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Launching Marianna Baer's The Inconceivable Life of Quinn
Quinn Cutler is sixteen and the daughter of a high-profile Brooklyn politician. She's also pregnant, a crisis made infinitely more shocking by the fact that she has no memory of ever having sex. Before Quinn can solve this deeply troubling mystery, her story becomes public. Rumors spread, jeopardizing her reputation, her relationship with a boyfriend she adores, and her father's campaign for Congress. Religious fanatics gather at the Cutlers' home, believing Quinn is a virgin, pregnant with the next messiah. Quinn's desperate search for answers uncovers lies and family secrets—strange, possibly supernatural ones. Might she, in fact, be a virgin?
"In a suspenseful and thought-provoking novel, Baer (Frost) tackles the illusiveness of memory (especially in regard to trauma), media firestorms, fear of the unknown, and the complexities of faith, without ever turning didactic or allowing Quinn's story to fall into melodrama... It's a delicate, complicated, and engrossing exploration of the collision between real life and the inexplicable."
-Publishers Weekly (starred review)
Marianna Baer has an MFA from Vermont College of Fine Arts and is the author of the young adult novel Frost. According to Kirkus Reviews, "Baer has a knack for dialogue and creating creepy situations that will intrigue teens." She lives in Brooklyn and works as a freelance editor.
Maggie Lehrman (moderator) is a writer and editor living in Brooklyn, NY. During her decade of working as an editor of books for children, she also earned an MFA in Writing for Children and Young Adults from Vermont College of Fine Arts. | {
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Q: How to get Support Action Bar inside a Master/Detail Fragment to set NAVIGATION_MODE_TABS? Right now I have a typical master/detail pattern, inside detail area I am supposed to display some kind of Tabs structure but with a ViewPager in order to implement the swipe functionality to change between those tabs.
In a Single-Pane-Mode (when using devices) inside detail area, I deploy AreaDetailActivity. Inside of this activity I call the supportActionBar (v7), set the navigation mode, and set the tabs, like this:
public class AreaDetailActivity extends ActionBarActivity implements ActionBar.TabListener {
private ViewPagerAdapter viewPagerAdapter;
private ViewPager viewPager;
@SuppressWarnings("deprecation")
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView( R.layout.view_pager );
String idTipoAreaActual = getIntent().getStringExtra( AreaDetailFragment.ARG_ITEM_ID );
final ActionBar actionBar = getSupportActionBar();
actionBar.setNavigationMode( ActionBar.NAVIGATION_MODE_TABS );
//Adapter that returns the selected fragment
viewPagerAdapter = new ViewPagerAdapter( this, idTipoAreaActual, getSupportFragmentManager() );
viewPager = (ViewPager) findViewById( R.id.view_pager );
viewPager.setAdapter( viewPagerAdapter );
viewPager.setOnPageChangeListener( new ViewPager.SimpleOnPageChangeListener(){
@Override
public void onPageSelected( int position ){
actionBar.setSelectedNavigationItem( position );
}
});
actionBar.addTab( actionBar.newTab().setText("Tab1").setTabListener(this) );
actionBar.addTab( actionBar.newTab().setText("Tab2").setTabListener(this) );
}
@Override
public void onTabReselected(Tab tab, FragmentTransaction fragmentTransaction) {
}
@Override
public void onTabSelected(Tab tab, FragmentTransaction fragmentTransaction) {
viewPager.setCurrentItem( tab.getPosition() );
}
@Override
public void onTabUnselected(Tab arg0, FragmentTransaction arg1){
}
}
The above code works perfectly, but I need a way to implement the same functionality in Two-Pane-Mode (actually Tablet Screen). In Two-Pane the detail area deploys a fragment, so I need to call the support action bar and implements the swipe functionality using the ViewPager, but when I do the cast the ActionBar to AreaDetailActivity (thet extends from ActionBarActivity), it shows a Casting error. This is my AreaDetailFragment displayed in Two-Pane-Mode:
public class AreaDetailFragment extends Fragment implements ActionBar.TabListener {
private Activity activity;
private ViewPagerAdapter viewPagerAdapter;
private ViewPager viewPager;
public AreaDetailFragment( Activity activity ) {
this.activity = activity;
}
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setHasOptionsMenu( true );
}
@Override
public void onCreateOptionsMenu(Menu menu, MenuInflater menuInflater){
super.onCreateOptionsMenu(menu, menuInflater);
menuInflater.inflate(R.menu.ccmaction_bar, menu);
}
@Override
public View onCreateView(LayoutInflater inflater, ViewGroup container, Bundle savedInstanceState) {
View view = inflater.inflate( R.layout.view_pager, container, false );
return view;
}
@Override
public void onViewCreated(View view, Bundle savedInstanceState){
super.onViewCreated(view, savedInstanceState);
String idTipoAreaActual = getArguments().getString( AreaDetailFragment.ARG_ITEM_ID );
final ActionBar actionBar = ( (AreaDetailActivity) getActivity()).getSupportActionBar();
actionBar.setNavigationMode( ActionBar.NAVIGATION_MODE_TABS );
viewPagerAdapter = new ViewPagerAdapter( getActivity().getApplicationContext(), idTipoAreaActual, getChildFragmentManager() );
viewPager = (ViewPager) view.findViewById( R.id.view_pager );
viewPager.setAdapter( viewPagerAdapter );
viewPager.setOnPageChangeListener( new ViewPager.SimpleOnPageChangeListener(){
@Override
public void onPageSelected( int position ){
actionBar.setSelectedNavigationItem( position );
}
});
}
@Override
public void onTabReselected(Tab tab, FragmentTransaction fragmentTransaction) {
}
@Override
public void onTabSelected(Tab tab, FragmentTransaction fragmentTransaction) {
viewPager.setCurrentItem( tab.getPosition() );
}
@Override
public void onTabUnselected(Tab arg0, FragmentTransaction arg1){
}
}
Any Ideas??? Thanks for your Time!!!
A: Nice tutorial for SWIPE/TAP/PAGES: http://indyvision.net/2014/08/android-tutorial-screen-with-tabs-and-swipe-part-i/
Technical information about ACTION BAR: http://developer.android.com/guide/topics/ui/actionbar.html
Un buen tutorial sobre SWIPE/TAP/PAGES: http://indyvision.net/2014/08/android-tutorial-screen-with-tabs-and-swipe-part-i/
Información técnica sobre ACTION BAR: http://developer.android.com/guide/topics/ui/actionbar.html
| {
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Jennifer and Kieron Deeney were a young, newly married couple, looking forward to starting a family.
Life couldn't have been better, until Kieron was killed in a tragic workplace accident, just thirteen weeks after their wedding day.
Now Jennifer is beginning to devote her time to making people's working lives safer, using Kieron's story as a powerful and sobering reminder of the fragility of our everyday lives and the vital role that workplace safety play in securing all our futures.
To watch a free online preview of the film and find out more, click here. | {
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} | 6,708 |
De Ware Tijd (vaak afgekort tot DWT) is een dagblad in Suriname. De krant werd opgericht in 1957. De Ware Tijd was vroeger een avondblad: het verscheen 's avonds rond 20.00 uur en droeg toen de datum van de daarop volgende dag. Tegenwoordig is het echter een ochtendblad.
De krant stond tot midden jaren negentig onder hoofdredacteurschap van Leo Morpurgo, die werd opgevolgd door Nita Ramcharan, die op haar beurt weer werd opgevolgd door Desi Truideman. In 2007 volgde Ricardo Carrot Desi Truideman op als nieuwe hoofdredacteur. De familie Jong Tjien Fa geeft het ochtendblad uit.
In Nederland verschijnt sinds begin 2005 ook een weekeditie van deze krant, met het Surinaamse nieuws van de afgelopen week, gebaseerd op eerder verschenen berichten in de Surinaamse editie. Daarnaast bracht De Ware Tijd tot medio 2005 een maandblad uit, het opinieblad Paramaribo Post.
De Ware Tijd bevat dagelijks verschillende bijlagen. Zo verschijnt in de zaterdageditie De Ware Tijd Literair, dat sinds 1986 de belangrijkste wekelijkse bron van en over literatuur is voor het land.
De dictatoriale periode
Tijdens de dictatoriale periode was er een actief censuurbeleid voor alle Surinaamse media. Na de Decembermoorden was De Ware Tijd de enige krant die nog mocht verschijnen. Deze stond toen actief onder curatele van het militair bewind.
Zie ook
Lijst van Surinaamse nieuwsmedia
Dagbladpers van Suriname
Externe links
Officiële website
dWTblog, de officiële weblog van de Ware Tijd (gearchiveerd)
Surinaamse krant | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,976 |
HomeHealth
US Hopeful for New Treatment Method for Coronavirus
HealthNewsWorld
Photo Credit: Nypost.com
The United States is expected to announce a new treatment method for the novel coronavirus. White House Chief of Staff Mark Meadows said the announcement of a new treatment therapy for Covid-19 will come in a few days.
"We are hopeful we will be able to announce new therapies for coronavirus in the next few days," said Mark Meadows. This new therapy-based treatment will be done through some groundbreaking technologies," he said in an exclusive interview with ABC News.
Despite the message of hope for Covid-19 treatment, Mark Meadows did not disclose what kind of therapy or groundbreaking technology has been used. He just said, the announcement of the new treatment is coming in a few days.
Meanwhile, 173 initiatives are underway around the world in the race to develop the coronavirus vaccine. Of these, three have reached the third stage test, two of which are from China. In such a situation, the United States is going to announce a new medical treatment.
So far, more than 44.33 million people have tested positive for the virus in the United States. Of them, more than 150,000 people have died. Meanwhile, more than 2,137,000 have recovered from the infection, according to Worldometers.
Donald Trump's Adviser Robert O'Brien Tests Positive for Covid-19
A Breakthrough in Covid-19 Drug Makes 3 UK Professors Millionaires | {
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} | 7,884 |
_Iced_ is a work of fiction. Names, characters, places, and incidents are the products of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is entirely coincidental.
Copyright © 2012 by Karen Marie Moning
Excerpt from _Burned_ by Karen Marie Moning copyright © 2012 by Karen Marie Moning, LLC
All rights reserved.
Published in the United States by Delacorte, an imprint of Random House, a division of Random House LLC, a Penguin Random House Company, New York.
DELACORTE PRESS and the HOUSE colophon are registered trademarks of Random House LLC.
Originally published in hardcover in the United States by Delacorte Press, an imprint of Random House, a division of Random House LLC, in 2012.
Library of Congress Cataloging-in-Publication Data
Moning, Karen Marie.
Iced : a novel / Karen Marie Moning.
p. cm.
eBook ISBN: 9780440339809 Hardcover ISBN: 9780385344401 Mass Market ISBN: 9780440246411
1. Teenage girls—Ireland—Dublin—Fiction. 2. Fairies—Fiction. I. Title.
PS3613.O527I34 2012
813'.6—dc23 2012025239
www.bantamdell.com
Cover design: Eileen Carey
Cover illustration: © Les and Dave Jacobs/cultura/Corbis (man)
v3.1_r5
# Contents
_Cover_
_Title Page_
_Copyright_
Part I
Prologue
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-one
Chapter Twenty-two
Chapter Twenty-three
Part II
Chapter Twenty-four
Chapter Twenty-five
Chapter Twenty-six
Chapter Twenty-seven
Chapter Twenty-eight
Chapter Twenty-nine
Chapter Thirty
Chapter Thirty-one
Chapter Thirty-two
Chapter Thirty-three
Chapter Thirty-four
Chapter Thirty-five
Chapter Thirty-six
Part III
Chapter Thirty-seven
Chapter Thirty-eight
Chapter Thirty-nine
Chapter Forty
Chapter Forty-one
Chapter Forty-two
Chapter Forty-three
Chapter Forty-four
_Dedication_
_Excerpt from_ Burned
_Other Books by This Author_
_About the Author_
#
# PROLOGUE
# _Dublin, you had me at "Hello"_
Imagine a world that doesn't know its own rules. No cell phones. No Internet. No stock market. No money. No legal system. A third of the world's population wiped out in a single night and the count rising by millions every day. The human race is an endangered species.
A long time ago the Fae destroyed their world and decided to take ours. History says they moved in on us between 10,000 and 6,000 B.C., but historians get a lot wrong. Jericho Barrons says they've been here since the dawn of time. He should know, because I'm pretty sure he has, too.
For a long time there was a wall between our worlds. With the exception of a few cracks, it was a solid barricade, especially the prison that held the Unseelie.
That barricade is gone now and the prison walls are dust.
All of the Fae are free: the deadly Dark Court and the imperious Light Court, who are every bit as deadly, just prettier. A Fae is a Fae. Never trust one. We're being hunted by voracious monsters that are nearly impossible to kill. Their favorite food? People.
As if that's not bad enough, there are fragments of Faery reality drifting around that swallow up anything in their path. They're tricky to spot; you can drive right into one, if you're not careful. The night the walls fell, Faery itself was fractured. Some say even the inimical Hall of All Days was changed, and opened new portals onto our world. The drifting is the part that really gets me. You can go to sleep in your own bed and wake up in a completely different reality. If you're lucky, the climate won't kill you instantly and the inhabitants won't eat you. If you're really really lucky, you'll find your way home. Eventually. If you're superlucky, time will pass at a normal rate while you're gone. Nobody's that lucky. Folks vanish all the time. They just disappear and are never seen again.
Then there are the amorphous Shades that lurk in the dark and consume every living thing in their path, right down to the nutrients in the soil. When they're done, all that's left is dirt that an earthworm couldn't live in—not that they leave those either. It's a minefield outside that door. Walk lightly. Your parents' rules don't apply. _Do_ be afraid of the dark. And if you're thinking there might be a monster under your bed or in your closet, there probably is. Get up and check.
Welcome to Planet Earth.
This is our world now—one that doesn't know its own rules. And when you've got a world that doesn't know its own rules, everything dark and nasty that was once held in check comes slithering out of the cracks to try to take a shot at whatever it wants. It's a free-for-all. We're back to being cavemen. Might is right. Possession is nine-tenths of the law. The bigger and badder you are, the better your odds of surviving. Get a gun or learn to run. Fast. Preferably both.
Welcome to Dublin, AWC—After the Wall Crash—where we're all fighting for possession of what's left of the planet.
The Fae have no king, no queen, no one in charge. Two psychotic, immortal Unseelie princes battle for dominion over both races. Humans have no government. Even if we did, I doubt we'd listen to them. It's complete chaos.
I'm Dani "Mega" O'Malley.
I'm fourteen.
The year was just officially declared 1 AWC, and the streets of Dublin are my home. It's a war zone out there. No two days are alike.
And there's no place else I'd rather be.
# ONE
# _"Ding-dong! The witch is dead": subtitled Rowena who?_
"I say we take Mac's suggestion and pump the room full of concrete," Val says.
I wince. Just hearing her name makes my stomach hurt. Me and Mac used to be two peas in the Mega pod, close as sisters. She'd kill me in a heartbeat now.
Well, she'd try.
I'm faster.
"Exactly how do you expect us to get concrete trucks down into the catacombs beneath the abbey?" Kat demands. "To say nothing of how much it would take to seal that chamber. It's three times the size of Inspector Jayne's training green, with a ceiling as high as any cathedral!"
I shift position, tucking my knees up, careful to be real quiet. My legs are cramped from sitting with them crossed beneath me. I'm in the cafeteria at the abbey, high up on a beam in the ceiling rafters where nobody can see me, munching a Snickers bar and eavesdropping. It's one of my favorite perches for scoping out the details. I'm a good climber, fast and agile. Since I'm still just a kid in most people's opinions, folks rarely let me in on the scoop. No worries there. I became a pro at letting myself in years ago.
"What are you suggesting we do, then, Kat?" Margery says. "Leave the most powerful Unseelie prince ever created frozen in a little ice cube beneath our home? That's crazy!" The cafeteria is full of _sidhe-_ seers. Most of them murmur agreement but they're like that. Whoever's talking loudest at the moment is the person they agree with. Sheep. Half the time I'm spying, it's all I can do not to jump down there, waggle my ass and say _Baaaa_ , see if any of them catch my drift.
I've been at the abbey most of the night, waiting for people to wake up and wander in for breakfast, impatient for those who've been up all night like me to tell everyone else the news and start discussing it. I don't need as much sleep as other people, but when I do finally crash, I'm as good as dead. It's dangerous to lose consciousness as hard as I do, so I'm always careful about where I sleep—behind a lot of locked doors, with booby traps in place. I know how to take care of myself. I've been on my own since I was eight.
"It's hardly an ice cube," Kat says. "The Unseelie king himself imprisoned Cruce. You saw the bars shoot up from the floor around him."
I've got no family. When my mom was killed, Ro made me move into the abbey with the other _sidhe_ -seers—those of us who can see the Fae, and could even before the walls fell. Some of us have unique gifts, too. We used to think of ourselves in terms of us and them, humans and Fae, until we learned that the Unseelie King tampered with us way back, mixing his blood with the bloodlines of six ancient Irish houses. Some say we're tainted, that we have the enemy within. I say anything that makes you stronger, duh, makes you stronger.
"The alarm's not set," Margery counters. "And none of us can figure out how to arm the grid that keeps people from getting in. Worse, we can't even get the door closed. Mac tried for hours."
I don't puke the bite of chocolate and peanuts I'm trying to swallow but it's close. I got to get over my reaction to her name. Every time I hear it, I see the look on her face when she learned the truth about me.
Feck that! I knew what would happen if she found out I killed her sister. Got no business being mopey about it. If you know what's coming and don't do anything to stop it, you got no right to act all surprised and pissy when the crap hits the fan. Rule #1 in the Universe: the crap always hits the fan. It's the nature of crap. It's a fan magnet.
"She said it won't respond to her," Margery says. "She thinks the king did something to it. Barrons and his men tried to muscle it closed, but no luck. It's stuck open."
"Just anyone can wander in," says Colleen. "We found the Meehan twins standing down there this morning, hands around the bars, staring up at him like he was some kind of angel!"
"And what were _you_ doing down there this morning?" Kat says to Colleen. Colleen looks away.
Tainted blood or not, I've got no complaints about being a _sidhe_ -seer. I got the best gifts of all. None of the other _sidhe_ -seers know how to deal with me. I'm superfast, superstrong, have superhearing, supersmell, and wicked sharp eyesight. I don't know if I taste better or not. Since I can't taste with anyone else's tongue, I guess I'll never know. The superfast part is the best. I can whiz through a room without people even seeing me. If they feel the breeze of me passing, they usually blame it on an open window. I open windows everywhere I go. It's my camo. If you walk into a room with a lot of open windows, look sharp at breezes that seem contrary to what's coming in from the outside.
"That's because he _looks_ like an angel," Tara says.
"Tara Lynn, don't you go there for even a second," Kat says sharply. "Cruce would have destroyed us all if he'd thought he had something to gain by it, and that was _before_ he read the Book and absorbed its power. Now, he _is_ the _Sinsar Dubh_ —the darkest, most twisted magic of the Fae race. Have you forgotten what it did to Barb? Don't you remember how many people the Book massacred when it _didn't_ have a body? Now it has one. And it's beneath our abbey. And you think it looks like an angel? That it's pretty? Have you lost your mind?"
I wasn't beneath the catacombs last night so I didn't get to see what happened with my own eyes. I'd been keeping a distance from that person whose name I'm not saying. I heard what happened, though. It's all anyone's talking about.
Dude, V'lane is Cruce!
He isn't even Seelie. He's the worst of all the Unseelie princes.
I can hardly believe it. I had the wickedest crush on him! I thought he was the one who was going to save us all, fighting the good fight, on the human side of the war. Turns out he _was_ war—literally, as in the Four Horsemen of the Apocalypse's War, riding alongside his three Unseelie prince brothers: Death, Pestilence, and Famine. Sure enough our myths were right. When they rode our world again everything went straight to hell. Nobody even knew he was alive. Cruce was supposed to have been killed three-quarters of a million years ago. Instead he was masquerading as V'lane all that time, disguising himself with glamour, infiltrating the Seelie court, manipulating events, orchestrating the prime opportunity to take what he wanted—dominion over both races.
Fae have patience like beaches have sand. 'Course, I guess patient is easy to be when you live, like, for-fecking-ever.
I also heard he was one of the four who raped M—that person whose name I'm not thinking—that day at the church when the Lord Master turned the princes loose on her.
And I'd told him I was going to give him my virginity one day! He'd brought me chocolates, been all flirty-flirty!
V'lane is Cruce. _Dude_. Sometimes that's all you can say.
Tara holds Kat's glare defiantly. "That doesn't mean I want to set him free. I'm just saying he's beautiful. Nobody can argue with that. He has wings like an angel."
He _is_ beautiful. And we have big, big problems. I went down to the catacombs last night, the instant everyone finally cleared out. I made my way through the underground maze until I found the chamber that once held the _Sinsar Dubh_. And still holds it—just in another skin.
V'lane doesn't look like V'lane anymore. He's sealed in the center of a block of ice, surrounded by a cage of glowing bars. His head is back, his eyes are iridescent fire, he's roaring, and his enormous black-velvet wings are spread wide. Brilliant tattoos snake beneath skin that shimmers like gold dust. And he's naked. If I hadn't seen other penises in movies, I'd be worried about losing my virginity.
"Black wings, Tara," Kat says. "As in black magic, as in 'deadly.' He was dangerous before. He's a thousand times worse now. The King never should have let him read the whole Book. He should have stopped him."
"Mac said the King didn't want to leave the _Sinsar Dubh_ split up," says Colleen. "He was worried we wouldn't be able to keep it locked down in two places."
I dig around in a pocket of the backpack I always got over a shoulder—you never know what you might need when, and I'm always on the go—and pull out another Snickers bar. There's that fecking name again. Eating soothes the bruise I'm getting from repeated sucker-punches to my belly.
"We couldn't keep it locked down when it was in only one place," Kat says.
"Because Rowena _let_ it out," Val says.
I learned that part of the story earlier this morning, listening to _sidhe_ -seers talking in the showers. When the _Sinsar Dubh_ took possession of Rowena last night, that person I'm not naming killed her. But not before Ro bragged about how she set the _Sinsar Dubh_ free. And still, some folks are talking about having a service for the old bat! I say the Grand Mistress of the _sidhe-_ sheep is dead. Hoo-fecking-rah! Break out the cake and party hats!
"It weakened Rowena," Kat says.
Rowena was _born_ weak. Power-hungry witch.
"Maybe Cruce will weaken us," Kat says.
I plaster a sigh around a bite of candy bar and swallow it. The new temporary leader of the abbey and interim Grand Mistress of _sidhe_ -seers around the world just made a big mistake. I learned a thing or two from that unnamed person when we used to hang together. _Sidhe_ -sheep need a firm hand. Not firm like Ro's, which was bullying, belittling, and tyrannical, but firm in a way that doesn't make the herd stampede. Fear and doubt are major stampeders. Kat should have said something like what a good thing it was they were all so much stronger than Rowena. Even a kid can see what's going on in the room down there. The _sidhe_ -seers are afraid. Rowena is dead. Dublin is a riot-ravaged mess filled with monsters. One of the good guys turned out to be the bad guy. Their lives changed too quickly in too many ways for them to deal with. They're easy targets to be swayed by the most persuasive, strongest leader, and that means Kat needs to become one, fast.
Before somebody a lot less capable and kind does.
Somebody like Margery, who's even now watching the crowd through narrowed eyes, like she's got a thermometer up its butt, taking its temperature. She's a year older than Kat, and was part of Ro's inner circle when the old witch was alive. She's not going to put up with a changing of the guard that doesn't include her. She'll make trouble every chance she gets. I hope Kat knows how treacherous she can be. Anyone that was ever close to Ro for longer than like—one second—has something seriously scary about her. I know. I was closest to her of all. _Sidhe_ -sheep politics. Dude, I hate them. They tangle you up like sticky spiderwebs. I love living on my own!
Still, I miss the abbey every now and then. Especially when I think about them baking cookies and stuff. Hearing voices in the background when you doze is nice. Knowing even if you are misunderstood, you aren't totally alone in the world isn't the worst thing.
Kat's right: the _Sinsar Dubh_ we used to have locked up and magicked down beneath our abbey is nothing compared to what we've got under our floorboards now.
The problem is it doesn't look like the _Sinsar Dubh_ anymore.
All of the darkest magic and power of the Fae race is no longer trapped between the covers of a book. It's in the body of a Fae prince in all his naked, winged glory. And if you've never seen a Fae prince before, that's one jaw-dropping, eye-popping, mind-scrambling amount of glory.
It's only a matter of time before somebody sets him free.
Kat hasn't even made her way around to the killer-critical fact yet: lots of people know he's down there now, crammed to the gills with every last bit of the deadly magic of the Fae race.
I know people. I've seen all the shapes and sizes they come in. Somebody's going to be stupid enough to believe they can control him. Somebody's going to find a way through that ice.
Jericho Barrons is only one of a lot of different folks that hunted the _Sinsar Dubh_ for thousands of years. None of them ever knew where it was. If they had, they'd have descended on our abbey back in the dark ages when a rough-piled, round stone tower was all that concealed the entrance to our underground city. And they would have pulled it, stone from stone, into rubble, until they got what they came for.
Now a whole bunch of humans and Fae know exactly where the most powerful weapon ever created is being stored.
Folks talk.
Soon the whole world is going to know it's here.
I snort, imagining hordes descending on us, rioting, raging, brandishing weapons. Stupid _sidhe_ -sheep too busy squabbling about the best way to fight back, to get around to fighting back. I sigh.
Kat glances up.
I stop breathing, hug my knees tight to my chest and stay perfectly still.
After a moment Kat shakes her head and goes back to the conversation.
I sigh again but softer.
She just made her second mistake.
Confronted by something she couldn't explain, she pretended it wasn't there. Dude, ostrich much?
Oh, yeah. Just a matter of time.
I wait a few minutes for things to get heated again, take advantage of the commotion and freeze-frame out.
I love moving the way I do.
I can't imagine life any other way.
Whenever something is bugging me, all I need to do is zoom around the city, spy on all the slo-mo Joes trudging through, and I instantly feel a million times better.
I've got the coolest gig in the world.
I'm a superhero.
Until recently, I was the only one I knew of.
According to my mom, I didn't make the normal toddler transition from crawling to walking. I went from lolling on my back, counting pudgy toes and cooing happily while she changed my diapers (I've never seen any reason to cry when someone is cleaning poop off you), to what she initially thought was teleporting. One second I was on the living room floor, the next I'd vanished. She was afraid the Fae had taken me—they used to do that to _sidhe_ -seers if they discovered them—until she heard me rummaging around in the pantry trying to get a jar of baby food open. It was creamed corn. I remember. I still love creamed corn. Not much fuel-power there though. I burn through the punch of sugar-energy in no time.
I never got to go to school.
You don't want to know how she kept me from leaving the house. There aren't many options with a kid who can move faster than you can blink. And none of them are PC.
I'm not the only superhero in Dublin anymore, which annoys the feckity-feck out of me, but I'm slowly coming around to seeing it might be a good thing.
I was getting complacent. And that turns into sloppy if you're not careful. Bored, too. It's not much fun always being the best and fastest. A little competition keeps you on your toes, makes you try harder, live larger.
I'm all about that: living large.
I want to go out in a blaze of glory while I'm young. I don't want to break piece by piece, lose my mind and die wrinkly and old. Given the current state of our world, I'm not sure any of us have to worry about that anymore.
Top on my list of dudes to beat are Jericho Barrons and his men. Like me, they're superfast and superstrong. Much as I hate to admit it—they're faster. But I'm working on it.
Barrons can pluck me right out of thin air (dude, why isn't it thick air? The things people say!) while I'm freeze-framing, which is what I call the way I get around. I start at point A, lock down a mental snapshot of everything around me, hit the gas, and in a blink I'm at point B. It's only got a couple of downsides. One, I'm constantly bruised from running into things at top speed because some of the things I lock down on my mental grid aren't stationary, like people and animals and Fae. Two, freeze-framing requires a ton of food for fuel. I have to eat constantly. It's a pain in the butt collecting and carrying that much food. If I don't eat enough, I get limp and wobbly. It's pathetic. I'm a gas tank that's either full or empty. There's no half tank with me. You know those movies where folks wear rounds of ammo on their body? I wear protein bars and Snickers.
At least once a night I whiz over to Chester's, Dublin's underground hot spot for partying and scoring whatever your fantasy is and angling for a shot at immortality, owned and operated by Barrons's go-to dude Ryodan, and I start killing every Fae hanging around outside it. It usually takes all of five seconds for his men to show up, but I can do a lot in five seconds.
Chester's is a safe-zone. Killing Fae is prohibited there, no matter what they do. And they do some sick stuff.
Killing humans, however, isn't prohibited at Chester's. That's a major issue with me, so I keep giving Ryodan grief and I'm not about to stop.
One of these nights I'm going to be faster than him, faster than all of them.
Then I'm going to slay every Fae in Chester's.
Second on my list of competition are the Fae I hunt. Some of them _can_ teleport. They call it "sifting." I don't understand the physics of it. I just know it's faster than freeze-framing. Which would worry me more if I didn't have the Sword of Light, one of two weapons that can exterminate their immortal asses, so they leave me alone for the most part. She-who-isn't-getting-named has the other weapon, the Spear.
My stomach hurts again. As I peel open a protein bar, I decide to start thinking of her as "That Person," abbreviated to TP. Then maybe my mind will slide over thoughts of "TP" without hitching and kicking me in the stomach.
Last are the Unseelie princes. There used to be four. Cruce is out of the picture for now. Two are at large, in Dublin, no longer under the Lord Master's rule, which makes them way more dangerous than they used to be. They've begun fighting with each other and are striking out on their own. There's major trouble coming from those two. Not only can they sift, just looking at them makes you weep blood. And if you have sex with them... well don't! Enough said. Already cults are forming around them. Sheep are always looking for a new shepherd when the terrain gets rocky.
I don't test myself against the princes. I keep my distance. I sleep with my sword in my hand. I shower with it. I never let anyone else touch it. I love my sword. It's my best friend.
I killed the other Unseelie prince. I'm the only person who ever has. Dani Mega O'Malley slayed an Unseelie prince! Gotta love it. Only problem is, now the two that are left have a wicked hate-on for me. I'm hoping they'll be too busy fighting with each other to come after me.
My life consists mainly of watching my city. Keeping tabs on all that's changing. I love knowing the details, spreading the important news around. I don't know what Dublin would do without me.
I run a newspaper called _The Dani Daily_ that I put out three times a week. Sometimes I'll do a special edition if something big comes up. I collect messages at what's left of the General Post Office, from folks who are having problems with tough-to-kill Fae. I like to swoop in and save the day! I take my beat seriously, like Inspector Jayne and the Guardians who patrol the streets at night. Dublin needs me. I'm not about to let her down.
I just published my first book, _Dani Does Dublin: the ABCs of the AWC_. Dancer helps me print and distribute it. The reviews have been great. Only problem is, whenever I learn new stuff, which is like constantly, I have to put out a revised edition. I'm on the fifth already.
Some of the folks I help are real basket cases, afraid of their own shadow. I can tell just by looking at them they won't survive long. It makes me sad but I do all I can.
I decide to pop over to the General Post Office now, see if anybody left notes for me.
I polish off my protein bar in two gulps and pocket the wrapper. Don't know why I can't bring myself to litter, considering the streets are covered with debris from the riot the night Dublin fell, but adding to it feels wrong.
I narrow my eyes, look down the street far as I can see, plot each obstacle on my mental grid until it all snaps into place: abandoned cars with open doors just waiting to slam me if I'm off by an inch, streetlamps ripped from the pavement with chunks of concrete attached at the base and strips of metal sticking out that are going to kill my shins if I'm not careful, tables flung through pub windows blocking the sidewalks. You get the idea.
I take a deep breath and give in, set that _sidhe_ -seer place in my head free and slide into a different way of being. Ro used to try to get me to explain it to her, like maybe she could figure out how to do it if she tried hard enough. The best I can come up with is this: it's like picking your whole self up mentally and shoving it sideways, till suddenly you're... just something else. I shift Danigears, I guess. The rush is megaintense and, well... I can't imagine life without it because there's no such thing as life without it.
I do it now, shift hard and fast, and then I'm whole and free and perfect. Wind in my hair! Freeze-framing! Can't even feel my feet, because I got wings on them! I scrunch up my face in concentration and push harder, faster, every nanosecond is going to count if I'm going to beat—
I slam into a wall.
Where the feck did _that_ come from?
How could I have missed it on my grid?
My whole face is numb and I can't see. The impact snaps me out of freeze-framing and sends me into a blind stumble. When I finally get my balance, I'm still not able to focus. I hit the wall so hard it temporarily blinded me. My face is going to be black and blue for days, eyes swollen to slits. How embarrassing! I hate walking around with all my mistakes on my face, right there for anybody to see!
I waste precious seconds trying to recover and all I can think is: good thing it was a wall, not an enemy. I'm a sitting duck right now and it's my own fault. I know better than to lead with my head when I'm freeze-framing. You can kill yourself that way. The body can take a much harder impact than the face. You'll drive your nose up into your brain, if you're not careful.
"Sloppy, Mega," I mutter. I still can't see. I wipe my bloody nose on my sleeve and reach out to feel what I hit.
"That's my dick," Ryodan says.
I snatch my hand away. "Gah!" I choke out. I can feel my face again—because, like, it's going up in flames. What kind of universe makes me reach out at exactly that fecking level to feel what I think is a wall and puts my hand on a penis?
Then I remember this is Ryodan and scowl. "You did that on purpose!" I accuse. "You saw my hand go out and you stepped right into it!"
"I'd do that why, kid?"
Ryodan has the most infuriating way of asking questions without the proper inflection at the end. His voice doesn't rise at all. I don't know why it annoys me so much. It just does. "To embarrass me and make me feel stupid! Always angling for the advantage, aren't you?" Ryodan makes me totally crazy. I can't stand him!
"Sloppy is an understatement," Ryodan says. "I could have killed you. Pull your head out, kid. Watch where you're going."
My vision is finally starting to clear. "I. Was. Watching," I say pissily. "You stepped into my way."
I look up at him. Dude is tall. The only streetlamp that works is smack behind his head, casting his face in shadow, but that's the way he likes it. I swear he stages every place he goes in order to keep the light at his back for some reason. He's wearing that faint half smile he usually has on, as if he's perpetually amused by us lesser mortals.
"I am _not_ a lesser mortal," I say testily.
"Didn't say you were. In fact, it's precisely because you're not lesser that you're on my radar."
"Well, get me off it."
"Can't."
I get a sinking feeling. Not too long ago Ryodan tracked me down where I was hanging out up on top of my favorite water tower and told me he had a job for me. I refused, of course. Since then I've been telling myself he filled whatever vacancy he had with someone else.
I don't want to fall in with Ryodan and his men. I get the feeling you don't ever get to fall back out. You just keep falling.
Of course, that doesn't stop me from snooping around Chester's. You have to know your competition, know what they're up to. Dude wants something from me, I want to know what. Last week I found a back way into his club that I bet nobody but me and his men know about. I think they thought it was so well hidden they didn't need to bother protecting it. Did I ever see some things! My face gets hot again, remembering.
"I've been waiting for you to report for work, Dani. You must have encountered a problem I don't know about."
Report for work, my ass. I don't answer to anyone. The way he says that last part makes it sound like he's been keeping major tabs on me and knows every problem I have and don't have. "I'll say this one more time. Never going to happen."
"You don't understand. I'm not giving you a choice."
"You don't understand. I'm taking it. You're not the boss of me."
"You better hope I am, kid, because you're a risk in my city. And there are only two ways I deal with uncontrolled variables. One of them is to offer you a job."
The look he gives me makes it clear I don't want to know what the second option is. I wipe more blood from my nose and puff myself up. "Thought it was Barrons's city," I say.
He ignores my jibe. "A risk I won't take. You're too fast, too strong, and too stupid."
"There's nothing stupid about me. I _am_ fast and strong, though." I preen. "Best of the best. Dani Mega O'Malley. That's what they call me. The Mega. Nobody's got nothing on me."
"Sure they do. Wisdom. Common sense. The ability to differentiate between a battle worth fighting and the posturing of adolescent hormones."
Gah! I don't posture! I don't have to! I'm the real thing, one hundred percent superhero! Ryodan knows just how to get under my skin but I'm not giving him the satisfaction of showing it. "Hormones don't interfere with my thought processes," I say coolly. "And as fecking _if_ my 'adolescent hormones' are any different than yours. Talk about the pot calling the kettle black." After my clandestine visit last week, I know a thing or two about Ryodan.
"You're human. Hormones will undermine you at every turn. And you're way too young to know shit about me."
"I'm not too young to know anything. I know you and the other dudes are all sex all the time. I saw those women you keep—" I clamp my mouth shut.
"You saw."
"Nothing. Didn't see nothing." I don't slip often. At least I didn't used to. But things are weird lately. My mood changes like a chameleon in a kaleidoscope. I get touchy and end up saying things I shouldn't. Especially when someone keeps calling me "kid" and ordering me around. I'm unpredictable, even to myself. It bites.
"You've been on level four." His eyes are scary. Then again, this is Ryodan. His eyes are scary a lot.
"What's level four?" I say innocently, but he's not buying it for a minute. Level four is like something out of a porn movie. I know. I was watching a lot of them until recently, until somebody who doesn't give one little tiny ounce of crap about me read me the riot act, like TP cared. It's stupid to think just because somebody yells at you like they worry about how you're growing up and who you're becoming that they care about you.
He smiles. I hate it when he smiles. "Kid, you're flirting with death."
"You'll have to catch me first."
We both know it's empty bravado. He can.
He locks gazes with me. I refuse to look away even though it feels like he's sifting through my retinal records, reviewing everything I've seen. Long seconds pass. I notch up my chin, shove a hand in my jeans pocket and cock my hip. Jaunty, flippant, bored, my body says. 'Case he's not getting the message from the look on my face.
"I felt a breeze in the private part of my club last week," he says finally. "Somebody passing by fast. I thought it had to be Fade not wanting to be seen for some reason, but it wasn't. It was you. Not cool, Dani. Way not cool. Am I speaking your language well enough to penetrate that rock-hard, suicidal, adolescent head of yours."
I roll my eyes. "Gah, old dude, please don't try to talk like me. My ears'll fall off!" I flash him a cocky, hundred-megawatt grin. "It's not my fault you can't focus on me when I pass. And what's with all this adolescent bunk? I know how old I am. _You_ the one needs reminding? Is that why you keep throwing it at me like some kind of insult? It isn't, you know. Fourteen is on top of the world."
The next thing I know he's in my space, swallowing it up. Barely leaving me room to be. I'm not about to stick around for it.
I freeze-frame around him.
Or I try to.
I crash, full frontal into him, smacking my forehead on his chin. Not hard either. Freeze-framing into him should have split my head again, not tickled like a stumble.
I slam it into Mega-reverse.
I succeed in backpedaling a pansy foot or two. I don't even make it out of arm's reach.
What the feck?
I'm so discombobulated by failure that I just stand there like an idiot. Until this precise moment, I wasn't even sure I knew how to _spell_ the F-word, much less do it. Fail, with a big fat F. Me.
He grabs my shoulders and starts pulling me to him. I don't know what he thinks he's doing but I'm not getting anywhere _near_ close to Ryodan. I explode into a Dani-grenade, all fists and teeth, and ten kinds of you-don't-want-to-hold-me-when-the-pin-is-out.
At least I try to.
I noodle off one limp punch before I stop myself so I won't telegraph any more catastrophic news to a dude that doesn't miss a trick and won't hesitate to use any weakness against me.
What the feck is wrong with me?
Did slamming into him do something to me? Like break me?
Superspeed—gone.
Superstrength—gone.
I'm as weak as a Joe and... ew! Stuck in Ryodan's arms. Close. Like we're about to slow dance, or get all kissy.
"Dude, you _like_ me or something? Get off me!"
He looks down at me. I can see the mind working behind his eyes. I don't like Ryodan's mind working when he's looking at me.
"Fight, kid."
I tilt my nose up at a defiant angle, jut my jaw at my best "feck you" slant. "Maybe I don't feel like it. You said there's no point. You keep telling me how large and in-charge you are."
"Never stopped you before."
"Maybe I don't want to break a nail," I toss out all nonchalant-like, to cover up that I just tried fighting. _And_ fleeing. And for the first time in, well— _ever_ —I'm... norm—
The word sticks like a hard, spiky burr in the back of my throat. I can't cough it up. I can't swallow it.
It's okay. I don't need to be able to say it. It's not true. It never will be.
I've never been that word. It's not part of my reality. I probably just forgot to eat enough. I take a hasty mental tally of my fuel consumption over the past few hours: eleven protein bars, three cans of tuna, five cans black beans, seven Snickers. Okay, so my menu's coming up a little light, but not enough to drain my gas tank. I step on the freeze-frame pedal again.
I still don't move. Motionless is me. That and way freaked out.
He's holding my hand, looking at my short nails that TP painted black the night she found out the truth about me. I don't know why I haven't taken it off yet. It chips like crazy in no time with all the fighting I do.
"You don't have nails to break. Try again."
"Let go of my hand."
"Make me."
Before I can snap off a pithy, brilliant reply, my head is back, my spine is arched like a bow, and Ryodan's face is in my neck.
He bites me.
The fecker _bites_ me!
Right on the neck!
Fangs bracket my jugular. I feel them, sharp and deep, sinking into me. It hurts.
Ryodan _does_ have fangs! I didn't imagine what I thought I saw on the rooftop the other night when he was telling me he had a job for me!
"What the feck you doing? You a vamp or something? You turning me?" I'm horrified. I'm... intrigued. How much stronger might I get? Are vampires real? Fairies are. I suppose that flings the closet door wide open. Everything's going to be springing out now. Does TP know about this? Is Barrons a vampire? What's going on here? Dude, my world just got so much more interesting!
Suddenly I'm staggering for footing, resisting nothing and looking like a drunken pinwheel doing it. It pisses me off, Ryodan making me look clumsy in front of him. I wipe a smear of blood from my neck and glare at it. When was the last time somebody spilled my blood? Like never. Sure, I bang myself up. But nobody else does. Not anymore.
Bleeding? Clumsy? Slow? Who _am_ I?
"I know your taste now, kid. I know your scent like I know my own. You will never be able to pass me again without me knowing it's you. And if I ever catch you on the lower levels of Chester's... or anywhere in my club for that matter..."
I jerk my glare from my hand to his face.
He smiles at me. There's blood on his teeth.
Fact: it's just wrong to be smiled at by someone who has your blood on his teeth. It offends to the bone. Where were his fangs? Did he _have_ fangs? Natural or cosmetic implants? You never know with folks these days. They didn't retract with a smoothly audible _snick_ like on TV or I would have heard it. I have superhearing. Well, sometimes I do. Like when I also have superspeed and superstrength. Which used to be all the time. Until exactly now.
"... don't let me..."
His gaze does that unnerving flickery thing it does sometimes. I think it's because he looks me up and down so quick that I can't focus on his eyes changing directions, I just see a kind of ocular shiver. I wonder if I can do it, too, superspeed a single part of me, like maybe tap a finger hyperfast. I need to practice. Assuming I can superspeed again at _all_. What the feck is wrong with me? Did I stall? How could I stall? I don't stall!
"... unless you're working for me and there at my direction. That's the deal." He's cold. Ice cold. And I know without him even saying what the second option is: die. Work for me or die. It pisses me off big-time.
"Are you giving me an ultimatum? Because that is so not cool." I don't emote disdain. I _become_ disdain. I flash him number seventeen of my thirty-five Looks of Death. Grown-ups! They see a teenager with a little more stuff going on than they know what to do with, so they try to lock them down, box them up, make them feel bad just for being what they are. Like I can even help it. Dancer's right, adults are afraid of the kids they're raising.
"If growing up means turning out like you," I say, "I'm never doing it. I know who I am and I like it. I'm not changing for anybody."
"One day, kid, you'll be willing to mortgage your fucking soul for somebody."
"I don't think you should say 'fucking' around me. In case you forgot, I'm only fourteen. And news flash, dude, I've got no soul. There aren't any banks. And there isn't any currency. Ergo. Never. Going. To. Happen."
"I'm not sure you could be any more full of yourself."
I cut him a smug look. "I'm willing to try."
Ryodan laughs. The instant he does, I flash back to what I saw on level four the other night. He was laughing then, too. The look on the woman's face and the noise she was making when he did that thing he was doing—Gah! Old dude! Gross! What's wrong with me?
He's looking at me hard.
It makes me want to blink out of existence.
Ryodan looks at people different than anybody else I know. Like he has X-ray vision or something and knows exactly what's happening inside people's skulls.
"No mystery there, kid. If you live long enough, you do know what they're thinking," he says. "Humans are predictable, cut from patterns. Few evolve beyond them."
Huh? He did _not_ just answer my thought. No fecking way.
"I know your secret, Dani."
"Got no secrets."
"Despite all the swaggering you do, you don't want anybody to see you. Not really see you. Invisa-girl. That's who you want to be. I wonder why."
I flip him off with both hands and freeze-frame with everything I've got.
It works this time! Fecking-A, it's good to be me! Wind in my hair! Mega on the move! Leaps tall buildings in a single bound!
Well, maybe that last part's a little exaggeration, but still...
_Zoooooooom!_ I freeze-frame through the streets of Dublin. When I slam into the next wall, it knocks me out cold.
# TWO
# _"Ice ice baby"_
Since I sleep like the dead, I come to hard. It doesn't matter whether I've fallen asleep or been knocked out. I'm always broody at first because I can't shake off slumber as fast as most folks. My dreams get tangled up with the real world and it takes a while for them to melt away, like icicles dripping off gutters in the morning sun.
Not this time.
I come up from unconsciousness like a live wire: flat on my back one second, the next on all fours, then I've got my sword at Ryodan's throat.
He knocks it away. It flies out of my hand and crashes into the wall of his office.
I lunge after it and crash into the wall myself, but who cares? My sword's in my hand again. I spine up to the wall, blade straight out in front of me, never taking my eyes off him, waiting for him to try to take it from me again. It's going through his heart if he does.
"We can do this all day if you like," he says.
"You knocked me out," I say through clenched teeth. I'm spitting mad, my face is throbbing and my teeth hurt. It's a wonder I have any left.
"Correction. I got in your way. You knocked yourself out. I told you to watch where you're going."
"You're faster than me. That means you're supposed to yield right of way."
"Like we're cars. Cute. I don't yield. Ever." He hooks a foot around a chair and kicks it toward me. "Sit."
"Feck you."
"I'm stronger than you, faster than you, and lack the human emotion that drives you. That makes me your worst nightmare. Sit. Or I'll make you sit."
"I can think of a couple worse," I mutter.
"You want to play games. I don't think you'll like mine."
I think it over. I'm worried because of earlier, when I stalled. What if it happens again and he figures it out? I'm double worried because he knocked me out cold, mid freeze-frame. It's obvious I can't escape if he doesn't want to let me go. I'm in Chester's, on his turf, with all his men in the vicinity. Even if Barrons is around, he's not going to help me. I'm pretty sure TP has him hating me now.
I take stock of the room. I've never been in his office before. LED screens serve as cove moldings, lining the entire perimeter of the ceiling, flashing from one zone to the next. From here Ryodan watches everything. I'm in the guts of his club.
"How'd I get here?" There's one possible answer. I'm just trying to buy more time to orient myself. Gingerly I touch my nose, feel the tip. It's alarmingly bulbous and squishy.
"I carried you."
It makes me so mad I almost can't breathe. He knocked me out, picked me up like a sack of potatoes, toted me through the streets of Dublin and hauled me through the middle of all the skeevy folks and fairies that hang at Chester's, probably with everybody staring at me and smirking. I haven't been helpless for a long time.
Fact: he could do it again if he felt like it. Over and over. This dude standing in front of me could chain me down worse than anything my mom or Ro ever did to me.
I decide the wisest thing is to humor him until he lets me leave. Then I'll eat everything I can get my hands on, test myself, make sure I'm working right, hole up somewhere safe and lie low for a while. I'll spend my time in hiding, working on getting faster and stronger, so I never have to put up with a moment like this again. I thought these kinds of days were gone for good.
I sit.
He doesn't look all smug like I would have. He gives me... like a look of approval or something.
"Don't need your approval," I say irritably. "Don't need anybody's."
"Stay that way."
I scowl at him. I don't get Ryodan at all. "Why am I here? Why'd you bring me to Chester's? Get to the point. I got stuff to do. Busy schedule, you know. I'm in demand."
I look around. The office is made of solid glass, walls, ceiling, and floor. Nobody can see in, but you can see out. It's freaky walking on a glass floor. Like the bottom's dropping out of your world with every step you take. Even sitting, you feel a kind of vertigo.
I look down. There are acres of dance floor beneath me. The club has multiple tiers, maybe a hundred subclubs on split levels, each with its own theme. Seelie, Unseelie, and humans hang together and strike who knows what kind of deals. Here in post-wall Dublin, anything you want can be had at Chester's, for a price. For a second I forget he's there, fascinated by watching it all between my high-top sneakers. I could sit here for days, study stuff, get smarter. Itemize every caste of Fae, spread the word around the city, what they are, how they can be defeated, or at least escaped from or restrained until I can get there to kill them with my sword. That's a big part of the reason I've been so determined to get inside Chester's. How can I protect my city if I can't warn everyone about all its dangers? I got a job to do. I need all the intel I can get.
There's a Seelie male on the dance floor, blond and beautiful like V'lane was before he dropped his glamour and revealed himself as an Unseelie. In the next subclub over is a lower caste of dark Fae that I've never seen before, shiny wet and segmented, with— _Ew!_ The many segments are coming apart and scurrying off into a hundred different directions like roaches! I hate roaches. They begin to disappear up people's pants legs. I pick my feet up off the floor and sit cross-legged on the chair.
"You watch everything."
It's not a question so I don't answer. I look at him, fold my arms and wait.
There's that smile again.
I poke out my lower lip defiantly. "What am I? Like a walking joke to you? Why do you always smile when you look at me?"
"You'll figure it out." He moves to his desk, opens a drawer, pulls out a sheet of paper and hands it to me. "Complete and sign this."
I take it and look at it. It's a job application. I give him a look. "Dude. Post-apocalyptic world. Who does job applications anymore?"
"I do."
I squint at it, then him. "What are you paying me?" I angle.
"Dude. Post-apocalyptic world. Who does money anymore."
I snicker. First sign of any sense of humor he's shown. Then I remember where I am and why. I wad it up and throw it at him. It bounces off his chest.
"You're wasting time, kid. The sooner you do what I tell you, the sooner you can get out of here." He goes to his desk, gets another and hands it to me with a pen.
I relax. He plans to let me leave. Maybe even soon.
I skim the application. It has the usual blanks: name, address, date of birth, education, prior job history, places for signature and date. Fanciest application I've ever seen, with the name CHESTER'S worked into an ornate border that frames the page.
Everybody clings to something when the world melts down. I suppose Ryodan likes having his business details all squared up, no matter the chaos at his door. It's not like it'll kill me to fill out the stupid thing, agree to do whatever he wants, then get the feck out of here and go into deep hiding. I sigh. Hiding. Me. I pine for the days when I was the only superhero in town.
"If I fill this out, you'll let me leave?"
He inclines his head.
"But I have to do some kind of job for you?"
He inclines his head again.
"If I do that job, are we through? For good? Just one job, right?" I have to make this convincing or he'll figure out I plan to disappear.
Once more he gives me that imperial nod that's hardly a nod, like he's stooping to acknowledge my puny existence.
I don't ask him what the job is because I have no intention of ever doing it. I'm never going to be anyone's solution to folks' problems again. I crossed lines for Ro. Big lines. Deep lines. She's dead. I'm free. Life starts now. I study him. He's perfect stillness, with the light behind his face as usual, features in shadow.
Cats get still like him. Before they pounce.
Something's going on here, bigger than I'm seeing.
My face hurts. My eyes are puffy and the left one's trying to swell shut. "You got any ice?" I need to buy time to figure out what's going on. Plus, if he leaves for ice I can snoop through his office.
He gives me a look I've seen men do before, especially to women: chin down, looking up from beneath his brows, with a faintly mocking smile. There's something in that look I don't get but the challenge is unmistakable. "Come here," he says. "I'll heal you." He's sitting behind his desk, watching me. Still, so still. It's like he's not even breathing.
I look at him. I don't know what to make of him. Part of me wants to get up, go around that desk and find out what he's talking about. "You could do that? Make my bruises and cuts go away?" I'm always beat up and my muscles are constantly strained from overuse. Sometimes I burn through my shoes and scrape the skin right off my feet. It gets old.
"I can make you feel better than you've ever felt in your life."
"How?"
"There are some secrets, Dani O'Malley, that you learn only by participating."
I consider that. "So. You got any ice?"
He laughs and presses a button on his desk. "Fade. Ice. Now."
"Gotcha, boss."
A few minutes later I'm sitting with an ice pack on half my face, squinting around it to fill out Ryodan's stupid application. I'm almost done and ready to sign when I get the strangest feeling in my hand, the one holding the page.
It's my left hand, my sword hand, the one that turned black a little while ago, the night I stabbed a Hunter through the heart and killed it. Or rather, the night I _thought_ I killed a Hunter. Truth is, I'm not actually sure I did but I'm not about to print a retraction. The public needs to believe in certain things. When I went back to take pictures of it for _The Dani Daily_ to show folks it was gone, completely. Not a trace remained. Not a single drop of black blood anywhere. Barrons says they can't be killed. After the incident I thought I was going to lose my hand. My veins turned black and my whole hand went cold as a block of ice. I had to wear a glove for days. Told the _sidhe_ -sheep I got poison sumac. Rare around these parts but there used to be some. Don't know if the Shades ate it all. Wonder if they did, if they got itchy bellies inside.
Now it's all tingly and weird. I study it, wondering what might go wrong with me next. Maybe stabbing the Hunter did something to me. Maybe that's why I stalled. Maybe there are worse things on the horizon.
That is so not me! Optimism is me. Tomorrow's my day. You never know what grand adventures wait around the next corner!
"Kid, you going to sit there all day daydreaming, or sign the fucking thing."
That's when I see it. I'm so stunned my mouth opens, and hangs there catching flies for a minute.
I almost signed it!
He must have been sitting over there, laughing his butt off inside, congratulating himself.
My head snaps up. "So, what exactly does the spell in the border of this thing do?" I've never seen anything like it. And I've seen a lot of spells. Ro was a pro at them. Some really nasty ones. Now that I'm seeing it, I can't believe I missed it. Cleverly tucked into the ornate black border are shimmering shapes and symbols, slithering, in constant motion. One of them is trying to crawl off the page and onto my lap.
I wad it up and throw it at him. "Nice try. _Not_."
"Ah, well. It was possible you would sign. It was the simplest solution."
He's completely unperturbed. I wonder, does anything shake him up, make him lose his cool, get hot about something, scream and yell? I can't see it. I think Ryodan glides through life in the same coolly amused mood all the time. "What would it have done to me if I'd signed it?" I ask. Curiosity. I have it in spades. Mom swore it was going to be the death of me. Something's got to be. There are worse things.
"Some secrets—"
"Yeah, yeah, blah blah, participating and all that bunk. Got it."
"Good."
"Didn't want to know anyway."
"Yes you did. You can't stand not knowing things."
"So, what now?" We're at an impasse, him and me. I suspect his "application" was really a contract. A binding contract, the kind that knits up your soul and tucks it in someone else's pocket. I heard of them but never believed they were real. If anybody had a way to sew up a soul in a business deal, it would be Ryodan. Jericho Barrons is an animal. Pure lawless beast. Not so Ryodan. Dude's a machine.
"Congratulations, kid," he says. "You passed my first test. You may just get the job yet."
I sigh. "This is going to be a long day, isn't it? You serve lunch around here? And I'm going to need more ice."
A door I didn't even know was there in the glass wall of his office opens, revealing a glass elevator.
Chester's is way bigger than I thought. As we ride the elevator down, I'm riveted by the view.
And a little worried.
That he's letting me see so much means that whether I signed his stupid application or not, he thinks he has me buttoned up.
Ryodan's glass office isn't the only place he can watch things. It's the tip of the iceberg, and, dude, I do mean iceberg, as in megatons of stuff hidden beneath the surface. The central club part of Chester's—the interior half, a dozen levels the public sees—is barely a tenth of it. That main part where everybody hangs out and dances and makes deals with the devil is constructed inside a much larger structure. Ryodan and his dudes live _behind_ the walls of that club in what's beginning to look like a vast underground city, from where I am. All the walls are two-way glass. They can go to any level, by elevator or catwalk, and watch anything that's happening at any time. Serious thought went into designing this place. There's no way they built it all since the walls fell last Halloween. I wonder how long it's all been here, beneath the polished, glitzy, glamorous Chester's that used to exist, hot spot for movie stars and models and the überrich. I wonder if, like our abbey, their underground world has been beneath a changing exterior for millennia.
I couldn't be more impressed. It's so brilliant I'm jealous. This is snooping elevated to a whole new techno-nerd level of expertise.
"Like what you see, kid."
I pick at my cuticles, pretending to be bored.
The elevator stops and the doors swish open. I figure we must be at least half a mile beneath Dublin.
First thing that hits me is the cold. I pull my coat tighter but it doesn't do a lot of good. Love the look of leather. Hate the insulation of it.
Second thing that hits me is the quiet. In most parts of Chester's you can hear faint strains of some kind of music or conversation, 24/7. At least some kind of white noise. This level is still as death.
Third thing is how dark it is.
Ryodan is waiting for me outside the elevator.
"Can you actually see out there?" Does he have another superpower on me? I see good in the dark, but not in pitch-black.
He nods.
I hate Ryodan. "Well, I can't. So, turn on some fecking lights. Besides, Shades much?"
"They don't bother me."
The Shades don't bother him. Shades eat everything. They don't discriminate. "Bully for you. They bother me. Lights. Pronto."
"The lights aren't working down here."
Before I can dig one out, he removes a flashlight from his pocket and hands it to me. Coolest one I ever seen, shaped like a bullet. It's tiny, sleek, silver, and when I turn it on lights up the hallway beyond the elevator like the sun came out.
"Dude," I say reverently, "you got the best toys."
"Off the elevator, kid. We've got work to do."
I follow him, my breath frosting the air.
I used to think there were only six levels in Chester's. Now I know there are at least twenty; I counted on the way down. The level we're on holds three very different subclubs. I glimpse things through the open doors of clubs that no fourteen-year-old should see. But then, that's been the story of my life.
The cold is getting worse the farther down the hall we go, as we make for a pair of tall doors. It slices through my long coat, cutting into my skin. I shiver and my teeth start to chatter.
Ryodan glances at me. "How cold can you get before you die."
Blunt and to the point. That's Ryodan for you. "Dunno. I'll tell you when I think I'm pushing it."
"But colder than most humans."
As usual with him, it's not a question, but I nod anyway. I can take more of everything than most humans.
Still, by the time we stop outside the pair of closed doors at the end of the hall, I'm hurting. I've been stamping my feet with every step for fifty yards. I begin to jog in place, to keep the blood from icing in my veins. My throat and lungs burn with each breath I take. I can feel the cold pressing at the other side of those doors like a presence. I look at Ryodan. His face is frosted. When he raises a brow, ice shatters and hits the floor.
I shake my head. "Can't." No way I'm going in there.
"I think you can."
"Dude, I'm awesome. I'm even All That sometimes. But I have limits. Think my heart's getting sludgy."
Next thing I know his hand is on my chest like he's feeling me up.
"Get off me!" I say, but he's manacled his other hand around my wrist. I shake my head and slant my face away like I can't even stand to look at him. I can't stop him. Not with words or actions. I may as well let him do it, and get it over with.
"You're strong enough." He drops his hand.
"Am not." It's been a rough morning. Sometimes I like to test myself. Now isn't one of them. Not after my earlier stutter.
"You'll survive."
I look up at him. Weird thing is, as mad as he makes me, as unpredictable as he is, I believe him. If Ryodan thinks I can take it, who am I to argue? Like he's infallible or something. Figures I'd put more faith in the devil than any god.
"But you'll have to do it at your top speed."
"Do what?"
"You'll see." The double doors are tall and ornately carved. They look heavy. When he touches the knob and pushes the door open, his fingers are instantly encased in ice. When he takes his hand away, chunks of frozen skin are left on the handle. "Don't stop once you're in there. Not even for a second. Your heart will last only as long as you're moving. Stop and you're dead."
He could figure all that out from a palm on my chest? "And I'm going to go in there _why_?" I can't see a single reason to take such a risk. I like living. I like it a lot.
"Kid, Batman needs Robin."
Dude. I go all soft and melty inside and swallow a dreamy sigh. Robin to his Batman! Superhero partners. There are lots of versions where Robin gets way stronger. He could have had me at hello if he'd said that first. "You don't want me to work for you. You want a superhero _partner_. That's a whole different story. Why didn't you just _say_ so?"
He steps into the room and I hate to admit it but I'm awed that he can do it. I couldn't and I know it. The blast of killing cold coming through the open door makes me want to cry from the sheer pain of it, makes me want to turn and run the other way as fast as I can, but he just pushes forward into it. He doesn't move fluid, as usual. It's like he's shoving himself into concrete, by sheer force of will. I wonder why he doesn't go fast, the way he's telling me to.
That he can do it at all provokes me. Am I going to be a chicken? Let myself be outdone? This is Ryodan. If I'm ever going to be able to beat him, I have to take risks.
"What am I looking for?" I say through chattering teeth, psyching myself up to freeze-frame. I _really_ don't want to go in there.
"Anything and everything. Absorb all details. Look for any clue. I need to know who did this to the patrons of my club. I guarantee protection. I deliver it. If word of this gets out..."
He doesn't finish the sentence. He doesn't have to. It can't get out. Chester's has to be safe ground with no exceptions or he'll lose business. And Ryodan isn't one of those men who will ever tolerate losing anything that's his, for any reason. "You want me to play detective for you."
He looks back at me. His face is coated with ice. It cracks at the seam of his lips when he speaks. "Yes."
I can't help but ask. "Why me?"
"Because you see everything. You aren't afraid to do what it takes and not breathe a word of it to anyone."
"Talking like you know a thing or two about me."
"I know everything about you."
The chill I get from those quietly delivered words is almost worse than what's coming out of the club. I know people. Ryodan doesn't talk big. Doesn't blow smoke up other people's tushes or bluff. He can't know everything. No fecking way he knows everything. "Quit talking. I need to concentrate if you want me to put both my superbody and my superbrain to work at the same time. That's a whole lot of Mega-nitude."
He laughs, I think. The sound is flat and tinkles like ice in his throat.
I shine my flashlight into the darkened club. A hundred or so humans are frozen, mid-gyration, mid-sex, mid-dying, mixed in with a caste of Unseelie I've only seen a time or two: the caste that served as the Lord Master's imperial guard. The room is decorated in tribute to their rank, all red and black, with frosted red velvet drapes and ice-dusted black velvet chaises, red leather sofas and padded racks and lots of chains on every piece of furniture. Leather straps. Sharp blades. There are puddles of black ice on the floor. Human blood.
Torture. Murder. People slaughtered.
It sinks in and I just stare a second, trying to get a grip on my temper. "You let this happen. You _let_ people be killed by those monsters!"
"They come here of their own volition. The line into my club last night wrapped around two city blocks."
"They're confused! Their whole world just melted down!"
"You sound like Mac. This isn't new, kid. The weak have always been food for the strong."
Her name is a kick in my stomach. "Yeah, well Mom taught me not to play with my food before I ate it. Dude, you're a fecking psychopath."
"Careful, Dani. You've got a glass house of your own."
"I got no place like Chester's."
"It's a famous quote."
"Not too famous if I don't know it."
"People who live in glass houses shouldn't throw stones. Maybe you want to talk about your mother."
I look away. I'll pocket my stones for a little while. At least until I know for sure exactly what he knows about me.
I turn my attention back to the room and my tension melts away, replaced by an anticipatory thrill. I love mysteries. Way to test my brain! Dancer and me do logic puzzles. He beats me sometimes. Dancer's the only person I've ever met that I think might be smarter than me. What's with this place? What happened? "You got cameras in here?" I say.
"They stopped working while everything was still normal."
As if anything was ever "normal" in this torture chamber. Now it's even weirder.
Each person and Fae in the room is frozen solid, silent, white, iced figurines. Twin plumes of diamond-ice crystals extend from many of their nostrils; exhales frozen. Unlike Cruce, who is contained inside a solid block of ice, these folks look like they somehow got frozen right where they stood. I wonder if I pinged one of the Fae it would shatter.
"You think it was the Unseelie King did this?"
"No reason I can see," Ryodan says. "He's not the kind to waste time on small stuff. Hurry up, kid. Standing in here is no picnic."
"Why are you?"
"I take nothing for granted."
He means he thinks it's possible one of them isn't completely frozen. "You're watching my back."
"I watch all my employees' backs."
"Partner," I correct, and I don't even like that. I was flattered when he called me Robin to his Batman, but I'm over it already. This is who he is: someone who runs a place where humans get killed for the amusement of the Fae.
I save them. He damns them. That's a gulf between us no bridge will ever span. I'll look into this. But not for him. For humans. Sides have to be taken. I know which one I'm on.
I go all cool inside, thinking about how many folks in Dublin need a little help to survive, and just like that I'm perfect and on fire and free, and I slip sideways into freeze-framing like gliding into a dream.
Moving like I do makes seeing things a little difficult. That's why I stood at the door, looking in so long, collecting observations from a distance. Even freeze-framing, the chill causes intense pain in every bone in my body. As I whiz past him I say, "What's the temp in here?" planning to get the answer on my way back around.
"No thermometer can take it," he says by my ear, and I realize he's freeze-framing, too. He's right beside me. "Don't touch anything. It's too cold to risk."
I circle a Fae guard at top speed. Around and around, looking for clues. If the Unseelie King did this, why would he choose here? Why ice his own guards?
"Is this the only cl-club that g-got iced?" I stutter with cold.
"Yes."
"Wh-When?" I stamp my foot in hyperspeed, pissed that I'm stuttering. Doesn't matter that it's from the cold, it makes me sound pansy. Next thing you know, I'll lisp.
"Eight days ago."
A few days after Ryodan cornered me on my water tower. I cock my head. I just heard a sound in a completely frozen room. I whiz back to where I was when I heard it and go in tight circles, listening hard.
Silence.
"D-Did you hear th-th-that?" I manage to spit out. My face is going numb and it's getting harder to move my lips. I circle a human woman, frozen mid-coitus. It's not hoar frost that turned her white. She's covered with hard rime, the kind of ice that builds up on a cold foggy night. Over it all is a layer of clear ice a good inch thick.
"Yes." Ryodan flashes past me. Warily, we circle the room on opposite ends, watching everything real careful-like.
It's hard to listen good when you got so much wind in your ears from moving like we do. Ryodan and I have been practically shouting at each other the whole time we've been talking. "Like a high-p-p-pitched whine," I say. I'm not going to be able to stay in the room much longer. There it was again! Where was it coming from? I whiz though the subclub faster and faster. Ryodan and I do figure eights between the frozen figurines, trying to isolate it.
"You f-feel that?" I ask. Something's happening... I feel a vibration, like the floor has the tremors, like everything is... _changing_.
"Fuck!" Ryodan explodes. Then his hands are on my waist, and he's tossing me over his shoulder like that stupid sack of potatoes again, and moving faster than I've ever managed to move in my whole life.
That's when they begin to pop, going off like firecrackers. Fae and humans explode, filling the air with icy, flesh-colored shrapnel.
One after the next, they blow violently, and with each new explosion, the next one blows harder. The furniture is popping now, too. Sofas erupt into icy splinters of wood and rock-hard chunks of stuffing. Racks get blasted into smithereens of metal shards. It sounds like a thousand machine guns going off.
A pair of knives whiz by, chased by a dozen ice picks.
I bury my nose in Ryodan's back. My face has taken enough of a beating for the day. I'm not in the mood for anything sharp in it. Something slams me in the back of my head and I wrap my arms around my skull. I hate being over his shoulder but he's faster than me. I tense, pelleted by chunks, waiting for one of those nasty-looking blades or picks to sink into me.
We're halfway down the hall, almost to the elevator. The other two clubs have begun blowing up, too. I hear an enormous, deep, rumbling sound and realize the floor is cracking beneath us.
Chunks of ceiling begin to fall.
At the elevator, Ryodan flings me from his shoulder into the compartment in one smooth motion.
I explode right back out. "Fecking thing is going to blow and you want me _on_ it?"
"It'll last long enough to get you out of here."
"Bull-fecking- _crikey_! I give you fifty-percent odds I'll make it!"
"I'll take them."
I'm in the air, over his shoulder, slammed back into the elevator again. The whole ceiling of the hallway is coming down now, crown moldings, drywall, steel girders. He'll be crushed. Not that I care. "What about you?"
His smile is fanged. Creeps me out. "What, kid, you care?"
He slams the doors closed with his bare hands and I swear he gives the thing a push from below.
I shoot up into Chester's.
# THREE
# _"When the cat's away..."_
Under normal circumstances I'd have snooped through Ryodan's office, but my day hadn't been normal and I was in a pissy mood.
Two things were on my mind: get as far away from Ryodan as possible while he was busy dying (hopefully), and kill as many Fae inside Chester's as I could on my way out.
The club "proper" was unprotected. Hoo-fecking-rah.
His dudes had whizzed past me so fast my hair shot straight up in the air five, six, seven times, minus Barrons, who doesn't much leave TP's side. No doubt they were heading down to the iced level, to save their boss. Keep him from being crushed. With any luck, the whole club would collapse into a pile of rubble and kill them all.
Somehow I doubted it.
They were like Barrons. I wasn't even sure they _could_ be killed. If so, it was probably only by a single weapon, hidden inside an invisible box, on an invisible planet, with an atmosphere that would burn up any living thing instantly, like a gazillion light-years away.
But I knew a few things that _could_ be killed.
And my sword hand has a permanent itch.
Slaying Unseelie gives me a rush that's almost as intense as freeze-framing. The only thing missing is TP at my back, but I know if I ever have TP at my back again, she'll be trying to shove a spear through my heart.
Supercharged on adrenaline and anger, I slice and dice my way through the subclub that bugs me the most: the one where the waitresses dress like school kids, in short, pleated plaid skirts and white socks, and crisp white blouses with starched collar points.
Kids. They're the worst victims of the fall. There are so many of them hiding in the streets, with no clue how to survive.
At Chester's, grown women are dressing like kids to trade favors for pieces of Unseelie flesh, the latest drug on the market. It has epic healing powers, and temporarily gives humans extra strength and stamina. I hear it makes sex really intense, too. The things people are willing to do for a quick high—eat pieces of our enemies' flesh! Makes me want to knock heads together.
So I do.
I get a few good elbow jabs in on the waitresses, too. Half of them are those stupid See-You-in-Faery chicks who chirp the stupid phrase at each other every time they part, like going to Faery is something to aspire to instead of something to avoid like ten variations of the black plague.
They should be out in the streets, helping us fight and rebuild our world. Instead they're in here, consorting with the enemy, selling themselves for a shot at immortality. I don't buy that bunk. I think the Unseelie made that part up—that if you eat enough Unseelie flesh, eventually you become immortal, too, and you can hang with them in Faery all social-like.
I slay every last one of the Fae in the kiddie subclub, ignoring the waitresses screaming at me to stop. Some people just don't know what's good for them.
There's black blood on my hands, goop in my hair, and my eyes are so swollen from my earlier collisions that I can barely see, but I don't need to see much. I've got a homing device where Fae are concerned. I sense Unseelie. I slay.
I feel a big bad one behind me, worse than any of the ones I've killed so far, oozing all kinds of power. Sword back, poised for the killing blow, I whirl and bring my blade slashing down—
And miss!
The Unseelie ducks, rolls, and springs lightly to his feet half a dozen tables away. He flips his long black hair over a muscled, tattooed shoulder and hisses at me.
I lunge after him without even thinking and am about to slam into him when I realize what he is.
I change direction mid-lunge and scramble back, feet pedaling air. Feck, feck, feck, one of the Unseelie princes found me!
This is a battle I'm not up to today! I wasn't expecting this because I never heard of any of the princes strolling into Chester's!
I crash into a table, fall over backward, roll onto all fours and launch myself away. I'm about to find out if I can freeze-frame faster than it can sift. I rip open a power bar, shove half of it in my mouth and start shifting gears when the Unseelie prince says, "Lass, what the bloody hell are you doing? Have you taken a look around?"
I'm seeing through slits from all the swelling in my face, and my vision is a little dim, but I scan the place quick-like. All activity in the club has stopped. Fae and humans are lined up at balconies, staring at me from every level.
I tune in to what they're saying.
"Crazy. The kid's nuts."
"Somebody needs to put that bitch down."
"I'm not going near her. Did you see her move? Do you see what she's holding?"
"The Sword of Light," a Fae says icily. " _Our_ sword."
"Take it from her!"
"How dare she?"
"Kill her now."
"I bet I can sift faster than she can slay," one growls.
I toss my hair from my eyes, on all fours, every muscle tense, waiting. We'll sure as feck find out.
"Who permitted that... that revolting... human... _thing_ in here? Where is our host? This is neutral ground!"
"He swore an oath to us. He has failed us!"
I can't help but smirk. Assuming Ryodan survives the collapse, he's going to be seriously pissed. I just accomplished exactly what he'd tried to "hire" me to prevent. Ruined his rep. The whole club now knows Ryodan can't guarantee safety at Chester's. It'll be all over Dublin within an hour. I might as well print up a special edition of _The Dani Daily_ , broadcasting it. Good. If fewer folks come to Chester's, fewer folks will die.
I glance back at the dude I initially thought was an Unseelie prince. The moment he'd spoken, I'd relaxed. Now that I'm slo-mo again, I see the differences.
I almost killed a human. Well, a human that's in the process of becoming something else. If he hadn't spoken up, I still might not be sure who he was, but I've never heard a Fae call anybody "lass." I don't think they'd stoop to it, not even to fake someone out.
It's the Scot who crashed my water tower party the same night Ryodan did.
They'd faced off with each other, all bristling hostility, giving me time to escape. It had seemed he was there either to help me or to feck with Ryodan. Whichever—that makes him good for me.
This dude has problems as big as mine, maybe bigger. I consider him. He doesn't like Ryodan. And he's got some serious mojo. I can feel it shivering in the air around him. He could be a valuable ace in my hidey-hole. If he can be trusted.
"You're a MacKeltar, right?"
"Christian," he says.
"Aren't your uncles some kind of warlocks or something? They helped hunt the _Sinsar Dubh_."
"Druids, lass. Not warlocks."
"Can you fight?"
He gives me a mocking look. "I don't need to. I can walk you out of here without lifting a finger."
Big talk. I decide to let him try.
He flanks me and we head for the door. Between what he looks like and my sword, every last occupant of Chester's draws back as we pass. I can't help but swagger a little.
Hisses, jeers, threats follows us.
But no one makes a move.
I could get used to this. Who needs TP? I got what looks like an Unseelie prince at my side and nobody, but nobody—not even the Unseelie—mess with their princes. Oh, yeah, this guy's going to be a major plus in my column. I take a sidewise glance at him.
If I can get past that he looks like the most terrifying of all the Unseelie.
Beyond him I catch a glimpse of myself in a mirror. Between the bruises, swollen eyes, cuts, and blood of all colors, I'm not looking so hot myself.
Sword up, I squint through puffy eyelids and memorize faces on the way out.
Out in the streets, in the thick of battle, sometimes you have to make hard choices. Sometimes you can't save everyone.
Humans that hang at Chester's are never going to be at the top of my list.
# FOUR
# _"I want a girl with a mind like a diamond"_
I'm attracted to her.
She's fourteen. And I'm attracted to her.
I'm eight years older than she is. Eleven if you count the three years I spent trying to escape the Fae Silvers. Eight or eleven: what's the difference? It makes me one seriously fucked-up Highlander.
Or whatever the hell I am.
She's a bloody mess, literally. Covered with guts and gore from killing, her nose is crusted with dried blood, she's bruised, and she's going to have two fierce black eyes before nightfall. It's too late for ice to knock down the swelling.
And she's on fire.
Light shines out of her delicate, battered face, blazes in her green eyes. She's got a head of curly red hair that falls halfway down her back. Everything about her is brilliant and intense. She's aware and invested in the world in ways most adults never get around to being. I know. I was once, too. Back when I thought hearing the truth in everyone's lies was my biggest problem. She does everything one hundred and ten percent, with all her heart.
That's what gets me.
Attraction isn't always about sex. Sometimes it's about something far subtler, and far bigger.
I watched her fight.
And something stirred inside me that I thought was dead.
Not my dick. That's working great. Better than ever. Always hard. Always ready.
What stirred was like gentle rain on a warm summer day. Sweet. Tender. Something I used to be. With my clan. With my nieces and nephews.
She reminds me of my Highlands—to which I can never return.
I know exactly what she's going to be one day. Bloody hell is she ever.
Worth. Waiting. For.
Too bad I won't be here anymore.
_Take her now_.
"Fourteen," I growl. I've gotten good at arguing with the voice inside my head. I get a lot of practice. An Unseelie prince wouldn't give a second thought about her age. An Unseelie prince would see only that she has the right parts, and temper to spare. The bigger the fight, the better the feast.
"Why the feck does everybody keep saying that like it's some kind of insult? Like, maybe I managed to forget for a minute?" she says crossly. "Geez! I've never seen so many people obsessed with my age!"
Dani bristling is something to see. I smile.
She takes a wary step away from me. "Dude, you planning to eat me or something?"
My smile vanishes. I look away.
I wear a mask. A face that isn't mine.
I used to have what women called a killer smile.
Now I have a killer's smile.
" 'Cause, like Ryodan already bit me once today. I'm not in the mood for any more teeth in me anywhere."
Ryodan bit her? One more reason to kill him. I look back at her, my face void of all expression. There's no point in trying to look reassuring. This face can't pull it off. "No biting. I promise."
She squints at me suspiciously. "Dude, what are you? Unseelie or human? What happened to you?"
"Mac happened to me." She flinches when I say it, and I wonder why. I blame Jericho Barrons, too. If I survive what I'm turning into, I'll kill them both. Hate ripples through me, dense and black and suffocating. If not for them, I'd still be me. Then again, if Mac hadn't done what she'd done, I wouldn't be here at all. Then again, if Barrons hadn't done what he'd done, or rather failed to do, what Mac did might not have turned me into this. Barrons didn't check my tattoos before we performed a dangerous Druid ritual, then he abandoned me in the Silvers to die. When Mac found me in the Silvers, she fed me Unseelie to keep me alive. It's impossible to decide which one of them I blame the most. So I blame both and I'm getting happier about that every day.
I saw Mac a few nights ago, across the club at Chester's, looking blond and beautiful and happy. I want to take all that shiny-happy-blondness, twist it into a garrote, and strangle her with it. Hear her beg, and kill her anyway, love every minute of it.
Later that night, I'd stared at myself in the mirror for a long time. Arm bent behind my head, scratching my back with a knife—it itches all the time now—relishing the slide of warm blood on my skin as it ran down my spine into my jeans. I used to hate blood. Now I could bathe in it. Mother's milk.
"Yeah, she does that," Dani agrees with a sigh. "She happened to me, too."
"What did she do to you?"
"It's more like what she _will_ do to me if she catches me," she says. "Don't want to talk about it. You?"
"Don't want to talk about it."
"Better things to talk about anyway. So, what were you doing at Chester's?"
Good question. I have no bloody clue. I think the sheer number of Unseelie gathered calls to something in my blood. I don't know why I go half the places I go anymore. Sometimes I don't even remember the hours leading up to it. I just become aware that I'm someplace new with no memory of when I decided to go or how I got there. "I wanted a beer. Not many choices left in Dublin anymore."
"No shit," she agrees. "Not just for beer, for everything. Which side are you on?" she says bluntly. "Human or Fae?"
It's a good question. I don't have a good answer.
I can't tell her I don't discriminate. I despise everyone. Well, almost. There's this fourteen-year-old redhead with a mind like a diamond. "If you're asking if I've got your back, lass, I do."
She narrows her eyes and peers at me. We're standing outside Chester's in a pool of light. The sky is so overcast it looks like dusk at three in the afternoon. I get a sudden image of us from above: slim, delicate-faced young girl in a long black leather coat, hands on her hips, staring up at a Highlander-going-Unseelie prince. The image is painful. I should be a good-looking twenty-two-year-old college student with a killer smile and a bright future ahead of me. We'd plot and plan and fight the good fight together. That version of me would watch out for her. Make sure nobody does to her what the voice in my head tells me the first Unseelie that catches her without her sword is going to do. What a part of me wants to do, too. Fury fills me. At them. At me. At everything. "You never take that sword off your body, right?"
She backs up a step, hands going to her ears. "Dude, my hearing works great. You don't need to yell."
I didn't know I was. But a lot of things come out differently than I mean them to now. "Sorry. I'm just saying, you _do_ realize what will happen to you if one of the Unseelie catches you. Right?"
"Never going to happen," she says smugly.
"With that attitude, it will. Fear is healthy. Fear is good. It keeps you on your toes."
"Really? 'Cause I think it's a waste of time. Bet you don't fear nothing," she says admiringly.
Every time I look in the mirror. "Sure I do. That you'll get sloppy and slip up and one of them will grab you. Snuff you out."
She tilts her head, eyes narrowed on my face. Not many people look me full in the face anymore. Not for long anyway. "Maybe you aren't all Unseelie prince yet. Maybe we can, like, work out some kind of arrangement."
"What do you have in mind?"
"I want to shut down Chester's. Torch it. Exterminate it."
"Why?"
She cuts me a look of scorn and disbelief. "You saw it in there! They're fecking monsters! They hate humans. They use them and eat them and kill them. And Ryodan and his men let them!"
"Say we do close down the place, say we burn it to the ground. They'll just find another place to go."
"No they won't," she insists. "They'll pull their heads out. They'll smell the coffee percolating and see we saved them!"
A rush of emotion, cloyingly sweet as funeral lilies, floods me, swells my tongue with a taste both familiar and sickening. She's tough, smart, capable, a stone-cold killer when she needs to be.
And she's so bloody naïve.
"They're at Chester's because they _want_ to be at Chester's. Make no mistake about that, lass."
"No. Fecking. Way."
"Yes fecking way."
"They're confused!"
"They know exactly what they're doing."
"I thought you were different but you're not! You're just like Ryodan! Just like everyone. Ready to write them all off. You don't see that some people need saving."
"You don't see that most people are beyond saving."
"Nobody's beyond saving! Nobody! Ever!"
"Dani." I say her name tenderly, savoring the pain she makes me feel.
I turn and walk away. There's nothing for me here.
"So, that's it, then?" she yells after me. "You won't help me fight either? Gah! Sheep! You're all big fat fecking sheep waggling big fat fecking sheep asses!"
She's too young. Too innocent.
Too human. For what I'm becoming.
# FIVE
# _"Our house is a very very very fine house"_
"Hungry?" Dancer says as I bang in the door and throw my backpack and MacHalo on the couch.
"Starving."
"Cool. Went shopping today."
Me and Dancer love to go "shopping," aka looting. When I was a kid, I used to dream that I got forgotten inside a department store after it closed with nobody around, which meant I could have anything I wanted.
That's the world now. If you're tough enough to brave the streets, and got balls enough to go into the dark stores, anything you can carry out is yours. First thing I did when the walls went down was hit a sporting goods store and cram a duffel bag full of high-top sneakers. I burn through them quick.
"Found some canned fruit," he says.
"Dude!" It's getting harder to find. Plenty of the ick-stuff on the shelves. "Peaches?" I say hopefully.
"Those weird little oranges."
"Mandarin." Not my favorite but better than nothing.
"Found some ice cream toppings, too."
My mouth instantly waters.
One of the things I miss most is milk and all the things it made possible. A while back, a couple of counties to the west, some folks had three milk cows that the Shades didn't get, but then other people tried to steal them and they all shot each other. And the cows. I never did get that part of it. Why shoot the cows? All that milk and butter and ice cream re _-moo_ -ved from our world forever! I snicker, cracking myself up. Then I see the table and the spread of food and it cracks me up more. "You expecting an army?"
"Of one. I know how you eat."
And he's fascinated by it. Sometimes he just sits and watches me. Used to freak me out but not so much anymore.
