
The hospital had given him directions from the Florida Turnpike but Jack chose US 1 instead. He figured it would take longer. The red-vested guy at the Avis desk gave him a map and highlighted the way to Route 1.
He was on his way.
All around him South Florida lay flat as a tabletop under a merciless sun, bright in a cloud-dappled sky, blazing through a haze of humidity that hugged the land. Someone somewhere had called Florida an oversized sandbar hanging off the continent like a vestigial limb. Jack couldn't see anything to contradict that.
He'd expected more lushness, but the fronds of the palms along the side of the road hung limp and dull atop their trunks, their tips a dirty gray-brown. The grass and brush around them looked burned out. No doubt the result of the drought Abe had mentioned.
He reached Route 1—also known as Dixie Highway according to the signs—and ran into some traffic at the southbound merge. People rubber-necking an accident on the northbound side slowed him for a while. He saw the strobing police and ambulance lights and felt a flash of resentment, wondering if people had rubbernecked his father's accident like these yokels.
As soon as they passed the crash, the road speeded up again.
For a while the view along US 1 threatened to devolve into Anytown, USA—at least an Anytown warm enough for palm trees—with a parade of Denny's and Wendy's and McDonalds, and Blockbusters and Chevrons and Texacos. Further proof of the depressing homogenization of America, its terror of the untried, its angst of the unique.
But then he started noticing taquerias and tapas joints, and billboards in Spanish. The Cubano and Mexican influence. He passed a place offering "fishes." Okay, this wasn't Anytown. This had a flavor all its own.
The colors of the buildings struck him between the eyes. Standard granite gray had been banished. The palette here was way heavy on the pastels, especially turquoise and coral. The buildings looked like molded sherbet—orange, raspberry, key lime, lemon, watermelon, casaba, and maybe a few as yet untried flavors. He spotted a mall done up in what might be called rotten-lemon-rind yellow.
Further south he passed one car dealership after another, every make from every nation that exported cars, all interspersed with AutoZones and Midas Mufflers, Goodyear Tire Centers, and dozens of no-name auto parts shops. People must be nuts about cars down here.
He realized he was hungry. He saw a place called Joanie's Blue Crab Café and pulled off the road. The place was pretty much empty—this was off-season, after all—and decorated with local crafts. Paintings by local artists studded the wall. The other three patrons were glued to the TV where the Weather Channel was showing green, yellow, and orange swirls that were supposed to be tropical storm Elvis. They were asking when the hell they were going to get rain.
An air conditioner or two might have expanded the comfort zone in Joanie's, but that would have detracted from the funky Florida ambiance. Jack hung in there under the twirling ceiling fans and asked the waitress for a local brew. She brought him something called Ybor Gold and it tasted so damn good he had another along with a crabcake sandwich that was out of this world. This lady could open on the Upper East Side and clean up.
Belly full, Jack stepped outside. Elvis might be dumping tons of water on Jacksonville and the rest of north Florida, but down here, though the sky was speckled with clouds, none of them looked like the raining kind. The forecast was bone dry. Dry at least as far as precipitation went, but the air itself lay thick with humidity and clung to his skin like a sloppy wet kiss from a least-favorite aunt.
Back in the car he searched around the radio dial for some music—rock, preferably—but all he found was country or folks speaking Spanish or sweaty-voiced preachers shouting about Jay-sus.
If you want to believe in Jay-sus, he thought, fine. If you want me to believe in Jay-sus, fine too; you can want anything you wish. But do you have to shout?
He finally found a rock station but it was playing Lou Reed. He quick-hit SCAN. Through the years Jack had come to the conclusion that Lou Reed was a brilliant performance artist whose act was a lifelong portrayal of a singer-songwriter who couldn't carry a tune or write a melody.
The tuner stopped on a dance station. Jack didn't dance, the beat was monotonous, and he'd arrived in the middle of a woman doing a double-time version of "Boys of Summer." He bailed when a cheesy organ attempted to duplicate Kootch Kortchmar's riffs from the original. What had Don Henley ever done to deserve that?
Next stop, one of the country stations—"Gator Country One-Oh-One Point Nine!" He liked some country, mostly the Hank Williams—Senior, preferably—Buck Owens, Mel Tillis brand of mournful nobody-loves-mebut-my-dog-and-he's-got-fleas-so-pass-that-whiskey-bottle-over-here-if-you-please ballad. He lasted maybe fifteen minutes on 101.9. Three songs, three singers, and they all sounded exactly the same. Was that the awful truth about modern country music? The one they'd kill to keep? One lead singer performing under a gazillion different names? Jack wasn't sure about that part, but he had no doubt that the same guy had been singing backup harmony on all three songs.
Okay. Can the radio.
He saw a sign for Novaton and hung a right off US 1 onto a road that ran due west, straight as a latitude line. Looked like someone had given a guy a compass and a paver filled with asphalt and said, "Go west, young man! Go west!" It made sense. No hills or valleys to skirt. The only rises in the road he'd seen since leaving the airport had been overpasses.
He checked out the sickly palms and pines flanking the road. He'd worked with a landscaper as a teen and knew northeast greenery, but even healthy these trees would be a mystery to him. Dead gray fronds lay on the shoulder like roadkill while some skittered onto the pavement when the breeze caught them.
All the houses along the road were squat little ranches in overgrown yards, with carports instead of garages; they hunkered against the earth as if hiding from something. Every once in a while a warehouse would soar to one-and-a-half stories, but that was an aberration. The favored exterior shade seemed to be a sick green like oxidized copper, and here and there a pizza-size DTV dish would poke up from a roof. He'd been expecting lots of red-tile roofs but they seemed a rarity; most were standard asbestos shingles, pretty threadbare in many cases. Oddly, the shabbiest houses seemed to sport the most magnificent palms in their front yards.
Even if he didn't know much about tropical or subtropical trees, he did know banyans; their distinctive aerial roots gave them away. The road to Novaton was loaded with them. In some stretches banyan phalanxes lined each side of the street and interwove their branches above the pavement, transforming a bumpy secondary road into a wondrous, leafy green tunnel.
He recognized a couple of coconut palms, only because of the yellowing nuts hanging among the fronds. Plants that in New York grew only indoors in carefully watered and fertilized pots flourished like weeds down here.
He passed a tall white water tower emblazoned with the town name and shaped like one of those old WWI potato-masher hand grenades the Germans used to toss at the Allies. At its base lay a dusty soccer field flanked by a high school, a middle school, and a senior center.
He passed a feed store. Feed what? He hadn't seen any cattle.
Abruptly he was in Novaton and quickly found the center of town—the whole four square blocks of it. The directions from the hospital told him how to find it from there. Two right turns off Main Street and he came to a three-story cantaloupe-colored brick building of reasonable vintage. The sign out front told him he'd reached his destination.
NOVATON COMMUNITY HOSPITAL A MEMBER OF DADE COUNTY MEDICAL SYSTEMS
He parked in a corner of the visitor lot next to some sad looking cacti and headed through the stifling late-afternoon heat toward the front door. An arthritic old man in the information kiosk gave him his father's room number on the third floor.
Minutes later Jack was standing outside room 375. The door stood open. He could see the foot of the bed, the twin tents of the patient's feet under the sheet. The rest was obscured by a privacy curtain. He sensed no movement in the room, no one there besides the patient.
The patient...his father...Dad.
Jack hesitated, advancing one foot across the threshold, then drawing it back.
What am I afraid of?
He knew. He'd been putting this off—not only his arrival, but thinking about this moment as well—since he'd started the trip. He didn't want to see his father, his only surviving parent, laid out like a corpse. Alive sure, but only in the bodily sense. The man inside, the sharp-though-nerdy-middle-class mind, the lover of gin, sticky-sweet desserts, bad puns, and ugly Hawaiian shirts, was unavailable, walled off, on hold, maybe forever. He didn't want to see him like that.
Yeah, well that's just too damn bad for me, isn't it, he thought as he stepped into the room and marched to the foot of the bed. And stared.
Jeez, what happened to him? Did he shrink?
He'd expected bruises and they were there in abundance: a bandage on the left side of his head, a purple goose egg on his forehead, and a pair of black eyes. What shocked him was how small his father looked in that bed. He'd never been a big man, maintaining a lean and rangy build even through middle age, but now he looked so flat and frail, like a miniature, two-dimensional caricature tucked into a bed-shaped envelope.
Besides the IV bag hanging over the bed, running into him, another bag hung below the mattress, catching the urine coming out of him. Spikes marched in an even progression along the glowing line on the cardiac monitor.
Maybe this wasn't him. Jack looked for familiar features. He couldn't see much of the mouth as it hung open behind the transparent green plastic of the oxygen mask. The skin was tanned more deeply than he'd ever remembered, but he recognized the age spots on his forehead, and the retreating gray hairline. His blue eyes were hidden behind closed lids, and his steel-rimmed glasses—the only time his father took off his glasses was to sleep, shower, or trade them for prescription sunglasses—were gone.
But yeah, this was him.
Jack felt acutely uncomfortable standing here, staring at his father. So helpless...
They'd seen very little of each other in the past fifteen years, and when they had, it was all Dad's doing. His earliest memories of home were ones of playing catch in the backyard when he'd been all of five years old and the mitt was half the size of his torso, standing in a circle with his father and sister Kate and brother Tom, tossing the ball back and forth. Dad and Kate would underhand it to him so he could catch it; Tom always tried to make him miss.
His lasting, growing-up impressions were of a slim, quiet man who rarely raised his voice, but when he did, you listened; who rarely raised his hand, but when he did, a single, quick whack on the butt made you see the error of your ways. He'd worked as a CPA for Arthur Anderson, then moved—decades before the Enron scandal—to Price Waterhouse where he stayed until retirement.
He wasn't a showy sort, never the life of the party, never had a flashy car—he liked Chevys—and never moved from the west Jersey house he and Mom had bought in the mid-fifties. Then, without warning, he'd up and sold it last fall and moved to Florida. He was a middle-class man with a middle-class income and middle-class mores. He hadn't changed history and no one but the surviving members of his family and steadily diminishing circle of old friends would note or mourn his passing, yet Jack would remember him as a man who always could, as Joel McCrea had put it in Ride the High Country, enter his house justified.
Jack stepped around to the left side of the bed, the one opposite the IV pole. He pulled up a chair, sat, and took his father's hand. He listened to his breathing, slow and even. He felt he should say something but didn't know what. He'd heard that some people in comas can hear what's going on around them. It didn't make much sense, but it couldn't hurt to try.
"Hey, Dad. It's me. Jack. If you can hear me, squeeze my hand, or move a finger. I—"
His father said something that sounded like "Brashee!" The word startled Jack.
"What'd you say, Dad? What'd you say?"
He caught movement out of the corner of his eye and saw a heavyset young woman in a white coat enter with a clipboard in her hand. She had a squat body, café au lait skin, short dark hair; a stethoscope was draped around her neck.
"Are you a relative?" she said.
"I'm his son. Are you his nurse?"
She smiled briefly—very briefly. "No, I'm his doctor." She put out her hand. "Dr. Huerta. I was the neurologist on call when your father was brought to the ED last night."
Jack shook her hand. "Jack. Just call me Jack." He pointed to his father. "He just spoke!"
"Really? What did he say?"
"Sounded like 'brashee.'"
"Does that mean anything to you?"
"No."
And then he thought, Maybe he heard my voice and was trying to say, Black sheep.
"He's been vocalizing gibberish. It's not unusual in his state."
He studied Dr. Huerta for a few seconds. She didn't look old enough to be in med school, let alone a specialist.
"What is his state? How's he doing?"
"Not as well as we'd like. His coma score is seven."
"Out of ten?"
She shook her head. "We use the Glasgow Coma Score here. The lowest, or worst score, is three. That's deep coma. The best is fifteen. We go by eyes, verbalization, and movement. Your father scores a one on his eyes—they remain closed at all times—and a two on vocalization, which means he makes meaningless sounds like you just heard now and then."
"That's a total of three," Jack said.
This wasn't sounding too good.
"But his motor response is a four, meaning he withdraws from painful stimuli."
"What kind of painful stimuli? I won't be finding cigarette burns on his soles, will I?"
Dr. Huerta's eyes widened. "Good heavens, no! What on earth do you think—?"
"Sorry, sorry." Jeez, lady. Chill. "Just kidding."
"I should hope so," she said with an annoyed look. "We use a special pin to test motor responses. Your father's score of four brings his total to seven. Not great, but it could be worse." She checked her clip board. "His reflexes, however, are intact, his vitals are good, so are his labs. His brain MRI showed no stroke or subdural hemorrhage, and his LP was negative for blood."
"LP?"
"Lumbar puncture. Spinal tap."
"No blood. That's good, right?"
She nodded. "No signs of intracranial bleeding. His heart's been acting up, though."
"Whoa," Jack said, jolted by the remark. "His heart? He's always had a good heart."
"Well, he went into atrial fibrillation last night—that's a chaotically irregular heartbeat—and again this morning. I called for a cardiology consult and Dr. Reston saw him. Both times your father converted back to normal rhythm spontaneously, but it does indicate some level of heart disease."
"How bad is this atrial fibrillation?"
"The main worry is a clot forming in the left atrium and shooting up to the brain and causing a stroke."
"Swell," Jack said. "As if a coma isn't bad enough."
"Dr. Reston started him on a blood thinner to prevent that. But tell me about his medical history. I've been working in the dark, knowing nothing about him beyond the address and date of birth we got off his license. Has he been treated for any illnesses or heart problems in the past? Does he take any medications?"
"I think he once mentioned taking an aspirin a day, but beyond that..."
"Do you know if he's been seeing a doctor down here, for checkups and the like?"
Jack was embarrassed. He knew no more about what his father had been doing down here than what he'd been doing in Jersey before the move. He knew his father's new address but had never seen the place. Truth was, he knew nothing about his father's life down here or anywhere else, and even less about his health.
But he was getting a crash course this afternoon.
How to put this...
"He wasn't much for talking to me about his health."
Dr. Huerta smiled. "That's a switch. Most people his age talk about nothing but."
"Is he going to be okay?"
"I wish I could say. If his cardiac rhythm stabilizes, I believe he'll come out of this with little permanent damage. He won't remember a thing about the accident, but—"
"What about the accident?" Jack said. "What happened?"
She shrugged. "I have no idea. All I know is that he was brought in unconscious from head trauma. You'll have to ask the police."
The police...swell. The last people Jack wanted to talk to.
She fished in her pocket. "I'll be looking in on him again in the morning. If you learn anything about his medical history, give me a call." She handed him a card.
Jack slipped it into his pocket.
## 11
After the doctor bustled out of the room, Jack turned back to his father. As he stepped toward the bed—
"So, you're one of Thomas's sons."
Jack jumped at the sound of the voice, raspy, like someone who'd been gargling with kerosene. Startled because he hadn't heard anyone come in, he looked around and found the room empty.
"Who—?"
"Over here, honey."
The voice came from behind the curtain. Jack reached out and pulled it back. A thin, flat-chested old woman sat in a chair in a shadowed corner. Her black hair was pulled back in a tight bun and her skin was dark, made even darker by the sleeveless canary yellow blouse and bright pink Bermuda shorts she wore, but in the shadows he couldn't tell her race. A large straw shopping bag sat on the floor beside her.
"When did you come in?"
"I've been here the whole time." She pronounced it "Oy've been here the whole toym." The accent was from somewhere on Long Island—Lynn Samuels to the Nth degree. But that cinderblock-dragging-behind-a-truck voice...how many packs of cigarettes had it taken to achieve that tone?
"Since before I came in?"
She nodded.
That bothered Jack. He wasn't usually so careless. He'd have sworn the room was empty.
"You know my father?"
"Thomas and I are next-door neighbors. We moved in the same time and became friends. He's never mentioned me?"
"We, um, don't talk a lot."
"He's mentioned you, many times."
"You must be thinking of Tom."
She shook her head and spoke at jackhammer speed. "You don't look old enough to be Tom, Jr. You must be Jack. And he did talk about you. Hell, sometimes I couldn't get him to shut up about you." She rose and stepped forward, extending a gnarled hand. "I'm Anya."
Jack took her hand. He saw now that she was white—or maybe Caucasian was a better term, because she was anything but white. Her skin was deeply tanned and had that leathery quality that only decades of dedicated sunbathing can give. Her skinny arms and legs had the shape and texture of Slim Jims. Her hair was mostly jet black except for a mist of gray roots hugging her scalp.
Jack heard a faint yip from behind her. He looked and saw a tiny dog head with huge dark eyes poking over the edge of the straw shopping bag.
"That's Oyving," she said. "Say hello, Oyv."
The Chihuahua yipped again.
"Oyving? How do you spell that?" Jack said.
She looked at him. "I-R-V-I-N-G. How else would you spell it?"
He released her hand. "Oyving it is. I didn't know they allowed dogs in hospitals."
"They don't. But Oyv's a good dog. He knows how to behave. What they don't know won't hurt them. And if they find out, fuck 'em."
Jack laughed at the unexpected expletive. This didn't seem like the kind of woman his father would hang out with—she couldn't be more unlike his mother—but he liked her.
He told her so.
Her bright dark eyes fixed on him as she smiled, revealing too-bright teeth that were obviously caps.
"Yeah, well, I'll probably like you too if you hang around long enough for me to get to know you." She turned back to the bed. "I do like your father. I've been sitting with him for most of the day."
Jack was touched. "That's very kind of you."
"That's what friends are for, hon. The benison of a neighbor like your father you don't take for granted."
Benison? He'd have to look that up.
He cleared his throat. "So...he's mentioned me?"
Jack was curious how his father had depicted him but didn't want to ask.
He didn't have to.
"He speaks of all his children. He loves you all. I remember how he cried when he heard about your sister. A terrible thing, to outlive a child. But he speaks of you the most."
"Really?" That surprised Jack.
She smiled. "Perhaps because you so vex him."
Vex...another word you don't hear every day.
"Yeah, I guess I do that." In spades.
"I don't think he understands you. He wants to know you but he can't get near enough to find out who you are."
"Yeah, well..."
Jack didn't know what to say. This conversation was sidling into uncomfortable territory.
"But he loves you anyway and worries about you." Her eyes bored into his. "Sad, isn't it: The father doesn't know his son, and the son doesn't know his father."
"Oh, I know my father."
"You may think you do, hon," she said with a slow shake of her head, "but you don't."
Jack opened his mouth to correct her—no way this woman who'd met Dad less than a year ago could know more about the man he'd grown up with—but she held up a hand to cut him off.
"Trust me, kiddo, there's more to your father than you ever dreamed. While you're here, maybe you should try to get to know him better. Don't miss this opportunity."
Jack glanced at the still form pressed between the hospital sheets. "Maybe I already have."
She waved a dismissive hand at the bed. "Thomas will be fine. He's too tough for a little bump on the head to put him down."
More than a little bump on the head, Jack thought.
"The doctors don't seem to think so."
"Doctors." Another dismissive flip of her hand. "What do they know? Most of them have their heads up their tuchuses. Listen to Anya. Anya knows. And Anya says your father's going to be fine."
Foyn? Jack thought, taking on her accent. He's gonna be foyn because you say so, lady? Let's hope so.
She looked up at him. "Where are you staying tonight?"
"Not sure. Passed a Motel 6 on the way—"
"Nonsense. You'll stay at your father's place."
"I...I don't think so."
"Don't argue with Anya. He'd want you to. He'd be very upset if you didn't."
"I don't have a key. I don't even know how to get there."
"I'll show you."
She walked over to the bed and took his father's hand. "Jack and I are going now, Thomas. You rest. We'll be back tomorrow." Then she turned to Jack and said, "Let's go. Where's your car?"
"In the lot. Where's yours?"
"Oh, I don't drive. Trust me, hon, you wouldn't want to be on the same road as me. You're taking me and Oyv home."
## 12
As soon as Anya got in the car she placed Oyv on her lap and lit up an unfiltered Pall Mall.
"Mind if I smoke?"
A little late to object now, Jack thought.
"Nah. Go ahead." He lowered all the windows.
"Want one?"
"Thanks, no. Tried it a few times but never picked up the habit."
"Too bad," Anya said, blowing a stream out the window. "And if you're going to tell me to stop, save your breath."
"Wouldn't think of it. It's your life."
"Damn right. Over the years I've had five doctors tell me to stop. I've outlived every one of them."
"Now I definitely won't say a word."
She smiled and nodded and directed Jack onto a road leading west of town.
The sinking sun knifed through his dark glasses and stabbed at his eyes as he drove westward. He watched what passed for civilization in these parts fall away behind them. The land became progressively swampier, yet somehow managed to retain that burnt-out look.
They passed a freshly tilled field of rich brown earth and wondered what had been growing there all summer. Most of the cultivation seemed given over to palm tree nurseries. Odd to pass successive acre plots, each packed with successively larger palms, all of equal height within their own acre.
Anya pointed a crooked finger at a twin-engine outboard motorboat in someone's front yard.
"'For Sale By Owner'?" she said. "I should hope so. Who else would be selling it? Do they make 'For Sale By Thief' signs?"
A few turns later, past stands of scrub pines, they came to a block of concrete with a blue-and-white-tiled mosaic across its front.
GATEWAYS SOUTH GATEWAY TO THE FINEST IN MATURE LIFESTYLES
The droopy plants and palms framing the sign looked like they were on their last legs.
"Here we are," Anya said. "Home sweet home."
"This is it? This is where he lives?"
"Where I live too. Turn already or you'll miss it."
Jack complied and followed a winding path past a muddy pit with a metal pipe standing in its center.
"That used to be a pond with a fountain," Anya said. "It was beautiful."
All of Gateways South must have been beautiful when it was green, but it looked like it had been particularly hard hit by the drought. All the grass lining the road had been burned to a uniform beige. Only the pines—which probably pre-dated the community—seemed to be holding their own.
They came to a checkpoint divided into VISITORS and RESIDENT arches, each blocked with a red-and-white-striped crossarm. Jack began to angle left toward the visitor gate where a guard sat in an air-conditioned kiosk.
"No," Anya said, handing him a plastic card. "Use this at the other gate. Just wave it in front of the whatchamacallit."
The whatchamacallit turned out to be a little metal box atop a curved pole. Jack waved the card before the sensor and the striped crossarm went up.
"I feel like I'm entering some sort of CIA installation," he said. "Or crossing a border."
"Welcome to one of the retirement Balkans. Seriously though, as we all get on in our years, and become more frail than we like to admit, sometimes this is what it takes to let us feel secure when we turn out the lights."
"Well, as the song says, whatever gets you through the night. But I can't see this place as much of a crime risk. It's in the middle of nowhere."
"Which is exactly why we like a security force guarding the gate and patrolling the grounds." She pointed straight ahead. "Just take this road to its end."
Jack shook his head as he followed the asphalt path that wound past what looked like a par-three golf course. The grass was sparse and brown and the ground looked rock hard. That wasn't deterring the hardcore hackers; he spotted half a dozen golf carts bouncing along the fairways.
"Can't they even water the greens?"
Anya shook her head. "Drought emergency restrictions. No watering at all in South Florida now, even if you have your own well."
He drove on, passing tennis courts—at least their Har-Tru surfaces were still green—and shuffleboard areas, all busy.
"There's the assisted living facility," she said, pointing to a three-story building done up in coral shades. Then she pointed to a one-story structure. "That's the nursing home."
"I don't get it."
"The drought?"
"No. Why my father moved down here."
"Warmth is a factor. You get old, you feel the cold. But the main reason people come to Gateways and other places like it is so they'll never be a burden on their children."
"You talk like you're not one of them."
"I don't have anybody to burden, hon. I'm here for the sun." She held up an arm to show off her wafer-thin, beef-jerky skin. "As you can tell, I love to sit and soak up the rays. I used to sunbathe in the nude when I was younger. If I didn't know how the community board would squawk, I'd do it now."
Jack tried not to picture that.
"But I can't see my father being a burden on anyone."
"Maybe you don't, kiddo, but he can. That's why he's here instead of in some West Palm condo."
"I'm not following you."
"Gateways South—and North and East, for that matter—is a graduated care community that provides for us through the final stages of our lives. We start off in our own little bungalows; when we become more frail we move to assisted living where we have a suite and they provide meals and housekeeping services; and when we can no longer care for ourselves, we move into the nursing home."
"All it takes is money, I suppose."
She snorted a puff of smoke out her nose. "It's not cheap, I can tell you that. You buy your house, you buy a bond, you pay monthly maintenance fees, but your future care is assured. That's important."
"Important enough to hide yourself away down here?"
She shrugged and lit another cigarette—her third since leaving the hospital. "I'm just telling you what I've heard my neighbors say. Me, I'm here because I've got no one to care for me when I start losing it. But the rest, they're all terrified of ending up in diapers in a son or daughter's home."
"Some children might not see that as a burden."
"But what of the parents? They don't want to be remembered like that. Would you?"
"No, I guess not. I know not."
He didn't even want to remember his father as that flattened man pressed between the hospital sheets today. He wanted even less to remember him as an empty-eyed drooler in diapers, a lifetime's store of dignity vanishing like a gambler's paycheck.
He said, "Getting old sucks, doesn't it."
"For some, yes, but not all. The body begins to remind you in ways big and small that you ain't the maidel or boychick you used to be, but you find ways to adjust. It's largely a matter of acceptance." She pointed to the right. "Turn here."
Jack saw a sign for White Ibis Lane as he made the turn. At the end of the short road stood two small, identical houses. The four parking spots in the little cul-de-sac were empty. Jack pulled into one and stepped out of the car. Anya opened her door and let Oyv hop to the ground. The Chihuahua immediately trotted to the nearest palm and let loose a tiny yellow stream against its trunk.
Jack smiled. "That tree looks so dry, I bet it's grateful even for that."
Anya laughed as she straightened slowly from the passenger seat to a standing position. "You'd win. Take a look around while I go in and get the key to your father's place."
Jack felt his eyebrows jump. "He gave you a key?"
She waved a hand at him and laughed. "Nothing like that, kiddo. We traded keys as a precaution. In case of, you know, an emergency."
Jack couldn't resist. He winked at her. "You're sure that's all?"
"What? Thomas with an old skinny-assed crone like me when he has all those other women chasing him? Don't be silly."
Jack held up a hand. "Whoa. Rewind that. My father's got women chasing him?"
"Like vultures, they circle. Let me tell you, Thomas could have his pick of scores—scores."
Jack had to laugh. "I don't believe this. My father, the stud."
"It's not that. It's just that there's four widows for every widower down here. Thomas is an able-bodied man with a good mind and a nice personality. And best of all, he can drive himself. Such a catch, you wouldn't believe."
She reminded him a little of Abe. "Speaking of catches, Anya, if you ever decide to move back north, have I got a guy for you."
She waved her cigarette at him. "Forget about it. My balling days are over."
Jack shook his head. "My father, the catch. Wow." He smiled at her. "So if you're not one of the circling vultures you mentioned, can I ask how you two spend your time together?"
"It's none of your beeswax, hon, but I'll tell you anyway: Mostly we play mahjongg."
Another shock. "My father plays mahjongg?"
"See? I told you there were things you didn't know about him. I'm teaching him and he's getting very good." She tapped her temple. "That accountant's mind, you know."
"My father, the mahjongg maven. I think I need a drink."
"So do I. Come over after you've settled in. We'll knock back a few and I'll give you your first mahjongg lesson."
"I don't know..."
"You have to give it a try. And once you learn, it'll give you and your father something to do together."
When there's frost on hell's pumpkins, Jack thought.
"Anyway," Anya said, pointing to the house on the right, "this one's your father's. Look around. I'll be back in a minute."
She headed toward the house on the left with Oyv trotting behind. Her place was painted...what would they call that color? He'd never heard of white zinfandel pink as a paint shade, but if there were such a thing, that would be the color of Anya's house. Dad's was a more masculine sky blue.
Jack realized he was facing the rear of the house. He tried the door to the jalousied back porch but it was locked. It would have taken all of twenty seconds for him to open it but why bother if Anya had a key.
He strolled the slate walk between the houses. The grass around the stones was as dead and brown as the rest of Gateways South; the foundation plantings along the base of the smooth stucco exterior of his father's place looked thirsty but not as wilted as what he'd seen along the way. Jack suspected him of sneaking them a little water during the night.
Then again, maybe not. His father was such a stickler for rules that he just might watch all his plants die before breaking one.
Jack tried to peek through the windows but the shades were drawn. As he backed away from a window he glanced over at Anya's and stopped dead in his tracks.
Her place looked like a rain forest. Lush greens and reds and yellows of every imaginable tropical plant concealed most of the side of her house, not merely surviving, but thriving. A grapefruit tree, heavy with fruit, stood at a corner. And her grass...a rich, thick, pool-table green.
A little surreptitious sprinkling was one thing, but Anya seemed to be thumbing her nose at the water restrictions.
He noticed a small forest of ornaments dotting her lawn: the usual elves and pink flamingos and pinwheels of various models, but in among them were strange little things that looked homemade, like painted tin cans and bits of cloth on slim tree branches that had been stuck into the ground.
He spotted a name plaque on the side of the house. He stepped closer until he could read it. MUNDY.
He walked on to the front of his father's place. The front yards of the two bungalows sloped down to a pond, roughly round, maybe fifty feet in diameter. As he approached for a look he heard a number of splashes as frogs leaped off the bank for the safety of the water. A black bird stood on the far bank, its chevroned wings spread and held toward the sun as if storing up solar power. The pond stood full and clear, its perimeter rimmed with healthy looking grass and reeds. Beyond it lay a grassy marsh that seemed to stretch forever north and south, but ended at a stand of tall cypresses about a mile due west. Jack knew it was west because the sun was dipping behind the treetops.
He turned and checked out the front of his dad's place. A front porch, covered but open, held a small round table and a pair of chairs, all white. Some sort of flowering vine was trying to crawl up the supporting columns. The floor of the front porch was bluestone slate. A picture window dominated the wall to the left of the door, but vertical blinds hid the interior. He pulled open the screen and tried the front door. Locked, just like the rear.
"Here's the key," Anya said.
Jack turned to find her bustling from her green lawn across his father's brown one, a key held up in her left hand, a cigarette in her right. Oyv paced her.
"Your last name's Mundy?" Jack said. "Any relation to Talbot?"
"The author? Possibly."
"King of the Khyber Rifles was one of my favorite books as a kid."
"Never read it. Here's the key." She pressed it into his palm.
He waved his arm at the vista. "Looks like you two landed prime locations."
"Yes, quite a view. Of course, I was one of the earliest residents so I had my pick. I'm such a part of the scenery they hire me for temp work when they need help. Mostly it's just stuffing envelopes or applying address stickers to advertising brochures. At minimum wage, I won't get rich, but it gets me out of the house. It lets me pull a few strings, too. I helped Tom get this place when it went up for sale."
"Really?" He wanted to ask her why she'd do that for a stranger but didn't know quite how to put it. "I guess he owes you for that."
"He owes me more than he knows." She pointed to the jeweled watch on her wrist. "Don't forget, hon: drinks at my place in an hour."
"I'll have to take a rain check on that," Jack said.
"So, you don't want to drink with an old lady? I understand."
"Hey, come on. That's not it at all. I just want to check with the police on my dad's accident. You know, find out how it happened, if it was his fault, that sort of thing."
She frowned. "Why?"
"Because I want to know."
"Go tomorrow."
He shook his head. "I want to know now."
"Why?"
"Because that's the way I am."
She shrugged and began to turn away. "Suit yourself."
"Can I ask you a question?" Jack said. "Two questions, actually."
"Ask away, hon. Doesn't mean I'll answer."
"Okay. First thing is, how come that pond's full and all the rest are empty?"
"That one's fed by an underground channel from the Everglades."
"The Everglades?"
She gestured to the grassy marsh and the distant cypresses. "There it is. Thomas's place and mine are just about as close as you can legally build to the Everglades. Next question? I don't mean to hurry you, hon, but there's a bottle of wine chilling on my kitchen counter and it's calling my name."
"Sorry. I just want to know how you keep your grass so green in this drought."
"Just a knack, I guess. You could say I've got what they call a green thumb."
"Sure it's not just a wet thumb?"
She frowned and jabbed an index finger at him. "And if I do, so what?"
"Nothing, nothing." Jack held up his hands in a defensive gesture. "I just don't want to see a good friend of my dad's getting in trouble."
She relaxed and puffed her cigarette. "Well, okay. I guess it's natural to think I'm watering. I'm not, but no one'll believe me. Would you believe a couple of members of the board came by and threatened to turn me in if I didn't stop watering."
"What did you tell them?"
"Honey, I said if they catch me with a hose in my hand, they can slap the cuffs on. But until then, they can kiss my wrinkled tuchus!"
Oyv yipped in seeming agreement as Anya turned and marched off.
My kind of gal, Jack thought as he watched her go.
## 13
Jack unlocked his father's front door and stepped into the cool, dark interior. The shades were pulled, probably to keep it cooler during the day and cut down on the electric bill. His father had never been cheap, but he hated waste.
He closed the door behind him and stood in the darkness, listening, feeling the house. Somewhere ahead and to the left a refrigerator kicked on. He sniffed. Onions...a hint of sautéed onions lingered in the air. Dad's doing? He'd always been something of a chef, probably more so out of necessity after Mom's death, and had this thing for onions; liked them on just about everything. Jack remembered one Sunday morning as a kid when he'd sautéed a bunch and put them on pancakes. Everyone had started out complaining but they turned out to taste pretty good.
Jack stepped over to the picture window and pulled the blinds, letting in the fading sunlight. Dust motes gleamed in the air. He pulled up the rest of the shades and started exploring.
The front area was a large multipurpose living room/dining room angling into a small kitchen. That was what Jack wanted. He opened the fridge and found a six-pack and a half of Havana Red Ale. He checked the label: brewed in Key West. Another local brand. Why not? He popped the top and took a pull. A little bitter, not as good as Ybor Gold, but it would do.
He spotted a bottle of Rose's lime juice on a door shelf. On a hunch he opened the freezer and there it was: a frosty bottle of Bombay Sapphire. Looked like Dad still liked a gimlet now and then.
He wandered through the front room and recognized some of the paintings from the family home in Jersey. He noticed a trophy shelf on the south wall and moved in for a closer look. First place in the men's doubles in tennis—no surprise there—but what was this? A plaque for second place in the men's bocce tournament?
My father, the bocce champ. Jeez.
He called Gia to give her the medical report on his father. She said how sorry she was that the news wasn't better. Jack said hello to Vicks, then told them he'd call back later.
After he hung up he stepped into one of the bedrooms. This looked like a guest room/office: a bed, a dresser, and a desk with a computer and a printer. Jack saw a list of buy-sell confirmations in the printer tray. Looked like Dad was still day trading. He'd started it way before it became the rage in the nineties and had made enough to retire on. He'd tried to get Jack into it once, saying that if you were vigilant and knew the ropes, it didn't matter if the market was up or down, you could make money every day.
Not if you don't have a real Social Security Number, Dad.
He moved on to the other bedroom, more cluttered and obviously Dad's. He stopped in the doorway, taken aback by the photos filling the walls. Mostly Mom, Tom, and Kate at various ages, salted with a few of Jack as a kid. Here were the five of them as they embarked on their one and only family camp-out...what a disaster that had been.
Memories flooded back, especially of Kate—as his teenaged big sister, looking out for him...as an adult, dying in front of him.
He quickly turned away and checked the closet. There they were: Dad's ugly Hawaiian shirts. He pulled one out and looked at it: huge bulge-eyed goldfish swimming in a green fluid that could only be bile. Jack tried to imagine himself wearing this and failed. People would...notice him.
As he replaced the shirt he noticed a gray metal box on the shelf above the rod. He reached for it, hesitated, then took it down. He thumbed the latch but it was locked. He shook it. Papers and other things shuffled and rattled inside.
Locked...that piqued his curiosity. But this was his father's, not his, and probably locked for a good reason. He should put it back, he knew he should, but...
What would his father keep locked up when he was the only one in the house?
Jack looked at the little keyhole. Eminently pickable. All it would take was—
No. Mind your own business.
He put it back on the shelf and returned to the main room. He repressed a shudder. Time to visit the cops.
Jack found the phone book and looked up the address of the local police station. He'd planned to call them for directions, but why not see if he could learn what he wanted over the phone. Anything to avoid setting foot in a police station.
He dialed the number and was shuffled around until he wound up with Anita Nesbitt, a pleasant-sounding secretary who said she'd see what she could do for him.
"I'm assuming I'll need a copy of the accident report for the insurance," he told her. "You know, to get the car fixed."
"Okay. Here it is. I'll put a copy aside and you can pick it up."
"Any way you can mail it?"
"I suppose. We have his address on the report. How is your father, by the way? I heard he was pretty banged up."
"Still in a coma." A thought struck him. "Was anyone else injured?"
"Not that we know of," she said. "It was hit and run."
Jack swallowed. Those last three words sent a wave of unease through his gut.
"Hit and run?"
"Yes. It's under investigation."
"Save your stamp and envelope," Jack told her. "I'm coming down to pick up that report."
## 14
Dusk had arrived and the air was cooling enough to bring out the mosquitoes as Jack reached the mustard-yellow building with a two-story center flanked by single-story wings that served as Novaton City Hall. A skeletal clock tower, too modern for the rest of the building, loomed over the high-columned entrance. A green roof, front portico, and awnings completed the picture. A sign said the police station was toward the rear on the left side.
Steeling himself, he stepped inside and asked for Ms. Nesbitt. The desk sergeant directed him to her office. Walking down the hall, passing cops moving this way and that, he felt like Pee Wee Herman at a Klan rally. If anyone peeked under the sheet...
He hoped no one asked for ID to prove his relationship. His father's last name was not Tyleski.
Ms. Nesbitt turned out to be a plump and pleasant little woman with glossy black skin, short curly hair tight against her scalp, and a radiant smile.
"Here's the accident report," she said, handing him a sheet of paper.
Jack took a quick look at it; he meant to read it later but his eyes were drawn to the diagram of the accident site.
"Where's this intersection?" he said, pointing to the sheet. "Pemberton Road and South Road?"
She frowned. "They cross in the swamps on the fringe of the Everglades, way out in the middle of nowhere."
"What was my father doing out in the middle of nowhere?"
"That's what we're hoping you could tell us," said a voice behind him.
Jack turned to see a young, beefy cop with buzz-cut hair. His massive biceps stretched the seams of the short sleeves of his uniform shirt. His expression was neutral.
"This is Officer Hernandez," Anita said. "He took the call and found your father."
Jack stuck out a hand he hoped wasn't too sweaty. "Thanks. I guess you saved my father's life."
He shrugged. "If I did, great. But I hear he's not out of the woods yet."
"You've been keeping track?"
"We'd like to talk to him, get some details on the accident. Any idea what he was doing out there at that hour?"
Jack glanced down at the report. "What hour?"
"Around midnight."
Jack shook his head. "I can't imagine."
"Could your father have been mixed up in something he shouldn't have been?"
"My dad? Into something shady? He's like..."
Like who? Jack tried to think of a public figure who was a true straight shooter, whose integrity was beyond reproach, but came up blank. There had to be somebody. But no one came to mind. He almost said Mr. Deeds but Adam Sandler had screwed up that reference.
"He's like Casper Milquetoast." Jack saw no hint of recognition in Hernandez's face. "He's a regular everyday Joe who minds his own business and doesn't take chances. My dad is not a risk taker." Jack didn't want to call him timid, because he wasn't. Once he took a position he could be a bulldog about defending it. "He lived in Jersey most of his life, not fifty miles from Atlantic City, and in all that time I don't think he once visited the casinos. So the idea of him being involved in something even remotely criminal is, well, crazy."
Hernandez shrugged. "Doesn't have to be criminal. He could have been fooling around with the wrong guy's wife or—"
Jack held up his hands. "Wait. Stop. Not him. I promise you. No way."
Hernandez was studying him.
Uh-oh. Here it comes.
"Do you live around here?"
"No. I'm still in Jersey." Where did Tyleski live? All these identities...after a while they ran together in his head. "In Hoboken."
"How often do you see your father? How many times a year do you visit him?"
"He hasn't been here that long. Less than a year."
"And?"
"And this is my first visit."
"Do you talk often?"
"Uh, no."
"Then you really don't know that much about your father's life down here."
Jack sighed. There it was again. "I guess not. But I know what kind of man he is, and he's not a sneak or a liar, and people who are have no place in his life."
But how much more do I know? he wondered. What do you know about anyone, even someone who raised you, beyond how they act and what they've told you about themselves?
Anya's comment from this afternoon stole back to him: Trust me, kiddo, there's more to your father than you ever dreamed.
He hadn't paid much attention to it then, but now with Dad the victim of a hit-and-run accident in the middle...
"Say, if he got hit in the middle of nowhere..." He turned to Anita. "Didn't you say a call came in?"
She nodded. "It's in the report."
"But that means someone must have witnessed it."
"That's the obvious conclusion but..." Hernandez's macho cop persona wavered. Just a little.
"But what?"
"Well, it took me about twenty minutes to reach the intersection, and when I got there, your father's car was the only vehicle at the scene and it looked like the accident had just happened. The car was sitting across Pemberton Road. From the debris spray I reckoned your father had been proceeding west on Pemberton. He had a stop sign at South. Looked like he was almost halfway across when he got hit. Maybe he hadn't been paying attention, maybe he ran the stop sign, maybe he was having a little stroke. All I know is that something hit him hard enough to spin the car ninety degrees, and there was no one else in sight when I got there."
"Then who called in?" Jack said. "Man or woman?"
"Tony, the desk sergeant took it. I asked him but he couldn't tell. Said the person was whispering, real quick like. Said, 'Bad accident at Pemberton and South. Hurry.' That was it."
"Did they ID the number?"
Hernandez glanced at Anita. "That's another thing we can't figure out. The call came from a pay phone outside the Publix."
"Publix? What's a Publix?"
"Like a Winn-Dixie."
"I'm sorry." Was this another language they were speaking? "I'm from up north and I still don't—"
"Publix is a chain of grocery stores down here," Anita said. "It's like..." She snapped her fingers. "I've been up your way. What's it called...? A&P. That's right. Like an A&P."
"Okay. And where's this Publix?"
"About three blocks from here."
"What? But how? That's..."
"Impossible?" Hernandez said. "Not really. The hit-and-run driver might have been into something illegal and that's why he didn't stop. But he might have had an attack of conscience and called a friend and told him to call it in from a public phone so we couldn't ID him."
"Thank God for attacks of conscience," Anita said.
Hernandez nodded. "Amen to that. All I can say is it's a good thing we got the call when we did, otherwise your father might have been DOA."
## 15
Jack's mind raced as he drove toward the south end of Novaton.
After telling Hernandez where he was staying and promising not to leave without checking in with him—in case the cops had more questions—he'd left the police station in something of a daze. But not before getting directions to the impound lot where his dad's car had been towed.
A hit-and-run driver damn near kills his father but has enough Good Samaritan in him to arrange for the cops to be notified. A mixture of bad luck and good.
But the big question still remained: What the hell was Dad doing out there in the swamp at that hour?
The light had pretty well faded by the time Jack reached the south end of town. As Hernandez had told him, he passed an old limestone quarry, then a trailer park, then came to the impound lot.
It turned out to be a combination junkyard/used-car lot called Jason's. The place was closed. Jack could have climbed the chain-link fence but didn't want to risk an encounter with a guard dog, so he wandered the perimeter, squinting at the wrecked cars within.
The accident report said the make was—what else?—a silver Mercury Grand Marquis, the unofficial state car of Florida, and gave the plate number. Jack found it near the gate. He clutched the fence and gaped at the front end. The bumper was gone, the right front fender was a memory, the windshield was a caved-in, spider-webbed mess, the engine block was tilted and canted and twisted to the left.
Had he run into a tank?
Jack's fingers squeezed the chain-linked wire, making it squeak. Who'd done this and run off? Maybe Dad had been thinking of something else and hadn't seen the stop sign. Okay. His bad, not the other driver's. But still...what the hell had the other guy been driving?
## 16
Jack's stomach started to growl as he left Jason's. He realized he hadn't eaten anything since the crabcake sandwich at Joanie's. He'd seen a Taco Bell on the way in and couldn't help thinking of little Oyv. He stopped for a couple of burritos and a Mountain Dew to go.
As he ate and drove, he decided to swing by the hospital on his way back to Gateways South and have another look at his dad.
On the third floor, Jack met Dr. Huerta coming out of the room, followed by a red-haired nurse. Her picture ID badge read C. MORTENSON, RN.
"How is he? Any change?"
Dr. Huerta shook her head and brushed back a vagrant strand of hair. She looked tired.
"The same. Still a score of seven. No better but, thankfully, no worse."
Jack supposed that was good. But he hadn't come here tonight just to see his father.
"Where are his personal effects?"
"Effects?"
"You know, his clothes, his wallet, any papers he had on him."
Dr. Huerta glanced at Nurse Mortenson who said, "They're in a locker by the nurses' station. I'll get them for you."
Dr. Huerta moved on and Jack stepped into his father's room. He stood by the bed, watching him breathe, feeling helpless and confused. This wasn't right. His father should be at Anya's place, drinking gimlets and playing mahjongg instead of lying here unconscious with tubes running in and out of him.
Mortenson came in with a clipboard and clear plastic bag.
"You'll have to sign for this," she said. As Jack made an illegible scrawl across the sheet, she added, "We couldn't keep his clothes. The blood, you know."
"But you emptied his pockets first, right?"
"I assume so. That's done in the ER, long before he gets to us."
Jack handed back the clipboard and took the bag. Not much in it: a wallet, a watch, some keys, and maybe a buck's worth of change.
When the nurse was gone, Jack checked the wallet: an AmEx and a MasterCard, AARP and AAA cards, a Costco card, seventy-some dollars in cash, and a couple of restaurant receipts.
Jack dropped it back into the bag. What had he been hoping for? A note with a cryptic message? A scrap of paper with a hastily scribbled address he could check out?
Watching too many mystery movies, he told himself.
Maybe there is no mystery. Maybe it was just an accident. Maybe Dad was simply out for a drive and wound up in the wrong place at the wrong time...got clocked by accident by someone who wasn't quite legit and couldn't hang around to explain himself to the police.
Jack understood that. Perfectly.
Just an accident...a random collision...
But his gut wasn't buying. Not yet at least.
Jack looked down at his father.
"Have you been holding out on me, Dad?"
No response, of course. He patted his father's knee through the sheet.
"See you tomorrow."
## 17
Fortunately Anya had left her gate passcard in Jack's car. He used it to breeze through the resident's arch. The old lady's lights were out by the time Jack reached the house. Her lawn ornaments clinked and clanked and whirred in the dark.
Once inside, he went straight to his father's room and took out the metal lockbox.
"Sorry, Dad," he muttered as he carried it to the kitchen.
He hated invading his father's privacy, but this box might hold an explanation as to why he'd been out in the swamps after midnight instead of home in bed.
First, a beer. He grabbed another Havana Red from the fridge, then searched the bathroom for a pair of tweezers. He found one, and twenty seconds later the lid popped open. Jack hesitated. Maybe there were things in here his father didn't want anyone to know about. And maybe Jack wouldn't want to know about them once he saw them. Maybe parents should be able to keep their secrets.
All fine and good when they weren't the comatose victim of a hit and run.
Jack lifted the lid.
Not much there. A handful of black-and-white photos, now sepiaed with age, and something that looked like a small jewelry case. He checked the photos first. Mostly soldiers. He recognized his dad in a few of them—he didn't recall him ever having that much hair—but most were of other uniformed guys in their late teens or early twenties posing awkwardly for the camera against unfamiliar landscapes. Jack spotted a pagoda-like building in the background of one.
Korea. Had to be. He knew his dad had been in the war, in the Army, but he'd never wanted to talk about it. Jack remembered pressing him for war stories but getting nowhere. "It's not something I care to remember," he'd always say.
The last photo was a posed shot of eight men in fatigues, four kneeling in front, four standing behind, grinning at the camera. His father was second from the left, standing. It looked like a plaque had been set up in the right foreground but that corner of the photo was missing. It appeared to have been torn off.
Jack studied the other seven men, looking for a connection to his father. Who were they? They all looked so young. Like a high school varsity basketball team. It looked like a graduation photo. But from what?
Maybe he'd never know.
He put down the photos and picked up the jewelry case. Something rattled within. He snapped it open and found two medals. He didn't know much about military decorations but one he immediately recognized.
A Purple Heart.
His father's? That meant he'd been wounded. But where? The only scar he'd ever seen on his father was from his appendectomy. Maybe this belonged to someone else...a dead war buddy that his father wanted to remember?
Nah. Purple Hearts tended to be kept by the loved one's family.
Which meant this was probably his father's.
He checked the other medal: a gold star hanging on a red-white-and-blue ribbon; a smaller silver star was set at its center. This could be a Silver Star. Wasn't that for extraordinary bravery in battle?
Trust me, kiddo, there's more to your father than you ever dreamed.
I guess you got that right, lady. Maybe I should have stayed in touch more.
Funny...just a few months ago he wouldn't have felt this way. But after reconnecting with Kate...
With frustration wriggling under his skin like an itch he couldn't scratch, Jack replaced the contents to the box in roughly the same order that he'd found them. He'd wanted answers, but all this damn box had provided was more questions.
He returned it to the closet shelf, then headed back to the kitchen for another beer. Along the way he spotted his father's watch on the table. He hadn't noticed the cracked crystal when he'd brought it home from the hospital. He checked it out. An old Timex. No, not old—ancient. The wind-up type. Typical of him: If the old one still works, why get a new one? This Timex had taken a licking but hadn't kept on ticking. It had stopped at 12:08.
Wait a sec...
Jack pulled the accident report out of his pocket and unfolded it. He'd scanned through Officer Hernandez's report. He'd mentioned a call coming in to the station at...where was it? Here.
11:49 P.M.
But that would mean the accident had been reported before it happened. No way. His father's watch must have been set ahead. Some people did that. Or maybe he'd forgotten to wind it.
But not his father. He'd always been a stickler for the correct time, down to the minute. And he'd always wound his watch at breakfast. Jack had seen him do it a million times.
Hernandez was mistaken about the time of the call. Had to be. But for all his brawn the cop had seemed like a pretty tight, spit-shine type. And hadn't he said that even though it took him twenty minutes to reach the accident, it looked like it had just happened?
Shaking his head, Jack went to the fridge. He decided against another beer. Right now he needed a gimlet.
## WEDNESDAY
## 1
Jack awoke with a buzzing in his ears. At first he thought it was a mosquito, but this was lower pitched. Then he thought it might be gimlet-related, but he'd had only two. Finally he realized it was coming from outside the window. He lifted his head and looked around, momentarily disoriented by the unfamiliar room.
Oh, yeah. He was at Dad's place. In the front room. Must have fallen asleep on the couch. He'd found Rio Bravo playing on TNT or some such station and had watched it for about the thirtieth time—not for John Wayne or Dean Martin, and certainly not for Ricky Nelson, but for Walter Brennan. Hands down, Stumpy was his best part, best job, ever—except maybe for his Old Man Clanton in My Darling Clementine. Old Walt made the movie for Jack.
But where was that buzzing coming from?
He rolled off the couch, padded to the kitchen, and squinted through the window.
A groundskeeper was running a weed whacker along the edge of the dead grass bordering the foundation plantings. Was that a long-sleeved flannel shirt he was wearing? In this weather? Where Jack came from a long-sleeved shirt in the summer meant one thing: junkie.
But the weed whacker...he blinked and shook his head...it looked like it was coming out of the guy's right sleeve.
The rest of Jack's clothes were still in the car so he had to go out anyway. Maybe he could get a closer look along the way.
The heat and humidity hit him like a wave as he stepped outside. Barely 8:30 and already it was cooking. As he rounded the corner, the groundsman stopped working and stared at him, then turned off his weed whacker.
"You ain't Tom. Whatta you doin here?"
"I'm his son."
And yes, that was a flannel shirt he had on. He wore green work pants and a tattered olive drab boonie cap. His eyes were a piercing blue, but the left angled to the outside—the kind of eye known on the street as a bent lamp. Yet even this close Jack couldn't see his right hand. The weed whacker seemed to be growing out of the sleeve. Jack thrust out his own right hand in hopes of getting a look.
"My name's Jack."
The groundsman used his left hand to give Jack's a squeeze. "Carl."
So much for that strategy.
"How come you're out here so early?" Jack said. "You can't have much to do with this drought."
"Be surprised," Carl said. "Grass won't grow, tropical plants get all curly and dried up, but the weeds...the weeds do just fine. Never able to figure that out."
"Maybe they should all cultivate weeds," Jack said.
Carl nodded. "Fine with me. Green is green." He glanced at Jack. "Miss Mundy told me about your daddy. How's the old guy doin?"
"Still in a coma."
Jack fought the urge to sidle to his right to put himself in line with Carl's left eye.
"Yeah?" He shook his head. "Too bad, too bad. Nice guy, your daddy. He was one of the good uns."
"'Was'? Hey, he's not gone yet."
"Oh, yeah. Right, right. Well, let's hope he pulls through. But bein so close to the Glades and all..."
"The Everglades? What's wrong with that?"
Carl looked away. "Nothin. Forget I said it."
"Hey, don't leave me hanging. If you're going to start a thought, finish it."
He kept his gaze averted. "You'll think I'm loco." You don't know loco like I know loco, Jack thought.
"Try me."
"Well, all right. Gateways here is too close to the Glades. It's been mistreated for years and years now. All the freshwater runoff it's upposed to get from upstate, you know, from Lake Okeechobee, it's mostly been channeled away to farms and funeral-parlor waitin rooms like Gateways. Everywhere you look someone's filling in acres of lowlands and paving it over to build a bunch of houses or condos. The Glades been hurtin for years and years, but this year's the worst because of the drought. Summer's upposed to be our rainy season but we ain't had barely a lick."
"There's still water out there, though, isn't there?"
"Yep, there's water, but it's low. Lower than it's ever been in anyone's memory. And that could be bad. Bad for all of us."
"Bad how?"
"Well, maybe things that always used to be underwater ain't under no more."
Where was this going? Was it going anywhere?
"Carl—"
He stared toward the Everglades. "The good thing bout your daddy's and Miss Anya's places here on the pond is you never have to look into someone else's backyard..."
Jack glanced out at the endless expanse of grass. "Yeah. A panoramic view."
"Pan-o-ramic?" Carl said carefully. "What's that?"
Jack wondered how to explain it. He spread his arms. "It means wide angle...a wide view."
"Pan-o-ramic...I like that."
"Fine. The panoramic view is the good thing, but I've got a feeling you were about to tell me a down side."
"I was. The bad part is...they's real close to the Glades and the Glades ain't happy these days. You might even say it's kinda pissed. And if it is, we'd all better watch out."
Jack stared across the mile or so of grass at the line of trees. He'd seen a bunch of weird things lately, but an angry swamp...?
You were right, Carl, he thought. I do think you're loco.
## 2
Semelee stood on the lagoon bank with Luke and watched the small dredgin barge suck wet sand out of the sinkhole and deposit it into one of the even smaller, flat-bottomed boats it had towed along behind it. Excess water ran out the gunwales and into the lagoon. The clan had moved the houseboats aside to give the barge access to the hole.
"I still can't believe you done this, Semelee," Luke said. "You of all people."
Semelee had been surprised herself. She didn't like outsiders gettin anywheres near the clan's lagoon, and especially near the sinkhole, but these folks had offered too much money to turn down.
"You been sayin that for two weeks now, Luke. Every time the barge shows up you say the same thing. And every time I give you the same answer: We can use the money. People're pretty tight with their spare change these days, in case you ain't noticed."
"Oh, I noticed, all right. Probably cause they ain't got all that much to spare. But I still don't like it, specially this time of year."
"Don't worry. They'll be outta here before the lights come. The deal I made with them was they had to finish up their business before this weekend. The lights'll start comin Friday night. Told them Friday was a stone-solid deadline. Didn't care how much they offered me, by sundown on Friday, they're gone."
"Still don't like it. This is our home. This is where we was born."
"I know, Luke," she said, rubbing his back and feeling the sharp tips of the fins through the cloth. "But just think. The top of the sinkhole is above water for the first time anyone remembers. Maybe for the first time ever. When the lights come this time, they won't have to shine through the water. They'll shine straight out into the night. That's never happened before, at least not in anyone's memory."
"I ain't so crazy about that neither." He rubbed a hand over his face. "My daddy said them lights made us the way we is, twisted us up, just like it's twisted the trees and the fish and the bugs around here. And that's from when they was just shining up through the water. What happens this year when there ain't no water?"
Semelee felt a thrill at the prospect. "That's what I want to see."
The lights had been comin twice a year—at the spring and fall equinoxes—for as long as anyone could remember. Her momma had told her they'd kept that schedule every year since she'd been born, and her momma had told her the same thing.
But Semelee's momma'd said that years back the lights started gettin stronger and brighter. And it wasn't long after that, maybe a few years, that the people livin around the lagoon started noticin changes in the plants and the fish and things around the sinkhole. It started with the frogs missin legs or growin extra ones. Then the fish started lookin weird and the plants started gettin twisted up.
All that was bad enough, but when the lagooners' kids started bein born dead or strange lookin, the lagooners moved out. Not as a group to the same place, but piecemeal like, in all different directions. Some stayed as close as Homestead, some as far as Louisiana and Texas. After they moved away, they stopped havin strange kids and they was happy about that.
But the strange kids they already had wasn't happy. Not one bit. Not because they was all mistreated by people as they was growin up—Semelee hadn't been alone in that—but because when they all finally growed up they felt like somethin was missin in their lives.
One by one they all—all the misshapen ones—found their way back here to the lagoon and learned that this was where the itch stopped, this was where they felt whole, where they belonged. This was home.
And home was where your family lived. They came to call themselves a clan, and all decided to stay here on the lagoon.
Yet even with this big family-type gang around her, Semelee still felt a yearning emptiness within. She wanted more, needed more.
"Why do they hafta take our sand? There's plenty of sand around. Why they want ours?"
"Don't rightly know," Semelee said.
"Who is they, anyway?"
"Blagden and Sons. You know that."
"Yeah, I know the name, but that's all it is: a name. Who are they? Where do they come from?"
"Don't know, Luke, but their money's good. Cash up front. That's bout as good as it gets."
"Do they know about the lights?"
"That one I can answer: Yeah, they know about the lights."
Some guy named William somethin from this company called Blagden and Sons come around in a canoe a few weeks ago askin if anyone'd been seein funny lights about this time of the year. The clan folk he talked to sent him to Semelee since she was sorta the leader round here. Not that she'd ever looked to be the leader, but it seemed whenever somethin needed decidin, she wound up the one who did it.
Semelee played it cagey with this William fellow until she was pretty sure he wasn't no tour-guide type or scientist or anything like that, and wouldn't be bringing boatloads of strangers or teams of pointy heads to peek or poke at the clan and the sinkhole. Nope, all William wanted was to haul off the dirt and sand from around where they'd seen the lights.
When Semelee had told him they'd been comin up through this sinkhole that used to be underwater but was now gettin dry, he got all excited and wanted to know where it was. Semelee pretended she wasn't gonna tell him, and held off even when he offered money. So he offered more money and more money until Semelee had to say yes. Maybe she could've held out for even more, but there weren't no sense in gettin all greedy about it.
When she'd took him to the sinkhole she thought he was gonna pee his pants. He danced around it, callin it a senn-oaty or somethin like that. When she asked him what he was talkin about he spelled it for her: C-E-NO-T-E. Told her it was a Mex word and you said it like coyote. Semelee liked sinkhole better.
The dredgin was all hush-hush, of course. The clan wasn't upposed to be livin here on the lagoon, this bein a National Park and all, and Blagden and Sons wasn't upposed to be takin the sand.
"Matter of fact," she told Luke, "I'm pretty sure they want the sand because of the lights."
"That's kinda scary, dontcha think? Them lights ain't natural. They changed us and everythin around them. Probably even changed the sand in that hole."
"Probably did."
Luke looked uneasy. "What on earth could they want it for? I mean, what're they gonna do with it?"
"Can't rightly say, Luke. And I don't rightly care. That ain't our worry. What I do know is that our little sinkhole is gonna be a lot deeper without all that sand. And that just may mean that the lights'll be brighter than ever. When the time comes maybe someone can even look down into that hole and see where they're comin from."
"Who's gonna do that?" Luke said.
Semelee kept her eyes on the rim of the deepening hole. "Me."
Luke grabbed her arm. "Uh-uh! You ain't! That's crazy! I won't let you!"
She let Luke have sex with her once in a while when she felt the need, and that probably was a mistake. She'd told him flat out from the git-go that it didn't mean nothin, that they was just now-and-again fuck buddies and that was all there was to it, but she'd probably made a mistake lettin it get started. Still, every so often she needed to get laid and Luke was the least ugly of anyone else in the clan. Trouble was, it let him feel like he owed her, like he had to protect her or somethin.
If anyone needed protectin, it wasn't her.
"You got nothin to say about it, Luke," Semelee told him as she wrenched her arm free of his grasp. "Now lemme be. I gotta get to town."
"What for?"
She flashed him a sly smile. "I'm joinin the nursin profession."
He shook his head. "What? Why?"
Semelee felt the smile melt away in a blaze of anger. "To finish your half-assed job from the other night!"
## 3
As Jack stepped out of the elevator on the hospital's third floor, he spied Dr. Huerta waiting to get in.
"Any change in my father?"
She shook her head. "Stable, but still level seven."
"How long can this go on?" he said. "I mean, before we start thinking about feeding tubes and all that?"
She stepped into the elevator. "That's a bit premature. I know it must seem like a long time to you, but it's been less than seventy-two hours. The IVs are perfectly adequate for now."
"But—"
The elevator doors slid shut.
Jack walked down the hall to his father's room, wondering if Anya would be there. He'd stopped by her place before leaving this morning, threading his way through the gizmos crowding her lawn, to offer her a ride to the hospital if she needed it. But she hadn't answered his knocks.
Normally that wouldn't have bothered him, but with old folks...well, you never knew. She could have had a stroke or something. Jack had peered through the front door glass but hadn't seen anyone on the floor or slumped in a chair. Then he'd remembered Oyv. The little dog would have been barking up a storm by then if he'd been around.
But Anya wasn't in his father's room either—he checked the corners and behind the curtains, just to be sure. Empty except for the patient.
He stepped to the side of the bed and gripped the limp right hand. "I'm back, Dad. Are you in there? Can you hear me? Give a squeeze, just a little one, if you can. Or move just one finger so I know."
Nothing. Just like yesterday.
Jack pulled up a chair and sat at the bedside, talking to his father as if the old guy could hear him. He kept his voice low—pausing when the nurses buzzed in and out—and discussed what he'd learned about the accident and the conflicting information, dwelling on the time discrepancies between the report and his father's watch. He'd hoped talking it out would clarify the incident for him, but he was as confused afterward as before.
"If only you could tell me what you were doing out there at that hour, it would clear up a whole lot of questions."
Once off the subject of the accident, he thought he'd run out of things to say. Then he remembered the pictures in his father's room and decided to use them as launch pads.
"Remember the family camping trip? How it never stopped raining...?"
## 4
After an hour or so of talking, Jack's mouth was dry and his vocal cords felt on fire. He stepped into the bathroom to get a drink of water. As he was finishing his second cupful his peripheral vision caught a flash of white. He turned to see a nurse approaching his dad's bed. She hadn't been around before; he was sure he would have noticed her if she had. She was pretty in an odd way. Very slim, almost to the point of boyishness, and with her dark skin—made all the darker by the contrast of her white uniform—prominent nose, and glossy black hair trailing most of the way down her back in a single braid, Jack thought she might be part Indian—not the Bombay kind, the American kind.
She had her hand in the pocket of her uniform—little more than a white shift, really—and seemed to be gripping something.
Jack was about to step out of the bathroom and say hello when he noticed something strange about her. Her movements were odd, jerky. She'd slowed her progress toward the bed and seemed to be straining to move forward, as if the air was holding her back. He saw sweat break out on her forehead, watched her face flush and then go pale as she forced herself forward another step. He watched her throat working, as if she was trying to keep from vomiting.
Jack stepped out and approached her.
"Miss, are you all—?"
She jumped, twisted toward him, staring with wide, confused, onyx eyes. Her hand darted from her pocket to a thong tied around her neck, and Jack thought he saw something move in the pocket.
She shook her head, pulling on the slim leather thong around her neck. It snapped but she barely seemed to notice. She was drenched in sweat.
"Who—?"
Before Jack could reply she turned and staggered out of the room. He started to go after her but heard a groan from the bed.
"Dad?" He rushed over to the bed and grabbed his father's hand again. "Dad, was that you?"
He squeezed the fingers—gently at first, then harder. His father winced, but Dr. Huerta had said he was responsive to pain. After shaking his father's shoulder and calling to him, all with no response, he backed off. Nothing happening here.
He went out to check on that nurse. Something wrong about her...besides looking sick.
At the nursing station he found a big, brawny, gray-haired nurse who seemed to be in charge. Her photo ID badge read R. SCHOCH, RN.
"Excuse me," he said. "A nurse just came into my father's room, then turned and ran out. She looked kind of sick and I was wondering if she was okay."
Nurse Schoch frowned—or rather, her frown deepened. It seemed to be her only expression. "Sick? No one said anything." She looked around at the assignment board. "Three-seventy-five, right? What was her name?"
"I didn't get a look at her badge. Come to think of it, I don't think she was wearing one."
"Oh, she had to be. What did she look like?"
"Slim, dark, maybe five-three or so."
Schoch shook her head. "No one like that here. Not on my shift, anyway. You sure she was a nurse?"
"I'm not sure of a lot of things," Jack muttered, "and that's just been added to my list."
"She could have been from housecleaning, but then she would have been in gray instead of white—and she'd still have to have a badge." She picked up a phone. "I'll call security."
Jack wished she wouldn't—he didn't want rent-a-cops messing into this—but couldn't think of a reason he could tell Schoch.
"Yeah, okay. I'll be back in my father's room."
He'd been keeping an eye on the door, making sure no one else went in there. When he returned, he checked his father to see if he'd moved—he hadn't—then went to the window and looked out at the parking lot. He saw a slim woman in white walking away through the lot. Heat from the late-morning sun made her shimmer like a mirage.
It was her. Couldn't mistake that long braid. And now she was climbing into the passenger side of a battered old red pickup.
Jack dashed into the hall in time to see the elevator doors closing. Too slow anyway. He found the stairs and raced down to the first floor. By the time he hit the parking lot, the pickup was gone. But he kept moving, running to his Buick and gunning out to the street. He flipped a mental coin and turned right, telling himself he'd give this ten minutes and then call it quits.
He'd traveled about half a mile when he spotted the truck, stopped at a red light two blocks ahead.
"Gotcha," he said.
When the light changed he followed the truck out of town and into the swamps. Somewhere along the way the pavement ended, replaced by a couple of sandy ruts flanked by tall, waving reeds. He lost sight of the truck for a while but wasn't going to worry about that unless he came to a fork. Better to stay out of sight. Luckily there were no forks, and before too long he was pulling into a clearing at the edge of a small, slow-moving stream.
The red pickup sat there, idling, while the woman in white rode downstream in a small, flat-bottomed motor boat piloted by a hulking man in a red, long-sleeved shirt. Jack jumped out of his car and ran to the bank, waving his arms, calling after them.
"Hey! Come back! I want to ask you something!"
The woman and the man turned and stared at him, surprise evident on their faces. The woman said something to the man, who nodded, then they both turned away and kept moving. He saw the name on the stern: Chicken-ship.
"Hey!" Jack shouted.
"Whatchoo wanner for?" said a voice from behind.
Jack turned and saw a man with a misshapen head leaning out the driver window of the pickup. With his bulbous forehead, off-center eyes, and almost non-existent nose he reminded Jack of Leo G. Carroll from the opening scenes of Tarantula. This guy made Rondo Hatton look handsome.
"I want to talk to her, ask her a few questions."
"Looks to me like she don't wanna talk to you." His voice was high and nasal.
"Where does she live?"
"In the Glades."
"How do I find her?"
"You don't. Whatever it is, mister, leave it be."
Suddenly another guy, thinner and only marginally better looking, jumped into the pickup's passenger seat.
Where'd he come from?
The new guy slapped the driver on the shoulder and nodded. Neither looked too bright. If someone suggested playing Russian roulette with a semi-automatic, they'd probably say, "Cool!"
The driver gave Jack a little two-finger salute. "Welp, nice talkin to ya. Gotta go now."
Before Jack could say anything the guy threw the truck into gear and roared off. Jack raced back to his car. If he couldn't follow the girl, then he'd tail these two. Sooner or later they had to—
He skidded to a halt when he saw the Buick's flat front tire, and the gash in its side wall.
"Swell," he muttered. "Just swell."
## 5
"I don't get it," Luke said as he piloted the Chicken-ship deeper into the swamp. "Who was that guy?"
Semelee pulled off the black wig and shook out her silver white hair. She didn't feel like talkin. Her stomach still wasn't right.
"He saw me in the room. I think he might be the old guy's kin."
"That why he was trailin you?"
"Maybe. I don't know. All I know is I felt so strange in that room. It started as soon as I stepped through the door and got worse and worse the closer I got to the old man's bed. I started feeling sick and weak, and the air got so thick I could barely breathe. I tell you, Luke, all I wanted to do was get out of there and get far, far away as fast I could."
"Think it was the guy?"
"Could've been, but I don't think so."
This man wasn't just a guy, wasn't just one of the old man's kin. This man was the one she'd sensed coming for the past two days, and he was special. She sensed something about him...a destiny, maybe. She didn't know exactly what, she just knew he was special.
So am I, she thought. But in a different way.
Maybe she and this new man was destined to be together. That would be wonderful. She liked the way he looked, liked his hair, his build—not too beefy, not too slight—liked his brown eyes and hair. She especially liked his face, his regular, normal face. Hangin round the clan like she did, she didn't see too many of those.
Maybe he'd been sent to her. Maybe he was here for her. Maybe they was meant to share their destinies. She sure hoped so. She needed someone.
"Well, if you don't think it's him made you sick," Luke said, "what was it?"
Semelee pulled the white dress off over her head, leaving her wearing nothin but a pair of white panties. She looked down at her small, dark-nippled breasts. Losers in the size sweepstakes, maybe, but at least they didn't sag. One of the guys she'd screwed in high school had called them "perky." They were that, she guessed.
Keepin her back to Luke—she didn't want him gettin all hot and bothered out here on the water—she slipped into her cut-offs and a green T-shirt.
"I don't know. It was like..." She shuddered as she remembered that awful sick feelin runnin through her body, like she was being turned inside out..."like nothin I ever felt before. And I hope I don't never feel it again."
She turned and whacked Luke on the leg as hard as she could.
He jumped. "Hey, what—?"
"And I wouldna had to feel it in the first place if you and Corley had done the job you was upposed to!"
"Hey, we did just what we was upposed to. You was there."
"I wasn't there."
"Well, you was watchin. You saw what happened. The sacrifice was goin exactly accordin to plan when that cop showed up outta nowhere. I said all along we shoulda just flattened the old guy inside his car and have done with it."
She hit him again. "Don't you never learn? The old man had to be done in by somethin from the swamp or else it ain't a sacrifice, it's just a killin. And we ain't about just killin. We got a purpose to what we're doin, a duty. You know that."
"Awright, awright. I know that. But I still can't figure why that cop had to come along just then. We never seen him out there before."
"Maybe he was sent," Semelee said as the thought struck her.
"Whatchoo mean?"
"I mean maybe whoever was protectin the old man today was protectin him the other night as well."
"How can that be? We was the only ones who knew we'd be out there."
"I don't know how and I don't know why, but someone's protectin that old man."
"You mean like with magic?"
"Maybe."
Lotsa people'd see what Semelee could do as magic, so why couldn't there be someone else out there who could do somethin different but just as magical? Might be all sorts of magical people out there no one ever dreamed of.
"I ain't got no idea who right now, but I'm gonna find out. And when I do..."
She reached down and removed a palm-sized toad from the pocket of the discarded white dress. She held it up and stroked its back. This little feller was a relative to the big African marine toads some fool had brought into Florida sometime in the last century. It had only three legs—its left arm was nothing but a nubbin—but it had these swollen glands startin behind each eye and runnin down its back in a pair of lines. Those glands was full of poison. Every so often a dog would lick or bite one of its bigger cousins and die. This little guy came from the clan's lagoon where his family had bathed in the glow of the lights for generations, and he was even more poisonous. Just a little drop on a tongue was enough to stop a grown man's heart.
That had been Semelee's plan: sneak into the room, press the toad's back against the old man's lips, then get out. A minute or so later he'd be on his way to his maker and the job would be done.
She'd have to think of another plan now.
After she set the toad on the front seat of the boat, where it squatted and watched her with its big black eyes, her hand instinctively went to her breastbone to touch—
She stiffened. What? Where is it?
Then she remembered—the thong had broken in the hospital room. As she'd fled the terrible feelin, she recalled stuffin it into a pocket.
She rummaged in the uniform's other pocket and heaved a sigh of relief when she felt the slim thong. She pulled it out, expecting to see the pair of black freshwater clam shells she wore around her neck. She gasped when she saw only one.
"What's wrong?" Luke said.
Semelee didn't answer him. Instead she lifted the uniform and pawed through one pocket then the other.
"Oh, no! It's gone!"
"What's gone?"
"One of my eye-shells is missin!"
"Check around your feet. Maybe it fell out when you was gettin changed."
She checked, running her fingers along the slimy bottom through the inch or so of water.
"It's gone!" she cried, feeling panic rising like a tide. "Oh, Luke, what am I gonna do? I need them!"
She'd had the eye-shells ever since she was twelve. She'd never forget that moment. Her mother'd taken her to her daddy's funeral. That was the first time she'd ever seen him...or at least remembered seein him. He'd up and left Momma when Semelee was just a baby, soon after they moved to Tallahassee. He was Miccosukee Indian, banished from the tribe for somethin Momma never knew. She'd hooked up with him at the lagoon—lotsa people livin round the lagoon back then was on the run from somethin or other—and the three of them moved outta there along with everyone else shortly after Semelee was born.
Her daddy—or rather the man who'd knocked up her momma—had been killed in a bar fight. Some of his Miccosukee kin had decided to give him a proper Indian send-off and his wife and child was invited.
She'd been scared of the whole idea of lookin at a dead man, so she'd hung back, as far away from the body as she could. Just getting her first period the day before and feelin sick and tired didn't help none. That was when she spotted the old Indian woman in a beaded one-piece dress starin at her from across the room. She had eyes black as a bird's and hair like Semelee's, but also the wrinkles to go with it. She remembered how the old lady'd come close and sniffed her. Semelee'd shrunk back, scared, embarrassed. Did her period smell?
The old woman'd nodded and showed her gums in a toothless smile. "You wait right here, child," she'd whispered. "I've got something for you."
And then she'd gone away. Semelee'd hoped she wouldn't come back but she did. And when she did she came carryin two black freshwater clam shells. They'd been drilled through near their hinges and was strung on a leather thong.
She took Semelee's hand, pried open her tight-clenched fingers, and pressed the shells into her palms. "You got the sight, child. But it's no good without these. You take them and keep them close. Always keep them close. You'll need them when you're ready, and you'll be ready soon."
Then she'd walked away.
Semelee's first thought had been to throw them away, but she changed her mind. Nobody hardly ever gave her anything, so she kept them. She didn't know what the old lady had been talkin about—"You got the sight," and all that—but it made her feel special. Till that time in her life she'd never run into nothin that had made her feel special. As for "the sight"...maybe someday she'd find out what that meant.
And one day she did find out. And it had changed her life.
"Now just relax, Semelee," Luke was sayin. "It's got to be somewheres. Probably fell out while you was sittin in the truck. We'll find it."
"We got to!"
She needed those eye-shells to do her magic. She'd kept them slung around her neck so's they'd never be away from her. But now...
Those eye-shells'd saved her life...or rather, stopped her from killing herself.
It had been a day, a Tuesday in May in her sixteenth year, when everything that could go wrong did. She'd tried new hair dye the night before. Every other one she'd ever tried in the past—and she'd tried them all—didn't take. The dye just ran off her hair like water off wax. This one was touted as different, and promised to turn her hair a luxurious chestnut brown. And it looked like it might work. It didn't run off like the others.
But when Semelee looked in the bathroom mirror that morning she saw that instead of chestnut brown her hair had turned fire-engine red. Worse, it wouldn't wash out.
Maybe the color woulda been okay for the dopers and weirdoes who just wanted attention or wanted to show how they were rejecting their parents or society or whatever, but it was awful for Semelee. She'd spent her whole life bein rejected. She wanted to belong.
After crying for a few minutes—she would have liked to scream but Momma and her new boyfriend Freddy were in the bedroom down the other end of the trailer—she tried to figure what to do. She would've liked to call out sick and spend the day washin her hair, but that would leave her alone with Freddy, and the way she kept catchin him lookin at her gave her the creeps. Not that she was a virgin or nothin—she was havin plenty of sex—but Freddy...yuck.
So she dried her bright red hair, jammed a cap over it, and headed for school. Not a good start to the day but it got worse as soon as Suzie Lefferts spotted her. She'd had it in for Semelee since grammar school and never passed up a chance to torment her. She yanked off Semelee's cap just for sport, but when she saw the color of her hair she raised a holler and called all the other girls over, sayin look who's here: Lucy Ricardo!
Their laughter and cries of "Luuuuceeeeee!" chased her down the hall, right into the arms of Jesse Buckler. She was Jesse's latest squeeze—or rather, he was hers. Depended on how you looked at it. Semelee had discovered that the way to a boy's heart was through his fly. Dates for her had been as few as turtle teeth until she turned fifteen and started puttin out. After that it was a different story. She knew she had a rep but so what? She liked screwin, and durin sex was the only time she was sure she had a boy's undivided attention.
Jesse pulled her into the boy's room and for a minute she thought they was gonna have sex there—screw in school, how cool. But when she saw Joey Santos and Lee Rivers standin there with their flies open and their peckers at attention, she got scared. She tried to run but Lee grabbed her and said Jesse told them how she gave the best blow job in school and they wanted a sample. She said no and how she'd report them and they laughed and said who'd believe the school slut? They called her "Granny" and Jesse said how he got off doin it to an old lady.
The words shocked Semelee. She'd thought of herself as somethin of a goodtime gal, of easy virtue maybe, but not the school slut. And it wasn't like Semelee loved Jesse or nothin, or ever even entertained the idea that he loved her, but...he'd been talkin about her like she was a pull of chewin tobacco that he was gonna pass around between his friends.
With some kickin and clawin she broke free and ran out—not just out of the boys room, but out of the school as well. She could've gone to the principal, but it would be the word of three of the football stars against the school slut, and besides, nothin had happened.
So she'd run home. And there was Freddy. Alone. Drinkin a beer. And horny. He offered her a brew, then started touchin her. Semelee just snapped. She started screamin and throwin things and the next thing she knew Freddy was out the door and headin for his car.
He musta called Momma because half an hour later she came stormin in, started slappin at Semelee, callin her a little whore for playin hooky so she could come on to Freddy. Now look what she'd done! Freddy was gone, sayin he wasn't stayin in no house with a freaky piece of jailbait tryin to get him in trouble.
Momma wouldn't listen to her, and Semelee'd been hurt that her own momma was takin Freddy's side over hers. But then Momma crushed her, sayin she wished Semelee'd never been born, wished she'd died like all the other girls been born to the lagoon folk round that time, that she'd been a weight around her neck ever since, draggin her down, her white hair scarin off the men interested in Momma.
That did it. Semelee busted out through the door with no direction in mind and kept goin. She wound up on the beach where she collapsed on the sand. Her momma, who she'd thought of as her best friend, her only true friend, hated her, had always hated her. She wanted to die.
She thought about drowning herself but didn't have the energy to jump in the water. The tide was out so she decided to just lie here on the sand and let the water come to her, wash her out to sea, and that would be the end of it. No more hassles, no more names like "Granny," no more heartbreak, no nothin.
She lay there on her back in the sand with her eyes closed. The sun was so bright it blazed through her lids, botherin her. She didn't have her sunglasses on her but she did have those two shells around her neck. They was just the right size to go over her eyes. It'd be like layin in a tannin booth.
As she sat up to untie the thong, she saw the gulls glidin overhead and wished she had wings like them so she could fly away.
She lay back on the sand and fitted a shell over each eye—
What?
She snatched the shells away from her eyes and levered back up to sittin.
What just happened?
She'd put the shells over her eyes expectin to see black. But she'd seen white instead...white sand...and she'd been above it, lookin down on a girl lyin in the sand...a girl with shells over her eyes.
Semelee put those shells over her eyes again and suddenly she was lookin down on a girl sitting in the stand—a girl with fire-engine hair.
That's me!
She pulled off the shells again and looked up. A seagull hovered above, looking down at her, probably wondering if she had a sandwich and might throw it a crust or two.
She started experimentin and found she could look through the eyes of any bird on the beach. She could soar, she could hover, she could spot a fish near the surface of the water and dive for it. Then she discovered she could see through fishes' eyes, swim around the rocks and coral and stay underwater as long as she pleased without comin up for air.
It was wonderful. She spent the rest of the day testin her powers. Finally, after the sun had set, she headed home. She didn't want to go there, didn't want to see her momma's face, but she had no place else to go.
When Semelee opened the door to the trailer Momma was all tears and apologies, sayin she hadn't really meant what she'd said, that she was just upset and talkin crazy. But Semelee knew the truth when she heard it. Momma had said what was deep in her heart and meant every word of it.
But Semelee didn't care now. She'd thought her world had ended but now she knew it was just beginnin. She knew she was special. She could do somethin no one else could do. They could make fun of her, call her names, but no one could hurt her now.
She was special.
But now she'd lost one of her shells. She'd lose all her specialness without them. She'd be a nobody again.
Semelee gripped the edges of the canoe in white-knuckled panic. "I just had a terrible thought, Luke. What if I dropped it back in that hospital room?"
## 6
When Jack returned to his father's room, almost an hour after he'd left, he was in a foul mood. He could have called the rental agency to come and change the tire, but had canned that course of action. He'd had no idea where he was, so how could he tell them where to find him?
So he'd changed the tire himself. No biggee. He'd changed a lot of tires in his day, but usually on pavement. Today the jack had kept slipping in the sand, fraying his patience. Then the clouds wandered off to let the sun out so it could cook him. But all that wouldn't have been so bad if the mosquitoes hadn't declared his skin a picnic ground. Never in all his life had he seen so many mosquitoes. Now his forearms looked like pink bubble wrap and the itching was driving him nuts.
Felt like a jerk for letting those yokels sandbag him like that.
The TV was on and some news head was talking about Tropical Storm Elvis. It had lost a lot of steam crossing northern Florida but was now in the Gulf where it was gaining strength again, stoking itself over the warm waters. Elvis had not entirely left the building.
He went to the bed and checked his father. No change that he could see. He stepped to the window and looked out again at the parking lot. Who were they, the girl and those odd-looking people? From the way the girl had approached the bed—or at least started to—she'd come here with a purpose. But what?
As he turned back to the bed he spotted something on the floor, something glossy black and oblong. He squatted beside it, wondering if it was some sort of Florida bug, a roach maybe. But no, it looked like a shell. He bent closer. It was curved like a mussel but flatter. Some kind of clam, maybe.
As he reached to pick it up, something under the bed caught his eye. Not under the bed exactly—more like behind the headboard. Looked like a slim tree branch standing on its end.
Jack picked up the shell and stepped to the head of the bed. He peeked behind the headboard and found a tin can painted with odd little squiggles sitting atop the branch. He'd seen something like this before, then remembered Anya's yard—it was full of them.
He smiled. The old lady must think they're good luck or something. Probably put it here for him when she visited the other day. Might as well leave it. Sure as hell wasn't doing Dad any harm. And who knew? Maybe it would help him. Jack had seen a lot stranger things these past few months.
As he straightened he noticed a glistening design on the back of the headboard. He slid the bed a few inches away from the wall for a better look. Someone had painted a pattern of black squiggles and circles there. No question as to who, because they were very similar to the squiggles on the can. But how had that skinny old lady moved the bed? It was damn heavy.
Jack decided to ask her later. He pushed the bed back, then placed the shell on the nightstand. Maybe one of the staff had dropped it. If so, they could reclaim it here. At least this way no one would step on it.
Scratching his arms, Jack said goodbye to his father and headed back to the car. He hoped his father had some calamine lotion at home.
## 7
Back at Gateways Jack found another car parked in the cul-de-sac. Maybe Anya had company. But when he went around to the front of his father's place he found the front door open and heard voices inside.
He stepped into the front room and found a young woman in a jacket and skirt showing an elderly couple through the house.
"Who the hell are you?" Jack said.
The old folks jumped and the young woman clutched her looseleaf notebook defensively against her chest. Jack figured he might have had a little too much edge on his voice, but that was the kind of mood he was in.
"I-I'm with Gateways," the woman said. "I'm showing this couple the house." She squared her shoulders defiantly. "And just who are you?"
"The owner's son. What are you doing here?"
The woman blinked. "Oh. I'm so sorry for your loss, but—"
"Loss? What loss? You talk as if my father's dead."
Another blink—a double this time. "You mean he's not?"
"Damn right, he's not. I just came from the hospital. He's not too healthy at the moment, but he's not dead."
The old couple were looking uncomfortable now. They stared at the ceiling, at the rug, anywhere but at Jack.
"Oh, dear," the younger woman said. "I was told he was."
"Even if he was, so what? What are you doing here?"
"I was showing it to these—"
Fury hit him like a kick in the gut. Vultures!
"Showing it? Where do you get off showing this place to anyone? It's his until he sells it."
Another squaring of the shoulders, this time with a defiant lift of the chin. "Apparently you don't know the arrangement in Gateway communities."
"Apparently I don't. But I'm going to find out. As for now"—he jerked a thumb over his shoulder—"out."
"But—"
"Out!"
She strode out the door with her head high. The old couple shuffled out behind her.
"I'm sorry," the old woman said, pausing as she passed.
"Not your fault," Jack told her.
She put a wrinkled hand on his arm. "I hope your father gets well soon."
"Thank you," he said, feeling suddenly deflated.
He closed the door after them and leaned against it. He'd overreacted. He told himself it was the frustration of all these questions with no answers. Not one goddamn answer.
Bad day. And it was only noon.
He was just turning away from the door when he heard a knock. He counted to three, promised he'd be more genteel this time about telling the sales lady where she could stick her commission, and pulled open the door.
But Anya stood there instead. She held out a familiar taped-over FedEx box.
"This came while you were out," she said. "I signed for it."
Ah. His Glock and his backup. Now he could feel whole again.
"Thanks."
"Heavy," she said. "What've you got in there? Lead?"
"You might say. Come on in where it's cool."
"I can't stay. You were by the hospital already?"
Jack nodded. "No change." He debated whether or not to ask her about the can on the stick behind his father's headboard but decided to save it for later. "Are you going over?"
She nodded. "I thought I'd sit with him for a while."
What a grand old lady. "I'll give you a lift."
She waved him off. "I've already called a cab." She turned to go. "I'll be back later. Cocktails at five, if you're available."
He couldn't turn her down twice. "It's a date." Jack thought of something. "By the way, who's the head honcho around here?"
"You mean Gateways?"
"Yeah. The general manager or acting director or chairman of the board of whatever you call him. Who runs the show?"
"That would be Ramsey Weldon. You can find him at the administration building. You can't miss it. It's mostly glass and right on the golf course. Why?"
"We need to have a little tête-à-tête," Jack said.
## 8
The administration building was pretty much as Anya had described it: a small, cubical structure sheathed in mirrored glass. As Jack got out of his car he saw a tall, distinguished-looking man unlocking the door to a classic-looking four-door sedan. He looked fiftyish, had longish black hair, graying at the temples, and wore a milk-chocolate brown lightweight silk suit that perfectly matched the color of his beautifully restored car: two-tone—white over brown—with wide whitewall tires.
"Am I dreaming," Jack said, "or is that a 1956 Chrysler Crown Imperial?"
The man's smile was tolerant, and his tone carried a hint of impatience.
"It's a Crown Imperial, all right, but not a Chrysler. Everyone makes that mistake. Chrysler spun off the Imperial into its own division in 1954. This baby came out two years later."
"It's beautiful," Jack said, meaning it.
He ran a hand along the crest of the rear fender to one of the stand-alone taillights, sticking up like a miniature red searchlight. The chrome of the split grille gleamed like a gap-toothed grin; the flawless finish threw back his reflection.
God, he wished he could use something like this for his wheels. But it was too conspicuous. The last thing he wanted was people to notice him as he drove around. That was why he'd finally given up Ralph, his old '63 Corvair convertible. People kept stopping him and asking about it.
"You restore this yourself?"
"Yes, it's a hobby of mine. Took me two years. Fewer than eleven thousand Imperials were made in '56 and only a hundred and seventy were Crowns. This one has the original engine, by the way—a 354-cubic-inch Hemi V-8."
"So it cranks."
"Yes, indeed. It cranks." He looked at Jack. "Visiting, I assume?"
"Yeah, in a way. My father's in the hospital in a coma and—"
"You're Tom's son? Poor man. How is he?"
Jack was surprised at the instant recognition. "Not great. You know him?"
He stuck out his hand. "Ramsey Weldon. I'm director of Gateways South."
"Isn't that something," Jack said, shaking his hand. "I came here looking for you."
"I bet I know why, too. I got a call from one of our sales team. It seems she was given false information about your father. The initial word from the hospital was that he was DOA. I'm terribly sorry about the misunderstanding."
"Okay," Jack said. "I can see somebody getting the wrong information, but where did she get off showing the place to prospective buyers?"
"Because she thought—erroneously—that the place belonged to Gateways."
"Where would she get an idea like that?"
Weldon's eyebrows rose. "Upon the death of the owner—or owners—the house reverts to Gateways."
"You're kidding."
He shook his head. "That's the arrangement. It's not unique. Plenty of graduated-care senior communities have similar arrangements."
"I can't believe my father signed on for that."
"Why not? His purchase of the home and the bond guarantees him not only a place to live, but quality care from the moment he signs to the moment he goes to meet his maker, no matter how long it takes. Members of a Gateways community will never be a burden on their families. 'What do we do with Papa?' or 'Who's going to take care of Mom?' are questions that will never arise in their families."
A smooth pitch, delivered with the timing and conviction of a lifelong salesman. Jack could see how powerful that pitch could be to someone like his father who had a lot of pride and had always been an independent sort.
"At no point," Weldon went on, "will your father be a burden on his children. And at no point will you have to feel guilty about him, because you can rest assured that he's being well cared for."
"Maybe it's not so much guilt I'm feeling as—pardon me if I sound paranoid, but it seems to be to your advantage to have a quick turnover in housing."
Weldon laughed. "Please, please, we're asked that all the time. But you have to remember, this isn't a Robin Cook novel. This is real life. Trust me, it's all been amortized and insured and reinsured. You can check our financials. Gateways is a public company that posts an excellent bottom line every year."
He noticed that Weldon was starting to sweat. But then, so was Jack. It was like a steam bath out here on the macadam.
"Then I'm not the first to raise the question."
"Of course not. Our society is conspiracy crazy, seeing dark plots wherever it looks. I assure you, Gateways takes excellent care of its citizens. We do care. And our caring is what makes our citizens recommend Gateways to their friends and relatives. That's why we have waiting lists all over the country and can't build these communities fast enough. Just one example is the availability of free annual exams I instituted last year to catch medical problems early when they're most treatable."
"Really? Where are they done?"
"Right there in the clinic." He pointed to a one-story structure a hundred yards away across a dead lawn. "It's attached to the skilled nursing facility."
Jack guessed that was Gateways-speak for nursing home.
"Do you think I could speak to the doctor about my father?"
"Please. Go right ahead." He glanced at his watch. "Oops. Going to be late for my meeting." He thrust out his hand again. "Nice meeting you, and good luck to your father. We're all pulling for him."
He slipped into his car and started it up. Jack listened to the throaty roar of its V-8 and, again, wanted one.
He watched him drive away. During all that talk he'd tried to get a bead on Ramsey Weldon but couldn't get past the smooth all-business, all-for-the-company exterior. If his father's accident hadn't been hit and run, he wouldn't have bothered. But since it was...
He shook his head. Maybe he was just looking for something that wasn't there. He knew there was plenty going on out there where no one could see. He didn't need to be inventing a conspiracy around here.
## 9
The doctor working the clinic today was named Charles Harris. He wasn't too busy at the moment so Jack got to see him after only a short wait.
A nurse led him into a walnut-paneled consultation room with a cherry wood desk and lots of framed diplomas on the walls. Harris wasn't the only name Jack saw, so he assumed other doctors rotated through the clinic. Dr. Harris turned out to be a young, dark, curly-haired fellow with bright blue eyes. Jack introduced himself by his real surname—a name he hadn't used in so long it tasted foreign on his lips—and then added: "Tom's son."
Dr. Harris hadn't heard about the accident but offered his wishes for a speedy recovery. Then he wanted to know what he could do for Jack.
"First off I'd like to know if my father had a physical here recently."
Dr. Harris nodded. "Yes, just a couple of months ago."
"Great. Dr. Huerta is his neurologist at the hospital—"
"I know Inez. Your father's in good hands."
"That's comforting. But I'm wondering about his medical condition before the accident."
Jack thought he sensed Dr. Harris recede about half a dozen feet. "Such as?"
"Well, anything that might have contributed to the accident, or might explain what he was doing driving around at that hour."
Dr. Harris leaned forward and thrust his hand across the desk, palm up. "Could I see some ID?"
"What?" Jack hadn't seen this coming. "What for?"
"To prove you're who you say you are."
Jack knew he couldn't. All his ID was in the name of John Tyleski. He owned nothing with his own surname.
"I've got to prove I'm my father's son? Why on earth—?"
"Patient privilege. Normally I wouldn't under any circumstances discuss a medical file without the patient's permission, even with a spouse. But since this particular patient is incapable of giving permission, I'm willing to make an exception for a close relative—if that's what you are."
Since Jack couldn't show ID, maybe he could talk his way around this.
"If I wasn't his son, why would I care?"
"You could be a lawyer or someone hired by a lawyer looking for an angle to sue."
"Sue? What the hell for?"
"On behalf of someone injured in the accident."
"But my father was the only one injured."
Dr. Harris shrugged. "I don't know that. I know nothing about the accident. I do know that people in these parts sue at the drop of a hat. They're caught up in some sort of lottery mentality. Malpractice insurance is through the roof. People may not be able to figure out a presidential ballot but they damn sure know what lawyer to call if they stub a toe."
He could see Dr. Harris was getting steamed just talking about it.
"Look, I assure you I'm not a lawyer. I can't even remember the last time I spoke to one—that is, if you don't count my brother who's a judge in Philadelphia."
Maybe that'll mollify him, Jack thought.
It didn't.
"On the other hand," Dr. Harris said, "you could be a con man looking to pull some kind of slimy scam."
"Like what?" Jack was interested in hearing this.
He shrugged. "I don't know, but Florida's got more con men per square mile than any other state in the union."
"I'm not a con man"—at least not today—"and I'm concerned about my father. In fact, you've got me worried now. What's wrong with him that you won't tell me? What are you hiding?"
"Not a thing." Dr. Harris wiggled the fingers on his still outstretched hand. "We're wasting time. Just show me some ID and I'll tell you what I know."
Shit.
"I don't have it with me. I left it at my father's place."
Dr. Harris's features hardened. He shook his head and stood up. "Then I'm afraid I can't do anything for you." He hit a buzzer. "I'll have the nurse show you out."
"All right," Jack said, rising. "But will you at least call Dr. Huerta and tell her what you know?"
Dr. Harris obviously hadn't expected that one.
"I...well, of course. I can do that. I'll call her this afternoon."
As frustrated and worried as he was, Jack had to respect this guy's ethics. He forced a smile and thrust out his hand.
"Thanks. Nice to meet you, doc. You could be classified as a real pain in the ass, but I'm glad my dad has someone like you looking after his privacy. My doc at home is the same way."
Of course, Doc Hargus was a different case. His license to practice had been pulled, so no one was supposed to know he even had patients.
Jack didn't wait for the nurse. He left the thoroughly befuddled Charles Harris, MD behind and headed for the clinic exit.
Along the way he paid close attention to the windows and the walls—especially the upper corners near the ceiling—and the door frame as he stepped through it. No alarm contacts or release buttons, no motion detectors.
Good.
## 10
"Is it workin?" Luke said. "Can you see?"
Semelee sat on a bench in the galley of the Bull-ship. Some of the clan was in town, beggin, while others was ashore, dozin in the shade. She and Luke were the only ones aboard. She wished he'd get away and stop hangin over her shoulder and leave her be. But his heart was in the right place and so she bit her lip and kept her voice low.
"Just give me a minute here, Luke," she said as she adjusted her one remaining shell over her right eye. "Just give me a little space so's I can see if I can get this to work."
It was so different with only one shell. With two she could focus right in. With one...
With only one eye-shell she could still get into the heads of higher forms like Dora, but the lower forms...they were hard even with two. They didn't have much goin for them brainwise, and that meant she had to concentrate all the harder. If only she had that other shell.
"I could take a few of the guys and hop the fence and watch him ourselfs. We—"
"Just hesh up, will you? I think I'm gettin it."
"Yeah?"
She could hear the hope, the excitement in his voice.
She didn't see any way she or one of her clansmen could sneak into the hospital to hunt down that other eye-shell, but if she could keep an eye on the old guy's son, the special one who'd been sent to her, maybe she'd find out if he had it.
But she had to get control here.
Control...back in her teens she'd thought her power was limited to only seein through a critter's eyes, but she soon learned that was just part of the story. She found out in her junior year when Suzie Lefferts paid her a visit on the beach.
Semelee had been comin down to the ocean almost every day, except for the rainy ones, to put on her eye-shells and fly, soar, and dive with the flocks, or swim and dart through the depths with the schools. She could even get into a crab and crawl along the sandy bottom. These was the only times she felt truly alive...truly free...like she belonged.
The sudden sound of a too-familiar voice behind her jarred her back to the beach.
"So this is where you spend all your time."
Suzie must have realized that she was no longer getting to Semelee, that her taunts and tiny tortures weren't having their usual effect. So she'd followed her to see why.
"I thought you might've had a new boyfriend or something," Suzie said, "but all you do is sit here with those stupid shells over your eyes. You were always a loser, Semelee, but now you've totally lost it."
When Semelee didn't even remove the shells from her eyes or bother to reply, Suzie flew into a rage. She grabbed the shells and put them over her own eyes.
"What is it with these things anyway?"
Oh, no! She'd see! She'd know!
But Suzie mustn't've seen anything. She called them junk and tossed them toward the surf.
Terrified they might wash out to sea, Semelee screamed and ran down to the tide's edge. She found what she thought was them—they were freshwater clamshells after all—but wasn't sure. As Suzie walked up the dune laughing, Semelee wanted to choke her, but she couldn't go after her, not until she made sure she had the right shells...to see if they still worked...
They did. She put them on and there she was, glidin high over the beach, watching Suzie strutting toward her car. The bitch!
Suddenly she was divin toward Suzie, beak open, screechin. She plowed into the back of her neck, staggerin the bitch. And then she was peckin at her head, cuttin her scalp and tearin out her teased blond hair in chunks.
Semelee was so surprised she dropped her shells. She watched the squawkin gull leave Suzie's head and flap away while Suzie ran screamin for her car. The truth smacked Semelee right between the eyes then: She couldn't just get inside things and look through their eyes, she could control them, make them do what she wanted.
This cool feelin of power surged through her. She wasn't just a tiny bit special, she was really special.
But was she all that special with only one shell?
She clapped a hand over her left eye and focused all her will, all her concentration through her right. Something was coming into focus. A blade of grass, dry and brown, loomed huge in her vision, like the trunk of a tree.
"I'm there!" she cried. "I got one. Now I got to get another."
And another after that, and another, and another...
This was going to take time and effort. Lots of effort.
"I got to spread myself around the old guy's house and get in if I can."
"You really think he has it?"
"Don't know. But I'm gonna do my damnedest to find out."
"And if he got it, then what?"
"We ain't come to that bridge yet, Luke. When we do, we'll figure somethin out."
And maybe in the meantime I'll just test this guy's inner stuff, she thought. See if he's worthy of me.
## 11
Jack's head was spinning. Not from the wine he'd been drinking but from this damn game he was trying to learn.
He'd spent the latter part of the afternoon in his father's hospital room with Anya—and Oyv, of course. No change in Dad's condition—still the same random, involuntary movements and incomprehensible sounds. He'd been hoping to see Dr. Huerta and find out if Dr. Harris had contacted her. He figured he might be able to get her to tell him what the doc was hiding about his father's pre-accident condition.
But she didn't show, and finally he drove Anya and Oyv back to Gateways. She didn't let up on his joining her for a drink, so after a shower and a call to Gia to reassure himself that she, Vicky, and the baby were fine, he ambled next door.
He found Anya outside on her front lawn, cigarette in one hand, wineglass in the other, reclining face up on a chaise lounge next to a big liter-and-a-half bottle of red wine chilling in an ice bucket. She wore huge sunglasses with turquoise frames. Her flat breasts were encased in a pink halter top over skimpy black shorts. She'd coated the exposed areas of her wrinkled, leathery brown skin with some sort of sun-tanning oil and lay marinating in the sun.
Oyv was curled up next to her. He barked once when Jack stepped across the line of dry brown grass onto Anya's lush green lawn, then settled down again.
"I started without you, hon," she said. "Pull up a chair and pour yourself a glass."
"Chilled red wine," Jack said. "I don't think I've ever had that."
"Don't tell me you're a wine snob."
Jack shook his head. "A bit of a beer snob, maybe, but I wouldn't know a cabernet from a merlot without the label."
"Glad to hear it. You've probably had people tell you that the only wine you should drink cold is white or blush or rosé. Trust me, kiddo, they're talking out their tuchuses. This is a Côtes du Rhone. That's French, by the way."
"Really?"
"You probably expect an old broad like me to be a whiskey sour or Manhattan drinker, but as far as I'm concerned, on a hot summer day like this, a glass of chilled Côtes du Rhone or Beaujolais hits the spot. Try it and see if you like it. If you don't, sorry, but that's what we serve at Casa Mundy. You want beer, you'll have to bring your own. I'm not into that fizzy hops-and-malt drek."
So Jack poured himself a glass and damn if it didn't, as Anya had said, hit the spot.
"Not bad."
He pulled up a chaise lounge on the other side of the table with the ice bucket.
"How come you're the only one visiting my father? Doesn't he have any other friends?"
"He has lots. But they probably don't know. I think I'm the only one who knows, and I don't talk to many people."
"How did you find out?"
"When I saw his car was missing Tuesday morning, I called the police and asked if there'd been any serious accidents. They sounded pretty suspicious until I told them why I was calling. They told me about your father so I went right over to the hospital to see."
"Shouldn't you let people know?"
"Why? So they can send dead flowers and come in and stare at him? Tom wouldn't want that."
No, he wouldn't. Jack guessed she did know his father after all.
Together they sat and sipped and watched the sun settle in the west.
"Maybe we'd better go in," Jack said as it sank below the distant treetops. He checked his watch. 7:10. "The Wehrmacht mosquito squadrons will be launching soon."
"So?"
"You like mosquito bites?"
"You like to deny those poor females their sustenance?"
"Females?"
"Only the female mosquito bites. The males suck nectar."
"Male or female, I'm not keen on being a mosquito buffet."
She waved a hand at him. "Not to worry. They won't bother you here."
"Why not?"
"Because I won't let them."
Ooookay, lady, Jack thought. If that's what you want to believe.
But damn if they didn't sit there well into the dusk without a single mosquito bite.
When the magnum of Côtes du Rhone was done, Anya draped a fuchsia blouse over her shoulders, rose, and faced him.
"Come on inside, hon. I'll fix you dinner."
Not having a better offer, Jack accepted.
He stopped short as he crossed the threshold. He'd thought the outside was lush, but inside was a mini jungle of potted plants and trees lining the perimeter and clustered here and there on the floor, with vines growing among them and climbing the walls. He could identify a ficus here, a bird of paradise and a rubber plant there, but the rest were a mystery: potted palms of all sorts—were those baby bananas on the big one in the corner?—and smaller plants with leaves mixing reds and yellows and even silver on a couple. Reminded Jack of one of the plant shops on Sixth Avenue.
Anya turned to him and said, "I'm going to change into something more appropriate for dinner."
"What's wrong with what you're wearing?"
"I want something more haute couture," she said with a wink.
"Not necessary, but this is your party..."
As she threaded her way through the plants toward the master bedroom, Jack decided to take a look around. Oyv, curled like a cat on a worn yellow easy chair, watched him with his big dark eyes as he wandered the front room.
He realized that her layout was the mirror image of his father's—whatever was on the right here, was on the left there. But where his father's walls sported some artwork—mostly south Florida beachscapes—and some photos, Anya's walls were bare except for the vines. Not a shell, not a fishnet, not a knick knack. Nada.
She'd said she had no family. Jack guessed she was right. But how about a painting of something? Even Elvis or a tiger on black velvet would say something about her.
And the furniture...a nondescript mishmash. Jack knew his talents for interior décor were on a par with his ability to fly a 747, but this stuff looked like secondhand junk. Fine if Anya didn't care, but he was struck by the lack of personality. He'd been in motel rooms with more personal touches than this. It was as if she lived in a vacuum.
Except for the plants. Maybe they were her personal statement. Her family. Her children.
Anya reentered and struck a pose with one arm held aloft. "What do you think?"
She'd wrapped herself in some sort of psychedelic kimono which made her skinny figure seem even thinner. She looked like a Rainbow Pop that had been left out in the sun too long.
"Woo-woo," Jack said.
It was the best he could do on such short notice.
Dinner turned out to be as idiosyncratic as the chef. She mixed up a wok of walnuts, peanuts, peas, jalapeño peppers, and corn seasoned with, among other things, ashes falling from her ever-present cigarette, all rolled up in big flour tortillas. Despite Jack's initial reservations, the mélange proved very tasty.
"Can I hazard a guess that you're a vegetarian?" he said.
They were into their second magnum of Côtes du Rhone. Anya kept refilling his glass, and Jack noticed that she was putting away two or three glasses to every one he had, but showing no effects.
Anya shook her head. "Heavens, no. I don't eat vegetables at all. Only fruits and seeds."
"There's corn in this," Jack said around a mouthful. "Corn's a vegetable."
"Sorry, no. It's a fruit, just like the tomato."
"Oh. Right." He remembered hearing that somewhere. "Well, how about the peas?"
"Peas are seeds—legumes. Nuts are seeds too."
"No lettuce, no broccoli—?"
"No. Those require killing the plant. I don't approve of killing. I eat only what a plant intends to discard."
"What about Oyv?" He glanced at the little Chihuahua chowing down on something in his bowl. "He needs meat."
"He does perfectly well on soy burgers. Loves them, in fact."
Poor puppy.
"So I guess if I stop by with a craving for a bacon cheeseburger—"
"You can just keep on going, hon. There's a Wendy's not too far down the road toward town."
Gia would be right at home here, Jack thought. She wasn't a vegan or anything, but she'd stopped eating meat.
Whatever. This dish was delicious. Jack wound up having four burritofuls.
He helped clear the dishes, then Anya brought out the mahjongg tiles, saying, "Come, I'll teach you."
"Oh, I don't know..."
"Don't be afraid. It's easy."
She lied.
Mahjongg was a four-person game played with illustrated tiles, but Anya was teaching him a two-player variant. The images on the tiles swam before his eyes—circles, bamboo stalks, ideograms that were supposed to represent dragons or the four winds—while terms such as chow and pong and chong searched for purchase in his brain. He didn't have any references for this stuff. Why couldn't the tiles have spades and hearts or jacks and queens and kings?
The constant stream of smoke from the chimney that was Anya didn't help. Neither did her plants. They seemed to be watching the game, like a gaggle of curious spectators crowding around a high-stakes poker table in Las Vegas. One strand of vine with broad green and yellow leaves kept falling off a palm frond and draping across his shoulder. Jack would put it back, but it wouldn't stay up.
"That's Esmeralda," Anya said.
"Who?" Jack replied, thinking she was referring to some new tile or rule in the game.
"The gold-net honeysuckle behind you." She smiled. "I think she likes you."
"I'm not fond of clingy women," he said, reaching once again to remove the vine from his shoulder. But when he saw Anya's frown he changed his mind and let it stay where it was. "But in this case I'll make an exception."
She smiled and Jack thought, Sweet lady, but nutso, nutso, nutso.
In addition to the green, leafy distractions, all the wine he'd consumed wasn't exactly helping his learning curve. Anya lifted the bottle—she'd opened a third magnum—to give him a refill. Jack put his hand over his glass.
"I'm flagging myself."
"Don't be silly, hon. It's not as if you have to drive home."
"I have something I want to do tonight."
"Oh? And that would be...?"
"Just getting some answers to a few questions."
"Answers are a good thing," she said. Her voice was clear, her hand steady as she refilled her glass almost to the rim. No doubt about it: The woman had a hollow leg. "Just make sure you're asking the right questions."
## 12
Even in his slightly inebriated state, Jack had no trouble entering the clinic. All it took was a flat-head screwdriver from his father's toolbox to pop the window lock and he was in.
He'd managed to extricate himself gracefully from the mahjongg lesson with a promise to return for another real soon. He wasn't big into board games, although he'd played Risk a lot as a kid. He liked video games, though. Not so much the first-person shooters that were mostly reflexes; he did well in those but preferred role-playing games that involved strategy. He liked trying to outwit the designers.
After leaving Anya's he'd gone back to his father's place and doused himself with a mosquito repellent spray he'd found on a shelf with the tennis racquets and balls. Then he'd walked around some to clear his head and get the lay of the land. Here it was 9:30 and no one was out. This was good. An occasional car drove by but he'd duck into the bushes as soon as he saw its lights. One set of lights had turned out to be a cruising security patrol jeep.
A couple of times he'd stayed in the bushes longer than he had to because of the faint feeling that he was being watched. He couldn't find a trace of anyone following him, though, and wrote it off to his being on unfamiliar ground.
He'd approached the clinic building from the rear, where there was less light, and held his breath as he lifted the window, ready to run in case it was armed with an alarm system he hadn't spotted. But nothing sounded.
Made sense when he thought about it. Why spring for the extra expense of alarming all the buildings when you had a real live security force manning the gates and patrolling the streets?
He crawled through, closed the window behind him, and began searching about. He used the penlight he'd found in his father's top drawer, flicking it on and off as he moved. He found the small file room to the right of the receptionist area. He'd been hoping it would be windowless, but it wasn't, so he had to search the files with his penlight.
Again that feeling of being watched, but he was the only one here. He sneaked to the window but saw no one outside.
A few minutes later he found his father's slim chart. Holding it in his hand, he hesitated before opening it. What was the bad news Dr. Harris had been hiding? He knew the question—did he want the answer?
Again, the matter of his father's privacy. The information inside could be pretty intimate. Did he have a right to peek this far into the man's life?
Probably not. But the guy was in a coma, and Jack needed answers.
Taking a breath, he opened the file and flipped through it. He found two pages of lab test results. He didn't know what all these numbers meant but noted that the "Abnormal" column was blank on both sheets. Good enough. An EKG had a typewritten reading at its top: "Normal resting EKG." Even better.
But hadn't Dr. Huerta said something about his father developing an abnormal rhythm in the hospital? Maybe from the stress of the injuries. Everyone had heard of the patient with the normal EKG who has a heart attack on the way out of the doctor's office.
He checked the handwritten notes but couldn't read much of Dr. Harris's scribbling. The last entry was fairly legible though.
Reviewed labs w pt. All WNL. Final assess: excellent health.
Excellent health. Well, that was a relief.
But damn it, doc, why couldn't you have just said so in the first place? Would have saved me a whole lot of trouble.
## 13
Jack fished the house key out of his pocket as he walked down the slope toward his father's place. The good news was that the man was in excellent health. The bad news was that Jack didn't know one damn thing more than he had when he woke up this morning.
Nearing the house, he passed a beat-up old rustbucket Honda Civic parked in the deep shadows on the grass adjacent to the cul-de-sac. Hadn't been there when he passed by before.
On alert now, Jack slowed his pace. Before rounding the rear corner of the house, he peeked first. He froze when he saw the silhouette of someone squatting beside one of the trees between his house and Anya's. Was this who'd been watching him?
Dropping into a crouch he hugged the jalousied back porch and crept toward the figure. The wash of light from the parking area of the cul-desac cast long shadows across the space, but not enough light for Jack to make out his features. Could be one of those weird-looking characters from the pickup truck this morning.
Then the figure flicked a flashlight off and on—only for a second, but that was enough for Jack to identify him.
He straightened and walked up behind him.
"What's up, Carl?"
The man jumped and let out a little yelp. He wore a lightweight, long-sleeved camouflage suit—if nothing else, it protected him from mosquitoes—but a screwdriver instead of a hand protruded from the right cuff. He looked up at Jack and held his left hand over his heart.
"Oh, it's you. Tom's son..." He seemed to be fumbling for the name.
"Jack."
"Right. Jack. Boy, I gotta tell you, Jack, you shouldn't come up on a body like that. You just bout scared the life outta me."
Jack noticed something metallic with a silver finish on the grass before Carl. He couldn't tell what it was, but he knew it was too bulky to be a gun.
"I've found that people tend to get jumpy when they're doing something they shouldn't. You doing something you shouldn't, Carl?"
Still in a squat, Carl looked away. "Well, yeah, I guess so. Sorta. But not really."
Now there's a clear-cut answer, Jack thought.
"And what would that be?" When Carl hesitated Jack said, "Share, Carl. It's good to share."
"Oh, all right. Might as well tell you since you caught me in the act." He looked up at Jack. "I'm doin a job for Dr. Dengrove."
"Who's he? Your therapist?"
"Naw. He lives three houses back, near the beginnin of the cul-desac. He wants me to catch Miss Mundy in the act of waterin her stuff and all."
"Why would he want to do that?"
"Because it's makin him crazy that his grass and his flowers is all dead and wilty while Miss Mundy's is all green and growin like a jungle."
"So you're supposed to hang out here all night and catch her in the act?"
Carl nodded. "Sorta. He's been after me for weeks, offering me money to do it, but I keep tellin him no."
"Because you don't want to get Miss Mundy in any trouble, right?"
"Well, yeah, there's that, but also on account of how I gotta be up bright an early ever mornin for my job. That don't stop him from offerin me more money, though. But I just kept on tellin him no."
"'Kept'?" Jack said. "I guess your being here tonight means he made you an offer you couldn't refuse."
"In a way, yeah." He motioned Jack down. "Here. Take a look at this."
Jack glanced around to see if anyone else was lurking about. He sensed Carl was exactly what he seemed to be: just a cracker working as a groundsman. But still...after having one of his tires slashed by another cracker this morning, he wasn't taking any chances.
It looked like they were alone out here, so Jack squatted beside Carl.
"What've you got?"
"Somethin really cool." He picked up the metal object and held it toward Jack. "Dr. Dengrove lent it to me. Ain't it somethin?"
Jack took it and turned it over in his hands. A digital minicam. He noticed two slim wires trailing from the casing.
"What do you think you're going to do with this?"
"Get pictures. Dr. Dengrove wants me to get a movie of Miss Mundy waterin her stuff."
Jack shook his head. "In this light, Carl, I'm afraid all you're going to get is a dark screen."
"Nuh-uh. Nuh-uh." Jack detected a certain note of nyah-nyah glee in Carl's tone as he reached over and pressed a button above the camera's pistol-grip handle. "Take a look."
Jack raised the viewfinder to his eye and blinked as the walls of Anya's house and the grass and plants surrounding it leaped into view.
"Whoa," he said. "A night-vision camera."
He could make out the palms and the larger flowers—not the colors, of course, because everything was either green or black, just the shapes—along with her array of crazy lawn ornaments. As he swung the view past a lighted window the image flared, losing all detail. As he kept moving, the light from the window left a wavering smear across the tiny screen that quickly faded, allowing him to make out details again.
"Yeah," Carl said. "Almost like I'm runnin a Big Brother show, dontcha think?"
"I suppose."
Jack had never watched a single episode. His own life was more interesting than any reality-TV show. He couldn't resist tuning into The Anna Nicole Show now and again, but that couldn't be classified as reality. At least he hoped not.
"These don't come cheap," he said as he lowered the camera and turned it over in his hands. "What's this Dr. Dengrove doing with it?"
"Ask me, I think he bought it just so's he can catch Miss Mundy in the act. He don't seem to be hurtin none for bucks, but he's sure hurtin bad for a green lawn." He snorted a laugh at this little turn of phrase. "Hurtin so bad he's near about crazy."
"Crazy enough to drop a bundle on a night-vision video camera and hire you to run it?"
Carl grinned. "You betcha."
Jack shook his head. Some people. "I think Dr. Dengrove should get a life."
"Mostly I think he eats. You should see the gut and butt on him—real pan-o-ramic."
"Pano—?"
"You know." He spread his arms. "Like you told me: wide."
A panoramic butt...Jack opened his mouth, then shut it again. Let it ride.
"He's like most of the folks here, I guess. They got too much time on their hands so they worry about all the wrong things. That's why I liked your daddy so much—"
"Like, Carl. He's still alive, so you can still like him."
"Oh, yeah. Right. Well, anyway, he didn't just sit around and complain. He kept busy. Always seemed to have somethin to do, someplace to go."
"Speaking of going places...the accident happened out on a swamp road in the dead of night. You have any idea what he was doing out there?"
Jack couldn't make out Carl's expression but saw him shake his head.
"Nope. I go home at night and I stay there."
"Where's home?"
"Got me a real nice little trailer in a park just south of town. Me and the guy next door share a satellite dish. For bout thirty bucks a month each we got us a zillion channels. No reason to go out. And even if there was, you wouldn't catch me out in the Glades at night. I told you: It's angry these days."
"Right. You did. But you're out tonight—nice camo suit, by the way."
"These here are my jammies."
"They're you, Carl. So the plan is, you're going to sit out here all night and wait for Miss Mundy to show?"
"Nup. Don't hafta. At first I figured I'd just set the camera up and let her run, but that wasn't going to work. Even if the battery would last, the memory wouldn't. But then I came up with this real smart idea to solve all my problems. Lookit here."
He held up a little circuit board.
"What's that do?"
"It's a motion detector."
This Carl was full of surprises. "Did Dr. Dengrove give you that too?"
"Nup. Got it myself. Took it out of a singin fish."
"I'm sorry," Jack said, poking a finger in his right ear. "I thought you just said you took it out of a singing fish."
"That's right. That's what I did. Actually, I took it out of the board the fish sits on."
"You're losing me."
"Big Mouth Billy Bass...the singin fish. He bends out from the board and sings 'Don't Worry, Be Happy,' and some other song I never heard before."
"Oh, right. I know what you're talking about."
Jack had seen one in a store once and couldn't imagine why anyone would want one. But a clerk had told him he couldn't keep them in stock.
"Course you do. I bought mine years ago. Was one of the first around here to get one. Hung it by my front door and anytime someone came in it started singin. Pretty soon everyone in the trailer park had one, but I was first." He shook his head. "Haven't used it much lately, though. Got pretty tired of havin to listen to those same two songs every time I walked by. So I let the batteries run out. But just the other night I remembered that it had a motion detector inside that set it off every time you passed." He waved the circuit board. "And here it is."
"I get it," Jack said. "You're going to attach the motion detector to the camera, and when Anya comes out to water, you'll catch her."
"That's the plan. I made sure I popped off the speaker, though." He chuckled. "Wouldn't do to have that fish voice start singin 'Don't Worry, Be Happy' in the middle of the night, now would it."
"I guess not. You think this'll work?"
"Oh, it works. I checked it out at home."
"You really think you'll catch her?" Jack didn't like the idea of Anya getting in trouble.
"Nup. But don't tell Dr. Dengrove that, and don't you go tellin her I'm doin this. I don't want her mad at me."
"And you also don't want her tipped off that she's being watched." He nudged Carl with his elbow. "Won't you feel bad if you get her in trouble?"
"I would, cept that's not gonna happen. Like I told Dr. Dengrove, all this work's gonna be for nothin. We ain't never gonna catch Miss Mundy waterin."
"Why not?"
"Because she don't. All she does is sit and watch TV all night. Just like everbody else. Reruns of either Matlock or Golden Girls. That and the Weather Channel's all anybody round here ever seems to watch." He licked his lips. "But there's somethin else."
"What?"
"She looks dead when she's watchin TV."
"How do you know?"
"I peeked in last night while I was settin up, and I thought she was dead. I seen my share of dead folks—I'm the one found Mr. Bass dead in a chair on his front porch awhile back, and Miss Mundy looked just like him. Boy, was I glad to see her up and about this mornin."
"Didn't you call anyone?"
"Hey, I wasn't supposed to be there. And if she was as gone as she looked, there wasn't nothin nobody could do anyhow. Tonight I looked in again, just a few minutes ago, and it was the same thing. Gwon. Look for yourself."
Jack shook his head. "I don't think so."
"Gwon. See if I ain't lyin. It's creepy, I tell you."
The last thing Jack needed was to get caught acting like a Peeping Tom, but his curiosity was piqued. He crept up to the lighted window that looked in on the front room and peeked through the lower right corner.
Still in her kimono, Anya lay back in her recliner, mouth slack, eyes half open and staring straight ahead. A Law and Order episode was playing—Jack recognized the music—but Anya wasn't watching it. Her gaze was fixed on a spot somewhere above the TV. Oyv was stretched across her lap, looking equally dead.
Jack watched her for signs of breathing but she was still as, well, death. His comatose father showed more signs of life. Jack straightened and was about to head around front to knock on her door, when he saw her chest move. She took a breath. Oyv took a breath too, at exactly the same time. Just one each. Then they went dead still again.
Okay. So she was alive. Maybe it was all that wine—she must have put away three liters—that put her into such a deep sleep.
Shaking his head, he returned to where Carl waited.
"You weren't kidding," he said. "But I saw her breathe. She's okay." He put a hand on Carl's shoulder. "But you haven't explained how she can have such a healthy lawn without watering."
"Magic," Carl said, looking around as if someone besides Jack might be close enough to hear. "You may think I'm loco, but that's the only explanation."
Jack remembered Abe telling him about Occam's razor earlier in the year. It went something like: the simplest, most direct explanation—the one that requires the fewest assumptions—is usually the right one. Magic required a lot of assumptions. Water didn't.
"I like water better as an explanation."
"Nuh-uh. Not when you look at where her green grass ends and the brown begins. It runs in a perfect line twenty feet from her house all the way around in a big circle. And when I say line, I mean it's got sharp edges. I know, cause I cut it. I may not know much about lotsa things, but I know you can't water like that."
Jack couldn't see the line in the low light. He figured Carl was exaggerating. Had to be.
"I think it's them doohickies she's got all over her yard," Carl said. "And that writin on her walls."
"Writing?" Jack didn't remember seeing anything on Anya's walls.
"Yup. You can't see it lookin at it reglar, but—here." He handed Jack the camera again. "You look through that while I put my flashlight on. Now I'm only goin to put it on for a second so you look real hard."
Jack peered through the viewfinder at the blank wall, avoiding the glare of the lighted window. A section of the wall lit as Carl's flashbeam hit it. And there, flaring to life, a collection of arcs and angles and squiggles, very much like the symbols on the homemade ornaments dotting her lawn.
And like the symbols he'd found behind his father's headboard.
"Y'see em? Didja see em?"
"Yeah, Carl. I saw them." But what did they mean? He'd never seen anything like them. On a hunch, Jack did a one-eighty turn. "Flash that on my father's place, will you?"
When Carl complied, the same symbols appeared.
Dumfounded, Jack lowered the camera. "He's got them too."
"Hmmm," Carl said. "They sure ain't doin nothin for his lawn. Wonder what they's for?"
"Let's do a little research," Jack said.
With Carl in tow, Jack used the same procedure to check out three other nearby houses, but their walls were blank.
Returning to Carl's original spot, he handed back the camera. That feeling of being watched was back and stronger than ever. He scanned the area and spotted a bunch of dead leaves scattered across the remains of his father's lawn. Hadn't noticed them before. Not unexpected, though. He'd seen trees drop leaves in a hot dry spell.
While Carl attached the motion detector to the camera—still no sign of a right hand, just a screwdriver poking from the cuff—Jack turned toward Anya's house.
He had to admit he was baffled. That strange old lady was the common factor here: She lived next door to his father...visited him in the hospital...the symbols on her house were also on his dad's place. Jack knew his father hadn't painted them on his hospital bed. Not while comatose. So that left Anya.
She must have painted them with some sort of clear lacquer so they'd be invisible. But what did they mean? And what did she think she was accomplishing with them?
Maybe he should just ask her. But then he'd have to explain how he knew.
He glanced around again and noticed even more leaves on the lawn. Their number had doubled or tripled since his last look. Where the hell were they coming from? They were small, maybe three inches long; light from the parking area glinted off their shiny, reddish brown surfaces. Odd...dead leaves usually lost their gloss.
Jack looked around for the source but couldn't see any trees in the vicinity with that kind of leaf.
"There," Carl said. Jack turned and saw him on his feet, dusting off his knees. He'd duct taped the camera to the slender trunk of a young palm. "All set."
"Tell me something, Carl," Jack said, jerking a thumb over his shoulder. "Where'd all those leaves come from?"
Carl was facing the light when he glanced past Jack. Jack saw his expression change from curiosity, to puzzlement, to shock. He turned and looked and knew his expression must be mirroring Carl's.
No grass was visible. The leaves had multiplied till they now covered every square inch of the lawn.
"Those ain't leaves," Carl said in a hushed, awed tone. "Them's palmettos!"
"What's a palmetto?"
"A bug! A Florida roach!"
"You mean like a cockroach?"
"Yeah. But I can't remember ever seein more'n half a dozen palmettos in one spot at the same time."
Jack had encountered his share of cockroaches—couldn't live in New York without seeing them—but never this size. These were cockroaches on steroids. His skin crawled. He wasn't the squeamish type, but these were big, and there had to be thousands of them, all just a few feet away. If they started scuttling his way...
"What're they doing here?" Jack said.
"Dunno. There ain't nothin for them to eat on that lawn, that's for sure." He looked over his shoulder. "Tell you what I'm gonna do. My car's parked in the shadows on the other side of your daddy's place. I'm gonna head around the front of the house and get to it that way."
"Why don't you just shine your flashlight at them. Cockroaches hate light. Turn one on and they disappear."
"Not Palmettos. Light don't bother them ay-tall. They actually like the light." He turned and took a step away. "Be back tomorrow."
That step seemed to trigger the bugs. With a chittering whir of wings they took to the air in a cloud.
"They fly?" Jack shouted as he started backing away. "Cockroaches don't fly!"
"Palmettos do!" Carl broke into a run.
Jack felt a surge of fear and didn't know why. They were just roaches; not as if they were going to eat him alive or anything. But his adrenaline was kicking in, pushing his heart rate up a few notches. He quickened his backpedal.
At that instant the churning mass of bugs turned as one and swept toward him in a swirling cloud. Jack whirled and dashed after Carl.
"Here they come!" he shouted.
Carl didn't even turn his head; instead he put it down and upped his speed.
But neither stood a chance of outrunning the bugs. The palmettos were too fast. They swirled around Jack, engulfing him, clinging to his face, his arms, his hair, buzzing in his ears, scratching at his eyelids, wiggling their antennaed heads into his nostrils, digging at his lips. The clatter of their wings sounded like a million tiny hands applauding. He felt countless little nips all over his exposed skin. Were they biting him? Did they have teeth?
He swept a mass of them from his face but they poured back in on him. He couldn't see and he was afraid to open his mouth to breathe—they might crawl down his throat. He tore them again from his face and stole a quick look ahead. The last thing he needed now was to run into a wall or tree trunk and knock himself silly.
He saw that he'd reached the corner of the house. Carl was still ahead, waving his arms wildly about, all but unrecognizable under a swarming mass of palmettos, but still maintaining a stumbling run. Jack cupped a hand over his mouth, took a quick, bug-free breath, and shouted.
"Carl! Forget the car! Go into the house!"
But Carl either didn't hear the muffled advice or chose to ignore it. Jack had to close his eyes again against the storm of palmettos. He angled to his right—the front door was somewhere in that direction—and hoped he wouldn't trip over one of the front porch chairs.
He slammed into a wall and heard some of the bugs crunch against the siding. He felt to his left, found the handle to the screen door, and pulled it open.
The front door—had he locked it? He hoped to hell not. This being a gated community and all, why would he bother? But he was a New Yorker, and New Yorkers never—
He fumbled around, found the knob, turned it, pushed it open, and leaped inside. As he moved he was trying to think of ways to kill the bugs that made it through the door with him, but then he realized that wouldn't be necessary. They were peeling off of him at the threshold line, like vacuum wrap being stripped from a piece of meat. Jack stopped two feet inside the door and looked down at his arms, his clothes—not a single bug had made it in with him.
He turned and stared through the door as the screen banged shut. The palmettos were buzzing off in all directions, scattering like...like the leaves he'd first mistaken them for.
What the hell was going on here?
## 14
"Semelee! Semelee, answer me! Are you all right?"
Semelee opened her eyes and saw Luke's big face and hulking form hangin before her. No...hangin above her. She shook her head, propped herself up on her elbows, and looked around.
"What happened?"
"You was usin the shell, had it over your eye, and you was smilin and laughin and then all of a sudden you yelled and fell back on the floor. What happened?"
Good question. Real good question. But it was startin to come back to her now.
She'd spotted the old man's kid, the special one, outside his daddy's house and followed him through palmetto eyes to one of the buildings in the old folks' village. She'd been hopin he'd show her that he had her other eye-shell but he surprised her by breakin into the building. She tried to follow him inside but he closed the window too quick. She peeked through the windows and saw him lookin at some papers. She had no idea what they were and didn't care. She was lookin for her eye-shell.
Pretty soon he was out again. She followed him back to the house where he met someone outside. She thought there was somethin familiar about the stranger but couldn't place him.
It was about then that she'd started feelin the strain of controllin mindless little creatures like palmettos with just one eye-shell. She had to make somethin happen, get the special one into the house where she could have a look around for her eye-shell.
So she'd gathered as many as she could and attacked. She'd been havin a good time chasin him and seein what he was made of, and was gonna follow him into the house and give him a good scare—maybe have the bugs gather in the air and spell out somethin spooky—so he'd leave and let her search the place. But as she approached the front door she started feelin strange, a little sick even. And then when she tried to follow him inside it was like runnin into a wall. She was slammed back and things got a little fuzzy after that.
"It's him," she told Luke. "It's him made me sick in the hospital room this mornin."
"How you know that?"
"Cause I felt the same way just now tryin to follow him into his daddy's house."
She'd sensed he was special, but she hadn't known just how special.
"You think he's got your other eye-shell then?"
"I'm willin to bet on it."
"What're we gonna do?"
"I don't know." She rolled over and buried her face in her arms. "Let me think on it."
She had no experience in this sort of thing. Sometimes she wished she didn't have to make all the decisions. She was only twenty-three. Wasn't being special and having a destiny enough? Did she have to lead too?
And worse was realizing that the man, the special one, might not be here for her...the way she'd been stopped dead at his doorstep tonight made her suspect he might be against her.
People against her paid a price, a high one, for treatin her bad.
Suzie Lefferts found that out. In spades.
After Semelee had experimented with her control powers for a while, she decided to put them to the test. She chose prom night. No one had asked her to go, of course. Like, big surprise. And guess who Jesse Buckler asked: big-haired Suzie Lefferts.
So Semelee had sat in her bedroom—another thing she'd discovered was she didn't have to be on the beach to fly with her birds—and got together a flock of big fat seagulls and followed Jesse's car from Suzie's house to the prom. When they was both out of the car, she arranged the gulls into a low circle. As each one got near them it let loose with a big load of bird shit. Suzie started screamin as the big white globs landed in her hair, on her dress. Same with Jesse. They both jumped back in the car and drove away. Toward home, most likely. Semelee was sure Suzie wasn't goin into the prom lookin like that.
Semelee lay on her bed and near split her sides laughing. But she realized how a few of her gulls hadn't done their thing yet, so she chased after the car, droppin big white splotches all over Jesse's nice new wax job. He kept goin faster, trying to outrun them, but that wasn't gonna happen. Then a particularly big glob landed on his windshield. She saw the wipers come on but they just smeared it all over the glass. That was when Jesse missed the curve and smashed into the utility pole. The two of them'd been in such a rush to get away from the bombardment that they never buckled up. Jesse wound up dead; Suzie survived but with a broke neck. Doctors said she'd never walk again.
Semelee had been shook up somethin terrible. She put her shells away, but only for a little while...she couldn't stay away from them too long. But she used them only for flyin and swimmin. She didn't try to control no more critters.
Leastways not while she was still in Jacksonville.
But that was then. The now Semelee thought the then Semelee was a dork. Don't make no sense to waste a special power. You don't use it, you ain't special no more. You're just like everybody else.
Besides, people tend to get what they deserve.
Semelee lay on the deck a moment longer, till the stink of the floorboards—the spilled drink and bits of old food rubbed into them over the years—became too much. She climbed to her feet.
"Well?" Luke said. "You gotta plan?"
She told him the truth. "No. Not yet, anyways. I'll figure something out." She turned to him. "There was somebody with him tonight. Somebody I think I seen before."
"Who?"
"If I knew that, I'd tell you his name. But I know I seen him. It'll come to me."
"Well, in the meantime we got unfinished business. That old man—"
"Yeah. We're gonna have to finish him. That's number one on the list."
But how? She wished she knew.
"If his kid is standin in the way, I can take care of that. Me and Corley can go out and catch him alone and—"
"No! Don't you touch him!"
"Why the hell not? He's in the way, and he's even makin you sick. He..." Luke squinted at her. "Hey. You ain't sweet on him, are you?"
"Course not." She couldn't let on about the connection she felt between her and the special one. Luke might go off and do somethin really stupid. "But like I told you before, we ain't killers. We do what needs to be done but we don't go past that. This guy's only protectin his kin. Can't blame a body for that."
...protectin his kin...
Of course. It wasn't a matter of him fightin against her, he was simply doin a son's duty. That thought gave Semelee a surge of hope. Suddenly she felt better.
"I can too blame him if he's gettin in our way and makin you sick and knockin you to the floor!"
"Just don't do anything unless I tell you, okay? Are you listenin to me, Luke? Nothin until I say so."
Luke looked away. "Awright."
Semelee didn't know whether she could believe him or not. She knew Luke would do anything to protect her, whether she needed protectin or not. And that worried her.
## 15
After watching the cloud of palmettos disperse into the night, Jack slammed the door and ducked into the rear bedroom. He peered through the window in time to see a bug-free Carl getting into his old Honda and roaring off. Obviously the bugs had lost interest in Carl as well.
Jack rubbed his arms and face as he returned to the front room. He could still feel them crawling on him. What had made them attack like that? And what had made them quit just as suddenly?
What was happening around here? Odd ornaments on lawns and behind beds, invisible symbols painted on walls, flying killer cockroaches...what had he stepped into? It didn't smell of the Otherness, but that didn't mean the Otherness wasn't lurking behind these weird goings on.
Bigger question: Where did Anya fit in? She was involved, no way around it. Whether peripherally or centrally, he couldn't say. But she seemed to be on his father's side, and that gave him a little comfort. Very little. If she weren't dead to the world in her recliner, he might go over and ask her for an explanation.
And say what? I was just attacked by palmetto bugs. Know anything about that?
Maybe she did, maybe she didn't. He was pretty sure she didn't cause it. But at the very least she could explain the symbols on her house and his father's, and how they'd got there.
Jack decided to let it go until tomorrow.
He paced the front room a couple of times. He was still feeling the after-buzz of the bug-induced adrenaline surge. It had burned away the alcohol from the wine and he was thirsty. Right now he needed a beer.
He grabbed a couple from the fridge—getting low; he'd have to pick up some tomorrow—and settled himself in front of the TV. After listening to the latest on T. S. Elvis, now drifting south in the Gulf and threatening to become a hurricane, he surfed around until he chanced upon his favorite Woody Allen film, Zelig, playing on TCM. He always envied Zelig's talent for blending in with any group; it would be so handy in Jack's fix-it business back home.
He sat and watched with the lights on. He wasn't about to let any bugs sneak up on him.
## THURSDAY
## 1
A soft clattering noise woke Jack. He lifted his head from the pillow on the guest room bed and squinted at the clock. The red LED numbers swam for a second, then came into sharp focus: 8:02.
He rolled out of bed and went to the window for a peek outside. There he was: Carl, dressed in the same shirt and work pants as yesterday, but this morning a set of electric hedge clippers protruded from his right sleeve as he trimmed away at dry-looking bushes that didn't need it.
Jack pulled on a pair of shorts from his open gym bag on the floor and went outside.
Carl Scissorhands looked up and jumped at Jack's approach. He shook his head and stopped the clippers.
"Mornin," he said. "Man, that gang of palmettos was somethin last night, wasn't it. Never seen nothin like that in all my born days. Never heard of it neither. How'd you finally do with them?"
"Soon as I got inside the house they just flew off. How about you?"
"Same. I was halfway to my car when they suddenly lost interest. Pretty weird, huh?"
"Very weird."
"I had trouble sleepin. I kept feelin like they was still on me." He shivered inside his flannel shirt. "Gives me the willies just thinkin about it. And then my car wouldn't start this mornin. My luck's runnin pretty bad and pan-o-ramic these days."
Jack glanced over to where Carl had set up his camera last night. The spot was empty now.
"How did the video surveillance go?"
Carl shook his head. "Nada. I come by real early this mornin to pick it up, you know, before anyone else found it." He winked and jerked his thumb at a tattered backpack sitting among his gardening tools. "I quick-checked the playback but the only thing on it was me bendin over it and picking it up. Least ways I know the motion detector's workin. Told Dr. Dengrove and he wasn't too happy, but wants me to try again tonight."
"You going to?"
"Sure." He grinned. "He wants to keep payin me, I'll keep settin up the camera. It's his money, and I could sure use some of it."
"Fine, as long as you don't catch Miss Mundy doing anything that'll cause her trouble."
"Told you: no worry bout that."
"Speaking of Miss Mundy..." Jack turned and looked at Anya's place. No signs of life there. Considering how she looked last night..."maybe I should go over and see how she's doing."
"Oh, she's doin fine. She was up bright and early this mornin, waitin for a cab. It picked her up a little before seven."
"Oh? Well, it's good to know she's all right."
Jack wondered where she'd be going at that hour. Hardly anything open then except the convenience stores.
The idea of a convenience store got him thinking about coffee. He needed a couple of cups, but he didn't feature the idea of winding all through Gateways twice, then back and forth through the security gates, and hunting down a store in between. Oh, for the Upper West Side where he could walk around the corner and have his choice of coffee spots.
He remembered his father had always been a big coffee drinker. He'd seen a can in the refrigerator.
"I'm going to make some coffee," he told Carl. "Want some?"
Carl shook his head. "Had some at home. Besides, I gotta keep lookin busy otherwise they'll lay me off. Not a lotta gardenin to do when nothin's growin."
As Jack turned away he glanced again at the clippers protruding from Carl's right sleeve. What was holding them? Maybe he didn't want to know.
## 2
Back inside Jack pulled the can of coffee from the fridge. Brown Gold—"100% Colombian Coffee." Sounded good. But he couldn't find a coffee pot. Just a miniature French press. Jack remembered seeing a big version of this in a restaurant where he once waited tables, but had never worked one.
And he needed coffee. Now.
He flipped on his father's computer, did a Google search for "French press," sifted through sites about French newspapers and other sites wanting to sell him a press until he found one telling how to use one: two scoops of coffee into a small press, followed by near-boiling water at about 195–200 degrees—were they kidding? Stir after one minute. After a total of three minutes, put on the cap and push the plunger to the bottom.
Jack followed the directions using boiling water—like he was going to check the temperature, right?—and finally had his coffee. A damn good cup of coffee, he admitted, but who had time for all this rigamarole every time you wanted some?
Retired people, that's who. And his father was one of them.
He flipped on the Weather Channel while he was waiting the required three minutes and learned that Elvis was still drifting south in the Gulf. Its sustained winds had reached seventy-eight miles per hour. That meant it had graduated from a tropical storm to a Category I hurricane. Whoopee.
Coffee in hand, he searched through the front-room desk until he found a couple of Florida maps. One was a roadmap of the state, but the other was Dade County only. That was the one he needed.
He found Pemberton Road and followed it till it intersected with South Road...the site of the accident. Out in the boonies. Way out.
Time for a road trip.
He was halfway through refolding the map—these things never wanted to go back to their original state—when a knock on the front door interrupted him. He found Anya, dressed in a bright red-and-yellow house dress, standing outside with Oyv cradled in her arms.
"Good morning," she said. Hot, steamy air flowed around her.
Jack motioned her inside. "Come on in where it's cool. If you've got half an hour, I can make you a cup of coffee."
She shook her head as she stepped in. "No thanks, hon."
"Sure? It's made from beans." He winked at her. "And on the label it says that no plants were killed during the making of Brown Gold coffee."
She winked back. "I'll have to try some another time." She gestured to the map in Jack's hand. "Planning a trip?"
"Yeah. Out to where my father got hurt."
"I'll come with you."
"That's not necessary." Jack had planned to do a little aimless reconnoitering after checking out the intersection and didn't know if he wanted an old lady and a yip-yip dog along.
"No trouble at all," she told him. "Besides, you're a newcomer and I've lived around here awhile. I can keep you from getting lost."
Well...on that score, maybe she'd be more of a help than a hindrance.
"Okay. Thanks. But I want to stop at the hospital and check up on my father before we head out to the swamps."
"That can wait till you get back," she told him. "I was just there."
"You were?" He was touched by her devotion. "That's awfully nice of you. How was he?"
"When I left he was just the same as yesterday and the day before."
"No progress?" Bummer. "How long can this go on?"
"Not much longer, hon," Anya said with a smile. "I have a feeling he'll be taking a turn for the better soon. Just give it a little time. But as for exploring the hinterlands, we should get started before it gets too hot."
She had a point. "Okay. Just let me throw a few things together and I'll be right out."
"Oyv and I will meet you at the car."
Jack figured he'd bring his backup .38 along—just in case. And mosquito repellent. Lots of mosquito repellent.
## 3
A voice had called him from his long dream of the war and he'd responded. He was glad to leave the dream...so many dead men, with pierced skulls and ruptured chests...staring at him with mournful eyes...
And then he was out of the dream and awake. He sat up. He was in a bed, in a barrack. But where were the other beds, the other soldiers? No one here but him.
Then he saw a little woman, a thin bird of a woman in some sort of uniform, mopping the floor. He spoke to her. Not volitionally. The words seemed to pop out of his mouth. He didn't even hear them. But the woman did. Her head snapped up. Her eyes widened. Then she hurried from the room.
Where am I? he wondered.
Was this still part of the dream? If not, how did he get here?
## 4
Jack tried to draw Anya out during the trip but she wasn't very responsive. He told her about the palmetto attack last night but she didn't seem horrified or even concerned. Her only remark was that it was "very unusual."
"How about you?" he said, shifting the subject from him to her. He wanted to know more about her. "Where are you from?"
"I moved here from Queens," she said.
"I'd have thought you came from Long Island."
"Well, I've lived there too."
"What about your childhood? Where'd you grow up?"
"Just about everywhere, it seems." She sighed. "It was so long ago it seems like a dream."
This was getting nowhere. "Where haven't you lived?"
"On the moon." She smiled at him. "So what's with all the questions?"
"Just curious. You seem to know a lot about me and my father, and you two seem close, so don't you think it's natural for me to want to know a little about you?"
"Not to worry. We're not involved. We never will be. We're just friends. Isn't that enough?"
"I suppose it is," Jack said.
He supposed it would have to be.
He took Pemberton Road southwest with Anya following on the map and acting as navigator. Oyv lay stretched out in the sun on the deck under the rear window. A drainage ditch paralleled the road, sometimes on the left, other times on the right. Probably served as a canal of some sort in normal times, but now it was mostly a succession of intermittent pools of stagnant water.
"They're called borrow pits," Anya said, as if reading his mind. "They're where the dirt and limestone came from when they were building up these roads. This time of year they should be filled with water, with turtles and little alligators and jumping fish. Now..."
He could see what filled them now: beer cans, Snapple bottles, old tires, and hunks of algae-encrusted Styrofoam.
Coarse brown grass stretched away to either side. He spotted three white-tailed deer—a doe and two fawns—grazing near a stand of trees. As the car approached they leaped over a bush and disappeared.
He saw a sign that read PANTHER CROSSING.
"Panthers?"
Anya nodded. "They still have some around here."
The idea of wild panthers about was a little unsettling even when in a car. Imagine seeing that sign while on foot.
"I've driven through here with your father a couple of times. Every time we pass that sign he says some rhyme about a 'panther' and 'anther.'"
Jack had to laugh. "Ogden Nash!"
"Who?"
"He was a very clever, down-to-earth poet. No airs about his stuff. Wrote a lot for kids. Dad loved him."
Jack remembered his father's nightly ritual of doling out a few of Nash's animal poems at bedtime.
He'd forgotten about those times. He made a mental note to check the bookstores when he got home and see what was still in print. Vicky would love Nash's wordplay.
He was jarred back to the present as they passed a burnt-out area where some asshole probably had flicked a cigarette out the window. Up ahead, a sign displaying a goofy-looking alligator informed them that this was a "South Florida Water Management District."
"Not much water to manage at the moment," Jack said as the pavement ran out and became a dusty, rutted dirt road bed.
"Even when there is they mismanage it. All the development north of here, it's screwed up the Everglades—screwed it royally."
Jack sensed anger in Anya's voice. And something else...
"You sound as if you're taking it personally."
"I am, kiddo. I am. No decent person can feel otherwise."
"Pardon my saying so, but isn't it really just a big swamp?"
"Not a swamp at all. Swamps are stagnant; there's constant flow through the Everglades. It's a prairie—a wet, saw grass prairie. This whole part of the state runs downhill from Lake Okeechobee to the sea. The overflow from the lake travels all those miles in sloughs—"
"Whose?"
"Slough. It's spelled S-L-O-U-G-H but pronounced like it's S-L-E-W. The sloughs are flows of water through these prairies that keep things wet. We're near the Taylor Slough here. The Miccosukee Indians call the Everglades Pa-hay-okee: river of grass or grassy waters. But look what's been done in the past fifty years: Canals have been cut and farms have been put in the way, leaking all their chemicals into the water—or should I say, whatever water reaches here. What the farms don't take is 'managed' by so many canals and dikes and dams and levies and flood gates that you've got to wonder how any of it gets where it naturally wants to go. It's amazing anything at all has survived here. Just pure dumb luck that the whole area's not a complete wasteland." She glanced at him. "Sorry, kiddo. End of lecture."
"Hey, no. I learned something. But I'd think that since Florida is just an overgrown sandbar, all of the water in the sloughs would just seep into the ground."
"Sandbar? Where'd you get that idea?"
"I heard somebody describe it that way, so—"
She wagged a finger at him. "He was talking out his tuchus. Florida is mostly limestone. It's not an overgrown sandbar; if anything, it's a huge reef. There's sand, sure, but dig down and you hit the calcified corpses of countless little organisms who built up this mound back in the days when all this was under water. That's why the water runs downhill to the Everglades: Because it must."
"How'd you manage to learn so much about these problems?"
"It's no secret. You just have to read the papers. Supposedly the government is going to spend billions to correct the mess. We'll see. Shouldn't have let it happen in the first place." She glanced down at the map. "We should be coming up on it soon."
"On what?"
"The intersection." She pointed through the windshield. "There. That must be it."
Jack saw a stop sign ahead. He slowed the car to a stop a dozen feet before the intersection. The crossing road was unmarked. Jack took the map from Anya and stared at the intersection he'd circled.
"How do we know this is the place?"
"It is," Anya said.
"But nothing says that's South Road."
"Trust me, kiddo. It is."
Jack looked at the map again. Not too many crossroads out here. This had to be it.
Leaving the engine running to keep the AC going, he got out and walked to the stop sign. It sported a couple of bullet holes—.45 caliber, maybe—but the rime of rust along their ragged edges said they were old. A sour breeze limped from the west. He stepped into the intersection and looked left and right. He checked the ground. Little pieces of glass glittered in the dust. This was where it had happened.
"What are you hoping to find?"
He turned and found Anya approaching. Oyv trotted behind her, weaving back and forth as he sniffed the ground.
"Don't know," he said. "It's just that a lot of things don't add up, especially with the timing and the assumption that my father ran a stop sign."
"I imagine a lot of people do that out here. Look around. Here we are, midmorning on a Thursday and not a car in sight. You think maybe there were more in the early A.M. Tuesday?"
"No. I guess not. But he was—is—such a by-the-book guy, and not a risk taker, that I can't see him doing it. And I can't see what he was doing out here in the first place."
"Oh, I can tell you that: He was driving."
Jack tried not to show his irritation. "I know he was driving. But where to?"
"To nowhere. Many nights he had trouble sleeping, so he'd go out for a drive."
"How do you know?"
"He told me. Asked me if I wanted to come along some night. I said he should include me out. I don't know from insomnia. Like the dead I sleep."
So I noticed, Jack thought.
"Where did he go?"
"Out here. He said he always took the same route. He'd drive with his windows open. He said he liked the silence, liked to stop and look at the stars—you can see so many out here—or watch an approaching storm. That would have been back when we had storms, of course." She sighed. "Such a long while since we've heard thunder around here."
"All right. So he's out here on his nightly drive and—"
"Not nightly. Two, maybe three times a week."
"Okay, so Monday night or early Tuesday morning, he's out here and somehow he winds up in the middle of an intersection when something else is coming along. Something big enough to total his car and keep on rolling."
"A truck then. Sounds as if he pulled out in front of a truck."
Jack looked up and down the road. His father's Marquis had been hit on the right front fender. That meant...
"A truck? It would have to have been coming from the west...from the Everglades. Maybe he had a little stroke or something."
"Dr. Huerta said his brain scans showed no damage."
"Then it's a mystery."
"I don't like mysteries, especially when they involve someone I know. And speaking of mysteries, I'm still trying to find out how someone reported the accident from downtown Novaton—"
Anya shook her head. "You call that a downtown?"
"Okay, from the local supermarket—before it happened."
Anya peered at him through her huge sunglasses. "How do you know when it happened?"
"From my father's watch. It's cracked and broken, and the time on the face is something like twenty minutes after the accident call. How is that possible?"
"Clocks," Anya said with a shrug. "Who can trust them? One's set too fast, one's set too slow—"
"My father was always a tightass about having the right time."
"'Was,'" Anya said. She tsked and pointed a gnarled finger at him. "What do you know about his watch lately?"
Jack looked away. She had him there.
"Not much."
"Right. And—"
Oyv started barking. He was standing at the edge of the ditch with his head down and his ears drawn back flat against his head.
"What is it, my sweet doggie?" Anya said. "What have you found?"
Jack followed Anya over to where Oyv was still making his racket.
"Oh, my!" she said.
Jack came up beside her. "What?"
"Look at these tracks."
Jack saw five-toed impressions in the damp mud at the bottom of the ditch. They spanned about a foot across. Whatever had made them was big. And pigeon-toed.
"Got to be a crocodile."
Anya looked at him and made a face. "Crocodile? The Florida crocodile likes brackish water. These are alligator tracks. See that wavy line running between them? That was left by his tail. Look at the size of those feet. This is a big alligator."
Jack did a slow turn. With all the reeds and saw grass around, it could be hiding anywhere.
Now he knew how Captain Hook felt.
"How big?"
"Judging from the size of these prints, I'd say twenty feet long, maybe more."
Jack couldn't imagine how she'd know that, but wasn't going to call her on it. This lady knew an awful lot about Florida.
"Twenty-plus, huh? Why don't we get back in the car."
"Not to worry. These look old. See how the mud is dry? They were probably made days ago."
"That doesn't mean the maker isn't still nearby."
The tiny Chihuahua was down in the canal sniffing at the tracks. He showed no fear. Jack half expected him to start cooing, Heeere, leezard, leezard, leezard...
His right hand drifted to the small of his back where his little AMT backup rested in its holster under his T-shirt. He wondered if a .38 caliber frangible would stop a gator that size. Probably break up on its head. But he alternated them with FMJs in the magazine. They might do some damage.
"Anyway, I've seen what I came to see."
"Which was?"
"Nothing in particular. I just thought I should come out and see where it happened."
What had he been hoping for? A mystery-solving clue, like in the movies? It hadn't happened. Wasn't going to happen. The whole thing was just a stupid accident.
But still...he wished he knew who'd been barreling along South Road out of the swamp in something big and heavy early Tuesday morning.
Back at the car, Jack played the gentleman and held the door for Anya—and Oyv—as she settled herself in the passenger seat, then he walked around to the other side. Physically he was heading for the driver seat; mentally he was miles away, thinking about giant gators and heavy rolling equipment. He was reaching for the door handle when Oyv started barking again. He looked up and saw a red truck racing toward him—for him.
No time to get in the car so he back-rolled onto the hood and got his feet up and out of the way just as the truck sideswiped the Buick.
Jack's heart pounded. That son of a bitch almost—
The truck...an old red pickup he'd seen before. Jack couldn't make out who was driving but he'd bet he wasn't pretty. Coughing in the trailing dust cloud, he slid off the hood, pulled open the door, and jumped inside.
"What was that?" Anya said as Oyv kept barking.
Thanks little guy, Jack thought. Bark all you want.
"That was an attempted hit and run."
He slammed the car into gear and spun the tires as he started pursuit.
Anya looked worried. "What do you think you're doing?"
"Going after them."
"And if you catch them, then what?"
"As the saying goes, I'm going to kick ass and take names—in a very literal sense."
The bogus nurse in his father's room yesterday had driven away from the hospital in that truck, and now that truck had tried to drive into him. It wasn't big enough to cause the damage that had befallen his father's Grand Marquis without totaling itself, but it was connected. Oh, yes. Definitely connected.
Jack followed the pickup's dust cloud along Pemberton. He was gaining on it when it suddenly braked and hung a hard right. Jack skidded to a halt, almost missing the turn. He nosed onto a pair of sandy ruts that curved to the right. He accelerated but the dust was so thick that he missed the path and slid off into the brush. It took a few back-and-forth maneuvers to get moving again, and by the time he made it back to the road—that pair of ruts was nothing more than an arc that curved back to South Road—the truck was nowhere to be seen.
Jack drove to the intersection and got out. He scanned the roads up and down in search of a tell-tale dust cloud, but saw nothing. The truck had either slowed or pulled off the road to hide in the brush.
Frustration set his teeth on edge as he swung back into the driver seat. He pounded once on the steering wheel.
"Not to worry," Anya said. "I have a feeling you'll be seeing that truck again."
"So do I," Jack said. "That's the problem."
## 5
Jack needed to pick up some beer and a few munchies. Anya said she needed to do some food shopping as well. So, following her directions, he drove them to the Publix in downtown Novaton. On the way he saw a number of homeless types begging on the sidewalks. He hadn't noticed them on past trips through.
A fellow with a cauliflower nose and a lumpy face that looked like he'd stuffed his cheeks with marbles stood near the door. He held a Styrofoam cup and shook the change within, looking for more.
As Jack slowed, trying not to stare but wondering if this guy was related to the two in the pickup, Anya grabbed his arm and pulled him through the automatic door.
"Give him nothing. His type are up to no good."
Inside, he and Anya split, she rolling her cart toward the produce section while he headed for the snack aisle. There he found more varieties of fried pork rinds and pork cracklins than he'd ever imagined possible. He'd heard of them but never tried any. He passed them by and stocked up on healthier fare—tubes of cheese Pringles, one of his household staples. On his way back past the pork rinds he gave in to an impulse and picked up a bag. He'd try anything once. Couldn't tell Gia, though. She'd be grossed out.
He found the beer section on the left side of the store where it took up the whole wall. But nowhere on that wall could he find Ybor Gold. He saw a stock boy who didn't look old enough to drink stacking twelve-packs of Bud Light in the cooler; he had late acne and an early goatee. His brown hair was gelled into shiny spikes. Jack asked him where they hid the Ybor Gold.
"I don't think we carry that one anymore," he said.
Damn. He'd enjoyed those two he'd had on the way down.
"Why not? It's a local beer."
"That's not local. It's made in Tampa."
Exasperated, Jack started waving his arms. "If you can stock Sapporo Draft from the other side of the world, how come you can't stock something from the other side of the state?"
"Wait a minute," the kid said. "Come to think of it..."
He went over to the imported section, shuffled some stock around, and pulled out a six pack of Ybor Gold. He held it up, grinning.
"Knew I'd seen this somewhere."
"My hero," Jack said.
"There's one more back there. Do you—?"
"Sold!"
As the kid put the two six packs in the cart, Jack handed him a five-dollar bill.
"Naw, that's okay," he said. "Just doing my job."
Jack shoved it into the breast pocket of the kid's shirt. "Yeah, but you deserve a raise."
He hunted up Anya and followed her around as she picked out what she wanted. This involved playing touchy-feely with almost every piece of fruit in the store. Finally she was done and they checked out. Jack qualified for an Express Lane and cooled his heels by the door as her order was rung up.
Out in the parking lot, he was loading everything into the trunk when he spotted a battered red pickup parked against the far curb half a block down. Anya and Oyv were already in the car; it was running with the AC on. Jack leaned in the driver door.
"Can you spare a few minutes?" he said. "I want to check something out."
She glanced at her watch. "Don't be too long. I'd like to stop in on your father before we head home."
That was on Jack's to-do list as well. But first...
He angled across the parking lot, then crossed the street. As he approached the truck—no question now that it was the same one—he noticed a slim young woman with a dark complexion and wild hair a startling silver white. She leaned against a nearby wall. She wore white Levis and a tight black vest over a long-sleeved white shirt buttoned up to the collar.
He stared at her. Something familiar about her. Not the hair, but that face, those black eyes...
And then he knew. Stuff that hair under a black wig, put her in a nurse's uniform, and she'd be the mystery woman who'd fled his father's room yesterday.
First she's a brunette in the hospital, now she's white haired and hanging out on the street. What the hell?
Next to her stood a hulking man Jack recognized as the one who'd ferried the mystery nurse away yesterday.
The woman's eyes met his and he saw an instant of recognition there. She hid it immediately and slid her gaze off him, but he'd caught it.
Jack stepped back and edged toward the truck. The guy with the bulging forehead was leaning against it. Couldn't forget him. He'd been driving when Jack's tire was slashed. Had he been driving an hour ago?
Time to find out. Time to see if he could provoke a little something out of this clown.
Jack lidded his anger and sidled up to him. The man's misaligned eyes were fixed on the crowd. Jack got his attention by giving his right shoulder a none-too-gentle shove. The guy bumped against the truck's passenger door and whirled on Jack.
"Hey! What—?"
Whatever he was going to say never got out. Jack saw his eyes widen with recognition and knew he had his man.
"Almost nailed me out there, didn't you," Jack said, stepping closer and getting in his face.
"Luke?" the guy said in a high, quavering voice.
Jack gave him another shove. "Whose bright idea was that? Yours? Or somebody else's?"
"Luke?" he said again, louder this time, his eyes darting back and forth. "Luke!"
Jack was about to give him another shove when the big burly guy who'd been next to the woman came up. His little pig eyes fixed on Jack.
"What's goin on?"
"This your truck?"
"What if it is?"
"It sideswiped me out in the boonies a little while ago."
Luke shook his head. "No way. It's been sittin here all day. Ain't that right, Corley?"
Corley missed a beat, then nodded his misshapen head. "Yeah. That's right. Here all day."
"Really?" Jack stepped over to the right front fender and ran his hand along the beige-streaked dent there. "I bet if the police compare the paint on these scrapes to the paint on my car they'll come up with a perfect match."
He had no intention of getting the cops involved, but they didn't know that.
Luke's eyes shifted from the scrapes, to Corley, to Jack. "What if it does? Don't prove nothin."
"I think the cops will see it differently, and then I won't be the only one wanting to know why you tried to run me down."
"Somebody tried to run you down?" said a woman's voice behind him.
It was the girl.
"Do I know you?" Jack said.
She stuck out her hand. "My name's Semelee. What's yours?" Her dark eyes were alive with interest as she looked at him.
"Jack," he said as he shook her hand. Her skin was soft, like a baby's. He nodded his head toward Luke and Corley. "You connected to them?" He knew the answer but wanted to see how she'd respond.
"They're kin. You think they tried to run you down?"
"I don't know who was driving, but I know it was that truck."
Her expression darkened. "Oh, it was, was it?" She turned and glared at her "kin."
"Get in the truck."
Luke spread his hands. "But Semelee..."
"In the truck," she said through her teeth. "Now!"
The two of them moved off like whipped dogs. If nothing else, Jack had learned who ruled the roost.
She was all smiles when she turned back to him. A nice smile. The first he'd seen. It lit up her face and made her almost pretty.
"I'm sure it was just an accident. Those boys drive a little crazy sometimes. Why don't I buy you a drink and we can talk it over. Maybe—"
"What were you doing in my father's room?"
"Your father?" Her brow furrowed. "I don't think I—"
"His hospital room. You were in it yesterday, wearing a wig and dressed like a nurse."
She snapped her fingers. "I knew I seen you before."
Yeah, right. She'd known him the instant she saw him.
"What were you doing there?"
"Oh, that. I been thinkin bout becomin a nurse, so I dressed up like one and went to the hospital to see what it was like. It didn't work out. Made me feel kinda sickish. I guess nursin ain't for me."
"I guess it ain't."
Good story. It fit nicely with what he'd seen, but Jack wasn't buying a word.
She smiled again. "Now, about that drink...?"
He hesitated. A little face time with her and he might get a handle on what was going on between his father and Semelee and her "kin." But he had Anya back in the car and he hadn't seen his father yet today. But maybe he could catch her later.
"Have to take a rain check," he told her. "Got to get to the hospital."
"Oh, yeah. Your daddy. Is he bad sick?"
"He's been better."
Another battered pickup, this one blue, pulled up beside the first. For a moment he thought it was filled with migrant workers, but then Jack saw their misshapen heads and bodies. If they were any sort of workers, they looked like they might be extras for Wes Craven if he was doing a new sequel to The Hills Have Eyes. He recognized the marble-cheeked guy from the Publix. All the funny-looking street people he'd seen begging on his trip through town were gathered in these two trucks.
"Well," Semelee said, "we'll try for that drink some other time."
Jack tore his eyes away from the blue truck. "We sure will. When?"
"Whenever you want."
"How do I reach you?"
"Don't worry." Her smile broadened as she opened the passenger door of the pickup and climbed in. "Just say the word and I'll know."
Something in her tone sent an icy trickle down Jack's spine.
## 6
Jack walked into the hospital room and froze just inside the door. His father, dressed in an open-back hospital gown with little booties on his feet, was sitting up on the edge of the bed eating a plate of green Jell-O.
"Christ! Dad...you're awake!"
His father looked up. He looked fresh and rested. He might have been sitting on his front porch having a gimlet.
"Jack? You're here? You?"
His blue eyes were clear and bright through his steel-rimmed glasses. His hair was damp and combed, his face looked freshly scrubbed. If not for the facial bruises and the bandage on the side of his head, there was no evidence that he'd been seriously hurt.
"Yeah. Me." He shook his head. "I can't believe this. Last night you were still level-seven coma and today..."
"They told me one of my sons had been visiting. I assumed it was Tom. But come to think of it, I seem to remember hearing your voice."
"I was talking to you a lot."
"You were? Maybe that's what brought me out. I couldn't believe you were here so I had to see for myself." He sighed and looked at Jack. "Is this what I have to do to get you to visit?"
"Such a thing to say!" Anya said, bustling around Jack and heading for the bed. She'd hung back at the doorway, making Oyv comfortable, she'd said, and had waved Jack ahead. "Be nice, Thomas."
"Anya!" his father said, eyes lighting at the sight of her. "What are you doing here?"
"Jack brought me. We've become fast friends." She took his right hand in both of hers. "How are you?"
"I'm fine. Better every minute, especially since they took that catheter out of me." He shuddered. "That's not something—"
"There she is!" said a heavily accented woman's voice. Jack turned and saw a thin little Hispanic woman, dressed like a nurse's aide, standing next to the hulking form of Nurse Schoch, pointing at Anya. "She's the one I told you about."
Nurse Schoch, looking as stern as ever, glanced down at the aide and spoke in a rumbling voice. "You want to tell me again what you saw?"
"I was in the bathroom, washing the sink, when she come in and hold his hand and say, 'Okay, Tom. You've been asleep long enough. Today's the day you get up.' That's what she say."
Anya laughed and waved a hand at her. "How do you know I don't say that to him every day?"
The little woman shook her head. "Right after she leave, he sit up in bed and ask me if he miss breakfast."
"Did I?" his father said, smiling. "I don't remember. I was a little groggy after I first woke up, but I'm fine now." The smile faded. "So many things I don't remember. They tell me I had an accident but I don't remember a thing about it."
The aide was still pointing at Anya. "Bruja!"
Jack knew enough Spanish to know she was calling Anya a witch.
"Enough of that," Schoch said. "Go clean something. Git."
After one last fearful look at Anya, the little woman scurried off. Nurse Schoch stepped over to his father's side and took his blood pressure. She nodded and wrote on a clipboard.
"How am I doing?" he said.
"Fine." Schoch smiled and, surprisingly, it didn't break her face. "Amazingly fine. Dr. Huerta's coming up to see you."
"Who's he?"
"She. She's been taking care of you since you were brought in to the ED."
"Well, she'd better get here fast, because as soon as I finish this Jell-O, I'm going home."
Jack and Schoch began talking at the same time, telling him he couldn't, that he'd just had a serious injury, and so on and so on. Didn't faze him.
"I don't like hospitals. I feel fine. I'm going home."
Jack recognized the note of finality in his father's voice. He'd heard it as a kid. It meant Dad had made up his mind and that was that.
"You can't," Schoch told him.
He peered at her through his glasses. "I guess I'm a little confused. When did I become the hospital's property?"
Schoch blinked and Jack guessed no one had ever asked her that.
"You're certainly not the hospital's property, but you became its responsibility when you were wheeled through the doors."
"I appreciate that," he said. "Really, I do. And from the way I feel right now, you've all done a wonderful job. But I no longer need a hospital, so I'm going home. Where's the problem?"
"The problem, Dad...," Jack said, feeling his patience slipping. His father was acting dumb. "The problem is that you had a serious accident—"
"So I'm told. Can't remember a thing about it so I guess I'll have to take people's word for it."
"It happened," Jack told him. "I've seen the car. Totaled."
He winced. "Not even a year old." He shook his head. "I wish I could remember."
Jack watched his father's expression. Was that fear in his eyes? Was he afraid? Of what?
"That's not the point," he told him. "The point is you've been in a coma for three days and how do we know you won't lapse back into one in the next minute or hour or day?"
His smile was thin. "We don't. But if I do, you can bring me back here." He held out his arm—the one with the IV running into it—to Schoch. "Would you remove that, please?"
She shook her head. "Not without doctor's orders."
"Okay, then. I'll do it myself."
"Christ, Dad," Jack said as his father began peeling off the tape that held the line in place.
"All right, all right," Schoch said. "I'll take it out for you. Just let me get a tray."
As she lumbered out, Jack looked at Anya. She hadn't said a word through all this. He looked at his father who had lowered the top of his hospital gown and was peeling off the cardiac monitor leads.
"Can't you convince him?" he said to her. "I obviously can't."
Oyv popped his head out of her big straw bag as Anya shook hers. "I should be making his decisions? He's not crazy."
"He's acting crazy."
"He wants to leave the hospital because he feels fine. What's so crazy about that?"
Thanks for the help, he thought. He'd feel a lot better if his father would stay just one more day, to make sure his condition was stable. He had to find a way around his reckless stubbornness.
Anya was staring at him. "Switch places. What would you do in his situation?"
I'd get the hell out of here and go home, he thought. But he couldn't say that.
"I'm lots younger and—"
Oyv dropped back down into the bag as an anxious looking Nurse Schoch came charging into the room, carrying a tray. She stopped at the foot of the bed and shook her head as she stared at the cardiac leads scattered across the sheet.
"I figured that was what you were doing when the monitor flatlined, but I had to be sure."
A few minutes later, Dad had a gauze patch taped over the spot where the IV had been. He stood and looked around.
"All I need now are my clothes."
"They had to throw them out." Here was the angle Jack had been looking for. "They were too bloody to keep. You know what? Why don't you hang out here one more night and I'll come back first thing in the morning with some of your clothes. How does that sound?"
"Terrible. I'll wear this if I have to."
Jack thought of refusing to drive him home, but what would that accomplish? All he had to do was call a cab.
He caught a glimpse of his father's skinny white buttocks through the back of the hospital gown as he walked to the tiny closet.
"Well, will you look at this!" he said as he opened the door. He held up a white golf shirt and tan Bermuda shorts. "Just what the doctor ordered."
"Somehow I doubt that," Jack said. He looked at Anya. "Where'd they come from? You were here this morning. Did you—?"
"You think I go snooping in closets?"
His father headed for the bathroom. "I'll be out in a minute."
"Dad, those aren't your clothes."
"I'm claiming them for the moment. I'll bring them back tomorrow."
I give up, Jack thought. I'm licked. He's going home.
While he was changing, Anya puttered around the room, opening and closing drawers, filling a little plastic bag with the soaps, mouthwash, toothpaste, and other necessities the hospital had supplied.
"No sense in letting any of this go to waste," she said. "He's paid for it, after all—probably through the nose, if I know hospitals."
Jack watched as her hand darted behind the headboard. She pulled something out and quickly shoved it into the plastic bag. He didn't see it, but he could guess what it was. She was taking back her painted tin can totem.
Dad, still wearing his hospital booties, stepped out of the bathroom and spread his arms to show off his new duds.
"Would you believe it? A perfect fit."
"Imagine that."
Jack looked at Anya but she wouldn't make eye contact. What was her part in all this? Was that nurse's aide right? Could Anya have had something to do with his father's miraculous recovery? That would be strange, but he was becoming used to strange.
"Are we ready?" his father said. "Then let's go!"
## 7
On the ride back to Gateways—Jack driving, his father in the passenger seat, Anya and Oyv in the back—he told his father what he knew about the accident, including the anonymous call to the police that appeared to have been made before the crash.
"I wish I could remember," he said. "The last thing I recall is leaving the house and driving out the front gate. And that's it. What happened during the drive? Why can't I remember?"
"It's called retrograde amnesia," Jack told him. "You can't retrieve memories of events right before you got hit. There's a good chance over time your brain will sort them out, but then again, it may never."
His father stared at him. "How do you know so much about it?"
Oops. "A sort of lecture I listened to once. Very interesting."
The speaker had been Doc Hargus. Jack had been knocked cold in a fall from a fire escape. After coming to he'd known enough to get to Hargus to have his scalp sewn up, but couldn't remember why he'd been on the fire escape in the first place. The doc had explained about post-traumatic memory loss, both antegrade and retrograde. It had taken a few days, but Jack finally remembered how he'd got there. And who'd shoved him off.
"Well, I hope mine comes back soon. As for the accident being reported before it happened..." He shook his head. "Impossible. So we can forget that. Somebody's watch was way off. That's the only explanation. Wasn't it Sherlock Holmes who said, 'When you eliminate the impossible, whatever remains, however improbable, must be the truth'?"
Jack was sure he'd heard Basil Rathbone state that a hundred times.
"Yeah, I think so."
Except, considering the course of Jack's life these past months, the impossible was not as easy to eliminate as he'd once assumed.
After Jack parked the car in the cul-de-sac, his father insisted—over his son's protests—on helping carry Anya's groceries into her house. They left her there with a promise to return for cocktail hour.
As he preceded Jack into the front room of his house, he said, "I guess I should be saying, Boy, it's great to be home. But I can't. I may have been in that hospital bed for days, but I feel as if I left here only a few hours ago."
He lowered himself into the recliner and stared into space. Jack watched him and realized he was scared. He'd never seen his father scared, or imagined he could be. He knew he couldn't leave him like this.
"I'm going to stay a few days," he told him. "If that's all right with you."
His father looked up at him. "You? Acting like you're a member of a family? What gives?"
The remark stung, and that must have shown in Jack's face because his father's voice abruptly softened.
"I'm sorry. I shouldn't have said that. I'm glad you're here. You don't know how glad. It's just..."
"Just what?"
"Kate's funeral. Why weren't you there? I still can't believe you didn't show up."
"I couldn't."
"Like hell. A hundred, maybe two hundred people showed up. Mothers bringing the children she'd treated, people she'd treated as kids bringing their own children. All those strangers made it to her funeral, but not her own brother. She touched a lot of lives in her life, Jack, but yours most of all. She practically raised you. You brought out the nurturer in her. When you needed changing or needed to be sung to sleep, she'd take over, she'd say she'd do it. She'd all but fight with your mother to take care of you."
"I know," Jack whispered through a constricting throat. "Don't you think I'd have been there if it had been possible—any way possible?"
"Then why weren't you?"
How could he tell him it was because BATF and FBI people were there too? Taking pictures. Because of the way Kate died, and the events leading up to and connected to her death, they'd camped outside the funeral home and cemetery with their telefoto lenses. Jack had spotted them just as he was about to turn into the funeral home parking lot. He'd driven on. He couldn't let them take his picture and have it end up pinned to a corkboard wall with a question mark beneath it. Who he was was a question he didn't want them even asking, let alone answering.
"It wasn't...it just wasn't possible."
"Why not? Were you in jail? In a hospital in a coma? Those reasons I'll accept. Anything less..."
"I was there. I couldn't make it to the ceremonies, but I visited her grave after the funeral."
"If you could show up then, why couldn't you show up before?"
Jack remembered the anger he'd felt at spotting the feds outside the funeral home. But it had been an anger tinged with guilty relief. Their presence meant he wouldn't have to face Kate's kids, her ex-husband, and his father. Because there'd be too many questions about Kate's last days and he couldn't tell them anything because there was so much she hadn't wanted them to know. But most of all because he felt in some ways responsible for her death. In her last moments he'd soothed her while she bled, held her cooling hand after she died.
"Through the whole ordeal," his father said, "everyone kept asking if the long-lost Jack would show, and I said of course you would, especially since she'd just been taking care of you while you were sick."
"You know about that?"
"She called Ron the night she died...told him. She was still looking after you, even after you'd grown up." Tears filled his eyes. "She brought Kevin and Lizzie down for Easter week last spring. I didn't know it would be the last time I'd see her alive. I was supposed to go up and stay with her awhile in July. Instead I went up for her funeral." His voice hovered on the edge of a sob. "I miss her, Jack. Even though I moved down here we still talked. We phoned each other two or three times a week."
Jack took a step closer. He reached out a hand to put on the old man's shoulder, hesitated halfway there—would he shrug it off?—then pushed past the doubt. He gave his father's bony shoulder a gentle squeeze.
"Kate was a wonderful person, Dad. You can always be proud of her. You and Mom deserve a lot of credit for that."
He looked up at Jack. "I wonder. Kate turned out great, but you and Tom...where is he, anyway?"
That reminded him: He should call Tom and let him know Dad was out of the coma. Not that he seemed too worried. He'd yet to call for an update.
"He couldn't make it. He told me he's tied up with some legal thing in Philly."
He shook his head. "Figures. Tom's always got something else to do; we all know who's number one in his life. And then there's you...the vanished son. I suppose your mom and I deserve credit for the two of you as well as Kate, don't we."
He sounded so bitter. Maybe he had a right to be. Jack started to slide his hand off the shoulder but his father grabbed it and squeezed.
"I'm sorry, Jack. I had to let this out. It's been eating at me since the funeral. And since you never returned my calls..."
"Yeah, sorry about that." Again, he hadn't known what he could say.
"...I never had a chance to get this off my chest. I still don't understand, and I guess I never will. You're holding back on me. I don't know why but I hope someday you'll tell me the real story." He released Jack's hand and slapped his palms against his thighs. "Until then, I'm through with this kind of talk. It's putting me in a funk."
He sat in silence for a moment, Jack standing beside the chair, trying to come up with something to say. But he didn't have to. His father broke the silence by rising from the chair and heading for the kitchen.
"I'm going to have a beer. Want one?"
"Do you think you should? I mean, you were in a coma this morning and—"
"Do you want one or not?" he snapped.
If you can't beat him, Jack thought, join him.
"Yeah, okay. Pop me one."
His father opened the refrigerator door and pulled out an amber bottle. "What's this?"
"Oh, that's an Ybor. It's a Florida brew I discovered."
His father gave him a hard look. "What did you do? Move in while I was out cold?"
"Well, Anya said you'd want it that way."
"She did, did she?"
These mood swings between friendly and hostile were getting to be a bit too much. "Look, if you want me to move out—"
"I wouldn't hear of it."
He popped the caps off a pair and handed one to Jack. They clinked the bottles.
Jack said, "To letting bygones be bygones?" At least for now.
"Not always as easy as it sounds, but I'll drink to that." His father took a sip and then studied the label. "Ybor Gold, ay? I like it."
Jack took a long pull. "Yeah. But they should have named it Ygor Gold. Then they could have had this sneaky-looking hunchback on the label. Would have been very cool."
His father stared at him. "Now why on earth would you think of that? Why would anyone think of that? You know, I used to worry that all those monster movies you watched as a kid would warp you. Now I can see they did. I swear they did."
"Hey, I've watched lots of romantic films too, Dad, but they didn't make me romantic. And I know I must have seen hundreds, maybe a thousand comedies, but they didn't make me funny. I haven't committed stand-up yet and, trust me, I'm not the life of the party."
His father laughed for the first time since he'd come out of the coma. That was a good thing.
## 8
They hung around the front room for about twenty minutes or so, sipping their brews and making small talk, then his father dozed off in his recliner. At first Jack worried that he'd lapsed back into coma, but he responded when Jack shook his shoulder. He left him sleeping in his chair and went outside.
Through the late afternoon haze he spotted Carl working three houses down. When he saw Jack he hurried toward him across the dry grass. A small garden spade protruded from his right sleeve.
"I heard about your daddy," he said, flashing a yellow grin. "Real glad he's okay. That's pan-o-ramic!"
"Sorry?"
He shrugged. "I just like the word. Anyways, I'm glad he's back."
"Thanks, Carl. He's napping now."
"Good. Real good. Looks like the list don't get more pan-o-ramic."
Wishing he'd never uttered that word, Jack said, "What list?"
"The list of Gateways folks who've gone before their time—not that 'before their time' means a whole helluva lot round a place like this. Funeral home waiting rooms is what they is."
"I'm not following you."
"Had a bunch of strange deaths real recent like."
Jack felt a crawly sensation in his gut. "Like what? Hit and runs?"
"Nup. Nothin like that. I mean strange. Like Mrs. Borger bein attacked by about a dozen pelicans last year—right before Christmas, it was. Pecked her to death. I hear tell one of them bit into her neck and there was blood shootin everwhere. Been in Florida all my life and I ain't never heard of no one bein attacked by no pelicans. Then back in March there was Mr. Leo, all bitten up by a bunch of spiders. Brown recluses, they say." He shuddered. "If I was ever on Fear Factor, that's what would set me to runnin. Anyways, Doc Harris said he's never heard of someone gettin bit more'n once, but there you go. Poor old guy died in the hospital."
"Jeez."
"Then just last June, Mr. Neusner trips and falls into a whole nest of coral snakes. He was DOA like the others. Come to think of it, your daddy was the only accident that made it to the hospital alive. I guess that's a good sign."
"Let's hope so."
"Funny thing about Mr. Neusner and the coral snakes. We got a sayin down here: 'red touch yellow—kill a fellow.'"
"What's that mean?"
"Well, there's coral snakes, which got red, yellow, and black stripes, and they's poisonous as all get out. And then there's the scarlet snake and the scarlet king snake which got similar stripes but they're harmless. The way you tell 'em apart is by the order of their stripes."
"You mean people hang around long enough to check out the stripe order?"
"Sure. If it's got a red stripe next to a yellow stripe, it's a coral snake. If it don't, then you're okay. You may get bit, but you won't get poisoned." He pronounced it "pie-zund."
Jack said, "I'm a city boy. I see any snake, striped or plaid, I'm gone."
He much preferred dealing with human snakes than the legless kind.
"But the thing is," Carl added, "I seen one of them snakes, the one Mr. Neusner stomped on before he keeled over. Don't know bout the other ones that bit him, but this one didn't have no red touchin yellow. It shouldn't have been poisonous, but it was." He shook his head. "Kinda scary when somethin you always depended on turns out not to be true anymore."
Tell me about it, Jack thought. He'd seen the pins kicked from under more than one Cherished Truth lately.
"You said there was a nest of them? Right here at Gateways? How? The place looks so...manicured."
"I can't figure that one neither. I run the mower over that spot every week and I ain't never seen no snake nest. I think a buncha them just coiled theirselfs all together durin the night and was still there when Mr. Neusner come by like he did every mornin." Carl looked away, toward the Everglades. "Almost like..."
"Almost like what?"
"Like they was waitin for him."
Jack's gut crawled again. "You don't really believe that, do you?"
A shrug. "Just a thought."
"I'm having a thought too," Jack said as the crawling sensation increased. "December, March, June...every three months someone buys it. And three months from June is—"
"September," Carl said. "You're thinkin of your daddy, right? But the others was done in right here at Gateways by things like birds and spiders and snakes—all natural like. Your daddy had a car accident and he wasn't here at Gateways like the others."
But the regularity of the fatal mishaps to Gateways residents, the steady three-month intervals between them, bothered Jack. Especially since his father had almost bought it at the end of another three-month cycle.
Something might be going on, but it sure as hell wasn't the Everglades seeking revenge.
Jack feared something less substantial but far more real might be behind it.
## 9
Tom awoke from his nap and looked around. Where was Jack? Or had he only dreamed he was here? That might mean that the whole coma thing was a dream too.
Then Jack walked in the front door and he felt a strange mix of emotions: up that his prodigal son had come home, even if only for a few days, and down because it meant the accident and coma were all real.
"Oh," Jack said. "You're awake. Short nap."
"The short ones are the best. They don't leave you groggy."
Jack headed toward the kitchen. "I'm going to have another beer. Want one?"
"No, thanks. But you go ahead."
Tom watched him twist off the top of an Ybor Gold and thought how much he looked like his mother. He had Jane's brown hair and brown eyes. And he moved with her grace, her economy of motion.
Tom hadn't seen his younger son in over a year, not since that father-son tennis match he'd roped him into last summer. He'd changed in that time. He didn't look older, but his eyes held a different look. He couldn't call it a hunted look. Maybe haunted? Haunted by Kate's death? Or was it something else? Guilt, maybe. Well, he should feel guilty about missing Kate's funeral. Damn guilty.
He didn't know what to make of his younger son. He'd thought they'd been close. He'd made a special effort to spend time with Jack while he was growing up. An unplanned baby. He and Jane had their boy and their girl and were content with that. But Jack showed up eight years after Kate, and neither Tom nor Jane had quite the energy they'd had with the first two. But Tom hadn't wanted to shortchange the little guy, thus the special effort.
But then Jane was killed; and less than a year later Jack disappeared. He'd called home once to say he was okay and not to worry, but wouldn't say any more. In the space of less than a year Tom lost his wife and one of his sons. He'd never imagined he could hurt so much. He thought his world had come apart.
He blamed himself at first—what had he done, where had he gone wrong? But then he came to realize that disappearing was in keeping with Jack's character as he'd come to know it.
He'd realized early on how bright Jack was, brighter than either Tommy or Kate, but he was also something of a loner. Okay, more than something of a loner. He did well enough gradewise, but all his teachers said he'd do better if he applied himself. That and "Does not play well with others" were constants during his early schooling.
Although a natural athlete, he never seemed to care for sports. At least not team sports. It was his father's urging rather than any desire to compete that drove him to sign up for a couple of the high school teams. He joined the track team, but as a cross-country runner where he was competing with the terrain and himself as much as the opposing school's team. He also spent two years on the swim team. Both loner sports.
Even his first summer job—cutting lawns in the neighborhood—was a solitary enterprise. He borrowed the family lawnmower and went into business for himself. As a college student he needed more cash so he went to work for one of the local landscapers.
But what he really seemed to enjoy most was reading far-out fiction—if it had a monster or a spaceship on the cover he bought it—and watching old sci-fi and monster movies.
He'd worried about Jack, urging him into more social activities. It's a beautiful Saturday. Go down to the park and get into one of the ball games! Jack would reluctantly get on his bike and pedal off. Later, as Tom was riding through town, he'd spot Jack's bike chained to a standpipe outside the local theater that was showing a Saturday afternoon monster double feature.
He'd worried then, he worried now. Jack earned his living, at least as far as Tom could tell, as an appliance repairman. In the few times during the past fifteen years that he'd seen his son—times he could number on the fingers of one hand—and had a chance to ask him about it, he'd always seemed evasive. Maybe because he sensed his father's disappointment. Nothing wrong in being a repairman in and of itself; the world needed people who could fix the mechanical and electronic conveniences of modern life. Fine. But he wanted more for his son. Jack had three and a half years of college behind him that he wasn't using. What was he going to do when his eyes got bad and his fingers got arthritic? Did he think he was going to get by on that Ponzi scheme called Social Security? Tom hoped not.
But what bothered him more was that Jack seemed rootless, disconnected, adrift. Not exactly a ne'er-do-well, but...
But what? Why was he so secretive about his life? Tom was a believer in everyone's right to privacy, but really...it was almost as if Jack were hiding something.
Earlier this year Tom had gathered the courage to ask if he was gay. Jack had denied it, and his easy laugh as he'd assured him that he was attracted only to women had convinced him he was telling the truth. Tom wouldn't deny that that had been a relief. But if Jack had said yes, well, Tom would have tried to find a way to accept it. He was glad that wouldn't be necessary.
So if it wasn't that, what? Was he using drugs? Or worse, dealing them? He prayed not. And for some reason, thought not.
He supposed Jack's unused education rankled him the most. Education wasn't something Tom took lightly. He'd fought and killed to get his.
He slid back along the lines of his life to his childhood. He'd been born during the Great Depression, the son of a truck farmer outside Camden who'd been scraping by before the economy crashed, and continued to scrape by after. At least they always had food on the table, even if it was only vegetables they picked or pulled from the ground themselves.
Tom's father had been just old enough to see a little action in the First World War, and just a little bit too old to fight in the Second, although that hadn't stopped him from trying to enlist after hearing news of what the Japs did to Pearl Harbor. Tom remembered being afraid that they'd soon see hordes of yellow men running wild through the streets of America. He'd read numerous scenarios describing just that during the late thirties in the pages of the Operator 5 magazines he borrowed from a kid in school.
But his father was rejected and the Japs never set foot one on North America. So much for that worry.
But when Tom hit eighteen there was no money for college. He'd done well in high school but not well enough for a scholarship. So he enlisted in the Army. It was peacetime so it seemed a safe place to be: earn a little money, save what he could, and maybe see some of the world in the bargain. But most importantly, it offered a chance to get off that farm.
A year after he enlisted he was seeing the world, all right. Shipped to Japan and then to South Korea to fight in a UN "police action." Even now, he ground his teeth every time he heard that phrase. It had been a full-blown war. He'd fought from sunny Seoul to the frozen hills of North Korea where he witnessed firsthand the Red Chinese human-wave assaults. For years after, he awoke sweating and shaking with the memory. At least he was alive to have nightmares, unlike too many in his unit who came back in boxes.
When he returned to the States he found a day job and used the GI Bill to put himself through night school. He graduated with an accounting degree and soon qualified as a CPA. He joined Price Waterhouse and spent the rest of his working life with the firm. He was able to provide his wife and children with all the things his own father had been unable to give him. To Tom, the most important of those was a higher education. Tom Jr. had made good use of it, so had dear Kate. The result was a lawyer and a doctor in the family.
And then there was Jack...
The man in question dropped into a chair opposite Tom.
"Can I ask you something, Dad?"
"Sure."
"What were you doing out on those back roads at that hour?"
Tom almost told him it was none of his business but bit it back. He had to put this anger behind him, forget what happened before and be glad for the now.
Could he do that? He had to try.
"Just driving. I have trouble sleeping lately. I lie there in bed and I close my eyes but it won't come. They tell you not to stay in bed if you can't get to sleep, so I go out for a drive."
"And do what?"
"Not much. Lots of times I stop the car and sit on the hood and watch the sky. Jack, you wouldn't believe it. You can cruise those back roads at night and not see another soul. You stop the car and turn off the headlights and get out and above you are stars like you've never seen, stars like I haven't seen since I was a kid in the Jersey sticks, when the air was still clean enough to see the Milky Way smeared across the top of the sky. It's breathtaking."
"You always drive the same route?"
"Pretty much. There aren't many roads to choose from out there."
"So you have a pretty set pattern?"
"I guess so. Why are you asking?"
Jack took a sip from his bottle. "Just trying to put some pieces together. Since there's no one out there, do you bother to stop at stop signs?"
"Well, yes. Of course I do. It may not make sense but...I guess it's just habit. And it's not as if I'm going anywhere, or in a hurry to get there."
"The cops think you might have blown through a stop sign and got tagged by something speeding along South Road. Something big."
Tom shook his head. "I wish I could remember."
It disturbed him no end that a piece of his life was missing—an important piece, one that had put him in a coma for days. It scared him a little...no, it scared him a lot not knowing any of the details. That was why he couldn't stay in the hospital. If he had to be in the dark as to what had happened to him, he'd rather be in the dark here, in familiar surroundings...where he felt he was in control. Or felt he had at least some modicum of control, even if illusory.
"Do you remember a woman attacked by pelicans last year?"
"Sure. Adele Borger. Terrible thing. I heard she was walking with two other women whom the pelicans ignored. They attacked just her. They say she was a terrible mess."
"And the guy bitten by the snakes?"
"Ed Neusner. Where'd you hear about him and Adele?"
"From Carl."
Tom had to smile. "Telephone, telegraph, tell Carl. He's the Gateways gossipmonger. Not the brightest bulb in the box, but a good man. Hard worker. He's got some wild ideas, though. Has he told you his theory about the angry Everglades yet?"
Jack nodded. "Yeah. Maybe it's not so far-out. What about the guy killed by spiders?"
"Joe Leo? What about him?"
"Hasn't anyone noticed a pattern to these deaths—like every three months?"
"No." Was he right? Every three months? "No one's ever mentioned it. But why would they? It can't be anything other than coincidence."
"Do you realize your accident falls right into the pattern?"
Good Lord, Jack was right. The muscles along the back of his neck tightened, but only for a second. Coincidence. That was all it was, all it could be.
Tom forced a smile. "Is this what you do in your spare time—invent conspiracies?"
Jack looked at him. "As a matter of fact, yes."
"Don't tell me you believe in UFOs. Please don't."
"The kind with aliens inside? Hardly. But I've had to stop believing in coincidences."
Tom wondered at the bleakness in his son's tone. "What does that mean?"
"Nothing." Jack shook his head. "Maybe I'm reading too much into this. For a minute I had this wild idea that the Gateways honchos might be offing some of their healthier residents in order to have their houses revert to them."
"That is a wild idea."
He sighed. "I know. Especially when I realized that the houses would stay with the spouses. So there goes the motive for that scenario."
"Except...," Tom said as that tightening sensation crept again into the back of his neck, stronger this time. "Except that Adele was a widow and Joe and Ed were widowers."
"Oh, Jeez," Jack said as they stared at each other.
## 10
In Semelee's vision, at least in the eye covered by the shell, she moved at a height varyin from one foot to almost two foot above the ground. Clumps of saw grass whipped past at eye level. Then she was splashin through a shallow pond, and now back up into the grass again. The goin was tougher than it shoulda been. In September of any other year, she—or rather, Devil—woulda been able to stay in the wet for the whole trip. This year, though, was different. Still, the drought wasn't gonna keep Devil from goin where she wanted him to.
The goin was rougher for another reason: She had to stay on course and find her landmarks with only one eye.
At last she came to the pond she'd been searchin for. The level was down, but not as much as most others. She slid into the water and dove deep. Devil's underwater vision was good, better than any human could claim, and soon enough she found the mouth of the tunnel.
She entered a dark place, so dark that even Devil's eyes was no good here. Sometime long ago, when all this land was formed, something happened hereabouts that left a channel through the limestone. Its width was enough to allow Devil to swim through, but just barely. She had to go mostly by feel.
The channel branched and Semelee guided Devil to the left. It seemed to go on forever, but eventually she saw a glimmer of light ahead. Devil surged forward. She could feel his hunger, but she held him back, slowin him to a stop a few feet below the surface. She made him hover there for a few heartbeats, then started a slow float toward the surface. She let only his eyes and the top of his snout break the surface. An egret wading at the pond edge saw him and took flight. Smart bird. As Devil took a breath through the nostrils atop his snout, Semelee focused on the old man's house.
She'd been watchin the place through a frog's eyes, waitin for the son to come home. After bein so close to him in town this afternoon, she had to see him again. She'd felt somethin click between them. Like magic. She sensed destiny there. No doubt about it.
But as she'd been watchin she saw him arrive with that old crone from next door and his daddy! Semelee was so shocked she almost dropped her eye-shell. She thought this was bad at first, but then changed her mind. She realized that somethin must be helpin her, somethin big and powerful, maybe even the Glades itself must be guidin events. Because now that the old man was out of the hospital, he was closer to her. Comin home put him within strikin distance.
And strikin was just why she'd guided Devil here. She had to get this finished. And it had to be this old man. He'd been offered, and had to go before the time of the lights.
As a bonus, after the old man was gone, there'd be nothin standin between her and the son. They could get together, just like they was meant to.
She watched the front door. She wondered when the old man would come out...or if he'd come out. Might be a long wait.
She heard voices. Good thing a gator's ears was atop his head, just behind the eyes, otherwise she woulda missed it. A swish of Devil's tail angled him around so she could see who was talkin and...
Semelee blinked—her own eyes, not Devil's—and stared. There he was: the old man—wearing one of the ugliest Hawaiian shirts she'd ever seen—and his son sittin in the neighbor lady's front yard. This was too good to be true.
She made Devil sink toward the bottom of the pond, and then had him back up to the far end. When she and Devil made their strike, he had to be movin fast. He had to come out of the water at full speed and charge right at the old man. The big gator was hungry so she couldn't let him get distracted and go for anyone else—not that skinny old lady and especially not the son. She had to keep him on course. Not such an easy thing because when a gator opened his mouth, it blocked his straight-ahead vision. To make up for that, nature made it so that if anything touches the lower jaw, the upper snaps down like a bear trap. That meant she had to aim just right so that nothin—not furniture and not the wrong people—got in Devil's way.
Once he got his teeth set in the old man, nothin was gonna break his hold. Semelee would have Devil drag him into the pond and take him to the bottom. The tunnel was too narrow to fit both gator and prey, so once the old man was drowned, she'd let Devil chow down a little before high-tailin it back to the lagoon.
Back at the far end of the pond now, she surfaced for another look. Yes...there he was, talkin and drinkin...if she angled herself just right, she'd have a clear shot at the old buzzard. She'd sink, use Devil's powerful tail to propel them through the water, then hit the land a-runnin. The old man wouldn't know what hit him. And finally she'd finish off what she, Luke, and Devil had begun the other night.
## 11
Tom watched the sunset. He and Anya did this a lot. Not every afternoon, but often enough to approach the status of a tradition. He was wearing one of his favorite shirts, the one with Mauna Loa in full eruption on the back with bright orange lava flows trailing around to the front. As usual, Anya was sipping her wine. He'd brought over a few beers. Often he'd supply a stainless-steel shaker of gimlets that he put in the ice bucket, but the Sapphire supply seemed lower than he remembered. Had Jack been nipping at it?
Jack had called his brother to tell him their father was up and about, then handed him the phone. His older son had made a stab at sounding overjoyed, but what he really sounded was distracted. He said everything was fine but Tom sensed that something was bothering him.
Did this mean he now had two secretive sons?
Jack had come along for the sunset tonight, and Tom learned that he and Anya had done the watch last night.
They really seemed to have hit it off, those two. He felt a twinge of...what? Jealousy? No, that was ridiculous. He liked Anya—loved her, in fact—but in a brotherly way. He felt no sexual attraction to her. She was a friend, a confidante, a drinking buddy. He could talk to her, confide in her. She'd lent him an ear when he'd talked about his self doubts and his wayward children, she'd held him when he'd cried after receiving word about Kate's death. What sexual urges he had—and they seemed to be diminishing—were more than satisfied by a couple of the horny widows populating Gateways South. They weren't looking for long-term relationships—what an alien concept in this environment—and neither was he. The couplings were Viagra fueled, but a lot of the pleasure was in the snuggling and cuddling and having someone else in bed with you.
He turned on the battery-powered CD-player-radio he always brought along. But instead of the usual gentle music from the AM station he kept it tuned to, rap burst from the speakers.
"What the hell?" He checked the dial and, sure enough, it was tuned to the right band. "What's going on here?"
"They changed the format while you were in the hospital, hon," Anya said.
"No!"
"Afraid so. Sorry."
He jabbed at the off switch. "What's happening to the world? Used to be I'd drive behind women and they'd be doing eye makeup and fixing their hair in the rearview mirror. Now it's men who can't take their eyes off themselves—staring at themselves and primping. Christ, everything's going to hell in a hand basket."
"Yeah," Jack said, "and you can bet it's got a Fendi or Gucci logo on it."
"Very funny." He pointed at his son's T-shirt. "Look at that. 'Hilfiger' all across the front of your shirt. They sell you the shirt, then turn you into a free walking advertisement for their product. You should be charging them to wear it."
"It's the way of the world, Dad," Jack said. "Everybody does it."
"And that makes it right? Since when do you of all people want to look like everybody else?"
"Long story, Dad."
"I'll bet."
What's the matter with me? he thought. Why am I so cross? I sound like a crotchety old man.
He smiled to himself. Hell, I am a crotchety old man. But not without reason, not—
Anya's dog started yipping. The little Chihuahua was standing at the edge of the pond barking at the water. Crazy little dog. Tom had noticed a snowy egret there a few moments ago but it was gone now. Probably scared off by the pooch. Nothing in sight but placid water.
He noticed another sound. A chorus of clanking rattles from all around him. The homemade ornaments—the painted cans on sticks salted in among the leprechauns, bunnies, turtles, and flamingoes—were shaking and rattling on their sticks. Funny...he didn't feel a breeze.
The dog increased the pitch and volume of his yipping.
Tom turned to Anya. "What's wrong with him? He hardly ever barks."
"He must sense something out of the ordinary," she said. "Oyv! Get away from there and stop that racket. A migraine I'm getting already. Go back to—"
Suddenly the water erupted and something huge and bellowing exploded from the pond. Tom dropped his beer and his mind blanked in shock for an instant. What the hell was it? All he saw at first was a wide-open set of jaws bordered with daggerlike teeth, the delicate pink membranes lining the maw, and the long, tapered, slightly darker tongue waggling within. Then he saw the dark green scaly legs and the thick undulating tail behind.
An alligator, bigger than any he'd ever seen in all the gator parks he'd visited. And it was racing right for him.
The only thing between Tom and those jaws was Anya's Chihuahua. The little dog held its ground for a second, then charged the gator, leaping at it with a high-pitched growl. The onrushing jaws scooped up the dog and snapped closed.
"Oyv!" Anya cried.
"Holy shit!" Jack was out of his chair and reaching for the small of his back.
Without breaking stride, the alligator made one convulsive swallow and the dog was gone, devoured like a canapé.
The monster gator was still lunging forward. Tom started to leap up but his foot slipped on the grass and suddenly he was falling backward in his chair. Before the gator opened its jaws again, Tom got a look at its head. He caught a flash of two scaly protrusions, gray-green like the rest of its hide, each about six inches long, on either side just behind and below the large brown eyes with their vertical-slit pupils. They looked like horns.
Something twisted in his chest...something familiar about this alligator. But what? How could he ever forget a creature like this?
As he and his chair hit the ground, Tom rolled to the side and started to scramble to his feet. He heard Jack mutter a curse and saw his hand coming out from under his shirt at the small of his back. Jack moved quickly, like a pouncing cat, grabbing the back of his chair and holding it out legs first, like a shield. To Tom's shock, he leaped between him and the gator.
"Dad! Get back!"
Tom regained his feet and backed away, but Jack hung in there, facing the big gator down.
"Jack! Anya!" Tom cried. "Into the house!"
"Not to worry," Anya said.
Tom looked her way and saw that she was still on her recliner. She'd straightened so that she was off the back rest, but she still held her wineglass.
"Anya!" he said. "Get up! It's—"
She glanced at him. Her eyes and expression were unreadable, but her voice was calm, almost serene.
"No creature on earth will harm you here."
"Tell that to Oyv!" Jack said, backing away from the onrushing gator, but keeping himself between it and Tom and Anya.
His son's courage and protective stance amazed Tom. He'd known guys like that in the service—most of them long gone, sadly—but had seen little of it in today's every-man-for-himself world.
And then, incredibly, the gator halted its charge. One second it was roaring toward them, the next it stopped as if it had run into a wall. It stood on the border of Anya's emerald sward and the brown grass that typified the rest of Gateways. It closed its jaws and shook its head as if confused. It tried again to cross the line but then quickly retreated.
It turned left and stalked along the margin of green, thrashing its huge tail as it looked for a way in, and that was when Tom saw something dangling from its right flank. He squinted in the failing light and saw that it was an extra leg. But it looked vestigial. It didn't move and didn't touch the ground. It simply hung there.
The gator then turned and stalked the other way. Tom saw another vestigial limb on its left flank. But far more puzzling was its inability to cross onto Anya's lawn. It made no sense.
And then it occurred to him that the situation might be only temporary. If only he had a gun!
"Call the cops!" he cried. "Call security! Get someone here to either drive this thing off or kill it before it kills someone!"
"No need," Anya said from her recliner. "It will be leaving soon."
The alligator stopped its stalking and bellowed. It shook its head and whipped its tail back and forth. It seemed confused. It bellowed again, and this time it sounded as if it was in pain. Then it rolled onto its side, and from there onto its back, swinging its head back and forth, thrashing its tail and clawing at the air with its taloned feet.
With another throaty bellow it rolled back onto its feet but didn't charge. Instead it made a slow turn and began a limping retreat toward the pond. As it moved away Tom noticed a fist-sized bulge in its left flank, just ahead of the vestigial limb. Not so much a bulge as a pulsation.
The gator roared again as the bulge ruptured, spewing blood along the hide, a crimson splash along the gray-green scales. Something moved within that opening, something red and snouted. The hide split further and—
"Holy shit!" Jack shouted. "It's Oyv!"
Dear God, he was right! The little Chihuahua was chewing its way out of the gator. It squeezed through the ragged opening like a baby being born. Once the upper half of his body was clear, the rest of him slid out. He landed on all fours and shook himself, then started barking at the retreating gator, chasing after it, nipping at its tail until it slid into the water and disappeared below the surface.
The dog dove into the water, repeatedly dipping its head under as it paddled in a small circle, then emerged with the blood washed away. He shook off the water with an almost epileptic shudder, then trotted back toward Anya with his tail wagging, his little head held high, and his black eyes shining. Proud, and very pleased with himself.
"Good boy," Anya said, patting her lap. "Come to Momma."
"What?" Jack started to laugh and Tom thought he heard an hysterical edge to his voice. "What the—? This is impossible! Just plain...." his voice trailed off to a whisper "...impossible."
Jack turned and stared at Anya and she stared right back. Tom would have asked what was going on between them, but he couldn't speak. He had to sit down. He quickly righted his chair and dropped into it, panting for air as his chest tightened.
He remembered now where he'd seen that horned alligator before.
## 12
Semelee dropped the eye-shell and fell to the floor, clutchin her left side. She felt as if someone had shoved a spear halfway through her. Never in her life had she felt pain like this.
"It hurts, Luke. Oh, God, it hurts!"
He hovered over her, hands reachin toward her, then pullin back. "What happened? What's wrong?"
"Not sure." The pain was easin off now. "Don't know how, but Devil got hurt. Hurt bad."
"Did you finish the old man?"
"No. I couldn't get to him."
"That old guy?" Luke's tone said he didn't believe a word of it. "He hurt Devil?"
"No-no. It was the same like in the hospital, only ten times worse. There was this line I couldn't cross without feelin like I was gonna be sick or explode or both. I couldn't push Devil past it." Truth was, she couldn't push herself past it. "And then this pain in Devil's side that I felt too. Like he was bein stabbed, but from the inside."
"The old guy's kid?"
"I don't think so. This wasn't even at the old man's house. It was at the old lady's next door. It's her. Gotta be her. She's the one that's been messin us up."
"Whatta we do?"
"I don't know. I'll worry about that later. First thing I gotta do is get Devil home. He's hurt bad, and he won't know where he is. I gotta bring him in."
She looked down at her eye-shell. She knew that if she put it on she'd feel that pain again. But she had to. She couldn't leave Devil hangin. Had to bring him back to his gator hole where he could wallow and heal up.
How'd that skinny old hag do it? How'd she hurt Devil whose hide was like armor plate?
Semelee didn't know but she was gonna find out. And when she did, that old lady was gonna pay for what she'd done to Devil. That bitch was gonna hurt like Devil. Maybe even worse.
## 13
"Dad? Are you okay?"
Tom looked up from his chair and found Jack staring at him, a worried look on his face.
I must look like hell, he thought. He tried to respond but all he could do was shake his head and sweat.
"Is it your heart?"
"No." Finally he could speak. "Not my heart. It's my head. I remember what happened Monday night."
"You mean, Tuesday morning?"
"Whenever I had the accident. That...that alligator was there."
"That same one?" Jack said.
"You think I could forget those horns and those extra legs?"
Anya was watching him from her recliner. "Don't go out at night like you do—how many times did I tell you that?"
"Countless times." He shook his head. "I should have listened."
Jack dropped into his own chair, opposite. "But how does that alligator figure into your accident? Or doesn't it?"
"Oh, it does. I remember it now. I was driving south along Pemberton, taking my time..."
No hurry, no place to go, no timetable to hew to on that warm yet unseasonably cool night. Cool enough to drive with the windows open, not worrying about the mosquitoes because even that easy pace was too fast for them. He remembered the hum of his tires on the pavement, the soft feel of the wind swirling through the car and the mix of fragrances riding it: the sour smell of the saw grass yearning for water, the sweetness of the flowering roadside bushes.
"...and as I came to the stop sign on South Road, I slowed to a stop—well, maybe not a complete stop, but a sort of rolling stop. I was taking my foot off the brake as the car eased into the intersection, but before I could give it gas again I saw something crawl onto the road ahead of me. I hit the brakes hard and came to a dead stop maybe three-quarters of the way through the intersection."
"An alligator?" Jack said. "The one we just saw?"
Tom nodded. "No question. I couldn't keep going. Something that size—I mean it must be twenty feet long—doesn't leave you any room to go around it. And truth be known, I didn't want to go around it. I felt safe in the car—especially after I put the windows up. It wasn't threatening me, just staring at me. I put on the high beams for a better look at it, and I must have been so fascinated by the sight of this horned gator that I didn't hear the truck until it was practically on top of me. My closed windows and its off headlights didn't help either."
"Wait," Jack said. "The guy was driving out there in the dark with no lights? Not even running lights?"
"Nothing. I heard a rumble to my right and looked and saw this dark shape roaring down at me from the west. It was practically on top of me. I didn't have time to react—or maybe I froze in shock. Whatever the reason, I couldn't move out of its way and it rammed me hard. I saw a big bumper smash into my right front fender and then the car was jerked around like...like I don't know...like it had been punched by God. My head hit something and everything went dark for a while, I don't know how long, and then I was back again, but the world was blurry and full of steam. My ruptured radiator, maybe."
"Did you see any part of the truck? I mean, was it an old red pickup, by chance?"
Tom shook his head. "No. This was a big rig, and seemed to be in good shape. At least its bumper was. I remember seeing what looked like a wall of shiny chrome slamming into me. Why did you think it was a pickup?"
"Just a thought." Somehow Jack looked disappointed.
"Getting hit wasn't the worst part. The really frightening part came after the impact. I was lying there, feeling sick, hurt, bleeding, barely able to move, but alive and so thankful I'd worn my seat belt, when I heard these voices, growing louder as they got closer. I remember hearing someone sounding mad, cursing, saying something about hitting me too hard and what if they'd killed me. And then the door was pulled open and I almost fell out of the car. That was when I heard someone say, 'Look! He's moving! You damn well better thank your lucky stars he's still alive!'"
"That sounds like they meant to hit your car."
"They did." Tom repressed a shudder. He glanced at Anya who was watching him impassively, her expression neutral. "It didn't click then, but now I'm sure they did."
"Sure?" Jack said. "What makes—?"
"By what came next. They unbuckled my seat belt and pulled me out and laid me on the road. I thought they were being awful rough with a man who might have a spine injury. As I was lying there I saw the big truck pulled over down along the side of South Road."
"Wait," Jack said. "The truck pulled over? But the police said it was a hit and run."
"In a very real way, it was. It's just that the run part was delayed a bit. Let me finish, will you?"
"Okay," Jack said. "Just trying to keep all this straight in my head."
"Forget about the truck for now. I know I did as soon as I saw that big alligator start to waddle toward me. I couldn't be sure, but I thought the men who'd pulled me from the car were waving it forward. Like they wanted it to maul me...kill me...eat me." This time he couldn't repress the shudder. "It was within ten feet of me when I heard a siren. I couldn't see any flashing lights but I could hear the two men start cursing about a cop car and what was he doing out here. That sort of thing."
"Officer Hernandez," Jack said.
"You know him?"
"Met him. Remember I told you that a call about your accident came in twenty minutes before it happened?" He glanced at Anya but she didn't react. "He's the one who went out to investigate. Sounds like that call saved your life."
But that didn't make sense, Tom thought. How could anyone have known about the accident before it happened? Yet something with a siren had been coming down the road.
"I don't know who or what was heading my way. All I know is that it scared off the two men who'd pulled me from the car, because they started calling to the alligator as if it was human, as if it could understand. I heard one yell, 'There's a cop on the way! Get out of sight. We'll meet you back at the lagoon!' And then they started running back toward the truck."
"Did you notice anything about them?" Jack said. "Like did one have a funny-shaped head?"
"Funny-shaped head? Why—?"
"Anything distinguishing," Jack added quickly.
"No. Not that I could tell. I didn't take my eyes off that alligator until it slithered off the road and into the grass, and by then they were almost to the truck."
"Do you remember anything at all about the truck? Like what kind? Was it a semi or a big van or what?"
"A semi, maybe, but it didn't have the usual big rectangular trailer. This had an odd shape, like those trucks that carry gravel or something."
"What about a name or a sign?"
"None that I could see. I had only moonlight and starlight to go by and..." Something flashed in his memory.
Jack leaned closer. "What?"
"On its rear panel...I think I saw something that looked like a flower, but all black. At least it looked black in the moonlight. After that, I remember flashing lights and then I didn't see anything until I woke up this morning."
A sudden realization hit him like...like an onrushing truck. He looked at Jack and then at Anya.
"Someone tried to kill me."
"Not necessarily," Jack said. "From what you heard them say...'thank your lucky stars he's still alive...that sounds like they didn't want to kill you."
He sensed that Jack didn't believe a word of it, that he was just trying to make him feel better. But it wasn't working.
"They wanted to hit my car. And I have a feeling they were going to feed me to that alligator."
"Maybe you were just in the wrong place at the wrong time."
No...that didn't wash. No question in Tom's mind: Someone wanted him dead.
The thought sickened him. When he'd been in Korea, the NKs and the Chinese Reds had wanted him dead, but that was war, that was to be expected. This was Florida. He'd been here just a little over a year. He'd made a number of new friends but couldn't imagine how he could have made an enemy.
Yet someone had tried to kill him.
Suddenly Tom felt exposed out here on Anya's lawn. He wanted walls around him. He rose unsteadily from the chair.
"I think I'll head home."
"You okay?" Jack said.
"Yeah. Sure. I'll just go inside and lie down. Excuse me, Anya."
"Go, Tom," she said. She was still in her recliner, the wet dog curled up on her lap. "You should rest."
"I'll come with you," Jack said.
"That's okay. I can find my own way."
"That's not the point," his son said, rising and gripping his arm. "Come on. I'll walk you back. I know how you feel."
No, you don't, Tom thought. And I hope you never do.
A good kid, Jack. No, not a kid. A man, and a pretty gutsy one at that, placing himself between a ferocious gator and the old folks with only a lightweight resin chair as a weapon. But Jack couldn't know what it was like to fear for his life, to have someone wanting him dead. That took a war. It had been Tom's great hope for his sons that neither would have to go to war as he did and know that kind of fear. And it had worked out. Both boys had been too young for Vietnam, and a volunteer army had been in place by the time the Gulf Wars rolled around.
"Wait," he said, turning. "We should call the cops or the wildlife control or something, shouldn't we?"
"Why?" Anya said.
"To let them know there's a monster gator in our pond."
"Not to worry," Anya said with a wave of her hand. "He's gone. And after such a reception as he got here today, I doubt he'll be back."
"Where'd he go?" Jack said.
"There's an underground tunnel that leads from the pond back into the Everglades."
"Really?" Tom said. "I didn't know that."
Jack stared at her. "How do you know, Anya?"
She shrugged. "I've been around here a long time. I shouldn't know things?"
He saw Jack stare at her again for a moment, then point a finger her way. "We need to talk."
She raised her wineglass. "I'll be here."
Tom wondered at that exchange. As soon as they were in the house he turned to Jack. "Why did you say that to Anya?"
"What?"
"'We need to talk.' About what? What does that mean?"
"I've got some questions for her."
"About what?"
"Things. Tell you about it later."
Why didn't Tom believe that? What was going on between those two? He was about to press him when Jack grabbed the pen and notepad from the counter by the phone.
"Just thought of something. Give me the names again of those three people who were killed."
"Why?" And then he knew. "Oh, no. You don't think—"
"I don't know what to think, Dad. When Carl told me about the others he said you didn't fit the pattern because the others were killed by birds and spiders and snakes. You were different because you were hurt in a car accident. But if what you remember is correct, you weren't going to be the victim of a hit-and-run accident, you were going to be a meal for that alligator. And that does fit the pattern."
Tom shook his head. "A few hours ago you were implicating Gateways in a scheme to get properties reverted. Now you think it's...what? How, just how, do you get birds and snakes to attack someone?"
Jack stared at him. "How do you get an alligator to attack someone? Twice. Because, Dad, that gator was coming for you. He was aimed at you like an arrow shot from a bow."
Tom wanted to deny it—tried to deny it—but couldn't. Jack was right. Those open jaws had been coming straight at him.
"But it's crazy," he said. Even crazier was how the gator had stopped at the edge of Anya's lawn. He was suddenly too tired to think about that now. Another question was far more pressing. "Why me?"
"That's what I intend to find out," Jack said.
Tom noticed a fierce look in his eyes. There was fire in Jack, a heat and a resolve he'd never expected in his appliance-repairman son.
And something else. He had a sense that Jack already knew the answer, or at least where to look. But how was that possible? He'd been here barely two days.
"Give me those three names," Jack said with the pencil poised over the pad.
## 14
His father had said good night and retreated to his bedroom. Jack heard the shower run, then the mutter of the TV through the closed door. Maybe Dad was watching it, maybe just zoned out in front of it.
Jack was grateful for the solitude. It gave him time to think. He grabbed a beer from the fridge and paced the front room, mulling what had happened, and what had almost happened. He'd been unarmed. Well, why not? Just visiting a neighbor lady for some conversation and a few sundown drinks. Who needs to be armed?
He'd know better next time. If there was going to be a next time. A few rounds into that gator's eyes or its open mouth...that would have stopped it. Or at least he was pretty sure it would have.
But a gun would have been superfluous because the gator hadn't been able to cross the line into Anya's yard.
Jack was getting used to the surreal, but still...
Could someone or—worse—something be controlling the wildlife around here? This whole situation had Otherness written all over it. He was convinced the Otherness had taken Kate from him, then it had made an attempt at Gia and Vicky and the unborn baby. Was it after his father now?
Giaand Vicky...
He pulled out his Tracfone and punched in Gia's number. She was delighted to hear that his father was out of his coma. Jack left out the other details, like attempted murder by alligator—twice—and told her he'd be hanging around a few days more, just to make sure he was okay.
Then Vickie got on the phone. She wanted him to bring her back a pet alligator. Jack shuddered at the thought but told her he'd see if he could catch one for her. A little one. Right.
Then Gia again. She was feeling good; she thought she'd felt the baby move but wasn't sure. All quiet on Sutton Square.
After I-love-yous and goodnights, he hung up and made another call to Manhattan. This time to Abe.
When Abe picked up, Jack said, "Hey. It's me."
Jack's Tracfone was untraceable, but he could never rule out that the BATF had taken an interest in Abe—linked him to an illegal weapon, perhaps—and were eavesdropping. So for his own sake and for Abe's, he never mentioned his name or anyone else's, even Abe's.
"Good evening, Me. How's the vacation going?"
"Could be better. You know how I thought I'd have an easy time at the tournament? It's not turning out that way. The competition is a lot stiffer than I dreamed possible."
"Is that so? As I recall, you weren't expecting any competition."
"Turned out I was wrong. Imagine that. But here's the thing. I need bigger and better equipment. Some new tennis clothes, for sure. Large size."
"How large? X? Double-X? Triple-X?"
"Big as you've got. Think elephant when you pick it out."
"Elephant?"
"Mastodon. Oh, and maybe some new racquets."
"Any particular model?"
"You pick them out. I need something with a nice sweet spot and lots more power than what I've got."
"So it's a power player you're up against?"
"Yeah. Back court all the way until today's round. That was when he started coming to the net. I don't think I've seen his best stuff yet, so I want to be prepared."
"I should say so. I'll send you a nice selection of racquets that you should be able to adjust to your needs. You want I should include extra strings in case you break some?"
"Definitely. The more the better. You know how I break strings."
"Do I. Anything else?"
"Some tennis balls."
"Balls? I'm not following you here. Surely they have tennis balls where you are?"
"Not like the brand you carry. Yours always seem fresher. And make sure they're yellow. A pale yellow."
"Pale yellow..."
Jack detected a note of uncertainty in Abe's voice. "Yeah, pale yellow. Like the color of my favorite fruit."
"A lemon?"
"No! Pineapple, my man. Pineapple. You know how I love pineapple."
"Oy, of course. How could I have forgotten? Yes, well, I'll check to see if I have any of that shade in stock. I should send you how many?"
"Let's see...I don't want to run short. How about a dozen?"
"A dozen. Sounds to me like you'll be playing a lot of tennis."
"I hope not. The longer you play, the greater the chance of injury. As you know, I like to rip right through the matches without much wear and tear, but you never know. Best to be prepared, don't you think?"
"Definitely. You want I should send them to that address you left with me?"
"That's the place. And make it quick, okay? Who knows what I'll be facing tomorrow."
"I'll pack it up right away and get it out tonight. I'll use my special carrier. If all goes well you should have them by tomorrow afternoon."
"Swell. Put it on my tab and we'll settle up when I get back. I owe you one."
"I'll add this to the 'owe' list."
"Do that. Oh, and by the way. Have I got a girl for you. She's an older woman, but she could be a soul mate."
"Now you're a matchmaker?"
"Just trying to enrich your life, my friend."
"Okay. I'll humor you. First question: Is she on the thin side or the heavy side?"
"She makes Olive Oyl look like a sumo wrestler."
"Sorry. Not interested. I need a woman with some meat on her, enough bulk so that we don't look like Mr. and Mrs. Sprat when we go out together. Someone who won't frown when I put extra cream cheese on my bagel. Someone, in fact, who'll ask me if I want seconds, or even thirds. An anorexic woman is the last thing I need."
"Okay. Just thought I'd ask."
"Find a Sophie Tucker for me and then we'll talk. But back to the tennis matches: Listen, be careful. Watch your footwork. Sounds like even a minor misstep could take you out of the game."
"Ain't that the truth. Talk to you later."
"Stay in touch. Let me know the scores."
"Will do."
Jack smiled as he cut the connection, but it faded as he turned toward his father's bedroom. He knocked softly on the door. When he received no answer, he pushed it open and peeked in. His father lay in bed, snoring softly, the remote in his hand, the Weather Channel playing on the TV.
Jack turned and headed for the front door. Time to visit Ms. Mundy. He had a few questions he wanted answered. Hell, he had lots of questions, and he knew she had answers to some of them.
## 15
Anya's front yard was deserted. The furniture was as he'd left it but she and Oyv were gone. So were the glasses, the wine, and the beer Jack and his father had brought over.
Jack knocked on the door. Anya, wearing another garish kimono with bright red sampans sailing across her flat chest, answered almost immediately.
"You're back. That must mean your father's okay."
"Shaken up but he's all right, I think. We need to talk."
"As you wish," she said, moving away from the door. "Come in."
Jack stepped into the greenhouse interior.
"I put your beer in the refrigerator so it wouldn't get warm," she said on her way to the kitchen. "Do you want one?"
"Thanks, no. I'm not here to drink."
She stopped at the kitchen counter where the wine bottle waited. An empty glass stood next to one half filled. Not dainty little claret glasses but big glass balloons that held eight to ten ounces if they held a drop. She topped off both and held out the fresh one to Jack.
"Here. Try this. It's Italian. Valpolicella."
"No, really. I—"
She locked eyes with him. "I don't like to talk to people who won't share a glass with me."
Jack shrugged and took the glass. He'd done worse things to get someone to talk. He took a sip.
"It's good." There. Was she happy? "Now, can I ask you a few questions?"
"If you wish." She seated herself on the sofa overhung with plants and vines. She lit a cigarette and began shuffling a deck of cards. She pointed him toward the recliner. "Sit. You want to ask me about a Russian woman with a malamute, don't you."
Jack felt his jaw drop. "I—I—"
"And an Indian woman with a German shepherd. The one who told you to stay away from that house in Astoria. The one you foolishly ignored."
"How did you know?" Jack said, finding his voice.
She blew smoke and shrugged as she began laying out the cards in a classic solitaire tableau. "Lucky guess."
"Since June I've been running into women who know too much—women with dogs. You're the third. Two isn't a trend. But three..."
"Not to worry. You have nothing to fear from them. Or me."
Jack took a deep breath and let it out. He'd expected denials or, at the very least, evasions. To have her come right out and confirm his suspicions...it knocked him off balance.
He took a gulp of his wine. Maybe this was why she'd insisted he take a glass.
"Who are you people?"
She finished laying out the cards and began to play, flipping them over with sharp little snaps. "No one in particular."
"I don't buy that. You know too much. Back in June, when I was sick, the Russian lady came to my room"—he saw her in his mind, salt-and-pepper hair, gray jogging suit, big white malamute—"and told me things about a war I'd been drafted into. 'Is war and you are warrior,' she said. I don't know if she mentioned it directly or not, but I'm pretty sure she was going on about something called the Otherness and—"
Anya stopped her card play and looked up at him. "You'd already heard of the Otherness by then."
"Yeah."
Although he wished he hadn't. The first mention had been earlier in the year, in the spring at a—surprise—conspiracy convention. Since then his life hadn't seemed quite his own.
According to what he'd been told, two vast, unimaginably complex cosmic forces have been at war forever. The prize in the war is all existence—all the dimensions, all the realities, all the parallel dimensions up for grabs. Earth and humanity's corner of reality is a minor piece on the game board, of no special importance. But if one is going to declare itself winner, one has to take all the pieces. Even the inconsequential ones.
One side—a force, a state of being, whatever—is inimical to humankind. It has no name but through the ages came to be called the Otherness by people aware of its existence. If the Otherness takes over, it will transform Earth's reality into a place toxic to all known life. Fortunately, Earth and its attendant reality are currently in the portfolio of the other side, the force known only as the Ally. From what Jack had learned, "Ally" was a misnomer. This force was not a friend, merely an enemy of humanity's enemy. The most Earth could expect from it was benign neglect.
"At the time I thought the Russian lady was some sort of fever dream, but then she showed up again and told me..."
"That there would be no more coincidences in your life."
Jack nodded. The words still chilled him. The implications were devastating.
"Was she right?"
Anya went back to her game, flipping and arranging the cards in the tableau, moving some aces and deuces up to the foundation.
"I'm afraid so, hon."
"Then it means that my life is being manipulated. Why?"
"Because you are involved."
"Not by choice."
"Choice means nothing in these matters."
"Well, if someone or something thinks I'm its standard bearer, it had better think again."
"You are not the standard bearer. Not yet."
If true, that was a relief. A small one.
"Then who is?"
Anya was dealing to herself from the stock now, and Jack couldn't help but notice that the cards were falling her way, more and more finding places in the tableau or the foundation.
"One who preceded you," she said. "He preceded the twins as well. You remember the twins, don't you."
Jack had a flash of two men in identical black suits and dark glasses, with identical pale, expressionless faces.
"How could I forget?"
"They were meant to replace their predecessor. But when you dispatched them—"
"They didn't leave me much choice. It was them or me. And I tried to help them at the end, but they refused."
"They did what they had to do, but their passing left a void. One that you were tapped to fill."
"But you said there's someone else."
Anya nodded as she laid the final card from her stock on the solitaire tableau. All the cards were face up. She'd won. Without bothering to shift all the tableau cards to the foundation, she gathered them up and began shuffling.
"There is. A mensch of mensches, that one. But he's old now, and may die before he's needed again."
"'Again'?"
"He was the Ally's champion for a long time."
"How long?"
"Very long. So long you wouldn't believe. But now his days are numbered. After ages in the Ally's service—too long, I think, but who listens to an old woman—he was freed. But it seems his liberation was premature. Even though he has aged, he may be needed again. But if he doesn't live till that day..." Her eyes met Jack's.
"Then it'll be me?"
"You."
Against all reason, Jack believed her. With an effort, he shelved his dismay. Maybe that day would never come. Or maybe he'd have died of old age when it did.
But he hadn't come here about himself. He'd come about his father.
"Is the Otherness involved in what's been happening to my father?"
She nodded as she finished shuffling and began to lay out another solitaire tableau.
"The Ally is involved here as well, though tenuously."
"But I can assume, at least from what I've seen, that you and your ladies are on the Ally's side, right?"
She shook her head. "No. I oppose the Otherness, but I've no connection to the Ally."
"Then whose side are you on?"
"Yours."
"But I'm stuck with the Ally, so that means—"
Anya grimaced with irritation and stopped her card play.
"I didn't say the Ally's side, did I? No. I said, yours. That means you, separate and distinct from the Ally."
"But why?"
"Because the Ally can be as ruthless as the Otherness. It opposes the Otherness for its own reasons, none of which involves our health and happiness. It will use you and anyone else it can to fend off the Otherness, and not care a whit what happens to you. Humanity's well-being is not on its agenda. It is, however, on mine."
"Why? What's your stake in this?"
She began rearranging the cards in the tableau.
"My stake is your stake. Everyone here on this planet is in the same boat—Earth is a boat, when you think of it—and we all deserve to be free of both these meddling powers. This planet, in this subdivision of reality, is inhabited by sentient beings, which makes it all the more valuable in the struggle. But it's more than mere property that can be won or lost or traded at will. If it must belong to one of them, then I'd far prefer the Ally over the Otherness. But why belong to either? Why not be shut of both of them?"
"Sounds good to me," Jack said. He leaned back, trying to get a handle on what she was saying, and what it meant. "But what I'm getting here...what you're telling me...is that there's a third force involved in all this."
"I suppose you could put it that way."
"And you...you and those other women...you're part of that?"
"So it would appear."
"But how can you hope to compete with the other two players?"
"Because I must."
"But who are you? What are you? Where do you come from?"
"We come from everywhere. We're all around you. You simply never see us."
Jack shook his head to clear it. He didn't want to deal with this now. He'd had trouble enough buying into the cosmic tug-of-war scenario. But now Anya was telling him that a third party had entered the fray—or maybe had always been in the fray but no one had told him. Whatever the case, he'd get to that later. Right now he had to stay focused on his father.
"Why my father? Why would—?"
And then he had a chilling thought. What had she said to him that first day in the hospital room?
Trust me, hon, there's more to your father than you ever dreamed.
"Oh, no! You're not telling me that this 'predecessor champion' you've been telling me about is my father!"
"Tom?" Anya laughed. "Oy! Such a thought! You think you're living in a fairy tale? How can you even consider such a thing!"
"That's not a exactly a 'no.'"
"All right then. You want a 'no'? Here's a 'no.' Your father has no direct connection to the Ally or the Otherness. Never did, never will."
She laughed again and continued her card play.
Jack too had to smile. All right, yeah, it was a ridiculous thought. The pen might be mightier than the sword, but an accountant as defender of humanity against the Otherness? Crazy.
Yet...for a moment there...
"Wait. You said no direct connection. Does he have an indirect connection?"
"Of course. Isn't it obvious?"
"Because he's my father?"
Anya nodded. "A blood relative."
Jack closed his eyes. This was what he'd suspected, what he'd feared.
"That alligator, then...it was sent by the Otherness."
"Sent? No, that was someone else's idea. I can tell you that the creature was created by the Otherness, but whether intentionally or accidentally is hard to say."
"Why? You seem to know everything else. Why don't you know that?"
"I don't know everything, kiddo. If I did, maybe the two of us could send the Otherness and the Ally packing."
"Why do I get this feeling you're holding back? You don't know everything? Fine. Nobody does. But why don't you just come out and tell everything you do know?"
"Because sometimes it's best that you learn things on your own. But I can tell you about the connection between the Otherness and that alligator."
Jack leaned back and took another slug of wine. "I'm all ears."
"It was born near a nexus point."
"And that is...?"
"A place. A very special place. In various locales around the globe there are spots where the veil between our world and the Otherness is thin. Occasionally the veil attenuates to the point where a little of the Otherness can enter our sphere. But only briefly. Rarely do beings from the other side pass through. But influence...ah, that's another matter."
"Let me guess a location," Jack said. "Washington, DC, maybe? Say, near the Capitol Hill or the White House?"
Anya smiled as she gathered up her cards. She'd won again.
"I'm afraid those gonifs have no such excuses for their behavior, hon. But one is near here, and another near where you live."
"Where?" Somehow Jack wasn't surprised.
"In the New Jersey Pine Barrens. At a place called Razorback Hill."
Jack had gone into the Barrens last spring, and almost hadn't come out. "It must be pretty well hidden. I mean, don't you think someone would have stumbled across it by now?"
"There are places in the Pine Barrens that no human eyes have seen. But even so, the nexus points manifest themselves directly only twice a year—at the equinoxes. But their indirect effects can be viewed every day."
"Like what?"
"Mutations. Something leaks through from the other side around the time of the equinox; whatever it is changes the cells of the living things around it—plants, animals, trees...and people."
"You'd think someone would have noticed that by now."
Anya shook her head. "The nexus points are located in unpopulated areas."
"How convenient."
"Not so. When you consider that these leaks have been occurring for ages, and that most people experience a sense of uneasiness when they near a nexus point, it makes sense. Nexus points don't occur in places that people avoid. Just the opposite: People—most people, that is—instinctively avoid nexus points."
Jack was thinking, nexus point...mutations...a humongous horned alligator...
"There's a nexus point out there in the swamp, isn't there."
"I told you, it's not a swamp, it's—"
"A river of grass. Right. Okay. But am I right that there's a nexus point nearby in the Everglades?"
Anya nodded. "In a lagoon within one of the hardwood hummocks."
"How do you know all this?"
Anya shrugged. "Like I said before, hon, I've been around here longer than you."
"How long?"
"Long enough."
"All right, then." He sensed a certain timelessness about Anya, and was convinced she was more than she pretended to be. He took a chance and asked her flat out: "How long have you and these other women been around?"
"I should tell you my age?"
She lit another cigarette and gathered up her cards. She'd won another game. That made three in a row. More than luck there. Had to be. She was either cheating or...
Let it go.
"All right, don't tell me. Maybe if I see that Indian woman again"—he remembered her orange sari and long braid, and her German shepherd—"maybe I'll ask her. She looked young."
Anya laughed. "Never ask a woman her age!"
Thinking of the other women with dogs reminded Jack of something one of them had said.
"The Russian woman mentioned someone called the Adversary. Who's that? She said I'd met him."
Anya leaned back and stared at him.
"You have. Remember my telling you about the aging one who once spearheaded the Ally's cause? Well, the Otherness has its own champion. He's very dangerous. He's ancient. He's been killed more than once but each time he's been reborn."
"And I've met him? I—"
And then Jack knew. The strange, strange man who'd first explained the Otherness to him, the man he suspected of being ultimately responsible for Kate's death...
"Roma," he whispered. "Sal Roma. At least that was what he told me his name was. I later learned that was a lie."
"Always you must expect lies where he is concerned—unless the truth will hurt you. He feeds on pain."
"Yeah. That was what your Russian friend told me: human misery, discord, and chaos. But who is he, really?"
"More like what is he. He used to be a man just like you, but now he is more. He is destined to become something else, but he hasn't reached that state yet. He can do things that humans can only dream of, but he is still in the process of becoming. He's known as 'the Adversary' to those who oppose the Otherness, and 'the One' to those aligned with it."
"Why would people work for the Otherness when they know it means the end of everything?"
Anya shrugged. "Who can explain people? Some are so filled with hate that they want to see everything destroyed, some believe their efforts toward bringing the Otherness apocalypse will be rewarded afterward, some believe packages of lies they've been fed, and some are simply mad. The Adversary orchestrates their movements from afar."
"But what's his name?"
"He uses many. He has many identities, many different looks, but he never uses his True Name."
"Do you know it?"
Anya nodded. "But I will not tell you."
"Why the hell not?"
"Because he would hear you. And you do not want to attract his attention."
"Says who?" Jack said, feeling the heat of the rage he'd been carrying around for months now. "I've got a score to settle with him and—"
"No!" Anya was leaning forward in her seat, eyes ablaze. "You stay away from him! Whatever you do, you must not antagonize him. He will snuff you out like a match if it suits him."
"We'll see about that. Just tell me his name and let me worry about the rest."
Anya shook her head. "Speaking his name would lead him here—and he's looking for me."
"You? Why?"
"To kill me."
Her words shocked Jack. And the matter-of-fact way she said it, as if she'd been dealing with this threat for so long she'd grown used to it, made it all the more believable.
But could it be true? If so, he'd lay off pressing her for Roma's real name.
"Because you oppose the Otherness?"
"More than that. I stand in its way—in his way."
Jack wanted to say, You're a little old lady...how can you stand in anyone's way? But he hadn't forgotten how that alligator had been unable to enter her yard. Perhaps she and the others were keeping out the Otherness just as she'd kept out the gator, but on a far greater scale.
This little old lady was a lot more than she seemed. She had power...but from where?
Jack wasn't going to waste his time asking. She'd already made it damn clear there were things about her and her friends she didn't want known.
"You stand in his way to...what?"
"To opening the gates to the Otherness. The Adversary will remain in a state of becoming until he succeeds. If he does, he will be transformed and life, reality, existence as we know it will end. He thought he'd found a shortcut earlier this year. You were there and—"
"How do you know this stuff? Or was one of your ladies watching?"
"You might say that."
Jack remembered gazing down into a bottomless hole...into an abyss glowing with strange lights...a steadily enlarging hole that he feared might devour him and the rest of the world.
Anya said, "The Adversary failed then because he acted prematurely. That shows me he's anxious to finish his becoming. Since then he and those he has manipulated have doubled and redoubled their efforts to open those gates. But to achieve final success he must kill me or hurt me so severely that I can no longer oppose them."
Apprehension tightened his shoulders. If the Adversary or the One or Roma or whatever the hell he was called was as dangerous as Anya said, she could be in big trouble. Jack hadn't known her long, but he'd taken a real liking to this old broad.
"But if he doesn't know where you are, he can't hurt you, right?"
She shook her head. "No. He can hurt me. He hurts me all the time."
"But how—?"
Anya stiffened and grimaced with pain as she sucked air through her teeth with a hiss. She arched her back and reached around to touch her right shoulder blade. Oyv jumped up and started barking.
"See?" she gasped. "Even now he does it! He's hurting me again!"
Jack was up and around the chair, looking at her back.
"What? What's happening?"
"Oh!" She was taking quick, shallow, panting breaths. "He stabs me! It hurts!"
"What can I do?"
"Nothing. It will pass."
Jack thought he saw a small spot of red—blood red—appear on the back of her kimono, but couldn't be sure because it was within the hull of one of the bright red sampans.
"Are you bleeding?"
She leaned back against the chair, hiding her back from view.
"I'll be all right."
Her color was better and her breathing, though not normal yet, was easing in the right direction.
"Should I get a doctor?"
She shook her head. "No doctor can help with this. I'll be fine. This isn't the first time he's hurt me, and it won't be the last. He's moving closer and closer to his goal. A strange season is upon us, and it will grow stranger."
"Damn it, Anya, tell me his name. I'll put an end to this."
She shook her head. "No, Jack. He's immune to your methods. He's more than you can handle."
"Then how do we stop him?"
Anya looked up at him and Jack saw fear in her eyes. "I don't know. We can only hope that he makes a fatal mistake—he's not perfect you know—or that the Ally steps in on our side. Otherwise, I don't know if he can be stopped."
## 16
After Anya's pain had subsided, she shooed Jack out of the house. He felt he should stay but he could see that she wanted to be alone.
He stood in her front yard among the ornaments, staring at the rising moon, and wondering at how his life had changed since a year ago last summer when he'd accepted the seemingly simple, straightforward job of finding a stolen necklace. Now it seemed that every time he turned around, a new revelation leaped at him, tearing a jagged rent in the fabric of the snug, familiar worldview he'd been wrapped in for the first thirty-five years of his life.
A year ago he'd have written Anya off as a loon. But no more.
He popped into his father's house and peeked again into his bedroom. The old guy was still sleeping peacefully with the TV going. Jack found the screwdriver and flashlight he'd used last night, then stepped outside and headed for the clinic.
Although he'd broken in once before, he didn't take for granted that it would be as easy the second time. He was just as careful about approaching the building, keeping to the bushes and watching for the security patrols. About halfway there he realized he'd forgotten the mosquito repellent. They'd declared his arms and neck an all-night deli and were ordering takeout.
Slapping and scratching, he picked up his pace and made it to the clinic faster than last night. He popped the window latch again and slid inside. After reclosing the window, he killed a couple of mosquitoes that were still drilling into his skin, then got to work.
Straight to the record room where he began flipping through the charts. He had the list of names his father had given him and though it was a long shot that they'd all had recent physicals, he had to check.
He started at the top of the alphabet and worked his way down, pulling the charts as he came across them: Adele Borger...Joseph Leo...Edward Neusner....
All here.
No second guessing the ethics of invading privacy this time. These folks weren't his father, and they were dead.
Inside the charts, Jack knew where to look. He went to the bottom of the final page of the complete physical. Each one read the same: Final assess: excellent health.
A prickling sensation ran along the back of his neck. Seemed like being single at Gateways South and passing your free physical with flying colors was not a good thing. In their cases, it appeared to be a death sentence.
The pattern was obvious: The healthiest single members of Gateways South were dying by mishap. An early demise meant that, instead of having to wait many years for these healthy folk to go, the management was able to resell their homes immediately.
Jack had a pretty good idea as to the why and the who, and a wild idea as to the how.
He wondered if the doc was in on it. Probably not. He seemed like too much of a straight shooter.
Besides, you didn't need the doc to get a look at the files. Jack's presence here proved that. But there was an even easier way. If you were someone with an official position at Gateways South, and if you had a key to the clinic, you could stroll in here at night, check out the names of those who'd had a complete physical lately, and peruse their files to your heart's content.
Jack decided that he and Gateways South director Ramsey Weldon were going to have a little heart-to-heart chat tomorrow.
## FRIDAY
## 1
Jack jogged along the asphalt walking/bicycle path that wound through the pines lining the eastern limits of Gateways South. A thin morning mist wound between the trunks; brown needles, shedding early due to the drought, littered the path. The scent of pine lay thick in the air.
He'd awakened to silence for a change. Carl must have been trimming someone else's hedges this morning. His father was just starting to stir, so Jack had come out for a run. He'd been too sedentary the past few days. Needed to get the blood flowing. He'd thought about checking on Anya but it was too early. He'd swing by on the way back.
He chugged along in a Boneless T-shirt and gym shorts, building a sweat; he wore his leather belt under the loose shirt to hold the small-of-the-back holster for his Glock 19; the way it bounced against the base of his spine as he ran was annoying, but no way he was going unarmed around this place.
An eight-foot chain-link fence ran along the Gateways border to his right. The links of the par-3 golf course lay to his left. He noticed a lone, vaguely familiar figure hunched over a putter on a rise ahead. As he neared he recognized him: Carl.
Jack veered to his left and found Carl on a putting green, working with a club that protruded from his right sleeve. Jack had thought he was a righty, but he was using a lefty stance.
He waited until Carl had hit the ball—he just missed, rimming the cup—before speaking.
"When did you join the community?"
Carl jumped and spun. "Oh, it's you! You scared me again! You gotta learn to make more noise when you come up on people."
"Sorry," Jack said. "I'll work on it. Say, did your video camera catch any signs of Ms. Mundy watering her lawn?"
"Zilch again." He grinned. "And I hope it don't. Wouldn't mind keepin this up the rest of the year, long as old Doc Dengrove keeps payin me."
Jack glanced down at the balls Carl had arranged on the grass before him, sitting in a line, waiting for the putter. "Is a golfing membership one of the perks of your job?"
He shook his head. "Only on weekdays, and only on my day off, and only if I stay out of everbody's way. I ain't much with the drivers—I mean, my scores for eighteen holes are pretty pan-o-ramic—but I like to putt. I ain't a bad miniature golf player."
"No kidding." This was fascinating, simply fascinating. Jack waved and turned away. "Got to keep moving. Good luck. Sink those putts. Make those birdies."
But he never got restarted. The sight of a beat-up red pickup cruising the dirt road on the far side of the fence stopped him cold. It slowed as a pair of mismatched eyes peered at him from under the brim of a dirty John Deere cap, then picked up speed again.
A thought struck Jack. He turned back to Carl, intending to ask him if he knew them, but the half-sick look on his face as he watched the pickup bounce away into the trees said it all.
"You know those guys, don't you."
Carl swallowed. His left eye was already looking away; the right followed. "Why you say something like that?"
"Because I think you do. Who are they?"
"Nobody to mess with. You don't want to know em."
"Yeah, I do." Especially after what his father had told him last night about the accident. Jack gave him a hard stare. "Who are they, Carl?"
Carl looked like he was going to try to float some bullshit, then his shoulders sagged and he shook his head.
"They live out in the Glades. On a lagoon in one of the hardwood hummocks."
"I thought no one was supposed to live out there except maybe some local Indians."
"Well, I think you know that what's upposed to be and what is ain't necessarily the same thing."
Yeah. Jack knew that.
"You know where this lagoon is?"
Carl nodded. "I guess so."
"How do I get there?"
"You don't, not unless you know the way."
"Can you show me on a map?"
Another shake of his head. "It ain't marked on no maps. It's pretty well hid."
"Then how come you know where it is?"
Carl looked away. "I was born there."
This didn't surprise Jack. He'd seen what the folks connected to the red pickup looked like, and figured there had to be something wicked strange about Carl's right arm. Add that to what Anya had said about the mutating effects of the Otherness leak at the nexus point in the Glades, and the connection looked obvious.
He remembered other misshapen people he'd met earlier in the year...Melanie Ehler and Frayne Canfield...both had attributed their deformities to "a burst of Otherness" during their gestations. Carl's story was most likely the same.
"All right then," Jack said, "take me there."
Carl backed away a step, holding up his hand. "Nuh-uh. No way. I left there years ago and I ain't goin back."
"Well, if it's not on a map, and you can't tell me how to get there, and you won't take me there, how am I supposed to find it?"
"You ain't. That's the whole point."
As if to say he was through talking, Carl bent over his putter and lined up a shot. He tapped the ball and it went wide.
"I've good reason to believe they caused my father's accident and were setting him up to be eaten by an alligator when the police interrupted them."
Carl straightened and looked at him. "Alligator? That woulda meant your daddy'd go the same way as the others, killed by a swamp critter."
"Well, this wasn't no ordinary swamp critter." Jeez, Jack thought. A couple of conversations with this guy and I'm starting to talk like him. "This gator was huge, with what looked like horns sticking out of its head."
Carl visibly shuddered. "Devil. That could only be Devil."
"Who's Devil?"
"Big freaky bull gator that hangs around the lagoon. But how on earth did they get him out of the swamp?"
"Couldn't say. But it seems Devil gets around. He visited Gateways last night."
"No way!"
"Way."
Jack gave him a Reader's Digest version of the attack, leaving out Oyv's amazing feat and the gator's inability to cross into Anya's yard. He remembered what his father had said about Carl being the community gossip.
"I want to get a look at this lagoon, Carl. I've already met the people, now I want to see where they live."
"You met them?"
"In town yesterday. Met that woman, too. The one with the white hair."
"Semelee."
"Right. What do you know about her? Is she as spooky as she looks?"
"Can't rightly say. I left the clan about—"
"Whoa! Are we talking Kluxers here?"
"Naw. That's just what we call ourselfs. We're all kinda related in a way."
"Yeah? How?"
Carl's good eye shifted away again. "Not by blood or anything like that. More like we was all in the same situation. Anyway, it was just us guys, maybe twenty of us, when she showed up a couple years ago. I'd been kinda plannin on leavin anyway, but when she showed up I took it as a sign and skeedaddled outta there."
"A sign of what?"
"That things in the clan was gonna head south real soon. I mean, you got eighteen-twenty guys and one woman, that's trouble."
"They seemed pretty tight when I saw them in town yesterday."
"Yeah, well, maybe. I seen em from a distance a couple times. We always done some panhandlin, but now they's become like professionals. I stay away from em cause we ain't exactly on good terms."
"Why not?"
"They was kinda pissed I left. Luke—he was sorta kinda like the leader—he called me a traitor and all sortsa stuff like that. But that don't matter to me. I'm glad I got out. I didn't wanta live like them no more. Y'know, like gypsies. They live on the boats or in what's left of a bunch of old Indian huts on the shore. No runnin water, no lectricity, no TV." He shook his head. "Man, I sure do love TV. Anyways, I wanted my own place where I didn't have to sleep next to nobody cept myself."
"A room of one's own," Jack murmured. He knew the feeling.
Carl grinned. "Hell, I got more than just a room, I got me a whole trailer."
"But do you have any money in the bank?" Jack said as an idea hit him.
"Naw. Pretty much everthing gets spent just for livin."
"Okay, then. What say I pay you a thousand bucks to take me to this lagoon?"
"A thousand?" Carl laughed. "You're shittin me, right?"
"Nope. Five hundred when we leave, and another five when we get back. That sound fair to you?"
Carl licked his lips. "Yeah, but..."
"But what?"
"But they's gonna be awful mad if they find I brung an outsider to the lagoon."
"Don't worry about that." Jack flipped up the back of his shirt to show Carl the Glock. "I'll get you back home. I promise. And anyway, if we go in the afternoon, won't they all be in town, begging?"
"Come to think of it, yeah. Specially this bein Friday."
"What's so special about Friday?"
He shrugged. "Lotsa people round here get paid on Thursdays, and on Fridays they're happy the work week's over, so they're looser with their change. Saturday's pretty much the same. But Sunday's usually a bust."
"Spent too much on Saturday night, right?"
"Yeah. Or they just come from church and did some givin there. Monday's even worse." He scratched his jaw. "So yeah. We should have the lagoon pretty much to ourselfs this afternoon."
"Then that's when we'll go. A quick trip for a quick look-see. In and out. Easiest thousand you ever made."
Carl took a breath. "Okay. But since my car ain't workin, you gotta drive me down to the waterside." He began picking up his golf balls. "Guess I better get movin. Gotta get home, gotta find us a boat."
"How'd you get here without a car?"
"Bike. How else?"
More power to you, pal, Jack thought. Maybe the thousand would let Carl repair his junker Honda.
He got directions to Carl's trailer park—it was the one Jack had seen between the auto body place and the limestone quarry—and continued his jog.
## 2
Semelee stood with Luke a couple dozen feet from Devil's gator hole and watched. The big gator lay half sunk in the water at the shady end, his eyes closed. The water around his left flank wound was tinged red. At first she thought he was dead, but then she saw his sides pull in a little as he took a breath.
"He's still bleeding," Luke said.
"I know," she said through her clenched teeth. "I got eyes."
She felt so on edge this morning she wanted to take a bite out of somebody.
Devil was the biggest gator anybody'd ever seen, so it made sense he'd have the biggest gator hole in the Glades. Like all gators, as the winter dry season began, he'd scrape out all the vegetation from this low spot in the limestone floor and create a big wallowing hole. Fish would work their way into it, turtles and frogs too, and even some birds would come around to see if they could snag a quick meal. Sometimes those birds and turtles became gator snacks.
In the wet summers gators left their holes and spread out through the Glades, but not this year. The dry spell made gator holes more important than ever.
The edges of Devil's hole were piled high with muck he'd scraped out. This provided rootin soil for things like cattails, swamp lilies, ferns and arrowleaf. Yellow-flowered spatterdock lilies floated on the surface of the blood-tinged water.
Devil lifted his head and let out a hoarse, rumbling bellow, then let it flop back down into the water as if it was too heavy to hold up.
"He's hurtin, Luke. Hurtin bad."
Because of me, she thought.
Guilt scalded her. She'd considered Devil indestructible, invincible, almost supernatural. But he wasn't. He was just a big, misshapen gator who would have been happy spendin his days doin what gators do: lolling in his hole, eatin this and that, waitin for the rains.
But no. Semelee couldn't let him be. She had to roust him out of his comfy hole and lead him out of the Glades into the outer world where he didn't belong. The result was he got hurt. Hurt bad.
"He can't die," she said. "He just can't."
She had this terrible feeling that if Devil died, part of the spirit of the Glades would die with him. And it would be all her fault.
"It was that guy," Luke said. "That city guy you been takin a shine to. He done this."
"No, he didn't. I already told you that. He didn't have nothin to do with hurtin Devil. It was the old lady. She's the one. She's some sorta witch. So's her dog."
In a way Semelee was secretly glad that the old witch's spell, or whatever it was, had kept Devil out of her yard. Because she'd seen her man, the special one, place himself between Devil and his father. She'd've had to go through him to get to the old man, and that would've meant hurtin him, maybe even killin him, somethin she definitely didn't want to do. But it had showed her that he was made of good stuff. That was important.
"I say we do all three of them—old lady, father, and son—and have done with it."
"No. I told you: The son ain't to be touched."
Luke grumbled. "All right. We'll have another go at the old guy, but the lady...what're you gonna do about her?"
"Don't know yet. We can't do her unless we can get to her. I'll think of somethin. But it'll have to wait till the lights is done. I ain't lettin nothin get between me and the lights."
"Awright. But what do we do till the lights come? We goin panhandlin as usual?"
"Not durin the lights. We'll just hang out. Besides, we don't need to go beggin cause we'll be gettin a hunk of cash from those dredgin guys when they finish at noon."
"What if they try to stiff us?"
"They won't. They ain't gettin out of the lagoon less'n they pay up."
But Semelee didn't want to think about dredgin or money or nothin cept the lights. Anticipation thrummed through her like she was a plucked guitar string. The lights'd start tonight and run for three days. But this year would be like no other. This time they wouldn't be underwater, which meant they'd be bigger and brighter and better than ever before.
Starting tonight, everything in her life would change. She sensed it, she knew it.
## 3
Tom had been watching the Weather Channel's reports on Hurricane Elvis. It continued to move south off Florida's west coast; although its winds had increased to 90 miles an hour, it was still a Category I. And no threat to Florida at this point.
He was just finishing his cup of coffee when Jack came through the door, dripping with sweat.
"I was wondering where you were." He'd been a little anxious after awakening to finding the house empty and Jack's car still parked outside. Obviously he'd been out jogging. "I don't suppose you'd care for a cup of hot coffee right now."
"After my shower I'd love one. Never turn down coffee."
As Jack ducked into the bathroom, Tom rinsed out the French press and began to make another serving. He noticed his hand shaking a little as he spooned the ground coffee. He touched the fresh bandage on his head. The stitches were still a little tender under there. He'd been shocked at the sight of his bruised, black-eyed face in the mirror this morning. He felt so good he'd almost forgotten about the accident.
Now he couldn't get it out of his head. Someone wanted him dead. Why?
Last week his life had been safe and sane, prosaic, maybe even a little dull. Now...
What was happening? He didn't live the sort of life where he got on people's wrong side. Was it a mistake? Had he been mistaken for somebody else? Who on earth would want to kill him?
He pondered those imponderables until Jack returned, in fresh shorts and T-shirt, his wet hair combed straight back.
"Hey, good coffee," he said after sipping the cup Tom had made for him.
"Colombian. I was thinking of scrambling some eggs. Want some?"
"Sure. And some hash browns and toast, and maybe some grits with extra butter. Oh, and while you're at it, a side of biscuits and gravy."
Tom gave him a dour look.
Jack shrugged and smiled. "Hey, we're in the south so I figured one of their traditional, artery-clogging breakfasts would be in order."
"What do you know about southern cooking?"
"There's a place called Down Home a few blocks from where I live. In New York you can eat any style you want."
"Right now," Tom said, "I don't feel like eating at all. Hard to be hungry when there's someone out to get you. If I knew who or why, maybe it wouldn't be so bad. I'd still be scared, but..."
"Maybe I can help there," Jack said softly.
"You? How?"
The phone rang. It was the front gate, wanting to know if he was expecting any packages.
"Not that I know of. Wait." He turned to Jack. "Are you expecting a delivery of some sort?"
"Yeah!" He grinned. "It's here already? Great. Good old Abe."
Tom told the gate to send the truck through, then turned back to Jack.
"You were saying something...?"
Jack cleared his throat. "I checked out the medical records on Borger, Leo, and Neusner last night and—"
"How on earth did you do that?"
"I got in through one of the clinic's windows."
"What?"
"No biggee. I popped the lock on one and crawled through. Don't worry. You'd have to look pretty close to the underside of the sash to even suspect someone was there."
Tom couldn't believe this. His own son breaking and entering—and the clinic of all places.
"Dear God, why?"
"Stay calm. I wanted to see if any of them had had physicals recently—the answer turned out to be yes to all three, by the way—and to see how they did."
"What if it had an alarm, or what if you were caught on camera? You could go to jail for something like that!"
"Only if I got caught, which I didn't. No alarm, no surveillance cameras. I checked that out first. But I found what I was looking for: Each one of them passed their physical with flying colors."
"A lot of good it did them. They're all dead."
"I think they died because they passed with flying colors."
"Oh, you're not going back to that Gateways conspiracy thing you were talking about yesterday, are you?"
"Follow the money, Dad. Whenever you wonder if something funny might be going on, follow the money. And the money leads to Gateways."
Had he gone completely paranoid?
"Jack—"
"Think about it: It's only younger, healthy widows and widowers being attacked—the ones who stand the best chance for holding on to their houses the longest. Coincidence?"
"You're talking about a billion-dollar corporation, Jack. This is penny-ante stuff. Imagine the impact of four extra resales in a year on a nine-digit bottom line. Meaningless!"
"It may be meaningless globally, but what about locally? What if someone in Gateways South needs to boost his bottom line and this is a way—just one of a number of ways, say—to do it?"
Tom didn't know what to say. Breaking into offices, digging up "clues"...he had to admire Jack's initiative, and was touched that he'd go to all that trouble for him, but...Jack seemed to think he was Philip Marlowe or Sam Spade. And he wasn't. He was an appliance repairman, and he was going to get in over his head and in deep trouble if he kept this up.
"I suppose you can make a circumstantial case for it, but it just doesn't add up. You're implying that Ramsey Weldon or someone at his level of management went out and hired those men to smash up my car and then have me eaten by an alligator. It's preposterous."
Jack scratched his head. "I know it seems that way, but so far he and Gateways South are the only ones I can see benefiting from your passing. I'll have to go with Weldon for the time being."
Tom felt a surge of acid in his stomach. "'Go with'? What does that mean?"
"Oh, I don't know," Jack said with a smile that did nothing to relieve Tom's anxiety. "Have a little tête-à-tête or something like that."
"Don't. Please, don't. You're just going to get yourself in trouble."
"Don't worry. I'll be discreet. The very soul of discretion."
Somehow Tom doubted that. But before he could say anything else, the doorbell rang.
"I'll get it," Jack said.
A delivery man stood at the door holding a cardboard carton.
"I've got four packages for 'Jack.'"
"That's me." Jack took the box and placed it on the floor. "I'll help you with the others."
As Jack followed the man outside to his truck, Tom stepped over and looked at the return address: Bammo Toy Co.
Toys?
He noticed too that the shipping label was addressed to "Jack" at this address. No last name, just "Jack." Odd.
When all four cartons were inside the door, Jack tipped the driver, then lifted one of the boxes.
"I'm going to put these in the spare bedroom, if you don't mind."
"Sure. Go ahead."
As Jack headed for the bedroom, Tom lifted one of the packages to help. He hefted it...heavier than he'd expected.
Jack had already relocated the first box and almost ran into Tom in the bedroom doorway. He took the package from him—rather quickly, Tom thought.
"Hey, no, Dad. Thanks, but that's okay. I don't want you hurting your back."
"Don't be silly. They're not that heavy."
He returned to the living room and picked up another package. Jack was right behind him, hovering like a mother hen.
"Dad, really—"
Tom ignored him and carried the carton into the bedroom.
When all four were piled against the wall, he said, "It says they're from a toy company. What kind of toys are we talking about? Toy robots? I mean, they're heavy enough."
"Just toys." Jack seemed tense.
"Do you mind showing me one?"
A heartbeat's hesitation, then Jack said, "I guess not. But we'll need a knife to cut the tape."
"I'll get one."
Tom found an old serrated steak knife in the kitchen drawer, but by the time he'd returned, Jack had the smallest box already open.
He held up a folder with a curved blade. "I forgot I had one in my pocket."
Inside, Tom saw an odd-looking stuffed toy, some unidentifiable little animal a little bigger than a football. "What's that?"
"It's a Pokemon. This one's Pikachu. They were all the rage with kids a few years ago."
"But why are you buying them?"
"I'll probably wind up giving them to a local kids' charity."
Tom shook his head. What an odd man his son had turned out to be.
## 4
Jack found Carl waiting on the street outside his trailer park in knee-high green rubber boots; a short wooden paddle protruded from his right sleeve.
"Where's the boat?" Jack said as Carl slid into the passenger seat.
"It's waitin. A guy I know's lettin me borrow it." He stuck out his hand. "My money?"
Jack handed him an envelope. "As promised."
He'd come down with about a thousand in cash. His deal with Carl was going to leave him short, so he'd stopped at an ATM for an advance on the John L. Tyleski Visa card. Another envelope with the balance of the fee was tucked into a back pocket.
Carl checked the contents. Didn't take long to count five bills. The reverent way he touched them made Jack wonder if Carl had ever seen that much money at once.
"I hope I ain't makin a big mistake," he said, still staring into the envelope.
"Don't worry. A few hours from now you'll be sitting in front of your TV with another one of those in your pocket."
He sighed and folded the envelope. "Okay. Let's go."
As they pulled away, Jack noticed high chain-link fencing disappearing into the foliage; a rusted length of chain with a beat-up NO TRESPASSING sign spanned a gap that looked like an entrance.
"That the quarry I've heard about?" Jack said.
Carl nodded. "Some company carved a mess of limestone blocks outta there, then went outta business."
"What's it like down there?"
Carl shrugged. "Just a big hole in the ground. Used to have a big pool of water in its bottom, but not this year."
"Much security?"
"None I ever seen. You can't steal a hole in the ground. Kids sneak in there at night to drink, smoke dope, and screw. Never seen anyone kick em out. Why you so interested?"
"Just curious."
Jack hoped it wouldn't be necessary, but if worse came to worst, he might have use for the quarry.
He followed Carl's directions, turning this way and that, heading in a generally northwest direction. Along the way he saw a black bird with a red head pecking at something on the side of the road.
"Christ, that's an ugly bird."
"That's a turkey vulture—'TV,' for short. Right homely, aren't they. Good thing about them is they clean up roadkill. They do such a good job that round here we call roadkill 'TV dinners.'" He snickered. "'TV dinners.' Get it?"
"I get it, Carl."
The vegetation became reedier as they rolled along. Finally Carl pointed to a small building with a big AIR BOATS sign. Another, smaller sign—not much more than a slim board with a handwritten message—had been tacked to the bottom.
CLOSED DO TO DROWT.
Jack wondered what the owners were doing with all this extra spare time. Playing Scrabble maybe?
"We're going on an air boat?" He'd seen them whizzing across the Everglades in movies and nature shows and had always wanted to ride one. "Cool."
"Can't use no air boats when it's this dry. There's enough water in the big channels, but the little ones—forget it."
Jack followed Carl around to the rear of the shack where a beached canoe waited on the mud.
"That's our boat?"
"That's her," Carl said with a grin. "She ain't too pan-o-ramic, but she's got a motor."
Jack looked at the tiny, odd-shaped hunk of steel clamped to the right rear stern.
"You call that a motor? I've seen bigger eggbeaters."
"Don't knock it. It's better'n paddlin the whole way."
Carl stepped into the water and pulled the canoe off the mud. He hopped into the stern seat and used the paddle jutting from his right sleeve to steady the boat. Jack had no choice but to wade in, sneakers and all, after him.
"Didn't you bring no boots?"
"Ain't got no boots." There I go, talking like him again.
Jack was calf high in water before he reached the canoe and eased himself onto the forward bench. Carl primed the motor, then: a couple of quick pulls on the cord, a cloud of smoke, a bubbling clatter, and—hi-yo, Silver—they were off.
Jack looked down at the sodden legs of his jeans, and his once white sneakers, now tinted brown with mud. His feet squished and squeaked inside them.
Swell.
"This channel's usually so much deeper and wider this time of year. And most of this saw grass is half underwater." He shook his head. "Man, we really need us some rain."
Jack looked up. A lid of clouds had moved in, hiding the sun and the sky, but none of them looked like rain clouds.
"What you need down here, Carl, is a big storm, a hurricane to dump a load of water. Maybe Elvis will take care of your drought."
"I'd go for a tropical storm, okay. You know—thirty-five-or forty-mile-an-hour winds and a ton of rain. I could handle that. But no hurricane, thank you. I was here when Andrew came through and I don't never want to see the likes of that again."
As they slid along, Jack heard a call and response of throaty roars from either side.
"Those alligators?"
"Yep. Bulls callin from their gator holes."
"What are those grunting sounds? The females?"
Carl laughed. "Naw. Them's pig frogs. Got the name cause they grunt like pigs."
Jack noticed lots of snails, with shells maybe an inch to an inch and a half across, floating near the surface. The tops of some of their shells broke the surface as they clung to underwater growths. He saw little pristine white beads lined up on blades of saw grass and asked Carl about them.
"Those're snail eggs. Cormorants love the snails. Use the hook on the end of their beak to yank them from their shells."
A goose-necked turtle with a smooth brown shell and an uncircumcised nose stuck its head above the water and looked at him.
"Hello," Jack said.
The turtle ducked away.
"That's a soft-shelled turtle. Gators just love to catch those. Gobble them up like crunchy tacos."
Jack slapped at his neck. He didn't have a long-sleeved shirt so he'd sprayed on lots of repellent, but it didn't seem to be helping much.
"How can you stand all these mosquitoes?"
"All? You kidding? This is a good year, a great year for mosquitoes. The drought dried up most of their little breedin pools."
If this is a good year, Jack thought, remind me to stay far away in a bad one.
He reached out a hand to grab a few of the long thin blades of grass brushing the side of the canoe. A sharp sting made him snatch it back. He looked and found long scratches across his palm.
Carl laughed. "Now you know why they call it saw grass." He swept his paddle around in a wide arc. "Pa-hay-okee."
Jack remembered Anya using that word.
"Indian, right? Means 'river of grass' or something?"
Carl grinned. "Hey, you been studyin."
A river of grass...sea of grass was more like it. An ocean of browned saw grass swept away in all directions, dotted here and there by islandlike hummocks of cypress, oak, and pine that looked like giant green mushrooms sprouting from a dead lawn. He hoped it wasn't dead. Just sleeping.
So flat, so like he'd envisioned Kansas might be. Too open for Jack. He was used to living in steel-, concrete-, and glass-lined canyons. The horizon seemed so far away here. Who needed a horizon anyway? Horizons gave him the creeps. He could live very well without one. In fact, back home he did.
Why on earth would anyone want to live out here? No deli, no pizza delivery, no electricity to keep beer cold. Like living in the Dark Ages.
Carl said, "I got Miccosukee blood in me, you know. At least that's what my momma told me. They've got a reservation north of here off Route 41, and even a casino, but I ain't never been to neither. The Miccosukee's on my momma's side. Don't know bout my dad. My momma met him at the lagoon. I hear he didn't hang around after he seen me. Just took off and we never heard from him again."
Jack flicked a glance at Carl's covered right arm. Should he ask about it?
Maybe some other time.
Instead he said, "So there's been people living around this lagoon for generations?"
"Yeah and no," Carl said. "The only people livin there now are the kids of the ones who used to live there. Everybody moved away when we was itty-bitty babies because they thought the lagoon was makin us all strange. But we kids came back."
"Why?"
"Cause I guess we didn't seem to fit no other place."
Jack tried to think of a delicate way to say this. "Because of the way you all looked?"
Carl shrugged. "Some of that, maybe. But mostly because the lagoon seemed right for us. It felt like...home."
"You moved out, though."
"Yeah. But not far. That's why I wasn't too excited bout goin back. I'm afraid I might get sucked in again."
"So how many live there?"
"Bout twenty. We're all bout the same age too, give or take a couple years."
Jack ducked as a big bird with an enormous wingspan swooped above them.
"What the hell is that?"
"Just a big ol' heron."
"Oh."
For a moment there Jack had thought it was a pterodactyl. Or maybe a pteranodon. Whatever. The one with the tail.
They began to pass alligators of various sizes sunning themselves on the banks, but none came even close in size to the monster from yesterday.
Jack heard a scraping sound from the bottom of the canoe.
"That's all for the motor for a while," Carl said.
They used their paddles until the channel grew too shallow even for that.
"What do we do now?"
Carl rose and stepped out of the boat. "We carry her till the water gets deeper."
Easy for you to say, Jack thought. You've got boots.
The hauling itself wasn't so bad—only about thirty yards before the water deepened again—but the knowledge that a gator might step out of the surrounding greenery at any second upped Jack's pace until he was fairly dragging Carl behind him.
"Too bad they don't do a Survivor down here," Carl said. "Survivor: Everglades...they'd never let me on, but I know I could win that million."
Another reality show. Carl did like his TV.
Jack looked over his shoulder. "If you did win, what's the first thing you'd do?"
"Get me a new TV." He grinned. "One of them big sixty-inch models. Oh, and a new easy chair, an electric one that massages your back while you're sittin in it. And get my car fixed."
"How about travel?"
"What for? I've already been all over the world watchin Survivor and Celebrity Mole and the Travel Channel."
"But it's not the same as being there."
Listen to me, Jack thought. The guy who never leaves New York.
"Is for me," Carl said. "Oh, yeah, and I'd probably give some money to Mrs. Hansen. She's havin a hard time. Might lose her trailer."
"That's a nice thought, Carl."
He shrugged. "Just bein neighborly."
Back in the water and putt-putting along again, Jack saw larger plants starting to crowd the saw grass off the banks. Ferns and trees fought for space. Jack spotted a fruit-bearing tree.
"What's that?"
"Pond apple. Don't even think about eatin one less you're partial to the taste of kerosene."
He went on to point out willows that didn't look like willows, live oaks that didn't look like oaks, and trees with exotic names like cocoa plum and Brazilian pepper.
Jack pointed to the tall, scraggly, droopy-needled, cedarlike pines that loomed ahead.
"What are those?"
Carl looked at him as if he'd asked if the sun rose in the east or the west.
"Them's cypresses."
"They look like pines."
"Yeah, I guess they do. But they drop their needles come winter. Pines don't do that."
Jack noticed that the leaves on some of the live oaks were turning red or orange, as if it were fall. The drought, he guessed.
As they glided nearer the cypresses, Jack saw long, gray-brown Merlin beards of moss hanging from the limbs and swaying in the breeze.
He spotted other trees. He knew a Nelson pine when he saw one; royal palms had that distinctive smooth sleeve of green at the upper end of the trunk, and of course coconut palms and banana palms were identifiable by their fruit. But the rest were mysteries.
Carl pointed to a couple of dragonflies, one riding on the back of another.
"Looky there. Makin baby dragonflies."
"And in public," Jack said. "Have they no shame?"
Carl laughed. "Hey, don't knock it. Dragonflies eats up tons of mosquito babies."
"Yeah?" Jack raised a fist in salute. "Go for it, you two!"
Carl shut off the motor.
"What's up?" Jack said. "More shallows?"
Carl shook his head and pointed. "We're getting close now. See that big hardwood hummock dead ahead?"
Jack saw a rise studded with trees of all different sizes and shapes that blocked most of the western horizon.
"The lagoon's in there," Carl said. "So we got to go real quiet now."
"I thought the place was going to be deserted."
"Y'never know. Sometimes somebody's feelin poorly and they don't go to town."
Jack pulled the Glock from its SOB holster, worked the slide to chamber a round, then tucked it away again.
They paddled ahead to where the channel ran into a dense green tunnel of vegetation. Speaking softly, Carl pointed out gumbo limbo trees, aerial plants, orchids, ferns, banyan trees with their dangling aerial roots, coffee plants, vines trailing from tree to tree, and every imaginable variety of palm.
"Looks like a rain forest," Jack whispered.
Carl nodded. "Yeah. Even now, when there ain't no rain. It stays wetter here cause the sun can't get through."
As they paddled around a few more bends in the channel Jack started noticing subtle changes in the greenery, most obvious in the royal palms. Every one Jack had seen till now had had a ramrod-straight trunk. These were bent here and there at odd intervals along their lengths.
Was this the first evidence of the mutation effects of Anya's so-called nexus point?
Then Carl turned to him and put a finger to his lips. He nodded and made a hooking motion with his arm.
Jack got the message: almost there...around the next bend.
And then they rounded that bend and the right bank fell away, opening into a wide pond, 150, maybe 200 feet across. The surface lay smooth and placid, but the surrounding vegetation was anything but.
The willows, oaks, cypresses, and palms lining the banks had been twisted into grotesque, unnatural shapes, as if they'd been frozen mid-step in some epileptic ballet. And in one area they all appeared to be leaning away from an opening on the edge of the bank, as if trying to escape it.
That had to be it—the nexus point, where a little of the Otherness slipped through a couple of times a year. Anya hadn't been exaggerating about the mutations. The vegetation looked like it had been designed by someone with PCP for blood.
All we need to make this scene complete, Jack thought, is the Creature from the Black Lagoon rearing its ugly head.
A large, skiff-style boat, Bull-ship across its stern, rocked gently against the far bank. Its crude, ramshackle superstructure looked like it had been built by someone with only rudimentary carpentry skills. Another smaller, equally rundown skiff, the Horse-ship—cute—lay directly to their right. They looked like floating tenements.
As he and Carl glided toward the center of the lagoon, Jack searched the banks for stray members of Carl's clan. Just as predicted, the place was deserted.
Well, it looked deserted. Somehow it didn't feel deserted.
"That's funny," Carl whispered, pointing to a small fleet of canoes beached on the far bank. "All the boats is here. If they went into town—"
"Well, well, well," said a gruff voice from behind and to the right. "Look who's here."
Jack started at the sound and swiveled to see half a dozen men standing on the deck of the Horse-ship. As he watched, the snow-haired Semelee emerged from the superstructure and smiled at him.
"Hi, Jack," she said.
Jack noticed the color draining from Carl's face. "Oh, shit!"
Jack faced front again and saw another dozen or so men gathering on the deck of the bigger Bull-ship.
"Paddle!" Carl cried as he began yanking on the little motor's starter cord. "We gotta get outta here!"
Jack thought that might not be a bad idea. He reversed his oar stroke to turn the canoe around, but then noticed that the men in the Horse-ship were poling it across the lagoon entrance, blocking their escape route.
He laid a hand on Carl's shoulder. "Forget it, Carl. Looks like we're staying awhile."
"Long time, no see, Carl," said the big guy Jack had run into in town. His grin was feral. "I knew you'd be back someday."
"Hey, Luke," Carl said in a faint voice. His shoulders slumped. He looked defeated.
Jack checked the comforting weight of the Glock at the small of his back. Not the right time to reveal what he was carrying, especially when they were such sitting ducks out here on the water. Better to wait and see what happened, wait till these guys got closer, or things got ugly.
Who knew? Maybe he wouldn't need artillery. Maybe he'd even come away with some answers. Like, what do you have against my father? Or, who hired you to kill him?
"Knew I shouldn'ta come," Carl muttered. His good eye veered right and left like a frightened rabbit on the run.
"Easy," Jack whispered. "I promised I'd get you back to your trailer, and I will. Let's just go with the flow here for a bit."
"Don't see's we got much choice."
Luke pointed to the row of canoes on the bank. "Why dontcha beach it over there with the others," he called, "and we'll all get real friendly like."
Jack started paddling. "Let's do like the man says."
Carl hesitated a few heartbeats—he seemed frozen in place—then shook himself and joined in.
## 5
When they reached the far bank, some of the men from the Bull-ship helped pull its nose onto the dirt. Jack recognized the flat-bottomed motorboat he'd seen Semelee ride away in—the Chicken-ship. Next to it was a canoe labeled No-ship. Someone in the clan was a regular Shecky Green.
He managed to step ashore without resoaking his sneakers, but Carl got out and waded.
They all seemed to know Carl. A few acted genuinely glad to see him but most were standoffish, some even hostile.
As Jack and Carl stood together and waited for the Horse-ship to be poled over, Jack looked around. Close up, the vegetation looked even more demented. Back from the banks, maybe a hundred feet, stood half a dozen hutlike structures with open sides. Each seemed to be little more than half a dozen wobbly poles, three to a side, topped by a pitched roof of dried palm fronds. A small fire smoldered between two of the nearest. When they weren't on the boats, Jack guessed they lived there.
Crooked men in crooked houses. He had little doubt that each contained at least one crooked mouse.
"Old Indian huts," Carl said, following his gaze. "Been there forever."
When the smaller boat arrived, Semelee was the first to step off, followed by Luke, bulge-browed Corley, and the rest. Soon the whole clan was assembled behind her, facing Jack and Carl in a semicircle.
Circe and her pigs.
A single woman with—Jack had made a quick count—eighteen men.
One scary looking bunch, Jack thought, eyeing their misshapen heads, mismatched limbs, and twisted bodies. Looked like they'd suffered an algae bloom in their gene pool. But he knew that, just like the trees, it must be due to the nexus point. The trees had no choice about where they grew, but these folks...why did they stay?
Only Semelee and Luke looked reasonably normal...if you discounted her wild white Medusa hair. Storm from the X-Men had nothing on Semelee in the hair department. She wore the same Levi's and tight black vest as yesterday, but her long-sleeved shirt was red this time, with the top two buttons left open.
"Who's this one?" she said, pointing to Carl. "He's one of us, ain't he."
Luke flashed his nasty grin at Carl. "He sure is. He just don't act like it."
"How come I ain't never seen him before?"
"You probably did but just don't remember. Carl decided to leave right after you showed up. I don't think we're good enough for him no more." He stepped closer. "Ain't that right, Carl? Ain't that right? But that was okay. This ain't no prison. You can come and go as you please." He got into Carl's face. "But that don't mean you can bring outsiders. You know the rule about outsiders."
He reached to grab the front of Carl's shirt and Jack laid a hand on his arm—gently but firmly. He wasn't looking for a fight, not against these impossible odds, but he was not about to let Carl be manhandled.
"Don't," Jack said.
Luke's fingers stopped inches from Carl's shirtfront. "What?"
Jack kept his voice low but gave Luke a hard look, hoping he'd think twice. He didn't have a plan—he'd been expecting an empty lagoon—but he was willing to ad lib, maybe do something quick and very nasty to make a point and throw the crowd off balance.
"Just...don't."
Luke glared at him, then glanced toward the water. "Back off or you'll be goin for a swim."
"Doesn't sound so bad to me."
"Yeah?" He grinned. "Look who you'll be swimmin with."
Jack turned and saw what appeared to be a giant turtle gliding toward shore. Its head was down but its mossy, four-foot long shell looked like a relief map of the Himalayas.
Then it raised its head—and then its other head. Christ, it had two—big, ugly, rough-hewn things—both of which were now angled up, their beaked, sharp-edged jaws agape, showing huge mouths that could fit a regulation NFL football with room to spare. Its four beady black eyes were fixed on Semelee as it reached the bank and waited with its long, snakelike tail thrashing back and forth in the water behind it.
Luke grabbed a fallen tree branch and shouted, "Show time!" He stepped closer and lowered the branch toward the waiting jaws. "This here's a alligator snapper. When you take your swim—and we'll see that you do—here's what's gonna happen to your arms and legs."
The branch came to within a foot of the left head and in a flash the neck telescoped out and the jaws chomped, breaking it in half with a loud crunching crack, as easily as Jack might snap a toothpick. One of the halves tumbled into the right head's strike zone and suffered a similar fate. Three pieces of branch floated on the water.
Jack's tongue tasted dusty.
"'When'?" Jack said, knowing this many guys would have no trouble tossing him into the water. But he couldn't back down. "You mean 'if,' don't you?"
Luke stepped toward him. "No, I mean—"
"Just hold on there," Semelee said, wedging herself between them. "Ease up. This ain't no way to treat company." She turned to Jack. Her eyes locked on his, displaying none of the animosity radiating from Luke. "What're you doing here?"
Jack had his reply ready. "You suggested we have a drink together. Well, here I am."
"Bullshit!" Luke said.
This guy had one helluva chip on his shoulder.
Semelee ignored him and smiled. "Yeah. I can see you're here. But I meant back in town."
"I guess I misunderstood. I happened to mention you to Carl and—"
"You did?" Her face lit as her smile broadened. "You were talking about me?"
Jack realized with a start that she was infatuated with him. He couldn't fathom why. She'd had a couple of glimpses of him and they'd exchanged a few sentences; she didn't know anything about him.
Or did she?
Jack debated playing to her infatuation, then discarded the idea. It could backfire too easily, especially with the jealousy he sensed seething in Luke. It was plain that he wanted Semelee looking at him like she was looking at Jack.
"Yeah, sort of," Jack said, keeping it neutral. "When Carl said he knew where you lived, I convinced him to take me there."
"And here you are."
"Right. But I wasn't expecting such an unfriendly reception."
"Oh, don't take Luke too serious. He's been right cranky lately." She patted his arm. "Ain't that right, Luke."
The big guy only glowered at Jack.
"Hey," said Carl, pointing along the bank with his oar. "Don't tell me that's the lights hole!"
"It sure is," Semelee said. "Want to see?"
Lights hole? Jack wondered. What's a lights hole?
Semelee led the way toward a patch of ground completely bare of vegetation. Jack followed Carl. The crowd parted to let them pass. The center of the bald area was pierced by a roughly oval opening, maybe eight feet across. It ran straight down into the limestone like a well. Jack even knew what it was called: a cenote.
He stopped next to Carl at the edge and peered down. Deep. Deeper than he'd expected. He could just barely make out the pool at the bottom.
Carl gasped. "It wasn't never this deep. What happened to the sand?"
Luke grunted. "Semelee sold it. Some guys came here and sucked a whole lot of it out. You just missed them."
"Got a pretty penny for it too," she said.
Carl looked from Luke to Semelee. "Looks like I ain't the first to break the no-outsiders rule."
Score one for you, Carl, Jack thought.
"That was different," Semelee said.
Carl didn't seem to hear. His eyes were fixed on the hole.
"I was a-fearin this," he said, "what with the drought'n all. The lights hole ain't never been above water before. That's bad enough. But then you went'n had sand sucked out."
"Why's that bad?" Semelee said. "I think that's good."
"Good? How can it be good? The light used to have to come up through the sand and the water, and even then, look what it did to us. Now there ain't hardly nothin in the way."
Semelee grinned. "Ain't it cool?"
"Nuh-uh. That ain't cool. That's scary."
Jack knelt at the edge and peered into the depths. He didn't like deep holes, at least not since the spring when he'd had a bad experience with one out on Long Island. But that one hadn't had a bottom. This one...
He found a thumb-size stone and dropped it. He heard a satisfying plop, saw ripples on the water far below.
...this one definitely had a bottom.
But for how long?
"What are these lights like?" he asked.
Semelee squatted close beside him. He glanced up briefly and noticed the others wandering off. The two of them had the hole to themselves.
"Like nothin you ever seen in your life." Her voice was full of hushed wonder as she spoke. "I mean, whoever heard of lights comin outta the ground?"
Jack had seen light shooting up from a hole in the earth...just last spring.
"What color are these lights?"
"Sorta like pinkish orange, but that ain't right. Every time you think you got the color pinned, it melts into somethin else just a teeny bit different. I can't describe it. You gotta see it to believe it."
Jack believed. He'd seen a light just like she described.
"How often do they come?" Jack asked, knowing the answer.
"Twice a year."
"No kidding. When's the next show?"
"Tonight."
"But—" Jack caught himself. Anya had said the nexus points opened during the equinoxes, but that wasn't until tomorrow night. He knew; he'd checked. But if he admitted that, Semelee would realize that he knew way more than he should.
She frowned at him. "But what?"
What to say? "But that's too soon!" he blurted. "I won't be able to get my cameras set up for—"
"Who said anything bout cameras?"
"Well, it's obvious, isn't it? I take some pictures of the lights and we sell them to the papers, to National Geographic, to—"
"Wait-wait-wait," she said, waving her hands in front of his face. "What makes you think you're gonna take pictures? Nobody takes no pictures of the lights."
"No exceptions?"
"No way, no how. As a matter of fact, I can't even let you see them, cause then you'd talk about them."
"No, I wouldn't."
Jack had no desire to see these lights, but he didn't want to appear anxious to leave. Maybe the way to get out of here was to pretend to want to stay.
Semelee shook her head. "Maybe you wouldn't, but I can't risk it. Not yet, anyways. But maybe when I get to know you better..."
Jack noted how she said "when" instead of "if."
"What's wrong with getting to know each other now? We could go back to town, have that drink, maybe two or three, and do some serious talking."
"Not tonight, or tomorrow or the next night, for that matter."
"Why not?"
"The lights run for three nights. I gotta be here for that. But after Sunday..." She leaned closer and he caught her pleasant, musky scent. ".... We got all the time in the world."
That's what you think, sister.
But he had to be careful here...hell hath no fury and all that.
Then he noticed the black shell dangling from the thong around her neck. The same size and shape as the one he'd found in his father's hospital room. Even had a hole drilled at the hinge end. Had to be the same.
He pointed to the shell. "How'd you get that back?"
Semelee started and clutched the shell. Jack figured from the sudden widening of her eyes that she hadn't wanted him to see it. Because that meant she'd visited the room a second time—and he didn't like that one bit.
But if that were the case, why had she worn it around her neck and left the collar loose?
"What do you mean?" she said.
"I found it by my father's bed in the hospital, right after you were there. When did you go back for it?"
"I...I didn't." She kept the shell wrapped in her fist. "I had two."
"Oh." That made Jack feel a little better—if she was telling the truth. "I guess I saw the other one then."
"Where?" She grabbed his wrist. "Where'd you see it last?"
Jack was about to shrug and say he'd left it on the bedside table and assumed the housekeeping staff had chucked it out, but her tight grip on his arm and the intensity in her eyes made him hold off.
"I'm not sure. Let me think..."
Why was a damn shell so important?
He glanced around and noticed Carl was missing.
"Carl?" Jack broke Semelee's grip on his arm as he rose to his feet and scanned the lagoon banks. "Hey, Carl! Where are you?"
"Never mind him," Semelee said, rising with him. "What about that shell?"
Jack left her behind. He skirted the edge of the cenote and headed in the direction of the huts where he saw a number of the men sitting around the little fire, smoking, drinking, but Carl wasn't among them.
Shit! Where was he?
He called his name a few more times but got no response. He asked the group by the fire where he was but they ignored him.
Jack's gut began a little crawl. If they'd done anything to Carl it would be Jack's fault for inducing him to come back here.
Luke strolled up to the fire. The men around it looked up, their mismatched eyes questioning, and he nodded to them.
"Where is he, Luke?" Jack said.
Luke didn't look up, didn't turn, didn't acknowledge Jack's existence.
Jack's concern boiled over into anger. He pulled the Glock and sent a round into the fire. The mini-explosion of ash and flaming embers scattered the men, sending them rolling and tumbling. Luke ducked away and faced him.
Now he had their attention.
"I'm going to say it once more, and this time I'd better get an answer: Where...is...Carl?"
"Right where he belongs," Luke said. "With us."
"He doesn't want to be with you. He left, remember?"
"Maybe. But he's had a change of heart. He's gonna stay."
Jack sensed movement around him. His peripheral vision caught about a dozen clan members scurrying toward him, armed with rifles and shotguns. Should have figured they'd be armed—couldn't live out here and not do some hunting.
The newcomers didn't seem to give Luke much of a boost in confidence, especially when Jack pointed the Glock at the center of his chest. "I want to hear that from him."
Luke's eyes darted left and right. He seemed about to say something when Semelee spoke up.
"Don't worry, Luke. He ain't gonna kill you."
Jack glanced left and saw her standing a few feet away, smiling at him.
"Right, mister," Luke said, licking his lips. "That's because you'll be full of holes if you do."
"That won't make you any less dead."
"You won't," Semelee said to Jack. "I know it, and you know it."
She was right. This wasn't a killing situation. He lowered the pistol a few inches.
"Maybe not. But one of these hollowpoints can mess up a knee like you wouldn't believe."
Luke was sweating now. Taking one in the knee seemed to bother him more than one in the chest.
"Semelee...?"
"You won't do that neither. Because we ain't hurt Carl and we ain't keepin him here but for a few days."
"You've got no right to keep him a minute."
"Yeah, we do," Luke said, emboldened by the fact that Jack hadn't pulled the trigger again. "He's kin. He's blood."
"I promised I'd get him back home. I intend to keep that promise."
"It's only gonna be three days," Semelee said. "We want him to stay for the lights. But I tell you what: You find my other shell and we'll do a trade...the shell for Carl."
"Semelee," Luke said. "You got no right. Carl belongs here."
She turned on him, eyes flashing. "What's more important—givin Carl a light show or gettin my eye-shell back?"
Luke looked away and said nothin.
Semelee turned to Jack. "So that's the deal. How's it set with you?"
"Lady, I don't know where this shell of yours is. If I'd known it was going to matter, I'd have kept track of it."
She pointed to Carl's borrowed canoe. "Maybe you'd better start lookin."
Keeping his pistol trained on Luke, Jack considered his options. He had a few, but didn't like any of them.
He could do a little shooting, but he could see how that could turn counterproductive. He could do his own search for Carl, but he'd be a stranger looking for someone who'd been stashed away by folks who knew every nook and cranny of the terrain. He could head back and take one of these guys with him, then trade him back for Carl; but Jack had no place to stash him.
Or he could go back and find the shell, which was one tall order.
Going back...there was another challenge. He wasn't Woodsman Jack. The closest he ever wanted to get to outdoor life was a copy of Field & Stream.
"I don't know the Everglades," he said. "I'll get lost out there."
Semelee laughed, a musical sound, void of harshness or derision.
"No, you won't. The drought ain't left too many wet channels. Every time you come to a fork, just take the eastmost. It ain't all that far."
"And if I do find this shell, how will I let you know?"
"Easy. Just stand outside you daddy's house and say, 'I found the shell.' I'll hear you."
Jack didn't think she was lying, and that gave him the creeps.
"All right," he said. "I'll go." He hated to leave without Carl, but he'd be back. He also hated leaving without satisfying the reason he'd come here in the first place. "But I want to know something before I go: What have you got against my father?"
Semelee looked away, then back to him. "Nothin."
"Like hell. You folks tried to kill him the other night, and somehow sicced that freaky alligator on him yesterday. Let me ask you: What did he ever do to you?"
"We ain't after him," she said.
Jack caught Luke giving her a sharp look, but she didn't see it.
"Why don't I believe that?" Jack said.
She shrugged. "That's up to you. But I tell you true, your daddy ain't got nothin to fear from us."
"How about me?" Jack said. "What happens when I turn my back on you and your clan?"
"Nothin. You can't find me my shell if you're dead, now, can you." She turned to the clan. "Ain't that right, fellers." They looked at one another but didn't say much. Semelee's expression turned fierce. "Ain't that right? Cause I hate to think what would happen to anybody who stopped this man from doin what I need him to do."
Jack saw a lot of uneasy, fearful expressions as the men nodded and lowered their weapons.
What kind of hold did she have on them? What could that slim little woman threaten them with?
Taking a breath and hoping he wasn't making a mistake, Jack holstered the Glock and walked back to the canoe. He stepped into the water, pushed off, and slid in. A couple of pulls got the engine going. He putted away, propelled by the weight of dozens of eyes on his back.
## 6
"Why'd you let him go?" Luke said.
Semelee stood on the bank and watched Jack's retreating form as he turned the canoe left and disappeared around the bend.
"Told you why."
"You believe him?"
She could hear lots of anger in his voice. She knew he was jealous, but she figgered his pride had got hurt bein on the wrong end of Jack's gun.
"Yeah, I do."
She wasn't sure why, but she had the feeling that he'd thought there was only one shell until she told him otherwise.
"You're actin like a fool, Semelee. We coulda gone lookin for that eye-shell ourselfs."
"Yeah? Like where? Like how? We can't go to the hospital and ask about it when I wasn't supposed to be there in the first place. We can't search his daddy's place like he can."
"We coulda tried. Way it stands now we ain't never gonna see him or your other eye-shell again."
"Oh, we'll see him again...one way or the other."
"What's that supposed to mean?"
"Didn't you hear him? He said he promised Carl he'd get him back home. If he finds the shell he'll be back to make the trade. But even if he don't find it he'll be comin for Carl. He'll take him home again or spread a whole lotta hurt tryin."
Luke snorted. "What makes you think you know so much about him? You ain't spoke to him but twice."
She turned to him. "Let me tell you somethin, Luke. That's a man who keeps his promises."
She'd seen that in his eyes. Not a lick of fear, just stubborn as all get out. And that made him all the more special. Brave and loyal, two traits any woman wanted in a man. But Jack wasn't just any woman's man. He was destined to be hers.
The way things was fallin into place...it was like it was all part of a plan. His daddy gets chosen to die, but he don't. He lives and that brings Jack down here where he and Semelee can meet and be together. She lost an eye-shell, but now Jack was gonna find it, and that was gonna bring them even closer together.
"What do you need that other eye-shell so bad for anyway?" Luke said. "You been doin all right with just the one."
"No, I ain't. Ain't the same. Much harder to keep control and see where I'm goin. I need the two of them."
"Awright awright. But you was kiddin bout layin off his daddy, right?"
"Wrong. We ain't interested in his daddy no more."
"But Seme—"
"We got us a new target."
She didn't know how, but Jack had somehow connected her and the clan to what had happened to his daddy. If his daddy got killed, he'd blame her, and that might keep them apart and wreck their destiny. No, she had a better victim, someone who needed killin.
Luke was starin at her. "Who?"
"The old lady. She'll be takin daddy's place."
## 7
How was he going to find that damn shell?
The question plagued Jack as he drove toward Novaton.
Semelee had been right: It hadn't been all that hard to find his way back to the real world. He'd left the canoe beached by the air-boat dock and headed toward town. The clouds persisted but hadn't dumped drop one of rain.
Where to start? The hospital was the obvious place, but Dad had checked himself out almost twenty-four hours ago. Jack was sure the room had been stripped and scrubbed by now. Probably even had a new occupant. That meant he might have to go pawing through the hospital's Dumpsters.
He shook his head. Maybe if he had half a dozen people helping him they might—just might—come up with that shell. He doubted it.
He decided that before he gave the hospital another thought, he'd check out his dad's place. Maybe by some freaky turn of good luck the shell had wound up there. But again, the chances—
If nothing else, he could get out of these sodden sneakers.
He'd stopped at a red light. A dump truck was turning in front of him, going the opposite way. He wouldn't have given it a second thought except for the insignia on the door of the cab. It looked like a black sun...a shape that might be mistaken for the head of a black flower.