I decimate the feast, then we sack out on the couch and watch movies. Dancer's got everything wired for power, with the quietest generators I've ever seen. He's smart. He survived the fall without a single superpower, no family, and no friends. He's seventeen and all alone in the world. Well, technically he has family but they're somewhere in Australia. With splinters of Faery reality slicing everything up, no planes flying and nobody about to take a boat out, they may as well be dead.
If they aren't.
Nearly half the world is. I know he thinks they're dead. We don't talk about it. I know it from the things he doesn't say.
Dancer was in Dublin checking out Trinity College's Physics Department, trying to decide where he wanted to go to grad school when the walls fell, leaving him cut off and alone. Home-schooled by multiple tutors and smarter than anybody I ever met, he finished college six months ago, speaks four languages fluently and can read three or four more. His folks are humanitarians, über-rich from old money. His dad is or was some kind of ambassador, his mom a doctor who spent her time organizing free medical care for third world countries. Dancer grew up all over the world. I have a hard time wrapping my brain around his kind of family. I can't believe how well he adapted. He impresses me.
I watch him sometimes when he's not watching me. He catches me now.
"Thinking how hot I am, Mega?" he teases.
I roll my eyes. That kind of stuff isn't between us. We just hang together.
"Speaking of hot..."
I roll my eyes bigger, because if he's finally about to say something about how much prettier I am since the Gray Woman took my looks then gave me back a little extra, I'm out of here. He's been cool so far about not commenting. I like it that way. Dancer's... well, Dancer. He's my safety zone. There's no pressure here. It's just two kids in a fecked-up world.
"... try some hot water. Mega, you're a mess. I got the shower working again. Go take one."
"It's just a little blood—"
"It's a bucket. Maybe two."
"—and a few bruises."
"You look like you got hit by a truck. And you smell."
"I do not," I say indignantly. "I would know. I have supersmell."
He looks at me hard. "Mega, I think you have guts in your hair."
I reach up, dismayed. I thought I got them all out on the way over. I root around in my curls and pull out a long slimy piece.
I stare at it, revolted, thinking how maybe I should cut my hair really short or start wearing a ball cap all the time, then I look at him and he's looking at me like he's going to toss his cookies, then all the sudden we both start cracking up.
We laugh so hard we can't breathe. We're on the floor, holding our sides.
Guts in my hair. What kind of world am I living in? Even though I was always different, and saw things other people didn't see, I never thought I'd be sitting on a sofa, in a virtual bomb shelter underground, with security cams and trapdoors and booby traps all around us, hanging with a seventeen-year-old (hot!) genius who makes sure I eat more than protein and candy bars (he says I'm not getting the right vitamins and minerals for proper bone health) _and_ knows how to get a shower running in post-wall Dublin.
He plays a mean game of chess, too.
He pauses the movie when I head for the shower. I grab a change of clothes on the way in.
This is Dancer's place, not mine. But he keeps things stocked for me in case I come by. Like me, he's got lots of other digs, too. You have to keep moving in this city to increase your odds of survival, and set things real careful when you leave, so you know if somebody's invaded your turf while you were gone. It's a dog-eat-dog world. People kill each other over milk.
The hot water lasts four glorious minutes. I scrub my hair, wrap it in a towel and study my face in the steamed-up mirror. Bruises are me. I know the progression: black turns purple, purple goes green, then you get all jaundiced-looking for a while. I look past the bruises. I lock eyes with my reflection and don't look away. The day you look away you start to lose yourself. I'm never going to lose myself. You are what you are. Deal with it or change.
I toss the towel, finger-comb my hair, tug on jeans, a tee, and consider a pair of combat boots. Dancer picked them out for me. Said I won't burn through the soles as fast. I decide to give them a try.
I grab another bowl of puny orange slices on the way back to the sofa, pop open a jar of marshmallow cream and slather it on, then coat it all with hard-shell chocolate.
Dancer and me get down to business. He starts the movie again while I get out the game board. He kicked my butt at Go Bang for hours the last time I dropped in, but I'm feeling lucky tonight. I even magnanimously accept a restricted second move when I win the flip for opening play.
I do something I haven't done in a long time. I let my guard down. I'm drunk on fruit and marshmallow cream and the thrill of winning at Go Bang. I was up all night last night, and my day was long and eventful.
Besides, Dancer's got killer booby traps around his place, almost as good as mine.
I push my backpack out of the way and fall asleep on his couch, fist under my cheek, sword in my hand.
I don't know what wakes me but something does and I lift my head a few inches, slit my eyes and peer around.
Big, scary-looking men surround me.
I blink, trying to clear my vision. It's hard to do when my eyes are even more swollen than they were when I went to sleep.
Dimly I realize I'm the focal point of a circle of machine guns.
I shoot up to sitting and I'm just about to freeze-frame when a hand slams me back into the couch so hard the wood frame cracks behind my shoulder blades.
I lunge up, and get slammed right back down again.
One of the men laughs. "Kid doesn't know when to stay down."
"She'll learn."
"Bet your ass she will. _If_ he lets her live."
"He sure as fuck shouldn't. Not after what she did."
"Dani, Dani, Dani."
I flinch. I've never heard anyone say my name so gently. It creeps me all kinds of out.
He's towering over me, arms crossed over his chest, scarred forearms dark against the rolled-up sleeves of a crisp white shirt. Heavy silver cuffs glint at both wrists. The light is smack behind his head, as usual.
"You didn't really think I'd let you get away with it," Ryodan says.
# SIX
# _"I will break these chains that bind me"_
"Hurt's a funny thing," Ryodan says.
I say nothing. It's taking all my energy to stand, despite the chains holding me. I'm somewhere in Chester's, in a room with stone walls. I feel the distant beat of rhythmic bass behind me, in the soles of my feet. If I didn't have supersenses, I wouldn't be able to pick it up at all. Because it's so faint, I know I'm far beneath the public part of the club, probably at the bottom. That means the lower levels didn't get as badly damaged in the explosion yesterday as I hoped.
They put a bag over my head when they brought me in. Wherever I am, they didn't want me to be able to find my way back. It's a logical deduction that they plan to let me live. You don't bag the head of somebody who's never going to see anything again. A single low-watt lamp illuminates the room behind him—or fails to. There's barely enough light to see him standing a dozen feet away.
"Some people fall apart when they get hurt," he says. "Puddle into apathy and despair and never recover. They wait all their lives for someone to come along and rescue them." He moves in that strangely fluid way—not freeze-framing but not walking like a Joe either—a ripple of muscle and cascade of wind. Then he's standing in front of me. "But others... well, they don't go from hurt to pain. They flash from insult to fury. They raze everything in sight, which usually succeeds in obliterating the very thing that hurt them. However, it causes collateral damage."
I hang my head so he can't see the fire in my eyes. "Dude. Bored. If I'd ever been hurt, I'd give a shit. But I haven't."
He pushes the hair out of my face with both his hands, sliding his palms over my cheeks. It takes all I've got to conceal a shiver. He forces my chin up. I flash him my best hundred-Megawatt smile.
We lock eyes. I'm not looking away first.
"It didn't hurt you when your mother left you in a cage like a dog, and forgot you for days while she was off with one of her endless string of boyfriends."
"You've got a seriously wild imagination."
He grabs a handful of my hair close to the scalp and uses it to keep me from looking away, as if I fecking planned to. When he reaches into one of my coat pockets and pulls out a Snickers bar, my mouth waters. I fought him and his men so hard back at Dancer's place that I'm drained. I pretend my spine is a broomstick so I don't sag into the chains holding me to the wall. Pretending is a game I'm good at.
He rips it open with his teeth. I smell chocolate and my stomach hurts.
"How many times did you curl in that cage, chained by a collar around your neck, waiting, wondering if she was going to remember you this time. Wondering what would kill you first: hunger or dehydration. What was it—five days she left you sometimes. No food or water. You slept in your own—"
"You want to shut up now."
"When you were eight, she died while you were locked up. Rowena didn't find you for a week."
That's the story. I don't say anything. There's nothing to say. Things got real simple in that cage. There are only two things to worry about in life: either you're free or you're not. If you're free, there's nothing to worry about. If you're not, you kick the shit out of everything around you until you are.
"Sometimes her boyfriends played with you."
Not that way. Never that way. I'm a virgin and I take it seriously. I'm going to lose it in a really epic way someday, when I'm ready. I'm all about gathering up some fan-fecking-tastic experiences to compensate for the crappy ones I had as a kid. That's why I wanted to give it to V'lane or maybe Barrons when I was old enough. Someone stellar. I want it to be with someone who will make it a night to remember.
"Are we like swapping philosophies, Ryodan? 'Cause if so, here's one of mine. Feck you. Past is past."
"It carves you."
"Vanishes. Means nothing," I say.
"You can never outrun it."
"I can outrun the wind."
"The wound you refuse to dress is one that will never heal. You gush lifeblood and never even know why. It will make you weak at a critical moment when you need to be strong."
"I get it, all right? You're going to torture me to death by talking. Kill me now. Get it over with. But use something quick and clean. Like a chain saw. Maybe a grenade."
He touches my cheek. "Dani."
"Is that pity, Ryodan? 'Cause I don't need it. Thought you were tougher than that."
His thumb brushes my mouth and he gives me a look I don't understand. I head-butt his hand away.
"You think you're going to chain me to a wall then stand here and tell me why it's okay that I am the way I am? That because of all the crap folks put me through when I was young it's all right that I turned out like this? Dude, I don't have a problem with how I turned out. I like me."
"Rowena made you kill your first human when you were nine years old."
How the feck does he _know_ this stuff? She made it a game. Told me she wanted to know if I could whiz in and dump extra milk in Maggie's cereal bowl without her seeing me. Of course I could. Maggie died, sitting there at the breakfast table. Ro told me it was a coincidence, that she was old and had a heart attack. When I was eleven, I found out the truth. Ro hated Maggie because she'd been rallying _sidhe_ -seers to elect a new Grand Mistress. I found the old witch's journals. She chronicled everything she did, like she thought one day she'd be immortalized and people would want to read her private memoirs. I have all those journals now, tucked away in a safe place. I'd poisoned Maggie that day with the "milk" I'd added to her bowl. I'd done a lot of other things, too, that I hadn't understood.
"Significant words there: Rowena made me. I got over it a long time ago."
"Funny, your speech is changing, kid. Getting all grown-up-like."
"Dude," I add.
"You're going to be a tough one to crack."
"Let me give you a clue: substitute the word 'impossible' for 'tough.' "
He peels the wrapper back from the Snickers. Offers me a bite.
I turn my head away. I won't eat like a chained-up animal.
"When we find your little boyfriend, you'll change your mind."
My guts unknot and I almost slump into the chains with relief but I lock my knees so I can't. He said "when" we find, which means they haven't. I don't telegraph unless I slip. I was afraid they had Dancer. He must have left while I was sleeping. He keeps odd hours, goes off sometimes until he feels like coming back. I can't always find him when I want to. Sometimes I don't see him for days. It's good to know he's safe somewhere. They didn't get him. They only got me. I can handle this kind of stuff. I cut my teeth on it. Dancer... well, until the walls fell, he lived a charmed life. I never want him to have to deal with these men.
"He isn't my boyfriend."
"How long will you make me keep you here, Dani?"
"Until you figure out it isn't going to do you any good."
He smiles faintly and turns away. At the door, he pauses and puts his hand on the light switch like he's giving me a choice. As if all I have to do is give him a look that says "Please don't leave me in the dark" and he won't.
I flip him off big and showy, with both hands chained over my head.
He leaves me without my sword, in the dark.
I don't worry.
I know Ryodan. If anyone is going to kill me, it'll be him. That means he's got this place protected from Shades and Fae or he'd never have left me here.
I'm hungry and tired. I close my eyes and play an old game with myself, one I learned young.
I pretend I have a giant, cushy pillow in my stomach, filling it up softly, absorbing the acid that boils from extreme hunger. I pretend that I'm stretched out in a downy soft bed in a perfectly safe place where nobody can hurt me.
Hanging by manacles around my wrists, I sleep.
"What did you _think_ was going to happen, Dani?" Mac says.
I squint my eyes open a slit and groan. TP is here, standing right in front of me.
I do a quick scan. I don't see her spear but I know it's on her somewhere. She doesn't go anyplace without it.
"Not fair," I say. "You can't kill me while I'm chained up. Dude, you have to at least give me a fighting chance. Unchain me." I won't fight her. But I will run. I can outrun TP till the end of days.
"I don't understand, Dani," she says. "You had to know when you killed all those Fae in front of thousands of witnesses that it would put you on the shit-list of every person and Fae with any power in this city, with Ryodan and his men first in line. Were you _trying_ to become Dublin's most wanted?"
"Not like you weren't for a while, and you survived."
"I had Barrons at my back. You pissed off your potential version of Barrons."
I'm deliberately obtuse. "Christian MacKeltar? He's not pissed at me."
"Ryodan."
"Ryodan isn't Barrons and never will be!"
"Agreed. But he could have your back, if you'd let him. Instead, you not only blatantly antagonized him, you put him in a position where he has to punish you. You defied him in front of the entire city. _Dani, Dani_."
"Who the feck's side are you on? And why aren't you trying to kill me?"
"I don't need to. You've got the whole city lined up waiting to do that. _Dani! Dani!_ "
"They have to catch me first. Why do you keep saying my name like that?"
" _Wake up_. You're caught," TP says. "I know you're not stupid. What are you doing? _Dani! Dani!_ "
"Same thing you always did. Taking a stand. Not backing down. Even if I don't have all the answers and can't predict how I'll get out of this one, I _will_ get out of this one."
I'm still waiting for a spear through my gut. Instead TP smiles and says, "Hold on to that thought."
"Wake _up_ , Dani!"
My face stings like somebody slapped me. I squint my eyes open when I thought they already were.
Jo's standing in front of me. My cheek stings. I'd rub it but I'm chained.
"Where did TP go?" I say, confused.
"What?" Jo says.
I lick my lips, or try to. My mouth is so dry my tongue doesn't make any difference. My lower lip is split and crusted with dried blood. The base of my skull hurts. I must have banged myself a good one passing out, or got hit in the back of my head when I was fighting Ryodan's men.
"I'm sorry I hit you but I was afraid you were... oh, Dani! What did he do to you? He beat you! Then I hit you, too!" She looks like she might cry. She touches my face gently and I flinch.
"Get off me!"
"I'm going to kill him," she whispers, and something in the softly spoken words surprises me. Like she's turning all bloodthirsty, becoming like me.
I try to figure out if TP was the dream or Jo is, or they both are. I have the weirdest dreams sometimes. As if TP would actually bother trying to give me advice. I should have known it was a dream instantly by the fact that she wasn't killing me.
"I ran into him," I tell her. "As in collided. Twice. That's why my face is so beat up." Well, it's most of the reason.
"Are you _defending_ Ryodan? Look what he's done to you! Dani, has he brainwashed you? Are you getting Stockholm syndrome?"
"What the feck's Stockholm got to do with any of this? Ain't that some city in Sweden?"
She wraps her arms around me and gets all in my space. It's awkward with my hands chained above my head and my ankles shackled to the floor. She sort of hugs me and I can't get her off me because I'm stuck.
"Dude!" I give a whole body shrug, trying to dislodge her. She's tenacious, lopping all over me. "What are you doing?"
When she pulls back I see she's crying. I must look pretty bad.
"Why did you do it?" She sniffs and wipes her nose with the back of her hand. "We talked and talked about it, and can't figure it out. You didn't just wave a red flag at a bull. You sauntered right up to it, punched it in the face then tried to dance on its horns. Dani, what were you thinking?"
I sigh. People ask the stupidest questions. Sometimes you _don't_ think. You just do. Some moments are too golden to pass up. You play—you pay. I've always been okay with that.
I peer at her suspiciously. Jo can't be here. Not in the guts of Chester's. "You're not real," I say.
She feels my forehead. "You're running a fever."
I know. I'm dripping sweat and freezing cold. I always get a fever if I get dangerously hungry. It's another fecking weakness. So many superstrengths. So many limits. I don't let folks know about them. "Must have caught a cold," I tell her. I have food stuffed in every pocket, but with my hands chained above my head I can't get to one bite of it.
"Get a protein bar out of my pocket and feed it to me." If this is really happening, I'll get strong again and my body temp will drop back to normal. If this is a dream, at least I'll get to dream the taste of food. I've got nothing to lose and everything to gain. "I don't suppose you've seen a key to these manacles lying around somewhere convenient?" I say with no hope. Ryodan's not sloppy.
Four protein bars later I know I'm not dreaming. My head is still throbbing but starting to clear. TP wasn't real.
But Jo is.
She tells me word spread everywhere that I'd single-handedly taken on a bunch of Fae in Chester's then sauntered out all cocky-like with an Unseelie prince. Margery insisted the Unseelie prince had killed me, and managed to convince a lot of _sidhe_ -sheep to write me off, taking up right where Rowena left off, smearing my name.
Kat had seen things differently. She'd done some investigating before making her decision. According to onlookers, the "prince" who'd walked me out hadn't been wearing a torque. The Unseelie princes have silver torques around their necks that glow like they're radioactive. The necklace seems to be part of them, inseparable like their tattoos and wings. That told Kat all she needed to know: if the prince wasn't wearing a torque, it had to be Christian who'd escorted me out.
I'm not sure how she made the next deductive leap, but I'm glad she did. She sent a group of girls to Chester's to search for me, believing Ryodan had gone after me and captured me.
I'm amazed by how speedily she acted. Maybe Kat's going to do all right by the _sidhe_ -seers. "How did she figure out I was missing so quickly?"
"You've been gone for three days, Dani."
I'm stunned. I've been chained down here for three days? No wonder I'm starving.
"How the feck did you find me? I figured I was like, buried in the dungeon of Chester's or something."
"You are. I saw Ryodan get off an elevator hidden in the wall outside the retroclub. The door didn't close all the way and I slipped in when nobody was looking."
I close my eyes and sigh.
There were three mistakes in that sentence. (1) Ryodan doesn't get seen if he doesn't want to. (2) The doors around this place don't stay slightly open. (3) Nobody slips into them without being noticed.
The only way Jo saw Ryodan get off an elevator was if he let her.
Which means he hadn't been able to find my "little boyfriend" over the past three days. But he'd sure found somebody else to use against me.
On the insides of my eyelids I see Jo chained, beaten.
Ryodan hadn't even had to leave his club. He just sat back and waited for whoever showed up first, looking for me.
I open my eyes. "Get out of here, Jo," I say. "Now."
"Neither of you are going anywhere," Ryodan says as he steps from the shadows.
# SEVEN
# _"I fall to pieces"_
I'm absurdly easy to break if you know the right buttons to push.
If you've read any comics, you know superheroes have a critical vulnerability: the society they protect.
Jo's part of my society. Fact is, any _sidhe_ -sheep chained up next to me would have me singing a new tune. Well, maybe not Margery.
Actually, probably even her, too.
The hard thing for me is knowing I can take more than everyone else. Like that stupid bunny that used to be in commercials all the time, I take a licking and keep on kicking. And punching. And breathing.
Not true other folks. They die so easily.
Besides, I'm not afraid of the big sleep. I figure it's just another adventure.
I try to talk Ryodan out of chaining Jo up.
He doesn't listen to me.
Jo goes ballistic when he grabs her. Screaming and yelling and kicking. I'm kind of impressed by how hard she fights.
I think watching Dublin get destroyed on Halloween, seeing our friend Barb get taken by the _Sinsar Dubh_ and ridden as a machine-gun-toting bitch to massacre so many of us, plus living in a world where you have to shake your shoes out before you put them on to make sure you don't get eaten by a Shade faster than you can say "Aw, shit" is messing with Jo's head.
She used to be like Kat, all even-tempered and cautious with decisions, didn't have a sharp word for anyone.
"I'm going to kill you, you bastard, you won't get away with this!" she's shouting. "Let me go! Get your hands off me, you son of a bitch!"
Ryodan chains her next to me. She struggles but it's like watching a fly batting at a window, trying to get outside. You know it's never going to work.
I give her a look. "Got any more bright ideas, Jo? Try bringing a few babies for him to torture next time."
She gives her chains a violent jerk. We're bolted to a stone wall.
"Good luck with that." If I couldn't break them with my superstrength, she's got a snowball's chance in hell. I think he has the metal spelled. I think he has everything spelled. I want to know where he learns his spells so I can sign up for a crash course. If I've been down here three days, I should be, well, messier than I am. How did he keep me unconscious for three days? Put me in some kind of suspended animation? I seriously have to pee.
"I was trying to help," she says.
"You should have just taken a baseball bat to my head. Put me out of my misery." I could have held out down here forever until she went and served herself up to Ryodan as a weapon.
Ryodan stands in front of us, legs apart, arms folded over his chest. He's a big dude. I wonder if Jo knows he has fangs. I wonder what he is. I wonder why she's staring at him like that. She hates him.
I trash my pointless wonderings and cut to the chase. Procrastinating is number three on my Stupid List. You still end up exactly where you didn't want to be, doing exactly what you didn't want to do, with the only difference being that you lost all that time in between, during which you could have been doing something fun. Even worse, you probably stayed in a stressed-out, crappy mood the whole time you were avoiding it. If you know something is inevitable, do it and get it over with. Move on. Life is short.
If he tortures Jo, I'll cave.
I know it.
He knows it.
Ergo, torturing her is a great big fat waste of time. His. Mine. Hers.
"What do you want from me, Ryodan?" I say.
"It's decision time, Dani."
"Deaf much? I said, what do you want from me?"
"You owe me compensation."
"Dude, the bush is ready. Why you still beating around it?"
"I've lived a long time, kid, and I've never heard anyone mutilate the English language quite like you."
"How long is that?" Jo says.
I yawn, big and dramatic. "Still beating. And me all bush-like." I give an all-body, bushy bristle.
His eyes narrow on me like he's thinking. Like maybe he hasn't decided exactly what he wants from me yet. That worries me. It should be real simple: he wants me to work for him. I know he's not as bright as I am, so I help him out.
"I'll look into your little ice mystery, Ryodan. I'll put it at the top of my priority list. Unchain us already."
"It's not that simple anymore. You complicated the fuck out of things when you decided to defy me publicly. Nobody does that and lives."
"Breathing here," I say.
"Do you have to keep saying 'fuck' around her? She's barely thirteen," Jo says.
"Fourteen," I correct irritably.
"My men want you dead. They're pushing for a dramatic execution, in the club. They say it's the only way to appease the patrons of Chester's."
"I always wanted to go out in a big way," I say. "Maybe we could do some fireworks, huh? I think there are some left up at that old petrol station on O'Clare."
"Nobody's executing anyone," Jo says. "She's a child."
"I'm not a fecking child. I don't think I was even born that way."
"I told them I believe you can be useful," Ryodan says. "That I can control you."
I bristle and rattle my chains. Nobody controls me. Not anymore.
"They say you'll never answer to anyone. Not even Barrons is on my side."
No doubt because TP was telling Barrons to tell Ryodan to kill me. Or let her do it.
"It's eight against one," he says.
"It's eight against two," Jo says. "If you count her sister _sidhe-_ seers—and you'd better—it's eight against thousands."
"Your numbers have been severely diminished," Ryodan says.
"Worldwide, we're over twenty thousand."
"I didn't know that," I say to Jo. "Why didn't I know that?" To Ryodan, I say, "Dude, kill me or free me."
"If you kill her," Jo says, "you'll incur the wrath of every _sidhe_ -seer in the world. They'll hunt you. Dani's a legend among us. We won't lose her."
"If I decide to kill her," Ryodan says, "no one will ever know what happened to either of you."
I blink, mentally replaying what Jo said again and again, but I can't hear it enough. "Really? I'm a legend? Like, around the whole world they know of me? Say it again!" I preen. I had no idea. There might be a little swagger left in my body after all. I cock a jaunty hip.
"Let her go," Jo says to Ryodan, "and I'll stay in her place."
"The feck you'll stay!" I explode.
"You're offering to stay here. Chained up. With me. In exchange for her." A smile plays at his lips.
"As long as you have me as a hostage, she'll behave."
"The feck you'll stay!" I say again since nobody reacted like they were supposed to, like, by obeying me. Or paying any attention to me at all.
"I haven't forgotten what you did to my cell phone, _sidhe-_ seer," Ryodan says.
"You were taking pictures on our property. It's private," Jo says.
"You're on my property. It's private."
"I'm not taking pictures. I came to take back something that's ours. Something you had no right to take."
"I'm not a something. Or a child," I say.
"She had no right to kill the patrons of my club. She'd been warned. Repeatedly."
"And you know how well she listens. You shouldn't have brought her into your club and left her alone with a sword. Could you possibly be that stupid?"
"Dudes, quit talking about me like I'm not here!"
" _Sidhe_ -seer, tread lightly," he says to Jo, and his voice goes real soft. Soft from Ryodan is never good.
"Let me stay in her place. She's just a kid."
"I'm _not_ a kid! And she's not fecking staying here. Nobody's staying here! Except maybe me!"
"You do understand what it would mean," he says to Jo, like I'm not even having a violent, noisy fight with a wall and four chains. "If she makes a single misstep, you're dead."
I feel the blood drain from my face. I _always_ misstep. Misstep is my middle name, right after Mega. I can't _not_ misstep. I have feet.
"I understand."
"She doesn't mean it!" I shout. "She doesn't even know what she's talking about! She doesn't have any clue what you dudes are really like. Besides, I don't really even care about her at all. You can kill her. So, you may as well let her go."
"Shut up, Dani," Jo says.
"You'll have to sign an employment application," Ryodan tells Jo.
"Don't sign it, Jo! He's got some kind of spell on it."
"Am I being held hostage or applying for a job?" Jo says.
"I'm short a few waitresses. Some of them were—" Ryodan gives me a look. "—collateral damage the other day."
"I didn't kill any humans."
"Two of them had enough Unseelie in them that apparently you couldn't tell the difference," Ryodan says.
I killed humans? How much Unseelie had they eaten?
"You want me to be a _waitress_?" Jo says, horrified, like it's a fate worse than death. "I tried to wait tables in high school. I can't. I drop plates. I spill drinks. I'm a researcher. A linguist. I live in my head. I don't wait tables."
"Conveniently, I have two applications handy." Ryodan withdraws a folded packet of papers from his pocket.
"Why two? I ain't waiting tables," I say belligerently.
"I have to serve _Fae_? As in take orders and _fill_ them? And bring things to their tables?" Jo can't seem to wrap her brain around it. Like she'd rather stay chained to the wall than wait tables.
"And my men. Occasionally, I imagine, even me. With a smile." He looks her up and down, slo-mo. "You'll look good in the uniform. Do we have a deal." In typical Ryodan fashion, his voice doesn't rise at the end of the question. He knows they have a deal. He can read Jo like a book with see-through covers.
My chains rattle as I test them with everything I've got. He is _not_ putting Jo to work in the kiddie subclub. She's got the kind of face that's so delicate and pretty that she can wear really short hair like she does and look totally hot. Even those stupid glasses she wears when she reads just make her look good because they make her bones seem even more dainty. She has something ethereal. She is _not_ wearing a short plaid skirt, tight white blouse, socks, and baby doll heels. She will _not_ be waiting on him and his men! Chester's will swallow her up like a tasty morsel and spit out blood and gristle.
"No, Jo," I say flatly. "Don't you dare."
"We have a deal," Jo says.
He unchains Jo, hands her the "application" and a pen.
She flattens it out on the wall and signs it without even reading it.
He folds it up and hands it back to her. "Take the elevator back up the way you came. Lor is waiting for you there. He'll get you a uniform. You start tonight. You have a single priority—make my patrons happy."
"Lor is waiting for me," Jo says. She pushes a hand through her short dark hair and gives him a look that kind of surprises me, it's got so much balls in it. "I thought you said your men expected you to kill us."
"If you don't hand him the signed application, he will. I suggest you make sure he sees it the instant you get off the elevator."
"What about Dani?"
"She'll be up soon."
"She comes with me now," Jo says.
"Never. Tell. Me. What. To. Do." Ryodan's talking soft again, and I don't know about Jo but it gives me a shiver when he speaks like that.
"Get out of here, you stupid fecking _sidhe_ -sheep!" I say. "I'll be fine. I'd have been finer if you'd never showed up!" He owns her now. He's got some kind of spell on her. It pisses me off so bad I'm shaking.
After Jo leaves, Ryodan glides toward me in that weird fluid way he has. He didn't move that way in front of Jo. He walked all slow-mo when she was here.
I see the glint of a silver knife in his hand.
"Dude, no need to cut me. I'll sign the fecking application. Just give me a pen." I have to get out of here. I have to save Jo. She put herself on the line for me. I can't stand it.
"Kid, when will you learn."
"You'd be amazed the things I know."
"You might be able to thrash your way out of a spiderweb, but thrashing in quicksand doesn't work. The harder you fight, the more ground you lose. Struggling merely expedites your inevitable defeat."
"Never been defeated. Never will be."
"Rowena was a spiderweb." He touches my cheek with the hand holding the knife. The silver glints an inch from my eye. "Do you know what I am."
"A great big pain in my ass."
"Quicksand. And you're dancing on it."
"Dude, what's with the knife?"
"I'm not interested in ink anymore. You're going to sign my contract in blood."
"Thought you said it was an application," I say pissily.
"It is, Dani. To a very exclusive club. What's Mine."
"Ain't nobody's."
"Sign."
"You can't make me."
"Or Jo dies. Slowly and painfully."
"Dude, why are you still talking? Unchain me and give me the fecking contract already."
There's a guillotine above my neck. I hear it swishing as it slices through the air. There's a name carved into the shiny blade: JO. I see it in my periphery with every step I take. It's going to make me nuts.
After I sign his fecking contract—I got a paper towel in my fist because my palm's still bleeding where he cut me—he lets me go. Just like that. Unchains my other arm and legs, offers to heal me, to which I say a great big kiss-my-booty, then escorts me to the elevator and tells me to go wherever my current version of home is.
I expect him to tell me I have to move into Chester's so he can watch my every move, like Barrons did with M—TP.
I expect him to go all control-freak on me.
I don't expect him to give me my sword back and send me on my way with a casual reminder to show up for "work" tomorrow at eight P.M. He says there's something else he wants me to see.
I hate this.
He's not reeling off one thousand and one Ryodan commandments like I thought he would.
He's giving me all kinds of rope to hang myself with. I tie knots with rope. And I move really fast. It's inevitable I'll get tangled up in all that rope somehow, with a loop or two around my neck.
How am I going to get Jo out of this?
Four of his big scarred dudes are waiting for me when I get off the elevator. I glance warily around for Barrons and TP as I wave my contract big and noisy at Ryodan's men so they don't give me any grief before they take it from me to put it wherever it is Ryodan plans to keep it and I'm going to have to eventually steal it back from. I'm out of protein bars and not in the mood for a pissing contest. Fortunately, TP is nowhere to be seen.
I hit the bathroom under heavy guard. What do they think I'll do? Blow the place up? I can't. I don't have my backpack. No MacHalo either. They didn't bring it when they nabbed me at Dancer's. I'd look out a window but there aren't any in the club. My bones tell me it's night. I don't take chances with Shades. I refuse to die so stupidly. "I need flashlights," I say, blowing out of the bathroom.
One of the dudes grunts and walks away. The rest of them escort me through the subclubs. I get stared at by every Fae we pass. There's murder in their eyes.
Something weird happens to me on the way out.
Freeze-framing feels like picking myself up mentally and shifting sideways into a different way of being, and I like it.
Now, as I walk out and see all the pissed-off faces, human and Fae, a completely different part of me gets picked up and shifted sideways without me even trying—in fact, I'm pretty sure I'm resisting—and I don't like it one bit, because all the sudden I'm seeing my world with what feels like totally different eyeballs.
I don't like these eyeballs. They see things wrong.
The Fae hate me. A lot of the humans do, too.
Ryodan's men want me dead and I have no idea why he's keeping me alive.
TP—oh, feck it—Mac, the best friend I ever had, _Mac_ —who made me a birthday cake and hung with me and treated me cool, and sold a piece of her soul to the Gray Woman to save me, hates me, too. She wants to kill me because I killed her sister on Rowena's orders before I ever even knew Mac existed.
Jo's life dangles on a thread held by my completely unreliable hands.
And I have a thought that I've never had in my entire fourteen years of life (and I've had a lot of thoughts!), and it's a little muffled (probably because I'd rather not hear it) and it goes something like this:
Geez, Dani, what the feck have you done?
I've always been a speedboat blasting across the whitecaps, thriving on sensation, wind in my hair, salt spray on my face, having the time of my life. Never looking back. Never seeing what happens around or behind me.
These new eyeballs see my wake. They see what I leave behind when I've passed.
Boats capsized. People flailing in the waves.
People I care about. I'm not talking about Dublin, my city that I always keep cool and impersonal with no real face. These people have faces.
We pass Jo. She's already dressed and at her new post, paired with another waitress, being trained. She _does_ look good in the uniform. She gives me a look as I pass, part exasperation, part plea to behave. Her trainer stares daggers at me. I wonder if the waitresses I killed were her friends.
"They shouldn't have eaten so much Unseelie," I mutter in my defense.
I try to shift back to the way I was before I got off the elevator, back to Dani "the Mega" who doesn't give a crap.
Nothing happens.
I try it again.
Still feeling the breeze from that guillotine.
One of Ryodan's dudes, Lor, hands me a flashlight. "Gee," I say, "thanks. A whole flashlight against a city of Shades."
"They moved on. Mostly."
I roll my eyes. " 'Mostly' might be okay with you 'cause, like, they don't eat whatever you dudes are. Why is that?"
Lor doesn't answer me, but I didn't expect him to.
The second we reach the door, I freeze-frame.
I can outrun anything.
Even myself.
# EIGHT
# _"And I'm hungry like the wolf"_
I click on a flashlight and head for the nearest store I know of that still has Snickers on the shelves so I can replenish my supplies. I have a bottomless stomach and it hurts from hunger. That's a feeling I take pains to avoid. Especially when my head's still throbbing so bad. I'd put ice on it, but if I've been out for three days, it's too late. Ice only works if you use it right away. I root through my hair, find the swollen, bruised patch at my nape that's causing so much pain, and sigh, wondering what I hit and when. Some folks think since I'm always banged up I'm a glutton for pain. I'm not. It's just the way my life is.
Like I thought, it's night, so the streets are pretty much deserted. Folks do their "shopping" during the day. Those that do hunt at night, do just that—hunt. They come out in packs, armed to the gills, and go after any Unseelie they can find.
A lot of the night-hunters have a death wish. They don't know how to live in the world the way it is now, so they take crazy risks. I end up bailing out vigilantes left and right. Sometimes they run into Jayne, and before anybody can say, "Don't shoot, we're human," there's casualties. Everybody's got jumpy trigger fingers.
Things sure have changed since the walls fell last October. Seven months ago the streets were easy. Hit the night, kill some Fae, then kill some more. The Unseelie were simple to take by surprise because they had such a low opinion of humans. They didn't see us as a serious threat.
They do now.
They're on guard, more dangerous, harder to trap, and impossible to kill unless you're me or Mac or a Shade. Shades are cannibals. Life is life. They don't discriminate. We have humans fighting Fae, humans fighting humans, Fae fighting each other, and all of us trying to get rid of the Shades.
I slow to a Joe-walk, running out of steam. I need food fast. I already ate everything I had stuffed in my pockets. Three days of starvation does a number on me. Swinging my sword around my wrist (it took me months to perfect that move—and it is _smooooth_!), I duck into a convenience store with broken-out windows, shelves spilled sideways, cash register open and overturned. I can't see why anybody would bother stealing money. It doesn't get you anything. People's eyes are finally open, money's as worthless as it always really was. Used to amaze me when I was little how everybody passed around pieces of paper that they all agreed to pretend meant the same thing when everybody knew it didn't mean anything. It was the first adult conspiracy I became aware of. Made me think maybe no adults should ever be the boss of me. I'm the smartest person I know. Except maybe for Dancer. Not bragging. It's a real pain in the ass a lot of the time.
"Buying" nowadays operates on something solid and real: the barter system. Ryodan has the bartenders and waitresses at Chester's coached to take certain items he either wants for himself or can turn around for something else he wants. If you have a big item he's interested in, he'll give you a line of credit. I hear he gets favors from the Fae in exchange for making them a place where they can prey on humans. Though I hate Jo working at Chester's, in a way I'm glad because I'll get more inside scoop now. Figure out what motivates Ryodan, what his weaknesses are. Dude's got to have some chink in his armor. Everybody's got their kryptonite.
I circle a pile of clothes and husks (fecking Shades, I hate them!) and head for my candy rack.
It's empty.
Not a single Snickers.
Not a single anything for that matter.
I head down the cracker aisle.
The shelves are bare.
My stomach growls. Pissily. My knees aren't wobbling yet but they're close.
I turn my flashlight to wide beam and sweep it around the store.
The place has been cleaned out.
I'd blow out a melodramatic sigh but it's an expenditure of energy I suddenly can't afford. I'm no longer swinging my sword or bouncing from foot to foot the way I do a lot. I'm not moving a hair I don't have to. My life just got harder. When you're a supercar like me, you either need a huge gas tank, which I don't have at five feet two and three-quarters first thing in the morning, or you need to live in a city with a lot of gas stations.
My gas stations are drying up.
It's okay. I saw this coming. Dancer did, too. I squirreled away stashes of food, water, and medical supplies, in lots of hidey-holes around Dublin months ago. Me and Dancer have been building on those reserves in our spare time over the past few weeks. He doesn't know where all my hideouts are, and I don't know where he keeps all his stuff. That way if somebody tries to torture one of us to tell, we can't totally wipe each other out. I tried to tell the _sidhe_ -sheep to do it, but they thought I was crazy. They said that with more than half the population gone there was plenty of stuff in the stores to last a good long while. I said somebody was going to try to monopolize food distribution. Dude, barter system—food and water are the premium. They said everyone was too busy trying to survive. I said that wouldn't last long and didn't they read _A Canticle for Leibowitz_ , see how things trend? They said what did _A Canticle for Leibowitz_ have to do with food? And I said should I start calling you _sidhe-_ simpletons instead of _sidhe_ -sheep? Do I have to spell out everything? Can't we metaphor some things?
_I hate always being right_ , I mutter in my head. Talking takes breath and breathing takes gas I don't have.
I Joe-walk out of the store and nearly have a fecking heart attack when I see the Unseelie prince standing there, half in the shadows. The half-out part of him is splashed with moonlight, but the moon doesn't glow the same way it used to before the Fae came. It's rarely the same color from night to night. Tonight it has a silvery purple luminosity, making half of him a black silhouette, the other half lavender-metallic. He's tattooed and beautiful and eerie and exotic, and gets my heart thumping in a way that has nothing to do with fear.
My sword flashes up. My blade is long and alabaster. I lock my elbow so my arm doesn't wobble.
"Easy, lass."
"Fecking _stop_ sneaking _up_ on me like that!" How can I not hear him? Him and Ryodan can both get the jump on me. It makes me crazy. I have superhearing. My hearing is so good that I can hear air displacement when other people move, for feck's sake. Nobody sneaks up on me. Both of them managed to do it that night on the water tower, and Christian just did it again. Got within five feet of me without me even knowing it. "Sword. Lower."
"Why should I do that?" He's turning erotic, like the other UPs. My used-to-be best friend Mac calls them death-by-sex Fae because they can kill with sex. And that's the best-case scenario. Worst case? They turn you Pri-ya like they did Mac. They leave you alive, totally addicted to sex, insatiable and out of your mind. The other UPs corralled me once, kept me between them, and did things to me I don't like to think about. I don't want sex to be that way. Like you're some kind of helpless animal. I've had helpless animal up to my eyebrows already in my life. What Christian is throwing off isn't a tenth of what the other UPs have, but it's bad.
"I'll never hurt you, lass."
"Says the Unseelie prince." But I lower my sword, prop it against my leg. I wasn't sure how much longer I was going to be able to hold it up anyway.
The muscles in his face ripple, like they're competing to shape an expression, and rage is looking like the victor, and I get the feeling calling him an Unseelie prince might just have been a tiny error of judgment on my part. Been making a few of those lately.
"Say my name, lass."
I cover my ears and look at him like what the feck? His voice just came out as big as a house.
"Say my fucking name!" Thunder rolls in the sky. I wrap my arms around my head to mute his voice. Times like this, I hate my superhearing. I look up. There's no storm moving in. It's him. Influencing the weather, just like Fae royalty. I look back down. A veneer of ice coats the sidewalk around him, a shimmer of crystals dusts his black boots and frosts halfway up his jeans.
"Christian," I say.
He inhales sharp, like something hurts in him somewhere just from me saying his name, and closes his eyes. His face ripples, goes smooth like Silly Putty just out of the egg then ripples again. I wonder if I touched it, I could mold it into shape, maybe stamp some funnies from the comic section of the newspaper on it. Cracking myself up again!
"Say it again, lass."
If it keeps him from turning all UP on me, fine. "Christian. Christian. Christian."
He smiles faintly. I think. Feck if I can figure out what's going on with his face. No more than I can figure out how he keeps sneaking up—
"Holy flour chunks!" It dawns on me. "You can _sift_! You really are turning total UP. Like with all the superpowers. Dude! What else are you getting?"
If it was a smile, it just disappeared. He doesn't look as happy as I'd be if I was getting all that juice. I bet _his_ fuel tank doesn't run out of gas. I'm so jealous I could spit. But that, too, would require energy.
He moves forward, steps from the shadows, and I see he's carrying a box under his arm.
"I'm going to kill Ryodan," he says.
I unwrap my arms from around my head. We're doing normal conversational tones again. I tuck the sword beneath my coat.
"Good luck with that. You figure out how to, you let me know, okay?"
"Here, take this." He shoves the box at me.
I fumble for it, clumsy from hunger. It's slippery with a coating of ice. I catch it as it hits the ground. Sloppy! I recognize the color and shape now that it's in my hands, and light up like a Christmas tree. "Christian!" I beam. I'll say his name however many times he wants. I'll crow it from the top of water towers. What the feck, I'll compose a jaunty ditty for him and sing it as I whiz around Dublin!
He just handed me a whole box of Snickers! I rip open a wrapper, break the half-frozen bar in half and cram it in my mouth sideways.
When I toss my hair out of my face and look up to thank him around a mouthful, he's gone.
Three candy bars later what just happened sinks in.
I sit on the curb, stow the candy bars away in my pockets and pack, and say, "Aw, bugger."
Christian knew how bad I needed food. He watches me. I wonder why. I wonder how often. I wonder if he's out there right now, looking at me from somewhere and I don't even know it. Dude, I got an Unseelie prince spying on me. Great.
Tank full again, I swing by Dublin Castle. Three days was a long time to be out of commission. I got a job to do. A beat to walk. A superhero's work is never done. Between patrolling my city, printing and distributing the _Daily_ , slaying Unseelie, keeping an eye on Jo and the other _sidhe_ -seers—and now working for Ryodan all night every night—there aren't going to be enough hours in the day!
"Where the bloody hell have you been?" Inspector Jayne says the instant he sees me. "I've got Unseelie spilling out of every cage. We agreed that you would come by three times a week and slay them with the sword—and that's barely enough as it is. I haven't seen you in five days! Five bloody days! If you won't take your responsibilities seriously, my men will relieve you of that weapon."
He stares at the pucker of leather, where my sword's tucked beneath a long coat that brushes the laces of my high-top sneakers. It's May and almost too warm to be wearing my favorite black leather. Soon I'm going to have to sling the sword over my back and deal with everybody staring at it, coveting it. At least now lots of folks don't know I have it. Then again my rep _is_ starting to precede me. Jo said I was a legend!
"You just try that, dude." I swagger onto the training green, between him and his men. A few dozen of them are in full armor, sweating up a stink-storm. Supersmell is a pain in the butt sometimes. He's been working them hard. I wonder what's with that. It's night. He usually has his men out hunting at night, patrolling, keeping the streets safe.
We glare at each other.
He softens. He always does. He has a hard time looking at me and staying mad. He sees his own kids in my face. Jayne's got a supersoft spot for children. He and his wife have been taking in orphans left and right. I don't know how he feeds them all. But Jayne's no dummy. I suspect he's got stores stashed away, too. Till tonight, it seemed like most of us were playing by the same rules. Take a lot—but leave some.
No rules anymore. Somebody's cleaning the shelves. That's just not civilized.
"Damn it all to hell, Dani, I was worried about you!"
"Get over it, Jayne. I take care of myself just fine. Always have."
He gets that look in his eyes that always makes me uncomfortable, like he's about to put a fatherly arm around me or wipe a smudge of blood off my cheek. I shudder. My sword hand's itching and I'm all about scratching it. "I'm here now. Quit wasting time. Which Unseelie you want dead first?"
"Do you know why we're not out hunting tonight?"
I don't like being cued to speak so I just look at him.
"No room in the cages. Go make me all of it. And don't leave until you have."
He glances again at the pucker of sword under my coat then does something he does a lot. He looks at his men, and looks back at me again, all cool and speculative-like. He's not seeing a kid when he does it. He's seeing an obstacle.
I know Jayne real well. He doesn't even know he does it.
He's wondering if they could take my sword. Wondering if he'd let his men kill me to get it. If I told him that, he'd deny it to the end of days. He thinks he really cares about me, and on a level he does. He thinks he'd like to take me home to his wife and make me part of their family, give me the kind of life he's pretty sure I didn't have.
But there's four feet of a shiny metal problem between us, and it's four feet of immense power. And it changes everything. I'm not a kid. I'm what stands between him and something he wants for all the right reasons. And he isn't so sure he wouldn't do something very wrong for all the right reasons.
My sword and Mac's spear are the only two weapons that can kill Fae. That makes them hands-down the hottest Big Ticket items in—not just Dublin—but the world. A part of Jayne is like Barrons. He wants to kill Fae—and I have the weapon he needs to do it. He can't help himself. He's a leader. And a good one. Every time he sees me, he will instinctively assess whether he thinks he can take it from me. And one day he might make a move.
I don't hold it against him.
I'd do the same.
I see when he decides it's not a risk worth taking because he's still not sure I won't kill some of his men, maybe even him. I keep those doubts in his mind. The subconscious part where all this stuff takes place.
He says something nice to me, but I don't absorb it. Jayne's a good man, good as they come. It doesn't make him any less dangerous. Some folks think I'm a little psychic along with my other superpowers. I'm not. I just see the ways folks telegraph. Pick up on tiny clues other folks don't, like the way their muscles tense in their fingers when they look at my sword like they're imagining how it would feel to hold it, or how their gaze darts to the side when they say they're glad it's _my_ responsibility not theirs. Funny thing to me is how their conscious and subconscious seem to be so split, like they aren't talking to each other at all. Like competing feelings can't possibly coexist inside you. Dude, they do all the time. I'm an emotional Ping-Pong ball between paddles: one day I can't wait to have sex, the next I think semen's the grossest thing in the world. Monday I'm crazy about Dancer, Tuesday I hate him for mattering to me. I just go with it, focus on whichever feeling I have most often and try to keep my mouth shut when it's the other. But most folks got Id and Ego living on different floors in their head's house, in different rooms, and they've locked all the doors between them, and nailed sheets of plywood over that, because they think they're, like, sworn enemies that can't hang together.
Ro thought the whole subconscious/conscious issue had something to do with why I am the way I am. She said I have the neurological condition synesthesia out the ass, with all kinds of cross regions of my brain talking to each other. Old witch was always psychoanalyzing me (as in she was the psycho and I was being analyzed). She said my Id and Ego are best buds, they don't just live on the same floor, they share a bed.
I'm cool with that. Frees up space for other stuff.
I take off, tune out, and do what I do best.
Kill.
# NINE
# _And it all goes boom, chicka boom,
boom-boom, chicka boom_
"What is this place?" I ask Ryodan.
"You got lots of places around the city, kid."
I don't say "Yes." Lately, everybody seems to know everything about me anyway. And he doesn't say, "Well, I do, too." When Ryodan wastes words, he does it in the worst possible way. He gets all philosophical. Yawn the feck out of me. There's observation of fact that keeps you alive like understanding Jayne, and there's philosophizing. Way different things. The former is my gig.
We're standing on a concrete loading dock, outside commercial doors at an industrial warehouse on the north side of Dublin. Ryodan drove us here in a military Humvee. It's parked behind us, barely visible in the night, black on black, wheels and everything, with black windows. It's something I would've driven. If I'd found one. But I didn't. It's pure badass. And I thought Barrons's cars were cool.
I begin my investigation. There are no lights on around the building. "Dude, got Shade protection?"
"Don't need it. Nothing alive inside."
"What about the folks that come and go?"
"Only during daylight."
"Dude. Night. I'm here."
He looks at me, looks at my head, and his lips twitch like he's trying not to bust out laughing. "You don't need that... whatever the fuck it is."
"Ain't dying by Shade. It's a MacHalo." First thing I did this morning was swing by Dancer's and grab my stuff.
The MacHalo is a brilliant invention. In Dublin alone it's saved thousands of lives. It's named after my used-to-be best friend Mac, the person who invented the bike helmet covered with LED lights, front, sides, rear. I added a few brackets to mine for better coverage in fast-mo. (Though I've always wondered if I could fast-mo through a Shade even without it.) It's the ultimate in Shade protection. I heard they're going gangbusters around the world. Everybody in Dublin's got one. For a while there I was making and delivering them to survivors every day. Some folks say the Shades have left Dublin. Moved on for greener pastures. But Shades are sneaky and it only takes one to kill you instantly. I'm not taking any chances.
"What does this place have in common with your club?" I say.
He gives me a look that says, "Dude, if I knew that do you think I'd have enlisted your puny help?"
I snicker.
"Something funny here."
"You. All prickly and pissed 'cause there's something you don't know. Got to call on the megaservices of the Mega."
"Ever occur to you I'm using you for reasons your inferior human brain can't begin to understand."
It's another of his questions that doesn't sound like a question. It's such an irritating tactic, I wish I'd thought of it myself. Now if I start doing it, I'll look like a copycat. Of course it had occurred to me that he had ulterior motives. Everyone does. Now I'm the one feeling all prickly and pissy. I go into observation mode, ruffle my feathers back down into a duck-coat so I'm more likely to quack up than get pissy. Humor is a girl's best friend. The world's a funny place.
I estimate the double doors of the warehouse at thirty feet, with an entrance nearly twice as wide if you slide back all four panels of the doors. The corrugated metal is throwing off such intense cold that my breath freezes a few puffs from my face and hangs in the air like small frosty clouds. When I punch one, it tinkles to the ground in a dusting of ice and my mind attaches a pattern to a pattern: I see the dusting of ice up Christian's jeans. I consider it for a moment then decide no way. Fae royalty can minorly affect the weather around them. Key word there is "minorly." This is major stuff. And Christian's not even full-blooded Fae.
The doors are coated with clear ice. I reach for my sword.
Ryodan's front is against my back, and his hand is on my hand on the sword hilt before I even process that he moved. I go totally still, don't even breathe. He's touching me. I don't think when he gets this close to me. I just turn on static in my head real loud and focus on trying to get away as fast as possible. Riding in a car with him sucked. Closed compartment. Electrified sardine can. Rolling down the windows hadn't helped a bit. This is a gazillion times worse.
"Dude." I pump up the volume on my static station.
"What are you doing, Dani?"
His face feels real close to my neck. If he bites me again, I'm going to kick his ass. "I was thinking about poking the ice, seeing how thick it is."
"Two and one-sixteenth inches."
"Get off me."
"Get off your sword. Or I won't continue to let you keep it."
Fecker can take my sword away like Jayne never could. Like only the UPs can. One more reason I can't stand Ryodan. "Can't get off my sword till you get your hand off mine. Pressure much?" I say testily.
We both sort of let go at the same time. I glare at him, or where I think he is, but he's not there. I find him twenty feet away, near a small, normal-size door. He opens it. His face instantly frosts. "Ready?" he says.
"You don't move that way in front of Jo."
"What I do with Jo is none of your business."
"You better not be doing nothing with Jo. I'm staying in line like a good little soldier." And fecking-A does it ever chafe. Report to work at eight P.M. Gah. Report. Like I don't have plans of my own. Like I didn't spend hours hunting for Dancer and I'm not two _Dani Dailies_ behind and haven't spent most of my fecking day working on one, after whizzing out to the abbey to make sure Jo's okay. She had some seriously sick scoop for me about the new, segmented Unseelie, but other than that she hadn't wanted to talk much. I think she's pretty upset with me. Nothing new there. If there weren't any _sidhe_ -sheep upset with me, I wouldn't know who I was, or if the Earth was still orbiting the sun. "I'm behaving. She's safe. You just leave her alone."
He smiles faintly. "Or what, kid?"
"You know something, dude, if you don't put a question mark at the end of your questions, I'm not answering them anymore. It's rude."
He laughs. I hate it when he laughs. It tries to put me right back on the porno level of Chester's and that just grosses me out, so I do the static-thing in my head again.
I freeze-frame past him so fast his hair blows straight up. I make sure to go through a pile of dust, and give it a little extra twist with my heel as I whiz by so it shoots straight up his nose (a trick I perfected at the abbey!). He sneezes. Just like a real person. I'm half surprised to find he actually breathes.
The cold slams into me like a brick wall and for a second I can't inhale.
Then I feel him at my back, an inch from my figurative rear tire like he's drafting off my freeze-frame. It sets my teeth on edge. Makes my temper hot and breathing is easy again.
Like the first scenario he showed me, a frozen hush fills the space like those mornings in fresh, new-fallen snow when no one else is awake and the world is stiller than you ever thought it could be until you take that first step that squeaks in the drift. I always wanted to have a wicked snowball fight with somebody on mornings like those but nobody else has ever been able to keep up. Lobbing snowballs at folks is like picking tin cans off a fence with a BB gun.
I flash through the warehouse, checking it all out, fascinated in spite of being ordered here and bossed around. I love a good puzzle. What's freezing these places and why?
A few dozen Unseelie are iced in the entry bay.
Ryodan has lower-caste grunts working for him. There are lots of Rhino-boys iced in mid-action. Like the subclub in Chester's, the place is killingly cold. It makes my heart feel dull and tight. I don't stop moving, won't stop moving for anything.
Rhino-boys are frozen loading and unloading pallets and crates, gray skin coated white, shellacked by a clear layer of ice. Whatever happened to them happened fast. They had no warning. Their frosted expressions are completely normal.
Well... as normal as Unseelie ever look. I think.
I whiz around two beefy ones, studying their bumpy rhino faces, gashed mouths bared on tusks, analyzing that thought.
It occurs to me that maybe their expressions aren't normal. I'm basing my assumptions on what I know of humans, of how our faces react. Christian is proof that I can't do that. I can't even figure out when Christian is smiling.
Logic demands I eliminate my assumption that the Rhino-boys had no warning. Can a Rhino-boy look terrified? I don't know. Perhaps they show fear by something so small and weirdly Fae as a tiny rainbow-hued glint in their beady little eyes, and the white frost is concealing it. I've never noticed what their faces look like when I kill them. I'm usually too busy looking at the next one I plan to stab. I'm suddenly looking forward to finding one tonight and performing a test. Any excuse to kill an Unseelie is an awesome one.
What would do something like this?
And why?
It has to be a Fae because I just can't see a human managing to build a freeze-ray gun that works on this scale only to go vigilante.
Then again.... I can't eliminate that possibility either.
So far, both places I've seen iced are exactly the kind of places I would ice myself. If I had such a wicked cool weapon.
Most folks wouldn't believe that someone who can move like me, fight and hear like me, could exist. Ergo, I can't rule out the possibility that someone else might be so smart they figured out how to build a massive freeze-ray gun that's capable of reducing the temperature of places to the frigidity of objects in space. Given enough time, I think Dancer could manage it. He's that smart!
Bugger. I have facts and no connections. I can deduce nothing. Yet.
Suddenly I see past the frozen figures.
The warehouse is packed full of boxes, crates, and pallets, piled everywhere. There's a pie of iced electronic stuff that looks like audio equipment of some kind. I guess maybe for the club. Crates are stacked to the ceiling, and more stuff was being brought in when whatever happened did.
I make one crystal clear deduction: Ryodan's the dude emptying the stores! Preying on humans just like the Unseelie. Stealing our ability to survive so he can sell it back to us at whatever cost he decides to demand.
It's all iced. Every bit of it.
I wonder if any of the edible stuff can be thawed and saved. People are going to die because he's such a greedy pig.
I'm so pissed that I smash open a crate as I go whizzing by. "Oops," I say, all innocent and accident-like. Wood splinters, two-by-fours, go flying in all directions.
Automatic weapons explode from the wreckage and skid across the iced floor, where they smash into frozen Unseelie who shatter like little glass goblins.
Okay, so that crate had guns in it. It just means I kicked open the wrong crate. I'm so sure he's the prick stockpiling the food that I kick another, not even pretending it was by accident this time. More guns.
I go on a smashing rampage. Each time I smash a box or crate open that holds ammo or guns, I get madder. Figures he'd hide the food from me before he brought me here. I'm about to kick open my fifth crate when Ryodan suddenly has me hanging in midair by the collar of my coat, manhandles me into potato-sack-girl over his shoulder again, superspeeds me out the door, slams me into a telephone pole and says, "What the fuck is wrong with you?" at the precise moment the whole building blows up.
"Dude, are you arming these places to blow?" I say on the way back to Chester's. "Is this another of your stupid tests? I have to solve your little mystery in the whopping three seconds I get to study it before the scene gets blown to smithereens?" The whole building had exploded outward, for a city block. We'd barely freeze-framed from the shrapnel zone in time.
"I lost a great deal of personal property in both explosions. I sacrifice nothing that is mine from which I might profit."
"Which translates into as long as I'm useful, since you think I'm yours, I'm not going to get the—" I drag a finger across my neck.
"Kid, you might just annoy me into killing you."
"Right back at you, boss."
He smiles and I feel myself starting to smile back and it pisses me off so I look out the window and get real intent on what scenery I can make out in the pinkish moonlight, which isn't much because the Shades took everything worth looking at out here. Got three hidey-holes down this way and a big stash. Didn't know Ryodan was holing up here, too. I'll vacate this district as soon as I get time to relocate.
"Observations," he says.
"Four imperial Unseelie guards were the only commonality I was able to isolate endemic to both scenes." They'd been standing, armed, at the dock doors, overseeing the delivery.
He gives me a sidewise look. "Wow. That was, like, a whole sentence. With nouns and verbs and connective tissue. Endemic. Fancy word."
"Sloppy, dude. Should have omitted the connective tissue part."
"Nothing else."
I give him a look. I hate his statement-questions. I'm not answering them anymore.
He laughs. "Nothing _else_." His voice rises on _else_ about one one-hundredth of a note higher than the word "nothing," a concession only someone like me with superhearing would ever be able to pick up. Still, it's a concession. From Ryodan. Rarer than water in the desert.
"The ice was layered the same. Maybe hoar frost. Definitely hard rime. Clear ice on top of it all. The hard rime's weird. White ice comes from fog freezing. What's fog doing inside both these buildings?"
"How did the place blow?"
I think back. It happened so fast and we were outside, and he was blocking my view, and I was more focused on getting him off me than anything else. I hate to, but I admit, "I can draw no conclusions, circumstances being what they were."
He looks sidewise at me again.
"Talking like you, dude, thinking it might get all this stupid fecking stuff over with sooner. Communication is hard enough when everybody's trying."
"Isn't that the truth. Give me your hand."
"No."
"Now."
There's no way I'm giving him my hand.
He says something soft in a language I don't understand. My arm jerks up. I watch in horror as my hand passes to his side of the Humvee, palm up.
He drops a Snickers in it, murmurs something, and my hand is my own again. I wonder when, how, and why my fecking appetite became everyone else's business.
"Eat."
I think about throwing the candy bar back in his face or out the window. I refuse to let my fingers close around it.
But I sure could use it.
He brakes, comes to a stop in the middle of the road, turns toward me, grabs the collar of my coat, pulls me across the expanse between our seats and leans in. Locks eyes. We're maybe eight inches apart, and I think the only reason my nose ain't touching his is because one of the brackets on my MacHalo is just about touching his forehead. My butt's no longer touching the seat.
I've never seen such clear eyes as Ryodan's got. Most folks are crammed full of emotions, with lines around them like battle scars. I can tell by looking at grown-ups if they've spent their years laughing or crying or resenting the whole world. I hear moms say to their kids when they make faces, "Careful, your face will stick like that." And it really does. By middle age most folks wear whatever they felt the most in their lives smack on their kisser for all the world to see. Dude, so many of them should be embarrassed! It's why I laugh so much. If my face is going to stick, I'm going to like looking at it.
Looking at Ryodan is like staring the devil in the face. It's obvious what he's felt the most—nothing. Ruthless. Cold dude.
"I won't ever hurt you unless you make me, Dani."
"You being the one who gets to decide what constitutes the definition of 'make.' Big fat lot of wiggle room in there."
"I don't need wiggle room."
"Because you annihilate."
"Another of those fancy words."
"Dude. What did you just do to me?"
"Gave you what you needed but were too stubborn to take." He closes my fingers around the candy bar with his. I can't shake him off fast enough. "Eat, Dani."
He drops me back into my seat, puts the Humvee in gear again and takes off.
I munch the candy bar despite the sour taste in my mouth, thinking how I used to be invisible.
"Superheroes are never invisible," he says. "They're just deluded."
Turning my head toward the buildings flashing by, I screw up my face and stick out my tongue.
He laughs. "Sideview mirror, kid. And careful. Your face'll stick like that."
I head out into the streets with boxes of freshly printed dailies (I love the smell of new ink!) in a battered grocery cart the minute my time is my own again. I can run with a cart and slap my papers up on poles faster than I can do it on my crotch rocket. My bike's for pleasure, for pure downtime, when I got nothing else weighing down on me, like always saving the world. I don't get to ride it much.
Ryodan's reminder that I'm to report to work every single night at eight P.M. on the dot is still ringing in my ears, making me nuts. What the feck can he possibly have to torture me with every night? Is he icing these stupid scenarios himself just for an excuse to mess with me?
I head west and begin my usual route. It's a little after midnight. It shouldn't take me more than a couple hours, then I'll start hunting for Dancer again. I'm getting a little worried about him. Most times he goes somewhere else without telling me, he's only gone a few days. I don't know all his haunts any more than he knows all mine but I'll keep checking those I do.
I've got certain posts and poles and benches that folks frequent, like regular newspaper stands, waiting for my latest updates. Folks have probably been a little worried with my paper being late and all. I've got important info to share tonight.
I glance down at my rag, proud of it. The ink is crisp and clean, and it looks real professional.
The Dani Daily
May 21, 1 AWC
New Unseelie Caste!
Update your DDD Manual!
**BROUGHT TO YOU EXCLUSIVELY BY** **_TDD_** **YOUR ONLY SOURCE FOR THE LATEST NEWS IN & AROUND DUBLIN!**
Dudes, I discovered a brand new kind of Unseelie hanging at Chester's!
Calling this one Papa Roach, and I don't mean the band! Take notes: it's three to four feet tall, with a shiny brownish-purplish segmented body, six arms, two legs, and the smallest head you ever saw, like the size of a walnut, with little fish-egg eyes. It can break down into segments that are the size of roaches that crawl inside your clothes, and get under your skin—LITERALLY!
If you see this thing coming, run like heck because I haven't figured out a way to kill it yet. You want to carry a can of hair spray or fill a spray bottle with gas and _always_ have some matches on you (I got a blowtorch myself). That way if you get cornered, you can spray them and set them on fire. It doesn't kill them but it sure keeps them busy while you run.
I'll keep you posted, Dublin!
Dani out!
I don't tell them the worst part is what Jo told me this morning—that some of the waitresses at Chester's _encourage_ the bugs to get under their skin. I don't want to give them any ideas. This Unseelie has a specialty: it feeds on human fat. Presto—tiny waist! Hello bug—goodbye cellulite! Don't like those dimpled thighs? Bug up. The walls haven't been down long enough for folks to get dystopian-thin, and with the amped-up sexuality of so much Fae royalty walking around dangling the promise of potential immortality, the focus on fashion and beauty has never been more extreme.
Jo told me that a couple of the waitresses are real proud to have one. It's becoming a status symbol or something, like hair extensions or boob jobs. Jo said the waitresses claim they don't kill humans, they just eat their fat, and they can hardly feel them in their skin at all.
I think that's bull. I think they hitch a ride because they're getting more from humans than fat. I think they experience everything their "host" experiences: pleasure, pain, whatever. The Unseelie are bugging us and we let them. They invade our bodies and gather intel from the inside, then report back to Papa, who probably reports back to the Unseelie princes, the better to prey on us. What do these idiot waitresses think? That the bug will eventually return to its own body and leave them all pretty and thin, no harm no foul?
Dude, it's an Unseelie! There's always a catch.
I zip around the corner to my first pole, grocery cart rattling.
When I see one of my papers from last week still hanging up, gleaming pinkish-white in the rosy moonlight, it surprises me. Folks always take them down, and take them home, wherever that is. Darn few get left behind.
As I get closer, I realize it's not my paper.
What the feck? What's on my pole? Folks know to leave me notes at the General Post Office.
I slip into fast-mo, get nose-to-nose with it.
I'm so flabbergasted my jaw about hits the pavement.
**The Dublin Daily**
May 20, 1 AWC
**YOUR** **_ONLY_** **SOURCE FOR** **_CREDIBLE_** **NEWS IN** **AND AROUND NEW DUBLIN** **BROUGHT TO YOU BY** **_WECARE_** **WE BRING YOU ALL THE NEWS THAT MATTERS**.
**WE WILL HELP YOU SURVIVE!**
** _WECARE_**
"Gah, dudes! Plagiarize much?" I pluck the offending matter from my pole and almost drop the thing, my eyeballs are so freaked out. " _The Dublin Daily_ not _The Dani Daily_? Like, maybe they could have an original thought? Holy mimicking monkeys, they aped my intro! Hardly even changed any fecking words!"
I scan it, quick-like.
Don't be fooled by IMITATION dailies. _The Dublin Daily_ is the ONLY daily you'll ever need. We can help you TURN YOUR POWER AND WATER BACK ON!!!
Join us now!
Unlike IMITATION dailies, **WeCARE** delivers all the important news direct to your door, no matter how difficult your "door" is to reach.
DON'T subject yourself to terrible threats in the streets in order to read OVERINFLATED JUVENILE BOASTS that advise you to indulge in DANGEROUS fireworks and battles!
**WeCARE** will come to YOU.
**WeCARE will** fight your battles FOR YOU.
**WeCARE** will keep you safe and IN THE LIGHT.
**Who cares about you? WE do**.
**_WeCARE_**.
"Buh!" It's all I can come up with. "Buh!" I say again. I can't even stand to keep reading. I ball it up and crush it into a tiny hard wad. Finally I manage, "Imitation?" I'm so perturbed I can't even cuss. I can barely talk. "Overinflated? Who's _writing_ this drivel?"
I been keeping Dublin safe and in the light since last October! Months of delivering food and supplies to folks too scared to leave their hidey-holes. Months of fighting monsters, of finding and collecting little kids that got orphaned on Halloween when their folks were out celebrating and never came home because they got devoured by Shades or some other Unseelie. Months of rounding up people and taking them to Inspector Jayne so they could learn to fight.
Nobody else ever bothered to step forward and help folks survive.
Now this?
I'm getting dissed by some paper that's pretending _I'm_ the pretender?
"There is some serious ass-kicking going to happen," I mutter. As soon as I find out who the feck We-the-feck-Care is.
I spend the next few hours whizzing around my city, tearing the stupid things off my posts and putting up _The Dani Daily_.
They used my posts. Couldn't even find their own places to put them up.
Reaching out to MY market by taking MY posts. Stupid fecking copycats. I'm so mad, I'm steaming. If anybody was watching from above, all they'd see is a blur of motion leaving two plumes of pure pissed-offedness trailing out of my ears.
I figure tomorrow's got to be a better day.
Lately, it seems all I ever figure is wrong.
# TEN
# _"Cat scratch fever"_
Four nights he's come to me, murmuring my name.
_Kat_ , he says and he makes of that one syllable an exquisite melody with which not even the divine orchestral choir of all the angels in heaven could compete.
He chimes my name in the language of the Unseelie and it makes my ears ring until my mind is emptied of all thought, until my eyes are incapable of beholding any vision other than him. He is so beautiful that merely looking at him makes me weep, and when I brush tears from my cheeks, my hands come away tainted red by blood.
He wakes me but doesn't wake me.
He takes me to a place that is so perfect and serene and free of worry that I want to stay there forever.
_Kat_ , he says, _my name is Cruce. Not V'lane. I was so weary of wearing his golden shining face. He was never half the Fae I am. I have you in the Dreaming, is it not beautiful? Do you not feel divine here with me? You need not fear me. I am not what I seem_.
I am in danger.
Terrible danger.
And I cannot tell a soul because they are all looking to me to lead, to be strong and show them the way.
I am their hope.
I am afraid "their hope" will soon be beyond all hope.
They judged Rowena so harshly! They have no idea what she faced. God knows how many years she withstood similar torment before she succumbed! Who knows what caliber of person she was before the _Sinsar Dubh_ tampered with her mind. Did it happen to her every night like it does to me? Did the darkness beneath our stone fortress beeline straight for her head, her heart, her bed, the moment she lay down and tried to relinquish for a few stolen hours the heavy mantle of rule?
I cannot help but wonder if this hasn't been going on for millennia. If the Unseelie king knew when he interred his deadly alter ego beneath our sacred ground and charged us with guarding it then infused our blood with his own to make us strong—or is it that very kiss of evil in our veins that makes us weak?—how much hell on earth he was going to cause. How many women's lives he would ruin. How many humans would one day die.
I wonder if thousands of times before me a woman stepped into the position I occupy, assumed leadership of our Order, and was instantly subjected to the harshest test of will imaginable: besieged by the insidious seduction of the _Sinsar Dubh_.
_Take me, free me, be invincible, save the world_.
Oh, the siren song of power. Even I who care nothing for power am not immune.
I do not believe it was ever quiet down there. Not for a moment!
I do not believe any Grand Mistress was ever spared.
Remarkable we kept it hidden so long!
He came to me that first night the Unseelie king imprisoned him beneath our home. I slept, and while I was vulnerable, he came to me in my dreams. He has come to me each night since.
I tried sleeping pills. They only drugged me, rendering me more vulnerable to the pleasures of temptation.
He shows himself to me, in all his glory. He shows me how much more beautiful Cruce is and always was. V'lane was a pale imitation of the real thing. Cruce is black and white and brilliant and hard and strong and perfect. He wraps velvet wings around me and makes me feel things I've never imagined.
I agree with Margery.
I want that chamber pumped full of concrete or lead or iron, or anything that might bar the path between him and me.
I do not know a tenth of the spells Rowena knew. And still she failed.
I can't even get the door closed!
The night the Book was laid to rest, I left the chamber celebratory, with my heart feeling lighter than it had in a long time. The _Sinsar Dubh_ was finally off the streets, and although the method of confinement was not all I'd hoped, I'd envisioned a reprieve. A time of rest and rebuilding, precious, necessary time to come to terms with the many changes in our lives, the endless killing, time to grieve the loss of our many sisters.
It was not to be.
He comes to me with his promises and his lies, with his beauty and unchained desires, and he says that I am all that he needs. He says I and I alone can rule at his side and that my special gift of emotional empathy makes me the only woman capable of ever truly understanding him to the deepest degree, on that rare and uncompromising level of emotional bonding an Unseelie prince must have, or will go mad without. He says I am his only possible mate and he has waited an eternity to have me.
He claims he is being wrongly accused, and we are all being tricked. He says he is _not_ the _Sinsar Dubh_. He claims the moment he was imprisoned in his block of ice, the King took it all back.
He says we are being played by a clever, cunning, mad ruler who cares nothing for his children, who never has, who loves only his concubine, and once he had her in his arms again, reclaimed the power of the _Sinsar Dubh_ , too. He says the concubine still isn't fully Fae, and the King retrieved his spells so he might resume his work, that it was all sleight of hand in the chamber that night.
He tells me he was made out to look like the villain again so we wouldn't search too hard for the Unseelie king, so we would worry instead about containing the only prince capable of stopping him when he decides our world is expendable, which Cruce assures me the King will one day do—and not too far in the future.
He tells me I must be humanity's savior. When I am ready, he will show me the way to free him. He says that only I am strong enough, level-headed enough, to see the truth when it stands before me, wise enough to make the hard decisions.
He speaks with forked tongue and I know it!
And I am _still_ losing the battle.
I wake in the morning smelling of him. Tasting him in my mouth, feeling his tongue on my skin. Filled with him, as no man has ever filled me: body, mind, soul. He makes love to me and I resist but somehow I'm not resisting. In my dreams I say no but do it anyway and love each exquisite, soul-charring moment of it. I wake up coming over and over again from my invisible lover. Shuddering with heat.
And need.
And shame.
My sisters count on me. I am their leader.
How will I survive this? How do I stop him from coming to me? There must be spells to block him, wards, runes to place around my bed! Maybe I should leave the abbey, now, before it's too late. Can I leave my sisters? _Dare_ I leave my sisters? If I don't leave right now, will I ever again have the strength of will to go, or will I find myself down there one night, trembling hands on the bars, willing to do anything it takes to set Cruce free?
How many died the night Rowena let the _Sinsar Dubh_ out, how many murders weighed on her conscience? Did she even have a conscience left by then or had it been corrupted completely?
Who will step up if I leave?
There's no guarantee the next woman will be any stronger than me, or more capable of resisting his seduction. How long would Margery last, in the face of such temptation? How cruel might she become with the power of the _Sinsar Dubh_ blackening her heart?
God help me, I must stay.
I must win this silent, invisible war, with no one the wiser.
God help me.
# ELEVEN
# _"Trouble ahead, trouble behind"_
"There you are," Jo says as I saunter past the kiddie subclub. "It's almost eight-thirty. I thought you were supposed to be here at eight." She's got on makeup. She never wears makeup. And she did something sparkly on her eyelids and between her boobs. It makes me mad. I don't know why she changed. She was just fine the way she was.
The words "supposed to be here" chafe me raw. They're insult heaped on injury. I had a crappy day. It's already taking every ounce of my self-control to hide how much it kills me to see Jo waitressing, wearing a short kicky skirt, serving Fae. But I choke it down because if I let an ounce of it show, who knows what Ryodan might do? The dude's as predictable as an Interdimensional Fairy Pothole, those pieces of fractured Fae reality drifting around that you never know what's inside of till you're ass-deep in alligators.
"Mac's looking for you," she says.
I rubberneck wildly, trying to search every subclub in Chester's at once. "She here?"
"What?" Jo looks at me blankly, and I realize I must have spoken in fast-mo. That happens sometimes when I get agitated. I start to vibrate, and I think all other people hear is the high-pitched whine of a mosquito.
"Is she here?" I slow down for a sec to talk then speed up the rubbernecking.
"No. She left with Barrons half an hour ago. You're going to give yourself whiplash if you don't slow down your head, Dani. It's creepy when you do that. You just missed each other. If you'd been on time, you wouldn't have. What's wrong? You just went as white as a sheet."
If I'd been on time.
Did Mac come here looking for me? Was she hunting me? Does she know I'm supposed to show up for "work" at eight?
I feel woozy. I need to get the blood back in my head. Sometimes I think my heart and veins go into fast-mo without the rest of me, prepping my body for flight or fight, sending all the juice to my sword hand or my feet, and away from my brain. It's the only thing that explains how stupid I go when I get mad or worried. But then, guys work the same way with their dicks, and they can't fast-mo, so maybe it's just a human design flaw. Intense feeling? Ha! Instant brain death.
"Where the fuck is my drink, bitch? You want a piece of me or what?" an Unseelie at a nearby table growls. It means it, literally.
"Tell me you're not eating Unseelie," I say.
"Ew! Never!" Jo says like she can't believe I asked.
"Did you get highlights in your hair?"
She touches it, with a self-conscious smile. "A few."
"You never have highlights. And you don't wear makeup."
"Sometimes I do."
"Like, not once in the whole time I've known you. And I ain't never seen you with sparkly stuff on your boobs."
She starts to say something then shakes her head.
"You dressing up for these creeps?"
"Bitch, I said where's my drink?"
I look at the Unseelie. It's looking Jo up and down, licking thin, nasty lips like she's its next meal. Way too personal-like.
An Unseelie just called Jo a bitch. Pressure builds behind my sternum. My hand goes to the hilt of my sword. Before I can close a finger around it, I'm hemmed in by a mountain range of men with attitudes as big as avalanches. Being in the middle of four of Ryodan's dudes is sort of like standing on a glacier while being gently electrocuted. Never felt anything like it, except from the dude himself, and Barrons.
"That Unseelie called Jo a bitch," I say. Clearly, the Unseelie deserves to die.
"Boss says if you kill a Fae in his protected area, the waitress dies in front of you, real slow," Lor says. "Then we kill you. We'll never remind you of this again. We'll never intervene again. It's on your head, kid. Control your temper or you'll kill her. _You_. We're merely the weapon by which she'll die. And we're inventive as fuck when it comes to slow killing."
Jo's eyes are huge. She sees their faces. Knows how moody I am.
I sigh and let go of my sword. "Wow, dude, I've never heard you string so many complete sentences all together in, like, ever. You're downright loquacious tonight." Brute force is Lor's usual way of dealing with things. His idea of seduction is capture-and-abduct. You don't want to catch this dude's eye. You end up in his bed whether you want to or not. I give him a baleful glare. He's telling me to control myself, and the only way I see to do that inside Chester's is maybe beat myself over the head with a riot baton a few times and knock myself out.
"Bitch, I said where the fuck is my drink?"
Temper nearly pops my skull. My brain empties. My sword hand swells, full of blood and eagerness.
Jo gives me a look and turns away.
Then she goes to play fetch and deliver to an Unseelie. Who isn't respecting her. I'm never going to survive this.
But she has to. So I have to.
I turn away, shoulder through the dudes, making sure to pop Lor a good one with an elbow as I go.
He snarls.
I bat my lashes at him.
He says, "Kid, you need to grow your ass up in a hurry."
"Funny. I think everybody else needs to grow their ass down."
"Like a horse, honey, somebody's going to break you."
"Never. Going. To. Happen."
I'm bored off my gourd, sitting in Ryodan's office. I thought we were going to go out investigating, hunt for clues about what's icing these places. So far the only commonality I see is Ryodan. Both places that got iced were his, like someone's targeting him and the dregs of the society I protect: Fae and Fae-loving humans. It occurs to me if enough of his places get iced, and word gets around, folks will start avoiding Chester's. The club could die from lack of patrons. "One can always hope," I say pissily. Ryodan doesn't even acknowledge that I've spoken. I shift in my chair and glare at the top of his head.
He's doing paperwork.
He's been doing paperwork for over an hour. What kind of paperwork can possibly need to be done in this kind of fecked-up world?
He didn't say anything when I walked in, so I didn't say anything.
We've been sitting here in total silence for one hour, seven minutes, and thirty-two seconds.
I tap a pen on the edge of his desk.
I'm not about to say the first word.
"So, why the feck am I here again?" I say.
"Because I told you to be," he says, without raising his head from whatever stupid thing he's working on.
"Are you going to make me do your filing next? Am I Robin to your Batman, or some stupid temp assistant here to help you sharpen pencils? Don't we have better things to do, like solve a mystery? Do you _want_ more of your places to get iced? We just hanging around waiting for it to happen?"
"Robin and a stupid temp assistant would have been on time."
I sit up straight from my bored slump, tapping faster. " _That's_ what this is all about? You're punishing me because I was late?"
"Bright girl. Stop tapping that pen. You're driving me bugfuck."
I tap faster. He's driving me bugfuck, too. "So, like if I'm on time next time, I won't have to sit here and watch you do stupid stuff I can't believe you even do?"
Half the pen—the part not in my fist—is suddenly plastic powder. I blink at it.
I didn't see him move, he crushed the pen so fast. Now I see little crumbles of blue plastic on the blade of his hand, ink smeared on the paper he's working on. I sit up even straighter. I have a lot to compete with if I'm ever going to be as fast as him.
"I do what I do, Dani, because the mundane makes the world go around. Whoever controls the daily grind controls everyone else's reality."
" _That's_ why you're stealing all the food?"
"Ah, that's why you had your crate-smashing fit. No. I hoard weapons. Someone else is stockpiling food. That's too mundane even for me. I arm the swarm, feed the greed. Someone else is getting ready to starve them."
I give him an admiring look in spite of myself. "You know it's been going on." He's known for longer than I have.
"Someone started clearing the stores a while back. Where've you been?"
"Like, chained in somebody's dungeon. Dude, can we _please_ go do something before I die of boredom? We got a mystery to solve!"
He looks at me. How did I ever think his face was impassive? It says whole sentences.
I roll my eyes. "You've got to be kidding me."
He inclines his head, waiting.
"You're actually going to make me say it?"
He folds his arms over his chest.
I nearly choke on my tongue trying to get it out. But I'll do anything to not have to sit in this office all night. Watching the Unseelie between my high-tops is getting old. I've taken mental notes out the wazoo. My young body needs to see some action. There's a live wire inside me, sizzling beneath my skin. If I don't discharge, I'll die. Bring on the night! There's stuff happening out there and I'm stuck in here!
"I'll. Be. On. Time. Next time."
"Good. Next time you won't have to sit in my office all night."
I shoot up from the chair. "Awesome, let's go!"
He pushes me back down. "But tonight you screwed up. So, tonight you do."
Seven hours later it occurs to me that Lor might be right. I might be breakable. Seven hours of boredom and I'm a puddle of willingness, ready to do virtually anything guaranteed to result in a change of scenery. Chains I can deal with. Boredom, no way. My brain gets ahead of my feet and I don't like to think about where I'm going. I just go.
At six A.M. on the dot Ryodan looks up and says, "Tonight at eight, Dani."
I glare murder at him and head for the door. It doesn't open. I glare at it. A whole night wasted. More seconds ticking by as I wait for my jailer to set me free.
There aren't many crimes in my book. Not many sins either.
But top on both of those lists is killing time. Have fun with it, make something cool, play video games, work hard if you feel like it, but _do_ something. Killed time is an abortion, life that never gets lived, gone, just gone. A cage and a collar killed way too much of mine.
Just when I'm about to blow, he does something and the door retracts into the smooth glass wall.
As I storm out I hear him say, "You wasted my time, Dani. I wasted yours."
I whirl on him, fists at my waist. "That's bullshit! It wasn't even proportionate!"
"It rarely will be."
"Thirty little fecking minutes cost me nine and a half hours?"
"The way you treat me is the way I will treat you. Since I'm bigger and older, I imagine it will always be worse."
"Oh, _now_ you get all proportionate. If you're going to be as much of a dickhead as you are big and old, dude, that's some serious dickheadedness. That's not fair. You can't be totally disproportionate one minute and then all quid pro quo the next."
"I can be anything I want."
"Oh, whose fecking comic book _is_ this?" I explode. "That's _my_ line."
He laughs and his face changes. All the sudden he doesn't look so old. He looks happy. Free. Totally different. I see lines around his eyes from laughing that I never noticed before. My mind flashes straight back to level four and I see him behind that woman again and he groans like he did that night, then he laughs, and I feel almost sick to my stomach remembering. I don't know what's wrong with me. I wish I'd never fecking gone down to level four! I stand there and gape at him.
The door slides shut in my face.
"You're early."
I give him a mutinous look. Of course he thinks my being early is about him. It's not. Mac was at Chester's last night at eight. I think she's hunting me. Since I can't be late to avoid her, I have to be early. "Watch broke. Thought I was on time."
"You don't wear a watch."
"See? I knew I had a problem. I'll just dash out and get one. Be back tomorrow. On time." Jewelry gets caught on things in battle. The only concession I make is a bracelet Dancer gave me that I wear snug on my arm. Besides, without him around, giving orders, I might actually some make progress in the investigation.
"Don't even think about it."
I drop into a chair in his office, dangle a leg over the side. "What are we doing tonight?" I say just like him. No inflection at the end.
"Ah, Dani, if only you took instruction in all things so well."
"You'd be bored."
"So would you. There are three other iced places in Dublin."
"Three!" I sit up straight in my chair. "Are they all yours?"
"Local places. Unrelated to me in any way."
Bugger, there goes my theory about him being the target, along with my hope that Chester's might die a slow death. "Casualties?"
"About fifty between the three."
"Humans or Fae?"