Jack would have hung a U right there if he'd been in the left lane. Instead he had to cut through two parking lots to turn himself around. By the time he was heading north, the truck was out of sight. Racing along as best he could in the Friday afternoon traffic, trying to catch up, he almost missed the truck parked in a Burger King lot.
Jack pulled in next to it and got out. It had been backed diagonally across two spaces at the rear of the lot where it was out of the way. The cab was empty but the big diesel engine was running. He checked out the logo—definitely a black sun. And beneath it: Wm. Blagden & Sons, Inc.
He walked around it. It sure as hell looked big enough to inflict heavy damage on any car, even a Grand Marquis. He wondered what the left end of the front bumper looked like.
Jack stopped and stared at the dent in the fender...and the streaks of silver paint ground into its black surface.
"Can I help you with something?" said a voice behind him.
Jack turned to find a prototypical truck driver—big cowboy hat, big gut, big belt buckle, big boots—walking his way with a bag of burgers in one hand and a travel mug of coffee in the other.
"Yeah," Jack said. "Just admiring the ding in your fender here." A euphemism; the "ding" was a deep dent. "Looks pretty fresh."
"It is. Best I can figure it must've happened Monday night when the truck was stolen."
"Stolen? No kidding? By who?"
The driver unlocked the door to the cab, put the burgers and coffee inside, then shrugged.
"Damned if I know." He rubbed his weather-beaten face. "Never happened to me before. After she got the first part of her load Monday evening, I locked her up and hit the hay. I got up the next morning and she was gone. Couple hours after I reported her missing the cops found her in a liquor store parking lot. I was so glad to get her back—I mean, you don't know what kind of shit was gonna come down on me if she was gone for good—that I didn't notice the ding till later."
"You report it to the cops?"
"No. Why?"
"Because your rig might have been involved in a hit and run."
His eyes narrowed. "You a cop or something?"
"Nope. Just an interested party." He saw the questioning look on the trucker's face. "My dad's car took a wallop early Tuesday morning."
"He okay?"
"Luckily, yeah."
"Good." He hauled himself into the cab. "Because I can't hang around for no investigation. I ain't running or nothing, but I got a schedule to keep."
"I hear you," Jack said.
He thought about stopping him but decided against it. If his story was true—and Jack sensed it was—what good would it do? If he hadn't reported his truck stolen, Jack could call Hernandez and the Novaton cops would pick him up.
Of course, the reported theft could have been a cover, but Jack doubted that.
As the cab door slammed shut, Jack said, "What're you hauling?"
"Sand."
"Where to?"
"North Jersey."
Jersey? Jersey was loaded with sand.
"What the hell for?"
The driver shrugged. "I don't set up the jobs or choose the loads; I just get it where it wants to go."
Then Jack remembered Luke saying something about Semelee sucking all the sand out of the cenote and selling it. Could this be...?
"Where'd you get the sand?"
Another shrug. "It got boated in from somewheres in the swamp. That's all I know."
With that he threw the truck into first and headed for the exit.
Jack watched him go. He made a mental note of the company name. Wm. Blagden & Sons. He might look them up when he got back north, maybe find out who'd hired them. Shipping sand from a Florida nexus point to New Jersey...he couldn't imagine the reason, but it couldn't be good.
He started back toward his car. At least now he knew what had hit his father's Marquis. And he had a pretty good idea who had been driving it.
But he still didn't know why. Had a pretty good idea about that too, and hoped to nail that down this afternoon.
## 8
By the time Jack reached Gateways South he'd stopped at a local hardware store for a roll of duct tape, then called the Novaton Police where he reached Anita Nesbitt. After a quick check she told him that, yes, on Tuesday morning a dump truck had been reported stolen during the night and was found shortly thereafter.
Okay. So Wm. Blagden & Sons, Inc., was covered.
Jack parked in the cul-de-sac and hurried into his father's place.
His father was watching TV. Classic ESPN was running the 1980 Wimbledon slugfest between Borg and McEnroe. McEnroe was screaming at himself for missing a bullet passing shot.
He looked up at Jack and grinned. "Right about now I bet McEnroe wishes Borg had never been Bjorn."
Normally Jack would have groaned, but a bad pun was a good sign. His father loved puns. He was getting back to normal.
He looked down at Jack's muddy sneakers and still-wet jeans. "What happened to you?"
"Took a little boat trip."
"You went boating? Why didn't you tell me? I would have—"
"It wasn't exactly a pleasure trip. Look, Dad, do you remember seeing a little black shell in your hospital room?"
He frowned. "No. When would this have been?"
"I found it the day before you woke up. It was black, oblong, had a little hole drilled in the hinge."
Please remember. Please...
Dad was shaking his head. "Sorry. Never saw anything like that."
Jack suppressed a groan. He'd have to try the hospital next.
Hospital...Jack remembered the plastic bag of sundries that Anya had thrown together as his father was signing himself out. He knew it wasn't in his car. Had he brought it in?
"Did you see a bag of goodies from the hospital? You know, toothpaste, mouthwash—"
"Oh, that. I threw it out."
"You didn't see a shell in there?"
"I didn't really look. I mean, I glanced inside but I don't use any of those brands so I tossed it out."
Maybe...maybe...Jack didn't want to get his hopes up.
"Where? In the kitchen?"
"Well, yes, at first. But this morning I tossed the kitchen bag into the can out back. Look, what's so important—?"
Jack didn't wait for him to finish. He dashed outside and around to the back porch. The green plastic garbage can sat to the left on a small concrete slab. Just his luck, Friday would be garbage pickup day and the shell—if it was in there—was on its way to the county dump.
But no. The can was empty except for one white plastic bag. Jack untied the top and poked around until he found the bag from the hospital. He yanked it out and pawed through the sample-size toiletries. He sent out a silent prayer to the patron saint of garbage that he'd find the shell within, but it wasn't looking good...
And then he reached the bottom and felt something hard and rough edged. He pulled it out—
"Yes!"
He had it. Now Carl could come home. But first Jack had to arrange an exchange. He shook his head. A shell for a human being...what kind of a deal was that?
What had Semelee told him to do? Stand outside his father's house and announce that he'd found it. Riiiight. But she'd said she'd hear him, and she probably could. Jack's Doubting Thomas days were over. Anything goes.
"Okay," he said aloud, feeling foolish but forcing himself to go on. "I've found the shell. Did you catch that? I've found it. Tell me how we make the trade."
Now what? He supposed he'd have to wait until Semelee got in touch with him.
Pocketing the shell, he turned and found Dad staring at him through the back porch jalousies. He wore the same perplexed expression as when Jack had unpacked those stuffed animals from Abe. Maybe more perplexed this time.
Probably thinks I'm doing drugs.
"Hi, Dad."
"Are you okay, Jack?"
No, he thought. I'm not. Someday I'd like to be, but at the moment...
"I'm fine."
His father pushed open the porch door. "Come back in this way. It's shorter."
Jack took a step toward the porch, then remembered again that it was Anya who'd packed up the bag. Had she known...?
He glanced toward her place and noticed a figure stretched out on a lounge in the front yard.
"Be with you in a minute," he said. "I want to say hello to Anya."
As Jack crossed onto the green grass, Oyv trotted up to meet him, wagging his tail in welcome. The dog escorted him toward Anya, but Jack slowed, letting Oyv pull ahead as he noticed that Anya was topless.
She lay face down on a towel on the lounge cushion, dressed only in lime-colored Bermuda shorts, baking her bare back in the afternoon sun. He was about to turn away when he noticed a pattern of red marks on her exposed skin. He took a step closer and...
Jack bit his upper lip. They looked like burn marks...and crisscrossing her skin between them were thin, angry red lines, as if someone had been stubbing out cigarettes on her back and then whipping her with a fine lash.
Jack wanted to turn away, but couldn't. He had to stay and stare, horrified, yet fascinated.
Anya's voice startled him.
"A map of my pain," she said without looking up. "See what he does to me?"
"Who?"
"You know. The Adversary. The One."
Oh, yeah. The One...whose True Name Jack wasn't supposed to know.
"But how? Why?"
"I've told you the why: Because I hinder his path. As to the how...he has many ways, and they are all written here, on my back."
"But how do those burns, those cuts get there?"
"They simply appear. They map his efforts to destroy me."
Jack shook his head to clear it. "I'm not following. What is he doing to destroy you?"
"Help me with this towel," she said. "Fold the ends over my back."
Jack did as she asked, allowing her to wrap the towel around her upper torso as she rose to a sitting position.
"Talk to me," Jack said.
Anya shook her gray head. "You have your own concerns. Those you should be worrying about. And besides, what can you do to help? Nothing. This I must face on my own."
"Try me."
He liked this old lady. He wanted to help her, do something to lighten her load.
"It's all right, Jack. The sun makes it feel better. The rays don't heal me, but they lessen the pain." She rose to her feet. "I'm going in to lie down."
"Are you okay?"
"I'm better than I was this morning and I'll be even better by tonight."
"Will you be up for drinks later? We'll do it at my father's place this time."
She shook her head. "Not tonight. But tomorrow definitely."
Jack watched her and Oyv enter the leafy interior of her house, then, feeling sad and angry and helpless, he turned away.
## 9
Jack had lounged around with his father, dodging questions about the toys and the shell until his father nodded off in his recliner. An afternoon nap—one of the great pleasures in life. But Jack couldn't indulge today. He had to wait for word from Semelee.
But that wasn't the only matter on his afternoon schedule.
He stepped into his father's bedroom and dialed Ramsey Weldon's office. He learned from the receptionist that Mr. Weldon was on another line. Would he care to leave a message?
"No. When can I call back?"
"Well, he'll probably be leaving in a half hour or so."
Jack thanked her, hung up, then went out to his car.
The duct tape he'd bought earlier sat on the front seat in a flimsy white plastic bag emblazoned with the Novaton Hardware logo. He snatched it up, bag and all. As he was closing the door he spotted an envelope on the floor by the passenger seat. He picked it up and checked the contents.
Carl's five hundred dollars.
He'd trusted Jack enough to leave it in the car for safekeeping. He'd also trusted Jack to bring him back.
"I've got your damn shell," Jack said aloud. "I'm ready to trade."
He glanced at his watch. Couldn't wait around here any longer. He set off on a stroll toward the administration building.
This time he could walk in the open and say hello to passers-by instead of ducking into the bushes every time someone approached. When he reached the parking lot, his heart gave a kick when he didn't see Weldon's Crown Imperial, but eased back when he spotted a '57 DeSoto in Weldon's space. This guy had some neat cars.
Jack strolled over to it. A four-door Firedome with a glossy turquoise body, white roof and side panels, big chrome bumpers, white-wall tires, and those fins—humongous wedge-shaped projections, each fitted with a vertical row of three rocketlike red lights that made the car look like a spaceship. Jack peered inside. White-and-turquoise upholstery and a dash-mounted rearview mirror.
What was wrong with Detroit—or Japan or Germany, for that matter? Why the hell didn't they make cars like this anymore?
He hung around the DeSoto, studying it from every angle for what seemed like forever before Weldon showed up. He wore a pale beige silk suit today, so pale it was almost white.
"Another beauty, Mr. Weldon," he said.
Weldon grinned. "Tom's son, right? Jack?"
"You've got a good memory."
"And you've got excellent taste in cars. How's your father?"
"Doing great. He came home yesterday."
Weldon's cheek twitched. "Really? I had no idea. Why didn't anyone tell me?"
"I don't think anyone else knows." Jack ran his fingers lightly along the DeSoto's right front fender. "Say, would you mind giving me a little ride in this baby?"
Weldon shook his head. "I'd love to, but I've got to get straight home."
Jack opened the door and slipped into the passenger seat. "That's okay. Just drive me to the front gate and I'll walk back. I need the exercise."
Weldon didn't look happy about it, but Jack hadn't left him much choice.
The interior was like a furnace. Jack cranked down his window as Weldon fired her up and backed out of his space.
"Smooth ride," Jack said once they were rolling.
"Torsion-Air suspension."
Jack watched him closely as he asked the next question. "You ever hear of a woman named Semelee?"
Weldon's hands tightened on the steering wheel, whitening the knuckles. His right cheek twitched as it had before.
"No, can't say as I have. Is she one of our residents?"
"Nope. Too young for Gateways. Lives out in the Glades with a bunch of funny looking guys. She's got this snow white hair. You'd remember her if you ever met her. You sure you don't know her?"
Weldon looked ready to jump out of his skin and his forehead was beaded with sweat. It was hot in the car, but not that hot.
"Quite sure," he said.
"You're sure you're sure?"
"Yes! How many times do I have to tell you that?" He began to brake. "Well, here's the gate. I hope you enjoyed—"
"Keep driving."
"I told you. I have to—"
Jack pulled out the Glock and held it in his lap, pointed in the general direction of Weldon's gut.
"You'll be in a world of gut-shot hurt if this happens to go off. Think Reservoir Dogs. So keep driving. We haven't finished our chat. Smile and wave to the nice guard. That's right. Now...let's head out to where my father had his accident."
"Where's that?" Now Weldon was really sweating.
"You don't know? Pemberton and South Road."
"But there's nothing out there."
"I know."
"This is illegal, this is carjacking, it's kidnapping, it's—"
"It's happening. Relax. Don't fight it and we'll have a nice ride."
"If you want the car, take it."
"I don't want the car."
"Then...then why are you doing this?"
Jack let him stew in his juices for a while before responding.
"Just wanted to ask you what you know about people who've been dying at Gateways South." Weldon opened his mouth to reply but Jack held up a hand to stop him. "I don't want to hear any bullshit about them being elderly and what can you expect. I'm talking about three spouseless people in excellent health—your own doctor said so—who've suffered death by mishap over the past nine months. At a rate of one every three months. I'm sure you know their names: Adele Borger, Joseph Leo, and Edward Neusner."
Weldon had turned pale. He looked as if he might be getting sick.
"Of course I know their names. Those were terrible tragedies."
"My father would have made number four, and right on schedule. Know anything about that, Mr. Weldon?"
"No, of course not. How could I?"
That did it. Jack looked around, saw no other cars in sight. This was as good a place as any.
He made Weldon pull over, then he got out and made him slide to the passenger side—easy with the bench seat.
"Now, put your hands behind your back."
"W-w-what are you going to do?"
"I'm g-g-gonna tape your wrists together."
"No!"
Jack grabbed a handful of Weldon's longish dark hair. "Look. We can do this the easy way—which is you doing what I tell you—or the hard way, which means I have to shoot you in the hip or through the thigh or something equally messy and bloody and keep on doing that until you cooperate. Me, I don't like getting splattered with blood. The stains are almost impossible to get out. So I prefer neat and easy to messy and bloody. How about you?"
Weldon sobbed and put his hands behind his back.
Jack duct taped his wrists together, then his knees, then his ankles. That done, he took over the driver seat and put the DeSoto back in motion. He pointed it toward town and kept hammering at Weldon about the three dead folks, his father, and Semelee. Weldon kept stonewalling him. Finally Jack pulled up before the locked gates to the limestone quarry.
"So," he said. "You don't know nuttin' 'bout nuttin', is that it?"
"Please. I don't. Really. You've got to believe me."
Jack didn't.
"This is going to hurt me almost as much as it hurts you."
With that he gunned the DeSoto and rammed it against the gates. Weldon cried out as the chain snapped and the gates flew back.
"The bumpers! The chrome!"
Jack turned the car left onto the steep grade of the narrow road that ran down into the pit. A rough limestone wall loomed to his left. He didn't want to do it—he hated himself for doing it—but forced his hands to turn the steering wheel and drag the left side of the car against the stone.
"My God, no!" Weldon cried.
"Sorry." And he was.
As they reached the bottom of the quarry Jack didn't quite make the turn, ramming the front end into an outcropping of stone. The impact stopped the car short, hurling Weldon off the seat and into the dashboard. Without a seat belt or his hands to protect him, he hit hard, then flopped back against the seat.
"Whoa," Jack said. "That must have hurt. But probably just a fraction of what my father felt when that truck clocked his car out on South Road." He looked around. "Let's see. We've remodeled the left side, let's see what we can do with the right."
Between getting a taste of what his dad had gone through that night and realizing what he was doing to this beautiful, classic, innocent car, Jack was having trouble keeping his tone light.
"No, please!" Weldon screamed.
Jack accelerated and rammed the right front end against another outcropping. Once again Weldon went flying forward, this time hard enough to catch his chest on the dashboard and his head against the windshield. He wound up on the floor instead of the seat.
Weldon was sobbing now. "Okay, okay. I'll tell you about it, but you're not going to believe it."
"Try me." Jack threw the on-the-column automatic shift into neutral and set the emergency brake. "You'd be amazed at what I can believe."
Weldon struggled back into his seat. A blue-black goose egg was swelling under the hair that hung over on his forehead. He held his back-tied hands toward Jack.
"Please?"
Jack pulled out his Spyderco folder and slit the tape. He left the knife open and in hand.
"Don't get any ideas. Now talk."
Weldon sagged back. His neck bowed against the top of the backrest as he looked at the ceiling.
"It was just about this time last year that the white-haired woman you mentioned, Semelee, called me with this crazy story, a demand that Gateways make sacrifices to the Everglades. Figuring this was some clumsy sort of local shakedown I asked her what kind of sacrifices. She said...human."
He glanced at Jack. If he was expecting to see shock or incredulity, he was disappointed. Jack had half expected something like this.
"And you laughed her off."
"Of course. Wouldn't you? It was ridiculous. Or so I thought then. But she wouldn't quit. She kept calling me, at the office, at home, on my cell phone, going on about how Gateways South had encroached too near the 'lagoon'—I still don't know what lagoon she was talking about—and that the Everglades was angry and demanded sacrificial victims. Four a year. Ridiculous, right? But she kept after me, saying that I, as head of Gateways, must make the offering. By that she meant, choose the victim. All I had to do was point out a resident and the lagoon would do the rest. If I didn't, the lagoon would choose one for me—from my own family."
"And so you caved."
"No. At least not yet. As soon as she threatened my family, I went to the police. Since I had only a voice on the phone, and couldn't tell them what she looked like or where she lived, all they could do was keep an eye out for her and do regular patrols past my house."
"And I take it that didn't work."
Weldon shook his head. "That same night, my son was bitten by a brown recluse spider and had to be rushed to the hospital—he was only three and almost lost his arm. And right there, in Kevin's hospital room, the woman calls me on my cell phone and says this was just a warning. Had I changed my mind? I hung up but she called right back and asked me if my daughter was afraid of snakes. And if not, she should be." Weldon rubbed a hand over his face. "I've got to tell you, that spooked me. I don't know how she knew about the spider bite, I don't know how she got a brown recluse close enough to my son to bite him, but I was really spooked."
Jack couldn't blame him. He knew how he'd felt when Vicky had been threatened.
"Did you go back to the cops?"
"What for? I couldn't tell them any more then than before. So I took matters into my own hands. I packed up my wife and both kids and sent them to stay with my in-laws in Woodstock, right outside Atlanta. I figured putting them hundreds of miles away in a different town, a different state, would keep them safe." He shook his head. "The very first day there Laurie was bitten by a copperhead and almost died. After spending a week up north, waiting for Laurie to be released from the hospital, I finally returned home—alone, because I couldn't bear the thought of bringing them back here until I'd dealt with this woman."
"Obviously you didn't succeed."
"Not for lack of trying. When I got home I found this young woman with white hair waiting in my backyard. She was sitting with her back to me, holding her hands up to her face, and in an instant I knew who she was. I grabbed the revolver I keep in the top of our bedroom closet and went out to her. I was going to shoot her, so help me, I was, but as soon as I raised the pistol I was attacked by a swarm of bees and—"
"Killer bees?"
Weldon nodded. "Only they didn't sting me enough to kill me. They concentrated on my face and my gun hand and didn't let up until I'd dropped it. Then she turned and I saw her face for the first time. I was surprised that she was so young. From her white hair I'd assumed she'd be some old witch, but she was young and—"
"Not bad looking. I know."
"You've met her then. How did you—?"
"Let's stick to you. What did you do then?"
"What could I do? She told me I already had two strikes against me. I still remember her words: 'Strike three and your wife is out.' What else could I do? Tell me you would have done any different."
"My approach to settling problems differs a bit from the average."
"I don't know how, but this woman somehow controls snakes, insects, birds, and who knows what else? Don't you see the position I was in?"
Jack stared at Weldon. No question, the guy had been thrust into an appalling situation: Finger a relative stranger for death or lose a family member. A no-brainer, but also a no-win.
"I see that a man has to put his family before strangers, which is regrettably acceptable. But when one of those strangers is my father, we have a problem." Jack jabbed the knife blade at Weldon's face, stopping the point an inch from his nose. "We have even more of a problem when it becomes clear that you took an awful predicament and used it to turn a quick buck."
"I did no such thing!"
Weldon cowered back, pressing himself against the door as the knife point touched the tip of his nose.
"Now's not the time for lies, bozo." Jack was doing his best to check his flaring rage. "I could go along with you doing what you had to if you'd picked out the sickest Gateways folks, the ones with the shortest life expectancy. But you didn't do that. Instead you picked ones who were not only the healthiest, but were unattached, guaranteeing that their homes would go back on the market years, maybe even a decade or two before their natural time."
"No!"
"Yes!" The word hissed through Jack's teeth. "Yes, you son of a bitch! You fingered people whose deaths would turn you a profit! And one of them was my father!"
Weldon's face crumpled. His eyes squeezed shut and he began to sob.
"I'm sorry, I'm sorry..."
"Three innocent people are dead and my father was put in a coma, and that's all you can say?" He wanted to drop the knife and throttle him. "Get out!"
Weldon looked at him. "What?"
"Get out, you pathetic bastard. Out before I cut you."
Weldon fumbled behind him for the latch. As the door swung open, Jack raised his right leg and kicked him. Hard.
"Out!"
Weldon fell out the door and landed on his back in the limestone powder and rubble. Without bothering to close the door, Jack threw the DeSoto into gear and hit the gas. He gunned the car into a tire-spinning turn, then raced back toward where Weldon was staggering to his feet. He let him scramble out of the way. Despite Jack's dark urge to maim, maybe even kill the man, Weldon wasn't worth the hassle.
He tore up the steep roadway out of the pit and onto the street. He knew Weldon wouldn't be going to the police about this; he'd fear it would draw a loot of unwanted attention to the deaths at Gateways. Let him find his own way home.
As he passed the trailer park he pulled in. An impulse. He spotted Carl's junker parked by a mildewed trailer. He got out and checked the door. Locked. He lifted the lid of a garbage can by the steps and found takeout containers—KFC, Chinese, Domino's. He pulled out his wallet as he scoped the area. No one about so he slipped the door latch with his MasterCard. Inside he closed the door behind him and looked around. He wasn't sure why he was here. Just an urge to know a little more about Carl.
The air conditioner was off and the trailer smelled faintly of old food and sweat. The kitchen, bathroom, and bedroom lay to the left, the main room to the right. He noticed the disassembled remnants of Big Mouth Billy Bass, the singing fish, on the kitchen counter, neatly stored in a little box. Jack was struck by how clean the place was. Carl had said he loved his little trailer, and it showed.
In the main room sat a good-size TV. It looked like at least a twenty-seven-incher—pan-o-ramic, one might say. A battered Naugahyde recliner sat before it. The thick Direct TV program guide for September lay open on the seat, marked up with a yellow pen. Jack picked it up and saw that Carl had highlighted Survivor, Fear Factor, Boot Camp, Big Brother...secondhand living.
But that seemed good enough for Carl.
Jack shrugged. Whatever gets you through the night...
But nowhere in the trailer was there a sign of who Carl was. No family pictures, no sign that he had a past. Maybe his past wasn't anything he wanted to remember.
Jack stepped out, locked the door, and drove back toward Gateways. He turned off the road and parked in the trees next to the security fence. He noticed other tire tracks nearby. After wiping down the steering wheel, gearshift, door and window handles, he stepped up on the hood and went over the fence.
Easy. Too easy. Semelee's clan could do the same with their pickup.
Semelee...As he walked back to his father's house he ran the Semelee situation back and forth and sideways through his head, looking for a solution.
He agreed with Weldon on one point: Semelee seemed to be able to control the swamp creatures. How, Jack didn't know, but he'd bet it had something to do with the nexus point at the lagoon. She'd used that power to commit perfect murders—"sacrifices," as she'd put it to Weldon—in plain view without anyone suspecting that a human agent lay behind the attacks. No question in Jack's mind that she was behind the palmetto swarm and the alligator attack as well.
She had to be stopped, that much was clear. He had no idea how, but he'd worry about that later. The first thing he had to do was put Carl back in his trailer...his home.
## 10
"There you are," Dad said as Jack stepped through the door. He'd obviously awakened from his nap. Looked like he'd showered and shaved too. "Where have you been?"
"Here and there. Did anyone call or come by while I was out?"
He shook his head. "No. All quiet. You're expecting someone?"
Jack hid his frustration. "Yeah. Sort of."
"Well, I need to do some grocery shopping. How about driving me down to the Publix so I can stock up?"
"How about I give you the keys and stay here? In case that call comes, or someone shows up."
"Are you in some sort of trouble, Jack? Because if you are, maybe I can help."
Jack laughed and hoped it didn't sound as forced as it felt. "Trouble? No, not me. But someone I know might be in a little."
"What kind?"
Jack knew he'd been acting strange—at least in his father's eyes—but he wasn't used to all these questions, or having his comings and goings noted and commented on.
This is why I live alone.
"You might say it's a kind of family thing."
"Do those toys have anything to do with it?"
"It might come down to that."
Dad sighed and dropped into his recliner. "You are the hardest person to talk to, Jack. You were a great kid, but now you're a stranger. It's like you don't want to know me or me to know you. You've got this wall around you. Is that my fault? Did I do something...?"
This was painful. Jack could see the hurt in his father's troubled eyes.
"Absolutely not. It's me. It's just the way I am."
"But it's not the way you were."
Jack shrugged. "People change. You must know that."
"No. I don't. Most people don't change. Kate didn't change. And Tom didn't—although it might not be such a bad thing if he had. But you—you're a completely different person."
Jack could only shrug again. He wanted off this uncomfortable topic.
"Enough about me. How about you, Dad? How are you getting on down here?"
His father gave him a long, baffled stare, then shook his head.
"Me? I guess I'm doing pretty well. I like the climate enough, but..."
"But?"
"I don't know. Sometimes I think I made a mistake moving down here. Sometimes I wonder why I ever left Jersey."
"I'd wondered the same thing. So did Kate."
"I've never been the impulsive sort, but this was an impulse. A Gateways South brochure came in the mail one day and that was it. I took one look and had to be here. The graduated care aspect and the idea of never being a burden appealed to me...appealed to me so much it became an obsession that took hold and wouldn't let go. I couldn't get it out of my head that this was the place for me. I sold the old house and reinvested some of the money in this place and..." He spread his hands. "Here I am."
"From what Anya told me while you were in your coma, it sounds as if you've gotten into the swing of things down here."
"I have. I've had to. I had it in my head that Kate and Tom would jump at the chance to gather up the grandkids and come down to Florida to visit. But only Kate did that. And only once. Everyone's so busy these days. So I made a choice: I can sit before the TV and ossify, or get up and about and do things while I still can. I figure I'd rather be a moving target than a stationary one."
Target, Jack thought. Helluva word choice, Dad. If you only knew...
Dad was shaking his head. "But as nice as it is, I still can't believe I sold the family home and left my kids and grandkids up north to move down here. I know not being a burden was a big part of it, but really...what was I thinking?"
Something in the words sent a chill through Jack. His father had done something he didn't quite understand...developed a compulsion to move down here, to this particular development, right outside the Everglades, close to the lagoon where Semelee and her clan lived...
...close to a nexus point.
Hadn't Carl told him that he'd developed a yearning—an "ache," as he put it—to get back to the place where he'd been born, back to the lagoon...?
Back to that same nexus point.
Coincidence?
He'd been told there'd be no more coincidences in his life.
Was someone or some thing moving pieces around the board—Jack's board?
But wait...Anya had said she'd done part time work addressing brochures. Had she sent one to his father? Had she influenced him to come down here? So she could—what?—protect him?
Jack's head spun. One thing he knew was he wanted his father out of here, out of Gateways, out of the whole damn state.
"Nothing says you can't go back. In fact, I think you should. I'm sure Jersey's got a load of graduated care places, if that's what you want."
Dad stood silent a moment, then, "I don't know. I'd feel like an old fool."
"Which is more foolish: admitting you made a mistake and rectifying it, or hanging around a place you don't like?"
"When you put it that way..." He shook his head. "I'll have to think about it." He clapped his hands. "But no matter what I decide, we have to eat tonight. I'll run out and get eggs and cheese and some ham. I make a mean omelet. How's that sound for dinner?"
"Perfect."
With a pang of reluctance, Jack gave him the keys to his rental. He had an urge to go with him, to not let him out on his own unprotected, but Semelee had said he wasn't a target, and he believed her. She'd had Jack at her mercy—outnumbered and outgunned—when she'd said it, so she'd had no reason to lie.
## 11
As soon as he was alone, Jack pulled out the toys. He inspected them for repaired seams, found one on each, and slit it open. He removed the sundry weapons Abe had sent him and, armed with a screwdriver and an adjustable wrench, hid them around the house.
Then he called Gia. She and Vicky and the baby were doing fine.
"When are you coming home, Jack?" Vicky asked. "I miss you."
"I miss you too, Vicks, and I'll be home as soon as I can. As soon as I know my dad's okay."
He seemed okay now, but it would take a little doing to make sure he stayed that way.
Still no word from the clan. Jack stepped outside and looked around. The sun lay low over the Everglades, brushing the fringe of the far-off hardwood hummock. He wondered if that was the same hummock that housed the lagoon and his nexus point. If so, he might see these mysterious lights tonight.
"I've got your damn shell!" he shouted into the fading light. "Let's do this!"
Then he waited, not really expecting anything, but hoping. After a moment of listening to frogs and crickets, he turned to go back inside. He noticed a light on at Anya's. Maybe she'd like to come over for dinner.
His knocks went unanswered, even by Oyv, so Jack stepped around to the side window. There he saw her and Oyv sleeping in front of the TV, in the same positions they'd been in Wednesday night. Again, they looked dead. But he kept watching until he caught Anya taking a breath.
He was halfway back to the house when he saw his rental car pull into the parking area. He angled that way and arrived in time to carry a couple of the grocery sacks.
"I picked up some scallions," Dad said as they were unpacking. "I figured that would add a little extra flavor."
"You've become a regular Chef Boyardee."
"Had to learn some cooking. When you live alone, you can get awful tired of frozen dinners and fast food. And it gives me something to do at night." He looked at Jack. "Nights are always the hardest."
Jack wasn't sure what to say. He wanted to tell him he was sorry about that but sensed his father wasn't looking for pity. He'd merely been stating a fact.
So Jack ducked it. "Hey, want me to slice those scallions?"
"Sure," Dad said with a grin. "Think you can slice them nice and fine?"
He washed them off, then handed Jack a slim knife and a cutting board. Jack positioned himself on the other side of the counter and began slicing.
"Hey," Dad said. "You're pretty handy with that blade."
"I'm a super sous chef." He'd picked up a lot from helping Gia cook.
"While you're doing that, I'll open this bottle of Chardonnay I've had in the fridge. Been saving it for a special occasion."
"Omelets are a special occasion?"
"Company is a special occasion, especially when it's one of my sons."
Jack realized then with a pang how lonely his father was.
"Can I ask you something, Dad?"
"Sure." He'd pulled a pale bottle from the refrigerator and was twisting a corkscrew into its top. "Go ahead."
"Why didn't you ever remarry?"
"Good question. Kate always asked me that, always encouraged me to get into a new relationship. But..." He grabbed two glasses and half filled them. "There's more where this came from, by the way."
Jack got the feeling he was trying to stall, or maybe even evade an answer. He wasn't going to let that happen.
"You were saying about not remarrying?"
He sighed. "Having your mother taken away like that—one moment she's sitting next to me in the car, next moment there's blood all over her and no one can save her. She's...gone. You were there. You knew what it was like."
Jack nodded. His knife picked up speed, slicing the scallions faster, harder, thinner.
Dad shook his head. "I never got over it. Your mother was special, Jack. We were a team. We did everything together. The bond was more than love, it was..." He shook his head. "I don't know how to describe it. 'Soul mate' is such a hackneyed term, but that pretty well describes what she was to me."
He pulled a carving knife from a drawer and started dicing the thick slice of cured ham he'd bought.
"And let me tell you, Jack, the grief over losing someone that close to you, it doesn't just go away, you know. At least it didn't for me. Something like that happens and people pepper you with all sorts of platitudes—it got to the point where I wanted to punch out the next person who said, 'She's in a better place.' I almost committed murder on that one. Then there was, 'At least you had her for a little while.' I didn't want her for a little while. I wanted her forever."
Jack was moved by the depth of his feeling. This was a side his father kept hidden.
"If I can use an equally hackneyed phrase: She wouldn't have wanted you to spend the rest of your life alone."
"I haven't been completely alone. I've allowed myself short-term relationships, and I've taken comfort in them. But a long-term relationship...that would be like telling your mother she can be replaced. And she can't."
Heavy going here. Jack tossed off the rest of his wine and poured them both some more, all the while trying to think of an adequate response.
His Dad saved him by pointing the carving knife at Jack's chest.
"Your mother," he said. "That's it, isn't it. I've always suspected that it made you a little crazy, but now I want to hear it from you. I remember you at the wake and the funeral. Like a zombie, hardly speaking to anyone. You were never a momma's boy. Far from it. You were closest to Kate. But to see your mother killed by violence, to have her bleeding and dying in your arms...there's no shame in having a breakdown after what happened. No one should have to go through that. No one."
Jack gulped more of his wine. He could feel it hitting him. He'd had nothing to eat since breakfast and the alcohol was jumping directly into his bloodstream. So what? And why not?
"I agree that no one should have to go through that. But it wasn't Mom's death that put me on the road."
"What then? It's driven me crazy for the past fifteen years. What made you disappear?"
"Not her death. Another death."
"Whose?"
"I was pissed at everyone back then for not finding the guy who'd dropped that cinder block. The state cops were going on about keeping an eye on the overpasses, but it takes a lot of effort to track down someone who commits a random act of violence. And they had better things to do—like ticketing speeders on the Turnpike. God forbid we drive above the limit. And you, you weren't doing anything but talking about what should happen to the murdering bastard when they caught him. Only it wasn't a 'when,' it was an 'if'—an 'if' that was never going to happen."
Jack finished the glass and poured himself some more, killing the bottle.
Dad looked up from the ham. "What the hell was I supposed to do?"
"Something. Anything."
"Like what? Go out and track him down myself?"
"Why not?" Jack said. "I did."
Oh, shit, he thought. Did I just say that?
"You what?"
Jack raced through his options here. Say never mind and stonewall it? Or go ahead and tell all. Abe was the only other person on earth who knew.
But now the wine and a cranky, don't-give-a-shit mood pushed him to let it roll. He sucked in a deep breath.
Here goes.
"I tracked him down and took care of him."
Jack thought he saw Dad's hand tremble as he put down the carving knife. His expression was tight, his eyes bright and wide behind his glasses.
"Just how...I'm not sure I want to hear this but...just how did you take care of him?"
"I saw to it that he never did anything like that again."
Dad closed his eyes. "Tell me you broke his arms, or smashed his elbows."
Jack said nothing.
Dad opened his eyes and stared at him. His voice dropped to a whisper. "Jack...Jack, you didn't..."
Jack nodded.
Dad sidled left to one of the counter stools and slumped on it. He cradled his head in his hands, staring down at the pile of sliced scallions.
"Oh, my God." His voice was a moan. "Oh, my God."
Here it comes, Jack thought. The shock, the outrage, the revulsion, the moral repugnance. He wished now he could take it back, but he couldn't, so...
He walked around the counter, past his father's bent back, opened the refrigerator, and took out another bottle of wine.
"How did you know it was him?" Dad said. "I mean, how could you be sure?"
Without bothering to remove the black lead foil, Jack wound the screw through it and into the cork.
"He told me. Name was Ed, and he bragged about it."
"Ed...so, the shit had a name."
Jack blinked. Other than hell and damn, his father had always been scrupulous about four-letter words. At least when Jack was a kid.
He lifted his head but didn't look at Jack. "How?" He licked his lips. "How did you do it?"
"Tied him up and dangled him by his feet off the same overpass. Made him a human piñata for the big trucks going by below."
The cork popped from the bottle as Jack remembered seeing Ed swinging over the road, the meaty thunk! as the first truck hit him, then the second.
Music. Heavy metal.
Dad was finally looking at him. "That's why you left, isn't it. Because you'd committed murder. You should have stayed, Jack. You should have come to me. I would have helped you. You didn't have to spend all those years dealing with that guilt alone."
"Guilt?" Jack said, pouring more wine for both of them. "No guilt. What did I have to feel guilty about? No guilt, no remorse. Send me back in time to relive that night and I'd do the same thing."
"Then why on earth did you just take off like that?"
Jack shrugged. "You want an eloquent, thoughtful, soul-searching answer? I don't have one. It seemed to make sense at the time. From that moment on the world looked different, seemed like another place, and I didn't belong. Plus I was disgusted with just about everything. I wanted out. So I got out. End of story."
"And this creep, this Ed...why didn't you call the police?"
"That's not the way I work."
Dad squinted at him. "Work? What does that mean?"
Jack didn't want to go there.
"Because they'd have carted him off and then let him out on bail, and then let him plead down to a malicious mischief charge."
"You're exaggerating. He'd have done hard time."
"Hard time wouldn't cut it. He needed killing."
"So you killed him."
Jack nodded and sipped his wine.
Dad started waving his arms. "Jack, do you have any idea what could have happened to you? The chance you took? What if somebody saw you? What if you'd been caught?"
Jack opened his mouth to reply, but something in his father's words and tone stopped him. He was going on about...he seemed more concerned about the possible consequences of the killing rather than the killing itself. Where was the outrage, the middle-class repugnance for deliberate murder?
"Dad? Tell me you wish I hadn't killed him."
His father pressed a hand over his eyes. Jack saw his lips tremble and thought he was going to sob.
Jack put a hand on his shoulder. "I never should have told you."
Dad looked at him with wet eyes. "Never? I wish you'd told me back then! I've spent the last fifteen years thinking he was still out there, unnamed, unknown, some kind of wraith I'd never get my hands on. You don't know how many nights I've lain awake and imagined my hands around his throat, squeezing the life out of him."
Jack couldn't hide his shock. "I thought you'd be horrified if you knew what I'd done."
"No, Jack. The real horror was losing you all those years. Even if you'd been caught, you could have pled temporary insanity or something like that and got off with a short sentence. At least then I'd have known where you were and could have visited you."
"Better for you, maybe."
A jolt in the joint, even a short one...unthinkable.
"I'm sorry. I'm not thinking straight."
Jack still couldn't believe it. "I killed a man and you're okay with that?"
"With killing that man, yes, I'm okay. I'm more than okay, I'm—" He threw his arms around Jack. "I'm proud of you."
Whoa.
Jack wasn't into hugs, but he did manage to give his dad a squeeze, all the while thinking, Proud? Proud? Christ, how could I have read him so wrong?
Once again Anya's words from that first day came back to him.
Trust me, kiddo, there's more to your father than you ever dreamed.
They broke the clinch and backed off a couple of feet.
Jack said, "If I'd have known you felt that way, I might have asked you for help. I could have used some. And you would have been doing something instead of waiting for the police to do it for you."
Dad looked offended. "How do you know I wasn't doing something? How do you know I didn't take a rifle and sit in the bushes, watching that overpass, waiting to see if someone would try again."
Jack managed to suppress a laugh but not a smile. "Dad, you don't own a rifle. Not even a pistol."
"Maybe not now, but I could have back then."
"Yeah, right."
They stood facing each other, his father staring at him as if seeing a new person. Finally he thrust out his hand. Jack shook it.
Dad looked around and said, "I don't know about you, but I'm starving. Let's get going on these omelets."
"You start the eggs," Jack said, "and I'll finish dicing the ham."
A good night. A surprising, shocking, revelatory night. Like nothing he could have anticipated.
He might have enjoyed it even more if he'd managed to bring Carl home. He wondered how the poor guy was doing.
## 12
Carl looked up at the starry sky, at the misshapen shadows of the surroundin trees, at the water in the lagoon, anywhere but at the lights. Leastways he tried not to look. But as much as he wanted to stop it, his gaze kept driftin back to the sinkhole...and the lights.
They'd set him here on the ground, his back against one of the Indian hut support posts. They'd been ready to tie his hands behind him when they remembered that he only had one, so they lashed him to the post with coils of thick rope around his arms and body.
He'd overheard Semelee mention that Jack had found her shell but how's it would have to wait till tomorrow. Tonight was too important.
The air was warm and wet and thick enough to choke a frog—maybe that was why they weren't peepin. Even the crickets had shut up. The lagoon and its surroundins was quiet as a grave.
The lights had started flashin a little after dark, strange colors and mixes of colors he never seen nowheres else. That was when it really got crowded around the hole. But there'd been lots goin on before that. Luke and Corley and Udall and Erik had been settin up some sort of steel tripod over the mouth. It had a pulley danglin from the top center where the three legs came together. They threaded a good, long length of half-inch rope through it, then tied that to some sorta chair.
He kept telling himself, Naw, she ain't really gonna do that. She ain't that crazy.
But come full dark, when the crazy flashin colors was lightin up the trees and the water, sure enough, Semelee put herself into the chair. She was danglin over the hole, with the lights reflectin even stranger colors off that silver hair of hers, and then Luke and a couple other guys Carl couldn't recognize cause their pan-o-ramic backs was to him started lowerin her down into the hole.
After she disappeared he could hear her voice echoin up from below.
"What're you stoppin for? Keep me goin!"
Luke called out, "You're deeper'n you should be already. How much to go till you hit the water?"
"Can't see no water. Looks like it all dried up."
"Then where's the bottom?"
"Can't see no bottom, just the lights."
"That's it," Luke said. "I'm haulin you up now."
"Luke, you do that and I ain't never gonna speak to you again! You hear that? Never! It's like nothin I could ever dream down here. The lights...so bright...all around me...feels like they're goin through me. This is so cool. You keep on lettin out that rope. I want to see where they come from."
Carl wasn't sure of a whole lotta things in life, but he was damn sure that was a real bad idea. He was glad he was back here, away from the lights. He would've liked to be even farther, like in his trailer watchin TV. He was missin all his Friday night shows. But he couldn't worry about that now. He had to get outta here.
He'd been usin his hand, workin at the knot behind his back, but this was one good knot. When you lived out here in the wilds, specially on the water, you learned how to tie a good knot. But that didn't keep him from tryin to loosen it up.
"Keep a-goin!" he heard Semelee call up from the hole, her voice faint and all echoey like.
Luke shouted, "We're almost outta rope!"
"Take me down to the end! As much as you got!"
Good, Carl thought. They's all concentrated on her.
If he could just get this knot loose, he could sneak down to the water and steal a canoe and slip away real quiet like. He could be long gone before anyone noticed. Then he'd—
He jumped at the sound of a scream, a long tortured sound like someone havin their skin tore off—not just a piece, but the whole thing.
Everbody around the hole started shoutin and callin and movin this way and that. Four-five guys was haulin on that rope as fast as they could. Finally they got to the end. Carl caught a peek between the shufflin bodies and saw Semelee still in the chair. But she was all slumped over like a piece of fish bait and not movin a muscle.
She looked dead.
## SATURDAY
## 1
Semelee heard herself scream and woke up all sweaty and thrashin.
Where am I?
"Semelee! Semelee, are you okay?"
Luke's voice...and then his face appeared, hovering over her.
She sat up, recognized her corner of the Bull-ship, then flopped back.
"Here," Luke told her. "Drink this."
He tipped a bottle over her mouth and she gulped. Water. Lord, that tasted good.
She looked around again. "How'd I get here? I don't remember going to bed. I—"
"You was down in the lights," he said.
The lights! Of course.
She remembered now. She'd been down in the hole, baskin in them strange weird lights like a sun worshipper. But she hadn't felt strange. She'd felt welcome, more welcome than she'd ever felt in her own home. She remembered wantin to tear off her clothes so the rays could go straight to her skin. But she didn't get the chance...
Because that was when the voices began.
Whispers at first, so soft she could barely make them out. Not sounds, really. More like voices in her head, like she was a mental case or somethin. She wasn't even sure they was talkin to her. Maybe they was jawin at each other and their words was passin through her head, but she had a feelin they was talkin to her. She wanted them to be talkin to her.
"What happened to you down there?" Luke said. "You screamed like I ain't never heard nobody scream, and when we pulled you up you was out cold. I thought you was a goner."
Out cold...she jammed her hands against her temples. Damn, she wished she could remember what had happened, and remember more of what them voices had said. She did know she kept hearin about 'the One.' All sorts of yammerin about the One, repeatin it over and over again. The One what?
Suddenly she realized they was talkin about a person. The One was preparing the way, everything depended on the One because the One was special.
Wait, she thought, stiffening as a thrill ran through her. I'm special. I got a power like no one else. And then there's my name...
She levered up to a sittin position and crossed her legs, Indian style. "Yes!"
"What is it?"
"Luke, do you know what my name means?"
"Y'mean Semelee? It means...it means 'Semelee.' Just like Luke means 'Luke.'"
"All names mean somethin. I ain't got no idea what Luke means, but my momma told me that Semelee means 'one and only.' She said she named me that because I was her first and I was a real hard birth, and she wasn't goin through that again. She said I was her first and last kid, her one and only."
Luke frowned. "Okay. So?"
"I heard voices down in that hole and they was talkin about 'the One.' That has to be me. They was talking about me." She closed her eyes. Excitement flashed like lectric shocks through her body. "And they kept on sayin somethin else too."
What was it? It was right there, just out of reach...started with an R...but what was the rest?
And then she had it! The name popped into her head like she'd known it all along.
A strange name. She'd never heard nothin like it before. But then she'd never heard nothin like those voices before neither. Was that strange word their name for her, their name for the One? Had to be.
But who were the voices and what did they mean about "preparing the way"? What was the "everything" that depended on her, the One?
She had to find out. Maybe she'd learn tonight. But she had to do a couple of things before then. One of them was gettin her other eye-shell back. But first...
"I'm changin my name, Luke."
He laughed. "That's crazy! You can't just change your name anytime you feel like it."
"No. I got to. That's why I was called back here. I thought the lagoon was talking to me when it said it wanted sacrifices, but it wasn't. It was the lights—or at least the things that live in the lights."
"Lay back down, Semelee. You're talkin outta your head."
"No." She pushed him away. "Don't you see? It was all to bring me here, to this place, at this time—to teach me my True Name. And now that I know it, I'm gonna use it." She rose to her feet and looked out at the lights still flickering up from the hole into the early morning darkness. "Big changes comin, Luke, and I'm gonna be part of them, I'm gonna be right at their heart. And if you and the rest of the clan stick by me, we'll have our day. Oh, yes, Luke, we'll have our day."
"Semelee—"
"Told you: I ain't Semelee no more. From this moment on you call me—"
The name died on her lips. She realized that she mustn't tell no one her True Name. It was only for her and those closest to her. Luke was close, but not close enough. The man called Jack, the special one...she could tell him maybe, but not right away. He'd have to prove himself worthy first.
"Call you what?" Luke said.
"Semelee."
Luke stared at her. "Wasn't you just tellin me—?"
"Changed my mind. I'm goin to change my name inside, but outside you can keep callin me Semelee." She rubbed her stomach. "We got anything to eat round here?"
Luke straightened. "I'll go check by the fire."
As soon as he was gone, Semelee stepped out onto the deck and looked up at the stars wheelin above her.
"Rasalom," she whispered, lovin the way it rolled off her tongue. That was her new name. "Rasalom."
## 2
The man who was something more than a man opened his eyes in the darkness.
His name...someone had spoken his name. Not one of the many he used in the varied identities he assumed for various purposes. No, this had been his True Name.
He'd been reveling in the continued corporal mutilation of a teenage girl named Suzanne and the spiritual ruination of the family that tortured her.
Poor Suzanne had been chained to the other side of the wall of this Connecticut home for eleven days now. She had been raped and defiled and tortured and mutilated beyond the point of her endurance. Her mind had snapped. She had no more to give. She was dying. Her brain had shut down all but the most basic functions. She barely felt the corkscrew being wound into the flesh of her thigh.
But what was so delicious here was the nature of the one twisting the corkscrew: an eight-year-old boy. For it was not simply the pains of the tortured that nourished this man who was something more than a man; the depravity and self-degradation of the torturers were equally delicious.
He'd returned to this house to bask in the dying embers of a young life's untimely end.
But now that was ruined, the delicious glow fading, cooled by a growing anger and—he admitted it—concern.
Someone had spoken his True Name.
But who? Only two beings in this sphere knew that name: one was listening for it, and the other dared not speak it. They—
There! There it was again!
Why? Was someone calling him? No. This time he sensed that the speaker was not merely saying his True Name, but trying to usurp it.
Rage bloomed in his brain like a blood-red rose. This was intolerable!
Where was it coming from? He rose to his feet and turned in a slow circle—once, twice—then stopped. The source of the outrage...it came from there...to the south. He would find the misbegotten pretender there.
All his plans were progressing smoothly now. After all these centuries, millennia, epochs, he was close, closer than he'd ever been. Less than two years from now—barring interference from those who knew he was the One—his hour, his moment, his time would be at hand.
But now this. Someone usurping his True Name...
Never!
The man who was something more than a man strode away from the house through the dissipating darkness. He had no time to waste. He must head south immediately, trace his True Name to the lips that were speaking it, and silence them.
He paused at the curb. But what if that was just what someone wanted him to do?
This could be a trap, set by the one man he feared in this sphere, the only man he must hide from until the Time of Change.
Back in the days of his first life, when he was closer to the source, he had enormous power; he could move clouds, call down lightning. Even in his second life he could control disease, make the dead walk. But here in this third life his powers were attenuated. Yet he wasn't helpless. Oh, no. Far from that. And he could not allow anyone to use his True Name.
He must proceed with caution. But he must proceed. This could not go on.
## 3
Jack stepped into the front room and found his father fiddling with the French press.
"Don't bother, Dad," Jack told him. "I'll pick up some coffee and donuts in town."
He'd seen a Dunkin' Donuts the other day and had awakened with a yen for some of their glazed crullers.
"Donuts? That sounds good. But I don't mind making coffee. After all, the job has its perks."
Jack groaned. "What kind do you like?"
"A couple of chocolate glazed would be great."
Jack headed outside, trying to concentrate on donuts in the hope that would help take his mind off Carl and how he was going to bring him back. The air seemed less humid. Felt like a cool front had come through.
About time. The relentless heat day after day had been wearing him out. Maybe this was Elvis's doing. If so, thank you, Big E.
A mist lay over the saw grass sea stretching away to the distant hummock. The egret was back in the pond, black legs shin deep in the water by the edge, waiting like a snowy statue for breakfast to move and give itself away.
He headed around the side of his house toward the car. He stopped when he rounded the corner. A woman was seated on the hood of his car. She wore cutoffs and a green tank top. Her white hair had been wound into a single braid. The companion to the shell Jack had found hung at her throat.
Semelee.
"About time you showed up," he said, moving toward her, wary, eyes scanning the surroundings. Had she come alone? "I've been standing out here like some kind of nut announcing to the air that I've found your shell. I thought you said you'd know."
She smiled. "I did know. That's why I'm here."
Jack couldn't pin it down but she looked different. Her hair was just as white as ever, but her eyes held a strange look, as if she'd peeked through someone's window and seen something she wasn't supposed to know.
That was it. She looked like she'd discovered some sort of secret no one else knew. Or thought she had.
"Took you long enough."
Her smile remained. "I had other things to do."
Jack tensed. "Like what? You better not have hurt Carl."
"Carl's fine." She held out her hand, palm up. "My shell, please."
Now it was Jack's turn to smile. "You're kidding, right?"
"No. You give me the shell and I'll send Carl back."
"Not likely."
The smile vanished. "You don't trust me?"
"Tell you what: You send Carl back, and I'll give you the shell."
"No way."
"What? You don't trust me?"
Semelee glared at him. "The One don't lie."
Jack stiffened. The One? She'd just mentioned the One.
"What did you say?"
"Nothin."
"You called yourself the One. What did you mean by that?"
"Told you: nothing. Now leave it be."
Anya had talked about the One, but she'd indicated that Sal Roma was the One. Was he involved in what was going down here?
"Do you know a guy named Roma?"
She shook her head. "Ain't never heard of him."
"Is he the one who got you started on this sacrifice-to-the-swamp kick?"
Semelee's eyes widened. She slid off the hood and stepped toward him. "How do you know about that?"
"Not important. Just tell me: Was it Roma?"
"Told you: Don't know no Roma."
Jack believed her. "Then who? Who gave you such a crazy idea?"
"Wasn't no 'who.' It came from the lagoon its own self. If you listen, the lagoon'll talk to you. Leastways, it talks to me. Told me in a dream that it was pissed off and that Gateways had to pay. Said it would exact a price of four Gateways lives a year and—"
"Wait-wait. That's what it said? 'Exact'?"
That didn't sound like it belonged in Semelee's vocabulary—at least not as a verb.
"Yeah. 'Exact.' Pretty weird kind of talk, doncha think?"
Jack wondered if it had been a dream at all. It sounded as if someone or something had been influencing her, and he doubted very much it was her lagoon. Much more likely it was an influence from that nexus point within the cenote.
He said, "You ever hear of something called the Otherness?"
"Don't reckon I have," she said, shaking her head. "Should I?"
"Never mind." Just because she hadn't heard of the Otherness didn't mean she wasn't working for it, knowingly or unknowingly. "But why Gateways people? There must be other folks living even closer to your lagoon."
"There is, but the lagoon wants Gateways folks. Don't ask me why, it just does."
Jack jerked a thumb over his shoulder. "There's one Gateways folk in there it's not going to get. We clear on that?"
She nodded. "Absolutely. The lagoon's already done what it set out to do with the sacrifices. There's still maybe a score to settle, but the sacrifice thing is over."
"What score?"
"That's between me and the lagoon, but don't you worry. Your daddy ain't a part of it."
Jack believed her this time, and found relief in the fact that his father was no longer in the clan's crosshairs. But that was tempered by the knowledge that he'd been replaced by someone else.
"He'd better not be. And I'd better see Carl pretty soon or I might just lose that shell. Or it might slip out of my pocket as I'm crossing a street downtown. Wouldn't take long for the traffic to reduce it to powder."
Semelee went pale beneath her tan. "Don't even joke about that."
"What's so important about that shell?"
Her hand went to the one around her neck. "I've had em since I was a kid, is all. I just want it back."
"And I want Carl back."
She sighed. "Looks like we'll have to put together a swap meet. Bring the shell to the lagoon and—"
Jack shook his head. "Uh-uh. Bring Carl here."
Jack watched Semelee's hands open wide, then close into tight fists.
"You're makin this awful hard." She looked up at the hazy sky, then back to him. "Guess we'll have to meet somewheres in the middle. You got any ideas?"
Jack reviewed his trip with Carl and remembered the dry stretch where they'd had to carry their canoe. He mentioned it to Semelee and she knew where it was.
"Okay," she said. "We'll meet there in an hour."
Jack looked out at the Everglades and the clinging haze. Semelee seemed on the level but he didn't know about the rest of the clan. And because of that, he wanted maximum visibility.
"What say we make it noonish?" he said.
"Why're you makin me wait so long?"
"I need the time."
"All right. See you then. And don't be late."
She turned and walked off. Jack watched the sway of her hips as she moved away. He missed Gia.
He was still watching her, wondering how she was going to get out of Gateways, when his father's voice interrupted him.
"I hope you're not really thinking of going through with this."
Jack turned to find Dad standing on the porch, staring at him through the jalousies.
"You heard the whole thing?"
"Just the end. Enough to know that she's connected to what happened to me, and probably to the others who've been killed. But what was that about Carl? Carl the gardener?"
"One and the same."
Jack gave him a quick overview of what had happened—about the trip to the lagoon, and Semelee and her clan.
Dad was shaking his head. "You've only just got here, Jack. How did you manage to get involved in something like this in just a couple of days?"
"Lucky, I guess."
"I'm serious, Jack. You've got to take this to the police and the Park Service."
"That's not the way I do things."
"What's that supposed to mean? This is the second time you've said something like that."
"It's plain and simple, Dad: I promised Carl I'd get him back safely. Me. Not the cops, not the park rangers. Me. So that's how it's going down."
"But you didn't know the odds against you when you made that promise. He can't hold you to it."
"He's not," Jack said. He shook his head. "You wouldn't understand."
Dad rubbed his jaw. "I understand perfectly. And you know, Jack...the better I know you, the more I like you. Carl's not holding you to your promise...you are. I can respect that. It's damn foolish, but I have to respect that."
"Thanks."
How about that? Dad did understand.
"But you can't go out there alone. You're going to need backup."
"Tell me about it. Know where I can find any?"
"You're looking at him."
Jack laughed. Dad didn't.
"I'm not kidding, Jack."
"Dad, you're not cut out for that."
"Don't be so sure." He pushed open the porch door. "Come inside. I need to tell you some things you don't know."
"About what?"
No matter what he was told, Jack wasn't taking an accountant in his seventies as backup, especially if that accountant in his seventies was his father.
"About me."
## 4
Inside, Dad handed him a cup of coffee, then, before Jack could ask him what this was about, disappeared into his bedroom. He returned a minute later carrying the gray metal lockbox Jack had found back on Tuesday. He hadn't expected to see it again, but he was more surprised by what his father was wearing.
"Dad, are you kidding with that sweater?"
His father pulled the front of the ancient brown mohair cardigan closer about him. "It's cold! The thermometer outside my window says sixty-nine degrees."
Jack had to laugh. "The Sasquatch look. It's you, Dad."
"Never mind the sweater." He set the box on the coffee table. "Have a seat."
Jack sat across from him. "What've you got there?" he said, already knowing the answer.
Dad unlocked the box and flipped it open. He pulled out an old photo and passed it to Jack: Dad and six other young guys in fatigues.
Jack pretended to study it, as if seeing it for the first time.
"Hey. From your Army days."
"Army?" His father made a face. "Those clods? These are Marines, Jack. Semper fi and all that."
Jack shrugged. "Army, Marines, what's the diff?"
"You wouldn't say that if you'd ever been in the Corps."
"Hey, you were all fighting the same enemy, weren't you?"
"Yeah, but we fought them better." He tapped the photo. "These were my wartime buddies." His expression softened. "And I'm the only one left."
Jack looked at those young faces. He pointed to the photo. "What are you all smiling about?"
"We'd just graduated Corps-level scout-sniper school."
Jack looked up from the photo. "You were a sniper?" He'd learned to believe in the unbelievable, but this was asking too much. "My father was a sniper?"
"Don't say it like it's a dirty word."
"I didn't. I'm just...shocked."
"Lots of people look on sniping with disdain, even in the military. And after that pair of psychos killed all those folks in the DC area a while back, so does just about everybody else. But those two weren't sniping. They were committing random murder, and that's not what sniping is about. A sniper doesn't go out and shoot anything that moves, he goes after specific targets, strategic targets."
"And you did that in Korea."
Dad nodded slowly. "I killed a lot of men over there, Jack. I'm sure there's plenty of soldiers walking around today who've killed more of the enemy—Germans, Japs, North Koreans, Chinese, Vietnamese—in their tours of duty than I did, but they were just shooting at the faceless foreign bodies who were trying to kill them. We snipers were different. We positioned ourselves in hiding and took out key personnel. We could have a hundred, a thousand soldiers milling around just five hundred yards away, but we weren't interested in the grunts. We were after the officers, the NCOs, the radio men, anyone whose death would diminish the enemy's ability to mount or sustain an attack."
Jack was watching his dad's face. "Sounds almost...personal."
"It does. And that's what makes people uncomfortable. They feel there's something cold-blooded about picking out a specific individual in, say, a bivouac area, sighting down on him, and pulling the trigger." He sighed. "And maybe they're right."
"But if it saves lives..."
"Still pretty cold-blooded, though, don't you think. When I started out, if I couldn't nail an officer or NCO, I'd go after radio men and howitzer crews. But I noticed that whenever I took a guy out, another would pick up the radio or jump in and start reloading the howitzer, and then I'd have to take them out as well."
Jack started nodding. "So you began going after their equipment."
"Exactly. Know what a .30 caliber hardball will do to a radio? Or to the sights on a howitzer?"
"I can imagine." Jack had a very good idea of the damage it could do. "Good for the junk pile and nothing else. You guys were using M1s back then, right?"
"Not us snipers. I was trained on the M1903A1 with an eight-power Unertl scope, and that's what I used. Made a couple of thousand-yard kills with that."
A thousand yards...three thousand feet...killing someone more than half a mile away. Jack couldn't imagine that. He tried to keep guns out of his fix-its whenever possible, but when the need arose he had no qualms about using them. Usually it was up close and personal, and never more than twenty-five feet.
A thousand yards...
"What kind of round were you shooting?"
"I got hold of a cache of Match M72s and I hoarded them."
Jack wasn't familiar with the round. "How many grains?"
Dad's eyes narrowed. "You shoot?"
Jack shrugged. "A little. Mostly range stuff."
"Mostly?"
"Mostly." He didn't want to get into that. "Grains?"
"One-seventy-five point five."
Jack whistled.
"Yeah," Dad said, nodding. "Penetrated eleven inches of oak. Nice little accuracy radius. I loved that round."
"Don't think I'm morbid, but...how many did you kill?"
Dad closed his eyes and shook his head. "I don't know. I stopped counting at fifty."
Fifty-plus kills...jeez.
"I thought I was hot stuff," Dad said, "really making a difference in the fighting, so I kept count at first. But by the time I reached fifty or so it stopped mattering. I just wanted to go home."
"How long were you there?"
"Not terribly long—most of the latter half of 1950. I was shipped into Pusan in August and what a major screw-up that was, mainly because the Army units didn't do their job. Mid September I was shipped to Inchon where I landed with the Fifth Regiment. By the end of the month we'd fought through to Seoul, recaptured it, and handed it back to the South Koreans. We thought that was it. We'd freed up the country, kicked those NK commies back above the thirty-eighth parallel. Job done, time to go home. But no."
Dad drew out that last word in a way that reminded Jack of John Belushi. He rubbed a hand across his face to hide a smile.
"No, MacArthur had the bright idea of pushing into North Korea so we could reunite the country. And there we found ourselves facing the Red Chinese. What a bunch of crazies they were. No respect for life, their own or anyone else's, just hurling themselves at us in human waves."
"Maybe what was facing them at the rear if they didn't do as ordered was worse than charging you guys."
"Maybe," Dad said softly. "Maybe." He seemed to shiver inside his cardigan. "If there's a colder place on Earth than the mountains of North Korea, I don't want to know about it. It was chilly in October, but when November rolled around...temperatures in the days would be in the thirties but at night it would drop to minus-ten with a howling thirty-to forty-mile-an-hour wind. You couldn't get warm. So damn cold the grease that lubricated your gun would freeze up and you couldn't shoot. Fingers and toes and noses were falling off left and right from frostbite." He looked up at Jack. "Maybe that's the deep psychological reason I moved down here: so I'd never be cold again."
Christ, it sounded like a nightmare. Jack could see this talk was disturbing his father, but he needed answers to a few more questions. He pointed to the medal case resting in the bottom of the box.
"What's in there?"
Dad looked embarrassed. "Nothing."
Jack reached in and snatched up the case. "Then you won't mind if I open it." He did, and then held up the two medals. "Where'd you get these?"
Dad sighed. "The same time and place: November 28th, 1950, at the Chosin Reservoir, North Korea. The Chinese commies were knocking the crap out of us. There seemed no end to the men they were throwing our way. I had a good position when what looked like a couple of companies of reds made a flanking move on the fifth. I'd brought lots of ammo and I took out every officer I could spot. Anyone who made an arm motion or looked like he was shouting an order went down. Every radio I spotted took a hit. Pretty soon they were in complete disarray, all but bumping into one another. It might have been funny if it had been warmer and if my whole division wasn't being chopped to pieces. Still, they told me I saved a lot of lives that day."
"By yourself...you faced down a couple of Chinese companies by yourself?"
"I had a little help at first from my spotter, but Jimmy took one in the head early on and then it was just me."
Dad didn't seem to take all that much pride in it, but Jack couldn't help being impressed. This soft-spoken, slightly built man he'd known all his life, who he'd thought of as the epitome of prosaic middle-classdom, had been a stone-cold military sniper.
"You were a hero."
"Not really."
Jack held up the Silver Star. "This medal says different. You had to have been scared."
"Of course I was. I was ready to wet my pants. I'd been good friends with Jimmy and he was lying dead beside me. I was trapped. They weren't taking prisoners there, and if I surrendered, God knows what they'd have done to me for killing their officers. So I hung in and figured I'd take as many of them with me as I could." He shrugged. "And you know, I wasn't that scared of dying, not if I could go as quickly as Jimmy. I hadn't met your mother, I had no kids depending on me for support. And at least I wouldn't be cold anymore. At that moment, dying did not seem like the worst thing in the world."
Fates worse than death...Jack understood that. But there was still the Purple Heart to be explained. Jack held it up.
"And this one?"
Dad pointed to his lower left abdomen. "Took a piece of shrapnel in the gut."
"You always told me that scar was from appendicitis!"
"No. I told you that's where I had my appendix taken out. And that's what they did. When they went in after the shrapnel they discovered it had nicked my appendix, so they removed it along with the metal fragments. Somehow they got me to Hungnam alive, put me on penicillin for a week, and that was the war for me."
Jack looked at his father. "Why'd you keep all this hidden? Or am I the only one who doesn't know?"
"No, you're the only one who does know."
"Why didn't you tell me sooner, like when I was eight, or ten?"
As a kid it would have been so cool to know he had a father who'd been a Marine sniper. And even as an adult, he'd have had a whole different perspective on his Dad.
My father, the sniper...my father, the war hero...yow.
Dad shrugged. "I don't know. When I was finally sent home, I realized how many of my buddies weren't going with me. Their families would never see them again. And then I got to thinking about all the NKs and Red Chinese I'd killed who wouldn't be going home to their families, and it made me a little sick. No, make that a lot sick. And the worst of it was, beyond getting a lot of good men killed, we didn't accomplish a goddamn thing by pushing north of the thirty-eighth. So I just put it all behind me and tried not to think about it."
"But you kept the medals."
"You want them? Keep them. Or throw them away. I don't care. It was the photos I kept—I didn't want to forget those guys. Somebody should remember them. The rest just happened to come along for the ride."
Jack dropped the medals into the little case and returned it to the strong box.
"You keep them. They're part of who you were."
"And you might say they're part of who I still am. That's why I'll be backing you up when you go out there to get Carl back."
"No way."
"Jack, you can't go out there alone."
"I'll think of something."
Dad sat silent a moment, then said, "What if I can prove to you that I still have it? Please, Jack. I want to do this with you."
His father was practically begging Jack to take him along. But damn...it could turn ugly, and then what? He'd never forgive himself if the old guy got hurt.
Still, he felt he owed him a chance.
"Okay, Dad. You're on—for a test run. How are we going to work this?"
His father's eyes were bright behind his glasses. "I think I know a way."
## 5
The sign shouted DON'S GUNS & AMMO in big red letters—peeling red letters—with Shooting Range below it in smaller black print.
"This must be the place," Jack said as they pulled into the sandy lot on a rural road in Hendry County.
Only one other car, an old Mercedes diesel sedan, in sight. Probably the owner's. Opening time was 9:00 A.M. and it was after ten now. Jack figured there probably would be lots more activity once hunting season started, but at the moment he and Dad seemed like the only customers.
They went inside. Behind the counter they found a slim guy with salt-and-pepper hair and mustache. His lined face made him look sixtyish, maybe even older.
"Are you Don?" Dad said, extending his hand.
"That's me."
"We called about the M1C."
They'd made a lot of calls to a lot of gun shops—amazing how many there were in Florida—and not one of them had a M1903A1. But this place said it had an old M1C. Close enough, Dad had said. Hendry County was a good ways north of Gateways, but they'd had no other options.
Don smiled as he lifted the rifle leaning against the wall behind him and laid it on its side, bolt handle up.
"One M1C Garand, coming up. Heavy sucker. Gotta weigh a dozen pounds. But it's fully rigged—still has the original scope and flash hider."
"I see that," Dad said.
Jack was seeing a beat-up piece of junk: The dried-out wooden stock was scratched and dinged and gouged, the metal finish worn, and the whole thing looked like it had just received its first dusting in years.
Dad picked up the rifle and hefted it. In one seamless move he raised it to his shoulder and sighted down the scope.
"Never liked the M82 scope. Never liked the way it was mounted, and only two-and-a-half power. The Unertl I used was an eight." He looked at Jack. "This was the Army's sniper rifle for a while. Couldn't hold a candle to the M1903A1, if you ask me."
"If you really want to shoot that thing," Don said, "I can sell you a much better scope."
Dad shook his head. "I qualified on this as well as the 1903. It'll have to do. But will it shoot?"
Don shrugged. "Got me there. I'd forgotten I had it until you called. That thing's been here so long, I can't remember when I bought it or who from."
"What do you want for it?"
Don pursed his lips. "I'll let it go for twenty-five hundred."
"What?" Jack said.
Dad laughed. "Let it go? That's way overpriced for Army surplus junk."
"A fully outfitted M1C like this is a collector's item. If this baby was in better shape it'd go for twice that at auction."
"Hey, Dad, you can get a better rifle for a lot less."
"But not one I'm used to."
"Yeah, but twenty-five hundred bucks..."
"Hell, it's only money." He looked at Don. "I tell you what: You can have your asking price on the condition that it still fires. That means you've got to let me clean it and fire a few test rounds. Do you have a bench where I can spruce it up?"
Don pursed his lips again. "Okay. I've got a cleaning set-up in the back you can use. Go ahead. But give me a picture ID and your Social Security Number so I can background you while you're doing that."
"Background?" Jack said.
"Yeah. Instant background check. It's the law. I've got to place a call to the FDLE to make sure he hasn't got a criminal record, a domestic violence conviction, or under a restraining order. If he comes through clean, he gets the rifle. If not, no deal."
"Might as well quit now, Dad," Jack said gravely. "You are so busted."
"Very funny." He looked at Don. "No waiting period?"
He shook his head. "Not for rifles, but there's a mandatory three-day 'cooling-off period' for pistols."
Jack was glad he didn't have to buy his guns through legal channels.
Dad fished out his wallet and handed his Florida driver license to Don, saying, "What about ammo? Have any match grade?"
Don nodded. "Got a box of thirty-ought-six Federals. I'll throw in half a dozen rounds to let you check it out."
Dad smiled. "You're on."
## 6
"Jesus, Dad," Jack said as he stared through the field glasses.
"Not bad for an old fart, ay?"
Dad was down on his right knee, left elbow resting on his left thigh, eye glued to his scope.
"Not bad? It's fantastic!"
Earlier he'd watched with amazement as his father's wrinkled old hands disassembled the M1C like it was a tinker toy. He'd inspected the firing pin, wiped the scope lenses, cleaned and oiled all the works, scoured the inside of the barrel with a long-handled brush, then reassembled it with a precision and an efficiency that left Jack in awe.
Dad had explained that it was like riding a bike: Do it enough times and you never forget how. Your hands know what to do.
Then it was time for the test firing. Don had a two-hundred-yard rifle range behind his shop with acres of open country beyond it. Dad's targets—large paper sheets with concentric black circles at their centers—were set against a rickety wooden fence.
His first shots had been grouped wide to the left, but as he made progressive adjustments on the sight, the holes in the target crept inexorably toward the heart of the bull's-eye. He'd punched the last three shots through a one-and-a-half-inch circle.
"Not so fantastic," Dad said. "It's only two hundred yards." He patted the stock. "Definitely worth the price."
"A hundred yards is all we'll need, I hope. And by the way, I'm paying."
The Tyleski Visa had a five-thousand-dollar credit limit. Still plenty of slack there.
"Like hell."
"No, the least a guy can do for his backup is arm him." Jack extended his hand toward his father. "You've still got it, Dad."
The flash of his father's smile as they shook hands warmed him.
## 7
As Jack beached the motorized canoe on the bank of the channel shallows, he got his sneakers soaked yet again. This was getting to be a habit. The clouds had blown off and the sun was cooking his shoulders.
The shell lay nestled in the right front pocket of his jeans. Now where was Semelee?
"You're late," she said.
Jack looked right and saw her rounding a bend on the far side of the shallows. She stood in the front of a small, flat-bottomed boat and—
What the hell? She held a shell over her left eye and had her hand clapped over her right. As Jack watched, she lowered the shell and the hand and smiled at him.
Carl and Corley sat amidships directly behind her; Luke operated the little outboard motor mounted on the stern and glowered at Jack.
Carl grinned and waved the oar protruding from his sleeve. Jack was relieved that he looked pretty much the same as he'd left him.
"Sorry," Jack said. "Had some things to do and everything down here seems to take longer than it does up north. Ever notice that?"
"I wouldn't know," Semelee said. "I ain't never been up north."
Luke pulled up the motor; the hull of the boat scraped the sandy bottom as he let it run aground in the shallows. All four stepped out. Corley stayed by the boat while the other three approached—Semelee and Luke first, Carl behind them.
Jack gave Corley a quick look, noted a knife in his belt, but no gun. Same with Luke: a hunting knife with a six-inch blade in a leather scabbard strapped to his belt, but again, no gun. Good. Jack wanted to keep an eye on that knife, though.
They stopped in front of him. Luke stood with his arms folded across his barrel chest.
"Well," he said with a belligerent edge to his voice, "you can see plain and simple we got Carl. Time for you to show us the shell."
Jack dug into his pocket, all the while keeping an eye on Luke's hunting knife. If he made a move toward it, Jack would go for the Glock.
He fished out the shell and handed it to Semelee. As she took it and clutched it between her breasts, Luke's right hand moved, not going for the knife but flicking toward Jack's face. He heard a metallic click and found himself face to face with a three-inch, semi-serrated, tanto-style blade. Sunlight gleamed off the stainless steel surface.
Jack cursed himself for not guessing Luke might be palming a folder.
"Luke!" Semelee cried. "What're you doin?"
"Taking care of business."
"I've got the shell! Put that away!"
Luke shook his head. "Uh-uh. We're leavin with Carl and the shell. None of this trade shit."
Jack started creeping his free hand around toward his back while they argued, taking his time, moving a few millimeters at a time.
"Luke," Semelee said, "we told him we'd trade and that's what we're gonna do."
Luke shook his head, never taking his eyes off Jack. "I'm callin the shots here, Semelee. This is man's work."
"You better put that knife away, Luke," Semelee told him. "His daddy's over there in that willow thicket with a rifle trained on us."
Jack stiffened. The little stand of trees where he'd stationed his father was about a hundred and fifty yards away. How did she know?
Luke's gaze snapped past Jack's shoulder, then back. He grinned. "That old coot? What's he gonna do?"
"Think about that," Semelee said. "He's got a rifle and he's been watchin this spot since before any of us arrived."
How did she know?
"Yeah? So? He ain't gonna hit nothin from that distance. But if he's watchin, maybe he'd like to watch me cut his little boy's face."
As Luke drew back his arm for a slash, Jack reached for his Glock and raised his free arm to block the thrust, but didn't have to.
Everything seemed to happen at once—red sprayed from Luke's head, something whizzed by Jack's ear, a rifle cracked from somewhere behind him, though not necessarily in that order.
Semelee screamed as Luke staggered back, spun, and crashed face first into the water. A bright red stain began to drift away from him in the barely existent current.
Jack drew the Glock and turned to stare at the thicket.
Jesus, Dad! You didn't have to go for a kill shot.
This was going to make for big trouble—police, coroner's inquests, the whole legal ball of wax—shit!
"Luke! Luke!" Corley cried as he splashed toward him.
Jack kept the Glock trained on him; to his left, Semelee hadn't moved; she stood with her hands pressed against her mouth. Carl was in a squat, looking around like a cat who'd just heard thunder for the first time.
And then, miraculously, Luke jerked his face out of the water and coughed. He shook his head and sat up. Blood still streamed down his forehead, but Jack could see now that it was from a front-to-back furrow along the center of his scalp.
Jack had to laugh. Dad, you pisser! You pisser!
"He only parted your hair, big boy," Jack said. "Next time, he parts your tiny brain." He waved his pistol at Corley. "Get him back to the boat." Jack motioned Carl toward the canoe. "Welcome back, Carl. Get that thing turned around and ready to go."
Carl grinned. "You got it."
"Wait," Semelee said as Jack turned to go.
"Sorry. Gotta go. We're finished with this bullshit."
"No." She reached out and touched his arm. Gently. "I need to talk to you."
"Sorry."
"Please?"
## 8
Jack waited. Semelee looked around as if checking to make sure Luke was out of earshot.
She lowered her voice. "You gotta believe I didn't know Luke was gonna pull somethin like that."
Jack looked into her eyes and did believe. "Okay. But that wouldn't have made much difference if I was the one bleeding now instead of your pal there."
"Please don't be mad at me."
The plaintive note in her voice, the fawnlike look in her huge dark eyes...Jack couldn't fathom what she was up to.
"Lady, you've got to be kidding." He went to jab a finger at her and realized he still had the Glock in his hand. So he pointed with his left. "This is all your doing. We're all here because of you. You kidnapped Carl. You're behind the deaths of three innocent folks, and it was only by luck that my dad didn't wind up the fourth."
"You gonna tell the cops?"
"Maybe."
A slow smile stretched her lips. "No, you ain't. I can tell."
Well, she had that right. Jack couldn't see any point of bringing cops into the picture. The Dade County DA was going to charge Semelee with what? Murder by coral snake? Murder by bird? Yeah, right.
"You can't blame me," she said. "Don't you see? It wasn't really me. It was all part of the plan."
"Plan?" Jack felt the weight of the pistol in his hand. I should put one in her right now, he thought. Who knew how many lives he'd save if she never got back to her lagoon. "Well, you'd better come up with another plan, because I'm declaring this one over, done, finis."
"Ain't my plan."
That caught Jack off guard. "Then whose?"
"The lights'."
Oh, boy, Jack thought. Here we go.
"You mean the lights—the ones that supposedly come out of your sinkhole—are behind all this?"
She beamed. "Yeah. I didn't see it before, but then I got the big picture. It's all been part of a plan, one big, beautiful plan."
"Okay. The lights have a plan." The lights...if they were connected to the nexus point, then, according to Anya, they were connected to the Otherness. "Tell me about it."
Her smile widened. "Can't tell you all of it, but I can tell you some. I can tell you that the lights drew me back here so's I could find out who I was."
"Really. And who would that be?"
"Oh, I can't tell you that. Leastways not yet. Only someone real close to me can know that."
"Well, I'm only a foot or two away."
"Not that kind of close. The other kind of close...the way you're gonna be with me real soon."
Oooooh, lady, I wouldn't count on that, Jack thought.
"Really."
"Yeah. Which brings me to another part of the big picture: the sacrifices. They was done for a purpose."
"Like what?"
"To get you down here."
Jack's mouth went dry. All along he'd had a niggling suspicion, a creeping fear that his father hadn't been a random victim; but having it laid out before him like this was unnerving.
I'm responsible.
But he saw a problem.
He licked his lips. "Wait. That doesn't make sense. You say your lights figured if they killed my father I'd come down here. But I might not have come. My brother might have come instead. And why the other three deaths before him?"
Semelee shrugged. "Who can explain how the lights think? Maybe they liked the sacrifices, maybe they knew Mr. Weldon would get to your daddy sooner or later and so they just let things happen. Maybe your daddy's name came up when only you could come. Don't much matter none. You're here, ain't you."
Yeah, he thought. I'm here all right.
"Why would your lights want me here?"
Semelee smiled. "For me."
"For you? What do you want with me? What do you even know about me?"
"I know you're special. And I know we was meant to be together."
"Yeah? Well, sorry. You and your lights are a little late. I'm taken."
"Don't matter. It's gonna be you and me. Can't stop it. It's like...like..."
"Kismet?"
"Kiss what?"
"Destiny?"
"Yeah, that's it. Destiny. You and me's destined to be together. You're gonna bring me in, take me back with you, make me belong, and then together we're gonna rule the roost."
Make you belong? he thought. Boy, sister, have you picked the wrong guy. "Listen, if you're an outsider, the last guy you want to hook up with is me."
"Lemme be the judge of that." She stepped closer until her lips were barely an inch from his. "I'll meet you tonight at—"
"Sorry," Jack said, backing away. "Game over. Hang out with your lights and your buddies here, do whatever floats your boat, but stay away from Gateways, especially from my father." He raised the Glock and held it beside his head, muzzle skyward. "I see you or any of your clan within a hundred yards of my father, you're dead. Not figuratively dead, not virtually dead, not merely dead, but clearly and sincerely dead. Got it?"
She stared at him with her big, suddenly sad eyes. Her lower lip trembled.
"No...you can't..."
"Got it?"
Jack turned and sloshed over toward where Carl waited with the boat in the deeper water.
"You can't!" she screamed behind him.
Watch me.
## 9
"He needs killin, Semelee," Luke said. "He needs killin real bad."
They had the deck of the Horse-ship all to theirselfs. Semelee sat with her legs danglin over the side, starin at her reflection in the water. Luke crouched next to her.
His head had stopped bleedin. Finally. For a while there she'd thought he was gonna lose every drop of blood in his body. He'd refused to go to the hospital, sayin he'd heal up just fine without no damn fool doctors stickin him with needles. Maybe he was right, but he sure looked stupid with that red bandanna tied across his head and under his chin.
"You're right," Semelee told him. "For once, I ain't got no argument with you."
Luke stared at her with shocked eyes. "You mean it?"
"Damn right I do."
"But I thought you was sweet on him."
"Wasn't never sweet on him. I thought he was special but that don't matter none now. He hurt you and—"
"His daddy did the shootin."
"I know that. But his daddy only pulled the trigger. It was him, it was Jack who put him up to it. Probably told his daddy to blow your head off but the old boy only creased you. Can't have that, Luke. Can't have nobody, no matter how special they are, hurtin someone in the clan."
"So then it's okay with you if I take Corley and a couple—"
Semelee shook her head. "Uh-uh. I'm gonna handle this my own self. For you, Luke. It'll be a present from me to you."
The shock in Luke's eyes melted into something like love.
Don't be gettin no ideas, she thought.
Because this had nothin to do with Luke. She was just lettin him think that. He'd been too far away and too busy with his bleedin head to pay any attention to what had gone on between her and Jack in the shallows. Didn't hurt none though to let him think he was the reason she was gonna go after Jack.
But this was gonna be all for her.
She'd wanted to cry all the way back from the shallows. Her heart still felt like it'd been tore right out of her chest. He'd turned her down, turned his back and walked away. He said it was because he was taken, but that was a lie. Semelee had seen it all through her life and she knew the real truth: Jack thought he was too good for her.
But as she'd returned to the lagoon she realized it was the other way around.
Jack...how could she've thought he was special and meant for her? What was she thinking? He obviously wasn't so special and definitely not for her. She saw that now. Her visit to the lights in the sinkhole had changed everything. She knew her True Name now, knew that she'd been brought here for a purpose. She wasn't sure what that was yet, but she would. She just knew she would.
She'd been special before—her powers proved that—but now she was even more special. Much too special for Jack.
Yeah, but if that was true, why was she still hurtin? Why this cold hard lump where her stomach used to be?
She knew of only one way to make it better.
"Leave me be for now," she told Luke. "I gotta work on this. I'm gonna fix a big fat surprise for our friend Jack."
He got up and backed away. "Okay, Semelee. Sure. Sure. Maybe I'll go check on Devil. See how he's doin."
Despite how bad she was feeling, Semelee had to smile. Luke'd always been sorta like her puppy dog, but now he was actin like her slave.
But she was okay with that. Every girl should have a slave.
## 10
"I think this calls for a drink," Dad said as they stepped into the house.
They'd dropped Carl—with his thousand dollars—off at the trailer park. All the way home he'd been so effusive in his thanks for rescuing him from the clan and the lights that Jack had had to shut him up by getting him to describe what he'd seen last night. He'd found Carl's description of Semelee being lowered into the hole particularly unsettling. If the lights, filtered through sand and water, had caused the clan's deformities, what would direct exposure do? Make you crazy? The cenote must have been where she'd learned—how had she put it? Who I am. Who was she if not Semelee?
"That was one hell of a shot, Dad. One hell of a shot."
Jack kept reliving the emotional swings of that moment.
"Wasn't it? Wasn't it, though?"
Dad had darted into the kitchen and was searching through the bottles in a cabinet above the sink. His speech came in staccato bursts, his movements were quick, jittery, as if he'd mainlined caffeine.
He's higher than the proverbial kite, Jack thought.
"I wasn't looking to kill him, you know, and prayed I wouldn't, but I was also thinking, if it's his life or Jack's, then I can live just fine with a kill shot. All the skills came back as I was sitting in that tree, Jack. Suddenly I was back at the Chosin Reservoir, and I was on autopilot and really, really relaxed because no one was shooting at me out in the Glades. It was just me and the rifle, and control of the situation was mine for the taking. I—here it is." He pulled a dark green bottle from the cabinet and held it aloft. "Wait till you taste this."
"Scotch? I think I'll go for a beer."
"No-no. You've got to try this. Remember Uncle Stu?"
Jack nodded. "Sure."
Uncle Stu wasn't a real uncle, just a close friend of the family. Close enough to earn "Uncle" status.
"He belongs to a single malt scotch club. He let me try this once and I had to get a bottle. Aged in old sherry casks—amontillado, I believe."
"And discovered with a skeleton behind a brick wall?" When Dad gave him a questioning look, Jack said, "Never mind."
"You drink this neat." Dad poured two fingers' worth into a couple of short tumblers. "Adding ice, water, or soda is punishable by death." He handed Jack a glass and clinked his own against it. "To the best day of my life in the last fifteen years."
Jack was pierced by an instant of sadness. The best? Really?
Not a Scotch drinker, Jack took a tentative sip and rolled it around on his tongue. It had a sweetness and a body he'd never tasted in any other Scotch. And the finish was...fabulous.
"For the love of God, Montresor!" he said. "That is good!"
"Isn't it?" Dad said, grinning. "Isn't that the best you ever had?"
"No question. Potent stuff."
"That's what I hear, but I haven't seen any proof."
Jack let that one slide. "Where can I get a bottle?"
"You can't. It's all gone. They produce only so many casks and this batch is long sold out."
Jack lifted his glass for another sip. "Then we'd better nurse this one."
"I don't care if we empty the bottle. This is a special day. It's been a long, long time since I've felt this alive." He looked at Jack. "But I have to ask you something."
"Shoot."
"Where'd that pistol come from, the one you pulled after I parted the big guy's hair?"
Jack felt very close to his father at the moment, closer than he could ever remember. The father-son slope had been leveled. They were eye to eye now. Equals. Friends. He didn't want anything to get in the way of that, but he couldn't very well tell Dad he'd imagined the Glock.
So he pulled it from the small of his back and laid it on the kitchen counter.
"You mean this?"
"Yes. That." His father picked it up and hefted it. Jack noted with approval how he kept the muzzle directed down and away from both of them. "What's it made of? Feels almost like..."
"Plastic? That's because most of it is. Not the barrel and firing pin, of course, but pretty much all the rest."
He turned it back and forth in his hand, staring at it. "Amazing." He raised his eyes to Jack. "But what's an appliance repairman doing with something like this?"
How to handle this...
"Sometimes I wind up in bad neighborhoods and I feel more comfortable knowing I'm carrying."
"But how did you get it down here? I know you didn't carry it aboard the plane."
Jack shrugged. "There are ways."
Dad continued to stare at him. "Tell me the truth: You're not really a repairman, are you."
"Oh, but I am. That's the truth."
"Okay, but what else are you?" He waggled the Glock. "I saw how you handled this out there. I saw plenty of people handle guns in the war, and you could always tell the ones who knew what they were doing and were comfortable with them, just as you could tell the ones who weren't. You fall into the first category, Jack."
Despite the closeness he felt to his father at this moment, despite the combat-zone bond they'd formed, Jack couldn't bring himself to tell him.
"You're pretty comfortable too, Dad. Maybe it just runs in the family."
"All right. Keep your secrets. For now. But promise me that someday, before I die, you'll tell me. Promise?"
Jack knew a trap when he heard one. This one was a cousin of "When did you stop beating your wife?" If he promised, he'd be admitting there was something to tell.
"Let's not talk about you dying, Dad."
He sighed. "I'm not going to get anywhere, am I?" He poured more Scotch into Jack's glass. "Maybe this will loosen your tongue."
Jack laughed. "No one's ever tried to ply me with liquor before. Bring it on!"
## 11
The shadows was gettin long by the time Semelee was ready to make her move. Even usin both eye-shells, it had took her a while to get Dora in place. Like any other alligator snapper, she was slow and kinda clumsy. Nothin like Devil.
Poor Devil. Luke said he was doin right poorly and looked like he was fixin to die. That made her feel bad.
But she shook off the sadness and fixed on what she aimed to do. Now that she finally had Dora where she wanted her, Semelee was ready for the next step.
She moved away from the lagoon and walked through the hummock until she came to the bees' nest. She didn't get too close. These was killer bees and once they got mad they'd swarm and wouldn't stop stingin. They didn't know how.
She fixed the shells over her eyes and concentrated......and sees the inside of the hive. Her vision's all weird, like she's lookin through dozens of eyes at once...
Semelee lowered the shells and picked up the rock she'd brought along. She tossed it at the hive, then put the shells back over her eyes, real quick like.
...and once again she's inside the hive with that weird way of looking at things. But the hive's different now. It's filled with angry buzzing—real angry. They're movin toward the opening, hittin the air and the sunlight, and then she's flyin, movin right with them.
She sees herself, standin in the shadows with the shells over her eyes. The swarm homes in on her like she's the absolute worst thing in their world, like they gotta protect the hive from her or die tryin. Sweat breaks out all over her body. Maybe she shouldn't have done this. Maybe she should have thought of another way. Cause if she can't turn them, they're gonna kill her.
She pulls at them, pushes at them, there's somethin worse than her, somethin that's a bigger threat to the hive and they've gotta get him, gotta stop him or the hive'll be destroyed.
It doesn't seem to be working. They're still comin at her. Somethin inside her is screamin to run but she knows that won't do no good. Ain't nobody gonna outrun these bees.
Gotta turn em, gotta turn em, gotta—
There! They're turnin, veerin away from her and turnin east. She did it. She's in control now and her own rage adds fuel to the bees'.
## 12
With his father noisily engaged in an exploration of the deep, dark recesses of napland, Jack wandered outside. Square-foot-wise, Dad's place was bigger than Jack's apartment back in New York, but it felt smaller. Maybe because he didn't have to share his place with anyone. He needed some fresh air.
With the comforting weight of the Glock at the small of his back, he scanned his surroundings as he yawned and stretched, looking for signs of the clan. Semelee had said Dad was no longer a target, but she'd been acting pretty weird out there in the Glades. What was to prevent her from changing her mind?
He started to circle the house, as much inspecting as trying to walk off the Scotch. He hadn't had all that much but it had made him a little drowsy. Not drowsy enough for a nap, though.
No white-haired girl sitting on his car hood this time. No one at all in sight. As he walked around to the left side he heard a faint buzzing, like a far-off chainsaw, filtering through the air. He looked around for the source but saw nothing. Maybe someone was using one on the far side of one of the houses. One thing he knew, it wasn't Carl. He was taking the rest of the day off—although he'd told Jack he'd return briefly tonight to set up the Anya-cam again.
The buzzing grew louder and Jack did another slow turn. What—?
Then he saw the man-size cloud sweeping toward him from the Glades and knew with a sick, cold dread what it was and who had sent them. All his instincts urged him to turn and run but he forced himself forward, toward them. Because that was where the front door was. He sprinted with everything he had, but the bees got there first.
He staggered back as they swarmed over him and began stinging. Their angry buzzing and pain like dozens of red-hot ice picks stabbing into his flesh became Jack's world. He needed both hands to bat the bees away from his face but that left the rest of him vulnerable—his neck, his scalp, his bare arms. He could feel them stinging him through his T-shirt. He tried for the door again but they drove him back.
Through the cloud he caught a glint of water—the pond. He stumbled in that direction, picking up speed. When he reached the bank he leaped blindly in a headlong dive. As he knifed through the surface he felt most of the swarm back off—but not all. Some still clung to him, stinging as he—
His outstretched hands hit the rough, hard surface of an underwater rock. He clung to it to keep himself submerged. He was safe for the moment, but he was going to need air soon. Very—
The rock moved, twisting under him. Through the murky water he saw that it had scalloped edges and a tail and he didn't need to see the two big heads rearing up, hooked jaws agape, to know what was sharing the pond with him.
He clung to the edges of the shell as the big alligator snapper surged toward the surface, twisting this way and that as it tried to shake him off. The ridged surface was slimy and his fingers were losing their grip. Jack was running out of air as he raced through his options. The pond was clearly a no-win. Had to get out and take his chances with the bees. With the snapper surfacing, he was going to have to deal with them anyway.
As his lungs screamed for air, he drew his legs up under him, folding them till his sneaker soles were on the shell. As soon as his head broke water, the bees were on him again. He kept his face submerged until the last possible instant, then sprang off the shell, leaping for land. His right sneaker slipped, robbing him of the distance he needed, and the breath he'd taken while airborne was knocked out of him when he belly flopped onto the edge of the bank. His legs were still in the water and, for a panicky instant as he heard the splash of the snapper coming for him, he remembered what those jaws could do to a broomstick. A flashing vision of himself crawling the rest of the way out of the water with a bloody stump where a foot used to be threw him into a twisting roll that left him clear of the water. As he batted at the relentless bee swarm, he glimpsed the two heads stretched to the limits of their thick necks snapping at empty air where his legs had been.
Could an alligator snapper move on land? Jack wasn't waiting around to find out, especially with the bees stinging him again. He realized he'd emerged on Anya's side of the pond, so he scrambled to his feet and raced toward her front door. It was closed but maybe it was unlocked.
Please be unlocked!
But he didn't need the shelter of her house. As soon as he crossed into her circle of green lawn, the killer bees peeled off him the same way the palmettos had the other night when he'd jumped through his father's door.
He heard their enraged buzzing rise in pitch and volume as they hurled themselves at him, only to be turned back as soon as they crossed the line into Anya's space.
"Go!" he heard a voice cry behind him.
Jack turned and saw Anya crossing the lawn in his direction. She was waving both arms in a shooing motion.
"Go!" she shouted again. "Back where you came from!" She pointed to the snapper's two heads, watching from the pond. "You too! Go!"
The bees swarmed in random confusion, then gathered into an oblong cloud and buzzed away. When Jack looked at the pond again, the snapper was gone.
He dropped to his knees, panting. His skin felt aflame, his stomach threatened to heave.
"Thank you," he gasped. "I don't know how you did that, but thanks."
"Didn't I tell you that nothing on earth can hurt you here?"
"I guess you did." He looked up at her. "Who are you? Really."
Anya smiled. "Your mother."
The familiar words chilled Jack.
"That's what the Russian lady said to me by my sister's grave. And that Indian woman in Astoria said the same thing to Gia. What's it mean?"
Anya shook her head. "Don't worry about it, hon. There's no need for you to know. Not yet. Hopefully not ever."
"Then why say it to me?"
Anya had turned and started walking away. Over her shoulder she said, "Because it's true."
## 13
Semelee stumbled pantin and sweatin along the path through the palms. She stopped and leaned against a gumbo limbo tree to catch her breath.
That same old lady...doin it again...causin trouble, gettin in the way...
She was stronger than Semelee. Somehow she'd just waved her hand and told the bees and Dora to get home and that was that. Semelee's power got canceled like turnin off a light. Everything went black. When she come to, the sun was pretty much down and she was flat on her back in the ferns with the shells off her eyes but still in her hands.
She had to be stopped. But how? How do you stop someone with that kind of power?
Where did she come from? Who was she that she could protect herself from Dora and a swarm of bees—not only keep them out but give them orders?
Maybe she couldn't be hurt. Maybe she was beyond Semelee's special power.
She stumbled up to the bank of the lagoon and saw Luke sitting on the deck of the Bull-ship.
He looked up at her with sad eyes. "Bad news, Semelee. Devil's dead."
A wave of sadness washed over her. Feelin weak, she lowered herself to the ground and rested her back against a palm.
Poor Devil...her fault...if she hadn't—
No, wait. It was that old bitch and her dog. They were the ones killed Devil. Not her.
She ground her teeth. Had to be a way to get back at her.
She glanced to her left toward the sinkhole and saw the glow of the lights seepin up through the darkenin air. Pullin herself to her feet she walked over. She stopped at the edge, then stretched herself out flat on her belly with her head pokin over the rim. She gazed into the flashin deeps and tried to remember more of what happened down there. But nothin came back to her.
She gave up tryin to remember and was just startin to get to her feet when she had an idea. She still had the eye-shells in her hands and figured, Why not? She put them over her eyes. For an instant they blotted out the lights, then suddenly she was seein them again. But they looked different.
Then Semelee realized she wasn't seein the lights from above, she was seein them from within. She was inside some kinda creature down there and was seein things through its eyes. She looked around and saw wings and jaws and teeth—lots of long, sharp teeth.
An idea crept into her head, an idea so wonderful she started to laugh out loud.
## 14
"I still say we should take you to the emergency room," Dad said.
Jack shook his head as he shivered under the blanket. "I'll be fine, Dad. No doctors."
At least not yet.
He sat on the sofa and shook despite the dark blue wool blanket wrapped around him. Most of his sting-lumped skin was crusted pink with calamine lotion and he was dopey from the Benadryl his father had picked up for him in town. The stings themselves—he hadn't counted them, but Pinhead had nothing on Jack—itched and burned, and now his muscles were aching. The chills and fever had started about an hour after the attack. He figured he had so much bee venom in his system that he was having a reaction. He felt as if he had the flu.
At least he wasn't vomiting; his stomach was queasy but he was holding down the orange juice Dad kept pushing at him.
He'd shown his father how to break down the Glock and wipe it dry. Here was where its mostly plastic construction was a blessing. Dad didn't have any gun oil, but substituted a little 3-in-1 to lubricate the few metal parts.
And now his father paced back and forth between Jack and the TV as the Weather Channel showed a satellite photo of Hurricane Elvis picking up speed and power as it looped southward through the Gulf of Mexico. It had graduated to Category II and was expected to brush South Florida and the Keys sometime tomorrow, then continue on toward Cuba.
"We've got to call the cops," Dad said.
Dad always seemed to want to call the cops.
"And what—tell them about this woman in the Glades who sent a swarm of bees and a two-headed snapping turtle after me? They'll take you away in a straitjacket."
"We've got to do something! We can't just sit here like targets and let her take potshots at us!"
"I can't think right now, Dad."
Jack hauled himself unsteadily to his feet and shuffled toward the guest bedroom.
He'd planned to drop in on Anya tonight. He'd cut her too much slack, let her evade straight answers for too long. He was going to get nose to nose with her and find out exactly who she was, how she could keep giant alligators and bees and mosquitoes from trespassing on her property, and have them obey her when she told them to take off. He wasn't going to leave until he had some answers.
But that was all changed now. Christ, he felt awful. If he'd been sitting on the hood of Dad's car when it got clocked by that truck, he didn't think he'd feel much worse.
"I'm going to hit the rack. In the meantime, don't do anything I wouldn't do."
"That's all fine and dandy," Dad said with a touch of acid in his voice, "except I don't know what you wouldn't do."
"Well, for one thing, I wouldn't leave the house tonight, that's what I wouldn't do. As for what I would do"—he pointed to the reassembled Glock resting on a section of the Novaton Express—"I'd keep that handy. See you in the morning."
## 15
Jack awoke bathed in sweat. He threw back the covers, sat up, and pulled off his undershirt.
What time was it? The clock's LED display was angled away from him. No light filtered through the curtains. Still night. He ran a hand over a tender, bumpy arm. God, he felt like hell.
As he flopped back and pulled the sheet up over him, he thought he heard a dog barking—high-pitched yips that could only belong to Oyv. They had an almost hysterical edge. Jack wondered what was bothering him. Not that the little guy couldn't take care of himself—look at what he'd done to that big ugly gator—but he hadn't struck Jack as the kind of pooch to bark at nothing.
Jack was ready to force himself out of bed to go have a look when the barking stopped. Whatever had set off Oyv must have passed.
Jack closed his eyes and drifted back to sleep.
## SUNDAY
## 1
I've got to get back to New York, Jack thought.
Not just because he missed Gia and Vicky, but here it was Sunday afternoon and instead of watching the Jets kick Dolphin butt up at Giants Stadium, he was sitting here with his father and staring at the Weather Channel.
Trouble was, he found it mesmerizing.
The Weather Channel as a way of life...scary.
I stay much longer I'll be as addicted as everybody else around here.
He excused his present fascination by the fact that the weather was about to have significant personal impact: Hurricane Elvis had reentered the building. In fact he was announcing his presence with a chorus of gusts that hurled sheets of rain against the outside of this little building.
Satellite tracking of Elvis showed how it had made a sharp eastward turn during the night and homed in on the Everglades like a cruise missile. At this moment its eye was making landfall on South Florida's west coast. Elvis wasn't a monster; it was a tight little storm with sustained winds now in the 120-mile-an-hour neighborhood, making it a Category III. Multiple waterspouts had been spotted among the Ten Thousand Islands, wherever they were. But apparently it was a very wet storm and everyone was happy that it was going to dump a lot of much needed rain onto the Everglades.
But how many times could you watch the same graphic and listen to the same Storm Center report?
Gia apparently had been watching the weather too. She'd called to tell him to stay inside. Not that he had any intention of venturing out into this mess, but he appreciated her concern. He hadn't told her about the bee stings. They were still swollen; not as much as last night, but still itchy and tender.
He was about to ask his father to switch the channel for half a minute—not a second more than that, God forbid—to check the score of the Jets game, when he heard a frantic knocking on the door. As his father peeled himself away from the tube to see who it was, Jack slipped the Glock from where he'd stowed it under his sofa cushion.
"Better let me get it, Dad."
But before either of them could reach the door, it blew open. Jack had the pistol up and aimed at the figure standing in the doorway, his finger tightening on the trigger, when he recognized Carl.
"Come quick!" he cried as wind swirled around him and scattered sections of the Sunday paper. He wore a dripping, dark green poncho, had a screwdriver sticking out of his right sleeve, and a plastic shopping bag clutched in his left. "Y'gotta see this, y'just gotta!"
"See what?" Dad said.
"Miss Mundy's place! It's all tore up!"
Carl turned and started to lead the way, but once they were outside in the slashing wind and rain, Jack broke into a trot and pulled ahead of him. The sudden memory of Oyv's barking last night sent a cold spike of unease through his chest. It speared down through his gut when he saw her doorway.
"Oh, shit!"
The screen had been shredded; gray, mosslike tatters fluttered within the frame. The wooden door behind it stood open.
"Anya!" Jack shouted as he pulled open what was left of the screen door and stepped inside.
He stopped suddenly, just beyond the threshold, causing Dad to bump into him, pushing him forward.
"Oh, dear God!" he heard his father gasp.
"Didn't I tell you?" Carl said. "Didn't I?"
The place was a shambles. That was the only word for it. The furniture had been torn apart, the carpet gouged up, and the plants...they'd been torn from their pots, their roots savaged, and every leaf had been torn from the ravaged branches.
Jack forced himself to move forward, calling Anya's name as he checked both bedrooms and behind the kitchen counter. He found a small spatter of darkening red fluid, and something that looked like a severed finger on the floor.
Jack knelt for a closer look. It was pale, the size of a finger, but it was covered with fur.
What the—?
And then he knew: Oyv's tail.
Christ! The blood...Oyv had to be dead—died defending Anya no doubt. A slow wave of sadness settled over him. But what could have killed that preternaturally tough little dog? It had to be something bigger and meaner than a giant alligator. But what? And where was the rest of him?
Jack noticed something glittering on the floor. He bent closer: three little slivers of glass. He looked around for a broken window but didn't see one. Maybe a glass had been knocked off the counter and shattered.
He was pushing himself to his feet when he noticed that all three shards appeared identical. Each about an inch and a half long, with the same curve, and the same taper from thicker base to needle-fine point. He picked one up and rotated it in the light. Its edges were smooth, rounded. If he didn't know better, he'd have said it was a fang of some sort. But he didn't know anything that had glass teeth.
He touched the point with the tip of his finger and it slipped through the skin like a bird's beak dipping into water.
Damn! He started to toss it back to the floor, then decided not to. Maybe he should find out what it was before he threw it away.
He rose and grabbed a paper towel from the roll suspended from the underside of a cabinet. He rolled the needle within and used it to blot the drop of blood oozing from his fingertip.
He turned to his father and Carl, still standing in the doorway.
"What the hell happened here?"
Dad could give him only a stunned look, but Carl held up his plastic shopping bag.
"It's all here!"
"What's all there?"
"What happened. The camera caught it all. Or at least most of it."
## 2
"When I picked up the camera this morning," Carl said, "I was in a hurry so it just sat in the bag till after I got home. Long after I got home."
They'd all hurried back to Dad's place to set up the camera for playback.
"You didn't check it right away?"
"Nuh-uh. I figured, what for? I mean, I ain't never seen nothin before and figured this wouldn't be no different. So I just left it be until I was watchin the Dolphins game. That's when I checked it and found the battery didn't have no charge left. That ain't happened before. So I recharged it and took a look to see if somethin'd set it off."
"What's this camera about?" Dad said.
Jack ran through a quick explanation of Dr. Dengrove's attempts to catch Anya watering her yard.
"Dengrove," Dad said. "Cheats at golf but God forbid anyone sneaks a little water onto their lawn. What an ass."
Jack had the two-inch LCD screen flipped open. He hit PLAY and started to watch. Dad hung over his shoulder, Carl crouched farther back. The screen lit with green and black blobs that quickly stretched and coalesced into recognizable shapes—the side of Anya's house, her plantings, the doo-dads, the lawn furniture in her front yard. And then a set of legs went by. Then more.
"Doesn't this thing have any sound?" Dad said.
"If you hook it up to your TV you can get sound. Want me—?"
"We can do that later if we need to," Jack said. He had a sick cold feeling in his gut that they'd be listening to the high-pitched barking he'd ignored last night. "First let's see what's to see."
Carl jabbed a finger toward the little screen. "There they are! See?"
Jack saw. A crowd was gathering in an irregular semicircle around the edge of Anya's lawn. Light from the front windows lit their faces. His intestines began to writhe as he recognized Luke and Corley and a couple of the others. Looked like the whole gang had shown up.
"The clan," he said.
"All cept Semelee. I didn't see her nowheres when I watched."
Jack stared at the tiny screen. He now wished they'd hooked it to the TV. Probably would have lost some resolution, but maybe he'd have a better view of their faces. Beyond a few grins, he couldn't make out much in the way of expressions. He could read their postures, though, and they radiated something between revulsion and avid fascination, as if they wanted to press forward for a better look, but fear held them in check.
He kept watching, waiting for the clan to do something. He searched for Semelee but couldn't find her. That white hair of hers would be hard to miss. Why were all the men there? What did they have against—?
Oh, right. The big ugly alligator...her dog had chewed a hole in its side. And the bees yesterday...Anya had chased them off. Yeah, he could see where Semelee could have a bone or two to pick with Anya. But how was she going to get her if Anya's promise—Nothing on earth can harm you here—was true?
Obviously it wasn't. Someone had got to her—and to poor little Oyv. What had Semelee—?
"There!" Carl cried. "Didja see that?"
"No." Jack's attention had been wandering. "What?"
"I saw something too," Dad said, "but I don't know what."
Jack found the reverse button and backed up the recording. Again he watched Luke and the rest of the men standing in their semicircle, eyes fixed on the front of the house. The camera angle didn't include the front door, but they were staring like there was a stripper doing her thing there. And then something—maybe three somethings, two feet long at most—suddenly streaked out of the house and over their heads. The way the men ducked and covered made it pretty obvious that they were afraid of the things, whatever they were. More flew out. Once they were gone, the clan came to life. Luke swung an arm and they all charged toward the house.
For a good five to seven minutes, nothing happened, and then the clan reemerged. A group of them seemed to be carrying something but the way they were bunched together prevented him from seeing exactly what. He didn't have to see. He knew.
"They've got Anya."
"The sons of bitches," Dad said, straightening and reaching for the phone. "I'm calling the cops."
Jack grabbed his arm. "Hold on a sec. I want to see this again—on the TV."
"Fine. And while you're setting that up, I'll be calling—"
"Just wait, okay? Just let me see it again before we get officialdom involved."
Dad reluctantly agreed, grumbling about wasting time as Jack wired the camera to the audio-visual inputs on the backside of the TV.
"This happened at least twelve hours ago, Dad. Maybe more. Another ten minutes isn't going to matter."
He finished plugging in the wires, then reran the movie. The TV screen offered over one hundred times the viewing area of the camera's LCD. It offered sound as well. The movie began with the rattle of the lawn-ornament cans and Oyv's barking, but that broke off with a high-pitched squeal just as the last of the clan reached the front of the house. A couple of minutes later the things streaked away. Jack was ready with his finger on the PAUSE button.
"Got 'em!" he said. He leaned closer to the screen. "But what the hell are they?"
The camera's image intensification coupled with the speed of the things left little more than amorphous, blurry streaks on the screen, but there was enough resolution to reveal five shapes instead of three in the first batch. He'd missed the other two because they were farther from the camera and hadn't caught as much light. He could see that the three in front had slightly curved bodies that reflected light like a shell might; their wings were fuzzy blurs.
"Y'ask me," Carl said. "They look like flyin lobsters."
Not a bad characterization, Jack thought. But lobsters didn't fly, so what on earth were these?
Jack felt his neck muscles tighten. On earth...
Nothing on earth can harm you here.
But what if those flying lobsters weren't from anywhere on earth? What if they were somehow from the Otherness? Semelee had gone down into that sinkhole. Maybe she'd found something down there that she could control like she did the creatures in the Glades.
Jack pulled the rolled-up paper towel out of his jeans pocket and unwrapped the little crystal shard.
"What have you got there?" Dad said.
"Not sure." He handed it to him on the towel. "Careful. It's sharp. Ever seen anything like it?"
"I did," Carl said. "Saw one just like it stickin outta the tore-up wood on Miss Mundy's door. I just figgered it was glass."
Dad was holding it up, rotating it back and forth in the light. "You know, it almost looks like some weird sort of fang."
Carl laughed. "Glass teeth! That's funny!"
Dad lifted the beer bottle he'd been sipping at during the endless weather reports and scratched the fang's point along the glass. It gave out a faint, high-pitched squeak as it scored the surface.
Dad frowned. "Not glass. Much harder. The only thing I know that can scratch glass like that is a diamond."
"If it is a tooth," Jack said, "that means that Anya was attacked by things with diamond teeth."
They all sat silent for a moment, then Jack restarted the movie. They watched more of the things fly out, then the clan crowd into the house. When they emerged this time he kept freezing the frames but got no better view of what they were carrying than before. What else could it be but Anya?
But alive or dead?
As the movie ended again, Dad slapped his thighs. "That does it. Time to call 9-1-1."
"Don't bother, Dad."
"Why on earth not?"
Jack pulled the Glock from the SOB holster and checked the magazine: full. "Because I'm going after her and I don't want them getting in the way."
## 3
Tom could only stare at his son. He'd sensed that the Jack who had gone into Anya's ruined house was a different Jack from the one who'd come out. But now he'd changed further. His mild brown eyes had turned to stone; he seemed remote, as if he'd left the room without moving his body.
"After her? Are you crazy? We trumped a couple of them once because it was a controlled situation and we had surprise on our side. But all that's changed now. You can't expect to stroll in there alone and—"
"Won't be alone," Carl said. "I'll come along."
Tom noticed Jack's cold eyes warm briefly at this simple man's unadorned courage. And in that moment he wished Jack were looking at him like that.
"Not necessary, Carl," Jack said.
"'Tis. She's a good lady. Lotsa people look at me funny, some don't even want me around. But she always smiled at me and when it was hot she gave me lemonade and cookies and stuff like that. My own mother never treated me that good. And besides, the clan ain't got no right to do that to her. Semelee's gone crazy. Ever since she come up outta those lights she's been different. Scary. Who knows what she's got in mind for Miss Mundy. We gotta get her back."
"But that's what we have police for!" Tom cried.
He'd resisted the urge to chime in and say he'd go along too. Anya was a friend, a good one, and his blood curdled at the thought of her in the hands of a bunch of swampland inbreds. But it was just because he cared about her that he had to stop this craziness. Jack's gung-ho plan might wind up putting Anya in greater danger. Might even get her killed.
"And in case you two would-be vigilantes haven't noticed," he added, "there's a Category-Three hurricane blowing out there."
"Exactly why we've got to take care of this," Jack said. "Who're you going to call? The Novaton police? Their whole department, along with every other cop south of Miami, is going to be tied up with the hurricane emergency. They'll be busy with evacuation, shelters, looting prevention. You know the drill. A missing-person problem will be put on a back burner till the storm's passed. Hell, we don't even have proof she was taken."
"But the movie—"
"—will be great in court. But do you think it will get a bunch of cops running around in boats out in the Everglades looking for a particular hummock in the middle of a hurricane?"
Tom had to admit he doubted it—but only to himself. Under no circumstances did he want Jack going out there—not even with Carl, who Tom couldn't see as much help.
"Carl," Jack said, pointing to the screwdriver protruding from his sleeve. "Do me a favor and use that to take the medicine cabinet out of the wall in the bathroom."
Carl gave him a strange look—imagine that—then shrugged and nodded and said, "Okay."
"Medicine cabinet?" Tom said. "What—?"
Jack turned his back and headed for the hall closet.
"Look, Dad," he said as he knelt by the toolbox and began rummaging through its contents. "I don't know for sure, but I think that taking Anya has something to do with the lights. But the lights only last a couple of days. By tonight or early tomorrow morning they'll be gone for another six months."
"What lights?"
"Oh, yeah. Right. I forgot." He pulled a socket wrench from the toolbox and headed for the dinette table. "You don't know about the lights."
"Care to enlighten me?" Tom said, following. "And what do you think you're going to do with that wrench?"
"You'll see. As for the lights, forget about them for now. Take too long to explain. What matters is that after the lights go out, Semelee and Company will have no more need to hang around their lagoon. Good chance they'll be gone by sunup tomorrow."
"And take Anya with them?"
Jack gave him a stony look before he crouched under the table and began loosening the nuts that fastened it to its support pillar.
"I doubt it. She's the one whose dog chewed a hole in the side of that big mutant gator, remember? I'm worried they'll feed her to it before they go—if they haven't already."
Tom felt his knees go rubbery. "No...they couldn't."
"Let's hope not."
"Hey!" Carl called from the bathroom. "They's only one screw holdin this cabinet in place and that's only halfway in."
"I know," Jack called back. "Just twist it out."
One screw? Tom brushed aside questions about his medicine cabinet. The thought of Anya being hurt overshadowed all that.
"Jack, we've got to call the police. Or the Coast Guard, or the Park Service."
Jack stuck his head out from under the table and gave him a you've-got-to-be-kidding look.
"She's a friend, Dad. A better friend than you know. And I owe her."
"For what?"
"For you being alive."
"What are you talking about?"
"She's the one who reported your accident to the police twenty minutes before it happened."
"That's as crazy as going out in this storm. She told you that?"
"She didn't. But I've no question in my mind that's what happened. She knows things, Dad. All sorts of things. And now she needs help. When a good friend needs help, you don't call on somebody else. You go yourself."
The words struck a chord deep within Tom. Yes, he knew that. He'd been taught that. He'd lived that. But where had Jack come by it?
And yet he couldn't allow himself to bend here, couldn't let Jack go out into that storm against twenty men.
"Where's that written?"
Jack slipped out from under the table and rose to his feet, his face barely a foot away. He tapped a finger on the center of his forehead.
"In here. Right in here."
Yes...that was where it would be. But not the only place.
He tapped his son's chest, over the heart. "In there too."
Jack nodded. "Yeah. There too."
And as they stood staring at each other, Tom flashed back to Korea. That had been the Marine code: Nobody gets left behind. At least nobody still breathing. Sometimes you had to leave your dead, but you never left your living. If someone was stranded, or hurt and unable to get out on his own, you went in and got him.
And you didn't call on anyone else because there wasn't anyone better. You were US Marines, the toughest sons of bitches on earth. It was a matter of pride. If you couldn't do it, no one could.
Back at Chosin, when Tom took that piece of shrapnel in the gut, he'd radioed in that he'd been hit and couldn't make it out. He'd expected his buddies to want to come and get him, but figured there was no way with all the shit coming down on the Fifth. But damned if three of them hadn't shown up after dark and carried him out.
"Help me lift off this top," Jack said.
"What on earth for?"
"Let's just do it."
Tom grabbed one side, Jack the other. They lifted it, tilted it, and leaned it against the kitchenette counter. Then Jack reached into the hollow interior of the post and came up with a black plastic bag. Its lumpy contents clunked together as he laid it on the counter.
"What the hell? How'd that get in there?"
"I put it in the other day. Let me tell you, I had one hell of a time maneuvering that tabletop around on my own."
"But what've you got in there?"
Jack reached in and came out with a fist-size lump of metal that he flipped over the counter. Tom caught it, saw what it was—a smooth metal sphere the size of a tennis ball, with a key ring at the top attached to a safety clip—and felt his heart trip over a beat.
"A grenade?"
"M-67s. I had a dozen sent down after seeing that gator."
"Sent down when? I never saw any—" And then it hit him. "The toys. They were in the toys, right?"
Jack gave him a tight smile. "Right. I also—"
"Hey!" Carl called from the bathroom. "You got a gun in this wall!"
"What?"
A gun? In his wall? Tom started toward the bathroom but Jack got there first. Carl had pulled the medicine cabinet from the wall, exposing the studs and the unfinished backside of the Sheetrock of the opposite wall. The end of an empty metal tube jutted a couple of inches up from the lower end of the space. It had a blued-steel finish and looked like an open plumbing pipe until Tom spotted the bead sight on the end and realized this was the business end of a shotgun barrel.
Jack fished it out and handed it to Carl. Its black polymer stock barely reflected the overhead lights.
"Ever use a shotgun?"
Carl laughed. "You kiddin? Fed myself mostly by fishin and huntin before I came to work here. If'n I wasn't no good, I'da starved." He took it from Jack and hefted it. "But I ain't never see one like this before."
Neither had Tom. He saw a breechlock, a magazine tube, but where was the slide handle?
"It's a Benelli—an M1 Super 90, to be exact. I think the semi-auto action will work best for you."
"A semi-auto shotgun?" Tom said. "I didn't even know they made such a thing."
"She's a beauty," Carl said. "I like the rubber grip. Kinda like a pistol."
"Very much like a pistol. Will you be able to handle it?"
"Sure. I told you—"
"I mean"—Jack glanced at Carl's right sleeve—"will you need to modify the stock or anything?"
"Nuh-uh. I'll be fine."
"Great. Excuse me, Dad," he said as he turned and edged by Tom into the front room. "Be back in a minute."
Without another word he ran out into the storm. Two minutes later he returned, dripping, carrying an oblong object wrapped in a blanket Tom had last seen in the linen closet. He pulled it off to reveal another shotgun.
"I'll use this one," Jack said.
With its ridged slide handle riding under the barrel, this one was more like how Tom pictured a shotgun. Its polymer stock was done up with standard camouflage greens and browns.
"It looks military," Tom said.
"It is. It's a Mossberg 590, made to military specs. Very reliable." He started across the front room. "Now...one last thing and we'll be set to go."
Tom followed Jack around to the guest bedroom where Jack pulled out the bottom drawer on the dresser and laid it on the floor. Tom watched in shock as his son reached into the space beneath and produced one box of shells, then another, then another...
"Jesus, Jack! Did you think you were going to war?"
"After I saw that gator, I figured a little old 9mm pistol wasn't going to do the job, so I ordered up some heavy artillery."
"But two shotguns?"
"Well, yeah. One for here and one for the car, in case something happened while we were out."
Carl stepped into the doorway, carrying the Benelli. "What you got this loaded with?"
"With what's known as a 'Highway Patrol cocktail'—alternating shells of double-ought buckshot and rifled slugs." He held up one of the boxes. "Here are our reloads."
Tom felt a tightening in his chest. He didn't know if it was his heart or dismay at what was happening here. He slipped past Carl, went to his own bedroom, and pulled the M1C from the closet. He carried it back to Jack and Carl.
"What are you doing with that?" Jack said.
"Well, since I can't talk you out of this insanity, I guess I'll have to come along."
"No way, Dad."
Tom felt his anger flare. "Aren't you the one who just gave me a lecture on going out for a friend in trouble?"
"Yeah, but—"
"And have either of you ever been in a firefight?" He didn't wait for an answer. "No, of course not. Well, I have. And that's what you could very easily wind up in. You're going to need me."
"Dad—"
Tom jabbed a finger at him. "Who put you in charge anyway? Besides, your mother would never forgive me if I let you go out there without backing you up. I'm in."
Jack stared at him a moment, then sighed. "All right." He held out the Mossberg. "But put away that antique and take this."
"But I'm more comfortable with—"
"Dad, it's going to be dark with all sorts of wind and rain. Let's hope we can pull this off without any gunplay, but if it comes to that, we'll be working close—maybe twenty-five feet, fifty max. A sniper rifle's no good in that situation."
Tom had to admit he was right. He reluctantly took the shotgun.
"But what are you going to use?"
"I'll have the grenades. But I'll also have..." Jack reached back into the space below the drawer and pulled out a huge revolver. It had a gray finish and was well over a foot in length. The barrel alone looked to be about ten inches long.
"Oh, man!" Carl said. "What's that?"
"Took the words right out of my mouth," Tom said.
"A Ruger Super Redhawk chambered for .454 Casull rounds. I do believe this will stop that gator if he shows up again."
"Looks like it'll stop a elephant," Carl said.
A discomforting thought started worming through Tom's brain.
"Jack...you're not in one of these right-wing paramilitary groups, are you?"
He laughed. "You mean like the Posse Comitatus or Aryan Nation? Not a chance. I'm not a joiner, and even if I were, I wouldn't join them."
"Then what are you? Some sort of mercenary?"
"Why are you asking all this?"
"Why do you think? Because of all these guns!"
Jack looked around. "Not so many."
"You didn't answer my question, Jack. Are you a mercenary?"
"If you mean one of those soldiers of fortune, no. But people do hire me to, well, fix things. I guess that might make me a mercenary. But—"
Just then the TV started emitting high-pitched beeps. They all hustled into the front room. A red banner took up the lower quarter of screen, announcing that a hurricane-spawned tornado had set down in Ochopee.
"Where's Ochopee?" Jack said.
"Other side of the state," Carl replied. "Way out Route 41."
Jack looked at Tom. "Anyone wants to back out, now's the time. No explanation required, no questions asked."
Carl grinned. "Hey, I live in a trailer park. You know how tornadoes zero in on them places. I figure I gotta be safer out in the Glades."
Just then, lightning lit the windows, followed by a rumble of thunder.
Tom's gut crawled, but he said, "Let's get moving."
And God help us all.
## 4
Jack drove his paddle into the water to keep the canoe moving against the wind and driving rain. He had a terrible feeling that it might already be too late for Anya, but if not, then the sooner they reached her, the better.
Carl sat in the stern, working the little motor, steering them along the channel. Dad had the front, Jack the middle seat. When the channel nosed them into the wind, the engine didn't have what it took to keep them moving; that was when he and Dad put their paddles to use.
He'd never seen rain like this. He'd expected it to be cold, but it was almost warm. When it wasn't lashing them with horizontal cascades that would put Niagara Falls to shame, it pelted them with huge, marble-size drops that did drum rolls on the hood of his poncho. The rest of the Glades had gone away; the world had narrowed to a short length of the channel's rippling water with only occasional glimpses of its banks. Everything else, including the sky, had been swallowed by dark gray sheets of wet. Only the ever more frequent flashes of lightning and roars of thunder hinted that there might be a world beyond.
Good thing the hardware store had been open so he and Dad could pick up ponchos—dark green, like Carl's—and a hand pump. He didn't want to imagine what this trek would have been like without the ponchos. Jack had his hood pulled tight around his head, the drawstring knotted at his throat. Still he was getting wet.
And the hand pump—they wouldn't have got even this far without it. Into the wind, they paddled; when the twisting channel put the wind to their backs, Jack let Dad rest while he worked the pump to rid them of the rainwater that kept accumulating around their feet.
The canoe had been flooded when they found it. They'd flipped it to empty it, then wasted precious time trying to get the little motor to turn over. Carl finally got it going and they were off.
Jack cupped his hands around his mouth and leaned back toward Carl.
"Did we get to the shallows yet?" he shouted above the din of the rain.
Carl nodded. "Just passed them."
And we didn't have to get out and walk, Jack thought. Testament to the amount of water falling out of that sky.
"Let me know when we're almost to the lagoon."
Ahead of him Jack noticed that his father had stopped paddling. His oar rested across his lap as he rubbed his left shoulder.
"You okay?" he said, leaning forward.
Dad turned sideways. All Jack could see was his profile; the rest of his head was tucked into the poncho hood.
"I'm okay. Just not used to this sort of thing. At least I don't have to worry about the lightning."
"Why not?"
"I tried to lead an orchestra once and found out I was a poor conductor."
Jack gave him a gentle shove. "One more of those and we toss you overboard!" He could see Dad was exhausted, but not too exhausted to come up with a rotten pun. He gripped his shoulder. "Take a breather. We're almost there."
Dad gave a silent nod.
Jack bent his back into paddling, forcing the canoe ahead into the wind. And as he sweated, he planned. They'd reach the lagoon soon. He tried to picture the layout...the houseboats, the huts on the bank. Would the clan be on the boats or ashore? Would they be at the lagoon at all?
Had to be. The lights would keep them there.
Light...it was fading fast. Somewhere on the far side of Elvis the sun was crawling toward the horizon, but the storm swallowed up its light, leaving Jack and company in growing darkness.
Good. The lower the light, the longer it would take the clan to figure out how much backup Jack had brought along.
He felt a tap on his shoulder: Carl.
"We'll be getting to the hummock soon."
The storm seemed to let up as they fought their way into the rainforestlike tunnel of green at the edge of the hummock. The palms, banyans, and gumbo limbo trees seemed to hang lower under the weight of the rain; aerial roots and vines brushed against their ponchos.
"Couple more turns and we'll be in the lagoon," Carl said.
Jack leaned back. "Should we shut off the motor?"
At that moment a bolt of lightning struck close enough for Jack to hear its buzz and sizzle; the almost simultaneous blast of thunder hit him like a fist.
He could just barely hear Carl through the ringing in his ears: "I don't think that'll be a problem. You?"
"Probably not, but shut down anyway."
No telling what kind of vibrations the little motor might set up in the hulls of those ships. Why risk tipping them off?
Wind and rain blasted them again as the canoe slipped out of the tree tunnel and into the relative open. Straight for a while, then around a bend and they were in the lagoon.
At least he thought it was the lagoon. The water was wider and he could see only the near bank on his right, but where were the houseboats? He had a bad moment as he looked around and couldn't find them, then a flash of lightning lit up the area and he saw both boats through the rain, floating straight ahead. The Bull-ship sat to the left, the Horse-ship to the right.
Dad must have spotted them too because he turned and started motioning toward the right bank.
"Put it in over there!" he said.
Jack figured he must have his reasons—and he was, after all, the only one with military training—so he passed the message to Carl.
When the canoe nosed into the bank, Dad hopped out and motioned Jack and Carl ashore. He led them to the lee side of a stand of twisted palms where they could converse without shouting.
"If they're here," Dad said, "they're on those boats. Agreed?"
Jack nodded. "Agreed."
"Okay. Then we need to deploy ourselves around the bank at wide intervals along a hundred-fifty-degree arc, no bigger."
"Why not?" Jack asked.
"Because when you get much closer to one-eighty you run the risk of shooting at each other. Ideally we want all three of us to have line of sight to both boats, but if that doesn't work, then the two flanking guns will concentrate their fire on the nearer boat; the gun in the center can fire on either—wherever it's most needed."
"Dad, I'm looking to get this done without turning the lagoon into the OK Corral."
"Amen to that, but we have to be prepared for a worst-case scenario." Dad patted the Mossberg through his poncho. "To get the most out of shotguns in this rain and low light, we'll need to set up about fifty to seventy-five feet from the boats. That's closer than I'd like, and lots closer than I'm used to, but these conditions don't leave us much choice."
Dad's takeover of the tactics impressed Jack. He seemed to be talking from experience, so Jack deferred to his judgment.
"Just don't set up too near the cenote," Jack warned him. "You might see some lights shining up from it, but don't get curious. Just stay away."
"You mean the sinkhole?" Carl said. "I'll take that spot. The lights've already done what they're gonna do to me."
Dad said, "Speaking of lights, if we do get into a firefight, don't stay in one spot. We can hide pretty well in the rain and the dark, but our guns don't have flash suppressers, so once we start firing, the muzzle flash will give away your position. Fire and move, fire and move. Unless of course you can time your shot to a lightning flash, but that's a lot easier said than done."
Jack swung the plastic bag with the grenades and the big Ruger over his shoulder. "Carl, you take the north position, near the cenote; Dad, you set up on the south end, I'll take the middle; that way I can lob a grenade at either boat should the need arise."
Which he hoped wouldn't. He didn't feature being shot at, and liked his father being shot at even less. The old guy had the experience, and he had the skills, but he also had a body that didn't move or react like it did in its heyday.
"Anyone see any problems with that?"
Dad and Carl shook their heads.
"Good. Okay, once we're all in position, I'll fire a couple of shots to get their attention, then tell them I'm from the Novaton Police Department and demand they release Anya or else."
Dad grinned. "Novaton Police Department? You're planning to kill them with laughter...is that the plan? Better off saying you're from the Miami-Dade Sheriff's Department."
"What if they don't buy it?" Carl said. "What if they start shooting?"
"Then we'll have to shoot back—unless of course they bring Anya on deck."
"Then what?" Carl asked.
"Then we improvise."
Lifting his poncho to reveal the Mossberg, Dad spoke to Carl. "Since these are loaded with alternating slugs and double-ought, I suggest we aim the buckshot at the decks and the slugs at the waterline, preferably near the bow. Anywhere but the superstructure. At this range the boat walls will, I hope, stop most of the shot, but the slugs will go through them like paper, and Anya could be in there."
Carl nodded. "Gotcha. Easy. Those boats is too pan-o-ramic to miss."
Dad looked at Carl, then Jack.
"Don't ask, Dad." Jack gestured ahead. "Let's go."
"And look out for that alligator along the way," Dad said.
Carl shook his head. "I heard Semelee and Luke talkin while I was stuck here and they was sayin Devil was hurt bad. The way they was talking, I don't think he'll be up for chasin us."
"Be on the lookout anyway," Jack said. "Even if he's not, there's still that two-headed snapping turtle."
"Oh, yeah," Carl said. His lips tightened. "Dora."
"Two-headed snapping turtle?" Dad said. "What—?"
"Later, Dad. Just don't get too close to the water."
"Haven't you both forgotten about something else to look out for? What about those flying things that gobbled up Anya's dog and made such a mess of her place? I don't want to run into them."
"A snootful of double-ought buck will clip their wings, don't you think?" Jack said.
Dad frowned. "If you can hit them. The ones I saw in the movie were moving pretty damn fast."
On that reassuring note, Jack turned and led them away from the canoe. Heads down against the wind and rain, they sloshed through the oaks, palms, and cypresses, keeping a good ten feet from the water's edge, heading toward the cenote. Well before they reached it, even through the driving rain, Jack could see the lights flashing up from its depths.
As they arrived at the rim, now only an inch or so above the waterline, Dad leaned close to Jack and spoke in a low voice, barely audible above the storm. "Now isn't this a helluva thing?" He peered down into the flashing depths. "What on earth is going on down there?"
"Not sure," Jack told him. "But you want to avoid too much exposure to those lights."
Dad took a quick step back. "Why? Radioactive?"
Worse, Jack wanted to say, but that would stimulate a lot of questions he didn't have time to answer. So he settled for, "Could be."
Carl stepped ahead and crouched behind the head of a newly fallen royal palm. "This here looks like a good spot. Gives me a good bead on the Horse-ship. I'll park here."
Jack nodded and motioned his father southward. Dad followed, but kept glancing over his shoulder at the lights from the cenote. They seemed to fascinate him.
Along the way they passed the clan's little boats—the Chicken-ship, the No-ship, and others—pulled up, turned over, and tied down on the bank. Jack spied a spot near the old Indian huts to take cover, but he kept walking. He wanted to see Dad as fully protected as possible.
He found him a spot behind the wide trunk of a cypress where he had a good angle on the Bull-ship.
Jack gave the old man's shoulder a gentle squeeze and leaned in close. "Keep your head down, Dad. And if all hell breaks loose, be careful."
His father patted his hand. "I'm the soldier here, remember? You just take care of yourself and don't worry about me."
Jack had a sudden urge to pull everyone out and head back to Novaton. A dark premonition stole over him, a feelin that something terrible was about to happen, that fewer would be leaving here than arrived. But he couldn't turn back now, and he knew neither his dad nor Carl would go. They'd come too far. And Anya needed them.
One more squeeze of his father's shoulder and then he hurried back to the ruins of the Indian huts. He found himself a spot behind a thick support post. He wouldn't have thought it possible, but it began to rain harder.
Jack squatted and spread his poncho like an umbrella over the plastic bag. He removed a few of the grenades and stuck the safety clips into his belt. He pulled out the big Ruger and checked the cylinder. He didn't have a holster big enough to hold it so he stuck it in his waistband. The nine-plus-inch barrel was cold and not a comfortable fit. If Semelee got a look at him she'd probably think he was very glad to see her.
But he wouldn't be. It would be just fine with Jack if he never saw her again.
He rose and started to cup his hands around his mouth when he sensed movement behind him. He whirled, pawing at his poncho, trying to get his hand under its flapping hem, but stopped when he saw what it was: a small towel, tacked to one of the hut posts, was flapping in the wind.
Jack waited to let his racing heart slow—for a second there he'd thought he'd walked into an ambush—then turned back to the water.
He cupped his hands around his mouth and shouted.
"Hello the boats!"
He repeated this three times at top volume before deciding that they weren't going to hear him over the storm. He pulled out the Ruger and pointed it skyward. He'd never fired one of these, and had only heard of the .454 Casull round. He knew it was a monster so he was ready for a loud report and a wrist-jolting kick when he fired two shots in the air. Even so, the boom surprised him.
That ought to wake them up.
He replaced the two rounds as he began calling again.
## 5
"You'll never guess who's out there," Luke said, grinnin and drippin as he came in from the deck. He wore a yellow slicker and a Devil Rays cap. Corley and a couple of the other men trooped in behind him, shakin the water off theirselfs like dogs.
Semelee didn't feel like guessin—specially if she'd 'never' guess the answer—so she waited for him to tell her.
Everybody in the Bull-ship had jumped at the sound of those two shots a moment ago. It'd sounded like a cannon goin off. Luke and the others went out to see what was up. Semelee had heard some shoutin back and forth but couldn't make nothin out of it due to the poundin of the rain on the roof and sides of the boat.
Finally Luke told her: "It's your boyfriend."
Boyfriend? Semelee thought. What's Luke—? Oh, shit.
"You mean that Jack guy? He ain't no boyfriend of mine. I hate him."
She did. Sort of. But that didn't keep her heart from flutterin for a second at the passin thought that he'd come all the way out here in this for her. But that thought flew out the window soon as it came. He'd made it awful clear he wasn't interested in the likes of her.
"Good," Luke said. "Cause I hate him too. I hate anybody who thinks I'm stupid, and he must think we're pretty damn stupid. Know what he said? Said he was from the Miami-Dade Sheriff's office and that he's got a whole passle of cops out there in the dark with him."
"You sure it's him?"
"Sure I'm sure. Recognized his voice, even through the rain. Couldn't see him, but it's him."
"What's he want?"
"Says he wants the old lady back. Callin her 'Anya' or somethin like that."
Semelee felt her stomach plummet. "Then he knows we was there."
She went to one of the little rectangles of glass that served as windows on Bull-ship's deckhouse and looked real hard into the storm. The rain splashin against the glass and runnin down its outside kept her from seein even an inch beyond it.
"He knows somethin," Luke said, "but he don't know everthing, that's for sure."
"But how's he know we was there?" She couldn't imagine Jack just watchin from a window. He and his daddy woulda come out sure, probably with guns a-blazin.
"Don't know, don't care," Luke said.
She turned and saw that Luke had opened a closet and was handin out rifles and shotguns. He pointed to Corley.
"Get below and haul everbody up here."
"What you gonna do?"
He smiled at her again. "Gonna give him a nice warm lagoon-style welcome and make sure he don't leave the Glades—least not alive."
"That really necessary?"
As Semelee watched the men start pilin up from below decks, grabbin guns, and headin for the deck, she felt a little somethin stir in her chest. Like sadness. Like guilt. She'd taken a change of heart about Jack since yesterday afternoon. She'd tried to make him die then, but afterwards she was a little glad she'd failed. Yeah, he'd turned her down right to her face, but he'd only been tellin the truth: I'm taken meant he had someone else he liked better. End of story. He could've lied and then used her like she'd been used before, then dump her like she'd been dumped before. That would've been worse. That didn't make her heart hurt any less, but at least he'd been straight with her.
"I think when he don't get what he's askin for—and he ain't gonna—then I got a feelin there may be some shootin. So I figure we'll shoot first."
"What if you're wrong?" Semelee said. "What if that really is a buncha deputies out there?"
"Ain't wrong. It's him, I tell you."
"All right. Say it is. What if he ain't alone?"
Luke's smile turned real ugly. "I hope he ain't. I hope he brought Daddy along." He lifted his cap and ran a hand over his scabbed-up head. "I got me a score or two to even with that old coot."
Semelee stepped back to the window. Why did he come? This storm's tearin up the place and yet here he comes, loaded for bear, lookin for an old lady he only met a couple days ago. What sort of man does that?
She ducked away from the window as the gunfire started outside.
Whatever sort of man Jack is, she thought with a sting of sadness, he's gonna be a dead one pretty soon.
## 6
Jack had taken cover behind an old fallen trunk at the first sight of a rifle on the Bull-ship's deck. Good thing too, because they'd opened up without warning. Dad and Carl had responded immediately. The element of surprise allowed them to take down a couple of the clan before the rest of them dropped to the deck to take cover behind the gunwales. The Horse-ship crew had their guns out now and the air was filled with wind and water and lightning and bullets and shot.
Most of the fire from the Bull-ship seemed concentrated on Jack's position. Semelee's idea, probably...or Luke's...or both. He'd definitely put himself on the wrong side of those two. When Jack dared raise his head, he fired back with the Ruger. He wanted Luke. If he could take him out, the rest of the clan would lose their steam. But Jack couldn't identify him through the dim light and the rain. And even if he did, he'd be hard to hit. Jack wished he were a better marksman, but knew if by some chance he did hit Luke he'd be a goner. He was firing Cor-Bon .454 Casulls, hard-cast, flat-point, 335-grain rounds that jerked the barrel high every time he pulled the trigger. Which was okay in a way. If he missed, he wanted to miss high. He didn't want one of those big rounds to plow through the hull and hit Anya.
The fire on Jack's position became so intense he didn't dare raise his head to return it. These guys were good shots. When a lull came, he belly-crawled back to the old huts and took a position behind a post. Maybe from back here he'd be able to take the time to aim and make his shots count. He glanced back at that towel flapping in the rain, thinking it ought to be one damn clean piece of cloth by the time this storm is done.
Lightning flashed as he turned back to the boat, revealing a design on the fabric that caught the corner of his eye. Something familiar about that pattern of lines and dots...
Whatever it was caused a ripple of nausea, and a chill, as if something has crawled under his hood and whispered across his neck on spider legs.
Jack fixed his gaze on the cloth, waiting for the next flash, and when it came he saw the pattern again and knew where he'd seen it before.
On Anya's back.
With his blood sludging in his veins, Jack rose and stepped over to the cloth, ignoring the lead whistling around him, because it had to be a cloth, a cloth someone had drawn on, copying the pattern they'd seen cut and burnt and punctured into Anya's back. He reached out and touched it, and when his fingers flashed the message that this was too thick and entirely the wrong texture for cloth, he slumped to his knees in the mud. Somehow he managed to hold on to the Ruger.
A sob burst from his lips, but the grief that spawned it lasted only a few heartbeats before a black frenzy boiled out of the vault where he stored it and took over. Repressing a howl of rage, he rolled back to the post and found his plastic bag of grenades. Breath hissing through bared teeth, he snatched one from within, pulled the pin, popped the safety clip, and waited, counting...
One thousand and one...
The note Abe had included with the grenades said the M-67 fuse gave a four-to-five-second delay between release of the clip and detonation.
...one thousand and two...
It also said each grenade had a kill radius of fifteen feet and a casualty radius of about fifty. Dad and Carl weren't much beyond that but he was only peripherally aware of the risk. His focus was tunneled in on the Bull-ship and nothing was going to pull it away.
...one thousand and three!
As soon as he hit three, he lobbed the grenade up and out, then ducked behind the pole. If it hit the deck and exploded, great; if it exploded above the deck, even better.
But he didn't wait for it to hit before pulling another from the bag. He was popping the clip when the first went off. He poked his head up as he started counting. His throw had been short by maybe half a dozen feet, but not a complete loss. It had exploded at deck level and the screams of the wounded and frightened shouts of the rest were music.
...three!
This one sailed toward Horse-ship—no need for them to feel left out—and it too fell short, but not without doing some damage to hull and human alike.
It looked so much easier in movies.
Jack was ready to pop the clip on a third when he heard someone thrashing through the underbrush to his right. The fact that whoever it was made no attempt at stealth left him pretty sure it was his father, but he raised the Ruger anyway. Sure enough, seconds later, Dad burst from a stand of ferns in a crouch and dropped down beside him.
"What the hell are you doing, Jack?" His eyes were wide; rain ran down his face in rivulets. "Anya's in one of those boats!"
"No, she's not, Dad," he said through a constricting throat. "She's dead."
He frowned. "How can you know that?"
"I found a big piece of her skin hanging back there."
"No!" he gasped. Jack couldn't see his complexion but was sure it had gone waxy. "You can't mean it!"
"I wish I was wrong, but I saw her back the other day and the same marks are on that piece of skin. They skinned her, Dad. They fucking skinned her and hung it out to dry."
Dad placed a trembling hand over his eyes and was silent a moment. Then he lowered the hand and thrust it toward Jack's sack of grenades. His voice was taut, strained.
"Give me one of those."
## 7
Semelee lay tremblin on the floor, head down, hands over her ears. It sounded as if war had broken out. Those weren't just guns firin out there. With the explosions and the way the windows was shatterin, it felt like they was bein bombed.
Luke fell through the door, grabbin onto a bleedin shoulder.
"They got grenades, Semelee! They're killin us out there! Corley's dead and Bobby's leg's bleedin real bad! Y'gotta do somethin!"
"What can I do? Devil's dead and Dora's no good on land."
"The things from the sinkhole, the ones you brought up last night...we need em now. We need em bad!"
"I can't! I told you before—they won't come up till after sundown."
No matter how she'd tried yesterday, she couldn't get those awful winged monsters to come out of the hole while the sun was up. But as soon as it went down, they were hers—or so she'd thought.
She'd almost lost it when she first saw them. She hadn't been able to get a good look at them while they was down in the lights, but once they was up in the air, in the twilight, what she saw scared her so much she almost dropped her eye-shells.
The most horrible lookin critters she'd ever seen.
They was the size of lobsters—not the crawdadlike things around these parts; no, these was thick and heavy, like the big-clawed ones from up north. These things had shells and claws too, but that's where the likeness ended. Their bodies was waisted, like a wasp's, and they had wings, two big transparent ones on each side, sproutin from the top of the body like a dragonfly's.
Chew wasps—that was the name that popped into her head, and it seemed to fit them perfect.
Plus they had teeth. Oh God did they have teeth—each had big jaws that opened wide as a cottonmouth's, and they was filled to overflowin with long sharp transparent fangs that looked like slivers of glass. One of the weirdest touches was the rows of little blue dots of lights along their sides that glowed like neon. They looked like they'd been drug up from the bottom of the sea where the sun don't shine, a place so deep and dark that even God's forgot about them.
God...he must've been havin a real bad day when he made those things. She had to wonder what kind of a world they came from, and how anything else survived with them roamin free.
"It's dark as night out there now! Give it a try! You gotta! They're putting holes in the hull. They're tryin to sink us!"
"But why're they tryin to do that? Why're they throwin grenades, Luke? If they think we got the old lady and they want her back, ain't they afraid of killin her along with us?"
"Who knows why, damn it!" Luke shouted. "They've gone crazy!"
But Semelee caught a look in his eyes, like he was hidin somethin.
"What is it, Luke? What changed their minds? What makes them think she's not here, or that she's dead? You didn't open your big mouth, did you?"
"No. Course not. What kinda fool you take me for?"
"Well, then what? What, Luke?"
Luke looked away. "I guess they found her skin."
"What? How could they do that? You buried it." Luke still kept lookin away. "You did bury it like I told you to, didn't you, Luke?"
He shook his head. "Nuh-uh. I hung it up to let the rain clean it off, then I was gonna tan it...you know, like a hide."
Semelee closed her eyes. If she had a gun right now she'd've shot Luke—right through his stupid, brainless head.
Her thoughts flashed back to last night...
She'd been in a frenzy, completely out of control...so pissed at that old lady for killin Devil and then ruinin her plans for Jack that she just...lost it. All the trouble she had gettin those things to come out of their hole didn't help matters none either. By the time she realized that they wouldn't come out in the day, she was all but frothin at the mouth.
When sunset came, so did the things. She had trouble controllin them from the git-go. Soon as they came out they wanted to run wild, but she managed to gather them into a group and herd them toward the old woman's house. When they got there, they went crazy, rippin through the screen and gnawing through the front door.
Their ferocity frightened the hell outta Semelee, and she remembered thinkin, Oh, God what have I got myself into now? And, bein inside them, she was beginnin to feel some of their bloodlust.
When they got through the door, there was the old lady, standin in the middle of her livin room, all done up in one of them funny Japanese dresses. She just stood there smokin a cigarette. Smokin! It was like she knew she was gonna die. She didn't scream, she didn't cry, she didn't even fight back.
But her plants did. They lashed out at the chew wasps and tried to entangle them with her branches. The wasps splintered them and striped off all their leaves.
But they still couldn't get to the lady because of her little dog. Semelee especially wanted to even the score with that mongrel for killin Devil, but he wasn't going quietly. She'd wondered how such a little thing could've killed the biggest gator she'd ever seen, and last night she found out. That tiny dog fought like a full-grown Rottweiler. He brought down two of the chew wasps before three of them ganged up on him and tore him to pieces.
And then there was nothing between the chews and the old lady. She didn't try to run, she just stood there, like she was acceptin what was comin.
That was when Semelee had second thoughts. She sensed somethin special about this lady—something extra special—and had a feeling she'd be losing somethin precious if she killed her.
Maybe it was the way she was just standin there. She had to be scared outta her mind but she wasn't showing it, not one bit.
But the thing that most made Semelee want to hold off was knowin that this lady wasn't just gonna be killed, she was gonna be torn apart. Much as Semelee hated her for messin with her plans, she didn't know if she could go through with that. The other folks she'd sacrificed here at Gateways had been stung or bit or pecked up, and they'd died later...not right in front of her.
Semelee was gonna have to watch this and she didn't have the stomach for it. Maybe gettin her house wrecked and her dog killed would be enough for the old lady. Maybe she'd learn her lesson and stop messin where she didn't belong. Maybe she'd even have a heart attack and die later. A lot better'n bein torn to pieces.
But when Semelee tried to turn the chew wasps around and bring them home, they wouldn't go. They smelled blood and there was no stoppin them. They lit into the old lady. And what did she do? She stood there and raised her arms straight out from her sides and just let them come.
Semelee wasn't sure if it was the bravest or craziest thing she'd ever seen, but she did know it was horrible to watch.
More than watch. Semelee was in close with the wasps, inside them as they gouged the old lady's flesh, crunched her bones. She could almost taste it, and gagged now at the memory. They was so fierce they didn't even let her body fall to the ground. They ate her upright, even slurped sprays of blood right out of the air. And no matter what Semelee did she couldn't pull them away. She wanted to drop the eye-shells but was afraid the chew wasps would turn on the clan who'd gone there just to see what these ugly-lookin things could do.
Finally, when they were through, there was nothin left of the old lady but the skin of her back. For some reason, the chew wasps wasn't interested in it. They gobbled her up from head to toe, but left that rectangle of skin.
And when they was finished they started listenin to Semelee again. She quick got them outta there and back to the sinkhole. Soon as they was back where they belonged, Semelee yanked off the eye-shells and got real sick.
Back at the old lady's house, Luke did two things, one smart and one dumb. The smart thing was pickin up the two dead chew wasps and bringin them back to the lagoon. If people came lookin for the old lady and found those, it'd be in all the papers and everyone'd assume they came from the Glades. Soon there'd be scientists and hunters and cops and thrill seekers all over the place, including the lagoon. The clan's whole way of life'd be messed up.
The dumb thing Luke did was bring back the old lady's skin. He—
The boom of another grenade—sounded like it must've exploded over by Horse-ship—yanked Semelee back to the here and now.
"Why, Luke?" She finally opened her eyes and stared real hard at him. "Why'd you do such a fool thing?"
"I wanted to keep it. You know, kinda like a souvenir. I like all those marks. They're almost like a map. But never mind that. Y'gotta try those wasp things again, Semelee! You just gotta!"
She didn't want to tell him that she was afraid to. She hated the way they made her feel...like all dark and ugly inside, with this endless hunger. Even with the gunfire, the explosions, the howlin wind, the leakin roof, the thunder and lightnin all around her, this seemed like a better place than where she'd been last night.
But she couldn't just sit around and do nothin while the whole clan got massacred. She had to do somethin...and there was only one thing she could do.
Her gorge rose as she pulled the eye-shells out of her pocket.
"You're gonna do it?" Luke said, a grin spreadin cross his face.
She nodded. "Yeah, but you gotta get outta here."
The grin collapsed. "But Semelee...there's all sorts of shootin out there."
"Then get out there and shoot back. Just leave me alone so I can save our asses."
"Okay, okay."
He headed for the door in a crouch, then crawled out onto the deck.
Taking a deep breath, Semelee pressed the shells over her eyes and went searchin for some chew wasps...
## 8
"We're not doing a whole helluva lot of damage with these things," Dad said after they'd watched the latest grenade sail through the air and explode off the bow of the Bull-ship.
Jack had to agree. He would have thought that something that small and weighing almost a pound wouldn't get tossed around by the wind. But this was no ordinary wind. He'd tried compensating for it by adjusting his throw but the trouble was you couldn't wing these things like a baseball; you had to lob them, and the wind kept changing direction.
"We've caused some hurt, though."
"Not enough," Dad said, his expression grim. "After what they did to Anya, they..." He swallowed and shook his head. "They shouldn't be allowed to live."
"I don't think we'll be able to kill all twenty guys."
Dad gave him a strange look. "I said they shouldn't be allowed to live. I didn't say we should do the killing."
Oops. "Oh. Guess I misunderstood."
"You're scaring me, Jack."
"Sometimes I scare myself."
Just then Jack heard something that sounded like a scream. He looked over toward Carl but couldn't find him in the dark. Then lightning flashed and he saw him rolling on the ground as he fought something that had clamped onto his right shoulder. Jack couldn't get a good look at it, but whatever it was, it wasn't alone. More of them were lifting out of the cenote and weaving toward Carl. The one that had him was too close for Carl to shoot at, so he was using the shotgun as a bat. But Jack could see that he wasn't getting anywhere.
He slapped his father on the back. "Stay here and keep firing at the boats. Keep them pinned down. When you reload, forget the slugs and fill up on shot. I think we're going to need it."
"Where are you going?"
"Carl needs a little help."
Rising to a crouch, Jack pulled the Ruger from under the poncho and ran through the rain. Lightning flashes lit the scene, and as he neared Carl and got a better look at what was attacking him, it almost stopped him in his tracks. The thing clinging to his shoulder had the head and saber-toothed jaws of a viper fish, the shelled body of a lobster on steroids, and two pairs of long, diaphanous wings. Another of its kind was gliding in for its own piece of Carl.
Jack stopped, knelt, took aim with the Ruger and fired. He scored a hit. The big Casull slug tore into the flying thing, leaving only a spray of greenish blood and a pair of still-flapping wings. Then Jack leaped next to Carl, rammed the Ruger's muzzle against the eye of the thing chewing on him, and pulled the trigger. This time, not even the wings remained.
Carl groaned. "It hurts, Jack!" His left hand was covered with blood where it clutched his shoulder through the shredded poncho. "Oh, God, it hurts!"
Jack took only a quick look, wincing at what looked like exposed bone and a dozen crystalline teeth still buried in the ragged flesh, then turned back to the cenote. Three more of the things were up and coming their way. He grabbed the Benelli and started firing. The semiautomatic action let him get off four shots quickly. They weren't all direct hits but the shot tore up the wings of the ones it didn't dismember.
"Where are your shells?" Jack shouted.
Carl jutted his chin toward a box on the ground. His teeth were bared in agony. He seemed in too much pain to speak.
Jack started reloading the Benelli's magazine. If he'd known he'd be facing these things he would have had Abe send down flechette rounds.
"Think you can walk?"
Carl nodded.
"Okay, then. Get over to where my dad is. I'll cover you from the rear."
Spreading out had been a good idea against the clan, but it meant certain death against these things. Time to circle the wagons.
"It's Semelee," Carl gritted as he lurched to his feet. "She's controllin them." Then he staggered off.
Jack turned back to the cenote and found half a dozen more of the things hovering over the opening in a cluster. He ducked behind a palm trunk and fired once into their center, knocking down two. They fell into the abyss but were replaced by four more.
Jack felt his stomach knot. This wasn't good. He hadn't brought enough ammo. But he'd brought his father and Carl. That made him responsible for them.
In the background he heard his father firing methodically, rhythmically, at the boats.
Save some of that ammo, Dad, he thought. We're gonna need it.
And now another four joined the flock. But they didn't swarm his way...their movements were sluggish and they didn't seem to know he was there. They milled about, looking confused. What were they waiting for? Reinforcements?
If more were coming up from the cenote, maybe Jack could ambush them along the way. He unclipped a grenade from his belt—only a couple left—pulled the pin, and lofted it toward the cenote. It passed through the swarm and down into the opening. A few seconds later he saw a flash, heard a boom, but that was it. The ones fluttering over the hole didn't even react.
If this were a movie like Rio Bravo, he'd stumble onto a crate of dynamite, conveniently left behind by a construction company, and use it to seal the cenote. But this was Jack's world, not Howard Hawks's. Things never seemed to work out that way for him.
He heard a scream behind him and recognized the voice this time: Carl again. He looked around and saw him staggering in a circle at the water's edge. One of those things had its fangs buried in the back of his neck...and it was chewing...
Where'd that one come from?
Jack leaped to his feet and took off on a run. He couldn't use the shotgun without hitting Carl too, so he pulled the Ruger. But before he could use it, Carl pitched over backward into the water.
That wasn't all bad. The cenote thing didn't seem to like water. It loosed it's grip and buzzed back into the air, banking and gliding toward Jack. He already had the Ruger up. He waited until it was close, then fired at it head on. It dissolved in an explosion of green. As its wings fluttered to the ground, Jack dropped the Benelli and the Ruger and jumped into the water to help Carl, who wasn't doing too well.
The water was waist deep and cool, its surface churning and bubbling from the wind and rain. The muddy bottom was slippery and sloped off on a steady decline. A bullet whizzed by, then another. Someone on the Horse-ship had spotted them. Jack heard Dad's Mossberg boom, then a cry from the boat, and the bullets stopped coming.
"Carl!" Jack shouted as he leaned forward and stretched out his arm. "Give me your hand!"
Carl, with his poncho floating around him like a lily pad, thrashed and splashed and kicked his way shoreward. Jack grabbed his outstretched left hand and began hauling him in.
Suddenly Carl was jerked back. He let out a scream of pain and Jack was barely able to hold on to him as something pulled him back toward the center of the lagoon.
"Oh, my leg!" he wailed. "My leg! It's Dora! She's got me! Don't let her have me, Jack!"
"I won't, Carl."
He started sobbing. "I don't wanna die, Jack. Please don't let her—"
And then his head plunged below the surface. Jack tried to dig in his heels but the bottom was too slippery. Another powerful tug pulled Jack forward so hard he went face first into the water. He was only under for a few seconds, but during that time he lost his grip on Carl's hand. His feet found the bottom and he stood again, shaking the water from his face and eyes. He was shoulder deep now.
"Carl!"
Nothing. No reply, nothing but empty, wind-and rain-swept water stretching before him. He shouted the name again and thought he saw a hand break the surface and claw the air maybe fifty feet away. But it was there for only a second—if it was there at all—and then it was gone.
"Oh, Carl," he said softly, staring at the spot. "You poor bastard. I'm sorry. So sorry..."
A lump formed in his throat. A good, simple man was gone. Jack had known him just a couple of days, but he'd come to respect him. He still didn't know what had gone wrong with Carl's right arm, but that didn't matter. Carl hadn't let it stop him from leading a useful life. He'd adjusted, with no apologies, no excuses.
A bullet whizzed by Jack and he realized he was a sitting duck out here.
My fault, he thought as he quickly waded ashore. If I hadn't bribed him to take me to the lagoon, if I'd just said no tonight when he wanted to come along, he'd still be alive. Probably be sitting in his trailer right now watching his TV.
My fault. But not all my fault.
It's Semelee...she's controllin them.
Right. Semelee.
Jack reached the bank and climbed up onto the mud. He looked toward the cenote and saw maybe twenty of the winged things clustered over the opening. As he watched, they began to fan out and glide toward him.
His blood cooled at the sight. No way he and Dad could bring them all down, even standing back to back with shotguns. Some of them would get through. And once they got you down, you were finished.
Couldn't stop the winged things...but maybe he could stop the one controlling them.
With the things trailing him, Jack ran back to where his father was still firing at the boats. He heard cheering from the decks as the clan spotted the winged things on Jack's tail. They didn't shoot. Probably thought it would be more fun to watch him get gobbled up like Anya.
"Behind me, Dad! Incoming!"
Dad was crouched behind a tree, with the trunk between him and the boats. Jack dove for the ground, sliding through the mud on his belly as his father looked around.
"Where?"
"Right behind me!"
Lightning flashed and he saw his father's jaw drop.
"Dear God! What are—?"
"Don't talk, shoot!"
And shoot he did, pumping round after round out of the Mossberg into the air behind Jack. Jack didn't look around to see what effect he was having. He assumed it was about as good as it got. He laid the Benelli across Dad's knees for when the Mossberg ran dry, then seated himself back to back with his father and turned to the Bull-ship. If Semelee was anywhere, that would be the place.
He wiped the rain from his eyes and took aim at the superstructure. The big Casulls would rip through it, in one plywood side wall and out the other. He couldn't be sure he'd hit Semelee, but at least he could distract her...
## 9
This was so hard...
Semelee crouched in the dark of the cabin and pressed the shells tighter against her eyes. The chew wasps hadn't wanted to leave the sinkhole until the sun was down, but she'd forced them. She'd tried that yesterday and it hadn't worked, but this time she was able to coax them up. Maybe it was the storm or the nightlike darkness up here. Whatever the reason, they came. But so slowly...like only one or two at a time.
Then, once she got them outta the hole, she could barely see. Had to be because of the sun. Even though it was hidden behind mountains of storm clouds, it was still above the horizon; she guessed that whatever was filterin through was enough to affect the eyes of the chew wasps.
But she'd been able to see Carl who was right close to the hole and shootin at the boats. Traitor to his kin! She set a couple of the wasps on him, then went back to draggin others up.
Suddenly one of the ones on Carl got blowed up. And then the other. She seen it was Jack doin the shootin, and though she didn't hate him like before, she couldn't let this stand. She had to end it between them. One of them had to go. Semelee preferred Jack.
She had a whole bunch of the chew wasps up by then but couldn't get them organized. They wanted to go here and there and it was just about all she could do to keep them together. Jack blasted a couple of them out of the air and then got four more with a grenade in the hole as she was pushin them up.
She had to attack with what she had, but couldn't get the swarm to move. She could control one of them, though, so she sent it after Jack. Somehow it wound up on Carl instead. The wasps seemed attracted to sound and movement, and Carl had been makin plenty of both.
But she didn't have to send Dora after Carl when he went in the water—Dora did that on her own.
Good-bye, Carl.
Finally she'd got the swarm to move. She didn't know why she suddenly had more control. Maybe cause the sun got closer to settin while she was chasin Jack. Didn't know, didn't care, all she knew now was she was on the hunt. And though her stomach turned at the thought of havin to go through another chew-up with these things, it had to be done. The survival of the whole clan depended on her stoppin Jack and whoever was with him—probably his daddy.
As she guided the wasps after the runnin Jack, she heard the guys on the deck start to yellin. She wished they'd shut up. The chew wasps kept wantin to turn toward the noise. The voices pulled at them. She had to keep forcin them to stay on Jack's trail.
Suddenly a piece of the wall exploded and showered her with splinters as something whizzed by her head. She was already crouched on the floor in a corner. Now she dropped flat, and just in time too. Another big bullet smashed through the cabin, low this time, just about singeing her butt.
He's tryin to kill me!
She had to move those chew wasps in on Jack and his daddy. Now!
The old man was shotgunnin them, so Semelee split the swarm into two groups. She veered one left over the water, and the other around back. She'd catch em in the middle and—
A third big slug blasted into the cabin then, but this one didn't go all the way through. It plowed into one of the benches of the picnic table and sent it flyin against her. She cried out as it conked her on the head. She didn't think—she put her hands up to protect herself and dropped the eye-shells.
"Oh no!" She started feelin around on the floorboards, real frantic like. But it was so dark in here. "Where'd they go?"
She couldn't control the chew wasps without em. They'd all go flyin back to the sinkhole if she wasn't there to hold them.
Or maybe they wouldn't.
Semelee wasn't sure which would be worse.
## 10
"Jack!" Dad shouted. "Look!"
Jack was reloading the Ruger, readying to riddle the Bull-ship's superstructure with a few more Casulls. He'd been leaning against his father's back, getting rocked forward whenever Dad's shotgun went off, rocking back with the recoil from the Ruger.
He half turned, not sure of what he'd heard. His ears were ringing from the thunder and the booms of the weapons.
"What?"
"Those things. They were all clustered together at first, then they started dividing into two groups, and now..."
Jack turned further and squinted through the rain. He watched for a moment as the cenote things buzzed around in disarray, practically bumping into one another in midair. It looked like they didn't know where they were, but the men from the boats were still cheering them on.
One of the things veered out over the water; two more followed it; then the whole swarm was making a beeline for the boats. Suddenly the cheering stopped, replaced a couple of heartbeats later by the reports of rifles and shotguns. Jack saw the clan knock a few down, but then the swarm was upon them. The shooting stopped, replaced by screams of pain and panic.
## 11
Semelee waited for the lightnin to flash again. That was the only time she could see what she was doin. Here! A new flash, coming through the broke windows—where was they? She crouched on her hands and knees, searchin the floor. Where was those damn eye-shells?
At least the big bullets had stopped poundin through the walls. Not for long she bet. Probably just reloadin. In another minute—
Somebody started screamin outside. Then another. She recognized Luke's voice among the hellish choir. He sounded like he was bein tortured. She jumped to the door and peeked out.
The chew wasps! They was attackin the clan. Oh shit oh shit oh shit! What was she gonna do?
Another lightnin flash, this time through the doorway. She looked around just in time to see the shells, lyin on the floor right up against the wall to her right. She jumped on them and clutched them tight in her fists.
Thank God! She had them. Now she could turn the chew wasps away and get them headed back to where they should be—on Jack and his daddy. But as she raised them to her eyes the door burst open and somethin came staggerin into the cabin.
Semelee screamed as it lurched to the left, then the right, then stumbled toward her. Whatever it was, it didn't look human. It let out a muffled screech and then the lightnin flashed and Semelee screamed again. It was a man with three of the chew wasps hangin on him. One on his leg, the other with its head buried in his flank, and the third with its teeth worryin his face. He screeched again, then spun and collapsed onto his belly. He twitched a few times, then lay still.
Another flash of lightnin gave her another look at him. Through the rips in his shirt Semelee saw scales and finny spines on his back and knew who it was.
"Luke!"
Her eye-shells. She could use them to get Luke free of the wasps. But before she could get them up, the one on Luke's leg let go and buzzed straight at Semelee's face. She stumbled back and fell out the door onto the deck and into a hell on earth. Chew wasps and blood-soaked men everywhere—and the men who wasn't screamin wasn't movin.
Semelee's arrival got their instant attention. The chew wasp that chased her out of the cabin was still comin, but so were others from the deck. The only place to go was the water.
She slipped in blood and banged her knee as she tried to get up, then broke into a low run and dove into the water. As she kicked toward shore she knew it would take her right into the sights of Jack and his daddy. She pressed the shells over her eyes. She had to get back control of the chew wasps and give those two somethin else to worry about before she came up for air.
## 12
During a lightning flash Jack caught a glimpse of someone—someone small and slim with dead white hair—leaping off the Bull-ship and diving head-first into the water. He watched a couple of cenote things chase after her and hover a couple of feet over the water, waiting for her to surface.
He tapped Dad on the arm. His father was watching the strobe-lit carnage on the boat decks in horrid fascination. Jack had to tap him again.
"Hey, Dad. Which one of those is loaded?"
Dad shook himself free of the spectacle. "Both now."
"Give me one, will you?"
Dad handed him the Benelli. Jack took aim at the nearest winged thing, not so much from a desire to protect Semelee—she deserved just about anything that happened to her—but because he wasn't up for watching someone being eaten alive.
The shotgun boomed, rocking his shoulder, and the nearest thing blew apart. But its companion, instead of retreating or continuing to hover, darted straight for Jack.
He fell back, raising the Benelli. Good thing it was semiautomatic—those things could move. His shot went a little high, missing the body but dissolving the right pair of wings. It went into a spin and landed on the edge of the bank, vibrating its remaining wings and gnashing its teeth in fury as it made circles in the mud.
Movement on the surface of the lagoon caught Jack's eye. He saw a white head begin to emerge from the water. He took aim with the Benelli but hesitated. He wasn't sure why. Maybe because he felt responsible. Maybe if he'd let her down a little more easily she wouldn't have attacked him, then Anya. Maybe something about her pathetic desire to fit in touched him. Or maybe he couldn't bring himself to blow holes in a young woman, no matter how sick and twisted she was.
Whatever the reason, he dropped the shotgun, grabbed the cenote thing by the roots of its remaining wings, and lifted it. It looked heavy but he found it surprisingly light. It writhed in his grasp, trying to twist around and gouge him with those diamond teeth, but its carapace limited its agility.
Jack leaped off the bank and into the water.
"Jack!" he heard his father cry. "What in God's name are you doing?"
Jack didn't answer. Holding the cenote thing high, he splashed toward where Semelee was emerging from the water. He noticed she was holding two shells over her eyes.
The shells—that was what she'd wanted them for. Somehow they let her control these things.
And I helped complete her set.
He also noticed the other winged things rising from their feasts on the decks of the two boats and heading his way. He put everything he had into forcing himself through the water.
When he reached her he grabbed the back of her hair. He yanked downward, hard, stretching her throat, and held the crystalline teeth of the cenote thing inches from her skin. The twisting, gnashing jaws reminded him of a wood router.
"Drop the shells! Drop them now, Semelee, or this thing gets a free lunch! Don't think I'm bluffing! You may have been right about me not shooting Luke the other day, but this is different. After what you've pulled in the last twenty-four hours, I'm more than ready for payback."
"Okay, okay," she said, but kept the shells over her eyes. "Just let me send the chew wasps back to the sinkhole."
Chew wasps...a perfect name.
"You do that."
The approaching chew wasps veered away and headed for the cenote, its lights faintly visible through the rain. Jack watched them fade into the mist, then, with his free hand, pulled Semelee's hands away from her face. He hadn't forgotten about Dora. He took her by the upper arm and guided her toward the bank.
As Jack pulled her up on land, he heard Dad call his name. He glanced over and saw him pointing toward the lagoon.
"Who or what is that?"
Jack turned and stared. He saw nothing at first, then the lightning flashed and he spotted a man in a suit standing at the center of the lagoon. Not in the lagoon—on it. No, not just standing on the water, walking on it. His stride was long and purposeful, moving him along at a good pace, yet without the slightest hint of hurry.
Jack tossed the partially dewinged chew wasp into the lagoon where it sank like a mob hit. He squinted through the storm. Couldn't make out the man's features, but as he neared, Jack noticed that he seemed to be moving in a bubble—not something with a membrane, simply an area around him, a dry area. The rain driving at him from all directions didn't touch him. And it didn't sluice away, it simply...went away.
"Oh, God!" Semelee cried, cringing against Jack. "It's Jesus come to get me for my sins!"
"You've got a lot of things to answer for, but I don't think that's Jesus."
Not unless he's taken to wearing Armani, Jack thought.
Of course he hadn't a clue as to the designer—if an Armani suit introduced itself, he'd have to ask it for ID—but it looked expensive, maybe silk, charcoal gray, perfectly tailored, worn over a black shirt buttoned to the collar. Very Euro, this water strider.
When the man moved close enough for Jack to make out his face, he felt his blood congeal. He knew that face, that supercilious expression. He raised the Benelli and roared.
"Roma!"
Jack held him accountable for Kate's death—at least indirectly—and for a lot of other things that had gone wrong in his life since they'd met at that conspiracy convention last spring. He'd called himself Sal Roma then. Who knew what he was calling himself now. He'd tried to kill Jack then and almost succeeded. Either he or the Otherness or the two in league had tried to kill Gia and their baby just last month. Now it was payback time. No hesitation—he wasn't sighting down on a waifish woman, this was the "Adversary" Anya had mentioned, the One whose True Name she refused to speak.
"Good-bye, whoever you are," he whispered, and pulled the trigger.
Or tried to. It wouldn't budge. Jammed!
And then Roma glanced at him and Jack felt himself lifted through the air and slammed back against a palm trunk. The pain of the impact on his spine blew all the air out of him and blurred his vision for a few heartbeats. His knees turned to jelly and he slid earthward to end up sitting in the mud, propped against the palm.
"Jack!" he heard his father cry from what seemed like the end of a long hallway. "Jack, are you all—?"
Jack's vision cleared in time to see his father tumble back into the brush and disappear from view.
He wanted to shout to him but his voice wouldn't work.
Fear spiked his chest. Was Dad hurt? Was he even alive?
Jack tried to get to his feet but couldn't move. For a panicky instant he thought he was paralyzed from a broken spine, then realized that something was holding him in place, something he couldn't see or feel but powerful enough to press on him so effectively that all he could do was breathe. He tried to shout to Roma but couldn't do even that. He was at Roma's mercy.
But Roma didn't seem interested in him, didn't even glance toward Jack as he casually stepped onto the bank to stand not two feet away, facing Semelee.
Semelee cringed back as he stared at her.
"So," Roma said. Jack heard him clearly. The rain and wind seemed to be easing up, although lightning still flashed all around them. "You're the one who's trying to usurp my name."
"Name? What name?"
"You know...the one that doesn't belong to you."
"You mean Rasalom? It does belong to me. I'm Rasalom."
He slapped her face. The move was so quick Jack would have wondered what had happened if not for the sound of flesh hitting flesh, and the sight of Semelee staggering back a step as her face jerked to the right. Jack could almost feel the sting.
And then it hit him—Rasalom. That was the fuck's True Name.
"Never," Rasalom said softly, with no show of emotion, "ever refer to yourself by my name."
"Who says it's your name?" Semelee cried, baring her teeth.
Jack had to hand it to her—she wasn't cowed. And the way she took the blow...clearly she'd been slapped around before.
"I do," Roma said softly. "And the only reason I haven't pulled your limbs and head from your torso is that you somehow—through pure dumb luck, I'm sure—managed to find a way to kill the Lady. For that I am in your debt. But don't press your luck, little girl."
"Ain't luck," she said. "And I ain't no little girl! I was down in that hole, in the lights, and I heard the voices. They told me I was the One and that my name was Rasalom."
He slapped her again, harder, and this time she went down. She lay in the mud, rubbing her reddened cheek. A few minutes ago the rain might have soothed it, but it was clearly easing up.
"This is your last warning," he said. "You are not the One. What you heard was talk about me, not you."
"No!" she screamed, struggling to her feet and backing away. "I'm the One, and my name is Rasalom! Rasalom-Rasalom-Rasalom!" She raised the shells and pressed them over her eyes. "And now you're gonna pay. Nobody pushes me around anymore! Nobody!"
Jack knew what was coming and found himself rooting for her.
Enemy of my enemy...
He looked over toward the cenote and saw half a dozen chew wasps rising from the opening. He guessed they hadn't been too far down.
Oh, yes...Rasalom was in for one messy, bloody, and—Jack hoped—painful death. He was glad for a front row seat.
The wasps arranged themselves in V formation and charged, homing in on Rasalom.
Jack braced himself. This was going to be ugly, but he wanted to watch every second of it.
Rasalom remained facing Semelee, his back to the cenote. When the wasps were almost upon him, Rasalom gestured with his left hand—little more than a wrist-flick, like a diner signaling a waiter that the amount in the wineglass was quite sufficient, thank you—and they stopped, hovering around him like bees guarding a hive.
Jack heard a low-pitched screech from Semelee. Her teeth were clenched and bared as she struggled for control of the chew wasps. Jack could tell by the vaguely amused twist of Rasalom's lips that he was enjoying the struggle and that she didn't have a chance.
Finally he seemed to tire of the game. Another flick of his hand and the wasps were on her like ants on a sugar cube. She dropped her shells and tried to bat them away but they attacked from all sides and she went down in sprays of red, kicking, thrashing, writhing. Her screams as they tore her flesh were awful to hear. Jack couldn't help wonder if Anya had wailed like that.
Jack looked away, toward Rasalom, and almost worse than the screams was the avid look on his face as he stood over her and watched her death agonies.
If he could move an arm, just one arm, he could pull out one of the grenades still clipped to his belt and frag this bastard. But his body wouldn't respond.
As soon as Semelee's screaming died away in a gurgling moan, Rasalom seemed to lose interest. He sauntered to where Jack sat propped against the tree trunk and stood over him.
Now it's my turn, he thought as his bladder clenched.
He hoped he didn't go out screaming like Semelee, but the pain of being eaten alive had to be...his imagination failed him.
The rain died to a drizzle and the sky lightened fractionally as Rasalom stared down at him. Again Jack tried to speak but his voice was locked.
Then he gave Jack's foot a dismissive kick.
"My instincts tell me to kill you now, that you'll be a stone upon my path. But I can't see you ever being too much of a stone for me to kick aside any time I wish. Besides, killing you now might be something of a favor. It would spare you so much pain in the months to come. And why should I do you a favor? Why should I spare you that pain? I don't want you to miss one iota of what is coming your way."
The words drove a cold spike through Jack.... so much pain in the months to come...
What did that mean? What was going to cause it? And how did he know? Jack wanted to shout the questions but couldn't even whisper.
He struggled to move. He wanted at this smug son of a bitch, wanted to smash his jaw and rip out his tongue.
Rasalom glanced back to where Semelee had been. A partially flayed skull and a twisted mass of blood-matted white hair were all that remained of her. The chew wasps milling above her seemed confused; two of them bumped in midair and started to fight. Was it the increasing light? Was that what was bothering them?
Rasalom made another of his little gestures and the wasps darted for the cenote. He pointed toward what was left of Semelee.
"Physical pain is mere sustenance. But a strong man slowly battered into despair and hopelessness...that is a delicacy. In your case, it might even approach ecstasy. I don't want to deprive myself of that." He frowned. "Of course there's always the risk that what's coming will only make you stronger. But it's a gamble I'm willing to take. So for now, you live on. But as soon as you stop amusing me..."
He let the words hang as he turned and stepped off the bank onto the water.
As Rasalom strode away, Jack felt the pressure against him ease, but slowly. He wasn't able to regain his feet until Rasalom was out of sight. His first urge was to go after him, but that dissolved in a blast of anxiety about his father. He rushed over to where he'd last seen him and found him sprawled in a clump of ferns, his legs and arms splayed in all directions.
Jack rushed toward him. "Dad!"
Was this the sort of pain Rasalom was talking about? He'd lost Kate, now he was going to lose his father?
But as Jack reached him, he moved.
## 13
Tom sat up and ran his hands over his arms and legs.
I can move! I can feel!
Dear God, I thought—
He looked up and saw Jack skid to a stop before him.
"Dad—you okay?"
"I thought I'd had a stroke! One moment I was standing by that tree. I saw you fly backwards, then the next thing I knew I was on my back and couldn't speak or move a finger."
Jack reached a hand down to him. "Can you get up?"
Tom let his son help him to his feet. He brushed himself off and looked around. He felt shaky and a little weak. Well, why not? He was seventy-one and had just experienced the firefight of his life. He'd been in battle before, but against other men, other soldiers. This time...
"Jack! What happened here? Who was that? Was he really walking on water?"
"That's what it looked like."
Jack's eyes were flat. Not hard and cold like before when he looked like murder personified, but Tom sensed that he'd put up a wall.
"What's going on, Jack? A girl who can control snakes and birds and even flying things from hell—and I'm sure that sinkhole goes straight to hell—and a guy who walks on water...what's happening to the world?"
"Nothing that hasn't been going on for a long, long time. Nothing's changed except you got a peek behind the curtain."
"What curtain?"
What was he talking about? Had Jack snapped under the stress of what he'd been through...or had he been through something like this before...something even worse?
"It's over, Dad."
"What's over?"
"Semelee, the chew wasps, the guy on the water—"
"But you knew him. You called him by name—Roma, wasn't it?"
"Just let it go, Dad. Tuck it away and forget about it. It's over." He looked up. "Even Hurricane Elvis is over."
Tom realized then that it had stopped raining. He could still hear the rumble of thunder, but the wind had died, leaving the air deathly still. He followed Jack's gaze, and through the partially denuded tree branches he saw clear sky, light blue, tinged with orange from the sinking sun.
Over...for a while there he'd thought the storm would never end.
He looked around...at the fallen palms and cypresses, at the slowly sinking houseboats canted in the leaf-and debris-strewn water, at their red decks and the mutilated bodies littering them like jackstraws.
Tom's mouth went dry. "Did we do that?"
"Some of it." He didn't seem the least bit fazed. "We can take credit for the holes in the hulls and some of the blood, but Semelee bears the freight for the rest. She's the one who called those chew wasps out of the cenote and lost control of them. Good thing too. Otherwise they'd be standing here looking at what was left of us."
Jack picked up one of the shotguns and hurled it far out into the lagoon.
"What—?"
"Evidence."
The second shotgun followed the first. He saw Jack pull the pistol from his belt, look at it, then tuck it back in.
Tom glanced once more at the carnage on the boat decks, then looked again. Had one of the bodies moved?
"I think someone's still alive out there."
"Probably not for long."
"Do you think we should—?"
Jack turned on him. "You've got to be kidding. A few moments ago they were trying to kill us."
"In the Corps we always treated enemy wounded."
"This isn't the Corps, and this isn't war. This is a street fight that just happened to take place where there aren't any streets." His face twisted, almost into a snarl. "What do you think we're going to do? Paddle a couple of them back and lug them to a hospital? How do you explain their wounds? How do you explain the double-ought buckshot in their hides? In this system, you'll wind up behind bars while they lounge around a hospital. And when they're all fixed up, some ambulance chaser will hook up with them and file civil suits to clean you out of everything you own, every penny you've saved up your whole life."
Tom was seeing another side of Jack and wasn't sure he liked this one.
"But—"
"But nothing!"
He turned and stomped off to one of the old huts and returned a moment later with something dangling from his hand. He stopped before Tom and held it up.
"See this?"
It was rectangular and looked a little like parchment, but it was too supple for that. It was patterned with crisscrossing scars and round, punctate depressions the size of a pencil eraser. When Tom realized what it was he took an involuntary step back.
"Right," Jack said. "This is all they left of Anya, and then they hung it up to cure. Now tell me how much you want to risk to help one of those bastards."
Tom felt a rising fury. Anya...what they'd done to Anya...a part of him wanted to paddle out there and finish off any survivors. But he couldn't allow himself to step over that line.
He shook his head. "Nothing. They're on they're own."
"Damn right."
Jack stared at the grisly remnant in his hands, then looked around. He didn't seem to know what to do with it. He appeared to come to a decision as he rolled up the skin and tucked it inside his shirt.
"What are you going to do with that?"
"It's all that's left of her. I think she deserves some sort of burial ceremony, don't you?"
Here was still another side of Jack. Tom sensed it could be a living nightmare to be his son's enemy, but a very good thing to be his friend.
He nodded. "Most definitely. Now that the storm's over, we'll take her home and find a place to lay her to rest."
Jack looked up at the sky. "Good thing it ended when it did. I thought we were in for a much longer blow."
"So did I."
Then an awful thought struck him. He turned and started pushing through the ferns and brush.
"Where are you going?" Jack called from behind him.
"To high ground. I want the highest point on this hummock."
It wasn't far—these islands in the saw grass sea weren't all that large. Just a few minutes walk and he was standing atop the crest of the hummock.
But he still didn't have the view he needed. He hurried to a nearby live oak that somehow had weathered the storm intact. He stretched for the lowest branch but couldn't reach it.
"Give me a boost," he said to Jack, who had followed him.
"What do you think you're doing?"
"Just help me up, damn it. I need to see."
He was sorry for the sharp tone, but he was worried. He crawled onto the limb, then, hanging on to a nearby branch, straightened until he was standing. When he saw the wall of cloud and rain less than a mile away to the west, his fears were confirmed.
"Jack, the hurricane isn't over. We're in its eye. It's going to hit us again. Maybe even worse than what we've been through. We've got to—oh, hell!"
"What?" Jack said from below.
Tom watched a pale funnel cloud skating back and forth inside the edge of the onrushing eye wall. Another snaked down a short way north of the first.
"Tornadoes!" He turned and slid down the trunk. "We have to get off this hummock!"
"Tornadoes?" As soon as Tom landed on the ground, Jack started climbing. "I've always wanted to see a tornado." He reached the limb and peered west. "I'll be damned. Three of them."
"Three? There were only two before! Get down from there and get moving!"
Jack stared a few heartbeats longer, then joined Tom on the ground.
Jack led the way back to the lagoon on a run. As they passed the sinkhole, Tom slowed and peered into the depths. The lights had faded to a dim glow and the lagoon had risen to the level where water was beginning to trickle over the edge.
"This thing should be sealed up," he said. "Maybe after all this is over we should come back and—"
Jack spoke over his shoulder. "Don't worry about it. It's closing itself down until the spring. Keep moving."
Closing itself down...how could he know that?
Tom was winded, with a dull ache squeezing his chest by the time they reached the bank. He hunched over, hands on knees, panting while Jack inspected the clan's boats. He pointed to a water-filled flat-bottom dinghy at the edge of the lagoon with Chicken-ship across its stern.
"This one's got a bigger motor than the canoe. We'll make better time. Help me tip it up to get rid of this water." He stared at him. "You okay?"
"Yeah," Tom said. "Just not conditioned for this."
Tipping a boat was the last thing Tom felt like doing right now, but he didn't think Jack could handle it alone. Jack pulled off his poncho and positioned himself at the aft end of the starboard side. As Tom moved to join him, something splashed near Jack's foot. Tom saw him jump and scramble away from the water.
Tom too backed away when he saw what was crawling up the bank. He'd heard mention of a two-headed snapping turtle, and hadn't quite believed it, but here it was—and much larger than he would have imagined. The shell had to be at least four feet long. It's gaping hooked jaws closed with loud clacks and they snapped at Jack.
Jack yanked a grenade from his belt, pulled the pin, and popped the clip.
"This is for Carl," he said, and lobbed it toward the creature.
Tom stood paralyzed for a moment. Carl...dear God, he'd all but forgotten about poor Carl...
He saw the right head snatch the grenade on the fly and swallow it, then Jack was rushing him, pushing him to the ground.
"Down!"
Tom hit the mud and covered his head with his hands. The explosion was muffled but he could still feel the impact through the ground. And then bloody turtle meat and bits of shell began to rain around them.
When it stopped, Jack helped him to his feet, then stepped back to the boat. The remains of the snapper were sinking into the water, trailing a red cloud. Jack froze, then hurried to the stern.
"Christ! Can't we get a break here?"
"What's wrong?"
"The explosion sheared off the propeller!" He kicked the side of the boat. "Damn! Okay. Looks like it's the canoe."
They hurried along the bank to where they'd left it. Jack slipped into the rear and started yanking on the little motor's pull cord. After a couple of dozen quick pulls, he spewed a string of curses and gave up. The motor hadn't even coughed.
"Won't start. Who knows what was blown or washed into it during the storm. We'll have to power it ourselves."
"Jack..." Tom hated to admit it, but he was all in. "I don't know if I can."
Jack stared at him a moment, then said, "It's okay, Dad. I'll handle it. You take the rear, maybe use the outboard as a rudder while I paddle us out of here."
Feeling unsteady, Tom stepped into the canoe and dropped into the rear seat. His chest felt funny, as if his heart was flailing wildly against his sternum. The chaotic rhythm left him drained. But not too drained to grab the tiller of the motor as Jack began paddling.
The canoe nosed out of the lagoon and soon they were gliding along the swollen channel. They hadn't gone too far before the light began to die as the clouds closed in again. Then the wind and rain returned with a vengeance.
Tom still wore his poncho but Jack had shed his a while back. His T-shirt was plastered to his skin and Tom watched the play of muscles across his son's back as he worked the paddle. Not bulky steroidal clumps, but sleek efficient bands, close to the skin. He hadn't noticed Jack's muscles till now. Where had they come from? He'd been such a skinny kid, even in college. Now...well, he reminded Tom of a few guys he'd known in the service, lean, quiet types who didn't look like much until someone tried to push them around. He'd seen a guy built like Jack take down someone twice his size.
He'd been angry with Jack all these years for disappearing, and never more angry than when he didn't show up for Kate's funeral. But all that seemed ancient history now. Despite Jack's secretiveness, his reclusiveness, his quirky behavior, Tom realized he loved, even admired the strange, enigmatic man his son had grown into. He sensed a strength, a resolve, a simple decency about him. He'd worried for so long that he must have made terrible mistakes raising Jack—why else would he turn his back on his family the way he had?—but now he sensed that maybe he'd done all right. Not that anyone should take full credit or full blame for how another person turns out; everyone makes their own choices. But as a parent he had to think he'd had some input.
More than anything he wanted Jack to survive this storm. He didn't care about himself so much, though of course he wasn't looking to die, but he sensed somehow that it was important for Jack to live—not simply important to his father, but for other, larger reasons. He couldn't pinpoint what those were; they hovered just out of reach, but they were there. Somewhere along the way, Jack was going to matter.
Tom's heart had resumed a more sedate rhythm but it jumped again as a lightning bolt speared the saw grass ahead of them. He looked around in the near-night darkness. They were out in the open, begging to be struck by lightning; but staying among the trees of the hummock, especially with this wind and tornadoes, seemed even riskier.
They rounded a bend in the channel and the canoe kicked ahead as the wind roared from behind. Tom spread his flapping poncho to give the wind something more to blow against. It worked. The canoe picked up speed.
He was feeling pretty proud of himself until another bolt of lightning lit up a funnel cloud reaching for the ground a few hundred yards to his left. It hadn't touched down, which meant it wasn't—
Another flash showed it on the ground, kicking up mud and grass and water. It was now officially a tornado.
He leaned forward and tapped Jack on the shoulder. "Look left!"
Jack did so, and of course the lightning chose just that moment to hold off; but then a double flash lit up the funnel, whiter than before, and closer. It was coming this way.
"Fuck!" Jack shouted and started paddling even harder.
Fuck...Tom had rarely if ever used the word since leaving the Marines. He didn't believe it belonged within the walls of a family home, and certainly not in mixed company. But looking at that swirling, swaying mass of wind and debris heading their way...fuck.
Yes, fuck indeed.
During storms on trips to the Keys, he'd witness an occasional waterspout—long, pale, wispy, short-lived things more beautiful than threatening. Even though there was plenty of water about, this thing to the left wasn't a waterspout, nor was it one of those quarter-mile-wide monsters the Weather Channel liked to show. Its base seemed to be only fifty feet or so across—
Only? Tom thought. What am I thinking? That thing is plenty big enough to kill us both.
He tried to gauge its intensity. He knew about the Fujita scale—he'd learned a few things during all those hours in front of the Weather Channel—and hoped this one didn't clock in at more than an F2. They wouldn't survive a direct hit by an F2, but they might handle a close encounter. If they wound up near anything higher up the scale, that would be it.
No matter what its scale, Tom prayed it would head in the other direction.
He pulled a paddle from the sloshing bottom of the canoe and did what he could to speed the boat along. He kept glancing to his left. He could hear a growing roar—that was the damn tornado getting closer, running on an erratic diagonal that was sure to intersect their course. At least that was how it looked. The way it was weaving back and forth made avoidance a crap shoot.
The big question: Stay in the boat or get out? In the boat seemed worse than being in a trailer. They were too exposed; if that funnel came even close, flying debris could cut them to shreds. But to get out...
Jack was looking around too.
"Let's dump the boat!" he shouted over the growing roar.
"And go where?"
He pointed to the right. "I saw something over there."
Tom squinted through the rain and darkness. A flash revealed the dark splotch of a willow thicket sitting like an island in the saw grass sea. The willows tended to be small in these thickets, little more than a dozen feet tall. They'd provide some shelter, something to hold on to without worrying it would crush them if it toppled over.
A glance in the opposite direction showed the tornado even closer.
"Let's do it!" Tom shouted.
"What about gators?"
"If they're smart they're on the bottom of the deepest channel they can find."
He didn't mention snakes. He had no idea what snakes did in weather like this. He hoped they didn't head for higher ground...like hummocks and thickets...
Jack jumped out of the canoe, Tom followed. The water was thigh high in the channel. Tom slipped only once climbing the slope to the saw grass where the water was only ankle deep. Jack pulled the canoe up behind him and left it on its side in the grass.
Lightning lit their way as they sloshed toward the thicket, Jack in the lead, while the roar of the twister grew behind them...no, not behind them...to the left...
A flash revealed the swaying, writhing funnel less than a hundred yards away, flanking them. Tom gasped for breath as his heart writhed like the twister. How had it caught up so fast? Another flash showed it veering this way. Almost seemed as if it was chasing them, homing in on them. But that was ridiculous.
Then again, after all he'd seen today...
"Crawl in here!" Jack shouted as they reached the thicket. His voice was barely audible over the roar of the onrushing funnel. Tom saw that he was holding aside a patch of underbrush. "Find a trunk and hang on!"
Tom dropped to his hands and knees as he ducked into the leafy mesh, feeling ahead of him in the dark until he found a sturdy-feeling trunk maybe six inches across.
"You take this one!" he shouted to Jack who was close behind. "I'll take the next."
He heard a garbled protest from Jack but kept moving. Half a dozen feet farther on he found another, more slender trunk, maybe half the size of the first. He dropped prone and wrapped his arms around it. His lungs struggled for air. God, it was good to lie still. He felt his heart ramming at his chest wall as he lay in the mud.
"You okay, Jack?" he shouted. He could barely hear himself above the tornado's roar. "Jack?"
That roar...it had to be at least an F2...any higher, they were goners.
Frantic, he looked around for Jack and saw nothing but darkness. And then the tree began to shake and the ground to tremble; he ducked his head against the wind and the saw grass blades whistling through the underbrush like knives.
Thank God they weren't trying to weather this back at the lagoon. The flying debris from the boats and the huts would be lethal. Here it was only grass and mud and water. Not that any of that would matter if the funnel passed directly over them.
The wind scythed at him from all angles as he clung to the trunk. He could hear the twister grinding through the saw grass on the far edge of the thicket, roaring like a freight train—he'd always heard tornado survivors describe the sound that way, and now he knew it was true...like a train...in a tunnel...
Tom felt the underbrush around him being twisted and yanked from the mud. And then his tree started to tilt, first to the left, then the right, then—
Dear God, it was coming out of the ground, ripping free of the mud, rising into the air!
Tom had to let go or rise with it. As he released his grip the willow ripped free with an agonized crunch and sailed off. He tried to cling to the rootlets left in the hole but the deluge of water made them slick and they slipped through his fingers. Then he felt his legs lift as he was pulled backward. He clutched for grass or weeds or ferns—anything!—but they came free in his grasp. His body angled off the ground and he clawed at mud that had no more consistency than beef stew. He was losing his last contact with the ground when he felt a hand grab his right ankle and yank him down.
Jack!
Another set of fingers wound around his left ankle and started hauling him backward. He heard Jack's enraged voice shouting above the storm. "You got away with this once, but not again. No fucking way!"
Who was he talking to? The twister? But he'd said "again." Tom doubted Jack had ever even seen a twister, let alone dealt with one. Who, then?
He'd worry about that later. Right now he wanted to know how Jack was hanging on. If both hands were holding Tom, who was holding Jack?
He felt one of Jack's hands grab his belt and haul him farther back. Tom craned his neck to look over his shoulder and saw that Jack had locked his legs around a willow trunk. He kept dragging Tom back until he could wrap his arms around the larger tree.
And with that...the roaring began to fade. After brushing the thicket, the twister was moving on, probably carving a new channel through the saw grass as it traveled.
Jack rolled away from the tree and lay on his back.
"Thought I was going to lose you there, Dad."
As his heart regained a normal rhythm, Tom watched Jack lie there with closed eyes as rain pounded his face.
"I thought I was a goner too. Thanks."
"De nada."
Nothing? No, it wasn't nothing. It was something...something very special. He owed his life to Jack.
He couldn't think of anyone he'd more like to be indebted to.
Tom swallowed the lump in his throat. "Come on. Let's see if we can find that canoe and get to someplace dry."
## TUESDAY
## 1
"I've decided to move back north," Dad said as Jack packed his duffel bag for the trip home.
Jack studied his face, still bruised from the accident, and weathered from the storm. "You're sure about that?"
Dad nodded. "Very. I'll never be able to look at Anya's house without remembering what...what we saw there...what happened to her. And I can't see me ever looking out my front door at the Everglades without thinking of the other night...all that blood spilled, especially Carl's...and that sinkhole and the things that came out of it. And the storm, that tornado..." He shook his head. "We damn near died out there."
"But we didn't," Jack told him. "That's all that counts."
It hadn't been easy getting back. The canoe had been far enough from the twister to come through in one piece, but the subsequent battle through the storm had been an ordeal. With the smaller channels filling up, and no way to judge east or west, Jack had become disoriented and made a few wrong turns. It took nearly two hours of paddling before they arrived at the air-boat dock and gratefully collapsed in the shelter of the car.
Monday had been spent recuperating. Muscles Jack didn't even know he had protested every time he moved. The groundsmen—sans Carl—were out in force cleaning up the mess left by the storm. They must have seen Anya's shredded screen door but probably attributed it to the storm.
Late in the afternoon, after the crews had finished for the day and no one was about, Jack and his father buried Anya's remains in her garden, among the plants she'd loved. Since she kept pretty much to herself, no one had discovered yet that she was missing.
Jack dug a two-foot hole in the wet soil—deeper than any dog or coon would go—and then Dad reverently placed the quarter-folded skin within. He'd chosen not to wrap it in anything. Better to let it decompose quickly and recycle its nutrients back to her plants.
And then a quiet night of mourning, Dad looking for answers to a long list of questions, Jack doing his best not to answer them. Dad didn't need to know more than he already did and, despite what he'd been through, probably wouldn't accept the truth as Jack understood it. So Jack told him only what he'd gleaned from Anya and let him assume that the rest of the answers had died with her. It never occurred to either of them to turn on the Monday night football game.
"Besides," Dad was saying on this bright morning, "what am I doing down here while my sons and all my grandchildren are up north? It makes no sense. I don't know what I was thinking."
Maybe you weren't thinking, Jack thought. Maybe you were being manipulated. Maybe everything that's happened down here was part of a plan—a plan that, thanks to Anya, didn't go quite the way it was supposed to.
And then again, maybe not.
But with the Otherness so obviously involved, Jack couldn't help but think that his father had been scheduled to die last Tuesday morning.
"Maybe I'll come south for just a month or two a year," Dad went on, "say February and March. Statistics say that an American male who reaches age sixty-five can expect to live another sixteen years. That leaves me ten more. Makes no sense to spend them fifteen hundred miles from the most important people in my life."
"You're right. It doesn't."
Jack had a feeling he'd better watch over his father. He was sure the Otherness wasn't through with him yet. Rasalom's words kept haunting him:
...a strong man slowly battered into despair and hopelessness...that is a delicacy. In your case, it might even approach ecstasy...
How was this battering into despair and hopelessness going to happen? By destroying everyone he cared about?
He was glad his father would be closer to home, but right now he wanted to get back to Gia and Vicky. Worry for them was a knife point in his back, urging him home. And he had to get working on a way to become a citizen before March, when the baby was due.
Yesterday he'd overnighted the Ruger back to one of his mail drops. He'd pick it up after it was forwarded to another drop. All he had to do now was pack up his clothes and head for the airport.
The phone rang.
"That should be the sales office," Dad said. "I phoned them first thing this morning about putting the place on the market."
As he left, Jack reminded himself to check out Blagden & Sons once he got home. See if he could find out why they wanted that sand from the cenote. He had a feeling it wasn't for mixing concrete for back porches.
He scooped the last of his things out of the bureau and froze: The rectangle of Anya's skin lay in the bottom of the drawer.
His mouth went dry. This couldn't be. They'd buried it yesterday, yet here it was, without a speck of dirt.
Jack walked out to the main room where his father was discussing prices and commissions with the sales office; he went directly to the back porch and grabbed the shovel he used yesterday. He headed for Anya's garden.
The burial spot was just as they'd left it. Jack dug into the loose soil and quickly reached the two-foot level.
No skin.
He dug down another six inches—he knew he hadn't gone this deep yesterday—and still nothing but dirt.
Anya's skin was gone.
No, wait, not gone. It was lying in a drawer in his Dad's guest bedroom. But how...?
Jack didn't waste time with unanswerable questions—how it had gotten out of the hole and into the house, why it was there. Either he'd find out later or he wouldn't.
He quickly refilled the hole and hurried back to the house. Dad was still on the phone. He looked up with a questioning expression as Jack passed but Jack waved him off. Back in the room he went directly to the bureau and froze again. Now the drawer was empty.
What the hell?
He turned and saw a now familiar pattern through the open top of his duffel bag. He stretched the zippered mouth and stared.
There it lay. Apparently Anya, or at least this piece of her, wanted to go home with him.
Jack sighed. Again, he wouldn't ask why, he'd just go with the flow and trust that sooner or later this would all make sense.
He covered the skin with his remaining clothes and zipped the bag closed.
All right, Anya, he thought. You want to come along, be my guest.
He lifted the bag and headed for the front room. Dad hung up as he entered.
"Well, just a few papers to sign and the place is officially on the market."
"Great. I hear they've got people lined up to get in here, so it shouldn't take long."
"Yeah."
A silence grew between them. Jack knew he had to go, but he was reluctant to leave his father here alone.
Finally Dad said, "It's been wonderful getting to know you, Jack. There's so much about you I still don't know, but what I've learned...I'm surprised, but pleasantly so."
"You're pretty full of surprises yourself."
"But you know all mine now. I get the feeling—no, I know you've still got quite a few left."
Here we go. "Probably not as many as you think. But who knows what you'll find out once you get back north?"
Dad nodded. "Right. Who knows?"
As if there'd been some unspoken signal, they embraced.
"Good to have you back, son," his father whispered. "Really, really good."
They broke the clinch, but still gripped each other's arms.
"Good to know the real you, Dad. You can take my back any time." He broke free and grabbed his duffel. "See you back home."
"Call me when you get in."
"You're kidding, right?"
"No. I've always worried about you, but after what I've learned about you down here, I'll really, really worry about you."
Jack laughed as he pushed through the door and headed for the car and the airport and the plane home to Gia and Vicky.
www.repairmanjack.com
## AFTERWORD
South Floridians will know I played fast and loose with some of the geography in Gateways. Joanie's Blue Crab Café is not on US 1, but on the other side of the state, on Route 41 in Ochopee. But the crab cakes and softshell crab sandwiches are just as good as I described. While researching the Glades I'd often drive twenty or thirty miles out of my way to grab a bite and an Ybor Gold at Joanie's.
As for Gator Country FM 101.9, it's hard to pull in if you're on US 1, but travel a little ways west and there it is. A good station for modern country and it kept me company during the drives.
Novaton may seem like Homestead, but it's an amalgam of a number of towns I stayed in during my research sorties.
One thing I did not make up or overstate is the shameful neglect, mismanagement, and outright abuse suffered by the Everglades during the twentieth century. It's a fragile, fascinating environment, sui generis, that's been damn near ruined by rampant overdevelopment. There's lots of talk lately of restoring the Everglades; let's hope the folks talking the talk will walk the walk before it's too late.
F. Paul Wilson
The Jersey Shore
March, 2003
www.repairmanjack.com
This is a work of fiction. All the characters and events portrayed in this novel are either fictitious or are used fictitiously.
GATEWAYS: A REPAIRMAN JACK NOVEL
Copyright © 2003 by F. Paul Wilson
All rights reserved, including the right to reproduce this book, or portions thereof, in any form. This book is printed on acid-free paper.
Edited by David G. Hartwell
A Forge Book
Published by Tom Doherty Associates, LLC
175 Fifth Avenue
New York, NY 10010
www.tor.com
Forge® is a registered trademark of Tom Doherty Associates, LLC.
Library of Congress Cataloging-in-Publication Data
Wilson, F. Paul (Francis Paul)
Gateways / F. Paul Wilson.—1st ed.
p. cm.
"A Forge book"—T.p. verso.
ISBN: 978-0-7653-0690-6
1. Repairman Jack (Fictitious character)—Fiction. 2. Retirement communities—Fiction. 3. Everglades (Fla.)—Fiction. 4. Fathers and sons—Fiction. 5. Coma—Patients—Fiction. 6. Sinkholes—Fiction. 7. Florida—Fiction. I. Title.
PS3573.I45695G38 2003
813'.54—dc21
2003049142
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 6,713
|
Acacia mooreana är en ärtväxtart som beskrevs av William Vincent Fitzgerald. Acacia mooreana ingår i släktet akacior, och familjen ärtväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Externa länkar
Akacior
mooreana
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 3,921
|
Thank you for your interest in becoming Chief Executive of Brightside. We are a social enterprise that seeks to raise disadvantaged young people's aspirations and awareness about education and career pathways and enhance their capability to achieve those aspirations. We believe that no-one should be stopped from achieving their potential, and that what counts towards your success should be how hard you work and the skills and talents you have, not background.
We create, develop and manage online mentoring projects, and other online tools and resources, to connect, inform and inspire more disadvantaged young people to achieve their full potential through education and careers. Our mentoring places human relationships at the heart of the provision of career and transition support for young people and we are recognised as the clear leader in online mentoring, working with over 100 different organisations and supporting over 45,000 young people since 2003.
Our current partners range from higher education institutions including King's College London, University of Birmingham and Canterbury Christ Church University; major private sector employers such as Allen and Overy and PwC; the NHS; and bodies such as the Royal Academy of Engineering. Examples of current projects include: Big Deal Blogs, working with the University of York –where around 160 fourteen to fifteen year olds working in teams over the course of ten weeks are set the task of developing a product and selling it to a live "Dragon's Den" panel. Another example is the provision of e-mentoring software and the training of mentors for universities' widening participation work.
Most of our funding is through our contracts with partners, but we also have some Trust funding and are able to invest in future developments through an endowment.
Our current Chief Executive, Dr Tessa Stone, has been a leading light in the world of widening participation and social mobility, and is well connected within government and higher education. She has helped establish Brightside's pre-eminence in the world of online mentoring and built our profile.
Her departure comes at a watershed moment for Brightside: our future strategy is in development but we are clear that we want to grow substantially to enable us to increase the number of disadvantaged young people we work with, and increasingly look at how we can support more young people into work as well as higher education. We want to develop more national projects with corporate or major public sector partners, for example tapping into the current focus on apprenticeships. And we want to look at developing more "second tier projects" with partners to help get young people into careers such as medicine or engineering, even when those individuals may not join the company we partner with.
We have a great team of young, enthusiastic and talented individuals, and a supportive board. I have a supportive but fairly hands-off approach and am looking for a Chief Executive who will enjoy working with substantial autonomy.
If you believe you have the skills, experience and passion to assist us in our next stage of our growth, enabling us to reach many more disadvantaged young people and have an even bigger impact on social mobility; then I would very much like to hear from you.
© Peridot Partners 2013. All rights reserved.
|
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| 2,478
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Barabás János (Sárvár, 1885. február 2. – Szombathely, 1967. október 22.) igazgató
Élete
Atyja Barabás György. Tanári oklevelet a Budapesti Tanárképző Főiskolán szerzett. Utána a lipcsei egyetem hallgatója lett, ahol dr. Wundt világhírű filozófus laboratóriumában a psycho-fizikában képezte tovább magát. Az I. világháborúban a 83. gyalogezred kötelékében részt vett a San folyó és a Lublin melletti ütközetekben, majd Bichawa mellett súlyosan megsebesült. 1915-ben Bécsben teljesített katonai szolgálatot. Pályafutását Iglón kezdte, mint nevelőtanár, utána két évig a Szenicei áll. polgári fiúiskolában, négy évig az Iglói polgári fiúiskolában tanár, majd igazgató volt. 1922-ben a Balatonfüredi Polgári Iskola igazgatójává nevezték ki. Az általa vezetett iskola 1930-ban ünnepelte fennállásának 70-ik évfordulóját. 1932-ben nyugalomba vonult. Nyugdíjasként Veszprémben élt.
Munkatársa volt a Szepesi Lapoknak. Elnöke volt a füredi népművelési bizottságnak, vezetője a Balaton cserkészcsapatnak, tagja volt a Felsőmagyarországi Irodalmi Társaságnak, és a Felvidéki Magyar Írók és Újságírók Egyesületének, a Kisfaludy Társaskörnek , Balatonfüredi Yacht Clubnak, a Balatoni Szövetségnek, az Országos Polgáriiskolai Tanáregyesületnek, az Országos Vöröskereszt Szövetségnek, a Természettudományi Társaságnak, a Földrajzi Társaságnak.
Munkái
Írói munkássága során cikkeket írt a Felvidéki Magyar Írók és Újságírók Egyesülete szerkesztésében megjelent lapban.
Könyve: "A magyar költészet legújabb iránya" címmel jelent meg 1911-ben celldömölki Dinkgreve Nándor könyvnyomdája gondozásában.
Jegyzetek
Források
Bertalan Vincze: Barabás György Életrajza (Egri Nyomda-Részvénytársaság Nyomása). Eger 1916
Zalai Közlöny 1930. június 12.
Zala Vármegye Feltámadása Trianon Után (Zalai fejek). (Budapest, 1930. Hungária Hírlapnyomda Részvénytársaság Kiadása)
Eötvös Károly Veszprém Megyei Könyvtár - Évfordulók 2017-ben
Magyar pedagógusok
Sárváriak
Szombathelyiek
1885-ben született személyek
1967-ben elhunyt személyek
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 457
|
Aertal is a nonsteroidal anti-inflammatory agent.
Available drug in the form of oral tablets, powder for suspension for oral administration and 1.5% cream for external use.
Sepifilm 752 is white, containing titanium dioxide, microcrystalline cellulose, hypromellose, and macrogol stearate (film coating composition).
Implemented tablets of 10 pcs. in blister packs packed in cartons of 1, 2, 3, 4, 6, or 9 pcs.
Excipients: hypromellose, sodium saccharinate, titanium dioxide, aspartame, sorbitol, colloidal silicon dioxide, caramel, milky and cream flavors.
Sold powder 3 g in bags, packaged in 20 pcs. in cardboard packs.
Excipients: propyl parahydroxybenzoate, methyl parahydroxybenzoate, liquid paraffin, emulsion wax and water.
Implemented cream in aluminum tubes of 60 g.
For symptomatic treatment of ankylosing spondylitis, rheumatoid arthritis and osteoarthritis.
In the form of a solution prepared from powder, the use of Aertal is advisable in inflammatory diseases of the musculoskeletal system, including ankylosing spondylarthritis, osteoarthrosis, as well as in juvenile, psoriatic, gouty and rheumatoid arthritis.
All patients with hypersensitivity to aceclofenac or any auxiliary component of the drug.
Orally, Aertal is taken in 1 tablet or 1 packet, dissolved in 40–60 ml of water immediately before use, twice a day, in the morning and evening.
Cream Aertal rubbing movements applied to the affected area three times a day. The amount of funds depends on the area of the proposed lesion. 1.5-2 g (the size of a pea) is enough for processing 5-7 sq. Cm. surface.
Leukopenia, in individual cases - hemolytic anemia, agranulocytosis, thrombocytopenia, aplastic anemia.
Also Aertal, reviews, can cause allergic reactions that manifest skin rash and itching, rarely - urticaria, bronchospasm, and systemic anaphylactic reactions, rarely - pneumonitis, vasculitis, erythema multiforme, toxic epidermal necrolysis, erythroderma, anaphylactic shock, Stevens-Johnson syndrome .
Structural analogue of Aertal is Asinak.
Aertal is a prescription drug. It should be stored in a dry place at temperatures up to 25 ºС. Shelf life of tablets and powder - 4 years, cream - 3 years.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,647
|
package com.intellij.psi.stubs;
import com.intellij.diagnostic.PluginException;
import com.intellij.openapi.diagnostic.Logger;
import com.intellij.psi.templateLanguages.TemplateLanguage;
import com.intellij.psi.tree.IStubFileElementType;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;
import java.util.Objects;
class StubBuilderType {
private static final Logger LOG = Logger.getInstance(StubBuilderType.class);
private final IStubFileElementType myElementType;
private final BinaryFileStubBuilder myBinaryFileStubBuilder;
private final Object myBinarySubBuilder;
StubBuilderType(@NotNull IStubFileElementType elementType) {
myElementType = elementType;
myBinaryFileStubBuilder = null;
myBinarySubBuilder = null;
}
StubBuilderType(@NotNull BinaryFileStubBuilder binaryFileStubBuilder) {
myElementType = null;
myBinaryFileStubBuilder = binaryFileStubBuilder;
myBinarySubBuilder = null;
}
StubBuilderType(@NotNull BinaryFileStubBuilder.CompositeBinaryFileStubBuilder binaryFileStubBuilder,
@Nullable Object binarySubBuilder) {
myElementType = null;
myBinaryFileStubBuilder = binaryFileStubBuilder;
myBinarySubBuilder = binarySubBuilder;
}
BinaryFileStubBuilder getBinaryFileStubBuilder() {
return myBinaryFileStubBuilder;
}
String getVersion() {
if (myElementType != null) {
if (myElementType.getLanguage() instanceof TemplateLanguage &&
myElementType.getStubVersion() < IStubFileElementType.getTemplateStubVersion()) {
PluginException.logPluginError(LOG, myElementType.getLanguage() + " stub version should call super.getStubVersion()",
null, myElementType.getClass());
}
return myElementType.getClass().getName() + ":" + myElementType.getStubVersion();
} else {
assert myBinaryFileStubBuilder != null;
String baseVersion = myBinaryFileStubBuilder.getClass().getName() + ":" + myBinaryFileStubBuilder.getStubVersion();
if (myBinaryFileStubBuilder instanceof BinaryFileStubBuilder.CompositeBinaryFileStubBuilder) {
return baseVersion + ":" + ((BinaryFileStubBuilder.CompositeBinaryFileStubBuilder)myBinaryFileStubBuilder).getSubBuilderVersion(myBinarySubBuilder);
} else {
return baseVersion;
}
}
}
@Override
public boolean equals(Object o) {
if (this == o) return true;
if (o == null || getClass() != o.getClass()) return false;
StubBuilderType type = (StubBuilderType)o;
return Objects.equals(myElementType, type.myElementType) &&
Objects.equals(myBinaryFileStubBuilder, type.myBinaryFileStubBuilder) &&
Objects.equals(myBinarySubBuilder, type.myBinarySubBuilder);
}
@Override
public int hashCode() {
return Objects.hash(myElementType, myBinaryFileStubBuilder, myBinarySubBuilder);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,263
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shanegarrison
A FellowTraveler in God's Kingdom
The CRC & Dr. Ted Taylor
April 3, 2008 / shanegarrison
I get the privilege to serve on a advisory committee for Campbellsville University called the "Church Relations Council." The CRC is a group of men and women from all over the US who are active in church leadership and have some affiliation with Campbellsville.
As a former alum and current associate pastor, they invited me to participate. Actually, I ask them if I could join…and they said "yes." I have served on the CRC for four years.
One special treat for this CRC meeting is hearing my mentor, Dr. Ted Taylor, give a presentation on the FIRST CLASS leadership institute. Dr. Taylor was the professor who really introduced me to Christian ministry and how to live out my calling from God. He taught me to think critically about the practice of ministry and how to keep the message the same, but to use new methods. He is the one that led me down a path toward this calling in Christian education. Without him, I would not be where I am today.
He invested in my life outside of the classroom as well. I have spent hours talking with Dr. Taylor over the years, seeking his council and insight. He probably has written more letters of reference for me than anyone else. He even wrote one recommendation letter to someone saying "Garrison wants my job and I would give it to him."
I am one of the thousands of students he has impacted over the years. I will forever be indebted to Dr. Taylor for his godly, Christian influence in my life.
Taylor…thank you.
← Pastor Calvin Perry & the Reds
Library Daydreaming →
One thought on "The CRC & Dr. Ted Taylor"
Ashley Taylor Jessie
I think the man is as wonderful as you say he is. But then again he is my dad!!!!! Enjoyed reading the blog!!!
Leave a Reply to Ashley Taylor Jessie Cancel reply
My name is Dr. Shane Garrison. I am married to Jennifer and we have two boys, Isaac and Ethan.
I am an associate professor of Educational Ministries in the Campbellsville University School of Theology and Vice President for Enrollment Services.
I am a follower of Christ, a husband, a father, a pastor, and a book nerd.
One Year in Review at YCreekBC
Unusual Yet Blessed: A Different Approach to Church Staffing
8 Things I Am Thankful For at YCreek BC
The Difference Between an Interim and Transitional Pastor
Three Types of Transitional Pastorates
Each sheet represents a small group led by one of our @CUTheology students this semester. Our goal is to teach, tra… twitter.com/i/web/status/1… 14 hours ago
@corrieshull Thank you Bro. Corrie. Blessing to you and the great folks at the Ave. 6 days ago
@PastorDwayneN Love it. Maybe I should use that for an intro. 6 days ago
Beginning new message series Sun at ImmanuelBC- Near and Far. God is infinite and intimate. Bigger than our biggest… twitter.com/i/web/status/1… 6 days ago
Would you rather ? 1 week ago
Pulpit, Podium, Music Stand or Bistro Table
Discipling Head, Heart and Hands
Its a Great Pumpkin, Charlie Brown vs. The Charlie Brown Christmas Special
How Many Seeds Are In a Pomegranate? - A Spiritual Question for the Produce Aisle
A Chilly Baptism in a Country Church
shanegarrison on The Biblical Gentiles Were All…
Jerry E. Money on The Biblical Gentiles Were All…
Beth Carlisle on The Difference Between an Inte…
Gary on How Many Seeds Are In a Pomegr…
anupama on Pulpit, Podium, Music Stand or…
Archives Select Month November 2017 May 2017 April 2017 November 2016 October 2016 August 2016 April 2016 February 2016 January 2016 October 2015 July 2015 March 2015 January 2015 December 2014 November 2014 October 2014 September 2014 August 2014 July 2014 June 2014 May 2014 April 2014 March 2014 February 2014 January 2014 December 2013 November 2013 October 2013 September 2013 August 2013 July 2013 June 2013 May 2013 April 2013 March 2013 February 2013 January 2013 December 2012 November 2012 October 2012 September 2012 August 2012 July 2012 June 2012 May 2012 April 2012 March 2012 February 2012 January 2012 December 2011 October 2011 September 2011 August 2011 July 2011 June 2011 May 2011 April 2011 March 2011 February 2011 January 2011 December 2010 November 2010 October 2010 September 2010 August 2010 July 2010 June 2010 May 2010 April 2010 March 2010 February 2010 January 2010 December 2009 November 2009 October 2009 September 2009 August 2009 July 2009 June 2009 May 2009 April 2009 March 2009 February 2009 January 2009 December 2008 November 2008 October 2008 September 2008 August 2008 July 2008 June 2008 May 2008 April 2008 March 2008 February 2008 January 2008 December 2007 November 2007 October 2007 September 2007 August 2007 July 2007 June 2007 May 2007 April 2007 March 2007
Follow shanegarrison on WordPress.com
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,880
|
<?php
Namespace DesignPatterns\Behavioral\Observer\Observer;
/**
* Interface Observable
*/
interface iObservable
{
/**
* Attach observer to a certain object events
* @param iObserver $observer
* @param array $events
*/
public function attach(iObserver $observer, $events);
/**
* Detach observer from a certain events
* @param iObserver $observer
* @param array $events
*/
public function detach(iObserver $observer, $events);
/**
* Notify observers
* @param int|string $event
* @param array $data
*/
public function notify($event, $data);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,202
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osCmax v2.5.0 PL1 is a patch update to correct several issues that were discovered after the 2.5.0 official release that we felt needed to be fixed before the official 2.5.1 release at the end of March.
This patch fixes four issues with store administration that could cause significant inconvenience to workflow.
There are no database changes. The patch is easy to perform - simply upload the files in the patch, overwriting your existing osCmax files. If you have made any code changes to osCmax, first make sure the files in the patch do not overwrite any of your customized files!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 8,035
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Robert Garrett ist der Name folgender Personen:
* Robert Garrett (Leichtathlet) (1875–1961), amerikanischer Leichtathlet
Robert Garrett (Basketballspieler) (* 1977), deutscher Basketballspieler
Robert Garrett (Fußballspieler) (* 1988), nordirischer Fußballspieler
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,954
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This is an Opel steering joint, it is a new old stock part made by Opel, with the original Opel box and bearing part number 90112334. Preliminary research would suggest that it is for the Kadett D series or Vauxhall Astra Mk1, see here for a list with the corresponding part number.
Given the widespread sharing of parts between so many Vauxhall and Opel models it may be suitable for a variety of 1970s models, but the Kadett D series is the only one I have seen listed for this part number. However I tried a test fit with a spare Opel Manta A series steering column shaft and it fitted perfectly.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,860
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Sonia Robatto (Salvador, Bahia, 1937) é uma atriz, bibliotecária e escritora de livros infantis.
Biografia
Sonia nasceu em Salvador e era filha do dentista e cineasta Alexandre Robatto Filho, primeiro cineasta da Bahia. Foi aluna da primeira turma da Escola de Teatro da Ufba, onde atuou em espetáculos como, As Três Irmãs (1958), A Almanjarra (1958), A Via Sacra (1958). Destacou-se, também, como membro-fundadora da Sociedade Teatro dos Novos, primeira companhia profissional de teatro da Bahia (1959) e criou o Teatro Vila Velha (1964). Após seu casamento, morou no Rio de Janeiro e em São Paulo, onde começou a escrever para crianças.
Em 1969,Sonia Robatto, ao apresentar seu original, História da Sapa Cristina, à Editora Abril, acabou selecionada para criar e ser a editora da Revista Recreio na sua primeira fase, que, além das histórias, teve o mérito de propôr exercícios que desenvolviam a motricidade, baseados na doutrina de Jean Piaget. Sonia foi a responsável, através da revista, pelo lançamento de autores consagrados como Ana Maria Machado, Ruth Rocha, Joel Rufino dos Santos, Magui e outros.
Nos anos 80, a escritora tornou-se responsável pelo projeto da Coleção Taba (revistas em fascículos, acompanhadas por discos infantis, que reuniram grandes nomes da MPB, sob a coordenação musical de Tom Zé). Para esta publicação, Robatto criou, entre outras histórias: O Vaqueiro Misterioso, O Bicho Folhagem, A Ratinha Ritinha e Marte Invade a Terra (já publicada anteriormente pela Revista Recreio edição nº 05).
Em sua carreira de escritora, escreveu e publicou em revistas, fascículos e livros, mais de 400 histórias infantis.
Em 2001, sua obra mais importante, Pé de Guerra, foi adaptada para o teatro por Márcio Meirelles. Encenada em Salvador, acabou recebendo o prêmio Copene de melhor montagem em 2001.
Atualmente, Sonia reside na capital baiana, ainda escrevendo e atuando.
Principais Obras
História da Sapa Cristina
A Menina Sem Jeito
Pé de Guerra
Uma nuvem chamada Fofinha e outras histórias
A Casa Barriga - Memórias de um Bebê
A Viagem de Retalhos
Auto da Miséria Divina
O segredo do curumim
A Ciranda do Medo
Natal com lua cheia
O Vaqueiro Misterioso
O Bicho Folhagem
Nana Nenê: Uma História para cada dia
Lilica, a Formiga
A Mágica das Plantas
Lia, a Centopeia
Lucas, o Menino que Descobriu o Tempo
Ligações Externas
A Tarde on Line - Entrevista com Sonia Robatto
O Tempo on Line- Entrevista com Ruth Rocha
Escritores contemporâneos do Brasil
Mulheres romancistas do Brasil
Autores de literatura infantojuvenil do Brasil
Escritores da Bahia
Alunos da Escola de Teatro da Universidade Federal da Bahia
Naturais de Salvador
Bibliotecários da Bahia
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,479
|
While the rest of the World and other parts of Canada were unsettled by the change in the world economy, British Columbia remained steady and was seen as an oasis of calm. Its resilience and economic strength stems from its business diversification program with direct access to Asian markets, and above all good governance.
Fri Oct 07 2016 06:29:43 GMT+0000 (UTC)
A Look At The Economy And Governance Of British Columbia
While the rest of the World and other parts of Canada were unsettled by the change in the world economy, British Columbia remained steady and was seen as an oasis of calm. Its resilience and economic strength stems from its business diversification program with direct access to Asian markets, and above all good governance. This success may be due to the port generating unprecedented amounts of income and the provincial government making the sensible decision of reinvesting it right away.
The port has a lot of activities going on, and the government has made million-dollar decisions to invest in port capacity, that in turn has helped the creation of a huge amount of construction jobs that help propel the economy of Canada as a whole.
With oilseed, Canadian grain, and wood products on their way to the lucrative markets of Asia, creating 700,000 jobs in areas like; Lower Mainland, North shore, and the East side of the Burrard Inlet.
Tourists visiting the province continue to grow annually and Vancouver International Airport has introduced new flight routes to Asia while still luring travellers and vacationers from other parts of Canada.
The provincial government currently has plans and is negotiating with concerned stakeholders to make Vancouver the regional base for Asian firms and companies. The trade and business relationship between Canada and the U.S.A will always be big, but the emergence of Asia as a business partner cannot be ignored and British Columbia is well placed to take full advantage of it.
Predicted debt-free future
As strong and resilient as the British Columbia economy is, it is not immune to global economic difficulties like; increase in commodity prices, and sharp drops in oil and coal prices. The provincial government's support of a much-lauded decision to export liquefied natural gas (LNG), to Asia and China specifically has faced some challenges, although this was expected and the forecast is good.
The British Columbia government will soon post $277 million surplus in the 2015-2016 financial year, this represents the provinces third consecutive balanced budget. It emphasises the point that even though the economy of the province hasn't exactly gone through much growth lately, it is stable and controlled unlike other parts of the country and world.
The governance of the British Columbia government has contributed tremendously to the unprecedented economic stability of the province. The decision to diversify in terms of economic production and the supply to new markets that were previously thought of as too risky has proved to be the key, and other provinces have looked to learn from them.
Another contributing factor to the good economy of the province is the arrival of new residents from other parts of Canada and especially the increase of new immigrants moving to the province. 1, 118 New residents moved to the province in the first quarter of 2016, and more than half of them were immigrants while the rest came from Alberta, after the oil slump experienced there. With the ambition of the British Columbia government to increase production and export of liquid natural gas, there will be many job opportunities for skilled workers who will help construct bigger export terminals.
Visitors and Tourists
The city of Victoria has been vancoureceiving an increased number of visitors from other parts of Canada as well as from the United States, Australia, and Europe. On the other hand, Vancouver has also experienced a boost in its tourism industry, helped mostly by the improved air connections.
The British Columbia government saw an improvement of the air travel industry as one of its core areas for a healthy economy and duly invested in it. In the second half of 2015, Vancouver International Airport handled a record 19.4 million passengers, showcasing an incredible growth of 8 percent from the year before.
Air Canada has chosen Vancouver as one of its major flight routes for the new Boeing 787 Dreamliner, which has helped entice a new market of travellers from as far as Tokyo, Seoul, Shanghai and Beijing. A major shipping company from Singapore will soon open an office in Vancouver, a move that has been made possible by the newly created Vancouver International Maritime Centre, whose main aim was to attract investment to the province of British Columbia. The port of Prince Rupert is currently going through an extensive expansion program and construction work to enable it to handle large cargo such as wood chips and cement powder.
The economic success of the province has been made possible by the good governance and detailed plans that have seen British Columbia take advantage of its natural resources and foraying into untapped new markets. The province also aims to improve agriculture and forestry sectors to produce enough products for international exports as well as to other parts of Canada.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,031
|
<?xml version="1.0" encoding="utf-8"?>
<selector xmlns:android="http://schemas.android.com/apk/res/android">
<item android:state_focused="true" android:state_pressed="false" android:drawable="@drawable/oorlogoselected" />
<item android:state_focused="true" android:state_pressed="true" android:drawable="@drawable/oorlogoselected" />
<item android:state_focused="false" android:state_pressed="true" android:drawable="@drawable/oorlogoselected" />
<item android:drawable="@drawable/oorlogo" />
</selector>
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{
"redpajama_set_name": "RedPajamaGithub"
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| 5,284
|
\section{Introduction}
\label{sec:introduction}
Theories with extra dimensions have attracted enormous attention in particle physics
in the last decade. They not only provide new avenues for theoretical
explorations, but also offer the exciting prospect of playing an active role in the upcoming
collider experiments. Starting with the revelation that extra dimensions could be
as large as the submillimeter distance and the scale of quantum gravity could be at TeV
\cite{Arkani-Hamed:1998rs,Antoniadis:1998ig,Arkani-Hamed:1998nn}, it was realized
that warped extra dimensions \cite{Randall:1999ee,Randall:1999vf} could have novel
features to address issues ranging from electroweak to gravitational physics. Ever since
there has been an explosion in the number of extra dimensional models inspired by either
the large or warped extra dimensions.
From the four-dimensional (4D) perspective extra dimensions, if there, manifest themselves through a series
of KK modes for every particle that propagates in the bulk. Typically
one starts with a given metric in the extra dimensions, assumes some boundary conditions for the bulk
fields, and then computes the KK masses from the metric. Sometimes it is assumed that all the standard
model (SM) fields are confined on a three-brane, in which case only the graviton would have a massive
spin-2 KK tower, while sometimes all or part of the SM could live in the extra dimensions. In
most cases the size of the extra dimensions is at TeV scale or higher, suggesting the first KK mass
in the TeV order as well. If this is the case, it seems that the second or higher KK mode might lie beyond the
reach of the Large Hadron Collider (LHC).
An interesting exception is the Universal Extra Dimensions (UEDs) \cite{Appelquist:2000nn, Cheng:2002ab},
in which not only
all the SM particles propagate in the compactified bulk, giving the hope that one may be able to
observe another copy of
SM in the first KK level, but also the lower bound on the size of the extra dimensions is only
at 300 GeV or
so, raising the attractive
possibility that more than one KK level can be discovered at the LHC.
On a separate front, given the imminent start of the LHC, there are recently strong interests in the
inverse problem of interpreting the underlying physics from LHC data \cite{Arkani-Hamed:2005px}.
The goal is to study the map from the signature space of LHC to the parameter space of theoretical
models. In the context of extra dimensional models, the traditional forward approach is to study
phenomenological consequences of a given model in a particular spacetime background, such as computing
the KK masses from the postulated metric. In this paper we consider the
LHC inverse problem in the extra dimensional context. We will focus on the most
obvious 4D observable, the KK masses, and ask how one can extract geometrical properties of the
extra dimensions. We are interested in questions like, if one assumes a compactified extra dimension,
what can we learn from the mass of the first KK mode? What about the second KK mass? Even if we knew
all the KK masses, would the shape of the extra dimension be uniquely determined?
It is our purpose to study the aforementioned questions in this paper,
which is organized as follows. In section II we set the stage by
considering a five-dimensional $U(1)$ Yang-Mills
theory on a finite interval, as well as its KK decompositions. By transforming
the eigenvalue equation into a Schr\"odinger equation,
in section III we use the time-independent
perturbation theory, as well as the reflection symmetry of the background geometry,
to study the inverse problem assuming the Neumann boundary
conditions (BCs). In section IV we consider a general background geometry
without assuming any symmetry property. The treatment here closely follows
the mathematical technique of solving the Dirichlet
inverse eigenvalue problem for flat background in Ref.~\cite{ist}. The approach in this section
also allows us to extend the study to other types of BCs, which
is done in section V.
Then in section VI we conclude with some discussion.
\section{$U(1)$ gauge theory on an interval}
We start with assuming 4D Lorentz invariance and one finite extra dimension which can be thought of
as an interval. Without loss of generality, the metric can be written in warped form
\begin{equation}
ds^2 = g_{MN}\ dx^M dx^N= e^{2A(z)}\left( \eta_{\mu\nu} dx^\mu dx^\nu - dz^2\right) , \quad 0 \le z \le L,
\end{equation}
where $\eta_{\mu\nu}={\rm diag}(1,-1,-1,-1)$ and we call
$A(z)$ the warp factor. We also use the convention that the capital Roman letters $M,N=0,1,2,3,z$
are contracted with $g_{MN}$ whereas the Greek letters $\mu,\nu=0,1,2,3$ are contracted with
$\eta_{\mu\nu}$.
When the warp factor is constant, the resulting space is
a flat extra dimension compactified on a circle, $z \simeq z + 2L$, with the projection,
$z \simeq -z$,
which is the $S^1/Z_2$ orbifold. Because of the orbifold projection all the fields living in the bulk
must be either even or odd under $z \to -z$. Even (odd) fields have Neumann
(Dirichlet) BCs at the boundaries $z=0, L$.
Usually this is the main
motivation for considering an orbifold compactification, because the
zero modes for the odd fields are projected out, which is crucial in terms of getting
a chiral zero mode for the fermion if the SM is to live in the bulk. We wish to consider the possibility
that the warp factor is non-trivial.
As an illustration let us consider an abelian vector
field $A_M(x,z)$ propagating on an interval $0 \le z \le L$ \cite{Davoudiasl:1999tf,Muck:2001yv}:
\begin{eqnarray}
S &=& \int \sqrt{g}\, d^4x dz\, \frac{-1}{4g_5^2} F_{MN} F^{MN} \\
&=& \int d^4x \int_0^L dz\, \frac{e^{A(z)}}{g_5^2}\, \left( -\frac14 F_{\mu\nu} F^{\mu\nu} + \frac{1}2
(\partial_z A_\mu - \partial_\mu A_z)^2 \right).
\end{eqnarray}
In this paper we will only be concerned with classical physics and neglect the gauge-fixing
and ghost terms.
We will also choose a gauge where $A_z = 0$ and the Neumann BCs for $A_\mu$
\begin{equation}
A_\mu^\prime(x,0) = A^\prime_\mu(x,L)=0,
\end{equation}
where $^\prime$ denotes $\partial_z$. The Neumann BCs are normally chosen to ensure a massless
zero mode. However, there could be other types of BCs if there are bulk mass terms or
scalars at the boundaries of the interval \cite{Csaki:2003dt}, so later we will generalize the results
to Dirichlet as well mixed BCs.
Performing the KK expansion
\begin{equation}
A_\mu(x,z) = \sum_n A_\mu^n(x) f_n(z),
\end{equation}
the action becomes diagonal in the KK basis
\begin{equation}
S= \int d^4x \sum_n \left( -\frac14 F_{\mu\nu}^n F^{n\, \mu\nu} + \frac12 m_n^2 A_\mu^n A^{n\, \mu}
\right),
\end{equation}
if the KK profiles $f_n(z)$ satisfy the equations
\begin{eqnarray}
\label{eom1}
&& \partial_z \left( e^{A(z)} \partial_z f_n \right) + m_n^2 e^{A(z)} f_n = 0, \quad
f_n^\prime(0) = f_n^\prime(L) = 0\,; \\
&& \frac1{g_5^2} \int_0^L dz \, e^{A(z)}f_n f_m = \delta_{mn}.
\end{eqnarray}
From now on we will ignore the $1/g_5^2$ factor in the orthogonality condition as it is irrelevant to
our analysis.
Eq.~(\ref{eom1}) is an equation of Sturm-Liouville type with Neumann BCs. The
question we are interested is essentially the inverse eigenvalue problem of the above equation:
given the mass eigenvalues, what can we learn about the warp factor $A(z)$?
To proceed, it is convenient to transform Eq.~(\ref{eom1}) into a non-relativistic Schr\"odinger
equation \cite{Randall:1999vf}:
\begin{eqnarray}
\label{eig0}
&& f_n(z) = e^{-A(z)/2} \psi_n(z), \\
&& -\psi_n^{''} + V(z) \psi_n = m_n^2 \psi_n \, , \\
\label{kkp0}
&& V(z) = \frac12 A^{\prime\prime} + \frac14 ( A^\prime)^2,
\end{eqnarray}
We will call
$V(z)$ the KK potential associated with the warp factor $A(z)$.
It is worth noting that in Eq.~(\ref{eig0}) the BCs of the original KK wave functions $f_n(z)$ do not
translate simply into the BCs of the new $\psi_n(z)$; the boundary values of $A(z)$ are involved as well.
For example, Neumann BCs for
the $f_n$ translate into Neumann BCs for the $\psi_n$ only if one further assumes
$A^\prime(0) = A^\prime(L)=0$. On the other hand, in solving for the warp factor from a given
KK potential in Eq.~(\ref{kkp0}), the assumed BCs for the warp factor would presumably give a unique solution.
For example, in the flat space case, $V=0$, the BCs $A^\prime(0) = A^\prime(L)=0$
give a unique, albeit trivial, answer $A(z)=0$. Otherwise the general solution for $V(z)=0$ in
Eq.~(\ref{kkp0}) looks like $A(z)=c_1 + 2 \log(z+c_2)$, where $c_1$ and $c_2$ are integration constants.
Therefore from now on we will assume suitable BCs for $A(z)$ so that
the BCs translate in Eq.~(\ref{eig0}).
\section{Neumann Inverse problem}
In this section we study the Neumann inverse spectral problem of the Schr\"odinger equation
\begin{equation}
-\psi^{\prime\prime} + V \psi = \lambda \psi,\quad \psi^\prime(0)=\psi^\prime(L) = 0.
\end{equation}
The idea is that when the KK potential $V(z)$ can be considered as perturbations on the flat
background, $V=0$, we can use time-independent perturbation theory of the Schr\"odinger
equation.
In the flat space limit, the unperturbed solutions are
\begin{equation}
\label{eig00}
\lambda_n^{(0)} = \frac{n^2\pi^2}{L^2},
\quad \psi_n^{(0)} = \sqrt{\frac2{L}} \cos\left(\frac{n\pi}{L}z\right),
\end{equation}
from which we see there is a massless zero mode with constant wave function.
To the first order in perturbation, the KK masses and wave functions are,
for $n>1$,
\begin{eqnarray}
\label{eig2}
\lambda^{(1)}_n &=& \int_0^L dz \left[\psi_n^{(0)}(z) \right]^2 V(z) \nonumber \\
&=& \frac1{L} \int_0^L dz\,V(z) + \frac1{L} \int_0^L dz \cos\left(\frac{2n\pi}{L}z\right) V(z), \\
\label{eigf0}
\psi_n^{(1)}(z) &=& \sum_{m \neq n} \frac{\psi_m^{(0)}(z)}{\lambda^{(0)}_n - \lambda^{(0)}_m}
\int_0^L dt\, \psi_n^{(0)}(t) \psi_m^{(0)}(t) V(t).
\end{eqnarray}
The zero mode $n=0$ is a special case and needs to be singled out from perturbation because
its masslessness is guaranteed by the 4D gauge invariance.\footnote{I am grateful
to Yuri Shirman and Arvind Rajaraman for bringing this issue to my attention.}
Indeed, the constant wavefuction is always a solution with zero eigenmass in
Eq.~(\ref{eom1}), which implies the exact zero mode wavefunction $\psi_0(z) = \exp(A/2)$.
On the other hand, if we had chosen differen BCs, there would have been no massless zero mode
and no need to single it out in perturbation.
From Eq.~(\ref{eig2}) we immediately see that the first-order corrections
to flat-space KK masses are related to coefficients of the Fourier cosine
series of the KK potential. In particular, the correction to the $n$th eigenmass
is related to the sum of the average of the KK potential and the $n$th coefficient
of Fourier cosine series. Therefore,
higher KK masses probe the metric at shorter distances, in accordance with usual intuition.
One important observation following from Eq.~(\ref{eig2}) is the fact that the KK masses
are only sensitive to the even part of the KK potential with
respect to reflections on the mid-point of the interval $z \to L-z$. Unless this $Z_2$ reflection
is a symmetry of the extra dimension, KK masses alone are not sufficient to uniquely determine
the KK potential. Nevertheless,
such a geometric $Z_2$ reflection is none other than the KK parity
in UEDs \cite{Appelquist:2000nn,Cheng:2002ab}. In UEDs with one extra dimension the SM propagate
in 5D compactified on the orbifold $S^1/Z_2$. For theories compactified on a circle $S^1$, momentum
conservation in the 5th direction implies conservation of the KK number at each interaction vertex.
When considering $S^1/Z_2$, however, the orbifold has fixed points at the boundaries which break
the translational invariance, and hence momentum conservation, in the 5th direction. Moreover,
quantum corrections in the bulk induces divergent terms on the two boundaries that renormalize
localized 4D interactions there \cite{Georgi:2000ks,Cheng:2002iz}. In the end only a $Z_2$ subgroup of the
translational invariance, that is reflections with respect to the mid-point $z=L/2$, is preserved,
which is called KK parity \cite{Cheng:2002ab}. For phenomenological considerations,
the KK parity is defined as a flip of the line interval about the center $z=L/2$ combined with a
$Z_2$ transformation that changes the sign of all fields odd under the orbifold projection, which
are fields that have Dirichlet BCs. This is so that all the even number
KK modes are invariant, while the odd number KK modes change sign, under the KK parity. Our finding
is that for extra dimensional models that have the KK parity, the shape of the extra dimension is
completely determined by measurements of KK masses.
A new $Z_2$ parity for theories beyond SM, under which the SM is even and
(some of) the new particles are odd, is in fact very well-motivated phenomenologically.
Perhaps the most
prominent feature
of such a $Z_2$ parity is suppressions of precision electroweak contributions from the new particles
\cite{Cheng:2003ju},
rendering their masses light at or below 1 TeV and allowing for a solution to the little hierarchy
problem. Another important feature is the existence a stable particle, that is the
lightest particle charged under the parity, which is a good candidate for dark matter if it is
electrically neutral.
Examples of such a parity, other than the KK parity, are the $R$ parity in supersymmetry and the $T$ parity
in little Higgs models \cite{Cheng:2003ju,Cheng:2004yc,Low:2004xc}.
Without KK parity, it is
natural to ask is how to determine the coefficients of the Fourier sine series of the KK potential from
four-dimensional data. To this end we notice that the coefficient of the Fourier sine series
\begin{equation}
\label{sine0}
s_n=\frac{2}L \int_0^L dz\, V(z) \sin \left(\frac{2n\pi z}{L} \right)
\end{equation}
only depends on the KK odd part of $V(z)$, and as such is a measure of the breaking of the KK parity.
Therefore any quantity that is sensitive to violations of KK parity will be related to the Fourier
sine coefficients. One such quantity is the absolute value of the ratio $|\psi_n(z)/\psi_n(L-z)|$.
If KK parity is a good symmetry and the geometry is symmetric with respect to reflections about
the mid-point of the interval, then the ratio should be unity. The above argument suggests
the definition
\begin{equation}
\label{kap0}
\kappa_n(z) = \log \left|\frac{\psi_n(z)}{\psi_n(L-z)}\right|,
\end{equation}
which vanishes when KK parity is conserved. Using Eqs.~(\ref{eig00}) and (\ref{eigf0}),
we can derive an expression for $\kappa_n(z)$ in perturbation:
\begin{eqnarray}
\label{kap1}
\kappa_n(z) &=& \sum_{m\neq n} \frac{1-(-1)^{m-n}}{\lambda_n^{(0)}-\lambda_m^{(0)}}
\ \frac{\psi_m^{(0)}(z)}{\psi_n^{(0)}(z)} \int_0^L dt\, \psi_m^{(0)}(t) \psi_n^{(0)}(t) V(t)
\\
\label{kap2}
&=& \sum_{m\neq n} \frac{1-(-1)^{m-n}}{(n^2-m^2)(\pi^2/L^2)}
\ \frac{\cos \frac{m \pi z}{L}}{\cos \frac{n \pi z}{L}}
\frac2L \int_0^L dt\, \cos\left(\frac{m\pi t}L \right) \cos\left(\frac{n\pi t}L \right) V(t).
\end{eqnarray}
In the above, because of the coefficient $1-(-1)^{m-n}$, the summation effectively only runs over
those $m$'s for which $m+n$ is an odd integer. For this case, the cosines in the integrand
has odd parity under $z\to L-z$ and the integral is non-vanishing only if $V(t)$ has a KK odd component.
In principle, one could work out the Fourier sine series of the integrand in Eq.~(\ref{kap2})
\begin{equation}
\cos\left(\frac{m\pi t}L \right) \cos\left(\frac{n\pi t}L \right) = \sqrt{\frac2L}\sum_k a_k
\ \sin \frac{2\pi k t}L,
\end{equation}
from which a relation between $\kappa_n(z)$ and the $s_n$ in Eq.~(\ref{sine0}) follows. However,
following a suggestion for a similar quantity for the Dirichlet inverse problem in \cite{ist}, one
can show that $\kappa_n(0)$ is directly proportional to the Fourier sine coefficients $s_n$,
\begin{eqnarray}
\label{kap3}
\kappa_n(0) &=& \sum_{m\neq n} \frac{1-(-1)^{m-n}}{(n^2-m^2)(\pi^2/L^2)}
\
\frac2L \int_0^L dt\, \cos\left(\frac{m\pi t}L \right) \cos\left(\frac{n\pi t}L \right) V(t)
\\
&=& \frac{L}{2\pi n} \, \int_0^L dz\, V(z) \sin \left(\frac{2n\pi z}{L} \right),
\end{eqnarray}
if one uses the identity
\begin{equation}
\label{idd0}
\frac{L}{n\pi} \sin \frac{n\pi z}L =
\sum_{m\neq n} \frac2L \cos \frac{m \pi z}L\frac{1-(-1)^{m-n}}{(n^2-m^2)(\pi^2/L^2)}, \quad 0 \le z \le L.
\end{equation}
One way to derive the above identity is to use the Green's function with Neumann BCs
\begin{eqnarray}
&& G(z,z') = \sum_n \frac{\psi_n^{(0)}(z)\psi_n^{(0)}(z')}{\lambda - \lambda_n^{(0)}}, \\
&& -\partial_z^2 G(z,z') - \lambda G(z,z') = \delta(z-z'),
\end{eqnarray}
and then plug into
\begin{equation}
\sin \sqrt{\lambda}z' = \int_0^L dz\ \delta(z-z') \sin \sqrt{\lambda} z,
\end{equation}
which is just the expansion of the sine
function in the complete basis $\{\psi_n^{(0)}(z), 0 \le z \le L\}$.
In terms of the original eigenfunctions in Eq.~(\ref{eig0}),
\begin{equation}
\log \left| \frac{f_n(0)}{f_n(L)} \right| = -\frac12(A(0)-A(L))
+ \frac{L}{2\pi n} \, \int_0^L dz\, V(z) \sin \left(\frac{2n\pi z}{L} \right), \quad n>0.
\end{equation}
That is $\kappa_n(0)$ measures the difference in the
boundary values of the warp factor $A(z)$, as well as
the Fourier sine coefficients of the KK potential. Unfortunately, it appears that the ratio
of the boundary values of the wave functions is not easily accessible from the experimental
perspective; one needs to be able to resolve the extra dimension and make a comparison at two
opposite points. What is worse,
as mentioned earlier quantum corrections in the bulk will induce
logarithmically divergent contributions to the gauge kinetic terms that are localized on the
boundaries \cite{Georgi:2000ks,Cheng:2002iz}. Thus from the viewpoint of 4D effective field theories,
the values of the wave functions on the orbifold fixed points may even be arbitrary and theoretically
incalculable due to their
UV sensitivity.
On the other hand, there are certainly low-energy
observables that probe the breaking of KK parity. Suppose
we extend the $U(1)$ gauge theory to a non-abelian theory, then there are three-point couplings
$g_{lmn}$ as well
as four-point couplings $g_{klmn}$ of different KK modes, where the indices denote the KK numbers.
If KK parity is a good quantum number, $g_{lmn}=0$ for odd integral $l+m+n$ and $g_{klmn}=0$
for odd integral $k+l+m+n$. A non-zero value for either of them would indicate breaking of
KK parity and potentially probe the KK odd part of $V(z)$.
Nevertheless, the relations between the three/four-point couplings and the Fourier
sine coefficients are contaminated by the warp factor itself.
As an example, consider the three-point couplings
\begin{eqnarray}
\label{3pt}
g_{lmn} &\propto& \int_0^L dz\, e^{A(z)} f_l(z) f_m(z) f_n(z) \\
&=& \int_0^L dz\, e^{-A(z)/2} \psi_l(z) \psi_m(z) \psi_n(z).
\end{eqnarray}
The product of the $\psi(z)$'s in the integrand could be computed in perturbation using $\psi^{(0)}(z)$.
It is also possible to express the product
in the Fourier sine series, which however would involve an infinite number of terms. Unfortunately,
the warp factor also goes into the integrand. Thus
without knowing the warp factor a priori, it seems difficult, if not impossible, to actually perform
the integration and extract the desired Fourier coefficients from the three-point couplings. It is
in fact possible to eliminate the warp factor in the integrand in Eq.~(\ref{3pt}) by taking
advantage of the fact that the zero mode wavefunction is constant. For example,
choosing $l=0$ we have
\begin{equation}
g_{0mn} \propto \int_0^L dz\, \psi_m(z) \psi_n(z)
\end{equation}
which does not involve the warp factor explicitly. Nevertheless, it is simple to check in perturbation
that the terms linear in the KK potential all cancel and only ${\cal O}(V^2)$ terms survive. Again
it is very difficult to extract the Fourier sine coefficients this way.
To sum up, the three/four-point couplings probe the KK odd part of the warp factor as well
as the KK potential, and in general it seems very difficult to disentangle these two effects
in the couplings.
On the
other hand, if empirically it is found that all these KK odd $g_{lmn}$'s and $g_{klmn}$'s are
vanishingly small, as would be preferred from precision electroweak constraints, then one could just
use the eigenmasses to extract the Fourier cosine coefficients of the KK potential
to reconstruct the metric.
\section{General Background}
In this section we discuss the situation when the warp factor cannot be considered as
perturbations on flat spacetime $V(z)=0$. One example is the Anti-di Sitter
(AdS) background employed in \cite{Randall:1999ee,Randall:1999vf}, for which the metric is
\begin{equation}
ds^2 = \left(\frac{1}{k z}\right)^2 \left( \eta_{\mu\nu} dx^\mu dx^\nu - dz^2\right) ,
\end{equation}
where $k$ is the AdS curvature scale. The KK potential for the AdS background is
\begin{equation}
V_{AdS}(z) = \frac3{4z^2} .
\end{equation}
The KK potential for the AdS space apparently does not respect KK parity, and therefore
the first KK mass is generally required to be heavier than 1 TeV or higher. Moreover,
it does not appear proper to consider the above KK potential as a perturbation on the
flat background $V=0$ because of the singularity at $z=0$; the integral of $V_{AdS}$
diverges in the interval $0 \le z \le L$. On the other hand, if the warp
factor is exactly AdS, then the KK spectrum is given by roots of Bessel functions
\cite{Pomarol:1999ad,Gherghetta:2000qt}
and should
be identifiable. Therefore in the following we assume a KK spectrum that can be
roughly, but not exactly, identified with that coming from a known background such as
the AdS, suggesting that the real
geometry only slightly deviates from the known background and could be considered as perturbations.
Our construction in the following
is adapted from that in \cite{ist}, which specifically considers Dirichlet
inverse spectral problem for $V(z)$ that is regular on the interval.
Assuming the warp factor in the metric to be of the form
\begin{equation}
A(z) = A_0(z) + A_1(z), \quad 0 \le z \le L,
\end{equation}
where $A_0(z)$ is a known background and $A_1(z)$ is a small fluctuation. The KK potential is then
\begin{eqnarray}
V(z) &=& V_0(z) + V_1(z) , \\
V_0 &=& \frac12 A_0'' + \frac14 (A_0')^2 ,\\
V_1 &=& \frac12 (A_0' A_1'+A_1'') + \frac14 (A_1')^2 .
\end{eqnarray}
That is,
we would like to consider the Neumann inverse eigenvalue problem of the following differential equation
\begin{equation}
\label{eom2}
-y^{\prime\prime} + (V_0 + V_1) y = \lambda y, \quad 0 \le z \le L,
\end{equation}
when $V_1$ can be considered as perturbations.
One first considers the unperturbed
equation
\begin{equation}
\label{eom20}
-y^{\prime\prime} + V_0\, y = \lambda y,
\end{equation}
and constructs the eigensystem $\{\lambda_n^{(0)}, \psi_n^{(0)}\}$
satisfying the Neumann BCs. Then as before the first-order perturbed
eigenvalues are
\begin{eqnarray}
\label{gen0}
\lambda_n &=& \lambda_n^{(0)} + \lambda_n^{(1)} \nonumber \\
&=& \lambda_n^{(0)} + \int_0^L dz [\psi_n^{(0)}(z)]^2 V_1(z).
\end{eqnarray}
Therefore, once the first $N$ KK masses are measured, the above equation leaves
$N$ constraints on the KK potential.
In general,
the eigenvalues $\lambda_n$ and eigenfunctions $\psi_n$ are both functionally dependent
on the perturbation $V_1$. One can compute $\delta \lambda_n/\delta V_1$ by taking
the functional derivative $\delta/\delta V_1$ of the equation\footnote{Note that for the purpose of
taking functional derivative $V_0$ and $V_1$ are independent variables.}
\begin{equation}
\label{eom6}
-\psi_n'' + (V_0 + V_1) \psi_n = \lambda_n \psi_n, \quad \psi_n'(0)=\psi_n'(L)=0.
\end{equation}
Interchanging differentiations with respect to $z$ and $V_1$, multiplying both sides by $\psi_n$, and
integrating we find
\begin{eqnarray}
&& \int_0^L dz\, \psi_n(z) \left[ -\frac{d^2}{dz^2} + (V_0+V_1)\right]
\frac{\delta \psi_n(z)}{\delta V_1(z')} = \nonumber \\
&& \quad \quad -[\psi_n(z')]^2 + \frac{\delta \lambda_n(V_1)}{\delta V_1(z')} +
\int_0^L dz\, \lambda_n \psi_n(z) \frac{\delta \psi_n(z)}{\delta V_1(z')}.
\end{eqnarray}
Since the differential equation (\ref{eom6}) is self-adjoint, we arrive at
\begin{equation}
\label{id21}
\frac{\delta \lambda_n(V_1)}{\delta V_1(z)} = [\psi_n(z)]^2,
\end{equation}
which is indeed satisfied by Eq.~(\ref{gen0}) in perturbation. This result will be useful later.
The lesson learned from the previous section, is that
the eigenmasses only give limited information on $V_1$; in the flat space case
only the Fourier cosine coefficients are given by the eigenmasses. More information can be
extracted by looking
at the eigenfunctions. To do so we need to consider solutions
$Y=\{y_1,y_2\}$ of Eq.~(\ref{eom2}) satisfying
the following initial conditions
\begin{eqnarray}
\label{bc1}
y_1(0,\lambda;V_1) = y_2^\prime(0,\lambda;V_1)=1 \ ,\\
\label{bc2}
y_1^\prime(0,\lambda;V_1)=y_2(0,\lambda;V_1) = 0 \ ,
\end{eqnarray}
where we have emphasized the dependence of the solutions on $\lambda$ and $V_1$.
It is clear that every solution of Eq.~(\ref{eom2})
can be written as $y(z)=y(0) y_1(z) + y'(0) y_2(z)$. For example, with Neumann BCs we
have
\begin{equation}
\label{bc4}
\psi_n(z) = \frac{y_1(z,\lambda_n;V_1)}{||y_1(z,\lambda_n;V_1)||}, \quad
y_1(z,\lambda_n;V_1) = \frac{\psi_n(z)}{\psi_n(0)}.
\end{equation}
where $||\cdot||$ means the norm of the function in the Hilbert space.
In addition, Eq.~(\ref{id21}) becomes
\begin{equation}
\label{id3}
\frac{\delta \lambda_n(V_1)}{\delta V_1(z)} = \frac{[y_1(z,\lambda_n;V_1)]^2}
{||y_1(z,\lambda_n;V_1)||^2}.
\end{equation}
The fundamental solution $Y$ also has the
property that the Wronskian determinant
is unity
\begin{equation}
W(Y) = \det \left| \begin{array}{cc}
y_1 & y_2 \\
y_1' & y_2'
\end{array} \right| = 1 .
\end{equation}
This can be proven by showing that $dW/dz=0$, which follows from the fact that $Y$
satisfies Eq.~(\ref{eom2}), and using $W(0)=1$.
Then the solution of the inhomogeneous equation
\begin{equation}
\label{eom3}
-y^{\prime\prime} + (V_0+V_1) y = \lambda y - f(z)
\end{equation}
is given by
\begin{equation}
\label{eom4}
y(z, \lambda;V_1) = \int_0^z dt \left[y_1(t)y_2(z)-y_1(z)y_2(t)\right]f(t).
\end{equation}
Our objective is to show that the quantity
\begin{equation}
\label{kap5}
\kappa_n(V_1) = -\log \left|y_1(L,\lambda_n;V_1) \right|
\end{equation}
provides additional information on $V_1$. In the flat space case, using Eq.~(\ref{bc4}), we see that
the above definition agrees with Eq.~(\ref{kap0}) and is related to the Fourier sine coefficient of
the KK potential.
We need three identities to
complete the proof. The first one is Eq.~(\ref{id21}). For the second we need to compute
the functional derivative of $Y=\{y_1,y_2\}$ with respect to $V_1$. Taking
$\delta/\delta V_1(z')$
in Eq.~(\ref{eom2}) and interchanging the functional derivative with $d/dz$, we have
\begin{equation}
-\left(\frac{\delta Y(z)}{\delta V_1(z^\prime)} \right)^{\prime\prime} +
(V_0+V_1) \frac{\delta Y(z)}{\delta V_1(z^\prime)} =
\lambda \frac{\delta Y(z)}{\delta V_1(z^\prime)} - \delta(z-z^\prime) Y(z),
\end{equation}
which is of the form in Eq.~(\ref{eom3}). Utilizing Eq.~(\ref{eom4}) we obtain
\begin{eqnarray}
\label{id1}
\frac{\delta Y(z)}{\delta V_1(z^\prime)} &=&
\int_0^z dt \left[y_1(t)y_2(z)-y_1(z)y_2(t)\right] \delta(z^\prime - t) Y(t) \nonumber \\
&=& \left[y_1(z^\prime) y_2(z) - y_1(z) y_2(z^\prime) \right] Y(z^\prime)\ {\cal I}_{[0,z]}(z^\prime),
\end{eqnarray}
where the indicator function ${\cal I}$ is such that
\begin{eqnarray}
{\cal I}_{[0,z]}(z^\prime) &=& 1 \quad {\rm if }\ \ z^\prime < z, \nonumber \\
&=& 0 \quad {\rm if} \ \ z^\prime > z.
\end{eqnarray}
From Eq.~(\ref{id1}) we could derive a similar expression for $\delta Y'(z)/\delta V_1$.
The last identity is $\partial Y / \partial \lambda$. Again differentiating Eq.~(\ref{eom2})
with respect to $\lambda$, interchanging the derivatives, and making use of Eq.~(\ref{eom4}), we have
\begin{equation}
\label{id2}
\frac{\partial Y(z)}{\partial \lambda} =
-\int_0^z dt \left[ y_1(t)y_2(z)-y_1(z)y_2(t) \right] Y(t).
\end{equation}
Using Eqs.(\ref{id3}, \ref{id1}, \ref{id2}), it only takes some algebra to show that
\begin{eqnarray}
\label{eom5}
\frac{\delta \kappa_n(V_1)}{\delta V_1(z^\prime)} &=& -
\frac1{y_1(L)}\left.\left( \frac{\partial}{\partial \lambda} y_1(L,\lambda;V_1)
\frac{\delta \lambda}{\delta V_1(z^\prime)} +
\frac{\delta}{\delta V_1(z^\prime)} y_1(L,\lambda;V_1)\right)\right|_{\lambda=\lambda_n}
\nonumber\\
&=& y_1(z^\prime,\lambda_n;V_1) y_2(z^\prime,\lambda_n;V_1) - [\psi_n(z^\prime)]^2
\int_0^L dt\, y_1(t,\lambda_n;V_1) y_2(t,\lambda_n;V_1).
\end{eqnarray}
We then have
\begin{eqnarray}
\label{kappa1}
\kappa_n(V_1) - \kappa_n(0) &=& \int_0^1 dt\, \frac{d}{dt} \kappa_n(tV_1) \nonumber \\
&=& \int_0^1 dt \int_0^L dz \frac{\delta \kappa_n(tV_1)}{\delta (tV_1)} V_1(z) \nonumber \\
&=& \int_0^L dz\, \frac{\delta \kappa_n(0)}{\delta V_1} V_1(z) + {\cal O}(V_1^2),
\end{eqnarray}
where we have used the definition of total derivative on a functional in the Hilbert space:
\begin{equation}
\left. \frac{d}{d\epsilon} F[q+\epsilon v] \right|_{\epsilon=0}= \int_0^L dz \frac{\delta F}{\delta q} v .
\end{equation}
Now $\kappa_n(V)$ can be computed in perturbation:
\begin{eqnarray}
\label{kap6}
\kappa_n(V_1)-\kappa_n(0) &=& \int_0^L dz \, y_1(z,\lambda_n^{(0)};0)y_2(z,\lambda_n^{(0)};0)\, V_1(z)\nonumber \\
\quad \quad && -\lambda_n^{(1)}
\int_0^L dt\, y_1(t,\lambda_n^{(0)};0)y_2(t,\lambda_n^{(0)};0) .
\end{eqnarray}
For the flat space case $V_0=0$, $\lambda_n = n^2\pi^2/L^2$ and
\begin{equation}
y_1(z,\lambda;0) = \cos \sqrt{\lambda} z, \quad y_2(z,\lambda;0) = \frac{\sin \sqrt{\lambda} z}{\sqrt{\lambda}}.
\end{equation}
Furthermore, $\kappa_n(0)=0$ and
Eq.~(\ref{kap6}) gives the Fourier sine coefficients of $V_1$.
\section{Other types of boundary conditions}
In this section we extend the results so far to other types of BCs, the Dirichlet
and mixed BCs, with a focus on the flat space background. These BCs might be useful for
bulk scalars or fermions on an interval.
The mixed BCs actually do not arise in orbifold compactification for its fields do not have a
definite parity under the orbifold projection $z\to -z$. Nevertheless, if we are only concerned
with a field theory on an interval, then it could be consistent.
The identities we derived
so far, Eqs.~(\ref{id21}, \ref{id1}, \ref{id2}), do not depend on the BCs
we choose. However, the eigenvalues and eigenfunctions in Eq.~(\ref{eig00}), as well
as Eq.~(\ref{id3}), do depend on the Neumann BCs chosen. To generalize to other types
of BCs, the important
observation is the following:
\begin{itemize}
\item If $ \psi_n(z) = y_1(z,\lambda_n)/||y_1(z,\lambda_n)||$,
\begin{eqnarray}
&&
- \frac{\delta}{\delta V_1(z')} \log |y_1(L,\lambda_n;V_1)| =
- \frac{\delta}{\delta V_1(z')} \log |y_1'(L,\lambda_n;V_1)| = \nonumber \\
&& \quad \quad y_1(z^\prime,\lambda_n;V_1) y_2(z^\prime,\lambda_n;V_1) - [\psi_n(z^\prime)]^2
\int_0^L dt\, y_1(t,\lambda_n;V_1) y_2(t,\lambda_n;V_1) .
\end{eqnarray}
\item If $\psi_n(z) = y_2(z,\lambda_n)/||y_2(z,\lambda_n)||$,
\begin{eqnarray}
&&
- \frac{\delta}{\delta V_1(z')} \log |y_2(L,\lambda_n;V_1)| =
- \frac{\delta}{\delta V_1(z')} \log |y_2'(L,\lambda_n;V_1)| = \nonumber \\
&& \quad \quad - y_1(z^\prime,\lambda_n;V_1) y_2(z^\prime,\lambda_n;V_1) + [\psi_n(z^\prime)]^2
\int_0^L dt\, y_1(t,\lambda_n;V_1) y_2(t,\lambda_n;V_1) .
\end{eqnarray}
\end{itemize}
These equations can be proven along the line of proving Eq.~(\ref{eom5}).
Now we can summarize the results for the Dirichlet as well mixed BCs in the flat background,
using the notation $\lambda_n = n^2\pi/L^2$
and $\lambda_{n+1/2} = (n+1/2)^2\pi^2/L^2$,
\begin{itemize}
\item Dirichlet BCs $\psi(0)=\psi(L)=0$:
\begin{eqnarray}
&& m_n^2=\lambda_n , \quad
\psi_n^{(0)}(z) = \sqrt{\frac2L} \sin \sqrt{\lambda_n} z
= \frac{y_2(z,\lambda_n;0)}{||y_2(z,\lambda_n;0)||} ,\\
&& \lambda_n^{(1)} = \frac1{L} \int_0^L dz\, V(z) -
\frac1{L} \int_0^L dz\, V(z)\, \cos\, 2\sqrt{\lambda_n}z , \\
&& \kappa_n(V)\equiv -\log |y_2^\prime(L,\lambda_n;V)| = -
\log \left|\frac{\psi_n^{\prime}(L)}{\psi_n^{\prime}(0)}\right| \nonumber \\
&& \phantom{\kappa_n(V)} = \kappa_n(0)
-\frac{L}{2\pi n} \int_0^L dz\,V(z)\, \sin\,2\sqrt{\lambda_n} z , \\
&& \kappa_n(0)=-\log |y_2^\prime(L,\lambda_n;0)| = 0.
\end{eqnarray}
\item Mixed BCs (I) $\psi(0)=\psi^\prime(L)=0$:
\begin{eqnarray}
&& m_n^2=\lambda_{n+\frac12}, \quad
\psi^{(0)}_n(z) = \sqrt{\frac2L} \sin \sqrt{\lambda_{n+\frac12}} z
= \frac{y_2(z,\lambda_{n+\frac12};0)}{||y_2(z,\lambda_{n+\frac12};0)||} ,\\
&& \lambda_n^{(1)} = \frac1{L} \int_0^L dz\, V(z) -
\frac1{L} \int_0^L dz\, V(z)\, \cos\, 2\sqrt{\lambda_{n+\frac12}}z , \\
&& \kappa_{n}(V)\equiv -\log |y_2(L,\lambda_{n+\frac12};V)| = -
\log \left|\frac{\psi_n(L)}{\psi_n^{\prime}(0)}\right| \nonumber \\
&& \phantom{\kappa_{n}(V)} =
\kappa_n(0)- \frac{L}{2\pi (n+1/2)} \int_0^L dz\,V(z)\, \sin\,2\sqrt{\lambda_{n+\frac12}} z , \\
&& \kappa_{n}(0)= -\log |y_2(L,\lambda_{n+\frac12};0)| = \frac12 \log \lambda_{n+\frac12}.
\end{eqnarray}
\item Mixed BCs (II) $\psi^\prime(0)=\psi(L)=0$:
\begin{eqnarray}
&& m_n^2=\lambda_{n+\frac12}, \quad
\psi_n^{(0)}(z) = \sqrt{\frac2L} \cos \sqrt{\lambda_{n+\frac12}} z
= \frac{y_1(z,\lambda_{n+\frac12};0)}{||y_1(z,\lambda_{n+\frac12};0)||} ,\\
&& \lambda_{n+\frac12}^{(1)} = \frac1{L} \int_0^L dz\, V(z) +
\frac1{L} \int_0^L dz\, V(z)\, \cos\, 2\sqrt{\lambda_{n+\frac12}}z , \\
&& \kappa_{n}(V)\equiv -\log |y_1^\prime(L,\lambda_{n+\frac12};V)| = -
\log \left|\frac{\psi_n^\prime(L)}{\psi_n(0)}\right| \nonumber \\
&& \phantom{\kappa_{n}(V)} = \kappa_n(0)
+ \frac{L}{2\pi (n+1/2)} \int_0^L dz\,V(z)\, \sin\,2\sqrt{\lambda_{n+\frac12}} z , \\
&& \kappa_{n}(0)=- \log | y_1^\prime(L,\lambda_{n+\frac12};0)| = -\frac12 \log \lambda_{n+\frac12}.
\end{eqnarray}
\end{itemize}
\section{Discussion and conclusion}
In this paper we studied the problem of reconstructing the metric of the extra dimension using
four-dimensional data. When the geometry can be considered as perturbations in a flat background,
we showed that the deviation of each KK mass from the exact flat space limit
gives the Fourier cosine coefficient of a KK potential, which
is related to the warp factor through a non-linear second-order differential equation.
If the KK parity, reflections about the mid-point of the extra dimension, is a good symmetry of the
theory, then the Fourier sine coefficient of the KK potential vanishes and the metric can be
determined by measuring KK masses alone. On the other hand, if KK parity is not a symmetry, then
the boundary values of each wave function are necessary to determine the Fourier sine coefficient
of the KK potential. Such information, nonetheless, seems challenging to obtain experimentally for
one needs to resolve the size of the extra dimension first and then make comparison of the
wave function at two opposite ends. If there are brane localized interactions at the boundaries,
as required by quantum corrections coming from bulk fields, then it might be possible
to probe the values of the wave functions at the boundaries. However, because of the UV sensitivity
of these brane localized terms, their strength is not calculable within the low-energy effective
theory. There are averaged quantities sensitive to the breaking of KK parity such as the three-
and four-point couplings of the non-abelian gauge fields. However these couplings generally involve
many different Fourier sine coefficients of the KK potential. Moreover, they also probe
the KK odd part of the warp factor and, therefore, do not provide direct
access to Fourier coefficients of the KK potential without prior knowledge of the warp factor.
A general background geometry other than the flat space is also considered in this paper. The
possibility arises when the KK potential of the geometry has non-integrable
singularities on the
interval and cannot be considered as perturbations in the flat background. One example is the
AdS geometry whose KK potential grows like $1/z^2$ as $z\to 0$. In this situation
three types of BCs: Neumann, Dirichlet,
and the mixed, are considered. Generically information on the behavior of the wave function at the
boundaries of the extra dimension provides constraints on the KK potential in addition to those
coming from the KK mass.
To implement the idea in this paper in the real world, one needs to first identify the background
spacetime on which the geometry can be considered fluctuations. For example whether the KK spectrum
roughly fits the flat space spectrum, which is evenly-spaced, or the AdS spectrum, which is the root
sequence of Bessel functions. Obviously this would require measurements of several KK masses,
even though
realistically it is not clear one would be able to measure more than one KK level, if at all,
in the near future,
as the KK mode
is generally expected to be heavier than 1 TeV from various constraints. An exception in this regard
is the UEDs, for which the compactification scale can be as low as 300 GeV, raising the prospect of
observing several KK levels.
In UEDs this is possible because of the KK parity, a $Z_2$ reflection
about the mid-point of the extra dimension, which is strongly suggested by precision electroweak
measurements. If KK parity is indeed a good symmetry, even for non-flat geometry, then the measurement
of $N$ KK masses could provide useful information on the first $N$ Fourier cosine coefficients of the
KK potential, if the KK spectrum fits approximately that from a flat extra dimension.
However, because it is the deviation from $n^2\pi^2/L^2$,
the flat space limit, that gives the sum of the average as well as the $n$th Fourier cosine
coefficient (see Eq.~(\ref{eig2})), and the size of the extra dimension $L$ is unlikely to be known a priori, in reality
one could probably only hope for an $(N+2)$-parameter fit using $N$ measured KK masses. On the other
hand,
it will be important to understand the extent of KK parity violation through KK odd
processes like the
decay of the first KK mode into two zero modes, or inelastic scattering of two zero modes into
one first KK mode and one zero mode. These information will be an indication on the size of
the Fourier sine coefficients of the KK potential.
Another approach to the inverse problem discussed in this paper is to discretize the Sturm-Liouville
equation and turn the problem into the matrix inverse eigenvalue problem. Physically speaking this
amounts to using deconstruction \cite{Arkani-Hamed:2001ca,Hill:2000mu} to approximate the continuous
extra dimension. However, there is some subtlety due to the mismatch of eigenmasses in the high
energy between
the deconstruction and the continuous case. Such an approach is currently under investigation
\cite{low}.
\begin{acknowledgments}
This work was supported by the Department
of Energy under grant DE-FG02-90ER40542. It is a pleasure to acknowledge
correspondence with H. C. Cheng, J. Erlich, G. Kribs, and M. Wise. This work was completed while
visiting the particle theory group at University of California at Irvine, whose hospitality is appreciated.
In addition, I am grateful to J. A. R. Cembranos, A. Rajaraman, and Y. Shirman for helpful discussion.
\end{acknowledgments}
\section*{Note Added}
After this work was completed, Ref.~\cite{Rabadan:2002wb} came to my awareness, which
also considered the problem of reconstructing the geometrical properties of a compact manifold from
KK masses, albeit with a very different approach. I would like to thank M. Kleban for bringing
this reference to my attention.
|
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"redpajama_set_name": "RedPajamaArXiv"
}
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Molchanov reached the quarterfinals of the ATP Prague Open By Moneta tournament
Ukrainian tennis player Denis Molchanov won the right to perform in the quarterfinals of the ATP Prague Open By Moneta challenger tournament in the Czech capital with a prize fund of 137,560 euros.
Kiev together with Arthur Rinderknecht from France defeated the Czech tandem Michal Vrbensky / Jonas Foreitek — 6: 2, 6: 2, Ukrinform reports.
Opponents spent 57 minutes on the court.
Read also: Stakhovsky covered the racket in the third round of the APR tournament in Prague
In the quarterfinals, the Ukrainian-French pair will meet the 4th seeded Dutch duo Sander Arends / David Pal.
Photo: tennisua.org.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,807
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\section{Introduction}
Low-mass stars of less than 1~M$_{\odot}$ are the dominant stellar component of the Milky Way. They constitute 70\% of all stars \citep[][Bochanski et
al. 2010]{Reid1997}\nocite{Bochanski2010} and 40\% of the total stellar mass of the Galaxy \citep{Chabrier03,Chabrier05}. Our understanding of the
Galaxy therefore relies upon the description of this faint component. Indeed, M dwarfs have been employed in several Galactic studies as they carry
the fundamental information regarding the stellar physics, galactic structure and its formation and dynamics. Moreover, M dwarfs are now known to host
exoplanets, including super-Earth exoplanets \citep[][Udry et al. 2007]{Bonfils2007,Bonfils2012}\nocite{Udry2007}. The determination of accurate
fundamental parameters for M dwarfs has therefore relevant implications for both stellar and Galactic astronomy. Because of their intrinsic faintness
and difficulties in getting homogeneous samples with respect to age and metallicity, their physics is not yet well understood.
Their atmosphere has been historically complex to model with the need for computed and ab initio molecular line lists accurate and complete to high
temperatures. But for over ten years already, water vapor \citep[][Barber et al. 2006]{AMESH2O}\nocite{Barber2006} and titanium oxide
\citep{Plez1998} line lists, the two most important opacities in strength and spectral coverage, have become available meeting these conditions.
And indeed, the \texttt{PHOENIX} model atmosphere synthetic spectral energy distribution improved greatly from earlier studies \citep[Hauschildt
et al. 1999]{AH95}\nocite{Hauschildt1999} to the more recent models by \cite{Allard2001,Allard2011,Allard2012} and by \cite{Witte2011} using the
most recent water vapor opacities.
The $T_\mathrm{eff}$-scale of M dwarfs remain to this day model-dependent to some level. Many efforts have been made to derive the effective temperature
scale of M dwarfs. Due to the lack of very reliable model atmosphere, indirect methods such as blackbody fitting techniques have historically been
used to estimate the effective temperature. The \cite{Bessell1991} $T_\mathrm{eff}$-scale was based on black-body fits to the near-IR (JHKL) bands by
\cite{Pettersen1980} and \cite{Reid1984}. The much cooler black-body fits shown from \cite{Wing1979} and \cite{Veeder1974} were fits to the optical.
Their fitting line
was a continuation of the empirical $T_\mathrm{eff}$ relation for the hotter stars through the \cite{Pettersen1980} and \cite{Reid1984} IR fits for the
cooler stars. The work by \cite{Veeder1974}, \cite{Berriman1987}, \cite{Berriman1992} and \cite{Tinney1993} also used the blackbody fitting
technique to estimate the $T_\mathrm{eff}$. \cite{Tsuji1996b} provide good $T_\mathrm{eff}$ using infrared flux method (IRFM). \cite{Casagrande2008} provides a
modified IRFM $T_\mathrm{eff}$ for dwarfs including M dwarfs. These methods tend to underestimate $T_\mathrm{eff}$ since the blackbody carries little flux compared
to the M dwarfs in the Rayleigh Jeans tail redwards of $2.5~\mu$m. Temperature derived from fitting to model spectra \citep{Kirkpatrick1993} are
systematically $\sim$ 300~\,K warmer than those empirical methods. These cooler $T_\mathrm{eff}$-scale for M dwarfs was corrected recently by
\cite{Casagrande2012} bring it close to the \cite{Bessell1991,Bessell1995} $T_\mathrm{eff}$-scale.
\cite{Tinney1998} determined an M dwarf $T_\mathrm{eff}$-scale in the optical by ranking the objects in order of TiO, VO, CrH and FeH equivalent widths.
\cite{Delfosse1999} pursued a similar program in the near-infrared (hereafter NIR) with H$_2$O indices. \cite{Tokunaga1999} used a spectral color
index based on moderate dispersion spectroscopy in the K band. \cite{Leggett1996} used observed NIR low resolution spectra and photometry to
compare with the AMES-Dusty models \citep{Allard2001}. They found radii and effective temperature which are consistent with the estimates based on
photometric data from interior model or isochrone results. \cite{Leggett1998,Leggett2000} revised their results by comparing the spectral energy
distribution and NIR colors of M dwarfs to the same models. Their study provided for the first time a realistic temperature scale of M dwarfs.
In this paper, we present a new version of the BT-Settl models using the TiO line list by \cite[][and B. Plez, private communication]{Plez1998}
which is an important update since
TiO accounts for the most important features in the optical spectrum.
Compared to the version presented in \cite{Allard2012} that was using \cite{Asplund2009} solar abundances, this new BT-Settl model uses
also the latest solar elemental abundances by \cite{Caffau2011}. We compare the revised BT-Settl synthetic spectra with the observed spectra of
152 M dwarfs using spectral synthesis and $\chi^2$ minimization techniques, and color-color diagrams to obtain the atmospheric parameters
(effective temperature, surface gravity and metallicity).
We determine the revised effective temperature scale along the entire M dwarfs spectral sequence and compare these results to those obtained by
many authors. Observations and spectral classification are presented in section~\ref{S_obs}. Details of the the model
atmospheres are described in section ~\ref{S_mod} and the $T_\mathrm{eff}$ determination is explained in section~\ref{S_teffd}. The comparison between
observations and models is done in section~\ref{S_comp} where spectral
features and photometry are compared. The effective temperature scale of M dwarfs is presented in this section. Conclusions are given in
section~\ref{S_concl}.
\section{Observations}
\label{S_obs}
We carried out spectroscopic observations on the 3.6m New Technology Telescope (NTT) at La Silla Observatory (ESO, Chile) in November 2003.
Optical low-resolution spectra were obtained in the Red Imaging and Low-dispersion spectroscopy (RILD) observing mode with the EMMI instrument.
The spectral dispersion of the grism we used is 0.28 nm/pix, with a wavelength range 385-950 nm. We used an order blocking filter to avoid
the second order overlap that occurs beyond 800 nm. Thus the effective wavelength coverage ranges from 520 to 950 nm. The slit was 1 arcsec wide
and the resulting resolution was 1\,nm. The seeing varied from 0.5 to 1.5 arcsec. Exposure time ranged from 15~s for the brightest to 120~s
for the faintest dwarf ($I$ = 15.3). The reduction of the spectra was done using the context \textit{long} of MIDAS. Fluxes were calibrated with
the spectrophotometric standards LTT~2415 and Feige~110.
We obtained spectra for 97 M dwarfs along the entire spectral sequence. They are presented in \cite{Reyle2006,Phan-Bao2005,Crifo2005,Martin2010}.
The list of stars, their spectral types and their optical and NIR photometry are given in Table 1. The photometry has been compiled using the
Vizier catalog access through the Centre de Donn\'{e}es astronomiques de Strasbourg. It comes from the NOMAD catalogue \citep{Zacharias2005}, the
Deep Near-Infrared Survey \citep[DENIS,][] {epchtein1997}, the Two Micron All Sky Survey \citep[2MASS,][]{Skrutskie2006},
\cite{Reid2004,Reid2007}, \cite{Koen2002,Koen2010}.\\
The observations of 55 additional M dwarfs at Siding Spring Observatory (hereafter SSO) were carried out using the Double Beam Spectrograph (DBS) that uses a dichroic
beamsplitter to separate the blue
(300--630 nm) and red (620--1000 nm) light. The blue camera with a 300 l/mm grating provided a 2 pixel resolution of 0.4 nm and the red camera
with a 316 l/mm grating provided a 2 pixel resolution of 0.37 nm. The detectors were E2V 2048x512 13.5 micron/pixel CCDs. The observations were
taken on Mar 27 2008. The spectrophotometric standards used were HD44007, HD45282, HD55496, HD184266, and HD187111 from the STIS Next Generation
Spectral Library (NGSL, version 1)\footnote{http://archive.stsci.edu/prepds/stisngsl/index.html}, and L745-46a and EG131 from
http://www.mso.anu.edu.au/$\sim$bessell/FTP/Spectrophotometry/. The list of stars with their photometry are given in table 2.
Spectral types for the NTT sample are obtained by visual comparison with a spectral template of comparison stars, observed together with the
targets stars at NTT as explained in \cite{Reyle2006}. For comparison, we also derive spectral types using the classification scheme based on the
TiO and CaH bandstrength \citep{Reid1997}. However no comparison stars have been observed with the DBS at
SSO. Thus spectral types for the SSO sample are computed from TiO and CaH bandstrength. Although the instrument is different, we allow to use the
comparison stars observed with EMMI on the NTT as a final check. The results agree within 0.5 subclass.
\begin{table*}[h!]
\caption{Observable and physical quantities for our sample of stars observed at NTT with EMMI.}
\begin{tabular}{llllllllllll}
\hline
Name &Spectral Type &$T_\mathrm{eff}$& $T_\mathrm{eff}$ $^b$ & log\ g &$V$ &$R$ &$I$ &$J$& $H$& $K$ \\
& & (K) &(K)&($cm s^{-2}$) && &&&& \\
\hline
Gl143.1$^a$ &K7 &3900 &--- & 5.0 &10.03 &9.15 &--- &--- &--- &---\\
LHS141 &M0 &3900 &--- &5.0 &10.15 &9.35 &8.38 &7.36 &6.76 &6.58\\
LHS3833$^a$ &M0.5 &3800 &--- & 5.0 &10.06 &9.33 &--- &--- &--- &---\\
HD42581$^a$ &M1 &3700 &--- & 5.0 &8.12 &7.16 &6.12 &--- &--- &---\\
LHS14$^a$ &M1.5 &3600 &---&5.0 &10.04 &9.09 &7.99 &--- &--- &---\\
LHS65$^a$ &M1.5 &3600&3567 & 5.0 &10.86 &10.31 &10.64 &--- &--- &---\\
L127-33 &M2 &3500 &--- & 5.0 &14.19 &14.04 &12.41 &11.17 &10.58 &10.32 \\
NLTT10708 &M2 &3500 &---& 5.0 &11.16 &10.31 &9.17 &7.86 &7.28 &6.98 \\
LP831-68 &M2 &3500 &---& 5.0 &11.02 &10.02 &8.80 &7.61 &6.95 &6.69 \\
NLTT83-11 &M2 &3500&--- & 5.0 &12.90 &12.25 &11.00 &9.68 &9.01 &8.78 \\
APMPMJ0541-5349 &M2 &3500 &---& 5.0 &13.30 &12.84 &11.77 &10.64 &10.17 &9.89\\
LHS1656 &M2.5 &3400 &---& 5.0 &13.30 &12.44 &10.75 &9.52 &8.94 &8.65\\
LP763-82 &M2.5 &3400 &---& 5.0 &12.19 &11.25 &9.86 &8.55 &7.97 &7.69\\
LP849-55 &M2.5 &3400 &---& 5.0 &13.32 &13.25 &11.48 &9.97 &9.36 &9.14\\
LHS5090 &M3 &3300 &---& 5.0 &--- &14.97 &12.85 &11.58 &11.04 &10.84\\
LHS3800 &M3 &3300 &---&5.0 &--- &--- &12.23 &10.93 &10.39 &10.15 \\
LHS3842 &M3 &3300&---& 5.0 &13.80 &12.95 &11.30 &9.88 &9.29 &9.04 \\
LHS1293 &M3 &3300&--- & 5.0 &13.65 &12.66 &11.36 &9.94 &9.35 &9.07 \\
LP994-114 &M3 &3300 &---& 5.0 &--- &11.59 &10.36 &9.00 &8.37 &8.15\\
LTT9783 &M3 &3300&---& 5.0 &--- &12.11 &10.56 &9.17 &8.59 &8.34 \\
LP715-39 &M3 &3300 & 3161&5.0 &12.65 &11.53 &10.09 &8.67 &8.11 &7.82 \\
LHS1208$^a$ &M3 &3300&--- & 5.0 &9.85 &8.97 &--- &--- &--- &---\\
LEHPM4417 &M3 &3300&---& 5.0 &13.73 &13.06 &11.37 &10.09 &9.43 &9.20\\
LP831-45 &M3.5 &3200& 3125&5.0 &12.54 &11.51 &9.90 &8.49 &7.88 &7.62\\
2MASSJ04060688-0534444 &M3.5 &3200&--- & 5.0 &13.29 &12.28 &--- &9.13 &8.55 &8.30\\
LP834-32 &M3.5 &3200 &3108& 5.0 &12.38 &11.24 &9.74 &8.24 &7.65 &7.41\\
LHS502$^a$ &M3.5 &3200&--- & 5.0 &11.49 &10.43 &9.11 &--- &--- &---\\
LEHPM 1175 &M3.5 &3200&--- & 5.0 &--- &13.08 &11.51 &10.01 &9.47 &9.17\\
LEHPM1839 &M3.5 &3200&--- & 5.0 &--- &13.32 &12.11 &10.55 &9.95 &9.71\\
L130-37 &M3.5 &3200 &--- &5.0 &13.04 &11.97 &10.37 &8.94 &8.34 &8.01\\
LEHPM6577 &M3.5 &3200&--- & 5.0 &--- &13.03 &11.79 &10.34 &9.73 &9.47\\
L225-57 &M4 &3200&--- & 5.0 &--- &11.70 &9.79 &8.23 &7.61 &7.31 \\
LP942-107 &M4 &3200 &3052& 5.0 &13.93 &12.73 &11.13 &9.63 &9.08 &8.77 \\
LP772-8 &M4 &3200&--- & 5.0 &14.11 &13.43 &11.52 &10.05 &9.48 &9.20\\
LP1033-31 &M4 &3200&--- & 5.0 &--- &12.12 &10.54 &9.10 &8.46 &8.21 \\
L166-3 &M4 &3200&--- & 5.0 &--- &12.76 &11.33 &9.83 &9.28 &9.00 \\
LP877-72 &M4 &3200&--- &5.0 &--- &11.--- &10.22 &8.86 &8.24 &8.00 \\
LP878-73 &M4 &3200&--- & 5.0 &14.55 &14.22 &12.63 &10.86 &10.27 &10.00 \\
LP987-47 &M4 &3200&--- &5.0 &--- &--- &10.82 &9.41 &8.78 &8.55 \\
LP832-7 &M4 &3200&--- & 5.0 &14.09 &13.45 &--- &9.87 &9.24 &8.98 \\
LHS183 &M4 &3200&--- & 5.0 &12.79 &11.51 &--- &8.57 &8.00 &7.75 \\
LHS1471 &M4 &3200 &---& 5.0 &--- &13.22 &11.56 &9.94 &9.37 &9.08\\
APMPMJ2101-4125 &M4 &3200&--- & 5.0 &--- &13.34 &11.47 &9.96 &9.38 &9.09\\
APMPMJ2101-4907 &M4 &3200&--- & 5.0 &--- &--- &10.52 &9.12 &8.48 &8.19\\
LEHPM3260 &M4 &3200&--- & 5.0 &--- &12.53 &10.60 &9.13 &8.54 &8.19\\
LEHPM3866 &M4 &3200&--- & 5.0 &--- &--- &11.82 &10.21 &9.58 &9.29\\
LEHPM5810 &M4 &3200&--- & 5.0 &--- &13.58 &11.66 &9.91 &9.33 &9.05\\
LHS5045 &M4.5 &3100&--- & 5.0 &--- &--- &10.78 &9.17 &8.60 &8.24\\
LP940-20 &M4.5 &3100 &--- & 5.0 &--- &14.87 &12.65 &10.92 &10.32 &10.01\\
L170-14A &M4.5 &3100&--- & 5.0 &--- &12.86 &11.50 &9.76 &9.13 &8.88\\
LHS1201 &M4.5 &3100&--- & 5.0 &17.55 &15.52 &12.90 &11.12 &10.52 &10.25\\
LHS1524 &M4.5 &3100&--- &5.0 &--- &14.45 &12.65 &10.98 &10.45 &10.17\\
LTT1732 &M4.5 &3100&--- & 5.0 &--- &13.19 &11.27 &9.69 &9.11 &8.80\\
LP889-37 &M4.5 &3100& 2923 & 5.0 &14.52 &13.21 &11.46 &9.77 &9.16 &8.82\\
LHS5094 &M4.5 &3100&--- & 5.0 &14.02 &12.72 &10.97 &9.30 &8.72 &8.41\\
LP655-43 &M4.5 &3100 &2924&5.0 &14.44 &13.14 &11.41 &9.73 &9.14 &8.82\\
LHS138$^a$ &M4.5 &3100&--- & 5.0 &12.07 &10.70 &8.94 &--- &--- &---\\
APMPMJ1932-4834 &M4.5 &3100 &---& 5.0 &--- &14.38 &12.37 &10.63 &10.02 &9.72\\
2MASSJ23522756-3609128 &M4.5 &3100 &--- & 5.0 &--- &17.27 &--- &13.09 &12.57 &12.28\\
LEHPM640 &M4.5 &3100 &---& 5.0 &17.74 &14.26 &12.30 &10.76 &10.14 &9.90\\
LEHPM1853 &M4.5 &3100 &--- & 5.0 &--- &12.77 &11.03 &9.46 &8.85 &8.61\\
LEHPM3115 &M4.5 &3100 &--- & 5.0 &--- &13.94 &12.10 &10.49 &9.92 &9.63\\
LEHPM4771 &M4.5 &3100 &--- & 5.0 &17.74 &13.79 &11.29 &9.54 &8.95 &8.63\\
LEHPM4861 &M4.5 &3100 &--- & 5.0 &--- &13.28 &11.75 &10.13 &9.60 &9.34\\
L291-115 &M5 &2900&--- & 5.0 &15.88 &14.90 &12.26 &10.44 &9.83 &9.54 \\
LP904-51 &M5 &2900&--- & 5.0 &--- &15.32 &12.84 &11.04 &10.44 &10.16 \\
LHS168 &M5 &2900&--- & 5.0 &13.78 &12.60 &--- &8.77 &8.21 &7.83 \\
\hline
\end{tabular}
\end{table*}
\begin{table*}[t]
{\bf \small Table 1.} Continued.\vspace*{0.15cm}\\
\begin{tabular}{llllllllllll}
\hline
Name &Spectral Type &$T_\mathrm{eff}$& $T_\mathrm{eff}$ $^b$ & log\ g &$V$ &$R$ &$I$ &$J$& $H$& $K$ \\
& &(K) &(K)&($cm s^{-2}$) && &&&& \\
\hline
LP829-41 &M5.5 &2800&--- & 5.0 &16.10 &15.95 &13.21 &11.31 &10.76 &10.40 \\
LP941-57 &M5.5 &2800&--- & 5.0 &--- &14.88 &12.98 &11.06 &10.47 &10.13 \\
LHS546 &M5.5 &2800&--- & 5.0&14.69 &--- &--- &9.15 &8.50 &8.18\\
LP714-37 &M5.5 &2800&--- & 5.5 &16.26 &15.02 &12.99 &11.01 &10.37 &9.92 \\
LHS1326 &M6 &2800&--- & 5.5 &15.61 &14.49 &--- &9.84 &9.25 &8.93 \\
2MASSJ12363959-1722170 &M6 &2800&--- &5.0 &17.56 &15.86 &13.91 &11.67 &11.09 &10.71\\
2MASSJ21481595-1401059 &M6.5 &2700&--- & 5.0 &--- &20.20 &17.15 &14.68 &14.11 &13.65\\
2MASSJ05181131-3101519 &M6.5 &2700&--- &5.0 &17.74 &16.85 &14.17 &11.88 &11.23 &10.90\\
LP788-1 &M6.5 &2700 &---& 5.0 &--- &16.66 &13.36 &11.07 &10.47 &10.07 \\
APMPMJ1251-2121 &M6.5 &2700&--- & 5.0 &--- &16.65 &13.78 &11.16 &10.55 &10.13\\
APMPMJ2330-4737 &M7 &2700&--- &5.0 &--- &--- &13.70 &11.23 &10.64 &10.28 \\
LP789-23 &M7 &2700&--- & 5.0 &--- &17.90 &14.55 &12.04 &11.39 &10.99\\
LHS292 &M7 &2700&--- & 5.5 &15.60 &14.40 &11.25 &8.86 &8.26 &7.93 \\
2MASSJ03144011-0450316 &M7.5 &2600&--- &5.0 &--- &19.43 &--- &12.64 &12.00 &11.60 \\
LHS1604 &M7.5 &2600&--- & 5.0 &18.02 &16.52 &13.75 &11.30 &10.61 &10.23 \\
LP714-37 &M7.5 &2600&--- & 5.5 &16.26 &15.52 &12.99 &11.01 &10.37 &9.92 \\
LP655-48 &M7.5 &2600 &2250& 5.0 &17.86 &15.95 &13.35 &10.66 &9.99 &9.54 \\
LP851-346 &M7.5 &2600&--- & 5.5 &--- &16.79 &13.77 &10.93 &10.29 &9.88 \\
LHS1367 &M8 &2600&--- & 5.0 &--- &17.34 &14.18 &11.62 &10.95 &10.54 \\
2MASSJ05022640-0453583 &M8 &2600&---& 5.0 &--- &20.39 &17.35 &14.52 &13.95 &13.58 \\
LHS132 &M8 &2600&--- & 5.0 &--- &17.14 &13.83 &11.13 &10.48 &10.07 \\
2MASSJ22062280-2047058 &M8 &2600&--- & 5.0 &--- &18.93 &15.09 &12.37 &11.69 &11.31 \\
2MASSJ22264440-7503425 &M8 &2600&--- & 5.0 &--- &18.95 &15.20 &12.35 &11.70 &11.25 \\
2MASSJ04103617-1459269 &M8.5 &2500&--- & 5.5 &--- &--- &16.68 &13.94 &13.24 &12.81\\
2MASSJ05084947-1647167 &M8.5 &2500&--- & 5.5 &--- &--- &16.46 &13.69 &12.96 &12.53 \\
2MASSJ04362788-4114465 &M8.5 &2500&--- & 5.5 &--- &19.96 &16.04 &13.10 &12.43 &12.05 \\
2MASSJ10481463-3956062 &M9 &2500&--- &5.5 &--- &15.93 &12.67 &9.54 &8.90 &8.45 \\
2MASSJ20450238-6332066 &M9.5 &2500&--- & 5.5 &--- &19.24 &16.05 &12.62 &11.81 &11.21\\
2MASSJ09532126-1014205 &M9.5 &2500&--- & 5.5 &--- &19.58 &16.82 &13.47 &12.64 &12.14 \\
\hline
\end{tabular}
$^a$ Saturation in NIR bands.\\
$^b$ $T_\mathrm{eff}$ from \cite{Casagrande2008}.
\end{table*}
\begin{table*}[h!]
\caption{Observable and physical quantities for our sample of stars observed at SSO.}
\begin{tabular}{llllllllllll}
\hline
Name &Spectral Type &$T_\mathrm{eff}$& $T_\mathrm{eff}$ $^b$ & log\ g &$V$ &$R$ &$I$ &$J$& $H$& $K$ \\
& & (K) &(K)&($cm s^{-2}$) && &&&& \\
\hline
HIP49986 & M1.5 & 3700&3445 &5.0 & 9.07 & 8.21 & 7.08 & 5.89 & 5.26 & 5.01 \\
HIP82256 & M1.5 & 3700&3470 &5.0 & 11.38 & 10.39 & 9.24 & 8.04 & 7.48 & 7.22 \\
HIP56528 & M1.5 & 3600 &3472 &5.0 &9.81 & 8.85 & 7.66 & 6.47 & 5.86 & 5.62 \\
NLTT19190 & M1.5 & 3600&3456 &5.0 &11.49 & 10.57 & 9.34 & 8.11 & 7.47 & 7.20 \\
NLTT42523 & M2 & 3600 &3444 & 5.0 &12.08 & 11.06 & 9.81 & 8.60 & 8.01 & 7.80 \\
HIP80229 & M2 & 3600 &3486 & 5.0 &11.91 & 10.90 & 9.65 & 8.48 & 7.87 & 7.64 \\
LP725-25 & M2 & 3600 &3476 &5.0 &11.76 & 10.82 & 9.59 & 8.36 & 7.68 & 7.44 \\
HIP61413 & M2 & 3500 &3454 &5.0 & 11.49 & 10.48 & 9.17 & 7.99 & 7.37 & 7.15 \\
LP853-34 & M2 & 3500 & 3339&5.0 &12.32 & 11.31 & 9.99 & 8.69 & 8.10 & 7.83 \\
LP859-11 & M2 & 3500 &3433 & 5.0 &12.00 & 10.97 & 9.69 & 8.49 & 7.88 & 7.63 \\
LP788-49 & M2 & 3500 &3356 & 5.0 &11.81 & 10.85 & 9.55 & 8.30 & 7.74 & 7.49 \\
HIP42762 & M2 & 3500 &3302 & 5.0 &11.75 & 10.76 & 9.42 & 8.12 & 7.49 & 7.28 \\
HIP51317 & M2 & 3500 &3403 &5.0 &9.67 & 8.67 & 7.34 & 6.18 & 5.60 & 5.31 \\
HIP60559 & M2 & 3500 &3382 & 5.0 &11.30 & 10.29 & 8.99 & 7.73 & 7.25 & 6.95 \\
HIP47103 & M2 & 3500 &3319 &5.0 &10.87 & 9.89 & 8.58 & 7.34 & 6.74 & 6.47 \\
HIP93206 & M2.5 & 3500 &3366 &5.0 & 11.23 & 10.18 & 8.80 & 7.52 & 6.93 & 6.70 \\
LP834-3 & M2.5 & 3500 &--- & 5.0 &--- & --- &--- &--- &--- &--- \\
HIP84521 & M2.5 & 3500 &3345 &5.0 &11.57 & 10.53 & 9.22 & 7.93 & 7.39 & 7.11 \\
HIP91430 & M2.5 & 3500 &3352 & 5.0 &11.32 & 10.26 & 8.92 & 7.66 & 7.06 & 6.85 \\
HIP50341 & M2.5 & 3500&3314 & 5.0 &11.02 & 10.01 & 8.62 & 7.32 & 6.71 & 6.45 \\
LP672-2 & M2.5 & 3400 &--- &5.0 &12.58 & 11.54 & 10.12 & 8.80 & 8.14 & 7.93 \\
NLTT24892 & M2.5 & 3400 &3244 &5.0 & 12.52 & 11.47 & 10.05 & 8.73 & 8.118 & 7.84 \\
NLTT34577 & M2.5 & 3400 & 3254&5.0 &12.44 & 11.40 & 9.99 & 8.64 & 8.00 & 7.80 \\
LP670-17 & M3 & 3400 &3226 & 5.0 &12.14 & 11.08 & 9.63 & 8.28 & 7.68 & 7.39 \\
HIP59406 & M3 & 3400 &3226 & 5.0 &11.75 & 10.69 & 9.25 & 7.89 & 7.36 & 7.04 \\
HIP74190 & M3 & 3400 &3258 &5.0 &11.55 & 10.48 & 9.05 & 7.72 & 7.13 & 6.86 \\
NLTT46868 & M3.5 & 3400 &3221 &5.0 & 12.23 & 11.08 & 9.61 & 8.26 & 7.73 & 7.44 \\
HIP62452 & M4 & 3300 & 3095 & 5.0 &11.46 & 10.31 & 8.71 & 7.19 & 6.67 & 6.36 \\
NLTT25488 & M4 & 3200 & 2986& 5.0 &15.66 & 14.46 & 12.73 & 11.09 & 10.52 & 10.21 \\
NLTT29087 & M4 & 3200&2971 &5.0 & 14.79 & 13.57 & 11.84 & 10.22 & 9.62 & 9.35 \\
NLTT29790 & M4 & 3200 & 2987 & 5.0 &14.73 & 13.54 & 11.85 & 10.22 & 9.64 & 9.34 \\
LP734-32 & M4 & 3200 &3024 & 5.0 &12.15 & 10.99 & 9.35 & 7.77 & 7.14 & 6.86 \\
LP739-2 & M4 & 3100 &2939 &5.0 &14.44 & 13.18 & 11.40 & 9.73 & 9.17 & 8.89 \\
LP735-29 & M4 & 3100&2940 &5.0 & 14.18 & 12.95 & 11.18 & 9.52 & 8.97 & 8.67 \\
GJ1123 & M4 & 3100&--- & 5.0& 13.14 & 11.90 & 10.10 & 8.33 & 7.77 & 7.45 \\
GJ1128 & M4 & 3100 & ---& 5.0 &12.66 & 11.40 & 9.61 & 7.95 & 7.38 & 7.04 \\
NLTT35266 & M4.5 & 3100 & 2942& 5.0& 15.15 & 13.88 & 12.05 & 10.41 & 9.94 & 9.66 \\
NLTT41951 & M4.5 & 3100 & & 5.0 &15.06 & 13.77 & 11.99 & 10.36 & 9.80 & 9.51 \\
NLTT21329 & M4.5&3000&2949 &5.0 & 13.75 & 12.38 & 10.42 & 8.60 & 8.07 & 7.73 & \\
LP732-35 & M5 & 3100 &2901 & 5.0&14.10 & 12.78 & 10.94 & 9.36 & 8.76 & 8.49 \\
NLTT18930 & M5 & 3100 &2903 & 5.0 &15.34 & 13.93 & 12.03 & 10.31 & 9.76 & 9.44 \\
2MASS J14221943-7023371 & M5 & 3000&--- & 5.0& --- & --- &--- &--- &--- &--- \\
NLTT22503 & M5 & 3000& 2785 &5.0 &13.66 & 12.32 & 10.39 & 8.50 & 7.92 & 7.60 \\
NLTT28797 & M5 & 3000 &2826 & 5.0 &15.62 & 14.24 & 12.32 & 10.54 & 9.99 & 9.64 \\
NLTT30693 & M5.5 & 3000 &2785 &5.5 &15.32 & 13.86 & 11.85 & 9.95 & 9.36 & 9.00 \\
LHS288 & M5.5 & 3000&2770 &5.0 &13.87 & 12.42 & 10.31 & 8.48 & 8.05 & 7.73 \\
GJ551 & M5.5& 2900 &--- & 5.0 &3.63 & 2.08 & 5.36 & 4.83 & 4.38 &--- \\
LHS2502 & M6 & 2900 & 2468&5.5 &19.36 & 17.54 & 15.33 & 12.75 & 12.07 & 11.79 \\
NLTT20726 & M6.5 & 2800& 2464 & 5.0 &16.11 & 14.24 & 11.85 & 9.44 & 8.84 & 8.44 \\
GJ406 & M6.5 & 2800&--- & 5.5 &13.57 & 11.81 & 9.51 & 7.08 & 6.48 & 6.08 \\
LHS2351 & M7 & 2800& 2346 &5.5 & 19.22 & 17.39 & 14.91 & 12.33 & 11.72 & 11.33 \\
SCR J1546-5534 & M7.5 & 2700 &---&5.5 & --- &--- &--- &--- &--- &--- \\
GJ752b & M8 & 2700 & ---& 5.5& 5.01 &--- &--- & 9.91 & 9.23 & 8.76 \\
GJ644c & M7 & 2700 &--- & 5.5 &16.90 & 14.78 & 12.24 & 9.78 & 9.20 & 8.82 \\
LHS2397a & M8 & 2700 &---& 5.5 &19.66 & 17.42 & 14.86 & 11.93 & 11.23 & 10.73 \\
\hline
$^b$ $T_\mathrm{eff}$ from \cite{Casagrande2008}.
\end{tabular}
\end{table*}
\section{Model atmospheres}
\label{S_mod}
For this paper, we use the most recent BT-Settl models partially published in a review by \cite{Allard2012} and described by \cite{Allard2012b}.
These model atmospheres are computed with the \texttt{PHOENIX} multi-purpose atmosphere code version 15.5 \citep[][Allard et al.
2001]{Phoenix97}\nocite{Allard2001} solving the radiative transfer in 1D spherical symmetry, with the classical assumptions: hydrostatic
equilibrium, convection using the Mixing Length Theory, chemical equilibrium, and a sampling treatment of the opacities. The models use
a mixing length as derived by the Radiation HydroDynamic (hereafter RHD) simulations of \cite{Ludwig2002,Ludwig2006} and \cite{Freytag2012} and a
radius as determined by the \cite{BCAH98} interior models as a function of the atmospheric parameters ($T_\mathrm{eff}$, log $g$, [M/H]).
The BT-Settl grid extends from $T_\mathrm{eff} = 300 - 7000$\,K, log$g = 2.5 - 5.5$ and [M/H]$= -2.5 - 0.0$ accounting for alpha element enrichment. The
reference solar elemental abundances used in this version of the BT-Settl models are those defined by \cite{Caffau2011}. The synthetic colors and
spectra are distributed with a spectral resolution of around R=100000 via the \texttt{PHOENIX} web
simulator\footnote{http://phoenix.ens-lyon.fr/simulator}.
\label{S_teffd}
\begin{figure*}
\centering
\includegraphics[width=9.0cm,height=12cm]{Rajpurohit_fig1b.ps}
\includegraphics[width=9.0cm,height=12cm]{Rajpurohit_fig1a.ps}
\caption{$T_\mathrm{eff}$ vs near-infrared colors (left panel) and color-color plot (right panel) for observed M dwarfs (open and filled circle) compared
to the values obtained with the 5 Gyrs isochrones from \cite{BCAH98} at various metallicities.}
\label{Fig:1}
\end{figure*}
Hot temperature grains have been shown to form in the uppermost layers of M dwarfs with effective temperatures below 3000\,K, but clear effects
observable at the spectral resolution considered in this paper are only apparent below 2600\,K i.e. for later spectral type than those considered in
this paper. These grains produce a "veiling" by dust scattering over the optical band of the latest type M dwarfs. The BT-Settl models use therefore
a slightly revised version of the \cite{Rossow1978} cloud model. See \cite{Allard2012,Allard2012c}, \cite{Allard2012b} and \cite{Rajpurohit2012a} for
details on the model construction.
Relative to previous models by \cite{Allard2001}, the current version of the BT-Settl model atmosphere is using the BT2 water vapor line list computed
by \cite{Barber2006}, TiO, VO, CaH line lists by \cite{Plez1998}, MgH by \cite[Story et al. 2003]{Weck2003}\nocite{Story2003}, FeH and CrH by
\cite[Chowdhury et al. 2006]{Dulick2003}\nocite{Chowdhury2006}, NH$_3$ by \cite{Yurchenko2011}, CO$_2$ \citep{Tashkun2004}, and H$_2$ Collision
Induced Absorption (CIA) by \cite{Borysow2001} and \cite{Abel2011}, to mention the most important. We use the CO line list by
\cite{Goorvitch94a,Goorvitch94b}. Detailed profiles for the alkali lines are also used \citep{AllardN2007}.
In general, the \cite{Unsold1968} approximation is used for the atomic damping constants with a correction factor to the widths of 2.5 for the
non-hydrogenic atoms \citep{Valenti1996}. More accurate broadening data for neutral hydrogen collisions by \cite{Barklem2000} have been included for
several important atomic transitions such as the alkali, Ca\,I and Ca\,II resonance lines. For molecular lines, we have adopted average values
(e.\,g.\ $\langle\gamma_6^{HIT}(T_0, P_0\footnote{Standard temperature 296\,K and pressure 1 atm})\rangle_{H_2O} = 0.08 \,\, P_{\rm gas} \,
[\mathrm{cm}^{-1}\mathrm{atm}^{-1}]$ for water vapor lines) from the HITRAN database \citep{HITRAN2008},
which are scaled to the local gas pressure and temperature
\begin{equation}
\gamma_6(T) = \langle\gamma_6^{HIT}(T_0,P_0)\rangle \left(\frac{296\,K}{T}\right)^{0.5}\, \left(\frac{P}{1\, {\rm atm}} \right) \enspace,
\end{equation}
with a single temperature exponent of 0.5, to be compared to values ranging mainly from 0.3 to 0.6 for water transitions studied by
\citet{Gamache1996}. The HITRAN database gives widths for broadening in air, but \citet{VSTAR2012} find that these agree in general within 10\,--\,20\% with
those for broadening by a solar composition hydrogen-helium mixture.
\section{$T_\mathrm{eff}$ determination}
\label{S_teffd}
We use a least-square minimization program using the new BT-Settl model atmospheres
to derive a revised effective temperature scale of M dwarfs. The stars in our samples most probably belong to the thin disc of our Galaxy
\citep{Reyle2002,Reyle2004}. Thus we determine the $T_\mathrm{eff}$ of our targets assuming solar metallicity. This is a reasonable assumption as can be
seen in Fig.~\ref{Fig:1} where we compare our two samples to three 5 Gyrs isochrones with solar, [M/H]=~-0.5 and -1.0~dex. The samples are clearly
compatible with solar metallicity.
Both theory and observation indicate that M dwarfs have log\,$g~=~5.0 \pm 0.2$ \citep{Gizis1996,Casagrande2008} except for the latest-type M
dwarfs. We therefore restrict our analysis to log\,$g~=~5.0 - 5.5$ models.
Each synthetic spectrum was convolved to the observed spectral resolution \and a scaling factor is applied to normalize the average flux to unity.
We then compare each of the observed spectra with all the synthetic spectra in the grid by taking the difference between the
flux values of the synthetic and observed spectra at each wavelength point. We interpolated the model spectra on the wavelength grid of the observed spectra. The sum of the squares of these differences is obtained for each model
in the grid, and the best model for each object is selected. The best models were finally inspected visually by comparing them with the
corresponding observed spectra.
Due to the lower signal-to-noise ratio in the SSO 2.3~m spectra bluewards of
500~nm (see Fig.~3), especially for spectral types later than M4, we have excluded this region below 500~nm from the $\chi^2$ computation.
We have also checked the variation in effective temperature of the best fit as a function of the spectral type of the observed dwarfs. We found
generally good agreement and conclude that our model fitting procedure can be used to estimate the effective temperature with an uncertainty of
$\sim$100\,K. The purpose of this fit is to determine the effective temperature by fitting the overall shape of the optical spectra.
No attempt has been made to fit the individual atomic lines such as the K\,I and Na\,I resonance doublets. With the available resolution we
cannot constrain the metallicity; high resolution spectra would be necessary (Rajupurohit et al. in prep.). In addition, we checked the influence of the spectral resolution to our derived temperatures. We
degraded the resolution of the spectra of SSO 2.3~m down to 1~nm and redid the procedure. No systematic difference in $T_\mathrm{eff}$ was found.
The results are summarized in Table~1 and ~2.
\section{Comparison between models and observations}
\label{S_comp}
\subsection{Spectroscopic confrontation}
The optical spectrum of M dwarfs is dominated by molecular band absorption, leaving no window onto the continuum \citep{AllardPhDT90}. The major
opacity sources in the optical regions are due to titanium oxide (TiO) and vanadium oxide (VO) bands, as well as to MgH, CaH, FeH hydrides
bands and CAOH hydroxide bands in late-type M dwarfs. In M dwarfs of spectral type later than M6, the outermost atmospheric layers fall below the
condensation temperature of silicates, giving rise to the formation of dust clouds \citep[][Allard et al.
1997]{Tsuji1996b,Tsuji1996a}\nocite{Allard1997}.
\begin{figure*}[ht!]
\centering
\includegraphics[width=18cm,height=18cm]{Rajpurohit_fig2.ps}
\caption{Optical to red SED of M dwarfs from M0 to M9.5 observed with the NTT at a spectral resolution of 10.4~\AA\ compared to the best fit
BT-Settl synthetic spectra (red lines). The models displayed have a surface gravity of log $g$ = 5.0 to 5.5. Telluric features near
7600\,\AA\ have been ignored from the chi-square minimization.
}
\label{Fig:2}
\end{figure*}
\begin{figure*}[ht!]
\centering
\includegraphics[width=18cm,height=18cm]{Rajpurohit_fig3.ps}
\caption{Optical to red SED of M dwarfs from M1 to M8 observed with the SSO 2.3~m at a spectral resolution of 1.4~\AA\
compared to the best fitting (chi-square minimization) BT-Settl synthetic spectra (red lines). The models displayed have a surface
gravity of log $g$=5.0 to 5.5. At blue wavelengths ($<$ 5000$\AA$) the instrumentals noise dominate the late-type M dwarfs.
}
\label{Fig:3}
\end{figure*}
We compared the two samples of M dwarfs with the most recent BT-Settl synthetic spectra in Fig.~\ref{Fig:2} and \ref{Fig:3} through the entire
M dwarf spectral sequence. The synthetic spectra reproduce very well the slope of the observed spectra across the M dwarfs regime. This is a
drastic improvement compared to previous comparisons of earlier models \citep[e.g.][]{Leggett1998}.
However, some indications of missing opacities
persist in the blue part of the late-type M dwarf such as the B' $^{ 2}${$\Sigma$}$^{+}$$<$-- X $^{ 2}${$\Sigma$}$^{+}$
system of MgH \citep{Story2003}, as well as TiO and VO opacities around 8200 \AA. Opacities are totally
missing for the CaOH band at 5570\,\AA. The missing hydride bands of AlH and NaH between 3800 and 4600~\AA\ among others could be responsible for
the remaining discrepancies. Note that chromospheric emission fills the Na\,I\,D transitions in the latest-type M dwarfs displayed here.
We see in this spectral regime no signs of dust scattering or of the weakening of features due to sedimentation onto grains until the M8 and later
spectral types where the spectrum becomes flat due to the sedimentation of TiO and VO bands and to the veiling by dust scattering.
\begin{figure*}[ht!]
\centering
\subfloat{\label{fig:4}\includegraphics[width=12.5cm,height=10cm]{Rajpurohit_fig4a.ps}}
\qquad
\subfloat{\label{fig:4}\includegraphics[width=12.5cm,height=10cm]{Rajpurohit_fig4b.ps}}
\caption{Optical and NIR colors obtained with the 5 Gyrs isochrones from \cite{BCAH98} at solar metallicity compared with the
two observation samples (filled circles for the NTT sample and open circle for the SSO 2.3~m spectra). Typical error bars are
comparable or smaller than the size of the symbols.}
\label{Fig:4}
\end{figure*}
\begin{figure*}[ht!]
\ContinuedFloat
\centering
\subfloat{\label{fig:4}\includegraphics[width=12.5cm,height=10cm]{Rajpurohit_fig4c.ps}}
\qquad
\subfloat{\label{fig:4}\includegraphics[width=12.5cm,height=10cm]{Rajpurohit_fig4d.ps}}
\caption{Continued.}
\label{Fig:4}
\end{figure*}
\subsection{Photometric confrontation}
The models can be validated by comparing published isochrones interpolated into the new BT-Settl synthetic color tables with observed photometry.
We have taken the log g and $T_\mathrm{eff}$ for the fixed age of 5 Gyrs from \cite{BCAH98} isochrones and calculated the colors of the star
according to the BT-Settl models.
.
The models are compared to observations in color-color diagrams in Fig. ~\ref{Fig:4} for our two samples. The compiled photometry in the
NTT sample is less homogeneous, translating to a larger spread in particular for colors including the $V$ and $R$-band. This dispersion becomes dramatical for the coolest, and faintest, stars.
except for lowest mass objects at very young ages. The isochrone reproduces the two samples over the entire M dwarf spectral range in most colors.
In particular, the models reproduce the $V$-band colors of M
dwarfs, as illustrated by the $V-I$, $V-J$ and $V-K$ colors.
An increasing offset to the latest types persists in the $H-K$ and $V-R$ color indices. The observations suggest also a flattening and possibly a rise in $J-H$ and $J-K$ to the latest types which is not reproduced by the model. These inadequacies at the coolest temperature could be linked to missing opacities.
\subsection{The $T_\mathrm{eff}$-scale of M dwarfs}
\label{S_teff}
The effective temperature scale versus spectral type is shown in Fig.~\ref{Fig:5}. The $T_\mathrm{eff}$-scale determined using the NTT sample (filled
circles) is in agreement with the SSO sample (filled triangles) but we found systematically 100\,K higher $T_\mathrm{eff}$ for SSO sample for spectral
type later than M5. The relation shows a saturation trend for spectral types later than M8. This illustrate the fact that the optical spectrum no
longer change sensibly with $T_\mathrm{eff}$ in this regime due to dust formation.
In the following we compare our scale to other works. \cite{Bessell1991} determined the temperatures by comparing blackbodies to the NIR photometry
of their sample. They used the temperature
calibration of \cite{Wing1979} and \cite{Veeder1974}. These calibrations were identical between $ 2700 \le T_\mathrm{eff} \le 3500\,K$.
Their scale agrees with the modern values for M dwarfs earlier than M6, but becomes gradually too cool with later spectral type and too hot for
earlier M types.
\cite{Leggett1996} used the Base grid by \cite{AH95} covering the range of parameter down to the coolest known M dwarfs, M subdwarfs and
brown dwarfs. They obtained the $T_\mathrm{eff}$ of M dwarfs by comparing the observed spectra to the synthetic spectra. They perform their comparison
independently at each of their four wavelength regions: red, $J$, $H$, and $K$. The different wavelength regions gave consistent values of
$T_\mathrm{eff}$ within 300\,K.
\cite{Gizis1997} used the NextGen model atmosphere grid by \cite{Allard1997}. These models include more molecular lines from ab
initio simulations (in particular for water vapor) than the previous Base model grid. \cite{Leggett2000} used the
more modern AMES-Dusty model atmosphere grid by \cite{Allard2001}. They obtained a revised $T_\mathrm{eff}$ scale which is 150-200\,K cooler for
early-Ms, and 200\,K hotter for late-Ms than the scale presented in Fig.~\ref{Fig:5}. \cite{Testi2009} determine the $T_\mathrm{eff}$ by fitting the
synthetic spectra to the observations. They used three classes of models: the AMES-Dusty, AMES-Cond and the BT-Settl models. With some individual
exceptions they found that the BT-Settl models were the most appropriate for M type and early L-type dwarfs.
Finally, for spectral type later than M0, \cite{Luhman2003} adopted the effective temperature which is based on the NextGen and AMES-Dusty
evolutionary models of \cite{BCAH98} and \cite{Chabrier2000} respectively. They obtained the $T_\mathrm{eff}$ by comparing the H-R diagram from
theoretical isochrones of \cite{BCAH98} and \cite{Chabrier2000}. For M8 and M9, \cite{Luhman2003} adjusted the temperature scale from
\cite{Luhman1999} so that spectral sequence fall parallel to the isochrones. Their $T_\mathrm{eff}$ conversion is likely to be inaccurate at some level,
but as it falls between the scales for dwarfs and giants, the error in $T_\mathrm{eff}$ are modest.
The different $T_\mathrm{eff}$ scales are in agreement within 250-300\,K. But the \cite{Gizis1997} relation shows the largest differences, with the largest
$T_\mathrm{eff}$-values (up to 500\,K). This is due to the incompleteness of the TiO and water vapor line lists used in the NextGen model atmospheres. Note
also how the \cite{Luhman2003} $T_\mathrm{eff}$ scale is gradually overestimating $T_\mathrm{eff}$ towards the bottom of the main sequence for spectral types later
than M4.
\begin{figure*}[ht!s]
\centering
\includegraphics[width=15cm,height=10cm]{Rajpurohit_fig5.ps}
\caption{
Spectral type - $T_\mathrm{eff}$ relation obtained with the NTT sample (filled circles) and the SSO 2.3~m sample (open circles) compared
to relations by \cite{Bessell1991}, \cite{Gizis1997}, \cite{Leggett1996}, \cite{Leggett2000}, \cite{Testi2009}, and \cite{Luhman1999}.
}
\label{Fig:5}
\end{figure*}
\medskip
\begin{figure*}[ht!]
\centering
\subfloat{\label{fig:6}\includegraphics[width=12.8cm,height=7.5cm]{Rajpurohit_fig6a.ps}}
\qquad
\subfloat{\label{fig:6}\includegraphics[width=12.8cm,height=7.5cm]{Rajpurohit_fig6b.ps}}
\qquad
\subfloat{\label{fig:6}\includegraphics[width=12.8cm,height=7.5cm]{Rajpurohit_fig6c.ps}}
\caption{Color-$T_\mathrm{eff}$ plots in different bands from the NTT sample (filled circles) and the SSO 2.3~m sample (open circles).
Spectral types are also indicated. The predictions from BT-Settl (solid line), NextGen (dotted line) and AMES-Dusty (das-dotted) for
solar metallicities are over plotted. Theoretical masses in solar mass are indicated. Predictions from other authors are shown for comparison when
available.}
\label{Fig:6}
\end{figure*}
\begin{figure*}[ht!]
\ContinuedFloat
\centering
\subfloat{\label{fig:6}\includegraphics[width=12.5cm,height=7.5cm]{Rajpurohit_fig6d.ps}}
\qquad
\subfloat{\label{fig:6}\includegraphics[width=12.5cm,height=7.5cm]{Rajpurohit_fig6f.ps}}
\qquad
\subfloat{\label{fig:6}\includegraphics[width=12.5cm,height=7.5cm]{Rajpurohit_fig6g.ps}}
\caption{Continued.}
\label{Fig:6}
\end{figure*}
\begin{figure*}[ht!]
\ContinuedFloat
\centering
\subfloat{\label{fig:6}\includegraphics[width=12.5cm,height=7.5cm]{Rajpurohit_fig6h.ps}}
\qquad
\subfloat{\label{fig:6}\includegraphics[width=12.5cm,height=7.5cm]{Rajpurohit_fig6i.ps}}
\qquad
\subfloat{\label{fig:6}\includegraphics[width=12.5cm,height=7.5cm]{Rajpurohit_fig6j.ps}}
\caption{Continued.}
\label{Fig:6}
\end{figure*}
$T_\mathrm{eff}$ versus color relations are shown in Fig.~\ref{Fig:6} in various photometric bands. The photometry of our NTT sample (filled circles) is
compiled from the literature, causing a large spread particularly in the $V$ and $R$-band. The SSO 2.3~m sample (filled triangles) in
comparison is more uniform. Our relations are compared to the predictions from BT-Settl isochrones at 5 Gyrs. It shows that the model is able to
reproduce quite
properly the colors of M-dwarfs, even in the $V$-band. There is a slight offset visible in the R-band due to missing molecular opacities (see above).
These relations are compared to previously published relations when available.
\cite{Berriman1992} derive the $T_\mathrm{eff}$ by matching the blackbody flux anchored at $K$ band (2.2~$\mu$m) to the total bolometric flux including
both the spectroscopic and photometric observed data points. They estimated the uncertainties in $T_\mathrm{eff}$ to be $\pm$ 4$\%$. \cite{Leggett1996} used
the synthetic $I-K$ and $I-J$ colors to estimate $T_\mathrm{eff}$.
\cite{Leggett1996} used synthetic broadband colors from the preliminary version of AMES-Dusty model produced by \cite{Allard1994}. They used the $V-K$, $I-K$, $J-H$ and $H-K$ colors assuming log$g = 5.0$ and solar metallicity, and found a hotter $T_\mathrm{eff}$-scale (by on average of 130\,K) than that of \cite{Berriman1992}.
More recently, \cite{Casagrande2008} used the \texttt{PHOENIX} Cond-GAIA model atmosphere grid (P. H. Hauschildt, unpublished) to determine the atmospheric parameters of their sample of 343 nearby M dwarfs with high-quality optical and IR photometry. These models are similar to those
published by \cite{Allard2001} with the exception that they were computed by solving the radiative transfer in spherical symmetry. The authors
determined the $T_\mathrm{eff}$ using a version of the multiple optical-infrared method (IRFM) generalized to M dwarfs, and elaborated by \cite{Blackwell1977}
and \cite{Blackwell1979,Blackwell1980}. Fig.~\ref{Fig:6} shows that the \cite{Casagrande2008} $T_\mathrm{eff}$-scale is systematically, and progressively with
decreasing $T_\mathrm{eff}$, cooler than the BT-Settl isochrones. Given that a large number of stars are common with \cite{Casagrande2008} sample, we did a
star-by-star comparison of the $T_\mathrm{eff}$ determination. The values are given in Table~1 and 2. It confirms the systematic offset in the temperature
scale.
For cooler stars with $T_\mathrm{eff} < 3000$\,K, the $T_\mathrm{eff}$ determinations diverge by 100 to 300\,K.
This is due, among other things, to the use of the \cite{GNS93} solar elemental abundances (see Allard et al. 2012\nocite{Allard2012} for a comparison
of the different solar elemental abundance determinations and their effects on model atmospheres).
\section{Conclusion}
\label{S_concl}
We have compared a revised version of the BT-Settl model atmospheres \citep{Allard2012} to the observed NTT and SSO 2.3~m spectra and colors.
This new version uses the \cite{Caffau2011} solar elemental abundances, updates to the atomic and molecular line broadening and the TiO line
list from \cite[][and B. Plez, private communication]{Plez1998}. This list provides a more accurate description on the TiO bands in the M dwarfs. The
systematic discrepancy
between the delta- and epsilon-bands found by \cite{Reiners2005}, which seriously affected the effective temperature determination, is largely
alleviated by using the \cite[][and B. Plez, private communication]{Plez1998} TiO line list although discrepancies remain for the coolest stars.
The BT-Settl models reproduce the spectral energy distribution and observed colors across the M dwarfs spectral regime to an unprecedented
quality, as well as the colors. The $V$ band is also well reproduced by the models. Some
discrepancies remain in the strength of some and missing other molecular absorption bands in particular in the ultraviolet spectral range.
Effective temperatures were determined by using a least-square minimization routine which gives accurate temperatures within 100\,K uncertainty. We
compare our temperature versus color relations using multi-wavelength photometry with the predictions from BT-Settl isochrones, assuming an age of 5
Gyrs. In general, the
BT-Settl isochrones are in good agreement with the observed colors, even at temperatures below 2800\,K affected by dust-treatment in the BT-Settl
models. We found that the \cite{Casagrande2008} $T_\mathrm{eff}$-scale is systematically cooler than the BT-Settl isochrones due, among other things,
to the \cite{GNS93} solar elemental abundances adopted in the GAIA-Cond model atmosphere grid used for that work.
The \cite{Luhman2003} $T_\mathrm{eff}$-scale is on the contrary progressively too hot towards the bottom of the main sequence. New interior and evolution
models are currently being prepared, based on the BT-Settl models.
We provide and compare temperature versus color relations in the Optical and Infrared which matches well the BT-Settl isochrones and can be
further used for large photometric datasets. We determined the effective temperature scale for the M dwarfs in our samples. Our effective
temperature scale extended down to the latest-type M dwarfs where the dust cloud begins to form in their atmosphere.
\begin{acknowledgements}
The use of Simbad and Vizier databases at CDS, as well as the
ARICNS database was very helpful for this research.
The research leading to these results has received funding from the
French ``Agence Nationale de la Recherche'' (ANR), the ``Programme
National de Physique Stellaire'' (PNPS) of CNRS (INSU), and the
European Research Council under the European Community's Seventh
Framework Programme (FP7/2007-2013 Grant Agreement no. 247060).
It was also conducted within the Lyon Institute of Origins under
grant ANR-10-LABX-66.
The computations were performed at
the {\sl P\^ole Scientifique de Mod\'elisation Num\'erique} (PSMN) at
the {\sl \'Ecole Normale Sup\'erieure} (ENS) in Lyon, and at the {\sl
Gesellschaft f{\"u}r Wissenschaftliche Datenverarbeitung
G{\"o}ttingen} in collaboration with the Institut f{\"u}r Astrophysik
G{\"o}ttingen.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction and Statement of Results}
In this paper, we present a new and simpler proof of a fundamental result concerning cycles of random permutations which
gives some intuition for the connection between Touchard polynomials and the Poisson distribution.
We also introduce a rather novel permutation statistic and study its distribution.
This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation.
This leads to a new
and simpler derivation of the exponential generating function for the number of covers of certain multisets.
We begin by recalling some basic facts concerning Bell numbers and Touchard polynomials, and their
connection to Poisson distributions. The facts noted below without proof can be found in many books
on combinatorics; for example, in \cite{S}, \cite{W}.
The Bell number $B_n$ denotes the number of partitions of a set of $n$ distinct elements. Elementary combinatorial reasoning yields the recursive formula
\begin{equation}\label{Bell}
B_{n+1}=\sum_{k=0}^n\binom n k B_k,\ n\ge0,
\end{equation}
where $B_0=1$.
Let
\begin{equation}\label{Eseries}
E_B(x)=\sum_{n=0}^\infty \frac{B_n}{n!}x^n,
\end{equation}
denote the exponential moment generating function of $\{B_n\}_{n=0}^\infty$. Using \eqref{Bell} it is easy to show that
$E_B'(x)=e^xE_B(x)$, from which it follows that
\begin{equation}\label{E_B}
E_B(x)=e^{e^x-1}.
\end{equation}
A random variable $X$ has the Poisson distribution Pois$(\lambda)$, $\lambda>0$, if
$P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\ k=0,1,\dots$.
Let $M_\lambda(t)=Ee^{tX}$ denote the moment generating function of $X$, and let $\mu_{n;\lambda}=EX^n$ denote the $n$th moment of $X$.
Since $M_\lambda^{(n)}(0)=\mu_{n;\lambda}$, we have
\begin{equation}\label{Mseries}
M_\lambda(t)=\sum_{n=0}^\infty \frac{\mu_{n;\lambda}}{n!}t^n.
\end{equation}
However a direct calculation gives
\begin{equation}\label{MGF}
M(t)=\sum_{k=0}^\infty e^{tk}e^{-\lambda}\frac{\lambda^k}{k!}=e^{\lambda(e^t-1)}.
\end{equation}
From \eqref{Eseries}-\eqref{MGF}, it follows that the $n$th moment $\mu_{n;1}$ of a Pois$(1)$-distributed random variable satisfies
\begin{equation}\label{Bellpois}
\mu_{n;1}=B_n.
\end{equation}
Since
$$
\mu_{n;1}=EX^n=\sum_{k=0}^\infty (e^{-1}\frac1{k!})k^n,
$$
we conclude that
\begin{equation}\label{Dobinski}
B_n=\frac1e\sum_{k=0}^\infty \frac{k^n}{k!},
\end{equation}
which is known as \it Dob\'inski's formula\rm.
The Stirling number of the second kind $\stirling nk$ denotes
the number of partitions of a set of $n$ distinct elements into $k$ nonempty sets.
Elementary combinatorial reasoning yields the recursive formula
$$
\stirling{n+1}k=k\stirling nk+\stirling n{k-1}.
$$
From this, it is not hard to derive the formula
\begin{equation}\label{stirpoly}
x^n=\sum_{j=0}^n\stirling nj(x)_j,
\end{equation}
where $(x)_j=x(x-1)\cdots (x-j+1)$ is the falling factorial, and one defines $(x)_0=\stirling00=1$.
Now using \eqref{stirpoly} and the fact that $(k)_j=0$ for $j>k$, we can write the $n$th moment $\mu_{n;\lambda}$ of a Pois$(\lambda)$-distributed random variable as
\begin{equation}\label{Touchardmoment}
\begin{aligned}
&\mu_{n;\lambda}=\sum_{k=0}^\infty (e^{-\lambda}\frac{\lambda^k}{k!})k^n=\sum_{k=0}^\infty (e^{-\lambda}\frac{\lambda^k}{k!})\Big(\sum_{j=0}^n\stirling nj(k)_j\Big)=\\
&\sum_{k=0}^\infty (e^{-\lambda}\frac{\lambda^k}{k!})\Big(\sum_{j=0}^{k\wedge n}\stirling nj(k)_j\Big)=e^{-\lambda}\sum_{j=0}^n\stirling nj\sum_{k=j}^\infty\lambda^k\frac{(k)_j}{k!}=\\
&\sum_{j=0}^n\stirling nj\lambda^j.
\end{aligned}
\end{equation}
The Touchard polynomials $T_n(x)$, $n\ge0$, are defined by
$$
T_n(x)=\sum_{j=0}^n\stirling njx^j.
$$
Thus, \eqref{Touchardmoment} gives the formula
\begin{equation}\label{Touchardpois}
\mu_{n;\lambda}=T_n(\lambda).
\end{equation}
Since
$$
\mu_{n;\lambda}=\sum_{k=0}^\infty (e^{-\lambda}\frac{\lambda^k}{k!})k^n,
$$
we conclude from \eqref{Touchardpois} that
\begin{equation}\label{Dobinski-gen}
T_n(x)=e^{-x}\sum_{k=0}^\infty\frac{k^n}{k!}x^k.
\end{equation}
Since $T_n(1)=\sum_{j=0}^n\stirling nj=B_n$, the Dob\'inski formula \eqref{Dobinski} is contained in \eqref{Dobinski-gen}.
Let $S_n$ denote the set of permutations of $[n]=:\{1,\ldots, n\}$. For $\sigma\in S_n$, let $C^{(n)}_m(\sigma)$ denote the number of cycles of length $m$ in $\sigma$.
Let $P_n$ denote the uniform probability measure on $S_n$. We can now think of $\sigma\in S_n$ as random, and of $C^{(n)}_m$ as a random variable.
Using generating function techniques and/or inclusion-exclusion formulas,
one can show that under $P_n$, the distribution of the random variable $C^{(n)}_m$ converges weakly to $Z_\frac1m$, where $Z_{\frac1m}$ has the Pois$(\frac1m)$-distribution; equivalently;
\begin{equation}\label{wkconvm}
\lim_{n\to\infty}P_n(C^{(n)}_m=j)=e^{-\frac1m}\frac1{m^jj!},\ j=0,1\ldots.
\end{equation}
More generally, we consider the Ewens sampling distributions, $P_{n;\theta}$, $\theta>0$, on $S_n$
as follows. Let $N^{(n)}(\sigma)$ denote the number
of cycles in the permutation $\sigma\in S_n$,
and let $s(n,k)=|\{\sigma\in S^n:N^{(n)}(\sigma)=k\}|$ denote the number of permutations in $S_n$ with $k$ cycles.
It is known that the polynomial $\sum_{k=1}^n s(n,k)\theta^k$ is equal to the rising factorial
$\theta^{(n)}$, defined by $\theta^{(n)}=\theta(\theta+1)\cdots(\theta+n-1)$.
For $\theta>0$, define the probability measure
$P_{n;\theta}$ on $S_n$ by
$$
P_{n;\theta}(\{\sigma\})=\frac{\theta^{N^{(n)}(\sigma)}}{\theta^{(n)}}.
$$
Of course, $P_{n;1}$ reduces to the uniform measure $P_n$.
The following theorem can be proven; see for example, \cite{A}, \cite{P}.
\begin{C}\label{C}
Under $P_{n,\theta}$, the random vector $(C^{(n)}_1,C^{(n)}_2,\cdots ,C^{(n)}_m,\ldots)$ converges weakly to $(Z_{\theta},Z_\frac\theta2,\cdots, Z_{\frac\theta m},\cdots)$, where the random variables $\{Z_\frac\theta m\}_{m=1}^\infty$ are independent,
and $Z_\frac\theta m$ has the Pois$(\frac\theta m)$-distribution:
\begin{equation}\label{wkconvall}
(C^{(n)}_1,C^{(n)}_2,\cdots C^{(n)}_m,\ldots)\stackrel{w}{\Rightarrow}(Z_\theta,Z_\frac\theta 2,\cdots, Z_{\frac{\theta}m}\cdots);
\end{equation}
equivalently,
\begin{equation}\label{wkconvallexplicit}
\begin{aligned}
&\lim_{n\to\infty}P_n(C^{(n)}_1=j_1,C^{(n)}_2=j_2,\ldots, C^{(n)}_m=j_m)=\prod_{k=1}^me^{-\frac\theta k}\frac{(\frac{\theta}k)^{j_k}}{j_k!},\\
&\text{for all}\ m\ge1 \ \text{and}\ j_1,\ldots j_m\in \mathbb{Z}_+.
\end{aligned}
\end{equation}
\end{C}
We will use the method of moments to give a new and simpler proof of Theorem C, which will give
intuition for \eqref{Touchardpois}, or equivalently,
for \eqref{Dobinski-gen}; that is for the connection between the moments of Poisson random variables and Touchard
polynomials.
We now consider a permutation statistic that hasn't been studied much.
(Indeed, it was only after completing the first version of this paper that we were directed to any papers on this subject.)
For $\sigma\in S_n$ and $A\subset[n]$, define $\sigma(A)=\{\sigma_j:j\in A\}$.
If $\sigma(A)=A$, we will say that $\sigma$ fixes $A$.
Let $\mathcal{E}^{(n)}_m(\sigma)$ denote the number of sets of cardinality $m$ that are fixed by $\sigma$. (Note that $\mathcal{E}^{(n)}_1(\sigma)=C^{(n)}_1(\sigma)$, the number of fixed points
of $\sigma$.) A little thought reveals that
\begin{equation}\label{equ-en}
\mathcal{E}^{(n)}_m(\omega)=\sum_{\stackrel{(l_1,\ldots, l_m):\sum_{j=1}^mjl_j=m}{l_j\le C^{(n)}_j, j\in[m]}}\ \
\prod_{j=1}^m \binom{C^{(n)}_j(\omega)}{l_j}.
\end{equation}
For example, if $\sigma\in S_9$ is written in cycle notation as $\sigma=(379)(24)(16)(5)(8)$, then $\mathbb{E}^{(9)}_4(\omega)=5$, with the sets $A\subset[9]$ for which $|A|=4$ and
$\sigma(A)=A$ being $\{3,5,7,9\}, \{3,7,8,9\}, \{1,2,4,6\}, \{2,4,5,8\}, \{1,5,6,8\}$.
We consider the uniform measure $P_n=P_{n;1}$ on $S_n$. From Theorem C and \eqref{equ-en} it follows that
the random variable $\mathcal{E}^{(n)}_m$ under $P_n$ converges weakly as $n\to\infty$ to the random variable
\begin{equation}\label{Einfty}
\mathcal{E}_m=:\sum_{\stackrel{(l_1,\ldots, l_m):\sum_{j=1}^mjl_j=m}{l_j\le Z_\frac1j, j\in[m]}}\ \ \prod_{j=1}^m \binom{Z_\frac1j}{l_j},
\end{equation}
where $\{Z_\frac1j\}_{j=1}^m$ are independent and $Z_\frac1j$ has the Pois$(\frac1j)$-distribution.
\medskip
\bf \noindent Remark.\rm\
Note that $\mathcal{E}_1=Z_1$, $\mathcal{E}_2=\binom{Z_1}2+Z_\frac12$,
$\mathcal{E}_3=Z_\frac13+Z_\frac12Z_1+\binom{Z_1}3$.
\medskip
For $k,m\in\mathbb{N}$, consider the multiset consisting of $m$ copies of the set $[k]$. A
collection $\{\Gamma_l\}_{l=1}^r$ such that each $\Gamma_l$ is a nonempty subset of $[k]$, and such that each
$j\in[k]$ appears in exactly $m$ from among the $r$ sets $\{\Gamma_l\}_{l=1}^r$,
is called an \it $m$-cover of $[k]$ of order $r$.\rm\
Denote the total number of $m$ covers of $[k]$, regardless of order, by
$v_{k;m}$. Note that when $m=1$, we have $v_{k;1}=B_k$, the $k$th Bell number, denoting the number of partitions of a set of $k$ elements.
Also, it's very easy to see that $v_{1;m}=1$ and $v_{2,m}=m+1$.
By calculating directly the moments of $\mathcal{E}^{(n)}_m$, we will prove the following theorem.
\begin{theorem}\label{1}
For $m,k\in\mathbb{N}$,
$$
E\mathcal{E}_m^k=v_{k;m}.
$$
In particular, $E\mathcal{E}_m=1$ and $E\mathcal{E}_m^2=m+1$; thus, Var$(\mathcal{E}_m)=m$.
\end{theorem}
\bf \noindent Remark.\rm\ It is natural to suspect that $\mathcal{E}_m$ converges weakly to 0 as $m\to\infty$;
that is, $\lim_{n\to\infty}P(\mathcal{E}_m\ge1)=0$. This is in fact a hard problem.
In \cite{LP} it was shown that $P_n(\mathcal{E}_m^{(n)})\le Am^{-\frac1{100}}$, for $1\le m\le \frac n2$ and $n\ge2$.
Thus indeed, $\mathcal{E}_m$ converges weakly to 0 as $m\to\infty$. A lower bound on
$P(\mathcal{E}_m\ge1)$ of the form $A\frac{\log m}m$ was obtained in \cite{DFG}. These results were dramatically
improved in \cite{PPR} where it was shown that $P(\mathcal{E}_m\ge1)=m^{-\delta+o(1)}$ as $m\to\infty$,
where $\delta=1-\frac{1+\log\log2}{\log2}\approx0.08607$.
And very recently, in \cite{EFG}, this latter bound has been refined to $A_1m^{-\delta}(1+\log m)^{-\frac32}\le P(\mathcal{E}_m\ge1)\le
A_2m^{-\delta}(1+\log m)^{-\frac32}$.
\medskip
Let
$$
V_m(x)=\sum_{k=1}^\infty \frac{v_{k;m}}{k!}x^k
$$
denote the exponential generating function of the sequence $\{v_{k;m}\}_{k=1}^\infty$.
Of course, by Theorem \ref{1} $V_m$ is also the moment generating function of the random variable
$\mathcal{E}_m$: $V_m(x)=Ee^{x\mathcal{E}_m}$.
Using \eqref{Einfty} and Theorem \ref{1}, we will give an almost immediate proof of the following representation theorem for $V_m(x)$. We use the notation $[z^m]P(z)=a_m$, where $P(z)=\sum_{m=0}^\infty a_mz^m$.
\begin{theorem}\label{2}
\begin{equation}
V_m(x)=Ee^{x\mathcal{E}_m}= e^{-\sum_{j=1}^m\frac1j}\sum_{u_1,\ldots, u_m\ge0}\big(\prod_{j=1}^m\frac{j^{-u_j}}{u_j!}\big)~e^{x\gamma_m(u)},
\end{equation}
where
$$
\gamma_m(u)=[z^m]\prod_{j=1}^m(1+z^j)^{u_j}.
$$
\end{theorem}
\noindent\bf Remark.\rm\ When $m=2,3$, the above formula reduces to
$$
\begin{aligned}
&V_2(x)=e^{-\frac32}e^{\frac12e^x}\sum_{r=0}^\infty \frac{e^{\binom r2x}}{r!};\\
&V_3(x)=e^{-\frac{11}6}e^{\frac{e^x}3}\sum_{r=0}^\infty\frac{e^{\binom r3x+\frac12e^{rx}}}{r!}.
\end{aligned}
$$
The formula for $m=2$ was proved by Comtets \cite{C} and the formula for $m=3$ was proved by Bender \cite{B}.
The case of general $m$ was proved by Devitt and Jackson \cite{DJ}.
They also prove that there exists a number $c$ such that the extraction of the coefficient $v_{k;m}$ from the exponential generating function $V_m(x)$ can be done
in no more than $ck^m\log k$ arithmetic operations.
In section \ref{ProofC} we will give our new proof of Theorem C via the method of moments. In section \ref{Proofmulti}
we prove Theorems \ref{1} and \ref{2}.
\section{ A proof of Theorem C via the method of moments}\label{ProofC}
If a sequence of nonnegative random variables $\{X_n\}_{n=1}^\infty$ satisfies
$\sup_{n\ge1}EX_n<\infty$, then the sequence is tight, that is, pre-compact with respect to weak convergence.
Let $X$ be distributed as one of the accumulation points.
If for some $k\in \mathbb{N}$, $\lim_{n\to\infty}EX_n^k$ exists and equals $\mu_k$, and $\sup_{n\ge1}EX_n^{k+1}<\infty$,
then the $\{X_n^k\}_{n=1}^\infty$ are uniformly integrable, and thus $EX^k=\mu_k$. Thus,
if
\begin{equation}\label{moments}
\mu_k=:\lim_{n\to\infty}EX_n^k\ \text{exists for all}\ k\in\mathbb{N},
\end{equation}
then $EX^k=\mu_k$, for all
$k$. The Stieltjes moment theorem states that if
\begin{equation}\label{Stie}
\sup_{k\ge1}\frac{\mu_k^\frac1k}k<\infty,
\end{equation}
then
the sequence $\{\mu_k\}_{k=1}^\infty$ uniquely characterizes the distribution \cite{D}.
We conclude then that if a sequence of nonnegative random variables $\{X_n\}_{n=1}^\infty$ satisfies
\eqref{moments} and \eqref{Stie}, then the sequence is weakly convergent to a random variable $X$ satisfying
$EX^k=\mu_k$.
An extremely crude argument shows that the Bell numbers satisfy $B_k\le k^k$; thus
\begin{equation}\label{Bell-Stie}
\sup_{k\ge1}\frac{B_k^\frac1k}k<\infty.
\end{equation}
By \eqref{Touchardpois}, the $k$th moment $\mu_{k;\frac\theta m}$ of the Pois$(\frac\theta m)$-distributed random variable
$Z_\frac\theta m$ is equal to $T_k(\frac\theta m)$. Now $T_k(\frac\theta m)$ is bounded from above by
$T_k(\theta)$, for all $m\ge1$, and $T_k(\theta)\le B_k\max(1,\theta^k)$.
Thus, in light of \eqref{Bell-Stie} and the previous paragraph, if we prove
that
\begin{equation}\label{momentmethodCm}
\lim_{n\to\infty}E_{n;\theta}(C^{(n)}_m)^k=T_k(\frac\theta m),\ k,m\in\mathbb{N},
\end{equation}
where $E_{n;\theta}$ denotes the expectation with respect to $P_{n;\theta}$, then we will have proved that $C^{(n)}_m$ under $P_{n;\theta}$ converges weakly to $Z_\frac\theta m$, for all $m\in\mathbb{N}$.
And if we then prove that
\begin{equation}\label{momentmethodCms}
\lim_{n\to\infty}E_{n;\theta}\prod_{j=1}^m(C^{(n)}_j)^{k_j}=\prod_{j=1}^mT_{k_j}(\frac\theta j),\ m\ge2, k_j\in\mathbb{N}, j=1,\ldots, m,
\end{equation}
then we will have completed the proof of Theorem C.
We first prove \eqref{momentmethodCm}.
In fact, we will first prove \eqref{momentmethodCm} in the case of the uniform measure, $P_n=P_{n;1}$.
Once we have this, the case of general $\theta$ will follow after a short explanation.
Assume that $n\ge mk$.
For $D\subset[n]$ with $|D|=m$,
let $1_D(\sigma)$ be equal to 1 or 0 according to whether or not $\sigma\in S_n$ possesses an $m$-cycle consisting of
the elements of $D$. Then we have
\begin{equation}\label{Cnrep}
C^{(n)}_m(\sigma)=\sum_{\stackrel{D\subset[n]}{|D|=m}}1_D(\sigma),
\end{equation}
and
\begin{equation}\label{Cnmkthmoment}
E_n(C^{(n)}_m)^k=\sum_{\stackrel{D_j\subset[n],|D_j|=m}{j\in[k]}}E_n\prod_{j=1}^k1_{D_j}.
\end{equation}
Now $E_n\prod_{j=1}^k1_{D_j}\neq0$ if and only if for some $l\in[k]$, there exist
disjoint sets $\{A_i\}_{i=1}^l$ such that $\{D_j\}_{j=1}^k=\{A_i\}_{i=1}^l$.
If this is the case, then
\begin{equation}\label{ldistinct}
E_n\prod_{j=1}^k1_{D_j}=\frac{(n-lm)!((m-1)!)^l}{n!}.
\end{equation}
(Here we have used the assumption that $n\ge mk$, since otherwise $n-ml$ will be negative for certain $l\in[k]$.)
The number of ways to construct $l$ disjoint, ordered sets $\{A_i\}_{i=1}^l$, each of which consists of $m$ elements from $[n]$,
is
$\frac{n!}{(m!)^l(n-lm)!}$.
Given the $\{A_i\}_{i=1}^l$, the number of ways to choose the sets $\{D_j\}_{j=1}^k$ so that $\{D_j\}_{j=1}^k=\{A_i\}_{i=1}^l$ is
equal to the Stirling number $\stirling kl$, the number of ways to partition a set of size $k$ into
$l$ nonempty parts.
From these facts along with \eqref{Cnmkthmoment} and \eqref{ldistinct},
we conclude that for $n\ge mk$,
\begin{equation}\label{finaltouch}
\begin{aligned}
&E_n(C^{(n)}_m)^k=\sum_{l=1}^k\big(\frac{(n-lm)!((m-1)!)^l}{n!}\big)\big(\frac{n!}{(m!)^l(n-lm)!}\big)\stirling kl=\\
&\sum_{l=1}^k\frac1{m^l}\stirling kl=T_k(\frac1m),
\end{aligned}
\end{equation}
proving \eqref{momentmethodCm} in the case $\theta=1$.
For the case of general $\theta$, we note that the only change that must be made in the above proof is
in \eqref{ldistinct}. Recalling that $s(n,k)$ denotes the number of permutations in $S_n$ with $k$ cycles,
we have
\begin{equation}\label{ldistincttheta}
\begin{aligned}
&E_{n;\theta}\prod_{j=1}^k1_{D_j}=\frac{((m-1)!)^l\sum_{k=1}^{n-ml}s(n-lm,k)\theta^{k+l}}{\theta^{(n)}}=
\frac{\theta^{(n-ml)}((m-1)!)^l}{\theta^{(n)}}\theta^l\sim\\
&\frac{(n-ml)!((m-1)!)^l}{n!}\theta^l=\theta^lE_n\prod_{j=1}^k1_{D_j},\ \text{as}\ n\to\infty.
\end{aligned}
\end{equation}
Thus, instead of \eqref{finaltouch}, we have
$$
E_n(C^{(n)}_m)^k\sim\sum_{l=1}^k(\frac\theta m)^l\stirling kl=T_k(\frac\theta m), \ \text{as}\ n\to\infty.
$$
We now turn to \eqref{momentmethodCms}. The method of proof is simply the natural
extension of the one used to prove \eqref{momentmethodCm}; thus, since
the notation is cumbersome we will suffice with illustrating the method by proving that
\begin{equation}\label{forfinalproof}
\lim_{n\to\infty}E_n(C^{(n)}_{m_1})^{k_1}(C^{(n)}_{m_2})^{k_2}=T_{k_1}(\frac1{m_1})T_{k_2}(\frac1{m_2}).
\end{equation}
Let $n\ge m_1k_1+m_2k_2$.
By \eqref{Cnrep}, we have
\begin{equation}\label{productexp}
E_nC^{(n)}_{m_1})^{k_1}(C^{(n)}_{m_2})^{k_2}=\sum_{\stackrel{D^{1}_j\subset[n],|D^{1}_j|=m_1}{j\in[k_1]}}
\sum_{\stackrel{D^{2}_j\subset[n],|D^{2}_j|=m_2}{j\in[k_2]}}E_n\prod_{j=1}^{k_1}1_{D^{1}_j}
\prod_{j=1}^{k_2}1_{D^{2}_j}.
\end{equation}
Now $E_n\prod_{j=1}^{k_1}1_{D^{1}_j}
\prod_{j=1}^{k_2}1_{D^{2}_j}\neq0$
if and only if for some $l_1\in[k_1]$ and some $l_2\in[k_2]$, there exist
disjoint sets $\{A^{1}_i\}_{i=1}^{l_1}$, $\{A^{2}_i\}_{i=1}^{l_2}$ such that $\{D^{r}_j\}_{j=1}^{k_r}=
\{A^{r}_i\}_{i=1}^{l_r},\ r=1,2$.
If this is the case, then
\begin{equation}\label{l12distinct}
E_n\prod_{j=1}^{k_1}1_{D^{1}_j}
\prod_{j=1}^{k_2}1_{D^{2}_j}=\frac{(n-l_1m_1-l_2m_2)!((m_1-1)!)^{l_1}(m_2-1)!)^{l_2}}{n!}.
\end{equation}
The number of ways to construct disjoint, ordered sets $\{A^1_i\}_{i=1}^l, \{A^2_i\}_{i=1}^l$, with
the $A^1_i$ each consisting of $m_1$ elements from $[n]$ and the $A^2_i$ each consisting of $m_2$ elements from $[n]$, is
$\frac{n!}{(m_1!)^{l_1}(m_2!)^{l_2}(n-l_1m_1-l_2m_2)!}$.
Given $\{A^1_i\}_{i=1}^l, \{A^2_i\}_{i=1}^l$,
the number of ways to choose the ordered sets $\{D^1_j\}_{j=1}^{k_1}, \{D^2_j\}_{j=1}^{k_2}$ so that $\{D^1_j\}_{j=1}^{k_1}=
\{A^1_i\}_{i=1}^{l_1}$ and $\{D^2_j\}_{j=1}^{k_2}=
\{A^2_i\}_{i=1}^{l_2}$ is
equal to $\stirling {k_1}{l_1}\stirling{k_2}{l_2}$.
We have
$$
\frac{(n-l_1m_1-l_2m_2)!((m_1-1)!)^{l_1}(m_2-1)!)^{l_2}}{n!}\frac{n!}{(m_1!)^{l_1}(m_2!)^{l_2}(n-l_1m_1-l_2m_2)!}=
\frac1{m_1^{l_1}m_2^{l_2}}.
$$
From these fact along with \eqref{productexp} and \eqref{l12distinct}, we conclude that for $n\ge m_1k_1+m_2k_2$,
$$
E_nC^{(n)}_{m_1})^{k_1}(C^{(n)}_{m_2})^{k_2}=
\sum_{l_2=1}^{k_2}\sum_{l_1=1}^{k_1} \frac1{m_1^{l_1}m_2^{l_2}}\stirling {k_1}{l_1}\stirling{k_2}{l_2}=T_{k_1}(\frac1{m_1})
T_{k_2}(\frac1{m_2}),
$$
proving \eqref{forfinalproof}.\hfill $\square$
\section{Proofs of Theorems \ref{1} and \ref{2}}\label{Proofmulti}
\noindent \it Proof of Theorem \ref{1}.\rm\
Since $\mathcal{E}^{(n)}_m$ converges weakly to $\mathcal{E}_m$, it follows from the discussion in the first paragraph of section \ref{ProofC} that it suffices to show that
\begin{equation}
\lim_{n\to\infty}E_n(\mathcal{E}^{(n)}_m)^k=v_{k;m}.
\end{equation}
Let $n\ge km$.
For $D\subset[n]$, let $1_D(\sigma)$ equal 1 or 0 according to whether or not $\sigma\in S_n$ induces an embedded
permutation on $D$.
Then we have
\begin{equation}\label{repembedded}
\mathcal{E}^{(n)}_m(\omega)=\sum_{\stackrel{D\subset[n]}{|D|=m}}1_D(\omega),
\end{equation}
and
\begin{equation}\label{Ekthmom}
E_n(\mathcal{E}^{(n)}_m)^k=\sum_{\stackrel{D_j\subset[n],|D_j|=m}{j\in[k]}}E_n\prod_{j=1}^k1_{D_j}.
\end{equation}
There is a one-to-one correspondence between collections $\{D_j\}_{j=1}^k$, satisfying $D_j\subset[n]$ and $|D_j|=m$,
and collections $\{A_I\}_{I\subset[k]}$ of disjoint sets satisfying $A_I\subset[n]$
and satisfying
\begin{equation}\label{theequation}
\sum_{I:i\in I}l_I=m,\ \text{for all}\ i\in[k],
\end{equation}
where
\begin{equation}\label{thels}
l_I=|A_I|, \ I\subset[k].
\end{equation}
The correspondence is through the formula
\begin{equation}\label{theAs}
A_I=\big(\cap_{i\in I}D_i\big)\cap\big(\cap_{i\in[k]-I}([n]-D_i)\big),\ I\subset[k].
\end{equation}
Now $\prod_{j=1}^k1_{D_j}(\omega)=1$ if and only if
for all $I\subset[k]$,
$\sigma$ induces an embedded permutation on $A_I$.
Thus,
we have
\begin{equation}\label{expprodD}
E_n\prod_{j=1}^k1_{D_j}=\frac{(n-\sum_{I\subset[k]}l_I)!\prod_{I\subset[k]}l_I!}{n!}.
\end{equation}
Given the values $l_I=|A_I|$, $I\subset[k]$, the number of ways to construct the disjoint sets $\{A_I\}_{I\subset [k]}$
is $\frac{n!}{(n-\sum_{I\subset[k]}l_I)!\prod_{I\subset[k]}l_I!}$. Using this with
\eqref{Ekthmom}-\eqref{expprodD}, it follows that
$E_n(\mathcal{E}^{(n)}_m)^k$ equals the number of solutions
$\{l_I\}_{I\subset[k]}$ to \eqref{theequation}.
To complete the proof, we will
show that the number of solutions to \eqref{theequation} is $v_{k;m}$.
Consider the set $\cup_{j=1}^kD_j\subset[n]$. Label the elements of this set by $\{x_i\}_{i=1}^r$.
Of course, $m\le r\le km$. Now construct the sets $\{\Gamma_i\}_{i=1}^r$ by
$\Gamma_i=\{j:x_i\in D_j\}$. By construction,
the sets $\{\Gamma_i\}_{i=1}^r$ form an $m$-cover of $[k]$ (of order $r$).
There is a one-to-one correspondence between solutions to \eqref{theequation}
and $m$ covers of $[k]$; indeed, $l_I=|\{i\in [k]:\Gamma_i=I\}|$.\hfill $\square$
\bigskip
\noindent \it Proof of Theorem \ref{2}.\rm\
We use \eqref{Einfty} to calculate
$V_m(x)=Ee^{x\mathcal{E}_m}$. We have
\begin{equation}
\begin{aligned}
&V_m(x)=E\exp(x\mathcal{E}_m)=E\exp\big(x\sum_{\stackrel{(l_1,\ldots, l_m):\sum_{j=1}^mjl_j=m}{l_j\le Z_{\frac1j}, j\in[m]}}
\prod_{j=1}^m\binom{Z_\frac1j}{l_j}\big)=\\
&\sum_{u_1\ge0,\cdots, u_m\ge0}\prod_{j=1}^m(\frac1j)^{u_j}\frac1{u_j!}e^{-\frac1j}
\exp\big(x\sum_{\stackrel{(l_1,\ldots, l_m):\sum_{j=1}^mjl_j=m}{l_j\le u_j, j\in[m]}}\prod_{j=1}^m\binom{u_j}{l_j}\big).
\end{aligned}
\end{equation}
Thus, to complete the proof, we only need to show that
\begin{equation}\label{lastformula}
\sum_{\stackrel{(l_1,\ldots, l_m):\sum_{j=1}^mjl_j=m}{l_j\le u_j, j\in[m]}}\prod_{j=1}^m\binom{u_j}{l_j}=\gamma_m(u),
\end{equation}
where
\begin{equation}\label{gammadef}
\gamma_m(u)=[z^m]\prod_{j=1}^m(1+z^j)^{u_j}.
\end{equation}
Expanding with the binomial formula, we have
\begin{equation}\label{binomexp}
\prod_{j=1}^m(1+z^j)^{u_j}=\prod_{j=1}^m\big(\sum_{l_j=0}^{u_j}\binom{u_j}{l_j}z^{jl_j}\big).
\end{equation}
From \eqref{binomexp} and \eqref{gammadef}, it follows that \eqref{lastformula} holds.
\hfill $\square$
\medskip
\noindent\bf Acknowlegement.\rm\ The author thanks Ron Holzman and Roy Meshulam for a discussion and references with regard to multisets, and Ron Holzman for the reference \cite{EFG}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,282
|
Space Oddity (2012) continues Jai McKenzie's ongoing research into un-built, propositional architecture of the mid 20th Century. Space Oddity features lighting elements and a large hand-made net, the pattern of this net is based on repetitive patterns the artist found in architectural drawings by Superstudio. By working with forms that were never actualised McKenzie activates Superstudio's models for experience. Similar to minimalist practices of the 20th century, McKenzie employs systems, seriality and sequence to create an illusory plane or structure. However, unlike minimalists who used seriality to remove subjective decisions, here the surface of the net is imperfect and made by hand; instead repetition serves as a means for meditation for both the artist while making the work and the viewer while observing the final piece. It is intended that the viewer get lost in this pattern and open themselves to perceptual observation which is aided by the addition of light. Here light is used to dematerialise form and create an opening for various ways of seeing.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,480
|
Q: Bootstrap carousel not displaying newly created item I have a div that uses a bootstrap carousel with ng-repeat. I also have a way to add items the current list of items.
So if I am in index 0 and I click the add button, I want to be able to slide to the new item.
Currently with what I have, the indexes seem to be set correctly but it just isn't sliding to the new item
$scope.items.splice($scope.currentIndex+1, 0, newItem);
$scope.currentIndex = $scope.currentIndex+1;
$('#myCarousel').carousel($scope.currentIndex);
It just shows the data from item 1 still. But when I do try and click next, it then moves to that new item.
I have also tried it with $('#myCarousel').carousel('next'); which results in the same thing
Edit:
<div id="myCarousel" class="carousel slide" data-interval="false">
<div class="carousel-inner">
<div class="item" ng-class="{active:!$index}" ng-repeat="item in items">
// rest of the html
</div>
</div>
</div>
A: Can you post the html part? also begin with more than 1 slide so that you have at least 3 slides at the end just for debug.
Following your code i compiled something like this, don't know if it's what you are looking for
<body ng-app="cApp" ng-controller="myslides">
<div id="myCarousel" class="carousel slide" data-interval="false">
<div class="carousel-inner">
<div ng-repeat="item in items" class="item" ng-class="{active:$index==currentIndex}" >
<div>{{item}}</div>
</div>
<h1>Hello Plunker!</h1>
</div>
<a class="left carousel-control" href="#myCarousel" ng-click="currentIndex=currentIndex!=0?currentIndex-1:items.length-1" data-slide="prev">
<span class="glyphicon glyphicon-chevron-left"></span>
<span class="sr-only">Previous</span>
</a>
<a class="right carousel-control" href="#myCarousel" ng-click="currentIndex=currentIndex<items.length-1?currentIndex+1:0" data-slide="next">
<span class="glyphicon glyphicon-chevron-right"></span>
<span class="sr-only">Next</span>
</a>
</div>
<button class="btn btn-default" ng-click="addslide('another1')">add slide</button>
</body>
with the script
var app = angular.module('cApp', []);
app.controller('myslides', function($scope){
$scope.items = ['this is 1','this is 2'];
$scope.currentIndex = 0;
$scope.addslide=function(newItem){
$scope.items.splice($scope.currentIndex+1, 0, newItem);
$scope.currentIndex = $scope.currentIndex+1;
};
});
Aldo if you are going to use angular, i personally prefer to use angularui that has a simple carousel with prebuild functions and you can still do some customizing.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,388
|
\section{How to Use this Template}
\section{Introduction}
Quantum technologies promise to revolutionize the way we communicate and process information by giving us the ability to experimentally manipulate quantum states of light and matter at the single-particle level \cite{ion,circuitqed,nanoresonator}. To this end, it is necessary to isolate these systems from the interaction with their surroundings in such a way that it might be possible, for example, to cool atoms close to absolute zero or to maintain the fragile quantum correlations between these systems. Likewise, this degree of control of quantum systems also enables their use for more efficient information processing or as quantum simulators of complex dynamics. In this context, it is necessary to understand different aspects such as the system dynamics of many interacting quantum systems; the possible decoherence processes that these devices may undergo, and the thermodynamics of systems on these scales.
A natural question has emerged about whether it is possible to use new technologies to produce quantum machines. The novelty comes from the fact that these small systems can exhibit quantum properties that could potentially be exploited to get an advantage over classical machines or present new obstacles to their operation. These questions constitute the backbone of a new area of physics that has come to be called quantum thermodynamics, a fruitful crucible of research fields where the foundations of physics, information science and statistical mechanics merge.
In most cases, a finite-time operation causes the emergence of coherence in the state of the system that results in an efficiency loss \cite{Otto1, Otto2, Otto_nos}. However, in many cases, it is possible to implement protocols named shortcuts to adiabaticity (STAs), that evolve the initial state into the final state that would have been obtained with an adiabatic evolution, but in a finite time \cite{berry,STA1,STA2,STA3}.
STAs are powerful quantum control methods, allowing quick evolution into target states of otherwise slow adiabatic dynamics. Such methods have widespread applications in quantum technologies, and various shortcuts to adiabaticity protocols have been demonstrated in closed systems. These protocols typically require a full control of the quantum system and end up being extremely challenging from an experimental standpoint.
Another area where an STA might be extremely useful is relativistic quantum information (RQI). Fundamental questions have arisen on how the motion of different observers affect shared quantum information and how to distribute and process it \cite{RQI1,RQI2,RQI3,RQI4,RQI5,RQI6,RQI7}. Recent works have shown that the entanglement shared between two moving cavities is diminished as observers accelerate \cite{RQI2,RQI3}. This is due to the fact that their motion causes a nonadiabatic evolution of the quantum system that generates excitations that affect the entanglement \cite{RQI_nos}. Hence, if one can find an STA that achieves a fast adiabatic evolution of the field inside a moving cavity, it would be possible to exactly preserve the entanglement solving a fundamental problem in RQI.
In previous works, STAs have been considered from a theoretical and/or an experimental point of view for different physical systems: trapped ions \cite{Palmero}, cold atoms \cite{Torrontegui}, ultracold Fermi gases \cite{Dowdall}, Bose--Einstein condensates in atom chips \cite{Amri}, etc. In Ref. \cite{STAqft1}, we showed how to implement shortcuts to adiabaticity for the case of a massless scalar field inside a cavity with a moving wall, in (1 + 1) dimensions. The approach was based on the known solution to the problem that exploited the conformal symmetry, and the shortcuts took place whenever the solution matched the adiabatic Wentzel--Kramers--Brillouin (WKB) solution \cite{calzetta}, i.e., when there was no dynamical Casimir effect (DCE) \cite{Moore,reviewsDCE1,reviewsDCE2,reviewsDCE3,reviewsDCE4}. We obtained a fundamental limit for the efficiency of an Otto cycle with the quantum field as a working system, which depended on the maximum velocity that the mirror could attain. We also described possible experimental realizations of the shortcuts using \mbox{superconducting circuits.}
In this paper, we generalize the results of \cite{STAqft1} to the case of a quantum scalar field in a one-dimensional optomechanical cavity with two moving mirrors. We show that, given the trajectories for the left ($L_{\rm ref}(t)$) and right ($R_{\text{ref}}(t)$) mirrors, we can find a shortcut to adiabaticity ruled by the effective trajectories ($L_{\text{eff}}(t)$) and ($R_{\text{eff}}(t)$) that, when implemented in finite time, result in the same state as if the original ones had been evolved adiabatically. This protocol has the advantage that it can be easily implemented experimentally using either an optomechanical cavity or superconducting circuits, since it does not require additional exotic potentials. Moreover, the effective trajectory can be computed from the original one quite simply, paving the way for more efficient quantum field thermal machines. Besides its intrinsic interest, this generalization may have useful applications in the area of RQI.
In the next Section we discuss that for a quantum field, STAs are not as simple as for a nonrelativistic quantum system with a finite number of degrees of freedom. Section~\ref{sec3} is dedicated to the study of an optomechanical cavity with two moving mirrors and, in Section~\ref{sec4}, we show how to find STAs in these cavities. Section~\ref{sec5} is dedicated to the numerical analysis of the STA for different reference trajectories such as a contraction, expansion or a rigid motion of the cavity. In Section~\ref{sec6}, we complete the work with a discussion of our results
\section{STA in Quantum Field Theory}\label{sec2}
When a quantum field is subjected to time-dependent external conditions, the phenomenon of particle creation seems unavoidable. However, as already mentioned, in some particular situations this phenomenon can be avoided. We shall discuss some examples in the following.
\subsection{Electromagnetic Cavity: Single-Mode Approximation}
Let us consider an electromagnetic cavity with time-dependent properties (variable length and/or time-dependent electromagnetic properties). It is usual to describe the physics inside the cavity using a single-mode approximation for the quantum electromagnetic field. The dynamics of the mode is that of a harmonic oscillator with a \mbox{time-dependent frequency}
\begin{equation}
\ddot Q_{\mathbf k} +\omega_{\mathbf k}^2(t) Q_{\mathbf k} =0\, ,
\end{equation}
where ${\mathbf k}$
is the index that identifies the mode. The frequency $\omega_{\mathbf k}(t)$ depends on time if, for instance, the length of the cavity $d(t)$ is time-dependent.
Assuming that the frequency is constant for $t\to\pm\infty$, and that the mode is in the ground state $\vert 0_{IN}\rangle$
for $t\to -\infty$, in the case of a nonadiabatic evolution the electromagnetic mode will be excited for $t\to +\infty$, that is $\vert\langle 0_{OUT}\vert 0_{IN}\rangle\vert \neq 1 $. The Bogoliubov transformation that connects the $IN$ and $OUT$ Fock spaces, when nontrivial, is an indication of particle creation and describes the presence of photons inside the cavity.
The adiabatic WKB solution for the operator associated with the mode, $\hat Q_{\mathbf k}(t)$, can be written in terms of annihilation and creation operators as
\begin{equation}
\hat Q_{\mathbf k}(t)= \hat a \frac{e^{- i \int^t\omega_{\mathbf k\, \rm ref}(t')dt'}}{\sqrt{2\omega_{\mathbf k\,\rm ref}(t)}}+\hat a^\dagger \frac{e^{ i \int^t\omega_{\mathbf k\,\rm ref}(t')dt'}}{\sqrt{2\omega_{\mathbf k\,\rm ref}(t)}}.
\end{equation}
This
is an approximate solution for the oscillator
with a reference frequency $\omega_{\mathbf \, \rm ref}(t)$, valid if it is slowly varying, but an
exact solution of a system with an effective frequency \cite{calzetta}
\begin{equation}
\omega_{\mathbf k\,\rm eff}^2(t)=\omega_{\mathbf k\, \rm ref}^2+\frac{1}{2}\left(\frac{\ddot\omega_{\mathbf k\,\rm ref}}{\omega_{\mathbf k\, \rm ref}}-\frac{3}{2}\left(\frac{\dot\omega_{\mathbf k\, \rm ref}}{\omega_{\mathbf k\, \rm ref}}\right)^2\right)\, .
\end{equation}
From the effective frequency, one can read the effective time-dependent length of the cavity $d_{\mathbf k\, \rm eff} (t)$ which leads to no particle creation, and therefore constitutes an STA. It is important to remark that the evolution at intermediate times is in general nonadiabatic, but the system returns to the initial state when the effective length becomes constant
at $t\to +\infty$. Particles are created and subsequently absorbed.
The STA described above cannot be generalized beyond the single-mode approximation since the effective frequency and length are different for each mode, and therefore it is not possible to avoid particle creation in all modes. Moreover, for this system, an electromagnetic field inside a time-dependent cavity, the modes are coupled.
In the rest of the paper, we consider a physical system in which it is possible to find a nontrivial STA for a full quantum field. By nontrivial we mean that, although there is no particle creation at the end of the evolution, the dynamics is nonadiabatic at intermediate times, that is, there is creation and absorption of particles. Before doing this, we mention some examples of quantum fields in time-dependent backgrounds in which there is no particle creation at all, that is, the modes of the fields are oscillators with \mbox{time-independent frequency. }
\subsection {Quantum Fields in Curved Space-Times}
Assuming a Robertson--Walker metric
\begin{equation}\label{modes}
ds^2= a^2(\eta) (-d\eta^2+ d\mathbf x^2)\, ,
\end{equation}
the modes of a free quantum scalar field satisfy \cite{Birrel1,Birrel2}
\begin{equation}
\ddot\chi_k + (k^2+m^2 a^2+(\xi-1/6) R a^2)\chi_k =0\, ,
\end{equation}
where $m$ is the mass of the field, $R$ the scalar curvature and $\xi$ the coupling to the curvature. We are describing the dynamical equations in terms of the conformal time $\eta$ {and a($\eta$) is the scale factor}.
The equations for the modes correspond to those of harmonic oscillators with time-dependent frequency. As mentioned above, for each mode, one can find particular evolutions of the scale factor such that there is no particle creation. However, as the time-dependent frequency depends on the momentum $k$, it is not possible to find an STA for the full quantum field, but only for a given mode.
There are some particular situations in which the frequency of all modes is time-independent, for an arbitrary time dependence of the scale factor.
This is the case when there is conformal invariance $m=0$ and $\xi=1/6$. Another possibility is to consider a massless field in a radiation-dominated universe, for which $R=0$ (for another example in the context of non-Abelian field theories see \cite{Vachaspati2022}).
The relevance of conformal invariance can be reinforced by another example.
Let us consider now a massless quantum scalar field in an almost flat metric
\begin{equation}
ds^2= (\eta_{\mu\nu}+h_{\mu\nu}) dx^\mu dx^\nu\, ,
\end{equation}
with $\vert h_{\mu\nu}\vert\ll 1$.
The probability of pair creation reads \cite{Frieman}
\begin{equation}
P= \frac{1}{960\pi}\int d^4x[60 (\xi-1/6)R^2 + C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}]\, ,
\end{equation}
where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor. Once again, for a conformal field $(\xi=1/6)$ in a conformally flat metric $(C_{\mu\nu\rho\sigma}=0)$, the pair creation probability vanishes.
There are some subtle points in these examples. On the one hand, particle creation vanishes when one chooses the conformal vacuum as the vacuum state of the system. For Robertson--Walker metrics, this corresponds to the choice of the mode functions
\begin{equation}
\chi_k=\frac{1}{\sqrt{2k}} e^{- i k\eta}\, ,
\end{equation}
that solve Equation \eqref{modes} when $m=0$ and $\xi=1/6$. This choice is natural if the metric is asymptotically flat for $\eta\to -\infty$.
The mean value of the energy--momentum tensor vanishes in that region.
Even with this choice, it is known that conformal invariance is broken at the quantum level, producing a nonvanishing trace for the mean value of the energy--momentum tensor (that is traceless for a conformal field at the classical level). While each mode of the quantum field evolves in a trivial way, the mean value of the energy--momentum tensor may depend on time during the evolution. This dependence is, however, local in the metric and its derivatives, and therefore, the energy--momentum tensor returns to its vanishing value if the scale factor tends to a constant for $\eta\to +\infty$.
From the previous discussion, we see that for a quantum field, STAs are not as simple as for a quantum system with a finite number of degrees of freedom. The renormalization, which is unavoidable even for free fields in external backgrounds, is an additional ingredient that should be taken into account. On the other hand, we also see that while conformal invariance simplifies the dynamical equations for the modes, its quantum anomaly may introduce nontrivial effects. We see all these aspects at work in the optomechanical cavity with moving mirrors.
\section{The Optomechanical Cavity}\label{sec3}
The system we are now considering is a scalar field, $\Phi(x,t)$, inside a cavity formed by two moving mirrors to the left and right whose position are given by $L(t)$ and $R(t)$, respectively {(see Figure \protect\ref{fig:schematic})}. The evolution of the field is determined by the wave equation inside the cavity
\begin{equation}
(\partial_x^2-\partial_t^2)\Phi(x,t)=0,
\end{equation}
and Dirichlet boundary conditions on each mirror
\begin{equation}
\Phi(L(t),t)=\Phi(R(t),t)=0.
\end{equation}
\begin{figure*}
\includegraphics[width=8 cm]{schematic.pdf}
\caption{Schematics
of the one dimensional cavity with a scalar quantum field $\Phi(x,t)$ inside and two moving mirrors with trajectories $L(t)$ and $R(t)$. The red and green curves illustrate two modes of the field in the cavity.\label{fig:schematic}}
\end{figure*}
It is important to remark that we are considering units where $c=\hbar=k_B=1$, which we use throughout the rest of the paper.
It is known that the time evolution of the field is solved by expanding the field in modes
\begin{equation}
\Phi(x,t)=\sum_{k=1}^{\infty}\left[a_k\psi_k(x,t)+a_k^\dagger\psi_k^*(x,t)\right],
\end{equation}
such that the modes are given by \cite{Dalvit2mirrors}
\begin{equation}
\psi_k(x,t)=\frac{i}{\sqrt{4\pi k}}[e^{-ik\pi G(t+x)}+e^{ik\pi F(t-x)}],
\end{equation}
where $F(z)$ and $G(z)$ are functions determined by Moore's equations
\begin{align}
\label{eq:MooreEqL}
G(t+L(t))-F(t-L(t))&=0\\
G(t+R(t))-F(t-R(t))&=2 \label{eq:MooreEqR}
.
\end{align
The functions $F(z)$ and $G(z)$ implement the conformal transformation
\begin{equation}\label{conftransf}
\bar t +\bar x =G(t+x)\quad \bar t -\bar x = F(t-x)
\end{equation}
such that in the new coordinates, the left and right mirrors are static at $\bar x_L=0$ and
$\bar x_R=1$.
Finding the evolution of the field given the motion of the mirrors is therefore reduced to solving Moore's equations. Once this is achieved, the renormalized energy density of the field can be found \cite{Dalvit2mirrors}
\begin{align}
\label{eq:dens_en}
\langle T_{tt}(x,t)\rangle_{\text{ren}} = f_G(t+x)+f_F(t-x),
\end{align}
where
\begin{align}
f_G &= -\frac{1}{24\pi}\left[\frac{G'''}{G'}-\dfrac{3}{2}\left(\frac{G''}{G'}\right)^2\right]+\dfrac{(G')^2}{2}\left[-\dfrac{\pi}{24}+Z(Td_0)\right] \nonumber\\
f_F &= -\frac{1}{24\pi}\left[\frac{F'''}{F'}-\dfrac{3}{2}\left(\frac{F''}{F'}\right)^2\right]+\dfrac{(F')^2}{2}\left[-\dfrac{\pi}{24}+Z(Td_0)\right] ,
\end{align}
and $d_0=|R_0-L_0|$ is the initial length of the cavity. We are considering the state of the field to be initially in a thermal state at temperature $T$, and $Z(Td_0)$ is related to the initial mean energy
\begin{equation}
Z(Td_0)=\sum_{n=1 }^\infty\frac{n\pi}{\exp\left(\frac{n\pi}{Td_0}\right)-1}.
\end{equation}
The expression for the renormalized energy--momentum tensor above can be obtained using the standard approach based on point-splitting regularization
(see for instance \cite{FullingDavies}). It can also be derived using the conformal anomaly associated with the conformal transformation Equation~\eqref{conftransf}~\cite{Birrel1,Birrel2}.
Finally, it is important to note that for a static cavity with $L(t)=0$, $R(t)=d_0$, we have
$F(z)=G(z)= z/d_0$, and the renormalized energy density reduces to the static Casimir energy density. The phenomenon of particle creation appears when $F(z)$ and $G(z)$ are nonlinear functions.
\section{STA for the Field}\label{sec4}
In this case, it is particularly challenging to find an STA since the only parameters that we can control and that affect the time evolution of the field are the positions of the left and right walls, $L(t)$ and $R(t)$, respectively.
However, we achieve this by finding the adiabatic Moore functions which correspond to the infinitely slow evolution of the field for reference trajectories $L_{\text{ref}}(t)$ and $R_{\text{ref}}(t)$. Then, we look for effective trajectories $L_{\text{eff}}(t)$ and $R_{\text{eff}}(t)$ such that they give rise to the adiabatic Moore functions previously found. The effective trajectories obtained produce an adiabatic evolution of the field in finite time, hence they constitute a shortcut to adiabaticity.
\subsection{Adiabatic Evolution of the Field}
We start by looking for functions $F$ and $G$ that satisfy Equations~\eqref{eq:MooreEqL} and \eqref{eq:MooreEqR}.
We can take the derivative of the above set of equations
\begin{equation}
G^{\prime}\left[t+L(t)\right]\left[1+\dot{L}(t)\right]-F^{\prime}\left[t-L(t)\right]\left[1-\dot{L}(t)\right]=0
\end{equation}
\begin{equation}
G^{\prime}\left[t+L(t)\right]\left[1+\dot{R}(t)\right]-F^{\prime}\left[t-R(t)\right]\left[1-\dot{R}(t)\right]=0
\end{equation}
and define
\begin{equation}
A(z):=F^{\prime}(z)
\end{equation}
\begin{equation}
B(z):=G^{\prime}(z).
\end{equation}
Then, it is easy to rewrite the previous equations as
\begin{equation}
B\left[t+L\right]\left[1+\dot{L}\right]-A\left[t-L\right]\left[1-\dot{L}\right]=0
\end{equation}
\begin{equation}
B\left[t+R\right]\left[1+\dot{R}\right]-A\left[t-R\right]\left[1-\dot{R}\right]=0.
\end{equation}
Further, we can expand the functions in a Taylor series
\begin{equation}
B\left[t+x\right]=\sum_{n}\frac{d^{n}B(t)}{dt^{n}}\frac{x^{n}}{n!}
\end{equation}
\begin{equation}
A\left[t+x\right]=\sum_{n}\frac{d^{n}A(t)}{dt^{n}}\frac{x^{n}}{n!}
\end{equation}
which results in the following equations up to the third order
\begin{widetext}
\begin{equation}
\left[B+\frac{dB(t)}{dt}L+\frac{1}{2}\frac{d^{2}B(t)}{dt^{2}}L^{2}+\frac{1}{3!}\frac{d^{3}B(t)}{dt^{3}}L^{3}\right]\left[1+\dot{L}\right]
-\left[A-\frac{dA(t)}{dt}L+\frac{1}{2}\frac{d^{2}A(t)}{dt^{2}}L^{2}-\frac{1}{3!}\frac{d^{3}A(t)}{dt^{3}}L^{3}\right]\left[1-\dot{L}\right]=0
\end{equation}
\begin{equation}
\left[B+\frac{dB(t)}{dt}R+\frac{1}{2}\frac{d^{2}B(t)}{dt^{2}}R^{2}+\frac{1}{3!}\frac{d^{3}B(t)}{dt^{3}}R^{3}\right]\left[1+\dot{R}\right]-\left[A-\frac{dA(t)}{dt}L+\frac{1}{2}\frac{d^{2}A(t)}{dt^{2}}L^{2}-\frac{1}{3!}\frac{d^{3}A(t)}{dt^{3}}L^{3}\right]\left[1-\dot{R}\right]=0.
\end{equation}
\end{widetext}
These can be rewritten as
\begin{eqnarray}
&& B-A+\left((B+A)L\right)^{\prime}+\frac{1}{2}\left((B^{\prime}-A^{\prime})L^{2}\right)^{\prime}\nonumber\\
&& +\frac{1}{6}\left((B^{\prime\prime}+A^{\prime\prime})L^{3}\right)^{\prime}=0
\end{eqnarray}
\begin{eqnarray}
&& B-A+\left((B+A)R\right)^{\prime}+\frac{1}{2}\left((B^{\prime}-A^{\prime})R^{2}\right)^{\prime}
\nonumber\\
&& +\frac{1}{6}\left((B^{\prime\prime}+A^{\prime\prime})R^{3}\right)^{\prime}=0,
\end{eqnarray}
by discarding the terms $B^{\prime\prime\prime}L^{\prime}$, $A^{\prime\prime\prime}L^{\prime}$, $B^{\prime\prime\prime}R^{\prime}$ and $A^{\prime\prime\prime}R^{\prime}$ because they involve third derivatives of time.
At this point, we can expand these functions in different timescales
\begin{equation}
A=A_{0}+A_{1}+A_{2}+A_{3}+\ldots
\end{equation}
\begin{equation}
B=B_{0}+B_{1}+B_{2}+B{}_{3}+\ldots
\end{equation}
where the subindices indicate how many temporal derivatives are involved in each contribution. Using this expansion, the previous equation results for order 0 in
\begin{equation}
A_{0}=B_{0}
\end{equation}
\\
For the first order, we have
\begin{equation}
B_{1}-A_{1}+(2A_{0}L)^{\prime}=0
\end{equation}
\begin{equation}
B_{1}-A_{1}+(2A_{0}R)^{\prime}=0
\end{equation}
and therefore
\begin{equation}
\left(2A_{0}(R-L)\right)^{\prime}=0
\quad \implies A_{0}=\frac{1}{R-L}.
\end{equation}
With this result, $B_{1}-A_{1}$ can be calculated by replacing it in the previous equations.
The second order gives
\begin{equation}
B_{2}-A_{2}+((A_{1}+B_{1})L)^{\prime}=0
\end{equation}
\begin{equation}
B_{2}-A_{2}+((A_{1}+B_{1})R)^{\prime}=0,
\end{equation}
where we have used that $A_{0}=B_{0}$. Subtracting, we obtain
\begin{equation}
\left[(A_{1}+B_{1})(L-R)\right]^{\prime}=0\implies A_{1}+B_{1}=\frac{k}{L-R},
\end{equation}
where $k$ is some constant. However, we must note that, by definition, $A_{1}$ and $B_{1}$ should have one and only one time derivative. Therefore
\begin{equation}
k=0\implies A_{1}=-B_{1}.
\end{equation}
Replacing this result in the equation for $B_{1}-A_{1}$, we find that
\begin{equation}
A_{1}=-B_{1}=(A_{0}R)^{\prime}=\left(\frac{R}{R-L}\right)^{\prime}=\left(\frac{1}{2}\frac{R+L}{R-L}\right)^{\prime}.
\end{equation}
The Moore functions are then given by
\begin{equation}
F(t)=\int dtA(t)=\int dtA_{0}(t)+\int dtA_{1}(t)+\int dtA_{2}(t)+....
\end{equation}
\begin{equation}
G(t)=\int dtB(t)=\int dtB_{0}(t)+\int dtB_{1}(t)+\int dtB_{2}(t)+....,
\end{equation}
where $A_{j}(t)$ and $B_{j}(t)$ include $j$ time derivatives. If the timescale in which the mirror moves is given by $\tau$ then $\int dtA_{j}(t)\propto \tau^{j-1}$, and in the adiabatic limit ($\tau\to\infty$), only the first two terms are relevant. Therefore, the adiabatic Moore functions for a cavity with two moving mirrors are given by
\begin{align}
\label{eq:adMoore}
F_{\text{ad}}(t)&=\int dt\frac{1}{R(t)-L(t)}+\frac{1}{2}\frac{R(t)+L(t)}{R(t)-L(t)}\\
G_{\text{ad}}(t)&=\int dt\frac{1}{R(t)-L(t)}-\frac{1}{2}\frac{R(t)+L(t)}{R(t)-L(t)}.
\end{align}
Following this procedure, one can also compute the higher adiabatic orders, generalizing to the case of two mirrors the results in Ref. \cite{Moore}. However, the above results are enough for our purposes.
\subsection{Shortcut to Adiabaticity}
Given the reference trajectories for the right, $R_{\text{ref}}(t)$, and left, $L_{\text{ref}}(t)$, mirrors, it is possible to find effective trajectories, $R_{\text{eff}}(t)$ and $L_{\text{eff}}(t)$, such that the evolution of the field from start to finish is exactly the adiabatic evolution for the reference trajectories.
A way to find such effective trajectories is to select them such that the Moore functions for the field are those of the adiabatic evolution produced by the reference ones, that is
\begin{equation}
\label{eq:Leff}
G_{\text{ad}}(t+L_{\text{eff}}(t))-F_{\text{ad}}(t-L_{\text{eff}}(t))=0
\end{equation}
\begin{equation}
\label{eq:Reff}
G_{\text{ad}}(t+R_{\text{eff}}(t))-F_{\text{ad}}(t-R_{\text{eff}}(t))=2,
\end{equation}
where $G_{\text{ad}}(t)$ and $F_{\text{ad}}(t)$ are given by Equation (\ref{eq:adMoore}) with $L(t)=L_{\text{ref}}(t)$ and $R(t)=R_{\text{ref}}(t)$. Thus, knowing the reference trajectories, we can solve Equations (\ref{eq:Leff}) and (\ref{eq:Reff}) independently to find effective trajectories that evolve the field in a way that exactly matches the adiabatic evolution for the reference trajectories.
\subsection{Limit of Effective Trajectories}
We would like to obtain some analytical understanding of the effective trajectories that produce the STAs. In order to do this, we exactly solve the Moore equations for the case where the reference trajectories are given by an instantaneous motion
\begin{equation}
R_{\text{ref}}(t)=R_{0}\theta(-t)+R_{f}\theta(t)
\end{equation}
\begin{equation}
L_{\text{ref}}(t)=L_{0}\theta(-t)+L_{f}\theta(t).
\end{equation}
We are looking for the limit effective trajectories $L_{\text{lim}}(t)$ and $R_{\text{lim}}(t)$ such that
\begin{equation}
G_{\text{ad}}(t+L_{\text{lim}}(t))-F_{\text{ad}}(t-L_{\text{lim}}(t))=0
\end{equation}
\begin{equation}
G_{\text{ad}}(t+R_{\text{lim}}(t))-F_{\text{ad}}(t-R_{\text{lim}}(t))=2.
\end{equation}
For these reference trajectories the adiabatic Moore functions are given by
\begin{align}
\label{eq:adMooreLimF}
F_{\text{ad}}(t)&=\frac{t+L_{0}}{R_{0}-L_{0}}\theta(-t)+\frac{t+L_{f}}{R_{f}-L_{f}}\theta(t)\\
G_{\text{ad}}(t)&=\frac{t-L_{0}}{R_{0}-L_{0}}\theta(-t)+\frac{t-L_{f}}{R_{f}-L_{f}}\theta(t), \label{eq:adMooreLimG}
\end{align}
which means that the Moore functions are linear functions before and after $t=0$.
We can use this result to analyze the Moore equations one by one. If $t<-R_{0}$, then
\begin{equation}
t-R_{\text{eff}}(t)<0,\,\quad t+R_{\text{eff}}(t)<0
\end{equation}
and the solution is
\begin{equation}
R_{\text{lim}}(t<-R_{0})=R_{0}.
\end{equation}
In addition, if $t>R_{f}$, then
\begin{equation}
t-R_{\text{lim}}(t)>0,\,\quad t+R_{\text{lim}}(t)>0
\end{equation}
and the solution is
\begin{equation}
R_{\text{lim}}(t>R_{f})=R_{f}.
\end{equation}
However, if $-R_0<t<R_f$, then
\begin{equation}
t+R(t)>0,\quad t-R(t)<0
\end{equation}
and the Moore equation is given by
\begin{equation}
\frac{(t+R_{\text{lim}}(t))-L_{f}}{R_{f}-L_{f}}-\frac{(t-R_{\text{lim}}(t))+L_{0}}{R_{0}-L_{0}}=2.
\end{equation}
Solving this equation for $R_{\text{lim}}(t)$, we get
\begin{eqnarray}
\label{eq:R_lim}
R_{\text{lim}}(t)&=&\frac{2(R_{0}-L_{0})(R_{f}-L_{f})+L_{f}(R_{0}-L_{0})+L_{0}(R_{f}-L_{f})}{(R_{0}-L_{0})+(R_{f}-L_{f})} \nonumber \\
&-& t\frac{(R_{0}-L_{0})-(R_{f}-L_{f})}{(R_{0}-L_{0})+(R_{f}-L_{f})}
= R_{c}+v_{\text{lim}}t.
\end{eqnarray}
Similarly, we have $L_{\text{lim}}(t<-L_0)=L_0$, $L_{\text{lim}}(t>L_f)=L_f$ and
\begin{align}
\label{eq:L_lim}
L_{\text{lim}}(-L_0<t<L_F)=\frac{L_{f}(R_{0}-L_{0})+L_{0}(R_{f}-L_{f})}{(R_{0}-L_{0})+(R_{f}-L_{f})}+v_{\text{lim}}t.
\end{align}
In simple words, the limit effective trajectories, at early and late times, coincide with the constant position from the reference trajectory. For intermediate time values, say between these initial and final positions, the motion of the limit trajectories is simply a uniform motion with the same velocity, $v_{\text{lim}}$, for the left and right mirrors. This velocity is determined only by the initial and final lengths of the cavity, being negative for a contraction, positive for an expansion and zero if the cavity moves rigidly.
In addition, it is possible to notice that, in general, these trajectories are not continuous functions. This is related to the fact that if the reference motion occurs in a timescale $\tau$, there exists a critical $\tau_c$ (which depends on the precise reference motion) for which the effective trajectories cease to be physically achievable since the speed should be greater than the speed of light at some time.
However, by enforcing continuity for the functions,
\begin{align}
R_{0}=R_{\text{lim}}(-R_{0}),\quad R_{f}=R_{\text{lim}}(R_{f}),\\
L_{0}=L_{\text{lim}}(-L_{0}), \quad L_{f}=L_{\text{lim}}(L_{f}),
\end{align}
we find that if $L_{f}R_{0}=L_{0}R_{f}$, the limit trajectories are actually continuous. Two simple cases where this is verified is either when there is a trivial reference motion (that is $L_0=L_f$ and $R_0=R_f$) or in the case when one of the walls is at rest at the origin, $L_0=L_f=0$, and the other moves freely.
\section{Numerical Analysis of the STA}\label{sec5}
We now consider a particular set of reference trajectories for which we find the associated effective trajectories by numerically solving Equations (\ref{eq:Leff}) and (\ref{eq:Reff}). We consider different types of motions for the mirrors, such as a contraction, an expansion and a rigid translation, and we compare the results of the obtained trajectories and energies between the reference and effective trajectories.
Before proceeding, we need to establish a magnitude to decide whether an STA has been achieved and measure how far we are from one. Hence, we define the adiabaticity coefficient
\begin{equation}
Q(t):=\frac{E(t)}{E_{\text{ad}}(t)},
\end{equation}
where $E(t)$ is the total energy in the cavity
\begin{align}
\label{eq:E}
E(t)=\int_{L(t)}^{R(t)}dx\langle T_{00}(x,t)\rangle_{\text{ren}},
\end{align}
while the adiabatic energy is given by
\begin{align}
E_{\text{ad}}(t)=-\frac{\pi}{24d}+\frac{Z(TL_0)}{d},
\end{align}
where $d=|R(t)-L(t)|$ is the length of the cavity.
Notice that the adiabaticity parameter equals one if the field evolves in an adiabatic manner. However, due to the static Casimir energy (the first term of $E_{\text{ad}}$), $Q$ can either be lower than one for low temperatures or bigger than one for high temperatures, if the cavity is static.
Once the effective trajectories are obtained, the Moore functions are given by \mbox{Equation (\ref{eq:adMoore})}. The energy and adiabaticity coefficients can then be calculated using \mbox{Equations (\ref{eq:dens_en}) and (\ref{eq:E})}. However, it is useful to contrast these results with the energy and adiabaticity parameters corresponding to the original reference trajectories. In order to do this, we need to obtain the functions $F(t)$ and $G(t)$ by numerically solving Moore's Equations (\ref{eq:MooreEqL}) and (\ref{eq:MooreEqR}). We dedicate the next section to develop an algorithm for solving this system of coupled functional equations.
\subsection{Algorithm for Moore's Equations}
In the following, we derive an algorithm for solving Moore's equations for $F(z)$ and $G(z)$ which can be used for arbitrary trajectories $L(t)$ and $R(t)$ of the mirrors. This algorithm is a generalization of the one used for a single moving mirror in Ref. \cite{ColeSchieve}.
In order to find $G(z_1)$, we look for $t_1$ such that
\begin{equation}
z_{1}=t_{1}+R(t_{1}),
\end{equation}
which can be done simply by solving an algebraic equation. Then, from Equation (\ref{eq:MooreEqR}) we know that
\begin{equation}
G(t_{1}+R(t_{1}))=F(t_{1}-R(t_{1}))+2
\end{equation}
and, solving for $t_{1}^{*}$, such that
\begin{equation}
t_{1}-R(t_{1})=t_{1}^{*}-L(t_{1}^{*}),
\end{equation}
we find, using Equation (\ref{eq:MooreEqL}),
\begin{equation}
F(t_{1}^{*}-L(t_{1}^{*}))=G(t_{1}^{*}+L(t_{1}^{*}))
\end{equation}
\begin{equation}
\implies G(t_{1}+R(t_{1}))=F(t_{1}^{*}-L(t_{1}^{*}))+2=G(t_{1}^{*}+L(t_{1}^{*}))+2.
\end{equation}
Hence, given
\begin{equation}
z_{2}:=t_{1}^{*}+L(t_{1}^{*}),
\end{equation}
we have
\begin{equation}
G(z_{1})=G(z_2)+2.
\end{equation}
If we assume $z_2$ to be our starting point and iterating this $n$ times, we obtain
\begin{equation}
G(z_{1})=G(z_{n+1})+2n.
\end{equation}
Note that if $L(t)<R(t)$ for all $t$, then
\begin{align}
&t_{1}-R(t_{1})=t_{1}^{*}-L(t_{1}^{*})\nonumber\\
&\implies t_{1}-t_{1}^{*}=R(t_{1})-L(t_{1}^{*})>0\implies t_{1}>t_{1}^{*}
\end{align}
\begin{align}
&t_{1}^{*}+L(t_{1}^{*})=t_{2}+R(t_{2})\nonumber\\
&\implies t_{1}^{*}-t_{2}=R(t_{2})-L(t_{1}^{*})>\implies t_{1}>t_{1}^{*}>t_{2},
\end{align}
which in turn means that $z_1>z_2>...>z_n$. This means we have reduced the original problem of finding the value of the function for a given time to knowing it at a previous temporal value. We can iterate this procedure going back in time until $G(z_{n})$ is known, which eventually happens since we know the solution for static mirrors (Equation (\ref{eq:adMooreLimG})).
Analogously, for the other Moore function, if we wish to find $F(w_1)$, we can search for a $t_1$ such that
\begin{equation}
w_{1}=t_{1}-L(t_{1}).
\end{equation}
By means of Moore's Equation (\ref{eq:MooreEqL}), we know that
\begin{equation}
F(t_{1}-L(t_{1}))=G(t_{1}+L(t_{1})).
\end{equation}
If we solve for $\tilde{t}_{1}$
\begin{equation}
\tilde{t}_{1}+R(\tilde{t}_{1})=t_{1}+L(t_{1}),
\end{equation}
we obtain
\begin{equation}
G(\tilde{t}_{1}+R(\tilde{t}_{1}))=2+F(\tilde{t}_{1}-R(\tilde{t}_{1}))
\end{equation}
\begin{align}
\implies F(t_{1}-L(t_{1}))&=G(t_{1}+L(t_{1}))=G(\tilde{t}_{1}+R(\tilde{t}_{1}))\nonumber\\
&=2+F(\tilde{t}_{1}-R(\tilde{t}_{1}))
\end{align}
\begin{equation}
F(w_{1})=2+F(\tilde{t}_{1}-R(\tilde{t}_{1})).
\end{equation}
Finally, defining
\begin{equation}
w_{2}:=\tilde{t}_{1}-R(\tilde{t}_{1})
\end{equation}
we can express the value of the function at point $w_1$ in terms of the value of the function at $w_2$
\begin{equation}
F(w_{1})=2+F(w_2).
\end{equation}
In general, by iterating, we get
\begin{equation}
F(w_{1})=2n+F(w_{n+1}),
\end{equation}
and, once more, if we go back enough times, we eventually reach the point where the mirrors are static and the function $F(z_n)$ can be evaluated using Equation (\ref{eq:adMooreLimF}). In the end, we obtain an iterative algorithm to evaluate Moore's functions $F(z)$ and $G(z)$ at any point.
\subsection{Reference Trajectories}
We performed a numerical analysis and compared the reference trajectories with their effective trajectories.
We also computed the adiabaticity parameter for different temperatures. In order to do this, we needed a well-defined continuous energy density for the field. Since $\langle T_{tt}(x,t)\rangle_{\text{ren}}$ involves third derivatives of the Moore functions, which in turn involve third derivatives of the reference trajectories, we chose these reference trajectories to have continuous derivatives up to the third order. We considered the motion of the wall to be restricted to a finite-time interval. In order to fulfill these conditions, we chose the reference trajectories to be
\begin{align}
\label{eq:Lref}
L_{\text{ref}}(t)&=
(L_{f}-L_0)\delta(t/\tau)+L_0\\
R_{\text{ref}}(t)&=R_{0}\left[1-\epsilon\delta(t/\tau)\right]\label{eq:Rref}
\end{align}
where $\delta(x)=35x^{4}-84x^{5}+70x^{6}-20x^{7}$ satisfies $\delta(0)=\delta^{\prime}(0)=\delta^{\prime\prime}(0)=\delta^{\prime\prime\prime}(0)=\delta^{\prime}(1)=\delta^{\prime\prime}(1)=\delta^{\prime\prime\prime}(1)=0$ and $\delta(1)=1$.
In the following sections, we use these trajectories to analyze different types of motions, such as a contraction, expansion and rigid motion, their effective trajectories and whether they achieve an STA.
\subsection{Contraction}
We first analyzed a symmetric contraction of the cavity, meaning that both mirrors performed the same reference motion at the same time, but in opposite directions. We represented this by considering the reference functions Equations (\ref{eq:Lref}) and (\ref{eq:Rref}) and solving numerically Equations (\ref{eq:Leff}) and (\ref{eq:Reff}) for the effective trajectories, $R_{\text{eff}}(t)$ and $L_{\text{eff}}(t)$.
In Figure \ref{fig:tray_cont}a, we show the reference and corresponding effective trajectories for the left (dashed lines) and right mirrors (solid lines) in a symmetric contraction. We note that the effective trajectory for the right mirror starts moving first close to $t=-R_0$, while the left mirror moves at later times near $t=-L_0$, as pointed out by our analysis for the limit trajectories. If we look at $R_{\text{eff}}$, we also note the local minimum and maximum around these points develop into discontinuities for very small $\tau$. This can also be seen in Figure \ref{fig:tray_lim_cont}, where we compare the effective trajectories for an asymmetric contraction with $\tau/R_0=0.4$ and the limit effective trajectories analytically (Equations (\ref{eq:R_lim}) and (\ref{eq:L_lim})). Therein, the discontinuities are more evident. It is also noticeable that the right trajectory converges faster than the left one and that the slope of the curve, i.e., the velocity, is negative, which is consistent with our analytical results.
In Figure \ref{fig:tray_cont}b, we show the resulting Moore functions for the reference and effective trajectories. In the case of the effective trajectory, the functions are linear at the early and late times. On the other hand, Moore's functions for the reference trajectory are linear plus an oscillation at late times, which is the manifestation of particle creation.
\begin{figure*}
\subfloat[]{\includegraphics[width=7 cm]
{tray_eff_ref_cont1b-eps-converted-to.pdf}\label{fig:tray_conta}}%
\subfloat[]{\includegraphics[width=7 cm]
{FG_eff_ref_cont1b-eps-converted-to.pdf}\label{fig:tray_contb}}%
\caption{(\textbf{a}) Reference
and corresponding effective trajectories for the left and right mirrors in the case of a symmetric contraction. (\textbf{b}) Resulting Moore's functions for reference and effective trajectories. The parameters used for this calculation were $\tau/R_0=1.2$, $\epsilon=0.3$, $L_0/R_0=0$, $L_f/R_0=0.3$ and $R_f/R_0=0.7$. \label{fig:tray_cont}}
\end{figure*}
\begin{figure*}
\subfloat[]{\includegraphics[width=7 cm]
{tray_Llim_cont-eps-converted-to.pdf}\label{fig:tray_lim_conta}}%
\subfloat[]{\includegraphics[width=7 cm]
{tray_Rlim_cont-eps-converted-to.pdf}\label{fig:tray_lim_contb}}%
\caption{(\textbf{a}) Effective and limit trajectories for the left mirror for an asymmetric contraction. (\textbf{b})~Effective and limit effective trajectories for the right mirror. The parameters used for this calculation were $\tau/R_0=0.4$, $\epsilon=-0.3$, $L_0/R_0=0$, $L_f/R_0=0.5$ and $R_f/R_0=1.3$. \label{fig:tray_lim_cont}}
\end{figure*}
Further, we analyzed the adiabaticity parameter for different initial temperatures as shown in Figure \ref{fig:Q_cont}. We note that the adiabaticity parameter is initially one. However, for both the reference and effective trajectories at later times, the reference trajectory is very far from unity, while the effective trajectory returns to one, indicating that an adiabatic evolution has been achieved by reabsorbing the emitted photons. It is also noticeable that as the temperature increases, the curves become smoother.
\begin{figure*}
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_cont1c-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_cont1T1c-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_cont1T5c-eps-converted-to.pdf}}%
\caption{Adiabaticity parameter for a symmetric contraction for three different temperatures: (\textbf{a})~$TR_0=0$, (\textbf{b}) $TR_0=1$ and (\textbf{c}) $TR_0=5$. The parameters used for this calculation were $\tau/R_0=1.2$, $\epsilon=0.3$, $L_0/R_0=0$ and $L_f/R_0=0.3$, $R_f/R_0=0.7$. \label{fig:Q_cont}}
\end{figure*}
\subsection{Expansion}
We then analyzed our proposed STA for a reference trajectory given by a symmetric expansion of the cavity. {To achieve this, we used the reference trajectories given by Equations (\protect{\ref{eq:Lref}) and (\ref{eq:Rref}}) with $\epsilon<0$ and $L_f=\epsilon R_0$.}
In Figure \ref{fig:tray_exp}a, we show the reference and corresponding effective trajectories for the left (dashed lines) and right (solid lines) mirrors in a symmetric expansion.
We notice that the effective trajectory of the right mirror has a local minimum and maximum close to the point where a discontinuity will develop for $\tau\to0$, which is in agreement with the limit effective trajectories calculated. On the other hand, the effective trajectory of the right mirror is very similar to the reference one. This is because, as we have previously seen, the convergence of the effective trajectory of the left mirror to the limit is slower. The Moore functions, however, have a similar behavior to that of the previous case.
\begin{figure*}
\subfloat[]{\includegraphics[width=7 cm]
{tray_eff_ref_exp1b-eps-converted-to.pdf}\label{fig:tray_expa}}%
\subfloat[]{\includegraphics[width=7 cm]
{FG_eff_ref_exp1b-eps-converted-to.pdf}\label{fig:tray_expb}}%
\caption{(\textbf{a}) Reference
and corresponding effective trajectories for the left and right mirrors in the case of a symmetric expansion. (\textbf{b}) Resulting Moore functions for reference and effective trajectories. The parameters used for this calculation were $\tau/R_0=1.2$, $\epsilon=-0.3$, $L_0/R_0=0$, $L_f/R_0=-0.3$ and $R_f/R_0=1.3$. \label{fig:tray_exp}}
\end{figure*}
In Figure \ref{fig:Q_exp}, we present the adiabaticity parameter for a symmetric expansion for three different temperatures.
In this case, the adiabaticity parameter again confirms that we have obtained an STA, as it is equal to one for late times for the effective trajectories. We can also see that the effect of the temperature on this parameter is to smooth the curve as the temperature increases. The STA allows us to save more energy for higher temperatures.
\begin{figure*}
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_exp1b-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_exp1T1b-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_exp1T5b-eps-converted-to.pdf}}%
\caption{Adiabaticity
parameter for a symmetric expansion for three different temperatures: (\textbf{a})~$TR_0=0$, (\textbf{b})$TR_0=1$ and (\textbf{c}) $TR_0=5$. The parameters used for this calculation were $\tau/R_0=1.2$, $\epsilon=-0.3$, $L_0/R_0=0$, $L_f/R_0=-0.3$ and $R_f/R_0=1.3$. \label{fig:Q_exp}}
\end{figure*}
\begin{figure*}
\subfloat[]{\includegraphics[width=7 cm]
{tray_eff_ref_rig1b-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=7 cm]
{FG_eff_ref_rig1b-eps-converted-to.pdf}}%
\caption{(\textbf{a}) Reference
and corresponding effective trajectories for the left and right mirrors in the case of a rigid translation. (\textbf{b}) Resulting Moore functions for reference and effective trajectories. The parameters used for this calculation were $\tau/R_0=1.2$, $\epsilon=-0.3$, $L_0/R_0=0$ and $L_f/R_0=0.3$ $R_f/R_0=1.3$. \label{fig:tray_rig}}
\end{figure*}
\begin{figure*}
\subfloat[]{\includegraphics[width=7 cm]
{tray_Llim_rigb-eps-converted-to.pdf}\label{fig:tray_lim_riga}}%
\subfloat[]{\includegraphics[width=7 cm]
{tray_Rlim_rigb-eps-converted-to.pdf}\label{fig:tray_lim_rigb}}%
\caption{(\textbf{a}) Effective
and limit trajectories for the left mirror for a rigid motion. (\textbf{b}) Effective and limit trajectories for the right mirror for a rigid motion. The parameters used for this calculation were $\tau/R_0=0.4$, $\epsilon=-0.4$, $L_0/R_0=0$, $L_f/R_0=0.4$ and $R_f/R_0=1.3$ .\label{fig:tray_lim_rig}}
\end{figure*}
\begin{figure*}
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_rig1b-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_rig1T1b-eps-converted-to.pdf}}%
\subfloat[]{\includegraphics[width=4.7 cm]
{Q_eff_ref_rig1T5b-eps-converted-to.pdf}}%
\caption{Adiabaticity
parameter for a rigid translation for three different temperatures: (\textbf{a}) $TR_0=0$, (\textbf{b})~$TR_0=1$ and (\textbf{c}) $TR_0=5$. The parameters used for this calculation were $\tau/R_0=1.2$, $\epsilon=-0.3$, $L_0/R_0=0$, $L_f/R_0=0.3$ and $R_f/R_0=1.3$. \label{fig:Q_rig}}
\end{figure*}
\subsection{Rigid Motion}
The final type of reference trajectory that we considered was a rigid translation. {To achieve this, we used the reference trajectories given by Equations (\protect{\ref{eq:Lref}) and (\ref{eq:Rref}}) with $\epsilon<0$ and $L_f=-\epsilon R_0$.} There were several motivations for this from the fact that the limit effective trajectories were qualitatively different from fundamental questions on relativistic quantum information tasks.
In Figure \ref{fig:tray_rig}, we show the reference and corresponding effective trajectories for the left (dashed lines) and right (solid lines) mirrors in a rigid translation.
We can see that the effective trajectory of the left mirror is very similar to the reference trajectory. However, the effective trajectory for the right mirror is extremely different from the other two cases studied previously. We observe that the right mirror moves half of the way while the left one is static, then it stops, and the left mirror moves and stops, and then it moves again up to the final position. Although this motion can look strange at first sight, it is very well described by the limit effective trajectory in Figure \ref{fig:tray_lim_rig}, which predicts that the speed of the motion should be zero for a reference trajectory that does not change the length of the cavity. The Moore functions also have a similar behavior as in the previous cases.
Finally, we studied the adiabaticity parameter for the reference rigid motion as shown in Figure \ref{fig:Q_rig}. We see that the effective trajectories given by Equations (\ref{eq:Lref}) and (\ref{eq:Rref}) result in a successful shortcut to adiabaticity, since for late times, $Q_{\text{eff}}=1$. The energy saved by using this motion protocol is greatly enhanced for high temperatures for the initial state of the quantum field.
\section{Discussion}\label{sec6}
As technology improves and quantum systems can be operated at smaller timescales, it becomes increasingly important to consider the nonadiabatic effects of these operations and to develop new ways of mitigating of even avoiding them entirely. With this motivation in mind, in this manuscript we found a shortcut to adiabaticity for a scalar quantum field in a one-dimensional cavity with two moving mirrors. This allowed an extremely efficient way to manipulate microwave resonators very rapidly. Moreover, our results gave an explicit protocol to find STAs for any initial and final state of the mirrors, which can be implemented in an experimental setup by choosing adequately effective trajectories for the mirrors calculated from a given reference trajectory.
We analytically analyzed the properties of these effective trajectories and found that the limit effective trajectories, for infinitely fast reference trajectories, are in general not continuous functions, which signaled that there was a critical timescale beyond which the resulting shortcut ceased to be physical, since the mirror should not
move faster than the speed of light.
In addition, we solved numerically the effective trajectories for three different types of reference motions: a contraction, an expansion and a rigid translation of the cavity. Our numerical analysis confirmed the analytical results, showing that our protocol successfully implemented a shortcut to adiabaticity and that the effective trajectories were very well described by the limit effective trajectories found analytically.
These findings call for further studies that analyze in more depth the experimental implementation in superconducting circuits or optomechanical cavities. It would also be interesting to better understand the instantaneous energetic cost of this STA and their utilization in more efficient quantum heat engines.
\section*{Acknowledgments} This research was supported by Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), Consejo Nacional de Investigaciones Científicas y Técnicas (CON- ICET), Universidad de Buenos Aires (UBA) and Univer- sidad Nacional de Cuyo (UNCuyo). P.I.V. acknowledges ICTP-Trieste Associate Program.
\vspace{6pt}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Classification is a major subject in all areas of mathematics and has attracted the attention of many talented mathematicians. In the category
of C*-algebras, the program of classifying all amenable C*-algebras was initiated by Elliott, first with the classification of AF-algebras,
and later with the classification of certain simple C*-algebras of real rank zero. His work was followed by many other classification results for
nuclear C*-algebras, both in the stably finite and the purely infinite case.
The classification theory for von Neumann algebras precedes the classification program initiated by Elliott. In fact, the classification of
amenable von Neumann algebras with separable pre-dual, which is due to Connes, Haagerup, Krieger and Takesaki, was completed more than 30 years
ago. Connes moreover classified automorphisms of the type II$_1$ factor up to cocycle conjugacy in \cite{Connes-I}. This can be regarded as the
first classification result for actions on von Neumann algebras, and it was followed by his own work on the classification of pointwise outer
actions of amenable groups on von Neumann algebras in \cite{Connes-II}.
Several people have since then tried to obtain similar classification results for actions on C*-algebras. Early results in this direction
include the work of Herman and Ocneanu in \cite{Herman-Ocneanu} on integer actions with the Rokhlin property on UHF-algebras, the work of Fack and
Mar\'echal in \cite{Fack-Marechal-I} and \cite{Fack-Marechal-II} for cyclic groups actions on UHF-algebras, and the work of Handelman and
Rossmann \cite{Handelman-Rossmann} for locally representable compact group actions on AF-algebras. Other results have been obtained by Elliott
and Su in \cite{Elliott-Su} for direct limit actions of $\mathbb{Z}_2$ on AF-algebras, and by Izumi in \cite{Izumi-I} and \cite{Izumi-II}, where he
proved a number of classification results for actions of finite groups on arbitrary unital separable C*-algebras with the Rokhlin property, as well as for approximately representable actions. The classification result of Izumi regarding actions with the Rokhlin property has been extended recently by Nawata in \cite{Nawata} to cover actions on certain not-necessarily unital separable C*-algebras (specifically for algebras $A$ such that $A\subseteq \overline{\mathrm{GL}(\widetilde{A})}$). It should
be emphasized that the classification of group actions on C*-algebras is a far less developed subject than the classification of C*-algebras
and even farther less developed than the classification of group actions on von Neumann algebras.
In this paper we extend the classification results of Izumi and Nawata of finite group actions on C*-algebras with the Rokhlin property to actions of finite groups with the Rokhlin property on arbitrary separable C*-algebras. This is done by first obtaining a classification result for equivariant *-homomorphism between C*-dynamical systems associated to actions of finite groups with the Rokhlin property, and then applying Elliott's intertwining argument. In this paper we also obtain obstructions on the Cuntz semigroup, the Murray-von Neumann semigroup, and the K-groups of a C*-algebra allowing an action of a finite group with the Rokhlin property. These results are used together with the classification result of actions to obtain an equivariant UHF-absorption result.
This paper is organized as follows. In Section 2, we collect a number of definitions and results that will be used throughout the paper. In Section 3, we give an abstract classification for equivariant *-homomorphism between C*-dynamical systems associated to actions of finite groups with the Rokhlin property, as well as, a classification for the given actions. These abstract classification results are used together with known classification results of C*-algebras to obtain specific classification of equivariant *-homomorphisms and actions of finite groups on C*-algebras that can be written as inductive limits of 1-dimensional NCCW-complexes with trivial K$_1$-groups and for unital simple AH-algebras of no dimension growth.
In Section 4, we obtain obstructions on the Cuntz semigroup, the Murray-von Neumann semigroup, and the K$_*$-groups of a C*-algebra allowing an action of a finite group with the Rokhlin property. Then using the Cuntz semigroup obstruction we show that the Cuntz semigroup of a C*-algebra that admits an action of finite group with the Rokhlin property has certain divisibility property. In this section we also compute the Cuntz semigroup, the Murray-von Neumann semigroup, and the K$_*$-groups of the fixed-point and crossed product C*-algebras associated to an action of a finite group with the Rokhlin property.
In Section 5, we obtain divisibility results for the Cuntz semigroup of certain classes of C*-algebras and use this together with the classification results for actions obtained in Section 3 to prove an equivariant UHF-absorbing result.
\section{Preliminary definitions and results}
Let $A$ be a C*-algebra. We denote by $\mathrm{M}(A)$ its multiplier algebra, by $\widetilde{A}$ its unitization (that is, the C*-algebra obtained by adjoining a unit to $A$, even if $A$ is unital). If $A$ is unital, we denote by $\mathrm{U}(A)$ its unitary group. We denote by $\mathrm{Aut}(A)$ the automorphism group of $A$. The identity map of $A$ is denoted $\mathrm{id}_A$.
Topological groups are always assumed to be Hausdorff. If $G$ is a locally compact group and $A$ is a C*-algebra, then an \emph{action} of $G$ on $A$ is a strongly continuous group homomorphism $\alpha\colon G\to\mathrm{Aut}(A)$. Strong continuity for $\alpha$ means that for each $a$ in $A$, the map from $G$ to $A$ given by $g\mapsto\alpha_g(a)$ is continuous with respect to the norm topology on $A$.
We denote by $\mathcal{K}$ the C*-algebra of compact operators on a separable Hilbert space. We take $\mathbb{N}=\{1,2,\ldots\}$, $\mathbb{Z}_+=\{0,1,2,\ldots\}$, and $\overline{\mathbb{Z}_+}=\mathbb{Z}_+\cup \{\infty\}$.
\subsection{The Rokhlin property for finite group actions}\label{Rokhlin}
Let us briefly recall the definition of the Rokhlin property, in the sense of \cite[Definition 2]{SantiagoRP}, for actions of
finite groups on (not necessarily unital) C*-algebras. Actions with the Rokhlin property are the main object of study of this work.
\begin{definition}\label{def: Rokhlin}
Let $A$ be a C*-algebra and let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a finite group $G$ on $A$. We say that $\alpha$ has
the \emph{Rokhlin property} if for any $\varepsilon>0$ and any finite subset $F\subseteq A$ there exist mutually orthogonal positive
contractions $r_g$ in $A$, for $g\in G$, such that
\begin{itemize}
\item[(i)] $\|\alpha_g(r_h)-r_{gh}\|<\varepsilon$ for all $g,h\in G$;
\item[(ii)] $\|r_ga-ar_g\|<\varepsilon$ for all $a\in F$ and all $g\in G$;
\item[(iii)] $\|(\sum_{g\in G} r_g)a-a\|<\varepsilon$ for all $a\in F$.
\end{itemize}
The elements $r_g$, for $g\in G$, will be called {\it Rokhlin elements} for $\alpha$ for the choices of $\varepsilon$ and $F$.
\end{definition}
It was shown in \cite[Corollary 1]{SantiagoRP} that Definition \ref{def: Rokhlin} agrees with \cite[Definition 3.1]{Izumi-I} whenever
the C*-algebra $A$ is unital. It is also shown in \cite[Corollary 2]{SantiagoRP} that Definition \ref{def: Rokhlin} agrees with
\cite[Definition 3.1]{Nawata} whenever the C*-algebra $A$ is separable.
If $A$ is a C*-algebra, we denote by $\ell^\infty(\mathbb{N},A)$ the set of all bounded sequences $(a_n)_{n\in\mathbb{N}}$ in $A$ with the supremum norm $\|(a_n)_{n\in\mathbb{N}}\|=\sup\limits_{n\in\mathbb{N}}\|a_n\|$,
and pointwise operations. Then $\ell^\infty(\mathbb{N},A)$ is a C*-algebra, and it is unital when $A$ is (the unit being the constant sequence $1_A$). Let
$$c_0(\mathbb{N},A)=\left\{(a_n)_{n\in\mathbb{N}}\in\ell^\infty(\mathbb{N},A)\colon \lim\limits_{n\to\infty}\|a_n\|=0\right\}.$$
Then $c_0(\mathbb{N},A)$ is an ideal in $\ell^\infty(\mathbb{N},A)$, and we denote the quotient
$$\ell^\infty(\mathbb{N},A)/c_0(\mathbb{N},A)$$
by $A^\infty$, which we call the \emph{sequence algebra} of $A$.
Write $\pi_A\colon \ell^\infty(\mathbb{N},A)\to A^\infty$ for the quotient map, and identify $A$ with the subalgebra of $\ell^\infty(\mathbb{N},A)$
consisting of the constant sequences, and with the subalgebra of $A^\infty$ by taking its image under $\pi_A$. We write $A_\infty=A^\infty\cap A'$
for the relative commutant of $A$ inside of $A^\infty$, and call it the \emph{central sequence algebra} of $A$.
Let $G$ be a finite group and let $\alpha\colon G\to\mathrm{Aut}(A)$ be an action of $G$ on $A$. Then there are actions of $G$ on $A^\infty$ and
$A_\infty$ which, for simplicity and ease of notation, and unless confusion is likely to arise, we denote simply by $\alpha$.
The following is a characterization of the Rokhlin property in terms of elements of the sequence algebra $A^\infty$ (\cite[Proposition 1]{SantiagoRP}):
\begin{lemma}\label{lem: Rokhlin equivalence}
Let $A$ be a C*-algebra and let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a finite group $G$ on $A$. Then the following are equivalent:
\begin{itemize}
\item[(i)] $\alpha$ has the Rokhlin property.
\item[(ii)] For any finite subset $F\subseteq A$ there exist mutually orthogonal positive contractions $r_g$ in $A^\infty\cap F'$, for $g\in G$, such that
\begin{itemize}
\item[(a)] $\alpha_g(r_h)=r_{gh}$ for all $g,h\in G$;
\item[(b)] $(\sum_{g\in G} r_g)b=b$ for all $b\in F$.
\end{itemize}
\item[(iii)] For any separable C*-subalgebra $B\subseteq A$ there are orthogonal positive contractions $r_g$ in $A^\infty\cap B'$ for $g\in G$
such that
\begin{itemize}
\item[(a)] $\alpha_g(r_h)=r_{gh}$ for all $g,h\in G$;
\item[(b)] $(\sum_{g\in G} r_g)b=b$ for all $b\in B$.
\end{itemize}
\end{itemize}
\end{lemma}
The first part of the following proposition is \cite[Theorem 2 (i)]{SantiagoRP}. The second part follows trivially from the definition of the
Rokhlin property.
\begin{proposition}\label{Rp properties}
Let $G$ be a finite group, let $A$ be a C*-algebra, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be an action with the Rokhlin property.
\begin{enumerate}
\item If $B$ is any C*-algebra and $\beta\colon G\to\mathrm{Aut}(B)$ is any action of $G$ on $B$, then the action
$$\alpha\otimes\beta\colon G\to\mathrm{Aut}(A\otimes_{\mathrm{min}} B)$$
defined by $(\alpha\otimes\beta)_g=\alpha_g\otimes\beta_g$ for all $g\in G$, has the Rokhlin property.
\item If $B$ is a C*-algebra and $\varphi\colon A\to B$ is an isomorphism, then the action $g\mapsto \varphi\circ\alpha_g\circ\varphi^{-1}$ of $G$
on $B$ has the Rokhlin property.
\end{enumerate}
\end{proposition}
The following example may be regarded as the ``generating" Rokhlin action for a given finite group $G$. For some classes of C*-algebras, it can be shown that every action of $G$ with the Rokhlin property tensorially absorbs the action we construct below. See \cite[Theorems 3.4 and 3.5]{Izumi-II} and Theorem \ref{thm: Rp action absorbs model action} below.
\begin{example}\label{eg: model action} Let $G$ be a finite group. Let $\lambda\colon G\to U(\ell^2(G))$ be the left regular representation, and identify $\ell^2(G)$ with $\mathbb{C}^{|G|}$. Define an action $\mu^G\colon G\to \mathrm{Aut}(\mathrm M_{|G|^\infty})$ by
$$\mu^G_g=\bigotimes_{n=1}^\infty \mathrm{Ad}(\lambda_g)$$
for all $g\in G$. It is easy to check that $\alpha$ has the Rokhlin property. Note that $\mu^G_g$ is approximately inner for all $g\in G$. \end{example}
It follows from part (i) of Proposition \ref{Rp properties} that any action of the form $\alpha\otimes\mu^G$ has the Rokhlin property.
One of our main results, Theorem \ref{thm: Rp action absorbs model action}, states that in some circumstances, every action with the Rokhlin
property has this form.
\subsection{The category $\mathbf{Cu}$, the Cuntz semigroup, and the $\mathrm{Cu}^\sim$-semigroup}
In this subsection, we will recall the definitions of the Cuntz and $\mathrm{Cu}^\sim$ semigroups, as well as the category $\mathbf{Cu}$, to which these semigroups naturally belong.
\subsubsection{The category $\mathbf{Cu}$}
Let $S$ be an ordered semigroup and let $s,t\in S$. We say that $s$ is \emph{compactly contained} in $t$, and denote this by $s\ll t$,
if whenever $(t_n)_{n\in\mathbb{N}}$ is an increasing sequence in $S$ such that $t\le \sup\limits_{n\in\mathbb{N}} t_n$, there exists $k\in \mathbb{N}$ such that $s\le t_k$.
A sequence $(s_n)_{n\in\mathbb{N}}$ is said to be \emph{rapidly increasing} if $s_n\ll s_{n+1}$ for all $n\in \mathbb{N}$.
\begin{definition}\label{def: CCu} An ordered abelian semigroup $S$ is an object in the category $\mathbf{Cu}$ if it has a zero element and it satisfies the following properties:
\begin{itemize}
\item[(O1)] Every increasing sequence in $S$ has a supremum;
\item[(O2)] For every $s\in S$ there exists a rapidly increasing sequence $(s_n)_{n\in\mathbb{N}}$ in $S$ such that $s=\sup\limits_{n\in\mathbb{N}} s_n$.
\item[(O3)] If $(s_n)_{n\in\mathbb{N}}$ and $(t_n)_{n\in\mathbb{N}}$ are increasing sequences in $S$, then
$$\sup\limits_{n\in\mathbb{N}} s_n +\sup\limits_{n\in\mathbb{N}} t_n=\sup\limits_{n\in\mathbb{N}} (s_n+t_n);$$
\item[(O4)] If $s_1,s_2,t_1,t_2\in S$ are such that $s_1\ll t_1$ and $s_2\ll t_2$, then $s_1+s_2\ll t_1+t_2$.
\end{itemize}
Let $S$ and $T$ be semigroups in the category $\mathbf{Cu}$. An order-preserving semigroup map $\varphi\colon S\to T$ is a morphism in the category $\mathbf{Cu}$ if it preserves the zero element and it satisfies the following properties:
\begin{itemize}
\item[(M1)] If $(s_n)_{n\in\mathbb{N}}$ is an increasing sequence in $S$, then
$$\varphi\left(\sup\limits_{n\in\mathbb{N}} s_n\right)=\sup\limits_{n\in\mathbb{N}} \varphi(s_n);$$
\item[(M2)] If $s, t\in S$ are such that $s\ll t$, then $\varphi(s)\ll \varphi(t)$.
\end{itemize}
\end{definition}
It is shown in \cite[Theorem 2]{Coward-Elliott-Ivanescu} that the category $\mathbf{Cu}$ is closed under sequential inductive limits. The following description of inductive limits in the category $\mathbf{Cu}$ follows from the proof of this theorem.
\begin{proposition}\label{prop: inductivelimitCu}
Let $(S_n,\varphi_n)_{n\in\mathbb{N}}$, with $\varphi_n\colon S_n\to S_{n+1}$, be an inductive system in the category $\mathbf{Cu}$. For $m, n\in \mathbb{N}$
with $m\geq n$, let $\varphi_{n,m}\colon S_n\to S_{m+1}$ denote the composition $\varphi_{n,m}=\varphi_m\circ\cdots\circ\varphi_n$. A pair $(S,(\varphi_{n,\infty})_{n\in\mathbb{N}})$, consisting of a semigroup $S$ and morphisms $\varphi_{n,\infty}\colon S_n\to S$ in the category $\mathbf{Cu}$ satisfying $\varphi_{n+1,\infty}\circ\varphi_{n}=\varphi_{n,\infty}$ for all $n\in \mathbb{N}$, is the inductive limit of the system $(S_n,\varphi_n)_{n\in\mathbb{N}}$ if and only if:
\begin{itemize}
\item[(i)] For every $s\in S$ there exist elements $s_n\in S_n$ for $n\in\mathbb{N}$, such that $\varphi_n(s_n)\ll s_{n+1}$ for all $n\in \mathbb{N}$ and
$$s=\sup\limits_{n\in\mathbb{N}} \varphi_{n,\infty}(s_n);$$
\item[(ii)] Whenever $s,s',t\in S_n$ satisfy $\varphi_{n,\infty}(s)\le\varphi_{n,\infty}(t)$ and $s'\ll s$, there exists $m\ge n$ such that $\varphi_{n,m}(s')\le\varphi_{n,m}(t)$.
\end{itemize}
\end{proposition}
\begin{lemma}\label{lem: sup}
Let $S$ be a semigroup in $\mathbf{Cu}$, let $s$ be an element in $S$ and let $(s_n)_{n\in\mathbb{N}}$ be a rapidly increasing sequence in $S$ such that $s=\sup\limits_{n\in\mathbb{N}} s_n$. Let $T$ be a subset of $S$ such that every element of $T$ is the supremum of a rapidly increasing sequence of elements in $T$. Suppose that for every $n\in \mathbb{N}$ there is $t\in T$ such that $s_n\ll t\le s$. Then there exists an increasing sequence $(t_n)_{n\in\mathbb{N}}$ in $T$ such that $s=\sup\limits_{n\in\mathbb{N}} t_n$.
\end{lemma}
\begin{proof}
It is sufficient to construct an increasing sequence $(n_k)_{k\in\mathbb{N}}$ of natural numbers and a sequence $(t_k)_{k\in\mathbb{N}}$ in $T$ such
that $s_{n_k}\le t_k\le s_{n_{k+1}}$ for all $k\in \mathbb{N}$, since this implies that $s=\sup\limits_{k\in\mathbb{N}} t_k$.\\
\indent For $k=1$, set $n_1=1$ and $s_{n_1}=0$. Assume inductively that we have constructed $n_j$ and $t_j$ for all $j\le k$ and let us
construct $n_{k+1}$ and $t_{k+1}$. By the assumptions of the lemma, there exists $t\in T$ such that $s_{n_k}\ll t\le s$. Also by assumption,
$t$ is the supremum of a rapidly increasing sequence of elements of $T$. Hence there exists $t'\in T$ such that $s_{n_k}\le t'\ll s$. Use
that $s=\sup\limits_{n\in\mathbb{N}} s_n$ and $t'\ll s$, to choose $n_{k+1}\in\mathbb{N}$ with $n_{k+1}>n_k$ such that $t'\le s_{n_{k+1}}\ll s$. Set $t_{k+1}=t'$.
Then $s_{n_k}\le t_{k+1}\le s_{n_{k+1}}$. This completes the proof of the lemma.
\end{proof}
\begin{definition} \label{df: S gamma}
Let $S$ be a semigroup in the category $\mathbf{Cu}$. Let $I$ be a nonempty set and let $\gamma_i\colon S\to S$ for $i\in I$, be a family of
endomorphisms of $S$ in the category $\mathbf{Cu}$. We introduce the following notation:
$$S^\gamma=\left\{s\in S\colon \exists \ (s_t)_{t\in(0,1]} \mbox{ in } S \colon
\begin{aligned}
& s_r\ll s_{t} \mbox{ if } r<t,\, s_t=\sup\limits_{r<t} s_r \ \forall \ t\in (0,1],\\
& s_1=s, \mbox{ and } \gamma_i(s_t)=s_t \ \forall \ t\in (0, 1] \mbox{ and } \forall\ i\in I
\end{aligned}
\right\},$$
and
$$S^\gamma_\mathbb{N}=\left\{s\in S\colon \exists \ (s_n)_{n\in\mathbb{N}} \mbox{ in } S\colon
\begin{aligned}
& s_n\ll s_{n+1} \ \forall \ n\in\mathbb{N}, \, s=\sup\limits_{n\in\mathbb{N}} s_n,\\
& \mbox{ and } \gamma_i(s_n)=s_n \ \forall \ n\in\mathbb{N} \mbox{ and } \forall\ i\in I
\end{aligned}
\right\}.$$
\end{definition}
\begin{lemma}\label{lem: closure}
Let $S$ be a semigroup in the category $\mathbf{Cu}$. Let $I$ be a nonempty set and let $\gamma_i\colon S\to S$ for $i\in I$, be a family of
endomorphisms of $S$ in the category $\mathbf{Cu}$. Then
\begin{itemize}
\item[(i)] $S^\gamma_\mathbb{N}$ is closed under suprema of increasing sequences;
\item[(ii)] $S^\gamma$ is an object in $\mathbf{Cu}$.
\end{itemize}
\end{lemma}
\begin{proof} (i). Let $(s_n)_{n\in\mathbb{N}}$ be an increasing sequence in $S_{\mathbb{N}}^\gamma$. For each $n\in \mathbb{N}$, choose a rapidly increasing
sequence $(s_{n,m})_{m\in\mathbb{N}}$ in $S$ such that $s_n=\sup\limits_{m\in\mathbb{N}} s_{n,m}$ and $\gamma_i(s_{n,m})=s_{n,m}$ for all $i\in I$
and $m\in \mathbb{N}$. By the definition of the compact containment relation, there exist increasing sequences $(n_j)_{j\in\mathbb{N}}$ and $(m_j)_{j\in\mathbb{N}}$
in $\mathbb{N}$ such that $s_{k,l}\le s_{n_j,m_j}$ whenever $1\le k,l\le j$, and such that $( s_{n_j,m_j})_{j\in \mathbb{N}}$ is increasing. Let $s$ be the supremum of $(s_{n_j,m_j})_{j\in\mathbb{N}}$ in $S$. Then
$s\in S_\mathbb{N}^\gamma$, and it is straightforward to check, using a diagonal argument, that $s=\sup\limits_{n\in\mathbb{N}} s_n$, as desired.
(ii). It is clear that $S^\gamma$ satisfies O2, O3 and O4.
Now let us check that $S^\gamma$ satisfies axiom O1. Let $(s^{(n)})_{n\in\mathbb{N}}$ be an increasing sequence in $S^\gamma$ and let $s$ be its supremum in $S$. It is sufficient to show that $s\in S^\gamma$.
For each $n \in \mathbb{N}$, choose a path $(s_t^{(n)})_{t\in (0,1]}$ as in the definition of $S^\gamma$ for $s^{(n)}$.
Using that $s_t^{(n)}\ll s^{(n+1)}$ for all $n\in\mathbb{N}$ and all $t\in (0,1)$, together with a diagonal argument,
choose an increasing sequence $(t_n)_{n\in\mathbb{N}}$ in $(0,1]$ converging to $1$, such that
$$s_{t_n}^{(n)}\ll s_{t_{n+1}}^{(n+1)} \quad\forall \ n\in \mathbb{N}, \mbox{ and } s=\sup\limits_{n\in\mathbb{N}}s_{t_n}^{(n)}.$$
This implies, using the definition of the compact containment relation, that for each $n\in \mathbb{N}$ there exists $t_{n+1}'$ such that
$t_n<t_{n+1}'<t_{n+1}$ and
$$s_{t_n}^{(n)}\ll s_t^{(n+1)}\le s_{t_{n+1}}^{(n+1)} \mbox{ for all } t\in (t_{n+1}', t_{n+1}].$$
Choose an increasing function $f\colon (0,1]\to (0,1]$ such that
$$f\left(\left(1-\frac{1}{n}, 1-\frac{1}{n+1}\right]\right)=(t'_{n+1}, t_{n+1}]$$
for all $n\in \mathbb{N}$.
Define a path $(s_t)_{t\in (0,1]}$ in $S$ by taking $s_1=s$ and
$$s_t=s_{f(t)}^{(n+1)} \mbox{ for } t\in \left(1-\frac{1}{n}, 1-\frac{1}{n+1}\right].$$
Then $\gamma_i(s_t)=s_t$ for all $t\in (0,1]$ and all $i\in I$, so $s\in S^\gamma$. It is clear that this path satisfies the conditions in the definition of $S^\gamma$ for $s$.
\end{proof}
\subsubsection{The Cuntz semigroup}
Let $A$ be a C*-algebra and let $a, b\in A$ be
positive elements. We say that $a$ is \emph{Cuntz subequivalent} to $b$, and denote this by $a\precsim b$, if there exists a sequence $(d_n)_{n\in\mathbb{N}}$
in $A$ such that $\lim\limits_{n\to \infty} \|d_n^*bd_n- a\|=0$. We say that $a$ is \emph{Cuntz equivalent} to $b$, and denote this by $a\sim b$,
if $a\precsim b$ and $b\precsim a$. It is clear that $\precsim$ is a preorder relation in the set of positive elements of $A$, and thus
$\sim$ is an equivalence relation. We denote by $[a]$ the Cuntz equivalence class of the element $a\in A_+$.
The first conclusion of the following lemma was proved in \cite[Proposition 2.2]{Rordam} (see also \cite[Lemma 2.2]{Kirchberg-Rordam}).
The second statement was shown in \cite[Lemma 1]{Robert-Santiago}.
\begin{lemma}\label{lem: Cuntz relation}
Let $A$ be a C*-algebra and let $a$ and $b$ be positive elements in $A$ such that $\|a-b\|<\varepsilon$. Then $(a-\varepsilon)_+\precsim b$.
More generally, if $r$ is a non-negative real number, then $(a-r-\varepsilon)_+\precsim (b-r)_+$.
\end{lemma}
The Cuntz semigroup of $A$, denoted by $\mathrm{Cu}(A)$, is defined as the set of Cuntz equivalence classes of positive elements of $A\otimes \mathcal{K}$.
Addition in $\mathrm{Cu}(A)$ is given by
$$[a]+[b]=[a'+b'],$$
where $a',b'\in (A\otimes\mathcal{K})_+$ are orthogonal and satisfy $a'\sim a$ and $b'\sim b$. Furthermore, $\mathrm{Cu}(A)$ becomes an ordered semigroup when
equipped with the order $[a]\le [b]$ if $a\precsim b$. If $\phi\colon A\to B$ is a *-homomorphism, then $\phi$ induces an order-preserving
map $\mathrm{Cu}(\phi)\colon \mathrm{Cu}(A)\to \mathrm{Cu}(B)$, given by $\mathrm{Cu}(\phi)([a])=[(\phi\otimes \mathrm{id}_\mathcal{K})(a)]$ for every $a\in (A\otimes \mathcal{K})_+$.
\begin{remark}
Let $A$ be a C*-algebra, let $a\in A$ and let $\varepsilon>0$. It can be checked that $[(a-\varepsilon)_+]\ll [a]$ and that
$[a]=\sup\limits_{\varepsilon>0} [(a-\varepsilon)_+]$, thus showing that $\mathrm{Cu}(A)$ satisfies Axiom O2.
\end{remark}
It is shown in \cite[Theorem 1]{Coward-Elliott-Ivanescu} that $\mathrm{Cu}$ is a functor from the category of C*-algebras to the category $\mathbf{Cu}$.
\begin{lemma}\label{lem: morphism}
Let $A$ and $B$ be C*-algebras and let $\rho\colon \mathrm{Cu}(A)\to \mathrm{Cu}(B)$ be an order-preserving semigroup map. Suppose that for all $a\in (A\otimes\mathcal{K})_+$ one has
\begin{itemize}
\item[(i)] $\rho([a])=\sup\limits_{\varepsilon>0}\rho([(a-\varepsilon)_+])$,
\item[(ii)] $\rho([(a-\varepsilon)_+])\ll \rho ([a])$ for all $\varepsilon>0$.
\end{itemize}
Then $\rho$ is a morphism in the category $\mathbf{Cu}$; that is, it preserves suprema of increasing sequences and the compact containment relation.
\end{lemma}
\begin{proof}
We show first that $\rho$ preserves suprema of increasing sequences. Let $a$ be a positive element in $A\otimes\mathcal{K}$ and
let $(a_n)_{n\in\mathbb{N}}$ be an increasing sequence of positive elements in $A\otimes\mathcal{K}$ such that $\sup\limits_{n\in\mathbb{N}}[a_n]=[a]$.
Then $\rho([a_n])\le \rho([a])$ for all $n\in\mathbb{N}$. Suppose that $b\in (B\otimes\mathcal{K})_+$ is such that $\rho([a_n])\le [b]$ for
all $n\in \mathbb{N}$ and let $\varepsilon>0$. By the definition of the compact containment relation and the fact that $[(a-\varepsilon)_+]\ll [a]$,
there exists $n_0\in \mathbb{N}$ such that $[(a-\varepsilon)_+]\le [a_{n_0}]$. By applying $\rho$ to this inequality we get
$$\rho([(a-\varepsilon)_+])\le \rho([a_{n_0}])\le [b].$$
By taking supremum in $\varepsilon>0$ and applying (i) we get
$$\rho([a])=\sup\limits_{\varepsilon>0}\rho([(a-\varepsilon)_+])\le [b].$$
This shows that $\rho([a])$ is the supremum of $(\rho([a_n]))_{n\in \mathbb{N}}$, as desired.
We proceed to show that $\rho$ preserves the compact containment relation. Let $a$ and $b$ be positive elements in $A\otimes\mathcal{K}$ such
that $[a]\ll [b]$. Choose $\varepsilon>0$ such that $[a]\le [(b-\varepsilon)_+]\le [b]$. It follows that
$$\rho([a])\le \rho([(b-\varepsilon)_+])\le \rho([b]).$$
By (ii) applied to $[b]$ we get $\rho([a])\ll \rho([b])$, which concludes the proof.
\end{proof}
The following lemma is a restatement of \cite[Lemma 4]{Robert-Santiago}.
\begin{lemma}\label{lem: interpolation}
Let $A$ be a C*-algebra, let $(x_i)_{i=0}^n$ be elements of $\mathrm{Cu}(A)$ such that $x_{i+1}\ll x_i$ for all $i=0,\ldots,n$, and let $\varepsilon>0$.
Then there exists $a\in (A\otimes \mathcal{K})_+$ such that
\begin{align*}
x_n\ll [(a-(n-1)\varepsilon)_+]&\ll x_{n-1}\ll [(a-(n-2)\varepsilon)_+]\ll\cdots\\
\cdots\ll &x_3\ll [(a-2\varepsilon)_+]\ll x_2\ll [(a-\varepsilon)_+] \ll x_1\ll [a]=x_0.
\end{align*}
\end{lemma}
\subsubsection{The $\mathrm{Cu}^\sim$-semigroup}
Here we define the $\mathrm{Cu}^\sim$-semigroup of a C*-algebra. This semigroup was introduced in \cite{Robert} in order to classify certain inductive limits of 1-dimensional NCCW-complexes.
\begin{definition}\label{df: Cu sim}
Let $A$ be C*-algebra and let
$\pi\colon \widetilde{A} \to \widetilde{A}/A\cong\mathbb{C}$ denote the quotient map.
Then $\pi$ induces a semigroup homomorphism
$$\mathrm{Cu}(\pi)\colon \mathrm{Cu}(\widetilde{A})\to \mathrm{Cu}(\mathbb{C})\cong \overline{\mathbb{Z}_+}.$$
We define the semigroup $\mathrm{Cu}^\sim(A)$ by
\[
\mathrm{Cu}^\sim(A)=\{([a], n) \in \mathrm{Cu}(\widetilde{A})\times \mathbb{Z}_+ \mid \mathrm{Cu}(\pi)([a])=n\}/\sim,
\]
where $\sim$ is the equivalence relation defined by
\[
([a], n)\sim ([b],m) \quad\text{ if }\quad [a]+m[1]+k[1]= [b]+n[1]+k[1],
\]
for some $k\in \mathbb{N}$. The
image of the element $([a], n)$ under the canonical quotient map is denoted by $[a]-n[1]$.
Addition in $\mathrm{Cu}^\sim(A)$ is induced by pointwise addition in $\mathrm{Cu}(\widetilde{A})\times\mathbb{Z}_+$.
The semigroup $\mathrm{Cu}^\sim(A)$ can be endowed with an order: we say that $[a]-n[1]\le [b]-m[1]$ in $\mathrm{Cu}^\sim(A)$ if there exists $k$ in $\mathbb{Z}_+$ such that
$$[a]+(m+k)[1]\le [b]+(n+k)[1]$$
in $\mathrm{Cu}(\widetilde{A})$.
The assignment $A\mapsto \mathrm{Cu}^\sim(A)$ can be turned into a functor as follows.
Let $\phi\colon A\to B$ be a *-homomorphism and let $\widetilde{\phi}\colon \widetilde{A}\to \widetilde{B}$ denote the unital extension of
$\phi$ to the unitizations of $A$ and $B$. Let us denote by $\mathrm{Cu}^\sim(\phi)\colon \mathrm{Cu}^\sim(A)\to \mathrm{Cu}^\sim(B)$ the map defined by
$$\mathrm{Cu}^\sim(\phi)([a]-n[1])=\mathrm{Cu}(\widetilde{\phi})([a])-n[1].$$
It is clear that $\mathrm{Cu}^\sim(\phi)$ is order-preserving, and thus $\mathrm{Cu}^\sim$ becomes a functor from the category of C*-algebras to
the category of ordered semigroups.
\end{definition}
It was shown in \cite{Robert} that the $\mathrm{Cu}^\sim$-semigroup of a C*-algebra with stable rank one
belongs to the category $\mathbf{Cu}$, that $\mathrm{Cu}^\sim$ is a functor from the category of C*-algebras of
stable rank one to the category $\mathbf{Cu}$, and that it preserves inductive limits of sequences.
\section{Classification of actions and equivariant *-homomorphisms}
In this section we classify equivariant *-homomorphisms whose codomain C*-dynamical system have the Rokhlin property. We use this results to classify actions of finite groups on separable C*-algebras with the Rokhlin property.
Our results complement and extend those obtained by Izumi in \cite{Izumi-I} and \cite{Izumi-II} in the unital setting, and by Nawata in \cite{Nawata} for C*-algebras $A$ that satisfy $A\subseteq \overline{\mathrm{GL}(\widetilde{A})}$.
\subsection{Equivariant *-homomorphisms}
Let $A$ and $B$ be C*-algebras and let $G$ be a compact group. Let $\alpha\colon G\to \mathrm{Aut}(A)$ and $\beta\colon G\to \mathrm{Aut}(B)$ be (strongly continuous) actions. Recall that a *-homomorphism $\phi\colon A\to B$ is said to be \emph{equivariant} if $\phi\circ\alpha_g=\beta_g\circ\phi$ for all $g\in G$.
\begin{definition}\label{df: B^G}
Let $A$ and $B$ be C*-algebras and let $\alpha\colon G\to \mathrm{Aut}(A)$ and $\beta\colon G\to \mathrm{Aut}(B)$ be actions of a compact group $G$. Let $\phi, \psi\colon A\to B$ be equivariant *-homomorphisms. We say that $\phi$ and $\psi$ are \emph{equivariantly approximately unitarily equivalent}, and denote this by $\phi\sim_{\mathrm{G-au}}\psi$, if for any finite subset $F\subseteq A$ and for any $\varepsilon>0$ there exists a unitary $u\in \widetilde{B^\beta}$ such that
$$\|\phi(a)-u^*\psi(a)u\|<\varepsilon,$$
for all $a\in F$.
\end{definition}
Note that when $G$ is the trivial group, this definition agrees with the standard definition of approximate unitary equivalence of *-homomorphisms. In this case we will omit the group $G$ in the notation $\sim_{\mathrm{G-au}}$, and write simply $\sim_{\mathrm{au}}$.
The following lemma can be proved using a standard semiprojectivity argument. Its proof is left to the reader.
\begin{lemma}\label{lem: unitaries can be lifted}
Let $A$ be a unital C*-algebra and let $u$ be a unitary in $A_\infty$. Given $\varepsilon>0$ and given a finite subset $F\subseteq A$, there exists a unitary $v\in A$ such that $\|va-av\|<\varepsilon$ for all $a\in F$. If moreover $A$ is separable, then there exists a sequence $(u_n)_{n\in\mathbb{N}}$ of unitaries in $A$ with
$$\lim\limits_{n\to\infty} \|u_na-au_n\|=0$$
for all $a\in A$, such that $\pi_A((u_n)_{n\in\mathbb{N}})=u$ in $A_\infty$.
\end{lemma}
\begin{proposition}\label{prop: uniqueness}
Let $A$ and $B$ be C*-algebras and let $\alpha\colon G\to \mathrm{Aut}(A)$ and $\beta\colon G\to \mathrm{Aut}(B)$ be actions of a finite group $G$ such that $\beta$ has the Rokhlin property. Let $\phi,\psi\colon (A,\alpha) \to (B,\beta)$ be equivariant *-homomorphisms such that $\phi\sim_{\mathrm{au}}\psi$. Then $\phi\sim_{G-\mathrm{au}}\psi$.
\end{proposition}
\begin{proof}
Let $F$ be a finite subset of $A$ and let $\varepsilon>0$. We have to show that there exists a unitary $w\in \widetilde{B^\beta}$ such that
$$\|\phi(a)-w^*\psi(a)w\|<\varepsilon,$$
for all $a\in F$. Set $F'=\bigcup\limits_{g\in G}\alpha_g(F)$, which is again a finite subset of $A$. Since $\phi\sim_{\mathrm{au}}\psi$, there exists a unitary $u\in \widetilde{B}$ such that
\begin{equation}\label{au} \|\phi(b)-u^*\psi(b)u\|<\varepsilon\end{equation}
for all $b\in F'$. Choose $x\in B$ and $\lambda\in \mathbb{C}$ of modulus 1 such that $u=x+\lambda 1_{\widetilde{B}}$. Then equation (\ref{au}) above
is satisfied if one replaces $u$ with $\overline{\lambda}u$. Thus, we may assume that the unitary $u$ has the form $u=x+1_{\widetilde{B}}$ for some $x\in B$.
Fix $g\in G$ and $a\in F$. Then $b=\alpha_{g^{-1}}(a)$ belongs to $F'$. Using equation (\ref{au}) and the fact that
$\phi$ and $\psi$ are equivariant, we get
$$\|\beta_{g^{-1}}(\phi(a))-u^*\beta_{g^{-1}}(\psi(a))u\|<\varepsilon.$$
By applying $\beta_g$ to the inequality above, we conclude that
$$\|\phi(a)-\beta_g(u)^*\psi(a)\beta_g(u)\|<\varepsilon$$
for all $a\in F$ and $g\in G$
Choose positive orthogonal contractions $(r_g)_{g\in G}\subseteq B_\infty$ as in the definition of the Rokhlin property for $\beta$, and set $v=\sum\limits_{g\in G}\beta_g(x)r_g+1_{\widetilde{B}}$. Using that $x_g+1_{\widetilde{B}}$ is a unitary in $\widetilde{B}$, one checks that
\begin{align*}
& v^*v=\sum\limits_{g\in G} \left(\beta_g(x^*x)r_g^2+\beta_g(x)r_g+\beta_g(x)r_g\right)+1_{\widetilde{B}}=1_{\widetilde{B}}.
\end{align*}
Analogously, we have $vv^*=1_{\widetilde{B}}$, and hence $v$ is a unitary in $\widetilde{B}$. For every $b\in B$, we have
$$v^*bv=\sum\limits_{g\in G}r_g\beta_g(u)^*b\beta_g(u).$$
Therefore,
$$\|\phi(a)-v^*\psi(a)v\|=\left\|\sum\limits_{g\in G}r_g\phi(a)-\sum\limits_{g\in G}r_g\beta_g(u)^*\psi(a)\beta_g(u)\right\|<\varepsilon,$$
for all $a\in F$ (here we are considering $\phi$ and $\psi$ as maps from $A$ to $(\widetilde B)^\infty$, by composing them with the natural inclusion of $B$ in $(\widetilde B)^\infty$). Since $v=\sum\limits_{g\in G}\beta_g(xr_e)+1_{\widetilde{B}}$, we have $v\in (\widetilde{B^\beta})^\infty\subseteq(\widetilde{B})^\infty$. By Lemma \ref{lem: unitaries can be lifted}, we can choose a unitary $w\in \widetilde{B^\beta}$ such that
$$\|\phi(a)-w^*\psi(a)w\|<\varepsilon,$$
for all $a\in F$, and the proof is finished.
\end{proof}
\begin{lemma}\label{lem: limit of homomorphisms}
Let $A$ and $B$ be C*-algebras and let $\psi\colon A\to B$ be a *-homomorphism. Suppose there exists a sequence $(v_n)_{n\in\mathbb{N}}$ of unitaries in $\widetilde{B}$ such that the sequence $(v_n\phi(x)v_n^*)_{n\in\mathbb{N}}$ converges in $B$ for all $x$ in a dense subset of $A$. Then there exists a *-homomorphism $\psi\colon A\to B$ such that
$$\lim\limits_{n\to\infty} v_n\phi(x)v_n^*=\psi(x)$$
for all $x\in A$.
\end{lemma}
\begin{proof} Let
$$S=\{x\in A\colon (v_n\phi(x) v_n^*)_{n\in\mathbb{N}}\mbox{ converges in }B\}\subseteq A.$$
Then $S$ is a dense *-subalgebra of $A$. For each $x\in S$, denote by $\psi_0(x)$ the limit of the sequence $(v_n\phi(x)v_n^*)_{n\in\mathbb{N}}$. The map $\psi_0\colon S\to B$ is linear, multiplicative, preserves the adjoint operation, and is bounded by $\|\phi\|$, so it extends by continuity to a *-homomorphism $\psi \colon A\to B$. Given $a\in A$ and given $\varepsilon>0$, use density of $S$ in $A$ to choose $x\in S$ such that $\|a-x\|<\frac{\varepsilon}{3}$. Choose $N\in\mathbb{N}$ such that $\|v_N\phi(x)v_N^*-\psi(x)\|<\frac{\varepsilon}{3}$. Then
\begin{align*} \|\psi(a)-v_N\phi(a)v_N^*\|&\leq \|\psi(a-x)\|+\|\psi(x)-v_N\phi(x)v_N^*\|+\|v_N\phi(x)v_N^*-v_N\phi(a)v_N^*\|\\
&< \frac{\varepsilon}{3} + \frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\varepsilon,\end{align*}
It follows that $\psi(a)=\lim\limits_{n\to\infty} v_n\phi(a)v_n^*$ for all $a\in A$, as desired.
\end{proof}
The unital case of the following proposition is \cite[Lemma 5.1]{Izumi-I}. Our proof for arbitrary C*-dynamical systems follows similar ideas.
\begin{proposition}\label{prop: existence}
Let $A$ and $B$ be C*-algebras and let $\alpha\colon G\to \mathrm{Aut}(A)$ and $\beta\colon G\to \mathrm{Aut}(B)$ be actions of a finite group $G$. Suppose that $A$ is separable and that $\beta$ has the Rokhlin property. Let $\phi\colon A\to B$ be a *-homomorphism such that $\beta_g\circ \phi\sim_{\mathrm{au}}\phi\circ \alpha_g$ for all $g\in G$. Then:
\begin{itemize}
\item[(i)] For any $\varepsilon>0$ and for any finite set $F\subseteq A$ there exists a unitary $u\in \widetilde{B}$ such that
\begin{align}\label{eq: phiphi}
\begin{aligned}
&\|(\beta_g\circ\mathrm{Ad}(w)\circ \phi)(x)-(\mathrm{Ad}(w)\circ\phi\circ \alpha_g)(x)\|<\varepsilon, \quad \forall g\in G,\, \forall x \in F,\\
&\|(\mathrm{Ad}(w)\circ\phi)(x)-\phi(x)\|<\varepsilon+\sup\limits_{g\in G}\|(\beta_g\circ\phi\circ\alpha_{g^{-1}})(x)-\phi(x)\|, \quad \forall x\in F.
\end{aligned}
\end{align}
\item[(ii)] There exists an equivariant *-homomorphism $\psi\colon A\to B$ that is approximately unitarily equivalent to $\phi$.
\end{itemize}
\end{proposition}
\begin{proof}
(i) Let $F$ be a finite subset of $A$ and let $\varepsilon>0$. Set $F'=\bigcup\limits_{g\in G}\alpha_g(F)$, which is a finite subset of $A$. Since $\beta_g\circ \phi\sim_{\mathrm{au}}\phi\circ\alpha_g$ for all $g\in G$, there exist unitaries $(u_g)_{g\in G}\subseteq \widetilde{B}$ such that
$$\left\|(\beta_g\circ\phi)(a)-(\mathrm{Ad}(u_g)\circ\phi\circ \alpha_g)(a)\right\|<\frac{\varepsilon}{2},$$
for all $a\in F'$ and $g\in G$. Upon replacing $u_g$ with a scalar multiple of it, one can assume that there are $(x_g)_{g\in G}\subseteq B$ such that $u_g=x_g+1_{\widetilde{B}}$ for all $g\in G$. For $a\in F$ and $g, h\in G$, we have
\begin{align*}
\|(\mathrm{Ad}(u_g)&\circ\phi\circ\alpha_h)(a)-(\beta_h\circ\mathrm{Ad}(u_{h^{-1}g})\circ\phi)(a)\| \\
&=\left\|(\mathrm{Ad}(u_g)\circ\phi\circ\alpha_g)(\alpha_{g^{-1}h}(a))-(\beta_h\circ\mathrm{Ad}(u_{h^{-1}g})\circ\phi\circ\alpha_{h^{-1}g})(\alpha_{g^{-1}h}(a))\right\|\\
&\le \left\|(\mathrm{Ad}(u_g)\circ\phi\circ\alpha_g)(\alpha_{g^{-1}h}(a))-(\beta_g\circ\phi)(\alpha_{g^{-1}h}(a))\right\|\\
& \ \ \ \ +\left\|(\beta_g\circ\phi)(\alpha_{g^{-1}h}(x))-(\beta_h\circ\mathrm{Ad}(u_{h^{-1}g})\circ\phi\circ\alpha_{h^{-1}g})(\alpha_{g^{-1}h}(x))\right\|\\
&\le \frac\varepsilon 2+\frac\varepsilon 2=\varepsilon.
\end{align*}
Choose positive orthogonal contractions $(r_g)_{g\in G}\subseteq B_\infty$ as in the definition of the Rokhlin property for $\beta$, and set
$$u=\sum\limits_{g\in G}r_gx_g+1_{\widetilde{B}}\in (\widetilde{B})^\infty.$$
Using that $x_g+1_{\widetilde{B}}$ is a unitary in $\widetilde{B}$, one checks that
$$u^*u=1_{\widetilde{B}}+\sum\limits_{g\in G}(r_g^2x_g^*x_g+r_gx_g+r_gx_g^*)=1_{\widetilde{B}}.$$
Analogously, one also checks that $uu^*=1_{\widetilde{B}}$, thus showing that $u$ is a unitary in $(\widetilde{B})^\infty$. The map $\mathrm{Ad}(u)$ can be written
in terms of the maps $\mathrm{Ad}(u_g)$ and the contractions $(r_g)_{g\in G}$, as follows:
$$(\mathrm{Ad}(u))(x)=uxu^*=\sum\limits_{g\in G}(u_gxu_g^*) r_g=\sum\limits_{g\in G}(\mathrm{Ad}(u_g))(x)r_g,$$
for all $x\in A$. Now for $a\in F$ and considering $\phi$ as a map from $A$ to $(\widetilde B)^\infty$ by composing it with the natural inclusion of $B$ in $(\widetilde B)^\infty$, we have the following identities
\begin{align*}
&(\beta_h\circ\mathrm{Ad}(u)\circ\phi)(a)=\sum\limits_{g\in G}r_{hg}(\beta_h\circ\mathrm{Ad}(u_g)\circ\phi)(a)=\sum\limits_{g\in G}r_g(\beta_h\circ\mathrm{Ad}(u_{h^{-1}g})\circ\phi)(a),\\
& (\mathrm{Ad}(u)\circ\phi\circ\alpha_h)(a)=\sum\limits_{g\in G}r_g(\mathrm{Ad}(u_g)\circ\phi\circ\alpha_h)(a).
\end{align*}
Therefore,
\begin{align*}
\|(\beta_h\circ\mathrm{Ad}(u)\circ\phi)(a) &- (\mathrm{Ad}(u)\circ\phi\circ\alpha_h)(a)\|\\
& \le \sup\limits_{g\in G}\left\|(\mathrm{Ad}(u_g)\circ\phi\circ\alpha_h)(a)-(\beta_h\circ\mathrm{Ad}(u_{h^{-1}g})\circ\phi)(a)\right\|<\varepsilon.\end{align*}
This in turn implies that
\begin{align*}
&\|(\mathrm{Ad}(u)\circ\phi)(a)-\phi(a)\|=\left\|\sum\limits_{g\in G}r_g((\mathrm{Ad}(u_g)\circ\phi)(a)-\phi(a))\right\|\\
&\le \sup\limits_{g\in G}\left\|(\mathrm{Ad}(u_g)\circ\phi)(a)-\phi(a)\right\|\\
&\le\sup\limits_{g\in G} \left(\left\|(\mathrm{Ad}(u_g)\circ\phi\circ\alpha_g)(\alpha_{g^{-1}}(a))-(\beta_g\circ\phi)(\alpha_{g^{-1}}(a))\right\|+\left\|(\beta_g\circ\phi\circ\alpha_{g^{-1}})(a)-\phi(a)\right\|\right)\\
&\le \varepsilon + \sup\limits_{g\in G}\left\|(\beta_g\circ\phi\circ\alpha_{g^{-1}})(a)-\phi(a)\right\|.
\end{align*}
We have shown that the inequalities in \eqref{eq: phiphi} hold for a unitary $u\in (\widetilde{B})^\infty$. By Lemma \ref{lem: unitaries can be lifted}, we can replace $u$ with a unitary in $w\in \widetilde{B}$ in such a way that both inequalities still hold for $w$ in place of $u$.
(ii) Let $(F_n)_{n\in \mathbb{N}}$ be an increasing sequence of finite subsets of $A$ whose union is dense in $A$. Upon replacing each $F_n$ with $\bigcup\limits_{g\in G}\alpha_g(F_n)$, we may assume that $\alpha_g(F_n)=F_n$ for all $g\in G$ and $n\in \mathbb{N}$. Set $\phi_1=\phi$ and find a unitary $u_1\in \widetilde{B}$ such that the conclusion of the first part of the proposition is satisfied with $\phi_1$ and $\varepsilon=1$. Set $\phi_2=\mathrm{Ad}(u_1)\circ \phi_1$, and find a unitary $u_2\in \widetilde{B}$ such that the conclusion of the first part of the proposition is satisfied with $\phi_2$ and $\varepsilon=\frac{1}{2}$. Iterating this process, there exist *-homomorphisms $\phi_n\colon A\to B$ with $\phi_1=\phi$ and unitaries $(u_n)_{n\in \mathbb{N}}$ in $\widetilde{B}$ such that $\phi_{n+1}=\mathrm{Ad}(u_n)\circ \phi_{n}$, for all $n\in \mathbb{N}$, which moreover for all $n\in \mathbb{N}$ satisfy
\begin{align*}
&\|(\beta_g\circ\phi_n)(x)-(\phi_n\circ\alpha_g)(x)\|<\frac{1}{2^n}\end{align*}
for all $g\in G$ and for all $x\in F_n$, and
\begin{align*}
\|\phi_{n+1}(x)-\phi_{n}(x)\|<\frac{3}{2^n}
\end{align*}
for all $x\in F_n$. For each $n\in \mathbb{N}$ set $v_n=u_n\cdots u_1$. Then the sequence of unitaries $(v_n)_{n\in\mathbb{N}}$ in $\widetilde{B}$ and the *-homomorphism
$\phi\colon A\to B$ satisfy the hypotheses of Lemma \ref{lem: limit of homomorphisms}, so it follows that the sequence $(\phi_n)_{n\in \mathbb{N}}$
converges to a *-homomorphism $\psi\colon A\to B$ that satisfies $\beta_g\circ\psi=\psi\circ\alpha_g$ for all $g\in G$; that is, $\psi$ is equivariant. Since each $\phi_n$ is unitarily equivalent to $\phi$, we conclude that $\phi$ and $\psi$ are approximately unitarily equivalent.
\end{proof}
\subsection{Categories of C*-dynamical systems and abstract classification}
Let $G$ be a second countable compact group and let $\mathbf A$ denote the category of separable C*-algebras. Let us denote by $\mathbf A_G $ the category whose objects are $G$-C*-dynamical systems $(A, \alpha)$, that is, $A$ is a C*-algebra and $\alpha\colon G\to \mathrm{Aut}(A)$ is a strongly continuous action, and whose morphisms are equivariant *-homomorphisms. We use the notation $\phi\colon (A,\alpha)\to (B,\beta)$ to denote equivariant *-homomorphisms $\phi\colon A\to B$. Approximate unitary equivalence of maps in this category is given in Definition \ref{df: B^G}.
If $\mathbf{B}$ is a subcategory of $\mathbf A$, we denote by $\mathbf B_G $ the full subcategory of $\mathbf A_G $ whose objects are C*-dynamical systems $(A,\alpha)$ with $A$ in $\mathbf B$, and whose morphisms are given by
$$\mathrm{Hom}_{\mathbf B_G }((A,\alpha),(B,\beta))=\mathrm{Hom}_{\mathbf A_G }((A,\alpha),(B,\beta)).$$
\begin{definition}
Let $\mathbf B$ be a subcategory of $\mathbf A$. Let $\mathrm F\colon \mathbf B_G \to \mathbf C$ be a functor from the category $\mathbf B_G$ to a category $\mathbf C$.
We say that the functor $\mathrm F$ {\it classifies homomorphisms} if:
\begin{itemize}
\item[(a)] For every pair of objects $(A,\alpha)$ and $(B,\beta)$ in $\mathbf B_G $ and for every morphism
$$\lambda\colon \mathrm F(A, \alpha)\to \mathrm F(B,\beta)$$
in $\mathbf C$, there exists a homomorphism $\phi\colon (A, \alpha)\to (B,\beta)$ in $\mathbf B_G $ such that $\mathrm F(\phi)=\lambda$.
\item[(b)] For every pair of objects $(A,\alpha)$ and $(B,\beta)$ in $\mathbf B_G $ and every pair of homomorphisms
$$\phi,\psi\colon (A, \alpha)\to (B,\beta),$$
one has $\mathrm F(\phi)=\mathrm F(\psi)$ if and only if $\phi\sim_{\mathrm{G-au}}\psi$.
\end{itemize}
We say that the functor $\mathrm F$ \emph{classifies isomorphisms} if it satisfies (a) and (b) above for ismorphisms instead of homomorphisms (such a functor is a \emph{strong classifying functor} in the sense of Elliott (see \cite{Elliott})).
\end{definition}
Let $\mathbf C_1$ and $\mathbf C_2$ be two categories. Recall that a functor $\mathrm F\colon \mathbf C_1\to \mathbf C_2$ is said to be \emph{sequentially continuous} if whenever $C=\varinjlim (C_n,\theta_n)$ in $\mathbf C_1$ for some sequential direct system $(C_n,\theta_n)_{n\in\mathbb{N}}$ in $\mathbf C_1$, then the inductive limit $\varinjlim (\mathrm F(C_n),\mathrm F(\theta_n))$ exists in $\mathbf C_2$, and one has
$$\mathrm F(\varinjlim (C_n,\theta_n))=\varinjlim (\mathrm F(C_n),\mathrm F(\theta_n)).$$
The following theorem is a consequence of \cite[Theorem 3]{Elliott}.
\begin{theorem}\label{thm: isomorphism}
Let $G$ be a second countable compact group, let $\mathbf B$ be a subcategory of $\mathbf A$, let $\mathbf B_G$ be the associated category of C*-dynamical systems, and
let $\mathbf C$ be a category in which inductive limits of sequences exist.
Let $\mathrm F\colon \mathbf B_G \to \mathbf C$ be a sequentially continuous functor that classifies homomorphisms. Then $\mathrm F$ classifies isomorphisms.
\end{theorem}
\begin{proof}
Let us briefly see that the conditions of \cite[Theorem 3]{Elliott} are satisfied for the category $\mathbf B_G$. First, using that the algebras in $\mathbf B_G$ are separable and that the group is second countable we can see that the set of equivariant *-homomorphisms between two C*-algebras in $\mathbf B_G$ is metrizable. Also, by taking the inner automorphisms of a C*-dynamical systems in $\mathbf B_G$ to be conjugation by unitaries in the unitization of the fixed point algebra of the given dynamical system, one can easily see that these automorphisms satisfy the conditions of \cite[Theorem 3]{Elliott}. Finally note that the category $\mathbf D$ whose objects are objects of $\mathbf C$ of the form $\mathrm F(A,\alpha)$ for some C*-dynamical system $(A, \alpha)$ in $\mathbf B_G$, and whose morphisms between two objects $\mathrm F(A,\alpha)$ and $\mathrm F(B,\alpha)$ are all the maps of the form $\mathrm F(\phi)$ for some equivariant *-homomorphism $\phi\colon (A, \alpha)\to (B, \beta)$, is just the classifying category of $B_G$ (in the sense of \cite{Elliott}), since $\mathrm F$ classifies homomorphisms by assumption. Therefore, by \cite[Theorem 3]{Elliott} the functor $\mathrm F$ is a strong classifying functor; in other words, it classifies *-isomorphisms.
\end{proof}
\begin{definition}\label{df: C^G, RB^G}
Let $G$ be a compact group. Let $\mathbf C$ be a category and let $\mathbf C_G $ denote the category whose objects are pairs $(C,\gamma)$,
where $C$ is an object in $\mathbf C$ and $\gamma\colon G\to \mathrm{Aut}(C)$ is a group homomorphism, also called an action of $G$ on $C$.
(We do not require any kind of continuity for this action since $C$ does not a priori have a topology.) The morphisms of $\mathbf C_G $ consist of
the morphisms of $\mathbf{C}$ that are equivariant.
Let $\mathbf B$ be a subcategory of $\mathbf A$ and let $\mathbf B_G$ be the associated category of C*-dynamical systems. Let $\mathrm F \colon \mathbf{B} \to \mathbf C$ be a functor.
Then $\mathrm F$ induces a functor $\mathrm F_G\colon \mathbf B_G \to \mathbf C_G $ as follows:
\begin{itemize}
\item[(i)] For an object $(A,\alpha)$ in $\mathbf B_G $, define an action $\mathrm F(\alpha)\colon G\to \mathrm{Aut}(\mathrm F(A))$ by
$(\mathrm F(\alpha))_g=\mathrm F(\alpha_g)$ for all $g\in G$. We then set $\mathrm F_G (A, \alpha)=(\mathrm F(A),\mathrm F(\alpha))$;
\item[(ii)] For a morphism $\phi\in \mathrm{Hom}_{\mathbf B_G }((A,\alpha),(B,\beta))$, we set $\mathrm F_G(\phi)= \mathrm F (\phi)$.
\end{itemize}
If $G$ is a finite group, we let $\mathbf{RB}_G$ denote the subcategory of $\mathbf B_G $ consisting of those C*-dynamical systems $(A,\alpha)$ in $\mathbf B_G $ with the Rokhlin property.
\end{definition}
The next theorem is a restatement, in the categorical setting, of Proposition \ref{prop: uniqueness} and Proposition \ref{prop: existence} (ii).
\begin{theorem}\label{thm: mainclassification}
Let $G$ be a finite group. Let $\mathbf B$, $\mathbf B_G$, $\mathbf{RB}_G$, $\mathbf C$, and $\mathbf C_G $ be as in Definition \ref{df: C^G, RB^G}. Let $\mathrm F\colon \mathbf B\to \mathbf C$ be a functor that classifies homomorphisms.
\begin{itemize}
\item[(i)] Let $(A, \alpha)$ be an object in $\mathbf B_G $ and let $(B,\beta)$ be an object in $\mathbf{RB}_G$.
\begin{itemize}
\item[(a)] For every morphism $\gamma\colon (\mathrm F(A), \mathrm F(\alpha))\to (\mathrm F(B), \mathrm F(\beta))$ in $\mathbf C_G $, there exists a morphism $\phi\colon (A, \alpha)\to(B, \beta)$ in $\mathbf B_G $ such that $\mathrm F_G(\phi)=\gamma$.
\item[(b)] If $\phi, \psi\colon (A, \alpha)\to(B, \beta)$ are morphisms in $\mathbf B_G $ such that $\mathrm F_G(\phi)=\mathrm F_G(\psi)$, then $\phi\sim_{G-\mathrm{au}}\psi$.
\end{itemize}
\item[(ii)] The restriction of the functor $\mathrm F_G$ to $\mathbf{RB}_G$ classifies homomorphisms.
\end{itemize}
\end{theorem}
\begin{proof}
(i) Let $(A, \alpha)$ be an object in $\mathbf B_G $ and let $(B,\beta)$ be an object in $\mathbf{RB}_G$.
(a) Let $\gamma\colon (\mathrm F(A), \mathrm F(\alpha))\to (\mathrm F(B), \mathrm F(\beta))$ be a morphism in $\mathbf C_G $. Using that $\mathrm F\colon \mathbf B\to \mathbf C$ classifies homomorphisms, choose a *-homomorphism $\psi\colon A\to B$ such that $\mathrm F(\psi)=\gamma$. Note that
$$\mathrm F(\psi\circ\alpha_g)=\mathrm F(\psi)\circ\mathrm F(\alpha_g)=\mathrm F(\beta_g)\circ \mathrm F(\psi)=\mathrm F(\beta_g\circ\psi),$$
for all $g\in G$. Using again that $\mathrm F$ classifies homomorphisms, we conclude that $\psi\circ\alpha_g$ and $\beta_g\circ\psi$ are approximately unitarily equivalent for all $g\in G$. Therefore, by part (ii) of Proposition \ref{prop: existence} there exists an equivariant *-homomorphism $\phi\colon (A,\alpha)\to (B,\beta)$ such that $\phi$ and $\psi$ are approximately unitarily equivalent. Thus $\phi$ is a morphism in $\mathbf B_G $ and
$$\mathrm F_G(\phi)=\mathrm F(\phi)=\mathrm F(\psi)=\gamma,$$
as desired.
(b) Let $\phi, \psi\colon (A,\alpha)\to(B,\beta)$ be morphisms in $\mathbf B_G $ such that $\mathrm F_G(\phi)=\mathrm F_G(\psi)$. Then $\phi\sim_{\mathrm{au}}\psi$ because $\mathrm F$ classifies homomorphisms and $\mathrm F$ agrees with $\mathrm F_G$ on morphisms. It then follows from Proposition \ref{prop: uniqueness} that $\phi\sim_{G-\mathrm{au}}\psi$.
Part (ii) clearly follows from (i).
\end{proof}
\begin{lemma}\label{lem: categories}
Let $G$ be a compact group, let $\Lambda$ be a directed set and let $\mathbf C$ be a category where inductive limits over $\Lambda$ exist.
Let $\mathbf C_G $ be the associated category as in Definition \ref{df: C^G, RB^G}. Then:
\begin{itemize}
\item[(i)] Inductive limits over $\Lambda$ exist in $\mathbf C_G $.
\item[(ii)] If $\mathbf D$ is a category where inductive limits over $\Lambda$ exist and $\mathrm F\colon \mathbf C\to \mathbf D$ is a functor that preserves direct limits over $\Lambda$, then the associated functor $\mathrm F_G\colon \mathbf C_G \to \mathbf D_G $ also preserves direct limits over $\Lambda$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) Let $((C_\lambda,\alpha_\lambda)_{\lambda\in\Lambda}, (\gamma_{\lambda,\mu})_{\lambda,\mu\in\Lambda, \lambda<\mu})$ be a direct system in $\mathbf C_G $ over $\Lambda$, where $\gamma_{\lambda,\mu}\colon (C_\lambda,\alpha_\lambda)\to (C_\mu,\alpha_\mu, )$, for $\lambda<\mu$, is a morphism in $\mathbf C_G $. Let $(C,(\gamma_{\lambda,\infty})_{\lambda\in\Lambda})$, with $\gamma_{\lambda,\infty}\colon C_\lambda\to C$, be its direct limit in the category $\mathbf C$. Then
$$(\gamma_{\mu,\infty}\circ\alpha_\mu(g))\circ \gamma_{\lambda,\mu}=\gamma_{\lambda,\infty}\circ\alpha_\lambda(g)$$
for all $\mu\in \Lambda$ with $\lambda<\mu$. Hence, by the universal property of the inductive limit $(C,(\gamma_{\lambda,\infty})_{\lambda\in\Lambda})$, there exists a unique $\mathbf C$-morphism $\alpha(g)\colon C\to C$ that satisfies $\alpha(g)\circ \gamma_{\lambda,\infty}=\gamma_{\lambda,\infty}\circ \alpha_\lambda(g)$ for all $\lambda\in \Lambda$. Note that for $g,h\in G$, one has
$$(\alpha(g)\circ\alpha(h))\circ \gamma_{\lambda,\infty}= \gamma_{\lambda,\infty}\circ\alpha_\lambda(g)\circ\alpha(h)=\gamma_{\lambda,\infty}\circ\alpha_\lambda(gh)$$
for all $\lambda\in\Lambda$. By uniqueness of the morphism $\alpha(gh)$, it follows that $\alpha(g)\circ\alpha(h)=\alpha(gh)$ for all $g, h\in G$. This implies that $\alpha(g)$ is an automorphism of $C$ and that $\alpha\colon G\to \mathrm{Aut}(C)$ is an action. Thus $(C, \alpha)$ is an object in $\mathbf C_G $.
We claim that $(C, \alpha)$ is the inductive limit of $((C_\lambda,\alpha_\lambda)_{\lambda\in\Lambda}, (\gamma_{\lambda,\mu})_{\lambda,\mu\in\Lambda, \lambda<\mu})$ in the category $\mathbf C_G $. For $\lambda\in \Lambda$, The map $\gamma_{\lambda,\infty}$ is equivariant since $\gamma_{\lambda,\infty}\circ\alpha_\lambda(g)=\alpha(g)\circ\gamma_{\lambda,\infty}$ for all $g\in G$ and $\lambda\in\Lambda$. Let $(D, \beta)$ be an object in $\mathbf C_G $ and for $\lambda\in\Lambda$, let $\rho_\lambda\colon (C_\lambda,\alpha_\lambda)\to (D, \beta)$ be an equivariant morphism. By the universal property of the inductive limit $C$, there exists a unique morphism $\rho\colon C\to D$ satisfying $\rho_\lambda=\rho\circ \gamma_{\lambda,\infty}$ for all $\lambda\in\Lambda$. We therefore have
\begin{align*}
(\beta(g)^{-1}\circ \rho\circ\alpha(g))\circ \gamma_{\lambda,\infty}&=\beta^{-1}(g)\circ \rho\circ \gamma_{\lambda,\infty}\circ \alpha_\lambda(g)\\
&=\beta^{-1}(g)\circ\rho_{\lambda,\infty}\circ\alpha_\lambda(g)\\
&=\rho_{\lambda,\infty},
\end{align*}
for all $g\in G$ and $\lambda\in\Lambda$. Hence by uniqueness of $\rho$, we conclude that
$$\beta^{-1}(g)\circ \rho\circ \alpha(g)=\rho$$
for all $g\in G$. In other words, $\rho$ is equivariant. We have shown that $(C, \alpha)$ has the universal property of the inductive limit in $\mathbf C_G $, thus proving the claim and part (i).
(ii) Let $((C_\lambda,\alpha_\lambda)_{\lambda\in\Lambda}, (\gamma_{\lambda,\mu})_{\lambda,\mu\in\Lambda, \lambda<\mu})$ be a direct system in $\mathbf C_G $ and let $(C, \alpha)$ be its inductive limit in $\mathbf C_G $, which exists by the first part of this lemma. We claim that $(\mathrm F(C), \mathrm F(\alpha))$ is the inductive limit of
$$\left((\mathrm F(C_\lambda),\mathrm F(\alpha_\lambda))_{\lambda\in\Lambda}, (\mathrm F(\gamma_{\lambda,\mu}))_{\lambda,\mu\in\Lambda, \lambda<\mu}\right)$$
in the category $\mathbf D_G $. Let $(D, \delta)$ be an object in $\mathbf D_G $ and for $\lambda\in\Lambda$, let
$\rho_\lambda\colon (\mathrm F(C_\lambda),\mathrm F(\alpha_\lambda))\to (D, \delta)$
be an equivariant morphism satisfying $\rho_\mu=\mathrm F(\gamma_{\lambda,\mu})\circ \rho_\lambda$ for all $\mu\in\Lambda$ with $\lambda<\mu$. Since $\mathrm F$ is continuous by assumption, we have
$$\mathrm F(C)=\varinjlim \left((\mathrm F(C_\lambda))_{\lambda\in\Lambda}, (\mathrm F(\gamma_{\lambda,\mu}))_{\lambda,\mu\in\Lambda, \lambda<\mu}\right)$$
in $\mathbf D$. By the universal property of the inductive limit $\mathrm F(C)$ in $\mathbf D$, there exits a unique morphism $\rho\colon \mathrm F(C)\to D$ in the category $\mathbf D$ satisfying $\rho\circ \mathrm F(\gamma_{\lambda,\infty})=\rho_\lambda$. It follows that
$$(\delta(g)^{-1}\circ \rho\circ \mathrm F(\alpha(g)))\circ \mathrm F(\gamma_{\lambda,\infty})=\delta(g)^{-1}\circ \rho_\lambda\circ \mathrm F(\alpha_\lambda(g))=\rho_\lambda,$$
for all $g\in G$ and $\lambda\in\Lambda$. By the uniqueness of the morphism $\rho$, we conclude that $\delta(g)^{-1}\circ \rho\circ \mathrm F(\alpha)(g)=\rho$ for all $g\in G$. That is, $\rho\colon (\mathrm F(C),\mathrm F(\alpha))\to (D,\delta)$ is equivariant. This shows that $(\mathrm F(C), \mathrm F(\alpha))$ has the universal property of inductive limits in $\mathbf D_G $.
\end{proof}
\begin{theorem}\label{thm: mainclassification1}
Let $G$ be a finite group, let $\mathbf B$ be a subcategory of $\mathbf A$, and let $\mathbf C$ be a category where inductive limits of sequences exist. Let $\mathbf{B}_G$, $\mathbf{RB}_G$, and $\mathbf C_G $ be as in Definition \ref{df: C^G, RB^G}. Let $\mathrm F\colon \mathbf{B}\to \mathbf C$ be a sequentially continuous functor that classifies homomorphisms and let $\mathrm{F}_G\colon \mathbf{B}_G\to \mathbf{C}_G$ be the associated functor as in Definition \ref{df: C^G, RB^G}. Then the restriction of $\mathrm{F}_G$ to $\mathbf{RB}_G$ classifies isomorphisms. In particular, if $(A,\alpha)$ and $(B,\beta)$ are C*-dynamical systems in $\mathbf{RB}_G$, then $\alpha$ and $\beta$ are conjugate if and only if there exists an isomorphism $\rho\colon \mathrm F_G(A,\alpha)\to \mathrm F_G(B,\beta)$ in $\mathbf C_G$.
\end{theorem}
\begin{proof}
Since by Theorem \ref{thm: mainclassification} the restriction of the functor $\mathrm{F}_G$ to $\mathbf{RB}_G$ classifies homomorphisms, it is sufficient to show that the conditions of Theorem \ref{thm: isomorphism} are satisfied. First note that sequential inductive limits exists in $\mathbf{B}_G$ since $G$ is finite and they exists in $\mathbf{B}$ by assumption. Now by \cite[Theorem 2 (v)]{SantiagoRP} the same is true for $\mathbf{RB}_G$. Given that sequential inductive limits exist in the category $\mathbf C$ and $\mathrm F\colon \mathbf{B}\to \mathbf C$ is sequentially continuous, it follows from Lemma \ref{lem: categories} applied to $\Lambda=\mathbb{N}$ that sequential inductive limits exist in $\mathbf{C}_G$ and the functor $\mathrm F_G\colon \mathbf{B}_G\to \mathbf{C}_G$ is sequentially continuous. In particular, it follows that the restriction of $\mathrm{F}_G$ to $\mathbf{RB}_G$ is sequentially continuous. This shows that the conditions of Theorem \ref{thm: isomorphism} are met. The last statement of the theorem follows from the definition of a functor that classifies isomorphisms.
\end{proof}
The following result was proved by Izumi in \cite[Theorem 3.5]{Izumi-I} for unital C*-algebras, and more recently by Nawata in \cite[Theorem 3.5]{Nawata} for C*-algebras with \emph{almost stable rank one} (that is, C*-algebras $A$ such that $A\subseteq\overline{\mathrm{GL}(\widetilde{A})}$).
\begin{theorem}\label{thm: isoexistence}
Let $G$ be a finite group, let $A$ be separable C*-algebra and let $\alpha$ and $\beta$ be actions of $G$ on $A$ with the Rokhlin property. Assume that $\alpha_g\sim_{\mathrm{au}}\beta_g$ for all $g\in G$. Then there exists an approximately inner automorphism $\psi$ of $A$ such that $\psi\circ\alpha_g=\beta_g\circ\psi$ for all $g\in G$.
\end{theorem}
\begin{proof}
Let $\mathbf C$ be the category whose objects are separable C*-algebras and whose morphisms are given by
$$\mathrm{Hom}(A, B)=\{[\phi]_{\mathrm{au}}\colon \phi\colon A\to B \mbox{ is a *-homomorphism}\},$$
where $[\phi]_{\mathrm{au}}$ denotes the approximate unitary equivalence class of $\phi$. (It is easy to check that composition of maps is well defined in $\mathbf{C}$, and thus $\mathbf{C}$ is indeed a category.) Let $\mathrm F\colon \mathbf{A}\to \mathbf{C}$ be the functor given by $\mathrm{F}(A)=A$ for any C*-algebra $A$ in $\mathbf A$, and $\mathrm F(\phi)=[\phi]_{\mathrm{au}}$ for any *-homomorphism $\phi$ in $\mathbf A$. It is straightforward to check that sequential inductive limits exist in $\mathbf C$ and that $\mathrm F$ is sequentially continuous. Moreover, by the construction of $\mathbf C$ and $\mathrm F$ it is clear that $\mathrm F$ classifies *-homomorphisms. Therefore, by Theorem \ref{thm: mainclassification1} the restriction of the associated functor $\mathrm{F}_G$ to $\mathbf{RA}_G$ classifies isomorphisms.
Let $A$ be a separable C*-algebra (that is, a C*-algebra in $\mathbf A$), and let $\alpha$ and $\beta$ be as in the statement of the theorem. Since $\alpha_g\sim_{\mathrm{au}}\beta_g$ for all $g\in G$, we have
$$\mathrm{F}(\mathrm{id_A})\circ\mathrm F(\alpha_g)=\mathrm{F}(\mathrm{id_A}\circ\alpha_g)=\mathrm F(\beta_g\circ\mathrm{id_A})=\mathrm F(\beta_g)\circ\mathrm{F}(\mathrm{id_A}),$$
for all $g\in G$.
In other words, the map $[\mathrm{id}_A]_{\mathrm{au}}$ is equivariant. Also, note that this map is an automorphism. Therefore, it is an isomorphism in the category $\mathbf{C}_G$. Since by the previous discussion, the restriction of $\mathrm{F}_G$ to $\mathbf{RA}_G$ classifies isomorphisms, it follows that that there exists an equivariant *-automorphism
$\psi\colon (A, \alpha)\to (A,\beta)$ such that $\mathrm{F}_G(\psi)=[\mathrm{id}_A]_{\mathrm{au}}$. In particular, $\alpha$ and $\beta$ are conjugate. Using that $\mathrm{F}(\psi)=\mathrm{F}_G(\psi)=[\mathrm{id}_A]_{\mathrm{au}}=\mathrm F(\mathrm{id}_A)$ and that $\mathrm F$ classifies homomorphisms, we get that $\psi\sim_{\mathrm{au}}\mathrm{id}_A$. In other words, $\psi$ is approximately inner.
\end{proof}
\begin{remark}
In view of \cite[Remark 3.6]{Nawata}, it may be worth pointing out that one can directly modify the proof of \cite[Lemma 3.4]{Nawata} to get rid of the assumption that $A$ has almost stable rank one. Indeed, one just needs to replace the element $w$ in the proof by the unitary $w'=\sum\limits_{g\in G}(v_g-\lambda_g 1_{\widetilde{A}})f_g+1_{\widetilde{A}}$, where $\lambda_g\in\mathbb{C}$ is such that $v_g-\lambda_g 1_{\widetilde{A}}\in A$.
\end{remark}
\subsection{Applications}
In this section we apply Theorems \ref{thm: mainclassification}, \ref{thm: mainclassification1}, and \ref{thm: isoexistence}, and known classification results, to obtain classification of equivariant *-homomorphisms and finite group actions on certain classes of 1-dimensional NCCW-complexes and AH-algebras.
\subsubsection{1-dimensional NCCW-complexes}
Let $E$ and $F$ be finite dimensional C*-algebras, and for $x\in [0,1]$, denote by $\mathrm{ev}_x\colon \mathrm{C}([0,1], F)\to F$ the
evaluation map at the point $x$. Recall that a C*-algebra $A$ is said to be a \emph{one-dimensional non-commutative CW-complex}, abbreviated 1-dimensional
NCCW-complex, if $A$ is given by a pullback diagram of the form:
\begin{equation*}
\xymatrix{A \ar[d] \ar[rr] & & E\ar[d] \\ \mathrm C([0,1], F)\ar[rr]_-{\mathrm{ev}_0\oplus \mathrm{ev}_1} & & F\oplus F.}
\end{equation*}
\begin{theorem}\label{thm: classification-h-RP-Cutilde}
Let $G$ be a finite group. Let $(A, \alpha)$ and $(B, \beta)$ be separable C*-dynamical systems such that $A$ can be written as an inductive limit of 1-dimensional NCCW-complexes with trivial $\mathrm K_1$-groups and such that $B$ has stable rank one. Assume that $\beta$ has the Rokhlin property.
\begin{itemize}
\item[(i)] Fix strictly positive elements $s_A$ and $s_B$ of $A$ and $B$, respectively. Let $\rho\colon \mathrm{Cu}^\sim(A)\to \mathrm{Cu}^\sim(B)$ be a morphism in the category $\mathbf{Cu}$ such that
$$\rho([s_A])\le [s_B] \ \mbox{ and } \ \rho\circ\mathrm{Cu}^\sim(\alpha_g)=\mathrm{Cu}^\sim(\beta_g)\circ \rho$$
for all $g\in G$. Then there exists an equivariant *-homomorphism
$$\phi\colon (A, \alpha)\to (B, \beta) \ \mbox{ such that } \ \mathrm{Cu}^\sim(\phi)=\rho.$$
\item[(ii)] If $\phi, \psi\colon (A, \alpha)\to (B, \beta)$ are equivariant *-homomorphisms, then $\mathrm{Cu}^\sim(\phi)=\mathrm{Cu}^\sim(\psi)$ if and only if $\phi\sim_{G-\mathrm{au}}\psi$.
\end{itemize}
Moreover, if $A$ is unital, or if it is simple and has trivial $\mathrm K_0$-group, or if it can be written as an inductive limit of punctured-trees algebras, then the functor $\mathrm{Cu}^\sim$ can be replaced by the Cuntz functor $\mathrm{Cu}$ in the statement of this theorem.
\end{theorem}
\begin{proof}
(i) Let $\rho\colon \mathrm{Cu}^\sim(A)\to \mathrm{Cu}^\sim(B)$ be as in the statement of the theorem. By \cite[Theorem 1]{Robert}, there exists a *-homomorphism $\psi\colon A\to B$ such that $\mathrm{Cu}^\sim(\psi)=\rho$. Using that $\rho$ is equivariant, we get $\mathrm{Cu}^\sim(\beta_g\circ\psi)=\mathrm{Cu}^\sim(\psi\circ\alpha_g)$ for all $g\in G$. By the uniqueness part of \cite[Theorem 1]{Robert}, it follows that $\beta_g\circ\psi\sim_{\mathrm{au}}\psi\circ\alpha_g$ for all $g\in G$. By Proposition \ref{prop: existence}, there exists an equivariant *-homomorphism $\phi\colon A\to B$ such that $\phi\sim_{\mathrm{au}}\psi$. Since $\mathrm{Cu}^\sim$ is invariant under approximate unitary equivalence,
we conclude $\mathrm{Cu}^\sim(\phi)=\mathrm{Cu}^\sim(\psi)$, as desired.
(ii) The ``if" implication is clear. For the converse, let $\phi$ and $\psi$
be as in the statement of the theorem. By the uniqueness part of \cite[Theorem 1]{Robert}, we have $\phi\sim_{\mathrm{au}}\psi$. It now follows from Proposition \ref{prop: uniqueness} that $\phi\sim_{G-\mathrm{au}}\psi$.
It follows from \cite[Remark 3 (ii)]{Robert}, and by \cite[Corollary 4 (b)]{Robert}, \cite[Corollary 6.7]{Elliott-Robert-Santiago}, and \cite[Corollary 8.6]{Tikuisis}, respectively, that the functors $\mathrm{Cu}^\sim$ and $\mathrm{Cu}$ are equivalent when restricted to the class of C*-algebras that are inductive limits of 1-dimensional NCCW-complexes which are either unital or simple and with trivial $\mathrm K_0$-group. Hence, for these classes of C*-algebras, the theorem holds when $\mathrm{Cu}^\sim$ is replaced by $\mathrm{Cu}$. For C*-algebras that are inductive limits of punctured-trees algebras, one can use \cite[Theorem 1.1]{Ciuperca-Elliott-Santiago} instead of \cite[Theorem 1]{Robert} in the proof above to obtain the desired result. Finally, since $B$ has stable rank one, the results in
\cite{Robert} show that $\mathrm{Cu}(B)$ is a subsemigroup of $\mathrm{Cu}^\sim(B)$. In particular, for a homomorphism
$\phi\colon A\to B$, the range of the induced map $\mathrm{Cu}^\sim(\phi)\colon \mathrm{Cu}^\sim(A)\to \mathrm{Cu}^\sim(B)$ is
contained in $\mathrm{Cu}(B)\subseteq \mathrm{Cu}^\sim(B)$.
\end{proof}
\begin{theorem}\label{classif Rp on NCCW}
Let $G$ be a finite group, and let $(A, \alpha)$ and $(B, \beta)$ be separable dynamical systems such that $A$ and $B$ can be written as inductive limits of 1-dimensional NCCW-complexes with trivial $\mathrm K_1$-groups. Suppose that $\alpha$ and $\beta$ have the Rokhlin property.
\begin{itemize}
\item[(i)] Fix strictly positive elements $s_A$ and $s_B$ of $A$ and $B$ respectively. Then the actions $\alpha$ and $\beta$ are conjugate if and only if there exists an isomorphism $\gamma\colon \mathrm{Cu}^\sim(A)\to \mathrm{Cu}^\sim(B)$ with $\gamma([s_A])=[s_B]$, such that
$$\gamma\circ\mathrm{Cu}^\sim(\alpha_g)=\mathrm{Cu}^\sim(\beta_g)\circ\gamma \mbox{ for all } g\in G.$$
\item[(ii)] Assume that $A=B$. Then the actions $\alpha$ and $\beta$ are conjugate by an approximately inner automorphism of $A$ if and only if $\mathrm{Cu}^\sim(\alpha_g)=\mathrm{Cu}^\sim(\beta_g)$ for all $g\in G$.
\end{itemize}
Moreover, if both $A$ and $B$ are unital, or if they are simple and have trivial $\mathrm K_0$-groups, or if they can be written as inductive limits of punctured-trees algebras, then the functor $\mathrm{Cu}^\sim$ can be replaced by the Cuntz functor $\mathrm{Cu}$.
\end{theorem}
\begin{proof}
Part (ii) clearly follows from (i). Let us prove (i).
Let $\mathbf B$ denote the subcategory of the category $\mathbf A$ of C*-algebras consisting of those C*-algebras that can be written as an inductive limit of 1-dimensional NCCW-complexes with trivial $\mathrm K_1$-groups. By \cite[Theorem 1]{Robert}, the functor $(\mathrm{Cu}^\sim(\cdot), [s_{\,\cdot\,}])$, where $s_{\,\cdot\,}$ is a strictly positive element of the given algebra, restricted to $\mathbf B$ classifies homomorphisms. Therefore, by Theorem \ref{thm: mainclassification1}, the associated functor $(\mathrm{Cu}^\sim_G(\cdot), [s_{\,\cdot\,}])$ restricted to $\mathbf{RB}_G$ classifies isomorphisms, which implies (i).
The last part of the theorem follows from the same arguments used at the end of the proof of Theorem \ref{thm: classification-h-RP-Cutilde}.
\end{proof}
Let $G$ be a finite group. Recall that the action $\mu^G\colon G\to \mathrm{Aut}\left(\mathrm{M}_{|G|^\infty}\right)$ constructed in Example \ref{eg: model action} has the Rokhlin property, and that $\mu^G_g$ is approximately inner for all $g\in G$.
In the next corollary, we do not assume that either $\alpha$ or $\beta$ has the Rokhlin property.
\begin{corollary}\label{cor: tensor with mu-G}
Let $G$ be a finite group and let $(A, \alpha)$ and $(A, \beta)$ be C*-dynamical systems such that $A$ can be written as an inductive limit of 1-dimensional NCCW-complexes with trivial $\mathrm K_1$-groups. Suppose that $\mathrm{Cu}^\sim(\alpha_g)=\mathrm{Cu}^\sim(\beta_g)$ for all $g\in G$. Then $\alpha\otimes \mu^G$ and $\beta\otimes \mu^{G}$ are conjugate.
Moreover, if $A$ belongs to one of the classes of C*-algebras described in the last part of Theorem \ref{classif Rp on NCCW}, then the statement of the corollary holds for the functor $\mathrm{Cu}$ in place of the functor $\mathrm{Cu}^\sim$.
\end{corollary}
\begin{proof}
The actions $\alpha\otimes \mu^G$ and $\beta\otimes \mu^{G}$ have the Rokhlin property by part (i) of Lemma \ref{Rp properties}. Note that $\mu^G_g$ is approximately inner for all $g\in G$. Thus,
$$\mathrm{Cu}^\sim(\alpha\otimes\mu^G_g)=\mathrm{Cu}^\sim(\alpha\otimes \mathrm{id}_{\mathrm M_{|G|^\infty}})=\mathrm{Cu}^\sim(\beta\otimes \mathrm{id}_{\mathrm M_{|G|^\infty}})=\mathrm{Cu}^\sim(\beta\otimes\mu^G_g)$$
for all $g\in G$. It follows from Theorem \ref{classif Rp on NCCW} (ii) that $\alpha\otimes \mu^G$ and $\beta\otimes \mu^{G}$ are conjugate.
\end{proof}
\subsubsection{AH-algebras}
Recall that a C*-algebra $A$ is approximate homogeneous (shortly AH) if it can be written as an inductive limit $A=\varinjlim (A_n, \phi_{n,m})$, with
$$A_n=\oplus_{j=1}^{s(n)}P_{n,j}\mathrm{M}_{n,j}(\mathrm{C}(X_{n,j}))P_{n,j},$$
where $X_{n,j}$ is a finite dimensional compact metric space, and $P_{n,j}\in \mathrm{M}_{n,j}(\mathrm{C}(X_{n,j}))$ is a projection for all $n$ and $j$. The C*-algebra $A$ is said to have \emph{no dimension growth} if there exists an inductive limit decomposition of $A$ as an AH-algebra such that
$$\sup_n\max_j\dim X_{n,j}<\infty.$$
Let $A$ be a unital simple separable C*-algebra and let $\mathrm T(A)$ denote the metrizable compact convex set of tracial states of $A$. Denote by $\mathrm T$ the induced contravariant functor from the category of unital separable simple C*-algebras to the category of metrizable compact convex sets. It is not difficult to check that $\mathrm T$ is continuous, meaning that it sends inductive limits to projective limits.
Let $T$ be a metrizable compact convex set and let $\mathrm{Aff}(T)$ denote the set of real-valued continuous affine functions on $T$. Let $\mathrm{Aff}$ denote the induced contravariant functor from the category of metrizable compact convex sets to the category of normed vector spaces. Denote by $\rho_A\colon \mathrm{K}_0(A)\to \mathrm{Aff}(\mathrm T(A))$ the map defined by
\begin{align}\label{eq: rho}
\rho_A([p]-[q])(\tau)=(\tau\otimes \mathrm{Tr}_n)(p)-(\tau\otimes \mathrm{Tr}_n)(q)
\end{align}
for $p, q\in \mathrm{M}_n(\mathbb{C})$, where $\mathrm{Tr}_n$ denotes the standard trace on $\mathrm{M}_n(\mathbb{C})$.
Let $A$ be a unital C*-algebra. Denote by $\mathrm U(A)$ the unitary group of $A$ and by $\mathrm{CU}(A)$ the closure of the normal subgroup generated by the commutators of $\mathrm U(A)$. We denote the quotient group by
$$\mathrm H(A)=\mathrm U(A)/\mathrm{CU}(A).$$
(see \cite{Thomsen} and \cite{Nielsen-Thomsen} for properties of this group).
The set $\mathrm H(A)$, endowed with the distance induced by the distance in $\mathrm U(A)$, is a complete metric space.
We denote by $\mathrm H$ the induced functor from the category of C*-algebras to the category of complete metric groups. Also, if $A$ is a simple unital AH-algebra of no dimension growth (or more generally, a simple unital C*-algebra of tracial rank no greater than one), then there exists an injection
\begin{align}\label{eq: lambda}
\lambda_A\colon \mathrm{Aff}(\mathrm T(A))/\overline{\rho_A(\mathrm K_0(A))}\to \mathrm H(A).
\end{align}
(See \cite{Thomsen} and \cite{Huaxin}.)
For a C*-algebra $A$, denote by $\underline{\mathrm K}(A)$ the sum of all K-groups with $\mathbb{Z}/n\mathbb{Z}$ coefficients for all $n\ge 1$. Let $\Lambda$ denote the category generated by the Bockstein operations on $\underline{\mathrm K}(A)$ (see \cite{Dadarlat-Loring}). Then $\underline{\mathrm K}(A)$ becomes a $\Lambda$-module and it induces a continuous functor $\underline{\mathrm K}$ from the category of C*-algebras to the category of $\Lambda$-modules.
Let $A$ and $B$ be unital simple AH-algebras and let $\mathrm{KL}(A, B)$ denote the group defined in \cite{RordamKL}. By the Universal Coefficient Theorem and the Universal Multicoefficient Theorem (see \cite{Dadarlat-Loring}), the groups $\mathrm{KL}(A,B)$ and $\mathrm{Hom}_{\Lambda}(\underline{\mathrm K}(A),\underline{\mathrm K}(B))$ are naturally isomorphic. Let $\mathrm{KL}^{++}_e(A, B)$ be as in \cite[Definition 6.4]{Huaxin}. By the previous isomorphism, the group $\mathrm{KL}^{++}_e(A,B)$ is naturally isomorphic to
$$\{\kappa\in \mathrm{Hom}_{\Lambda}(\underline{\mathrm K}(A),\underline{\mathrm K}(B))\colon \kappa(\mathrm{K}_0(A)_+\setminus \{0\})\subseteq \mathrm{K}_0(B)_+\setminus \{0\},\, \kappa([1_A])=\kappa([1_B])\}.$$
Let us define a functor $\underline{\mathrm K}^{++}$ from the category of separable, unital, simple, finite C*-algebras to the category whose objects are 4-tuples $(M, N, E, e)$, where $M$ is a $\Lambda$-module, $N$ is a subgroup of $M$, $E$ is a subset of $N$, and $e$ is an element of $N$; and whose morphisms $\kappa\colon (M, N, E, e)\to (M', N', E', e')$ are $\Lambda$-module maps $\kappa\colon M\to M'$ such that $\kappa(N)\subseteq N'$, $\kappa(E)\subseteq E'$, and $\kappa(e)=e'$. The functor $\underline{\mathrm K}^{++}$ is defined as follows:
$$\underline{\mathrm K}^{++}(A)=(\underline{\mathrm K}(A), \mathrm K_0(A), \mathrm K_0(A)_+\setminus \{0\}, [1_A]), \quad\mbox{ and }\quad \underline{\mathrm K}^{++}(\phi)=\underline{\mathrm K}(\phi).$$
Note that if $A$ and $B$ are unital AH-algebras then $\mathrm{KL}^{++}_e(A,B)$ is isomorphic to $\mathrm{Hom}(\underline{\mathrm K}^{++}(A), \underline{\mathrm K}^{++}(B))$.
Let $\mathbf C$ denote the category whose objects are tuples
$$\left((M, N, E, e), T, H, \rho, \lambda\right),$$
where $(M, N, E, e)$ is as above, $T$ is a metrizable compact convex set, $H$ is a complete metric group, $\rho\colon N\to \mathrm{Aff}(T)$ is a group homomorphism, and $\lambda\colon \mathrm{Aff}(T)/\overline{\rho(N)}\to H$ is an injective continuous group homomorphism. The maps in $\mathbf C$ are triples
$$(\kappa, \eta, \mu)\colon \left((M, N, E, e), T, H, \rho, \lambda\right)\to \left((M', N', E', e'), T', H', \rho', \lambda'\right),$$
where $\kappa\colon (M, N, E, e)\to (M', N', E', e')$, $\eta\colon T'\to T$, and $\mu\colon H\to H'$ are maps in the corresponding categories that satisfy the compatibility conditions:
$$\rho'\circ\kappa|_N=\mathrm{Aff}(\eta)\circ \rho, \quad \mbox{and}\quad \lambda'\circ \mu=\overline{\mathrm{Aff}}(\eta)\circ\lambda,$$
where
$$\overline{\mathrm{Aff}}(\eta)\colon \mathrm{Aff}(T)/\overline{\rho(N)}\to \mathrm{Aff}(T')/\overline{\rho(N')}$$
is the map induced by $\mathrm{Aff}(\eta)$.
Using that inductive limits of sequences exist in each of the categories that form $\mathbf C$, it is not difficult to show that $\mathbf C$ is also closed under taking inductive limits of sequences. Also, it is easy to see that $\mathrm F=(\underline{\mathrm K}^{++}, \mathrm T,\mathrm H)$ is a functor from the category of unital, simple, separable, finite C*-algebras to the category $\mathbf C$. Moreover, since the functors that form $\mathrm F$ are continuous, $\mathrm F$ is also continuous.
\begin{theorem}\label{thm: AH homomrphisms}
Let $G$ be a finite group. Let $(A, \alpha)$ and $(B, \beta)$ be dynamical systems such that $A$ and $B$ are unital simple AH-algebras of no dimension growth. Assume that $\beta$ has the Rokhlin property.
\begin{itemize}
\item[(i)] Let
\begin{align*}
\kappa\colon \underline{\mathrm K}^{++}(A)\to \underline{\mathrm K}^{++}(B),\quad \eta\colon \mathrm T(B)\to \mathrm T(A),\quad \mbox{and}\quad
\mu\colon \mathrm H(A)\to\mathrm H(B),
\end{align*}
be maps in the corresponding categories that satisfy the compatibility conditions
\begin{align*}
\rho_B\circ\kappa|_{\mathrm{K_0(A)}}=\mathrm{Aff}(\eta)\circ \rho_A, \quad \mbox{and}\quad \lambda_B\circ \mu=\overline{\mathrm{Aff}}(\eta)\circ\lambda_A,
\end{align*}
where $\rho_A$, $\rho_B$, $\lambda_A$, and $\lambda_B$ are as in \eqref{eq: rho} and \eqref{eq: lambda}. Suppose that
\begin{align*}
\kappa\circ\underline{\mathrm K}(\alpha_g)= \underline{\mathrm K}(\beta_g)\circ \kappa,
\quad\eta\circ\mathrm T(\beta_g)=\mathrm T(\alpha_g)\circ \eta,\quad \mu\circ \mathrm H(\alpha_g)=\mathrm H(\beta_g)\circ \mu,
\end{align*}
for all $g\in G$. Then there exists an equivariant *-homomorphism $\phi\colon (A, \alpha)\to (B, \beta)$ such that
$$\underline{\mathrm K}^{++}(\phi)=\kappa, \quad \mathrm T(\phi)=\eta,\quad \mbox{and}\quad \mathrm H(\phi)=\mu.$$
\item[(ii)] Let $\phi, \psi\colon A\to B$ be equivariant *-homomorphisms such that
$$\underline{\mathrm K}(\phi)=\underline{\mathrm K}(\psi),\quad \mathrm T(\phi)=\mathrm T(\psi), \quad \mbox{and}\quad \mathrm H(\phi)=\mathrm H(\psi).$$
Then $\phi\sim_{G-\mathrm{au}}\psi$.
\end{itemize}
\end{theorem}
\begin{proof}
It is shown in \cite{Gong} that every unital simple AH-algebra of no dimension growth has tracial rank almost one. By \cite[Theorems 5.11 and 6.10]{Huaxin} applied to the algebras $A$ and $B$, and using the computations of $\mathrm{KL}^{++}_e(A,B)$ given in the paragraphs preceding the theorem, we deduce that the functor $\mathrm F$ (defined above) restricted to the category of unital simple AH-algebras of no dimension growth classifies *-homomorphisms. The theorem now follows from Theorem \ref{thm: mainclassification}.
\end{proof}
\begin{theorem}\label{thm: ismomorphismAH}
Let $G$ be a finite group and let $A$ and $B$ be unital simple AH-algebras of no dimension growth. Let $\alpha$ and $\beta$ be actions of $G$ on $A$ and $B$ with the Rokhlin property.
\begin{itemize}
\item[(i)]
The actions $\alpha$ and $\beta$ are conjugate if and only if there exist isomorphisms
\begin{align*}
\kappa\colon \underline{\mathrm K}^{++}(A)\to \underline{\mathrm K}^{++}(B), \quad\eta\colon \mathrm T(B)\to \mathrm T(A),\quad
&\mu\colon \mathrm H(A)\to\mathrm H(B),
\end{align*}
in the corresponding categories, that satisfy the compatibility conditions of the previous theorem, and such that
\begin{align*}
\kappa\circ\underline{\mathrm K}(\alpha_g)= \underline{\mathrm K}(\beta_g)\circ \kappa,\quad
\eta\circ\mathrm T(\beta_g)=\mathrm T(\alpha_g)\circ \rho,\quad \mu\circ \mathrm H(\alpha_g)=\mathrm H(\beta_g)\circ \lambda,
\end{align*}
for all $g\in G$.
\item[(ii)] Assume that $A=B$. Then the actions $\alpha$ and $\beta$ are conjugate by an approximately inner automorphism if and only if
\begin{align*}
\underline{\mathrm K}(\alpha_g)=\underline{\mathrm K}(\beta_g),\quad \mathrm T(\alpha_g)=\mathrm T(\beta_g), \quad \mbox{and}\quad \mathrm H(\alpha_g)=\mathrm H(\beta_g),
\end{align*}
for all $g\in G$.
\end{itemize}
\end{theorem}
\begin{proof}
Part (ii) clearly follows from (i) and part (ii) of Theorem~\ref{thm: AH homomrphisms}.
Let us prove (i). As in the proof of Theorem \ref{thm: AH homomrphisms}, the functor $\mathrm F$ restricted to the category of unital simple AH-algebras of no dimension growth classifies homomorphisms. The statements of the theorem now follows from Theorem \ref{thm: mainclassification1}.
\end{proof}
\begin{corollary}
Let $G$ be a finite group and let $A$ be a unital simple AH-algebra of no dimension growth. Let $(A, \alpha)$ and $(A, \beta)$ be C*-dynamical systems. Suppose that
$$\underline{\mathrm K}^{++}(\alpha_g)=\underline{\mathrm K}^{++}(\beta_g), \quad \mathrm T(\alpha_g)=\mathrm T(\beta_g), \quad \mbox{and} \quad \mathrm H(\alpha_g)=\mathrm{H}(\beta_g),$$
for all $g\in G$. Then $\alpha\otimes \mu^G$ and $\beta\otimes \mu^{G}$ are conjugate.
\end{corollary}
\begin{proof}
The proof of this corollary follows line by line the proof of Corollary \ref{cor: tensor with mu-G}, using the functor $\mathrm F$ instead of the functor $\mathrm{Cu}^\sim$ and Theorem \ref{thm: ismomorphismAH} instead of Theorem \ref{classif Rp on NCCW}.
\end{proof}
\section{Cuntz semigroup and K-theoretical constraints}
In this section, a Cuntz semigroup obstruction is obtained for a C*-algebra to admit an action with the Rokhlin property. Also, the Cuntz semigroup of the fixed-point C*-algebra and the crossed product C*-algebra associated to an action of a finite group with the Rokhlin property are computed in terms of the Cuntz semigroup of the given algebra. As a corollary, similar results are obtained for the Murray-von Neumann semigroup and the K-groups.
Let $G$ be a group and let $(S,\gamma)$ be an object in the category $\mathbf{Cu}_G$, this is, $S$ is a semigroup in the category $\mathbf{Cu}$ and
$\gamma\colon G\to \mathrm{Aut}(S)$ is an action of $G$ on $S$. Let $S^\gamma$ and $S^\gamma_\mathbb{N}$ be the subsemigroups of $S$ defined in Definition \ref{df: S gamma}.
It was shown in Lemma \ref{lem: closure} that $S^\gamma$ belongs to the category $\mathbf{Cu}$ and that $S^\gamma_\mathbb{N}$ is closed under suprema
of increasing sequences. We do not know in general whether $S^\gamma_\mathbb{N}$ is an object in $\mathbf{Cu}$. However, if $\alpha\colon G\to\mathrm{Aut}(A)$ is
an action of a finite group $G$ on a C*-algebra $A$ with the Rokhlin property, it will follow by the next theorem that
$\mathrm{Cu}(A)_\mathbb{N}^{\mathrm{Cu}(\alpha)}$ coincides with $\mathrm{Cu}(A)^{\mathrm{Cu}(\alpha)}$, and with the Cuntz semigroup of $A^\alpha$, so in
particular belongs to $\mathbf{Cu}$.
For use in the proof of the next theorem, if $\phi\colon A\to B$ is a *-homomorphism between C*-algebras $A$ and $B$, we denote by $\phi^s\colon A\otimes\mathcal{K} \to B\otimes \mathcal{K}$ the stabilized *-homomorphism $\phi^s=\phi\otimes\mathrm{id}_\mathcal{K}$.
\begin{theorem}\label{thm: Rokhlin Constraint}
Let $A$ be a C*-algebra and let $\alpha$ be an action of a finite group $G$ on $A$ with the Rokhlin property. Let $i\colon A^\alpha\to A$
be the inclusion map. Then:
\begin{itemize}
\item[(i)] The map $\mathrm{Cu}(\widetilde i)\colon\mathrm{Cu}(\widetilde{A^\alpha})\to \mathrm{Cu}(\widetilde A)$ is an order embedding;
\item[(ii)] The map $\mathrm{Cu}(i)\colon\mathrm{Cu}(A^\alpha)\to \mathrm{Cu}(A)$ is an order embedding and
\begin{align*}
\begin{aligned}
\mathrm{Im}(\mathrm{Cu}(i))&=\overline{\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)\right)}=\mathrm{\mathrm{Cu}(A)}_\mathbb{N}^{\mathrm{Cu}(\alpha)}
=\mathrm{\mathrm{Cu}(A)}^{\mathrm{Cu}(\alpha)};
\end{aligned}
\end{align*}
\end{itemize}
\end{theorem}
\begin{proof}
In the proof of this theorem, we will denote the action induced by $\alpha$ on $\widetilde{A}\otimes \mathcal{K}$ again
by $\alpha$.
(i) Let $a,b\in \widetilde{A^\alpha}\otimes \mathcal{K}$ satisfy $a\precsim b$ in $\widetilde A\otimes \mathcal{K}$. We want to
show that $a\precsim b$ in $\widetilde{A^\alpha}\otimes \mathcal{K}$. Let $\varepsilon>0$. By Lemma \ref{lem: Cuntz relation}, there exists $d\in \widetilde A\otimes \mathcal{K}$ such that $(a-\varepsilon)_+=dbd^*$. Apply $\alpha_g$ to this equation to get $(a-\varepsilon)_+=\alpha_g(d)b\alpha_g(d^*)$ for all $g\in G$.
Let $\pi\colon \widetilde A\to \mathbb{C}$ be the quotient map and let $j\colon \mathbb{C}\to \widetilde A$ be the inclusion $j(\lambda)=\lambda 1_{\widetilde A}$ for all $\lambda\in \mathbb{C}$. It is clear that $\pi\circ j=\mathrm{id}_{\mathbb{C}}$. Set
\begin{align*}
& a_1=(j^s\circ \pi^s)((a-\varepsilon)_+)\in \mathbb{C} 1_{\widetilde A}\otimes \mathcal{K}, \qquad a_2=(a-\varepsilon)_+-a_1\in A^\alpha\otimes \mathcal{K},\\
& b_1=(j^s\circ \pi^s)(b)\in \mathbb{C} 1_{\widetilde A}\otimes \mathcal{K}, \qquad \qquad\quad \, b_2=b-b_1\in A^\alpha\otimes \mathcal{K},\\
& d_1=(j^s\circ \pi^s)(d)\in\mathbb{C} 1_{\widetilde A}\otimes \mathcal{K}, \qquad \qquad\quad d_2=d-d_1\in A\otimes \mathcal{K}.
\end{align*}
Then $a_1=d_1b_1d_1^*$. Set
$$F=\{\alpha_g(d_2)b\alpha_g(d_2)\colon g\in G\}\cup \{d_1b\alpha_g(d_2^*)\colon g\in G\}\cup \{a_2-d_1b_2d_1^*\}\subseteq
\widetilde{A}\otimes\mathcal{K}.$$
Use Lemma \ref{lem: Rokhlin equivalence} (ii) to choose orthogonal positive contractions $(r_g)_{g\in G}$ in
$(A\otimes \mathcal{K})^\infty\cap F'\subseteq (\widetilde A\otimes \mathcal{K})^\infty\cap F'$ such that $\alpha_g(r_g)=r_{gh}$ for all $g,h\in G$, and $(\sum\limits_{g\in G}r_g)x=x$ for all $x\in F$. Set
$$f=\sum\limits_{g\in G}r_g\alpha_g(d_2)+d_1\in (\widetilde A\otimes \mathcal{K})^\infty.$$
In the following computation, we use in the first step the identities $r_gx=xr_g$ for all $g\in G$ and $x\in F$,
$r_gr_h=0$ for all $g\neq h$, and $(r_g^2-r_g)x=x$ for all $g\in G$ and $x\in F$;
in the second step the definition of $d_2$;
in the fourth step that $d_1\in (\widetilde A\otimes \mathcal{K})^\alpha$ and the identity $(a-\varepsilon)_+=\alpha_g(d)b\alpha_g(d^*)$ for all $g\in G$;
in the fifth step the identity $a_1=d_1b_1d_1^*$;
and in the last step the identity $(\sum\limits_{g\in G}r_g)x=x$ for all $x\in F$:
\begin{align*}
&fbf^*=\left(\sum\limits_{g,h\in G}r_g\alpha_g(d_2)b\alpha_h(d_2^*)r_h+\sum\limits_{g\in G}r_g\alpha_g(d_2)bd_1^*+\sum\limits_{g\in G}d_1b\alpha_g(d_2^*)r_g\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\alpha_g(d_2)b\alpha_g(d_2^*)+\sum\limits_{g\in G}r_g\alpha_g(d_2)bd_1^*+\sum\limits_{g\in G}r_gd_1b\alpha_g(d_2^*)\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\left(\alpha_g(d-d_1)b\alpha_g(d^*-d_1^*)+\alpha_g(d_2)bd_1^*+d_1b\alpha_g(d_2^*)\right)\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\left(\alpha_g(d)b\alpha_g(d^*)-\alpha_g(d-d_2)bd_1^*-d_1b\alpha_g(d^*-d_2^*)+d_1bd_1^*\right)\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\left((a-\varepsilon)_+-d_1bd_1^*-d_1bd_1^*+d_1bd_1^*\right)\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\left((a-\varepsilon)_+-d_1bd_1^*\right)\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\left(a_1+a_2-d_1b_1d_1^*-d_1b_2d_1^*\right)\right)+d_1bd_1^*\\
&=\left(\sum\limits_{g\in G}r_g\left(a_2-d_1b_2d_1^*\right)\right)+d_1bd_1^*\\
&=a_2+d_1b_1d_1^*\\
&=(a-\varepsilon)_+.
\end{align*}
Shortly, $(a-\varepsilon)_+=fbf^*$ in $(\widetilde A\otimes \mathcal{K})^\infty$. Since
$$f=\sum\limits_{g\in G}r_g\alpha_g(d_2)+d_1=\sum\limits_{g\in G}\alpha_g(r_ed_2)+d_1,$$
it follows that $\alpha_g(f)=f$ for all $g\in G$. This implies that $f$ is the image of a sequence $(f_n)_{n\in\mathbb{N}}$ in $\ell^\infty(\mathbb{N},\widetilde{A^\alpha}\otimes \mathcal{K})$, which satisfies
$$\lim_{n\to\infty} f_nbf_n^*=(a-\varepsilon)_+.$$
Thus, $(a-\varepsilon)_+\precsim b$ in $\widetilde{A^\alpha}\otimes \mathcal{K}$. Since $\varepsilon>0$ is arbitrary, we conclude that $[a]\le [b]$ in $\mathrm{Cu}(\widetilde{A^\alpha})$, as desired.
(ii) Since $A$ is an ideal in $\widetilde A$, the semigroup $\mathrm{Cu}(A)$ can be identified with the subsemigroup of $\mathrm{Cu}(\widetilde A)$ given by
$$\{[a]\in \mathrm{Cu}(\widetilde A)\colon a\in (A\otimes \mathcal{K})_+\}.$$
Using this identification, it is clear that the restriction of $\mathrm{Cu}(\widetilde i)$ to $\mathrm{Cu}(A)$ is $\mathrm{Cu}(i)$. Therefore, it follows from the first part of the theorem that $\mathrm{Cu}(i)$ is an order embedding.
Let us now proceed to prove the equalities stated in the theorem. It is sufficient to show that
\begin{align}\label{eq: inclusions}
\mathrm{Im}(\mathrm{Cu}(i))\subseteq\overline{\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)\right)}\subseteq \mathrm{Cu}(A)^{\mathrm{Cu}(\alpha)}\subseteq \mathrm{Cu}(A)^{\mathrm{Cu}(\alpha)}_\mathbb{N}\subseteq \mathrm{Im}(\mathrm{Cu}(i)).
\end{align}
The third inclusion is immediate and true in full generality. The second inclusion follows using that for $[a]\in \mathrm{Cu}(A)$, the element $\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)([a])$ is $\mathrm{Cu}(\alpha)$-invariant, that $\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)([a])$ is the supremum of the path
$$t\mapsto \sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)([(a+t-1)_+]),$$
and that $\mathrm{Cu}(A)^{\mathrm{Cu}(\alpha)}=\overline{\mathrm{Cu}(A)^{\mathrm{Cu}(\alpha)}}$ by part (ii) of Lemma \ref{lem: closure}.
We proceed to show the first inclusion. Fix a positive element $a\in A^\alpha\otimes \mathcal{K}$ and let $\varepsilon>0$. Using the Rokhlin property for $\alpha\otimes \mathrm{id}_\mathcal{K}$ with $F=\{a\}$, choose orthogonal positive contractions $(r_g)_{g\in G}\subseteq A\otimes \mathcal{K}$ such that
\begin{align}\label{eq: inequalities}
\left\|a-\sum\limits_{g\in G}r_gar_g\right\|<\varepsilon \ \ \mbox{ and } \ \ \left\|\alpha_g(r_ear_e)-r_gar_g\right\|<\varepsilon,
\end{align}
for all $g\in G$. Using the first inequality above and Lemma \ref{lem: Cuntz relation}, we obtain
\begin{align*}
\left[\left(a-4\varepsilon\right)_+\right]\le \left[\left(\sum\limits_{g\in G}r_gar_g-3\varepsilon\right)_+\right]\le
\left[ \left(\sum\limits_{g\in G}r_gar_g-\varepsilon\right)_+\right]\le [a].
\end{align*}
Furthermore, using the second inequality in \eqref{eq: inequalities} and again using
Lemma \ref{lem: Cuntz relation}, we deduce that
$$\left[\left(r_gar_g-3\varepsilon\right)_+\right]\le \left[\left(\alpha_g(r_ear_e)-2\varepsilon\right)_+\right]\le \left[\left(r_gar_g-\varepsilon\right)_+\right].$$
Take the sum of the previous inequalities, add them over $g\in G$, and use that $\mathrm{Cu}(\alpha_g)[(r_ear_e-2\varepsilon)_+]=[(\alpha_g(r_ear_e)-2\varepsilon)_+]$,
to conclude that
$$\left[\left(a-4\varepsilon\right)_+\right]\ll \sum\limits_{g\in
G}\mathrm{Cu}(\alpha_g)\left[\left(r_ear_e-2\varepsilon\right)_+\right]\le [a].$$
We have shown that for every $\varepsilon>0$, there is an element $x$ in $\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)\right)$ such that
$$[(a-\varepsilon)_+]\ll x\le [a].$$
By Lemma \ref{lem: sup} applied to $[a]=\sup\limits_{\varepsilon>0} [(a-\varepsilon)_+]$ and to the set $S=\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)\right)$, it follows that $[a]$ is the supremum of an increasing sequence in $\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)\right)$, showing that the first inclusion in (\ref{eq: inclusions}) holds.
In order to complete the proof, let us show that the fourth inclusion in \eqref{eq: inclusions} is also true. Fix $x\in \mathrm{Cu}(A)^{\mathrm{Cu}(\alpha)}_\mathbb{N}$. Choose a rapidly increasing sequence $(x_n)_{n\in\mathbb{N}}$ in $\mathrm{Cu}(A)$ such that $\mathrm{Cu}(\alpha_g)(x_n)=x_n$ for all $n\in \mathbb{N}$ and all for all $g\in G$. Fix $m\in \mathbb{N}$ and consider the elements $x_n$ with $n\geq m$. Note that $x_m\ll x_{m+1}\ll \cdots \ll x$. By Lemma \ref{lem: interpolation}, there is a positive element $a\in A\otimes\mathcal{K}$ such that
$$x_m\ll [(a-3\varepsilon)_+]\ll x_{m+1}\ll (a-2\varepsilon)_+\ll x_{m+2}\ll (a-\varepsilon)_+\ll x= [a].$$
Note that this implies that
$$[\alpha_g(a)]=\mathrm{Cu}(\alpha_g)[a]=\mathrm{Cu}(\alpha_g)(x)=x=[a]\le [a]$$
and
$$[(a-2\varepsilon)_+]\le x_{m+2}=\mathrm{Cu}(\alpha_g)(x_{m+2})\le\mathrm{Cu}(\alpha_g)[(a-\varepsilon)_+]=[\alpha_g((a-\varepsilon)_+)]$$
for every $g\in G$. By the definition of Cuntz subequivalence, there are elements $f_g, h_g\in A\otimes \mathcal{K}$ for $g\in G$ such that
$$\|\alpha_g(a)-f_gaf_g^*\|<\frac{\varepsilon}{|G|}$$
and
$$\|(a-2\varepsilon)_+-h_g\alpha_g((a-\varepsilon)_+)h_g^*\|<\frac{\varepsilon}{|G|}.$$
Using the Rokhlin property for $\alpha$, with
$$F=\{\alpha_g(a), \alpha_g((a-\varepsilon)_+), f_g, h_g\colon g\in G\}\cup \{(a-2\varepsilon)_+\},$$
choose positive orthogonal contractions $(r_g)_{g\in G}\subseteq (A\otimes \mathcal{K})^\infty\cap F'$ as in (ii) of Lemma \ref{lem: Rokhlin equivalence}. Set $f=\sum\limits_{g\in G}f_gr_g$ and $h=\sum\limits_{g\in G}h_gr_g$. Then
\begin{align*}
&\left\|\sum\limits_{g\in G} r_g\alpha_g(a)r_g-faf^*\right\|=\left\|\sum\limits_{g\in G}r_g(\alpha_g(a)-f_gaf_g^*)\right\|<|G|\cdot \frac{\varepsilon}{|G|}=\varepsilon,
\end{align*}
in $(A\otimes \mathcal{K})^\infty$.
Similarly,
\begin{align*}
\left\|(a-2\varepsilon)_+-h\left(\sum\limits_{g\in G}\alpha_g((a-\varepsilon)_+)\right)h^*\right\|<\varepsilon.
\end{align*}
Using that $r_g$ commutes with $\alpha_g(a)$ and that $r_g^2\alpha_g(a)=r_g\alpha_g(a)$ for all $g\in G$, one easily shows that
$$\sum\limits_{g\in G} r_g\alpha_g(a)r_g=\sum\limits_{g\in G}\alpha(r_ear_e), \quad \text{ and }\quad r_g(\alpha_g((a-\varepsilon)_+))r_g=(r_g\alpha_g(a)r_g-\varepsilon)_+,$$
for all $g\in G$. Thus, we have
\begin{align*}
\sum\limits_{g\in G}r_g(\alpha_g((a-\varepsilon)_+))r_g&=\sum\limits_{g\in G}r_g(\alpha_g(a)-\varepsilon)_+r_g\\
&=\sum\limits_{g\in G}(r_g\alpha_g(a)r_g-\varepsilon)_+\\
&=\left(\sum\limits_{g\in G}r_g\alpha_g(a)r_g-\varepsilon\right)_+\\
&=\left(\sum\limits_{g\in G} \alpha_g(r_ear_e)-\varepsilon\right)_+.
\end{align*}
Therefore, we conclude that
$$\left\|\sum\limits_{g\in G}\alpha_g(r_ear_e)-faf^*\right\|<\varepsilon, \quad \text{and}\quad\left\|(a-2\varepsilon)_+-h\left(\sum\limits_{g\in G} \alpha_g(r_ear_e)-\varepsilon\right)_+h^*\right\|<\varepsilon.$$
Let $(r_n)_{n\in \mathbb{N}}$, $(f_n)_{n\in \mathbb{N}}$, and $(h_n)_{n\in \mathbb{N}}$ be representatives of $r_e$, $f$, and $h$ in $\ell^\infty(\mathbb{N}, A\otimes \mathcal{K})$, with $r_n$ positive for all $n\in \mathbb{N}$. By the previous inequalities, there exists $k\in \mathbb{N}$ such that
$$\left\|\sum\limits_{g\in G}\alpha_g(r_kar_k)-f_kaf_k^*\right\|<\varepsilon, \ \text{ and } \ \left\|(a-2\varepsilon)_+-h_k\left(\sum\limits_{g\in G} \alpha_g(r_kar_k)-\varepsilon\right)_+h_k^*\right\|<\varepsilon$$
hold in $A\otimes \mathcal{K}$. By Lemma \ref{lem: Cuntz relation} applied to the elements $\sum\limits_{g\in G}\alpha_g(r_kar_k)$ and $f_kaf_k^*$, and to the elements $(a-2\varepsilon)_+$ and $h_k\left(\sum\limits_{g\in G}\alpha(r_kar_k)-\varepsilon\right)_+h_k^*$, we deduce that
$$[(a-3\varepsilon)_+]\le \left[\left(\sum\limits_{g\in G}\alpha_g(r_kar_k)-\varepsilon\right)_+\right]\le [a].$$
Therefore,
$$x_m\ll \left[\left(\sum\limits_{g\in G}\alpha_g(r_kar_k)-\varepsilon\right)_+\right]\ll x.$$
Note that the element $\left(\sum\limits_{g\in G}\alpha_g(r_kar_k)-\varepsilon\right)_+$ belongs to $(A\otimes\mathcal{K})^{\alpha}$ and so it is in the image of the inclusion map $i^s=i\otimes\mathrm{id}_\mathcal{K}\colon (A\otimes\mathcal{K})^{\alpha}\to A\otimes \mathcal{K}$. Since $m$ is arbitrary, we deduce that $x$ is the supremum of an increasing sequence in $\mathrm{Im}(\mathrm{Cu}(i))$ by Lemma \ref{lem: sup}. Choose a sequence $(y_n)_{n\in\mathbb{N}}$ in $\mathrm{Cu}(A^\alpha)$ such that $(\mathrm{Cu}(i)(y_n))_{n\in\mathbb{N}}$ is increasing in $\mathrm{Cu}(A)$ and set $x=\sup\limits_{n\in\mathbb{N}} (\mathrm{Cu}(i)(y_n))$. Since $\mathrm{Cu}(i)$ is an order embedding, it follows that $(y_n)_{n\in \mathbb{N}}$ is itself increasing in $\mathrm{Cu}(A^\alpha)$. Set $y=\sup\limits_{n\in\mathbb{N}} y_n$. Then $\mathrm{Cu}(y)=x$ since $\mathrm{Cu}(i)$ preserves suprema of increasing sequences.
\end{proof}
\begin{corollary}\label{cor: Ctz smgp of cp}
Let $A$ be a C*-algebra and let $\alpha$ be an action of a finite group $G$ on $A$ with the Rokhlin property. Then $\mathrm{Cu}(A\rtimes_\alpha G)$ is order-isomorphic to the semigroup:
\begin{align*}
\left\{x\in \mathrm{Cu}(A)\colon \exists \ (x_n)_{n\in\mathbb{N}} \mbox{ in } \mathrm{Cu}(A)\colon
\begin{aligned}
& x_n\ll x_{n+1} \ \forall n\in\mathbb{N} \mbox{ and } x=\sup\limits_{n\in\mathbb{N}} x_n,\\
& \ \mathrm{Cu}(\alpha_g)(x_n)=x_n \ \forall g\in G, \forall n\in\mathbb{N}
\end{aligned}
\right\}.
\end{align*}
\end{corollary}
\begin{proof} Since $\alpha$ has the Rokhlin property, the fixed point algebra $A^\alpha$ is Morita equivalent to the crossed product $A\rtimes_\alpha G$ by \cite[Theorem 2.8]{Phillips-Freeness-of-actions}. Therefore, there is a natural isomorphism $\mathrm{Cu}(A\rtimes_\alpha G)\cong \mathrm{Cu}(A^\alpha)$. Denote by $i\colon A^\alpha\to A$ the natural embedding. By Theorem \ref{thm: Rokhlin Constraint}, the semigroup $\mathrm{Cu}(A^\alpha)$ can be naturally identified with its image under the order embedding $\mathrm{Cu}(i)$, which is $\mathrm{Cu}(A)_\mathbb{N}^{\mathrm{Cu}(\alpha)}$ again by Theorem \ref{thm: Rokhlin Constraint}. The result follows.\end{proof}
\begin{corollary}\label{cor: n-divisible}
Let $A$ be a C*-algebra, let $\alpha$ be an action of a finite group $G$ on $A$ with the Rokhlin property, and set $n=|G|$. Suppose that $\mathrm{Cu}(\alpha_g)=\mathrm{id}_{\mathrm{Cu}(A)}$ for every $g\in G$, and that the map multiplication by $n$ on $\mathrm{Cu}(A)$ is an order embedding (in other words, whenever $x,y\in\mathrm{Cu}(A)$ satisfy $nx\leq ny$, one has $x\leq y$.) Then the map multiplication by $n$ in $\mathrm{Cu}(A)$ is an order-isomorphism.
\end{corollary}
\begin{proof} It suffices to show that for all $x\in\mathrm{Cu}(A)$, there exists $y\in \mathrm{Cu}(A)$ such that $x=ny$. By Theorem \ref{thm: Rokhlin Constraint} (ii), we have
$$\overline{\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)\right)}=\mathrm{Cu}(A)_\mathbb{N}^{\mathrm{Cu}(\alpha)}.$$
Since $\mathrm{Cu}(\alpha_g)=\mathrm{id}_{\mathrm{Cu}(A)}$ for all $g\in G$, this identity can be rewritten as
$$\overline{n\mathrm{Cu}(A)}=\mathrm{Cu}(A).$$
In particular, if $x$ is an element in $\mathrm{Cu}(A)$, then there exists a sequence $(y_k)_{k\in\mathbb{N}}$ in $\mathrm{Cu}(A)$ such that $(ny_k)_{k\in\mathbb{N}}$ is increasing and $x=\sup\limits_{k\in\mathbb{N}}(ny_k)$. Since $(ny_k)_{k\in\mathbb{N}}$ is increasing, it follows from our assumptions that $(y_k)_{k\in\mathbb{N}}$ is increasing as well. Set $y=\sup\limits_{k\in\mathbb{N}}y_k$. Then
$$x=\sup\limits_{k\in\mathbb{N}} (ny_k)=n\sup\limits_{k\in\mathbb{N}} y_k=ny,$$
and the claim follows.
\end{proof}
Let $A$ be a C*-algebra and let $p$ and $q$ be projections in $A$.
We say that $p$ and $q$ are \emph{Murray-von Neumann equivalent}, and
denote this by $p\sim_{\mathrm{MvN}}q$, if there exists $v\in A$ such that $p=v^*v$ and $q=vv^*$.
We say that $p$ is \emph{Murray-von Neumann subequivalent} to $q$, and denote this by $p\precsim_{\mathrm{MvN}} q$, if there is a projection $p'\in A$ such that $p\sim_{\mathrm{MvN}}p'$ and $p'\le q$.
The projection
$p$ is said to be \emph{finite} if whenever $q$ is a projection in $A$ with $q\leq p$ and $q\sim_{\mathrm{MvN}}p$, then $q=p$.
If $A$ is unital, then $A$ is said to be \emph{finite} if its unit is a finite projection. Moreover, $A$ is said to be
\emph{stably finite} if $\mathrm{M}_n(A)$ is finite for all $n\in \mathbb{N}$. If $A$ is not unital, we say that $A$ is (stably) finite if so
is its unitization $\widetilde{A}$.
\begin{lemma}\label{lem: projections}
Let $A$ be a stably finite C*-algebra and let $p\in A\otimes \mathcal{K}$ be a projection. Suppose that there are positive elements $a,b\in A\otimes\mathcal{K}$ such that $[p]=[a]+[b]$ in $\mathrm{Cu}(A)$. Then $a$ and $b$ are Cuntz equivalent to projections in $A\otimes\mathcal{K}$ (see the comments before Lemma~2.4 for the definition of Cuntz equivalence).
\end{lemma}
\begin{proof}
Let $a$ and $b$ be elements in $A\otimes\mathcal{K}$ as in the statement. By the comments before Lemma \ref{lem: morphism}, we have
$$[a]=\sup\limits_{\varepsilon>0} [(a-\varepsilon)_+] \ \ \mbox{ and } \ \ [b]=\sup\limits_{\varepsilon>0} [(b-\varepsilon)_+].$$
Since $[p]\ll [p]$, there exists $\varepsilon>0$ such that $[p]=[(a-\varepsilon)_+]+[(b-\varepsilon)_+]$. Choose a function
$f_\varepsilon\in \mathrm C_0(0,\infty)$ that is zero on the interval $[\varepsilon, \infty)$, nonzero at every point of $(0,\varepsilon)$
and $\|f_\varepsilon\|_\infty\leq 1$. Then
$$[p]+[f_\varepsilon(a)]+[f_\varepsilon(b)]= [(a-\varepsilon)_+]+[f_\varepsilon(a)]+[(b-\varepsilon)_+]+[f_\varepsilon(b)]\le [a]+[b]=[p].$$
Hence,
$[p]+[f_\varepsilon(a)]+[f_\varepsilon(b)]=[p]$.
Choose $c\in (A\otimes \mathcal{K})_+$ such that $[c]=[f_\varepsilon(a)]+[f_\varepsilon(b)]$ and $cp=0$. Then $p+c\precsim p$. By
\cite[Lemma 2.3 (iv)]{Kirchberg-Rordam}, for every $\delta>0$ there exists $x\in A\otimes \mathcal{K}$ such that
$$p+(c-\delta)_+=x^*x, \quad xx^*\in p(A\otimes \mathcal{K})p.$$
Fix $\delta>0$ and let $x$ be as above. Let $x=v|x|$ be the polar decomposition of $x$ in the bidual of $A\otimes \mathcal{K}$. Set $p'=vpv^*$ and
$c'=v(c-\delta)_+v^*$. Then $p'$ is a projection, $p'$ and $c'$ are orthogonal, $p$ and $p'$ are Murray-von Neumann equivalent, and
$p'+c'\in pAp$. Using stable finiteness of $A$ we conclude that $p=p'$ and $c'=0$. It follows that $(c-\delta)_+=0$ for all $\delta>0$,
and thus $c=0$. Hence, $f_\varepsilon(b)=f_\varepsilon(a)=0$ and in particular, $a$ and $b$ have a gap in their spectra. Therefore, they
are Cuntz equivalent to projections.
\end{proof}
Recall that the \emph{Murray-von Neumann semigroup} of $A$, denoted by $\mathrm V(A)$, is defined as the quotient of the set of projections of $A\otimes \mathcal{K}$ by the Murray-von Neumann equivalence relation.
Note that $p\precsim_{\mathrm{Cu}} q$ if and only if $p\precsim_{\mathrm{MvN}} q$.
On the other hand, $p\precsim_{\mathrm{MvN}} q$ and $q\precsim_{\mathrm{MvN}} p$ do not in general imply that $p\sim_{\mathrm{MvN}} q$, although this is the case whenever $A$ is finite.
In particular, if $A$ is finite, then $p\sim_{\mathrm{Cu}} q$ if and only if $p\sim_{\mathrm{MvN}} q$.
Hence, if $A$ is stably finite, then the semigroup $\mathrm{V}(A)$ can be identified with the ordered subsemigroup of $\mathrm{Cu}(A)$ consisting of the Cuntz equivalence classes of projections of $A\otimes \mathcal{K}$.
Recall that if $S$ is a semigroup in $\mathbf{Cu}$ and $x$ and $y$ are elements of $S$, we say that $x$ is \emph{compactly contained} in $y$, and denote this by $x\ll y$, if for every increasing sequence $(y_n)_{n\in\mathbb{N}}$ in $S$ such that $y=\sup\limits_{n\in\mathbb{N}}y_n$, there exists $n_0\in\mathbb{N}$ such that $x\leq y_n$ for all $n\geq n_0$.
\begin{definition}\label{df: compact} Let $S$ be a semigroup in $\mathbf{Cu}$ and let $x$ be an element of $S$. We say that $x$ is \emph{compact} if $x\ll x$. Equivalently, $x$ is compact if whenever $(x_n)_{n\in\mathbb{N}}$ is a sequence in $S$ such that $x=\sup\limits_{n\in\mathbb{N}}x_n$, then there exists $n_0\in\mathbb{N}$ such that $x_n=x$ for all $n\geq n_0$.\end{definition}
It is easy to check that the Cuntz class $[p]\in\mathrm{Cu}(A)$ of any projection $p$ in a C*-algebra $A$ (or in $A\otimes\mathcal{K}$) is a compact element in $\mathrm{Cu}(A)$. Moreover, when $A$ is stably finite, then every compact element of $\mathrm{Cu}(A)$ is the Cuntz class of a projection in $A\otimes \mathcal{K}$ by \cite[Theorem 3.5]{Brown-Ciuperca}. In particular, $\mathrm V(A)$ can be identified with the semigroup of compact elements of $\mathrm{Cu}(A)$ if $A$ is a stably finite C*-algebra.
When studying stably finite C*-algebras in connection with finite group actions with the Rokhlin property, the following lemma is often times useful. The result may be interesting in its own right, and could have been proved in \cite{Osaka-Phillips} since it is a direct application of their methods.
\begin{lemma}\label{lem: stable finiteness preserved}
Let $G$ be a finite group, let $A$ be a unital stably finite C*-algebra and let $\alpha\colon G\to\mathrm{Aut}(A)$ be an action with the Rokhlin property. Then the crossed product $A\rtimes_\alpha G$ and the fixed point algebra $A^\alpha$ are stably finite.\end{lemma}
\begin{proof} The fixed point algebra $A^\alpha$, being a unital subalgebra of $A$, is stably finite. On the other hand, the crossed product $A\rtimes_\alpha G$, being stably isomorphic to $A^\alpha$ by \cite[Theorem 2.8]{Phillips-Freeness-of-actions}, must itself also be stably finite.
\end{proof}
For unital, simple C*-algebras, part (ii) of the theorem below was first proved by Izumi in \cite{Izumi-I}.
The proof in our context follows completely different ideas.
\begin{theorem}\label{thm: image of inclusion on K-theory}
Let $A$ be a stably finite C*-algebra and let $\alpha$ be an action of a finite group $G$ on $A$ with the Rokhlin property. Let $i\colon A^\alpha\to A$ be the inclusion map.
\begin{itemize}
\item[(i)] The map $\mathrm V(i)\colon \mathrm V(A^\alpha)\to \mathrm V(A)$ is an order embedding and
\begin{align*}
&\mathrm{Im}(\mathrm V(i))=\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm V(\alpha_g)\right)=\left\{x\in \mathrm V(A)\colon \mathrm V(\alpha_g)(x)=x, \, \forall g\in G\right\}.
\end{align*}
\item[(ii)] If $A$ has an approximate identity consisting of projections, then $\mathrm K_0(i)\colon \mathrm K_0(A^\alpha)\to \mathrm K_0(A)$ is an order embedding and
\begin{align*}
\mathrm{Im}(\mathrm K_0(i))=\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm K_0(\alpha_g)\right)=\left\{x\in \mathrm K_0(A)\colon \mathrm K_0(\alpha_g)(x)=x, \, \forall g \in G\right\}.
\end{align*}
\end{itemize}
\end{theorem}
\begin{proof}
(i) The fact that $\mathrm V(i)$ is an order embedding is a consequence of Theorem \ref{thm: Rokhlin Constraint} and the remarks before and after Definition~\ref{df: compact}. Let us now show the inclusions
\begin{equation}\label{inclusions} \mathrm{Im}(\mathrm V(i))\subseteq\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm V(\alpha_g)\right)\subseteq\left\{x\in \mathrm V(A)\colon \mathrm V(\alpha_g)(x)=x \ \forall \ g\in G\right\}\subseteq \mathrm{Im}(\mathrm V(i)).\end{equation}
Let $p\in A^\alpha\otimes \mathcal{K}$ be a projection. By Theorem \ref{thm: Rokhlin Constraint}, there exists a sequence $(a_n)_{n\in\mathbb{N}}$ in $(A\otimes\mathcal{K})_+$ such that $\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)([a_n])\right)_{n\in\mathbb{N}}$ is increasing and
$$[i(p)]=\sup\limits_{n\in\mathbb{N}}\left(\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)([a_n])\right).$$
Since $[i(p)]$ is a compact element in $\mathrm{Cu}(A)$, it follows that there exists $n_0\in \mathbb{N}$ such that $[i(p)]=\sum\limits_{g\in G}\mathrm{Cu}(\alpha_g)([a_n])$ for all $n\geq n_0$. Fix $m\geq n_0$. It is easy to check that if $S$ is a semigroup in the category $\mathbf{Cu}$, then a sum of elements in $S$ is compact if and only if each summand is compact. It follows that $\mathrm{Cu}(\alpha_g)([a_m])$ is compact for all $g\in G$. In particular, and denoting the unit
of $G$ by $e$, we deduce that $[a_m]=\mathrm{Cu}(\alpha_e)([a_m])$ is compact. Since $A$ is stably finite by assumption, there exists a projection $q\in A\otimes\mathcal{K}$ such that $[q]=[a_m]$. Thus
$$\mathrm V(i)([p])=\sum\limits_{g\in G}\mathrm V(\alpha_g)([q])\in \mathrm{Im}\left(\sum\limits_{g\in G}\mathrm V(\alpha_g)\right),$$
showing that the first inclusion in (\ref{inclusions}) holds.
Using the fact that $\alpha_h\circ \left(\sum\limits_{g\in G}\alpha_g\right)=\sum\limits_{g\in G}\alpha_g$ for all $h\in G$, it is easy to check that
$$\mathrm{Im}\left(\sum\limits_{g\in G}\mathrm V(\alpha_g)\right)\subseteq\left\{x\in \mathrm V(A)\colon \mathrm V(\alpha_g)(x)=x, \, \forall g\in G\right\},$$
thus showing that the second inclusion also holds.
We proceed to prove the third inclusion. Let $x\in \mathrm{V}(A)$ be such that $\mathrm{V}(\alpha_g)(x)=x$ for all $g\in G$. Note that $x$ is compact as an element in $\mathrm{Cu}(A)$. It follows that $\mathrm{Cu}(\alpha_g)(x)=x$ for all $g\in G$ and hence by Theorem \ref{thm: Rokhlin Constraint} there exists $a\in (A^\alpha\otimes\mathcal{K})_+$ such that $\mathrm{Cu}(i)([a])=x$. Since the map $\mathrm{Cu}(i)$ is an order embedding again by Theorem \ref{thm: Rokhlin Constraint}, one concludes that $[a]$ is compact.
Finally, the fixed point algebra $A^\alpha$ is stably finite by Lemma \ref{lem: stable finiteness preserved} and thus there is a projection $p\in A^\alpha\otimes\mathcal{K}$ such that $[p]=[a]$ in $\mathrm{Cu}(A^\alpha)$. It follows that $\mathrm{Cu}(i)([p])=x$, showing that the third inclusion in (\ref{inclusions}) is also true.
(ii) Follows using the first part, together with the fact that the $\mathrm K_0$-group of a C*-algebra containing an approximate identity consisting of projections, agrees with the Grothendieck group of the Murray-von Neumann semigroup of the algebra; see Proposition~5.5.5 in \cite{Blackadar-book}.
\end{proof}
In the following corollary, the picture of $\mathrm V(A\rtimes_\alpha G)$ is valid for arbitrary $A$.
\begin{corollary}\label{cor: K-thy of cp} Let $A$ be a stably finite C*-algebra containing an approximate identity
consisting of projections, and let $\alpha$ be an action of a finite group $G$ on $A$ with the Rokhlin property. Then there are isomorphisms
\begin{align*}
\mathrm V(A\rtimes_\alpha G)&\cong \left\{x\in \mathrm V(A)\colon \mathrm V(\alpha_g)(x)=x, \, \forall g\in G\right\},\\
\mathrm K_\ast(A\rtimes_\alpha G)&\cong \left\{x\in \mathrm K_\ast(A)\colon \mathrm K_\ast(\alpha_g)(x)=x, \, \forall g\in G\right\}.
\end{align*}\end{corollary}
\begin{proof} Recall that if $\alpha$ has the Rokhlin property, then the fixed point algebra $A^\alpha$ and the crossed product $A\rtimes_\alpha G$ are Morita equivalent, and hence have isomorphic $\mathrm K$-theory and Murray-von Neumann semigroup. The isomorphisms for $\mathrm V(A\rtimes_\alpha G)$ and $\mathrm K_0(A\rtimes_\alpha G)$ then follow from Theorem \ref{thm: image of inclusion on K-theory} above.
Denote $B=A\otimes C(S^1)$ and give $B$ the diagonal action $\beta=\alpha\otimes\mathrm{id}_{C(S^1)}$ of $G$. Note that $B$ is stably finite and has an approximate identity
consisting of projections, and that $\beta$ has the Rokhlin property by part (i) of Proposition \ref{Rp properties}. Moreover, there is a natural isomorphism $B\rtimes_\beta G\cong (A\rtimes_\alpha G) \otimes C(S^1)$. Applying the K\"unneth formula in the first step, together with the conclusion of this proposition for $\mathrm K_0$ (which was shown to hold in the paragraph above) in the second step, and again the K\"unneth formula in the fourth step, we obtain
\begin{align*}
\left\{x\in \mathrm K_\ast(A)\colon \mathrm K_\ast(\alpha_g)(x)=x, \ \forall \ g\in G\right\}&\cong \left\{x\in \mathrm K_0(B)\colon \mathrm K_0(\beta_g)(x)=x, \ \forall \ g\in G\right\}\\
&\cong \mathrm K_0(B\rtimes_\beta G)\\
&\cong \mathrm K_0((A\rtimes_\alpha G)\otimes C(S^1))\\
&\cong \mathrm K_\ast(A\rtimes_\alpha G),\end{align*}
as desired.
\end{proof}
\section{Equivariant UHF-absorption}
In this section, we study absorption of UHF-algebras in relation to the Rokhlin property. We show that for a certain class of C*-algebras, absorption of a UHF-algebra of infinite type is equivalent to existence of an action with the Rokhlin property that is pointwise approximately inner. (The cardinality of the group is related to the type of the UHF-algebra.) Moreover, in this case, not only the C*-algebra absorbs the corresponding UHF-algebra, but also the action in question absorbs the model action constructed in Example \ref{eg: model action}. Thus, Rokhlin actions allow us to prove that certain algebras are \emph{equivariantly} UHF-absorbing.
\subsection{Unique $n$-divisibility.}
The goal of this section is to show that for certain C*-algebras, absorption of the UHF-algebra of type $n^\infty$ is equivalent to its Cuntz semigroup being $n$-divisible. Along the way, we show that for a C*-algebra $A$, the Cuntz semigroups of $A$ and of $A\otimes \mathrm{M}_{n^\infty}$ are isomorphic if and only if $\mathrm{Cu}(A)$ is uniquely $n$-divisible.
We point out that some of the results of this section, particularly Theorem~\ref{Ctz smgp and unique n divis},
were independently obtained in the recent preprint \cite{APT}, as applications of their theory of tensor
products of Cuntz semigroups. On the other hand, the proofs we give here are direct and elementary.
Additionally, our techniques also apply to other functors, for instance the functor $\mathrm{Cu}^\sim$.
We begin defining the main notion of this section. Recall that if $S$ and $T$ are ordered semigroup and $\varphi\colon S\to T$ is a semigroup homomorphism, we say that $\varphi$ is an \emph{order embedding} if $\varphi(s)\leq \varphi(s')$ implies $s\leq s'$ for all $s, s'\in S$. A semigroup isomorphism is called an \emph{order preserving} semigroup isomorphism if it is an order embedding.
\begin{definition}\label{df: uniquely n div}
Let $S$ be an ordered semigroup and let $n$ be a positive integer.
\begin{enumerate}
\item We say that $S$ is \emph{$n$-divisible}, if for every $x$ in $S$ there exists $y$ in $S$ such that $x=ny$.
\item We say that $G$ is \emph{uniquely $n$-divisible}, if multiplication by $n$ on $S$ is an order preserving semigroup isomorphism.
\end{enumerate}
\end{definition}
Recall that the category $\mathbf{Cu}$ is closed under sequential inductive limits.
\begin{lemma}\label{lem: n-div-inductivelim}
Let $n\in \mathbb{N}$ and let $S$ be a semigroup in the category $\mathbf{Cu}$. Denote by $\rho\colon S\to S$ the map given by $\rho(s)=ns$ for
all $s\in S$. Let $T$ be the semigroup in $\mathbf{Cu}$ obtained as the inductive limit
of the sequence
\begin{align*}
\xymatrix{
S\ar[r]^-{\rho} &S\ar[r]^-{\rho} & S\ar[r]^-{\rho} & \cdots.
}
\end{align*}
Then $T$ is uniquely $n$-divisible.
\end{lemma}
\begin{proof}
Let $S$ and $T$ be as in the statement. To avoid any confusion with the notation, we will denote the map between the $k$-th and
$(k+1)$-st copies of $S$ by $\rho_k$, so we write $T$ as the direct limit
\begin{align*}
\xymatrix{
S\ar[r]^-{\rho_1} &S\ar[r]^-{\rho_2} & S\ar[r]^-{\rho_3} &\ar[r] \cdots & T.
}
\end{align*}
For $k,m\in\mathbb{N}$ with $m>k$, we let $\rho_{k,m}\colon S\to S$ denote the composition $\rho_{m-1}\circ\rho_{m-2}\circ\cdots\circ\rho_k$, and we let
$\rho_{k,\infty}\colon S\to T$ denote the canonical map from the $k$-th copy of $S$ to $T$.
Let $s, t\in T$ satisfy $ns\le nt$. By part (i) of Proposition \ref{prop: inductivelimitCu}, there exist sequences $(s_k)_{k\in\mathbb{N}}$ and
$(t_k)_{k\in \mathbb{N}}$ in $S$ such that
\begin{align*}
\rho_k(s_k)\ll s_{k+1} \mbox{ for all } k\in\mathbb{N} \ \ &\mbox{ and } \ \ s=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(s_k)\\
\rho_k(t_k)\ll t_{k+1} \mbox{ for all } k\in\mathbb{N} \ \ &\mbox{ and } \ \ t=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(t_k).
\end{align*}
It follows that $\rho_{k,\infty}(s_k)\ll \rho_{k+1,\infty}(s_{k+1})$ and $\rho_{k,\infty}(t_k)\ll \rho_{k+1,\infty}(t_{k+1})$ for all $k\in \mathbb{N}$.
Let $k\geq 2$ be fixed. Since
$$\rho_{k,\infty}(n s_k)\ll n s\le n t=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(n t_k),$$
there exists $l\in \mathbb{N}$ such that $\rho_{k,\infty}(n s_k)\le \rho_{l,\infty}(n t_l)$. Use part (ii) of Proposition
\ref{prop: inductivelimitCu} and $\rho_{j-1}(s_{j-1})\ll s_j$ for all $j\in \mathbb{N}$, to choose $m\ge k,l$ such that
$\rho_{k-1,m}(n s_{k-1})\le \rho_{l,m}(n t_l)$. Therefore,
\begin{align*}
\rho_{k-1,\infty}(s_{k-1})&=\rho_{m+1,\infty}(\rho_{m+1}(\rho_{k-1,m}(s_{k-1})))\\
&=\rho_{m+1,\infty}(n\rho_{k-1,m}(s_{k-1}))\\
&=\rho_{m+1,\infty}(\rho_{k-1,m}(ns_{k-1}))\\
&\le \rho_{m+1,\infty}(\rho_{l,m}(n t_l))\\
&=\rho_{m+1,\infty}(n\rho_{l,m}(t_l))\\
&=\rho_{m+1,\infty}(\rho_{m+1}(\rho_{l,m}(t_l)))\\
&=\rho_{l,\infty}(t_l)\\
&\le t,
\end{align*}
this is, $\rho_{k-1,\infty}(s_{k-1})\le t$. Since this holds for all $k\ge 2$, we conclude that
$$s=\sup\limits_{k\geq 2} \rho_{k-1,\infty}(s_{k-1})\le t.$$
We have shown that $ns\le nt$ in $T$ implies $s\le t$. In other words, multiplication by $n$ on $T$ is an order embedding, as desired.
To conclude the proof, let us show that $T$ is $n$-divisible. Fix $t\in T$ and choose a sequence $(t_k)_{k\in\mathbb{N}}$ in
$T$ satisfying
$$\rho_k(t_k)\ll t_{k+1} \mbox{ for all } k\in\mathbb{N} \ \ \mbox{ and } \ \ t=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(t_k).$$
For each $k\in\mathbb{N}$ we have
\[
\rho_{k,k+2}(t_k)=n^2t_k=n\rho_{k+1,k+2}(t_k).
\]
With $x_k=\rho_{k+1,\infty}(t_k)$, it follows that $\rho_{k,\infty}(t_k)= nx_k$. Since $(\rho_{k,\infty}(t_k))_{k\in\mathbb{N}}$ is an increasing
sequence in $T$, we deduce that $(nx_k)_{k\in\mathbb{N}}$ is an increasing sequence in $T$ as well. Since we have shown in the first part of this
proof that multiplication by $n$ on $T$ is an order
embedding, we conclude that $(x_k)_{k\in\mathbb{N}}$ is also increasing. With $x$ denoting the supremum of $(x_k)_{k\in\mathbb{N}}$, we have
\[
t=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(t_k)=\sup nx_k=n\sup\limits_{k\in\mathbb{N}} x_k=nx,
\]
which completes the proof.
\end{proof}
We point out that the functor $\mathrm{Cu}^\sim$ does not distinguish between *-homomorphisms that are approximately unitarily equivalent (with unitaries taken in the unitization). On the other hand, the corresponding statement for approximate unitary equivalence with unitaries taken in the multiplier algebra is not known in general.
The following proposition, of independent interest, implies that this is the case whenever the codomain has stable rank one. This will be used in the proof of Lemma~\ref{lem: n} to
deduce that certain *-homomorphisms are trivial at the level of $\mathrm{Cu}^\sim$.
\begin{proposition}\label{prop: multiplierau}
Let $A$ and $B$ be C*-algebras with $B$ stable, and let $\phi, \psi\colon A\to B$ be *-homomorphisms. Suppose that $\phi$ and $\psi$ are
approximately unitarily equivalent with unitaries taken in the multiplier algebra of $B$. Then $\phi$ and $\psi$ are approximately unitarily
equivalent with unitaries taken in the unitization of $B$.
\end{proposition}
\begin{proof}
Denote by $\iota\colon B\to \mathrm{M}(B)^\infty$ the canonical inclusion as constant sequences. We will identify $B$ with a
subalgebra of $\mathrm{M}(B)$, and suppress $\iota$ from the notation. Hence we will denote the maps
$\iota\circ\phi,\iota\circ\psi\colon A\to \mathrm{M}(B)^\infty$ again by $\phi$ and $\psi$, respectively.
Let $F\subseteq A$ be a finite set. Then there exists a unitary $u=\pi_{\mathrm{M}(B)}((u_n)_{n\in\mathbb{N}})$ in $\mathrm{M}(B)^\infty$ such that
$\phi(a)=u\psi(a)u^*$ for all $a\in F$. Choose a sequence $(s_n)_{n\in \mathbb{N}}$ of positive contractions in $B$ such that
$$\lim\limits_{n\to \infty} s_n\psi(a)=\lim\limits_{n\to \infty} \psi(a)s_n=\psi(a)$$
for all $a\in F$. Let $s=\pi_{\mathrm M(B)}((s_n)_{n\in\mathbb{N}})$ denote the image of $(s_n)_{n\in\mathbb{N}}$ in $B^\infty\subseteq \mathrm M(B)^\infty $. Then
$$s\phi(a)=\phi(a)s=\phi(a)$$
for all $a\in F$. Since $B$ is stable, we have $B\subseteq \overline{\mathrm{GL}(\widetilde B)}$ by \cite[Lemma 4.3.2]{B-R-T-T-W}. Hence, elements in $B$ have
approximate polar decompositions with unitaries taken in $\widetilde B$. This implies that there exists a sequence $(v_n)_{n\in \mathbb{N}}$
of unitaries in $\widetilde B$ such that $\lim\limits_{n\to \infty}\|u_ns_n-v_ns_n\|=0$. Let $v=\pi_{\widetilde{B}}((v_n)_{n\in \mathbb{N}})$ denote the image
of $(v_n)_{n\in \mathbb{N}}$ in $(\widetilde{B})^\infty$. Then $us=vs$ and
$$\phi(a)=u\psi(a)u^*=us\psi(a)su^*=vs\psi(a)sv^*=v\psi(a)v^*$$
for all $a\in F$. This implies that $\lim\limits_{n\to \infty}\|\phi(a)-v_n\psi(a)v_n^*\|=0$ for all $a\in F$.
Since $v_n$ is a unitary in $\widetilde{B}$ for all $n\in\mathbb{N}$, we conclude that $\phi$ and $\psi$ are approximately unitarily equivalent
with unitaries taken in the unitization of $B$.
\end{proof}
Let $n,k\in \mathbb{N}$. We let $\left(f_{i,j}^{(n^k)}\right)_{i,j=0}^{n^k-1}$ denote the set of matrix units of $\mathrm{M}_{n^k}(\mathbb{C})$. Recall that
if $A$ and $B$ are C*-algebras and $\phi,\psi\colon A\to B$ are *-homomorphisms with orthogonal ranges, then $\phi+\psi$ is also
a *-homomorphism and $\mathrm{Cu}(\phi+\psi)=\mathrm{Cu}(\phi)+\mathrm{Cu}(\psi)$.
\begin{lemma}\label{lem: n}
Let $A$ be a C*-algebra and let $n, k\in \mathbb{N}$. Let $\iota_k\colon A\to \mathrm{M}_{n^k}(A)$ be the map given by $\iota_k(a)=a\otimes f^{(n^k)}_{0,0}$ for all $a\in A$, and let $j_k\colon \mathrm{M}_{n^k}(A)\to \mathrm{M}_{n^{k+1}}(A)$ be the map given by $j_k(a)=a\otimes 1_n$ for all $a\in \mathrm{M}_{n^k}(A)$.
Then the map
\begin{align*}
\mathrm{Cu}(\iota_{k+1})^{-1}\circ \mathrm{Cu}(j_k)\circ \mathrm{Cu}(\iota_k)\colon \mathrm{Cu}(A)\to \mathrm{Cu}(A),
\end{align*}
is the map multiplication by $n$.
\end{lemma}
\begin{proof}
Since $\mathrm{Cu} $ is invariant under stabilization, we may assume that the algebra $A$ is stable.
Fix $k$ in $\mathbb{N}$. For each $0\le i\le n-1$, let $j_{k,i}\colon \mathrm{M}_{n^k}(A)\to \mathrm{M}_{n^{k+1}(A)}$ be the map defined by
$j_{k,i}(b)=b\otimes f_{i,i}^{(n)}$
for all $b\in \mathrm{M}_{n^k}(A)$. Then the maps $(j_{k,i})_{i=0}^{n-1}$ have orthogonal ranges and $j_k=\sum\limits_{i=1}^{n-1} j_{k,i}$. By the comments
before this lemma, we have
$$\mathrm{Cu}(j_k)=\sum\limits_{i=0}^{n-1}\mathrm{Cu}(j_{k,i}).$$
Since $f_{i,i}^{(n)}$ and $f_{\ell,\ell}^{(n)}$ are unitarily equivalent in $\mathrm{M}_n(\mathbb{C})$ for all $i,\ell=0,\ldots,n-1$, we conclude that the
maps $j_{k,i}$ and $j_{k,\ell}$ are unitarily equivalent with unitaries in the multiplier algebra of $\mathrm{M}_{n^{k+1}}(A)$. By Proposition
\ref{prop: multiplierau}, this implies that the maps $j_{k,i}$ and $j_{k,\ell}$ are approximately unitarily equivalent (with unitaries taken in the unitization of $\mathrm M_{n^{k+1}}(A)$). Since approximate
unitary equivalent maps yield the same morphism at the level of the Cuntz semigroup, we deduce that $\mathrm{Cu}(j_{k,i})=\mathrm{Cu}(j_{k,\ell})$ for all
$i,\ell=0,\ldots,n-1$.
Given a positive element $a$ in $A\otimes\mathcal{K}$,we have
\begin{align*}
(\mathrm{Cu}(\iota_{k+1})^{-1}\circ\mathrm{Cu}(j_k)\circ \mathrm{Cu}(\iota_k))([a])&=(\mathrm{Cu}(\iota_{k+1})^{-1}\circ\mathrm{Cu}(j_k))\left(\left[a\otimes f_{0,0}^{(n^k)}\right]\right)\\
&=\mathrm{Cu}(\iota_{k+1})^{-1}\left(\sum\limits_{i=0}^{n-1}\mathrm{Cu}(j_{k,i})\left(\left[a\otimes f_{0,0}^{(n^k)}\right]\right)\right)\\
&=\mathrm{Cu}(\iota_{k+1})^{-1}\left(n\mathrm{Cu}(j_{k,0})\left(\left[a\otimes f_{0,0}^{(n^k)}\right]\right)\right)\\
&=n\mathrm{Cu}(\iota_{k+1})^{-1}\left(\left[a\otimes f_{0,0}^{(n^k)}\otimes f_{0,0}^{(n)}\right]\right)\\
&=n\mathrm{Cu}(\iota_{k+1})^{-1}\left(\left[a\otimes f_{0,0}^{(n^{k+1})}\right]\right)\\
&=n [a]
\end{align*}
We conclude that $\mathrm{Cu}(\iota_{k+1})^{-1}\circ \mathrm{Cu}(j_k)\circ \mathrm{Cu}(\iota_k)$ is the map multiplication by $n$.
\end{proof}
\begin{theorem}\label{Ctz smgp and unique n divis}
Let $A$ be a C*-algebra and let $n\in\mathbb{N}$ with $n\ge 2$. Then $\mathrm{Cu}(A)$ is uniquely $n$-divisible if and only if $\mathrm{Cu}(A)\cong \mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ as order semigroups.
\end{theorem}
\begin{proof}
Assume that there exists an isomorphism $\mathrm{Cu}(A)\cong \mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ as ordered semigroups. Using
the inductive limit decomposition $\mathrm{M}_{n^\infty}=\varinjlim \mathrm{M}_{n^k}$ with connecting maps
$j_k\colon \mathrm{M}_{n^{k-1}}(A)\to \mathrm{M}_{n^{k}}(A)$ is given by
$j_k(a)=a\otimes 1_{n}$ for all $a\in \mathrm{M}_{n^{k-1}}(A)$, we can write $A\otimes \mathrm{M}_{n^\infty}$ as the inductive limit
\[
\xymatrix{
A\ar[r]^-{j_1} &\mathrm{M}_n(A)\ar[r]^-{j_2} & \mathrm{M}_{n^2}(A)\ar[r]^-{j_3} &\ar[r] \cdots & A\otimes\mathrm{M}_{n^\infty}.
}
\]
By continuity of the functor $\mathrm{Cu}$ (see \cite[Theorem 2]{Coward-Elliott-Ivanescu}),
the semigroup $\mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ is isomorphic to the inductive limit in the category $\mathbf{Cu}$ of the sequence
\begin{align}\label{A_UHF}
\xymatrix{
\mathrm{Cu}(A)\ar[r]^-{\mathrm{Cu}(j_0)} &\mathrm{Cu}(\mathrm{M}_n(A))\ar[r]^-{\mathrm{Cu}(j_1)} & \mathrm{Cu}(\mathrm{M}_{n^2}(A))\ar[r]^-{\mathrm{Cu}(j_2)} & \cdots.
}
\end{align}
By \cite[Appendix]{Coward-Elliott-Ivanescu}, the inclusion $i_{k}\colon A\to \mathrm{M}_{n^k}(A)$ from $A$ into the upper left corner
of $\mathrm{M}_{n^k}(A)$ induces an isomorphism between the Cuntz semigroup of $A$ and that of $\mathrm{M}_{n^k}(A)$. For $k\in \mathbb{N}$,
let $\varphi_k\colon \mathrm{Cu}(A)\to\mathrm{Cu}(A)$ be given by
$$\varphi_k=\mathrm{Cu}(i_{k+1})^{-1}\circ\mathrm{Cu}(j_{k})\circ\mathrm{Cu}(i_k).$$
The sequence \eqref{A_UHF} implies that $\mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ is the inductive limit of the sequence
\begin{align}\label{A_UHF_2}
\xymatrix{
\mathrm{Cu}(A)\ar[r]^-{\varphi_1} &\mathrm{Cu}(A)\ar[r]^-{\varphi_2} & \mathrm{Cu}(A)\ar[r]^-{\varphi_3} & \cdots.
}
\end{align}
By Lemma \ref{lem: n}, each $\varphi_k$ is the map multiplication by $n$. It follows from Lemma \ref{lem: n-div-inductivelim}
that $\mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ is uniquely $n$-divisible. This shows the ``if" implication.
Conversely, assume that $\mathrm{Cu}(A)$ is uniquely $n$-divisible and adopt the notation used above.
The map $\varphi_k$ is the map
multiplication by $n$ on $\mathrm{Cu}(A)$ by Lemma \ref{lem: n}, so it is an order-isomorphism by assumption.
By the inductive limit expression of $\mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ in \eqref{A_UHF_2}, we conclude that
$\mathrm{Cu}(A)\cong \mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$, as desired.
\end{proof}
\begin{remark}
Let $\mathcal{Q}$ denote the universal UHF-algebra. Using the same ideas as in the proof of the previous theorem, one can show that $\mathrm{Cu}(A)\cong\mathrm{Cu}(A\otimes \mathcal{Q})$ if and only if $\mathrm{Cu}(A)$ is uniquely $p$-divisible for every prime number $p$.
\end{remark}
We now turn to direct limits of one-dimensional NCCW-complexes. The following lemma will allow us to reduce to the case where the
algebra itself is a one-dimensional NCCW-complex when proving that multiplication by $n$ is an order embedding at the level of the Cuntz semigroup.
\begin{lemma}\label{lem: cancellation}
Let $(S_k,\rho_k)_{k\in\mathbb{N}}$ be an inductive system in the category $\mathbf{Cu}$, and let $S=\varinjlim (S_k,\rho_k)$ be its inductive
limit in $\mathbf{Cu}$. Let $n\in \mathbb{N}$. If multiplication by $n$ on $S_k$ is an order embedding for all $k$ in $\mathbb{N}$, then the same holds for $S$.
\end{lemma}
\begin{proof}
For $l\geq k$, denote by $\rho_{k,l}\colon S_k\to S_{l+1}$ the composition $\rho_{k,l}=\rho_l\circ\cdots\circ\rho_k$, and denote by
$\rho_{k,\infty}\colon S_k\to S$ the canonical map as in the definition of the inductive limit. Let $s,t\in S$ satisfy $ns\le nt$.
By part (i) of Proposition \ref{prop: inductivelimitCu}, for each $k\in\mathbb{N}$ there exist $s_k, t_k\in S_k$ such that
\begin{align*}
\rho_k(s_k)\ll s_{k+1} \ \ &\mbox{ and } \ \ s=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(s_k),\\
\rho_k(t_k)\ll t_{k+1} \ \ &\mbox{ and } \ \ t=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(t_k).
\end{align*}
Note in particular that $\rho_{k,\infty}(s_k)\ll\rho_{k+1,\infty}(s_{k+1})$ and $\rho_{k,\infty}(t_k)\ll\rho_{k+1,\infty}(t_{k+1})$ for all $k\in \mathbb{N}$.
Fix $k\in \mathbb{N}$. Then
$$\rho_{k,\infty}(ns_k)\ll\rho_{k+1,\infty}(ns_{k+1})\ll \sup\limits_{j\in\mathbb{N}}\rho_{j, \infty}(nt_j).$$
By the definition of the compact containment relation, there exists $j\in\mathbb{N}$ such that
$$\rho_{k,\infty}(ns_k)\ll\rho_{k+1,\infty}(ns_{k+1})\le\rho_{j, \infty}(nt_j).$$
By part (ii) of Proposition \ref{prop: inductivelimitCu}, there exists $l\in\mathbb{N}$ such that
$$n\rho_{k,l}(s_k)=\rho_{k,l}(ns_k)\le\rho_{j, l}(nt_j)=n\rho_{j, l}(t_j).$$
Using that multiplication by $n$ on $S_k$ is an order embedding, we obtain $\rho_{k,l}(s_k)\le \rho_{j, l}(t_j)$. In particular,
$$\rho_{k,\infty}(s_k)\le \rho_{j, \infty}(t_j)\le t.$$
Since $k\in\mathbb{N}$ is arbitrary and $s=\sup\limits_{k\in\mathbb{N}} \rho_{k,\infty}(s_k)$, we conclude that $s\le t$.
\end{proof}
\begin{proposition}\label{pro: NCCW complexes}
Let $A$ be a C*-algebra that can be written as the inductive limit of 1-dimensional NCCW-complexes. Then the endomorphism of $\mathrm{Cu}(A)$ given by multiplication by $n$ is an order embedding.
\end{proposition}
\begin{proof}
By Lemma \ref{lem: cancellation}, it is sufficient to show that the proposition holds when $A$ is a 1-dimensional NCCW-complex. Let $E=\bigoplus_{j=1}^r\mathrm{M}_{k_j}(\mathbb{C})$ and $F=\bigoplus_{j=1}^s\mathrm{M}_{l_j}(\mathbb{C})$ be finite dimensional C*-algebras, and for $x\in [0,1]$ denote by $\mathrm{ev}_x\colon \mathrm{C}([0,1], F)\to F$ the evaluation map at the point $x$. Assume that $A$ is given by the pullback decomposition
\begin{equation*}
\xymatrix{A \ar[d] \ar[rr] & & E\ar[d] \\ \mathrm C([0,1], F)\ar[rr]_-{\mathrm{ev}_0\oplus \mathrm{ev}_1} & & F\oplus F,}
\end{equation*}
By \cite[Example 4.2]{Antoine-Perera-Santiago}, the Cuntz semigroup of $A$ is order-isomorphic to a subsemigroup of
$$\mathrm{Lsc}\left([0,1], \overline{\mathbb Z_+}^s\right)\oplus (\overline{\mathbb Z_+})^r.$$
Since multiplication by $n$ on this semigroup is an order embedding, the same holds for any subsemigroup;
in particular, it hold for $\mathrm{Cu}(A)$.
\end{proof}
\begin{corollary}\label{cor: Minfty absorbing}
Let $A$ be a C*-algebra in one of the following classes: unital algebras that can written as inductive limits 1-dimensional NCCW-complexes with trivial $\mathrm K_1$-groups; simple algebras with trivial $\mathrm K_0$-groups that can be written as inductive limits 1-dimensional NCCW-complexes with trivial $\mathrm K_1$-groups; and algebras that can written as inductive limits of punctured-tree algebras. Let $n\in \mathbb{N}$.
Suppose that the map multiplication by $n$ on $\mathrm{Cu}(A)$ is an order-isomorphism. Then $A\cong A\otimes \mathrm{M}_{n^\infty}$.
\end{corollary}
\begin{proof}
By Proposition \ref{pro: NCCW complexes} together with the assumptions in the statement, it follows that the map multiplication by $n$ on $\mathrm{Cu}(A)$ is an order-isomorphism. Hence, it is an isomorphism in the category $\mathrm{Cu}$. By part (ii) of Theorem \ref{Ctz smgp and unique n divis}, there is an isomorphism $\mathrm{Cu}(A)\cong \mathrm{Cu}(A\otimes \mathrm{M}_{n^\infty})$ in $\mathbf{Cu}$.
The same arguments used at the end of the proof of Theorem \ref{thm: mainclassification} show that the classes of C*-algebras in the statement can be classified up to stable isomorphism by their Cuntz semigroup. Therefore, we deduce that
$$A\otimes \mathcal{K}\cong A\otimes \mathrm{M}_{n^\infty}\otimes \mathcal{K}.$$
Using that $\mathrm{M}_{n^\infty}$-absorption is inherited by hereditary C*-subalgebras (\cite[Corollary 3.1]{Toms-Winter}), we conclude that $A\cong A\otimes \mathrm{M}_{n^\infty}$.
\end{proof}
\subsection{Absorption of the model action}
We now proceed to obtain an equivariant UHF-absorption result (compare with \cite[Theorems 3.4 and 3.5]{Izumi-II}).
\begin{theorem}\label{thm: Rp action absorbs model action}
Let $G$ be a finite group and let $A$ be a C*-algebra belonging to one of the classes of C*-algebras described in Corollary \ref{cor: Minfty absorbing}. Then the following statements are equivalent:
\begin{enumerate} \item The C*-algebra $A$ absorbs the UHF-algebra $\mathrm{M}_{|G|^\infty}$.
\item There is an action $\alpha\colon G\to\mathrm{Aut}(A)$ with the Rokhlin property such that $\mathrm{Cu}(\alpha_g)=\mathrm{id}_{\mathrm{Cu}(A)}$ for all $g\in G$.
\item There are actions of $G$ on $A$ with the Rokhlin property, and for any action $\beta\colon G\to\mathrm{Aut}(A)$ with the Rokhlin property and for any action $\delta\colon G\to\mathrm{Aut}(A)$ such that $\mathrm{Cu}(\beta_g)=\mathrm{Cu}(\delta_g)$ for all $g\in G$, one has
$$(A,\beta)\cong (A\otimes \mathrm{M}_{|G|^\infty},\delta\otimes\mu^G),$$
that is, there is an isomorphism $\varphi\colon A\to A\otimes \mathrm{M}_{|G|^\infty}$ such that
$$\varphi\circ\beta_g=(\delta\otimes\mu^G)_g\circ\varphi$$
for all $g$ in $G$.\end{enumerate}
In particular, if the above statements hold for $A$, and if $\alpha\colon G\to\mathrm{Aut}(A)$ is an action with the Rokhlin property such that $\mathrm{Cu}(\alpha_g)=\mathrm{id}_{\mathrm{Cu}(A)}$ for all $g\in G$, then $(A,\alpha)\cong (A\otimes \mathrm{M}_{|G|^\infty},\mathrm{id}_A\otimes\mu^G)$.\end{theorem}
\begin{proof}
(i) implies (ii). Fix an isomorphism $\varphi\colon A\to A\otimes \mathrm{M}_{|G|^{\infty}}$ and define an action $\alpha\colon G\to\mathrm{Aut}(A)$ by $\alpha_g=\varphi^{-1}\circ(\mathrm{id}_A\otimes\mu^G)_g\circ\varphi$ for all $g$ in $G$. For a fixed group element $g$ in $G$, the automorphism $\mathrm{id}_A\otimes\mu^G_g$ of $A\otimes\mathrm{M}_{|G|^\infty}$ is approximately inner, and hence so is $\alpha_g$. It follows that $\mathrm{Cu}(\alpha_g)=\mathrm{id}_{\mathrm{Cu}(A)}$ for all $g$ in $G$, as desired.
(ii) implies (i). Assume that there is an action $\alpha\colon G\to\mathrm{Aut}(A)$ with the Rokhlin property such that $\mathrm{Cu}(\alpha_g)=\mathrm{id}_{\mathrm{Cu}(A)}$ for all $g\in G$. Then $A\cong A\otimes \mathrm{M}_{|G|^\infty}$ by Proposition \ref{pro: NCCW complexes}, Corollary \ref{cor: Minfty absorbing} and Corollary \ref{cor: n-divisible}.
(i) and (ii) imply (iii). Let $\beta$ and $\delta$ be actions of $G$ on $A$ as in the statement. Since $\mathrm{M}_{|G|^{\infty}}$ is a strongly self-absorbing algebra, there exists an isomorphism $\phi\colon A\to A\otimes \mathrm{M}_{|G|^{\infty}}$ that is approximately unitarily equivalent to the map $\iota\colon A\to A\otimes \mathrm{M}_{|G|^{\infty}}$ given by $\iota(a)=a\otimes 1_{\mathrm{M}_{|G|^{\infty}}}$ for $a$ in $A$. In particular, one has $\mathrm{Cu}(\phi)=\mathrm{Cu}(\iota)$. Hence, for every $a\in (A\otimes\mathcal{K})_+$ we have
$$(\mathrm{Cu}(\phi)\circ \mathrm{Cu}(\beta_g))([a])=\mathrm{Cu}(\iota)[(\beta_g\otimes \mathrm{id}_{\mathcal{K}})(a)]=\left[((\beta_g\otimes \mathrm{id}_{\mathcal{K}})(a))\otimes 1_{\mathrm{M}_{|G|^\infty}}\right]$$
and
\begin{align*}
(\mathrm{Cu}(\delta_g\otimes \mu^G)\circ\mathrm{Cu}(\phi))([a])&=\mathrm{Cu}(\delta_g\otimes \mu^G)\left(\left[a\otimes 1_{\mathrm{M}_{|G|^\infty}}\right]\right)\\
&=\left[((\delta_g\otimes \mathrm{id}_{\mathcal{K}})(a))\otimes 1_{\mathrm{M}_{|G|^\infty}}\right].
\end{align*}
Since $\mathrm{Cu}(\beta_g)=\mathrm{Cu}(\delta_g)$ for all $g\in G$, it follows that
$$\mathrm{Cu}(\phi)\circ \mathrm{Cu}(\beta_g)=\mathrm{Cu}(\delta_g\otimes \mu_g)\circ\mathrm{Cu}(\phi)$$
for all $g$ in $G$. In other words, the $\mathbf{Cu}$-isomorphism $\mathrm{Cu}(\phi)\colon \mathrm{Cu}(A)\to \mathrm{Cu}(A\otimes \mathrm{M}_{|G|^\infty})$ is equivariant. Therefore, by the unital case of Theorem \ref{classif Rp on NCCW}, there exists an isomorphism $\varphi\colon A\to A \otimes \mathrm{M}_{|G|^\infty}$ such that $\varphi\circ\beta_g=(\delta\otimes\mu^G)_g\circ\varphi$ for all $g\in G$, showing that $\beta$ and $\delta\otimes\mu^G$ are conjugate.
(iii) implies (i). The existence of an action $\beta\colon G\to\mathrm{Aut}(A)$ with the Rokhlin property implies the existence of an isomorphism $A\to A\otimes \mathrm{M}_{|G|^\infty}$, simply by taking $\delta=\beta$.
The last claim follows immediately from (iii).
\end{proof}
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\end{document}
|
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"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Flutter http post data when click the button I want to send post data "1" or "2" int value when the button is pressed with flutter (http package). How can I do that?
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'http://test.com/control.php');
var response = await http.post(url, body: {'status': 2});
print('Response status: ${response.statusCode}');
print('Response body: ${response.body}');
},
child: Text('Close'),
),
],
),
),
Sorry for my english and Thank you for your helping!