"Humans."
" _All_ humans?"
He nods.
I let out a low whistle. Fifty more people dead. The human race just keeps getting hammered with blow after blow. "Then why do you care? It didn't happen on your turf. Nothing of yours was damaged or destroyed."
"I have other reasons for wanting it stopped."
"Like what? You move fast like me. You can outrun anything. You can steal more stuff to replace what got iced. So what's the deal?" What motives does a dude like him have?
"The walls between our realms were destroyed on Halloween. Since then things have changed. Human laws of physics are no longer laws, they're wishful thinking. It's possible parts of Faery are manifesting spontaneously, bleeding through into our reality. It's possible it's happening randomly, instantly, and without warning. I didn't see surprise on anyone's face at either of my properties. Put the big picture together, even for people who can move like you and me."
I snap up straight to full attention, both feet on the floor, not liking that at all. "You mean if it happened in the place I was standing, I'd be alive one second, dead the next. I wouldn't even know it. I'd just be gone!" My hands fist. I'm so freaked I want to fight something right now.
"Exactly. Instant death. No warning. No awareness. I don't know about you, but that offends the fuck out of me."
No blaze of glory, no epic battle! I'd die a totally meaningless death. Worse, I wouldn't even get to experience it. How much would that suck, to go through my whole life waiting to die, and then not even know it happened? I think Death is like the final stage of a video game. And if what Ryodan is saying is true, and I get iced, I'll never reach that final stage. I'll get wiped right out of existence on the second-to-last level. I want to _play_ that last level when it's time. I want to taste it all, even the dying.
I'm suddenly one hundred and ten percent invested in solving this mystery. Fifty more folks dead coupled with the possibility of a completely meaningless death is powerful motivation. You don't get a big write-up in the history books unless you go out in a big way. I crunch thoughts and regurgitate them. "Well, first of all, the humans in your subclub were a little preoccupied with things like getting tortured and dying so it's understandable if they didn't notice that they were about to die in some other unexpected and surprising way, and second, I can't say for certain what surprise looks like on an Unseelie's face but I got a great idea: I'll go downstairs and kill a few right now and we'll collect some empirical data." I don't bother to mention I already hunted and killed half a dozen different kinds this morning after I left but I still couldn't decide what their expressions meant. Their faces just don't work like ours.
When he doesn't bother to dignify my dig with a response, I say, " _Three_ new places?" What if the "bleeding through" starts to speed up? There could be dozens of iced spots soon. Assuming that's what's happening, how the feck are we going to stop it?
"All iced last night within a few hours of each other. Two of them have already exploded."
I shoot to my feet. "Dude, we got to get to the third, before it goes, too!"
# TWELVE
# _"Life is a highway, I wanna ride it all night long"_
I slo-mo Joe it across Halfpenny Bridge.
We didn't learn a single new thing at the latest ice sculpture. Like the others, it blew shortly after we arrived. I freeze-framed out of there through flesh-colored shrapnel I pretended _wasn't_ parts of fingers and faces I'd failed to save.
The new places that got iced have nothing in common that I can see. There were two of those small underground pubs that've been springing up all over the city, and a fitness center where three people were frozen doing yoga in the middle of a bunch of crystal bowls. How weird is that? People doing yoga in times like these!
So far I've got an underground club at Chester's, a warehouse on the outskirts of the city, two inner-city small pubs, and a fitness center. Humans, Unseelie, and Imperial guards all at some places but not others, so whatever's happening doesn't appear to be targeting a certain person like Ryodan or group of victims. It's looking more like a random, spontaneous event with each scene I see.
I'm trudging, which I don't usually do, because I'm thinking hard and when I'm thinking hard plus freeze-framing I run into things a lot. My bruises are fading and sometimes I try to be my normal-colored self for like a whole day. I'm too wired for sleep. I get like that sometimes and can't do anything about it but ride it out. I need something to do or I'm going to drive myself nuts.
I find Dancer in his favorite corner penthouse on the south side of the river Liffey. The two outer walls are solid floor-to-ceiling windows that look out over the streets. When I get there, he's stretched out on a rug in the sunshine with his shirt off, eyes closed, glasses on the floor beside him.
Dancer's going to be a big guy one day, if he ever gains weight. Last time we measured ourselves, he was fourteen inches taller than me, lanky and lean. He forgets to eat. His hair is dark with some wave and he never cuts it until it gets in his way, then he asks me to trim it. It's soft. I like it to his chin as it is now, falling away from his face. When he wears his glasses, which is pretty much every minute he's awake because he's so nearsighted (he hates them and before the walls fell he was going to get Lasik), he looks like a hunky geek. I'd never tell him that! I like his hands. His feet are ginormous! His eyes aren't green or blue, they're aqua, like they're Fae-brushed. He's got better eyelashes than me.
When I see him I don't say, "Dude where you been, I was starting to worry," because me and Dancer don't do that to each other. He survived the walls going down all by himself. So did I. And I don't say, "What happened the night Ryodan showed up and took me, where'd you disappear to?" It doesn't matter. We're here now. It's like somehow we know in our guts that it'll never be too long, the other is always going to walk through the door one day, eventually.
He props up on an elbow when the door closes. He knows it's me because I had to disarm ten booby traps before I got to the door. Nobody else could make it through one of his gauntlets without tripping some alarm. Well, except for Ryodan, who seems to be the exception to every fecking rule.
My heart squinches a little when I look at him. I never had siblings but I think he's like a brother to me. I can never wait to see him again, tell him all the ideas I've been thinking, the things I've seen, and get his take on it all. Sometimes when we see each other we can't stop talking for hours and hours and we get so excited we start to stumble over our words trying to say it all so fast. I consider telling him about the iced scenes and the mystery I'm looking into but I don't want Dancer to be any bigger on Ryodan's radar than he already is. That Ryodan even knows he exists makes me nuts. I want Dancer safe. And I know him. If he got the tiniest hint of a big mystery like this, he'd start poking into all kinds of places that could get him killed. It doesn't matter how über-impressed I am with how smart he is. Ryodan's _worse_ than walls falling or the world melting down. You don't survive if he doesn't want you to.
"Mega, I've been thinking—"
"Stop the presses! Do I need to put out a special edition of _The Dani Daily_?"
"Might."
He grins and I grin back. Dancer thinking has stellar results. You wouldn't believe the bombs he can build. We blow up things sometimes just for fun. You know, things that need to be blown up anyway like places where a lot of Shades used to hide that maybe they would return to one day like birds along a migration route, if it was still there.
"You got me wondering about Papa Roach's babies," he says.
"Yeah?" I stretch out in the sun next to him, prop up on an elbow, too, facing him. I love being able to see his eyes without his glasses in the way. It's a rare treat.
"Do you know how long they can stay separate from a body, either Papa or human?"
"Dunno. Dancer, I finally found _Scream 4_. Want to watch it tonight?"
"Watched it last night," he says absently, running a hand through his hair, making it stand up funny in a totally hot way, and I can tell by the way his eyes are unfocused that he's lost in thought and not aware of stuff around him. He gets that way a lot.
"You watched it without me?" I'm hurt. Me and Dancer love horror flicks. We gorge on them because they make us laugh. They have a way of putting the world in perspective. We'd been hunting for _Scream 4_ for a while, planning to watch it. Dancer doesn't usually watch movies alone, least not that I know of.
"But I'll watch it again. It was cool."
"Cool." I still feel hurt, even though there's no reason for it. He's watching it with me tonight. So what if he saw it last night, too? And so what if he saw it with someone else? I don't care about stuff like that. What happens when I'm not around ain't got nothing to do with me. "What about Papa Roach?"
"Blowing them up doesn't work. Torching them is no good either. But what if we keep them from returning to a body? Any body. Human or their own. Wouldn't that solve the problem? Our goal is to keep them from getting inside more people. They're immortal, and your time is too important to waste running around after thousands of them with your sword. So, I started thinking what about a tough, impossible-to-escape spray-plastic? Encase them and keep them from being able to reattach to anything. I've been working on a formula. Once it's done, we can fill those small fertilizer tanks we swiped from the hardware store and test it out. I already rigged up a couple of sprayers to fit."
So, that's where he'd been. And when he got done working last night he watched a movie to chill. No big.
"I've got something that sets hard at a quarter-inch thick. I'm still trying to get it to gel to the perfect degree of solidity. I think I've figured out a way to add iron to the mix without making it too rigid. How do the segments attach to Papa? Tentacles? Suckers? How do they get under human skin? Can you catch me a couple to test it on?"
"You're the Shit, you know that," I say.
"No, _you're_ the Shit," he says and grins, and we say it back and forth a couple of times. He thinks I'm the Shit because I can actually catch them. I was born with my gifts. Dancer is always thinking, trying to find ways to do things better. Surviving the fall with no special powers and no friends wows the feck out of me.
We relax on the floor because sunshine in Dublin is rare, and we talk about anything and everything except things like where I was when he was wherever he was. I don't tell him I was in a dungeon for almost four days and he doesn't ask. I like that about him. Friends don't build cages for each other.
We watch the sun move across the sky, and sometimes he gets up to get me things to eat. He tells me he's been checking stores and nearly every single one has been wiped clean. I have to stop myself three times from almost spilling the beans about the iced stuff I've been seeing.
When it's getting near seven o'clock, I start getting antsy and it makes me mad because I don't want to have to leave but somebody else is pulling my strings and I've got to go. I have to get to Chester's early enough to avoid Mac but not so early Ryodan gets all cocky about it.
I sigh.
"Something worrying you, Mega?" Dancer says.
"Just got to go take care of some things."
"I thought we were going to watch a movie. I found a whole box of Skittles at the airport. And jerky. The hot stuff."
I smack myself in the forehead. Skittles, jerky, and a movie. What was I thinking, saying hey, let's watch a movie tonight. My nights don't belong to me anymore. Somebody else owns them. That's not just a bitter pill to swallow. For someone like me it's a suicide tooth. It's irrelevant that I _want_ to go work on the ice mystery and keep more innocent folks from dying. I can't handle that Ryodan gets to dictate when, how, and where I do it. It almost makes me not want to work on it at all. I hate being controlled.
I can't not go to Chester's because I don't know what Ryodan will do to Jo if I don't show, and there's no way I'm running the risk of finding out. I don't know if he'd hunt me down here, smash up the TV and DVD player, and take Dancer and put him in his dungeon. I never know what that dude will do next.
But I'm crystal clear about one thing he's doing.
Ruining my life.
I bang into Ryodan's office. "I been in enough cages in my life," I say. I got worked up on the way over, talking to myself in my head about the unfairness of it all.
He glances up from his paperwork.
"Paperwork! Holy replicating reams! Is that all you ever do? It's no wonder you want me coming around so much. Got to liven up your boring life with the superexcitement of the Mega." I'm so mad, I'm vibrating and the papers on his desk flutter in the breeze. When I get really mad, I cause a kind of air displacement that does on a tiny scale what the Fae do on a massive scale, except I can't affect the temperature. I do it sometimes to freak people out, get them off balance. It used to bug the crap out of Ro.
He catches a paper before it flies off the desk. "Something wrong."
How does he _do_ that? Say questions without them sounding like questions at all? I been practicing and it's not easy. Vocal cords want to go up at the end of an interrogatory. I been trying to reprogram myself. Not because I plan to start acting like him (at least not around him) but because I think it's good to test yourself, override compulsion. Learn more self-control.
My hair's flying around my head in a cloud, getting in my eyes. I shove it back with both hands, wishing me and Dancer were eating jerky and hanging cool. "Yeah! Like, I might just have a life! Like I might just have plans for things that conflict with your stupid report-to-work-every-night-at-eight rule! Nobody else has to work every single night! Maybe I could get a couple of nights off to do something _I_ want to do. Is that too fecking much to ask?"
"You have a date."
Another nonquestion, but the word "date" in the same thought with Dancer makes me say, "Huh?"
Ryodan stands and dwarfs me. I live in a world of people who are taller than me, but Jo says she thinks I'm going to grow more. I measure myself a lot. I don't want to be stuck at five foot two and three-quarters forever.
"You mentioned plans. You didn't say what they were."
"None of your fecking business."
"Everything is my business."
"Not my personal life. That's why they call it personal."
"This is about your little boyfriend."
"Don't talk about him. Don't even think about him. And he's not little. Stop calling him little. One day he's going to be bigger than you. You just wait and see."
"This isn't the time to play house and get clumsy with a kid that doesn't know what to do with his own dick."
He just made me think about Dancer's dick. The thought is so uncomfortable I start bouncing from foot to foot. "Who said anything about dicks? I just want to watch a movie tonight!"
"Which one."
"How could that possibly _matter_?"
He gives me a look.
" _Scream 4_. Happy?"
"Wasn't very good."
"Dancer said it was," I say crossly. Has everybody seen it but me?
"Shows what he knows."
"You got a problem with Dancer?"
"Yes. He's the reason you're in a shit mood tonight and I have to put up with it. So fix the shit mood or I'll fix Dancer."
My hand goes to the hilt of my sword. "Don't you even think about trying to take anything from me that's mine."
"Don't make me."
His fangs just slid out. I shake my head and whistle. "Dude, what are you?"
He looks at me long and hard and I see something in his eyes that I almost get but don't. It's a look that I feel like I should know but just can't make sense of. There's more of a breeze in the small, closed office than I usually manage to generate, and I realize he's vibrating, too—and he makes wind, too. I'm beyond annoyed. Is there anything I can do that he can't do? When I look down through the glass floor, I see that everyone beneath us is moving slo-mo. We're both freeze-framing. I didn't realize I'd shifted all the way up.
He drops back into slo-mo first.
It takes me a sec longer to get ahold of my temper. When I manage to shift down, I flop into a chair and sling a leg over the side. I speak belligerence in every language known to man. Sign language is my native tongue.
Ryodan is like the ocean. He is what he is. And he's not about to change. There's no point in fighting the tide. It ebbs. It flows. You ride it. He's got me by the short hairs and he's not about to let go.
"So, what are we doing tonight? Boss." I put all my aggravation into the last word.
There's that look again. Mystery to me. Sometimes I can read him like a book, other times the only things I see on his face are two eyes, a nose, and a mouth.
I roll my eyes. "What?"
"Something's come up. I was going to tell you." He goes back to his paperwork, dismissing me. "You can go."
I sit up straight. "Really? You mean it?"
"Get out of my office, kid. Go watch your movie."
I can't get to the door fast enough. I yank it open.
"But watch out for icy spots. I hear they're deadly."
I pause on the threshold, getting mad all over again. I had a happy feeling for all of one stinking second before he went and squashed it. "You just had to say that. You can't help yourself, can you? You think the only thing to do with a parade is rain on it. Some people know to enjoy the parade because, dude, the rain always comes back."
"The wise man ensures his survival before enjoying it. The fool dies enjoying it."
Skittles, jerky, and Dancer are calling my name. I rip open a candy bar, bouncing from foot to foot. "But what if the wise man never gets _around_ to the enjoying part?" I got a lot of unlived experiences waiting for me. Sometimes I want to be just what I am. Fourteen and free.
"Perhaps the wise man knows being alive _is_ the enjoying part."
"Have more places gotten iced since last night?" I should have kept my mouth shut. I shouldn't have asked. Responsibility adds weight and years to my shoulders when he nods.
He rubs salt in the wound. "But maybe you'll get lucky, watching a movie with your little boyfriend, and nothing will happen. Bright side of it is, if something does, you'll never know."
'Cause I'd like, be dead instantly. Bright side, my ass. Ryodan knows just how to push my buttons.
I roll my eyes, close the door and sit back down. I'll be fourteen later. Like probably next year. When I'm fifteen.
Without looking up, he says, "I said get out of here, kid."
"Cancel your plans, dude. Folks are dying. We've got work to do."
This one takes the cake, way out on the south side of Dublin, where things get rural.
Behind a shack that's barely managing to stay upright, with a swayback porch and a roof that looks like a really old person's mouth without dentures in, a man, a woman, and a little boy are frozen, doing laundry the old-fashioned way that Ro used to wash her Grand Mistress robes. She said it kept her humble. There wasn't a humble bone in that porky old witch's body, not even a nice hair anywhere.
The man's hands are iced to an antique washboard and he has some weird kind of metal thing iced on his shoulders like part of a frame that holds your head still if you broke your neck. The child is frozen, banging a spoon against the bottom of a battered pot. I don't let myself look at the kid long. It slays me when they die. He never even got to have a life. The woman got iced while she was lifting a shirt from a bucket of soapy water. I stand at the edge of the lawn, shivering, absorbing as many details as I can from a distance, getting ready to freeze-frame in. If this scene behaves anything like the others, it's going to explode soon.
"How did you even hear of this one?" The pubs I understand, even the fitness center because it was in Dublin and Ryodan knows everything that goes on in the city. But these are farmers doing laundry out in the country.
"I hear everything."
"Yeah, but how?"
"That was supposed to terminate your line of questioning."
"Dude, news flash. 'Supposed to' never works with me."
"Observations."
"They knew it was coming, whatever it was." Which makes me feel a whole lot better. I can stop worrying about dying with no warning. Although the boy was looking down at the pot he was holding, the grown-ups' mouths were open, their faces contorted. "They saw it and screamed. But why didn't they run? Why didn't she drop the shirt she was washing? It doesn't make sense. Does it freeze them mildly before it totally ices them? Could they have a small reaction but not be able to fully move? Did it sneak up on the other folks at the other scenes from behind?"
"I need answers, kid, not questions."
I puff out a breath. It gets foggy but doesn't ice. "It's not as cold as the other scenes."
"It's older. It's thawing."
"How do you know that?"
"There's a drop of condensation on the end of the man's nose that's about to fall."
I squint. "I don't see no stinking drop. You can't see that far that clearly." I have supereyes and I can't see it.
"Jealous, kid." He lets the last word rise that one-hundredth of a note that he does sometimes when he's humoring me. There's a smile in his voice. It pisses me off more.
"There is no fecking way you can see a drop of water from here!"
"There's another sliding down between the woman's breasts. Just above the mole on her left one."
"Dude, you can't outsee me by that much!"
"I can out-everything you." He gives me a look that I usually see in the mirror.
Just like that I'm in a total snit. "Then I guess you don't need me, and I'm wasting my time." I turn around and stomp back to the Humvee. But before I make it five steps, he's in my way, looming over me, arms crossed, looking at me weird. "Not in the mood, Ryodan. Get out of my way!"
"Being needed is toxic."
"It's good to be needed. Means you're important."
"It means there's an imbalance of power. There was no shortage of life-suckers before the walls fell. You're not responsible for the world just because you're more capable."
" 'Course I am. That's what more capable folks do."
"You could ask me to teach you."
"Huh?" This night is getting weird in a hurry. "Teach me like you're teaching a class or something? What are you going to call it: 'You Too Can Be a Sociopath 101'?"
"It would be more like a graduate-level class."
I start to snicker. His sense of humor sneaks up on you. Then I remember who's talking and bite it off.
"You want to be faster, stronger, smarter. Ask me to teach you."
"I ain't asking you for nothing. And you might be faster and stronger. For now. No way you're smarter."
"Your choice. But turn around because you're not leaving. It's night, and you know what that means."
"Like, it's dark?"
"You're with me until dawn."
"Why dawn? You a vamp or a zombie or something that can't stand the light?"
He freeze-frames away, moves in on the scene. "I like sex for breakfast, kid. I eat early and often."
There I am thinking normal thoughts about iced people and how much he bugs me, then he slams me in the eyeballs with sex for breakfast stuff, and just like that my hormones do that crazy thing they do sometimes, where they start slapping up pictures all over the inside of my head and each one is more embarrassing than the last. And I can't close my internal eyes because they don't really exist and hormones are more stubborn and unpredictable than even me.
I wish I'd never watched porn movies or seen Ryodan "eating breakfast" because then the pictures wouldn't be so vivid and hard to get rid of.
But there he is, in graphic detail because I know exactly what he looks like naked, I saw him. I know how his body moves. He's got a lot of muscle. Scars, too. I know that when he has sex he laughs like the world is a perfect place. And when he did that, my hands curled into fists because I thought about touching his face like maybe I could catch joy in my hands and hold it. I had all kinds of fecking strange and stupid thoughts standing there on level four. I could so kick the shit out of myself for watching. I don't get hormones. I don't understand why the horny little buggers would even notice an old dude like him.
"You coming?"
I shake myself mentally, pick up and shift sideways.
Nothing happens.
"You've got to be kidding me," I mutter.
"Kid, why are you still standing there?" He's freeze-framing around the frozen trio. "It could blow any second."
I don't move, thinking how much I hope it will, so he won't figure out I've lost my superpowers again.
"I have to, uh, use the, uh—" I gesture to the woods behind me. "Need a little privacy. Be right back."
Just like I hoped, while I'm in the shrubbery, pretending to pee, the laundry people blow.
The ride back to Dublin is a long and silent one.
# THIRTEEN
# _"The very worst part of you is me"_
I'm on the roof of a building, across the street from the pile of concrete, twisted metal, and broken glass that once was Chester's. The club is deep underground now. Usually there's a line for blocks, but it's four in the morning and everyone who wanted to be inside got inside about an hour ago. I guess that means enough people died to open up additional standing room because I didn't see anyone come out.
A black Humvee pulls up.
It's what I've been waiting for.
I used to hate being up high, which is ironic, considering I'm a Highlander. Or I was.
I'm getting used to heights. The view's better. You see more and you might as well be invisible. People don't look up much, not even in times like these, when they should because you never know what's in the sky above you, getting ready to feed on you, maybe a Hunter, or a Shade. Or me.
I watch her get out of the Humvee. She's bouncing from foot to foot between steps, moving sideways and forward at the same time, eating a candy bar. I've never seen anyone with so much energy. Her hair is auburn fire in the moonlight. Her skin is luminous. She has sweet young curves and long legs. Her features are bone china fine, and expressions rush across her skin like my new Unseelie tattoos rush beneath mine.
But it's the heart of the girl that gets me.
He's big and towers over her. Hard face. Hard body. Hard walk. They look so wrong together. They're talking. She keeps looking up at him like he gets on her last nerve. Good. Her hand hovers near the hilt of her sword and I know what she's thinking. She despises Chester's. She can barely stand to be in the same place with Fae without killing them. She hates them. All of them.
It's a category that will soon include me.
The owner of Chester's looks up.
I'm deep in shadows on the roof, throwing a light glamour, a new power I've been testing, trying to make my face more palatable to her.
I focus on projecting a general blanket of night and emptiness so he can't see me.
His gaze stops right where I am and he gets a smug-ass look on his face, but that's his look most of the time. I've nearly decided that while he might sense a disturbance in the night up here, he can't actually see me when he inclines his head in that arrogant, imperial way so characteristic of the dickhead.
Rage washes over me, thick and intense and smothering, and for a few seconds I drift in a black place where everything's icy and wasted and evil and I _like_ it. I'm glad I'm going Unseelie prince. I say bring on the power.
I say let there be war.
I throw back my head and slide a mane of hair over my shoulders. Cutting it doesn't do a bloody thing. I sleep, I wake up, it's there again. I turn my face up to the moon and inhale greedily. I want to drop to all fours and bay like a wild thing drunk on being hungry and strong, a beast that could fuck for days without cease if I could only find something that could take it as hard and long as I can give it. I want to chime to the moon in Unseelie, and hear it chime back. I can smell death in the city, everywhere, and it's intoxicating. I can smell need and sex and hunger and it's so bloody sweet—humanity ripe for the plucking and playing and eating! I shift my dick in my jeans. It's painfully hard. And the Earth is round.
I look back down, my eyes narrow. My boots are crusted with ice. The roof has gone white in a circle of snow and glittering ice in a fifteen-foot radius around me. I lope lightly along the edge of the roof, crunching snow, following as they go around back. This is going to be so much easier when I don't have to use my feet.
He isn't what he's pretending to be with her.
I watch him all the time. I'm going to be there when he stops pretending. I'm going to be her bulletproof vest, her shield, her fallen fucking angel whether she wants one or not. He's pretending he's almost human. He's no more human than me. He's pretending to be nice, like he's safe to be around, like he doesn't have fangs for a reason. He's pretending the term the "Gavel Effect" wasn't coined about him, meaning you're fine with him. Right up until you're not.
Right up until you're dead.
The devil in a businessman's suit, he bides his time, gathers information, processes it, and when he makes a decision, the gavel falls and everyone that pissed him off or offended him or just breathed wrong dies.
She won't be given a stay of execution. No one gets one. The only things that matter to him are others of his kind.
She thinks he's not an animal like Barrons. That he's more civilized. She's right, he is more polished. But it only makes him more dangerous. With Barrons you _expect_ to get fucked up royally. With Ryodan you don't see it coming.
He's treating her like she's fourteen and he's a normal adult, acting like he's taken her under his wing. Like he needs her detecting skills, same as Barrons did to Mac, and she's falling for it, same as Mac. He's lining up his dominoes, so they fall more easily when he feels like pushing them over, conserving energy so he doesn't have to hunt her when he's ready to kill her.
A bastard like him has one use for women. And she's not old enough. Yet. I can't decide which would be worse, if he killed her before she was old enough or waited and made her one of his endless string of women.
She's not that kind of girl, the endless string type. You get a shot at something like her once in a lifetime. And if you screw it up there's a special place in hell for you.
She breaks away from him suddenly and stomps off ahead. She's pissed. I smile.
I pull out my knife, twist my arm over my shoulder and scratch my back with it. Blood trickles. I sigh with relief, but it doesn't last long. Sleeping is a real bitch. My back itches all the time and human drugs don't work on me. I twist to get a better scratch.
My blade hits bone with a dull clunk. I saw at it with the serrated tip of the blade but can't get the angle right. I don't have any friends that are glad to see me, nobody to lend a helping hand. I tried to get Dad to cut them out of my back. He said they're attached to my spine and it would kill me. I don't believe that. Nothing kills me. They itch. I want them gone almost as much as I'm beginning to want them.
Fucking wings.
Funny how things work out. Dani killed an Unseelie prince to save Mac, and I end up turning into the replacement for the prince Dani killed. But it's not the lass's fault. It's Mac's. For needing saving. Later, for forcing me to eat something I would never have eaten if I'd been in my right mind.
I wonder if my wings will get as big as Cruce's. I wonder what it would feel like to fly the night sky with him and the other two. I see a vision in my head sometimes of the four of us, swooping down over the city, black wings beating air, filling the sky, owning the world. I can hear the sound we make as the four of us chime deep in our bodies. There's a special, bloodcurdling song the Unseelie princes sing, sometimes it plays in my head while I sleep. The call to the Wild Hunt burns in my blood.
I back up to the corner of a small brick building on the roof that houses heat pumps, lean against it and drag my back from side to side across the edge, scratching, watching as they move toward a metal door in the ground.
He catches up with her and they walk together again.
She glides through the night. He punches into it, a boxing glove with razor blades for knuckles. When she passes, the world is a better place. He leaves bloody footprints in a graveyard of bones.
He lifts the door, light blazes up from a hole in the ground, and she descends, my angel into a sordid hell.
He squats at the edge and watches her go and, for a split second, I see an unguarded expression on his face.
It chills even a creature as cold as me.
I know that look. I've seen it on my own face.
Then the son of a bitch looks up at me and, this time, there's no question in my mind that he sees me. He looks straight at me and inclines his head with a mocking smile. I return it coolly. My nod says, "Yes, yes, I see you, too. Be very careful."
I can't decide if what he just let me see was real—or another of his games. They don't call him the master of manipulation for nothing. Barrons breaks heads. Ryodan turns them inside out. Barrons fucks you up. Ryodan makes you fuck yourself up. He pushes buttons and rearranges things according to his own private, coolly sociopathic plan.
I liked it better when I thought he was going to kill her.
I stop scratching.
I want those wings. They'll make the fight that's coming easier.
He's a walking dead man.
If he wasn't serious about what he just showed me, and he's gaming me, he gamed the wrong Unseelie prince. I'll kill him long before he gets around to killing her. I know how men like him work. I'm becoming one.
If he _was_ serious about what he just showed me, he showed it to the wrong Unseelie prince. Because what he showed me is that he sees the same things in her I do.
He knows she's worth waiting for.
And when it's time, he intends to be the one. That's why he's keeping her close. To those of us who live forever, a few years isn't long to wait.
Not for something worth waiting for. Not for a once-in-a-lifetime girl.
A few years are a mere blink of an eye to men like us, for whom women crush sweetly like rotting pumpkins after Halloween. Sex isn't easy for me anymore. I'm always holding back. Human women are breakable.
Not this one.
He sees her like I do: at seventeen, twenty, thirty. Superimposed over the fourteen-year-old, he sees the woman she'll become.
And he's staking his claim.
Over my. Dead. Fucking. Body.
And I can't die.
But I know one of his kind that recently did, and I know how. I hear there's a Hunter up there in the night sky that likes Unseelie royalty.
Soon I'll have the wings to find him.
My superpowers come back three blocks from Chester's. I know because I've been trying to tap a finger in hyperspeed on my thigh the whole way back. Finally did it. I still haven't managed to make only my eyes move like Ryodan but I've been practicing and can get certain parts of my body to speed up for short amounts of time. Only problem is, the place where the part connects to my body gets a little sore, like I stressed out the muscles where the slow-mo and fast-mo parts are having a kind of what-the-feck-are-we-doing-here battle with each other.
But it's not like I could sit in the Humvee with the dude, who would love to know sometimes I'm helpless, and practice trying to freeze-frame my whole body. If he stopped sudden, I could go shooting straight through the windshield and then I'd be all cut up for days on top of my usual bruises.
I look at him, irritated. "Why are _you_ never bruised?" What is he? Like the exception to everything? And if so, where do I apply?
"Participating and all that bunk," he says. In other words, I don't get to know because I'm not in whatever his inner circle is. Fine. Don't want to be there anyway.
"You got some kind of magic salve, dude? Because it's only fair to share stuff like that."
He pulls up to the curb out front of Chester's. I hop out of the Humvee the second he parks and instantly start bouncing from foot to foot, sideways, in between steps forward, to make sure I'm working right again. No way I'd go inside Chester's with no superpowers. I whip out a candy bar, devour it then munch three more in quick succession, stockpiling energy. "Aren't we done for the night? What else have you got for me to do?" I just spent an hour in an electrified sardine can with Ryodan, after losing my powers. He saturates confined spaces, like he's got ten people's stuff crammed into his body. He's pissed at me for not inspecting the scene before it blew. I'm pissed at me, too, but it wasn't like I had any choice. Without superpowers, I'm not getting anywhere near one of those scenes. It was a sucky drive. I want some time alone, or time with Dancer. He recharges me. Hanging with him is simple and pretty much perfect.
He doesn't answer me and I look at him. He's staring up at the roof of a building across the street and he's got an amused, smug look on his face. I search the shadows of the roofline but I can't figure out what he's checking out. There's nothing up there. "Dude, you listening to me? Hello? Do you even know I'm here?"
He continues looking at the roof like he's seeing something I can't see. Like that stupid drop of condensation I'm still not sure I believe was there.
"I always know you're there, Dani. I tasted your blood. I feel you all the time."
Okay, that's disturbing.
"You mean like when I'm around," I clarify for him.
"How do you think I found you at your little boyfriend's place."
"You need to look at him harder if you think he's little."
"And so breakable."
"Stop talking about him. He's none of your business. Just what are you saying? That you could find me, like, anywhere, anytime?" There's a right answer and wrong answer to that question.
"Yes."
That was the wrong answer. I get so mad I'm breathless. "Bull. Liar."
He laughs and looks at me. "Want to play hide and seek, little girl?" He purrs it in a voice I've never heard him use before, and he actually makes it a question.
His fangs are out.
"Dude, you are one weird... whatever you are." I'm nearly at a loss for words.
He laughs again and I can't even stand to look at him so I charge off to the door in the ground that is the new entrance to Chester's.
He holds the door up for me. I sigh gustily as I descend the ladder. I hate Ryodan.
So I'm walking across the dance floor, cutting a beeline straight for the stairs to head up to Ryodan's office to do whatever it is he wants me to do, when I see _her_.
She's moving across the main dance floor with Jericho Barrons behind her, and it looks like they're heading for one of the subclubs, though I can't figure out why. Mac doesn't like it here any more than I do.
I freeze.
I hate seeing her. I hate not knowing what's going on in her life. I hate what I've done. Can't change it, though, so no point in feeling it.
Ryodan slams into my back, knocking me forward into the crowd. "Walk much?" I say testily as I careen off of a hulking Rhino-boy that gnashes yellowed tusks at me.
As usual, he doesn't miss a trick. His gaze does that ocular-shiver thing all over my face. "I thought you and Mac were friends."
"We are friends," I lie.
"Then go say hi."
Shit, I hate how much he notices. "We might have had a tiny tiff."
"Tiny, my ass."
"Quit nosing into my business."
"Learn not to wear it on your face, kid. Except in private, with me and no one else. You need some serious training. Telegraph like that, it's only a matter of time before somebody hoists you on your own entrails."
"Dude, who _uses_ words like entrails? Or hoists?"
"Tell me what happened."
I fist my hands at my waist. "It's none of your business and that's the beginning and end of it. Some things you can horn into. Some things you can't. Stay the fuck out of it."
He looks at me weird. "You said fuck. Not feck."
"That's all you got from what I just said?"
"You want privacy on this. I'll give it to you. See how easy that was. If you want something, ask me for it. You'll find I can be a generous man. When you treat me right. If you ever figure out what that is."
He moves past me and heads for his office.
I can't help myself, I look back at Mac. I grin and kick myself inside for doing it but there was a time when I loved waking up every day in Dublin, different than I do now, because I knew _she_ was there at Barrons Books & Baubles and we were going to go do something cool that day and then she baked me a birthday cake and picked me out presents and we watched movies and we fought back-to-back and I ain't never had anything like that before and sometimes I feel like a homeless dog out in the rain and thunder and I'm muddy and cold and I'm staring in the window at the pretty collie sleeping on a doggie bed close to the fire, and there's a name on the bowl that's next to her, and I wonder what it would be like to—
"Gah! Get over yourself, wussy-girl." I got big-dog teeth and a big-dog bite and I know the rules: you stay inside, you get collared and spayed. I pick myself up and start to freeze-frame after Ryodan when a commotion in Mac's general direction makes me stop, stay in slow-mo and glance back.
There's a new type of Unseelie in Chester's tonight and they're something out of a horror flick. They look like anorexic wraiths that might drift around graveyards, breaking open coffins and feeding on rotting corpses. They're draped in black cloaks with hoods so you can't see their faces, and they don't walk, they hover and glide just above the floor. I glimpse a flash of bone at the sleeves. In their hoods I catch a quick hint of pale, bloodless skin and something black. There are twenty or so of them in the subclub Mac and Barrons are just entering. They make me think of carrion crow that sense the coming of a storm and perch in treetops everywhere, waiting for the destruction to begin so they can swoop down on the dying and tear flesh from bone with sharp beaks. I'm suddenly certain they don't have normal mouths. And equally certain I'd rather never see what they do have.
They turn toward Mac like they're a single unit or something, which is totally creepy, and begin making a chittering noise that sets every nerve in my body on edge. There are no snakes in Ireland. Not because St. Patrick banished them like folks like to tell, but because of the island and climate issues. When I was a kid I was fascinated by snakes because I'd never seen one. I took a holiday after Mom died and Ro freed me, before she started controlling me, too, and went to a bunch of museums and zoos. I saw a rattlesnake. When it moved its tail, it had the same effect on me as these hooded Unseelie when they chitter. The dry, dusty rattle elicited some kind of atavistic response in me and got me thinking maybe genetic memory really does exist and certain sounds just make you want to run like hell.
What are they? How come I've never seen them before? What's their unique prey? How do they feed? How can they be killed? Better yet, why are they all peeling away from Mac like she has the Unseelie version of the bubonic plague?
There are too many people on the dance floors between us. I can't get a good view. I slide sideways into fast-mo, blow past Lor and Fade guarding the stairs at the bottom, making sure I catch Lor a good one with an elbow and snicker when he grunts, then stop at the top of the stairs and look down. Much better view.
The wraiths are chittering even louder, gliding back from Mac and Barrons, but it's Mac all those dark hoods are turned toward.
"Interesting," Ryodan says close to my ear. "You have to wonder why they can't get out of her way fast enough. I've never seen them do that before." Ryodan doesn't like Mac. He never has. She got between him and his best boy-bud.
I give him a look. "I'll tell you a secret, Ryodan. You mess with her, Barrons'll kill you." I drag a finger across my neck. "Just like that. You aren't all that. Barrons'll stomp your ass, hands down."
He smiles faintly. "I'll be damned. You have a crush on Barrons."
"I do not have a crush—"
"You do, too. It's all over your face. Anybody could see it."
"Sometimes, boss, you're just wrong."
"I'm never wrong. You might as well take out a billboard. 'Dani O'Malley thinks Jericho Barrons is hot.' My offer to teach you is still open. Save you from future embarrassment. If I can see it on your face, he can, too."
"He never figured it out before," I grumble, then realize I just admitted it. Ryodan has a tricky way of wording things that makes you say things you didn't mean to say. "Maybe I'll ask Barrons to teach me," I mutter, and turn away from the stairs, heading for his office. I run smack into his chest. "Dude, move. Trying to get somewhere here."
"No one but me is ever going to teach you, Dani."
He touches me before I see it coming, has his hand under my chin, turning my face up. My shiver is instant and uncontrollable.
"That's non-negotiable. You signed a contract with me that grants exclusivity. You won't like it if you try to break it."
I glare at him, wondering what the heck I actually signed. Kind of hoping I never find out.
"What are we doing here? Pansy talk or work? You got something else for me to do or not?" I glance over my shoulder one more time as I push past him. Barrons is standing in front of Mac like a shield, and I allow myself a quick flash of a smile. Ryodan is right, I need to learn to hide what I'm feeling. She's safe. She'll always be safe with Barrons in the picture. I never have to worry about Mac. Just about what she might do to me one day. I'd rather worry about that than Mac, so essentially all is right with my world.
# FOURTEEN
# _"Knock, knock, knockin' on heaven's door"_
It turns out Ryodan didn't have diddly-squat for me to do. There were no other iced scenes to visit so he made me hang around his office with him.
I wanted to go back out and examine the debris of the warehouse scene that exploded the other night, pick through it more thoroughly for clues (thinking I could move my hidey-holes at the same time), but he told me to study all the folks and Fae through the glass floor and see if I thought any of them might be responsible for what was happening.
I said, dude, you said you think it's happening spontaneously, like some part of Faery is bleeding through. Now you want me checking out individuals like they might be doing it. Which one is it?
He said both and went back to his paperwork. I don't think he feels the same sense of urgency I do, since it's only been humans getting iced lately and none of them on his turf. If he doesn't start showing me some investigative action, I'll be forced to work on it on my own time, and I don't know how to squeeze everything in, plus sleep every few days or so.
Mac left pretty quick. She seemed to get real nervous about what was happening with the ZEWs. That's Zombie Eating Wraiths for short, because that's what they look like. They had dirt and cobwebs on their cloaks, clues to where they hole up. I relaxed once she was gone. Then I got tense again having to watch Jo down there in the kiddie subclub, showing off so much leg to the Unseelie, and there's no question they were liking it. I'd like to have legs like Jo some day, all curvy and smooth-skinned and pretty.
No bruises!
She kept looking up at Ryodan's office with a weird look on her face, all longing-like, like she must have known I was up there. I didn't know she missed me so much! It made me feel bad for not spending more time with her. Sometimes she'd look real hard at the stairs like she was hoping maybe I'd come down.
I watched, sword hand itching all the while, because there were so many things in the club preying on humans that needed killing. By dawn I was a seething knot of repressed, homicidal _sidhe_ -seer thoughts, and not one bit wiser about who or what was behind the icings.
Two good things came from the hours I sat there till he finally let me leave. I learned about four new types of Unseelie and I composed my next _Dani Daily_. I plan to clean it up a little visually, make it even more professional-like before printing.
Now, sitting up on my favorite water tower perch, I read through my handwritten copy one more time, proofing it before I go to press.
The Dani Daily
May 24, 1 AWC
**Brought to you exclusively by**
**DANI MEGA O'MALLEY aka**
**I Give a Rat's Ass**
**and unlike IMITATION**
**newcomers**
**I always have been**
**YOUR ONLY CREDIBLE SOURCE FOR THE LATEST NEWS IN & AROUND DUBLIN!**
Who's been bringing you the facts about what's what ever since the walls fell? Me.
Who searched you out and brought food and news to your hidey-holes when you were too afraid to leave them? Me. Who carried messages, hunted for missing family members, and brought them home to you if they were alive? Dani Mega O'Malley.
Who dug through the rubble for wallets and IDs, and gave you back their things so you could grieve? It wasn't some fly-by-night organization that got most of their whole first paper's "news" out of being snarky about me. That's not news. That's slander. I give you facts you can use.
Who's been killing your enemies and teaching you how to fight for the past seven months? Who rounded up the children and took them to safety? Don't forget what you know is true just because somebody else suddenly pops up, imitating MY paper, making crazy fecked-up claims. I haven't seen any power or water running yet that isn't generator-powered, and, folks, I can hook that up for you.
ICare
Always will, Dublin.
Dani out!
I don't do rebuttals and I ain't got no love letters in me, so this'll have to do. Once I print and post it, I'm going to hole up and sleep like the dead for ten hours. Been up two or three days now. I always forget until I'm about to keel over.
I've been sitting on my water tower, looking down over the city, watching the sun come up. The air is clean like it never was before the walls fell. It's foggy but not smoggy like it used to be. I love living in a harbor town. Once, when I was nine, I stowed away on a fishing boat. They couldn't get rid of me until the end of the day because they needed the full day's catch. They finally ended up letting me ride up front, wind in my hair, salt spray in my face. The docks have always fascinated me with big ships coming and going places, tales of adventure and excitement stuck to their hulls like barnacles! Now they just sit, dead in the water like so much else. I've got a cool hidey-hole on one of them. I decide I ain't been there in a while and I'll catch some z's there later.
The sky is platinum, the sea slate, and the river Liffey is sliding down through the city, metallic. Fog spills silver lace over it all. Takes my fecking breath away!
I could admire it for hours but I got a job to do.
People got short memories. They get fear-blind and easily dazzled. Especially during times of war when the world starts looking so dark and gritty that shiny things start looking shinier. I got to keep reminding them of the things they know are true.
Me and Dublin, we're peas in the Mega pod. This is my city and my paper and I don't give up nothing that's mine without a fight.
I've never lost a fight yet.
Well, only to that fecker Ryodan. And there's no way he's behind WeCare. He's like, the antithesis of WeCare. He's, like, We-Don't-Fucking-Care all wrapped up with We'll-Eat-You-for-Lunch, too.
There goes my mood again. That's all it takes. One little thought about him. I have to go to "work" again tonight like some fecking slow-mo Joe, trudging along with the masses, and the unfairness of it all is now that the world has melted down, _nobody_ has to go to work anymore. Except me.
I bristle, realizing I can't go sleep like the dead once I get my rag up because I have to set an alarm. Me. I have to get up at a certain time!
I've never paid any attention to time. Dancer says I've enjoyed a luxury most people never have. He hates clocks and watches and everything that has to do with time. He says people already have too many lost days and that most folks live in the past or the future but never the present, always saying stuff like "I'm unhappy because 'X' happened to me yesterday, or I'll be happy again when 'Y' happens to me tomorrow." He says time is the ultimate villain. I don't really get that but that's probably because until this very frigging moment I never had to look at a clock for anything. I woke up when I felt like it. I went to sleep when I felt like it.
If I'm lucky, I'll be able to squeeze in five whole hours of sleep before I have to go back to "work."
I'm aghast at the horrificness of it all. Clock hands are ticking away my life at someone else's direction.
It's so wrong.
I wake up slow and careful, don't even stretch. I lay still, feeling the boat rock gentle on the waves. I love sleeping on my ship. Got it booby-trapped to the nines. _I_ got caught by one of them today, they're so good! I don't open my eyes because it takes me a while to get moving. Sometimes it can take me a half hour. That's why I set my alarm for seven instead of seven-thirty.
My alarm.
Was that what just woke me?
I don't remember turning it off.
I fumble for my cell phone. Signal might be dead but it still plays music and games. And has a stupid alarm clock.
I encounter an obstacle between me and my phone that feels like—
"Aiy _-eeeeeeee_!" I make a sound I didn't know I could make, part gasp, part shriek, and shoot straight up in bed, eyes flying open. What just came out of my mouth is so girly it makes me cringe so I grab my sword and swing it.
He knocks it out of my hand and it clatters across the floor.
I can't even say anything for a sex. I mean sec.
This is like my worst nightmare ever in the whole world! This is worse than all the ZEWs coming after me plus the devil and all the Unseelie princes, too!
Ryodan is in _bed_ next to me!
Sitting there, cool as you please! We're in bed together! He's giving me that faint smile and mocking stare. Guess he was watching me sleep. Did I snore? Was I flopped flat on my back with my mouth hanging open? I have no idea how long he's been here! How'd he get in? How the heck did he get past all my booby traps? Obviously I'm going to have to come up with some new ones!
I try to push him off the bed. It's like trying to budge a mountain. I hit him. Like a girl. Not even using my superpowers. Assuming I have them at the moment, the fickle fecking things. What good is it to be a superhero if you only are some of the time and you never get to know when?
He catches my fist and holds it.
I can't get my fist out of his hand. "Dude, give me some space here! I need room when I wake up! I can't breathe! Move!"
He laughs and I want to crawl under the covers and burrow deep and hide and pretend this is just a really bad nightmare and it'll be over soon.
"Get _off_ my bed!"
When he lets me go and stands, the mattress rises four inches on his side. I can't believe I didn't feel him sit down. Yes I can. I sleep hard.
"You're late for work, kid."
"What time is it?" I glance wildly around for my cell phone. I'm so sleep-discombobulated I can barely function. I spot it on the end table next to the bed. It's smashed into a gazillion pieces. "You broke my cell phone!"
"It was smashed when I got here. You must have done it when the alarm went off."
"It's not like it's my fault," I say crossly, shoving my hair out of my face with both hands. "I've never had to use an alarm before."
"Am I giving you shit."
"You're like, here!"
"That's because you're late for work, kid. Get dressed."
Clothes hit me in the chest.
I realize I have on my favorite pjs. They're flannel and have ducks on them. Maybe he didn't notice. I can't stand it. This is my place. It's supposed to be private.
"Captain's quarters. Pretty plush. Get moving. We've got things to do." He walks to the door and heads for the deck.
"Nice pjs, kid."
He takes me to a church.
Churches crack me up. They're like money, a conspiracy of faith. Like everyone agreed to believe that not only is there a God, but he comes down and checks on folks, so long as they hang in certain places, put up altars, burn lots of candles and incense, and perform sit-stand-kneel and other wacky rituals that'd make a coven of witches look not OCD. Then to further complicate it, some folks perform rituals, subset A, and other folks perform rituals, subset B, C, or D, and so on into an infinity of denominations, and call themselves different things then deny everyone else's right to heaven if they're not performing the same rituals. Dude. Weird. I figure if there is a God, he or she isn't paying attention to what we build or if we follow some elaborate rules, but copping a ride on our shoulders, watching what we do every day. Seeing if we took this great big adventure called life and did anything interesting with it. I figure the folks that are the most interesting get to go to heaven. I mean, if I was God, that's who I'd want there with me. I also figure being eternally happy would be eternally boring so I try not to be _too_ interesting, even though it's hard for me. I'd rather be a superhero in hell, kicking all kinds of demon ass, than an angel in heaven, wafting around with a beatific smile on my face, playing a pansy harp all day. Dude, give me drums and big cymbals! I like the crash and bang.
So, Ryodan takes me to a church and I stand outside looking in, stymied.
I mentally review the places I've seen so far that got iced: Chester's subclub, a warehouse on the outskirts of town, two small underground pubs, a fitness center, the rural Laundromat-family, and now a small congregation in a church.
I linger at the tall, double-door entrance, absorbing details because I'm in no hurry to rush in. The cold emanating from the interior is brutal, worse than any scene yet. My breath burns all the way down into my lungs, even with a good fifty yards between me and the front of the church where the folks are gathered at the altar in a frosty nativity scene. There are eight men, three women, a priest, a dog standing there, and an old man sitting at the organ. I hear more men than women survived Halloween, and in a lot of rural places women have become a wicked hot commodity with men tripping all over themselves to score one. The pipes of the organ behind the altar are covered with icicles, and the ceiling drips enormous stalactites. There's a frozen fog hanging around the entire interior. The priest is standing behind the altar, facing the others, his arms raised, like he was in the middle of a sermon.
"It's colder than any of the rest, which suggests it happened more recently, ambient temperature and all factored in," I say, and when I talk, my breath crystallizes in little clouds that hang in the air. I jerk with a sudden uncontrollable shiver. "Feck, it's cold!"
"Too cold for you."
I look at him. There was nearly a question mark at the end of that one. "Dude, you worried about me? I'm indestructible. When did you find out about this one?"
"Fade found it about forty minutes ago. He'd passed the church ten minutes earlier, and it wasn't iced. On his way back it was."
"So it _is_ the freshest one we've seen so far." I notice he's not pressing into the church in slow-mo like he has at prior scenes. Guess it's a little cold even for him.
I breathe in and out, fast and hard, bellowing my lungs, priming my adrenaline pump. "Let's do it."
I mentally pick myself up, shift gears and freeze-frame in.
There's cold and then there's something worse. This cold knifes into me and twists, catching gristle and bone. It slices down through muscle and tendon, razoring my nerves. But this scene is the freshest of them all, and if there's anyplace I'm going to find clues, it's here, before the temperature starts to rise and things change. If things do. I just don't know enough.
I circle the small gathering, shivering. I've stuttered with cold at other scenes but never shivered while freeze-framing. I think shivering is cool because it's the body's way of freeze-framing on a molecular level. Your cells sense the temperature is too cold for you, and your brain makes you vibrate minutely all over to generate heat. So I'm, like, freeze-framing twice right now, on a cellular level and on my feet. The body is a brilliant thing.
I look at their faces first.
They're frozen with their mouths open, faces contorted, screaming, same as the outdoor Laundromat-people. These folks saw it coming, too. All except for the priest who's looking startled at the folks standing there, which tells me whatever it was, it came from behind the priest and it came _fast_ because his head isn't even turning. He must have been reacting to the looks on their faces. It must have appeared and iced simultaneously, or he'd have had time to begin to look behind him.
I feel a little better about whatever's happening because twice now people saw it coming. That means I have a chance of getting out of its way if it comes in my direction.
"Save your. Observations and breath," Ryodan says at my ear. "Gather. Info and. Get out."
I look at him because of how he just spoke. Soon as I do I understand why he kept stopping and starting. His face is iced solid. It cracks when he adds, "Hurry the. Fuck. Up."
My face isn't iced. Why is his? I reach out without thinking, like I'm going to touch him or something, and he knocks my hand away. "Don't. Fucking touch. Anything. Not. Even me." Ice shatters and re-forms on his face four times before he completes the sentence.
Embarrassed, I whiz away, snap my mind up tight and focus on the details. I have no clue why I almost touched him. There's no explanation for my behavior. I think he put some kind of spell on me with his application.
What's happening at these iced places? Why is it happening? Is some inhumanly cold part of Faery really bleeding through? I understand why Ryodan thinks it is. At each scene, nothing appears to have been taken. I see no common denominators. Nothing was eaten. No one was harmed. Then why did it happen? I consider each of these iced scenes a crime. People are dead. Crimes require motive. I whiz back and forth, trying to discern some inkling of a motive, a hint of a sentient mind behind this. Looking close, for tiny injuries, say from something like needle-thin teeth. Are they drained of bodily fluids certain sick Fae consider tasty? The thought makes me think of a few Fae I should have killed. If I had, everything would be fine between me and Mac. She'd never have known. Still don't know why I didn't. Wasn't like I _wanted_ to get caught.
I see no signs of harm or foul play of any kind.
Then I see her and it's an instant heart punch.
"Aw, bugger!" I say.
I don't mind so much when adults get killed because I know they had a life. They lived. They had their chance. And hopefully they died fighting. But kids... well, kids just slay me. They didn't even get to know what a crazy, wonderful, amazing place the world is! They didn't even get to have hardly any adventures.
This one didn't get any adventures at all. She never even got passed the "Gee, I'm glad I got milk" stage.
One of the women is holding a baby girl with a halo of curly red hair just like mine, nestled in the crook of her arm. She has a tiny fist wrapped around her mom's finger and is frozen staring up at her mom like she's the most beautiful, magical angel in the world, which is exactly how I felt about mine before everything got so... yeah, well. So.
And something nuts happens to me that I don't understand, but I'm going to start doing what the rest of the world does and blame everything on my hormones because I used to be the coolest of the cool until I started having periods.
I get all mushy inside like some kind of wimp that buys into those greeting card commercials, and I think about Mom, and even though she did things to me that other people would think were awful, I understand why she kept me in a cage. There weren't many choices and she didn't have much money and she wasn't _always_ mean to me. She did it to keep me safe. I never blamed her for keeping me in a cage with a collar.
I just wished she'd stop forgetting me.
Like she didn't want to remember me.
Or maybe she wished she'd never had me.
But it wasn't always like that with us. I remember feeling crazy-loved. I remember when it was different. I just never could get it back.
And all the sudden there's like this stupid fecking thing so cold at the corner of my eyes on the insides like I tried to _cry_ or something and I don't fecking cry, and it froze the second it started and my head hurts and I reach out and I touch the tiny fist wrapped around her mommy's finger and my heart squinches and then I have this horrible pressure in my ears and then something inside me gives with a soft squishing sound, and all the sudden I can't breathe and I'm so cold I guess it must be like getting dumped naked in space.
The cold knifes into me, flays me, slays me, glacierizes me.
Cold takes on new meanings and just about when I think I understand it, like it's some complex state of being that I could exist inside of, it flips all around on me, and I burn everywhere and I'm hot, and I'm hot, and I'm so fecking unbelievably hot that I start tearing off my clothes and I can't do it fast enough because I feel thick and slow and stupid and I realize somehow I've dropped back down into slow-mo!
Was it when I touched her? Was that why he told me not to touch anything? Does touching something so cold knock you down from fast-mo? How does he know that, if it's true? Did it knock him down once somewhere, is that how he knew? Then why didn't it kill him?
It's too cold down in slow-mo, seriously like outer space.
I try to freeze-frame back up.
I stumble to my knees. I must have waited too long. Maybe the instant I dropped down was too long.
God, the floor is cold! It hurts, it hurts, it hurts! I just thought "God." I don't use that word. Do I believe? Have I found faith here, on my knees, now, at the end? That seems kind of hypocritical-like to me. Ain't dying a hypocrite. I start to snicker. I'm not shivering. I'm hot. I'm so hot.
Even now I try to absorb more details. Curiosity. Cat dying. May as well. It's a vacuum here. Something's wrong, something's missing that I couldn't feel missing in fast-mo but I don't understand what. The stuff around me, the people and everything feel... somehow flat, void of an essential ingredient that would give it multidimensionality.
"Ry—" I can't get his name out.
I hear him yelling, but I can't understand the words and it sounds weird. Like he's talking muffled into a pillow.
I try to skinny off my jeans. Need them off. They're cold, so cold. Have to get everything off. It's so cold it's burning my skin. He's fighting me, trying to keep them on me. _Get out of my way_ , I try to say but nothing comes out. I need them _off_. If I can get them _off_ I might be okay.
And all I can think is—
_Help me!_ I scream inside my head.
My heart is going. It summons up the energy for one last violent feck-you pump but only manages a soft squish.
I can't die like this. I have things to do. My adventure has hardly begun. Everything goes black. I see Death. Ain't so fascinating. It's a sledgehammer.
Aw, shit. I know what rigor mortis is. I know my face is going to stick. I'm choosing how.
I belly up a laugh from way down deep where I'm always half laughing anyway because being alive—dude!—it's the greatest adventure in the world. What a ride it's been. Short but stupendous. Ain't nobody can say Dani Mega O'Malley didn't live while she was here.
No regrets!
_Dani out_.
# FIFTEEN
# _"Hot child in the city"_
I lose track of them for one minute, distracted by a female Unseelie down in the streets that has what the Highlander in me considers revolting parts but the prince in me thinks are all the right ones. Sex has become bloody weird. Incredible. But weird. She's a few blocks south of the church, and she's throwing off pheromones that make my dick go flat to my stomach, and by the time I realize what's happened to Dani, I have one more reason to hate Ryodan and the whole fucking world, as if I needed one.
"No!" I roar as I rush for the edge of the roof. That's the bad thing about being a half-breed. The Highlander in me wants to take the stairs. The Unseelie in me wants to use wings.
Too bad I don't have any yet.
My heart makes the decision without me and tries to get to her the fastest way possible.
I jump.
I curse as I plummet four stories and brace for impact. Contrary to what she thinks, I can't sift yet so I can't cut out of this fall. What kind of idiot breaks all his bones at the precise moment his damsel needs him the most? Up to now I've been glad I can't sift yet. I think it's the point of no return. The day I can blink out of existence and back in at a mere thought, I'm no longer human.
I twist in midair, trying to land on my feet.
I'm astonished when it works. I discover new things about myself every day, most of which disgust me, but this is a welcome change. My center of balance has shifted. I pivot and realign flawlessly. My bones seem to have developed an incredible rubbery resilience. My knees bend slightly, bowing in a distinctly inhuman way to absorb the impact. I land like a graceful cat. I stare down at my feet, which are intact and functioning perfectly, and all I can think is bloody hell, I just fell four—
"Bring her OUT here! NOW, you buggering idiot!"
My head whips up.
Some teenage guy wearing glasses is standing outside the church, looking in, screaming at Ryodan. I have no idea who he is or where he came from. But he just said _my_ line, although I'd've done it minus the buggering part and with a lot more "fucks."
The kid's hands are fists and he's plastered up against the door-jamb of the church. His face and hair are frosted and he's shivering violently.
I push past him, shouldering him aside. "She doesn't need you. Worthless human. Get lost."
He snarls at me.
I laugh. Looking me in the face and snarling takes major balls. "Kudos to you, kid. Now take yourself off somewhere and die before I decide to cram those big balls you think you have down your throat." I shove into the church, so I can rescue Dani and kill Ryodan for taking a hothouse flower into the arctic zone.
The cold hits me like a brick wall and stops me in my tracks. A solid shell of ice forms on my skin. When I flex my muscles, the ice cracks and falls in a tinkle of crystals to the floor. I take another step and ice, mid-step this time, while I'm still moving.
I spent a small eternity in the Unseelie prison and never had this problem, and it was inhumanly cold there. I'm half Unseelie prince. I didn't think there _was_ anyplace too cold for me. How can that dickhead Ryodan tolerate it, if I can't?
I take another step, ice again, crack it and step back. It won't do me any good to freeze up like the tin man and become useless to her. I don't understand how this is happening. The cold in the Unseelie King's kingdom iced my soul and made me hate being alive. This is worse. I wouldn't have believed there was anything worse. There's something familiar about this place, this scene, this cold. Déjà vu. I despise this cold. It makes me feel bad in the center of my bones. Empty, hollow, somehow... flawed. I narrow my eyes, looking around.
_Dani!_
She's on the floor and it's not the cold that takes my breath away. Her jeans are tangled around her knees. She has on a black bra and underwear with little white skulls and crossbones all over them. She's thrashing her arms and legs and crying incoherently.
And I can't get to her. My girl is half naked and dying and I can't get to her!
I push forward.
I ice solid.
I crack it and pull back.
_Fuck!_
She's trying to kick off her jeans the rest of the way and he's fighting her, trying to keep them on. He needs to get her out of there. Why is he wasting time trying to keep her clothes on?
"Bring her to me!" I demand.
"Don't freeze-frame with her!" the kid on the steps bellows. He's got some lungs. "If you move fast, you'll kill her!"
"What the fuck do _you_ know," Ryodan says.
"Everything there is to know about hypothermia! And I'm willing to bet neither of you can warm her. Bring her to me if you want her to live! Stop trying to put her clothes back on. It's not going to help!"
"Fuck you, kid," Ryodan says, but he quits trying to dress her and scoops her up. Her jeans fall to the floor. She's mostly naked in his arms. I can't see past the red rage in my eyes.
"Don't move her any more than you have to! It'll force cold blood to her heart and she'll have afterdrop!" the kid yells.
Ryodan walks with her real slow and easy.
She's stopped flailing.
She's not making any noise now either. She's gone limp. Her arms and legs flop like a rag doll with each step he takes. If he killed her I'm going to beat him bloody and eat him piece by piece, slowly, with steak sauce.
It's all I can do to keep my feet rooted where I am and not attack him as he passes. Glorious, beautiful scenes of death and destruction, battlefields and torture chambers, crowd my mind, enticing, sexual, egging me on to smash and crash and raze everything in my path with no care for the consequences because there are no consequences for what I'm becoming.
When he walks past me, my fists drip blood. But I don't fight for her. If I fight for her, I could kill her. That would turn me into something worse than an Unseelie prince.
"You!" The kid stabs a finger at me. "I need sleeping bags, an aluminum blanket, and hot packs. Outdoor store on Ninth and Central. Get me sugar, Jell-O, and water if you can find it. Don't waste time if you can't. Same goes for a generator. Now!"
"I don't fetch for humans!"
But I'd cut the fucking moon out of the sky for her.
When I return with blankets and hot packs, she's on the sidewalk on the opposite side of the street from the church.
The kid with glasses is in his underwear. Apparently dickhead doesn't wear any.
Rage chokes me. I fight for control. The human part of my brain knows exactly why they took their clothes off. So they could bundle her in them. She needed everything they had. She's curled in a fetal ball, packed in their pants and shirts and jackets. The Unseelie part of my brain comprehends nothing but that two male dicks are way too close to something that's mine.
The kid is on top of her, on his hands and knees, with his face brushing hers like he's kissing her.
Ryodan looks like he's about to rip his head off. As I get closer, I see the kid is breathing just over her nose and mouth, letting his breath drift up her nostrils. I'm shaking with rage. My hands are fists again, bleeding from clenching them so tight.
"She keeps curling up," Ryodan says.
"Burrowing instinct. Freezing people do it when they're about to die."
"You let her die," I say to the kid, "I'll kill you every way a human can get killed, bring you back and do it all over again."
"Did you get what I need?" The kid thrusts a hand behind him, ignoring my threat. "Aluminum blanket. Now. And easy when you move her," he says over his shoulder, like he doesn't even know two homicidal maniacs are watching his every move and want him dead just for being so close to her. "Nothing sudden."
"Why aluminum?" I want to know exactly what he's doing so I can do it myself when there's a next time. I'd say that there's not going to be one, but since the walls fell there's always a next time.
"Superinsulation. Traps in heat. Keeps out everything else."
Ryodan and I place her gently on the blanket, then the kid stretches over her again. She's motionless. I can't even see her chest rising and falling. She's pale and still as death. It's a disturbing turn-on. I've never seen an Unseelie princess but I suspect they're like this: white and cold and beautiful. "Is she breathing?"
"Barely. Her body is using everything it's got just to keep her brain and organs functioning. She needs to urinate."
"You can't fucking know that," Ryodan says.
The kid doesn't turn his head or look at him, just talks straight up her nose. "She eats and drinks constantly. Her bladder is always at least partially full. Her body is wasting precious energy trying to keep the urine in her bladder from freezing. We need that energy directed at her heart. Ergo, she needs to piss. The sooner the better. We need her conscious to do that, unless you have a handy catheter."
"Get her conscious," Ryodan snarls.
"You're not putting a catheter in her," I growl.
"I'll do whatever I need to do to save her life. You. Bloody. Idiots," the kid says.
He pops open heat packs and shoves them in her armpits and groin. Then he stretches out next to her. "Roll us up in sleeping bags."
I look at Ryodan and he looks at me and for a second I think we might both kill the kid. Ryodan's more stone-faced than usual, if that's possible without turning to concrete, and his fangs are out. I look down. Ryodan's dick is as big as mine. "Why the bloody hell don't you wear underwear?" To an Unseelie prince, an exposed male dick is a call to battle.
"They chafe. Too small and confining."
"Fuck you," I say.
"Dudes. Get over yourselves," the kid says. "Roll us up. Do you want her to die?"
"You should never have taken her in there. I'm going to kill you for that," I say to Ryodan as I help roll up a nearly naked kid with my girl.
"I told her not to touch anything," Ryodan says. "I knew it would drop her out of fast-motion. I reminded her at every scene we went to. And bring it on, Highlander. Any time you think you're ready."
"And we all know how well she listens," the kid says dryly.
Ryodan gives him a look that would make grown, armed, psychopathic men shut up. "There was no reason for her to touch anything."
"Obviously she thought otherwise," the kid says, completely unperturbed.
"I was right there with her. I figured I could get her out."
"You figured wrong, dickhead," I say.
"I didn't think it would affect her so quickly if she did. It didn't do that to me when I tried it."
"She's not like you. And shut up, both of you," the kid says, and puts his face on hers again, breathing, cupping his hands around their faces to keep the warm air in.
"Why are you doing that?" I say.
"Warm air. Hypothalamus. Regulates internal temperature and will help raise her consciousness. I need her conscious so she can piss."
"I would have rubbed her down to warm her. Restored her circulation."
"Brilliant. You would have killed her. Her blood is too cold. It would have stopped her heart."
"I don't understand why she stripped," Ryodan says. I look at him. He's doing the same thing I am. Learning what to do if it happens again. Both of us would have sped off with her, trying to get her somewhere warm. And according to this kid, we both would have killed her.
"Blood vessels widen. She thought she was hot. Hikers get found all the time dead in the mountains, naked with their clothes folded nearby. They get confused. Brain tries to make order out of chaos."
"How do you know all this?" I despise that he knows it and I don't. Makes him the better man for her in this situation. I want to be the better man for her in every situation.
"Mom was a doctor. I nearly died of hypothermia in the Andes once."
"I almost killed you," Ryodan says.
"She can't hear you," the kid tells him.
"I wasn't talking to her."
"Give me more hot packs," the kid says. "Bugger, she's cold!"
"A few weeks back. I almost killed you."
The kid gives him a look. I think, what the fuck gives a kid this young the balls it takes to snarl at me and give dickhead a look like that?
Ryodan says, "I stood in the shadows of an alley you were walking down. You wouldn't have seen me coming. She would have died tonight if I'd killed you."
"Is that, like, an apology?" I mock.
"Does she gasp in horror every time she sees you, Highlander?"
I unfurl wings that aren't there yet and hiss.
"You both talk too much," the kid says. "Shut up. Don't make me tell you again."
We shut up, which I find hysterically funny.
I suddenly see us from above. I do that all the time now. I think it's because I'm losing my humanity and it's my way of marking my descent into hell. I observe that there's only one human male at this scene and it's not me.
I see a radiant woman-child who has more curves under her clothes than I guessed, and from the way Ryodan is looking at her, he didn't guess it either. She's bloodless, blue-tinged, rolled up tight in the arms of a half-naked teenager that could have been, should have been, me. Keeping vigil over her are two monsters of very different breeds but monsters just the same.
Death on her left.
Devil on her right.
The kid looks like I did at his age, except for the glasses and a few inches of height he has on me. Dark hair, great smile, wide shoulders, the kid's going to be good-looking.
If he survives past next week.
At the moment I'd wager strongly against it.
He's in a sleeping bag with her, holding her. She has skulls and crossbones on her underwear. It charms me beyond reason.
The way I see it, if it's not Ryodan in that next dark alley, it's going to be me.
# SIXTEEN
# _I fight authority and authority always wins probably always will_
I make a new discovery that totally blows.
Dying is the easy part.
It's coming back to life that sucks.
One second I'm gone. I don't even exist.
The next second, I'm on fire with pain.
I hear voices talking but I feel like somebody stacked weights on my eyes and don't even try to open them. I hurt so bad I _want_ to lose consciousness again. I groan, miserable.
"You said we could move her, so let's do it. Now. We'll take her to my place."
It's Christian. I wonder what he's doing here.
"She's not going anywhere with you. She's coming with me. If you're wrong and it's not safe now, kid, you're dead."
That's Ryodan. But who'd he call kid? The only person I know that he calls "kid" is me.
"I don't take chances with her. It's safe."
"D-D-D-Dancer?" I chatter.
"Easy, Mega. You almost died." He closes his hand around mine and I hold on. I like his hand. It's big and holds easy but sure. It's the kind of hold that says, _I got you if you want me, but I'll let go if you feel like running for a while_. "She's not going anywhere with either of you. She's coming with me," he says.
"The fuck she is!" Christian explodes, and I see flashing lights behind my eyelids from the hugeness of his voice and the pain I'm in.
Ryodan says, "She's weak, and you don't have what it takes to protect her."
"I'm n-not weak," I mutter. "I'm n-never weak." I slit my eyes open and the faint light in the street nearly splits my head. I close them again. Feck, I'm weak.
"The hell I don't."
"I sauntered right into your place and took her from you."
"I wasn't there at the time. Or you wouldn't have."
Ryodan laughs. "Puny human."
"She comes with me," Christian says.
"D-Dudes, I feel really s-sick," I say. "What's closest?"
"My place," Christian says.
"The hell it is," Ryodan says.
"You don't even know where it is," Christian says.
"I know everything."
Dancer says, "Chester's."
To him I say, "Take me there. And h-hurry. I'm starving and f-f-freezing."
When we walk into Chester's the noise just about splits my skull from temple to temple. I'm so sick I'm wobbly. Ryodan tells Lor to get blankets warmed and take them to a room somewhere upstairs. I hope it's soundproofed. Knowing Ryodan, it is. Like Batman, he has all the best toys. I don't care where I go right now. I just need to lie down. I want them to stop making me walk but I insisted that they let me walk, because I hate being carried so I'm faking. Every muscle I've got is burning and cramping. I can't think straight.
"Get the kid out of here," Ryodan says to another of his men.
Two men move in, close their hands on his arms.
"Leave Dancer alone!" I say.
"It's okay, Mega. I've got things to do anyway. You take care, you hear?" He looks at me hard and for a second I want everyone to go away and leave me alone with him. Life is so easy with Dancer. I want to ask him how he ended up in the street with me. I want to know what happened. Someone saved my life tonight. I want to know who and all the details.
But I don't want him here. Not in Chester's. I don't want the stain of it on him. "See you tonight?" I say.
He grins. "Hope so, Mega. Got a movie to watch."
"Get him out of here. Now," Ryodan barks.
Dancer impresses the feck out of me when he shakes their hands off his arms and says real calm, "I can see myself out." He doesn't shake testosterone off his skin like a wet dog. He doesn't turn into a stupid bull, throwing his horns around. He just takes care of himself.
I'd watch him go but Ryodan is suddenly turning me away, steering me like I'm a go-cart. He snaps an order for warm water and Jell-O and tells Christian to get the feck out of his club.
Christian laughs and settles on a bar stool in the subclub closest to the stairs.
As I hobble up the stairs, I see a funny thing. Ryodan pauses for a sec and I look back. He's looking out over the dance floor, down at the kiddie subclub, and like she can feel him or something, Jo looks up, straight at him. Almost like she's been waiting for this moment. Like there's some kind of rubber band between them and she can feel him if he tugs on it. I think her highlights are even more dramatic than they were a couple days ago, gold in her dark hair. She's sparkly between the boobs again—I wouldn't notice except the sparkly makes you look there!—and wearing pretty bangles on her arms. She never wears jewelry. Even sick as I feel, I think Jo looks _good_. Ryodan gives her an imperceptible nod and she goes real still and wipes her hands on her skirt and swallows so hard I see her throat work from here. They look at each other and neither looks away.
After a long moment Jo nods back.
And I think what the feck? Is she an empath like Kat? How did she know what he was saying? And what _was_ he saying anyway? And why is she turning her tray over to somebody else?
Then my legs are going out from under me because I faked as long as I could, and he's got me before I hit the floor, carrying me, and I don't even fight it because I'm too miserable.
They take me to a room a few doors down from Ryodan's office and put me in bed. I burrow deep into the soft mattress, sigh with relief and pass out cold. Ryodan pisses me off what can't be more than three minutes later by waking me back up and forcing me to drink warm Jell-O water.
At first I don't want it but it tastes like heaven.
"What happened?" I say. "Did I, like, die and come back?" What an adventure! I wonder if this'll get put into the legend of me when I do die. I wonder how many times I might kick Death's ass in my life. How wicked cool is that?
"Drink."
"Where'd Dancer come from?" My stomach cramps. "Aw, it's hurting my stomach."
"Stop gulping. Take small sips."
I see another funny thing when he pours a second glass of warm Jell-O water. "Dude, shake much?"
"I got too cold."
Lor laughs and gives him a look. "Or too hot. Get out of here. I've got it."
Ryodan looks at my empty glass. I've drained the pitcher already and I want more.
"I'll get it," Lor says. "Go do what you need to do, boss."
I wonder what he needs to do, why he's shaking. If this is his weakness, I want to know all about it. Too bad I'm about to pass out again.
Ryodan stands up. "Take care of her." He walks out.
Lor says, "Sleep, kid. I'll be back before you know it. With candy bars."
I slump into the pillows, curl up and sigh. Candy bars. Life is sweet. All I have to do is lie here where it's cozy and warm and wait for them. They heated blankets for me. Someone's bringing me candy bars in bed.
I'm going to sleep for days.
I wonder what happened. Dying to talk to Dancer. But it'll have to wait.
I'm drifting, just about to pass out again when I suddenly get wired, struck by a certainty that pisses me all kinds of off.
I know why Ryodan gave Jo that look!
Because they're in his office right now, talking about me! Conspiring, with Jo all worried about me because I almost died.
And they're trying to figure out what to do with me since I don't follow rules and almost got myself killed tonight. I hate it when adults have their stupid powwows about me! They always end with me getting read the riot act and handed a whole new list of rules that nobody in their right mind could possibly obey, most of which aren't even logical or smart.
How the feck was I supposed to know if I touched one tiny little thing it would snap me out of freeze-frame? Why couldn't he have just told me that? I would never have done it!
Thinking about how I didn't almost get myself killed tonight, really _he_ did, I start to steam from the inside and warm right up from sheer temper. I crawl out from under my huddle of blankets, get my sword, stumble to the door and wobble out into the hall. I look up and down but don't see anybody. 'Cause, like everybody's probably already in his office, dissing me.
I careen down the hall, stumbling from wall to wall, using them to steady me until I make it to his door, then I slap my palm where I always see him put his, and the door slides open. I don't even wait for it to finish opening before I begin airing my gripes.
"It is _not_ my fault I almost got killed, dude. It's _your_ fault and here's ho— _ooooww_ — _Ew_!" I shake my head, horrified and... and... and...
Horrified.
My mouth hangs open, with nothing coming out.
Ryodan looks over his shoulder at me.
He's got Jo in there but they're not talking. She's bent over his desk with her skirt up. And he's doing that thing I wish I'd never seen him doing. Holy travel agent! Did I, like, go through a time warp or something? How long did it take me to get here? Don't grown-ups do _other_ things before they get to this point? Like maybe hug and kiss, make out for a little while? I move fast and all but, dude! Kind of thinking some things'd be nice, a little slow, like maybe give you a chance to get ready for stuff that's happening!
Jo gasps and turns bright red. "Oh! Dani! Get _out_ of here!"
I'm seeing more of Jo than I ever wanted to.
They aren't talking about me.
They weren't even _thinking_ about me.
Like I wasn't even lying a few doors down the hall on my deathbed with obviously nobody worrying about me at all!
"You are such a traitor! Sleeping with the enemy! What's wrong with you? This is just too gross for my eyeballs!"
"Go back to bed, Dani," Ryodan says, looking at me funny.
I hate him and I hate her and I hate his stupid retracting door.
I can't even slam it on the way out.
I wake up feeling amazing. Usually I wake up confused and cross. I'm thinking maybe I should almost get killed more often. I have no clue why I feel so good but I love it so I stretch, milking it for all I can get. My muscles are totally smooth and happy and relaxed, and I don't feel a bruise anywhere, which is impossible. My muscles are always knotted somewhere. Bruises are me. This feels like a brand-new body! I figure I must be in some kind of pre-waking state I never been in before, where the brain's been turned on but the body's still numb. I feel candy bars in bed with me, melty in my warm nest. One's mashed between my cheek and the pillow, I feel another plastered to my butt. I scootch them both out, tear one open and eat it without opening my eyes, blissfully happy. I could get used to this. No pain from assorted bumps and bruises, breakfast in bed.
Then I remember where I am.
Chester's.
And I remember what I saw before I fell asleep.
Ryodan doing the dirty with Jo.
On his desk.
_Gah!_
Like I'm ever going to be able to look at his desk again! How am I supposed to sit in his office now?
I'm so pissed off I shoot bolt upright in bed and swallow the last half of my candy bar so fast it gets stuck in my throat.
I start choking and all the sudden a fist slams into my back. My mouth pops open and half a mangled candy bar goes flying into the glass wall with a gooey chocolate splat. It's too gross for me so early in the morning. My stomach heaves and I double over trying to keep it down.
Yeah, this is more like how I wake up. All screwed up and confused. When I lived at the abbey, Ro told me I have growing pains and that superheroes have them worse than most people. She said that's why I need to sleep so hard and deep, and wake up so slow, because my body has to do more work to repair me on a cellular level. Makes scientific sense.
"Might help, kid," Lor says behind me, "if you chew more than once before you swallow."
"I never chew more than once. I wouldn't be able to eat fast enough if I did. I'd have to spend my whole day chewing. I'd get jaw muscles the size of Popeye's biceps."
"You're too young to know who Popeye is."
When you spent most of your childhood in a cage in front of a TV, you know who everybody is. I can sing the songs for _Green Acres_ and _Gilligan's Island_. I even know who _That Girl_ was. I learned everything I know about the world from watching TV. There's a whole lot of psychology in there if you're paying attention, and I was a captive audience. Ro said I got all my melodrama from growing up that way. That I think folks are supposed to be larger than life like they are in shows. Dude, of course I do! But I didn't need TV to tell me that. Life's a choice: you can live in black and white, or you can live in color. I'll take every shade of the rainbow and the gazillion in between! I push up from the bed, grab my sword and head for the door.
Lor's in front of it, arms folded over his chest. "Boss didn't say you could leave."
"I didn't say your boss could boink Jo," I say real calm-like, but inside I'm seething. I don't know why I feel so betrayed. Why do I care? They're grown-ups. Grown-ups never make sense. Jo doesn't even like him. And I know he doesn't give a shit about Jo.
"Honey, boss don't ask nobody who he fucks."
"Well, he ain't going to do Jo again. Get out of my way. Move." I'm going to tell her I'm never talking to her if she has sex with him ever again. I'll make her choose and she'll choose me.
"So you can start some shit?"
"Yep." I don't even try to deny it. I'm ready to knock heads and I'm not going to feel better until I make somebody else as miserable as I am.
He looks down at me. I slant my jaw at a jauntier angle, and I can tell he's trying not to laugh.
"What? You think I'm funny?" I'm so sick of people smiling at me like that. My hand goes to the hilt of my sword. It closes on his hand. They're all faster than me. "I'm not funny. I'm dangerous. You just wait and see. I'm not full grown yet, but when I am, I'm going to kick your ass from one end of Chester's to the other. You just wait and see."
He lets go of my sword and moves out of my way, laughing. "Go ahead, kid. Raise some hell. Been boring around here lately."
On my way out the door I decide maybe I could like Lor. He lives in color, too.
When I blow past Ryodan's office, I think I feel a breeze and spin around real fast, ready to fight him if I have to, but nobody's there. I shake my head and bounce down the stairs, freeze-framing sideways in between steps because I have so much energy this morning, checking out the dance floor as I go. It's packed and the place is rocking. Looks like I either didn't sleep long or I slept a whole day until the next night, because there's Jo, waiting tables in the kiddie subclub, looking all long-legged and... Geez! I squint over the railing at her. Happy. She's, like, glowing! What does she think? That this is some kind of fairy tale she's living? It ain't. These fairies maim and kill, and the dude she's having sex with lets them. How can she glow about that? There wasn't even any romance or anything. Just... Gah! I don't even want to think about it. I can't scrape that memory off the inside of my skull fast enough!
I freeze-frame through the club, hyperfast, knocking folks out of my way left and right. Hearing grunts all around makes me feel better 'bout stuff.
When I stop in front of her, she looks startled then mad. What the feck does she have to be mad at _me_ about?
She removes the last drink from her tray, sits it on a napkin in front of a Rhino-boy then holds the tray to her chest, her arms around it like it's a shield or something.
"Traitor."
"Dani, don't do this. Not here. Not now."
"You did _that_ up _there_ ," I say, flinging my arm up toward Ryodan's office, "without worrying for one tiny little sec about _my_ here and now. The whole time I was practically dying, you were having sex two doors down with the dude you came to rescue me from. From his _dungeon_. Like, where he was holding me _prisoner_. Remember?"
"It's not like that."
"What? I wasn't in the dungeon? Or you didn't come to rescue me from him? Don't tell me you weren't having sex. I saw what I saw."
"I didn't believe he'd hurt you and he didn't. He didn't hurt either of us."
"He's got us both working like dogs for him! You're waiting on Fae, and I'm running around on his fecking leash! He feeds people to the Fae, Jo. He kills them!"
"He does not. He runs a club. It's not his fault if people want to die. What is he supposed to do? Talk them out of it? Start a Chester's counseling service? What do you expect of him, Dani?"
I stare at her in disbelief. "You've got to fecking be kidding me! You're going to defend him? Stockholm syndrome much, Jo?" I mock.
She moves to an empty table and begins to clear it, stacking dirty dishes on her tray. It makes me madder that she's cleaning up after these monsters. Doubly mad that she looks so good doing it. Jo's making herself prettier. I don't understand it. She used to be perfectly happy wearing jeans and a T-shirt and no makeup and just hanging with the girls. We had pj parties and watched movies. Now she's all superglam Jo. I hate it.
"I thought you didn't know what that was."
"I looked it up and, dude, you got it bad. You're letting him screw you every which way. How long do you think it's going to last? You think he's going to bring you flowers? You think you're going to, like, go steady with the owner of Chester's?"
She stacks a small tower of glasses on her tray and gives me an exasperated look. "Can we just not do this right now?"
"Sure. If you tell me you'll never have sex with him again, I'll go away. Right now. End of conversation."
Her mouth tightens. As she wipes the table off with a damp cloth, she glances up at his office. It pisses me off how soft her face goes when she looks up. The tension fades and she looks like a woman in love. I hate it. I hate him.
She looks back at me.
"No, Dani. I won't. And stay out of this. It's none of your business. This is grown-up stuff between grown-ups." She turns away and heads for the bar with her cluttered tray. Distantly I hear Fae calling orders, trying to get her attention, but I don't care. _I_ want her attention.
I freeze-frame in behind her hard and fast, causing a wicked breeze in the subclub and nearly knocking the tray from her hands. She has to work hard to catch it. Almost doesn't. Ryodan's not the only one that can screw with people and things.
"Don't walk away from me. I'm not done yet."
"Yes, you are."
I hiss in her ear, "Don't you get it? Dude's never going to love you. He's not wired that way. He's just using you and he's going to throw you away, and then there you're going to be like a dirty piece of toilet paper he doesn't want anymore."
She sucks in a breath and gives me a look over her shoulder that just fecking slays me.
I drown in instant self-hate for saying what I just said. And I hate him because I know it's true. Jo will never be able to keep Ryodan's interest. She's too good. Clean and nice inside. She doesn't have an ounce of malice or deceit or unkind feelings or anything bad in her. She's not complicated enough for him. He's twisted like that. I chose the wrong person to chew out. I should have chosen him. He's going to hurt her and I'm never going to forgive him. So, here I am, hurting her first. Dude, stupid much?
"Do you really think I don't know that?" If we weren't in Chester's, I'm pretty sure the wet in her eyes would start to slide down her cheeks.
All the sudden I'm miserable I said anything about any of it. I want to hug her. I want to run away. I don't want Jo to hurt. I should have kept my mouth shut. I can't keep my mouth shut. Grown-ups are so strange. But I don't understand! "Then why? Why would you do something that you know is going to end up bad? Why would anyone ever do something they know is going to hurt them?"
"You're too young to be talking about this kind of stuff."
"Aw, c'mon, Jo, it's me. I was never young. Life didn't happen that way in my world. Tell me."
"It's complicated."
"Like everything isn't. Try."
She doesn't say anything so I just stand and wait. A long silence usually makes people fill it up with something.
It stretches. Finally she looks away like she's embarrassed and, so soft it's almost like she's talking to herself, not me at all, she says, "Every morning he comes to the top of the stairs and looks down over the club and he stands there, so big and powerful and beautiful and..." She swallows hard like her mouth just went totally dry. "Sexy. God, so unbelievably sexy." Her eyes get a weird, intense look like she's remembering something, then she makes a soft noise and doesn't say anything for a second. "And he's funny. Do you know he's funny? You must know that. You spend a lot of time with him."
I fist my hands. Sure I do. I didn't know she did. What do they do? Crack jokes with each other like Dancer and me?
Her expression is far off, seeing a memory. "Every morning when the night shift ends, he singles out a woman in the crowd and he nods at her. She goes upstairs and when she eventually shows up in the club again she looks like..." She shivers like she just got goose bumps. "And you wonder what he did that made her look like that. You watch her walking around, smiling, moving different than she moved before she went to him, and you know something happened up there that made her feel more alive than she ever felt before, that she got to be the way you hope you'll get to be with a man, even if it's just once in your life. A man has to see women a certain way for it to be that way. You try not to think about him, but it doesn't work. I swore if he ever gave me that nod, I wouldn't go."
"Dude, wake-up call. You went."
"I know."
She's glowing again like she won some kind of prize instead of got picked by a class-A sociopath to be his disposable lube.
"Why _him_?" I don't understand and I want to. I don't want to feel like Jo's a traitor. I lost Mac. I don't want to lose Jo, too. "You know what he's like!"
"He's not a bad man, Dani."
"Bullshit."
"Everything isn't black and white like you want it to be."
Some things are, and Ryodan's blacker than black. He's one of the bad guys, period, end of subject. I'm pissed. She needs to wake up and smell the coffee burning before the whole fecking coffeemaker goes up in flames. "And when he comes to those stairs tomorrow and chooses someone else?" I say. "It's only a matter of time, Jo. You know he will. You'll be standing here looking all dreamy like you do right now and it'll be the waitress _next_ to you that he chooses and you'll never go upstairs again because a dude like that don't press the replay button. When he's done he's done. How are you going to feel then?"
She turns away.
I go after her, grab her elbow, make her stop. "Well? What do you think, Jo? That you're special? That you'll be the one that changes him? Give me a fecking break! You think you and him are going to go pick out china patterns together? Register for flatware?"
She inhales like she forgot to breathe, then when she remembered couldn't get air fast enough. "I know what I'm doing, Dani."
"Good, then you can explain it to me! 'Cause it sure looks like every shade of stupid from where I'm standing!"
She's distant again, talking soft, like I'm not even here. Even with my superhearing I lean in to catch it.
"There are men you build a future with, Dani. And then there are men that you know, going in, that you're only making a memory with. I know the difference."
Doesn't look like it to me.
"Some memories are worth the price. I'll deal with it."
But she won't. I know she won't. I know Jo. She's brilliant and kind and has the heart of a warrior but she doesn't have ice and razor blades inside where your soul is supposed to be. She loves. And she doesn't know how to take it back when you have to, because sometimes you sure as feck have to. Got to grab it up with both hands and pull it back before somebody turns it into knives and uses it to cut you to pieces. She's not going to be able to deal with it good at all. And I'm going to have to clean up the mess he made, and kill him. I suck in a breath. "You're too stupid to live and I'm not talking to you anymore. You need to pull your head out."
"You need to quit judging everyone."
"You don't know shit about me. And I'd rather judge people than be a pansy-ass that can't make her mind up about anyone or anything and gets sucked into all kinds of stupid shit."
"Dani, please don't—"
"My ears are full. I can't hear anymore!" I turn away and start to slip into freeze-frame. I have no idea what makes me look up. Kind of like a rubber-band feeling, like it's fused into my gut, and like something at the top of the stairs is pulling at my opposite end.
Ryodan is standing at the top of the stairs looking down at me. And I think what Jo said about him being big and powerful and beautiful.
We lock eyes.
Mine say, "Don't you ever choose her again. You leave her alone."
His say something I don't get at all. Then he does that ocular-shiver thing all over me and I get a real clear: "Go home, kid."
He looks right past me at Jo.
And he nods.
# SEVENTEEN
# _"These girls fall like dominoes"_
_We're not so different, you and I_ , Cruce says as he moves inside me. _Both born to lead_.
I try desperately to wake myself. I'm in the Dreaming and he has me in his wings. The moment I fell asleep, he was there, waiting for me at the end of a white marble path in a garden of exquisite blood roses. He lays me on them, with a crush of velvet petals. I brace for the thorns.
_You must not rue it, Kat. The sun does not rue that it rises_.
He goes deep, filling me completely, making every nerve ending in my body vibrate with erotic ecstasy. I arch my back and hiss with pleasure.
_We will rule the world and they will love us. We will save them_.
"Dreaming of me, are you now, my sweet Kat?"
Like a dropped snow globe, my dream world shatters and I remember why I asked Sean to stay the night with me at the abbey. Why I slipped him around back and into my suite of rooms. To save me from Cruce. To ground me to the world I know and love.
I roll into Sean's arms and press against him, shuddering with fear that I pretend is desire. We make love quick and hard and fast. He never knows I'm trying to erase someone else.
Someone that makes me come harder. Better. More.
Sean, my love, my childhood friend, teenage sweetheart, mate to my soul. I've never known life without him. We shared a playpen and went to our first day of school together. We got the measles the same week, swapped our first flu snuggled in blankets in front of the TV. We got pimples and got rid of them. He was there the night I started my period, and I was there the day his voice began to change. We know everything about each other. Our history is rich and long. I love his dark eyes, his black hair and fair Irish skin. I love the way he wears a fisherman's sweater with faded jeans and always has a quick smile. I love how strong his arms are from years of pulling fishing nets, and the way his long-limbed body moves, how he looks when he's lost in a good book, the way he feels moving inside me.
"Are you all right, love?" He brushes a tangle of hair from my face.
I lay my head on his chest and listen to his heart beating, solid and sure. Sometimes I think he has a touch of my _sidhe_ -seer gift, he reads me so well. He's known about my emotional empathy since we were children. Nothing about me disturbs him, a rare gift from those who fully understand what I do. Few can lie to me. I sense their inner conflict, unless they suffer no guilt or scruple about anything, and I've been blessed to encounter only a handful of those in my life—all of them in or near Chester's, recently. I don't know the truth, only that there is a lie. It takes a scrupulously honest man to love me. That's my Sean. We learned to trust each other completely before we were old enough to have learned suspicion.
"What if I can't do it?" I say. I don't elaborate. With Sean few words are necessary. We've been finishing each other's sentences since we were young. We were virgins when we made love the first time. There's never been anyone else for either of us.
Now I have an invisible lover violating everything I hold dear. Making me want _him_ and not my Sean.
He laughs. "Kat, sweetheart, you can do anything."
My heart feels like a rock in my chest. I burn with shame, and deceit. I've made love in my dreams in exquisite detail with another man, and have done so every night for over a week. I've taken him in my mouth, felt him at the entrance of my womb, places that are Sean's alone. "But what if I can't? What if I make mistakes that cost lives?"
He rolls onto his side and pulls me back into him, spooning. I press in. We fit together perfectly. Like we were carved from the same piece of wood, from the same tree.
"Hush, sweet Kat. I'm here. I'll always be here. Together we can do anything. You know that. Remember our vows."
I pull his arms tighter around me. We were young, so young. Everything was simple then. We were fifteen, deliriously and passionately in love, delighted by our developing bodies, growing up together into one. We stole off to Paradise Point out by the lighthouse, dressed up like it was our wedding day, and took vows with each other. We came from broken families, temperamental fighting families, and we learned from watching them. Too much passion burns. Tenderness fuses. We knew what it took to stay together. It was nothing fancy. Common sense, really.
_If you weaken, I'll be strong. If you get lost, I'll be your way home. If you despair, I'll bring you joy. I will love you until the end of time_.
"I love you, Sean O'Bannion. Never leave me."
"Wild horses, Kat. Couldn't drag me an inch. You're the only one for me. Always." There's a smile in his voice.
We make love again, and this time, when dark wings try to shadow me, they fail. There's no one else in bed with me but my Sean.
I watch him dress while dawn paints pale white rectangles around the heavy drapes. I have young charges at the abbey and we are not wed. We'd begun making plans to marry before the walls fell but our families interfered. The O'Bannions tried to stop it. When they realized Sean was having none of it, they tried to take over and turn it into the spectacle of the decade.
An O'Bannion marries a McLaughlin!
It would have been a grand step up for my family. We were small-time criminals. His family controlled nearly all Dublin's mob underbelly. I grew up with Sean because my mother was his nanny.
We'd been fighting bitterly with our parents for months before the walls fell and billions died.
Including our families. Where else would they have been than out in the riots, watching the chaos, trying to profit from the lawlessness?
I can't pretend that I'm sorry for their deaths, and I won't feel ashamed that I'm not. The only deaths I rue are those of my two half brothers that survived the fall, only to be killed by Shades. Rowena didn't teach us about eating Unseelie in time for me to save them. My parents and other siblings were corrupt to the core. Sometimes people are born into the wrong family. Sean and I turned our backs on them years ago. But our families never stopped pressuring us and never accepted letting us go. I used to worry so much about what they would do to Sean, how they might try to force him into the family ways, but now such worries are a thing of the past.
It's today and we're free!
As soon as we get a quiet moment, and a priest, we plan to wed. Some of the girls are hoping we'll decide to make a lovely ceremony of it here at the abbey. A wedding in times like these can be an uplifting thing, but I'll not make my wedding into something for another. It's between Sean and God and me.
When he holds my face in his hands and kisses me, I feel his heart, both against my chest and with my gift. He's happy. It's all I need.
He asks if I'll have him again tonight and I smile and kiss him.
"Aye, and every night thereafter and well you know it. If you're fishing for a compliment, my bonny Sean, I've thousands for you."
But as he slips out, my laughter dies and I stare at the bed.
I should tell him what's happening. I would wish it from him. I would fight for him at night against my invisible foe. We would stand together as one. I would know all the secrets of his tormenting succubus, the better to defeat her.
But I can't. I just can't. It happened before I could stop it the first time. I've had intimate carnal knowledge of another man. I've felt things with Cruce I've never felt with Sean. And I hate myself and I can't tell him. I just can't.
So I'm walking home slow-mo Joe style, pissed off but having a hard time focusing on being pissed off because my body feels so good. My mind's grumpy, but my body's saying, "Hey, dude, let's _play_!"
I kick a can down the alley and send it flying into a wall, and I do mean _into_ it. It flattens and gets impacted in the brick, and I crack up. Someday somebody is going to see it and be like, dude, what happened here? I leave clues about me all over the city, bending sculptures and broken streetlamps into twisty D's for Dani and Dude and Dangerous, leaving my calling card for folks to see. It's my Bat Signal, letting the world know somebody is out there, watching, caring.
I got a whole day stretching ahead of me and almost can't believe it! It feels like old times. I think about what to do with myself. Stupid as it is, I resent working on the ice mystery during the day because Ryodan's taking such a big chunk of my time every night. But I don't have the luxury of being stupid when folks' lives are at stake. It sure would be nice if I could get Dancer's superbrain in on it!
Trouble is, I should also head out to the abbey for a checkup. It's been a while since I was out there and _sidhe_ -sheep can get in trouble faster than I can waggle my ass and say _baa_. I got a worried feeling about them I ain't been able to shake.
Then there's Inspector Jayne. I'm pretty sure I'm due for a cage-cleaning session.
I mosey through Temple Bar, taking my time, drinking my city in, trying to decide how to prioritize my day. Kind of reveling in the simple fact that the choice is mine for a change! I used to love this part of town before the walls fell, so much cool stuff happening every night with tourists and pubs and new Fae to spy on and kill. I found out what it was like to live in these streets after mom died. No collar, no cage. Just a crazy old witch I learned to keep a little afraid of me all the time.
Then Mac came and the streets got even cooler. There's nothing like having a sidekick superhero to pal around with. Especially one that was part sister, part mom, and all best friend.
Now, like the rest of my city, Temple Bar is a mess. Abandoned cars, wrecked and stripped, are shoved up on the sidewalks, opening a tight lane down the middle of the street for traffic. There's broken glass everywhere from shattered windows and streetlamps; you can hardly take a step that doesn't crunch. Newspapers and trash and husks of what used to be people blow down the streets. On a gray, rainy day it can look real grim, if you don't superimpose a bright future over it. Mac's mom is heading up some kind of Green-up Program, and I hear her dad is working on a Cleanup Program, as well as hearing disputes and stuff, and one day Dublin's going to be rocking and full of _craic_ again.
I saunter past the bright red facade of _the_ Temple Bar of the district and feel it before I even turn the corner. I stop instantly.
It's like a breeze blowing down on me from a glacier.
I consider not turning the corner. I haven't investigated one of these scenes alone. I could nudge Ryodan this way tonight and pretend we just found it. It's not like they change too much between "recently iced" and "iced for a few days." Besides, if I turn the corner and find kids dead, it'll totally ruin my day.
Almost dying is fresh in my mind. If I'd been alone at the church last night... That's a weird thought. I can't imagine me dead. I look around, and up. As far as I can tell, I'm by myself. Christian can't be spying on me all the time. So, like, if I leave, nobody will know I'm not always a superhero. If I stay and something bad happens to me, well, my heart could stop, and there'd be nobody around to save me.
"Wussy girl! Get your cool back!" I just disgusted myself. I don't walk away and I don't need backup. Never have. A superhero isn't something you play at sometimes—it's something you _are_. Full-time, all the time, every day.
I flip back my long coat, liking the crisp leathery sound it makes, draw my sword and turn the corner, ready for action. My sword frosts white and my fingers get stiff with an instant chill.
In the middle of the street is one of those fancy cars Mac likes so much, totally iced, glittering diamond-crusted in the sunlight. An iced arm is sticking out the open window on the driver's side. A dude is hanging half out the passenger side, like he tried to climb out or something, mouth open on a scream, eyes closed, fist up in the air like he was trying to fight something off. No kids. That's a relief. Looks like only two casualties this time. That's another relief.
I study it, absorbing the details.
This scene's not so cold. Brutal, but nothing like the church or the subclub at Chester's. More like the laundry scene. I figure being outside, the frosted vignettes warm up faster. Piece of cake!
I take a couple of deep breaths, locking everything down on my mental grid, psyching myself up to freeze-frame in.
Just as I've nearly got it perfect, right exactly when I've almost got everything snapped into precise place and I'm preparing to shift gears smooth and easy, folks start shouting behind me and guns start going off.
Bullets can hurt me. I'm not _that_ superhero. It spooks me and I startle into freeze-frame before I mean to. That's even more dangerous than leading with your head!
I blast off wild, and try to get control of myself, but it's hard to do once I'm moving so fast. I whirl dizzyingly like a drunken Tasmanian devil and smash into the side of the iced car.
It knocks me out of freeze-frame but either doesn't catch me so much by surprise this time or the cold isn't as deadly as it was in the church or a little of both, because I manage to shove myself right back up into freeze-frame almost as fast as I dropped down. I can't get my feet under control, though, because I didn't get off on the right foot to begin with, and I slam into the car again and this time the people inside it blow like supercharged grenades into a gazillion shards of ice and I get sprayed by icy pink shrapnel.
Diamond-hard splinters of frozen flesh pierce every inch of my exposed skin. A thick dagger of ice as big around as a hot dog punctures my jeans and sinks into my thigh, and another impales my shoulder.
I get knocked out of freeze-frame again and push myself back up, and when I do, the ice splinters shove deeper into my body from the pressure of how fast I'm moving and it hurts so fecking bad that I drop back down instantly without thinking. Reflexive, just trying to stop the pain.
I start to freeze to death.
I push back up.
Ow! Shit, shit, shit, it _hurts_!
Down, I'll die.
Up, I'll only wish I would.
I stay in freeze-frame, stumble into the stupid car again, bounce back, careen off another car, and give it everything I've got in a violent effort to get out of the cold zone. I can't feel my hands. I can't feel my feet. Feck, I can't believe I did this! Who was yelling and why were they shooting?
I push, push, _push_ with all my might!
I collapse facedown in the street. Ice daggers bite deep. But I don't care. I'm out. I'm back around the corner where it's warm enough to live. I made it. At least the hundreds of splinters in me will melt now. Either they're already starting to or I'm bleeding a lot, because something warm and wet is trickling all over my skin.
I'm out of immediate mortal danger. I won't freeze to death. Now I just have to worry about bleeding to death.
It takes me three tries to manage to roll over on my back, and by the time I get there I'm panting worse than I do when I've freeze-framed for an hour, and shaking like a leaf. There's blood in my eyes. I try to blink it away. Dude, that was a grand debacle! How embarrassing! Glad nobody saw it!
I assess my situation without moving. I'm severely cut up. My skin burns where I can feel myself. The biggest threats to my survival are the holes in my thigh and shoulder, or what _will_ be holes when the ice finishes melting. I'll need to get them bandaged fast. The problem is, I can't feel my hands. I close my eyes, trying to focus on moving my fingers. Nothing happens.
"Ah, Dani."
I look up to see Inspector Jayne bending over me. I've never been gladder to see him in my whole life.
"You've certainly done it now, haven't you?"
"C-C-Candy b-b-bar," I manage.
He smiles but it doesn't reach his eyes.
"In m-m-m-my p-puh..." I trail off. I don't even have the energy to say _pocket_. I give him a longing, starving look and I know he gets the picture.
He looks across me. I realize I'm surrounded by Guardians. Good, they can carry me to Chester's and help me get patched up!
"Have you got it?" Jayne says.
"Got it, Captain."
I go ice cold in a way that has nothing to do with cars or frozen people. I try to lunge to my feet but succeed only in flopping on the pavement like a beached fish. "D-D-Don't you d-d-d-d-dare—"
"It's been six days, Dani."
Six days? How long did I sleep at Chester's?
"You should have come. If you'd kept your word, I might have continued trying to put up with it. But I won't allow the fate of our city to rest in whimsical hands. The sword is ours now, for the good of Dublin. We take far more of them off the streets than you do. In time you'll understand it always should have been this way."
"Y-Y-You—"
"Don't try to take it back. Your first warning is your final one. I won't treat you like a child if you do."
"K-K- _Kill_ y-you!" I explode. I still can't feel my hands or feet but I feel my head. It's about to explode. He has no right. It's _my_ sword!
"Don't make it war, Dani. You won't win."
I try to tell him he better kill me right here and now because there's no way they can keep my sword from me. I'll take it back the second I'm on my feet again. There's no place on Earth, feck, there's no place in all of heaven or hell that they'll ever be safe from me again! But I'm too light-headed to talk. Dizzy. My vision's getting weird.
"She's awful bloody, Captain. She gonna live?"
"She's tough," Jayne says.
"Maybe we should do something."
"We can't help her, not even a little, or she'll be able to take it back."
I flop on the pavement, unable to do a thing to stop them. I'm vulnerable, completely at his mercy.
And he's not having any.
I won't have any for him when the time comes.
He's leaving me here, to live or die on my own. I'll never forgive. I'll never forget.
They walk away. Just like that they leave me in the middle of a dirty street like a dog that got hit by a car, bleeding and helpless and alone. Dead if another car comes along. I'll remember that, too, when I see him again. Dude, they could have at least moved me to the sidewalk, balled up a shirt or something for a pillow beneath my head.
Something really bad happens to me then. Worse even than everything that's already happened to me in the past few days.
I feel woozy and strange and all the sudden it's like I'm outside of my body, watching me. But the me lying in the street has long blond hair and is looking up at the redheaded me with tears in her eyes and telling me she can't die yet because she's got people to protect. She's got a sister named Mac back home in Georgia and she just left her a message, and if she dies, Mac will come over to hunt her killer because she's stubborn and idealistic, and she'll die, too. But I don't seem to be able to feel anything about what's happening, and none of it seems real, so I walk away just like Jayne did.
My stomach heaves and I puke my guts out right there in the middle of the street. I can't even get on my hands and knees to do it. Lying on my back, I get sick all over myself. Not the blond-haired me that's the ghost of Alina, but the real, red-haired Dani that's really lying in the street wondering if she's going to bite it this time. And if there's something wet on my face that isn't blood or vomit... Nah. Ain't.
Eventually I get the feeling back in my hands and feet. I guess they thaw. I fumble for a candy bar. I curl in a ball in the street and eat every candy bar I've got and plot revenge.
Don't make it war, he said, and I won't.
I don't have to.
He already did.
# EIGHTEEN
# _"I can be your hero, baby"_
I find her stumbling through the streets, bleeding to death. If not for all that hair, I might not have recognized her. She's covered with blood, it's on her clothes, matted in her curls, crusted on her face. Her long coat is flayed and hangs in tatters from her shoulders. It looks like she went through a dicer.
I don't see her sword anywhere. I look around, nothing shiny in the streets but her.
I roar and she clamps her arms around her head and falls to her knees, and I remember how much noise I'm capable of making and kick myself. I deafened a human woman I recently had sex with. I broke her arm, too. I didn't mean to. I can't get used to what's happening to me. Try living your whole life one way then abruptly being something else. It's not easy to remember what you are every single bloody fucking second.
Except enraged. That, I'm aware of all the time. It never diminishes, never stops. The blackouts where I lose chunks of time are getting more frequent, lasting longer.
She topples in the street. I throw myself from the roof, land on the balls of my feet, and gather her in my arms. Where was I when she needed me? Fucking another faceless woman. Trying to defuse the constant lust.
She feels so slight against my chest.
I'm not surprised to feel myself trembling. I'm touching my goddess.
"Och, lass, what've you done to yourself now?" I push hair from her face. There's so much blood that I can't see what's causing it. How is she even walking? It makes me crazy that she's in this city, without a guardian or consort, always getting into trouble. I want to lock her up somewhere I can keep her safe forever. Someplace white and shining and beautiful, where nothing ever goes wrong.
Her brain has more muscles than her body, and less sense. Her passion for life pushes her limbs further than they were meant to go. She's going to burn herself to ash if she doesn't find someone or something that takes her all the way down to ground zero and recharges her. She needs to crash as hard as she lives or she'll die young. I can't stand the thought of her dying. If I knew how, I would make her Fae so she would never die. Doesn't matter that I hate being it myself, or that she would, too. Immortality is immortality.
I run with her, careful to move easy. I take her where I've pictured her a thousand times but knew better. I still know better. I'm doing it anyway.
Just one time before I turn into the villain of this piece, just one time before I become the fourth and final Unseelie prince, I want to be her Highlander. And her hero.
She'll remember, when there's nothing else left of me worth remembering.
I can't wait till I grow up enough that I stop having superhero growing pains. Waking up confused and cross all the time sucks. My hair is in my face and it makes me so mad for a sec that I almost tear it out at the scalp, trying to get it out of my eyes, and it's matted, then my bracelet gets stuck in it and there's something crusty—"Ew," I say irritably, then somebody else has their hands in my hair, trying to gently disentangle my wrist from my hair.
Who? What? Where?
I'm always doing a mental check when I first wake up, trying to remember what happened before I fell asleep so I can connect to where I am and how I got there. When I first got run of the abbey (dude, like a million times bigger than my cage with Mom!) I was constantly knocking myself out because I couldn't get over how far and fast I was able to run and giddiness made me a freeze-framing train wreck. I'm never real sure when I first wake up if I went to sleep or just managed to brain myself unconscious again. Then that fecker Ryodan knocked me out, too, and now I have to add that in to my worries when I wake.
Memory slams me upside the head. I get so mad I yank my bracelet off my head along with a good chunk of hair, and feel frantically for my sword even though I know it's not at my hip or anywhere else.
A man curses. My eardrums vibrate painfully and my head feels like it might split.
I open my eyes. "Christian, mute it!" I shove my hair out of my face and look up. I'm lying in bed and he's sitting next to me, looking down at me. Something's different. He doesn't look quite so scary. I take that back. Yes, he does, but either I'm getting better at reading his expressions or he's getting better at making them, because there's like an ounce of remorse in his iridescent eyes. Dude. His eyes are full Fae now! They weren't last time I saw him.
"Sorry, lass. But I almost had the bracelet free. You tore some of your hair out. You might have waited a second longer." He picks up the thatch of hair I yanked out at the root and flattens it straight between his fingers. Curl springs back instantly. "Stubborn as the head it came from," he murmurs. Then he does the weirdest thing. He puts it in his pocket. Maybe the dude collects hair. I got bigger worries on my mind.
"He took my sword! The fecker actually took my sword!" I can't believe it. I have no way to kill my enemies. I could hunt them all day long—and do a great big fat nothing with them when I catch them. It makes me so nuts I can't stand it. I try to push up from the bed but my legs aren't a hundred percent.
"Who took your sword?"
"Inspector Jayne. I'm going to kill him."
"HE DID THIS TO YOU?"
I get an instant migraine, flop down, cover my head with my arms and burrow beneath the pillows.
He sighs so loud I still hear it, even with my ears covered.
"Sorry, lass. Did he?"
I'm not taking my hands from my ears. I think about telling him yes so he'll go after Jayne for me, but I don't like to lie unless the payoff is huge. Lies are horny little buggers, they breed like rabbits and bound around just as insanely and then you have to try to keep track of them. "I got cut up myself but it's his fault. He startled me and I freeze-framed too soon." Speaking of cuts, I don't feel as bad as I did and I don't seem to be bleeding anymore.
"You want help killing him?"
He sounds a little too eager. Like homicidal maniac eager. "Don't need no stinking help," I say crossly. My eardrums hurt. "I mean not that your help is stinking or anything. Your help is totally cool. I just want to do it myself."
"Would you come out from under there, lass?"
"Would you, like, never yell again? You're slaying me. I got superhearing." I poke my head out. "Where am I?" I'm in a cloud of down pillows and comforters, in a high bed, in the corner of a huge room.
"My place."
I look around. Cool digs. He's holed up in a rehabbed industrial warehouse, one of those kinds with a giant living area that has no walls except for the ones you make with furniture and stuff. There's lots of brick and wood floors and exposed heating ducts, tons of light spilling in tall windows and a massive flat-screen 3D TV in front of a ginormous comfy-looking couch. There's a pool table and some old video game machines and an awesome bar, a kitchen with stainless steel appliances, and no racks or torture instruments visible anywhere. It's just the kind of place a college guy would die for—too bad he's not one anymore, but hey, we all have the things we need to pretend. No scary-looking knife collections. No red, no black, their favorite colors. The place is totally _not_ Unseelie prince.
A shaft of rosy light is shining on me and I look up. The bed is under a skylight, the sun's setting and it's got one of those new, strange Fae hues to it, brilliant orangey pink. I could sprawl in bed and watch the stars at night. I like it being pushed in the corner, giving me wall at my right and back, leaving only two sides to defend. Feels snug. Makes me think about rearranging some of my own rooms. I'm fascinated by the way other folks live, and love to look in other people's houses. "Aw, man, you ever move out, I'm moving in!"
"Like it, do you, lass?" he says, and his voice sounds funny. Thick and weird.
I look down at him and jerk. "Something on my face?" He's staring at me hard, eyes intense, and what's looking out of them doesn't look like it belongs in this place of brick and wood and sunshine at all. It belongs in the dark somewhere, with razor blades, about to do something real nasty.
"No. Your face is lovely, lass. The sunset looks good on you." He reaches a hand out toward my face and I go real still.
"Dude, you're scaring me."
He looks at me but it's like he's not seeing me at all, so I sit there with his hand about an inch from my face and look back and think about wild animals. About how they'll attack if they smell fear, not that I feel it, but when you're staring at an Unseelie prince, even though you know he started out human, it's kind of hard to predict what might happen next. This isn't a scenario I can lock down on my mental grid and freeze-frame through. This obstacle course has too many unknown variables.
He drops his hand without touching me, pushes up from the bed and goes to the kitchen. He braces his fists on the island and leans there with his back to me. He's bigger than he was when I first saw him up on my water tower. The back of his shirt is stained with blood and his spine presses knobby and weird against it. It's creepy.
I scoot to the exit side of the bed, thinking I'll just be moseying along now, when I realize I'm not wearing enough to get up. I only got on a bra and underwear. I sink back down and tuck my knees up. Not that I want to draw attention to the fact, but looking around yields no results. "Where are my clothes?"
"Destroyed."
He undressed me! He must have washed me, too, because I'm not covered with blood. Holy high wire! An Unseelie death-by-sex Fae that's having all kinds of temper problems undressed and cleaned me up. "Do you, like, have any other clothes I could wear?"
"Don't use that tone with me."
"What tone?"
"The one that thinks I'm some kind of freak predator that molests children. I'm not a freak and you're not a child. I undressed you, lass. I cleaned you up. I healed you. I will never hurt you."
" _How_ did you heal me?"
"Fed you my blood."
My gag reflex is instant and uncontrollable. I retch dry and noisy. Unlike a lot of other folks I know, drinking blood doesn't sound cool to me at all. It grosses me out. Same with eating Unseelie flesh; I've never done it and never will. I'm staying an Unseelie-flesh virgin all my life. I'm not even tempted by the possibility that it could make me stronger and faster than I already am. Dude, you got to draw your lines in the sand somewhere and hold them. It's especially important when the sand keeps shifting beneath your feet.
"It's potent. Works better than Unseelie flesh. A few drops in your mouth and..." He turns around and smiles at me. I think. Tattoos rush beneath the skin of his face, shadowing the planes and valleys, making it hard to decide just what that twist of his lips means. "There's really only one question: would you rather have died?"
That's an easy one for me. I'd never rather that. Not under any circumstances. I'll take survival at any price. Always. "No. Thanks for the blood, dude. Means a lot." I hate admitting this next part but I'm pretty sure it's true. "You saved my life. I won't forget it." I smile back at him, then I just sit there trying not to gape at his reaction. He totally changes and I see the Highlander he was. His eyes go brown and playful and he looks college-guy hunky again; the tattoos recede from his face. Even his muscles change, smooth, and suddenly his body is more human.
He tosses me a candy bar. I catch it, rip it open and munch it, and begin making plans to get my sword back. I know Jayne. He knows if I survive I'll come after it, so he'll take it somewhere he thinks I can't get to it. He won't want to waste too many of his men guarding it because he wants them out on the street, fighting. I waste a couple secs trying to figure out where he'd take it, then realize I don't have to. All I have to do is spy on him and follow him back to wherever he takes the Fae he catches to be slaughtered. I can't believe he's so stupid he really thinks he'll be able to keep it!
"Stay there and I'll get you some clothes," Christian says.
He lopes off, moving long-limbed and easy, not gliding in the weird way the princes do. Down the other end of the long room he rummages through an armoire, and comes back with a pair of flannel pj bottoms with a drawstring waist and a huge cream fisherman's sweater.
I suit up under the covers, tying off my waist tight and rolling up the legs and arms about a hundred times. When he tosses me a pair of balled-up socks as he heads off to the kitchen, I'm distracted, still thinking about Jayne, and miss them. They go sailing past me, hit the wall and fall down in the crack. I roll over and reach down, rooting around for them.
It takes me a second to figure out what I close my hand on.
Hair. Attached to a head. There's a head in the narrow space between the bed and the wall. I freeze, totally horrified and massively grossed out.
I jerk my hand back and just sit there, swallowing the creeped-out sound trying to claw its way out of my throat, then look over my shoulder at him. He's humming a weird song under his breath that sounds a lot like the music they play at Chester's and disappearing into a pantry off the kitchen.
I force myself to reach back down and pat around, never taking my eyes off the pantry door. "I'm hungry, Christian," I call. When he answers, I can make a good guess at how deep the pantry is, how far in it he's gone. How much time I have to figure out what the feck is going on here.
The head has a neck attached to it, and sure enough there's a body, too. It's naked and female and human. She's stiff with rigor mortis and ice cold.
I barely let myself breathe. I hear boxes being moved around on shelves.
"Sorry, lass, I'll have more for you in a second. I thought I had some Snickers in here but so far I'm only finding Almond Joy."
I yank my hand out of the crack and scoot back to the middle of the bed, and when I answer him I sound relaxed, playful. "Aw, dude, keep looking. You know how I love my Snickers."
The boxes stop moving. "Something wrong, lass?"
There's a dead woman wedged between Christian's bed and the wall. Normally I'd say that's a whole lot of wrong and I'd get real vocal about it, but I'm in the killer's apartment, wearing his pjs with no shoes and I got no fecking sword that kills Fae because that fecker Jayne took it, so I'm in no hurry to do that right now.
There's no way I tipped him off. My delivery was perfect. "No, nothing. Just starving out here!" Another flawless lie. I may not do it often but I shine at it like I do most things.
He steps out of the pantry and looks at me. The Highlander is gone. He's full Unseelie prince, iridescent eyes tinged with crimson. "Och, lass, Mac never told you, did she?"
"Told me what?"
"I'm a walking lie detector, Dani, my darling."
"Nobody is."
"It's inherited, like your _sidhe_ -seer gifts."
"Which I'm going to use to kick your ass from here to next week."
"And that was one big fat fuck of a lie. You found her, didn't you? I knew I should have put her away. But you were here, and bleeding so much, and I needed her off the bed. Saving you was all that mattered."
"So you shoved her off the side of the bed and thought I wouldn't notice? You stuffed her in a crack!" My hands are fists. The ignominy of it. Dead and disposed of like a used condom. If I hadn't missed the socks, I'd never have known. I'd have left thinking Christian was wicked cool for saving me and not been one ounce the wiser that I'd been in bed next to a dead woman, eaten and dressed without even seeing her two feet away. "Dude, you are one sick feck."
"Och, Dani, my love," he says, gliding toward the bed, "you've really no idea."
# NINETEEN
# _"I stand alone"_
I kick up into freeze-frame without even thinking. I don't lock one thing down on my mental grid. I hope I do a lot of damage, break everything I hit and just don't knock myself out, because I have a feeling if I do, I'll wake up strapped down to a rack with an insane ex-Highlander about to do seriously fecked-up things to me.
If he can sift, I'm dead meat.
I make it to the door but he's there in front of me, arms spread, crouched low, looking like he's about to bum-rush me and take me off my feet. His face is contorted with anger, kaleidoscopic tattoos rush beneath his skin. His eyes are full black. Only thing that's missing to complete the Unseelie prince picture is a radioactive torque and huge black wings spreading, getting ready to crush me in a deadly embrace. I backpedal frantically and he lunges.
Then I'm on the floor and he's on top of me, and I know the second he hits me that Christian is so much stronger than me that I don't stand a chance of taking him. I can't believe the strength I feel in his body! The Unseelie part of him has kicked in with a vengeance. It's not just power oozing off him. He's turning into pure sex just like the rest of them. I shake my head, trying to keep it clear. I think about horrible things like the dead woman stuffed into the space between his bed and the wall, and how I don't want to end up like her.
I'm flat on my back and he's got my wrists and he's stretching my hands above my head. I curse and struggle and kick but it's like fighting a concrete wall. Nothing, and dude, I do mean _nothing_ , seems to have any impact on him. I head-butt him. He laughs and drops his face into my shoulder, and sniffs me!
I bite his ear, try to tear it off his head. Blood fills my mouth, gagging me, and I let go.
"Dani, Dani, Dani," he says like he doesn't even feel it. "Don't fight me. You don't need to fight me. I'll never hurt you. Not you. You're my brightest shining star."
I ain't nobody's bright shiny nothing! He's a certifiable lunatic! "Get off me!" Up close the death-by-sex Fae part of him is doing wicked bad things to me. Things I don't like feeling. My mouth is dry and I'm seeing those graphic images plastered inside my skull. Christian. Naked. Doing the things I seen Ryodan doing. And I want to watch and I don't want to watch and I have to get the feck out of here now! "Can you even feel? Or are you as dead inside as that woman? Why did you even bother saving me? So you could kill me slower?"
"It's not like that. Would you hold still and listen to me for a second?"
"There's nothing you can say that matters!"
"It's hard to talk to you when I'm touching you."
"Dude, quit sniffing me! That's just rude. Get off me!"
"I can't. You'll run."
"If you really didn't kill her, you'll let me go. You'll trust me to come around. Give me room to breathe."
"If I let you go, will you sit down and hear me out, lass?"
He's relaxed since we're kind of negotiating but he's a lie detector and I know I can't answer that last question, so I take my best shot and knee him in the balls with everything I've got. There's no such thing as a dirty fight when you're fighting to win.
He roars so loud my head just about blows apart. Then he's off me and curled in a ball, howling. I've kneed a few dudes before. Got to out there on the streets sometimes. Never seen one react so bad. I wonder if it's because he was hard as a rock when I hit him, so I had to twist really hard to get to his balls and I came up on them from underneath and probably mashed his... uh, yeah, Mega, now's a good time to run.
I blow out the door so hard I blast it off the hinges.
This morning when I left Chester's after almost dying and coming back to life—I think it was this morning, I spend so much time unconscious lately that I'm never sure if I've been out for a few hours or a couple of days—I was trying to decide what to do with the rarity of a whole day of free time. But then I nearly got killed again, this time by exploding frozen people, then Jayne took my sword, then I passed out from blood loss, got cleaned up by an Unseelie prince and drank his blood, found a dead woman practically in bed with me, and now I'm out on the streets again and feck if it's not time for me to report to work again!
I can't decide which of those things is the worst.
Dude, sucky day. Free time, my ass. I barely survived it.
I bounce from bare foot to bare foot, freeze-framing with all I've got, skinning up my heels something fierce, waiting for Christian to sift into the spot in front of me. Knowing if he does, I'm going so fast I'll knock myself out cold on him and probably wake up dead. Knowing I don't have the one weapon that would protect me from him because Jayne took it.
Mac has one, though.
Bet I could take her.
Way I see things, I got three options.
Go to Chester's, use Ryodan as a shield against Christian while making him help me get my sword back.
Go straight after Jayne myself, knowing Christian is hot on my heels.
Go after Mac and take the spear. Barrons might be in the way. Who am I kidding? Barrons would definitely be in the way, and even if he wasn't and I took her spear, he'd come after me. Then I'd have Christian hunting me, Ryodan pissed at me for missing work, and Barrons breathing down my neck.
A day in the life of me. The stuff I have to put up with.
I'm always thinking things are as bad as they can get and they get worse. I nearly crash into something in the street, one of those fecking variables that move off my predicted grid, like people, animals, and Fae.
"Stay out of my way, human!" it hisses.
I want to drop out of freeze-frame and kick this monster's ass all the way to dead. I haven't seen her since the night Mac saved me from her, and forced her to give me back my good looks. I almost died that night, too. I almost die a lot. Superheroes do.
"You stay out of mine, you ugly old bitch!" I hiss back at the Gray Woman.
Then she's gone on her way and me on mine. She's off to hunt and kill and I've got an itch I can't scratch. My hand closes on nothing at my waist.
I need my sword like I need to breathe.
I detour into a sporting goods store, jam shoes on my feet, grab an oversized fleece pullover to pull on over my sweater because its fecking cold for May, and dash off again, heading for my best shot at success. Trying to take on Jayne and his men with Christian trying to kill me is a weak shot. I don't have any idea where he's taken my sword. There are times, like Ryodan said, when Batman needs Robin. Well, I don't _need_ Ryodan, but he sure will make it easier. He can watch my back like Mac used to. I've got no time for pride. I want results and I know how to get them. He's always telling me to ask. Tonight I'm asking.
I feel naked without my sword.
I feel exposed. It's throwing me all kinds of off balance like I don't even know who I am anymore without it.
When I blast into Ryodan's office, I'm going a million miles a minute, feet and mouth. Every one of his dudes scowls at me on the way in, even Lor, and I have no idea why. Guess Ryodan told them to be pissy to me or something. You never know what's coming next with him.
I blurt out what happened with the frozen car and Jayne and tell him how we have to go get my sword back like right now, like this very instant.
"Drop down, kid," he says without raising his head from his stupid paperwork. "You're messing up my office." Papers are flying around his head.
I drop down from hypermode and he looks up. He's looking at me weird. Takes me a sec to figure it out. It's like he's looking at a stranger. One he doesn't like and is thinking about killing. Why the feck is he mad at _me_?
"You reek of Highlander. The whole club can smell him on you. You're wearing his clothes."
I don't think I've ever heard him talk so soft. "Dude, who cares? Didn't you hear anything I said? Inspector Jayne took my sword!"
"Explain why you're wearing his clothes."
Softer still. If I wasn't so hot with temper I'd get a chill. I don't understand him. What does what I'm wearing have to do with anything that's actually, like, relevant? How could it possibly matter? I don't even understand why it registers! But I can tell by the look on his face he's not budging until I explain, and if I don't get my sword back soon, I'm going to go crazy. I also know if I tell him Christian killed some woman and I was next, he won't pay any attention to the problem of my sword, he'll go after Christian, when I need him to go after Jayne. I'm not sure he can take Christian. Not with what he's turning into. But with my sword, I know I can.
"The explosion cut up my clothes. He gave me some of his."
"You were together at the explosion."
"He found me after."
"And you changed in the street."
"Huh?" Stymied is me. This isn't where I expect the conversation to go at all.
"Elucidate upon where you changed."
"What the feck does that have to do with anything?"
"Answer me."
"I ducked into a convenience store. That's why they call them that. So they can be, like, convenient."
His gaze shivers up and down me. "If ice splinters tore up your clothes so badly that you needed to change, I'd think your injuries would be greater."
I gape at him, baffled. Somebody took my sword and he wants to talk about what I'm wearing and where I got dressed, and that he doesn't think I look hurt bad enough!
"He healed me. I was bleeding a lot. Holy hurrying hurricane, how'd you get next to me so fast?" Ryodan isn't behind his desk anymore. He's standing practically on my toes. I didn't even see him move. Or feel a breeze or anything! "Give me some personal space!"
He drops his head forward and smells me. "Healed you how?"
What is the deal with everyone sniffing me? If Dancer starts doing it, too, I'm so out of here. "I drank his blood. Got a problem with that?"
"Three."
"Huh?"
"I have three problems with that."
"That was a rhetorical question. Maybe you can't hear me talking or something so I'll say it again: Jayne has my fecking sword. I'm in deep shit without it and need it back. You going to do something or not?"
Just like that he's back behind his desk, head bent over his paperwork, all but ignoring me. "No."
I'm incredulous. "What? Why? You know I'll go after it myself! Is that what you want?"
"Jayne stopped by a few hours ago."
"That took a fecking lot of nerve! He left me for dead. In the middle of a street. Wouldn't even give me a fecking candy bar. Did he tell you how bad off I was? Why didn't you come help me?"
"You look fine to me."
"Whose side are you on?"
"He told me why he took the sword, and agreed not to kill any Fae within five blocks of my club. That's more than you do."
"Why would he agree to that? Jayne hates all the Fae!"
"He knew you'd come to me and ask me to help you get it back."
"And you're on _his_ side?" How dare Jayne predict my moves and avert them while I'm busy dying and then being chased by a homicidal maniac! All of which was his fault to begin with!
"Truth is, kid, I prefer you without it."
"Why?"
"You can't kill my patrons. And now maybe you'll start exercising caution. Or at least learn how to spell it."
I glare at his bent head. "I'm asking for your help here, boss. You keep telling me to, and I'm asking."
"I also said how you treat me is how I'll treat you."
"What am I doing wrong?"
"The answer is no."
"You've got to be kidding me!" I tap my foot hyperfast, hoping maybe I'll crack his stupid floor.
He doesn't say anything. Just keeps working on whatever it is he works on.
"You know what, dude? If you don't help me get my sword back, you and me are through! You solve the ice mystery yourself," I bluff, not about to give it up. "I'm not working for you. You don't help me, I don't help you."
"Jo." He doesn't even raise his head. Just murmurs her name.
"I don't care if you keep boinking her! Just get me my sword back! And don't be making any more deals with folks about me behind my back!"
"That's not our arrangement. You signed a contract. Jo's life is only one of many prices should you renege. There are repercussions for your actions. You can't walk away from me, Dani. Not tonight. Not ever. You're not the one calling the shots. Sit down." He's standing again, and again I didn't see him move. He kicks a chair at me. "Now."
Sometimes I think everybody else in the world knows something I don't know. Like they're all in on some kind of conspiracy and if I just knew that one secret thing, too, the things adults do that baffle me would make perfect sense.
Other times I think I know something extra that the whole rest of the world doesn't know and that's why nothing they do makes sense. 'Cause they don't know it and all their actions stem from flawed logic. Unlike mine.
I told Mac that once and she said it wasn't something everyone else knew; the missing ingredient was that I didn't yet understand my own emotions. They were new and I was just learning them for the first time. She said I was never factoring other people's feelings into things, so of course everything grown-ups did seemed mysterious and weird.
I said, dude, you just said I don't understand them, so how can I factor them in?
She said you can't, so just accept that teenage years are a great big clusterfuck of insecurity and confusion and hunger. Try to survive them without getting yourself killed.
A-fecking-men to that. Except for the insecurity part. Well, without my sword, plus the insecurity part.
As soon as I sit down, Ryodan says, "Get out of here."
"Bipolar much?"
"Go take a shower and change your clothes."
"I don't smell that bad," I say crossly.
He writes something, then turns the page in whatever-the-heck-stupid-thing he's reading.
"Dude, where do you want me to go? I can't go anywhere without my sword. I can't outrun the sifters. Every Fae in your club has a hard-on for killing me. You want me dead? Just do it yourself and get it over with."
He stabs a button on his desk. "Lor, get in here."
Lor blows in like he was plastered to the other side of the door.
"Escort the kid to clean the fuck up and get that stench off her."
"Sure thing, boss." He scowls at me.
I scowl right back.
Lor points through the glass floor. "See that blonde down there with the big tits? I was about to get laid."
"One, I'm too young to hear that kind of stuff, and two, I don't see you carrying a club to knock her over the head with, so how were you going to accomplish that?"
Behind me, Ryodan laughs.
"You're ruining my night, kid."
"Ditto. Ain't life at Chester's grand."
# TWENTY
# _"I've got soul but I'm not a soldier"_
_I am not the Sinsar Dubh, Kat. He has tricked all of you. You will need me to save you_.
Each night Cruce has taken me into the Dreaming, he has made the same claim. His lies hold the polish and consistency of truth. If my emotional empathy works on Fae—a test I've not yet had the opportunity to perform to my satisfaction—I get such conflicting signals from him that my gift is of no avail.
Now, wide-awake after another night of diabolical dreams, I pass through double doors a hundred feet tall, several feet thick, with unfathomable tonnage, but do not afford them a second glance. My eyes are only for him. It does not seem odd to me we cannot close such doors. The oddity is that we were ever able to open them: tiny mortals tampering with chariots of the gods.
I find myself in the position the Meehan twins recently occupied, hands fisted on the glowing bars of Cruce's cage, staring in at the iced vision.
He is War. Divisiveness. Brutality. Heinous crimes against humanity. As an event on the battlefield, and the personification of it in a cage, he is all that and more. How many humans fell before the murderous hooves of this sly horseman of the apocalypse?
Nearly half the world's population, by last count.
Cruce brought down the walls between our races. If not for him, it would never have happened. He arranged the players, nudged them where and when necessary, set the game in motion, then galloped about the board in the guise of an avenging angel, agitating here and stirring up there, until World War III began.
I should not be here with him.
Yet I am.
I told myself white lies as I made my way beneath the abbey, deep into our hidden city, picking through a misleading maze of corridors and crypts and dead-end and pigtailing tunnels. I told myself I must ascertain the cage is secure and he is still in it. That I will see him and realize he is but a pale imitation of my dreams; that I will gaze upon him and scoff at the thrall in which his dream-self holds me. That somehow coming down to check on him might set not him—but me—free.
My knees tremble. Desire parches my mouth and thickens my tongue.
There is no freedom for me here.
This close to him, I long to strip where I stand, dance wildly around his cage and keen the notes of an inhuman melody I do not even know how I know. This close to him I must bite my tongue to prevent myself from moaning with need.
This close to him I feel like an animal.
I stare at my hands on the bars, pale and white, with slender fingers clutching the glowing columns, and in my mind's eye I can only see them wrapped around that part of Cruce that has made of me an adulteress. Curled as they were last night and the night before and the night before. I see the curve of my lips as I smile. I see the soft roundness of my mouth as I take him inside it.
I find my fingers dancing lightly over the pearl buttons of my blouse and snatch them away. I see a shameful vision of my girls discovering their new Grand Mistress cavorting naked around Cruce's cage. It is erotic. It is horrific.
_Freedom terrifies you because you never permit yourself any_ , Cruce said last night in my dreams. _I am not the only one in a cage. The shame you feel is not about me but that you know you stand in a cage, too, and it is of your own making. You have felt the darkest emotions of others since you were a child, you know what monsters crouch inside them, and you confuse your passions with their monsters. They are not the same, my beloved Kat. Not the same at all_.
He says I repress passion. That I do not permit myself to feel any of it. He says my love for Sean is a lie. That I seek comfort and safety and do not know what love is. He says I choose Sean because he, too, feels no passion. He says we are not running toward each other in love, but away from things in fear. _Set yourself free_ , he says. _Come to me. Choose me_.
_God help me. I walk in a valley of darkness and I need your light to guide me_.
I unwrap my hands and back away. I must never come here again.
I will build a blockade of mental tricks in my mind, as I did when I was young and needed to protect myself from the wild, hurtful emotions of my family.
As I turn away I hear a noise so small I nearly overlook it. I don't want to turn back. It is nearly impossible for me to force myself to leave this place.
Yet I turn. I am the Grand Mistress here. The cavernous chamber, lit by a skein of torches on the walls, appears empty. There is nothing in it but a stone slab, Cruce's cage, and me. If I share this chamber with another, they are either behind the slab or on the far side of his cage. Hiding. Quiet. Waiting for me to leave.
Cognizant of my position at the abbey, I avert my gaze from the iced prince and sedately walk the circumference of his cage, head straight, shoulders squared.
I turn the corner. "Margery," I say. She is directly opposite where, moments ago, I stood. Had she made no sound, I would have left none the wiser.
"Kat."
Hostility buffets me in hot waves. The emotions of others have temperature and color, and when intense, texture as well.
Margery is red, fevered, and complexly crafted as a honeycomb, with hundreds of tiny deceits and angers and resentments tucked into each small nook. I know a thing about resentment: it is a poison you drink yourself, expecting others to die.
I've been classifying emotions into categories all my life. Navigating the hearts of those around me is a minefield. There are people I stand near a single time and skirt forevermore. Margery's emotions are deeply conflicted, dangerous.
I wonder if I could feel my own, I would also be hot, red, a honeycomb of lies and resentments. _But I do not want to lead!_ my soul is crying.
"I was wondering if we overlooked something about the grid," she says. "I fear he is not securely contained."
"As was I. As do I."
"Great minds." She offers a tight smile. Her hands clench the bars, white-knuckled.
I do not add the cued "think alike" because she and I do not. She hungers for power. I long for simplicity. I would have made a fine fisherman's wife, in a cottage by the sea, with five children, cats and dogs. She would make a grand Napoleon.
We assess each other warily.
Does he visit her?
Does he make love to her?
I cannot ask if she is dreaming of him and if that is what has brought her down here on this rainy, cold morning. Whether she is or not, she will claim she is not then tell the entire abbey that I am, that I am being corrupted and must be replaced.
She will use anything against me to take control of the abbey. At the very core of my first cousin Margery Annabelle Bean-McLaughlin is a great, sucking need. It was there when we were children, playing together, and she broke the knees of my dolls and stole small treasures from me. I have never understood it. I observe her white knuckles. She clenches the bars of his cage as if she is squeezing the life from something. "Your thoughts?"
She moistens her lower lip, looks as if she's about to speak, then stops. I wait and after a moment she says, "What if the King took the book? I mean, took it from Cruce before he iced him."
"Do you think that's possible?" I say, as if it's a perfectly reasonable question. As if I don't know in that instant we are both being fed the same lies.
She looks at Cruce then back at me. Her eyes are billboards, advertising her emotions. She regards Cruce with tender, private communion. She looks at me as if I could not possibly begin to understand the first thing about her, him, or the world we live in. "You are not gifted," she hissed at me when we were nine and she heard her parents praising me for saving the family from a traitor in the endless plots and plans and betrayals that were our life. My parents used to take me to "business" meetings with Dublin's seediest, and watch me carefully to see who made me most uncomfortable. "You are cursed and flawed and no one is ever going to love you!"
All these years later I see the same taunt in her eyes. Oh, yes, he is attending her nightly, too.
I am not only an adulteress, I am a cheap one. I shape that realization into a brick around my heart and slather it with mortar so it is ready for the next brick I can use. It will be in his way when he comes tonight. My Sean will be in bed beside me.
She shrugs. "Perhaps we don't know what really happened down here that night. What if the king tricked us?"
"Why would he do that?" I say.
"How could I presume to divine his motives?"
I need to know how deep her corruption goes. "Are you thinking perhaps we should free Cruce?"
A hand floats to her chest as if in alarm. "Do you think we should?" A crafty gleam enters her eye. "Do you know how?"
She has always been weaker than me. He is merely a blacker stain in her already corrupt blood.
"I think we need to figure out how to get the grid the Unseelie King created back up and functioning. I think the chamber should be filled with concrete, the grid reactivated, the doors closed, and the entire city beneath our abbey filled with lead."
I nearly stagger from the crippling fury of her emotional reply, although her lips shape sweetly the lie, "You are right, Katarina. As always, as everyone knows, you are right."
I offer my hand and she takes it as she did when we were children, lacing our fingers together. When we jumped rope, she would always pull it short. She had strong conflicting emotions about me when she was young that made her hard to read. I chipped four teeth before I stopped thinking the next time she would be different.
We walk from the chamber hand in hand, as if strengthening one another with love instead of keeping the enemy close.
# TWENTY-ONE
# _"I'm a cowboy, on a steel horse I ride. I'm wanted..."_
I ain't afraid of nothing. Never have been.
But there are some things that would be plain stupid to do. Got nothing to do with fear. It's all about logic and practicality. You look at the world, assess your odds of survival in light of current circumstances, and choose the course that offers the best chance for whatever it is you want.
Like, say, continuing to breathe.
I stand outside Chester's, staring up at a streetlamp in the scant light of dawn. The sky is one big bank of thunderclouds. It's going to be a dismal, wet day. Happy fecking May in Dublin. Cold, too. I'm starting to wonder if summer's ever going to get here.
Hanging on the side of the streetlamp is a poster. At first when I walked out of the club, I thought We-the-feck-Care had posted another paper in the few hours I spent cleaning up then sitting in Ryodan's office doing a great big fat nothing but glaring at the top of his head while he worked, trying not to think about what stupid purpose his stupid desk served so recently—like, did he disinfect it or what? Whole time I was there he wouldn't even look at me. Not even when he finally told me I could leave. I know I look bizarre in the clothes Lor gave me after my shower, but c'mon, get over it already. He didn't have to not look at me the whole time and make me feel even more stupid than I already do.
Back to the poster... despite what I'm wearing and despite not having my sword, I was going to freeze-frame around the city and tear them all down.
Except WTFC didn't post this paper.
Something worse did.
The flyer tacked on the lamppost is poster quality. Staring out from it, my face looks back at me in living color, full frontal and profile.
And I think: when did they take pictures of me? I study it, trying to remember the last time I wore that shirt. I think it was four or five days ago. There's no mistaking who it is. Anybody would recognize me in a heartbeat. They were either really close to me and I somehow didn't know it, which is inconceivable, or someone else took the pictures for them, or they had one heck of a good lens. I look pretty good. Well, except for the black eye and cut-up lip, but I hardly see those kinds of things on my face anymore. I'm used to the terrain, who notices trees in the forest? I squint. "Bugger. You kidding me?" There were guts in my hair whenever it was. I sigh. One day I'm going to have clean hair and no bruises. Right. And one day Ryodan will apologize for being such a total dickhead all the time.
The message is direct and to the point.
WANTED
**Alive
If you are human immortality
is the reward
If you are fae you will rule
beside us
she no longer has the sword
she is** **_defenseless_**
There's info on where to bring me when I'm found.
To the Unseelie princes. The fecking feckers have taken out a hit on me. I always wanted everybody to know my face, but not this way!
"Defenseless, my ass." Oh, yeah, they're pissed at me. And they aren't too busy fighting each other to hunt me. Or to be keeping constant tabs on me.
I look down the street.
A poster flaps on every single lamppost left standing, as far as I can see. I imagine they wallpapered the city with them.
"Aw, feck it."
Then I brighten. Dude, I'm worth immortality and co-rule! They put a wicked high price on my head! 'Cause I'm, like, wicked dangerous!
I want to go hang with Dancer, enlist his help getting my sword back. It took me nearly an hour to shake Lor. Ryodan's got him trailing me, making like my protective shadow. If I had my sword, Lor and me wouldn't have to put up with each other. I finally managed to get him distracted with what he likes best: blonde with boobs.
I tear down the poster and ball it up. If these hadn't been up, I would have already sped off into the morning, sword or no sword, taking my chances. This was a rude and unwanted wake-up call.
_She no longer has the sword_.
Gah, feckers! They just had to broadcast that, didn't they? I guess Jayne is already using it, and word got back to the princes.
_She is defenseless_.
Did they have to underline that word, make it bigger than all the rest _and_ red, too? I mean, what part of defenseless needs emphasizing? The word is bad enough! The whole bloody city is going to be gunning for me soon. Every big bad out there I ever beat up on, everyone I threatened or just irritated is about to learn I can no longer kill them. They already know I can't outrun the sifters. But having the sword always tipped the balance in my favor. Kept them all from trying.
I feel exposed, standing in the street. Anything could sift in behind me, grab me, and the fight would be on. Would I win? What if there were a dozen of them? What if humans come for me in a small army? What if the princes themselves come?
Gah, I'm what-iffing! I don't what-if! What-iffing is for grownups. They what-if themselves right into doing nothing, and die without ever living.
I turn around and look back at Chester's.
Then I turn back around and look down the street.
In front of me, high odds of death. Behind me, a cage.
I hate cages. For most folks, they're built from fear and they do it to themselves. Not me. Mine were forged of helplessness. Most kids' are.
So this is what it comes down to: death or a cage.
I grin. Dude, I'm a superhero. No contest.
I flip the street both birds and slide sideways into freeze-framing, ripping down the posters as I go.
I go hunting for Dancer and find him hunting for me in, like, no time at all. It cracks me up because what are the odds we could go looking for each other in the hugeness that's Dublin and actually find each other? But we always do. Like magnets.
When I see him, I grin. He's walking down the street in the gray dawn, glowing like a star going supernova. I can't look straight at him. I have to take quick looks at him from the corner of my eye. There's a bubble of light around him so bright it's blinding. He's wearing sunglasses over his glasses and looks like some kind of glowing Mutant X guy with a superpower of his own, like say Super Brain.
"Dude!" I say.
"Like it? Hold on, let me turn it down." He fiddles with something near his waist and the light dims to something closer to what my MacHalo throws off.
I check him out. His clothes are shiny. Shiny jeans, shiny shirt, even shiny ball cap. Clothes hang on his tall, lanky frame like something out of one of those glossy magazines, casual perfection. His hair's getting long again. He's going to ask me to cut it soon. I like those times. We take care of each other like two monkeys picking each other's nits. Folks underestimate a good nit-pick. "New fashion statement?" I tease.
"Thinking about your wardrobe, Mega," he says. "I was working on the spray for Papa Roach when all the sudden I got this idea for Shade protection. I need to spray your clothes with a reflective base, then I designed a harness of lights for you that runs off a battery system, and get this: it self-charges with motion!" He fiddles with a gizmo at his waist, wearing the rapt expression of a boy genius playing with electronics. All the sudden his head whips up and he grins and I just grin back because when Dancer grins like that all my worries disappear.
"Because of the way you move, it'll never go out. I've been testing it and it stays charged off even my movements for days. I figure one good freeze-frame will juice it up for a week. That means when you go to Shade-town, you can sleep easy, wearing it."
I'm speechless. Dancer was thinking about me, pondering the ins and outs of my life, so he could make it better. He spent his time working on something, not to save Dublin, like the Papa Roach spray, but just me. I fiddle with the bracelet on my wrist. He gave me that, too. It weirded me out when he did because I was afraid he was going to get mushy on me but that was way back in the beginning of us hanging together when I didn't know that Dancer never gets mushy. We don't let that kind of stupid stuff get between us. Using some of your own time to make someone else's life better is, like, the nicest thing you can do for anybody. I almost can't stand it, it makes me so happy.
"You're the Shit," I tell him.
And this time he doesn't say it right back at me, he says, "You think so?" like he wants to hear it again, so I say it again and his grin gets even bigger.
After a sec he notices the wad of posters I forgot I was holding.
He makes a sound of disgust. "Mega, I been tearing those things down for hours. I stumbled on one of the crews putting them up and followed them around, ripping them down. They've got a bunch of Rhino-boys hanging them. Is it true? Did somebody take your sword?" He looks me up and down, searching for it. He blinks like he just noticed me for the first time and I get so embarrassed it's all I can do to not freeze-frame right out of there. I feel so stupid!
I forgot what I was wearing!
My jaw juts and I say, stiff-like, "It's all they had that fit me. Ryodan made me change. I didn't have nothing to do with this getup. I wouldn'ta picked it in a million years!"
Dancer's looking at me like I'm an alien from outer space. I could just sink into the street, yank the concrete and trash over my head and hide. I hug my arms over my chest, cross my feet at the ankles and turn sideways a little, trying to make myself narrower so there won't be so much of me to see.
"I know I look stupid, okay? It's been a real sucky day for me and I got bigger problems on my mind than what I'm wearing so quit looking at me like I'm some kind of geek dressed up for Halloween, because I didn't have a choice since Christian gave me his stupid pajamas and Ryodan said they smelled—"
"Christian gave you his pajamas and they smelled? Wait a minute, Christian _wears_ pajamas?"
"I only needed his pjs because I woke up in his bed with only my bra and underwear on and all my clothes destroyed, otherwise I never would have worn them," I clarify when I realize how weird the first part sounded.
"Well. That explains things."
I love that about Dancer. He always gets me without me having to go on and on telling how point A got to point B. "All I'm saying is this ain't my fashion statement, so don't hold it against me."
"S'cool, Mega. You look cool."
"I look stupid." I'm so mortified I could expire of mortification.
"You look older. Sixteen or seventeen. If you had on makeup you'd probably look eighteen."
I think I'm nonplussed. I've never been nonplussed before but I know the definition and I imagine this must be what it feels like. It's not quite flummoxed, or bewildered. Words have subtle nuances. A year or two ago I might have been flabbergasted. This is a slightly different kind of stymied. Yes. I think it's nonplussed. "Well," I say, and smooth my skirt.
Gah! Feck! What are my hands doing to me? I actually just smoothed my skirt! Am I turning into some kind of sissy? I don't even wear skirts! But when Ryodan made me change the only thing they could find that fit me was the waitress uniform from the kiddie subclub and I was so pissed off about the posters, then so glad to see Dancer, that I totally forgot I was wearing a short skirt, snug blouse, and baby doll heels that suck to freeze-frame in, but I had more important things to do than dash into a store and switch shoes, like tear my face off every fecking lamppost in the city. Feet are feet sometimes; if they're working that's good enough.
"Who took your sword, Mega? And how did anybody even get it from you?"
My mood darkens instantly. I get so mad I get lockjaw and I can't talk for a sec. "Jayne," I finally grind out through my teeth. I rub my jaw muscles a sec and loosen my mouth back up. Superstrength sucks sometimes when it's in every single muscle in your body. When you get a muscle cramp, it's a big deal. It can go on for a good long while. "That fecker Jayne took it and left me for dead. I got hurt by one of those..." All Dancer knows about the iced scenes is what he saw the other night and it still hadn't exploded by the time they got me out of there. At least I don't think it had. It occurs to me I'm not sure. I need to ask somebody later. "I got hurt and Jayne took it while I couldn't do anything to stop him. I went to Ryodan and told him we needed to go get my sword back and he refused. Said he liked me better without it."
"Dude!"
In a single word Dancer just gave me all the righteous indignation and pissed-offedness that the situation deserves. "I know, right?"
"What is he thinking? You're the Mega. You don't take Wolverine's claws!"
"I know, right?"
"Dude," he says again.
We look at each other commiserating, because grown-ups are so fecked up and we're never going to turn out like them.
Then he grins. "What are we waiting for? Let's go take it back."
Since the walls fell, Dublin feels a lot like a movie set to me.
It's the quiet. The city is a ghost town with squatters hiding in the wreckage, rifles cocked. Sometimes I see whites of eyes gleaming at me through boarded-up windows. If they're human, I try to talk to them. Not all of them are receptive. There are some real nuts out there, as creepy as some of the Unseelie.
Before the walls crashed, back when I used to pedal around the districts on my courier bike, back when the _sidhe_ -seers were masquerading as an international messenger service run by Ro, the city was filled with a constant white noise. It was hard, even with my superhearing, to distinguish between the congestion of cars and buses, folks' heels on pavers and cement, planes landing and taking off, boats docking in the bay. Cell phones drove me crazy. There were days when all I heard was a blur of text message alerts, e-mail alerts, rings, songs, games.
Still, as annoying as it could be, it was music to my ears, the complex chords of the city I love. Now there are only the flat notes of soldiers marching, monsters hunting, and the occasional plaintive trill of something dying.
Dancer and I race through the streets, telling each other jokes, laughing our heads off. Hanging with him is the only time I can totally forget myself.
We round a corner and belly up to a contingent of Rhino-boys.
When they see us, one of them grunts into a radio, "Got her, boss, she's at Dame and Trinity."
I glance over my shoulder, lock everything down on my grid, grab Dancer, slide sideways and freeze-frame us out of there.
A short time later we're skulking around outside Dublin Castle, quiet as two mice sneaking around the kitchen looking for cheese.
Dancer's eyes are bright with excitement. I'd never freeze-framed him before. He said it was the coolest thing he'd ever done and wants to do it again. It used to make Mac almost puke when I did it to her.
After I hit a department store and changed into a cooler outfit of jeans, tennis shoes, and a new leather coat, we stopped in one of his digs I didn't even know he had and got some explosives. Some of the best plans are the simplest, less room for error. He's going to make me a distraction by blowing something up while I go in after my sword. I'll grab it, grab him, and we're gone. Then I'll swagger into Chester's tonight at eight and everybody will see you don't mess with the Mega. Ryodan'll see I don't need him for nothing.
"You were right," Dancer says, "the cages are crammed full of Unseelie waiting to be killed."
I snicker. "Jayne didn't know what he was getting into when he took my sword. I knew he didn't have enough time to kill six days' worth. Only way I can is I do it in hyperspeed."
Covered trucks are parked near the training green. We circle behind them. Fresh Unseelie bodies are piled in the back of one, still dripping. That means somebody is currently using my sword, and it's nearby. My fingers curl, aching for it. I don't know where Jayne disposes of the bodies. He has them trucked somewhere. I know his routine. I've been a part of it for a long time. His men patrol the streets, capture every Unseelie they can get their hands on and imprison them in iron holding cells in buildings behind Dublin Castle. The facilities are guarded, because several times in the past one Fae faction or another has hired humans to try to break somebody—or all of them—out.
Whenever the cages started getting full and I had free time I zoomed in, sliced and diced Unseelie, then loaded the bodies and trucked them out. It ran fast and efficient.
But only because I kill in superspeed. No slow-mo Joe can walk into a cage filled with Unseelie armed only with a single weapon, whether it's the Sword of Light or not. He'd be torn to pieces while he was still stabbing his first Fae.
Now, Jayne is being forced to separate out each Unseelie, take it out of the cage, kill it, separate the next, kill it, and so on for days. He'll need a full-time contingent to run it. It will take dozens of his men to replace me. And he was already short-handed.
"Mega, I know where the sword is," Dancer says.
"Me, too."
When I slay Unseelie, I do it so fast that there's not much time for the Unseelie standing nearby to react. They die quickly. Most of them before they even know what's happening.
But the way Jayne's doing it, they have to be standing around, watching the others get slaughtered for hours, watching Death inch closer.
I hate Fae. But there's something about knowing that they're just standing there, locked up, watching their buddies die a few feet away, waiting to be killed, that makes me feel... queasy. It's not like we owe them mercy—they don't show us any—but I figure if you're going to kill something you should do it quick and painless or you're just as sick as whatever you're killing.
I don't need my sword back just for me. I need it back because I'm the best person to do this job. Jayne needs to pull his head out and see that. This is fecked up, this drawn-out protracted slaughter.
Dancer's eyes aren't shining anymore. He looks as somber as I feel. I decide I'll make a show of good faith when I get my sword back.
I'll stay and slay, and put everything out of its misery fast and clean.
Then me and Jayne are going to sit ourselves down and have a serious talk.
I look at Dancer and he nods.
We head for the screaming.
The corrugated steel dock doors are wide-open on the warehouse, making room enough for two semis to back in and unload if they wanted to. Seeing into the building where Jayne is killing all the Unseelie isn't the hard part.
It's not being seen if someone looks out that's tricky.
The concrete dock is five feet high, and I've crept along it until I'm standing real close to the entrance, with just my eyes and hair sticking up above the side while I assess the scene and start building my mental grid. Even that small slice of me showing makes me feel too exposed. Having red hair is like wearing a neon sign sometimes. Dirty blond would blend with the background, mouse brown would merge nicely with the murky dawn, but my hair never fades into obscurity unless I'm backdropped by a crimson sky.
Dancer's off somewhere up high, laying explosives. Times like these I wish I had a clone so I could do the cool stuff I'm doing plus hang with him. I love blowing up things. But my part of the job is to whiz in, grab my sword and blast us out of here.
I was right about it taking a contingent to handle the slaying, although Jayne would probably keep that many around the sword at all times just to protect it from me.
As if that's enough to protect it from me!
Jayne's got two dozen men with him, toting automatic weapons, draped in ammo. They're standing inside the entrance at full alert, watching every move being made. I hate guns. Automatic weapons can dump a spray of bullets that's nearly impossible for me to avoid.
That's why I need the distraction. I need most of them gone before I'm willing to freeze-frame in, smash into Jayne and weave a zigzag path out of there, making it as hard as possible for anyone to shoot me.
I look up, scanning the rooftops around me. No snipers up there. If I were Jayne, I would have had at least six men up on the rooftops, watching for me. But that's why I'm the Mega and he's not.
I glance back inside and see my sword. Used to be, Ro took it from me sometimes, when I was younger. But when all the shit started hitting the fan with Mac, I took it back and never let anyone touch it again. Once, in battle, I saw Mac toss her spear to Kat to use. Dude, she's a bigger man than me. Ain't never sharing my weapon. It's my second skin. I can't stand seeing someone else touching it, holding it, using it. It's mine and he took it and he had no right to. I won't feel like me again until I have it back.
The screaming isn't so bad right now because Jayne isn't currently killing a Fae. But as I watch, his men bring a Rhino-boy up to the front of the warehouse near the dock and shove it to its stumpy knees on the floor in front of him.
Jayne draws back his arm, swings my sword and neatly decapitates it.
Not. I snicker.
Like maybe in his dreams. I see what's going to go wrong before it even does. "Holy webbed feet, it's going to duck," I mutter.
The Rhino-boy twists and ducks at the last second and my sword gets lodged in one of its yellow tusks.
I sigh. What does Jayne think their tusks are for besides blocking blows to their heads? Well, they use them for impaling, too, but mostly to protect their skulls and necks.
The Rhino-boy is enraged. Squealing and grunting, it nearly breaks free. Somebody shoots it, then Jayne's men wrestle it back down to the floor.
He yanks the sword from its tusk and when it comes out he stumbles. Somewhere an Unseelie guffaws.
Jayne regains his balance, raises his arm and swings again.
I wince.
Jayne is strong. But Unseelie are made of sinew and gristle and weird bony structure where you least expect to find it, and cutting their head off isn't as easy as it looks like it should be.
Now the sword is halfway through its thick neck and the Rhino-boy is gushing green goo, squealing like a stuck pig, flopping stumpy arms and legs, and hundreds of caged Unseelie start screaming again.
Jayne saws at the thing's head with the sword and I almost puke. His men don't look any happier. The noise is deafening. Rhino-boys are emitting one continuous high-pitched squeal, tiny winged Fae (the ones that make you laugh yourself to death!) are chiming with fury and making dazzling light displays as they try to escape their iron pens, slithering multilegged Unseeliepedes writhe between their pen-mates, and the sound they make is like several tons of gravel dumped onto metal sheets, getting dragged across it. Gaunt, slender wraiths flicker in and out of solidity, emitting a high-pitched whine. The sound is so huge I feel the vibration of it in the concrete dock beneath my palms.
Jayne finally manages to kill the Rhino-boy he's hacking away at, and turns to one of his men for a towel to wipe away the goo and blood. He looks back at the cages, his expression bleak. I snicker mirthlessly. No doubt he's got a new appreciation for the speedy services of moi! It isn't easy walking into a warehouse full of condemned monsters and killing them all. But each one that gets back out on the streets will ultimately kill dozens, maybe hundreds or thousands of humans, in its immortal existence. It's what they do. It's us or them.
I check my cell phone. My timer's counting down. I got seven minutes before Dancer sets off the charges. I would have gone for a single explosion but Dancer wanted multiple sites, the better to divide Jayne's men and increase our odds of getting in and out smooth and easy.
I stare at my sword. I'm fixated. I know it. I don't care. There are worse things to be fixated with. Like, say, Jo with Ryodan. Duh. How fecked up is that?
Jayne's men have emptied a cage of all but the tiny, death-by-laughter Fae. Now they net the brilliant little harpies and toss the nets on the floor in front of Jayne. Dainty, pretty Fae scream and shake their fists as Jayne swings again and again. Making the scene even more macabre, the men in the immediate area, including the good inspector, laugh helplessly, many of them doubled over with mirth, until the last one is dead.
The caged Unseelie roar and howl.
Because I'm a _sidhe_ -seer, I can sense Fae in my bones, in my marrow, in that strange hot/cold center of my brain other folks don't have.
Before the walls fell, when there were fewer of them in our world, my "spidey-sense" was a crystal clear beacon, warning me if one of them got too close to me long before it was close enough to be a threat. But ever since the walls fell, there are so many of them around me that my Fae-alarm is constantly going off, 24/7. Like every other _sidhe_ -seer that wants to remain sane—or just get some much-needed sleep—I've learned to mute it. If you don't figure out how to turn the volume down, you'll go crazy. It's not just an inner alarm saying, "Warning, a Fae is near." It rides tandem with a flash of pure rage, of prime directive to kill, kill, kill, and do it right now this very instant even if you have to use your bare hands to tear it apart. It's not something you can suppress. It's too strong. The older women at the abbey say it's like having the worst, most bloodthirsty hot flash imaginable, a hormone surge of pure homicidal fury. I don't want to live long enough to get hot flashes. Puberty is bad enough.
My Fae-sensor is on full mute right now. And even with it completely shut down, I feel it: a very powerful Unseelie is close to me, too close for comfort.
In order for it to be penetrating the barricade of silence I've erected around myself, its power has to be enormous. I nudge my volume up a hair, trying to figure out the who, what, and where. With so many Unseelie in the warehouse it takes me a few seconds to isolate the new arrival.
There it is!
I expand my awareness, taking its measure.
Ancient. Deadly.
Sex. Hunger. Rage. Hunger. Sex. Hunger. Rage. Hunger.
I feel it but I can't see it.
The short hairs on the back of my neck are tiny needles in my skin.
Suddenly a shadow moves in the gloomy, humid dawn and it's there, on the other side of the dock, hair and eyes barely visible. We're directly across from each other, no more than thirty feet of concrete between us.
It's not Christian this time. It's one of the full-blood Unseelie princes. Then again, after finding the dead naked woman stuffed between his bed and the wall, I'm not sure that's a meaningful distinction.
I go as still as Christian's dead woman.
It's not looking at me. It's watching Jayne. It appears to be completely unaware of me. I consider slinking down out of sight and cowering on my knees, focus hard on trying not to see all those graphic sex pictures on the inside walls of my skull like I'm seeing now.
_Hunger. Need. Sex_.
But I can't slink down because I don't dare take my eyes off it. I'm too dangerous to let a prince capture me, turn me Pri-ya and control me! _That's_ the argument I should have made to Ryodan! Without my sword the princes can take me hostage, turn me into one of their mindless sex-crazed slaves and use me as a weapon against him. I bet he'd have listened if I'd said that, but I didn't think of it because I was too pissed off.
I scan the edge of the dock but I see just the one prince. Where is the other? Holding my head perfectly still, slanting only my eyes, I peer down at my timer. I've got over four minutes before our first explosion goes off.
How did it find me so fast? Well, it hasn't found me yet, but apparently it knew where to look. Did we pass more Rhino-boys without realizing it, and they phoned my whereabouts in?
I stare, holding my breath, trying to decide if I should drop to my knees now or just try to continue not breathing or moving. I watch it while it watches Jayne, who's killing another Unseelie, and all the sudden I get this total epiphany: it didn't come here looking for me!
It came for my sword.
Now that I'm no longer the sword's guardian, the princes actually have a chance to take it and destroy it. It can't resist the opportunity to eliminate one of only two weapons that can kill Fae. It couldn't take it from me because I'm the Mega, but it thinks it can steal it from Jayne because he doesn't have any special powers. He's just a man.
Worst part is, it's probably right. It probably _can_ sift in and grab it before Jayne even knows what happened. It's an Unseelie, which means it won't actually be able to touch it because the Dark Fae can't touch the Light Hallows and vice versa, but I'm willing to bet it's got some kind of plan for that.
I'm fast but I can't beat a sifter. That's the whole reason I need my sword back so bad. With all the sifters I've pissed off, I'm a walking dead girl without it.
I envision the possible scenarios, starting with the worst first. I like to do it that way so I can end on the happy thought and aim for it.
One: the Unseelie prince sifts in, and kills everyone. He has one of his Pri-ya chick groupies with him whose head is currently not visible because it's somewhere lower doing something totally disgusting, and she picks up the sword, and he sifts out with her holding the booty.
Two: the Unseelie prince spots me, sifts over and kills me.
Three: the Unseelie prince spots me, sifts over, captures me and turns me Pri-ya. I refuse to follow that thought further. Bottom line: any version with the Unseelie prince spotting me ends badly.
Four: I drop to my knees and hide. It never knows I'm here. Dancer's bombs go off in quick succession. I freeze-frame in and take my sword while everyone is discombobulated. I kill the Unseelie prince in a dazzlingly display of dexterity and grace. Sonnets are composed about me.
I grin. I like that one.
I return my attention to the situation at hand and realize Reality—the impatient bitch—has made my decision for me. She does that a lot. You get busy planning your life, then it has the nerve to just go ahead and happen to you before you're ready. Before you even get the chance to aim yourself right!
It's one of the bad scenarios.
The Unseelie prince spotted me.
# TWENTY-TWO
# _"Your mind's in disturbia, it's like the darkness is light"_
The most scared I get is the most alive I ever feel.
I should collapse into a puddle of terror but adrenaline shoves a broomstick up my spine.
If the Unseelie prince gets within a few feet of me, I'll collapse anyway whether I've got a supercharged backbone or not. Nobody's immune to Fae royalty. Nobody's got any protection against them. The Seelie royalty keep their deadly eroticism mostly muted around humans as a courtesy. The Unseelie revel in using it on us full force. The princes have already turned hundreds of women Pri-ya. Nobody knows what to do with them. Folks can't decide whether to lock them up or mercy-kill them. Last I heard, they were keeping them locked up in what used to be a psych ward.
My superpowers are useless against the princes. All that sex and need and hunger wipes your mind clean of everything but lust that you're willing to die for. I saw Mac at her worst, when she was Pri-ya. She's the only person anybody knows of that's ever been brought back from the mentally shattered condition. It's one thing to have your body caged. I can't think of anything worse than losing your mind. I glance in at Jayne, desperate for my sword. He's currently using it to hack another Unseelie to death in front of a screaming, snarling, roaring audience. Without Dancer's diversion there's no way I'll make it past all those Guardians and guns. I glance at my watch. Still three and a half minutes to go!
"Hey, dude, what's up?" I say all nonchalant-like to the Unseelie prince, while I pull the pin out of one of the grenades Dancer altered, months ago, to cause a blinding, delayed explosion. I use them as Shade-grenades, tucked into a ball of immortal flesh. While we were at his digs earlier, I stuffed my pockets with all kinds of things. I cram a candy bar in my mouth with my other hand and say, "Check this out. It came off the sword before Jayne took it. What do you think it is?"
I lob it high, straight over the dock. The prince does exactly what I was pretty sure it would do: catches it. A human would recognize what I threw, but I'm betting it won't. Whether or not it does, its reaction isn't at all what I expected. I figured worst-case scenario, it would pitch it over its shoulder.
The fecker tosses it right back at me!
Like an idiot, I catch it, too. I think there are two kinds of people in life: those you can throw something at that will instinctively duck and bat it away and those that will instinctively grab it. I've always been a grabber. I'll take offense over defense any day. I rubberneck in freeze-frame and assess my situation: Jayne is clueless that we're here because he can't hear us over the racket the caged monsters are making. The grenade in my hand is going to explode in five, four, three—
"No, _you_ take it," I say, and lob it high, right back at the prince.
He catches it, closes a hand around it, and I see a flash of light in his fist. Then he opens his hand and black dust falls to the ground. If I could make out its expression, it might have just given me a total smirk.
Well, feck me. What is it made of? Galvanized steel?
Suddenly I know where the second prince is because the temperature in the space behind me just dropped forty or so degrees. The hairs on the back of my neck frost and I shiver.
I lunge into reverse freeze-frame but it blocks me and I slam back into its icy, powerful body.
Feck feck feck! I slam it into forward. It's there before me. I twist and duck but crash into it sideways. We do this whiz/block thing about ten more times, with me cramming candy bars in my mouth. We're moving like an orchestrated dance. It seems to read my body's smallest cues, anticipate my moves. The thing is wicked fast! All I can make out is a tangle of long black hair and the brilliant flash of kaleidoscopic tattoos rushing beneath its dark skin.
I drop low and roll past it, then spring up to flee, but it grabs me from behind and yanks me back into it. I can't stop shivering. I have to get away. It makes a sound against my ear that I heard a lot of Ryodan's men making on level four when they were all having sex, low and rough and strained. I hear myself making a noise I didn't even know I knew how to make.
I turn into a Dani-grenade, fighting with all I've got. It ain't happening to me like this!
I punch, I kick, I bite. It doesn't fight back. It bands its arms around me from behind and keeps me pulled back hard against its body, waiting for my fury to turn into something else.
And it does.
I'm losing myself!
I can feel my mind slipping away!
I'm changing into something I don't want to be and I can't stop it! Is this what Mac went through? How did she stand it? Three princes at once, then Cruce, too!
I don't want this! It isn't supposed to be like this! I'm supposed to lose my virginity in some awesome, superspectacular, sensational way. Not like this!
But everything inside me is going gooey like rich, warm, velvety chocolate fondue that's so thick and sweet and scrumptious that I want to swim in it, let it cover my head, take me down deep into a place where I don't have to think anymore and I don't have to fight anymore and I can just be without always having to struggle to stay on top and protect myself and win all the time.
I want to get naked here in the street. I want to do it every which way, standing up and lying down, doggy style and reverse cowgirl. Long black hair is tangling around my neck, sliding like hot silk. It's arms around me feel like the best slow dance I ever imagined, not that I imagine wussy things like slow dancing with Dancer or anything but I'm having a hard time breathing right, it's getting all shallow and caught in my throat.
It makes a sound like dark wind chimes caught in a storm, beautiful and brittle. The haunting melody scrapes across my nerve endings, turning each one into a tiny mass of orgasmic tissue.
I'm lost. I press back against it. It's hard where I'm soft and pretty much perfect in every way.
"Och, Dani my darling, you're not giving me a single reason to wait for you to grow up. You're giving me a thousand reasons not to."
It's Christian! I'm so glad it's him, not one of the other princes! I turn around in his arms and tip my head back. "Hi, Christian!" I beam at him. He's hotter than the other princes. I'm glad I got him. I'll take the others, too, but I want him first. "I _want_ to grow up. Now. Hurry."
"Not. Like. This."
I reach up and pull Christian's head down for kisses but he knocks my hands away. It makes me mad. I grab him again. He shoves me and I stumble.