A: You can do it like this.
final body = jsonEncode({'status': 1});
final url = Uri.http('http://test.com', 'control.php');
final response = await http.post(
url,
headers: {'Content-Type': 'application/json'},
body: body
);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,794
|
package cognitosync_test
import (
"context"
"testing"
"time"
"github.com/aws/aws-sdk-go/aws"
"github.com/aws/aws-sdk-go/aws/awserr"
"github.com/aws/aws-sdk-go/aws/request"
"github.com/aws/aws-sdk-go/awstesting/integration"
"github.com/aws/aws-sdk-go/service/cognitosync"
)
var _ aws.Config
var _ awserr.Error
var _ request.Request
func TestInteg_00_ListIdentityPoolUsage(t *testing.T) {
ctx, cancelFn := context.WithTimeout(context.Background(), 5*time.Second)
defer cancelFn()
sess := integration.SessionWithDefaultRegion("us-west-2")
svc := cognitosync.New(sess)
params := &cognitosync.ListIdentityPoolUsageInput{}
_, err := svc.ListIdentityPoolUsageWithContext(ctx, params, func(r *request.Request) {
r.Handlers.Validate.RemoveByName("core.ValidateParametersHandler")
})
if err != nil {
t.Errorf("expect no error, got %v", err)
}
}
func TestInteg_01_DescribeIdentityPoolUsage(t *testing.T) {
ctx, cancelFn := context.WithTimeout(context.Background(), 5*time.Second)
defer cancelFn()
sess := integration.SessionWithDefaultRegion("us-west-2")
svc := cognitosync.New(sess)
params := &cognitosync.DescribeIdentityPoolUsageInput{
IdentityPoolId: aws.String("us-east-1:aaaaaaaa-bbbb-cccc-dddd-eeeeeeeeeeee"),
}
_, err := svc.DescribeIdentityPoolUsageWithContext(ctx, params, func(r *request.Request) {
r.Handlers.Validate.RemoveByName("core.ValidateParametersHandler")
})
if err == nil {
t.Fatalf("expect request to fail")
}
aerr, ok := err.(awserr.RequestFailure)
if !ok {
t.Fatalf("expect awserr, was %T", err)
}
if len(aerr.Code()) == 0 {
t.Errorf("expect non-empty error code")
}
if len(aerr.Message()) == 0 {
t.Errorf("expect non-empty error message")
}
if v := aerr.Code(); v == request.ErrCodeSerialization {
t.Errorf("expect API error code got serialization failure")
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,651
|
Факультет прикладної математики та комп'ютерних технологій, ФПМКТ - організаційний, навчально-науковий структурний підрозділ Хмельницького національного університету.
Історія створення
Вимоги часу, активний розвиток інформаційних технологій як самостійної індустрії, спонукали створення в університеті освітнього напряму підготовки інженерів-математиків, програмістів, спеціалістів з комп'ютерного дизайну, Інтернет-дизайну, спеціалістів з інформаційних технологій проектування в різних галузях економічної діяльності: машинобудування, легкої промисловості, економіки, банківської сфери тощо. Це викликало необхідність створення у грудні 2003 року факультету прикладної математики та комп'ютерних технологій. На факультеті здійснюється підготовка фахівців за чотирма напрямами:
Прикладна математика
Інформатика
Комп'ютерні науки
Інженерія програмного забезпечення
Студенти факультету навчаються у семи дисплейних класах, які оснащені сучасними персональними комп'ютерами та підключені до мережі Інтернет, мультимедійних лекційних аудиторіях. Локальна комп'ютерна мережа, яка створена на базі сучасного обладнання, об'єднує всі комп'ютерні аудиторії і забезпечує вихід в мережу Інтернет. В аудиторіях реалізовано Wi-Fi підключення персональних комп'ютерів студентів до мережі. Комп'ютерні аудиторії забезпечують проведення лабораторних робіт та самостійної роботи студентів. При необхідності використовуються комп'ютерні класи інформаційно-комп'ютерного центру університету. Студенти, які проживають у гуртожитку, працюють у спеціально обладнаних аудиторіях для самостійної роботи. На факультеті створено електронну бібліотеку спеціальної сучасної літератури з інформаційних технологій та прикладної математики, яка налічує більше 14 тисяч джерел.
Студенти факультету активно залучаються до наукової роботи. Результати студентських наукових досліджень опубліковані у збірниках студентських наукових конференцій різного рівня від університетських до всеукраїнських та міжнародних.
Серед доробок студентів факультету перемоги на міжнародних та всеукраїнських олімпіадах з програмування та САПР і комп'ютерного моделювання, ярмарках програмних продуктів та Вебдизайнерів.
Керівництво факультету факультету
Декан факультету: кандидат технічних наук, доцент Ковальчук Сергій Станіславович.
Структура факультету
Кафедра прикладної математики та соціальної інформатики
Зав. кафедри: д-р біол. наук., проф. Чернишенко Сергій Вікторович
Адреса: м. Хмельницький, вул. Тернопільська, 14, ТБЛ
Викладацький склад кафедри
Професори: Пелещишин А.М., Радченко В.М., Савула Я.Г.
Доценти: Бедратюк Л.П., Григорук С.С., Грипинська Н.В., Горбатюк К.В., Драч І.В., Кисіль Т.М., Кучерук О.Я., Праворська Н.І., Романюк В.В.
Викладачі: Атаманюк А.В., Самігулін І.В., Ярмолюк Р.С., Яровий А.В.
Напрямки дослідження кафедри:
Дослідження в галузях: теорії звичайних диференціальних рівнянь; застосувань чисельних методів; педагогічних наук; математичного моделювання та системного аналізу застосувань методів оптимізації
Кафедра інформаційних технологій проектування
Зав. кафедри: д-р техн. наук, проф. Сорокатий Руслан Володимирович
Адреса: м. Хмельницький, вул. Інститутська, 11, ауд. 3-401а, 3-404а
Викладацький склад кафедри
Професори: Кривий С.Л.
Доценти: Бармак О.В., Ковальчук С.С., Манзюк Е.А., Міхалевський В.Ц., Пасічник О.А., Петровський С.С., Свірневський М.С.
Викладачі Багрій Р.О., Борячок Р.О., Лищук О.А., Лясковський І.О., Мазурець О.В., Скрипник Т.К.
Напрямки дослідження кафедри:
Теоретичні аспекти дослідження властивостей програмного та апаратного забезпечення комп'ютерних систем. Розробка та чисельна реалізація математичних моделей систем автоматизованого проектування та обробки інформації в галузях машинобудування. Вдосконалення систем обробки інформації в базах даних та знань
Кафедра інженерії програмного забезпечення
Зав. кафедри: д.ф-м.н., доц. Бедратюк Л.П.
Адреса: м. Хмельницький, вул. Інститутська, 11, ауд. 1-203,1-204
Викладацький склад кафедри
Доценти Бедратюк Л.П., Бурлаков А.А., Длугунович Н.А., Радельчук Г.І., Форкун Ю.В.
Викладачі Бедратюк Г.І., Кравчук О.А., Яшина О.М., Гурман І.В., Омельчук С.С.
Напрямки дослідження кафедри:
Методи і засоби колаборативного документування інформаційних систем та процесів на інфраструктурі мережі Інтернет, розробка методів та алгоритмів автоматизованих інформаційно-пошукових систем роботи з технічною інформацією, математичне та комп'ютерне моделювання, теорія оптимізації, теорія диференціальних, інтегро-диференціальних та інтегральних рівнянь, психолого-педагогічні аспекти розвитку вищої освіти, менеджмент ІТ-проектів, формування інформаційної інфраструктури підприємства.
Кафедра вищої математики та комп'ютерних застосувань
Зав. кафедри: д-р мат. наук, проф. Рудницький Вадим Броніславович
Адреса: м. Хмельницький, вул. Інститутська, 11, ауд. 3-316, 3-316а
Викладацький склад кафедри
Доценти: Бедратюк Л.П., Діхтярук М.М., Лесюк І.І., Марчук Р.А., Міхалевська Г.І., Рамський А.О., Самарук Н.М., Стопень Г.Я., Трасковецька Л.М.
Викладачі: Войтков В.Г., Герасимчук Л.О., Зварич А.М., Криворучко Н.І., Курінєнко О.В., Максимчук Д.М., Лисова Л.О., Луньо Н.Б., Марчук А.Р., Марчук Л.Д., Мороз В.В., Поплавська О.А., Судик Л.І., Троян Г.Ф., Ярецька Н.О., Ярош О.М.
Напрямки дослідження кафедри:
Дослідження в галузі механіки деформованого твердого тіла, контактних задач трибології, теорії звичайних диференціальних рівнянь, теорії груп та інтегральних перетворень. Нові інформаційні технології та засоби навчання.
Джерела і посилання
Офіційна вебсторінка університету
Офіційна вебсторінка кафедри ПМ та СІ
Університети Хмельницького
Хмельницький національний університет
Факультети за алфавітом
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,384
|
La taïga des basses-terres de l'intérieur de l'Alaska et du Yukon est une écorégion terrestre nord-américaine du type forêts boréales, taïga du World Wildlife Fund
Répartition
La taïga des basses-terres de l'intérieur de l'Alaska et du Yukon recouvre une bonne partie de l'intérieur de l'Alaska et du nord du Yukon. Elle est enclavée par les Monts Richardson à l'est, la chaîne Brooks au nord et la chaîne d'Alaska et s'étend jusqu'à la mer de Béring à l'ouest.
Climat
Les précipitations annuelles varient entre et , sauf dans les upper Yukon flats où elles ne sont que de et de dans certains secteurs de l'ouest de l'écorégion. La moyenne quotidienne hivernale varie entre et . La moyenne quotidienne estivale varie entre et .
Géomorphologie
Le relief se compose surtout de plaines ou de collines ondoyantes de faible altitude. L'inclinaison des pentes dépassent rarement 5 %. L'altitude varie généralement entre et et le plus haut sommet atteint .
Caractéristiques biologiques
Les forêts sont principalement composées d'épinettes. L'Épinette blanche se rencontre sur les sols bien drainés, sur les collines et les versants exposés au soleil. L'Épinette noire occupe surtout les zones mal drainées et le fond des vallées. Les rives sinueuses des cours d'eau sont soumises à une perpétuelle colonisation par le saule et l'aulne, suivis par le Peuplier baumier et le Peuplier faux-tremble, ensuite remplacés par l'épinette. Les lieux récemment perturbés, les zones dénudées près de la limite des arbres, le versant nord des pentes et les milieux plus humides supportent des fruticées dominées par le saule, l'aulne et le bouleau nain. Les tourbières au fond des vallées et les milieux humides supportent des communautés de buissons et de graminées comprenant le saule, le bouleau nain, le Ledum decumbens, la Potentilla fruticosa, l'Eriophorum vaginatum et les carex. Les feux sont fréquents et entretiennent dans le paysage une mosaïque de stades successionnels.
La faune et la sauvagine sont particulièrement abondantes dans cette écorégion. On y trouve quelques espèces végétales rares telles Cryptantha shacklettiana, Erysimum asperum et Eriogonum flavum.
Conservation
On estime cette écorégion relativement intacte.
Notes et références
Environnement en Alaska
Environnement au Yukon
Écorégion au Canada
Écorégion aux États-Unis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,375
|
'It's going to be devastating.' Lawmakers are reluctantly readying for the no-talking, no-electronics impeachment trial of President Trump
Police in Paris were forced to call for backup on Friday as dozens of protesters outside a theater tried to storm the building and reach President Emmanuel Macron.
Up to 12 inches of snow is forecast to fall in the Midwest to the Northeast
Heavy snow will fall across the Midwest as tens of millions remain in the path of a dangerous winter storm through the weekend.
Virginia pro-gun rally reveals extremist tactics
On Thursday, the FBI arrested three men, Patrik J. Mathews, 27, Brian M. Lemley Jr., 33, and William G. Bilbrough IV, 19, with firearms charges, and they had plans, an official said, to attend a Virginia pro-gun rally. This followed Virginia Gov. Ralph Northam's declaration of a temporary state of emergency after authorities learned that extremists hoped to use the anti-gun control rally planned next Monday -- Martin Luther King, Jr. Day -- to incite a violent clash.
1,723 people likely infected by mystery virus
The number of cases in an outbreak of a new strain of coronavirus in China is likely to have been grossly underestimated, according to a new study, which warns that human-to-human transmission of the mysterious virus may be possible.
Analysis: The cruelty of losing civil rights icons in the Trump era
Dashcam footage shows a pickup truck lose control and hit a delivery vehicle in western Iowa. There were no serious injuries.
4 killed, 1 wounded in Utah shooting
Four people were shot to death in a residence in Grantsville, Utah, Friday night, according to police.
What you need to know about today's Women's March
The FBI believes that the Kingdom of Saudi Arabia officials "almost certainly" help their US-based citizens flee the country to avoid legal issues, according to a recently declassified intelligence bulletin.
The US operation in Iraq could come to a humiliating end as Trump's Iran moves backfire
Baseball is a game of legend. Babe Ruth. "Casey at the Bat." Hammerin' Hank Aaron. "A League of Their Own:" In every great and true baseball story, there's a little bit of fiction, and its greatest myths are just believable enough to be true.
At a time when almost everything is politicized, vaccination has planted itself squarely on the national stage.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,168
|
require_relative 'appinit'
require_relative 'emailhelper'
require_relative 'environments'
require_relative 'firsttimesetup'
require_relative 'tickets'
require_relative 'profilehelper'
require_relative 'tickets'
require_relative 'webcontext'
require_relative 'personnel'
require_relative 'parthelper'
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,884
|
<?xml version="1.0" encoding="utf-8"?>
<TableLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="match_parent" android:layout_height="match_parent">
<TableRow
android:layout_weight="1"
android:layout_width="0dp"
android:layout_height="0dp">
<ImageButton
android:layout_weight="1.5"
android:layout_width="0dp"
android:layout_height="230dp"
android:id="@+id/imageButton"
android:layout_column="0"
android:src="@drawable/multiple_choice_pp_v1"
android:scaleType="centerInside"
android:layout_marginLeft="20dp"
android:layout_marginTop="20dp"
android:layout_marginRight="10dp"
android:layout_marginBottom="10dp" />
<!-- hello -->
<ImageButton
android:layout_weight="1.5"
android:layout_width="0dp"
android:layout_height="230dp"
android:id="@+id/imageButton3"
android:layout_column="1"
android:src="@drawable/yes_no_pp_v1"
android:scaleType="centerInside"
android:layout_marginLeft="10dp"
android:layout_marginTop="20dp"
android:layout_marginRight="20dp"
android:layout_marginBottom="10dp" />
</TableRow>
<TableRow
android:layout_weight="1"
android:layout_width="0dp"
android:layout_height="0dp">
<ImageButton
android:layout_weight="1"
android:layout_width="0dp"
android:layout_height="230dp"
android:id="@+id/imageButton2"
android:layout_column="0"
android:scaleType="fitCenter"
android:src="@drawable/discrete_pp_v1"
android:layout_marginLeft="20dp"
android:layout_marginTop="10dp"
android:layout_marginRight="10dp"
android:layout_marginBottom="20dp" />
<ImageButton
android:layout_weight="1"
android:layout_width="0dp"
android:layout_height="230dp"
android:id="@+id/imageButton4"
android:layout_column="1"
android:scaleType="centerInside"
android:src="@drawable/pictures_pp_v3"
android:layout_marginLeft="10dp"
android:layout_marginTop="10dp"
android:layout_marginRight="20dp"
android:layout_marginBottom="20dp" />
</TableRow>
</TableLayout>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,748
|
Produced by Paul Marshall and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive)
Transcriber's Notes:
Underscores "_" before and after a word or phrase indicate _italics_
in the original text.
Equal signs "=" before and after a word or phrase indicate =bold=
in the original text.
Small capitals have been converted to SOLID capitals.
Old or antiquated spellings have been preserved.
Typographical errors have been silently corrected but other variations
in spelling and punctuation remain unaltered.
Where double quotes have been repeated at the beginnings of
consecutive lines, they have been omitted for clarity.
_The Fourth Edition of_
=ELIZABETH BARRETT BROWNING'S POEMS.=
With numerous Additions. Three Vols. Foolscap 8vo.
=MEN AND WOMEN.=
=BY ROBERT BROWNING.=
Two Vols. Foolscap 8vo. 12_s._
_A New Edition of_
=ROBERT BROWNING'S POEMS.=
Two Vols. Foolscap 8vo. 16_s._
ALSO,
=CHRISTMAS-EVE AND EASTER-DAY.=
A POEM.
Foolscap 8vo. 6_s._
CHAPMAN AND HALL, 193, PICCADILLY.
=AURORA LEIGH.=
BY
=ELIZABETH BARRETT BROWNING.=
LONDON:
CHAPMAN AND HALL, 193, PICCADILLY.
1857.
LONDON:
BRADBURY AND EVANS, PRINTERS, WHITEFRIARS.
DEDICATION TO JOHN KENYON, ESQ.
THE words 'cousin' and 'friend' are constantly recurring in this poem,
the last pages of which have been finished under the hospitality of
your roof, my own dearest cousin and friend;—cousin and friend, in a
sense of less equality and greater disinterestedness than 'Romney''s.
Ending, therefore, and preparing once more to quit England, I venture
to leave in your hands this book, the most mature of my works, and the
one into which my highest convictions upon Life and Art have entered:
that as, through my various efforts in literature and steps in life,
you have believed in me, borne with me, and been generous to me, far
beyond the common uses of mere relationship or sympathy of mind, so you
may kindly accept, in sight of the public, this poor sign of esteem,
gratitude, and affection, from
your unforgetting
E. B. B.
39, DEVONSHIRE PLACE,
_October_ 17, 1856.
AURORA LEIGH.
FIRST BOOK.
OF writing many books there is no end;
And I who have written much in prose and verse
For others' uses, will write now for mine,—
Will write my story for my better self,
As when you paint your portrait for a friend,
Who keeps it in a drawer and looks at it
Long after he has ceased to love you, just
To hold together what he was and is.
I, writing thus, am still what men call young;
I have not so far left the coasts of life
To travel inland, that I cannot hear
That murmur of the outer Infinite
Which unweaned babies smile at in their sleep
When wondered at for smiling; not so far,
But still I catch my mother at her post
Beside the nursery-door, with finger up,
'Hush, hush—here's too much noise!' while her sweet eyes
Leap forward, taking part against her word
In the child's riot. Still I sit and feel
My father's slow hand, when she had left us both,
Stroke out my childish curls across his knee;
And hear Assunta's daily jest (she knew
He liked it better than a better jest)
Inquire how many golden scudi went
To make such ringlets. O my father's hand,
Stroke the poor hair down, stroke it heavily,—
Draw, press the child's head closer to thy knee!
I'm still too young, too young, to sit alone.
I write. My mother was a Florentine,
Whose rare blue eyes were shut from seeing me
When scarcely I was four years old; my life,
A poor spark snatched up from a failing lamp
Which went out therefore. She was weak and frail;
She could not bear the joy of giving life—
The mother's rapture slew her. If her kiss
Had left a longer weight upon my lips,
It might have steadied the uneasy breath,
And reconciled and fraternised my soul
With the new order. As it was, indeed,
I felt a mother-want about the world,
And still went seeking, like a bleating lamb
Left out at night, in shutting up the fold,—
As restless as a nest-deserted bird
Grown chill through something being away, though what
It knows not. I, Aurora Leigh, was born
To make my father sadder, and myself
Not overjoyous, truly. Women know
The way to rear up children, (to be just,)
They know a simple, merry, tender knack
Of tying sashes, fitting baby-shoes,
And stringing pretty words that make no sense,
And kissing full sense into empty words;
Which things are corals to cut life upon,
Although such trifles: children learn by such,
Love's holy earnest in a pretty play,
And get not over-early solemnised,—
But seeing, as in a rose-bush, Love's Divine,
Which burns and hurts not,—not a single bloom,—
Become aware and unafraid of Love.
Such good do mothers. Fathers love as well
—Mine did, I know,—but still with heavier brains,
And wills more consciously responsible,
And not as wisely, since less foolishly;
So mothers have God's licence to be missed.
My father was an austere Englishman,
Who, after a dry life-time spent at home
In college-learning, law, and parish talk,
Was flooded with a passion unaware,
His whole provisioned and complacent past
Drowned out from him that moment. As he stood
In Florence, where he had come to spend a month
And note the secret of Da Vinci's drains,
He musing somewhat absently perhaps
Some English question ... whether men should pay
The unpopular but necessary tax
With left or right hand—in the alien sun
In that great square of the Santissima,
There drifted past him (scarcely marked enough
To move his comfortable island-scorn,)
A train of priestly banners, cross and psalm,—
The white-veiled rose-crowned maidens holding up
Tall tapers, weighty for such wrists, aslant
To the blue luminous tremor of the air,
And letting drop the white wax as they went
To eat the bishop's wafer at the church;
From which long trail of chanting priests and girls,
A face flashed like a cymbal on his face,
And shook with silent clangour brain and heart,
Transfiguring him to music. Thus, even thus,
He too received his sacramental gift
With eucharistic meanings; for he loved.
And thus beloved, she died. I've heard it said
That but to see him in the first surprise
Of widower and father, nursing me,
Unmothered little child of four years old,
His large man's hands afraid to touch my curls,
As if the gold would tarnish,—his grave lips
Contriving such a miserable smile,
As if he knew needs must, or I should die,
And yet 'twas hard,—would almost make the stones
Cry out for pity. There's a verse he set
In Santa Croce to her memory,
'Weep for an infant too young to weep much
When death removed this mother'—stops the mirth
To-day, on women's faces when they walk
With rosy children hanging on their gowns,
Under the cloister, to escape the sun
That scorches in the piazza. After which,
He left our Florence, and made haste to hide
Himself, his prattling child, and silent grief,
Among the mountains above Pelago;
Because unmothered babes, he thought, had need
Of mother nature more than others use,
And Pan's white goats, with udders warm and full
Of mystic contemplations, come to feed
Poor milkless lips of orphans like his own—
Such scholar-scraps he talked, I've heard from friends,
For even prosaic men, who wear grief long,
Will get to wear it as a hat aside
With a flower stuck in't. Father, then, and child,
We lived among the mountains many years,
God's silence on the outside of the house,
And we, who did not speak too loud, within;
And old Assunta to make up the fire,
Crossing herself whene'er a sudden flame
Which lightened from the firewood, made alive
That picture of my mother on the wall.
The painter drew it after she was dead;
And when the face was finished, throat and hands,
Her cameriera carried him, in hate
Of the English-fashioned shroud, the last brocade
She dressed in at the Pitti. 'He should paint
No sadder thing than that,' she swore, 'to wrong
Her poor signora.' Therefore very strange
The effect was. I, a little child, would crouch
For hours upon the floor, with knees drawn up,
And gaze across them, half in terror, half
In adoration, at the picture there,—
That swan-like supernatural white life,
Just sailing upward from the red stiff silk
Which seemed to have no part in it, nor power
To keep it from quite breaking out of bounds:
For hours I sate and stared. Assunta's awe
And my poor father's melancholy eyes
Still pointed that way. That way, went my thoughts
When wandering beyond sight. And as I grew
In years, I mixed, confused, unconsciously,
Whatever I last read or heard or dreamed,
Abhorrent, admirable, beautiful,
Pathetical, or ghastly, or grotesque,
With still that face ... which did not therefore change,
But kept the mystic level of all forms
And fears and admirations; was by turns
Ghost, fiend, and angel, fairy, witch, and sprite,—
A dauntless Muse who eyes a dreadful Fate,
A loving Psyche who loses sight of Love,
A still Medusa, with mild milky brows
All curdled and all clothed upon with snakes
Whose slime falls fast as sweat will; or, anon,
Our Lady of the Passion, stabbed with swords
Where the Babe sucked; or, Lamia in her first
Moonlighted pallor, ere she shrunk and blinked,
And, shuddering, wriggled down to the unclean;
Or, my own mother, leaving her last smile
In her last kiss, upon the baby-mouth
My father pushed down on the bed for that,—
Or my dead mother, without smile or kiss,
Buried at Florence. All which images,
Concentred on the picture, glassed themselves
Before my meditative childhood, ... as
The incoherencies of change and death
Are represented fully, mixed and merged,
In the smooth fair mystery of perpetual Life.
And while I stared away my childish wits
Upon my mother's picture, (ah, poor child!)
My father, who through love had suddenly
Thrown off the old conventions, broken loose
From chin-bands of the soul, like Lazarus,
Yet had no time to learn to talk and walk
Or grow anew familiar with the sun,—
Who had reached to freedom, not to action, lived,
But lived as one entranced, with thoughts, not aims,—
Whom love had unmade from a common man
But not completed to an uncommon man,—
My father taught me what he had learnt the best
Before he died and left me,—grief and love.
And, seeing we had books among the hills,
Strong words of counselling souls, confederate
With vocal pines and waters,—out of books
He taught me all the ignorance of men,
And how God laughs in heaven when any man
Says 'Here I'm learned; this, I understand;
In that, I am never caught at fault or doubt.'
He sent the schools to school, demonstrating
A fool will pass for such through one mistake,
While a philosopher will pass for such,
Through said mistakes being ventured in the gross
And heaped up to a system.
I am like,
They tell me, my dear father. Broader brows
Howbeit, upon a slenderer undergrowth
Of delicate features,—paler, near as grave;
But then my mother's smile breaks up the whole,
And makes it better sometimes than itself.
So, nine full years, our days were hid with God
Among his mountains. I was just thirteen,
Still growing like the plants from unseen roots
In tongue-tied Springs,—and suddenly awoke
To full life and its needs and agonies,
With an intense, strong, struggling heart beside
A stone-dead father. Life, struck sharp on death,
Makes awful lightning. His last word was, 'Love—'
'Love, my child, love, love!'—(then he had done with grief)
'Love, my child.' Ere I answered he was gone,
And none was left to love in all the world.
There, ended childhood: what succeeded next
I recollect as, after fevers, men
Thread back the passage of delirium,
Missing the turn still, baffled by the door;
Smooth endless days, notched here and there with knives;
A weary, wormy darkness, spurred i' the flank
With flame, that it should eat and end itself
Like some tormented scorpion. Then, at last,
I do remember clearly, how there came
A stranger with authority, not right,
(I thought not) who commanded, caught me up
From old Assunta's neck; how, with a shriek,
She let me go,—while I, with ears too full
Of my father's silence, to shriek back a word,
In all a child's astonishment at grief
Stared at the wharfage where she stood and moaned,
My poor Assunta, where she stood and moaned!
The white walls, the blue hills, my Italy,
Drawn backward from the shuddering steamer-deck,
Like one in anger drawing back her skirts
Which suppliants catch at. Then the bitter sea
Inexorably pushed between us both,
And sweeping up the ship with my despair
Threw us out as a pasture to the stars.
Ten nights and days we voyaged on the deep;
Ten nights and days, without the common face
Of any day or night; the moon and sun
Cut off from the green reconciling earth,
To starve into a blind ferocity
And glare unnatural; the very sky
(Dropping its bell-net down upon the sea
As if no human heart should scape alive,)
Bedraggled with the desolating salt,
Until it seemed no more that holy heaven
To which my father went. All new, and strange—
The universe turned stranger, for a child.
Then, land!—then, England! oh, the frosty cliffs
Looked cold upon me. Could I find a home
Among those mean red houses through the fog?
And when I heard my father's language first
From alien lips which had no kiss for mine,
I wept aloud, then laughed, then wept, then wept,—
And some one near me said the child was mad
Through much sea-sickness. The train swept us on.
Was this my father's England? the great isle?
The ground seemed cut up from the fellowship
Of verdure, field from field, as man from man;
The skies themselves looked low and positive,
As almost you could touch them with a hand,
And dared to do it, they were so far off
From God's celestial crystals; all things, blurred
And dull and vague. Did Shakspeare and his mates
Absorb the light here?—not a hill or stone
With heart to strike a radiant colour up
Or active outline on the indifferent air!
I think I see my father's sister stand
Upon the hall-step of her country-house
To give me welcome. She stood straight and calm,
Her somewhat narrow forehead braided tight
As if for taming accidental thoughts
From possible pulses; brown hair pricked with grey
By frigid use of life, (she was not old,
Although my father's elder by a year)
A nose drawn sharply, yet in delicate lines;
A close mild mouth, a little soured about
The ends, through speaking unrequited loves,
Or peradventure niggardly half-truths;
Eyes of no colour,—once they might have smiled,
But never, never have forgot themselves
In smiling; cheeks, in which was yet a rose
Of perished summers, like a rose in a book,
Kept more for ruth than pleasure,—if past bloom,
Past fading also.
She had lived, we'll say,
A harmless life, she called a virtuous life,
A quiet life, which was not life at all,
(But that, she had not lived enough to know)
Between the vicar and the county squires,
The lord-lieutenant looking down sometimes
From the empyreal, to assure their souls
Against chance-vulgarisms, and, in the abyss,
The apothecary looked on once a year,
To prove their soundness of humility.
The poor-club exercised her Christian gifts
Of knitting stockings, stitching petticoats,
Because we are of one flesh after all
And need one flannel, (with a proper sense
Of difference in the quality)—and still
The book-club, guarded from your modern trick
Of shaking dangerous questions from the crease,
Preserved her intellectual. She had lived
A sort of cage-bird life, born in a cage,
Accounting that to leap from perch to perch
Was act and joy enough for any bird.
Dear heaven, how silly are the things that live
In thickets, and eat berries!
I, alas,
A wild bird scarcely fledged, was brought to her cage,
And she was there to meet me. Very kind.
Bring the clean water; give out the fresh seed.
She stood upon the steps to welcome me,
Calm, in black garb. I clung about her neck,—
Young babes, who catch at every shred of wool
To draw the new light closer, catch and cling
Less blindly. In my ears, my father's word
Hummed ignorantly, as the sea in shells,
'Love, love, my child.' She, black there with my grief,
Might feel my love—she was his sister once—
I clung to her. A moment, she seemed moved,
Kissed me with cold lips, suffered me to cling,
And drew me feebly through the hall, into
The room she sate in.
There, with some strange spasm
Of pain and passion, she wrung loose my hands
Imperiously, and held me at arm's length,
And with two grey-steel naked-bladed eyes
Searched through my face,—ay, stabbed it through and through,
Through brows and cheeks and chin, as if to find
A wicked murderer in my innocent face,
If not here, there perhaps. Then, drawing breath,
She struggled for her ordinary calm,
And missed it rather,—told me not to shrink,
As if she had told me not to lie or swear,—
'She loved my father, and would love me too
As long as I deserved it.' Very kind.
I understood her meaning afterward;
She thought to find my mother in my face,
And questioned it for that. For she, my aunt,
Had loved my father truly, as she could,
And hated, with the gall of gentle souls,
My Tuscan mother, who had fooled away
A wise man from wise courses, a good man
From obvious duties, and, depriving her,
His sister, of the household precedence,
Had wronged his tenants, robbed his native land,
And made him mad, alike by life and death,
In love and sorrow. She had pored for years
What sort of woman could be suitable
To her sort of hate, to entertain it with;
And so, her very curiosity
Became hate too, and all the idealism
She ever used in life, was used for hate,
Till hate, so nourished, did exceed at last
The love from which it grew, in strength and heat,
And wrinkled her smooth conscience with a sense
Of disputable virtue (say not, sin)
When Christian doctrine was enforced at church.
And thus my father's sister was to me
My mother's hater. From that day, she did
Her duty to me, (I appreciate it
In her own word as spoken to herself)
Her duty, in large measure, well-pressed out,
But measured always. She was generous, bland,
More courteous than was tender, gave me still
The first place,—as if fearful that God's saints
Would look down suddenly and say, 'Herein
You missed a point, I think, through lack of love.'
Alas, a mother never is afraid
Of speaking angerly to any child,
Since love, she knows, is justified of love.
And I, I was a good child on the whole,
A meek and manageable child. Why not?
I did not live, to have the faults of life:
There seemed more true life in my father's grave
Than in all England. Since _that_ threw me off
Who fain would cleave, (his latest will, they say,
Consigned me to his land) I only thought
Of lying quiet there where I was thrown
Like sea-weed on the rocks, and suffer her
To prick me to a pattern with her pin,
Fibre from fibre, delicate leaf from leaf,
And dry out from my drowned anatomy
The last sea-salt left in me.
So it was.
I broke the copious curls upon my head
In braids, because she liked smooth-ordered hair.
I left off saying my sweet Tuscan words
Which still at any stirring of the heart
Came up to float across the English phrase,
As lilies, (_Bene_ ... or _che ch'è_) because
She liked my father's child to speak his tongue.
I learnt the collects and the catechism,
The creeds, from Athanasius back to Nice,
The Articles ... the Tracts _against_ the times,
(By no means Buonaventure's 'Prick of Love,')
And various popular synopses of
Inhuman doctrines never taught by John,
Because she liked instructed piety.
I learnt my complement of classic French
(Kept pure of Balzac and neologism,)
And German also, since she liked a range
Of liberal education,—tongues, not books.
I learnt a little algebra, a little
Of the mathematics,—brushed with extreme flounce
The circle of the sciences, because
She misliked women who are frivolous.
I learnt the royal genealogies
Of Oviedo, the internal laws
Of the Burmese empire, ... by how many feet
Mount Chimborazo outsoars Himmeleh,
What navigable river joins itself
To Lara, and what census of the year five
Was taken at Klagenfurt,—because she liked
A general insight into useful facts.
I learnt much music,—such as would have been
As quite impossible in Johnson's day
As still it might be wished—fine sleights of hand
And unimagined fingering, shuffling off
The hearer's soul through hurricanes of notes
To a noisy Tophet; and I drew ... costumes
From French engravings, nereids neatly draped,
With smirks of simmering godship,—I washed in
From nature, landscapes, (rather say, washed out.)
I danced the polka and Cellarius,
Spun glass, stuffed birds, and modelled flowers in wax,
Because she liked accomplishments in girls.
I read a score of books on womanhood
To prove, if women do not think at all,
They may teach thinking, (to a maiden-aunt
Or else the author)—books demonstrating
Their right of comprehending husband's talk
When not too deep, and even of answering
With pretty 'may it please you,' or 'so it is,'—
Their rapid insight and fine aptitude,
Particular worth and general missionariness,
As long as they keep quiet by the fire
And never say 'no' when the world says 'ay,'
For that is fatal,—their angelic reach
Of virtue, chiefly used to sit and darn,
And fatten household sinners,—their, in brief,
Potential faculty in everything
Of abdicating power in it: she owned
She liked a woman to be womanly,
And English women, she thanked God and sighed,
(Some people always sigh in thanking God)
Were models to the universe. And last
I learnt cross-stitch, because she did not like
To see me wear the night with empty hands,
A-doing nothing. So, my shepherdess
Was something after all, (the pastoral saints
Be praised for't) leaning lovelorn with pink eyes
To match her shoes, when I mistook the silks;
Her head uncrushed by that round weight of hat
So strangely similar to the tortoise-shell
Which slew the tragic poet.
By the way,
The works of women are symbolical.
We sew, sew, prick our fingers, dull our sight,
Producing what? A pair of slippers, sir,
To put on when you're weary—or a stool
To stumble over and vex you ... 'curse that stool!'
Or else at best, a cushion, where you lean
And sleep, and dream of something we are not,
But would be for your sake. Alas, alas!
This hurts most, this ... that, after all, we are paid
The worth of our work, perhaps.
In looking down
Those years of education, (to return)
I wonder if Brinvilliers suffered more
In the water-torture, ... flood succeeding flood
To drench the incapable throat and split the veins ...
Than I did. Certain of your feebler souls
Go out in such a process; many pine
To a sick, inodorous light; my own endured:
I had relations in the Unseen, and drew
The elemental nutriment and heat
From nature, as earth feels the sun at nights,
Or as a babe sucks surely in the dark.
I kept the life, thrust on me, on the outside
Of the inner life, with all its ample room
For heart and lungs, for will and intellect,
Inviolable by conventions. God,
I thank thee for that grace of thine!
At first,
I felt no life which was not patience,—did
The thing she bade me, without heed to a thing
Beyond it, sate in just the chair she placed,
With back against the window, to exclude
The sight of the great lime-tree on the lawn,
Which seemed to have come on purpose from the woods
To bring the house a message,—ay, and walked
Demurely in her carpeted low rooms,
As if I should not, harkening my own steps,
Misdoubt I was alive. I read her books,
Was civil to her cousin, Romney Leigh,
Gave ear to her vicar, tea to her visitors,
And heard them whisper, when I changed a cup,
(I blushed for joy at that)—'The Italian child,
For all her blue eyes and her quiet ways,
Thrives ill in England: she is paler yet
Than when we came the last time; she will die.'
'Will die.' My cousin, Romney Leigh, blushed too,
With sudden anger, and approaching me
Said low between his teeth—'You're wicked now?
You wish to die and leave the world a-dusk
For others, with your naughty light blown out?'
I looked into his face defyingly.
He might have known, that, being what I was,
'Twas natural to like to get away
As far as dead folk can; and then indeed
Some people make no trouble when they die.
He turned and went abruptly, slammed the door
And shut his dog out.
Romney, Romney Leigh.
I have not named my cousin hitherto,
And yet I used him as a sort of friend;
My elder by few years, but cold and shy
And absent ... tender, when he thought of it,
Which scarcely was imperative, grave betimes,
As well as early master of Leigh Hall,
Whereof the nightmare sate upon his youth
Repressing all its seasonable delights,
And agonising with a ghastly sense
Of universal hideous want and wrong
To incriminate possession. When he came
From college to the country, very oft
He crossed the hills on visits to my aunt,
With gifts of blue grapes from the hothouses,
A book in one hand,—mere statistics, (if
I chanced to lift the cover) count of all
The goats whose beards are sprouting down toward hell,
Against God's separating judgment-hour.
And she, she almost loved him,—even allowed
That sometimes he should seem to sigh my way;
It made him easier to be pitiful,
And sighing was his gift. So, undisturbed
At whiles she let him shut my music up
And push my needles down, and lead me out
To see in that south angle of the house
The figs grow black as if by a Tuscan rock,
On some light pretext. She would turn her head
At other moments, go to fetch a thing,
And leave me breath enough to speak with him,
For his sake; it was simple.
Sometimes too
He would have saved me utterly, it seemed,
He stood and looked so.
Once, he stood so near
He dropped a sudden hand upon my head
Bent down on woman's work, as soft as rain—
But then I rose and shook it off as fire,
The stranger's touch that took my father's place,
Yet dared seem soft.
I used him for a friend
Before I ever knew him for a friend.
'Twas better, 'twas worse also, afterward:
We came so close, we saw our differences
Too intimately. Always Romney Leigh
Was looking for the worms, I for the gods.
A godlike nature his; the gods look down,
Incurious of themselves; and certainly
'Tis well I should remember, how, those days,
I was a worm too, and he looked on me.
A little by his act perhaps, yet more
By something in me, surely not my will,
I did not die. But slowly, as one in swoon,
To whom life creeps back in the form of death,
With a sense of separation, a blind pain
Of blank obstruction, and a roar i' the ears
Of visionary chariots which retreat
As earth grows clearer ... slowly, by degrees,
I woke, rose up ... where was I? in the world;
For uses, therefore, I must count worth while.
I had a little chamber in the house,
As green as any privet-hedge a bird
Might choose to build in, though the nest itself
Could show but dead-brown sticks and straws; the walls
Were green, the carpet was pure green, the straight
Small bed was curtained greenly, and the folds
Hung green about the window, which let in
The out-door world with all its greenery.
You could not push your head out and escape
A dash of dawn-dew from the honeysuckle,
But so you were baptised into the grace
And privilege of seeing....
First, the lime,
(I had enough, there, of the lime, be sure,—
My morning-dream was often hummed away
By the bees in it;) past the lime, the lawn,
Which, after sweeping broadly round the house,
Went trickling through the shrubberies in a stream
Of tender turf, and wore and lost itself
Among the acacias, over which, you saw
The irregular line of elms by the deep lane
Which stopped the grounds and dammed the overflow
Of arbutus and laurel. Out of sight
The lane was; sunk so deep, no foreign tramp
Nor drover of wild ponies out of Wales
Could guess if lady's hall or tenant's lodge
Dispensed such odours,—though his stick well-crooked
Might reach the lowest trail of blossoming briar
Which dipped upon the wall. Behind the elms,
And through their tops, you saw the folded hills
Striped up and down with hedges, (burly oaks
Projecting from the lines to show themselves)
Through which my cousin Romney's chimneys smoked
As still as when a silent mouth in frost
Breathes—showing where the woodlands hid Leigh Hall;
While, far above, a jut of table-land,
A promontory without water, stretched,—
You could not catch it if the days were thick,
Or took it for a cloud; but, otherwise
The vigorous sun would catch it up at eve
And use it for an anvil till he had filled
The shelves of heaven with burning thunderbolts,
And proved he need not rest so early:—then,
When all his setting trouble was resolved
To a trance of passive glory, you might see
In apparition on the golden sky
(Alas, my Giotto's background!) the sheep run
Along the fine clear outline, small as mice
That run along a witch's scarlet thread.
Not a grand nature. Not my chestnut-woods
Of Vallombrosa, cleaving by the spurs
To the precipices. Not my headlong leaps
Of waters, that cry out for joy or fear
In leaping through the palpitating pines,
Like a white soul tossed out to eternity
With thrills of time upon it. Not indeed
My multitudinous mountains, sitting in
The magic circle, with the mutual touch
Electric, panting from their full deep hearts
Beneath the influent heavens, and waiting for
Communion and commission. Italy
Is one thing, England one.
On English ground
You understand the letter ... ere the fall,
How Adam lived in a garden. All the fields
Are tied up fast with hedges, nosegay-like;
The hills are crumpled plains,—the plains, parterres,—
The trees, round, woolly, ready to be clipped;
And if you seek for any wilderness
You find, at best, a park. A nature tamed
And grown domestic like a barn-door fowl,
Which does not awe you with its claws and beak,
Nor tempt you to an eyrie too high up,
But which, in cackling, sets you thinking of
Your eggs to-morrow at breakfast, in the pause
Of finer meditation.
Rather say,
A sweet familiar nature, stealing in
As a dog might, or child, to touch your hand
Or pluck your gown, and humbly mind you so
Of presence and affection, excellent
For inner uses, from the things without.
I could not be unthankful, I who was
Entreated thus and holpen. In the room
I speak of, ere the house was well awake,
And also after it was well asleep,
I sate alone, and drew the blessing in
Of all that nature. With a gradual step,
A stir among the leaves, a breath, a ray,
It came in softly, while the angels made
A place for it beside me. The moon came,
And swept my chamber clean of foolish thoughts.
The sun came, saying, 'Shall I lift this light
Against the lime-tree, and you will not look?
I make the birds sing—listen!... but, for you,
God never hears your voice, excepting when
You lie upon the bed at nights and weep.'
Then, something moved me. Then, I wakened up
More slowly than I verily write now,
But wholly, at last, I wakened, opened wide
The window and my soul, and let the airs
And out-door sights sweep gradual gospels in,
Regenerating what I was. O Life,
How oft we throw it off and think,—'Enough,
Enough of life in so much!—here's a cause
For rupture;—herein we must break with Life,
Or be ourselves unworthy; here we are wronged,
Maimed, spoiled for aspiration: farewell Life!'
—And so, as froward babes, we hide our eyes
And think all ended.—Then, Life calls to us
In some transformed, apocryphal, new voice,
Above us, or below us, or around....
Perhaps we name it Nature's voice, or Love's,
Tricking ourselves, because we are more ashamed
To own our compensations than our griefs:
Still, Life's voice!—still, we make our peace with Life.
And I, so young then, was not sullen. Soon
I used to get up early, just to sit
And watch the morning quicken in the grey,
And hear the silence open like a flower,
Leaf after leaf,—and stroke with listless hand
The woodbine through the window, till at last
I came to do it with a sort of love,
At foolish unaware: whereat I smiled,—
A melancholy smile, to catch myself
Smiling for joy.
Capacity for joy
Admits temptation. It seemed, next, worth while
To dodge the sharp sword set against my life;
To slip down stairs through all the sleepy house,
As mute as any dream there, and escape
As a soul from the body, out of doors,—
Glide through the shrubberies, drop into the lane,
And wander on the hills an hour or two,
Then back again before the house should stir.
Or else I sate on in my chamber green,
And lived my life, and thought my thoughts, and prayed
My prayers without the vicar; read my books,
Without considering whether they were fit
To do me good. Mark, there. We get no good
By being ungenerous, even to a book,
And calculating profits ... so much help
By so much reading. It is rather when
We gloriously forget ourselves, and plunge
Soul-forward, headlong, into a book's profound,
Impassioned for its beauty and salt of truth—
'Tis then we get the right good from a book.
I read much. What my father taught before
From many a volume, Love re-emphasised
Upon the self-same pages: Theophrast
Grew tender with the memory of his eyes,
And Ælian made mine wet. The trick of Greek
And Latin, he had taught me, as he would
Have taught me wrestling or the game of fives
If such he had known,—most like a shipwrecked man
Who heaps his single platter with goats' cheese
And scarlet berries; or like any man
Who loves but one, and so gives all at once,
Because he has it, rather than because
He counts it worthy. Thus, my father gave;
And thus, as did the women formerly
By young Achilles, when they pinned the veil
Across the boy's audacious front, and swept
With tuneful laughs the silver-fretted rocks,
He wrapt his little daughter in his large
Man's doublet, careless did it fit or no.
But, after I had read for memory,
I read for hope. The path my father's foot
Had trod me out, which suddenly broke off,
(What time he dropped the wallet of the flesh
And passed) alone I carried on, and set
My child-heart 'gainst the thorny underwood,
To reach the grassy shelter of the trees.
Ah, babe i' the wood, without a brother-babe!
My own self-pity, like the red-breast bird,
Flies back to cover all that past with leaves.
Sublimest danger, over which none weeps,
When any young wayfaring soul goes forth
Alone, unconscious of the perilous road,
The day-sun dazzling in his limpid eyes,
To thrust his own way, he an alien, through
The world of books! Ah, you!—you think it fine,
You clap hands—'A fair day!'—you cheer him on,
As if the worst, could happen, were to rest
Too long beside a fountain. Yet, behold,
Behold!—the world of books is still the world;
And worldlings in it are less merciful
And more puissant. For the wicked there
Are winged like angels. Every knife that strikes,
Is edged from elemental fire to assail
A spiritual life. The beautiful seems right
By force of beauty, and the feeble wrong
Because of weakness. Power is justified,
Though armed against St. Michael. Many a crown
Covers bald foreheads. In the book-world, true,
There's no lack, neither, of God's saints and kings,
That shake the ashes of the grave aside
From their calm locks, and undiscomfited
Look stedfast truths against Time's changing mask.
True, many a prophet teaches in the roads;
True, many a seer pulls down the flaming heavens
Upon his own head in strong martyrdom,
In order to light men a moment's space.
But stay!—who judges?—who distinguishes
'Twixt Saul and Nahash justly, at first sight,
And leaves king Saul precisely at the sin,
To serve king David? who discerns at once
The sound of the trumpets, when the trumpets blow
For Alaric as well as Charlemagne?
Who judges prophets, and can tell true seers
From conjurors? The child, there? Would you leave
That child to wander in a battle-field
And push his innocent smile against the guns?
Or even in the catacombs, ... his torch
Grown ragged in the fluttering air, and all
The dark a-mutter round him? not a child!
I read books bad and good—some bad and good
At once: good aims not always make good books:
Well-tempered spades turn up ill-smelling soils
In digging vineyards, even: books, that prove
God's being so definitely, that man's doubt
Grows self-defined the other side the line,
Made atheist by suggestion; moral books,
Exasperating to license; genial books,
Discounting from the human dignity;
And merry books, which set you weeping when
The sun shines,—ay, and melancholy books,
Which make you laugh that any one should weep
In this disjointed life, for one wrong more.
The world of books is still the world, I write,
And both worlds have God's providence, thank God,
To keep and hearten: with some struggle, indeed,
Among the breakers, some hard swimming through
The deeps—I lost breath in my soul sometimes,
And cried, 'God save me if there's any God,'
But, even so, God saved me; and, being dashed
From error on to error, every turn
Still brought me nearer to the central truth.
I thought so. All this anguish in the thick
Of men's opinions ... press and counterpress,
Now up, now down, now underfoot, and now
Emergent ... all the best of it, perhaps,
But throws you back upon a noble trust
And use of your own instinct,—merely proves
Pure reason stronger than bare inference
At strongest. Try it,—fix against heaven's wall
Your scaling ladders of high logic—mount
Step by step!—Sight goes faster; that still ray
Which strikes out from you, how, you cannot tell,
And why, you know not—(did you eliminate,
That such as you, indeed, should analyse?)
Goes straight and fast as light, and high as God.
The cygnet finds the water; but the man
Is born in ignorance of his element,
And feels out blind at first, disorganised
By sin i' the blood,—his spirit-insight dulled
And crossed by his sensations. Presently
We feel it quicken in the dark sometimes;
Then, mark, be reverent, be obedient,—
For those dumb motions of imperfect life
Are oracles of vital Deity
Attesting the Hereafter. Let who says
'The soul's a clean white paper,' rather say,
A palimpsest, a prophet's holograph
Defiled, erased and covered by a monk's,—
The apocalypse, by a Longus! poring on
Which obscene text, we may discern perhaps
Some fair, fine trace of what was written once,
Some upstroke of an alpha and omega
Expressing the old scripture.
Books, books, books!
I had found the secret of a garret-room
Piled high with cases in my father's name;
Piled high, packed large,—where, creeping in and out
Among the giant fossils of my past,
Like some small nimble mouse between the ribs
Of a mastodon, I nibbled here and there
At this or that box, pulling through the gap,
In heats of terror, haste, victorious joy,
The first book first. And how I felt it beat
Under my pillow, in the morning's dark,
An hour before the sun would let me read!
My books!
At last, because the time was ripe,
I chanced upon the poets.
As the earth
Plunges in fury, when the internal fires
Have reached and pricked her heart, and, throwing flat
The marts and temples, the triumphal gates
And towers of observation, clears herself
To elemental freedom—thus, my soul,
At poetry's divine first finger-touch,
Let go conventions and sprang up surprised,
Convicted of the great eternities
Before two worlds.
What's this, Aurora Leigh,
You write so of the poets, and not laugh?
Those virtuous liars, dreamers after dark,
Exaggerators of the sun and moon,
And soothsayers in a tea-cup?
I write so
Of the only truth-tellers, now left to God,—
The only speakers of essential truth,
Opposed to relative, comparative,
And temporal truths; the only holders by
His sun-skirts, through conventional grey glooms;
The only teachers who instruct mankind,
From just a shadow on a charnel-wall,
To find man's veritable stature out,
Erect, sublime,—the measure of a man,
And that's the measure of an angel, says
The apostle. Ay, and while your common men
Build pyramids, gauge railroads, reign, reap, dine,
And dust the flaunty carpets of the world
For kings to walk on, or our senators,
The poet suddenly will catch them up
With his voice like a thunder ... 'This is soul,
This is life, this word is being said in heaven,
Here's God down on us! what are you about?'
How all those workers start amid their work,
Look round, look up, and feel, a moment's space,
That carpet-dusting, though a pretty trade,
Is not the imperative labour after all.
My own best poets, am I one with you,
That thus I love you,—or but one through love?
Does all this smell of thyme about my feet
Conclude my visit to your holy hill
In personal presence, or but testify
The rustling of your vesture through my dreams
With influent odours? When my joy and pain,
My thought and aspiration, like the stops
Of pipe or flute, are absolutely dumb
If not melodious, do you play on me,
My pipers,—and if, sooth, you did not blow,
Would no sound come? or is the music mine,
As a man's voice or breath is called his own,
Inbreathed by the Life-breather? There's a doubt
For cloudy seasons!
But the sun was high
When first I felt my pulses set themselves
For concords; when the rhythmic turbulence
Of blood and brain swept outward upon words,
As wind upon the alders, blanching them
By turning up their under-natures till
They trembled in dilation. O delight
And triumph of the poet,—who would say
A man's mere 'yes,' a woman's common 'no,'
A little human hope of that or this,
And says the word so that it burns you through
With a special revelation, shakes the heart
Of all the men and women in the world,
As if one came back from the dead and spoke,
With eyes too happy, a familiar thing
Become divine i' the utterance! while for him
The poet, the speaker, he expands with joy;
The palpitating angel in his flesh
Thrills inly with consenting fellowship
To those innumerous spirits who sun themselves
Outside of time.
O life, O poetry,
—Which means life in life! cognisant of life
Beyond this blood-beat,—passionate for truth
Beyond these senses,—poetry, my life,—
My eagle, with both grappling feet still hot
From Zeus's thunder, who has ravished me
Away from all the shepherds, sheep, and dogs,
And set me in the Olympian roar and round
Of luminous faces, for a cup-bearer,
To keep the mouths of all the godheads moist
For everlasting laughters,—I, myself,
Half drunk across the beaker, with their eyes!
How those gods look!
Enough so, Ganymede.
We shall not bear above a round or two—
We drop the golden cup at Heré's foot
And swoon back to the earth,—and find ourselves
Face-down among the pine-cones, cold with dew,
While the dogs bark, and many a shepherd scoffs,
'What's come now to the youth?' Such ups and downs
Have poets.
Am I such indeed? The name
Is royal, and to sign it like a queen,
Is what I dare not,—though some royal blood
Would seem to tingle in me now and then,
With sense of power and ache,—with imposthumes
And manias usual to the race. Howbeit
I dare not: 'tis too easy to go mad,
And ape a Bourbon in a crown of straws;
The thing's too common.
Many fervent souls
Strike rhyme on rhyme, who would strike steel on steel
If steel had offered, in a restless heat
Of doing something. Many tender souls
Have strung their losses on a rhyming thread,
As children, cowslips:—the more pains they take,
The work more withers. Young men, ay, and maids,
Too often sow their wild oats in tame verse,
Before they sit down under their own vine
And live for use. Alas, near all the birds
Will sing at dawn,—and yet we do not take
The chaffering swallow for the holy lark.
In those days, though, I never analysed
Myself even. All analysis comes late.
You catch a sight of Nature, earliest,
In full front sun-face, and your eyelids wink
And drop before the wonder of't; you miss
The form, through seeing the light. I lived, those days,
And wrote because I lived—unlicensed else:
My heart beat in my brain. Life's violent flood
Abolished bounds,—and, which my neighbour's field,
Which mine, what mattered? It is so in youth.
We play at leap-frog over the god Term;
The love within us and the love without
Are mixed, confounded; if we are loved or love,
We scarce distinguish. So, with other power.
Being acted on and acting seem the same:
In that first onrush of life's chariot-wheels,
We know not if the forests move or we.
And so, like most young poets, in a flush
Of individual life, I poured myself
Along the veins of others, and achieved
Mere lifeless imitations of live verse,
And made the living answer for the dead,
Profaning nature. 'Touch not, do not taste,
Nor handle,'—we're too legal, who write young:
We beat the phorminx till we hurt our thumbs,
As if still ignorant of counterpoint;
We call the Muse.... 'O Muse, benignant Muse!'—
As if we had seen her purple-braided head
With the eyes in it, start between the boughs
As often as a stag's. What make-believe,
With so much earnest! what effete results,
From virile efforts! what cold wire-drawn odes,
From such white heats!—bucolics, where the cows
Would scare the writer if they splashed the mud
In lashing off the flies,—didactics, driven
Against the heels of what the master said;
And counterfeiting epics, shrill with trumps
A babe might blow between two straining cheeks
Of bubbled rose, to make his mother laugh;
And elegiac griefs, and songs of love,
Like cast-off nosegays picked up on the road,
The worse for being warm: all these things, writ
On happy mornings, with a morning heart,
That leaps for love, is active for resolve,
Weak for art only. Oft, the ancient forms
Will thrill, indeed, in carrying the young blood.
The wine-skins, now and then, a little warped,
Will crack even, as the new wine gurgles in.
Spare the old bottles!—spill not the new wine.
By Keats's soul, the man who never stepped
In gradual progress like another man,
But, turning grandly on his central self,
Ensphered himself in twenty perfect years
And died, not young,—(the life of a long life,
Distilled to a mere drop, falling like a tear
Upon the world's cold cheek to make it burn
For ever;) by that strong excepted soul,
I count it strange, and hard to understand,
That nearly all young poets should write old;
That Pope was sexagenarian at sixteen,
And beardless Byron academical,
And so with others. It may be, perhaps,
Such have not settled long and deep enough
In trance, to attain to clairvoyance,—and still
The memory mixes with the vision, spoils,
And works it turbid.
Or perhaps, again,
In order to discover the Muse-Sphinx,
The melancholy desert must sweep round,
Behind you, as before.—
For me, I wrote
False poems, like the rest, and thought them true,
Because myself was true in writing them.
I, peradventure, have writ true ones since
With less complacence.
But I could not hide
My quickening inner life from those at watch.
They saw a light at a window now and then,
They had not set there. Who had set it there?
My father's sister started when she caught
My soul agaze in my eyes. She could not say
I had no business with a sort of soul,
But plainly she objected,—and demurred,
That souls were dangerous things to carry straight
Through all the spilt saltpetre of the world.
She said sometimes, 'Aurora, have you done
Your task this morning?—have you read that book?
And are you ready for the crochet here?'—
As if she said, 'I know there's something wrong;
I know I have not ground you down enough
To flatten and bake you to a wholesome crust
For household uses and proprieties,
Before the rain has got into my barn
And set the grains a-sprouting. What, you're green
With out-door impudence? you almost grow?'
To which I answered, 'Would she hear my task,
And verify my abstract of the book?
And should I sit down to the crochet work?
Was such her pleasure?' ... Then I sate and teased
The patient needle till it spilt the thread,
Which oozed off from it in meandering lace
From hour to hour. I was not, therefore, sad;
My soul was singing at a work apart
Behind the wall of sense, as safe from harm
As sings the lark when sucked up out of sight,
In vortices of glory and blue air.
And so, through forced work and spontaneous work,
The inner life informed the outer life,
Reduced the irregular blood to settled rhythms,
Made cool the forehead with fresh-sprinkling dreams,
And, rounding to the spheric soul the thin
Pined body, struck a colour up the cheeks,
Though somewhat faint. I clenched my brows across
My blue eyes greatening in the looking-glass,
And said, 'We'll live, Aurora! we'll be strong.
The dogs are on us—but we will not die.'
Whoever lives true life, will love true love.
I learnt to love that England. Very oft,
Before the day was born, or otherwise
Through secret windings of the afternoons,
I threw my hunters off and plunged myself
Among the deep hills, as a hunted stag
Will take the waters, shivering with the fear
And passion of the course. And when, at last
Escaped,—so many a green <DW72> built on <DW72>
Betwixt me and the enemy's house behind,
I dared to rest, or wander,—like a rest
Made sweeter for the step upon the grass,—
And view the ground's most gentle dimplement,
(As if God's finger touched but did not press
In making England!) such an up and down
Of verdure,—nothing too much up or down,
A ripple of land; such little hills, the sky
Can stoop to tenderly and the wheatfields climb;
Such nooks of valleys, lined with orchises,
Fed full of noises by invisible streams;
And open pastures, where you scarcely tell
White daisies from white dew,—at intervals
The mythic oaks and elm-trees standing out
Self-poised upon their prodigy of shade,—
I thought my father's land was worthy too
Of being my Shakspeare's.
Very oft alone,
Unlicensed; not unfrequently with leave
To walk the third with Romney and his friend
The rising painter, Vincent Carrington,
Whom men judge hardly, as bee-bonnetted,
Because he holds that, paint a body well,
You paint a soul by implication, like
The grand first Master. Pleasant walks! for if
He said ... 'When I was last in Italy' ...
It sounded as an instrument that's played
Too far off for the tune—and yet it's fine
To listen.
Ofter we walked only two,
If cousin Romney pleased to walk with me.
We read, or talked, or quarrelled, as it chanced:
We were not lovers, nor even friends well-matched—
Say rather, scholars upon different tracks,
And thinkers disagreed; he, overfull
Of what is, and I, haply, overbold
For what might be.
But then the thrushes sang,
And shook my pulses and the elms' new leaves,—
And then I turned, and held my finger up,
And bade him mark that, howsoe'er the world
Went ill, as he related, certainly
The thrushes still sang in it.—At which word
His brow would soften,—and he bore with me
In melancholy patience, not unkind,
While, breaking into voluble ecstacy,
I flattered all the beauteous country round,
As poets use ... the skies, the clouds, the fields,
The happy violets hiding from the roads
The primroses run down to, carrying gold,—
The tangled hedgerows, where the cows push out
Impatient horns and tolerant churning mouths
'Twixt dripping ash-boughs,—hedgerows all alive
With birds and gnats and large white butterflies
Which look as if the May-flower had caught life
And palpitated forth upon the wind,—
Hills, vales, woods, netted in a silver mist,
Farms, granges, doubled up among the hills,
And cattle grazing in the watered vales,
And cottage-chimneys smoking from the woods,
And cottage-gardens smelling everywhere,
Confused with smell of orchards. 'See,' I said,
'And see! is God not with us on the earth?
And shall we put Him down by aught we do?
Who says there's nothing for the poor and vile
Save poverty and wickedness? behold!'
And ankle-deep in English grass I leaped,
And clapped my hands, and called all very fair.
In the beginning when God called all good,
Even then, was evil near us, it is writ.
But we, indeed, who call things good and fair,
The evil is upon us while we speak;
Deliver us from evil, let us pray.
SECOND BOOK.
TIMES followed one another. Came a morn
I stood upon the brink of twenty years,
And looked before and after, as I stood
Woman and artist,—either incomplete,
Both credulous of completion. There I held
The whole creation in my little cup,
And smiled with thirsty lips before I drank,
'Good health to you and me, sweet neighbour mine,
And all these peoples.'
I was glad, that day;
The June was in me, with its multitudes
Of nightingales all singing in the dark,
And rosebuds reddening where the calyx split.
I felt so young, so strong, so sure of God!
So glad, I could not choose be very wise!
And, old at twenty, was inclined to pull
My childhood backward in a childish jest
To see the face of't once more, and farewell!
In which fantastic mood I bounded forth
At early morning,—would not wait so long
As even to snatch my bonnet by the strings,
But, brushing a green trail across the lawn
With my gown in the dew, took will and way
Among the acacias of the shrubberies,
To fly my fancies in the open air
And keep my birthday, till my aunt awoke
To stop good dreams. Meanwhile I murmured on,
As honeyed bees keep humming to themselves;
'The worthiest poets have remained uncrowned
Till death has bleached their foreheads to the bone,
And so with me it must be, unless I prove
Unworthy of the grand adversity,—
And certainly I would not fail so much.
What, therefore, if I crown myself to-day
In sport, not pride, to learn the feel of it,
Before my brows be numb as Dante's own
To all the tender pricking of such leaves?
Such leaves! what leaves?'
I pulled the branches down,
To choose from.
'Not the bay! I choose no bay;
The fates deny us if we are overbold:
Nor myrtle—which means chiefly love; and love
Is something awful which one dares not touch
So early o' mornings. This verbena strains
The point of passionate fragrance; and hard by,
This guelder-rose, at far too slight a beck
Of the wind, will toss about her flower-apples.
Ah—there's my choice,—that ivy on the wall,
That headlong ivy! not a leaf will grow
But thinking of a wreath. Large leaves, smooth leaves,
Serrated like my vines, and half as green.
I like such ivy; bold to leap a height
'Twas strong to climb! as good to grow on graves
As twist about a thyrsus; pretty too,
(And that's not ill) when twisted round a comb,'
Thus speaking to myself, half singing it,
Because some thoughts are fashioned like a bell
To ring with once being touched, I drew a wreath
Drenched, blinding me with dew, across my brow,
And fastening it behind so, ... turning faced
... My public!—cousin Romney—with a mouth
Twice graver than his eyes.
I stood there fixed—
My arms up, like the caryatid, sole
Of some abolished temple, helplessly
Persistent in a gesture which derides
A former purpose. Yet my blush was flame,
As if from flax, not stone.
'Aurora Leigh,
The earliest of Auroras!'
Hand stretched out
I clasped, as shipwrecked men will clasp a hand,
Indifferent to the sort of palm. The tide
Had caught me at my pastime, writing down
My foolish name too near upon the sea
Which drowned me with a blush as foolish. 'You,
My cousin!'
The smile died out in his eyes
And dropped upon his lips, a cold dead weight,
For just a moment.... 'Here's a book, I found!
No name writ on it—poems, by the form;
Some Greek upon the margin,—lady's Greek,
Without the accents. Read it? Not a word.
I saw at once the thing had witchcraft in't
Whereof the reading calls up dangerous spirits;
I rather bring it to the witch.'
'My book!
You found it'....
'In the hollow by the stream,
That beech leans down into—of which you said,
The Oread in it has a Naiad's heart
And pines for waters.'
'Thank you.'
'Rather _you_,
My cousin! that I have seen you not too much
A witch, a poet, scholar, and the rest,
To be a woman also.'
With a glance
The smile rose in his eyes again, and touched
The ivy on my forehead, light as air.
I answered gravely, 'Poets needs must be
Or men or women—more's the pity.'
'Ah,
But men, and still less women, happily,
Scarce need be poets. Keep to the green wreath,
Since even dreaming of the stone and bronze
Brings headaches, pretty cousin, and defiles
The clean white morning dresses.'
'So you judge!
Because I love the beautiful, I must
Love pleasure chiefly, and be overcharged
For ease and whiteness! Well—you know the world,
And only miss your cousin; 'tis not much!—
But learn this: I would rather take my part
With God's Dead, who afford to walk in white
Yet spread His glory, than keep quiet here,
And gather up my feet from even a step,
For fear to soil my gown in so much dust.
I choose to walk at all risks.—Here, if heads
That hold a rhythmic thought, must ache perforce,
For my part, I choose headaches,—and today's
My birthday.'
'Dear Aurora, choose instead
To cure such. You have balsams.'
'I perceive!—
The headache is too noble for my sex.
You think the heartache would sound decenter,
Since that's the woman's special, proper ache,
And altogether tolerable, except
To a woman.'
Saying which, I loosed my wreath,
And, swinging it beside me as I walked,
Half petulant, half playful, as we walked,
I sent a sidelong look to find his thought,—
As falcon set on falconer's finger may,
With sidelong head, and startled, braving eye,
Which means, 'You'll see—you'll see! I'll soon take flight—
You shall not hinder.' He, as shaking out
His hand and answering 'Fly then,' did not speak,
Except by such a gesture. Silently
We paced, until, just coming into sight
Of the house-windows, he abruptly caught
At one end of the swinging wreath, and said
'Aurora!' There I stopped short, breath and all.
'Aurora, let's be serious, and throw by
This game of head and heart. Life means, be sure,
Both heart and head,—both active, both complete,
And both in earnest. Men and women make
The world, as head and heart make human life.
Work man, work woman, since there's work to do
In this beleaguered earth, for head and heart,
And thought can never do the work of love!
But work for ends, I mean for uses; not
For such sleek fringes (do you call them ends?
Still less God's glory) as we sew ourselves
Upon the velvet of those baldaquins
Held 'twixt us and the sun. That book of yours,
I have not read a page of; but I toss
A rose up—it falls calyx down, you see!...
The chances are that, being a woman, young,
And pure, with such a pair of large, calm eyes, ...
You write as well ... and ill ... upon the whole,
As other women. If as well, what then?
If even a little better, ... still, what then?
We want the Best in art now, or no art.
The time is done for facile settings up
Of minnow gods, nymphs here, and tritons there;
The polytheists have gone out in God,
That unity of Bests. No best, no God!—
And so with art, we say. Give art's divine,
Direct, indubitable, real as grief,—
Or leave us to the grief we grow ourselves
Divine by overcoming with mere hope
And most prosaic patience. You, you are young
As Eve with nature's daybreak on her face;
But this same world you are come to, dearest coz,
Has done with keeping birthdays, saves her wreaths
To hang upon her ruins,—and forgets
To rhyme the cry with which she still beats back
Those savage, hungry dogs that hunt her down
To the empty grave of Christ. The world's hard pressed;
The sweat of labour in the early curse
Has (turning acrid in six thousand years)
Become the sweat of torture. Who has time,
An hour's time ... think!... to sit upon a bank
And hear the cymbal tinkle in white hands?
When Egypt's slain, I say, let Miriam sing!—
Before ... where's Moses?'
'Ah—exactly that!
Where's Moses?—is a Moses to be found?—
You'll seek him vainly in the bulrushes,
While I in vain touch cymbals. Yet, concede,
Such sounding brass has done some actual good,
(The application in a woman's hand,
If that were credible, being scarcely spoilt,)
In colonising beehives.'
'There it is!—
You play beside a death-bed like a child,
Yet measure to yourself a prophet's place
To teach the living. None of all these things,
Can women understand. You generalise
Oh, nothing!—not even grief! Your quick-breathed hearts,
So sympathetic to the personal pang,
Close, on each separate knife-stroke, yielding up
A whole life at each wound; incapable
Of deepening, widening a large lap of life
To hold the world-full woe. The human race
To you means, such a child, or such a man,
You saw one morning waiting in the cold,
Beside that gate, perhaps. You gather up
A few such cases, and, when strong, sometimes
Will write of factories and of slaves, as if
Your father were a <DW64>, and your son
A spinner in the mills. All's yours and you,—
All, with your blood, or otherwise
Just nothing to you. Why, I call you hard
To general suffering. Here's the world half blind
With intellectual light, half brutalised
With civilisation, having caught the plague
In silks from Tarsus, shrieking east and west
Along a thousand railroads, mad with pain
And sin too!... does one woman of you all,
(You who weep easily) grow pale to see
This tiger shake his cage?—does one of you
Stand still from dancing, stop from stringing pearls,
And pine and die, because of the great sum
Of universal anguish?—Show me a tear
Wet as Cordelia's, in eyes bright as yours,
Because the world is mad! You cannot count,
That you should weep for this account, not you!
You weep for what you know. A red-haired child
Sick in a fever, if you touch him once,
Though but so little as with a finger-tip,
Will set you weeping; but a million sick ...
You could as soon weep for the rule of three,
Or compound fractions. Therefore, this same world
Uncomprehended by you, must remain
Uninfluenced by you.—Women as you are,
Mere women, personal and passionate,
You give us doating mothers, and chaste wives,
Sublime Madonnas, and enduring saints!
We get no Christ from you,—and verily
We shall not get a poet, in my mind.'
'With which conclusion you conclude'....
'But this—
That you, Aurora, with the large live brow
And steady eyelids, cannot condescend
To play at art, as children play at swords,
To show a pretty spirit, chiefly admired
Because true action is impossible.
You never can be satisfied with praise
Which men give women when they judge a book
Not as mere work, but as mere woman's work,
Expressing the comparative respect
Which means the absolute scorn. 'Oh, excellent!
What grace! what facile turns! what fluent sweeps!
What delicate discernment ... almost thought!
The book does honour to the sex, we hold.
Among our female authors we make room
For this fair writer, and congratulate
The country that produces in these times
Such women, competent to ... spell.'
'Stop there!'
I answered—burning through his thread of talk
With a quick flame of emotion,—'You have read
My soul, if not my book, and argue well
I would not condescend ... we will not say
To such a kind of praise, (a worthless end
Is praise of all kinds) but to such a use
Of holy art and golden life. I am young,
And peradventure weak—you tell me so—
Through being a woman. And, for all the rest,
Take thanks for justice. I would rather dance
At fairs on tight-rope, till the babies dropped
Their gingerbread for joy,—than shift the types
For tolerable verse, intolerable
To men who act and suffer. Better far,
Pursue a frivolous trade by serious means,
Than a sublime art frivolously.'
'You,
Choose nobler work than either, O moist eyes,
And hurrying lips, and heaving heart! We are young
Aurora, you and I. The world ... look round ...
The world, we're come to late, is swollen hard
With perished generations and their sins:
The civiliser's spade grinds horribly
On dead men's bones, and cannot turn up soil
That's otherwise than fetid. All success
Proves partial failure; all advance implies
What's left behind; all triumph, something crushed
At the chariot-wheels; all government, some wrong:
And rich men make the poor, who curse the rich,
Who agonise together, rich and poor,
Under and over, in the social spasm
And crisis of the ages. Here's an age,
That makes its own vocation! here, we have stepped
Across the bounds of time! here's nought to see,
But just the rich man and just Lazarus,
And both in torments; with a mediate gulph,
Though not a hint of Abraham's bosom. Who,
Being man and human, can stand calmly by
And view these things, and never tease his soul
For some great cure? No physic for this grief,
In all the earth and heavens too?'
'You believe
In God, for your part?—ay? that He who makes,
Can make good things from ill things, best from worst,
As men plant tulips upon dunghills when
They wish them finest?'
'True. A death-heat is
The same as life-heat, to be accurate;
And in all nature is no death at all,
As men account of death, as long as God
Stands witnessing for life perpetually,
By being just God. That's abstract truth, I know,
Philosophy, or sympathy with God:
But I, I sympathise with man, not God,
I think I was a man for chiefly this;
And when I stand beside a dying bed,
It's death to me. Observe,—it had not much
Consoled the race of mastodons to know
Before they went to fossil, that anon
Their place should quicken with the elephant;
They were not elephants but mastodons:
And I, a man, as men are now, and not
As men may be hereafter, feel with men
In the agonising present.'
'Is it so,'
I said, 'my cousin? is the world so bad,
While I hear nothing of it through the trees?
The world was always evil,—but so bad?'
'So bad, Aurora. Dear, my soul is grey
With poring over the long sum of ill;
So much for vice, so much for discontent,
So much for the necessities of power,
So much for the connivances of fear,—
Coherent in statistical despairs
With such a total of distracted life, ...
To see it down in figures on a page,
Plain, silent, clear ... as God sees through the earth
The sense of all the graves!... that's terrible
For one who is not God, and cannot right
The wrong he looks on. May I choose indeed
But vow away my years, my means, my aims,
Among the helpers, if there's any help
In such a social strait? The common blood
That swings along my veins, is strong enough
To draw me to this duty.'
Then I spoke.
'I have not stood long on the strand of life,
And these salt waters have had scarcely time
To creep so high up as to wet my feet.
I cannot judge these tides—I shall, perhaps.
A woman's always younger than a man
At equal years, because she is disallowed
Maturing by the outdoor sun and air,
And kept in long-clothes past the age to walk.
Ah well, I know you men judge otherwise!
You think a woman ripens as a peach,—In
the cheeks, chiefly. Pass it to me now;
I'm young in age, and younger still, I think,
As a woman. But a child may say amen
To a bishop's prayer and see the way it goes;
And I, incapable to loose the knot
Of social questions, can approve, applaud
August compassion, christian thoughts that shoot
Beyond the vulgar white of personal aims.
Accept my reverence.'
There he glowed on me
With all his face and eyes. 'No other help?'
Said he—'no more than so?'
'What help?' I asked.
'You'd scorn my help,—as Nature's self, you say,
Has scorned to put her music in my mouth,
Because a woman's. Do you now turn round
And ask for what a woman cannot give?'
'For what she only can, I turn and ask,'
He answered, catching up my hands in his,
And dropping on me from his high-eaved brow
The full weight of his soul,—'I ask for love,
And that, she can; for life in fellowship
Through bitter duties—that, I know she can;
For wifehood ... will she?'
'Now,' I said, 'may God
Be witness 'twixt us two!' and with the word,
Meseemed I floated into a sudden light
Above his stature,—'am I proved too weak
To stand alone, yet strong enough to bear
Such leaners on my shoulder? poor to think,
Yet rich enough to sympathise with thought?
Incompetent to sing, as blackbirds can,
Yet competent to love, like HIM?'
I paused:
Perhaps I darkened, as the light-house will
That turns upon the sea. 'It's always so!
Anything does for a wife.'
'Aurora, dear,
And dearly honoured' ... he pressed in at once
With eager utterance,—'you translate me ill.
I do not contradict my thought of you
Which is most reverent, with another thought
Found less so. If your sex is weak for art,
(And I who said so, did but honour you
By using truth in courtship) it is strong
For life and duty. Place your fecund heart
In mine, and let us blossom for the world
That wants love's colour in the grey of time.
With all my talk I can but set you where
You look down coldly on the arena-heaps
Of headless bodies, shapeless, indistinct!
The Judgment-Angel scarce would find his way
Through such a heap of generalised distress,
To the individual man with lips and eyes—
Much less Aurora. Ah, my sweet, come down,
And, hand in hand, we'll go where yours shall touch
These victims, one by one! till, one by one,
The formless, nameless trunk of every man
Shall seem to wear a head, with hair you know,
And every woman catch your mother's face
To melt you into passion.'
'I am a girl,'
I answered slowly; 'you do well to name
My mother's face. Though far too early, alas,
God's hand did interpose 'twixt it and me,
I know so much of love, as used to shine
In that face and another. Just so much;
No more indeed at all. I have not seen
So much love since, I pray you pardon me,
As answers even to make a marriage with,
In this cold land of England. What you love,
Is not a woman, Romney, but a cause:
You want a helpmate, not a mistress, sir,—
A wife to help your ends ... in her no end!
Your cause is noble, your ends excellent,
But I, being most unworthy of these and that,
Do otherwise conceive of love. Farewell.'
'Farewell, Aurora? you reject me thus?'
He said.
'Why, sir, you are married long ago.
You have a wife already whom you love,
Your social theory. Bless you both, I say.
For my part, I am scarcely meek enough
To be the handmaid of a lawful spouse.
Do I look a Hagar, think you?'
'So, you jest!'
'Nay so, I speak in earnest,' I replied.
'You treat of marriage too much like, at least,
A chief apostle; you would bear with you
A wife ... a sister ... shall we speak it out?
A sister of charity.'
'Then, must it be
Indeed farewell? And was I so far wrong
In hope and in illusion, when I took
The woman to be nobler than the man,
Yourself the noblest woman,—in the use
And comprehension of what love is,—love,
That generates the likeness of itself
Through all heroic duties? so far wrong,
In saying bluntly, venturing truth on love,
Come, human creature, love and work with me,'—
Instead of, 'Lady, thou art wondrous fair,
And, where the Graces walk before, the Muse
Will follow at the lighting of their eyes,
And where the Muse walks, lovers need to creep:
Turn round and love me, or I die of love.'
With quiet indignation I broke in.
'You misconceive the question like a man,
Who sees a woman as the complement
Of his sex merely. You forget too much
That every creature, female as the male,
Stands single in responsible act and thought,
As also in birth and death. Whoever says
To a loyal woman, 'Love and work with me,'
Will get fair answers, if the work and love,
Being good themselves, are good for her—the best
She was born for. Women of a softer mood,
Surprised by men when scarcely awake to life,
Will sometimes only hear the first word, love,
And catch up with it any kind of work,
Indifferent, so that dear love go with it:
I do not blame such women, though, for love,
They pick much oakum; earth's fanatics make
Too frequently heaven's saints. But _me_, your work
Is not the best for,—nor your love the best,
Nor able to commend the kind of work
For love's sake merely. Ah, you force me, sir,
To be over-bold in speaking of myself,—
I, too, have my vocation,—work to do,
The heavens and earth have set me, since I changed
My father's face for theirs,—and, though your world
Were twice as wretched as you represent,
Most serious work, most necessary work,
As any of the economists'. Reform,
Make trade a Christian possibility,
And individual right no general wrong;
Wipe out earth's furrows of the Thine and Mine,
And leave one green, for men to play at bowls,
With innings for them all!... what then, indeed,
If mortals were not greater by the head
Than any of their prosperities? what then,
Unless the artist keep up open roads
Betwixt the seen and unseen,—bursting through
The best of your conventions with his best,
The speakable, imaginable best
God bids him speak, to prove what lies beyond
Both speech and imagination? A starved man
Exceeds a fat beast: we'll not barter, sir,
The beautiful for barley.—And, even so,
I hold you will not compass your poor ends
Of barley-feeding and material ease,
Without a poet's individualism
To work your universal. It takes a soul,
To move a body: it takes a high-souled man,
To move the masses ... even to a cleaner stye:
It takes the ideal, to blow a hair's-breadth off
The dust of the actual.—Ah, your Fouriers failed,
Because not poets enough to understand
That life develops from within.——For me,
Perhaps I am not worthy, as you say,
Of work like this!... perhaps a woman's soul
Aspires, and not creates! yet we aspire,
And yet I'll try out your perhapses, sir;
And if I fail ... why, burn me up my straw
Like other false works—I'll not ask for grace,
Your scorn is better, cousin Romney. I
Who love my art, would never wish it lower
To suit my stature. I may love my art.
You'll grant that even a woman may love art,
Seeing that to waste true love on anything,
Is womanly, past question.'
I retain
The very last word which I said, that day,
As you the creaking of the door, years past,
Which let upon you such disabling news
You ever after have been graver. He,
His eyes, the motions in his silent mouth,
Were fiery points on which my words were caught,
Transfixed for ever in my memory
For his sake, not their own. And yet I know
I did not love him ... nor he me ... that's sure....
And what I said, is unrepented of,
As truth is always. Yet ... a princely man!—
If hard to me, heroic for himself!
He bears down on me through the slanting years,
The stronger for the distance. If he had loved,
Ay, loved me, with that retributive face, ...
I might have been a common woman now,
And happier, less known and less left alone;
Perhaps a better woman after all,—
With chubby children hanging on my neck
To keep me low and wise. Ah me, the vines
That bear such fruit, are proud to stoop with it.
The palm stands upright in a realm of sand.
And I, who spoke the truth then, stand upright,
Still worthy of having spoken out the truth,
By being content I spoke it, though it set
Him there, me here.—O woman's vile remorse,
To hanker after a mere name, a show,
A supposition, a potential love!
Does every man who names love in our lives,
Become a power for that? is love's true thing
So much best to us, that what personates love
Is next best? A potential love, forsooth!
We are not so vile. No, no—he cleaves, I think,
This man, this image, ... chiefly for the wrong
And shock he gave my life, in finding me
Precisely where the devil of my youth
Had set me, on those mountain-peaks of hope
All glittering with the dawn-dew, all erect
And famished for the morning,—saying, while
I looked for empire and much tribute, 'Come,
I have some worthy work for thee below.
Come, sweep my barns, and keep my hospitals,—
And I will pay thee with a current coin
Which men give women.'
As we spoke, the grass
Was trod in haste beside us, and my aunt,
With smile distorted by the sun,—face, voice,
As much at issue with the summer-day
As if you brought a candle out of doors,—
Broke in with, 'Romney, here!—My child, entreat
Your cousin to the house, and have your talk,
If girls must talk upon their birthdays. Come,'
He answered for me calmly, with pale lips
That seemed to motion for a smile in vain.
'The talk is ended, madam, where we stand.
Your brother's daughter has dismissed me here;
And all my answer can be better said
Beneath the trees, than wrong by such a word
Your house's hospitalities. Farewell.'
With that he vanished. I could hear his heel
Ring bluntly in the lane, as down he leapt
The short way from us.—Then, a measured speech
Withdrew me. 'What means this, Aurora Leigh?
My brother's daughter has dismissed my guests?'
The lion in me felt the keeper's voice,
Through all its quivering dewlaps: I was quelled
Before her,—meekened to the child she knew:
I prayed her pardon, said, 'I had little thought
To give dismissal to a guest of hers,
In letting go a friend of mine, who came
To take me into service as a wife,—
No more than that, indeed.'
'No more, no more?
Pray Heaven,' she answered, 'that I was not mad.
I could not mean to tell her to her face
That Romney Leigh had asked me for a wife,
And I refused him?'
'Did he ask?' I said;
'I think he rather stooped to take me up
For certain uses which he found to do
For something called a wife. He never asked.'
'What stuff!' she answered; 'are they queens, these girls?
They must have mantles, stitched with twenty silks,
Spread out upon the ground, before they'll step
One footstep for the noblest lover born.'
'But I am born,' I said with firmness, 'I,
To walk another way than his, dear aunt.'
'You walk, you walk! A babe at thirteen months
Will walk as well as you,' she cried in haste,
'Without a steadying finger. Why, you child,
God help you, you are groping in the dark,
For all this sunlight. You suppose, perhaps,
That you, sole offspring of an opulent man,
Are rich and free to choose a way to walk?
You think, and it's a reasonable thought,
That I besides, being well to do in life,
Will leave my handful in my niece's hand
When death shall paralyse these fingers? Pray,
Pray, child,—albeit I know you love me not,—
As if you loved me, that I may not die!
For when I die and leave you, out you go,
(Unless I make room for you in my grave)
Unhoused, unfed, my dear, poor brother's lamb,
(Ah heaven,—that pains!)—without a right to crop
A single blade of grass beneath these trees,
Or cast a lamb's small shadow on the lawn,
Unfed, unfolded! Ah, my brother, here's
The fruit you planted in your foreign loves!—
Ay, there's the fruit he planted! never look
Astonished at me with your mother's eyes,
For it was they, who set you where you are,
An undowered orphan. Child, your father's choice
Of that said mother, disinherited
His daughter, his and hers. Men do not think
Of sons and daughters, when they fall in love,
So much more than of sisters; otherwise,
He would have paused to ponder what he did,
And shrunk before that clause in the entail
Excluding offspring by a foreign wife,
(The clause set up a hundred years ago
By a Leigh who wedded a French dancing-girl
And had his heart danced over in return);
But this man shrunk at nothing, never thought
Of you, Aurora, any more than me—
Your mother must have been a pretty thing,
For all the coarse Italian blacks and browns,
To make a good man, which my brother was,
Unchary of the duties to his house;
But so it fell indeed. Our cousin Vane,
Vane Leigh, the father of this Romney, wrote
Directly on your birth, to Italy,
'I ask your baby daughter for my son
In whom the entail now merges by the law.
Betroth her to us out of love, instead
Of colder reasons, and she shall not lose
By love or law from henceforth'—so he wrote;
A generous cousin, was my cousin Vane.
Remember how he drew you to his knee
The year you came here, just before he died,
And hollowed out his hands to hold your cheeks,
And wished them redder,—you remember Vane?
And now his son who represents our house
And holds the fiefs and manors in his place,
To whom reverts my pittance when I die,
(Except a few books and a pair of shawls)
The boy is generous like him, and prepared
To carry out his kindest word and thought
To you, Aurora. Yes, a fine young man
Is Romney Leigh; although the sun of youth
Has shone too straight upon his brain, I know,
And fevered him with dreams of doing good
To good-for-nothing people. But a wife
Will put all right, and stroke his temples cool
With healthy touches'....
I broke in at that.
I could not lift my heavy heart to breathe
Till then, but then I raised it, and it fell
In broken words like these—'No need to wait.
The dream of doing good to ... me, at least,
Is ended, without waiting for a wife
To cool the fever for him. We've escaped
That danger ... thank Heaven for it.'
'You,' she cried,
'Have got a fever. What, I talk and talk
An hour long to you,—I instruct you how
You cannot eat or drink or stand or sit,
Or even die, like any decent wretch
In all this unroofed and unfurnished world,
Without your cousin,—and you still maintain
There's room 'twixt him and you, for flirting fans
And running knots in eyebrows! You must have
A pattern lover sighing on his knee:
You do not count enough a noble heart,
Above book-patterns, which this very morn
Unclosed itself, in two dear fathers' names,
To embrace your orphaned life! fie, fie! But stay,
I write a word, and counteract this sin.'
She would have turned to leave me, but I clung.
'O sweet my father's sister, hear my word
Before you write yours. Cousin Vane did well,
And cousin Romney well,—and I well too,
In casting back with all my strength and will
The good they meant me. O my God, my God!
God meant me good, too, when he hindered me
From saying 'yes' this morning. If you write
A word, it shall be 'no.' I say no, no!
I tie up 'no' upon His altar-horns,
Quite out of reach of perjury! At least
My soul is not a pauper; I can live
At least my soul's life, without alms from men;
And if it must be in heaven instead of earth,
Let heaven look to it,—I am not afraid,'
She seized my hands with both hers, strained them fast,
And drew her probing and unscrupulous eyes
Right through me, body and heart. 'Yet, foolish Sweet,
You love this man. I have watched you when he came,
And when he went, and when we've talked of him:
I am not old for nothing; I can tell
The weather-signs of love—you love this man.'
Girls blush, sometimes, because they are alive,
Half wishing they were dead to save the shame.
The sudden blush devours them, neck and brow;
They have drawn too near the fire of life, like gnats,
And flare up bodily, wings and all. What then?
Who's sorry for a gnat ... or girl?
I blushed.
I feel the brand upon my forehead now
Strike hot, sear deep, as guiltless men may feel
The felon's iron, say, and scorn the mark
Of what they are not. Most illogical
Irrational nature of our womanhood,
That blushes one way, feels another way,
And prays, perhaps, another! After all,
We cannot be the equal of the male,
Who rules his blood a little.
For although
I blushed indeed, as if I loved the man,
And her incisive smile, accrediting
That treason of false witness in my blush,
Did bow me downward like a swathe of grass
Below its level that struck me,—I attest
The conscious skies and all their daily suns,
I think I loved him not ... nor then, nor since....
Nor ever. Do we love the schoolmaster,
Being busy in the woods? much less, being poor,
The overseer of the parish? Do we keep
Our love, to pay our debts with?
White and cold
I grew next moment. As my blood recoiled
From that imputed ignominy, I made
My heart great with it. Then, at last, I spoke,—
Spoke veritable words, but passionate,
Too passionate perhaps ... ground up with sobs
To shapeless endings. She let fall my hands,
And took her smile off, in sedate disgust,
As peradventure she had touched a snake,—
A dead snake, mind!—and, turning round, replied,
'We'll leave Italian manners, if you please.
I think you had an English father, child,
And ought to find it possible to speak
A quiet 'yes' or 'no,' like English girls,
Without convulsions. In another month
We'll take another answer ... no, or yes.'
With that, she left me in the garden-walk.
I had a father! yes, but long ago—
How long it seemed that moment. Oh, how far,
How far and safe, God, dost thou keep thy saints
When once gone from us! We may call against
The lighted windows of thy fair June-heaven
Where all the souls are happy,—and not one,
Not even my father, look from work or play
To ask, 'Who is it that cries after us,
Below there, in the dusk?' Yet formerly
He turned his face upon me quick enough,
If I said 'father.' Now I might cry loud;
The little lark reached higher with his song
Than I with crying. Oh, alone, alone,—
Not troubling any in heaven, nor any on earth,
I stood there in the garden, and looked up
The deaf blue sky that brings the roses out
On such June mornings.
You who keep account
Of crisis and transition in this life,
Set down the first time Nature says plain 'no'
To some 'yes' in you, and walks over you
In gorgeous sweeps of scorn. We all begin
By singing with the birds, and running fast
With June-days, hand in hand: but once, for all,
The birds must sing against us, and the sun
Strike down upon us like a friend's sword caught
By an enemy to slay us, while we read
The dear name on the blade which bites at us!—
That's bitter and convincing: after that,
We seldom doubt that something in the large
Smooth order of creation, though no more
Than haply a man's footstep, has gone wrong.
Some tears fell down my cheeks, and then I smiled,
As those smile who have no face in the world
To smile back to them. I had lost a friend
In Romney Leigh; the thing was sure—a friend,
Who had looked at me most gently now and then,
And spoken of my favourite books ... 'our books' ...
With such a voice! Well, voice and look were now
More utterly shut out from me, I felt,
Than even my father's. Romney now was turned
To a benefactor, to a generous man,
Who had tied himself to marry ... me, instead
Of such a woman, with low timorous lids
He lifted with a sudden word one day,
And left, perhaps, for my sake.—Ah, self-tied
By a contract,—male Iphigenia, bound
At a fatal Aulis, for the winds to change,
(But loose him—they'll not change); he well might seem
A little cold and dominant in love!
He had a right to be dogmatical,
This poor, good Romney. Love, to him, was made
A simple law-clause. If I married him,
I would not dare to call my soul my own,
Which so he had bought and paid for: every thought
And every heart-beat down there in the bill,—
Not one found honestly deductible
From any use that pleased him! He might cut
My body into coins to give away
Among his other paupers; change my sons,
While I stood dumb as Griseld, for black babes
Or piteous foundlings; might unquestioned set
My right hand teaching in the Ragged Schools,
My left hand washing in the Public Baths,
What time my angel of the Ideal stretched
Both his to me in vain! I could not claim
The poor right of a mouse in a trap, to squeal,
And take so much as pity, from myself.
Farewell, good Romney! if I loved you even,
I could but ill afford to let you be
So generous to me. Farewell, friend, since friend
Betwixt us two, forsooth, must be a word
So heavily overladen. And, since help
Must come to me from those who love me not,
Farewell, all helpers—I must help myself,
And am alone from henceforth.—Then I stooped,
And lifted the soiled garland from the ground,
And set it on my head as bitterly
As when the Spanish king did crown the bones
Of his dead love. So be it. I preserve
That crown still,—in the drawer there! 'twas the first;
The rest are like it;—those Olympian crowns,
We run for, till we lose sight of the sun
In the dust of the racing chariots!
After that,
Before the evening fell, I had a note
Which ran,—'Aurora, sweet Chaldean, you read
My meaning backward like your eastern books,
While I am from the west, dear. Read me now
A little plainer. Did you hate me quite
But yesterday? I loved you for my part;
I love you. If I spoke untenderly
This morning, my beloved, pardon it;
And comprehend me that I loved you so,
I set you on the level of my soul,
And overwashed you with the bitter brine
Of some habitual thoughts. Henceforth, my flower,
Be planted out of reach of any such,
And lean the side you please, with all your leaves!
Write woman's verses and dream woman's dreams;
But let me feel your perfume in my home,
To make my sabbath after working-days;
Bloom out your youth beside me,—be my wife.'
I wrote in answer—'We, Chaldeans, discern
Still farther than we read. I know your heart,
And shut it like the holy book it is,
Reserved for mild-eyed saints to pore upon
Betwixt their prayers at vespers. Well, you're right,
I did not surely hate you yesterday;
And yet I do not love you enough to-day
To wed you, cousin Romney. Take this word,
And let it stop you as a generous man
From speaking farther. You may tease, indeed,
And blow about my feelings, or my leaves,—
And here's my aunt will help you with east winds,
And break a stalk, perhaps, tormenting me;
But certain flowers grow near as deep as trees,
And, cousin, you'll not move my root, not you,
With all your confluent storms. Then let me grow
Within my wayside hedge, and pass your way!
This flower has never as much to say to you
As the antique tomb which said to travellers, 'Pause,'
'Siste, viator.' Ending thus, I signed.
The next week passed in silence, so the next,
And several after: Romney did not come,
Nor my aunt chide me. I lived on and on,
As if my heart were kept beneath a glass,
And everybody stood, all eyes and ears,
To see and hear it tick. I could not sit,
Nor walk, nor take a book, nor lay it down,
Not sew on steadily, nor drop a stitch
And a sigh with it, but I felt her looks
Still cleaving to me, like the sucking asp
To Cleopatra's breast, persistently
Through the intermittent pantings. Being observed,
When observation is not sympathy,
Is just being tortured. If she said a word,
A 'thank you,' or an 'if it please you, dear,'
She meant a commination, or, at best,
An exorcism against the devildom
Which plainly held me. So with all the house.
Susannah could not stand and twist my hair,
Without such glancing at the looking-glass
To see my face there, that she missed the plait:
And John,—I never sent my plate for soup,
Or did not send it, but the foolish John
Resolved the problem, 'twixt his napkined thumbs,
Of what was signified by taking soup
Or choosing mackerel. Neighbours, who dropped in
On morning visits, feeling a joint wrong,
Smiled admonition, sate uneasily,
And talked with measured, emphasised reserve,
Of parish news, like doctors to the sick,
When not called in,—as if, with leave to speak,
They might say something. Nay, the very dog
Would watch me from his sun-patch on the floor,
In alternation with the large black fly
Not yet in reach of snapping. So I lived.
A Roman died so; smeared with honey, teased
By insects, stared to torture by the noon:
And many patient souls 'neath English roofs
Have died like Romans. I, in looking back,
Wish only, now, I had borne the plague of all
With meeker spirits than were rife in Rome.
For, on the sixth week, the dead sea broke up,
Dashed suddenly through beneath the heel of Him
Who stands upon the sea and earth, and swears
Time shall be nevermore. The clock struck nine
That morning, too,—no lark was out of tune;
The hidden farms among the hills, breathed straight
Their smoke toward heaven; the lime-tree scarcely stirred
Beneath the blue weight of the cloudless sky,
Though still the July air came floating through
The woodbine at my window, in and out,
With touches of the out-door country-news
For a bending forehead. There I sate, and wished
That morning-truce of God would last till eve,
Or longer. 'Sleep,' I thought, 'late sleepers,—sleep,
And spare me yet, the burden of your eyes.'
Then, suddenly, a single ghastly shriek
Tore upwards from the bottom of the house.
Like one who wakens in a grave and shrieks,
The still house seemed to shriek itself alive,
And shudder through its passages and stairs
With slam of doors and clash of bells.—I sprang,
I stood up in the middle of the room,
And there confronted at my chamber-door,
A white face,—shivering, ineffectual lips.
'Come, come,' they tried to utter, and I went;
As if a ghost had drawn me at the point
Of a fiery finger through the uneven dark,
I went with reeling footsteps down the stair,
Nor asked a question.
There she sate, my aunt,—
Bolt upright in the chair beside her bed,
Whose pillow had no dint! she had used no bed
For that night's sleeping ... yet slept well. My God,
The dumb derision of that grey, peaked face
Concluded something grave against the sun,
Which filled the chamber with its July burst
When Susan drew the curtains, ignorant
Of who sate open-eyed behind her. There,
She sate ... it sate ... we said 'she' yesterday ...
And held a letter with unbroken seal,
As Susan gave it to her hand last night:
All night she had held it. If its news referred
To duchies or to dunghills, not an inch
She'd budge, 'twas obvious, for such worthless odds:
Nor, though the stars were suns, and overburned
Their spheric limitations, swallowing up
Like wax the azure spaces, could they force
Those open eyes to wink once. What last sight
Had left them blank and flat so,—drawing out
The faculty of vision from the roots,
As nothing more, worth seeing, remained behind?
Were those the eyes that watched me, worried me?
That dogged me up and down the hours and days,
A beaten, breathless, miserable soul?
And did I pray, a half hour back, but so,
To escape the burden of those eyes ... those eyes?
'Sleep late' I said.—
Why now, indeed, they sleep.
God answers sharp and sudden on some prayers,
And thrusts the thing we have prayed for in our face,
A gauntlet with a gift in't. Every wish
Is like a prayer ... with God.
I had my wish,—
To read and meditate the thing I would,
To fashion all my life upon my thought,
And marry, or not marry. Henceforth, none
Could disapprove me, vex me, hamper me.
Full ground-room, in this desert newly made,
For Babylon or Balbec,—when the breath,
Just choked with sand, returns, for building towns!
The heir came over on the funeral day,
And we two cousins met before the dead,
With two pale faces. Was it death or life
That moved us? When the will was read and done,
The official guest and witnesses withdrawn,
We rose up in a silence almost hard,
And looked at one another. Then I said,
'Farewell, my cousin.'
But he touched, just touched
My hatstrings tied for going, (at the door
The carriage stood to take me) and said low,
His voice a little unsteady through his smile,
'Siste, viator.'
'Is there time,' I asked,
'In these last days of railroads, to stop short
Like Cæsar's chariot (weighing half a ton)
On the Appian road, for morals?'
'There is time,'
He answered grave, 'for necessary words,
Inclusive, trust me, of no epitaph
On man or act, my cousin. We have read
A will, which gives you all the personal goods
And funded monies of your aunt.'
'I thank
Her memory for it. With three hundred pounds
We buy in England even, clear standing-room
To stand and work in. Only two hours since,
I fancied I was poor.'
'And, cousin, still
You're richer than you fancy. The will says,
_Three hundred pounds, and any other sum
Of which the said testatrix dies possessed_.
I say she died possessed of other sums.'
'Dear Romney, need we chronicle the pence?
I'm richer than I thought—that's evident.
Enough so.'
'Listen rather. You've to do
With business and a cousin,' he resumed,
'And both, I fear, need patience. Here's the fact.
The other sum (there _is_ another sum,
Unspecified in any will which dates
After possession, yet bequeathed as much
And clearly as those said three hundred pounds)
Is thirty thousand. You will have it paid
When?... where? My duty troubles you with words.'
He struck the iron when the bar was hot;
No wonder if my eyes sent out some sparks.
'Pause there! I thank you. You are delicate
In glosing gifts;—but I, who share your blood,
Am rather made for giving, like yourself,
Than taking, like your pensioners. Farewell.'
He stopped me with a gesture of calm pride.
'A Leigh,' he said, 'gives largesse and gives love,
But gloses neither: if a Leigh could glose,
He would not do it, moreover, to a Leigh,
With blood trained up along nine centuries
To hound and hate a lie, from eyes like yours.
And now we'll make the rest as clear; your aunt
Possessed these monies.'
'You will make it clear,
My cousin, as the honour of us both,
Or one of us speaks vainly—that's not I.
My aunt possessed this sum,—inherited
From whom, and when? bring documents, prove dates.'
'Why now indeed you throw your bonnet off,
As if you had time left for a logarithm!
The faith's the want. Dear cousin, give me faith,
And you shall walk this road with silken shoes,
As clean as any lady of our house
Supposed the proudest. Oh, I comprehend
The whole position from your point of sight.
I oust you from your father's halls and lands,
And make you poor by getting rich—that's law;
Considering which, in common circumstance,
You would not scruple to accept from me
Some compensation, some sufficiency
Of income—that were justice; but, alas,
I love you ... that's mere nature!—you reject
My love ... that's nature also;—and at once,
You cannot, from a suitor disallowed,
A hand thrown back as mine is, into yours
Receive a doit, a farthing, ... not for the world!
That's etiquette with women, obviously
Exceeding claim of nature, law, and right,
Unanswerable to all. I grant, you see,
The case as you conceive it,—leave you room
To sweep your ample skirts of womanhood;
While, standing humbly squeezed against the wall,
I own myself excluded from being just,
Restrained from paying indubitable debts,
Because denied from giving you my soul—
That's my misfortune!—I submit to it
As if, in some more reasonable age,
'Twould not be less inevitable. Enough.
You'll trust me, cousin, as a gentleman,
To keep your honour, as you count it, pure,—
Your scruples (just as if I thought them wise)
Safe and inviolate from gifts of mine.'
I answered mild but earnest. 'I believe
In no one's honour which another keeps,
Nor man's nor woman's. As I keep, myself,
My truth and my religion, I depute
No father, though I had one this side death,
Nor brother, though I had twenty, much less you,
Though twice my cousin, and once Romney Leigh,
To keep my honour pure. You face, to-day,
A man who wants instruction, mark me, not
A woman who wants protection. As to a man,
Show manhood, speak out plainly, be precise
With facts and dates. My aunt inherited
This sum, you say—'
'I said she died possessed
Of this, dear cousin.'
'Not by heritage.
Thank you: we're getting to the facts at last.
Perhaps she played at commerce with a ship
Which came in heavy with Australian gold?
Or touched a lottery with her finger-end,
Which tumbled on a sudden into her lap
Some old Rhine tower or principality?
Perhaps she had to do with a marine
Sub-transatlantic railroad, which pre-pays
As well as pre-supposes? or perhaps
Some stale ancestral debt was after-paid
By a hundred years, and took her by surprise?—
You shake your head my cousin; I guess ill.'
'You need not guess, Aurora, nor deride,—
The truth is not afraid of hurting you.
You'll find no cause, in all your scruples, why
Your aunt should cavil at a deed of gift
'Twixt her and me.'
'I thought so—ah! a gift.'
'You naturally thought so,' he resumed.
'A very natural gift.'
'A gift, a gift!
Her individual life being stranded high
Above all want, approaching opulence,
Too haughty was she to accept a gift
Without some ultimate aim: ah, ah, I see,—
A gift intended plainly for her heirs,
And so accepted ... if accepted ... ah,
Indeed that might be; I am snared perhaps,
Just so. But, cousin, shall I pardon you,
If thus you have caught me with a cruel springe?'
He answered gently, 'Need you tremble and pant
Like a netted lioness? is't my fault, mine,
That you're a grand wild creature of the woods,
And hate the stall built for you? Any way,
Though triply netted, need you glare at me?
I do not hold the cords of such a net;
You're free from me, Aurora!'
'Now may God
Deliver me from this strait! This gift of yours
Was tendered ... when? accepted ... when?' I asked.
'A month ... a fortnight since? Six weeks ago
It was not tendered. By a word she dropped,
I know it was not tendered nor received.
When was it? bring your dates.'
'What matters when?
A half-hour ere she died, or a half-year,
Secured the gift, maintains the heritage
Inviolable with law. As easy pluck
The golden stars from heaven's embroidered stole,
To pin them on the grey side of this earth,
As make you poor again, thank God.'
'Not poor
Nor clean again from henceforth, you thank God?
Well, sir—I ask you ... I insist at need, ...
Vouchsafe the special date, the special date.'
'The day before her death-day,' he replied,
'The gift was in her hands. We'll find that deed,
And certify that date to you.'
As one
Who has climbed a mountain-height and carried up
His own heart climbing, panting in his throat
With the toil of the ascent, takes breath at last,
Looks back in triumph—so I stood and looked:
'Dear cousin Romney, we have reached the top
Of this steep question, and may rest, I think.
But first,—I pray you pardon, that the shock
And surge of natural feeling and event
Had made me oblivious of acquainting you
That this, this letter ... unread, mark,—still sealed,
Was found enfolded in the poor dead hand:
That spirit of hers had gone beyond the address,
Which could not find her though you wrote it clear,—
I know your writing, Romney,—recognise
The open-hearted _A_, the liberal sweep
Of the _G_. Now listen,—let us understand;
You will not find that famous deed of gift,
Unless you find it in the letter here,
Which, not being mine, I give you back.—Refuse
To take the letter? well then—you and I,
As writer and as heiress, open it
Together, by your leave.—Exactly so:
The words in which the noble offering's made,
Are nobler still, my cousin; and, I own,
The proudest and most delicate heart alive,
Distracted from the measure of the gift
By such a grace in giving, might accept
Your largesse without thinking any more
Of the burthen of it, than King Solomon
Considered, when he wore his holy ring
Charáctered over with the ineffable spell,
How many carats of fine gold made up
Its money-value. So, Leigh gives to Leigh—
Or rather, might have given, observe!—for that's
The point we come to. Here's a proof of gift,
But here's no proof, sir, of acceptancy,
But rather, disproof. Death's black dust, being blown,
Infiltrated through every secret fold
Of this sealed letter by a puff of fate,
Dried up for ever the fresh-written ink,
Annulled the gift, disutilised the grace,
And left these fragments.'
As I spoke, I tore
The paper up and down, and down and up
And crosswise, till it fluttered from my hands,
As forest-leaves, stripped suddenly and rapt
By a whirlwind on Valdarno, drop again,
Drop slow, and strew the melancholy ground
Before the amazèd hills ... why, so, indeed,
I'm writing like a poet, somewhat large
In the type of the image,—and exaggerate
A small thing with a great thing, topping it!—
But then I'm thinking how his eyes looked ... his,
With what despondent and surprised reproach!
I think the tears were in them, as he looked—
I think the manly mouth just trembled. Then
He broke the silence.
'I may ask, perhaps,
Although no stranger ... only Romney Leigh,
Which means still less ... than Vincent Carrington ...
Your plans in going hence, and where you go.
This cannot be a secret.'
'All my life
Is open to you, cousin. I go hence
To London, to the gathering-place of souls,
To live mine straight out, vocally, in books;
Harmoniously for others, if indeed
A woman's soul, like man's, be wide enough
To carry the whole octave (that's to prove)
Or, if I fail, still, purely for myself.
Pray God be with me, Romney.'
'Ah, poor child,
Who fight against the mother's 'tiring hand,
And choose the headsman's! May God change his world
For your sake, sweet, and make it mild as heaven,
And juster than I have found you!'
But I paused.
'And you, my cousin?'—
'I,' he said,—'you ask?
You care to ask? Well, girls have curious minds,
And fain would know the end of everything,
Of cousins, therefore, with the rest. For me,
Aurora, I've my work; you know my work;
And, having missed this year some personal hope,
I must beware the rather that I miss
No reasonable duty. While you sing
Your happy pastorals of the meads and trees,
Bethink you that I go to impress and prove
On stifled brains and deafened ears, stunned deaf,
Crushed dull with grief, that nature sings itself,
And needs no mediate poet, lute or voice,
To make it vocal. While you ask of men
Your audience, I may get their leave perhaps
For hungry orphans to say audibly
'We're hungry, see,'—for beaten and bullied wives
To hold their unweaned babies up in sight,
Whom orphanage would better; and for all
To speak and claim their portion ... by no means
Of the soil, ... but of the sweat in tilling it,—
Since this is now-a-days turned privilege,
To have only God's curse on us, and not man's.
Such work I have for doing, elbow-deep
In social problems,—as you tie your rhymes,
To draw my uses to cohere with needs,
And bring the uneven world back to its round;
Or, failing so much, fill up, bridge at least
To smoother issues, some abysmal cracks
And feuds of earth, intestine heats have made
To keep men separate,—using sorry shifts
Of hospitals, almshouses, infant schools,
And other practical stuff of partial good,
You lovers of the beautiful and whole,
Despise by system.'
'_I_ despise? The scorn
Is yours, my cousin. Poets become such,
Through scorning nothing. You decry them for
The good of beauty, sung and taught by them,
While they respect your practical partial good
As being a part of beauty's self. Adieu!
When God helps all the workers for his world,
The singers shall have help of Him, not last.'
He smiled as men smile when they will not speak
Because of something bitter in the thought;
And still I feel his melancholy eyes
Look judgment on me. It is seven years since:
I know not if 'twas pity or 'twas scorn
Has made them so far-reaching: judge it ye
Who have had to do with pity more than love.
And scorn than hatred. I am used, since then,
To other ways, from equal men. But so,
Even so, we let go hands, my cousin and I,
And, in between us, rushed the torrent-world
To blanch our faces like divided rocks,
And bar for ever mutual sight and touch
Except through swirl of spray and all that roar.
THIRD BOOK.
'TO-DAY thou girdest up thy loins thyself,
And goest where thou wouldest: presently
Others shall gird thee,' said the Lord, 'to go
Where thou would'st not.' He spoke to Peter thus,
To signify the death which he should die
When crucified head downwards.
If He spoke
To Peter then, He speaks to us the same;
The word suits many different martyrdoms,
And signifies a multiform of death,
Although we scarcely die apostles, we,
And have mislaid the keys of heaven and earth.
For 'tis not in mere death that men die most;
And, after our first girding of the loins
In youth's fine linen and fair broidery,
To run up hill and meet the rising sun,
We are apt to sit tired, patient as a fool,
While others gird us with the violent bands
Of social figments, feints, and formalisms,
Reversing our straight nature, lifting up
Our base needs, keeping down our lofty thoughts,
Head downward on the cross-sticks of the world.
Yet He can pluck us from that shameful cross.
God, set our feet low and our forehead high,
And show us how a man was made to walk!
Leave the lamp, Susan, and go up to bed.
The room does very well; I have to write
Beyond the stroke of midnight. Get away;
Your steps, for ever buzzing in the room,
Tease me like gnats. Ah, letters! throw them down
At once, as I must have them, to be sure,
Whether I bid you never bring me such
At such an hour, or bid you. No excuse.
You choose to bring them, as I choose perhaps
To throw them in the fire. Now, get to bed,
And dream, if possible, I am not cross.
Why what a pettish, petty thing I grow,—
A mere, mere woman,—a mere flaccid nerve,—
A kerchief left out all night in the rain,
Turned soft so,—overtasked and overstrained
And overlived in this close London life!
And yet I should be stronger.
Never burn
Your letters, poor Aurora! for they stare
With red seals from the table, saying each,
'Here's something that you know not.' Out alas,
'Tis scarcely that the world's more good and wise
Or even straighter and more consequent
Since yesterday at this time—yet, again,
If but one angel spoke from Ararat,
I should be very sorry not to hear:
So open all the letters! let me read.
Blanche Ord, the writer in the 'Lady's Fan,'
Requests my judgment on ... that, afterwards.
Kate Ward desires the model of my cloak,
And signs, 'Elisha to you.' Pringle Sharpe
Presents his work on 'Social Conduct,' ... craves
A little money for his pressing debts ...
From me, who scarce have money for my needs,—
Art's fiery chariot which we journey in
Being apt to singe our singing-robes to holes,
Although you ask me for my cloak, Kate Ward!
Here's Rudgely knows it,—editor and scribe—
He's 'forced to marry where his heart is not,
Because the purse lacks where he lost his heart.'
Ah,—— lost it because no one picked it up!
That's really loss! (and passable impudence.)
My critic Hammond flatters prettily,
And wants another volume like the last.
My critic Belfair wants another book
Entirely different, which will sell, (and live?)
A striking book, yet not a startling book,
The public blames originalities,
(You must not pump spring-water unawares
Upon a gracious public, full of nerves—)
Good things, not subtle, new yet orthodox,
As easy reading as the dog-eared page
That's fingered by said public, fifty years,
Since first taught spelling by its grandmother,
And yet a revelation in some sort:
That's hard, my critic Belfair! So—what next?
My critic Stokes objects to abstract thoughts;
'Call a man, John, a woman, Joan,' says he,
'And do not prate so of humanities:'
Whereat I call my critic, simply Stokes.
My critic Jobson recommends more mirth,
Because a cheerful genius suits the times,
And all true poets laugh unquenchably
Like Shakspeare and the gods. That's very hard.
The gods may laugh, and Shakspeare; Dante smiled
With such a needy heart on two pale lips,
We cry, 'Weep rather, Dante.' Poems are
Men, if true poems: and who dares exclaim
At any man's door, 'Here, 'tis probable
The thunder fell last week, and killed a wife,
And scared a sickly husband—what of that?
Get up, be merry, shout, and clap your hands,
Because a cheerful genius suits the times—'?
None says so to the man,—and why indeed
Should any to the poem? A ninth seal;
The apocalypse is drawing to a close.
Ha,—this from Vincent Carrington,—'Dear friend,
I want good counsel. Will you lend me wings
To raise me to the subject, in a sketch
I'll bring to-morrow—may I? at eleven?
A poet's only born to turn to use;
So save you! for the world ... and Carrington.'
'(Writ after.) Have you heard of Romney Leigh,
Beyond what's said of him in newspapers,
His phalansteries there, his speeches here,
His pamphlets, pleas, and statements, everywhere?
He dropped _me_ long ago; but no one drops
A golden apple—though indeed, one day,
You hinted that, but jested. Well, at least,
You know Lord Howe, who sees him ... whom he sees,
And _you_ see, and I hate to see,—for Howe
Stands high upon the brink of theories,
Observes the swimmers, and cries 'Very fine,'
But keeps dry linen equally,—unlike
That gallant breaster, Romney. Strange it is,
Such sudden madness seizing a young man,
To make earth over again,—while I'm content
To make the pictures. Let me bring the sketch.
A tiptoe Danae, overbold and hot;
Both arms a-flame to meet her wishing Jove
Halfway, and burn him faster down; the face
And breasts upturned and straining, the loose locks
All glowing with the anticipated gold.
Or here's another on the self-same theme.
She lies here—flat upon her prison-floor,
The long hair swathed about her to the heel,
Like wet sea-weed. You dimly see her through
The glittering haze of that prodigious rain,
Half blotted out of nature by a love
As heavy as fate. I'll bring you either sketch.
I think, myself, the second indicates
More passion.'
Surely. Self is put away,
And calm with abdication. She is Jove,
And no more Danae—greater thus. Perhaps
The painter symbolises unawares
Two states of the recipient artist-soul;
One, forward, personal, wanting reverence,
Because aspiring only. We'll be calm,
And know that, when indeed our Joves come down,
We all turn stiller than we have ever been.
Kind Vincent Carrington. I'll let him come.
He talks of Florence,—and may say a word
Of something as it chanced seven years ago,—
A hedgehog in the path, or a lame bird,
In those green country walks, in that good time,
When certainly I was so miserable ...
I seem to have missed a blessing ever since.
The music soars within the little lark,
And the lark soars. It is not thus with men.
We do not make our places with our strains,—
Content, while they rise, to remain behind,
Alone on earth instead of so in heaven.
No matter—I bear on my broken tale.
When Romney Leigh and I had parted thus,
I took a chamber up three flights of stairs
Not far from being as steep as some larks climb,
And, in a certain house in Kensington,
Three years I lived and worked. Get leave to work
In this world,—'tis the best you get at all;
For God, in cursing, gives us better gifts
Than men in benediction. God says, 'Sweat
For foreheads;' men say 'crowns;' and so we are crowned,—
Ay, gashed by some tormenting circle of steel
Which snaps with a secret spring. Get work, get work;
Be sure 'tis better than what you work to get.
So, happy and unafraid of solitude,
I worked the short days out,—and watched the sun
On lurid morns or monstrous afternoons,
Like some Druidic idol's fiery brass,
With fixed unflickering outline of dead heat,
In which the blood of wretches pent inside
Seemed oozing forth to incarnadine the air,—
Push out through fog with his dilated disk,
And startle the slant roofs and chimney-pots
With splashes of fierce colour. Or I saw
Fog only, the great tawny weltering fog,
Involve the passive city, strangle it
Alive, and draw it off into the void,
Spires, bridges, streets, and squares, as if a spunge
Had wiped out London,—or as noon and night
Had clapped together and utterly struck out
The intermediate time, undoing themselves
In the act. Your city poets see such things,
Not despicable. Mountains of the south,
When, drunk and mad with elemental wines,
They rend the seamless mist and stand up bare,
Make fewer singers, haply. No one sings,
Descending Sinai: on Parnassus mount,
You take a mule to climb, and not a muse,
Except in fable and figure: forests chant
Their anthems to themselves, and leave you dumb.
But sit in London, at the day's decline,
And view the city perish in the mist
Like Pharaoh's armaments in the deep Red Sea,—
The chariots, horsemen, footmen, all the host,
Sucked down and choked to silence—then, surprised
By a sudden sense of vision and of tune,
You feel as conquerors though you did not fight,
And you and Israel's other singing girls,
Ay, Miriam with them, sing the song you choose.
I worked with patience which means almost power.
I did some excellent things indifferently,
Some bad things excellently. Both were praised,
The latter loudest. And by such a time
That I myself had set them down as sins
Scarce worth the price of sackcloth, week by week,
Arrived some letter through the sedulous post,
Like these I've read, and yet dissimilar,
With pretty maiden seals,—initials twined
Of lilies, or a heart marked _Emily_,
(Convicting Emily of being all heart);
Or rarer tokens from young bachelors,
Who wrote from college (with the same goosequill,
Suppose, they had just been plucked of) and a snatch
From Horace, 'Collegisse juvat,' set
Upon the first page. Many a letter signed
Or unsigned, showing the writers at eighteen
Had lived too long, though every muse should help
The daylight, holding candles,—compliments,
To smile or sigh at. Such could pass with me
No more than coins from Moscow circulate
At Paris. Would ten roubles buy a tag
Of ribbon on the boulevard, worth a sou?
I smiled that all this youth should love me,—sighed
That such a love could scarcely raise them up
To love what was more worthy than myself;
Then sighed again, again, less generously,
To think the very love they lavished so,
Proved me inferior. The strong loved me not,
And he ... my cousin Romney ... did not write.
I felt the silent finger of his scorn
Prick every bubble of my frivolous fame
As my breath blew it, and resolve it back
To the air it came from. Oh, I justified
The measure he had taken of my height:
The thing was plain—he was not wrong a line;
I played at art, made thrusts with a toy-sword,
Amused the lads and maidens.
Came a sigh
Deep, hoarse with resolution,—I would work
To better ends, or play in earnest. 'Heavens,
I think I should be almost popular
If this went on!'—I ripped my verses up,
And found no blood upon the rapier's point;
The heart in them was just an embryo's heart,
Which never yet had beat, that it should die;
Just gasps of make-believe galvanic life;
Mere tones, inorganised to any tune.
And yet I felt it in me where it burnt,
Like those hot fire-seeds of creation held
In Jove's clenched palm before the worlds were sown,—
But I—I was not Juno even! my hand
Was shut in weak convulsion, woman's ill,
And when I yearned to loose a finger—lo,
The nerve revolted. 'Tis the same even now:
This hand may never, haply, open large,
Before the spark is quenched, or the palm charred,
To prove the power not else than by the pain.
It burns, it burnt—my whole life burnt with it,
And light, not sunlight and not torchlight, flashed
My steps out through the slow and difficult road.
I had grown distrustful of too forward Springs,
The season's books in drear significance
Of morals, dropping round me. Lively books?
The ash has livelier verdure than the yew;
And yet the yew's green longer, and alone
Found worthy of the holy Christmas time.
We'll plant more yews if possible, albeit
We plant the graveyards with them.
Day and night
I worked my rhythmic thought, and furrowed up
Both watch and slumber with long lines of life
Which did not suit their season. The rose fell
From either cheek, my eyes globed luminous
Through orbits of blue shadow, and my pulse
Would shudder along the purple-veined wrist
Like a shot bird. Youth's stern, set face to face
With youth's ideal: and when people came
And said, 'You work too much, you are looking ill,'
I smiled for pity of them who pitied me,
And thought I should be better soon perhaps
For those ill looks. Observe—'I,' means in youth
Just _I_ ... the conscious and eternal soul
With all its ends,—and not the outside life,
The parcel-man, the doublet of the flesh,
The so much liver, lung, integument,
Which make the sum of 'I' hereafter, when
World-talkers talk of doing well or ill.
_I_ prosper, if I gain a step, although
A nail then pierced my foot: although my brain
Embracing any truth, froze paralysed,
_I_ prosper. I but change my instrument;
I break the spade off, digging deep for gold,
And catch the mattock up.
I worked on, on.
Through all the bristling fence of nights and days
Which hedges time in from the eternities,
I struggled, ... never stopped to note the stakes
Which hurt me in my course. The midnight oil
Would stink sometimes; there came some vulgar needs:
I had to live, that therefore I might work,
And, being but poor, I was constrained, for life,
To work with one hand for the booksellers,
While working with the other for myself
And art. You swim with feet as well as hands,
Or make small way. I apprehended this,—
In England, no one lives by verse that lives;
And, apprehending, I resolved by prose
To make a space to sphere my living verse.
I wrote for cyclopædias, magazines,
And weekly papers, holding up my name
To keep it from the mud. I learnt the use
Of the editorial 'we' in a review,
As courtly ladies the fine trick of trains,
And swept it grandly through the open doors
As if one could not pass through doors at all
Save so encumbered. I wrote tales beside,
Carved many an article on cherry-stones
To suit light readers,—something in the lines
Revealing, it was said, the mallet-hand,
But that, I'll never vouch for. What you do
For bread, will taste of common grain, not grapes,
Although you have a vineyard in Champagne,—
Much less in Nephelococcygia,
As mine was, peradventure.
Having bread
For just so many days, just breathing room
For body and verse, I stood up straight and worked
My veritable work. And as the soul
Which grows within a child, makes the child grow,—
Or as the fiery sap, the touch from God,
Careering through a tree, dilates the bark,
And roughs with scale and knob, before it strikes
The summer foliage out in a green flame—
So life, in deepening with me, deepened all
The course I took, the work I did. Indeed,
The academic law convinced of sin;
The critics cried out on the falling off,
Regretting the first manner. But I felt
My heart's life throbbing in my verse to show
It lived, it also—certes incomplete,
Disordered with all Adam in the blood,
But even its very tumours, warts, and wens,
Still organised by, and implying life.
A lady called upon me on such a day.
She had the low voice of your English dames,
Unused, it seems, to need rise half a note
To catch attention,—and their quiet mood,
As if they lived too high above the earth
For that to put them out in anything:
So gentle, because verily so proud;
So wary and afeared of hurting you,
By no means that you are not really vile,
But that they would not touch you with their foot
To push you to your place; so self-possessed
Yet gracious and conciliating, it takes
An effort in their presence to speak truth:
You know the sort of woman,—brilliant stuff,
And out of nature. 'Lady Waldemar,'
She said her name quite simply, as if it meant
Not much indeed, but something,—took my hands,
And smiled, as if her smile could help my case,
And dropped her eyes on me, and let them melt.
'Is this,' she said, 'the Muse?'
'No sybil even,'
I answered, 'since she fails to guess the cause
Which taxed you with this visit, madam.'
'Good,'
She said, 'I like to be sincere at once;
Perhaps, if I had found a literal Muse,
The visit might have taxed me. As it is,
You wear your blue so chiefly in your eyes,
My fair Aurora, in a frank good way,
It comforts me entirely for your fame,
As well as for the trouble of my ascent
To this Olympus.'
There, a silver laugh
Ran rippling through her quickened little breaths
The steep stair somewhat justified.
'But still
Your ladyship has left me curious why
You dared the risk of finding the said Muse?'
'Ah,—keep me, notwithstanding, to the point,
Like any pedant. Is the blue in eyes
As awful as in stockings, after all,
I wonder, that you'd have my business out
Before I breathe—exact the epic plunge
In spite of gasps? Well, naturally you think
I've come here, as the lion-hunters go
To deserts, to secure you, with a trap,
For exhibition in my drawing-rooms
On zoologic soirées? Not in the least.
Roar softly at me; I am frivolous,
I dare say; I have played at lions, too,
Like other women of my class,—but now
I meet my lion simply as Androcles
Met his ... when at his mercy.'
So, she bent
Her head, as queens may mock,—then lifting up
Her eyelids with a real grave queenly look,
Which ruled, and would not spare, not even herself,—
'I think you have a cousin:—Romney Leigh.'
'You bring a word from _him_?'—my eyes leapt up
To the very height of hers,—'a word from _him_?'
'I bring a word about him, actually.
But first,'—she pressed me with her urgent eyes—
'You do not love him,—you?'
'You're frank at least
In putting questions, madam,' I replied.
'I love my cousin cousinly—no more.'
'I guessed as much. I'm ready to be frank
In answering also, if you'll question me,
Or even with something less. You stand outside,
You artist women, of the common sex;
You share not with us, and exceed us so
Perhaps by what you're mulcted in, your hearts
Being starved to make your heads: so run the old
Traditions of you. I can therefore speak,
Without the natural shame which creatures feel
When speaking on their level, to their like.
There's many a <DW7> she, would rather die
Than own to her maid she put a ribbon on
To catch the indifferent eye of such a man,—
Who yet would count adulteries on her beads
At holy Mary's shrine, and never blush;
Because the saints are so far off, we lose
All modesty before them. Thus, today.
'Tis _I_, love Romney Leigh.'
'Forbear,' I cried.
'If here's no Muse, still less is any saint;
Nor even a friend, that Lady Waldemar
Should make confessions'....
'That's unkindly said.
If no friend, what forbids to make a friend
To join to our confession ere we have done?
I love your cousin. If it seems unwise
To say so, it's still foolisher (we're frank)
To feel so. My first husband left me young,
And pretty enough, so please you, and rich enough,
To keep my booth in May-fair with the rest
To happy issues. There are marquises
Would serve seven years to call me wife, I know:
And, after seven, I might consider it,
For there's some comfort in a marquisate
When all's said,—yes, but after the seven years;
I, now, love Romney. You put up your lip,
So like a Leigh! so like him!—Pardon me,
I am well aware I do not derogate
In loving Romney Leigh. The name is good,
The means are excellent; but the man, the man—
Heaven help us both,—I am near as mad as he,
In loving such an one.'
She slowly swung
Her heavy ringlets till they touched her smile,
As reasonably sorry for herself;
And thus continued,—
'Of a truth, Miss Leigh,
I have not, without struggle, come to this.
I took a master in the German tongue,
I gamed a little, went to Paris twice;
But, after all, this love!... you eat of love,
And do as vile a thing as if you ate
Of garlic—which, whatever else you eat,
Tastes uniformly acrid, till your peach
Reminds you of your onion. Am I coarse?
Well, love's coarse, nature's coarse—ah, there's the rub!
We fair fine ladies, who park out our lives
From common sheep-paths, cannot help the crows
From flying over,—we're as natural still
As Blowsalinda. Drape us perfectly
In Lyons' velvet,—we are not, for that,
Lay-figures, look you! we have hearts within,
Warm, live, improvident, indecent hearts,
As ready for distracted ends and acts
As any distressed sempstress of them all
That Romney groans and toils for. We catch love
And other fevers, in the vulgar way.
Love will not be outwitted by our wit,
Nor outrun by our equipages:—mine
Persisted, spite of efforts. All my cards
Turned up but Romney Leigh; my German stopped
At germane Wertherism; my Paris rounds
Returned me from the Champs Elysées just
A ghost, and sighing like Dido's. I came home
Uncured,—convicted rather to myself
Of being in love ... in love! That's coarse you'll say.
I'm talking garlic.'
Coldly I replied.
'Apologise for atheism, not love!
For me, I do believe in love, and God.
I know my cousin: Lady Waldemar
I know not: yet I say as much as this—
Whoever loves him, let her not excuse
But cleanse herself, that, loving such a man,
She may not do it with such unworthy love
He cannot stoop and take it.'
'That is said
Austerely, like a youthful prophetess,
Who knits her brows across her pretty eyes
To keep them back from following the grey flight
Of doves between the temple-columns. Dear,
Be kinder with me. Let us two be friends.
I'm a mere woman,—the more weak perhaps
Through being so proud; you're better; as for him,
He's best. Indeed he builds his goodness up
So high, it topples down to the other side,
And makes a sort of badness; there's the worst
I have to say against your cousin's best!
And so be mild, Aurora, with my worst,
For his sake, if not mine.'
'I own myself
Incredulous of confidence like this
Availing him or you.'
'I, worthy of him?
In your sense I am not so—let it pass.
And yet I save him if I marry him;
Let that pass too.'
'Pass, pass! we play police
Upon my cousin's life, to indicate
What may or may not pass?' I cried. 'He knows
What's worthy of him; the choice remains with _him_;
And what he chooses, act or wife, I think
I shall not call unworthy, I, for one.'
''Tis somewhat rashly said,' she answered slow.
'Now let's talk reason, though we talk of love.
Your cousin Romney Leigh's a monster! there,
The word's out fairly; let me prove the fact.
We'll take, say, that most perfect of antiques,
They call the Genius of the Vatican,
Which seems too beauteous to endure itself
In this mixed world, and fasten it for once
Upon the torso of the Drunken Fawn,
(Who might limp surely, if he did not dance,)
Instead of Buonarroti's mask: what then?
We show the sort of monster Romney is,
With god-like virtues and heroic aims
Subjoined to limping possibilities
Of mismade human nature. Grant the man
Twice god-like, twice heroic,—still he limps,
And here's the point we come to.'
'Pardon me,
But, Lady Waldemar, the point's the thing
We never come to.'
'Caustic, insolent
At need! I like you'—(there, she took my hands)
'And now my lioness, help Androcles,
For all your roaring. Help me! for myself
I would not say so—but for him. He limps
So certainly, he'll fall into the pit
A week hence,—so I lose him—so he is lost!
And when he's fairly married, he a Leigh,
To a girl of doubtful life, undoubtful birth,
Starved out in London, till her coarse-grained hands
Are whiter than her morals,—you, for one,
May call his choice most worthy.'
'Married! lost!
He, ... Romney!'
'Ah, you're moved at last,' she said.
'These monsters, set out in the open sun,
Of course throw monstrous shadows: those who think
Awry, will scarce act straightly. Who but he?
And who but you can wonder? He has been mad,
The whole world knows, since first, a nominal man,
He soured the proctors, tried the gownsmen's wits,
With equal scorn of triangles and wine,
And took no honours, yet was honourable.
They'll tell you he lost count of Homer's ships
In Melbourne's poor-bills, Ashley's factory bills,—
Ignored the Aspasia we all dare to praise,
For other women, dear, we could not name
Because we're decent. Well, he had some right
On his side probably; men always have,
Who go absurdly wrong. The living boor
Who brews your ale, exceeds in vital worth
Dead Cæsar who 'stops bungholes' in the cask;
And also, to do good is excellent,
For persons of his income, even to boors:
I sympathise with all such things. But he
Went mad upon them ... madder and more mad,
From college times to these,—as, going down hill,
The faster still, the farther! you must know
Your Leigh by heart: he has sown his black young curls
With bleaching cares of half a million men
Already. If you do not starve, or sin,
You're nothing to him. Pay the income-tax,
And break your heart upon't ... he'll scarce be touched;
But come upon the parish, qualified
For the parish stocks, and Romney will be there
To call you brother, sister, or perhaps
A tenderer name still. Had I any chance
With Mister Leigh, who am Lady Waldemar,
And never committed felony?'
'You speak
Too bitterly,' I said, 'for the literal truth.'
'The truth is bitter. Here's a man who looks
For ever on the ground! you must be low
Or else a pictured ceiling overhead,
Good painting thrown away. For me, I've done
What women may, (we're somewhat limited,
We modest women) but I've done my best.
—How men are perjured when they swear our eyes
Have meaning in them! they're just blue or brown,—
They just can drop their lids a little. In fact,
Mine did more, for I read half Fourier through,
Proudhon, Considerant, and Louis Blanc,
With various others of his socialists;
And if I had been a fathom less in love,
Had cured myself with gaping. As it was,
I quoted from them prettily enough,
Perhaps, to make them sound half rational
To a saner man than he, whene'er we talked,
(For which I dodged occasion)—learnt by heart
His speeches in the Commons and elsewhere
Upon the social question; heaped reports
Of wicked women and penitentiaries,
On all my tables, with a place for Sue;
And gave my name to swell subscription-lists
Toward keeping up the sun at nights in heaven,
And other possible ends. All things I did,
Except the impossible ... such as wearing gowns
Provided by the Ten Hours' movement! there,
I stopped—we must stop somewhere. He, meanwhile,
Unmoved as the Indian tortoise 'neath the world,
Let all that noise go on upon his back:
He would not disconcert or throw me out;
'Twas well to see a woman of my class
With such a dawn of conscience. For the heart,
Made firewood for his sake, and flaming up
To his very face ... he warmed his feet at it;
But deigned to let my carriage stop him short
In park or street,—he leaning on the door,
With news of the committee which sate last
On pickpockets at suck.'
'You jest—you jest.'
'As martyrs jest, dear, (if you've read their lives)
Upon the axe which kills them. When all's done
By me, ... for him—you'll ask him presently
The colour of my hair—he cannot tell,
Or answers 'dark' at random,—while, be sure,
He's absolute on the figure, five or ten,
Of my last subscription. Is it bearable,
And I a woman?'
'Is it reparable,
Though _I_ were a man?'
'I know not. That's to prove.
But, first, this shameful marriage.'
'Ay?' I cried,
'Then really there's a marriage?'
'Yesterday
I held him fast upon it. 'Mister Leigh,'
Said I, 'shut up a thing, it makes more noise.
The boiling town keeps secrets ill; I've known
Yours since last week. Forgive my knowledge so:
You feel I'm not the woman of the world
The world thinks; you have borne with me before,
And used me in your noble work, our work,
And now you shall not cast me off because
You're at the difficult point, the _join_. 'Tis true
Even I can scarce admit the cogency
Of such a marriage ... where you do not love,
(Except the class) yet marry and throw your name
Down to the gutter, for a fire-escape
To future generations! it's sublime,
A great example,—a true Genesis
Of the opening social era. But take heed;
This virtuous act must have a patent weight,
Or loses half its virtue. Make it tell,
Interpret it, and set in the light,
And do not muffle it in a winter-cloak
As a vulgar bit of shame,—as if, at best,
A Leigh had made a misalliance and blushed
A Howard should know it.' Then, I pressed him more—
'He would not choose,' I said, 'that even his kin, ...
Aurora Leigh, even ... should conceive his act
Less sacrifice, more appetite.' At which
He grew so pale, dear, ... to the lips, I knew
I had touched him. 'Do you know her,' he enquired,
'My cousin Aurora?' 'Yes,' I said, and lied,
(But truly we all know you by your books)
And so I offered to come straight to you,
Explain the subject, justify the cause,
And take you with me to St. Margaret's Court
To see this miracle, this Marian Erle,
This drover's daughter (she's not pretty, he swears)
Upon whose finger, exquisitely pricked
By a hundred needles, we're to hang the tie
'Twixt class and class in England,—thus, indeed,
By such a presence, yours and mine, to lift
The match up from the doubtful place. At once
He thanked me, sighing ... murmured to himself,
'She'll do it perhaps; she's noble,'—thanked me twice,
And promised, as my guerdon, to put off
His marriage for a month.'
I answered then.
'I understand your drift imperfectly.
You wish to lead me to my cousin's betrothed,
To touch her hand if worthy, and hold her hand
If feeble, thus to justify his match.
So be it then. But how this serves your ends,
And how the strange confession of your love
Serves this, I have to learn—I cannot see.'
She knit her restless forehead. 'Then, despite,
Aurora, that most radiant morning name,
You're dull as any London afternoon.
I wanted time,—and gained it,—wanted _you_,
And gain you! You will come and see the girl,
In whose most prodigal eyes, the lineal pearl
And pride of all your lofty race of Leighs
Is destined to solution. Authorised
By sight and knowledge, then, you'll speak your mind,
And prove to Romney, in your brilliant way,
He'll wrong the people and posterity
(Say such a thing is bad for you and me,
And you fail utterly,) by concluding thus
An execrable marriage. Break it up,
Disroot it—peradventure, presently,
We'll plant a better fortune in its place.
Be good to me, Aurora, scorn me less
For saying the thing I should not. Well I know
I should not. I have kept, as others have,
The iron rule of womanly reserve
In lip and life, till now: I wept a week
Before I came here.'—Ending, she was pale;
The last words, haughtily said, were tremulous.
This palfrey pranced in harness, arched her neck,
And, only by the foam upon the bit,
You saw she champed against it.
Then I rose.
'I love love! truth's no cleaner thing than love.
I comprehend a love so fiery hot
It burns its natural veil of august shame,
And stands sublimely in the nude, as chaste
As Medicean Venus. But I know,
A love that burns through veils, will burn through masks,
And shrivel up treachery. What, love and lie!
Nay—go to the opera! your love's curable.'
'I love and lie?' she said—'I lie, forsooth?'
And beat her taper foot upon the floor,
And smiled against the shoe,—'You're hard, Miss Leigh,
Unversed in current phrases.—Bowling-greens
Of poets are fresher than the world's highways;
Forgive me that I rashly blew the dust
Which dims our hedges even, in your eyes,
And vexed you so much. You find, probably,
No evil in this marriage,—rather good
Of innocence, to pastoralise in song:
You'll give the bond your signature, perhaps,
Beneath the lady's mark,—indifferent
That Romney chose a wife, could write her name,
In witnessing he loved her.'
'Loved!' I cried;
'Who tells you that he wants a wife to love?
He gets a horse to use, not love, I think:
There's work for wives as well,—and after, straw,
When men are liberal. For myself, you err
Supposing power in me to break this match.
I could not do it, to save Romney's life;
And would not, to save mine.'
'You take it so,'
She said; 'farewell then. Write your books in peace,
As far as may be for some secret stir
Now obvious to me,—for, most obviously,
In coming hither I mistook the way.'
Whereat she touched my hand, and bent her head,
And floated from me like a silent cloud
That leaves the sense of thunder.
I drew breath
As hard as in a sick room. After all
This woman breaks her social system up
For love, so counted—the love possible
To such,—and lilies are still lilies, pulled
By smutty hands, though spotted from their white;
And thus she is better, haply, of her kind,
Than Romney Leigh, who lives by diagrams,
And crosses out the spontaneities
Of all his individual, personal life,
With formal universals. As if man
Were set upon a high stool at a desk,
To keep God's books for Him, in red and black,
And feel by millions! What, if even God
Were chiefly God by living out Himself
To an individualism of the Infinite,
Eterne, intense, profuse,—still throwing up
The golden spray of multitudinous worlds
In measure to the proclive weight and rush
Of His inner nature,—the spontaneous love
Still proof and outflow of spontaneous life?
Then live, Aurora!
Two hours afterward,
Within St. Margaret's Court I stood alone,
Close-veiled. A sick child, from an ague-fit,
Whose wasted right hand gambled 'gainst his left
With an old brass button, in a blot of sun,
Jeered weakly at me as I passed across
The uneven pavement; while a woman, rouged
Upon the angular cheek-bones, kerchief torn,
Thin dangling locks, and flat lascivious mouth,
Cursed at a window, both ways, in and out,
By turns some bed-rid creature and myself,—
'Lie still there, mother! liker the dead dog
You'll be to-morrow. What, we pick our way,
Fine madam, with those damnable small feet!
We cover up our face from doing good,
As if it were our purse! What brings you here,
My lady? is't to find my gentleman
Who visits his tame pigeon in the eaves?
Our cholera catch you with its cramps and spasms,
And tumble up your good clothes, veil and all,
And turn your whiteness dead-blue.' I looked up;
I think I could have walked through hell that day,
And never flinched. 'The dear Christ comfort you,'
I said, 'you must have been most miserable
To be so cruel,'—and I emptied out
My purse upon the stones: when, as I had cast
The last charm in the cauldron, the whole court
Went boiling, bubbling up, from all its doors
And windows, with a hideous wail of laughs
And roar of oaths, and blows perhaps ... I passed
Too quickly for distinguishing ... and pushed
A little side-door hanging on a hinge,
And plunged into the dark, and groped and climbed
The long, steep, narrow stair 'twixt broken rail
And mildewed wall that let the plaster drop
To startle me in the blackness. Still, up, up!
So high lived Romney's bride. I paused at last
Before a low door in the roof, and knocked;
There came an answer like a hurried dove—
'So soon? can that be Mister Leigh? so soon?'
And as I entered, an ineffable face
Met mine upon the threshold. 'Oh, not you,
Not you!' ... the dropping of the voice implied,
'Then, if not you, for me not any one.'
I looked her in the eyes, and held her hands,
And said, 'I am his cousin,—Romney Leigh's;
And here I'm come to see my cousin too.'
She touched me with her face and with her voice,
This daughter of the people. Such soft flowers,
From such rough roots? the people, under there,
Can sin so, curse so, look so, smell so ... faugh!
Yet have such daughters?
No wise beautiful
Was Marian Erle. She was not white nor brown,
But could look either, like a mist that changed
According to being shone on more or less.
The hair, too, ran its opulence of curls
In doubt 'twixt dark and bright, nor left you clear
To name the colour. Too much hair perhaps
(I'll name a fault here) for so small a head,
Which seemed to droop on that side and on this,
As a full-blown rose uneasy with its weight,
Though not a breath should trouble it. Again,
The dimple in the cheek had better gone
With redder, fuller rounds: and somewhat large
The mouth was, though the milky little teeth
Dissolved it to so infantine a smile!
For soon it smiled at me; the eyes smiled too,
But 'twas as if remembering they had wept,
And knowing they should, some day, weep again.
We talked. She told me all her story out,
Which I'll re-tell with fuller utterance,
As and confirmed in aftertimes
By others, and herself too. Marian Erle
Was born upon the ledge of Malvern Hill
To eastward, in a hut, built up at night
To evade the landlord's eye, of mud and turf,
Still liable, if once he looked that way,
To being straight levelled, scattered by his foot,
Like any other anthill. Born, I say;
God sent her to his world, commissioned right,
Her human testimonials fully signed,
Not scant in soul—complete in lineaments;
But others had to swindle her a place
To wail in when she had come. No place for her,
By man's law! born an outlaw, was this babe.
Her first cry in our strange and strangling air,
When cast in spasms out by the shuddering womb,
Was wrong against the social code,—forced wrong.
What business had the baby to cry there?
I tell her story and grow passionate.
She, Marian, did not tell it so, but used
Meek words that made no wonder of herself
For being so sad a creature. 'Mister Leigh
Considered truly that such things should change.
They _will_, in heaven—but meantime, on the earth,
There's none can like a nettle as a pink,
Except himself. We're nettles, some of us,
And give offence by the act of springing up;
And, if we leave the damp side of the wall,
The hoes, of course, are on us.' So she said.
Her father earned his life by random jobs
Despised by steadier workmen—keeping swine
On commons, picking hops, or hurrying on
The harvest at wet seasons,—or, at need,
Assisting the Welsh drovers, when a drove
Of startled horses plunged into the mist
Below the mountain-road, and sowed the wind
With wandering neighings. In between the gaps
Of such irregular work, he drank and slept,
And cursed his wife because, the pence being out,
She could not buy more drink. At which she turned,
(The worm) and beat her baby in revenge
For her own broken heart. There's not a crime
But takes its proper change out still in crime,
If once rung on the counter of this world;
Let sinners look to it.
Yet the outcast child,
For whom the very mother's face forewent
The mother's special patience, lived and grew;
Learnt early to cry low, and walk alone,
With that pathetic vacillating roll
Of the infant body on the uncertain feet,
(The earth being felt unstable ground so soon)
At which most women's arms unclose at once
With irrepressive instinct. Thus, at three,
This poor weaned kid would run off from the fold,
This babe would steal off from the mother's chair,
And, creeping through the golden walls of gorse,
Would find some keyhole toward the secresy
Of Heaven's high blue, and, nestling down, peer out—
Oh, not to catch the angels at their games,
She had never heard of angels,—but to gaze
She knew not why, to see she knew not what,
A-hungering outward from the barren earth
For something like a joy. She liked, she said,
To dazzle black her sight against the sky,
For then, it seemed, some grand blind Love came down,
And groped her out, and clasped her with a kiss;
She learnt God that way, and was beat for it
Whenever she went home,—yet came again,
As surely as the trapped hare, getting free,
Returns to his form. This grand blind Love, she said,
This skyey father and mother both in one,
Instructed her and civilised her more
Than even the Sunday-school did afterward,
To which a lady sent her to learn books
And sit upon a long bench in a row
With other children. Well, she laughed sometimes
To see them laugh and laugh, and moil their texts;
But ofter she was sorrowful with noise,
And wondered if their mothers beat them hard,
That ever they should laugh so. There was one
She loved indeed,—Rose Bell, a seven years' child,
So pretty and clever, who read syllables
When Marian was at letters; _she_ would laugh
At nothing—hold your finger up, she laughed,
Then shook her curls down on her eyes and mouth
To hide her make-mirth from the schoolmaster.
And Rose's pelting glee, as frank as rain
On cherry-blossoms, brightened Marian too,
To see another merry whom she loved.
She whispered once (the children side by side,
With mutual arms entwined about their necks)
'Your mother lets you laugh so?' 'Ay,' said Rose,
'She lets me. She was dug into the ground
Six years since, I being but a yearling wean.
Such mothers let us play and lose our time,
And never scold nor beat us! don't you wish
You had one like that?' There, Marian breaking off
Looked suddenly in my face. 'Poor Rose,' said she,
'I heard her laugh last night in Oxford Street.
I'd pour out half my blood to stop that laugh,—
Poor Rose, poor Rose!' said Marian.
She resumed.
It tried her, when she had learnt at Sunday-school
What God was, what he wanted from us all,
And how, in choosing sin, we vexed the Christ,
To go straight home and hear her father pull
The Name down on us from the thunder-shelf,
Then drink away his soul into the dark
From seeing judgment. Father, mother, home,
Were God and heaven reversed to her: the more
She knew of Right, the more she guessed their wrong;
Her price paid down for knowledge, was to know
The vileness of her kindred: through her heart,
Her filial and tormented heart, henceforth,
They struck their blows at virtue. Oh, 'tis hard
To learn you have a father up in heaven
By a gathering certain sense of being, on earth,
Still worse than orphaned: 'tis too heavy a grief,
The having to thank God for such a joy!
And so passed Marian's life from year to year.
Her parents took her with them when they tramped,
Dodged lanes and heaths, frequented towns and fairs,
And once went farther and saw Manchester,
And once the sea, that blue end of the world,
That fair scroll-finis of a wicked book,—
And twice a prison,—back at intervals,
Returning to the hills. Hills draw like heaven,
And stronger sometimes, holding out their hands
To pull you from the vile flats up to them;
And though, perhaps, these strollers still strolled back,
As sheep do, simply that they knew the way,
They certainly felt bettered unawares
Emerging from the social smut of towns
To wipe their feet clean on the mountain-turf.
In which long wanderings, Marian lived and learned,
Endured and learned. The people on the roads
Would stop and ask her how her eyes outgrew
Her cheeks, and if she meant to lodge the birds
In all that hair; and then they lifted her,
The miller in his cart, a mile or twain,
The butcher's boy on horseback. Often, too,
The pedlar stopped, and tapped her on the head
With absolute forefinger, brown and ringed,
And asked if peradventure she could read;
And when she answered 'ay,' would toss her down
Some stray odd volume from his heavy pack,
A Thomson's Seasons, mulcted of the Spring,
Or half a play of Shakspeare's, torn across:
(She had to guess the bottom of a page
By just the top sometimes,—as difficult,
As, sitting on the moon, to guess the earth!)
Or else a sheaf of leaves (for that small Ruth's
Small gleanings) torn out from the heart of books,
From Churchyard Elegies and Edens Lost,
From Burns, and Bunyan, Selkirk, and Tom Jones.
'Twas somewhat hard to keep the things distinct,
And oft the jangling influence jarred the child
Like looking at a sunset full of grace
Through a pothouse window while the drunken oaths
Went on behind her; but she weeded out
Her book-leaves, threw away the leaves that hurt,
(First tore them small, that none should find a word)
And made a nosegay of the sweet and good
To fold within her breast, and pore upon
At broken moments of the noontide glare,
When leave was given her to untie her cloak
And rest upon the dusty roadside bank
From the highway's dust. Or oft, the journey done,
Some city friend would lead her by the hand
To hear a lecture at an institute:
And thus she had grown, this Marian Erle of ours,
To no book-learning,—she was ignorant
Of authors,—not in earshot of the things
Out-spoken o'er the heads of common men,
By men who are uncommon,—but within
The cadenced hum of such, and capable
Of catching from the fringes of the wind
Some fragmentary phrases, here and there,
Of that fine music,—which, being carried in
To her soul, had reproduced itself afresh
In finer motions of the lips and lids.
She said, in speaking of it, 'If a flower
Were thrown you out of heaven at intervals,
You'd soon attain to a trick of looking up,—
And so with her.' She counted me her years,
Till _I_ felt old; and then she counted me
Her sorrowful pleasures, till I felt ashamed.
She told me she was almost glad and calm
On such and such a season; sate and sewed,
With no one to break up her crystal thoughts;
While rhymes from lovely poems span around
Their ringing circles of ecstatic tune,
Beneath the moistened finger of the Hour.
Her parents called her a strange, sickly child,
Not good for much, and given to sulk and stare,
And smile into the hedges and the clouds,
And tremble if one shook her from her fit
By any blow, or word even. Out-door jobs
Went ill with her; and household quiet work,
She was not born to. Had they kept the north,
They might have had their pennyworth out of her,
Like other parents, in the factories;
(Your children work for you, not you for them,
Or else they better had been choked with air
The first breath drawn;) but, in this tramping life,
Was nothing to be done with such a child,
But tramp and tramp. And yet she knitted hose
Not ill, and was not dull at needlework;
And all the country people gave her pence
For darning stockings past their natural age,
And patching petticoats from old to new,
And other light work done for thrifty wives.
One day, said Marian,—the sun shone that day—
Her mother had been badly beat, and felt
The bruises sore about her wretched soul,
(That must have been): she came in suddenly,
And snatching, in a sort of breathless rage,
Her daughter's headgear comb, let down the hair
Upon her, like a sudden waterfall,
And drew her drenched and passive, by the arm,
Outside the hut they lived in. When the child
Could clear her blinded face from all that stream
Of tresses ... there, a man stood, with beast's eyes,
That seemed as they would swallow her alive,
Complete in body and spirit, hair and all,—
With burning stertorous breath that hurt her cheek,
He breathed so near. The mother held her tight,
Saying hard between her teeth—'Why wench, why wench,
The squire speaks to you now—the squire's too good;
He means to set you up, and comfort us.
Be mannerly at least.' The child turned round,
And looked up piteous in the mother's face,
(Be sure that mother's death-bed will not want
Another devil to damn, than such a look) ...
'Oh, mother!' then, with desperate glance to heaven,
'God, free me from my mother,' she shrieked out,
'These mothers are too dreadful.' And, with force
As passionate as fear, she tore her hands
Like lilies from the rocks, from hers and his,
And sprang down, bounded headlong down the steep,
Away from both—away, if possible,
As far as God,—away! They yelled at her,
As famished hounds at a hare. She heard them yell,
She felt her name hiss after her from the hills,
Like shot from guns. On, on. And now she had cast
The voices off with the uplands. On. Mad fear
Was running in her feet and killing the ground;
The white roads curled as if she burnt them up,
The green fields melted, wayside trees fell back
To make room for her. Then, her head grew vexed,
Trees, fields, turned on her, and ran after her;
She heard the quick pants of the hills behind,
Their keen air pricked her neck. She had lost her feet,
Could run no more, yet, somehow, went as fast,—
The horizon, red 'twixt steeples in the east,
So sucked her forward, forward, while her heart
Kept swelling, swelling, till it swelled so big
It seemed to fill her body; then it burst,
And overflowed the world and swamped the light,
'And now I am dead and safe,' thought Marian Erle—
She had dropped, she had fainted.
When the sense returned,
The night had passed—not life's night. She was 'ware
Of heavy tumbling motions, creaking wheels,
The driver shouting to the lazy team
That swung their rankling bells against her brain;
While, through the waggon's coverture and chinks,
The cruel yellow morning pecked at her
Alive or dead, upon the straw inside,—
At which her soul ached back into the dark
And prayed, 'no more of that.' A waggoner
Had found her in a ditch beneath the moon,
As white as moonshine, save for the oozing blood.
At first he thought her dead; but when he had wiped
The mouth and heard it sigh, he raised her up,
And laid her in his waggon in the straw,
And so conveyed her to the distant town
To which his business called himself, and left
That heap of misery at the hospital.
She stirred;—the place seemed new and strange as death.
The white strait bed, with others strait and white,
Like graves dug side by side, at measured lengths,
And quiet people walking in and out
With wonderful low voices and soft steps,
And apparitional equal care for each,
Astonished her with order, silence, law:
And when a gentle hand held out a cup,
She took it, as you do at sacrament,
Half awed, half melted,—not being used, indeed,
To so much love as makes the form of love
And courtesy of manners. Delicate drinks
And rare white bread, to which some dying eyes
Were turned in observation. O my God,
How sick we must be, ere we make men just!
I think it frets the saints in heaven to see
How many desolate creatures on the earth
Have learnt the simple dues of fellowship
And social comfort, in a hospital,
As Marian did. She lay there, stunned, half tranced,
And wished, at intervals of growing sense,
She might be sicker yet, if sickness made
The world so marvellous kind, the air so hushed,
And all her wake-time quiet as a sleep;
For now she understood, (as such things were)
How sickness ended very oft in heaven,
Among the unspoken raptures. Yet more sick,
And surelier happy. Then she dropped her lids,
And, folding up her hands as flowers at night,
Would lose no moment of the blessed time.
She lay and seethed in fever many weeks,
But youth was strong and overcame the test;
Revolted soul and flesh were reconciled
And fetched back to the necessary day
And daylight duties. She could creep about
The long bare rooms, and stare out drearily
From any narrow window on the street,
Till some one, who had nursed her as a friend,
Said coldly to her, as an enemy,
'She had leave to go next week, being well enough,'
While only her heart ached. 'Go next week,' thought she,
'Next week! how would it be with her next week,
Let out into that terrible street alone
Among the pushing people, ... to go ... where?'
One day, the last before the dreaded last,
Among the convalescents, like herself
Prepared to go next morning, she sate dumb,
And heard half absently the women talk,
How one was famished for her baby's cheeks—
'The little wretch would know her! a year old,
And lively, like his father!' one was keen
To get to work, and fill some clamorous mouths;
And one was tender for her dear goodman
Who had missed her sorely,—and one, querulous ...
'Would pay those scandalous neighbours who had dared
To talk about her as already dead,'—
And one was proud ... 'and if her sweetheart Luke
Had left her for a ruddier face than hers,
(The gossip would be seen through at a glance)
Sweet riddance of such sweethearts—let him hang!
'Twere good to have been as sick for such an end.'
And while they talked, and Marian felt the worse
For having missed the worst of all their wrongs,
A visitor was ushered through the wards
And paused among the talkers. 'When he looked,
It was as if he spoke, and when he spoke
He sang perhaps,' said Marian; 'could she tell?
She only knew' (so much she had chronicled,
As seraphs might, the making of the sun)
'That he who came and spake, was Romney Leigh,
And then, and there, she saw and heard him first.'
And when it was her turn to have the face
Upon her,—all those buzzing pallid lips
Being satisfied with comfort—when he changed
To Marian, saying 'And _you_? you're going, where?'—
She, moveless as a worm beneath a stone
Which some one's stumbling foot has spurned aside,
Writhed suddenly, astonished with the light,
And breaking into sobs cried, 'Where I go?
None asked me till this moment. Can I say
Where _I_ go? when it has not seemed worth while
To God himself, who thinks of every one,
To think of me, and fix where I shall go?'
'So young,' he gently asked her, 'you have lost
Your father and your mother?'
'Both,' she said,
'Both lost! my father was burnt up with gin
Or ever I sucked milk, and so is lost.
My mother sold me to a man last month,
And so my mother's lost, 'tis manifest.
And I, who fled from her for miles and miles,
As if I had caught sight of the fires of hell
Through some wild gap, (she was my mother, sir)
It seems I shall be lost too, presently,
And so we end, all three of us.'
'Poor child!'
He said,—with such a pity in his voice,
It soothed her more than her own tears,—'poor child!
'Tis simple that betrayal by mother's love
Should bring despair of God's too. Yet be taught;
He's better to us than many mothers are,
And children cannot wander beyond reach
Of the sweep of his white raiment. Touch and hold!
And if you weep still, weep where John was laid
While Jesus loved him.'
'She could say the words,'
She told me, 'exactly as he uttered them
A year back, ... since, in any doubt or dark,
They came out like the stars, and shone on her
With just their comfort. Common words, perhaps;
The ministers in church might say the same;
But _he_, he made the church with what he spoke,—
The difference was the miracle,' said she.
Then catching up her smile to ravishment,
She added quickly, 'I repeat his words,
But not his tones: can any one repeat
The music of an organ, out of church?
And when he said 'poor child,' I shut my eyes
To feel how tenderly his voice broke through,
As the ointment-box broke on the Holy feet
To let out the rich medicative nard.'
She told me how he had raised and rescued her
With reverent pity, as, in touching grief,
He touched the wounds of Christ,—and made her feel
More self-respecting. Hope, he called, belief
In God,—work, worship ... therefore let us pray!
And thus, to snatch her soul from atheism,
And keep it stainless from her mother's face,
He sent her to a famous sempstress-house
Far off in London, there to work and hope.
With that, they parted. She kept sight of Heaven,
But not of Romney. He had good to do
To others: through the days and through the nights,
She sewed and sewed and sewed. She drooped sometimes,
And wondered, while, along the tawny light,
She struck the new thread into her needle's eye,
How people, without mothers on the hills,
Could choose the town to live in!—then she drew
The stitch, and mused how Romney's face would look,
And if 'twere likely he'd remember hers,
When they two had their meeting after death.
FOURTH BOOK.
THEY met still sooner. 'Twas a year from thence
When Lucy Gresham, the sick sempstress girl,
Who sewed by Marian's chair so still and quick,
And leant her head upon the back to cough
More freely when, the mistress turning round,
The others took occasion to laugh out,—
Gave up at last. Among the workers, spoke
A bold girl with black eyebrows and red lips,—
'You know the news? Who's dying, do you think?
Our Lucy Gresham. I expected it
As little as Nell Hart's wedding. Blush not, Nell,
Thy curls be red enough without thy cheeks;
And, some day, there'll be found a man to dote
On red curls.—Lucy Gresham swooned last night,
Dropped sudden in the street while going home;
And now the baker says, who took her up
And laid her by her grandmother in bed,
He'll give her a week to die in. Pass the silk.
Let's hope he gave her a loaf too, within reach,
For otherwise they'll starve before they die,
That funny pair of bedfellows! Miss Bell,
I'll thank you for the scissors. The old crone
Is paralytic—that's the reason why
Our Lucy's thread went faster than her breath,
Which went too quick, we all know. Marian Erle!
Why, Marian Erle, you're not the fool to cry?
Your tears spoil Lady Waldemar's new dress,
You piece of pity!'
Marian rose up straight,
And, breaking through the talk and through the work,
Went outward, in the face of their surprise,
To Lucy's home, to nurse her back to life
Or down to death. She knew, by such an act,
All place and grace were forfeit in the house,
Whose mistress would supply the missing hand
With necessary, not inhuman haste,
And take no blame. But pity, too, had dues:
She could not leave a solitary soul
To founder in the dark, while she sate still
And lavished stitches on a lady's hem
As if no other work were paramount.
'Why, God,' thought Marian, 'has a missing hand
This moment; Lucy wants a drink, perhaps.
Let others miss me! never miss me, God!'
So Marian sate by Lucy's bed, content
With duty, and was strong, for recompense,
To hold the lamp of human love arm-high
To catch the death-strained eyes and comfort them,
Until the angels, on the luminous side
Of death, had got theirs ready. And she said,
When Lucy thanked her sometimes, called her kind,
It touched her strangely. 'Marian Erle, called kind!
What, Marian, beaten and sold, who could not die!
'Tis verily good fortune to be kind.
Ah, you,' she said, 'who are born to such a grace,
Be sorry for the unlicensed class, the poor,
Reduced to think the best good fortune means
That others, simply, should be kind to them.'
From sleep to sleep while Lucy slid away
So gently, like the light upon a hill,
Of which none names the moment that it goes,
Though all see when 'tis gone,—a man came in
And stood beside the bed. The old idiot wretch
Screamed feebly, like a baby overlain,
'Sir, sir, you won't mistake me for the corpse?
Don't look at _me_, sir! never bury _me_!
Although I lie here, I'm alive as you,
Except my legs and arms,—I eat and drink,
And understand,—(that you're the gentleman
Who fits the funerals up, Heaven speed you, sir,)
And certainly I should be livelier still
If Lucy here ... sir, Lucy is the corpse ...
Had worked more properly to buy me wine:
But Lucy, sir, was always slow at work,
I shan't lose much by Lucy. Marian Erle,
Speak up and show the gentleman the corpse.'
And then a voice said, 'Marian Erle.' She rose;
It was the hour for angels—there, stood hers!
She scarcely marvelled to see Romney Leigh.
As light November snows to empty nests,
As grass to graves, as moss to mildewed stones,
As July suns to ruins, through the rents,
As ministering spirits to mourners, through a loss,
As Heaven itself to men, through pangs of death,
He came uncalled wherever grief had come.
'And so,' said Marian Erle, 'we met anew,'
And added softly, 'so, we shall not part.'
He was not angry that she had left the house
Wherein he placed her. Well—she had feared it might
Have vexed him. Also, when he found her set
On keeping, though the dead was out of sight,
That half-dead, half-live body left behind
With cankerous heart and flesh,—which took your best
And cursed you for the little good it did,
(Could any leave the bedrid wretch alone,
So joyless, she was thankless even to God,
Much less to you?) he did not say 'twas well,
Yet Marian thought he did not take it ill,—
Since day by day he came, and, every day,
She felt within his utterance and his eyes
A closer, tenderer presence of the soul,
Until at last he said, 'We shall not part.'
On that same day, was Marian's work complete:
She had smoothed the empty bed, and swept the floor
Of coffin sawdust, set the chairs anew
The dead had ended gossip in, and stood
In that poor room so cold and orderly,
The door-key in her hand, prepared to go
As _they_ had, howbeit not their way. He spoke.
'Dear Marian, of one clay God made us all,
And though men push and poke and paddle in't
(As children play at fashioning dirt-pies)
And call their fancies by the name of facts,
Assuming difference, lordship, privilege,
When all's plain dirt,—they come back to it at last;
The first grave-digger proves it with a spade,
And pats all even. Need we wait for this,
You, Marian, and I, Romney?'
She, at that,
Looked blindly in his face, as when one looks
Through driving autumn-rains to find the sky.
He went on speaking.
'Marian, I being born
What men call noble, and you, issued from
The noble people,—though the tyrannous sword
Which pierced Christ's heart, has cleft the world in twain
'Twixt class and class, opposing rich to poor,—
Shall _we_ keep parted? Not so. Let us lean
And strain together rather, each to each,
Compress the red lips of this gaping wound,
As far as two souls can,—ay, lean and league,
I, from my superabundance,—from your want,
You,—joining in a protest 'gainst the wrong
On both sides!'—
All the rest, he held her hand
In speaking, which confused the sense of much;
Her heart, against his words, beat out so thick,
They might as well be written on the dust
Where some poor bird, escaping from hawk's beak,
Has dropped, and beats its shuddering wings,—the lines
Are rubbed so,—yet 'twas something like to this,
—'That they two, standing at the two extremes
Of social classes, had received one seal,
Been dedicate and drawn beyond themselves
To mercy and ministration,—he, indeed,
Through what he knew, and she, through what she felt,
He, by man's conscience, she, by woman's heart,
Relinquishing their several 'vantage posts
Of wealthy ease and honourable toil,
To work with God at love. And, since God willed
That, putting out his hand to touch this ark,
He found a woman's hand there, he'd accept
The sign too, hold the tender fingers fast,
And say, 'My fellow-worker, be my wife!''
She told the tale with simple, rustic turns,—
Strong leaps of meaning in her sudden eyes
That took the gaps of any imperfect phrase
Of the unschooled speaker: I have rather writ
The thing I understood so, than the thing
I heard so. And I cannot render right
Her quick gesticulation, wild yet soft,
Self-startled from the habitual mood she used,
Half sad, half languid,—like dumb creatures (now
A rustling bird, and now a wandering deer,
Or squirrel against the oak-gloom flashing up
His sidelong burnished head, in just her way
Of savage spontaneity,) that stir
Abruptly the green silence of the woods,
And make it stranger, holier, more profound;
As Nature's general heart confessed itself
Of life, and then fell backward on repose.
I kissed the lips that ended.—'So indeed
He loves you, Marian?'
'Loves me!' She looked up
With a child's wonder when you ask him first
Who made the sun—a puzzled blush, that grew,
Then broke off in a rapid radiant smile
Of sure solution. 'Loves me! he loves all,—
And me, of course. He had not asked me else
To work with him for ever, and be his wife.'
Her words reproved me. This perhaps was love—
To have its hands too full of gifts to give,
For putting out a hand to take a gift;
To love so much, the perfect round of love
Includes, in strict conclusion, the being loved;
As Eden-dew went up and fell again,
Enough for watering Eden. Obviously
She had not thought about his love at all:
The cataracts of her soul had poured themselves,
And risen self-crowned in rainbow: would she ask
Who crowned her?—it sufficed that she was crowned.
With women of my class, 'tis otherwise:
We haggle for the small change of our gold,
And so much love, accord, for so much love,
Rialto-prices. Are we therefore wrong?
If marriage be a contract, look to it then,
Contracting parties should be equal, just;
But if, a simple fealty on one side,
A mere religion,—right to give, is all,
And certain brides of Europe duly ask
To mount the pile, as Indian widows do,
The spices of their tender youth heaped up,
The jewels of their gracious virtues worn,
More gems, more glory,—to consume entire
For a living husband! as the man's alive,
Not dead,—the woman's duty, by so much,
Advanced in England, beyond Hindostan.
I sate there, musing, till she touched my hand
With hers, as softly as a strange white bird
She feared to startle in touching. 'You are kind.
But are you, peradventure, vexed at heart
Because your cousin takes me for a wife?
I know I am not worthy—nay, in truth,
I'm glad on't, since, for that, he chooses me.
He likes the poor things of the world the best;
I would not therefore, if I could, be rich.
It pleasures him to stoop for buttercups;
I would not be a rose upon the wall
A queen might stop at, near the palace-door,
To say to a courtier, 'Pluck that rose for me,
'It's prettier than the rest,' O Romney Leigh!
I'd rather far be trodden by his foot,
Than lie in a great queen's bosom.'
Out of breath
She paused.
'Sweet Marian, do you disavow
The roses with that face?'
She dropt her head,
As if the wind had caught that flower of her,
And bent it in the garden,—then looked up
With grave assurance. 'Well, you think me bold!
But so we all are, when we're praying God.
And if I'm bold—yet, lady, credit me,
That, since I know myself for what I am,
Much fitter for his handmaid than his wife,
I'll prove the handmaid and the wife at once,
Serve tenderly, and love obediently,
And be a worthier mate, perhaps, than some
Who are wooed in silk among their learned books;
While _I_ shall set myself to read his eyes,
Till such grow plainer to me than the French
To wisest ladies. Do you think I'll miss
A letter, in the spelling of his mind?
No more than they do, when they sit and write
Their flying words with flickering wild-fowl tails,
Nor ever pause to ask how many _t_s,
Should that be _y_ or _i_—they know't so well:
I've seen them writing, when I brought a dress
And waited,—floating out their soft white hands
On shining paper. But they're hard sometimes,
For all those hands!—we've used out many nights,
And worn the yellow daylight into shreds
Which flapped and shivered down our aching eyes
Till night appeared more tolerable, just
That pretty ladies might look beautiful,
Who said at last ... 'You're lazy in that house!
'You're slow in sending home the work,—I count
I've waited near an hour for't.' Pardon me,—
I do not blame them, madam, nor misprize;
They are fair and gracious; ay, but not like you,
Since none but you has Mister Leigh's own blood
Both noble and gentle,—and, without it ... well,
They are fair, I said; so fair, it scarce seems strange
That, flashing out in any looking-glass
The wonder of their glorious brows and breasts,
They are charmed so, they forget to look behind
And mark how pale we've grown, we pitiful
Remainders of the world. And so, perhaps,
If Mister Leigh had chosen a wife from these,
She might ... although he's better than her best,
And dearly she would know it ... steal a thought
Which should be all his, an eye-glance from his face,
To plunge into the mirror opposite,
In search of her own beauty's pearl: while _I_....
Ah, dearest lady, serge will outweigh silk
For winter-wear, when bodies feel a-cold,
And I'll be a true wife to your cousin Leigh.'
Before I answered, he was there himself.
I think he had been standing in the room,
And listened probably to half her talk,
Arrested, turned to stone,—as white as stone.
Will tender sayings make men look so white?
He loves her then profoundly.
'You are here,
Aurora? Here I meet you!'—We clasped hands.
'Even so, dear Romney. Lady Waldemar
Has sent me in haste to find a cousin of mine
Who shall be.'
'Lady Waldemar is good.'
'Here's one, at least, who is good,' I sighed, and touched
Poor Marian's happy head, as, doglike, she
Most passionately patient, waited on,
A-tremble for her turn of greeting words;
'I've sate a full hour with your Marian Erle,
And learnt the thing by heart,—and, from my heart,
Am therefore competent to give you thanks
For such a cousin.'
'You accept at last
A gift from me, Aurora, without scorn?
At last I please you?'—How his voice was changed!
'You cannot please a woman against her will,
And once you vexed me. Shall we speak of that?
We'll say, then, you were noble in it all,
And I not ignorant—let it pass. And now,
You please me, Romney, when you please yourself;
So, please you, be fanatical in love,
And I'm well pleased. Ah, cousin! at the old hall,
Among the gallery portraits of our Leighs,
We shall not find a sweeter signory
Than this pure forehead's.'
Not a word he said.
How arrogant men are!—Even philanthropists,
Who try to take a wife up in the way
They put down a subscription-cheque,—if once
She turns and says, 'I will not tax you so,
Most charitable sir,'—feel ill at ease,
As though she had wronged them somehow. I suppose
We women should remember what we are,
And not throw back an obolus inscribed
With Cæsar's image, lightly. I resumed.
'It strikes me, some of those sublime Vandykes
Were not too proud, to make good saints in heaven;
And, if so, then they're not too proud to-day
To bow down (now the ruffs are off their necks)
And own this good, true, noble Marian, ... yours,
And mine, I'll say!—For poets (bear the word)
Half-poets even, are still whole democrats,—
Oh, not that we're disloyal to the high,
But loyal to the low, and cognisant
Of the less scrutable majesties. For me,
I comprehend your choice—I justify
Your right in choosing.'
'No, no, no,' he sighed,
With a sort of melancholy impatient scorn,
As some grown man, who never had a child,
Puts by some child who plays at being a man;
—'You did not, do not, cannot comprehend
My choice, my ends, my motives, nor myself:
No matter now—we'll let it pass, you say.
I thank you for your generous cousinship
Which helps this present; I accept for her
Your favourable thoughts. We're fallen on days,
We two, who are not poets, when to wed
Requires less mutual love than common love,
For two together to bear out at once
Upon the loveless many. Work in pairs,
In galley-couplings or in marriage-rings,
The difference lies in the honour, not the work,—
And such we're bound to, I and she. But love,
(You poets are benighted in this age;
The hour's too late for catching even moths,
You've gnats instead,) love!—love's fool-paradise
Is out of date, like Adam's. Set a swan
To swim the Trenton, rather than true love
To float its fabulous plumage safely down
The cataracts of this loud transition-time,—
Whose roar, for ever, henceforth, in my ears,
Must keep me deaf to music.'
There, I turned
And kissed poor Marian, out of discontent.
The man had baffled, chafed me, till I flung
For refuge to the woman,—as, sometimes,
Impatient of some crowded room's close smell,
You throw a window open, and lean out
To breathe a long breath in the dewy night,
And cool your angry forehead. She, at least,
Was not built up, as walls are, brick by brick;
Each fancy squared, each feeling ranged by line,
The very heat of burning youth applied
To indurate forms and systems! excellent bricks,
A well-built wall,—which stops you on the road,
And, into which, you cannot see an inch
Although you beat your head against it—pshaw!
'Adieu,' I said, 'for this time, cousins both;
And, cousin Romney, pardon me the word,
Be happy!—oh, in some esoteric sense
Of course!—I mean no harm in wishing well.
Adieu, my Marian:—may she come to me,
Dear Romney, and be married from my house?
It is not part of your philosophy
To keep your bird upon the blackthorn?'
'Ay,'
He answered, 'but it is:—I take my wife
Directly from the people,—and she comes,
As Austria's daughter to imperial France,
Betwixt her eagles, blinking not her race,
From Margaret's Court at garret-height, to meet
And wed me at St. James's, nor put off
Her gown of serge for that. The things we do,
We do: we'll wear no mask, as if we blushed.'
'Dear Romney, you're the poet,' I replied,—
But felt my smile too mournful for my word,
And turned and went. Ay, masks, I thought,—beware
Of tragic masks, we tie before the glass,
Uplifted on the cothurn half a yard
Above the natural stature! we would play
Heroic parts to ourselves,—and end, perhaps,
As impotently as Athenian wives
Who shrieked in fits at the Eumenides.
His foot pursued me down the stair. 'At least,
You'll suffer me to walk with you beyond
These hideous streets, these graves, where men alive,
Packed close with earthworms, burr unconsciously
About the plague that slew them; let me go.
The very women pelt their souls in mud
At any woman who walks here alone.
How came you here alone?—you are ignorant.'
We had a strange and melancholy walk:
The night came drizzling downward in dark rain;
And, as we walked, the colour of the time,
The act, the presence, my hand upon his arm,
His voice in my ear, and mine to my own sense,
Appeared unnatural. We talked modern books,
And daily papers; Spanish marriage-schemes,
And English climate—was't so cold last year?
And will the wind change by to-morrow morn?
Can Guizot stand? is London full? is trade
Competitive? has Dickens turned his hinge
A-pinch upon the fingers of the great?
And are potatoes to grow mythical
Like moly? will the apple die out too?
Which way is the wind to-night? south-east? due east?
We talked on fast, while every common word
Seemed tangled with the thunder at one end,
And ready to pull down upon our heads
A terror out of sight. And yet to pause
Were surelier mortal: we tore greedily up
All silence, all the innocent breathing-points,
As if, like pale conspirators in haste,
We tore up papers where our signatures
Imperilled us to an ugly shame or death.
I cannot tell you why it was. 'Tis plain
We had not loved nor hated: wherefore dread
To spill gunpowder on ground safe from fire?
Perhaps we had lived too closely, to diverge
So absolutely: leave two clocks, they say,
Wound up to different hours, upon one shelf,
And slowly, through the interior wheels of each,
The blind mechanic motion sets itself
A-throb, to feel out for the mutual time.
It was not so with us, indeed. While he
Struck midnight, I kept striking six at dawn,
While he marked judgment, I, redemption-day;
And such exception to a general law,
Imperious upon inert matter even,
Might make us, each to either, insecure,
A beckoning mystery, or a troubling fear.
I mind me, when we parted at the door,
How strange his good-night sounded,—like good-night
Beside a deathbed, where the morrow's sun
Is sure to come too late for more good-days:—
And all that night I thought.... 'Good-night,' said he.
And so, a month passed. Let me set it down
At once,—I have been wrong, I have been wrong.
We are wrong always, when we think too much
Of what we think or are; albeit our thoughts
Be verily bitter as self-sacrifice,
We're no less selfish. If we sleep on rocks
Or roses, sleeping past the hour of noon
We're lazy. This I write against myself.
I had done a duty in the visit paid
To Marian, and was ready otherwise
To give the witness of my presence and name
Whenever she should marry.—Which, I thought,
Sufficed. I even had cast into the scale
An overweight of justice toward the match;
The Lady Waldemar had missed her tool,
Had broken it in the lock as being too straight
For a crooked purpose, while poor Marian Erle
Missed nothing in my accents or my acts:
I had not been ungenerous on the whole,
Nor yet untender; so, enough. I felt
Tired, overworked: this marriage somewhat jarred;
Or, if it did not, all the bridal noise ...
The pricking of the map of life with pins,
In schemes of ... 'Here we'll go,' and 'There we'll stay,'
And 'Everywhere we'll prosper in our love,'
Was scarce my business. Let them order it;
Who else should care? I threw myself aside,
As one who had done her work and shuts her eyes
To rest the better.
I, who should have known,
Forereckoned mischief! Where we disavow
Being keeper to our brother, we're his Cain.
I might have held that poor child to my heart
A little longer! 'twould have hurt me much
To have hastened by its beats the marriage-day,
And kept her safe meantime from tampering hands,
Or, peradventure, traps? What drew me back
From telling Romney plainly, the designs
Of Lady Waldemar, as spoken out
To me ... me? had I any right, ay, right,
With womanly compassion and reserve
To break the fall of woman's impudence?—
To stand by calmly, knowing what I knew,
And hear him call her _good_?
Distrust that word.
'There is none good save God,' said Jesus Christ.
If He once, in the first creation-week,
Called creatures good,—for ever, afterward,
The Devil only has done it, and his heirs,
The knaves who win so, and the fools who lose;
The word's grown dangerous. In the middle age,
I think they called malignant fays and imps
Good people. A good neighbour, even in this,
Is fatal sometimes,—cuts your morning up
To mince-meat of the very smallest talk,
Then helps to sugar her bohea at night
With your reputation. I have known good wives,
As chaste, or nearly so, as Potiphar's;
And good, good mothers, who would use a child
To better an intrigue; good friends, beside,
(Very good) who hung succinctly round your neck
And sucked your breath, as cats are fabled to do
By sleeping infants. And we all have known
Good critics, who have stamped out poet's hopes;
Good statesmen, who pulled ruin on the state;
Good patriots, who, for a theory, risked a cause;
Good kings, who disembowelled for a tax;
Good popes, who brought all good to jeopardy;
Good Christians, who sate still in easy chairs,
And damned the general world for standing up.—
Now, may the good God pardon all good men!
How bitterly I speak,—how certainly
The innocent white milk in us is turned,
By much persistent shining of the sun!—
Shake up the sweetest in us long enough
With men, it drops to foolish curd, too sour
To feed the most untender of Christ's lambs.
I should have thought ... a woman of the world
Like her I'm meaning,—centre to herself,
Who has wheeled on her own pivot half a life
In isolated self-love and self-will,
As a windmill seen at distance radiating
Its delicate white vans against the sky,
So soft and soundless, simply beautiful,—
Seen nearer ... what a roar and tear it makes,
How it grinds and bruises!... if she loves at last,
Her love's a re-adjustment of self-love,
No more; a need felt of another's use
To her one advantage,—as the mill wants grain,
The fire wants fuel, the very wolf wants prey;
And none of these is more unscrupulous
Than such a charming woman when she loves.
She'll not be thwarted by an obstacle
So trifling as ... her soul is, ... much less yours!—
Is God a consideration?—she loves _you_,
Not God; she will not flinch for Him indeed:
She did not for the Marchioness of Perth,
When wanting tickets for the birthnight-ball.
She loves you, sir, with passion, to lunacy;
She loves you like her diamonds ... almost.
Well,
A month passed so, and then the notice came;
On such a day the marriage at the church.
I was not backward.
Half St. Giles in frieze
Was bidden to meet St. James in cloth of gold,
And, after contract at the altar, pass
To eat a marriage-feast on Hampstead Heath.
Of course the people came in uncompelled,
Lame, blind, and worse—sick, sorrowful, and worse,
The humours of the peccant social wound
All pressed out, poured out upon Pimlico,
Exasperating the unaccustomed air
With hideous interfusion: you'd suppose
A finished generation, dead of plague,
Swept outward from their graves into the sun,
The moil of death upon them. What a sight!
A holiday of miserable men
Is sadder than a burial-day of kings.
They clogged the streets, they oozed into the church
In a dark slow stream, like blood. To see that sight,
The noble ladies stood up in their pews,
Some pale for fear, a few as red for hate,
Some simply curious, some just insolent,
And some in wondering scorn,—'What next? what next?'
These crushed their delicate rose-lips from the smile
That misbecame them in a holy place,
With broidered hems of perfumed handkerchiefs;
Those passed the salts with confidence of eyes
And simultaneous shiver of moiré silk;
While all the aisles, alive and black with heads,
Crawled slowly toward the altar from the street,
As bruised snakes crawl and hiss out of a hole
With shuddering involutions, swaying slow
From right to left, and then from left to right,
In pants and pauses. What an ugly crest
Of faces, rose upon you everywhere,
From that crammed mass! you did not usually
See faces like them in the open day:
They hide in cellars, not to make you mad
As Romney Leigh is.—Faces!—O my God,
We call those, faces? men's and women's ... ay,
And children's;—babies, hanging like a rag
Forgotten on their mother's neck,—poor mouths,
Wiped clean of mother's milk by mother's blow,
Before they are taught her cursing. Faces!... phew,
We'll call them vices festering to despairs,
Or sorrows petrifying to vices: not
A finger-touch of God left whole on them;
All ruined, lost—the countenance worn out
As the garments, the will dissolute as the acts,
The passions loose and draggling in the dirt
To trip the foot up at the first free step!—
Those, faces! 'twas as if you had stirred up hell
To heave its lowest dreg-fiends uppermost
In fiery swirls of slime,—such strangled fronts,
Such obdurate jaws were thrown up constantly,
To twit you with your race, corrupt your blood,
And grind to devilish colours all your dreams
Henceforth, ... though, haply, you should drop asleep
By clink of silver waters, in a muse
On Raffael's mild Madonna of the Bird.
I've waked and slept through many nights and days
Since then,—but still that day will catch my breath
Like a nightmare. There are fatal days, indeed,
In which the fibrous years have taken root
So deeply, that they quiver to their tops
Whene'er you stir the dust of such a day.
My cousin met me with his eyes and hand,
And then, with just a word, ... that 'Marian Erle
Was coming with her bridesmaids presently,'
Made haste to place me by the altar-stair,
Where he and other noble gentlemen
And high-born ladies, waited for the bride.
We waited. It was early: there was time
For greeting, and the morning's compliment;
And gradually a ripple of women's talk
Arose and fell, and tossed about a spray
Of English _s_s, soft as a silent hush,
And, notwithstanding, quite as audible
As louder phrases thrown out by the men.
—'Yes, really, if we've need to wait in church,
We've need to talk there.'—'She? 'Tis Lady Ayr,
In blue—not purple! that's the dowager.'
—'She looks as young.'—'She flirts as young, you mean!
Why if you had seen her upon Thursday night,
You'd call Miss Norris modest.'—'_You_ again!
I waltzed with you three hours back. Up at six,
Up still at ten: scarce time to change one's shoes.
I feel as white and sulky as a ghost,
So pray don't speak to me, Lord Belcher.'—'No,
I'll look at you instead, and it's enough
While you have that face.' 'In church, my lord! fie, fie!'
—'Adair, you stayed for the Division?'—'Lost
By one.' 'The devil it is! I'm sorry for't.
And if I had not promised Mistress Grove' ...
—'You might have kept your word to Liverpool.'
'Constituents must remember, after all,
We're mortal.'—'We remind them of it.'—'Hark,
The bride comes! Here she comes, in a stream of milk!'
—'There? Dear, you are asleep still; don't you know
The five Miss Granvilles? always dressed in white
To show they're ready to be married.'—'Lower!
The aunt is at your elbow.'—'Lady Maud,
Did Lady Waldemar tell you she had seen
This girl of Leigh's?' 'No,—wait! 'twas Mrs. Brookes,
Who told me Lady Waldemar told her—
No, 'twasn't Mrs. Brookes.'—'She's pretty?'—'Who?
Mrs. Brookes? Lady Waldemar?'—'How hot!
Pray is't the law to-day we're not to breathe?
You're treading on my shawl—I thank you, sir.'
—'They say the bride's a mere child, who can't read,
But knows the things she shouldn't, with wide-awake
Great eyes. I'd go through fire to look at her.'
—'You do, I think.'—'And Lady Waldemar
(You see her; sitting close to Romney Leigh;
How beautiful she looks, a little flushed!)
Has taken up the girl, and organised
Leigh's folly. Should I have come here, you suppose,
Except she'd asked me?'—'She'd have served him more
By marrying him herself.'
'Ah—there she comes,
The bride, at last!'
'Indeed, no. Past eleven.
She puts off her patched petticoat to-day
And puts on May-fair manners, so begins
By setting us to wait.'—'Yes, yes, this Leigh
Was always odd; it's in the blood, I think;
His father's uncle's cousin's second son
Was, was ... you understand me—and for him,
He's stark!—has turned quite lunatic upon
This modern question of the poor—the poor:
An excellent subject when you're moderate;
You've seen Prince Albert's model lodging-house?
Does honour to his Royal Highness. Good!
But would he stop his carriage in Cheapside
To shake a common fellow by the fist
Whose name was ... Shakspeare? no. We draw a line,
And if we stand not by our order, we
In England, we fall headlong. Here's a sight,—
A hideous sight, a most indecent sight!
My wife would come, sir, or I had kept her back.
By heaven, sir, when poor Damiens' trunk and limbs
Were torn by horses, women of the court
Stood by and stared, exactly as to-day
On this dismembering of society,
With pretty troubled faces.'
'Now, at last.
She comes now.'
'Where? who sees? you push me, sir,
Beyond the point of what is mannerly.
You're standing, madam, on my second flounce—
I do beseech you.'
'No—it's not the bride.
Half-past eleven. How late. The bridegroom, mark,
Gets anxious and goes out.'
'And as I said ...
These Leighs! our best blood running in the rut!
It's something awful. We had pardoned him
A simple misalliance, got up aside
For a pair of sky-blue eyes; our House of Lords
Has winked at such things, and we've all been young.
But here's an inter-marriage reasoned out,
A contract (carried boldly to the light,
To challenge observation, pioneer
Good acts by a great example) 'twixt the extremes
Of martyrised society,—on the left,
The well-born,—on the right, the merest mob,
To treat as equals!—'tis anarchical!
It means more than it says—'tis damnable!
Why, sir, we can't have even our coffee good,
Unless we strain it.'
'Here, Miss Leigh!'
'Lord Howe,
You're Romney's friend. What's all this waiting for?'
'I cannot tell. The bride has lost her head
(And way, perhaps!) to prove her sympathy
With the bridegroom.'
'What,—you also, disapprove!'
'Oh, _I_ approve of nothing in the world,'
He answered; 'not of you, still less of me,
Nor even of Romney—though he's worth us both.
We're all gone wrong. The tune in us is lost:
And whistling in back alleys to the moon,
Will never catch it.'
Let me draw Lord Howe;
A born aristocrat, bred radical,
And educated socialist, who still
Goes floating, on traditions of his kind,
Across the theoretic flood from France,—
Though, like a drenched Noah on a rotten deck,
Scarce safer for his place there. He, at least,
Will never land on Ararat, he knows,
To recommence the world on the old plan:
Indeed, he thinks, said world had better end;
He sympathises rather with the fish
Outside, than with the drowned paired beasts within
Who cannot couple again or multiply:
And that's the sort of Noah he is, Lord Howe.
He never could be anything complete,
Except a loyal, upright gentleman,
A liberal landlord, graceful diner-out,
And entertainer more than hospitable,
Whom authors dine with and forget the port.
Whatever he believes, and it is much,
But no-wise certain ... now here and now there, ...
He still has sympathies beyond his creed,
Diverting him from action. In the House,
No party counts upon him, and all praise
All like his books too, (he has written books)
Which, good to lie beside a bishop's chair,
So oft outreach themselves with jets of fire
At which the foremost of the progressists
May warm audacious hands in passing by.
—Of stature over-tall, lounging for ease;
Light hair, that seems to carry a wind in it,
And eyes that, when they look on you, will lean
Their whole weight half in indolence, and half
In wishing you unmitigated good,
Until you know not if to flinch from him
Or thank him.—'Tis Lord Howe.
'We're all gone wrong,'
Said he, 'and Romney, that dear friend of ours,
Is no-wise right. There's one true thing on earth;
That's love! He takes it up, and dresses it,
And acts a play with it, as Hamlet did,
To show what cruel uncles we have been,
And how we should be uneasy in our minds,
While he, Prince Hamlet, weds a pretty maid
(Who keeps us too long waiting, we'll confess)
By symbol, to instruct us formally
To fill the ditches up 'twixt class and class,
And live together in phalansteries.
What then?—he's mad, our Hamlet! clap his play,
And bind him.'
'Ah Lord Howe, this spectacle
Pulls stronger at us than the Dane's. See there!
The crammed aisles heave and strain and steam with life—
Dear Heaven, what life!'
'Why, yes,—a poet sees;
Which makes him different from a common man.
_I_, too, see somewhat, though I cannot sing;
I should have been a poet, only that
My mother took fright at the ugly world,
And bore me tongue-tied. If you'll grant me now
That Romney gives us a fine actor-piece
To make us merry on his marriage-morn,
The fable's worse than Hamlet's, I'll concede.
The terrible people, old and poor and blind,
Their eyes eat out with plague and poverty
From seeing beautiful and cheerful sights,
We'll liken to a brutalised King Lear,
Led out,—by no means to clear scores with wrongs—
His wrongs are so far back, ... he has forgot;
All's past like youth; but just to witness here
A simple contract,—he, upon his side,
And Regan with her sister Goneril
And all the dappled courtiers and court-fools,
On their side. Not that any of these would say
They're sorry, neither. What is done, is done,
And violence is now turned privilege,
As cream turns cheese, if buried long enough.
What could such lovely ladies have to do
With the old man there, in those ill-odorous rags,
Except to keep the wind-side of him? Lear
Is flat and quiet, as a decent grave;
He does not curse his daughters in the least.
_Be_ these his daughters? Lear is thinking of
His porridge chiefly ... is it getting cold
At Hampstead? will the ale be served in pots?
Poor Lear, poor daughters! Bravo, Romney's play!'
A murmur and a movement drew around;
A naked whisper touched us. Something wrong!
What's wrong? The black crowd, as an overstrained
Cord, quivered in vibrations, and I saw ...
Was that _his_ face I saw?... his ... Romney Leigh's ...
Which tossed a sudden horror like a sponge
Into all eyes,—while himself stood white upon
The topmost altar-stair, and tried to speak,
And failed, and lifted higher above his head
A letter, ... as a man who drowns and gasps.
'My brothers, bear with me! I am very weak.
I meant but only good. Perhaps I meant
Too proudly,—and God snatched the circumstance
And changed it therefore. There's no marriage—none.
She leaves me,—she departs,—she disappears,—
I lose her. Yet I never forced her 'ay,'
To have her 'no' so cast into my teeth,
In manner of an accusation, thus.
My friends, you are all dismissed. Go, eat and drink
According to the programme,—and farewell!'
He ended. There was silence in the church;
We heard a baby sucking in its sleep
At the farthest end of the aisle. Then spoke a man,
'Now, look to it, coves, that all the beef and drink
Be not filched from us like the other fun;
For beer's spilt easier than a woman is!
This gentry is not honest with the poor;
They bring us up, to trick us.'—'Go it, Jim,'
A woman screamed back,—'I'm a tender soul;
I never banged a child at two years old
And drew blood from him, but I sobbed for it
Next moment,—and I've had a plague of seven.
I'm tender; I've no stomach even for beef,
Until I know about the girl that's lost,
That's killed, mayhap. I did misdoubt, at first,
The fine lord meant no good by her, or us.
He, maybe, got the upper hand of her
By holding up a wedding-ring, and then ...
A choking finger on her throat, last night,
And just a clever tale to keep us still,
As she is, poor lost innocent. 'Disappear!'
Who ever disappears except a ghost?
And who believes a story of a ghost?
I ask you,—would a girl go off, instead
Of staying to be married? a fine tale!
A wicked man, I say, a wicked man!
For my part I would rather starve on gin
Than make my dinner on his beef and beer.'—
At which a cry rose up—'We'll have our rights.
We'll have the girl, the girl! Your ladies there
Are married safely and smoothly every day,
And _she_ shall not drop through into a trap
Because she's poor and of the people: shame!
We'll have no tricks played off by gentlefolks;
We'll see her righted.'
Through the rage and roar
I heard the broken words which Romney flung
Among the turbulent masses, from the ground
He held still, with his masterful pale face—
As huntsmen throw the ration to the pack,
Who, falling on it headlong, dog on dog
In heaps of fury, rend it, swallow it up
With yelling hound-jaws,—his indignant words,
His piteous words, his most pathetic words,
Whereof I caught the meaning here and there
By his gesture ... torn in morsels, yelled across,
And so devoured. From end to end, the church
Rocked round us like the sea in storm, and then
Broke up like the earth in earthquake. Men cried out
'Police'—and women stood and shrieked for God,
Or dropt and swooned; or, like a herd of deer,
(For whom the black woods suddenly grow alive,
Unleashing their wild shadows down the wind
To hunt the creatures into corners, back
And forward) madly fled, or blindly fell,
Trod screeching underneath the feet of those
Who fled and screeched.
The last sight left to me
Was Romney's terrible calm face above
The tumult!—the last sound was 'Pull him down!
Strike—kill him!' Stretching my unreasoning arms,
As men in dreams, who vainly interpose
'Twixt gods and their undoing, with a cry
I struggled to precipitate myself
Head-foremost to the rescue of my soul
In that white face, ... till some one caught me back,
And so the world went out,—I felt no more.
What followed, was told after by Lord Howe,
Who bore me senseless from the strangling crowd
In church and street, and then returned alone
To see the tumult quelled. The men of law
Had fallen as thunder on a roaring fire,
And made all silent,—while the people's smoke
Passed eddying slowly from the emptied aisles.
Here's Marian's letter, which a ragged child
Brought running, just as Romney at the porch
Looked out expectant of the bride. He sent
The letter to me by his friend Lord Howe
Some two hours after, folded in a sheet
On which his well-known hand had left a word.
Here's Marian's letter.
'Noble friend, dear saint,
Be patient with me. Never think me vile,
Who might to-morrow morning be your wife
But that I loved you more than such a name.
Farewell, my Romney. Let me write it once,—
My Romney.
''Tis so pretty a coupled word,
I have no heart to pluck it with a blot.
We say 'my God' sometimes, upon our knees,
Who is not therefore vexed: so bear with it ...
And me. I know I'm foolish, weak, and vain;
Yet most of all I'm angry with myself
For losing your last footstep on the stair,
That last time of your coming,—yesterday!
The very first time I lost step of yours,
(Its sweetness comes the next to what you speak)
But yesterday sobs took me by the throat,
And cut me off from music.
'Mister Leigh,
You'll set me down as wrong in many things.
You've praised me, sir, for truth,—and now you'll learn
I had not courage to be rightly true.
I once began to tell you how she came,
The woman ... and you stared upon the floor
In one of your fixed thoughts ... which put me out
For that day. After, some one spoke of me,
So wisely, and of you, so tenderly,
Persuading me to silence for your sake ...
Well, well! it seems this moment I was wrong
In keeping back from telling you the truth:
There might be truth betwixt us two, at least,
If nothing else. And yet 'twas dangerous.
Suppose a real angel came from heaven
To live with men and women! he'd go mad,
If no considerate hand should tie a blind
Across his piercing eyes. 'Tis thus with you:
You see us too much in your heavenly light;
I always thought so, angel,—and indeed
There's danger that you beat yourself to death
Against the edges of this alien world,
In some divine and fluttering pity.
'Yes,
It would be dreadful for a friend of yours,
To see all England thrust you out of doors
And mock you from the windows. You might say,
Or think (that's worse), 'There's some one in the house
I miss and love still.' Dreadful!
'Very kind,
I pray you mark, was Lady Waldemar.
She came to see me nine times, rather ten—
So beautiful, she hurts me like the day
Let suddenly on sick eyes.
'Most kind of all,
Your cousin!—ah, most like you! Ere you came
She kissed me mouth to mouth: I felt her soul
Dip through her serious lips in holy fire.
God help me, but it made me arrogant;
I almost told her that you would not lose
By taking me to wife: though, ever since,
I've pondered much a certain thing she asked ...
'He loves you, Marian?' ... in a sort of mild
Derisive sadness ... as a mother asks
Her babe, 'You'll touch that star, you think?'
'Farewell!
I know I never touched it.
This is worst:
Babes grow, and lose the hope of things above;
A silver threepence sets them leaping high—
But no more stars! mark that.
I've writ all night,
And told you nothing. God, if I could die,
And let this letter break off innocent
Just here! But no—for your sake ...
Here's the last:
I never could be happy as your wife,
I never could be harmless as your friend,
I never will look more into your face,
Till God says, 'Look!' I charge you, seek me not,
Nor vex yourself with lamentable thoughts
That peradventure I have come to grief;
Be sure I'm well, I'm merry, I'm at ease,
But such a long way, long way, long way off,
I think you'll find me sooner in my grave,
And that's my choice, observe. For what remains,
An over-generous friend will care for me,
And keep me happy ... happier....
There's a blot!
This ink runs thick ... we light girls lightly weep ...
And keep me happier ... was the thing to say, ...
Than as your wife I could be!—O, my star,
My saint, my soul! for surely you're my soul,
Through whom God touched me! I am not so lost
I cannot thank you for the good you did,
The tears you stopped, which fell down bitterly,
Like these—the times you made me weep for joy
At hoping I should learn to write your notes
And save the tiring of your eyes, at night;
And most for that sweet thrice you kissed my lips
And said 'Dear Marian.'
'Twould be hard to read,
This letter, for a reader half as learn'd,
But you'll be sure to master it, in spite
Of ups and downs. My hand shakes, I am blind,
I'm poor at writing, at the best,—and yet
I tried to make my _g_s the way you showed.
Farewell—Christ love you.—Say 'poor Marian' now.'
Poor Marian!—wanton Marian!—was it so,
Or so? For days, her touching, foolish lines
We mused on with conjectural fantasy,
As if some riddle of a summer-cloud
On which one tries unlike similitudes
Of now a spotted Hydra-skin cast off,
And now a screen of carven ivory
That shuts the heavens' conventual secrets up
From mortals over-bold. We sought the sense:
She loved him so perhaps, (such words mean love,)
That, worked on by some shrewd perfidious tongue,
(And then I thought of Lady Waldemar)
She left him, not to hurt him; or perhaps
She loved one in her class,—or did not love,
But mused upon her wild bad tramping life,
Until the free blood fluttered at her heart,
And black bread eaten by the road-side hedge
Seemed sweeter than being put to Romney's school
Of philanthropical self-sacrifice,
Irrevocably.—Girls are girls, beside,
Thought I, and like a wedding by one rule.
You seldom catch these birds, except with chaff:
They feel it almost an immoral thing
To go out and be married in broad day,
Unless some winning special flattery should
Excuse them to themselves for't, ... 'No one parts
Her hair with such a silver line as you,
One moonbeam from the forehead to the crown!'
Or else ... 'You bite your lip in such a way,
It spoils me for the smiling of the rest'—
And so on. Then a worthless gaud or two,
To keep for love,—a ribbon for the neck,
Or some glass pin,—they have their weight with girls.
And Romney sought her many days and weeks:
He sifted all the refuse of the town,
Explored the trains, enquired among the ships,
And felt the country through from end to end;
No Marian!—Though I hinted what I knew,—
A friend of his had reasons of her own
For throwing back the match—he would not hear:
The lady had been ailing ever since,
The shock had harmed her. Something in his tone
Repressed me; something in me shamed my doubt
To a sigh, repressed too. He went on to say
That, putting questions where his Marian lodged,
He found she had received for visitors,
Besides himself and Lady Waldemar
And, that once, me—a dubious woman dressed
Beyond us both. The rings upon her hands
Had dazed the children when she threw them pence;
'She wore her bonnet as the queen might hers,
To show the crown,' they said,—'a scarlet crown
Of roses that had never been in bud.'
When Romney told me that,—for now and then
He came to tell me how the search advanced,
His voice dropped: I bent forward for the rest:
The woman had been with her, it appeared,
At first from week to week, then day by day,
And last, 'twas sure ...
I looked upon the ground
To escape the anguish of his eyes, and asked
As low as when you speak to mourners new
Of those they cannot bear yet to call dead,
'If Marian had as much as named to him
A certain Rose, an early friend of hers,
A ruined creature.'
'Never,'—Starting up
He strode from side to side about the room,
Most like some prisoned lion sprung awake,
Who has felt the desert sting him through his dreams.
'What was I to her, that she should tell me aught?
A friend! was _I_ a friend? I see all clear.
Such devils would pull angels out of heaven,
Provided they could reach them; 'tis their pride;
And that's the odds 'twixt soul and body-plague!
The veriest slave who drops in Cairo's street,
Cries, 'Stand off from me,' to the passengers;
While these blotched souls are eager to infect,
And blow their bad breath in a sister's face
As if they got some ease by it.'
I broke through.
'Some natures catch no plagues. I've read of babes
Pound whole and sleeping by the spotted breast
Of one a full day dead. I hold it true,
As I'm a woman and know womanhood,
That Marian Erle, however lured from place,
Deceived in way, keeps pure in aim and heart,
As snow that's drifted from the garden-bank
To the open road.'
'Twas hard to hear him laugh.
'The figure's happy. Well—a dozen carts
And trampers will secure you presently
A fine white snow-drift. Leave it there, your snow!
'Twill pass for soot ere sunset. Pure in aim?
She's pure in aim, I grant you,—like myself,
Who thought to take the world upon my back
To carry it o'er a chasm of social ill,
And end by letting slip through impotence
A single soul, a child's weight in a soul,
Straight down the pit of hell! yes, I and she
Have reason to be proud of our pure aims.'
Then softly, as the last repenting drops
Of a thunder-shower, he added, 'The poor child;
Poor Marian! 'twas a luckless day for her,
When first she chanced on my philanthropy.'
He drew a chair beside me, and sate down;
And I, instinctively, as women use
Before a sweet friend's grief,—when, in his ear,
They hum the tune of comfort, though themselves
Most ignorant of the special words of such,
And quiet so and fortify his brain
And give it time and strength for feeling out
To reach the availing sense beyond that sound,—
Went murmuring to him, what, if written here,
Would seem not much, yet fetched him better help
Than, peradventure, if it had been more.
I've known the pregnant thinkers of this time,
And stood by breathless, hanging on their lips,
When some chromatic sequence of fine thought
In learned modulation phrased itself
To an unconjectured harmony of truth.
And yet I've been more moved, more raised, I say,
By a simple word ... a broken easy thing,
A three-years infant might say after you,—
A look, a sigh, a touch upon the palm,
Which meant less than 'I love you' ... than by all
The full-voiced rhetoric of those master-mouths.
'Ah dear Aurora,' he began at last,
His pale lips fumbling for a sort of smile,
'Your printer's devils have not spoilt your heart:
That's well. And who knows but, long years ago,
When you and I talked, you were somewhat right
In being so peevish with me? You, at least,
Have ruined no one through your dreams! Instead,
You've helped the facile youth to live youth's day
With innocent distraction, still perhaps
Suggestive of things better than your rhymes.
The little shepherd-maiden, eight years old,
I've seen upon the mountains of Vaucluse,
Asleep i' the sun, her head upon her knees,
The flocks all scattered,—is more laudable
Than any sheep-dog trained imperfectly,
Who bites the kids through too much zeal.'
'I look
As if I had slept, then?'
He was touched at once
By something in my face. Indeed 'twas sure
That he and I,—despite a year or two
Of younger life on my side, and on his,
The heaping of the years' work on the days,—
The three-hour speeches from the member's seat,
The hot committees, in and out the House,
The pamphlets, 'Arguments,' 'Collective Views,'
Tossed out as straw before sick houses, just
To show one's sick and so be trod to dirt,
And no more use,—through this world's underground
The burrowing, groping effort, whence the arm
And heart come bleeding,—sure, that he and I
Were, after all, unequally fatigued!
That he, in his developed manhood, stood
A little sunburnt by the glare of life;
While I ... it seemed no sun had shone on me,
So many seasons I had forgot my Springs;
My cheeks had pined and perished from their orbs,
And all the youth-blood in them had grown white
As dew on autumn cyclamens: alone
My eyes and forehead answered for my face.
He said ... 'Aurora, you are changed—are ill!'
'Not so, my cousin,—only not asleep!'
I answered, smiling gently. 'Let it be.
You scarcely found the poet of Vaucluse
As drowsy as the shepherds. What is art,
But life upon the larger scale, the higher,
When, graduating up in a spiral line
Of still expanding and ascending gyres,
It pushes toward the intense significance
Of all things, hungry for the Infinite?
Art's life,—and where we live, we suffer and toil.'
He seemed to sift me with his painful eyes.
'Alas! you take it gravely; you refuse
Your dreamland, right of common, and green rest.
You break the mythic turf where danced the nymphs,
With crooked ploughs of actual life,—let in
The axes to the legendary woods,
To pay the head-tax. You are fallen indeed
On evil days, you poets, if yourselves
Can praise that art of yours no otherwise;
And, if you cannot, ... better take a trade
And be of use! 'twere cheaper for your youth.'
'Of use!' I softly echoed, 'there's the point
We sweep about for ever in argument;
Like swallows, which the exasperate, dying year
Sets spinning in black circles, round and round,
Preparing for far flights o'er unknown seas.
And we ... where tend we?'
'Where?' he said, and sighed.
'The whole creation, from the hour we are born,
Perplexes us with questions. Not a stone
But cries behind us, every weary step,
'Where, where?' I leave stones to reply to stones.
Enough for me and for my fleshly heart
To harken the invocations of my kind,
When men catch hold upon my shuddering nerves
And shriek, 'What help? what hope? what bread i' the house,
What fire i' the frost?' There must be some response,
Though mine fail utterly. This social Sphinx,
Who sits between the sepulchres and stews,
Makes mock and mow against the crystal heavens,
And bullies God,—exacts a word at least
From each man standing on the side of God,
However paying a sphinx-price for it.
We pay it also if we hold our peace,
In pangs and pity. Let me speak and die.
Alas! you'll say, I speak and kill, instead.'
I pressed in there; 'The best men, doing their best,
Know peradventure least of what they do:
Men usefullest i' the world, are simply used;
The nail that holds the wood, must pierce it first,
And He alone who wields the hammer, sees
The work advanced by the earliest blow. Take heart.'
'Ah, if I could have taken yours!' he said,
'But that's past now,' Then rising ... 'I will take
At least your kindness and encouragement.
I thank you. Dear, be happy. Sing your songs,
If that's your way! but sometimes slumber too,
Nor tire too much with following, out of breath,
The rhymes upon your mountains of Delight.
Reflect, if Art be, in truth, the higher life,
You need the lower life to stand upon,
In order to reach up unto that higher;
And none can stand a-tiptoe in the place
He cannot stand in with two stable feet.
Remember then!—for Art's sake, hold your life.'
We parted so. I held him in respect.
I comprehended what he was in heart
And sacrificial greatness. Ay, but _he_
Supposed me a thing too small to deign to know:
He blew me, plainly, from the crucible,
As some intruding, interrupting fly
Not worth the pains of his analysis
Absorbed on nobler subjects. Hurt a fly!
He would not for the world: he's pitiful
To flies even. 'Sing,' says he, 'and teaze me still,
If that's your way, poor insect.' That's your way!
FIFTH BOOK.
AURORA LEIGH, be humble. Shall I hope
To speak my poems in mysterious tune
With man and nature,—with the lava-lymph
That trickles from successive galaxies
Still drop by drop adown the finger of God,
In still new worlds?—with summer-days in this,
That scarce dare breathe, they are so beautiful?—
With spring's delicious trouble in the ground
Tormented by the quickened blood of roots,
And softly pricked by golden crocus-sheaves
In token of the harvest-time of flowers?—
With winters and with autumns,—and beyond,
With the human heart's large seasons,—when it hopes
And fears, joys, grieves, and loves?—with all that strain
Of sexual passion, which devours the flesh
In a sacrament of souls? with mother's breasts,
Which, round the new-made creatures hanging there,
Throb luminous and harmonious like pure spheres?—
With multitudinous life, and finally
With the great out-goings of ecstatic souls,
Who, in a rush of too long prisoned flame,
Their radiant faces upward, burn away
This dark of the body, issuing on a world
Beyond our mortal?—can I speak my verse
So plainly in tune to these things and the rest,
That men shall feel it catch them on the quick,
As having the same warrant over them
To hold and move them, if they will or no,
Alike imperious as the primal rhythm
Of that theurgic nature? I must fail,
Who fail at the beginning to hold and move
One man,—and he my cousin, and he my friend,
And he born tender, made intelligent,
Inclined to ponder the precipitous sides
Of difficult questions; yet, obtuse to _me_,—
Of _me_, incurious! likes me very well,
And wishes me a paradise of good,
Good looks, good means, and good digestion!—ay,
But otherwise evades me, puts me off
With kindness, with a tolerant gentleness,—
Too light a book for a grave man's reading! Go,
Aurora Leigh: be humble.
There it is;
We women are too apt to look to one,
Which proves a certain impotence in art.
We strain our natures at doing something great,
Far less because it's something great to do,
Than, haply, that we, so, commend ourselves
As being not small, and more appreciable
To some one friend. We must have mediators
Betwixt our highest conscience and the judge;
Some sweet saint's blood must quicken in our palms,
Or all the life in heaven seems slow and cold:
Good only, being perceived as the end of good,
And God alone pleased,—that's too poor, we think,
And not enough for us, by any means.
Ay—Romney, I remember, told me once
We miss the abstract, when we comprehend!
We miss it most when we aspire, ... and fail.
Yet, so, I will not.—This vile woman's way
Of trailing garments, shall not trip me up.
I'll have no traffic with the personal thought
In art's pure temple. Must I work in vain,
Without the approbation of a man?
It cannot be; it shall not. Fame itself,
That approbation of the general race,
Presents a poor end, (though the arrow speed,
Shot straight with vigorous finger to the white,)
And the highest fame was never reached except
By what was aimed above it. Art for art,
And good for God Himself, the essential Good!
We'll keep our aims sublime, our eyes erect,
Although our woman-hands should shake and fail;
And if we fail.... But must we?—
Shall I fail?
The Greeks said grandly in their tragic phrase,
'Let no one be called happy till his death.'
To which I add,—Let no one till his death
Be called unhappy. Measure not the work
Until the day's out and the labour done;
Then bring your gauges. If the day's work's scant,
Why, call it scant; affect no compromise;
And, in that we have nobly striven at least,
Deal with us nobly, women though we be,
And honour us with truth, if not with praise.
My ballads prospered; but the ballad's race
Is rapid for a poet who bears weights
Of thought and golden image. He can stand
Like Atlas, in the sonnet,—and support
His own heavens pregnant with dynastic stars;
But then he must stand still, nor take a step.
In that descriptive poem called 'The Hills,'
The prospects were too far and indistinct.
'Tis true my critics said, 'A fine view, that!'
The public scarcely cared to climb the book
For even the finest; and the public's right,
A tree's mere firewood, unless humanised;
Which well the Greeks knew, when they stirred the bark
With close-pressed bosoms of subsiding nymphs,
And made the forest-rivers garrulous
With babble of gods. For us, we are called to mark
A still more intimate humanity
In this inferior nature,—or, ourselves,
Must fall like dead leaves trodden underfoot
By veritabler artists. Earth, shut up
By Adam, like a fakir in a box
Left too long buried, remained stiff and dry,
A mere dumb corpse, till Christ the Lord came down,
Unlocked the doors, forced open the blank eyes,
And used his kingly chrisms to straighten out
The leathery tongue turned back into the throat:
Since when, she lives, remembers, palpitates
In every limb, aspires in every breath,
Embraces infinite relations. Now,
We want no half-gods, Panomphæan Joves,
Fauns, Naiads, Tritons, Oreads and the rest,
To take possession of a senseless world
To unnatural vampire-uses. See the earth,
The body of our body, the green earth,
Indubitably human, like this flesh
And these articulated veins through which
Our heart drives blood! there's not a flower of spring,
That dies ere June, but vaunts itself allied
By issue and symbol, by significance
And correspondence, to that spirit-world
Outside the limits of our space and time,
Whereto we are bound. Let poets give it voice
With human meanings; else they miss the thought,
And henceforth step down lower, stand confessed
Instructed poorly for interpreters,—
Thrown out by an easy cowslip in the text.
Even so my pastoral failed: it was a book
Of surface-pictures—pretty, cold, and false
With literal transcript,—the worse done, I think,
For being not ill-done. Let me set my mark
Against such doings, and do otherwise.
This strikes me.—If the public whom we know,
Could catch me at such admissions, I should pass
For being right modest. Yet how proud we are,
In daring to look down upon ourselves!
The critics say that epics have died out
With Agamemnon and the goat-nursed gods—
I'll not believe it. I could never dream
As Payne Knight did, (the mythic mountaineer
Who travelled higher than he was born to live,
And showed sometimes the goitre in his throat
Discoursing of an image seen through fog,)
That Homer's heroes measured twelve feet high.
They were but men!—his Helen's hair turned grey
Like any plain Miss Smith's, who wears a front;
And Hector's infant blubbered at a plume
As yours last Friday at a turkey-cock.
All men are possible heroes: every age,
Heroic in proportions, double-faced,
Looks backward and before, expects a morn
And claims an epos.
Ay, but every age
Appears to souls who live in it, (ask Carlyle)
Most unheroic. Ours, for instance, ours!
The thinkers scout it, and the poets abound
Who scorn to touch it with a finger-tip:
A pewter age,—mixed metal, silver-washed;
An age of scum, spooned off the richer past;
An age of patches for old gaberdines;
An age of mere transition, meaning nought,
Except that what succeeds must shame it quite,
If God please. That's wrong thinking, to my mind,
And wrong thoughts make poor poems.
Every age,
Through being beheld too close, is ill-discerned
By those who have not lived past it. We'll suppose
Mount Athos carved, as Persian Xerxes schemed,
To some colossal statue of a man:
The peasants, gathering brushwood in his ear,
Had guessed as little of any human form
Up there, as would a flock of browsing goats.
They'd have, in fact, to travel ten miles off
Or ere the giant image broke on them,
Full human profile, nose and chin distinct,
Mouth, muttering rhythms of silence up the sky,
And fed at evening with the blood of suns;
Grand torso,—hand, that flung perpetually
The largesse of a silver river down
To all the country pastures. 'Tis even thus
With times we live in,—evermore too great
To be apprehended near.
But poets should
Exert a double vision; should have eyes
To see near things as comprehensively
As if afar they took their point of sight,
And distant things, as intimately deep,
As if they touched them. Let us strive for this.
I do distrust the poet who discerns
No character or glory in his times,
And trundles back his soul five hundred years,
Past moat and drawbridge, into a castle-court,
Oh not to sing of lizards or of toads
Alive i' the ditch there!—'twere excusable;
But of some black chief, half knight, half sheep-lifter,
Some beauteous dame, half chattel and half queen,
As dead as must be, for the greater part,
The poems made on their chivalric bones.
And that's no wonder: death inherits death.
Nay, if there's room for poets in the world
A little overgrown, (I think there is)
Their sole work is to represent the age,
Their age, not Charlemagne's,—this live, throbbing age,
That brawls, cheats, maddens, calculates, aspires,
And spends more passion, more heroic heat,
Betwixt the mirrors of its drawing-rooms,
Than Roland with his knights, at Roncesvalles.
To flinch from modern varnish, coat or flounce,
Cry out for togas and the picturesque,
Is fatal,—foolish too. King Arthur's self
Was commonplace to Lady Guenever;
And Camelot to minstrels seemed as flat,
As Regent Street to poets.
Never flinch,
But still, unscrupulously epic, catch
Upon the burning lava of a song,
The full-veined, heaving, double-breasted Age:
That, when the next shall come, the men of that
May touch the impress with reverent hand, and say
'Behold,—behold the paps we all have sucked!
That bosom seems to beat still, or at least
It sets ours beating. This is living art,
Which thus presents, and thus records true life.'
What form is best for poems? Let me think
Of forms less, and the external. Trust the spirit,
As sovran nature does, to make the form;
For otherwise we only imprison spirit,
And not embody. Inward evermore
To outward,—so in life, and so in art,
Which still is life.
Five acts to make a play.
And why not fifteen? why not ten? or seven?
What matter for the number of the leaves,
Supposing the tree lives and grows? exact
The literal unities of time and place,
When 'tis the essence of passion to ignore
Both time and place? Absurd. Keep up the fire,
And leave the generous flames to shape themselves.
'Tis true the stage requires obsequiousness
To this or that convention; 'exit' here
And 'enter' there; the points for clapping, fixed,
Like Jacob's white-peeled rods before the rams;
And all the close-curled imagery clipped
In manner of their fleece at shearing-time.
Forget to prick the galleries to the heart
Precisely at the fourth act,—culminate
Our five pyramidal acts with one act more,—
We're lost so! Shakspeare's ghost could scarcely plead
Against our just damnation. Stand aside;
We'll muse for comfort that, last century,
On this same tragic stage on which we have failed,
A wigless Hamlet would have failed the same.
And whosoever writes good poetry,
Looks just to art. He does not write for you
Or me,—for London or for Edinburgh;
He will not suffer the best critic known
To step into his sunshine of free thought
And self-absorbed conception, and exact
An inch-long swerving of the holy lines.
If virtue done for popularity
Defiles like vice, can art for praise or hire
Still keep its splendor, and remain pure art?
Eschew such serfdom. What the poet writes,
He writes: mankind accepts it, if it suits,
And that's success: if not, the poem's passed
From hand to hand, and yet from hand to hand,
Until the unborn snatch it, crying out
In pity on their fathers' being so dull,
And that's success too.
I will write no plays.
Because the drama, less sublime in this,
Makes lower appeals, defends more menially,
Adopts the standard of the public taste
To chalk its height on, wears a dog-chain round
Its regal neck, and learns to carry and fetch
The fashions of the day to please the day;
Fawns close on pit and boxes, who clap hands,
Commending chiefly its docility
And humour in stage-tricks; or else indeed
Gets hissed at, howled at, stamped at like a dog,
Or worse, we'll say. For dogs, unjustly kicked,
Yell, bite at need; but if your dramatist
(Being wronged by some five hundred nobodies
Because their grosser brains most naturally
Misjudge the fineness of his subtle wit)
Shows teeth an almond's breadth, protests the length
Of a modest phrase,—'My gentle countrymen,
There's something in it, haply, of your fault,'—
Why then, besides five hundred nobodies,
He'll have five thousand, and five thousand more,
Against him,—the whole public,—all the hoofs
Of King Saul's father's asses, in full drove,—
And obviously deserve it. He appealed
To these,—and why say more if they condemn,
Than if they praised him?—Weep, my Æschylus,
But low and far, upon Sicilian shores!
For since 'twas Athens (so I read the myth)
Who gave commission to that fatal weight,
The tortoise, cold and hard, to drop on thee
And crush thee,—better cover thy bald head;
She'll hear the softest hum of Hyblan bee
Before thy loud'st protesting.—For the rest,
The risk's still worse upon the modern stage:
I could not, in so little, accept success,
Nor would I risk so much, in ease and calm,
For manifester gains; let those who prize,
Pursue them: _I_ stand off.
And yet, forbid,
That any irreverent fancy or conceit
Should litter in the Drama's throne-room, where
The rulers of our art, in whose full veins
Dynastic glories mingle, sit in strength
And do their kingly work,—conceive, command,
And, from the imagination's crucial heat,
Catch up their men and women all a-flame
For action, all alive, and forced to prove
Their life by living out heart, brain, and nerve,
Until mankind makes witness, 'These be men
As we are,' and vouchsafes the kiss that's due
To Imogen and Juliet—sweetest kin
On art's side.
'Tis that, honouring to its worth
The drama, I would fear to keep it down
To the level of the footlights. Dies no more
The sacrificial goat, for Bacchus slain,—
His filmed eyes fluttered by the whirling white
Of choral vestures,—troubled in his blood,
While tragic voices that clanged keen as swords,
Leapt high together with the altar-flame,
And made the blue air wink. The waxen mask,
Which set the grand still front of Themis' son
Upon the puckered visage of a player;—
The buskin, which he rose upon and moved,
As some tall ship, first conscious of the wind,
Sweeps slowly past the piers;—the mouthpiece, where
The mere man's voice with all its breaths and breaks
Went sheathed in brass, and clashed on even heights
Its phrasèd thunders;—these things are no more,
Which once were. And concluding, which is clear,
The growing drama has outgrown such toys
Of simulated stature, face, and speech,
It also, peradventure, may outgrow
The simulation of the painted scene,
Boards, actors, prompters, gaslight, and costume;
And take for a worthier stage the soul itself,
Its shifting fancies and celestial lights,
With all its grand orchestral silences
To keep the pauses of the rhythmic sounds.
Alas, I still see something to be done,
And what I do falls short of what I see
Though I waste myself on doing. Long green days,
Worn bare of grass and sunshine,—long calm nights,
From which the silken sleeps were fretted out,—
Be witness for me, with no amateur's
Irreverent haste and busy idleness
I've set myself to art! What then? what's done?
What's done, at last?
Behold, at last, a book.
If life-blood's necessary,—which it is,
(By that blue vein athrob on Mahomet's brow,
Each prophet-poet's book must show man's blood!)
If life-blood's fertilising, I wrung mine
On every leaf of this,—unless the drops
Slid heavily on one side and left it dry.
That chances often: many a fervid man
Writes books as cold and flat as grave-yard stones
From which the lichen's scraped; and if St. Preux
Had written his own letters, as he might,
We had never wept to think of the little mole
'Neath Julie's drooping eyelid. Passion is
But something suffered, after all.
While Art
Sets action on the top of suffering:
The artist's part is both to be and do,
Transfixing with a special, central power
The flat experience of the common man,
And turning outward, with a sudden wrench,
Half agony, half ecstasy, the thing
He feels the inmost: never felt the less
Because he sings it. Does a torch less burn
For burning next reflectors of blue steel,
That _he_ should be the colder for his place
'Twixt two incessant fires,—his personal life's,
And that intense refraction which burns back
Perpetually against him from the round
Of crystal conscience he was born into
If artist-born? O sorrowful great gift
Conferred on poets, of a twofold life,
When one life has been found enough for pain!
We, staggering 'neath our burden as mere men,
Being called to stand up straight as demi-gods,
Support the intolerable strain and stress
Of the universal, and send clearly up
With voices broken by the human sob,
Our poems to find rhymes among the stars!
But soft!—a 'poet' is a word soon said;
A book's a thing soon written. Nay, indeed,
The more the poet shall be questionable,
The more unquestionably comes his book!
And this of mine—well, granting to myself
Some passion in it, furrowing up the flats,
Mere passion will not prove a volume worth
Its gall and rags even. Bubbles round a keel
Mean nought, excepting that the vessel moves.
There's more than passion goes to make a man,
Or book, which is a man too.
I am sad.
I wonder if Pygmalion had these doubts,
And, feeling the hard marble first relent,
Grow supple to the straining of his arms,
And tingle through its cold to his burning lip,
Supposed his senses mocked, and that the toil
Of stretching past the known and seen, to reach
The archetypal Beauty out of sight,
Had made his heart beat fast enough for two,
And with his own life dazed and blinded him!
Not so; Pygmalion loved,—and whoso loves
Believes the impossible.
And I am sad:
I cannot thoroughly love a work of mine,
Since none seems worthy of my thought and hope
More highly mated. He has shot them down,
My Phœbus Apollo, soul within my soul,
Who judges, by the attempted, what's attained,
And with the silver arrow from his height,
Has struck down all my works before my face,
While _I_ said nothing. Is there aught to say?
I called the artist but a greatened man;
He may be childless also, like a man.
I laboured on alone. The wind and dust
And sun of the world beat blistering in my face;
And hope, now for me, now against me, dragged
My spirits onward,—as some fallen balloon,
Which, whether caught by blossoming tree or bare,
Is torn alike. I sometimes touched my aim,
Or seemed,—and generous souls cried out, 'Be strong,
Take courage; now you're on our level,—now!
The next step saves you!' I was flushed with praise,
But, pausing just a moment to draw breath,
I could not choose but murmur to myself
'Is this all? all that's done? and all that's gained?
If this then be success, 'tis dismaller
Than any failure.'
O my God, my God,
O supreme Artist, who as sole return
For all the cosmic wonder of Thy work,
Demandest of us just a word ... a name,
'My Father!'—thou hast knowledge, only thou,
How dreary 'tis for women to sit still
On winter nights by solitary fires,
And hear the nations praising them far off,
Too far! ay, praising our quick sense of love,
Our very heart of passionate womanhood,
Which could not beat so in the verse without
Being present also in the unkissed lips,
And eyes undried because there's none to ask
The reason they grew moist.
To sit alone,
And think, for comfort, how, that very night,
Affianced lovers, leaning face to face
With sweet half-listenings for each other's breath,
Are reading haply from some page of ours,
To pause with a thrill, as if their cheeks had touched,
When such a stanza, level to their mood,
Seems floating their own thought out—'So I feel
For thee,'—'And I, for thee: this poet knows
What everlasting love is!'—how, that night,
A father, issuing from the misty roads
Upon the luminous round of lamp and hearth
And happy children, having caught up first
The youngest there until it shrunk and shrieked
To feel the cold chin prick its dimples through
With winter from the hills, may throw i' the lap
Of the eldest, (who has learnt to drop her lids
To hide some sweetness newer than last year's)
Our book and cry, ... 'Ah you, you care for rhymes;
So here be rhymes to pore on under trees,
When April comes to let you! I've been told
They are not idle as so many are,
But set hearts beating pure as well as fast:
It's yours, the book; I'll write your name in it,—
That so you may not lose, however lost
In poet's lore and charming reverie,
The thought of how your father thought of _you_
In riding from the town.'
To have our books
Appraised by love, associated with love,
While _we_ sit loveless! is it hard, you think?
At least 'tis mournful. Fame, indeed, 'twas said,
Means simply love. It was a man said that.
And then, there's love and love: the love of all
(To risk, in turn, a woman's paradox,)
Is but a small thing to the love of one.
You bid a hungry child be satisfied
With a heritage of many corn-fields: nay,
He says he's hungry,—he would rather have
That little barley-cake you keep from him
While reckoning up his harvests. So with us;
(Here, Romney, too, we fail to generalise!)
We're hungry.
Hungry! but it's pitiful
To wail like unweaned babes and suck our thumbs
Because we're hungry. Who, in all this world,
(Wherein we are haply set to pray and fast,
And learn what good is by its opposite)
Has never hungered? Woe to him who has found
The meal enough! if Ugolino's full,
His teeth have crunched some foul unnatural thing:
For here satiety proves penury
More utterly irremediable. And since
We needs must hunger,—better, for man's love,
Than God's truth! better, for companions sweet,
Than great convictions! let us bear our weights,
Preferring dreary hearths to desert souls.
Well, well! they say we're envious, we who rhyme;
But I, because I am a woman perhaps,
And so rhyme ill, am ill at envying.
I never envied Graham his breadth of style,
Which gives you, with a random smutch or two,
(Near-sighted critics analyse to smutch)
Such delicate perspectives of full life;
Nor Belmore, for the unity of aim
To which he cuts his cedarn poems, fine
As sketchers do their pencils; nor Mark Gage,
For that caressing colour and trancing tone
Whereby you're swept away and melted in
The sensual element, which, with a back wave,
Restores you to the level of pure souls
And leaves you with Plotinus. None of these,
For native gifts or popular applause,
I've envied; but for this,—that when, by chance,
Says some one,—'There goes Belmore, a great man!
He leaves clean work behind him, and requires
No sweeper up of the chips,' ... a girl I know,
Who answers nothing, save with her brown eyes,
Smiles unaware, as if a guardian saint
Smiled in her:—for this, too,—that Gage comes home
And lays his last book's prodigal review
Upon his mother's knees, where, years ago,
He had laid his childish spelling-book and learned
To chirp and peck the letters from her mouth,
As young birds must. 'Well done,' she murmured then,
She will not say it now more wonderingly;
And yet the last 'Well done' will touch him more,
As catching up to-day and yesterday
In a perfect chord of love; and so, Mark Gage.
I envy you your mother!—and you, Graham,
Because you have a wife who loves you so,
She half forgets, at moments, to be proud
Of being Graham's wife, until a friend observes,
'The boy here, has his father's massive brow,
Done small in wax ... if we push back the curls.'
Who loves _me_? Dearest father,—mother sweet,—
I speak the names out sometimes by myself,
And make the silence shiver: they sound strange,
As Hindostanee to an Ind-born man
Accustomed many years to English speech;
Or lovely poet-words grown obsolete,
Which will not leave off singing. Up in heaven
I have my father,—with my mother's face
Beside him in a blotch of heavenly light;
No more for earth's familiar, household use,
No more! The best verse written by this hand,
Can never reach them where they sit, to seem
Well-done to _them_. Death quite unfellows us,
Sets dreadful odds betwixt the live and dead,
And makes us part as those at Babel did,
Through sudden ignorance of a common tongue.
A living Cæsar would not dare to play
At bowls, with such as my dead father is.
And yet, this may be less so than appears,
This change and separation. Sparrows five
For just two farthings, and God cares for each.
If God is not too great for little cares,
Is any creature, because gone to God?
I've seen some men, veracious, nowise mad,
Who have thought or dreamed, declared and testified,
They've heard the Dead a-ticking like a clock
Which strikes the hours of the eternities,
Beside them, with their natural ears,—and known
That human spirits feel the human way,
And hate the unreasoning awe which waves them off
From possible communion. It may be.
At least, earth separates as well as heaven.
For instance, I have not seen Romney Leigh
Full eighteen months ... add six, you get two years.
They say he's very busy with good works,—
Has parted Leigh Hall into almshouses.
He made an almshouse of his heart one day,
Which ever since is loose upon the latch
For those who pull the string.—I never did.
It always makes me sad to go abroad;
And now I'm sadder that I went to-night
Among the lights and talkers at Lord Howe's.
His wife is gracious, with her glossy braids,
And even voice, and gorgeous eyeballs, calm
As her other jewels. If she's somewhat cold,
Who wonders, when her blood has stood so long
In the ducal reservoir she calls her line
By no means arrogantly? she's not proud;
Not prouder than the swan is of the lake
He has always swum in;—'tis her element,
And so she takes it with a natural grace,
Ignoring tadpoles. She just knows, perhaps,
There _are_ men, move on without outriders,
Which isn't her fault. Ah, to watch her face,
When good Lord Howe expounds his theories
Of social justice and equality—
'Tis curious, what a tender, tolerant bend
Her neck takes: for she loves him, likes his talk,
'Such clever talk—that dear, odd Algernon!'
She listens on, exactly as if he talked
Some Scandinavian myth of Lemures,
Too pretty to dispute, and too absurd.
She's gracious to me as her husband's friend,
And would be gracious, were I not a Leigh,
Being used to smile just so, without her eyes,
On Joseph Strangways, the Leeds mesmerist,
And Delia Dobbs, the lecturer from 'the States'
Upon the 'Woman's question.' Then, for him,
I like him ... he's my friend. And all the rooms
Were full of crinkling silks that swept about
The fine dust of most subtle courtesies.
What then?—why then, we come home to be sad.
How lovely One I love not, looked to-night!
She's very pretty, Lady Waldemar.
Her maid must use both hands to twist that coil
Of tresses, then be careful lest the rich
Bronze rounds should slip:—she missed, though, a grey hair,
A single one,—I saw it; otherwise
The woman looked immortal. How they told,
Those alabaster shoulders and bare breasts,
On which the pearls, drowned out of sight in milk,
Were lost, excepting for the ruby-clasp!
They split the amaranth velvet-boddice down
To the waist, or nearly, with the audacious press
Of full-breathed beauty. If the heart within
Were half as white!—but, if it were, perhaps
The breast were closer covered, and the sight
Less aspectable, by half, too.
I heard
The young man with the German student's look—
A sharp face, like a knife in a cleft stick,
Which shot up straight against the parting line
So equally dividing the long hair,—
Say softly to his neighbour, (thirty-five
And mediæval) 'Look that way, Sir Blaise.
She's Lady Waldemar—to the left,—in red—
Whom Romney Leigh, our ablest man just now,
Is soon about to marry.'
Then replied
Sir Blaise Delorme, with quiet, priestlike voice,
Too used to syllable damnations round
To make a natural emphasis worth while:
'Is Leigh your ablest man? the same, I think,
Once jilted by a recreant pretty maid
Adopted from the people? Now, in change,
He seems to have plucked a flower from the other side
Of the social hedge,'
'A flower, a flower,' exclaimed
My German student,—his own eyes full-blown
Bent on her. He was twenty, certainly.
Sir Blaise resumed with gentle arrogance,
As if he had dropped his alms into a hat,
And had the right to counsel,—'My young friend,
I doubt your ablest man's ability
To get the least good or help meet for him,
For pagan phalanstery or Christian home,
From such a flowery creature,'
'Beautiful!'
My student murmured, rapt,—'Mark how she stirs!
Just waves her head, as if a flower indeed,
Touched far off by the vain breath of our talk.'
At which that bilious Grimwald, (he who writes
For the Renovator) who had seemed absorbed
Upon the table-book of autographs,
(I dare say mentally he crunched the bones
Of all those writers, wishing them alive
To feel his tooth in earnest) turned short round
With low carnivorous laugh,—'A flower, of course!
She neither sews nor spins,—and takes no thought
Of her garments ... falling off.'
The student flinched,
Sir Blaise, the same; then both, drawing back their chairs
As if they spied black-beetles on the floor,
Pursued their talk, without a word being thrown
To the critic.
Good Sir Blaise's brow is high
And noticeably narrow: a strong wind,
You fancy, might unroof him suddenly,
And blow that great top attic off his head
So piled with feudal relics. You admire
His nose in profile, though you miss his chin;
But, though you miss his chin, you seldom miss
His golden cross worn innermostly, (carved
For penance, by a saintly Styrian monk
Whose flesh was too much with him,) slipping through
Some unaware unbuttoned casualty
Of the under-waistcoat. With an absent air
Sir Blaise sate fingering it and speaking low,
While I, upon the sofa, heard it all.
'My dear young friend, if we could bear our eyes
Like blessedest St. Lucy, on a plate,
They would not trick us into choosing wives,
As doublets, by the colour. Otherwise
Our fathers chose,—and therefore, when they had hung
Their household keys about a lady's waist,
The sense of duty gave her dignity:
She kept her bosom holy to her babes;
And, if a moralist reproved her dress,
'Twas, 'Too much starch!'—and not, 'Too little lawn!''
'Now, pshaw!' returned the other in a heat,
A little fretted by being called 'young friend,'
Or so I took it,—'for St. Lucy's sake,
If she's the saint to curse by, let us leave
Our fathers,—plagued enough about our sons!'
(He stroked his beardless chin) 'yes, plagued, sir, plagued:
The future generations lie on us
As heavy as the nightmare of a seer;
Our meat and drink grow painful prophecy:
I ask you,—have we leisure, if we liked,
To hollow out our weary hands to keep
Your intermittent rushlight of the past
From draughts in lobbies? Prejudice of sex,
And marriage-laws ... the socket drops them through
While we two speak,—however may protest
Some over-delicate nostrils, like your own,
'Gainst odours thence arising.'
'You are young,'
Sir Blaise objected.
'If I am,' he said
With fire,—'though somewhat less so than I seem,
The young run on before, and see the thing
That's coming. Reverence for the young, I cry.
In that new church for which the world's near ripe,
You'll have the younger in the Elder's chair,
Presiding with his ivory front of hope
O'er foreheads clawed by cruel carrion-birds
Of life's experience.'
'Pray your blessing, sir,'
Sir Blaise replied good-humouredly,—'I plucked
A silver hair this morning from my beard,
Which left me your inferior. Would I were
Eighteen, and worthy to admonish you!
If young men of your order run before
To see such sights as sexual prejudice
And marriage-law dissolved,—in plainer words,
A general concubinage expressed
In a universal pruriency,—the thing
Is scarce worth running fast for, and you'd gain
By loitering with your elders.'
'Ah,' he said,
'Who, getting to the top of Pisgah-hill,
Can talk with one at bottom of the view,
To make it comprehensible? Why, Leigh
Himself, although our ablest man, I said,
Is scarce advanced to see as far as this,
Which some are: he takes up imperfectly
The social question—by one handle—leaves
The rest to trail. A Christian socialist,
Is Romney Leigh, you understand.'
'Not I.
I disbelieve in Christian-pagans, much
As you in women-fishes. If we mix
Two colours, we lose both, and make a third
Distinct from either. Mark you! to mistake
A colour is the sign of a sick brain,
And mine, I thank the saints, is clear and cool:
A neutral tint is here impossible.
The church,—and by the church, I mean, of course,
The catholic, apostolic, mother-church,—
Draws lines as plain and straight as her own wall;
Inside of which, are Christians, obviously,
And outside ... dogs.'
'We thank you. Well I know
The ancient mother-church would fain still bite,
For all her toothless gums,—as Leigh himself
Would fain be a Christian still, for all his wit;
Pass that; you two may settle it, for me.
You're slow in England. In a month I learnt
At Göttingen, enough philosophy
To stock your English schools for fifty years;
Pass that, too. Here, alone, I stop you short,
—Supposing a true man like Leigh could stand
Unequal in the stature of his life
To the height of his opinions. Choose a wife
Because of a smooth skin?—not he, not he!
He'd rail at Venus' self for creaking shoes,
Unless she walked his way of righteousness:
And if he takes a Venus Meretrix,
(No imputation on the lady there)
Be sure that, by some sleight of Christian art,
He has metamorphosed and converted her
To a Blessed Virgin.'
'Soft!' Sir Blaise drew breath
As if it hurt him,—'Soft! no blasphemy,
I pray you!'
'The first Christians did the thing;
Why not the last?' asked he of Göttingen,
With just that shade of sneering on the lip,
Compensates for the lagging of the beard,—
'And so the case is. If that fairest fair
Is talked of as the future wife of Leigh,
She's talked of, too, at least as certainly,
As Leigh's disciple. You may find her name
On all his missions and commissions, schools,
Asylums, hospitals,—he has had her down,
With other ladies whom her starry lead
Persuaded from their spheres, to his country-place
In Shropshire, to the famed phalanstery
At Leigh Hall, christianised from Fourier's own,
(In which he has planted out his sapling stocks
Of knowledge into social nurseries)
And there, they say, she has tarried half a week,
And milked the cows, and churned, and pressed the curd,
And said 'my sister' to the lowest drab
Of all the assembled castaways; such girls!
Ay, sided with them at the washing-tub—
Conceive, Sir Blaise, those naked perfect arms,
Round glittering arms, plunged elbow-deep in suds,
Like wild swans hid in lilies all a-shake.'
Lord Howe came up. 'What, talking poetry
So near the image of the unfavouring Muse?
That's you, Miss Leigh: I've watched you half an hour,
Precisely as I watched the statue called
A Pallas in the Vatican;—you mind
The face, Sir Blaise?—intensely calm and sad,
As wisdom cut it off from fellowship,—
But _that_ spoke louder. Not a word from _you_!
And these two gentlemen were bold, I marked,
And unabashed by even your silence.'
'Ah,'
Said I, 'my dear Lord Howe, you shall not speak
To a printing woman who has lost her place,
(The sweet safe corner of the household fire
Behind the heads of children) compliments,
As if she were a woman. We who have clipt
The curls before our eyes, may see at least
As plain as men do: speak out, man to man;
No compliments, beseech you.'
'Friend to friend,
Let that be. We are sad to-night, I saw,
(—Good night, Sir Blaise! Ah, Smith—he has slipped away)
I saw you across the room, and stayed, Miss Leigh,
To keep a crowd of lion-hunters off,
With faces toward your jungle. There were three;
A spacious lady, five feet ten and fat,
Who has the devil in her (and there's room)
For walking to and fro upon the earth,
From Chipewa to China; she requires
Your autograph upon a tinted leaf
'Twixt Queen Pomare's and Emperor Soulouque's;
Pray give it; she has energies, though fat:
For me, I'd rather see a rick on fire
Than such a woman angry. Then a youth
Fresh from the backwoods, green as the underboughs,
Asks modestly, Miss Leigh, to kiss your shoe,
And adds, he has an epic, in twelve parts,
Which when you've read, you'll do it for his boot,—
All which I saved you, and absorb next week
Both manuscript and man,—because a lord
Is still more potent than a poetess,
With any extreme republican. Ah, ah,
You smile at last, then.'
'Thank you.'
'Leave the smile,
I'll lose the thanks for 't,—ay, and throw you in
My transatlantic girl, with golden eyes,
That draw you to her splendid whiteness, as
The pistil of a water-lily draws,
Adust with gold. Those girls across the sea
Are tyrannously pretty,—and I swore
(She seemed to me an innocent, frank girl)
To bring her to you for a woman's kiss,
Not now, but on some other day or week:
—We'll call it perjury; I give her up.'
'No, bring her.'
'Now,' said he, 'you make it hard
To touch such goodness with a grimy palm.
I thought to tease you well, and fret you cross,
And steel myself, when rightly vexed with you,
For telling you a thing to tease you more.'
'Of Romney?'
'No, no; nothing worse,' he cried,
'Of Romney Leigh, than what is buzzed about,—
That _he_ is taken in an eye-trap too,
Like many half as wise. The thing I mean
Refers to you, not him.'
'Refers to me.'
He echoed,—'Me! You sound it like a stone
Dropped down a dry well very listlessly,
By one who never thinks about the toad
Alive at the bottom. Presently perhaps
You'll sound your 'me' more proudly—till I shrink.'
'Lord Howe's the toad, then, in this question?'
'Brief,
We'll take it graver. Give me sofa-room,
And quiet hearing. You know Eglinton,
John Eglinton, of Eglinton in Kent?'
'Is _he_ the toad?—he's rather like the snail;
Known chiefly for the house upon his back:
Divide the man and house—you kill the man;
That's Eglinton of Eglinton, Lord Howe.'
He answered grave. 'A reputable man,
An excellent landlord of the olden stamp,
If somewhat slack in new philanthropies;
Who keeps his birthdays with a tenants' dance,
Is hard upon them when they miss the church
Or keep their children back from catechism,
But not ungentle when the aged poor
Pick sticks at hedge-sides; nay, I've heard him say,
'The old dame has a twinge because she stoops:
'That's punishment enough for felony.''
'O tender-hearted landlord! May I take
My long lease with him, when the time arrives
For gathering winter-<DW19>s!'
'He likes art,
Buys books and pictures ... of a certain kind;
Neglects no patent duty; a good son'....
'To a most obedient mother. Born to wear
His father's shoes, he wears her husband's too:
Indeed, I've heard it's touching. Dear Lord Howe,
You shall not praise _me_ so against your heart,
When I'm at worst for praise and <DW19>s.'
'Be
Less bitter with me, for ... in short,' he said,
'I have a letter, which he urged me so
To bring you ... I could scarcely choose but yield;
Insisting that a new love passing through
The hand of an old friendship, caught from it
Some reconciling perfume.'
'Love, you say?
My lord, I cannot love. I only find
The rhymes for love,—and that's not love, my lord.
Take back your letter.'
'Pause: you'll read it first?'
'I will not read it: it is stereotyped;
The same he wrote to,—anybody's name,—
Anne Blythe, the actress, when she had died so true,
A duchess fainted in a private box:
Pauline, the dancer, after the great _pas_,
In which her little feet winked overhead
Like other fire-flies, and amazed the pit:
Or Baldinacci, when her F in alt
Had touched the silver tops of heaven itself
With such a pungent soul-dart, even the Queen
Laid softly, each to each, her white-gloved palms,
And sighed for joy: or else (I thank your friend)
Aurora Leigh,—when some indifferent rhymes,
Like those the boys sang round the holy ox
On Memphis-road, have chanced, perhaps, to set
Our Apis-public lowing. Oh, he wants,
Instead of any worthy wife at home,
A star upon his stage of Eglinton!
Advise him that he is not overshrewd
In being so little modest: a dropped star
Makes bitter waters, says a Book I've read,—
And there's his unread letter.'
'My dear friend,'
Lord Howe began....
In haste I tore the phrase.
'You mean your friend of Eglinton, or me?'
'I mean you, you,' he answered with some fire.
'A happy life means prudent compromise;
The tare runs through the farmer's garnered sheaves;
But though the gleaner's apron holds pure wheat,
We count her poorer. Tare with wheat, we cry,
And good with drawbacks. You, you love your art,
And, certain of vocation, set your soul
On utterance. Only, ... in this world we have made,
(They say God made it first, but, if He did,
'Twas so long since, ... and, since, we have spoiled it so,
He scarce would know it, if He looked this way,
From hells we preach of, with the flames blown out,)
In this bad, twisted, topsy-turvy world,
Where all the heaviest wrongs get uppermost,—
In this uneven, unfostering England here,
Where ledger-strokes and sword-strokes count indeed,
But soul-strokes merely tell upon the flesh
They strike from,—it is hard to stand for art,
Unless some golden tripod from the sea
Be fished up, by Apollo's divine chance,
To throne such feet as yours, my prophetess,
At Delphi. Think,—the god comes down as fierce
As twenty bloodhounds! shakes you, strangles you,
Until the oracular shriek shall ooze in froth!
At best it's not all ease,—at worst too hard:
A place to stand on is a 'vantage gained,
And here's your tripod. To be plain, dear friend,
You're poor, except in what you richly give;
You labour for your own bread painfully,
Or ere you pour our wine. For art's sake, pause.'
I answered slow,—as some wayfaring man,
Who feels himself at night too far from home,
Makes stedfast face against the bitter wind.
'Is art so less a thing than virtue is,
That artists first must cater for their ease
Or ever they make issue past themselves
To generous use? alas, and is it so,
That we, who would be somewhat clean, must sweep
Our ways as well as walk them, and no friend
Confirm us nobly,—'Leave results to God,
But you, be clean?' What! 'prudent compromise
Makes acceptable life,' you say instead,
You, you, Lord Howe?—in things indifferent, well.
For instance, compromise the wheaten bread
For rye, the meat for lentils, silk for serge,
And sleep on down, if needs, for sleep on straw;
But there, end compromise. I will not bate
One artist-dream, on straw or down, my lord,
Nor pinch my liberal soul, though I be poor,
Nor cease to love high, though I live thus low.'
So speaking, with less anger in my voice
Than sorrow, I rose quickly to depart;
While he, thrown back upon the noble shame
Of such high-stumbling natures, murmured words,
The right words after wrong ones. Ah, the man
Is worthy, but so given to entertain
Impossible plans of superhuman life,—
He sets his virtues on so raised a shelf,
To keep them at the grand millennial height,
He has to mount a stool to get at them;
And, meantime, lives on quite the common way,
With everybody's morals.
As we passed,
Lord Howe insisting that his friendly arm
Should oar me across the sparkling brawling stream
Which swept from room to room,—we fell at once
On Lady Waldemar. 'Miss Leigh,' she said,
And gave me such a smile, so cold and bright,
As if she tried it in a 'tiring glass
And liked it; 'all to-night I've strained at you,
As babes at baubles held up out of reach
By spiteful nurses, ('Never snatch,' they say,)
And there you sate, most perfectly shut in
By good Sir Blaise and clever Mister Smith,
And then our dear Lord Howe! at last, indeed,
I almost snatched. I have a world to speak
About your cousin's place in Shropshire, where
I've been to see his work ... our work,—you heard
I went?... and of a letter, yesterday,
In which, if I should read a page or two,
You might feel interest, though you're locked of course
In literary toil.—You'll like to hear
Your last book lies at the phalanstery,
As judged innocuous for the elder girls
And younger women who still care for books.
We all must read, you see, before we live:
But slowly the ineffable light comes up,
And, as it deepens, drowns the written word,—
So said your cousin, while we stood and felt
A sunset from his favourite beech-tree seat:
He might have been a poet if he would,
But then he saw the higher thing at once,
And climbed to it. I think he looks well now,
Has quite got over that unfortunate ...
Ah, ah ... I know it moved you. Tender-heart!
You took a liking to the wretched girl.
Perhaps you thought the marriage suitable,
Who knows? a poet hankers for romance,
And so on. As for Romney Leigh, 'tis sure
He never loved her,—never. By the way,
You have not heard of _her_ ...? quite out of sight,
And out of saving? lost in every sense?'
She might have gone on talking half-an-hour,
And I stood still, and cold, and pale, I think,
As a garden-statue a child pelts with snow
For pretty pastime. Every now and then
I put in 'yes' or 'no,' I scarce knew why;
The blind man walks wherever the dog pulls,
And so I answered. Till Lord Howe broke in;
'What penance takes the wretch who interrupts
The talk of charming women? I, at last,
Must brave it. Pardon, Lady Waldemar!
The lady on my arm is tired, unwell,
And loyally I've promised she shall say
No harder word this evening, than ... goodnight;
The rest her face speaks for her.'—Then we went.
And I breathe large at home. I drop my cloak,
Unclasp my girdle, loose the band that ties
My hair ... now could I but unloose my soul!
We are sepulchred alive in this close world,
And want more room.
The charming woman there—
This reckoning up and writing down her talk
Affects me singularly. How she talked
To pain me! woman's spite!—You wear steel-mail;
A woman takes a housewife from her breast,
And plucks the delicatest needle out
As 'twere a rose, and pricks you carefully
'Neath nails, 'neath eyelids, in your nostrils,—say,
A beast would roar so tortured,—but a man,
A human creature, must not, shall not flinch,
No, not for shame.
What vexes, after all,
Is just that such as she, with such as I,
Knows how to vex. Sweet heaven, she takes me up
As if she had fingered me and dog-eared me
And spelled me by the fireside, half a life!
She knows my turns, my feeble points.—What then?
The knowledge of a thing implies the thing;
Of course, she found _that_ in me, she saw _that_,
Her pencil underscored _this_ for a fault,
And I, still ignorant. Shut the book up! close!
And crush that beetle in the leaves.
O heart,
At last we shall grow hard too, like the rest,
And call it self-defence because we are soft.
And after all, now, ... why should I be pained,
That Romney Leigh, my cousin, should espouse
This Lady Waldemar? And, say, she held
Her newly-blossomed gladness in my face, ...
'Twas natural surely, if not generous,
Considering how, when winter held her fast,
I helped the frost with mine, and pained her more
Than she pains me. Pains me!—but wherefore pained?
'Tis clear my cousin Romney wants a wife,—
So, good!—The man's need of the woman, here,
Is greater than the woman's of the man,
And easier served; for where the man discerns
A sex, (ah, ah, the man can generalise,
Said he) we see but one, ideally
And really: where we yearn to lose ourselves
And melt like white pearls in another's wine,
He seeks to double himself by what he loves,
And make his drink more costly by our pearls.
At board, at bed, at work, and holiday,
It is not good for man to be alone,—
And that's his way of thinking, first and last;
And thus my cousin Romney wants a wife.
But then my cousin sets his dignity
On personal virtue. If he understands
By love, like others, self-aggrandisement,
It is that he may verily be great
By doing rightly and kindly. Once he thought,
For charitable ends set duly forth
In Heaven's white judgment-book, to marry ... ah,
We'll call her name Aurora Leigh, although
She's changed since then!—and once, for social ends,
Poor Marian Erle, my sister Marian Erle,
My woodland sister, sweet maid Marian,
Whose memory moans on in me like the wind
Through ill-shut casements, making me more sad
Than ever I find reasons for. Alas,
Poor pretty plaintive face, embodied ghost,
He finds it easy, then, to clap thee off
From pulling at his sleeve and book and pen,—
He locks thee out at night into the cold,
Away from butting with thy horny eyes
Against his crystal dreams,—that, now, he's strong
To love anew? that Lady Waldemar
Succeeds my Marian?
After all, why not?
He loved not Marian, more than once he loved
Aurora. If he loves, at last, that Third,
Albeit she prove as slippery as spilt oil
On marble floors, I will not augur him
Ill luck for that. Good love, howe'er ill-placed,
Is better for a man's soul in the end,
Than if he loved ill what deserves love well.
A pagan, kissing, for a step of Pan,
The wild-goat's hoof-print on the loamy down,
Exceeds our modern thinker who turns back
The strata ... granite, limestone, coal, and clay,
Concluding coldly with, 'Here's law! Where's God?'
And then at worse,—if Romney loves her not,—
At worst,—if he's incapable of love,
Which may be—then indeed, for such a man
Incapable of love, she's good enough;
For she, at worst too, is a woman still
And loves him ... as the sort of woman can.
My loose long hair began to burn and creep,
Alive to the very ends, about my knees:
I swept it backward as the wind sweeps flame,
With the passion of my hands. Ah, Romney laughed
One day ... (how full the memories come up!)
'—Your Florence fire-flies live on in your hair,'
He said, 'it gleams so.' Well, I wrung them out,
My fire-flies; made a knot as hard as life,
Of those loose, soft, impracticable curls,
And then sat down and thought.... 'She shall not think
Her thought of me,'—and drew my desk and wrote.
'Dear Lady Waldemar, I could not speak
With people round me, nor can sleep to-night
And not speak, after the great news I heard
Of you and of my cousin. May you be
Most happy; and the good he meant the world,
Replenish his own life. Say what I say,
And let my word be sweeter for your mouth,
As you are _you_ ... I only Aurora Leigh.'
That's quiet, guarded! though she hold it up
Against the light, she'll not see through it more
Than lies there to be seen. So much for pride;
And now for peace, a little! Let me stop
All writing back.... 'Sweet thanks, my sweetest friend,
'You've made more joyful my great joy itself,'
—No, that's too simple! she would twist it thus,
'My joy would still be as sweet as thyme in drawers,
However shut up in the dark and dry;
But violets, aired and dewed by love like yours,
Out-smell all thyme! we keep that in our clothes,
But drop the other down our bosoms, till
They smell like' ... ah, I see her writing back
Just so. She'll make a nosegay of her words,
And tie it with blue ribbons at the end
To suit a poet;—pshaw!
And then we'll have
The call to church; the broken, sad, bad dream
Dreamed out at last; the marriage-vow complete
With the marriage-breakfast; praying in white gloves,
Drawn off in haste for drinking pagan toasts
In somewhat stronger wine than any sipped
By gods, since Bacchus had his way with grapes.
A postscript stops all that, and rescues me.
'You need not write. I have been overworked,
And think of leaving London, England even,
And hastening to get nearer to the sun,
Where men sleep better. So, adieu.'—I fold
And seal,—— and now I'm out of all the coil;
I breathe now; I spring upward like a branch,
A ten-years school-boy with a crooked stick
May pull down to his level, in search of nuts,
But cannot hold a moment. How we twang
Back on the blue sky, and assert our height,
While he stares after! Now, the wonder seems
That I could wrong myself by such a doubt.
We poets always have uneasy hearts;
Because our hearts, large-rounded as the globe,
Can turn but one side to the sun at once.
We are used to dip our artist-hands in gall
And potash, trying potentialities
Of alternated colour, till at last
We get confused, and wonder for our skin
How nature tinged it first. Well—here's the true
Good flesh-colour; I recognise my hand,—
Which Romney Leigh may clasp as just a friend's,
And keep his clean.
And now, my Italy.
Alas, if we could ride with naked souls
And make no noise and pay no price at all,
I would have seen thee sooner, Italy,—For
still I have heard thee crying through my life,
Thou piercing silence of extatic graves,
Men call that name!
But even a witch, to-day,
Must melt down golden pieces in the nard
Wherewith to anoint her broomstick ere she rides;
And poets evermore are scant of gold,
And, if they find a piece behind the door,
It turns by sunset to a withered leaf.
The Devil himself scarce trusts his patented
Gold-making art to any who make rhymes,
But culls his Faustus from philosophers
And not from poets. 'Leave my Job,' said God;
And so, the Devil leaves him without pence,
And poverty proves, plainly, special grace.
In these new, just, administrative times
Men clamour for an order of merit. Why?
Here's black bread on the table, and no wine!
At least I am a poet in being poor;
Thank God. I wonder if the manuscript
Of my long poem, if 'twere sold outright,
Would fetch enough to buy me shoes, to go
A-foot, (thrown in, the necessary patch
For the other side the Alps)? it cannot be:
I fear that I must sell this residue
Of my father's books; although the Elzevirs
Have fly-leaves over-written by his hand,
In faded notes as thick and fine and brown
As cobwebs on a tawny monument
Of the old Greeks—_conferenda hæc cum his_—
_Corruptè citat_—_lege potiùs_,
And so on, in the scholar's regal way
Of giving judgment on the parts of speech,
As if he sate on all twelve thrones up-piled,
Arraigning Israel. Ay, but books and notes
Must go together. And this Proclus too,
In quaintly dear contracted Grecian types,
Fantastically crumpled, like his thoughts
Which would not seem too plain; you go round twice
For one step forward, then you take it back,
Because you're somewhat giddy! there's the rule
For Proclus. Ah, I stained this middle leaf
With pressing in't my Florence iris-bell,
Long stalk and all: my father chided me
For that stain of blue blood,—I recollect
The peevish turn his voice took,—'Silly girls,
Who plant their flowers in our philosophy
To make it fine, and only spoil the book!
No more of it, Aurora.' Yes—no more!
Ah, blame of love, that's sweeter than all praise
Of those who love not! 'tis so lost to me,
I cannot, in such beggared life, afford
To lose my Proclus. Not for Florence, even.
The kissing Judas, Wolff, shall go instead,
Who builds us such a royal book as this
To honour a chief-poet, folio-built,
And writes above, 'The house of Nobody:'
Who floats in cream, as rich as any sucked
From Juno's breasts, the broad Homeric lines,
And, while with their spondaic prodigious mouths
They lap the lucent margins as babe-gods,
Proclaims them bastards. Wolff's an atheist;
And if the Iliad fell out, as he says,
By mere fortuitous concourse of old songs,
We'll guess as much, too, for the universe.
That Wolff, those Platos: sweep the upper shelves
As clean as this, and so I am almost rich,
Which means, not forced to think of being poor
In sight of ends. To-morrow: no delay.
I'll wait in Paris till good Carrington
Dispose of such, and, having chaffered for
My book's price with the publisher, direct
All proceeds to me. Just a line to ask
His help.
And now I come, my Italy,
My own hills! Are you 'ware of me, my hills,
How I burn toward you? do you feel to-night
The urgency and yearning of my soul,
As sleeping mothers feel the sucking babe
And smile?—Nay, not so much as when, in heat,
Vain lightnings catch at your inviolate tops,
And tremble while ye are stedfast. Still, ye go
Your own determined, calm, indifferent way
Toward sunrise, shade by shade, and light by light;
Of all the grand progression nought left out;
As if God verily made you for yourselves,
And would not interrupt your life with ours.
SIXTH BOOK.
THE English have a scornful insular way
Of calling the French light. The levity
Is in the judgment only, which yet stands;
For say a foolish thing but oft enough,
(And here's the secret of a hundred creeds,—
Men get opinions as boys learn to spell,
By re-iteration chiefly) the same thing
Shall pass at last for absolutely wise,
And not with fools exclusively. And so,
We say the French are light, as if we said
The cat mews, or the milch-cow gives us milk:
Say rather, cats are milked, and milch-cows mew;
For what is lightness but inconsequence,
Vague fluctuation 'twixt effect and cause,
Compelled by neither? Is a bullet light,
That dashes from the gun-mouth, while the eye
Winks, and the heart beats one, to flatten itself
To a wafer on the white speck on a wall
A hundred paces off? Even so direct,
So sternly undivertible of aim,
Is this French people.
All, idealists
Too absolute and earnest, with them all
The idea of a knife cuts real flesh;
And still, devouring the safe interval
Which Nature placed between the thought and act,
With those too fiery and impatient souls,
They threaten conflagration to the world
And rush with most unscrupulous logic on
Impossible practice. Set your orators
To blow upon them with loud windy mouths
Through watchword phrases, jest or sentiment,
Which drive our burley brutal English mobs
Like so much chaff, whichever way they blow,—
This light French people will not thus be driven.
They turn indeed; but then they turn upon
Some central pivot of their thought and choice,
And veer out by the force of holding fast.
—That's hard to understand, for Englishmen
Unused to abstract questions, and untrained
To trace the involutions, valve by valve,
In each orbed bulb-root of a general truth,
And mark what subtly fine integument
Divides opposed compartments. Freedom's self
Comes concrete to us, to be understood,
Fixed in a feudal form incarnately
To suit our ways of thought and reverence,
The special form, with us, being still the thing.
With us, I say, though I'm of Italy
By mother's birth and grave, by father's grave
And memory; let it be,—a poet's heart
Can swell to a pair of nationalities,
However ill-lodged in a woman's breast.
And so I am strong to love this noble France,
This poet of the nations, who dreams on
And wails on (while the household goes to wreck)
For ever, after some ideal good,—
Some equal poise of sex, some unvowed love
Inviolate, some spontaneous brotherhood,
Some wealth, that leaves none poor and finds none tired,
Some freedom of the many, that respects
The wisdom of the few. Heroic dreams!
Sublime, to dream so; natural, to wake:
And sad, to use such lofty scaffoldings,
Erected for the building of a church,
To build instead, a brothel ... or a prison—
May God save France!
However she have sighed
Her great soul up into a great man's face,
To flush his temples out so gloriously
That few dare carp at Cæsar for being bald,
What then?—this Cæsar represents, not reigns,
And is no despot, though twice absolute;
This Head has all the people for a heart;
This purple's lined with the democracy,—
Now let him see to it! for a rent within
Must leave irreparable rags without.
A serious riddle: find such anywhere
Except in France; and when it's found in France,
Be sure to read it rightly. So, I mused
Up and down, up and down, the terraced streets,
The glittering boulevards, the white colonnades
Of fair fantastic Paris who wears boughs
Like plumes, as if man made them,—tossing up
Her fountains in the sunshine from the squares,
As dice i' the game of beauty, sure to win;
Or as she blew the down-balls of her dreams,
And only waited for their falling back,
To breathe up more, and count her festive hours.
The city swims in verdure, beautiful
As Venice on the waters, the sea-swan.
What bosky gardens, dropped in close-walled courts,
As plums in ladies' laps, who start and laugh:
What miles of streets that run on after trees,
Still carrying the necessary shops,
Those open caskets, with the jewels seen!
And trade is art, and art's philosophy,
In Paris. There's a silk, for instance, there,
As worth an artist's study for the folds,
As that bronze opposite! nay, the bronze has faults;
Art's here too artful,—conscious as a maid,
Who leans to mark her shadow on the wall
Until she lose a 'vantage in her step.
Yet Art walks forward, and knows where to walk:
The artists also, are idealists,
Too absolute for nature, logical
To austerity in the application of
The special theory: not a soul content
To paint a crooked pollard and an ass,
As the English will, because they find it so,
And like it somehow.—Ah, the old Tuileries
Is pulling its high cap down on its eyes,
Confounded, conscience-stricken, and amazed
By the apparition of a new fair face
In those devouring mirrors. Through the grate,
Within the gardens, what a heap of babes,
Swept up like leaves beneath the chestnut-trees,
From every street and alley of the town,
By the ghosts perhaps, that blow too bleak this way
A-looking for their heads! Dear pretty babes;
I'll wish them luck to have their ball-play out
Before the next change comes.—And, farther on,
What statues, poised upon their columns fine,
As if to stand a moment were a feat,
Against that blue! What squares! what breathing-room
For a nation that runs fast,—ay, runs against
The dentist's teeth at the corner, in pale rows,
Which grin at progress in an epigram.
I walked the day out, listening to the chink
Of the first Napoleon's dry bones, as they lay
In his second grave beneath the golden dome
That caps all Paris like a bubble. 'Shall
These dry bones live,' thought Louis Philippe once,
And lived to know. Herein is argument
For kings and politicians, but still more
For poets, who bear buckets to the well,
Of ampler draught.
These crowds are very good
For meditation, (when we are very strong)
Though love of beauty makes us timorous,
And draws us backward from the coarse town-sights
To count the daisies upon dappled fields,
And hear the streams bleat on among the hills
In innocent and indolent repose;
While still with silken elegiac thoughts
We wind out from us the distracting world,
And die into the chrysalis of a man,
And leave the best that may, to come of us,
In some brown moth. Be, rather, bold, and bear
To look into the swarthiest face of things,
For God's sake who has made them.
Seven days' work;
The last day shutting 'twixt its dawn and eve,
The whole work bettered, of the previous six!
Since God collected and resumed in man
The firmaments, the strata, and the lights,
Fish, fowl, and beast, and insect,—all their trains
Of various life caught back upon His arm,
Reorganised, and constituted MAN,
The microcosm, the adding up of works;
Within whose fluttering nostrils, then, at last,
Consummating Himself, the Maker sighed,
As some strong winner at the foot-race sighs
Touching the goal.
Humanity is great;
And, if I would not rather pore upon
An ounce of common, ugly, human dust,
An artisan's palm, or a peasant's brow,
Unsmooth, ignoble, save to me and God,
Than track old Nilus to his silver roots,
And wait on all the changes of the moon
Among the mountain-peaks of Thessaly,
(Until her magic crystal round itself
For many a witch to see in)—set it down
As weakness,—strength by no means. How is this,
That men of science, osteologists
And surgeons, beat some poets, in respect
For nature,—count nought common or unclean,
Spend raptures upon perfect specimens
Of indurated veins, distorted joints,
Or beautiful new cases of curved spine;
While we, we are shocked at nature's falling off,
We dare to shrink back from her warts and blains,
We will not, when she sneezes, look at her,
Not even to say 'God bless her'? That's our wrong;
For that, she will not trust us often with
Her larger sense of beauty and desire,
But tethers us to a lily or a rose
And bids us diet on the dew inside,—
Left ignorant that the hungry beggar-boy
(Who stares unseen against our absent eyes,
And wonders at the gods that we must be,
To pass so careless for the oranges!)
Bears yet a breastful of a fellow-world
To this world, undisparaged, undespoiled,
And (while we scorn him for a flower or two,
As being, Heaven help us, less poetical)
Contains, himself, both flowers and firmaments
And surging seas and aspectable stars,
And all that we would push him out of sight
In order to see nearer. Let us pray
God's grace to keep God's image in repute;
That so, the poet and philanthropist,
(Even I and Romney) may stand side by side,
Because we both stand face to face with men
Contemplating the people in the rough,—
Yet each so follow a vocation,—his
And mine.
I walked on, musing with myself
On life and art, and whether, after all,
A larger metaphysics might not help
Our physics, a completer poetry
Adjust our daily life and vulgar wants,
More fully than the special outside plans,
Phalansteries, material institutes,
The civil conscriptions and lay monasteries
Preferred by modern thinkers, as they thought
The bread of man indeed made all his life,
And washing seven times in the 'People's Baths'
Were sovereign for a people's leprosy,—
Still leaving out the essential prophet's word
That comes in power. On which, we thunder down,
We prophets, poets,—Virtue's in the _word_!
The maker burnt the darkness up with His,
To inaugurate the use of vocal life;
And, plant a poet's word even, deep enough
In any man's breast, looking presently
For offshoots, you have done more for the man,
Than if you dressed him in a broad-cloth coat
And warmed his Sunday potage at your fire.
Yet Romney leaves me....
God! what face is that?
O Romney, O Marian!
Walking on the quays
And pulling thoughts to pieces leisurely,
As if I caught at grasses in a field,
And bit them slow between my absent lips,
And shred them with my hands....
What face is that?
What a face, what a look, what a likeness! Full on mine
The sudden blow of it came down, till all
My blood swam, my eyes dazzled. Then I sprang—
It was as if a meditative man
Were dreaming out a summer afternoon
And watching gnats a-prick upon a pond,
When something floats up suddenly, out there,
Turns over ... a dead face, known once alive—
So old, so new! It would be dreadful now
To lose the sight and keep the doubt of this.
He plunges—ha! he has lost it in the splash.
I plunged—I tore the crowd up, either side,
And rushed on,—forward, forward ... after her.
Her? whom?
A woman sauntered slow, in front,
Munching an apple,—she left off amazed
As if I had snatched it: that's not she, at least.
A man walked arm-linked with a lady veiled,
Both heads dropped closer than the need of talk:
They started; he forgot her with his face,
And she, herself,—and clung to him as if
My look were fatal. Such a stream of folk,
And all with cares and business of their own!
I ran the whole quay down against their eyes;
No Marian; nowhere Marian. Almost, now,
I could call Marian, Marian, with the shriek
Of desperate creatures calling for the Dead.
Where is she, was she? was she anywhere?
I stood still, breathless, gazing, straining out
In every uncertain distance, till, at last,
A gentleman abstracted as myself
Came full against me, then resolved the clash
In voluble excuses,—obviously
Some learned member of the Institute
Upon his way there, walking, for his health,
While meditating on the last 'Discourse;'
Pinching the empty air 'twixt finger and thumb,
From which the snuff being ousted by that shock,
Defiled his snow-white waistcoat, duly pricked
At the button-hole with honourable red;
'Madame, your pardon,'—there, he swerved from me
A metre, as confounded as he had heard
That Dumas would be chosen to fill up
The next chair vacant, by his 'men _in us_.'
Since when was genius found respectable?
It passes in its place, indeed,—which means
The seventh floor back, or else the hospital:
Revolving pistols are ingenious things,
But prudent men (Academicians are)
Scarce keep them in the cupboard, next the prunes.
And so, abandoned to a bitter mirth,
I loitered to my inn. O world, O world,
O jurists, rhymers, dreamers, what you please,
We play a weary game of hide-and-seek!
We shape a figure of our fantasy,
Call nothing something, and run after it
And lose it, lose ourselves too in the search;
Till, clash against us, comes a somebody
Who also has lost something and is lost,
Philosopher against philanthropist,
Academician against poet, man
Against woman, against the living, the dead,—
Then home, with a bad headache and worse jest!
To change the water for my heliotropes
And yellow roses. Paris has such flowers.
But England, also. 'Twas a yellow rose,
By that south window of the little house,
My cousin Romney gathered with his hand
On all my birthdays for me, save the last;
And then I shook the tree too rough, too rough,
For roses to stay after.
Now, my maps.
I must not linger here from Italy
Till the last nightingale is tired of song,
And the last fire-fly dies off in the maize.
My soul's in haste to leap into the sun
And scorch and seethe itself to a finer mood,
Which here, in this chill north, is apt to stand
Too stiffly in former moulds.
That-face persists.
It floats up, it turns over in my mind,
As like to Marian, as one dead is like
The same alive. In very deed a face
And not a fancy, though it vanished so;
The small fair face between the darks of hair,
I used to liken, when I saw her first,
To a point of moonlit, water down a well:
The low brow, the frank space between the eyes,
Which always had the brown pathetic look
Of a dumb creature who had been beaten once,
And never since was easy with the world.
Ah, ah—now I remember perfectly
Those eyes, to-day,—how overlarge they seemed,
As if some patient passionate despair
(Like a coal dropt and forgot on tapestry,
Which slowly burns a widening circle out)
Had burnt them larger, larger. And those eyes
To-day, I do remember, saw me too,
As I saw them, with conscious lids astrain
In recognition. Now, a fantasy,
A simple shade or image of the brain,
Is merely passive, does not retro-act,
Is seen, but sees not.
'Twas a real face,
Perhaps a real Marian.
Which being so,
I ought to write to Romney, 'Marian's here.
Be comforted for Marian.'
My pen fell,
My hands struck sharp together, as hands do
Which hold at nothing. Can I write to _him_
A half truth? can I keep my own soul blind
To the other half, ... the worse? What are our souls,
If still, to run on straight a sober pace
Nor start at every pebble or dead leaf,
They must wear blinkers, ignore facts, suppress
Six tenths of the road? Confront the truth, my soul!
And oh, as truly as that was Marian's face,
The arms of that same Marian clasped a thing
... Not hid so well beneath the scanty shawl,
I cannot name it now for what it was.
A child. Small business has a cast-away
Like Marian, with that crown of prosperous wives,
At which the gentlest she grows arrogant
And says, 'my child.' Who'll find an emerald ring
On a beggar's middle finger, and require
More testimony to convict a thief?
A child's too costly for so mere a wretch;
She filched it somewhere; and it means, with her,
Instead of honour, blessing, ... merely shame.
I cannot write to Romney, 'Here she is,
Here's Marian found! I'll set you on her track:
I saw her here, in Paris, ... and her child.
She put away your love two years ago,
But, plainly, not to starve. You suffered then;
And, now that you've forgot her utterly
As any last year's annual, in whose place
You've planted a thick flowering evergreen,
I choose, being kind, to write and tell you this
To make you wholly easy—she's not dead,
But only ... damned.'
Stop there: I go too fast;
I'm cruel like the rest,—in haste to take
The first stir in the arras for a rat,
And set my barking, biting thoughts upon't.
—A child! what then? Suppose a neighbour's sick
And asked her, 'Marian, carry out my child
In this Spring air,'—I punish her for that?
Or say, the child should hold her round the neck
For good child-reasons, that he liked it so
And would not leave her—she had winning ways—
I brand her therefore, that she took the child?
Not so.
I will not write to Romney Leigh.
For now he's happy,—and she may indeed
Be guilty,—and the knowledge of her fault
Would draggle his smooth time. But I, whose days
Are not so fine they cannot bear the rain,
And who, moreover, having seen her face,
Must see it again, ... _will_ see it, by my hopes
Of one day seeing heaven too. The police
Shall track her, hound her, ferret their own soil;
We'll dig this Paris to its catacombs
But certainly we'll find her, have her out,
And save her, if she will or will not—child
Or no child,—if a child, then one to save!
The long weeks passed on without consequence.
As easy find a footstep on the sand
The morning after spring-tide, as the trace
Of Marian's feet between the incessant surfs
Of this live flood. She may have moved this way,—
But so the star-fish does, and crosses out
The dent of her small shoe. The foiled police
Renounced me; 'Could they find a girl and child,
No other signalment but girl and child?
No data shown, but noticeable eyes
And hair in masses, low upon the brow,
As if it were an iron crown and pressed?
Friends heighten, and suppose they specify:
Why, girls with hair and eyes, are everywhere
In Paris; they had turned me up in vain
No Marian Erle indeed, but certainly
Mathildes, Justines, Victoires, ... or, if I sought
The English, Betsies, Saras, by the score.
They might as well go out into the fields
To find a speckled bean, that's somehow specked,
And somewhere in the pod.'—They left me so.
Shall _I_ leave Marian? have I dreamed a dream?
—I thank God I have found her! I must say
'Thank God,' for finding her, although 'tis true
I find the world more sad and wicked for't.
But she—
I'll write about her, presently;
My hand's a-tremble as I had just caught up
My heart to write with, in the place of it.
At least you'd take these letters to be writ
At sea, in storm!—wait now....
A simple chance
Did all. I could not sleep last night, and, tired
Of turning on my pillow and harder thoughts,
Went out at early morning, when the air
Is delicate with some last starry touch,
To wander through the Market-place of Flowers
(The prettiest haunt in Paris), and make sure
At worst, that there were roses in the world.
So, wandering, musing, with the artist's eye,
That keeps the shade-side of the thing it loves,
Half-absent, whole-observing, while the crowd
Of young vivacious and black-braided heads
Dipped, quick as finches in a blossomed tree,
Among the nosegays, cheapening this and that
In such a cheerful twitter of rapid speech,—
My heart leapt in me, startled by a voice
That slowly, faintly, with long breaths that marked
The interval between the wish and word,
Inquired in stranger's French, 'Would _that_ be much,
That branch of flowering mountain-gorse?'—'So much?
Too much for me, then!' turning the face round
So close upon me, that I felt the sigh
It turned with.
'Marian, Marian!'—face to face—
'Marian! I find you. Shall I let you go?'
I held her two slight wrists with both my hands;
'Ah Marian, Marian, can I let you go?'
—She fluttered from me like a cyclamen,
As white, which, taken in a sudden wind,
Beats on against the palisade.—'Let pass,'
She said at last. 'I will not,' I replied;
'I lost my sister Marian many days,
And sought her ever in my walks and prayers,
And, now I find her ... do we throw away
The bread we worked and prayed for,—crumble it
And drop it, ... to do even so by thee
Whom still I've hungered after more than bread,
My sister Marian?—can I hurt thee, dear?
Then why distrust me? Never tremble so.
Come with me rather, where we'll talk and live,
And none shall vex us. I've a home for you
And me and no one else'....
She shook her head.
'A home for you and me and no one else
Ill-suits one of us: I prefer to such,
A roof of grass on which a flower might spring,
Less costly to me than the cheapest here;
And yet I could not, at this hour, afford
A like home, even. That you offer yours,
I thank you. You are good as heaven itself—
As good as one I knew before.... Farewell.'
I loosed her hands.—'In _his_ name, no farewell!'
(She stood as if I held her.) 'For his sake,
For his sake, Romney's! by the good he meant,
Ay, always! by the love he pressed for once,—
And by the grief, reproach, abandonment,
He took in change'....
'He, Romney! who grieved _him_?
Who had the heart for't? what reproach touched _him_?
Be merciful,—speak quickly.'
'Therefore come,'
I answered with authority,—'I think
We dare to speak such things, and name such names,
In the open squares of Paris!'
Not a word
She said, but, in a gentle humbled way,
(As one who had forgot herself in grief)
Turned round and followed closely where I went,
As if I led her by a narrow plank,
Across devouring waters, step by step,—
And so in silence we walked on a mile.
And then she stopped: her face was white as wax.
'We go much farther?'
'You are ill,' I asked,
'Or tired?'
She looked the whiter for her smile.
'There's one at home,' she said, 'has need of me
By this time,—and I must not let him wait.'
'Not even,' I asked, 'to hear of Romney Leigh?'
'Not even,' she said, 'to hear of Mister Leigh.'
'In that case,' I resumed, 'I go with you,
And we can talk the same thing there as here.
None waits for me: I have my day to spend.'
Her lips moved in a spasm without a sound,—
But then she spoke. 'It shall be as you please;
And better so—'tis shorter seen than told.
And though you will not find me worth your pains,
_That_ even, may be worth some pains to know,
For one as good as you are.'
Then she led
The way, and I, as by a narrow plank
Across devouring waters, followed her,
Stepping by her footsteps, breathing by her breath,
And holding her with eyes that would not slip;
And so, without a word, we walked a mile,
And so, another mile, without a word.
Until the peopled streets being all dismissed,
House-rows and groups all scattered like a flock,
The market-gardens thickened, and the long
White walls beyond, like spiders' outside threads,
Stretched, feeling blindly toward the country-fields
Through half-built habitations and half-dug
Foundations,—intervals of trenchant chalk,
That bite betwixt the grassy uneven turfs
Where goats (vine-tendrils trailing from their mouths)
Stood perched on edges of the cellarage
Which should be, staring as about to leap
To find their coming Bacchus. All the place
Seemed less a cultivation than a waste:
Men work here, only,—scarce begin to live:
All's sad, the country struggling with the town,
Like an untamed hawk upon a strong man's fist,
That beats its wings and tries to get away,
And cannot choose be satisfied so soon
To hop through court-yards with its right foot tied,
The vintage plains and pastoral hills in sight!
We stopped beside a house too high and slim
To stand there by itself, but waiting till
Five others, two on this side, three on that,
Should grow up from the sullen second floor
They pause at now, to build it to a row.
The upper windows partly were unglazed
Meantime,—a meagre, unripe house: a line
Of rigid poplars elbowed it behind,
And, just in front, beyond the lime and bricks
That wronged the grass between it and the road,
A great acacia, with its slender trunk
And overpoise of multitudinous leaves,
(In which a hundred fields might spill their dew
And intense verdure, yet find room enough)
Stood, reconciling all the place with green.
I followed up the stair upon her step.
She hurried upward, shot across a face,
A woman's on the landing,—'How now, now!
Is no one to have holidays but you?
You said an hour, and stay three hours, I think,
And Julie waiting for your betters here?
Why if he had waked, he might have waked, for me.'
—Just murmuring an excusing word she passed
And shut the rest out with the chamber-door,
Myself shut in beside her.
'Twas a room
Scarce larger than a grave, and near as bare;
Two stools, a pallet-bed; I saw the room:
A mouse could find no sort of shelter in't,
Much less a greater secret; curtainless,—
The window fixed you with its torturing eye,
Defying you to take a step apart,
If peradventure you would hide a thing.
I saw the whole room, I and Marian there
Alone.
Alone? She threw her bonnet off,
Then sighing as 'twere sighing the last time,
Approached the bed, and drew a shawl away:
You could not peel a fruit you fear to bruise
More calmly and more carefully than so,—
Nor would you find within, a rosier flushed
Pomegranate—
There he lay, upon his back,
The yearling creature, warm and moist with life
To the bottom of his dimples,—to the ends
Of the lovely tumbled curls about his face;
For since he had been covered over-much
To keep him from the light-glare, both his cheeks
Were hot and scarlet as the first live rose
The shepherd's heart-blood ebbed away into,
The faster for his love. And love was here
As instant! in the pretty baby-mouth,
Shut close as if for dreaming that it sucked;
The little naked feet drawn up the way
Of nestled birdlings; everything so soft
And tender,—to the little holdfast hands,
Which, closing on a finger into sleep,
Had kept the mould of't.
While we stood there dumb,—
For oh, that it should take such innocence
To prove just guilt, I thought, and stood there dumb;
The light upon his eyelids pricked them wide,
And, staring out at us with all their blue,
As half perplexed between the angelhood
He had been away to visit in his sleep,
And our most mortal presence,—gradually
He saw his mother's face, accepting it
In change for heaven itself, with such a smile
As might have well been learnt there,—never moved,
But smiled on, in a drowse of ecstasy,
So happy (half with her and half with heaven)
He could not have the trouble to be stirred,
But smiled and lay there. Like a rose, I said:
As red and still indeed as any rose,
That blows in all the silence of its leaves,
Content, in blowing, to fulfil its life.
She leaned above him (drinking him as wine)
In that extremity of love, 'twill pass
For agony or rapture, seeing that love
Includes the whole of nature, rounding it
To love ... no more,—since more can never be
Than just love. Self-forgot, cast out of self,
And drowning in the transport of the sight,
Her whole pale passionate face, mouth, forehead, eyes,
One gaze, she stood! then, slowly as he smiled,
She smiled too, slowly, smiling unaware,
And drawing from his countenance to hers
A fainter red, as if she watched a flame
And stood in it a-glow. 'How beautiful,'
Said she.
I answered, trying to be cold.
(Must sin have compensations, was my thought,
As if it were a holy thing like grief?
And is a woman to be fooled aside
From putting vice down, with that woman's toy,
A baby?)—— 'Ay! the child is well enough,'
I answered. 'If his mother's palms are clean,
They need be glad, of course, in clasping such:
But if not,—I would rather lay my hand,
Were I she,—on God's brazen altar-bars
Red-hot with burning sacrificial lambs,
Than touch the sacred curls of such a child.'
She plunged her fingers in his clustering locks,
As one who would not be afraid of fire;
And then, with indrawn steady utterance, said,—
'My lamb, my lamb! although, through such as thou,
The most unclean got courage and approach
To God, once,—now they cannot, even with men,
Find grace enough for pity and gentle words.'
'My Marian,' I made answer, grave and sad,
'The priest who stole a lamb to offer him,
Was still a thief. And if a woman steals
(Through God's own barrier-hedges of true love,
Which fence out licence in securing love)
A child like this, that smiles so in her face,
She is no mother, but a kidnapper,
And he's a dismal orphan ... not a son;
Whom all her kisses cannot feed so full
He will not miss hereafter a pure home
To live in, a pure heart to lean against,
A pure good mother's name and memory
To hope by, when the world grows thick and bad,
And he feels out for virtue.'
'Oh,' she smiled
With bitter patience, 'the child takes his chance,—
Not much worse off in being fatherless
Than I was, fathered. He will say, belike,
His mother was the saddest creature born;
He'll say his mother lived so contrary
To joy, that even the kindest, seeing her,
Grew sometimes almost cruel: he'll not say
She flew contrarious in the face of God
With bat-wings of her vices. Stole my child,—
My flower of earth, my only flower on earth,
My sweet, ray beauty!' ... Up she snatched the child,
And, breaking on him in a storm of tears,
Drew out her long sobs from their shivering roots,
Until he took it for a game, and stretched
His feet, and flapped his eager arms like wings,
And crowed and gurgled through his infant laugh:
'Mine, mine,' she said; 'I have as sure a right
As any glad proud mother in the world,
Who sets her darling down to cut his teeth
Upon her church-ring. If she talks of law,
I talk of law! I claim my mother-dues
By law,—the law which now is paramount;
The common law, by which the poor and weak
Are trodden underfoot by vicious men,
And loathed for ever after by the good.
Let pass! I did not filch ... I found the child.'
'You found him, Marian?'
'Ay, I found him where
I found my curse,—in the gutter, with my shame!
What have you, any of you, to say to that,
Who all are happy, and sit safe and high,
And never spoke before to arraign my right
To grief itself? What, what, ... being beaten down
By hoofs of maddened oxen into a ditch,
Half-dead, whole mangled ... when a girl, at last,
Breathes, sees ... and finds there, bedded in her flesh,
Because of the overcoming shock perhaps,
Some coin of price!... and when a good man comes
(That's God! the best men are not quite as good)
And says, 'I dropped the coin there: take it, you,
And keep it,—it shall pay you for the loss,'—
You all put up your finger—'See the thief!
Observe that precious thing she has come to filch!
How bad those girls are!' Oh, my flower, my pet,
I dare forget I have you in my arms,
And fly off to be angry with the world,
And fright you, hurt you with my tempers, till
You double up your lip? Ah, that indeed
Is bad: a naughty mother!'
'You mistake,'
I interrupted; 'if I loved you not,
I should not, Marian, certainly be here.'
'Alas,' she said, 'you are so very good;
And yet I wish, indeed, you had never come
To make me sob until I vex the child.
It is not wholesome for these pleasure-plats
To be so early watered by our brine.
And then, who knows? he may not like me now
As well, perhaps, as ere he saw me fret,—
One's ugly fretting! he has eyes the same
As angels, but he cannot see as deep,
And so I've kept for ever in his sight
A sort of smile to please him,—as you place
A green thing from the garden in a cup,
To make believe it grows there. Look, my sweet,
My cowslip-ball! we've done with that cross face,
And here's the face come back you used to like.
Ah, ah! he laughs! he likes me. Ah, Miss Leigh,
You're great and pure; but were you purer still,—
As if you had walked, we'll say, no otherwhere
Than up and down the new Jerusalem,
And held your trailing lutestring up yourself
From brushing the twelve stones, for fear of some
Small speck as little as a needle-prick,
White stitched on white,—the child would keep to _me_,
Would choose his poor lost Marian, like me best,
And, though you stretched your arms, cry back and cling,
As we do, when God says it's time to die
And bids us go up higher. Leave us, then;
We two are happy. Does _he_ push me off?
He's satisfied with me, as I with him.'
'So soft to one, so hard to others! Nay,'
I cried, more angry that she melted me,
'We make henceforth a cushion of our faults
To sit and practise easy virtues on?
I thought a child was given to sanctify
A woman,—set her in the sight of all
The clear-eyed Heavens, a chosen minister
To do their business and lead spirits up
The difficult blue heights. A woman lives,
Not bettered, quickened toward the truth and good
Through being a mother?... then she's none! although
She damps her baby's cheeks by kissing them,
As we kill roses.'
'Kill! O Christ,' she said,
And turned her wild sad face from side to side
With most despairing wonder in it—'What,
What have you in your souls against me then,
All of you? am I wicked, do you think?
God knows me, trusts me with the child! but you,
You think me really wicked?'
'Complaisant,'
I answered softly, 'to a wrong you've done,
Because of certain profits,—which is wrong
Beyond the first wrong, Marian. When you left
The pure place and the noble heart, to take
The hand of a seducer'....
'Whom? whose hand?
I took the hand of'....
Springing up erect,
And lifting up the child at full arm's length,
As if to bear him like an oriflamme
Unconquerable to armies of reproach,—
'By _him_' she said, 'my child's head and its curls,
By those blue eyes no woman born could dare
A perjury on, I make my mother's oath,
That if I left that Heart, to lighten it,
The blood of mine was still, except for grief!
No cleaner maid than I was, took a step
To a sadder end,—no matron-mother now
Looks backward to her early maidenhood
Through chaster pulses. I speak steadily:
And if I lie so, ... if, being fouled in will
And paltered with in soul by devil's lust,
I dared to bid this angel take my part, ...
Would God sit quiet, let us think, in heaven,
Nor strike me dumb with thunder? Yet I speak:
He clears me therefore. What, 'seduced''s your word?
Do wolves seduce a wandering fawn in France?
Do eagles, who have pinched a lamb with claws,
Seduce it into carrion? So with me.
I was not ever, as you say, seduced,
But simply, murdered.'
There she paused, and sighed,
With such a sigh as drops from agony
To exhaustion,—sighing while she let the babe
Slide down upon her bosom from her arms,
And all her face's light fell after him,
Like a torch quenched in falling. Down she sank,
And sate upon the bedside with the child.
But I, convicted, broken utterly,
With woman's passion clung about her waist,
And kissed her hair and eyes,—'I have been wrong,
Sweet Marian' ... (weeping in a tender rage)
'Sweet holy Marian! And now, Marian, now,
I'll use your oath although my lips are hard,
And by the child, my Marian, by the child,
I'll swear his mother shall be innocent
Before my conscience, as in the open Book
Of Him who reads for judgement. Innocent,
My sister! let the night be ne'er so dark,
The moon is surely somewhere in the sky;
So surely is your whiteness to be found
Through all dark facts. But pardon, pardon me,
And smile a little, Marian,—for the child,
If not for me, my sister.'
The poor lip
Just motioned for the smile and let it go:
And then, with scarce a stirring of the mouth,
As if a statue spoke that could not breathe,
But spoke on calm between its marble lips,—
'I'm glad, I'm very glad you clear me so.
I should be sorry that you set me down
With harlots, or with even a better name
Which misbecomes his mother. For the rest,
I am not on a level with your love,
Nor ever was, you know,—but now am worse,
Because that world of yours has dealt with me
As when the hard sea bites and chews a stone
And changes the first form of it. I've marked
A shore of pebbles bitten to one shape
From all the various life of madrepores;
And so, that little stone, called Marian Erle,
Picked up and dropped by you and another friend,
Was ground and tortured by the incessant sea
And bruised from what she was,—changed! death's a change,
And she, I said, was murdered; Marian's dead.
What can you do with people when they are dead,
But, if you are pious, sing a hymn and go,
Or, if you are tender, heave a sigh and go,
But go by all means,—and permit the grass
To keep its green feud up 'twixt them and you?
Then leave me,—let me rest. I'm dead, I say.
And if, to save the child from death as well,
The mother in me has survived the rest,
Why, that's God's miracle you must not tax,—
I'm not less dead for that: I'm nothing more
But just a mother. Only for the child,
I'm warm, and cold, and hungry, and afraid,
And smell the flowers a little, and see the sun,
And speak still, and am silent,—just for him!
I pray you therefore to mistake me not,
And treat me, haply, as I were alive;
For though you ran a pin into my soul,
I think it would not hurt nor trouble me.
Here's proof, dear lady,—in the market-place
But now, you promised me to say a word
About ... a friend, who once, long years ago,
Took God's place toward me, when He draws and loves
And does not thunder, ... whom at last I left,
As all of us leave God. You thought perhaps,
I seemed to care for hearing of that friend?
Now, judge me! we have sate here half-an-hour
And talked together of the child and me,
And I not asked as much as, 'What's the thing
You had to tell me of the friend ... the friend?'
He's sad, I think you said,—he's sick perhaps?
It's nought to Marian if he's sad or sick.