He slaps me then. Hard, across the face. My ears ring from the violence of his blow. I wet my lips and give him a look. Pain isn't what I need. I need him to _ease_ my pain. I might be a virgin but my body knows how to move, what to do. It's kind of embarrassing, but at the same time I like it. Sex is powerful. It makes all your cells feel hyperalive. How didn't I know that? I want to explore it. I want to learn it inside and out like everything else I do. I feel amazing! Like I'm about to learn stuff I got no idea about and it's going to change me forever. When this is done, I'll be a woman. Not a child anymore. I'm fascinated by the idea.
I'm not ready for it!
I'm racing toward it, can't get there fast enough.
He slaps me again. "Stop looking at me like that. Get mad at me. Hate me for what I would do to you. I'll kill you if you keep looking at me like that! I'll fuck you until you die!" he hisses.
Suddenly the Unseelie prince who was across the dock from me is standing shoulder-to-shoulder with him. They begin to argue in Unseelie and I can't understand a word they're saying but I get the tone. The other prince is pissed.
A third prince sifts in. Or a second, if you're counting Christian as an almost-prince. They look so much alike, I wonder if maybe he's already gone through the final change in the short time since I saw him last. Yesterday, being so close to him didn't mess me up this badly. Did something happen to him overnight? Is it because there's more than one prince here and they amp each other up? Did he really just say something weird about waiting for me? My brain is a mess. None of my circuits are working.
I can't stand up to the princes. For all my superhero powers, here I'm nothing. I'm as weak and helpless and doomed as any other person. I'm a willing victim, ready, waiting, eager to be destroyed. I know with one part of my mind how horrifying that is, but with another part of my mind—a much larger one—I don't care. Being a victim to eternal pleasure sounds like the most perfect state of existence I could ever imagine.
I stare at them. My cheeks are wet. I want to look away but I can't. I wipe my face and my hands come away bloody from my tears. I try to back up but there's superglue on the bottom of my boots. The spell Christian had begun to shatter is weaving itself around me again and I can't do anything to stop it. I'm standing ten feet away from three death-by-sex Fae and I don't see any way to get out of this one. Could Christian actually protect me from them if I don't want him to? Because if they move even one inch closer, I'm not going to want him to.
"Get behind me, kid," Lor growls from somewhere behind me. It seems mere thoughts of Ryodan conjured his men. If I could move, I'd go limp with relief. I can't. I stand there.
Lor grabs me and shoves me behind him. He's got half a dozen of his dudes flanking him and they circle in around me.
They face off with the princes, and just when all hell is about to break loose between them, one of Jayne's men barks a sharp command because they spotted us, and the Guardians swing their guns our way.
Then the Unseelie trapped in the cages must see their princes standing outside because they start roaring and howling at the top of their lungs, I guess trying to get them to set them free.
That's when the first of Dancer's bombs goes off.
# TWENTY-THREE
# _"My pretty pretty thing. Do you want to freeze?... The Iceman cometh"_
Dancer planted the bombs up on the top floors because we try not to destroy whole buildings unless they're nests and need to be demolished.
When the charges start going off, the roofs blow sky high, one after the next, and rubble rains down on us.
Glass and bricks and chunks of drywall spray the street. The air is so full of dust that I can't see for a couple seconds.
We all start scrambling and ducking, covering our heads, even the Unseelie princes. I guess being immortal doesn't make you like getting slammed by a slab of concrete any more than the next guy. So we all start looking for cover, except Jayne and his men, who are already standing inside the warehouse and don't need it.
The bombs detonating didn't work as I'd planned. The Guardians were supposed to look outside when they went off, and see no one because I'd be hiding. Then they were supposed to go hunting for whoever was setting the bombs in the surrounding buildings, and I was going to take on Jayne and whoever was left.
Instead, they look out and see all of us because we're dodging falling debris and we're all doing it in slow-mo because you can't fast-mo through an unpredictable, unmappable rain of bomb shrapnel.
The Guardians start trying to line us up in their sights and bark orders for us to freeze and drop our weapons, which is ridiculous, like anybody's actually going to listen, but I guess old habits die hard. Nobody freezes or drops anything. And I wonder, doesn't Jayne get that me and the Unseelie princes want what's in his hand and we'll kill for it? Dude, if I was him, I'd drop it and run.
Once I'm pretty sure the biggest chunks of roof have hit the ground, I freeze-frame past Lor to take back my sword from Jayne. Only I slam into Lor on the way there because the fecker's faster than me.
Then we both crash into two Unseelie princes that weren't there two seconds ago and my head starts getting screwed up with sex thoughts again. Lor grabs me and we freeze-frame away. The princes take one look at Lor and vanish, too, leaving Jayne a sitting duck for me. I try to freeze-frame around Lor again, and again I slam right back up into his chest. Then we're all scrambling for cover, because a chimney just crashed to the ground and exploded in a spray of bricks.
"Why does everything leave you alone?" I say pissily as we crouch behind the dock. "You got some kind of Fae-repellent spray? Ever hear of sharing, dude?"
He gives me a look. His face is gray with grime. I taste mortar dust on my tongue. Stuff is still falling but the shower of debris is slowing. Dancer makes killer awesome bombs!
"Why don't you just let me get my fecking sword?" I make the argument with him I should have made with Ryodan. "If the princes turn me Pri-ya they can use me against you guys."
"All the more reason he should have killed you. But no, he bloody 'hires' you."
"I didn't ask to be hired. Fact is, I asked _not_ to."
"Then he makes me fucking babysit you."
"I didn't ask to be babysat either." I poke my head up over the dock. The princes are trying to get to Jayne! I try to freeze-frame around Lor again but I don't even make it two feet. I slam into him. Dude's a wall. No holes anywhere. This is getting old. "Get. Out. Of. My. Way."
"I'll get it for you."
"Why would you do that?" I say suspiciously. More like it he'll take it to Ryodan, who will use it as leverage to boss me around.
"Boss says I've got to keep you safe. He's had me shadowing you constantly."
"Nuh-uh. I would have noticed."
"You never see him when he's shadowing you either. And he's been doing it a lot longer than you think."
"Bull-crikey."
"I'll never get laid trying to keep you safe. You're a train wreck on steroids."
"Am not." Usually I'm cooler than cool. It's been a rough couple of days. "So, like, if you get it, you're going to give it back to me right now this very instant?"
"Didn't I just say so? Go hide somewhere and shut up, kid."
The Mega doesn't hide. "My ass."
"Can't possibly be worth what he thinks."
I have no idea what he's talking about but it doesn't have anything to do with me so I dismiss it.
I freeze-frame back toward Jayne the second he lets go of my arm. This time he's not expecting it because he thinks I'll just wait around like a wuss for somebody else to get my sword. Not. I snicker when I hear him cursing behind me.
Then I slam into Christian halfway up the stairs to the warehouse, blocking my way to Jayne.
Then Lor has me again and I kind of melt over his shoulder because the death-by-sex Fae punch Christian's packing is doing funny things to me, but it fades as soon as we get away from him, so I bite Lor because I hate being carried around like a sack of potatoes. If he feels it, he has no reaction.
"Stay the fuck away from the Unseelie princes."
"I'm just trying to get my sword. He got in the way."
"I said I'll get it for you."
"I want to get it myself!" I want to look Jayne in the eyes when I take it from him. He left me to die like a dog in the street. No mercy. Not one drop.
Lor dumps me and shoves me up against a wall. "Fade, Kasteo, get over here and keep her out of my fucking hair."
Then I've got two of his dudes on me, one on each arm, and I freeze-frame or try to but they weigh so much I end up buzzing around in drunken circles like a bug dying on its back because I can't get all four of their feet off the ground at the same time. One or the other keeps digging a heel in. We slam into the wall then stumble all over each other and the whole time I'm trying to watch what's going on with Jayne and the sword. "Let go of me!"
They don't. In fact, they don't even acknowledge that I'm speaking, much less breathing. They hang on my arms like deadweights and eventually I wise up enough to stop trying. Exercises in futility aren't me. They could hold me till I run my gas tank out and there I'd be. A noodle, and somebody would no doubt fecking toss me over his shoulder again and tote me around rather than give me a candy bar.
After a few minutes I end up standing there, pissed as all get-out, just watching.
And that's how I have a front row seat when the real circus begins.
The two original Unseelie princes keep sifting in, trying to get close to Jayne. Each time they do, Lor or one of his men is there, blocking their way.
Christian keeps trying to get to Jayne, too, and I realize he can't sift yet. He's moving at just under full sift mode. Still, he's faster than me. Fecker. Lately, seems everybody is.
Jayne is spinning in a circle with my sword out in front of him, trying to keep everyone from taking it.
The Guardians are spinning in circles, pointing their guns, trying to get a fix on something. Good luck with that. They can't even see anything that's happening, just feel the wind of everyone freeze-framing past them.
The hundreds of caged Unseelie are grunting and howling, stomping and rattling bars and making a deafening noise, and there's some kind of Unseelie in there that starts making a sound I've never heard before. It's enormous and dissonant and it sets my teeth on edge, crawls under my skin and makes me want to crawl right out of it. I'm not the only one it bothers.
"What the hell is making that noise?" Fade snarls.
"I know, right?" I want to cover my ears but they've got my arms so I clench my teeth and begin to hum real loud instead.
An Unseelie prince materializes in the middle of the whole shebang, Lor pops in directly in front of him, they smash into each other and careen off, then go slamming into a half-dozen Guardians who slam into Jayne, and all the sudden everyone is stumbling off the edge of the dock.
When Jayne falls, my sword goes flying straight up in the air, end over end, an alabaster column of light. I close my fingers like I'm catching it.
It's there, right there for the taking! I can almost feel the perfect weight of it slapping into my palm.
"Let _go_ of me!" I nearly yank my arms out of my sockets but they don't let go. I'm forced to stand there and watch as the princes, Lor, a dozen Guardians, and the latest intended Unseelie victim all try to position themselves to catch my sword when it comes down. One of the princes tries to spread his wings but the quarters are too close and he can't lift off. The other sifts into the air, and Lor lunges in a totally inhuman way and they collide in midair with my sword still going up.
Like I said, a total circus.
And that's when the freak show part of it begins.
I'm standing, wrists manacled by Kasteo and Fade, not going anywhere without somebody losing an arm, and since I don't have anything to cut theirs off with, I'm stuck like a fly in superglue, when all the sudden the air in front of the dock starts to shimmer, and I get this feeling I've never had before in my life. I've been worried on occasion. A time or two, like when the Gray Woman got me, I was actually a smidge scared. She was sucking the life out of me and I could feel it. Nothing wrong with admitting when you know you're in a scary place, so long as you don't let it mess with your head. I stayed cool, even tried to talk Mac out of not making any deals with the fecker, because most of the time deals made under duress come back and bite you in the butt with saber-toothed tiger teeth.
But this is different. I'm feeling panic with a capital P. Crazy, dumb, blind panic. All the sudden, for no reason I can figure, I'm ducking like a rabbit in the middle of a huge, open field with no cover for miles and the sky just went dark with hawks, flying wingtip to wingtip. Death seems that certain. One swoop, a rustle of wings, and I'm gone. All because of some weird spot in the air. What the feck? I'm panicking because of a shimmer in the air? Dude, what's it going to do to me? Give me a Twilight moment, make me all shimmery, too?
I'm torn between fighting to run and staying put so I can see what's happening because I can't conceive of anything that could panic me so bad and I need to see it! I'm tired of these eyeballs missing all the exciting stuff lately!
I realize I'm not the only one freaking out. Everybody that was trying to get my sword is suddenly scrambling away from the dock like they're running for their lives, which I take it to mean we're all in agreement about not liking unexplained shimmery spots in the air. I see my blade is still flying up, but it's moving slow like it's about to come back down. If I could just get Fade and Kasteo off my fecking arms, I'd rush in and catch it... well, maybe I would. I'm not real sure about that because my feet aren't obeying a thing I'm telling them about moving forward. Much to my annoyance, they're inching me backward.
The princes vanish.
Jayne and the Guardians are rushing straight for us.
Christian, Lor, and his men freeze-frame out, then Lor's replaced the other two dudes and has my arm, and he's dragging me away from the dock.
Then we're all retreating and I grin when I realize we're backing together, shoulder-to-shoulder, in tight formation. Jayne's next to Kasteo, who's next to Christian, who's next to a Guardian, and way down at the end are the full-blood princes, which totally freaks me because I can't figure anything they'd back away from. There are more balls in twenty feet of street here than there are in all of Dublin, and I'm proud to be swaying in the nut sack. We might fight each other, but in times of danger, we'll fight together. Dude!
A dark slit appears in the center of the shimmering spot. My panic increases exponentially. I'd turn and run but I'm anchored by two dudes that could hold the _Titanic_ during a tsunami.
The slit widens and belches thick fog. I shiver. Frozen fog becomes hard rime. Hard rime coated every person that got iced and died.
The caged Unseelie howl like banshees, and the one making that horrible screeling noise finally nails its hellish crescendo. The windows that didn't shatter when Dancer's bombs went off blow out now—not in slivers and chunks—they're literally pulverized, spraying the streets with glass dust.
The slit widens. More fog puffs out, milky and cold. The temperature plummets.
"Hold!" Jayne shouts, and we stop.
Fade says, "What the—"
Sound ceases.
The world goes silent.
Utterly.
Still.
Did I lose my hearing? Did the Unseelie's crescendo deafen me? I can't even hear my own breath in my ears like when I'm swimming underwater. I look at Lor. He's looking at me and pointing to his ears. I point to my own and nod. Everybody is doing the same thing. Least if I lost my hearing, we all did.
I look back at the widening slit and the oppressive silence grows.
It's worse than a vacuum.
It's. Awful. It's. Messing with my. Head. It's...
Void.
Disconnect.
Feels like being dead.
But there's something...
I slide into my _sidhe_ -seer center and extend curious tentacles...
I get a mishmash of impressions but I can't find words for them because what I'm feeling is beyond my ability to comprehend. Like I'm three-dimensional and what I'm feeling is six or seven dimensions. It's...
Complicated.
Ancient.
Sentient.
I try to get a read on its... well, mind for lack of a better word, and all I get is a weird flash of... calculation?
Something missing. Something being searched for.
I look at Lor and see an expression on his face I've never seen before and never thought I'd see.
Fear.
It worries me. A lot.
He looks at Fade and Kasteo and they nod. He tightens his grip on my arm.
The slit widens and it comes.
Holy fecking crikey, it comes!
#
# TWENTY-FOUR
# _"And the beat goes on"_
Cruce came tonight as he always does, stealing my sleep, parting my lips and thighs, leaving me near dawn in tangled bed linens, soaking with sweat from sex and shame.
In the few moments of rest I snatch before rising, I have a terrifying dream.
I walk to the hidden entry of the catacombs with the shuffling, mindless gait of a woman dead and risen from the grave.
Margery blocks the door of stones that looks like any other wall unless you are privy to its secret. She is voluptuously nude, hair and eyes wild, smelling of him—a scent I know too well. A banshee, she shows sharp teeth in a cackle and tells me he is gone. I am too late.
With violence of which I did not believe myself capable, I shove her aside, and when she slams into the wall, she slumps down it and is still. Blood blossoms behind her head, staining red daisy petals on the wall.
Bemused by the hostility in my heart, I pass through the door and shuffle on.
The tunnels are pitch, forcing me to feel sightless passage along damp stone walls. This is not the Underneath I know: dry and well lit, with everything in its place. In this dark, moist maze, moss grows thickly on walls and bones crunch beneath my feet. The odor of decay couples with some fecund scent on the breeze. There is nothing down here to generate wind unless a thing stirs that cannot possibly be stirring.
I pull my wrapper closer and thrust myself forward on hobbled feet with stunted, stumbling steps, blind in eye but not purpose. I pray, and with the whimsy of dreams the gold cross I wear upon my neck begins to glow. I do not deserve such comfort in this dark night of my soul!
I shuffle for time uncounted through darkness, until finally I reach the chamber wherein the erotic, deadly prince is iced.
There, no darkness preys, no moss grows, no water trickles. There are no bones in this forbidden place. Only flesh. Extraordinary, exquisite flesh.
The walls have been gold-leafed in my absence. The chamber is radiant with brilliant light.
Cruce is still caged!
Nude, towering, wings unfurled, he snarls with animalistic rage.
Iced solid.
I weep with joy. My fears were for naught!
Upon trembling legs I hurry to his cage, celebrating that it holds.
One of the bars is missing.
"Stop. Vibrating." Ryodan plucks a paper out of the air and slaps it back down on his desk.
I wonder if he cleans it. How many tushes have been on that thing? I'm never touching it again. "Can't help it," I say around a mouthful of candy bar. I know what I look like: a smudge of black leather and hair. "It happens when I get really excited. The more excited I get, the more I vibrate."
"Now there's a thought," Lor says.
"If you mean what I think you mean, you want to shut the fuck up and never think it again," Ryodan says.
"Just saying, boss," Lor says. "You can't tell me you didn't think it, too."
Five of Ryodan's dudes are in his office and it's like standing in the middle of heat lightning, closed in with them. Jayne is here, too, but I'm totally ignoring him because, like, if I didn't, I'd have to kill him with my bare hands and that would get messy, then Ryodan would probably make me mop his fecking office.
I never understand half of what these dudes are talking about and don't care. "You can touch me if you want to," I say to Lor magnanimously. I'm so pumped on adrenaline and excitement that I'm feeling downright sociable. I poke one of my shoulders toward him. "Check me out. It feels really cool."
All heads swivel my way, then they look back at Ryodan.
"He doesn't own my fecking shoulder. Why you looking at him?"
Lor guffaws but doesn't reach for my shoulder.
I don't know why. I like touching myself when I'm vibrating like this. It vibrates me twice. If I was really cold and started to shiver, I'd be vibrating three times! "So, what the feck are we going to do to stop this thing?" I beam. We got plans to make and implement. I thrive on times like these! They bring out my best! I'm a riser-to-the-occasion kind of girl. I'm feeling so excited and generous about having such a wicked cool adventure to live that I'm finding it hard to be mad at folks right now. We got an enemy that's bigger and badder than anything I've seen. Fecking-A, it's good to be alive! 'Cause, like, for a sec there by the dock, I wasn't sure I was going to be. I wasn't sure any of us were!
Speaking of back by the dock...
My mood shifts and I glower. I still don't have my sword. It got iced. The warehouse is now filled with iced Unseelie, the ceiling covered with stalactites, the floor deep with stalagmites. My sword got frozen in a stalactite way up high, and the place was way too deadly cold for anyone to enter, like freeze-you-to-death-instantly cold. We had to leave it stuck there, in an enormous icicle. Lor ordered Kasteo and Fade to stand guard until the scene thawed enough to retrieve it. Last I saw, the two Unseelie princes were still hanging around, too. If Christian was there, he was staying out of sight. No sign of Dancer. I didn't want to leave but Lor threatened to potato-sack me over his shoulder, and seeing how he can do it as easily as Ryodan, I didn't see any point in making myself miserable.
"It's your fault," I tell Ryodan. "You should never have let Jayne keep my sword. Now who knows what's going to happen to it! If the scene explodes like the others..." I trail off because I can't stand the thought of my sword blowing up into alabaster smithereens.
"That's the least of our problems," Ryodan says. "Tell me exactly what happened."
"Lor just told you," I say crossly. "What else do you want to know?"
"I want to hear it from you."
I tear open another candy bar and around mouthfuls repeat most of what Lor said about the fog and the slit widening. The feeling of panic we all experienced. How all the sudden none of us could hear a thing, like we'd gone deaf. "Then this... this... _thing_ that was twice the size of your office sailed out."
"Thing."
"Dude, Lor didn't describe it any better. I mean, come on, 'dark mass about the size of two semis, side by side'?"
"Try."
I frown, thinking, then brighten. "Did you ever see the movie _The Blob_? It was like that. Only it was oblong. And I don't know if it was slimy and it levitated instead of rolled. And I don't know if it was dense. But it didn't look like a Shade. It was nothing like a Shade."
_"The Blob."_
"Old movie, from way back in silent movie times."
"It's not that old," Jayne says. "I saw it when I was a kid."
"Which was like, way back during silent movie times, right? You shouldn't even be talking to me. Don't talk to me. You shouldn't even be here. I should kill you. You're lucky I'm not killing you right now. You left me for dead." I look at Ryodan. "And you let him. Feckers. All of you."
"I went straight to Chester's and told Ryodan where you were," Jayne says. "I wasn't going to let you die. I didn't like leaving you. I needed the sword. I couldn't afford to pass up the chance."
He told Ryodan where I was? "I said don't talk to me. And that worked out real well for you, didn't it? How many years do you think it might have taken you to kill a few hundred Unseelie?" I glare at Ryodan. "And you didn't say nothing about him telling you where I was. You didn't come neither." Didn't he care that I might've died?
"Boss sent me for you the second Jayne showed up," Lor says. "You were gone by the time I got there. I was following your blood trail but it disappeared."
"Texture," Ryodan says to me.
"You mean did it have any? Not that I could see."
"Then what happened."
"It moved into the warehouse, all ponderous-like, and belched white fog out everywhere and we couldn't see a thing. It iced the whole place, worse than anyplace we've seen yet. I mean, dude, the ceiling sprouted stalactites and the floor is covered with stalagmites so thick you can't even walk in there! We never seen anything like that at the other scenes."
"Postulate on why it got iced worse."
I'd pondered that on the way back. There was only one significant difference I'd been able to isolate. "There were a lot more people and Fae at this scene than any we've investigated. There were hundreds of Unseelie in cages and they all got frosted. It's possible more ice was necessary. Or maybe the thing had more juice today for some reason. We got iced, too, but it was only a thin layer, and once we moved, it cracked. We kept re-icing the second we stopped moving so I started doing jumping jacks and, like sheep can't think for themselves, everybody copied me, then there we were all standing in the street doing jumping jacks. I got worried the commotion might make it turn around and come after us but the thing never even noticed us. It was like we were fish and it wanted chips. Or maybe we weren't even noticeable as food. Then it vanished. Another of those slits opened inside the warehouse, all the white fog got sucked into it and the thing followed. Once it closed, we could hear again. Sort of."
"Clarify."
"There wasn't any noise. Nothing. You'd think the ice on all those Unseelie might have popped or cracked a little like ice does when it settles because they were warm before they got iced but they didn't. When we walked, our shoes didn't sound right on the pavement. When we talked it was... flat. It was worse than flat. There was a feeling to the silence. A really bad feeling."
"Elucidate," Ryodan says.
"I just did. I think you mean speculate."
Lor snorts. Ryodan gives me a look. Don't even know why I bother answering him sometimes. Maybe I like to hear myself talk. I've got a lot of interesting things to say. "You know how sound is really movement, and vibration is what makes noise? Which is, like, total contradiction to its effect on things because when the world went dead quiet, it was still moving while making absolutely no noise. But what I'm saying is, after it departed, things never got back to normal. It's like things weren't vibrating all the way. Or maybe the sound waves weren't bouncing off things the way they should. Or maybe the things the waves were bouncing off weren't right."
"Narrow it down."
I shrug. "Got insufficient data to form conclusive deductions."
"How long from the time it appeared to the time it disappeared."
"That was the weird thing. It felt like it was happening in slow-mo but I figure two seconds from start to finish. It came. It iced. It vanished." Sometimes I don't have the most accurate sense of time because I'm in a kind of in-between fast-mo and slow-mo and don't even realize it, which makes things around me seem to be happening more slowly. I'm pretty sure when it came, I was so wound up I was half freeze-framing. I look at Lor, who nods.
"Two to three seconds at most, boss. The fog rushed out, the thing came, the fog sucked back in and it was gone."
"I assume it was Fae," Ryodan says.
"Unequivocally," I say.
"You're a _sidhe_ -seer. That means you should be able to get a read on it like Mac did with the _Sinsar Dubh_."
"I could to a degree."
"Intelligence."
"Enormous sentience. Stupefying." I wish I'd felt the Unseelie king. I'd have something to compare it to.
"Emotion."
"None discernible. No malevolence. I got the impression destruction was a by-product, not a goal." I notice everyone's looking at me funny. "Dude," I add and flash my best street-urchin grin at the room in general. "Fecking-A, was it ever cool!" Got to watch my tendency to geek out when I get excited.
"You think it had a goal."
"There was... I don't know... _purpose_ to what it was doing. I could feel it. Some Fae feel simple when I focus on them with my _sidhe_ -seer sense. They're dumb, acting on instinct, capable of random destruction. Then there are things like Papa Roach, that Fae that breaks down into little parts," I remind him, in case he missed that especially scintillating edition of my _Dani Daily_. "Papa feels... structured. It has plans. So does the Ice Monster. But there's a big difference between Papa and the Ice Monster. Papa has a beady little mind. This thing is... vast in construct and purpose."
"Motive."
I sigh. "No clue. Just that it had one."
"No idea why it chose that place, or why it ices things."
"None," I say. "It didn't even touch anything, as far as I could see. It just kind of hovered above it all. Unless the fog is like its fingers or something and it sucks folks' life force up with it and inadvertently ices them in the process. No way around it, I need more time with it. Got to feel it out longer."
Jayne starts cursing and says nobody is going to be spending more time with it because it's too dangerous, and even Lor looks disturbed by the idea of another encounter with the Iceman. Which reminds me...
"Why were _you_ afraid?" I say. "I didn't think anything scared you dudes."
Lor gives me a cool look, like I didn't see what I saw, and says, "What are you talking about, kid? Only thing I was worried about was how pissed Boss was going to be if the thing killed you."
Bull. I know these dudes. They don't care about anything but themselves and he was freaked, which means it was a threat to him somehow. I want to know how. I want to know what Ryodan's Kryptonite is. I know a few universals like: he who can destroy a thing controls it. Not that I get off on destroying things but when you get backed against a wall, coming out with both guns blazing is pretty much your only choice. I want enough power to void a contract, enough to quit my job, permanent-like. I'm ready for retirement. I want enough leverage to get Jo out of the kiddie subclub, assuming she ever wants to leave.
She will. The day he chooses someone else.
In my estimation that won't be long at all.
Half an hour later I'm outside Chester's, skidding holes in the rain-slicked pavement, pacing in hyperspeed, munching power bars to keep myself juiced. I'm waiting for Ryodan to get done with whatever the feck business he said he had to take care of that couldn't wait, so we can get on with our investigation. He told me to sit tight inside the club, but me hanging around inside Chester's without my sword ain't real likely and he should know better than to expect it.
Then again, without my sword I'm not straying far either. I hate waiting for backup but I want it. The Unseelie princes freaked me out. I got a major thought brewing about them, an idea I despise but have to consider for its end result. Right now I'm keeping it back-burnered where it can't burn me too much.
I got so many other thoughts exploding inside my head that I expect a few are poking out my ears. One second I'm so excited to be living in these times I almost can't stand it, the next I'm a nervous wreck because my people are out here in these streets and they don't have any clue we got a big scary Ice Monster turning parts of our world into a deep freezer! I got to get the word out fast, but what do I tell them? If you see a shimmery spot in the air—run? That's assuming they even _notice_ the shimmery spot before they're iced!
Trouble is, I know folks. You can tell them to run all you want but there aren't a lot of people that will, until they believe they're in major danger—which is usually too late. They gape like cows, and if you don't know it, cows gape a lot. There used to be a big herd out by the abbey where I tested my speed and navigating abilities after Ro took me in and I was drunk on my new freedom. The cow pasture was a great place to practice freeze-framing on a dime because (a) cows move and are unpredictable, therefore hard to map like the real world, and (b) if I hit a cow it usually hurt me more than the cow. I had a riveted bovine audience the entire time. They'd chew their cud and swing their big heads back and forth, watching me like I was cow TV. If all I had to do was chew regurgitated food and watch other cows all day, I'd be riveted by me, too. Heck, that bored, I'd be enraptured by a fly battle on a cow pie.
But back to folks, shimmery spots aren't terrifying enough to make them flee. And there are some folks, like the see-you-in-Faery chicks that hang at Chester's 24/7, sporting new Papa Roach bugged-sized waistlines, trading sex, competing with each other to see who can eat enough Unseelie to turn immortal and get to hang with the Fae first, that would deliberately linger if they saw a shimmery spot, just because it was, like, pretty. Gah, some chicks should be shot. Put out of everyone else's reproduction pool.
I need a couple of pictures to put in my daily, to show peeps the gruesome finality of what the Ice Monster does. I need to get over by Dublin Castle, shoot a few frames. Then I need to get to _TDD_ headquarters and fire up the presses! I love printing my dailies. I got double the reason to get one out fast now. After seeing the Unseelie Princes WANTED posters, folks are no doubt worrying their butts off about me! I need to reassure them, let them know I'm still on the job.
"You must be Dani!"
I turn.
And pop up on my heel and keep turning. I'd turn all the way to bum-feck-China if I could. I spin in a full rotation so she's at my back again and stand there, trying to compose myself. I don't want to look at her. I don't want her to see my face. I didn't expect this. Wasn't ready for it. Feck, I'll never be ready for this. It's one thing to know she's out there somewhere, along with Mac. It's another thing to have to face her.
Feck, feck, feck.
I paste on a mask, turn around and begin the pretending game.
"And you're Rainey Lane," I say. Same beautiful blond hair as her daughters, even though they were both adopted. Same pretty demeanor: classy feminine Deep South. She's walking around in chilly, gloomy-ass late afternoon Dublin, dressed like somebody's going to care she's color-coordinated and accessorized. I guess Jack Lane does. Unlike most married folks I've seen—not that I've seen a lot—they seem to be crazy about each other. I saw them in Alina's photo albums. I saw them in Mac's. I've seen pictures of this woman holding her daughters when they were small. I've pored over photos of her beaming at their sides when they were grown.
Like she's beaming at me now.
Like she doesn't know I killed her daughter. I guess she doesn't. I guess the last time Mac talked to her was before she found out it was me that took Alina to that alley to die.
For a second I get this stupid vision of how she'd be looking at me right now if she knew, and it kicks the breath right out of my lungs and leaves me standing there dumb. I have to clamp down all my insides so I don't puke. She'd hate me, despise me, stare at me like I was the most disgusting, horrible thing on the face of the Earth. She'd probably try to claw my face off.
Instead of this... this... mom-love-bullshit thing glowing in her eyes like I'm her daughter's best friend or something, not her other daughter's murderer. I thought Mac was the worst thing I'd have to face one day on these streets.
I'm smothered in a hug before I can dodge it, which shows how discombobulated I am. On a good day I can dodge raindrops! I forget myself for a sec, because she's got soft mom arms and hair and a neck you want to cling to. Worries melt on mom bosoms. She smells good. I'm enveloped in a cloud that's part perfume, part something she baked lingering on her clothes, and part some indefinable thing I think are mother-hormones that a woman's skin doesn't smell like until she's raised babies. It all combines to make one of the best scents in the world.
After my mom was dead and Ro took me to the abbey, I used to whiz by the house every couple days. I'd go into Mom's bedroom to smell her on her pillow. She had a yellow pillowcase embroidered with little ducks along the edges like my favorite pajamas. One day the smell was just gone. Every vestige of it vanished without a trace. Not one tiny little sniff left for my supersniffer. That's the day I knew she was never coming back.
"Get off me!" I eject myself violently from her embrace and back away, scowling at her.
She beams like one of Ryodan's supercharged flashlights.
"And stop beaming at me! You don't even know me!"
"Mac told me so much about you that I feel like I do."
"Well that's just stupid on your part."
"I read the latest _Dani Daily_. Jack and I hadn't heard of those bugs. You've been doing a wonderful job keeping everyone informed. I bet that's a lot of work for you."
"So?" I say suspiciously. I hear a "but" coming.
"But you really don't need to anymore, honey. You can relax and let the adults take over."
"Yeah, right. Weren't adults in charge when the walls fell? And haven't they been in charge since? Doing a real bang-up job, aren't you all?"
She laughs, and the sound is music to my ears. Mom laughter. Melts me like nothing else can. Guess because I heard it so rarely from my own. I think I made my mom laugh three times. All before I "transported" for the first time. Maybe it happened once or twice after that. I tried. I'd memorize funny things I saw on TV while she was gone. I'd watch musicals, learn cheery songs. Nothing I did was right. Rainey Lane is looking at me with more approval than my mom ever did.
"Go. Away. No, wait. Don't. You can't be out here alone. I'll find somebody to take you back wherever you go. What are you doing walking around Dublin alone? Don't you know nothing? There's all kinds of monsters in the streets! It's going to be dark soon!" Somebody needs to knock some sense into her.
"Aren't you the sweetest, to worry about me? But you don't need to. Jack's just around the corner, parking, honey. There's too much debris in the streets to park any closer. I keep telling Mr. Ryodan he needs to clean up outside his club but he hasn't gotten around to it yet. I suppose we may have to help him out with that. He's a busy man, you know, a lot on his plate."
"Crime _is_ time-consuming, isn't it?"
She laughs and I get my first suspicion she might just be totally clueless. "Aren't you funny? Mr. Ryodan a criminal. That nice man." She shakes her head, smiling like I'm just the funniest thing. Yep, clueless. "Dani, honey, I've been hoping to run into you. Mac has been, too. Why don't you come have dinner with us tomorrow night?"
Yeah, right. Skewered Dani on the menu, served up with a side of veggies. Not. Would all three of them take turns beating me to death, once Mac ratted me out?
"There are some people I'd love for you to meet. There's a wonderful new organization in the city that's been doing fabulous things, bringing about some real changes."
I shoot a big, melodramatic, beleaguered look at the heavens, then back at her. "You can't be talking about WeCare. Please tell me you're not talking about WeCare."
"Why, yes, I am. You've heard of us!" She's beaming again.
"Us? Gah! Please tell me you're not part of them! You can't be part of them! Do you know they hate me?"
"No we don't. WeCare doesn't hate anyone. We're all about rebuilding and helping. Whatever gave you that idea?"
"We." She's slaying me. Is Mac part of them, too? "Dude, like, maybe the way they copied my paper, took over my posts, and printed all kinds of lies about me."
"I happen to know for a fact that top individuals at WeCare are eager to meet you. They think as much of you as Mac does."
Gee, great, so they want me dead, too. Top individuals. Lovely. They can get in line behind Christian. Who's behind the Unseelie princes.
"They think you could be a tremendous asset. I think so, too."
I give her a look. "You might want to recheck your facts. I think you're missing a few. Folks in charge of organizations don't consider me an asset. Never have, never will." I hate organizations. Folks got to be free, able to breathe, and make up their own minds about stuff, not be fed a party line. Ritual numbs the brain. Repetition is grass for sheep.
"Mrs. Lane, so nice to see you again," Ryodan says, and I almost fall over. Not only didn't I hear him approach us, he's being polite. Ryodan is never polite.
I squinch up my face, studying him. "Dude, you feeling okay?"
"Never better."
"Why are you, like, pretending to be nice?"
"Mr. Ryodan is always nice. He was a lovely host while we stayed at Chester's."
"You didn't _stay_ at Chester's, you were _hostages_." What is wrong with everyone that they can't see things for what they are?
"He and his men were keeping us safe, Dani. The _Sinsar Dubh_ was targeting people Mac loved."
"Was the door to your room locked? Dude, that makes you a hostage," I say.
"Our door was never locked."
Huh? "Yeah, but did you even know how to get out? He's got those tricky panels."
"Mr. Ryodan showed Jack and me both how to operate the doors."
Huh? "Yeah, but there were guards outside. Keeping you in."
"For our protection. We were free to come and go. We chose to stay. The city was dangerous when the Book was loose. Jack and I are very grateful for Mr. Ryodan's help during those difficult times."
I scowl at Ryodan, who's wearing a smug-ass smile. He probably worked some kind of spell on them, like the kind of thing he did to me in the Hummer when he forced me to take the candy bar from him by muttering strange words. He makes people puppets. Empty-headed slaves. Not me.
"Do you know he's forcing me to work for him by holding Jo hostage?" I tell Rainey. She needs to wake up and smell the coffee.
"You mean that lovely young waitress? I've seen the way she looks at him. She's crazy about him," Rainey says.
And that pisses me off even more. Mac's mom can tell Jo's stupid crazy about this psychopath just by looking at her? Gah! Just gah! On top of it all, said psychopath has Rainey so fooled there's no point in even talking to her anymore! Not that lack of a point would shut me up. "Do you know he has private clubs under Chester's where—"
"I just spoke with Barrons," Ryodan cuts me off. "Mac's on her way to meet you, Mrs. Lane. She should be here any second now."
I give him a suspicious look. He's probably lying. And he knows perfectly well I won't risk finding out.
Rainey gives me a warm smile. "Dani, she'll be so glad to see you! She's been looking for you for weeks."
I'm sure she has.
I lock down my mental grid to freeze-frame, make like folks on _Gunsmoke_ and get the feck out of Dodge.
# TWENTY-FIVE
# _"I don't know who he is behind that mask"_
"What are you doing."
"Why do you care?" I got belligerence stuck in my craw and I don't even know why. Sometimes just standing next to Ryodan makes me feel that way.
"Because if there's no point in what you're doing, you're wasting my time."
"Dude, got eyes? I'm collecting evidence." Finally! I been trying to get out for a second look at the exploded scenes for just about fecking ever but things keep coming up, like me almost getting killed. Oh, and me almost getting killed again. There's never a dull moment in the Mega-verse. The Ice Monster would freak me out a lot more if my world hadn't been jam-packed with monsters of all kinds since pretty much my birth: big, small, human, not.
"In Ziploc bags."
"I think they're Glad."
"They look impartial to me."
I start to snicker then stop myself. This is Ryodan. I hate Ryodan. Lying deceitful dickhead. Tricking folks into thinking he's really nice so I look stupid. "Think my sword's unfrozen yet?"
"No."
I stoop and scoop. I know a thing or two about myself. I see a lot. But sometimes there are small things going on that even I miss. Ergo my impartial ziplocks. I'll fill one at each scene. Go deep into the frigid center of the exploded debris, scoop up handfuls of icy detritus, stuff it in, and label it all neat and tidy-like. Later, me and Dancer will sift through the ziplock bags and look for clues. I pull a Sharpie from my pocket and write on the white strip "Warehouse, North Dublin." Then I tuck it carefully away in a backpack slung over my shoulder. Collecting my ziplocks makes perfect sense to me.
"It doesn't make sense. You could examine the detritus thoroughly right here at the scene."
"Dude, do I ask you to explain yourself?"
"Kid, are you ever not prickly."
I root around in the rubble, making sure I got some of everything, keeping my back to him because sometimes looking at him is more than I can stand. "Sure. Like, when I'm not around a _prick_. We investigating or having a conversation all personal-like? 'Cause I got business to take care of today and you're wasting my time. It's going to be dark soon."
"Observations."
"I got two. The scene blew to smither-fecking-reens and everything's still cold."
"Give me something I can use."
"I wish I could, boss, but this is... well, this is a mess." I rock back on my heels, shove hair out of my face and look up at him. The sun's nearly level with the horizon, right behind his head, making this weird halo effect around his face—as if! I'm surprised he doesn't smell like brimstone. He probably has a red pitchfork and hides horns under his hair. Making it weirder, the sun's got a sparkly gold tint to it—thank you fairies for changing everything in our world—and he looks—oh, who cares how he looks? Why am I even noticing?
I look away, focusing on my investigation. We got a Fae that appears out of a slit and arrives with a lot of fog. It ices everything in its path then disappears back into another slit. Sometime after that the scene explodes. But why? That's the big question. Why is it icing what it ices, and why does the scene explode afterward? And why does it take varying amounts of time for the different places to explode?
I feel the ground with my palm. It's freezing. There's a chill that hasn't dissipated. I wonder if it ever will. Might be kind of cool if it didn't. You could clear the ground, build a house and never need air-conditioning. It'd suck in the winter, though.
I survey the scene. Where the warehouse used to be are piles of crumbled bricks and mortar and splintered framing, with twisted girders from steel racking everywhere, some bent, some poking straight up at the sky. Chunks of Unseelie flesh are plastered to pretty much every—
I smack myself in the forehead. "Holy priceless collection of Etruscan snoods, they're not moving!" I exclaim.
There's a choking noise over my head somewhere. "Etruscan snoods?"
I glow quietly inside. Some accomplishments mean more than others. I am officially the Shit. Now and forever. "Dude, watch your question marks. I just pried one out of you."
"I have no idea what you're talking about."
"Admit it, you lost your eternal fecking composure."
"You have an obsession with a delusion about how I end my sentences. What the fuck are Etruscan snoods?"
"Dunno. It's just another of Robin's sayings. Like, 'Holy strawberries, Batman, we're in a jam!' "
"Strawberries."
"Or, 'Holy Kleenex, Batman, it was right under our nose and we blew it!' "
There's another choking noise above my head. I could go on for hours.
"Check out this one, it's one of my faves! 'Holey rusted metal, Batman! The ground. It's all metal. It's full of holes. You know, holey.' " I snicker. Gotta love the dudes that wrote Batman. They had to sit around cracking themselves up all the time. "Or, 'Holy crystal ball, Batman, how did you see that coming?' " I look up at him.
He's staring at me like I have three heads.
The truth dawns on me. "Holy prostrate rugs, you lied! You've never even _read_ Batman, have you? Like not one single issue. You never even watched an episode on TV! That was, like, your only redeeming quality and it wasn't even true. You been pretending we're superhero partners and you don't even know the first thing about Robin!" No wonder Ryodan's no fun to hang with. I'm so disgusted I can't stand it!
I skirt my irritation and get back to the important stuff. "The Unseelie parts are motionless. Dead as the humans. Look at them. Unseelie don't die. Nothing but my sword and Mac's spear kill them that dead. Unseelie are immortal. You can slice and dice them with human weapons, and the pieces will flop around forever. These ain't flopping. This thing is killing them dead. And we never even noticed." Preconceptions. They trip you up every time. When something explodes, you expect to see dead things. Maybe there's something to my idea it's after folks' life force. Kind of like the Shades, sucking them empty but instead of leaving husks, it leaves the whole shell of their bodies iced. "And notice something else: none of the pieces, human or Unseelie, are rotting. Why is that?"
"I'll be damned."
"I know, right?"
"And you didn't notice this before."
I glare at him. "You didn't either. And I tried to recheck scenes twice but you made me sit in your office while you did paperwork. The third time I was thinking about rechecking a scene, I stumbled on a fresh one and almost got exploded myself." I stand up and walk away to get a good bird's-eye view of the destruction. I pull out the new phone I grabbed to replace the one I smashed and snap a couple pictures. "So," I say crossly, "where to next?"
As we head for the church where I almost died, I realize Ryodan's been keeping me so busy asking the questions he wants answered that I never get around to asking any questions I want answered. "So, what happened to me when I got frozen that night? When I came to, Dancer was there with you and Christian. Talk about unexpected. How'd Dancer get there? Who saved me?"
"I got you out of the church or you would have died right there on the floor."
"You're the one who took me into the church to begin with and didn't warn me what would happen if I touched something. You're _why_ I almost died, dude. So, who saved me?"
"I had to take you out slow or you would have had afterdrop."
"Yeah, but did Dancer tell you about afterdrop? 'Cause that sounds like something he would know."
"Why did you laugh right before you lost consciousness."
"Death's an adventure. I lived big. Rigor mortis makes your face stick. So, who knew how to thaw me?"
"Death's an insult."
"At least an affront," I agree. "Think my sword's unfrozen yet? Maybe we should go check."
"You're too young to laugh when you're dying. And no. I don't think your sword is unfrozen. Focus."
"Ain't too young for nothing."
"In some societies that would be true. Different places. Different times. You'd be old enough to be a wife and mother."
"That's a horrible thought. So, Dancer saved me."
"I didn't say that."
"That's how I know. Maybe we could use hair dryers to melt the ice around my sword."
"You need to get rid of him. He's a liability. Forget about the fucking sword. I'm taking care of it."
I whirl on him, fists at my waist. "He's an asset! He's my best friend! You don't know nothing about Dancer!"
" 'Nothing' is the key word there. Because that's what he is. Nothing. He's just human."
"Bull-crikey, Dancer's the Shit!"
"He wears glasses. I bet that works out real well for him in battle. No, wait, he doesn't battle. Never will. Too fragile. One poke with a sharp stick and his guts would spill all over the street. Sayonara, human."
"His guts aren't spilling anywhere. He's supersmart and... and... and he's super, _super_ smart—"
"What the fuck kind of name is Dancer, anyway."
"—and he can build anything. He made my Shade-grenades and he made me this net of lights that charges just off me moving, and it totally outperforms the MacHalo! Besides, all Batman had was a cool costume and the best toys and the smartest ideas, and everybody knows he's the greatest superhero of all time! Besides, I'm just human, too."
All the sudden Ryodan's standing one inch away from me, hand under my chin, holding my face up to his. "You'll never be just anything. A tsunami can never be 'just' a wave."
"Get off my chin."
"I like that about you. Waves are banal. Tsunamis reshape the Earth. Under the right circumstances, even entire civilizations."
I blink.
"You're going to be one hell of a woman one day, Dani."
I never knew my jaw was flexible enough to hit the pavement. My arms aren't even long enough to pick it back up. Catch flies in it, my butt, you could drive a truck in my mouth right now. Did Ryodan just, like, compliment me? Has hell frozen over? Are birds flying backward? It makes me so uncomfortable in my own skin, I feel like skinning myself. A three-quarter moon is behind his head, and his face is all shadows. "Fecking-A, dude, I know that. Everybody knows that. I'm the Mega. As in, short for 'Alpha and O.' " I shrug him off me and push past him.
He laughs. "You might have to fight somebody else for that title."
"Get a move on," I say crossly. I'm so behind on work I can't stand it. "You only got me for a limited time tonight. I need to get a _Daily_ out. Folks need to know about the Iceman." I lock down my grid and slip into freeze-frame.
"You're going to get the boy killed one day, Dani," Ryodan says behind me.
"Rot in purgatory, dude. Batman never dies. Dancer won't either."
When we arrive at the church, I roll my eyes.
Five Seelie are standing in front of the demolished cathedral, amid rubble, shredded hymn books with pages everywhere like they rained down from heaven, chunks of organ, and miscellaneous debris. "Think my sword's unfrozen yet?" I say, making a fist around the empty space where my sword hilt should be. I see sifting Fae, and all I can think of is how I don't have my sword. 'Course, I have that thought pretty much every other second anyway.
"Kid, you're a broken record."
"Well, it might be."
The Seelie are talking, and although they know we're here, they completely ignore us. I ignore them, too. Despite them being so beautiful I have to pry my eyeballs off their faces. I'm not making the same mistake I made with V'lane. Getting sucked in by how gorgeous they are. Thinking they're any different than the Unseelie. Just because they're gold and velvet and iridescent-eyed and hunky. Christian's hunky too. He keeps dead women by his bed.
I'm feeling major juice coming from at least one of them but they're muting it. That worries me. Fae don't mute themselves unless they're up to no good, trying to pretend to be something they're not to make us less worried when we should be really, really worried. "Fecking Fae. I wish they'd all just go away."
"Then what would we do for excitement."
I snicker. He's got a point. I pull out my phone and snap a picture of the scene, planning to get my ziplock out next, skirt the fairies and go to work.
All the sudden there's a disturbance in the air in front of me. It takes a sec for the dust to settle in my brain. One of the Fae just tried to sift over to me to do who-knows-what. Ryodan beat it to its destination and they collided. The Fae looks like a pissed cat, eyes narrowed, spine twitching, iridescent eyes flashing fire. I've seen this one in Chester's. He has a taste for human women and the stupid sheep are nuts about him, with his tight leather pants and open shirts and sleek golden hair and skin.
Ryodan's standing between it and me, legs spread, arms folded. He's a mountain. Nothing's getting past him that he doesn't want past him. It pisses me off I need him there. With my sword, no Fae would dare bum-rush me! I'm used to more respect than this. This bites.
The Fae says all stiff-like, "His highness does not permit his likeness to be captured in small human boxes. The runt will give me the box."
Runt? _Moi?_ I'm at least five-foot-three with my tennis shoes on! "I'm not a runt. I'm young and still growing. And we call them cameras, dickhead."
"Whose highness," Ryodan says.
"Ours. Yours. All he suffers to live. Give me the box or the runt dies."
"You just try," I say. "Better fairies than you have. Worse ones, too. They all tasted delicious. With catsup. And mustard. And a side of onion rings."
"Should have left it at catsup," Ryodan says. "Less is more sometimes, kid." To the Fae, he says, "Queen Aoibheal."
"Was never our true queen. She is gone. We have a new leader. Our sacred light, King R'jan."
"The Fae are matriarchal," Ryodan says.
"Were. We have decided it is time for a new rule. If not for the flaws of a woman, so many of our race would not have died, and still be dying. If not for her idiocy, the abominations would not have been freed. She was not even Fae," he sneers. "She began her life as one of you! The indignity of it, to have been ruled by a mortal masquerading—"
"Enough, Velvet," R'jan says. "We do not explain ourselves to humans. Kill the runt and bring me the box."
"I'm not a runt." My hand closes where my sword hilt used to be.
"Missing something, runt?" one of the courtiers standing with the new "king" says and they all laugh. Guess everybody has seen the fecking Wanted posters. I take a mental snapshot of its face and mark it for death. Someday, somewhere, fairy.
Velvet was just getting started airing his grievances. "She forced us to grant humans rights to which they were never entitled. No more. It is a new rule. A new age. We are no longer weakened by a weak queen."
"I said 'enough,' " R'jan says. "If I must tell you again it will be the last thing you hear for ten thousand years. You will not enjoy where you pass them."
I give R'jan a conspiratorial wink. "You going to give him a 'time out,' dude?"
Velvet looks horrified. "If you are fool enough to address King R'jan, you will do it thus and in no other manner! 'My King, Liege, Lord, and Master, your servant begs you grant it leave to speak.' "
"Wow. Totally delusionary there."
"Good luck with that," Ryodan says. "She doesn't beg to speak, or do anything else. You can lock her up, down, and sideways and it's never going to happen."
I beam at him. I had no idea he thought so highly of me.
Then he's gone. So is Velvet.
I stand there a little uncertain because Ryodan didn't telegraph a single intention before he and the Fae disappeared. I'm not even sure who took who. Or if one took off and the other chased. All I know is both of them are gone.
I shift from foot to foot, looking at R'jan and his remaining three cohorts, and he looks at me and I try to think of something to say. Best I come up with is: "So, why are you guys here, anyway?"
"Kill the runt," R'jan says.
I yank out two candy bars and cram them in my mouth, wrapper and all, and give them a superstrength chew that makes the wrapper explode so I can swallow some chocolate and get a rush fast, because I've got no sword and who the feck knows where Ryodan went. I crunch, swallow, spit out the wrappers, and lock down my grid to freeze-frame when all the sudden Ryodan's back.
He's standing right in front of R'jan.
"In these streets," he says so cool-like I almost expire from the sheer coolness of it, " _I'm_ King, Liege, Lord, and Master. You are the 'it.' "
Then he dumps Velvet's dead body at his feet.
# TWENTY-SIX
# _"It's the hard-knock life"_
"You did me a favor. Velvet was an annoyance," R'jan says. "He spoke too often and too much, saying little of consequence."
Ryodan looks at the King's remaining courtiers and says, "I'll do you three more 'favors.' Just say the word. Wrong one or right one. Doesn't matter to me."
The courtiers sneer at him. Uneasily. We might have postured for hours and never gotten to the position of strength Ryodan established with a single action. I'm learning from him. I'd never tell him that, though.
R'jan opens his mouth then closes it, not entirely sure Ryodan didn't just say that he was going to kill the other three courtiers if he said even one more word. Smart dude. I'm not sure Ryodan didn't mean that, too. How the feck did he kill Velvet? I study the Fae corpse but see no obvious wounds. No cuts or... wait a minute, is that a few drops of blood on his shirt? I sidle left for a better view but Ryodan moves like there's a tether between us, conveniently blocking it. I got no doubts he left so I wouldn't know. He's so fecking secretive!
Does he have my sword somewhere? Did Mac loan him the spear? Never! Obviously he's got some other weapon that kills Fae, and I want it. The prick. He's been holding out on me big-time. When I lost my sword he could have given me whatever he just used. I'm so pissed I could spit. He knows how to kill Fae. No wonder he's so fearless. He's faster than me, stronger, and has a Fae-killing weapon. I pine for the days I was the biggest, baddest superhero in town!
Abruptly, I got graphic sex images in my brain! I'm hot and uncomfortable in my jeans. Bugger it all! R'jan is a prince, a death-by-sex Fae. He's the one I sensed muting himself so as not to draw attention to his little entourage, but now that the crap's hitting the fan, he's going to use any weapon at his disposal. I guess he figures to mess me up to get to Ryodan.
But R'jan is staring at Ryodan like he expects it to be working on him. Huh? I thought they were hetero and their killer eroticism only worked on the opposite sex. I realize that was a stupid assumption. It's just that I never saw the Unseelie princes around men and V'lane always kept it muted around humans. There's no reason, whatever the mechanism, that it wouldn't work on both genders.
"On your knees, human." R'jan tosses his golden mane imperiously. "You will crawl before your king."
Ryodan laughs. "Is that all you've got."
I hang back, listening, not about to get closer. It's all I can do to not start stripping. Aw, bugger, I am! My coat's on the ground and I'm pulling up my shirt! I make a sound of protest but it doesn't come out like that at all.
"Turn it off," Ryodan says without even looking at me. "You're distressing Dani. No one distresses Dani but me."
"I said 'kneel,' " R'jan says, like he can't believe Ryodan is still standing there.
"And I said 'fuck you.' Turn it off or die."
R'jan cuts it off so suddenly I'm shivering, cold and miserable, like I was just sunning by a pool then got an iceberg dropped on me.
"Why are you here," Ryodan says.
R'jan says tightly, "What the fuck are you?"
Darn good question. I wonder it myself.
"If you answer me wrong one more time, your death." He kicks Velvet's lifeless body.
R'jan grimaces. Unlike Unseelie, Seelie expressions make sense to me. They're similar to ours, I guess because they've spent so much time preying on us. "Something is killing our people."
"I didn't know you counted the Unseelie as yours."
"It has visited... places other than Dublin. It has killed Seelie, too."
"It's been in Faery."
"Twice. How dare an abomination enter our realm? Never has an Unseelie been suffered in Faery!"
The temperature drops and I tense, searching for a shimmer in the air. It was already colder near the church than in the rest of Dublin, but now the pages of the hymnals scattered around the street glisten with a thin sheen of ice. I see Ryodan looking around, too. Snow starts to fall. I realize R'jan's temper is doing it at the same time Ryodan does. I brush snow off my bare shoulders, then I jerk, embarrassed. I was so riveted by stuff happening that I didn't realize I'm only wearing my bra. I scoop up my clothes and yank my shirt over my head. I hate Fae.
To R'jan, I say, "Cruce lived in Faery for hundreds of thousands of years and you guys never figured it out. There's an Unseelie in Faery for you, sitting right next to your queen. Wait!" I snicker. "I forgot. She wasn't your queen either. She was human. Dudes, stupid much?"
"I will speak with you," R'jan says to Ryodan, "when you make the runt be silent."
I puff myself up, waiting for Ryodan's defense.
"Be quiet, kid."
I deflate.
"You're certain it's Unseelie," Ryodan says to R'jan.
" _I_ said it was," I say indignantly.
"Unequivocally."
"Like, I even used that exact word!"
"What is this 'abomination?' " Ryodan says.
"We do not know. We have never needed to know about our foul brethren."
"Yet you're worried enough about it that you're here. In a dark Dublin street. The new king of the Seelie himself."
It seems to mollify R'jan to hear himself called the new king of the Seelie. He looks away and doesn't say anything for a second. Then he shivers. "It brings final death to our kind."
"Like the spear and the sword," I say.
"I told you to shut her up."
"Answer her."
"She cannot understand what it is to be Fae."
Ryodan doesn't say a word. He takes one step forward and R'jan immediately takes one step back all smooth, like they're doing a choreographed dance.
"One day, human—"
"You might want to rethink what you're calling me."
"—I will crush you beneath my heel and—"
"Until that fictitious day, you will answer me when I speak." He steps over Velvet's body, closing the distance between them.
R'jan steps back.
"How does 'final death' differ from what the sword does," Ryodan says.
"Your puny brains were not fashioned to grasp the greatness of being D'Anu."
Ryodan crosses his arms, waiting. Dude's got some serious presence. I want to be like him when I grow up. "You'll have no brain at all in three seconds. Two."
R'jan says tightly, "The spear and sword end immortal life. They sever the connection that binds our matter together and scatter it to the wind."
"Tell me something I don't know."
"Even if we die, that of which we are fashioned is still out there, blowing. We feel all our kindred through all time, impressions in the fabric of the universe. We are individual yet a skein, vast and glorious. You cannot know what it is to belong to such an enormous, divine entity. This... this... thing... whatever it is, is pruning our tree. It does more than merely unbind our matter. It scatters nothing to the wind. _Nothing_. It is as if those it takes have never been. Its victims are... erased. You cannot begin to perceive how painful that is for us. Death, even by the sword and spear, leaves us connected. This abomination is amputating our race, limb by limb!"
The Ice Monster is stripping away Fae existence on the deepest level. There's something to my "life force" theory!
"You have strong incentive to see it stopped."
I interpret R'jan's expression as a royal "Duh."
"Which makes it worth a lot to you."
R'jan gives him an incredulous look. "You could not hope to terminate it nor do I barter with pigs and fools."
"I will terminate it. You will pay me handsomely for services rendered when and how I choose to invoice you. And, one day, you will kneel before me and swear your fealty. At Chester's. Before an audience of Fae."
"We could do fireworks," I say excitedly.
"Never," R'jan says.
"I'm a patient man," Ryodan says.
I think about that later, as we dig through the rubble, fill my ziplock and tuck it into my backpack. I munch a candy bar to make more room in my bag. "You're not patient. You zero in on something and lock on like a missile. You're the most pushy, manipulative person I know. And I knew Rowena."
"Patience and persistence aren't mutually exclusive. You have no idea how patient I am. When I want something."
"What does somebody like you want? More power? More toys? More sex?"
"All of the above. All the time."
"Greedy bugger."
"Kid, let me tell you something. Most people spend their short time in this world less than half alive. They wander through their days in a haze of responsibility and resentment. Something happens to them not long after they're born. They get conflicted about what they want and start worshipping the wrong gods. Should. Mercy. Equality. Altruism. There's nothing you _should_ do. Do what you want. Mercy isn't Nature's way. She's an equal opportunity killer. We aren't born the same. Some are stronger, smarter, faster. Never apologize for it. Altruism is an impossible concept. There's no action you can make that doesn't spring from how you want to feel about yourself. Not greedy, Dani. Alive. And happy about it every single fucking day."
"Are we done here yet? I got a paper to get out." I roll my eyes when I say it so he doesn't see how much what he just said got to me. I think it might be the smartest thing I ever heard anyone say. "Hey, you think my sword's—"
"For fuck's sake _no_."
"Geez, dude. Just asking."
We stop by two more scenes in Dublin that got iced, first the fitness center, then one of the small underground pubs. It's a gaping hole in the pavement, with chunks of concrete listing in at dangerous angles. There's nobody around to cordon it off and make sure wandering kids don't fall in. Fortunately there aren't as many wandering kids as there were right after Halloween. We've gotten most of them off the streets. Some of them refused to come in, chose to go underground instead. Got to respect that. It sucks being taken pity on by someone else's family, knowing you're not really part of it. I wonder how wild they'll be in a few years. I can't wait to see what they become. I think in a few years they'll make a heck of an army. Growing up alone makes you tough.
Until the walls fell, I never knew there were so many places beneath Dublin. I used to think there were only a few underground rivers, a couple of crypts like the ones at Christ Church and St. Patrick's, and maybe the occasional cellar. Dublin keeps a lot of secrets. Since the walls came down, I've discovered all kinds of places down under. We Irish are a canny lot, we like multiple ways out of a tight spot. And why shouldn't we? Look at how many folks have tried to be the boss of us, and for how long!
I peer into the rubble-filled hole. "Dude, how am I going to get my ziplock?"
"Boss, we got a problem."
I glance over my shoulder. One of Ryodan's men is standing there, looking pissed. It's a dude I don't often see. I've never heard anyone say his name. I think of him as Shadow because he glides into rooms barely disturbing the air. You almost overlook him, which is a feat considering he's a foot and a half taller than me and got to be three hundred pounds. Watches everything like me. Doesn't speak much, unlike me. Tall and muscled like the rest of them, scarred like the rest of them, hair like night and eyes like whiskey in a glass.
"Listening."
"Fucking half-breed Highlander took the sword."
"What?" I explode. "Christian took my sword? I told you and _told_ you it was probably unfrozen! I kept saying that we needed to go check! What the feck is wrong with you dudes? Can't you guard a measly little sword from a measly little half-human?"
Shadow gives me a look. "He's damn-near full Unseelie prince and he had a flamethrower, kid." To Ryodan, he adds, "Lor and Kasteo are badly burned."
A fecking flamethrower! Why didn't _I_ think of that? Best I came up with was a measly hair dryer. I need to start thinking on grander scales! I return the look. I'm so pissed off my head is mean with pure pissed-offedness. "You don't understand, when I was in his bed, I found a dead woman stuffed between it and the wall! Now he wants _me_ dead and you let him get my sword! What am I supposed to do now? Ryodan won't share whatever the feck weapon he has! How am I supposed to protect myself? Can't you guys do anything right? One little sword! That's all you had to watch over! And why didn't _we_ think of a flamethrower? Anybody got a brain among you dudes? Flamethrower! Brilliant! Did it hurt my sword?"
"When were you in Christian's bed," Ryodan says softly.
I gape. "Dude, you got a serious case of selective hearing, the kind that bleeps out all the important stuff! Who cares when I was in his stupid bed? How the feck did you kill Velvet? You been holding out on me! You need to learn to share your weapons!"
"When."
There's something in the way he utters that single word that makes me shiver, and I'm hard to rattle. "So, I didn't change in a convenience store! So, shoot me! I need my sword. What are you going to do to get it back?"
I've never seen Ryodan's face go so smooth. It's like it got iced blank of all expression. I've never heard him talk so soft and silky either. "Take her back to Chester's and lock her down. I'll get the sword."
Shadow looks grim. Like my own personal grim reaper. Not.
I slip a hand in my pocket. Pull the pin on a grenade. Start counting because I got to time it just right. I'm not getting locked down anywhere. No more cages for Dani Mega O'Malley. A split second before it goes off, I lob the bomb to the pavement in front of them. It detonates with the brilliant, Shade-killing flash of light Dancer rigged up for me. "My ass, you will."
I freeze-frame out of there with everything I've got.
# TWENTY-SEVEN
# _" 'Cause I'm one step closer to the edge and I'm about to break"_
I think I set a personal best.
I had a lot of incentive. The look on Ryodan's face was like nothing I've ever seen. Worse than when I killed all those Fae in Chester's and he locked me in his dungeon. Way worse.
While I'm freeze-framing, I think about how he's been screwing up my life since the sec he stepped foot on my water tower and told me he had a job for me. I think I got him figured out. I think the reason he's so pissy about both Christian and Dancer is because he's worried I'll get a superhero boyfriend who will kick his ass from one end of Dublin to the other and tear up that nasty little contract he made me sign. He doesn't want any other dudes too close to me because it would interfere with his ability to use me for his own purposes. Christian's a physical competitor. Dancer could brainiac him dead.
He doesn't get that I'm not interested in a superhero boyfriend.
I'm going to _be_ the superhero that can kick his ass from one end of Dublin to the other.
"Oh, sweet fecking day," I sigh raptly around a mouthful of chocolate, anticipating it. Peanuts and chocolate get stuck in my throat and I almost can't swallow. I been eating too many candy bars lately because I'm on the go so much and it's all I got handy. I'm having a major salt craving. Sometimes when I eat too much sugar I start obsessing about my mom's corned beef and cabbage with her fresh rosemary bread and potatoes and chives and—Holy Ashleagh Falls, my mouth is watering!
I whiz into a grocery store. Empty. I head north three blocks to Paddy's Stop 'n' Go. Empty. I dash ten blocks south to Porter's. Also completely cleaned out. What I wouldn't give for a bag of chips! Useless for an energy punch but a hopping St. Patty's Day Parade on the tongue! I'm practically drooling I'm so hungry for something besides chocolate. A can of beans. Crimeny, even tuna would cheer me up!
I get over it. Wasted energy. There _is_ no other food right now, and one thing I learned in a cage is you either pretend you have what you want or you don't think about it. And if you pretend, make it real, milk it for every nuance, every succulent taste, scent, touch. I don't have time for that kind of indulgence right now. I got a crazy Unseelie prince gunning for me with my own sword. I got a nutty nightclub owner out there who thinks he has a point to make with me and wants to lock me up to do it. I got a bloodthirsty ex–best friend who's after my ass. I got an Ice Monster killing all kinds of innocents.
I can deal with the first three. Dublin needs to know about the last one!
I got several places in the city I can print a daily. It won't take Ryodan long to find them all, so I know I don't have much time. If I can print off even just a thousand and get them up, word will spread fast. Then I'll get down to the business of figuring out how to get my sword back from Christian.
I head for the old Bartlett Building on the south side of the river Liffey, whizzing over Ha'Penny Bridge, freeze-framing parallel to the water. Stars are twinkling on it, ice crystals on a silver slide. It's all kissed with the new lavender-metallic shade the Fae brought with them.
A few seconds later I blast in through double doors, dump my backpack on a table, and fire up the presses, blowing on my hands to warm them. I set up my little miniprinter and hook up my phone to print out the photos I been taking all day. My hands are clumsy with cold. I think the Iceman is starting to screw up our weather or something. Usually in May we run a low of forty, high of sixty. And I run hotter because, well, I _run_ everywhere. But I been cold all day. Feels like no more than twenty-five or thirty outside right now. I wish this place had a fireplace like Mac has at Barrons Books & Baubles. I been avoiding that part of town for weeks. Can't stand the thought of seeing her coming and going, knowing I'm dead to her. Knowing I'll never step foot inside BB&B again and laugh with her, feel like I fit somewhere. I wish I had a place like Mac has at Barrons Books & Baubles.
"Wishes. Horses. Fecking waste of time." I was alone a lot as a kid, and at night sometimes there was nothing on TV and the silence would get ten times as big as our house. I used to talk to myself to fill it up. I was scintillating, too, always up on the latest news and stuff because I was stuck in a cage watching it all the time. Maybe that's where I got my love of broadcasting it. I had so much to say and nobody to say it to. Now I got the whole city! I keep up a running monologue as I work on my rag, mostly venting my irritation with current circumstances.
I don't have time to write anything real entertaining, something I try hard to do whenever I put a _Dani Daily_ out because any writer worth their salt knows you got to give folks bread and circuses along with the information they need to save their own asses. Otherwise they won't read it. There was this whole series on TV when I was nine about how to write and keep folks reading and I was riveted by it because I knew I'd be writing my memoirs one day.
I had no idea I'd start running a paper when I was still only thirteen and get a book published when I was fourteen!
**The Dani Daily**
**NEW MONSTER LOOSE IN DUBLIN!!!!**
**The ICEMAN slays hundreds!!!**
**READ ALL ABOUT IT!**
**AND BY THE WAY I'M NOT DEFENSELESS. DUDES, YOU THINK I'M DEFENSELESS, YOU JUST TRY. BRING IT ON! I GOT ALL KINDS OF SECRET WEAPONS UP MY SLEEVE!**
You heard it from me first and nobody else!
There's some kind of big, bad Unseelie loose in Dublin, icing folks to death. You hardly get any warning that it's about to be in your space. It's hit churches, pubs, gyms, warehouses, rural yards, and spots smack in the middle of the street. No place is off its grid! You got to watch for it real careful. At best, if you're paying close attention, you'll see a kind of shimmery spot in the air then a slit opens, fog spills out, and the monster comes. In like, just two seconds it ices everything in its path TO DEATH INSTANTLY then disappears.
Lie low, stay off the streets! I'll keep you posted, Dublin.
Oh, and if you stumble across one of its iced scenes, steer clear—they explode!
"They're not worth it."
I just about squirt right out of my skin like a dab of toothpaste in a tube squeezed too tight. I expected Ryodan to find me first.
I freeze-frame.
And slam into Christian.
"I'm a full sifter now, lass. You'll never outrun me again. It was driving me bugfuck that you could get away from me. No more." His hands close on my waist and I try to twist free but it's like I got steel vises biting into my body, clamping down on bone. I look up at him. The faint outline of a torque is luminous at his neck. His eyes are iridescent fire. If insanity has a color, it's swirling in there.
"Humans," he says coldly, and his face is like chiseled ice. Pale against midnight hair. Brilliant tattoos rush up his neck, around his jaw, back down his body, a kaleidoscopic storm just beneath his skin. "Puny. Stupid. Frightened of their shadow. Why do you bother with them? Why waste your time? You're worth so much more than that."
"Dude. I _am_ one. Give me my fecking sword. It ain't yours."
"No you're not. You're beyond human. You're what the race should aspire to be." He leans in, sniffs my hair and sighs. "Stay the fuck away from Ryodan. I bloody hate it when you smell like him. It turns my stomach."
I search my brain for a way out of this one. With my sword. Is it on him somewhere? I let my lids drift down, peek at his lower body. Don't want to telegraph. I don't see it anywhere. Jeans, hiking boots, cream fisherman's sweater that strains across shoulders way wider than they used to be. To support the wing structure he's developing? Does he miss who he was? Is that why he's dressed like this? No visible sign of any weapons on him anywhere, but then he's so far past needing a weapon. He _is_ a weapon. There's blood all over his sweater. I don't even want to know why. "You're human, too, remember?" Obviously with some part of his brain he does. The Unseelie princes rarely wear clothing.
"Not anymore, Dani, my sweet darling. You know how I'm so sure? I'm a lie detector. I said, 'I'm human' and heard my own lie." He laughs, and there's madness in it.
"You are what you choose to be," I say. All the sudden I can't breathe because his hands have slipped up over my ribs and he's squeezing me so tight I think they're going to crack.
"I would NEVER have fucking chosen this!"
"Ow! Volume control, Christian! And you're hurting me!"
He releases his grip instantly. "Are you all right, lass? Are your ears bleeding? I made the last woman's ears bleed. Nose, too. And her... well, that's neither here nor there."
"Let me go. I got stuff to do."
"No."
"Look, if you're going to try to kill me, get it over with." I put both of my fists in front of my face. "Put up your dukes!"
He stares at me. "Why would I do that?"
"Hello—Mister I Keep Dead Women Stuffed Down the Side of My Bed!"
"I tried to explain that to you. You wouldn't listen. You ran away from me. Why did you run away from me? Don't I keep telling you I'll never hurt you?"
"Did you kill her?"
"No."
I give him a look. I don't need to be a lie detector to see through that one. It was there in the shifty slide of his eyes. "Try again."
"Fine. Okay. I killed her. But I didn't mean to. And I didn't _kill_ her, kill her."
"Oh, I see. As long as you didn't _kill_ her, kill her, then that's okay."
"I knew you'd understand," he says, like I'm not being totally facetious. I'm not sure he gets human nuances anymore. I think he's too far gone.
"All ears here."
He shrugs. "There's not much to tell. We were having sex and all the sudden she was dead."
"Just like that?"
"Just like. It was the bloody weirdest thing. I don't even know what I did."
"Your hands weren't, like, around her throat or holding a knife or anything?"
"No. That's why I kept her. I wanted to examine her to figure out what I did so I don't do it again. It's not like I can go without sex for the rest of my life. I can barely make it a few bloody hours. One second she was having a great time and so was I, and she was making this really hot noise while I was—sorry, you probably don't want to hear about that. I'm not trying to make you jealous, lass. Then she just wasn't moving and you have no idea how disturbing that was. Well, mostly. But not entirely. I think the Unseelie I'm becoming was aroused because once she stopped moving it was like—"
"Too much information! I can't hear you!" I start humming to tune him out. Jealous? What is he talking about?
"I got distracted and left her on the bed to look at later, then I found you bleeding to death and brought you back to my place. I didn't want you to see her and get upset. I was going to figure out what I did to her after you were gone."
"Did you?"
"Still no clue. There wasn't a mark on her anywhere. I thought I must have been too rough and bruised her from the inside, but if I did, you'd think there'd be external bruises somewhere, and there aren't any. Maybe you'd take a look at her. I've been considering an autopsy but I don't know any morticians. Do you?"
He asks it like it's a normal question. Like he's the person investigating a murder, not the one who committed it. "Nope." I wonder how crazy he is. "Does it bother you that you killed her?"
He looks aghast. "Of course it does! I don't want to kill anything. Well... actually that's not entirely true. I do want to kill things. Lots of things. Especially Ryodan lately. I can lose myself for hours in a soothing haze of murderous intent about that dickhead."
"Won't argue with you there," I commiserate.
"But I don't. At least I didn't until now. And if I can't figure out what I did this time, I can't stop myself from doing it in the future."
"Where's my sword." I say it like Ryodan, with no question mark at the end. I'm beginning to understand why he does it. It's a subtle demand instead of a question. Folks answer instinctively, against their better judgment. That's Ryodan, always playing the odds, stacking them in his favor.
Christian smiles and for a second I see a hint of who he was. Now that his face has completed most of the transition to Unseelie prince, his expressions are more readable. I guess the muscles aren't always at war, trying to shape a look. He has a dazzling smile, almost a killer smile, but not quite. It's the smile of a man who could get any woman he wanted into bed, but might just kill her while she's there.
"You have to admit, the flamethrower was bloody brilliant, wasn't it? I blasted the thing right out of the stalagmite and fried Ryodan's men. They didn't even think of it. Fucking idiots. You want something, take it."
"Did you hurt my sword? Wait a minute!" I realize something I can't believe it took me so long to realize. "You're not making me feel like I'm turning Pri-ya!"
"I figured out how to mute it. It's just as easy to turn back on. All I have to do is this."
Horniness slams into me, and I hear myself making such an embarrassing sound I could die of embarrassment.
He keeps me from sinking to the pavement, physically holding me up, hands around my waist. "Lass, doona be looking up at me like that. On the other hand, do. Yes. Yes. Exactly like that. Princess, you're slaying me."
"Turn it off, Christian! I want to choose my first time!"
I collapse on the floor, blinking, dazed.
Christian is gone.
Without his hands holding me, I slumped in on myself like a wet cardboard box. I sit there, looking around but seeing nothing, trying to clear my head. Either he's completely gone or he's muting himself again. But the aftereffects linger.
His voice floats down from somewhere in the rafters above my head "First time, is it now? I was fair certain, lass, but I like hearing it from you. I'll wait. I want you to choose your first time, too. It'll be chocolate and roses. Music and sweet kisses. Everything a lass dreams of. I want it to be perfect for you."
I turn beet red. Nobody, but _nobody_ talks about my virginity but me! "Butt out of my virginity-losing plans! They're none of your business."
"They're my business and mine alone. But we don't have to talk about them. Yet."
I feel like I just got brained upside the head with a frying pan. Is he kidding me? Has Christian decided in his half-mad Unseelie prince mind that he's going to be, like, my boyfriend and be, like my first? Dude, I'm fourteen and he's an Unseelie prince! And he's like ten years older than me! I open my mouth to read him the fecking riot act and set things straight between us when I think about how an Unseelie prince with a crush on me might not be an entirely bad thing to have, and close my mouth again. He might be tricky to handle but all weapons are good weapons, and Christian on a leash would be like, the ultimate weapon. Especially against Ryodan.
The question is: can I leash him? And if I manage to, will I be able to hang on to his collar when it counts?
I choose my words carefully. Prince and a lie detector to boot. If I can collar this dude, I can do anything! It'll be like dancing on a minefield. I'm fascinated by the prospect. What a way to test myself. "Thanks for understanding, Christian," I say.
"No problem. Well, it is. But I'll deal with it. For now."
"The other Unseelie princes scare me."
"They should. They're walking nightmares! You wouldn't believe some of the sick shit they do."
The irony that's lost on him isn't on me. One second he's aware of himself as Unseelie prince, the next he acts like he's the furthest thing from it. I don't say _Yes I would because, dude, you do sick stuff, too_. Casting aspersions won't win me any points. "I feel so unsafe without my sword." I squint up toward the ceiling. The Bartlett Building used to be an old warehouse before it was converted. They left the steel beams and girders exposed when they moved in. I don't see him up there anywhere.
Then he's in front of me, sweeping low in a formal bow.
"Your sword, my lady. I would have razed heaven and earth to get it back for you." He's holding it across both hands, presenting it to me. He looks at me and I look straight back, measuring the madness in his eyes. I feel moisture pressing at the corners of mine, like they're going to start bleeding. I pinch the bridge of my nose, hard. I can't stop staring at him. It's like his eyes are made of liquid silver on top of rainbows, like the kaleidoscopic tattoos beneath his skin run like a river at the bottom of them, like I could tumble in and dive down deep. I feel woozy.
"Don't look me straight in the eyes, lass. Stop it!" He chucks me under the chin, jarring me into breaking eye contact. He trails his fingers across my cheek and when his hand comes away bloody, he licks it. "Never look in my eyes too long. It hurts people." Then he smiles. "You'll notice I can touch the hallow. I worried I wouldn't be able to."
I look down at the Seelie hallow across his hands, one of four Fae talismans that only humans and those of the Light Court can touch. I could take it, drive the blade through his heart and be free of him forever.
I reach for my sword.
He pulls it back. "A little thanks might be nice."
"Christian, you're the Shit," I say. "First, you saved my life and now you're giving me back my sword when nobody else would even help me."
"Dickhead sure didn't."
"He sure didn't," I agree and reach for my sword again. "Nobody cares about me like you do."
"Och, lass, you've no bloody idea," he says in a near-whisper. "I see you from the inside."
"Can I have it now?" I want it so bad my palms itch.
He cocks his head and looks at me then swivels his head just like an Unseelie prince, like his head and neck aren't put together quite right. It gives me chills. "You wouldn't be thinking of killing me with it, now would you, lass?"
"Of course I would. But I'm not going to." Not right now, anyway.
His smile is blinding. "Good, because I have another gift for you tonight. I know you like to save humans, so I'm going to help you. You may consider it one of many early wedding gifts."
I blink. Huh? Either I manage to mask my astonishment or he doesn't even notice the look on my face, because he just keeps talking.
"The Unseelie princes know the thing that came through the slit at the warehouse. They called it Gh'luk-ra d'J'hai."
"What the heck does that mean?" Also, wedding gifts? Has he completely lost his marbles?
"It's hard to translate. Unseelie have forty-nine words for ice, and there's a nuance to d'J'hai I'm not sure I understand. Loosely, I'd call it the Hoar Frost King."
"The Hoar Frost King," I echo. "What is it? How do you kill it? Will the sword work?" Assuming anyone could even get close enough without freezing to death.
"I don't know. But I know a place where we might find out. If there are answers anywhere, they'll be there. Take the sword, lass. I don't like you being unprotected. And I know you don't want me hanging around all the time. Not that I blame you with the monster I'm becoming."
I reach for it with both hands. I almost can't contain myself. I'm trembling with excitement.
He leans in and lays the sword across my palms.
I close my eyes and sigh with ecstasy. The heft of cool steel in my hands is... well, better than I think sex must be! It's like having both arms amputated and thinking you'll have to learn to live without them, then getting them put back on completely fine again. I love my sword. I'm invincible with it. I don't know one fecking ounce of fear with this thing in my hands. Deep inside where my blood runs a little stranger than other folks', gears shift and slide back into perfect alignment. I am one with my blade. I'm complete.
"Och, and there's the woman you'll be one day," Christian murmurs. "Passion enough to rule an army. Not that I have one. Yet."
I might want an Unseelie prince on a leash but we need to get one thing straight. "I ain't ever getting married."
"Who said anything about getting married?"
"Dude, early wedding gifts."
He looks at me like _I'm_ nuts. "Who said anything about wedding gifts?"
"And I don't want an Unseelie army to rule."
"Army? Dani, my bonny will o' the wisp, what are you talking about? I was telling you about the Hoar Frost King. Are you coming or not? It's a fine night to be alive. We've a monster to catch." He winks at me. "And tonight it's not me."
Dude. Sometimes that's all you can say.
# TWENTY-EIGHT
# _"I walk up on high and I step to the edge to see my world below"_
A good leader knows her world.
I know nothing of my world.
Well, that's not entirely true.
I know that 152 paces beyond where I stand looking out of Rowena's dressing room window there is a serene arbor of shaped topiary with a tiled pavilion, stone benches, and a reflecting pool that centuries-dead Grand Mistress Deborah Siobhan O'Connor built for meditation in times of turmoil. Far enough from the abbey to grant privacy, near enough to be used often, the silvery pool was long ago usurped by fat frogs on lily pads, and on a gentle summer night, in my old room three floors above Rowena's and two to the south, they charmed me to sleep with their lazy baritone _ah-uuups_ for many years.
I know also that there are 437 rooms in the abbey, in common-knowledge use. I know of an additional twenty-three on the main floor alone, with more on the other three, and undoubtedly countless more of which I know nothing at all. The rambling fortress is a hive of concealed passageways and hidden panels, stones and floorboards and fireplaces that move, if you've the secret to their operation. Then there is the Underneath. That is how I have always seen the abbey: the Upstairs proper where sunshine glitters on windowpanes and we bake and clean and are normal women, and the Underneath where a dark city twists and turns, with passageways and catacombs and vaults, and the sweet Lord only knows what all. There, those of us in the Haven become something else sometimes, something ancient in our blood.
I know that a quarter of a mile behind the abbey is a barn with 282 stalls where cows and horses and pigs were once penned. I know a brisk walk beyond it is a dairy that housed forty-odd milk cows, with a chilled larder where we made butter and cream. I know that there are seventeen rows of five beds making eighty-five tiered vegetable garden beds behind the dairy that once grew enough to sustain the abbey's thousands of occupants plus more to sell in the village for a tidy sum.
All these things I know belonged to a different world.
The world I live in is no longer a world I know.
It is four-thirty in the morning. I pull my wrapper more closely around me and stare out the window at gnarled oaks casting long shadows, and moonbeams crisscrossing the lawn through latticed branches. My comforting view of the familiar shaped topiary is blocked by one of those dangerous aberrations of physics Mac calls Interdimensional Fairy Potholes; IFPs for short, an expedient abbreviation. This one has the funnel shape of a crystalline tornado and shines milky-lilac, its dull, faceted exterior reflecting the moonlight. In the light of day those pellucid facets are difficult to distinguish from the surrounding countryside, compounded by their extreme variations in shape, texture, and size. I have seen IFPs larger than our back field and smaller than my hand. This one is taller than a four-story building and as wide.
When first she told me her name for them, I laughed. That was back when my family had recently been killed and I was drunk on freedom. For the first time in my life, when everyone around me felt anxious about the many new monsters on the loose, I felt gloriously, deliriously safe. My monsters were gone. They'd been trying to pry me from the abbey again, my mother evidencing a triumphant gleam in her eye at Sunday-supper-last, and I was certain she and Father had finally struck upon something Rowena wanted badly enough to give me up. For years, the diminutive Grand Mistress had commanded my blind devotion merely for standing bulwark between them and me.
IFPs are no longer a laughing matter. They never were. This one was discovered a week ago, heading straight for our abbey. We wasted days tracking its progress, trying to devise ways to divert it. Nothing worked. It is not as if an IFP can be blown off course with a giant fan. I am the leader of this enclave, yet I'm unable to do something so simple as protect it from being swallowed up by a fractured piece of Faery! The IFP is not even a sentient enemy. It is merely an accident of circumstance.
Then there are the sentient enemies I have to worry about. The thinking, coveting ones whose Upstairs never matches their Underneath, who are no doubt even now talking about the repository of endless knowledge and power the world now knows we have locked beneath our fortress, guarded by a snortingly inept 289 women ranging in age from seven all the way up to Tanty Anna, at one hundred and two.
These are my charges. Trusted to my care.
I see no end for them that does not involve their hapless slaughter!
I need more _sidhe_ -seers. I need to strengthen our numbers.
Last night I gathered my girls around the IFP when it was a mere mile from the abbey. We'd plotted its course with ninety-nine percent certainty: it would enter our home. The only questions were how much of the south chapel next to Rowena's chambers would it instantly engulf, and would it raze every square inch of our abbey or leave the occasional pile of rubble, perhaps a glowing, red-hot wall standing here and there?
Given its rate of locomotion, it would take nearly an hour for it to complete its passage from end to end. We were able to plot the time and trajectory of its destruction so accurately because it had already left hundreds of miles of fine, sooty ash in its wake. Dirt fields were emblazoned with deep ruts of scorched earth. Large buildings were reduced to small mountains of postapocalyptic embers.
A drifting crematorium, the IFP on a crash course with our abbey contained a fire-world fragment, a roaring inferno capable of instantly reducing concrete to cinders. Were it to enter our walls, it would leave us homeless. To say nothing of what such heat might do to a certain ice cube beneath our fortress.
We tried to spell it, divert it, destroy it, bind it into place. I'd spent the entire day scouring old books Rowena kept in her bedchamber library, although I was fairly certain it was useless. I have yet to find her real "library." This is another thing I know, because I saw her carrying books at times of crisis that are nowhere to be found. Yet.
My girls wept at the end. We were hot and tired and soon to be homeless. We'd tried everything we knew.
Then a black Humvee drove up and three of Ryodan's men got out.
With Margery.
The men bade us retreat to a safe perimeter. Using dark magic that mystified us, they tethered the IFP to the earth a mere twenty yards from our walls, where it has remained stationary since. Where, they assured me, it shall continue to remain stationary for all time.
"But I don't want it there," I told them. "What am I to do with it? Can we not move it?"
They looked at me as if I had five heads. "Woman, we saved you from certain destruction and you want to critique how we did it? Use the bloody thing as a trash compactor. Incinerate your dead and enemies. Boss'd love to have something like this near Chester's. It's a fire that will never go out."
"Take it there, then!"
"Only way to get it there is cut the tether. Do that and it goes straight through your abbey. Be glad he hasn't decided he wants it or this place would be forfeit. Dublin is on the other side of your walls. Keep your door open. Ryodan will be by in a few days to tell you what you owe him."
After they left, Margery pumped her fist in the air and called for celebration that the danger was averted and we lived to fight another day. My girls rallied around her, jubilant, cheering. I stood jostled and forgotten in the melee.
Ryodan will be by in a few days.
To tell me what I owe him.
For years I have hidden behind these walls, trying to be as unimportant as possible. Unassuming. Overlooked. I was happy to walk the fields, daydreaming of Sean and the future we would have, studying _sidhe_ -seer magic and occasionally guiding the girls with gentle wisdom, praising God for my blessings.
I love this abbey. I love these girls.
I turn and walk past the transparent vision of Cruce, who has been sitting on the divan in my dressing room watching me ever since the bells chimed the witching hour, four and a half hours ago, winged and naked as only he could be. I dab my brow with a handkerchief, blotting the sheen of perspiration that is constant of late. As Sean was unable to come last night, I have not slept in two days. Not to be deterred, Cruce found a waking way to torment me. Fortunately all he is capable of at the moment is a weak transmission of his appearance. He cannot speak or touch me. Or he surely would have. I slide my gaze over him with only the smallest hitch.
I begin to dress.
Last night my first cousin was a better leader than I.
Because I don't know my world.
The time has come to change that.
The drive to Dublin is long and silent. There are no longer any radio stations to listen to and I don't carry a phone or iPod.
The day was arduous, with Margery presiding over the abbey as if she were in charge, riding the wave of adulation for her lastminute save, peppering her salted commentary on my many failings with inflammatory phrases calculated to incite the girls and make them feel as if I am restricting them as Rowena did. I watch her and think: Am I to take less than three hundred children, young girls, and aging women to war? Later, I tell her. We must fight smart and hard, not fearlessly.
Smart and hard would have left us homeless, she retorted. Fearless is why the abbey stands today.
On that score she is correct, but here, between us and for the fate of my girls, is a deeper problem. She does not _care_. In order to gain control, Margery would lead the _sidhe_ -seers to their deaths, because for her, leadership is not about their well-being, only hers. Ironically, her very self-engrossment makes her charismatic where I am not. On my way into the city I ponder the need for charm in my management of the girls. It is clear that a decision looms: I must either abdicate leadership or change in more ways than I am certain I can survive.
I arrive at Chester's just after ten, stunned to find a line spanning three demolished city blocks. I had no idea so many young people were alive in Dublin or that I might find them lined up as if it were a common Tuesday night, as if this were the new Temple Bar. Do they not know the world is infected and dying? Do they not feel the pounding hooves of the Horsemen of the Apocalypse riding? One has been unhorsed for now, although he smiled seductively at me from my divan before I left. Another is being remade. Soon there will be four again.
I leave my car in an alley and walk to the end of the line, resigned to turning what will inevitably be an all-night wait into a lesson about my new world.
I have barely begun to say hellos to my new companions when a hand closes on my upper arm from behind.
"Ryodan will see you now."
It is one of his men, tall, muscled, scarred as the rest. He escorts me to the front of the line, over protests and promises, from the flirtatious to the grotesque. As we descend into the club, I raise barriers to shield my empath heart.
Music hammers me, pounding, visceral. Emotions bite deep despite my efforts to deflect them. Such naked hunger, such anguished desire for connection and relevance! But they are going about it the wrong way. I see here the very definition of insanity: attending Chester's, looking for love. Why not go to the desert, expecting to find water?
They would fare better to loot a hardware store and hope to meet another looter in the process; at least they'd know he was a responsible, capable man intent on rebuilding something. Or pilfer a library! Any man who reads is a fine one. Find a prayer group; they've sprung up all over the city.
On the surface, each person we pass seems happier than the next, but I feel it all: pain, insecurity, isolation, and fear. Most of them have no idea how they will survive past this night. Some have lost so many loved ones they no longer care. They live in isolated pockets of abandoned houses and buildings with no televisions and no way to keep up on the threats in the world, which are constantly evolving. Their prime directive is simple: to not sleep alone tonight. These are people who only recently could find out anything they wanted to know at the mere touch of a screen. Now, stripped of their outer hulls, defenses breached, they are adrift and listing badly.
And I cannot help but wonder...
Could I reach them? Could I somehow gather them into a single place and shape them for a purpose? I feel light-headed at the thought. They are not _sidhe_ -seers... but they are young and strong, and impressionable.
A woman dances, her head back in mock ecstasy, smiling, surrounded by men and Unseelie. I get a flash of her heart as we pass and know that she believes a man will never love her unless she always makes him feel good. She has relinquished her right to be a person with needs and desires, and become a receptacle for filling the needs of a lover. If she is bright as a butterfly and sexual as a lioness in mating season, she will be cherished.
"That is not love," I say as we pass. "That is a bargain. You should charge for it. You should get something in return."
When I was young, I began ranking people by a number system: one to ten—how broken are they? She is a seven. Her heart could be healed but it would take an intensely committed man and much time. Few are so lucky. Fewer still are soul mates like me and my Sean.
As we ascend to the second floor, I look out over the subclubs and see Jo, dressed as a Catholic, school-age child. I dislike the mockery of my faith and am still uneasy with her decision to work here, but she argued passionately for it, strongly committed to her mission to gather intelligence at the richest source. She has yet to tell me anything that makes me feel subjecting her to this cesspool is worthwhile. I know a thing about people: who and what we surround ourselves with is who and what we become. In the midst of good people, it is easy to be good. In the midst of bad people, it is easy to be bad.
As we top the stairs, I find my eyes drawn back to the subclub where the waiters wear only tight black leather pants and a bow tie, revealing vast expanses of tanned, muscled skin, or in other cases generous bare bosoms. Only the stunning are employed there. I catch my breath. One of the male waiters has a beautiful back, a lovely long-limbed way of moving. I could watch him walk away for hours. I am a woman and I appreciate a fine-backed man. I am relieved because it is not Cruce. He has not so thoroughly perverted me that I no longer find human men attractive.
My escort guides me down a hallway of smooth glass walls to my left and my right, unbroken but by nearly nonexistent seams. The rooms up here are all made of two-way glass. Depending on how the lighting is adjusted in each room, you can either see into it from the outside but not out or out from the inside but not in. I had heard from Dani a description of the upper levels of Chester's so I knew to expect a see-through glass floor, but expecting it and walking on it are two very different things. People do not like to see what lies beneath. Yet here at Chester's the owner forces you to view it with every step you take in his demesne. He is a calculating man, and a dangerous one. And so I have come here tonight to determine my debt, pay it, and get it over with.
My escort stops before a seemingly seamless glass wall and places his palm against it. A glass panel whisks aside with a hydraulic hiss. The weight of his palm on my neck guides me into the darkened room.
"Boss'll be with you in a minute."
I can see out on all sides, up and down. From Ryodan's glass aerie, he studies his world by naked eye and camera. The perimeter of the room, at the ceiling, is lined with hundreds of small monitors, three rows deep. I scan them. There are cameras focused on every room, from nearly every angle. There are rooms that are sordid beyond my awareness of such activities. This is the world I must learn if I am to lead my girls.
The door hisses open behind me and I say nothing, wait for him to speak. When he does not, I expand my empath gift to get a feel for him. There is no one in the room with me. I realize someone must have opened the door, seen me, not him, and walked on. I continue with my observations of the screens, turning slowly as I absorb the faces, the actions, the offerings. I must learn people as I have never learned them before.
The hand on my shoulder draws from me a small, involuntary scream.
I whirl, frightened, and I'm against Ryodan's chest, with his arms gently around me. I would speak but I know I would only stammer. There was no one in this room with me. I did not hear the door open again. How, then, is he in the room?
"Easy, Katarina. I did not save you from harm last night to harm you tonight."
I look into a face that is unreadable. It is said of this man that he wears three expressions and three alone: amused mockery, urbane aloofness, or anger. It is said if you see anger, you are dead.
I open my empath gift.
I am in this room alone.
I can find no words to say. I decide to use the ones I have. "I am in this room alone."
"Not quite."
"You don't exist."
"Touch me, Katarina. Tell me I don't exist." He brushes my cheek with a kiss and I shiver. "Turn your head for me and I'll kiss you as a woman should be kissed." He waits, his mouth brushing my cheek, for me to turn ever-so-slightly, part my lips and take his tongue. I shiver again. This man would not kiss me as I like to be kissed but as _he_ does. His way is too hard, demanding, dangerous. His way is not love. It is passion and it burns. Incinerates. It leaves only embers as surely as the IFP his men tethered at my abbey last eve.
When I pull back, he laughs and drops his loose embrace. I give him a level look. "Thank you for sending your men to tether the fragment of Faery. They spoke of payment. We do not have much. What can our abbey offer in return for such generous aid?"
He smiles faintly. "Ah, so that is how we are to be. You speak eloquently for one who spoke no words at all until she was nearly five."
I will not be rattled. So, he knows I was without voice for years after I was born. Many know the story. The pain of the world's emotions overwhelmed me upon birth. I was a terrible baby, an awful infant. I cried incessantly. I never spoke. I curled in a ball and tried to escape the pain of the world. They called me autistic. "Thank you."
"Until Rowena came and offered your family a deal."
"I did not come to speak of myself, but of how I may repay you."
"She would draw you from your autistic shell, but at eighteen years of age you were hers. You would come live at the abbey. Your parents leapt at the opportunity. They despaired of ever silencing your weeping."
Sometimes, even then, Sean had been there. Sometimes in the delirium of my pain he had curled beside me and said, "Girl, why do you cry?" I remember moments of silence then. He would put his chubby arms around me, and for a short time the pain would go away.
"How would they make a grand alliance with larger and nastier criminals if their only marriageable daughter was defective?" I say dryly.
He laughs. "There you are, behind that eternal serenity. The woman that feels. Funny thing is, I, too, thought I was in this room alone. Until you said that. The dearth of emotion here is not mine alone." His smile fades and he looks straight into my eyes with a stare so penetrating, direct, and uncomfortable that I feel I am an insect pinned to a board, prepared for dissection. "You owe me nothing further."
I blink. "But I haven't paid you yet."
"You have."
"No, I haven't. I've given nothing."
"The price was not required of you."
I get a chill and almost can't get my breath. This man is dangerous. Clever. Terrifying to me. "Of whom was it required? I am the one responsible. I am the one who failed. I am the one who should have led them to safety, therefore it should be me and me alone who pays a price!"
"Funny thing about payment is that it isn't the buyer of the goods or services that gets to set it. It's the seller. That's me." His face is hard and cold now.
"What price did you set?" I school my breath slow and even, waiting for his reply,
He moves to my side, guides me to the glass and directs my attention below. "I have had difficulty staffing lately. My servers keep dying on me."
The skin of my spine begins to crawl.
"One club in particular is hard to keep staffed. The Tuxedo Club is constantly requiring replacements."
It is the subclub where the servers dress in tight black leather pants and bow ties, and serve topless.
"Your Sean was good enough to fill in for a time."
Bile rises in the back of my throat. "My Sean does not belong here."
"Perhaps. But even you have to admit he looks good in the uniform."
I look where he's pointing. The back I admired on my way up the stairs has known my hands on its shoulder blades as he moved inside me. I have tickled it many nights as he drifted to sleep. I have massaged it when he worked the nets overlong. I have kissed each and every muscle and curve. It is, indeed, a beautiful back.
"How long?"
"I haven't decided."
"Don't do this to me."
"Why."
"He is..." I stop and sigh. This man would understand nothing of what I would say.
"Go on."
"Sean is my soul mate."
"Soul mate."
He mocks me. He mocks God. "Such things are sacred."
"To who? Your god may love soul mates but man does not. Such a couple is vulnerable, particularly if they are fool enough to let the world see how shiny and happy they are. Their risk rises tenfold during times of war. There are two courses a couple in such circumstances can chart: go deep into the country and hide as far from humanity as possible, hoping like hell nobody finds them. Because the world _will_ tear them apart."
He is wrong. He knows nothing of soul mates. Still I cannot help but ask, "The other?"
"Sink up to their necks in the stench and filth and corruption of their war-torn existence—"
"You mean behave like common criminals. Would you prefer us ruthless animals? Why are you doing this?"
"I mean look at it, Katarina. See things for what they are. Drop your blinders and raise the sewer to eye level; admit you're swimming in shit. If you don't acknowledge the turd hurtling down the drain toward you, you can't dodge it. You have to face every challenge together. Because the world _will_ tear you apart."
"You are manipulative, cynical, and base."
"Guilty as charged."
"Life is not as you see it. You don't know anything about love."
"I am intimately acquainted with the vagaries of fate in times of war. They've been my worst and best centuries."
"That is not love."
"I didn't say it was." He flashes a smile, white teeth gleaming in shadow. "I prefer war. The colors run more brilliant; food and drink are more rare, and the sweeter for it. People are so much more interesting. More alive."
"And more dead," I say sharply. "We lost nearly half the world and you find it 'interesting'? You are a pig. Barbaric and cruel." I turn away. I have had enough. If this is his price then I am free to go. There is nothing more I owe him. He has already taken it all.
I move for the door.
"You must tell him, Katarina. If you are to have any hope at all."
I stop. He cannot know. There is no way that he could know. "Tell who, what?"
"Sean. About Cruce. You must tell him."
I whirl, hand fluttering to my throat. "What in God's name are you talking about?"
I search his eyes and I see there that somehow he knows my deepest shame. They hold a secret smile and a certain amused resignation. As if he has watched humanity's idiocies play out in front of him so many times that they have begun to... not pain but perhaps perturb him. As if he wearies of watching the rats in the maze run into the same walls over and over. I expand my empath gift, I push with all I've got, and still I can't even sense that he is in the room with me. There is _nothing_ where he stands.
"If you don't tell Sean that Cruce is fucking you while you sleep, it will destroy what you have with him more certainly than any job in my club could. That, down there"—he points to Sean serving a drink to a pretty, nearly naked Seelie—"is a bump in the road, a test of temptation and fidelity. If your Sean loves you, he will pass it with flying colors. Cruce is a test of your fucking soul."
I don't bother arguing with him. He knows. Somehow he knows. Perhaps he can read thoughts like I read emotions. It is a terrifying idea. "Why can't I feel you?"
"Perhaps the lack is not mine. Perhaps it is within you."
"No." Of this I am certain. "There's something wrong with you."
Again he flashes that smile. "Or something right."
Perhaps I take the coward's way. Perhaps I take the honorable path. I cannot decide. My head is a muddle. But I give the Tuxedo Club wide berth and pull up the hood of my cloak. I do not confront my Sean as I leave. If he tells me, we will discuss it. If he does not, we will not. I tell myself I am respecting his boundaries, preserving his dignity. This is where he will be instead of in my bed in coming nights.
The price of saving my abbey is a piece of my heart and the lion's share of my spine. That is what Ryodan called due.
My Sean will face temptation alone every night at Chester's, and I will face it alone at the abbey, in my bed.
This is not a world I ever wanted to know.
# TWENTY-NINE
# _"In the white room"_
One night when Mac and me were killing Unseelie back-to-back, she had a kind of meltdown and started crying and yelling while she sliced and diced. She said that she was going to send them all straight back to hell because they stole everything from her that mattered. She said she used to know her sister, everything about her, and that was where love was, in the knowing and sharing, but it turned out Alina had a boyfriend she'd never mentioned and a whole other life she knew nothing about, and not only didn't Alina love her, her entire existence to date had been one great big fat lie. Her parents weren't her parents, her sister probably wasn't her sister, nobody was what they seemed, not even her.
In Rowena's stash of journals chronicling her nasty, evil reign, I found Mac's sister's diary. I have over four hundred journals locked away with the Grand Mistresses emblem emblazoned on dark green kidskin leather. She was eighty-eight when she died, though she didn't look a day over sixty. She had a Fae she'd been nibbling on for decades locked in a vault beneath the abbey. I killed it when I found out about it.
When I discovered Alina's diary, I tore out pages and got them to Mac on the sly, trying to make up for silencing her sister's voice and show her she'd meant everything in the world to Alina.
"Why the feck are we here?" I say crossly. I wouldn't even be thinking about Mac if we weren't. Christian's been sifting me around the city, helping me plaster my _Dailies_ on lampposts. I been letting him touch my pinky finger to do it. He keeps trying to put his arms around me. His last sift deposited us catty-corner to Barrons Books & Baubles, with the street between us.
I feel like puking.
I ain't been here since the night Mac found out the truth about me. The night she baked me a cake and painted my fingernails and saved me from the Gray Woman, only to end up ready to kill me herself a few minutes later.
In the middle of a ruined city, Barrons Books & Baubles stands untouched. I think a silent benediction: May it always. There's something about this place. As if its mere existence means the world will always have hope. I can't explain why I feel that way but all the folks I know that have ever visited it, all the other _sidhe-_ seers, feel the same. There's something different, something extraordinary on this island, in this city, on this street, in this precise spot. It feels almost like once, a very long ago time, something terrible nearly happened here at this longitude and latitude, and somebody put BB&B on the gash to keep the possibility from ever occurring again. As long as the walls stand and the place is manned, we're okay. I snicker, picturing it looking just like it does right here and now, in prehistoric times. It doesn't seem so improbable.
To the left and right the cobbled street is swept clean. There's no riot-detritus outside Barrons's establishment. No husks left from Shades gorging. No trash. Planters line the cobbled street, and there are small plants trying to grow in them, valiantly fighting the uncommon chill. The entry to the tall, deep brick building is drenched in dark cherry and brass and polished to a high gloss. The place is Old World and urbane as the dude himself, with pillars and wrought-iron latticework and a great big heavy door with fancy sidelights and a transom that I used to bang through, and sometimes I'd go in and out, in and out, just to hear the bell above the door tinkle. It sounded really cool in fast-mo, used to crack me up.
A hand-painted shingle hangs perpendicular to the sidewalk, suspended by an elaborate brass pole bolted into the brick above the door alcove, swaying in a light breeze.
Amber lights glow behind glass panes tinged with a hint of green.
It's all I can do not to go banging in that door, say, "Dude, what's up?"
I'm never going to bang in that door again.
"Get us out of here," I say crossly.
"Can't. This is where we need to be. And what the bloody hell is up with that?"
I look at him. He's looking up at the roof of BB&B, where dozens of enormous floodlights shine down into the street. I have to back up a few steps to see past them and see what he's seeing because I'm so much shorter. I gape. "What the feck are ZEWs doing here?" The entire roof of BB&B is covered with Zombie Eating Wraiths. Hulking anorexic vultures, with creepily hunched bodies and a gaunt grimness that defies description, they huddle in their voluminous black robes, dusted with dirt and cobwebs, unmoving. Carrion-eaters, packed shoulder-to-shoulder, they're as fixedly still as a deathwatch. I'm not sure I would have even noticed them if Christian hadn't pointed them out. They're not chittering and it's somehow worse that they're silent. "Why they hanging out on Mac's roof like that?"
"How the fuck would I know? Sorry, lass. I mean, how would I know?"
"You can say 'fuck' around me. Everybody does. And you'd know because you're Unseelie."
"Not completely, not yet and not originally. That's a lot of nots. And just because the rest of the men in this city are pigs doesn't mean I am. There's another 'not' for you. I'm bloody well made of nots tonight. I'm not the monster being hunted either."
I give him a look. His eyes are wild. This is a dude on serious edge, teetering, arms pinwheeling. "So, what are we doing here?" I try to bring some focus back to the conversation.
He doesn't answer me. Just stalks off, straight toward the bookstore, and right when I'm about to freeze-frame it out of there because there's no way I'm going inside, even if nobody's home, he turns sharp and heads down the alley between BB&B and the neighboring Dark Zone.
"If you want to stop the Hoar Frost King, you'll have to come with me, lass. I'm taking you to the Unseelie King's library. If there are answers to be had, they'll be found there."
The Unseelie King's library! "Holy borrowing bibliophile, let's book!" I take one last look up at the ZEWs and freeze-frame to catch up. If Mac's in the bookstore, she won't notice the blur that just passed her door. I shiver as I chase after him. It's fecking cold tonight. I more than _want_ to stop the Hoar Frost King. I've got to. It's getting downright frigid in Dublin and I got a terrible feeling it's going to get a lot worse.
When Christian pushes into the brick wall of the building catty-corner to the rear of BB&B—first left on the Dark Zone side—and disappears, I melt down in a fit of the giggles. I toss a rock at the spot where he vanished. It bounces off the brick and clatters to the cobblestone. I'm feeling twenty shades of Harry Potter's train station, especially when he pokes his head back out of the wall and says impatiently, "Come on, lass. This is hardly my favorite place to be."
I approach the wall and study it, trying to decide if I'd be able to find the spot again without knowing exactly where it was. His head disappears. I wouldn't. I want to chalk a big X on it, in case I need it again, but that would betray its location to everyone else, too, being as "X marks the spot" and all, so I back up partway down the alley and lock the scene down on my mental grid, permanent-like. I got that kind of memory. If I deliberately file something, I can always find it again. Hard part is remembering to deliberately file it. I'm usually so excited by the life I'm living I forget to take pictures.
Then I follow him in. Dude! I step _into_ a brick wall! It's the freakiest thing I've ever felt. Like it's a sponge and I'm a sponge and for a second there all our sponge parts are one and I don't just have square pants, everything about me is squarish because I'm part of a wall, then I'm me again and the wall kind of squirts me out on the other side in a completely white room.
White floor, white ceiling, white walls. Inside the white room are ten mirrors. Just like that. Standing there, in thin air. You can circle all the way around them. Nothing is holding them up that I can see. They're all different sizes and shapes, in different frames. Some of the glass surfaces are dark as pitch and you can't see a thing. Others swirl with silver fog but the things that move in their cloudy shadows are too fast and strange to define.
"Good," he says. "They're where I left them."
"Where else would they be?"
"They used to hang on the wall. I shuffled them around so if anyone else knew where they went, they'd lose track. Used to be the one we're taking was fourth from the left. Now it's second from the right."
I take one last look around, I don't know, maybe looking for tired starlings, but there aren't any, and push into the mirror behind him. I get all spongy again and this time it's like I pass through a lot of things and just when I'm starting to get a little tense about it, wondering if all my parts are definitely going to come back together, I squirt out into Christian's back. "Ooof! What are you doing, standing there blocking the mirror?"
"Hush, I thought I heard something."
I perk up my superhearing. "I don't hear nothing and I can hear everything."
"There are things in here," he says. "You never know what you might find."
"Bad things?"
"Depends on your definition. And who you are. Being a prince has its advantages."
I look around. "Where are we?"
"The White Mansion."
"Duh, like I might never have figured that out," I say, because we're in yet another white room. "Is the whole place this boring? Don't the Fae ever use paint, maybe a little wallpaper?"
He chimes softly.
"Dude, you're ringing like a bell."
He stops abruptly and I realize he was laughing. I'm beginning to understand how to interact social-like with an Unseelie prince.
"The White Mansion isn't boring, lass. Never boring. It's the grand demesne the Unseelie King built for his concubine. It's a living, breathing love story, testament to the brightest passion that ever burned between our races. You can follow the scenes through if you've time enough and are willing to risk getting lost for a few centuries."
I heard of the White Mansion from eavesdropping but never paid much attention to the talk. I was always more interested in the _Sinsar Dubh_. "What do you mean, you can follow the scenes through?"
"Their residue is still here. They loved so intensely that moments of their life have been etched into the very fabric of the mansion. Some say the king designed it that way, so if one day he lost her he could come live with her residue. Some say the mansion was built of memory-tissue and is a living creature, with a great brain and heart hidden somewhere in the house. I've no wish to believe it's true because that would mean the White Mansion can be killed, and she must never die. The record of the greatest love in the history of History would be lost, along with countless artifacts from myriad universes that could never be collected together again. This place is home, love story, and museum all in one."
"So, where's the library?"
"You see, lass," he says tenderly, like I never even just opened my mouth, like I'm looking for a lesson in love, and I ain't, "the Unseelie King fell in love with a mortal woman. She was his reason for being. His every defining moment occurred because of her, and only in her presence did he know peace. She was his brightest shining star. She made him a better man, and to men who know how fundamentally and deeply they're flawed, such a woman is irresistible. The idea that she would live less than a single century was more than he could bear, so he resolved to make her Fae like himself that they might live forever together. While he worked in his laboratory, trying to perfect the Song of Making, he needed to keep her safe and alive. He knew it might take him eons to learn to wield the power of creation."
If he was human I might call that funny glint in Christian's iridescent eyes speculative as it rests on me. I can't look too long trying to decide because one short lock with his gaze and my eyes are already leaking blood. Dude's getting more potent by the minute. And weirder. Like he's thinking him and me are like the Unseelie King and his concubine, some kind of star-crossed lovers. "And where did you say the library was?"
"He built his beloved a playground of infinite proportions, tucked away in a safe pocket of reality where she could stay for all time, unchanging. Unaging. She would be safe. Nothing and no one could ever hurt her. He would never have to worry that he might lose her." His voice sinks to a whisper, as if he's forgotten I'm even here. "They would be together always. Soul mates. He would never be alone. Never get lost in madness, for she would never fail to find him and bring him back."
"Dude, your story's fascinating and all, but where's the library? Time's wasting. We got the Hoar Frost King to stop."
"If you stayed here, Dani, my light o' love, you'd never die. I'd never have to worry about anyone hurting you. Ever."
"Yeah, and I'd, like, be fourteen forever. I'd kind of like to grow a few more inches," I say irritably. In more than a few places. He tries to keep me here out of some lunatic thought that I'm his queen, we'll be staining this place with a whole new residue: it'll be war in the White Mansion.
"I'd forgotten that." He sighs. "Come, lass. Shall we go find the library?"
"Dude, thought you'd never ask."
We exit the white room on white marble floors and enter a sparkling white hallway with floor-to-ceiling windows that stretch to domed ceilings forty feet high. There I see my first residue. Beyond tall windows is a beautiful woman in a snowy garden, silken folds of a bloodred gown spilling over a white marble bench. Face pressed into her hands, she weeps.
"It's the king's concubine," he says.
"I thought you said they were crazy in love. Why is she crying?"
"She wearied of being alone while the king labored at his experiments. She waited hundreds of thousands of years for him, alone except for those few creatures he trusted with her, and his occasional visits."
Christian tells me the rest of the story while we twist and turn down hallways and corridors. I'm riveted in spite of myself. Who'd have ever thought such fantastical places existed side by side with our world, accessible through hidden portals and mirrors? My life is so fecking interesting I almost can't stand it!
We pass over lemon marble floors in sunny wings with tall windows that frame brilliant summer days, down rose quartz floors that reflect violet hues of the sunset beyond, across bronze tiles that wind through rooms that have no windows, only stately, enormous, kingly chairs and couches and beds. There are fireplaces here as tall as a small house, with ceilings higher than the spires on cathedrals.
"How big is this place?"
"Some say it goes on forever, that the king created a house that constantly grows itself."
"How do you find anything?"
"Och, and there's the rub, lass. It's difficult. Things move. It doesn't help that the king created decoys. To better protect his dangerous journals, he seeded multiple libraries within the house. Barrons thinks he found the true repository. He didn't. I saw the books he pilfered. They came from the king's Green Study."
"How do _you_ know where the true library is?"
He hesitates. "Something in it calls to me," he says finally. "I was trapped for a while in the king's boudoir, and I could feel the pull of the house beyond it. The residue in his chambers was so strong that reality and illusion blurred for a time. Sometimes I would hear whispers as I fell asleep, and those times I would dream I was the king, walking my halls. I knew where everything was, as if it was I who'd fashioned this house. I even understood how things shifted. A few of those memories remain. Others aren't so trustworthy. Still, I know that down a crimson hall that will always be found off a bronze corridor is a music room with thousands of instruments that play themselves when you twist a key inside the door, like a giant music box. I know there is a vast arena in the cobalt wing with no gravity, and stars painted all around where sometimes he took his beloved and created universes in the air for her amusement. And I know that because he feared other Fae would find the journals he kept, filled with notes about his experiments, he brought them into the White Mansion. It is said that he locked away the recipe for every Unseelie he ever created, and countless more unborn, that he chiseled a warning above the entry when he left. It is by that inscription you can know it's the true library."
"What does it say?"
He stops. "See for yourself, lass."
I look up, and up some more. We're standing outside doors that are nearly identical to those in our abbey, at the entrance to the chamber where Cruce is trapped. Alien symbols glow with eerie blue-black fire, chiseled into the stone all around the doors, with much larger symbols carved across the arch.
"I can't read it. It's not in English."
Christian moves from side to side of the archway, pressing various symbols, and after a moment the doors swing open silently. "It says, 'Read them and weep.' Come, lass. We've a needle in a haystack to find."
The king's library is the craziest place I've ever seen.
Christian disappears the second we're in the door. Me, I stand in the doorway, catching flies in my open mouth. The view seems to go on forever, between jagged, zigzagging bookcases, dwindling to a tiny black point that seems miles away. I step inside, fascinated.
Despite how ginormous the doors are, I can spread my arms wide as they go and my fingertips brush the walls of books on both sides. Lined with shelves and cubbyholes and built-in desktops that drop down on invisible hinges and are covered with more books and jars and knickknacks, every horizontal surface perches at skewed, absurd angles that defy physics. The things on those shelves shouldn't be staying on them. The bookcases lean in, and close over me in places, which means the books should be falling on my head. The walls soar to a ceiling beyond my line of vision. It's like being at the bottom of a jagged chasm of books, and there are millions of them in all colors, shapes, and sizes.
Here, the passage between the shelves widens to twenty feet, there it narrows barely wide enough for me to turn sideways and force myself through. I munch candy bar after candy bar as I move deeper into the nutty place.
There are bookshelves that branch off, perpendicular to the main passageway with only an inch of space between them. "Nobody could even get a book off some of these!" I say irritably. "How are we supposed to search?"
"A Fae could." His voice floats down from somewhere above me. I guess he's sifting up and down the shelves.
I pass through a low-hanging doorway, the top of which is a shelf of upside-down books. They should be dropping on my head as I walk under them. There's a bronze plaque on the ceiling near them, I suppose saying what that section is, but I can't read the language. I reach up and pluck one from the shelf. I have to tug, like the book is set in glue or something, and it comes off with a wet _pop_. The pale green cover is soft and mossy, and the book smells like the woods after a spring rain. I open it and realize it was pointless to bring me here. I can't read a word. It's all in some other language and I have no idea what it is. I don't think even Jo could translate this stuff.
I'm about to close it when the sentence at the top of the page gets up and starts crawling across the page like a centipede. I snicker until it pauses at the edge of the page like its psyching itself up for something then flings itself off the book with a mighty leap and starts wriggling up my arm. I jerk back my hand to shake it off, but it digs in by pointy letters and holds on. I pinch the sentence's butt with my other hand and tug it from my skin like a leech, smack it back on the page and clamp the book shut. Part of it's hanging out, and it waves jerkily at me with what appears to be blatant hostility. I stick the book back on the upside-down shelf over my head, pissed-off sentence first, counting on the gluey base to hold it in. All I need is a badly mangled, irate sentence stalking me.
I open the next one I pull down more cautiously. Same thing happens, only this time a whole paragraph leaps off the page the sec I open it and lands on my stomach. I swipe at it but the words are sticky like cobwebs and I only succeed in smearing them around on my shirt. Then they all start to separate and I spend the next few minutes trying to catch them all and put them back in the book, but every time I open it, something else gets out.
"You aren't messing with the Boora-Boora books, are you, Dani?" Christian says from somewhere far away. "You're awfully quiet down there."
"What are the Boora-Boora books?"
"The ones where the words crawl off the pages. They're named after their home world. Nothing works like it's supposed to there." He makes a sound that is suspiciously like a choked laugh. "You have to watch out, they sting like fire ants if they get pissed."
"Ow! You could have told me that sooner!" No sooner did he say the word "sting" than they started doing it. I swat at them with the book they're supposed to be in. They scurry under a pile of teetering manuscripts and disappear. I sigh, hoping they weren't a critical part that someone comes looking for in a few hundred years, and stick the tome back up on its upside-down shelf. "So, not all of the words are self-propelled like that?"
"Some of the books are just books. Bloody few, though."
"Found anything up there?"
"Not yet."
"Dude, I can't read a thing. I'm useless here."
I wait but there's no reply. I squint up at the ceiling. He could be anywhere, sifting from shelf to shelf. When he said he was taking me to the Unseelie King's library, I expected something like the one we got at the abbey. Even if I could read whatever languages the Unseelie King's books are written in, it would take an eternity to search this place, not to mention a couple of gazillion-foot ladders. It was stupid to come here. I don't regret it, though, because now I know how to get in the White Mansion. Dude! What a perfect place to hide out for a while if I need to. And there's so much to explore. Who knows what kinds of useful things I might find in here!
I wander the passage between shelves, periodically calling for Christian. He doesn't answer. Books are piled in haphazard stacks along the sides and I have to be careful not to bump into them. I get the feeling that if I knock over a stack and half a dozen come open at once, not even my speedy freeze-framing will be able to keep up with all that comes out. I open a few more books along the way, curiosity and me being best buds and all. One puffs out acrid smoke the sec I lift the cover, making me sneeze, and I slam it shut again. Another has fat brown spiders with hairy legs that spring from the pages! I squash the ones that make it out. Yet another has videos instead of words but the images are so alien I can't make sense of them.
I find a little mini-laboratory amid the stacks, covered with petri-like-dishes and stoppered bottles and jars. "Christian!" I call again as I study the contents visible through thick wavy glass.
I get a reply this time but it's so far away I can't make it out.
"Dude, unless you're finding something, this is a total waste of time! I'd rather be back in Dublin, investigating."
"Hang on, lass," comes his far-away reply. "I think I'm onto something."
One of the stoppered bottles has a dab of crimson at the bottom. I pick it up and turn it in my hand, watching the crimson liquid ripple. Rainbow colors skitter across the surface in kaleidoscopic designs. It's so beautiful I almost can't take my eyes off it. I turn the bottle upside down and study the label on the bottom. No clue what the glyphlike symbols mean. As I turn the bottle back upright, I must have nudged the stopper a little because I get a whiff of the scent of its contents and it's like sticking your nose right up into heaven. It's night jasmine and fresh-baked bread, homemade fish and chips and salt air, it's the smell of my mom's neck, fresh-washed pjs, and sunshine on Dancer's skin. It's the scent of all my favorite things rolled up into one. I swear my hair lifts on the breeze of it. I groan and pull out a candy bar, abruptly ravenous.
There's curiosity and there's cats.
You'd think I'd learn.
I unplug the bottle while I chew.
# THIRTY
# _In the court of the crimson king hag_
"What the hell is that smell?" Christian says.
"Fecking awesome, isn't it?" I say dreamily. Crimson smoke swirls in the glass bottle, poking tentative tendrils at the rim. The amazing aroma fills the library, making me giddy. I want to stretch out, fold my arms behind my head, be lazy and bask in the fragrance. I want to share it with Dancer. I've never smelled anything so scrumptious.
"Bloody fucking noxious," he says from much nearer than he's been in a while.
"How can you say that?"
"Because it is."
Crimson strands puff from the bottle and swirl above it. After a moment they begin to dart toward each other, circle around and dart back, slender red strands knitting themselves into a smoky shape.
"Dude, it smells like heaven! There's something wrong with your nose. Maybe you only like Unseelie smells now." I can't wait to see what awesome thing comes out of this!
"It smells," he says from directly above me, "like rotting intestines. What did you open? A book?" He drops down beside me, carrying a stack of books beneath his arm. I'm glad to see he found something. "A bottle? Christ, lass, you can't be randomly unplugging bottles in this place! Give me that. Let's see what you've done."
The hint of a face is forming in the crimson smoke; delicate, pointy chin, enormous eyes slanted up at the corners. I try to turn my head to look at Christian but my head isn't taking orders. It's stuck, still staring at the materializing face. I can't force myself to look away no matter how hard I try. It's got me mesmerized. I've never seen a face so beautiful, smelled a smell so good. I want to stand in the middle of it and breathe it deep into my lungs.
When he plucks the bottle from my hand, the spell is broken. When he turns it on its side to read the label on the bottom, a cloud of crimson smoke gushes out, obscuring the passageway between the shelves. Tendrils lick at me, rough as tiny cat tongues.
Suddenly, everything changes.
Now that I'm no longer holding the bottle, I can smell what he smelled. Saliva floods my mouth, my stomach heaves, and I just about puke the candy bars I just ate. The face in the smoke isn't so beautiful anymore. It's morphing into something monstrous before my eyes. Long fangs slide from thin lips, bloody hair writhes like snakes. "Dude, what the feck did I open?" I say, aghast.
The bottle clatters to the floor.
My blood goes cold when Christian utters a single word.
"RUN."
There are a few absolute no-brainer rules in my world. Real close to the top of this list is: if an Unseelie prince runs from it, I'm going to run from it, too. I'm not even going to ask any questions. I'm just going to vamoose with all my might.
Still... I can't help but try to steal a peek over my shoulder. I'm the one that let it out. I have to know what it is so I can hunt it down and kill it.
"DON'T LOOK BACK!" Christian roars.
I cradle my head with my arms, trying to hold my skull together until the instant headache subsides. "Stop yelling at me and sift us, dude!" I'm freeze-framing, trying to keep up with him, but I don't know these halls. They're a maze that isn't on any of my maps. I have to keep dropping down, lock my grid into place and kick back up again. The stench of rotting meat behind me is getting stronger. The skin on the back of my neck is crawling. I keep waiting for whatever is chasing us to close icy talons on my nape, rip my head off my shoulders, and kill me. All those scary movies I watched with Dancer aren't making me laugh now. They're filling my head with a million gruesome deaths, each more horrible than the last. It'd help if I knew what was chasing us. The unknown is always scarier than the known. I got a Mega-sized imagination, and it can do a real number on me.
"Sifting doesn't work inside the White Mansion. Take my hand. I know these halls."
I grab his hand, ignoring the groaning sound he makes. He laces his fingers with mine and I'm blasted by a wave of horniness. "Mute it, Christian. This ain't the time to go death-by-sex Fae on me."
"Sorry, lass. It's just that it's your hand and there's danger, and danger always—"
"Off it now!"
I can breathe again. Not that I want to. The stench is suffocating and closing in on us fast.
"What's chasing us?"
"Loosely translated, the Crimson Hag."
"How does it kill?"
"Hope you never find out."
"Could it kill even you, an Unseelie prince?"
"She prefers us alive. She once held two princes captive for nearly a hundred thousand years before the king stopped her. Among other foul things, she tried to breed with us. I had no idea he'd stored her in his library. Everybody figured he'd destroyed the bitch."
"Why would she take you captive?"
"Because we're immortal, and once she takes what she wants from us, our bodies grow it back. Then she takes it again. We're a never-ending supply. She can just keep us chained up, sit and knit."
Knit? The idea of an Unseelie monster knitting is more than I can wrap my brain around. "What does she want from you?" A cloud of red smoke slithers over my shoulder. "Hurry, Christian! We've got to go faster! Get us out of here!"
We barrel down bronze halls, twist and turn through lemon wings, until finally we skid onto white marble. I swear I can feel the Hag breathing down my neck.
Then we're in the white room, rushing into the mirror, and I can't help myself, I look back as I turn all spongy.
The Crimson Hag is the most revolting creature I've ever seen. Worse than the Gray Woman, worse than the Unseelie princes, worse even than Papa Roach, and I have a special hatred for roaches. Roaches hang out on floors. My cage was on the floor.
Bloody, matted hair frames an ice-white face with black holes for eyes. She licks crimson fangs when she sees me looking. But the truly disturbing thing about her is what she's wearing. Her upper body is voluptuous and encased in a corset of bone and sinew. She has no lower body that I can see. A tattered, incomplete crimson gown streams behind her.
And now I know why she smells of rotting meat.
Her unfinished gown is made of guts.
My stomach heaves again. "It collects Unseelie prince guts?"
"Among others. She'd take yours, too. Though yours would rot sooner."
"Can't you go any _faster_?" I like my guts. I want to keep them for a long time.
We explode from the mirror into the second white room and leap headfirst into the next mirror. We pass through multiple mirrors, chased by the scent of rotting meat. "Uh, Christian, she's going to get out."
"Good. More prey in Dublin. She'll go after someone else."
"We can't have her loose in my city!"
"You're the one that opened the bottle."
I screwed up. Big-time. But I'll figure it out. I'll trap and kill her and make my city safe again. _Before_ she hurts anyone. I can't stand the thought of innocent folks getting killed because of my stupid curiosity. "You should have warned me not to open stuff!"
"I did. Then there was the whole 'read them and weep' thing chiseled over the door. Which warning didn't you get?"
"That was about the books, not bottles!"
"Some warnings are unilateral."
Then we're out and the cold slams into me like the brick wall we just exploded from. It takes my breath away, and when I get it back, it comes out in frosty puffs on the air. I go skidding across the alley on snow and ice and crash into the building opposite. Christian slams into me. We steady each other and I look around disbelievingly. The ground's covered with six inches of snow!
Did the Hoar Frost King ice something in this alley in the few hours we were gone? It can't be more than ten degrees and the windchill is killer. It _never_ gets cold like this at night! And never over the space of a few hours. I look around for an ice sculpture nearby.
"Aw, crap," I say, because it's about to hit the fan. Snow's not the only thing in the alley.
Ryodan and Barrons are behind BB&B, getting out of Barrons's Bugatti Veyron. They both stare at me a sec, like they can't believe their eyes, then Ryodan's gaze fixes on where I'm holding Christian's hand. I drop it like a hot potato, but the look on his face doesn't improve. "It's not what you think! He's not going to be my superhero boyfriend and kick your—"
"Yes, I am," Christian says.
"No, you're not," Ryodan says. "And where the fuck have you been. Do you know the problems you've caused me."
"Dude, I only been gone, like, two hours. And we got bigger problems right now," I say.
"No shit. This whole city is turning to ice."
"What the bloody hell were you doing in the White Mansion?" Barrons demands. "Who told you how to get in there?"
"You will never go anywhere without me again," Ryodan says to me. "If you do, I'll lock you in my dungeon until you rot."
"Speaking of rotting, I think—"
"No more. From this moment forward, I'm going to be doing all your thinking for you."
I bristle. "My ass, you will."
"Seal the wall," he says to Barrons. "And get her the fuck out of here. It's time for the Highlander to die."
"You just try," Christian says.
"I ain't going nowhere. Well," I amend, "actually I am and you need to, too. We all have to get out of here." I start trying to freeze-frame but I crash into Barrons and bounce off. What happens next happens so fast I almost can't process it.
The stench of rotten meat fills the air, and Christian and me duck and split off in opposite directions because we know what's coming, then the Crimson Hag explodes from the wall, holding what looks like six-foot-long knitting needles made from bone, like lances at her sides.
She pierces Barrons and Ryodan with them then shoots straight up in the air, trailing their guts behind her.
# THIRTY-ONE
# _"I'm swimming in the smoke of bridges I have burned"_
I stand there like an idiot.
I should run before she turns on me, too, but my feet seem to sprout icy roots.
Barrons and Ryodan are lying in the alley on their backs, blood staining the snow in widening circles around them, and I gape, thinking: They can't die! Superheroes don't die!
Misguided beliefs aside, they sure look like they're dying to me. Nothing could get that mutilated and survive.
The Crimson Hag didn't just puncture them, she flayed them from groin to neck, and split them clean through bone. In one quick yank she scrapped all their intestines and internal organs from their bodies. It's a move she's had hundreds of thousands of years to perfect. Puncture, fly, yank. Their chest and abdomen cavities are open and scraped empty. The only way the treacherous bitch could have done this to them was to catch them by surprise.
What the feck was I thinking, standing there saying anything besides "Run"? Bickering as usual, like we had forever and always would!
"I thought you guys would duck at the last minute," I mutter at their bodies. Or freeze-frame away, faster than me. Or maybe Ryodan would use whatever secret weapon he used against Velvet against her. Never in a gazillion years did I think anything could actually get the jump on them!
But she exploded from the wall and her lances were through them before any of us could even react. Their bodies are still moving but I think it's just the final twitches a body makes when it gets traumatized so abruptly and completely.
I hear a weird clicking sound that affects me the same way the ZEWs' chittering does, terrifies me on a primitive level. Is she coming for me now? I grab my sword and whirl. It takes me a second to spot her. I follow the trail of blood.
Up.
The Crimson Hag is perched on the roof of the building behind BB&B, with ropes of entrails dangling over the side in long glistening strands, dripping on the sidewalk. The bony needles she used to flay Barrons and Ryodan are actually her legs, which bend weirdly, sort of like a praying mantis's front legs, and have curved hooks on the ends.
With insectlike appendages, she's knitting their guts into the hem of her dress. As her bony legs click and clack together, the guts sway over the edge, shortening, inch by inch, smearing blood up the brick.
It's so disturbing that my stomach heaves and my body tries to burst into tears and puke at the same time. I swallow it all and choke.
I hear a guttural sound followed by a weak sigh and look back at the bodies.
"I'm going to kill the kid," Barrons says faintly.
Ryodan makes a burbling sound like a bloody laugh. I don't think he even has the parts left to laugh with. "Get in line."
They both deflate and go still.
I stare dumbly.
They die like superheroes: cracking a joke. Like they're just going to get up tomorrow and fight another day. No fear. Balls to the wall until the bloody end.
I feel like somebody ripped my guts out, too. I can't stand to look at them anymore so I drop my head, and squinch my eyes up tight. My head's a muddle. How did I get here? How did deciding to go to the Unseelie King's library end up with Ryodan and Barrons dead? I can't make sense of it. I mean, I can, because duh, I can follow the chain of events, but who the feck could have foreseen such a bizarre and preposterous outcome? How am I supposed to make small decisions when they can have such large, unforeseeable results?
"Well, that was fortuitous." Christian skirts their bodies and moves toward me, laughing. "Two down, seven to go. I wonder if we can just point the bitch at the rest of them. Mac, too."
My head whips up. He's laughing. They died and he's _laughing_. I start to shake. "Stay. Away. From. Me."
"What did I do, lass?"
"You took me in there, that's what you did! You didn't warn me enough. I'm only fourteen! I don't know everything! I can't know everything! You're older! You're supposed to warn me about stuff! And now you act like it's good that they're dead!"
"I thought you wanted Ryodan out of the picture."
"I just wanted him to leave me alone! And I never wanted Barrons to die! Aw, crap, Mac!" I wail. I look at the back of the bookstore, now even more miserable than before. Mac's in there. How long before she comes out and finds Barrons in the alley, bled out in the snow? How long before she discovers my complicity in this, too? I can see her, finding him, flinging herself over his body, weeping. One more tragic loss in her life.
Because I opened a fecking bottle.
Because I was curious.
The night Alina died, I felt like I wasn't... really there. I never been able to shake the feeling there was something wrong with me. I searched Ro's journals from beginning to end but she never wrote a single fecking word about me. Never. Makes me think maybe she had other journals I ain't found yet.
But tonight I'm all here.
I suffer that unpleasant shift I felt once before, the night I got Jo stuck working in the club. The one where I move sideways into a different way of being me, see myself different, and I don't like it. It's the shift where I'm a boat and there are all kinds of people capsized in my wake. No, not a boat. What did Ryodan say I was? A tsunami. That's it. Crashing into things and leveling them. When he said that, he had no idea he'd be one of those things I leveled. Or that he wouldn't live to see the one hell of a woman I'm going to become.
Above my head bony needles clack away. I hear the wet slap of intestines against the wall as they're drawn up the side. I should be terrified. I should be running for my life so she doesn't do to me what she did to them. Should I hide their bodies so Mac won't find them and figure out what I did?
"Come, lass. We have to get out of here while she's busy. The Hag gets obsessed with her knitting but she'll be done soon," Christian says.
My legs are made of cement and I have concrete blocks for feet. I just keep looking from Barrons and Ryodan to the bookstore and back. First Alina. Now Barrons. There isn't going to be any place on the face of this planet Mac won't hunt me down when she finds out what happened here tonight.
I look at Ryodan. How can he be dead? Who's going to run Chester's? Who's going to keep the loser Fae and humans in line? With both Barrons and Ryodan gone, is there any safe place in Dublin? Will BB&B and Chester's get abandoned?
A hand closes on my shoulder and I just about jump out of my skin.
"We've got to get out of here, Dani. She's finishing up."
I shake him off violently. "Don't you ever touch me again, Christian MacKeltar!"
He exhales sharp and sudden like I punched him in the gut. "You don't mean that."
"Try me." I fist my hand around the hilt of my sword.
"I'm the one that gave it back to you, lass. I'm the one that watches out for you."
"You're the one that took me somewhere I didn't know was as dangerous as it was. Folks got killed because of it. Did you at least manage to bring out the books you found?"
"I had other things on my mind. You were in danger."
It was all for nothing. The books got dropped, forgotten. I look at the wall. Sure, I could go back in, but I can't read any of the stuff in the library, so what's the point? And who knows what else I might set free by opening anything else in there?
I look up. Blood drips down the side of the building. As the gruesome Hag knits away, she plucks a small bone from the mess of entrails and organs and tucks it into her corset, taking a moment to rearrange her obscenely human-looking breasts. Then she stops abruptly and looks down at me as if she's suddenly realized there's more prey in the alley and it's watching her. After a moment she dismisses me and returns to her stitching, but I feel... marked somehow. Like she filed me away in her Unseelie-insect brain.
"How do I kill her? Will my sword work?"
"Might. But you'd never get close enough. Her needles are longer than your sword. She'd have your guts in her dress before you even managed to swing it."
"You said she gets obsessed while she's knitting."
"Not that obsessed."
The ambience in the back alley changes abruptly and it takes me a minute to figure out why. A light just came on in the back of BB&B and is spilling out the window, across the bloodstained snow.
I know what that means. Mac's moving around inside, looking for Barrons. I imagine it won't be long before she looks out back to see if his car's out there.
If Mac walked out that door and tried to kill me right now, I'm not sure how well I'd fight.
I take one last look at Barrons and Ryodan. I have to make this right somehow. I have to balance the scales and there's a lot weighing in against me.
"Come near me again and I'll kill you," I say, soft like Ryodan used to talk.
I freeze-frame into the night.
# THIRTY-TWO
# _"If I stay lucky then my tongue will stay tied"_
I spend the next two days slapping up terse _Dani Dailies_ that describe the Crimson Hag and her M.O., hunting for Dancer, collecting the rest of the ziplocks I need from the other iced scenes (except for the club beneath Chester's, which I'm in no hurry to go near), and packing my backpack full of samples. They're some of the most miserable days of my life. I go up and down like a fecking psychotic elevator being controlled by some fecking psychotic little kid, punching random floor buttons. One second I'm swaggering, the next I'm drooping.
One minute I'm elated because I never have to go to work again. My life is my own. Jo can quit the subclub. She'll stop wearing sparkly stuff between her boobs and boinking Ryodan. The next minute I remember that if Ryodan's remaining men learn that I played even one tiny little part in their boss's death, I'm deader than every doornail in Dublin. On top of that, the Crimson Hag is loose, the Hoar Frost King is still out there, Dublin is slowly turning into Ant-fecking-arctica, Christian and me are on the outs, and now Mac has double the reasons to kill me, assuming she knows.
I can't decide if she knows. One minute I think she does, the next I don't.
The bodies are gone. I went back in the middle of the night to hide them. I should have hidden them right away but I wasn't thinking clear. Aside from blood in the alley and up the brick wall, no trace of them remained.
At first I thought Mac must have found and taken them somewhere for a proper burial, but then I decided she didn't, because yesterday I saw her hurrying down the street toward Chester's, all bundled up and shivering in the cold, and she didn't look sad. I've seen Mac sad. I know what it looks like. She looked a little tense but otherwise normal. She had a trail of ZEWs behind her, chittering away. I wonder if, like crows, the ZEWs are harbingers of death. It worries me they're following Mac. Her tension is probably because of what's happening to Dublin. Everybody I'm seeing is tense. And shivering. It's ten degrees in Dublin during the day, even colder at night. Snow's been falling, piling up. The city isn't set up to handle this kind of weather. Lots of folks don't have power where they're staying. They won't survive these conditions long.
I wonder if the Crimson Hag ate Barrons's and Ryodan's bodies. Stitched up their entrails then dined on the rest. I'd think she'd have spit up a few bones but maybe she needed them all to spruce up her corset. Then I figured Christian probably went back to tidy up and hide the evidence. Trying to get on my good side again or something.
I wonder where the heck Dancer is! I need his superbrain to help me crunch the facts so I can save my city from turning into an iceberg. So then I can save folks from getting knitted up into a dress.
I got two more places to check for him, then I'm out of places to hunt.
I freeze-frame up O'Connell, yanking WeCare posters off streetlamps as I go. Stupid fecking stupid feckers are trying to take advantage of people not having power, encouraging them to come into prayer meetings, to get warm and "take the white." I didn't know what that meant till I saw a couple folks coming out of one of the churches the WeCare people have designated as their own, wearing long white robes over their clothes.
They were carrying bags of canned goods and smiling. In my experience, anybody besides your mom that feeds you is going to want something in exchange for it.
I whiz up to Dancer's penthouse, where we like to stretch out in the sun, disarm his booby traps and poke my head in the door, calling for him. The place is silent and empty. I decide to see if he's got any food in the pantry 'cause I'm starving. When I get there, I crack up. There's a note taped to a stack of cans sitting smack in the middle of the floor. It's a cryptogram. It's how we leave messages for each other.
I pop open can after can of beanie weenies and gorge while I solve the puzzle that tells me where he is.
There's a lot hidden away in Dublin, just like out at the abbey. When I first started hanging out in the city, I got one of those sightseeing books and visited all the hot spots like any tourist. I was embarrassed to be a stranger in my own town, never having been out of my cage much. I wanted to know everything everybody else knew, see it all with my own eyeballs instead of watching it on TV or reading about it in a book.
I went to Trinity College and toured all the cool stuff there. I never got to go to school so it was neat to see the classrooms and labs and libraries and folks being all social instead of being kept by themselves all the time. I couldn't wrap my brain around growing up that way. Mom taught me to read. I taught myself the rest.
I hit up the museums, dropped by the brewery, hung out in Temple Bar, visited the catacombs beneath Christ Church Cathedral and St. Michan's Church, and eventually hunted down the underground rivers. I listened when college kids raved about their favorite places and went there, too. I paid attention when old folks talked on the streets about things that used to be.
That's how I found Dublin-down. Couple of wrinkly old dudes playing checkers by River Liffey used to work for a mob family and knew some interesting stuff. Beneath a restaurant run by a dude name of Rocky O'Bannion, this big-time mobster that disappeared last year in the craziness of the walls coming down, I found it. A honeycomb of tunnels and hidden crypts beyond a pile of rubble and a series of grated entrances so complex only someone as curious as me or a criminal trying to hide bodies and booty would ever have gone through. Dancer and me mapped out parts but we still got a lot to explore.
That's where I find him now, in one of the underground catacombs, down a collapsed tunnel (unless you knew how to find the hidden detour) beyond bolted steel doors, hinged into stone, all booby-trapped.
The room he's in is long and narrow and made completely of stone, with those old vaulted fornix ceilings, supported by massive columns, like I only ever seen in ancient crypts and the library at the abbey. He's got lights set up that I figure have to be battery-powered 'cause I don't hear a generator, and setting one up to vent down here would take a lot of work. He's standing behind a stone slab that used to hold a corpse but is now covered with notebooks and envelopes, laptops, bottles, beakers, and burners. Yep, this place is Dancer, just missing a TV to watch movies on, a fridge and shower, and knowing him, he probably has a hidey-hole rigged up nearby with all the conveniences. Another slab is crammed with bottled water and food. His head is down and he's working on something, deep in thought.
"Dude, this is fecking awesome!" I say as I step inside.
Dancer looks up and the grin he gives me is blinding. His whole body changes, like he was strung up on wires hanging him from the ceiling and they just got cut. His shoulders ride lower, his limbs slide smoother, the hard planes of his face relax into the Dancer I know. "Mega!" he says. Then he says it again, "Mega!"
"That's my name, dude. Don't wear it out." I swagger into the chamber and see he's been collecting things from the scenes, too. Behind him is the pièce de résistance: a mystery board! He blew up maps and pieced together an enormous topographical survey of Dublin and the outlying areas and has pins and notes plastered all over it. I beam. I couldn't have done better myself. "This place is the Shit," I say.
"Thought you'd like it." He picks his glasses up off the slab, pushes them back on his nose and grins at me. His eyes are red like he's been studying too long. He's tall and lanky and pretty much perfect. I grin back and we just grin at each other for a few seconds, 'cause we're so happy to see each other again. It's a big city. Sometimes I feel alone in it. Then I see Dancer.
I toss my backpack on a nearby folding table and pull out my ziplocks and photos to add to his board. He comes over and we sort through them in happy silence, brushing shoulders and grinning at each other. He keeps looking at me like he can't exactly believe I'm there. Dude's acting like he really missed me. We're always glad to see each other, but something's different today.
I go to start pinning my photos of the scenes on the board, and I look back at him, 'cause something don't make sense to me, besides how strange he's acting. "There ain't this many iced places in Dublin!" I gesture at the pins on the board.
"There weren't a few weeks ago. It's been escalating."
"Dude, there were only ten. You got, like, twenty-five pins on this board! You telling me fifteen more places got iced in the past few days?"
"Mega, the last time I saw you was nearly a month ago. The day we tried to get your sword back from Jayne."
I gape. "That wasn't a month ago. That was a couple days ago!"
"Nope. I haven't seen you for three weeks, four days, and"—he looks at his watch—"seventeen hours."
I let out a low whistle. I knew time moved different in Faery but it didn't occur to me that the White Mansion was part of Faery. No wonder Ryodan was so pissed at me! I missed work for _weeks_. I snicker. It must have been driving him crazy. My snicker dies. I forgot for a sec that he was dead. I feel sick all the sudden so I tear open a candy bar and eat it.
"I was worried."
I look at him. He's looking me right in the eyes, more serious than I ever seen him. It makes me uncomfortable. Like I'm supposed to say something and I don't know what.
I stare back and we just look at each other for a few seconds. I root around in my repertoire and come up with: "Dude, get over yourself. I'm the Mega. You never got to worry about me. I been on my own forever. I like it that way." I flash him my trademark grin.
I get a faint smile in return. "Got the message, Mega. Loud and clear." He turns around, walks back to the slab. He's not moving smooth anymore. Some of those wires are back. I don't like those wires. They look... I don't know, grown up to me.
"Just saying, don't worry about me. Stupid to worry about me. I can take care of myself."
"Now I'm stupid."
"I didn't say _you_ were stupid. I say _it_ was stupid to worry about me."
"And it—the act of worrying—isn't to be confused with the person doing it."
"Exactly. I'm the Mega, remember? I kick butt all over Dublin!" I don't know what's wrong with him. He's not responding right to anything I'm saying!
"Ability to defend oneself has absolutely no bearing on or relevance to deportment or emotional comportment of others."
"Huh?"
"Don't tell me what I can and can't feel. If I feel like worrying about you, I buggering well will."
"Dude, no need to get snippy."
"I'm not snippy. I'm offended. You were gone nearly a month. Between dodging the psychotic jackass that stalks you day and night, analyzing evidence, and trying to save this city, I've been haunting every iced scene that pops up. Visiting them two and three times a day. Do you know why?"
"To collect more evidence?"
"I've been waiting for them to melt enough that I could see if you were in there. Dead. Never to be talking to me again."
I stare at him. We never talk about stuff like this. It reeks of a cage to me. Like there's one more person I'm supposed to check in with now. Like my life isn't already owned by too many other folks. "I got my sword back now," I say stiffly. "I'm not going to get iced."
"Invalid. Those two statements have no relevance to each other. None. Zip. Zilch. Nada. The sword won't protect you from getting iced. I left notes for you in the pantry of every hideout I've got and all of yours I could find. Do you know what I heard? Nothing. For almost a month."
"Dude, I got the picture. You didn't like not being able to find me. Too bad you can't put a leash on me, huh? Maybe stick me in a cage somewhere?" He's pissing me off. I think we're having our first ever fight. It makes me feel sick to my stomach.
"Excuse the crap out of me for caring about you."
"Dude, what's wrong with you? This ain't us. Why are you ruining us?"
"Caring about you is ruining us?"
"Caring is one thing. Trying to lock me up is another."
He gives me a look that I just don't get. Like I'm being obtuse when he's the one being obtuse. I thought our way of hanging was clear and well-defined. We're superheroes. He's not sticking to the script. If he keeps deviating, I'm jumping comic books.
"My mistake. I won't make it again." Just like that, he goes back to being Dancer, all business. "That day at the castle was the first time I got a look at what's been freezing things. A lot's happened since then. It freezes a new place just about every day. Ryodan and his men have been tearing this city apart looking for you. He raided half my stoops. I moved down here to get the bloody feck away from him. He's going to kill you when he finds you."
"Not if I kill him first," I mutter around a mouthful of candy bar, pretending I didn't already. When you have a secret that folks would kill you for, you sit mum on it. From everyone. 'Course, if I'm learning from my mistakes, I should kill Christian like I didn't kill those stupid lisping fairies that ate Alina and ratted me out to Mac. I'm a little irked that Dancer's back to talking about stuff like we never even had our first spat, because it's a big deal to me. It's going to take me hours to stop feeling nauseous and confused inside. I eat when I get confused. I stuff another candy bar in my mouth.
"Even Barrons got in on the hunt. So did those abbey girls you sometimes hang with. The city keeps getting colder with each new place that's iced. People are falling apart. Nobody knows what to do, how to stop it, or even where it's safe to be anymore." He steps back and looks at the map. "So far I haven't been able to discern the pattern. We've got to figure out what it's looking for."
"What do you mean 'looking for'?" That was exactly the feeling I picked up with my _sidhe-_ seer senses, but Dancer doesn't have those. I start to feel a little less sick. I don't know if it's the candy bars in my stomach or thinking about work.
"Unless it's behaving in a random, illogical manner driven by absolutely no biological imperative—which I postulate is antithetical for any sentient life form—it has purpose."
I beam, our fight forgotten. Got to love a dude that says stuff like "postulate" and "antithetical"! "I love hanging with you!" I tell him.
He gives me a look that's vintage Dancer but a little wary, so I turn up the wattage on my grin till he grins back.
"That purpose may be alien enough," he goes on, "to elude ready detection, but it's there. It's our methods that are lacking. We have to step outside our box and process the facts with no preconceptions. This thing isn't from our world. It doesn't follow our rules or any laws of physics. It appears capable of opening a portal wherever it feels like it. I've seen it do it twice now."
"You saw it _again_?" I'm so fecking jealous I could spit.
"I've been keeping an eye on WeCare, trying to figure who the head honcho is. Nobody seems to know who started the organization up. A few nights ago I went to check out one of their prayer meetings. The church where they were holding it got iced when I was half a block away. One minute they were singing, the next I couldn't hear a thing. Seemed like the whole world went still or I went deaf. I stood in the street and watched. It did exactly the same thing it did at Dublin Castle. Came out of a portal, fogged everything, iced it, opened another portal, and vanished."
I flinch. He was half a block away! What if he'd been, like, one single minute earlier? Then I have a worse thought. What if I hadn't been able to find him for a month? Would I have been freeze-framing from one ice sculpture to the next, waiting for them to melt, wondering if I'd lost my best friend?
I'm suddenly ashamed. "Dude. Sorry I was gone so long."
His head whips up and he gives me a grin that fecking slays me. "Dude. Thanks. Glad you're back."
"I hear you saved my life at the church that night. You're the Shit."
"No, you are."
We grin at each other for what feels like an hour of heaven, and just like that everything is okay between us again.
We start yakking our heads off like nothing was ever wrong. He tells me scoop about new gangs forming in the city. I tell him about the Unseelie King's library. I can't keep that kind of fascinating stuff to myself. I can tell by how bright his eyes get that he's dying to see it.
He tells me a huge fire-world IFP almost burned the abbey to the ground! It evaporated iron and concrete and if it had made it to the abbey, nothing would have been left. But Ryodan's men stopped it by tethering it stationary to the ground somehow. I don't like that it's out there by the abbey, tied up or not. It makes me nervous.
I tell him about the Boora-Boora books and he laughs his butt off about me chasing unruly sentences down. He fills me in on how WeCare started painting buildings white to let folks know it's one of theirs, and if you go in and sign up, and attend meetings, they give you all kinds of food and stuff. I tell him about R'jan trying to take over as king of the Fae and that the ice monster has a name: the Hoar Frost King. I think it's the most we've ever told each other about the daily details of our lives. He tells me food is getting really hard to find. I tell him about the Fae being totally inert at the scenes and about what R'jan said about it killing Seelie and Unseelie deader than dead, wiping them out of all record of existence. "I think it might be after folks' life force," I tell him.
"But why those scenes? How does it select the ones it chooses and why does it ice them? And if it wanted people's life force, why wouldn't it go where the greatest number were gathered? At some of these scenes, there were only a few people."
"You mean like why would it ice the small club beneath Chester's when it could have iced the whole place?"
"It iced part of Chester's?"
"That was the first place it iced that I know of. It's the reason Ryodan dragged me into this mess."
"It can't be after life force. It took out a church spire, too. There wasn't a single person or Fae at that scene."
"Maybe it was just flying over that spire and accidentally iced it. Or maybe there was a tiny life force, like a mouse there, and it was feeling peckish."
He grins. "Maybe."
"I kind of doubt it, though. I think we should label them by order of occurrence. Maybe that'll help us see something."
"What bites," he says, "is we can't even tell people something so simple as: stay in small groups and you'll be okay. People are scared of their own shadows, Mega. The whole city is on edge, tempers are high, and people are getting into fights over nothing. We've got to figure out what's going on because if they don't freeze to death first, they'll kill each other. They've lost too much and been afraid too long. While you were gone, there were no _Dani Dailies_ getting out, and in times like these, no news _isn't_ good news. People need to believe someone is in the streets watching out for them."
"What about WeCare? Aren't they taking their job seriously? Dude, when I was gone, they should have stepped up to the plate, put out more issues! A newspaper's got a responsibility to the people!"
"The only thing WeCare is telling people is that they need to 'take the white' and everything will be all right. Half the city is rushing in to blindly embrace faith; the other half isn't buying it. Toss in the shortage of food and water, and the brutal cold, and we're going to have riots on our hands any day now."
I push my hair out of my face and stare at the mystery board. I count twenty-four pins. My nine ziplocks are no longer representative of the scenes. "Did you collect debris?"
He gives me a what-kind-of-idiot-do-you-think-I-am look, and a grin, and picks a box up off the floor that's crammed with more yellow envelopes like the ones on the slab. "I've been analyzing samples from the scenes, categorizing and isolating commonalities. I took photos, too."
I grin back because great minds think alike and it's so fecking cool to be peas in the Mega-pod.
While he opens envelopes, I get back to pinning pictures of the scenes where they go on our mystery board. I thought my life force idea was right until he pointed out two glaring flaws. Feck. It's a good thing I got my "impartial" ziplocks of evidence. I start to snicker then remember again that Ryodan's dead. It's hard for me to remember for some reason. Like I thought he was eternal or something. I got no clue why it feels like such a kick in the teeth every time I think about it. Sure, I let the Hag out, but he's the dude that failed to dodge her. I don't move as fast as him, and _I_ managed to get away.
Eight hours later I can hardly see straight. Alternately staring at bits of debris then studying the map is making me bug-eyed.
I been awake for three days, juiced on a constant sugar rush from candy bars, sodas, and the pall of something hanging over me that makes me nuts. Guilt. Guilt is for losers. Guilt is for folks who have stupid things like regrets. I contemplate the notion that maybe regrets are a process of accumulation of time, as unavoidable as a closet full of clothes and more bags of them in the attic. Is accumulated baggage what makes people get old? If so, they need to clean out their fecking attics, send the stuff to consignment shops and remember how to walk around naked like kids, little bellies sticking out, always ready for a good laugh. The second I kill the Crimson Hag, I'm sending my guilt straight to hell where it can burn. Problem is, I'm stuck with it till then and it's making me even testier than hormones. I don't like feeling responsible for crap. Like little anchors holding me still on my happy sea that's got an even grander adventure waiting just beyond the next wave.
There's a bit of everything in the plastic bags. Splinters of wood from church pews, stained glass, hair, bits of bone and carpet and leather, dirt, plastic, food, human parts, Unseelie parts. There are chunks of white crystal and shreds of yoga mats, parts of phones, teeth, jewelry, fragments of various electronics, bits of iron bars, a piece of a washboard, metal racking. There's paper and plastic wrappers, part of a fingernail with a finger bone fused to it, a hearing aid, half a driver's license, and so on. We make a list of each scene's contents, tack it to the murder board, and cross off anything that wasn't in every single bag.
We're left with "mystery debris," which is what we decide to call the dirt stuff at the bottom of each ziplock, metal and plastic.
"Does this stuff feel... I don't know, weird to you, Mega?"
I scoop a chunk of crystal into my palm and hold it a second. "It's colder than it should be, like it's still partly iced. It doesn't warm up no matter how long you hold it."
"No, there's something else. I can't put my finger on it."
I wait. I didn't go to school and I'm a little in awe of how much stuff Dancer knows. If he says there's something else, there is.
He muses aloud. "If it's not after life forces, how is it selecting its scenes? It might not be metal or plastic that the thing is after, which is at every scene in some form, but an ingredient _in_ metal or plastic. The thing could be hunting infinitesimal traces of something."
I push a pile of old bones to the edge of a stone slab, stretch out next to them, fold my arms behind my head and begin mentally rebuilding the scenes to what they were before they blew, thinking it might be easier to find a commonality before they were reduced to rubble. "Like some kind of theoretical vitamin or mineral it needs in order to accomplish something it wants to do?"
"Or a common element at the scenes that makes it think what it wants _might_ be at that scene," Dancer says.
"Huh?"
"It could be like a fisherman, going wherever there's salt water, because he's looking for a whale. We wouldn't necessarily ever find a whale. But we would always find saltwater. If we can figure out what draws it, we're halfway to stopping it."
"We still got three scenes we don't have samples from. The two that R'jan said got iced in Faery and the one under Chester's."
"Can you ask Ryodan to help us get samples? From what I hear, pretty much everybody owes that dude something."
All my mental pictures shatter when Dancer says his name, and suddenly I'm seeing two images at the same time: Ryodan on level four laughing, having sex, more alive than anybody I ever met 'cept of course me, and Ryodan, bled out in the alley, guts draping down the side of the building, cracking a joke while he dies, and I'm thinking the most fecked-up thought—I hardly even got to know him! "Yes, I did," I mutter, pushing myself up because if I'm going to puke my candy bar, I'm not going to be on my back while I do it.
"Did what?" Dancer says.
I always fought with him and kept saying I hated him. "He deserved it. He was the most arrogant, irritating fecker I've ever known!"
"Deserved what? Who was?"
Looks like I'll have to start calling Ryodan TP, too. 'Cause he's making my stomach cramp. I don't like him not being in the world. "Does this mean my contract expired, or can one of the other dudes call it in?" You just never know with dudes like them. I don't ever want to go into Chester's again, and I don't want to go into BB&B again, assuming I could, because it's just B&B now and the critical ingredients that made both places so exciting and larger than life had nothing to do with the places themselves.
"What contract?"
Now that those critical ingredients are gone forever I got a bad feeling about Dublin, about the whole world. Like I might have tilted the planet on its axis into some strange, new, and not nearly as safe a position by eliminating them.
"Mega." Dancer's standing in front of me. "Talk to me."
"We can't ask Ryodan nothing," I tell him.
"Why not?"
I rub my eyes and sigh. "I killed him."
I wake up with my neck all crinked and a ziplock bag stuck to my cheek by drool. I lift my head an inch or two and peek out from under my hair, hoping Dancer isn't looking at me, and when I find him staring at the mystery board, I swallow a sigh of embarrassed relief.
I peel the bag from my face, wipe the drool off with my shirt, and rub at the grooves in my cheek. I can feel part of a ring indent in it plus a couple of those zipper lines. I don't even remember falling asleep. But somewhere along the way I just dropped my head on the stuff I was examining and passed out. A few hours? More? "What time is it?"
"What day, you mean."
"Dude, tell me I didn't sleep that long!"
"You needed it. I'm not sure you're going to be able to move, though. I've never seen anyone sit on a stool, drop her head on a slab and not move for fifteen hours. I thought about getting you to stretch out somewhere more comfortable. You changed my mind." He turns around and grins at me. He's got a busted lip. "You had no intention of being moved. You decked me in your sleep."
"Aw, dude, sorry!" I have no memory of it.
"No worries, Mega."
My stomach growls loud enough to wake the dead, and he says, "I got something I been saving for you." He rummages around in one of his bags on the floor, pulls out a box and tosses it to me.
I light up like a Christmas tree. "Fecking-A! Pop-Tarts! Where did you find Pop-Tarts? I haven't seen any for months!" Even before the walls fell, they could be tough to find. "And they're my favorites—chocolate with frosting!" I rip open a pack and munch happily away. I polish off the first two in a quick inhale then slow down to savor every delicious, preservative-packed, sugar-crammed crumb of the remaining six. When the walls fell, all the good stuff—which is the bad-for-you stuff—got taken off the shelves first. Soda and liquor went real fast. Candy, cake, cookies, pies, things like that were next. Pop-Tarts, all the sugary cereal, flew off the shelves, too. I'm as guilty as the next person. Funny thing is, nowadays I'd just about give my right arm for a hot meal of fresh slow-cooked pot roast, carrots, peas, bread, and gravy.
Still, Pop-Tarts are close to heaven and Dancer got them for me, which makes them taste twice as good. I eat, and he tells me everything he considered and discarded while I was sleeping so I can poke holes in his theories if there are any. When he's done talking, we're no closer to conclusions than we were before I fell asleep.
"So, all we've still got is that every scene has dirt, some kind of plastic, and metal at it."
"Actually it's dirt, plastic, and iron. The metal in every one of the ziplocks is largely iron."
"Iron is what we use to imprison the Fae."
"I know. Remember how much worse the Unseelie at Dublin Castle got iced?"
I nod. "I thought it was because there were so many of them."
"It also happens to be the location with the most iron. Tons of the stuff was used to build those cages."
"Where was the iron at the other scenes?"
"Old railroad tracks run right next to where they were washing clothes out in the country. I checked maps, and discovered railroad tracks run past four other scenes. I found iron bullets in two of the bags. The church steeple had enormous cast-iron bells. The fitness center had part of a cast-iron teapot and iron chime fragments. At another scene there were several older cars that had frames of iron. They don't make them like that anymore. At Dublin Castle there are all those iron cages. The racking in one of the old warehouses was made of iron." He goes on, detailing location after location.
"Why iron? Why not say... steel. Isn't steel iron?"
"Iron gets turned into steel. What I'm seeing is a preponderance of _unworked_ iron, like the railroad tracks, bells, and bars. Old stuff. You don't see a lot of iron anymore. You see composites. Steel is stronger and iron rusts. You know how old railroad tracks are almost always red with it?"
"You think we need to go back to the scenes and see if it took the iron?"
"No. I'm wondering if iron is in the salt water. If that's what draws it."
"But what is it after?"
He shrugs. "Who knows? Who cares? I only want to know two things: how to lure it to us and how to get rid of it. Its goals are irrelevant."
"But Fae hate iron."
"I know. That's what makes me wonder if it's drawing it somehow. I'm not saying it's coming to iron because it likes it. Maybe it's trying to destroy the iron by icing it. Maybe one of the Fae summoned it to destroy the only means we have of imprisoning them. Maybe trying to understand something that can open a multidimensional portal, sail across the sky, open another portal and vanish, is as much an exercise in futility as trying to divine the motives of God."
"You believe in God?"
"Dude. Only God could have created physics."
I snicker. "Or Pop-Tarts."
He grins. "See. There you are. Proof of the divine. All in the chocolate smudge around your mouth."
"I got chocolate on my face?"
"Kind of hard to see with all those ziplock lines but yeah."
I sigh. Someday I'm going to be around Dancer with no guts in my hair, no weird clothes on, no black eyes or blood, and no food on my face. He probably won't recognize me. "But what about those two places in Faery?" I say.
"What about them?"
"There's no way there's iron in Faery."
"Assumption. Potentially erroneous. The walls came down. Everything got fractured and Faery has been bleeding through to our world. Maybe parts of our world are bleeding through to Faery, and there are railroad ties or bells in those parts. We need samples from Faery."
"And how the feck are we going to get those? Why don't we just try to lure it with iron and see what happens?"
"That's plan B. Let's try to get samples first and I'll keep analyzing this stuff. There's something I'm missing. I can feel it in my gut. I need more time with the evidence. Besides, even if we got it to come, what would we do with it then? We need to know what draws it _and_ how to stop it. You get the samples. I'll work on the rest. If there's no iron in Faery, we know we're back at square one without having to round up tons of iron and find a place to stack it all up where nobody will get hurt."
I push up and head for the door.
As I'm leaving, he says, "Don't go to Faery yourself, Mega. Make a sifter do it for you. We can't lose another month. I got a bad feeling about these iced places."
" 'Cause they keep exploding?"
He takes off his glasses and rubs his eyes. "No. Like there's something worse about them. A lot worse. I can't explain. It's just a hunch."
I know Dancer. When he gets a hunch, what that really means is his subconscious is seeing something he hasn't wrapped his conscious brain around yet. Every time he's ever told me he had a hunch, he's worked his way around an epiphany. I trust him like I never trusted anybody. If he wants samples and more time, he's got it.
I head up and out into the Dublin night. A light snow is falling. The moon has a blood red ring.
There's one sure place to find a sifting Fae. Conveniently, it's also the third place we need a sample from. With luck, I'll be back here in a few hours with the final three ziplocks to complete our evidence chain.
My luck ain't been so good lately.
# THIRTY-THREE
# _"Who's your daddy?"_
Chester's. Feckin-A, I hate this place even more than I used to. The line outside tonight is nuts. It's zero degrees in Dublin, snow's begun to fall in earnest, there's a killer wind kicking up and still five blocks of folks are shivering outside, bundled in layers of clothing, huddled together waiting to get in.
I blast past them in fast-mo, skidding on an icy spot, whiz around one of Ryodan's human bouncers who's got his hands too full controlling the crowd to stop me, jump the ladder down to the main entrance and explode through tall black doors into the club.
It's rocking tonight same as always: music thumping, lights flashing, folks partying up a storm. We got something icing our city, killing innocents everywhere, turning it into an arctic zone in June, and this is what folks are doing about it. Dancing, laughing, getting drunk, getting laid, acting like the walls didn't fall, the world didn't lose half the human race, and nothing's changed.
I stand on the platform inside the door that overlooks it all for a sec, scowling, blowing on my hands, trying to warm them up. I need gloves. And a scarf and earmuffs. The scowl doesn't last long because I get distracted from being pissed by the song that's playing. It's one of my oldie faves from a few decades back, heavy on bass, and it's so loud it vibrates the soles of my combat boots, all the way up my legs and into my belly. My bones rumble with resonance. I love music because it's so fecking brilliant. Music is math, and math is the structure of everything and pretty much perfect. Before everything got so crazy, Dancer was teaching me stuff about math that dazzled me.
My scowl comes back.
Jo's in the kiddie subclub, dressed all sexy, laughing at something some skanky waitress said, moving sleek and pretty with the music as she goes from table to table, chatting up the customers and occasionally looking around, like she's keeping an eye on things in general, or watching for someone. She's still got those highlights and sparkly boobs. I'll be real glad when that stuff's gone and she's the Jo I know again.
I'm going to make her quit tonight. We don't owe a dead man anything, and if the other dudes think to try to enforce our contracts, well, we're walking out anyway and they can just try.
I groan and roll my eyes, realizing I can't make her quit tonight because I can't tell her he's dead. I can't tell nobody he's dead. Only me, Christian, and whoever moved their bodies—assuming it wasn't Christian—know they got killed. It's only been three days. Folks might not decide he's dead for a while yet. Knowing her, she'll stick around for weeks, hoping he comes back!
I feel a little perturbed. I been gone almost a month and she doesn't look sad at all. Didn't she miss me? Worry about me?
I shove that thought away and look up at the ceiling, eyeing the girders, wondering what kind of metal was used in the construction of Chester's. If this place is as old as it seems, I'd think it'd have to be iron because I don't think the method of making steel was figured out till recent times. Well, recent in terms of how old this place is. Then I wonder how old iron is. Then I wonder if Ryodan and his dudes just spelled the whole mess together. Or maybe they created their own kind of metal or brought it with them from whatever planet they were born on.
I wonder who's in charge now that I killed Barrons and Ryodan. Lor?
As if my thoughts conjured him, I hear him say behind me, real close to my ear: "Aw, honey, you got some nerve coming here."
I turn around to say suspiciously, "What do you mean by that?" but he's not there by the time I complete my rotation. I wonder if I imagined him, a product of my guilty conscience. Then I decide if I really did hear him say what I thought he said, he was only referring to how Ryodan's been looking for me for a month and now I waltz in like I never been gone, and he thinks Ryodan's going to toast my ass for missing work so long. Because, like, he doesn't know Ryodan's dead either.
This is exactly why I hate lies. The second you tell one, you know something everybody else doesn't know and you have to constantly keep reminding yourself to behave like you don't know it, so they don't decide you're acting weird and figure out you know something they don't. If they do, they'll back you against a wall and demand to know why you're acting weird and you'll say something stupid and they'll use it to trip you up with. Then everything comes spilling out and you're in ten kinds of trouble! It's so much easier not to tell any lies to begin with.
This is going to be a tough pretending gig. Reminders of Ryodan are everywhere in here. Heck, Ryodan _is_ Chester's! It's, hands-down, the hardest place to pretend he's not dead that I could possibly be. But I need those samples. The HFK is icing something practically every day, and Dancer thinks things are going to get worse.
I spot a sifter down in the Tuxedo Club and grin. The Gray Bitch. This is one I'm going to love laying the flat of my sword against and ordering around. Mac promised not to hunt her but I never took no such stupid oath, and besides, I'm not hunting her, I'm just going to threaten her into doing something for me. Hand hovering over the hilt of my sword, I map out the grid as best I can, considering most things on it are moving—not that I mind jabbing all these idiots with my elbows—and freeze-frame down the stairs. At the last minute I detour from the Tuxedo Club and head for Jo. I want to see her face when she sees me. See how glad she gets to know I'm alive. She must have been as worried about me as Dancer and it's only right to put her mind at ease.
"Dani! What are you doing here?" Jo goes white as a sheet when I whiz to a stop in front of her. "Are you _crazy_?"
Not the reaction I expected. Where's the look of relief, the big hug, the excitement to see me alive and back here again? "What are you talking about?"
"Ryodan's been looking for you for a month! You broke your contract with him!"