Another would have crawled beside your foot
And prayed your words out. Why, a beast, a dog,
A starved cat, if he had fed it once with milk,
Would show less hardness. But I'm dead, you see,
And that explains it.'
Poor, poor thing, she spoke
And shook her head, as white and calm as frost
On days too cold for raining any more,
But still with such a face, so much alive,
I could not choose but take it on my arm
And stroke the placid patience of its cheeks,—
Then told my story out, of Romney Leigh,
How, having lost her, sought her, missed her still,
He, broken-hearted for himself and her,
Had drawn the curtains of the world awhile
As if he had done with morning. There I stopped,
For when she gasped, and pressed me with her eyes,
'And now ... how is it with him? tell me now,'—
I felt the shame of compensated grief,
And chose my words with scruple—slowly stepped
Upon the slippery stones set here and there
Across the sliding water. 'Certainly,
As evening empties morning into night,
Another morning takes the evening up
With healthful, providential interchange;
And, though he thought still of her,'—
'Yes, she knew,
She understood: she had supposed, indeed,
That, as one stops a hole upon a flute,
At which a new note comes and shapes the tune,
Excluding her would bring a worthier in,
And, long ere this, that Lady Waldemar
He loved so' ...
'Loved,' I started,—'loved her so!
Now tell me' ...
'I will tell you,' she replied:
'But since we're taking oaths, you'll promise first
That he, in England, he, shall never learn
In what a dreadful trap his creature here,
Round whose unworthy neck he had meant to tie
The honourable ribbon of his name,
Fell unaware, and came to butchery:
Because,—I know him,—as he takes to heart
The grief of every stranger, he's not like
To banish mine as far as I should choose
In wishing him most happy. Now he leaves
To think of me, perverse, who went my way,
Unkind, and left him,—but if once he knew ...
Ah, then, the sharp nail of my cruel wrong
Would fasten me for ever in his sight,
Like some poor curious bird, through each spread wing
Nailed high up over a fierce hunter's fire,
To spoil the dinner of all tenderer folk
Come in by chance. Nay, since your Marian's dead,
You shall not hang her up, but dig a hole
And bury her in silence! ring no bells.'
I answered gaily, though my whole voice wept;
'We'll ring the joy-bells, not the funeral-bells,
Because we have her back, dead or alive.'
She never answered that, but shook her head;
Then low and calm, as one who, safe in heaven,
Shall tell a story of his lower life,
Unmoved by shame or anger,—so she spoke.
She told me she had loved upon her knees,
As others pray, more perfectly absorbed
In the act and aspiration. She felt his,
For just his uses, not her own at all,
His stool, to sit on, or put up his foot,
His cup, to fill with wine or vinegar,
Whichever drink might please him at the chance,
For that should please her always: let him write
His name upon her ... it seemed natural;
It was most precious, standing on his shelf,
To wait until he chose to lift his hand.
Well, well,—I saw her then, and must have seen
How bright her life went, floating on her love,
Like wicks the housewives send afloat on oil,
Which feeds them to a flame that lasts the night.
To do good seemed so much his business,
That, having done it, she was fain to think,
Must fill up his capacity for joy.
At first she never mooted with herself
If _he_ was happy, since he made her so,
Or if _he_ loved her, being so much beloved:
Who thinks of asking if the sun is light,
Observing that it lightens? who's so bold,
To question God of His felicity?
Still less. And thus she took for granted first,
What first of all she should have put to proof,
And sinned against him so, but only so.
'What could you hope,' she said, 'of such as she?
You take a kid you like, and turn it out
In some fair garden; though the creature's fond
And gentle, it will leap upon the beds
And break your tulips, bite your tender trees:
The wonder would be if such innocence
Spoiled less. A garden is no place for kids.'
And, by degrees, when he who had chosen her,
Brought in his courteous and benignant friends
To spend their goodness on her, which she took
So very gladly, as a part of his,—
By slow degrees, it broke on her slow sense,
That she, too, in that Eden of delight
Was out of place, and, like the silly kid,
Still did most mischief where she meant most love.
A thought enough to make a woman mad,
(No beast in this, but she may well go mad)
That, saying 'I am thine to love and use,'
May blow the plague in her protesting breath
To the very man for whom she claims to die,—
That, clinging round his neck, she pulls him down
And drowns him,—and that, lavishing her soul,
She hales perdition on him. 'So, being mad,'
Said Marian ...
'Ah—who stirred such thoughts, you ask?
Whose fault it was, that she should have such thoughts?
None's fault, none's fault. The light comes, and we see:
But if it were not truly for our eyes,
There would be nothing seen, for all the light;
And so with Marian. If she saw at last,
The sense was in her,—Lady Waldemar
Had spoken all in vain else.'
'O my heart,
O prophet in my heart,' I cried aloud,
'Then Lady Waldemar spoke!'
'_Did_ she speak,'
Mused Marian softly—'or did she only sign?
Or did she put a word into her face
And look, and so impress you with the word?
Or leave it in the foldings of her gown,
Like rosemary smells, a movement will shake out
When no one's conscious? who shall say, or guess?
One thing alone was certain,—from the day
The gracious lady paid a visit first,
She, Marian, saw things different,—felt distrust
Of all that sheltering roof of circumstance
Her hopes were building into with clay nests:
Her heart was restless, pacing up and down
And fluttering, like dumb creatures before storms,
Not knowing wherefore she was ill at ease.'
'And still the lady came,' said Marian Erle,
'Much oftener than _he_ knew it, Mister Leigh.
She bade me never tell him that she had come,
She liked to love me better than he knew,
So very kind was Lady Waldemar:
And every time she brought with her more light,
And every light made sorrow clearer ... Well,
Ah, well! we cannot give her blame for that;
'Twould be the same thing if an angel came,
Whose right should prove our wrong. And every time
The lady came, she looked more beautiful,
And spoke more like a flute among green trees,
Until at last, as one, whose heart being sad
On hearing lovely music, suddenly
Dissolves in weeping, I brake out in tears
Before her ... asked her counsel ... 'had I erred
In being too happy? would she set me straight?
For she, being wise and good and born above
The flats I had never climbed from, could perceive
If such as I, might grow upon the hills;
And whether such poor herb sufficed to grow,
For Romney Leigh to break his fast upon 't,—
Or would he pine on such, or haply starve?'
She wrapt me in her generous arms at once,
And let me dream a moment how it feels
To have a real mother, like some girls:
But when I looked, her face was younger ... ay,
Youth's too bright not to be a little hard,
And beauty keeps itself still uppermost,
That's true!—Though Lady Waldemar was kind,
She hurt me, hurt, as if the morning-sun
Should smite us on the eyelids when we sleep,
And wake us up with headache. Ay, and soon
Was light enough to make my heart ache too:
She told me truths I asked for ... 'twas my fault ...
'That Romney could not love me, if he would,
As men call loving; there are bloods that flow
Together, like some rivers, and not mix,
Through contraries of nature. He indeed
Was set to wed me, to espouse my class,
Act out a rash opinion,—and, once wed,
So just a man and gentle, could not choose
But make my life as smooth as marriage-ring,
Bespeak me mildly, keep me a cheerful house,
With servants, broaches, all the flowers I liked,
And pretty dresses, silk the whole year round' ...
At which I stopped her,—'This for me. And now
'For _him_.'—She murmured,—truth grew difficult;
She owned, ''Twas plain a man like Romney Leigh
Required a wife more level to himself.
If day by day he had to bend his height
To pick up sympathies, opinions, thoughts,
And interchange the common talk of life
Which helps a man to live as well as talk,
His days were heavily taxed. Who buys a staff
To fit the hand, that reaches but the knee?
He'd feel it bitter to be forced to miss
The perfect joy of married suited pairs,
Who, bursting through the separating hedge
Of personal dues with that sweet eglantine
Of equal love, keep saying, 'So _we_ think,
It strikes _us_,—that's _our_ fancy.''—When I asked
If earnest will, devoted love, employed
In youth like mine, would fail to raise me up,—
As two strong arms will always raise a child
To a fruit hung overhead? she sighed and sighed ...
'That could not be,' she feared. 'You take a pink,
You dig about its roots and water it,
And so improve it to a garden-pink,
But will not change it to a heliotrope,
The kind remains. And then, the harder truth—
This Romney Leigh, so rash to leap a pale,
So bold for conscience, quick for martyrdom,
Would suffer steadily and never flinch,
But suffer surely and keenly, when his class
Turned shoulder on him for a shameful match,
And set him up as nine-pin in their talk,
To bowl him down with jestings.'—There, she paused;
And when I used the pause in doubting that
We wronged him after all in what we feared—
'Suppose such things should never touch him, more
In his high conscience, (if the things should be,)
Than, when the queen sits in an upper room,
The horses in the street can spatter her!'—
A moment, hope came,—but the lady closed
That door and nicked the lock, and shut it out,
Observing wisely that, 'the tender heart
Which made him over-soft to a lower class,
Could scarcely fail to make him sensitive
'To a higher,—how they thought, and what they felt.'
'Alas, alas,' said Marian, rocking slow
The pretty baby who was near asleep,
The eyelids creeping over the blue balls,—
'She made it clear, too clear—I saw the whole!
And yet who knows if I had seen my way
Straight out of it, by looking, though 'twas clear,
Unless the generous lady, 'ware of this,
Had set her own house all a-fire for me,
To light me forwards? Leaning on my face
Her heavy agate eyes which crushed my will,
She told me tenderly, (as when men come
To a bedside to tell people they must die)
'She knew of knowledge,—ay, of knowledge, knew,
That Romney Leigh had loved _her_ formerly;
And _she_ loved _him_, she might say, now the chance
Was past ... but that, of course, he never guessed,—
For something came between them ... something thin
As a cobweb ... catching every fly of doubt
To hold it buzzing at the window-pane
And help to dim the daylight. Ah, man's pride
Or woman's—which is greatest? most averse
To brushing cobwebs? Well, but she and he
Remained fast friends; it seemed not more than so,
Because he had bound his hands and could not stir:
An honourable man, if somewhat rash;
And she, not even for Romney, would she spill
A blot ... as little even as a tear ...
Upon his marriage-contract,—not to gain
A better joy for two than came by that!
For, though I stood between her heart and heaven,
She loved me wholly.'
Did I laugh or curse?
I think I sate there silent, hearing all,
Ay, hearing double,—Marian's tale, at once,
And Romney's marriage-vow, '_I'll keep to_ THEE,'
Which means that woman-serpent. Is it time
For church now?
'Lady Waldemar spoke more,'
Continued Marian, 'but, as when a soul
Will pass out through the sweetness of a song
Beyond it, voyaging the uphill road,—
Even so, mine wandered from the things I heard,
To those I suffered. It was afterward
I shaped the resolution to the act.
For many hours we talked. What need to talk?
The fate was clear and close; it touched my eyes;
But still the generous lady tried to keep
The case afloat, and would not let it go,
And argued, struggled upon Marian's side,
Which was not Romney's! though she little knew
What ugly monster would take up the end,—
What griping death within the drowning death
Was ready to complete my sum of death.'
I thought,—Perhaps he's sliding now the ring
Upon that woman's finger....
She went on:
'The lady, failing to prevail her way,
Upgathered my torn wishes from the ground,
And pieced them with her strong benevolence;
And, as I thought I could breathe freer air
Away from England, going without pause,
Without farewell,—just breaking with a jerk
The blossomed offshoot from my thorny life,—
She promised kindly to provide the means,
With instant passage to the colonies
And full protection,—'would commit me straight
'To one who once had been her waiting-maid
And had the customs of the world, intent
On changing England for Australia
Herself, to carry out her fortune so.'
For which I thanked the Lady Waldemar,
As men upon their death-beds thank last friends
Who lay the pillow straight: it is not much,
And yet 'tis all of which they are capable,
This lying smoothly in a bed to die.
And so, 'twas fixed;—and so, from day to day,
The woman named, came in to visit me.'
Just then, the girl stopped speaking,—sate erect,
And stared at me as if I had been a ghost,
(Perhaps I looked as white as any ghost)
With large-eyed horror. 'Does God make,' she said,
'All sorts of creatures, really, do you think?
Or is it that the Devil slavers them
So excellently, that we come to doubt
Who's strongest, He who makes, or he who mars?
I never liked the woman's face, or voice,
Or ways: it made me blush to look at her;
It made me tremble if she touched my hand;
And when she spoke a fondling word, I shrank,
As if one hated me, who had power to hurt;
And, every time she came, my veins ran cold,
As somebody were walking on my grave.
At last I spoke to Lady Waldemar:
'Could such an one be good to trust?' I asked.
Whereat the lady stroked my cheek and laughed
Her silver-laugh—(one must be born to laugh,
To put such music in it) 'Foolish girl,
'Your scattered wits are gathering wool beyond
The sheep-walk reaches!—leave the thing to me.'
And therefore, half in trust, and half in scorn
That I had heart still for another fear
In such a safe despair, I left the thing.
'The rest is short. I was obedient:
I wrote my letter which delivered _him_
From Marian, to his own prosperities,
And followed that bad guide. The lady?—hush,—
I never blame the lady. Ladies who
Sit high, however willing to look down,
Will scarce see lower than their dainty feet:
And Lady Waldemar saw less than I,
With what a Devil's daughter I went forth
The swine's road, headlong over a precipice,
In such a curl of hell-foam caught and choked,
No shriek of soul in anguish could pierce through
To fetch some help. They say there's help in heaven
For all such cries. But if one cries from hell ...
What then?—the heavens are deaf upon that side.
'A woman ... hear me,—let me make it plain,—
A woman ... not a monster ... both her breasts
Made right to suckle babes ... she took me off,
A woman also, young and ignorant,
And heavy with my grief, my two poor eyes
Near washed away with weeping, till the trees,
The blessed unaccustomed trees and fields,
Ran either side the train, like stranger dogs
Unworthy of any notice,—took me off,
So dull, so blind, and only half alive,
Not seeing by what road, nor by what ship,
Nor toward what place, nor to what end of all.—
Men carry a corpse thus,—past the doorway, past
The garden-gate, the children's playground, up
The green lane,—then they leave it in the pit,
To sleep and find corruption, cheek to cheek
With him who stinks since Friday.
'But suppose;
To go down with one's soul into the grave,—
To go down half dead, half alive, I say,
And wake up with corruption, ... cheek to cheek
With him who stinks since Friday! There it is,
And that's the horror of 't, Miss Leigh.
'You feel?
You understand?—no, do not look at me,
But understand. The blank, blind, weary way
Which led ... where'er it led ... away, at least;
The shifted ship ... to Sydney or to France ...
Still bound, wherever else, to another land;
The swooning sickness on the dismal sea,
The foreign shore, the shameful house, the night,
The feeble blood, the heavy-headed grief, ...
No need to bring their damnable drugged cup,
And yet they brought it! Hell's so prodigal
Of devil's gifts ... hunts liberally in packs,
Will kill no poor small creature of the wilds
But fifty red wide throats must smoke at it,—
As HIS at me ... when waking up at last ...
I told you that I waked up in the grave.
'Enough so!—it is plain enough so. True,
We wretches cannot tell out all our wrong,
Without offence to decent happy folk.
I know that we must scrupulously hint
With half-words, delicate reserves, the thing
Which no one scrupled we should feel in full.
Let pass the rest, then; only leave my oath
Upon this sleeping child,—man's violence,
Not man's seduction, made me what I am,
As lost as ... I told _him_ I should be lost;
When mothers fail us, can we help ourselves?
That's fatal!—And you call it being lost,
That down came next day's noon and caught me there
Half gibbering and half raving on the floor,
And wondering what had happened up in heaven,
That suns should dare to shine when God himself
Was certainly abolished.
'I was mad,—
How many weeks, I know not,—many weeks.
I think they let me go, when I was mad,
They feared my eyes and loosed me, as boys might
A mad dog which they had tortured. Up and down
I went by road and village, over tracts
Of open foreign country, large and strange,
Crossed everywhere by long thin poplar-lines
Like fingers of some ghastly skeleton Hand
Through sunlight and through moonlight evermore
Pushed out from hell itself to pluck me back,
And resolute to get me, slow and sure;
While every roadside Christ upon his cross
Hung reddening through his gory wounds at me,
And shook his nails in anger, and came down
To follow a mile after, wading up
The low vines and green wheat, crying 'Take the girl!
'She's none of mine from henceforth,' Then, I knew,
(But this is somewhat dimmer than the rest)
The charitable peasants gave me bread
And leave to sleep in straw: and twice they tied,
At parting, Mary's image round my neck—
How heavy it seemed! as heavy as a stone;
A woman has been strangled with less weight:
I threw it in a ditch to keep it clean
And ease my breath a little, when none looked;
I did not need such safeguards:—brutal men
Stopped short, Miss Leigh, in insult, when they had seen
My face,—I must have had an awful look.
And so I lived: the weeks passed on,—I lived.
'Twas living my old tramp-life o'er again,
But, this time, in a dream, and hunted round
By some prodigious Dream-fear at my back
Which ended, yet: my brain cleared presently,
And there I sate, one evening, by the road,
I, Marian Erle, myself, alone, undone,
Facing a sunset low upon the flats,
As if it were the finish of all time,—
The great red stone upon my sepulchre,
Which angels were too weak to roll away.
SEVENTH BOOK.
'THE woman's motive? shall we daub ourselves
With finding roots for nettles? 'tis soft clay
And easily explored. She had the means,
The monies, by the lady's liberal grace,
In trust for that Australian scheme and me,
Which so, that she might clutch with both her hands,
And chink to her naughty uses undisturbed,
She served me (after all it was not strange;
'Twas only what my mother would have done)
A motherly, unmerciful, good turn.
'Well, after. There are nettles everywhere,
But smooth green grasses are more common still;
The blue of heaven is larger than the cloud;
A miller's wife at Clichy took me in
And spent her pity on me,—made me calm
And merely very reasonably sad.
She found me a servant's place in Paris where
I tried to take the cast-off life again,
And stood as quiet as a beaten ass
Who, having fallen through overloads, stands up
To let them charge him with another pack.
'A few months, so. My mistress, young and light,
Was easy with me, less for kindness than
Because she led, herself, an easy time
Betwixt her lover and her looking-glass,
Scarce knowing which way she was praised the most.
She felt so pretty and so pleased all day
She could not take the trouble to be cross,
But, sometimes, as I stooped to tie her shoe,
Would tap me softly with her slender foot,
Still restless with the last night's dancing in't,
And say, 'Fie, pale-face! are you English girls
All grave and silent? mass-book still, and Lent?
And first-communion colours on your cheeks,
Worn past the time for't? little fool, be gay!'
At which she vanished, like a fairy, through
A gap of silver laughter.
'Came an hour
When all went otherwise. She did not speak,
But clenched her brows, and clipped me with her eyes
As if a viper with a pair of tongs,
Too far for any touch, yet near enough
To view the writhing creature,—then at last;
'Stand still there, in the holy Virgin's name,
Thou Marian; thou'rt no reputable girl,
Although sufficient dull for twenty saints!
I think thou mock'st me and my house,' she said;
'Confess, thou'lt be a mother in a month,
Thou mask of saintship.'
'Could I answer her?
The light broke in so: it meant _that_ then, _that_?
I had not thought of that, in all my thoughts,—
Through all the cold, numb aching of my brow,
Through all the heaving of impatient life
Which threw me on death at intervals,—through all
The upbreak of the fountains of my heart
The rains had swelled too large: it could mean _that_?
Did God make mothers out of victims, then,
And set such pure amens to hideous deeds?
Why not? He overblows an ugly grave
With violets which blossom in the spring.
And _I_ could be a mother in a month!
I hope it was not wicked to be glad.
I lifted up my voice and wept, and laughed,
To heaven, not her, until it tore my throat.
'Confess, confess!' what was there to confess,
Except man's cruelty, except my wrong?
Except this anguish, or this ecstasy?
This shame, or glory? The light woman there
Was small to take it in: an acorn-cup
Would take the sea in sooner.
'Good,' she cried;
Unmarried and a mother, and she laughs!
These unchaste girls are always impudent.
Get out, intriguer! leave my house, and trot:
I wonder you should look me in the face,
With such a filthy secret.'
'Then I rolled
My scanty bundle up, and went my way,
Washed white with weeping, shuddering head and foot
With blind hysteric passion, staggering forth
Beyond those doors. 'Twas natural, of course,
She should not ask me where I meant to sleep;
I might sleep well beneath the heavy Seine,
Like others of my sort; the bed was laid
For us. But any woman, womanly,
Had thought of him who should be in a month,
The sinless babe that should be in a month,
And if by chance he might be warmer housed
Than underneath such dreary, dripping eaves.'
I broke on Marian there. 'Yet she herself,
A wife, I think, had scandals of her own,
A lover, not her husband.'
'Ay,' she said,
'But gold and meal are measured otherwise;
I learnt so much at school,' said Marian Erle.
'O crooked world,' I cried, 'ridiculous
If not so lamentable! It's the way
With these light women of a thrifty vice,
My Marian,—always hard upon the rent
In any sister's virtue! while they keep
Their chastity so darned with perfidy,
That, though a rag itself, it looks as well
Across a street, in balcony or coach,
As any stronger stuff might. For my part,
I'd rather take the wind-side of the stews
Than touch such women with my finger-end!
They top the poor street-walker by their lie,
And look the better for being so much worse:
The devil's most devilish when respectable.
But you, dear, and your story.'
'All the rest
Is here,' she said, and signed upon the child.
'I found a mistress-sempstress who was kind
And let me sew in peace among her girls;
And what was better than to draw the threads
All day and half the night, for him, and him?
And so I lived for him, and so he lives,
And so I know, by this time, God lives too.'
She smiled beyond the sun, and ended so,
And all my soul rose up to take her part
Against the world's successes, virtues, fames.
'Come with me, sweetest sister,' I returned,
'And sit within my house, and do me good
From henceforth, thou and thine! ye are my own
From henceforth. I am lonely in the world,
And thou art lonely, and the child is half
An orphan. Come,—and, henceforth, thou and I
Being still together, will not miss a friend,
Nor he a father, since two mothers shall
Make that up to him. I am journeying south,
And, in my Tuscan home I'll find a niche,
And set thee there, my saint, the child and thee,
And burn the lights of love before thy face,
And ever at thy sweet look cross myself
From mixing with the world's prosperities;
That so, in gravity and holy calm,
We two may live on toward the truer life.'
She looked me in the face and answered not,
Nor signed she was unworthy, nor gave thanks,
But took the sleeping child and held it out
To meet my kiss, as if requiting me
And trusting me at once. And thus, at once,
I carried him and her to where I lived;
She's there now, in the little room, asleep,
I hear the soft child-breathing through the door;
And all three of us, at to-morrow's break,
Pass onward, homeward, to our Italy.
Oh, Romney Leigh, I have your debts to pay,
And I'll be just and pay them.
But yourself!
To pay your debts is scarcely difficult;
To buy your life is nearly impossible,
Being sold away to Lamia. My head aches;
I cannot see my road along this dark;
Nor can I creep and grope, as fits the dark,
For these foot-catching robes of womanhood:
A man might walk a little ... but I!—He loves
The Lamia-woman,—and I, write to him
What stops his marriage, and destroys his peace,—
Or what, perhaps, shall simply trouble him,
Until she only need to touch his sleeve
With just a finger's tremulous white flame,
Saying, 'Ah,—Aurora Leigh! a pretty tale,
A very pretty poet! I can guess
The motive'—then, to catch his eyes in hers,
And vow she does not wonder,—and they two
To break in laughter, as the sea along
A melancholy coast, and float up higher,
In such a laugh, their fatal weeds of love!
Ay, fatal, ay. And who shall answer me
Fate has not hurried tides; and if to-night
My letter would not be a night too late,—
An arrow shot into a man that's dead,
To prove a vain intention? Would I show
The new wife vile, to make the husband mad?
No, Lamia! shut the shutters, bar the doors
From every glimmer on thy serpent-skin!
I will not let thy hideous secret out
To agonise the man I love—I mean
The friend I love ... as friends love.
It is strange,
To-day while Marian told her story, like
To absorb most listeners, how I listened chief
To a voice not hers, nor yet that enemy's,
Nor God's in wrath, ... but one that mixed with mine
Long years ago, among the garden-trees,
And said to _me_, to _me_ too, 'Be my wife,
Aurora!' It is strange, with what a swell
Of yearning passion, as snow of ghosts
Might beat against the impervious doors of heaven,
I thought, 'Now, if I had been a woman, such
As God made women, to save men by love,—
By just my love I might have saved this man,
And made a nobler poem for the world
Than all I have failed in.' But I failed besides
In this; and now he's lost! through me alone!
And, by my only fault, his empty house
Sucks in, at this same hour, a wind from hell
To keep his hearth cold, make his casements creak
For ever to the tune of plague and sin—
O Romney, O my Romney, O my friend!
My cousin and friend! my helper, when I would,
My love, that might be! mine!
Why, how one weeps
When one's too weary! Were a witness by,
He'd say some folly ... that I loved the man,
Who knows?... and make me laugh again for scorn.
At strongest, women are as weak in flesh,
As men, at weakest, vilest, are in soul:
So, hard for women to keep pace with men!
As well give up at once, sit down at once,
And weep as I do. Tears, tears! _why_, we weep?
'Tis worth enquiry?—That we've shamed a life,
Or lost a love, or missed a world, perhaps?
By no means. Simply, that we've walked too far,
Or talked too much, or felt the wind i' the east,—
And so we weep, as if both body and soul
Broke up in water—this way.
Poor mixed rags
Forsooth we're made of, like those other dolls
That lean with pretty faces into fairs.
It seems as if I had a man in me,
Despising such a woman.
Yet indeed,
To see a wrong or suffering moves us all
To undo it, though we should undo ourselves;
Ay, all the more, that we undo ourselves;
That's womanly, past doubt, and not ill-moved.
A natural movement, therefore, on my part,
To fill the chair up of my cousin's wife,
And save him from a devil's company!
We're all so,—made so—'tis our woman's trade
To suffer torment for another's ease.
The world's male chivalry has perished out,
But women are knights-errant to the last;
And, if Cervantes had been greater still,
He had made his Don a Donna.
So it clears,
And so we rain our skies blue.
Put away
This weakness. If, as I have just now said,
A man's within me,—let him act himself,
Ignoring the poor conscious trouble of blood
That's called the woman merely. I will write
Plain words to England,—if too late, too late,—
If ill-accounted, then accounted ill;
We'll trust the heavens with something.
'Dear Lord Howe,
You'll find a story on another leaf
That's Marian Erle's,—what noble friend of yours
She trusted once, through what flagitious means
To what disastrous ends;—the story's true.
I found her wandering on the Paris quays,
A babe upon her breast,—unnatural
Unseasonable outcast on such snows
Unthawed to this time. I will tax in this
Your friendship, friend,—if that convicted She
Be not his wife yet, to denounce the facts
To himself,—but, otherwise, to let them pass
On tip-toe like escaping murderers,
And tell my cousin, merely—Marian lives,
Is found, and finds her home with such a friend,
Myself, Aurora. Which good news, 'She's found,'
Will help to make him merry in his love:
I send it, tell him, for my marriage gift,
As good as orange-water for the nerves,
Or perfumed gloves for headaches,—though aware
That he, except of love, is scarcely sick;
I mean the new love this time, ... since last year.
Such quick forgetting on the part of men!
Is any shrewder trick upon the cards
To enrich them? pray instruct me how it's done.
First, clubs,—and while you look at clubs, it's spades;
That's prodigy. The lightning strikes a man,
And when we think to find him dead and charred ...
Why, there he is on a sudden, playing pipes
Beneath the splintered elm-tree! Crime and shame
And all their hoggery trample your smooth world,
Nor leave more foot-marks than Apollo's kine,
Whose hoofs were muffled by the thieving god
In tamarisk-leaves and myrtle. I'm so sad,
So weary and sad to-night, I'm somewhat sour,—
Forgive me. To be blue and shrew at once,
Exceeds all toleration except yours;
But yours, I know, is infinite. Farewell.
To-morrow we take train for Italy.
Speak gently of me to your gracious wife,
As one, however far, shall yet be near
In loving wishes to your house.'
I sign.
And now I'll loose my heart upon a page,
This—
'Lady Waldemar, I'm very glad
I never liked you; which you knew so well,
You spared me, in your turn, to like me much.
Your liking surely had done worse for me
Than has your loathing, though the last appears
Sufficiently unscrupulous to hurt,
And not afraid of judgment. Now, there's space
Between our faces,—I stand off, as if
I judged a stranger's portrait and pronounced
Indifferently the type was good or bad:
What matter to me that the lines are false,
I ask you? Did I ever ink my lips
By drawing your name through them as a friend's,
Or touch your hands as lovers do? thank God
I never did: and, since you're proved so vile,
Ay, vile, I say,—we'll show it presently,—
I'm not obliged to nurse my friend in you,
Or wash out my own blots, in counting yours,
Or even excuse myself to honest souls
Who seek to touch my lip or clasp my palm,—
'Alas, but Lady Waldemar came first!'
''Tis true, by this time, you may near me so
That you're my cousin's wife. You've gambled deep
As Lucifer, and won the morning-star
In that case,—and the noble house of Leigh
Must henceforth with its good roof shelter you:
I cannot speak and burn you up between
Those rafters, I who am born a Leigh,—nor speak
And pierce your breast through Romney's, I who live
His friend and cousin!—so, you are safe. You two
Must grow together like the tares and wheat
Till God's great fire.—But make the best of time.
'And hide this letter! let it speak no more
Than I shall, how you tricked poor Marian Erle,
And set her own love digging her own grave
Within her green hope's pretty garden-ground;
Ay, sent her forth with some one of your sort
To a wicked house in France,—from which she fled
With curses in her eyes and ears and throat,
Her whole soul choked with curses,—mad, in short,
And madly scouring up and down for weeks
The foreign hedgeless country, lone and lost,—
So innocent, male-fiends might slink within
Remote hell-corners, seeing her so defiled!
'But you,—you are a woman and more bold.
To do you justice, you'd not shrink to face ...
We'll say, the unfledged life in the other room,
Which, treading down God's corn, you trod in sight
Of all the dogs, in reach of all the guns,—
Ay, Marian's babe, her poor unfathered child,
Her yearling babe!—you'd face him when he wakes
And opens up his wonderful blue eyes:
You'd meet them and not wink perhaps, nor fear
God's triumph in them and supreme revenge,
So, righting His creation's balance-scale
(You pulled as low as Tophet) to the top
Of most celestial innocence! For me
Who am not as bold, I own those infant eyes
Have set me praying.
'While they look at heaven,
No need of protestation in my words
Against the place you've made them! let them look!
They'll do your business with the heavens, be sure:
I spare you common curses.
'Ponder this.
If haply you're the wife of Romney Leigh,
(For which inheritance beyond your birth
You sold that poisonous porridge called your soul)
I charge you, be his faithful and true wife!
Keep warm his hearth and clean his board, and, when
He speaks, be quick with your obedience;
Still grind your paltry wants and low desires
To dust beneath his heel; though, even thus,
The ground must hurt him,—it was writ of old,
'Ye shall not yoke together ox and ass,'
The nobler and ignobler. Ay, but you
Shall do your part as well as such ill things
Can do aught good. You shall not vex him,—mark,
You shall not vex him, ... jar him when he's sad,
Or cross him when he's eager. Understand
To trick him with apparent sympathies,
Nor let him see thee in the face too near
And unlearn thy sweet seeming. Pay the price
Of lies, by being constrained to lie on still;
'Tis easy for thy sort: a million more
Will scarcely damn thee deeper.
'Doing which,
You are very safe from Marian and myself:
We'll breathe as softly as the infant here,
And stir no dangerous embers. Fail a point,
And show our Romney wounded, ill-content,
Tormented in his home, ... we open mouth,
And such a noise will follow, the last trump's
Will scarcely seem more dreadful, even to you;
You'll have no pipers after: Romney will
(I know him) push you forth as none of his,
All other men declaring it well done;
While women, even the worst, your like, will draw
Their skirts back, not to brush you in the street;
And so I warn you. I'm ... Aurora Leigh.'
The letter written, I felt satisfied.
The ashes, smouldering in me, were thrown out
By handfuls from me: I had writ my heart
And wept my tears, and now was cool and calm;
And, going straightway to the neighbouring room,
I lifted up the curtains of the bed
Where Marian Erle, the babe upon her arm,
Both faces leaned together like a pair
Of folded innocences, self-complete,
Each smiling from the other, smiled and slept.
There seemed no sin, no shame, no wrath, no grief.
I felt, she too, had spoken words that night,
But softer certainly, and said to God,—
Who laughs in heaven perhaps, that such as I
Should make ado for such as she.—'Defiled'
I wrote? 'defiled' I thought her? Stoop,
Stoop lower, Aurora! get the angels' leave
To creep in somewhere, humbly, on your knees,
Within this round of sequestration white
In which they have wrapt earth's foundlings, heaven's elect!
The next day, we took train to Italy
And fled on southward in the roar of steam.
The marriage-bells of Romney must be loud,
To sound so clear through all! I was not well;
And truly, though the truth is like a jest,
I could not choose but fancy, half the way,
I stood alone i' the belfry, fifty bells
Of naked iron, mad with merriment,
(As one who laughs and cannot stop himself)
All clanking at me, in me, over me,
Until I shrieked a shriek I could not hear,
And swooned with noise,—but still, along my swoon,
Was 'ware the baffled changes backward rang,
Prepared, at each emerging sense, to beat
And crash it out with clangour. I was weak;
I struggled for the posture of my soul
In upright consciousness of place and time,
But evermore, 'twixt waking and asleep,
Slipped somehow, staggered, caught at Marian's eyes
A moment, (it is very good for strength
To know that some one needs you to be strong)
And so recovered what I called myself,
For that time.
I just knew it when we swept
Above the old roofs of Dijon. Lyons dropped
A spark into the night, half trodden out
Unseen. But presently the winding Rhone
Washed out the moonlight large along his banks,
Which strained their yielding curves out clear and clean
To hold it,—shadow of town and castle blurred
Upon the hurrying river. Such an air
Blew thence upon the forehead,—half an air
And half a water,—that I leaned and looked;
Then, turning back on Marian, smiled to mark
That she looked only on her child, who slept,
His face towards the moon too.
So we passed
The liberal open country and the close,
And shot through tunnels, like a lightning-wedge
By great Thor-hammers driven through the rock,
Which, quivering through the intestine blackness, splits,
And lets it in at once: the train swept in
Athrob with effort, trembling with resolve,
The fierce denouncing whistle wailing on
And dying off smothered in the shuddering dark,
While we, self-awed, drew troubled breath, oppressed
As other Titans, underneath the pile
And nightmare of the mountains. Out, at last,
To catch the dawn afloat upon the land!
—Hills, slung forth broadly and gauntly everywhere,
Not crampt in their foundations, pushing wide
Rich outspreads of the vineyards and the corn,
(As if they entertained i' the name of France)
While, down their straining sides, streamed manifest
A soil as red as Charlemagne's knightly blood,
To consecrate the verdure. Some one said,
'Marseilles!' And lo, the city of Marseilles,
With all her ships behind her, and beyond,
The scimitar of ever-shining sea,
For right-hand use, bared blue against the sky!
That night we spent between the purple heaven
And purple water: I think Marian slept;
But I, as a dog a-watch for his master's foot,
Who cannot sleep or eat before he hears,
I sate upon the deck and watched all night,
And listened through the stars for Italy.
Those marriage-bells I spoke of, sounded far,
As some child's go-cart in the street beneath
To a dying man who will not pass the day,
And knows it, holding by a hand he loves.
I, too, sate quiet, satisfied with death,
Sate silent: I could hear my own soul speak,
And had my friend,—for Nature comes sometimes
And says, 'I am ambassador for God.'
I felt the wind soft from the land of souls;
The old miraculous mountains heaved in sight,
One straining past another along the shore,
The way of grand dull Odyssean ghosts
Athirst to drink the cool blue wine of seas
And stare on voyagers. Peak pushing peak
They stood: I watched beyond that Tyrian belt
Of intense sea betwixt them and the ship,
Down all their sides the misty olive-woods
Dissolving in the weak congenial moon,
And still disclosing some brown convent-tower
That seems as if it grew from some brown rock,—
Or many a little lighted village, dropt
Like a fallen star, upon so high a point,
You wonder what can keep it in its place
From sliding headlong with the waterfalls
Which drop and powder all the myrtle-groves
With spray of silver. Thus my Italy
Was stealing on us. Genoa broke with day;
The Doria's long pale palace striking out,
From green hills in advance of the white town,
A marble finger dominant to ships,
Seen glimmering through the uncertain grey of dawn.
But then I did not think, 'my Italy,'
I thought, 'my father!' O my father's house,
Without his presence!—Places are too much
Or else too little, for immortal man;
Too little, when love's May o'ergrows the ground,—
Too much, when that luxuriant wealth of green
Is rustling to our ankles in dead leaves.
'Tis only good to be, or here or there,
Because we had a dream on such a stone,
Or this or that,—but, once beings wholly waked,
And come back to the stone without the dream,
We trip upon't,—alas! and hurt ourselves;
Or else it falls on us and grinds us flat,
The heaviest grave-stone on this burying earth.
—But while I stood and mused, a quiet touch
Fell light upon my arm, and, turning round,
A pair of moistened eyes convicted mine.
'What, Marian! is the babe astir so soon?'
'He sleeps,' she answered; 'I have crept up thrice,
And seen you sitting, standing, still at watch.
I thought it did you good till now, but now' ...
'But now,' I said, 'you leave the child alone.'
'And _you're_ alone,' she answered,—and she looked
As if I, too, were something. Sweet the help
Of one we have helped! Thanks, Marian, for that help.
I found a house, at Florence, on the hill
Of Bellosguardo. 'Tis a tower that keeps
A post of double-observation o'er
The valley of Arno (holding as a hand
The outspread city) straight toward Fiesole
And Mount Morello and the setting sun,—
The Vallombrosan mountains to the right,
Which sunrise fills as full as crystal cups
Wine-filled, and red to the brim because it's red.
No sun could die, nor yet be born, unseen
By dwellers at my villa: morn and eve
Were magnified before us in the pure
Illimitable space and pause of sky,
Intense as angels' garments blanched with God,
Less blue than radiant. From the outer wall
Of the garden, dropped the mystic floating grey
Of olive-trees, (with interruptions green
From maize and vine) until 'twas caught and torn
On that abrupt black line of cypresses
Which signed the way to Florence. Beautiful
The city lay along the ample vale,
Cathedral, tower and palace, piazza and street;
The river trailing like a silver cord
Through all, and curling loosely, both before
And after, over the whole stretch of land
Sown whitely up and down its opposite <DW72>s,
With farms and villas.
Many weeks had passed,
No word was granted.—Last, a letter came
From Vincent Carrington:—'My dear Miss Leigh,
You've been as silent as a poet should,
When any other man is sure to speak.
If sick, if vexed, if dumb, a silver-piece
Will split a man's tongue,—straight he speaks and says,
'Received that cheque.' But you!... I send you funds
To Paris, and you make no sign at all.
Remember I'm responsible and wait
A sign of you, Miss Leigh.
'Meantime your book
Is eloquent as if you were not dumb;
And common critics, ordinarily deaf
To such fine meanings, and, like deaf men, loth
To seem deaf, answering chance-wise, yes or no,
'It must be,' or 'it must not,' (most pronounced
When least convinced) pronounce for once aright:
You'd think they really heard,—and so they do ...
The burr of three or four who really hear
And praise your book aright: Fame's smallest trump
Is a great ear-trumpet for the deaf as posts,
No other being effective. Fear not, friend;
We think, here, you have written a good book,
And you, a woman! It was in you—yes,
I felt 'twas in you: yet I doubted half
If that od-force of German Reichenbach
Which still from female finger-tips burns blue,
Could strike out, as our masculine white heats,
To quicken a man. Forgive me. All my heart
Is quick with yours, since, just a fortnight since,
I read your book and loved it.
'Will you love
My wife, too? Here's my secret, I might keep
A month more from you! but I yield it up
Because I know you'll write the sooner for't,—
Most women (of your height even) counting love
Life's only serious business. Who's my wife
That shall be in a month? you ask? nor guess?
Remember what a pair of topaz eyes
You once detected, turned against the wall,
That morning, in my London painting-room;
The face half-sketched, and slurred; the eyes alone!
But you ... you caught them up with yours, and said
'Kate Ward's eyes, surely.'—Now, I own the truth,
I had thrown them there to keep them safe from Jove;
They would so naughtily find out their way
To both the heads of both my Danaës,
Where just it made me mad to look at them.
Such eyes! I could not paint or think of eyes
But those,—and so I flung them into paint
And turned them to the wall's care. Ay, but now
I've let them out, my Kate's! I've painted her,
(I'll change my style, and leave mythologies)
The whole sweet face; it looks upon my soul
Like a face on water, to beget itself.
A half-length portrait, in a hanging cloak
Like one you wore once; 'tis a little frayed;
I pressed, too, for the nude harmonious arm—
But she ... she'd have her way, and have her cloak;
She said she could be like you only so,
And would not miss the fortune. Ah, my friend,
You'll write and say she shall not miss your love
Through meeting mine? in faith, she would not change:
She has your books by heart, more than my words,
And quotes you up against me till I'm pushed
Where, three months since, her eyes were! nay, in fact,
Nought satisfied her but to make me paint
Your last book folded in her dimpled hands,
Instead of my brown palette, as I wished,
(And, grant me, the presentment had been newer)
She'd grant me nothing: I've compounded for
The naming of the wedding-day next month,
And gladly too. 'Tis pretty, to remark
How women can love women of your sort,
And tie their hearts with love-knots to your feet,
Grow insolent about you against men,
And put us down by putting up the lip,
As if a man,—there _are_ such, let us own,
Who write not ill,—remains a man, poor wretch,
While you——! Write far worse than Aurora Leigh,
And there'll be women who believe of you
(Besides my Kate) that if you walked on sand
You would not leave a foot-print.
'Are you put
To wonder by my marriage, like poor Leigh?
'Kate Ward!' he said. 'Kate Ward!' he said anew.
'I thought ...' he said, and stopped,—'I did not think....'
And then he dropped to silence.
'Ah, he's changed.
I had not seen him, you're aware, for long,
But went of course. I have not touched on this
Through all this letter,—conscious of your heart,
And writing lightlier for the heavy fact,
As clocks are voluble with lead.
'How weak,
To say I'm sorry. Dear Leigh, dearest Leigh!
In those old days of Shropshire,—pardon me,—
When he and you fought many a field of gold
On what you should do, or you should not do,
Make bread or verses, (it just came to that)
I thought you'd one day draw a silken peace
Through a golden ring. I thought so. Foolishly,
The event proved,—for you went more opposite
To each other, month by month, and year by year,
Until this happened. God knows best, we say,
But hoarsely. When the fever took him first,
Just after I had writ to you in France,
They tell me Lady Waldemar mixed drinks
And counted grains, like any salaried nurse,
Excepting that she wept too. Then Lord Howe,
You're right about Lord Howe! Lord Howe's a trump;
And yet, with such in his hand, a man like Leigh
May lose, as _he_ does. There's an end to all,—
Yes, even this letter, though the second sheet
May find you doubtful. Write a word for Kate:
Even now she reads my letters like a wife,
And, if she sees her name, I'll see her smile,
And share the luck. So, bless you, friend of two!
I will not ask you what your feeling is
At Florence, with my pictures. I can hear
Your heart a-flutter over the snow-hills;
And, just to pace the Pitti with you once,
I'd give a half-hour of to-morrow's walk
With Kate ... I think so. Vincent Carrington.'
The noon was hot; the air scorched like the sun,
And was shut out. The closed persiani threw
Their long-scored shadows on my villa-floor,
And interlined the golden atmosphere
Straight, still,—across the pictures on the wall,
The statuette on the console, (of young Love
And Psyche made one marble by a kiss)
The low couch where I leaned, the table near,
The vase of lilies, Marian pulled last night,
(Each green leaf and each white leaf ruled in black
As if for writing some new text of fate)
And the open letter, rested on my knee,—
But there, the lines swerved, trembled, though I sate
Untroubled ... plainly, ... reading it again
And three times. Well, he's married; that is clear.
No wonder that he's married, nor much more
That Vincent's therefore, 'sorry.' Why, of course,
The lady nursed him when he was not well,
Mixed drinks,—unless nepenthe was the drink,
'Twas scarce worth telling. But a man in love
Will see the whole sex in his mistress' hood,
The prettier for its lining of fair rose;
Although he catches back, and says at last,
'I'm sorry.' Sorry. Lady Waldemar
At prettiest, under the said hood, preserved
From such a light as I could hold to her face
To flare its ugly wrinkles out to shame,—
Is scarce a wife for Romney, as friends judge,
Aurora Leigh, or Vincent Carrington,—
That's plain. And if he's 'conscious of my heart' ...
Perhaps it's natural, though the phrase is strong;
(One's apt to use strong phrases, being in love)
And even that stuff of 'fields of gold,' 'gold rings,'
And what he 'thought,' poor Vincent! what he 'thought,'
May never mean enough to ruffle me.
—Why, this room stifles. Better burn than choke;
Best have air, air, although it comes with fire,
Throw open blinds and windows to the noon
And take a blister on my brow instead
Of this dead weight! best, perfectly be stunned
By those insufferable cicale, sick
And hoarse with rapture of the summer-heat,
That sing like poets, till their hearts break, ... sing
Till men say, 'It's too tedious.'
Books succeed,
And lives fail. Do I feel it so, at last?
Kate loves a worn-out cloak for being like mine,
While I live self-despised for being myself,
And yearn toward some one else, who yearns away
From what he is, in his turn. Strain a step
For ever, yet gain no step? Are we such,
We cannot, with our admirations even,
Our tip-toe aspirations, touch a thing
That's higher than we? is all a dismal flat,
And God alone above each,—as the sun
O'er level lagunes, to make them shine and stink,—
Laying stress upon us with immediate flame,
While we respond with our miasmal fog,
And call it mounting higher, because we grow
More highly fatal?
Tush, Aurora Leigh!
You wear your sackcloth looped in Cæsar's way,
And brag your failings as mankind's. Be still.
There _is_ what's higher, in this very world,
Than you can live, or catch at. Stand aside,
And look at others—instance little Kate!
She'll make a perfect wife for Carrington.
She always has been looking round the earth
For something good and green to alight upon
And nestle into, with those soft-winged eyes
Subsiding now beneath his manly hand
'Twixt trembling lids of inexpressive joy:
I will not scorn her, after all, too much,
That so much she should love me. A wise man
Can pluck a leaf, and find a lecture in 't;
And I, too, ... God has made me,—I've a heart
That's capable of worship, love, and loss;
We say the same of Shakspeare's. I'll be meek,
And learn to reverence, even this poor myself.
The book, too—pass it. 'A good book,' says he,
'And you a woman.' I had laughed at that,
But long since. I'm a woman,—it is true;
Alas, and woe to us, when we feel it most!
Then, least care have we for the crowns and goals,
And compliments on writing our good books.
The book has some truth in it, I believe:
And truth outlives pain, as the soul does life.
I know we talk our Phædons to the end
Through all the dismal faces that we make,
O'er-wrinkled with dishonouring agony
From any mortal drug. I have written truth,
And I a woman; feebly, partially,
Inaptly in presentation, Romney'll add,
Because a woman. For the truth itself,
That's neither man's nor woman's, but just God's;
None else has reason to be proud of truth:
Himself will see it sifted, disenthralled,
And kept upon the height and in the light,
As far as, and no farther, than 'tis truth;
For,—now He has left off calling firmaments
And strata, flowers and creatures, very good,—
He says it still of truth, which is His own.
Truth, so far, in my book;—the truth which draws
Through all things upwards; that a twofold world
Must go to a perfect cosmos. Natural things
And spiritual,—who separates those two
In art, in morals, or the social drift,
Tears up the bond of nature and brings death,
Paints futile pictures, writes unreal verse,
Leads vulgar days, deals ignorantly with men,
Is wrong, in short, at all points. We divide
This apple of life, and cut it through the pips,—
The perfect round which fitted Venus' hand
Has perished utterly as if we ate
Both halves. Without the spiritual, observe,
The natural's impossible;—no form,
No motion! Without sensuous, spiritual
Is inappreciable;—no beauty or power!
And in this twofold sphere the twofold man
(And still the artist is intensely a man)
Holds firmly by the natural, to reach
The spiritual beyond it,—fixes still
The type with mortal vision, to pierce through,
With eyes immortal, to the antetype
Some call the ideal,—better called the real,
And certain to be called so presently
When things shall have their names. Look long enough
On any peasant's face here, coarse and lined,
You'll catch Antinous somewhere in that clay,
As perfect-featured as he yearns at Rome
From marble pale with beauty; then persist,
And, if your apprehension's competent,
You'll find some fairer angel at his back,
As much exceeding him, as he the boor,
And pushing him with empyreal disdain
For ever out of sight. Ay, Carrington
Is glad of such a creed! an artist must,
Who paints a tree, a leaf, a common stone,
With just his hand, and finds it suddenly
A-piece with and conterminous to his soul.
Why else do these things move him, leaf or stone?
The bird's not moved, that pecks at a spring-shoot;
Nor yet the horse, before a quarry, a-graze:
But man, the two-fold creature, apprehends
The two-fold manner, in and outwardly,
And nothing in the world comes single to him,
A mere itself,—cup, column, or candlestick,
All patterns of what shall be in the Mount;
The whole temporal show related royally,
And built up to eterne significance
Through the open arms of God. 'There's nothing great
Nor small,' has said a poet of our day,
(Whose voice will ring beyond the curfew of eve
And not be thrown out by the matin's bell)
And truly, I reiterate, ... nothing's small!
No lily-muffled hum of a summer-bee,
But finds some coupling with the spinning stars;
No pebble at your foot, but proves a sphere;
No chaffinch, but implies the cherubim:
And,—glancing on my own thin, veinéd wrist,—
In such a little tremour of the blood
The whole strong clamour of a vehement soul
Doth utter itself distinct. Earth's crammed with heaven,
And every common bush afire with God:
But only he who sees, takes off his shoes;
The rest sit round it, and pluck blackberries,
And daub their natural faces unaware
More and more, from the first similitude.
Truth, so far, in my book! a truth which draws
From all things upwards. I, Aurora, still
Have felt it hound me through the wastes of life
As Jove did Io: and, until that Hand
Shall overtake me wholly, and, on my head,
Lay down its large unfluctuating peace,
The feverish gad-fly pricks me up and down,
It must be. Art's the witness of what Is
Behind this show. If this world's show were all,
Then imitation would be all in Art;
There, Jove's hand gripes us!—For we stand here, we,
If genuine artists, witnessing for God's
Complete, consummate, undivided work:
—That not a natural flower can grow on earth,
Without a flower upon the spiritual side,
Substantial, archetypal, all a-glow
With blossoming causes,—not so far away,
That we, whose spirit-sense is somewhat cleared,
May not catch something of the bloom and breath,—
Too vaguely apprehended, though indeed
Still apprehended, consciously or not,
And still transferred to picture, music, verse,
For thrilling audient and beholding souls
By signs and touches which are known to souls,—
How known, they know not,—why, they cannot find,
So straight call out on genius, say, 'A man
Produced this,'—when much rather they should say,
''Tis insight, and he saw this.'
Thus is Art
Self-magnified in magnifying a truth
Which, fully recognised, would change the world
And shift its morals. If a man could feel,
Not one day, in the artist's ecstasy,
But every day, feast, fast, or working-day,
The spiritual significance burn through
The hieroglyphic of material shows,
Henceforward he would paint the globe with wings,
And reverence fish and fowl, the bull, the tree,
And even his very body as a man,—
Which now he counts so vile, that all the towns
Make offal of their daughters for its use
On summer-nights, when God is sad in heaven
To think what goes on in his recreant world
He made quite other; while that moon He made
To shine there, at the first love's covenant,
Shines still, convictive as a marriage-ring
Before adulterous eyes.
How sure it is,
That, if we say a true word, instantly
We feel 'tis God's, not ours, and pass it on
As bread at sacrament, we taste and pass
Nor handle for a moment, as indeed
We dared to set up any claim to such!
And I—my poem;—let my readers talk;
I'm closer to it—I can speak as well:
I'll say, with Romney, that the book is weak,
The range uneven, the points of sight obscure,
The music interrupted.
Let us go.
The end of woman (or of man, I think)
Is not a book. Alas, the best of books
Is but a word in Art, which soon grows cramped,
Stiff, dubious-statured with the weight of years,
And drops an accent or digamma down
Some cranny of unfathomable time,
Beyond the critic's reaching. Art itself,
We've called the higher life, still must feel the soul
Live past it. For more's felt than is perceived,
And more's perceived than can be interpreted,
And Love strikes higher with his lambent flame
Than Art can pile the <DW19>s.
Is it so?
When Jove's hand meets us with composing touch,
And when, at last, we are hushed and satisfied,—
Then, Io does not call it truth, but love?
Well, well! my father was an Englishman:
My mother's blood in me is not so strong
That I should bear this stress of Tuscan noon
And keep my wits. The town, there, seems to seethe
In this Medæan boil-pot of the sun,
And all the patient hills are bubbling round
As if a prick would leave them flat. Does heaven
Keep far off, not to set us in a blaze?
Not so,—let drag your fiery fringes, heaven,
And burn us up to quiet! Ah, we know
Too much here, not to know what's best for peace;
We have too much light here, not to want more fire
To purify and end us. We talk, talk,
Conclude upon divine philosophies,
And get the thanks of men for hopeful books;
Whereat we take our own life up, and ... pshaw!
Unless we piece it with another's life,
(A yard of silk to carry out our lawn)
As well suppose my little handkerchief
Would cover Samminiato, church and all,
If out I threw it past the cypresses,
As, in this ragged, narrow life of mine,
Contain my own conclusions.
But at least
We'll shut up the persiani, and sit down,
And when my head's done aching, in the cool,
Write just a word to Kate and Carrington.
May joy be with them! she has chosen well,
And he not ill.
I should be glad, I think,
Except for Romney. Had _he_ married Kate,
I surely, surely, should be very glad.
This Florence sits upon me easily,
With native air and tongue. My graves are calm,
And do not too much hurt me. Marian's good,
Gentle and loving,—lets me hold the child,
Or drags him up the hills to find me flowers
And fill those vases, ere I'm quite awake,—
The grandiose red tulips, which grow wild,
Or else my purple lilies, Dante blew
To a larger bubble with his prophet-breath;
Or one of those tall flowering reeds which stand
In Arno like a sheaf of sceptres, left
By some remote dynasty of dead gods,
To suck the stream for ages and get green,
And blossom wheresoe'er a hand divine
Had warmed the place with ichor. Such I've found
At early morning, laid across my bed,
And woke up pelted with a childish laugh
Which even Marian's low precipitous 'hush'
Had vainly interposed to put away,—
While I, with shut eyes, smile and motion for
The dewy kiss that's very sure to come
From mouth and cheeks, the whole child's face at once
Dissolved on mine,—as if a nosegay burst
Its string with the weight of roses overblown,
And dropt upon me. Surely I should be glad.
The little creature almost loves me now,
And calls my name ... 'Alola,' stripping off
The _r_s like thorns, to make it smooth enough
To take between his dainty, milk-fed lips,
God love him! I should certainly be glad,
Except, God help me, that I'm sorrowful,
Because of Romney.
Romney, Romney! Well,
This grows absurd!—too like a tune that runs
I' the head, and forces all things in the world,
Wind, rain, the creaking gnat or stuttering fly,
To sing itself and vex you;—yet perhaps
A paltry tune you never fairly liked,
Some 'I'd be a butterfly,' or 'C'est l'amour:'
We're made so,—not such tyrants to ourselves,
We are not slaves to nature. Some of us
Are turned, too, overmuch like some poor verse
With a trick of ritournelle: the same thing goes
And comes back ever.
Vincent Carrington
Is 'sorry,' and I'm sorry; but _he_'s strong
To mount from sorrow to his heaven of love,
And when he says at moments, 'Poor, poor Leigh,
Who'll never call his own, so true a heart,
So fair a face even,'—he must quickly lose
The pain of pity in the blush he has made
By his very pitying eyes. The snow, for him,
Has fallen in May, and finds the whole earth warm,
And melts at the first touch of the green grass.
But Romney,—he has chosen, after all.
I think he had as excellent a sun
To see by, as most others, and perhaps
Has scarce seen really worse than some of us,
When all's said. Let him pass. I'm not too much
A woman, not to be a man for once,
And bury all my Dead like Alaric,
Depositing the treasures of my soul
In this drained water-course, and, letting flow
The river of life again, with commerce-ships
And pleasure-barges, full of silks and songs.
Blow, winds, and help us.
Ah, we mock ourselves
With talking of the winds! perhaps as much
With other resolutions. How it weighs,
This hot, sick air! and how I covet here
The Dead's provision on the river's couch,
With silver curtains drawn on tinkling rings!
Or else their rest in quiet crypts,—laid by
From heat and noise!—from those cicale, say,
And this more vexing heart-beat.
So it is:
We covet for the soul, the body's part,
To die and rot. Even so, Aurora, ends
Our aspiration, who bespoke our place
So far in the east. The occidental flats
Had fed us fatter, therefore? we have climbed
Where herbage ends? we want the beast's part now,
And tire of the angel's?—Men define a man,
The creature who stands front-ward to the stars,
The creature who looks inward to himself,
The tool-wright, laughing creature. 'Tis enough:
We'll say instead, the inconsequent creature, man,—
For that's his specialty. What creature else
Conceives the circle, and then walks the square?
Loves things proved bad, and leaves a thing proved good?
You think the bee makes honey half a year,
To loathe the comb in winter, and desire
The little ant's food rather? But a man—
Note men!—they are but women after all,
As women are but Auroras!—there are men
Born tender, apt to pale at a trodden worm,
Who paint for pastime, in their favourite dream,
Spruce auto-vestments flowered with crocus-flames:
There are, too, who believe in hell, and lie:
There are, who waste their souls in working out
Life's problem on these sands betwixt two tides,
And end,—'Now give us the beast's part, in death.'
Alas, long-suffering and most patient God,
Thou need'st be surelier God to bear with us
Than even to have made us! thou, aspire, aspire
From henceforth for me! thou who hast, thyself,
Endured this fleshhood, knowing how, as a soaked
And sucking vesture, it would drag us down
And choke us in the melancholy Deep,
Sustain me, that, with thee, I walk these waves,
Resisting!—breathe me upward, thou for me
Aspiring, who art the way, the truth, the life,—
That no truth henceforth seem indifferent,
No way to truth laborious, and no life,
Not even this life I live, intolerable!
The days went by. I took up the old days
With all their Tuscan pleasures, worn and spoiled,—
Like some lost book we dropt in the long grass
On such a happy summer-afternoon
When last we read it with a loving friend,
And find in autumn, when the friend is gone,
The grass cut short, the weather changed, too late,
And stare at, as at something wonderful
For sorrow,—thinking how two hands, before,
Had held up what is left to only one,
And how we smiled when such a vehement nail
Impressed the tiny dint here, which presents
This verse in fire for ever! Tenderly
And mournfully I lived. I knew the birds
And insects,—which look fathered by the flowers
And emulous of their hues: I recognised
The moths, with that great overpoise of wings
Which makes a mystery of them how at all
They can stop flying: butterflies, that bear
Upon their blue wings such red embers round,
They seem to scorch the blue air into holes
Each flight they take: and fire-flies, that suspire
In short soft lapses of transported flame
Across the tingling Dark, while overhead
The constant and inviolable stars
Outburn those lights-of-love: melodious owls,
(If music had but one note and was sad,
'Twould sound just so) and all the silent swirl
Of bats, that seem to follow in the air
Some grand circumference of a shadowy dome
To which we are blind: and then, the nightingales,
Which pluck our heart across a garden-wall,
(When walking in the town) and carry it
So high into the bowery almond-trees,
We tremble and are afraid, and feel as if
The golden flood of moonlight unaware
Dissolved the pillars of the steady earth
And made it less substantial. And I knew
The harmless opal snakes, and large-mouthed frogs,
(Those noisy vaunters of their shallow streams)
And lizards, the green lightnings of the wall,
Which, if you sit down still, nor sigh too loud,
Will flatter you and take you for a stone,
And flash familiarly about your feet
With such prodigious eyes in such small heads!—
I knew them, though they had somewhat dwindled from
My childish imagery,—and kept in mind
How last I sate among them equally,
In fellowship and mateship, as a child
Will bear him still toward insect, beast, and bird,
Before the Adam in him has foregone
All privilege of Eden,—making friends
And talk, with such a bird or such a goat,
And buying many a two-inch-wide rush-cage
To let out the caged cricket on a tree,
Saying, 'Oh, my dear grillino, were you cramped?
And are you happy with the ilex-leaves?
And do you love me who have let you go?
Say _yes_ in singing, and I'll understand.'
But now the creatures all seemed farther off,
No longer mine, nor like me; only _there_,
A gulph between us. I could yearn indeed,
Like other rich men, for a drop of dew
To cool this heat,—a drop of the early dew,
The irrecoverable child-innocence
(Before the heart took fire and withered life)
When childhood might pair equally with birds;
But now ... the birds were grown too proud for us!
Alas, the very sun forbids the dew.
And I, I had come back to an empty nest,
Which every bird's too wise for. How I heard
My father's step on that deserted ground,
His voice along that silence, as he told
The names of bird and insect, tree and flower,
And all the presentations of the stars
Across Valdarno, interposing still
'My child,' 'my child.' When fathers say 'my child,'
'Tis easier to conceive the universe,
And life's transitions down the steps of law.
I rode once to the little mountain-house
As fast as if to find my father there,
But, when in sight of't, within fifty yards,
I dropped my horse's bridle on his neck
And paused upon his flank. The house's front
Was cased with lingots of ripe Indian corn
In tesselated order, and device
Of golden patterns: not a stone of wall
Uncovered,—not an inch of room to grow
A vine-leaf. The old porch had disappeared;
And, in the open doorway, sate a girl
At plaiting straws,—her black hair strained away
To a scarlet kerchief caught beneath her chin
In Tuscan fashion,—her full ebon eyes,
Which looked too heavy to be lifted so,
Still dropt and lifted toward the mulberry-tree
On which the lads were busy with their staves
In shout and laughter, stripping all the boughs
As bare as winter, of those summer leaves
My father had not changed for all the silk
In which the ugly silkworms hide themselves.
Enough. My horse recoiled before my heart—
I turned the rein abruptly. Back we went
As fast, to Florence.
That was trial enough
Of graves. I would not visit, if I could,
My father's, or my mother's any more,
To see if stone-cutter or lichen beat
So early in the race, or throw my flowers,
Which could not out-smell heaven, or sweeten earth.
They live too far above, that I should look
So far below to find them: let me think
That rather they are visiting my grave,
This life here, (undeveloped yet to life)
And that they drop upon me, now and then,
For token or for solace, some small weed
Least odorous of the growths of paradise,
To spare such pungent scents as kill with joy.
My old Assunta, too, was dead, was dead—
O land of all men's past! for me alone,
It would not mix its tenses. I was past,
It seemed, like others,—only not in heaven.
And, many a Tuscan eve, I wandered down
The cypress alley, like a restless ghost
That tries its feeble ineffectual breath
Upon its own charred funeral-brands put out
Too soon,—where, black and stiff, stood up the trees
Against the broad vermilion of the skies.
Such skies!—all clouds abolished in a sweep
Of God's skirt, with a dazzle to ghosts and men,
As down I went, saluting on the bridge
The hem of such, before 'twas caught away
Beyond the peaks of Lucca. Underneath,
The river, just escaping from the weight
Of that intolerable glory, ran
In acquiescent shadow murmurously:
And up, beside it, streamed the festa-folk
With fellow-murmurs from their feet and fans,
(With _issimo_ and _ino_ and sweet poise
Of vowels in their pleasant scandalous talk)
Returning from the grand-duke's dairy-farm
Before the trees grew dangerous at eight,
(For, 'trust no tree by moonlight,' Tuscans say)
To eat their ice at Doni's tenderly,—
Each lovely lady close to a cavalier
Who holds her dear fan while she feeds her smile
On meditative spoonfuls of vanille,
He breathing hot protesting vows of love,
Enough to thaw her cream, and scorch his beard.
'Twas little matter. I could pass them by
Indifferently, not fearing to be known.
No danger of being wrecked upon a friend,
And forced to take an iceberg for an isle!
The very English, here, must wait to learn
To hang the cobweb of their gossip out
And catch a fly. I'm happy. It's sublime,
This perfect solitude of foreign lands!
To be, as if you had not been till then,
And were then, simply that you chose to be:
To spring up, not be brought forth from the ground,
Like grasshoppers at Athens, and skip thrice
Before a woman makes a pounce on you
And plants you in her hair!—possess, yourself,
A new world all alive with creatures new,
New sun, new moon, new flowers, new people—ah,
And be possessed by none of them! no right
In one, to call your name, enquire your where,
Or what you think of Mister Some-one's book,
Or Mister Other's marriage, or decease,
Or how's the headache which you had last week,
Or why you look so pale still, since it's gone?
—Such most surprising riddance of one's life
Comes next one's death; it's disembodiment
Without the pang. I marvel, people choose
To stand stock-still like fakirs, till the moss
Grows on them, and they cry out, self-admired,
'How verdant and how virtuous!' Well, I'm glad:
Or should be, if grown foreign to myself
As surely as to others.
Musing so,
I walked the narrow unrecognising streets,
Where many a palace-front peers gloomily
Through stony vizors iron-barred, (prepared
Alike, should foe or lover pass that way,
For guest or victim) and came wandering out
Upon the churches with mild open doors
And plaintive wail of vespers, where a few,
Those chiefly women, sprinkled round in blots
Upon the dusky pavement, knelt and prayed
Toward the altar's silver glory. Oft a ray
(I liked to sit and watch) would tremble out,
Just touch some face more lifted, more in need,
Of course a woman's—while I dreamed a tale
To fit its fortunes. There was one who looked
As if the earth had suddenly grown too large
For such a little humpbacked thing as she;
The pitiful black kerchief round her neck
Sole proof she had had a mother. One, again,
Looked sick for love,—seemed praying some soft saint
To put more virtue in the new fine scarf
She spent a fortnight's meals on, yesterday,
That cruel Gigi might return his eyes
From Giuliana. There was one, so old,
So old, to kneel grew easier than to stand,—
So solitary, she accepts at last
Our Lady for her gossip, and frets on
Against the sinful world which goes its rounds
In marrying and being married, just the same
As when 'twas almost good and had the right,
(Her Gian alive, and she herself eighteen).
And yet, now even, if Madonna willed,
She'd win a tern in Thursday's lottery,
And better all things. Did she dream for nought,
That, boiling cabbage for the fast-day's soup,
It smelt like blessed entrails? such a dream
For nought? would sweetest Mary cheat her so,
And lose that certain candle, straight and white
As any fair grand-duchess in her teens,
Winch otherwise should flare here in a week?
_Benigna sis_, thou beauteous Queen of heaven!
I sate there musing, and imagining
Such utterance from such faces: poor blind souls
That writhed toward heaven along the devil's trail,—
Who knows, I thought, but He may stretch his hand
And pick them up? 'tis written in the Book,
He heareth the young ravens when they cry;
And yet they cry for carrion.—O my God,—
And we, who make excuses for the rest,
We do it in our measure. Then I knelt,
And dropped my head upon the pavement too,
And prayed, since I was foolish in desire
Like other creatures, craving offal-food,
That He would stop his ears to what I said,
And only listen to the run and beat
Of this poor, passionate, helpless blood—
And then
I lay, and spoke not. But He heard in heaven.
So many Tuscan evenings passed the same!
I could not lose a sunset on the bridge,
And would not miss a vigil in the church,
And liked to mingle with the out-door crowd
So strange and gay and ignorant of my face,
For men you know not, are as good as trees.
And only once, at the Santissima,
I almost chanced upon a man I knew,
Sir Blaise Delorme. He saw me certainly,
And somewhat hurried, as he crossed himself,
The smoothness of the action,—then half bowed,
But only half, and merely to my shade,
I slipped so quick behind the porphyry plinth,
And left him dubious if 'twas really I,
Or peradventure Satan's usual trick
To keep a mounting saint uncanonised.
But I was safe for that time, and he too;
The argent angels in the altar-flare
Absorbed his soul, next moment. The good man!
In England we were scarce acquaintances,
That here in Florence he should keep my thought
Beyond the image on his eye, which came
And went: and yet his thought disturbed my life:
For, after that, I oftener sate at home
On evenings, watching how they fined themselves
With gradual conscience to a perfect night,
Until the moon, diminished to a curve,
Lay out there, like a sickle for His hand
Who cometh down at last to reap the earth.
At such times, ended seemed my trade of verse;
I feared to jingle bells upon my robe
Before the four-faced silent cherubim:
With God so near me, could I sing of God?
I did not write, nor read, nor even think,
But sate absorbed amid the quickening glooms,
Most like some passive broken lump of salt
Dropt in by chance to a bowl of œnomel,
To spoil the drink a little, and lose itself,
Dissolving slowly, slowly, until lost.
EIGHTH BOOK.
ONE eve it happened, when I sate alone,
Alone, upon the terrace of my tower,
A book upon my knees, to counterfeit
The reading that I never read at all,
While Marian, in the garden down below,
Knelt by the fountain (I could just hear thrill
The drowsy silence of the exhausted day)
And peeled a new fig from that purple heap
In the grass beside her,—turning out the red
To feed her eager child, who sucked at it
With vehement lips across a gap of air
As he stood opposite, face and curls a-flame
With that last sun-ray, crying, 'give me, give,'
And stamping with imperious baby-feet,
(We're all born princes)—something startled me,—
The laugh of sad and innocent souls, that breaks
Abruptly, as if frightened at itself;
'Twas Marian laughed. I saw her glance above
In sudden shame that I should hear her laugh,
And straightway dropped my eyes upon my book,
And knew, the first time, 'twas Boccaccio's tales,
The Falcon's,—of the lover who for love
Destroyed the best that loved him. Some of us
Do it still, and then we sit and laugh no more.
Laugh _you_, sweet Marian! you've the right to laugh,
Since God himself is for you, and a child!
For me there's somewhat less,—and so, I sigh.
The heavens were making room to hold the night,
The sevenfold heavens unfolding all their gates
To let the stars out slowly (prophesied
In close-approaching advent, not discerned),
While still the cue-owls from the cypresses
Of the Poggio called and counted every pulse
Of the skyey palpitation. Gradually
The purple and transparent shadows slow
Had filled up the whole valley to the brim,
And flooded all the city, which you saw
As some drowned city in some enchanted sea,
Cut off from nature,—drawing you who gaze,
With passionate desire, to leap and plunge,
And find a sea-king with a voice of waves,
And treacherous soft eyes, and slippery locks
You cannot kiss but you shall bring away
Their salt upon your lips. The duomo-bell
Strikes ten, as if it struck ten fathoms down,
So deep; and fifty churches answer it
The same, with fifty various instances.
Some gaslights tremble along squares and streets;
The Pitti's palace-front is drawn in fire;
And, past the quays, Maria Novella's Place,
In which the mystic obelisks stand up
Triangular, pyramidal, each based
On a single trine of brazen tortoises,
To guard that fair church, Buonarroti's Bride,
That stares out from her large blind dial-eyes,
Her quadrant and armillary dials, black
With rhythms of many suns and moons, in vain
Enquiry for so rich a soul as his,—
Methinks I have plunged, I see it all so clear....
And, oh my heart, ... the sea-king!
In my ears
The sound of waters. There he stood, my king!
I felt him, rather than beheld him. Up
I rose, as if he were my king indeed,
And then sate down, in trouble at myself,
And struggling for my woman's empery.
'Tis pitiful; but women are so made:
We'll die for you, perhaps,—'tis probable;
But we'll not spare you an inch of our full height:
We'll have our whole just stature,—five feet four,
Though laid out in our coffins: pitiful!
—'You, Romney!—— Lady Waldemar is here?'
He answered in a voice which was not his.
'I have her letter; you shall read it soon:
But first, I must be heard a little, I,
Who have waited long and travelled far for that,
Although you thought to have shut a tedious book
And farewell. Ah, you dog-eared such a page,
And here you find me.'
Did he touch my hand,
Or but my sleeve? I trembled, hand and foot,—
He must have touched me.—'Will you sit?' I asked,
And motioned to a chair; but down he sate,
A little slowly, as a man in doubt,
Upon the couch beside me,—couch and chair
Being wheeled upon the terrace.
'You are come,
My cousin Romney?—this is wonderful.
But all is wonder on such summer-nights;
And nothing should surprise us any more,
Who see that miracle of stars. Behold.'
I signed above, where all the stars were out,
As if an urgent heat had started there
A secret writing from a sombre page,
A blank last moment, crowded suddenly
With hurrying splendours.
'Then you do not know'—
He murmured.
'Yes, I know,' I said, 'I know.
I had the news from Vincent Carrington.
And yet I did not think you'd leave the work
In England, for so much even,—though, of course,
You'll make a work-day of your holiday,
And turn it to our Tuscan people's use,—
Who much need helping since the Austrian boar
(So bold to cross the Alp by Lombardy
And dash his brute front unabashed against
The steep snow-bosses of that shield of God
Who soon shall rise in wrath and shake it clear,)
Came hither also,—raking up our vines
And olive-gardens with his tyrannous tusks,
And rolling on our maize with all his swine,'
'You had the news from Vincent Carrington,'
He echoed,—picking up the phrase beyond,
As if he knew the rest was merely talk
To fill a gap and keep out a strong wind,—
'You had, then, Vincent's personal news?'
'His own,'
I answered. 'All that ruined world of yours
Seems crumbling into marriage. Carrington
Has chosen wisely.'
'Do _you_ take it so?'
He cried, 'and is it possible at last' ...
He paused there,—and then, inward to himself,
'Too much at last, too late!—yet certainly' ...
(And there his voice swayed as an Alpine plank
That feels a passionate torrent underneath)
'The knowledge, if I had known it, first or last,
Had never changed the actual case for _me_.
And best, for _her_, at this time.'
Nay, I thought,
He loves Kate Ward, it seems, now, like a man,
Because he has married Lady Waldemar.
Ah, Vincent's letter said how Leigh was moved
To hear that Vincent was betrothed to Kate.
With what cracked pitchers go we to deep wells
In this world! Then I spoke,—'I did not think,
My cousin, you had ever known Kate Ward.'
'In fact I never knew her. 'Tis enough
That Vincent did, before he chose his wife
For other reasons than those topaz eyes
I've heard of. Not to undervalue them,
For all that. One takes up the world with eyes.'
—Including Romney Leigh, I thought again,
Albeit he knows them only by repute.
How vile must all men be, since _he's_ a man.
His deep pathetic voice, as if he guessed
I did not surely love him, took the word;
'You never got a letter from Lord Howe
A month back, dear Aurora?'
'None,' I said.
'I felt it was so,' he replied: 'Yet, strange!
Sir Blaise Delorme has passed through Florence?'
'Ay,
By chance I saw him in Our Lady's church,
(I saw him, mark you, but he saw not me)
Clean-washed in holy water from the count
Of things terrestrial,—letters and the rest;
He had crossed us out together with his sins.
Ay, strange; but only strange that good Lord Howe
Preferred him to the post because of pauls.
For me I'm sworn to never trust a man—
At least with letters.'
'There were facts to tell,—
To smooth with eye and accent. Howe supposed ...
Well, well, no matter! there was dubious need;
You heard the news from Vincent Carrington.
And yet perhaps you had been startled less
To see me, dear Aurora, if you had read
That letter.'
—Now he sets me down as vexed.
I think I've draped myself in woman's pride
To a perfect purpose. Oh, I'm vexed, it seems!
My friend Lord Howe deputes his friend Sir Blaise,
To break as softly as a sparrow's egg
That lets a bird out tenderly, the news
Of Romney's marriage to a certain saint;
To _smooth with eye and accent_,—indicate
His possible presence. Excellently well
You've played your part, my Lady Waldemar,—
As I've played mine.
'Dear Romney,' I began,
'You did not use, of old, to be so like
A Greek king coming from a taken Troy,
'Twas needful that precursors spread your path
With three-piled carpets, to receive your foot
And dull the sound of't. For myself, be sure,
Although it frankly ground the gravel here,
I still could bear it. Yet I'm sorry, too,
To lose this famous letter, which Sir Blaise
Has twisted to a lighter absently
To fire some holy taper with: Lord Howe
Writes letters good for all things but to lose;
And many a flower of London gossipry
Has dropt wherever such a stem broke off,—
Of course I know that, lonely among my vines,
Where nothing's talked of, save the blight again,
And no more Chianti! Still the letter's use
As preparation ... Did I start indeed?
Last night I started at a cockchafer,
And shook a half-hour after. Have you learnt
No more of women, 'spite of privilege,
Than still to take account too seriously
Of such weak flutterings? Why, we like it, sir,—
We get our powers and our effects that way.
The trees stand stiff and still at time of frost,
If no wind tears them; but, let summer come,
When trees are happy,—and a breath avails
To set them trembling through a million leaves
In luxury of emotion. Something less
It takes to move a woman: let her start
And shake at pleasure,—nor conclude at yours,
The winter's bitter,—but the summer's green.'
He answered, 'Be the summer ever green
With you, Aurora!—though you sweep your sex
With somewhat bitter gusts from where you live
Above them,—whirling downward from your heights
Your very own pine-cones, in a grand disdain
Of the lowland burrs with which you scatter them.
So high and cold to others and yourself,
A little less to Romney, were unjust,
And thus, I would not have you. Let it pass:
I feel content, so. You can bear indeed
My sudden step beside you: but for me,
'Twould move me sore to hear your softened voice,—
Aurora's voice,—if softened unaware
In pity of what I am.'
Ah friend, I thought,
As husband of the Lady Waldemar
You're granted very sorely pitiable!
And yet Aurora Leigh must guard her voice
From softening in the pity of your case,
As if from lie or licence. Certainly
We'll soak up all the slush and soil of life
With softened voices, ere we come to _you_.
At which I interrupted my own thought
And spoke out calmly. 'Let us ponder, friend,
Whate'er our state, we must have made it first;
And though the thing displease us, ay, perhaps
Displease us warrantably, never doubt
That other states, thought possible once, and then
Rejected by the instinct of our lives,—
If then adopted, had displeased us more
Than this, in which the choice, the will, the love,
Has stamped the honour of a patent act
From henceforth. What we choose, may not be good;
But, that we choose it, proves it good for _us_
Potentially, fantastically, now
Or last year, rather than a thing we saw,
And saw no need for choosing. Moths will burn
Their wings,—which proves that light is good for moths,
Or else they had flown not, where they agonise,'
'Ay, light is good,' he echoed, and there paused.
And then abruptly, ... 'Marian. Marian's well?'
I bowed my head, but found no word. 'Twas hard
To speak of _her_ to Lady Waldemar's
New husband. How much did he know, at last?
How much? how little?—— He would take no sign,
But straight repeated,—'Marian. Is she well?'
'She's well,' I answered.
She was there in sight
An hour back, but the night had drawn her home;
Where still I heard her in an upper room,
Her low voice singing to the child in bed,
Who restless with the summer-heat and play
And slumber snatched at noon, was long sometimes
At falling off, and took a score of songs
And mother-hushes, ere she saw him sound.
'She's well,' I answered.
'Here?' he asked.
'Yes, here.'
He stopped and sighed. 'That shall be presently,
But now this must be. I have words to say,
And would be alone to say them, I with you,
And no third troubling.'
'Speak then,' I returned,
'She will not vex you.'
At which, suddenly
He turned his face upon me with its smile,
As if to crush me. 'I have read your book,
Aurora.'
'You have read it,' I replied,
'And I have writ it,—we have done with it.
And now the rest?'
'The rest is like the first,'
He answered,—'for the book is in my heart,
Lives in me, wakes in me, and dreams in me:
My daily bread tastes of it,—and my wine
Which has no smack of it, I pour it out;
It seems unnatural drinking.'
Bitterly
I took the word up; 'Never waste your wine.
The book lived in me ere it lived in you;
I know it closer than another does,
And that it's foolish, feeble, and afraid,
And all unworthy so much compliment.
Beseech you, keep your wine,—and, when you drink,
Still wish some happier fortune to your friend,
Than even to have written a far better book.'
He answered gently, 'That is consequent:
The poet looks beyond the book he has made,
Or else he had not made it. If a man
Could make a man, he'd henceforth be a god
In feeling what a little thing is man:
It is not my case. And this special book,
I did not make it, to make light of it:
It stands above my knowledge, draws me up;
'Tis high to me. It may be that the book
Is not so high, but I so low, instead;
Still high to me. I mean no compliment:
I will not say there are not, young or old,
Male writers, ay, or female,—let it pass,
Who'll write us richer and completer books.
A man may love a woman perfectly,
And yet by no means ignorantly maintain
A thousand women have not larger eyes:
Enough that she alone has looked at him
With eyes that, large or small, have won his soul.
And so, this book, Aurora,—so, your book.'
'Alas,' I answered, 'is it so, indeed?'
And then was silent.
'Is it so, indeed,'
He echoed, 'that _alas_ is all your word?'
I said,—'I'm thinking of a far-off June,
When you and I, upon my birthday once,
Discoursed of life and art, with both untried.
I'm thinking, Romney, how 'twas morning then,
And now 'tis night.'
'And now,' he said, ''tis night.'
'I'm thinking,' I resumed, ''tis somewhat sad
That if I had known, that morning in the dew,
My cousin Romney would have said such words
On such a night, at close of many years,
In speaking of a future book of mine,
It would have pleased me better as a hope,
Than as an actual grace it can at all.
That's sad, I'm thinking.'
'Ay,' he said, ''tis night.'
'And there,' I added lightly, 'are the stars!
And here, we'll talk of stars, and not of books.'
'You have the stars,' he murmured,—'it is well:
Be like them! shine, Aurora, on my dark,
Though high and cold and only like a star,
And for this short night only,—you, who keep
The same Aurora of the bright June day
That withered up the flowers before my face,
And turned me from the garden evermore
Because I was not worthy. Oh, deserved,
Deserved! That I, who verily had not learnt
God's lesson half, attaining as a dunce
To obliterate good words with fractious thumbs
And cheat myself of the context,—_I_ should push
Aside, with male ferocious impudence,
The world's Aurora who had conned her part
On the other side the leaf! ignore her so,
Because she was a woman and a queen,
And had no beard to bristle through her song,—
My teacher, who has taught me with a book,
My Miriam, whose sweet mouth, when nearly drowned
I still heard singing on the shore! Deserved,
That here I should look up unto the stars
And miss the glory' ...
'Can I understand?'
I broke in. 'You speak wildly, Romney Leigh,
Or I hear wildly. In that morning-time
We recollect, the roses were too red,
The trees too green, reproach too natural
If one should see not what the other saw:
And now, it's night, remember; we have shades
In place of colours; we are now grown cold,
And old, my cousin Romney. Pardon me,—
I'm very happy that you like my book,
And very sorry that I quoted back
A ten years' birthday; 'twas so mad a thing
In any woman, I scarce marvel much
You took it for a venturous piece of spite,
Provoking such excuses, as indeed
I cannot call you slack in.'
'Understand,'
He answered sadly, 'something, if but so.
This night is softer than an English day,
And men may well come hither when they're sick,
To draw in easier breath from larger air.
'Tis thus with me; I've come to you,—to you,
My Italy of women, just to breathe
My soul out once before you, ere I go,
As humble as God makes me at the last,
(I thank Him) quite out of the way of men,
And yours, Aurora,—like a punished child,
His cheeks all blurred with tears and naughtiness,
To silence in a corner. I am come
To speak, beloved'....
'Wisely, cousin Leigh,
And worthily of us both!'
'Yes, worthily;
For this time I must speak out and confess
That I, so truculent in assumption once,
So absolute in dogma, proud in aim,
And fierce in expectation,—I, who felt
The whole world tugging at my skirts for help,
As if no other man than I, could pull,
Nor woman, but I led her by the hand,
Nor cloth hold, but I had it in my coat,—
Do know myself to-night for what I was
On that June-day, Aurora. Poor bright day,
Which meant the best ... a woman and a rose, ...
And which I smote upon the cheek with words,
Until it turned and rent me! Young you were,
That birthday, poet, but you talked the right:
While I, ... I built up follies like a wall
To intercept the sunshine and your face.
Your face! that's worse.'
'Speak wisely, cousin Leigh.'
'Yes, wisely, dear Aurora, though too late:
But then, not wisely. I was heavy then,
And stupid, and distracted with the cries
Of tortured prisoners in the polished brass
Of that Phalarian bull, society,—
Which seems to bellow bravely like ten bulls,
But, if you listen, moans and cries instead
Despairingly, like victims tossed and gored
And trampled by their hoofs. I heard the cries
Too close: I could not hear the angels lift
A fold of rustling air, nor what they said
To help my pity. I beheld the world
As one great famishing carnivorous mouth,—
A huge, deserted, callow, black, bird Thing,
With piteous open beak that hurt my heart,
Till down upon the filthy ground I dropped,
And tore the violets up to get the worms.
Worms, worms, was all my cry: an open mouth,
A gross want, bread to fill it to the lips,
No more! That poor men narrowed their demands
To such an end, was virtue, I supposed,
Adjudicating that to see it so
Was reason. Oh, I did not push the case
Up higher, and ponder how it answers, when
The rich take up the same cry for themselves,
Professing equally,—'an open mouth
A gross want, food to fill us, and no more!'
Why that's so far from virtue, only vice
Finds reason for it! That makes libertines:
That slurs our cruel streets from end to end
With eighty thousand women in one smile,
Who only smile at night beneath the gas:
The body's satisfaction and no more,
Being used for argument against the soul's,
Here too! the want, here too, implying the right.
—How dark I stood that morning in the sun,
My best Aurora, though I saw your eyes,—
When first you told me ... oh, I recollect
The words ... and how you lifted your white hand,
And how your white dress and your burnished curls
Went greatening round you in the still blue air,
As if an inspiration from within
Had blown them all out when you spoke the same,
Even these,—'You will not compass your poor ends
Of barley-feeding and material ease,
Without the poet's individualism
To work your universal. It takes a soul,
To move a body,—it takes a high-souled man,
To move the masses ... even to a cleaner stye:
It takes the ideal, to blow an inch inside
The dust of the actual: and your Fouriers failed,
Because not poets enough to understand
That life develops from within.' I say
Your words,—I could say other words of yours;
For none of all your words has been more lost
Than sweet verbena, which, being brushed against,
Will hold you three hours after by the smell,
In spite of long walks on the windy hills.
But these words dealt in sharper perfume,—these
Were ever on me, stinging through my dreams,
And saying themselves for ever o'er my acts
Like some unhappy verdict. That I failed,
Is certain. Stye or no stye, to contrive
The swine's propulsion toward the precipice,
Proved easy and plain. I subtly organised
And ordered, built the cards up high and higher,
Till, some one breathing, all fell flat again;
In setting right society's wide wrong,
Mere life's so fatal! So I failed indeed
Once, twice, and oftener,—hearing through the rents
Of obstinate purpose, still those words of yours,
'_You will not compass your poor ends, not you!_'
But harder than you said them; every time
Still farther from your voice, until they came
To overcrow me with triumphant scorn
Which vexed me to resistance. Set down this
For condemnation,—I was guilty here:
I stood upon my deed and fought my doubt,
As men will,—for I doubted,—till at last
My deed gave way beneath me suddenly,
And left me what I am. The curtain dropped,
My part quite ended, all the footlights quenched,
My own soul hissing at me through the dark,
I, ready for confession,—I was wrong,
I've sorely failed; I've slipped the ends of life,
I yield; you have conquered.'
'Stay,' I answered him;
'I've something for your hearing, also. I
Have failed too.'
'You!' he said, 'you're very great;
The sadness of your greatness fits you well:
As if the plume upon a hero's casque
Should nod a shadow upon his victor face.'
I took him up austerely,—'You have read
My book, but not my heart; for recollect,
'Tis writ in Sanscrit, which you bungle at.
I've surely failed, I know; if failure means
To look back sadly on work gladly done,—
To wander on my mountains of Delight,
So called, (I can remember a friend's words
As well as you, sir,) weary and in want
Of even a sheep-path, thinking bitterly....
Well, well! no matter. I but say so much,
To keep you, Romney Leigh, from saying more,
And let you feel I am not so high indeed,
That I can bear to have you at my foot,—
Or safe, that I can help you. That June-day,
Too deeply sunk in craterous sunsets now
For you or me to dig it up alive;
To pluck it out all bleeding with spent flame
At the roots, before those moralising stars
We have got instead,—that poor lost day, you said
Some words as truthful as the thing of mine
You care to keep in memory: and I hold
If I, that day, and, being the girl I was,
Had shown a gentler spirit, less arrogance,
It had not hurt me. Ah, you'll not mistake
The point here. I but only think, you see,
More justly, that's more humbly, of myself,
Than when I tried a crown on and supposed....
Nay, laugh, sir,—I'll laugh with you!—pray you, laugh.
I've had so many birthdays since that day,
I've learnt to prize mirth's opportunities,
Which come too seldom. Was it you who said
I was not changed? the same Aurora? Ah,
We could laugh there, too! Why, Ulysses' dog
Knew _him_, and wagged his tail and died: but if
I had owned a dog, I too, before my Troy,
And, if you brought him here, ... I warrant you
He'd look into my face, bark lustily,
And live on stoutly, as the creatures will
Whose spirits are not troubled by long loves.
A dog would never know me, I'm so changed;
Much less a friend ... except that you're misled
By the colour of the hair, the trick of the voice,
Like that Aurora Leigh's.'
'Sweet trick of voice!
I would be a dog for this, to know it at last,
And die upon the falls of it. O love,
O best Aurora! are you then so sad,
You scarcely had been sadder as my wife?'
'Your wife, sir! I must certainly be changed,
If I, Aurora, can have said a thing
So light, it catches at the knightly spurs
Of a noble gentleman like Romney Leigh,
And trips him from his honourable sense
Of what befits' ...
'You wholly misconceive,'
He answered.
I returned,—'I'm glad of it;
But keep from misconception, too, yourself:
I am not humbled to so low a point,
Nor so far saddened. If I am sad at all,
Ten layers of birthdays on a woman's head,
Are apt to fossilise her girlish mirth,
Though ne'er so merry: I'm perforce more wise,
And that, in truth, means sadder. For the rest,
Look here, sir: I was right upon the whole,
That birthday morning. 'Tis impossible
To get at men excepting through their souls,
However open their carnivorous jaws;
And poets get directlier at the soul,
Than any of your œconomists:—for which,
You must not overlook the poet's work
When scheming for the world's necessities.
The soul's the way. Not even Christ Himself
Can save man else than as He holds man's soul;
And therefore did He come into our flesh,
As some wise hunter creeping on his knees
With a torch, into the blackness of some cave,
To face and quell the beast there,—take the soul,
And so possess the whole man, body and soul.
I said, so far, right, yes; not farther, though:
We both were wrong that June-day,—both as wrong
As an east wind had been. I who talked of art,
And you who grieved for all men's griefs ... what then?
We surely made too small a part for God
In these things. What we are, imports us more
Than what we eat; and life, you've granted me,
Develops from within. But innermost
Of the inmost, most interior of the interne,
God claims his own, Divine humanity
Renewing nature,—or the piercingest verse,
Prest in by subtlest poet, still must keep
As much upon the outside of a man,
As the very bowl, in which he dips his beard.
—And then, ... the rest. I cannot surely speak.
Perhaps I doubt more than you doubted then,
If I, the poet's veritable charge,
Have borne upon my forehead. If I have,
It might feel somewhat liker to a crown,
The foolish green one even.—Ah, I think,
And chiefly when the sun shines, that I've failed.
But what then, Romney? Though we fail indeed,
You ... I ... a score of such weak workers, ... He
Fails never. If He cannot work by us,
He will work over us. Does He want a man,
Much less a woman, think you? Every time
The star winks there, so many souls are born,
Who all shall work too. Let our own be calm:
We should be ashamed to sit beneath those stars,
Impatient that we're nothing.'
'Could we sit
Just so for ever, sweetest friend,' he said,
'My failure would seem better than success.
And yet, indeed, your book has dealt with me
More gently, cousin, than you ever will!
The book brought down entire the bright June-day,
And set me wandering in the garden-walks,
And let me watch the garland in a place,
You blushed so ... nay, forgive me; do not stir:
I only thank the book for what it taught,
And what, permitted. Poet, doubt yourself;
But never doubt that you're a poet to me
From henceforth. Ah, you've written poems, sweet,
Which moved me in secret, as the sap is moved
In still March-branches, signless as a stone:
But this last book o'ercame me like soft rain
Which falls at midnight, when the tightened bark
Breaks out into unhesitating buds,
And sudden protestations of the spring.
In all your other books, I saw but _you_:
A man may see the moon so, in a pond,
And not be nearer therefore to the moon,
Nor use the sight ... except to drown himself:
And so I forced my heart back from the sight;
For what had _I_, I thought, to do with _her_,—
Aurora ... Romney? But, in this last book,
You showed me something separate from yourself,
Beyond you; and I bore to take it in,
And let it draw me. You have shown me truths,
O June-day friend, that help me now at night,
When June is over! truths not yours, indeed,
But set within my reach by means of you:
Presented by your voice and verse the way
To take them clearest. Verily I was wrong;
And verily, many thinkers of this age,
Ay, many Christian teachers, half in heaven,
Are wrong in just my sense, who understood
Our natural world too insularly, as if
No spiritual counterpart completed it
Consummating its meaning, rounding all
To justice and perfection, line by line,
Form by form, nothing single, nor alone,—
The great below clenched by the great above;
Shade here authenticating substance there;
The body proving spirit, as the effect
The cause: we, meantime, being too grossly apt
To hold the natural, as dogs a bone,
(Though reason and nature beat us in the face);
So obstinately, that we'll break our teeth
Or ever we let go. For everywhere
We're too materialistic,—eating clay,
(Like men of the west) instead of Adam's corn
And Noah's wine; clay by handfuls, clay by lumps,
Until we're filled up to the throat with clay,
And grow the grimy colour of the ground
On which we are feeding. Ay, materialist
The age's name is. God himself, with some,
Is apprehended as the bare result
Of what his hand materially has made,
Expressed in such an algebraic sign,
Called God;—that is, to put it otherwise,
They add up nature to a naught of God
And cross the quotient. There are many, even,
Whose names are written in the Christian church
To no dishonour,—diet still on mud,
And splash the altars with it. You might think
The clay, Christ laid upon their eyelids when,
Still blind, he called them to the use of sight,
Remained there to <DW44> its exercise
With clogging incrustations. Close to heaven,
They see, for mysteries, through the open doors,
Vague puffs of smoke from pots of earthenware;
And fain would enter, when their time shall come,
With quite a different body than St. Paul
Has promised,—husk and chaff, the whole barley-corn,
Or where's the resurrection?'
'Thus it is,'
I sighed. And he resumed with mournful face.
'Beginning so, and filling up with clay
The wards of this great key, the natural world,
And fumbling vainly therefore at the lock
Of the spiritual,—we feel ourselves shut in
With all the wild-beast roar of struggling life,
The terrors and compunctions of our souls,
As saints with lions,—we who are not saints,
And have no heavenly lordship in our stare
To awe them backward! Ay, we are forced, so pent,
To judge the whole too partially, ... confound
Conclusions. Is there any common phrase
Significant, when the adverb's heard alone,
The verb being absent, and the pronoun out?
But we, distracted in the roar of life,
Still insolently at God's adverb snatch,
And bruit against Him that his thought is void,
His meaning hopeless;—cry, that everywhere
The government is slipping from his hand,
Unless some other Christ ... say Romney Leigh ...
Come up, and toil and moil, and change the world,
For which the First has proved inadequate,
However we talk bigly of His work
And piously of His person. We blaspheme
At last, to finish that doxology,
Despairing on the earth for which He died.'
'So now,' I asked, 'you have more hope of men?'
'I hope,' he answered: 'I am come to think
That God will have his work done, as you said,
And that we need not be disturbed too much
For Romney Leigh or others having failed
With this or that quack nostrum,—recipes
For keeping summits by annulling depths,
For learning wrestling with long lounging sleeves,
And perfect heroism without a scratch.
We fail,—what, then? Aurora, if I smiled
To see you, in your lovely morning-pride,
Try on the poet's wreath which suits the noon,—
(Sweet cousin, walls must get the weather-stain
Before they grow the ivy!) certainly
I stood myself there worthier of contempt,
Self-rated, in disastrous arrogance,
As competent to sorrow for mankind
And even their odds. A man may well despair,
Who counts himself so needful to success.
I failed. I throw the remedy back on God,
And sit down here beside you, in good hope.'
'And yet, take heed,' I answered, 'lest we lean
Too dangerously on the other side,
And so fail twice. Be sure, no earnest work
Of any honest creature, howbeit weak,
Imperfect, ill-adapted, fails so much,
It is not gathered as a grain of sand
To enlarge the sum of human action used
For carrying out God's end. No creature works
So ill, observe, that therefore he's cashiered.
The honest earnest man must stand and work;
The woman also; otherwise she drops
At once below the dignity of man,
Accepting serfdom. Free men freely work:
Whoever fears God, fears to sit at ease.'
He cried, 'True. After Adam, work was curse;
The natural creature labours, sweats and frets.
But, after Christ, work turns to privilege;
And henceforth one with our humanity,
The Six-day Worker, working still in us,
Has called us freely to work on with Him
In high companionship. So, happiest!
I count that Heaven itself is only work
To a surer issue. Let us work, indeed,—
But, no more, work as Adam ... nor as Leigh
Erewhile, as if the only man on earth,
Responsible for all the thistles blown
And tigers couchant,—struggling in amaze
Against disease and winter,—snarling on
For ever, that the world's not paradise.
Oh cousin, let us be content, in work,
To do the thing we can, and not presume
To fret because it's little. 'Twill employ
Seven men, they say, to make a perfect pin:
Who makes the head, content to miss the point,—
Who makes the point, agreed to leave the join:
And if a man should cry, 'I want a pin,
And I must make it straightway, head and point,'—
His wisdom is not worth the pin he wants.
Seven men to a pin,—and not a man too much!
Seven generations, haply, to this world,
To right it visibly, a finger's breadth,
And mend its rents a little. Oh, to storm
And say,—'This world here is intolerable;
I will not eat this corn, nor drink this wine,
Nor love this woman, flinging her my soul
Without a bond for't, as a lover should,
Nor use the generous leave of happiness
As not too good for using generously'—
(Since virtue kindles at the touch of joy,
Like a man's cheek laid on a woman's hand;
And God, who knows it, looks for quick returns
From joys)!—to stand and claim to have a life
Beyond the bounds of the individual man,
And raze all personal cloisters of the soul
To build up public stores and magazines,
As if God's creatures otherwise were lost,
The builder surely saved by any means!
To think,—I have a pattern on my nail,
And I will carve the world new after it,
And solve so, these hard social questions,—nay,
Impossible social questions,—since their roots
Strike deep in Evil's own existence here,
Which God permits because the question's hard
To abolish evil nor attaint free-will.
Ay, hard to God, but not to Romney Leigh!
For Romney has a pattern on his nail,
(Whatever may be lacking on the Mount)
And not being overnice to separate
What's element from what's convention, hastes
By line on line, to draw you out a world,
Without your help indeed, unless you take
His yoke upon you and will learn of him,—
So much he has to teach! so good a world!
The same, the whole creation's groaning for!
No rich nor poor, no gain nor loss nor stint,
No potage in it able to exclude
A brother's birthright, and no right of birth,
The potage,—both secured to every man;
And perfect virtue dealt out like the rest,
Gratuitously, with the soup at six,
To whoso does not seek it.'
'Softly, sir,'
I interrupted,—'I had a cousin once
I held in reverence. If he strained too wide,
It was not to take honour, but give help;
The gesture was heroic. If his hand
Accomplished nothing ... (well, it is not proved)
That empty hand thrown impotently out
Were sooner caught, I think, by One in heaven,
Than many a hand that reaped a harvest in
And keeps the scythe's glow on it. Pray you, then,
For my sake merely, use less bitterness
In speaking of my cousin.'
'Ah,' he said,
'Aurora! when the prophet beats the ass,
The angel intercedes.' He shook his head—
'And yet to mean so well, and fail so foul,
Expresses ne'er another beast than man;
The antithesis is human. Harken, dear;
There's too much abstract willing, purposing,
In this poor world. We talk by aggregates,
And think by systems; and, being used to face
Our evils in statistics, are inclined
To cap them with unreal remedies
Drawn out in haste on the other side the slate.'
'That's true,' I answered, fain to throw up thought,
And make a game of't; 'Oh, we generalise
Enough to please you. If we pray at all,
We pray no longer for our daily bread,
But next centenary's harvests. If we give,
Our cup of water is not tendered till
We lay down pipes and found a Company
With Branches. Ass or angel, 'tis the same:
A woman cannot do the thing she ought,
Which means whatever perfect thing she can,
In life, in art, in science, but she fears
To let the perfect action take her part
And rest there: she must prove what she can do
Before she does it,—prate of woman's rights,
Of woman's mission, woman's function, till
The men (who are prating, too, on their side) cry,
'A woman's function plainly is ... to talk.'
Poor souls, they are very reasonably vexed!
They cannot hear each other speak.'
'And you,
An artist, judge so?'
'I, an artist,—yes,
Because, precisely, I'm an artist, sir,
And woman,—if another sate in sight,
I'd whisper,—Soft, my sister! not a word!
By speaking we prove only we can speak;
Which he, the man here, never doubted. What
He doubts, is whether we can _do_ the thing
With decent grace, we've not yet done at all:
Now, do it; bring your statue,—you have room!
He'll see it even by the starlight here;
And if 'tis e'er so little like the god
Who looks out from the marble silently
Along the track of his own shining dart
Through the dusk of ages,—there's no need to speak;
The universe shall henceforth speak for you,
And witness, 'She who did this thing, was born
To do it,—claims her license in her work.'
—And so with more works. Whoso cures the plague,
Though twice a woman, shall be called a leech:
Who rights a land's finances, is excused
For touching coppers, though her hands be white,—
But we, we talk!'
'It is the age's mood,'
He said; 'we boast, and do not. We put up
Hostelry signs where'er we lodge a day,—
Some red colossal cow, with mighty paps
A Cyclops' fingers could not strain to milk;
Then bring out presently our saucer-full
Of curds. We want more quiet in our works,
More knowledge of the bounds in which we work;
More knowledge that each individual man
Remains an Adam to the general race,
Constrained to see, like Adam, that he keep
His personal state's condition honestly,
Or vain all thoughts of his to help the world,
Which still must be developed from its _one_,
If bettered in its many. We, indeed,
Who think to lay it out new like a park,
We take a work on us which is not man's;
For God alone sits far enough above,
To speculate so largely. None of us
(Not Romney Leigh) is mad enough to say,
We'll have a grove of oaks upon that <DW72>
And sink the need of acorns. Government,
If veritable and lawful, is not given
By imposition of the foreign hand,—
Nor chosen from a pretty pattern-book
Of some domestic idealogue, who sits
And coldly chooses empire, where as well
He might republic. Genuine government
Is but the expression of a nation, good
Or less good,—even as all society,
Howe'er unequal, monstrous, crazed, and cursed,
Is but the expression of men's single lives,
The loud sum of the silent units. What,
We'd change the aggregate and yet retain
Each separate figure? Whom do we cheat by that?
Now, not even Romney.'
'Cousin, you are sad.
Did all your social labour at Leigh Hall
And elsewhere, come to nought then?'
'It _was_ nought,'
He answered mildly. 'There is room indeed,
For statues still, in this large world of God's,
But not for vacuums,—so I am not sad:
Not sadder than is good for what I am.
My vain phalanstery dissolved itself;
My men and women of disordered lives,
I brought in orderly to dine and sleep,
Broke up those waxen masks I made them wear,
With fierce contortions of the natural face;
And cursed me for my tyrannous constraint
In forcing crooked creatures to live straight;
And set the country hounds upon my back
To bite and tear me for my wicked deed
Of trying to do good without the church
Or even the squires, Aurora. Do you mind
Your ancient neighbours? The great book-club teems
With 'sketches,' 'summaries,' and 'last tracts' but twelve,
On socialistic troublers of close bonds
Betwixt the generous rich and grateful poor.
The vicar preached from 'Revelations,' (till
The doctor woke) and found me with 'the frogs'
On three successive Sundays; ay, and stopped
To weep a little (for he's getting old)
That such perdition should o'ertake a man
Of such fair acres,—in the parish, too!
He printed his discourses 'by request;'
And if your book shall sell as his did, then
Your verses are less good than I suppose.
The women of the neighbourhood subscribed,
And sent me a copy bound in scarlet silk,
Tooled edges, blazoned with the arms of Leigh:
I own that touched me.'
'What, the pretty ones?
Poor Romney!'
'Otherwise the effect was small.
I had my windows broken once or twice
By liberal peasants, naturally incensed
At such a vexer of Arcadian peace,
Who would not let men call their wives their own
To kick like Britons,—and made obstacles
When things went smoothly as a baby drugged,
Toward freedom and starvation; bringing down
The wicked London tavern-thieves and drabs,
To affront the blessed hillside drabs and thieves
With mended morals, quotha,—fine new lives!—
My windows paid for't. I was shot at, once,
By an active poacher who had hit a hare
From the other barrel, tired of springeing game
So long upon my acres, undisturbed,
And restless for the country's virtue, (yet
He missed me)—ay, and pelted very oft
In riding through the village. 'There he goes,
Who'd drive away our Christian gentlefolks,
To catch us undefended in the trap
He baits with poisonous cheese, and lock us up
In that pernicious prison of Leigh Hall
With all his murderers! Give another name,
And say Leigh Hell, and burn it up with fire.'
And so they did, at last, Aurora.'
'Did?'
'You never heard it, cousin? Vincent's news
Came stinted, then.'
'They did? they burnt Leigh Hall?'
'You're sorry, dear Aurora? Yes indeed,
They did it perfectly: a thorough work,
And not a failure, this time. Let us grant
'Tis somewhat easier, though, to burn a house
Than build a system:—yet that's easy, too,
In a dream. Books, pictures,—ay, the pictures! what,
You think your dear Vandykes would give them pause?
Our proud ancestral Leighs with those peaked beards,
Or bosoms white as foam thrown up on rocks
From the old-spent wave. Such calm defiant looks
They flared up with! now, nevermore they'll twit
The bones in the family-vault with ugly death.
Not one was rescued, save the Lady Maud,
Who threw you down, that morning you were born,
The undeniable lineal mouth and chin,
To wear for ever for her gracious sake;
For which good deed I saved her: the rest went:
And you, you're sorry, cousin. Well, for me,
With all my phalansterians safely out,
(Poor hearts, they helped the burners, it was said,
And certainly a few clapped hands and yelled)
The ruin did not hurt me as it might,—
As when for instance I was hurt one day,
A certain letter being destroyed. In fact,
To see the great house flare so ... oaken floors,
Our fathers made so fine with rushes once,
Before our mothers furbished them with trains,—
Carved wainscoats, panelled walls, the favourite slide
For draining off a martyr, (or a rogue)
The echoing galleries, half a half-mile long,
And all the various stairs that took you up
And took you down, and took you round about
Upon their slippery darkness, recollect,
All helping to keep up one blazing jest;
The flames through all the casements pushing forth,
Like red-hot devils crinkled into snakes,
All signifying,—'Look you, Romney Leigh,
We save the people from your saving, here,
Yet so as by fire! we make a pretty show
Besides,—and that's the best you've ever done.'—
—To see this, almost moved myself to clap!
The 'vale et plaude' came, too, with effect,
When, in the roof fell, and the fire, that paused,
Stunned momently beneath the stroke of slates
And tumbling rafters, rose at once and roared,
And wrapping the whole house, (which disappeared
In a mounting whirlwind of dilated flame,)
Blew upward, straight, its drift of fiery chaff
In the face of Heaven, ... which blenched, and ran up higher.'
'Poor Romney!'
'Sometimes when I dream,' he said,
'I hear the silence after; 'twas so still.
For all those wild beasts, yelling, cursing round,
Were suddenly silent, while you counted five!
So silent, that you heard a young bird fall
From the top-nest in the neighbouring rookery
Through edging over-rashly toward the light.
The old rooks had already fled too far,
To hear the screech they fled with, though you saw
Some flying on still, like scatterings of dead leaves
In autumn-gusts, seen dark against the sky:
All flying,—ousted, like the House of Leigh.'
'Dear Romney!'
'Evidently 'twould have been
A fine sight for a poet, sweet, like you,
To make the verse blaze after. I myself,
Even I, felt something in the grand old trees,
Which stood that moment like brute Druid gods
Amazed upon the rim of ruin, where,
As into a blackened socket, the great fire
Had dropped,—still throwing up splinters now and then,
To show them grey with all their centuries,
Left there to witness that on such a day
The house went out.'
'Ah!'
'While you counted five
I seemed to feel a little like a Leigh,—
But then it passed, Aurora. A child cried;
And I had enough to think of what to do
With all those houseless wretches in the dark,
And ponder where they'd dance the next time, they
Who had burnt the viol.'
'Did you think of that?
Who burns his viol will not dance, I know,
To cymbals, Romney.'
'O my sweet sad voice,'
He cried,—'O voice that speaks and overcomes!
The sun is silent, but Aurora speaks.'
'Alas,' I said; 'I speak I know not what:
I'm back in childhood, thinking as a child,
A foolish fancy—will it make you smile?
I shall not from the window of my room
Catch sight of those old chimneys any more.'
'No more,' he answered. 'If you pushed one day
Through all the green hills to our fathers' house,
You'd come upon a great charred circle where
The patient earth was singed an acre round;
With one stone-stair, symbolic of my life,
Ascending, winding, leading up to nought!
'Tis worth a poet's seeing. Will you go?'
I made no answer. Had I any right
To weep with this man, that I dared to speak?
A woman stood between his soul and mine,
And waved us off from touching evermore
With those unclean white hands of hers. Enough.
We had burnt our viols and were silent.
So,
The silence lengthened till it pressed. I spoke,
To breathe: 'I think you were ill afterward.'
'More ill,' he answered, 'had been scarcely ill.
I hoped this feeble fumbling at life's knot
Might end concisely,—but I failed to die,
As formerly I failed to live,—and thus
Grew willing, having tried all other ways,
To try just God's. Humility's so good,
When pride's impossible. Mark us, how we make
Our virtues, cousin, from our worn-out sins,
Which smack of them from henceforth. Is it right,
For instance, to wed here, while you love there?
And yet because a man sins once, the sin
Cleaves to him, in necessity to sin;
That if he sin not _so_, to damn himself,
He sins _so_, to damn others with himself:
And thus, to wed here, loving there, becomes
A duty. Virtue buds a dubious leaf
Round mortal brows; your ivy's better, dear.
—Yet she, 'tis certain, is my very wife;
The very lamb left mangled by the wolves
Through my own bad shepherding: and could I choose
But take her on my shoulder past this stretch
Of rough, uneasy wilderness, poor lamb,
Poor child, poor child?—Aurora, my belov'd,
I will not vex you any more to-night;
But, having spoken what I came to say,
The rest shall please you. What she can, in me,—
Protection, tender liking, freedom, ease,
She shall have surely, liberally, for her
And hers, Aurora. Small amends they'll make
For hideous evils (which she had not known
Except by me) and for this imminent loss,
This forfeit presence of a gracious friend,
Which also she must forfeit for my sake,
Since, ... drop your hand in mine a moment, sweet,
We're parting!—— Ah, my snowdrop, what a touch,
As if the wind had swept it off! you grudge
Your gelid sweetness on my palm but so,
A moment? angry, that I could not bear
_You_ ... speaking, breathing, living, side by side
With some one called my wife ... and live, myself?
Nay, be not cruel—you must understand!
Your lightest footfall on a floor of mine
Would shake the house, my lintel being uncrossed
'Gainst angels: henceforth it is night with me,
And so, henceforth, I put the shutters up;
Auroras must not come to spoil my dark.'
He smiled so feebly, with an empty hand
Stretched sideway from me,—as indeed he looked
To any one but me to give him help,—
And, while the moon came suddenly out full,
The double-rose of our Italian moons,
Sufficient, plainly, for the heaven and earth,
(The stars, struck dumb and washed away in dews
Of golden glory, and the mountains steeped
In divine languor) he, the man, appeared
So pale and patient, like the marble man
A sculptor puts his personal sadness in
To join his grandeur of ideal thought,—
As if his mallet struck me from my height
Of passionate indignation, I who had risen
Pale,—doubting, paused, ... Was Romney mad indeed?
Had all this wrong of heart made sick the brain?
Then quiet, with a sort of tremulous pride,
'Go, cousin,' I said coldly. 'A farewell
Was sooner spoken 'twixt a pair of friends
In those old days, than seems to suit you now:
And if, since then, I've writ a book or two,
I'm somewhat dull still in the manly art
Of phrase and metaphrase. Why, any man
Can carve a score of white Loves out of snow,
As Buonarroti down in Florence there,
And set them on the wall in some safe shade,
As safe, sir, as your marriage! very good;
Though if a woman took one from the ledge
To put it on the table by her flowers,
And let it mind her of a certain friend,
'Twould drop at once, (so better,) would not bear
Her nail-mark even, where she took it up
A little tenderly; so best, I say:
For me, I would not touch so light a thing,
And risk to spoil it half an hour before
The sun shall shine to melt it: leave it there.
I'm plain at speech, direct in purpose: when
I speak, you'll take the meaning as it is,
And not allow for puckerings in the silks
By clever stitches. I'm a woman, sir,
And use the woman's figures naturally,
As you, the male license. So, I wish you well.
I'm simply sorry for the griefs you've had—
And not for your sake only, but mankind's.
This race is never grateful: from the first,
One fills their cup at supper with pure wine,
Which back they give at cross-time on a sponge,
In bitter vinegar.'
'If gratefuller,'
He murmured,—'by so much less pitiable!
God's self would never have come down to die,
Could man have thanked him for it.'
'Happily
'Tis patent that, whatever,' I resumed,
'You suffered from this thanklessness of men,
You sink no more than Moses' bulrush-boat,
When once relieved of Moses; for you're light,
You're light, my cousin! which is well for you,
And manly. For myself,—now mark me, sir,
They burnt Leigh Hall; but if, consummated
To devils, heightened beyond Lucifers,
They had burnt instead a star or two, of those
We saw above there just a moment back,
Before the moon abolished them,—destroyed
And riddled them in ashes through a sieve
On the head of the foundering universe,—what then?
If you and I remained still you and I,
It would not shift our places as mere friends,
Nor render decent you should toss a phrase
Beyond the point of actual feeling!—nay,
You shall not interrupt me: as you said,
We're parting. Certainly, not once or twice,
To-night you've mocked me somewhat, or yourself;
And I, at least, have not deserved it so
That I should meet it unsurprised. But now,
Enough: we're parting ... parting. Cousin Leigh,
I wish you well through all the acts of life
And life's relations, wedlock, not the least;
And it shall 'please me,' in your words, to know
You yield your wife, protection, freedom, ease,
And very tender liking. May you live
So happy with her, Romney, that your friends
May praise her for it. Meantime, some of us
Are wholly dull in keeping ignorant
Of what she has suffered by you, and what debt
Of sorrow your rich love sits down to pay:
But if 'tis sweet for love to pay its debt,
'Tis sweeter still for love to give its gift;
And you, be liberal in the sweeter way,—
You can, I think. At least, as touches me,
You owe her, cousin Romney, no amends;
She is not used to hold my gown so fast,
You need entreat her now to let it go:
The lady never was a friend of mine,
Nor capable,—I thought you knew as much,—
Of losing for your sake so poor a prize
As such a worthless friendship. Be content,
Good cousin, therefore, both for her and you!
I'll never spoil your dark, nor dull your noon,
Nor vex you when you're merry, nor when you rest:
You shall not need to put a shutter up
To keep out this Aurora. Ah, your north
Can make Auroras which vex nobody,
Scarce known from evenings! also, let me say,
My larks fly higher than some windows. Right;
You've read your Leighs. Indeed 'twould shake a house,
If such as I came in with outstretched hand,
Still warm and thrilling from the clasp of one ...
Of one we know, ... to acknowledge, palm to palm,
As mistress there ... the Lady Waldemar.'
'Now God be with us' ... with a sudden clash
Of voice he interrupted—'what name's that?
You spoke a name, Aurora.'
'Pardon me;
I would that, Romney, I could name your wife
Nor wound you, yet be worthy.'
'Are we mad?'
He echoed—'wife! mine! Lady Waldemar!
I think you said my wife.' He sprang to his feet,
And threw his noble head back toward the moon
As one who swims against a stormy sea,
And laughed with such a helpless, hopeless scorn,
I stood and trembled.
'May God judge me so,'
He said at last,—'I came convicted here,
And humbled sorely if not enough. I came,
Because this woman from her crystal soul
Had shown me something which a man calls light:
Because too, formerly, I sinned by her
As, then and ever since, I have, by God,
Through arrogance of nature,—though I loved ...
Whom best, I need not say, ... since that is writ
Too plainly in the book of my misdeeds;
And thus I came here to abase myself,
And fasten, kneeling, on her regent brows
A garland which I startled thence one day
Of her beautiful June-youth. But here again
I'm baffled!—fail in my abasement as
My aggrandisement: there's no room left for me,
At any woman's foot, who misconceives
My nature, purpose, possible actions. What!
Are you the Aurora who made large my dreams
To frame your greatness? you conceive so small?
You stand so less than woman, through being more,
And lose your natural instinct, like a beast,
Through intellectual culture? since indeed
I do not think that any common she
Would dare adopt such fancy-forgeries
For the legible life-signature of such
As I, with all my blots: with all my blots!
At last then, peerless cousin, we are peers—
At last we're even. Ah, you've left your height;
And here upon my level we take hands,
And here I reach you to forgive you, sweet,
And that's a fall, Aurora. Long ago
You seldom understood me,—but, before,
I could not blame you. Then, you only seemed
So high above, you could not see below;
But now I breathe,—but now I pardon!—nay,
We're parting. Dearest, men have burnt my house,
Maligned my motives,—but not one, I swear,
Has wronged my soul as this Aurora has,
Who called the Lady Waldemar my wife.'
'Not married to her! yet you said' ...
'Again?
Nay, read the lines' (he held a letter out)
'She sent you through me.'
By the moonlight there,
I tore the meaning out with passionate haste
Much rather than I read it. Thus it ran.
NINTH BOOK.
EVEN thus. I pause to write it out at length,
The letter of the Lady Waldemar.—
'I prayed your cousin Leigh to take you this,
He says he'll do it. After years of love,
Or what is called so,—when a woman frets
And fools upon one string of a man's name,
And fingers it for ever till it breaks,—
He may perhaps do for her such a thing,
And she accept it without detriment
Although she should not love him any more.
And I, who do not love him, nor love you,
Nor you, Aurora,—choose you shall repent
Your most ungracious letter, and confess,
Constrained by his convictions, (he's convinced)
You've wronged me foully. Are you made so ill,
You woman—to impute such ill to _me_?
We both had mothers,—lay in their bosom once.
Why, after all, I thank you, Aurora Leigh,
For proving to myself that there are things
I would not do, ... not for my life ... nor him ...
Though something I have somewhat overdone,—
For instance, when I went to see the gods
One morning on Olympus, with a step
That shook the thunder in a certain cloud,
Committing myself vilely. Could I think,
The Muse I pulled my heart out from my breast
To soften, had herself a sort of heart,
And loved my mortal? He, at least, loved her;
I heard him say so; 'twas my recompence,
When, watching at his bedside fourteen days,
He broke out ever like a flame at whiles
Between the heats of fever.... 'Is it thou?
Breathe closer, sweetest mouth!' and when at last
The fever gone, the wasted face extinct
As if it irked him much to know me there,
He said, ''Twas kind, 'twas good, 'twas womanly,'
(And fifty praises to excuse one love)
'But was the picture safe he had ventured for?'
And then, half wandering ... 'I have loved her well,
Although she could not love me.'—'Say instead,'
I answered, 'that she loves you.'—'Twas my turn
To rave: (I would have married him so changed,
Although the world had jeered me properly
For taking up with Cupid at his worst,
The silver quiver worn off on his hair.)
'No, no,' he murmured, 'no, she loves me not;
Aurora Leigh does better: bring her book
And read it softly, Lady Waldemar,
Until I thank your friendship more for that,
Than even for harder service.' So I read
Your book, Aurora, for an hour, that day:
I kept its pauses, marked its emphasis;
My voice, empaled upon rhyme's golden hooks,
Not once would writhe, nor quiver, nor revolt;
I read on calmly,—calmly shut it up,
Observing, 'There's some merit in the book.
And yet the merit in't is thrown away
As chances still with women, if we write
Or write not: we want string to tie our flowers,
So drop them as we walk, which serves to show
The way we went. Good morning, Mister Leigh;
You'll find another reader the next time.
A woman who does better than to love,
I hate; she will do nothing very well:
Male poets are preferable, tiring less
And teaching more.' I triumphed o'er you both,
And left him.
'When I saw him afterward,
I had read your shameful letter, and my heart.
He came with health recovered, strong though pale,
Lord Howe and he, a courteous pair of friends,
To say what men dare say to women, when
Their debtors. But I stopped them with a word;
And proved I had never trodden such a road,
To carry so much dirt upon my shoe.
Then, putting into it something of disdain,
I asked forsooth his pardon, and my own,
For having done no better than to love,
And that, not wisely,—though 'twas long ago,
And though 'twas altered perfectly since then.
I told him, as I tell you now, Miss Leigh,
And proved I took some trouble for his sake
(Because I knew he did not love the girl)
To spoil my hands with working in the stream
Of that poor bubbling nature,—till she went,
Consigned to one I trusted, my own maid,
Who once had lived full five months in my house,
(Dressed hair superbly) with a lavish purse
To carry to Australia where she had left
A husband, said she. If the creature lied,
The mission failed, we all do fail and lie
More or less—and I'm sorry—which is all
Expected from us when we fail the most,
And go to church to own it. What I meant,
Was just the best for him, and me, and her ...
Best even for Marian!—I am sorry for't,
And very sorry. Yet my creature said
She saw her stop to speak in Oxford Street
To one ... no matter! I had sooner cut
My hand off (though 'twere kissed the hour before,
And promised a pearl troth-ring for the next)
Than crush her silly head with so much wrong.
Poor child! I would have mended it with gold,
Until it gleamed like St. Sophia's dome
When all the faithful troop to morning prayer:
But he, he nipped the bud of such a thought
With that cold Leigh look which I fancied once,
And broke in, 'Henceforth she was called his wife.
His wife required no succour: he was bound
To Florence, to resume this broken bond:
Enough so. Both were happy, he and Howe,
To acquit me of the heaviest charge of all—'
—At which I shot my tongue against my fly
And struck him; 'Would he carry,—he was just,—
A letter from me to Aurora Leigh,
And ratify from his authentic mouth
My answer to her accusation?'—'Yes,
If such a letter were prepared in time.'
—He's just, your cousin,—ay, abhorrently.
He'd wash his hands in blood, to keep them clean.
And so, cold, courteous, a mere gentleman,
He bowed, we parted.
'Parted. Face no more,
Voice no more, love no more! wiped wholly out
Like some ill scholar's scrawl from heart and slate,—
Ay, spit on and so wiped out utterly
By some coarse scholar! I have been too coarse,
Too human. Have we business, in our rank,
With blood i' the veins? I will have henceforth none;
Not even to keep the colour at my lip.
A rose is pink and pretty without blood;
Why not a woman? When we've played in vain
The game, to adore,—we have resources still,
And can play on at leisure, being adored:
Here's Smith already swearing at my feet
That I'm the typic She. Away with Smith!—
Smith smacks of Leigh,—and, henceforth, I'll admit
No socialist within three crinolines,
To live and have his being. But for you,
Though insolent your letter and absurd,
And though I hate you frankly,—take my Smith!
For when you have seen this famous marriage tied,
A most unspotted Erle to a noble Leigh,
(His love astray on one he should not love)
Howbeit you should not want his love, beware,
You'll want some comfort. So I leave you Smith;
Take Smith!—he talks Leigh's subjects, somewhat worse;
Adopts a thought of Leigh's, and dwindles it;
Goes leagues beyond, to be no inch behind;
Will mind you of him, as a shoe-string may,
Of a man: and women, when they are made like you,
Grow tender to a shoe-string, footprint even,
Adore averted shoulders in a glass,
And memories of what, present once, was loathed.
And yet, you loathed not Romney,—though you've played
At 'fox and goose' about him with your soul:
Pass over fox, you rub out fox,—ignore
A feeling, you eradicate it,—the act's
Identical.
I wish you joy, Miss Leigh.
You've made a happy marriage for your friend;
And all the honour, well-assorted love,
Derives from you who love him, whom he loves!
You need not wish _me_ joy to think of it,
I have so much. Observe, Aurora Leigh;
Your droop of eyelid is the same as his,
And, but for you, I might have won his love,
And, to you, I have shown my naked heart,—
For which three things I hate, hate, hate you. Hush,
Suppose a fourth!—I cannot choose but think
That, with him, I were virtuouser than you
Without him: so I hate you from this gulf
And hollow of my soul, which opens out
To what, except for you, had been my heaven,
And is instead, a place to curse by! LOVE.'
An active kind of curse. I stood there cursed—
Confounded. I had seized and caught the sense
Of the letter with its twenty stinging snakes,
In a moment's sweep of eyesight, and I stood
Dazed.—'Ah!—not married.'
'You mistake,' he said;
'I'm married. Is not Marian Erle my wife?
As God sees things, I have a wife and child;
And I, as I'm a man who honours God,
Am here to claim them as my child and wife.'
I felt it hard to breathe, much less to speak.
Nor word of mine was needed. Some one else
Was there for answering. 'Romney,' she began,
'My great good angel, Romney.'
Then at first,
I knew that Marian Erle was beautiful.
She stood there, still and pallid as a saint,
Dilated, like a saint in ecstasy,
As if the floating moonshine interposed
Betwixt her foot and the earth, and raised her up
To float upon it. 'I had left my child,
Who sleeps,' she said, 'and, having drawn this way,
I heard you speaking, ... friend!—Confirm me now.
You take this Marian, such as wicked men
Have made her, for your honourable wife?'
The thrilling, solemn, proud, pathetic voice.
He stretched his arms out toward the thrilling voice,
As if to draw it on to his embrace.
—'I take her as God made her, and as men
Must fail to unmake her, for my honoured wife.'
She never raised her eyes, nor took a step,
But stood there in her place, and spoke again.
—'You take this Marian's child, which is her shame
In sight of men and women, for your child,
Of whom you will not ever feel ashamed?'
The thrilling, tender, proud, pathetic voice.
He stepped on toward it, still with outstretched arms,
As if to quench upon his breast that voice.
—'May God so father me, as I do him,
And so forsake me as I let him feel
He's orphaned haply. Here I take the child
To share my cup, to slumber on my knee,
To play his loudest gambol at my foot,
To hold my finger in the public ways,
Till none shall need inquire, 'Whose child is this,'
The gesture saying so tenderly, 'My own'.'
She stood a moment silent in her place;
Then, turning toward me, very slow and cold—
—'And you,—what say you?—will you blame me much,
If, careful for that outcast child of mine,
I catch this hand that's stretched to me and him,
Nor dare to leave him friendless in the world
Where men have stoned me? Have I not the right
To take so mere an aftermath from life,
Else found so wholly bare? Or is it wrong
To let your cousin, for a generous bent,
Put out his ungloved fingers among briars
To set a tumbling bird's-nest somewhat straight?
You will not tell him, though we're innocent
We are not harmless?... and that both our harms
Will stick to his good smooth noble life like burrs,
Never to drop off though you shake the cloak?
You've been my friend: you will not now be his?
You've known him, that he's worthy of a friend;
And you're his cousin, lady, after all,
And therefore more than free to take his part,
Explaining, since the nest is surely spoilt,
And Marian what you know her,—though a wife,
The world would hardly understand her case
Of being just hurt and honest; while for him,
'Twould ever twit him with his bastard child
And married harlot. Speak, while yet there's time:
You would not stand and let a good man's dog
Turn round and rend him, because his, and reared
Of a generous breed,—and will you let his act,
Because it's generous? Speak. I'm bound to you,
And I'll be bound by only you, in this.'
The thrilling, solemn voice, so passionless,
Sustained, yet low, without a rise or fall,
As one who had authority to speak,
And not as Marian.
I looked up to feel
If God stood near me, and beheld his heaven
As blue as Aaron's priestly robe appeared
To Aaron when he took it off to die.
And then I spoke—'Accept the gift, I say,
My sister Marian, and be satisfied.
The hand that gives, has still a soul behind
Which will not let it quail for having given,
Though foolish worldlings talk they know not what,
Of what they know not. Romney's strong enough
For this: do you be strong to know he's strong:
He stands on Right's side; never flinch for him,
As if he stood on the other. You'll be bound
By me? I am a woman of repute;
No fly-blow gossip ever specked my life;
My name is clean and open as this hand,
Whose glove there's not a man dares blab about,
As if he had touched it freely:—here's my hand
To clasp your hand, my Marian, owned as pure!
As pure,—as I'm a woman and a Leigh!—
And, as I'm both, I'll witness to the world
That Romney Leigh is honoured in his choice,
Who chooses Marian for his honoured wife.'
Her broad wild woodland eyes shot out a light;
Her smile was wonderful for rapture. 'Thanks,
My great Aurora.' Forward then she sprang,
And dropping her impassioned spaniel head
With all its brown abandonment of curls
On Romney's feet, we heard the kisses drawn
Through sobs upon the foot, upon the ground—
O Romney! O my angel! O unchanged,
Though, since we've parted, I have past the grave!
But Death itself could only better _thee_,
Not change thee!—_Thee_ I do not thank at all:
I but thank God who made thee what thou art,
So wholly godlike.'
When he tried in vain
To raise her to his embrace, escaping thence
As any leaping fawn from a huntsman's grasp,
She bounded off and 'lighted beyond reach,
Before him, with a staglike majesty
Of soft, serene defiance,—as she knew
He could not touch her, so was tolerant
He had cared to try. She stood there with her great
Drowned eyes, and dripping cheeks, and strange sweet smile
That lived through all, as if one held a light
Across a waste of waters,—shook her head
To keep some thoughts down deeper in her soul,—
Then, white and tranquil as a summer-cloud
Which, having rained itself to a tardy peace,
Stands still in heaven as if it ruled the day,
Spoke out again—'Although, my generous friend,
Since last we met and parted, you're unchanged,
And, having promised faith to Marian Erle,
Maintain it, as she were not changed at all;
And though that's worthy, though that's full of balm
To any conscious spirit of a girl
Who once has loved you as I loved you once,—
Yet still it will not make her ... if she's dead,
And gone away where none can give or take
In marriage,—able to revive, return
And wed you,—will it, Romney? Here's the point;
O friend, we'll see it plainer: you and I
Must never, never, never join hands so.
Nay, let me say it,—for I said it first
To God, and placed it, rounded to an oath,
Far, far above the moon there, at His feet,
As surely as I wept just now at yours,—
We never, never, never join hands so.
And now, be patient with me; do not think
I'm speaking from a false humility.
The truth is, I am grown so proud with grief,
And He has said so often through his nights
And through his mornings, 'Weep a little still,
Thou foolish Marian, because women must,
But do not blush at all except for sin,'—
That I, who felt myself unworthy once
Of virtuous Romney and his high-born race,
Have come to learn, ... a woman, poor or rich,
Despised or honoured, is a human soul;
And what her soul is,—that, she is herself,
Although she should be spit upon of men,
As is the pavement of the churches here,
Still good enough to pray in. And, being chaste
And honest, and inclined to do the right,
And love the truth, and live my life out green
And smooth beneath his steps, I should not fear
To make him, thus, a less uneasy time
Than many a happier woman. Very proud
You see me. Pardon, that I set a trap
To hear a confirmation in your voice ...
Both yours and yours. It is so good to know
'Twas really God who said the same before:
For thus it is in heaven, that first God speaks,
And then his angels. Oh, it does me good,
It wipes me clean and sweet from devil's dirt,
That Romney Leigh should think me worthy still
Of being his true and honourable wife!
Henceforth I need not say, on leaving earth,
I had no glory in it. For the rest,
The reason's ready (master, angel, friend,
Be patient with me) wherefore you and I
Can never, never, never join hands so.
I know you'll not be angry like a man
(For _you_ are none) when I shall tell the truth,—
Which is, I do not love you, Romney Leigh,
I do not love you. Ah well! catch my hands,
Miss Leigh, and burn into my eyes with yours,—
I swear I do not love him. Did I once?
'Tis said that women have been bruised to death,
And yet, if once they loved, that love of theirs
Could never be drained out with all their blood:
I've heard such things and pondered. Did I indeed
Love once? or did I only worship? Yes,
Perhaps, O friend, I set you up so high
Above all actual good or hope of good,
Or fear of evil, all that could be mine,
I haply set you above love itself,
And out of reach of these poor woman's arms,
Angelic Romney. What was in my thought?
To be your slave, your help, your toy, your tool.
To be your love ... I never thought of that.
To give you love ... still less. I gave you love?
I think I did not give you anything;
I was but only yours,—upon my knees,
All yours, in soul and body, in head and heart,—
A creature you had taken from the ground,
Still crumbling through your fingers to your feet
To join the dust she came from. Did I love,
Or did I worship? judge, Aurora Leigh!
But, if indeed I loved, 'twas long ago,—
So long! before the sun and moon were made,
Before the hells were open,—ah, before
I heard my child cry in the desert night,
And knew he had no father. It may be,
I'm not as strong as other women are,
Who, torn and crushed, are not undone from love.
It may be, I am colder than the dead,
Who, being dead, love always. But for me
Once killed, ... this ghost of Marian loves no more,
No more ... except the child!... no more at all.
I told your cousin, sir, that I was dead;
And now, she thinks I'll get up from my grave,
And wear my chin-cloth for a wedding-veil,
And glide along the churchyard like a bride,
While all the dead keep whispering through the withes,
'You would be better in your place with us,
You pitiful corruption!' At the thought,
The damps break out on me like leprosy,
Although I'm clean. Ay, clean as Marian Erle:
As Marian Leigh, I know, I were not clean:
I have not so much life that I should love,
... Except the child. Ah God! I could not bear
To see my darling on a good man's knees,
And know by such a look, or such a sigh,
Or such a silence, that he thought sometimes,
'This child was fathered by some cursed wretch' ...
For, Romney,—angels are less tender-wise
Than God and mothers: even _you_ would think
What _we_ think never. He is ours, the child;
And we would sooner vex a soul in heaven
By coupling with it the dead body's thought,
It left behind it in a last month's grave,
Than, in my child, see other than ... my child.
We only, never call him fatherless
Who has God and his mother. O my babe,
My pretty, pretty blossom, an ill-wind
Once blew upon my breast! can any think
I'd have another,—one called happier,
A fathered child, with father's love and race
That's worn as bold and open as a smile,
To vex my darling when he's asked his name
And has no answer? What! a happier child
Than mine, my best,—who laughed so loud to-night
He could not sleep for pastime? Nay, I swear
By life and love, that, if I lived like some,
And loved like ... _some_ ... ay, loved you, Romney Leigh,
As some love (eyes that have wept so much, see clear),
I've room for no more children in my arms;
My kisses are all melted on one mouth;
I would not push my darling to a stool
To dandle babies. Here's a hand, shall keep
For ever clean without a marriage-ring,
To tend my boy, until he cease to need
One steadying finger of it, and desert
(Not miss) his mother's lap, to sit with men.
And when I miss him (not he me) I'll come
And say, 'Now give me some of Romney's work,
To help your outcast orphans of the world,
And comfort grief with grief.' For you, meantime,
Most noble Romney, wed a noble wife,
And open on each other your great souls,—
I need not farther bless you. If I dared
But strain and touch her in her upper sphere,
And say, 'Come down to Romney—pay my debt!'
I should be joyful with the stream of joy
Sent through me. But the moon is in my face ...
I dare not,—though I guess the name he loves;
I'm learned with my studies of old days,
Remembering how he crushed his under-lip
When some one came and spoke, or did not come:
Aurora, I could touch her with my hand,
And fly, because I dare not.'
She was gone.
He smiled so sternly that I spoke in haste.
'Forgive her—she sees clearly for herself:
Her instinct's holy,'
'_I_ forgive?' he said,
'I only marvel how she sees so sure,
While others' ... there he paused,—then hoarse, abrupt,—
Aurora! you forgive us, her and me?
For her, the thing she sees, poor loyal child,
If once corrected by the thing I know,
Had been unspoken; since she loves you well,
Has leave to love you:—while for me, alas,
If once or twice I let my heart escape
This night, ... remember, where hearts slip and fall
They break beside: we're parting,—parting,—ah,
You do not love, that you should surely know
What that word means. Forgive, be tolerant;
It had not been, but that I felt myself
So safe in impuissance and despair,
I could not hurt you though I tossed my arms
And sighed my soul out. The most utter wretch
Will choose his postures when he comes to die,
However in the presence of a queen;
And you'll forgive me some unseemly spasms
Which meant no more than dying. Do you think
I had ever come here in my perfect mind,
Unless I had come here, in my settled mind,
Bound Marian's, bound to keep the bond, and give
My name, my house, my hand, the things I could,
To Marian? For even _I_ could give as much;
Even I, affronting her exalted soul
By a supposition that she wanted these,
Could act the husband's coat and hat set up
To creak i' the wind and drive the world-crows off
From pecking in her garden. Straw can fill
A hole to keep out vermin. Now, at last,
I own heaven's angels round her life suffice
To fight the rats of our society,
Without this Romney: I can see it at last;
And here is ended my pretension which
The most pretended. Over-proud of course,
Even so!—but not so stupid ... blind ... that I,
Whom thus the great Taskmaster of the world
Has set to meditate mistaken work,
My dreary face against a dim blank wall
Throughout man's natural lifetime,—could pretend
Or wish ... O love, I have loved you! O my soul,
I have lost you!—but I swear by all yourself,
And all you might have been to me these years,
If that June-morning had not failed my hope,—
I'm not so bestial, to regret that day
This night,—this night, which still to you is fair;
Nay, not so blind, Aurora. I attest
Those stars above us, which I cannot see ...'
'You cannot'....
'That if Heaven itself should stoop,
Remix the lots, and give me another chance,
I'd say, 'No other!'—I'd record my blank.
Aurora never should be wife of mine.'
'Not see the stars?'
''Tis worse still, not to see
To find your hand, although we're parting, dear.
A moment let me hold it, ere we part;
And understand my last words—these, at last!
I would not have you thinking, when I'm gone,
That Romney dared to hanker for your love,
In thought or vision, if attainable,
(Which certainly for me it never was)
And wish to use it for a dog to-day,
To help the blind man stumbling. God forbid!
And now I know He held you in his palm,
And kept you open-eyed to all my faults,
To save you at last from such a dreary end.
Believe me, dear, that if I had known, like Him,
What loss was coming on me, I had done
As well in this as He has.—Farewell, you,
Who are still my light,—farewell! How late it is:
I know that, now: you've been too patient, sweet.
I will but blow my whistle toward the lane,
And some one comes ... the same who brought me here.
Get in—Good night.'
'A moment. Heavenly Christ!
A moment. Speak once, Romney. ''Tis not true.
I hold your hands, I look into your face—
You see me?'
'No more than the blessed stars.
Be blessed too, Aurora. Ah, my sweet,
You tremble. Tender-hearted! Do you mind
Of yore, dear, how you used to cheat old John,
And let the mice out slily from his traps,
Until he marvelled at the soul in mice
Which took the cheese and left the snare? The same
Dear soft heart always! 'Twas for this, I grieved
Howe's letter never reached you. Ah, you had heard
Of illness,—not the issue ... not the extent:
My life long sick with tossings up and down;
The sudden revulsion in the blazing house,—
The strain and struggle both of body and soul,
Which left fire running in my veins, for blood:
Scarce lacked that thunderbolt of the falling beam,
Which nicked me on the forehead as I passed
The gallery-door with a burden. Say heaven's bolt,
Not William Erie's; not Marian's father's; tramp
And poacher, whom I found for what he was,
And, eager for her sake to rescue him,
Forth swept from the open highway of the world,
Road-dust and all,—till, like a woodland boar
Most naturally unwilling to be tamed,
He notched me with his tooth. But not a word
To Marian! and I do not think, besides,
He turned the tilting of the beam my way,—
And if he laughed, as many swear, poor wretch,
Nor he nor I supposed the hurt so deep.
We'll hope his next laugh may be merrier,
In a better cause.'
'Blind, Romney?'
'Ah, my friend,
You'll learn to say it in a cheerful voice.
I, too, at first desponded. To be blind,
Turned out of nature, mulcted as a man,
Refused the daily largesse of the sun
To humble creatures! When the fever's heat
Dropped from me, as the flame did from my house,
And left me ruined like it, stripped of all
The hues and shapes of aspectable life,
A mere bare blind stone in the blaze of day,
A man, upon the outside of the earth,
As dark as ten feet under, in the grave,—
Why that seemed hard.'
'No hope?'
'A tear! you weep,
Divine Aurora? tears upon my hand!
I've seen you weeping for a mouse, a bird,—
But, weep for me, Aurora? Yes, there's hope.
Not hope of sight,—I could be learned, dear,
And tell you in what Greek and Latin name
The visual nerve is withered to the root,
Though the outer eyes appear indifferent,
Unspotted in their chrystals. But there's hope.
The spirit, from behind this dethroned sense,
Sees, waits in patience till the walls break up
From which the bas-relief and fresco have dropt:
There's hope. The man here, once so arrogant
And restless, so ambitious, for his part,
Of dealing with statistically packed
Disorders, (from a pattern on his nail,)
And packing such things quite another way,—
Is now contented. From his personal loss
He has come to hope for others when they lose,
And wear a gladder faith in what we gain ...
Through bitter experience, compensation sweet,
Like that tear, sweetest. I am quiet now,—
As tender surely for the suffering world,
But quiet,—sitting at the wall to learn,
Content, henceforth, to do the thing I can:
For, though as powerless, said I, as a stone,
A stone can still give shelter to a worm,
And it is worth while being a stone for that:
There's hope, Aurora.'
'Is there hope for me?
For me?—and is there room beneath the stone
For such a worm?—And if I came and said ...
What all this weeping scarce will let me say,
And yet what women cannot say at all,
But weeping bitterly ... (the pride keeps up,
Until the heart breaks under it) ... I love,—
I love you, Romney'....
'Silence!' he exclaimed.
'A woman's pity sometimes makes her mad.
A man's distraction must not cheat his soul
To take advantage of it. Yet, 'tis hard—
Farewell, Aurora.'
'But I love you, sir;
And when a woman says she loves a man,
The man must hear her, though he love her not,
Which ... hush!... he has leave to answer in his turn;
She will not surely blame him. As for me,
You call it pity,—think I'm generous?
'Twere somewhat easier, for a woman proud
As I am, and I'm very vilely proud,
To let it pass as such, and press on you
Love born of pity,—seeing that excellent loves
Are born so, often, nor the quicklier die,—
And this would set me higher by the head
Than now I stand. No matter: let the truth
Stand high; Aurora must be humble: no,
My love's not pity merely. Obviously
I'm not a generous woman, never was,
Or else, of old, I had not looked so near
To weights and measures, grudging you the power
To give, as first I scorned your power to judge
For me, Aurora: I would have no gifts
Forsooth, but God's,—and I would use _them_, too,
According to my pleasure and my choice,
As He and I were equals,—you, below,
Excluded from that level of interchange
Admitting benefaction. You were wrong
In much? you said so. I was wrong in most.
Oh, most! You only thought to rescue men
By half-means, half-way, seeing half their wants,
While thinking nothing of your personal gain.
But I who saw the human nature broad,
At both sides, comprehending, too, the soul's,
And all the high necessities of Art,
Betrayed the thing I saw, and wronged my own life
For which I pleaded. Passioned to exalt
The artist's instinct in me at the cost
Of putting down the woman's,—I forgot
No perfect artist is developed here
From any imperfect woman. Flower from root,
And spiritual from natural, grade by grade
In all our life. A handful of the earth
To make God's image! the despised poor earth,
The healthy odorous earth,—I missed, with it,
The divine Breath that blows the nostrils out
To ineffable inflatus: ay, the breath
Which love is. Art is much, but love is more.
O Art, my Art, thou'rt much, but Love is more!
Art symbolises heaven, but Love is God
And makes heaven. I, Aurora, fell from mine:
I would not be a woman like the rest,
A simple woman who believes in love,
And owns the right of love because she loves,
And, hearing she's beloved, is satisfied
With what contents God: I must analyse,
Confront, and question; just as if a fly
Refused to warm itself in any sun
Till such was _in leone_: I must fret
Forsooth, because the month was only May;
Be faithless of the kind of proffered love,
And captious, lest it miss my dignity,
And scornful, that my lover sought a wife
To use ... to use! O Romney, O my love,
I am changed since then, changed wholly,—for indeed,
If now you'd stoop so low to take my love,
And use it roughly, without stint or spare,
As men use common things with more behind,
(And, in this, ever would be more behind)
To any mean and ordinary end,—
The joy would set me like a star, in heaven,
So high up, I should shine because of height
And not of virtue. Yet in one respect,
Just one, beloved, I am in no wise changed:
I love you, loved you ... loved you first and last,
And love you on for ever. Now I know
I loved you always, Romney. She who died
Knew that, and said so; Lady Waldemar
Knows that; ... and Marian: I had known the same
Except that I was prouder than I knew,
And not so honest. Ay, and, as I live,
I should have died so, crushing in my hand
This rose of love, the wasp inside and all,—
Ignoring ever to my soul and you
Both rose and pain,—except for this great loss,
This great despair,—to stand before your face
And know I cannot win a look of yours.
You think, perhaps, I am not changed from pride,
And that I chiefly bear to say such words,
Because you cannot shame me with your eyes?
O calm, grand eyes, extinguished in a storm,
Blown out like lights o'er melancholy seas,
Though shrieked for by the shipwrecked,—O my Dark,
My Cloud,—to go before me every day
While I go ever toward the wilderness,—
I would that you could see me bare to the soul!—
If this be pity, 'tis so for myself,
And not for Romney: _he_ can stand alone;
A man like _him_ is never overcome:
No woman like me, counts him pitiable
While saints applaud him. He mistook the world:
But I mistook my own heart,—and that slip
Was fatal. Romney,—will you leave me here?
So wrong, so proud, so weak, so unconsoled,
So mere a woman!—and I love you so,—
I love you, Romney.'
Could I see his face,
I wept so? Did I drop against his breast,
Or did his arms constrain me? Were my cheeks
Hot, overflooded, with my tears, or his?
And which of our two large explosive hearts
So shook me? That, I know not. There were words
That broke in utterance ... melted, in the fire;
Embrace, that was convulsion, ... then a kiss ...
As long and silent as the ecstatic night,—And
deep, deep, shuddering breaths, which meant beyond
Whatever could be told by word or kiss.
But what he said ... I have written day by day,
With somewhat even writing. Did I think
That such a passionate rain would intercept
And dash this last page? What he said, indeed,
I fain would write it down here like the rest,
To keep it in my eyes, as in my ears,
The heart's sweet scripture, to be read at night
When weary, or at morning when afraid,
And lean my heaviest oath on when I swear
That, when all's done, all tried; all counted here,
All great arts, and all good philosophies,—
This love just puts its hand out in a dream,
And straight outreaches all things.
What he said,
I fain would write. But if an angel spoke
In thunder, should we, haply, know much more
Than that it thundered? If a cloud came down
And wrapt us wholly, could we draw its shape,
As if on the outside, and not overcome?
And so he spake. His breath against my face
Confused his words, yet made them more intense,—
As when the sudden finder of the wind
Will wipe a row of single city-lamps
To a pure white line of flame, more luminous
Because of obliteration; more intense,—
The intimate presence carrying in itself
Complete communication, as with souls
Who, having put the body off, perceive
Through simply being. Thus, 'twas granted me
To know he loved me to the depth and height
Of such large natures, ever competent
With grand horizons by the land or sea,
To love's grand sunrise. Small spheres hold small fires:
But he loved largely, as a man can love
Who, baffled in his love, dares live his life,
Accept the ends which God loves, for his own,
And lift a constant aspect.
From the day
I had brought to England my poor searching face,
(An orphan even of my father's grave)
He had loved me, watched me, watched his soul in mine,
Which in me grew and heightened into love.
For he, a boy still, had been told the tale
Of how a fairy bride from Italy,
With smells of oleanders in her hair,
Was coming through the vines to touch his hand;
Whereat the blood of boyhood on the palm
Made sudden heats. And when at last I came,
And lived before him, lived, and rarely smiled,
He smiled and loved me for the thing I was,
As every child will love the year's first flower,
(Not certainly the fairest of the year,
But, in which, the complete year seems to blow)
The poor sad snowdrop,—growing between drifts,
Mysterious medium 'twixt the plant and frost,
So faint with winter while so quick with spring,
So doubtful if to thaw itself away
With that snow near it. Not that Romney Leigh
Had loved me coldly. If I thought so once,
It was as if I had held my hand in fire
And shook for cold. But now I understood
For ever, that the very fire and heat
Of troubling passion in him, burned him clear,
And shaped to dubious order, word and act:
That, just because he loved me over all,
All wealth, all lands, all social privilege,
To which chance made him unexpected heir,—
And, just because on all these lesser gifts,
Constrained by conscience and the sense of wrong
He had stamped with steady hand God's arrow-mark
Of dedication to the human need,
He thought it should be so too, with his love;
He, passionately loving, would bring down
His love, his life, his best, (because the best)
His bride of dreams, who walked so still and high
Through flowery poems as through meadow-grass,
The dust of golden lilies on her feet,
That _she_ should walk beside him on the rocks
In all that clang and hewing out of men,
And help the work of help which was his life,
And prove he kept back nothing,—not his soul.
And when I failed him,—for I failed him, I—
And when it seemed he had missed my love,—he thought,
'Aurora makes room for a working-noon;'
And so, self-girded with torn strips of hope,
Took up his life, as if it were for death,
(Just capable of one heroic aim,)
And threw it in the thickest of the world,—
At which men laughed as if he had drowned a dog:
No wonder,—since Aurora failed him first!
The morning and the evening made his day.
But oh, the night! oh, bitter-sweet! oh, sweet!
O dark, O moon and stars, O ecstasy
Of darkness! O great mystery of love,—
In which absorbed, loss, anguish, treason's self
Enlarges rapture,—as a pebble dropt
In some full wine-cup, over-brims the wine!
While we two sate together, leaned that night
So close, my very garments crept and thrilled
With strange electric life; and both my cheeks
Grew red, then pale, with touches from my hair
In which his breath was; while the golden moon
Was hung before our faces as the badge
Of some sublime inherited despair,
Since ever to be seen by only one,—
A voice said, low and rapid as a sigh,
Yet breaking, I felt conscious, from a smile,—
'Thank God, who made me blind, to make me see!
Shine on, Aurora, dearest light of souls,
Which rul'st for evermore both day and night!
I am happy.'
I flung closer to his breast,
As sword that, after battle, flings to sheathe;
And, in that hurtle of united souls,
The mystic motions which in common moods
Are shut beyond our sense, broke in on us,
And, as we sate, we felt the old earth spin,
And all the starry turbulence of worlds
Swing round us in their audient circles, till
If that same golden moon were overhead
Or if beneath our feet, we did not know.
And then calm, equal, smooth with weights of joy,
His voice rose, as some chief musician's song
Amid the old Jewish temple's Selah-pause,
And bade me mark how we two met at last
Upon this moon-bathed promontory of earth,
To give up much on each side, then take all.
'Beloved,' it sang, 'we must be here to work;
And men who work, can only work for men,
And, not to work in vain, must comprehend
Humanity, and, so, work humanly,
And raise men's bodies still by raising souls,
As God did, first.'
'But stand upon the earth,'
I said, 'to raise them,—(this is human too;
There's nothing high which has not first been low;
My humbleness, said One, has made me great!)
As God did, last.'
'And work all silently,
And simply,' he returned, 'as God does all;
Distort our nature never, for our work,
Nor count our right hands stronger for being hoofs.
The man most man, with tenderest human hands,
Works best for men,—as God in Nazareth.'
He paused upon the word, and then resumed;
'Fewer programmes; we who have no prescience.
Fewer systems; we who are held and do not hold.
Less mapping out of masses, to be saved,
By nations or by sexes. Fourier's void,
And Comte is dwarfed,—and Cabet, puerile.
Subsists no law of life outside of life;
No perfect manners, without Christian souls:
The Christ himself had been no Lawgiver,
Unless He had given the life, too, with the law.'
I echoed thoughtfully—'The man, most man,
Works best for men: and, if most man indeed,
He gets his manhood plainest from his soul:
While, obviously, this stringent soul itself
Obeys our old rules of development;
The Spirit ever witnessing in ours,
And Love, the soul of soul, within the soul,
Evolving it sublimely. First, God's love.'
'And next,' he smiled, 'the love of wedded souls,
Which still presents that mystery's counterpart.
Sweet shadow-rose, upon the water of life,
Of such a mystic substance, Sharon gave
A name to! human, vital, fructuous rose,
Whose calyx holds the multitude of leaves,—
Loves filial, loves fraternal, neighbour-loves,
And civic, ... all fair petals, all good scents,
All reddened, sweetened from one central Heart!'
'Alas,' I cried, 'it was not long ago,
You swore this very social rose smelt ill.'
'Alas,' he answered, 'is it a rose at all?
The filial's thankless, the fraternal's hard,
The rest is lost. I do but stand and think,
Across dim waters of a troubled life
The Flower of Heaven so vainly overhangs,—
What perfect counterpart would be in sight,
If tanks were clearer. Let us clean the tubes,
And wait for rains. O poet, O my love,
Since _I_ was too ambitious in my deed,
And thought to distance all men in success,
Till God came on me, marked the place, and said,
'Ill-doer, henceforth keep within this line,
Attempting less than others,'—and I stand
And work among Christ's little ones, content,—
Come thou, my compensation, my dear sight,
My morning-star, my morning! rise and shine,
And touch my hills with radiance not their own;
Shine out for two, Aurora, and fulfil
My falling-short that must be! work for two,
As I, though thus restrained, for two, shall love!
Gaze on, with inscient vision toward the sun,
And, from his visceral heat, pluck out the roots
Of light beyond him. Art's a service,—mark:
A silver key is given to thy clasp,
And thou shalt stand unwearied, night and day,
And fix it in the hard, slow-turning wards,
And open, so, that intermediate door
Betwixt the different planes of sensuous form
And form insensuous, that inferior men
May learn to feel on still through these to those,
And bless thy ministration. The world waits
For help. Beloved, let us love so well,
Our work shall still be better for our love,
And still our love be sweeter for our work,
And both, commended, for the sake of each,
By all true workers and true lovers born.
Now press the clarion on thy woman's lip
(Love's holy kiss shall still keep consecrate)
And breathe the fine keen breath along the brass,
And blow all class-walls level as Jericho's
Past Jordan; crying from the top of souls,
To souls, that they assemble on earth's flats
To get them to some purer eminence
Than any hitherto beheld for clouds!
What height we know not,—but the way we know,
And how by mounting aye, we must attain,
And so climb on. It is the hour for souls;
That bodies, leavened by the will and love,
Be lightened to redemption. The world's old;
But the old world waits the hour to be renewed:
Toward which, new hearts in individual growth
Must quicken, and increase to multitude
In new dynasties of the race of men,—
Developed whence, shall grow spontaneously
New churches, new œconomies, new laws
Admitting freedom, new societies
Excluding falsehood. He shall make all new.'
My Romney!—Lifting up my hand in his,
As wheeled by Seeing spirits toward the east,
He turned instinctively,—where, faint and fair,
Along the tingling desert of the sky,
Beyond the circle of the conscious hills,
Were laid in jasper-stone as clear as glass
The first foundations of that new, near Day
Which should be builded out of heaven, to God.
He stood a moment with erected brows,
In silence, as a creature might, who gazed:
Stood calm, and fed his blind, majestic eyes
Upon the thought of perfect noon. And when
I saw his soul saw,—'Jasper first,' I said,
'And second, sapphire; third, chalcedony;
The rest in order, ... last, an amethyst.'
THE END.
BRADBURY AND EVANS, PRINTERS, WHITEFRIARS.
End of Project Gutenberg's Aurora Leigh, by Elizabeth Barrett Browning
***
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{
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Myzomorphus sparsimflabellatus is een keversoort uit de familie van de boktorren (Cerambycidae). De wetenschappelijke naam van de soort werd voor het eerst geldig gepubliceerd in 1963 door Zajciw.
Boktorren
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mawdesigns
Topic: architecture | authors | design education | design entrepreneurship | engineering & technology | fashion & textile | furniture & interior | general | graphic | human resources | industrial & product | landscape | people | reviews | university research | web
https://architectureau.com/articles/design-challenge-for-the-future-of-senior-living/ | Architecture AU, 22 jan 2020
Seven sofas that deserve to be centre of attention | Dezeen, 22 jan 2020
Design Shows Take On the Future. And It's Not Pretty. | The New York Times, 21 jan 2020
In Arizona, a case study in how architecture can adapt to climate change | Fast Company, 21 jan 2020
Finding solutions in design thinking | Business Mirror, 21 jan 2020
New coat design saving lives, launching new careers for the homeless | Fox6Now, 21 jan 2020
How to Incorporate Gardens in Home Design | ArchDaily, 20 jan 2020
Daniel Weil: 'Designers are the ones who show change' | Design Week, 20 jan 2020
Learn Graphic Design and Launch Your Own Business | Entrepreneur, 17 jan 2020
7 web design principles that are crucial to know for 2020 | TechGenix, 15 jan 2020
Researchers from IIT-Madras (Tamil Nadu, India), Prof. Asokan Thondiyath and research scholar Nagamanikandan Govindan, have designed and developed a multimodal robotic system, termed as 'Grasp Man', that has good grasping, manipulation and locomotion abilities. Their research, 'Design and Analysis of a Multimodal Grasper Having Shape Conformity and Within-Hand Manipulation With Adjustable Contact Forces', is recently published in ASME Journal of Mechanisms and Robotics. The robot is fitted with a pair of graspers that provide morphological adaptation, enabling it to conform to the geometry of the object being grasped, and allowing it to hold objects securely and manipulate them much like the human hand. The two graspers are equipped with a robotic platform that provides behavioural adaptation. The robot will have various industrial applications such as pipe inspection, search-and-rescue operations, and others that involve climbing, holding, and assembling. Prof. Asokan says, 'The motivation behind this research is to realise a robot with a minimalistic design that can overcome the need for task-specific robots that are capable of navigating and manipulating across different environments without increasing the system complexity.' Read on...
YourStory: IIT-Madras researchers design robot with graspers that function like the human hand
Author: Teja Lele Desai
Research study, 'Onboard Evolution of Understandable Swarm Behaviors', published in Advanced Intelligent Systems by researchers from University of Bristol (Simon Jones, Sabine Hauert) and University of the West of England (Alan F. Winfield, Matthew Studley), brings development of a new generation of swarming robots which can independently learn and evolve new behaviours in the wild a step closer. Researchers used artificial evolution to enable the robots to automatically learn swarm behaviours which are understandable to humans. This could create new robotic possibilities for environmental monitoring, disaster recovery, infrastructure maintenance, logistics and agriculture. This new approach uses a custom-made swarm of robots with high-processing power embedded within the swarm. In most recent approaches, artificial evolution has typically been run on a computer which is external to the swarm, with the best strategy then copied to the robots. Prof. Jones says, 'Human-understandable controllers allow us to analyse and verify automatic designs, to ensure safety for deployment in real-world applications.' Researchers took advantage of the recent advances in high-performance mobile computing, to build a swarm of robots inspired by those in nature. Their 'Teraflop Swarm' has the ability to run the computationally intensive automatic design process entirely within the swarm, freeing it from the constraint of off-line resources. Prof. Hauert says, 'This is the first step towards robot swarms that automatically discover suitable swarm strategies in the wild. The next step will be to get these robot swarms out of the lab and demonstrate our proposed approach in real-world applications.' Prof. Winfield says, 'In many modern AI systems, especially those that employ Deep Learning, it is almost impossible to understand why the system made a particular decision...An important advantage of the system described in this paper is that it is transparent: its decision making process is understandable by humans.' Read on...
Engineering.com: Robots Learn Swarm Behaviors, Aim to Escape the Lab
People with the twin passion of design and development of new products can transform into design entrepreneurs. They are able to control both the design and business processes. Vijayant Bansal, founder of World University of Design (India), explains what it takes to be a design entrepreneur and explores the shifting landscape of design entrepreneurship in India. He says, 'We are in the midst of a design revolution and increasingly design is gaining a lot of focus...But it's not easy starting from ground zero and working yourself towards achieving credibility, recognition and last but not the least, generating demand. This involves having to create a balance between what we want to create with what the customer wants; what is possible technically and how much of a resource pull will it involve.' Contemporary design entrepreneurship includes new product development, restoring crafts, innovating existing products and providing design services based on new & emerging technologies. Explaining the design revolution, he says, 'Designing is undergoing a metamorphosis, aided by new technologies and digital transformation of today. New and disruptive technologies like Artificial intelligence, IoT, Machine learning etc., are the biggest enablers, disrupting traditional processes and systems, enabling out of the box thinking and new ideas, which in turn reshape the entire user experience.' Universities can play an important role in guiding and mentoring students to pursue design entrepreneurship. Industry experts can also play a role in this and enable students to participate in hands-on training. Virtual products have also expanded the scope of design entrepreneurship with designers engaged in designing and developing games and apps. Design entrepreneurship is the new career paradigm. Mr. Bansal suggests, 'Today the scenario has undergone a sea change, with almost every industry, be it apparel, automobiles, film making, animation, product design or gaming, with design playing an intrinsic role in the entire process from an idea to the end product. It's worth the challenge if financial security and stability are not foremost on your mind and you have the patience and inclination to see through the entire process of making the design-centric idea into a successful venture.' Read on...
Entrepreneur: The Rise of the Contemporary Indian Design Entrepreneur
Author: Vijayant Bansal
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{"url":"http:\/\/blog.brucemerry.org.za\/","text":"Friday, July 18, 2014\n\nIOI 2014 day 2 analysis\n\nI found day 2 much harder than day 1, and I still don't know how to solve all the problems (I am seriously impressed by those getting perfect scores). Here's what I've managed to figure out so far.\n\nUpdate: I've now solved everything (in theory), and the solutions are below. The official solutions are now also available on the IOI website. I'll try coding the solutions at some point if I get time.\n\nGondola\n\nThis was the easiest of the three. Firstly, what makes a valid gondola sequence? In all the subtasks of this problem, there will be two cases. If you see any of the numbers 1 to n, that immediately locks in the phase, and tells you the original gondola for every position. Otherwise, the phase is unknown. So, the constraints are that\n\u2022 if the phase is known, every gondola up to n must appear in the correct spot if it appears;\n\u2022 no two gondolas can have the same number.\nNow we can consider how to construct a replacement sequence (and also to count them), which also shows that these conditions are sufficient. If the phase is not locked, pick it arbitrarily. Now the \"new gondola\" column is simply the numbers from n+1 up to the largest gondola, so picking a replacement sequence is equivalent to deciding which gondola replaces each broken gondola. We can assign each gondola greater than n that we can't see to a position (one where the final gondola number is larger), and this will uniquely determine the replacement sequence. We'll call such gondolas hidden.\n\nFor the middle set of subtasks, the simplest thing is to assign all hidden gondolas to one position, the one with the highest-numbered gondola in the final state. For counting the number of possible replacement sequences, each hidden gondola can be assigned independently, so we just multiply together the number of options, and also remember to multiply by n if the phase is unknown. In the last subtask there are too many hidden gondolas to deal with one at a time, but they can be handled in batches (those between two visible gondolas), using fast exponentiation.\n\nFriend\n\nThis is a weighted maximum independent set problem. On a general graph this is NP-hard, so we will need to exploit the curious way in which the graph is constructed. I haven't figured out how to solve the whole problem, but let's work through the subtasks:\n1. This is small enough to use brute force (consider all subsets and check whether they are independent).\n2. The graph will be empty, so the sample can consist of everyone.\n3. The graph will be complete, so only one person can be picked in a sample. Pick the best one.\n4. The graph will be a tree. There is a fairly standard tree DP to handle this case: for every subtree, compute the best answer, either with the root excluded or included. If the root is included, add up the root-excluded answers for every subtree; otherwise add up the best of the two for every subtree. This takes linear time.\n5. In this case the graph is bipartite and the vertices are unweighted. This is a standard problem which can be solved by finding the maximum bipartite matching. The relatively simple flow-based algorithm for this is theoretically $$O(n^3)$$, but it is one of those algorithms that tends to run much faster in most cases, so it may well be sufficient here.\nThe final test-case clearly requires a different approach, since n can be much larger. I only managed to figure this out after getting a big hint from the SA team leader, who had seen the official solution.\n\nWe will process the operations in reverse order. For each operation, we will transform the graph into one that omits the new person, but for which the optimal solution has the same score. Let's say that the last operation had A as the host and B as the invitee, and consider the different cases:\n\u2022 YourFriendsAreMyFriends: this is the simplest: any solution using B can also use A, and vice versa. So we can collapse the two vertices into one whose weight is the sum of the original weights, and use it to replace A.\n\u2022 WeAreYourFriends: this is almost the same, except now we can use at most one of A and B, and which one we take (if either) has no effect on the rest of the graph. So we can replace A with a single vertex having the larger of the two weights, and delete B.\n\u2022 IAmYourFriend: this is a bit trickier. Let's start with the assumption that B will form part of the sample, and add that to the output value before deleting it. However, if we later decide to use A, there will be a cost to remove B again; so A's weight decreases by the weight of B. If it ends up with negative weight, we can just clamp it to 0.\nRepeat this deletion process until only the original vertex is left; the answer will be the weight of this vertex, plus the saved-up weights from the IAmYourFriend steps.\n\nHoliday\n\nConsider the left-most and right-most cities that Jian-Jia visits. Regardless of where he stops, he will need to travel from the start city to one of the ends, and from there to the other end. There is no point in doing any other back-tracking, so we can tell how many days he spends travelling just from the end-points. This then tells us how many cities he has time to see attractions in, and obviously we will pick the best cities within the range.\n\nThat's immediately sufficient to solve the first test case. To solve more, we can consider an incremental approach. Fix one end-point, and gradually extend the other end-point, keeping track of the best cities (and their sum) in a priority queue (with the worst of the best cities at the front). As the range is extended, the number of cities that can be visited shrinks, so items will need to be popped. Of course, the next city in the range needs to be added each time as well. Using a binary heap, this gives an $$O(n^2\\log n)$$ algorithm: a factor of n for each endpoint, and the $$\\log n$$ for the priority queue operations. That's sufficient for subtask 3.\u00a0 It's also good enough for subtask 2, because the left endpoint will be city 0, saving a factor of n.\n\nFor subtask 4, it is clearly not possible to consider every pair of end-points. Let's try to break things up. Assume (without loss of generality) that we move first left, then back to the start, then right. Let's compute the optimal solution for the left part and the right part separately, then combine them. The catch is that we need to know how we are splitting up our time between the two sides. So we'll need to compute the answer for each side for all possible number of days spent within each side. This seems to leave us no better off, since we're still searching within a two-dimensional space (number of days and endpoint), but it allows us to do some things differently.\n\nWe'll just consider the right-hand side. The left-hand side is similar, with minor changes because we need two days for travel (there and back) instead of one. Let f(d) be the optimal end-point if we have d days available. Then with a bit of work one can show that f is non-decreasing (provided one is allowed to pick amongst ties). If we find f(d) for d=1, 2, 3, ... in that order, it doesn't really help: we're only, on average, halving the search space. But we can do better by using a divide-and-conquer approach: if we need to find f for all $$d \\in [0, D)$$ then we start with $$d = \\frac{D}{2}$$ to subdivide the space, and then recursively process each half of the interval on disjoint subintervals of the cities. This reduces the search space to $$O(n\\log n)$$.\n\nThis still leaves the problem of efficiently finding the total number of attractions that can be visited for particular intervals and available days. The official solution uses one approach, based on a segment tree over the cities, sorted by number of attractions rather than position. The approach I found is, I think, simpler. Visualise the recursion described above as a tree; instead of working depth-first (i.e., recursively), we work breadth-first. We make $$O(\\log n)$$ passes, and in each pass we compute f(d) where d is an odd multiple of $$2^i$$ (with $$i$$ decreasing with each pass). Each pass can be done in a single incremental process, similar to the way we tackled subpass 2. The difference is that each time we cross into the next subinterval, we need to increase $$d$$, and hence bring more cities into consideration. To do this, we need either a second priority queue of excluded cities, or we can replace the priority queue with a balanced binary tree. Within each pass, d can only be incremented $$O(n)$$ times, so the total running time will be $$O(n\\log n)$$ per pass, or $$O(n\\log n \\log n)$$ overall.\n\nIOI 2014 day 1\n\nSince there is no online judge, I haven't tried actually coding any of these. So these ideas are not validated yet. You can find the problems here.\n\nRails\n\nI found this the most difficult of the three to figure out, although coding it will not be particularly challenging.\n\nFirstly, we can note that distances are symmetric: a route from A to B can be reflected in the two tracks to give a route from B to A. So having only $$\\frac{n(n-1)}{2}$$ queries is not a limitation, as we can query all distances. This might be useful in tackling the first three subtasks, but I'll go directly to the hardest subtask.\n\nIf we know the position and type of a station, there is one other that we can immediately locate: the closest one. It must have the opposite type and be reached by a direct route. Let station X be the closest to station 0. The other stations can be split into three groups:\n1. d(X, Y) < d(X, 0): these are reached directly from station X and of type C, so we can locate them exactly.\n2. d(0, X) + d(X, Y) = d(0, Y), but not of type 1: these are reached from station 0 via station X, so they lie to the left of station 0.\n3. All other stations lie to the right of station X.\nLet's now consider just the stations to the right of X, and see how to place them. Let's take them in increasing distance from 0. This ensures that we encounter all the type D stations in order, and any type C station will be encountered at some point after the type D station used to reach it. Suppose Y is the right-most type D station already encountered, and consider the distances for a new station Z. Let $$z = d(0, Z) - d(0, Y) - d(Y, Z)$$. If Z is type C, then there must be a type D at distance $$\\frac{z}{2}$$ to the left of Y. On the other hand, if Z is of type D (and lies to the right of Y), then there must be a type C station at distance $$\\frac{z}{2}$$ to the left of Y. In the first case, we will already have encountered the station, so we can always distinguish the two cases, and hence determine the position and type of Z.\nThe stations to the left of station zero can be handled similarly, using station X as the reference point instead of station 0.\nHow many queries is this? Every station Z except 0 and X accounts for at most three queries: d(0, Z), d(X, Z) and d(Y, Z), where Y can be different for each Z. This gives $$3(n-2) + 1$$, which I think can be improved to $$3(n-2)$$ just by counting more carefully. Either way, it is sufficient to solve all the subtasks.\n\nWall\n\nThis is a fairly standard interval tree structure problem, similar to Mountain from IOI 2005 (but a little easier). Each node of the tree contains a range to which its children are clamped. To determine the value of any element of the wall, start at the leaf with a value of 0 and read up the tree, clamping the value to the range in each node in turn. Initially, each node has the range [0, inf). When applying a new instruction, it is done top-down, and clamps are pushed down the tree whenever recursion is necessary.\n\nAn interesting aspect of the problem is that it is offline, in that only the final configuration is requested and all the information is provided up-front. This makes me think that there may be an alternative solution that processes the data in a different order, but I can't immediately see a nicer solution than the one above.\n\nGame\n\nI liked this problem, partly because I could reverse-engineer a solution from the assumption that it is always possible to win, and partly because it requires neither much algorithm\/data-structure training (like Wall) nor tricky consideration of cases (like Rails). Suppose Mei-Yu knows that certain cities are connected. If there are any flights between the cities that she has not asked about, then she can win simply by saving one of these flights for last, since it will not affect whether the country is connected. It follows that for Jian-Jia to win, he must always answer no when asked about a flight between two components that Mei-Yu does not know to be connected, unless this is the last flight between these components?\n\nWhat if he always answers yes to the last flight between two components? In this case he will win. As long as there are at least two components left, there are uncertain edges between every pair of them, so Mei-Yu can't know whether any of them is connected any other. All edges within a component are known, so the number of components can only become one after the last question.\n\nWhat about complexity? We need to keep track of the number of edges between each pair of components, which takes $$O(N^2)$$ space. Most operations will just decrement one of these counts. There will be $$N - 1$$ component-merging operations, each of which requires a linear-time merging of these edge counts and updating a vertex-to-component table. Thus, the whole algorithm requires $$O(N^2)$$ time. This is optimal given that Mei-Yu will ask $$O(N^2)$$ questions.\n\nMonday, June 30, 2014\n\nICPC Problem H: Pachinko\n\nProblem A has been covered quite well elsewhere, so I won't discuss it. That leaves only problem H. I started by reading the Google translation of a Polish writeup by gawry. The automatic translation wasn't very good, but it gave me one or two ideas I borrowed. I don't know how my solution compares in length of code, but I consider it much simpler conceptually.\n\nThis is a fairly typical Markov process, where a system is in some state, and each timestep it randomly selects one state as a function of the current state. One variation is that the process stops once the ball reaches a target, whereas Markov processes don't terminate. I was initially going to model that as the ball always moving from the target to itself, but things would have become slightly complicated.\n\nGawry has a nice way of making this explicitly a matrix problem. Set up the matrix M as for a Markov process i.e., $$M_{i,j}$$ is the probability of a transition from state j to state i. However, for a target state j, we set $$M_{i,j}=0$$ for all i. Now if $$b$$ is our initial probability vector (equal probability for each empty spot in the first row), then $$M^t b$$ represents the probability of the ball being in each position (and the game not having previously finished) after $$t$$ timesteps. We can then say that the expected amount of time the ball spends in each position is given by $$\\sum_{t=0}^{\\infty} M^t b$$. The sum of the elements in this vector is the expected length of a game and we're told that it is less than $$10^9$$, so we don't need to worry about convergence. However, that doesn't mean that the matrix series itself converges: Gawry points out that if there are unreachable parts of the game with no targets, then the series won't converge. We fix that by doing an initial flood-fill to find all reachable cells and only use those in the matrix. Gawry then shows that under the right conditions, the series converges to $$(I - M)^{-1} b$$.\n\nThis is where my solution differs. Gawry dismisses Gaussian elimination, because the matrix can be up to 200,000 square. However, this misses the fact that it is banded: by numbering cells left-to-right then top-to-bottom, we ensure that every non-zero entry in the matrix is at most W cells away from the main diagonal. Gaussian elimination (without pivoting) preserves this property. We can exploit this both to store the matrix compactly, and to perform Gaussian elimination in $$O(W^3H)$$ time.\n\nOne concern is the \"without pivoting\" caveat. I was slightly surprised that my first submission passed. I think it is possible to prove correctness, however. Gaussian elimination without pivoting is known (and easily provable) to work on strictly column diagonally dominant matrices. In our case the diagonal dominance is weak: columns corresponding to empty cells have a sum of zero, those corresponding to targets have a 1 on the diagonal and zeros elsewhere. However, the matrix is also irreducible, which I think is enough to guarantee that there won't be any division by zero.\n\nEDIT: actually it's not guaranteed to be irreducible, because the probabilities can be zero and hence it's possible to get from A to B without being able to get from B to A. But I suspect that it's enough that one can reach a target from every state.\n\nSaturday, June 28, 2014\n\nICPC Problem L: Wires\n\nWhile this problem wasn't too conceptually difficult, it requires a lot of code (my solution is about 400 lines), and careful implementation of a number of geometric algorithms. A good chunk of the code comes from implementing a rational number class in order to precisely represent the intersection points of the wires. It is also very easy to suffer from overflow: I spent a long time banging my head against an assertion failure on the server until I upgraded my rational number class to use 128-bit integers everywhere, instead of just for comparisons.\n\nThe wires will divide the space up into connected regions. The regions can be represented in a planar graph, with edges between regions that share an edge. The problem is then to find the shortest path between the regions containing the two new end-points.\n\nMy solution works in a number of steps:\n1. Find all the intersection points between wires, and the segments between intersection points. This just tests every wire against every other wire. The case of two parallel wires sharing an endpoint needs to be handled carefully. For each line, I sort the intersection points along the line. I used a dot produce for this, which is where my rational number class overflowed, but would probably have been safer to just sort lexicographically. More than two lines can meet at an intersection point, so I used a std::map to assign a unique ID to each intersection point (I'll call them vertices from here on).\n2. Once the intersection points along a line have been sorted, one can identify the segments connecting them. I create two copies of each segment, one in each direction. With each vertex A I store a list of all segments A->B. Each pair is stored contiguously so that it is trivial to find its partner. Each segment is considered to belong to the region to its left as one travels A->B.\n3. The segments emanating from each vertex are sorted by angle. These comparisons could easily cause overflows again, but one can use a handy trick: instead of using the vector for the segment in an angle comparison, one can use the vector for the entire wire. It has identical direction but has small integer coordinates.\n4. Using the sorted lists from the previous step, each segment is given a pointer to its following segment from the same region. In other words, if one is tracing the boundary of the region and one has just traced A->B, the pointer will point to B->C.\n5. I extract the contours of the regions. A region typically consists of an outer contour and optionally some holes. The outermost region lacks an outer contour (one could add a big square if one needed to, but I didn't). A contour is found by following the next pointers. A case that turns out to be inconvenient later is that some segments might be part of the contour but not enclose any area. This can make a contour disappear completely, in which case it is discarded. Any remaining contours have the property that two adjacent segments are not dual to each other, although it is still possible to both sides of an edge to belong to the same contour.\n6. Each contour is identified as an outer contour or a hole. With integer coordinates I could just measure the signed area of the polygon, but that gets nasty with rational coordinates. Instead, I pick the lexicographically smallest vertex in the contour and examine the angle between the two incident segments (this is why it is important that there is a non-trivial angle between them). I also sort the contours by this lexicographically smallest vertex, which causes any contour to sort before any other contours it encloses.\n7. For each segment I add an edge of weight 1 from its containing region to the containing region of its dual.\n8. For each hole, I search backwards through the other contours to find the smallest non-hole that contains it. I take one vertex of the hole and do a point-in-polygon test. Once again, some care is needed to avoid overflows, and using the vectors for the original wires proves useful. One could then associate the outer contour and the holes it contains into a single region object, but instead I just added an edge to the graph to join them with weight 0. In other words, one can travel from the boundary of a region to the outside of a hole at no cost.\n9. Finally, I identify the regions containing the endpoints of the new wire, using the same search as in the previous step.\nAfter all this, we still need to implement a shortest path search - but by this point that seems almost trivial in comparison.\n\nWhat is the complexity? There can be $$O(M^2)$$ intersections and hence also $$O(M^2)$$ contours, but only $$O(M)$$ of them can be holes (because two holes cannot be part of the same connected component). The slowest part is the fairly straightforward point-in-polygon test which tests each hole against each non-hole segment, giving $$O(M^3)$$ time. There are faster algorithms for doing point location queries, so it is probably theoretically possible to reduce this to $$O(M^2\\log N)$$ or even $$O(M^2)$$, but certainly not necessary for this problem.\n\nICPC Problem J: Skiing\n\nI'll start by mentioning that there is also some analysis discussion on the Topcoder forums, which includes alternative solutions that in some cases I think are much nicer than mine.\n\nSo, on to problem J - which no team solved, and only one team attempted. I'm a little surprised by this, since it's the sort of problem where one can do most of the work on paper while someone else is using the machine. A possible reason is that it's difficult to determine the runtime: I just coded it up and hoped for the best.\n\nIt is a slightly annoying problem for a few reasons. Firstly, there are a few special cases, because $$v_y$$ or $$a$$ can be zero. We also have to find the lexicographically smallest answer.\n\nLet's deal with those special cases first. If $$v_y = 0$$ then we can only reach targets with $$y = 0$$. If $$a \\not= 0$$ then we can reach all of them in any order, otherwise we can only reach those with $$x = 0$$. To get the lexicographically smallest answer, just go through them in order and print out those that are reachable. I actually dealt with the $$v_y \\not= 0$$, $$a = 0$$ case as part of my general solution, but is also easy enough to deal with. We can only reach targets with $$x = 0$$, and we can only reach them in increasing order of y. One catch is that targets are not guaranteed to have distinct coordinates, so if two targets have the same coordinates we must be careful to visit them in lexicographical order. I handled this just by using a stable sort.\n\nSo, onwards to the general case. Before we can start doing any high-level optimisation, we need to start by answering this question: if we are at position P with X velocity $$v$$ and want to pass through position Q, what possible velocities can we arrive with? I won't prove it formally, but it's not hard to believe that to arrive with the smallest velocity, you should start by accelerating (at the maximum acceleration) for some time, then decelerate for the rest of the time. Let's say that the total time between P and Q is $$T$$, the X separation is $$x$$, the initial acceleration time is $$T - t$$ and the final deceleration time is $$t$$. Some basic integration then gives\n$x = v(T-t)+\\tfrac{1}{2}a(T-t)^2 + (v+a(T-t))t - \\tfrac{1}{2}at^2.$\nThis gives a quadratic in $$t$$ where the linear terms conveniently cancel, leaving\n$t = \\sqrt{\\frac{vT+\\tfrac{1}{2}aT^2-x}{a}}$\nThe final velocity is just $$v+aT-2at$$. We can also find the largest possible arrival velocity simply by replacing $$a$$ with $$-a$$. Let's call these min and max velocity functions $$V_l(v, T, x)$$ and $$V_h(v, T, x)$$.\n\nMore generally, if we can leave P with a given range of velocities $$[v_0, v_1]$$, with what velocities can we arrive at Q? We first need to clamp the start range to the range from which we can actually reach Q i.e., that satisfy $$|vT - x| \\le \\frac{1}{2}aT^2$$. A bit of calculus shows that $$V_l$$ and $$V_h$$ are decreasing functions of $$v$$, so the arrival range will be $$[V_l(v_1), V_h(v_0)]$$.\n\nFinally, we are ready to tackle the problem as a whole, with multiple targets. We will use dynamic programming to answer this question, starting from the last target (highest Y): if I start at target i and want to hit a total of j targets, what are the valid X velocities at i? The answer will in general be a set of intervals. We will make a pseudo-target at (0, 0), and the answer will then be the largest j for which 0.0 is a valid velocity at this target.\n\nComputing the DP is generally straightforward, except that the low-level queries we need are the reverse of those we discussed above i.e. knowing the arrival velocities we need to compute departure velocities. No problem, this sort of physics is time-reversible, so we just need to be careful about which signs get flipped. For each (i, j) we consider all options for the next target, back-propagate the set of intervals from that next target, and merge them into the answer for (i, j). Of course, for $$j = 1$$ we use the interval $$(-\\infty, \\infty)$$.\n\nThe final inconvenience is that we must produce a lexicographical minimum output. Now we will see the advantage of doing the DP in the direction we chose. We build a sequence forwards, keeping track of the velocity interval for our current position. Initially the position will be (0, 0) and the velocity interval will be [0.0, 0.0]. To test whether a next position is valid, we forward-propagate the velocity interval to this next position, and check whether it has non-empty intersection with the set of velocities that would allow us to hit the required number of remaining targets. We then just take the next valid position with the lowest ID.\n\nWhat is the complexity? It could in theory be exponential, because every path through the targets could induce a separate interval in the interval set. However, in reality the intervals merge a lot, and I wouldn't be surprised if there is some way to prove that there can only be a polynomial number of intervals to consider. My solution still ran pretty close to the time limit, though.\n\nFriday, June 27, 2014\n\nMore ICPC analysis\n\nNow we're getting on to the harder problems. Today I've cover two that I didn't know how to solve. Some of the others I have ideas for, but I don't want to say anything until I've had a chance to code them up.\n\nFirstly, problem I. This is a maximum clique problem, which on a general graph is NP-complete. So we will need to use the geometry in some way. Misof has a nice set of slides showing how it is done: http:\/\/people.ksp.sk\/~misof\/share\/wf_pres_I.pdf. I had the idea for the first step (picking the two points that are furthest apart), but it didn't occur to me that the resulting conflict graph would be bipartite.\n\nNow problem G. I discovered that this is a problem that has had a number of research papers published on the topic, one of which achieves $$O(N^3)$$. Fortunately, we don't need to be quite that efficient. Let's start by finding a polynomial-time solution. Let's suppose we've already decided the diameters of the two clusters, D(A) and D(B), and just want to find out whether this is actually possible. For each shipment we have a boolean variable that says whether it goes into part A (false) or part B (true). The constraints become boolean expressions: if d(i, j) > D(A) then we must have i || j, and if d(i, j) > D(B) then we must have !i || !j. Determining whether the variables can be satisfied is just 2-SAT, which can be solved in $$O(N^2)$$ time.\n\nNow, how do we decide which diameters to test? There are $$O(N^2)$$ choices for each, so the naive approach will take $$O(N^6)$$ time. We can reduce this to $$O(N^4\\log N)$$ by noting that once we've chosen one, we can binary search the other (it's also possible to eliminate the log factor, but it's still too slow). So far, this is what I deduced during the contest.\n\nThe trick (which I found in one of those research papers) is that one can eliminate most candidates for the larger diameter. If there is an odd-length cycle, at least one of the edges must be within a cluster, and so that is a lower bound for the larger diameter. What's more, we can ignore the shortest edge of an even cycle (with some tie-breaker), because if the shortest edge lies within a cluster, then so must at least one other edge.\n\nWe can exploit this to generate an O(N)-sized set of candidates: process the edges from longest to shortest, adding each to a new bipartite graph (as for constructing a maximum spanning tree with Kruskal's algorithm). There are three cases:\n1. The next edge connects two previously disconnected components. Add the edge to the graph (which will remain bipartite, since one can always re-colour one of the components. This edge length is a candidate diameter.\n2. The next edge connects two vertices in the same component, but the graph remains bipartite. This edge is thus part of an even cycle, and so can be ignored.\n3. The next edge connects two vertices, forming an odd cycle. This edge length is a candidate diameter, and the algorithm terminates.\nThe edges selected in step 1 form a tree, so there are only O(N) of them.\n\nWednesday, June 25, 2014\n\nACM ICPC 2014\n\nACM ICPC 2014 is over. The contest was incredibly tough: solving less than half the problems was enough to get a medal, and 20 teams didn't solve any of the problems (unfortunately including the UCT team, who got bogged down in problem K).\n\nI managed to solve 5 problems during the contest (in 4:30, since I didn't wake up early enough to be in at the start), and I've solved one more since then. Here are my solutions, in approximately easy-to-hard order.\n\nEDIT: note that I've made follow-up blog posts with more analysis.\n\nD: game strategy\n\nThis was definitely the easiest one, and this is reflected in the scores. We can use DP to determine the set of states from which Alice can force a win within i moves. Let's call this w[i]. Of course, w[0] = {target}. To compute w[i+1], consider each state s that is not already a winning state. If one of Alice's options is the set S and $$S \\subseteq w[i]$$, then Alice can win from this state in i+1 moves.\n\nWe can repeat this N times (since no optimal winning sequence can have more than N steps), or just until we don't find any new winning states. We don't actually need to maintain all the values of w, just the latest one.\n\nK: surveillance\n\nLet's first construct a slower polynomial-time algorithm. Firstly, for each i, we can compute the longest contiguous sequence of walls we can cover starting at i. This can be done in a single pass. We sort all the possible cameras by their starting wall, and as we sweep through the walls we keep track of the largest-numbered ending wall amongst cameras whose starting wall we have passed. The camera with $$a > b$$ are a bit tricky: we can handle them by treating them as two cameras $$[a - N, b]$$ and $$[a, b + N]$$.\n\nLet's suppose that we know we will use a camera starting at wall i. Then we can use this jump table to determine the minimum number of cameras: cover as much wall starting from i as possible with one camera, then again starting from the next empty slot, and so on until we've wrapped all the way around and covered at least N walls. We don't know the initial i, but we can try all values of i. Unfortunately, this requires $$O(N^2)$$ time.\n\nTo speed things up, we can compute accelerated versions of the jump table. If $$J_c[i]$$ is the number of walls we can cover starting from i by using c cameras, then we already have $$J_1$$, and we can easily calculate $$J_{a+b}$$ given $$J_a$$ and $$J_b$$. In particular, we can compute $$J_2, J_4, J_8$$ and so on, and them combine these powers of two to make any other number. This can then be used in a binary search to find the smallest c such that $$J_c$$ contains a value that is at least N. The whole algorithm thus requires $$O(N\\log N)$$ time.\n\nOne catch that broke my initial submission is that if the jump table entries are computed naively, they can become much bigger than N. This isn't a problem, until they overflow and become negative. Clamping them to N fixed the problem.\n\nC: crane balancing\n\nThis is a reasonably straightforward geometry problem, requiring some basic mechanics. Let the mass of the crane be $$q$$, let the integral of the x position over the crane be $$p$$, let the base be the range $$[x_0, x_1]$$, and let the x position of the weight be $$x$$. The centre of mass of the crane is $$\\frac p q$$, but with a weight of mass $$m$$, it will be $$\\frac{p+mx}{q+m}$$. The crane is stable provided that this value lies in the interval $$[x_0, x_1]$$. This turns into two linear inequalities in $$m$$. Some care is then needed to deal with the different cases of the coefficient being positive, negative or zero.\n\nComputing the area and the integral of the x value is also reasonably straightforward: triangulate the crane by considering triangles with vertices at $$(0, i, i+1)$$ and computing the area (from the cross product) and centre of mass (average of the vertices) and then multiply the area by the x component of the centre of mass to get the integral.\n\nI prefer to avoid floating point issues whenever possible, so I multiplied all the X coordinates by 6 up front and worked entirely in integers. This does also cause the masses to be 6 times larger, so they have to be adjusted to compensate.\n\nE: maze reduction\n\nThis is effectively the same problem I set in Topcoder SRM 378 (thanks to ffao on Topcoder for reminding me where I'd seen this). If you have an account on Topcoder you can read about an alternative solution in the match analysis, in which it is easier to compute worst-case bounds.\n\nThe approach I used in this contest is similar in concept to D, in that you incrementally update a hypothesis about which things are distinguishable. We will assign a label to every door of each chamber. Two doors are given the same label if we do not (yet) know of a way to distinguish them. Initially, we just label each door with the degree of the room it is in, since there is nothing else we can tell by taking zero steps.\nNow we can add one inference step. Going clockwise from a given door, we can explore all the corridors leading from the current chamber and note down the labels on the matching doors at the opposite end of each corridor. This forms a signature for the door. Having done this for each door, we can now assign new labels, where each unique signature is assigned a corresponding label. We can repeat this process until we achieve stability, i.e., the number of labels does not change. We probably ought to use each door's original label in the signature too, but my solution passed without requiring this.\nFinally, two rooms are effectively identical if their sequence of door labels is the same up to rotation. I picked the lexicographically minimum rotation for each room and used this as a signature for the room.\nI'm not sure what the theoretical work-case performance is, but I suspect it would be quite large. For a start, by algorithm requires $$O(NE\\log N)$$ time per iteration, which I suspect could be improved by using a hash table with some kind of rolling hash to rotation in O(1) time. I was surprised when my first submission ran in time.\n\nB: buffed buffet\n\nThis one was deceptively sneaky, but the principles are reasonably simple. It's not too hard to guess that one will solve the continuous and discrete problems separately, and then consider all partitions between the two.\nLet's start with the continuous problem, since it is a little easier. I don't have a formal proof, but it shouldn't be too hard to convince yourself that a greedy strategy works. We start by eating the tastiest food. Once it degrades to being as tasty as the second-tastiest food, we then each both, in the proportion that makes them decay at the same rate. In fact, at this point we can treat them as a combined food with a combined (slower) decay rate. We continue eating this mix until tastiness decays to match the third-tastiest, and so on. There are some corner cases that need to be handled if there are foods that don't decay.\nNow what about the discrete case? Because the items have different weights, a greedy strategy won't work here, as with any knapsack problem. There is a reasonably straightforward $$O(DW^2)$$ dynamic programming, where you consider each possible number of serving of each discrete food type, but this will be too slow. But there is some structure in the problem, so let's try to exploit it. Let's say that $$f(i)$$ is the maximum tastiness for a weight of $$i$$ using only the foods we've already considered, and we're now computing an updated $$f'$$ by adding a new discrete food with weight $$w$$, initial tastiness $$t$$ and decay rate $$d$$. For a given i, $$f'(i)$$ clearly only depends on $$f(j)$$ where $$i - j$$ is a multiple of $$w$$, so let's split things up by the remainder modulo $$i$$. Fix a remainder $$r$$ and let $$g(i) = f(iw + r)$$ and $$g'(i) = f'(iw + r)$$. Now we can say that\n\\begin{aligned} g'(i) &= \\max_{0 \\le j \\le i}\\big\\{ g(j) + \\sum_{n=1}^{i-j}(t - (n-1)d\\big\\}\\\\ &= \\max_{0 \\le j \\le i}\\big\\{ g(j) + (i-j)t - \\frac{(i-j-1)(i-j)}{2}\\cdot d\\big\\}\\\\ &= \\max_{0 \\le j \\le i}\\big\\{ g(j) + (i-j)t - \\frac{i(i-1)+j(j+1)-2ij}{2}\\cdot d\\big\\}\\\\ &= it - \\frac{i(i-1)d}{2} + \\max_{0 \\le j \\le i}\\big\\{ g(j)-\\frac{j(j+1)d}{2} - jt + ijd\\big\\}\\\\ &= it - \\frac{i(i-1)d}{2} + \\max_{0 \\le j \\le i}\\big\\{ h(j) + ijd \\big\\} \\end{aligned}\nHere we have defined $$h(j) = g(j)-\\frac{j(j+1)d}{2} - jt$$, which can be computed in constant time per $$j$$.\n\nThe key observation is that we have turned the expression inside the maximum into a linear function of $$j$$. Thus, if we plot $$h$$ on a graph, only the upper convex hull is worth considering. We can maintain this upper hull as we increase $$i$$ (remember, $$j \\le i$$). The second observation is that the optimal choice of $$j$$ will increase as $$i$$ increases, because increasing $$i$$ will increase the $$ijd$$ more for larger values of $$j$$. It follows that we can incrementally find the optimal $$j$$ for each $$i$$ by increasing the previous $$j$$ along the upper hull until increasing it further would start decreasing the objective function. There are a few corner cases: the upper hull might not have any valid entries (because there might be no way to make up a weight of exactly $$jw+r$$), and we might have popped off the previous $$j$$ as part of updating the hull. These just need careful coding. Another snag to watch out for is that if all the foods decay very fast, the optimal result may in fact overload a 32-bit integer in the negative direction.\n\nThe discrete part of the algorithm now requires O(DW), and the continuous part requires O(D\\log D + W).\n\nF: messenger (solved after contest)\n\nThis is a nasty problem mostly because of numerical stability problems. The problem itself is numerically unstable: a rounding error in measuring the total lengths of the paths is sufficient to change the answer between possible and impossible. It requires the black art of choosing good epsilons. I'll only describe the ideal case in which none of these nasty rounding problems occur.\n\nSuppose we pick a point on both Misha and Nadia's path. In general this won't work, because the courier will arrive too late or too early at this point to intercept Nadia. Let L(A, B) be the number of time units that the courier is late when travelling from A to B. L can be late if the courier is early. Valid solutions are those where L = 0.\n\nFirst, let's prove that L is an increasing function of A (where \"increasing\" means that A moves further along Misha's path). Suppose that the courier decided to, instead of moving straight to B, instead kept pace with Misha for a while, then went to B. Obviously this would mean arriving later (or at the same time). But this is equivalent to the arrival time if A increases. A similar argument shows that L is a decreasing function of B. In both cases, this is not strict.\n\nThis means that there can be at most $$O(N)$$ pairs of segments for which L=0, rather than $$O(N^2)$$, because we cannot move forward on one path and backwards on another. We can iterate over the candidate pairs by successively advancing either one segment or the other, by measuring L at pairs of segment endpoints.\n\nNow we have reduced the problem to a single segment from each path. Given a point on one path, we can find the corresponding point on the other by solving what looks like a quadratic but which decays into a linear equation. Again, there are corner cases where the linear coefficient is also zero, in which case we have to break ties so as to minimise the travel time. Using this mapping function, we can identify the range of positions on Misha's path that correspond to valid positions on Nadia's path. I then used ternary search to find the optimal position along this segment. I didn't actually prove that the function is convex, but it seemed to work.\n\nThe solution I implemented that passed is actually rather messier than what I've described, and ran close to the time limit. I tried implementing it as described above but it hits assertions in my code due to numeric instabilities, but I think it should still be possible to fix it.\n\nMonday, June 02, 2014\n\nNew website\n\nI've finally gotten around to replacing my ugly nineties-looking personal web page that was cobbled together with raw HTML and m4, with a much prettier version that nevertheless still contains roughly the same content. It is generated using Sphinx, with the individual pages written in reStructuredText. 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Q: How can I solve this question? Compute the value of the following improper integral. If it is divergent, type "Diverges" or "D".
$$\int_0^2 \frac{dx}{\sqrt{4-x^2}}$$
Do I make $u= 4-x^2$ then $du= -2x \, dx$
Not exactly sure..
A: The limit of integration for which our integrand is undefined is $x = 2$. So, what we really need to compute is:
\begin{align}
\lim \limits_{t \to 2} \int_0^t\dfrac{dx}{\sqrt{4-x^2}}
\end{align}
First, let's worry about the integral. Consider the substitution $x=2\sin(\theta)$ (do a bit of trigonometry to see why this might be useful). So, $dx=2\cos(\theta)d\theta$. So, the integral we're dealing with is:
\begin{align}
\int \dfrac{2\cos(\theta)d\theta}{\sqrt{4-4\sin^2(\theta)}}\\
= \int \dfrac{2\cos(\theta)d\theta}{2\sqrt{1-\sin^2(\theta)}}\\
= \int \dfrac{\cos(\theta)d\theta}{\cos(\theta)}\\
= \int d\theta\\
= \theta + C
\end{align}
Now, we need to put things back in terms of $x$. Solving for $\theta$, we see that this is equal to $\sin^{-1}(\tfrac{x}{2}) + C$.
Now, back to what we were originally dealing with. We now have:
\begin{align}
\lim \limits_{t \to 2} \left(\sin^{-1}(\tfrac{x}{2}) \Big|_0^t\right)\\
= \sin^{-1}(\tfrac{2}{2}) - \sin^{-1}(\tfrac{0}{2})\\
= \tfrac{\pi}{2} - 0 = \boxed{\tfrac{\pi}{2}}
\end{align}
We now see that this improper integral converges to the value $\tfrac{\pi}{2}$.
A: Consider the substitution $x = 2\sin\theta$. From this we have $dx = 2\cos\theta\cdot d\theta$. Substitute these in and see what happens:
$$\int\frac{2\cos\theta \cdot d\theta}{\sqrt{4 - 4\sin^2\theta}}\\\\
= \int\frac{2\cos\theta \cdot d\theta}{2\cos\theta}\\
= \int d\theta\\
= \theta + C\\
= \sin^{-1}{\frac{x}{2}} + C$$
Hence,
$$\int_0^2\frac{dx}{\sqrt{4 - x^2}} = \left[\sin^{-1}{\frac{x}{2}}\right]_0^2\\
= \frac{\pi}{2}$$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,111
|
Q: Crud for users via an admin controller I'm currently developing a .net MVC app using the built in authentication system for users. What I want is for a site administrator to be able to log in, list all registered users and have the ability to edit and create users. Each user will have a role.
I have the following action:
public ActionResult Index()
{
var users = Membership.GetAllUsers();
return View(users);
}
which lets me display the users but it does not provide access to a userID to pass through to an edit/create action. Nor does it give access to a role (I have set up a role the configuration tool). I'm a .MVC noob so please point me in the right direction.
Thanks,
James
A: Is it necessary for you to build an app to manage web site administration? There are apps such as http://wsat.codeplex.com/ which could achieve your objective.
If you need to roll out a custom app, I'd make it role-based. That is check if current user has an admin role and then redirect user to perform the CRUD operations. Something akin to http://msdn.microsoft.com/en-us/library/ff647401.aspx#paght000013_step4
BTW, you may need to consider that GetAllUsers would slow up your app, if there are a lot of users to manage.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,763
|
Q: How can I use SignalR in an Azure? I'm having one azure account and I want to use signalR performance counter in azure web.
If any one has used if before please help me.
A: It is very well documented in the following link.
using-signalr-performance-counters-in-an-azure-web-role
Hope this will solve your problem.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 799
|
{"url":"https:\/\/gmatclub.com\/forum\/a-cherry-pie-with-a-radius-of-8-inches-is-cut-into-six-equalslices-wh-289426.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 19 Mar 2019, 20:13\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# A cherry pie with a radius of 8 inches is cut into six equalslices. Wh\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMath Expert\nJoined: 02 Sep 2009\nPosts: 53709\nA cherry pie with a radius of 8 inches is cut into six equalslices. Wh\u00a0 [#permalink]\n\n### Show Tags\n\n25 Feb 2019, 02:20\n00:00\n\nDifficulty:\n\n(N\/A)\n\nQuestion Stats:\n\n100% (00:44) correct 0% (00:00) wrong based on 8 sessions\n\n### HideShow timer Statistics\n\nA cherry pie with a radius of 8 inches is cut into six equalslices. What is the area, in square inches, of each slice?\n\nA. $$\\frac{8\\pi}{3}$$\n\nB. $$\\frac{16\\pi}{3}$$\n\nC. $$\\frac{32\\pi}{3}$$\n\nD. $$\\frac{64\\pi}{3}$$\n\nE. $$\\frac{128\\pi}{3}$$\n\n_________________\nSVP\nJoined: 18 Aug 2017\nPosts: 2370\nLocation: India\nConcentration: Sustainability, Marketing\nGPA: 4\nWE: Marketing (Energy and Utilities)\nRe: A cherry pie with a radius of 8 inches is cut into six equalslices. Wh\u00a0 [#permalink]\n\n### Show Tags\n\n25 Feb 2019, 02:25\nBunuel wrote:\nA cherry pie with a radius of 8 inches is cut into six equalslices. What is the area, in square inches, of each slice?\n\nA. $$\\frac{8\\pi}{3}$$\n\nB. $$\\frac{16\\pi}{3}$$\n\nC. $$\\frac{32\\pi}{3}$$\n\nD. $$\\frac{64\\pi}{3}$$\n\nE. $$\\frac{128\\pi}{3}$$\n\narea of pie ; 64 pi\nso total 6 pieces will make 64 * pi \/ 6 ;\n\n$$\\frac{32\\pi}{3}$$\nIMO C\n_________________\n\nIf you liked my solution then please give Kudos. Kudos encourage active discussions.\n\nRe: A cherry pie with a radius of 8 inches is cut into six equalslices. Wh \u00a0 [#permalink] 25 Feb 2019, 02:25\nDisplay posts from previous: Sort by","date":"2019-03-20 03:13:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7252042889595032, \"perplexity\": 5394.743496159313}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912202199.51\/warc\/CC-MAIN-20190320024206-20190320050206-00554.warc.gz\"}"}
| null | null |
\section{Introduction}
There are many independent observations hinting at the relevance of a high scale
for particle physics. The smallness of neutrino masses has a simple explanation
in terms of the see-saw mechanism~\cite{Minkowski:1977sc} and relies on the
existence of heavy singlet neutrinos. Stabilizing the electroweak scale against
the see-saw scale seems to require supersymmetry; remarkably, the simplest
supersymmetric extension of the standard model, the MSSM, realizes the
compelling scenario of gauge unification~\cite{Dimopoulos:1981yj} at a scale
$M_\mathrm{GUT}\simeq2\cdot10^{16}\,\mathrm{GeV}$, which is suspiciously close
to the see-saw scale. Both scales are not too far from $M_\mathrm{P}\simeq2\cdot
10^{18}\,\mathrm{GeV}$, which is set by Newton's constant.
The question of how to incorporate all scales in a coherent scheme has been
addressed for more than 30 years. From a bottom-up perspective one is led to the
scheme of grand unified theories (GUTs). Although this scheme exhibits various
very appealing features, there are three major obstacles. First, there is the
so-called doublet-triplet splitting, and related to it, the MSSM $\mu$ problem.
Second, even if this problem is solved, unified models typically are in conflict
with dimension five proton decay \cite{Sakai:1981pk,Dimopoulos:1981dw}
operators. (It is well known that dimension four proton decay can be forbidden
by matter parity.\footnote{Matter parity is sometimes also known as ``$R$
parity''. We choose not to use this terminology as matter parity is non-$R$,
i.e.\ it commutes with supersymmetry, and one point of this paper is to discuss
a true discrete $R$ symmetry.}) Third, in four-dimensional models of grand
unification there is no relation between the GUT and Planck scales,
$M_\mathrm{GUT}$ and $M_\mathrm{P}$.
String theory is believed to provide us with such a relation. However, if string
theory is to describe the real world, it should also provide us with solutions
to the first and second problems. In fact, as known for a long time, the
doublet-triplet splitting problem has a simple solution in theories with extra
dimensions in which the GUT symmetry is broken in the process of
compactification \cite{Witten:1985xc,Breit:1985ud}, which also avoids the most
stringent problems with dimension five proton decay \cite{Altarelli:2001qj}.
However, in concrete string compactifications (see
\cite{Ibanez:1987sn,Font:1988mm,Font:1989aj} for early attempts and
\cite{Cleaver:1998saa,Cleaver:1999mw} for a different approach) very often the problem is
reintroduced; this applies also to the models discussed more recently
\cite{Buchmuller:2005jr,Buchmuller:2006ik,Lebedev:2006kn,Lebedev:2007hv}. On the
other hand, one cannot rule out these constructions as their vacua are not
completely understood. That is, the analysis of potentially realistic string
models is a non-trivial task since a given model exhibits a plethora of vacua
with very different features. The role of discrete symmetries in identifying and
analyzing such vacua has been stressed recently \cite{Buchmuller:2008uq}. One of
these symmetries is matter parity, which has been successfully embedded in string
theory~\cite{Lebedev:2007hv}. This study is devoted to a discussion of the role
of further discrete symmetries in such models and their phenomenological implications.
Specifically, we will focus on a proposed $\Z4^R$ symmetry
\cite{Babu:2002tx}
which has recently been shown in~\cite{Lee:2010gv} to be the unique anomaly-free possibility
with the following properties:
\begin{enumerate}
\item it forbids the $\mu$ term at the perturbative level;
\item it allows the MSSM Yukawa couplings and the effective neutrino mass operator;
\item it commutes with SO(10) in the matter sector.
\end{enumerate}
This symmmetry has the appealing feature that it forbids automatically dimension
four and five proton decay operators. We will discuss how to identify string
vacua exhibiting this symmetry and present a globally consistent string-derived
model with the exact MSSM spectrum realizing this symmetry.
We start in section~\ref{sec:GeneralPicture} with a short description of the
general picture. In section~\ref{sec:the_model} we present an explicit
string-derived model in which an anomalous $\Z4^R$
symmetry explains a suppressed vacuum expectation value of the superpotential,
provides us with a solution of the $\mu$ problem and suppresses dimension five
proton decay operators. Section~\ref{sec:Summary} contains our conclusions. In
various appendices we collect details of our calculations.
\section{General picture}
\label{sec:GeneralPicture}
String theory compactifications provide us with a plethora of vacuum
configurations, each of which comes with symmetries and, as a consequence, with
extra massless degrees of freedom whose mass terms are prohibited by these symmetries.
Simple examples for such compactifications include heterotic
orbifolds~\cite{Dixon:1985jw,Dixon:1986jc}, where the rank of the gauge group
after compactification equals that of $\E8\times\E8$, i.e.\ 16. A few hundreds
of orbifold models are known in which $\E8\times\E8$ gets broken to the standard
model gauge symmetry
$G_\mathrm{SM}=\SU3_C\times\SU2_\mathrm{L}\times\U1_Y$ (with
hypercharge in GUT normalization) times $\U1^n$ times a hidden sector group and
the chiral spectra of the MSSM \cite{Lebedev:2006kn,Lebedev:2008un}. They also
exhibit exotics which are vector-like with respect to $G_\mathrm{SM}$ and which can be
decoupled when the extra gauge symmetries are broken. Each of these models
contains many vacua, i.e.\ solutions of the supersymmetry conditions
$V_F=V_D=0$. Typically these vacua exhibit flat directions before supersymmetry
breaking.
At the orbifold point, where the vacuum expectation values (VEVs) of all fields
are zero, we have discrete $R$ as well as continuous and discrete non-$R$
symmetries. Typically one of the \U1 symmetries appears anomalous, which is
conventionally denoted by $\U1_\mathrm{anom}$. Also some of the discrete
symmetries may appear anomalous \cite{Ibanez:1992hc,Araki:2008ek}. After
assigning VEVs to certain fields, some of the symmetries are spontaneously
broken and others remain. We shall be mainly interested in remnant discrete
symmetries, which can be of $R$ or non-$R$ type and be either anomalous or
non-anomalous. We will discuss examples of all kinds in
section~\ref{sec:the_model}.
Clearly, one cannot assign VEVs to the fields at will. Rather, one has to
identify field configurations which correspond to local minima of the
(effective) scalar potential. Let us briefly describe the first steps towards
identifying such vacuum configurations. Consider a configuration in which
several fields attain VEVs. We focus on ``maximal vacua'' (as
in~\cite{Buchmuller:2008uq}), i.e.\ we assume that all fields which are neutral
under the remnant gauge and discrete symmetries, called $\phi^{(i)}$ $(1\le i\le
N)$ in what follows, attain VEVs (if these are consistent with
$D$-flatness). All fields without expectation value, denoted
by $\psi^{(j)}$ $(1\le j\le M)$, therefore transform non-trivially under some of
the remnant symmetries.
\subsection{Discrete non-$\boldsymbol{R}$ symmetries}
The case of vacua with non-$R$ discrete symmetries has
been discussed in detail in \cite{Buchmuller:2006ik,Buchmuller:2008uq}.
In this case, the superpotential has the form
\begin{equation}
\mathscr{W}~=~\Omega(\phi^{(1)},\dots\phi^{(N)})+(\text{terms at least
quadratic in the}~\psi^{(j)})\;.
\end{equation}
Therefore, the $F$-term equations for the $\psi^{(j)}$ fields trivially vanish and we are
left with $N$ $F$-term equations for the $N$ $\phi^{(i)}$ fields, which generically
have solutions. Hence, if all $\phi^{(i)}$ enter gauge invariant monomials
composed of $\phi^{(i)}$ fields only, we will find supersymmetric vacua, i.e.\
solutions to the $F$- and $D$-term equations.
Because of the above arguments it is sufficient to look at the system of
$\phi^{(i)}$ fields only, which has been studied in the literature. Consider the
case of a \emph{generic} superpotential $\mathscr{W}$. It is known that the
solutions to the $D$- and $F$-term equations intersect \emph{generically} in a
point~\cite{Luty:1995sd}. That is, there are point-like field configurations
which satisfy
\begin{equation}\label{eq:GenericVacuum}
D_a~=~F_i~=~0\quad\text{at}~\phi^{(i)}~=~\langle\phi^{(i)}\rangle\;,
\end{equation}
where, as usual,
\begin{subequations}
\begin{eqnarray}
D_a & = & \sum_i (\phi^{(i)})^*\,\mathsf{T}_a\,\phi^{(i)}\;,\\
F^{(i)} & = & \frac{\partial\mathscr{W}}{\partial\phi^{(i)}}\;.
\end{eqnarray}
\end{subequations}
The term `point-like' means that there are no massless deformations of the
vacuum \eqref{eq:GenericVacuum}. The reason why these vacua are point-like is
easily understood: generically the $F$-term equations constitute as many gauge
invariant constraints as there are gauge invariant variables. However, this also
means that, at least generically,
\begin{equation}
\mathscr{W}|_{\phi^{(i)}~=~\langle\phi^{(i)}\rangle}~\ne~0\;.
\end{equation}
If the fields attain VEVs $\langle\phi^{(i)}\rangle$ of the order of the
fundamental scale, one hence expects to have too large a VEV for $\mathscr{W}$.
One possible solution to the problem relies on approximate $R$
symmetries~\cite{Kappl:2008ie}, where one obtains a highly suppressed VEV of the
superpotential. In what follows, we discuss an alternative: in settings with a
residual $R$ symmetry the above conclusion can be avoided as well.
\subsection{Discrete $\boldsymbol{R}$ symmetries}
Let us now discuss vacua with discrete $R$ symmetries. To be
specific, consider the order four symmetry $\Z4^R$, under which the
superpotential $\mathscr{W}$ has charge 2, such that
\begin{equation}
\mathscr{W}~\xrightarrow{\zeta}~-\mathscr{W}
\end{equation}
under the $\Z4^R$ generator $\zeta$. Superspace coordinates transform as
\begin{equation}
\theta_\alpha~\to~\mathrm{i}\,\theta_\alpha
\end{equation}
such that the $F$-term Lagrangean
\begin{equation}
\mathscr{L}_F~=~\int\!\mathrm{d}^2\theta\,\mathscr{W}+\text{h.c.}
\end{equation}
is invariant. Chiral superfields will have $R$ charges $0,1,2,3$.\footnote{A
special role is played by the dilaton $S$, whose imaginary part $a=\im
S|_{\theta=0}$ shifts under $\Z4^R$.} Both the fields of the type $\psi_1$ and
$\psi_3$ with $R$ charges $1$ and $3$, respectively, can
acquire mass as the $\psi_1^2$ and $\psi_3^2$ terms have $R$ charge $2\mod4$ and
thus denote allowed superpotential terms.
The system of fields $\phi_0^{(i)}$ and $\psi_2^{(j)}$ with $R$ charges 0 and 2,
respectively, is more interesting. Consider first only one field $\phi_0$ and
one field $\psi_2$. The structure of the superpotential is
\begin{equation}\label{eq:superpotential0+2}
\mathscr{W}~=~\psi_2\cdot f(\phi_0)+\mathcal{O}(\psi_2^3)
\end{equation}
with some function $f$. The $F$-term for $\phi_0$ vanishes trivially as long as
$\Z4^R$ is unbroken,
\begin{equation}
\frac{\partial\mathscr{W}}{\partial\phi_0}
~=~
\psi_2\cdot f'(\phi_0)
~=~0\;.
\end{equation}
Note that due to the $\Z4^R$ symmetry the superpotential vanishes in the
vacuum. Thus it is sufficient to look at the global supersymmetry $F$-terms. On the
other hand, the $F$-term constraint (at $\psi_2=0$)
\begin{equation}
\frac{\partial\mathscr{W}}{\partial\psi_2}
~=~f(\phi_0)~\stackrel{!}{=}~0
\end{equation}
will in general fix $\phi_0$ at some non-trivial zero $\langle\phi_0\rangle$ of
$f$. Indeed, there will be a supersymmetric mass term, which can be seen by
expanding $\phi_0$ around its VEV, i.e.\ inserting
$\phi_0=\langle\phi_0\rangle+\delta\phi_0$ into
\eqref{eq:superpotential0+2},
\begin{equation}
\mathscr{W}~=~f'(\langle\phi_0\rangle)\,\delta\phi_0\,\psi_2
+\mathcal{O}(\delta\phi_0^2,\psi_2^3)\;.
\end{equation}
The supersymmetric mass $f'(\langle\phi_0\rangle)$ is generically
different from 0.
Repeating this analysis for $N$ $\phi_0^{(i)}$ and $M$ $\psi_2^{(j)}$ fields
reveals that the $F$-terms of the $\psi_2^{(j)}$ lead to $M$, in general
independent, constraints on the $\phi_0^{(i)}$ VEVs. For $N=M$ we therefore
expect point-like vacua with all directions fixed in a supersymmetric way.
To summarize, systems with a residual $R$ symmetry ensure, unlike in the case
without residual symmetries, that $\langle\mathscr{W}\rangle=0$. However, in
systems which exhibit a linearly realized $\Z4^R$ somewhere in field space it
may not be possible to find a supersymmetric vacuum that preserves $\Z4^R$. In
the case of a generic superpotential this happens if there are more, i.e.\
$M>N$, fields with $R$ charge 2 than with 0. On the other hand, if there are
more fields with $R$ charge 0 than with 2, i.e.\ for $M<N$, one expects to have
a Minkowski vacuum with $N-M$ flat directions. For $N=M$ one can have
supersymmetric Minkowski vacua with all directions fixed in a supersymmetric
way.
An important comment in this context concerns the moduli-dependence of
couplings. As we have seen, in the case of discrete $R$-symmetries one might
obtain more constraint (i.e.\ $F$-term) equations than $R$-even `matter' fields.
Specifically, in string vacua one should, however, carefully take into account
all $R$-even fields, also the K\"ahler and complex structure moduli, $T_i$ and
$U_j$, on whose values the coupling strengths depend.
\section{An explicit string-derived model}
\label{sec:the_model}
In order to render our discussion more specific, we base our analysis on a
concrete model. We consider a $\Z2\times\Z2$ orbifold compactification with an
additional freely acting $\Z2$ of the $\E8\times\E8$ heterotic string. Details
of the model including shift vectors and Wilson lines can be found in
appendix~\ref{app:details_model}.
In \cite{Blaszczyk:2009in} a vacuum configuration of a very similar
$\Z2\times\Z2$ model with matter parity and other desirable features was
presented. However, the vacuum configuration discussed there has the unpleasant
property that, at least generically, all Higgs fields attain large masses. In
what follows we discuss how this can be avoided by identifying vacuum
configurations with enhanced symmetries. In \cite{Lee:2010gv} another vacuum
with the $\Z4^R$ symmetry discussed in the introduction was found by using the
methods presented in this paper. In both models the GUT symmetry
is broken non-locally. This may be advantageous from the point of view of
precision gauge unification \cite{Hebecker:2004ce}. It also avoids fractionally
charged exotics, which appear in many other compactifications (cf.\ the
discussion in \cite{GatoRivera:2010xn}).
\paragraph{Labeling of states.}
We start our discussion with a comment on our notation. In a first step, we
label the fields according to their
$G_\mathrm{SM}\times[\SU3\times\SU2\times\SU2]_\mathrm{hid}$ quantum numbers. In
particular, we denote the standard model representations with lepton/Higgs and
$d$-quark quantum numbers as
\begin{subequations}
\begin{eqnarray}
L_i & : & (\boldsymbol{1},\boldsymbol{2})_{-1/2}\;,\\
\bar L_i & : & (\boldsymbol{1},\boldsymbol{2})_{1/2}\;,\\
D_i & : & (\boldsymbol{3},\boldsymbol{1})_{-1/3}\;,\\
\bar D_i & : & (\boldsymbol{\overline{3}},\boldsymbol{1})_{1/3}\;.
\end{eqnarray}
\end{subequations}
In the next step we identify $\Z4^R$ such that the $\bar L_i$/$L_i$ decompose in
lepton doublets $\ell_i$ with odd $\Z4^R$ charges and Higgs candidates
$h_d$/$h_u$ with even $\Z4^R$ charges etc. The details of labeling states are
given in appendix~\ref{app:details_model}.
\paragraph{Searching for $\boldsymbol{\Z4^R}$.}
How can one obtain vacua with $\Z4^R$ in practice? We found
the following strategy most efficient:
\begin{enumerate}
\item In a first step we switch on a random sample of SM singlets in such a way
that all unwanted gauge factors are spontaneously broken.
\item With these VEVs at hand, the original gauge and discrete symmetries at the
orbifold point get broken to a discrete subgroup, which can be determined
unambiguously with the methods described in \cite{Petersen:2009ip}. Details of
the automatization of these methods are explained in \cite{Schieren:2010pt}.
\item We only keep configurations in which there is a residual $\Z4^R$ symmetry
with precisely three generations of matter having $R$-charge 1; details of how
to identify such configurations are given in appendix~\ref{app:IdentifyingZ4R}.
\item From these configurations we select those exhibiting the following properties:
\begin{itemize}
\item $F$- and $D$-flat;
\item all exotics decouple;
\item one pair of massless Higgs, i.e.\ $\mu$ term forbidden to all orders (at
the perturbative level);
\item Yukawa couplings have full rank.
\end{itemize}
\end{enumerate}
One of the main achievements of this study is a considerable simplification in
the verification of the four items listed in step 4. In order to check
$D$-flatness of a given configuration we use the Hilbert basis method, which is
described in detail in appendix~\ref{app:HilbertBasis}. The other three
properties can be verified by inspecting the remnant discrete symmetries only.
In earlier studies
\cite{Buchmuller:2005jr,Buchmuller:2006ik,Lebedev:2006kn,Lebedev:2007hv} we had
to explicitly identify couplings that are consistent with the string selection
rules in order to show that all exotics decouple and the Yukawa couplings have
full rank. In our new approach the remnant symmetries will tell us immediately
whether an entry of a mass or Yukawa matrix will or will not appear. We have
cross-checked this method extensively by explicitly computing the couplings
between the charged and the VEV fields, and were always able to find a coupling
which fills in an entry of a matrix, albeit sometimes at very high orders. Note,
we assume that all couplings allowed by string selection rules appear in the superpotential.
\paragraph{VEV configuration.}
Following the above steps, we obtained a promising
configuration in which the fields
\begin{eqnarray}
\widetilde{\phi}^{(i)}
& = &
\{\phi _1, \phi _2, \phi _3, \phi _4, \phi _5, \phi _6, \phi _7, \phi _8,
\phi _9, \phi _{10}, \phi _{11}, \phi _{12}, \phi _{13}, \phi _{14}, \nonumber\\
& & \hphantom{\{}{}
x_1,x_2,x_3,x_4,x_5,\bar x_1,\bar x_3 ,\bar x_4 ,\bar x_5,y_3,y_4,y_5,y_6\}
\label{eq:phitildefields}
\end{eqnarray}
attain VEVs. The full quantum numbers of these fields are given
in table~\ref{tab:fullspectrum} in appendix~\ref{app:details_model}. In order
to ensure $D$-flatness with respect to the hidden sector gauge factors, in a given basis
not all components of the $x_i$/$\bar x_i$ and $y_i$ attain VEVs. Details are
given in equations \eqref{eq:SU2vacuum} and \eqref{eq:SU3vacuum} in
appendix~\ref{app:SUSYVacuum}.
\paragraph{Remnant discrete symmetries.} By giving VEVs to the
$\widetilde{\phi}^{(i)}$ fields in \eqref{eq:phitildefields}, we arrive at a
vacuum in which, apart from $G_\mathrm{SM}$ and a `hidden' \SU2, all gauge
factors are spontaneously broken. The vacuum exhibits a
$\Z4^R$ symmetry, whereby the superpotential $\mathscr{W}$
has $\Z4^R$ charge 2.
The $\Z4^R$ charges of the matter fields are shown in
table~\ref{tab:DiscreteChargesMatterZ2xZ2_1-1}. The detailed origin of the $\Z4^R$
symmetry is discussed later. Given these charges, we confirm
by a straightforward field-theoretic calculation
(cf.~\cite{Ibanez:1991hv,Araki:2008ek}) that $\Z4^R$ appears indeed anomalous with
universal $\SU2_\mathrm{L}-\SU2_\mathrm{L}-\Z4^R$ and $\SU3_C-\SU3_C-\Z4^R$
anomalies (see \cite{Lee:2010gv} and
appendix~\ref{app:DiscreteAnomalyCalculation}). The statement
that $\Z4^R$ appears anomalous
means, as we shall discuss in detail below, that the
anomalies are cancelled by a Green-Schwarz (GS) mechanism.
On the other hand, the $\Z4^R$ has a, by the traditional
criteria, non-anomalous
$\Z2^\mathcal{M}$ subgroup which is equivalent to matter
parity~\cite{Lee:2010gv}.
\begin{table}[htb]
\centering
\subtable[Quarks and leptons.]{
\begin{tabular}{cccccc}
\toprule[1.3pt]
& $q_i$ & $\bar u_i$ & $\bar d_i$ & $\ell_i$ & $\bar e_i$ \\
$\Z4^R$ & 1 & 1 & 1 & 1 & 1\\
\bottomrule[1.3pt]
\end{tabular}
}
\\
\subtable[Higgs and exotics.]{
\begin{tabular}{ccccccccccccccccccc}
\toprule[1.3pt]
& $h_1$ & $h_2$& $h_3$& $h_4$& $h_5$& $h_6$ & $\bar h_1$ & $\bar h_2$ & $\bar h_3$ & $\bar h_4$ & $\bar h_5$ & $\bar h_6$ & $\delta_1$ & $\delta_2$ & $\delta_3$ & $\bar\delta_1$ & $\bar\delta_2$ & $\bar\delta_3$ \\
$\Z4^R$ & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 2 & 0 & 2 & 2 & 2 & 0 & 0\\
\bottomrule[1.3pt]
\end{tabular}
}
\caption{$\Z4^R$ charges of the (a) matter fields and (b) Higgs and exotics. The
index $i$ in (a) takes values $i=1,2,3$.}
\label{tab:DiscreteChargesMatterZ2xZ2_1-1}
\end{table}
\paragraph{$\boldsymbol{D}$-flatness.}
As already discussed, we cannot switch on the $\widetilde{\phi}^{(i)}$ fields at will; rather we have to show that there are
vacuum configurations in which all these fields acquire VEVs. This requires to
verify that the $D$- and $F$-term potentials vanish.
With the Hilbert basis method (see appendix~\ref{app:HilbertBasis}) we could
identify a complete set of $D$-flat directions composed of $\widetilde{\phi}^{(i)}$
fields. We compute the dimension of the $D$-flat moduli space
using Singular \cite{GPS05} and the STRINGVACUA \cite{Gray:2008zs} package; the
result is that there are $18$ $D$-flat directions; the details of the
computation are collected in appendix~\ref{app:SUSYVacuum}.
\paragraph{$\boldsymbol{F}$-term constraints.} Next we consider the $F$-term
constraints. As discussed in section~\ref{sec:GeneralPicture}, the $F$-term
conditions come from the fields with $R$-charge 2. We compute the number of
independent conditions in appendix~\ref{app:SUSYVacuum}. The result is that
there are 23 independent conditions on $18+6=24$ $D$-flat directions, where we
included the K\"ahler and complex structure moduli. We therefore expect to find
supersymmetric vacuum configurations in which all the $\widetilde{\phi}^{(i)}$
acquire VEVs. In this configuration, almost all singlet fields, including the
geometric moduli are fixed in a supersymmetric way. It will be interesting to
compare this result to similar results found recently in the context of smooth
heterotic compactifications \cite{Anderson:2010mh}. We expect a significantly
different, i.e.\ healthier, phenomenology than in the case in which a large
number of singlets acquire mass only after supersymmetry breaking~\cite{Dundee:2010sb,Parameswaran:2010ec}.
Notice that there are two possible caveats. First, the analysis
performed strictly applies only to superpotentials which are, apart from all
the symmetries we discuss, generic. Second, it might happen that there are
supersymmetric vacua, but they occur at large VEVs of some of the
fields, i.e.\ in regions of field space where we no longer control our
construction. Both issues will be addressed elsewhere.
\paragraph{Higgs vs.\ matter.}
The $\Z2^\mathcal{M}$ subgroup of the $\Z4^R$ symmetry allows us to discriminate between
\begin{itemize}
\item 3 lepton doublets, $\ell_i=\{L_4, L_6, L_7\}$,
\item 3 $d$-type quarks,
$\bar d_i=\{\bar D_1, \bar D_3, \bar D_4\}$,
\end{itemize}
on the one hand, and
\begin{itemize}
\item Higgs candidates, $h_i=\{L_1,L_2,L_3,L_5,L_8,L_9\}$ and
$\bar h_i=\{\bar L_1,\bar L_2,\bar L_3,\bar L_4,\bar L_5,\bar L_6\}$,
\item exotic triplets, $\delta_i=\{D_1,D_2,D_3\}$ and
$\bar\delta_i=\{\bar D_2,\bar D_5,\bar D_6\}$
\end{itemize}
on the other hand.
\paragraph{Decoupling of exotics.}
With the charges in table~\ref{tab:DiscreteChargesMatterZ2xZ2_1-1} we can
readily analyze the structure of the mass matrices.
We crosscheck these structures by explicitly computing the
couplings allowed by the string selection rules (cf.\ \cite{Blaszczyk:2009in}). Note there is a caveat: our results are based on the
assumption that all couplings that are allowed by the selection
rules will appear with a non-vanishing coefficient.
A $\widetilde{\phi}^{n}$ in the matrices
represents a known polynomial of order $n$ in the $\widetilde{\phi}$ fields which we
have calculated using string selection rules. A zero entry in the matrices means
that the corresponding coupling is not present in the perturbative
superpotential. The $\bar h_i-h_j$ Higgs mass matrix is
\begin{equation}
\mathcal{M}_h ~=~
\begin{pmatrix}
0 & \phi _6 & 0 & \phi _4 & 0 & 0 \\
\phi _7 & 0 & \phi _2 & 0 & \phi _{13} & \phi _{14} \\
0 & \phi _1 & 0 & \widetilde{\phi}^3 & 0 & 0 \\
0 & \widetilde{\phi}^3& 0 & \widetilde{\phi}^5 & 0 & 0 \\
\widetilde{\phi}^3 & 0 & \phi _{11} & 0 & \phi _8 &\widetilde{\phi}^3 \\
\widetilde{\phi}^3 & 0 & \phi _{12} & 0 & \widetilde{\phi}^3 & \phi _8
\end{pmatrix}\;.
\end{equation}
Here we omit coefficients, which depend on the three K\"ahler
moduli $T_i$ and complex structure moduli $U_i$.
Clearly, this mass matrix has rank five, such that there is one massless Higgs
pair
\begin{subequations}
\begin{eqnarray}
h_u & = & a_1\, \bar h_1 +a_2\, \bar h_3 + a_3 \bar h_4\;,\\
h_d & = & b_1\,h_1 +b_2\,h_3 +b_3\,h_5 +b_4\,h_6
\end{eqnarray}
\end{subequations}
with $a_i$ and $b_j$ denoting coefficients.
The $\bar \delta-\delta$ mass matrix is
\begin{equation}
\mathcal{M}_\delta ~=~
\left(
\begin{array}{ccc}
\widetilde{\phi}^5 & 0 & 0 \\
0 & \phi _8 & \widetilde{\phi}^3\\
0 & \widetilde{\phi}^3 & \phi _8
\end{array}
\right)\;.
\end{equation}
Hence, the matrix has full rank and all exotics decouple. Note that the block
structure of $\mathcal{M}_\delta$ is not a coincidence but a consequence of the
fact that $\delta_2/\delta_3$ and $\bar \delta_2/\bar \delta_3$ form $D_4$
doublets (see below). Altogether we see that all exotics with
Higgs quantum numbers, and all but one pair of exotic triplets, decouple at the
linear level in the $\widetilde{\phi}^{(i)}$ fields. This leads to the
expectation that all but one pair of exotics get mass of the order of the GUT
(or compactification) scale $M_\mathrm{GUT}$ while one pair of triplets might be
somewhat lighter. We also note that the presence of colored states somewhat
below $M_\mathrm{GUT}$ can give a better fit to MSSM gauge coupling unification
(cf.\ \cite{Dundee:2008ts}). However, a crucial property of the $\delta$- and
$\bar\delta$ triplets is that, due to the $\Z4^R$ symmetry, they do not mediate
dimension five proton decay.
\paragraph{Effective Yukawa couplings.}
The effective Yukawa couplings are defined by
\begin{equation}
\mathscr{W}_Y~=~
\sum_{i=1,3,4}\left[ (Y_u^{(i)})^{fg}\,q_f\,\bar u_g\,\bar h_i \right]
+\sum_{i=1,3,5,6}
\left[(Y_d^{(i)})^{fg}\,q_f\,\bar d_g\,h_i
+(Y_e^{(i)})^{fg}\,\ell_f\,\bar e_g\,h_i\right]\;.
\label{eq:definition_yukawas}
\end{equation}
The Yukawa coupling structures are
\begin{subequations}\label{eq:YukawaMatricesZ2xZ2_1-1}
\begin{eqnarray}
Y_u^{(1)} & = &
\left(
\begin{array}{ccc}
\widetilde{\phi} ^2 & \widetilde{\phi} ^4 & \widetilde{\phi} ^6 \\
\widetilde{\phi} ^4 & \widetilde{\phi} ^2 & \widetilde{\phi} ^6 \\
\widetilde{\phi} ^6 & \widetilde{\phi} ^6 & 1
\end{array}
\right) \;,
\quad
Y_u^{(3)} ~=~
\left(
\begin{array}{ccc}
1 & \widetilde{\phi} ^6 & \widetilde{\phi} ^4 \\
\widetilde{\phi} ^6 & 1 & \widetilde{\phi} ^4 \\
\widetilde{\phi} ^4 & \widetilde{\phi} ^4 & \widetilde{\phi} ^2
\end{array}
\right)\;,
\label{eq:Yu}\\
Y_e^{(5)} = (Y_d^{(5)})^T & = &
\left(
\begin{array}{ccc}
\widetilde{\phi} ^6 & \widetilde{\phi} ^6 & \widetilde{\phi} ^6 \\
\widetilde{\phi} ^6 & \widetilde{\phi} ^6 & 1 \\
\widetilde{\phi} ^6 & 1 & \widetilde{\phi} ^4
\end{array}
\right) \;,
\\
Y_e^{(6)} = (Y_d^{(6)})^T & = &
\left(
\begin{array}{ccc}
\widetilde{\phi} ^6 & \widetilde{\phi} ^6 & 1 \\
\widetilde{\phi} ^6 & \widetilde{\phi} ^6 & \widetilde{\phi} ^6 \\
1 & \widetilde{\phi} ^6 & \widetilde{\phi} ^4
\end{array}
\right) \;.
\label{eq:YdYe}
\end{eqnarray}
\end{subequations}
$Y_d$ and $Y_e$ coincide at tree-level, i.e.\ they exhibit \SU5
GUT relations, originating from the non-local GUT breaking due to the freely acting Wilson line.
There are additional contributions to $Y_u$ from couplings to $\bar h_4$ and to $Y_e$/$Y_d$
from couplings to $h_{1,3}$ which can be neglected if the VEVs of the
$\widetilde{\phi}^{(i)}$ fields are small.
Because of the localization of the matter fields, we expect the renormalizable
(1,3) and (3,1) entries in $Y_e^{(6)}$ to be exponentially suppressed.
\paragraph{Gauge-top unification.} The $(3,3)$ entry of $Y_u$ is related to the
gauge coupling. More precisely, in an orbifold GUT limit in which the first
$\Z2$ orbifold plane is larger than the other dimensions there is an \SU6 bulk
gauge symmetry, and the ingredients of the top Yukawa coupling $h_u$ (i.e.\ the
fields $\bar h_{1,3,4}$), $\bar u_3$ and $q_3$ are bulk fields of this plane,
i.e.\ hypermultiplets in the $N=2$ supersymmetric description. As discussed in
\cite{Hosteins:2009xk}, this implies that the top Yukawa coupling $y_t$ and the
unified gauge coupling $g$ coincide at tree-level. Moreover, localization
effects in the two larger dimensions \cite{Lee:2003mc} will lead to a slight
reduction of the prediction of $y_t$ at the high scale such that realistic top
masses can be obtained.
\paragraph{$\boldsymbol{D_4}$ flavor symmetry.} The block structure of the
Yukawa matrices is not a coincidence but a consequence of a $D_4$ flavor
symmetry \cite{Ko:2007dz}, related to the vanishing Wilson line in the $e_1$
direction, $W_1 = 0$ (cf.\ e.g.\ \cite{Kobayashi:2006wq}). The first two
generations transform as a $D_4$ doublet, while the third generation is a $D_4$
singlet.
\paragraph{Neutrino masses.} In our model we have 11 neutrinos,
i.e.\ SM singlets whose charges are odd under $\Z4^R$ meaning that
they have odd $\Z2^\mathcal{M}$ charge, where
$\Z2^\mathcal{M}$ is the matter parity subgroup of $\Z4^R$. Their
mass matrix has rank 11 at the perturbative level. The neutrino
Yukawa coupling is a $3\times 11$ matrix and has full rank. Hence
the neutrino see-saw mechanism with many neutrinos
\cite{Buchmuller:2007zd} is at work.
\paragraph{Proton decay operators.} The $\Z4^R$ symmetry forbids
all dimension four and five proton decay operators at the
perturbative level~\cite{Lee:2010gv}. In addition, the
non-anomalous matter parity subgroup $\Z2^\mathcal{M}$ forbids all
dimension four operators also non-perturbatively. The dimension
five operators like $q\,q\,q\,\ell$ are generated
non-pertubatively, as we will discuss below.
\paragraph{Non-perturbative violation of $\boldsymbol{\Z4^R}$.}
Once we include the terms that are only
forbidden by the $\Z4^R$ symmetry, we obtain further couplings.
An example for such an additional term is the dimension five proton decay operator,
\begin{equation}\label{eq:Wnp1}
\mathscr{W}_{np}
~\supset~
q_1\,q_1\,q_2\,\ell_1\,
\mathrm{e}^{-a\,S}\, (x_4 \bar x_5 + x_5 \bar x_4)
\left[\begin{pmatrix}\phi_{11}\\\phi_{12}\end{pmatrix} \cdot \begin{pmatrix}\phi_{11}\\\phi_{12}\end{pmatrix}\right]^3
\phi_4\,\phi_7^2
\left[\begin{pmatrix}\phi_{9}\\\phi_{10}\end{pmatrix} \cdot \begin{pmatrix}\phi_{9}\\\phi_{10}\end{pmatrix}\right]
\end{equation}
where we suppressed coefficients. The bracket structure between the
$\phi_{11}$/$\phi_{12}$ and $\phi_{9}$/$\phi_{10}$ is a consequence of the
non-Abelian $D_4$ symmetry, where these fields transform as a doublet.
The dot `$\,\cdot\,$' indicates the standard scalar product. Note that there are
invariants with more than two $D_4$ charged fields which cannot be written in
terms of a scalar product. Further,
$S$ is the dilaton and the coefficient $a=8\pi^2$ in
$\mathrm{e}^{-a\,S}$ is such that $\mathrm{e}^{-a\,S}$ has positive anomalous
charge with respect to the normalized generator of the `anomalous' \U1. This generator
is chosen such that it is the gauge embedding of the anomalous space group
element\footnote{See \cite{Araki:2008ek} for the discussion in a more general
context. Note that we can always bring the anomalous space group element to the
form $(\theta^{k}\,\omega^\ell,0)$ by redefining the model input appropriately.
This amounts to a redefinition of the `origin' of the orbifold.} (cf.\
\Eqref{eq:anomalous_space_group_element}),
\begin{equation}
\mathsf{t}_\mathrm{anom}~=~W_3+\E8\times\E8~\text{lattice vectors}\;.
\end{equation}
The discrete Green-Schwarz mechanism is discussed in detail in \cite{LRRRSSV2}.
\paragraph{Solution to the $\boldsymbol{\mu}$ problem.} The $\Z4^R$ anomaly has
important consequences for the MSSM $\mu$ problem. The $\mu$ term is forbidden
perturbatively by $\Z4^R$, however, it appears at the non-perturbative level.
Further, this model shares with the mini-landscape models the property that any
allowed superpotential term can serve as an effective $\mu$ term (cf.\ the
discussion in \cite{Kappl:2008ie}). This fact can be seen from
higher-dimensional gauge invariance \cite{Brummer:2010fr}. Therefore, the
(non-perturbative) $\mu$ term is of the order of the gravitino mass,
\begin{equation}
\mu~\sim~\langle\mathscr{W}\rangle~\sim~m_{3/2}
\end{equation}
in Planck units. If some `hidden' sector dynamics induces a
non-trivial $\langle\mathscr{W}\rangle$, the $\mu$ problem is
solved.
In our model, we have only a `toy' hidden sector with an unbroken \SU2 gauge
group and one pair of massless doublets whose mass term is prohibited by
$\Z4^R$. This sector has the structure discussed by Affleck, Dine and Seiberg
(ADS) \cite{Affleck:1983mk}. We find that the ADS superpotential is $\Z4^R$
covariant. However, the hidden gauge group is probably too small for generating
a realistic scale of supersymmetry breakdown. Yet there are alternative ways,
such as the one described in \cite{Kappl:2008ie}, for generating a
hierarchically small $\langle\mathscr{W}\rangle$.
\paragraph{Origin of $\boldsymbol{\Z4^R}$.}
In the orbifold CFT description the $\Z4^R$ originates from the so-called
$H$-momentum selection rules \cite{Dixon:1986qv} (see also
\cite{Kobayashi:2004ya,Buchmuller:2006ik}). These selection rules appear as
discrete $R$ symmetries in the effective field theory description of the model.
We would like to stress that in large parts of the literature the order of these
symmetries was given in an unfortunate way. This criticism also applies to the
papers by some of the authors of this study. For instance, the $\Z2$ orbifold
plane was said to lead to a $\Z2^R$ symmetry, but it turned out that there are
states with half-integer charges. We find it more appropriate to call this
symmetry $\Z4^R$, and to deal with integer charges only. In our model we have
three $\Z4^R$ symmetries at the orbifold point, stemming from the three $\Z2$
orbifold planes.
$H$-momentum corresponds to angular momentum in the compact space; therefore the
discrete $R$ symmetries can be thought of as discrete remnants of the Lorentz
symmetry of internal dimensions. That is to say that the orbifold
compactification breaks the Lorentz group of the tangent space to a discrete
subgroup. In this study we content ourselves with the understanding that these
symmetries appear in the CFT governing the correlators to which we match the
couplings of our effective field theory. The precise geometric interpretation of
this symmetry in field theory will be discussed elsewhere.
The actual $\Z4^R$ charges of $[\SU3\times\SU2\times\SU2]_\mathrm{hid}$
invariant expressions in this model are given by
\begin{equation}\label{eq:Z4Rcharge}
q_{\Z4^R}~=~q_X+R_2+2n_3\;,
\end{equation}
where $q_X$ is the \U1 charge generated by
\begin{equation}
\mathsf{t}_X~=~
\left(4, 0, 10, -10, -10, -10, -10, -10\right)\,
\left(-10, 0, 5, 5, -5, 15, -10, 0\right)\;,
\end{equation}
$R_2$ denotes the $R$ charge with respect to the second orbifold plane and $n_3$ is the
localization quantum number in the third torus. The relevant quantum numbers are
given in table~\ref{tab:fullspectrum}. The expression \eqref{eq:Z4Rcharge} for
$q_{\Z4^R}$ is not unique, there are 17 linear combinations of \U1 charges and
discrete quantum numbers which can be used to rewrite the formula without
changing the $\Z4^R$ charges. Also the \U1 factors contained in
$[\SU3\times\SU2\times\SU2]_\mathrm{hid}$ can be used to redefine
$\mathsf{t}_X$. We refrain from spelling this out as we find it more convenient
to work with invariant monomials (cf.\ the discussion in
appendix~\ref{app:SUSYVacuum}). It is straightforward to see that all monomials
we switch on have $R$ charge 0.
\section{Summary}
\label{sec:Summary}
We have re-emphasized the important role of discrete symmetries in string model
building. As an application, we discussed an explicit string model which
exhibits MSSM vacua with a $\Z4^R$ symmetry, which has recently been shown to be
the unique symmetry for the MSSM that forbids the $\mu$ term at the perturbative
level, allows Yukawa couplings and neutrino masses, and commutes with \SO{10}.
This $\Z4^R$ has a couple of appealing features. First, the
$\mu$ term and dangerous dimension five proton decay operators are forbidden at
the perturbative level and appear only through (highly suppressed)
non-perturbative effects. Second, at the perturbative level, the expectation
value of the superpotential is zero; a non-trivial expectation value is
generated by non-perturbative effects. These two points imply that $\mu$ is of
the order of the gravitino mass $m_{3/2}$, which is set by the expectation value
of the superpotential (in Planck units).
The model is a $\Z2\times\Z2$ orbifold compactification of the $\E8\times\E8$
heterotic string. We discussed how to search for field configurations which
preserve $\Z4^R$ and how to find supersymmetric vacua within such
configurations. The Hilbert basis method allowed us to construct a basis for
all gauge invariant holomorphic monomials, and therefore to survey the
possibilities of satisfying the $D$-term constraints.
As we have seen, in the case of residual $R$ symmetries it may
in principle happen that the $F$-term equations overconstrain the system. We have
explicitly verified that this is not the case in our model, i.e.\ there are
supersymmetric vacua with the exact MSSM spectrum and a residual $\Z4^R$
symmetry. Let us highlight the features of the model:
\begin{itemize}
\item exact MSSM spectrum, i.e.\ no exotics;
\item almost all singlet fields/moduli are fixed in a
supersymmetric way;
\item non-local GUT breaking, i.e.\ the model is consistent with MSSM precision
gauge unification;
\item dimension four proton decay operators are completely absent as $\Z4^R$
contains the usual matter parity as a subgroup;
\item dimension five proton decay operators only appear at the non-perturbative
level and are completely harmless;
\item the gauge and top-Yukawa couplings coincide at tree level;
\item see-saw suppressed neutrino masses;
\item $\mu$ is related to the vacuum expectation value of the superpotential
and therefore of the order of the gravitino mass;
\item there is an \SU5 GUT relation between the $\tau$ and bottom masses.
\end{itemize}
There are also two drawbacks: first, there are also \SU5 relations for the light
generations and second the hidden sector gauge group is only \SU2 and therefore
probably too small for explaining an appropriate scale of
dynamical supersymmetry breaking.
Although we have obtained a quite promising string vacuum, the main focus of
this paper was on developing new methods rather than working out the
phenomenology of a model. We have discussed in detail how to determine the
residual discrete symmetries of a given VEV configuration. As a consequence, we
could immediately understand the features of such a configuration. For instance,
in earlier studies
\cite{Buchmuller:2005jr,Buchmuller:2006ik,Lebedev:2006kn,Lebedev:2007hv} we had
to explicitly identify couplings that are consistent with the string selection
rules in order to show that all exotics decouple and the Yukawa couplings have
full rank. This is a very time-consuming task in practice. With the new methods
we could obtain this information by just looking at the remnant symmetries. We
have performed extensive cross-checks in order to show that both methods yield
the same results. We have also shown how to search for vacua with a given
symmetry. Further, we presented the Hibert basis method which allows us to
survey all $D$-flat directions comprised of a selected set of fields in very
short time. It will be interesting to apply, and to extend, our methods to other
examples.
\subsection*{Acknowledgements}
We thank James Gray, Thomas Grimm, Arthur Hebecker, Raymond
Hemmecke, Christoph L\"udeling, Graham Ross and Timo Weigand for
valuable discussions. This research was supported by the DFG cluster
of excellence Origin and Structure of the Universe, the
\mbox{SFB-Transregio} 27 ``Neutrinos and Beyond'', LMUExcellent, the
Graduiertenkolleg ``Particle Physics at the Energy Frontier of New
Phenomena'' by Deutsche Forschungsgemeinschaft (DFG), and by the
National Science Foundation under Grant No.\ PHY05-51164. We would
like to thank the Aspen Center for Physics and the KITP in Santa
Barbara, where some of this work has been carried out, for
hospitality and support. S.R. acknowledges partial support from DOE
Grant DOE/ER/01545-890.
|
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{"url":"http:\/\/latex.org\/forum\/viewtopic.php?f=45&t=30740&p=103740&sid=0308ce2837edfcc9873d12b7a35ce617","text":"## LaTeX forum \u21d2 Graphics, Figures & Tables \u21d2 Table Caption and Footnote\n\nInformation and discussion about graphics, figures & tables in LaTeX documents.\nPosts: 5\nJoined: Thu Dec 14, 2017 5:42 am\n\n### Table Caption and Footnote\n\nHi everyone,\n\nHow to make the table caption of longtable to begin from the left? and same thing for footnote, how can I move it to the left as well?\n\nIs there any solution?\n\nThanks!\nMoh\n\nStefan Kottwitz\nPosts: 8596\nJoined: Mon Mar 10, 2008 9:44 pm\nLocation: Hamburg, Germany\nContact:\nHi Moh,\n\nuse the caption package:\n\n`\\usepackage[singlelinecheck=off]{caption}`\n\nStefan","date":"2018-01-17 20:19:27","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9987754821777344, \"perplexity\": 11894.747848695359}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084886964.22\/warc\/CC-MAIN-20180117193009-20180117213009-00057.warc.gz\"}"}
| null | null |
Q: if \ else vs. switch \ case в C# Что лучше использовать: if\else или switch\case?
int a = 3;
if (a == 1)
{
...
}
else if (a == 2)
{
...
}
else
{
...
}
switch(a)
{
case 1:
...
break;
case 2:
...
break;
default:
...
break;
}
*
*Какая из этих конструкций работает быстрее? Почему?
*Когда лучше использовать то или другое?
A: Вы задаётесь неправильным вопросом.
Если ваша задача — ускорить вашу программу, то решением этой задачи никогда не будет «заменить свитч на последовательность ифов» или наоборот. Ускорение программы проводится на уровне используемых алгоритмов и структур данных.
Тот факт, что один или другой вариант в тех или иных условиях, на той или иной версии компилятора выполняется на несколько наносекунд быстрее, не стоит потраченных на него байт текста.
Во-первых, вы должны помнить, что ваша главная задача — не выиграть 10 тактов процессора (несчастное переключение контекста стоит намного больше!), а сделать вашу программу понятной. Поэтому вы должны использовать switch там, где это прибавляет читаемости коду, там, где это соответствует тому, что именно вы хотите сказать этим кодом. И использовать if там, где это лучше отражает вашу мысль.
Например, если вы устраиваете перебор значений enum'а, то чаще всего вам нужен switch. А если вы реально проверяете несколько условий, особенно с длинными вычислениями, то, возможно, лучше цепочка if'ов.
Ну и во-вторых, то, что на одной версии компилятора один из методов выражения одной и той же идеи компилируется в более быстрый код, чем другой метод — штука временная и преходящая. Помните, что в C++ сначала рекомендовалось возвращать значения по ссылке для скорости, а затем пришёл тренд «want speed? pass by value!», и наверняка это поменялось сейчас ещё раз.
Поскольку код имеет одинаковый смысл, то в любом из его вариантов рано или поздно он будет компилироваться в одно и то же. JIT-компилятор C# покамест не генерирует одинаковый объектный код для этих двух случаев, а вот компиляторы C++ уже умеют.
Итак, то, какая из имплементаций скорее, а какая медленнее — это не абсолютное понятие.
Вы не должны жертвовать читаемостью кода ради сиюминутной мизерной выгоды. Пишите не «как скорее», а «как правильнее». Расходы на поддержку плохо написанного кода несравнимо больше выгод от микрооптимизаций.
A: Ответ выше не совсем корректный, даже учитывая комментарий.
Если значения лежат рядом, например:
switch (x)
{
case 0: /* */ break;
case 1: /* */ break;
case 2: /* */ break;
case 3: /* */ break;
case 4: /* */ break;
case 5: /* */ break;
}
В таком случае компилятор генерирует таблицу переходов (jump table) (IL-код):
switch (IL_0043, IL_004e, IL_0059, IL_0064, IL_006f, IL_007a)
Время работы такой таблицы O(1). Не нужно волноваться, если несколько значений упущено, и они не идут один за другим, в таком случае компилятор все равно генерирует таблицу переходов:
switch (x)
{
case 0: /* */ break;
case 1: /* */ break;
case 2: /* */ break;
case 3: /* */ break;
case 4: /* */ break;
case 5: /* */ break;
case 9: /* */ break;
case 13: /* */ break;
}
IL:
switch (IL_0043, IL_004e, IL_0059, IL_0064, IL_006f, IL_007a, IL_009a, IL_009a, IL_009a, IL_0085, IL_009a, IL_009a, IL_009a, IL_0090)
Как видно, "дырки" заполнились переходами за конец конструкции switch: IL_009a
Если расстояние между значениями слишком большое для построение одной таблицы, компилятор может разделить их на 2+ таблицы. Например:
switch (x)
{
case 0: /* */ break;
case 1: /* */ break;
case 2: /* */ break;
case 3: /* */ break;
case 2000000000: /* */ break;
case 2000000001: /* */ break;
case 2000000002: /* */ break;
case 2000000003: /* */ break;
case 2000000004: /* */ break;
}
IL:
IL_0004: ldloc.0
IL_0005: switch (IL_0037, IL_0042, IL_004d, IL_0058)
IL_001a: ldloc.0
IL_001b: ldc.i4 2000000000
IL_0020: sub
IL_0021: switch (IL_0063, IL_006e, IL_0079, IL_0084)
Как видно из примера выше, компилятор выделил 2 таблицы переходов. Для другой таблицы из значения вычитается 2,000,000,000. В общем случае скорость выполнения O(1) или O(m), где m - количество групп.
Посмотрим что будет в общем случае. Я взял 20 случайных чисел:
switch (x)
{
case 955626049: /* */ break;
case 1732096473: /* */ break;
case 1416261125: /* */ break;
case 901627868: /* */ break;
case 713886433: /* */ break;
case 289231272: /* */ break;
case 598309392: /* */ break;
case 1284278823: /* */ break;
case 1517161566: /* */ break;
case 1548994144: /* */ break;
case 1854092737: /* */ break;
case 1725885116: /* */ break;
case 1984539276: /* */ break;
case 558794563: /* */ break;
case 337975821: /* */ break;
case 1687544575: /* */ break;
case 1048183578: /* */ break;
case 102389471: /* */ break;
case 1017837673: /* */ break;
case 1012360293: /* */ break;
}
В таком случае компилятор их сортирует, и ищет бинарным поиском:
IL_0004: ldloc.0
IL_0005: ldc.i4 1017837673
IL_000a: bgt IL_0099
IL_000f: ldloc.0
IL_0010: ldc.i4 598309392
IL_0015: bgt.s IL_0058
IL_0017: ldloc.0
IL_0018: ldc.i4 289231272
IL_001d: bgt.s IL_0036
IL_001f: ldloc.0
IL_0020: ldc.i4 102389471
IL_0025: beq IL_016e
IL_002a: ldloc.0
IL_002b: ldc.i4 289231272
IL_0030: beq IL_0126
IL_0035: ret
IL_0036: ldloc.0
IL_0037: ldc.i4 337975821
IL_003c: beq IL_015c
IL_0041: ldloc.0
IL_0042: ldc.i4 558794563
IL_0047: beq IL_0156
IL_004c: ldloc.0
IL_004d: ldc.i4 598309392
IL_0052: beq IL_012c
IL_0057: ret
IL_0058: ldloc.0
IL_0059: ldc.i4 901627868
IL_005e: bgt.s IL_0077
IL_0060: ldloc.0
IL_0061: ldc.i4 713886433
IL_0066: beq IL_0120
IL_006b: ldloc.0
IL_006c: ldc.i4 901627868
IL_0071: beq IL_011a
IL_0076: ret
IL_0077: ldloc.0
IL_0078: ldc.i4 955626049
IL_007d: beq IL_0108
IL_0082: ldloc.0
IL_0083: ldc.i4 1012360293
IL_0088: beq IL_017a
IL_008d: ldloc.0
IL_008e: ldc.i4 1017837673
IL_0093: beq IL_0174
IL_0098: ret
IL_0099: ldloc.0
IL_009a: ldc.i4 1548994144
IL_009f: bgt.s IL_00d6
IL_00a1: ldloc.0
IL_00a2: ldc.i4 1284278823
IL_00a7: bgt.s IL_00bd
IL_00a9: ldloc.0
IL_00aa: ldc.i4 1048183578
IL_00af: beq IL_0168
IL_00b4: ldloc.0
IL_00b5: ldc.i4 1284278823
IL_00ba: beq.s IL_0132
IL_00bc: ret
IL_00bd: ldloc.0
IL_00be: ldc.i4 1416261125
IL_00c3: beq.s IL_0114
IL_00c5: ldloc.0
IL_00c6: ldc.i4 1517161566
IL_00cb: beq.s IL_0138
IL_00cd: ldloc.0
IL_00ce: ldc.i4 1548994144
IL_00d3: beq.s IL_013e
IL_00d5: ret
IL_00d6: ldloc.0
IL_00d7: ldc.i4 1725885116
IL_00dc: bgt.s IL_00ef
IL_00de: ldloc.0
IL_00df: ldc.i4 1687544575
IL_00e4: beq.s IL_0162
IL_00e6: ldloc.0
IL_00e7: ldc.i4 1725885116
IL_00ec: beq.s IL_014a
IL_00ee: ret
IL_00ef: ldloc.0
IL_00f0: ldc.i4 1732096473
IL_00f5: beq.s IL_010e
IL_00f7: ldloc.0
IL_00f8: ldc.i4 1854092737
IL_00fd: beq.s IL_0144
IL_00ff: ldloc.0
IL_0100: ldc.i4 1984539276
IL_0105: beq.s IL_0150
IL_0107: ret
В таком случае скорость выполнения будет O(log n).
Аналогичная if then else конструкция выполняется за O(n). А это значит switch почти всегда быстрее if then else.
Все примеры IL скомпилированы для .NET версии 4.6
Дополнение для String:
Для типа string работает немного по-другому, так как таблицу переходов не построить, а искать бинарным поиском (перебирая каждый символ) - неоптимально. В таком компилятор считает хеш всех строк, а потом во время выполнения считает хеш от входящей строки, и ищет бинарным поиском. Пример:
switch (x)
{
case "ff": return 10;
case "f1": return 11;
case "f2": return 12;
case "f3": return 13;
case "f4": return 14;
case "f5": return 15;
case "f6": return 16;
case "f7": return 17;
case "f8": return 18;
case "f9": return 19;
case "10": return 20;
case "11": return 21;
case "12": return 22;
default:
return 0;
}
IL-код:
IL_0001: call uint32 '<PrivateImplementationDetails>'::ComputeStringHash(string)
IL_0006: stloc.0
IL_0007: ldloc.0
IL_0008: ldc.i4 89242279
IL_000d: bgt.un.s IL_0063
IL_000f: ldloc.0
IL_0010: ldc.i4 38909422
IL_0015: bgt.un.s IL_003d
IL_0017: ldloc.0
IL_0018: ldc.i4 5354184
IL_001d: beq IL_0147
IL_0022: ldloc.0
IL_0023: ldc.i4 22131803
IL_0028: beq IL_0159
IL_002d: ldloc.0
IL_002e: ldc.i4 38909422
IL_0033: beq IL_011d
IL_0038: br IL_01da
IL_003d: ldloc.0
IL_003e: ldc.i4 55687041
IL_0043: beq IL_0132
IL_0048: ldloc.0
IL_0049: ldc.i4 72464660
IL_004e: beq IL_00f3
IL_0053: ldloc.0
IL_0054: ldc.i4 89242279
IL_0059: beq IL_0108
IL_005e: br IL_01da
IL_0063: ldloc.0
IL_0064: ldc.i4 1972548895
IL_0069: bgt.un.s IL_008b
IL_006b: ldloc.0
IL_006c: ldc.i4 122797517
IL_0071: beq.s IL_00de
IL_0073: ldloc.0
IL_0074: ldc.i4 810679896
IL_0079: beq.s IL_00c9
IL_007b: ldloc.0
IL_007c: ldc.i4 1972548895
IL_0081: beq IL_01a4
IL_0086: br IL_01da
IL_008b: ldloc.0
IL_008c: ldc.i4 2006104133
IL_0091: bgt.un.s IL_00ae
IL_0093: ldloc.0
IL_0094: ldc.i4 1989326514
IL_0099: beq IL_0195
IL_009e: ldloc.0
IL_009f: ldc.i4 2006104133
IL_00a4: beq IL_0186
IL_00a9: br IL_01da
IL_00ae: ldloc.0
IL_00af: ldc.i4 -28201054
IL_00b4: beq IL_0168
IL_00b9: ldloc.0
IL_00ba: ldc.i4 -11423435
IL_00bf: beq IL_0177
IL_00c4: br IL_01da
После этого идет сравнение методом System.String::op_Equality (так как хеш может совпать и у разных строк). Этот механизм не является хеш-таблицей, так как в хеш-таблицы скорость доступа к элементу O(1), а здесь просто бинарный поиск за O(log n)
Я попытался создать очень большой switch на 60 строк, но ничего не изменилось.
A: Оба условных оператора работают с одинаковой скоростью. Так как на уровне машинных команд они преобразуются в одни и те же инструкции.
Выбирать имеет смысл только по вкусу и по задаче.
switch может работать только с одной переменной. У if таких ограничений нет.
На мой взгляд, язык C# слишком высокоуровневый для такого рода оптимизаций.
Пишите читаемый код ;)
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"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
\setlength\baselineskip{15pt}
The original A.A. Markov inequality states that
$ || P'||_{ L^{ \infty }(I) } \leq n^{2} || P ||_{ L^{ \infty }(I) } $
for any algebraic polynomial $P$ of degree $n$. Here, $ I= [-1,1] $.
This inequality becomes an equality if $P$ is the Chebyshev polynomial
$ P(x) = \cos nt $ where $ x = \cos t $. The reader may find the details of this
in page 40 of \cite{lorentz}.
Upper estimates of the derivative norm by that of the function itself are usually termed
Markov-Bernstein inequalities.
There is an extensive literature on such inequalities, which
play an important role in inverse theorems, where
smoothness of a function is deduced from rates of convergence of polynomial approximations.
For an excellent survey on Markov-Bernstein and related inequalities, the reader may consult
the book \cite{borwein} of P. Borwein and T. Erd\'elyi.
By imposing additional assumptions on the zeros of the polynomials, one can obtain
estimates which give lower estimates for the norm of a
derivative in terms of the norm of the function. These results are usually
termed inverse Markov-Bernstein inequalities or Tur\'an type inequalities.
For instance, Tur\'an \cite{turan} proved that
\[ || P'||_{ L^{ \infty }(I) } \geq \frac{ \sqrt{n}}{ 6 } || P ||_{ L^{ \infty }(I) } \]
for any polynomial $P$ of degree $n$, provided that all of its zeros lie in the interval
$I = [-1,1]$. We also refer the reader to a valuable paper of Er\"od \cite{erod}.
There is an upsurge of interest in such estimates, with a number of
recent results dealing with the topic (\cite{erdelyi2}, \cite{levenberg},
\cite{szilard}, \cite{zhou}).
For instance, in \cite{zhou}, Zhou showed that if $ 0 <r \leq q \leq \infty $ and
$ 1 \geq 1/r - 1/q$, then
\[ || P'||_{ L^{ r }(I) } \geq C n^{\alpha} || P ||_{ L^{ q }(I) } \]
for every polynomial $P$ whose zeros lie in the interval $I$. Here,
$ \alpha = \frac{1 }{2 } - \frac{ 1}{2r } + \frac{1 }{ 2q} $.
More related to our work are results of Erd\'elyi and Nevai \cite{erdelyi}
where they obtained
\[ \lim_{ n \rightarrow \infty} \frac{ ||p_{n}'||_{X} }{ ||p_{n} ||_{Y} } = \infty \]
for sequences of polynomials $ p_{n}$ whose zeros satisfy cetain conditions .
Markov-Bernstein inequalities have also been obtained for other classes of functions such as
Gaussian networks. For instance, in \cite{mhaskar}, Mhaskar showed that
for some constant $c$, $ ||g' ||_{p} \leq c m || g ||_{p} $ for any function
$g$ defined on the real line of the form
\[ g(x) \ = \ \sum_{k=1}^{N} a_{k} \exp ( - ( x-x_{k} )^{2} ), \]
where $ | x_{j} - x_{k} | \geq 1/m $ for $ j \neq k $, and $ \log N =
{\mathcal O} (m^{2}) $.
One of our goals in this note is to show that under certain conditions on
an integrable function $ P :{\bf R} \longrightarrow {\bf R} $ having a
real-valued Fourier transform $ \hat{ P }$ with $ P(0) = 0 $,
\begin{equation}
r \geq C \frac{\| P'\|_{ \infty } }{\| P\|_{ \infty }} \ \
\Longrightarrow \ \ \int_{ -r}^{r } ( \hat{P })_{ \pm } \ \geq
\frac{\sqrt{2\pi}}{4}\ || P ||_{ \infty } .
\label{eqn:oscillation}
\end{equation}
Here, we can take $C= 8^{3} / \pi $. This estimate not only tells us that
$ \hat{P} $ will have a zero in the interval $ [-r,r] $, but also provides
an effective estimate on how it oscillates in the interval.
For a fixed function $ \phi $, let
\begin{equation}\label{eqn:Endef}
E_n(\lambda):=\left\{ \sum_{k= -n}^{n } b_{k} \phi (x + \lambda k)
~:~ b_k\in {\bf R},~k=-n,\dots,-1,0,1,\dots,n \right\}.
\end{equation}
The estimate in (\ref{eqn:oscillation}) allows us to construct
$ P_{ \lambda } \in E_n( \lambda ) $ for each $ \lambda > 0$
and for sufficiently large positive integers $n$ (depending on $ \lambda$)
such that $ \| P_{ \lambda} '\|_{ \infty } > c
\| P_{ \lambda} \|_{ \infty } / \lambda $ with some absolute constant $c>0$.
In particular, our construction proves sharpness of the above-mentioned
inequality of Mhaskar \cite{mhaskar} for Gaussian networks.
\section{Notations and preliminaries}
For any integrable function $f$ on the real line, we write for its
Fourier transform
\[ \hat{ f }( \omega ) \ = \ \frac{1}{ \sqrt{ 2 \pi}}
\int_{ {\bf R}} f(x) e^{ -i \omega x} \ dx \ . \]
Given a real number $x$, its positive and negative parts are
$x_{+} = \max \{ x,0 \} $ and $ x_{-} = \max \{-x,0 \} $
respectively.
We will write $h$ for the Fej\'er kernel, that is
\[ h( x ) \ := \ \frac{1}{\sqrt{ 2 \pi } }
\left( \frac{ \sin x/2 }{ x/2 } \right)^{2} \ . \]
Its Fourier transform is given by
\[ \hat{ h } ( \omega ) \ = \
\max \{ 1 - | \omega |, \ 0 \} \ . \]
For the rest of the paper we fix an auxiliary function $H$. We could use
any even, $ 2 \pi $-periodic, and e.g. twice continuously differentiable
function \\ $ H: {\bf R} \longrightarrow {\bf R} $, not identically one, such
that $ H(x) = 1 $ if $ |x| \leq \pi/2$. The special constants and values
in the following choice are not relevant, only some order is essential.
Nevertheless, for definiteness and more explicit calculation we take e.g.
\begin{equation}
H(x)= \left\{
\begin{array}{ll}
1, \qquad\qquad & \mbox{if} \ |x| \leq \pi /2; \\
\sin^2x, & \mbox{if } \pi/2 < |x| \leq \pi .
\end{array}
\right. \label{eqn:Hdef}
\end{equation}
Then $H$ has the Fourier cosine series development
\[ H(x) \ = \ \sum_{k=0}^{\infty } a_{k} \cos kx \]
where $a_k$ are the Fourier cosine coefficients of $H$. Although
precise values are not needed here, a calculation leads to
$a_0=3/4$, $a_1=4/(3\pi)$, $a_2=\frac{-1}{4}$ and
\begin{equation}
a_k= \frac{-4\sin\frac{k\pi}{2}}{\pi k(k^2-4)} =
\left\{
\begin{array}{ll}
0, & k \mbox{ even;} \\
\frac{-4}{\pi k(k^2-4)}, & k \equiv 1 \mbox{ mod } 4; \\
\frac{4}{\pi k(k^2-4)}, & k\equiv 3 \mbox{ mod } 4.
\end{array}
\right. ~~~~ \hbox{for}~k \geq 3, ~ k\in {\bf N}.
\label{eqn:akdef}
\end{equation}
It is immediate that $|a_k| \leq k^{-2} $ for all $k \in {\bf N}$;
moreover, a direct calculation yields
\begin{equation}
\sum_{k=1}^{\infty} |a_k| = 1 +\frac{5}{3\pi} = 1.530516...<1.6
\qquad \textrm{and}\qquad \sum_{k=1}^{\infty} a_k^2 = \frac{9}{8}.
\label{eqn:akabssum} \end{equation}
\section{Oscillation of Fourier transforms}
\begin{lemma}\label{l:oscillation}
Let $ P: {\bf R} \longrightarrow {\bf R} $ be bounded,
differentiable, and integrable such that $ \hat{ P } $ is
real-valued. Suppose $ P(0) = 0$ and let
\begin{equation}
r\ > \ \frac{8^{3} \ ||P'||_{\infty }}{\pi\ || P||_{\infty}} \ ,
\label{eqn:lambda}
\end{equation}
then
\begin{equation}
\frac{4}{\sqrt{ 2 \pi}} \int_{ -r}^{r } ( \hat{P })_{\pm} \ \geq
\ || P ||_{ \infty }
\end{equation}
\end{lemma}
{\bf Proof of Lemma \ref{l:oscillation}}: There is nothing to
prove if $ || P' ||_{ \infty} = \infty $. Hence, we assume $ || P'
||_{ \infty} < \infty $. Fix $ r $ satisfying (\ref{eqn:lambda})
and define
\[ f(x) \ = \ P \star h_{ r } (x) \ = \ \frac{1}{ \sqrt{ 2 \pi }}
\int_{ {\bf R}} P( x-t ) h_{ r } (t) dt \]
where $ h_{ r } (t) = r h ( r t) $.
Since $ ( 2 \pi )^{ -1/2} \int_{ {\bf R}} h_{ r } =1 $,
for any real number $x$,
\begin{equation}
f(x) - P(x) = S(x) + L(x)
\label{eqn:split}
\end{equation}
where
\[ S(x) = \frac{1}{ \sqrt{ 2 \pi }} \int_{ |t| < \delta } ( P(x-t ) - P(x) )
h_{ r } (t) \ dt \ , \]
\[ L(x) = \frac{1}{ \sqrt{ 2 \pi }} \int_{ |t| \geq \delta } ( P(x-t ) - P(x) )
h_{ r } (t) \ dt \ \]
and $ \delta > 0 $ is chosen such that $ 8 \delta q = 1 $ with
$ q = || P'||_{ \infty } / || P ||_{ \infty } $.
Combining the inequalities
\[ | S(x)| \ \leq \ \delta \ || P' ||_{ \infty } \ = \
\frac{ || P ||_{ \infty }}{ 8}
\ \ \mbox{ and } \ \ | L(x) | \ \leq \
\frac{ 8 \ || P||_{ \infty } }{ \pi r \delta } \ < \
\frac{|| P||_{ \infty }}{ 8 } \]
with (\ref{eqn:split}), we obtain for any real number $x$,
\begin{equation}
|f(x) - P(x) | < || P||_{ \infty } / 4 \ .
\label{eqn:main}
\end{equation}
Since $f$ and $ \hat{f} $ are both integrable, the inversion
formula for the Fourier transform shows that
\[ \sqrt{ 2 \pi } || f ||_{ \infty } \ \leq \ \int_{ {\bf R}} | \hat{ f } |
\ = \ \int_{ {\bf R}} \left( \hat{ f } + 2 ( \hat{ f } )_{-} \right)
\ = \ \sqrt{ 2 \pi } f(0) + 2 \int_{ {\bf R}}( \hat{ f } )_{-} . \]
Applying (\ref{eqn:main}) with $ x= 0$ and noting that $ P(0) = 0
$, we conclude that
\[
|| f ||_{ \infty } \ \leq \ \frac{1}{4}|| P ||_{ \infty } +
\frac{2}{ \sqrt{2 \pi } } \int_{ {\bf R}}( \hat{ f } )_{-} \ .
\]
Making use once more of (\ref{eqn:main}) and the last inequality gives
\[
|| P ||_{ \infty } \ \leq \ || f ||_{ \infty } + \frac{1}{4}|| P
||_{ \infty } \ \leq \ \frac{1}{2}|| P ||_{ \infty } + \frac{2}{
\sqrt{ 2 \pi } } \int_{ {\bf R}}( \hat{ f } )_{-}
\]
and therefore
\[ || P ||_{ \infty } \ \leq \
\frac{4}{\sqrt{ 2 \pi}} \int_{ {\bf R}}( \hat{ f } )_{-} \ . \]
Finally, we observe that $ \hat{f} ( \omega ) = \hat{P} ( \omega ) \hat{h} ( r^{-1} \omega )$,
$ 0 \leq \hat{h} \leq 1 $ and $ \hat{h} = 0 $ outside $ [-1,1]$. These
imply that $ ( \hat{ f } )_{-} = 0 $ outside $ [-r,r] $
and $ ( \hat{ f } )_{-} \leq ( \hat{ P})_{- } \ $.
Therefore
\[ || P ||_{ \infty } \ \leq \
\frac{4}{\sqrt{ 2 \pi}} \int_{ -r}^{r } ( \hat{ P } )_{-} \ . \]
A similar argument leads to the same inequality for $ ( \hat{ P } )_{+} \ $.
$ \Box $
\section{Construction of sums of translates with \\ large oscillation}
\begin{theorem}
Let $ \phi : {\bf R} \longrightarrow {\bf R}$ be an even, continuous,
integrable function such that $ \phi (0) =1 $. In addition, suppose that
its Fourier transform $ \hat{ \phi } $ is nonnegative, integrable and
analytic on $ {\bf R} $. Given $ \lambda > 0 $, then there exist a
positive integer $n$ and $P\in E_n(\lambda)$, with $E_n(\lambda)$ defined
in (\ref{eqn:Endef}), such that
\[ \frac{|| P '||_{\infty }}{|| P ||_{\infty }} \ \geq
\frac{ C}{ \lambda} . \] Here, we could take $ C = \pi^{2}/2^{10} $.
\end{theorem}
{\bf Proof:} For each positive integer $n$ and for each real number $x$,
we define
\begin{equation}\label{eqn:pndef}
P_{n}(x) = 2 A_{n} \phi (x) \ + \ \sum_{ k=1}^{n} a_{k} ( \phi ( x
+ \lambda k) + \phi ( x - \lambda k) )
\end{equation}
and also
\begin{equation}
P_{ \infty } (x) \ := \ \lim_{ n \rightarrow \infty} P_{n} (x) =2
A_{\infty}(\lambda)\phi (x) \ + \ \sum_{ k=1}^{\infty} a_{k} (
\phi ( x + \lambda k) + \phi ( x - \lambda k) ),
\label{eqn:pinfty}
\end{equation}
where the coefficients $a_k$ are the Fourier cosine coefficients
of $H$ in (\ref{eqn:akdef}), and
\begin{equation}\label{eqn:Andef}
A_{n} := A_n(\lambda):=- \sum_{k=1}^{n} a_{k} \phi ( \lambda k),
\qquad A_{\infty} := A_\infty(\lambda):=- \sum_{k=1}^{\infty}
a_{k} \phi (\lambda k).
\end{equation}
We start with showing that $ P_{ \infty } $ is not identically zero.
\begin{lemma}\label{l:Pnotvanish}
Under the assumptions of Theorem 1, we have $ || P_{ \infty } ||_{
\infty } > 0 $.
\end{lemma}
{\bf Proof of lemma \ref{l:Pnotvanish}:} For each $ \omega \in
{\bf R} $ and $n\in{\bf N}$ we define
\begin{equation}
T_{n} ( \omega ) \ := \ \sum_{k=1}^{n} a_{k} ( \cos ( k \lambda
\omega ) - \phi ( \lambda k ) ) \label{eqn:trig}
\end{equation}
and also
\begin{equation}
T_{ \infty }( \omega ) \ := \ \lim_{n \rightarrow \infty} T_{n} (
\omega ) = \sum_{k=1}^{\infty} a_{k} ( \cos ( k \lambda \omega ) -
\phi ( \lambda k ) ). \label{eqn:triginfty}
\end{equation}
Thus, $ \hat{ P}_{ \infty } ( \omega ) \ = \hat{\phi}(\omega) 2 T_{
\infty }( \omega ) = 2 \hat{ \phi } ( \omega ) ( H( \lambda \omega ) - F(
\lambda ) ) $, where
\begin{equation}\label{eqn:Fdef}
F( \lambda) := \sum_{ k=0}^{\infty} a_{k} \phi(\lambda k ) =
a_0-A_{\infty}(\lambda)~
\end{equation}
is a uniformly convergent sum of bounded functions of $\lambda$.
By the Fourier inversion formula, $ P_{ \infty } \equiv 0 $ if and
only if $ \hat{P}_{ \infty} \equiv 0 $. Thus, it suffices to show
that for any given $ \lambda
> 0$, $ \hat{ \phi } ( \omega ) ( H( \lambda \omega ) - F( \lambda
) ) $ does not vanish identically.
Note that for any $ \lambda > 0$, $ H ( \lambda \omega) \neq 1 $
for $ \ \omega \in {\mathcal I} = ( \frac{ \pi }{ 2 \lambda } ,
\frac{ 3\pi }{ 2 \lambda } )
+ (2 \pi /\lambda) {\bf Z} $ while
$ H ( \lambda \omega) = 1 $ for
$ \ \omega \in {\mathcal J} = [ - \frac{ \pi }{ 2 \lambda } ,
\frac{ \pi }{ 2 \lambda } ]
+ (2 \pi /\lambda) {\bf Z} $.
Therefore, if $ F( \lambda) =1 $, then $ F( \lambda ) \neq H(
\lambda \omega) $ for $ \omega \in {\mathcal I} $, while if $ F(
\lambda) \neq 1 $, then $ F( \lambda ) \neq H( \lambda \omega) $
for $ \omega \in {\mathcal J} $. In any case, $ H( \lambda \omega
) - F( \lambda ) \neq 0 $ for $ \omega $ in a union of non-empty
open intervals. If $ \hat{P}_{ \infty} \equiv 0$, then $ \hat{
\phi } $ would have to be zero on these intervals, which is
impossible since $ \hat{ \phi } $ is assumed to be analytic on $
{\bf R}$. This completes the proof of lemma 2. $ \Box $
To finish the proof of the theorem it suffices to show the next
assertion.
\begin{lemma}\label{claim:punchline} If a positive integer $n$ is chosen
such that $ 20 \sum_{ k > n } | a_{k}| < || P_{ \infty} ||_{ \infty} $,
then
\[ \frac{ || P_{n} '||_{ \infty }}{ || P_{n} ||_{ \infty }} \ \geq
\ \frac{ \pi^{2} }{ 2^{10} \lambda }. \]
\end{lemma}
{\bf Proof of Lemma \ref{claim:punchline}:} Recall $\hat{P}_{n} (
\omega ) = 2 \hat{ \phi} ( \omega ) T_{n} ( \omega ) $ with
$T_{n} $ in (\ref{eqn:trig}). We also define $ \Delta_{n} ( \omega
) = T_{n} ( \omega ) - H ( \lambda \omega ) + F( \lambda) $ for $
\omega\in {\bf R}$, with $F(\lambda)$ in (\ref{eqn:Fdef}).
Meanwhile, in view of the assumptions that $ \hat{ \phi } \geq 0 $
and $ \hat{ \phi } \in L^{1} $, the inversion formula for the
Fourier transform shows that $ ||\phi||_{\infty} = \phi(0) = 1$.
With this in mind, we obtain for every positive integer $n$
\begin{equation}
|| \Delta_{n} ||_{ \infty } \leq \ 2 \sum_{ k > n } | a_{k}|~,
\qquad \mbox{and} \qquad
||P_{\infty} - P_{n} ||_{ \infty } \leq \ 4 \sum_{k > n } |a_{k}|.
\label{eqn:error}
\end{equation}
Suppose $ 0 < r \leq \pi / ( 2 \lambda ) $. Then $ H( \lambda \omega ) =1 $
for $ | \omega | \leq r$.
Therefore, if $ F( \lambda ) \geq 1 $, then
\[ \int_{ -r}^{r } ( \hat{ P}_{n} )_{ +} \ = \ 2 \int_{ -r}^{r } \hat{ \phi
}~
( 1 - F( \lambda )+ \Delta_n )_{ +}
\ \leq \ 4 \sqrt{ 2 \pi } \sum_{ k > n } | a_{k}| . \]
Here, we've again made use of the conditions $ \hat{ \phi } \geq 0$ and
$ \phi (0 ) =1 $.
Similarly, if $ F( \lambda ) < 1 $, we also obtain
\[ \int_{ -r}^{r } ( \hat{ P}_{n} )_{- } \ \leq \ 4
\sqrt{ 2 \pi } \sum_{ k > n } | a_{k}| . \]
Thus, we've shown that if $ 0 < r \leq \pi / ( 2 \lambda ) $, then
for each positive integer $n$,
\begin{equation}
\min \left( \int_{ -r}^{r } ( \hat{ P}_{n} )_{- }, \int_{ -r}^{r }
( \hat{ P}_{n} )_{+ } \right)
\ \leq \ 4 \sqrt{ 2 \pi } \sum_{ k > n } | a_{k}|.
\label{eqn:summary} \end{equation}
On the other hand, lemma \ref{l:oscillation} together with the
second inequality in (\ref{eqn:error}) asserts that if $r > (8^{3}
/ \pi) || P_{n}' ||_{ \infty } / || P_{n}||_{ \infty }$, then
\begin{equation}\label{eqn:lower-bound}
\frac{ 4}{ \sqrt{ 2 \pi} } \min \left( \int_{ -r}^{r } ( \hat{
P}_{n} )_{- }~, \int_{ -r}^{r } ( \hat{ P}_{n} )_{+ } \right)
\geq || P_{n}||_{ \infty} \geq || P_{\infty}||_{ \infty} - 4
\sum_{ k
> n } | a_{k}|.
\end{equation}
Combining (\ref{eqn:summary}) and (\ref{eqn:lower-bound}) we
conclude that if $ (8^{3} / \pi) || P_{n}'||_{ \infty} / || P_{n}
||_{ \infty} < \pi /(2 \lambda)$, then $ || P_{ \infty } ||_{
\infty }- 4\sum\limits_{ k > n } | a_{k}| \leq 16 \sum\limits_{ k
> n } | a_{k}|$ and therefore
$ || P_{ \infty } ||_{ \infty } \leq 20 \sum\limits_{ k > n } | a_{k}| $.
This proves the lemma which gives the conclusion of the theorem.
$\Box$
\section{Application to Gaussian networks}
Our goal in this section is to prove sharpness of an inequality of
Mhaskar (mentioned in the introduction of this paper) for Gaussian
networks. We shall apply Theorem 1 (in particular, lemma 1 in the
proof) with $ \phi (x ) = \exp ( -x^{2} ) $. In this section
$E_n(\lambda)$ is defined according to (\ref{eqn:Endef}) with our
above given Gaussian $\phi$.
The following theorem is the main result of this section.
\begin{theorem}
Let $ n \in {\bf N} $ and $ \lambda \in (0,1) $ satisfy
\begin{equation}\label{eqn:Nzerocond}
n > N_0:= C_0 \lambda \exp\left( \frac{\pi^2}{2\lambda^2}
\right)\qquad \qquad \left( C_0:=\frac{1280}{3\pi}\right).
\end{equation}
Then there exists $P\in E_n(\lambda)$, such that
\[
\frac{|| P '||_{ \infty }}{|| P ||_{ \infty }}
\geq \frac{ \pi^{2} }{ 2^{10} \lambda }.
\]
\end{theorem}
{\bf Remark.} Note $\log N_0 = O (1/\lambda^2)$, in complete
agreement with the above mentioned result of H. N. Mhaskar. Thus
the result proves sharpness of the result in \cite{mhaskar} for an
arithmetic progression of shifts $x_k:=\lambda k$ with separation
$1/m=\lambda$.
\[
\]
We retain the function $H$ from (\ref{eqn:Hdef}) and its Fourier coefficients
$a_k$ in (\ref{eqn:akdef}) also in this section. With these Fourier
coefficients $a_k$ and for each $ \lambda > 0 $ and $x \in {\bf R}$,
$P_{\infty}(\lambda,x)$ will again be as in (\ref{eqn:pinfty}) with $
A_{\infty} ( \lambda ) $ defined in (\ref{eqn:Andef}). However, in
contrast to the proof of Theorem 1, $ \lambda $ is no longer fixed.
As we are dealing with the Gaussian function
$\phi(x):=\exp(-x^2)$, a number of properties are immediate.
First of all, the fact that $ \phi $ is even and decreasing on
$[0, \infty )$ implies that for each $ \lambda > 0 $ and for any
real number $x$,
\begin{equation} \ \sum_{ k \in {\bf Z}} \phi ( k \lambda - x )
\ \leq \ 1 + \frac{1}{\lambda } \int_{ {\bf R}} \phi = 1 +
\frac{\sqrt{\pi}}{\lambda}. \label{eqn:riemann}
\end{equation}
Indeed, all values of $\phi ( k \lambda - x )$ can be replaced by
the $\int$ over the interval of length $\lambda$ from $k \lambda - x$
towards $0$, except perhaps the function value at the (single, if $x\ne
\pm\lambda/2$) point which is closest to $0$ (and thus is estimated by 1).
Also, the Fourier transform of $\phi$ is given by $ \hat{ \phi} ( \omega )
= (1/\sqrt{ 2 }) \exp ( - \omega ^{2} / 4) $. Keeping only the term with
maximal absolute value, we easily obtain
\begin{equation}\label{eqn:phihatest}
\sum_{l=-\infty}^{\infty} \left|\hat{\phi} \left( \frac{\omega +
2\pi l} {\lambda} \right) \right|^{2} \geq \hat{\phi}^{2}
\left(\frac{\pi}{\lambda}\right) \qquad\qquad \left( \forall
~\omega \in {\bf R}\right).
\end{equation}
\begin{lemma} For the function (\ref{eqn:pinfty}) we have
\begin{equation} |P_{\infty}(
\lambda, x)| \leq \frac{24}{1+x^2} \qquad \qquad \left( x\in {\bf
R}\right), \label{eqn:decay-of-p}
\end{equation} uniformly for all $\lambda \in (0,1)$.
\label{lemma:unidecay}
\end{lemma}
{\bf Proof of Lemma \ref{lemma:unidecay}:} Using $\phi(\lambda k)\leq 1$
and (\ref{eqn:akabssum}) we obtain
$$
|A_\infty(\lambda)| \leq \sum_{k=1}^\infty |a_k| \leq 1.6~.
$$
As $\max\limits_{\bf R}(1+x^2)\phi(x)= \max\limits_{[0,\infty)}
(1+t)e^{-t}=1$, we get
\begin{equation}\label{Aestim}
|2A_\infty(\lambda)\phi(x)| \leq \frac{3.2}{1+x^2}.
\end{equation}
It follows that we indeed have
\begin{equation} |P_{\infty}( \lambda, x)| \ \leq \frac{3.2}{1+x^2} +
\ \sum_{k \in {\bf Z}\setminus 0} |a_k| \phi ( x - \lambda k),
\label{eqn:decroiss}
\end{equation}
where $ a_{k} = a_{-k} $ if $ k <0$. As $\|\phi\|_{\infty} = \phi(0)= 1$,
in case $|x|\le 2$ this immediately leads to $|P_{\infty}( \lambda, x)|
\leq {3.2}/(1+x^2) + 3.2 < 20/(1+x^2)$, hence (\ref{eqn:decay-of-p}).
Because the right hand side of (\ref{eqn:decroiss}) is even, it
remains to take $x>2$.
Now let $\mathcal A$ be the set of all nonzero integers $k$ such that
$|x-\lambda k | < x /2 $. Observe that for $k\in\mathcal A~$, $\lambda |k|
\geq x/2 $ and thus $|k| \geq x/(2\lambda)$, which gives by $|a_k|\leq
1/k^2$, also $|a_k| \leq 4\lambda^2/x^2 \leq 5 \lambda^{2}/(1+x^2)$ for
$x>2$. Therefore, taking into account also (\ref{eqn:riemann}) and $x>2$,
we are led to
\begin{equation}
\sum_{ k \in {\mathcal A}} |a_k| \phi ( x - \lambda k) \leq \frac{5
\lambda^2}{1+x^2} \left( 1 + \frac{\sqrt{\pi}}{\lambda} \right) \leq
\frac{5\lambda^2+5 {\sqrt{\pi}}{\lambda}}{1+x^2} . \label{eqn:nearx}
\end{equation} On the other hand, in view of (\ref{eqn:akabssum}) and
\[ \max\limits_{[2,\infty)} ( 1+x^{2} ) \phi(x/2)=
\max\limits_{[4,\infty)}(1+t)e^{-t/4}=5/e , \]
we have
\begin{equation}
\sum_{ k \not\in {\mathcal A} } |a_k|\phi ( x - \lambda k) \leq \
\phi\left(\frac{x}{2}\right) 2 \sum_{k=1}^{\infty} |a_k| \leq
\frac{10}{e(1+x^{2})} \sum_{k=1}^{\infty} |a_k| <
\frac{6}{1+x^2}. \label{eqn:farx}
\end{equation}
Recalling $0<\lambda<1$ a combination of (\ref{eqn:decroiss}),
(\ref{eqn:nearx}) and (\ref{eqn:farx}) gives the result of the
lemma. $ \Box$
\[ \]
We shall also make use of an explicit lower bound for the
$L^{2}$-norm of $ P_{ \infty } ( \lambda , \cdot ) $ in terms of
the $ l^{2}$-norm of its coefficients. Actually, a more general
phenomenon can be observed here.
\begin{lemma} Let $ \lambda > 0 $ be fixed and $c_k\in{\bf C}$
$(k\in{\bf Z})$ be arbitrary coefficients satisfying $\sum_{k\in{\bf Z}}
|c_k|^{2} < \infty$, i.e., $(c_k)\in\ell_2({\bf Z})$. Consider the
function $f(\lambda,x):=\sum_{k=-\infty}^{\infty} c_k \phi(x-\lambda k)$.
We then have
\begin{equation}\label{eqn:independence}
|| f (\lambda, \cdot ) ||^{2}_{2}
\ \geq \ \mu ( \lambda ) \sum_{ k =-\infty}^{ \infty} |c_{k}|^{2}
\end{equation}
where
\begin{equation}
\mu ( \lambda ) : = \ \frac{ 2 \pi }{ \lambda } \inf_{ \omega \in {\bf R}}
\sum_{ l \in {\bf Z}} \left| \hat{\phi} \left( \frac{\omega + 2 \pi
l}{\lambda } \right) \right|^{2}. \label{eqn:riesz-bound}
\end{equation}
\label{lemma:independence}
\end{lemma}
{\bf Proof of Lemma
\ref{lemma:independence}:} First of all, for a fixed $ \lambda > 0$, the
series defining $ f:=f (\lambda , \cdot ) $ converges in $ L^{2} ( {\bf
R}) $. To see this, we consider its sequence
$ f_{n} ( \lambda , x ) = \sum_{|k| \leq n} c_k \phi(x-\lambda k)$
of partial sums. The Fourier transform of $ f_{n}:=f_n (\lambda ,
\cdot ) $ is given by $ \hat{f}_{n} ( \lambda , t) = \hat{ \phi
}(t) \sum_{|k| \leq n } c_{k} e^{ - ik \lambda t}$. Applying
Plancherel's theorem to
$ || f_{n}(\lambda , \cdot ) - f_{m}(\lambda , \cdot )||^{2}_{2} $ and
writing the resulting integral as a sum of integrals over the
intervals $ [ 2 \pi \lambda^{-1}l, 2 \pi \lambda^{-1}(l+1) ] , \
l \in {\bf Z} $, we obtain
\[ || f_{n}(\lambda , \cdot ) - f_{m}(\lambda , \cdot ) ||^{2}_{2} \ = \
\frac{1}{ \lambda } \sum_{ l = - \infty}^{ \infty }
\int_{ 0}^{ 2 \pi } \left| \hat{ \phi }
\left( \frac{ \omega + 2 \pi l}{ \lambda } \right)
\sum_{ m < |k| \leq n } c_k e^{ ik \omega} \right|^{2}
d \omega \]
for $m < n $.
The rapid decay of $ \hat{ \phi }$ assures the
finiteness of
\[ M(\lambda) \ := \ \frac{2 \pi}{ \lambda } \
\sup\limits_{ \omega \in {\bf R}} \
\sum_{ l = - \infty }^{ \infty} \left| \hat{ \phi }
\left( \frac{ \omega + 2 \pi l}{ \lambda } \right)
\right|^{2} \]
and therefore by Parseval's theorem,
\[ || f_{n}(\lambda , \cdot ) - f_{m}(\lambda , \cdot ) ||^{2}_{2}
\ \leq \ M_{ \lambda } \sum\limits_{m< |k| \leq n }
| c_{k}|^{2} \ \longrightarrow 0 \]
as $ n > m \longrightarrow \infty $.