"And according to that," I say irritably, "you should be dead. But you're not. Fact is, you look pretty darn good to me. Guess boinking him kept you alive, huh? You been doing it all this time? Didn't he get tired of you?"
She flushes. "He said it wasn't fair to take out his displeasure at you on me. Ryodan's a smart man. He makes good decisions. He's not impulsive like some people." She gives me a pointed stare.
I'm disgusted. "Oh, he was just a... uh, _is_ a fecking saint, now, huh?"
"He's a fine man. You should give him a chance."
"He's a dead man, is what he is!" I blurt, because I can't fecking stand to hear her defending him.
"Would you quit making threats about him every time you turn around? It's getting old." She lowers her voice. "You need to get out of here before he catches you. I've never seen him like he's been since he hasn't been able to find you."
"I ain't scared of Ryodan." Gah, I wish I could just tell her!
"You should be. You pushed him too far this time, Dani. I don't know what he's going to do when he sees you, and I'm not sure I can stop him. I don't think he'll listen to even me about you."
He's never going to find out because he's dead, but that's not what I fixate on. "What do you mean 'even you,' like you're some kind of special to him?"
She blushes and gets this soft-eyed look on her face like a sap in love. "We're a couple, Dani. It's been over a month and we're exclusive. All the waitresses are talking about it. They never thought anybody would... you know, get a man like him to settle down."
I just stare at her, blinking. Ryodan ain't exclusive with nobody. Settle down? Tornados _touch_ down. They don't settle. They leave destruction in their wake. Not shiny, happy people. I feel sick inside, at the idea of him and Jo setting up house together, making plans for the future. As fecking if. What am I going to be? Their little fetch-it dog? I shake my head, reminding myself again that Ryodan's dead. How does she keep getting me all distracted? Talking like he's alive is confusing me.
"I ain't talking to you anymore. I got things to do. Maybe you noticed Dublin is turning into the North Pole?"
"Of course I have. You're the one that took off for a month and didn't tell anyone that you were going to Faery with Christian."
"Huh?" I gape at her. "How'd you know that?"
"Christian told me."
"Scary-Unseelie-prince-Christian dropped in and told you I was okay?"
"I don't know why he came, but he overheard me talking with Cormac yesterday in the Tux Club about how worried I was about you and he said the two of you had just gotten back and you were fine. I'm not going to breathe a word to Ryodan even though we tell each other everything. But I don't appreciate you putting me in a position where I have to lie to him. Now get out of here before he comes down! Things are calm tonight. I'd like them to stay that way."
Tell each other everything? She's wrong on all counts. Ryodan was the most keep-it-to-yourself dude I ever met. Things aren't calm in here; as usual they're a catastrophe waiting to happen. And he ain't ever coming down again.
So I'm walking away from Jo, heading toward the Tuxedo Club to commandeer the Gray Bitch's services, when somebody crashes into me from behind so hard I go flying into one of the fluted columns at the exit of the kiddie subclub. I end up hugging it, to keep myself from puddling to the floor. I hit it so hard I'm going to have another black eye and the whole left side of my face is already working itself into the mother of all contusions. I think: Who the feck would dare attack me when I'm carrying so blatantly? Mac? 'Cause she hates me so much it made her stupid? I didn't hide my sword when I came in. I peeled my leather coat back so everybody could see it's mine again!
I stumble away from the column and am about to turn when I get slammed into it again. This time I swear I see stars and hear cuckoo birds whistling. My hand falls off the hilt of my sword, I'm so dazed. I hear Jo yelling behind me. "Stop it! Don't hurt her! Stop it!"
I get slammed again as soon as I start to move. This time I bust my lip against the column. It pisses me off so bad that I shift up into fast-mo, grab my sword and yank it out. If it's Mac, I don't want to hurt her. I just want to run. But she's really got to stop pushing me around in front of the whole fecking club. I got a reputation to consider.
It's gone from my hand before I even can turn around. I get slammed again and I bite the fecking column a fourth time.
"You move one more time, I'll rip your fucking heart out."
I go still as the slayed Unseelie chunks at the iced scenes. That was _not_ Ryodan that just spoke behind me because he, like, got gutted and died. Apparently I'm having hallucinations. Either that or a ghost is haunting me. It would figure the dude would come back from the dead just to make my life miserable. He was such a pro at it when he was alive.
I'm crushed so tight between the column and whatever's behind me I almost can't breathe.
"You can't be here," I say. "You're dead."
He slams me into the column again and I make an involuntary squeak.
"I first learned of your existence when you were nine years old," he says. "Fade told me he'd seen a human child on the streets that could move like us. He advocated, as did the rest of my men, killing you immediately. I have rarely found it necessary to kill human infants. They don't live long anyway."
That sure sounds like Ryodan. Cold. Void of inflection. Maybe Ryodan had a twin brother I knew nothing about. If not, I've gone completely nuts and being tormented by a guilty conscience in a weird and incredibly real way. He died. I watched it happen. There was no mistaking it. I try to move my hand, thinking to wipe blood from my face. He crushes it in his fist so hard my bones grind together.
"I said don't fucking move. Not a hair on your head. Got it."
Another Ryodan characteristic. No question mark. I hate being cued so I don't say anything. A bone snaps in my little finger. Gently. Precisely. Like he's showing me he could break them all, one at a time, if he felt like it. I grit my teeth. "Got it."
"When you were ten, Kasteo told me you'd somehow gotten the sword. Again my men advocated I take it and kill you. Again, I felt the mewling pup would die soon enough."
"I'm not a pup and I don't mewl. Ow! You said don't move. I didn't. I spoke!"
"Don't. And you _will_ mewl before the night is over. In a moment I'm going to step back and let you go. You will turn around and follow me, walking behind me. You will not speak to anyone. You will not look at anyone. If anyone but me speaks to you, you will not answer. You will not move any part of your body that is not absolutely necessary to get you up the stairs and into my office. If you deviate from my orders in any way, I will break your left leg in front of the entire club. If you piss me off while I'm doing it, I will break your right leg. Then I'll carry you up the stairs I'm currently giving you the choice to walk up and break both your arms. I trust I've made myself clear. Answer me."
"Clear as the floor of your office." He can't be alive. I watched the Hag scrape his guts out and sew them up into her dress. Surely he wouldn't really break all my arms and legs. Would he?
The presence behind my back is gone and I'm floored for a second by how cold I am. I hadn't realized how much heat he was throwing off until he was gone.
There's no way he's alive. It can't be Ryodan behind me. Is Barrons alive, then, too? How could they be? I know they're tough to kill and all but folks don't survive being gutted! Where did they get new guts from? Did somebody take them back from the Hag and sew them both up again? Will he look like Frankenstein's monster?
I don't want to turn around. I don't like any of the possibilities confronting me. If it's not Ryodan, I've gone nuts. If it is Ryodan, dude, I'm dead.
"Turn around, kid."
I can't make my feet move. I can't wrap my brain around that he's standing behind me. I'm shaking like a leaf. Me! What the feck is wrong with me? I'm tougher than tough! I ain't scared of nothing.
"Now."
I take a deep breath and turn around. I absorb his face, his body, the way he stands, the look in his eyes, the arrogant, faint smile.
It's either Ryodan or a perfect clone.
I do something I can't believe I do. I hate hormones, I hate Chester's, and I bloody fecking hate Ryodan. I'm never going to be able to live this down!
I burst into tears.
Ryodan turns and stalks off for the stairs.
I trail miserably behind him. The whole fecking club is watching Dani Mega O'Malley cry and walk behind Ryodan without saying a word, like a dog brought to heel. I can't fecking believe it. I hate my life. I hate myself. I hate my stupid face. I want to snap, "He broke my ribs and I'm crying from the pain of one them puncturing my lung but I'm tough and I'll kick his ass and be okay and then I'll kick all of your asses, too!" to save face, but I'm pretty sure if I say a word he really _will_ break my leg. I wipe angrily at my eyes. My stupid, pansy, betraying eyes with their stupid, pansy, betraying tear ducts.
The whole club has gone silent. Folks and Fae part a wide path to let us walk through. I've never taken a long walk of shame before and it chafes real bad. Jo's standing there, white-faced, looking from me to Ryodan's back, and back at me again. She might be his flavor of the month but I can tell by the look on her face that she's afraid of pushing him. She mouths, _Apologize! Bend. Or he'll break you!_
Over my dead body. The Mega doesn't bend. I pass Lor at the bottom of the steps to the upper level. I turn my face away because I can't stand him to see me being such a baby. He leans in close and says soft-like against my ear, "Honey, you might just have saved your life with those tears. I thought you had too much ego and too little common sense to know when to turn on the waterworks. He can't stand a woman crying. It fucks him up every time."
I look at him. He winks at me.
I flash fire at him with my eyes because I ain't allowed to use my tongue. They say: _I ain't a woman and I ain't crying and I ain't afraid of nothing_.
"He can deal with not being able to control you as long as you let the world believe he does. He's king here, honey. Kings can't be challenged publicly."
_Nobody controls me. Ever_ , my eyes snarl. _And I challenge whoever the feck I want wherever I fecking feel like doing it!_
He grins. "I hear you, kid. Loud and clear. Just remember what I said."
I jut my jaw and follow Ryodan up the stairs.
He turns on me the second I close the door.
"Turn it off. You don't cry. I expect you not to cry. Stop it. This fucking instant."
"I'm not crying! I got stuff in my eyes when you slammed me into the column. And I expect dead people to stay dead! So, I guess we both got disappointed, huh?"
"Is that what you are? Disappointed? You watched me get gutted and die and now that I'm standing in front of you alive you feel disappointed?"
"Did I just hear, like, three question marks?"
"Do _not_ fuck with me right now!" He slams me back into the wall so hard I feel the pane rumble behind my back.
"You don't care what I feel! You never have. You just order me around and expect me to obey and get pissy if I don't. I'm nothing to you so don't pretend you give a royal rat's ass what I feel!"
"Loyalty stems from what you feel. Or don't. You aren't on thin ice, kid. You're underwater and my hand is on your head, holding you down. So choose well: 'D' is for disappointed to see me. And Death. 'L' is for loyalty. And Life. Convince me I should let you live."
His face is an inch from mine. He's breathing hard and I feel violence in him. Lor said I should use my tears to manipulate him. There's no way I'm stooping to such wussy-girl depths. I'm just as big and bad as he is.
He's alive. He's here. Bullying me. No doubt getting ready to eventually—after he's done killing me—order me to report to work again.
We're back to being us. Robin to his Batman.
He's alive.
Tears stream from my eyes.
"Stop it!" He slams me back into the wall again so hard my teeth clatter but the idiotic tears just keep coming.
I bounce off and use the ricochet to smash into him as hard as I can. He grabs my wrist when I hit him and when he goes flying back, takes me with him. We crash into his desk. I go flying up on it, roll over it and leap to my feet, tossing my hair from my eyes.
I slam my palms against the desk and snarl across it, "Don't you think I would if I could! Do you think I liked looking all sissified in front of your whole fecking club? In front of you? You stupid fecking stupid fecker! What were you doing outside that wall anyway! Why did you have to be right there in that exact spot when we came out? I mean, who has that kind of crap luck? Ever since I started to hang with you, my life has been a total fecking nightmare! Couldn't you just stay dead?"
He slams his hands down on the desk so hard it cracks down the middle. "Not. Convincing. Me."
I glare through my tears. "Not trying to! I don't convince nobody of nothing. You take me or leave me just the way I am! But I ain't changing for you or nobody else and I ain't faking either, and if you think breaking my bones one by one is going to accomplish a thing besides, like, breaking my bones, good luck with that!"
I'm sobbing now and don't have any clue why. Just that it feels like ever since I came out of the wall with the Crimson Hag and watched it kill Barrons and Ryodan, I've been all trussed up in one great big painful knot, and the second I looked at him and realized he was alive, really, truly alive, and I wasn't going to have to walk around for the rest of my life with his death on my head, never seeing his smug-ass smile again, that knot relaxed, and when it let go, everything in me came apart and my whole self heaved a sigh of relief and somewhere I guess I got a well of tears in me, like maybe everybody has a certain allotment of them and if you never let them out, the second a single one sneaks out, it opens a floodgate and you can't shut it again. Why doesn't anyone ever tell me the rules of life? If I'd known it worked this way, I would have taken myself off somewhere private and cried until I'd use up my quota! This is worse than getting off on the wrong foot when I'm freeze-framing. This is emotional careening with no control.
I look at him and I think, Crimeny, if only Alina could have stood back up from what I did to her. Mac could have had her sister back. And I wouldn't have to walk around all the time, every single minute of every single day, hating myself because even though I'm pretty sure Ro did something to me that night that made me some kind of automaton that didn't have a will of her own, I was there. I was _there_! I led her to the spot where she died by lying to her and telling her I had something really important to show her and I'm just a kid so she trusted me! I stood in that alley and I watched Mac's sister get killed by Fae that I could have stopped with one flick of my sword and I can never undo it and I can never scrape it out from behind my eyes. It's seared into my soul for the rest of my life, if I've even got one after all the shit I've done!
I hurt Mac worse than anything in her life ever did and I can never undo it.
Still... there's a silver lining to this cloud: if Ryodan isn't dead, Barrons isn't either. At least Mac still has Barrons.
"You killed Mac's sister," Ryodan says. "I'll be damned."
I didn't say that. "Stay the feck out of my head!"
He's across the desk and practically on top of me. He shoves me back against the wall, clamps my head between his hands and forces me to look up at him. "How did you feel when you thought you'd killed me."
He's looking in my eyes like he doesn't need me to answer, just think it. I try to double over so he can't poke around in my thoughts but he won't let me. He's holding me firm, but almost gentle now. I hate gentle from him. I prefer fighting. I know exactly where we stand then.
"Answer me."
I don't answer him. I'm never going to answer him. I hate him. Because when I thought I'd killed him, I felt more alone than I've felt in a long time. Like I couldn't stand walking through this city knowing he wasn't in it. Like somehow, as long as he was out there somewhere, if I was ever really in trouble, I knew where I could go and while maybe he wouldn't do exactly what I wanted him to do, he'd keep me alive. He'd get me through whatever it was to live another day. I think that's the kind of feeling you get from parents when you're a kid, if you're lucky. I didn't get that feeling. I curled in a cage and every time she put on her perfume and makeup and hummed while she got dressed, I worried that she was going to kill me this time by forgetting me. I hoped her new boyfriend would suck so she'd come home sooner. I know that no matter what fecked-up things Ryodan does, he'll never forget me. He's meticulous. There's a lot to be said for detail-oriented. Least in my world there is. Especially when I'm one of the details.
I can't look away. How the heck is he alive? I feel like he's stirring around in my brain. Watching the light go out of his cool, clear eyes in the alley behind BB&B had just about slayed me. I missed him. I bloody fecking missed him.
Ryodan says real soft, "Disappointed or loyal."
I got no intention of dying. "Loyal," I say.
He lets me go and walks away. I slump down the wall, scrubbing tears from my face. I hurt everywhere, face, hands, chest, ribs. "But you're going to have to—"
"Do _not_ try to barter with me right now."
"But it's not fair that I—"
"Life isn't."
"But I can't stand working every night!"
"Deal with it."
"You're making me nuts! A person needs some time off!"
"Kid, you just never give up."
"I'm like, alive. How could I?" I stand up and dust myself off. My tears are gone as mysteriously as they came.
He kicks a chair at me. "Sit. There are new house rules. Take notes. Violate one and you're dead. Acknowledge."
I roll my eyes and toss myself into the chair, slinging a leg over the side. Belligerence is me. "I'm listening," I say irritably.
I hate rules. They always screw me up.
# THIRTY-FOUR
# _"Where do you think you're going?_
_Don't you know it's dark outside?"_
I slow-mo Joe it down the corridor cussing Ryodan but keeping it under my breath since he's walking right next to me.
The new house rules are the biggest pile of BS I ever heard. It's going to kill me to follow them. Literally result in my death because there's no way I'll remember to do everything he wants me to do while also keeping track of everything I'm not allowed to do. In addition to "Report to work at eight every night" is the most offensive rule of all: "You will never leave Chester's unaccompanied by one of my people again."
"So, I never get to be alone, like, ever?" I exploded, flabbergasted. "Dude, I need my private time." I been alone most of my life. Too many people in my personal space start to chafe me after a while. I get edgy and weird. And tired, too, like they wear me out just being there. I have to get off by myself, or be with one person like Dancer to recharge.
He didn't answer me.
Another one that really gets me is that I'm supposed to never question or argue with him in public! I'm going to be dead by morning. Only way I have a snowball's chance in hell of succeeding there is if I start wearing a muzzle or cut out my own tongue.
"You can say anything you want to me in private," he said. "Which is way the fuck more than I permit anyone else."
"I don't want no private time with you."
"Too bad," he said. "Plan on a lot of it."
"Why do you dick with me? Why don't you just forget about me and let me live my life." It's weird to think he's been watching me since I was nine. I never even noticed him. He's noticed me probably more than anybody else ever has, including my mom.
Again, he doesn't answer.
I walk with him to the end of a hallway on the third floor. He stops at a glass panel that's smoked black and pulls a cloth hood out of his pocket. When he reaches for me, I duck back and say, "You're kidding, right?"
He just looks at me until I snatch the hood from his hand, put it on myself, and let him guide me by an arm.
I suffer the indignity of being blinded in silence, and focus on absorbing every detail I can. I count steps. I sniff through the heavy fabric. I listen hard. When we get on an elevator and go down, I count seconds so I can figure out what floor he's taking me to when I finally get some time alone, and I will. He can't have someone on me every second of every day. He'll get tired of it. I need to get back to Dancer! I need to talk to Ryodan about getting samples but when I brought up the Ice Monster he told me to stow it.
When we arrive at our destination and he pulls the hood off, I'm floored to see Ryodan's got his own War Room, and of course it's top-of-the-line, technological perfection, and makes ours look stupid! Once again I'm jealous. There are computers everywhere. CPUs and monitors and keyboards and I don't know what half the stuff in the room is, and I know a lot. Dancer would go crazy in here!
He's got a map up, too, but unlike our paper one, his is electronic, on a glass panel suspended from the ceiling, about twenty feet wide and ten feet tall. It's something out of a futuristic movie. It's got lots of lines and dots and triangulated areas marked out in different colors.
"Sit."
I drop down in a chair behind an enormous slab table that faces the map. There are nine chairs at the table. I wonder how long this room has been here, how many centuries these dudes who don't seem to be able to die have sat in this room and plotted things. I wonder what kind of things guys like them plot. Coups? Economic catastrophes? World wars?
"So, Barrons is alive, too," I fish.
"Yes."
"Dude, what the feck? I don't know what your superpower is, but I want whatever you've got."
"You think."
"I know."
"You don't even know what it is. Yet you'd take it sight unseen."
"To, like, never die? Fecking-A I would!"
"And if there's a price."
"Dude, we're talking immortality. There ain't no price too high!"
He gives me a faint smile. "Ask me again when you're older."
"Huh?" I say. "Really? When I'm older I can have whatever you got? Like, how much older? Fifteen?"
"I didn't say you could have it. I said you could ask me. And no, not fifteen."
"Dude, give me a little hope here."
"I just did."
He taps something in on a remote device and all the sudden I'm not looking at Dublin on the grid anymore. He's zoomed out and I'm seeing a map of surrounding countries. There are dots pegged in England, Scotland, France, Germany, Spain, Poland, Romania, and Greece. He zooms out farther and I see two in Morocco and one in Norway.
I let out a low whistle, horrified. Dancer and me were only seeing the little picture. "There's more than one Ice Monster."
"Not necessarily. I think if there was more than one, we'd be hearing reports of it all over the world and we're not. So far, it's confined to this region."
"I need samples from Faery and the first place it iced in Chester's."
"Elaborate."
"Dancer and me went through all the evidence. There's iron in every bag and—"
"No."
"You didn't let me finish."
"I don't have to. Iron has nothing to do with it."
"How can you know that?"
"Because there's not a single drop of iron anywhere in or near Chester's."
"Well, what the feck is this place built from?"
"Irrelevant. Besides," he says, "if it was after iron, it would have taken the cages at Dublin Castle and it didn't. It iced the place and vanished. We've been studying the map and scenes for weeks. There's no pattern, no commonality. I put my best man on it, a linchpin pro. He can't find a tipping point, sees no order in this chaos."
"Who's your linchpin pro?" I want to talk to him. I'm fascinated by linchpin theory. If you know where to make the dominoes start toppling, you own the dominoes! Of course, Ryodan doesn't answer that question either so I tell him Dancer's theory about salt water and whales and that maybe it's drawn by something because it's looking for something else.
"Possible. But not iron."
"You dudes been hosting fairies for, like, millennia, haven't you? That's the only reason for a place like this having no iron!"
"There are other things that don't like iron. Not just Fae. A smart person might find a lot of things missing in Chester's." A faint smile plays at his lips, and I almost get the idea he's challenging me to figure something out.
"Dude, if I'm stuck here long enough, I will." I gesture at the map. "Show me Dublin again." When he resets the map, I say, "I need the remote."
He punches numbers in on it, no doubt locking systems off from me, then hands it over.
"Let me stare at the map a while."
When he leaves, he locks me in.
I'm still staring hours later, no closer to an epiphany, when I start smelling the most fecking awesome smell in the world. I try to concentrate on the map but I can't. I shove a candy bar in my mouth. It tastes like Styrofoam. I haven't smelled fresh-cooked beef in longer than I can remember. I never got it at the abbey! Somewhere in Chester's, some spoiled person is feasting. My mouth fills with saliva. I slide down in my chair, drop my head back and inhale real deep and slow, making lip-smacking noises, pretending I'm the lucky recipient. I smell all kinds of spices! I think whatever the meat is, it's accompanied by mashed potatoes and some kind of greens. I smell garlic, salt and pepper, butter! I smell onion and oregano and rosemary! It's almost enough to make me cry, thinking about that kind of food. I'm beyond sick of candy bars and protein bars and canned stuff. I'm so home-cooked-meal starved that not even my chocolate Pop-Tarts hit the spot like they used to.
When the door slides open and Lor comes in, pushing a cart like you see in hotels for room service, I just sit there and stare, thinking: Is this a new way to torture me? I don't move a muscle. I'm not going to make an idiot of myself. Ryodan's probably on his way to eat in front of me just to make me suffer.
Lor rolls the cart to a stop a few inches from the toes of my shoes. I have to grip the arms of my chair so I don't jump out of it and attack whatever's in those covered dishes.
"Boss says eat."
He takes the lid off the biggest plate and sure enough there's meat sizzling like it just came off a grill with a side of mashed potatoes, plus a mixed veggie medley! There's a bowl with bread, hot from the oven. And butter! I almost expire from the sheer excitement of it. Like, the real stuff and a whole carafe of milk! It's the most beautiful sight I think I've ever seen. I stare, holding my breath.
"You're scrawny," he adds.
"That's for me?" I say wonderingly. I still don't move. It's got to be a trick. The meat is a rib-eye steak, perfectly marbled. It's thick and has grill marks on it and looks like it's cooked to perfection. I've only ever had it twice in my life. Once when Mom got engaged—it didn't work, the dude ditched her, they all did eventually—and another time when she got a new job that she thought would get us out of Ireland for good if she saved everything she made for three years. She got fired after a month and cried herself to sleep every night for weeks. I think she thought if she could just get us out of Ireland, everything would be easier. I know other _sidhe_ -seer families ran. Mac's did.
Lor nods.
I'm out of the chair and on the cart in fast-mo.
"Kid, slow down. You might want to taste it."
My hands shake when I pick up the fork. I go straight for the steak, slicing a big chunk off. The first bite explodes in my mouth, full of meaty juices and sheer succulent beefy perfection. I slump back into my chair and close my eyes, chewing slowly, delicately milking it for every single taste. I fork up a pile of fluffy mashed potatoes and they're fecking heaven! The bread is tender and warm inside, crusty outside, and kissed with rosemary just like Mom's. I wonder who cooks around here. I wonder where their kitchen is. I'm going to rob them blind if I find it. I slather butter on the bread then lick it off and slather more. I pour a long cool drink of milk down my gullet. I force myself to count to five between each drink and bite. It occurs to me I've never seen Ryodan eat. He probably pigs out in private. Probably eats steak and milk every day!
"The snow's piling up and the temperature's dropping," Lor says. "People are lined up for five blocks, trying to get inside. Generators and gas have gotten scarce. People are freezing to death. It's June in Dublin. Who'd fucking believe it?"
I chew reverently, listening to him and staring at nothing. "Maybe it's not after an element like iron or something. Maybe it's after a feeling. Maybe someone was having sex at every scene, or... eating at every scene, or fighting or praying or... something."
"Doesn't hold water. There was no life at the steeple."
I knew that. I just forgot for a sec. "So we're back to the inanimate."
"Looks like."
All too soon my meal is over. I've got the best taste ever on my tongue. I won't eat again until I absolutely have to, and I'm not about to brush my teeth for a while. I want to relish the residue from my taste buds till there's nothing left. I may never get this kind of a meal again. After I sop up every drop of beef juice with the last few bites of bread, Lor takes the cart and leaves.
I could almost pass out from the overload of rich food. Digesting it stupefies me for a while and I stretch out on the floor, staring up at the map.
I can't shake the feeling that I'm still not seeing the big picture. I'm lying here, staring at an enormous map, and I know there's something about these scenes I'm missing or reading wrong. I can feel it. Like Dancer, I get hunches and I listen to them. Used to be, when I was little, I couldn't concentrate because of all the things I could hear around me. When Ro took me in, she taught me to plug my ears, shut out the din and focus. Old witch passed on a few good things but they'll never counter all the evil she did.
I dig earplugs out of my backpack. Dancer made them for me out of some kind of stuff that absorbs noise way better than the standard plugs. I wedge them in, tune out the world, and begin sorting through my facts.
One: It's not after iron. There's none at Chester's. I need to get that info to Dancer ASAP.
Two: It's not after life force because one of the scenes had no life forms and I seriously doubt a mouse would be enough.
Three: Dirt, metal, and plastic are the only physical elements all the scenes had in common.
I start mentally rebuilding every scene I visited, labeling and depositing them in one of the more readily accessible drawers in my brain's filing cabinet, right next to where Dancer and me play chess sometimes without a board. It's an important part of your brain to exercise if you want to stay sharp. Being smart is handy, but if you aren't mentally agile, it doesn't get you anywhere but stuck in your own fact-ruts.
First up is the subclub. There were over a hundred humans and Fae engaged in various social and sexual activities. I visualize the room in detail, from the torture racks to the sofas, the sexual couplings to the band that was playing in the corner, the food that was on a table, the tapestries and mirrors on the walls. I look for something in the club that I can easily spot at every other scene. Maybe it's hunting for a tapestry or a special mirror. It sounds stupid, but who can say what might draw a creature like that? Maybe it was cursed and it needs some hallowed Fae object to free itself. You never know with the Fae.
Next up is the warehouse that got iced, populated only by Unseelie and filled with crates and boxes of guns. What was in this place that was also in the club? No tapestries or mirrors that I saw, but maybe there was one in a crate somewhere behind all the audio equipment and electronics.
Then there were two underground pubs with the usual stuff: wood bar, bottles, drinks, stools, a huge mirror behind the bar, folks dancing, a few shooting pool in the corner of one place, playing darts in the other. The wood could have come from anywhere: the stools, the bar, the framed pictures on the walls, the floor. The plastic also could have come from anything: bottle toppers, chairs, plates, phones, the list goes on and on.
The fitness center had three people in a building filled with treadmills and ellipticals and all kinds of weight machines and twenty or so of those milky-crystal meditation bowls. I guess the wood at that scene must have come from the framing of the building. I go back and begin mentally breaking down the structure of each scene, too, so I can add all that stuff into the mix.
"This is impossible," I mutter. It's worse than looking for a needle in a haystack. I'm looking for a dozen needles in dozens of different haystacks that are no longer even there because they all exploded. It could be after a red Solo cup for all I know! Do they have red Solo cups in Morocco?
I go through the rest of the scenes and realize I need more info on the ones that happened while I was gone in order to visualize them. Ryodan might have a kick-ass War Room but Dancer's got lists already put together.
Too bad I'm locked in.
I look at the door. I don't remember hearing Lor lock it. Lor likes to stir things up, keep them hopping.
I freeze-frame over to it, test the knob and grin.
"Dani, I don't think this is a good idea," Jo says.
"He said I couldn't leave without one of his people. Listening to you talk, you and him are, like, peas in the Jo-pod. That makes you one of his people. Are you or aren't you? 'Cause the way I figure it, if the dude's banging you every day and doesn't consider you one of his people, you're not just getting screwed, you're stupid." I hate manipulating Jo. When her heart's involved, it's way too easy. And her heart's dangling off her sleeve where Ryodan's concerned. "Dude, you been outside lately?" I push. We have to go _now_. It took me twenty minutes to find my way back to the main part of Chester's from the War Room. I got a bad feeling Ryodan doesn't plan to leave me alone in there too long, with all those computers. I wouldn't. If I really _was_ stuck in there, that's what I'd be messing with right now, trying to hack into his systems. "The world is falling apart. Folks are dying! I just want to run a quick errand. That's all. One tiny little errand. It won't hardly take any time at all."
"I'll go ask him if it's okay first."
"You got any idea where he is? 'Cause I ain't seen him in hours. Isn't it morning? Did he come to the top of the stairs yet? Is he still summoning you that way for a quickie over his desk, or have you graduated to, like, getting banged in a bed and everything? What's he got, some kind of progressive ranking system? If you last a whole week, you get to do it in a chair, and if you make it two—"
"Now you're just being mean," she says. "Stop it."
"Just saying. I'd like to see you get some real romance, Jo. You deserve it. You're the prettiest girl in here and everybody'd love to date you. Do you know he has steak and milk and bread and stuff? I had the best meal today. Does he feed you like that?"
She tries to mask her surprise but doesn't succeed. "Isn't he still mad at you?"
"Don't look like it from where I'm sitting."
"Steak?"
I lick my lips, still tasting it. "Rib eye."
"Milk?"
"Dude." I nod. "Look, all I want to do is run by Dancer's and get the lists."
"He really gave you steak and milk today?"
I'd laugh but it's sad. We're all so fecking hungry for a home-cooked meal. When spring started to green things up out at the abbey, the girls started talking about growing veggies again. All the produce was gone within a month of the walls falling. If you want to bake something, you have to run a generator to power the oven. Either that or have whatever the feck kind of setup Ryodan's got here at Chester's, and even then you can only bake stuff that doesn't require butter or milk or eggs. Jo's almost as upset that he gave me good food as she is about him not romancing her.
"I'd call and ask Dancer to courier it over but, dude, no phones and no couriers. Can we just go? We'll be back before anybody knows we're gone. And if you and Ryodan really are a 'thing,' he ain't going to give you any guff. He's going to appreciate a woman with a little spine and independence!" Yeah, right. Ryodan despises spine and independence. He likes good little robots.
"Did he give you anything else?"
If I was having sex with somebody and they gave someone besides me awesome food, I'd be ten kinds of furious. The way I see it, intimacy should entitle you to privileges. If it don't, it's just skintimacy like on TV with folks always swapping partners and hurting each other. "Fresh strawberries and ice cream," I lie.
"Ice cream? Are you kidding me? What kind?"
It's sleeting when we get outside. Abandoned cars are shiny with a layer of ice. Skeletal trees shimmer like they're crusted with diamonds. Snowdrifts are piling up. There's a group of people outside Chester's but it's a somber, quiet crowd and I realize these ain't partiers trying to get inside, these are folks looking to survive what's coming. I guess all the partiers have already been let in. Wrapped in blankets, wearing hats, earmuffs, and gloves, these are folks that got no generators at home, and the weather has turned dangerously cold, sending them out into the streets to look for a source of heat before it's too late.
Jo and me look at the folks as we pass.
"Let us in," they say. "We just want to get warm."
You can tell there's heat in the club—and a lot of it—because the area above Chester's is bare of accumulation. The pavement is an underinsulated roof, and the heat radiating up keeps melting the snow. Even that nominal sign of warmth is enough to keep folks standing around, hoping, waiting.
There's old people here, with nothing to trade for food or drink or the privilege of hanging at Chester's. The big, brawny human bouncers Ryodan uses outside the club turn them back at the door, and a crowd has moved into the snow-free ruin of stone and wood that used to be the club aboveground. They got fires going in cans. They've gathered wood from surrounding buildings and piled it up. They look like they plan to stay a good long while. Like until they get let in. They look too defeated to fight. A cluster has begun to sing "Amazing Grace." Before long fifty voices lift in song.
"Maybe you could talk some sense into your 'boyfriend' and get him to let those folks inside," I say.
"I will," she says. "Or we could bus them to the abbey."
"What about WeCare? Don't they fecking care? Aren't they supposed to be giving away generators left and right?"
"Even if they are," Jo says, "some of these people are too old to get out and hunt down enough gas to keep one running. You've been gone for weeks. A lot changed in that time. The weather is all anybody talks about anymore. Making it through last winter wasn't as hard because the stores were all still stocked and the nights were mild. But supplies have been wiped out. We didn't expect winter in June. All the generators are gone. People are changing. They're fighting each other to survive. We need a long warm summer to give us enough time to grow and stockpile food before winter comes again. We need to get out and hunt for supplies in other towns."
"They're going to die, Jo. If we don't stop the Hoar Frost King, we're going to lose the other half of our world." I look back at the crowd huddled around the fire cans above Chester's. A mom is helping her kids get closer to one of the barrels so they can rub their hands together over the flames. Old folks that look too frail to be hiking through this ice and snow watch the kids with weary eyes that have seen three-quarters of a century of change but never anything like what's been happening since last Halloween. Men that look like they were office workers at desk jobs before the walls fell hold the perimeter, encircling the women, kids, and old folks. They're all displaced now. No jobs. No paychecks. None of the rules they used to live by. They look exhausted. Desperate. It fecking slays me. They've moved on to a new song, another hymn. Folks need faith in times like these. You can't give somebody faith. They either got it or they don't. But you sure can try to give them hope.
She gives me a bleak look. "If there was ever a time for you to dazzle us with your brilliance, it's now."
"I'm working on it. But I need stuff. Let's go. We'll make it back before anybody even knows we're gone."
We turn and begin walking down the street. I'm going to have to leave her aboveground. I'm not about to give away Dublin-down's secrets. But I'll take her as close as I can and leave her someplace sheltered. The snow crunches beneath my boots twice, as I sink through snow then ice, snow then ice. I hear Jo going through three layers because she weighs more than me. The sky is white with thick flakes swirling down in a dizzying display if you look up at them too long. They melt on my face, the only part of me exposed. We raided Chester's coatroom before we left, bundling in layers, tugging on hats and mittens and boots. If this weather keeps up, we could end up with ten feet of ice and drifts in the next day or two and it will totally shut the city down. Folks that didn't think to come out somewhere for warmth will freeze, snowed into their hidey-holes. If the sun doesn't start shining soon, this stuff'll never melt. It'll just keep piling. Time is getting more critical with each passing day. I can't believe I lost almost a whole month in the White Mansion with Christian! Speaking of which, I look around warily, checking all the rooftops, making sure the Hag isn't sitting on one of them, knitting away, or worse, getting ready to swoop down on us. The crazy blood and guts bitch creeps me out. I shiver. "We need to freeze-frame, Jo. Take my hand."
She gives me a look like I'm deranged. "There's no way you're doing that to me! Especially not on ice. Half your face is a bruise and the other half is recovering from one. Have you looked in a mirror lately?"
"That ain't because I'm a sloppy freeze-framer. It's because of stupid jerk-ass Ryodan."
"Stupid jerk-ass Ryodan is going to break both your legs if you take one more step," Ryodan says right behind us.
I whirl on him. "Why are you always stalking me?"
"You're always making me."
"How do you keep finding me?" Do I have a blinking beacon on my forehead that sends a signal straight to him every time I disobey an order? I refuse to believe since he bit me, he can track me wherever I go. That's a suffocating thought. It's wrong and unfair.
"Get back inside. Now."
"You didn't find me in the White Mansion." A lightbulb goes off in my head. I been busy with other worries, or I'd have clued into it sooner. "You can't track me in Faery!" That's why he was so mad. I almost punch air I'm so happy. I have a safety zone. If I ever need to hide from him, Faery's the place to go. "And you're the one who's always making me do stuff that makes me have to do other stuff that ain't what you want me to do. It's not my fault. I'm just reacting to you."
"There's your first mistake. Learn to act, kid."
"I _am_ acting. I'm trying to do something about our problems."
"And you, Jo," he says soft, "you should have known better."
"Leave her out of this," I say.
"She helped you disobey me."
"She did not. 'Cause, see, I didn't disobey you. You said I could leave with one of 'your people.' You're boinking her every day, and if that doesn't make her one of your people then you need to quit boinking her. Either she is or she ain't, and you can't have it both ways. You don't get to have sex with folks then discount them. So. Is Jo one of your people? Or just another piece of booty in your endless lineup?"
"Dani, stop it," Jo warns.
"Feck no, I'm not stopping it." I'm so pissed, I'm vibrating. "He doesn't deserve you and you deserve so much better!" It doesn't help that behind Ryodan the fire-can folks have switched songs again and are now booming out a rousing rendition of "Hail Glorious St. Patrick," clapping their hands and banging on cans with pieces of wood, getting all rambunctious. The louder they sing, the hotter my temper gets. "He's always pushing everybody else around but nobody ever calls him out on the carpet. I say it's way past time. Either you matter to him or you don't, and he needs to say which one it is. I want to know which one it is."
"She matters," Ryodan says.
Jo looks stunned.
It pisses me off even more. She's looking all dreamy-eyed and in love again. Anybody can see she ain't his type. "You liar, she does not!"
"Dani, stow it," Jo says.
I know him. I know how he tricked me. He's splitting verbal hairs. Of course she matters. But he didn't say "to me." She matters to the club, for mercenary reasons, because she's a waitress. "Does she, like, matter to you emotionally? Do you love her?"
"Dani, stop it right now!" Jo says, horrified. To Ryodan she says, "Don't answer her. I'm sorry. Just ignore her. This is so embarrassing."
"Answer me," I say to Ryodan. The hymn folks are really rocking it now, dancing and swaying, and I'm almost having to yell to be heard. But that's okay. I feel like yelling.
"For fuck's sake," Ryodan growls over his shoulder, "can't they go sing somewhere else."
"They want in," I say. "They're going to die on your doorstep because you're too much of a prick to save them."
"The world is not my responsibility."
"Obviously." I put twenty kinds of verbal condemnation in the single word.
"She just wanted to find Dancer," Jo says. "I think it's important. Sometimes you have to trust her."
"Do you love her?" I push.
Jo groans likes she's going to die of embarrassment. "Oh God, Dani, shut _up_!"
I expect him to scoff at me, say something bullying, throw an insult back in my face, but he just says, "Define love."
I stare straight into those clear, cool eyes. There's some kind of challenge there. I don't get this dude. But the definition he wants is easy. I had a lot of time in a cage to think about it. I saw a TV show once that gave the perfect definition, and I say it to him now: "The active caring and concern for the health and well-being of another person's body and heart. Active. Not passive." In a nutshell, you remember that person all the time. You never forget them. You factor their existence into yours every single hour of every single day. No matter what you're doing. And you never leave them locked up somewhere to die.
"Think about what that entails," he says. "Providing food. Shelter. Protection from one's enemies. A place to rest and heal."
"You forgot about the heart part. But I didn't expect anything else. 'Cause you ain't got one. All you got are rules. Oh, and yeah, more rules."
Jo says, "Dani, can we just—"
Ryodan cuts her off. "Those rules keep people alive."
Jo tries again. "Look, guys, I think—"
"Those rules strangle folks who need to breathe," I say, talking right over her. Nobody's listening to her anyway.
All the sudden he has me by the collar, hanging in the air, my feet dangling off the ground, our noses touching.
"By your own definition," he says, "you don't love anyone either. An argument could be made that you only ever do one of three things to the people closest to you: make enemies of them, kill the people they love, or get them killed. Careful. You're on thinner ice than you've ever been with me."
"Because I'm asking if you love Jo?" I say coolly, like I'm not hanging helpless by my shirt. Like he didn't just take a mean shot at me below the belt.
"It's not your business, Dani," Jo says. "I can take care of my—"
"Pull your head out of your ass and see the world," Ryodan says.
"I _do_ see the world," I say. "I see it better than most folks and you know it. Put me down."
"—self just fine." Jo is sounding kind of pissed now, too.
"And for that very reason, you're blinder than most," Ryodan says.
"That doesn't make sense. Still dangling here, dude." I try to toe the ground by pointing my foot but I think I'm a few feet above it.
"You don't see the forest for the trees."
"Ain't no forest. Shades ate it. Let me go. You don't get to just dangle folks in the air when you feel like it."
He drops me so abruptly I stumble on the ice and almost fall, but he catches me and puts me back on my feet. I shake his hand off my arm.
"There doesn't need to be love," Jo says. "Sometimes it's not about that."
"Then you shouldn't be boinking him!"
"It's my own business who I boink," Jo says.
"I don't 'boink' anyone. I fuck," Ryodan says.
"Thank you for that much-needed clarification," I say with saccharine pissiness. "Hear that, Jo? You get fucked by him. Not even the decency of a boinking. Screwed. Plain and simple." I'm beyond irate. I'm seeing through a red haze. The fecking fire-can folks are singing so loud they're hampering my ability to think straight. I want Dancer. Ryodan drives me insane. Jo's a hopeless case. Dublin's dying.
I can't stand things anymore so I punch Ryodan in the nose.
We all kind of freeze for a second and even _I_ can't believe I just hauled off and decked Ryodan with no warning and no real provocation. At least no more than he's always giving me.
Then Ryodan manacles my arm and starts dragging me back toward Chester's, looking madder than I ever seen him, but Jo gets my other arm, trying to make him stop, yelling at him and yelling at me. I'm slipping and skidding on the ice, trying to get them both off me.
We stumble across snowdrifts, fighting each other, when all the sudden the day gets foggy and I can't hear a sound any of us are making. My mouth's moving and nothing's coming out. I can't hear the fire-can folks either. I can't even hear my breath in my ears. Panic compresses my chest.
Me and Ryodan look at each other and have a moment of perfect communion like me and Dancer do sometimes. No words necessary. We're made of the same stuff. In battle there's nobody else I'd rather be hanging with. Not even Christian or Dancer.
I grab Ryodan and he grabs me and we sandwich Jo between us.
Then we freeze-frame the hell out of there like the devil is on our heels.
Or more precisely, the Hoar Frost King.
# THIRTY-FIVE
# _"She blinded me with science"_
Like we're chained together or something, Ryodan and me stop about three-quarters of the way down the block. We go just far enough to escape danger, while staying close enough to get a look back at Chester's.
By the time we glance back, it's too late. The temperature where we're standing just plummeted a good thirty degrees. The Hoar Frost King is vanishing into a slit in the air just above the street about a hundred yards away. The fog sucks in, the dark blob glides into a portal, the slit vanishes and noise returns to the world.
Sort of. Jo's crying but it sounds like she's doing it in a paper bag beneath a pile of blankets.
One day, in the field near the abbey, a cow head-butted me in the stomach because I freeze-framed into her and woke her up, startling her. I feel exactly the same way now: I can't get a breath into my lungs. I keep trying to inflate them but they stay flat as pancakes glued together. When I finally do manage to breathe, it's with a great sucking wheeze that sounds hollow and wrong and it's so cold it burns going down.
I stare bleakly down the street.
They're all dead.
Every last one of them is dead.
Chester's topside is a sculpture of frozen statues shrouded in ice and silence.
"Feck, no!" I explode and wail all at the same time.
Where, moments ago, people were talking and singing, worrying and planning, living, for feck's sake, _living_ , not a spark of life remains. Every man, woman, and child we were standing in the middle of is dead.
The human race is down by another few hundred.
Hoar Frost King: 25. Human Race: 0.
Dublin's going to be a fecking ghost town if this keeps up.
I stare. White bumps and knobs and pillars, folks are coated with hoar frost then glazed with a thick shiny layer of ice. Icicles hang from their hands and elbows. Breaths are frozen plumes of frosted crystals on the air. The cold the scene radiates is painful, even from here, like part of Dublin just got dunked into outer space. Kids are frozen, huddled around the fire cans, warming their hands above them. Adults are frozen, arms around each other, some swaying, some clapping. It's eerily silent, too silent. Like the whole scene is heavily baffled and all noise is being absorbed.
Beside me, Jo is crying soft and pretty. It's the only noise in the night, heck, it sounds like it's the only noise in the whole world! Figures she even cries like a dainty cat. Me, I blubber like a snot-nosed hound with big wet, gulping sounds, not tiny sighs and mews. I stand in silence, shaking, gritting my teeth and fisting my hands, to keep from blubbering.
I retreat like I do when things are too much for me to deal with. I pretend they aren't people under all that milky frost and ice. I refuse to let what happened touch me because grief isn't going to save Dublin. I pretend they're puzzle pieces. Nothing but evidence. They're the way to keep it from happening again, if I can interpret the clues they left. Later, they'll be folks to me again, and I'll make some kind of memorial here.
They just wanted to get warm.
"You should have let them inside," I say.
"Speculate why it came to this spot at this moment." Ryodan says.
"Speculate, my ass. Dude, you're colder than they are! And ain't that the million-dollar question?" I can't look at him. If he'd let them inside, they wouldn't be dead. If I hadn't stood there arguing about stupid stuff and spent more time talking him into letting them inside, they wouldn't be dead. I shiver and button the top button of my coat, right up under my neck, and scrub frost from the tip off my nose. "Do our voices sound wrong to you?"
"Everything sounds wrong. This whole street feels wrong."
"That's because it _is_ wrong," Dancer says behind me. "Massively wrong."
I turn. "Dancer!"
He gives me a faint smile but it doesn't light up his face like usual. He looks tired, pale, and there are dark circles under his eyes. "Mega. Good to see you. I thought you were coming back." He looks at Ryodan then me with a quizzical expression.
I slice my head once to the side and shrug. Last thing I want him to do is bring up that I told him Ryodan was dead. He reads me well, like always. Later we'll chew over how the heck Ryodan survived a gutting. "I _was_ coming back—"
"No, you weren't," Ryodan says. "You live at Chester's now."
"Do not."
"I had to go to somewhere," Dancer says, "and thought maybe you came looking for me but missed the note I left."
I try to flash him a grin that says how happy I am to see him but it comes out wobbly.
"Me, too, Mega."
I do grin then, because we're always on the same wavelength.
"She lives with me," Christian says from somewhere above us. "I'm the only one that can take care of her."
I look up but don't spot him. "I take care of myself. I ain't living with nobody. Got my own digs. What are you doing up there?"
"Tracking the Hag. Trying to devise a way to trap her. She's fast but she's not a sifter."
I jerk, and look around warily. That's all we need right now. "Is she here?"
"If you brought that crazy bitch near me again." Ryodan doesn't finish his sentence. He doesn't need to.
"I left her south of the city. Knitting. She'll be busy awhile."
There's a sudden, flat whoosh of air and it instinctively makes me duck, hare to a hawk. I think the noise made by the winged fliers of the Wild Hunt is branded into a _sidhe_ -seer's subconscious. I'm dusted with black snow. "Christian, you got your wings!" They're huge. They're incredible. He can fly. I'm so jealous I almost can't stand it.
He cocks his head and looks at me. I don't see anything human left in his face at all. "Don't say it like it's a wonderful fucking life. You didn't hear any bells tinkling. What you heard was the sound of a demon, not an angel, recently born. And like any other newborn, it needs colostrum." He gives me a look that I think is supposed to be a smile. "Och, and you, sweet lass, are mother's milk."
All the sudden he looks like the most gorgeous, hunky dude I've ever seen, and I blink. He's standing there, nearly six and a half feet of black-haired, bronze-skinned Unseelie prince with gigantic wings, terrifying iridescent eyes, and brilliant tattoos moving like a storm beneath his skin, but I'm seeing a good-looking Highlander. Sort of. This is new. This isn't a blast of his death-by-sex Fae nature. This is a controlled...
"You're throwing a glamour!" He hits me with a blast of eroticism that almost buckles my knees. He's learning control, fast. Way too fast for my comfort. I reach for my sword. "Off it!"
"For you. Today. Not always. And remember who gave you that back, lass."
"Touch her, I'll cut off your wings and use them to sweep the floor at Chester's," Ryodan says.
"Oh, I'll touch her. And when I do, you won't be able to do a bloody thing to stop me," Christian says.
"Nobody's going to be touching me," I say. "Unless I say so. I'm not public property."
"What is _wrong_ with all of you?" Jo says. "People just got murdered in front of us and you're all too busy arguing to—"
"Humans," Christian cuts in. "Waste of space anyway." He looks at Ryodan. "You're alive. Pity. I was hoping the Hag did you in for good."
"Not a chance."
"You should have let them in," Jo says to Ryodan. "Then they wouldn't all be dead."
"Don't tell me what to do," Ryodan says, soft.
"She's right," I say. "You should have let them in." The flash of hurt in Jo's eyes makes me mad. "And don't you snap at her."
"Right, dickhead," Christian says. "You should have let them in." When I give him a look, he shrugs. "Being supportive, lass. Part of a healthy relationship."
I roll my eyes. "We're not having a relationship and I don't need your support."
"If I'd let them in, the thing might have come inside after whatever it was that drew it to them in the first place, and iced the whole fucking club," Ryodan says.
He has a point but I'm not about to admit it. "Don't you snap at her," I say again. "You be nice to Jo."
"I can take care of myself, Dani," Jo says.
"Difficult though you all might find it to believe," Dancer says, "we've got bigger problems than your egos. Listen up. We need to talk. Let's go inside. It's bloody cold out here."
Ryodan looks at him hard a sec and I can tell by the look on his face he doesn't like what he's seeing with his weird X-ray vision. "Whatever you have to say can be said here. Now."
"You're such an asshole," Dancer says. "Periodically I suffer the brief delusion you might wise up. Brief."
Jo and Christian look at Dancer like they think he must have a death wish. I snicker but keep it under my breath. Ryodan looks majorly pissed and I'm in no mood to be noodled over a shoulder. I want to hear what Dancer has to say because for him to hunt me down, it's important. I look back at the iced scene and sober in a hurry. All those folks dead make me feel sick to my stomach. They died in a second, for no reason. Death is bad enough. Dying for nothing adds insult to injury.
I look at the ice sculpture. This evidence is as fresh as it's ever going to be. The morning all those Unseelie got iced at Dublin Castle, I didn't get to examine the scene. I want to get as close as I can today, without freeze-framing because that night in the church when I got bumped down into slow-mo and almost died, it seemed I could feel things better.
I move down the street, knowing they'll follow: Dancer because he wants to tell me stuff; Jo because she's... well, Jo; Ryodan and Christian because they got some kind of ownership issues with me like I'm a supercar they got the title to. They're so deluded it's laughable.
I open my _sidhe_ -seer senses. I'm nearly suffocated by a feeling of... wrongness. Like the stuff that got iced is missing some essential ingredient, like they're no longer three-dimensional, just cardboard cutouts stood up in the street.
"Talk, kid," Ryodan says to Dancer.
I know Ryodan irritates him because he makes it clear he's talking to me. "After you left, Mega, I sat there for hours, staring. I knew I was missing something. I wasn't looking at things right. I started thinking about how I came to Dublin last fall to check out Trinity and see what I thought of their Physics Department. I wanted to know if I liked their professors and labs, if they had good enough equipment for the kind of research I planned to specialize in. Not that any of that's relevant anymore. It's just a hobby now. I never got around to checking the place out because two days after I arrived, the walls fell and going to college became a moot point."
"For fuck's sake, do you think I care who you are," Ryodan says.
"Dude's as bad as you say, Mega," Dancer says.
I stop about fifty feet from the frozen folks and look around. Jo and Dancer stopped about ten feet back and are shivering miserably. Ryodan and Christian flank me. I'm pretty sure Ryodan could go farther than any of us but he doesn't. When I exhale, my breath frosts in a suspended plume. My bones hurt with cold and my lungs burn. I can't make it another step without freeze-framing. I shiver, taking it all in. What element is present at this scene that was also present at every other scene that got iced? The answer is right here, staring me in the face, if I can peel my preconception-blinders back and see it.
There's wood, plastic, metal, and dirt everywhere. But I know it's not that simple.
There are no mirrors. No tapestries. No walls. No carpet. No real furnishings of any kind. No Unseelie. Pretty simple scene, really. Folks huddled around fire cans to keep warm. Was there fire at the other scenes? Like the ugly Gray Woman that's drawn by the one thing she was created without—beauty—is the Ice Monster drawn to the warmth it can never have? "So you finally went and checked the college out?" I say.
"Yep. I went to their optical analysis lab. Place is a dream. I wanted to know what was happening to the stuff that got iced on a molecular level. Why it was still cold. Why it felt wrong."
I consider and discard the fire theory swiftly. Off the top of my head I can think of five scenes with no fire present. I dredge my memories, find the file where I put the reconstructed images of the scenes and slap them up on an imaginary screen inside my head. I flash through them as I listen, back and forth, breaking them down, analyzing. "What did you find?"
"Trinity was pretty much untouched. Seems people don't pilfer things that don't address immediate needs. I padlocked everything I wanted for myself before I left. They have ultrafast Femtosecond laser systems! The setup is sweet. Pretty much everything I ever wanted to play with is there. Dude, they've got an FT-IR connected to a Nicolet Continuum Infrared Microscope!"
"Dude," I say appreciatively, though I have no idea what he just said. I look beyond my mental screen at the scene in front of me again, wondering if these folks saw it coming, too, like a lot of the others did. They must have. Beneath the ice their mouths are open, faces contorted. They were screaming at the end. Soundlessly but screaming all the same.
"With enough generators running, there's no kind of spectroscopy I couldn't perform," Dancer says happily.
"What the hell is spectroscopy?" Christian says.
"The study of the interaction of matter and radiated energy," Dancer says. "I wanted to excite molecules so I could study them."
"How... exciting," Ryodan says.
"I prefer to excite women," Christian says.
"It excites the feck out of me," I say. "Don't make fun of Dancer. He can think circles around you. He could probably figure how to excite _your_ molecules and short them out."
"Excitation," Dancer says, "can be accomplished by a number of means. I was specifically interested in temperature and velocity, curious about the kinetic energy of our ziplock detritus. I thought the base state of the atoms might tell me something."
Got to love a dude that says things like "kinetic" and "detritus."
"What's kinetic energy?" Jo says.
"Everything vibrates, all the time. Nothing is motionless. Atoms and ions are constantly deviating from their equilibrium position," Dancer explains. "Kinetic energy is the energy an object possesses due to its motion."
"Sound is a type of kinetic energy," I say. I've often wondered about the properties of my ability to freeze-frame, why I can use energy the way I can, where I get it from, how my body manufactures it but another person's doesn't. I'm fascinated by the different types of energy, what they can do, how everything around us is constantly in motion on some minuscule level. "When a guitar is strummed, molecules are disturbed and vibrate at whatever frequency they'll vibrate at under those circumstances. Their kinetic energy creates sound."
"Exactly," Dancer says. "Another example of kinetic energy is when you crack a whip in one of several specific modes of motion, the sound it makes is because a portion of the whip is moving faster than the speed of sound, and it creates a small sonic boom."
"I didn't know that." Now I'm jealous of a whip. The speed of sound is over seven hundred miles per hour! I don't make sonic booms. I want a whip. I like the idea of walking around making sonic booms everywhere. I can't believe he never told me this before.
"This better be going somewhere," Ryodan says.
"It is," I say. "Dancer doesn't waste time."
"He's wasting mine."
Something's gnawing at the edge of my brain. I'm relieved to realize these folks died quickly and without suffering, because I just calculated the most likely trajectory of the Hoar Frost King's path, from where I saw it disappear, and I was wrong about my first assumption. There's no way these folks saw it coming. None of them were facing the direction it came from. They died instantly, with no awareness of what killed them. I'm relieved. Unlike me, most folks don't seem to want to live their death in slow-mo. Mom always used to say she hoped she'd die in her sleep, easy and without pain. She didn't.
"You're never going to believe what I discovered," Dancer says. "I was staring straight at the results and still refused to accept it. I kept checking, running different tests, testing different objects. I went back and grabbed more ziplocks and tested that stuff, too. The results were the same over and over. You know what absolute zero is, right, Mega?"
"Like, where the feck I'm standing?" I say, but I don't mean it, because if it was, I wouldn't standing here. I'd be dead. I frown, studying the scene, trying to make sense of something. If they didn't see it coming, why were they screaming? Did they feel the same suffocating panic I felt at Dublin Castle before it arrived?
"Isn't absolute zero theoretical?" Christian says.
"Technically, yes, because all energy can never be removed. Ground state energy still exists, although laser cooling has managed to produce temperatures less than a billionth of a Kelvin."
"Again, what the hell is your point?" Ryodan says. "Are you saying these scenes are being cooled to absolute zero?"
"No. The only reason I brought that up was to illustrate the connection between extreme cold and molecular activity, and the fact that even at the most extreme cold possible, all objects still have energy of some type."
"And?" Jo says.
"On a molecular level, the debris left by the Hoar Frost King has absolutely no energy. None."
"That's impossible!" I say.
"I know. I ran the tests over and over. I tested multiple samples from every scene. I went to Dublin Castle, dug pieces of Unseelie from the snow and tested them, too," he says. "They're inert, Mega. No energy. No vibrations. Nothing. They're motionless. Deader than dead. The things I was testing can't exist, yet there I was holding them in my hands! Physics as I know it is being reinvented. We're standing in the doorway of a new world."
"So, you think it's drawn by energy, and eats it? Like fuel, maybe it uses it so it can move through dimensions?" Jo says.
Dancer shakes his head. "I don't think it's that simple. Most of the scenes it iced didn't have an impressive stockpile of energy. If it was after energy, there are an infinite number of richer places to fuel up. I speculate the absence of energy when it vanishes is a secondary and perhaps a completely unintended effect of whatever it's doing, tangential to its primary purpose."
I got the same impression with my _sidhe_ -seer senses at Dublin Castle, that it wasn't malevolent or intentionally destructive. I sensed it was enormously intelligent and hunting for something.
"What _is_ its primary purpose?" Ryodan says.
Dancer shrugs. "Wish I knew. I haven't been able to figure that out. Yet. I'm working on it."
"Well, what are we supposed to do?" Jo says, looking around. "There has to be something!"
"Stand around, hoping the bloody thing decides to appear while we're looking, then hit it with whatever we've got handy in the two seconds it's actually here in our world?" Christian says disgustedly. "At least I know what the Crimson Hag wants. Guts, preferably immortal ones." He gives Ryodan a look. "And I know what to use for bait."
"So do I," Ryodan says.
"What are you talking about?" Jo says, looking between Christian and Ryodan. "What's the Crimson Hag?"
I realize she hasn't seen my _Dani Daily_. Nor does she know Ryodan was ever dead. She has no clue her "boyfriend" is immortal. I decide to save that bombshell for the perfect moment. I also decide I'm going to be spending a lot of time with Christian and Ryodan, hoping the Hag comes after them. I let her loose. I'm the one that has to send her back to hell.
Ryodan says to Dancer, "Work faster. Get back in your lab and find me an answer. Dublin's turning into Siberia and the thing just deposited a pile of frozen shit on top of my club."
"At least it didn't ice the door," I say. " 'Cause then we couldn't get back in."
Ryodan gives me a look that says he knows I know the back way in.
"Try a flamethrower," Christian says. "Does the trick. Till everything blows."
"Speaking of which, any ideas what makes the scenes blow up?" I ask Dancer.
"I think it creates a kind of energy vacuum where things get unstable. Like I said, physics aren't working right. It's possible objects reduced to no energy are brittle, and when disturbed by vibrations of objects around them, they explode. The lack of energy may also be the lack of 'glue' necessary to hold matter together. Except in these cases, they're shellacked in ice. Once that shell is compromised, everything comes apart. The larger the disturbance of molecules surrounding the scene, the more violent the explosion. You freeze-framing in to study the scene would generate a significant vibrational disturbance."
Sometimes I miss the most obvious things. How many scenes exploded when Ryodan and me were fast-mo-ing through them and I never put two and two together? I ponder what Dancer just told me, crunch it with a few other facts, mix it all up good to see what I get.
The Hoar Frost King leaves no energy behind when he vanishes. It's stripped from everything he ices.
R'jan said that when the HFK iced places in Seelie, the Fae weren't just killed, they were erased like they'd never been.
Both times I saw the HFK appear, all sound vanished. None of us could hear a thing. Dancer confirmed a third case of similar silence and hollow-sounding aftereffects at the WeCare event he witnessed.
Why would sound vanish? Because everything stopped vibrating the instant the HFK appeared? Why would things stop vibrating? Because it was sucking energy? What exactly is the HFK doing? What attracts it to where it's being attracted? What is the fecking commonality? Until we figure it out, we have no hope of stopping it. We're sitting ducks.
I examine the icy tableau before me. I need answers and I need them now. Before I went into the White Mansion I might have had a little time to play with, but since I've been gone, things in my city have gotten critical. There's too much snow and the cold's getting too extreme, and if the HFK doesn't kill folks, cold alone will.
How many hundreds, even thousands more people will die before we figure out how to stop it? What if it goes to the abbey next? What if it takes Jo from me? What if everybody's generators run out of gas and they all die holed up, alone?
I sigh and close my eyes.
I shiver. What I need to see is right here in front of me. I can feel it. I'm just not looking with the right eyes, the clear eyes that suffer no conflicts. I need a brain like mine and eyes like Ryodan's.
I focus on the backs of my lids, take the grayness of them and cocoon it around me. I make a bland womb where I can begin the process of erasing myself, detaching from the world; the one where I exist and I'm part of reality and everything I see is colored by my thoughts and feelings.
I strip away all that I know about myself, all that I am, and sink into a quiet cavern in my head where there is no corporeality, no pain.
In that shadowy cave, I don't wear a long black leather coat, or skull-and-crossbones panties, or crack jokes. I don't love being a superhero. I don't think Dancer is hot and I'm not a virgin, because I don't really even exist.
In that cave, I was never born. I won't die.
All things are distilled to their essence.
I go inside my head and become that other me, the one I don't tell anybody about.
The observer.
_She_ can't feel hunger in her belly or cramped muscles from being in a cage for days on end. She isn't Dani. _She_ can survive anything. Feel nothing. See what's in front of her for exactly and only what it is. Her heart doesn't break a little every time her mom leaves, and she holds no price too high for survival.
I don't let go of myself and seek her often because once I got stuck there and she took over and the things she did...
I live in terror that one day I won't get to be Dani again.
But, fecking-A, she's one smart cookie! Tough, too. She sees everything. It's hard to see like she does. Makes me feel like a freak. She thinks I'm a wuss. But she never refuses me when I come.
I open her eyes and study the scene. She's a receiver. Things go in and come out. She processes. No ego or id. Nothing but a puzzle here, and all puzzles can be solved, all codes decoded, all prisons escaped. No price too high for success. There is an end and there are means, and all means are justified.
The facts, void of emotion, look completely different.
Folks bang cans. Fist-pump the air. Some clap. Others warm themselves. I pick and discard. I strip to bare essence.
Their bodies are bent and moving in ways that suggest intended, even relaxed motion, not the instinctual, tense muscular and skeletal flexion of panic. Everyone whose mouth is frozen open appears to be making an elongated E. Their eyes are nearly closed and the cords are tight in their necks.
I couldn't see it, but she can.
It's right there, in front of us. It was there the whole time. She thinks it's obvious and I'm stupid. I think she's a sociopathic nut job.
I have my answer but can't rejoice in it because she doesn't feel. I close my eyes to detach but she won't let me. She wants to stay. She thinks she's better equipped than me. I try to leave the cave but she hides all the doors. I visualize brilliant lights in it, like those on top of BB&B. She turns them off.
I open her eyes because I can't stand the darkness.
Ryodan is staring at me, hard. "Dani," he says. "Are you okay?"
He uses a whole, unadulterated question mark, a bona-fecking-fide interrogatory that rises just like a normal person, and that simple thing penetrates. It surprises me the things that rattle her. It loosens her hold on me and I slip free. I guess my sense of humor is more Dani, not her, than anything else about us because when he cracks me up, just like that, she's gone. For a few fleeting seconds I know I'm going to forget her again. I think she makes me forget her and I won't remember until I need her or I get pushed too far.
Then I don't even know that anymore.
I replay all my filed scenes, looking for—and finding—that single commonality it took me so long to see. It was right in front of me all this time but I couldn't drop my preconceptions. I saw what I expected to see and that wasn't what it was at all. "Holy frozen frequencies, Dancer," I say softly. "It's drinking sound Slurpees!"
"What?" Dancer says.
None of them were screaming. All the folks I thought were yelling in fear and horror at the end were _singing_.
The music changes beneath my feet. A heavy metal song just came on in Chester's and the vibrations increase in tempo and intensity. I feel the blood drain from my face.
If I'm right...
And I _am_ right.
There are thousands of people below us, in Chester's, and although I'm not real impressed with their choice of a lifestyle, the race we're in now needs all the humans we've got left.
"We've got to turn it off!" I say. "We've got to turn everything off right now! Dude, we've got to shut Chester's down!"
# THIRTY-SIX
# _"Oh the weather outside is frightful"_
Beyond the frost-etched window of my bedroom, fat snow-flakes drift lazily to the ground. Unlike me, they know no urgency. At the abbey, snow obeys a simple prime directive: fall without cease. It began two days after Sean went to work at Chester's, and has not stopped for twenty-three.
My heart suffers a similar accumulation, with chill piling in treacherous drifts and valleys. Despite our efforts to beat it back, winter claims more of our world with each passing day. Ours has dwindled to paths shoveled between waist-high white walls crusted by ice. I do not know how to navigate this new terrain. I fear my nana's snow goblins lurk in these drifts, waiting to carry off those who stray into the blinding wintry white.
Sean has not been able to reach the abbey nor have I been able to leave for fifteen days. We venture into the countryside with hatchets and saws only to procure timber from hard-iced, felled trees so that we may keep our fires burning bright. We have run dry of gasoline; generators squat in silent reminder of auspicious times we no longer enjoy. We have precious few candles and lack ingredients to make more. If not for the batteries Dani spent obsessive weeks stockpiling as protection against the Shades months past, it is possible we would all be dead, unable to protect ourselves from the amorphous apparitions that may yet lurk within our walls, although we've yet to glimpse one since the night Cruce was interred in his subterranean sepulcher. Some say the Unseelie King took them with him when he left. One can hope.
Night sees us gathered in common rooms to conserve supplies. It is impossible to say when this snow will stop. The sky is night-black or storm-leaden but for an occasional shaft of brilliant sunshine piercing clouds. If we do not soon remove the weight of accumulation from the roof of our chapel, we will lose both roof and interior supports. Ice will crush our altar and drifts will take our pews. Early this morning the rafters creaked and groaned a somber hymn as I prayed: God, grant me serenity, wisdom, strength, courage, and fortitude.
But all is not snow at our abbey. Oh, no.
All is not chill within or without our walls.
My wing of the abbey is a temperate sixty-five degrees, with not one fire burning.
My chambers are nearer eighty, sweltering for one born and raised on the Emerald Isle. I mop my brow and tuck damp tendrils behind my ears. I unfasten the top button of my blouse and dab at my skin.
Beyond the window, the sharp-shaved crystalline fire-world funnel towers over the abbey, glittering bright as diamonds in a capricious ray of sun. Between it and the perimeter wall of my bedchamber, snow is conspicuously absent.
In that narrow boundary, grass grows.
Grass, by the saints, green as St. Patrick's clover! Kelly green as the misshapen shamrock that symbolizes the mission and integrity of our order to See, Serve, and Protect.
Against the crumbling mortar flush to my bedroom wall, sultry flowers in every shade of boysenberry and orchid, cerise and Byzantium bend and sway with blossoms so heavy on delicate stems they droop and nod, deceptively dulcet on a breeze as conflicted as my soul; temperate one moment, frigid the next.
Were I to crank the window and part the leaded glass, the scent that drifted in would intoxicate me. The blossoms reek of spices that make me think of Persian carpets and far-off lands where hookahs are smoked for breakfast and sultans keep harems, and life is lazy, licentious, and short-lived.
But _well-_ lived, Cruce would say.
I blot sweat from my palms and smooth a blueprint on Rowena's stately desk. I must know and I do not want to know if what I have begun to suspect is true.
Although the IFP is tethered to a piece of earth that has been fired to a kiln-smooth, porcelain black gloss, were one to approach it, one would feel no heat. The fire world is contained.
Yet, between the IFP and our abbey grows that loathsome grass despite the snow, that grass upon which Cruce lays me gently back in my dreams, amid fragrant blooms where he makes me feel things for which I despise myself come dawn.
I am not wise in the ways of geography. I know east when the sun rises. I know west when it sets.
Rowena protected many secrets, clanking keys in the bracelet of power that remained on her wrist, held over our heads, until the day she died. I discovered a cache in her bedchamber four nights ago when, desperate to resist another torturous slumber, I occupied myself by studying every inch of the Grand Mistresses' apartment, seeking telltale clues of false panels or retractable floorboards. In the faux bottom of a centuries-old armoire I found maps, sketches, and plans, many of places that baffle me, in which I am unable to divine her interest.
Also therein I found blueprints of the abbey on scrolls and bound in large flat volumes, both Upstairs and Underneath. It is the blueprint of the subterranean chamber and adjoining passages wherein the _Sinsar Dubh_ was once entombed, over which I now place the transparent sketch I have prepared of my wing.
I smooth them together so they meet, corner-to-corner, and press my tongue to the roof of my mouth in silent protest, a technique I perfected when young to keep from crying out when lambasted by another's intolerable emotion.
Cruce's chamber is beneath my bedroom!
Begging the question: does the false summer that makes grass grow and flowers bloom come from the fire world adjacent or the iced prince below?
I decide maybe I can stand Ryodan, at least today, because when I say shut Chester's down, the dude doesn't even ask me another question!
He skirts the ice sculpture's perimeter and heads straight for the metal door in the ground. The ice ends some fifteen feet from it, about which I'm real glad because the back way in that I'm not supposed to know about is a long way from here. Takes a lot of underground navigating. And knowing him, since he heard I knew of it, he probably shut it down and had his men make him another one. But I'll find that one, too. It's like a game with me. Him trying to hide stuff just makes me more determined to find it.
I follow, happy he takes my word for things. Jo and Christian sure don't. They're behind me, peppering me with questions that Dancer isn't answering either, I think because he's still busy putting together all the ramifications of what we just figured out. Either that or he's as obsessed as I am about getting every single thing in our general vicinity turned off ASAP.
I'm still missing a few facts that I don't think I can gather since the scenes all blew up. Speculation may be all we got to work with. I know the Hoar Frost King likes ice cream but I don't know what flavor. And I'm pretty sure he's picky. Or else we'd all have been iced months ago.
I follow Ryodan to his office, where he cuts the power to the subclubs. With each tap of the computer screen, one more subclub dies and it's all I can do not to hoot and holler, especially when the kiddie subclub goes dark and still.
Lights dim. Music stops.
People—the fecking sheep who should have pulled their heads out of their asses weeks ago and banded together to save our city—protest vociferously. Some just keep dancing like nothing ever happened, like they're hearing music in their heads.
Others shrug and get back to practically doing the dirty on the dance floor, clothes half off, like everybody wants to see their Baby-Roach-slimmed butts!
"Can I talk to all the clubs at once?" I say. "You got some kind of PA system in here?"
He gives me a look that says: nice try, like I'd ever let you address my patrons en masse.
I snicker. He has a point. I could rant at these folks for hours. "You got to explain," I say. "They need to understand what they're up against. You got to tell them about the Hoar Frost King and that they can't go outside and make noise or else they might die. And you got to tell them how the scenes explode, so if anybody leaves they don't do nothing stupid with the frozen folks up there and get all cut up on shrapnel! And don't forget to tell them that even in here they need to stay as quiet as they can and—"
Ryodan presses a button on his desk. "There will be no lights or music until further notice." He releases the button.
"That's it?" I say. Fecking good thing he ain't writing the Ryodan Rag! Through the glass floor I watch folks rustle angrily. Many are drunk and don't like this new development. They want their bread and circuses. That's why they come here. "Boss, what the feck was that? Maybe you could, like, tell them not to leave or they'll die?"
He presses the button again. "Don't leave or you'll die."
There's a pregnant hush then, like they all think he's God or something, and folks and Fae stop everything they're doing and sit down. Only after a long moment do they begin talking again.
"I think you should lock the doors," Jo says. "Don't let them out for their own good."
"I'd prefer they leave. Less to draw it here."
"If you want me to tell you what to do to keep this place safe," I say, "you better keep _them_ safe."
"I thought you were disgusted by the people that come to my club."
"They're still people."
He presses the button again. "If you go outside, you will be killed. If you make noise, you will be sent outside. Don't piss me off."
Just like that, Chester's goes completely silent.
#
# THIRTY-SEVEN
# "The sound of silence"
I call my _sidhe_ -seers to gather in the chapel beneath creaking eaves.
Our sanctum could once scarce contain the half of us. Seated now between marching rows of majestic ivory pillars, those who remain are swallowed in voluminous, echoing silence save the groaning of rafters and the hollow resonance of my footfalls as I walk the center aisle that leads to the sanctuary at the liturgical east of the church.
Dull, despairing eyes follow my progress. My girls occupy the front eleven pews in the nave. The ghosts of cherished friends fill the rest. It was a hard winter followed by the tease of stillborn spring.
Now this incessant snow!
I feel stronger in the chapel.
Here, the divine defies the devil at our door. Faith is an unquenchable flame in my heart. Although twice Cruce has followed me here, these hallowed floors remain inviolate. He has not been able to enter.
Reliquaries of polished ivory and gold, adorned with precious gems, attend the altar. More are sheltered at shrines where once candles flickered, until we were obliged to purloin them for other purposes. These urns and boxes hold sacrosanct bones and bits of cloth from saints canonized not by the Holy See but a more ancient church. I suffer no conflict that they reside beside acceptably venerated bones. Bones are bones and good people are good people. I beseech them all to watch over us in our time of need.
I enter the raised chancel in the sanctuary and approach the lectern. We have no power for the microphone but it is no longer necessary, as my voice will carry clearly to the few occupied front rows.
Two hundred eighty-nine of us remain.
I would weep if I had tears but they are drained dry each dawn when I awaken, exhausted, stained by semen that is not mine by right and guilt that is. Semen from one who has just dipped his fingers in the stoup of holy water and now traces a cross at his forehead, his lips, his heart!
He violates my sanctum. He mocks my rituals.
His fingers do not burst into flame nor is he struck by bolts of celestial retribution and banished to hell as Satan should be. I believed him barred at the door. Was he amused to deceive me or has he gained strength to project himself?
He winks at me as he walks the center aisle. Near the rood screen he pauses and unfurls his wings.
Dark angel. Black-winged and black-souled.
In my church.
_In my church!_
The girls rustle. I become aware my gaze is fixed on Cruce, exquisite, naked Cruce, standing in the center of my chapel, wings spanning the aisle, stretching half to heaven, and my first emotion is panic. I must not let them know I see him or Margery will stand in my stead!
I sweep my gaze over the pews and lower my barriers so that I may know the state of their hearts. I've been muffling their emotions for months, for they have known such anger, grief, and fear of late that I cannot tender the daily inundation.
Anxiety slams into me. Shame steals my breath. I press shaking fingertips to the hollow of my throat as if to release a catch hidden there that controls my inhalations.
I see clearly for the first time in more than a month.
If I am the only one who sees Cruce, I _should_ be deposed.
If I am not, if others see him, too, and I have kept my silence this long, I should be damned.
For what is war renowned?
He divides. He carves down the middle and makes enemies of even brothers and sisters, parents and offspring. War has been dividing my family since birth. Perhaps, indeed, he has been paying me uncommon attention.
How best to divide?
Sean's cousin Rocky kept a watch of gold and diamonds etched with his credo. He vowed, despite education, pedigree, or wealth, all prey fell indiscriminate to this simple strategy: isolate the mark.
Silence is the ultimate isolator.
Have I played into his hands?
He stands smugly certain of me, assured of our private complicity. How pleased he must be when each morning I remain an isolated berg in this winter that has claimed our world!
I turn back to the women in my care. "Who among you sees Cruce standing in the aisle?"
Ryodan calls a meeting in one of the rooms on the second floor. I never seen such quiet in the club. Folks sit alone, not talking. The lights are dim and all music is off. I can't feel the tiniest vibration in my feet. A soft glow radiates at ceiling and floor level. He's got some kind of illuminated tubing behind the moldings. I always assumed he had giant generators somewhere and I just couldn't feel the vibration over the pounding, incessant music. If not generators, what's keeping the lights on?
"Dude, I thought you were turning everything off."
"Everything is off."
"What's powering the lights that are still on?"
"The bulk of Chester's runs on geothermal power."
I smack myself in the forehead with the butt of my palm. Of course. He's got all the best toys. Why wouldn't he dig all the way to the center of the Earth and harness planetary power? The dude, like, lives forever!
Me, Jo, Dancer, and Christian are joined by six of Ryodan's dudes. Every time Jericho Barrons doesn't walk into the room with me, I heave a sigh of relief. One of these days it's going to happen. It's inevitable. And one of these days it will probably be with Mac at his side. S'cool. I've lived most of my life under threat of "one of these days" for one reason or another. Superheroes do.
Ryodan sends three of his men down to the club to keep order, and sends the other three into the icy day to track what noise they find and shut it down. Jo tempers his orders with: "And bring any people you discover back to Chester's so we can keep them alive."
I watch him real careful when she adds to his commands like she has the right. Like she's his girlfriend and they're a team, out to save the world together or something. We'll see if his dudes obey her. If they come back with a band of ragtag survivors, I might just be impressed. I can't read his face. It's like he's got it totally closed to me.
He refuses to let me fire up a press and get a _Dani Daily_ out. I argue but Jo makes a point: nobody is venturing out unless they absolutely have to anyway, so the time wasted printing and posting would be better used bringing everyone up to speed so we can make a plan. When did she become Ms. Voice of Reason? Oh, and Glam Girl! When she slips off her coat and unwinds her scarf, her boobs aren't sparkly but she's sure got a push-up bra on!
"Sound Slurpees? Dani, what's going on?" Jo says.
"It's being drawn by music," I say. "At first I thought it was attracted to singing, but it's not. It's a component of music it's after. Sound waves. Frequencies. Who knows, maybe a single note. And the sound doesn't need to be made by a person. It can come from a stereo, a musical instrument, church bells, a car radio, even an Unseelie screaming a note high enough to shatter glass."
"Like at Dublin Castle, the night it iced the cages," Christian says. He's been quiet but I can feel temper rolling off the dude. He's barely keeping his cool.
"Exactly. Or it could be drawn by the chiming of crystal bowls."
"The fitness center," Ryodan says.
"Right. Or playing a washboard, banging on a pot and singing."
"The Laundromat folks," Dancer says.
"And the weird wire contraption around the dude's head wasn't a medical device for an injured neck. It was a harmonica holder," I say. "With their primitive band, the small family managed to make whatever noise draws the Hoar Frost King."
"The band in my subclub must have made it, too."
"So why didn't it ice the entire club?" Christian says.
"I'm guessing it's drawn to a specific sound. The same way I like Life cereal but not Chex. They're both little squares of crunchy goodness but they sure as feck ain't equal to my taste buds. And all the audio equipment in your warehouse must have been hooked up and turned on. At the church where I almost died, they were singing and playing the organ. At all the underground pubs there was a band or a stereo playing."
"The WeCare folks were singing and playing the organ, too," Dancer says.
"So how do we figure out what noise it likes?" Jo says. "All the scenes got blown up, didn't they?"
"I don't think we need to," Dancer says. "We just need to set up somewhere and make an enormous variety of sounds. Wait for it to come."
"Great idea, kid," Christian says. "Then we all bloody get iced!"
"Not necessarily," Ryodan says.
"What do you mean? What are you thinking?" Jo's sloe-eyed puppy-dog expression says she thinks he's the smartest person she's ever met. Gag me! Dancer's the smartest person she ever met, and I'm second.
When he tells us I just shake my head. "It won't work," I say.
"Actually, Mega," Dancer says, "it might."
"Bull-fecking-crikey. He's assuming a lot of things."
"I think it's worth a try," Dancer says.
"Are you defending him?" I say.
"Only the idea, Mega."
"Are you sure you can pull this off?" I ask Ryodan. "You know how many things could go wrong?"
Ryodan gives me a look.
Jo's gone white. "You're crazy. You're talking about setting one monster free to destroy another."
"The world is turning to ice," Ryodan says to Jo. "If this continues, the Hoar Frost King will finish what Cruce started: the destruction of the world. Sometimes you plug the hole any way you can, and worry about fixing the boat later. If the choices are sinking today or tomorrow, I'll take tomorrow."
Him and me think alike a lot of times. I'd never tell him that.
To me, he says, "You and the kid get what we need. I want to be ready by nightfall."
I am blasted by the crimson complexity of Margery's rage.
She surges to her feet to demand my immediate resignation as Grand Mistress, but before she can incite the hue and cry upon which she so thrives, one by one heads bow and hands rise. White flags of surrender are hoisted until each woman has her arm above her head save one. My cousin reclaims her seat in the pew, fists clenched in white-knuckled balls on her lap.
I open myself with a tight, narrow focus. Her fury is bottomless, directed in its entirety at me. She believed she was his only one. She castigates _me_ for the wanton ways of our enemy. She is a fool in too many ways to number: in affairs of infidelity, if a man strays, it is not the fault of the woman with whom he lays. A worthy heart eschews temptation, despite the magnitude. Clearly my heart is not worthy.
I dismiss her and regard my girls with regret and resolve.
In my silence, I failed my charges. It was not merely myself I isolated. I cut them off from one another.
"Did any of you tell someone else?"
I hear no replies and need none. I can tell from their faces that not one of them spoke of it. We became a group of close-huddled islands in our shame, eating and working and living side by side, in complete disconnect. For more than a month each of us waged the same hellish battle, and rather than sharing that burden, suffered it alone.
"We permitted him to separate us," I say. "It was exactly what he wanted. But it is over. We have called his bluff and are now united against him."
Cruce's enormous wings rustle. It is the only sound I have ever heard the projected image of him make. Oh, yes, our enemy is gaining strength with each passing day!
Again I wonder if it is Cruce or the presence of the IFP that causes the grass to grow. If it is the IFP, might its location above Cruce's cage also be weakening the integrity of those icy bars? I have not permitted myself to visit his chamber since last Sean and I made love. Failing my soul mate to anchor me, I risk nothing.
Did this clever, clever prince devise a way to summon a fire-world fragment to set him free? Were I to make the long descent into the bowels of this abbey today, what would I find?
Darkness, moss, and bones?
No bar where once one was?
"Must we leave the abbey?" Tanty Anna exclaims. "Is it the only way to escape him?"
"It's our home! We can't leave!" Josie protests.
"Where would we go? How would we get there? Dog sleds?" Margery says.
"There aren't any dogs left. The Shades ate them all," Lorena says.
"That was a joke. The point is we can't leave," Margery says. "Under any circumstances. This is our home. I will let no one drive me from it!"
Again I turn a tight focus on her. She wishes we would vanish, doesn't care the how or why of it, so long as she gets him to herself. She has been in no way dissuaded by the fickleness of his affection.
I dab at my neck, my brow. The temperature in the chapel is rising. I smell blossoms, spicy and sweet.
I cannot move Cruce. But I can and will do something about the IFP.
I must find a way to contact Ryodan and his men. He already has my Sean. What more can he thieve from me?
We will move the fire world, send it back the way it came, and I will have my answer, if the grass dies. Fire world or ice prince; which is overheating our home? Did the Fates cackle when they stitched together the tapestry that froze our greatest enemy in our basement then parked a heater above it?
I do not believe fragments of Faery are one-way.
If it can be tethered, surely it can be towed.
# THIRTY-EIGHT
# _"Burning down the house"_
Our exodus from Dublin is a somber one.
It isn't easy to leave the city. It takes a small army of us to battle our way out.
Before we go, we set up sound decoys at the north, south, and west edges of the city, in abandoned neighborhoods where nobody hangs anymore. Dancer hooks them up, broadcasting from a central radio source. Even Ryodan is impressed, making me über-proud Dancer is my best friend! Hopefully it'll be enough to keep the Hoar Frost King from being drawn to all the noise we have to make in order to escape the snowy prison Dublin has become.