This proves convergence in $ L^{2} $ of the series defining
$ f (\lambda , \cdot ) $.
A similar argument furnishes the conclusion of the lemma except that we
take the infimum $\mu(\lambda)$ (as defined in (\ref{eqn:riesz-bound})),
instead of the supremum $M(\lambda)$ above. $ \Box$
{\bf Proof of Theorem 2:} First of all, we estimate $ ||
P_{\infty}( \lambda, \cdot ) ||_{ \infty } $ from below by $ ||
P_{\infty}( \lambda, \cdot ) ||_{ 2 } $. In view of lemma
\ref{lemma:unidecay} we have
\begin{equation}
|P_{\infty}( \lambda, x )| \leq \frac{C}{|x|} \qquad
(\textrm{with}~~C=12)\label{eqn:decay}
\end{equation}
for all real numbers $x\ne 0$ and for each $\lambda >0$.
Now let the parameter $\sigma$ be chosen so that
$$
\sigma := \frac{4C^{2}}{|| P_{\infty}(\lambda, \cdot ) ||_{ 2 }^2} .
$$
Note that $P_{\infty}$ does not vanish identically, hence
$ \sigma >0$.
We write $ || P_{\infty}( \lambda, \cdot ) ||_{ 2 }^{2} $ as a sum
of integrals over $ [- \sigma , \sigma ] $ and over $ {\bf R} \setminus
[- \sigma , \sigma ]$. Estimating trivially in
$[- \sigma, \sigma ]$ and applying (\ref{eqn:decay})
to the second integral yields
\[ || P_{\infty}( \lambda, \cdot ) ||_{ 2 }^{2} \leq
2 \sigma || P_{\infty}( \lambda, \cdot ) ||_{ \infty }^{2} +
2 C^{2} \sigma^{-1} . \]
Thus, a short calculation with the chosen value of $ \sigma$ leads for
each $ \lambda> 0$
\begin{equation}
|| P_{\infty}( \lambda, \cdot ) ||_{ 2 }^{2} \ \leq \ 4C
|| P_{\infty}( \lambda, \cdot ) ||_{ \infty }.
\label{eqn:infty-two}
\end{equation}
To evaluate $\|P_{\infty}(\lambda,\cdot)\|_2$ we note
$P_{\infty}(\lambda,x)= \sum_{ k \in {\bf Z} } \alpha_{k} \phi(x-
k \lambda)$, where $\alpha_{k} = a_{|k|}$ if $ k \neq 0$, and $
\alpha_{0} = 2 A_{\infty}(\lambda)$, with $A_{\infty}(\lambda)$
defined in (\ref{eqn:Andef}). For this function we clearly have
$\sum_{k\in{\bf Z}} |\alpha_k|^2 \geq 2 \sum_{k=1}^{\infty}
|a_k|^2=9/4$ in view of (\ref{eqn:akabssum}).
Meanwhile, we consider the function $ \mu(\lambda) $ defined in
(\ref{eqn:riesz-bound}). Recalling (\ref{eqn:phihatest}) and the explicit
form of $\widehat{\phi}$ provides for each $ \lambda > 0$ the estimate
\[
\mu( \lambda ) \geq \frac{\pi}{ \lambda } \exp \left( -
\frac{\pi^2}{2\lambda^2} \right).
\]
Combining this with lemma \ref{lemma:independence} and
(\ref{eqn:infty-two}) we obtain
\begin{equation}
|| P_{\infty}( \lambda, \cdot ) ||_{ \infty } \geq \frac{\pi}{4C\lambda}
\exp \left( - \frac{\pi^2}{2\lambda^2} \right) \sum_{k=-\infty}^{\infty}
|\alpha_k|^2 = \frac{9\pi}{16C\lambda} \exp \left( -
\frac{\pi^2}{2\lambda^2} \right)~. \label{eqn:l2}
\end{equation}
Now recalling $|a_k|\leq 1/k^2$ we obtain $ \sum_{ k>n} |a_{k}| < 1/n$ for
each positive integer $n$. Recalling also $C=12$, this and (\ref{eqn:l2})
yields that whenever (\ref{eqn:Nzerocond}) holds, then
\[
20 \sum_{k>n} |a_{k}| < 20/n < 20/N_0 = \frac{3\pi }{64 \lambda}
\exp\left(-\frac{\pi^2}{2\lambda^2}\right) < || P_{\infty}(
\lambda, \cdot ) ||_{ \infty }.
\]
Therefore, an application of lemma \ref{claim:punchline} concludes
the proof of Theorem 2. $ \Box $
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 330
|
{% extends "base.html" %}
{% load static %}
{% load custom_filters %}
{% block breadcrumbs %}
<li><a href="/assays/assaystudy/">Studies</a></li>
<li><a href="{{ object.get_absolute_url }}">{{ object }}</a></li>
<li><a href="{% url 'assays-assaystudy-data-index' object.id %}">Data</a></li>
<li class="active">Assay Plate Reader Maps</li>
{% endblock %}
{% block load_js %}
<script src="{% static "assays/assayplatereadermap_index.js" %}"></script>
{% endblock %}
{% block content %}
{% include 'tracking.html' with study_submit='true' %}
<div class="padded-bottom">
<div class="well">
<h1 class="text-center">
<em><b>Assay Plate Map List</b></em>
<br>
<br>
Study: {{ object }}
</h1>
<div class="row text-center small-padding-top">
<a href="{{ object.get_summary_url }}" class="btn btn-primary" role="button">Study Summary</a>
<a href="{% url 'assayplatereaderfile-index' object.id %}" class="btn btn-primary" role="button">Assay Plate Reader File List</a>
</div>
</div>
{# Hide setup add buttons if the study is marked reviewed #}
{% if user|is_group_editor:object.group.name and not object.signed_off_by %}
{% if not review %}
<div class="padded-bottom">
<a href="{% url 'assayplatereadermap-add' object.id %}" class="btn btn-success" role="button">Add Plate Map</a>
</div>
{% endif %}
{% endif %}
<br>
{% if assayplatereadermaps %}
<div>
<div><p id="plate_map_index_tooltip"></p>
<label for="plate_map_index_tooltip">Assay Plate Map List </label>
</div>
<table id="assayplatereadermaps" class="display table table-striped table-hover" cellspacing="0" width="100%">
<thead>
<tr>
<th>View</th>
<th>Edit/Calibrate</th>
<th>Assay Plate Map Name</th>
<th>Plate Size</th>
<th>Standard Unit</th>
<th>Target, Method, Unit</th>
<th>Time Unit</th>
<th>Volume Unit</th>
<th>Cell Count</th>
<th>Description</th>
<th>File Count</th>
<th>Data Block Count</th>
</tr>
</thead>
<tbody>
{% for plate in assayplatereadermaps %}
<tr>
<td>
<a class="btn btn-primary" href="{% url 'assayplatereadermap-view' plate.id %}">View </a>
</td>
<td>
{% if not object.signed_off_by %}
{% if plate.block_count > 0 %}
<a class="btn btn-primary" href="{% url 'assayplatereadermap-update' plate.id %}">Calibrate</a>
{% else %}
<a class="btn btn-primary" href="{% url 'assayplatereadermap-update' plate.id %}">Edit Map</a>
{% endif %}
{% endif %}
</td>
<td>{{ plate.name }}</td>
<td>{{ plate.device }}</td>
<td>{{ plate.standard_unit }}</td>
<td>{{ plate.new_study_assay }}</td>
<td>{{ plate.time_unit }}</td>
<td>{{ plate.volume_unit }}</td>
<td>{{ plate.cell_count }}</td>
<td>{{ plate.description }}</td>
<td>{{ plate.file_count }}</td>
<td>{{ plate.block_count }}</td>
</tr>
{% endfor %}
</tbody>
</table>
</div>
{% else %}
<div class="alert alert-warning" role="alert">
No assay plate maps exist for this study
</div>
{% endif %}
</div>
{% endblock %}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,140
|
The semantic web community is supported by an interdisciplinary research field in which academy and industry share their vision of the web of data and join the forces to make it mainstream. ESWC is a prominent conference on the topic and the 12th edition took place this year in Portoroz, Slovenia.
I presented WordLift at the developers' workshop held during the first day of ESWC2015. WordLift is a WordPress plugin which aims to enhance web publishing with semantic capabilities, allowing non technical people (e.g. a blogger or a journalist) to integrate linked data in their ordinary workflow. Designed and developed by InsideOut10, WordLift is a software module being used in the validation of MICO.
MICO provides cross-media analysis tools for online multimedia producers. WordLift will include MICO algorithmic efforts to offer semantic publishing capabilities (analysis, querying and recommendation) for images, text and videos. This will help reducing the time spent by online editors for bolstering their media contents by creating a context, detecting quality issues for online videos and supporting the interlinking between different media assets whether in textual or visual form.
This technology stack described above will be validated in a real-world scenario in collaboration with Greenpeace Italy starting this coming September. The online magazine dedicated to Greenpeace's Italian supporters will feature the newest WordLift version (3.0) and MICO to restructure and repurpose contents using semantic capabilities and cross-media analysis. This concretely will help the organisation to sustain its business model, retaining its followers and stimulating their environmental sensibility.
Here follows the slides presented at ESWC and a summary of the questions I received right after the talk from web experts attending the presentation.
– I think you are doing something really important, since every scientist is on the way to self-publish his research. Can you add researcher-friendly features?
WordLift offers many possibilities to enrich and organise a researcher's blog in a meaningful way. A researcher can create entities and relate them one with the other; he/she can visualize entity-related data via graphical widgets and make them browsable for the end-user.
– Can you recommend content from around the web?
WordLift already suggests concepts related to the post being edited, in the form of semantic entities with their description and properties, included thumbnails. This is done using the Redlink Semantic platform which is focused on textual analysis. With MICO the plugin will be able to do the same starting from images and videos.
– Is the user-created vocabulary stored in the blog?
For 'vocabulary' we intend the set of entities mentioned in the blog post, organised in categories such as Persons, Organizations, Places, Events (following schema.org definitions). WordLift stores the vocabulary both locally and in the cloud (via Redlink). The cloud exposes a SPARQL end-point as we've seen in the past that running a triple store on every website was far too inefficient .
In semantic web technologies the gap between state-of-the-art and real-world applications remains an issue. There is still a noticeable distance between W3C specifications and concrete implementations that common users can benefit from. We've seen at ESWC 2015 that this gap is shrinking and WordLift, Redlink and MICO can play a constructive role in supporting content producers reaching their audience using these technologies.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,390
|
\subsection{Self-mirror prediction for number of reflexive polytopes}
Let $N$ be the number of reflexive polytopes in fixed dimension $d$.
Then the number of (formal) duality assignments with $S$ self-mirrors, i.e.
the number of involutions with $S$ fixed points, is $N\choose S$
for the choice of $S$ self-mirrors times
$(N-S-1)\cdot(N-S-3)\cdot\ldots \cdot3\cdot 1$ for the possible selections of
the remaining dual pairs, hence
\BE
n_S
\frac{N!}{{S!}\cdot2^{\frac{N-S}2}\,(\frac{N-S}2)!}\qquad
\hbox{with}\quad S-N\in2\IZ.
\EE
\del
For large $N$ we can use
Stirling's formula
\BE \textstyle
N!=\sqrt{2\p N} N^Ne^{-N+\frac\th{12N}}
=\sqrt{2\p N}\left(\frac Ne\right)^n\left(1+\mathcal O(\frac1N)\right)
\EE
with $0<\th<1$
\BE \textstyle
\frac{\6\log(n_{2s})}{\6N}\approx \frac1{\sqrt{2N}}
\left(1+\frac sN+\mathcal O(\frac{s^2}{N^2})\right)
\EE
hence with $Z(N)=\sum_{0\le s\le N/2}\,2s\,n_{2s}$
\BE \textstyle
\6_N\log Z\approx\frac1{\sqrt{2N}}
\left(1+\frac{\langle S\rangle}N\right)
\EE
\BE \textstyle
\6_S n_{S}=
\EE
\BE
n_S=\frac {(\frac N2)!\;2^{S/2}} {S!\;(\frac{N-S}2)!} WRONG
\EE
where we assumed that $N$, and hence also the number $S$
of self-mirror polytopes, is even.
This yields the expectation value
\BE \textstyle
\langle S\rangle={\sum 2s\,n_{2s}\over\sum n_{2s}}\approx
{\sum 2s\,(2/N)^s/(2s)!\over\sum(2/N)^s/(2s)!}
=-2N\6_N\log(\textstyle{\sum{(2/N)^s\over(2s)!}})
\approx-2N\6_N\log\cosh\8{\sqrt{2\0N}}
\EE \enddel
Let $Z_N=\sum_{S\le N} n_S$ be the total number of involutions.
The asymptotic expansion for large $N$
can be derived from the generating function \cite[section 3.8]{genfun}
\BE
e^{x+\frac12x^2}=\sum_{N\ge0}\frac{Z_N}{N!}x^N.
\EE
Since it happens $Z_{N-1}$ times for the $Z_N$ involutive
permutations of $N$ objects that a given object is a fixed point,
we obtain the following formula for the expectation value \cite{ILugo}
\BE \textstyle
\big\langle S\big\rangle=\frac1{Z_N}\sum_Sn_S\,S=N\frac{Z_{N-1}}{Z_N}.
\EE
Assuming a uniform probability distribution of involutions
we thus expect
\BE \textstyle
\big\langle S\big\rangle=\sqrt N-\frac12+\frac13\frac1{\sqrt N}
+{\mathcal O}(\frac1N)
\EE
selfdual polytopes for
large $N$.
As shown in \tab tab:SD. this roughly explains the size but
underestimates the correct numbers for $d\le4$.
\begin{table}[h]
\begin{center}} \def\EC{\end{center}\begin{tabular}{||c||c|c|c|c||}\hline\hline
$d$ & 1&2&3&4\\\hline\hline
$N_R(d)$& 1&16&4319&473800776\\\hline
$N_{\rm selfdual}$&1&4&79&41710 \\\hline
$\big\langle S\big\rangle$
& 1 & 3.6 & 65.2 & 21766.5 \\\hline\hline
\end{tabular}\\[7pt]
\caption{
Numbers of selfdual polytopes and
probabilistic
expectation
.\label{tab:SD}\HS-259
} \vspace{-18pt}
\EC\end{table
\del
\BE \hspace*{-9pt}
\frac{\sum_{s\ge0} \,2s\,\, n_{2s}}{\sum_{s\ge0}n_{2s}}
\approx 2N\6_N\log\sum_{s\ge0} {N^s\over(2s)!}=2N\6_N\log\cosh\sqrt N
=\sqrt N\tanh\sqrt N\approx\sqrt N
\EE
With $s=S/2\in\IZ_{\ge0}$ this yields the expectation value
\BE \langle S\rangle={\sum 2s\,n_{2s}\over\sum n_{2s}}\approx
{\sum 2s\,(2/N)^s/(2s)!\over\sum(2/N)^s/(2s)!}
=-2N\6_N\log(\textstyle{\sum{(2/N)^s\over(2s)!}})
\approx-2N\6_N\log\cosh\8{\sqrt{2\0N}}
\EE
For large $N$ we thus find
\BE \langle S\rangle\approx 2N\tanh\8{\sqrt{2\0N}}\;\8{{\frac12}} \long\def\keep#1\endkeep{\sqrt{N\02}
{1\0N^2}}=1/N
\EE
\enddel
\del
The numbers $N_2=16$, $N_3=4\,319$ and $N_4=473\,800\,776$ of reflexive
polytopes in 2, 3 and 4 dimensions show that a complete enumeration of the
5-dimensional case is hopeless at present already because no existing
harddisk would be large enough to store the data. One may try to estimate
their number by extrapolation. The double-exponential ansatz
$N_d\approx 2^{2^{d+1}-4}$ due to Harald Skarke provides a good
fit to the known data and would predict $N_5\approx1.2\cdot10^{18}$ and
$N_6\approx2.1\cdot10^{37}$ in 5 and 6 dimensions, respectively.
For a random choice we would expect the first mirror pairs to show up
at $\sqrt{N_d}$, i.e. at $10^9$ polytopes ... this is about twice the upper
limit of the maximal content of one PALP database with
present computer architecture and amounts to 30GB of data.
Appears to be hard to get a reasonable statistics.
\enddel
As this result seems to indicate that a statistical approach makes sense we
now want to use similar considerations for predicting the number of
reflexive polytopes in 5 dimensions on the basis of incomplete lists.
In a random set of $p>\sqrt N$ reflexive polytopes in fixed dimension
the formula $\langle S\rangle\approx
\sqrt N$ implies that we expect
$s\approx\sqrt N \cdot p/N$ self-mirror polytopes.
This leads to the prediction $N\approx (p/s)^2$ if we find $s$ selfdual
polytopes in the sample.
Similarly, if we ignore the relatively small number of self-mirrors
for large $N$ and count the number $m$ of mirrors pairs in a sample of
$p$ polytopes we obtain the prediction $N\approx p^2/(2m)$.
If we increase the size of a random sample we expect $s$ to grow linearly
and $m$ quadratically with $p$, and more precisely $s\approx\sqrt{2m}$.
Actually, the formula $N\approx p^2/(2m)$
has been used already serval years ago when we enumerated the reflexive
polytopes in 4 dimensions \cite{c4d}, which required two years of program
improvements and computation time after the 3-dimensional case \cite{c3d}.
We thus could check the sufficiency of the implemented data structures for
the storage of the result at an early stage of the project.
The starting point of the calculation was the list of 308 weight matrices
(206 weight vectors and 102 matrices with $2\le\mao{rank}\le4$) of
maximal reflexive polytopes $\D_M$ for which the reflexive subpolytopes
were computed in the order of an increasing number of lattice points.
The first Newton polytope in this list is defined by the weight vector
$(3,3,4,4,10)$ with degree 24. It has 47 lattice points and 6 vertices.
After fetching and compiling PALP its subpolytopes can be computed
with the following commands,
\begin{equation}
\hspace*{-34mm}\hbox to 5cm {
\vbox{ \tiny\label{FetchCompile}\begin{verbatim}
$ Bdir=$HOME/bin # directory for binary files (check $PATH)
$ Wdir=$PWD # working directory
$ cd /tmp
$ wget hep.itp.tuwien.ac.at/~kreuzer/CY/palp/palp-1.1.tar.gz # fetch
$ gunzip palp-*.tar.gz; tar -xvf palp-*.tar; cd palp; # unpack
$ make; mv *.x $Bdir; cd $Wdir # and compile PALP
$
$ echo "24 3 3 4 4 10" | class.x -f -po /tmp/zbin.47
...
800kR-1658 11MB 4215kIP 1042kNF-23k 9_46 v17r17 f28r27 85b24 120s 120u 25n
24 3 3 4 4 10 R=798878 +1658sl hit=0 IP=4215623 NF=1042005 (23261)
... 1181m+14s 9851469b u7 pp/2m=2.68615e+08 pp/ss=3.25615e+09
\end{verbatim}\hss}}
\end{equation
Within two minutes we thus obtain
the values 269 million for $p^2/2m$ with $p\!=\!798878$ and $m\!=\!1181$
and 3.25 billion for $p^2/s^2$ with $s\!=\!14$ on a standard 3GHz PC
(cf.\;the last output line) and hence a good approximation of the correct value 473\,800\,776 with
a production rate of about $7000$ polytopes per second (back in 1998 the
CPU time was almost 1 hour).
\del
\begin{verbatim}
$ echo "15 2 2 3 3 5" | class.x -f -po /tmp/zbin56 # M:56 9 N:8 6
...
11202kR-3930 167MB 77MIP 14MNF-82k 7_55 v22r21 f32r29 61b30 47m 47u 8n
15 2 2 3 3 5 R=11198394 +3930sl hit=0 IP=77167925 NF=14872004 (82172)
... 11198394+3930sl 34804m+71s 160699062b u13 pp/2m=9.27455e+08 pp/ss=7.40524e+09
$ echo "4 1 1 1 1 0 0 6 0 0 1 1 2 2" |class.x -f -po /tmp/zbin # M:57 8 N:9 6
...
17791kR-2445 238MB 105MIP 27MNF-255k 7_55 v25r23 f36r36 46b31 68m 68u 15n
4 1 1 1 1 0 0 6 0 0 1 1 2 2 R=17789212 +2445sl hit=0 IP=105672169 NF=27019369 (255903)
... 17789212+2445sl 795011m+1482s 235323323b u23 pp/2m=1.86804e+07 pp/ss=4.55277e+06
echo "14 2 2 3 3 4" | class.x -f -po /tmp/zbin # M:57 10 N:10 7
wget http://quark.itp.tuwien.ac.at/~kreuzer/W/4dTransWH.gz;
gunzip 4dTransWH.gz ; mv 4dTransWH 4dT # cws.x -w4 -t > 4dT
cws.x -w3 -t > 3dT ; cws.x -w2 -t > 2dT ; cws.x -w1 -t > 1dT
wget http://quark.itp.tuwien.ac.at/~kreuzer/W/5dTransWH.all.gz
for((i=10; i<=27; i++));
do echo "$i:"; cat tc52* | grep "M:$i " >> tcm10-27.in; done
echo "24 3 3 4 4 10" | class.x -f -po /tmp/zbin
Writing zbin: 798878+1658sl 1181m+14s 9851469b u7 pp/2m=2.68615e+08 pp/ss=3.25615e+09
echo "24 3 3 4 4 10" | class.x -f -po zbin
798878+1658sl 1181m+14s 9851469b
pp/2m=2.68615e+08
pp/ss=3.25615e+09
for((i=28;i<=30;i++));do echo "$i:"; cat tc52*|grep "M:$i ">>tcm$i.in; done
beauty/tmp> echo "24 3 3 4 4 10" | class.x -f -po /tmp/zbin
Fri Mar 21 15:40:56 2008
100kR-415 1MB 453kIP 111kNF-2k 7_47 v16r15 f25r22 85b21 12s 12u 2n
200kR-1168 2MB 857kIP 227kNF-5k 6_37 v16r16 f26r24 85b21 24s 24u 5n
800kR-1658 11MB 4215kIP 1042kNF-23k 9_46 v17r17 f28r27 85b24 121s 121u 25n
24 3 3 4 4 10 R=798878 +1658sl hit=0 IP=4215623 NF=1042005 (23261)
Writing /tmp/zbin: 798878+1658sl 1181m+14s 9851469b u7 pp/2m=2.68615e+08 pp/ss=3.25615e+09 done: 1s
Fri Mar 21 15:42:58 2008
brane/palp> echo "24 3 3 4 4 10" | class.x -f -po /tmp/zbin
Fri Mar 21 15:42:01 2008
100kR-415 1MB 453kIP 111kNF-2k 7_47 v16r15 f25r22 85b21 15s 15u 3n
200kR-1168 2MB 857kIP 227kNF-5k 6_37 v16r16 f26r24 85b21 30s 30u 6n
300kR-1024 4MB 1327kIP 345kNF-7k 8_40 v16r16 f27r25 85b21 47s 47u 10n
700kR-1221 9MB 3553kIP 892kNF-17k 6_42 v17r17 f28r27 85b24 127s 126u 26n
800kR-1643 11MB 4208kIP 1041kNF-22k 7_46 v17r17 f28r27 85b24 150s 150u 31n
800kR-1658 11MB 4215kIP 1042kNF-23k 9_46 v17r17 f28r27 85b24 150s 150u 31n
24 3 3 4 4 10 R=798878 +1658sl hit=0 IP=4215623 NF=1042005 (23261)
Writing /tmp/zbin: 798878+1658sl 1181m+14s 9851469b u7 pp/2m=2.68615e+08 pp/ss=3.25615e+09 done: 0s
Fri Mar 21 15:44:31 2008
\end{verbatim}
\enddel
For 5 dimensions it is, of course, much harder to get a reliable statistics.
With an expectation of $N\approx 10^{18}$ according to Skarke's guess, the
storage of $\sqrt N$ polytopes would already required some 30GB of disk space
so that we can only go above that value by 1-2 orders of magnitude with
currently available hardware. Nevertheless, it should be possible
to either verify that the number is not much smaller or to get a reasonable
prediction if it is.
For a first attempt we defined data samples in terms of the Newton polytopes
of transversal reflexive weight vectors%
\footnote{~ cf.
\tt http://quark.itp.tuwien.ac.at/{\tiny$^\sim$}kreuzer/W/5dTR
.}
ordered according to increasing numbers of lattice points. For the increasing
series data samples consisting of all reflexive subpolytopes of transversal
Newton polytopes with $\le36$ \ldots $\le65$ points
the predictions are plotted in \fig fig:TM5.
The result is obviously inconclusive, but certainly compatible with
the guess $10^{18}$. Unfortunately the bias of the a priori independent
predictions $p^2/2m$ and $p^2/s^2$ is strongly correlated in our data samples.
For the $9.025\cdot10^{9}$ polytopes of the largest sample
with $m=86323$ and $s=354$, which occupies 239 GB of disk space, we find
$p^2/2m\approx4.7 \cdot10^{14}$ and $p^2/s^2\approx6.5 \cdot10^{14}$.
\bigskip
\Section{Conclusions}
In this note we determined all lattice polytopes with reflexive dimension
$\mao{rd}\le4$ and discussed enumeration problems and algorithmic aspects
with applications to algebraic geometry and string theory.
We pointed out the need for an efficient algorithm for the enumeration
of IP weight vectors $w$ with a bounded number of lattice points in
the convex hull of
the simplex
defined by the linear relations $\sum w_jv_j=0$. Such an algorithm could
be used for the enumeration of reflexive polytopes with fixed number of
points rather than fixed dimension.
We introduced the concept of \hbox{I\hspace{-.7pt}P}} \long\def\del#1\enddel{-confined polytopes, which are a subclass
of \hbox{I\hspace{-.7pt}P}} \long\def\del#1\enddel{\ polytopes, and extended the polar duality of reflexive polytopes
to IPC-closed polytopes. Maximal IPC-closed polytopes contain all reflexive
polytopes in arbitrary dimensions and hence lead to a simplification of
the classification program. In turn, we pointed out the existence of
\hbox{I\hspace{-.7pt}P}} \long\def\del#1\enddel{-simplices that are not \hbox{I\hspace{-.7pt}P}} \long\def\del#1\enddel{-confined and enumerated them for the case
of 3 dimensions. A constructive classification of such
simplices in higher dimensions is another interesting open problem.
We suggested a statistical
approach to the enumeration of reflexive polytopes which should at least
allow us to obtain probabilistic lower bounds, depending essentially on
the size of the available hard-disks for storage of the data.
As a first attempt we constructed a data-base containing about $9\cdot10^9$
pairs of reflexive 5-dimensional polytopes, which can also be used to
produce incomplete lists of polytopes with reflexive dimension 5.
Studies of correlations of polytope data like $f$-vectors (numbers of
faces) can thus be initiated and may be useful for selecting appropriate
data samples for statistical applications.
\del
In the remaining months before the workshop
``Information-theoretic aspects of integer-point enumeration in polyhedra''
{\footnotesize\verb+http://www.bio-complexity.com/ITSL/ITSL_index.html+}
we plan to create a complementary
sample based on weight matrices of rank two and to attempt an enumeration
of all IP weight vectors for dimension 5, which would enable
the enumeration of all maximal/minimal pairs of IPC-complete polytopes,
and presumably also of the maximal/minimal pairs of reflexive polytopes.
More detailed data will be made available on the internet \cite{inprog}.
\enddel
\vspace{-2pt}
\def{\Fsec References \hspace*{165pt}}{{\Fsec References \hspace*{165pt}}}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,245
|
Q: React Typescript Discriminate type if prop is missing I am trying to create a select with a clearable prop which allows a clearable button to be clicked, which when clicked calls the onChange handler with null as the value. When clearable is set as true I want onChange handler to be typed as T | null => void and when it is set to false I want it to be typed as just T => void. That part is not a problem, the problem arises when I want the default value for clearable to be true which allows the user to call the select component without specifying the clearable prop and have the onChange typed as if clearable was true. Please see code below and let me know how to type the SelectProps to set the onChange properly when clearable is not set.
type SelectProps<T> = {
options: T[];
value: T;
} & (
{
clearable: false;
onChange: (val: T) => void;
}
|
{
clearable?: true;
onChange: (val: T | null) => void;
}
);
const Select = <T extends string>({
options = [],
value,
clearable = true,
onChange
}: SelectProps<T>): React.ReactElement<SelectProps<T>> => <div>select</div>;
class App extends React.Component {
render() {
return (
<>
{/*Works, onChange val is typed to string */}
<Select clearable={false} options={[]} value="" onChange={val => console.log(val)} />
{/*Works, onChange val is typed to string OR null */}
<Select clearable={true} options={[]} value="" onChange={val => console.log(val)} />
{/* Doesn't Work, onChange val is typed to any */}
<Select options={[]} value="" onChange={val => console.log(val)} />
</>
);
}
}
Note, none of the following work
{
clearable: false;
onChange: (val: T) => void;
}
|
{
clearable: true | undefined;
onChange: (val: T | null) => void;
}
{
clearable: false;
onChange: (val: T) => void;
}
|
{
clearable: true;
onChange: (val: T | null) => void;
}
|
{
clearable: undefined;
onChange: (val: T | null) => void;
}
}
{
clearable: false;
onChange: (val: T) => void;
}
|
{
clearable: true;
onChange: (val: T | null) => void;
}
|
{
onChange: (val: T | null) => void;
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,023
|
Xestoblatta sancta es una especie de cucaracha del género Xestoblatta, familia Ectobiidae.
Distribución
Esta especie se encuentra en Ecuador.
Referencias
sancta
Insectos descritos en 1898
Insectos de Ecuador
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,173
|
This vehicle won't be on the lot long! Breathtaking performance, sumptuous accommodations and jaw dropping good looks! Turbocharger technology provides forced air induction, enhancing performance while preserving fuel economy. With just over 35,000 miles on the odometer, this 4 door sedan prioritizes comfort, safety and convenience. All of the following features are included: delay-off headlights, power front seats, and leather upholstery. It features an automatic transmission, front-wheel drive, and a 2 liter 4 cylinder engine. Our team is professional, and we offer a no-pressure environment. We'd be happy to answer any questions that you may have. We are here to help you.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,768
|
\section{Introduction}
A classical question in stochastic process theory is to understand the asymptotic behavior of a given stochastic process $X=(X_t)_{t\geq 0}$ on the level of paths. In the present work we consider general L\'evy processes and find Chung type LIL (laws of the iterated logarithm) at zero, that is, given the L\'evy process $X$, we aim at characterizing a norming function $b$ satisfying
\begin{align}\label{LIL}
\liminf_{t\rightarrow 0}\frac{||X||_t}{b(t)}=1,\qquad \text{where}\qquad ||X||_t:=\sup_{0\leq s\leq t}|X_s|.
\end{align}
The topic of large and small time fluctuations of L\'evy processes has been studied extensively in the past (see for instance \cite{D} for an overview and \cite{B96,sato}).
It is well-known that, via the Borel-Cantelli Lemma, Chung type LIL for a general stochastic process are connected to the so-called small deviation rate of the process, i.e.\
\begin{align}\label{small}
-\log \P(||X||_t\leq \eps),\qquad \text{as $\eps\to 0$ and $t\to0$}.
\end{align}
The main motivation for this paper originates from the recent work \cite{AD09}, where a framework for obtaining the small deviation rate (\ref{small}) for general L\'evy processes is provided. The difficulty in passing over from the small deviation estimate to the respective LIL concerns circumventing the independence assumption of the Borel-Cantelli lemma.
\medskip
In this paper we show how the asymptotics of (\ref{small}) imply explicit LIL. \medskip
Small deviation problems are studied independently of LIL and have connections to other fields, such as the approximation of stochastic processes, coding problems, the path regularity of the process, limit laws in statistics, and entropy numbers of linear operators. We refer to the surveys \cite{lishao,lif} for an overview of the field and to \cite{sdbib} for a regularly updated list of references, which also includes references to laws of the iterated logarithm of Chung type. The papers \cite{taylor,mogulskii,bormog,simon01,simonpvar,lindeshi,ls,lindezipfel,elena,elena2} provide a good source for earlier results on small deviations of L\'evy processes.\medskip
We now discuss LIL for special L\'evy processes that have already appeared in the literature. The norming function $b(t)=\sqrt{\pi^2 t/ (8\log|\log t|)}$ for a standard Brownian motion can be derived from the large time LIL, proved by Chung \cite{chung}, via time inversion. \\ For any L\'evy process with non-trivial Brownian component the recent result of \cite{BM09} shows that (\ref{LIL}) holds with the norming function for a standard Brownian motion. If $X$ is an $\alpha$-stable L\'evy process (\ref{LIL}) holds with norming function $b(t)= ( c_\alpha t/\log|\log t|)^{1/\alpha}$, which goes back to \cite{taylor}.\medskip
Of course it is natural to ask for the general structure of the norming function for arbitrary L\'evy processes not having the special features of the examples mentioned so far.
LIL for more general L\'evy processes were obtained by Wee in \cite{W88} (see \cite{wee2} for more examples). It was shown that if for some positive constant $\theta$
\begin{align} \label{eqn:weecondition}
\P(X_t>0)\geq \theta\quad \text{and}\quad \P(X_t<0)\geq \theta,\qquad\text{for all $t$ sufficiently small,}
\end{align}
holds, then upper and lower bounds in the LIL hold in the following sense: for $\lambda_1$ sufficiently small and $\lambda_2$ sufficiently large,
\begin{align*}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_t}{b_{\lambda_1}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_t}{b_{\lambda_2}(t)}\leq 1
\end{align*}
for norming functions $b_{\lambda}$ given by
\begin{align*}
b_\lambda(t):=f^{-1}\left(\frac{\log|\log t|}{\lambda t} \right),
\end{align*}
where $f$ is given by some explicit, but complicated expression depending on the L\'evy triplet.\medskip
Although the results of Wee are quite general, there are some drawbacks which we aim to overcome in the present work.\\
First, the proofs given in \cite{W88} and \cite{wee2} are rather obscure. This might be due to the use of the old-fashioned notation for the L\'evy measure. In particular, it does not become clear that actually the LIL follow from small deviation estimates of type (\ref{small}) and which behavior of the process is actually responsible for the correct norming function.
Secondly, even in the symmetric case, if the norming function $b_{\lambda}$ is not regularly varying at zero, the unspecified (and suboptimal) constants $\lambda_1$ and $\lambda_2$ do not only appear in a weaker limiting constant, but they influence the norming function essentially (see (\ref{eqn:constinexp}) below for an example of influence on the exponential level). In our approach, we keep track of the appearing constants in an optimal way.
Thirdly, although condition (\ref{eqn:weecondition}) looks innocent at a first glance it turns out to be quite delicate. It is certainly fulfilled for symmetric L\'evy processes. Unfortunately, only given the L\'evy triplet it seems to be unknown how (\ref{eqn:weecondition}) can be checked. On the contrary, our conditions are explicit in terms of the L\'evy triplet. We give a couple of examples showing that our conditions can be checked easily.
\medskip
This paper is structured as follows. In Section~\ref{sec:results}, we give the main results that manage the transfer between small deviations and LIL. Several examples of LIL for concrete L\'evy processes are collected in Section~\ref{sec:examples}. The proofs are given in Section~\ref{sec:proofs}.
\medskip
Let us finally fix some notation. In this paper we let $X$ be a L\'evy process with characteristic triplet $(\gamma, \sigma^2, \Pi)$, where $\gamma\in \R$, $\sigma^2\geq 0$, and the L\'evy measure $\Pi$ has no atom at zero and satisfies
\begin{align*}
\int (1\wedge x^2)\Pi(\d x)<\infty.
\end{align*}
For basic definitions and properties of L\'evy processes we refer to \cite{B96,sato}. As we are interested only in the behavior for small times we may truncate large jumps. In particular, we restrict ourselves to L\'evy processes involving only jumps of absolute value at most $1$. Hence, the characteristic exponent, $\E e^{i z X_t } =: e^{ t \psi(z)}$, has the form
\begin{align*}
\psi(z)=i\gamma z -\frac{\sigma^2z^2}{2}+\int_{-1}^{1}(e^{iz x}-1-iz x)\Pi(\d x), \qquad z\in\R.
\end{align*}
For later use we denote by $\Phi$ the Laplace exponent of a subordinator $A$, $\E e^{-u A_1}=e^{-\Phi(u)}$,
\begin{align*}
\Phi(u)= u \gamma_A + \int_0^\infty (1-e^{-ux}) \Pi_A(\d x).
\end{align*}
Further, we use the standard notation $\bar\Pi(\eps):=\Pi([-\eps,\eps]^c)$ for the tail of the L\'evy measure.
In the following, we denote by $f \sim g$ the strong asymptotic equivalence, i.e.\ $\lim f/g=1$, and by $f\approx g$ the weak asymptotic equivalence, i.e.\ $0<\liminf f/g \leq \limsup f/g < \infty$.
\section{Main results} \label{sec:results}
Our first theorem manages the transfer from small deviation rates to LIL under minimal loss of constants.
\begin{theorem}\label{t3}
Let $X$ be a L\'evy process and $F$ be a function increasing to infinity at zero such that with some $0<\lambda_1\leq \lambda_2<\infty$
\begin{align}\label{eqn:sdestmain}
\lambda_1 F(\eps)t\leq -\log \P(||X||_{t}<\eps) \leq \lambda_2 F(\eps)t, \qquad \text{for all $\eps<\eps_0$ and $t<t_0$.}
\end{align}
Further, define
\begin{align*}
b_\lambda(t):=F^{-1}\left(\frac{\log|\log t|}{\lambda t} \right)
\end{align*}
for $\lambda>0$ and assume that
\begin{align} \label{eqn:conditionM}
(n+1)^{-(n+1)^\beta} \left| \int_{|x|>b_{\lambda_2'}(n^{-n^\beta})} x \Pi(\d x) - \gamma\right|=o\big(b_{\lambda_2'}\big(n^{-n^\beta}\big)\big),\qquad \text{for all $\beta>1$ and $\lambda_2'>\lambda_2$.}
\end{align}
Then the LIL
\begin{align*}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_1'}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_2'}(t)}\leq 1
\end{align*}
hold almost surely for any $\lambda_1'<\lambda_1$ and $\lambda_2'>\lambda_2$.
\end{theorem}
Unfortunately, our proof forces us to assume condition (\ref{eqn:conditionM}) in order to prove the more delicate upper bound. This condition is clearly satisfied for symmetric processes and can be checked readily from the L\'evy triplet.
It is crucial that there is almost no loss of constants in the transfer from the small deviations to the LIL as in cases when $b_{\lambda}$ is not regularly varying, the constants $\lambda_1', \lambda_2'$ may influence the rate function drastically (see (\ref{eqn:constinexp}) for an extreme example).\\
If instead $b_{\lambda}$ only depends on $\lambda$ via a multiplicative constant, our approach allows to strengthen the previous theorem to the optimal limiting constants. Such examples occur for instance if the small deviation rate function $F$ is regularly varying.
\begin{corollary}\label{cor:t1corollary}
In the setting of Theorem~\ref{t3} assume additionally that $F$ is regularly varying at zero with non-positive exponent. Then the following LIL hold almost surely:
\begin{align} \label{eqn:lilnonre}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_1}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_2}(t)}\leq 1.
\end{align}
In particular, if there is $\lambda>0$ such that (\ref{eqn:sdestmain}) holds for all $\lambda_1<\lambda$ and all $\lambda_2>\lambda$ then
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda}(t)}=1\qquad\text{a.s.}
\end{align*}
\end{corollary}
In the setting of regularly varying rate function, say $F$ is regularly varying at zero with exponent $-\alpha$, $\alpha>0$, one can express (\ref{eqn:lilnonre}) as
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{1}(t)}\in \big[\lambda_1^{1/\alpha},\lambda_2^{1/\alpha}\big],\qquad\text{a.s.}
\end{align*}
This shows that only the quality of the small deviation estimate (\ref{eqn:sdestmain}) matters in order to obtain the limiting constant in the LIL. Recall that the Blumenthal zero-one law implies that the limit is almost surely equal to a deterministic constant, which in this case can be specified.\medskip
Theorem~\ref{t3} reduces the question of the right norming function for the LIL to the question of small deviations which is known precisely for many examples.
For general L\'evy processes those have been obtained in \cite{AD09} (their results were stated for $t=1$ only but hold in general as we discuss in Proposition~\ref{prop:ad} below). In particular, for symmetric L\'evy processes their main result states that the rate function is given by
\begin{align}\label{a2}
F(\eps)=\eps^{-2}U(\eps),
\end{align}
where $U(\eps)$ is the variance of $X$ with jumps larger than $\eps$ replaced by jumps of size $\eps$:
\begin{align}\label{eqn:defnU}
U(\eps):=\eps^2\bar\Pi(\eps)+\sigma^2+\int_{-\eps}^{\eps}x^2\Pi(\d x).
\end{align}
From these specific small deviations we can deduce the following corollary for symmetric processes.
\begin{corollary} \label{cor:sddirectsymmetric}
Let $X$ be a symmetric L\'evy process, then there are $0<\lambda_1\leq \lambda_2<\infty$ such that
almost surely,
\begin{align*}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_1}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_2}(t)}\leq 1,
\end{align*}
with
\begin{align*}
b_\lambda(t):=F^{-1}\left(\frac{\log|\log t|}{\lambda t} \right)
\end{align*}
and $F$ defined in (\ref{a2}). If additionally $F$ is regularly varying at zero with exponent $-\alpha$, $\alpha>0$, then the following general bounds hold:
\begin{align*}
\frac{1}{12}\frac{1}{2^{\alpha}}\leq \lambda_1\leq \lambda_2\leq 3^{\alpha}10.
\end{align*}
\end{corollary}
The loss of constants in the corollary is only due to the general formulation. For some examples we will see below that the small deviations are known in the strong asymptotic sense so that Theorem~\ref{t3} gives the precise law.\medskip
For strongly non-symmetric L\'evy processes we have to proceed differently, since here condition (\ref{eqn:conditionM}) does not hold. For this case, we provide another, different link between small deviation rates and LIL; we keep track of the constants in the norming function in an optimal way and only lose the limiting constant.
\begin{theorem}\label{t}
Let $X$ be a L\'evy process and $F$ be a function increasing to infinity at zero such that for $0<\lambda_1\leq \lambda_2<\infty$
\begin{align} \label{eqn:yetanothersdestimate}
\lambda_1 F(\eps)t\leq -\log \P(||X||_{t}<\eps) \leq \lambda_2 F(\eps)t, \qquad \text{for all $\eps<\eps_0$ and $t<t_0$.}
\end{align}
Furthermore, set
\begin{align} \label{eqn:quantitythebee}
b_\lambda(t):=F^{-1}\left(\frac{\log|\log t|}{\lambda t} \right)
\end{align}
and suppose that there is a constant $C>0$ such that
\begin{align} \label{eqn:regularityofb}
C b_{\lambda}(t)\leq b_{\lambda}(t/2),\qquad 0<t\leq t_0, \lambda\in(\lambda_1/2,2\lambda_2).
\end{align}
Then the LIL
\begin{align*}
0<\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_1'}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_2'}(t)}<\infty
\end{align*}
hold almost surely for all $\lambda_1'<\lambda_1$ and $\lambda_2<\lambda_2'$.
\end{theorem}
Again, if the rate function $F$ is regularly varying, then we can strengthen the result.
\begin{corollary}\label{cor:t1corollary2}
In the setting of Theorem~\ref{t} assume additionally that $F$ is regularly varying at zero with negative exponent. Then the following LIL hold almost surely:
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{1}(t)} \in (0,\infty).
\end{align*}
\end{corollary}
The theorems listed so far manage the transfer between small deviation order and LIL. Similarly to Corollary~\ref{cor:sddirectsymmetric}, we can combine them with the main results of \cite{AD09}. This looks more technical in the present case. We give an explanation of the role of the different terms after the result.
\begin{theorem}\label{t2}
Let $X$ be a L\'evy process with triplet $(\gamma, \sigma^2, \Pi)$. Assume that $u_\eps$ is the solution of the equation $\Lambda_\eps'(u)=0$, where $\Lambda_\eps$ is the following log Laplace transform:
\begin{align} \label{eqn:quantitylambda}
\Lambda_\eps(u)=\frac{\sigma^2}{2}\, u^2 + \left(\gamma-\int_{[-1,1]\setminus [-\eps,\eps]} x \Pi(\d x)\right) u + \int_{-\eps}^{\eps} (e^{u x}-1 - u x) \Pi(\d x).
\end{align}
Set
\begin{align} \label{eqn:quantitythef}
F(\eps):=\eps^{-2} U_\eps(\eps)-\Lambda_\eps(u_\eps),\qquad U_\eps(\eps):=\eps^2\bar\Pi(\eps)+\sigma^2+\int_{-\eps}^{\eps}x^2 e^{-u_\eps x} \Pi(\d x),
\end{align}
define $b$ as in (\ref{eqn:quantitythebee}) and assume that $b$ satisfies (\ref{eqn:regularityofb}).
If furthermore
\begin{align} \label{eqn:cond-esschervanishes}
\eps |u_\eps| = o( \log \log F(\eps) ), \qquad \text{as $\eps\to 0$,}
\end{align}
is satisfied then we have for some $\lambda_1, \lambda_2>0$
\begin{align*}
0< \liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_1}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{\lambda_2}(t)}<\infty\qquad \text{a.s.}
\end{align*}
\end{theorem}
Let us explain the quantities occurring in Theorem~\ref{t2} in more detail. The main observation is that the proof for the small deviation estimates in \cite{AD09} (Theorem~1.5) can be used directly for any $t>0$ to obtain the following proposition.
\begin{proposition} \label{prop:ad}
Let $\Lambda_\eps$ be as defined in (\ref{eqn:quantitylambda}) and assume that $u_\eps$ is the solution of $\Lambda_\eps'(u_\eps)=0$. Then, with $F$ as in (\ref{eqn:quantitythef}), we have for all $t>0$ and all $\eps<1$
\begin{align} \label{eqn:sdquantityadstrengthend}
\frac{1}{12}\, t\, F(2\eps) - \eps |u_{2\eps}| -1 \leq - \log \P\left( ||X||_t \leq \eps \right) \leq 10 t\, F\left(\frac{\eps}{3}\right) + \eps |u_{\eps/3}|+3.
\end{align}
\end{proposition}
The term $\bar\Pi(\eps)$ in (\ref{eqn:sdquantityadstrengthend}) (included in the $F$ term) comes from the requirement that there should be no jumps larger than $\eps$. After removing these jumps, the process may drift out of the interval $[-\eps,\eps]$, which is prevented by applying an Esscher transform to the process, whose `price' is given by the term $-\Lambda_\eps(u_\eps)$. The quantity $u_\eps$ is the drift that has to be subtracted in order to make the process a martingale. Then the remaining process is treated as in the symmetric case, and the same term $\eps^{-2} U_\eps(\eps)$ appears as in (\ref{a2}), but this time with respect to the L\'evy measure transformed by the change of measure.
Note that (\ref{eqn:sdquantityadstrengthend}) is almost the required estimate in (\ref{eqn:yetanothersdestimate}), except for the term $\eps |u_\eps|$, which may spoil the estimate. It is exactly condition (\ref{eqn:cond-esschervanishes}) that ensures that the term $\eps |u_\eps|$ can be neglected.\\
We stress that in some cases $\eps |u_\eps|$ does give an order that is larger than $t F(\eps)$ so that the function $b$ from (\ref{eqn:quantitythebee}) is not the right norming function. This effect can be observed in some examples below. In particular, this happens for processes of bounded variation with non-zero drift.
\begin{proposition} \label{prop:bvnodrift}
Let $X$ be a L\'evy process with bounded variation and non-vanishing effective drift, i.e.\ $\int_{[-1,1]} |x| \Pi(\d x)< \infty$ and $c:=\gamma-\int_{-1}^1 x\,\Pi (\d x)\neq 0$. Then
\begin{align*}
\lim_{t\rightarrow 0}\frac{||X||_t}{t} =|c|\qquad\text{a.s.}
\end{align*}
\end{proposition}
The proof of this proposition is based on classical arguments rather than any connection to small deviations. \medskip
\section{Explicit LIL for L\'{e}vy processes}\label{sec:examples}
In this section we collect concrete L\'evy processes for which we can transform small deviation results to an LIL. As we have seen, understanding the small deviation rates is crucial. \medskip
The first corollary gives us a useful variance domination principle for LIL that works for many examples.
\begin{corollary}\label{cor:domination}
Suppose $X^1$ and $X^2$ are independent symmetric L\'evy processes, then $X^1+X^2$ and $X^2$ fulfill precisely the same LIL if
\begin{align*}
\lim_{\eps\to0}\frac{U_{X^1}(\eps)}{U_{X^2}(\eps)}=0.
\end{align*}
\end{corollary}
\begin{proof}
This follows directly from Corollary~\ref{cor:sddirectsymmetric} noticing that $U_{X^1+X^2}=U_{X^1}+U_{X^2}$.
\end{proof}
In the same spirit the following corollary (recovering (3.2) in \cite{BM09}) displays the intuitive fact that a non-zero Brownian component dominates the jumps of a L\'evy process.
\begin{corollary}\label{cor:withbrownian}
If $X$ is a L\'evy process with $\sigma\neq 0$, then
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_t}{\sqrt{t/\log| \log t|}}=\frac{\pi\sigma}{\sqrt{8}}\qquad\text{ a.s.}
\end{align*}
\end{corollary}
\begin{proof}
Following precisely the proof of Corollary~2.6 of \cite{AD09} one can show that the small deviation rates of L\'evy processes with non-zero Brownian component is given by
\begin{align*}
-\log \P(||X||_{t}<\eps)\sim \frac{\pi^2\sigma^2}{8}\eps^{-2}t, \qquad\text{as $\eps\to 0$ and $t\to 0$.}
\end{align*}
Hence, the norming function follows from Theorem~\ref{t3}. As the process is not necessarily symmetric, condition (\ref{eqn:conditionM}) has to be checked: Since $b(t)=\sqrt{t\pi^2/ (8\log |\log t|)}$ and $\int_{|x|>\eps} |x| \Pi(\d x)=o(\eps^{-1})$, it remains to be seen that
$$ a_{n+1} \leq c b(a_n)^2= a_n / \log |\log a_n|$$ for $a_n=n^{-n^\beta}$ and $\beta>1$. This can be verified by simple computations.
\end{proof}
Similarly to L\'evy processes with non-zero Brownian component, symmetric processes of smaller small deviation order (e.g.\ stable processes of smaller index) are dominated by stable L\'evy processes.
\begin{corollary}\label{cor:stables}
Let $X$ be a symmetric $\alpha$-stable L\'evy process with $\alpha\in (0,2]$ and let $Y$ be symmetric with $U_Y(x)=o(x^{2-\alpha})$. Then there is a constant $0<c_\alpha<\infty$ such that
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X+Y||_t}{(t/\log|\log t|)^{1/\alpha}}=\liminf_{t\rightarrow 0}\frac{||X||_t}{(t/\log|\log t|)^{1/\alpha}}=c_\alpha^{1/\alpha}\qquad\text{a.s.}
\end{align*}
\end{corollary}
\begin{proof}
The small deviation rate is given by
\begin{align*}
-\log \P(||X||_{t}<\eps)\sim c_{\alpha} \eps^{-\alpha}t, \qquad\text{as $\eps\to 0$ and $t\to 0$},
\end{align*}
for some constant $c_{\alpha}>0$ (see e.g.\ page 220 in \cite{B96}). Hence, the LIL follow from Corollary~\ref{cor:t1corollary} and Corollary~\ref{cor:domination}.
\end{proof}
\begin{rem}
The constant $c_\alpha$ in the LIL of stable L\'evy processes is the unknown constant of the small deviations for respective $\alpha$-stable L\'evy processes (see \cite{taylor} and Proposition~3 and Theorem~6 in Chapter VIII of \cite{B96}). The results of \cite{AD09} entail the following concrete bounds:
\begin{align*}
\frac{2 C }{2^{\alpha}} \left( \frac{1}{\alpha} + \frac{1}{12(2-\alpha)}\right) < c_\alpha < 3^\alpha\cdot 2 C \left( \frac{1}{\alpha} + \frac{10}{2-\alpha}\right),
\end{align*}
where $C$ is the constant in the L\'evy measure: $\Pi(\d x)=C |x|^{-(1+\alpha)}\d x$. This implies $c_\alpha\sim 2 C/\alpha$, as $\alpha\to 0$. We remark that, contrary to the symmetric case, the constant $c_\alpha$ is known explicitly for completely asymmetric stable L\'evy processes, see \cite{bertoin96}.
\end{rem}
\medskip
If $\Pi$ behaves as a regularly varying function at zero and is symmetric the following LIL are satisfied.
\begin{corollary}
Let $X$ be a L\'evy process with triplet $(0,0,\Pi)$ with $\Pi$ being symmetric and
$$\bar\Pi(\eps)\approx \eps^{-\alpha} |\log\eps|^{-\gamma},\qquad\text{as $\eps\to 0$,}$$
with $0<\alpha<2$ or $\alpha=2, \gamma>1$. Then
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_{t}}{b(t)}\in(0,\infty)\qquad{a.s.}
\end{align*}
with
\begin{align*}
b(t)=\begin{cases}
\left(\frac{t|\log t|^{-\gamma}}{\log|\log t|}\right)^{1/\alpha}&:0<\alpha<2,\\
\left(\frac{t|\log t|^{1-\gamma}}{\log|\log t|}\right)^{1/2}&:\alpha=2, \gamma>1.
\end{cases}
\end{align*}
\end{corollary}
\begin{proof}
The corollary follows from Theorem~\ref{t2}. The required small deviation estimate,
\begin{align*}
-\log \P(||X||_{t}<\eps)\approx \begin{cases}
\eps^{-\alpha}|\log \eps|^{-\gamma} t&:0<\alpha<2,\\
\eps^{-2}|\log \eps|^{1-\gamma}t&:\alpha=2, \gamma>1,
\end{cases}
\end{align*}
as $\eps\to 0$ and $t\to 0$, is obtained from Proposition~\ref{prop:ad} (cf.\ Example~2.2 in \cite{AD09} for $t=1$). Since we deal with a symmetric process, condition (\ref{eqn:cond-esschervanishes}) is trivially satisfied due to $u_{\eps}=0$.
\end{proof}
Having discussed the $\alpha$-stable like cases, we now consider L\'evy processes with polynomial tails near zero of {\it different} exponents. The technique used for this example can be extended to any case with essentially regularly varying L\'{e}vy measure at zero. Let $X$ be a L\'evy process with triplet $(\gamma,0,\Pi)$, where $\Pi$ is given by \begin{equation} \frac{\Pi(\d x)}{\d x} = \frac{C_1 \ind_{(0,1]}(x)}{x^{1+\alpha_1}} + \frac{C_2 \ind_{[-1,0)}(x)}{(-x)^{1+\alpha_2}}, \label{eq:regularlm} \end{equation} with $2>\alpha_1\geq\alpha_2$ and $C_1,C_2\geq 0$, $C_1+C_2\neq 0$. We now analyze the pathwise behavior at zero in the cases when $\alpha_1>1$, $\alpha_1=1$, and $0<\alpha_1<1$, respectively. The second exponent $\alpha_2$ can be even negative.
\begin{corollary}\label{pol} Let $X$ be a L\'evy process with triplet $(\gamma,0,\Pi)$ with $\Pi$ as in (\ref{eq:regularlm}). Then the following holds:
\begin{enumerate}
\item If $\alpha_1\geq \alpha_2$ and $\alpha_1>1$, then
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_t}{(t / \log |\log t|)^{1/\alpha_1}}\in (0,\infty)\qquad\text{a.s.}
\end{align*}
\item If $\alpha_1=\alpha_2=1$ and $C_1=C_2$ then
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_t}{t / \log |\log t|}\in (0,\infty)\qquad\text{a.s.}
\end{align*}
\item If $1>\alpha_1\geq \alpha_2$ and the effective drift $c=\gamma-\int_{-1}^1 x\,\Pi(\d x)$ vanishes, then
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_t}{(t / \log |\log t|)^{1/\alpha_1}}\in (0,\infty)\qquad\text{a.s.}
\end{align*}
\item If $1>\alpha_1\geq \alpha_2$ and the effective drift does not vanish, then
\begin{align*}
\lim_{t\rightarrow 0}\frac{||X||_t}{t} = |c|\qquad\text{a.s.}
\end{align*}
\end{enumerate}
\end{corollary}
\begin{proof}
Parts (1), (2), and (3) follow from Theorem~\ref{t2}. The required small deviation estimates,
\begin{align*}
-\log \P(||X||_{t}<\eps)\approx \eps^{-\alpha_1}t
\end{align*}
for $\eps\to 0$ and $t\to 0$, are obtained from Proposition~\ref{prop:ad} (cf.\ Corollary~2.7,~2.8, and~2.9 of \cite{AD09} for $t=1$; note that $u_\eps\approx \eps^{-1}$ in all cases). One can easily check condition (\ref{eqn:cond-esschervanishes}).\\
In part (4), the process is of bounded variation, so that the claim is included in Proposition~\ref{prop:bvnodrift}.
\end{proof}
\medskip
We now come to L\'evy processes obtained from Brownian motion by subordination, i.e.\ $X_t=\sigma B_{A_t}$, where $B$ is a Brownian motion independent of the subordinator $A$. In this case, the resulting L\'evy process is symmetric and the small deviation asymptotics is governed by the truncated variance $U$ from (\ref{eqn:defnU}).
\begin{corollary}
Let $B$ be a Brownian motion independent of the subordinator $A$, where $A$ has Laplace exponent $\Phi$. For $\lambda>0$ we set $b_\lambda(t):=F^{-1}\left(\frac{\log|\log t|}{\lambda t} \right)$ with
\begin{align*}
F(\eps):= \Phi(\sigma^2\eps^{-2}) + \gamma_A \sigma^2 \eps^{-2}.
\end{align*}
Then for some $\lambda_1, \lambda_2>0$
\begin{align*}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_t}{b_{\lambda_1}(t)}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_t}{b_{\lambda_2}(t)}\leq 1\qquad\text{a.s.}
\end{align*}
In particular, if $\gamma_A=0$ and $\Phi$ is regularly varying with positive exponent, we have
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_t}{(\Phi^{-1}\left(\log | \log t| / t\right))^{-1/2}}\in(0,\infty)\qquad\text{a.s.}
\end{align*}
\end{corollary}
\begin{proof}
The corollary follows from Theorem~\ref{t3} with small deviation estimate from Proposition~\ref{prop:ad}
$$
-\log \P( ||X||_t \leq \eps) \approx (\Phi(\sigma^2\eps^{-2}) + \gamma_A \sigma^2 \eps^{-2})t,
$$
as $\eps\to 0$ and $t\to 0$ (cf.\ Example~2.13 of \cite{AD09} for $t=1$ and note the misprint there). Condition (\ref{eqn:conditionM}) is trivially fulfilled as the process is symmetric.
\end{proof}
For a more specific example, in particular exhibiting exotic small time behavior, we choose the subordinator $A$ to be a Gamma process. Then one defines the so called Variance-Gamma process as
\begin{align*}
X_t=\sigma B_{A_t}+\mu A_t,
\end{align*}
for some constants $\sigma\neq 0$ and $\mu\in\R$.
\begin{corollary
Let $X$ be a Variance-Gamma process, then for $\mu=0$ there are some constants $0<\lambda_1\leq \lambda_2<\infty$ such that
\begin{align} \label{eqn:constinexp}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_t}{ e^{ - \lambda_1 \log |\log t|/t}}\qquad\text{and}\qquad\liminf_{t\rightarrow 0}\frac{||X||_t}{ e^{- \lambda_2 \log |\log t|/t}}\leq 1\qquad\text{ a.s.},
\end{align}
whereas for $\mu\neq 0$
\begin{align*}
\liminf_{t\rightarrow 0}\frac{||X||_t}{t}=|\mu|\,\E A_1\qquad\text{ a.s.}
\end{align*}
\end{corollary}
\begin{proof}
The second part is included in Proposition~\ref{prop:bvnodrift}, since the process is of bounded variation with non-zero effective drift. In the first part, the effective drift is zero, and the claim follows from Theorem~\ref{t3}. The small deviation estimate,
\begin{align*}
-\log \P\left( ||X||_t \leq \eps\right) \approx t |\log \eps|, \qquad \text{as $\eps\to 0$ and $t\to 0$}
\end{align*}
follows from Proposition~\ref{prop:ad} (cf.\ Example~2.12 of \cite{AD09} for $t=1$).
\end{proof}
In the first case of the previous corollary the dependence of good small deviation estimates and good LIL becomes transperant. The fact that we cannot specify the constants $\lambda_1, \lambda_2$ in (\ref{eqn:constinexp}) is only caused by the weak asymptotics for the small deviation estimate as we do not lose any further constants in the transfer of small deviations to the LIL. If one does not have more control on the constants $\lambda_1, \lambda_2$, the understanding of the precise small time behavior of $X$ is far from optimal as the error enters exponentially.
\section{Proofs} \label{sec:proofs}
We start with a lemma which shows that the small deviation order is at least as large as the term induced by the variance, defined in (\ref{eqn:defnU}).
\begin{lemma}
Let $\eps>0$ and let $X$ be a L\'evy process with L\'evy measure concentrated on $[-\eps,\eps]$, then
\begin{align*}
\P( ||X||_t \leq \eps/2 )\leq e^{- \eps^{-2} \left(\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2\right) t/12+1}, \qquad\text{for $t\geq 0$.}
\end{align*}
\end{lemma}
\begin{proof}
We proceed similarly to Lemma~4.2 in~\cite{AD09}. Let $\tau$ be the first exit time of $X$ out of $[-\eps,\eps]$. Then, by Wald's identity,
\begin{align*}
4 \eps^2 & \geq \limsup_{t\to\infty} \E[ X^2_{t\wedge \tau}] \geq \limsup_{t\to\infty} {\rm var}[ X_{t\wedge \tau}] \\ &= \limsup_{t\to\infty} \left(\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2\right) \E[ t\wedge \tau] = \left(\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2\right) \E [ \tau].
\end{align*}
Therefore,
$$
\P\left(\tau\geq 8 \eps^2 / \left(\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2\right)\right) \leq \frac{\left(\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2\right) \E [\tau]}{8 \eps^2} \leq \frac12.
$$
Let $n:=\lfloor t(\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2)/(8 \eps^2) \rfloor$ and set $t_i := 8i \eps^2 / (\int_{-\eps}^\eps x^2 \Pi(\d x) + \sigma^2)$, $i=0,\ldots, n$. Then
$$
\P( ||X||_t \leq \eps/2) \leq \P\left( \forall i=0, \ldots, n-1 : \sup_{s\in[t_i,t_{i+1})} |X_s -X_{t_i}|\leq \eps\right) = \P( \tau \geq t_1)^n \leq 2^{-n}.
$$
\end{proof}
This shows that the small deviation order is always at least as large as the term induced by the truncated variance process. This fact will be needed later on.
\begin{lemma}\label{lem:flargeru}
Let $F$ be a function that increases to infinity at zero. If for some L\'evy process $X$ for $t\leq t_0$ and $\eps<\eps_0$
\begin{align*}
-\log \P( ||X||_t \leq \eps) \leq F(\eps) t
\end{align*}
then, for some absolute constant $c>0$ and all $\eps>0$ small enough,
\begin{align*}
\eps^{-2} U(\eps) \leq c (F(\eps)+1).
\end{align*}
\end{lemma}
\begin{proof}
We use the assumption together with the fact that if $||X||_t\leq \eps$ then $X$ must not have jumps larger than $2\eps$ and the previous lemma:
\begin{align*}
e^{-F(\eps)t} \leq \P(||X||_t\leq \eps) = e^{-\bar\Pi(2\eps) t} \P(||X'||_t\leq \eps)
\leq e^{-\bar\Pi(2\eps) t} e^{- (2\eps)^{-2} \left(\int_{-2\eps}^{2\eps} x^2 \Pi(\d x) + \sigma^2\right) t/12+1},
\end{align*}
where $X'$ has L\'evy measure $\Pi$ restricted to $[-2\eps,2\eps]$. Noting that Lemma~5.1 of \cite{AD09} implies that $U(\eps)/\eps^2 \approx U(2\eps)/(2\eps)^2$, the statement of the lemma is proved.
\end{proof}
The lower bound in the LIL comes from the following lemma.
\begin{lemma}\label{lem:lower}
Let $F$ be a function that increases to infinity at zero such that for all $t\leq t_0$ and $\eps\leq \eps_0$
\begin{align*}
\lambda F(\eps)t \leq -\log \P(||X||_t\leq \eps)
\end{align*}
and, for $\lambda>0$, we set $b_\lambda(t):=F^{-1}\left(\frac{\log|\log t|}{\lambda t} \right)$. Then, for any $\lambda'<\lambda$,
\begin{align*}
1\leq \liminf_{t\rightarrow 0}\frac{||X||_t}{b_{\lambda'}(t)}\qquad \text{a.s.}
\end{align*}
\end{lemma}
\begin{proof}
For any $\lambda'<\lambda$, we can find $0<r<1$ such that $1< \lambda r/\lambda'$. Note that
\begin{align*}
\sum_{n}\P\big(||X||_{r^{n+1}}\leq b_{\lambda'}(r^n)\big)<\infty
\end{align*}
since
\begin{align}\label{est}
-\log \P\big(||X||_{r^{n+1}}\leq b_{\lambda'}(r^n)\big)\geq \lambda F( b_{\lambda'}(r^n)) r^{n} r = \lambda \frac{r}{\lambda'} \log |\log r^n| = \log n^{r \lambda/\lambda'} + {\rm const.}
\end{align}
Hence, by the Borel-Cantelli lemma,
\begin{align*}
\big\{n ~:~||X||_{r^{n+1}}\leq b_{\lambda'}(r^{n})\big\}
\end{align*}
is almost surely a finite set. Thus, for each path $\omega$, we have that for any $n\geq n_{0}(\omega)$ and any $t\in[r^{n+1},r^{n})$
\begin{align*}
&\frac{||X||_{t}}{b_{\lambda'}(t)}\geq\frac{||X||_{r^{n+1}}}{b_{\lambda'}(r^{n})}\geq 1,
\end{align*}
as $b_{\lambda'}$ is an increasing function. We take $\liminf_{t\to 0}$ to obtain the statement.
\end{proof}
The proof of the upper bound in the LIL requires the following lemma.
\begin{lemma}\label{L2}
Let $F$ be a function that increases to infinity at zero such that for all $t\leq t_0$ and $\eps\leq \eps_0$
\begin{align*}
-\log \P(||X||_t\leq \eps) \leq \lambda F(\eps) t
\end{align*}
and, for $\lambda>0$, set $b_\lambda(t):=F^{-1}\left(\frac{\log|\log t|}{\lambda t} \right)$. Assume that
\begin{align}\label{mladen}
\limsup_{n\to\infty}\frac{|X_{(n+1)^{-(n+1)^{\beta}}}|}{b_{\lambda}\big(n^{-n^{\beta}}\big)}=0\qquad \text{ a.s.},
\end{align}
for all $\beta>1$. Then, for any $\lambda'>\lambda$,
\begin{align} \label{eqn:upper1ml}
\liminf_{t\to 0}\frac{||X||_{t}}{b_{\lambda'}(t)}\leq 1\qquad\text{a.s.}
\end{align}
\end{lemma}
\begin{proof}
For $\lambda'>\lambda$, we choose $\beta>1$ such that $\lambda'>\lambda \beta$. First note that (\ref{mladen}) implies
\begin{align} \label{eqn:suparg1}
\limsup_{n\to\infty}\frac{||X||_{(n+1)^{-(n+1)^{\beta}}}}{b_{\lambda'}\big(n^{-n^{\beta}}\big)}=0\quad\text{ a.s.},
\end{align}
as $b_{\lambda}(t)$ is an increasing function in $\lambda$ for fixed $t\geq 0$.
Using the L\'evy property we see the following:
\begin{align*}
&\,\,\,\,\,\,\,\sum_{n}\P\Big(\sup_{(n+1)^{-(n+1)^{\beta}}\leq t < n^{-n^{\beta}}}|X_{t}-X_{(n+1)^{-(n+1)^{\beta}}}|\leq b_{\lambda'}\big(n^{-n^{\beta}}\big)\Big)\\
&=\sum_{n}\P\big(||X||_{n^{-n^{\beta}}-(n+1)^{-(n+1)^{\beta}}}\leq b_{\lambda'}\big(n^{-n^{\beta}}\big)\big)\\
&\geq \sum_{n}\P\big(||X||_{n^{-n^{\beta}}}\leq b_{\lambda'}\big(n^{-n^{\beta}}\big)\big)=\infty.
\end{align*}
The last step follows as in (\ref{est}) since now $\lambda \beta/\lambda'<1$. The Borel-Cantelli lemma shows that the sequence of independent events
\begin{align*}
A_{n}=\Big\{\sup_{(n+1)^{-(n+1)^{\beta}}\leq t < n^{-n^{\beta}}}|X_{t}-X_{(n+1)^{-(n+1)^{\beta}}}|\leq b_{\lambda'}\big(n^{-n^{\beta}}\big)\Big\}
\end{align*}
satisfies $\P(A_{n}\,\,\text{i.o.})=1$. To reduce to the supremum note that
\begin{align*}
\frac{||X||_{n^{-n^{\beta}}}}{b_{\lambda'}(n^{-n^{\beta}})}\leq \frac{\sup_{(n+1)^{-(n+1)^{\beta}}\leq t < n^{-n^{\beta}}}|X_{t}-X_{(n+1)^{-(n+1)^{\beta}}}|}{b_{\lambda'}(n^{-n^{\beta}})}+\frac{2||X||_{(n+1)^{-(n+1)^{\beta}}}}{b_{\lambda'}(n^{-n^{\beta}})}
\end{align*}
and therefore by (\ref{eqn:suparg1})
\begin{align*}
\liminf_{n\to\infty}\frac{||X||_{n^{-n^{\beta}}}}{b_{\lambda'}(n^{-n^{\beta}})}\leq\liminf_{n\to\infty}\frac{\sup_{(n+1)^{-(n+1)^{\beta}}\leq t < n^{-n^{\beta}}}|X_{t}-X_{n^{-(n+1)^{\beta}}}|}{b_{\lambda'}(n^{-n^{\beta}})}\leq 1.
\end{align*}
This shows (\ref{eqn:upper1ml}).
\end{proof}
Now we are in position to prove Theorem~\ref{t3}. For a detailed analysis of the $\limsup$ case, we refer to~\cite{S09}.
\begin{proof}[Proof of Theorem~\ref{t3}:]
The claim follows from Lemmas~\ref{lem:lower} and~\ref{L2}. To verify the use of Lemma~\ref{L2} we still need to check that condition (\ref{mladen}) holds for all $\beta>1$.
We fix $\beta>1$ and $\lambda'_2>\lambda_2$. Since $\lambda'_2$ is fixed, we suppress the subscript $\lambda'_2$ in the definition of $b$ in order to increase readability. We define the auxiliary function
\begin{align*
h(t)=b(\phi(t)),
\end{align*}
where $\phi(t)$ is chosen such that $\phi\big((\frac{t}{t+1})^{(\frac{t+1}{t})^{\beta}}\big)=t^{\frac{1}{t^{\beta}}}$ and $\phi(0)=0$. Note that $\phi$ is increasing and that $\phi(s^{-s^\beta})=(s-1)^{-(s-1)^\beta}$. We also do not mark that $\phi$ and $h$ depend on $\beta$ and $\lambda_2'$.
{\it Step 1:} We show that
\begin{align}\label{int}
\int_0^{1/2} \bar \Pi( h(t))\,\d t<\infty.
\end{align}
First, by the definition of $h$ and a change of variables we obtain
\begin{align*}
&\,\,\,\,\,\,\,\int_0^{1/2} \bar \Pi( h(t))\,\d t\\
&=\int_0^{ C(\beta)} \bar \Pi\big( b\big(s^{s^{-\beta}}\big)\big)\frac{d\big(\frac{s}{s+1}\big)^{(\frac{s+1}{s})^{\beta}}}{\d s}\\
&=\int_0^{ C(\beta)} \bar \Pi\big( b\big(s^{s^{-\beta}}\big)\big)\big(\frac{s}{s+1}\big)^{(\frac{s+1}{s})^{\beta}}(\frac{s+1}{s})^{\beta-1}s^{-2}(1-\beta\log{(1-(s+1)^{-1}}))\,\d s,
\end{align*}
which can be estimated from above by
\begin{align*}
&\,\,\,\,\,\,\,\,C\int_0^{ C(\beta)} \frac{b^{2}(s^{s^{-\beta}})\bar\Pi( b(s^{s^{-\beta}}))}{b^{2}(s^{s^{-\beta}})}\Big(\frac{s}{s+1}\Big)^{(\frac{s+1}{s})^{\beta}}s^{-1-\beta} |\log s|\d s\\
&\leq C\int_0^{ C(\beta)} \frac{U( b(s^{s^{-\beta}}))}{b^{2}(s^{s^{-\beta}})}\Big(\frac{s}{s+1}\Big)^{(\frac{s+1}{s})^{\beta}}s^{-1-\beta} |\log s| \d s\\
&\leq C'\int_0^{ C(\beta)} F( b(s^{s^{-\beta}}))\Big(\frac{s}{s+1}\Big)^{(\frac{s+1}{s})^{\beta}}s^{-1-\beta}|\log s| \d s\\
&=\frac{C'}{\lambda}\int_0^{ C(\beta)}\frac{\log\Big|\log s^{s^{-\beta}}\Big|}{s^{s^{-\beta}}}\Big(\frac{s}{s+1}\Big)^{(\frac{s+1}{s})^{\beta}}s^{-1-\beta}|\log s| \d s\\
&\leq \frac{C'}{\lambda}\int_0^{ C(\beta)}s^{-1-\beta} \left( \log\Big|\log s^{s^{-\beta}}\Big| \right) s^{(\frac{s+1}{s})^{\beta}-\frac{1}{s^{\beta}}} |\log s| \d s<\infty,
\end{align*}
where we have used $x^{2}\bar\Pi(x)\leq x^{2}\bar\Pi(x)+\int_{-x}^{x}y^{2}\Pi(\d y) + \sigma^2=U(x)\leq c x^2F(x)$ for some absolute $c>0$ by Lemma~\ref{lem:flargeru} and the definition of $b$.
{\it Step 2:} We denote by
\begin{align} \label{eqn:defnAn}
A_{n}:=\Big\{\text{ there are jumps with modulus larger than $b(n^{-n^{\beta}})$ up to time $(n+1)^{-(n+1)^{\beta}}$}\Big\}
\end{align}
and show that
\begin{align}\label{ba}
\sum_n \P\big(A_{n}\big)<\infty.
\end{align}
This comes from (\ref{int}). Indeed, note that $h$ inherits the monotonicity of $b$ and $\phi$ and hence (\ref{int}) implies that
\begin{align} \label{eqn:nozerosu}
&\sum_{n} \big((n+1)^{-(n+1)^{\beta}}-(n+2)^{-(n+2)^{\beta}}\big) \bar\Pi\big( h\big((n+1)^{-(n+1)^{\beta}}\big)\big)\leq\sum_{n}\int_{(n+2)^{-(n+2)^{\beta}}}^{(n+1)^{-(n+1)^{\beta}}}\bar\Pi(h(t))\d t<\infty.
\end{align}
Using
\begin{align*}
(n+1)^{-(n+1)^{\beta}}-(n+2)^{-(n+2)^{\beta}}&\sim(n+1)^{-(n+1)^{\beta}},\\
b(n^{-n^{\beta}})&=h((n+1)^{-(n+1)^{\beta}}),
\end{align*}
and that the sequence $(n+1)^{-(n+1)^{\beta}}\bar\Pi( h((n+1)^{-(n+1)^{\beta}}))$ tends to zero by (\ref{eqn:nozerosu}), we obtain that
\begin{align*}
\P\big(A_{n}\big)=1-e^{-(n+1)^{-(n+1)^{\beta}}\bar\Pi( b(n^{-n^{\beta}}))}&\sim (n+1)^{-(n+1)^{\beta}}\bar\Pi\big(h\big((n+1)^{-(n+1)^{\beta}}\big)\big)
\end{align*}
is summable. Therefore (\ref{ba}) is proved.
{\it Step 3:} Let us now show how to use (\ref{ba}) to deduce (\ref{mladen}). Apparently, it suffices to show that
\begin{align*}
\limsup_{n\rightarrow \infty}\frac{|X_{(n+1)^{-(n+1)^{\beta}}}|}{b(n^{-n^{\beta}})}<\eps\,\,\,\text{a.s.},
\end{align*}
for any $\eps>0$ and, hence, by the Borel-Cantelli lemma it suffices to show that
\begin{align*}
\sum_n \P\big({|X_{(n+1)^{-(n+1)^{\beta}}}|}>\eps{b\big(n^{-n^{\beta}}\big)}\big)<\infty.
\end{align*}
Separating jumps of absolute value larger or smaller than $ b\big(n^{-n^{\beta}}\big)$ and using the definition of $A_n$ in (\ref{eqn:defnAn}), we obtain that
\begin{align*}
&\,\,\,\,\,\,\,\sum_n \P\big({|X_{(n+1)^{-(n+1)^{\beta}}}|}>\eps{b(n^{-n^{\beta}})}\big)\\
&=\sum_n \P\big({|X_{(n+1)^{-(n+1)^{\beta}}}|}>\eps{b(n^{-n^{\beta}})}\,;\,A^{c}_{n}\big) + \sum_n \P\big({|X_{(n+1)^{-(n+1)^{\beta}}}|}>\eps{b(n^{-n^{\beta}})}\,;\, A_{n}\big),
\end{align*}
which is bounded from above by
\begin{align*}
&\sum_n \P\left(\left.{|X_{(n+1)^{-(n+1)^{\beta}}}|}>\eps{b(n^{-n^{\beta}})}\,\right|A^{c}_{n}\right)\cdot \P\big(A^{c}_{n}\big)+\sum_n \P\big(A_{n}\big).
\end{align*}
The second term is finite by (\ref{ba}); and the first term is bounded by
\begin{align} \label{eqn:gaptoolarge}
\sum_n \P\left(\left.\left|X_{(n+1)^{-(n+1)^{\beta}}}\right|>\eps b(n^{-n^{\beta}})\,\right|A^{c}_{n}\right).
\end{align}
To estimate this sum note that conditionally on $A^{c}_{n}$, $X_t\stackrel{d}=X_{t}(n)$, where $X(n)$ differs from $X$ only by removing jumps of size larger than $|b(n^{-n^{\beta}})|$. Clearly, by Wald's identity,
\begin{align*}
{\rm var}(X_{t}(n))=t\left(\int_{-b(n^{-n^{\beta}})}^{b(n^{-n^{\beta}})}y^{2}\Pi(\d y)+\sigma^{2}\right) \leq t U(b(n^{-n^{\beta}})).
\end{align*}
Note that
\begin{align*}
\E X_{(n+1)^{-(n+1)^{\beta}}}(n) = (n+1)^{-(n+1)^{\beta}} \left| \int_{|x|>b(n^{-n^\beta})} x \Pi(\d x) - \gamma\right|.
\end{align*}
Therefore, by assumption (\ref{eqn:conditionM}),
\begin{align*}
\E X_{(n+1)^{-(n+1)^{\beta}}}(n) = o( b(n^{-n^{\beta}}) ).
\end{align*}
Using this (first step), Chebychev's inequality (second step), Lemma~\ref{lem:flargeru} (third step), and the definition of $b$ (fourth step), we are led to the upper bound of the term in (\ref{eqn:gaptoolarge}):
\begin{align*}
&\sum_n \P\left({|X_{(n+1)^{-(n+1)^{\beta}}}(n)|}> \eps{b(n^{-n^{\beta}})}\right)\\
&\leq \sum_n \P\left({|X_{(n+1)^{-(n+1)^{\beta}}}(n) - \E X_{(n+1)^{-(n+1)^{\beta}}}(n)|}> \frac{1}{2} \,\eps{b(n^{-n^{\beta}})}\right)\\
&\leq \sum_n\frac{{(n+1)^{-(n+1)^{\beta}}}U\big(b(n^{-n^{\beta}})\big)}{(\eps/2)^2b(n^{-n^{\beta}})^2}\\
&\leq \sum_n\frac{{(n+1)^{-(n+1)^{\beta}}}C\cdot F\big(b(n^{-n^{\beta}})\big)}{(\eps/2)^2}\\
&=\frac{C'}{\lambda \eps^2}\sum_n\frac{{(n+1)^{-(n+1)^{\beta}}}\log |\log n^{-n^{\beta}}|}{n^{-n^{\beta}}}<\infty,
\end{align*}
where we used the definition of $b$ in the last step. Thus, the term in (\ref{eqn:gaptoolarge}) is finite, as required.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:t1corollary}]
If $F$ is regularly varying so is $b_{\lambda}$, see \cite{bgt}, Proposition~1.5.7. Now note that if $F$ is regularly varying with exponent $-\alpha<0$, we have
\begin{align*}
b_{\lambda}(t)&=F^{-1}(\log|\log t|/\lambda t)\\
&\sim \lambda^{{1/\alpha}} F^{-1}(\log|\log t|/ t)\\
&=\lambda^{{1/\alpha}}b_1(t).
\end{align*}
Hence, the statement of Theorem~\ref{t3} reads
\begin{align*}
(\lambda'_1)^{{1/\alpha}}\leq \liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{1}(t)}\leq (\lambda'_2)^{{1/\alpha}}\qquad\text{ a.s.}
\end{align*}
for all $\lambda_1'<\lambda_1$ and $\lambda_2'>\lambda_2$. Taking the limits on both sides we obtain
\begin{align*}
(\lambda_1)^{{1/\alpha}}\leq \liminf_{t\rightarrow 0}\frac{||X||_{t}}{b_{1}(t)}\leq (\lambda_2)^{{1/\alpha}}\qquad\text{ a.s.}
\end{align*}
Applying the regular variation argument in the reverse direction yields the claim.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:sddirectsymmetric}]
This follows directly from Theorem~\ref{t3}. The bounds on the constants can be obtained from the absolute constants in Proposition~\ref{prop:ad}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t}]
Lemma~\ref{lem:lower} gives the lower LIL of the theorem. Unfortunately, the arguments for the proof of Theorem~\ref{t3} do not apply here. Hence, for the reverse direction we show more directly that the given norming function of the LIL implies the rate function of the small deviations. The following arguments go back to Kesten. The proof is via contradiction assuming that
\begin{align}\label{eqn:ass}
\liminf_{t\to 0}\frac{||X||_{t}}{b_{\lambda_2'}(t)}>\frac{2}{C}+\delta
\end{align}
for some $\delta>0$ and $\lambda_2'>\lambda_2$. We show that under this assumption we can derive the estimates
\begin{align}
1&\geq\sum_{n\geq l}\P\bigg(\frac{||X||_{r^{j}-r^{n}}}{b_{\lambda_2'}(r^{j}-r^{n})}>\frac{2}{C};\text{for all $l\leq j\leq n-1$}\bigg)\P\big(||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big)\label{eqn:schonwiedera}\\
&\geq\frac{1}{2}\sum_{n\geq l}\P\big(||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big)\label{eqn:schonwiederb}
\end{align}
which is a contradiction as, by the choice of $b_{\lambda_2'}$ and the small deviation rate (\ref{eqn:yetanothersdestimate}), the sum in (\ref{eqn:schonwiederb}) is infinite. First, let us derive estimate (\ref{eqn:schonwiedera}) for which Assumption (\ref{eqn:ass}) is not needed. For any fixed integer $l$ partitioning the probability space we obtain
\begin{align*}
1&\geq \sum_{n\geq l}\P\big(||X||_{r^{j}}>b_{\lambda_2'}(r^{j})\text{ for all $l\leq j\leq n-1$};||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big)\\
&\geq\sum_{n\geq l}\P\Big(\sup_{r^{n}\leq s<r^{j}}|X_{s}|>b_{\lambda_2'}(r^{j})\text{ for all $l\leq j\leq n-1$};||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\Big).
\end{align*}
In order to employ the independence of increments of $X$ we estimate from below by
\begin{align*}
\sum_{n\geq l}\P\Big(\sup_{r^{n}\leq s<r^{j}}|X_{s}-X_{r^{n}}|>2 b_{\lambda_2'}(r^{j})\text{ for all $l\leq j\leq n-1$};||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\Big)
\end{align*}
which equals
\begin{align*}
&\,\,\,\,\,\,\,\sum_{n\geq l}\P\big(||X||_{r^{j}-r^{n}}>2 b_{\lambda_2'}(r^{j})\text{ for all $l\leq j\leq n-1$}\big)\P\big(||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big)\\
&=\sum_{n\geq l}\P\Bigg(\frac{||X||_{r^{j}-r^{n}}}{b_{\lambda_2'}(r^{j}-r^{n})}>2 \frac{b_{\lambda_2'}(r^{j})}{b_{\lambda_2'}(r^{j}-r^{n})}\text{ for all $l\leq j\leq n-1$}\Bigg)\P\big(||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big).
\end{align*}
By the monotonicity of $b_{\lambda_2'}$ this yields the lower bound
\begin{align*}
\sum_{n\geq l}\P\bigg(\frac{||X||_{r^{j}-r^{n}}}{b_{\lambda_2'}(r^{j}-r^{n})}>2 \frac{b_{\lambda_2'}(r^{j})}{b_{\lambda_2'}(r^{j}-r^{j+1})};\text{for all $l\leq j\leq n-1$}\bigg)\P\big(||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big).
\end{align*}
Finally, we utilize the regularity of $b_{\lambda_2'}$ from (\ref{eqn:regularityofb}) to obtain the lower bound
\begin{align*}
\sum_{n\geq l}\P\bigg(\frac{||X||_{r^{j}-r^{n}}}{b_{\lambda_2'}(r^{j}-r^{n})}>\frac{2}{C};\text{for all $l\leq j\leq n-1$}\bigg)\P\big(||X||_{r^{n}}\leq b_{\lambda_2'}(r^{n})\big).
\end{align*}
As required we derived Estimate (\ref{eqn:schonwiedera}).
Assuming (\ref{eqn:ass}) we now derive Estimate (\ref{eqn:schonwiederb}). The assumption directly shows that
\begin{align*}
\lim_{t\to 0}\P\Big(\bigcap_{s\leq t}\big\{||X||_{s}\geq 2C^{-1} b_{\lambda_2'}(s)\big\}\Big)=1
\end{align*}
which implies that we may choose $l$ large enough such that
\begin{align*}
\P\Bigg(\frac{||X||_{r^{j}-r^{n}}}{b_{\lambda_2'}(r^{j}-r^{n})}>\frac{2}{C};\text{for all $l\leq j\leq n-1$}\Bigg)\geq\P\Big(\bigcap_{s\leq r^{l}}\big\{||X||_{s}\geq 2C^{-1} b_{\lambda_2'}(s)\big\}\Big)\geq \frac{1}{2}.
\end{align*}
Hence, we derived estimate (\ref{eqn:schonwiederb}) so that the proof is complete.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:t1corollary2}] This is completely analogous to the proof of Corollary~\ref{cor:t1corollary}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t2}] We use Proposition~\ref{prop:ad} and Theorem~\ref{t3}. In order to do so, we have to see that the term $\eps u_\eps$ in (\ref{eqn:sdquantityadstrengthend}) has no influence on the order. We apply Lemma~\ref{lem:lower} and the proof of Theorem~\ref{t} with the scaling
$$t=r^n \qquad\text{and}\qquad \eps=b(r^n)$$
and with the sequence $n^{-n^\beta}$, respectively. Therefore, it is sufficient to show that
$$\eps u_\eps =o( t F(\eps))$$
with the above scalings of $t$ and $\eps$. Since $\eps=b(t)$ and thus $t\sim F(\eps)^{-1} \log \log F(\eps)$, we need to show that
$$\eps u_\eps =o( \log \log F(\eps)).$$
As this is precisely what we stated in condition (\ref{eqn:cond-esschervanishes}), the proof is complete.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:bvnodrift}]
As $X$ is of bounded variation, the representation
\begin{align*}
X_t=A^1_t-A^2_t+ct
\end{align*}
holds with two independent pure jump subordinators $A^1, A^2$. Next, we use the simple observation
\begin{align*}
\frac{|X_t|}{t}\leq \frac{||X||_t}{t}\leq \frac{||A^1||_t+||A^2||_t+|c|t}{t}=\frac{A^1_t}{t}+\frac{A^2_t}{t}+|c|
\end{align*}
to conclude the proof. The left hand side converges to $|c|$ as $X$ has bounded variation (see Theorem~39 of \cite{D}). Finally, the right hand side converges to $|c|$ as $|A^i_t|/t$ converge at zero almost surely to their drift (see Proposition~5 of \cite{D}).
\end{proof}
\section*{Acknowledgment}
We thank Thomas Simon (Lille) for pointing out the reference \cite{W88} to the authors.
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\section{Reference datasets}
\label{sec:data}
We consider three real-world datasets: the BestBuy dataset, which represents a consumer product hierarchical classification task; the Reuters dataset, which contains news articles; and the DBPedia dataset with Wikipedia excerpts. The datasets have varying text lengths (34 to 212 words), differ in the number of overall data (51,000 to 800,000 records) and have also been used in related work on HTC without DP and MI.
\\
\\
\textbf{BestBuy.} The BestBuy dataset\footnote{\url{https://github.com/BestBuyAPIs/open-data-set}} contains $51,646$ unique products, each consisting of categorical features (e.g., SKU, type, manufacturer), numerical features such as price, textual features (e.g., name, description) and URLs, that are composed of one or more of the aforementioned features. Training a differentially private HTC model on datasets like BestBuy can therefore prevent leaking sensitive information about individual products of a company.
In our experiments, we concatenate the features \enquote{name}, \enquote{manufacturer} and \enquote{description} to a single string and ignore the other features for classification. This selection is based on empirically observed superior classification accuracy. On average, the resulting concatenated texts have a length of 34 words. Additionally, every product holds a special feature called \enquote{category} assigning the product to a \textit{single}, \textit{partial-depth} class label in the BestBuy product hierarchy.
The BestBuy product hierarchy is a \textit{tree} and consists of seven levels, each with a different number of classes, as shown in Table~\ref{tab:dataset-hierarchy}. As can be seen, level $L_4$ has the most classes. Also, we can see that even on the first level, not all of the existing $51,646$ products are assigned to a class. Particularly, we found that 256 products ($0.50$\%) are assigned to classes not contained in the BestBuy product hierarchy. We removed these products as the assigned classes did not fit into the given product hierarchy (e.g., \enquote{Other Product Categories} or \enquote{In-Store Only}). Furthermore, not every product is assigned to a class on every level, meaning the most specific class of many products is on a lower level than $L_7$. In our experiments, we only make use of the first three hierarchy levels. We decided to do so due to the long tail characteristic of the dataset. Thus, the predictions of our classifiers are less specific than potentially possible, but more robust due to a higher number of training examples in comparison to fine grained training for all hierarchies. 10\% of the overall data was used for testing.
All datasets have been \emph{tokenized}, which means that the text has to be split up into a sequence of smaller units called \emph{tokens}. A natural tokenization technique is splitting the text into a sequence of words, so that each token represents a word. After tokenization, the token sequence is converted into an integer sequence since ANNs only take numbers as input. During the conversion, a vocabulary is created that maps each token to a unique integer so that the same token is always converted into the exact same integer. The size of the vocabulary then represents the number of unique tokens in the text.
\begin{table}[ht!]
\centering
\begin{tabular}{c|c|c|c}
Hierarchy Level & Dataset & Classes & Assigned products \\ \hline
\multirow{3}{*}{Level $L_1$} & BestBuy & $19$ & $51,390$ \\ \cline{2-4}
& DBPedia & $9$ & $337,739$ \\ \cline{2-4}
& Reuters & $4$ & $804,427$ \\ \hline
\multirow{3}{*}{Level $L_2$} & BestBuy & $164$ & $50,837$ \\ \cline{2-4}
& DBPedia & $70$ & $337,739$ \\ \cline{2-4}
& Reuters & $55$ & $779,714$ \\ \hline
\multirow{3}{*}{Level $L_3$} & BestBuy & 612 & $44,949$ \\ \cline{2-4}
& DBPedia & 219 & $337,739$ \\ \cline{2-4}
& Reuters & 43 & $406,961$ \\ \hline
Level $L_4$ & BestBuy & $771$ & $26,138$ \\ \hline
Level $L_5$ & BestBuy & $198$ & $5,640$ \\ \hline
Level $L_6$ & BestBuy & $23$ & $346$ \\ \hline
Level $L_7$ & BestBuy & $1$ & $1$ \\
\end{tabular}
\caption{Classes and assigned records per level per dataset.}
\label{tab:dataset-hierarchy}
\end{table}
\\
\\
\textbf{Reuters Corpus Volume 1.} The \enquote{Reuters Corpus Volume 1} (RCV1) dataset\footnote{\url{https://trec.nist.gov/data/reuters/reuters.html}} is an archive of over $800,000$ manually categorized news articles~\cite{lewisRCV1NewBenchmark2004}. Per news article, a headline, text block and topic codes representing the classes in the hierarchy are provided. In our experiments we use the concatenation of headline and text block as input for the respective classifiers. The resulting texts have an average length of 237 words. Table~\ref{tab:dataset-hierarchy} shows the number of classes and assigned documents for each hierarchy level of the Reuters dataset. To ensure comparability with state of the art we follow the approach of Stein et al.~\cite{steinAnalysisHierarchicalText2019} and randomly assign $80,443$ texts to the test dataset and assign each news article to the least frequent topic code. This approach is based on the assumption that the least common topic code is the one that most specifically characterizes the document. A differentially private HTC model trained on a non-public article dataset such as RCV1 can prevent leaking sensitive information represented by individual articles.
\\
\\
\textbf{DBPedia.} DBPedia is a community project that extracts structured knowledge from Wikipedia and makes it freely available using linked data technologies~\cite{lehmannDBpediaLargescaleMultilingual2015}. The DBPedia dataset for HTC\footnote{\url{https://www.kaggle.com/danofer/dbpedia-classes}} is used as a reference dataset in many state-of-the-art publications~\cite{joulinBagTricksEfficient2016, yangXLNetGeneralizedAutoregressive2020, minaeeDeepLearningBased2020} on text classification. Overall, the dataset contains the introductions of $337,739$ Wikipedia articles, of which $240,942$ are pre-assigned to the training dataset and $60,794$ to the test dataset. Per article, the dataset contains a description of on average 102 words and three one-class label per hierarchy level ($L_1$, $L_2$, $L_3$). Table~\ref{tab:dataset-hierarchy} shows the number of classes on each level of the DBPedia dataset hierarchy and indicates that all texts are assigned to a class on all levels, which means that the labels are \textit{full depth}. Training a differentially private HTC model on a dataset like DBPedia prevents leaking information about participating institutions and people if the underlying encyclopedia is not public.
\section{Experimental Setup}
\label{sec:evaluation:setup}
For our experiments we split the datasets into training, validation and test data. Training data is used to learn the model parameters (i.e., weights), validation data to check the goodness of training hyperparameters and test data is used to assess generalization and real-world performance. Before target and attack model training, so called hyperparameters have to be set manually before training (e.g., learning rate, batch size). We used Bayesian hyperparameter optimization for all target model experiments to ensure that we found good hyperparameters that yield high accuracies on the respective models and data. Bayesian Optimization is more efficient than grid search since it considers past trials during the hyperparameter search. An overview of all hyperparameters, dataset size for training, validation and test, and the overall $\ensuremath{\epsilon}$ per training is provided in Table~\ref{tab:hyp}. For the attack model we reused the original hyperparameters of Nasr et al.~\cite{nasrComprehensivePrivacyAnalysis2018} which already performed well.
The majority of experiments were conducted on EC2\footnote{\url{https://aws.amazon.com/ec2/}} GPU optimized instances of type \enquote{p3.8xlarge} with the \enquote{Deep Learning AMI} machine image, building on a Linux~$4.14$ kernel, Python~$3.6$ and TensorFlow~$2.2$.
\begin{table*}[t!]
\centering
\small
\caption{Hyperparameters and \ensuremath{\epsilon}~per model and dataset. The hyperparameters were set with Bayesian optimization.}
\label{tab:hyp}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
\multicolumn{2}{c|}{\multirow{2}{*}{}} & \multicolumn{3}{c|}{BestBuy} & \multicolumn{3}{c|}{Reuters} & \multicolumn{3}{c}{DBPedia} \\ \cline{3-11}
\multicolumn{2}{c|}{} & BoW & CNN & Transformer & BoW & CNN & Transformer & BoW & CNN & Transformer \\ \hline
\multirow{2}{*}{learning rate} & Orig. & $0.001$ & $0.001$ & $0.005$ & $0.001$ & $0.001$ & $0.005$ & $0.001$ & $0.001$ & $0.005$ \\ \cline{2-11}
& DP & $0.01$ & $0.001$ & $0.015$ & $0.008$ & $0.001$ & $0.005$ & $0.016$ & $0.001$ & $0.01$ \\ \hline
\multirow{2}{*}{batch size} & Orig. & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ \\ \cline{2-11}
& DP & $64$ & $64$ & $64$ & $64$ & $64$ & $32$ & $64$ & $64$ & $32$ \\ \hline
$\mathcal{C}$ & DP & $0.19$ & $1.48$ & $2.07$ & $0.33$ & $6.28$ & $12.86$ & $0.03$ & $0.21$ & $1.6$ \\ \hline
microbatch size & DP & $1$ & $1$ & $1$ & $1$ & $1$ & $4$ & $1$ & $1$ & $4$ \\ \hline
\multirow{4}{*}{$\epsilon$} & $z=0.1$ & $30,253$ & $33,731$ & $5,902$ & $2,048$ & $1,091$ & $21,597$ & $4,317$ & $6,414$ & $24,741$ \\ \cline{2-11}
& $z=0.5$ & $11.5$ & $11.1$ & $6.58$ & $4.19$ & $4.11$ & $4.4$ & $5.1$ & $6.29$ & $4.88$ \\ \cline{2-11}
& $z=1.0$ & $1.51$ & $1.38$ & $1.04$ & $0.79$ & $0.79$ & $0.77$ & $0.87$ & $0.96$ & $0.81$ \\ \cline{2-11}
& $z=3.0$ & $0.26$ & $0.5$ & $0.29$ & $0.22$ & $0.22$ & $0.22$ & $0.2$ & $0.21$ & $0.21$ \\ \hline
\multicolumn{2}{c|}{Training records} & \multicolumn{3}{c|}{$41,625$} & \multicolumn{3}{c|}{$651,585$} & \multicolumn{3}{c}{$240,942$} \\ \hline
\multicolumn{2}{c|}{Validation records} & \multicolumn{3}{c|}{$4,626$} & \multicolumn{3}{c|}{$72,399$} & \multicolumn{3}{c}{$36,003$} \\ \hline
\multicolumn{2}{c|}{Test records} & \multicolumn{3}{c|}{$5,139$} & \multicolumn{3}{c|}{$80,443$} & \multicolumn{3}{c}{$60,794$} \\ \hline
\end{tabular}
\end{table*}
In all experiments we assume that the data owner would also want converging target models even when training with DP. Thus, all HTC models leverage early stopping with a patience of $3$ epochs to terminate the training process before overfitting. Furthermore, we set the DP parameter $C$ (i.e., the sensitivity $\Delta f$) in our experiments to the median of the norms of the unclipped gradients over the course of original training as suggested by Abadi et al.~\cite{abadiDeepLearningDifferential2016}. For all executions of the experiment, CDP noise is sampled from a Gaussian distribution (cf.~Definition~\ref{def:dp:gauss}) with scale $\ensuremath{\sigma}=\textit{noise multiplier}~z \times \textit{clipping norm}~\cali{C}$. According to McMahan et al.~\cite{mcmahanGeneralApproachAdding2019}, values of $z\approx 1$ will provide reasonable privacy guarantees. We evaluate increasing noise regimes per dataset by evaluating noise multipliers $z\in\{0.1, 0.5, 1.0, 3.0\}$ and calculate the resulting $\ensuremath{\epsilon}$~at a fixed $\ensuremath{\delta}=\frac{1}{n}$.
\section{Evaluation}
\label{sec:evaluation}
In this section, we first describe the experimental setup. Afterwards, we experimentally assess privacy and utility for the previously formulated HTC models and datasets\footnote{We publish all code and experiment scripts at \url{https://github.com/SAP-samples/security-research-dp-hierarchical-text}.}. Lastly, we present several experiment variations to illustrate the impact of different parameters.
\subsection{Empirical Privacy and Utility}
\label{sec:evaluation:results}
\noindent
A theoretic comparison of the CNN, BoW and Transformer models with respect to their robustness towards noise that is introduced by DP is only insightful to a limited extend, since their architectures and pre-training paradigms vary. However, in general the bias-variance trade-off for ANNs allows us to formulate high-level expectations. Simple ANNs will likely be prone to high bias and thus underfit in comparison to larger ANNs. Thus, the BoW model will potentially perform poorer on test data than the CNN or Transformer architecture, even in the presence of pre-training~\cite{ezen20}. In contrast, large ANNs will have high variance and thus require larger amounts of training data to generalize well. Thus, the Transformer model will likely perform poorer on small datasets. In general, the bias decreases and the variance increases with the ANN size~\cite{NMB+19}. In combination with DP, we expect high bias models such as the BoW to be less affected by the introduced noise. Additionally, Transformer models may be negatively affected by gradient explosion when using relu activation functions in combination with DP~\cite{papernot2020tempered}.
Figure~\ref{fig:utility} states the utility and Figure~\ref{fig:mi} the privacy scores over $\epsilon$ for the three datasets and model architectures. Furthermore, we additionally report the theoretical bound on $Adv$ by Yeom et al.~\cite{yeomPrivacyRiskMachine2018} to allow comparison of the theoretical and the empirical MI advantage. Notably, even if two classifiers were trained with the same noise multiplier $z$, they do not necessarily yield the same DP privacy parameter $\epsilon$ due to differing training epochs until convergence. All corresponding $\epsilon$ values per model and dataset were calculated for $\delta = \frac{1}{|\cali{D}_{target}^{train}|}$ per dataset and are stated in Table~\ref{tab:hyp}.
\begin{figure*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_cleaned_utility.pdf}
\caption{Target model test accuracy for BestBuy utility over $\epsilon$}
\label{fig:bestbuy_cleaned_utility}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_utility.pdf}
\caption{Target model test accuracy for Reuters utility over $\epsilon$}
\label{fig:reuters_utility}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/DBPEDIA_utility.pdf}
\caption{Target model test accuracy for DBPedia over $\epsilon$}
\label{fig:dbpedia_utility}
\end{subfigure}
\caption{Target model test accuracy per dataset over $\epsilon$.}
\label{fig:utility}
\end{figure*}
\begin{figure*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_cleaned_mi.pdf}
\caption{MI against BestBuy over $\ensuremath{\epsilon}$}
\label{fig:bestbuy_cleaned_mi}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_mi.pdf}
\caption{MI against Reuters over $\ensuremath{\epsilon}$}
\label{fig:reuters_mi}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/DBPEDIA_mi.pdf}
\caption{MI against DBPedia over $\ensuremath{\epsilon}$}
\label{fig:dbpedia_mi}
\end{subfigure}
\caption{MI AUC, $Adv$ and Bound on MI $Adv$ per dataset $\epsilon$.}
\label{fig:mi}
\end{figure*}
As expected, the model utility and adversary's success consistently decrease with stronger DP parameters for all models and all datasets. Figure~\ref{fig:bestbuy_cleaned_utility} shows that
for BestBuy the BoW model's utility is the most robust to the introduced noise, while the Transformer model's utility is most sensitive to the introduced noise. This observation becomes most evident when considering the flat accuracy $Acc$ (\textit{blue}), and is in line with our expectation for small datasets formulated at the beginning of this section. The hierarchical utility metrics $F_H$ and $F_{LCA}$ do not decrease as strongly as $Acc$, since they also account for partially correct predictions. Interestingly, for BestBuy, the CNN model's MI metrics in Figure~\ref{fig:bestbuy_cleaned_mi} already reach the baseline level at $\epsilon=33,731$ ($z=0.1$). The large $\epsilon$ points out that with respect to the upper bound a huge privacy loss is occurring (i.e., $e^\epsilon$) and the advantage should also be maximal (i.e., $e^\epsilon-1$~\cite{yeomPrivacyRiskMachine2018}). However, the empirical membership advantage lies far below this theoretical bound. In contrast to the CNN, the MI attack against the BoW and Transformer models is only reaching the baseline at $\epsilon=1$ and $\epsilon=1.5$, respectively.
The results for the Reuters dataset are provided in Figure~\ref{fig:reuters_utility} and \ref{fig:reuters_mi}. Compared to BestBuy, the decrease in model utility on Reuters is smaller for all three HTC models, which can be explained with a significantly higher amount of training examples and a smaller amount of hierarchical classes. The BoW classifier's utility is most robust to the addition of noise to the training process, yet closely followed by the Transformer model. However, the CNN model exhibits the most severe decrease in model utility. Figure~\ref{fig:reuters_mi} indicates that the MI adversary's advantage drops to the baseline level again for very weak DP guarantees of $\epsilon>10^2$ for all models. This behavior can be explained with the high amount of training examples and the smaller amount of hierarchical classes. Therefore, the gap between the empirically measured membership advantage and the upper bound on membership advantage diverge widely.
For DBPedia in Figure~\ref{fig:dbpedia_utility}, the BoW model is again the most robust, and the Transformer model is least robust to the added noise during the training process, similar to the observations made on the BestBuy dataset. This is in line with our formulated expectations. The only exception are the measured utility metrics for $\epsilon\approx10^{-1}$, for which the BoW model performs worse than the CNN and Transformer model. MI metrics for the DBPedia HTC models are provided in Figure~\ref{fig:dbpedia_mi}. We see that the MI metrics for the BoW and Transformer models drop to the baseline level for very weak DP guarantees, similar to the Reuters models. Therefore, our MI adversary does by far not reach the theoretical upper $Adv$ bound. Notably, the MI metrics for the CNN model do not drop to the baseline level for the considered range for $z$ and resulting $\epsilon$. Hence, the gap between the measured $Adv$ and theoretical upper bound on $Adv$ reaches its lowest value for this model.
Overall, the privacy and utility results support our expectation that the utility of a high bias model such as BoW is less affected by the introduced noise than models with high variance such as Transformer. On the other hand, Transformer models are less prone to the MI attack due to better generalization which has generally been demonstrated aside from MI in related work~\cite{vaswaniAttentionAllYou2017,devlinBERTPretrainingDeep2018}.
\begin{table*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{tabular}{cc|c|c|c|c|c|}
\cline{3-7}
& & $n=41,625$, & $n=41,625$, & $n=41,625$, & $n=4000$ & $n=400$ \\
& & $14$ epochs & $50$ epochs & $100$ epochs & $30$ epochs & $30$ epochs \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{$\cali{D}_{target}^{train}$}} & $L_1$ & $99.71\%$ & $99.94\%$ & $99.94\%$ & $99.94\%$ & $98.44\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_2$ & $99.20\%$ & $99.86\%$ & $99.92\%$ & $99.44\%$ & $85.16\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_3$ & $96.91\%$ & $99.74\%$ & $99.81\%$ & $96.07\%$ & $63.28\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & Loss & $0.18$ & $0.01$ & $0.01$ & $0.30$ & $3.15$ \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{$\cali{D}_{target}^{test}$}} & $L_1$ & $97.24\%$ & $96.93\%$ & $97.06\%$ & $93.47\%$ & $84.31\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_2$ & $95.00\%$ & $94.79\%$ & $94.73\%$ & $87.89\%$ & $59.23\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_3$ & $89.32\%$ & $91.11\%$ & $91.35\%$ & $78.53\%$ & $37.76\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & Loss & $0.89$ & $1.60$ & $1.87$ & $2.08$ & $3.97$ \\ \hline
\multicolumn{2}{|c|}{$L_3$ Gap} & $7.60\%$ & $8.63\%$ & $8.47\%$ & $17.54\%$ & $25.52\%$ \\ \hline
\multicolumn{2}{|c|}{Loss Ratio} & $5.2$ & $160$ & $187$ & $6.93$ & $1.26$ \\ \hline
\multicolumn{2}{|c|}{$Acc_{MI}$} & $53.06\%$ & $53.92\%$ & $54.01\%$ & $64.62\%$ & $75.00\%$ \\ \hline
\end{tabular}
\caption{Per-level accuracies and summarized loss for BestBuy Transformer network without DP.}
\label{tab:target-model-mods}
\end{table*}
\subsection{Drivers for Attack Performance}
\label{sec:evaluation:effect:parameters}
Our experiments show that state-of-the-art MI attacks are not very effective when run against the trained HTC models.
This leads to the question whether more HTC-specific attacks or less generalizing HTC target models would boost attack performance. In the following subsections, we first validate experimentally that the attack performance does not increase when introducing HTC-specific attack features. Second, we present several means that reduce target model generalization, where we show that especially reducing the number of training examples increases the vulnerability to MI attacks.
\subsubsection{HTC-specific attack model features}
\label{sec:evaluation:effect:attack}
We hypothesize that adapting the MI attack to exploit the hierarchical relation of the classes leads to an increased MI attack performance. Our approach to adapt the MI attack to HTC models is to extract additional attack features from the target model. We choose to evaluate two additional features in the following.
The former is a Boolean feature that we derive by checking if the HTC model's prediction before applying the post-processing step is consistent with the hierarchy of the HTC task. The latter is a scalar feature that is derived by multiplying the probabilities of the node with the highest softmax score on each level. The intuition behind this feature is to calculate a value that states how confident the target model is with the predicted label, since the assigned probability is obviously higher when the model outputs only small probabilities for the other labels. We refer to this attack feature as \textit{prediction confidence}.
We pass the features into the attack model with an additional FCN similar to the loss. We tested the effect of the additional features on all datasets and models. However, the results for this attack variant do not result in a significant change of the considered MI metrics (Figures~\ref{fig:bestbuy_mi_whf} to \ref{fig:dbpedia_mi_whf} in the appendix).
\subsubsection{Reduced Target Model Generalization}
\label{sec:evaluation:effect:target}
Next, we describe four approaches that we formulated to increase the attack model performance by reducing target model generalization in HTC. Each approach is first motivated and then evaluated based on experimental results.
First, we provoke an overfitted target model by training without early stopping for a fixed number of epochs, which is chosen significantly higher than the original number of epochs obtained with early stopping. In doing so, we deliberately force the model to overfit, i.e.,~adapt to the few samples in $\cali{D}_{target}^{train}$ instead of approximating the underlying distribution. We evaluated this approach based on the BestBuy Transformer classifier. Table~\ref{tab:target-model-mods} shows the metrics of the original model in the first column, which converged after 14 epochs. The second and third column reveal the metrics for overfit models, which are trained for 50 and 100 epochs, respectively. As expected, the overfit models achieve a smaller training loss and a higher test loss. However, surprisingly, the achieved test accuracy does not drop compared to the original model, while the training accuracy on $L_3$ increases to over $99\%$. The corresponding attack model accuracies rise from $53.06\%$ to $53.92\%$ and $54.01\%$ respectively. This insignificant change may appear counter-intuitive given the increased loss on $\cali{D}_{target}^{test}$. However, when analyzing the loss distribution of $\cali{D}_{target}^{train}$ and $\cali{D}_{target}^{test}$, we observe that the median losses decrease similarly as depicted in Figure~\ref{fig:loss_dist:orig} and \ref{fig:loss_dist:overfit}. The reason for the high average loss on $\cali{D}_{target}^{test}$ is due to the high loss value of a few outliers. Therefore, the loss ratio is not a consistently good indicator for MI attack effectiveness in practice and rather the accuracy gap should be taken into account.
\begin{figure}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.15\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/loss_boxplots_bestbuy_bert_orig.pdf}
\caption{Loss on original model after 14 epochs}
\label{fig:loss_dist:orig}
\end{subfigure}%
\begin{subfigure}{0.15\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/loss_boxplots_bestbuy_bert_overfit100.pdf}
\caption{Loss on overfit model after 100 epochs}
\label{fig:loss_dist:overfit}
\end{subfigure}
\caption{Loss distribution of members and non-members for BestBuy. Each boxplot is on a log scale depicting outliers (black), median (green) and mean (red) of the respective distribution.}
\label{fig:loss_dist}
\end{figure}
\begin{figure*}[h]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_bow_ratio.pdf}
\caption{Relation between the records/levels ratio and attack advantage for BestBuy}
\label{fig:bestbuy_cleaned_ratio}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_bow_ratio.pdf}
\caption{Relation between the records/levels ratio and attack advantage for Reuters}
\label{fig:reuters_ratio}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/dbpedia_bow_ratio.pdf}
\caption{Relation between the records/levels ratio and attack advantage for DBPedia }
\label{fig:dbpedia_ratio}
\end{subfigure}
\caption{Attack model advantage over ratio of records per hierarchy for an overall number of $\{0.25, 0.5, 0.75, 1\}\times n$ training records.}
\label{fig:ratio}
\end{figure*}
Second, we reduce the number of training examples in $\cali{D}_{target}^{train}$. With this adaption, the hierarchical classifier should not generalize as well as the original classifier due to two reasons. First, the training dataset is less representative of the problem domain, and second, underrepresented classes contain even fewer examples.
We again evaluate this approach based on the BestBuy transformer classifier, which originally contains $n=41,625$ training examples. Training the classifier with only $10\%$ of the training examples indeed leads to worse generalization with a maximum train-test gap of $17.54\%$ on the third level as shown in Table~\ref{tab:target-model-mods}. The trained attack model for this variation converged at $64.62\%$ accuracy, which is a significant increase compared to the original target model. Further reducing the training data to $n=400$ examples reduces the target model performance even more, with a maximum train-test gap of $25.52\%$ on the third level, as evident from Table~\ref{tab:target-model-mods}. For this target model with $n=400$, we observe an attack accuracy of $75\%$. In conclusion, reducing the number of training examples results in a significant MI attack improvement in comparison to changing the attack model features.
Third, we increase the number of hierarchy levels in the data, resulting in a more complex HTC task, which we again hypothesize to lead to worse generalization. Moreover, additional hierarchy levels lead to additional classification outputs and therefore additional attack features, which might further boost attack performance. And indeed, when increasing the number of levels from three to seven for the BestBuy BoW classifier, the corresponding MI attack accuracy rises from $56.15\%$ to $57.07\%$. To ensure that this effect was caused by noise, we also reduced the number of levels to one, resulting in an attack accuracy of $51.38\%$. This shows that an increased (decreased) number of hierarchy levels leads to higher (lower) attack performances.
Interestingly, when combining the effects of changing the number of hierarchy levels and decreasing the number of training examples, we made the observation that the ratio of the number of training examples to levels in the HTC task has an direct influence on the MI attack performance. We first noted this effect for BestBuy and then validated if we can also see it for the other datasets. The results confirm our expectations and are shown in Figure \ref{fig:ratio}.
Finally, we train the hierarchical classifier from scratch, without leveraging pre-trained weights to initialize the model. We hypothesize that this classifier variation might be more vulnerable to MI attacks, since a model without pre-training might tend to memorize more information about $\cali{D}_{target}^{train}$. Training the original BestBuy Transformer classifier from scratch did not converge to a useful HTC model, with only $18\%$ accuracy on the first level. This effect can be explained by the relatively small amount of training data compared to the large corpora the Transformer model is usually pre-trained on. The issue can be overcome by replacing the BERT-Base layers with BERT-Tiny layers, since tiny layer contain fewer weights to train. The hierarchical BERT-Tiny classifier trained from scratch yields a model with $75.17\%$ flat $Acc$. The trained attack model for this variation converged at $55.04\%$ accuracy, which is $\approx2\%$ higher than the attack on the original target model. This relatively small increase reveals that the use of pre-trained weights for the target models is not the reason for the relatively poor attack performance.
\section{Acknowledgements}
\label{sec:ack}
We would also like to thank the anonymous reviewers for their immensely helpful suggestions to improve the readability and contents of this paper.
This work has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 825333 (MOSAICROWN).
The project that gave rise to these results received the support of a fellowship from "la Caixa" Foundation (ID 100010434) and from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 847648. The fellowship code is LCF/BQ/PR20/11770009.
The work of Javier Parra-Arnau has been supported through an Alexander von Humboldt Post-Doctoral Fellowship.
\section{Discussion}
\label{sec:discussion}
\begin{figure*}[h]
\centering
\includegraphics[width=0.85\textwidth]{fig/TradeOff.pdf}
\caption[Privacy-utility trade-off per HTC model on each dataset.]{Privacy-utility trade-off per HTC model on each dataset. Privacy and utility are represented by MI $AUC$ and classification $Acc$, respectively. An optimal classifier would exhibit $100\%$ $Acc$ and no vulnerability to the MI adversary, expressed by $50\%$ $AUC$.}
\label{fig:tradeoff}
\end{figure*}
\textbf{Large values for privacy parameter $\epsilon$ are sufficient to completely mitigate MI attacks with moderate decrease in model utility.} In our experiments, we enforced the HTC models to satisfy DP guarantees by clipping and perturbing the computed gradients during the training process. As expected, the experiments showed that enforcing DP in this way reduces the effectiveness of the performed MI attacks but also harms model utility. Figure~\ref{fig:tradeoff} summarizes the trade-off between classification accuracy and MI $AUC$ for each dataset. As can be seen, for all examined datasets, it is possible to completely mitigate the MI attack while reducing classification $Acc$ by $<20\%$. For BestBuy, the CNN model yields the best model utility for $AUC=50\%$. In contrast, for Reuters and DBPedia, the Transformer yields the best model utility for $AUC=50\%$. This may be explained by the size and average text length of the datasets, since for small datasets with short texts (e.g., BestBuy) the CNN model is well suited, while for larger datasets with longer texts (e.g., Reuters, DBPedia) the Transformer architecture is suited better.
\textbf{Overfitting is a key driver for MI attack performance.}
Our experiments support the understanding that overfitting leads to higher MI attack performance. We therefore extend the findings of previous work for non-textual datasets to HTC~\cite{shokriMembershipInferenceAttacks2017, nasrComprehensivePrivacyAnalysis2018}.
To prevent overfitting, our analysis of drivers for MI attack performance suggests to gather as many training examples as possible and only predict as many hierarchy levels as needed. Adding HTC-specific features to the attack model did not increase MI attack performance, confirming the validity of our results also for stronger adversaries.
\textbf{Similar DP privacy parameters do not imply a similar MI attack effectiveness.} The experimental results show that the empirical MI risk for similar DP guarantees varies within each dataset but also within each model architecture. Therefore, we can again summarize that the MI attack effectiveness depends on the chosen model architecture and the dataset. Unfortunately, the results do not point to a model architecture that is strictly better suited to mitigate the MI attack. However, we recommend using a model with relatively few parameters such as the BoW model for smaller datasets, whereas for larger datasets models with a high number of parameters such as the Transformer classifier yields a favorable trade-off.
\textbf{The BoW model's utility was reduced least by the added DP noise.} Across all datasets, we observed that the CNN and Transformer model's utility scores were impacted more heavily compared to the BoW model's utility for similar DP guarantees. On two of the three datasets, the Transformer model's utility is impacted even more severely than the CNN model's utility. This finding suggests that a higher number of weights in an ANN might correlate with a stronger impact of DP training on the ANN utility. Specifically, the number of ANN weights is lowest in the BoW model and highest in the Transformer model. This insight should be taken into account when a data scientist wants to train an ANN based on a given formal DP guarantee.
\textbf{HTC ANNs exhibit a big gap between empirical and theoretical MI risk}. The obtained results support the conclusions by Jayaraman et al.~\cite{jayaramanEvaluatingDifferentiallyPrivate2019}, who found that there remains a big gap between what state-of-the-art MI attacks can infer and what is the maximum that can theoretically be inferred according to the bound presented by Yeom et al.~\cite{yeomPrivacyRiskMachine2018}. During evaluation, we measured the membership advantage and compared it to the theoretical membership advantage bound, which can be calculated given the respective DP guarantee. We showed that this conclusion also holds in the context of HTC.
\section{Reference datasets}
\label{sec:data}
We consider three real-world datasets: the BestBuy dataset, which represents a consumer product hierarchical classification task; the Reuters dataset, which contains news articles; and the DBPedia dataset with Wikipedia excerpts. The datasets have varying text lengths (34 to 212 words), differ in the number of overall data (51,000 to 800,000 records) and have also been used in related work on HTC without DP and MI.
\\
\\
\textbf{BestBuy.} The BestBuy dataset\footnote{\url{https://github.com/BestBuyAPIs/open-data-set}} contains $51,646$ unique products, each consisting of categorical features (e.g., SKU, type, manufacturer), numerical features such as price, textual features (e.g., name, description) and URLs, that are composed of one or more of the aforementioned features. Training a differentially private HTC model on datasets like BestBuy can therefore prevent leaking sensitive information about individual products of a company.
In our experiments, we concatenate the features \enquote{name}, \enquote{manufacturer} and \enquote{description} to a single string and ignore the other features for classification. This selection is based on empirically observed superior classification accuracy. On average, the resulting concatenated texts have a length of 34 words. Additionally, every product holds a special feature called \enquote{category} assigning the product to a \textit{single}, \textit{partial-depth} class label in the BestBuy product hierarchy.
The BestBuy product hierarchy is a \textit{tree} and consists of seven levels, each with a different number of classes, as shown in Table~\ref{tab:dataset-hierarchy}. As can be seen, level $L_4$ has the most classes. Also, we can see that even on the first level, not all of the existing $51,646$ products are assigned to a class. Particularly, we found that 256 products ($0.50$\%) are assigned to classes not contained in the BestBuy product hierarchy. We removed these products as the assigned classes did not fit into the given product hierarchy (e.g., \enquote{Other Product Categories} or \enquote{In-Store Only}). Furthermore, not every product is assigned to a class on every level, meaning the most specific class of many products is on a lower level than $L_7$. In our experiments, we only make use of the first three hierarchy levels. We decided to do so due to the long tail characteristic of the dataset. Thus, the predictions of our classifiers are less specific than potentially possible, but more robust due to a higher number of training examples in comparison to fine grained training for all hierarchies. 10\% of the overall data was used for testing.
All datasets have been \emph{tokenized}, which means that the text has to be split up into a sequence of smaller units called \emph{tokens}. A natural tokenization technique is splitting the text into a sequence of words,so that each token represents a word. After tokenization, the token sequence is converted into an integer sequence since ANNs only take numbers as input. During the conversion, a vocabulary is created that maps each token to a unique integer so that the same token is always converted into the exact same integer. The size of the vocabulary then represents the number of unique tokens in the text.
\begin{table}[ht!]
\centering
\begin{tabular}{c|c|c|c}
Hierarchy Level & Dataset & Classes & Assigned products \\ \hline
\multirow{3}{*}{Level $L_1$} & BestBuy & $19$ & $51,390$ \\ \cline{2-4}
& DBPedia & $9$ & $337,739$ \\ \cline{2-4}
& Reuters & $4$ & $804,427$ \\ \hline
\multirow{3}{*}{Level $L_2$} & BestBuy & $164$ & $50,837$ \\ \cline{2-4}
& DBPedia & $70$ & $337,739$ \\ \cline{2-4}
& Reuters & $55$ & $779,714$ \\ \hline
\multirow{3}{*}{Level $L_3$} & BestBuy & 612 & $44,949$ \\ \cline{2-4}
& DBPedia & 219 & $337,739$ \\ \cline{2-4}
& Reuters & 43 & $406,961$ \\ \hline
Level $L_4$ & BestBuy & $771$ & $26,138$ \\ \hline
Level $L_5$ & BestBuy & $198$ & $5,640$ \\ \hline
Level $L_6$ & BestBuy & $23$ & $346$ \\ \hline
Level $L_7$ & BestBuy & $1$ & $1$ \\
\end{tabular}
\caption{Classes and assigned records per level per dataset.}
\label{tab:dataset-hierarchy}
\end{table}
\\
\\
\textbf{Reuters Corpus Volume 1.} The \enquote{Reuters Corpus Volume 1} (RCV1) dataset\footnote{\url{https://trec.nist.gov/data/reuters/reuters.html}} is an archive of over $800,000$ manually categorized news articles~\cite{lewisRCV1NewBenchmark2004}. Per news article, a headline, text block and topic codes representing the classes in the hierarchy are provided. In our experiments we use the concatenation of headline and text block as input for the respective classifiers. The resulting texts have an average length of 237 words. Table~\ref{tab:dataset-hierarchy} shows the number of classes and assigned documents for each hierarchy level of the Reuters dataset. To ensure comparability with state of the art we follow the approach of Stein et al.~\cite{steinAnalysisHierarchicalText2019} and randomly assign $80,443$ texts to the test dataset and assign each news article to the least frequent topic code. This approach is based on the assumption that the least common topic code is the one that most specifically characterizes the document. A differentially private HTC model trained on a non-public article dataset such as RCV1 can prevent leaking sensitive information represented by individual articles.
\\
\\
\textbf{DBPedia.} DBPedia is a community project that extracts structured knowledge from Wikipedia and makes it freely available using linked data technologies~\cite{lehmannDBpediaLargescaleMultilingual2015}. The DBPedia dataset for HTC\footnote{\url{https://www.kaggle.com/danofer/dbpedia-classes}} is used as a reference dataset in many state-of-the-art publications~\cite{joulinBagTricksEfficient2016, yangXLNetGeneralizedAutoregressive2020, minaeeDeepLearningBased2020} on text classification. Overall, the dataset contains the introductions of $337,739$ Wikipedia articles, of which $240,942$ are pre-assigned to the training dataset and $60,794$ to the test dataset. Per article, the dataset contains a description of on average 102 words and three one-class label per hierarchy level ($L_1$, $L_2$, $L_3$). Table~\ref{tab:dataset-hierarchy} shows the number of classes on each level of the DBPedia dataset hierarchy and indicates that all texts are assigned to a class on all levels, which means that the labels are \textit{full depth}. Training a differentially private HTC model on a dataset like DBPedia prevents leaking information about participating institutions and people if the underlying encyclopedia is not public.
\section{Experimental Setup}
\label{sec:evaluation:setup}
For our experiments we split the datasets into training, validation and test data. Training data is used to learn the model parameters (i.e., weights), validation data to check the goodness of training hyperparameters and test data is used to assess generalization and real-world performance. Before target and attack model training, so called hyperparameters have to be set manually before training (e.g., learning rate, batch size). We used Bayesian hyperparameter optimization for all target model experiments to ensure that we found good hyperparameters that yield high accuracies on the respective models and data. Bayesian Optimization is more efficient than grid search since it considers past trials during the hyperparameter search. An overview of all hyperparameters, dataset size for training, validation and test, and the overall $\ensuremath{\epsilon}$ per training is provided in Table~\ref{tab:hyp}. For the attack model we reused the original hyperparameters of Nasr et al.~\cite{nasrComprehensivePrivacyAnalysis2018} which already performed well.
The majority of experiments were conducted on EC2\footnote{\url{https://aws.amazon.com/ec2/}} GPU optimized instances of type \enquote{p3.8xlarge} with the \enquote{Deep Learning AMI} machine image, building on a Linux~$4.14$ kernel, Python~$3.6$ and TensorFlow~$2.2$.
\begin{table*}[ht!]
\centering
\caption{Hyperparameters and composed \ensuremath{\epsilon}~per model and dataset. The hyperparameters were set with Bayesian optimization.}
\label{tab:hyp}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
\multicolumn{2}{c|}{\multirow{2}{*}{}} & \multicolumn{3}{c|}{BestBuy} & \multicolumn{3}{c|}{Reuters} & \multicolumn{3}{c}{DBPedia} \\ \cline{3-11}
\multicolumn{2}{c|}{} & BoW & CNN & Transformer & BoW & CNN & Transformer & BoW & CNN & Transformer \\ \hline
\multirow{2}{*}{learning rate} & Orig. & $0.001$ & $0.001$ & $0.005$ & $0.001$ & $0.001$ & $0.005$ & $0.001$ & $0.001$ & $0.005$ \\ \cline{2-11}
& DP & $0.01$ & $0.001$ & $0.015$ & $0.008$ & $0.001$ & $0.005$ & $0.016$ & $0.001$ & $0.01$ \\ \hline
\multirow{2}{*}{batch size} & Orig. & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ & $32$ \\ \cline{2-11}
& DP & $64$ & $64$ & $64$ & $64$ & $64$ & $32$ & $64$ & $64$ & $32$ \\ \hline
clipping norm $\mathcal{C}$ & DP & $0.19$ & $1.48$ & $2.07$ & $0.33$ & $6.28$ & $12.86$ & $0.03$ & $0.21$ & $1.6$ \\ \hline
microbatch size & DP & $1$ & $1$ & $1$ & $1$ & $1$ & $4$ & $1$ & $1$ & $4$ \\ \hline
\multirow{4}{*}{$\epsilon$} & $z=0.1$ & $30,253$ & $33,731$ & $5,902$ & $2,048$ & $1,091$ & $21,597$ & $4,317$ & $6,414$ & $24,741$ \\ \cline{2-11}
& $z=0.5$ & $11.5$ & $11.1$ & $6.58$ & $4.19$ & $4.11$ & $4.4$ & $5.1$ & $6.29$ & $4.88$ \\ \cline{2-11}
& $z=1.0$ & $1.51$ & $1.38$ & $1.04$ & $0.79$ & $0.79$ & $0.77$ & $0.87$ & $0.96$ & $0.81$ \\ \cline{2-11}
& $z=3.0$ & $0.26$ & $0.5$ & $0.29$ & $0.22$ & $0.22$ & $0.22$ & $0.2$ & $0.21$ & $0.21$ \\ \hline
\multicolumn{2}{c|}{Training records} & \multicolumn{3}{c|}{$41,625$} & \multicolumn{3}{c|}{$651,585$} & \multicolumn{3}{c}{$240,942$} \\ \hline
\multicolumn{2}{c|}{Validation records} & \multicolumn{3}{c|}{$4,626$} & \multicolumn{3}{c|}{$72,399$} & \multicolumn{3}{c}{$36,003$} \\ \hline
\multicolumn{2}{c|}{Test records} & \multicolumn{3}{c|}{$5,139$} & \multicolumn{3}{c|}{$80,443$} & \multicolumn{3}{c}{$60,794$} \\ \hline
\end{tabular}
\end{table*}
In all experiments we assume that the data owner would also want converging target models even when training with DP. Thus, all HTC models leverage early stopping with a patience of $3$ epochs to terminate the training process before overfitting. Furthermore, we set the DP parameter $C$ (i.e., the sensitivity $\Delta f$) in our experiments to the median of the norms of the unclipped gradients over the course of original training as suggested by Abadi et al.~\cite{abadiDeepLearningDifferential2016}. For all executions of the experiment, CDP noise is sampled from a Gaussian distribution (cf.~Definition~\ref{def:dp:gauss}) with scale $\ensuremath{\sigma}=\textit{noise multiplier}~z \times \textit{clipping norm}~\cali{C}$. According to McMahan et al.~\cite{mcmahanGeneralApproachAdding2019}, values of $z\approx 1$ will provide reasonable privacy guarantees. We evaluate increasing noise regimes per dataset by evaluating noise multipliers $z\in\{0.1, 0.5, 1.0, 3.0\}$ and calculate the resulting $\ensuremath{\epsilon}$~at a fixed $\ensuremath{\delta}=\frac{1}{n}$.
\section{Evaluation}
\label{sec:evaluation}
In this section, we first describe the experimental setup. Afterwards, we experimentally assess privacy and utility for the previously formulated HTC models and datasets\footnote{We publish all code and experiment scripts at \url{https://github.com/SAP-samples/security-research-dp-hierarchical-text}.}. Lastly, we present several experiment variations to illustrate the impact of different parameters.
\subsection{Empirical Privacy and Utility}
\label{sec:evaluation:results}
\noindent
A theoretic comparison of the CNN, BoW and Transformer models with respect to their robustness towards noise that is introduced by DP is only insightful to a limited extend, since their architectures and pre-training paradigms vary. However, in general the bias-variance trade-off for ANNs allows us to formulate high-level expectations. Simple ANNs will likely be prone to high bias and thus underfit in comparison to larger ANNs. Thus, the BoW model will potentially perform poorer on test data than the CNN or Transformer architecture, even in the presence of pre-training~\cite{ezen20}. In contrast, large ANNs will have high variance and thus require larger amounts of training data to generalize well. Thus, the Transformer model will likely perform poorer on small datasets. In general, the bias decreases and the variance increases with the ANN size~\cite{NMB+19}. In combination with DP, we expect high bias models such as the BoW to be less affected by the introduced noise. Additionally, Transformer models may be negatively affected by gradient explosion when using relu activation functions in combination with DP~\cite{papernot2020tempered}.
Figure~\ref{fig:utility} states the utility and Figure~\ref{fig:mi} the privacy scores over $\epsilon$ for the three datasets and model architectures. Furthermore, we additionally report the theoretical bound on $Adv$ by Yeom et al.~\cite{yeomPrivacyRiskMachine2018} to allow comparison of the theoretical and the empirical MI advantage. Notably, even if two classifiers were trained with the same noise multiplier $z$, they do not necessarily yield the same DP privacy parameter $\epsilon$ due to differing training epochs until convergence. All corresponding $\epsilon$ values per model and dataset were calculated for $\delta = \frac{1}{|\cali{D}_{target}^{train}|}$ per dataset and are stated in Table~\ref{tab:hyp}.
\begin{figure*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_cleaned_utility.pdf}
\caption{Target model test accuracy for BestBuy utility over $\epsilon$}
\label{fig:bestbuy_cleaned_utility}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_utility.pdf}
\caption{Target model test accuracy for Reuters utility over $\epsilon$}
\label{fig:reuters_utility}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/DBPEDIA_utility.pdf}
\caption{Target model test accuracy for DBPedia over $\epsilon$}
\label{fig:dbpedia_utility}
\end{subfigure}
\caption{Target model test accuracy per dataset over $\epsilon$.}
\label{fig:utility}
\end{figure*}
\begin{figure*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_cleaned_mi.pdf}
\caption{MI against BestBuy over $\ensuremath{\epsilon}$}
\label{fig:bestbuy_cleaned_mi}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_mi.pdf}
\caption{MI against Reuters over $\ensuremath{\epsilon}$}
\label{fig:reuters_mi}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/DBPEDIA_mi.pdf}
\caption{MI against DBPedia over $\ensuremath{\epsilon}$}
\label{fig:dbpedia_mi}
\end{subfigure}
\caption{MI AUC, $Adv$ and Bound on MI $Adv$ per dataset $\epsilon$.}
\label{fig:mi}
\end{figure*}
As expected, the model utility and adversary's success consistently decrease with stronger DP parameters for all models and all datasets. Figure~\ref{fig:bestbuy_cleaned_utility} shows that
for BestBuy the BoW model's utility is the most robust to the introduced noise, while the Transformer model's utility is most sensitive to the introduced noise. This observation becomes most evident when considering the flat accuracy $Acc$ (\textit{blue}), and is in line with our expectation for small datasets formulated at the beginning of this section. The hierarchical utility metrics $F_H$ and $F_{LCA}$ do not decrease as strongly as $Acc$, since they also account for partially correct predictions. Interestingly, for BestBuy, the CNN model's MI metrics in Figure~\ref{fig:bestbuy_cleaned_mi} already reach the baseline level at $\epsilon=33,731$ ($z=0.1$). The large $\epsilon$ points out that with respect to the upper bound a huge privacy loss is occurring (i.e., $e^\epsilon$) and the advantage should also be maximal (i.e., $e^\epsilon-1$~\cite{yeomPrivacyRiskMachine2018}). However, the empirical membership advantage lies far below this theoretical bound. In contrast to the CNN, the MI attack against the BoW and Transformer models is only reaching the baseline at $\epsilon=1$ and $\epsilon=1.5$, respectively.
The results for the Reuters dataset are provided in Figure~\ref{fig:reuters_utility} and \ref{fig:reuters_mi}. Compared to BestBuy, the decrease in model utility on Reuters is smaller for all three HTC models, which can be explained with a significantly higher amount of training examples and a smaller amount of hierarchical classes. The BoW classifier's utility is most robust to the addition of noise to the training process, yet closely followed by the Transformer model. However, the CNN model exhibits the most severe decrease in model utility. Figure~\ref{fig:reuters_mi} indicates that the MI adversary's advantage drops to the baseline level again for very weak DP guarantees of $\epsilon>10^2$ for all models. This behavior can be explained with the high amount of training examples and the smaller amount of hierarchical classes. Therefore, the gap between the empirically measured membership advantage and the upper bound on membership advantage diverge widely.
For DBPedia in Figure~\ref{fig:dbpedia_utility}, the BoW model is again the most robust, and the Transformer model is least robust to the added noise during the training process, similar to the observations made on the BestBuy dataset. This is in line with our formulated expectations. The only exception are the measured utility metrics for $\epsilon\approx10^{-1}$, for which the BoW model performs worse than the CNN and Transformer model. MI metrics for the DBPedia HTC models are provided in Figure~\ref{fig:dbpedia_mi}. We see that the MI metrics for the BoW and Transformer models drop to the baseline level for very weak DP guarantees, similar to the Reuters models. Therefore, our MI adversary does by far not reach the theoretical upper $Adv$ bound. Notably, the MI metrics for the CNN model do not drop to the baseline level for the considered range for $z$ and resulting $\epsilon$. Hence, the gap between the measured $Adv$ and theoretical upper bound on $Adv$ reaches its lowest value for this model.
Overall, the privacy and utility results support our expectation that the utility of a high bias model such as BoW is less affected by the introduced noise than models with high variance such as Transformer. On the other hand, Transformer models are less prone to the MI attack due to better generalization which has generally been demonstrated aside from MI in related work~\cite{vaswaniAttentionAllYou2017,devlinBERTPretrainingDeep2018}.
\begin{table*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{tabular}{cc|c|c|c|c|c|}
\cline{3-7}
& & $n=41,625$, & $n=41,625$, & $n=41,625$, & $n=4000$ & $n=400$ \\
& & $14$ epochs & $50$ epochs & $100$ epochs & $30$ epochs & $30$ epochs \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{$\cali{D}_{target}^{train}$}} & $L_1$ & $99.71\%$ & $99.94\%$ & $99.94\%$ & $99.94\%$ & $98.44\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_2$ & $99.20\%$ & $99.86\%$ & $99.92\%$ & $99.44\%$ & $85.16\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_3$ & $96.91\%$ & $99.74\%$ & $99.81\%$ & $96.07\%$ & $63.28\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & Loss & $0.18$ & $0.01$ & $0.01$ & $0.30$ & $3.15$ \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{$\cali{D}_{target}^{test}$}} & $L_1$ & $97.24\%$ & $96.93\%$ & $97.06\%$ & $93.47\%$ & $84.31\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_2$ & $95.00\%$ & $94.79\%$ & $94.73\%$ & $87.89\%$ & $59.23\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & $L_3$ & $89.32\%$ & $91.11\%$ & $91.35\%$ & $78.53\%$ & $37.76\%$ \\ \cline{2-7}
\multicolumn{1}{|c|}{} & Loss & $0.89$ & $1.60$ & $1.87$ & $2.08$ & $3.97$ \\ \hline
\multicolumn{2}{|c|}{$L_3$ Gap} & $7.60\%$ & $8.63\%$ & $8.47\%$ & $17.54\%$ & $25.52\%$ \\ \hline
\multicolumn{2}{|c|}{Loss Ratio} & $5.2$ & $160$ & $187$ & $6.93$ & $1.26$ \\ \hline
\multicolumn{2}{|c|}{$Acc_{MI}$} & $53.06\%$ & $53.92\%$ & $54.01\%$ & $64.62\%$ & $75.00\%$ \\ \hline
\end{tabular}
\caption{Per-level accuracies and summarized loss for BestBuy Transformer network without DP.}
\label{tab:target-model-mods}
\end{table*}
\subsection{Drivers for Attack Performance}
\label{sec:evaluation:effect:parameters}
Our experiments show that state-of-the-art MI attacks are not very effective when run against the trained HTC models.
This leads to the question whether more HTC-specific attacks or less generalizing HTC target models would boost attack performance. In the following subsections, we first validate experimentally that the attack performance does not increase when introducing HTC-specific attack features. Second, we present several means that reduce target model generalization, where we show that especially reducing the number of training examples increases the vulnerability to MI attacks.
\subsubsection{HTC-specific attack model features}
\label{sec:evaluation:effect:attack}
We hypothesize that adapting the MI attack to exploit the hierarchical relation of the classes leads to an increased MI attack performance. Our approach to adapt the MI attack to HTC models is to extract additional attack features from the target model. We choose to evaluate two additional features in the following.
The former is a Boolean feature that we derive by checking if the HTC model's prediction before applying the post-processing step is consistent with the hierarchy of the HTC task. The latter is a scalar feature that is derived by multiplying the probabilities of the node with the highest softmax score on each level. The intuition behind this feature is to calculate a value that states how confident the target model is with the predicted label, since the assigned probability is obviously higher when the model outputs only small probabilities for the other labels. We refer to this attack feature as \textit{prediction confidence}.
We pass the features into the attack model with an additional FCN similar to the loss. We tested the effect of the additional features on all datasets and models. However, the results for this attack variant do not result in a significant change of the considered MI metrics (Figures~\ref{fig:bestbuy_mi_whf} to \ref{fig:dbpedia_mi_whf} in the appendix).
\subsubsection{Reduced Target Model Generalization}
\label{sec:evaluation:effect:target}
Next, we describe four approaches that we formulated to increase the attack model performance by reducing target model generalization in HTC. Each approach is first motivated and then evaluated based on experimental results.
First, we provoke an overfitted target model by training without early stopping for a fixed number of epochs, which is chosen significantly higher than the original number of epochs obtained with early stopping. In doing so, we deliberately force the model to overfit, i.e.,~adapt to the few samples in $\cali{D}_{target}^{train}$ instead of approximating the underlying distribution. We evaluated this approach based on the BestBuy Transformer classifier. Table~\ref{tab:target-model-mods} shows the metrics of the original model in the first column, which converged after 14 epochs. The second and third column reveal the metrics for overfit models, which are trained for 50 and 100 epochs, respectively. As expected, the overfit models achieve a smaller training loss and a higher test loss. However, surprisingly, the achieved test accuracy does not drop compared to the original model, while the training accuracy on $L_3$ increases to over $99\%$. The corresponding attack model accuracies rise from $53.06\%$ to $53.92\%$ and $54.01\%$ respectively. This insignificant change may appear counter-intuitive given the increased loss on $\cali{D}_{target}^{test}$. However, when analyzing the loss distribution of $\cali{D}_{target}^{train}$ and $\cali{D}_{target}^{test}$, we observe that the median losses decrease similarly as depicted in Figure~\ref{fig:loss_dist:orig} and \ref{fig:loss_dist:overfit}. The reason for the high average loss on $\cali{D}_{target}^{test}$ is due to the high loss value of a few outliers. Therefore, the loss ratio is not a consistently good indicator for MI attack effectiveness in practice and rather the accuracy gap should be taken into account.
\begin{figure}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.15\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/loss_boxplots_bestbuy_bert_orig.pdf}
\caption{Loss on original model after 14 epochs}
\label{fig:loss_dist:orig}
\end{subfigure}%
\begin{subfigure}{0.15\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/loss_boxplots_bestbuy_bert_overfit100.pdf}
\caption{Loss on overfit model after 100 epochs}
\label{fig:loss_dist:overfit}
\end{subfigure}
\caption{Loss distribution of members and non-members for BestBuy. Each boxplot is on a log scale depicting outliers (black), median (green) and mean (red) of the respective distribution.}
\label{fig:loss_dist}
\end{figure}
\begin{figure*}[h]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_bow_ratio.pdf}
\caption{Relation between the records/levels ratio and attack advantage for BestBuy}
\label{fig:bestbuy_cleaned_ratio}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_bow_ratio.pdf}
\caption{Relation between the records/levels ratio and attack advantage for Reuters}
\label{fig:reuters_ratio}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/dbpedia_bow_ratio.pdf}
\caption{Relation between the records/levels ratio and attack advantage for DBPedia }
\label{fig:dbpedia_ratio}
\end{subfigure}
\caption{Attack model advantage over ratio of records per hierarchy for an overall number of $\{0.25, 0.5, 0.75, 1\}\times n$ training records.}
\label{fig:ratio}
\end{figure*}
Second, we reduce the number of training examples in $\cali{D}_{target}^{train}$. With this adaption, the hierarchical classifier should not generalize as well as the original classifier due to two reasons. First, the training dataset is less representative of the problem domain, and second, underrepresented classes contain even fewer examples.
We again evaluate this approach based on the BestBuy transformer classifier, which originally contains $n=41,625$ training examples. Training the classifier with only $10\%$ of the training examples indeed leads to worse generalization with a maximum train-test gap of $17.54\%$ on the third level as shown in Table~\ref{tab:target-model-mods}. The trained attack model for this variation converged at $64.62\%$ accuracy, which is a significant increase compared to the original target model. Further reducing the training data to $n=400$ examples reduces the target model performance even more, with a maximum train-test gap of $25.52\%$ on the third level, as evident from Table~\ref{tab:target-model-mods}. For this target model with $n=400$, we observe an attack accuracy of $75\%$. In conclusion, reducing the number of training examples results in a significant MI attack improvement in comparison to changing the attack model features.
Third, we increase the number of hierarchy levels in the data, resulting in a more complex HTC task, which we again hypothesize to lead to worse generalization. Moreover, additional hierarchy levels lead to additional classification outputs and therefore additional attack features, which might further boost attack performance. And indeed, when increasing the number of levels from three to seven for the BestBuy BoW classifier, the corresponding MI attack accuracy rises from $56.15\%$ to $57.07\%$. To ensure that this effect was caused by noise, we also reduced the number of levels to one, resulting in an attack accuracy of $51.38\%$. This shows that an increased (decreased) number of hierarchy levels leads to higher (lower) attack performances.
Interestingly, when combining the effects of changing the number of hierarchy levels and decreasing the number of training examples, we made the observation that the ratio of the number of training examples to levels in the HTC task has an direct influence on the MI attack performance. We first noted this effect for BestBuy and then validated if we can also see it for the other datasets. The results confirm our expectations and are shown in Figure \ref{fig:ratio}.
Finally, we train the hierarchical classifier from scratch, without leveraging pre-trained weights to initialise the model. We hypothesize that this classifier variation might be more vulnerable to MI attacks, since a model without pre-training might tend to memorize more information about $\cali{D}_{target}^{train}$. Training the original BestBuy Transformer classifier from scratch did not converge to a useful HTC model, with only $18\%$ accuracy on the first level. This effect can be explained by the relatively small amount of training data compared to the large corpora the Transformer model is usually pre-trained on. The issue can be overcome by replacing the BERT-Base layers with BERT-Tiny layers, since tiny layer contain fewer weights to train. The hierarchical BERT-Tiny classifier trained from scratch yields a model with $75.17\%$ flat $Acc$. The trained attack model for this variation converged at $55.04\%$ accuracy, which is $\approx2\%$ higher than the attack on the original target model. This relatively small increase reveals that the use of pre-trained weights for the target models is not the reason for the relatively poor attack performance.
\section{Conclusion}
\label{sec:conclusion}
This work analyzed and compared the privacy-utility trade-off in DP HTC under a white-box MI adversary. Even without the use of DP, white-box MI attacks only posed a minor risk to the three HTC model architectures and reference dataset that we considered. In consequence, large privacy parameters were sufficient to fully mitigate the white-box attack. We particularly observed Transformer-based HTC models to be rather resistant to MI attacks without using DP at all.
However, the privacy-accuracy trade-off for full mitigation of the white-box MI attack is differing widely for all considered models and datasets. Our results suggest that the Transformer model is also favorable for large datasets with long texts when using DP, while the CNN model is favorable for smaller datasets with shorter texts. However, if hardware costs shall be minimized or the training examples shall be protected with a strong formal DP guarantee (i.e., small $\ensuremath{\epsilon}$ value), the fastText based BoW model is a good choice due to the high robustness against DP perturbation. Our experiments also confirm a large gap between the empirical membership advantage of the MI white-box attack and the theoretical DP membership advantage bound for HTC datasets and models.
\section{Related work}
\label{sec:related_work}
This work is related to HTC, DP in NLP and MI attacks for evaluating the privacy of DP ML models. Therefore, in this section, we briefly introduce the most relevant publications in the respective research fields.
Stein et al.~\cite{steinAnalysisHierarchicalText2019} analyze the performance of different hierarchical text classifiers on the Reuters (RCV1) dataset that we also use for the experiments in our work. The authors find that a fastText-based classifier works better than a CNN-based classifier initialized with the same embeddings. For evaluation, all possible types of metrics are used in the paper, namely flat, hierarchical, and LCA metrics.
Interestingly, the authors do not consider any Transformer-based HTC model, even though Transformer architectures represent state-of-the-art for text classification.
Abadi et al.~\cite{abadiDeepLearningDifferential2016} formulate an implementation of the DP stochastic gradient descent, which uses the Gaussian mechanism to perturb gradient descent optimizers for ANNs. As ANNs are widely used in modern NLP and natural language data in many cases contain sensitive data, there are various publications regarding DP in NLP. While Vu et al.~\cite{vuDpUGCLearnDifferentially2019} learn DP word embeddings for user generated content, so that the resulting word embeddings can be shared safely, we focus on safe sharing of whole ML models. Other works use DP for author obfuscation in text classification~\cite{fernandesGeneralisedDifferentialPrivacy2019, weggenmannSynTFSyntheticDifferentially2018}. In contrast, our work addresses the privacy-utility trade-off for perturbation of the gradient descent optimizer.
Carlini et al.~\cite{carliniSecretSharerEvaluating2019} successfully apply DP to prevent information leakage in a generative model, specifically an ANN generating text. They introduce the \textit{exposure} metric to measure the risk of unintentionally memorizing rare or unique training-data sequences in generative models. Our work does not consider generative models, but solely classification models.
Empirical MI attacks against machine learning models such as the attack used in this work were first formulated by Shokri et al.~\cite{shokriMembershipInferenceAttacks2017} in the form of black-box MI. The authors compare MI attacks with model inversion attacks, which abuse access to an ML model to infer certain features of the training data. In contrast to model inversion attacks, MI attacks target a specific training example instead of targeting all training examples for a specific class. Therefore, the authors argue that successful MI attacks indicate unintended information leakage. Misra~\cite{misraBLACKBOXATTACKS2019a} uses black-box MI attacks to assess the information leakage of generative models.
Nasr et al.~\cite{nasrComprehensivePrivacyAnalysis2018} showed that white-box MI attacks, that take the target model's internal parameters into account, are more effective than black-box MI attacks. Additionally, the authors assume that the adversary owns a fraction of the data owner's sensitive data. This stronger assumption about the adversary's knowledge increases the overall strength of the MI attack compared with black-box MI attacks.
While Rahman et al.~\cite{rahmanMembershipInferenceAttack2018} analyze the effect of different values for $\epsilon$ on the effectiveness of only black-box MI attacks, Bernau et al.~\cite{bernauAssessingDifferentiallyPrivate2020} take both black-box and white-box attacks into account. However, both publications mostly consider specifically crafted non-textual MI datasets. We consider real-world textual training data.
Yeom et al.~\cite{yeomPrivacyRiskMachine2018} introduce membership advantage to measure the success of an MI attack. Furthermore, they formulate a theoretical upper bound for the membership advantage that depends on the DP guarantees of the target model. Humphries et al.~\cite{Humphries} derive a bound for membership advantage that is tighter than the bound derived by Yeom et al. by analyzing the impact of giving up the i.i.d.~assumption. However, there is a gap between the theoretic upper bound for the membership advantage and the membership advantage of state-of-the-art MI attacks, as has been shown by Jayaraman et al.~on numeric and image data~\cite{jayaramanEvaluatingDifferentiallyPrivate2019}. In our work, we investigate this gap in the context of ANNs for HTC.
\section{Introduction}
\label{sec:intro}
Organizing large corpora of unstructured data such as text documents, news articles, emails, and support tickets in an automated manner is a considerable challenge due to the inherent ambiguity of natural languages~\cite{HFB19JPC}. However, the automated classification of unstructured data overcomes manual data labelling activities and thus is a key capability for organizing data at scale~\cite{TaylorWIRED13}. Due to the wide range of applications, Hierarchical Text Classification (HTC) has received particular interest by the Natural Language Processing (NLP) community in recent years~\cite{mao-etal-2019-hierarchical, qu2012evaluation, agrawal2013multi, peng2016deepmesh}. HTC leverages machine learning to automatically organize documents into taxonomies, predicting multiple labels in a predefined label hierarchy.
After data owners have trained HTC models on their data the models may be shared with data analysts such as contractors, customers, or even the general public.
However, sharing a model may leak information about the training data~\cite{shokriMembershipInferenceAttacks2017, nasrComprehensivePrivacyAnalysis2018, zhangUnderstandingDeepLearning2017, carliniSecretSharerEvaluating2019}. Perturbation with differential privacy (DP)\footnote{For conciseness,
throughout this work we use the acronym DP to refer to both ``differential privacy'' and its adjective form ``differentially private''.} limits information leakage by anonymizing the training data or model training function~\cite{dworkDifferentialPrivacy2006,abadiDeepLearningDifferential2016,Hayes2019}. DP introduces an inherent trade-off between privacy and utility, which means that a stronger privacy guarantee implies a decrease in informative value. Balancing this trade-off is especially hard when training Artificial Neural Networks (ANNs). On the one hand, ANN utility can only be assessed empirically after training and even small perturbation can have a high impact on utility~\cite{BPS19}.
On the other hand, DP anonymization parameter $\epsilon$ formulates a theoretic upper bound on information leakage that does not reflect the empirical information leakage for a concrete dataset. For ANNs empirical information leakage can be assessed with Membership Inference (MI) attacks~\cite{shokriMembershipInferenceAttacks2017}, which aim at identifying single instances of the training data by sole access to the trained model~\cite{nasrComprehensivePrivacyAnalysis2018, rahmanMembershipInferenceAttack2018}. A rather large gap has been observed between the high theoretical bound on information leakage under MI attacks that can be derived from DP guarantees and the empirical information leakage posed by MI attacks~\cite{jayaramanEvaluatingDifferentiallyPrivate2019}. Consequently, choosing the anonymization strength via privacy parameter $\varepsilon$ remains a challenging problem since a data owner can either choose to consider the theoretical or empirical information leakage.
Our work compares the empirical privacy-utility trade-off of multiple HTC models by quantifying privacy under MI. We hypothesize that there are preferable ANN architectures w.r.t.~the privacy-accuracy trade-off when applying DP to HTC. Unlike previous studies on the privacy-accuracy trade-off for numerical or image data~\cite{rahmanMembershipInferenceAttack2018,bernauAssessingDifferentiallyPrivate2020, jayaramanEvaluatingDifferentiallyPrivate2019}, our work focuses on textual data which requires different ANN architectures, such as Transformers~\cite{vaswaniAttentionAllYou2017}. The main contributions of this paper are:
\begin{itemize}
\item Empirically quantifying and comparing the privacy-utility trade-off for three widely used HTC ANN architectures on three reference datasets. In particular, we consider Bag of Words (BoW), Convolutional Neural Networks (CNNs) and Transformer-based architectures.
\item Connecting DP privacy guarantees to MI attack performance for HTC ANNs. In contrast to the adversary considered by DP, the MI adversary represents an ML specific threat model.
\item Recommending HTC model architectures and privacy parameters for the practitioner based on the privacy-utility trade-off under DP and MI.
\end{itemize}
This paper is structured as follows. Section~\ref{sec:preliminaries} recalls key aspects of DP, MI attacks and HTC.
Section~\ref{sec:methodology} formulates our approach for modelling the privacy-utility trade-off in HTC. Section~\ref{sec:data} introduces reference datasets and Section~\ref{sec:evaluation:setup} the experiment setup. Experiment results results are presented in Section~\ref{sec:evaluation} and subsequently discussed in Section~\ref{sec:discussion}. Section~\ref{sec:related_work} introduces related work. Conclusions are drawn in Section~\ref{sec:conclusion}.
\section{Quantifying Utility and Privacy in HTC}
\label{sec:methodology}
\begin{figure*}
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=\textwidth]{fig/htc-models.pdf}
\caption{Architecture of the HTC models used in the experiments. Pre-trained layers are marked \textit{grey}.}
\label{fig:htc-models}
\end{subfigure}%
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=\textwidth]{fig/attack-archietcture.pdf}
\caption{Attack model architecture and observed attack features from an HTC target model.}
\label{fig:htc-models:attack}
\end{subfigure}
\label{fig:models}
\caption{Architecture of target models and attack model.}
\end{figure*}
This section describes our methods for quantifying and comparing the privacy-utility trade-off in HTC for three relevant model architectures under several utility and MI metrics.
\subsection{HTC Model Architectures}
Architectures for HTC include training a single classifier predicting classes in the flattened hierarchy (\textit{flat}), training multiple classifiers predicting classes for a given level or node (\textit{local}) and training a single classifier that respects the class hierarchy (\textit{global}).
We chose a global HTC approach that features a single ANN with one output layer per level $L_n$ in the tree hierarchy of the given HTC task. Each output layer is a fully connected layer with a \textit{softmax} activation which uses the output of the last hidden layer to predict the class for the input text on $L_n$. The architecture thus consists of only one model and does not ignore the class hierarchy. This yields two main advantages. First, our HTC classifier exhibits a reduced training time and therefore also a lower overall privacy cost in comparison to local HTC approaches. Secondly, the data analyst can still retrieve the prediction probabilities per level in contrast to flat HTC approaches that only provide prediction probabilities of the individual nodes~\cite{Babbar13NIPS}.
Our HTC approach, however, has the disadvantage that a post-processing step is needed to obtain predictions consistent with the hierarchy, since each output layer makes predictions independently for its hierarchy level. We show an example in Figure~\ref{fig:lcl-incconsistencies} where the prediction for $L_2$ does not coincide with the prediction for $L_1$ and $L_3$, resulting in an undefined assignment. We suggest to resolve such inconsistencies by multiplying the softmax probabilities along the path from the root to each possible node. After comparing the multiplied probabilities, we output the path with the highest probability as prediction, leading to only consistent predictions.
Even though we defined the output layer architecture for our HTC beforehand, multiple options for the architecture of the input and intermediary layers exist. We consider architectures widely used in state-of-the-art text classification as the basis for formulating a DP hierarchical text classifier. In the sequel, we briefly describe the three architectures employed in our methodology.
\textbf{Bag-of-Words.} BoW models represent ANNs that ignore the word order within a text. BoW models comprise a single embedding layer in which the word vectors are added or averaged, and which is followed by one or more feed-forward layers with a softmax activation in the last layer. The BoW classifier used in this paper is based on the architecture of \textit{fastText}~\cite{joulinBagTricksEfficient2016}, which achieves high accuracy and is computationally efficient due to the usage of a single feed-forward layer for classification. Each token is first embedded and the mean of all embeddings before the output layer is computed afterwards. Prior to training, the embedding layer is initialized with the widely used GloVe embeddings\footnote{\url{https://nlp.stanford.edu/projects/glove/}} that are pre-trained on the \enquote{Wikipedia 2014} and \enquote{Gigaword~5} corpora~\cite{penningtonGloveGlobalVectors2014, kowsariTextClassificationAlgorithms2019}. For training we use the Adam optimizer. The classifier architecture is visualized in Figure~\ref{fig:htc-models}.
\textbf{Convolutional Neural Networks.} A CNN contains one or more convolutional layers that convolve the input with a filter of a given width to extract and detect patterns. Although CNNs are widely used for detecting patterns in images, CNNs have also been effectively used for detecting patterns in a text~\cite{kalchbrenner-etal-2014-convolutional, collobert2011natural} and do not ignore the word order. In this paper we use the original CNN classifier for text classification proposed by Kim~\cite{kimConvolutionalNeuralNetworks2014}. Their architecture first applies three convolutional layers to the embeddings, which are then concatenated and passed through a dropout layer to foster generalization. The architecture of the resulting HTC model is provided in Figure~\ref{fig:htc-models}. For training, we use the Adam optimizer and Glove embeddings. The training hyperparameters are taken from Kim~\cite{kimConvolutionalNeuralNetworks2014}: filter sizes of $3$, $4$, and $5$ for the three convolutional blocks with $100$ filters each and a dropout probability of $p_{do}=0.5$.
\textbf{Transformer Networks.} Transformer layers are ANN layers that are especially suited for processing longer texts due to a mechanism called \textit{self-attention}. Due to self-attention, a single Transformer layer can relate all tokens of a text to each other~\cite{vaswaniAttentionAllYou2017}. In contrast, CNNs require multiple convolutional layers to relate the information between two arbitrary tokens in a text. The transformer classifier we use in this paper is the BERT model as formulated by Devlin et al.~\cite{devlinBERTPretrainingDeep2018}.
BERT comprises twelve transformer layers, each consisting of two sub-layers. To lower the computational effort that is needed for training BERT we follow related work and initialize the BERT layers with pre-trained weights~\cite{devlinBERTPretrainingDeep2018, wolfHuggingFaceTransformersStateoftheart2020}. During training, we employ the Adam optimizer and a dropout probability of $p_{do}=0.1$. The BERT HTC architecture is illustrated in Figure~\ref{fig:htc-models}.
\subsection{Utility Metrics}
\begin{figure*}[ht!]
\centering
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\p1=(W.west),
\p2=(P3.north),
\p3=(E.east),
\p4=(P3.south)
in
($(\x1,\y2)+(-0.1,0.1)$) rectangle ($(\x3, \y4)+(0.1,-0.1)$) node[pos=1] {$L_3$};
\draw[black!30!green,thick] ($(P1.north west)+(-0.2,0.2)$) rectangle ($(P1.south east)+(0.2,-0.2)$);
\draw[black!30!green,thick] ($(P2.north west)+(-0.2,0.2)$) rectangle ($(P2.south east)+(0.2,-0.2)$);
\draw[black!30!green,thick] ($(P3.north west)+(-0.2,0.2)$) rectangle ($(P3.south east)+(0.2,-0.2)$);
\end{tikzpicture}
\caption{HTC predictions before postprocessing (\textit{green}).}
\label{fig:lcl-incconsistencies}
\end{subfigure}%
\begin{subfigure}{0.5\linewidth}
\centering
\begin{tikzpicture}[auto,
scale=0.7,
level 1/.style={sibling distance=3cm},
level 2/.style={sibling distance=15mm},
level 3/.style={sibling distance=15mm},
every node/.style = {circle, align=center, fill=blue!20, draw=black, very thick,
minimum size = 8mm, font=\scriptsize\bf\sffamily,
inner sep=0
}]
\node {R}
child { node {1} }
child { node {2}
child { node {2.1} } }
child { node {3}
child { node (P2) {3.1} }
child { node {3.2}
child { node (T) {3.2.1} }
child { node (P1) {3.2.2} } }
child { node {3.3} } };
\draw[black!30!green,thick] ($(T.north west)+(-0.2,0.2)$) rectangle ($(T.south east)+(0.2,-0.2)$);
\draw[red,thick] ($(P1.north west)+(-0.2,0.2)$) rectangle ($(P1.south east)+(0.2,-0.2)$);
\draw[red,thick] ($(P2.north west)+(-0.2,0.2)$) rectangle ($(P2.south east)+(0.2,-0.2)$);
\end{tikzpicture}
\caption{Two incorrect predictions (\textit{red}) for a fictional input belonging to another category (\textit{green}) .}
\label{fig:example_tree}
\end{subfigure}
\caption{Visualization for obtaining and evaluating HTC predictions.}
\end{figure*}
\label{sec:methodology:utility}
There are two approaches for evaluating the utility provided of the described
model architectures, namely, flat and hierarchical evaluation metrics~\cite{sillaSurveyHierarchicalClassification2011, kosmopoulosEvaluationMeasuresHierarchical2015}. To illustrate the difference between flat and hierarchical classification metrics consider the tree hierarchy depicted in Figure~\ref{fig:example_tree}. Assume that the true category for a given test example $x$ is $3.2.1$ (\textit{green}) and that two different classifiers output $3.2.2$ and $3.1$ as the predicted categories (\textit{red}). When flat evaluation metrics are used, both systems are penalized equally since both predictions are counted as false negatives for the true category $3.2.1$. However, the second classifier's error is more severe since its prediction is in an unrelated sub-tree of node $3$, which is considered in hierarchical evaluation metrics.
In this work, we assess the utility of HTC based on a mix of flat and hierarchical metrics: \textit{accuracy}, the hierarchical and lowest common ancestor (LCA) \textit{F-measure}. We report the (flat) accuracy $Acc$ due to its wide use in machine learning. We calculate the \textit{hierarchical and LCA F-measure} since they are considered the state of the art in the field of HTC. The hierarchical $F$-measure $F_H$ is calculated from \emph{hierarchical precision} $P_H$ and \emph{hierarchical recall} $R_H$, and defined as follows~\cite{sillaSurveyHierarchicalClassification2011}:
\begin{align*}
&P_H = \frac{\sum_i|Anc_i\cap\hat{Anc_i}|}{\sum_i|\hat{Anc_i}|},\\
&R_H = \frac{\sum_i|Anc_i\cap\hat{Anc_i}|}{\sum_i|Anc_i|},\\
&F_H = \frac{2P_HR_H}{P_H+R_H}.
\end{align*}
For a record $i$, $Anc_i$ is the set consisting of the \textit{true} class and all ancestors (except the root). Analogously, $\hat{Anc_i}$ is the set consisting of the \textit{predicted} class and all ancestors (except the root). In Figure~\ref{fig:example_tree} $Anc_i = \{3.2.1, 3.2, 3\}$ for the true class $3.2.1$. $\hat{Anc_i}$ is $\{3.2.2, 3.2, 3\}$ and $\{3.1, 3\}$ with $R_H = \frac{2}{3}$ and $R_H = \frac{1}{3}$, respectively.
The LCA metrics $P_{LCA}, R_{LCA}, F_{LCA}$ are differing to the previous hierarchical metrics only by not considering the nodes above the lowest common ancestor in $Anc_i$, and thus void overpenalization of errors for nodes with many ancestors~\cite{kosmopoulosEvaluationMeasuresHierarchical2015}. Again, in the example shown in Figure~\ref{fig:example_tree}, for the prediction $3.2.2$ we would have $LCA = 3.2$ with $Anc_i = \{3.2.1, 3.2\}$ and $\hat{Anc_i} = \{3.2.2, 3.2\}$ and therefore $R_{LCA} = \frac{1}{2}$. For the prediction $3.1$ we would have $LCA = 3$ with $Anc_i = \{3.2.1, 3.2, 3\}$ and $\hat{Anc_i} = \{3.1, 3\}$ and therefore $R_{LCA} = \frac{1}{3}$.
\subsection{Privacy Metrics and Bounds}
DP formulates a privacy bound on the ratio of probability distributions around $D$ and $D'$ resulting from a mechanism. The privacy bound holds for an adversary with auxiliary knowledge of up to all but one records in the dataset~\cite{Lee2012,BEGKK21}. Yeom et al.~\cite{yeomPrivacyRiskMachine2018} demonstrate that the privacy bound can be transformed into an upper bound on the membership advantage of an MI adversary. Membership advantage is calculated as follows from the True Positive Rate (TPR) and the False Positive Rate (FPR)~\cite{FAWCETT2006861}:
\begin{equation*}
Adv = TPR - FPR.
\end{equation*}
The upper bound is:
\begin{equation}
\mathit{Adv}\leq e^\epsilon - 1.
\end{equation}
Whether the resulting membership advantage upper bound is reached, i.e.~the empirically observed $Adv$ matches the upper bound, depends on whether the sensitivity of the training data during model training matches the assumed global sensitivity (i.e., clipping norm $\mathcal{C}$; cf.~Theorem~\ref{thm:gauss})~\cite{Nissim2007}. The gap between the lower and upper bound can be validated by implementing an MI adversary~\cite{jayaramanEvaluatingDifferentiallyPrivate2019,bernauAssessingDifferentiallyPrivate2020}.
Figure \ref{fig:htc-models:attack} visualizes the architecture of our implemented MI adversary's attack model, which is based on the attack model of Nasr et al.~\cite{nasrComprehensivePrivacyAnalysis2018}. We mainly extended their attack model to accept multiple labels, one per hierarchy level. The remaining components are unchanged. The attack model itself is represented by an ANN that learns to discriminate between training data and test data based on the attack features (e.g., losses).
In addition to $Adv$ we also quantify the area under the Receiver-Operating-Characteristic-Curve (AUC) of the attack model. The AUC is also providing insights on the MI attack performance w.r.t.~addressing members and non-members. The AUC is a general performance metric for evaluation of binary classifiers such as the MI attack model~\cite{FAWCETT2006861}.
\section{Preliminaries}
In the following we provide preliminaries on DP in Section~\ref{sec:prel:dp}, MI in Section~\ref{sec:prel:mi}, HTC in Section~\ref{sec:htc} and lastly HTC specific machine learning concepts in Section~\ref{sec:embeddings}. Throughout this paper we will use the abbreviations and variables denoted in Table~\ref{tab:Notation}.
\begin{table}[bt!]
\footnotesize
\caption{List of acronyms.}
\label{tab:Notation}
\centering
\begin{tabularx}{.48\textwidth}{ >{\bfseries }lX}
ANN & Artificial Neural Network \\
AUC & Area under the ROC Curve \\
BERT & Bidirectional Encoder Representations from Transformers \\
BoW & Bag of Words \\
CNN & Convolutional Neural Network \\
DAG & Directed Acyclic Graph \\
DP & Differential Privacy \\
FPR & False Positive Rate \\
HTC & Hierarchical Text Classification \\
LCA & Lowest Common Ancestor \\
LCL & Local Classifier per Level \\
LCPN & Local Classifier per Parent Node \\
LCN & Local Classifier per Node \\
MI & Membership Inference \\
NLP & Natural Language Processing \\
RCV1 & Reuters Corpus Volume 1 \\
RDP & Rényi Differential Privacy \\
RNN & Recurrent Neural Network \\
ROC & Receiver Operating Characteristic \\
TPR & True Positive Rate \\
\end{tabularx}
\end{table}
\label{sec:preliminaries}
\subsection{Differential Privacy}
\label{sec:prel:dp}
In DP~\cite{dworkDifferentialPrivacy2006} a statistical aggregation function $f(\cdot)$ is evaluated over a dataset $\mathcal{D}$, and the result is perturbed before being provided to the data analyst. By means of perturbation DP prevents an adversary with arbitrary auxiliary knowledge on all but one participant in a dataset $\mathcal{D}$ from confidently deciding whether $f(\cdot)$ was evaluated on $\mathcal{D}$, or some neighboring dataset $\mathcal{D'}$ differing in one element.
Assuming that every participant in $\mathcal{D}$ is represented by a single record $d\in\mathcal{D}$, privacy is intuitively provided to any individual. The perturbation strength is steered by setting privacy parameters $(\epsilon,\delta)$, and small privacy parameters will result in high perturbation. A formal definition of DP is provided in Definition~\ref{def:dp}.
\begin{definition}[$(\epsilon,\delta)$-DP~\cite{dworkAlgorithmicFoundationsDifferential2013}]
\label{def:dp}
A randomized mechanism $\mathcal{M}$ on a query function $f$ satisfies $(\epsilon,\delta)$-DP for $\delta>0$ if, for all pairs of neighboring databases $\mathcal{D},\mathcal{D'}$ and for all outputs $\mathcal{O}\subseteq$ \emph{range}($\mathcal{M}$),
\begin{equation}
\label{eq:DP}
\Pr[\mathcal{M}(\mathcal{D})\in \mathcal{O}] \leq e^{\epsilon} \Pr[\mathcal{M}(\mathcal{D'})\in \mathcal{O}] + \delta.
\end{equation}
\end{definition}
DP is enforced by mechanisms. Mechanisms for numerical data perturb the original query value $f(\mathcal{D})$ by adding numerical noise. DP mechanisms need to add noise scaled to the \textit{global sensitivity}. Global sensitivity is formally defined in Definition~\ref{def:gs}.
\begin{definition}[Global $\ell_1$-sensitivity~\cite{dworkDifferentialPrivacy2006}]
\label{def:gs}
Let $\mathcal{D}$ and $\mathcal{D'}$ be neighboring databases. The global $\ell_1$-sensitivity of a function $f$, denoted by $\Delta f$, is defined as
\begin{equation*}
\Delta f = \max_{\forall\mathcal{D},\mathcal{D'}}\|f(\mathcal{D}) - f(\mathcal{D'})\|_1.
\end{equation*}
\label{def:gs_eq}
\end{definition}
In this work, we use the Gaussian mechanism for gradient-perturbation to perturb the Adam optimizer for ANN training \footnote{The Tensorflow privacy package was used throughout this work: \url{https://github.com/tensorflow/privacy}.} as suggested by Abadi et al.~\cite{abadiDeepLearningDifferential2016}. For simplicity, we shall refer to the perturbed Adam optimizer as DP-Adam. A DP optimizer for ANN training, such as DP-Adam, uses a randomized mechanism $\mathcal{M}_{nn}$. The optimizer updates the weight coefficients $\theta_t$ of an ANN per training step $t\in\{1,\ldots,T\}$ with $\theta_t \leftarrow \theta_{t-1}-\alpha(\tilde g)$, where $\tilde g~=~\mathcal{M}_{nn}(\partial loss / \partial \theta_{t-1})$ denotes a perturbed gradient. $\alpha$ is a scaling function on $\tilde g$ to compute an update (i.e., learning rate). After $T$ steps, DP-Adam outputs a DP weight matrix $\theta$ that is used by the ANN prediction function. In case of DP-Adam, $\cali{M}_{nn}$ is a Gaussian mechanism as specified in Theorem~\ref{thm:gauss}.
\newline
\begin{theorem}[Gaussian Mechanism~\cite{dworkAlgorithmicFoundationsDifferential2013}]
\label{thm:gauss}~Let $\epsilon\in(0,1)$ be arbitrary. For $c^2>2ln(\frac{1.25}{\delta})$, the Gaussian mechanism with parameter $\sigma\ge c\frac{\Delta f}{\epsilon}$ satisfies $(\epsilon,\delta)$-DP, when adding noise scaled to the Normal distribution $\mathcal{N}(0,\sigma^2)$.
\label{def:dp:gauss}
\end{theorem}
DP-Adam bounds the sensitivity of the computed gradients by a clipping norm $\mathcal{C}$, based on which the gradients are clipped before perturbation.
Since weight updates are performed iteratively during training, a composition of mechanism executions is required until the training step $T$ is reached and the final private weights $\theta$ are obtained.
We use R{\'e}nyi DP as suggested by Mironov~\cite{mironovRenyiDifferentialPrivacy2017} to calculate the tight, overall privacy guarantee $\ensuremath{\epsilon}$ under composition. $(\alpha,\ensuremath{\epsilon}_{RDP})-$R{\'e}nyi DP (RDP), with $\alpha >1$ quantifies the difference in distributions $\cali{M}(\cali{D}), \cali{M}(\cali{D'})$ by their R{\'e}nyi divergence~\cite{vanErven2010}. For a sequence of $T$ mechanism executions each providing ($\alpha$, $\ensuremath{\epsilon}_{RDP, i}$)-RDP, the privacy guarantee composes to ($\alpha$, $\sum_{i}\ensuremath{\epsilon}_{RDP, i}$)-RDP. The ($\alpha$, $\ensuremath{\epsilon}_{RDP}$)-RDP guarantee converts to $(\ensuremath{\epsilon}_{RDP}-\frac{\ln\ensuremath{\delta}}{\alpha-1},\ensuremath{\delta})$-DP. The Gaussian mechanism is calibrated to RDP by:
\begin{equation}\label{eq:gaussrdp}
\ensuremath{\epsilon}_{RDP} = \alpha \cdot \Delta f^2/2\sigma^2
\end{equation}
\subsection{Membership Inference}
\label{sec:prel:mi}
Membership Inference attacks strive for identifying the presence or absence of individual records in the training data of a machine learning model. Throughout this paper we refer to the trained machine learning model as \textit{target model} and the data owner's training data as $\mathcal{D}_{target}^{train}$. We solely consider ANNs as target models in this paper. ANNs are structured in layers of neurons that are connected by weights. We denote the weights between a layer $l$ and its preceding layer $l-1$ as $w^{(l)}$. The output of the $l$-th layer is denoted as $o^{(l)}$. The ANN's final output is the output of the last layer.
This paper builds upon the white-box MI attack against ANNs proposed by Nasr et al.~\cite{nasrComprehensivePrivacyAnalysis2018}. Essentially, the white-box MI attack assumes an honest-but-curious adversary with access to the target model weight matrix $w^{(l)}$. The white-box MI adversary leverages this knowledge to calculate attack features for each record $(x, y)$ in the form of layer outputs $o^{(l)}(x;W)$, losses $L(o(x;w),y)$, and gradients $\frac{\partial L}{\partial w^{(l)}}$.
With the aforementioned data the white-box MI adversary trains a binary classifier, the \textit{attack model}. The attack model allows to classify records into members and non-members w.r.t.~the target model and training dataset. The adversary is assumed to know a portion of the training and test data $\mathcal{D}_{target}^{train}$ and $\mathcal{D}_{target}^{test}$, and generates features for training the attack model by passing the known records repeatedly through the trained target model.
Nasr et al.~\cite{nasrComprehensivePrivacyAnalysis2018} assumed the portion of known records at 50\% and we follow this assumption to allow comparison. The performance metrics of an MI attack model are typically evaluated on a balanced dataset including members (target model training data) and an equal number of non-members (target model test data). An illustration of the data preparation for the white-box MI attack and its evaluation is shown in Figure~\ref{fig:wb_mia}.
\begin{figure}
\centering
\begin{tikzpicture}[thick,scale=0.85, every node/.style={transform shape}]
\footnotesize
\node[] (aa) {};
\node[draw,cylinder,shape border rotate=90, minimum width=4.5em, dashed, above= 0.25ex of aa, shape aspect=.10, fill=violet!10] (a) {$\mathcal{D}_{target}^{train}$};
\node[minimum width=4em, above=0ex of a] (uu) {$\mathcal{D}_{target}$};
\node[draw,cylinder,shape border rotate=90, minimum width=4.5em, below= 0.25ex of aa, shape aspect=.10, fill=green!10] (b) {$\mathcal{D}_{target}^{test}$};
\node[draw,cylinder,shape border rotate=90, minimum width=4.5em, right= 10em of a, shape aspect=.10, fill=violet!10] (c) {$\mathcal{D}_{target}^{\var{in}}$};
\node[draw,cylinder,shape border rotate=90, minimum width=4.5em, right= 10em of b, shape aspect=.10, fill=green!10] (d) {$\mathcal{D}_{target}^{\var{out}}$};
\node[draw,cylinder,shape border rotate=90, minimum width=4.5em, right= 3em of c, shape aspect=.10, double color fill={violet!10}{green!10}, shading angle=45] (e) {$\mathcal{D}_{attack}^{train}$};
\node[draw,cylinder,shape border rotate=90, minimum width=4.5em, right= 3em of d, shape aspect=.10, double color fill={violet!10}{green!10}, shading angle=45] (f) {$\mathcal{D}_{attack}^{test}$};
\node[minimum width=4em, above=0ex of e] (uu) {$\mathcal{D}_{attack}$};
\node[draw,chamfered rectangle, dashed, align=center, right=5em of aa] (z) {target \\ model};
\node[draw,chamfered rectangle, align=center, right=17em of z] (y) {attack \\ model};
\draw [->] (a.east) -- node [near start, above=1.5ex,sloped]{\small training}(z.west);
\draw [->] (b.east) -- (z.west);
\draw [->] (z.east) -- node [near end, above=2ex,sloped]{\small inference}(c.west);
\draw [->] (z.east) -- (d.west);
\draw [->] (c.east) -- (e.west);
\draw [->] (c.east) -- (f.west);
\draw [->] (d.east) -- (e.west);
\draw [->] (d.east) -- (f.west);
\draw [->] (e.east) -- (y.west);
\draw [->] (f.east) -- (y.west);
\end{tikzpicture}
\caption{White-box MI with attack features.
DP perturbation is applied on the target model training (dashed). The data that was used by the data owner during target-model training is colored: training (violet) and validation (green).}
\label{fig:wb_mia}
\end{figure}
\subsection{Hierarchical Text Classification}
\label{sec:htc}
The task of \emph{text classification} consists in categorizing texts into a pre-defined set of categories that do not have any hierarchical structure. Text classification is crucial in NLP, \cite{manningFoundationsStatisticalNatural}, a subfield of linguistics, computer science, and artificial intelligence
that is concerned with how computers can be programmed to process and analyze natural language data.
On the other hand, HTC addresses the task of classifying a text document into a \emph{hierarchy of classes and sub-classes}. Text classification can therefore be regarded as a special case of HTC with only one hierarchy level and no sub-categories. Hierarchical classification problems can be categorized on the basis of their \emph{hierarchy structure}, \emph{label type} and \emph{label depth}. In this work, we shall consider \textit{tree} hierarchies with \textit{single} \textit{partial-depth} labels, which means that every text shall be assigned to a single label that can be any node in a tree hierarchy.
In many circumstances, hierarchical classification may be desirable as categorizing documents into varying levels of abstraction better fits the nature of certain applications, e.g., product categorization or support-ticket classification.
Also, as it has been consistently shown by numerous cognitive studies~\cite{murphy2004big}, people tend to favor categorization at different levels of abstraction.
Formally, HTC comprises a collection of text documents $x_1,\ldots,x_j\in \mathbb{X}$, where $\mathbb{X}$ is a document space; and a fixed set of classes $\mathbb{Y} = \{ y_1,y_2,\ldots,y_k \}$ belonging to some hierarchy. Given a training set of labeled documents $(x_1,y_1),\ldots,(x_n,y_n)$ on a hierarchy, where $(x_i,y_i) \in \mathbb{X} \times \mathbb{Y}$, we wish to learn a classifier or classification function $\gamma $ that maps documents to classes,
\begin{displaymath}
\gamma: \mathbb{X} \rightarrow \mathbb{Y},
\end{displaymath}
\noindent
so that each text document is only assigned to a single label, and a certain utility metric (see Sec.~\ref{sec:methodology:utility}) is maximized.
\subsection{Embeddings}
\label{sec:embeddings}
ANNs employ \emph{embeddings} in the first layer to capture the meaning of each token. An embedding is a token's vector representation of length $n$, that embeds the token into an $n$-dimensional vector space~\cite{goyalDeepLearningNatural2018}. Although embeddings have the ability to map semantically similar tokens to the same region in the vector space, a common shortcoming is that a word is always assigned to the same vector, ignoring previous and subsequent words (e.g., see Word2Vec~\cite{mikolovEfficientEstimationWord2013,penningtonGloveGlobalVectors2014}).
Word embeddings enable transfer learning, meaning that they can be trained on large unlabeled text corpora and afterwards be used in NLP systems for various tasks. In the case of ANNs, this is achieved by pre-initializing the first layer of an ANN with the pre-trained word embeddings.
\section*{Appendix: Additional Figures}
\label{app:figures}
\begin{figure*}[ht!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/bestbuy_cleaned_whf_mi.pdf}
\caption{MI against BestBuy over $\ensuremath{\epsilon}$}
\label{fig:bestbuy_mi_whf}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/reuters_whf_mi.pdf}
\caption{MI against Reuters over $\ensuremath{\epsilon}$}
\label{fig:reuters_mi_whf}
\end{subfigure}
\begin{subfigure}{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/DBPEDIA_whf_mi.pdf}
\caption{MI against DBPedia over $\ensuremath{\epsilon}$}
\label{fig:dbpedia_mi_whf}
\end{subfigure}
\caption{MI AUC, $Adv$ and Bound on MI $Adv$ per dataset with additional hierarchical attack features}
\label{fig:appendix}
\end{figure*}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Homepage > In the Public Interest > Commandments for Bureaucrats
Commandments for Bureaucrats
When Herb Denenberg was Pennsylvania's Insurance Commissioner during the year 1971-1974, the insurance companies dubbed him "Horrible Herb." There is even a story that some insurance executives, while out on the golf course, would roar "DENENBERG" instead of an expletive whenever they muffed a swing. Well, the former University of Pennsylvania insurance scholar now turned television consumer editor came to Washington recently to speak at our Public Citizen Forum on how former public interest advocates were doing in the Carter Administration. In his customary satiric fashion, he delivered to the gathering "An Action Guide for Consumer Advocates Who Become Government Officials."
The guide deserves a wider audience — like anyone who works
in government or who wants to see government work. (Denenberg is willing to send you a copy on receipt
of a stamped, self-addressed envelope sent to him at Station WCAUTV, Phila., Pa. 19131.)
DRAWING FROM his own pioneering experience as a state government official, he said that there was too much talk among bureaucrats about how complicated problems are. That excuse, he observed, means that they either "don't understand the problem or that they can't explain it to me. So why not try to solve the simple problems first?"
Denenberg drew knowing laughter when he asserted that the real challenge was not how to .get something done with a competent staff, but was how to accomplish things with an incompetent staff. In his regulatory position, he worked as if he were going to be fired tomorrow. That, he declared, gave him a sense of urgency and priority.
Indeed, Denenberg's tenure as insurance commissioner was innovative and highly productive for consumer justice. Many of his ideas have been carried into other state regulatory agencies around the country. So he knows what he is talking about.
Here are some of his tips to bureaucrats:
1. "Unleash a salvo of criticism (of the industry) immediately. A fast start is necessary to build public support to head off the inevitable calls for resignation. Only if the public understands what you are trying to accomplish and what the ties are will you be able to survive." He noted that for a century, the insurance industry thought that the state insurance regulator's sole function was to "whip through their requests for illegal and exorbitant rate increases."
2. "Tell the truth once in a while. As H. L. Mencken has said, injustice is relatively easy to bear. What stings is justice. You should describe what's going on in precise and nontechnical terms that will have the most impact."
3. "Do things in a hurry that ought to be done in a hurry." Once Denenberg cancelled on the spot at a public hearing the contract between Blue Cross and the hospitals because of the widespread waste allowed. The renegotiated contract saved the public millions of dollars and advanced the quality of medical care.
4. "Communicate effectively with the public. That's necessary, not only to survive and establish a supportive constituency, but it is also an efficient way of bringing about change." To illustrate this point, Denenberg noted the "Shopper's Guide Services" that he issued on life insurance, auto, homeowners and health insurance, hospitals, dentistry, and surgery. He used to rank companies by name according to the size of the premiums charged.
5. "Don't work quietly within the system. Translated, that means 'don't get conned and co-opted into doing nothing and doing it slowly.'
"THERE IS traditionally too much working within the system in the sense of back-room manipulation rather than public dialogue. It is the public, not the wheeler-dealers in smoke-filled back rooms, that are supposed to determine public policy. As Insurance Commissioner of Pennsylvania, I was always told by industry lobbyists that I could accomplish more by quietly working behind the scenes rather than by taking my case to the public. Undoubtedly, I could have accomplished more behind the scenes, but not for the consumers."
The wisdom of Denenberg's remarks is salted with humor and his actual experience. But he made one point with which I disagree: "If you follow my guide," he concluded, "your time as a government official will be eventful but you will be unappointable to any other post." The more present public servants behave the way Denenberg suggests, the more appointable people conscience and dedication will be throughout government.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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import sys
from django.core.management.color import color_style
from django.db import IntegrityError, migrations, transaction
from django.db.models import Q
WARNING = """
A problem arose migrating proxy model permissions for {old} to {new}.
Permission(s) for {new} already existed.
Codenames Q: {query}
Ensure to audit ALL permissions for {old} and {new}.
"""
def update_proxy_model_permissions(apps, schema_editor, reverse=False):
"""
Update the content_type of proxy model permissions to use the ContentType
of the proxy model.
"""
style = color_style()
Permission = apps.get_model("auth", "Permission")
ContentType = apps.get_model("contenttypes", "ContentType")
alias = schema_editor.connection.alias
for Model in apps.get_models():
opts = Model._meta
if not opts.proxy:
continue
proxy_default_permissions_codenames = [
"%s_%s" % (action, opts.model_name) for action in opts.default_permissions
]
permissions_query = Q(codename__in=proxy_default_permissions_codenames)
for codename, name in opts.permissions:
permissions_query |= Q(codename=codename, name=name)
content_type_manager = ContentType.objects.db_manager(alias)
concrete_content_type = content_type_manager.get_for_model(
Model, for_concrete_model=True
)
proxy_content_type = content_type_manager.get_for_model(
Model, for_concrete_model=False
)
old_content_type = proxy_content_type if reverse else concrete_content_type
new_content_type = concrete_content_type if reverse else proxy_content_type
try:
with transaction.atomic(using=alias):
Permission.objects.using(alias).filter(
permissions_query,
content_type=old_content_type,
).update(content_type=new_content_type)
except IntegrityError:
old = "{}_{}".format(old_content_type.app_label, old_content_type.model)
new = "{}_{}".format(new_content_type.app_label, new_content_type.model)
sys.stdout.write(
style.WARNING(WARNING.format(old=old, new=new, query=permissions_query))
)
def revert_proxy_model_permissions(apps, schema_editor):
"""
Update the content_type of proxy model permissions to use the ContentType
of the concrete model.
"""
update_proxy_model_permissions(apps, schema_editor, reverse=True)
class Migration(migrations.Migration):
dependencies = [
("auth", "0010_alter_group_name_max_length"),
("contenttypes", "0002_remove_content_type_name"),
]
operations = [
migrations.RunPython(
update_proxy_model_permissions, revert_proxy_model_permissions
),
]
|
{
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| 7,044
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Смычка — посёлок в Туринском городском округе Свердловской области, России.
Географическое положение
Посёлок Смычка муниципального образования «Туринского городского округа» расположен в 5 километрах к востоку от города Туринска (по автотрассе — 8 километров), на левом берегу реки Тура, в 2 километрах к востоку от озер-стариц. В окрестностях посёлка проходит автодорога Туринск – Тавда, а в 1,5 километре от посёлка находится железнодорожная станция «о.п. 260 км» Восточно-Уральской железной дороги.
История посёлка
В советское время в посёлке работала узкоколейка Смычка – Фабричное, в настоящий момент полуразобрана.
Население
Примечания
Населённые пункты Туринского городского округа
|
{
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| 4,407
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Jintana Resort Buriram a pure natural resort in the heart of Buriram city. Major attractions of Buriram are reachable within arresting distance and at very convenience.
Jintana resort is a contemporary and distinctive resort designed in a beautiful landscape. This resort is perfectly set in the natural beauty of the surrounding area with the additional charm of the garden and caters 23 clean and comfortable rooms with well equipped facilities to rejuvenate your stay.
This place is quite conveniently located close to the bus station about 500 metres. It is a nice and quiet place, with good facilities, and a decent breakfast. Plenty of restaurants on the road outside some 200 metres distant.
Pleasant surprise, very good value for money, clean and spacious..
Very impressed with the check in. Receptionist spoke very good English. The upgrade was much appreciated. Hotel very close to bus station although a bit further to train station. There is a lot of building work taking place at the moment with work starting at 8am however this didn t disturb me as I was up and out early myself. I will stay at this hotel again.
Imagine a quiet place, surround by beautiful landscape design. Jintana Resort sits in the middle of Buriram City, within 10 minutes range you can reach any major attractions in the city.
Our hotel is only 300 m. or 3 mins away by foot from Buriram Bus Terminal and only 2.5 Km. away from I-Mobile Stadium!
Resorts offers you 49 rooms with 7 different types range from Typical Thai to Modern with all you need facilities.
Deluxe Room is the best economy room that you can find featuring 25 sqm. with 32" LCD TV installed on the wall. Large size table bar makes your working time entirely convenient.
Be your own self in a separate and private area with balcony in front of the room; take beautiful view of the small pond right in front of the accommodation. The room is fully equipped with 29" television with cable network, air-conditioner, Wi-Fi system, hot shower, and refrigerator.
Introduce you to our newest Villa at Jintana Resort, Studio Villa featuring a high class small suite style room furnished with luxurious king size bed and bathroom. You can relax at the sofa or enjoy your working space at 90 degree working desk.
Equipped with fully integrated technologies that the world has to offer : LCD TV 43", Play movies from flash drive, Wireless Internet In Room, Refrigerator, Hot Shower, Built in Closet, Private Parking Space, Large Front Window and Lake Front View! Make your days special at Jintana Resort Hotel Buriram with new Panorama Luxury Rooms.
The most prestigious room of our resort beautifully designed with old fashion Thai style. Thai Chalet room features 35 Sq.m. the accommodation surrounded by balcony for relaxation. The room is equipped with full facilities; Wi-Fi system, 29" Television with cable network, hot shower, air-conditioner, and refrigerator.
Jintana Resort Buriram situates on the middle of Buriram City on the north east of Thailand, 400 km from Bangkok and 120 km from Nakonratchasim a, Buriram has a number of leading attraction, one that you cannot really miss is Phanom Rung stone castle which is only 1 hour driving from the resort.
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{
"redpajama_set_name": "RedPajamaC4"
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| 9,808
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«Злой город» () — американский телевизионный сериал, который вышел на ABC в сезоне 2015—2016 годов. Сериал транслировался по вторникам в десять вечера после сериала «Агенты «Щ.И.Т.»», начиная с 27 октября 2015 года. В центре сюжета находятся поиски Департаментом полиции Лос-Анджелеса пары серийных убийц в стиле Бонни и Клайда, которые терроризируют обитателей Сансет Стрип в 1982 году.
Сериал получил крайне негативные отзывы от критиков и дебютировал с низким демографическим рейтингом 0,9 в категории 18-49. 13 ноября 2015 года ABC закрыл проект и снял его с эфира после трех эпизодов. Оставшиеся пять эпизодов были выпущены на Hulu в декабре 2015 года.
Производство
Разработка
В сентябре 2014 года было объявлено, что Стивен Бейгелман и Mandeville Television продали сценарий криминальной антологии для ABC. 23 января 2015 года канал заказал съемки пилотного эпизода, режиссёром которого выступил Том Шенкленд. Съемки пилота начались 16 марта 2015 года Лос-Анджелесе, штат Калифорния и завершились 2 апреля 2015 года. 7 мая 2015 года канал утвердил пилот и заказал съемки первого сезона.
Кастинг
Объявления о подборе актёров начались в феврале 2015 года. 24 февраля было объявлено, что Эрика Кристенсен будет играть медсестру и мать, которая становится серийным убийцей. На следующий день Таисса Фармига и Даррелл Бритт-Гибсон получили роли журналистов, которые разведывают подробности о случаях убийств на Сансет Стрип. 3 марта 2015 года Каролина Выдра получила роль молодого детектива, а день спустя было объявлено, что ведущего детектива будет играть Адам Ротенберг, тогда как Холли Файн — его жены. 11 марта 2015 года Энн Уинтерс получила роль их дочери, а Габриэль Луна — ещё одного детектива и партнера Ротенберга. 12 марта 2015 года было объявлено, что роль серийного убийцы будет играть Эд Вествик.
После майских апфронтов ABC решил произвести небольшие изменения в актерском составе. 18 мая 2015 года проект покинули Ротенберг, Файн и Бритт-Гибсон. 6 июля 2015 года Джереми Систо сменил Ротенберга, а Эван Росс — Бритта-Гибсона. Файн в свою очередь сменила Джейми Рэй Ньюман 13 августа.
Актёры и персонажи
Основной состав
Эрика Кристенсен в роли Бетти Смит
Эд Вествик в роли Кента Гэллуэйя
Джереми Систо в роли детектива Джека Рота
Таисса Фармига в роли репортера Карен Макларен
Гэбриель Луна в роли детектива Пако Контрераса
Каролина Выдра в роли детектива Дайан Гиббонс
Эван Росс в роли Дивера Хоукса
Энн Уинтерс в роли Вики Рот
Джейми Рэй Ньюман в роли Эллисон Рот
Второстепенный состав
Дэвид Салливан в роли детектива Арнольда Буковски
У. Эрл Браун в роли капитана Уилкинсона
Примечания
Ссылки
Телесериалы США, запущенные в 2015 году
Программы телеканала American Broadcasting Company
Криминальные телесериалы США
Телесериалы-антологии США
Телесериалы, сюжет которых разворачивается в Лос-Анджелесе
Телесериалы на английском языке
Телесериалы США 2010-х годов
Телесериалы ABC Studios
Телесериалы США, завершённые в 2015 году
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 4,067
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\section{Introduction}
Given a simple graph $G$ with vertex set $V(G)$ and edge set $E(G)$, we use $N_G(v)$ to denote the set of neighbors of $v$ in $G$ and say that $d_G(v)=|N_G(v)|$ is the \emph{degree of $v$} in $G$. A \emph{planar graph} is a graph admitting a drawing in the plane with no crossing and typically we say such a drawing a \emph{plane graph}. By $F(G)$ we denote the face set of a plane graph $G$, and for any face $f\in F(G)$, we use $d_G(f)$ to denote the \emph{degree of $f$} in $G$, which is the number of edges that are incident with $f$ in $G$ (cut-edges are counted twice). By $V_G(f)$, we denote the set of vertices incident with a face $f$ in a plane graph $G$.
A $t$-, $t^+$-, or $t^-$-vertex (resp.\,face) is a vertex (resp.\,face) of degree $t$, at least $t$, or at most $t$, respectively.
A \emph{proper $\ell$-coloring} of a graph $G$ is a coloring on $V(G)$ using $\ell$ colors so that adjacent vertices receive distinct colors. If every vertex of degree at least two is incident with at least two colors, then we call this proper $\ell$-coloring a \emph{dynamic $\ell$-coloring}. The minimum integer $\ell$
such that $G$ has a proper (resp.\,dynamic) $\ell$-coloring is the \emph{chromatic number} (resp.\,\emph{dynamic chromatic number}) of $G$, denoted by $\chi(G)$ (resp.\,$\chi^d(G)$).
The well-known four color theorem states that $\chi(G)\leq 4$ for every planar graph $G$. In 2013,
Kim, Lee, and Park \cite{Kim20132207} proved that $\chi^d(G)\leq 5$ for every planar graph $G$
and the equality holds if and only if $G\cong C_5$, answering a conjecture of Chen \emph{et al.}\,\cite{Chen20121064}. Furthermore, the same conclusion holds even for $K_5$-minor-free graphs, which was proved by Kim, Lee and Oum \cite{Kim201681} in 2016. For other results on the dynamic coloring of graphs, we refer the reads to \cite{Ahadi20122579,Alishahi2011152,Borowiecki2012105,Bowler2017151,Chen20121064,Karpov2011601,Lai2003193,Loeb2018129,Meng20063,Montgomery2001,Saqaeeyan2016249,Vlasova201821}.
Imaging that each vertex $v\in V(G)$ is assigned a \emph{list} $L(v)$ of distinct candidate colors, our goal is to color the vertices of $G$ so that every vertex receives color from its list assignment and the resulting coloring of $G$ is a proper (resp.\,dynamic) coloring. If we win for a given list assignment $L$ to $V(G)$, then $G$ is \emph{$L$-colorable} (resp.\,\emph{dynamically $L$-colorable}). Furthermore, if we win for every given list assignment $L$ to $V(G)$ with $|L(v)|=\ell$ for each $v\in V(G)$, then $G$ is \emph{$\ell$-choosable} (resp.\,\emph{dynamically $\ell$-choosable}). The minimum integer $\ell$ so that $G$ is $\ell$-choosable (resp.\,dynamically $\ell$-choosable) is the \emph{list chromatic number} (resp.\,\emph{dynamic list chromatic number}) of $G$, denoted by ${\rm ch}(G)$ (resp.\,${\rm ch}^d(G)$).
Thomassen's theorem \cite{Thomassen1994180} states that ${\rm ch}(G)\leq 5$ for every planar graph $G$, and the sharpness of this upper bound $5$ was confirmed by Voigt \cite{Voigt1993215}, who constructed a planar graph $G$ with $\chi(G)=4$ and ${\rm ch}(G)=5$. This reminds us that $\chi(G)$ and ${\rm ch}(G)$ are not always the same, even for planar graphs. Similarly, Esperet \cite{Esperet20101963} showed that there is a planar bipartite graph $G$ with ${\rm ch}(G)=\chi^d(G)=3$ and ${\rm ch}^d(G)=4$, and moreover, there exists for every $k\geq 5$ a bipartite graph $G_k$ with ${\rm ch}(G_k)=\chi^d(G_k)=3$ and ${\rm ch}^d(G_k)\geq k$. Hence the gap between $\chi^d(G)$ (or ${\rm ch}(G)$) and ${\rm ch}^d(G)$ can be any large. For further interesting readings on the dynamic list coloring of graphs, we refer the readers to \cite{Alishahi2011152,Kim20132207,Kim2011156}.
A \emph{$2$-subdivision} of a graph $G$ is the graph derived from $G$ by inserting on each edge a new vertex of degree two, denoted by $G^{\star}$. One can see Figure \ref{K7} for an example of $K_7^{\star }$, which is 1-planar.
\begin{figure}
\centering
\includegraphics[width=8cm]{K7-subdivision}
\caption{A 1-planar drawing of $K_7^{\star }$}
\label{K7}
\end{figure}
\begin{fact}\label{fact-1}
For any graph $G$, $\chi(G)\leq \chi^d(G^{\star}) $ and ${\rm ch}(G)\leq {\rm ch}^d(G^{\star})$.
\end{fact}
\begin{proof}
Let $M: V(G) \rightarrow V(G^{\star})$ be a mapping that maps a vertex of $G$ to the vertex of $G^{\star}$ corresponding to it, and let $S\subset V(G^{\star})$ be the set of new added 2-vertices to $G$ while doing the 2-subdivision. Let $L$ be an arbitrary $\ell$-list assigment on $V(G)$, where $\ell={\rm ch}^d(G^{\star})$.
We extend $L$ to an $\ell$-list $L^\star$ on $V(G^\star)$, i.e, $L^\star(u)=L(M^{-1}(u))$ for any $u\in V(G^{\star})\backslash S$.
Since the two neighbors of a 2-vertex of $G^\star$ shall be colored with distinct colors in any dynamic coloring of $G^\star$, there is a dynamic coloring $c^\star$ of $G^\star$ so that $c^\star(M(u))\in L^\star(M(u))=L(u)$, $c^\star(M(v))\in L^\star(M(v))=L(v)$, and $c^\star(M(u))\neq c^\star(M(v))$ for any $uv\in E(G)$. Therefore, we construct an $L$-coloring $c$ of $G$ by letting $c(v)=c^\star(M(v))$ for any $v\in V(G)$. This implies that
${\rm ch}(G)\leq \ell={\rm ch}^d(G^{\star})$.
The proof for $\chi(G)\leq \chi^d(G^{\star})$ is similar (we just proceed by fixing every $\ell$-list used in the privious proof to be $\{1,2,\ldots,\ell\}$).
\end{proof}
Note that the equality in Fact \ref{fact-1} does not always hold. One easy example is the cycle $C_n$ on $n$ vertices. Since
$C_n^\star=C_{2n}$ and it is known \cite{Akbari20093005,Lai2003193,Montgomery2001} that
\begin{align*}
\chi^d(C_{2n})={\rm ch}^d(C_{2n})
&=
\begin{cases}
3
&\text{if } n \equiv 0 ~({\rm mod~3}),
\\
4
&\text{if } n \not\equiv 0 ~({\rm mod~3}),
\end{cases}
\end{align*}
we have
\begin{align*}
\chi^d(C_n^{\star})-\chi(C_n)={\rm ch}^d(C_n^{\star})-{\rm ch}(C_n)
&=
\begin{cases}
0
&\text{if } n \equiv 3 ~({\rm mod~6}),
\\
1
&\text{if } n \equiv 0,1,5 ~({\rm mod~6}),
\\
2
&\text{if } n \equiv 2,4 ~({\rm mod~6}).
\end{cases}
\end{align*}
A graph is \emph{$k$-planar} if it can be drawn in the plane so that each edge is crossed at most $k$ times. Specially, the 1-planarity was initially introduced by Ringel \cite{Ringel1965107} in 1965, who proved that $\chi(G)\leq 7$ for every 1-planar graph and conjectured that every 1-planar graph is 6-colorable. This conjecture was solved by Borodin \cite{Borodin1984} in 1984, who also gave a new proof \cite{Borodin1995507} in 1995. Due to the 1-planar graph $K_6$, the upper bound 6 for the chromatic number of the class of 1-planar graphs is sharp. Since 2006, the list coloring of 1-planar graphs was also investigated by many researchers including Albertson and Mohar \cite{Albertson2006289}, Wang and Lih \cite{Wang200827}. In particular, the second group \cite{Wang200827} proved that ${\rm ch}(G)\leq 7$ for every 1-planar graph $G$. Actually, the class of 1-planar graphs is among the most investigated graph families within the so-called ``beyond planar graphs", see \cite{Didimo2019}. For those who want to know more about 1-planar graphs, we refer them to a recent survey due to Kobourov, Liotta and Montecchiani \cite{Kobourov201749}.
Let $\mathcal{G}_k$ be the class of graphs that are $k$-planar and non-$(k-1)$-planar. By $\chi(\mathcal{G}_k)$ we denote the minimum integer $\ell$ so that $\chi(G)\leq \ell$ for each $G\in \mathcal{G}_k$. Similarly, we can define $\chi^d(\mathcal{G}_k)$, ${\rm ch}(\mathcal{G}_k)$, and ${\rm ch}^d(\mathcal{G}_k)$.
If $G\in \mathcal{G}_{k+1}$ with $k\geq 1$, then it is easy to see that the 2-subdivision of $G$ is $k$-planar.
Pach and Tóth \cite{Pach1997427} showed that $|E(G)|\leq 5|V(G)|-10$ for each 2-planar graph $G$. This implies that each 2-planar graph $G$ has a vertex of degree at most $9$ and thus $\chi(G)\leq {\rm ch}(G)\leq 10$. Since $K_7$ is a non-1-planar 2-planar graph, $7\leq \chi(\mathcal{G}_2)\leq 10$ and $7\leq {\rm ch}(\mathcal{G}_2)\leq 10$.
We now look back at Fact \ref{fact-1}.
If there is a 2-planar graph $G$ with $\chi(G)=\ell$ (resp.\,${\rm ch}(G)=\ell$), then $G^\star$ is a 1-planar graph with $\chi^d(G^\star)\geq \ell$ (resp.\,${\rm ch}^d(G^\star)\geq \ell$). This implies
\begin{fact}\label{fact-2}
$\chi^d(\mathcal{G}_1)\geq \chi(\mathcal{G}_2)\geq 7$ and ${\rm ch}^d(\mathcal{G}_1)\geq {\rm ch}(\mathcal{G}_2)\geq 7$.
\end{fact}
The aim of this paper is to give a reasonable upper bound, say 11, for ${\rm ch}^d(\mathcal{G}_1)$ (note that $\chi^d(\mathcal{G}_1)\leq {\rm ch}^d(\mathcal{G}_1)$). In other words, we prove the following.
\begin{thm}\label{main-thm}
If $G$ is a 1-planar graph, then ${\rm ch}^d(G)\leq 11$.
\end{thm}
\section{Dynamically Minimal Graphs}
A graph class $\mathcal{F}$ is \emph{hereditary} if $\mathcal{F}$ is closed by taking subgraphs.
A graph $G$ is \emph{dynamically $\ell$-minimal in a hereditary class $\mathcal{F}$} if $G\in \mathcal{F}$ is not dynamically $\ell$-choosable and any graph $H\in \mathcal{F}$ with $|V(H)|+|E(H)|<|V(G)|+|E(G)|$ is dynamically $\ell$-choosable.
In this section, we use $\mathcal{G}_{1}^-$ to stand the class of 1-planar graphs, i.e., $\mathcal{G}_{1}^-=\mathcal{G}_0\cup \mathcal{G}_1$.
Suppose that $G$ is a dynamically $\ell$-minimal graph in $\mathcal{G}_{1}^-$,
It follows that $G$ is a 1-planar graph with the smallest value of $|V(G)|+|E(G)|$ such that
there is an $\ell$-list assignment $L$ to the vertices of $G$ such that $G$ is not dynamically $L$-colorable. Moreover, we assume that $G$ is a \emph{$1$-plane graph} (i.e, a drawing of $G$ in the plane so that its 1-planarity is satisfied) that has the minimum number of crossings.
The \emph{associated plane graph} $G^{\times}$ of a 1-plane $G$ is the plane graph derived from $G$ by turning all crossings into new vertices of degree 4, and those 4-vertices in $G^{\times}$ are called \emph{false vertices}. If a vertex of $G^{\times}$ is not false, then it is a \emph{true vertex}. A face of the plane graph $G^{\times}$ is a \emph{false face} if it is incident with at least one false vertex, and is a \emph{true face} otherwise. Clearly, no two false vertices are adjacent in $G^{\times}$ by the definition of the 1-planarity and each face $f$ of $G^{\times}$ is incident with at most $d_{G^{\times}}(f)/2$ false vertices.
In the following statements or the proofs of the propositions, $\mathcal{F}$ stands for an arbitrary given hereditary graph class, and $L$ is the $\ell$-list assignment mentioned above.
\begin{prop} \label{min-deg}
If $G$ is a dynamically $\ell$-minimal graph in $\mathcal{F}$ with $\ell\geq 3$, then $\delta(G)\geq 2$.
\end{prop}
\begin{proof}
Suppose, to the contrary, that $G$ has an edge $uv$ with $d_G(u)=1$. By the minimality of $G$, $G'=G-u\in \mathcal{F}$ is dynamically $L$-colorable. Let $c$ be a dynamic $L$-coloring of $G'$. Coloring $u$ from $L(u)$ with a color different from the colors on $v$ and a neighbor of $v$ besides $u$, we obtain a dynamic $L$-coloring of $G$, a contradiction.
\end{proof}
\begin{prop} \label{no-two-two}
If $G$ is a dynamically $\ell$-minimal graph in $\mathcal{F}$ with $\ell\geq 5$, then no two $2$-vertices are adjacent in $G$.
\end{prop}
\begin{proof}
Suppose, to the contrary, that $G$ has an edge $uv$ with $d_G(u)=d_G(v)=2$. Let $N_G(u)=\{v,x\}$, $N_G(v)=\{u,y\}$, $x_1\in N_G(x)\backslash \{u\}$, and $y_1\in N_G(y)\backslash \{v\}$. By the minimality of $G$, $G'=G-\{u,v\}\in \mathcal{F}$ has a dynamic $L$-coloring $c$. Coloring $u$ with $c(u)\in L(u)\backslash \{c(x),c(y),c(x_1)\}$ and $v$ with $c(v)\in L(v)\backslash \{c(u),c(x),c(y),c(y_1)\}$, we get a dynamic $L$-coloring of $G$, a contradiction.
\end{proof}
Actually, Proposition \ref{no-two-two} can be generalized to the following.
\begin{prop} \label{edge-with-big-vertex}
If $G$ is a dynamically $\ell$-minimal graph in $\mathcal{F}$ with $\ell\geq 5$, then each edge of $G$ is incident with at least one $\ell^+$-vertex.
\end{prop}
\begin{proof}
We first claim that if $uv\in E(G)$ and $d_G(u)=2$, then $d_G(v)\geq \ell$.
Suppose, to the contrary, that $d_G(v)\leq \ell-1$. Let $N_G(u)=\{v,z\}$ and $N_G(v)=\{u,x_1,\ldots, x_{t},y_1,\ldots,y_s\}$, where $d_G(x_i)=2$ for each $1\leq i\leq t$ and $d_G(y_i)\geq 3$ for each $1\leq i\leq s$. Let $N_G(x_i)=\{v,x'_i\}$ for each $1\leq i\leq t$. Note that $t$ or $s$ may be 0, in which case $N_G(v)=\{u,y_1,\ldots,y_s\}$ or $N_G(v)=\{u,x_1,\ldots,x_t\}$, respectively.
By Proposition \ref{no-two-two}, $t+s\geq 2$,
$d_G(z)\geq 3$, and $d_G(x'_i)\geq 3$ for each $1\leq i\leq t$.
By the minimality of $G$, $G'=G-\{u,v,x_1,\ldots,x_t\}\in \mathcal{F}$ has a dynamic $L$-coloring $c$.
Color $v,x_1,\ldots,x_t,u$ in this order with colors
$c(v),c(x_1),\ldots,c(x_t),c(u)$ such that
$c(v)\in L(v)\backslash F(v)$, where $F(v)=\{c(z),c(x'_1),\ldots,c(x'_t),c(y_1),\ldots,c(y_s)\}$,
$c(x_i)\in L(x_i)\backslash \{c(v),c(x'_i)\}$ for each $1\leq i\leq t$, and
$c(u)\in L(u)\backslash \{c(z),c(v),c(x_1)\}$ if $t\neq 0$, or
$c(u)\in L(u)\backslash \{c(z),c(v),c(y_1)\}$ if $t=0$.
Note that $|F(v)|=1+t+s=d_G(v)\leq \ell-1$. It is easy to see that this results in a dynamic $L$-coloring of $G$, a contradiction.
We come back to the proof of Proposition \ref{edge-with-big-vertex}. Suppose, to the contrary, that $G$ has an edge $uv$ with $d_G(u)=a\leq d_G(v)=b\leq \ell-1$. Let $N_G(u)=\{v,u_1,\ldots, u_{a-1}\}$ and $N_G(v)=\{u,v_1,\ldots, v_{b-1}\}$. By the arguments in the first paragraph, $a,b\geq 3$ and $d_G(u_i),d_G(v_j)\geq 3$ for each $1\leq i\leq a-1$ and $1\leq j\leq b-1$.
By the minimality of $G$, $G'=G-\{u,v\}\in \mathcal{F}$ has a dynamic $L$-coloring $c$. Without loss of generality, assume that $c(v_1)\leq c(v_2)\leq \cdots \leq c(v_{b-1})$.
If $c(v_1)\neq c(v_{b-1})$, then we construct a dynamic $L$-coloring of $G$ by coloring $v$ and $u$ in order with $c(v)$ and $c(u)$ such that
$c(v)\in L(v)\backslash F(v)$ and $c(u)\in L(u)\backslash F(u)$, where $F(v)=\{c(v_1),\ldots,c(v_{b-1}),c(u_1)\}$ and $F(u)=\{c(u_1),\ldots,c(u_{a-1}),c(v)\}$.
If $c(v_1)= c(v_{b-1})$, then we construct a dynamic $L$-coloring of $G$ by coloring $u$ and $v$ in order with $c(u)$ and $c(v)$ such that
$c(u)\in L(u)\backslash F(u)$ and $c(v)\in L(v)\backslash F(v)$, where $F(u)=\{c(u_1),\ldots,c(u_{a-1}),c(v_1)\}$ and $F(v)=\{c(v_1),c(u),c(u_1)\}$.
In each of the above two cases we win since $|F(u)|=a\leq \ell-1$ and $|F(v)|\leq b\leq \ell-1$. So we have contradictions.
\end{proof}
\begin{prop}\label{true-3-face}
If $G$ is a dynamically $\ell$-minimal graph in $\mathcal{F}$ with $\ell\geq 5$ and $u$ is a vertex incident with a triangle,
then $d_G(u)\geq \ell$.
\end{prop}
\begin{proof}
Suppose, to the contrary, that $f=uvw$ is a triangle such that $d_G(u)\leq \ell-1$. By the minimality of $G$, $G'=G-u
\in \mathcal{F}$ has a dynamic $L$-coloring $c$. By Proposition \ref{edge-with-big-vertex}, $d_G(v),d_G(w)\geq \ell$.
Let $N_G(u)=\{v,w,x_1,\ldots, x_{t}\}$. Since $d_G(u)\leq k-1$, $d_G(x_i)\geq \ell$ for each $1\leq i\leq t$ by Proposition \ref{no-two-two}. Extending $c$ to a dynamic $L$-coloring of $G$ by coloring $u$ with a color $c(u)\in L(u)\backslash F(u)$, where $F(u)=\{c(v),c(w),c(x_1),\ldots, c(x_t)\}$, we find a contradiction. Note that $|F(u)|=t+2=d_G(u)\leq \ell-1$.
\end{proof}
\begin{prop}\label{false-3-face}
If $G$ is a dynamically $\ell$-minimal graph in $\mathcal{G}_{1}^-$ with $\ell\geq 5$ and $u$ is a vertex incident with a false $3$-face of $G^{\times}$, then either $u$ is false or $d_G(u)\geq \ell-2$.
\end{prop}
\begin{proof}
Suppose, to the contrary, that $u$ is true and $d_G(u)\leq \ell-3$. Let
$f=upw$ be the false 3-face that is incident with $u$, where $p$ is a false vertex. Basically we assume
$uv$ crosses $ww'$ in $G$ at a point $p$.
Let $N_G(u)=\{v,w,x_1,\ldots, x_{t}\}$. Since $d_G(u)\leq \ell-3$, by Proposition \ref{edge-with-big-vertex}, $d(v),d(w)\geq \ell$ and $d_G(x_i)\geq \ell$ for each $1\leq i\leq t$.
If $vw\in E(G)$, then let $G'=G-u$. If $vw\not\in E(G)$, then let $G'=G-u+vw$.
In any case, we can see that $G'$ is still 1-planar, i.e, $G'\in\mathcal{G}_{1}^-$.
Let $v'$ be another neighbor of $v$ in $G'$ that is not $u$ or $w$ or $w'$.
By the minimality of $G$, $G'$ has a dynamic $L$-coloring $c$. Extending $c$ to a dynamic $L$-coloring of $G$ by coloring $u$ with a color $c(u)\in L(u)\backslash F(u)$, where $F(u)=\{c(v),c(w),c(x_1),\ldots, c(x_t),c(v'),c(w')\}$, we get a contradiction. Note that $|F(u)|=t+4=d_G(u)+2\leq \ell-1$.
\end{proof}
\begin{prop}\label{big-face}
If $G$ is a dynamically $\ell$-minimal graph in $\mathcal{G}_{1}^-$ with $\ell\geq 5$ and $f=wuvy_1\cdots y_s$ is
a $4^+$-face of $G^{\times}$ with $d_G(u)\leq \ell-3$, where $s\geq 1$, then both $v$ and $w$ are false. \end{prop}
\begin{proof}
Suppose, to the contrary, that at least one of $v$ and $w$ is true. We divide the proof into two major cases.
First of all, we assume that both $v$ and $w$ are true. If $vw\in E(G)$, then let $G'=G-u$. If $vw\not\in E(G)$, then let $G'=G-u+vw$.
In any case, it is easy to see that $G'$ is still 1-planar, i.e, $G'\in \mathcal{G}_1^-$. By the minimality of $G$, $G'$ has a dynamic $L$-coloring $c$. Let $N_G(u)=\{v,w,x_1,\ldots,x_t\}$. By Proposition \ref{edge-with-big-vertex}, any neighbor of $u$ in $G$ has degree at least $\ell$ since $d_G(u)\leq \ell-3$. Let $v'$ be another neighbor of $v$ in $G$ that is not among $\{x_1,\ldots,x_t,u,w\}$, and let $w'$ be another neighbor of $w$ in $G$ that is not among $\{x_1,\ldots,x_t,u,v,v'\}$ (such vertices exist since the number of the excluded vertices are at most $t+3=d_G(u)+1\leq \ell-2$). Color $u$ with a color $c(u)\in L(u)\backslash F(u)$, where $F(u)=\{c(x_1),\ldots,c(x_t),c(v),c(w),c(v'),c(w')\}$. Since $|F(u)|=t+4=d_G(u)+2\leq \ell-1$, we obtain a dynamic $L$-coloring of $G$, a contradiction.
On the other hand, we assume, by symmetry, that $v$ is true and $w$ is false. Basically we assume that $uu'$ crosses $w'y_s$ in $G$ at the point $w$. If $u'v\in E(G)$, then let $G'=G-u$, and otherwise let $G'=G-u+u'v$. The 1-planarity of $G'$ is easy to be confirmed (note that the crossing point $w$ in $G$ is removed by the deletion of $u$, and if we have to add the edge $u'v$, it can be drawn so that it is only crossed by $w'y_s$ in $G'$). By the minimality of $G$, $G'\in \mathcal{G}_1^-$ has a dynamic $L$-coloring $c$. Let $N_G(u)=\{u',v,x_1,\ldots,x_t\}$. By Proposition \ref{edge-with-big-vertex}, any neighbor of $u$ in $G$ has degree at least $\ell$ since $d_G(u)\leq \ell-3$. Let $u''$ or $v'$ be another neighbor of $u'$ or $v$ in $G$ that is not among $\{x_1,\ldots,x_t,u,v\}$ or $\{x_1,\ldots,x_t,u,u''\}$, respectively.
Color $u$ with a color $c(u)\in L(u)\backslash F(u)$, where $F(u)=\{c(x_1),\ldots,c(x_t),c(v),c(u'),c(u''),c(y_1)\}$. Since $|F(u)|=t+4=d_G(u)+2\leq \ell-1$, we get a dynamic $L$-coloring of $G$, a contradiction.
\end{proof}
\section{Discharging: the Proof of Theorem \ref{main-thm}}
If Theorem \ref{main-thm} is false, then there is a dynamically $11$-minimal 1-planar graph $G$.
For every element $x\in V(G^{\times})\cup F(G^{\times})$, we assign an initial charge $c(x)=d_{G^{\times}}(x)-4$. By the well-known Euler formulae $|V(G^{\times})|+|F(G^{\times})|-|E(G^{\times})|=2$ on the plane graph $G^{\times}$, we have
$$\sum_{x\in V(G^{\times})\cup F(G^{\times})}c(x)=-8<0.$$
If there is a 4-face $f=uxvy$ in $G^{\times}$ such that $d_{G^{\times}}(u)\geq 11$, $2\leq d_{G^{\times}}(v):=d\leq 3$ and $x,y$ are false vertices, then we call $f$ a \emph{special $4$-face}.
Initially, we define the following discharging rules (also see Figure \ref{Rules}) so that the charges are transferred among the elements in
$V(G^{\times})\cup F(G^{\times})$.
\begin{enumerate}
\item[R1.] Every true 3-face in $G^{\times}$ receives $\frac{1}{3}$ from each of its incident $11^+$-vertices;
\item[R2.] Every false 3-face in $G^{\times}$ receives $\frac{1}{2}$ from each of its incident $9^+$ vertices;
\item[R3.] Every $11^+$-vertex incident with a special $4$-face $f$ sends $1$ to $f$, from which the special $3^-$-vertex on $f$ receives $1$;
\item[R4.] Every $5^+$-face in $G^{\times}$ sends 1 to each of its incident special 2-vertices if there are some ones;
\item[R5.] After applying R1--R4, every $5^+$-face in $G^{\times}$ redistributes its charge equitably to each of its incident non-special 2-vertices or (special or non-special) $3$-vertices if there are some ones.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[width=15cm]{Rules}
\caption{The discharging rules R1--R3}
\label{Rules}
\end{figure}
Let $c'(x)$ be the final charge of the element $x\in V(G^{\times})\cup F(G^{\times})$ after discharging. Clearly
$$\sum_{x\in V(G^{\times})\cup F(G^{\times})}c'(x)=\sum_{x\in V(G^{\times})\cup F(G^{\times})}c(x)=-8<0.$$
In the following, we show that $c'(x)\geq 0$ for every $x\in V(G^{\times})\cup F(G^{\times})$ by Claims \ref{clm:5-minus-face-nonneg}, \ref{clm:6-plus-face-nonneg}, \ref{clm:2-vertex-nonneg}, \ref{clm:3-vertex-nonneg}, and \ref{clm:4-plus-vertex-nonneg}. This contradiction completes the proof of Theorem \ref{main-thm}.
\begin{clm}\label{clm:5-minus-face-nonneg}
Every $5^-$-face in $G^{\times}$ has a nonnegative final charge.
\end{clm}
\begin{proof}
If $f$ is a true 3-face (i.e.,\,a triangle in $G$), then every vertex incident with $f$ is a $11^+$-vertex by Proposition \ref{true-3-face}, and thus $c'(f)=3-4+3\times\frac{1}{3}=0$ by R1.
If $f$ is a false 3-face, then $f$ is incident with two $9^+$-vertices by Proposition \ref{false-3-face}, which implies $c'(f)=3-4+2\times\frac{1}{2}=0$ by R2.
If $f$ is a non-special $4$-face, then no rule is valid for $f$ and thus $c'(f)= c(f)=0$.
If $f$ is a special $4$-face, then $c'(f)=4-4+1-1=0$ by R3.
If $f$ is a 5-face, then $f$ is incident with at most one 2-vertex by Proposition \ref{big-face}. Therefore, the remaining charge of $f$ after R1--R4 are applied to it is at least $5-4-1=0$, and thus $f$ has a nonnegative final charge by R5.
\end{proof}
\begin{clm}\label{clm:6-plus-face-nonneg}
Every $6$-face is incident with at most two special $2$-vertices in $G^{\times}$.
Therefore, every $6^+$-face in $G^{\times}$ has a nonnegative final charge.
\end{clm}
\begin{proof}
Suppose, to the contrary, that $f=uxvywz$ is a 6-face such that $u,v,w$ are special 2-vertices and $x,y,z$ are false vertices. According to the definition of the special 2-vertices, there are three $11^+$-vertices $u',v'$ and $w'$ such that $uv'$ (resp.\,$v'w$ and $u'w$) crosses $u'v$ (resp.\,$vw'$ and $uw'$) in $G$ at the crossing $x$ (resp.\,$y$ and $z$), see Figure \ref{6-face}(a). Pulling the vertex $v$ (resp.\,$w$) into the face of $G^{\times}$ that is incident with the path $u'zw'$ (resp.\,$u'xv'$), we get another one 1-planar drawing of $G$ with three less crossings, see Figure \ref{6-face}(b). This contradicts the initial assumption that the drawing of $G$ has the minimum number of crossings.
Therefore, every $6$-face in $G^{\times}$ has charge at least $6-4-2\times 1=0$ after R1--R4 are applied to it, and thus has nonnegative final charge by R5.
On the other hand, every $d$-face $f$ with $d\geq 7$ is incident with at most $d/2$ 2-vertices if $d$ is even, and at most $(d-3)/2$ 2-vertices if $d$ is odd, by Proposition \ref{big-face}. Therefore, after R1--R4 are applied to $f$, $f$ remains charge at least $d-4-d/2=(d-8)/2\geq 0$ if $d$ is even (i.e., $d\geq 8$), and at least $d-4-(d-3)/2=(d-5)/2\geq 1$ if $d$ is odd (i.e., $d\geq 7$). Hence $f$ has nonnegative final charge by R5.
\end{proof}
\begin{figure}
\centering
\includegraphics[width=15cm]{6-face}
\caption{A redrawing with three less crossings: the proof of the first part of Claim \ref{clm:6-plus-face-nonneg}}
\label{6-face}
\end{figure}
\begin{clm}\label{clm:5-plus-face-to-small-v}
Let $f$ be a $5^+$-face in $G^{\times}$ with $u,x,v,y$ and $w$ being five consecutive vertices on the boundary of $f$ such that $u$ is a $11^+$-vertex, $v$ is a non-special $2$-vertex or a (special or non-special) $3$-vertex, and $x,y$ are false vertices.
$(1)$ If $w$ is a $11^+$-vertex, then $f$ sends at least $2$ to $v$;
$(2)$ If $w$ is a $10^-$-vertex, then $f$ sends at least $1$ to $v$.
\end{clm}
\begin{proof}
Let $a$ (resp.\,$b$) be the number of special $2$-vertices (resp.\,non-special $2$-vertices and special or non-special $3$-vertices) that are incident with $f$.
(1) Suppose that $w$ is a $11^+$-vertex. By Proposition \ref{big-face}, there are at least $a+(b-1)+1$ false vertices in $V_{G^{\times}}(f)\backslash \{u,x,v,y,w\}$. This implies that
$$a+(b-1)+a+(b-1)+1+5=2a+2b+4\leq d.$$
Therefore, $f$ sends to $v$ at least
\begin{align*}
\frac{d-4-a}{b}\geq \frac{2a+2b+4-4-a}{b}\geq 2
\end{align*}
by R4 and R5.
(2) Suppose that $w$ is a $10^-$-vertex. By Proposition \ref{big-face}, there are at least $a+(b-1)$ false vertices in $V_{G^{\times}}(f)\backslash \{u,x,v,y\}$. This implies that
$$a+(b-1)+a+(b-1)+4=2a+2b+2\leq d.$$
Therefore, by R4 and R5, $f$ sends to $v$ at least
\begin{align*}
\frac{d-4-a}{b}\geq \frac{2a+2b+2-4-a}{b}\geq 1
\end{align*}
if $a+b\geq 2$.
On the other hand, if $a+b\leq 1$, then $a=0$ and $b=1$, since $b\geq 1$. Hence $f$ would send at least $d-4\geq 1$ to $v$ by R5.
\end{proof}
\begin{clm}\label{clm:2-vertex-nonneg}
Every $2$-vertex in $G^{\times}$ has a nonnegative final charge.
\end{clm}
\begin{proof}
By Propositions \ref{true-3-face} (applying it by choosing $\mathcal{F}$ as $\mathcal{G}_1^-$) and \ref{false-3-face}, every $2$-vertex $v$ in $G^{\times}$ is not incident with a 3-face in $G^{\times}$. By Proposition \ref{big-face}, the neighbors of $v$ in $G^{\times}$, say $x$ and $y$, are both false vertices.
If $v$ is a special 2-vertex, then $v$ receives 1 from its incident special 4-face, say $uxvy$, by R3. If the other face incident with $v$ in $G^{\times}$ is still a 4-face, say $wxvy$, then there would be two edges in $G$ connecting $u$ to $w$, one passing through the crossing $x$ and the other passing through the crossing $y$. This contradicts the fact that $G$ is a simple graph. Therefore, $v$ is incident with a $5^+$-face, from which $v$ receives another 1 by R4. Hence $c'(v)=2-4+1+1=0$.
So in the following, we assume that $v$ is a non-special 2-vertex.
If $v$ is incident with a 4-face, say $uxvy$, then $d_{G^{\times}}(u)\leq 10$ since $v$ is non-special. Let $u_1$ (resp.\,$u_2$) be the vertices in $G$ such that $uu_1$ (resp.\,$uu_2$) passes through the crossing $x$ (resp.\,$y$). Since $G$ is a simple graph, $u_1\neq u_2$, and moreover, $u_1$ and $u_2$ are $11^+$-vertices by Proposition \ref{edge-with-big-vertex}. Therefore, $v$ is incident with a $5^+$-face that satisfies the condition of Claim \ref{clm:5-plus-face-to-small-v}(1). Since such a face would send at least 2 to $v$ by Claim \ref{clm:5-plus-face-to-small-v}(1), $c'(v)\geq 2-4+2=0.$
If $v$ is incident with two $5^+$-faces $f_1$ and $f_2$, then let $u_1u_2$ and $w_1w_2$ be edges of $G$ that pass through the crossings $x$ and $y$, respectively, such that $u_i$ and $w_i$ are vertices on $f_i$, where $i=1,2$. By Proposition \ref{edge-with-big-vertex}, there are at least two $11^+$-vertices among $u_1,u_2,w_1$ and $w_2$. Therefore, either $f_1$ or $f_2$ satisfies the condition of Claim \ref{clm:5-plus-face-to-small-v}(1), or both $f_1$ and $f_2$ satisfy the condition of Claim \ref{clm:5-plus-face-to-small-v}(2). In each case $v$ receives at least 2 from $f_1$ and $f_2$, and thus $c'(v)\geq 2-4+2=0$.
\end{proof}
\begin{clm}\label{clm:3-vertex-nonneg}
Every $3$-vertex in $G^{\times}$ has a nonnegative final charge.
\end{clm}
\begin{proof}
By Propositions \ref{true-3-face} and \ref{false-3-face}, every $3$-vertex $v$ in $G^{\times}$ is not incident with a 3-face in $G^{\times}$. By Proposition \ref{big-face}, the three neighbors of $v$ in $G^{\times}$, say $x,y$ and $z$, are false vertices. Let $f_1,f_2$ and $f_3$ be the face that is incident with the path $xvy$, $yvz$ and $zvx$ in $G^{\times}$.
Let $x_1x_3$ be the edge of $G$ that passes through the crossings $x$, where $x_1\in V_{G^{\times}}(f_1)$ and $x_3\in V_{G^{\times}}(f_3)$. By Proposition \ref{edge-with-big-vertex}, either $x_1$ or $x_3$, say $x_1$, is a $11^+$-vertex. If $f_1$ is a $5^+$-face, then it satisfies the condition of Claim \ref{clm:5-plus-face-to-small-v}(1) or Claim \ref{clm:5-plus-face-to-small-v}(2). This implies that $v$ receives at least 1 from $f_1$, and thus $c'(v)\geq 3-4+1=0$. Hence we assume that $f_1$ is a 4-face. Actually, $f_1$ is a special $4$-face now, from which $v$ receives $1$ by R3. This implies that $c'(v)\geq 3-4+1=0$.
\end{proof}
\begin{clm}\label{clm:no-adj-special-4-faces}
No two special $4$-faces sharing a common $11^+$-vertex are adjacent in $G^{\times}$.
\end{clm}
\begin{proof}
Suppose, to the contrary, $f_1=vxuy$ and $f_2=vywz$ are two adjacent special 4-faces in $G^{\times}$ so that $d_{G^{\times}}(v)\geq 11$. By the definition of the special $4$-face, $u$ and $w$ are $3^-$-vertices and $y$ is a false vertex. This implies that $uw\in E(G)$, contradicting Proposition \ref{edge-with-big-vertex}.
\end{proof}
\begin{clm}\label{clm:giving-three-consecutive-faces}
If $v$ is a $11^+$-vertex and $f_1,f_2$ and $f_3$ are three consecutive faces that are incident with $v$ in $G^{\times}$, then $v$ totally sends to $f_1, f_2$ and $f_3$ at most $2$.
\end{clm}
\begin{proof}
If there is only one special 4-face among $f_1,f_2$ and $f_3$, then by R1--R3, $v$ totally sends to $f_1, f_2$ and $f_3$ at most $1+\frac{1}{2}+\frac{1}{2}=2$. If there are at least two special 4-faces among $f_1,f_2$ and $f_3$, then by Claim \ref{clm:no-adj-special-4-faces}, they are $f_1$ and $f_3$, and $f_2$ is a non-special $4^+$-face. In this case
$v$ totally sends $1+0+1=2$ to $f_1, f_2$ and $f_3$ by R3.
\end{proof}
\begin{clm}\label{clm:4-plus-vertex-nonneg}
Every $4^+$-vertex in $G^{\times}$ has a nonnegative final charge.
\end{clm}
\begin{proof}
Since vertices of degree between 4 and 8 are not involved in the discharging rules, their final charges are the same with their initial charges, which are nonnegative. Suppose that $v$ is a vertex of degree $d\geq 9$.
If $9\leq d\leq 10$, then $c'(v)\geq d-4-\frac{1}{2}d>0$ by R2.
If $d\geq 11$, then let $f_1,f_2,\ldots,f_d$ be the faces in this order around $v$. Let $\alpha_i$ with $1\leq i\leq d$ be the charge that $v$ sends to $f_i$ and let $\omega_i=\alpha_i+\alpha_{i+1}+\alpha_{i+2}$, where the subscripts are taken modular $d$. One can see that
$$\sum_{i=1}^d \alpha_i=\frac{1}{3} \sum_{i=1}^d \omega_i\leq \frac{2}{3}d,$$
where the second inequality holds by Claim \ref{clm:giving-three-consecutive-faces}.
Hence $c'(v)=d-4-\sum_{i=1}^d \alpha_i\geq \frac{1}{3}d-4\geq 0$ if $d\geq 12$.
We now consider the case when $d=11$ more carefully. If $v$ is incident with at most three special 4-faces, then $c'(v)\geq 11-4-3\times 1-8\times \frac{1}{2}=0$ by R1--R3. So we assume that $v$ is incident with at least four special 4-faces. This implies that there is an integer
$1\leq i\leq d$ such that $f_i$ and $f_{i+2}$ are special 4-faces, where the subscripts are taken modular $d$. Assume, without loss of generality, that $i=1$. In this case, $f_2$ shall be a non-special $4^+$-face and therefore $\alpha_2=0$. By Claim \ref{clm:no-adj-special-4-faces}, $f_d$ and $f_4$ cannot be special 4-faces, to each of which $v$ sends at most $\frac{1}{2}$ by R1 and R2. This implies that $\omega_{11}\leq \frac{1}{2}+1+0=\frac{3}{2}$ and $\omega_2\leq 0+1+\frac{1}{2}=\frac{3}{2}$. Hence by Claim \ref{clm:giving-three-consecutive-faces}, we conclude that
$$\sum_{i=1}^{11} \alpha_i=\frac{1}{3} \bigg(\omega_2+\omega_{11}+\sum_{i\leq 10, i\neq 2} \omega_i\bigg)
\leq \frac{1}{3}\times \bigg(\frac{3}{2}+\frac{3}{2}+2\times 9\bigg)=7.$$
This implies that $c'(v)=11-4-\sum_{i=1}^{11} \alpha_i\geq 11-4-7=0$.
\end{proof}
\section{Remarks and Open Problems}
In this paper we have proved
\[
7\leq \chi(\mathcal{G}_2)\leq \chi^d(\mathcal{G}_1)\leq {\rm ch}^d(\mathcal{G}_1)\leq 11.~~~~~~(\star)
\]
Hence a natural problem is to close the gap between the lower and the upper bounds in $(\star)$. In other words, we propose
\begin{pblm}\label{prob:1}
Determine the minimum integers $\ell_1$ and $\ell_2$ so that every 1-planar graph is dynamically $\ell_1$-colorable and dynamically $\ell_2$-choosable, respectively.
\end{pblm}
On the other hand, one can see that the first relationship between the proper coloring of 2-planar graphs and the dynamic coloring of 1-planar graphs has been established by Fact \ref{fact-1}. Actually, if we have a better lower bound for $\chi(\mathcal{G}_2)$, then we can improve 7 in $(\star)$ immediately. We think this may be a good motivation to study the proper coloring of 2-planar graphs. In view of this, we pose the following
\begin{pblm}\label{prob:2
Does there exist $2$-planar graph with chromatic number $8$ or $9$ ?
\end{pblm}
Note that $K_8$ is not 2-planar, which was very recently proved by Angelini, Bekos, Kaufmann and Schneck \cite{angelini2019efficient}.
On the other hand, Dmitry Karpov (personal communication) announced a proof of 9-colorability of 2-planar graphs (written in Russian).
\section*{Acknowledgements}
The first author would like to thank Fedor Petrov who reminded him at MathOverflow \cite{ZHANG} on an unpublished result of Dmitry Karpov that $\chi(\mathcal{G}_2)\leq 9$, and appreciate Dmitry Karpov for the personal communication with him on this topic. The supports provided by China Scholarship Council (CSC) and Institute for Basic Science (IBS, Korea) during a visit of the first author to Discrete Mathematics Group, IBS are acknowledged.
\bibliographystyle{abbrv}
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{"url":"https:\/\/robertying.com\/post\/the-slippery-slope-of-building-computers\/","text":"#### The Slippery Slope of Building Computers\n\n##### July 6, 2018\ncomputers servers family\n\n## Resurrecting GLaDOS\n\nAfter a little over five years, the small form factor desktop computer I built back in late 2012 refused to boot properly. For a couple of months, I resisted\u2013I have a work-issue Macbook and iPad Pro, and I don\u2019t play nearly as many computer games as I used to. It didn\u2019t feel all that necessary for me to have a desktop on top of all of that.\n\nBut, it turns out, it\u2019s really quite nice to have a computer setup of your own. I ended up cannibalizing many of the parts from that desktop into a new, mid-tower sized box with 8th-generation Intel technology. And it\u2019s actually really nice! At work, my Linux desktop is also running a circa 2013 CPU, so I legitimately didn\u2019t know how much faster and more power-efficient desktop computers have gotten in the last half-decade.\n\nNow, I have a near-silent, modern desktop capable of running whatever I want it to\u2026 and it only cost about $600 to build. GLaDOS has always been a bit of a Ship of Theseus, but even so, I expected the replacement cost of a desktop computer to be considerably higher. ## Building moralitycore The week after all the parts came in for GLaDOS\u2019 resurrection, I learned that my mom\u2019s Dell PowerEdge small business server was having some trouble. We\u2019d bought this machine in 2010, and it was primarily used as a Samba \/ SFTP server for a network of about a dozen hosts. That week, I got a frantic call saying that it had stopped responding to all requests\u2026 so I had someone reboot it and figured things would be okay for a while. A day later, I got another frantic call. I was getting a little suspicious at this point: what had changed after so many years of good performance? I didn\u2019t see anything interesting in the dmesg or \/var\/log\/syslog, and the reboot logs didn\u2019t record any intentional reboots. I eventually wised up and decided to check the hard drives\u2019 SMART data\u2026 and there it was. Somehow, the boot drive on the machine had managed to acquire two billion read errors and an incredible number of bad sectors. Wonderful. At this point, I\u2019d realized that replacing just the boot drive would probably make the immediate problems go away for a while. But I\u2019d just built a computer, and it seemed to me that the failing HDD would just be the first of many parts to die\u2026 so I decided to rebuild it. My sister Alina recently decided (during her first year of university) that she was going to study Computer Science. Unlike me, Alina didn\u2019t spend most of high school acquiring esoteric computing knowledge via Internet osmosis, so I thought it might be fun to have her learn how to assemble a working computer. Thanks to the wonder of Amazon Prime two-day shipping, all the parts were sitting in Palo Alto by the following weekend. The build went wonderfully. Alina got to see what all the parts of a computer look like first hand, and the computer passed POST on the first try. As is true of most of my machines, we decided to name it with a reference to the puzzle game Portal: moralitycore. At this point, we still didn\u2019t have a working operating system\u2026 but I didn\u2019t expect that to be particularly problematic. Eventually (two USB flash drives and many boot attempts later), we got the Ubuntu LiveUSB to successfully run and install Ubuntu to moralitycore\u2019s SSD. For funsies, we set up ZFS with automatic snapshotting on the data HDD, and then set up Samba and the other useful things in my mom\u2019s office. ## A digression into some programming Over the summer, Alina has also been learning how to program in C++ and to work with the OpenCV library, which is actually a pretty nontrivial task. Her laptop runs Windows, so I convinced her to boot Ubuntu in a VirtualBox virtual machine and to do her OpenCV experiments there by pointing out that she\u2019d just built a real Ubuntu server from its base components. We also went over the use of ssh \/ PuTTY and sftp \/ WinSCP, so that she could access moralitycore from her laptop. It\u2019s amazing how much someone can pick up once they learn how the sausage is made! With a little apt magic, we got OpenCV, CMake, g++, and all the other parts of the OpenCV toolchain installed inside of her VM (and also in moralitycore). While it\u2019s of course possible to build programs which depend on OpenCV in Windows, especially with Visual Studio 2017\u2019s new CMake support, the vast majority of examples and tutorials on the Internet assume a Linux-like system. Also, since angercore is dead, Alina\u2019s website (as well as most of my own) went down. Single points of failure are bad, and all that. But with the power of ssh and a handy moralitycore with a static IP address, Alina\u2019s homepage is back up for business! ## But wait! Two computers doesn\u2019t sound like a slippery slope? Two points make a line, but the real reason why building computers is a fun family activity slippery slope is that we got my dad to want to build one too! While I was helping Alina build moralitycore, our parents were hanging out in the background (and largely ignoring what we were doing). But, Dad used to work at Intel\u2013and he still vaguely remembers things from his pre-management days (!!). So at this point, he was already pretty intrigued, especially since the total bill of materials for moralitycore worked out to about$500\u2026\n\n\u2026and then he decided he would join in on the OpenCV fun. However, we couldn\u2019t successfully get his work laptop to run a Linux VM. The Maxim Integrated IT department managed to lock down enough of the laptop that it can\u2019t even run Dropbox! For a couple of weeks, he struggled his way through using PuTTY and WinSCP and VIM, developing remotely off of the OpenCV installed on moralitycore. But then, he hit a roadblock! A fair amount of debugging and tuning in computer vision requires that you can actually see the output of the processing that you have performed. But there isn\u2019t any easy way to send graphical elements from a remote server to a Windows laptop\u2026 (yes, I know, X tunneling is a thing, but it\u2019s unreliable and slow).\n\nWhich is all to say, Alina is going to be teaching Dad how to build his very own Ubuntu\/Intel-powered desktop computer sometime next week.","date":"2019-02-17 09:33:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18930388987064362, \"perplexity\": 2183.346157446208}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247481832.13\/warc\/CC-MAIN-20190217091542-20190217113542-00003.warc.gz\"}"}
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Q: How do I access this keyword In a different class am having a problem trying to access this keyword in a different class using Java programming. I have tried Context, class.this but no help yet...
I have created a project using NetBeans gui builder, I want when i click button the form to get disposed...
Main class contains the click event for disposing the JFrame Form
BestQSystems.java:
private void jButton1ActionPerformed(java.awt.event.ActionEvent evt) {
CloseWindow.closeWindow();
}
Class to close the JFrame: CloseWindow.java
import java.awt.Toolkit;
import java.awt.event.WindowEvent;
import javax.naming.Context;
/*
* To change this license header, choose License Headers in Project Properties.
* To change this template file, choose Tools | Templates
* and open the template in the editor.
*/
/**
*
* @author Benson
*/
public class CloseWindow {
public static void closeWindow(){
WindowEvent widnowEvent = new WindowEvent(this, WindowEvent.WINDOW_CLOSING);
Toolkit.getDefaultToolkit().getSystemEventQueue().postEvent(widnowEvent);
}
}
Am having an error in this line WindowEvent widnowEvent = new WindowEvent(this, WindowEvent.WINDOW_CLOSING); Please advise me on how to access this keyword in a different class.
A: You can pass a reference to this to the other method. For example:
BestQSystems.java
private void jButton1ActionPerformed(java.awt.event.ActionEvent evt) {
CloseWindow.closeWindow(this);
}
and in the CloseWindow.java
public class CloseWindow {
public static void closeWindow(BestQSystems ref){
WindowEvent widnowEvent = new WindowEvent(ref, WindowEvent.WINDOW_CLOSING);
}
}
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Feb. 28, 2014 – Pod Cast – Molly Friedenfeld – Violet Wisdom, Internet Radio interview.
May 6-8, 2013 – Warsaw, Poland. Release of Polish Translation. Irena Sendler School, Foreign Ministry of Poland, Museum of the History of Polish Jews.
Oct. 26, 2013 – Dessert Reception and book talk – home of Elizabeth Rosenberg and Jonathan Katz.
Author Jack Mayer is available for book signings, discussions, presentations, and special events.
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Der Maßholderbach ist ein fast fünf Kilometer langer Bach insgesamt etwa südwestlicher Laufrichtung im Stadtgebiet von Öhringen im Hohenlohekreis im nördlichen Baden-Württemberg. Mit seinem gegenüber dem linken längeren rechten Oberlauf Langwiesenbächle kommt er sogar fast auf sechs Kilometer Länge. Er mündet wenig unterhalb des geschlossenen städtischen Siedlungsgebietes von Öhringen von rechts in die untere Ohrn.
Name
Der Bach wird zumindest im Stadtteil Büttelbronn von Öhringen Masselbach oder vielleicht auch Maßelbach genannt.
Geographie
Verlauf
Der Maßholderbach beginnt nach den amtlichen topographischen Karten seinen Lauf auf etwa und ca. 0,4 km südlich des Wohnplatzes Schönau von Zweiflingen eben schon auf Öhringer Gemarkung an einer Waldspitze am Gewann Strüt. In der sich bergwärts weiter fortsetzenden Talmulde, deren Tiefenlinie dem Waldrand folgt, liegt jedoch etwa 0,4 km weiter aufwärts und auf etwa am Ostrand von Schönau ein nur knapp 0,1 ha großer Teich auf der Bergseite eines Wirtschaftsweges vom Ort in den Wald. An der Talseite des Weges steht die lokale Kläranlage. Je nachdem, ob deren Abwässer hier zunächst verdolt oder offen abfließen, wäre etwa hier der wirkliche Ursprung des anfangs wie unbeständig auch immer fließenden Baches zu sehen.
Der Maßholderbach fließt anfangs südsüdwestlich, ab der Waldspitze 1,1 km lang erst zwischen Feldern, dann teils von einem Grünstreifen begleitet, bis in den Öhringer Weiler Obermaßholderbach, wo er seinen längeren und einzugsgebietsreicheren rechten Oberlauf Langenwiesbächle aufnimmt. Dieser entsteht westlich des Wohnplatzes Schönau und des damit zusammenhängenden Dorfes Friedrichsruhe von Zweiflingen auf dem dortigen Golfplatz einem Teich. In diesem Terrain läuft er zunächst südwestlich, durchquert einen weiteren Teich, nimmt die Abflüsse weiterer naher Teiche auf und wendet sich dann nach links auf Südlauf zwischen Feldern, um nach einem weiteren Richtungsschwenk nach links sich schließlich 2,0 km nach seinem Ursprung in Obermaßholderbach mit dem namentlichen Oberlauf zu vereinen.
Der Maßholderbach fließt auf seinem Mittellauf nach Obermaßholderbach nun lange südwärts in einer schmalen Aue. Anders als bei den beiden am Ufer fast kahlen Oberlauf-Gräben begleitet ihn ab hier eine Galerie aus Bäumen und Büschen. Wenig nach dem Weiler mündet der über einen Kilometer lange Bach aus den Hungerwiesen, erster und einzugsgebietsreichster von vier sämtlich aus dem Osten kommenden Zuflüssen. Am rechten Hangfuß begleitet ihn hier schon die in Obermaßholderbach beginnende Kreisstraße K 2332 in Richtung Öhringen. Zwischen dem Zufluss aus den Hungerwiesen und dem nächsten, Gießgraben genannten Nebenbach weitet sich die Galerie sogar zu einem kleinen Wäldchen anfangs vor allem an der nahen linken Böschung, später in der linken Aue. Der etwa anderthalb Kilometer lange Gießgraben wie auch die beiden noch folgenden, unter einen Kilometer langen Zuflüsse sind lange von Feldwegen begleitete Gräben in natürlicher Mulde. Der eine mündet in Untermaßholderbach, nach welchem der naturnahe Abschnitt des Maßholderbachs beginnt, auf dem er in sich leicht schlängelndem Lauf in seinem halb- bis dreimeterbreitem Bett zunächst weiter südwärts läuft und dabei bald den noch fehlenden seiner vier Zuflüsse aufnimmt.
Weniger als einen halben Kilometer abwärts von Untermaßholderbach überspannt den Bach die Maßholderbachtalbrücke der A 6. Dort knicken Tal und Bach in westsüdwestlicher Richtung ab. Es begleiten ihn nun auf seinen letzten nicht ganz zwei Kilometern auf dem linken Hügelkamm die nördlichsten Häuserzeilen des geschlossenen Siedlungsgebietes der Stadt Öhringen, während rechts am Unterhang die Autobahn parallel läuft. Auf dem sonst siedlungsfreien Talgrund grenzen nur ein Schrebergärtengelände und eine Schule ans Ufer. Schließlich, nach 4,9 km ab seinem Ursprung an der Waldspitze und ca. 78 Höhenmeter tiefer, mündet er auf etwa wenig abwärts der Weidenmühle gleich nach Öhringen und nur wenige Schritt vor der Ohrntalbrücke der Autobahn über diesen Fluss von rechts in die untere Ohrn; weniger als hundert Meter flussabwärts mündet in diese ihr letzter größerer Zufluss Westernbach.
Das mittlere Sohlgefälle des Maßlensbachs liegt bei etwa 16 ‰.
Einzugsgebiet
Der Maßholderbach entwässert ein 8,6 km² großes Gebiet nördlich der Stadt Öhringen, das naturräumlich gesehen zum Unterraum Öhringer Ebene der Hohenloher und Haller Ebene gerechnet wird. Seine mit bis zu größten Höhen erreicht es am Nordrand an der Straße zwischen den Dörfern Friedrichsruhe und Pfahlbach der Gemeinde Zweiflingen.
Reihum konkurrieren die folgenden Nachbarbäche:
Im Norden läuft der Pfahlbach westwärts zum Kocher wenig vor der Ohrn-Mündung.
Im Nordosten läuft wenig jenseits der Wasserscheide der Hirschbach zur Sall, einem Kocherzufluss weiter oben.
Hinter der Ostgrenze entwässert der Weinsbach das angrenzende Gebiet zum Epbach, dem nächsten größeren Ohrn-Zufluss oberhalb des Maßholderbachs.
Im Südosten konkurriert diese Epbach selbst.
Im Süden fließt recht nahe die Ohrn westwärts, mit nur allein dem kleinen rechten Zufluss Ströllerbach.
Im Westen entwässert der weniger als 100 Meter nach dem Maßholderbach ebenfalls von rechts mündende Westernbach gleichfalls zur Ohrn.
Im größten Teil des Einzugsgebietes steht geologisch gesehen der Lettenkeuper (Erfurt-Formation) des Unterkeupers an, in welchem der auch Bach entspringt. Auf Kammlagen der begleitenden Hügel, rechts schon recht früh, links erst ab dem Talknick nach Untermaßholderbach, liegt darüber noch in Inseln der Gipskeuper (Grabfeld-Formation) des Mittelkeupers. Vor allem am Ostrand und im südlichen Einzugsgebiet sind diese tertiären Schichten auf der Höhe noch weithin überdeckt von Lösssediment aus quartärer Ablagerung. Der Bach mündet in der breiten, von ebenfalls quartärem Hochwassersediment bedeckten Ohrntalaue. Ab Obermaßholderbach gibt es im Tal und auf den Randhügeln einige eher kurze Störungslinien etwa herzynischer Richtung.
Das Einzugsgebiet ist eine kleinhügelige, weit überwiegend offene Landschaft mit weniger als 0,5 km² Wald, der fast zur Gänze am Nordostrand steht. Einen 0,7 km² großen Golfplatz ganz im Norden bei Friedrichsruhe ausgenommen, wird es sonst fast überall ackerbaulich genutzt. Die Besiedlung ist gering und umfasst Friedrichsruh mit dem baulich zusammenhängenden Wohnplatz Schönau am Nordnordostrand, die beide zur Gemeinde Zweiflingen gehören; die Weiler Ober- und Untermaßholderbach am Ober- und Mittellauf sowie die Wohnplätze Buschfeld, Platzfeld und Weidenhof östlich der Talmulde, alle in der Büttelbronner Stadtteilgemarkung von Öhringen; den Platzhof in der Eckartsweiler Teilgemarkung; außerdem einen schmalen Streifen am Nordrand der zentralen Öhringer Stadtbebauung im Süden.
Südwestlich von Obermaßholderbach betritt die aus dem Nordnordosten kommende Trasse des Obergermanisch-Raetischen Limes am Gewann Pfahläcker das Einzugsgebiet und quert das Bachtal etwa an der Maßholderbachtalbrücke der Bundesautobahn 6 abwärts von Untermaßholderbach, um das Einzugsgebiet dann durch das zentrale Öhringer Siedlungsgebiet südsüdostwärts zu verlassen.
Zuflüsse und Seen
Hierarchische Liste der Zuflüsse und Seen von der Quelle zur Mündung. Gewässerlänge, Seefläche, Einzugsgebiet und Höhe nach den entsprechenden Layern auf der Onlinekarte der LUBW. Andere Quellen für die Angaben sind vermerkt.
Ursprung des Maßholderbach auf etwa ca. 0,4 km südlich des Ortsrandes von Zweiflingen-Schönau an einer südwestlichen Waldspitze am Strüt. Der Bach fließt zunächst in einem Graben südwestlich.
Ca. 0,4 km weiter nordöstlich-aufwärts liegt in der Bachmulde am Waldrand auf etwa hinter einem Feldweg ein Teich nahe bei Schönau, knapp 0,1 ha. Auf der abwärtigen Wegseite liegt die lokale Kläranlage.
Langwiesenbächle, von rechts und Norden auf etwa in Öhringen-Obermaßholderbach, 2,0 km und ca. 1,5 km².Der Maßholderbach selbst ist bis zu diesem Zufluss 1,1 km lang und hat ein Teileinzugsgebiet von ca. 1,2 km².
Entfließt auf einem Teich auf dem Golfplatz bei Zweiflingen-Friedrichsruhe, 0,3 ha.
Durchfließt auf über einen weiteren Teich, 0,9 ha.
Einige kleinere Teiche von allenfalls etwas über 0,1 ha auf dem Golfplatz entwässern ebenfalls zum Langwiesenbächle.
(Bach aus den Hungerwiesen), von links und Osten auf etwa etwa 200 Meter unterhalb von Obermaßholderbach, 1,2 km und ca. 1,7 km².
Entfließt auf dem Platzhofweiher bei Öhringen-Platzhof, 0,8 ha.
(Bach aus den Seewiesen), von rechts und Nordosten auf etwa etwa 150 Meter unterhalb des Sees, ca. 0,6 kmref name="TK-abgemessen-Länge" group="LUBW" /> und ca. 0,3 km². Entsteht auf etwa an der L 1050 Friedrichsruhe–Öhringen Wohnplatzes Platzhof. Unbeständig.
(Graben aus der Ammerklinge), von rechts und Nordnordosten auf etwa weniger als einen Kilometer östlich von Maßholderbach, ca. 0,5 km und ca. 0,6 km². Entsteht auf etwa in einem kleinen Talwäldchen südwestlich des Öhringer Wohnplatzes Platzfeld. Unbeständig.
Gießgraben, von links und Osten auf etwa vor Öhringen-Untermaßholderbach, 1,4 km und ca. 0,5 km². Entsteht auf etwa am Straßengraben neben der L 1050 Friedrichsruhe–Öhringen.
(Graben aus dem Gehrn), von links und Osten auf etwa in Untermaßholderbach, 0,9 km und ca. 0,8 km². Entsteht auf etwa westlich des Öhringer Weidenhofs in einem Feldweggraben.
(Bach aus den Schwarzäckern), von links und Osten auf etwa zwischen Untermaßholderbach und der Maßholderbachtalbrücke der A 6, 0,8 km und ca. 0,4 km². Entsteht auf etwa .
Mündung des Maßholderbachs von rechts und Nordosten auf etwa an der Weidenmühle kurz nach Öhringen in die untere Ohrn. Der Bach ist vom Ursprung am Waldzipfel 4,9 km, mit stattdessen dem längeren rechten Oberlauf Langwiesenbächle aus dem Golfplatz bei Friedrichsruhe sogar 5,8 km lang und hat ein Einzugsgebiet von 8,6 km².
Schutzgebiet
Der etwas an der Nordgrenze des Einzugsgebietes hinausreichende Park des Jagdschloss Friedrichsruhe in Friedrichsruhe ist als nur knapp zehn Hektar großes Landschaftsschutzgebiet Schloßpark Friedrichsruhe ausgewiesen.
Einzelnachweise
LUBW
Amtliche Online-Gewässerkarte mit passendem Ausschnitt und den hier benutzten Layern: Lauf und Einzugsgebiet des Maßholderbachs
Allgemeiner Einstieg ohne Voreinstellungen und Layer:
Höhe:
Länge:
EZG:
Seefläche:
Sonstige:
Andere Belege
Literatur
Topographische Karte 1:25.000 Baden-Württemberg, als Einzelblatt Nr. 6722 Hardthausen am Kocher, Nr. 6723 Öhringen
Geologische Karte des Naturparks Schwäbisch-Fränkischer Wald 1:50.000, herausgegeben vom Landesamt für Geologie, Rohstoffe und Bergbau Baden-Württemberg, Freiburg i. Br. 2001. (Nur für das untere Einzugsgebiet.)
Weblinks
Karte von Lauf und Einzugsgebiet des Maßholderbachs auf:
Touristische Karte des oberen Maßholderbach-Einzugsgebietes auf:
Meßtischblätter in der Deutschen Fotothek:
6722 Brettach von 1933
6723 Öhringen von 1933
Fließgewässer im Hohenlohekreis
Gewässer in Öhringen
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\section{Introduction}
Driven by nanoscience interests it has become necessary to develop
tools for hydrodynamic calculations at the atomistic scale \cite{Nanohydrodynamics_Alder,DSMC_MPCD_Gompper,Microfluidics_Review,PolymerTumbling_Review,FluctuatingHydro_Coveney}.
Of particular interest is the modeling of polymers in a flowing {}``good''
solvent for both biological (e.g., cell membranes) and engineering
(e.g., micro-channel DNA arrays) applications \cite{PolymerDynamics_Review,PolymerTumbling_Review}.
The most widely studied polymer models are simple linear bead-spring;
freely-jointed rods; or worm-like chains. Such models have been parameterized
for important biological and synthetic polymers. Much theoretical,
computational, and experimental knowledge about the behavior of these
models has been accumulated for various representations of the solvent.
However, the multi-scale nature of the problem for both time and length
is still a challenge for simulation of reasonably large systems over
reasonably long times. Furthermore, the omission in these models of
the \emph{explicit} coupling between the solvent and the polymer chain(s)
requires the introduction of adjustable parameters (e.g., friction
coefficients) to be determined empirically. The algorithm presented
here overcomes this for a linear polymer chain tethered to a hard
wall and subjected to a simple linear shear flow \cite{TetheredPolymer_Experiment_PRL,TetheredPolymer_HybridMD,TetheredPolymer_FullMD,TetheredPolymer_Cyclic_PRL,TetheredDNA_FullMD}.
Of particular interest is the long-time dynamics of the polymer chain
\cite{TetheredPolymer_Experiment_PRL,TetheredPolymer_FullMD,PolymerTumbling_PRL,TetheredPolymer_Cyclic_PRL,TetheredPolymer_Cyclic_AIP}
and any effects of the polymer motion on the flow field.
Brownian dynamics is one of the standard methods for coupling the
polymer chains to the solvent \cite{BrownianDynamics_DNA,BrownianDynamics_OrderN}.
The solvent is only implicitly represented by a coupling between the
polymer beads and the solvent in the form of stochastic (white-noise)
forcing and linear frictional damping. The flow in the solvent is
not explicitly simulated, but approximated as a small perturbation
based on the Oseen tensor. This approximation is only accurate at
large separations of the beads and at sufficiently small Reynold's
numbers. Even algorithms that do model the solvent explicitly via
Lattice Boltzmann (LB) \cite{LatticeBoltzmann_Polymers}, incompressible
(low Reynolds number) CFD solvers \cite{FluctuatingHydro_FluidOnly,DNA_Laden_Flow,FluctuatingHydroMD_Coveney},
or multiparticle collision dynamics \cite{PolymerCollapse_Yeomans,Trebotich_HardRods,MultiparticleDSMC_Polymers,DSMC_MPCD_MD_Kapral},
typically involve phenomenological coupling between the polymer chain
and the flowing fluid in the form of a linear friction term based
on an effective viscosity (a notable exception being the algorithm
described in Ref. \cite{DSMC_MPCD_MD_Kapral}). Furthermore, solvent
fluctuations in the force on the polymer beads are often approximated
without fully accounting for spatial and temporal correlations. Finally,
the reverse coupling of the effect of the bead motion on the fluid
flow is either neglected or approximated with delta function forcing
terms in the continuum fluid solver \cite{Trebotich_Penalty}. More
fundamentally, continuum descriptions of flow at micro and nanoscales
are known to have important deficiencies \cite{Microfluidics_Review,Nanohydrodynamics_Alder}
and therefore it is important to develop an all-particle algorithm
that is able to reach the long times necessary for quantitative evaluation
of approximate, but faster, algorithms.
The most detailed (and expensive) modeling of polymers in flow is
explicit molecular dynamics (MD) simulation of both the polymer (solute)
and the surrounding solvent \cite{PolymerShear_MD,TetheredDNA_FullMD}.
Multi-scale algorithms have been developed to couple the MD simulation
to Navier-Stokes-based computational fluid dynamics (CFD) calculations
of the flow field \cite{TetheredPolymer_HybridMD}. However, the calculation
time still remains limited by the slow molecular dynamics component.
Thus the computational effort is wasted on simulating the structure
and dynamics of the solvent particles, even though it is the polymer
structure and dynamics (and their coupling to the fluid flow) that
is of interest. Our algorithm replaces the deterministic treatment
of the solvent-solvent interactions with a stochastic momentum exchange
operation, thus significantly lowering the computational cost of the
algorithm, while preserving microscopic details in the solvent-solute
coupling.
Fluctuations drive the polymer motion and must be accurately represented
in any model. Considerable effort has been invested in recent years
in including fluctuations directly into the Navier-Stokes (NS) equations
and the associated CFD solvers \cite{FluctuatingHydro_FluidOnly,FluctuatingHydro_Coveney,FluctuatingHydro_Garcia}.
Such fluctuating hydrodynamics has been coupled to molecular dynamics
simulations of polymer chains \cite{FluctuatingHydroMD_Coveney},
but with empirical coupling between the beads and the fluid as discussed
above. To avoid the empirical coupling, the solvent region could be
enlarged by embedding the atomistic simulations of the region around
the polymer chain (such as pure MD or our combined MD/DSMC algorithm)
in a fluctuating hydrodynamics region. The bidirectional coupling
between the continuum and particle regions has to be constructed with
great care so that both fluxes and fluctuations are preserved \cite{FluctuatingHydro_AMAR}.
A well-known problem with such multiscale approaches is that the finest
scale (atomistic simulation) can take up the majority of computational
time and thus slow down the whole simulation. By using DSMC the cost
of the particle region can be made comparable to that of the continuum
component.
The Stochastic Event-Driven Molecular Dynamics (SEDMD) algorithm presented
here combines Event-Driven Molecular Dynamics (EDMD) for the polymer
particles with Direct Simulation Monte Carlo (DSMC) for the solvent
particles. The polymers are represented as chains of hard spheres
tethered by square wells. The solvent particles are realistically
smaller than the beads and are considered as hard spheres that interact
with the polymer beads with the usual hard-core repulsion. The algorithm
processes true (deterministic, exact) binary collisions between the
solvent particles and the beads, without any approximate coupling
or stochastic forcing. However, for the purposes of the EDMD algorithm,
the solvent particles themselves do not directly interact with each
other, that is, they can freely pass through each other as for an
ideal gas. Deterministic collisions between the solvent particles
are replaced with stochastic DSMC collisions. Both asynchronous (event-driven)
and synchronous (time-driven) algorithmic ways of processing these
stochastic collisions will be discussed in the next section. Note
that our algorithm is similar to a recent algorithm developed for
soft interaction potentials combining time-driven MD with multiparticle
collision dynamics \cite{DSMC_MPCD_MD_Kapral}.
The fundamental ideas behind our algorithm are described next, and
further details are given in Section \ref{sec:Algorithmic-Details}.
Section \ref{sec:Tethered-Polymer} gives results from the application
of the algorithm to the tethered polymer problem, and some concluding
remarks are given in Section \ref{sec:Conclusions}.
\section{Hybrid Components}
In this section we briefly describe the two components of the SEDMD
algorithm: The stochastic handling of the solvent and the deterministic
handling of the solute particles. These two components are integrated
(i.e., \emph{tightly} coupled) into a single event-driven algorithm
in Section \ref{sec:Algorithmic-Details}.
\subsection{Solvent DSMC Model}
The validity of the (incompressible) Navier-Stokes continuum equations
for modeling microscopic flows has been well established down to length
scales of $10-100nm$ \cite{Microfluidics_Review}. However, there
are several issues present in microscopic flows that are difficult
to account for in models relying on a purely PDE approximation. Firstly,
it is not \emph{a priori} obvious how to treat boundaries and interfaces
well so as account for the non-trivial (possibly non-linear) coupling
between the flow and the microgeometry. Furthermore, fluctuations
are not typically considered in Navier-Stokes solvers, and they can
be very important at instabilities \cite{FluidMixing_DSMC} or in
driving polymer dynamics. Finally, since the grid cell sizes needed
to resolve complex microscopic flows are small, a large computational
effort (comparable to DSMC) is needed even for continuum solvers.
An alternative is to use particle-based methods, which are explicit
and unconditionally stable and rather simple to implement. The solvent
particles are directly coupled to the microgeometry, for example,
they directly interact with the beads of a polymer. Fluctuations occur
naturally with the correct spatio-temporal correlations. However,
as in continuum descriptions, the structure of the fluid is lost and
under certain conditions the high compressibility of the DSMC (ideal
gas) fluid can cause difficulties.
Several particle methods have been described in the literature, such
as MD \cite{PolymerShear_MD}, dissipative particle dynamics (DPD)
\cite{DPD_DNA}, and multi-particle collision dynamics (MPCD) \cite{DSMC_MPCD_Gompper,DSMC_MPCD_MD_Kapral}.
Our method is similar to MPCD (also called stochastic rotation dynamics
or the Malevanets-Kapral method), and in fact, both are closely related
to the Direct Simulation Monte Carlo (DSMC) algorithm of Bird \cite{DSMCReview_Garcia}.
The key idea behind DSMC is to replace deterministic interactions
between the particles with stochastic momentum exchange (collisions)
between nearby particles. Specifically, particles are propagated by
a fixed time step $\D{t}$, as in MD, moving ballistically along straight
lines during a time-step (advection step). At the end of each time
step, the particles are sorted into cells, each containing on the
order of ten particles, and then a certain number of random pairs
of particles that are in the same cell are chosen to undergo stochastic
collisions (collision step). These collisions do not take into account
the positions of the particles other than the fact they are in the
same cell (i.e., they are nearby). The collisions conserve momentum
and energy (but not angular momentum) exactly. Formally, DSMC can
be seen as a method for solving the Boltzmann transport equation for
a low-density gas, however, it is not limited to gas flows \cite{DSMC_DenseFluids,DSMC_CBA,DSMC_CBATheory}.
Our purpose for using DSMC is as a replacement for expensive MD, preserving
the essential hydrodynamic {}``solvent'' properties: local momentum
conservation, and linear momentum exchange on length scales comparable
to the particle size, and a similar fluctuation spectrum.
In the multiparticle collision variant of this algorithm originally
proposed by Kapral, the traditional DSMC collection of binary collisions
is replaced by a multi-particle collision in which the velocities
of all particles in the cell are rotated by a random amount around
the average velocity \cite{DSMC_MPCD_Gompper,DSMC_MPCD_MD_Kapral}.
This change improves efficiency but at the cost of some artificial
effects such as loss of Galilean invariance. These problems can be
corrected and the method has been successfully used in modeling polymers
in flow by including the beads, considered as (massive) point particles,
in the stochastic momentum exchange step \cite{MultiparticleDSMC_Polymers,PolymerCollapse_Yeomans,ShearThinning_Yeomans}.
We will employ traditional DSMC in our algorithm in order to mimic
the actual (deterministic) momentum exchange between solvent molecules
(as it would be in an MD simulation) and in order to avoid any possible
artifacts.
A fundamental deficiency of DSMC as a (micro or nano) hydrodynamic
solver is the large (ideal gas) compressibility of the fluid. For
subsonic flows this compressibility does not qualitatively affect
the results as the DSMC fluid will behave similarly to an incompressible
liquid, however, the (Poisson) density fluctuations in DSMC are significantly
larger than those in realistic liquids. Furthermore, the speed of
sound is small (comparable to the average speed of the particles)
and thus subsonic (Mach number less than one) flows are limited to
relatively small Reynolds numbers %
\footnote{For a low-density gas the Reynolds number is $Re=M/K$, where $M=v_{flow}/c$
is the Mach number, and the Knudsen number $K=\lambda/L$ is the ratio
between the mean free path $\lambda$ and the typical obstacle length
$L$. This shows that subsonic flows can only achieve high $Re$ flows
for small Knudsen numbers, i.e., large numbers of DSMC particles.%
}. The Consistent Boltzmann Algorithm (CBA) \cite{DSMC_CBA,DSMC_CBATheory},
as well as algorithms based on the Enskog equation \cite{DSMC_Enskog,DSMC_Enskog_Frezzotti},
have demonstrated that DSMC fluids can have dense-fluid compressibility.
A similar algorithm was recently constructed for MPCD \cite{MPCD_CBA}.
We are currently evaluating several DSMC variants in terms of their
efficiency and thermodynamic consistency under high densities %
\footnote{Note that the density fluctuations in the CBA fluid are identical
to those in an ideal gas and thus thermodynamically inconsistent with
the compressibility.%
} and will report our findings in future work.
\subsection{Polymer MD Model}
Polymer chains in a solvent are modeled using continuous pair potentials
and time-driven MD (TDMD), in which particles are synchronously propagated
using a time step $\D{t}$, integrating the equations of motion along
the way. For good solvents, the polymer beads are represented as spherical
particles that interact with other beads and solvent particles with
(mostly) repulsive pair potentials, such as the positive part of the
Lennard-Jones potential. Additionally, beads are connected via (usually
finitely-extensible FENE or worm-like) springs in order to mimic chain
connectivity and elasticity \cite{PolymerShear_MD}. Additionally,
stochastic forces may be present to represent the solvent. The time
steps required for integration of the equations of motion in the presence
of the strongly repulsive forces is small and TDMD cannot reach long
time scales even after parallelization. An alternative is to use hard
spheres instead of soft particles, allowing replacement of the FENE
springs with square-well tethers, thus avoiding the costly force evaluations
in traditional MD. Hard sphere MD is most efficiently performed using
event-driven molecular dynamics (EDMD) \cite{EventDriven_Alder,Event_Driven_HE,EDMD_Polymers_Hall,PolymerCollapse_EDMD}.
If the detailed structure and energetics of the liquid is not crucial,
such EDMD algorithms can be just as effective as TDMD ones but considerably
faster. The essential difference between EDMD and TDMD is that EDMD
is asynchronous and there is no time step, instead, collisions between
hard particles are explicitly predicted and processed at their exact
(to numerical precision) time of occurrence. Since particles move
along simple trajectories (straight lines) between collisions, the
algorithm does not waste any time simulating motion in between events
(collisions).
Hard-sphere models of polymer chains have been used in EDMD simulations
for some time \cite{EDMD_Polymers_Hall,EDMD_Polymers_Aggregation,EDMD_Polymer_Fibrils2}.
These models typically involve, in addition to the usual hard-core
exclusion, additional \emph{square well} interactions to model chain
connectivity. The original work by Alder \emph{et al.} on EDMD developed
the collisional rules needed to handle arbitrary square wells \cite{EventDriven_Alder}.
Infinitely high wells can model tethers between beads, and the tethers
can be allowed to be broken by making the square wells of finite height,
modeling soft short-range attractions. Recent studies have used square
well attraction to model the effect of solvent quality \cite{PolymerCollapse_EDMD}.
Even more complex square well models have been developed for polymers
with chemical structure and it has been demonstrated that such models,
despite their apparent simplicity, can successfully reproduce the
complex packing structures found in polymer aggregation \cite{EDMD_Polymer_Fibrils2,EDMD_Polymers_Aggregation}.
Recent work on coupling a Kramer bead-rod polymer to a NS solver has
found that the use of hard rods (instead of soft interactions) not
only rigorously prevents rod-rod crossing but also achieves a larger
time step, comparable to the time step of the continuum solver \cite{Trebotich_HardRods}.
This study is focused on the simplest model of a polymer chain, namely,
a linear chain of $N_{b}$ particles tethered by unbreakable bonds.
This is similar to the commonly-used freely jointed bead-spring FENE
model model used in time-driven MD. The length of the tethers has
been chosen to be $1.1D_{b}$, where $D_{b}$ is the diameter of the
beads %
\footnote{Note that the hard-sphere model rigorously prevents chain crossing
if the tether length is less than $\sqrt{2}D_{b}$ since two tethers
shorter than this length cannot pass through each other without violating
impenetrability.%
}. The implementation of square-well potentials is based on the use
of near-neighbor lists (NNLs) in EDMD, and allows for the specification
of square-well interactions for arbitrary pairs of near neighbors.
In particular, one can specify a minimal $L_{t}^{min}\geq D_{b}$
and maximal distance (tether length) $L_{t}^{max}>L_{t}^{min}$ for
arbitrary pairs of near neighbors %
\footnote{A value $L_{t}^{min}>D_{b}$ can be used to emulate chain rigidity
(i.e., a finite persistence length) by using second nearest-neighbor
interactions between chain beads.%
}.
\section{\label{sec:Algorithmic-Details}Details of Hybrid Algorithm}
In this section the hybrid EDMD/DSMC algorithm, which we name Stochastic
EDMD (SEDMD), is described in detail. Only a brief review of the basic
features of EDMD is given and the focus is on the DSMC component of
the algorithm and the associated changes to the EDMD algorithm described
in detail in Ref. \cite{Event_Driven_HE}. A more general description
of asynchronous event-driven particle algorithms is given in Ref.
\cite{AED_Serial}.
Asynchronous event-driven (AED) algorithms process a sequence of \emph{events}
(e.g., collisions) in order of increasing event time $t_{e}$. The
time of occurrence of events is predicted and the event is scheduled
to occur by placing it an \emph{event queue}. The simulation iteratively
processes the event at the head of the event queue, possibly scheduling
new events or invalidating old events. One \emph{impending event}
per particle $i$, $1\leq i\leq N$, is scheduled to occur at time
$t_{e}$ with partner $p$ (e.g., another particle $j$). The particle
position $\V{r}_{i}$ and velocity $\V{v}_{i}$ are \emph{only} updated
when an event involving particle $i$ is processed and the time of
last update $t_{i}$ is recorded (we will refer to this procedure
as a \emph{particle update}). We note that traditional \emph{synchronous
time-driven} (STD) algorithms with a \emph{time step} $\D{t}$ are
a trivial variant of the more general AED class. In particular, in
an STD algorithm events occur at equispaced times and each event is
a \emph{time step} requiring an update of all of the particles. The
AED algorithm processes a mixture of events involving single particles
or pairs of particles with time steps that involve the simultaneous
(synchronous) update of a large collection of particles.
Every particle $i$ belongs to a certain specie $s_{i}$. Particles
with species $s_{i}$ and $s_{j}$ may or may not interact with each
other (i.e., they may not be subject to the hard-particle non-overlap
condition). We focus on a system in which a large fraction of the
particles belong to a special specie $s_{DSMC}$ representing DSMC
particles (e.g., solvent molecules). These DSMC particles do not interact
with each other (i.e., they freely pass through each other), but they
do interact with particles of other species. We focus on the case
when the non-DSMC particles are localized in a fraction of the simulation
volume, while the rest of the volume is filled with DSMC particles.
This will enable us to treat the majority of DSMC particles sufficiently
far away from non-DSMC particles more efficiently than those that
may collide with non-DSMC particles.
Before describing the SEDMD algorithm in detail, we discuss the important
issue of efficiently searching for nearby pairs of particles.
\subsection{Near Neighbor Searches}
When predicting the impending event of a given particle $i$, the
time of potential collision between the particle and each of its \emph{neighbors}
(nearby particles) is predicted \cite{Event_Driven_HE,AED_Serial}.
The DSMC algorithm also requires defining neighbor particles, that
is, particles that may collide stochastically during the DSMC collision
step. For efficiency, geometric techniques are needed to make the
number of neighbors of a given particle $O(1)$ instead of $O(N)$.
In SEDMD we use the so-called \emph{linked list cell} (LLC) method
for neighbor searching in both the MD and DSMC components. The simulation
domain is partitioned into $N_{cells}$ cells as close to cubical
as possible. Each particle $i$ stores the cell $c_{i}$ to which
its centroid belongs, and each cell $c$ stores a list $\mathcal{L}_{c}$
of all the particles it contains, as well as the total number of particles
$N_{c}$ in the cell. For a given interaction range, neighbors are
found by traversing the lists of as many neighboring cells as necessary
to ensure that all particles within that interaction range are covered.
In traditional DSMC, only particles within the same cell are considered
neighbors and thus candidates for collision. There are also variants
of DSMC in which particles in nearby cells are included in order to
achieve a non-ideal equation of state \cite{DSMC_Enskog,DSMC_Enskog_Frezzotti}.
A more general implementation would use different cell meshes for
MD and DSMC neighbor searches, however, that would significantly complicate
the implementation.
\subsubsection{Cell Bitmasks}
In addition to the list of particles $\mathcal{L}_{c}$, each cell
$c$ stores a \emph{bitmask} $\mathcal{M}_{c}$ consisting of $N_{\mbox{bits}}>N_{s}+4$
bits (bitfields). These bits may be one (set) or zero (not set) to
indicate certain properties of the cell, specifically, what species
of particles the cell contains, whether the cell is event or time
driven, and to specify boundary conditions. In order to distinguish
the cells that contain non-DSMC particles (i.e., particles of specie
other than $s_{DSMC}$) from those that contain only DSMC particles,
bit $\gamma$ is set if the cell may contain a particle of specie
$\gamma$. The bit is set whenever a particle of specie $\gamma$
is added to the cell, and all of the masks are reset and then re-built
(i.e., refreshed) periodically. When performing a neighbor search
for a particle $i$, cells not containing particles of species that
interact with specie $s_{i}$ are easily found (by OR'ing the cell
masks with a specie mask) and are simply skipped. This speeds up the
processing of DSMC particles since cells containing only DSMC particles
will be skipped without traversing their lists of particles.
For the purposes of the combined MD/DSMC algorithm we will also need
to distinguish those cells that are nearby non-DSMC particles, that
is, that contain particles within the interaction range of some non-DSMC
particle. Such cells will be treated using a fully event-driven (ED)
scheme, while the remaining cells will be treated using a time-driven
or mixed approach. We use one of the bits in the bitmasks, bit $\gamma_{ED}$,
to mark \emph{event-driven (ED) cells} whenever a neighbor search
is performed for a non-DSMC particle. Specifically, bit $\gamma_{ED}$
is set for a given cell whenever the cell is traversed during a neighbor
search for a non-DSMC particle. This scheme correctly masks the cells
by only modifying the neighbor search routines without changing the
rest of the algorithm, at the expense of a small overhead. We also
mark the cells near hard-wall boundaries as ED cells. Cell bitmasks
should be refreshed (rebuilt) periodically so as to prevent the fraction
of ED cells from increasing. As will be seen shortly, it is necessary
to introduce at least one {}``sticky'' bit $\gamma_{st}$ that is
not cleared but rather persists (has memory), and is initialized to
zero (not set) at the beginning of the simulation.
\subsubsection{Near Neighbor Lists}
The cell size should be tailored to the DSMC portion of the algorithm
and can become much smaller than the size of non-DSMC particles. The
LLC method becomes inefficient when the interaction (search) range
becomes significantly larger than the cell size because many cells
need to be traversed. In this case the LLC method can be augmented
with the \emph{near-neighbor list} (NNL) method, and in particular,
the bounding sphere complexes (BSCs) method, as described in detail
for nonspherical hard particles in Ref. \cite{Event_Driven_HE}. We
have implemented the necessary changes to the algorithm to allow the
use of NNLs and BSCs (in addition to LLCs), and we used NNLs in our
simulations of polymer chains in solution. The use of BSCs is not
necessary for efficient simulations of polymer solutions if the size
of the polymer bead is comparable to the size of the cells, which
is the case for the simulations we report. We do not describe the
changes to the algorithm in detail; rather, we only briefly mention
the essential modifications.
For the purposes of DSMC it is important to maintain accurate particle
lists $\mathcal{L}_{c}$ for all cells $c$, so that it is known which
particles are in the same cell (and thus candidates for stochastic
collisions) at any point in time. Therefore, transfers of particles
between cells need to be predicted and processed even though this
is not done in the NNL algorithm described in Ref. \cite{Event_Driven_HE}.
Near-neighbor lists are \emph{only} built and maintained for DSMC
particles that are in event-driven cells (essentially exactly as described
in Ref. \cite{Event_Driven_HE}). For a DSMC particle $i$ that is
not in an ED cell $c_{i}$ we consider the smallest sphere enclosing
cell $c_{i}$ to be the (bounding) neighborhood (see Ref. \cite{AED_Serial})
of particle $i$ and \emph{only} update the (position of the) neighborhood
when the particle moves to another cell. This ensures that neighbor
searches using the NNLs are still exact without the overhead of predicting
and processing \emph{NNL update} events for the majority of the DSMC
particles.
\subsection{\label{Section_EDTD}The SEDMD Algorithm}
We have developed an algorithm that combines time-driven DSMC with
event-driven MD by splitting the particles between ED particles and
TD particles. Roughly speaking, only the particles inside event-driven
cells (i.e., cells for which bit $\gamma_{ED}$ is set) are part of
the AED algorithm. The rest of the particles are DSMC particles that
are not even inserted into the event queue. Instead, they are handled
using a time-driven (TD) algorithm very similar to that used in classical
DSMC.
It is also possible to implement DSMC as a fully asynchronous event-driven
(AED) algorithm and thus avoid the introduction of an external time
scale through the time step $\D{t}$. The algorithm introduces a novel
type of event we term \emph{stochastic} (DSMC) \emph{collisions},
and it is discussed in more detail in Appendix \ref{Appendix_AEDDSMC}.
Asynchronous processing has a few advantages over the traditional
(synchronous) time-driven approach, notably, no errors due to time
discretization \cite{DSMC_TimeStepError} and improved efficiency
at low collision rates. For high densities (i.e., high collision rates)
we have found that these advantages are outweighed by the (implementation
and run-time) cost of the increased algorithmic complexity. Additionally,
time-driven handling has certain important advantages in addition
to its simplicity, notably, the synchrony of the DSMC portion of the
algorithm allows for parallelization and easy incorporation of algorithmic
alternatives (e.g., multi-particle or multi-cell collisions, adaptive
open boundary conditions, etc.).
The main types of events in the SEDMD algorithm are:
\begin{description}
\item [{Update}] Move particle $i$ to the current simulation time $t$
if $t_{i}<t$.
\item [{Transfer}] Move particle $i$ from one cell to another when it
crosses the boundary between two cells (this may also involve a translation
by a multiple of the lattice vectors when using periodic BCs).
\item [{Hard-core~collision}] Collide a particle $i$ with a boundary
such as a hard wall or another particle $j$ with which it interacts.
\item [{Tether~collision}] Bouncing of a pair of tethered particles in
a polymer chain when the tether stretches (processed exactly like
usual hard-particle collisions \cite{EventDriven_Alder,EDMD_Polymers_Hall}).
\item [{Time~step}] Move all of the time-driven particles by $\D{t}$
and process stochastic collisions between them.
\end{description}
The position $\V{r}_{i}$ and time $t_{i}$ as well as the impending
event prediction of particle $i$ are updated whenever an event involving
the particle is processed.
Both the event-driven and the time-driven DSMC algorithms process
stochastic binary \emph{trial collisions}. Processing a trial collision
consists of randomly and uniformly selecting a pair of DSMC particles
$i$ and $j$ that are in the same cell. For hard spheres in the low-density
limit, the probability of collision for a particular pair $ij$ is
proportional to the relative velocity $v_{ij}^{\mbox{rel}}$, and
therefore the pair $ij$ is accepted with probability $v_{ij}^{\mbox{rel}}/v_{\mbox{rel}}^{\mbox{\mbox{max}}}$.
If a pair is accepted for collision than the velocities of $i$ and
$j$ are updated in a random fashion while preserving energy and momentum
\cite{DSMCReview_Garcia}. If a real collision involving an ED particle
$i$ occurs then that particle is updated to time $t_{TS}$, its previous
event prediction is invalidated (this may involve updating a third-party
particle $k$), and an immediate update event is scheduled for $i$
(and possibly $k$).
It is important to note that the division of the DSMC particles between
ED and TD handling is dynamic and does not necessarily correspond
to the partitioning of the cells into ED and TD cells (based on the
cell bitfield $\gamma_{ED}$). As non-DSMC particles move, time-driven
cells may be masked as event-driven. This does not immediately make
the DSMC particles in such cells event-driven. Rather, time-driven
DSMC particles are moved into the event queue only when a collision
with a non-DSMC particle is scheduled for them, when they move into
a TD cell following a time step, or when restarting the event handling.
Event-driven particles are removed from the event queue when they
undergo cell transfer events into time-driven cells.
\subsubsection{Time Step Events}
The hybrid ED/TD algorithm introduces a new kind of event (not associated
with any particular particle) called a \emph{time step event}. This
event is scheduled to occur at times $t_{TS}=n\D{t}$, where $n\in\mathcal{Z}$
is an integer. When such an event is processed, all of the DSMC particles
not in the event queue are moved %
\footnote{Note that this update may involve moving some particles by less than
$\D{t}$ since the time of the last update for such particles does
not have to be a time step event but could be, for example, a cell
transfer.%
} to time $t_{TS}$ and are then re-sorted into cells (recall that
the ED particles are already correctly sorted into cells). Particles
that change from ED to TD cells and vice-versa are removed or inserted
into the event queue accordingly. Then, in each cell $\Gamma_{c}\D{t}$
trial DSMC collisions are performed, where\begin{equation}
\Gamma_{c}=\frac{N_{C}(N_{C}-1)\sigma v_{max}}{V_{c}}\label{eq:Gamma_c}\end{equation}
is the DSMC collision rate. Here $\sigma=4\pi R_{DSMC}^{2}$ in three
dimensions and $\sigma=4R_{DSMC}$ in two dimensions is the collisional
cross-section, $V_{c}$ is the volume of the cell, and $v_{max}$
is an upper bound for the maximal particle velocity %
\footnote{More precisely, $2v_{max}$ is an upper bound on the maximal relative
velocity between a pair of particles. In our implementation we maintain
the maximal encountered particle velocity $v_{max}$ and update it
after every collision and also reset it periodically.%
}.
In order to ensure correctness of the AED algorithm, a TD particle
must not move by more than a certain distance $\D{l_{max}}$ when
it undertakes a time step. Otherwise, it may overlap with a non-DSMC
particle that could not have anticipated this and scheduled a collision
accordingly. Specifically, recall that the event-driven cells are
marked whenever a neighbor search is performed for a non-DSMC particle.
Our simulation uses \[
\D{l_{max}}=(w_{ED}L_{c}-D_{DSMC})/2,\]
where the masking width $w_{ED}$ is the minimal number of cells covered
by any neighbor search in any direction, $L_{c}$ is the (minimal)
cell length, and $D_{DSMC}$ is the diameter of the DSMC particles.
Any DSMC particle whose velocity exceeds $v_{max}=\D{l_{max}}/\D{t}$
is inserted into the event queue at the end of a time step, and similarly,
any particles that would have been removed from the event queue are
left in the queue if their velocity exceeds the maximum safe velocity.
Typically, only a small (albeit non-zero) fraction of the DSMC particles
falls into this category and the majority of the particles that are
not in ED cells are not in the event queue. In fact, we choose the
time step to be as large as possible while still keeping the number
of dangerously fast DSMC particles negligible. This typically also
ensures that DSMC particles do not jump over cells from one time step
to the next (given that typically $w_{B}=1-2$).
\subsection{\label{Section_OpenBCs}Adaptive Open Boundary Conditions}
In three dimensions, a very large number of solvent particles is required
to fill the simulation domain. The majority of these particles are
far from the polymer chain and they are unlikely to significantly
impact or be impacted by the motion of the polymer chain. It therefore
seems reasonable to approximate the behavior of the solvent particles
sufficiently far away from the region of interest with that of a quasi-equilibrium
ensemble in which the positions of the particles are as in equilibrium
and the velocities follow a local Maxwellian distribution (the mean
of which is equal to the macroscopic local velocity). These particles
do not need to be simulated explicitly, especially for a DSMC liquid
which has no spatial structure (ideal gas). Rather, we can think of
the polymer chain and the surrounding DSMC fluid as being embedded
into an infinite reservoir of DSMC particles which enter and leave
the simulation domain following the appropriate distributions.
Such \emph{open (Grand Canonical) boundary conditions} (BC) are often
used in multi-scale (coupled) simulations. It is not trivial to implement
them when coupling the {}``reservoir'' to an MD simulation, especially
at higher densities. An example of an algorithm that achieves such
a coupling for soft-particle systems is USHER \cite{TetheredPolymer_HybridMD}.
It is also non-trivial to account for the velocity distribution of
the particles entering the simulation domain \cite{DSMC_InflowDistribution},
as would be needed in a purely event-driven algorithm in which particles
are inserted at the surface boundary of the domain. However, the combination
of a partially time-driven algorithm and an unstructured (ideal gas)
DSMC fluid makes it very easy to implement open BCs by inserting DSMC
particles in the cells surrounding the simulation domain only at time-step
events, based on very simple distributions.
\subsubsection{Cell Partitioning}
For the purposes of implementing such non-trivial BCs, we classify
the cells as being \emph{interior, boundary, and external cells}.
Interior cells are those that are in the vicinity of non-DSMC particles,
specifically, cells that are within a window of half-width $w_{int}>w_{ED}$
cells around the centroid of a non-DSMC particle. The interior cells
are divided into event-driven and time-driven and are handled as described
previously. If a boundary or external cell is marked as an event-driven
cell the simulation is aborted with an error, ensuring that ED cells
are always interior. Boundary cells surround the interior cells with
a layer of cells of thickness $w_{B}\geq1$ cells, and they represent
cells in which particles may be inserted during time step events %
\footnote{Since ED cells are never boundary cells such insertions cannot lead
to overlaps with non-DSMC particles.%
}. External cells are non-interior cells that are not explicitly simulated,
rather, they provide a boundary condition around the interior and
boundary cells. This layer must be at least $w_{B}$ cells thick,
and the cells within a layer of $w_{B}$ cells around the simulation
domain (interior together with boundary cells) are marked as both
external and boundary cells. All of the remaining cells are purely
external cells and simply ignored by the simulation. Our implementation
uses bits in the cell bitmasks to mark a cell as being event-driven
(bit $\gamma_{ED}$), boundary (bit $\gamma_{B}$), or external (bit
$\gamma_{P}$). Note that a cell may be a combination of these, for
example, cells near hard walls might be both interior and boundary,
and some cells may be both external and boundary.
\begin{figure}
\includegraphics[width=0.4\paperwidth,keepaspectratio]{0_home_donev1_HPC_Papers_DSMC_graphics_TetheredPolymer_DSMC_2D_N=25_partitioning_labeled}\hspace{0.5cm}\includegraphics[width=0.3\paperwidth,keepaspectratio]{1_home_donev1_HPC_Papers_DSMC_graphics_TetheredPolymer_DSMC_3D_N=25_partitioning}
\caption{\label{TetheredPolymer.partitioning}The partitioning of the domain
into interior (I) {[}either event-driven (ED) or time-driven (TD)],
boundary (B), and external (E) cells in two (left) and three (right)
dimensions for a polymer chain of $25$ beads tethered to a hard wall.
The cells are shaded in different shades of gray and labeled in the
two-dimensional illustration ($w_{ED}=2$, $w_{int}=5$, $w_{B}=2$).
The DSMC particles are also shown.}
\end{figure}
Figure \ref{TetheredPolymer.partitioning} provides an illustration
of this division of the cells for the simulation of a tethered polymer
in two and three dimensions. Note that we do not require that the
domains of interior or non-external cells form a rectangular domain:
The final shapes and even contiguity of such domains depends on the
positions of the non-DSMC particles %
\footnote{If this is not appropriate one can always make the simulation regions
(unions of disjoint) rectangular domains simply by padding with interior
cells.%
}. Our implementation traverses each of the non-DSMC particles in turn
and masks the cells in a window of half-width $w$ cells around the
cell containing the non-DSMC particle as:
\begin{description}
\item [{Interior}] $0\leq w\leq w_{int}$ representing cells where the
non-trivial flow occurs ($w_{int}>w_{ED}$)
\item [{Boundary}] $w_{int}<w\leq w_{int}+2w_{B}$ representing cells where
particles may be inserted or propagated during time step events.
\item [{External}] $w>w_{int}+w_{B}$ representing cells that are not explicitly
simulated but rather only provide appropriate BCs.
\end{description}
The division of the cells into event-driven, interior, boundary and
external cells is rebuilt periodically during the simulation. This
rebuilding may only happen at the beginning of time steps, and requires
a synchronization of all of the particles to the current simulation
time, a complete rebuilding of the cell bitmasks, and finally, a re-initialization
of the event processing. Importantly, particles that are in purely
external cells are removed from the simulation and those that are
in event-driven cells are re-inserted into the event queue scheduled
for an immediate update event. During the process of rebuilding the
cell bitmasks cells that are masked as purely external cells are also
marked with the sticky bit $\gamma_{s}$. This indicates that these
cells need to be re-filled with particles later if they enter the
simulation domain again (due to the motion of the non-DSMC particles).
Once the cell bitmasks are rebuilt, a time step event is executed
as described next.
\subsubsection{Reservoir Particles}
At the beginning of a time step event, after possibly rebuilding the
cell masks, the time-driven DSMC particles are propagated as usual.
If there are external cells, (trial) \emph{reservoir particles} are
then inserted into the cells that are both external and boundary,
and also in cells whose sticky bit $\gamma_{s}$ is set (i.e., cells
that have not yet been filled with particles), after which the bit
$\gamma_{s}$ is reset. The use of the sticky bit to mark such cells
ensures that subsequent rebuilding of the masks will not erase the
flag (the sticky bit is \emph{only} reset once the cell is filled
with particles). The trial particles are thought to be at a time $t-\D{t}$,
and are propagated by a time step $\D{t}$ to the current simulation
time. Only those particles that move into a non-external cell are
accepted and converted into real particles. If the acceptance would
insert the particle into a non-boundary cell (i.e., the particle moved
by at least $w_{B}$ cells), the insertion is rejected and a count
of the number of rejected particles reported to aid in choosing $w_{B}$
sufficiently large so as to ensure that the tails of the velocity
distribution are not truncated (in our experience $w_{B}=2$ suffices
for reasonable choices of $\D{t}$). Following the insertion of reservoir
particles stochastic collisions are processed in each cell as usual.
For improved efficiency, it is possible to replace the volume-based
particle reservoir with a surface reservoir, and insert particles
only at the surface of the simulation domain \cite{DSMC_InflowDistribution}.
However, we have not implemented such an approach since the boundary
handling is not critical for the overall efficiency.
\subsubsection{Boundary Conditions}
In our current implementation the reservoir particles follow simple
local-equilibrium ideal gas distributions. The number of particles
to insert in a given cell $c$ is chosen from a Poisson distribution
with the appropriate density, the positions are uniformly distributed
inside the cell, and the velocities are drawn from a biased (local)
Maxwellian distribution. The mean velocity $\V{v}_{M}$ and temperature
$T_{M}$ for the local Maxwellian are chosen according to the specified
boundary conditions (presently only uniform linear gradients are implemented).
For example, if a uniform shear in the $xy$ plane is to be applied,
$\V{v}_{M}=\gamma y_{c}\hat{x}$, where $y_{c}$ is the $y$ position
of the centroid of the cell and $\gamma$ is the shear rate. Using
such biased local insertions allows one to specify a variety of boundary
conditions (for example, a free polymer chain in \emph{unbounded}
shear flow) without resorting to hard-wall boundaries or complicating
Lee-Edwards conditions.
It should be noted that in principle we should not use a local Maxwellian
velocity distribution for a system that is not in equilibrium. In
particular, for small velocity, temperature, and density gradients
the Chapman-Enskog distribution is the appropriate one to use in order
to avoid artifacts near the open boundaries at length scales comparable
to the mean free path $\lambda$ \cite{AMAR_DSMC}. We judge these
effects to be insignificant in our simulations since our boundary
conditions are fixed externally and are thus not affected by the possible
small artifacts induced in the DSMC fluid flow, and since $\lambda$
is small.
In the future, we plan to replace the particle reservoir with a PDE-based
(Navier-Stokes) simulation coupled to the DSMC/MD one. Such a flux-preserving
coupling has been implemented in the past for coupled DSMC/Euler hydrodynamic
simulations \cite{AMAR_DSMC,AMAR_DSMC_SAMRAI}. It is however important
for the coupling to also correctly couple fluctuations. This requires
the use of \emph{fluctuating hydrodynamics} in the coupled domain.
Such solvers and associated coupling techniques are only now being
developed \cite{FluctuatingHydro_Garcia,FluctuatingHydro_AMAR}.
\subsection{Further Technical Details}
In this section we discuss several technical details of the SEDMD
algorithm such as hard-wall boundary conditions and the choice of
DSMC parameters.
\subsubsection{\label{Section_NoSlip}Slip and Stick Boundary Conditions}
We have already discussed the open boundary conditions and their use
to specify a variety of {}``far-field'' flow patterns. Additionally,
there can also be hard-wall boundaries, i.e., flat impenetrable surfaces.
These surfaces can have a velocity of their own and here we discuss
how particles reflect from such walls in the frame that moves with
the hard wall. Regardless of the details of particle reflections,
the total change in linear momentum of all the particles colliding
with a hard wall can be used to estimate the friction (drag) force
acting on the wall due to the flow. This can give reliable and quick
estimates of the viscosity of a DSMC fluid, for example. We use the
classical no-slip BCs (i.e., zero normal and parallel velocity) for
smooth hard-wall surfaces. Molecular simulations have found some slip;
however, at length-scales significantly larger than the mean free
path and/or the typical surface roughness one may assume no-slip boundaries
if the hard-wall boundary position is corrected by a slip length $L_{slip}$
\cite{Microfluidics_Review}.
Our simulations of tethered polymers use \emph{thermal walls} (kept
at $kT=1$) \cite{DSMCReview_Garcia} to implement no-slip hard walls
at the boundaries of the simulation cell. Following the collision
of a particle with such a wall, the particle velocity is completely
randomized and drawn from a half Maxwell-Boltzmann distribution (other
biased distributions may be used as appropriate). This automatically
ensures a zero mean velocity at the wall boundary and also acts as
a thermostat keeping the temperature constant even in the presence
of shear heating. No-slip boundaries can also be implemented using
(athermal) \emph{rough walls} which reflect incoming particles with
velocity that is the exact opposite of the incoming velocity \cite{DSMC_MPCD_CylinderFlow}.
Similarly, slip boundary conditions (zero normal velocity) can be
trivially implemented by using \emph{specular walls} that only reverse
the normal component of the velocity (relative to the wall). A mixture
of the two can be used to implement partially rough walls, for example,
a roughness parameter $0\leq r_{w}\leq1$ can be used as the probability
of randomly selecting a rough versus a specular collision.
Similar considerations apply to the boundary conditions at the interface
of a hard particle such as a polymer bead. Most particle-based methods
developed for the simulation of particle suspensions consider the
solvent particles as point particles for simplicity, and only MD or
certain boundary discretization schemes \cite{SuspensionsDSMC_Reflection}
resolve the actual solvent-solute interface. Specular BCs are typical
of MD simulations and assume perfectly conservative collisions (i.e.,
both linear momentum and energy are conserved). However, if the polymer
beads are themselves composed of many atoms, they will act as a partially
thermal (and rough) wall and energy will not be conserved exactly.
In the simulations reported here we have used rough walls for collisions
between DSMC and non-DSMC particles. This emulates a non-stick boundary
condition at the surface of the polymer beads. Using specular (slip)
conditions lowers the friction coefficient %
\footnote{The Stokes friction force has a coefficient of $4\pi$ for slip BCs
instead of the well-known $6\pi$ for no-slip BCs.%
}, but does not appear to qualitatively affect the behavior of tethered
polymers.
\subsubsection{\label{Section_ConstantPressure}Constant Pressure Flows}
We note briefly on our implementation of constant pressure boundary
conditions, as used to simulate flow through open pipes. A constant
pressure is typically emulated in particle simulations via a constant
acceleration $\V{a}$ for the DSMC particles \cite{DSMC_PlanePoiseuille}
together with periodic BCs along the flow (acceleration) direction.
In time-driven algorithms, one simply increments the velocity of every
particle by $\V{a}\D{t}$ and the position by $\V{a}\D{t}^{2}/2$
at each time-step (before processing DSMC collisions). In SEDMD this
is not easily implemented, since the trajectory of the DSMC particles
becomes parabolic instead of linear and exact collision prediction
between the DSMC and the non-DSMC particles is complicated. We have
opted to implement constant pressure BCs by using a periodic delta-function
forcing on the DSMC particles. Specifically, the velocities of all
DSMC particles are incremented %
\footnote{Recall that the event prediction for any ED particle $i$ whose velocity
is changed must be updated, typically by scheduling an immediate update
event. %
} at the beginning of each time step by $\V{a}\D{t}$, and then stochastic
collisions are processed.
\subsubsection{\label{Section_Gamma_c}Choice of DSMC Collision Frequency}
The viscosity of the DSMC fluid is determined by the choice of collision
frequency $\Gamma_{c}$ and cell size $L_{c}$. Classical DSMC wisdom
\cite{DSMCReview_Garcia} is that cell size should be smaller than
the mean free path, $L_{c}\ll\lambda$, but large enough to contain
on the order of $N_{c}\approx20$ particles (in three dimensions).
It is obvious that both of these conditions cannot be satisfied for
denser liquids, where $\lambda$ is only a fraction of the particle
size. It is now well-known that it is not necessary to have many particles
per cell, so long as in Eq. (\ref{eq:Gamma_c}) we use $N_{c}(N_{c}-1)$
instead of the traditional (but wrong) $N_{c}^{2}$. Coupled with
the Poisson distribution of $N_{c}$ this gives a constant average
total collision rate. However, using very small cells leads to very
large variability of collision rates from cell to cell and thus spatial
localization of momentum transfer during each time step. Namely, with
very small cells one rarely has two particles and thus most of the
collisions will occur in the few cells that happen to be densely populated.
We have aimed at trying to mimic what would happen in an MD simulation
in the DSMC one. In an MD simulation particles collide if their distance
is equal to the particle diameter $D$. Therefore, we have aimed at
keeping the cell size at a couple of diameters, $L_{c}\approx2D$.
At typical hard-sphere liquid densities this leads to $N_{c}\approx5-10$,
which seems appropriate in that it allows enough collision partners
for most of the particles but still localized the momentum transfer
sufficiently. For very small mean free paths DSMC does not distinguish
velocity gradients at length scales smaller than the cell size and,
in a long-time average sense, localizes the velocity gradients at
cell interfaces \cite{DSMC_CellSizeError}. We will assume that, for
problems of interest to us, the structure of the fluid and flow at
length-scales comparable to $D$ (and thus $L_{c}$) is unimportant,
and verify this by explicit comparisons to MD.
When the cell size is chosen such that $N_{c}\approx5-10$ and the
time step is reasonable, $\D{t}\approx(0.1-0.2)L_{c}/\bar{v}$, Eq.
(\ref{eq:Gamma_c}) gives collision frequencies that are sufficiently
high so that almost all particles suffer at least one collision every
time step, and typically more than one collision. The effect of such
repeated collisions is to completely thermalize the flow to a local
equilibrium (i.e., local Maxwellian), and we have observed that further
increasing the collision frequency does not change the effective viscosity
(we do not have a theoretical understanding of this behavior \cite{DSMC_CellSizeError}).
For greater efficiency, we have chosen to use the lowest collision
rate (for a given timestep) that still achieves a viscosity that is
as high as using a very high collision rate. We find that this is
typically achieved when each particle suffers about half a collision
or one collision each timestep \cite{DSMC_GammaBounds}. Appendix
\ref{Appendix_TRMC} describes some multi-particle collision variants
that may be more appropriate under different conditions.
\subsubsection{\label{Section_noHI}DSMC without Hydrodynamics}
The solvent exerts three primary effects on polymers in flow: (1)
stochastic forces due to fluctuations in the fluid (leading to Brownian-like
motion), (2) (local) frictional resistance to bead motion (usually
assumed to follow Stokes law), and (3) hydrodynamic interactions between
the beads due to perturbations of the flow field by the motion of
the beads. Brownian dynamics, the most common method for simulating
the behavior of polymers in flow, coordinates the first two effects
via the fluctuation-dissipation theorem, and potentially adds the
third one via approximations based on the Oseen tensor (neglecting
the possibility of large changes to the flow field due to the moving
beads). By turning off local momentum conservation one can eliminate
all hydrodynamic interactions, and thus test the importance of the
coupling between polymer motion and flow.
Yeoman's \emph{et al.} \cite{PolymerCollapse_Yeomans,ShearThinning_Yeomans}
have implemented a no-hydrodynamics variant of the MPCD algorithm
by randomly exchanging the velocities between all particles at each
time step (thus preserving momentum and energy globally, but not locally).
In the presence of a background flow, such as shear, only the components
of the velocities relative to the background flow are exchanged. We
have implemented a no-hydrodynamics variant of DSMC by neglecting
momentum conservation in the usual stochastic binary collisions %
\footnote{In this implementation switching hydrodynamics off becomes an alternative
branch localized in the binary collision routine and the algorithm
is otherwise unchanged.%
}. Specifically, if particles $A$ and $B$ collide, the post-collisional
velocity of $A$ is set to be the same magnitude as that of $B$ but
with a random orientation, and vice versa (this conserves energy but
\emph{not} momentum). If the boundary conditions specify a background
flow such as a uniform shear the flow velocity is evaluated at the
center of the DSMC cell and the collisions are performed in the frame
moving with that velocity. This forces the average velocity profile
to be as specified by the boundary conditions, but does not allow
for perturbations to that profile due to hydrodynamic effects.
\section{\label{Section_Performance}Performance Improvement}
It is, of course, expected that the DSMC algorithm will give a performance
improvement over MD. However, to make an impact on real-world problems
this performance gain must be an order of magnitude or more improvement.
Indeed, we find that SEDMD with adaptive boundary conditions can be
up to two hundred times faster than EDMD under certain conditions.
Note also that it is well-known that EDMD is already significantly
faster than TDMD (depending on the density), although such a comparison
is somewhat unfair since the hard-core interaction potentials are
very simple by design. It is important to note that this algorithm
is serial, and we do not consider or use any parallelization. Because
of the inherent simplicity and thus efficiency of the algorithm, however,
it is possible to study time scales and system sizes as large or larger
than parallel simulations described in the literature so far. The
combined time-driven DSMC with event-driven MD algorithm can be parallelized
using traditional techniques from TDMD if proper domain partitioning
is constructed, so that each event-driven region is processed by a
single processor %
\footnote{Achieving good load balancing will be easiest for systems containing
multiple polymer chains.%
}.
As model problem we study a tethered polymer in three dimensions.
The solvent density was chosen to be typical of a moderately dense
hard-sphere liquid. The performance and optimal choice of parameters
depends heavily on the size of the beads relative to the size of the
solvent particles for both MD and the hybrid algorithm. Realistically,
beads (meant to represent a Kuhn segment) should be larger than the
solvent molecules%
\footnote{For example, in Ref. \cite{FluctuatingHydroMD_Coveney} an appropriate
bead size for polyethylene is estimated at $1.5nm$, and for DNA (a
much stiffer molecule with large persistence length) at $40nm$.%
}. This of course dramatically increases the computational requirements
due to the increase in the number of solvent particles (and also makes
neighbor searching more costly). For this reason, most MD simulations
reported in the literature use solvent particles that are equivalent,
except for the chain connectivity, to the solute particles.
Our first test problem is for a chain of $25$ large beads, each about
$10$ times larger (in volume and in mass) than the solvent particles,
in a box of size $2\times1.25\times1.25$ polymer lengths, for a total
of about $N=2.3\times10^{5}$ particles. For the DSMC simulations,
we did not use BSCs (bounding sphere complexes \cite{Event_Driven_HE}),
and therefore the neighbor search had to include next-nearest neighbor
cells as well (i.e., $w_{ED}=2$). For the corresponding MD simulations,
BSCs were used. Under these conditions, DSMC outperformed MD by a
factor of $35$. If adaptive open BCs were used with $w_{int}=5$,
giving about $N=3.2\times10^{4}$ particles (the exact number changes
with polymer conformation), the speedup was $180$. While this may
seem an unfair comparison, it is important to point out that we do
not even know how to implement an adaptive simulation domain in pure
EDMD.
The second test problem was for a chain of $30$ beads which were
identical to the solvent particles, except for the added chain tethers.
The number of particles in the simulation cell was thus much smaller,
$N=4.8\times10^{4}$, and $w_{ED}=1$. Adaptive BCs with $w_{int}=5$
reduce the simulation domain to $N=2.2\times10^{4}$ particles. For
these parameters DSMC with adaptive BCs was about $30$ times faster
than full MD. Table \ref{PerformanceGains} summarizes the large performance
gains of SEDMD relative to traditional EDMD.
\begin{table}
\begin{tabular}{|c|c|c|}
\hline
&
Standard BCs&
Adaptive BCs\tabularnewline
\hline
\hline
Large beads&
35&
180\tabularnewline
\hline
Small beads&
20&
30\tabularnewline
\hline
\end{tabular}
\caption{\label{PerformanceGains}Performance gains of SEDMD relative to EDMD
for a typical tethered polymer simulation.}
\end{table}
One of the fundamental problems with multi-scale modeling is that
typically the majority of the simulation time is spent in the finest
model since it is difficult to match the time scales of the coupled
components \cite{Trebotich_Penalty}. For example, MD simulations
are so expensive that coupling them to almost any meso- or macro-scopic
solver leads to simulation times limited by that of MD simulations
(albeit of a much smaller system). By virtue of the fast microscopic
algorithm (EDMD instead of TDMD) and the efficient coupling, our method
spends comparable amounts of computation on the solute (and immediately
surrounding solvent) and the solvent particles. For the DSMC run with
adaptive open BCs and large beads, about $50\%$ of the time was spent
in manipulation of near neighbor lists. Most of the remaining time
was spent inside the routine that takes a DSMC timestep, and actual
processing of DSMC collisions (both trial and real) occupied about
$20\%$ of the computation time. For small beads, the majority of
the time, $80\%$, was spent in the DSMC time-step routine, and processing
of DSMC binary collisions occupied about $35\%$ of the computation
time.
\section{\label{sec:Tethered-Polymer}Tethered Polymer in Shear Flow}
In this section results are presented for a tethered polymer chain
in uniform shear in three dimensions. The linear chain is in a good
solvent and is attached at one end to a hard wall, as represented
by the plane $y=0$. A linear velocity profile $v=\gamma y\hat{x}$
along the $x$ axis is imposed sufficiently far from the chain. This
problem was first studied experimentally by Doyle \emph{et al.} \cite{TetheredPolymer_Experiment_PRL}
and since then numerous computational studies have investigated various
aspects of the problem \cite{TetheredPolymer_HybridMD,TetheredPolymer_FullMD,TetheredPolymer_Cyclic_PRL,TetheredPolymer_Cyclic_AIP,TetheredDNA_FullMD}.
We will focus on the dynamics of the chain at low to medium flow rates
(Weissenberg numbers) because we wanted to verify that our polymer
and solvent model can correctly reproduce non-trivial dynamics.
\subsection{Background}
The properties of a linear polymer in shear flow can be related to
the dimensionless Weissenberg number $\mbox{Wi}=\gamma\tau_{0}$, where
$\tau_{0}=\tau(\gamma=0)$ is the relaxation time of the polymer chain
when there is no shear. When $\mbox{Wi}<1$ the flow barely affects the
polymer, contrary to when $\mbox{Wi}>1$. Different models have given similar
properties for the same Weissenberg number.
The original experimental study of tethered polymers \cite{TetheredPolymer_Experiment_PRL}
observed what was termed {}``cyclic dynamics'' of the chains. Specifically,
the following cycle was proposed. When the polymer moves too far from
the wall, presumably by an unusual fluctuation, it experiences a stronger
flow and is stretched. A torque develops that then pushes the chain
closer to the wall, where it can contract again due to the weaker
flow near the wall. The cycle then repeats. Experiments \cite{TetheredPolymer_Experiment_PRL}
did not identify clear periodicity of this motion. Subsequent computational
studies have looked for such a characteristic period for this cycling
motion.
The MD study in Ref. \cite{TetheredPolymer_FullMD} examined the cross-correlation
function $C_{X\phi}(t)$, where $X$ measures the extension of the
polymer along the flow, and $\phi$ measures the angle of the chain
with respect to the hard wall. No exact definitions of $X$ or $\phi$
were given even though there are several possibilities. One can use
the difference between the maximal and the minimal bead positions
as a measure of the extension along a given axes. Optionally, one
can simply use the maximal position, or one can use the position of
the last bead. Similarly, the angle of the polymer can be based on
a linear fit to the shape of the chain, on the position of the center
of mass, the asymmetry of the gyration tensor \cite{PolymerTumbling_PRL},
or the position of the last bead. We have examined various choices
and have found little qualitative difference between the different
choices. We have found the position of the end bead $\V{r}_{N_{b}}=(x,y,z)$
to be the best option and will also measure the angle $\phi=\tan^{-1}(y/x)$.
The authors of Ref. \cite{TetheredPolymer_FullMD} found that $C_{x\phi}(t)$
develops a peak at positive time $t^{*}$ for sufficiently large $\mbox{Wi}$
numbers ($\mbox{Wi}>10$). This was interpreted as supporting the existence
of a critical Weissenberg number $\mbox{Wi}$ where the flow effect on the
polymer dynamics changes qualitatively. It was also found that $t^{*}$
decreases with increasing $\mbox{Wi}$ and the height of the peak increases.
It is important to note that $t^{*}$ was found to be comparable to
the relaxation time of the polymer $\tau_{0}$. Additionally, the
internal relaxation time $\tau$ was found to decrease with increasing
$\mbox{Wi}$, in agreement with theoretical predictions.
A subsequent study which used a hybrid MD/CFD model, and also a (free-draining)
Brownian dynamics model, claimed to observe periodic oscillations
in the cross-correlation function between the extensions along the
flow and along the shear direction (i.e., perpendicular to the wall),
$C_{xy}(t)$ \cite{TetheredPolymer_Cyclic_PRL,TetheredPolymer_Cyclic_AIP}.
However, the period of oscillation was found to be an order of magnitude
larger than the internal relaxation time, as revealed by a small peak
in the power spectral density $PSD_{xy}(f)$ of $C_{xy}(t)$. A similar
claim was made in Ref. \cite{PolymerTumbling_PRL} based on $PSD_{\phi\phi}$
of the polymer angle autocorrelation function %
\footnote{The PSD is equivalent to the Fourier spectrum power of the angle trace
$\phi(t)$ based on the convolution theorem.%
} $C_{\phi\phi}(t)$ for both a free polymer in unbounded shear flow
and a tethered polymer in shear flow. No results for the short-time
cross-correlation functions were reported in either of these studies
making it difficult to reconcile the results obtained from PSDs with
those in Ref. \cite{TetheredPolymer_FullMD}.
Most experimental and computational studies of the dynamics of polymers
in shear flow have been for free chains in unbounded flow \cite{PolymerTumbling_Review}.
In that problem, for $\mbox{Wi}>1$, it is possible to identify a well-defined
{}``tumbling'' event as the polymer rotates. The frequency of such
tumbling times can be measured by visual inspection and have been
compared to the computed location of the peak in the PSDs \cite{PolymerTumbling_PRL,PolymerTumbling_PDFs}.
The good match has thus been taken as an indicator that PSDs peaks
can be used to determine characteristic tumbling times and the same
methodology has been applied to a tethered polymer as well. However,
for the case of a tethered chain it is not easy to identify a periodic
event such as a specific rare fluctuation. Therefore, it is not surprising
that we do not confirm the existence of a characteristic time that
is an order of magnitude larger than the internal relaxation time.
One must here distinguish between {}``cyclic'' (repetitive) events
and periodic events. A Poisson time process of rate $\Gamma$ has
a well-defined time scale $\Gamma^{-1}$, however, the occurrence
of such events is not periodic; the delay between successive events
is exponentially-distributed. In Ref. \cite{PolymerTumbling_PDFs}
such an exponential distribution is proposed even for the delay between
successive tumbling events for a free chain in unbounded flow. The
PSD of such a process is expected to be that of white noise (i.e.,
flat) for frequencies small compared to $\Gamma$, and typically a
power-law decay for larger frequencies (gray noise). The occurrence
and shape of any local maxima (peaks) or frequencies comparable to
$\Gamma$ depends on the exact nature of the correlations at that
time scale.
\subsection{Model Parameters}
As explained in \ref{Section_Performance}, we have made several runs
for different polymer lengths and also bead sizes. One set of runs
used either $N_{b}=25$ or $50$ large beads each about 10 times larger
than a solvent particle, using DSMC with or without hydrodynamics
(see Section \ref{Section_noHI}) for the solvent. Another set of
runs used either $N_{b}=30$ or $60$ small beads each identical to
a solvent particle, using DSMC or pure MD for the solvent. The beads
were rough in the sense that no-slip conditions were applied for the
solvent-solute interface (see Section \ref{Section_NoSlip}). All
of the runs used open boundary conditions (see Section \ref{Section_OpenBCs}),
and the typical half-width of the interior region was $w_{int}=5$
or $w_{int}=7$ cells around the polymer chain. The difference in
the results (such as relaxation times) between these runs and runs
using $w_{int}=10$ or runs using periodic BCs were negligible for
the chain sizes we studied %
\footnote{It is expected that using a small $w_{int}$ would truncate the (long-ranged)
hydrodynamic interactions and thus increase the relaxation time. We
observe such effects for the $N_{b}=50$ chains, however, the effect
is too small compared to the statistical errors to be accurately quantified.%
}. The solvent was a hard-sphere MD or DSMC fluid with volume fraction
$\phi\approx0.25-0.30$, which corresponds to a moderately dense liquid
(the melting point is $\phi_{m}\approx0.49$). The $N_{b}=30$ runs
were run for $T\approx6000\tau_{0}$ with $w_{int}=7$, and such a
run takes about 6 days on a single 2.4GHz Dual-Core AMD Opteron processor.
Even for such long runs the statistical errors due to the strong fluctuations
in the polymer conformations are large, especially for correlation
functions at long time lags $t>\tau$.
\subsection{Relaxation Times}
The \emph{relaxation time} of the polymer $\tau$ is well-defined
only for linear models. It is often measured by fitting an exponential
to the autocorrelation function of the end-to-end vector $\V{r}_{end}(t)=\V{r}_{N_{b}}-\V{r}_{1}$,
where $\V{r}_{i}$ denotes the position of the $i$-th bead \cite{PolymerDynamics_Review}.
We will separately consider the different components of the end-to-end
vector $\V{r}_{end}=(x,y,z)$ and fit an exponential %
\footnote{The initial relaxation of the various auto-correlation functions $C(t)$
is faster than exponential, and the statistical error at longer times
is large even for long runs. We therefore fit the exponentials to
the portion of the curves at small times, when $0.2\leq C(t)\leq0.8$.
The fits are not perfect and there are large statistical errors depending
on the length of the run and the number of samples used to average
$C(t)$, and the relaxation times (and thus Weissenberg numbers) we
quote should be taken as approximate.%
} to the $C_{xx}$, $C_{yy}$ and $C_{zz}$ auto-correlations functions
to obtain the relaxation times $\tau_{x}$, $\tau_{y}$ and $\tau_{z}$
as a function of $\mbox{Wi}$. We find that $\tau_{z}$ is always the largest,
especially for large $\mbox{Wi}$ (for $\mbox{Wi}=0$, $\tau_{z}=\tau_{x}$ by
symmetry), and $\tau_{y}$ is always smaller by at least a factor
of two %
\footnote{This is because of the constraint that the polymer chain must be above
the plane $y=0$ at all times, which reduces the available configuration
space.%
}, even for $\mbox{Wi}=0$, as illustrated in the inset in Fig. \ref{tau_Wi.XYZ}.
We take $\tau_{0}=\tau_{x}(\mbox{Wi}=0)=\tau_{z}(\mbox{Wi}=0)$ as the definition
of the polymer relaxation time.
\begin{figure}
\includegraphics[width=0.9\columnwidth,keepaspectratio]{2_home_donev1_HPC_Papers_DSMC_graphics_tau_Wi_XYZ}
\caption{\label{tau_Wi.XYZ}Dependence of the relaxation times of the different
components of the end-to-end displacement vector on the Weissenberg
number. The relaxation times have been renormalized to equal unity
for $\mbox{Wi}=0$ for direct comparison. For each $\mbox{Wi}$, $\tau_{x}$ is
shown with circles, $\tau_{y}$ with squares, and $\tau_{z}$ with
diamonds. Different textures of the symbols are used for the different
models, as indicated in the legend. The inset shows $\tau_{y}/\tau_{x}$
and $\tau_{z}/\tau_{x}$ for the different runs.}
\end{figure}
The relaxation times we observe for $\mbox{Wi}=0$ are consistent with what
is predicted from theoretical considerations, $\tau\approx0.9\eta b^{3}N_{b}^{1.8}/kT$,
where $\eta$ is the viscosity and $b$ is the effective bead radius.
Using the viscosity (based on Enskog theory) of the MD liquid %
\footnote{Direct measurements of the viscosity of the DSMC liquid show that
it has viscosity rather close to that of the corresponding MD liquid
for the specific parameters we use.%
} and the tether length as $b$, we calculated $\tau\approx19$ for
the case of $N_{b}=25$ with large beads, to be compared to the numerical
results from DSMC $\tau=25\pm5$. The MD runs for the case of large
beads are not long enough to determine the relaxation time accurately.
We expect that the difference between MD and DSMC will become more
pronounced for smaller beads, and indeed, for $N_{b}=30$ we obtain
$\tau_{MD}\approx3\tau_{DSMC}$. Turning hydrodynamics off in DSMC
extends the relaxation times (and also the collapse times for an initially
stretched polymer) by a factor of $3-5$, as already observed using
MPCD \cite{PolymerCollapse_Yeomans} and as predicted by Zimm theory
\footnote{It is difficult to directly compare DSMC with and without hydrodynamics
since switching hydrodynamics off, in our model, affects the friction
force between the beads and the solvent. This is unlike the models
were the friction force is an added phenomenological term that has
an adjustable coefficient. %
}. Figure \ref{tau_Wi.XYZ} illustrates the dependence of $\tau_{x}(\mbox{Wi})/\tau_{x}(\mbox{Wi}=0)$
on $\mbox{Wi}$, and similarly for the $y$ and $z$ directions. Quantitatively
similar (but not identical) results are observed independently of
the details of the polymer model and even the existence of hydrodynamic
relaxations.
\subsection{Cyclic Dynamics}
We now turn our attention to cross-correlations between polymer extensions
in the $x$ and $y$ directions. We have found that the cross-correlations
lags are most visible in the $x$ and $y$ positions of the last bead,
$C_{xy}(t)$. Our results for $C_{xy}(t)$ are shown in Fig. \ref{C_xy.N=3D30},
along with $C_{x\phi}(t)$ as an inset. The results for $C_{x\phi}(t)$
compare well with those in Ref. \cite{TetheredPolymer_FullMD}, although
we see the secondary peak developing at somewhat lower $\mbox{Wi}$. We
do not see any evidence for the existence of a critical $\mbox{Wi}$: There
are peaks at both positive and negative time in $C_{xy}(t)$ for all
$\mbox{Wi}$. Some cross-correlations, such as $C_{x\phi}(t)$, have a large
positive or negative cusp at the origin at $\mbox{Wi}=0$ and it is this
cusp that masks the peaks at non-zero lags for small $\mbox{Wi}$.
\begin{figure*}
\includegraphics[width=0.95\textwidth,keepaspectratio]{3_home_donev1_HPC_Papers_DSMC_graphics_C_xy_N=30_long_2_shear}
\caption{\label{C_xy.N=3D30}Cross correlation function $C_{xy}(t)$ for chains
of $N_{b}=30$ small beads in a DSMC solvent at different shear rates.
The inset shows the corresponding $C_{x\phi}(t)$ for comparison with
the soft-particle MD results in Ref. \cite{TetheredPolymer_FullMD}.
Peaks are visible in $C_{xy}(t)$ at all $\mbox{Wi}>0$, but are obscured
in $C_{x\phi}(t)$ due to the large negative dip at the origin for
$\mbox{Wi}=0$. There are large statistical errors at small $\mbox{Wi}$ making
it difficult to identify the peaks.}
\end{figure*}
In Fig. \ref{C_xy.Wi=3D2} we compare $C_{xy}(t)$ at $\mbox{Wi}\approx2$
for several different models %
\footnote{The Weissenberg numbers were calculated after the runs were completed
and therefore the different runs are not at the exact same $\mbox{Wi}$
number.%
} and see a good match, even for the DSMC runs ignoring hydrodynamics
(momentum conservation). This indicates that the dynamics of the chains
is primarily driven by the competition between the internal stochastic
motion (entropy) and the external forcing due to the shear, and not
hydrodynamic interactions between the beads or the effect of the motion
of the chain on the flow.
\begin{figure}
\includegraphics[width=0.75\columnwidth,keepaspectratio]{4_home_donev1_HPC_Papers_DSMC_graphics_C_xy_Wi=2_comparison}
\caption{\label{C_xy.Wi=3D2}Comparison of $C_{xy}(t)$ for Weissenberg number
of about $2$ for several different models, after the time axes has
been normalized. }
\end{figure}
We do not discuss the origin and locations of the peaks in the cross-correlation
functions in detail in this work. These peaks are indicative of the
existence of a correlated motion in the $xy$ plane, but do not uniquely
identify that motion. An important question to address is the existence
of a time scale other than the internal relaxation time $\tau(\mbox{Wi}$).
In Fig. \ref{C_xy.renormalized} we show a renormalized cross-correlation
function \[
\tilde{C}_{xy}=\frac{1}{\mbox{Wi}}C_{xy}\left[\frac{t}{\tau(\mbox{Wi})}\right]\]
in an unsuccessful attempt to collapse the data for different $\mbox{Wi}$.
While the match is not perfect the picture does not point to the existence
of a time scale shorter than $\tau(\mbox{Wi})$. We also do not see any
convincing evidence for coherent and reproducible correlations on
time scales significantly larger than $\tau$, even in various power
spectral densities. Our results do not rule out the possibility of
a repetitive motion of the chain with widely varying cyclic times
(e.g., exponential tail) but we have not observed any direct evidence
for such cycling either. We will report more detailed results on the
dynamics of tethered polymer chains along with comparisons with Brownian
dynamics and Lattice-Boltzmann in future work.
\begin{figure}
\includegraphics[width=0.75\columnwidth,keepaspectratio]{5_home_donev1_HPC_Papers_DSMC_graphics_C_xy_N=30_renormalized}
\caption{\label{C_xy.renormalized}Cross correlation function $\tilde{C}_{xy}$
for $N=30$ DSMC runs as in Fig. \ref{C_xy.N=3D30} but with time
renormalized by $\tau(\mbox{Wi})$ and the correlation magnitude scaled
by $\mbox{Wi}$. }
\end{figure}
\section{\label{sec:Conclusions}Conclusions}
We presented a stochastic event-driven molecular dynamics (SEDMD)
algorithm that combines hard-sphere event-driven molecular dynamics
(EDMD) with direct simulation Monte Carlo (DSMC), aimed at simulating
flow in suspensions at the microscale. The overall algorithm is still
event-driven, however, the DSMC portion of the algorithm can be made
time-driven for increased efficiency. The fundamental idea is to replace
the deterministic (MD-like) interactions between particles of certain
species with a stochastic (MC-like) collision process, thus preserving
the phase space dynamics and conservation laws but ignoring the liquid
structure. The SEDMD methodology correctly reproduces hydrodynamic
behavior at the macroscale but also correctly represents fluctuations
at the microscale. A similar algorithm has been proposed using time-driven
(soft-particle) MD and a multiparticle collision variant of DSMC \cite{DSMC_MPCD_MD_Kapral}.
As an application of such a methodology we have considered the simulation
of polymer chains in a flowing solution, and in particular, a polymer
tethered to a hard wall and subject to shear flow. We have implemented
open boundary conditions that adaptively adjust the simulation domain
to only focus on the region close to the polymer chain(s). The algorithm
is found to be efficient even though it is not parallelized, and it
is found to reproduce results obtained via molecular dynamics and
other algorithms in the literature, after adjusting for the correction
to transport coefficients and compressibility of the DSMC fluid relative
to the MD fluid.
We studied the dynamics of a tethered polymer subject to pure shear
and found consistent results between MD and DSMC and also previous
TDMD studies. We find that neither the size of the polymer beads relative
to the solvent particles, nor the correct representation of the hydrodynamic
interactions in the fluid, qualitatively alter the results. This suggests
that fluctuations dominate the dynamic behavior of tethered polymers,
consistent with previous studies. Our results do not find periodic
motion of the polymer and show that the cross-correlation between
the polymer extensions along the flow and shear directions shows a
double-peak structure with characteristic time that is comparable
to the relaxation time of the polymer. This is in contrast to other
works that claim the existence of a new timescale associated with
the cyclic motion of the polymer. We will investigate these issues
further and compare with Brownian dynamics and Lattice-Boltzmann simulations
in future work.
We expect that this and related algorithms will find many applications
in micro- and nano-fluidics. In particular, the use of DSMC instead
of expensive MD is suitable for problems where the detailed structure
and chemical specificity of the solvent do not matter, and more general
hydrodynamic forces and internal fluctuations dominate. Using a continuum
approach such as Navier-Stokes (NS) equations for the solvent is questionable
at very small length scales. Furthermore, the handling of singularities
and fluctuations is not natural in such PDE methods and various approximations
need to be evaluated using particle-based methods. Since the meshes
required by continuum solvers for microflows are very fine, it is
expected that the efficiency of particle methods will be comparable
to PDE solvers. Nevertheless, algorithms based on fluctuating hydrodynamics
descriptions will be more efficient when fluctuations matter. Comparisons
and coupling of DSMC to fluctuating NS solvers is the subject of current
investigations \cite{FluctuatingHydro_AMAR}.
\section{Acknowledgments}
This work was performed under the auspices of the U.S. Department
of Energy by the University of California Lawrence Livermore National
Laboratory under Contract No. W-7405-Eng-48 (UCRL-JRNL-233235).
\begin{appendix}
\section{\label{Appendix_AEDDSMC}AED variants of DSMC }
In this Appendix we discuss a fully asynchronous event-driven (AED)
implementation of DSMC. The advantage of asynchronous algorithms is
that they do not introduce any artificial time scales (such as a time
step) into the problem \cite{AED_Review}. We have validated that
the AED algorithm produces the same results as the time-driven one
by comparing against published DSMC results for plane Poiseuille flow
of a rare gas \cite{DSMC_PlanePoiseuille}. We have also implemented
traditional time-driven (TD) DSMC and find identical results when
the time step is sufficiently small. We find that the event-driven
algorithm is almost an order of magnitude slower than the time driven
one at higher densities, and only becomes competitive at very low
densities (which is the traditional domain of interest for DSMC).
The overhead of the AED algorithm comes from the need to re-predict
the next event and update the event queue whenever a particle suffers
a DSMC collision. This cost is in addition to the equivalent cost
in the time-driven algorithm, namely, moving the particles forward
in time and colliding them.
The AED algorithm introduces a new type of event, a \emph{stochastic
(trial) collision} between two DSMC particles that are in the same
cell (see Section \ref{Section_EDTD}). These trial collisions occur
in a given cell $c$ as a Poisson process (i.e., exponentially distributed
waiting time) with a rate given by Eq. (\ref{eq:Gamma_c}). There
are several approaches to scheduling and processing DSMC collisions
directly borrowed from algorithms for performing Kinetic Monte Carlo
simulations (which are \emph{synchronous} event-driven algorithms
\cite{AdvancedKMC_Tutorial}). The simplest, and in our experience,
most efficient, approach to AED DSMC is to use \emph{cell rejection}
to select a host cell for the stochastic collisions. The rate of DSMC
collisions is chosen according to the cell with maximal occupancy
$N_{c}^{max}$, $\Gamma=N_{cells}\Gamma_{c}^{max}$. The randomly
chosen cell $c$ of occupancy $N_{c}$ is accepted with probability
$N_{c}(N_{c}-1)/\left[N_{c}^{max}(N_{c}^{max}-1)\right]$ and a random
pair of particles $i$ and $j$ are chosen from $\mathcal{L}_{c}$.
Since the DSMC fluid is perfectly compressible, the maximal cell occupancy
can be quite high for very large systems, and this leads to decreasing
cell acceptance probability as the size of the system increases.
One can avoid cell rejections altogether. The first option is to associate
stochastic collisions with cells and schedule one such event per cell.
The event time is easily predicted at any point in time $t$ to occur
at time $t-\Gamma_{c}^{-1}\ln r$ , where $r$ is a uniform random
deviate in $(0,1)$. These event times are put in the event queue
(which may be separate from the particle one and then the two queues
may be merged only at the top). The event times (and thus the cell
queue) need to updated whenever a cell occupancy $N_{c}$ changes,
that is, whenever a cell transfer is processed. This makes this algorithm
inefficient. Another alternative is to recognize that the sum of a
set of independent Poisson processes is a Poisson process with a rate
that is the sum of the individual rates, $\Gamma=\sum_{c}\Gamma_{c}$.
That is, DSMC collisions occur in the system as a Poisson process
with rate $\Gamma$. When processing such an event one has to first
choose the cell with probability $\Gamma_{c}/\Gamma$, which requires
some additional data structures to implement efficiently \cite{AdvancedKMC_Tutorial}.
For example, the cells could be grouped in lists based on their occupancy
and then an occupancy chosen first (with the appropriate weight),
followed by selection of a cell with that particular occupancy.
Finally, it is also possible to use a mixture of the asynchronous
and time-driven variants of DSMC. The asynchronous algorithm (e.g.,
based on cell rejection) can be used for DSMC particles in event-driven
cells, and the time-driven one elsewhere. This may be useful in situations
where the time-scale of the event-driven component (i.e., the solute
and nearby solvent particles) is significantly smaller than the time
step $\D{t}$ and thus time stepping would lead to discretization
artifacts.
In the AED variant of DSMC constant pressure BCs (see Section \ref{Section_ConstantPressure})
can be implemented by adding a new type of \emph{acceleration event}.
When such an event is processed, all of the particles are brought
to the same point in time (synchronized), the velocities of each DSMC
particle $i$ is incremented by $\V{a}\D{t}_{i}$ (here $\D{t}_{i}$
is the elapsed time since the last acceleration event), and the event
handling is restarted. The acceleration events occur as a Poisson
process with a suitably chosen rate, for example, ensuring that the
average or maximal change in velocity is a fraction of the average
particle velocity. Note that the choice of this acceleration rate
introduces an artificial time constant in the algorithm similar to
the time step $\D{t}$ in time-driven DSMC.
\section{\label{Appendix_TRMC}Multi-Particle Collisions in DSMC}
Under dense liquid conditions, DSMC binary collisions are so numerous
(see Section \ref{Section_Gamma_c}) that the velocities of the particles
are effectively thermalized to the local Maxwell distribution. We
have implemented a variant DSMC algorithm in which at every time step
the velocities of all of the particles are redrawn from a local Maxwellian,
preserving the total linear momentum and energy in each cell \cite{DSMC_Pullin}.
We found that this variant of DSMC is less efficient than and behaves
similarly to the usual binary-collision DSMC. Reference \cite{DSMC_TRMC}
describes a more general algorithm (TRMC) that combines binary collisions
for a subset of the particles with drawing from a local Maxwellian
for the remainder of the particles, and under dense liquid conditions
this typically degenerates to complete randomization of all of the
velocities at every time step. Until a theoretical framework is established
for the behavior of DSMC-like algorithms at high densities the classical
DSMC algorithm seems to be the best alternative in terms of simplicity,
efficiency, and theoretical foundation. The effect of collision rules
and cell size on multiparticle collision dynamics has been studied
and it was found that increased collisional viscosity is desirable
for achieving realistic convection to diffusion ratios \cite{DSMC_MPCD_Gompper,MultiparticleDSMC_Polymers}.
We mention that, strictly speaking, we should use as $V_{c}$ in Eq.
(\ref{eq:Gamma_c}) not the volume of the cell, but the unoccupied
cell volume (that is, the portion of the cell not covered by non-DSMC
particles) %
\footnote{Our implementation makes the additional approximation that non-DSMC
particles are also counted in $N_{c}$ in Eq. (\ref{eq:Gamma_c}),
instead of keeping a separate count of just the DSMC particles inside
each cell. If the polymer beads are larger than a cell than this approximation
does not matter since no cell can contain the centroid of both a DSMC
and non-DSMC particle.%
}. It is however difficult to dynamically maintain an accurate estimate
of the cell coverage, and the complication does not appear to be worth
the implementation complexity. In particular, an approximation is
already made in neglecting the structure of the fluid near a polymer
bead (i.e., the solvation layer), and furthermore, the majority of
the cells that are partially covered by a polymer bead will be entirely
or almost entirely covered so that they would at most contain a single
DSMC particle, in which case the probability of a DSMC collision would
be very low anyway. Finally, as explained in Section \ref{Section_Gamma_c},
the exact collision frequency does not really matter. In the context
of multiparticle collision dynamics, Ref. \cite{DSMC_MPCD_CylinderFlow}
proposes the use of virtual particles filling the partially-filled
cells as a way to achieve more accurate stick boundary conditions.
\end{appendix}
\bibliographystyle{unsrt}
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{"url":"https:\/\/thoughtstreams.io\/jtauber\/swift-programming-language\/4787\/","text":"# Swift Programming Language\n\n31 thoughts\nlast posted Aug. 6, 2014, 9:41 p.m.\n\n23 earlier thoughts\n\n0\n\nOne of the interesting things about the if let with optionals is you can chain optionals together and handle a nil at any level with a single if.\n\n7 later thoughts","date":"2022-07-05 03:26:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9095848202705383, \"perplexity\": 10374.562132040071}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104512702.80\/warc\/CC-MAIN-20220705022909-20220705052909-00327.warc.gz\"}"}
| null | null |
Q: Singular or Plural nouns Could someone please help me with this sentence? I'm having problem with deciding if it's it should be plural or singular.
I have a blue and red car.
or
I have a blue and red cars.
Which one is correct? I think it should be car and not cars as we are only talking about one unit.
A: In your example, you will want to use the singular, I believe you are trying to say
I have a blue car and a red car.
I have a blue and a red car.
I have two cars, one is red and one is blue
The article "a" distinguishes the two different cars.
However, your example
I have a blue and red car.
can mean you have one car and it has blue and red coloring.
I have blue and red cars.
means you have at least one blue car and at least one red car, but may have several of each.
|
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Q: Visual Studio 2012 MSVSMON.EXE does not work I'm really tired of this error.
Most of the time, when I want to debug the program , I'm faced with this message.
Microsoft to offer solutions, but it is quite confusing:
1.
Make sure the Visual Studio Remote Debugging Monitor is installed and running on the remote machine.
2.
Make sure the Remote Server Name is correct in the Name box in the Project Properties dialog box.
3.
Verify that the remote machine is accessible on the network.
please someone help and guide me because all of my projects have been suspended and i need visual studio debugging.
meanwhile, when i dc from internet, visual studio will work.
A: Windows 7 x64, VS 2012, VB.NET
I fixed it like this:-
Create a shortcut on your desktop to "C:\Program Files (x86)\Microsoft Visual Studio 11.0\Common7\IDE\Remote Debugger\x64\msvsmon.exe".
Right-click shortcut and select "Properties" from the dropdown menu.
Select the "Compatibity" tab
Tick "Run this program as administrator"
Click OK
Create a shortcut on your desktop to "C:\Program Files (x86)\Microsoft Visual Studio 11.0\Common7\IDE\devenv.exe".
Right-click shortcut and select "Properties" from the dropdown menu.
Select the "Compatibity" tab
Tick "Run this program as administrator"
Click OK
To start VS2012:-
1) Double-click the msvsmon shortcut icon (that you created above, to launch msvsmon)
2) Double-click the "Visual Studio 2012 Professional" shortcut icon (that you created above, to launch VS2012)
3) In VS2012, ensure standard toolbar is visible.
4) In VS2012, ensure "Solution Platforms" dropdown (on standard toolbar) is visible and set to "x86".
and ... wowee ... debug works !
A: I had the same problem. I fixed changing in properties/compile/target platform to x86 instead of Any CPU. It was the problem in my case. Hope it helps.
A: I got this error when starting VS2012 as a normal user. When starting as an administrator it opened up a UserControl TestContainer window instead.
What have happend was that the startup project was changed. I changed back to my ordinary startup project by right-click on it and choose "Set as StartUp Project".
A: I have started vs 2012 as administrator and it is fixed
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\section{Introduction}
Let $(M, g)$ be a smooth compact $n$-dimensional Riemannian manifold without boundary and $\Sigma$ a $k$-dimensional embedded submanifold. We denote by $\Delta_g$ the associated negative Laplace-Beltrami operator on $M$ so that the spectrum of $-\Delta_g$ is discrete. If $e_\lambda$ is any $L^2$ normalized eigenfunction, then we write
\begin{align*}
\Delta_g e_\lambda = -\lambda^2 e_\lambda,\quad \|e_\lambda\|_{L^2(M)}=1, \quad \lambda\geq 0.
\end{align*}
Here $L^p (M)$ is the space of $L^p$ functions with respect to the Riemannian measure. There have been many ways of measuring possible concentrations of the eigenfunctions of the Laplace-Beltrami operator on a manifold so far. One of the ways of measuring the possible concentrations of $e_\lambda$ on a manifold is to study the possible growth of the $L^p$ norm of the restrictions of $e_\lambda$ to submanifolds of $M$. This article deals with the concentrations of the restrictions of $e_\lambda$ to a curve with nonvanishing geodesic curvatures of $2$-dimensional manifold $M$.
We first review the previous results. We consider the operator $\mathds{1}_{[\lambda, \lambda+h(\lambda)]}(\sqrt{-\Delta_g})$, which projects a function onto all eigenspaces of $\sqrt{-\Delta_g}$ whose corresponding eigenvalue lies in $[\lambda, \lambda+h(\lambda)]$, which are approximations to eigenfunctions, or quasimodes. Recall that the exact eigenfunctions can also be considered as quasimodes in that
\begin{align*}
\mathds{1}_{[\lambda, \lambda+h(\lambda)]} (\sqrt{-\Delta_g}) e_\lambda=e_\lambda.
\end{align*}
For $h(\lambda)\equiv 1$ case, there are well-known estimates of Sogge \cite{Sogge1988concerning} which state that, for a uniform constant $C>0$ depending only on $M$,
\begin{align}\label{Sog88}
\| \mathds{1}_{[\lambda, \lambda+1]} (\sqrt{-\Delta_g}) \|_{L^2 (M) \to L^p (M)} \leq C \lambda^{\delta(p, n)},\quad \lambda\geq 1,
\end{align}
where
\[
\delta(p, n)=
\begin{cases}
\frac{n-1}{2}-\frac{n}{p}, & \text{if}\quad p_c\leq p\leq \infty, \\
\frac{n-1}{2}\left(\frac{1}{2}-\frac{1}{p} \right), & \text{if}\quad 2\leq p\leq p_c,
\end{cases}\quad\quad
p_c=\frac{2(n+1)}{n-1}.
\]
It follows immediately that
\begin{align}\label{Sog88 exact eigfcns}
\| e_\lambda \|_{L^p (M)}\leq C\lambda^{\delta (p, n)}.
\end{align}
The exponent $p_c$ is a so-called ``critical'' exponent. The work of Sogge \cite{Sogge1988concerning} (see also \cite[pp.142-145]{Sogge1993fourier}) also showed that the estimates \eqref{Sog88} are sharp in that there exist a function $f$, or a quasimode, such that
\begin{align*}
\|\mathds{1}_{[\lambda, \lambda+1]} (\sqrt{-\Delta_g}) f \|_{L^p (M)}\geq c \lambda^{\delta(p, n)} \|f\|_{L^2 (M)},\quad \text{for some uniform } c>0.
\end{align*}
Sogge \cite{Sogge1986oscillatory} showed that the estimates \eqref{Sog88 exact eigfcns} are sharp for an infinite family of exact eigenfunctions $e_\lambda$ in that
\begin{align*}
\|e_\lambda \|_{L^p (\mathbb{S}^n)} \geq c \lambda^{\delta(p, n)},\quad \text{for some uniform } c>0,
\end{align*}
where $M$ is the round sphere. Specifically, the $p_c\leq p\leq \infty$ case is saturated by a sequence of the zonal harmonics on the sphere, whereas $2\leq p\leq p_c$ case is sharp due to the highest weight spherical harmonics on the sphere. The estimates \eqref{Sog88} or \eqref{Sog88 exact eigfcns} are sometimes called ``universal estimates'' since they are satisfied on any smooth compact Riemannian manifold. If one assumes nonpositive curvatures or no conjugate points on $M$, the phenomenas are a bit different. For example, the geodesic flow in negatively curved manifolds behave chaotically, and so, there may be smaller concentration of the restrictions of eigenfunctions of the Laplace-Beltrami operator to geodesics in the negatively curved manifolds.
If $(M, g)$ has nonpositive sectional curvatures, we have some estimates of the case $h(\lambda)=(\log \lambda)^{-1}$
\begin{align}\label{LogM}
\| \mathds{1}_{[\lambda,\lambda+(\log \lambda)^{-1}]} (\sqrt{-\Delta_g}) \|_{L^2(M)\to L^p (M)} \leq C_p \frac{\lambda^{\delta(p, n)}}{(\log \lambda)^{\sigma(p, n)}},
\end{align}
for some constant $\sigma(p, n)>0$. By using methods of B\'erard \cite{Berard1977onthewaveequation}, Hassell and Tacy showed in \cite{HassellTacy2015improvement} that the estimates \eqref{LogM} hold for $\sigma(p, n)=\frac{1}{2}$ with $p_c < p\leq \infty$. This case was also recently investigated by Canzani and Galkowski \cite{CanzaniGalkowski2020Growth} under more general hypotheses. The case $2<p\leq p_c$ was investigated by Blair and Sogge \cite{BlairSogge2017refined, BlairSogge2018concerning, BlairSogge2019logarithmic}, Sogge \cite{Sogge2011KakeyaNikodym}, and Sogge and Zelditch \cite{SoggeZelditch2014eigenfunction}.
There are analogues of \eqref{Sog88} and \eqref{LogM} when we replace $\mathds{1}_{[\lambda, \lambda+h(\lambda)]}$ by $\mathcal{R}_{\Sigma}\circ\mathds{1}_{[\lambda, \lambda+h(\lambda)]}$, where $\mathcal{R}_\Sigma$ denotes the restriction map as $\mathcal{R}_\Sigma f=\left. f\right|_\Sigma$. The metric $g$ endows $\Sigma$ with induced measures, and thus, we can also consider the Lebesgue spaces $L^p (\Sigma)$. Burq, G\'erard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp} studied estimates of the form
\begin{align}\label{BGT}
\| \mathcal{R}_\Sigma \circ \mathds{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta_g}) \|_{L^2 (M) \to L^p(\Sigma)} \leq C \lambda^{\rho_k(p, n)},\quad \lambda\geq 1,
\end{align}
where
\[
\rho_k (p, n)=
\begin{cases}
\frac{n-1}{4}-\frac{n-2}{2p}, & \text{if } k=n-1 \text{ and } 2\leq p\leq \frac{2n}{n-1},\\
\frac{n-1}{2}-\frac{n-1}{p}, & \text{if } k=n-1, \text{ and } \frac{2n}{n-1}\leq p\leq \infty,
\end{cases}
\]
which in turn implies that
\begin{align}\label{BGT exact eigfcns}
\| e_\lambda \|_{L^p (\Sigma)}\leq C\lambda^{\rho_k(p, n)}.
\end{align}
These estimates are also called universal estimates since they hold on any smooth compact Riemannian manifold. The exponent $\frac{2n}{n-1}$ is the critial exponent in this case. They also considered other cases $k\leq n-2$, but we focus on $k=n-1$ here and below, since we will talk about $(n, k)=(2, 1)$ mainly in this paper. Using methods in semiclassical analysis, Tacy \cite{Tacy2010Semiclassical} considered the same estimates as special cases of estimates for quasimodes. In \cite{BurqGerardTzvetkov2007restrictions}, Burq, G\'erard, and Tzvetkov also showed the estimates \eqref{BGT} are sharp by showing that, for all $\lambda \geq 1$, there exists a function $f=f_\lambda$ such that
\begin{align*}
\| \mathcal{R}_\Sigma \circ \mathds{1}_{[\lambda, \lambda+1]} (\sqrt{-\Delta_g}) f\|_{L^p (\Sigma)} \geq c \lambda^{\rho_k (p, n)} \| f\|_{L^2 (M)},\quad \text{for some uniform } c>0,
\end{align*}
on any compact Riemannian manifold, and the estimates \eqref{BGT exact eigfcns} are sharp by showing that
\begin{align*}
\| e_\lambda \|_{L^p (\Sigma)} \geq c \lambda^{\rho_k (p, n)}, \quad \text{for some } c>0,
\end{align*}
if the $e_\lambda$ are the zonal harmonics or the highest weight spherical harmonics on the round sphere $M=\mathbb{S}^n$.
Focusing on the case $(n, k, p)=(2, 1, 2)$ in \eqref{BGT}, they showed
\begin{align*}
\| \mathcal{R}_\Sigma \circ \mathds{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta_g}) \|_{L^2 (M) \to L^2(\Sigma)} \leq C \lambda^{\frac{1}{4}},\quad \lambda\geq 1.
\end{align*}
Burq, G\'erard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp} showed that if $\Sigma$ is a curve $\gamma$ with nonvanishing geodesic curvatures, then $\lambda^{1/4}$ can be replaced by $\lambda^{1/6}$.
\begin{theorem}[Theorem 2 in \cite{BurqGerardTzvetkov2007restrictions}, Theorem 2 in \cite{Hu2009lp}]\label{Theorem Universal Estimates}
Suppose $\dim M=2$ and the curve $\gamma$ is a unit-length curve having nonvanishing geodesic curvatures, that is, $g(D_t \gamma', D_t \gamma')\not=0$, where $D_t$ is the covariant derivatives along the curve $\gamma$. We then have that, for a uniform constant $C$,
\begin{align}\label{Universal Estimates for Nonvanishing Curvature}
\| \mathcal{R}_\gamma \circ \mathds{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta_g}) \|_{L^2 (M) \to L^2(\gamma)} \leq C \lambda^{\frac{1}{6}},\quad \lambda\gg 1.
\end{align}
If follows immediately from this that
\begin{align}\label{Universal Estimates exact eigfcns}
\| e_\lambda \|_{L^2(\gamma)} \leq C \lambda^{\frac{1}{6}},\quad \lambda\gg 1.
\end{align}
\end{theorem}
This estimate was generalized to a higher dimensional analogue in \cite[Theorem 1.4]{Hu2009lp}. Again, using semiclassical analytic methods, Hassell and Tacy \cite{HassellTacy2012CurvedHypersurfaces} obtained estimates generalized to quasimodes.
Again, Burq, G\'erard, and Tzvetkov \cite[Section 5.2 and Remark 5.4]{BurqGerardTzvetkov2007restrictions} showed that the estimate \eqref{Universal Estimates for Nonvanishing Curvature} is sharp by finding a function $f$ as above, and the estimate \eqref{Universal Estimates exact eigfcns} is also sharp when $M$ is the standard sphere $\mathbb{S}^2$, and $\gamma$ is any curve with nonvanishing geodesic curvatures. See also Tacy \cite{Tacy2018Constructing} for constructing sharp examples for exact eigenfunctions on $\mathbb{S}^n$ or quasimodes. We will prove Theorem \ref{Theorem Universal Estimates} again in this article in a different point of view, since we need estimates in our proof to prove Theorem \ref{Theorem Log Improvement}, which will be illustrated below.
Similarly, when $(M, g)$ has nonpositive (or constant negative) curvatures, it has been studied that
\begin{align}\label{LogSub}
\| \mathcal{R}_\Sigma \circ \mathds{1}_{[\lambda, \lambda+(\log \lambda)^{-1}]} (\sqrt{-\Delta_g}) \|_{L^2(M) \to L^p(\Sigma)} \leq C \frac{\lambda^{\rho_k (p, n)}}{(\log \lambda)^{\sigma_k(p, n)}},\quad \lambda\geq 1,
\end{align}
for some constant $\sigma_k(p, n)>0$ with the same constant $\rho_k (p, n)$ as in \eqref{BGT}. In \cite{Chen2015improvement}, Chen obtained $\sigma_k(p, n)=\frac{1}{2}$ in \eqref{LogSub} for the cases $k=n-1$ with $p>\frac{2n}{n-1}$.
For $k=1$, there also have been studies of critical or subcritical exponent. For subcritical cases, Sogge and Zelditch \cite{SoggeZelditch2014eigenfunction} showed that for any $\epsilon>0$ there exists a $\lambda (\epsilon)<\infty$ such that
\begin{align}\label{SoggeZelditch Result}
\sup_{\gamma \in \Pi} \left(\int_\gamma |e_\lambda|^2\:ds \right)^{1/2} \leq \epsilon \lambda^{\frac{1}{4}},\quad \lambda >\lambda(\epsilon),\; \dim M=2,
\end{align}
where $\Pi$ is the space of all unit-length geodesics in $M$, and $ds$ is the arc-length measure on $\gamma$. By using the methods in \cite{SoggeZelditch2014eigenfunction} with Toponogov's comparison theorem, Blair and Sogge \cite{BlairSogge2018concerning} obtained $\sigma_1 (2, 2)=\frac{1}{4}$, which is an improvement of $\epsilon$ in \eqref{SoggeZelditch Result}. The works of Blair \cite{Blair2018logarithmic}, and Xi and Zhang \cite{XiZhang2017improved} obtain $\sigma_1(4, 2)=\frac{1}{4}$ for (unit-length) geodesics, which is a critical exponent in that $p=\frac{2n}{n-1}$.
As in the universal estimates, for the case $(n, k, p)=(2, 1, 2)$ in \eqref{LogSub}, we can expect that $\lambda^{1/4}$ may be replaced by $\lambda^{1/6}$ if $\gamma$ has nonvanishing geodesic curvatures, analogous to \eqref{Universal Estimates for Nonvanishing Curvature}. Moreover, by \cite[Theorem 2]{BurqGerardTzvetkov2007restrictions} and \cite[Theorem 1.2]{Hu2009lp}, we know that
\begin{align*}
\| \mathcal{R}_\gamma \circ \mathds{1}_{[\lambda, \lambda+(\log \lambda)^{-1}]}(\sqrt{-\Delta_g}) \|_{L^2 (M) \to L^p(\gamma)} \leq C \lambda^{\frac{1}{3}-\frac{1}{3p}},\quad\lambda\geq 1, \quad 2\leq p\leq 4.
\end{align*}
We want to show the analogue of this for $2\leq p<4$ in the presence of nonpositive sectional curvatures.
\begin{theorem}\label{Theorem Log Improvement}
Let $(M, g)$ be a compact $2$-dimensional smooth Riemannian manifold (without boundary) with nonpositive sectional curvatures pinched between $-1$ and $0$. Also suppose that $\gamma$ is a fixed unit-length curve with $g(D_t \gamma', D_t \gamma')\not=0$. Then, for a uniform constant $C_p>0$ and $\lambda\geq 1$,
\begin{align}\label{Our aim in this paper}
\| \mathcal{R}_\gamma \circ \mathds{1}_{[\lambda, \lambda+(\log \lambda)^{-1}]}(\sqrt{-\Delta_g}) \|_{L^2 (M) \to L^p (\gamma)} \leq C_p \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{(\log \lambda)^{\frac{1}{2}}},\quad 2\leq p<4,
\end{align}
where $C_p \to \infty$ as $p\to 4$.
It follows from this that
\begin{align*}
\| e_\lambda \|_{L^p (\gamma)} \leq C_p \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{(\log \lambda)^{\frac{1}{2}}},\quad \lambda\geq 1,\quad 2\leq p<4.
\end{align*}
\end{theorem}
We remark that, by scaling the metric, the above theorem deals with manifolds with nonpositive sectional curvatures. Using Theorem \ref{Theorem Log Improvement}, we can show the following estimate at the critical exponent $p=4$.
\begin{corollary}\label{Cor: at p=4}
Let $(M, g)$ be a compact $2$-dimensional smooth Riemannian manifold (without boundary) with nonpositive sectional curvatures pinched between $-1$ and $0$. Also suppose that $\gamma$ is a fixed unit-length curve with $g(D_t \gamma', D_t \gamma')\not=0$. Then, for a uniform constant $C>0$ and $\lambda\gg 1$,
\begin{align*}
\|\mathcal{R}_\gamma \circ \mathds{1}_{[\lambda, \lambda+(\log \lambda)^{-1}]} (\sqrt{-\Delta_g}) \|_{L^2 (M)\to L^4 (\gamma)}\leq C \frac{\lambda^{\frac{1}{4}}}{(\log \lambda)^{\frac{1}{8}}}.
\end{align*}
It then follows that
\begin{align*}
\|e_\lambda \|_{L^4 (\gamma)}\leq C\frac{\lambda^{\frac{1}{4}}}{(\log \lambda)^{\frac{1}{8}} },\quad \lambda\gg 1.
\end{align*}
\end{corollary}
This corollary is a curved curve analogue to Blair \cite[Theorem 1.1]{Blair2018logarithmic}, and Xi and Zhang \cite[Theorem 1, Theorem 2]{XiZhang2017improved}.
\subsection*{Outline of the work}
Even though Theorem \ref{Theorem Universal Estimates} is already proved in \cite[Theorem 2]{BurqGerardTzvetkov2007restrictions} and \cite[Theorem 2]{Hu2009lp}, we go through a variation of the proof of Theorem \ref{Theorem Universal Estimates} in \S \ref{S:Proof of universal estimates}, since we need some results from the proof to show Theorem \ref{Theorem Log Improvement}.
In \S \ref{S:Reduction to Universal Estimates}, we introduce some tools to prove Theorem \ref{Theorem Universal Estimates}. We will use pseudo-differential cutoffs $Q_j$ as in \cite{BlairSogge2018concerning} to reduce our problem to Proposition \ref{Prop: I-Qj estimates}, \ref{Prop: QJ estimates}, and \ref{Prop: q plus minus estimates}. The support properties of the $Q_j$ in $\xi$ are similar to a partition of unity in \cite[Section 6]{BurqGerardTzvetkov2007restrictions}.
We will prove Theorem \ref{Theorem Universal Estimates} by showing Proposition \ref{Prop: I-Qj estimates}, \ref{Prop: QJ estimates}, and \ref{Prop: q plus minus estimates} in \S \ref{S:Proof of universal estimates}. Stationary phase arguments, Young's inequality, and Egorov's theorem (cf. \cite{Sogge2014hangzhou}, \cite{Zworski2012Semiclassical}) will be the key points in the section.
By using Proposition \ref{Prop: I-Qj estimates} and \ref{Prop: q plus minus estimates}, we reduce Theorem \ref{Theorem Log Improvement} to a simpler version in \S \ref{S:Prop for large j}. To show the reduced estimates, we lift the remaining problem to the universal cover of the given manifold by the Cartan-Hadamard theorem. We will use the Hadamard parametrix there to compute the remaining part. We will need Proposition \ref{Prop: SP results in Log Improvement} to convert our problem to oscillatory integral operator problems. To finish the proof of Theorem \ref{Theorem Log Improvement}, we may need support properties of the oscillatory integral operators. We will use the Hessian comparison there (cf. \cite[Theorem 11.7]{Lee2018secondEd}) to figure out the support properties.
Using Theorem \ref{Theorem Log Improvement}, and the strategies in Xi and Zhang \cite{XiZhang2017improved}, Sogge \cite{Sogge2017ImprovedCritical}, and Bourgain \cite{Bourgain1991Besicovitch}, we will show Corollary \ref{Cor: at p=4} in the last section.
\subsection*{Notation}
\begin{enumerate}
\item For nonnegative numbers $A$ and $B$, $A\lesssim B$ means $A\leq CB$ for some uniform constant $C>0$ which depends only on the manifold under consideration.
\item $A\approx B$ means $cB\leq A \leq CB$ for some uniform constants $c>0$ and $C>0$.
\item The constant $C>0$ may be assumed to be a uniform constant, if there is no further notice. The uniform constant $C>0$ can also be different from each other at any different lines.
\item For geometric terminologies, the notation draws largely from Lee \cite{Lee2018secondEd}.
\item For terminologies of the pseudodifferential operator theory and Egorov's theorem, the notation draws largely from Sogge \cite{Sogge1993fourier}, \cite{Sogge2014hangzhou}, and Zworski \cite{Zworski2012Semiclassical}.
\item We use $\rho (x, y)$ for the Riemannian distance between $x$ and $y$.
\item Certain variables may be redefined in different places when the arguments there are independent of each other. For example,
\begin{itemize}
\item $\nabla$ may represent the gradient of functions in some places, and may be the Levi-Civita connection in other places.
\item $\alpha$ may represent a multiindice in some places, and may be deck transformations in other places, defined in the context of the universal cover of the base manifold $M$.
\item $\partial$ usually represents partial derivative, but $\partial^2 \phi$ represents the Hessian of $\phi$.
\item $N$ may represent a unit normal vector to a given curve $\gamma$ in some places, but may represent integers $N=1, 2, 3, \cdots$ in other places.
\item Tildes over letters usually denote the corresponding letters in the universal cover of the base manifold, but sometimes, we also use letters with tildes (or bars) when changing variables if needed.
\item $\epsilon>0$ appears in many places, and the meanings of $\epsilon>0$ there may be slightly different, but all of them are sufficiently small but fixed at the end of the computations in each section,
\end{itemize}
and so on. However, the context in which we are using the notations will be clear.
\end{enumerate}
\subsection*{Acknowledgements}
The author would like to thank his Ph.D. advisor Matthew D. Blair for suggesting the problem, for helpful insights, for unlimited patience, and for his guide with numerous details in the course of this work. The author was supported in part by the National Science Foundation grants DMS-1301717 and DMS-1565436.
\section{Some Tools and Reductions for Theorem \ref{Theorem Universal Estimates}}\label{S:Reduction to Universal Estimates}
Let $P=\sqrt{-\Delta_g}$. For some $\epsilon_0>0$ sufficiently small, let $\chi\in \mathcal{S}(\mathbb{R})$ be an even function such that
\begin{align}\label{epsilon0 and chi}
\chi(0)=1,\quad \chi(t)>0 \text{ for } |t|\leq 1,\quad \mathrm{supp}(\widehat{\chi})\subset \{t: \epsilon_0/2\leq |t|\leq \epsilon_0\}, \quad \mathrm{supp}(\widehat{\chi^2})\subset [-2\epsilon_0, 2\epsilon_0],
\end{align}
so that
\begin{align*}
\chi(\lambda-P)e_\lambda =e_\lambda.
\end{align*}
Assume that $\gamma$ has a unit-speed (parametrized by arc-length). With this in mind, to prove \eqref{Universal Estimates for Nonvanishing Curvature}, we now want to show
\begin{align}\label{Universal Estimate Chi Reduction}
\|\chi(\lambda-P) f \|_{L^2(\gamma)}\lesssim \lambda^{\frac{1}{6}}\|f\|_{L^2 (M)},
\end{align}
that is, we can replace the spectral projector $\mathds{1}_{[\lambda, \lambda+1]} (P)$ by $\chi (\lambda-P)$. Indeed, the operator $\chi (\lambda-P)$ is invertible on the range of the spectral projector $\mathds{1}_{[\lambda, \lambda+1]} (P)$ and
\begin{align*}
\| \chi(\lambda-P)^{-1} \circ \mathds{1}_{[\lambda, \lambda+1]} (P) \|_{L^2 (M)\to L^2 (M)} \lesssim 1,
\end{align*}
and so, it suffices to show \eqref{Universal Estimate Chi Reduction}.
Fix $\chi_0\in C_0^\infty (\mathbb{R})$ satisfying $\chi_0(t)=1$ for $|t|\leq 1$ and $\chi_0(t)=0$ for $|t|\geq 2$. We also fix $\Tilde{\chi}_0 \in C_0^\infty (\mathbb{R})$ that satisfies $\Tilde{\chi}_0 (t)=1$ for $|t|\leq 3$ and $\Tilde{\chi}_0 (t)=0$ for $|t|\geq 4$. Choose a Littlewood-Paley bump function $\chi_1 \in C_0^\infty (\mathbb{R})$ that satisfies $\chi_1(t)=0$ if $t\not\in (1/2, 2)$ so that we write
\begin{align*}
\sum_{j=-\infty}^\infty \chi_1 ( 2^j t )=1,\quad \text{for } t\not=0.
\end{align*}
We will use Fermi coordinates frequently in the rest of this article. We recall basic properties of Fermi coordinates briefly here. Let $\gamma$ and $M$ be as above, let $N$ be an element of the normal bundle $N\gamma$, let $\mathcal{E}\subset TM$ be the domain of the exponential map of $M$, let $\mathcal{E}_p=\mathcal{E}\cap N\gamma$, let $E:\mathcal{E}_p \to M$ be the restriction of $\mathrm{exp}$ (the exponential map of $M$) to $\mathcal{E}_p$, and let $U\subset M$ be a normal neighborhood of $\gamma$ with $U=E(V)$ for an appropriate open subset $V\subset N\gamma$. If $(W_0, \psi)$ is a smooth coordinate chart for $\gamma$, we define $B:\psi(W_0)\times \mathbb{R}\to N\gamma|_{W_0}$ by
\begin{align*}
B(x_1, v_1)=(q, v_1 N|_q),
\end{align*}
by shrinking $W_0$ if necessary. Setting $V_0=V\cap N\gamma|_{W_0}\subset N\gamma$ and $U_0=E(V_0)\subset M$, we define a smooth coordinate map $\varphi: U_0\to \mathbb{R}^2$ by $\varphi=B^{-1}\circ (E|_{V_0})^{-1}$,
\begin{align*}
\varphi: E(q, v_1 N_q)\mapsto (x_1 (q), v_1).
\end{align*}
Coordinates of this form are called Fermi coordinates. We list here properties of Fermi coordinates from \cite[Proposition 5.26]{Lee2018secondEd}.
\begin{enumerate}
\item $\gamma\cap U_0$ is the set of points where $v_1=0$.
\item At each point $q\in \gamma\cap U_0$, the metric components satisfy that
\[
g_{ij}=g_{ji}=\begin{cases}
0, & i=1 \quad \text{and} \quad j=2, \\
1, & i=j=2.
\end{cases}
\]
\item For every $q\in \gamma\cap U_0$ and $v=v_1 E_1|_q\in N_q \gamma$, the geodesic $\gamma_v$ starting at $q$ with initial velocity $v$ is the curve with coordinate expression $\gamma_v (t)=(x_1 (q), tv_1)$.
\end{enumerate}
For detail, see \cite[Chapter 2]{Gray2004Tubes}, \cite[Chapter 5]{Lee2018secondEd}, etc. If we identify a covector $\xi$ with a vector, then, in Fermi coordinates, we have
\begin{align*}
|\xi|_{g(x)}=g^{11}(x)\xi_1^2+\xi_2^2,\quad \text{for } x\in \gamma,
\end{align*}
where $(g^{ij})=(g_{ij})^{-1}$. Also, we observe that $g^{11}(x_1, 0)=1$ for $x=(x_1, 0)\in \gamma$ in Fermi coordinates, by the arc-length parametrization.
Suppose $\xi$ is a covector and $N$ is a unit vector field normal to $\gamma$. Here, $\xi(N)$ means $\langle \xi^\#, N \rangle_g$, where $\xi^\#$ is the sharp of $\xi$ as a musical isomorphism. In Fermi coordinates, $N=\frac{\partial}{\partial x_2}$. Set $J=\lfloor \log_2 \lambda^{\frac{1}{3}} \rfloor$. We write
\begin{align*}
1=\sum_{j=-\infty}^{J-1} \chi_1 \left(2^j \frac{|\xi(N)|}{|\xi|_g} \right)+\Tilde{\chi}_J \left(\lambda^{\frac{1}{3}} \frac{|\xi(N)|}{|\xi|_g} \right),
\end{align*}
where
\begin{align*}
\Tilde{\chi}_J (t)=1-\sum_{j=-\infty}^{J-1} \chi_1 (t),\quad \Tilde{\chi}_J \in C_0^\infty (\mathbb{R}),\quad \mathrm{supp} (\Tilde{\chi}_J) \subset \{t: |t|\lesssim 1\}.
\end{align*}
Here, if $j\ll 0$, the term $\chi_1 (2^j |\xi(N)|/|\xi_g|)$ is zero, and thus, the sum is a finite sum, since $|\xi(N)|\lesssim |\xi|_g$.
We will consider decomposition using pseudodifferential cutoffs in Smith and Sogge \cite{SmithSogge2007Boundary}, and Blair \cite{Blair2013LowRegularity}. In Fermi coordinates, if $j\leq J-1$, we define the compound symbols
\begin{align}\label{Construction of the compound symbol qj}
q_j (x, y, \xi)=\chi_0 (\epsilon_0^{-1} \rho (x, \gamma)) \Tilde{\chi}_0 (\epsilon_0^{-1} \rho (y, \gamma)) \chi_1 (2^j |\xi_2|/|\xi|_g) \Upsilon (|\xi|_g/\lambda),
\end{align}
where $d_g=\rho$, and $\Upsilon \in C_0^\infty (\mathbb{R})$ satisfies
\begin{align*}
\Upsilon (t)=1,\; \text{for } t\in [c_1, c_1^{-1}],\quad \Upsilon (t)=0,\; \text{for } t\not\in \left[\frac{c_1}{2}, 2c_1^{-1}\right],
\end{align*}
with a small fixed number $c_1>0$. Invariantly, we can also define the compound symbols by
\begin{align*}
q_j (x, y, \xi)=\chi_0 (\epsilon_0^{-1} \rho (x, \gamma)) \Tilde{\chi}_0 (\epsilon_0^{-1} \rho (y, \gamma)) \chi_1 \left(2^j\frac{|\xi(N)|}{|\xi|_g}\right)\Upsilon(|\xi|_g/\lambda).
\end{align*}
If $j=J$, we define, in Fermi coordinates,
\begin{align*}
q_J (x, y, \xi)=\chi_0 (\epsilon_0^{-1} \rho (x, \gamma)) \Tilde{\chi}_0 (\epsilon_0^{-1} \rho (y, \gamma)) \Tilde{\chi}_J (\lambda^{\frac{1}{3}}|\xi_2|/|\xi|_g) \Upsilon(|\xi|_g/\lambda),\quad 0<\epsilon_0\ll 1,
\end{align*}
or invariantly,
\begin{align*}
q_J (x, y, \xi)=\chi_0 (\epsilon_0^{-1} \rho (x, \gamma)) \Tilde{\chi}_0 (\epsilon_0^{-1} \rho (y, \gamma)) \Tilde{\chi}_J \left(\lambda^{\frac{1}{3}}\frac{|\xi(N)|}{|\xi|_g}\right) \Upsilon(|\xi|_g/\lambda), \quad 0<\epsilon_0\ll 1.
\end{align*}
Let $Q_j$ be the pseudodifferential operator with compound symbol $q_j$ whose kernel $Q_j (x, w)$ is defined by
\begin{align}\label{Definition of Qj kernel}
Q_j(x, w)=\frac{1}{(2\pi)^2} \int e^{i(x-w)\cdot \eta} q_j (x, w, \eta) \:d\eta.
\end{align}
As in \cite{BlairSogge2018concerning}, in Fermi coordinates, we know from the homogeneity in $\xi$ and $|\xi|\approx \lambda$ that
\begin{align}\label{Symbol Q properties}
\begin{split}
& |D_{x, w}^\beta D_{\xi_1}^{\alpha_1} D_{\xi_2}^{\alpha_2} q_j (x, w, \xi)|\leq C_{\alpha_1, \alpha_2, \beta} 2^{j|\alpha_2|} \lambda^{-|\alpha_1|-|\alpha_2|},\quad \text{for all } \alpha_1, \alpha_2,\\
& |\partial_{x, w}^\beta Q_j (x, w)|\leq C_N 2^{-j} \lambda^{2+|\beta|} (1+\lambda |x_1-y_1|+\lambda 2^{-j}|x_2-y_2|)^{-N},\quad \text{for } N=1, 2, 3, \cdots, \\
& \sup_x \int |Q_j (x, w)|\:dw,\quad \sup_w \int |Q_j (x, w)|\:dx \lesssim 1.
\end{split}
\end{align}
Now, for \eqref{Universal Estimate Chi Reduction}, we are reduced to showing that
\begin{align}\label{Qj estimates}
\|\sum_{j\leq J} Q_j \circ \chi (\lambda-P) f \|_{L^2(\gamma)} \lesssim \lambda^{\frac{1}{6}} \|f\|_{L^2 (M)},
\end{align}
and
\begin{align}\label{I-Qj estimates}
\|(I-\sum_{j\leq J} Q_j)\circ \chi(\lambda-P) f\|_{L^2(\gamma)} \leq C_N \lambda^{-N} \|f \|_{L^2(M)},\quad N=1, 2, 3, \cdots.
\end{align}
The estimate \eqref{I-Qj estimates} follows from Young's inequality and the analysis of its kernel.
\begin{proposition}\label{Prop: I-Qj estimates}
The kernel $(I-\sum_{j\leq J} Q_j)\circ \chi(\lambda-P) (x, y)$ satisfies
\begin{align*}
(I-\sum_{j\leq J} Q_j)\circ \chi(\lambda-P) (x, y)=O(\lambda^{-N}),
\end{align*}
for any $N\geq 1$.
\end{proposition}
We will talk about this later in \S \ref{SS: I-Qj estimates}. To see \eqref{Qj estimates}, we consider $j=J$ separately.
\begin{proposition}\label{Prop: QJ estimates}
We have
\begin{align*}
\| Q_J \circ \chi(\lambda-P) f\|_{L^2 (\gamma)}\lesssim \lambda^{\frac{1}{6}} \|f\|_{L^2 (M)}.
\end{align*}
\end{proposition}
We will talk about this proposition in the next section. Assuming this proposition is true, we would have \eqref{Qj estimates} if we could show that
\begin{align*}
\sum_{j\leq J-1} \| Q_j \circ \chi(\lambda-P) f\|_{L^2 (\gamma)} \lesssim \lambda^{\frac{1}{6}} \|f\|_{L^2 (M)},
\end{align*}
which follows from
\begin{align*}
\| Q_j \circ \chi(\lambda-P) f\|_{L^2 (\gamma)}\lesssim 2^{\frac{j}{2}} \|f\|_{L^2 (M)},\quad j\leq J-1.
\end{align*}
To see this, we split $Q_j$ into two operators $Q_{j, \pm}$
\begin{align*}
Q_j =Q_{j, +}+Q_{j, -},
\end{align*}
where the compound symbols $q_{j, \pm}$ of the $Q_{j, \pm}$ are
\begin{align*}
q_{j, \pm} (x, y, \xi)=\chi_0 (x_2) \Tilde{\chi}_0 (y_2) \chi_1 (\pm 2^j \xi_2/|\xi|_g) \Upsilon (|\xi|_g/\lambda),
\end{align*}
in Fermi coordinates. We would have \eqref{Qj estimates} if we could show the following.
\begin{proposition}\label{Prop: q plus minus estimates}
If $j\leq J-1$, we have
\begin{align*}
\| Q_{j, +} \circ \chi(\lambda-P) f\|_{L^2 (\gamma)}\lesssim 2^{\frac{j}{2}} \|f\|_{L^2 (M)}, \quad \text{and} \quad \| Q_{j, -} \circ \chi(\lambda-P) f\|_{L^2 (\gamma)} \lesssim 2^{\frac{j}{2}} \|f\|_{L^2 (M)}.
\end{align*}
\end{proposition}
We will also prove this proposition in the next section.
\subsection{Notation for symbols of pseudodifferential operators}\label{SS: Notation for PDOs}
The pseudodifferential operator $Q_j$ above is defined by using the compound symbols $q_j (x, y, \xi)$, but sometimes we will identify the compound symbol $q_j (x, y, \xi)$ with the usual symbol $q_j (x, \xi)$, modulo smoothing errors, especially when we apply Egorov's theorem and the theorem is stated with usual symbols of pseudodifferential operators. Indeed, recall from the pseudodifferential operator theory (cf. \cite[p.97]{Sogge1993fourier} or \cite[pp.92-pp.93]{Sogge2014hangzhou}) that there exists a symbol $\Tilde{q}_j (x, \xi)$ such that
\begin{align*}
\int e^{i(x-y)\cdot \xi} q_j (x, y, \xi)\:d\xi-\int e^{i(x-y)\cdot \xi} \Tilde{q}_j (x, \xi)\:d\xi
\end{align*}
is smoothing of any order.
The same principle is applied to any other symbols of pseudodifferential operators unless otherwise specified.
\section{Proof of Theorem \ref{Theorem Universal Estimates}}\label{S:Proof of universal estimates}
Note that
\begin{align*}
\chi^2 (\lambda-P) f(x)=\frac{1}{2\pi} \int e^{i(\lambda -P)t} \widehat{\chi^2}(t)f(x)\:dt =\frac{1}{2\pi} \int e^{it\lambda} \widehat{\chi^2}(t) e^{-itP} f(x)\:dt.
\end{align*}
We first recall from \cite{Sogge1993fourier} or \cite{Zworski2012Semiclassical} that, by the Lax parametrix, there exist $\varphi$ and $a$ such that, up to smoothing errors,
\begin{align*}
e^{-itP} (x, w)=\int e^{i\varphi (t, x, \xi)-iw\cdot \xi} a(t, x, \xi)\:d\xi.
\end{align*}
Here, the phase $\varphi=\varphi(t, x, \xi)$ satisfies, for small enough $t$,
\begin{align}\label{varphi construction}
\begin{split}
& \kappa_t (d_\xi \varphi(t, x, \xi), \xi)=(x, d_x \varphi(t, x, \xi)), \quad (\text{or, } \kappa_t (y, \xi(0))=(x, \xi(t))) \\
& \partial_t \varphi+p(x, d_x \varphi)=0,\quad \varphi(0, x, \xi)=\langle x, \xi \rangle,
\end{split}
\end{align}
where $\kappa_t: \mathbb{R}^{4}\to \mathbb{R}^4$ is the Hamiltonian flow of $p(x, \xi)=|\xi|_{g(x)}$, and homogeneous in $\xi$. Also, the amplitude $a$ satisfies
\begin{align*}
|\partial_t^j \partial_x^\alpha \partial_\xi^\beta a(t, x, \xi)|\leq C_{j, \alpha, \beta} (1+|\xi|)^{-|\beta|},
\end{align*}
and so,
\begin{align}\label{Size estimates for a}
|\partial_t^j \partial_x^\alpha \partial_\xi^\beta a(t, x, \lambda \xi)|\leq C_{j, \alpha, \beta} \lambda^{|\beta|} (1+\lambda |\xi|)^{-|\beta|}.
\end{align}
Note that the right hand side is dominated by $C_{j, \alpha, \beta}\lambda^{|\beta|}(1+\lambda |\xi|)^{-|\beta|}\lesssim C_{j, \alpha, \beta}$ if $|\xi|\approx 1$.
In this section, we prove Proposition \ref{Prop: q plus minus estimates}, Proposition \ref{Prop: QJ estimates}, and \eqref{I-Qj estimates} in order.
\subsection{Proof of Proposition \ref{Prop: q plus minus estimates}}\label{SS: Proof of Proposition q+-}
By the $TT^*$ argument, Proposition \ref{Prop: q plus minus estimates} follows from
\begin{align}\label{Qj+ estimates}
\|Q_{j, \pm} \circ \chi^2 (\lambda-P) \circ Q_{j, +}^* f \|_{L^2(\gamma)} \lesssim 2^j \|f\|_{L^2(\gamma)},\quad j\leq J-1.
\end{align}
We focus on the operator $Q_{j, +} \circ \chi^2 (\lambda-P) \circ Q_{j, +}^*$. The argument for $Q_{j, -} \circ \chi^2 (\lambda-P) \circ Q_{j, -}^*$ is similar. We write
\begin{align*}
Q_{j, +} \circ \chi^2 (\lambda-P) \circ Q_{j, +}^* f(x)&=\left[ Q_{j, +} \circ \left(\frac{1}{2\pi} \int e^{it\lambda} \widehat{\chi^2}(t) e^{-itP}\:dt \right) \circ Q_{j, +}^* \right]f(x) \\
&=\frac{1}{2\pi} \int K_{j, +}(x, y)f(y)\:dy,
\end{align*}
where
\begin{align}\label{Kj+ kernel}
K_{j, +} (x, y)=\int e^{it\lambda} \widehat{\chi^2}(t) \left( Q_{j, +} \circ e^{-itP} \circ Q_{j, +}^* \right)(x, y)\:dt.
\end{align}
By Egorov's theorem (cf. \cite[Theorem 11.1]{Zworski2012Semiclassical}, and/or \cite[Chapter 4]{Sogge2014hangzhou}), we have
\begin{align*}
Q_{j, +} \circ e^{-itP}=e^{-itP}\circ B_{t, j, +},
\end{align*}
where $B_{t, j, +}$ has a symbol
\begin{align*}
b_{t, j, +}=\kappa_t^* q_{j, +} +b'.
\end{align*}
Here, $\kappa_t^* q_{j, +}$ is homogeneous of degree $1$ in $\xi$, and $|\partial^\alpha b'|\leq C_\alpha' \lambda^{-1} 2^{2j} 2^{j|\alpha|}\leq C_\alpha \lambda^{-\frac{1}{3}} 2^{j|\alpha|}$ for all $\alpha$. We will ignore the remainder $b'$. Indeed, let $B'$ be the operator whose symbol is $b'=O_S (\lambda^{-1}2^{2j})$ such that
\begin{align*}
B'(x, y)=\frac{1}{(2\pi)^2}\int e^{i(x-y)\cdot \xi} b'(x, y, \xi)\:d\xi.
\end{align*}
The size estimates $|\partial^\alpha b'|\leq C_\alpha \lambda^{-\frac{1}{3}} 2^{j|\alpha|}$ are better than the size estimates of $\kappa_t^* q_{j, +}$ and $q_{j, +}$. Also, the symbol $b'$ is compactly supported with $\mathrm{supp}(b')\subset \mathrm{supp}(\kappa_{-t}(\mathrm{supp}(q_j)))$. Thus, the arguments below will work when we replace $\kappa_t^* q_{j, +}$ by $b'$ but with better estimates. Hence, for simplicity, we write $b_{t, j, +}=\kappa_t^* q_{j, +}$.
Without loss of generality, we can assume that $a(t, x, \lambda\xi)$ is compactly supported in $\xi$, independent of $\lambda$. Indeed, first let $h_t (z, y, \eta)$ be the symbol of $B_{t, j, +}\circ Q_j^*$. By construction, $h_t (z, y, \eta)$ is supported near $|\eta|\approx \lambda^{-1}$. Let $\beta\in C_0^\infty$ be a bump function with $\beta \equiv 1$ in a neighborhood of $\mathrm{supp}(h_t (z, y, \lambda(\cdot)))$. We then have that
\begin{align*}
(e^{-itP}\circ B_{t, j, +}\circ Q_j^*)(x, y)=\frac{\lambda^4}{(2\pi)^2}\iiint e^{i\lambda [\varphi(t, x, \xi)-z\cdot \xi+(z-y)\cdot \eta]} a(t, x, \lambda \xi) h_t(z, y, \lambda \eta)\:dz\:d\xi\:d\eta.
\end{align*}
Integrating by parts in $z$, we have, for $N=1, 2, 3, \cdots$,
\begin{align*}
& \left|\iiint e^{i\lambda [\varphi(t, x, \xi)-z\cdot \xi+(z-y)\cdot \eta]} (1-\beta(\xi)) a(t, x, \lambda \xi) h_t(z, y, \lambda \eta)\:dz\:d\xi\:d\eta \right| \\
&\lesssim \iiint_{|\xi|\not\approx 1,\; |\eta|\approx 1} (1+\lambda|\xi-\eta|)^{-N} |(1-\beta(\xi)) a(t, x, \lambda \xi) h_t (z, y, \lambda\eta)|\:dz\:d\xi\:d\eta \\
&\lesssim \iint_{|\xi-\eta|\gtrsim 1,\; |\eta|\approx 1} (1+\lambda |\xi-\eta|)^{-N}\:d\xi\:d\eta \lesssim \int_{|\xi|\gtrsim 1} (\lambda|\xi|)^{-N}\:d\xi \lesssim \lambda^{-N}.
\end{align*}
Since we can ignore this contribution, we can assume that $a(t, x, \lambda\xi)$ is compactly supported in $\xi$.
We write
\begin{align*}
(e^{-itP}\circ B_{t, j, +}\circ Q_{j, +}^*)(x, y)=\frac{\lambda^6}{(2\pi)^4}\int e^{i\lambda(\varphi(t, x, \xi)-y\cdot \xi)} V_{j, +}(t, x, y, \xi)\:d\xi,
\end{align*}
where
\begin{align}\label{Notation for semiclassical SP}
\begin{split}
& V_{j, +}(t, x, y, \xi)=\iiiint e^{i\lambda \Phi(w, \eta, z, \zeta)} v_{j, +}(t, w, \eta, z, \zeta)\:dw\:d\eta\:dz\:d\zeta, \\
& \Phi(w, \eta, z, \zeta)=(y-w)\cdot \xi+(w-z)\cdot \eta+(z-y)\cdot \zeta, \\
& v_{j, +}(t, w, \eta, z, \zeta)=v_{j, +}(x, y; t, w, \eta, z, \zeta)=a(t, x, \lambda\xi) b_{t, j, +}(w, z, \lambda\eta) q_{j, +}(y, z, \lambda\zeta).
\end{split}
\end{align}
We first consider the kernel $e^{-itP} \circ B_{t, j, +}\circ Q_{j, +}^*$.
\begin{lemma}\label{Lemma Semiclassical version of SP q+}
Let $\Phi, v$ be as in \eqref{Notation for semiclassical SP}. We have
\begin{align}\label{Result of SP for q+}
\begin{split}
(e^{-itP}\circ B_{t, j, +}\circ Q_{j, +}^*)(x, y)&=\lambda^2 \int e^{i\lambda(\varphi(t, x, \xi)-y\cdot \xi)} \Tilde{a}_j (t, x, y, \xi)\:d\xi \\
&\hspace{50pt}+\frac{\lambda^6}{(2\pi)^4}R_N(t, y) \int e^{i\lambda (\varphi(t, x, \xi)-y\cdot \xi)} a(t, x, \lambda \xi)\:d\xi,
\end{split}
\end{align}
where
\begin{align}\label{b lambda set up}
\begin{split}
\Tilde{a}_j (t, x, y, \xi)=\sum_{l=0}^{N-1} \lambda^{-l} L_l \Big(v_{j, +}(x, y; t, y, \xi, y, \xi)\Big), \quad \text{and} \quad |\partial_t^\alpha R_N|\leq C_{N, \alpha} \lambda^{-\frac{N}{3}},
\end{split}
\end{align}
and the $L_l$ are the differential operators of order at most $2l$ with respect to the variables $w, \eta, z$, and $\zeta$, acting on $v_{j, +}$ at the critical point of $\Phi (w, \eta, z, \zeta)$.
\end{lemma}
\begin{proof}
Since
\begin{align*}
\nabla_{w, \eta, z, \zeta} \Phi=(\Phi_w', \Phi_\eta', \Phi_z', \Phi_\zeta')=(-\xi+\eta, w-z, \zeta-\eta, z-y),
\end{align*}
the critical point is $(w, \eta, z, \zeta)=(y, \xi, y, \xi)$. We consider the stationary phase argument. The Hessian of $\Phi$, denoted by $\partial^2 \Phi$, is
\begin{align*}
\partial^2 \Phi=
\begin{pmatrix}
\Phi_{ww}'' & \Phi_{w\eta}'' & \Phi_{wz}'' & \Phi_{w\zeta}'' \\
\Phi_{\eta w}'' & \Phi_{\eta\eta}'' & \Phi_{\eta z}'' & \Phi_{\eta\zeta}'' \\
\Phi_{zw}'' & \Phi_{z\eta}'' &\Phi_{zz}'' & \Phi_{z\zeta}'' \\
\Phi_{\zeta w}'' & \Phi_{\zeta \eta}'' & \Phi_{\zeta z}'' & \Phi_{\zeta \zeta}''
\end{pmatrix}=
\begin{pmatrix}
O & I & O & O \\
I & O & -I & O \\
O & -I & O & I \\
O & O & I & O
\end{pmatrix}.
\end{align*}
By standard properties of the determinant, we have $|\det(\partial^2\Phi)|=1$. We begin by computing the signum of $\Phi$. Let $e$ be an eigenvalue of $\partial^2\Phi$, that is, $\det(\partial^2\Phi-eI)=0$. If $e=0$, then $|\det(\partial^2\Phi-eI)|=1\not=0$, which is a contradiction, and so, $e\not=0$. With this in mind, using the properties of block matrices (cf. \cite{Powell2011DetOfBlockMatrices}, \cite{Silvester2000Determinants}, etc.), we have
\begin{align*}
\det(\partial^2\Phi-eI)&=e^4\left(\left(e-\frac{1}{e} \right)^2-1 \right)^2=\left((e^2-1)^2-e^2 \right)^2=(e^2-e-1)^2 (e^2+e-1)^2 \\
&=\left(e-\frac{1+\sqrt{5}}{2} \right)^2 \left(e-\frac{1-\sqrt{5}}{2} \right)^2 \left( e-\frac{-1+\sqrt{5}}{2} \right)^2 \left(e-\frac{-1-\sqrt{5}}{2} \right)^2.
\end{align*}
This gives us $\mathrm{sgn}(\partial^2 \Phi)=0$.
By construction and homogeneity, we have the size estimates for radial and generic derivatives
\begin{align*}
\left|\partial_{w, z}^\alpha (\eta\cdot \nabla_\eta)^{l_1} (\zeta\cdot \nabla_\zeta)^{l_2} \partial_{\eta, \zeta}^\beta \Big(b_{t, j, +}(w, z, \lambda\eta) q_j(y, z, \lambda\zeta)\Big)\right|\leq C_{\alpha, k, l_1, l_2, \beta} 2^{j|\beta|},
\end{align*}
which in turn implies
\begin{align*}
|\partial_{w, \eta, z, \zeta}^\alpha v_{j, +}(t, w, \eta, z, \zeta)|\leq C_\alpha 2^{j|\alpha|}.
\end{align*}
Here, we used the homogeneity of $b_{t, j, +}=\kappa_t^* q_j$, and the fact that the size estimates of $b_{t, j, +}=\kappa_t^* q_j$ are comparable to those of $q_j$ by \cite[Lemma 11.11]{Zworski2012Semiclassical} with small $t$.
By the method of stationary phase (cf. Theorem 7.7.5 and (3.4.6) in \cite{Hormander2003LinearVol1}), we have that
\begin{align*}
V_{j, +}(t, x, y, \xi)&=\iiiint e^{i\lambda\Phi(w, \eta, z, \zeta)} v_{j, +}(t, w, \eta, z, \zeta) \:dw\:d\eta\:dz\:d\zeta \\
&=e^{i\lambda \Phi(y, \xi, y, \xi)} \left(\frac{\lambda}{2\pi} \right)^{-\frac{8}{2}} |\det \partial^2 \Phi|^{-\frac{1}{2}} e^{\frac{\pi i}{4} \mathrm{sgn}(\partial^2 \Phi)} \sum_{l<N} \lambda^{-l} L_l v_{j, +}(t, y, \xi, y, \xi)+ R_N(t, y) a(t, x, \lambda \xi),
\end{align*}
for $N=1, 2, 3, \cdots$, where, at the critical point $(y, \xi, y, \xi)$,
\begin{align*}
|R_N|\leq C_N \lambda^{-N} \sup_{|\alpha|\leq 2N} |\partial^\alpha v_{j, +}| \leq \Tilde{C}_N \lambda^{-N}(2^j)^{2N} \lesssim \Tilde{C}_N \lambda^{-\frac{N}{3}}.
\end{align*}
Here, the $L_l$ are differential operators of order at most $2l$ acting on $v_{j, +}$ at the critical point $(y, \xi, y, \xi)$, and $2^{-j} \gtrsim \lambda^{-\frac{1}{3}}$. It follows that
\begin{align*}
V_{j, +}(t, x, y, \xi)=(2\pi)^4 \lambda^{-4} \left(\sum_{k=0}^{N-1} \lambda^{-k} L_k v_{j, +} (t, y, \xi, y, \xi) \right)+R_N (t, y) a(t, x, \lambda \xi),
\end{align*}
which in turn implies that
\begin{align*}
(e^{-itP}\circ B_{t, j, +}\circ Q_{j, +}^*)(x, y)&=\lambda^2 \int e^{i\lambda(\varphi(t, x, \xi)-y\cdot \xi)} \Tilde{a}_j (t, x, y, \xi)\:d\xi \\
&\hspace{50pt}+\frac{\lambda^6}{(2\pi)^4}R_N (t, y) \int e^{i\lambda (\varphi(t, x, \xi)-y\cdot \xi)} a(t, x, \lambda \xi)\:d\xi,
\end{align*}
where
\begin{align*}
\Tilde{a}_j (t, x, y, \xi)=\sum_{l=0}^{N-1} \lambda^{-l} L_l v_{j, +}(x, y; t, y, \xi, y, \xi).
\end{align*}
This completes the proof of Lemma \ref{Lemma Semiclassical version of SP q+}.
\end{proof}
\begin{remark}
Note that the proof of this lemma also works for $j=J$, since we only used the fact that $2^{-j}\gtrsim \lambda^{-\frac{1}{3}}$ for $j$, which is also satisfied for $j=J$. We will use this later to prove Proposition \ref{Prop: QJ estimates}.
\end{remark}
We first want to show that we can ignore the contribution of the second term in the right hand side of \eqref{Result of SP for q+}. If we replace $(Q_j \circ e^{-itP}\circ Q_j^*)$ by the second term modulo smoothing errors, by \eqref{Kj+ kernel}, the contribution of the second term in $K_{j, +}$ is
\begin{align*}
\frac{\lambda^6}{(2\pi)^4} \iint e^{i\lambda[t+\varphi(t, x, \xi)-y\cdot \xi]} \widehat{\chi^2}(t) R_N (t, y) a(t, x, \lambda \xi)\:d\xi\:dt.
\end{align*}
We can ignore this contribution.
\begin{lemma}\label{Lemma 2nd contribution small}
We have
\begin{align*}
\left|\frac{\lambda^6}{(2\pi)^4} \iint e^{i\lambda[t+\varphi(t, x, \xi)-y\cdot \xi]} \widehat{\chi^2}(t) R_N (t, y) a(t, x, \lambda \xi)\:d\xi\:dt \right|\leq C_N \lambda^{6-\frac{N}{3}}=O(1),\quad \text{when } N\geq 18.
\end{align*}
\end{lemma}
\begin{proof}
Recall that we may assume that $a(t, x, \xi)$ is compactly supported in $\xi$. The function $\widehat{\chi^2}(t)$ is also compactly supported in $t$. It then follows that
\begin{align*}
\left|\frac{\lambda^6}{(2\pi)^4} \iint e^{i\lambda[t+\varphi(t, x, \xi)-y\cdot \xi]} \widehat{\chi^2}(t) R_N (t, y) a(t, x, \lambda \xi)\:d\xi\:dt \right|\lesssim \lambda^6 \sup_{t, y} |R_N| \lesssim \lambda^{6-\frac{N}{3}}=O(1),
\end{align*}
when $N\geq 18$.
\end{proof}
By Lemma \ref{Lemma 2nd contribution small}, the contribution of the second term of the right hand side of \eqref{Result of SP for q+} is $O(1)$ by the generalized Young's inequality, which is better than what we need to show.
We thus focus on the first term in the right hand side of \eqref{Result of SP for q+}, that is, modulo $O(1)$ errors,
\begin{align*}
K_{j, +} (x, y)=\lambda^2 \iint e^{i\lambda(t+\varphi(t, x, \xi)-y\cdot \xi)} \widehat{\chi^2 }(t) \Tilde{a}_j (t, x, y, \xi)\:d\xi\:dt.
\end{align*}
Recall that
\begin{align*}
\Tilde{a}_j (t, x, y, \xi)=\sum_{l=0}^{N-1} \lambda^{-l} L_l \Big(v_{j, +}(x, y; t, y, \xi, y, \xi)\Big),
\end{align*}
and, by the discussion in \S \ref{SS: Notation for PDOs}, we write
\begin{align*}
v_{j, +}(t, y, \xi, y, \xi)=a(t, x, \lambda\xi) b_{t, j, +}(y, y, \lambda \xi) q_{j, +} (y, y, \lambda \xi) =a(t, x, \lambda\xi) q_{j, +} (\kappa_t (y, \lambda \xi)) q_{j, +} (y, \lambda \xi).
\end{align*}
We will consider the contribution of the first term of \eqref{Result of SP for q+} in Fermi coordinates. To do so, we consider the behavior of $\xi(t)$ in the Hamiltonian flow of $|\xi|_{g(x)}$. By \eqref{varphi construction}, for Hamiltonian $p(x, \xi)=|\xi|_{g(x)}$, we have Hamilton's equation
\begin{align*}
\dot{x}=d_\xi p,\quad \dot{\xi}=-d_x p,
\end{align*}
that is,
\begin{align}\label{Hamilton's eqn's}
\begin{split}
& \dot{x}_1 (t)=\frac{g^{11}(x) \xi_1}{|\xi|_{g(x)}},\quad \dot{\xi}_1 (t)=-\frac{\partial_{x_1} g^{11}(x)\xi_1^2}{|\xi|_{g(x)}}, \\
& \dot{x}_2 (t)=\frac{\xi_2}{|\xi|_{g(x)}},\quad \dot{\xi}_2 (t)=-\frac{\partial_{x_2}g^{11}(x)\xi_1^2}{|\xi|_{g(x)}}.
\end{split}
\end{align}
Here, we used the fact that $|\xi|_g^2=g^{11}(x)\xi_1^2+\xi_2^2$. Recall that we focus only on small $t$ by \eqref{epsilon0 and chi}. We show that $|\partial_{\xi_2} \varphi(t, x, \xi)|=|t|$, if $\xi_1 (s)=0$ for some small $s$.
\begin{lemma}\label{Lemma xi2 with vanishing xi1}
If $\xi_1 (s)=0$ for some small $s$, then we have that $|\partial_{\xi_2} \varphi(s, x, \xi)|=|s|$.
\end{lemma}
\begin{proof}
Suppose $\xi_1 (s_0)=0$ for some small $s_0$. Since we focus on small $t$, we have a unique solution of \eqref{Hamilton's eqn's} by the uniqueness of the solutions to the ODEs. Since $\xi_1 \equiv 0$ satisfies both the equations \eqref{Hamilton's eqn's} and $\xi_1 (s_0)=0$, by the uniqueness of the solution, we should have $\xi_1 \equiv 0$. This implies that
\begin{align*}
|\xi|_g =\sqrt{g^{11}(x)\xi_1^2+\xi_2^2}=|\xi_2|.
\end{align*}
If $(z(s), \zeta(s))$ is the curve with $z(t)=x$, $\zeta(0)=\xi$, $z_2 (t)=0$, and $\zeta(0)=(0, \xi_2)$, then
\begin{align*}
\dot{z}_2 (s)=\frac{\xi_2 (s)}{|\xi (s)|_{g(z(s))} }=\pm 1.
\end{align*}
By the mean value theorem, we have $0=z_2 (t)=z_2 (0)\pm t$, and thus, $z_2 (0)=\mp t$. We then have that
\begin{align*}
|\partial_{\xi_2} \varphi(t, x, \xi)|=|z_2 (0)|=|t|,
\end{align*}
as required.
\end{proof}
We next consider the case of $\xi_1 (s)\not=0$ for any small $s$.
\begin{lemma}\label{Lemma xi2 dot positive}
For $|s|\ll 1$, suppose $\xi(s)\in \mathrm{supp}(\Tilde{a}_j (s, x, y, \cdot))$. Let $\gamma$ be as above. If $\xi_1 (s)\not=0$ for any small $s$, then, for $x, y\in \gamma$, in Fermi coordinates, we have either $\dot{\xi}_2 (s)>0$ or $\dot{\xi}_2 (s)<0$.
\end{lemma}
\begin{proof}
We know from the curvature assumption of $\gamma$ that
\begin{align*}
|\nabla_{\partial_1} \partial_1 |_g \not=0,\quad \text{where }\partial_1=\frac{\partial}{\partial x_1}\text{ and } \partial_2=\frac{\partial}{\partial x_2},
\end{align*}
where $\nabla$ denotes the Levi-Civita connection. Note that
\begin{align*}
0=\frac{\partial}{\partial x_1} \langle \partial_1, \partial_2 \rangle_g=\langle \nabla_{\partial_1} \partial_1, \partial_2 \rangle_g+\langle \partial_1, \nabla_{\partial_1} \partial_2 \rangle_g.
\end{align*}
But since $[\partial_1, \partial_2]=0$ and the Levi-Civita connection is symmetric, we have that
\begin{align*}
\langle \partial_1, \nabla_{\partial_1} \partial_2 \rangle_g =\langle \partial_1, \nabla_{\partial_2} \partial_1 \rangle_g +\langle \partial_1, [\partial_1, \partial_2] \rangle_g =\frac{1}{2} \frac{\partial}{\partial x_2} |\partial_1|_g^2=\frac{1}{2}\frac{\partial}{\partial x_2} g_{11} (x_1, 0).
\end{align*}
Combining these two, we have that
\begin{align}\label{g11 x2 derivative}
\frac{\partial}{\partial x_2} g_{11} (x_1, 0)=-2\langle \nabla_{\partial_1} \partial_1, \partial_2 \rangle_g.
\end{align}
Since $|\partial_1|_g=1$ along $x_2=0$ by the arc-length parametrization, we have that
\begin{align*}
\langle \nabla_{\partial_1} \partial_1, \partial_1 \rangle_g=\frac{1}{2}\frac{\partial}{\partial x_1} |\partial_1|_g^2=0,
\end{align*}
and thus,
\begin{align*}
\nabla_{\partial_1} \partial_1=\langle \nabla_{\partial_1} \partial_1, \partial_1 \rangle_g \partial_1+\langle \nabla_{\partial_1} \partial_1, \partial_2 \rangle_g \partial_2=c\partial_2,
\end{align*}
for some $c\not=0$ due to the assumption $|\nabla_{\partial_1} \partial_1|_g\not=0$. By \eqref{g11 x2 derivative} with this, we have that
\begin{align*}
-\frac{\partial}{\partial x_2} g^{11}(x)\not=0, \quad \text{on } x_2=0,
\end{align*}
since $g^{11}=g_{11}^{-1}$ (cf. \cite[Proposition 5.26]{Lee2018secondEd}), and this also holds on a neighborhood of $x_2=0$. Since we are assuming $\xi_1\not=0$, by the above Hamilton's equation, we have that
\begin{align*}
\dot{\xi}_2 (s)=-\partial_2 g^{11}(x)\xi_1^2\not=0,\quad \text{along } x_2=0.
\end{align*}
This completes the proof.
\end{proof}
We next consider the $\xi_2$ derivative of $\varphi$.
\begin{lemma}\label{Lemma xi2 deriv of varphi}
Suppose $\xi \in \mathrm{supp}(q_{j, +}(x, y, \lambda(\cdot)))$, $\xi_1 \not=0$, and $d_x \varphi (t, x, \xi)\in \mathrm{supp}(q_{j, +}(x, y, \lambda(\cdot)))$ for some $x\in \gamma$, i.e., $x_2=0$ in Fermi coordinates. Then there exists a uniform constant $C>0$ such that $|\partial_{\xi_2} \varphi(t, x, \xi)|\geq C2^{-j} |t|$.
\end{lemma}
\begin{proof}
We use Fermi coordinates to prove this lemma. Suppose $(z(s), \zeta(s))$ is the curve such that $z(t)=x$ and $\zeta(0)=\xi$. Without loss of generality, by homogeneity we assume $|\xi|_g=1$ since $|\xi|_g\approx 1$ for $\xi\in \mathrm{supp}(q_{j, +}(x, y, \lambda(\cdot)))$. It follows from \eqref{varphi construction} that $d_x \varphi(t, x, \xi)=\zeta(t)$ and $d_\xi \varphi(t, x, \xi)=z(0)$, and thus,
\begin{align*}
\chi_1 (2^j \zeta_2 (0))\not= 0 \quad \text{and} \quad \chi_1 (2^j \zeta_2 (t))\not=0, \quad \text{i.e.,} \quad \zeta_2 (0)\approx2^{-j} \quad \text{and} \quad \zeta_2 (t)\approx 2^{-j}.
\end{align*}
Thus, if $0\leq s\leq t$, we have $\zeta_2 (s)\approx 2^{-j}$ since the map $s\mapsto \zeta_2 (s)$ is monotonic in $s$, due to the fact that either $\dot{\zeta}_2 (s)>0$ or $\dot{\zeta}_2 (s)<0$ by Lemma \ref{Lemma xi2 dot positive}. Similarly, the map $s\mapsto \zeta_2 (s)$ is monotonic in $s$ when $t\leq s\leq 0$. In any cases, we have $|\zeta_2 (s)|\geq C 2^{-j}$ when $s$ is between $0$ and $t$.
Now recall that $\dot{z}_2 (s)=\zeta_2 (s)$ since we are in Fermi coordinates. Thus, since $z(t)=x\in \gamma$, we have $z_2 (t)=0$, and so, the mean value theorem gives
\begin{align*}
0=z_2 (0)+t\dot{z}_2 (\Tilde{c}),
\end{align*}
for some $\Tilde{c}$ between $0$ and $t$. This gives
\begin{align*}
|\partial_{\xi_2} \varphi (t, x, \xi)|=|z_2 (0)|=|t\dot{z}_2 (\Tilde{c})|=|t\zeta_2 (\Tilde{c})| \geq C|t|2^{-j},
\end{align*}
for some uniform constant $C>0$.
\end{proof}
We now return to the kernels $K_{j, +} (x, y)$. By Lemma \ref{Lemma Semiclassical version of SP q+}, we write
\begin{align*}
K_{j, +} (x, y)=\lambda^2 \iint e^{i\lambda(t+\varphi(t, x, \xi)-y\cdot \xi)} \Tilde{a}_j (t, x, y, \xi) \widehat{\chi^2}(t)\:d\xi\:dt,
\end{align*}
where $\Tilde{a}_j (t, x, y, \xi)=0$ unless
\begin{align*}
\chi_1 \left(2^j \frac{\partial_{x_2} \varphi(t, x, \xi)}{|d_x \varphi(t, x, \xi)|_g} \right)\not=0, \quad \chi_1 \left(2^j \frac{\xi_2}{|\xi|_g} \right)\not=0,\quad \text{and} \quad |\xi|_g \in [\frac{c_1}{2}, 2c_1^{-1}],
\end{align*}
for some small constant $c_1 >0$. Moreover, we have
\begin{align}\label{Size estimates for a tilde}
|\partial_t^k \partial_{\xi_1}^l \partial_{\xi_2}^m \Tilde{a}_j |\leq C_{k, l, m} 2^{jm}.
\end{align}
Here, we used \eqref{Size estimates for a} and size estimates of $q_j$ and $\kappa_t^* q_j$, since $|\xi|_g \approx 1$ by the support properties of $\Upsilon$. Also, note that $y_2=0$ in Fermi coordinates if $y=(y_1, y_2)\in \gamma$. If we set
\begin{align*}
L_\xi=\frac{1-i\lambda(\partial_{\xi_1} \varphi(t, x, \xi)-y_1)\partial_{\xi_1}-i\lambda 2^{-2j} \partial_{\xi_2} \varphi(t, x, \xi) \partial_{\xi_2} }{1+\lambda^2 |\partial_{\xi_1} \varphi(t, x, \xi)-y_1|^2+\lambda^2 2^{-2j} |\partial_{\xi_2} \varphi(t, x, \xi)|^2 }
\end{align*}
then we have
\begin{align*}
L_\xi (e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)})=e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)}.
\end{align*}
Integration by parts gives us that
\begin{align*}
\left|\int e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)} \Tilde{a}_j (t, x, y, \xi)\:d\xi \right|=\left|\int e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)} (L_\xi^T)^N (\Tilde{a}_j (t, x, y, \xi))\:d\xi\right|,
\end{align*}
where $L_\xi^T$ is the transpose of $L_\xi$. For simplicity, we set
\begin{align*}
& w_1=\lambda(\partial_{\xi_1} \varphi-y_1),\quad w_2=\lambda 2^{-j} \partial_{\xi_2} \varphi, \\
& L=L_{\xi}=\frac{1-iw_1 \partial_{\xi_1}-iw_2 2^{-j} \partial_{\xi_2} }{1+w_1^2+w_2^2}.
\end{align*}
Here, $w_1$ and $w_2$ are functions of $\lambda, t, x, y, \xi$ and we suppress the arguments for convenience if necessary. We then write (up to signs)
\begin{align*}
L^T \Tilde{a}_j=A_0+A_1+A_2+A_3+A_4,
\end{align*}
where
\begin{align*}
& A_0=\frac{1}{1+w_1^2+w_2^2} \Tilde{a}_j,\quad A_1=\frac{i w_1}{1+w_1^2+w_2^2} \partial_{\xi_1} \Tilde{a}_j, \quad A_2=\frac{i w_2}{1+w_1^2+w_2^2} 2^{-j} \partial_{\xi_2} \Tilde{a}_j, \\
& A_3=\Tilde{a}_j \partial_{\xi_1} \left( \frac{i w_1}{1+w_1^2+w_2^2} \right),\quad A_4=\Tilde{a}_j 2^{-j} \partial_{\xi_2} \left(\frac{i w_2}{1+w_1^2+w_2^2} \right).
\end{align*}
By \eqref{Size estimates for a tilde}, we have
\begin{align}\label{A1A2 approx}
|A_0|,\; |A_1|,\; |A_2|\leq \frac{1}{(1+w_1^2+w_2^2)^{\frac{1}{2}}}.
\end{align}
We note that $A_3$ is
\begin{align*}
\Tilde{a}_j\partial_{\xi_1} \left(\frac{i w_1}{1+w_1^2+w_2^2} \right)=\Tilde{a}_j\partial_{w_1} \left(\frac{i w_1}{1+w_1^2+w_2^2} \right)\partial_{\xi_1} w_1+\Tilde{a}_j\partial_{w_2} \left(\frac{iw_1}{1+w_1^2+w_2^2} \right)\partial_{\xi_1} w_2,
\end{align*}
where $\partial_{\xi_1} w_1=\lambda \partial_{\xi_1}^2 \varphi$ and $\partial_{\xi_1} w_2=\lambda 2^{-j} \partial_{\xi_1 \xi_2}^2 \varphi$. Similarly, $A_4$ is
\begin{align*}
\Tilde{a}_j 2^{-j} \partial_{w_1}\left(\frac{iw_2}{1+w_1^2+w_2^2} \right) \partial_{\xi_2} w_1+\Tilde{a}_j 2^{-j}\partial_{w_2} \left(\frac{iw_2}{1+w_1^2+w_2^2} \right) \partial_{\xi_2} w_2,
\end{align*}
where $\partial_{\xi_2} w_1=\lambda \partial_{\xi_1 \xi_2}^2 \varphi$ and $\partial_{\xi_2} w_2=\lambda \partial_{\xi_2}^2 \varphi$. Both $A_3$ and $A_4$ contain terms of the form $\partial^\alpha \varphi$ for $|\alpha|\geq 2$, and we want to approximate these first. Recall that we are assuming $|t|\lesssim 1$, by the support properties of $\chi$.
\begin{lemma}\label{Lemma varphi derivative}
If $|\alpha|\geq 2$ for $\alpha=(\alpha_1, \alpha_2)$, then
\[
|\partial_\xi^\alpha \varphi|\lesssim \begin{cases}
|\xi_2|^2 |t|, & \text{if } \alpha_2=0, \\
|\xi_2| |t|, & \text{if } \alpha_2=1, \\
|t|, & \text{for any } |\alpha| \geq 2.
\end{cases}
\]
\end{lemma}
\begin{proof}
It follows from \eqref{varphi construction} that
\begin{align*}
\varphi(t, x, \xi)=x\cdot \xi-\int_0^t p(x, \nabla_x \varphi(s, x, \xi))\:ds,
\end{align*}
where $p(x, \xi)=|\xi|_{g(x)}$. For any $|\alpha|\geq 2$, we obtain
\begin{align}\label{varphi deriv for any alpha}
|\partial_{\xi}^\alpha \varphi|=\left|-\int_0^t \partial_\xi^\alpha (p(x, \nabla_x \varphi(s, x, \xi)))\:ds \right| \leq |t| \sup_\xi \big[\partial^\alpha (p(x, \nabla_x \varphi(s, x, \xi)))\big] \leq C_\alpha |t|.
\end{align}
We now focus on $\alpha_2=0$ or $\alpha_2=1$.
On the other hand, if $\Phi (\xi)$ is homogeneous of degree $-k$, then, by Euler's homogeneous theorem, we have
\begin{align}\label{Homogeneous result}
\xi_1 \partial_{\xi_1} \Phi +\xi_2 \partial_{\xi_2} \Phi=-k \Phi.
\end{align}
Since $|\xi| \approx 1$, we have either $|\xi_1|\approx 1$ or $|\xi_2| \approx 1$. The case of $|\xi_1|\leq |\xi_2|$ is simpler. Indeed, if $|\xi_1|\leq |\xi_2|$, then $|\xi_2|\approx 1$, and so, by \eqref{varphi deriv for any alpha}, we have $|\partial_\xi^\alpha \varphi|\leq C_\alpha |\xi_2|^l |t|$ for any nonnegative integer $l$.
Thus, we may assume that $|\xi_2|\leq |\xi_1|$, and so, $|\xi_1| \approx 1$. Taking $\Phi=\partial_{\xi_2} \varphi$ with $k=0$ in \eqref{Homogeneous result}, it follows from \eqref{varphi deriv for any alpha} that
\begin{align}\label{xi1 xi2 deriv of varphi}
|\partial_{\xi_1 \xi_2}^2 \varphi|=\left|\frac{\xi_2}{\xi_1} \partial_{\xi_2}^2 \varphi \right|\lesssim |\xi_2| |t|.
\end{align}
Using this, if we take $\Phi=\partial_{\xi_1} \varphi$ with $k=0$ in \eqref{Homogeneous result}, then we have that
\begin{align*}
|\partial_{\xi_1}^2 \varphi|=\left|\frac{\xi_2}{\xi_1} \partial_{\xi_2 \xi_1}^2 \varphi \right|\lesssim |\xi_2| (|\xi_2| |t|)=|\xi_2|^2 |t|.
\end{align*}
We can also compute $\partial_{\xi_1 \xi_1 \xi_2}^3 \varphi$ taking $\Phi=\partial_{\xi_1 \xi_2}^2 \varphi$ with $k=-1$
\begin{align*}
\partial_{\xi_1 \xi_1 \xi_2}^3 \varphi=-\frac{1}{\xi_1} \partial_{\xi_1 \xi_2}^2 \varphi-\frac{\xi_2}{\xi_1} \partial_{\xi_1 \xi_2 \xi_2}^3 \varphi.
\end{align*}
By \eqref{varphi deriv for any alpha} and \eqref{xi1 xi2 deriv of varphi}, we have $|\partial_{\xi_1 \xi_1 \xi_2}^3 \varphi|\lesssim |\xi_2| |t|$. Similarly, we can find the estimate for $\partial_{\xi_1 \xi_1 \xi_1}^3 \varphi$. The higher order derivatives of $\varphi$ are bounded by induction and repeated use of \eqref{xi1 xi2 deriv of varphi}.
\end{proof}
By Lemma \ref{Lemma xi2 deriv of varphi}, we have $|\partial_{\xi_2}\varphi|\gtrsim |\xi_2| |t|$. By this and Lemma \ref{Lemma varphi derivative}, we have that
\begin{align*}
& |\partial_{\xi_1} w_1|=|\lambda \partial_{\xi_1}^2 \varphi|\lesssim \lambda|\xi_2|^2 |t| \lesssim \lambda |\partial_{\xi_2} \varphi| |\xi_2| \lesssim \lambda |\partial_{\xi_2} \varphi| 2^{-j} \lesssim |w_2|, \\
& |2^{-j} \partial_{\xi_2} w_1|=|\lambda 2^{-j} \partial_{\xi_1 \xi_2}^2 \varphi|\lesssim \lambda 2^{-j} |\xi_2| |t| \lesssim \lambda 2^{-j} |\partial_{\xi_2} \varphi| \lesssim |w_2|, \\
& |\partial_{\xi_1} w_2|=|\lambda 2^{-j} \partial_{\xi_1 \xi_2}^2 \varphi|\lesssim \lambda 2^{-j} |\xi_2| |t| \lesssim \lambda 2^{-j} |\partial_{\xi_2} \varphi|= |w_2|, \\
& |2^{-j} \partial_{\xi_2} w_2|=|\lambda (2^{-j})^2 \partial_{\xi_2}^2 \varphi|\lesssim \lambda 2^{-j} |\xi_2| |t| \lesssim \lambda 2^{-j}|\partial_{\xi_2} \varphi|= |w_2|.
\end{align*}
We also have that
\begin{align*}
\partial_{w_l} \left(\frac{w_k}{1+w_1^2+w_2^2} \right) \lesssim \frac{1}{1+w_1^2+w_2^2},\quad l, k\in \{1, 2\}.
\end{align*}
Combining these together, we have that
\begin{align*}
|A_3|,\; |A_4| \lesssim \frac{|w_2|}{1+w_1^2+w_2^2}\leq \frac{(1+w_1^2+w_2^2)^{\frac{1}{2}}}{1+w_1^2+w_2^2}=\frac{1}{(1+w_1^2+w_2^2)^{\frac{1}{2}}}.
\end{align*}
By this and \eqref{A1A2 approx}, we have
\begin{align}\label{L transpose estimate}
|L^T \Tilde{a}_j|\lesssim \frac{1}{(1+w_1^2+w_2^2)^{\frac{1}{2}}}.
\end{align}
Inductively, we can obtain
\begin{align*}
|(L^T)^N \Tilde{a}_j|\lesssim (1+w_1^2+w_2^2)^{-\frac{N}{2}} \lesssim (1+|w_1|+|w_2|)^{-N}.
\end{align*}
Hence, integration by parts gives, for $x, y\in \gamma$,
\begin{align*}
\left|\int e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)} \Tilde{a}_j (t, x, y, \xi)\:d\xi \right| &=\left|\int (L_\xi)^{N} (e^{i\lambda (t+\varphi(t, x, \xi)-y_1 \xi_1)}) \Tilde{a}_j (t, x, y, \xi)\:d\xi\right| \\
&=\left|\int e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)} (L_\xi^T)^{N} (\Tilde{a}_j (t, x, y, \xi))\:d\xi \right| \\
&\lesssim \int (1+\lambda|\partial_{\xi_1} \varphi(t, x, \xi)-y_1|+\lambda 2^{-j} |\partial_{\xi_2} \varphi(t, x, \xi)|)^{-N}\:d\xi,
\end{align*}
and thus, we have that
\begin{align*}
|K_{j, +} (x_1, 0, y_1, 0)|\leq C_N \lambda^2 \iint_{\mathrm{supp}(q_j) }|\widehat{\chi^2}(t)| \left( 1+\lambda|\partial_{\xi_1 }\varphi(t, x, \xi)-y_1|+\lambda 2^{-j}|\partial_{\xi_2} \varphi(t, x, \xi)| \right)^{-N}\:d\xi\:dt.
\end{align*}
In Fermi coordinates, we can write $\gamma=\{(x_1, 0): |x_1|\leq \epsilon\}$ for some small $\epsilon>0$, and so, we may write $x=(x_1, 0)$ and $y=(y_1, 0)$. To show \eqref{Qj+ estimates}, we now want to show that
\begin{align*}
\int |K_{j, +} (x_1, 0, y_1, 0)|\:dx_1 \lesssim 2^j,\quad \text{and} \quad \int |K_j (x_1, 0, y_1, 0)|\:dy_1 \lesssim 2^j.
\end{align*}
To see these, first note that, by Lemma \ref{Lemma xi2 with vanishing xi1} and Lemma \ref{Lemma xi2 deriv of varphi}, we have $|\partial_{\xi_2} \varphi(t, x, \xi)|\gtrsim 2^{-j}|t|$ in both cases $\xi_1\not=0$ and $\xi_1=0$, and so, we have that
\begin{align*}
|K_{j, +} (x_1, 0, y_1, 0)|\leq C_N \lambda^2 \iint_{\xi_2 \approx 2^{-j},|\xi|\approx 1} |\widehat{\chi^2}(t)| (1+\lambda |\partial_{\xi_1} \varphi (t, (x_1, 0), \xi)-y_1|+\lambda 2^{-2j}|t|)^{-N}\:dt\:d\xi,
\end{align*}
and thus, the second inequality follows from
\begin{align*}
& \int |K_{j, +} (x_1, 0, y_1, 0)|\:dy_1 \\
& \leq C_N \lambda^2 \int_{\xi_2 \approx 2^{-j}, |\xi| \approx 1 } \bigg(\iint |\widehat{\chi^2}(t)| (1+\lambda |\partial_{\xi_1} \varphi(t, (x_1, 0), \xi)-y_1|+\lambda 2^{-2j}|t| )^{-N}\:dt\:dy_1 \bigg) d\xi \\
& \lesssim \lambda^2 (\lambda 2^{-2j})^{-1} \lambda^{-1} \mathrm{Vol}(\{\xi_2 \approx 2^{-j},\; |\xi| \approx 1\}) \\
& \lesssim 2^{2j} 2^{-j}=2^j.
\end{align*}
Here, we gained $\lambda^{-1}$ from $y_1$ integration, $\lambda 2^{-2j}$ from $t$ integration, and $\mathrm{Vol}(\{|\xi_2|\approx 2^{-j}, |\xi|\approx 1\})$ from $\xi_2$ integration.
The proof that
\begin{align*}
\int |K_{j, +} (x_1, 0, y_1, 0)|\:dx_1 \lesssim 2^j
\end{align*}
is similar, but it uses that $|\partial_{x_1 \xi_1}^2 \varphi (t, x, \xi)|\geq c>0$ for some small $c>0$, for $|\xi|_g \approx 1$ and $\xi_2 \approx 2^{-j}$, i.e., $|\xi_1|\approx 1$.
To see $|\partial_{x_1 \xi_1}^2 \varphi(t, x, \xi)|\gtrsim 1$, we recall that $\varphi$ satisfies $\varphi(0, x, \xi)=\langle x, \xi \rangle$ (cf. \cite[Lemma 10.5 (ii)]{Zworski2012Semiclassical}). By this, we have $|\partial_{x_1 \xi_1}^2 \varphi(t, x, \xi)|=1$ at $t=0$, and so, $|\partial_{x_1 \xi_1}^2 \varphi (t, x, \xi)|\gtrsim 1$ for small $t$ by continuity, but we can focus only on small $t$ by taking $\epsilon_0>0$ to be sufficiently small in \eqref{epsilon0 and chi}, and hence $|\partial_{x_1 \xi_1}^2 \varphi (t, x, \xi)|\gtrsim 1$ in the support of $K_{j, +}$.
This completes the proof of Proposition \ref{Prop: q plus minus estimates}.
\subsection{Proof of Proposition \ref{Prop: QJ estimates}}
In this subsection, by the $TT^*$ argument, we want to show that
\begin{align}\label{QJ TT* claim}
\|Q_J \circ \chi^2 (\lambda-P) \circ Q_J^* f\|_{L^2 (\gamma)} \lesssim \lambda^{\frac{1}{3}} \|f\|_{L^2 (\gamma)},\quad J=\lfloor \log_2 \lambda^{\frac{1}{3}} \rfloor.
\end{align}
We obtain $K_J$, $\Tilde{a}_J$, $v_J$, etc., by replacing $j$ by $J$ in the settings of the previous section. We also ignore the contribution of the remainder after using Egorov's theorem.
Using the proof of Lemma \ref{Lemma Semiclassical version of SP q+}, we have the following lemma.
\begin{lemma}\label{Lemma Semiclassical SP QJ}
We have
\begin{align}\label{Semiclassical SP QJ result}
\begin{split}
(e^{-itP} \circ B_{t, J} \circ Q_J^*)(x, y)&=\lambda^2 \int e^{i\lambda (\varphi(t, x, \xi)-y\cdot \xi)} \Tilde{a}_J (t, x, y, \xi)\:d\xi \\
&\hspace{50pt}+\frac{\lambda^6}{(2\pi)^4} \int e^{i\lambda (\varphi(t, x, \xi)-y\cdot \xi)} R_N (t, y) a(t, x, \lambda \xi)\:d\xi,
\end{split}
\end{align}
where
\begin{align*}
\Tilde{a}_J (t, x, y, \xi)=\sum_{l=0}^{N-1} \lambda^{-l} L_l v_J (x, y; t, y, \xi, y, \xi), \quad |\partial_t^\alpha R_N|\leq C_{N, \alpha} \lambda^{-\frac{N}{3}},
\end{align*}
and the $L_l$ are the differential operators with respect to $(w, \eta, z, \zeta)$ of order at most $2l$ acting on $v_J$ at the point $(w, \eta, z, \zeta)=(y, \xi, y, \xi)$.
\end{lemma}
By Lemma \ref{Lemma 2nd contribution small} and the generalized Young's inequality again, the contribution of the second term of the right hand side of \eqref{Semiclassical SP QJ result} is $O(1)$ with $N$ large, and so, we focus on the first term in \eqref{Semiclassical SP QJ result}.
Using the proof of Lemma \ref{Lemma xi2 dot positive}, we can also show that $\dot{\xi}_2$ is nonvanishing.
\begin{lemma}\label{Lemma xi2 dot positive qJ}
For $|s|\ll 1$, suppose $\xi(s)\in \mathrm{supp}(\Tilde{a}_J (s, x, y, \cdot))$. Let $\gamma$ be as above. If $\xi_1 (s)\not=0$ for any small $s$, then, for $x, y\in \gamma$, in Fermi coordinates, we have either $\dot{\xi}_2 (s)>0$ or $\dot{\xi}_2 (s)<0$.
\end{lemma}
With this in mind, we figure out the support properties of $\Tilde{a}_J$.
\begin{lemma}\label{Lemma xi2 derivative of varphi qJ}
Suppose $\xi\in \mathrm{supp}(q_J (x, y, \lambda(\cdot)))$, and $d_x \varphi(t, x, \xi)\in \mathrm{supp}(q_J (x, y, \lambda (\cdot)))$ for some $x\in \gamma$, i.e., $x_2=0$ in Fermi coordinates. If $|t|\gg \lambda^{-\frac{1}{3}}$, then $\Tilde{a}_J (t, x, y, \xi)=0$, and thus, $\Tilde{a}_J$ is supported where $|t|\lesssim \lambda^{-\frac{1}{3}}$.
\end{lemma}
\begin{proof}
Suppose $(z(s), \xi(s))$ is the curve such that $z(t)=x, \xi(0)=\xi$. It follows that
\begin{align*}
d_x \varphi (t, x, \xi)=\xi(t)=(\xi_1 (t), \xi_2 (t)),\quad d_\xi \varphi(t, x, \xi)=z(0)=(z_1 (0), z_2 (0)).
\end{align*}
By construction, we have $\Tilde{a}_J(t, x, y, \xi)=0$ in Fermi coordinates unless
\begin{align*}
\Tilde{\chi}_J\left( \lambda^{\frac{1}{3}} \frac{|\xi_2 (t)|}{|\xi (t)|_g} \right)\not=0,\quad \text{and} \quad \Tilde{\chi}_J \left(\lambda^{\frac{1}{3}} \frac{|\xi_2|}{|\xi|_g} \right)\not=0.
\end{align*}
By the support properties of $\Upsilon$, we have $|\xi|_g \approx 1$ and $|\xi(t)|_g\approx 1$, and so, we have $\Tilde{a}_J (t, x, y, \xi)=0$ unless
\begin{align*}
|\xi_2 (t)|\lesssim \lambda^{-\frac{1}{3}},\quad |\xi_2 (0)|\lesssim \lambda^{-\frac{1}{3}}.
\end{align*}
We want to show that we cannot have $|\xi_2 (t)|\lesssim \lambda^{-\frac{1}{3}}$ when $|t|\gg \lambda^{-\frac{1}{3}}$. We note that $\xi_1 (s)\not=0$ for any small $s$. Indeed, if $|\xi_2|\lesssim \lambda^{-\frac{1}{3}}$ and $|\xi|\approx 1$, then $|\xi_1|\gtrsim 1$.
By the mean value theorem, we have
\begin{align}\label{MVT result of xi2}
\xi_2 (t)=\xi_2 (0)+\dot{\xi}_2 (c_t) t,
\end{align}
where $c_t$ is between $0$ and $t$. Since $\widehat{\chi^2}$ is compactly supported in $[-2\epsilon_0, 2\epsilon_0]$ for small $\epsilon_0>0$ by \eqref{epsilon0 and chi}, by the proof of Lemma \ref{Lemma xi2 dot positive qJ}, there exists a $\Tilde{c}>0$ such that $|\dot{\xi}_2 (s)|\geq \Tilde{c}$. If $|\xi_2 (0)|\gg \lambda^{-\frac{1}{3}}$, then we have $\Tilde{a}_J$ vanishes automatically. If $|\xi_2 (0)|\lesssim \lambda^{-\frac{1}{3}}$ and $|t|\gg \lambda^{-\frac{1}{3}}$, then, by \eqref{MVT result of xi2} and $|\dot{\xi}_2 (s)|\geq \Tilde{c}$, we have
\begin{align*}
|\xi_2 (t)|\geq |\dot{\xi}_2 (c_t)||t|-|\xi_2 (0)|\gg \lambda^{-\frac{1}{3}}.
\end{align*}
Hence, the amplitude $\Tilde{a}_J$ is supported where $|t|\lesssim \lambda^{-\frac{1}{3}}$.
\end{proof}
In Fermi coordinates, by Lemma \ref{Lemma Semiclassical SP QJ}, modulo $O(1)$ errors, we write
\begin{align*}
K_J (x, y)&=\lambda^2 \iint e^{i\lambda[t+\varphi(t, x, \xi)-y\cdot \xi]} \widehat{\chi^2}(t) \Tilde{a}_J(t, x, y, \xi)\:d\xi\:dt,
\end{align*}
where, by Lemma \ref{Lemma xi2 derivative of varphi qJ}, $\Tilde{a}_J(t, x, y, \xi)$ is supported where $|t|\lesssim \lambda^{-\frac{1}{3}}$. Moreover, we have
\begin{align}\label{Size estimates of b lambda J}
|\partial_t^k \partial_{\xi_1}^l \partial_{\xi_2}^m \Tilde{a}_J|\leq C_{k, l, m} (\lambda^{\frac{1}{3}})^m.
\end{align}
As before, here we used \eqref{Size estimates for a} and size estimates of $q_j$ and $\kappa_t^* q_j$, since $|\xi|_g \approx 1$ by the support properties of $\Upsilon$. Note that $y_2=0$ in Fermi coordinates for $y=(y_1, y_2)\in \gamma$. As before, if we set
\begin{align*}
L_\xi=\frac{1-i w_1 \partial_{\xi_1} }{1+|w_1|^2 },\quad w_1=\lambda (\partial_{\xi_1} \varphi(t, x, \xi)-y_1),
\end{align*}
then we have
\begin{align*}
L_\xi (e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)})=e^{i\lambda(t+\varphi(t, x, \xi)-y_1 \xi_1)}.
\end{align*}
By Lemma \ref{Lemma varphi derivative}, we have
\begin{align*}
|\partial_{\xi_1}^k \varphi|\lesssim |\xi_2|^2 |t|,\quad \text{for } k\geq 2,
\end{align*}
which in turn implies that
\begin{align}\label{xi1 deriv of w1}
|\partial_{\xi_1}^k w_1|\lesssim \lambda |\xi_2|^2 |t|\lesssim \lambda (\lambda^{-\frac{1}{3}})^2 \lambda^{-\frac{1}{3}} \lesssim 1, \quad k\geq 1.
\end{align}
Integration by parts, as before, gives, for $x, y, \in \gamma$,
\begin{align*}
\left| \int e^{i\lambda [t+\varphi(t, x, \xi)-y_1 \xi_1]} \Tilde{a}_J (t, x, y, \xi)\:d\xi \right|=\left|\int e^{i\lambda [t+\varphi(t, x, \xi)-y_1 \xi_1]} (L_\xi^T)^N (\Tilde{a}_J(t, x, y, \xi)) \:d\xi \right|,
\end{align*}
where $L_\xi^T$ is the transpose of $L_\xi$. As above, we write (up to signs)
\begin{align*}
L_\xi^T \Tilde{a}_J=B_0+B_1+B_2,
\end{align*}
where
\begin{align*}
B_0=\frac{1}{1+w_1^2} \Tilde{a}_J,\quad B_1=\frac{i w_1}{1+w_1^2} \partial_{\xi_1} \Tilde{a}_J,\quad B_2=\Tilde{a}_J \partial_{\xi_1} \left(\frac{i w_1}{1+w_1^2} \right).
\end{align*}
By \eqref{Size estimates of b lambda J}, we have
\begin{align}\label{bound of B0 and B1}
|B_0|,\; |B_1|\lesssim \frac{1}{(1+w_1^2)^{\frac{1}{2}}}\lesssim \frac{1}{1+|w_1|}.
\end{align}
Since we have
\begin{align*}
B_2=\Tilde{a}_J \partial_{w_1}\left(\frac{i w_1}{1+w_1^2} \right)\partial_{\xi_1} w_1=\Tilde{a}_J \frac{i(1-w_1^2)}{(1+w_1^2)^2} \partial_{\xi_1} w_1,
\end{align*}
it follows from \eqref{xi1 deriv of w1} that
\begin{align*}
|B_2|\leq |\Tilde{a}_J|\frac{1+w_1^2}{(1+w_1^2)^2}|\partial_{\xi_1}w_1|\lesssim \frac{1}{1+w_1^2}\lesssim \frac{1}{(1+|w_1|)^2}.
\end{align*}
By this and \eqref{bound of B0 and B1}, we have
\begin{align*}
|B_0|,\; |B_1|,\; |B_2|\lesssim \frac{1}{1+|w_1|}.
\end{align*}
Hence, integration by parts gives, for $x, y, \in \gamma$,
\begin{align*}
|K_J (x, y)|\leq C_N \lambda^2 \iint_{|t|\lesssim \lambda^{-\frac{1}{3}}, |\xi_2|\lesssim \lambda^{-\frac{1}{3}}, |\xi|_g \approx 1 } |\widehat{\chi^2}(t)| (1+\lambda |\partial_{\xi_1}\varphi(t, x, \xi)-y_1|)^{-N}\:d\xi\:dt.
\end{align*}
In Fermi coordinates, we write $\gamma=\{(x_1, 0): |x_1|\leq \epsilon\}$ for $\epsilon>0$ small, and so, $x=(x_1, 0)$ and $y=(y_1, 0)$. We thus want to show that
\begin{align}\label{KJ Young's inequality}
\int |K_J (x_1, 0, y_1, 0)|\:dx_1 \lesssim \lambda^{\frac{1}{3}},\quad \int |K_J (x_1, 0, y_1, 0)|\:dy_1 \lesssim \lambda^{\frac{1}{3}}.
\end{align}
Indeed, this and Young's inequality imply \eqref{QJ TT* claim} immediately.
We first focus on $\int |K_J (x_1, 0, y_1, 0)|\:dy_1$. We take $\Tilde{C}>0$ sufficiently large, and bound
\begin{align*}
\int |K_J (x_1, 0, y_1, 0)|\:dy_1 &\leq C_N \lambda^2 \iiint_{|t|\lesssim \lambda^{-\frac{1}{3}}, |\xi_2|\lesssim\lambda^{-\frac{1}{3}}, |\xi|_g \approx 1 } |\widehat{\chi^2}(t)| (1+\lambda|\partial_{\xi_1} \varphi(t, x, \xi)-y_1|)^{-N}\:dy_1\:d\xi\:dt \\
&\lesssim \lambda^2 \lambda^{-1} \lambda^{-\frac{1}{3}} \mathrm{Vol}(\{|\xi_2|\lesssim \lambda^{-\frac{1}{3}}, |\xi|\approx 1\})\lesssim \lambda^{\frac{1}{3}}.
\end{align*}
Here, we gained $\lambda^{-1}$ from $y_1$ integration, $\lambda^{-\frac{1}{3}}$ from $t$ integration due to $|t|\lesssim \lambda^{-\frac{1}{3}}$, and $\mathrm{Vol}(\{|\xi_2|\lesssim \lambda^{-\frac{1}{3}}, |\xi|\approx 1\})$ from $\xi_2$ integration.
The proof of the second inequality in \eqref{KJ Young's inequality} is similar, but uses that $|\partial_{x_1 \xi_1}^2 \varphi (t, x, \xi)|\gtrsim 1$ for small $t$ as in the case $j\leq J-1$.
This completes the proof of Proposition \ref{Prop: QJ estimates}.
\subsection{Proof of Proposition \ref{Prop: I-Qj estimates}}\label{SS: I-Qj estimates}
As we promised before, we talk about Proposition \ref{Prop: I-Qj estimates} here. Let $\Tilde{Q}=I-\sum_{j\leq J} Q_j$. By the Fourier inversion formula, we write
\begin{align*}
\Tilde{Q} f(x)=\int \Tilde{Q}(x, y) f(y)\:dy,
\end{align*}
where
\begin{align*}
\Tilde{Q}(x, y)=\frac{1}{(2\pi)^2} \int e^{i(x-y)\cdot \xi} \Big(1-\sum_{j\leq J} q_j (x, y, \xi)\Big)\:d\xi.
\end{align*}
Setting
\begin{align*}
\Tilde{q}(x, y, \xi)=1-\sum_{j\leq J} q_j (x, y, \xi),
\end{align*}
we write
\begin{align*}
\Tilde{q}(x, y, \xi)&=1-\chi_0 (\rho(x, \gamma)) \Tilde{\chi}_0 (\rho (y, \gamma))\sum_{j\leq J} \Tilde{\chi}_j \left(2^j \frac{|\xi(N)|}{|\xi|_g} \right) \Upsilon(|\xi|_g/\lambda)\\
&=1-\chi_0 (\rho(x, \gamma)) \Tilde{\chi}_0 (\rho(y, \gamma)) \Upsilon(|\xi|_g/\lambda).
\end{align*}
Let $\Tilde{Q}$ be a pseudodifferential operator whose kernel is $\Tilde{Q}(x, y)$. Since $\chi_0, \Tilde{\chi}_0$, and $\Upsilon$ are compactly supported bump functions, we have
\begin{align*}
|\partial_{x, y, \xi}^\alpha \Tilde{q}(x, y, \lambda\xi)|\leq C_\alpha,
\end{align*}
and so, we can consider integration by parts below easily.
We write the kernel of $\Tilde{Q}\circ \chi(\lambda-P)$ as
\begin{align*}
(\Tilde{Q}\circ \chi(\lambda-P))(x, y)=\frac{\lambda^4}{(2\pi)^3 } \iiiint e^{i\lambda \Psi(t, z, \eta, \xi)} \widehat{\chi}(t) \Tilde{q}(x, z, \lambda \eta) a(t, z, \lambda \xi) \:dt\:dz\:d\eta\:d\xi,
\end{align*}
where
\begin{align*}
\Psi(t, z, \eta, \xi)=(x-z)\cdot \eta+ \varphi(t, z, \xi) -y\cdot \xi.
\end{align*}
We note that, on the support of $\Tilde{q}(x, z, \lambda \eta)$ in $\eta$,
\begin{align*}
|\nabla_{t, z}\Psi(t, z, \eta, \xi)|&=|(\Psi_t', \Psi_z')|=\sqrt{|1-|\nabla_z \varphi(t, z, \xi)|_{g(z)}|^2+|\nabla_z \varphi(t, z, \xi)-\eta|^2} \\
&\gtrsim |1-|\nabla_z \varphi(t, z, \xi)|_{g(z)}|+||\nabla_z \varphi(t, z, \xi)|_{g(z)}-|\eta||\geq |1-|\eta||\gtrsim 1+|\eta|.
\end{align*}
With this in mind, we first consider the integral
\begin{align}\label{Integral 1-Upsilon for 2.6}
\frac{\lambda^4}{(2\pi)^3 } \iiiint e^{i\lambda \Psi(t, z, \eta, \xi)} \widehat{\chi}(t) \Tilde{q}(x, z, \lambda \eta) a(t, z, \lambda \xi) (1-\Upsilon(|\xi|)) \:dt\:dz\:d\eta\:d\xi.
\end{align}
On the support of $1-\Upsilon(\xi)$ in $\xi$, we have that
\begin{align*}
|\nabla_{t, z} \Psi (t, z, \eta, \xi)|\gtrsim |1-|\nabla_z \varphi(t, z, \xi)|_{g(z)}|=|1-|\xi|_{g(\nabla_\xi \varphi(t, z, \xi))}|\approx |1-|\xi||\approx 1+|\xi|,
\end{align*}
when we choose $c_1>0$ small enough in \eqref{Construction of the compound symbol qj}. Integration by parts in $t$ and $z$ then gives us that the integral \eqref{Integral 1-Upsilon for 2.6} is dominated by
\begin{align*}
C\lambda^4 \lambda^{-N} \iiiint_{t\in \mathrm{supp}(\widehat{\chi}), |z|\lesssim 1} (1+|\eta|)^{-N'} (1+|\xi|)^{-N'}\:dt\:dz\:d\eta\:d\xi \lesssim \lambda^{4-N},
\end{align*}
when we take $N, N'$ large enough. Using the generalized Young's inequality, this satisfies the estimates \eqref{I-Qj estimates}, and thus, we focus on the integral
\begin{align*}
\frac{\lambda^4}{(2\pi)^3 } \iiiint e^{i\lambda \Psi(t, z, \eta, \xi)} \widehat{\chi}(t) \Tilde{q}(x, z, \lambda \eta) a(t, z, \lambda \xi) \Upsilon(|\xi|) \:dt\:dz\:d\eta\:d\xi.
\end{align*}
In this case, the amplitude of the integral is compactly supported in $\xi$, and so, we do not need to consider $|\Psi_t'|$ separately. Thus, integration by parts in $t$ and $z$, the integral is dominated by
\begin{align*}
C\lambda^4 \lambda^{-N}\iiiint_{t\in \mathrm{supp}(\widehat{\chi}), |z|\lesssim 1, |\xi|\approx 1 } (1+|\eta|)^{-N}\:dt\:dz\:d\eta\:d\xi \lesssim \lambda^{4-N},
\end{align*}
when we take $N$ large enough, which proves Proposition \ref{Prop: I-Qj estimates}.
This shows \eqref{I-Qj estimates} by using the generalized Young's inequality, and thus, completes the proof of Theorem \ref{Theorem Universal Estimates}.
\section{Proof of Theorem \ref{Theorem Log Improvement}}\label{S:Prop for large j}
In this section, assuming nonpositive sectional curvatures on $M$, we want to prove Theorem \ref{Theorem Log Improvement}. Let
\begin{align}\label{T Definition}
T=c_0 \log \lambda,
\end{align}
where $c_0>0$ is small but fixed, which will be specified later. Let $P=\sqrt{-\Delta_g}$ as before. As in Theorem \ref{Theorem Universal Estimates}, we would have Theorem \ref{Theorem Log Improvement}, if we could show that
\begin{align}\label{Thm2 chi reduction 1}
\| \chi(T(\lambda-P)) f\|_{L^p (\gamma)}\leq C_p \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}} } \|f\|_{L^2 (M)},\quad 2\leq p<4,
\end{align}
where $C_p \to \infty$ as $p\to 4$. Since $\chi(0)=1$ and $\chi\in \mathcal{S}(\mathbb{R})$, by the mean value theorem, we have, for some $\Tilde{c}$ between $0$ and $t$,
\begin{align*}
|(1-\chi(t)) \chi(Tt)|&=|(\chi(0)-\chi(t))\chi(Tt)| \\
&=|\chi'(\Tilde{c})t\chi(Tt)|\leq C_N T^{-1} (1+T|t|)^{-N}.
\end{align*}
As in \cite{Sogge2017ImprovedCritical}, \cite{XiZhang2017improved}, and \cite{BlairSogge2019logarithmic}, etc., by this and the universal estimates in Theorem \ref{Theorem Universal Estimates}, we have that
\begin{align*}
\| (I-\chi(\lambda-P)) \circ \chi(T(\lambda-P)) f\|_{L^2 (\gamma)}\lesssim \frac{\lambda^{\frac{1}{6}}}{T} \|f\|_{L^2 (M)}.
\end{align*}
Similarly, using \cite[Theorem 1]{BurqGerardTzvetkov2007restrictions} (see also \cite[Theorem 1.1]{Hu2009lp}) instead of Theorem \ref{Theorem Universal Estimates}, we have that
\begin{align*}
\| (I-\chi(\lambda-P)) \circ \chi(T(\lambda-P)) f\|_{L^4 (\gamma)}\lesssim \frac{\lambda^{\frac{1}{4}}}{T} \|f\|_{L^2 (M)}.
\end{align*}
By interpolation, we have that
\begin{align*}
\| (I-\chi(\lambda-P)) \circ \chi(T(\lambda-P)) f\|_{L^p (\gamma)}\lesssim \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T} \|f\|_{L^2 (M)},\quad 2\leq p\leq 4.
\end{align*}
We would therefore have \eqref{Thm2 chi reduction 1} if we could show
\begin{align*}
\| \chi(\lambda-P) \circ \chi(T(\lambda-P)) f\|_{L^p(\gamma)} \leq C_p \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}} } \|f\|_{L^2(M)},\quad 2\leq p<4.
\end{align*}
This follows from
\begin{align}\label{Qj chi chi T estimate}
\sum_{j\leq J} \| Q_j \circ \chi (\lambda-P)\circ \chi(T(\lambda-P)) f\|_{L^p (\gamma)}\leq C_p \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}} } \|f\|_{L^2 (M)}, \quad 2\leq p<4,
\end{align}
and, for $N=1, 2, 3, \cdots$,
\begin{align}\label{I-Qj chi chi T estimate}
\|(I-\sum_{j\leq J} Q_j) \circ \chi(\lambda-P)\circ \chi(T(\lambda-P)) f\|_{L^p (\gamma)}\lesssim \lambda^{-N} \|f\|_{L^2 (M)},\quad 2\leq p<4.
\end{align}
We first show \eqref{I-Qj chi chi T estimate}. Recall that
\begin{align}\label{chi T L2 to L2}
\|\chi(T(\lambda-P)) f\|_{L^2 (M)}\lesssim \|f\|_{L^2 (M)}.
\end{align}
By Proposition \ref{Prop: I-Qj estimates} and \eqref{chi T L2 to L2}, we have, for $2\leq p<4$ and $N=1, 2, 3, \cdots$,
\begin{align*}
\|(I-\sum_{j\leq J} Q_j)\circ \chi(\lambda-P)\circ \chi(T(\lambda-P)) f\|_{L^p (\gamma)} \leq C_N \lambda^{-N} \|\chi(T(\lambda-P)) f\|_{L^2 (M)}\lesssim \lambda^{-N} \|f\|_{L^2 (M)},
\end{align*}
which is better than \eqref{I-Qj chi chi T estimate}, and so, we are left to show \eqref{Qj chi chi T estimate}.
Before we proceed further, let us look at the $L^2 (M) \to L^4(\gamma)$ estimate of $Q_j \circ \chi(\lambda-P)$.
\begin{lemma}\label{Lemma L2 to L4}
For $j\leq J$, we have
\begin{align*}
\| Q_j \circ \chi(\lambda-P) f\|_{L^4 (\gamma)}\leq C \lambda^{\frac{1}{4}} \|f\|_{L^2 (M)}.
\end{align*}
\end{lemma}
It follows from \eqref{chi T L2 to L2} that
\begin{align}\label{Result of Lemma 4.1}
\|Q_j \circ \chi(\lambda-P)\circ \chi(T(\lambda-P)) f\|_{L^4(\gamma)}\lesssim \lambda^{\frac{1}{4}} \|\chi(T(\lambda-P)) f\|_{L^2 (M)}\lesssim \lambda^{\frac{1}{4}} \|f\|_{L^2 (M)}.
\end{align}
\begin{proof}
In Fermi coordinates as above, we write, for $\epsilon>0$ small,
\begin{align*}
\gamma=\{(r, 0): |r|\leq \epsilon \},\quad \gamma_c=\{(x_1, x_2): |x_1|\leq \epsilon,\; x_2=c \}.
\end{align*}
We first show that
\begin{align}\label{Restriction to gamma c estimate}
\| \mathcal{R}_\gamma \circ Q_j g\|_{L^4 (\gamma)}\lesssim \sup_{|c|\leq \epsilon} \|\mathcal{R}_{\gamma_c} g \|_{L^4 (\gamma_c)},
\end{align}
where $\mathcal{R}_\gamma g$ and $\mathcal{R}_{\gamma_c} g$ are the restrictions of $g$ onto $\gamma$ and $\gamma_c$, respectively.
We can write
\begin{align*}
(\mathcal{R}_\gamma \circ Q_j)(r, y)=\frac{1}{(2\pi)^2} \int e^{i[(r-y_1)\xi_1-y_2 \xi_2]} q_j (r, 0, \xi) \:d\xi.
\end{align*}
We may assume $|y_1|, |y_2|\leq \epsilon$ by a partition of unity if necessary. By \eqref{Symbol Q properties}, integration by parts then gives
\begin{align*}
|(\mathcal{R}_\gamma \circ Q_j)(r, y)|&\leq C_N \lambda^2 2^{-j} (1+\lambda|r-y_1|+\lambda 2^{-j}|y_2|)^{-2N} \\
&\leq C_N \lambda^2 2^{-j} (1+\lambda |r-y_1|)^{-N}(1+\lambda 2^{-j}|y_2|)^{-N},\quad N=1, 2, 3, \cdots.
\end{align*}
This implies that
\begin{align*}
\int |(\mathcal{R}_\gamma\circ Q_j)(r, y_1, y_2)|\:dr,\quad \int |(\mathcal{R}_\gamma\circ Q_j)(r, y_1, y_2)|\:dy_1 \lesssim C_N \lambda 2^{-j} (1+\lambda 2^{-j} |y_2|)^{-N}.
\end{align*}
By Young's inequality, we then have that
\begin{align*}
\| \mathcal{R}_\gamma \circ Q_j g (\cdot, y_2) \|_{L_r^4 ([-\epsilon, \epsilon])} \lesssim \lambda 2^{-j} (1+\lambda 2^{-j}|y_2|)^{-N} \|g(\cdot, y_2)\|_{L_{y_1}^4([-\epsilon, \epsilon])}.
\end{align*}
By this and Minkowski's inequality for integrals, we have that
\begin{align*}
\|\mathcal{R}_\gamma \circ Q_j g \|_{L^4 (\gamma)}&=\left(\int \left|\int\left[\int (\mathcal{R}_\gamma \circ Q_j)(r, y_1, y_2)g(y_1, y_2)\:dy_1\right]\:dy_2 \right|^4\:dr \right)^{\frac{1}{4}} \\
&\leq \int \| (\mathcal{R}_\gamma \circ Q_j) g(\cdot, y_2) \|_{L^4 ([-\epsilon, \epsilon])}\:dy_2 \\
&\lesssim \lambda 2^{-j}\int (1+\lambda 2^{-j}|y_2|)^{-N} \|g(\cdot, y_2)\|_{L^4 ([-\epsilon, \epsilon])}\:dy_2 \\
&\lesssim \sup_{|y_2|\leq \epsilon} \|g(\cdot, y_2)\|_{L^4 ([-\epsilon, \epsilon])}=\sup_{|c|\leq \epsilon} \|g(\cdot, c)\|_{L^4 ([-\epsilon, \epsilon])}=\sup_{|c|\leq \epsilon} \|\mathcal{R}_{\gamma_c} g\|_{L^4 (\gamma_c)},
\end{align*}
which proves \eqref{Restriction to gamma c estimate}.
By (the proof of) \cite[Theorem 1]{BurqGerardTzvetkov2007restrictions} and \cite[Theorem 1.1]{Hu2009lp}, we know that
\begin{align*}
\sup_{|c|\leq \epsilon} \|\mathcal{R}_{\gamma_c} \circ \chi(\lambda-P) f\|_{L^4 (\gamma_c)} \lesssim \lambda^{\frac{1}{4}} \|f\|_{L^2 (M)}.
\end{align*}
Combining this and \eqref{Restriction to gamma c estimate} with $g=\chi(\lambda-P)f$, we obtain that
\begin{align*}
\|\mathcal{R}_\gamma \circ Q_j \circ \chi(\lambda-P) f\|_{L^4(\gamma)} \lesssim \sup_{|c|\leq \epsilon} \|\mathcal{R}_{\gamma_c}\circ \chi(\lambda-P) f\|_{L^4 (\gamma_c)} \lesssim \lambda^{\frac{1}{4}}\|f\|_{L^2 (M)}.
\end{align*}
Here, the implicit constants are uniform, which are stable under $C^\infty$ perturbation of $\gamma$. This completes the proof.
\end{proof}
By Proposition \ref{Prop: q plus minus estimates} and Lemma \ref{Lemma L2 to L4}, we have that, for $j\leq J$,
\begin{align*}
& \|Q_j \circ \chi(\lambda-P) f\|_{L^2 (\gamma)} \leq C 2^{\frac{j}{2}} \|f\|_{L^2 (M)}, \\
& \|Q_j \circ \chi(\lambda-P) f\|_{L^4 (\gamma)} \leq C \lambda^{\frac{1}{4}} \|f\|_{L^2 (M)}.
\end{align*}
By interpolation, we have
\begin{align}\label{Qj chi L2 to Lp}
\| Q_j \circ \chi(\lambda-P) f\|_{L^p (\gamma)}\leq C 2^{\frac{j}{2}(\frac{4}{p}-1)} \lambda^{\frac{1}{4}(2-\frac{4}{p})} \|f\|_{L^2 (M)}, \quad 2\leq p<4.
\end{align}
Let $\epsilon>0$ be a fixed but small number, which will be specified later. By \eqref{Qj chi L2 to Lp} and \eqref{chi T L2 to L2}, if $2\leq p<4$, then
\begin{align*}
& \sum_{j\leq \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor} \|Q_j \circ \chi (\lambda-P) \circ \chi (T(\lambda-P))f \|_{L^p (\gamma)} \\
&\hspace{100pt}\leq C \sum_{j\leq \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor} 2^{\frac{j}{2}(\frac{4}{p}-1)} \lambda^{\frac{1}{4}(2-\frac{4}{p})} \|\chi(T(\lambda-P)) f\|_{L^2 (M)} \\
&\hspace{100pt}\leq \frac{2C}{1-2^{-\frac{1}{2}(\frac{4}{p}-1)}} \lambda^{\frac{1}{3}-\frac{1}{3p}-\frac{\epsilon}{2}(\frac{4}{p}-1)} \|f\|_{L^2 (M)} \\
&\hspace{100pt}\leq \frac{2C}{1-2^{-\frac{1}{2}(\frac{4}{p}-1)}} \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}}}\|f\|_{L^2 (M)},
\end{align*}
which satisfies \eqref{Qj chi chi T estimate}.
\begin{remark}
We note that we cannot relax the condition $C_p \to \infty$ as $p\to 4$ in our argument. Indeed, note that
\begin{align*}
\lim_{\lambda\to \infty} \lim_{\epsilon\to 0} \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}-\frac{\epsilon}{2}(\frac{4}{p}-1)}}{\lambda^{\frac{1}{3}-\frac{1}{3p}}/T^{\frac{1}{2}} } =\lim_{\lambda\to \infty} T^{\frac{1}{2}}=\infty.
\end{align*}
Also, if we set
\begin{align*}
C_p=\frac{2C}{1-2^{-\frac{1}{2}(\frac{4}{p}-1)}},
\end{align*}
then our argument gives
\begin{align*}
\sum_{j\leq \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor} \|Q_j \circ \chi (\lambda-P) \circ \chi (T(\lambda-P))f \|_{L^p (\gamma)} \leq C_p \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}}} \|f\|_{L^2 (M)},
\end{align*}
but we have that $\displaystyle \lim_{p\to 4} C_p=\infty$.
\end{remark}
If we set
\begin{align}\label{mu T definition}
\chi_T (\zeta)=\chi(\zeta/T),\quad \mu_T (\zeta)=\chi_T (\zeta) \chi(\zeta),
\end{align}
we have $\chi(\lambda-P) \chi(T(\lambda-P))=\mu_T (T(\lambda-P))$. Also, since $\widehat{\chi}_T (\zeta)=T\widehat{\chi}(T\zeta)$ and $\widehat{\mu}_T (t)=(2\pi)^{-1} \widehat{\chi}_T * \widehat{\chi} (t)$, we have, by \eqref{epsilon0 and chi},
\begin{align*}
\mathrm{supp}(\widehat{\mu}_T) \subset \mathrm{supp}(\widehat{\chi}_T)+\mathrm{supp} (\widehat{\chi})\subset [-\frac{\epsilon_0}{T}, \frac{\epsilon_0}{T}]+[-\epsilon_0, \epsilon_0]\subset [-2\epsilon_0, 2\epsilon_0],
\end{align*}
and so,
\begin{align}\label{mu support}
\mathrm{supp}(\widehat{\mu_T^2})\subset \mathrm{supp}(\widehat{\mu_T})+\mathrm{supp} (\widehat{\mu_T})\subset [-4\epsilon_0, 4\epsilon_0],
\end{align}
since $T=c_0 \log \lambda \gg 1$. We have shown that
\begin{align*}
\sum_{j\leq \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor} \|Q_j \circ \mu_T (T(\lambda-P))f \|_{L^p (\gamma)}\leq \frac{2C}{1-2^{-\frac{1}{2}(\frac{4}{p}-1)}} \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}}}\|f\|_{L^2 (M)},
\end{align*}
For the rest of \eqref{Qj chi chi T estimate}, we want to show that
\begin{align*}
\|Q_j \circ \mu_T (T(\lambda-P)) f\|_{L^p (\gamma)}\lesssim \frac{\lambda^{\frac{1}{4}}}{T^{\frac{1}{2}}} e^{C'T} (2^{-j})^{\frac{1}{p}} \|f\|_{L^2 (M)},\quad 2\leq p<4,\quad \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor \leq j\leq J,
\end{align*}
or
\begin{align*}
\|Q_j \circ \mu_T (T(\lambda-P)) f\|_{L^p(\gamma)} \lesssim \frac{2^{j\left(\frac{2}{p}-\frac{1}{2}\right)} \lambda^{\frac{1}{2}-\frac{1}{p}} }{T^{\frac{1}{2}}} \|f\|_{L^2 (M)},\quad 2\leq p<4.
\end{align*}
Indeed, we have
\begin{align*}
\sum_{\lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor \leq j\leq J} \frac{\lambda^{\frac{1}{4}}}{T^{\frac{1}{2}}} e^{C'T} (2^{-j})^{\frac{1}{p}}\lesssim \frac{\lambda^{\frac{1}{4}-\frac{1}{3p}+\frac{\epsilon}{p}+C'c_0}}{T^{\frac{1}{2}}} \epsilon \log_2 \lambda \lesssim \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}}},\quad \text{when } \lambda\gg 1,
\end{align*}
and
\begin{align*}
\sum_{\lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor \leq j\leq J} \frac{2^{j\left(\frac{2}{p}-\frac{1}{2}\right)} \lambda^{\frac{1}{2}-\frac{1}{p}} }{T^{\frac{1}{2}}} \lesssim \frac{\lambda^{\frac{1}{3}-\frac{1}{3p}}}{T^{\frac{1}{2}}}.
\end{align*}
Here, we take $\epsilon>0$ to be sufficiently small and choose a small $c_0>0$ in \eqref{T Definition}.
By the $TT^*$ argument, we would have \eqref{Qj chi chi T estimate} if we could show either
\begin{align}\label{Log improvement TT* Claim}
\|Q_j \circ \mu_T^2 (T(\lambda-P))\circ Q_j^* f\|_{L^p (\gamma)} \lesssim \frac{\lambda^{\frac{1}{2}}}{T} e^{CT} (2^{-j})^{\frac{2}{p}} \|f\|_{L^{p'} (\gamma)},\quad 2\leq p<4, \quad \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor \leq j\leq J,
\end{align}
or
\begin{align}\label{Log improvement TT* Claim 2}
\|Q_j \circ \mu_T^2 (T(\lambda-P))\circ Q_j^* f\|_{L^p (\gamma)}\lesssim \frac{2^{j\left(\frac{4}{p}-1 \right)}\lambda^{1-\frac{2}{p}}}{T} \|f\|_{L^{p'}(\gamma) },\quad 2\leq p<4.
\end{align}
We want to lift this problem to the universal cover of $M$. Let $\Tilde{M}$ be the universal cover of $M$ with the pullback metric $\Tilde{g}$ under the covering map $p:\Tilde{M}\to M$. By the Cartan-Hadamard theorem, $\Tilde{M}$ is diffeomorphic to $\mathbb{R}^2$ with the diffeomorphism $T_{x_0} M\cong \mathbb{R}^2 \to \Tilde{M}$ for any $x_0 \in M$, so that the map $p=\mathrm{exp}_{x_0}: T_{x_0} M \to M$ is a smooth covering map. Without loss of generality, we write $p:\mathbb{R}^2\cong \Tilde{M} \to M$.
Let $D\subset \mathbb{R}^2$ be a fundamental domain of the universal covering $p$ so that every point in $\mathbb{R}^2$ is the translate of exactly one point in $D$. Without loss of generality, we may assume that $\gamma$ and other amplitudes like $q_j$ are supported in $D^\circ$, where $D^\circ$ is the interior of $D$, i.e., $\gamma\subset D^\circ$, and $\mathrm{supp}(q_j)\subset D^\circ$, etc. We write tildes over letters to express that those letters are defined in $\mathbb{R}^2\cong \Tilde{M}$. For example, for any $x\in M$, let $\Tilde{x}\in D$ be the unique point so that $p(\Tilde{x})=x$, $p(\Tilde{\gamma})=\gamma$, and the metric $\Tilde{g}$ on $\mathbb{R}^2\cong \Tilde{M}$ is the pullback metric of $g$, $\Tilde{\rho}(\Tilde{x}, \Tilde{y})$ is the Riemannian distance $d_{\Tilde{g}}(\Tilde{x}, \Tilde{y})$, and so on. Let $\Gamma$ be the group of deck transformations $\alpha$'s, which are diffeomorphisms satisfying $p\circ \alpha =p$. With this in mind, if we have a function $\Tilde{f}$ on $D$, we can extend this $\Tilde{f}$ to $\mathbb{R}^2 \cong \Tilde{M}$ by setting
\begin{align*}
\Tilde{f}(\Tilde{x})=\Tilde{f}(\alpha(\Tilde{x})) \quad\text{for } \Tilde{x}\in D.
\end{align*}
Here, since $p:\mathbb{R}^2 \to M$ is a local diffeomorphism, abusing notation we write
\begin{align*}
& \Tilde{Q}_j (\Tilde{x}, \Tilde{w})=\frac{\lambda^2}{(2\pi)^2} \int e^{i\lambda (\Tilde{x}-\Tilde{w})\cdot \eta} \Tilde{q}_j (\Tilde{x}, \Tilde{w}, \lambda \eta)\:d\eta,\\
& \Tilde{Q}_j^* (\Tilde{z}, \Tilde{y})=\overline{\Tilde{Q}_j (\Tilde{y}, \Tilde{z})}=\overline{\Tilde{Q}_j (\alpha (\Tilde{y}), \alpha (\Tilde{z}))}=\frac{\lambda^2}{(2\pi)^2} \int e^{-i\lambda (\alpha (\Tilde{y})-\alpha(\Tilde{z}))\cdot \zeta} \Tilde{q}_j (\alpha(\Tilde{y}), \alpha (\Tilde{z}), \lambda \zeta)\:d\zeta.
\end{align*}
Recall that we know from \cite{SoggeZelditch2014eigenfunction} that
\begin{align*}
(\cos tP)(x, y)=\sum_{\alpha \in \Gamma} (\cos t \sqrt{-\Delta_{\Tilde{g}}}) (\Tilde{x}, \alpha (\Tilde{y})),\quad \Tilde{x}, \Tilde{y}\in D.
\end{align*}
Also recall that, by a counting argument and finite propagation speed as in \cite{SoggeZelditch2014eigenfunction}, there are at most $O(e^{Ct})$ many nonzero terms in the sum.
Using Euler's formula, we have, modulo $O(\lambda^{-N})$ errors,
\begin{align*}
\chi^2 (T(\lambda-P))(x, y)&=\frac{1}{\pi T} \int e^{it\lambda} \widehat{\chi^2}(t/T) \cos(tP)(x, y)\:dt-\chi^2 (T(\lambda+P))(x, y) \\
&=\frac{1}{\pi T}\sum_{\alpha\in \Gamma} \int e^{it\lambda} \widehat{\chi^2}(t/T) \cos(t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{x}, \alpha(\Tilde{y}))\:dt,
\end{align*}
since $\chi^2(T(\lambda+P))(x, y)=O(\lambda^{-N})$.
We want to show that the estimate for $\alpha=\mathrm{Id}$ satisfies \eqref{Log improvement TT* Claim 2}, and the estimate for $\alpha\not=\mathrm{Id}$ satisfies \eqref{Log improvement TT* Claim}.
\begin{lemma}\label{Lemma alpha=Id}
If $\alpha=\mathrm{Id}$ and $2\leq p\leq 4$, then
\begin{align*}
\left\|\frac{1}{\pi T} \iint e^{it\lambda} \widehat{\mu_T^2}(t/T) (\Tilde{Q}_j \circ\cos(t\sqrt{-\Delta_{\Tilde{g}}})(\cdot, \alpha(\cdot)) \circ \Tilde{Q}_j^*)(\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s))f(s)\:dt\:ds \right\|_{L^p (\gamma)}\lesssim \frac{2^{j\left(\frac{4}{p}-1 \right)}\lambda^{1-\frac{2}{p}}}{T} \|f\|_{L^{p'} (\gamma)},
\end{align*}
which satisfies the estimate \eqref{Log improvement TT* Claim 2}.
\end{lemma}
\begin{proof}
We choose $\beta\in C_0^\infty (\mathbb{R})$ satisfying
\begin{align}\label{beta support in Lemma alpha Id}
\beta(t)=1 \text{ for } |t|\leq c, \text{ and } \beta(t)=0 \text{ for } |t|\geq 2c,
\end{align}
for a small $c>0$. Since $\beta(t)\widehat{\mu_T^2}(t/T)$ is compactly supported in $t$ and
\begin{align*}
|\partial_t^k [\beta(t) \widehat{\mu_T^2}(t/T)]| \leq C_k,
\end{align*}
the term $\beta(t)\widehat{\mu_T^2}(t/T)$ plays the same role as $\widehat{\chi^2}(t)$ in \S \ref{S:Proof of universal estimates}. Thus, by the proof of Theorem \ref{Theorem Universal Estimates}, we have, for $\alpha=\mathrm{Id}$,
\begin{align*}
\left\|\frac{1}{\pi T}\iint e^{it\lambda} \beta(t)\widehat{\mu_T^2}(t/T) (\Tilde{Q}_j \circ\cos(t\sqrt{-\Delta_{\Tilde{g}}})(\cdot, \cdot)\circ \Tilde{Q}_j^*)(\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s))f(s)\:dt\:ds \right\|_{L^2 (\gamma)} \lesssim \frac{2^j}{T}\|f\|_{L^2 (\gamma)}.
\end{align*}
The difference between this and Theorem \ref{Theorem Universal Estimates} is that here we use the Hadamard parametrix about the cosine propagator $\cos(t\sqrt{-\Delta_{\Tilde{g}}})$, and we used the Lax parametrix about $e^{-itP}$.
Similarly, instead of using Theorem \ref{Theorem Universal Estimates}, by using the proof of \eqref{Result of Lemma 4.1} with a $TT^*$ argument, we can obtain, for $\alpha=\mathrm{Id}$,
\begin{align*}
\left\|\frac{1}{\pi T}\iint e^{it\lambda} \beta(t)\widehat{\mu_T^2}(t/T) (\Tilde{Q}_j \circ \cos(t\sqrt{-\Delta_{\Tilde{g}}}) (\cdot, \cdot) \circ \Tilde{Q}_j^*)(\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s))f(s)\:dt\:ds \right\|_{L^4 (\gamma)} \lesssim \frac{\lambda^{\frac{1}{2}}}{T}\|f\|_{L^{\frac{4}{3}} (\gamma)}.
\end{align*}
The desired estimate then follows from interpolation.
It then suffices to show that, for $\alpha=\mathrm{Id}$ and $N=1, 2, 3, \cdots$,
\begin{align*}
\left\|\frac{1}{\pi T}\iint e^{it\lambda} (1-\beta(t))\widehat{\mu_T^2}(t/T) (\Tilde{Q}_j \circ\cos(t\sqrt{-\Delta_{\Tilde{g}}})(\cdot, \cdot)\circ \Tilde{Q}_j^*)(\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s)) f(s)\:dt\:ds \right\|_{L^p (\gamma)}\lesssim \lambda^{-N}\|f\|_{L^{p'} (\gamma)}.
\end{align*}
We show this as in \cite[Lemma 3.1]{ChenSogge2014few}. We first consider the kernel of the integral operator inside the $L^2$ norm without $Q_j$ and $Q_j^*$ compositions
\begin{align*}
\frac{1}{\pi T} \int e^{it\lambda} (1-\beta(t))\widehat{\mu_T^2}(t/T) \cos(t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{x}, \Tilde{y})\:dt,\quad \Tilde{x}, \Tilde{y}\in D.
\end{align*}
We recall properties of the cosine propagator (cf. \cite[(5.14)]{BlairSogge2015OnKakeyaNikodym}, etc.)
\begin{align*}
\mathrm{sing}\: \mathrm{supp}(\cos t\sqrt{-\Delta_{\Tilde{g}}})(\cdot, \cdot)\subset \{(\Tilde{x}, \Tilde{z})\in \mathbb{R}^2\times \mathbb{R}^2: \Tilde{\rho} (\Tilde{x}, \Tilde{z})=|t|\},
\end{align*}
that is, $\cos(t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{x}, \Tilde{z})$ is smooth if $\Tilde{\rho}(\Tilde{x}, \Tilde{z})\not=|t|$. Since $1-\beta(t)=0$ for $|t|\leq c$ where $c>0$ is as in \eqref{beta support in Lemma alpha Id}, we may assume that $|t|\geq c>0$. Here, we choose a sufficiently small $c>0$, compared to the injectivitiy radius of $M$. For $\alpha=\mathrm{Id}$, by a partition of unity if necessary, we may assume that $\Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))\leq c/2$ for $\Tilde{x}, \Tilde{y}\in D$, and thus,
\begin{align*}
|t|\geq c>\Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y})),\; \alpha=\mathrm{Id}, \; \Tilde{x}, \Tilde{y}\in D,\quad \text{that is, } \Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))\not=|t|.
\end{align*}
This implies that $\cos(t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{x}, \Tilde{z})$ is smooth for $\Tilde{x}, \Tilde{z}\in \mathbb{R}^2$, and thus, integration by parts in $t$ implies that
\begin{align*}
\frac{1}{\pi T}\int e^{it\lambda} (1-\beta(t))\widehat{\mu_T^2}(t/T) \cos(t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{x}, \alpha(\Tilde{y}))\:dt=O(\lambda^{-N}),\quad \alpha=\mathrm{Id},\; \Tilde{x}, \Tilde{y}\in D.
\end{align*}
For the contribution after compositions of $Q_j$ and $Q_j^*$, by \eqref{Symbol Q properties}, we note that
\begin{align*}
& \left|\frac{1}{\pi T} \int e^{it\lambda} (1-\beta(t))\widehat{\mu_T^2}(t/T) (\Tilde{Q}_j \circ \cos(t\sqrt{-\Delta_{\Tilde{g}}})(\cdot, \alpha(\cdot))\circ \Tilde{Q}_j^*)(\Tilde{\gamma}(r), \Tilde{\gamma}(s))\:dt\right| \\
& \lesssim \frac{1}{T}\left| \iint \Tilde{Q}_j (\Tilde{\gamma}(r), z)\left(\int e^{it\lambda} (1-\beta(t))\widehat{\mu_T^2}(t/T) \cos (t\sqrt{-\Delta_{\Tilde{g}}}) (z, w)\:dt \right) \Tilde{Q}_j^* (w, \Tilde{\gamma}(s))\:dz\:dw \right| \\
& \lesssim \sup_{z, w}\left(\frac{1}{T}\int e^{it\lambda} (1-\beta(t))\widehat{\mu_T^2}(t/T) \cos (t\sqrt{-\Delta_{\Tilde{g}}}) (z, w)\:dt \right) \iint |\Tilde{Q}_j (\Tilde{\gamma}(r), z)| |Q_j^* (w, \Tilde{\gamma}(s))|\:dz\:dw \\
& \lesssim \lambda^{-N} \sup_{\Tilde{\gamma}(r)}\int|\Tilde{Q}_j (\Tilde{\gamma}(r), z)|\:dz\; \sup_{\Tilde{\gamma}(s)}\int |\Tilde{Q}_j^* (w, \gamma(s))|\:dw \lesssim \lambda^{-N}.
\end{align*}
This completes the proof.
\end{proof}
By Lemma \ref{Lemma alpha=Id}, we can ignore the contribution of $\alpha=\mathrm{Id}$. Using Euler's formula, we know
\begin{align*}
\mu_T^2 (T(\lambda-P))(x, y)=\frac{1}{\pi T} \int e^{it\lambda} \widehat{\mu_T^2}(t/T) (\cos tP)(x, y)\:dt -\mu_T^2 (T(\lambda+P))(x, y),
\end{align*}
and also know that $\mu_T^2 (T(\lambda+P))(x, y)=O(\lambda^{-N})$. As in \S \ref{S:Proof of universal estimates}, if we set
\begin{align*}
K_j (x, y)=\frac{1}{2\pi T} \int e^{it\lambda} \widehat{\mu_T^2}(t/T) (Q_j \circ e^{-itP}\circ Q_j^*) (x, y)\:dt,
\end{align*}
then, by Euler's formula, modulo $O(\lambda^{-N})$ errors, we have
\begin{align*}
K_j (x, y)=\frac{1}{\pi T} \int e^{it\lambda} \widehat{\mu_T^2}(t/T) (Q_j \circ \cos tP\circ Q_j^*)(x, y)\:dt,
\end{align*}
which is the kernel of $Q_j \circ \mu_T^2 (T(\lambda-P))\circ Q_j^*$ modulo $O(\lambda^{-N})$ errors. From now on, we focus on $\lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon}\rfloor \leq j< J$. Similar arguments will also work for $j=J$.
By a version of Egorov's theorem in \cite{BouzouinaRobert2002Duke} and the subsequent observation in \cite[Theorem 4.2.4]{Anantharaman2008Entropy}, we have
\begin{align}\label{Long time Egorov thm}
e^{-it\sqrt{-\Delta_{\Tilde{g}}}}\circ Q_j^*=\Tilde{B}_{t, j} \circ e^{-it\sqrt{-\Delta_{\Tilde{g}}} },
\end{align}
where $\Tilde{B}_{t, j}$ has a symbol
\begin{align*}
\Tilde{b}_{t, j}=\kappa_t^* \Tilde{q}_j^* +b',
\end{align*}
with the Hamiltonian flow $\kappa_t$, and $| b'|=O(\lambda^{-1+\frac{2}{3}+2\Lambda c_0})=O(\lambda^{-\frac{1}{3}+2\Lambda c_0})$ for some fixed $\Lambda>0$. Since $M$ is compact, we have $|b'|=O(\lambda^{-\frac{1}{3}+2\Lambda c_0})=O(\lambda^{-\frac{1}{3}+\epsilon'})$ for some small $\epsilon'>0$ when taking $c_0>0$ to be sufficiently small in \eqref{T Definition} for a uniform constant $\Lambda$. As before, we will ignore the contribution of the remainder $b'$, and write $b_{t, j}=\kappa_t^* q_j$. Using Euler's formula again, we can replace $e^{-it \sqrt{-\Delta_{\Tilde{g}}}}$ by $\cos t \sqrt{-\Delta_{\Tilde{g}}}$ modulo $O(\lambda^{-N})$ errors in \eqref{Long time Egorov thm}.
With this in mind, modulo $O(\lambda^{-N})$ errors, we have that
\begin{align}\label{Kj mod errors}
K_j (x, y)=\frac{1}{\pi T} \int e^{it\lambda}\widehat{\mu_T^2}(t/T) (Q_j\circ B_{t, j}\circ \cos tP)(x, y)\:dt,
\end{align}
where $\Tilde{B}_{t, j}$ is the lift of $B_{t, j}$. Since $p$ is a local isometry, we may assume $|\det p|=1$ in Riemannian measure. Using $w=p(\Tilde{w})$ and $z=p(\Tilde{z})$, we write
\begin{align*}
(Q_j \circ B_{t, j} \circ \cos t\sqrt{-\Delta_g})(x, y)&=\sum_\alpha \iint_{D^2} \Tilde{Q}_j (\Tilde{x}, \Tilde{w}) \Tilde{B}_{t, j}(\Tilde{w}, \Tilde{z}) (\cos t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{z}, \alpha(\Tilde{y})) |\det p|^2 \:d\Tilde{w}\:d\Tilde{z} \\
&=\sum_\alpha \iint_{D^2} \Tilde{Q}_j (\Tilde{x}, \Tilde{w}) \Tilde{B}_{t, j} (\Tilde{w}, \Tilde{z}) (\cos t\sqrt{-\Delta_{\Tilde{g}}}) (\Tilde{z}, \alpha (\Tilde{y}))\:d\Tilde{w}\:d\Tilde{z}.
\end{align*}
By the Hadamard parametrix (cf. \cite{Berard1977onthewaveequation}, \cite{Sogge2014hangzhou}, \cite{SoggeZelditch2014eigenfunction}, etc.), we write, for $\Tilde{x}\in D$ and $\Tilde{w}\in \alpha(D)$,
\begin{align*}
(\cos t\sqrt{-\Delta_{\Tilde{g}}})(\Tilde{x}, \Tilde{w})=\Tilde{K}_N (t, \Tilde{x}; \Tilde{w})+ \Tilde{R}_N (t, \Tilde{x}; \Tilde{w}),
\end{align*}
where
\[
\Tilde{K}_N (t, \Tilde{x};\Tilde{w})=\begin{cases}
\sum_{\nu=0}^N u_\nu (\Tilde{x}, \Tilde{w}) \partial_t E_\nu (t, \Tilde{\rho} (\Tilde{x}, \Tilde{w})), & t\geq 0, \\
-\sum_{\nu=0}^N u_\nu (\Tilde{x}, \Tilde{w}) \partial_t E_\nu (-t, \Tilde{\rho} (\Tilde{x}, \Tilde{w})), & t< 0.
\end{cases}
\]
We explain the $u_\nu$ and $E_\nu$ below.
For simplicity, from now on, we focus on $t\geq 0$. Similar arguments work for $t\leq 0$. Here, the $C^\infty$ functions $u_\nu$ are as in \cite[\S 2 in B and (10)]{Berard1977onthewaveequation} and \cite[p.35]{Sogge2014hangzhou}:
\begin{align*}
\begin{split}
& u_0 (\Tilde{x}, \Tilde{w})=\Theta^{-\frac{1}{2}} (\Tilde{x}, \Tilde{w}), \\
& u_\nu (\Tilde{x}, \Tilde{w})=\Theta^{-\frac{1}{2}} (\Tilde{x}, \Tilde{w})\int_0^1 s^{\nu-1} \Theta^{1/2} (\Tilde{x}, \Tilde{x}_s) \Delta_{\Tilde{g}, \Tilde{w}} u_{\nu-1}(\Tilde{x}, \Tilde{x}_s)\:dx,\quad \nu=1, 2, 3, \cdots, \\
&\Theta (\Tilde{x}, \Tilde{w})=|\det D_{\exp_{\Tilde{x}}^{-1} (\Tilde{w})} \exp_{\Tilde{x}}|,
\end{split}
\end{align*}
where $\Tilde{x}_s$ is the minimizing geodesic from $\Tilde{x}$ to $\alpha(\Tilde{w})$ parametrized by arc length and
\begin{align*}
\Theta=(\det (\Tilde{g}_{jk}))^{\frac{1}{2}}.
\end{align*}
As in \cite[Chapter 1]{Sogge2014hangzhou}, the distributions $E_\nu$ are, in $\mathbb{R}^n$,
\begin{align*}
E_\nu (t, x)=\lim_{\epsilon\to 0+} \nu! (2\pi)^{-n-1} \iint_{\mathbb{R}^{1+n}} e^{ix\cdot \xi+it\tau} (|\xi|^2-(\tau-i\epsilon)^2)^{-\nu-1} \:d\xi\:d\tau,\quad \nu=0, 1, 2, \cdots,
\end{align*}
and
\begin{align*}
& E_0 (t, x)=H(t)\times (2\pi)^{-n} \int_{\mathbb{R}^n} e^{ix\cdot \xi} \frac{\sin t|\xi|}{|\xi|}\:d\xi,\\
& \Box E_\nu=\nu E_{\nu-1},\quad -2\frac{\partial E_\nu}{\partial x}=xE_{\nu-1},\quad 2\frac{\partial E_\nu}{\partial t}=tE_{\nu-1},\quad \nu=1, 2, 3, \cdots,
\end{align*}
where $H(t)$ is the Heaviside function
\[
H(t)=\begin{cases}
1, & t\geq 0,\\
0, & t<0.
\end{cases}
\]
We have $n=2$ in our work. Here, $E_0 (t, x)$ is interpreted as
\begin{align*}
E_0 (t, x)=\begin{cases}
(2\pi)^{-2} \int_{\mathbb{R}^2} e^{ix\cdot \xi} \frac{\sin t |\xi|}{|\xi|}\:d\xi, & t\geq 0,\\
0, & t<0,
\end{cases}
\end{align*}
and
\begin{align*}
\langle E_0 (t, \cdot), f \rangle=(2\pi)^{-2}H(t)\int_{\mathbb{R}^2} \frac{\sin t|\xi|}{|\xi|}\hat{f}(\xi)\:d\xi,
\end{align*}
that is, the Fourier transform of $E_0 (t, \cdot)$ is $\frac{\sin t|\xi|}{|\xi|}$. Also, since the $E_\nu$ are radial in $x$, we may abuse notation, for example, $E_\nu (t, x)=E_\nu (t, |x|)$.
We will ignore the contribution of $\Tilde{R}_N$. We first recall a result in \cite{Berard1977onthewaveequation}, \cite[Theorem 3.1.5]{Sogge2014hangzhou}, and \cite[Proposition 3.1]{Keeler2019TwoPointWeylLaw}, adapted to our settings.
\begin{lemma}[\cite{Berard1977onthewaveequation}, \cite{Sogge2014hangzhou}, \cite{Keeler2019TwoPointWeylLaw}]\label{Lemma: remainder small}
For $|t|\leq T$, we have $\Tilde{R}_N \in C^{N-5}([-T, T]\times D\times D)$ and
\begin{align*}
|\partial_{t, x, y}^\beta \Tilde{R}_N (t, \Tilde{x}; \Tilde{w})|\lesssim e^{C_\beta T},\quad \text{if } |\beta|\ll N.
\end{align*}
\end{lemma}
Let $\Tilde{R}_N$ be the operator whose kernel is
\begin{align*}
\frac{1}{\pi T}\int e^{it\lambda} \widehat{\mu_T^2} (t/T) \Tilde{R}_N (t, \Tilde{x};\Tilde{w})\:dt.
\end{align*}
By Lemma \ref{Lemma: remainder small}, integration by parts in $t$ gives
\begin{align*}
\frac{1}{\pi T}\int e^{it\lambda} \widehat{\mu_T^2} (t/T) \Tilde{R}_N (t, \Tilde{x};\Tilde{w})\:dt=O(\pi^{-1} T^{-1} (2T) \lambda^{-N'} e^{C_{N'} T})=O(\lambda^{-N}),\quad N=1, 2, 3, \cdots.
\end{align*}
By \eqref{Symbol Q properties} again as in the proof of Lemma \ref{Lemma alpha=Id}, we can obtain
\begin{align*}
\left(\Tilde{Q}_j \circ \left(\frac{1}{\pi T} \int e^{it\lambda} \widehat{\mu_T^2}(t/T)\Tilde{R}_N (t, \cdot;\cdot)\:dt \right)\circ \Tilde{Q}_j^*\right) (\Tilde{\gamma}(r), \Tilde{\gamma}(s))=O(\lambda^{-N}),\quad N=1, 2, 3, \cdots,
\end{align*}
and thus, by Young's inequality, we ignore the contribution of $\Tilde{R}_N$, when we take $N\gg 1$.
We can also ignore the contribution of $E_\nu$ for $\nu\geq 1$.
\begin{lemma}[Theorem 3.4 in \cite{Chen2015improvement}]\label{Lemma: nu positive small}
We have, for $\Tilde{x}\in D$ and $\Tilde{w}\in \alpha(D)$,
\begin{align*}
\int e^{it\lambda} \widehat{\mu_T^2}(t/T) \partial_t E_\nu (t, \Tilde{\rho} (\Tilde{x}, \Tilde{w}))\:dt=O(\lambda^{1-2\nu}),\quad \nu=0, 1, 2, \cdots.
\end{align*}
\end{lemma}
By the same arguments as in $\Tilde{R}_\lambda$, Lemma \ref{Lemma: nu positive small} gives us that
\begin{align*}
\left(\Tilde{Q}_j \circ \left(\frac{1}{\pi T} \int e^{it\lambda} \widehat{\mu_T^2}(t/T) \partial_t E_\nu (t, \Tilde{\rho}(\cdot, \cdot))\:dt \right)\circ \Tilde{Q}_j^*\right) (\Tilde{\gamma}(r), \Tilde{\gamma}(s))=O(\lambda^{1-2\nu}),\quad \nu=0, 1, 2, \cdots.
\end{align*}
By Young's inequality, the contribution of this is better than \eqref{Log improvement TT* Claim} when $\nu\geq 1$, and so, we only need to consider $\nu=0$. With this in mind, we may write, modulo $O(\lambda^{-1})$ errors,
\begin{align}\label{Cosine propagator without errors}
\begin{split}
(\cos t\sqrt{-\Delta_{\Tilde{g}}}) (\Tilde{z}, \alpha(\Tilde{y}))&=u_0 (\Tilde{z}, \alpha(\Tilde{y}))\partial_t E_0 (t, \Tilde{\rho}(\Tilde{z}, \alpha(\Tilde{y}))) \\
&=\frac{1}{(2\pi)^2} u_0 (\Tilde{z}, \alpha(\Tilde{y})) \int e^{i\Phi(\Tilde{z}, \alpha(\Tilde{y}))\cdot \xi} \cos (t|\xi|)\:d\xi,
\end{split}
\end{align}
where $|\Phi(\Tilde{z}, \alpha(\Tilde{y}))|=\Tilde{\rho} (\Tilde{z}, \alpha(\Tilde{y}))$ (cf. \cite[p.4026]{Chen2015improvement}, \cite{Sogge2014hangzhou}, etc.). Using (orthogonal) coordinate changes if necessary, we may assume that
\begin{align*}
\Phi (\Tilde{z}, \alpha(\Tilde{y}))\cdot \xi= \Tilde{z}\cdot \xi, \quad \text{in normal coordinates at } \alpha(\Tilde{y}).
\end{align*}
Modulo $O(\lambda^{-1})$ errors, it follows from \eqref{Cosine propagator without errors} that
\begin{align*}
& (Q_j \circ B_{t, j}\circ \cos (t\sqrt{-\Delta_g}) )(x, y) \\
&=(2\pi)^{-2} \sum_\alpha \iint_{D^2} \int \Tilde{Q}_j (\Tilde{x}, \Tilde{w}) \Tilde{B}_{t, j} (\Tilde{w}, \Tilde{z}) u_0 (\Tilde{z}, \alpha(\Tilde{y})) e^{i\Phi(\Tilde{z}, \alpha(\Tilde{y}))\cdot \xi} \cos t|\xi|\:d\xi\:d\Tilde{w}\:d\Tilde{z} \\
&=\frac{1}{2(2\pi)^2}\sum_\alpha \sum_\pm \iiint \Tilde{Q}_j (\Tilde{x}, \Tilde{w}) \Tilde{B}_{t, j}(\Tilde{w}, \Tilde{z}) u_0 (\Tilde{z}, \alpha(\Tilde{y})) e^{i\Phi (\Tilde{z}, \alpha(\Tilde{y}))\cdot \xi \pm it|\xi|}\:d\xi\:d\Tilde{w}\:d\Tilde{z}.
\end{align*}
We now write
\begin{align*}
& (Q_j \circ B_{t, j}\circ \cos (t\sqrt{-\Delta_g})) (x, y) \\
&=\frac{\lambda^6}{2(2\pi)^6} \sum_{\alpha, \pm}\int e^{i\lambda[(\Tilde{x}-\Tilde{w})\cdot \eta+(\Tilde{w}-\Tilde{z})\cdot \zeta+\Phi(\Tilde{z}, \alpha(\Tilde{y}))\cdot \xi\pm t|\xi|]} \Tilde{q}_j (\Tilde{x}, \Tilde{w}, \lambda \eta) \Tilde{b}_{t, j} (\Tilde{w}, \Tilde{z}, \lambda \zeta) u_0 (\Tilde{z}, \alpha(\Tilde{y})) \:d\Tilde{w}\:d\eta\:d\Tilde{z}\:d\zeta\:d\xi.
\end{align*}
By \eqref{Kj mod errors}, modulo $O(\lambda^{-1})$ errors, we write
\begin{align*}
K_j (x, y)=\sum_{\alpha, \pm} U_{\alpha, j, \pm} (\Tilde{x}, \Tilde{y}),
\end{align*}
where
\begin{align*}
& U_{\alpha, j, \pm} (\Tilde{x}, \Tilde{y})=\frac{\lambda^6}{(2\pi)^7 T} \int e^{i\lambda \Tilde{\psi}_{\alpha, \pm} (t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)} a_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)\:d\Tilde{w}\:d\eta\:d\Tilde{z}\:d\zeta\:d\xi\:dt, \\
& a_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)=a_j (\Tilde{x}, \Tilde{y}; t, \Tilde{w}, \eta, \Tilde{z}, \zeta)=\widehat{\mu_T^2}(t/T) \Tilde{q}_j (\Tilde{x}, \Tilde{w}, \lambda \eta) \Tilde{b}_{t, j} (\Tilde{w}, \Tilde{z}, \lambda \zeta) u_0 (\Tilde{z}, \alpha (\Tilde{y})), \\
& \Tilde{\psi}_{\alpha, \pm} (t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)=t+(\Tilde{x}-\Tilde{w})\cdot \eta+(\Tilde{w}-\Tilde{z})\cdot \zeta+\Phi(\Tilde{z}, \alpha(\Tilde{y}))\cdot \xi\pm t|\xi|.
\end{align*}
In geodesic normal coordinates centered at $\alpha(\Tilde{y})$, using suitable orthogonal coordinate changes, we have
\begin{align*}
\Tilde{\psi}_{\alpha, \pm} (t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)=t+(\Tilde{x}-\Tilde{w})\cdot \eta+(\Tilde{w}-\Tilde{z})\cdot \zeta+\Tilde{z}\cdot \xi\pm t|\xi|.
\end{align*}
By Lemma \ref{Lemma alpha=Id}, we can focus only on $\alpha\not=\mathrm{Id}$. We would then have \eqref{Log improvement TT* Claim}, if we could show that
\begin{align}\label{Log imp TT^* U reduction}
\left\|\int U_{\alpha, j, \pm} (\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s)) f(s)\:ds \right\|_p \lesssim \frac{\lambda^{\frac{1}{2}}}{T} e^{CT} (2^{-j})^{\frac{2}{p}} \|f\|_{p'},\quad \alpha\not=\mathrm{Id},\quad 2\leq p<4,\quad \lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon} \rfloor \leq j\leq J.
\end{align}
We have the following analysis for $U_{\alpha, j, \pm}$.
\begin{proposition}\label{Prop: SP results in Log Improvement}
For $\alpha \not=\mathrm{Id}$ fixed, we have, modulo $O(\lambda^{-1})$ errors, that
\[
U_{\alpha, j, \pm}(\Tilde{\gamma}(r), \Tilde{\gamma}(s))=\begin{cases}
\frac{\lambda^{\frac{1}{2}}}{T}e^{i\lambda \Tilde{\rho}(\Tilde{\gamma}(r), \alpha(\Tilde{\gamma}(s)))} \Tilde{a}_{\alpha, j}(r, s), & \text{if } |d_{\Tilde{x}} \Tilde{\rho}(\Tilde{\gamma}(r), \alpha(\Tilde{\gamma}(s)))(\Tilde{N})|\approx 2^{-j} \\
&\hspace{10pt} \text{and } |d_{\Tilde{y}}\Tilde{\rho}(\Tilde{\gamma}(r), \alpha(\Tilde{\gamma}(s)))(\alpha_*(\Tilde{N}))|\approx 2^{-j}, \\
O(\lambda^{-N}), & \text{otherwise},
\end{cases}
\]
where $\lfloor \log_2 \lambda^{\frac{1}{3}-\epsilon} \rfloor \leq j\leq J$, $\Tilde{\rho}=d_{\Tilde{g}}$, and $|\Tilde{a}_{\alpha, j} (r, s)|\lesssim e^{CT}$.
\end{proposition}
\begin{proof}
In normal coordinates at $\alpha(\Tilde{y})$, we write $U_{\alpha, j, \pm} (\Tilde{x}, \Tilde{y})$ as $U_{\alpha, j, \pm} (\Tilde{x})$
\begin{align*}
U_{\alpha, j, \pm}(\Tilde{x})=\frac{\lambda^6}{(2\pi)^7 T}\int e^{i\lambda \Tilde{\psi}_{\alpha, \pm}(t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)} \Tilde{a}_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)\:d\Tilde{w}\:d\eta\:d\Tilde{z}\:d\zeta\:d\xi\:dt,
\end{align*}
where $\Tilde{a}_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)$ is the coordinate expression of $a_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)$ in normal coordinates at $\alpha(\Tilde{y})$. Let $\Tilde{\beta}\in C_0^\infty (\mathbb{R}^2)$ be such that $\mathrm{supp}(\Tilde{\beta}) \subset \{\xi:c_2\leq |\xi|\leq c_2^{-1}\}$ for a small fixed $c_2>0$. We write
\begin{align*}
U_{\alpha, j, \pm} (\Tilde{x})=U_{\alpha, j, \pm}^1 (\Tilde{x})+U_{\alpha, j, \pm}^2 (\Tilde{x}),
\end{align*}
where
\begin{align*}
& U_{\alpha, j, \pm}^1 (\Tilde{x})=\frac{\lambda^6}{(2\pi)^7 T} \int e^{i\lambda \Tilde{\psi}_{\alpha, \pm} (t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)} \Tilde{\beta}(\xi)\Tilde{a}_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)\:d\Tilde{w}\:d\eta\: d\Tilde{z}\:d\zeta\:d\xi\:dt, \\
& U_{\alpha, j, \pm}^2 (\Tilde{x})=\frac{\lambda^6}{(2\pi)^7 T} \int e^{i\lambda \Tilde{\psi}_{\alpha, \pm} (t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)} (1-\Tilde{\beta}(\xi))\Tilde{a}_j (t, \Tilde{w}, \eta, \Tilde{z}, \zeta)\:d\Tilde{w}\:d\eta\: d\Tilde{z}\:d\zeta\:d\xi\:dt.
\end{align*}
We note that, choosing $c_2>0$ small in the support of $\Tilde{\beta}$, we have $|\partial_t \Tilde{\psi}_\alpha|=|1\pm |\xi||\gtrsim 1+|\xi|$. Thus, integrating by parts in $t$ as in the proof of Lemma \ref{Lemma alpha=Id}, we can write $U_{\alpha, j, \pm} (\Tilde{x}, \Tilde{y})$ as $U_{\alpha, j, \pm}^1 (\Tilde{x})$ in normal coordinates at $\alpha(\Tilde{y})$, and we focus on $U_{\alpha, j, \pm}^1 (\Tilde{x})$.
We will focus on the minus sign in the phase function. Indeed, if we choose the plus sign, then we have $\partial_t \Tilde{\psi}_{\alpha, +} =1+|\xi|>0$, and thus, there is no critical point of the phase function. Hence, integration by parts in $t$ again, we have $U_{\alpha, j, \pm}^1 (\Tilde{x})=O(\lambda^{-N})$. Set $\rho_0 =\Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y}))$. Since $\alpha\not=\mathrm{Id}$, we know $\rho_0>0$, and thus, can consider the following change of variables.
\begin{align}\label{Change of variables bar}
\begin{split}
\Bar{t}=\frac{t}{\sqrt{\rho_0}},\quad \Bar{w}=\frac{\Tilde{w}}{\sqrt{\rho_0}},\quad \Bar{\xi}=\sqrt{\rho_0}\xi,\quad \Bar{\eta}=\sqrt{\rho_0}\eta,\quad \Bar{z}=\frac{\Tilde{z}}{\sqrt{\rho_0}}, \quad \Bar{x}=\frac{\Tilde{x}}{\sqrt{\rho_0}},\quad \Bar{\zeta}=\sqrt{\rho_0} \zeta.
\end{split}
\end{align}
This implies that
\begin{align*}
d\Tilde{w}\:d\eta\:d\Tilde{z}\:d\zeta\:d\xi\:dt=\frac{1}{\sqrt{\rho_0}} d\Bar{w}\:d\Bar{\eta}\:d\Bar{z}\:d\Bar{\zeta}\:d\Bar{\xi}\:d\Bar{t}.
\end{align*}
Since we choose the minus sign in the phase function, we set
\begin{align*}
\Tilde{\psi}_{\alpha, -} (t, \xi, \Tilde{w}, \eta, \Tilde{z}, \zeta)=\sqrt{\rho_0} \Bar{t}-\Bar{t}|\Bar{\xi}|+(\Bar{x}-\Bar{w})\cdot \Bar{\eta}+(\Bar{w}-\Bar{z})\cdot \Bar{\zeta}+\Bar{z}\cdot \Bar{\xi}=:\Bar{\psi}(\Bar{t}, \Bar{\xi}, \Bar{w}, \Bar{\eta}, \Bar{z}, \Bar{\zeta}).
\end{align*}
Note that
\begin{align*}
\nabla \Bar{\psi}&=(\partial_{\Bar{t}}\Bar{\psi}, \partial_{\Bar{\xi}} \Bar{\psi}, \partial_{\Bar{w}} \Bar{\psi}, \partial_\eta \Bar{\psi}, \partial_{\Bar{z}} \Bar{\psi}, \partial_{\Bar{\zeta}} \Bar{\psi} ) \\
&=(\sqrt{\rho_0}- |\Bar{\xi}|, \Bar{z}- \Bar{t} \frac{\Bar{\xi}}{|\Bar{\xi}|}, \Bar{\zeta}-\Bar{\eta}, \Bar{x}-\Bar{w}, -\Bar{\zeta}+\Bar{\xi}, \Bar{w}-\Bar{z} ),
\end{align*}
and thus, the critical point satisfies
\begin{align}\label{Stationary point condition}
\begin{split}
\sqrt{\rho_0} = |\Bar{\xi}|,\quad \Bar{z}=\Bar{t}\frac{\Bar{\xi}}{|\Bar{\xi}|},\quad \Bar{\zeta}=\Bar{\eta}, \quad \Bar{x}=\Bar{w},\quad \Bar{\zeta}=\Bar{\xi}, \quad \Bar{w}=\Bar{z}.
\end{split}
\end{align}
The Hessian $\partial^2 \Bar{\psi}$ is
\[
\partial^2 \Bar{\psi} =\begin{pmatrix}
O_{1\times 1} & \left(-\frac{\Bar{\xi}^T}{|\Bar{\xi}|} \right)_{1\times 2} & O_{1\times 2} & O_{1\times 2} & O_{1\times 2} & O_{1\times 2} \\
\left(-\frac{\Bar{\xi}}{|\Bar{\xi}|} \right)_{2\times 1} & A_{2\times 2} & O_{2\times 2} & O_{2\times 2} & I_{2\times 2} & O_{2\times 2} \\
O_{2\times 1} & O_{2\times 2} & O_{2\times 2} & -I_{2\times 2} & O_{2\times 2} & I_{2\times 2} \\
O_{2\times 1} & O_{2\times 2} & -I_{2\times 2} & O_{2\times 2} & O_{2\times 2} & O_{2\times 2} \\
O_{2\times 1} & I_{2\times 2} & O_{2\times 2} & O_{2\times 2} & O_{2\times 2} & -I_{2\times 2} \\
O_{2\times 1} & O_{2\times 2} & I_{2\times 2} & O_{2\times 2} & -I_{2\times 2} & O_{2\times 2}
\end{pmatrix}=:\begin{pmatrix}
B_{7\times 7} & C_{7\times 4} \\
(C^T)_{4\times 7} & D_{4\times 4}
\end{pmatrix},
\]
where $A_{2\times 2}=\Bar{\psi}_{\Bar{\xi}\Bar{\xi}}''$. By properties of determinants for block matrices (cf. \cite{Powell2011DetOfBlockMatrices}, \cite{Silvester2000Determinants}, etc.), we have, at the critical point,
\begin{align*}
\det(\partial^2\Bar{\psi})=\det(B-CD^{-1}C^T)\det D,
\end{align*}
provided the matrix $D$ is invertible. Since $\det D=1$, by properties of block matrix determinants again, we have, at the critical point,
\begin{align*}
\det(\partial^2 \Bar{\psi})=\det (B-CD^{-1}C^T) &=\det \begin{pmatrix}
O_{1\times 1} & \left(-\frac{\Bar{\xi}^T}{\sqrt{\rho_0}} \right)_{1\times 2} & O_{1\times 2} & O_{1\times 2} \\
\left(-\frac{\Bar{\xi}}{\sqrt{\rho_0}} \right)_{2\times 1} & A_{2\times 2} & I_{2\times 2} & O_{2\times 2} \\
O_{2\times 1} & I_{2\times 2} & O_{2\times 2} & -I_{2\times 2} \\
O_{2\times 1} & O_{2\times 2} & -I_{2\times 2} & O_{2\times 2}
\end{pmatrix}\\
&=\det \begin{pmatrix}
O_{1\times 1} & \left(-\frac{\Bar{\xi}^T}{\sqrt{\rho_0}} \right)_{1\times 2} & O_{1\times 2} & O_{1\times 2} \\
\left(-\frac{\Bar{\xi}}{\sqrt{\rho_0}} \right)_{2\times 1} & A_{2\times 2} & O_{2\times 2} & O_{2\times 2} \\
O_{2\times 1} & O_{2\times 2} & O_{2\times 2} & I_{2\times 2} \\
O_{2\times 1} & O_{2\times 2} & I_{2\times 2} & O_{2\times 2}
\end{pmatrix} \\
&=\det\begin{pmatrix}
0 & -\frac{\Bar{\xi}_1}{\sqrt{\rho_0}} & -\frac{\Bar{\xi}_2}{\sqrt{\rho_0}} \\
-\frac{\Bar{\xi}_1}{\sqrt{\rho}_0} & -\frac{\Bar{t}}{\sqrt{\rho_0}^3} \Bar{\xi_2}^2 & \frac{\Bar{t}}{\sqrt{\rho_0}^3} \Bar{\xi}_1 \Bar{\xi}_2 \\
-\frac{\Bar{\xi}_2}{\sqrt{\rho_0}} & \frac{\Bar{t}}{\sqrt{\rho_0}^3} \Bar{\xi}_1 \Bar{\xi}_2 & -\frac{\Bar{t}}{\sqrt{\rho_0}^3} \Bar{\xi}_1^2
\end{pmatrix}=\frac{\Bar{t} |\Bar{\xi}|^4 }{\rho_0^{\frac{5}{2}}} = 1.
\end{align*}
In the last equality, we used $\Bar{t}, |\Bar{\xi}|= \sqrt{\rho_0}$ at the critical point, since, by \eqref{Change of variables bar} and \eqref{Stationary point condition}, we have that, for $\Bar{t}>0$,
\begin{align*}
|\Bar{\xi}|= \sqrt{\rho_0}, \quad \Bar{t}=|\Bar{t}|=|\Bar{z}|=|\Bar{w}|=|\Bar{x}|=\frac{1}{\sqrt{\rho_0}}|\Tilde{x}|=\sqrt{\rho_0}.
\end{align*}
This gives us that $|\det \partial^2 \Bar{\psi}|=1$ at the critical point.
\begin{remark}
Since we have shown $\det (\partial^2 \Bar{\psi})=1$, we have
\begin{align*}
(\partial^2 \Bar{\psi})^{-1}=\frac{1}{\det (\partial^2 \Bar{\psi})}\mathrm{adj}(\partial^2 \Bar{\psi})=\mathrm{adj}(\partial^2 \Bar{\psi}).
\end{align*}
Each entry of the adjugate $\mathrm{adj}(\partial^2 \Bar{\psi})$ is a finite linear combination of multiplications of terms of the form
\begin{align*}
1,\; \frac{\Bar{\xi}_1}{\sqrt{\rho_0}},\; \frac{\Bar{\xi}_2}{\sqrt{\rho_0}},\; \frac{\Bar{t}}{\sqrt{\rho_0}^3}\Bar{\xi}_1^2,\; \frac{\Bar{t}}{\sqrt{\rho_0}^3}\Bar{\xi}_2^2,\; \frac{\Bar{t}}{\sqrt{\rho_0}^3} \Bar{\xi}_1 \Bar{\xi}_2.
\end{align*}
These are all $O(1)$ near the critical point, since we have $|\Bar{t}|, |\Bar{\xi}|\approx \sqrt{\rho_0}$. This implies that the matrix norm of $\partial^2 \Bar{\psi}$ is $O(1)$, and thus, we can use the method of stationary phase below easily.
\end{remark}
Continuing with our proof, in the normal coordinates at $\alpha(\Tilde{y})$, by the stationary phase argument, we have, modulo $O(\lambda^{-1})$ errors, at the critical point,
\begin{align*}
U_{\alpha, j, -}^1 (\Tilde{x})=\frac{\lambda^6}{(2\pi)^7 \sqrt{\rho_0} T } \bigg[ \left(\frac{\lambda}{2\pi} \right)^{-\frac{11}{2}} e^{i\lambda \sqrt{\rho_0}|\Bar{x}|}
e^{\frac{i\pi}{4}\mathrm{sgn}(\partial^2 \Bar{\psi})} \sum_{l<l_0} \lambda^{-l} L_l a_0 + O\left( \lambda^{-l_0} \sum_{|\beta|\leq 2 l_0} \sup|D^\beta a_0| \right) \bigg],
\end{align*}
where $a_0$ is defined by
\begin{align*}
a_0 (\Bar{t}, \Bar{\xi}, \Bar{w}, \Bar{\eta}, \Bar{z}, \Bar{\zeta})=\Tilde{\beta}\left(\frac{\Bar{\xi}}{\sqrt{\rho_0}} \right)\widehat{\mu_T^2}\left(\frac{\sqrt{\rho_0}\Bar{t}}{T} \right) \Tilde{q}_j \left(\sqrt{\rho_0}\Bar{x}, \sqrt{\rho_0}\Bar{w}, \lambda \frac{\Bar{\eta}}{\sqrt{\rho_0}}\right) \Tilde{b}_{\sqrt{\rho_0}\Bar{t}, j}\left(\sqrt{\rho_0}\Bar{w}, \sqrt{\rho_0} \Bar{x}, \lambda \frac{\Bar{\eta}}{\sqrt{\rho_0}}\right) u_0 (\sqrt{\rho_0}\Bar{z}),
\end{align*}
$u_0 (\sqrt{\rho_0}\Bar{z})$ is the coordinate expression of $u_0 (\sqrt{\rho_0}\Bar{z}, \alpha(\Tilde{y}))$ in normal coordinates at $\alpha(\Tilde{y})$, and the $L_l$ are the differential operators of order at most $2l$ acting on $a_0$ at the critical point. Recall that we can easily control the size estimates of $\Tilde{q}_j$ by construction, and the size estimates of $u_0$ by \cite[Lemma B.1]{Keeler2019TwoPointWeylLaw}. Also, by \cite{BouzouinaRobert2002Duke} and/or \cite[Lemma 11.11]{Zworski2012Semiclassical}, the size estimtaes for $\kappa_t^* q_j^*$ are the same as those for $q_j$, up to $e^{CT}$. Thus, the remainder is
\begin{align*}
O\left( \lambda^{-l_0} \sum_{|\beta|\leq 2 l_0} \sup|D^\beta a_0| \right)=O(\lambda^{-l_0} (\lambda^{\frac{1}{3}})^{2 l_0} e^{CT})=O(\lambda^{-\frac{l_0}{3}} e^{CT}).
\end{align*}
Taking $l_0$ large enough, we can ignore the contribution of the remainder.
As before, by \eqref{Change of variables bar} and \eqref{Stationary point condition}, at the critical point, we have that
\begin{align*}
\Bar{t}=|\Bar{z}|=|\Bar{w}|=|\Bar{x}|=\frac{1}{\sqrt{\rho_0}}|\Tilde{x}|,\quad |\Bar{\xi}|=\sqrt{\rho_0},\quad \Bar{\xi}=\frac{|\Bar{\xi}|}{\Bar{t}}\Bar{z},\quad \Bar{z}=\Bar{w}=\Bar{x}.
\end{align*}
This gives us that, in the geodesic normal coordinates,
\begin{align*}
\Bar{\xi}=\frac{|\Bar{\xi}|}{\Bar{t}} \Bar{z}=\frac{\sqrt{\rho_0}}{|\Bar{x}|} \Bar{x}=\sqrt{\rho_0} \frac{\Tilde{x}}{|\Tilde{x}|}=\sqrt{\rho_0}\frac{d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y}))}{|d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y}))|_{\Tilde{g}} }.
\end{align*}
We then have, modulo $O(\lambda^{-1})$ errors, that
\begin{align*}
U_{\alpha, j, -}^1 (\Tilde{x}, \Tilde{y})&=\frac{\lambda^{\frac{1}{2}}}{2\pi\sqrt{2\pi}T} \sum_{\alpha \not=\mathrm{Id}} \frac{1}{\sqrt{\rho_0}} e^{i\lambda \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y}))} e^{\frac{i\pi}{4}\mathrm{sgn}(\partial^2\Bar{\psi})} \\
& \hspace{5pt} \times \sum_{l<l_0} \lambda^{-l} L_l a_0 \bigg(\frac{|\Tilde{x}|}{\sqrt{\rho_0}}, \sqrt{\rho_0} d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})), \frac{\Tilde{x}}{\sqrt{\rho_0}}, \sqrt{\rho_0} d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})), \frac{\Tilde{x}}{\sqrt{\rho_0}}, \sqrt{\rho_0} d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) \bigg) \\
&=\frac{\lambda^{\frac{1}{2}}}{2\pi\sqrt{2\pi}T} \sum_{\alpha \not=\mathrm{Id}} \frac{1}{\sqrt{\rho_0}} e^{i\lambda \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y}))} e^{\frac{i\pi}{4}\mathrm{sgn}(\partial^2\Bar{\psi})} \\
& \hspace{5pt} \times \sum_{l<l_0} \lambda^{-l} L_l \bigg( \widehat{\mu_T^2}\left(|\Tilde{x}|/T \right) u_0 (\Tilde{x}, \alpha(\Tilde{y})) \Tilde{b}_{|\Tilde{x}|, j}(\Tilde{x}, \Tilde{x}, \lambda d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) ) \Tilde{q}_j (\Tilde{x}, \Tilde{x}, \lambda d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) ) \bigg).
\end{align*}
By the discussion in \S \ref{SS: Notation for PDOs} and the properties of the geodesic flow, we can write
\begin{align*}
\Tilde{b}_{|\Tilde{x}|, j}(\Tilde{x}, \Tilde{x}, \lambda d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) )&=\Tilde{q}_j (\kappa_{|\Tilde{x}|} (\Tilde{x}, \lambda d_{\Tilde{x}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) )) \\
&=\Tilde{q}_j (\alpha(\Tilde{y}), \alpha(\Tilde{y}), -\lambda d_{\Tilde{y}} \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) ).
\end{align*}
Hence, modulo $O(\lambda^{-1})$ errors, for $\alpha\not=\mathrm{Id}$,
\begin{align*}
U_{\alpha, j, -}^1 (\Tilde{x}, \Tilde{y})&=\frac{\lambda^{\frac{1}{2}}}{2\pi\sqrt{2\pi}T} e^{i\lambda \Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) } a_j (\Tilde{x}, \alpha(\Tilde{y})),
\end{align*}
where
\begin{align*}
a_j (\Tilde{x}, \alpha(\Tilde{y}))=\sum_{l<l_0} \frac{1}{(\Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y})))^{\frac{1}{2}}} \lambda^{-l} L_l \bigg( \widehat{\mu_T^2}(\Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y}))/T) u_0 (\Tilde{x}, \alpha(\Tilde{y})) \Tilde{q}_j (\alpha(\Tilde{y}), \alpha(\Tilde{y}), -\lambda d_{\Tilde{y}} \Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y})))\\
\times \Tilde{q}_j (\Tilde{x}, \Tilde{x}, \lambda d_{\Tilde{x}}\Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) ) \bigg).
\end{align*}
Since we have $\Tilde{\rho} (\Tilde{x}, \alpha(\Tilde{y})) \gtrsim 1$ for $\alpha\not=\mathrm{Id}$, we have, by construction,
\begin{align*}
|a_j (\Tilde{x}, \alpha(\Tilde{y}))| \leq e^{CT}.
\end{align*}
Recall that the $\xi$-support of $q_j (x, y, \xi)$ is $\frac{|\xi(N)|}{|\xi|_g} \approx 2^{-j}$ and $\xi (N)=\langle \xi^{\#}, N \rangle_{\Tilde{g}}$. Note that, in geodesic normal coordinates centered at $\alpha(\Tilde{y})$, we have, for $\Tilde{\rho}(\Tilde{x}, \Tilde{z})$,
\begin{align*}
|d_{\Tilde{x}} \Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))|_{\Tilde{g}}= 1 = |d_{\Tilde{z}} \Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))|_{\Tilde{g}}.
\end{align*}
By the support properties of $\Tilde{q}_j (\Tilde{x}, \Tilde{x}, \lambda d_{\Tilde{x}} \Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y})))$, $a_j (\Tilde{x}, \alpha(\Tilde{y}))$ is supported where
\begin{align*}
|d_{\Tilde{x}}\Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))(\Tilde{N})| \approx 2^{-j}.
\end{align*}
Here, $\Tilde{N}$ is a unit normal vector to $\Tilde{\gamma}$. We also observe that $\alpha_* (\Tilde{N})$ is normal to $\alpha \circ \Tilde{\gamma}$. By the support properties of
\begin{align*}
\Tilde{q}_j (\alpha(\Tilde{y}), \alpha(\Tilde{y}), -\lambda d_{\Tilde{y}} \Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))),
\end{align*}
for each $\alpha \not=\mathrm{Id}$, if $\Tilde{y}=\alpha\circ \Tilde{\gamma}(s)$ for $|s|\ll 1$, then $a_j (\Tilde{x}, \alpha(\Tilde{y}))$ is supported where
\begin{align*}
|d_{\Tilde{y}}\Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))(\alpha_*(\Tilde{N}))|=|d_{\Tilde{y}} \Tilde{\rho}(\Tilde{x}, \alpha \circ \Tilde{\gamma}(s)) (\alpha_* (\Tilde{N}))| \approx 2^{-j}.
\end{align*}
This completes the proof.
\end{proof}
We next consider the support properties of the amplitude of $U_{\alpha, j, \pm}$. Let $\Tilde{\rho}_\alpha (\Tilde{x}, \Tilde{y})=\Tilde{\rho}(\Tilde{x}, \alpha(\Tilde{y}))$ for $\alpha\not=\mathrm{Id}$. Fix $r_0, s_0 \in [0, 1]$ so that
\begin{align*}
|d_{\Tilde{x}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r_0), \Tilde{\gamma}(s_0))(\Tilde{N})|\approx 2^{-j} \quad \text{and} \quad |d_{\Tilde{y}}\Tilde{\rho}_\alpha(\Tilde{\gamma}(r_0), \Tilde{\gamma}(s_0))(\alpha_* (\Tilde{N}))|\approx 2^{-j}.
\end{align*}
We can assume such $r_0$ and $s_0$ exist, or otherwise, by the above proposition, we have $U_{\alpha, j, \pm}=O(\lambda^{-N})$ for any $N=1, 2, 3, \cdots$. Using a partition of unity, we may assume that
\begin{align*}
|d_{\Tilde{x}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r), \Tilde{\gamma}(s))(\Tilde{N})|\approx 2^{-j} \quad \text{and} \quad |d_{\Tilde{y}}\Tilde{\rho}_\alpha(\Tilde{\gamma}(r), \Tilde{\gamma}(s))(\alpha_* (\Tilde{N}))|\approx 2^{-j}.
\end{align*}
happens only near $(r_0, s_0)$. By Proposition \ref{Prop: SP results in Log Improvement}, we may assume that $U_{\alpha, j, \pm} (\Tilde{\gamma}(r), \Tilde{\gamma}(s))$ is supported where
\begin{align*}
|d_{\Tilde{x}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r), \Tilde{\gamma}(s))(\Tilde{N})|,\quad |d_{\Tilde{y}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r), \Tilde{\gamma}(s))(\alpha_*(\Tilde{N}))| \in [2^{-j-1}, 2^{-j+1}].
\end{align*}
Suppose $r, s\in [0, \epsilon_1]=I$ for some small $\epsilon_1>0$, and write $I=\cup_k I_k$, where $\{I_k\}_k$ is a collection of almost disjoint intervals with $|I_k|\approx e^{-CT}$ for some large $C>0$. Let $r_0$ and $s_0$ be fixed points in a subinterval $I_k$ and $I_{k'}$, respectively. We want to show the following.
\begin{lemma}\label{lemma Hessian operator}
Suppose $|d_{\Tilde{x}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r_0), \Tilde{\gamma}(s_0))(\Tilde{N})|\in [2^{-j-1}, 2^{-j+1}]$. Then, choosing $C>0$ sufficiently large with $|I_k|\approx e^{-CT}$, there exists a uniform constant $\Tilde{C}>0$ such that, for $r$ and $r_0$ in a same subinterval $I_k$,
\begin{align*}
|d_{\Tilde{x}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r), \Tilde{\gamma}(s_0))(\Tilde{N})|\not\in [2^{-j-1}, 2^{-j+1}],\quad \text{whenever } |r-r_0|\geq \Tilde{C} 2^{-j}.
\end{align*}
Similarly, if $|d_{\Tilde{y}} \Tilde{\rho}_\alpha (\Tilde{\gamma}(r_0), \Tilde{\gamma}(s_0))(\alpha_*(\Tilde{N}))|\in [2^{-j-1}, 2^{-j+1}]$, then, choosing $C>0$ sufficiently large with $|I_k|\approx e^{-CT}$, there exists a uniform constant $\Tilde{C}>0$ such that, for $s$ and $s_0$ in a same subinterval $I_k$,
\begin{align*}
|d_{\Tilde{y}} \Tilde{\rho}_\alpha(\Tilde{\gamma}(r_0), \Tilde{\gamma}(s))(\alpha_*(\Tilde{N}))|\not\in [2^{-j-1}, 2^{-j+1}],\quad \text{whenever } |s-s_0|\geq \Tilde{C} 2^{-j}.
\end{align*}
\end{lemma}
Before we prove this lemma, we review some basic properties of the Hessian operator $\mathcal{H}_r$ in \cite{Lee2018secondEd}: Suppose $(M, g)$ is an $n$-dimensional Riemannian manifold, $U$ is a normal neighborhood of a point $p\in M$, and $r:U\to \mathbb{R}$ is the radial distance function from the point $p$ defined by
\begin{align}\label{radial dist fcn for Hessian opr}
r(x)=\sqrt{(x_1)^2+\cdots + (x_n)^2},
\end{align}
where $(x_i)$ are normal coordinates on $U$ centered at $p$. We also define the radial vector field on $U\setminus \{p\}$, denoted by $\partial_r$, as
\begin{align*}
\partial_r=\sum_{i=1}^n \frac{x_i}{r(x)} \frac{\partial}{\partial x_i}=\mathrm{grad}\:r
\end{align*}
(cf. \cite[Corollary 6.10]{Lee2018secondEd}), where $\mathrm{grad} f=(d_{\Tilde{x}} f)^\#$ is the Riemannian gradient of $f$ and $\#$ is the musical isomorphism sharp. Then, the $(1, 1)$-tensor field $\mathcal{H}_r=\nabla (\partial_r)$, defined by
\begin{align*}
\mathcal{H}_r (w)=\nabla_w \partial_r, \quad \text{for all } w\in TM|_{U\setminus \{p\}},
\end{align*}
is called the Hessian operator of $r$, where $\nabla$ is the Levi-Civita connenction. By \cite[Lemma 11.1]{Lee2018secondEd}, $\mathcal{H}_r$ is self-adjoint, $\mathcal{H}_r (\partial_r)\equiv 0$, and the restriction of $\mathcal{H}_r$ to vectors tangents to a level set of $r$ is equal to the shape operator of the level set associated with the normal vector field $-\partial_r$.
\begin{proof}[Proof of Lemma \ref{lemma Hessian operator}]
We prove the second case in this lemma. Similar arguments will work on the first one. In this proof, $\nabla$ denotes the Levi-Civita connection. We write $\Tilde{\rho}_0 (\Tilde{y})=\Tilde{\rho}_{0, \alpha} (\Tilde{y})=\Tilde{\rho}_0 (\Tilde{\gamma}(r_0), \alpha(\Tilde{y}))$ so that $\Tilde{\rho}_0$ is the distance function as in \eqref{radial dist fcn for Hessian opr}, since $r_0$ is fixed. We also write the radial vector field as $\partial_{\Tilde{\rho}_0}=\frac{\partial}{\partial \Tilde{\rho}_0}$. Set
\begin{align*}
h(s)=\langle \mathrm{grad}_{\Tilde{y}} \Tilde{\rho}_0 (\Tilde{\gamma}(s)), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}},
\end{align*}
where $\mathrm{grad}_{\Tilde{y}}$ is the gradient for $\Tilde{y}$. By assumption, we have $|h(s_0)|\in [2^{-j-1}, 2^{-j+1}]$. We will work in geodesic normal coordinates centered at $\Tilde{\gamma}(r_0)$.
We want to show that $|h'(s_0)|\approx 1$. Let $\Tilde{\eta}=\Tilde{\eta}_\alpha=\alpha\circ \Tilde{\gamma}$. We then have that
\begin{align}\label{h(s) derivative}
\frac{d}{ds}(h(s))=\langle \nabla_{\dot{\Tilde{\eta}}(s)} \mathrm{grad}_{\Tilde{y}} \Tilde{\rho}_0 (\Tilde{\gamma}(s)), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}} +\langle \mathrm{grad}_{\Tilde{y}} \Tilde{\rho}_0 (\Tilde{\gamma}(s)), \nabla_{\dot{\Tilde{\eta}}(s)} \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}}.
\end{align}
For the first term in the right hand side, note that
\begin{align*}
\langle \nabla_{\dot{\Tilde{\eta}}(s_0)} \mathrm{grad}_{\Tilde{y}} \Tilde{\rho}_0 (\Tilde{\gamma}(s_0)), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}}=\langle \mathcal{H}_{\Tilde{\rho}_0} (\dot{\Tilde{\eta}}(s_0)), \alpha_*(\Tilde{N})\rangle_{\Tilde{g}},
\end{align*}
where $\mathcal{H}_{\Tilde{\rho}_0}$ is the Hessian operator of $\Tilde{\rho}_0$. Since $\mathcal{H}_{\Tilde{\rho_0}} (\partial_{\Tilde{\rho}_0})\equiv 0$, we have
\begin{align}\label{Hessian at rho0}
|\mathcal{H}_{\Tilde{\rho}_0} (\dot{\Tilde{\eta}}(s_0))|_{\Tilde{g}}=\left|\mathcal{H}_{\Tilde{\rho}_0} \left(\dot{\Tilde{\eta}}(s_0)-\frac{\partial}{\partial \Tilde{\rho}_0} \right) \right|_{\Tilde{g}}\leq \|\mathcal{H}_{\Tilde{\rho}_0}\|\left| \dot{\Tilde{\eta}}(s_0)-\frac{\partial}{\partial \Tilde{\rho}_0} \right|_{\Tilde{g}}\approx 2^{-j} \|\mathcal{H}_{\Tilde{\rho}_0}\|,
\end{align}
where $\|\mathcal{H}_{\Tilde{\rho}_0} \|$ denotes the operator norm of $\mathcal{H}_{\Tilde{\rho}_0}$ and $\frac{\partial}{\partial \Tilde{\rho}_0}$ is a radial vector at $\alpha \circ \Tilde{\gamma}(s_0)$. Here we used the fact that $\left| \dot{\Tilde{\eta}}(s_0)-\frac{\partial}{\partial \Tilde{\rho}_0} \right|_{\Tilde{g}}\approx 2^{-j}$, which follows from the assumption $|h(s_0)|\approx 2^{-j}$. For the first term in \eqref{h(s) derivative} on the right hand side, we continue to show that $\|\mathcal{H}_{\Tilde{\rho}_0}\|\lesssim 1$. Let
\[
s_c (t)=\begin{cases}
t, & \text{if } c=0, \\
R\sin \frac{t}{R}, & \text{if } c=\frac{1}{R^2}>0, \\
R\sinh \frac{t}{R}, & \text{if } c=-\frac{1}{R^2}.
\end{cases}
\]
If we call the curvature of our manifold $\kappa$, then we can assume $-1\leq \kappa \leq 0$ by scaling our metric if necessary. By the Hessian comparison (cf. Theorem 11.7 in \cite{Lee2018secondEd}), we have that
\begin{align}\label{Hess comparison result}
\frac{1}{\Tilde{\rho}_0}\pi_{\Tilde{\rho}_0} =\frac{s_0'(\rho)}{s_0(\rho)} \pi_{\Tilde{\rho}_0} \leq \mathcal{H}_{\Tilde{\rho}_0} \leq \frac{s_{-1}'(\rho)}{s_{-1}(\rho)}\pi_{\Tilde{\rho}_0}=\coth (\Tilde{\rho}_0)\pi_{\Tilde{\rho}_0},
\end{align}
where $\pi_{\Tilde{\rho}_0}$ is the orthogonal projection onto the tangent space of the level set of $\Tilde{\rho}_0$ as in \cite{Lee2018secondEd}. Here, $A\leq B$ means $\langle Av, v \rangle_{\Tilde{g}} \leq \langle Bv, v \rangle_{\Tilde{g}}$ for all vectors $v$. From the second inequality in \eqref{Hess comparison result}, we have
\begin{align*}
\langle \mathcal{H}_{\Tilde{\rho}_0} v, v\rangle_{\Tilde{g}} \leq \coth(\Tilde{\rho}_0)\langle \pi_{\Tilde{\rho}_0} v, v \rangle_{\Tilde{g}} \lesssim \langle \pi_{\Tilde{\rho}_0} v, \pi_{\Tilde{\rho}_0} v+(v-\pi_{\Tilde{\rho}_0} v)\rangle_{\Tilde{g}}=|\pi_{\Tilde{\rho}_0} v|_{\Tilde{g}}^2 \leq |v|_{\Tilde{g}}^2.
\end{align*}
Here, we used the fact that $\coth(\rho)=\frac{e^\rho +e^{-\rho}}{e^{\rho}-e^{-\rho}}$ with $1\lesssim\Tilde{\rho}_0\leq T$. We can make the same argument for the first inequality, and so, in summary, we have
\begin{align*}
0\leq \frac{1}{\Tilde{\rho}_0}|\pi_{\Tilde{\rho}_0} v|_{\Tilde{g}}^2 \leq \langle \mathcal{H}_{\Tilde{\rho}_0} v, v \rangle_{\Tilde{g}} \lesssim |v|_{\Tilde{g}}^2,
\end{align*}
from which it follows that $0\leq |\langle \mathcal{H}_{\Tilde{\rho}_0} v, v \rangle| \lesssim |v|_{\Tilde{g}}^2$. Since $\mathcal{H}_{\Tilde{\rho}_0}$ is self-adjoint (cf. Lemma 11.1 in \cite{Lee2018secondEd}), what we have shown is
\begin{align*}
\|\mathcal{H}_{\Tilde{\rho}_0}\|=\sup_{|v|_{\Tilde{g}}=1} |\langle \mathcal{H}_{\Tilde{\rho}_0} v, v \rangle_{\Tilde{g}}|\lesssim \sup_{|v|_{\Tilde{g}}=1} |v|_{\Tilde{g}}^2=1.
\end{align*}
Combining this with \eqref{Hessian at rho0}, \eqref{h(s) derivative} is translated into
\begin{align}\label{h(s0) derivative}
\left.\frac{d}{ds}(h(s))\right.\bigg|_{s=s_0}=O(2^{-j})+\langle \mathrm{grad}_{\Tilde{y}} \Tilde{\rho}_0 (\Tilde{\gamma}(s_0)), \nabla_{\dot{\Tilde{\eta}}(s_0)} \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}}=O(2^{-j})+\left\langle \frac{\partial}{\partial \Tilde{\rho}_0}, \nabla_{\dot{\Tilde{\eta}}(s_0)} \alpha_*(\Tilde{N}) \right\rangle_{\Tilde{g}}.
\end{align}
For the second term in \eqref{h(s) derivative}, we first note that
\begin{align*}
\nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}}(s)=\langle \nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}} (s), \alpha_* (\Tilde{N})\rangle_{\Tilde{g}} \alpha_* (\Tilde{N}).
\end{align*}
Indeed, since $\Tilde{\eta}$ can be parametrized by arc length, we have
\begin{align*}
\langle \nabla_{\dot{\Tilde{\eta}} (s)} \dot{\Tilde{\eta}}(s), \dot{\Tilde{\eta}}(s) \rangle_{\Tilde{g}}=\frac{1}{2}\nabla_{\dot{\Tilde{\eta}}(s)} (\langle \dot{\Tilde{\eta}}(s), \dot{\Tilde{\eta}}(s) \rangle_{\Tilde{g}})=\frac{1}{2}\nabla_{\dot{\Tilde{\eta}}(s)} 1=0,
\end{align*}
which in turn implies that
\begin{align*}
\nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}}(s)=\langle \nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}}(s), \dot{\Tilde{\eta}}(s) \rangle_{\Tilde{g}} \dot{\Tilde{\eta}}(s)+\langle \nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}} (s), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}} \alpha_* (\Tilde{N})=\langle \nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}} (s), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}} \alpha_* (\Tilde{N}).
\end{align*}
Since $\alpha_*(\Tilde{N})$ and $\dot{\Tilde{\eta}}(s)$ are orthogonal (at $\Tilde{\eta}(s_0)$), we have
\begin{align*}
|\nabla_{\dot{\Tilde{\eta}}(s_0)} \dot{\Tilde{\eta}}(s_0)|_{\Tilde{g}}=|\langle \nabla_{\dot{\Tilde{\eta}}(s_0)} \dot{\Tilde{\eta}}(s_0), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}}|&=|\langle \mathrm{\RomanNumeralCaps{2}} (\dot{\Tilde{\eta}}(s_0), \dot{\Tilde{\eta}}(s_0)), \alpha_*(\Tilde{N}) \rangle_{\Tilde{g}}| \\
&=|\langle \dot{\Tilde{\eta}}(s_0), W_{\alpha_*(\Tilde{N})}(\dot{\Tilde{\eta}}(s_0)) \rangle_{\Tilde{g}}|\\
&=|\langle \dot{\Tilde{\eta}}(s_0), \nabla_{\dot{\Tilde{\eta}}(s_0)} \alpha_*(\Tilde{N})\rangle_{\Tilde{g}}|\approx \left|\left\langle \frac{\partial}{\partial \Tilde{\rho}_0 }, \nabla_{\dot{\Tilde{\eta}}(s_0)} \alpha_*(\Tilde{N}) \right\rangle_{\Tilde{g}} \right|,
\end{align*}
where the map $W_N$ is the Weingarten map in the direction of $N$ and $\mathrm{\RomanNumeralCaps{2}}$ is the second fundamental form of $\alpha(\gamma)$ in the universal cover $(\mathbb{R}^2, \Tilde{g})$. In the last approximation, we used the fact $\left|\frac{\partial}{\partial \Tilde{\rho}_0}-\dot{\Tilde{\eta}}(s_0) \right|_{\Tilde{g}}\approx 2^{-j}$, where $j$ is large enough.
Since we know $|\nabla_{\dot{\Tilde{\eta}}(s)} \dot{\Tilde{\eta}}(s)|_{\Tilde{g}}\approx 1$ by the assumption on the curvature of the given curve $\gamma$, we have
\begin{align*}
\left|\left\langle \frac{\partial}{\partial \Tilde{\rho}_0}, \nabla_{\dot{\Tilde{\eta}}(s_0)} \alpha_*(N) \right\rangle_{\Tilde{g}}\right| \approx |\nabla_{\dot{\Tilde{\eta}}(s_0)} \dot{\Tilde{\eta}}(s_0)|_{\Tilde{g}} \approx 1.
\end{align*}
Combining this with \eqref{h(s0) derivative}, we have that $|h'(s_0)|\approx 1$.
By Taylor's formula,
\begin{align*}
h(s)=h(s_0)+h'(s_0)(s-s_0)+O(|h''|(s-s_0)^2).
\end{align*}
As a consequence of \cite[Lemma B.2]{Keeler2019TwoPointWeylLaw}, there exists $C'>0$ such that
\begin{align*}
h(s)=h(s_0)+h'(s_0)(s-s_0)+O(e^{C'T} (s-s_0)^2).
\end{align*}
Since we are assuming $|s-s_0|\approx e^{-CT}$, for a sufficiently large $C>0$, we have
\begin{align*}
h(s)=h(s_0)+(h'(s_0)+O(e^{(C'-C)T}))(s-s_0)\approx h(s_0)\pm|h'(s_0)|(s-s_0).
\end{align*}
Since we have shown $|h'(s_0)|\approx 1$, there exists $C_2>0$ such that if $|s-s_0|\geq C_2 2^{-j}$, then we have $|h(s)|\not\in [2^{-j-1}, 2^{-j+1}]$, which proves the lemma.
\end{proof}
By Lemma \ref{lemma Hessian operator}, we have, modulo $O(\lambda^{-1})$ errors, for $r\in I_k, s\in I_{k'}$,
\begin{align}\label{U alpha j after Hessian comp}
\begin{split}
U_{\alpha, j, \pm}(\Tilde{\gamma}(r), \Tilde{\gamma}(s))=
\begin{cases}
\frac{\lambda^{\frac{1}{2}}}{T}e^{i\lambda \Tilde{\rho}(\Tilde{\gamma}(r), \alpha(\Tilde{\gamma}(s)))} \Tilde{a}_{\alpha, j}(r, s), & \text{if } |r-r_0|\lesssim 2^{-j} \text{ and } |s-s_0|\lesssim 2^{-j}, \\
O(\lambda^{-N}), & \text{otherwise},
\end{cases}
\end{split}
\end{align}
where $|\Tilde{a}_{\alpha, j}(r, s)|\leq C e^{CT}$. Here, there is at most one cube of sidelength $C 2^{-j}$ in $(r, s)\in I_k \times I_{k'}\subset I\times I=[0, \epsilon_1]^2$ for small $\epsilon_1>0$ such that the amplitude $\Tilde{a}_{\alpha, j}(r, s)$ is nonzero, and $(r_0, s_0)$ is the center of the cube $I_k \times I_{k'}$.
\begin{remark}
We observe that the way to find support properties of $U_{\alpha, j}$ here is similar to that of $K_j, +$ or $K_J$ in the previous section. We used the assumption of nonvanishing geodesic curvatures on $\gamma$ in both cases. We also used the properties of the Hessian operator and the Taylor expansion here, whereas used the properties of the solution to the eikonal equation $\varphi$ and the mean value theorem there.
\end{remark}
It follows from \eqref{U alpha j after Hessian comp} that
\begin{align*}
\int |U_{\alpha, j, \pm}(\Tilde{\gamma}(r), \Tilde{\gamma}(s))|\:dr=\sum_k \int_{I_k} |U_{\alpha, j, \pm} (\Tilde{\gamma}(r), \Tilde{\gamma}(s))|\:dr \lesssim e^{C'T} \frac{\lambda^{\frac{1}{2}}}{T} 2^{-j}.
\end{align*}
Here, $e^{C'T}$ comes from the fact $|\Tilde{a}_{\alpha, j}(\Tilde{\gamma}(r), \Tilde{\gamma}(s))|\leq e^{C''T}$ and the fact that the number of $\{I_k\}$ is $e^{CT}$ up to some constant, and $2^{-j}$ comes from the support property $|r-r_0|\lesssim 2^{-j}$ in $I_k$ for some $k$. Similarly, we also have $\int |U_{\alpha, j, \pm}(\Tilde{\gamma}(r), \Tilde{\gamma}(s))|\:ds\lesssim e^{C'T} \frac{\lambda^{\frac{1}{2}}}{T} 2^{-j}$. By Young's inequality, we have
\begin{align*}
\left\|\int U_{\alpha, j, \pm} (\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s))f(s)\:ds \right\|_2 \lesssim \frac{\lambda^{\frac{1}{2}}}{T} e^{CT} 2^{-j} \|f\|_2.
\end{align*}
By \eqref{U alpha j after Hessian comp}, we also have that
\begin{align*}
\left\|\int U_{\alpha, j, \pm} (\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s))f(s)\:ds \right\|_\infty \lesssim \frac{\lambda^{\frac{1}{2}}}{T} e^{CT} \|f\|_1.
\end{align*}
By interpolation, we obtain
\begin{align*}
\left\|\int U_{\alpha, j, \pm} (\Tilde{\gamma}(\cdot), \Tilde{\gamma}(s))f(s)\:ds \right\|_p \lesssim \frac{\lambda^{\frac{1}{2}}}{T} e^{CT} (2^{-j})^{\frac{2}{p}} \|f\|_{p'},\quad 2\leq p\leq \infty,
\end{align*}
which proves \eqref{Log imp TT^* U reduction}. This completes the proof.
\section{Proof of Corollary \ref{Cor: at p=4}}\label{S: Cor: at p=4}
Let $P=\sqrt{-\Delta_g}$, $\chi\in \mathcal{S}(\mathbb{R})$, and $\gamma$ be as above. In this section, we heavily borrow arguments from Xi and Zhang \cite{XiZhang2017improved}, which was also motivated by Bourgain \cite{Bourgain1991Besicovitch} and Sogge \cite{Sogge2017ImprovedCritical}. We first have an analogue of \cite[Lemma 1]{XiZhang2017improved}.
\begin{lemma}[Lemma 1 in \cite{XiZhang2017improved}]\label{Lemma l subsegment}
We set $\lambda^{-1}\leq l\leq 1$. Let $\gamma_l$ be a fixed subsegment of $\gamma$ with length $l$. We then have that
\begin{align*}
\|\chi(\lambda-P) f\|_{L^2 (\gamma_l)} \lesssim \lambda^{\frac{1}{4}} l^{\frac{1}{4}} \|f\|_{L^2 (M)}.
\end{align*}
\end{lemma}
\begin{remark}\label{Remark for subseg of gamma}
\begin{enumerate}
\item In fact, \cite[Lemma 1]{XiZhang2017improved} focuses on the case where $\gamma$ is a geodesic segment, but the argument there can also be considered for our $\gamma$, by using $\rho (\gamma(r), \gamma(s))\approx |r-s|$, which comes from $|r-s|\ll 1$ by a partition of unity if necessary, and \cite[Lemma 4.5]{BurqGerardTzvetkov2007restrictions}.
\item A similar argument also works for a general curve $\gamma$, considering that $\rho (\gamma(r), \gamma(s))\approx |r-s|$ holds for any curve $\gamma$ by using a $C^\infty$ topology argument in \cite[Section 6]{BurqGerardTzvetkov2007restrictions}.
\item As observed in \cite[Remark 1]{XiZhang2017improved}, a similar argument gives the same estimate for $\chi(T_0(\lambda-P))$ if $T_0 \geq 1$.
\end{enumerate}
\end{remark}
Let $T$ be as in \eqref{T Definition}. We show a weak $L^4$ estimate.
\begin{proposition}\label{Prop: a weak L4 estimate}
Suppose $(M, g)$ is a $2$-dimensional compact Riemannian manifold with nonpositive curvatures. Then, for $\lambda\gg 1$, we have
\begin{align*}
\| \chi(T(\lambda-P)) \|_{L^2 (M) \to L^{4, \infty}(\gamma) }\lesssim \frac{\lambda^{\frac{1}{4}}}{(\log \lambda)^{\frac{1}{4}} }.
\end{align*}
\end{proposition}
To show this, we will need a result from B\'erard \cite{Berard1977onthewaveequation}.
\begin{lemma}[\cite{Berard1977onthewaveequation}]\label{Lemma: a bound of Berard}
Let $(M, g)$ be as above. Then there exists a constant $C=C(M, g)$ so that, for $T_0 \geq 1$ and $\lambda \gg 1$, we have that
\begin{align*}
|\chi^2 (T_0 (\lambda-P)) (x, y)|\leq C \left[T_0^{-1}\left(\frac{\lambda}{\rho (x, y)} \right)^{\frac{1}{2}} +\lambda^{\frac{1}{2}} e^{CT} \right].
\end{align*}
\end{lemma}
We now show Proposition \ref{Prop: a weak L4 estimate}.
\begin{proof}[Proof of Proposition \ref{Prop: a weak L4 estimate}]
Assuming $\|f\|_{L^2 (M)}=1$, it suffices to show that
\begin{align*}
|\{x\in \gamma: |\chi(T(\lambda-P)) f(x)|>\alpha\}|\leq C\alpha^{-4} \lambda (\log \lambda)^{-1}.
\end{align*}
By the Chebyshev inequality and Theorem \ref{Theorem Log Improvement}, we have
\begin{align*}
|\{x\in \gamma: |\chi(T(\lambda-P)) f(x)|>\alpha\}|\leq \alpha^{-2}\int_\gamma |\chi(T(\lambda-P))f|^2\:ds \leq \alpha^{-2} \lambda^{\frac{1}{3}} (\log \lambda)^{-1}.
\end{align*}
Note that, for large $\lambda$,
\begin{align*}
\alpha^{-2} \lambda^{\frac{1}{3}} (\log \lambda)^{-1} \ll \alpha^{-4} \lambda (\log \lambda)^{-1},\quad \text{if } \alpha^2 \leq \lambda^{\frac{2}{3}},\quad \text{i.e., } \alpha\leq \lambda^{\frac{1}{3}}.
\end{align*}
We are left to show that
\begin{align}\label{Weak L4 claim}
|\{x\in \gamma: |\chi(T(\lambda-P)) f(x)|>\alpha\}|\leq C \alpha^{-4} \lambda (\log \lambda)^{-1},\quad \text{when } \alpha\geq \lambda^{\frac{1}{3}}, \text{ and } \|f\|_{L^2 (M)}=1.
\end{align}
We set
\begin{align*}
A=A_\alpha=\{x\in \gamma: |\chi (T(\lambda-P)) f(x)|>\alpha\},\quad \text{and} \quad r=\lambda \alpha^{-4} (\log \lambda)^{-2}.
\end{align*}
We consider a disjoint union $A=\cup_j A_j$, where $|A_j|\approx r$. Replacing $A$ by a set of proportional measure, we may assume that $\mathrm{dist}(A_j, A_k)>C_1 r$, when $j\not= k$ for some $C_1>0$, which will be specified later.
Let $T_\lambda: \chi (T(\lambda-P)):L^2 (M) \to L^2 (\gamma)$, and let
\[
\psi_\lambda (x)=\begin{cases}
\frac{T_\lambda f(x)}{|T_\lambda f(x)|}, & \text{if } T_\lambda f(x)\not=0,\\
1, & \text{otherwise.}
\end{cases}
\]
We also write
\begin{align*}
S_\lambda= T_\lambda T_\lambda^*,\quad \text{and } a_j =\overline{\psi_\lambda \mathds{1}_{A_j}}.
\end{align*}
By the Chebyshev inequality and Cauchy-Schwarz inequality, we have
\begin{align*}
\alpha |A|\leq \left|\int_\gamma T_\lambda f \overline{\psi_\lambda \mathds{1}_A}\:ds \right|\leq \left|\int_\gamma \sum_j T_\lambda f a_j\:ds \right|=\left|\int_M \sum_j T_\lambda^* a_j f\:dV_g \right|\leq \left(\int_M |\sum_j T_\lambda^* a_j|^2\:dV_g \right)^{\frac{1}{2}}.
\end{align*}
We can then write
\begin{align*}
\alpha^2 |A|^2\leq \RomanNumeralCaps{1}+\RomanNumeralCaps{2},
\end{align*}
where
\begin{align*}
\RomanNumeralCaps{1}=\sum_j \int_M |T_\lambda^* a_j|^2 \:dV_g,\quad \RomanNumeralCaps{2}=\sum_{j\not=k} \int_\gamma S_\lambda a_j \overline{a_k}\:ds.
\end{align*}
By duality and Remark \ref{Remark for subseg of gamma}, we have that
\begin{align*}
\RomanNumeralCaps{1}\leq C r^{\frac{1}{2}}\lambda^{\frac{1}{2}} \sum_j \int_\gamma |a_j|^2\:ds=C r^{\frac{1}{2}} \lambda^{\frac{1}{2}} |A|=C \lambda \alpha^{-2} (\log \lambda)^{-1} |A|.
\end{align*}
For \RomanNumeralCaps{2}, by Lemma \ref{Lemma: a bound of Berard}, we note that the kernel $K_\lambda (s, s')$ of $S_\lambda$ satisfies
\begin{align*}
|K_\lambda (s, s')|\leq C\left[\frac{1}{T}\left(\frac{\lambda}{|s-s'|} \right)^{\frac{1}{2}}+\lambda^{\frac{1}{2}} e^{CT} \right]=C\left[\frac{1}{c_0 \log \lambda}\left(\frac{\lambda}{|s-s'|} \right)^{\frac{1}{2}} +\lambda^{\frac{1}{2}+Cc_0} \right],
\end{align*}
which in turn implies that
\begin{align*}
\RomanNumeralCaps{2}\leq C\left[\frac{1}{c_0 \log \lambda} \left(\frac{\lambda}{C_1 r} \right)^{\frac{1}{2}}+\lambda^{\frac{1}{2}+Cc_0} \right]\sum_{j\not=k} \|a_j\|_{L^1} \|a_k\|_{L^1} \leq \left[\frac{C}{c_0 C_1^{\frac{1}{2}}} \alpha^2+C\lambda^{\frac{1}{2}+Cc_0} \right] |A|^2.
\end{align*}
We now take $c_0$ to be sufficiently small so that $C\lambda^{\frac{1}{2}+Cc_0}\leq \frac{1}{4}\lambda^{\frac{2}{3}} \leq \frac{1}{4}\alpha^2$, since $\lambda\gg 1$ and $\alpha\geq \lambda^{\frac{1}{3}}$. Given the small $c_0>0$, we take $C_1\gg 1$ so that $\frac{C}{c_0 C_1^{\frac{1}{2}}}\leq \frac{1}{4}$. It then follows that
\begin{align*}
\RomanNumeralCaps{2}\leq \frac{1}{2}\alpha^2 |A|^2.
\end{align*}
Putting these all together, we have that
\begin{align*}
\alpha^2 |A|^2 \leq \RomanNumeralCaps{1}+\RomanNumeralCaps{2}\leq C\lambda \alpha^{-2} (\log \lambda)^{-1}|A|+\frac{1}{2}\alpha^2 |A|^2,
\end{align*}
and thus,
\begin{align*}
|A|\leq C\lambda\alpha^{-4}(\log \lambda)^{-1},\quad \text{if } \alpha\geq \lambda^{\frac{1}{3}},
\end{align*}
which proves \eqref{Weak L4 claim}. This completes the proof of Proposition \ref{Prop: a weak L4 estimate}.
\end{proof}
We are now ready to prove Corollary \ref{Cor: at p=4}. We first recall a special case of a result in Bak and Seeger \cite{BakSeeger2011Extensions}.
\begin{lemma}[\cite{BakSeeger2011Extensions}]\label{Lemma: Bak and Seeger}
Suppose $(M, g)$ is any $2$-dimensional Riemannian manifold. If $\gamma\subset M$ is a curve segment in $M$, then
\begin{align*}
\| \mathds{1}_{[\lambda, \lambda+1]} (P) f\|_{L^{4, 2}(\gamma)}\lesssim \lambda^{\frac{1}{4}} \|f\|_{L^2 (M)},\quad \lambda \geq 1.
\end{align*}
\end{lemma}
We recall some properties of the Lorentz space $L^{p, q}(\gamma)$ (see also Grafakos \cite{Grafakos2014Classical}, etc.). First, for a function $u$ on $M$, the corresponding distribution function $d_u (\alpha)$ with respect to $\gamma$ is defined by
\begin{align*}
d_u (\alpha)=|\{x\in \gamma: |u(x)|>\alpha\}|,\quad \alpha>0.
\end{align*}
The function $u^*$ is the nondecreasing rearrangement of $u$ on $\gamma$, defined by
\begin{align*}
u^* (t)=\inf \{\alpha: d_u (\alpha)\leq t\},\quad t\geq 0.
\end{align*}
For $1\leq p\leq \infty$ and $1\leq q\leq \infty$, the Lorentz space $L^{p, q}(\gamma)$ is then
\begin{align*}
L^{p, q} (\gamma)=\left\{u: \|u\|_{L^{p, q}(\gamma)}:=\left(\frac{q}{p}\int_0^\infty [t^{\frac{1}{p}} u^* (t)]^q\:\frac{dt}{t} \right)^{\frac{1}{q}} <\infty \right\}.
\end{align*}
It is also known that
\begin{align*}
\| \cdot \|_{L^{p, p}(\gamma)}=\| \cdot \|_{L^p(\gamma)},\quad \text{and } \sup_{t>0} t^{\frac{1}{p}} u^* (t)=\sup_{\alpha >0} [d_u (\alpha)]^{\frac{1}{p}}.
\end{align*}
We now take $u=\mathds{1}_{[\lambda, \lambda+(\log \lambda)^{-1}]} (P)f$ with $\|f\|_{L^2 (M)}=1$. By Proposition \ref{Prop: a weak L4 estimate}, we have that
\begin{align}\label{A weak L4 and sup of u}
\sup_{t>0} t^{\frac{1}{4}} u^* (t)\lesssim \|u\|_{L^{4, \infty}} \lesssim \frac{\lambda^{\frac{1}{4}}}{(\log \lambda)^{\frac{1}{4}} }.
\end{align}
Since $\mathds{1}_{[\lambda, \lambda+1]} (P)u=u$, by Lemma \ref{Lemma: Bak and Seeger}, we have that
\begin{align}\label{Est from Bak Seeger}
\|u\|_{L^{4, 2}(\gamma)}\lesssim \lambda^{\frac{1}{4}} \|u\|_{L^2 (M)}\lesssim \lambda^{\frac{1}{4}}.
\end{align}
By \eqref{A weak L4 and sup of u} and \eqref{Est from Bak Seeger}, we have
\begin{align*}
\|u\|_{L^4(\gamma)}=\left(\int_0^\infty [t^{\frac{1}{4}} u^* (t)]^4\:\frac{dt}{t} \right)^{\frac{1}{4}}\lesssim (\sup_{t>0} t^{\frac{1}{4}} u^* (t))^{\frac{1}{2}} \|u\|_{L^{4, 2}(\gamma)}^{\frac{1}{2}} \lesssim \left(\frac{\lambda^{\frac{1}{4}}}{(\log \lambda)^{\frac{1}{4}}} \right)^{\frac{1}{2}} \lambda^{\frac{1}{8}}=\frac{\lambda^{\frac{1}{4}}}{(\log \lambda)^{\frac{1}{8}}}.
\end{align*}
This completes the proof.
\bibliographystyle{amsalpha}
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"redpajama_set_name": "RedPajamaArXiv"
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Q: Maven project works locally but not on docker compose I have a java project that I can run successfully by using
./mvnw spring-boot:run
The same project run through docker compose using the following set of instructions will not run:
Inside docker-compose.yml:
project:
build: ./project
ports:
- 8080:8080
and inside the Dockerfile in the project directory:
ENTRYPOINT ["./mvnw","spring-boot:run"]
Now the interesting part is that I am getting the following exception message when using docker-compose:
InvocationTargetException: Invalid bean definition with name 'redisTemplate' defined in class path resource [****/****/****/****/****/config/RedisContextConfiguration.class]: Cannot register bean definition [Root bean: class [null]; scope=; abstract=false; lazyInit=false; autowireMode=3; dependencyCheck=0; autowireCandidate=true; primary=false; factoryBeanName=****.****.***.***.redis.config.RedisContextConfiguration; factoryMethodName=redisTemplate; initMethodName=null; destroyMethodName=(inferred); defined in class path resource [***/***/**/***/**/config/RedisContextConfiguration.class]] for bean 'redisTemplate': There is already [Root bean: class [null]; scope=; abstract=false; lazyInit=false; autowireMode=3; dependencyCheck=0; autowireCandidate=true; primary=false; factoryBeanName=appConfiguration; factoryMethodName=redisTemplate; initMethodName=null; destroyMethodName=(inferred); defined in class path resource [***/****/***/***/config/AppConfiguration.class]]
I was getting the exact same error before, but I managed to avoid it by adding the following line to the application.properties file:
spring.data.redis.repositories.enabled=false
Now this tells me that somehow the changes made to application.properties are not being seen by docker-compose. Since the container can't run I cannot use docker exec to peruse the files inside and see what's inside the application.properties.
Is there a way to access this .properties inside the image that is being used to create the container, so that I can confirm my suspicions about the property not being there?
Might there be another reason why locally the project runs but not using docker?
Thanks for any assistance.
A: You can try docker run -it your_image_name sh to get a shell that will help you expect your image content.
I see two environment variables, SERVER_PORT and APP_PROPERTIES in the docker command, but you did not declare them in docker-compose.yml.
A: It turns out that the Dockerfile was referencing docker-application.properties instead of the application.properties file.
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"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,758
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Q: Latest version of jQuery hotkeys plugin doesn't accept characters unless using keypress I downloaded the latest version of the jQuery plugin here. I noticed that it doesn't work if I bind it using keydown and pass in something with a character like 'ctrl+u'. I found that this piece of code seems to be preventing it.
character = event.type === "keypress" && String.fromCharCode( event.which ).toLowerCase(),
namely this segment
event.type === "keypress"
This prevents the character from being true and then later binding the modif plus the character further down.
if ( character ) {
possible[ modif + character ] = true;
possible[ modif + jQuery.hotkeys.shiftNums[ character ] ] = true;
// "$" can be triggered as "Shift+4" or "Shift+$" or just "$"
if ( modif === "shift+" ) {
possible[ jQuery.hotkeys.shiftNums[ character ] ] = true;
}
}
I've seen people use this plugin on their site and they don't have the piece where event.type === "keypress" nor the if (character) piece. Is the hotkeys plugin designed to only accept characters with the keypress event? If so, the documentation doesn't say so.
A: The code === 'keypress' is no longer in master - if you re-test and it is still a problem send me a Fiddle and i'll take a look.
|
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"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
Classical Novae (CNe) are cataclysmic variable stars whose outbursts
are due to a Thermonuclear Runaway (TNR) on the surface of a white
dwarf in an interacting binary system \citep[see
e.g.][]{sta08}. Recurrent Novae (RNe) are related to CNe, but have
been seen to undergo more than one recorded outburst and may contain
evolved secondary (mass-donating) stars \citep[see][for a
review]{anu08}. Recurrent novae have been proposed as one of the
primary candidates for the progenitors of Type Ia Supernovae
\citep[SNe - see e.g.][for a recent review]{kot08}.
At present we know of a total of only 10 RNe in the Galaxy with confidence (based on two or more nova outbursts being observed). These RNe
appear to fall into three main groups, {\em viz.}: \\{\em RS Oph/T
CrB} with red giant secondaries, consequent long orbital periods
($\sim$ several hundred days), rapid declines from outburst ($\sim
0.3$ mag day$^{-1}$), high initial ejection velocities ($\ga 4000$ km
s$^{-1}$) and strong evidence of the interaction of the ejecta with
the pre-existing circumstellar wind of the red giant \citep[from
observations of optical coronal lines, non-thermal radio emission
and hard X-ray development in RS Oph; see papers in][]{eb08};\\ The
more heterogeneous {\em U Sco} group with members' central systems
containing an evolved main sequence or sub-giant secondary with an
orbital period much more similar to that in CNe (of order hours to a
day), rapid optical declines (U Sco itself being one of the fastest
declining novae of any type), extremely high ejection velocities
($v_{ej} \sim 10,000$ km s$^{-1}$, from FWZI of emission lines for U
Sco) but no evidence of the extent of shock interactions seen in RS Oph
post-outburst \citep[their post-outburst optical spectra resemble the
`He/N' class of CNe --][]{wil92};\\ {\em T Pyx, CI Aql, IM Nor} are
again short orbital period systems and although their optical spectral
evolution post-outburst is similar, with their early time spectra
resembling the `Fe II' CNe, they show a very heterogenous set of
moderately fast to slow declines in their optical light curves. This
sub-group of RNe also seems to show ejected masses similar to
those at the lower end of the ejected mass range for CNe with $M_{\rm
ej} \sim 10^{-5}$ M$_{\odot}$ (which appears to be one to two orders of magnitude
greater than $M_{\rm ej} $ in the other two sub-groups of RNe).
The short recurrence periods of RNe require high mass WD accretors and
relatively high accretion rates \citep[e.g. ][]{sta88}. Indeed, both
RS Oph and U Sco appear to have WDs near to the Chandrasekhar mass
limit. The WD mass in both these objects has been proposed as growing
and therefore they are potential SN Ia progenitors \citep[see
e.g.][respectively]{sok06,kah99}
The study of RNe is thus important for several broader fields of
investigation including mass loss from red giants, the evolution of
supernova remnants and the progenitors of Type Ia SNe. Progress in
determining the latter association in particular, as well as exploring
the evolutionary history of these close binary systems, is hampered by
the relative rarity of Galactic RNe. However, since the time of Edwin
Hubble, CNe have been observed in extragalactic systems, in particular
M31 \citep[see][for a review]{sha08}. In total over 800 CN candidates
have been catalogued in M31 \citep{pie07a} and among these are thought
to lie several RNe \citep[see e.g.][]{del96, sha09b}. Indeed
\cite{pie07a} identified 4 candidates in their search for the X-ray
counterparts of optical novae in M31 \citep[see also][]{hen09}. In
this paper we present evidence for an object in M31 previously
classified as being a CN as in fact being a recurrent nova. We use a
combination of optical and X-ray observations to explore its more
detailed nature, emphasise the need for careful exploration of
archival material to confirm or rule out previous outbursts, and go on
to point the way to more extensive observational programs in the
future.
\section{Observations of the 2007 Outburst}
Nova M31N 2007-12b was discovered on 2007 December 9.53 UT (which we
take as $t = 0$) by K. Nishiyama and
F. Kabashima\footnote{http://www.cfa.harvard.edu/iau/CBAT\_M31.html}
at mag $= 16.1-16.2$ (unfiltered) and located at RA = 00h 43m
19s.94$\pm$0s.01, DECL $= +41\degr 13' 46''.6 \pm 0''.1$ (J2000). They
reported that no object had been visible at this position on 2007
December 8.574 UT. Fig.\ref{fig1} gives details of these and other
optical observations around peak. Broadband $i', V, B$ plus narrowband
H$\alpha$ photometry was subsequently obtained with the RATCam CCD
camera on the 2-m Liverpool Telescope \citep[LT; ][]{ste04}. LT
photometry is part of a larger program of photometry and spectroscopy
of novae in M31 \citep{sha09} and began on 2007 December 14.94 UT ($t
= 5.4$ days post-outburst) then continued for 23 days. The LT data
were reduced using standard routines within the IRAF\footnote{IRAF is
distributed by the National Optical Astronomy Observatory, which is
operated by the Association for Research in Astronomy, Inc. under
cooperative agreement with the National Science Foundation.} and
STARLINK packages, and calibrated against standard stars from
\cite{lan92} and by using the secondary standards in M31
\citep{mag92,hai94}. The resulting lightcurves are shown in
Fig.\ref{fig1}.
The astrometric position of M31N 2007-12b was measured from an LT Sloan
$i'$-band image taken on 2007 December 14.95 UT. This image was chosen as
a compromise between good seeing and nova brightness. An
astrometric solution was obtained using 21 stars from the 2MASS
All-Sky Catalogue \citep{2003tmc..book.....C} which are coincident with
resolved sources in the LT observation. We obtain a position for M31N
2007-12b of RA = 00h 43m 19s.97$\pm$0s.01 DECL $= +41\degr 13' 46''.3
\pm 0''.1$ (J2000; consistent with Nishiyama and Kabashima's measurement). It should be
noted that the astrometric uncertainty is dominated by uncertainties
in the plate solution.
Optical spectroscopy was obtained by us on 2007 December 15.2 UT ($t =
5.7$ days) with the 9.2-m Hobby Eberly Telescope (HET) using the Low
Resolution Spectrograph \citep [LRS;][]{hil98}. We used the $g$1
grating with a $1.0''$ slit and the GG385 blocking filter, which
covers $4150-11000$ \AA ~with a resolution of $R \sim 300$, although
we limit any analysis to the $4150-8900$ \AA ~range where the
effects of order overlap are minimal. Data reduction was performed
using standard IRAF packages and the resulting spectrum is shown in
Fig.\ref{fig2}.
\cite{kon07} reported the detection of a Super-Soft X-ray Source (SSS)
coincident with the position of the nova using the X-ray Telescope
(XRT) on board the $Swift$ satellite \citep{bur05}. The detection was
made serendipitously as part of a survey of SSSs in the M31 globular
cluster Bol 194 on 2008 January 13.74 UT ($t = 35.2$ days) with an
exposure time of 4 ks. They reported previous observations of the
field on 2007 December 16 and December 30 that had not detected any
source at that position. We have re-analysed the XRT data for these
epochs and also consulted the {\em Swift} data archive to review other X-ray
observations of this field from 2007 November to 2008 May (see Table
\ref{xrt} and also Fig.\ref{fig1}).
\begin{table}
\begin{center}
\caption{~{\em Swift} XRT data. Upper limits are at the 90\% confidence level; error on the detection is $1 \sigma$.}
\begin{tabular}{lccc}
\hline \hline
Date (day) & Obs ID & Exposure & count rate\\
& & time (ks) & (s$^{-1}$) \\
\hline
2007-11-24 (-15)& 00031027001 & 7.27 & $<$0.0017 \\
2007-12-02 (-7)& 00031027002 & 1.00 & $<$0.0073 \\
2007-12-03 (-6)& 00031027003 & 3.63 & $<$0.0037\\
2007-12-16 (+7)& 00031027004 & 3.89 & $<$0.0039 \\
2007-12-30 (+21)& 00031027005 & 4.02 & $<$0.0034 \\
2008-01-13 (+35)& 00031027006 & 3.99 & 0.015 $\pm$ 0.002\\
2008-05-26 (+169)& 00037719001 & 4.86 & $<$0.0023\\
\hline
\end{tabular}
\label{xrt}
\end{center}
\end{table}
\section{Results and Discussion}
Nova M31N 2007-12b lies within $1.7''$ of the quoted position of M31N
1969-08a (RA = 00h 43m 19s.9 $\pm 0s.3$, DECL $ = +41\degr 13' 45''
\pm3''$ (J2000); i.e. coincident within the quoted measurement errors)
which was discovered on 1969 August 16.0 UT \citep[see][]{sha91} and
lies $7.1'$ from the nucleus of M31. Peak visual magnitude was
observed one day after the start of the 1969 outburst at $V =
16.4$. Subsequently, the nova declined at a rate of $\ga 0.3$ mag
day$^{-1}$ making this a very fast nova \citep{war08}. Supposed
positional coincidence and similarities in their light curves led to the initial conclusion that the
outbursts were from the same object. However, consultation of the
original plate material for M31N 1969-08a showed that its position is
in fact RA = 00h 43m 19s.6 $\pm0s.1$, DECL $ = +41\degr 13' 44''\pm1'' $ (J2000, i.e. separated by $4.8''\pm1.5''$ from M31 2007-12b) and blinking of the 1969
and 2007 images confirmed they are indeed separate objects (see Fig.\ref{finder}).
Our optical spectroscopy on day 5.7, shown in Fig. \ref{fig2}, reveals
strong and very broad (FWHM H$\alpha \simeq 4500$ km s$^{-1}$) Balmer,
He I and N III 464.0 nm emission lines consistent with the spectra of
He/N CNe \citep{wil92}. High emission line velocities and fast optical
declines are associated with ejection from a high mass WD and are also
typical of both the RS Oph and U Sco sub-classes of RNe
\citep{anu08}. Of these two, the spectrum more closely resembles that
of RS Oph around 3 days after the 2006 outburst (see Fig. \ref{fig2}),
than that observed in U Sco or the U Sco sub-class RN V394 CrA at
similar phases after their outbursts in 1987
\citep{sek88,sek89,wil91}. However, the most striking spectral
similarity is to the early optical spectrum of nova V2491 Cyg (again,
see Fig. \ref{fig2}) for which, although only one outburst has been
observed, it has been suggested that it is a RN \citep{pag09} by
virtue of its very fast optical decay and high ejection velocities
together with its low outburst amplitude \citep[$\Delta V = 8.5$
mag,][]{jur08} and detection as an X-ray source pre-outburst
\citep{iba09}.
\subsection{Constraints from the X-ray data}
Turning now to the X-ray spectra, we re-analysed the {\em Swift} detection
on day 35.2 referenced by \cite{kon07} using the {\em Swift} software
version 2.9. Source spectra were extracted from the cleaned Photon
Counting mode event lists, using a 10 pixel extraction radius (1 pixel
$= 2.36''$). A total of 49 background-subtracted counts were found
with only one count at $\sim 0.9$ keV and the rest at lower
energies. As an initial guide this super-soft spectrum was then fitted
with an absorbed black body spectrum using XSPEC. We estimated the
absorbing column as follows. \cite{sta92} derive a Galactic
contribution to the column in this direction equivalent to $E_{B-V}$ =
0.1. At the position of M31N 2007-12b in M31, following the
methodology discussed in \cite{dar06} and Section 3.2 below, and
assuming that the nova is situated half way down the absorbing column
internal to M31, we get $E_{B-V}$ = 0.25. Thus the total extinction to
the nova is estimated to be equivalent to $E_{B-V}$ = 0.35, which in
turn is equivalent to $N_{\rm H} = 2.1\times10^{21}$ cm$^{-2}$. The
best fit to the data using this total column then gives $kT =
63^{+10}_{-8}$ eV (i.e. $T = 7.6\pm{1.2} \times 10^5$K) and for $d =
780$ kpc to M31 \citep{hol98, sta98} yields an absorption-corrected
luminosity $L = (4.5^{+1.9}_{-1.4}) \times10^{38}$ ergs s$^{-1}$
(i.e. around twice the Eddington luminosity for a 1.4 M$_\odot$
WD). We obtained non-detections at the source position in 2007
November/December and 2008 May as detailed in Table \ref{xrt}.
From the above observations with {\em Swift}, the SSS was not detected $\sim
15$, 7 and 6 days before outburst and at 7, 21 and 169 days
afterwards. We can estimate therefore that the SSS appeared between
$t \sim 21$ and 35 days post-outburst and had turned off again at $t <
169$ days. A caveat here is that the onset of the SSS phase has shown extreme variability in a few objects so far \citep[e.g. RS Oph; see][]{pag08} and there is the possibility that the first emergence was earlier than 21 days.
We can however compare the observed behavior of the M31N 2007-12b SSS with the properties of this phase in possibly
related Galactic novae. For example, we note that the appearance of
the SSS in U Sco was around 19-20 days after the peak of the optical outburst in February 1999 \citep{kah99}.
We used the model parameters for U Sco found by \cite{kah99} to generate a spectrum with the correct unabsorbed flux. In order to determine the predicted count rate in the {\em Swift} XRT if the source were placed in M31 at $d = 780$ kpc, the absorbing column was changed to $N_{\rm H} = 2.1\times10^{21}$ cm$^{-2}$ but the normalization and derived $kT$ were kept fixed. Finally, a new spectrum was generated to derive
the predicted count-rate. Table \ref{novae3} gives details of the parameters of this and other sources described below, together with their derived count rates.
It can be seen from Table \ref{novae3} that with the
spectral parameters given in \cite{kah99}, the SSS emission
seen in U Sco ($d = 14$ kpc) would not have been
detectable by $Swift$ at the distance of M31 in the exposure times used
for M31N 2007-12b, although of course the U Sco SSS may have
subsequently increased in brightness. In RS Oph ($d =1.6$ kpc) the SSS
emerged and then dominated the X-ray emission from $t \sim 29$ days and
turned off by $\sim 90$ days \citep{pag08}, i.e. consistent with the
timescales in M31N 2007-12b. However, as can be seen from Table \ref{novae3}, even at the peak of its SSS
emission, RS Oph would also have been undetected in M31 with
the $Swift$ XRT. In V2491 Cyg ($d =
10.5$ kpc), SSS emission became apparent after around 25 days
\citep{pag09} and was sharply peaked at around 40 days. At the
distance of M31, the {\em Swift} XRT observations reported here would have
detected the V2491 Cyg SSS for a few days around this peak (again, see Table \ref{novae3}).
\begin{table*}
\begin{center}
\caption{~Parameters used to derive predicted Swift count rates for the SSS phase in other novae if at the distance and absorbing column of M31N 2007-12b (see text for details).}
\begin{tabular}{lcccccl}
\hline \hline
& $d$ & $N_{\rm H}$ & kT$_{BB}$ & Unabsorbed Flux & Predicted Count Rate \\
& (kpc) & (cm$^{-2}$) & (eV) & (ergs s$^{-1}$ cm$^{-2}$) & (s$^{-1}$) \\
\hline
U Sco$^{\rm a}$ & 14 & $2.2\times10^{21}$ & 107 & $5.4\times10^{-10}$ & $1\times10^{-3}$ \\
RS Oph$^{\rm b}$ & 1.6 & $3.4\times10^{21}$ & 70 & $6.3\times10^{-8}$ & $1.1\times10^{-3}$ \\
V2491 Cyg$^{\rm c}$ & 10.5 & $3.4\times10^{21}$ & 52 & $2.4\times10^{-8}$ & $9.4\times10^{-3}$ \\
\hline
\end{tabular}
\label{novae3}
\end{center}
$^{\rm a}$ From \cite{kah99} and assuming the unabsorbed flux they quote is for the 0.1-10 keV energy range of the LECS/MECS of {\em BeppoSAX}.\\
$^{\rm b}$ Fit to the {\em Swift} XRT data from day 50.5 after outburst during the SSS `plateau' phase. The unabsorbed flux is for the 0.3-10 keV energy range of the XRT. The $N_{\rm H}$ value used in the fit includes both interstellar and circumstellar components \citep[see][]{pag08}.\\
$^{\rm c}$ Fit to the {\em Swift} XRT data from day 41.7 after outburst around the observed SSS peak count rate \citep{pag09}. The unabsorbed flux is for the 0.3-10 keV energy range of the XRT.
\end{table*}
It is well established that the SSS arises from continued nuclear
burning on the WD surface following the TNR which is gradually
unveiled as the ejecta move outwards \citep{kra08}. The deduced
temperature and luminosity of the SSS in the case of M31N 2007-12b are
consistent with this model. Simplistically, the timescale for
uncovering and observed onset of the SSS phase is given by $t_{on}
\propto M_{\rm H}^{1/2} v_{ej}^{-1}$ \citep{kra96} where $M_{\rm H}$
is the mass of H in the ejected envelope and $v_{ej}$ is the ejection
velocity. Thus for the low ejected masses and high ejecta velocities
found in RS Oph-type and U Sco-type RNe, $t_{on}$ would be expected
to be relatively short compared to that for the T Pyx sub-class of RNe or its value for most CNe.
The turn-off time since outburst for nuclear burning, $t_{\rm rem}$,
is a steep function of WD mass. \cite{mac96} finds for example
$t_{\rm rem} \propto M_{\rm WD}^{-6.3}$. Generally in CNe this
timescale is much longer than that observed in M31N 2007-12b
\citep[although][make the point that this might be ascribed in part to
a selection effect]{pie07a}. For example, in one of the best studied
cases, the moderately fast CN V1974 Cyg, $511 < t_{\rm rem} < 612$
days \citep{bal98} with $M_{\rm WD} \sim 1$ M$_\odot$
\citep{hac06}. From \cite{sta91}, $t_{\rm rem} < 169$ days implies
$M_{\rm WD} \ga 1.3~$M$_{\odot}$. Similarly, the timescale after
outburst for the onset of the SSS phase, $t_{\rm on}$, is also a
function of $M_{\rm WD}$ in the sense that $t_{\rm on}$ is likely to
be shorter for systems containing a high mass WD. As noted above, both
U Sco and RS Oph have very short observed $t_{\rm on}$ of $\lesssim
20$ days \citep{kah99} and $\lesssim 30$ days \citep{bod06}
respectively. In both cases, the WD mass is determined to be
approaching the Chandrasekhar mass limit of M$_{\rm Ch} \sim
1.4$ M$_{\odot}$ \citep{kah99,hac07}. Similarly, the WD mass in V2491
Cyg is estimated to be $> 1.3$ M$_{\odot}$ \citep{hac09,pag09}. In
addition, we note that envelope composition may also be important in
determining the duration of the SSS phase. However, with $t_{on} <
35$ days and $t_{rem} < 169$ days, $M_{\rm WD} > 1.3$ M$_\odot$ for
the range of envelope compositions presented in \cite{hac06} and in
particular, $M_{\rm WD} > 1.35$ M$_\odot$ for the cases of (fast)
Neon novae they present.
\subsection{A search for the progenitor system}
If M31N 2007-12b arose from a RN system of the RS Oph sub-type, it
would contain a red giant secondary. We thus explored its detection
at quiescence in archival {\it Hubble Space Telescope (HST)}
imagery. The {\it HST} is capable of resolving giant branch stars
within M31 (see Fig.\ref{fig4}). The positions of both M31N
2007-12b and M31N 1969-08a lie within a pair of archival {\it HST}
Advanced Camera for Surveys (ACS) Wide Field Channel (WFC) images
(prop. ID 10273) taken in August 2004 using the F814W ($\sim I$) and
the F555W ($\sim V$) filters. PSF fitting photometry was performed on
all detected objects in both {\it HST} pass-bands using DOLPHOT, a
photometry package based on HSTphot \citep{2000PASP..112.1383D}. We
used the relations given in \cite{2005PASP..117.1049S} to transform
from these filters to Johnson-Cousins $V$ and $I$.
To isolate the position of M31N 2007-12b within the {\it HST} data we
computed the spatial transformation between the {\it LT} and Gaussian
convolved {\it HST} data using 23 stars resolved and unsaturated in
both images. This approach is independent of the astrometric calibration of both fields and hence yields the most accurate results. The uncertainty in the derived transformation is small
when compared to the 0.22 pixel ($0.06^{\prime\prime}$) average positional
error of the nova in the {\it LT} data. This positional uncertainty
in the {\it LT} data equates to a 1.25 pixel positional uncertainty
($1\sigma$) within the {\it HST} data.
There is a resolved object just inside $1\sigma$ from the {\it LT}
position (separated 1.12 {\it HST} pixels or $0.89\sigma$) seen in the
{\it HST} F555W image (see inset of Fig.\ref{finder}). We find that this object has $V=24.61\pm0.09$
and $I=22.33\pm0.04$, hence a color of $V-I=2.3\pm0.1$. It should be
noted that there is a cosmic ray track very close to this object's position
in the F814W image, hence the $I$-band photometry may have been
adversely affected by the subtraction of the cosmic ray. There are no
other resolved stars within 1.90 {\it HST} pixels or $1.52\sigma$.
Shown in Fig.\ref{fig4} is the position on a color-magnitude diagram of the object spatially coincident with M31N
2007-12b. This object (assuming no additional internal M31 extinction)
lies in the M0/M2III (RS Oph secondary, purple and dark blue dots respectively) and M3III (T CrB secondary, light blue dots) region of the
Giant Branch. The probability of finding such a star ($20<I<23$,
$1.5<V-I<2.5$) at least as close to the predicted position by chance
is only 3.4\%. We note as an aside that we have explored the region around M31N 1969-08a and found no significant spatial coincidence with any pre-existing stellar source.
We estimate the mean $I$-band extinction across an Sb galaxy, such as M31,
to be $A(I)=0.8$ magnitudes, equivalent to $E_{B-V}=0.54$
\citep{2005AJ....129.1396H}. However, we can estimate that the
average extinction experienced by an object at this position in M31
would be $A(r')=0.7$ magnitudes, equivalent to $E_{B-V}=0.25$
\citep[][see above]{2005PhDT.........2D}.
We also calculate the position of a quiescent RS Oph system on this
diagram. We use the LT $V$ and $i'$ luminosities of RS Oph in the
time range of 400-1300 days following the 2006 outburst
\citep[see][for days 400-600]{2009ASPC..401..203D} to estimate the mean quiescent magnitudes,
$<V>=11.03\pm0.03$, $<i'>=9.34\pm0.02$. These magnitudes were then corrected for
the extinction towards RS Oph
\citep[$E_{B-V}=0.7\pm0.1$,][]{1987rorn.conf...51S} and the distance to RS Oph \citep[$d=1.6\pm0.3$\ kpc,][]{1987rorn.conf..241B}. The Sloan-$i'$ flux was transformed to the Johnson-Cousins system, the system
was placed at the distance of M31 and reddened by an amount equal to
the extinction towards that galaxy. We find that the expected mean
quiescent magnitude of an RS Oph-like system in M31 (without any
internal extinction) is $<I>=21.0\pm0.5$ with a color of
$<V-I>=1.3\pm0.4$. Further correcting for the expected average internal
M31 extinction yields, $<I>=21.5\pm0.5$, $<V-I>=1.8\pm0.4$. We note that these values of quiescent magnitudes and colors include contributions from other sources than the secondary (e.g. any accretion disk).
\section{Conclusions}
M31N 2007-12b shows several characteristics consistent with it being a recurrent nova. These include the rapidity of its optical decline, extremely high ejection velocities and early emergence of its SSS phase. The early post-outburst optical spectrum also shows some similarities to that of RS Oph, but most closely resembles that of the proposed RN V2491 Cyg. Furthermore, we have found a coincident pre-outburst stellar source from archival {\it HST} observations that resides in the same region of the color-magnitude diagram as RS Oph. If this is indeed the quiescent nova system, this is the first time that this has been identified in a nova in M31. This finding also implies an outburst amplitude of $\Delta V \simeq 8.5$ mag, very similar to that given by \cite{jur08} for nova V2491 Cyg, although around 1 mag greater than that for RS Oph. The observed flux from the SSS detected in M31N 2007-12b is also more consistent with that of the short-lived peak at around 40 days in V2491 Cyg than that of the SSS in RS Oph. On the other hand, the secondary maximum reported in the light curve of V2491 Cyg at around $t = 15$ days is not apparent in our LT data for M31N 2007-12b.
Among the Galactic RNe, both U Sco and
RS Oph sub-types have been proposed as progenitors of Type Ia SNe as $M_{\rm WD}
\sim ~$M$_{\rm Ch}$ and it has been concluded that there is a net
accumulation of mass on the WD over time \citep{kah99,hac07}.
Determination of the true nature of Type Ia progenitors is of course a
very important quest for contemporary astrophysics, but still remains
a controversial area. recurrent novae have been one of the favored
systems, but there are likely to be more problems in explaining the lack of H in SNIa spectra for RS Oph-like than for U Sco-like RNe. Certainly, the paucity of Galactic examples remains a hindrance to
further progress. We have shown that it is now possible to
identify RNe in M31, and even to determine their
sub-type, via a suitable set of complementary
observations. However, from our experience we caution that identifying RNe from positional (near) coincidence of two or more outbursts can be precarious \citep[e.g., see][]{sha09b}. All RN candidates should be thoroughly explored through precise astrometry of the original images, where available. Furthermore, ambiguities of distance and host stellar
population are negated for novae in M31, and the soft X-ray absorbing column is low, compared to their Galactic
counterparts. Thus the prospects are good for extending our studies of
RNe, and in particular exploring any relationship to supernovae, from
the Milky Way to potentially a much larger and better-defined sample
of objects in the Andromeda galaxy.
\acknowledgments
We are grateful to M. Shetrone and A. Westfall for help with
obtaining the HET spectroscopy and to S. Starrfield for suggesting use of the HST observations. The authors also wish to thank two anonymous referees for very helpful comments on the initial versions of the manuscript. The Liverpool Telescope is operated on the island of La
Palma by Liverpool John Moores University in the Spanish Observatorio
del Roque de los Muchachos of the Instituto de Astrofisica de Canarias
with financial support from the UK Science and Technology Facilities
Council. AWS acknowledges support through NSF grant AST-0607682.
{\it Facilities:} \facility{Hubble Space Telescope}, \facility{Hobby Eberley Telescope}, \facility{Liverpool Telescope}, \facility{Swift}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,012
|
We would like to take this time extend our warmest welcome to the our website. Whether you are looking for information on planning a funeral, grief support, or general information on funeral arrangements please feel free to browse our website. If you have any questions, we are available by phone, email or in person.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,288
|
require 'fetcher'
require 'net/http'
require 'webmock'
include WebMock::API
describe Fetcher do
describe 'on create' do
it 'takes an url and a destination on creation do' do
fetcher = Fetcher.new('http://localhost/', 'test_dir/test_file')
expect(fetcher).to be_a_kind_of(Fetcher)
end
end
describe 'retrives a remote gem' do
before(:each) do
@fetcher = Fetcher.new('http://localhost/', 'test_dir/test_file')
end
it 'returns the path to the file when it exists' do
File.stub(:exists?).and_return(true)
destination = @fetcher.fetch
expect(destination).to eql('test_dir/test_file')
end
it 'fetches the file and returns the file when it doesn\'t exist' do
pending
File.stub(:exists?).and_return(false)
end
end
describe '#fetch_gem' do
before(:each) do
@fetcher = Fetcher.new('http://localhost/', 'test_dir/test_file')
end
it 'returns the path to the gem on HTTPSuccess' do
pending
stub_request(:any, 'http://localhost/')
.to_return(
body: 'abc', status: 200, headers: { 'Content-Length' => 3 })
@fetcher.should_receive(:write_to_file).and_return 'path'
@fetcher.fetch_gem
end
end
describe '#write_to_file' do
before(:each) do
@destination = 'test_dir/test_file'
@fetcher = Fetcher.new('http://localhost/', @destination)
end
it 'opens the destination file in w+ mode and writes to it' do
file = double(File)
response = double
@fetcher.should_receive(:write).and_return @destination
File.should_receive(:open)
.and_yield(file)
expect(@fetcher.write_to_file(response)).to eql(@destination)
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,889
|
Math tutoring is available in the Learning Center (Room 3-203) for all levels of math offered at RCC. Individual appointments can be made at the Learning Center or through https://rcc.mywconline.net. Tutoring is available in Basic Math, Algebra, Statistics, Pre-calculus and Calculus. Additional tutoring is available for the math encountered in science courses, nutrition and calculations for nursing. Students can make individual appointments or come with a classmate to a math tutoring session.
Study Groups in Algebra, Statistics and Pre-Calculus are offered each semester and facilitated by math tutors from the Learning Center.
Students interested in an independent math boot camp for an assessment of math skills and a self-paced review of arithmetic and beginning algebra, please contact Joyce Atkinson, Coordinator of Learning Resources in the Learning Center (3-203) or send an email for a one hour appointment to jatkinson@rcc.mass.edu.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,264
|
Q: Is there redundancy in saying that something "can be a potential risk"? I often read of "potential risks". This moved further into the realm of uncertainty with "can be a potential risk" in a recent, scientific magazine. Given that measurements of risk incorporate the less than 100% likelihood (which could be arbitrarily low) of an outcome occurring, what differentiates a "potential risk" from an actual one?
A:
what differentiates a "potential risk" from an actual one
Adding 'potential' could indicate Knightian uncertainty (ignorance, unknowability), in addition to or instead of quantifiable risk. In other words, a 'potential risk' is one that is still unknown (could be zero probability), whereas a 'normal' risk has a known probability strictly greater than zero.
A: I've wondered this before, as well. I wasn't able to find anything on the internet that directly addressed this problem so I'm approaching intuitively. In my mind 'potential risk' is potential because the action is potential. If you are thinking about blowing the whistle on something illegal going on in your company, you could potentially be risking your career. The minute you do blow the whistle, you are risking your career. Until you actually go to the press you haven't taken the risk, so it's potential.
A: Oy gott. "Potential risk" and "potential danger" are grossly redundant. "Danger" denotes the possibility of harm. To say something is "potentially dangerous" means there is the possibility of the possibility of harm. To say "can be potential risk" means there's the possibility of the possibility of the possibility of harm. If this isn't the matruschka reality you're describing, just say "It's risky" or "It's dangerous."
A: In most cases it would be redundant, but in the example provided by Waywardeevee, and in other examples of that character, I agree that it is not.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,910
|
The 2016 Worlds is gearing up to be one of the best of all times. The black belt division alone features a wide array of big names, many of which still achieving awesome results in the adult division. Looking at the list of athletes in each division, we can only hope to witness some awesome matches.
We decided to list seven of the best that can take place in Las Vegas, this weekend.
Another talent-packed division with big names. Osvaldo Moizinho is on one side of the bracket and Frazatto and Braga on the other side. Either way, great matches are coming our way.
Barbosa is a household name in the Worlds Masters that also have great results in the adult division. Same with Marcelo Mafra. If they meet in the final, it should be great.
That's a full blown classic right there. Five times world champion Barral and world champion Humphreys could meet in the final. Gracie Barra's Roberto Alencar is also in the mix.
To close out with teammate Rafael Lovato Jr., Xande will have to go through Kelly in a possible semifinal. Watch out.
Alvarez could not be a more fierce competitor while Soca is always a sight to see on the mats. Should be a great final.
Saulo is a legend. On the other side, if you don;t know Lazzarini that well, you should. He is as good as you could be and a clash between the two would be awesome.
Two team leaders, Monteiro and Medeiros could not be more purebred when it comes to lineage in Jiu-Jitsu. Another great match up to be on the look out for.
There are many other great matches bound to happen this weekend in las Vegas. That's why you should follow the best coverage on Jiu-Jitsu Magazine's social media channels. Coverage brought to you by Gameness.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,749
|
Get to Know Altur Santos, Rising Reggaeton Urbano Talent, and Singer
Connect with Muzique Magazine @
The recent trend in reggaeton and Latin culture in North America has taken the music industry by storm. From aspirations of J Balvin, Shakira and Daddy Yankee, to the recent growth of mainstream Latin music to the breath-taking Super Bowl performances by Shakira and Jennifer Lopez shows an upward growth in the love and passion for this genre of music.
One can say, Latin culture is here to stay and the rise of Latin music is at the forefront. One of those artists is Robert Santos, professionally known as Altur Santos. Born and raised in the Dominican Republic, the 26-year-old singer, saxophone player, and songwriter has released his debut single Baby Love. A romantic song that encompasses passion, childhood love and the ability to love through the years. We sat down for a free-wheeling chat with this talented, upcoming music star.
What is the daily routine like?
From the moment, I wake up, I am making projects, designing, recording videos as a hobby, talking to my manager and planning the next moves.
What is your creative process?
There is no guideline, as I get in the mood, and I hear a beat that I like I will go for it. I make sure that I am in flow make sure I get music, but the most important thing is, have fun with it.
You're hugely popular in the Dominican Republic. Any fan encounter you'd like to share?
I was playing on the stage once, when I get off the stage, I got attacked by a few fans but I was playing saxophone, they were trying to kiss me but I had to run into the bus. They started shaking the bus and the bus driver was so afraid he had to shut the doors, there was no security there because we weren't expecting it to happen.
What has your experience in North America been like, so far?
Its been really tough for me because it's not my country, but I am breaking through it, I learned that no matter where you, you just have to follow your dream.
Connect with Altur Santos on social media:
Instagram:https://www.instagram.com/altursantos
Facebook: https://www.facebook.com/altursantos
Davalenco live at Viva Mexico!
Related Topics:Altur Santos, Best indie magazine, Best online music magazine, Contributor Alfred Munoz, Dominican reggaeton, Dominican republic, Featured, independent artist, Independent Music Promotions, latino, latino music, music magazine, music submissions, Music Submit, Muzique Magazine, reggaeton
Interview with Nu-Soul/Nu-Pop diva Ava Cherry
Aria Wunderland talks about her style, musical influences and more
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 577
|
\section{Introduction}
We collaborated with John A. Coleman for 20 years, since 1990 till his death
in 2010. Moreover, he was not merely our co-author, but during this time, he
became our very close friend, with whom we spent many hours discussing various
problems. We are keeping warmest memories of John and his wife Marie Jeanne.
With great respect, we devote this paper to the memory of our friend and
colleague John Coleman.
One of the interesting ideas, we developed with John Coleman, was the notion
of the order indices for density matrices, introduced in Ref. [1]. The
properties and applications of the order indices to different types of bulk
matter, considered in thermodynamic limit, were studied in Refs. [2-5] and
summarized in the book [6]. For infinite systems, however, it is possible to
define the standard order parameters (see, e.g., [7,8]), because of which the
use of the additional notion of the order indices could seem to be
unnecessary. Nevertheless, as has been shown [1-3], the order indices are
useful even for infinite systems, where a kind of mid-range order arises.
The principal difference of the order indices from the order parameters is that
the former can be introduced not only for infinite systems, but also for finite
systems. In recent years, the investigation of finite systems has become of high
importance due to the widespread technological applications of various finite
objects. As examples, we can mention quantum dots, metallic grains, different
granular materials, nanoclusters, trapped atoms, and a variety of macromolecules,
including biomolecules. For finite systems, as is well known, the order parameters
are not defined [7,8]. In that case, the order indices can become principally
important, as far as they can be defined for systems of any size. Then the order
indices could characterize the amount of order specific for finite systems. It is
the aim of the present paper to extend the definition and application of the order
indices for finite systems.
In Sec. 2, we introduce the notion of order indices for arbitrary operators, which
is specified for generalized density matrices in Sec. 3. The application to the
usual reduced density matrices of statistical systems is given in Sec. 4, without
invoking thermodynamic limit. In Sec. 5, we exemplify the consideration for
bosonic atoms trapped in a finite box. Calculations for the order index of the
first-order density matrix are given in Sec. 6, where the order index is treated
as a function of the number of particles and of atomic interactions. Section 7
concludes.
\section{Operator order indices}
Order indices can be introduced for operators of any nature [9]. Let $\hat{A}$
be an operator acting on a Hilbert space $\mathcal{H}$. The operator is assumed
to possess a norm $||\hat{A}||$ and a trace ${\rm Tr}\hat{A}$, with the trace
taken over the space $\mathcal{H}$. In all other aspects, it can be arbitrary.
The {\it operator order index} is
\be
\label{1}
\om(\hat A) \equiv \frac{\log||\hat A||}{\log|{\rm Tr}\hat A|} \; .
\ee
The logarithm can be taken with respect to any base, since
$\log_a z = \log_b z / \log_b a$. This definition connects the norm and trace of
the operator through the relation
\be
\label{2}
||\hat A|| = |{\rm Tr}\hat A|^{\om(\hat A)} \; .
\ee
An operator $\hat{A}_1$ is said to be better ordered than $\hat{A}_2$, if and
only if
\be
\label{3}
\om(\hat A_1) > \om(\hat A_2) \; .
\ee
Respectively, two operators, $\hat{A}_1$ and $\hat{A}_2$ are equally ordered,
provided that $\omega(\hat{A}_1) = \omega(\hat{A}_2)$.
The operator norm can be defined as a norm associated with the vector norm
$|\varphi|$ for a nonzero vector $\varphi \in \mathcal{H}$, so that
\be
\label{4}
||\hat A|| = \sup_\vp \; \frac{|\hat A\vp|}{|\vp| } \; .
\ee
Employing the scalar product $(\varphi,\varphi)$ for defining the norm yields
the Hermitian vector norm $|\varphi| \equiv \sqrt{(\varphi,\varphi)}$. The
corresponding {\it Hermitian operator norm} is
\be
\label{5}
||\hat A|| = \sup_\vp \; \left [ \frac{(\hat A\vp,\hat A\vp)}{(\vp,\vp)}
\right ]^{1/2} = \sup_\vp \left [ \frac{(\vp,\hat A^+\hat A\vp)}{(\vp,\vp)}
\right ]^{1/2} \; .
\ee
For an orthonormalized basis $\{\varphi_k\}$, labelled with an index $k$,
in $\mathcal{H}$, such that
\be
\label{6}
\cH = {\rm Span}_k \{ |\vp_k \rgl \} \; ,
\ee
the Hermitian norm (5) becomes
\be
\label{7}
||\hat A|| = \sup_k \left [ (\hat A\vp_k,\hat A\vp_k )
\right ]^{1/2} \; .
\ee
If the operator $\hat{A}$ is self-adjoint, then its Hermitian norm
simplifies to
\be
\label{8}
||\hat A|| = \sup_\vp \; \frac{|(\vp,\hat A\vp)|}{|\vp|} =
\sup_k | (\vp_k,\hat A\vp_k ) | \; .
\ee
The eigenfunctions of a self-adjoint operator, defined by the eigenproblem
$$
\hat A\vp_k = A_k \vp_k \; ,
$$
form an orthogonal basis that can be normalized. The space basis can be
chosen as the set of these eigenfunctions of the considered operator. Then
the Hermitian norm becomes the {\it spectral norm}
\be
\label{9}
||\hat A|| = \sup_k | A_k | \; .
\ee
When the operator is semi-positive, then $|A_k| = A_k$.
For a semi-positive operator,
$$
||\hat A|| \leq {\rm Tr}\hat A \qquad (\hat A \geq 0 ) \; .
$$
Therefore, for such an operator,
\be
\label{10}
\om(\hat A) \leq 1 \qquad (\hat A \geq 0) \; .
\ee
In that way, the order index (1) makes it possible to characterize the level
of order in operators and to compare the operators as being more or less
ordered. Instead of the Hermitian norm, one could employ some other types of
operator norms, for instance, the trace norm [9]. But the use of the Hermitian
norm is more convenient for physical and chemical applications.
\section{Generalized density matrices}
Self-adjoint operators play a special role in applications, defining the
operators of observable quantities. One can resort to the coordinate
representation, defining the physical coordinates through $x$, implying
the set of all variables characterizing a particle. These can include
Cartesian coordinates, spin, isospin, component-enumerating labels, and like
that. The arithmetic space of all admissible values of the physical
coordinates is denoted as $\cX \equiv \{x\}$.
Let $\hat{\mathcal{A}}(x)$ be an operator of a local observable from the
algebra of local observables ${\mathcal{A}} \equiv \{\hat{A}(x)\}$ given on
the Fock space $\mathcal{F}$. And let
${\mathcal{A}}_\psi \equiv \{\hat{\psi}(x), \hat{\psi}^\dagger(x)\}$ be the
algebra of field operators on $\mathcal{F}$, describing the system. The direct
sum of these algebras is an {\it extended local algebra}
\be
\label{11}
\cA_{ext} \equiv \cA \bigoplus \cA_\psi \; .
\ee
The set $x^n \equiv \{x_1,x_2,\ldots,x_n\}$ of the coordinates of $n$
particles pertains to the arithmetic space
$$
\cX^n \equiv \cX \times \cX \times \ldots \times \cX
$$
that is an $n$-fold tensor product. The differential measure on the above
space ${\mathcal{X}}^n$ is defined as
$$
dx^n \equiv dx_1 dx_2 \ldots dx_n \; .
$$
A function $\varphi(x^n)$ can be treated as a vector
\be
\label{12}
\vp_n \equiv [ \vp(x^n) ] \in \cH_n
\ee
in a Hilbert space ${\mathcal{H}}_n$, where the scalar product is given by
\be
\label{13}
\vp_n^+ \vp_n \equiv \int \vp^*(x^n) \vp(x^n) \; dx^n \; .
\ee
We consider a quantum system, whose state is given by a statistical operator
$\hat{\rho}$ being a semi-positive self-adjoint operator normalized as
\be
\label{14}
{\rm Tr}_\cF \hat\rho = 1 \qquad (\hat\rho^+ = \hat\rho \geq 0 ) \; .
\ee
Taking any representative $A(x)$ of the extended algebra (11), we can define
a matrix
\be
\label{15}
\hat D_A^n \equiv \left [ D_A(x^n,y^n) \right ] \; ,
\ee
which is a matrix with respect to the variables $x$, with the components
\be
\label{16}
\hat D_A(x^n,y^n) \equiv {\rm Tr}_\cF A(x_1) \ldots A(x_n) \hat\rho
A^+(y_n) \ldots A^+(y_1) \; .
\ee
The action of matrix (15) on vector (12) is defined as the vector with
the components
\be
\label{17}
\hat D_A^n \vp_n = \left [ \int D_A (x^n,y^n) \vp(y^n)\; dy^n
\right ] \; .
\ee
As is seen from construction, matrix (15) is self-adjoint and semi-positive,
because of which it can be termed the generalized density matrix.
The norm of matrix (15) is defined as
\be
\label{18}
|| \hat D_A^n || = \sup_{\vp_n} \;
\frac{\vp_n^+\hat D_A^n\vp_n}{\vp_n^+\vp_n} \; .
\ee
And the trace of this matrix is
\be
\label{19}
{\rm Tr} \hat D_A^n \equiv \int D_A(x^n,x^n)\; dx^n \; .
\ee
The order index of the generalized density matrix (15) is given by the
expression
\be
\label{20}
\om(\hat D_A^n) \equiv
\frac{\log||\hat D_A^n||}{\log|{\rm Tr}\hat D_A^n|} \; ,
\ee
with the norm and trace defined as above.
\section{Reduced density matrices}
A particular case of the generalized density matrices, being the most
important for applications, is that of reduced density matrices. An $n$-th
order reduced density matrix is the matrix
\be
\label{21}
\hat\rho_n = [ \rho(x^n,y^n) ] \; ,
\ee
with the components
\be
\label{22}
\rho(x^n,y^n) \equiv {\rm Tr}_\cF \hat\psi(x_1) \ldots \hat\psi(x_n)
\hat\rho \hat\psi^\dgr(y_n) \ldots \hat\psi^\dgr(y_1) \; .
\ee
The relation of the reduced density matrix with the generalized density
matrices is given by the equations
$$
\hat\rho_n = \hat D_\psi^n \; , \qquad
\rho(x^n,y^n) = D_\psi(x^n,y^n) \; .
$$
Similarly to definition (20), the order index of the reduced density
matrix (21) is
\be
\label{23}
\om(\hat\rho_n) =
\frac{\log||\hat\rho_n||}{\log{\rm Tr}\hat\rho_n} \; .
\ee
In this definition, we do not take thermodynamic limit, as in Refs. [1-3].
The eigenproblem
\be
\label{24}
\hat\rho_n \vp_{nk} = N_{nk}\vp_{nk}
\ee
yields the eigenvalues
\be
\label{25}
N_{nk} = \vp_{nk}^+\hat\rho_n\vp_{nk}
\ee
that define the spectral norm
\be
\label{26}
||\hat\rho_n|| = \sup_k N_{nk} \; .
\ee
And the trace of the $n$-th order density matrix is given [6] by the
normalization
\be
\label{27}
{\rm Tr}\hat\rho_n = \frac{N!}{(N-n)!} \; .
\ee
We keep in mind a finite system with a finite number of particles $N$.
This number is assumed to be sufficiently large, $N \gg 1$, but finite.
For a large $N$ and fixed $n \ll N$, we can invoke the Stirling formula,
yielding
\be
\label{28}
{\rm Tr}\hat\rho_n \simeq \left ( \frac{N}{e} \right )^n \; .
\ee
Taking the natural logarithm gives
$$
\ln{\rm Tr}\hat\rho_n = n(\ln N - 1 ) \; .
$$
Then the order index (23) reads as
\be
\label{29}
\om(\hat\rho_n) = \frac{\ln||\hat\rho_n||}{n(\ln N-1) } \; .
\ee
From the properties of the reduced density matrices [6], we know that,
in the case of Bose particles,
\be
\label{30}
||\hat\rho_n|| \leq ( b_n N)^n \; ,
\ee
where $b_n$ is a constant. This results in the inequality
\be
\label{31}
\om(\hat\rho_n) \leq \frac{\ln N + \ln b_n}{\ln N -1 } \qquad
(Bose) \; .
\ee
For Fermi particles, depending on the order of the density matrices, we have
\be
\label{32}
||\hat\rho_{2n-1} || \leq ( c_{2n-1}N )^{n-1} \; , \qquad
||\hat\rho_{2n} || \leq ( c_{2n}N )^{n} \; ,
\ee
because of which
$$
\om(\hat\rho_{2n-1}) \leq
\frac{(n-1)(\ln N + \ln c_{2n-1})}{(2n-1)(\ln N-1) } \; ,
$$
\be
\label{33}
\om(\hat\rho_{2n}) \leq \frac{\ln N + \ln c_{2n}}{2(\ln N-1) }
\qquad (Fermi) \; .
\ee
Taking into account large $N$ results in the inequalities for Bose particles,
\be
\label{34}
\om(\hat\rho_{n}) \leq 1 \qquad (Bose) \; ,
\ee
and for Fermi particles,
\be
\label{35}
\om(\hat\rho_{2n-1}) \leq \frac{n-1}{2n-1} \; , \qquad
\om(\hat\rho_{2n}) \leq \frac{1}{2} \qquad (Fermi) \; .
\ee
\section{Bose system with quasi-condensate}
To give a feeling how the order indices describe physical ordering in finite
systems, let us consider a cloud of $N$ Bose atoms trapped in a finite box
of volume $V$. Systems of trapped Bose atoms are nowadays intensively studied
both theoretically as well as experimentally [10-24]. We consider the case
of either spinless atoms or that where atomic spins are frozen by an external
magnetic field, so that the spin degrees of freedom are not important, but
only the spatial variables $\bf r$ are considered.
As is known, in an infinite Bose system at low temperature, under thermodynamic
limit, when $N \ra \infty$, there appears Bose-Einstein condensate. But if the
system is finite, no matter how large it is, there can be no well defined
phase transition, hence, no Bose-Einstein condensation [20,24]. In a finite
system, there can exist only a kind of quasi-condensate [25]. Below, we
demonstrate the calculation of the order index for a Bose system in a finite
box, at low temperature, when a quasi-condensate appears. For brevity, we
shall often use the term condensate, keeping in mind quasi-condensate.
We start with the standard energy Hamiltonian
\be
\label{36}
\hat H = \int \hat\psi^\dgr(\br) \left ( -\; \frac{\nabla^2}{2m}
\right ) \hat\psi(\br)\; d\br \; + \;
\frac{1}{2} \int \hat\psi^\dgr(\br)\hat\psi^\dgr(\br')
\Phi(\br-\br') \hat\psi(\br') \hat\psi(\br)\; d\br d\br' \; ,
\ee
expressed through the field operators $\hat{\psi}(\bf r)$ satisfying the Bose
commutation relations. Here and in what follows, the system of units is used,
where the Planck and Boltzmann constants are set to one,
$\hbar \equiv 1, k_B \equiv 1$.
To take into account the appearance of quasi-condensate, we employ the
Bogolubov shift [26] of the field operator
\be
\label{37}
\hat\psi(\br) = \eta(\br) + \psi_1(\br) \; ,
\ee
separating it into the condensate function $\eta$ and the operator of the
normal, uncondensed, particles $\psi_1$. To avoid double counting, the
condensate and normal degrees of freedom are assumed to be orthogonal,
\be
\label{38}
\int \eta^*(\br) \psi_1(\br) \; d\br = 0 \; .
\ee
The operator of uncondensed particles satisfies the conservation law
\be
\label{39}
\lgl \psi_1(\br) \rgl = 0 \; .
\ee
There are two normalization conditions, for the number of condensed particles,
\be
\label{40}
N_0 = \int |\eta(\br) |^2 d\br \; ,
\ee
and for the number of uncondensed particles,
\be
\label{41}
N_1 = \lgl \hat N_1 \rgl \; ,
\ee
with the number operator of uncondensed atoms
\be
\label{42}
\hat N_1 \equiv
\int \psi_1^\dgr(\br) \psi_1(\br) d\br \; .
\ee
The total number of atoms in the box is the sum
\be
\label{43}
N = N_0 + N_1 \; .
\ee
To guarantee the validity of the normalization conditions (40) and (41), as
well as the conservation law (39), it is necessary to introduce the grand
Hamiltonian
\be
\label{44}
H = \hat H - \mu_0 N_0 - \mu_1 \hat N_1 -\hat\Lbd \; ,
\ee
in which
\be
\label{45}
\hat\Lbd = \int \left [ \lbd(\br)\psi_1^\dgr(\br) +
\lbd^*(\br)\psi_1(\br) \right ] d\br \; ,
\ee
with $\mu_0, \mu_1$, and $\lambda(\bf r)$ being the Lagrange multipliers.
The equations of motion are given by the variational equations for the
condensate function
\be
\label{46}
i\; \frac{\prt}{\prt t} \; \eta(\br,t) = \left \lgl
\frac{\dlt H}{\dlt\eta^*(\br,t) } \right \rgl
\ee
and for the field operator of uncondensed atoms
\be
\label{47}
i\; \frac{\prt}{\prt t} \; \psi_1(\br,t) =
\frac{\dlt H}{\dlt\psi_1^\dgr(\br,t) } \; ,
\ee
where $t$ is time and the angle brackets imply statistical averaging. These
equations are equivalent to the Heisenberg equations of motion [24,27].
The first-order reduced density matrix is
\be
\label{48}
\rho(\br,\br') = \lgl \hat\psi^\dgr (\br') \hat\psi(\br) \rgl \; .
\ee
This, in view of the Bogolubov shift (37), reads as
\be
\label{49}
\rho(\br,\br') = \eta^*(\br')\eta(\br) +
\lgl \psi_1^\dgr(\br') \psi_1(\br) \rgl \; .
\ee
The eigenfunctions of the reduced density matrix (48) are called natural
orbitals [6]. If these eigenfunctions are denoted as $\varphi_k(\bf r)$,
with a labelling quantum multi-index $k$, then the spectrum of the density
matrix is defined by the quantities
\be
\label{50}
N_{1k} = \int \vp_k^*(\br)\rho(\br,\br') \vp_k(\br')\;
d\br d\br' \; .
\ee
According to expression (49), there are two terms in the latter integral.
One term,
\be
\label{51}
N_k \equiv \left | \int \eta^*(\br) \vp_k(\br) \; d\br
\right |^2 \; ,
\ee
characterizes the condensed atoms, while another term,
\be
\label{52}
n_k \equiv \int \vp_k^*(\br) \lgl \psi_1^\dgr(\br')\psi_1(\br) \rgl
\vp_k(\br')\; d\br d\br' \; ,
\ee
defines the distribution of uncondensed atoms. With the notation
\be
\label{53}
a_k \equiv \int \vp_k^*(\br) \psi_1(\br) \; d\br \; ,
\ee
this distribution takes the form
\be
\label{54}
n_k = \lgl a_k^\dgr a_k \rgl \; .
\ee
In this way, the spectral norm of the density matrix (48) can be represented
as
\be
\label{55}
|| \hat\rho_1 || = \sup_k N_{1k} = \sup_k (N_k + n_k) \; .
\ee
In a similar way, it is possible to find the norms of the higher-order
density matrices. Thus, for the second-order density matrix, we would have
to find out the pairon spectrum [28]. But here we concentrate on the
properties of the first-order density matrix.
\section{Order index behavior}
Let us study the behavior of the order index
\be
\label{56}
\om(\hat\rho_1) = \frac{\ln||\hat\rho_1||}{\ln N}
\ee
of the first-order density matrix (48). For atoms in a box of volume $V$,
the natural orbitals are the plane waves
$$
\vp_k(\br) = \frac{1}{\sqrt{V}} \; e^{i\bk\cdot\br }
$$
labelled by the wave vector quantum number $k$. The condensate wave
function reduces to a constant $\eta = \sqrt{N_0/V}$. Then the condensate
spectrum (51) is
$$
N_k = N_0 \dlt_{k0} \; .
$$
The matrix eigenvalues (50) read as
\be
\label{57}
N_{1k} = \dlt_{k0} N_0 + ( 1 - \dlt_{k0} ) n_k \; .
\ee
And norm (55) becomes
\be
\label{58}
|| \hat\rho_1 || = \sup \{ N_0 ,\; \sup_k n_k \} \; .
\ee
To accomplish explicit calculations, we need to fix the form of the
interaction potential entering Hamiltonian (36). We shall keep in mind
dilute Bose gas for which the interaction potential is well modelled by
the local form
\be
\label{59}
\Phi(\br) = \Phi_0 \dlt(\br) \; , \qquad
\Phi_0 \equiv 4\pi \; \frac{a_s}{m} \; ,
\ee
where $a_s$ is scattering length. We shall accomplish calculations
invoking the Hartree-Fock-Bogolubov approximation in the self-consistent
approach using representative ensembles [23,24,29].
We introduce the notations
$$
\rho_0 \equiv \frac{N_0}{V} \; , \qquad \rho_1 \equiv \frac{N_1}{V}
= \frac{1}{V} \sum_k n_k \; ,
$$
\be
\label{60}
\rho \equiv \frac{N}{V} = \rho_0 + \rho_1 \; , \qquad
\sgm_1 = \frac{1}{V} \sum_k n_k \; ,
\ee
defining the mean densities of condensed ($\rho_0$) and uncondensed
($\rho_1$) atoms, and also the so-called anomalous average $\sigma_1$,
whose modulus $|\sigma_1|$ describes the number of correlated atomic
pairs. The corresponding atomic distributions, at temperature $T$,
read as
\be
\label{61}
n_k = \frac{\om_k}{2\ep_k} \; \coth \left ( \frac{\ep_k}{2T} \right )
- \; \frac{1}{2} \; , \qquad
\sgm_k = -\; \frac{mc^2}{2\ep_k} \; \coth \left ( \frac{\ep_k}{2T}
\right ) \; ,
\ee
where
$$
\om_k \equiv mc^2 + \frac{k^2}{2m} \; , \qquad
\ep_k \equiv \sqrt{(ck)^2 + \left ( \frac{k^2}{2m} \right )^2 }\;.
$$
The sound velocity $s$ satisfies the equation
\be
\label{62}
mc^2 = \Phi_0 (\rho_0 + \sgm_1 ) \; .
\ee
Quasi-condensate can arise only at low temperature. For concreteness,
we take zero temperature $T = 0$. Then, from the above formulas, we have
$$
n_k = \frac{\om_k - \ep_k}{2\ep_k} \; , \qquad
\sgm_k = - \; \frac{mc^2}{2\ep_k} \; ,
$$
\be
\label{63}
\rho_1 = \frac{(mc)^3}{3\pi^2} \; , \qquad
\sgm_1 = \frac{(mc)^2}{\pi^2} \; \sqrt{m\rho_0\Phi_0} \; .
\ee
In the calculation of $\sgm_1$, we employ dimensional regularization [24,29].
It is convenient to introduce dimensionless quantities simplifying the
formulas. Atomic interactions are characterized by the gas parameter
\be
\label{64}
\gm \equiv a_s \rho^{1/3} \; .
\ee
Dimensionless sound velocity is
\be
\label{65}
s \equiv \frac{mc}{\rho^{1/3}} \; .
\ee
The condensate and anomalous fractions are
\be
\label{66}
n_0 \equiv \frac{N_0}{N} = \frac{\rho_0}{\rho} \; ,
\qquad \sgm \equiv \frac{\sgm_1}{\rho} \; .
\ee
In this dimensionless notation, the equation for the sound velocity (62)
reduces to
\be
\label{67}
s^2 = 4\pi\gm (n_0 + \sgm) \; ,
\ee
the condensate fraction becomes
\be
\label{68}
n_0 = 1 - \; \frac{s^3}{3\pi^2} \; ,
\ee
and the anomalous fraction is
\be
\label{69}
\sgm = \frac{2s^2}{\pi} \; \sqrt{\frac{\gm n_0}{\pi} } \; .
\ee
Numerical solution to these equations is shown in Fig. 1, where $s, \sigma$,
and $n_0$ are presented as functions of the gas parameter (64).
The distribution of the normal atoms $n_k$ increases as $k$ diminishes.
The minimal value of $k$ is prescribed by the box volume $V$, so that
\be
\label{70}
k_{min} = \frac{1}{V^{1/3} } = \frac{\rho^{1/3}}{N^{1/3}} \; .
\ee
This gives
$$
\sup_k n_k = \frac{s}{2} \; N^{1/3} \; .
$$
Therefore, norm (58) is represented as
\be
\label{71}
|| \hat\rho_1|| = \sup \left \{ n_0 N , \; \frac{s}{2} \; N^{1/3}
\right \} \; .
\ee
The order index (56), for an infinite system in the presence of condensate,
in thermodynamic limit $N \ra \infty$, is exactly one, which corresponds
to long-range order. But for a finite system, the order-index behavior is
not so trivial, varying between zero and one, depending on the system
parameters.
In the case of asymptotically weak atomic interactions, when $\gamma \ra 0$,
from Eqs. (67) to (69), it follows
$$
n_0 \simeq 1 \; - \; \frac{8}{3\sqrt{\pi}} \; \gm^{3/2} \; - \;
\frac{64}{3\pi} \; \gm^3 \; ,
$$
$$
\sgm \simeq \frac{8}{\sqrt{\pi}} \; \gm^{3/2} + \frac{32}{\pi}\; \gm^3 \; ,
$$
\be
\label{72}
s \simeq 2\sqrt{\pi} \; \gm^{1/2} + \frac{16}{3} \; \gm^2 +
\frac{32}{9\sqrt{\pi} } \; \gm^{7/2} \; .
\ee
Then the matrix norm is
$$
|| \hat\rho_1 || \simeq n_0 N \qquad (\gm \ra 0 ) \; ,
$$
and the order index tends to
$$
\om(\hat\rho_1) \simeq 1 + \frac{\ln n_0}{\ln N} \; .
$$
Using expansions (72), we find
\be
\label{73}
\om(\hat\rho_1) \simeq 1 - \; \frac{8\gm^{3/2}}{3\sqrt{\pi}\ln N}
\qquad ( \gm \ra 0 ) \; .
\ee
As is seen, the order index reduces to one when either the interaction
strength tends to zero or the system becomes infinite. But for a finite
system of interacting atoms, the order index is less than one.
In the opposite case of arbitrarily strong interactions, when
$\gamma \ra \infty$, we have
$$
n_0 \simeq \frac{\pi}{64}\; \gm^{-3} \; ,
$$
$$
\sgm \simeq \frac{(9\pi)^{1/3}}{4} \; \gm^{-1} \; - \;
\frac{\pi}{64}\; \gm^{-3} \; - \; \frac{1}{128} \left (
\frac{\pi^4}{3} \right )^{1/3} \gm^{-4} \; ,
$$
\be
\label{74}
s \simeq \left ( 3\pi^2 \right )^{1/3} \; - \; \frac{1}{64}
\left ( \frac{\pi^5}{9} \right )^{1/3} \gm^{-3} \; .
\ee
For a finite system, with a fixed number of atoms $N$, when the
interaction strength exceeds the value $0.317 N^{2/9}$, we get
$$
|| \hat\rho_1|| \simeq \frac{(3\pi^2)^{1/3}}{2} \; N^{1/3}
\qquad (\gm \ra \infty ) \; .
$$
Therefore, the order index behaves as
\be
\label{75}
\om(\hat\rho_1) \simeq \frac{1}{3} + \frac{0.436}{\ln N} \qquad
(\gm \ra \infty) \; ,
\ee
which is again less than one.
Thus, we see that, for a finite system, the order index varies between
unity, when the gas parameter tends to zero, and the limiting value (75),
when the gas parameter tends to infinity. That is, a finite system can
exhibit only a kind of quasi-long-range order, or mid-range order. The
overall behavior of the order index (56) as a function of the
interaction strength $\gamma$ and the number of atoms in the box $N$,
is shown in Figs. 2 and 3. When $N$ tends to infinity, the order index
tends to unity. In the scale of Fig. 3, this is not seen for $\gamma = 1$,
since $\om(\hat\rho_1)$ for this $\gamma$ decreases after $N = 11$. In
order to demonstrate that this is just a temporary decrease, we show the
behavior of the order index $\om(\hat\rho_1)$, in a larger scale of $N$,
in Fig. 4. As is seen, the order index diminishes till $N = 200$, after
which it increases. This increase is rather slow. Thus, for $N = 10^9$, the
order index reaches the value 0.9, and for $N = 10^{80}$, it becomes 0.98.
So that it tends to unity as $N \ra \infty$.
\section{Conclusion}
The notion of order indices for reduced density matrices is extended to
the case of finite systems. In such systems there can be no long-range
order and the standard order parameters are not defined. Contrary to this,
the order indices have the meaning for finite systems, characterizing the
level of ordering in these objects. Examples of finite systems are
quantum dots, metallic grains, nanoclusters, trapped atoms, and various
macromolecules, including biomolecules.
Generally, in finite systems, there can exist only a kind of mid-range
order. The level of such an order is well described by order indices.
The suggested approach is illustrated by calculating the order index for
the first-order reduced density matrix of bosonic atoms at zero
temperature, trapped in a finite box. The order index, depending on the
values of the number of atoms and the strength of their interactions,
varies between zero and one.
The order indices can be defined for finite systems composed of several
types of particles. For such a system of several components, it is
possible to define the order index for the total system, as well as the
partial order indices for some of the components. For example, it is
often of interest the investigation of the properties of a particular
kind of atoms entering a complex molecule [30]. Then the order indices
can be introduced for the studied atoms characterizing the level of
their ordering inside the molecule.
The order indices serve as a measure of internal ordering and
correlations and could be used as an additional characteristic for
finite quantum systems.
\vskip 5mm
{\bf Acknowledgement}
\vskip 3mm
Financial support from the Russian Foundation for Basic Research is
acknowledged.
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,937
|
{"url":"http:\/\/stats.stackexchange.com\/questions\/14914\/how-to-test-the-autocorrelation-of-the-residuals","text":"How to test the autocorrelation of the residuals?\n\nI have a matrix with two columns that have many prices (750). In the image below I plotted the residuals of the follow linear regression:\n\nlm(prices[,1] ~ prices[,2])\n\n\nLooking at image, seems to be a very strong autocorrelation of the residuals.\n\nHowever how can I test if the autocorrelation of those residuals is strong? What method should I use?\n\nThank you!\n\n-\nYou don't need to test for autocorrelation. It is there. The plot shows that. You could look at the autocorrelation function of these residuals (function acf()), but this will simply confirm what can be seen by plain eye: the correlations between lagged residuals are very high. \u2013\u00a0Wolfgang Aug 28 '11 at 20:23\n@Wolfgang, yes, correct, but I have to check it programmatically.. I will take a look at acf function. Thanks! \u2013\u00a0Dail Aug 28 '11 at 22:12\n@Wolfgang, I'm seeing acf() but I don't see a sort of p-value to understand if there is a strong correlation or not. How to interpret its result? Thanks \u2013\u00a0Dail Aug 28 '11 at 22:16\nWith H0: correlation (r) = 0, then r follows a normal\/t dist with mean 0 and variance of sqrt(number of observations). So you could get the 95% confidence interval using +\/- qt(0.75, numberofobs)\/sqrt(numberofobs) \u2013\u00a0Jim Mar 17 at 5:46\n\nThere are probably many ways to do this but the first one that comes to mind is based on linear regression. You can regress the consecutive residuals against each other and test for a significant slope. If there is auto-correlation, then there should be a linear relationship between consecutive residuals. To finish the code you've written, you could do:\n\nmod = lm(prices[,1] ~ prices[,2])\nres = mod$res n = length(res) mod2 = lm(res[-n] ~ res[-1]) summary(mod2) mod2 is a linear regression of the time$t$error,$\\varepsilon_{t}$, against the time$t-1$error,$\\varepsilon_{t-1}$. if the coefficient for res[-1] is significant, you have evidence of autocorrelation in the residuals. Note: This implicitly assumes that the residuals are autoregressive in the sense that only$\\varepsilon_{t-1}$is important when predicting$\\varepsilon_{t}$. In reality there could be longer range dependencies. In that case, this method I've described should be interpreted as the one-lag autoregressive approximation to the true autocorrelation structure in$\\varepsilon$. - thank you so much for the example. Only one doubt, How can I test if res[-1] is significant? \u2013 Dail Aug 28 '11 at 22:10 you'd test it the same way you would any other regression coefficient - look at the$t$-statistic and$p\\$-value \u2013\u00a0Macro Aug 28 '11 at 22:22\ndoing a fast test with: lm(rnorm(1000)~jitter(1:1000)) I get: Residual standard error: 1.006 on 997 degrees of freedom Multiple R-squared: 0.0003463, Adjusted R-squared: -0.0006564 F-statistic: 0.3454 on 1 and 997 DF, p-value: 0.5569 the p-value can't reject the null hypothesis \u2013\u00a0Dail Aug 28 '11 at 22:29\nMacro, I have tested the residuals of the chart I plotted above, and the result is: Residual standard error: 0.04514 on 747 degrees of freedom Multiple R-squared: 0.9241, Adjusted R-squared: 0.924 F-statistic: 9093 on 1 and 747 DF, p-value: < 2.2e-16, It doesn't seem very good, It is very strange because there is a strong autocorrelation, what could I do? \u2013\u00a0Dail Aug 28 '11 at 22:44\nThis is called a Breusch-Godfrey test for autocorrelation. \u2013\u00a0Charlie Sep 7 '11 at 16:53\n\nUse the Durbin-Watson test, implemented in the lmtest package.\n\ndwtest(prices[,1] ~ prices[,2])\n\n-\n very strange I get: p-value < 2.2e-16, How it is possible? the data seems very correlated! \u2013\u00a0Dail Aug 29 '11 at 7:50 The p-value is the probably of getting as much correlation as that observed if there is no real correlation. So if the p is very small, as it is, that suggests there is a lot of correlation present in the sample. \u2013\u00a0Rob Hyndman Aug 29 '11 at 10:03 Do you mean a p-value like this indicates that the residuals are very autocorrelated? \u2013\u00a0Dail Aug 29 '11 at 11:41 Yes............ \u2013\u00a0Rob Hyndman Aug 29 '11 at 12:56 hmm strange, take a look at: imageshack.us\/f\/59\/17671620.png how is it possible that the right image is not autocorrelated? \u2013\u00a0Dail Aug 29 '11 at 13:35","date":"2013-06-19 08:05:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7528846263885498, \"perplexity\": 959.3433297552489}, \"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-2013-20\/segments\/1368708144156\/warc\/CC-MAIN-20130516124224-00013-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
The Landau's theory of interacting Fermi systems \cite{pines,abri} provides an elegant description of
low--energy excitations in Fermi liquids, for different types of many--body systems and at different energy scales. The complex dynamics of the interacting particles is simplified through the concept of
quasiparticles having an effective mass $m^*$ induced by the interparticle interaction.
The study of $m^*$ meets a broad interest in several branches of many--body physics. Being $m^*$ related to the propagation of particles in a medium and, more specifically, to the density of states in many--body systems \cite{fetter}, it has an important impact on several observables such as, for instance, the energies of axial compression modes in atomic gases \cite{nasci} and in nuclei \cite{blaizot1,bohigas}, the specific heat of a low--temperature Fermi gas \cite{fetter}, or the
maximum mass of a neutron star (the maximum mass that an equation of state may generate taking into account gravity) \cite{gle}.
The effective mass is usually evaluated through the computation of the self--energy (see for instance Refs. \cite{hedin,fetter}). Numerically, it may be provided by Quantum Monte Carlo calculations, for instance as done in Refs. \cite{lobo,pilati} for the polaron in strongly imbalanced atomic gases.
Analogous calculations were also done in Refs. \cite{forbes,roggero} for nuclear systems.
It was also proposed in Ref. \cite{eich} to extract the effective mass from thermodynamical properties in an interacting electron liquid.
For ultracold atomic Fermi gases, the properties of the polaron quasiparticle have been extensively studied \cite{na1,sc,nasci,sca,koh,kos}. In the case of
very large population imbalance, the measurement of low--frequency axial breathing modes has been used to extract dynamically the polaron effective mass \cite{nasci}: based on the Landau theory of Fermi liquids, a relation was drawn in the local--density approximation between the frequency of the polaron $\omega^*$ and its effective mass $m^*$, such that $\omega^*$ is proportional to $\sqrt{m/m^*}$, $\omega^*/\omega = \sqrt{(1-A)m/m^*}$, where $\omega$ is the trapping--potential frequency and $A$ describes the attraction between the impurity and the other atoms. An analogous relation, based on the Landau theory of Fermi liquids, was employed several decades ago for atomic nuclei \cite{blaizot1,bohigas} to connect the centroid energies of isoscalar (IS) giant quadrupole resonances (GQRs) (which are the nuclear axial breathing modes) with the effective mass in nuclear matter. The local--density approximation can be applied in this case because the effective mass is a smooth function of the density. Such a relation was widely used in nuclear physics in the past decades to extract, from the measurement of IS GQRs, phenomenological constraints for the effective mass in matter (see Ref. \cite{bao} and references therein).
The effective mass $m^*$ is defined by the relation
\begin{equation}
\frac{1}{m^*}=\frac{dE}{dk} \frac{1}{\hbar^2k} \,
\label{defimstar}
\end{equation}
for a particle of energy $E$ and momentum $k$, with
\begin{equation}
E=\frac{\hbar^2 k^2}{2m} + \Sigma_k + \Sigma_{k,E}.
\label{ener}
\end{equation}
In Eq. (\ref{ener}), $ \Sigma_k + \Sigma_{k,E}$ is the self--energy, sum of the mean--field (MF) contribution $\Sigma_k$ (from the leading order of the Dyson equation in the perturbative many--body expansion) and of a beyond--mean--field (BMF) energy--dependent contribution $\Sigma_{k,E}$.
In the MF approximation, the self--energy does not have any energy
dependence and may have only a $k$ dependence. An explicit energy dependence is induced when the MF approximation is overcome going beyond the leading order of the Dyson expansion \cite{fetter}.
Using the definition of $m^*$ in Eq. (\ref{defimstar}), one can write
\begin{eqnarray}
\nonumber
\frac{m^*}{m}&=&\left(1-\frac{\partial \Sigma_{k,E}}{\partial E}\right) \cdot \left(1+\frac{m}{\hbar^2k}\frac{\partial \Sigma_k}{\partial k}\right)^{-1} \\
&=& \frac{m_E^*}{m} \cdot \frac{m_k^*}{m},
\label{mstar}
\end{eqnarray}
where the above expression defines the so--called $E$--mass $m_E^*/m$ and $k$--mass $m_k^*/m$, using the same notations as in Refs. \cite{blaizot1,jeu,ma,bla,bern}.
In cases where the MF self-energy does not have a $k$ dependence (for instance, with a zero--range interaction characterized only by a coupling constant, without any velocity--dependent terms) the $k$--mass is equal to 1.
In these cases, an effective mass is generated only with second--order calculations.
\begin{figure}
\includegraphics[scale=0.35]{mstar.eps}
\caption{Centroid energies of the IS GQRs for the nuclei $^{48}$Ca and $^{90}$Zr as a function of $\sqrt{m/m^*}$. The RPA centroids (black circles) are reported for four Skyrme parametrizations and associated to the corresponding MF effective masses in nuclear matter. A linear fit is done on these points (blue dotted lines). The SSRPA-SLy4 and SSRPA-SGII centroids are reported on the blue dotted lines (green triangles and magenta squares, respectively). The experimental values are also displayed by orange bands.}
\label{effem}
\end{figure}
The $E$-mass is equal to 1 in the MF approximation, where $m^*=m^*_k$.
Any BMF effect produces a modification of $m^*$ generated by the $E$-mass.
Being the effective mass related to the density of states \cite{erler}, BMF changes of its value induce a different single--particle spectrum, which is compressed if the effective mass is enhanced beyond the mean field.
This aspect is investigated here with the SSRPA model introduced in Ref. \cite{gamba2015}, where the MF approximation is overcome owing to the coupling of 1 particle-1 hole (1p1h) and 2 particle-2 hole (2p2h) configurations. The study is done in the framework of the energy--density--functional (EDF) theory \cite{bender} with Skyrme forces.
We base our analysis on the above--mentioned relation between the frequency of axial modes and $\sqrt{m/m^*}$.
We propose a new and original procedure to estimate BMF effects on the effective mass of nuclear matter. This is based on BMF predictions of axial breathing modes in nuclei and is connected with an induced BMF modification of single--particle spectra. This procedure is quite general and can be employed with other BMF models. Nevertheless, the SSRPA has an important advantage (compared to other BMF models) of being definitely safe against instabilities, divergences, and double counting of correlations \cite{gamba2015,tse} in the EDF framework. This guarantees the quantitative robustness of the obtained predictions.
We first performed random--phase--approximation (RPA) calculations for the medium--mass nucleus $^{48}$Ca and the heavier nucleus $^{90}$Zr by using four Skyrme parametrizations SkP \cite{skp}, SGII \cite{sgii}, SLy4 \cite{sly4}, and Ska \cite{ska} having, respectively, MF effective masses equal to 1, 0.79, 0.7, and 0.61 in nuclear matter.
We plot in Fig. \ref{effem} the obtained centroid energies for the IS GQR modes as a function of $\sqrt{m/m^*}$, associating to each centroid energy the corresponding MF effective mass in matter. A linear fit is performed on these four points for each nucleus (blue dotted lines) and the experimental values are also displayed (orange bands).
We choose two parametrizations, SLy4 and SGII, having MF effective masses between 0.7 and 0.8.
To estimate the modification of the effective mass produced beyond the mean field we use the linear fits performed on the RPA points and we report, on the blue dotted lines, the points corresponding to the SSRPA centroid energies obtained for the two nuclei and the two parametrizations. We observe that the centroids are located at lower energies for the SSRPA model with respect to the corresponding RPA values. Such a lowering of the energies implies that the associated effective mass increases with respect to the MF value.
We deduce that,
for $^{48}$Ca ($^{90}$Zr), the extracted effective mass for nuclear matter increases from 0.7 in the MF case to 0.834 (0.769) for the BMF calculations of the IS GQR with SLy4. With SGII, the effective mass for matter increases from 0.79 to 0.837 (0.842) from the calculations done for $^{48}$Ca ($^{90}$Zr).
Figure \ref{window} displays an estimation of the theoretical error bar associated to the spreading of the values of the effective mass in matter. Figure \ref{window}(a) shows the MF error bar (yellow area) which is induced by the dependence on the used interaction of the nuclear matter $m^*$ value (11\% of discrepancy between the SLy4 and SGII values). Figure \ref{window}(b) shows the four BMF values for the effective mass (blue circles). One notices that the maximum discrepancy is now of 9\% (yellow area), slightly reduced with respect to the MF case even if, in the
BMF case, the extracted $m^*$ value depends not only on the used interaction but also on the nucleus for which the axial excitation is computed. The two nuclei under study already offer the possibility to cover two different mass regions (nucleus dependence). To
extend our analysis we performed an additional SSRPA calculation for $^{48}$Ca (the nucleus for which we have found the largest modification of effective mass going from MF to BMF) with the parametrization Ska (MF $m^*=0.61$). The MF theoretical error increases correspondingly (maximum discrepancy of 23\%, yellow + grey area in Fig. \ref{window}(a)). We observe that, including the Ska BMF value in Fig. \ref{window}(b) (red square) the discrepancy window is now represented by the yellow plus grey area (21 \% of maximum discrepancy). We may thus deduce that such an extraction of a BMF effective mass for nuclear
matter does not produce an overall error larger than the one already induced with MF calculations and related to the dependence on the used interaction.
From Eq. (\ref{mstar}), we may extract the average values of the $E$-mass, equal to 1.19 (1.06) with SLy4 (SGII) for $^{48}$Ca and
to 1.10 (1.07) with SLy4 (SGII) for $^{90}$Zr. The $E$-mass is equal to 1.14 for $^{48}$Ca with Ska. BMF effects produce an increase of the $E$-mass ranging from 6 to 16 \%, the largest variation from 1 occurring for $^{48}$Ca and the SLy4 parametrization.
\begin{figure}
\includegraphics[scale=0.34]{bar.eps}
\caption{(a) Theoretical error associated to the MF effective mass for nuclear matter induced by two Skyrme parametrizations, SLy4 and SGII (yellow band) and three Skyrme parametrizations, SLy4, SGII, and Ska (yellow plus grey band); (b) Same as in panel (a) but for the BMF effective mass. The four blue circles represent the calculations done with SLy4 and SGII for $^{48}$Ca and $^{90}$Zr whereas the red square represents the calculation done with Ska for $^{48}$Ca.}
\label{window}
\end{figure}
We observe in Fig \ref{effem} that, for $^{48}$Ca, the SSRPA centroid energies obtained with the two parametrizations SLy4 and SGII are very similar, leading to very similar values of $m^*/m$. Since the MF effective mass is not the same for the two parametrizations, this implies a stronger BMF modification of the
$E$-mass for the case of SLy4 (where the MF effective mass is lower). On the other side, the SSRPA centroids obtained with SLy4 and SGII are slightly different for $^{90}$Zr, leading to a higher value of $m^*/m$ for the case of SGII. The two BMF $E$-masses are very similar one to the other for this nucleus.
It is interesting to mention that the lowering
of the excitation energies provided by SRPA--based models (with respect to the RPA spectrum) is a general
feature of the model that does not occur only in nuclear systems. The same type of effect was found for instance also for metallic clusters in Ref. \cite{gamba2009,gamba2010}. In all cases, one thus expects an increase of the
effective mass ($E$-mass larger than 1) and, consequently, an effective compression of the single--particle spectrum.
We analyze this aspect in the framework of the BMF SSRPA model.
Writing the SRPA equations as energy--dependent RPA equations, the RPA--type energy--dependent matrix elements contain the RPA matrix elements plus additional contributions given by the energy--dependent self--energy (see for instance Ref. \cite{papa}). For example, the energy--dependent $A$-type matrix elements may be written as
\begin{equation}
A^{SRPA}_{1,1'} (E) = A^{RPA}_{1,1'}+\sum_{2,2'} \frac{A_{12} A_{2'1'}}{E+i\eta-A_{2,2'}},
\label{asrpa}
\end{equation}
where $A^{RPA}_{1,1'}$ are standard RPA $A$ matrix elements, 1 and 1' are two 1p1h configurations, 2 and 2' are two 2p2h configurations, and $A_{12}$ and $A_{22}$ are beyond--RPA matrix elements coupling 1p1h with 2p2h configurations and 2p2h configurations among themselves, respectively.
For the sake of simplicity, we write the expressions for cases where the interaction is density independent, where rearrangement terms are not present. The expressions which are used in practice for the results presented in Fig. \ref{effem} are slightly more involved and contain additional contributions corresponding to the rearrangement terms \cite{rearra}.
The energy--dependent self--energy correction provides a renormalization of the diagonal matrix elements
$A_{1,1}$ (corrections to both the single--particle excitation energies and the interaction matrix elements). Since such matrix elements contain the single--particle excitation energies, this renormalization certainly induces a BMF renormalization of the 1p1h single--particle spectrum. One has
\begin{eqnarray}
\nonumber
&& A_{1,1}^{RPA} =
\left[\epsilon_p - \epsilon_h \right]_{MF} + \bar{V}_{phhp} \rightarrow A_{1,1}^{SRPA} (E) \\
&=&\left[\epsilon_p - \epsilon_h\right]_{MF} + \bar{V}_{phhp} + \sum_{2,2'} \frac{A_{ph,2} A_{2',ph}}{E+i\eta-A_{2,2'}}.
\label{reno}
\end{eqnarray}
The subtraction procedure used in Ref. \cite{gamba2015} was formulated in Ref. \cite{tse} so to avoid any double counting of correlations in extensions of RPA within the framework of EDF theories.
The subtraction procedure is based on the dielectric theorem \cite{bohigas}
stating that
the inverse energy--weighted moment of the strength evaluated in RPA is
equal to the static polarizability. In the SRPA, because of the energy
dependence introduced into the self--energy, this
equality is not fulfilled anymore. The subtraction procedure restablishes
the equality of the RPA and SSRPA inverse energy--weighted moments and, as
a consequence, the dielectric theorem.
This is achieved by imposing that the beyond--RPA energy--dependent matrix elements are equal to the corresponding RPA matrix elements in the static limit (self--energy calculated at zero energy). For details, the reader may refer to Refs. \cite{gamba2015,tse}.
\begin{widetext}
\begin{figure}
\includegraphics[scale=0.45]{confinew.eps}
\caption{Diagonal matrix elements $A_{1,1}$ calculated with the parametrization SLy4 for the nucleus $^{48}$Ca for the first three single--particle configurations (which are neutron configurations). RPA and SSRPA results are presented. The BMF results are calculated using in the energy--dependent matrix elements an energy value equal to the IS GQR centroid obtained in SSRPA.}
\label{ene48}
\end{figure}
\begin{figure}
\includegraphics[scale=0.45]{confinew2.eps}
\caption{Same as in Fig. \ref{ene48} but for $^{90}$Zr. One of the configurations is in this case a proton configuration.}
\label{ene90}
\end{figure}
\end{widetext}
By performing the subtraction procedure, the rescaling of the matrix element $A_{1,1}$ is thus further modified by an additional corrective term, which guarantees that $A^{SSRPA}_{1,1}(0)=A^{RPA}_{1,1}$,
\begin{eqnarray}
\nonumber
&& A_{1,1}^{RPA} \rightarrow A_{1,1}^{SSRPA} (E) = \left[\epsilon_p - \epsilon_h \right]_{MF}
+ \bar{V}_{phhp} \\ &+& \sum_{2,2'} \frac{A_{ph,2} A_{2',ph}}{E+i\eta-A_{2,2'}}
+ \sum_{2,2'} \frac{A_{ph,2} A_{2',ph}}{A_{2,2'}}.
\label{renosub}
\end{eqnarray}
As an illustration, we discuss the case of the parametrization SLy4.
Figure \ref{ene48} shows the diagonal matrix elements $A_{1,1}$ calculated for the nucleus $^{48}$Ca for the first three 1p1h configurations entering in the construction of the collective quadrupole excitations. In this case, the three configurations are neutron configurations. We present RPA and SSRPA results. In the case of SSRPA, to compute the rescaling effect induced by BMF calculations, the energy--dependent self--energy correction is calculated at an energy value given by the SSRPA centroid of the IS GQR. This guarantees that we make this estimation in the region of the GQR.
We notice that the BMF rescaling of the matrix element is more pronounced for the first configuration and becomes less important at increasing energies. We have observed that this effect is indeed strongly quenched for the highest--energy 1p1h configurations.
Also, the BMF modification produces in all cases a global reduction of the matrix element (and, consequently, a reduction of the single--particle excitation energy). Such a reduction implies an effective compression of the single--particle spectrum, coherent with the enhancement of the effective mass indicated by our previous analysis done on the values of the centroid energies.
Figure \ref{ene90} shows the same quantities as Fig. \ref{ene48}, but for the nucleus $^{90}$Zr. In this case, the third single--particle configuration is a proton configuration.
For the two nuclei, $^{48}$Ca and $^{90}$Zr, the third single--particle configuration entering in the construction of the quadrupole collective phonon is located in the same energy region as the IS GQR. One can thus in this case provide an intuitive physical interpretation of the BMF renormalization effects. Such effects may be regarded in the same spirit as in particle--phonon--coupling models: the unperturbed 1p1h excitation mode couples with the collective phonon because the two excitation energies are close one to the other. The resulting effect is the lowering of the centroid for the collective phonon, the formation of a spreading width for the collective excitation due to the mixing with 2p2h configurations, and the compression of the single--particle spectrum that we deduce from the fact that our effective single--particle excitation energies are systematically reduced.
Let us focus on this third 1p1h configuration and on the nucleus $^{48}$Ca.
Figure \ref{ene48} shows that the SSRPA model induces a reduction of 11.6 \% of the matrix element $A_{1,1}$ with respect to the RPA result.
To push further this analysis, we compute the SSRPA energy--dependent self--energy correction using the second line of Eq. (\ref{renosub}). We have done this calculation for the third single--particle configuration, for the nucleus $^{48}$Ca and the parametrization SLy4. We denote this self--energy by $\Sigma_{3}$. We show in Fig. \ref{sigma}, as a function of the energy, the quantity $1-\partial \Sigma_3/\partial E$, which should
correspond to an estimation of the $E$-mass for the third 1p1h configuration (see Eq. (\ref{mstar})). The derivative of $\Sigma_3$ with respect to the energy is negative, which leads to values for the $E$-mass larger than 1.
This quantity goes to 1 in the static limit where we
recover the MF result. Coherently with the previous extraction of the average $E$-mass from the centroid of the collective axial modes, we also notice that, in the energy region of the third 1p1h configuration (the red dashed line in the figure represents the value of the diagonal matrix element $A_{1,1}$ for the third 1p1h configuration), for which the computation is performed, this estimation of the $E$-mass provides the value of 1.16, which is very close to the one previusly found for $^{48}$Ca and SLy4.
\begin{figure}
\includegraphics[scale=0.35]{sigma.eps}
\caption{Estimation of the $E$-mass (for the third single--particle configuration) for the nucleus $^{48}$Ca and the parametrization SLy4. The red dashed line represents the value of the diagonal matrix element $A_{1,1}$ for the third 1p1h configuration. }
\label{sigma}
\end{figure}
Analogous results were obtained for the heavier nucleus $^{90}$Zr and the parametrization SGII. In general, we did not identify any particular difference between the results obtained for the two nuclei. Their different mass does not lead to any specific dissimilarities in the renormalization of the effective mass and of the single--particle spectrum.
In conclusion, we have discussed the enhancement of the nucleon effective
mass induced by BMF effects. The study is based on a new procedure that was applied within the EDF theory. The SSRPA model is employed, which allows for a
microscopic description of BMF effects. The average BMF effective mass is extracted from the
predictions of the centroid energies of axial breathing modes in two
nuclei, $^{48}$Ca and $^{90}$Zr. The compression of single--particle spectra
generated by the SSRPA self--energy correction is also investigated. The
two analyses, one based on the centroid energies of axial modes and the other based on the compression of
single--particle spectra, lead to a coherent estimation of the average $E$-mass value,
which increases from 6 to 16 \%, depending on the nucleus and on the effective interaction, with respect to the MF value, which is equal to 1.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,902
|
\section{Introduction}
The concept of steering can date back to 1930s, introduced by Schr\"odinger~\cite{Sch} as a generalization of the Einstein-Podolsky-Rosen (EPR) paradox~\cite{Ein}. For a bipartite state, steering infers that an observer on one side can affect the state of the other spatially separated system by local measurements. In quantum information processing, steering can be defined as the task for a referee to determine whether two parties share entanglement with an untrusted party~\cite{Wiseman1,JWD,sau}. In 2007, Wiseman, Jones and Doherty~\cite{Wiseman1} formally defined quantum steering as a type of quantum nonlocality that is logically distinct from inseparability~\cite{Guhne,Horos} and Bell nonlocality~\cite{Brunner}. In the modern view, quantum steering can be understood as the impossibility of describing the conditional states at one party by a local hidden state (LHS) model.
A fundamental property is that steering is inherently asymmetric with respect to the observers~\cite{bowles,Midgley}, which is quite different from the quantum nonlocality and entanglement. Actually, there are entangled states which are one-way steerable~\cite{bowles,Bow}. Besides its foundational significance in quantum information theory, steering has been found useful in many applications. For examples, steering has a vast range of information-theoretic applications in one-sided device-independent scenarios where the party being steered has trust on his or her own quantum device while the other's device is untrusted, such as one-sided device-independent quantum key distribution~\cite{Bran}, advantage in subchannel discrimination \cite{piani}, secure quantum teleportation \cite{Reid1,He}, quantum communication~\cite{Reid1}, detecting bound entanglement \cite{Mor}, one-sided device-independent randomness generation~\cite{law}, and one-sided device-independent self-testing of pure maximally as well as nonmaximally entangled state~\cite{supic}.
Meanwhile, the detection and characterization of steering, especially the steering inequlities, have been widely discussed. In 1989, the variance inequalities violated with EPR correlations for the continuous variable system were derived by Reid~\cite{eid}, and this was generalized to discrete variable systems~\cite{Caval}. EPR-steering inequalities were defined~\cite{can22}, where the violation of any such inequality implies steering. Following these works, further schemes have been proposed to signalize steering, for instance, the linear and nonlinear steering criteria~\cite{sau,wit,Pusey,Evan,mar,rut}, steering inequalities based on multiplicative variances~\cite{ReidRMD}, steering criteria from uncertainty relations~\cite{wa,schnee,Costaa,costab,jia,kri}, steering with Clauser-Horne-Shimony-Holt (CHSH)-like inequalities~\cite{Can3,Girdhar,cos,quan}, moment matrix approach~\cite{Kig,mo,chen00}, steering criteria based on local uncertainty relations~\cite{Ji,Zhen}, and the universal steering criteria~\cite{Zhu}. The discussed criteria or small variation thereof have been used in several experiments~\cite{sau,wit,Bennet,smith,weston}. Quite recently, the connection between quantum steering and quantum coherence was discussed~\cite{Mondal,Mondal1}.
Our work is originated from one of the open questions summarized in Ref.~\cite{rmd}: Though a complete characterization of quantum steerability has been obtained for two-qubit systems and projective measurements, it is still desirable to extend such a characterization to higher-dimensional systems. Although there is indication that such an extension is possible, much remains to be worked out. Here, we shall focus on finding the sufficient criteria for steerability with linearly steering inequalities (LSIs) \cite{can22, sau,Joness}. The LSIs have an advantage that they can work even when the bipartite state is unknown. They also have a deep relation with the joint measurement problem: If an LSI is violated, the state is steerable from Alice to Bob and the measurements performed by Alice are also verified to be incompatible~\cite{quint,ula,UULA,Kiukas,Wu}.
In this paper, according to the definitions of steering~\cite{Can1}, we develop a general scheme to design linear steering criteria for high-dimensional systems by introducing the averaged fidelity as the steering parameter. The content of this work is organized as follows. In Sec.~\ref{Sec2}, we give a brief review on the definition of steering from Alice to Bob, the Werner states and the isotropic states. In Sec.~\ref{Sec3}, a detailed introduction to the non-steering threshold is given. In Sec.~\ref{Sec4}, we address the problem of constructing linear criteria for high-dimensional systems, and an explicit LSI is constructed. Some applications of the LSI are discussed in Sec.~\ref{Sec5}. Finally, we end our work with a short conclusion.
\section{Preliminary}
\label{Sec2}
\subsection{Steering from Alice to Bob}
A bipartite state $W$ shared by Alice and Bob can be expressed by a pure (entangled) state and a one-sided linear map $\varepsilon$~\cite{horo1,Ruskai},
\begin{equation}
\label{decoposition}
W=\mathbb{I}_d\otimes \varepsilon(\vert \Psi\rangle\langle \Psi\vert),
\end{equation}
where $\mathbb{I}_d$ is an identity map. Let $\rho_{\mathrm{A}}$ be the reduced density matrix on Alice's side, and $\vert\Psi\rangle$ could be fixed as $\vert\Psi\rangle=\vert \sqrt{\rho_{\mathrm{A}}}\rangle \rangle$. The details about $\varepsilon$ and $\vert \sqrt{\rho_{\mathrm{A}}}\rangle \rangle$ are shown in Appendix~\ref{appA}. Usually, $\rho_{\mathrm{A}}$ has an eigen decomposition $\rho_{\mathrm{A}}=\sum_{i=1}^d\lambda_i\vert i\rangle\langle i\vert$, with $d$ the dimension of the Hilbert space, and $\vert\Psi\rangle=\sum _{i=1}^d \sqrt{\lambda_i}\vert i\rangle\otimes \vert i\rangle.$ In the field of quantum information and computation, entanglement is one of the most important quantum resources, and it is important to verify whether a bipartite state $W$ is entangled or not. Based on this decomposition, the state $W$ should be a mixture of products states if and only if $\varepsilon$ is entanglement-breaking (EB)~\cite{horo1,Ruskai}.
Before one can show how to demonstrate a state is steerable from Alice to Bob, some necessary conventions are required. First, Alice can perform $N$ projective measurements on her side, labeled by $\mu=1,2,...,N$, each having $d$ outcomes $a=0,1,...,d-1$, and the measurements are denoted by $\hat{\Pi}^{a}_{\mu}$, $\sum_{a=1}^{d}\hat{\Pi}^{a}_{\mu}=I_d$, with $I_d$ the identity operator on the $d$-dimensional Hilbert space. The unnormalized postmeasurement states for Bob are
\begin{equation}
\label{df}
\tilde{\rho}_{\mu}^a=\mathrm{Tr}_\mathrm{A}\left[(\hat{\Pi}^{a}_{\mu}\otimes I_d)W\right],
\end{equation}
and from the decomposition in Eq.~\eqref{decoposition}, it can be rewritten as
\begin{equation}
\label{unnormalized}
\tilde{\rho}_{\mu}^a=\varepsilon\left(\sqrt{\rho_{\mathrm{A}}}(\hat{\Pi}_{\mu}^a)^*\sqrt{\rho_{\mathrm{A}}}\right).
\end{equation}
where ``$*$'' represents the complex conjugate.
The set of unnormalized states, $\{\tilde{\rho}^{a}_{\mu}\}$, is usually called an \emph{assemblage}.
In 2007, Wiseman , Jones and Doherty formally defined quantum steering as the possibility of remotely generating ensembles that could not
be produced by \emph{a local hidden states} (LHS) model ~\cite{Wiseman1}. A LHS model refers to the case where a source sends a classical message $\xi$ to one of the two parties, say, Alice, and a corresponding quantum state $\rho_{\xi}$ to the other party, say Bob. Given that Alice decides to perform the $\mu$-th measurement, the variable $\xi$ instructs the output $a$ of Alice's apparatus with the probability $\mathfrak{p}(a\vert\mu,\xi)$. The variable $\xi$ also can be interpreted as a local-hidden-variable (LHV) and chosen according to a probability distribution $\Omega(\xi)$.
Bob does not have access to the classical variable $\xi$, and his final assemblage is composed by
\begin{equation}
\label{tilderho}
\tilde{\rho}^{a}_{\mu}=\int d\xi\Omega(\xi)\mathfrak{p}(a\vert\mu,\xi)\rho_{\xi},
\end{equation}
Note that ${\rm{Tr}}\tilde{\rho}^a_{\mu}$ is the probability that the outcome is $a$ when the $\mu$-th measurement is performed by Alice, and in the LHS model above,
there is $\mathrm{Tr}[\tilde{\rho}^a_{\mu}]=\int d\xi\Omega(\xi)\mathfrak{p}(a\vert\mu,\xi)$.
In this paper, the definition of steering is directly cited from the review article~\cite{Can1}: An assemblage is said to demonstrate steering if it does not admit a decomposition of the form in Eq.~\eqref{tilderho}. Furthermore, a quantum state $W$ is said to be steerable from Alice to Bob if the experiments in Alice's part produce an assemblage that demonstrates steering. On the contrary, an assemblage is said to be LHS if it can be written as in Eq.~\eqref{tilderho}, and a quantum state is said to be unsteerable if an LHS assemblage is generated for all local measurements.
For a given class of bipartite states, how to construct the necessary and sufficient condition of steerability is one of the elementary tasks in quantum steering. Until now, the necessary and sufficient criteria have been obtained only for the Werner states~\cite{Werner}, isotropic states~\cite{Wiseman1}, and $T$-states~\cite{jev}. In the present work, sufficient criteria for steerability will be constructed. Some known results in Refs.~\cite {Werner,Wiseman1} are required in the derivation of LSI for the continuous setting in the following. Therefore, for the convenience of readability, we would like to give a brief review of the the Werner states~\cite{Werner} and isotropic states~\cite{Wiseman1}. Especially, some formulas, which are very useful in this work, will be introduced below.
\subsection{Werner states}
The Werner states are defined as~\cite{Werner,JWD}
\begin{equation}
W^w_d=\frac{d-1+w}{d-1}\frac{I_d \otimes I_d}{d^2}-\frac{w}{d-1}\frac{\mathbf{V}}{d},
\end{equation}
with $0\leqslant w\leqslant 1$, and $\mathbf{V}$ is the ``flip" operator $\mathbf{V}\vert\psi\rangle\otimes\vert\phi\rangle=\vert\phi\rangle\otimes\vert\psi\rangle$. Werner states are nonseparable iff $w>1/(d+1).$
If Alice performs a projective measurement $\Pi^{A}_a=\vert a\rangle\langle a\vert,\ \forall a\in\{0,1,...,d-1\}$ on her side, the unnormalized conditional state on Bob's side is
\begin{equation}
\label{constate}
\tilde{\rho}^A_a=\frac{d-1+w}{d(d-1)}\frac{I_d}{d}-\frac{w}{d(d-1)}\vert a\rangle\langle a\vert.
\end{equation}
It was shown that the original derivation by Werner in Ref.~\cite{Werner} can be equivalently expressed in terms of steering~\cite{Wiseman1}. Denote the $d$-dimensional unitary group by $\mathrm{U}(d)$, and with a unitary operator $\hat{U}_{\omega}\in\mathrm{U}(d)$, an state $\vert\psi_{\omega}\rangle$ can be expressed as $\vert\psi_{\omega}\rangle=\hat{U}_{\omega}\vert 0\rangle$, where $\vert 0\rangle$ is a fixed state in the $d$-dimensional Hilbert space and $\omega$ represents the group parameters. The complete set of pure states in the $d$-dimensional system is denoted by $F^\star\equiv\{\vert\psi_{\omega}\rangle\langle\psi_{\omega}\vert d\mu_{\mathrm{Haar}}(\omega)\}$, with $d\mu_{\mathrm{Haar}}(\omega)$ the Harr measure on the group $\mathrm{U}(d)$. If Alice is trying to simulate the conditional state above, the optimal set of LHS should be $F^\star$ \cite{Wiseman1}. Formally, the simulation can be described as
\begin{equation}
\tilde{\rho}^A_a=\int d\omega\Omega(\omega)\mathfrak{p}(a\vert A, \psi_{\omega})\vert\psi_{\omega}\rangle\langle\psi_{\omega}\vert,
\end{equation}
with the constraint $\sum_{a=0}^{d-1}\mathfrak{p}(a\vert A, \psi_{\omega})=1$ and a probability distribution $\Omega(\omega)$. For an explicit conditional state in Eq.~\eqref{constate}, the optimal choice of $\{\mathfrak{p}(a\vert A, \psi_{\omega})\}$ is
\begin{equation}
\label{optp}
\mathfrak{p}^\star(a\vert A, \psi_{\omega})=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle\psi_{\omega}\vert\hat{\Pi}^A_a\vert\psi_\omega\rangle<\langle\psi_{\omega}\vert\hat{\Pi}^A_{a'}\vert\psi_{\omega}\rangle,~a\neq a'\\
0~~\mathrm{otherwise}
\end{array}\right..
\end{equation}
As shown by Werner \cite{Werner}, for any positive normalized distribution $\mathfrak{p}(a\vert A, \psi_{\omega})$, there should be
\begin{equation}
\label{inequality1}
\langle a\vert\int d\mu_{\mathrm{Harr}}(\omega)\vert \psi_{\omega}\rangle\langle \psi_{\omega}\vert \mathfrak{p}(a\vert A, \psi_{\omega})\vert a\rangle\geqslant\frac{1}{d^3}.
\end{equation}
The equality is attained for the optimal $\mathfrak{p}^\star(a\vert A, \psi_{\omega})$ specified in Eq.~\eqref{optp}. From it, it can be found that Alice cannot simulate the conditional state in Eq.~\eqref{constate} iff $(1-w)/d^2<1/d^3$.
\subsection{Isotropic states}
The isotropic states, which were introduced in Ref.~\cite{Horo}, can be parameterized similarly to the Werner states with a mixing parameter $\eta$,
\begin{equation}
\label{isotropic}
W^{\eta}_d=(1-\eta)\frac{I_d \otimes I_d}{d^2}+\eta \mathbf{P}_+.
\end{equation}
Here $\mathbf{P}_+=\vert\psi_+\rangle\langle\psi_+\vert$, where $\vert\psi_+\rangle=\sum_{i=1}^d\vert i\rangle\otimes\vert i\rangle/\sqrt{d}$ is a maximally entangled state. For $d=2$, the isotropic states are identical to Werner states up to local unitary transformations.
These states are entangled if $\eta>1/(d+1)$.
If Alice makes a projective measurement, the conditional state for Bob is
\begin{equation}
\label{constate2}
\rho^A_a=\frac{1-\eta}{d}\frac{I_d}{d}+\frac{\eta}{d}\vert a\rangle\langle a \vert.
\end{equation}
When Alice tries to simulate this state, the ensemble $F^\star:=\{\vert\psi_{\omega}\rangle\langle\psi_{\omega}\vert d\mu_{\mathrm{Haar}}(\omega)\}$ has been proved to be the optima one~\cite{Wiseman1}. Especially, the choice of the $\mathfrak{p}(a\vert A, \psi_{\omega})$
\begin{equation}
\label{optp2}
\mathfrak{p}^\star(a\vert A, \psi_{\omega})=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle\psi_{\omega}\vert\hat{\Pi}^A_a\vert\psi_\omega\rangle>\langle\psi_{\omega}\vert\hat{\Pi}^A_{a'}\vert\psi_{\omega}\rangle,~a\neq a'\\
0~~\mathrm{otherwise}
\end{array}\right.
\end{equation}
is optimal for Alice to simulate the conditional states in Eq.~\eqref{constate2}. It has been found that for any positive normalized distribution
$\{\mathfrak{p}(a\vert A, \psi_{\omega})\}$,
\begin{equation}
\label{inequality}
\langle a\vert\int d\mu_{\mathrm{Harr}}(\omega)\vert \psi_{\omega}\rangle\langle \psi_{\omega} \vert\mathfrak{p}(a\vert A, \psi_{\omega})\vert a\rangle\leqslant\frac{H_d}{d^2},
\end{equation}
where $H_d=\sum_{n=1}^d(1/n)$ is the Harmonic series and the equality is attained for the optimal $\mathfrak{p}^\star(a\vert A, \psi_{\omega})$ specified in Eq.~\eqref{optp2}. Therefore, Alice cannot simulate the conditional states iff $\eta/d+(1-\eta)/d^2> H_d/d^2$~\cite{Wiseman1}.
\section{Nonsteering threshold}
\label{Sec3}
\subsection{ Sufficient criteria for steering}
The conditional state $\tilde{\rho}^{a}_{\mu}$ on Bob's side can be measured with a set of rank-one projective operators $\{\hat{M}^a_{\mu}\}$,
$\hat{M}^a_{\mu}\equiv\hat{\Phi}^a_{\mu}=\vert \phi^a_{\mu}\rangle\langle\phi^a_{\mu}\vert,\langle\phi^a_{\mu}\vert\phi^b_{\mu}\rangle=\delta_{ab}$, $\sum_{a=0}^{d-1}\hat{M}_{\mu}^{a}=I_{d}$, and the fidelity for the $\mu$-th run of experiment is defined as
\begin{equation}
F_{\mu}=\sum_{a=0}^{d-1}\mathrm{Tr}\left[\tilde{\rho}^{a}_{\mu}\hat{\Phi}^a_{\mu}\right].
\end{equation}
Let $\langle A\otimes B \rangle =\mathrm{Tr}(A\otimes B W)$ be the expectation value of the operator $A\otimes B$, and in experiment, $F_{\mu}$ can be measured as
\begin{equation}
F_{\mu}=\sum_{a=0}^{d-1}\left\langle \hat{\Pi}^a_{\mu}\otimes \hat{\Phi}^a_{\mu}\right\rangle.
\end{equation}
Assume the probability of the $\mu$-th measurement performed by Alice is $q_{\mu}$, $\sum_{\mu=1}^N q_{\mu}=1$, and the averaged fidelity $\bar{F}$ can be defined
\begin{equation}
\bar{F}\equiv\sum_{\mu=1}^N q_{\mu}F_{\mu}.
\end{equation}
The averaged fidelity plays an important role in the detection of entanglement. Let us recall the decomposition in Eq.~\eqref{decoposition}, and now the channel $\varepsilon$ is restricted to be an EB one, denoted by $\varepsilon_{\mathrm{EB}}$. With the assemblage resulted from the EB channel, $\{\tilde{\rho }^a_{\mu}=\varepsilon_{\mathrm{EB}}(\sqrt{\rho_{\mathrm{A}}}(\hat{\Pi}_{\mu}^a)^*\sqrt{\rho_{\mathrm{A}}})\}$, and following the above definitions, one has the fidelity $F_\mu^{\mathrm{EB}}=\sum_{a=0}^{d-1}\mathrm{Tr}[\hat{\Phi}^a_{\mu}\varepsilon_{\mathrm{EB}}(\sqrt{\rho_{\mathrm{A}}}
(\hat{\Pi}_{\mu}^a)^*\sqrt{\rho_{\mathrm{A}}})]$ and the averaged fidelity $F^{\mathrm{EB}}_{\mathrm{avg}}=\sum_{\mu}q_{\mu}F_\mu^{\mathrm{EB}}$.
The classical fidelity threshold (CFT) can be defined as $\mathfrak{F}_{\mathrm{CFT}}=\max_{\varepsilon_{\mathrm{EB}}}F^{\mathrm{EB}}_{\mathrm{avg}}$, where the maximum is taken over the set of all EB channels $\{ \varepsilon_{\mathrm{EB}}\}$~\cite{Barnett1,Fuchs,Massar,Horo,Adesso1,Chir,Namiki3,Chir1,Namiki4}. If the experiment data $\bar{F}$ exceeds this threshold, $\bar{F}>\mathfrak{F}_{\mathrm{CFT}}$, one may conclude that the channel $\varepsilon$ in Eq.~\eqref{decoposition} cannot be a EB channel and the state $W$ is an entangled state.
The above idea to detect entanglement is heuristic and sheds light on the detection of steering. Similarly, by taking the averaged fidelity as the steering parameter, a steering inequality can also be constructed by just considering the measurement performed by Bob~\cite{can22,sau,Joness}. Assume that the assemblage $\{\tilde{\rho}^a_{\mu}\}$ introduced in Eq.~\eqref{df} has an LHS decomposition in Eq.~\eqref{tilderho}, and one can have $\mathrm{Tr}[\tilde{\rho}^{a}_{\mu}\hat{\Phi}^a_{\mu}]=\int d\xi\Omega(\xi)\mathfrak{p}(a\vert\mu,\xi)\mathrm{Tr}(\rho_{\xi}\hat{\Phi}^{a}_{\mu})$. With the definition of averaged fidelity above, one may
introduce
an averaged fidelity
\begin{equation}
F^{\mathrm{LHS}}_{\mathrm{avg}}\equiv\sum_{\mu=1}^N\sum_{a=0}^{d-1}q_{\mu}
\int d\xi\Omega(\xi)\mathfrak{p}(a\vert\mu,\xi)\mathrm{Tr}(\rho_{\xi}\hat{\Phi}^{a}_{\mu})
\end{equation}
for the case where the assemblage $\{\tilde{\rho}^{a}_{\mu}\}$ admits a LHS model.
Formally, it can be rewritten as $F^{\mathrm{LHS}}_{\mathrm{avg}}\equiv\int d\xi\Omega(\xi)\mathrm{Tr}(\rho_{\xi}\bar{\rho})$, with $\bar{\rho}$ defined as
\begin{equation}
\label{rhobar}
\bar{\rho}=\sum_{\mu=1}^N\sum_{a=0}^{d-1}q_{\mu}\mathfrak{p}(a\vert\mu,\xi)\hat{\Phi}^{a}_{\mu},
\end{equation}
with the probability $\mathfrak{p}(a\vert\mu,\xi)$ interpreted as the value of $\hat{\Pi}^a_{\mu}$ in the local hidden variable (LHV) model. $\bar{\rho}$ is a density matrix, and formally, can be expanded as $\bar{\rho }=\sum_{\nu}\lambda_{\nu}\vert \lambda_\nu\rangle\langle \lambda_{\nu}\vert$, with $\lambda_{\nu}$ the eigenvalues and $\vert \lambda_{\nu}\rangle$ the corresponding eigenvectors.
Defining $\lambda^{\max}=\max_{\mathfrak{p}(a\vert\mu,\xi)}\max_{\mu}\{\lambda_{\mu}\}$, and together with the facts $\mathrm{Tr}[\rho_{\xi}\bar{\rho}]\leqslant\lambda^{\max}$ and $\int \Omega(\xi)d\xi=1$, one can conclude that $\lambda^{\max}$ is an upper bound of $F^{\mathrm{LHS}}_{\mathrm{avg}}$, say, $\lambda^{\max}\geqslant F^{\mathrm{LHS}}_{\mathrm{avg}}$.
From the definition of unsteerable states, we know that the assemblage resulted from the unsteerable state always admits an LHS model. Therefore,
$\lambda^{\max}$ also can be interpreted as the upper bound of the averaged fidelity derived from the unsteerable states, when the measurement on Bob's has been fixed as $\{ q_{\mu},M^a_{\mu}\}$. To emphasize this property of $\lambda^{\max}$, we define it as the nonsteering threshold (NST) and denote it by the symbol $\mathfrak{F}_{\mathrm{NST}}^+$ hereafter,
\begin{equation}
\label{NST+}
\mathfrak{F}_{\mathrm{NST}}^+(\{q_{\mu},M^a_{\mu}\})=\max_{\vert\phi\rangle}\max_{\mathfrak{p}(a\vert\mu,\xi)}\langle\phi\vert\bar{\rho}\vert\phi\rangle.
\end{equation}
In a similar way, the minimum eigenvalue of $\bar{\rho}$ is the other NST, and denoted by $\mathfrak{F}_{\mathrm{NST}}^-$ hereafter
\begin{equation}
\label{NST-}
\mathfrak{F}_{\mathrm{NST}}^-(\{q_{\mu},M^a_{\mu}\})=\min_{\vert\phi\rangle}\min_{\mathfrak{p}(a\vert\mu,\xi)}\langle\phi\vert\bar{\rho}\vert\phi\rangle.
\end{equation}
With $ \mathrm{Tr}[\rho_{\xi}\bar{\rho}]\geqslant\mathfrak{F}^-_{\mathrm{NST}}$ and $\int \Omega(\xi)d\xi=1$, one can conclude that
$\mathfrak{F}^-_{\mathrm{NST}}$ is a lower bound of $F^{\mathrm{LHS}}_{\mathrm{avg}}$, say, $\mathfrak{F}^-_{\mathrm{NST}}\leqslant F^{\mathrm{LHS}}_{\mathrm{avg}}$. Therefore, an LSI can be defined
\begin{equation}
\label{LSI}
\mathfrak{F}_{\mathrm{NST}}^-(\{q_{\mu},M^a_{\mu}\})\leqslant\bar{F}\leqslant\mathfrak{F}_{\mathrm{NST}}^+(\{q_{\mu},M^a_{\mu}\}).
\end{equation}
Since the following two conditions---(a) the state is steerable from Alice to Bob and (b) the set of measurements $\{\Pi^a_{\mu}\}$ performed by Alice is incompatible---are necessary so that the assemblage $\{{\tilde{\rho}}^{a}_{\mu}\}$ does not admit an LHS model, one may conclude that the violation of the steering inequality, is a sufficient condition for Bob to make the statements (a) and (b).
To show a state $W$ is steerable from Alice to Bob, the extremal values of the averaged fidelity should be considered. For a fixed measurement $\{\hat{\Phi}^a_{\mu}\}_{a=0}^{d-1}$ performed by Bob, let $F_{\mu}^+$ ($F_{\mu}^-$) be the maximum (minimum) value of $F_{\mu}$ with the corresponding measurement $\{\hat{\Pi}^a_\mu\}_{a=0}^{d-1}$ performed by Alice. The extremal values of the fidelity are
\begin{equation}
\bar{F}^{\pm}(\{q_{\mu},M^a_{\mu}\})=\sum_{\mu=1}^N q_{\mu}F_{\mu}^{\pm},
\end{equation}
and obviously, $\bar{F}^-\leqslant\bar{F}\leqslant\bar{F}^+$. Now, two types of steering criteria can be introduced. For the reason which will be clarified in the following, one can define the Wiseman-Jones-Doherty (WJD) type criterion
\begin{equation}
\label{WJDtype}
\bar{F}^+(\{q_{\mu},M^a_{\mu}\})> \mathfrak{F}_{\mathrm{NST}}^+(\{q_{\mu},M^a_{\mu}\}),
\end{equation}
and the Werner-type one
\begin{equation}
\label{Wernertype}
\bar{F}^-(\{q_{\mu},M^a_{\mu}\})<\mathfrak{F}_{\mathrm{NST}}^-(\{q_{\mu},M^a_{\mu}\}).
\end{equation}
The two criteria above are independent, which means that if either is verified, the state $W$ is demonstrated to be steerable from Alice to Bob. It will be shown that both types of the criteria should be considered for the high-dimensional system ($d>2$).
\subsection{The optimal eigenvectors}
First, let us consider the probabilistic LHV model. For the $\mu$-th measurement $\{\Pi^{a}_{\mu}\}$, $\sum_{a=0}^{d-1}\Pi^{a}_{\mu}=I_d$, and
\begin{equation}
0\leqslant\mathfrak{p}(a\vert\mu,\xi)\leqslant 1,~\sum_{a=0}^{d-1}\mathfrak{p}(a\vert\mu,\xi)=1.
\end{equation}
From Eqs.~\eqref{rhobar} and~\eqref{NST+}, a quantity $f_{\mu}(\phi)$ can be introduced
\begin{equation}
\label{quantityf}
f_{\mu}(\phi)=\langle\phi\vert\sum_{a=0}^{d-1} \mathfrak{p}(a\vert\mu,\xi)\hat{\Phi}^{a}_{\mu}\vert\phi\rangle,
\end{equation}
as a function of $\vert\phi\rangle$, and for a fixed $\vert\phi\rangle$, its maximum value,
\begin{equation}
f^{\max}_{\mu}(\phi)=\max_a\langle\phi\vert\hat{\Phi}^{a}_{\mu}\vert\phi\rangle,
\end{equation}
can be obtained with the optimal choice of the probabilities $\{\mathfrak{p}(a\vert\mu,\xi)\}$,
\begin{equation}
\label{optpro}
\mathfrak{p}^\star(a\vert\mu,\xi)=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle\phi\vert\hat{\Phi}^a_{\mu}\vert\phi\rangle>\langle\phi\vert\hat{\Phi}^{a'}_{\mu}\vert\phi\rangle,~a\neq a'\\
0~~ \mathrm{otherwise}
\end{array}\right.,
\end{equation}
where $a, a'\in\{0,1, ..., d-1\}$. $\mathfrak{F}^+_{\mathrm{NST}}$ can be rewritten as
\begin{equation}
\mathfrak{F}^+_{\mathrm{NST}}=\max_{\vert\phi\rangle}\sum_{\mu=1}^N q_{\mu}f^{\max}_{\mu}(\phi).
\end{equation}
Next, one may seek the optimal state $\vert\phi_+\rangle$ corresponding to the largest eigenvalue of $\bar{\rho}$, and then, the result can be formally expressed as
\begin{equation}
\label{NST+2}
\mathfrak{F}^+_{\mathrm{NST}}=\sum_{\mu=1}^N\sum_{a=0}^{d-1}q_{\mu}\langle\phi_+\vert\mathfrak{p}^\star(a\vert\mu,\xi)\hat{\Phi}^a_{\mu}\vert\phi_+\rangle.
\end{equation}
On the other hand, $\mathfrak{F}^-_{\mathrm{NST}}$ can be derived similarly. With the optimal probabilities
\begin{equation}
\mathfrak{p}^\star(a\vert\mu,\xi)=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle\phi\vert\hat{\Phi}^a_{\mu}\vert\phi\rangle<\langle\phi\vert\hat{\Phi}^{a'}_{\mu}\vert\phi\rangle,~a\neq a'\\
0~~\mathrm{otherwise }
\end{array}\right.,
\end{equation}
the same quantity $f_{\mu}(\phi)$ in Eq.~\eqref{quantityf} achieves its minimum value
\begin{equation}
f^{\min}_{\mu}(\phi)=\min_a\langle\phi\vert\hat{\Phi}^{a}_{\mu}\vert\phi\rangle.
\end{equation}
Therefore,
\begin{equation}
\mathfrak{F}^-_{\mathrm{NST}}=\min_{\vert\phi\rangle}\sum_{\mu=1}^N q_{\mu}f^{\min}_{\mu}(\phi),
\end{equation}
and by choosing the optimal vector $\vert\phi_-\rangle$ corresponding to the minimum eigenvalue of $\bar{\rho}$, one can come to the final result
\begin{equation}
\label{NST-1}
\mathfrak{F}^-_{\mathrm{NST}}=\sum_{\mu=1}^N\sum_{a=0}^{d-1}q_{\mu}\langle\phi_-\vert\mathfrak{p}^\star(a\vert\mu,\xi)\hat{\Phi}^a_{\mu}\vert\phi_-\rangle.
\end{equation}
Until now, we have considered the case where the number of the experiment settings is finite, and the above conclusions can be easily generalized to the case where the experiment settings are continuous.
\subsection{Deterministic LHV model}
In the above discussion, a general protocol to calculate NSTs through finding the optimal eigenvectors has been constructed. From the optimal choice of $\{\mathfrak{p}(a\vert\mu,\xi)\}$, it is shown that the NSTs are unchanged if a deterministic LHV is applied. For the $\mu$-th measurement $\{\Pi^{a}_{\mu}\}_{a=0}^{d-1}$, $\sum_{a=0}^{d-1}\Pi^{a}_{\mu}=I_d$, and
\begin{equation}
\mathfrak{p}(a\vert\mu,\xi)\in\{0,1\},~~\sum_{a=0}^{d-1}\mathfrak{p}(a\vert\mu,\xi)=1.
\end{equation}
So, one may have another way to derive the NSTs, shown in the following. For the measurements $\{q_{\mu},\hat{\Phi}^a_{\mu}\}$, a density matrix can be introduced
\begin{equation}
\bar{\rho}_{k_1,k_2,...,k_N}=\sum_{\mu=1}^N q_{\mu}\hat{\Phi}_{\mu}^{k_{\mu}},
\end{equation}
where $k_\mu\in\{0,1,...,d-1\}$ for all $\mu=1,2,...,N$. There are totally $d^N$ matrices of such kind. For $\bar{\rho}_{k_1,k_2,...,k_N}$, the largest eigenvalue and the minimum eigenvalue are
\begin{eqnarray}
\lambda^{\max}_{k_1,k_2,...,k_N}&=&\max_{\vert\phi\rangle}\langle\phi\vert \bar{\rho}_{k_1,k_2,...,k_N}\vert\phi\rangle,\\
\lambda^{\min}_{k_1,k_2,...,k_N}&=&\min_{\vert\phi\rangle}\langle\phi\vert \bar{\rho}_{k_1,k_2,...,k_N}\vert\phi\rangle,
\end{eqnarray}
respectively. The NSTs can be expressed as
\begin{eqnarray}
\mathfrak{F}^+_{\mathrm{NST}}&=&\max _{k_1,k_2,...,k_N}\lambda^{\max}_{k_1,k_2,...,k_N},\\
\mathfrak{F}^-_{\mathrm{NST}}&=&\min_{k_1,k_2,...,k_N}\lambda^{\min}_{k_1,k_2,...,k_N}.
\end{eqnarray}
As an illustration, let us consider a two-settings case as a specific example. Two sets of orthogonal basis $\{\vert\phi^a_{1}\rangle\}$ and $\{\vert\phi^b_{2}\rangle\}$ with $a,b=0,1,...,d-1$ can be chosen, which are related by a unitary matrix $U$. $U_{ab}$ are matrix elements and $\vert\phi^b_2\rangle=\sum_{a=0}^{d-1} U_{ba}\vert\phi^a_1\rangle$. Fixing the probability for each setting as $q_1=q_2=1/2$ and with the deterministic LHV model, a series of states $\bar{\rho}_{a,b}=(\hat{\Phi}^{a}_1+\hat{\Phi}^b_2)/2$ can be introduced and NST can be obtained:
\begin{equation}
\mathfrak{F}^+_{\mathrm{NST}}=\max_{a,b}\lambda^{\max}_{a,b},~~\mathfrak{F}^-_{\mathrm{NST}}=\min_{a,b}\lambda^{\min}_{a,b}.
\end{equation}
For a mixed state $\rho=(\vert e_1\rangle\langle e_1\vert+\vert\varphi\rangle\langle \varphi\vert)/2$, where the state $\vert\varphi\rangle=s\vert e_1\rangle+\sqrt{1-\vert s\vert^2} \vert e_2\rangle$ with two orthogonal bases $\vert e_1\rangle$ and $\vert e_2\rangle$, its maximum eigenvalue is $\lambda^{\max}(\rho)=(1+\vert s\vert)/2$. Based on this fact, $\lambda^{\max}_{a,b}=(1+\vert U_{ab}\vert)/2$ and
\begin{equation}
\mathfrak{F}^+_{\mathrm{NST}}=\frac{1}{2}(1+\max_{a,b}\vert U_{ab}\vert).
\end{equation}
One can select out the optimal element $U^{\mathrm{opt}}_{ab}$, whose modulus $\vert U^{\mathrm{opt}}_{ab}\vert$ has the largest value, from all the unitary matrix elements. Then, $\mathfrak{F}^+_{\mathrm{NST}}=(1+ \vert U^{\mathrm{opt}}_{ab}\vert)/2$. Note that each $\bar{\rho}_{a,b}$ is a density matrix in $d$-dimensional system, and it has a total number of $d$ eigenvalues. From the definition above, $\bar{\rho}_{a,b}$ is composed of two pure states, it has two nonzero eigenvalues, $\frac{1}{2}(1\pm\vert U_{ab}\vert)$, and a number of $d-2$ eigenvalues to be zero. Therefore, $\lambda^{\min}_{a,b}=\frac{1}{2}(1-\vert U_{ab}\vert)$ for $d=2$, and $\lambda^{\min}_{a,b}=0$ for $d>2$. With the definition
$\mathfrak{F}^-_{\mathrm{NST}}=\min_{a,b}\lambda^{\min}_{a,b}$,
we have
\begin{equation}
\mathfrak{F}^-_{\mathrm{NST}}=\left\{\begin{array}{l}
\frac{1}{2}(1-\vert U^{\mathrm{opt}}_{ab}\vert)~~\mathrm{if}~d=2\\
0~~~~~~~~~~~~~~~~~~~~~~~\mathrm{if}~d>2
\end{array}\right..
\end{equation}
It is known that a set of mutually unbiased bases (MUBs) consists of two or more orthonormal bases $\{\vert\phi_{x}^a\rangle\}$ in a $d$-dimensional Hilbert space satisfying
\begin{equation}
\left\vert\langle\phi^a_x\right\vert\phi^b_y\rangle\vert^2=\frac{1}{d},~\forall a,b\in\{0,1,...,d-1\},~x\neq y,
\end{equation}
for all $x$ and $y$~\cite{mubs}. Formally, one can introduce a unitary matrix $U$ with $\vert\phi^a_x\rangle=\sum_{a=0}^{d-1}U_{ab}\vert\phi^b_y\rangle$. From the definition for MUBs, $\vert U_{ab}\vert=1/\sqrt{d},\ \forall a, b \in\{0,1, ...,d-1\}$. With $\mathfrak{F}^+_{\mathrm{NST}}=(1+1/\sqrt{d})/2$ and the averaged fidelity $\bar{F}=\frac{1}{2}\sum_{a=0}^{d-1}\sum_{\mu=1}^2\langle\hat\Pi_{\mu}^a\otimes\hat{\Phi}^{a}_{\mu}\rangle$, one can have the WJD-type steering criterion
\begin{equation}
\sum_{a=0}^{d-1}\sum_{\mu=1}^2\langle\hat\Pi_{\mu}^a\otimes\hat{\Phi}^{a}_{\mu}\rangle >1+\frac{1}{\sqrt{d}},
\end{equation}
with $\hat\Phi^a_{\mu}$ one of MUBs. This result has appeared in previous works with different approaches~\cite{Li,Zeng}.
\subsection{Geometric steering inequality}
Here, the geometric averaged fidelity, which is related to the averaged fidelity $\bar{F}$ in a simple way, can be defined as
\begin{equation}
\label{geoavg}
\bar{f}\equiv\frac{d\bar{F}-1}{d-1}.
\end{equation}
Correspondingly, one can define the so-called geometric NSTs,
\begin{equation}
\mathfrak{g}^{\pm}_{\mathrm{NST}}(\{q_{\mu},M^a_{\mu}\})=\frac{d\mathfrak{F}^{\pm}_{\mathrm{NST}}(\{q_{\mu},M^a_{\mu}\})-1}{d-1},
\end{equation}
and the criteria about Eq.~\eqref{LSI} can be equivalently expressed as the following: If the geometric inequality,
\begin{equation}
\mathfrak{g}^{-}_{\mathrm{NST}}\leqslant\bar{f}\leqslant\mathfrak{g}^{+}_{\mathrm{NST}},
\end{equation}
is violated, the state $W$ is steerable from Alice to Bob. This type of inequality is convenient for the qubit case. With $\sigma_x$, $\sigma_y$, and $\sigma_z$ the Pauli matrices and a three-dimensional Bloch vector $\mathbf{r}=r_y\mathbf{\hat{x}}+r_y\mathbf{\hat{y}}+r_z \mathbf{\hat{z}}$ ($\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ are unit vectors along coordinate axes), a density matrix can be expressed as $\rho=(I_2+\mathbf{r}\cdot\bm{\sigma})/2$, with $\mathbf{r}\cdot\bm{\sigma}=r_x\sigma_x+r_y\sigma_y+r_y\sigma_z$. The geometric length of $\mathbf{r}$ is $\vert \mathbf{r}\vert=\sqrt{r_x^2+r_y^2+r_z^2}$. Furthermore, the measurement results of Alice are usually denoted by $a=+,-$. Then, the measurements performed by Alice can be expressed as $\hat{\Pi}^{\pm}_{\mu}=(I_2\pm\mathbf{\hat{r}}_{\mu}\cdot\bm{\sigma})/2$, and the target states can be written as $\hat{\Phi}^{\pm}_{\mu}=(I_2\pm\mathbf{\hat{n}}_{\mu}\cdot\bm{\sigma})/2$, where $\mathbf{\hat{r}}_{\mu}$ and $\mathbf{\hat{n}}_{\mu}$ are unit vectors. Now, one can define a quantity $\mathfrak{A }(\mu,\xi)=\mathfrak{p}(+\vert\mu,\xi)-\mathfrak{p}(-\vert\mu,\xi)$, and by the constraint $\mathfrak{p}(+\vert\mu,\xi)+\mathfrak{p}(-\vert\mu,\xi)=1$, it can be obtained that $-1\leqslant\mathfrak{A }(\mu,\xi)\leqslant1$. In fact, $\mathfrak{A }(\mu,\xi)$ may be viewed as the predetermined value of the operator $\mathbf{\hat{ r}}_{\mu}\cdot\bm{\sigma}$ in an LHV model. With the vector $\mathbf{\bar{r}}=\sum_{\mu=1}^N q_{\mu} \mathfrak{A}(a\vert\mu,\xi)\mathbf{\hat{n}}_{\mu}$, the state $\bar{\rho}$ in Eq.~\eqref{rhobar} can be expressed as $\bar{\rho}=(I_2+\mathbf{\bar{\mathbf{r}}}\cdot\bm{\sigma})/2$, and introducing the optimal length of $\mathbf{\bar{r }}$,
\begin{equation}
\vert\mathbf{\bar{r }}\vert_{\mathrm{opt}}=\max_{-1\leqslant\mathfrak{A }(\mu,\xi)\leqslant1}\vert \bar{\mathbf{r}}\vert,
\end{equation}
the geometric NSTs can be obtained as follows
\begin{equation}
\mathfrak{g}^{\pm}_{\mathrm{NST}}=\pm\vert\mathbf{\bar{r }}\vert_{\mathrm{opt}}.
\end{equation}
With Eq.~\eqref{geoavg}, the geometric averaged fidelity can be expressed as $\bar{f}=\sum_{\mu=1}^Nq_{\mu}\langle \mathbf{\hat{r}}_{\mu}\cdot\bm{\sigma}\otimes \mathbf{\hat{n}}_{\mu}\cdot\bm{\sigma}\rangle$, and the geometric steering inequality for the qubit case becomes
\begin{equation}
\label{geoLSI}
-\vert\mathbf{\bar{r }}\vert_{\mathrm{opt}}\leqslant\sum_{\mu=1}^Nq_{\mu}\langle\mathbf{\hat{r}}_{\mu}\cdot\bm{\sigma}\otimes\mathbf{\hat{n}}_{\mu}\cdot\bm{\sigma}\rangle\leqslant\vert\mathbf{\bar{r }}\vert_{\mathrm{opt}}.
\end{equation}
For the deterministic LHV model, $\mathfrak{A}(\mu,\xi)\in\{-1,+1\}$, and introducing $2^N$ vectors $\mathbf{\bar{r}}_{\pm\pm...\pm}= \sum_{\mu=1}^N(\pm q_\mu\mathbf{\hat{n}}_\mu)$, the optimal length of $\mathbf{\bar{r }}$ can be expressed as
\begin{equation}
\vert\mathbf{\bar{r }}\vert_{\mathrm{opt}}=\max_{\pm\pm...\pm}\vert \mathbf{\bar{r}}_{\pm\pm...\pm}\vert.
\end{equation}
The known result in Ref.~\cite{sau} is recovered here.
As a simple example, let us consider the case where Bob's measurements are MUBs: $\mathbf{\hat{n}}\cdot\bm{\sigma}$ and $\mathbf{\hat{n}}_{\bot}\cdot\bm{\sigma}$, where $\mathbf{\hat{n}}$ and $\mathbf{\hat{n}}_{\bot}$ are two orthogonal unit vectors. With $q(\mathbf{\hat{n}})$
and $q(\mathbf{\hat{n}}_{\bot})$ the probabilities for each measurement, respectively, all the possible four vectors are $\mathbf{\bar{r}}_{\pm\pm}=\pm q(\mathbf{\hat{n}})\mathbf{\hat{n}}\pm q(\mathbf{\hat{n}}_{\bot})\mathbf{\hat{n}}_{\bot}$, and it is easy to calculate the geometric length for each vector $\vert \mathbf{\bar{r}}_{\pm\pm}\vert=\sqrt{ q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}$. Thus, the optimal length is
$\vert\mathbf{\bar{r }}\vert_{\mathrm{opt}}=\sqrt{ q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}$ and the geometric steering inequality above has a more explicit form
\begin{equation}
\label{explicit}
-1\leqslant\frac{q(\mathbf{\hat{n}})\langle \hat{\mathbf{a}}\otimes \mathbf{\hat{n}}\rangle}{\sqrt{q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}}
+\frac{q(\mathbf{\hat{n}}_{\bot})\langle \hat{\mathbf{b}}\otimes \mathbf{\hat{n}}_{\bot}\rangle}{\sqrt{q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}}\leqslant 1,
\end{equation}
where $\langle \hat{\mathbf{a}}\otimes \mathbf{\hat{n}}\rangle=\langle\hat{\mathbf{a}}\cdot\bm{\sigma}\otimes\mathbf{\hat{n}}\cdot\bm{\sigma}\rangle$.
If the CHSH inequality~\cite{chsh},
\begin{equation}
\label{CHSH}
-2\leqslant\langle \mathbf{\hat{a}}\otimes (\mathbf{\hat{n}}_1-\mathbf{\hat{n}}_2)\rangle +\langle \mathbf{\hat{b}}\otimes (\mathbf{\hat{n}}_1+\mathbf{\hat{n}}_2)\rangle\leqslant2
\end{equation}
is violated, the state is Bell-nonlocal. By some algebra shown in Refs.~\cite{pop,h3,Wu}, the CHSH inequality in Eq.~\eqref{CHSH} can take an equivalent form
\begin{equation}
\label{CHSH2}
-1\leqslant\vert\cos\theta\vert\langle \mathbf{\hat{a}}\otimes \mathbf{\hat{n}}\rangle +\vert\sin\theta\vert \langle \hat{\mathbf{b}} \otimes \mathbf{\hat{n}}_{\bot}\rangle\leqslant1,
\end{equation}
where
\begin{equation}
\mathbf{\hat{n}}_1-\mathbf{\hat{n}}_2=2\vert\cos\theta\vert\mathbf{\hat{n}},~~\mathbf{\hat{n}}_1+\mathbf{\hat{n}}_2=2\vert\sin\theta\vert\mathbf{\hat{n}}_\bot
\end{equation}
It could be found that Eq.~\eqref{CHSH2} is very similar to the criteria in Eq.~\eqref{explicit}. In fact, the two operators
\begin{eqnarray}
\hat{T}_{\mathrm{\mathrm{steer}}}&=&\frac{q(\mathbf{\hat{n}})\hat{\mathbf{a}}\cdot\bm{\sigma}\otimes\mathbf{\hat{n}}\cdot\bm{\sigma}
+q(\mathbf{\hat{n}}_{\bot})\hat{\mathbf{b}}\cdot\bm{\sigma}\otimes\mathbf{\hat{n}}_{\bot}\cdot\bm{\sigma}}
{\sqrt{q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}},\nonumber\\
\hat{T}_{\mathrm{CHSH}}&=&\vert \cos\theta\vert\mathbf{\hat{a}}\cdot\bm{\sigma}\otimes\mathbf{\hat{n}}\cdot\bm{\sigma}+\vert \sin\theta\vert \mathbf{\hat{b}}\cdot\bm{\sigma}\otimes\mathbf{\hat{n}}_{\bot}\cdot\bm{\sigma},\nonumber
\end{eqnarray}
are equal $\hat{T}_{\mathrm{\mathrm{steer}}}=\hat{T}_{\mathrm{CHSH}}$, under the following one-to-one mapping
\begin{equation}
\vert \cos\theta\vert=\frac{q(\mathbf{\hat{n}})}
{\sqrt{q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}},\vert \sin\theta\vert=\frac{q(\mathbf{\hat{n}}_{\bot})}
{\sqrt{q^2(\mathbf{\hat{n}})+q^2(\mathbf{\hat{n}}_{\bot})}}.\nonumber
\end{equation}
Based on the results above, one may conclude that if the geometric inequality in Eq.~\eqref{explicit} is violated, the state must be Bell- nonlocal. A similar result has also been found in \cite{Can3, Girdhar}.
\section{continuous settings}
\label{Sec4}
\subsection{Qubit case}
In the above sections, we have developed a general scheme for constructing LSIs for the discrete case. In this section, two explicit LSIs will be constructed for the case where the measurement performed by Bob has a continuous form. Before the LSIs for an arbitrary dimensional system can be derived, a detailed discussion about the qubit case is required first, and this is useful to show what are necessary to construct the LSIs.
Now, instead of the symbol $\mu$, a three-dimensional unit vector $\hat{\mathbf{n}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ with $0\leqslant\theta\leqslant\pi$ and $0\leqslant\phi<2\pi$, is employed to label Bob's measurement as $\hat{\Phi}^a_{\hat{\mathbf{n}}}$ with $a=+,-$ the outcomes. One can introduce the measure $\frac{1}{4\pi}d^2\hat{\mathbf{n}}\equiv\frac{1}{4\pi}\sin\theta d\theta d\phi$, and certainly, $\frac{1}{4\pi}\int\int d^2\hat{\mathbf{n}}\equiv\frac{1}{4\pi}\int^{2\pi}_0\int^{\pi}_0\sin\theta d\theta d\phi=1$. In general, the measurements $\hat{\Phi}^a_{\hat{\mathbf{n}}}$ have a probability distribution $q(\hat{\mathbf{n}})$, and in this work, we just consider the case that the experimental settings are equal-weighted, say, $q(\hat{\mathbf{n}})=1$. Now, the density matrix in Eq.~\eqref{rhobar} becomes
\begin{equation}
\bar{\rho}=\frac{1}{4\pi}\int\int d^2\hat{\mathbf{n}}\sum_{a}p(a\vert\hat{\mathbf{n}},\xi)\hat{\Phi}^a_{\hat{\mathbf{n}}}.
\end{equation}
Correspondingly, the expressions for its maximum and minimum eigenvalue can be obtained from Eq.~\eqref{NST+} and Eq.~\eqref{NST-}, respectively.
The set of measurements performed by Bob can be denoted by $\{\hat{\Phi}^a_{\hat{\mathbf{n}}},\frac{1}{4\pi} d^2\hat{\mathbf{n}}\}$. This set of measurements has a special property: The optimal vector $\vert\phi_+\rangle$ should be the eigenvector of the measurement which belongs to the set $\{\frac{1}{4\pi}d^2\hat{\mathbf{n}}, \hat{\Phi}^a_{\hat{\mathbf{n}}}\}$. Without loss of generality, one may fix it as an eigenvector of $\hat{\sigma}_z$, $\vert\phi_+\rangle\equiv\vert +\rangle$, where $\hat{\sigma}_z\vert \pm\rangle=\pm\vert\pm\rangle.$
As a comparison, one may recall the case where Bob's measurements are MUBs: $\mathbf{\hat{n}}\cdot\mathbf{{\bm{\sigma}}}$ and $\mathbf{\hat{n}}_{\bot}\cdot\mathbf{{\bm{\sigma}}}$, where $\vert\phi_+\rangle$ should be the eigenvector of $\bar{\mathbf{r}}_{\pm\pm}\cdot\mathbf{{\bm{\sigma}}}$. However, this property does not hold anymore. With $\hat{U}_{\hat{\mathbf{n}}}$ a unitary matrix transforming $\vert +\rangle$ to a state represented by a unit Bloch vector $\hat{\mathbf{n}}$, $\vert \phi_{\hat{\mathbf{n}}}\rangle=\hat{U}_{\hat{\mathbf{n}}}\vert +\rangle$, one may rewrite $\hat{\Phi}^a_{\hat{\mathbf{n}}}=\hat{U}_{\hat{\mathbf{n}}}^{\dagger}\vert a\rangle\langle a\vert \hat{U}_{\hat{\mathbf{n}}}$, and obtain a complete set of pure states $\{\vert \phi_{\hat{\mathbf{n}}}\rangle\}$. By some simply algebra, $\langle +\vert\hat{\Phi}^a_{\hat{\mathbf{n}}}\vert +\rangle=\langle a\vert \phi_{\hat{\mathbf{n}}}\rangle\langle
\phi_{\hat{\mathbf{n}}}\vert a\rangle$, and Eq.~\eqref{NST+2} becomes
\begin{eqnarray}
\label{NST+3}
\mathfrak{F}^+_{\mathrm{NST}}&=&\frac{1}{4\pi}[\langle +\vert(\int\int d^2\hat{\mathbf{n}}\mathfrak{p}^\star(+\vert \hat{\mathbf{n}},\xi)\vert \phi_{\hat{\mathbf{n}}}\rangle\langle\phi_{\hat{\mathbf{n}}}\vert) \vert +\rangle \nonumber\\
&&+\langle -\vert(\int\int d^2\hat{\mathbf{n}}\mathfrak{p}^\star(-\vert \hat{\mathbf{n}},\xi)\vert \phi_{\hat{\mathbf{n}}}\rangle\langle
\phi_{\hat{\mathbf{n}}}\vert) \vert -\rangle],
\end{eqnarray}
with the optimal probabilities
\begin{equation}
\label{opt1}
\mathfrak{p}^\star(a,\vert\hat{\mathbf{n}},\xi)=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle a\vert\hat{\Phi}_{\hat{\mathbf{n}}}\vert a\rangle>\langle a'\vert\hat{\Phi}_{\hat{\mathbf{n}}}\vert a'\rangle,~a\neq a'\\
0~~\mathrm{otherwise}
\end{array}\right.,
\end{equation}
where $a, a'\in\{+,-\}$, and $\hat{\Phi}_{\hat{\mathbf{n}}}=\vert\phi_{\hat{\mathbf{n}}}\rangle\langle \phi_{\hat{\mathbf{n}}}\vert$. Now, only the pure states on the northern hemisphere of the Bloch sphere ($0\leqslant\theta<\pi/2$) contribute to the first term in Eq.~\eqref{NST+3},
\begin{eqnarray}
\frac{1}{4\pi}[\langle &+&\vert(\int\int d^2\hat{\mathbf{n}}\mathfrak{p}^\star(+\vert \hat{\mathbf{n}},\xi)\vert \phi_{\hat{\mathbf{n}}}\rangle\langle\phi_{\hat{\mathbf{n}}}\vert) \vert +\rangle]\nonumber\\
&=&\frac{1}{2}\int_{0}^{\pi/2}\sin\theta d\theta\frac{1}{2}(1+\cos\theta)=\frac{3}{8},
\end{eqnarray}
while, only the pure states on the southern hemisphere of the Bloch sphere contribute to the second term in Eq.~\eqref{NST+3},
\begin{eqnarray}
\frac{1}{4\pi}[\langle &-&\vert(\int\int d^2\hat{\mathbf{n}}\mathfrak{p}^\star(-\vert \hat{\mathbf{n}},\xi)\vert \phi_{\hat{\mathbf{n}}}\rangle\langle\phi_{\hat{\mathbf{n}}}\vert) \vert -\rangle]\nonumber\\
&=&\frac{1}{2}\int_{\pi/2}^{\pi}\sin\theta d\theta\frac{1}{2}(1-\cos\theta)=\frac{3}{8}.
\end{eqnarray}
Collecting the results above together, one can obtain the NST
$\mathfrak{F}^+_{\mathrm{NST}}=3/4$.
With a suitable basis, the optimal vector $\vert\phi_-\rangle$ can be fixed as the eigenvector of $\hat{\sigma}_z$,
$\vert\phi_-\rangle\equiv\vert +\rangle, \hat{\sigma}_z\vert \pm\rangle=\pm\vert\pm\rangle$. In a similar way to the one for deriving $\mathfrak{F}^+_{{\mathrm{NST}}}$, and with the optimal probabilities,
\begin{equation}
\label{opt2}
\mathfrak{p}^\star(a,\vert\hat{\mathbf{n}},\xi)=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle a\vert\hat{\Phi}_{\hat{\mathbf{n}}}\vert a\rangle<\langle a'\vert\hat{\Phi}_{\hat{\mathbf{n}}}\vert a'\rangle,~a\neq a'\\
0~~\mathrm{otherwise}
\end{array}\right.
\end{equation}
where $a,a'\in\{+,-\}$, and one can have $\mathfrak{F}^-_{\mathrm{NST}}=1/4$.
From the derivation above, one can see that the optimal probabilities in Eqs.~\eqref{opt1} and~\eqref{opt2} play an important role in deducing the NSTs. Finally, one can come to a state-independent LSI for the qubit case,
\begin{equation}
\label{LSI1}
\frac{1}{4}\leqslant\bar{F}(\{\frac{1}{4\pi}d^2\hat{\mathbf{n}}, \hat{\Phi}^a_{\hat{\mathbf{n}}}\})\leqslant\frac{3}{4},
\end{equation}
where the measurements by Bob are fixed as $\{\frac{1}{4\pi} d^2\hat{\mathbf{n}}, \hat{\Phi}^a_{\hat{\mathbf{n}}}\}$.
\subsection{High-dimensional case}
With a set of basis vectors $\{\vert a \rangle,a=0,...,d-1\}$, the parameter $\omega$ can be used to label the experiment settings by Bob's measurements, $\hat{\Phi}^a_{\omega}=\hat{U}^{\dagger}_{\omega}\vert a\rangle\langle a \vert \hat{U}_{\omega}$, where $U_\omega$ can take all the unitary operators in the $d$-dimensional unitary group $\mathrm{U}(d)$, and $a$ represents the outcomes. It is assumed that the probability for each measurement is equal-weighted, and a Harr measure $d\mu_{\mathrm{Haar}}(\omega)$ on $\mathrm{U}(d)$ can be introduced,
$\int d\mu_{\mathrm{Haar}}(\omega)\sum_{a=0}^{d-1}\hat{\Phi}^a_{\omega}=I_d$. Formally, the measurements by Bob are denoted by $\{d\mu_{\mathrm{Haar}}(\omega),\hat{\Phi}^a_{\omega}\}$. Meanwhile, $\vert\phi_{\omega}\rangle=\hat{U}_{\omega}\vert 0\rangle$ is a pure state in the $d$-dimensional Hilbert space. Analogously as the qubit case, without loss of generality, the optimal eigenvector is chosen as $\vert\phi_+\rangle\equiv\vert 0\rangle$. Now, Eq.~\eqref{NST+2} may be rewritten into a form more appropriate for the continuous setting
\begin{equation}
\label{NST+4}
\mathfrak{F}^{+}_{\mathrm{NST}}=\sum_{a=0}^{d-1}\langle a\vert(\int d\mu_{\mathrm{Haar}}(\omega)\mathfrak{p}^\star(a\vert \omega,\xi) \vert \phi_{\omega}\rangle\langle \phi_{\omega}\vert)\vert a\rangle,
\end{equation}
where $\langle 0 \vert\hat{\Phi}^a_{\omega}\vert 0\rangle= \langle a \vert\phi_{\omega}\rangle\langle\phi_{\omega}\vert
a \rangle$ has been applied, and as a generalization of Eq.~\eqref{optpro}, the optimal probabilities are
\begin{equation}
\label{optp3}
\mathfrak{p}^\star(a\vert \omega,\xi)=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle \phi_{\omega}\vert a\rangle\langle a\vert\phi_\omega\rangle >\langle \phi_{\omega}\vert a'\rangle\langle a'\vert\phi_\omega\rangle,~a\neq a'\\
0~~\mathrm{otherwise}
\end{array}\right.
\end{equation}
with $a, a'\in\{0, 1, ...,d-1\}$.
Now, let us come back to the general results about the isotropic states in Sec.~\ref{Sec2}. One may easily verify that the result in Eq.~\eqref{optp3} is similar to the one in Eq.~\eqref{optp2}. As a direct application of the inequality in Eq.~\eqref{inequality}, the NST can be derived as
\begin{equation}
\mathfrak{F}^{+}_{\mathrm{NST}}(\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\})=\frac{H_d}{d}
\end{equation}
from Eq.~\eqref{NST+4}.
To drive the other NST, similarly, one can fix the optimal eigenvector $\vert\phi_-\rangle\equiv\vert 0\rangle$ and rewrite Eq.~\eqref{NST-1} as
\begin{equation}
\mathfrak{F}^{-}_{\mathrm{NST}}=\sum_{a=0}^{d-1}\langle a\vert(d\mu_{\mathrm{Haar}}(\omega)\mathfrak{p}^\star(a\vert \omega,\xi) \vert \phi_{\omega}\rangle\langle \phi_{\omega}\vert)\vert a\rangle,
\end{equation}
with the optimal probabilities
\begin{equation}
\mathfrak{p}^\star(a\vert \omega,\xi)=\left\{\begin{array}{l}
1~~\mathrm{if}~\langle \phi_{\omega}\vert a\rangle\langle a\vert\phi_\omega\rangle <\langle \phi_{\omega}\vert a'\rangle\langle a'\vert\phi_\omega\rangle,~a\neq a'\\
0~~\mathrm{otherwise}
\end{array}\right.
\end{equation}
where $a, a'\in\{0, 1, ...,d-1\}$. With the inequality in Eq.~\eqref{inequality1}, the other NST can be obtained:
\begin{equation}
\mathfrak{F}^{-}_{\mathrm{NST}}(\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\})=\frac{1}{d^2}.
\end{equation}
Collecting the above results together, an LSI for the continuous settings $\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\}$ takes the form
\begin{equation}
\frac{1}{d^2}\leqslant\bar{F}(\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\})\leqslant\frac{H_d}{d}.
\end{equation}
For any state $W$, if the LSI is violated, the state is verified to be steerable from Alice to Bob, and the measurement performed by Alice is also incompatible. As a special case, the LSI in Eq.~\eqref{LSI1} can be recovered from the general one above with $d=2$.
\section{applications}
\label{Sec5}
\subsection{T-state problem}
An arbitrary two-qubit state can be expressed in the standard form
\begin{equation}
W=\frac{1}{4}(I_2\otimes I_2+\mathbf{a}\cdot\bm{\sigma}\otimes I_2+I_2\otimes\mathbf{b}\cdot\bm{\sigma}+\sum_{jk}T_{jk}\mathbf{\sigma}_i\otimes \mathbf{\sigma}_j),
\end{equation}
where $\mathbf{a}$ and $\mathbf{b}$ are the Bloch vectors for Alice and Bob's reduced states, respectively, and $T$ is the correlation matrix. The T-state is a special class of two-qubit states,
\begin{equation}
W=\frac{1}{4}(I_2\otimes I_2+\sum_{j}t_j\mathbf{\sigma}_j\otimes \mathbf{\sigma}_j),
\end{equation}
where $\mathbf{a}=\mathbf{b}=\bm 0$ and $T$ is a diagonal matrix with $t_j$ the diagonal elements. In 2015, Jevtic~\emph{et. al.} gave a necessary condition of EPR steerability for $T$-states~\cite{jev},
\begin{equation}
\label{Tnec}
\frac{1}{2\pi}\int\int d^2 \hat{\mathbf{n}}\sqrt{\mathbf{\hat{n}}^{\mathrm{T}}T^{2}\hat{\mathbf{n}}}=1.
\end{equation}
The authors also conjectured that the derived condition was precisely the border between steerable and nonsteerable states, and this was later shown analytically~\cite{ngu}. Here, we shall revisit this problem from the view of LSIs.
When Bob's measurement is fixed as $\mathbf{\hat{n}}\cdot\bm{\sigma}$, the expectation $\langle \mathbf{\hat{a}}\otimes \mathbf{\hat{n}}\rangle_+\equiv\max_{\mathbf{\hat{a}}}\langle \mathbf{\hat{a}}\otimes \mathbf{\hat{n}}\rangle$ is the maximum one. Further assume that Bob's measurement is the continuous set $\{\frac{1}{4\pi} d^2\hat{\mathbf{n}}, \hat{\Phi}^a_{\hat{\mathbf{n}}}\}$, and from the definition of the geometric fidelity in Eq.~\eqref{geoavg}, one can have the maximum value of the geometric fidelity $\bar{f}^+\equiv\frac{1}{4\pi}\int\int d^2\hat{\mathbf{n}}\mathbf{\langle\hat{a}}\otimes \mathbf{\hat{n}}\rangle_+$. With the geometric NST $\mathfrak{g}^+_{\mathrm{NST}}=1/2$, which can be directly calculated from Eq.~\eqref{LSI1}, a WJD-type criterion now is constructed
\begin{equation}
\label{g2bit}
\frac{1}{2\pi}\int\int d^2\hat{\mathbf{n}}\mathbf{\langle\hat{a}}\otimes \mathbf{\hat{n}}\rangle_+ > 1.
\end{equation}
This criterion is suitable for any two-qubit state. For the $T$-state, the correlation $\langle \mathbf{\hat{a}}\otimes \mathbf{\hat{n}}\rangle=\mathbf{\hat{a}}\cdot \mathbf{\tilde{n}}$ is the inner product between the two vectors $\mathbf{\hat{a}}=(a_x, a_y, a_z)$ and
$\mathbf{\tilde{n}}=(t_x n_x, t_y n_y, t_z n_z)$. Via the Cauchy-Schwarz inequality, the optimal choice of $\mathbf{\hat{a}}$ could be $a_i=t_in_i/\sqrt{\sum_i t_i^2n_i^2}$ with $i=x,y,z$. Thus, $\langle \mathbf{\hat{a}}\otimes \mathbf{\hat{n}}\rangle_+=\sqrt{\sum_i t_i^2n_i^2}$, and obviously, $\langle \mathbf{\hat{a}}\otimes \mathbf{\hat{n}}\rangle_+=\sqrt{\mathbf{\hat{n}}^{\mathrm{T}}T^{2}\hat{\mathbf{n}}}$. Therefore, a sufficient condition for the $T$-state to be steerable from Alice to Bob becomes
\begin{equation}
\frac{1}{2\pi}\int \int d^2 \hat{\mathbf{n}}\sqrt{\mathbf{\hat{n}}^{\mathrm{T}}T^{2}\hat{\mathbf{n}}}>1,
\end{equation}
with the equality in Eq.~\eqref{Tnec} the border of it.
The T-state contains only three parameters $t_j$ $(j=1,2,3)$. Naturally, one may ask whether it is possible to obtain an analytical function $g(t_j)=\frac{1}{2\pi}\int \int d^2 \hat{\mathbf{n}}\sqrt{\mathbf{\hat{n}}^{\mathrm{T}}T^{2}\hat{\mathbf{n}}}$. This question has already been discussed in Ref.~\cite{jev}, where Eq.~\eqref{Tnec} has an equivalent form
\begin{equation}
2\pi N_{T}\vert \det T\vert =1,\nonumber
\end{equation}
with $N_{T}$ a surface integral~\cite{jev}. For the special case $t_1=t_2$, an analytical expression for $N_{T}$ has been found. However, for the general case, it is highly unlikely that one can obtain the desired analytical expression for $g(t_j)$. For the general two-qubit state, it contains more parameters than the T-states, and therefore, when the criterion in Eq.~\eqref{g2bit} is applied, some additional numerical techniques are required to calculate $\bar{f}^+$.
\subsection{Bounds of the general NSTs}
When the state is the isotropic state and a set of measurements $\{q_{\mu}, \hat{\Phi}^a_{\mu}\}$ is used by Bob to detect steering, there is a sufficient criterion that the isotropic state is steerable from Alice to Bob
\begin{equation}
\label{sufcon}
\bar{F}^+_{\eta}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})> \mathfrak{F}^+_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\}),
\end{equation}
where the subscript $\eta$ indicates that the isotropic states are considered. For the $\mu$-th setting of measurements by Bob $\{\hat{\Phi}_{\mu}^a\}_{a=0}^{d-1}$, the conditional states defined in Eq.~\eqref{constate2} can be expressed as
\begin{equation}
\tilde{\rho}^{a}_{\mu}=\frac{1-\eta}{d}\frac{I_d}{d}+\frac{\eta}{d} \hat \Psi^a_{\mu},~a\in\{0,1,...,d-1\}.
\end{equation}
The extreme values of $f_{\eta}\equiv\sum_{a=0}^{d-1} \mathrm{Tr} [\hat{\Phi}^a_{\mu}\tilde{\rho}^{a}_{\mu}]$ will be derived in the following. Obviously, the maximum value $f^{\max}_{\eta}$ can be attained if $\hat{\Phi}^a_{\mu}=\hat{\Psi}^a_{\mu}$, and $f^{\max}_{\eta}=\left[1+(d-1)\eta\right]/d$. The minimum value $f^{\min}_{\eta}$ can be attained by setting $\mathrm{Tr}(\hat{\Phi}^a_{\mu}\hat{\Psi}^{a}_{\mu})=0$, and $f^{\min}_{\eta}=(1-\eta)/d$. Moreover, these extremal values do not depend on the actual form of the measurements $\hat{\Phi}_{\mu}$, and therefore,
\begin{eqnarray}
\bar{F}^+_{\eta}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\}) &=& \frac{1+(d-1)\eta}{d},\nonumber\\
\bar{F} ^-_{\eta}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\}) &=& \frac{1-\eta}{d},\\
\bar{F}^{\pm}_{\eta} (\{q_{\mu}, \hat{\Phi}^a_{\mu}\}) &=& \bar{F}^{\pm}_{\eta} (\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\}).\nonumber
\end{eqnarray}
However, for the continuous-settings case, the criterion
\begin{equation}
\label{continuous}
\bar{F}^+_{\eta} (\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\})>\frac{H_d}{d}
\end{equation}
is different from the one in Eq.~\eqref{sufcon}. The WJD threshold $H_d/d$ has been proven to be a tight bound: If it is achieved, the conditional states should admit a LHS model~\cite{Wiseman1}. In other words, the equivalent form of Eq.~\eqref{continuous}
\begin{equation}
\frac{1+(d-1)\eta}{d}>\frac{H_d}{d},
\end{equation}
is a necessary and sufficient condition for the isotropic state to be steerable, while $[1+(d-1)\eta]/d>\mathfrak{F}^+_{\mathrm{NST}}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\})$ is just a sufficient one. Therefore, a state-independent relation does exist
\begin{equation}
\label{NST+5}
\mathfrak{F}^+_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})\geqslant\frac{H_d}{d},
\end{equation}
where the WJD threshold is a lower bound of the general $\mathfrak{F}^+_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})$. This is the reason why we call the criterion in Eq.~\eqref{WJDtype} the WJD-type one.
When the state is the Werner state and the same measurements $\{q_{\mu}, \hat{\Phi}^a_{\mu}\}$ are performed by Bob, there is a criterion which is sufficient for the Werner state to be steerable from Alice to Bob
\begin{equation}
\bar{F}^-_w( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})< \mathfrak{F}^-_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\}),
\end{equation}
where the subscript $w$ is used to indicate that only the Werner state is considered.
For the $\mu$-th run of experiment, the conditional states, as they are defined in Eq.~\eqref{constate}, can be expressed as
\begin{equation}
\tilde{\rho}^a_{\mu}=\frac{d-1+w}{d(d-1)}\frac{I_d}{d}-\frac{w}{d(d-1)}\hat{\Psi}^a_{\mu}.
\end{equation}
The extreme values of $f_{w}\equiv\sum_{a=0}^{d-1} \mathrm{Tr} [\hat{\Phi}^a_{\mu}\tilde{\rho}^{a}_{\mu}]$ can be derived as follows. The minimum value $f^{\min}_w$ can be attained if $\hat{\Phi}^a_{\mu}=\hat{\Psi}^a_{\mu}$, $f^{\min}_w=(1-w)/d$. The maximum value $f^{\max}_{w}$ is obtained by setting $\mathrm{Tr} [\hat{\Phi}^a_{\mu}\hat{\Psi}^{a}_{\mu}]=0$, and $f^{\max}_{w}=(d-1+w)/[d(d-1)]$. Furthermore, these extremal values do not depend on the actual form of $\hat{\Phi}_{\mu}^a$, and thus, there should be
\begin{eqnarray}
\label{avgf}
\bar{F}^+_{w}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\}) &=& \frac{d-1+w}{d(d-1)},\nonumber\\
\bar{F} ^-_{w}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\}) &=& \frac{1-w}{d},\\
\bar{F}^{\pm}_{w} (\{q_{\mu}, \hat{\Phi}^a_{\mu}\}) &=& \bar{F}^{\pm}_{w} (\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\}).\nonumber
\end{eqnarray}
The criterion for the continuous settings takes the form
\begin{equation}
\bar{F}^{-}_{w} (\{d\mu_{\mathrm{Haar}},\hat{\Phi}^a_{\omega}\})<\frac{1}{d^2}.
\end{equation}
The Werner threshold $1/d^2$ has been proven to be a tight bound: If it is achieved, the conditional states should admit a LHS model~\cite{Wiseman1}. In other words,
\begin{equation}
\frac{1-w}{d}<\frac{1}{d^2}
\end{equation}
is a necessary and sufficient condition for the Werner state to be steerable, while $\frac{1-w}{d}<\mathfrak{F}^-_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})$ is just a sufficient one. Therefore, one can have a state-independent relation,
\begin{equation}
\mathfrak{F}^-_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})\leqslant \frac{1}{d^2},
\end{equation}
where the Werner threshold $1/d^2$ is the upper bound of an arbitrary $\mathfrak{F}^-_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})$. So, it is reasonable that the criterion in Eq.~\eqref{Wernertype} is referred to as the Werner-type one.
\subsection{ Detecting the steerability of Werner state}
For the qubit case, one can easily verify that $\mathfrak{g}^-_{\mathrm{NST}}=-\mathfrak{g}^+_{\mathrm{NST}}$ and $\bar{f}^{-}=-\bar{f}^+$ for arbitrary measurements $\{q_{\mu}, \hat{\Phi}^{a}_{\mu}\}$. The WJD-type geometric criterion, $ \bar{f}^+>\mathfrak{g}^+_{\mathrm{NST}}$, and the Werner-type one, $\bar{f}^-<\mathfrak{g}^-_{\mathrm{NST}}$, are equivalent. Therefore, only one of the above two criteria, usually the WJD-type one, is required to detect the steerability of the two-qubits states including the Werner state for $d=2$. This equivalence can also be easily explained since $\mathfrak{F}^+_{\mathrm{NST}}+\mathfrak{F}^-_{\mathrm{NST}}=1$ holds for $d=2$. However, for the high-dimensional system, this equality does not hold any more. For the Werner state, the maximal value of the averaged fidelity is shown in Eq.~\eqref{avgf}, $\bar{F}^+_{w}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\})=(d-1+w)/[d(d-1)]$. Certainly, $\bar{F}^+_{w}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\})\leqslant1/(d-1)$. With the WJD bound $H_d/d=(1+1/2+...+1/d)/d$, one can easily check that $\bar{F}^+_{w}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\})<H_d/d$ when $d>2$. Using Eq.~\eqref{NST+5}, one can have
\begin{equation}
\bar{F}^+_{w}(\{q_{\mu}, \hat{\Phi}^a_{\mu}\})< \mathfrak{F}^+_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\}),~\mathrm{if} ~d>2.
\end{equation}
So, if $d>2$, the steerability of the Werner state cannot be detected by the WJD-type criterion $\bar{F}^+>\mathfrak{F}^+_{\mathrm{NST}}( \{q_{\mu}, \hat{\Phi}^a_{\mu}\})$. This is the reason why both types of the criteria in Eqs.~\eqref{WJDtype} and~\eqref{Wernertype} are required for high-dimensional systems.
\subsection{A criterion with entanglement fidelity}
In the discussions above, it has been shown that it is usually difficult to calculate the extremal values $\bar{F}^+$ of the fidelity, even for the two-qubit state. With a maximally entangled state $\vert \psi_+ \rangle =\frac{1}{\sqrt{d}}\sum_{k=1}^{d}\vert kk\rangle$, a more convenient criterion may be constructed for the kind of states
\begin{equation}
W_{\varepsilon}=\mathbb{I}_d\otimes \varepsilon(\vert \psi_+\rangle\langle \psi_+\vert),
\end{equation}
which have been used for the discussion about distillation protocols~\cite{Horo}. The entanglement fidelity of $\varepsilon$ is defined as~\cite{Schu}
\begin{equation}
f(\vert\psi_+\rangle, \varepsilon)=\langle\psi_+\vert W_{\varepsilon}\vert\psi_+\rangle,
\end{equation}
which provides a measure of how well the entanglement is preserved by $\varepsilon$. For the state $W_{\varepsilon}$, the measurement $\hat{\Pi}^a_{\omega}$ performed by Alice can be fixed as $(\hat{\Pi}^a_{\omega})^*=\hat{\Phi}^a_{\omega}$, and with $\hat{\Phi}^a_{\omega}=U^{\dagger}_{\omega}\vert a\rangle \langle a\vert U_{\omega}$, one can obtain $\hat{\Pi}^a_{\omega}\otimes\hat{\Phi}^a_{\omega}=(U_{\omega}^*\otimes U_{\omega})^{\dagger}(\hat{P}_{a}\otimes \hat{P}_a)U_{\omega}^*\otimes U_{\omega}$, where $\hat{P}_{a}=\vert a\rangle \langle a\vert$. Now, the averaged fidelity $\bar{F}$ becomes
\begin{equation}
\label{fen}
\bar{F}=\sum_{a=0}^{d-1}\mathrm{Tr}[\hat{P}_{a}\otimes \hat{P}_a\int d\mu_{\mathrm{Harr}}(\omega)U_{\omega}^*\otimes U_{\omega}W_{\varepsilon}(U_{\omega}^*\otimes U_{\omega})^{\dagger}].
\end{equation}
For a Hermitian operator $\hat{A}$ in a $d$-dimensional Hilbert space, one can define a depolarizing channel $\varepsilon_{\eta}$ as
\begin{equation}
\varepsilon_{\eta}(\hat{A})=\eta\hat{A}+(1-\eta)\mathrm{Tr}(\hat{A})\frac{I_d}{d},
\end{equation}
and the isotropic states in Eq.~\eqref{isotropic} can be expressed as
$W^{\eta}_d=\mathbb{I}_d\otimes \varepsilon_{\eta}(\vert\psi_+\rangle \langle \psi_+\vert)$. As shown in Ref.~\cite{Horo}, an isotropic state can be obtained from $W_{\varepsilon}$ with the twirling procedure
\begin{equation}
W^{\eta}_d=\int d\mu_{\mathrm{Harr}}U_{\omega}^*\otimes U_{\omega}W_{\varepsilon}(U_{\omega}^*\otimes U_{\omega})^{\dagger}.
\end{equation}
Putting this result into Eq.~\eqref{fen}, one can have $\bar{F}=\eta+(1-\eta)/d$. Moreover, with the entanglement fidelity of the depolarizing channel $f(\vert \psi_+\rangle, \varepsilon_{\eta})=\eta+(1-\eta)/d^2$, one can come to $(d+1)\bar{F}=d f(\vert \psi_+\rangle, \varepsilon_{\eta})+1$. With the fact that the entanglement fidelity is invariant under the twirling procedure, say, $f(\vert \psi_+\rangle, \varepsilon)=f(\vert \psi_+\rangle, \varepsilon_{\eta})$, finally,
\begin{equation}
\bar{F}=\frac{d f(\vert \psi_+\rangle, \varepsilon)+1}{d+1}.
\end{equation}
Now, the criterion $\bar{F}> H_d/d$ can be reexpressed as
\begin{equation}
\label{fc}
f(\vert \psi_+\rangle, \varepsilon)>\frac{1}{d}\left(\frac{d+1}{d}H_d-1\right),
\end{equation}
which is a sufficient condition for $W_{\varepsilon}$ to be steerable from A to B.
Here, we say that a channel $\varepsilon$ is entanglement preserving (EP) if it is not an EB channel. It is shown in Appendix~\ref{appB} that a sufficient condition for the EP channel is
\begin{equation}
f(\vert \psi_+\rangle, \varepsilon)>\frac{1}{d}.
\end{equation}
Noting that $[(d+1)H_d/d-1]/d>1/d$, one may also apply the criterion in~Eq.~\eqref{fc} as a sufficient condition for $\varepsilon$ to be an EP channel. However, not every EP channel can be applied for constructing a steerable $W_{\varepsilon}$. For example, when
\begin{equation}
\frac{1}{d}<f(\vert \psi_+\rangle, \varepsilon_{\eta})\leqslant \frac{1}{d}\left(\frac{d+1}{d}H_d-1\right),
\end{equation}
the depolarizing channel is EP but the isotropic state is un-steerable from A to B.
\section{Conclusions}
\label{Sec6}
According to the fundamental idea that a steering inequality can be constructed by just considering the measurements performed by Bob, proposed in Refs.~\cite{can22,sau,Joness}, and from the definitions of steering from Alice to Bob \cite{Can1}, we have developed a general scheme for designing linear steering criteria for high-dimensional systems. For a given set of measurements (on Bob's side), we have defined two quantities, the so-called nonsteering thresholds. If the measured averaged fidelity exceeds these thresholds, the state shared by Alice and Bob is steerable from Alice to Bob, and the measurements performed by Alice are also verified to be incompatible. Within the general scheme, we also constructed a LSI when the set of measurements performed by Bob has a continuous setting. In the derivation of this LSI, the results in Refs~\cite{Wiseman1,Werner} have been applied. Two kinds of criteria, the WJD type and Werner type, can be applied as the sufficient conditions of steerability for bipartite state. For the qubit case, it has been shown that the two types of steering criteria are equivalent to each other. However, when $d>2$, these criteria have different properties.
The LSI in this work is limited for the case where the set of measurements by Bob has a continuous and equal-weighted form. From the view of experiment, the LSIs with a finite number of experimental settings are required. We leave such kinds of LSIs, especially adapted to the Werner state, as our future works. We expect that the results in this work could lead to further theoretical or experimental consequences.
\acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants No.~12047576 and No.~11947404.
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– dawny powiat w Japonii, w prefekturze Fukuoka. W 2005 roku liczył mieszkańców.
Miejscowości
Hirokawa
Historia
Powiat został założony 1 kwietnia 1896 roku w wyniku połączenia powiatów Takeno (1 miejscowość, 6 wiosek) i większości powiatu Ikuha (1 miejscowość, 8 wiosek).
1 kwietnia 1929 – w wyniku połączenia wiosek 椿子村 i Ukiha powstała wioska Miyuki. (2 miejscowości, 13 wiosek)
1 stycznia 1951 – wioska Miyuki zdobyła status miejscowości. (3 miejscowości, 12 wiosek)
1 kwietnia 1951: (3 miejscowości, 8 wiosek)
miejscowość Miyuki zmieniła nazwę na po przyłączeniu wiosek Himeharu, Yamaharu i Ōishi.
w wyniku połączenia wiosek 川会村 i Shibakari powstałą wioska Chikuyō.
1 grudnia 1954 – miejscowość Tanushimaru powiększyła się o teren wiosek Mizuwake, Chikuyō, Mizunawa, Takeno oraz części wsi Funagoshi. (3 miejscowości, 4 wiosek)
1 stycznia 1955 – miejscowość Yoshii powiększyła się o teren wiosek Funagoshi, 江南村, Fukutomi i Chitose. (3 miejscowości)
5 lutego 2005 – miejscowość Tanushimaru została włączona w teren miasta Kurume. (2 miejscowości)
20 marca 2005 – w wyniku połączenia miejscowości Ukiha i Yoshii powstało miasto . W wyniku tego połączenia powiat został rozwiązany.
Przypisy
Dawne powiaty w prefekturze Fukuoka
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 3,338
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Isopropylnitrat är en ester av isopropanol och salpetersyra. Det är en färglös vätska som används som enkomponentsbränsle och för att höja cetantalet för diesel. Isopropylnitrat är mycket brandfarligt och brinner med osynlig låga. Det har tidigare använts som startbränsle för jetmotorer. På grund av det låga syreinnehållet så sönderfaller isopropylnitrat till en mängd olika ämnen när den används som enkomponentsbränsle, bland annat isopropylnitrit, acetaldehyd, nitrometan, koldioxid och kvävedioxid. Vid förbränning i luft förbränns även sönderfallsprodukterna till koldioxid, kvävedioxid och vattenånga.
Sedan 1970-talet har Ottobränsle 2 till stor del ersatt isopropylnitrat som enkomponentsbränsle.
Källor
Nitrater
Estrar
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Dania Flea Market moves to bigger venue
Moving the event to Dolphins Stadium will allow for 300 more vendors, additional parking, say organizers
The Dania Marine Flea Market, a South Florida institution, is moving to Dolphins Stadium from Dania Jai Alai, its home for 27 years.
This will give the marine flea market — one of the largest of its kind in the world — room to grow, says owner and founder Al Behrendt.
"We've had a waiting list [for vendor space] the last three years," he says. The move will enable him to add 300 vendor spaces — bringing the number to 1,300 — and ease the parking crunch, which had become critical. Behrendt says people were staying home because on the flea market's busiest days they were having to park up to a mile away.
"There will be unlimited free parking at Dolphins Stadium," he says.
The flea market, scheduled for March 30 - April 2, offers a treasure-trove of marine stuff — new and used marine parts and accessories, diving and fishing gear, clothing, and even some boats. Exhibitors range from boaters who are clearing out their garages to manufacturers who are clearing out inventory, old parts and products at discount prices. Behrendt says he expects for the first time at least two new-boat dealers at the market, as well as the usual used-boat sellers.
Behrendt started the flea market in 1979 with 56 vendor spaces on the grounds of Dania's jai alai fronton. It is moving about 15 miles to the grounds of the Miami Dolphins' football stadium in Miramar.
Behrendt says the atmosphere won't change with the new digs. It will remain relaxed and social, and a big draw for hard-core boaters.
He says the serious buyers go to the market with their parts list. "They've got the part number. They've got the model number," he says. "They know exactly what they're looking for, and nine times out of 10 they find it. As your boat gets older, as your engine gets older, it's a great place to find the parts you need."
Tickets are $10, $12 on opening day, Thursday.
Behrendt will continue to call it the Dania Marine Flea Market, even though the event is no longer in Dania. "The Miami International Boat Show is not really in Miami," he says. "It's in Miami Beach."
Maine boatbuilder moving to bigger home
Hunt Yachts moves; will build bigger boats
Marine flea market returns to South Florida
Florida marine flea market starts Thursday
Gigantic nautical flea market delivers bargains
Electronics - Navigating the electronics market
Gigantic Nautical Flea Market set to open
Florida marine flea market returns in March
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,089
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Montlaur is een plaats en voormalige gemeente in het Franse departement Aude, in het zuiden van Frankrijk. De gemeente ligt op 210 m hoogte en beslaat een oppervlakte van 3392 ha.
Ze ligt op 24 km van Carcassonne en 16 km van Lagrasse. De gemeente had in 1999 een inwonertal van 522.
Deze landelijke gemeente ligt in de streek Corbières in het Val de Dagne in een mooie vallei aan de voet van de bergketen Montagne d'Alaric. Het is een echte wijnbouwgemeente met nog tal van kleine productiehuizen.
Geschiedenis
Het dorp bestaat zeker sinds de achtste eeuw en was het terrein van veldslagen tussen Karel de Grote en de Saracenen. Van de plek van een van die veldslagen (de "Mont des Lauriers") is de naam van het dorp afgeleid.
Van de versterkte burcht die gebouwd werd in de 12e eeuw resten momenteel enkel nog de vestingen.
Tijdens de woelige tijd van de Katharen werd Montlaur in 1210 door Simon de Montfort ingenomen, die het gebied doorgaf aan een leenheer, die het op zijn beurt doorgaf aan de Franse koning. In 1283 verkocht ridder Simon de Melun zijn eigendommen aan de abdij van Lagrasse, die dit gebied gebruikte tot de Franse Revolutie.
Montlaur is op 1 januari 2019 gefuseerd met de gemeente Pradelles-en-Val tot de gemeente Val-de-Dagne.
Demografie
Onderstaande figuur toont het verloop van het inwonertal (bron: INSEE-tellingen).
Toerisme
In het dorp bevinden zich een park, horend bij een 19e-eeuws kasteel, de sporen van de middeleeuwse burcht en twee originele windmolens.
Plaats in Aude
Val-de-Dagne
Voormalige gemeente in Aude
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Shakespeare Quote - "There are more things in heaven and earth"
Although set in different times many of the most famous quotes about life and love by William Shakespeare are still relevant today. Did you know that William Shakespeare is credited by the Oxford English Dictionary with the introduction of nearly 3,000 words into the language. It's no wonder that expressions from his works in literature, including the "There are more things in heaven and earth" quote, are an 'anonymous' part of the English language. Many people continue to use this "There are more things in heaven and earth" quote by William Shakespeare in famous quotes about life.
"There are more things in heaven and earth"
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{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
A Brownian motor (BM) converts random fluctuations into deterministic work \cite{BM1 , BM2 , BM3}.
Brownian motors exits naturally as, \emph{e.g.}, protein motors and intra-cell motion \cite{bioBM}, and a general understanding of their mechanism is of fundamental interest. The rectification mechanism requires that the system is both brought out of thermal equilibrium \cite{BM1} and spatially or temporally asymmetric \cite{Curie , BM1}. Although it has not been proven, fulfilling these two requirements are generally sufficient to realise a BM.
These requirements can be met using ultra cold atoms trapped in optical lattices \cite{lin-p-lin1 , OL}. Generally, the symmetry is broken by using a spatially asymmetric or flashing potential. In this type of noise rectifier, the direction of the induced drift is often fixed for a given potential or controllable in just 1D \cite{BM1 , BM2 , BM3 , r4 , RenzoniBM , RenzoniBM2 , RenzoniBM3 , GrynbergBM}.
Our BM \cite{OurBM1 , newBM} is based on two symmetric potentials, with an asymmetry that originates from a combination of a relative spatial phase shift between the potentials and an unequal transfer rate between them. Our system possesses an inherent ability to induce drifts in an arbitrary direction in three dimensions for the ultra cold atoms interacting with the potentials. The direction is mainly controlled by the relative spatial phase between the two potentials. \revision{In \cite{OurBM1 , newBM} we demonstrated that our BM does work in more} \revision{than one dimension, but how this works was not investigated. This question is especially complicated since the four-beam configuration induces a coupling between the different spatial dimensions.
This coupling affects the 3D behaviour of our BM profoundly and is the key to any understanding and control of the three-dimensional aspects of our BM.}
\revision{In this paper, we present a study of the dimensional coupling between the relative spatial phases and its influence on the induced drift, in two dimensions}. This is done both experimentally and with numerical simulations. \revision{This study results in a better understanding of the multidimensionality of our BM, and in a good control to set the induced drift to a chosen speed and direction.}
This also renders a controlled dynamical BM possible, where a time-dependent phase can induce realtime drifts along any pre-chosen trajectory. \revision{It even allows to control the induced drift in the vertical direction by purely controlling the phase shift horizontally, and vice versa.}
By studying the phase dependence, we can also study the coupling between the dimensions of the two potentials. Information about what potentials the atoms really experience and the adiabaticy \cite{OL} of the interaction can therefore be investigated. \revision{The numerical simulations are done with a simplified, classical model, which mimics the main features of our BM. This qualitatively reproduce the main features of the results from the experiments.}
\section{System}
The potentials used are realised from a double optical lattice (DOL) \cite{DOL , setups}, interacting with $^{133}$Cs.
An optical lattice is a periodic potential where atoms can be cooled and trapped due to dissipation and light shifts \cite{lin-p-lin1}, and is created from the interference between two or more light fields. A DOL consists of two spatially overlapped state-dependent optical lattices \cite{DOL , setups}.
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{rat9.pdf}
\caption{(Colour online) Simplified, classical model of the BM mechanism. (a) $\varphi = 0$, the system is spatially symmetric and the particle undergoes Hamiltonian oscillation at the bottom of the wells. No drift is induced. (b) $\varphi \ne 0$, the spatial-temporal symmetry is broken and a drift $v$ to the left is induced. \revision{In both cases there is a large difference in the transfer rates $\gamma$ between the lattices, indicated with arrow of different thickness in the figure. Note that this a strongly simplified model, presented to provide an understanding of the phase dependence. Features such as diffusion and friction, vital for the BM, are not present in this model}.}
\label{idea}
\end{figure}
To qualitatively understand the induced drift dependence on the relative spatial phase, a simple classical model is used \cite{BMsim , ratide}, see figure \ref{idea}.
Consider a classical Brownian particle situated in either of two symmetric and periodic potentials \revision{($U_\mathrm{A}$ and $U_\mathrm{B}$)}, coupled with unequal transfer rates ($\gamma_\mathrm{A\rightarrow B}$ and $\gamma_\mathrm{B\rightarrow A}$). The particle will shift back and forth between the potentials, but on average spend longer time in one of them. When the potentials are in phase, the trapped particle will jump between potentials and undergo Hamiltonian oscillations near the bottom of them, see figure \ref{idea}a. If the potentials are given a non-zero relative spatial phase shift, see figure \ref{idea}b, with transfer rates properly chosen, the particle will spend long enough time in the long-lived potential in order to be located close to it{s} minimum. After a certain time the particle is transferred to the short-lived potential, where it will be located at a slope of the potential. After a short while, it will be pumped back to the long-lived potential. The particle will therefore, on average gain kinetic energy in a certain direction when jumping between the potentials \cite{ratide}. \revision{The} \revision{optimal ratio between the transfer rates has been experimentally determined to $\gamma_\mathrm{A\rightarrow B}: \gamma_\mathrm{B\rightarrow A} \simeq 9:1$ \cite{newBM}. This is also related to the time atoms spend in each potential. The time spent in potential A should optimally be of the order of the the inverse of this potential's oscillation frequency \cite{phd peder}, while the optimal time spent in potential B is much larger. The oscillation frequencies are typically of the order of 100 kHz.}
\section{Methodology}
Detailed descriptions of the general experimental setup are given in \cite{Johan2002 , Harald2002 , setups , DOL}. More details concerning the construction and the control of the BM are found in \cite{OurBM1 ,newBM , balance}. In short, cesium atoms are first accumulated in a magneto-optical trap, from where they are transfered to the double optical lattice.
Both lattices (A and B) have a 3D topography and are constructed from four beam configurations, in which two beams are propagating in the \emph{xz}-plane, and two in \emph{yz}-plane. All beams have a $45\,^{\circ}$ angle with respect to the vertical \emph{z}-axis with polarizations perpendicular to the plane of propagation ({``3D lin$\perp$lin configuration'' \cite{lin-p-lin1}}). The lattices are state-dependent and operate from different hyperfine ground states within the Cs D2 line (6s $^2$S$_{1/2}$ $\rightarrow$ 6p $^2$P$_{3/2}$). Lattice A operates close to the $F_\mathrm g = 3 \rightarrow F_\mathrm e = 4$ transition and lattice B operates close to the $F_\mathrm g = 4 \rightarrow F_\mathrm e = 5$ transition.
The irradiances and the frequencies of the four beams are chosen to generate pumping rates between the lattices that maximise the induced drift.
The optical path lengths are controllable in all beams, resulting in a control of the relative spatial phase. This dependence originates from a slight difference in wavelength between the lattices. The difference is small enough for the relative spatial phase to be effectively the same throughout the whole atomic cloud \cite{DOL , balance}.
The induced drift velocity is obtained by a ballistic time-of-flight
method (TOF), where the trapped atoms are released by quickly turning
off the lattice beams. The atoms then freely fall through a resonant
probe located approximately 5 cm below the trapping region. The
fluorescence from the atoms passing through the probe is detected and
gives a measurement of the arrival time $t$ of the atomic cloud
\cite{TOF , sisyphus}. Since the drift velocity is constant
\cite{OurBM1}, the arrival time can be converted into drift velocity
through
\begin{equation}
v_z = \frac{gt^2 - 2s_\mathrm{p}}{2(t+\tau)},
\end{equation}
where $g$ is the gravitational constant, $\tau$ is the interaction time of the atoms with the DOL and $s_\mathrm{p}$, is the distance to the probe.
\section{Results}
To investigate the relative spatial phase dependency experimentally, we change the phase along \emph z ($\varphi_z$) and \emph y ($\varphi_y$) by slightly more than 2$\pi$. This is done in about 20 steps in each direction, covering all steps in \emph{z} for each \emph{y} value. Here, $\varphi_{z,y} = 0$ corresponds to a perfect overlap of the two lattices. Since these intervals exceed $2\pi$, whole periods along both the \emph{z} and \emph{y} axes are covered.
For each combination phases, the TOF signal is measured. To improve statistics, an average over five TOF measurements is made for each combination. A compilation of the measurements is shown in figure \ref{2dratchet}. Also displayed is the temperature measured at each step, which is used to determine the phase.
\begin{figure}[htbp]
\centering
\includegraphics[width=8.7cm]{v17.pdf}
\\
\centering
\includegraphics[width=8.7cm]{t16.pdf}
\caption{(Colour online). (a) Measured induced drift, and (b) temperature, as a function of the relative spatial phase along the \emph{y}- and \emph{z}-axes. A difference in the pattern created by the induced drift maxima and minima is evident. The dashed lines indicates the cuts shown in figure \ref{cuts}.}
\label{2dratchet}
\end{figure}
The origin, where $\varphi_{y,z}$ is zero, is chosen at a minimum of the temperature, since a perfect spatial overlap of the lattices is assumed to correspond to a temperature minimum \cite{DOL , setups}. The scale is obtained by measuring the periodicity of the temperature and comparing it to the change of the optical path lengths. The measurement indicates that the temperature maxima occur when the diabatic potentials \cite{lin-p-lin1 , OL} are shifted with a relative spatial phase of $\pi$. Since the induced drift velocities and the temperature are measured simultaneously, the relative phase scale obtained from the temperature is applied to the BM.
Throughout the measurement, lattice B is red-detuned below resonance by $40\Gamma$ and has an irradiance of 0.2 mW/cm$^2$ in each beam and lattice A is red-detuned by $33\Gamma$ and has an irradiance of 0.7 mW/cm$^2$ in each beam, where $\Gamma$ is the natural line width of the excited state ($\Gamma/2\pi = 5.2$ MHz).
These parameter values are chosen to optimize the induced drift \cite{newBM} while still having a strong TOF signal with a good signal to noise ratio.
%
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{zcut20.pdf}
\includegraphics[width=8cm]{ycut20.pdf}
\includegraphics[width=8cm]{dcut20.pdf}
\caption{Cuts in the two dimensional plot of figure \ref{2dratchet}a. (a) Cut along \emph{y}-axis, at $\varphi_z \simeq \frac{1}{3}\pi$. A clear splitting of the minima is visible at $\varphi_y \simeq \pi$ (b) Cut along \emph{z}-axis, at $\varphi_y \simeq 0$. The induced drift shows a close to sinusoidal shape. (c) A diagonal cut, at $\varphi_z \simeq \varphi_y$. A doubled frequency is evident.}
\label{cuts}
\end{figure}
Even though the induced drift has the same periodicity, the topography along the \emph{z} and \emph{y} axes differ. Along the \emph{z} axis, the drift has a close to sinusoidal dependence on $\varphi_z$, for any choice of $\varphi_y$ ({see figure \ref{cuts}b), although the amplitude and the phase of the drift velocity strongly depend on the relative spatial phase along the \emph{y} axis.
Along the \emph y axis, however, the induced drift clearly deviates from a sinusoidal shape, having two minima per period (see fig \ref{cuts}a). \revision{Such a difference between the \emph z and \emph y intersections is expected since the lattice structure differs slightly along these directions \cite{lin-p-lin1}}.
By moving diagonally in the \emph{yz} phase plane, the periodicity is doubled (see fig \ref{cuts}(c)).
In order to understand the qualitative behaviour of our Brownian motor, we have performed simulations, using a simplified, \revision{classical} model of our system \cite{BMsim , ratchet2 , ratide}.
The two-dimensional numerical simulations are done for a classical Brownian particle situated in either of two potentials, corresponding to the lowest diabatic potential \cite{BMsim , DOL} of each lattice in a DOL configuration, identical to the one used in the experiments. The dynamics of the atom is obtained from the Fokker-Planck equation \cite{Fokker1 , Fokker2}.
The induced drifts in the \emph z and \emph y directions as a function of the relative spatial phase are shown in figure \ref{sim}.
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{sim7.pdf}
\includegraphics[width=8cm]{arrowsim5.pdf}
\caption{(Colour online) Simulation of the induced drift as a function of the relative spatial phases $\varphi_z$ and $\varphi_y$. The speed is given in mm/s. (a) False colour contour plot of induced drift in \emph z. (b) Arrow plot of the induced drift in \emph z and \emph y. The arrows indicate the size and the direction of the induced drift. \revision{The phase at wich the velocity reaches its maximum differs along the \emph z and \emph y directions.
This agrees with the result obtained in \cite{newBM}.}}
\label{sim}
\end{figure}
The simulations show a clear periodic behaviour in both the \emph z and \emph y directions, and the fact that the induced drift maxima in positive and negative directions do not lie on a straight line along the \emph y axis shows that the potentials in different directions are coupled. \revision{That is, it is not possible to go from a maxima to a minima by purely changing $\varphi_y$, $\varphi_z$} must vary as well. This is also visible in the equation describing the potentials,
\begin{eqnarray}
\lefteqn{U_\pm = \frac{4\hbar\Delta^{\prime}_0}{45}\{23[\cos^2(k_x x) + \cos^2(k_y y) ] {} }
\nonumber\\
& & {}\mp 44\cos(k_x x)\cos(k_y y)\cos(k_z z)\},
\end{eqnarray}
where $\Delta_0^{\prime}$ is the scattering rate and \emph{k} is the effective angular wave vectors along the axes \cite{lin-p-lin1}.
This equation also shows that the potentials are identical in \emph x and \emph y, which would give identical relative spatial phase dependencies in \emph x and \emph y.
The simulations agree with the measurement, where a clear spatial coupling between the potentials is evident. Both also show a clear pattern, and even though the pattern does differ, this confirms that we can control our BM via the phase in arbitrary directions. \revision{While an extension of the simulations to 3D is conceptually straightforward, it would be computationally demanding. The third dimension would, if anything, just enhance the coupling observed in 2D. Moreover, while this model contains the basic ingredients of our BM (diffusion, friction, coupled potentials, etc.), it neglects potentially important factors (such as the spatial dependence of diffusion and friction or the presence of a manifold of potentials due to the magnetic substates), so no far reaching conclusions should be drawn from the comparison of the simulations with the experimental results.}
\section{Discussion}
The induced drift of our BM can be controlled by the difference between the transfer rates between the lattices, the diffusion, the potential height and the relative spatial phase between the lattices. The dependence of the first three parameters is investigated in \cite{newBM , OurBM1 , ratchet2}. The phase dependency was there fully investigated for drifts measured along only in one dimension, \revision{and phases varied along the same direction. Drifts }\revision{in more than one dimension were demonstrated but drifts induced by a phase shift in a perpendicular direction were not investigated at all. Since the potentials couples the different directions, the non-trivial relation between the phase shift of the lattices and the directionality of the induced drift is at the heart of the multidimensionality of our BM. Such an investigation is therefore of fundamental interest and} \revision{necessary obtain any understanding or control of our BM.}
By surveying the vertically induced drift by $2\pi\times 2\pi$ in the \emph{yz} phase plane, \revision{the influence of the lattice topography on} the structure and periodicity of the drift can be determined. Due to the identical potentials in the \emph x and \emph y directions, a generalisation to three dimensions is possible. The induced drift dependence on the transfer rates, \emph{i.e.}, the irradiances and detunings, has earlier been investigated \cite{newBM , OurBM1}, and thus we can now fully describe our three-dimensional BM, and with good control set the induced drift velocity to any arbitrary direction, with a controllable magnitude in three dimensions. Such control is necessary in future experiments where fast phase shifts should render it possible to realise a dynamical Brownian motor, where the atoms can change direction in real time and therefore travel along more or less any pre-chosen trajectory, limited only by the lifetime of the DOL-caused diffusion. These fast phase shifts can be realised with electro-optical modulators (EOM). The implementation of these EOMs also makes it possible to scan the phase over a range of 2$\pi$ in all three dimensions simultaneously, which is not possible with the current set-up.
The multidimensional \revision{coupling between the} phase dependency \revision{of the lattice topography and the induced drift} render it possible to redirect the motion in a certain direction without changing the phase in that direction. The induced drift can, by changing the phase in an orthogonal direction, be enhanced, put to zero or even be inverted. \revision{This is clearly evident in figure \ref{2dratchet}(a), and has not been shown before.}
The measured periodicity of the temperature also confirms that the diabatic potentials are the relevant potentials in the system, as shown in \cite{newBM}.
The two potentials in the classical simulations are therefore chosen as the two lowest diabatic potentials of \revision{the two} state-dependent potentials of the DOL used in the experiment \cite{BMsim}. The simulation
qualitatively reproduces the main features of our BM although it is based on a classical model. A closer comparison with the experimental result show clear differences, especially the location of the maxima in induced drift velocity in the \emph z direction. This shows that the simple classical picture is inadequate for extracting more detailed information about our system, and that not just the lowest diabatic potential is relevant. In reality an optical lattice works on a manifold of potentials, each corresponding to a transition between different magnetic sub-states \cite{OL}.
This manifold of potentials is also of importance in the model explaining the relative spatial phase dependence of the temperature in a DOL \cite{setups}, seen in figure \ref{2dratchet}b. The dependence of the temperature on the phase also affects our BM, since a change in temperature reflects different diffusion and friction in the system. This coupling may indeed contribute to the double peak structure of the maxima in the induced drift in the negative \emph z-direction, since the splitting overlaps with the temperature maxima. Moreover, the phase in the \emph x direction is roughly set to zero at the beginning of the experiment by measuring the minimum in temperature. During the scan this value may change slightly but the effect is small, as long as the phase is small. Finally, a full physical understanding of the structure of our BM calls for a quantum model where the entire manifold of potentials that exist in the real DOL is accounted for. Such a model is under construction.
\section{Conclusion}
In summary, \revision{we have shown that the relative spatial phase of the lattices, due the the four beam configuration of the latter, couple dimensionally the induced drift. This affects the multidimensional behaviour of our Brownian motor profoundly. These effects have been investigated and we showed that the vertical induced drift can be controlled by either of the vertical or the horizontal phase, or a combination of the two.}
This was done by measuring the vertical drift velocity in a $2\pi\times 2\pi$ area in the \emph{yz} phase plane. From the measurement, the structure and periodicity of the drift was extracted. This gives us full control over our three dimensional Brownian motor.
\acknowledgments
We thank S. Jonsell for helpful discussions. This work was supported by Knut och Alice Wallenbergs stiftelse, Vetenskapsr\aa det / Swedish research council, Carl Tryggers stiftelse and Kempestiftelserna. A part of this research was conducted using the resources of the High Preformence Computing Centre North (HPC2N).
\bibliographystyle{prsty}
|
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4 BEADLE'S 4
DIME
[Illustration]
Song Book
No. 4.
A COLLECTION OF NEW AND POPULAR
COMIC AND SENTIMENTAL
SONGS.
[Illustration]
NEW YORK:
BEADLE AND COMPANY,
General Dime Book Publishers.
Books for the Hour!
MILITARY EXPLOITS
OF
Great Soldiers and Generals.
BEADLE'S
DIME BIOGRAPHICAL LIBRARY.
Each Issue Complete. 100 Pages. Price Ten Cents.
No. 6.--THE LIFE, MILITARY AND CIVIC SERVICES OF LIEUT.-GEN. WINFIELD
SCOTT. Complete up to the present period.
No. 4.--THE LIFE, TIMES AND SERVICES OF ANTHONY WAYNE (MAD ANTHONY):
Brigadier-General in the War of the Revolution, and Commander-in-Chief
of the Army during the Indian War.
No. 1.--THE LIFE OF JOSEPH GARIBALDI: The Liberator of Italy. Complete
up to the withdrawal of Garibaldi to his Island Home, after the
Neapolitan Campaign, 1860.
These brilliant books of the most brilliant Commanders and soldiers of
modern times possess remarkable interest at this moment. Each book
will be found to be a _full_ record of the men and events in which
they acted so splendid a part.
EVERY YOUNG MAN SHOULD READ THEM!
EVERY SOLDIER SHOULD READ THEM!
EVERY LOVER OF THE UNION SHOULD READ THEM!
For Sale at all News Depots.
BEADLE'S
DIME
[Illustration]
Song Book
No. 4.
A COLLECTION OF NEW AND POPULAR
COMIC AND SENTIMENTAL
SONGS.
NEW YORK:
IRWIN P. BEADLE & CO.,
NO. 137 WILLIAM STREET.
Entered according to Act of Congress, in the year 1860
BY IRWIN P. BEADLE & CO.,
in the Clerk's Office of the District Court of the United States
for the Southern district of New York.
CONTENTS OF DIME SONG BOOK NO. 4.
Page
Ain't I Glad to get out of the Wilderness, 22
A National Song, 11
Answer to Katy Darling, 42
A Merry Gipsy Girl Again, 47
A Parody on "Uncle Sam's Farm," 34
Ben Fisher and Wife, 9
Bonnie Jamie, 17
Broken-Hearted Tom, the Lover, 39
By the Sad Sea-Waves, 58
Columbia Rules the Sea, 29
Come Gang awa' wi' Me, 13
Commence you <DW54>s all, 28
Cottage by the Sea, 8
Daylight is on the Sea, 59
Don't Cry so, Norah, Darling, 6
Erin is my Home, 31
Gal from the South, 27
He Led Her to the Altar, 66
Home, Sweet Home, 53
I am a Freeman, 55
I'll Hang My Harp on a Willow-Tree, 18
I'm not Myself at All, 30
Indian Hunter, 50
I've been Roaming o'er the Prairie, 16
I Wish He would Decide, Mamma, 32
Jane Monroe, 69
Johnny is Gone for a Soldier, 19
Jolly Jack the Rover, 23
Kate was Once a Little Girl, 60
Kitty Tyrrel, 61
Let Me Kiss Him for His Mother, 48
Linda's Gone to Baltimore, 15
Maud Adair and I, 5
Molly Bawn, 51
My ain Fireside, 49
My Boyhood's Home, 53
Nora the Pride of Kildare, 51
O God! Preserve the Mariner, 46
Oh, Kiss, but never Tell, 21
Old Uncle Edward, 64
Paddy on the Canal, 68
Poor Old Maids, 45
Ship A-hoy! 56
Somebody's Courting Somebody, 24
Song of the Farmer, 37
Song of Blanche Alpen, 57
Sparking Sunday Night, 41
Sprig of Shilleleh, 43
Stand by the Flag, 36
The Farmer's Boy, 36
The Hazel Dell, 52
The Harp that once Through Tara's Hall, 31
The Indian Warrior's Grave, 50
The Little Low Room where I Courted my Wife, 25
The Low Backed Car, 44
The Old Brown Cot, 12
The Old Kirk-Yard, 54
The Railroad Engineer's Song, 14
They don't Wish Me at Home, 38
Tom Brown, 70
Terry O'Reilly, 40
Uncle Gabriel, 65
Uncle Tim, the Toper, 71
We were Boys and Girls Together, 33
We are all so Fond of Kissing, 20
We are Growing Old Together, 7
Where are now the Hopes I Cherished? 64
Within a Mile of Edinburg Town, 62
Would I were a Boy Again, 35
Would I were a Girl Again, 35
Would I were with Thee, 63
5
BEADLE'S
DIME SONG BOOK.
No. 4.
Maud Adair and I.
Copied by permission of FIRTH, POND & CO., 547 Broadway, owners of the
copyright.
One year ago were we sixteen,
Maud Adair and I,
With lightsome tread we tript the green,
Maud Adair and I;
But Maud Adair is lying low,
She left poor me three moons ago;
We ne'er shall meet again below,
Maud Adair and I.
_Chorus._--My Maud Adair! Sweet Maud Adair!
We'll meet again up in the sky,
Maud Adair and I.
One year ago, with hand in hand,
Maud Adair and I,
We roam'd the sunny hill and strand,
Maud Adair and I;
But one sad eve, with tearful eye,
She whisper'd low a last "Good-by,"--
We'll meet again up in the sky,
Maud Adair and I.
_Chorus._--My Maud Adair, &c.
How happy were we, and how true,
Maud Adair and I,
Like elm and ivy, upward grew
Maud Adair and I;
Oh, be thy spirit ever near
To whisper softly words of cheer!
While God doth guard, what can we fear,
Maud Adair and I?
_Chorus._--My Maud Adair, &c.
6
Don't You Cry so, Norah, Darling.
Copied by permission, of FIRTH, POND & CO., 547 Broadway, owners
of the copyright.
Don't you cry so, Norah, darling,
Wipe those tears away,
Don't you cry so, Norah, darling,
Smile on me to-day;
See the wind is freshly blowing,
And the ship longs for the sea,
Be to-day your smiles bestowing
Sweetly, love, on me.
_Chorus._--Don't you cry so, Norah, darling,
Wipe those tears away;
Don't you cry so, Norah, darling,
Smile on me to-day.
Though 'tis sad to leave you, darling,
I must no more stay,
Think of me, Norina, darling,
When I'm far away;
And, although to part brings sadness,
Keep your young heart light and free,
Your sweet face adorn with gladness,
Thinking still of me.
Don't you cry so, &c.
Don't you cry so, Norah, darling,
Wipe those tears away,
Don't you cry so, Norah, darling,
Smile on me to-day;
When from work I rest a-weary,
All my thoughts on you will be,
And my life will not seem dreary,
If you're true to me.
Don't you cry so, &c.
7
We are Growing Old Together.
Copied by permission of FIRTH, POND & CO., 547 Broadway, owners
of the copyright.
We are growing old together, thou dearest of the dear,
The morning of our life is past, and evening shades appear;
Some friends we loved are in their graves, and many are estranged,
But in sunshine or in shadow, our hearts are never changed.
We are growing old together, thou dearest of the dear,
The morning of our life is past, and evening shades appear.
We are growing old together, the ivy and the tree
A fitting emblem is dear, of the love 'twixt you and me;
To be worthy of each other in the past was all our aim,
And 'tis pleasant now to know, dear, our hearts are still the same.
We are growing old together, thou dearest of the dear,
The morning of our life is past, and evening shades appear.
We are growing old together, together may we die--
Together may our spirits soar to our home beyond the sky;
For we loved as few can love, dear, when life's flowery paths we
ranged,
And though we've wander'd long here, our hearts have never changed.
We are growing old together, thou dearest of the dear,
The morning of our life is past, and evening shades appear.
8
Cottage by the Sea.
Copied by permission of FIRTH, POND & CO., 547 Broadway, owners
of the copyright.
Childhood's days now pass before me
Forms and scenes of long ago,
Like a dream they hover o'er me,
Calm and bright as evening's glow,
Days that know no shade of sorrow,
There my young heart pure and free,
Joyful hail'd each coming morrow
In the Cottage by the Sea.
CHORUS.
In the Cottage by the Sea,
In the Cottage by the Sea,
Joyful hail'd each coming morrow
In the Cottage by the Sea.
Fancy sees the rose-trees twining,
Round the old and rustic door,
And below, the white beach shining,
Where I gather'd shells of yore.
Hears my mother's gentle warning,
As she took me on her knee;
And I feel again life's morning,
In the Cottage by the Sea.
In the Cottage by the Sea, &c.
What though years rolled above me,
Though 'mid fairer scenes I roam,
Yet I ne'er shall cease to love thee,
Childhood's dear and happy home!
And when life's long day is closing,
Oh! how pleasant it would be;
On some faithful heart reposing
In the Cottage by the Sea.
In the Cottage by the Sea, &c.
9
Ben Fisher and Wife.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, N. Y.,
publishers of the music.
Ben Fisher had finish'd his hard day's work,
And he sat at his cottage door;
His good wife Kate sat by his side,
And the moonlight danced on the floor--
The moonlight danced on the cottage floor,
Her beams were clear and bright,
As when he and Kate, twelve years before,
Talk'd love in her mellow light.
Talk'd love in her mellow light.
_Chorus._--The moonlight danced on the cottage floor,
Her beams were clear and bright,
As when he and Kate, twelve years before,
Talk'd love in her mellow light.
Ben Fisher had never a pipe of clay,
And never a dram drank he,
So he loved at home with his wife to stay,
And they chatted right merrily--
Right merrily they chatted on,
Her babe slept on her breast,
While a chubby rogue, with rosy smile,
On his father's knee found rest,
On his father's knee found rest.
Right merrily, &c.
Ben told her how fast the potatoes grew,
And the corn in the lower field,
And the wheat on the hills was grown to seed,
And promised a glorious yield.
A glorious yield in the summer-time,
And his orchard was doing fair,
His sheep and his flock were in their prime,
His farm all in good repair,
His farm all in good repair.
A glorious yield, &c.
10
Kate said that her garden look'd beautiful,
Her fowls and her calves were fat,
The butter that Tommy that morning had churn'd,
Would buy him a Sunday hat.
That Jenny for pa a new shirt had made,
And it was done, too, by the rule,
That Neddy nicely could the garden spade.
And Ann was up head at school.
And Ann was up head at school.
That Jenny for pa, &c.
Ben slowly raised his toil-worn hand,
Through his locks of grayish brown:
"I'll tell you, Kate, what I think," said he,
"We're the happiest folks in town."
"I know," said Kate, "that we all work hard
Work and health go together I've found,
For there's Mrs. Bell does not work at all,
And she's sick the whole year round,
And she's sick the whole year round.
I know," said Kate, &c.
"They are worth their thousands, so people say,
But I ne'er saw them happy yet;
'Twould not be me that would take their gold,
And live in a constant fret.
My humble home has a light within,
Mrs. Bell's gold could not buy--
Six lovely children, a merry heart,
And a husband's love-lit eye,
And a husband's love-lit eye.
My humble home, &c."
I fancied a tear was in Ben's fine eye,
The moon shone brighter and clearer,
I could not tell why the man should cry,
But he hitch'd up to Kate still nearer.
He lean'd his head on her shoulder there,
And he took her hand in his,
And I guess (though I look'd at the moon just then),
That he left on her lips a kiss,
That he left on her lips a kiss.
He lean'd his head, &c.
11
A National Song.
All hail! Unfurl the stripes and stars!
The banner of the free!
Ten times ten thousand patriots greet
The shrine of Liberty;
Come, with one heart, one hope, one aim,
An undivided band,
To elevate, with solemn rites,
The ruler of our land.
Not to invest a potentate,
With robes of majesty--
Not to confer a kingly crown,
Nor bend a supple knee.
We now beneath no scepter'd sway--
Obey no royal nod--
Columbia's sons, erect and free,
Kneel only to their God!
Our ruler boasts no titled rank,
No ancient, princely line--
No legal right to sovereignty,
Ancestral and divine.
A patriot--at his country's call
Responding to her voice
One of the people--he becomes
A sovereign by our choice.
And now, before the mighty pile
We've rear'd to Liberty,
He swears to cherish and defend
The charter of the free!
God of our country! seal his oath
With thy supreme assent.
God save the Union of the States!
God save the President!
12
The Old Brown Cot.
Among the scenes to memory dear,
To which my fancy oft returns,
And for those long-lost days of joy
My spirit in its sadness dreams.
There's none which seems so dear to me
As that where past life's early morn;
There's none for which I sigh so oft,
As for the cot where I was born.
CHORUS.
The old brown cot, the low brown cot,
The moss-grown cot beneath the hill;
Though years have pass'd since I was there,
I love it, oh, I love it still.
It stood beside the running brook
Whose waters turn'd the noisy mill;
And close beside the tall old oaks
That nodded on the sloping hill.
The woodbine creeping o'er the walls,
The sunshine on the grassy plot,
How beautiful were they to me,
When home was in that old brown cot!
The old brown cot, &c.
Though I may view the fairest land
On which the sun in glory beams,
And dwell in climes more beautiful
Than poets visit in their dreams,
Still will affection linger round
That loved and consecrated spot,
And tears will fall as I go back
To boyhood and the old brown cot.
The old brown cot, &c.
13
Come, gang awa' wi' me.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
Oh! come my love, the moon shines bright,
Across yon rippling sea,
Come let thy heart be gay and light,
And hasten love wi' me.
'Tis mony a night sin' first we met
Beneath the greenwood tree,
Then let thy heart be lighter yet,
Come, gang awa' wi' me.
'Tis mony a night sin' first we met,
Beneath the greenwood tree,
Then let thy heart be lighter yet,
Come gang awa' wi' me.
Oh! tarry not, my only love,
I've pledged myself to thee,
And by yon stars that shine above,
Forever thine I'll be;
'Tis mony a night sin' first we met
Beneath the greenwood tree,
Then say, ere yonder stars have set,
Thou'lt gang awa' wi' me.
'Tis mony a night sin' first we met
Beneath the greenwood tree,
Then say ere yonder stars have set,
Thou'lt gang awa' wi' me.
Thy features are so fair my love,
Thy mind is ever free,
Oh! let thy willing heart still prove
The love thou bear'st to me.
'Tis mony a night sin' first we met
Beneath the greenwood tree,
Then say ere yonder stars have set,
I'll gang awa' wi' ye.
'Tis mony a night sin' first we met,
Beneath the greenwood tree.
Then say, ere yonder stars have set,
I'll gang awa' wi' ye.
14
The Railroad Engineer's Song.
I love--oh, how I love to ride
The Iron Horse in his fiery pride!
All other joys seem dull and vain,
When I lay my hand on his misty mane.
Fear him not! with his ribs of steel,
His flaming throat, and his brushing wheel;
And his smoky crest, so black and tall,
Like a pillar cover'd with a funeral pall.
Though his stamping shakes the solid ground,
And he scatters fire-flakes all around,
He's gentle as jennet in lady's rein
When he feels my hand on his misty mane.
Set me astride of the Iron Horse!
Full of fierce fury, speed, and force;
And hark how he pants, and blows, and snorts,
While my skill his eager bounding thwarts.
But when I'm mounted on his back,
And you see him coming--clear the track!
Nothing can check him on his course,
As he thunders along--my Iron Horse!
Then huzza! the Iron Horse for me!
The eagle scarce flies as fast as he;
He skims the valley and scours the plain,
And shakes, like a cloud, his misty mane.
He tracks the prairie, climbs the hill,
The wild woods echo his neighing shrill;
And when the fierce tempest lashes the shores,
Louder than ever the storm he roars.
15
Linda's gone to Baltimore.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
Oh, Linda's gone to Baltimore,
To stay a week or two,
And till she comes safe home again,
I don't know what to do.
I take the banjo on my knee,
But can not hear to play,
For music only makes me sad,
When Linda's gone away,
When Linda's gone away.
CHORUS.
Oh, my heart am very lonely
All the night and day,
For every thing seems sad and drear,
When Linda's gone away.
I think of all the olden times
We've had when she was here,
I did not know 'till she was gone,
That she was half so dear.
The flowers are blooming all around
And all but me are gay,
For all the time I think or dream
Of Linda far away.
_Chorus._--Oh, my heart am very lonely, &c.
Though many years have pass'd and gone
Since we were in our prime,
I loved her more as on we roam'd
Adown the Vale of Time!
How very much she thinks of me,
I should not dare to say;
But oh, it always breaks my heart
When Linda's gone away.
_Chorus._--Oh, my heart am very lonely, &c.
16
I've been Roaming o'er the Prairies.
I've been roaming, roaming o'er the prairies wild
Plucking dewy blossoms, happy as a child;
Casting care and sadness very far away,
For the earth rejoices on this pleasant day.
I've been roaming, roaming where the lilies sleep,
On the tiny lakelet sparkling cool and deep,
Where the brooklet singeth o'er the pebbles white,
Making gladsome music glancing in the light;
Where the brooklet singeth o'er the pebbles white,
Making gladsome music glancing in the light.
I've been roaming, roaming through the wild wood deep
Searching for the flowrets when the prairies sleep;
In the tiny blossoms swaying to and fro,
Whispering to each other very soft and low.
I've been roaming, roaming o'er the dewy grass,
Gemm'd with fairy blossoms waving as I pass,
For the breeze was flitting o'er the grassy lea,
Whispering many a story to the flowers and me;
For the breeze was flitting o'er the grassy lea,
Whispering many a story to the flowers and me.
17
Bonnie Jamie.
The twilight hour is stealing,
The day is dying fast,
Neath the birken tree I'm kneeling,
Where Jamie met me last.
Where Jamie met me last;
While tears fell from mine e'e,
But my bonnie, bonnie Jamie
Has cross'd the stormy sea.
The war's alarms were sounding,
For soldiers brave and true,
My deary's heart was bounding,
He join'd the army too.
He join'd the army too,
To fight for liberty,
Oh, my bonnie, bonnie Jamie
Has gone to war to dee.
Sin e'er I was a bairnee,
My Jamie I ha' known,
The fire of his bright e'e,
His voice sae saft and low.
His voice sae saft and low,
So snood and braw look'd he,
Oh, my bonnie, bonnie Jamie,
Will I nae mair see thee?
I gave unto my dearie
A lock of my gowden hair,
His sword I buckled cheerie,
And kiss'd his brow sae fair.
And kiss'd his brow sae fair,
Which he gave back to me,
Oh, my bonnie, bonnie Jamie,
Is a' the world to me.
Brave Mars, thou God of Battle,
My heart now speaks to thee,
When cannons loudly rattle,
On my dearie keep thine e'e.
On my dearie keep thine e'e,
My prayers I'll gie to thee,
For my bonnie, bonnie Jamie,
He's a' the world to me.
18
I'll Hang my Harp on a Willow-Tree.
I'll hang my harp on a willow-tree,
I'll off to the wars again,
My peaceful home has no charms for me,
The battle-field no pain;
The lady I love will soon be a bride
With a diadem on her brow;
Oh, why did she flatter my boyish pride,
She's going to leave me now.
Oh, why, &c.
She took me away from my warlike lord,
And gave me a silken suit,
I thought no more of my master's sword,
When I play'd on my master's lute.
She seem'd to think me a boy above
Her pages of low degree;
Oh, had I but loved with a boyish love,
It would have been better for me;
Oh, had I, &c.
Then I'll hide in my breast every selfish care;
I'll flush my pale cheeks with wine;
When smiles awake the bridal pair
I'll hasten to give them mine;
I'll laugh and I'll sing, though my heart may bleed,
And I'll walk in the festal train,
And if I survive it I'll mount my steed,
And I'll off to the wars again.
And if I survive, &c.
But one golden tress of her hair I'll twine
In my helmet's sable plume,
And then on the field of Palestine,
I'll seek an early doom.
And if by the Saracen's hand I fall,
'Mid the noble and the brave,
A tear from my lady love is all
I ask for the warrior's grave.
A tear from, &c,
19
Johnny is Gone for a Soldier.
I'll trace these gardens o'er and o'er,
Meditate on each sweet flower,
Thinking of each happy hour,--
Oh, Johnny is gone for a soldier.
CHORUS.
Shool, Shool, Shool, agrah!
Time can only ease my woe,
Since the lad of my heart from me did go
Oh, Johnny is gone for a soldier.
Some say my love is gone to France,
There his fortune to advance,
And if I find him it's but a chance,--
Oh, Johnny is gone for a soldier,
Shool, Shool, &c.
I'll sell my frock, I'll sell my wheel,
I'll buy my love a sword of steel,
So in the battle he may reel,--
Oh, Johnny is gone for a soldier.
Shool, Shool, &c.
I wish I was on yonder hill,
It's there I'd sit and cry my fill,
So every tear may turn a mill,--
Oh, Johnny is gone for a soldier.
Shool, Shool, &c.
I'll dye my dress, I'll dye it red,
All over the world I'll beg my bread,
So my parents may think me dead--
Oh, Johnny is gone for a soldier.
Shool, Shool, &c.
20
We are all so Fond of Kissing.
Oh, kiss me quick and let me go,
Don't keep me here a waiting,
For if by chance we should be caught,
It would set the gals a talking.
I vow, I quite in passion get,
To see you act so silly,
I think I'll have to kiss you first,
For I'm getting very chilly.
CHORUS.
Oh, kiss me quick, and let me go,
Don't keep me here a waiting,
For if by chance we should be caught,
It would set the gals a talking.
She's fond of kissing, that I know,
So often as I meet her,
She says, "Kiss me quick, and let me go,
You'll love me all the better."
At evening when the room was dark,
And time was getting later,
I thought I'd steal a kiss from her,
And I kiss'd the <DW65> Waiter.
Oh, kiss me quick, &c.
Oh, now I'll give you good advice,
When you go a sparking,
Don't do your kissing in the dark,
For fear your lips of marking.
But choose the day and fear no shame,
If its not distressing,
I'm sure its nothing very new,
For we're all so fond of kissing.
Oh, kiss me quick, and let me go, &c.
21
Oh, Kiss but never Tell.
Copied by permission of FIRTH, POND & CO., 547 Broadway, owners
of the copyright.
When love grows warm there is a charm,
And oft a sacred bliss,
When fond hearts greet for lips to meet
In sweet affection's kiss;
But to reveal the sacred seal
Which hallows it so well,
May quench love's flame with breath of shame,
So kiss, but never tell.
CHORUS.
Oh, kiss, but never tell, oh never!
Breathing breaks the spell.
True lovers pledged to keep forever,
Kiss, but never tell.
At night, when eyes like stars beam bright,
And kindred souls commune,
And heart to heart love's vows impart,
Beneath the smiling moon:
At such an hour of magic power,
What hallow'd raptures dwell,
In each true breast by honor blest,
To kiss, and never tell.
CHORUS.
Then kiss but never tell,
Breathing breaks the spell,
True lovers pledged to keep forever,
Kiss, but never tell!
22
Ain't I Glad to Get Out of the Wilderness.
Music-- Tum, Tum, Tum, Tum.
Chorus.-- Ahaa--Ahaa--Ahaa--Ahaa.
Solo-- Way down south in Beaver Creek,
In Beaver Creek, in Beaver Creek,
De <DW65>s-dey grow about ten feet,
Way down in Alabam.
Chorus.-- Oh, ain't I glad we got out of the wilderness
Out of the wilderness,
Oh, ain't we glad we got out of the wilderness
And left old Alabam.
[Symphony with dance as above.]
Solo-- Dey wet the ground wid bacca smoke,
Wid bacca smoke, wid bacca smoke,
When out of de ground dar heads do poke.
Way down in Alabam,
Dance & Chorus--Oh, ain't I glad, etc.
Solo-- My wife's dead, an I'll get anuder one,
I'll get anuder one, I'll get anuder one,
My wife's dead, and I'll get anuder one,
Way down in Alabam.
Dance & Chorus--Oh, ain't I glad, etc.
Solo-- I met a cat-fish in the ribber.
In the ribber, in the ribber,
I golly, it made dis <DW65> shiver
Way down in Alabam.
Dance & Chorus--Oh, ain't I glad, etc.
Solo-- I steer'd right straight for de critter's snout
De critter's snout, de critter's snout,
Turned de cat-fish inside out,
Way down in Alabam.
Dance & Chorus--Oh, ain't I glad, etc.
Solo-- Oh, here we go now altogether,
All together, all together,
Nebber mind de wind or wedder,
Way down in Alabam,
Dance & Chorus--Oh, ain't I glad.
23
Jolly Jack the Rover.
Here I am one, and still will be,
Who spend their days in pleasure,
The tailor's bill is seldom fill'd,
For he's never took my measure.
_Chorus._--It must be while I do live,
And I must not give over,
Until old age doth me engage,
From being a jolly rover.
It's on my vamps, I take my tramps,
My shoes being in a bad order,
My stockings down into the groun,
For I seldom wears a garter.
It must be, &c.
If I would dress up in fine clothes,
The ladies would adore me,
The <DW2>s of beaux that wear fine clothes,
They think to go before me,
It must be, &c.
It's I can play at cards and dice,
Let me be drunk or sober,
Win or lose, I'll have my dues,
For I'm Jolly Jack the Rover.
It must be, &c.
Three tons of wool through a comb I pul
All in the neatest order,
As white as milk and soft as silk,
To please the farmer's daughter.
It must be, &c.
When my work's done and finish'd off,
I'll take it to the owner,
I have no doubt that she's found out,
That I'm Jolly Jack the Rover.
It must be, &c,
When I am old, if I have gold,
I'll set down by my table,
With you my dear, I'll toast good beer
And drink while I am able.
It must be, &c.
When I am dead, and in my grave,
It's then I must give over,
Let each jolly lass fill a parting glass,
And drink a health to Jack the Rover.
It must be, &c.
24
Somebody's Courting Somebody.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, owners
of the copyright.
Somebody's courting somebody
Somewhere or other to-night;
Somebody's whispering to somebody,
Under the clear moonlight,
Near the bright river's flow,
Running so still and slow;
Talking so soft and low,
She sits with somebody.
Somebody's courting somebody
Somewhere or other to-night;
Somebody's listening to somebody
Under the clear moonlight,
Under the clear moonlight.
Pacing the ocean shore,
Edged by the foaming roar,
Words never breathed before,
Sound sweet to somebody;
Under the maple-tree,
Deep though the shadow be,
Plain enough they can see,
Bright eyes has somebody.
Somebody's courting somebody
Somewhere or other to-night;
Somebody's listening to somebody
Under the clear moonlight,
Under the clear moonlight.
No one sits up to wait,
Though she is out so late,
All know she's at the gate
Talking with somebody;
Two sitting side by side,
Float with the ebbing tide,
"Thus, dearest, may we glide
Through life," says somebody.
Somebody's courting somebody
Somewhere or other to-night;
Somebody's listening to somebody
Under the clear moonlight,
Under the clear moonlight.
25
The Little Low Room where I Courted my Wife.
Copied by permission of FIRTH, POND & CO., 547 Broadway, publisher
of the music.
My brow is seam'd o'er with the iron of years,
And the snow threads are gleaming the furrows among,
My eyes have grown dim in the shadow of tears,
Where the flowers of my soul have died as they sprung,
But memory bears to me on its broad wings
Bright images true of my earliest life,
And there, 'mid the fairest of all that is seen,
Is the little low room where I courted my wife,
Is the little low room where I courted my wife.
That low, humble room seem'd a palace of light,
As love held his torch, and illumined the scene,
With glory of state and profusion bedight,
Where I was a monarch, my darling a queen;
Ourselves were our subjects, pledged loyal were each,
And which should love best was our heartiest strife;
What tales could it tell, if possessing a speech,
That little low room where I courted my wife,
That little low room where I courted my wife.
Warm vows has it heard, the warmest e'er spoke,
Where lips have met lips in holy embrace,
Where feelings that never to utterance woke,
It saw oft reveal'd in a duplicate face;
The sweet hours hasten'd, how quickly they flew,
With fervent devotion and ecstasy rife!
Our hearts throbb'd the hours, but how I ne'er knew,
In the little low room where I courted my wife,
In the little low room where I courted my wife.
The romance of youth lent its rapturous zest,
And fairydom knew no delight like our own;
Our words were but few, but they were the best,
A dialect sweet for ourselves all alone.
So anxious to hear what the other might say,
We neither could utter a word for our life;
Thus the hours, in silence, pass'd quickly away
In the little low room where I courted my wife,
In the little low room where I courted my wife.
26
Long years have since pass'd o'er my darling and I,
The roses have vanish'd away from her cheek,
But the merciless moments, as onward they fly,
Leave love still undimm'd in her bosom so meek;
That love is the light to our faltering feet,
Our comfort in hours with sorrowing rife,
Our blessings in joy, as with joy 'twas replete,
In the little low room where I courted my wife,
In the little low room where I courted my wife.
Stand by the Flag.
Copied by permission of FIRTH, POND & CO., 547 Broadway, owners
of the copyright.
Stand by the flag, its folds have stream'd in glory,
To foes a fear, to friends a festal robe,
And spread in rythmic lines the sacred story,
Of freedom's triumphs over all the globe;
Stand by the flag on land and ocean billow;
By it your fathers stood unmoved and true;
Living defended; dying, from their pillow,
With their last blessing, pass'd it on to you.
Stand by the flag, though death-shots round it rattle;
And underneath its waving folds have met,
In all the dread array of sanguine battle,
The quivering lance and glittering bayonet,
Stand by the flag, all doubt and treason scorning,
Believe with courage firm and faith sublime
That it will float until the eternal morning
Pales in its glories all the lights of time.
27
Gal from the South.
My Massa had a color'd gal--
He brought her from the South,
Her hair it curl'd so very tight,
She could not shut her mouth,
Her eyes they were so very small,
They both ran into one,
And when a fly lit in her eye,
'Twas like a June-bug in the sun.
CHORUS.
Ha, ha, ha, yah, yah, yah,
The gal from the South;
Her hair it curl'd so very tight,
She could not shut her mouth.
Her nose, it was so very long,
It turn'd up like a squash,
And when she got her dander up,
She made me laugh, by gosh!
Old Massa had no hooks or nails
Or nothing else like that,
So on this darkie's nose he used
To hang his coat and hat.
_Chorus._--Ha, ha, ha, yah, yah, yah, &c.
One morning Massa going away,
He went to get his coat,
But neither hat nor coat was there,
For she had swallow'd both.
He took her to a tailor shop,
To have her mouth made small,
The lady took in one long breath,
And swallow'd tailor and all!
_Chorus._--Ha, ha, ha, yah, yah, yah, &c.
28
Commence you <DW54>s all.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, owners
of the copyright.
White folks, I am goin' to sing
A song dat am quite new,
Ob myself an' banjo-string,
An' you, an' you, an' you!
Oh, Sam, don't laugh, I say,
Our strings will keep in tune,
Just listen to de banjo play
For de white folks 'round de room!
CHORUS.
Den commence you <DW54>s all,
As loud as you can bawl!
Commence you <DW54>s all, to-night.
Touch light de banjo-string,
An' rattle de ole jaw-bone,
Oh, merrily sound de tamborine,
An' make de fiddle hum;
An' make de fiddle hum, old dad;
De way dem bones will shake,
Am a caution to all living niggs,
An' a deff to rattlesnakes.
Den commence, &c.
"Oh, for a piano or guitar!"
I hear a fair one cry;
But when I hear dese instruments,
I tink I'd like to die.
I tink I'd like to die, I does,
I could lay me down to rest,
For music hab such 'lodious sounds
To soothe dis darkey's breast.
Den commence, &c.
When I go to promenade,
I look so fine an' gay,
I hab to take de dogs along
Te keep de gals away;
My busom am so full ob lub,
Dis darkey can not rest,
So I'll bid you all good-by, at last,
An' trabble to de West.
Den commence, &c.
29
Columbia Rules the Sea.
Copied by permission of FIRTH, POND & CO., 547 Broadway, owners
of the copyright.
The pennon flutters in the breeze,
The anchor comes a-peak,
"Let fall sheet home,
The briny foam and ocean's waste we seek;
The booming gun speaks our adieu.
Fast fades our native shore,
Columbia free shall rule the sea,
Britannia ruled of yore.
We go the tempest's wrath to dare,
The billows' madden'd play,
Now climbing high against the sky,
Now rolling low away;
While Yankee oak bears Yankee hearts,
Courageous to the core,
Columbia free shall rule the sea,
Britannia ruled of yore.
We'll bear her flag around the world,
In thunder and in flame,
The sea-girt isles a wreath of smiles
Shall form around her name;
The winds shall pipe her pæans loud,
The billowy chorus roar
Columbia free shall rule the sea,
Britannia ruled of yore.
30
I'm not Myself at all.
Oh! I'm not myself at all, Molly dear, Molly dear,
I'm not myself at all,
Nothing caring, nothing knowing, 'tis after you I'm going,
Faith your shadow 'tis I'm growing, Molly dear, Molly dear,
And I'm not myself at all.
Th'other day I went confessin', and I ask'd the father's blessin'
But says I, "Don't give me one entirely,
For I fretted so last year,
But the half o' me is here,
So give the other half to Molly Brierly
Oh! I'm not myself at all."
Oh! I'm not myself at all, Molly dear, Molly dear,
My appetite's so small,
I once could pick a goose, but my buttons are no use,
Faith my tightest coat is loose, Molly dear, Molly dear,
And I'm not myself at all.
If thus it is I waste, you'd better dear make haste
Before your lover's gone away entirely,
If you don't soon change your mind
Not a bit o' me you'll find,
And what 'ud you think 'o that Molly Brierly?
Oh! I'm not myself at all.
Oh! my shadow on the wall, Molly dear, Molly dear,
Isn't like myself at all.
For I've got so very thin, myself says 'tisn't him,
But that purty girl so slim, Molly dear, Molly dear,
And I'm not myself at all.
If thus I smaller grow, all fretting dear for you,
'Tis you should make me up the deficiency,
So just let Father Taaf
Make you my better half,
And you will not the worse for the addition be;
Oh! I'm not myself at all.
I'll be not myself at all, Molly dear, Molly dear,
'Till you my own I call.
Since a change o'er me there came, shure you might change your name,
And 'twould just come to the same, Molly dear, Molly dear,
Oh! 'twould just come the same;
For if you and I were one, all confusion would be gone,
And 'twould simplify the mather entirely,
And 'twould save us so much bother
When we'd both be one another,
So listen now to rayson, Molly Brierly,
Oh! I'm not myself at all.
31
Erin is my Home.
Oh, I have roam'd in many lands,
And many friends I've met;
Not one fair scene or kindly smile
Can this fond heart forget;
But I'll confess that I'm content,
No more I wish to roam;
Oh, steer my bark to Erin's isle,--
For Erin is my home.
Oh, steer my bark, &c.
If England were my place of birth,
I'd love her tranquil shore;
But if Columbia were my home,
Her freedom I'd adore.
Though pleasant days in both I pass'd,
I dream of days to come;
Oh, steer my bark to Erin's isle,--
For Erin is my home.
Oh, steer my bark, &c.
The Harp that once thro' Tara's Halls.
The harp that once through Tara's halls
The soul of music shed,
Now hangs as mute on Tara's walls,
As if that soul were fled.
So sleeps the pride of former days,
So glory's thrill is o'er,
And hearts that once beat high for praise,
Now feel that pulse no more.
No more to chiefs and ladies bright,
The harp of Tara swells;
The chord alone, that breaks at night,
Its tale of ruin tells.
Thus freedom now but seldom wakes;
The only throb she gives,
Is when some heart indignant breaks,
To show that still she lives.
32
I Wish he would Decide, Mamma.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
I wish he would decide, Mamma,
I wish he would decide,
I've been a bridesmaid many time,
When shall I be a bride;
My cousin Anne and sister Fan,
The nuptial knot have tied,
Yet come what will I'm single still,
Yet come what will I'm single still,
I wish he would decide.
When shall I be a bride,
When shall I be a bride,
For come what will I'm single still,
I wish he would decide.
He takes me to the play, Mamma,
And brings me pretty books,
He woos me with his eyes, Mamma,
Such speechless things he looks.
Where e'er I roam, abroad, at home,
He lingers by my side,
Yet come what will I'm single still,
Yet come what will I'm single still,
I wish he would decide.
When shall I be a bride,
When shall I be a bride,
For come what will I'm single still,
I wish he would decide.
I've thrown out many a hint, Mamma,
I've spoke of other beaux,
I've talk'd about domestic life,
And sung "They don't propose."
Then if he means to break, Mamma,
My passion and my pride,
Unconquer'd yet I'll scorn regret,
Unconquer'd yet I'll scorn regret,
Although he won't decide,
Although he won't decide,
Although he won't decide,
Unconquer'd yet I'll scorn regret,
Although he won't decide.
33
We were Boys and Girls Together.
We were boys and girls together
In that happy time,
When the spirit's light shone brightest
And the heart was in its prime;
Ere the morning light was clouded,
That beam'd upon our youth,
And chill of worldly knowledge
Had blighted childhood's truth.
We were boys and girls together,
When the step was firm and light,
When the voice was clear and ringing,
And the laughing eyes were bright;
Then our love sought no concealment,
And our bosoms knew no art,
And the sunshine of our childhood
Cast no shadow on the heart.
34
A Parody on "Uncle Sam's Farm."
Of all the reformations, in the east or in the west,
Oh, the temperance reformation is the greatest and the best,
We invite the whole creation our pledge to come and sign,
And leave off drinking brandy, rum, cider, beer, and wine.
CHORUS.
Then come along, come along, make no delay,
Come sign the temperance pledge, sign it right away,
For if you do but keep it, you need not fear alarm
But you will soon be rich enough to buy a handsome farm.
The temperance cause is spreading o'er this our native land,
And Alchy with his subjects know not where to make a stand.
His army is decreasing, and soon there'll be but few,
Who to oppose the temperance cause on Alchy's smiles get blue.
The drunkard is so foolish that he will money waste,
On liquor, when there's water more pleasant to the taste;
The water is much cheaper, and much more healthy too,
And never makes a man a fool--which liquors often do.
It never yet caused people to quarrel and to fight,
Or come home intoxicated at twelve o'clock at night.
Cold water never caused man in the gutter to be found,
And never, as I know of, to feel upward for the ground.
Now if you only hasten our pledge to come and sign,
To leave off drinking brandy, rum, cider, gin, and wine,
You can not help but prosper in your business through life,
Provided you have with you a nice teetotal wife.
35
Would I were a Boy again.
Oh, would I were a boy again,
When life seem'd form'd of sunny years,
And all the heart then knew of pain
Was swept away in transient tears,
Was swept away in transient tears.
When ev'ry dream hope whisper'd then,
My fancy deem'd was only truth;
Oh, would that I could know again,
The happy visions of my youth.
Oh, would I were a boy again, &c.
'Tis vain to mourn that years have shown
How false these fairy visions were,
Or murmur that mine eyes have known
The burden of a fleeting tear;
But still the heart will fondly cling
To hopes no longer prized as truth,
And memory still delights to bring
The happy visions of my youth.
Oh, would I were a boy again, &c.
Would I were a Girl again.
Oh, would I were a girl again,
With heart and spirit free,
To gayly rove the village plain,
Or singing o'er the lea.
Then can you wonder if I sigh
And sadly thus deplore,
To wish for days, alas! gone by,
And be a girl once more.
I gayly trod the mountain side,
Knew naught of care or gloom,
Its purple bells brought home with pride,
To deck my mother's room,
Then can you wonder if I sigh, &c.
36
The Farmer's Boy.
The sun had gone down behind yon hill,
And o'er yon dreary moor,
When, weary and lame, a boy there came
Up to a Farmer's door,--
Saying, can you tell me, if any there be,
Can give to me employ,
For to plow, for to mow, for to reap, for to sow,
For to be a Farmer's Boy.
My father is dead, my mother is left
With her five children small,
And what is worse, for mother still,
I'm the eldest of them all;
Though small I am, I fear no work,
If you will give me employ.
For to plow, &c.
One favor yet I ask, If you can not me employ,
That is to shelter me this one night
From the cold winter's blast;
At the break of day, I will trudge away,
Elsewhere to seek employ,
For to plow, &c.
The farmer says, "We will try the lad,
No further let him seek."
Oh, yes, dear father, his daughter cried,
While the tears rolled down her cheek;
For him that can labor it is hard to want,
Or elsewhere to seek employ
For to plow, &c.
At length of years this boy grew up,
This good old farmer died,
He left the boy the farm he had,
And his daughter for his bride.
The boy that was, is a farmer now,
And he oft times thinks with joy,
On the happy, happy day, he came that way,
For to be a Farmer's Boy.
37
Song of the Farmer.
I have cattle that feed in the valley,
And herds that graze on the hill,
And I pride in the fruits of my labor,
For I'm lord of the land that I till,
I have plow'd the rough hill and the meadow
Till feeble with age and with toil,
And I know before long that another
Shall reap the new fruits of the soil.
For the son that hath toil'd for me ever,
And faithfully stood by my side,
Hath a hand that shall gather the harvest,
When his feeble old father hath died.
And the daughter so kind to her mother,
Shall share with him all I possess,
For I feel that they love me as father,
And welcome my tender caress.
There's my faithful, my trusting companion,
My kind-hearted dear loving wife;
I have toil'd for her comfort with pleasure,
For such was the pride of my life.
And still in my manhood I love her,
For her kind and affectionate care,
And all that the earth can afford me,
With her I most willingly share.
38
They don't wish Me at Home.
They don't wish me at home, though they miss me,
'Twould be a great assurance, I fear,
To think for a moment some soft one
Would say, "I wish Toby were here."
Although the poor tom-cat at the fireside
May think of poor me as I roam,
Oh yes, I'd be green beyond measure
To think they do wish me at home.
Dark nights were my joy for this reason:
Some orchard I'd visit alone;
Next morning some farmer would mention
My name with some fruit that was gone.
But now fruits are safe from all danger,
None's miss'd since poor Toby's away;
And the neighbors all wish I may never
Return from the place where I stay.
I forgot not my place at the table,
When "grub-time" was fast drawing nigh;
Then the "vittles" that lay all around me
Disappear'd in the wink of an eye.
Now, when my poor supper is over,
I spread myself out for a snore,
Oh! I dream of the fruits in the garden,
And think myself happy once more.
Oh! I wish I was home, though they quiz me
And jaw me from morning till night;
I'd finger the peach-trees around me--
The farmers should stare with affright.
Although they would give me no welcome,
I'd not be less bold than before;
Their fruit they shall miss by the bushel.
Because I am with them once more.
39
Broken-Hearted Tom, the Lover.
I'm lonesome since I cross'd the saes,
My mind is never aisy;
No mortal sowl can give relaif--
In troth, I'm getting crazy.
The burning tears roll down me chakes,
In faith, they nearly blind me;
I weep and sigh, both night and day,
For the Girl I left behind me.
The lovely lass I courted long,
She lives in Tipperary;
Her eyes were like the diamonds bright,
And they call'd her black-eyed Mary.
In summer's night I took delight,
Her beauty so inclined me,
A thousand crowns I'd give to see
The Girl I left behind me.
In foreign lands compell'd to roam,
Yet often think of Mary:
The black-eyed lass that won my heart
That lives in Tipperary.
On distant shores I weep and sigh,
Without a friend to mind me;
Bad luck unto the ship that sail'd
And left the Girl behind me.
If e'er I land on Erin's shore,
I'll haste to Tipperary;
Within me arms I will embrace
Me lovely black-eyed Mary.
With her I'll dwell while life shall last,
For she'd roam the world to find me,
From Mary I'll not wander more,
The Girl I left behind me.
40
Terry O'Reilly.
Sure, Terry O'Reilly, I've waited, you know,
And sure you're not coming like my own thrue beau;
I've look'd through the windy till each little pane,
Is near hid by my tears like a shower of rain.
Och! hone! Terry, come soon!
Or else I'll get married some fine afternoon.
Sweet Terry O'Reilly, why keep me sighing?
If I tarry longer, of grief I'll be dying;
Now, Terry, pray haste, and this heart give relief,
Or faith, my dear Terry, I'll soon die with grief.
Och! hone! Terry, come soon,
Or else I'll get married some fine afternoon.
Dear Terry O'Reilly, I ne'er was a flirt,
Still Terence is handsome, and he'll gain my heart;
Sure some one I must have, whose kindness will prove,
He's devoted to me, and faith him I'll love.
Och! hone! Terry, come soon,
Or else I'll get married some fine afternoon.
Now, Terry O'Reilly, I am tired of sighing,
I'm wearied to death, sure, with fretting and crying;
I'll marry to spite you, ma cushla, and part,
With love for you, Terry, and so break my heart.
Och! hone! Terry, come soon,
Or else I'll get married some fine afternoon.
41
Sparking Sunday Night.
Sitting in a corner, on a Sunday eve,
With a taper finger resting on your sleeve;
Starlight eyes are casting on your face their light;
Bless me, this is pleasant--sparking Sunday night!
CHORUS.
Bless me, ain't it pleasant,
Bless me, ain't it pleasant,
Bless me, ain't it pleasant,
Sparking Sunday night?
How your heart is thumping 'gainst your Sunday vest,
How wickedly 'tis working on this day of rest!
Hours seem but minutes, as they take their flight,
Bless me, ain't it pleasant, sparking Sunday night?
Dad and Mam are sleeping, on their peaceful bed,
Dreaming of the things the folks in meeting said.
"Love ye one another," ministers recite;
Bless me, DON'T we do it--sparking Sunday night?
One arm with gentle pressure lingers round her waist,
You squeeze her dimpled hand, her pouting lips you taste,
She freely slaps your face, but more in love than spite;
Oh, thunder! ain't it pleasant--sparking Sunday night?
But hark! the clock is striking; it is two o'clock, I snum,
As sure as I'm a sinner, the time to go has COME.
You ask, with spiteful accents, if "that old clock is right!"
And wonder if IT ever--sparked on Sunday night!
One, two, three sweet kisses; four, five, six, you hook;
But, thinking that you rob her, give back those you took;
Then, as for home you hurry, from the fair one's sight,
Don't you wish EACH DAY was only Sunday night?
42
Answer of Katy Darling.
Oh, in heaven you will meet your Katy Darling,
There my smiles you may ever more behold.
I believed not you were false to Katy Darling.
Or that your love had ever grown cold.
Oh no, I could not believe
That my Dermot was untrue.
No love was like the love of Katy Darling,
Search the world you will find very few.
I'm ever near you, dearest.
When all is wrapp'd in slumber, Katy Darling
Is watching by her dear Dermot's side,
Your loving and beloved Katy Darling,
Her spirit will ever be your guide.
When you kneel by the grave of Katy Darling,
Katy's spirit will meet with you there,
Dear Dermot, weep no more for Katy Darling,
This bright world is free from all care.
By my grave I see you weeping
In the silent starry light,
I long to have you with your Katy Darling,
Happy you'd be with her this night.
I hear you dear Dermot.
And every night by the grave of Katy Darling,
She will meet you till you lie by her side,
Then in heaven you will meet your Katy Darling,
Dear Dermot and his much loved bride.
43
Sprig of Shillelah.
Och, love is the soul of a neat Irishman;
He loves all that is lovely, loves all that he can;
With a sprig of shillelah, and shamrock so green.
His heart is good-humor'd, 'tis honest and sound,
No malice or hatred is there to be found;
He courts and he marries, he drinks and he fights
For love--all for love--for in that he delights,
With his sprig of shillelah, and shamrock so green.
Who has e'er had the luck to see Donnybrook fair?
An Irishman all in his glory is there,
With his sprig of shillelah and shamrock so green;
His clothes spick and span new, without e'er a speck,
A neat Barcelona tied round his neck;
He goes to his tent, and spends his half-crown,
He meets with a friend who for love knocks him down,
With his sprig of shillelah and shamrock so green.
At evening returning, as homeward he goes,
His heart, soft with whiskey, his head soft with blows,
From a sprig of shillelah and shamrock so green.
He meets with his Shelah, who, blushing a smile,
Cries, "Get you gone, Pat!" yet consents all the while.
To the priest soon they go, and nine months after that,
A fine baby cries, "How d'ye do, Father Pat?"
With your sprig of shillelah and shamrock so green?
"Bless the country!" says I, "that gave Patrick his birth,
Bless the land of the oak, and its neighboring earth,
Where grows the shillelah and shamrock so green.
May the sons of the Thames, the Tweed, and the Shannon,
Thrash the sons that would plant on their confines a cannon.
United and happy, at liberty's shrine,
May the rose and the thistle long flourish and twine
Round a sprig of shillelah and shamrock so green."
44
The Low Back'd Car.
When first I saw sweet Peggy,
'Twas on a market day;
A Low Back'd Car she drove, and sat
Upon a truss of hay.
But when that hay was blooming grass,
And deck'd with flowers of spring,
No flowers were there that could compare
With the lovely girl I sing,
As she sat in the Low Back'd Car, the man at the turnpike bar,
Good-natured old soul, never ask'd for his toll,
But look'd after the Low Back'd Car.
In battle's wild commotion,
The proud and mighty Mars,
With hostile scythes, demands his tythes,
Of death in warlike scars;
But Peggy, peaceful goddess,
Has darts in her bright eye,
That knock men down in the market-town,
As right and left they fly;
As she sits in the Low Back'd Car, than battle more dangerous far,
For the doctor's art, cannot cure the heart
That is hit from the Low Back'd Car.
Sweet Peggy round her car, sir,
Has strings of ducks and geese;
But the scores of hearts she slaughters,
By far outnumber these.
While she among her poultry sits,
Just like a turtle-dove,
Well worth the cage, I do engage,
Of the blooming God of Love.
As she sits in her Low Back'd Car, the lovers come from afar,
And envy the chickens that Peggy is picking,
As she rides in her Low Back'd Car.
I'd rather own that car, sir,
With Peggy by my side,
Than a coach and four, and gold galore,
With a lady for my bride.
For the lady would sit forninst me,
On a cushion made with taste,
While Peggy would sit beside me,
With my arm around her waist.
As we rode in that Low Back'd Car, to be married by Father Magar,
Oh, my heart would beat high at each glance of her eye,
As we rode in the Low Back'd Car.
45
Poor Old Maids.
Fourscore and four of us, poor old maids,
What will become of us, poor old maids?
Fourscore and four of us,
Without a penny in our purse,
What the deuce then can be worse, poor old maids?
Dress'd in yellow, pink, and blue, poor old maids,
Dress'd in yellow, pink, and blue, poor old maids,
Dress'd in yellow, pink, and blue,
Nursing cats is all we do,
Nursing cats is all we do, poor old maids.
All alone we go to bed, poor old maids,
All alone we go to bed, poor old maids,
All alone we go to bed,
And not a word to us is said,
And not a word to us is said, poor old maids.
We're all in a willing mind, poor old maids,
We're all in a willing mind, poor old maids,
We're all in a willing mind,
If the men would be so kind,
As to wed the lame and blind, poor old maids.
And if there's any in this room, poor old maids,
And if there's any in this room, poor old maids,
And if there's any in this room,
I hope they'll marry very soon,
And enjoy life's honeymoon, poor old maids.
46
O God! Preserve the Mariner.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, publishers
of the music.
O God! preserve the mariner,
When o'er the troubled deep
The rolling thunder-lightning flash,
And howling tempests sweep;
When like a reed the tall mast shakes,
And human art is vain,
O God! restore the mariner
To home, dear home again.
The sailor's wife sinks down to rest,
But dreams disturb her sleep,
She starts to hear the hollow wind,
And turns aside to weep;
She clasps her baby, and she prays,
Through tears, like falling rain,
"O God! restore the mariner,
To home, dear home again."
The widow for her darling child,
Her bosom's only joy,
Invokes the Power that rules the storm,
For blessings on her boy.
When ruin lurketh in the cloud,
And death sweeps o'er the main,
O God! restore the mariner,
To home, dear home again.
47
A Merry Gipsy Girl again.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
A merry Gipsy girl again,
I'm free to rove at will:
The woodlands wild, the meadows sweet,
The valley and the hill
How poor the proudest roof ye boast
To that high-arched dome,
Whose boundless circle makes me think
The whole wide world my home.
Here none can bar the free fresh air,
Nor mete out heaven's light,
Nor make the glorious day appear
Too near akin to night.
Amid the beauties of the mead
My summer days are spent,
And joyfully the stars look down
Upon my Gipsy tent;
And joyfully the stars look down
Upon my Gipsy tent.
I wander freely as the fawn
Which hath not learnt to fear
The death-cry of the hunter's voice
Resounding far and near;
And bounding through the woods
I feel as if I too could soar,
Bird-like, upon the wings of joy,
And sing for evermore!
Come out, ye pent-up toilers!
Come, from city dark and drear,
And see what gladness Nature has
In all her beauties here;
And ere ye seek your homes, ye'll say,
Your time has well been spent,
And wish that all the world
Could be, one happy Gipsy tent;
And wish that all the world
Could be, one happy Gipsy tent.
48
Let Me Kiss Him for His Mother.
Let me kiss him for his mother,
Let me kiss his youthful brow;
I will love him for his mother,
And seek her blessing now.
Kind friends have soothed his pillow,
Have watch'd his every care;
Beneath the weeping willow,
Oh, lay him gently there.
CHORUS.
Sleep, dearest, sleep;
I love you as a brother;
Kind friends around you weep,
I've kiss'd you for your mother,
Let me kiss him for his mother,
What though left a stranger here?
She has loved him as none other,
I feel her blessing near.
Though cold that form lies sleeping,
Sweet angels watch around;
Dear friends are near thee weeping;
Oh, lay him gently down.
Sleep, dearest, sleep, &c.
Let me kiss him for his mother,
Or perchance a sister dear;
If a father or a brother,
I know their blessing's here.
Then kiss him for his mother:
'Twill soothe her after-years;
Farewell, dear stranger brother,
Our requiem, our tears.
Sleep, dearest, sleep, &c.
49
My ain Fireside.
Copied by permission of FIRTH, POND & CO., 547 Broadway, publishers
of the music.
Oh! I hae seen great anes, and sat in great ha'as,
'Mang Lords and mang Ladies, a' cover'd wi' braws,
At feasts made for Princes wi' Princes I've been,
Whar the grand shine o' splendor has dazzled my een,
CHORUS.
But a sight sae delightful I trow, I ne'er spied,
As the bonnie blithe blink o' my ain fireside,
My ain fireside, my ain fireside, oh! sweet is the blink o' my ain
fireside.
Ance mair, Heaven be praised, round my ain heart-some ingle,
Wi' the friends o' my youth, I cordially mingle;
Nae force now upon me to seem wae or glad,
I may laugh when I'm merry, and sigh when I'm sad.
_Chorus._--My ain fireside. &c.
Nae falsehood to dread, nae malice to fear,
But truth to delight me, and kindness to cheer;
O' a' the roads to pleasure that ever were tried,
There's nane half so sure as ain's ain fireside.
_Chorus._--My ain fireside. &c.
50
The Indian Warrior's Grave.
Green is the grave by the wild dashing river,
Where sleeps the brave with his arrows and quiver
Where in his pride he roved in his childhood
Fought he, and died, in the depths of the wildwood.
In the lone dell, while his wigwam defending,
Nobly he fell 'neath the hazel-boughs bending;
Where the pale foe and he struggled together,
Who from his bow tore his swift-arrow'd feather.
Ere the next noon the bold warrior was buried;
And ere a moon his tribe westward had hurried.
But a rude cross, with its rough-chiseled numbers,
Half hid in moss, tells the red warrior slumbers.
Indian Hunter.
Oh, why does the white man follow my path, like the hound on the
tiger's track?
Does the flush of my dark cheek waken his wrath? does he covet the
bow at my back?
He has rivers and seas, where the billows and breeze
Bear riches for him alone--
And the sons of the wood, never plunge in the flood,
Which the white man calls his own.
Yha, yha!
Then why should he come to the streams where none but the red skin
dare to swim?
Why, why should he wrong the hunter? one who never did harm to him!
Yha, yha, yha!
The Father above thought fit to give to the white man corn
and wine--
There are golden fields where he may live, but the forest shades
are mine.
The eagle hath its place of rest, the wild horse where to dwell,
And the spirit that gave the bird its nest, made me a home as well.
Yha, yha!
Then back! go back! from the red man's track, for the red man's
eyes are dim,
To find that the white man wrongs the one who never did harm to him.
Yha, yha, yha!
51
Molly Bawn.
Oh, Molly Bawn, why leave me pining,
Or lonely waiting here for you--
While the stars above are brightly shining,
Because they have nothing else to do.
The flowers late were open keeping,
To try a rival blush with you,
But their mother, Nature, kept them sleeping,
With their rosy faces wash'd in dew.
Oh, Molly, &c.
The pretty flowers were made to bloom, dear,
And the pretty stars were made to shine;
The pretty girls were made for the boys, dear,
And may be you were made for mine.
The wicked watch dog here is snarling--
He takes me for a thief, d'ye see?
For he knows I'd steal you, Molly, darling,
And then transported I should be.
Oh, Molly, &c.
Norah, the Pride of Kildare.
As beauteous as Flora is charming young Norah,
The joy of my heart and the Pride of Kildare,
I ne'er will deceive her, for sadly 'twould grieve her,
To find that I sigh'd for another less fair.
CHORUS.
Her heart with truth teeming, her eye with smiles beaming,
What mortal could injure a blossom so fair.
Oh, Norah, dear Norah, the Pride of Kildare.
Where e'er I may be, love, I'll ne'er forget thee, love,
Though beauties may smile and try to ensnare,
Yet nothing shall ever, my heart from thine sever,
Dear Norah, sweet Norah, the Pride of Kildare.
52
The Hazel Dell.
Copied by permission of WM. HALL & SON, 543 Broadway, N. Y.,
Publishers of the music and owners of the copyright.
In the Hazel Dell my Nelly's sleeping,
Nelly loved so long,
And my lonely, lonely watch I'm keeping,
Nelly lost and gone;
Here in moon-light often we have wandered,
Through the silent shade,
Now where leafy branches drooping,
Downward little Nelly's laid.
CHORUS.
All alone my watch I'm keeping,
In the Hazel Dell,
For my darling Nelly's near me sleeping,
Nelly dear, farewell.
In the Hazel Dell my Nelly's sleeping,
Where the flowers wave,
And the silent stars are nightly weeping,
O'er poor Nelly's grave,
Hopes that once my bosom fondly cherished,
Smile no more for me,
Every dream of joy alas has perished,
Nelly dear, with thee.
All alone my watch, &c.
Now I'm weary, friendless and forsaken,
Watching here alone,
Nelly, thou no more will fondly cheer me,
With thy loving tone,
Yet forever shall thy gentle image,
In my memory dwell,
And my tears thy lonely grave shall moisten,
Nelly dear, farewell.
All alone my watch, &c.
53
Home, Sweet Home.
'Mid pleasures and palaces, though we may roam,
Be it ever so humble, there's no place like home;
A charm from the skies seems to hallow us there,
Which, seek through the world, is ne'er met with elsewhere.
Home, home, sweet, sweet home,
There's no place like home.
I gaze on the moon, as I trace the drear wild,
And feel that my parent now thinks of her child;
She looks on that moon from our own cottage door,
Through woodbines whose fragrance shall cheer me no more.
Home, home, sweet, sweet home,
There's no place like home.
An exile from home, splendor dazzles in vain,
Oh, give me my lowly, thatch'd cottage again;
The birds singing gayly, that came at my call,
Give me them, with the peace of mind, dearer than all.
Home, home, sweet, sweet home,
There's no place like home.
My Boyhoods Home.
My boyhood's home! I see thy hills--
I see thy valley's changeful green,
And manhood's eye a tear-drop fills,
Though years have roll'd since thee I've seen.
I come to thee from war's dread school,
A warrior stern o'er thee to rule;
But while I gaze on each loved plain,
I feel I am a boy again.
To the war-steed adieu--to the trumpet farewell--
To the pomp of the palace--the proud, gilded dome;
For the green scenes of childhood, I bid ye farewell!
The soldier returns to his boyhood's loved home.
My boyhood's home, &c.
54
The Old Kirk-Yard.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, publishers
of the music.
Oh, come with me to the old kirk-yard,
I well know the path through the soft, green sward;
Friends slumber there we were wont to regard,
We'll trace out their names in the old kirk-yard.
Oh, mourn not for them, their grief is o'er,
Oh, weep not for them, they weep no more.
For deep is their sleep, though cold and hard,
Their pillow may be in the old kirk-yard.
I know it is vain when friends depart,
To breathe kind words to a broken heart;
I know that the joy of life seems marr'd;
When we follow them home to the old kirk-yard.
But were I at rest beneath yon tree,
Why should'st thou weep, dear love, for me?
I'm way-worn and sad, ah, why then <DW44>,
The rest that I seek in the old kirk-yard?
55
I am a Freeman.
I am a freeman! 'Tis my boast and my pride,
The blue sky is o'er me, the dark soil beneath,
And spreading around is the wilderness wide;
My bath is the lake, my couch is the heath,
My rod and my rifle my larder provide--
I am a freeman! 'Tis my boast and my pride,
I am a freeman! _True_ freedom is mine;
I slay when I choose, yet spare when I will;
For my food use the bullet, or cast out the line,
But never, like fools, from wantonness kill.
My "roof-tree" is lofty, my dining-hall wide--
I am a freeman! 'Tis my boast and my pride.
The eagle above me soars lofty and free,
He knows that I'll speed no bullet at him--
He is game for a tyrant, but _never_ for me,
While he sits on his nest on that old pine limb.
A life in the woods some men may deride,
But _freedom_ is there, my boast and my pride.
I roam through the wild wood o'er skim or the lake,
My wreaths are of laurel, my plumes never fade;
I sleep when the night falls, with the dawn am awake,
To hunt the red deer while they feed in the glade.
I'm joyous and free as a bird of the air,--
A son of the forest, a stranger to care.
56
Ship A-Hoy!
Copied by permission of FIRTH, POND & CO., 547 Broadway, publishers
of the music.
When o'er the silent seas alone,
For days and nights we've cheerless gone,
Oh! they who've felt it, know how sweet,
Some sunny morn a sail to meet,
Some sunny morn a sail to meet!
Sparkling on deck is every eye;
"Ship a-hoy! ship a-hoy!" our joyful cry.
When answering back we faintly hear,
"Ship a-hoy! what cheer! what cheer!'"
Now sails aback we nearer come,
Kind words are said of friends at home;
But soon, too soon, we part in pain,
To sail o'er silent seas again,
To sail o'er silent seas again.
When o'er the ocean's dreary plain,
With toil her destined port to gain,
Our gallant ship has near'd the strand,
We claim our own, our native land,
We claim our own, our native land;
Sweet is the seaman's joyous shout,
"Land ahead! land ahead! look out! look out!"
Around on deck we gayly fly,
"Land ahead! land ahead!" with joy we cry;
Yon beacon light directs our way,
While grateful vows to Heaven we pay,
And soon our long-lost joys renew,
And bid the boisterous main adieu,
And bid the boisterous main adieu.
57
Song of Blanche Alpen.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, N. Y.,
publishers of the music.
You speak of sunny skies to me--
Of orange grove and bower--
Of winds that wake soft melody
From leaf and blooming flower;
And you may prize those far-off skies,
But tempt not me to roam;
In sweet content my days are spent,
Then wherefore leave my home?
In sweet content my days are spent,
Then wherefore leave my home?
You tell me oft of rivers bright,
Where golden galleys float;
But have you seen our lakes by night,
Or sail'd in Alpine boat?
You speak of lands where hearts and hands
Will greet me as I come,
But though I find true hearts and kind,
They're kinder still at home.
But though I find true hearts and kind,
They're kinder still at home.
Had you been rear'd by Alpine hills,
Or lived in Alpine dells,
You'd prize, like me, our mountain rills,
Nor fear the torrent swells;
It matters not how drear the spot
How proud or poor the dome,
Love still retains some deathless chains,
That binds the heart to home.
Love still retains some deathless chains,
That binds the heart to home.
58
By the Sad Sea-Waves.
Copied by permission of FIRTH, POND & CO., 547 Broadway, publishers
of the music.
By the sad sea-waves
I listen, while they moan
A lament o'er graves
Of hope and pleasure gone.
I am young, I was fair,
I had once not a care
From the rising of the morn
To the setting of the sun.
Yet I pine like a slave,
By the sad sea-wave.
Come again bright days
Of hope and pleasure gone;
Come again, bright days,
Come again, come again.
From my care last night,
By holy sleep beguiled,
In the fair dream-light
My home upon me smiled.
Oh, how sweet 'mid the dew,
Every flower that I knew
Breathed a gentle welcome back
To the worn and weary child!
I wake in my grave
By the sad sea-wave;
Come again, dear dream,
So peacefully that smiled,
Come again, dear dream,
Come again, come again.
59
Daylight is on the Sea.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
Daylight is on the sea!
Love, do not stay;
Land is no place for me,
I must away!
My bark is on the waves,
My boat ashore;
The surge its broadside laves,
While sleeps each oar.
CHORUS.
Daylight is on the sea,
Land is no place for me;
Come away, love, come away, love,
I dare no longer stay;
Come away, love, away, love,
I dare no longer stay.
Come away, away, away, away, away,
I dare no longer stay,
Away, away, away, away, away,
I dare no longer stay.
Daylight plays o'er the deep,
Like childhood's smile;
Blue waves and hush'd winds sleep,
Enchain'd awhile!
My bark is on the waves,
My boat ashore,
The surge its broadside laves,
While sleeps each oar.
Daylight is on the sea, &c.
60
Kate was once a Little Girl.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
Kate was once a little girl,
Heigh ho! heigh ho!
Eyes of blue, and teeth of pearl,
Heigh ho! heigh ho!
In the spring, when school was done,
Full of life and full of fun,
O'er the hills away she'd run,
Heigh ho! heigh ho!
Gentle breezes all the day,
Heigh ho! heigh ho!
Through her sunny locks would play,
Heigh ho! heigh ho!
Still on her cheek as brightly plays
The sunshine of her youthful days,
And still as sweet her girlish ways,
Heigh ho! heigh ho!
Kate's a little older now,
Heigh ho! heigh ho!
Still as fair her radiant brow,
Heigh ho! heigh ho!
All her thoughts are pure and bright,
As the stars we see at night,
Shining with a joyous light,
Heigh ho! heigh ho!
Kate will always be the same,
Heigh ho! heigh ho!
She'll never change except in name,
Heigh ho! heigh ho!
So gently time shall steal away,
She'll always be as bright and gay,
As when she laugh'd in girlhood's day,
Heigh ho! heigh ho!
61
Kitty Tyrrell.
Copied by permission of FIRTH, POND, & CO., 547 Broadway, N. Y.,
publishers of the music.
You're looking as fresh as the morn, darling,
You're looking as bright as the day;
But while on your charms I'm dilating,
You're stealing my poor heart away.
But keep it and welcome, mavourneen,
Its loss I'm not going to mourn;
Yet one heart's enough for a body,
So pray give me yours in return.
Mavourneen, mavourneen,
Oh! pray give me yours in return.
I've built me a neat little cot, darling,
I've pigs and potatoes in store;
I've twenty good pounds in the bank, love,
And may be, a pound or two more.
It's all very well to have riches,
But I'm such a covetous elf,
I can't help still sighing for something,
And, darling, that something's yourself.
Mavourneen, mavourneen,
And that something, you know, is yourself.
You're smiling, and that's a good sign, darling;
Say "Yes," and you'll never repent;
Or, if you would rather be silent,
Your silence I'll take for consent.
That good-natured dimple's a tell-tale,
Now all that I have is your own,
This week you may be Kitty Tyrrell,
Next week you'll be Mistress Malone.
Mavourneen, mavourneen,
You'll be my own Mistress Malone.
62
Within a mile of Edinboro' Town.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
'Twas within a mile of Edinboro' town,
In the rosy time of the year,
Sweet flowers bloom'd and the grass was down,
And each shepherd woo'd his dear;
Bonny Jocky blithe and gay,
Kiss'd sweet Jenny makin' hay,
The lassie blush'd and frowning cried,
No no, it will not do, I can not, can not, wonnot, wonnot,
monnot buckle too.
Jocky was a wag that never would wed,
Though long he had follow'd the lass,
Contented she earn'd and eat her brown bread,
And merrily turn'd up the grass.
Bonny Jocky blithe and free,
Won her heart right merrily;
Yet still she blush'd and frowning cried,
No no, it will not do, I can not, &c.
But when he vow'd he would make her his bride.
Though his flocks and herds were not few
She gave him her hand and a kiss beside,
And vow'd she'd forever be true.
Bonny Jocky blithe and free;
Won her heart right merrily;
At church she no more frowning cried.
No, no, it will not do, I can not, &c.
63
Would I Were With Thee.
Would I were with thee, ev'ry day and hour
Which now I pass so sadly far from thee,
Would that my form possess'd the magic power
To follow where my heavy heart would be;
Whate'er thy lot o'er land or sea,
Would I were with thee eternally.
Would I were with thee, when the world forgetting
Thy weary limbs upon the turf are thrown,
While bright and red our evening sun is setting,
And all thy thoughts belong to heaven alone;
While happy dreams thy thoughts employ,
Would I were with thee in thy joy.
Would I were with thee, when no longer feigning
The hurried laugh, that stifles back a sigh,
When thy young lip pours forth its sweet complaining,
And tears have quench'd the light within thine eye;
When all seems dark and sad below,
Would I were with thee in thy woe.
Would I were with thee, when the day is breaking,
And when the moon has lit the lonely sea,
Or when in crowds some careless note awaking,
Speaks to thy heart in memory of me:
In joy, or pain, by sea, or shore,
Would I were with thee evermore.
64
Old Uncle Edward.
There formerly might have been seen an aged individual,
Whose cognomen was Uncle Edward,
He departed this life some time since, some time since,
And he had no capillary substance on the summit of his cranium,
On the place designed by nature for the capillary to vegetate.
CHORUS.
Then lay down the agricultural implements,
Allow the violin and the bow to be pendent on the wall.--
For there is no more physical energy to be displayed by indigent
aged Edward,
For he has departed to the abode designated by a kind Providence
for all pious, humane, and benevolent individuals.
Uncle Edward had digits equal in longitude to the
Bamboo formation which springs so spontaneously on the bank of the
Southern Mississippi,
And he had no oculars with which to observe
The beauties of nature,
And he had no dental formations with which to
Masticate the Indian meal cake,
Consequently he was forced to permit the
Indian meal cake to pass by with impunity.
_Chorus._--Then lay down, &c.
When Uncle Ned relinquished his hold on vitality,
His master was exceedingly grieved,
And the lachrymal poured down his cheeks similar to the rain from
heaven,
For he knew that the old man was laid beneath terra firma, terra
firma,
He would never have the pleasure of beholding the physiognomy of
the aged Edward any more.
_Chorus._--Then lay down, &c.
65
Uncle Gabriel.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
I was gwan to Sandy Point de oder arternoon,
Dis <DW65>'s heel cum'b out ob joint a running arter a <DW53>;
I thought I see'd him on a log, a lookin' mighty quar
Wen I cum'd up to de log, de <DW53> he wasn't dar.
CHORUS.
Oh, come along, my Sandy boy, now come along, oh do;
Oh, what will uncle Gabriel say? ya eh eh eh ya eh eh eh;
What will Uncle Gabriel say, why Jinny can't you come along too?
I blow'd de horn, I call'd de dog, and tell him for to bark;
I hunt all night in de hollow log, but de <DW53> he still keep dark;
At last I hear de ole <DW53> sneeze, de dog he fly around,
And on to him he den did freeze, and pull him to the ground.
_Chorus._--Oh, come along, my Sandy boy, &c.
De <DW53> he lay upon de ground, as stiff as any post;
I knock him den upon de head, and he gabe up de ghost;
I took him to de old log house, as soon as he suspire;
He look'd just like a little mouse, and we roast him on de fire.
_Chorus._--Oh, come along, my Sandy boy, &c.
De <DW65>s dey come all around, and kick up a debil of a splutter.
Dey eat de <DW53> and clar de ground, to dance de chicken flutter,
Dey dance all night till de broke of day, to a tune on de old banjo,
And den dey all did gwan away, before de chicken crow.
_Chorus._--Oh, come along, my Sandy boy, &c.
66
He led Her to the Altar.
Copied by permission of FIRTH, POND & CO., 547 Broadway, N. Y.,
publishers of the music.
He led her to the altar,
But the bride was not his chosen;
He led her with a hand as cold
As though its pulse had frozen.
Flowers were crush'd beneath his tread,
A gilded dome was o'er him;
But his brow was damp, and his lips were pale,
As the marble steps before him.
CHORUS.
He led her to the altar,
But the bride was not his chosen;
He led her with a hand as cold
As though its pulse had frozen.
His soul was sadly dreaming,
Of one he had hoped to cherish;
Of a name and form that the sacred rites,
Beginning, told must perish.
He gazed not on the stars and gems
Of those who circled round him;
But trembled as his lips gave forth
The words that falsely bound him.
He led her to the altar, &c.
Many a heart was praising,
Many a hand was proffer'd;
But mournfully he turn'd him
From the greeting that was offer'd.
Despair had fix'd upon his brow
Its deepest, saddest token,
And the bloodless cheek and stifled sigh
Betray'd his heart was broken.
He led her to the altar, &c.
67
Where are now the Hopes I Cherished?
Copied by permission of FIRTH, POND & CO., 547 Broadway, publishers
of the music.
Where are now the hopes I cherish'd?
Where the joys that once were mine?
Gone forever--all have perish'd,
And the blighter's hand was thine!
Look upon me, and remember
Thy Norma ere she was betray'd;
Look again, and look exulting,
On the ruin thou hast made;
Look again, and look exulting,
On the ruin thou hast made.
Canst thou think, as thou dost listen
To thy children's artless songs,
Of that moment when their fond hearts,
First shall feel their mother's wrongs?
Ha! thou shrinkest like the lightning,
To thy bosom fell remorse shall dart,
And thou yet shall know the anguish
Which hath broken my poor heart;
And thou yet shall know the anguish
Which hath broken my poor heart.
68
Paddy on the Canal.
When I landed in sweet Philadelphia, the weather was pleasant and
clear,
I did not stay long in the city, so quickly I shall let you hear.
I did not stay long in the city, for it happen'd to be in the fall,
I never reef'd a sail in my rigging, till I anchor'd out on the
canal.
CHORUS.--So fare you well, father and mother,
Likewise to old Ireland too;
So fare you well, sister and brother,
So kindly I'll bid you adieu.
When I came to this wonderful rampire, it fill'd me with the
greatest surprise,
To see such a great undertaking, on the like I never open'd my
eyes;
To see full a thousand brave fellows at work among mountains so
tall,
To dig through the valleys so level, through rocks for to cut a
canal.
So fare you well, &c.
I enter'd with them for a season, my monthly pay for to draw,
And being in very good humor, I often sang Erin Go Bragh.
Our provision it was very plenty, to complain we'd no reason at
all,
I had money in every pocket while working upon the canal.
So fare you well, &c.
I learnt to be very handy, to use both the shovel and spade,
I learnt the whole art of canalling--I think it an excellent trade.
I learned to be very handy, although I was not very tall,
I could handle the sprig of shillelah, with the best man on the
canal.
So fare you well, &c.
I being an entire stranger, be sure I had not much to say,
The boss came round in a hurry, says, "Boys, it is grog-time
a-day;"
We all marched up in good order, he was father now unto us all,
Sure I wish'd myself from that moment to be working upon the canal.
So fare you well, &c.
When at night we all rest from our labor, be sure but our rent is
all paid.
We lay down our pick and our shovel, likewise our axe and our
spade.
We all set a-joking together, there was nothing our minds to
enthrall
If happiness be in this wide world, I am sure it is on the canal.
So fare you well, &c.
69
Jane Monroe.
Copied by permission of WM. HALL & SON, 547 Broadway, N. Y.,
owners of the copyright.
It was down in Louisiana,
Not many years ago,
I fell in lub wid a pretty gal,
And her name was Jane Monroe;
Her eyes was bright as diamonds,
Her teeth was white as snow--
Oh, de prettiest gal I eber saw,
Was charming Jane Monroe!
CHORUS.
But now she is far, far away,
And we hear from her ebery day;
And if she was here we'd have nothing to fear,
For we <DW54>s all lub her so gay.
She was like a model,
From her head down to her toe,
And sprightly as de hopper grass,
Was charming Jane Monroe.
I'd rather be a slave for life,
And hab de corn to hoe,
Dan to be free, and lib widout
My charming Jane Monroe.
But now she is far, far away, &c.
A darkey trader came one day,
And bought my gal from me,
And left me here alone to mourn
Beneaf de cypress-tree;
It fill'd my heart wid grief an' pain,
To think dey'd treat me so,
But I live in hopes to meet again
My charming Jane Monroe.
But now she far, far away, &c.
70
Tom Brown.
The King will take the Queen,
And the Queen will take the Jack;
And now as we're together here,
We'll ne'er a one go back:
Here's to you, Tom Brown,
And with you I'll drink a quart;
Here's to you with all my heart,
And with you I'll spend a shilling or two,
And thus before we part,
Here's to you, Tom Brown. _Repeat._
The Jack will take the Ten,
And the Ten will take the Nine;
And now that we're together here,
We'll take a glass of wine.
Here's to you, Tom Brown, &c.
The Nine will take the Eight,
And the Eight will take the Seven;
And now that we're together here,
We'll stay 'till after eleven.
Here's to you, Tom Brown, &c.
The Seven will take the Six,
And the Six will take the Five;
And now that we're together here;
We'll drink while we're alive.
Here's to you, Tom Brown, &c.
The Five will take the Four,
And the Four will take the Trey, (three)
And now that we're together here,
We'll stay till the break of day.
Here's to you, Tom Brown, &c.
The Trey will take the Deuce, (two)
And the Deuce won't take the One;
And now that we're together here,
We'll quit where we've begun.
Here's to you, Tom Brown, &c.
71
Uncle Tim, the Toper.
There was an old toper, his name was Uncle Tim,
And he lived long ago, long ago;
And he spent all his money for whiskey and gin,
At the place where he hadn't ought to go.
CHORUS.
So, throw away the bottle and the jug!
Hang up the dipper and the mug!
There's no more hard drink for old Uncle Tim,
For he's thrown away the bottle and the jug!
Uncle Tim had a nose like a red woolen sack,
And the pimples on his face not a few;
And he had one eye that was very, very black,
And the other t'other one was blue!
The hair on his head was like a mop on a stick,
And he had but one leg for to go;
So you see he couldn't go for to come it very quick,
So he had to, and go it very slow.
Uncle Tim was a hard one, and he used to take his T,
And the way he used to take it wasn't slow;
And the kind he used to take it wasn't Bohee,
If it had a been it wouldn't have served him so.
Oh! he toddled, t'other day, into the William Tell
A noted loafer's cubby-hole, you know;
Where they sell for medicine the raw material,
And sea-turtles caught in the Ohio.
He drank and he spree'd till his money was all gone,
And he couldn't drink and spree it any more;
And then they kick'd him out, and he went zigzag home,
Just as he'd done many times before.
Then the Devil, with the poker, and all the evil ones,
Got after him and worried him full sore;
Says he, "Old joker, I'm going to join the Sons,
So you can't come it never any more!"
Now come, you liquor-sellers, and you liquor-drinkers too:
Give up the bad practice, and be men!
Come up and join the Sons, and stick to them, too,
And never touch the filthy stuff again!
CONTENTS
OF
Beadle's Dime Military Song Book
AND SONGS FOR THE WAR.
A Dragoon Song,
A Good Time Coming,
A Hero of the Revolution,
A National Song,
A Soldier Lad my Love Shall be,
A Steed, a Steed of Matchless Speed,
All do Allow it, March where we may,
America,
Annie Laurie,
Auld Lang Syne,
Battle Hymn, Columns, Steady!
Bruce's Address,
Burial of Sir John Moore,
Charge of the Light Brigade,
Hail Columbia,
Hail to the Chief,
Happy are we to-night, Boys,
Hohenlinden,
Hymn,
I'm Leaving Thee in Sorrow, Annie,
It is Great for Our Country to Die,
It is not on the Battle-field,
Light Sounds the Harp,
Mad Anthony Wayne,
Martial Elegy,
Merrily every Bosom Boundeth
My Soldier Lad,
National Song,
Our Flag,
Peace be to those who Bleed,
Prelude--The American Flag,
Red, White and Blue,
Soldier's Dirge,
Song,
Song for Invasion,
Song for the Fourth of July,
Star-Spangled Banner,
The American Boy,
The American Volunteer,
The Army and the Navy,
The Battle of Lexington,
The Dead at Buena Vista,
The Death of Napoleon,
The Dying Soldier to his Sword,
The Fallen Brave,
The Flag of our Union,
The Land of Washington,
The Marseilles Hymn,
The Mothers of our Forest Land,
The Myrtle and Steel,
The Origin of Yankee Doodle
The Rataplan,
The Revolutionary Battle of Eutaw,
The Soldier's Adieu,
The Soldier's Dream,
The Soldier's Farewell,
The Soldier's Return,
The Soldier's Wife,
The Sword Chant,
The Sword and the Staff,
The Sword of Bunker Hill,
The Triumph of Italian Freedom,
The Wounded Hussar,
Through Foemen Surrounding,
To the Memory of the Americans who bled at Eutaw Springs,
Uncle Sam's Farm,
Unfurl the Glorious Banner,
Up! March Away,
War Song,
Warren's Address,
Yankee Doodle.
CONTENTS
OF
Beadle's Dime Union Song Book,
No. 1.
A "Big Thing" Coming,
A Doleful Ballad,
All Hail to the Stars and Stripes,
America,
An Ode to Washington,
An Old Story with a New Moral,
Anthem,
Army Hymn,
A Yankee Skip and a Yankee Crew,
Banner Song,
Cairo,
Columbia Forever,
Columbia Rules the Sea,
Dixie's Farms,
Dixie for the Union,
Eighty-five Years Ago,
Enfield Gun,
Freedom's Light,
God Save our Native Land,
God Save the Union,
God Save the Volunteers,
Hail Columbia,
Heaven for the Right,
Her Own Brave Volunteer,
Hunting Song of the Chivalry,
Hurra for the Union,
Let Cowards Shrink,
Long Live the Great and Free,
March Away, Volunteers,
Marching,
March of the Loyal Stores,
My own Native Land,
On, Brothers, on,
One I left There,
Our Banner Chorus,
Our Country,
Our Country, Right or Wrong,
Our Flag,
Our Good Ship Sails To-night,
Our Union, Right or Wrong,
Our Whole Country,
Red, White and Blue,
Soldier's Tent Song,
Song for Battle,
Stand by the Union,
Star-Spangled Banner,
Step to the Front,
The Banner of the Nation,
The Bold Zouaves,
The Dead of the Battle-field,
The Flag of our Union,
The Irish Brigade,
The Michigan "Dixie,"
The Northern Boys,
The Northman's Marseilles,
The Old Union Wagon,
The Original Yankee Doodle,
The Patriot Flag,
The Rock of Liberty,
The Southrons are Coming,
The Stripes and Stars,
The Sword of Bunker Hill,
The Union--It must be Preserved,
The Union, Young and Strong,
The Yankee Boy,
The Zouave Boys,
The Zouave's Song,
To the Seventy-ninth, Highlanders,
Traitor, Beware our Flag,
Unfurl the Glorious Banner,
Viva l'America,
Yankees are Coming.
CONTENTS
OF
Beadle's Dime Union Song Book,
No. 2.
A Life in the Soldier's Camp,
A Mother's Hymn in Time of War,
A Soldier's Dream of Home,
A Yankee Volunteer,
Away to the Fray,
Battle Invocation,
Beautiful Union,
Begone, Secesh,
Blue Jackets, Fall in,
Draw the Sword, Northland,
Drummer Boy of the National Greys,
"E Pluribus Unum,"
Flag Song,
Following the Drum,
Gathering Song,
Give us Room,
Hail Columbia,
Hark! to the Tread,
Hurrah for the Land we Love,
Liberty,
Mustering Chorus,
My Love he is a Zou-zu,
Our Country, Now and Ever,
Our Flag,
Rally, Boys!
Remember Traitors,
Rule, Columbia,
Song of the Zouaves,
Song of Union,
Stand by the Union,
Summons to the North,
Sweet is the Fight,
Sweet Maid of Erin,
The Alarum,
The Banner of Stars,
The Birth of our Banner,
The Brave and Free,
The Delaware Volunteers,
The Flag and the Union,
The Flag of the Brave,
The Flag of the Free,
The Great Union Club,
The "Mud-Sills" Greeting,
The Nation of the Free,
The Northmen are Coming,
The Northern Hurrah,
The Past and Present,
The Patriot's Address,
The Patriot's Serenade,
The Patriot's Wish,
The Patriot Soldier,
The Star Flag,
The Star-Gemmed Flag,
The Star-Spangled Banner,
The Stripes and Stars,
The Union Gunning Match,
The Union Harvesting,
The Union Marseillaise,
The Union Sacrifice,
The Volunteer Yankee Doodle of '61.
Three Cheers for our Banner.
Traitor, Spare that Flag,
Union Forever,
Victory's Band,
Volunteer's Song,
Where Liberty dwells there is my Country,
Wife of my Bosom,
Words of Sympathy.
CONTENTS
OF
Beadle's Dime Song Book,
NO. 1.
All's for the Best,
Annie Laurie,
A National Song,
Answer to a Thousand a Year,
Answer to Kate Kearney,
A Thousand a Year,
Belle Brandon,
Ben Bolt,
Blind Orphan Boy's Lament,
Bob Ridley,
Bold Privateer,
Do They Miss me at Home?
Don't be Angry, Mother,
Down the River,
E Pluribus Unum,
Evening Star,
Faded Flowers,
Gentle Annie,
Gentle Jenny Gray,
Glad to Get Home,
Hard Times,
Have You Seen my Sister,
Heather Dale,
Home Again,
I am not Angry,
I Want to Go Home,
Juney at the Gate,
Kate Kearney,
Kiss me Quick and Go,
Kitty Clyde,
Little Blacksmith,
My Home in Kentuck,
My Own Native Land,
Nelly Gray,
Nelly was a Lady,
Old Dog Tray,
Our Mary Ann,
Over the Mountain,
Poor Old Slave,
Red, White, and Blue,
Root, Hog, or Die,
Root, Hog, or Die, No. 2,
Root, Hog, or Die, No. 3,
Root, Hog, or Die, No. 4,
Row, Row,
Shells of the Ocean,
Song of the Sexton,
Star-Spangled Banner,
The Age of Progress,
The Dying Californian,
The Hills of New England,
The Lake-Side Shore,
The Miller of the Dee,
The Marseilles Hymn,
The Old Folks we Loved Long Ago,
The Old Farm-House,
The Old Play-Ground,
The Rock of Liberty,
The Sword of Bunker Hill,
The Tempest,
There's a Good Time Coming,
Twenty Years Ago,
Twinkling Stars,
Uncle Sam's Farm,
Unfurl the Glorious Banner,
Wait for the Wagon,
Willie, we have Miss'd You,
Willie'll Roam no More.
CONTENTS
OF
Beadle's Dime Song Book,
NO. 2.
Alice Gray,
America,
Banks of the Old Mohawk,
Be Kind to Each Other,
Billy Grimes the Rover,
Bryan O'Lynn,
Come Sit Thee Down,
Cora Lee,
Crazy Jane,
Darling Nelly Moore,
Darling Old Stick,
Fireman's Victory,
Good News from Home,
Good-Night,
Grave of Lilly Dale,
Graves of a Household,
Home, Sweet Home,
I have no Mother Now,
I'm leaving Thee in Sorrow, Annie,
I miss Thee so,
I Shouldn't like to Tell,
I Wandered, by the Brook-Side,
Katy Darling,
Kathleen Mavourneen,
Little Katy; or, Hot Corn,
Mary of the Wild Moor,
Mable Clare,
Mary Alleen,
Mill May,
Minnie Moore,
Minnie Dear,
Mrs. Lofty and I,
Mr. Finagan,
My Eye and Betty Martin,
My Love is a Saileur Boy,
My Mother Dear,
My Grandmother's Advice,
My Mother's Bible,
New England,
Oh! I'm Going Home,
Oh! Scorn not thy Brother,
O! the Sea, the Sea,
Old Sideling Hill,
Our Boyhood Days,
Our Father Land,
Peter Gray,
Rory O'More,
Somebody's waiting for Somebody,
The Farmer Sat in his Easy Chair,
The Farmer's Boy,
The Irishman's Shanty,
The Old Folks are Gone,
The Post-Boy's Song,
The Quilting Party,
Three Bells,
'Tis Home where the Heart is,
Waiting for the May,
We Stand Here United,
What other Name than Thine, Mother?
Where the Bright Waves are Dashing,
What is Home without a Mother,
Widow Machree,
Willie's on the Dark Blue Sea,
Winter--Sleigh-Bell Song,
Nancy Bell; or, Old Pine Tree.
CONTENTS
OF
Beadle's Dime Song Book,
NO. 3.
Annie, Dear, Good-by,
A Sailor's Life for Me,
Bessy was a Sailor's Bride,
Bonny Jean,
Comic Katy Darling,
Comic Parody,
Darling Jenny Bell,
Darling Rosabel,
Death of Annie Laurie,
Ettie May,
Few Days,
Give 'em String and let 'em Went,
Go it while You're Young,
Hail Columbia,
Happy Hezekiah,
I'd Choose to be a Daisy,
I have Something Sweet to Tell You,
Isle of Beauty,
I Think of Old Ireland whereever I Go,
Jeannette and Jeannot,
John Jones,
Jordan is a Hard Road to Travel,
Kitty Kimo,
Lather and Shave,
Lager Bier Song,
Linda has Departed,
Lillie Bell,
Love Not,
Man the Life-Boat,
My Dear Old Mother,
My Girl with a Calico Dress,
My Heart's in Old Ireland,
My Poor Dog Tray,
Old Rosin the Bow,
Over the Left,
Old Dog Tray, No. 2.,
Parody on the West,
Pop Goes the Weasel,
Pretty Jane,
Rosa Lee,
Song of the Locomotive,
Sparking Sarah Jane,
The American Girl,
The American Boy,
The Boys of Kilkenny,
The Emigrant's Farewell,
The Fine Old English Gentleman,
The Fine Old Irish Gentleman,
The Fine Old Dutchman,
The Fireman's Death,
The Fireman's Boy,
The Girl I Left behind Me,
The Gold-Digger's Lament,
The Indian Hunter,
The Old Oaken Bucket,
The Old Whiskey Jug,
The Other Side of Jordan,
The Pirate's Serenade,
The Yellow Rose of Texas,
Ten O'Clock, or, Remember, Love, Remember,
Tilda Horn,
True Blue,
To the West,
Uncle Ned,
Unhappy Jeremiah,
Vilkins and his Dinah,
We Miss Thee at Home,
What Will Mrs. Grundy Say?
Woodman, Spare that Tree.
CONTENTS
OF
Beadle's Dime Song Book,
NO. 5.
A Dollar or Two,
A Man's a Man for a' That,
Angel's Whisper,
Auld Lang Syne,
A Yankee Ship, and a Yankee Crew,
Bashful Young Man,
Call Me Pet Names,
Camptown Races,
Charity,
Cheer, Boys, Cheer,
Comin' Thro' the Rye,
Der mot Astore,
Dilla Burn,
Down the Burn, Davy, Love,
Dumbarton's Bonnie Dell,
Ever of Thee,
Gum-Tree Canoe,
Hark! I hear an Angel Sing,
I'd Offer Thee this Hand of Mine,
In the Days when I Was Hard Up,
John Anderson, my Jo, John,
Johnny was a Shoemaker,
Kind Relations,
Last Week I took a Wife,
Mary of Argyle,
Meet Me by Moonlight,
Napolitaine,
Norah M'Shane,
Nothing Else to Do,
Och! Paddy, is it Yerself?
Oft in the Stilly Night,
Roll on Silver Moon,
<DW71>, I have Miss'd You,
Sammy Slap, the Bill-Sticker,
Simon the Cellarer,
Something to Love Me,
Some Love to Drink,
Sourkrout and Sausages,
Still so Gently o'er Me Stealing
The Gay Cavalier,
The Gambler's Wife,
The Grave of Uncle True,
The Grave of Bonaparte,
The Ingle Side,
The Irish Emigrant's Lament,
The Ivy Green,
The Lass that Loves a Sailor,
The Last Rose of Summer,
The Lily of the West,
The Minute Gun at Sea,
The Monks of Old,
The Musical Wife,
The Ocean Burial,
The Old Arm-Chair,
The Poor Little Fisherman's Girl,
The Rat-catcher's Daughter,
The Rose of Allendale,
The Tail iv Me Coat,
The Watcher,
Thou art Gone from my Gaze,
Thou hast Wounded the Spirit,
'Tis Midnight Hour,
Twilight Dews,
Umbrella Courtship,
Wake! Dinah, Wake!,
Washington, Star of the West,
We'll have a little Dance To-Night, Boys,
We Met by Chance,
When I Saw Sweet Nelly Home,
When the Swallows Homeward Fly,
Whoop de Doodle do,
William of the Ferry,
Will You Love Me Then as Now?
CONTENTS
OF
Beadle's Dime Song Book,
NO. 6.
Annie Lisle,
Beautiful World,
Be Kind to the Loved Ones,
Bobbin' Around,
Bonnie Dundee,
Courting in Connecticut,
Dearest Mae,
Dear Mother, I'll Come again,
Ella Ree,
Fairy Dell,
Far, far upon the Sea,
Gentle Hallie,
Gentle Nettie Moore,
Happy are we To-night,
Hattie Lee,
He Doeth All Things Well,
I can not Call her Mother,
I'll Paddle my own Canoe,
I'm Standing by thy Grave, Mother,
Is it Anybody's Business?
Jane O'Malley,
Jenny Lane,
Joanna Snow,
Johnny Sands,
Lilly Dale,
Little more Cider,
Lulu is our Darling Pride,
Marion Lee,
Meet me by the Running Brook,
Minnie Clyde,
Not for Gold,
Not Married Yet,
Oh, carry me Home to Die,
Oh! Silber Shining Moon,
Oh! Spare the Old Homestead,
Old Homestead,
Ossian's Serenade,
Over the River,
Riding on a Rail,
Sailor Boy's Last Dream,
"Say Yes, Pussy,"
Spirit Voice of Belle Brandon,
Squire Jones's Daughter,
The Bloom is on the Rye,
The Blue Junietta,
The Carrier Dove,
The Child's Wish,
The Cottage of my Mother,
The Female Auctioneer,
The Irish Jaunting Car,
The Lords of Creation shall Woman obey,
The Maniac,
The Merry Sleigh-Ride,
The Miller's Maid,
The Modern Belle,
The Mountaineer's Farewell,
The Old Mountain Tree,
The Strawberry Girl,
The Snow Storm,
The Song my Mother used to Sing,
Three Grains of Corn,
Washington's Grave,
What is Home without a Sister,
Where are the Friends?
Why Chime the Bells so Merrily?
Why don't the Men propose?
Will Nobody Marry Me?
Young Recruit.
HAND-BOOKS FOR HOUSEKEEPERS.
BEADLE'S DIME COOK-BOOK,
BEADLE'S DIME RECIPE-BOOK,
BEADLE'S DIME DRESS-MAKER AND MILLINER,
BEADLE'S DIME BOOK OF ETIQUETTE,
BEADLE'S DIME FAMILY PHYSICIAN.
The COOK-BOOK embraces Recipes, Directions, Rules and Facts relating
to every department of Housekeeping.
The RECIPE-BOOK is a perfect treasure house of knowledge, for the
kitchen, parlor, nursery, sick-room, the toilet, &c., &c.
The BOOK OF ETIQUETTE can truly be called a useful work. It embodies
all the information necessary to "post" the reader, old or young, male
or female, upon every point of etiquette or social usage.
The FAMILY PHYSICIAN is an invaluable hand-book for the family and an
indispensable aid to the thrifty housewife.
BOOKS FOR THE SCHOOL AND HOME STUDENTS.
BEADLE'S DIME SPEAKER Nos. 1 & 2,
BEADLE'S DIME DIALOGUES Nos. 1 & 2,
BEADLE'S DIME SCHOOL MELODIST,
BEADLE'S DIME LETTER-WRITER.
This series of educational works is designed to meet the wants of
every school, public or private--every scholar, male or female, in our
country.
MUSIC AND SONG.
Beadle's Dime Song Books, No's 1, 2, 3, 4, 5, 6 & 7
BEADLE'S DIME MILITARY SONG BOOK,
BEADLE'S DIME MELODIST--WORDS AND MUSIC.
GAMES, AMUSEMENTS, &C.
BEADLE'S DIME BASE-BALL PLAYER,
BEADLE'S DIME GUIDE TO CRICKET,
BEADLE'S DIME GUIDE TO SWIMMING,
BEADLE'S DIME BOOK OF DREAMS,
BEADLE'S DIME BOOK OF FUN, Nos. 1 & 2,
BEADLE'S DIME CHESS INSTRUCTOR.
BEADLE'S DIME BIOGRAPHICAL LIBRARY.
No. 1.--GARIBALDI: THE WASHINGTON OF ITALY.
No. 2.--DANIEL BOONE: THE HUNTER OF KENTUCKY.
No. 3.--KIT CARSON: THE ROCKY MOUNTAIN SCOUT AND GUIDE.
No. 4.--MAJOR-GENERAL ANTHONY WAYNE: THE REVOLUTIONARY PATRIOT
AND INDIAN CONQUEROR.
No. 5.--COL. DAVID CROCKETT: AND HIS ADVENTURES.
No. 6.--JOHN PAUL JONES: THE NAVAL HERO OF '76.
HAVE YOU A FRIEND IN THE ARMY?
Send Him The Military Hand-Book.
The great want of a MILITARY HAND-BOOK of General and Special
Information on all matters connected with a Soldier's Life and
Experience, has induced the publishers of the Dime Publications to
have prepared, by competent hands, a work which will fully answer the
requirements of the market. They have, therefore, to announce
THE
MILITARY HAND-BOOK,
AND
SOLDIERS' MANUAL OF INFORMATION.
Embracing Pay-Lists of Officers and Men--Rations--
Incidents of Camp-Life--Hints on Health and
Comfort--How to Prepare Good Food from
Poor Rations--Recipes--Wounds, and
How to Care for Them--All about
Weapons of War, etc.; also
Official Articles of War,
AND A COMPLETE
DICTIONARY OF MILITARY TERMS.
☞ This admirable volume is published in large 12mo., with a
beautifully Engraved and Cover, and can be had of all News
Dealers at the low sum of TWENTY-FIVE CENTS.
BEADLE AND COMPANY, Publishers,
141 William St., New York.
Transcriber's Note:
Dialect was not changed. Unprinted letters and punctuation were added,
as necessary. Quotation marks were adjusted, where necessary.
Changes:
"evneing's" to "evening's": ... bright as evening's glow,...
"stara" to "star"s: ... Then say ere yonder stars have set,...
"brookleth" to "brooklet": ... Where the brooklet singeth o'er...
"permisson" to "permission": ... permission of FIRTH, POND & CO.,
(3 occasions)
"shsll" to "shall": ... When shall I be a bride,...
"companton" to "companion": ... my trusting companion, ...
"pleaures" to "pleasure": ... for her comfort with pleasure,
"BAEDLE'S" to "BEADLE'S": ... BEADLE'S DIME MILITARY SONG BOOK,...
End of Project Gutenberg's Beadle's Dime Song Book No. 4, by Various
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 1,606
|
Nergis & Caroline | Pop! Wed Co.
Nergis and Caroline had a Styled Popup Wedding indoors at a rented space that we styled with some garland and curtains — so much fun! See their whole wedding and their story here.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,658
|
Q: Printing array value from a hash in perl I am new to perl and this thing is driving me nuts. I have a hash as below
%temp = (
a_collection => [\%first, \%second]
)
I want to get the array elements out as a string so i can use them as args in the loop. I have below code
foreach $item (@{$temp{'a_collection'}})
{
<convert to json> $item #convert each of the above hash to a json blob
<write to file> $file #write first blob to file "first.json" and so on
}
I got the convert to json part. I can print it to stdout. Now i want to write it to a file. Here the $file should have name "first" and "second". So the loop will create two files with names of the hash variables which are there in the above hash. I want the filenames to match so i can keep track of whats getting created.
Edit :
The basic premise is simple. Whatever i do, be it json encoding etc, i want the hash variable names as a string. So in the array above, i can have a hash with any name \%somename, in the loop i want the actual string "somename" in a different variable. As above, i can use this string as file name that gets created . I cannot change the above hash structure. Its just there, created by someone else, i can only access it.
Thanks
A: Given the following code:
use strict;
use warnings;
my %first = (foo => 2, bar => 3, bat => 5);
my %second = (baz => 7, quux => 11);
my %temp = (a_collection => [\%first, \%second]);
for my $href (@{$temp{a_collection}}) {
for my $key (keys(%$href)) {
print "$key: $href->{$key}\n";
}
}
This is the output produced:
bar: 3
foo: 2
bat: 5
quux: 11
baz: 7
Edit after new information was provided:
my %first = (foo => 2, bar => 3, bat => 5);
my %second = (baz => 7, quux => 11);
my %temp = (first => \%first, second => \%second);
for my $key (keys(%temp)) {
print "$key\n";
}
Edit after yet more new information was provided:
use JSON::XS;
my %first = (foo => 2, bar => 3, bat => 5);
my %second = (baz => 7, quux => 11);
my %temp = (first => \%first, second => \%second);
for my $key (keys(%temp)) {
open(my $fh, '>', "$key.json") or die $!;
print $fh encode_json($temp{$key});
close($fh);
}
Contents of first.json:
{"foo":2,"bat":5,"bar":3}
Contents of second.json:
{"quux":11,"baz":7}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,344
|
using System.Collections;
using System.Collections.Generic;
using System.Management.Automation;
namespace Cognifide.PowerShell.Core.Extensions
{
internal static class PowerShellExtensions
{
public static object BaseObject(this object obj)
{
while ((obj is PSObject))
{
obj = (obj as PSObject).ImmediateBaseObject;
}
return obj;
}
public static List<T> BaseList<T>(this object enumarable) where T : class
{
var newList = new List<T>();
if (enumarable is IEnumerable)
{
foreach (var val in enumarable as IEnumerable)
{
var newVal = val.BaseObject();
if (newVal is T)
{
newList.Add(newVal as T);
}
}
}
return newList;
}
public static object[] BaseArray(this object[] array)
{
var newArray = new object[array.Length];
for (var i = 0; i < array.Length; i++)
{
newArray[i] = array[i].BaseObject();
}
return newArray;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,310
|
Home MMA News CM Punk CM Punk suggests he could potentially fight outside of the UFC
CM Punk suggests he could potentially fight outside of the UFC
Former pro wrestler CM Punk is now 0-2 as a mixed martial artist. Both of those losses occurred in the UFC's Octagon, where he was first submitted in the first round by Mickey Gall, then walloped to a decision by Mike Jackson.
Since his loss to Jackson, which occurred at UFC 225 in June, Punk's future has been nebulous. More recently, however, CM Punk has suggested that he still plans to continue training in MMA at the very least. And speaking on Ariel Helwani's MMA Show this week, he suggested that he would consider fighting outside the UFC if the UFC is no longer interested in promoting his fights.
"I don't think it's UFC or bust," CM Punk said (transcript via MMAMania). "I'm definitely in a weird position, you know what I mean? There are other places out there, but would I go? Should I go?"
Whatever the case, CM Punk understands that his big name will likely propel him into lucrative spots wherever he ends up— though he also understands why the UFC might no longer be interested in him.
"Deserves got nothing to do with it," he said. "I still look at it like the world's my oyster. I'm fortunate enough to kinda pick and choose. I'd be the first to admit – you know me, I'm brutally honest – it probably doesn't make a lot of sense for me to fight in the UFC again. We'll have to wait and see what happens."
At present, CM Punk is still under contract with the UFC. But, to the best of his knowledge, his contract with the promotion will soon wrap up. This would certainly seem to open the door to his fighting in other promotions, such as Bellator.
"If there's a deadline [on that] it's coming up at the end of the year," he said.
Would you be interested in seeing CM Punk fight in promotions outside of the UFC in the future?
This article first appeared on BJPENN.COM on 9/11/2018.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,694
|
Watch Jennifer Lopez and Shakira's Epic Super Bowl Halftime Show
By Devon Black On 3 February 2020
Muy caliente.
Jennifer Lopez and Shakira showed up and showed out on the biggest stage on earth, the Pepsi Halftime Show, during Super Bowl LIV in Miami on Sunday (Feb. 2), which featured pole dancing, hip shaking, and a cameo from J.Lo's daughter.
The Latin superstars took the field at Hard Rock Stadium for a 14-minute spectacular that kicked off with Shakira. Dressed in a sparkly red midriff-baring outfit, Shakira—who celebrated her 43rd birthday today—launched into "She Wolf" followed by "Empire." After hypnotizing with her belly dancing, she went into "Wherever, Whenever." As fireworks lit up the night sky, she brought out Bad Bunny for his verse on Cardi B's "I Like It."
The Colombian songstress also paid homage to her roots with "Chantaje" before delivering her 2006 smash "Hips Don't Lie" while crowd surfing. "No fighting," she said at the end before making way for J.Lo.
The Bronx bombshell entered on a sculpture of the Empire State Building while performing her 1996 hit "Jenny From the Block." Dressed in a black cutout Versace catsuit, she continued with "Ain't It Funny" and "Get Right" before changing into a sparkly stunner for "Waiting for Tonight." As green laser beams shot across the field, J.Lo showed off her Hustlers moves on the pole.
J Balvin repped Latino gang while performing "Mi Gente" alongside J.Lo before her 11-year-old daughter Emme stole the spotlight while singing "Let's Get Loud." With Shakira on drums, J.Lo performed with the Puerto Rican flag draped across her back as her daughter sang "Born in the USA."
They capped it off with a fiery medley of "Waka Waka (This Time For Africa)" and "Let's Get Loud," complete with tribal dancing and salsa moves, before standing side by side for the triumphant finale as fireworks exploded over the stadium.
"Muchos gracias!" said Shakira, while J.Lo added, "Thank you so much."
The performance received rave reviews from fans and friends including Lady Gaga, Pink, Kim Kardashian, and Cardi B, who was seen singing along during J.Lo's performance. See some of their tweets below.
Cardi B singing along to @JLo #SuperBowl #PepsiHalftime #CardiFromTheBlock pic.twitter.com/Cqni3DZAG9
— Rap-Up (@RapUp) February 3, 2020
. @JLo and @shakira and all the special guests were so incredible!!! What a fun halftime show I danced and smiled the whole time. Such powerful y women!!!! On camera and off!!!!! Love you beautiful y talented women 💕💋 #SuperBowlHalftimeShow #SuperBowl
— Lady Gaga (@ladygaga) February 3, 2020
AMAZING!! She ABSOLUTELY CRUSHED IT! Wow, that was so fun!
I'm so proud of you, Jen! ❤️ @JLo pic.twitter.com/bD07MLcKYx
— Alex Rodriguez (@AROD) February 3, 2020
Damn son….. #PepsiHalftime ….they came for blood like a That was friggin impressive. Not mad at tall
— Ate. Too. Fore. (@questlove) February 3, 2020
OMG @shakira !!!!! she looks so beautiful!
— Kim Kardashian West (@KimKardashian) February 3, 2020
OMGGGGGG @JLo looks soooooo beyond beautiful! 🔥
Yes! Halftime was JOY!!!!!!!!! Yes!!!!! @shakira @JLo everything that's yes ❤️😍❤️
— P!nk (@Pink) February 3, 2020
Omg Shakira, Shakiraaaaa 😩😩😍😍😍😍😍😍 her & jlo killed it tonighttttttttt
— TEYANA M.J. SHUMPERT (@TEYANATAYLOR) February 3, 2020
Craig Mack Died of Congestive Heart Failure Says Erick Sermon
Best Empowering Latin Women Collabs, From Gloria Trevi & Alejandra Guzman to Natti Natasha & Kany Garcia
Machine Gun Kelly Announces 'Hotel Diablo' Album & World Tour Dates
Pop Singer-Songwriter JOHN.k Signs With Epic Records
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,281
|
import socket
from flask import Flask
from flask_admin import Admin, base
from flask_cache import Cache
from flask_wtf.csrf import CsrfProtect
import airflow
from airflow import models
from airflow.settings import Session
from airflow.www.blueprints import ck, routes
from airflow import jobs
from airflow import settings
from airflow import configuration
csrf = CsrfProtect()
def create_app(config=None):
app = Flask(__name__)
app.secret_key = configuration.get('webserver', 'SECRET_KEY')
app.config['LOGIN_DISABLED'] = not configuration.getboolean('webserver', 'AUTHENTICATE')
csrf.init_app(app)
#app.config = config
airflow.load_login()
airflow.login.login_manager.init_app(app)
cache = Cache(
app=app, config={'CACHE_TYPE': 'filesystem', 'CACHE_DIR': '/tmp'})
app.register_blueprint(ck, url_prefix='/ck')
app.register_blueprint(routes)
app.jinja_env.add_extension("chartkick.ext.charts")
with app.app_context():
from airflow.www import views
admin = Admin(
app, name='Airflow',
static_url_path='/admin',
index_view=views.HomeView(endpoint='', url='/admin', name="DAGs"),
template_mode='bootstrap3',
)
av = admin.add_view
vs = views
av(vs.Airflow(name='DAGs', category='DAGs'))
av(vs.QueryView(name='Ad Hoc Query', category="Data Profiling"))
av(vs.ChartModelView(
models.Chart, Session, name="Charts", category="Data Profiling"))
av(vs.KnowEventView(
models.KnownEvent,
Session, name="Known Events", category="Data Profiling"))
av(vs.SlaMissModelView(
models.SlaMiss,
Session, name="SLA Misses", category="Browse"))
av(vs.TaskInstanceModelView(models.TaskInstance,
Session, name="Task Instances", category="Browse"))
av(vs.LogModelView(
models.Log, Session, name="Logs", category="Browse"))
av(vs.JobModelView(
jobs.BaseJob, Session, name="Jobs", category="Browse"))
av(vs.PoolModelView(
models.Pool, Session, name="Pools", category="Admin"))
av(vs.ConfigurationView(
name='Configuration', category="Admin"))
av(vs.UserModelView(
models.User, Session, name="Users", category="Admin"))
av(vs.ConnectionModelView(
models.Connection, Session, name="Connections", category="Admin"))
av(vs.VariableView(
models.Variable, Session, name="Variables", category="Admin"))
admin.add_link(base.MenuLink(
category='Docs', name='Documentation',
url='http://pythonhosted.org/airflow/'))
admin.add_link(
base.MenuLink(category='Docs',
name='Github',url='https://github.com/airbnb/airflow'))
av(vs.DagRunModelView(
models.DagRun, Session, name="DAG Runs", category="Browse"))
av(vs.DagModelView(models.DagModel, Session, name=None))
# Hack to not add this view to the menu
admin._menu = admin._menu[:-1]
def integrate_plugins():
"""Integrate plugins to the context"""
from airflow.plugins_manager import (
admin_views, flask_blueprints, menu_links)
for v in admin_views:
admin.add_view(v)
for bp in flask_blueprints:
app.register_blueprint(bp)
for ml in menu_links:
admin.add_link(ml)
integrate_plugins()
@app.context_processor
def jinja_globals():
return {
'hostname': socket.gethostname(),
}
@app.teardown_appcontext
def shutdown_session(exception=None):
settings.Session.remove()
return app
app = None
def cached_app(config=None):
global app
if not app:
app = create_app(config)
return app
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,055
|
Фрумушика-Нова — село в Тарутинському районі Одеської області, знищене радянською владою 1946 року у зв'язку з облаштуванням військового полігону.
Фрумушика () — комуна у повіті Ботошані.
Фрумушика () — село у повіті Ботошані. Входить до складу комуни Фрумушика.
Фрумушика () — село у повіті Ясси. Входить до складу комуни Медиржак.
Фрумушика () — село у Кагульському районі.
Фрумушика () — село у Фалештському районі.
Фрумушика () — село у Флорештському районі.
Фрумушика () — село у Леовському районі.
Фрумушика () — село у Кишиневі.
Фрумушика-Ноуе () — село у Флорештському районі.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,584
|
Hannescamps ist eine französische Gemeinde im Département Pas-de-Calais in der Region Hauts-de-France. Sie gehört zum Arrondissement Arras und zum Kanton Avesnes-le-Comte.
Geographie
Hannescamps liegt etwa 21 Kilometer südwestlich von Arras an der Grenze zum Département Somme. Umgeben wird Gommecourt von den Nachbargemeinden Monchy-au-Bois im Norden, Bucquoy im Osten, Foncquevillers und Gommecourt im Süden und Bienvillers-au-Bois im Westen.
Geschichte
Hannescamps war während des Ersten Weltkriegs Schauplätz heftiger Kämpfe. Im benachbarten Gommecourt befand sich der westlichste Punkt der deutschen Westfront.
Sehenswürdigkeiten
In Hannescamps befinden sich mehrere Soldatenfriedhöfe der Commonwealth War Graves Commission.
Weblinks
Soldatenfriedhof Hannescamps auf der Website der Commonwealth War Graves Commission.
Soldatengräber im Kirchhof von Hannescamps auf der Website der Commonwealth War Graves Commission.
Ort in Hauts-de-France
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,809
|
Sumerian Gods and Goddess: Geshtinanna
Also, written Ngeshtin-ana, which is short for Ningeshtinanna, is at first reading a little confusing, she is reportedly a daughter of Enki and Ninhursag, an old goddess of agriculture, vegetation and thus fertility, and apparently an oracle of dreams, she was Dumuzid's sister and consort of Ningishzida (Ningisida?) but like so many of these myths, the truth lies buried in the sands of time and the dunes of unconsciousness.
The myth
According to the myth upon the death of her brothers consort Inanna, he has a dream in which the demons drag him to hades to take the place of Inanna who Enki her father has forced them to release. She helps him hide and when the demons arrive she refuses even under torture to reveal his whereabouts, but all is to no avail as an unnamed friend tells them exactly where he is, he has momentarily some respite as the sun god turns him into a gazelle, but eventually the demons catch him and he is fore-with dragged below. But Inanna soon begins to regret her decision, and along with his mother and sister, they all cry for his return, but no word of his whereabouts is forth coming, until one day a fly, passing by, reveals his location, Inanna soon finds him, but there is no easy way to gain his release, for the doors of hades open onto a one-way street, from which few escape. Inanna remembers that only by trading places with him was she allowed to return, so she decides that Geshtinanna his sister will take his place, and the deal done he returns, but the people lament, the rains stop, the vegetation dies, the land is barren and the ever hungry desert threatens to engulf it, thus Inanna realises a compromise is needed, and she decides that Dumuzid will spend half the year in his sister's place, and stay with her sister Ereshkigal in hades, and then they will swap places and his sister will stay with Ereshkigal, while Dumuzid spends time with her.
Is it not odd that Dumuzid is the consort of Inanna, but his sister is called Gesht- Inanna? -guest-(subsitute) Inanna, did he marry his sister, who can Guess the Inanna, answer?
The turning of Dumuzid into a gazelle, is not the same as Herakles pursuit of the Ceryneian hind, I know of other myths where a gazelle or hind was chased, and only caught when it was finally caught in a snow drift, when it comes to myth and astrology times change names but they always tell the same story, in Egyptian myth Hathor captures a gazelle, and milks it to restore the eyes of Horus.
What is the foundation of the story
Geshtinanna represents the time of year when thing grow, when shoots burst forth and vegetation green thing appear and grow, her brother Dumuzid is the shepherd who looks after the herds during the period when, the fresh grass and likes have ceased, guides them to the places where food is stored, even under the snow, Geshtinanna lives when things are growing, thus animals and men are fed, but Dumuzid looks after them when things do not grow, thus he is a shepherd who leads his flock, and keeps them from deaths door.
The myth is about the changing of the seasons, the six or seven months when things grow, and the five or six when they do not, and there is no great hidden secret in it.
⮞ 18 Geshtinanna
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,887
|
Q: repaint() is not called from run I have a Jpanel where i draw a level of game. My problem is that the method run not repaint the images. What am i doing wrong?
If i call repaint() in the keyLister the jpanel is repainted but runnable not have importance...
Thanks for the help!!!!
public class Board extends JPanel implements Runnable {
private Map map;
private Player player;
public Board(String path) {
this.map = new Map(path);
this.jugador = new Jugador();
addKeyListener(new Al());
this.setFocusable(true);
}
public void paint(Graphics g) {
super.paint(g);
for (int y = 0; y < 23; y++) {
for (int x = 0; x < 37; x++) {
if (map.getLab(x, y).equals("f")) {
g.drawImage(map.getImgFinish(), x * map.getTamImagen(), y * map.getTamImagen(), null);
}
if (map.getLab(x, y).equals("g")) {
g.drawImage(map.getImgGrass(), x * map.getTamImagen(), y * map.getTamImagen(), null);
}
if (map.getLab(x, y).equals("w")) {
g.drawImage(map.getImgWall(), x * map.getTamImagen(), y * map.getTamImagen(), null);
}
}
player.drawPlayer(g, player.getCoorX() * map.getTamImagen(), player.getCoorY() * map.getTamImagen());
}
}
public void run() {
while (true) {
repaint();
}
}
public class Al extends KeyAdapter {
public void keyPressed(KeyEvent e) {
int keyCode = e.getKeyCode();
if (keyCode == KeyEvent.VK_LEFT) {
if (!map.getLab(player.getCoorX() - 1, player.getCoorY()).equals("w")) {
player.move(-1, 0);
}
}
if (keyCode == KeyEvent.VK_RIGHT) {
if (!map.getLab(player.getCoorX() + 1, player.getCoorY()).equals("w")) {
player.move(1, 0);
}}}}}
A: That's a very tight while loop, one that risks hogging the CPU completely rendering your GUI useless. Instead of looping as you're doing, you're far better off responding to events when they occur. For instance, if you want to change views when a key is pressed, then call repaint() after receiving notification that the key has been pressed, not within a tight while loop. You could use a KeyListener for this as you're doing, but understand that these are very fidgity with respect to component focus, and for this (and other reasons) you're better off using Key Bindings.
If you need to run a game loop for other reasons, don't make one so tight -- without Thread.sleep or any break within it. For the interests of Swing thread safety, you're best off not using that while loop, but rather using a Swing Timer instead.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,684
|
#include "utils.h"
#include <stdlib.h>
#include <string.h>
namespace MinVR {
class Cxml
{
public:
Cxml();
~Cxml();
// parse a string containg xml code
bool parse_string(std::string xml_string);
private:
int m_cursor;
int m_length;
element* m_root_node;
bool get_node(std::string xml_string);
public:
element* get_root_element();
};
} // ending namespace MinVR
#endif // CXML_H
// TODO list -- Needed improvements to the Cxml parser.
//
// - Limited entity support. At least the basics: &, <, >, etc.
//
// - Make a "printXML() method like printStructure(), for debugging.
//
// - Support comments (again). That is, restore the comment-processing
// calls, and persuade the data index code to ignore them.
//
// - Real entity support, where they can be defined.
//
// - CDATA support (maybe have this already?)
//
// - Add hook for pre-processing with xsltproc, and validation to a schema.
//
// - Tidying and better documentation.
//
// - Better errors.
//
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 621
|
At Hearty Nutrition we provide professional support and effective nutrition advice to help you achieve your goals. Our approach is to understand your needs and expectations, and to support you. We recognise that the topic of diet can be a sensitive topic for many. Therefore we believe that it is fundamental that we provide a nurturing and understanding environment to assist you.
Joel Feren is the director and principal dietitian at Hearty Nutrition. He is an Accredited Practising Dietitian and Accredited Nutritionist with a background in the biomedical sciences. Joel undertook his Masters in Dietetics following his undergraduate degree in Behavioural Neuroscience and Honours in Exercise Physiology.
Joel is currently working in aged care where he manages the dietetic services of a number of residential care facilities across Melbourne. He has consulted to the Box Hill Hawks (Hawthorn Football Club affiliate) as well as various blue-collar businesses. Joel is a professional member of the Dietitians Association of Australia and Sports Dietitians of Australia. He has a passion for the role of diet in the prevention and management of chronic disease.
Joel is a self-described foodie who loves to cook. On the weekends he is often found perusing the food stalls at South Melbourne market and trialling new recipes in the kitchen. He's a keen a blogger and member of the Nutrition Blog Network and Storehouse Blog Directory. Joel is a media spokesperson for the Dietitians Association of Australia. And, he has also provided comment on nutrition issues to Fairfax Media, News Corp, Huffington Post, Australian Gluten Free Life magazine, Body and Soul and the ABC. For more on Joel's media work and featured articles, please visit The Nutrition Guy.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,334
|
Prehospital and Disaster Medicine (2)
World Association for Disaster and Emergency Medicine (2)
Terrorist Attacks in the Middle East: A Counter-Terrorism Medicine Analysis
Derrick Tin, Saleh Fares, Mobarak Al Mulhim, Gregory R. Ciottone
Journal: Prehospital and Disaster Medicine / Volume 37 / Issue 2 / April 2022
Published online by Cambridge University Press: 03 March 2022, pp. 212-216
Print publication: April 2022
The Middle East and North Africa (MENA) region has been, like many parts of the world, a hotbed for terrorist activities. Terrorist attacks can affect both demand for and provision of health care services and often places a unique burden on first responders, hospitals, and health systems. This study aims to provide an epidemiological description of all terrorism-related attacks in the Middle East sustained from 1970-2019.
Data collection was performed using a retrospective database search through the Global Terrorism Database (GTD). The GTD was searched using the internal database search functions for all events which occurred in Iraq, Yemen, Turkey, Egypt, Syria, West Bank and Gaza Strip, Israel, Lebanon, Iran, Saudi Arabia, Bahrain, Jordan, Kuwait, United Arab Emirates, North Yemen, Qatar, and South Yemen from January 1, 1970 - December 31, 2019. Primary weapon type, primary target type, country where the incident occurred, and number of deaths and injuries were collated and the results analyzed.
A total of 41,837 attacks occurred in the Middle East from 1970-2019 accounting for 24.9% of all terrorist attacks around the world. A total of 100,446 deaths were recorded with 187,447 non-fatal injuries. Fifty-six percent of all attacks in the region occurred in Iraq (23,426), 9.4% in Yemen (3,929), and 8.2% in Turkey (3,428). "Private Citizens and Properties" were targeted in 37.6% (15,735) of attacks, 15.4% (6,423) targeted "Police," 9.6% targeted "Businesses" (4,012), and 9.6% targeted "Governments" (4,001). Explosives were used in 68.4% of attacks (28,607), followed by firearms in 20.4% of attacks (8,525).
Despite a decline in terrorist attacks from a peak in 2014, terrorist events remain an important cause of death and injuries around the world, particularly in the Middle East where 24.9% of historic attacks took place. While MENA countries are often clustered together by economic and academic organizations based on geographical, political, and cultural similarities, there are significant differences in terrorist events between countries within the region. This is likely a reflection of the complexities of the intricate interplay between politics, culture, security, and intelligence services unique to each country.
Disaster Diplomacy: Current Controversies and Future Prospects
Eugene S. Yim, David W. Callaway, Saleh Fares, Gregory R. Ciottone
Journal: Prehospital and Disaster Medicine / Volume 24 / Issue 4 / August 2009
Print publication: August 2009
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,215
|
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