I make a quick pit stop in the Cock and Bull tavern and take something off the wall I been dying to have ever since Dancer mentioned it. It's the only place I could remember seeing a whip, mounted like art next to a set of giant bullhorns. I got no doubt it'll come in handy somehow. And if not, so what? I can't resist making something move faster than the speed of sound. Sonic booms are _so_ going to be me!
Truck engines roar, scraping a path so Humvees and buses can lumber between snowdrifts piled in enormous banks, iced solid as rock. Streets are impassable with the stuff, and still it falls, landing thick on our windshields. We got dudes up front driving snowplows and trucks that scatter chunks of salt. I got no clue where they found the equipment. We don't get this kind of snow. Knowing Ryodan, he's got all of it tucked away in a warehouse somewhere, prepared for any and every eventuality, even the seemingly impossible.
Got to admit, I like that about him. I'm used to feeling like I'm the only one sees the hard things coming, and I'm always angling to skew the odds in my favor. It's nice to know somebody else is preparing, too.
He's right. The hole has to be plugged because the boat is sinking. Another few days and I'm not sure our exodus would be possible. We'd be iced in. I hate the plan we're about to put in motion but we got to do it. Sometimes when all hell's breaking loose the only thing to do is to break more hell loose.
Before it's too late.
When we get to the abbey and tell her what we're going to do, Kat's going to have a total meltdown.
Night brings a violet aurora borealis to our home. Aubergine and gentian flames flicker on shiny ice-capped snow as if on the swells of an alabaster ocean.
We gather at the windows of the common room to watch the dance of violet vapors. I am appalled to realize how much time I've spent in my chambers the past month, so as not to betray Cruce's visitations. I did not see we were all going off alone for similar reasons. Our abbey had become hauntingly quiet and lonely with me, their leader, unaware. I will never again permit myself to forget that isolation is the first step to defeat.
Tonight our unwanted visitor is conspicuously absent. It is the first evening in weeks he has not dogged my step. He knows we are angry and his appearance would only further rile us. Margery, too, is absent. I will confront the wasp in our nest come morning. She and I will reach terms or she will leave.
Tonight we break into our precious stash of corn sealed airtight in jars late last summer, popping it with oil over flame. We make the evening a celebration, warmed by the last of the cider scalded over a fire, spiced with cinnamon and clove. Communion, warmth, good scents in the air, contribute to a feeling of thanksgiving and hope, and we reconnect into the family we once were with new appreciation. Now that we all know Cruce was plying his seduction upon each of us, we are no longer divided by guilt.
When I hear the roar of engines approaching the abbey, I fear for the safety of my girls and bid them retreat to the cafeteria while I see to the door. Three of those who served in the Haven, Rowena's inner circle, refuse to leave, and another three step forward to join them, Tanty Nana at the forefront, her eyes wise in twin nests of wrinkles. They infuse me with courage. I begin to understand the purpose of the chosen inner circle.
The seven of us bundle into cloaks, scarves, and mittens, and step out into the snow. Lavender lights wisp across a twilight terrain, evoking a surreal, dreamy ambience. We watch as trucks with enormous blades carve their way up our white-capped drive followed by four Humvees and two buses.
When Ryodan steps down from the driver's seat of one of the trucks, for the briefest of startled moments I think: But how serendipitous, I can ask him to tow the IFP away!
Common sense asserts itself and my heart grows chill.
Yes, I wanted to see him. But for this man to come here tonight, for him to use machines to bulldoze a path through mountains of ice to reach our home, means we have something he wants.
Badly.
Through narrowed eyes, I regard him. Lack of visible cloven hoof, tail, or horns does not disguise the devil at my door. He glides, long-limbed and sure-footed, through the snow. He is a beautiful man but unlike my Sean the impression is of animal grace, something not human. Coupled, of course, with the fact that he is not really here! No man stands where he walks. I sense nothing. It is shocking. It is sensational in that it is the very antithesis of sensation. Loath though I am to admit it, it is such a relief! I get nothing from him. Never have I been around anyone that affords me such blissful emotional silence.
He takes both my hands in greeting and leans in to kiss my cheek. I turn my face, press my lips to his ear and say softly, "You can't have it. Whatever it is, you're not taking it. The answer is no."
His breath is warm on my ear. "I have come for something of which you'd like to be quit."
I wonder if he always speaks in the manner he is spoken to. The devil is the master of assimilation. It is how he gains entry: he makes himself appear a friend.
"Again, no." I think perhaps we have something to trade. Perhaps I will give him whatever it is he wants for moving the IFP. But best to deny from the onset.
He slides his hands up my arms to my elbows and cups them lightly, drawing us closer. "We could barter."
Does he read thoughts or merely expressions so well? "Give me my Sean back," I whisper. The stubble on his cheek abrades my skin.
"Your beloved Sean has been free to leave for weeks," he murmurs against my ear.
I mask a tiny jerk and swallow a cry of protest. I do not know whether he speaks the truth. If it is a lie, it is a bitter and hurtful one.
"It is not a lie." He lets his hands fall from my arms and steps back. I am colder where he was touching me.
I see Dani exiting one of the buses. The clouds part on my troubled heart and I am suddenly buoyant. Her fiery hair is a halo of sunshine around her glowing, delicate, eternally battered face. Her smile of greeting is infectious. How I have missed her!
I open my arms, knowing she will never run into them as I wish. Knowing any hug I steal from the child will be just that—stolen. Beneath her hardy, bruised exterior shines pure gold. She is filled with light as no one I have encountered before. It makes me both harder and gentler on her. Though she is cross and grumpy and irritable as any teen, there is not one ounce of ill will in her and she has had reason to feel it. Indeed, reasons enough to fill a book, but she radiates only excitement and happiness to be alive. I realize Ryodan is watching me watch her, intently. I wonder again if he can pick up my thoughts, and if so, how clearly?
"Why have you come?" I demand.
Dani skids to a stop on the ice in front of me and blurts in a rush of breath, "Hey, Kat, what's up? Long time, no see, huh? Everything okay out here? You got enough to eat and stuff? Sorry I haven't been around to check on things but I got stuck in Faery. Dude! You're never going to believe all the stuff that's been happening! _Brrr_ , it's cold out here! Oh, and we think we know how to stop the Unseelie responsible for turning our world into an arctic zone! Hey, I'm freezing, you going to let us in?"
We are again in the common room watching out the window as the most peculiar confederacy I have ever seen collaborate upon a shared goal prepares to destroy us.
I cannot see it any other way. They are wrong. It won't work. It's far too dangerous.
Five men who do not exist, one violent, immensely powerful, sex-obsessed Unseelie prince who believes himself in love with Dani, an exceedingly radiant and happy Jo, and a young, good-looking boy with glasses for whom Dani is the sun, moon, and stars and reminds me of my Sean but harbors secrets so dark and deep that even my gifts cannot reach them, work together to unload equipment from the buses and carry it across mounds of snow and ice to the chosen location.
While Dani told me their plan to trap the Hoar Frost King with the IFP, Ryodan remained silent, and with good reason. He knew each of my objections and that there was no valid rebuttal for any of them. At the end, when permission should have been given or withheld by me—and it most certainly would have been withheld—he informed me that if I failed to cooperate in any way, he would destroy the abbey and continue with his plan.
"You're going to destroy it anyway," I said.
"No we're not. It's going to work, Kat!" Dani exclaimed.
"You don't know that. You don't even know if the Hoar Frost King _can_ be killed."
Ryodan's gaze reflected the same odds of success I perceive. He said simply, "How much longer do you think you and your charges will survive if this snow continues."
He has the most jarring way of never punctuating his questions.
They plan to free a monster.
I said, "Assuming it works and the Hoar Frost King is destroyed, how do you plan to tether the IFP again?"
Even Dani had the good grace to look away.
I cannot read Ryodan. I will never be able to. But I can read the rest of them.
Deep down, they do not believe they can.
# THIRTY-NINE
# _"Crystal world with winter flowers turn my day to frozen hours"_
I ain't never been to a heavy metal concert though I seen some on TV. Dancer's been to all kinds of shows. Growing up in a cage had serious disadvantages. By the time I got out, there were so many things I wanted to do that I couldn't get to them all. Now all the good bands are dead, and tonight is probably as close as I'm going to ever get. The violet lights flickering in the sky are perfect for a rock concert, like having our own laser show! I seen some on TV and they were über-cool.
It's crazy how many speakers and cables and stuff Dancer and me picked up. We might have gotten a little carried away. But the music store we looted was untouched and crammed full of equipment, with no windows broken, and a full cash register. I guess in times of war nobody's thinking, Gee, I want to go steal a stereo. In the end we filled both buses, figuring the louder the better.
We set up the sound stage close to the abbey, between the wall and the IFP.
It's freaky working close to it, knowing if anybody jostles you into it, you're instantly dead. Creeps me all kinds of out but I got a job to do hooking up speakers while Dancer gets everything else up and running. The long, wide, scorched black trail behind it is a constant reminder that it would char me to cinders if I so much as touched it. Although the IFP emits no actual heat, no snow accumulates on the barren soil, as if where it passed it has left the earth antithetical to cold.
The faceted funnel is taller than the abbey, at least a hundred feet wide at the top, and tapers to forty or so at the base—more than big enough to swallow one Hoar Frost King. The earth beneath it is baked to a slick, shiny black finish, though the fire-world fragment doesn't throw off heat. A ribbon of glowing wards twist around the base, securely tethered to a black loop on a black box etched with symbols about twenty feet away. I skirt the IFP, eyeing the black box suspiciously, thinking how the feck is that tiny thing that is roughly the size of a Rubik's cube keeping an IFP from drifting? It can't weigh more than half a pound. I kick it gently to see how far it moves and just about break my toe! I can't resist trying to pick it up.
I can't even budge it on the snow!
"What? You got some kind of ultradense metal I ain't never heard of?" I say grumpily, but if he hears me he doesn't respond. How does Ryodan always have the coolest stuff? Where the heck does he get it?
I look back up at the funnel. It's eerily beautiful, crystalline planes and angles reflecting the dazzling purples of the aurora borealis. I will a silent thought to the universe: Let this work. We've all had a tough gig lately. Let there be no casualties tonight.
Kat's outside again, watching us. Ryodan told her she has to move the _sidhe_ -seers out into the snow once we begin. It almost made her nuts! She took it as the equivalent of him saying the abbey was an accepted casualty, but I know him. He wasn't saying that. He just takes the possibles into consideration and knows that trying to move nearly three hundred women in the middle of a crisis is a nightmare. I've tried to move them during times of peace and quiet and had the luck of a broken mirror nailed beneath an upside-down horseshoe with a ladder nearby that a black cat just walked under. Like sheep, _sidhe_ -seers herd by nature, until you _want_ them to go somewhere. Then they're all fluffy bottoms and broken legs.
Since we're just about to start, I guess they're all inside bundling up. Freeze-framing the equipment into place is the only thing keeping me from shivering. Well, that and nervousness about overshooting my mark and ending up a crispy critter might be heating me up, too. Some of Ryodan's dudes are getting fires going, and a few _sidhe_ -seers begin to straggle out and gather around.
I look at Kat walking toward me from the abbey. She looks so slight with her hair blowing back from her face and her body like a reed that could be too easily broken. I worry about Kat. I know she didn't want to run the abbey but everybody insisted. Kat exudes something peaceful and strong that makes you feel at ease even when you probably shouldn't. She says faith is a rock and as long as you have your feet planted firmly on it you can't falter.
"Dani."
"Hey, Kat."
"It's too close to the abbey. Set it up closer to the IFP."
"Can't. When the Hoar Frost King comes, if the IFP is too close to the speakers, the funnel could get iced before we can cut the tether and use it."
"If it's not closer, the Hoar Frost King could appear, ice, and vanish before the IFP even gets to it."
I don't say nothing. I already took that into consideration when Dancer and me did our time calculations.
"Do you really believe this has any chance of working?"
I plug two speakers into a generator and begin piggybacking the connections. "Which part of it?"
"Any."
"Well, I'm pretty sure we're going to end up drawing it here. I don't know the exact sound it'll come for, but we'll make it eventually. By the time Dancer's done, the music is going to vie with arena-rock for sheer decibels. We shut everything down in and around the city and Dancer dropped the signal to our decoys once we got here. If sound _is_ the Hoar Frost King's Scooby-snacks—and I'm not wrong about this—it's going to be starving and we're leaving it only one source of food. I give attracting it ninety-nine percent odds."
"And destroying it?"
I ponder that. I been pondering that all the way out. "I hear the IFP incinerated everything in its path, even boulders and concrete buildings and stuff, right?"
She nods.
"The IFP is part of Faery so it's not like we're trying to incinerate a Fae monster with human fire. We're trying to burn it in fire from its own world. I think that increases the odds substantially."
"But who says fire will win over ice? You said it's not even made of ice. What if it's made of something fire has no effect on? What if you summon the Hoar Frost King here and it ices the IFP?"
I been trying not to consider that possibility. "Then we're all in a world of shit and probably dead, Kat."
She gives me a look.
I flash her a gamine grin. "But then at least we got rid of the IFP!"
She gives me another look.
I spread my hands, palms up. "What do you want me to say? I'm not going to lie to you. You're like Christian. You know anyway."
"You do realize it will require impeccable timing. You have to draw it to a precise location, cut the tether holding the IFP, and hope the Ice Monster gets trapped in the few seconds it's in our dimension. And whoever cuts that tether might get iced."
"Dude, a few secs are all we need! Most of us can freeze-frame and Christian can sift. We're darned fast! We're setting up practically on top of the IFP. The instant the Hoar Frost King appears, we cut the tether, and both sound stage _and_ Ice Monster get swallowed."
Her gaze spans the fifteen yards between the sound stage and the perimeter wall of what used to be Ro's chambers but are now hers. "And so does the abbey."
"We're going to retether it before that happens!"
"Again, that's going to require impeccable timing."
"Again, dude, we all freeze-frame. 'Sides, I heard the IFP wasn't moving all that fast. Ryodan says as long as he's got thirty seconds and gets it done before it enters the abbey, it's a no-brainer."
"And if it reaches the abbey?"
"It won't."
"If it does?" she presses.
"Look, we got at least a minute before it hits the abbey wall. We ain't going to let it take the abbey." I'm not about to tell her that Ryodan said if it entered the abbey he'd be unable to ward it off until it came out the other side. Something about a containment spell only working if you completely enclose on all four sides the object you want contained.
"One minute," Kat says quietly. "Do you realize if it destroys this place, we will lose everything our order has spent millennia gathering? All our books and sacred objects, our history, our home. Do you see the grass and flowers growing up against the wall? Do you realize if the IFP breaches the abbey, it may very well melt Cruce's prison and set him free? The _Sinsar Dubh_ will be loose in our world, walking around in the body of an Unseelie prince!"
"Look, Kat, I ain't saying it's a perfect plan. But if you don't have any better ideas, get out of the way and let us do our job." I look around at the swells of snow, the iced trees, the piles of crusted drifts. "How much longer do you think we'll survive like this?"
She sighs and says, "That's the only reason I haven't stopped you."
"You haven't stopped us because you can't!" I say heatedly. "You're just one person and we're all superheroes!"
"I won't let it take my abbey, Dani. I won't let these women be ripped from the only home they've ever known. Like you, I'm willing to risk a great deal for those things in which I believe."
I watch her as she walks away and think she's starting to worry me a little.
It's nearly eight when our concert begins. We laid out sheets of plywood on the snow to hold the audio equipment and made a second platform a short distance away for the generators we're using to power it all, then a third platform for the music source and so our tushes don't get cold. We put that one far enough away that we don't get iced when it shows up. We build a couple fires and stack wood nearby. My hair and clothes smell like outdoors and wood smoke, and for a sec it makes me feel like I'm on a family vacation or something. All these folks, including six that are fast enough—and I still don't get to have a decent snowball fight!
Me and Ryodan, Christian, and Jo gather on the platform, ready to dart in and cut the tether when it comes.
"Jo shouldn't be here," I say. "She can't freeze-frame."
"I'm not leaving," she says.
"Make her leave," I say to Ryodan. "Unless you want to be responsible for getting her killed."
"Ryodan won't let anything happen to me," she says.
I roll my eyes. "Dude," I say to Ryodan. "Get her out of here."
"She's her own woman," he says. "She can make up her own mind."
Jo glows.
I just about puke. "Fine. It's on your head." Bugger it all. Now I'm going to have to watch out for Jo _and_ worry about everything else, too.
Kat, the _sidhe_ -seers, and a couple of Ryodan's dudes are at the opposite end of the abbey, down by the lake, with fires burning, sitting in total silence. Conversation is forbidden. They're to make no noise whatsoever.
I get a bad feeling looking at them. "You sure they should be so far away?" I ask Ryodan.
"We need to be split up so, worst-case scenario, we don't all get iced."
"Are we ready?" Dancer walks up and joins us on the platform.
"Get out of here, kid. You got no fucking superpower," Ryodan says.
"Sure I do," Dancer says easily. "I'm the one who saves her life when you guys would have killed her. Remember?"
"If Jo stays," I say, cutting off my nose to spite my face, "Dancer stays." Great. Now I got two people who can't freeze-frame that I have to watch out for.
Dancer and me settle back against a couple of extra speakers we stacked up for something to lean against. "Crank it up," I say. "Let's get this party started." I hand Dancer my iPod, loaded especially for tonight's show. I got almost ten thousand songs on it! Motorhead to Mozart, Linkin Park and Liszt, Velvet Revolver to Wagner, Puscifer and Pavarotti and everything in between. I even got show tunes and cartoon soundtracks!
Ten minutes later Lor says, "What _is_ this crap? Who let her load the iPod?"
"Nobody else brought one," I say. "I chose _awesome_ music."
"Where the hell is Hendrix on this thing?" Lor takes it out of the sound dock and scrolls through it, looking pissed. "By whose definition is this music?"
Jo says, "Did you get any Muse? I love Muse."
"If I'd known you all had such crappy taste in songs, I would have brought more earplugs," I say. "Dissing my taste. Like Hendrix is even listenable. And Muse is something you do."
"Well, Disturbed," Jo says, "is something you are."
"And Godsmacked is something you get," Dancer says. "But hopefully not tonight."
"Don't you have any Mötley Crüe or Van Halen?" Lor says. "Maybe 'Girls, Girls, Girls'?"
"How about some Flogging Molly," Christian says. "Dani, my darling, how could you not like the 'Devil's Dance Floor'? And what about Zombie?"
"I got 'Dragula' and 'Living Dead Girl,' " I say defensively.
"Bloody hell, 'Living Dead Girl' is one of my favorites!" Christian says, and grabs the iPod from Lor and starts scrolling to it.
I snatch it and hold it behind my back. "Don't mess with my lineup. Nobody else thought to bring an iPod. That means I'm in charge."
Ryodan takes the iPod from me so fast it's there one sec, gone the next.
"Hey, give it back!"
He scrolls through the playlist. "What's the deal with all the Linkin Park, for fuck's sake."
"Dudes, we need noise. Quit taking the iPod off the dock." Dancer snatches the iPod from Ryodan and puts it back on the dock. "And Mega has a crush on Chester."
"I do not!"
"Do too, Mega."
"He's like, old!"
"How old?" Christian says.
"Like at least thirty or something!"
Lor laughs. "Fucking ancient, ain't it, kid?"
"Dude," I agree. I like Lor.
"You got any Adele?" Jo says hopefully.
"Not a single song," I say happily. "Got some Nicki Minaj, though."
"Somebody kill me now," Ryodan says and closes his eyes.
Four hours later I'm getting a headache.
Six hours later I _am_ a headache, my butt hurts, and I'm low on candy bars.
Eight hours later I'm sick of Nicki Minaj.
Nine hours later I'd give darn near anything for five fecking minutes of silence.
Me, Christian, and Dancer been passing around a bottle of aspirin and it's empty. I got earplugs in my pack but we can't use them because we might miss something and screw up.
Across the drive, way down at the other end of the abbey, the _sidhe_ -seers are wrapped in blankets. Dozing. Because, like, the music down there isn't rattling the bone plates in their skull! I'm so jealous I could spit. Dejected, I eat another fecking candy bar. I hate candy bars.
"You said you were sure this would work," Jo says testily.
I'm beat. I haven't slept in days. I rub my eyes and say irritably, "We may have to stick with it for a while."
"Like, how long?" Christian says, and his voice is weirdly guttural. I look at him. He's staring down past the abbey at the _sidhe_ -seers and the look on his face is pure, sex-starved Unseelie prince. Kaleidoscopic tattoos rush under his skin. His jeans are... wow. Okay. Don't look there.
I realize nine hours is probably the longest he's gone without sex in months. "Don't you be looking at my friends like that," I say. "They're off-limits to Unseelie princes, dude!"
He looks at me and I have to shift my gaze away fast. He's throwing off power like a volcano about to blow. I feel the wetness of blood on my cheeks from a bare glimpse at his eyes.
"How _long_?" he says hoarsely.
"Well, it only ever iced one of the clubs in Chester's. That must mean most music doesn't make whatever sound it's after. If you need to leave and find somebody to... you know, go. But try not to kill anybody, okay?"
He gives me a look. I'm not even _looking_ at him and I can feel it.
"How is that even possible? We've been listening to some of the weirdest shit I've ever heard," Lor says pissily. "How can this thing _not_ want to kill it? It should have been here hours ago! My head hurts. I don't get headaches."
"I'm not going anywhere until you're safe," Christian says to me, real quiet.
"Isn't that quaint. The chivalrous Unseelie prince with the dick of death," Ryodan mocks.
"I'll take that as a compliment," Christian says.
"I'm getting fecking sick of everybody picking on my music!" I say.
"Fine, then I'll just change it," Lor says.
"You touch my iPod, I'll break every one of your fingers!"
"Knock yourself out trying, honey." He scrolls to a new song.
I stick my fingers in my ears. "Gah, I hate Hendrix!"
"Then why do you have it on here?"
"I don't know! I just thought 'Purple Haze' was a cool title, then I listened to it and ain't had time to delete it. Who writes such stupid lyrics? ' 'Scuse me while I kiss this guy'?"
"Sky," Jo corrects.
"Huh? That don't make no sense either. What the feck is purple haze anyway?"
"She's going to delete Jimi," Lor says disbelievingly. "Sacrilege."
Dancer nudges the volume. Up.
"Traitor!"
"Sorry, Mega, but I have to agree with him on this one."
I look at Ryodan like I'm expecting him to help me out or something but he's just sitting there and I see that Jo's sort of snuggled into him under one of his big shoulders and his cuff is gleaming silver at her throat 'cause his arm is around her neck and it almost makes my head pop off and I don't even know why. Like he's a real person or something, with a girlfriend, instead of some savage beast that would pick his teeth with her bones if he felt like it, and she's falling for it and... Oh! I just can't even stand looking at them no more! "This ain't no fecking campfire and cuddle!" I say.
Ryodan gives me his vintage permanently amused look.
I'm so mad I stand up and turn away.
"Don't worry, Mega," Dancer says. "We baited the trap right. The monster _will_ come."
He's right.
Just then it does.
Too bad it's not the one we wanted.
# FORTY
# _"Is it the end, my friend?
Satan's coming 'round the bend"_
The Crimson Hag explodes from the night, slicing through lavender lights on a cloud of putrefaction, tattered hem of her gut-gown snaking out behind her, to the bizarre accompaniment of "Purple Haze." She swoops us then shoots straight up to the highest dormer on the abbey roof and perches there.
We're all on our feet. "How did she find us?" I say. "You think noise draws _her_ , too?"
She sways from side to side, moving only from the waist, creepily reptilian, surveying us with black empty holes where eyes should be.
"I think the bitch is after me," Christian says. "I'm the weakest Unseelie prince with immortal guts. At least for a while yet."
"She's like a bat, isn't she? It's not like we weren't making enough noise. She can't see so she uses echolocation!" I exclaim.
"Don't know, don't care. Let's bag the bitch," Christian says.
"How the fuck do we get past her legs," Ryodan says, and I look at him. I can see he's got a personal itch on to kill her.
I look at Jo when I say, "What? You don't feel like dying again today?"
Then Ryodan isn't standing next to Jo anymore. He's got me and he freeze-framed me twenty feet away before I could even blink. "If the Highlander says something to Jo about that, she'll think he's lying. She might believe you. My men will kill her if she knows. And I won't be able to stop them."
I look at him hard and realize for maybe the first time ever he's telling me a simple truth. "She's not allowed to know you can't be killed?"
"Never."
"Why am I?"
He's gone. Back to Jo. Got his arm around her, protecting her.
The Hag swoops!
It's like some weird rock-opera battle that gets even weirder when the next song Lor cued up comes on and Black Sabbath starts playing "Black Sabbath" at about a gazillion decibels. As if the Crimson Hag ain't disturbing enough, we need that freaky song in the background. Don't get me wrong, I put it on my playlist because sometimes I like to listen to it. But I got to be in a real mood, because, dude, the song makes me feel unsettled and disturbed and pretty much everybody I ever talked to feels the same way about it.
First thing on my mind is Dancer! I grab him and yell at him to hold on to me no matter what. When the Hag swoops us, we duck like we're one big wave then freeze-frame in different directions.
She veers at the last second toward Christian and I see he was right. It's him she wants. But when she just about nabs Lor with one of her bony lances, I realize she'll take anyone she can get those terrible knitting needles on.
We're all freeze-framing or sifting, ducking, and dodging. I'm trying to hold on to Dancer and keep an eye on Ryodan, who's got Jo, and it's making me nuts that she's even here, in the middle of this fight. She ain't got nothing special to protect her except Ryodan and that ain't enough for me.
I can't move fast enough, watch out for her, _and_ hold on to Dancer, so I freeze-frame him to the far side of the abbey and dump him with the _sidhe_ -seers.
"Mega, what are you _doing_?"
"You got no chance against her. I hardly do. Don't get me killed because I get stupid worrying about you!"
He snaps real cool, "Didn't mean to be a liability."
"Well you are, so don't be," I snap back. I'd die if something happened to him.
He shakes his head, disgusted, like he can't believe I'm such a traitor when I'm just trying to keep him safe.
"Take me back. Buy me time to rewire things. We can electrocute her with some of the stuff we brought, snake a live cable around her!"
"We don't even know if electrocution would work! Maybe she'd just suck it up and use it for fuel!"
"We don't know it won't!"
Me and him are nose-to-nose, yelling at each other.
Jo explodes from a blur and stumbles into us. "Hey!" she shouts at what I think is Ryodan's vanishing backside. "You can't just dump me here!"
"Don't you two fecking move!" I say.
Then I'm back down by the tower of sound equipment, where we're all whizzing around, trying to evade the bitch and figure out how to get past her bony lances!
Ozzy wails away. I ain't never heard this song through a hundred speakers and Black Sabbath this loud makes the hair on my arms stand up on end. I feel like I really am at a Black Mass and Aleister Crowley himself might spontaneously manifest. It's funny how songs can make you feel different ways. I wonder if whatever sound it is that the Hoar Frost King collects makes _him_ feel something and that's why he goes after it.
As I zig and zag, I think about how the things that the Unseelie king created turned out so ugly and incomplete when the Seelie are so beautiful and whole.
And I start thinking how all the Unseelie are after _something_ , and it seems to be whatever they don't have. Why would the Hoar Frost King be after sound? Things go totally silent when he appears. Because he takes the sound, or because his mere existence eradicates sounds?
Or is it more complex than that? What if the Hoar Frost King is after what _all_ the Unseelie lack on the basest, most profound level? What if he's the only Unseelie smart enough to go straight for the root of the problem and, unlike the simple-minded Gray Woman who spends her life trying to collect beauty that can never be hers, or the Hag who's trying to finish a gown that can never be completed, the Hoar Frost King is trying to collect the song they were created without? Is it after the Song of Making? Eating chunks of it, bit by bit?
"Duck, you fucking idiot!" Lor roars, and I roll and freeze-frame. Then folks slam into me from opposite sides and just about squish me flat. I hear a couple of my ribs make protesting noises.
"Dudes, get off me!" Christian and Ryodan are both trying to get me out of there. "I lost focus for a couple secs 'cause I was thinking hard! It won't happen again!"
"You bet your ass it won't," Ryodan says.
Then I'm over a shoulder and wind is whizzing through my hair, then I'm being dumped in the sheep pen!
Me! The Mega! Put out to pasture!
"You can't stick me down here!" I say, indignant as all get-out. I freeze-frame back toward the action the second I hit my feet but slam into Christian, who noodles me over a shoulder and tosses me back to Ryodan, who dumps me in the middle of the sheep pen again!
"Stop it!" My ribs hurt. They need to quit noodling me.
"Don't be a liability," Ryodan says, and is gone.
I blink.
"Feels real good, doesn't it, Mega?" Dancer gives me a chilly look.
"I ain't no liability!" I wait until they're all back down the other end then freeze-frame back to the action. I'm a fecking superhero. Superheroes don't sit on sidelines.
The Hag is trying to take out Christian.
And Lor and Ryodan ain't doing nothing to help him! In fact, I can't figure out what they _are_ trying to do. They're working hard to stay on opposite sides of her, one front, one back, and they keep whizzing in, only to get blocked by one of those deadly legs lancing out. They retreat, whiz back in, get blocked, retreat, whiz back in, get blocked. It's a cool, methodical attack, and if they had all the time in the world, it might eventually work.
Might. Eventually.
And so what if it does? How do they plan to kill her? Doesn't look like the best-thought-out plan to me. I don't see no weapons on them.
The Hag shoots, straight up and dive-bombs Christian. He stumbles on ice and goes down.
He sifts out then all the sudden he's right back where he was. Looking startled, like his sift didn't work the way it was supposed to.
That split-second screwup was all she needed.
The Hag's going to get him this time!
And nobody even cares. Nobody's trying to save him.
Black Sabbath sounds more evil with each second, and it's all getting on my last nerve. I yank out my sword and throw it straight at the bitch's head. She hears it slicing through the air, veers sharply to the side and blasts into Lor, who goes flying backward.
Then suddenly she's gone!
My sword lodges in a snowbank. Already my hand hurts from the absence of it.
Christian looks from it to me, his alien, iridescent eyes bright. "You threw your sword for me." He looks stupefied.
I _feel_ stupefied. I never let my sword go. Unlike Mac, I won't share in battle. Ever.
Ryodan has his head down, looking up at me from under his brows in a way I only ever seen him do once before, and Lor looks major pissed.
"Dude," I say, because I got no other clue what to say, "would you, like, toss it back now?"
Christian slides long black hair over his shoulder and flashes me a killer smile. "Princess, I'd build you a fucking White Mansion." My sword slices through the night, alabaster steel flashing violet fire.
"Where the fuck did the bloody bitch go?" Lor snarls. "I want a piece of her."
"No clue," I say, and we all look around warily.
_That's_ when the _sidhe_ -seers start screaming.
# FORTY-ONE
# _"You must whip it, whip it good"_
The Hag couldn't get anywhere with us so she went after weaker prey.
We all freeze-frame or sift. I'm the last one there.
When the feck did _I_ become the slowpoke?
Two _sidhe_ -seers die instantly, guts trailing up into the sky.
After a moment their entrails are dropped back to the snow in a wet glistening tangle.
My jaw locks and I get a muscle cramp in it the size of a walnut. My teeth clamp so hard they hurt.
The Hag isn't even knitting with them. She didn't even want them. She just killed and threw them away like trash!
She wants Christian. And it looks like she's ready to kill every last one of us to get him.
"Get inside!" I shout at the women, trying to herd them back toward the abbey.
_Sidhe_ -seers duck and scatter like a herd of gazelles running from cheetahs. Stupid sheep are supposed to be pack animals and that means, duh, run in a pack!
The Hag swoops and takes two more of my sisters! Blood sprays everywhere and folks are screaming like crazy.
I'm so mad I'm shaking. It's total chaos. Before, it was just us we had to watch out for. Now the Hag is dive-bombing hundreds of helpless humans and I don't know who to help first.
Ryodan's covering Jo, Kat, and a dozen others.
Lor's protecting a bunch of pretty blondes—figures!
Christian has like fifty women around him. I realize he's turned on his death-by-sex Fae lure and it's working like magnet-to-magnets. He's got a second skin of pretty _sidhe_ -seers. I wonder if he did it on purpose for a shield or if it's just taking everything he's got to stay out of her reach and he can't suppress it. If he did it for a shield, I'll kill him myself.
How are we going to kill the Hag? None of us can get close enough, past her lethal legs. Not even my sword is any good. I can throw it, but the bitch is faster than a witch on a quidditch broom! Dancer's idea of trying to snake a cable around her and electrocute her is starting to look like a good one. Too bad we don't have any cables handy down this end.
"Holy sonic booms!" I exclaim. I may not have a cable but I do have something that's long and thin, and Indiana Jones sure made good use of it in desperate times.
I yank out my whip, freeze-frame to the outer edge of the crowd for a good shot, and crack it straight up at the Hag!
It flails limply, puddles back down on my head and I get tangled up in it. I can't even get the stupid thing off me. I swear those black holes in her face regard me with amused contempt. Apparently there's some skill to cracking a whip and I don't have time to learn it. It never looked hard on TV.
"Mega!" Dancer yells. I see him in the crowd, jumping up, waving both hands in the air.
I ball it up, knot the cord around the handle for weight, and toss it to him. He catches it, unties it and snaps it at the swooping Hag.
It explodes within a foot of her lethal left leg and sets off a small sonic boom.
She inhales, a horrific, wet, screeching sound, and rockets straight up into the sky. I don't know if it's because she can't believe something got so close to her leg or if her hearing is so sensitive that the sonar explosion gave her a migraine. Whatever—she doesn't like it one bit.
When she dives again, Dancer goes for her head this time and sets off a sonic boom right next to her ear.
She reels backward and vanishes upward into purple lights.
Me and Dancer beam at each other.
He cracks the whip triumphantly.
But this time it doesn't crack. It makes no sound at all. Not even a tiny little hiss as it slices through the air.
Because, like, all sound just disappeared.
Figures that when the fog finally rolls in, every last one of us is on the wrong end of the playing field.
# FORTY-TWO
# _"Try to set the night on fire"_
I think the reason I didn't feel panic preceding the Hoar Frost King's arrival this time is because I was already feeling too much panic for more panic to penetrate. The Crimson Hag butchering _sidhe_ -seers had me so frantic, I forgot why we were even out in the snow to begin with.
Like, to summon the Hoar Frost King.
And he's here.
And somebody's got to cut that fecking tether because if we don't turn the IFP loose, the Hoar Frost King is going to ice the speakers and vanish and it'll all have been for nothing! Worse still, if it's as smart as I think, it won't fall for the same trick twice. The sentience I feel rolling off it is gigantic. This is no simple-minded Unseelie. I don't know 'cause I haven't seen them all yet but it could be the most complex one the King ever made. I wonder if he maybe swirled a dash of himself into its beaker.
What happens next feels like it happens in slow-mo though I know it doesn't take any time at all.
Ryodan and Lor vanish, fast-mo-ing it to the other end of the field. I look from the _sidhe_ -seers to the slit that's opening, stymied, trying to figure out how to protect the _sidhe_ -seers _and_ cut the tether at the same time. Do I save the women I care about who are standing right next to me or do I save the world? I may be a superhero but I got everyday Joe feelings.
I see Christian and he's looking at me hard. He says without making any sound at all, _You can't do both, Dani, my love_.
_I know that_ , I mouth pissily.
_It's me she's after_.
_Your point?_
He vanishes.
I can't find him anywhere for a sec.
Next thing I see is him standing, just standing there in the middle of the field between me and the other end, with his arms spread wide, head tossed back, wearing a "come and get me" expression.
_What are you doing?_ I scream, but not a peep comes out.
The Crimson Hag swoops.
I jerk violently, like I'm the one that got stuck when she guts him.
She doesn't flay him, though. She pierces him with one leg like he's a shish kebob and draws him up toward her skirt. As she folds him into her dripping embrace he gives me a look. I can't make sense of it. I don't get it. Why did he do that? I don't get it! Why would anybody _do_ such a stupid thing!
As he vanishes up into the sky, clutched in her hideous legs, I shut it out. Refuse to process what he did. I can leave the _sidhe-_ seers behind now in relative safety. I'll think about what he did later.
Assuming there is a later.
I freeze-frame toward the Hoar Frost King. It's major weird not being able to hear a sound. I'm not feeling any vibrations either. At least deaf people can feel vibrations. This is worse than a sound deprivation chamber, it's a sensory deprivation world with the HFK in it.
As I get close, I see Lor and Ryodan are pushing toward the black box in what looks like slow-mo. Both of them are covered with thick white ice that keeps cracking when they move. It's cold like the night I died at the church.
The Hoar Frost King is hovering silently over the mountain of speakers, icing them one by one. It's lingering longer than usual, I guess all those decibels make the food source richer, and I think maybe it's licking chocolate off its fingers.
When I freeze-frame in behind Ryodan he turns and roars silently: _Get the fuck out of here!_
Icy needles spear my lungs with each breath, my heart labors to pump. My head feels heavy and I realize it's 'cause my hair has iced. I toss it, and the stuff shatters, white crystals rain from my head.
_You're not going to make it in time!_ I yell back, eyeing the distance between ice monster and IFP. When it opened its slit and glided into our dimension, it appeared in the worst possible place—between the IFP and speakers, not between the speakers and the abbey. Although it didn't ice the IFP, it's too fecking cold in the vicinity of the box for us to get there to cut the tether.
I look at Ryodan. He can survive this cold like I can't. I don't know why. Guess it's something to do with him surviving a gutting, too. He's always been able to get closer to the iced scenes than me.
But I can freeze-frame in faster for some reason. He gets bogged down when he gets closer to the center of the cold. Like he's trudging though concrete.
I don't pause to think. It's possible, it's the only plan I got, and there ain't no time for second guesses.
I blast into Ryodan's back and force him forward. As we go fast-mo-ing toward the black box, he totally gets my wavelength: I'm his locomotive and he's my shield. I can push us, but he's got to steer and slice.
I feel him yanking my sword from my coat and drive us forward. He ices, and cracks a half-dozen times, shaking off the crystals like a dog shaking water. I die a thousand icy deaths and come to life again. My lungs feel bloody and raw with each breath so I hold it. My bones hurt. I swear my eyeballs have iced in my head. My vision is getting all fractal-like.
Still I push us into the pain because this is _my_ world and no fecking Fae is taking it from me. My mouth is open on a silent howl. Ryodan shakes violently as I force us to the icy epicenter.
He slices down with the sword and cuts the tether.
We're expecting the IFP to move real slow.
Based on the rate of movement Kat documented when the _sidhe_ -seers had been tracking its progress toward our home, it takes about a minute between cutting it loose and the fire-world fragment hitting the far wall of the abbey. Giving us plenty of time to retether it, because according to her figures, we really had at least _two_ minutes.
Her figures were wrong. Way the feck wrong.
Like a redlined supercar with stockpiled torque, the IFP explodes free and smashes into the Hoar Frost King.
I fast-up as fast as I can go so things transpire in the slowest motion possible.
The fire-world fragment swallows the Hoar Frost King.
It engulfs it.
Sound returns.
I hear ragged breaths. Gasping. Somewhere, folks are crying.
It's gone.
The Hoar Frost King is gone.
Just like that.
It worked so well I almost can't believe it. I stand there stunned, feeling wary. I'm not the only one out of sorts. Ryodan's got his eyes narrowed suspiciously. Lor is kind of hunched like he thinks the sky is going to fall on him. I'd snicker—because, dude, it's pretty sad when you can't just take a happy ending for what it is—but we still got major trubs. The IFP is devouring the mountain of iced speakers and heading straight for the abbey.
Kat, Dancer, and the other _sidhe_ -seers are running toward us. "Cruce is below the abbey!" Kat screams. "You've got to stop it!"
Ryodan and Lor begin chanting but I can tell from the look on Ryodan's face he's got no expectation of finishing in time. The ten or twelve seconds we got before it hits the wall isn't the thirty he needs to do the job.
Kat starts screaming at Ryodan because he's not going fast enough, and Jo starts screaming at Kat for screaming at Ryodan because he's doing everything he can. Then all the _sidhe_ -seers get in on it, and since Ryodan and Lor are looking down at the totem cord they're trying to ward, nobody's looking at the IFP and I'm the first one to see what's happening.
I _knew_ it died too easily!
Ice is forming at the base of the fire-world fragment.
The bottom of the funnel is turning blue, crusted with white hoar frost.
The IFP sure swallowed the Ice Monster, but now the Ice Monster is icing the fecking IFP!
As I watch, frost spreads rapidly upward.
"Uh, guys," I say.
"Are you bloody kidding me?" Dancer explodes. "It's coming back _out_?"
Lor looks up. "Aw, shit."
"Motherfucker," Ryodan agrees.
The Hoar Frost King freezes the IFP from the inside out.
I don't know if the fire world is a roaring inferno that makes the sound the HFK likes to eat or if they just had a big battle of fire and ice, and ice won.
But the IFP cracks and hisses, steams and pops, as superfire gets supercooled.
Ice weighs it down and it slows to a stop. As the giant funnel gains substance, it becomes too heavy to drift and crashes thunderously to the ground like an icicle dropping from a gutter, lodging in the snow.
We all just stare at the giant ice funnel rooted in the ground, trying to process the sudden reversal of events. First, the Ice Monster was dead but the abbey was in danger. Now the abbey is safe but the Ice Monster isn't dead.
We didn't succeed in killing it, and virtually everyone standing here that can't freeze-frame is going to die the instant it comes back out.
The walls of the IFP begin to shiver and shake like the Hoar Frost King is trying to find the weakest point to hatch from its icy eggshell.
I narrow my eyes.
Eggshells are delicate. Fragile. But it's not a shell. In fact, the entire interior of the fire world must be _solid ice_ right now.
Which means, at the moment, the Hoar Frost King is completely encased in one of its own ice sculptures.
Trapped in a moment of perfect vulnerability.
Perhaps the only moment of vulnerability it has ever known.
I know what happens when an iced scene gets vibrated.
It explodes.
"Dancer," I shout, "use the whip! Make sonic booms!" To Ryodan and Lor I say, "Freeze-frame around it!" To the _sidhe-_ seers, "Dudes, get the feck out of here now!"
Then I freeze-frame in myself, moving as fast as I can on a nearly empty gas tank.
Dancer cracks his whip and we freeze-frame like maniacs.
The frozen IFP trembles and the surface suddenly blossoms a million tiny fissures.
The ground shudders, then there's this rumble like galaxywide thunder rolling inside the IFP.
All the sudden I hear the most awful noise ever, like maybe all the sounds the Hoar Frost King ever collected erupt in one huge dissonant, fingernails-on-a-chalkboard belch and then—fecking-A, I love being a superhero!—just like I thought they would, the fused monsters explode!
# FORTY-THREE
# _"Celebrate good times, come on!"_
I'm glowing. There's no denying it. Beaming from every pore. I never had such an amazing adventure in my whole entire life, and I've had some whoppers.
We're hanging in the great room at the abbey, warming up in front of fires blazing on three sides. There's a kettle of instant cocoa (mixed with water, not milk) being warmed in the main hearth, smelling up the room like a chocolate factory, and Kat broke out a hidden stash of—stale, but who cares?—marshmallows and a tin of hard-as-a-rock biscuits she's been saving for a special occasion, and some scrumptious, weirdly gelatinous honey. It all tastes like heaven. Every time I eat, I'm acutely aware we might not have any more of this stuff soon.
We won! We engaged in battle against the biggest bad I ever seen and we won. Unlike the last big battle fought around these parts, I was there to see it all go down with my very own eyeballs. I didn't have to hear about it the next day secondhand from folks that were lucky enough to be there. And no all-powerful Unseelie King swooped in and bailed us out at the last sec either. We did it ourselves!
When the IFP holding the Hoar Frost King exploded, splinters of ice went sky-high, ground-low, and every place in between. We all ducked and dodged and grabbed someone slower, freeze-framing for the shelter of the abbey. Still, we're a pretty ragtag lot, all beat up with scrapes and cuts and bruises. There was no avoiding the fallout.
We waited inside until it was quiet for a few secs and it seemed the debris had settled, then headed back out to poke around in the chunks and convince ourselves the threats were really gone. Dancer studied the stuff for a good five minutes before flashing me a grin and pronouncing the debris inert. He plans to take samples back to Trinity's labs but he said he was ninety-eight percent certain nothing was going to rise up from the remains.
"How did you know it would work?" Jo says to me.
"I didn't," I say around a mouthful of sticky honey-slathered biscuit. I lick crumbs off my fingers. "But once I saw the Hoar Frost King was icing the fire world from the inside, I realized it was stuck in one of its own frozen scenes, like a bug in amber. And every time Ryodan and me ever freeze-framed near a frozen scene, it exploded into shrapnel-sized slivers." I shrug. "Who knows? Maybe it would have stayed stuck in there and exploded all by itself in time. But I sure thought it looked like it was coming back out."
"I thought so, too," Lor says, and everybody agrees with him.
"Bloody brilliant about the whip, Mega," Dancer says.
I preen.
"It was close. We got lucky," Kat says.
"Lucky, my ass! You got superheroes on the job!" Part of superheroness is precision timing and delicate maneuvering, and if she wants to pretend it's luck, I'm not going to waste breath I could be using to eat arguing.
"Today, Lady Luck had a name." Ryodan looks at me.
"No shit." Lor says. "Nice work, honey."
I just about lose my biscuits then. I glow so hard it almost hurts. I think my skin is leaking light.
I swagger over to the hearth and gulp three marshmallows in quick succession.
"Can you believe what that Unseelie prince did?" goth-chick Josie says.
I choke on the last marshmallow I'm trying to swallow whole. I kick up into fast-mo and try to fast-cough it out but it doesn't work. Belatedly it occurs to me fast-mo might not have been the brightest move. Friction and mucus expand the confection like a waterlogged tampon. It swells in my throat and shuts down my airway.
I thump myself on the chest with a fist. Doesn't help. I'm about to give myself the Heimlich over the back of a chair when Lor pounds the center of my back and the marshmallow splats out onto the coat of arms above the fireplace.
"Dude, no need to shove," Dancer says. "I give her the Heimlich all the time. She doesn't chew when she eats."
I turn around and Dancer's picking himself off the floor, looking irked. And tired. I wonder when he last slept. I forget he's not superpowered like the rest of us just because he's got a superbrain.
"Clean it up, Dani," Kat says. "It'll bake onto the medallion."
I grab a napkin off the biscuit tray, not feeling so cocky anymore. There were casualties. I'd managed to let myself forget that for a sec. "Christian sacrificed himself because I couldn't make up my mind."
"An Unseelie prince sacrificed himself," Kat echoes, like she has no clue what to make of it.
I don't either. Why did he give himself up just to make my decision easier? I would have made a decision in another sec or two. We would have lost a lot more _sidhe_ -seers to the Crimson Hag. Was it his way of proving he wasn't full Unseelie yet? Maybe he was trying to make up for killing the woman he had sex with, or it was his idea of another wedding present.
"It's pretty clear he's obsessed with you, honey," Lor says.
" _Was_ obsessed," Ryodan says. "The Hag took him out. Saved me the trouble and good fucking riddance."
Now I have double the reason to track down the bitch and kill her. I have to free Christian so we can be even and call it quits between us. "We lost _sidhe_ -seers," I say. "One of them was Tanty Nana. She was too old. She should never have been out there to begin with."
We're all quiet a sec, thinking about her and the others that died.
Then Ryodan stands up and says to me, "Come on, kid. Let's go."
"Huh? Where?"
"You live with me now."
"Bull-fecking-crikey!"
"She's moving back into the abbey," Kat says.
"Bull-fecking-crikey!"
"The Mega can take care of herself," Dancer says. "If you bloody idiots didn't just see that, you're blind. Give her room to breathe."
"Fecking crikey!" I agree totally. I adore Dancer. I shoot him a look that doesn't bother trying to pretend otherwise.
Ryodan says, "She needs rules."
Lor says, "Boss, all she needs is somebody to train with, vent some of that boundless fucking energy."
Kat says, "What she _needs_ is—"
While they're all busy discussing my needs that they don't know the first fecking thing about, I make like the wind and blow out, sure to bang the door loud on the way.
I steal Ryodan's Humvee and head for the city.
He'll never catch me in one of the buses or big trucks that are the only other vehicles at the abbey.
I wish I could have brought Dancer with me but I never would have escaped if I'd slowed myself down.
Ain't nobody knows what I need better than me. They're probably all back there, still bickering, trying to decide how to control me and run my life.
I snicker. "Dudes. Never. Going. To. Happen."
# FORTY-FOUR
# _"This is not the end, this is not the beginning"_
So I'm blowing through the streets of Dublin after ditching the Humvee on the main road where Ryodan or one of his dudes is sure to spot it on their way back to Chester's because no matter how bugfuck he makes me, I got no desire to take something of his permanent-like. He'd probably hunt me for for-fecking-ever, instead of just trying to boss me around for-fecking-ever. That dude's radar is something I don't want to be any bigger on.
Live at Chester's, my fecking petunia!
"Ass," I mutter with a scowl. Petunia is one of Mac's words. Her and Alina grew up so stupid and sweet they never even said "feck" until their early twenties when they started seeing Fae. Up till then they had their own cute little vocabulary for things. I hate cute. I hate thinking about Mac. I remember seeing her for the first time, sitting on a bench at Trinity looking all soft and pretty and useless, then finding out she was really made of steel like me and my sword. I remember feeling like my world was finally going to change and bygones might somehow miraculously turn into never-fecking-have-beens.
I miss her. I hate knowing she's in this city somewhere, walking up and down streets just like me, thinking thoughts about killing Fae and saving the world and killing me, and she's on one street and I'm on another and those streets can never meet or one of us is dead.
After how great my day was I can't believe I'm thinking broody thoughts. I hurry through Temple Bar, dodging cars and streetlamps and all kinds of stuff half buried in piles of ice-crusted snow. Now that the Hoar Frost King is gone, the snow should start melting. I couldn't be more ready for summer! I don't tan. I freckle. Dancer likes freckles.
"Summer," I say, grinning. It can't come fast enough! They'll plant gardens at the abbey, grow a mixed veggie medley like the one I had at Chester's. I'll definitely be dropping by more once stuff starts growing! I can load myself down with bars and take a few long freeze-framing trips around Ireland, looking for cows or goats or sheep. Maybe even pigs. "Fecking-A, bacon!" My mouth waters just thinking about it.
Right now I got a ton of stuff to do, and getting around in all this snow is tedious. I can't freeze-frame for long because I have to keep dropping down and replot my mental grid. There are too many drifts and piles of ice that weren't there yesterday. Every time I drop down I just about freeze my fingers and toes off. It's night, a wicked breeze is blowing in off the ocean, and I swear it's twenty below with the windchill.
I snap my grid into place, whiz a quarter of a block, stop, re-plan. Freeze-frame forty feet, turn the corner on a slide, smash into a drift, roll, and remap while sliding some more. I slam into the side of a building and my breath frosts the air in a sharp white gasp. I curse and rub my side. I'm going to be one big bruise tomorrow.
First up on my list of things to do is get a new _Dani Daily_ out there _before_ We-don't-really-fecking-Care does and totally skews the news. Folks need to know all the scoop: that the villain icing peeps is dead, they can start making noise again, the snow _is_ going to melt, and even though it don't look like it right now, summer _is_ going to come. They need to know I got my sword back and ain't defenseless anymore. I'm back on my beat 24/7, watching over things and hunting the Crimson Hag, who's going to bite the dust as soon as I can figure how to kill her and get Christian back.
Tomorrow I'll cruise around Dublin slow-mo Joe style, listening for survivors over the crunch of snow and taking them in for food and shelter, which means Ryodan is about to get a lot more folks at Chester's. Our city just keeps getting hammered with walls falling and riots happening, food getting stolen and stockpiled, and now this killing winter in spring. I'm thinking we better get used to things never being predictable again. I suspect we'll lose a lot more folks before the tide starts to turn. Change is hard for most people. Not me. I love re-creating myself. Change means you get to choose again. Become something new. Unless you're dead like Alina. Then you never get to choose again. That's why I'm going to make Ryodan give me whatever secret he's got so I can live forever.
I ease back down into slow-mo to skirt a mound of ice-crusted snow. I'm standing there, starting to get all broody again thinking about all the ghosts I see in these streets sometimes, when I feel the tip of something sharp and pointy in my back.
"Drop your sword, Dani," Mac says, real soft-like behind me.
"Yeah, right. Like I'm actually falling for this." I snicker. Me and my overactive imagination. Like Mac would actually be able to sneak up behind me without my superhearing tipping me off. Like she would ever walk around at night with no MacHalo on. I got mine on and I know exactly how bright it is. If she was standing behind me, we'd be making double the light I'm throwing.
I freeze-frame.
Or try to.
Nothing happens. Just like those two times with Ryodan when all the sudden I just didn't have any juice. No gas in the tank, no engine in the train.
I squeeze my eyes shut hard and try again.
Still standing there.
Still feeling the tip of a spear in my back.
"I said 'drop your fucking sword,' " Mac says.
_For the love of it_
#
Mac and Barrons are back with a vengeance in the seventh novel in the blockbuster Fever series from #1 _New York Times_ bestselling sensation Karen Marie Moning
BURNED
Read on for a sneak peek
"Ms. Lane."
Barrons's voice is deep, touched with that strange Old World accent and mildly pissed off. Jericho Barrons is often mildly pissed off. I think he crawled from the swamp that way, chafed either by some condition in it, out of it, or maybe just by the general mass incompetence he encountered in both places. He's the most controlled, capable man I've ever known.
After all we've been through together, he still calls me Ms. Lane, with one exception: When I'm in his bed. Or on the floor, or some other place where I've temporarily lost my mind and become convinced I can't breathe without him inside me that very instant. Then the things he calls me are varied and nobody's business but mine.
I reply: "Barrons," without inflection. I've learned a few things in our time together. Distance is frequently the only intimacy he'll tolerate. Suits me. I've got my own demons. Besides, I don't believe good relationships come from living inside each other's pockets. I believe divorce comes from that.
I admire the animal grace with which he enters the room and moves toward me. He prefers dark colors, the better to slide in and out of the night, or a room, unnoticed except for whatever he's left behind that you may or may not discover for some time, like, say a tattoo on the back of one's skull.
"What are you doing?"
"Reading," I say nonchalantly, rubbing the tattoo on the back of my skull. I angle the volume so he can't see the cover. If he sees what I'm reading, he'll know I'm looking for something. If he realizes how bad it's gotten, and what I'm thinking about doing, he'll try to stop me.
He circles behind me, looks over my shoulder at the thick vellum of the ancient manuscript. "In the first tongue?"
"Is that what it is?" I feign innocence.
He knows precisely which cells in my body are innocent and which are thoroughly corrupted. He's responsible for most of the corrupted ones. One corner of his mouth ticks up and I see the glint of beast behind his eyes, a feral crimson backlight, bloodstaining the whites.
It turns me on. Barrons makes me feel violently, electrically sexual and alive. I'd march into hell beside him.
But I will not let him march into hell beside me. And there's no doubt that's where I'm going.
I thought I was strong, a heroine. I thought I was the victor. The enemy got inside my head and tried to seduce me with lies.
It's easy to walk away from lies.
Power is another thing.
Temptation isn't a sin that you triumph over once, completely, and then you're free. Temptation slips into bed with you each night and helps you say your prayers. It wakes you in the morning with a friendly cup of coffee, and knows exactly how you take it.
He skirts the Chesterfield sofa and stands over me. "Looking for something, Ms. Lane?"
I'm eye level with his belt but that's not where my gaze gets stuck and suddenly my mouth is so dry I can hardly swallow and I know I'm going to want to. I'm Pri-ya for this man. I hate it. I love it. I can't escape it.
I reach for his belt buckle. The manuscript slides from my lap, forgotten. Along with everything else but this moment, this man. "I just found it," I tell him.
BY KAREN MARIE MONING
THE DANI O'MALLEY TRILOGY
Iced
THE FEVER SERIES
Darkfever
Bloodfever
Faefever
Dreamfever
Shadowfever
GRAPHIC NOVEL
Fever Moon
NOVELLA
Into the Dreaming
THE HIGHLANDER SERIES
Beyond the Highland Mist
To Tame a Highland Warrior
The Highlander's Touch
Kiss of the Highlander
The Dark Highlander
The Immortal Highlander
Spell of the Highlander
**ABOUT THE AUTHOR**
KAREN MARIE MONING is the #1 _New York Times_ bestselling author of the Fever series, featuring MacKayla Lane, and the award-winning Highlander series. She has a bachelor's degree in society and law from Purdue University and is currently working on the next novel featuring Dani O'Malley.
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"redpajama_set_name": "RedPajamaBook"
} | 245 |
Okres Bělostok (Białystok; ) je okres v polském Podleském vojvodství. Rozlohu má 2984,64 km² a v roce 2019 zde žilo 149 611 obyvatel. Sídlem správy okresu je město Bělostok, které však není jeho součástí, ale tvoří samostatný městský okres.
Gminy
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Supraśl
Suraż
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Wasilków
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Vesnické:
Dobrzyniewo Duże
Gródek
Juchnowiec Kościelny
Poświętne
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Reference
Externí odkazy
Bělostok | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,429 |
Q: Use local storage to remember jQuery modal popups when minimized, even if page is reloaded or changed thank you in advance and sorry for my bad english. I'm not a javascript developer and therefore it's difficult, to me, to improve all alone the code that I have on my website, so any specific help is very welcome.
Inside the file called footer.inc.php that - of course - is common to all pages, I have a script that open the url inside the main page as modal popups. I can close them or reduce them.
<script type="text/javascript">
function modalWindow(name, title, url, width, height) {
// per width ed height imposto dei valori di default così non occorre specificarli in ogni occasione
width = typeof width === 'undefined' ? 800 : width;
height = typeof height === 'undefined' ? 600 : height;
// verifichiamo se nel body non esiste il sorgente per la dialog
if (top.$('#dialog-'+name).length == 0) {
// in questo caso lo creiamo:
top.$('body').append('<div id="dialog-' +name+ '" title="' +title+ '" style="padding:0;"><iframe src="' + url + '" frameborder="no" style="position:absolute;width:100%;height:100%;" scrolling="yes"></div>');
} else {
// se il sorgente invece esiste già assegnamo la nuova url all´iframe:
top.$('#dialog-' +name).attr('title', title);
top.$('#dialog-' +name+ ' iframe').attr('src', url);
}
// Ok, adesso siamo pronti per lanciare la modale!
top.$('#dialog-' +name).dialog({width: width, height: height});
top.$('#dialog-' +name).dialog({width: width, height: height}).dialogExtend({"minimizable" : true}).dialog({ position: { my: "center", at: "center+20px", of: target } });
$("#dialog").dialog({ position: { my: "center", at: "top+30%", of: window } });
}
</script>
dialogExtend({"minimizable" : true}) allows me to minimize the popup windows, but if I refresh or change the page, the popups will be lost.
Can I ask you how to change and improve the code above so that the opened modal popups won't be lost, even if I change or reload the page?
I got and answer but, since I don't know javascript well enough, I cannot improve the code by myself.
Thank you again!
A: You have a function that you call with few parameters. I would save those parameters in localStorage, then when reloading the page I would read the content of LS and call the function.
var modals = localStorage.getItem('modals');
if (modals) {
modals = JSON.parse(modals);
} else {
modals = [];
}
modals.forEach(args => {
modalWindow(...args);
});
function modalWindow(name, title, url, width, height) {
...
modals.push([name, title, url, width, height]);
localStorage.setItem('modals', JSON.stringify(modals));
}
You probably also want to remove the modal from the list when you close it. You can come up with a better data structure that will allow you to do that.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,427 |
\section{Introduction}
\label{sect:intro}
Type Ia supernovae (SNe~Ia) originate from the thermonuclear explosion of at least one C-O White Dwarf (WD) in a binary system \citep[e.g.,\ ][]{Whelan73,Livne90, Iben84,Hoeflich:Khokhlov:96}. These luminous events follow empirical observational relationships which are fundamental for the use of SNe~Ia as extra galactic distance indicators \citep{Pskovskii84,Phillips93,Phillips99}. This led to the discovery of the accelerating expansion of the cosmos, or dark energy \citep[e.g.,\ ][]{Riess98,Perlmutter99}.
To date, there have been many sub-types of SNe~Ia discovered including: 1991bg-like SNe \citep[e.g.,\ ][]{Filippenko:91bg:1992,Leibundgut93}, transitional SNe \citep[e.g.,\ ][]{Hsiao15,Gall18}, 2002cx-like SNe \citep[e.g.,\ ][]{Li03,Foley13}, 2002es-like SNe \citep[e.g.,\ ][]{Ganeshalingam12}, 1991T-like SNe \citep[e.g.,\ ][]{Filippenko92,Phillips92} and 2003fg-like SNe \citep[e.g.,\ ][]{Howell06,Hicken07}. 2003fg-like SNe, which are also known as ``super-Chandrasekhar-mass" SNe~Ia, are among the most rare sub-type of SNe~Ia. It was previously thought that all 2003fg-like SNe\ were over-luminous and require more $^{56}$Ni\ to power the light curve than could be produced in a detonation of a non-rotating Chandrasekhar-mass ($M_{Ch}$) C-O WD \citep{Howell06, Hicken07}. Hence, they were named ``super-Chandrasekhar mass''.
However, it has since become evident that not all 2003fg-like SNe\ are over-luminous and their properties are more diverse \citep[e.g.,\ ][]{Taubenberger19,Lu21}.
Therefore, in this work, we follow the convention of naming the sub-type after the first SN discovered in the group, SN~2003fg \citep{Howell06}.
There have only been a handful of 2003fg-like SNe\ discovered and their observational traits are varied \citep{Howell06,Hicken07,Yamanaka09,Scalzo10,Yuan10,Silverman11,Taubenberger11,Chakradhari14,Taubenberger19,Hsiao20,Lu21}.
They do, however, share a few characteristics: They generally have broad light curves, slow expansion velocity gradient before maximum light, and a very strong $\lambda$6580\ C~{\sc ii}\ absorption feature which lasts well past $B$-band maximum.
They also peak in the $i$-band well after the time of $B$-band maximum \citep{Ashall20}.
Furthermore, 2003fg-like SNe\ do not show a distinct $H$-band break at +10~d past $B$-band maximum which is seen in normal SNe~Ia \citep{Taubenberger11, Hsiao19, Lu21}.
This $H$-band break is directly linked to the distribution and bulk of $^{56}$Ni\ in the explosion \citep{Wheeler98}.
SN~2007if and SN~2009dc show low continuum polarizations which suggest spherical explosions \citep{Tanaka10, Cikota19}.
SN~2012dn and LSQ14fmg show evidence of dense circumstellar medium (CSM) \citep{Nagao17,Hsiao20}.
The majority of 2003fg-like SNe\ occur in low metallicity, low surface brightness galaxies with high specific star formation rates \citep{Childress11,Hsiao20,Lu21,Galbany21}.
There are several theoretical models that have been proposed for 2003fg-like SNe. An early suggestion is the violent merger of two WDs that exceed the $M_{ch}$ \citep{Howell06,Scalzo10}. Alternatively, others have suggested these bright SNe experience interaction with dense CSM \citep{Hachinger12,Noebauer16}. This is also referred to as an envelope model \citep{Hoeflich:Khokhlov:96}. Such an explosion may occur from the explosion of a degenerate core of an Asymptotic giant branch (AGB) star in the core degenerate scenario \citep{Kashi11,Hsiao20, Lu21} or from the disruption of a C-O WD with surrounding circumstellar dust \citep{Nagao17,Nagao18}. Finally, the explosion of a C-O WD which exceeds the classical non-rotating $M_{ch}$ limit due to rapid rotation or high magnetic fields may also be a viable model \citep{Yoon05,Das13}.
The current dataset of 2003fg-like SNe\ is sparse, and it has not been possible to disentangle the effects predicted by these scenarios.
The \textit{Carnegie Supernova project}-I \& II (CSP-I \& II) ran two
observing campaigns between 2004 and 2015, during which we obtained
optical and NIR spectra and photometry of over 300 SNe~Ia
\citep{Krisciunas17,Phillips19,Hsiao19}. Nine 2003fg-like SNe\ were followed in
these two campaigns.
In this work, we combine this data set
with data from the literature to produce and analyze the first statistical and homogeneous sample 2003fg-like SNe.
In Section \ref{sect:sample} we present the observational sample, followed by
the data reduction in Section \ref{sect:reduc}. Host galaxy extinction is discussed in Section \ref{sect:Extinction}. Section \ref{sect:phot} presents the photometric observations, followed by the spectroscopic observations in Section \ref{sect:spec}. Important correlations and their implications are discussed in Section \ref{sect:correlations}. Finally the discussion of possible explosion models is given in Section \ref{sect:discussion}, followed by the conclusions in Section \ref{sect:conclusion}.
\section{Sample Characteristics}
\label{sect:sample}
The vast majority of the 2003fg-like SNe\ followed by CSP were observed during CSP-II (7 out of 9).
This reflects one of the main differences between the CSP-I and CSP-II campaigns:
while nearly 90\% of the SNe~Ia followed up by CSP-I came from targeted searches, 83\% of those followed up by CSP-II came from untargeted searches.
As 2003fg-like SNe\ preferentially explode in low-luminosity hosts, untargeted surveys have the advantage in detecting them.
While we strove for a complete and unbiased sample in CSP-II, the strategy also contributes to our success at following up on a statistically significant and uniform sample of 2003fg-like SNe, the only sample of its kind.
Furthermore, the sample optical light curves are obtained with nearly nightly cadence and are placed on a single well-understood photometric system of the Swope Telescope.
All of the 2003fg-like SNe\ observed by CSP-II came from untargeted searches.
Only 3 (SNe~2006gz, 2009dc, and 2012dn) out of the 13 2003fg-like SNe\ presented here were discovered by targeted searches. The discovery information on the objects not previously published can be see in Appendix \ref{sec:Discoveryinfo}.
2003fg-like SNe\ are generally characterized by:
\begin{itemize}
\item[$\bigstar$] A primary $i$-band maximum after that of the $B$-band maximum
\item[$\bigstar$] A lack of $H$-band break at +10\,d in the NIR spectra
\item[$\bigstar$] A low ionization state in nebular phase spectra
\item A broad light curve
\item A strong C~{\sc ii}\ \red{$\lambda \lambda$6578,6582} feature past maximum light
\item A lack of or weak $i$-band secondary maximum
\item Low ejecta velocity gradients before maximum light
\item A lack of Ti~{\sc ii}\ in the maximum light spectra
\end{itemize}
However, not all of these features are observed in every 2003fg-like SN, and only the timing of the $i$-band primary, the lack of an $H$-band break, and the low ionization nebular phase appear to be ubiquitous \citep[see e.g.,\ ][]{
Howell06,Silverman11,Taubenberger11,Taubenberger19,Chen19,Hsiao19,Ashall20,Lu21}.
In this work the 2003fg-like SNe\ were identified in the CSP-I and CSP-II through photometric criteria. As mentioned previously, 2003fg-like SNe\ have distinct photometric properties from the normal population as well as other peculiar sub-types \citep{Gonzalez14}.
Here we adopt the method of \citet{Ashall20} to identify 2003fg-like SNe:
For an object to be chosen for the sample, it must have its $i$-band primary maximum occurring after that in $B$ band.
Furthermore, the object must have slowly declining light curves as indicated by $s_{BV}\gtrsim0.8$ or $\Delta \rm{m}_{15}(B)\lesssim1.3$\,mag
\footnote{See section \ref{sect:LWR} for the definition of $s_{BV}$ and $\Delta \rm{m}_{15}(B)$.}.
These photometric selection criteria are then used in conjunction with the examination of its optical spectrum near maximum to look for the identifying properties of 2003fg-like SNe\ described above.
The spectroscopic criteria eliminate the peculiar SN~2006bt, as it contains Ti~{\sc ii}\ features.
Note that using spectra alone can lead to misleading results, such as those of SN 2011hr \citep{Zhang16} and LSQ12gdj \citep{Scalzo14}.
Note that CSS140126 is a border-line case between a 1991T-like and a 2003fg-like SNe, it has only one low S/N optical spectrum which is featureless, a primary $i$-band maximum which peaks after that of $B$ band, but displays a secondary $i$-band maximum.
We chose to keep it in the sample, however due to poor spectral temporal coverage it is difficult to ascertain if it is a true 2003fg-like SN. Nine SNe in the CSP samples and a further four in the literature were found to meet these criteria.
Table~\ref{table:prop} contains the basic information of all of the SNe used in this analysis, and Table~\ref{table:propphot} summarizes their photometric properties.
For ten objects $z_{helio}$ was determined using integral field spectroscopy (IFS) data which will be presented in \citep{Galbany21}. For the other objects $z_{helio}$ was determined from a spectrum of the host galaxy, or, in the case of previously published 2003fg-like SNe\ objects, it was taken from the literature.
In Fig.~\ref{fig:hist}, the sample characteristics are compared to that of the ``Cosmology'' SN~Ia sample of CSP-II \citep{Phillips19}, as 96\% of the SNe were discovered by untargeted searches.
The majority of the objects from the sample are in the Hubble flow, similar to the CSP-II Cosmology sample.
As our selection criteria dictate, the 2003fg-like SNe\ are slow decliners as indicated by $s_{BV}$ and $\Delta \rm{m}_{15}(B)$.
However, it should be noted that there is a wide range of light-curve properties that overlap with those of the normal populations.
\begin{deluxetable*}{ c c c c l c c}
\tablewidth{\textwidth}
\tablecaption{The properties of the SNe in the sample. \label{table:prop}}
\tablehead{
\colhead{SN}&
\colhead{$z_{helio}$}&
\colhead{RA}&
\colhead{DEC}&
\colhead{$\mu$\tablenotemark{a}}&
\colhead{$E(B-V)_{MW}$}&
\colhead{Discoverer}
\\
\colhead{}&
\colhead{}&
\colhead{}&
\colhead{}&
\colhead{Mag}&
\colhead{Mag}&
\colhead{}}
\startdata
2003fg & 0.2440 & 14:16:18.8 & +52:14:53.66 & 40.37 $\pm$ 0.01 & 0.011 & \citet{Howell06} \\
2006gz & 0.0237 & 18:10:26.3 & +30:59:44.40 & 34.95 $\pm$ 0.09 & 0.055 & \citet{Puckett06gz} \\
2012dn & 0.0100 & 20:23:36.3 & $-$28:16:43.40 & 33.28 $\pm$ 0.21\tablenotemark{b} & 0.052 & \citet{12dndisc} \\
ASASSN-15pz & 0.0148 & 03:08:48.4 & +35:13:50.89 & 33.89 $\pm$ 0.14 & 0.015 & \citet{15pzdisc} \\
2007if\tablenotemark{c} & 0.0742 & 01:10:51.4 & +15:27:39.90 & 37.51 $\pm$ 0.03 & 0.072 &\citet{Akerlof07if} \\
2009dc\tablenotemark{c} & 0.0214 & 15:51:12.1 & +25:42:28.50 & 34.79 $\pm$ 0.09 & 0.060 &\citet{09dcdisc} \\
LSQ12gpw\tablenotemark{c} & 0.0506 & 03:12:58.2 & $-$11:42:40.13 & 36.65 $\pm$ 0.04 & 0.062& \citet{LSQ} \\
2013ao\tablenotemark{c} & 0.0435 & 11:44:44.7 & $-$20:31:41.10 & 36.39 $\pm$ 0.04 & 0.034 &\citet{13aodisc} \\
CSS140126\tablenotemark{c~d} & 0.0772 & 12:03:06.9 & $-$01:01:31.70 & 37.67 $\pm$ 0.03 & 0.021 &\citet{Drake09} \\
CSS140501\tablenotemark{c~e} & 0.0797 & 17:04:13.7 & +17:48:39.40 & 37.74 $\pm$ 0.03 & 0.066 &\citet{Drake09} \\
LSQ14fmg\tablenotemark{c} & 0.0649 & 22:16:46.1 & +15:21:14.13 & 37.24 $\pm$ 0.03 & 0.046 &\citet{LSQ} \\
2015M\tablenotemark{c} & 0.0231 & 13:00:32.3 & +27:58:41.00 & 35.04 $\pm$ 0.08 & 0.009 &\citet{kiss15ndisc} \\
ASASSN-15hy\tablenotemark{c}& 0.0185 & 20:10:02.4 & $-$00:44:21.31 & 34.33 $\pm$ 0.11 & 0.130 & \citet{15hydisc} \\
\enddata
\tablenotetext{a}{Corrected to the CMB rest frame and calculated using $H_{0}$=73\,km\,s$^{-1}$\,Mpc$^{-1}$, $\Omega_{m}$=0.27 and $\Omega_{\Lambda}$=0.73, which is used throughout this work.}
\tablenotetext{b}{Corrected for the infall towards the Virgo cluster and the Great Attractor (recession velocity =3300km~s$^{-1}$; \citealt{Mould00}).}
\tablenotetext{c}{SN observed by CSP.}
\tablenotetext{d}{For convenience we shorten CSS140126:120307-010132 to CSS140126.}
\tablenotetext{e}{For convenience we shorten CSS140501-170414+174839 to CSS140501.}
\end{deluxetable*}
\begin{deluxetable*}{ c c c c c c c }
\tablewidth{\textwidth}
\tablecaption{The basic light curve parameters of the 2003fg-like SNe.
All parameters were obtained from direct measurements to the Gaussian process interpolations to the data. It should be noted that the t$^{i-B}_{max}$\ here has not been K-corrected. \red{These values have been corrected for foreground but not host galaxy extinction.} \label{table:propphot}}
\tablehead{
\colhead{SN}&
\colhead{$T(B)_{max}$}&
\colhead{$B_{max}$}&
\colhead{$\Delta{\rm m_{15}(B)}$}&
\colhead{s$_{BV}$}&
\colhead{$(B-V)_{Bmax}$}&
\colhead{t$^{i-B}_{max}$}
\\
\colhead{}&
\colhead{Days}&
\colhead{Mag}&
\colhead{Mag}&
\colhead{}&
\colhead{Mag}&
\colhead{Days}}
\startdata
2003fg & 2452760.18 $\pm$ 0.87 & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
2006gz & 2454021.84 $\pm$ 0.10 & 15.86 $\pm$ 0.06 & 0.84 $\pm$ 0.02 & 1.37 $\pm$ 0.03 & 0.03 $\pm$ 0.02 & 4.60 $\pm$ 1.65 \\
2012dn & 2456132.63 $\pm$ 0.98 & 14.16 $\pm$ 0.03 & 0.87 $\pm$ 0.03 & 1.25 $\pm$ 0.15 & 0.02 $\pm$ 0.03 & 2.04 $\pm$ 0.61 \\
ASASSN-15pz & 2457307.47 $\pm$ 0.93 & 14.18 $\pm$ 0.03 & 0.67 $\pm$ 0.09 & 1.37 $\pm$ 0.09 & $-$0.02 $\pm$ 0.02 & 0.04 $\pm$ 0.81 \\
2007if & 2454349.97 $\pm$ 1.09 & 17.55 $\pm$ 0.02 & 0.88 $\pm$ 0.09 & 1.26 $\pm$ 0.15 & 0.01 $\pm$ 0.03 & 3.03 $\pm$ 3.24 \\
2009dc & 2454947.07 $\pm$ 0.60 & 15.09 $\pm$ 0.02 & 0.70 $\pm$ 0.05 & 1.29 $\pm$ 0.07 & $-$0.03 $\pm$ 0.01 & 2.52 $\pm$ 0.34 \\
LSQ12gpw & 2456269.75 $\pm$ 0.46 & 17.35 $\pm$ 0.01 & 0.70 $\pm$ 0.03 & $\cdots$ & 0.01 $\pm$ 0.01 & $\cdots$ \\
2013ao & 2456362.53 $\pm$ 0.26 & 16.87 $\pm$ 0.01 & 0.99 $\pm$ 0.03 & 1.02 $\pm$ 0.13 & 0.10 $\pm$ 0.01 & 1.80 $\pm$ 0.26 \\
CSS140126 & 2456668.48 $\pm$ 0.41 & 18.23 $\pm$ 0.01 & 0.73 $\pm$ 0.05 & $\cdots$ & $-$0.06 $\pm$ 0.01 & 3.97 $\pm$ 0.73 \\
CSS140501 & 2456787.70 $\pm$ 1.60 & 18.09 $\pm$ 0.04 & 1.05 $\pm$ 0.18 & $\cdots$ & 0.03 $\pm$ 0.02 & 2.58 $\pm$ 2.85 \\
LSQ14fmg & 2456939.24 $\pm$ 0.72 & 17.35 $\pm$ 0.01 & 1.04 $\pm$ 0.09 & 1.20 $\pm$ 0.08 & 0.09 $\pm$ 0.01 & 1.51 $\pm$ 1.05 \\
2015M & 2457169.00 $\pm$ 0.82 & 15.54 $\pm$ 0.03 & 0.82 $\pm$ 0.04 & $\cdots$ & 0.14 $\pm$ 0.01 & 1.85 $\pm$ 2.22 \\
ASASSN-15hy & 2457151.63 $\pm$ 0.40 & 15.19 $\pm$ 0.01 & 0.73 $\pm$ 0.03 & 1.24 $\pm$ 0.18 & 0.19 $\pm$ 0.01 & 7.28 $\pm$ 0.47 \\
\enddata
\end{deluxetable*}
\begin{figure*}[htb]
\centering
\includegraphics[width=.95\textwidth]{CSPII_SuperC_hist2.pdf}
\caption{Histograms of $z_{helio}$ (left panel), $\Delta{\rm m_{15}(B)}$ (middle panel) and $s_{BV}$ (right panel) of the 2003fg-like SNe\ sample compared to the CSP-II SNe~Ia cosmology samples from \citet{Phillips19}. Nine 2003fg-like SNe\ followed up by CSP (including CSP-I and CSP-II) are marked with shaded red bars, and four not followed by CSP are stacked on top with non-stripped red bars. The SNe~Ia from \citet{Phillips19} are the gray shaded bars. }
\label{fig:hist}
\end{figure*}
\section{Data reduction}
\label{sect:reduc}
\subsection{Photometry}
Optical $uBVgri$-band imaging was obtained for nine
2003fg-like SNe\ using SITe-3 and e2v on the 1-m Swope telescope at Las Campanas
Observatory (LCO).
For a sub-sample of these, NIR $YJH$-band imaging was also
acquired using NIR imager called RetroCam which was installed on the Swope telescope during CSP-I and the 2.5-m du~Pont telescope during CSP-II. All of the photometry presented here is on the well-understood CSP natural system. This allows for systematic differences between SNe to be examined. The reduction and calibration procedures are described in \citet{Krisciunas17} and \citet{Phillips19}, and the final light curves can be found online\footnote{\href{https://csp.obs.carnegiescience.edu/data}{CSP data products.}}. The light curves of four of the nine SNe have been previous published by the CSP: SN\,2007if and SN\,2009dc \citep{Krisciunas17}, LSQ14fmg \citep{Hsiao20}, and ASASSN-15hy \citep{Lu21}. Finder charts of the remaining five 2003fg-like SNe\ are presented in Fig.~\ref{fig:finders}, and the light curves of all the SNe are presented in the natural system in Fig.~\ref{fig:lightcurves}. These light curves are tabulated in Appendix~\ref{sec:CSP_phot_spec}.
Data from five 2003fg-like SNe\ that have been published by other groups (SN\,2003fg; \citealt{Howell06}, SN\,2006gz; \citealt{Hicken07} SN\,2009dc; \citealt{Yamanaka09,Tanaka10,Silverman11,Taubenberger11}, SN\,2012dn; \citealt{Chakradhari14,Parrent16,Taubenberger19}, and ASASSN-15pz; \citealt{Chen19}) are also included in the sample. In the $BVgri$ bands, where possible, S-corrections were applied to transform the photometry to the CSP natural system. \red{Swift Ultraviolet/Optical Telescope (UVOT) data of SN~2015M was also obtained from the Supernova Archive (SOUSA; \citealt{Brown14a}) via the Swift Supernova website.\footnote{https://pbrown801.github.io/SOUSA/ }}
K-corrections were computed for $BVgri$ light curves using the same method presented in Appendix~B of \citet{Lu21}. In short, the spectral series of SN\,2009dc, SN\,2012dn, ASASSN-15hy, and the Hsiao template were all used independently as the SED to compute the corrections. The SEDs were mangled to match the interpolated observed photometric colors. A comparison was made between the K-correction values between the four SED template spectral series and those computed with the actual observed spectra of the SNe. The template with the smallest average residual K-correction compared to the observed spectra was then selected for each SN.
This was done because most of the SNe did not have adequate spectral coverage to compute K-corrections directly from the observed spectra. The K-correction process was carried out individually for each SN. As discussed in \citet{Lu21}, the average K-correction uncertainty obtained with this method is smaller than 0.01\,mag, which is consistent with the values obtained for normal SNe~Ia from \citet{Hsiao07}. Due to the lack of spectral data in the UV and NIR, and hence the ability to understand the SED, no S or K-corrections were applied in these regions.
\begin{figure*}
\setlength\arraycolsep{0pt}
\renewcommand{\arraystretch}{0}
\centering$
\begin{array}{ccc}
\includegraphics[width=.35\linewidth]{2013ao.png} &
\includegraphics[width=.35\linewidth]{csp13abs.png}&
\includegraphics[width=.35\linewidth]{2014abk.png}\\
\includegraphics[width=.35\linewidth]{KISS15N.png}&
\includegraphics[width=.35\linewidth]{LSQ12gpw.png}
\end{array}$
\caption{The $r$-band finding charts of the five 2003fg-like SNe\ observed by CSP-II. The plots are produced with Swope eV2 images. An inset of the exact SN location is provided in the top right section of each panel. }
\label{fig:finders}
\end{figure*}
\subsection{Spectra}
Optical and NIR spectra were obtained of nine 2003fg-like SNe\ by the CSP I \& II. Twenty-four optical and six NIR spectra are presented here for the first time, which are logged in Appendix~\ref{sec:CSP_phot_spec}.
The majority of these optical spectra were acquired at LCO using B\&C on the 2.5-m du~Pont telescope and LDSS3 and IMACS on the 6.5-m Magellan Baade and Clay Telescopes. Additional spectra were obtained with the ALFOSC on the Nordic Optical Telescope (NOT) at La Palma, EFOSC2 on the New Technology Telescope (NTT) at La Silla, and RSS on the Southern African Large Telescope (SALT) at the South African Astronomical Observatory. The spectra were reduced using the standard \textsc{iraf}\footnote{The Image Reduction and Analysis Facility (\textsc{iraf}) is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} packages using the method as described in \citet{Hamuy06} and \citet{Folatelli13}.
All the NIR spectra were observed with the Folded-port InfraRed Echellette \citep[FIRE;][]{Simcoe13}. The details in observing set up and data reduction are outlined by \citet{Hsiao19}. Along with six NIR spectra of ASASSN-15hy \citep{Lu21}, we have tripled the size of the sample of 2003fg-like SNe\ NIR spectra, as the previous sample includes only spectra of SN~2009dc \citep{Taubenberger11,Taubenberger13a}.
Plots of the previously unpublished CSP spectra of SN~2007if, SN~2009dc, LSQ12gpw, SN~2013ao, CSS140126, CSS140501, and SN~2015M are presented in Fig.~\ref{fig:spec} and Fig.~\ref{fig:specNIR}.
All of the spectra for analysis have been made available at the CSP website\footnote{\href{https://csp.obs.carnegiescience.edu/data}{CSP data products.}}.
\begin{figure*}
\centering
\includegraphics[width=.925\textwidth]{LCs_new.pdf}
\caption{Rest-frame UV to NIR photometry of all 2003fg-like SNe\ in the sample. The photometry in this plot has not been K-corrected or corrected for host galaxy extinction, but have been corrected for foreground extinction. Filled symbols are photometric measurements computed from Swope or du Pont telescope images and are in the CSP natural system. The open symbols are from other sources. SN~2003fg has been excluded from the plot due to poor photometric coverage. Individual light curve plots of these SN can be found in Appendix \ref{sec:CSP_phot_spec}}.
\label{fig:lightcurves}
\end{figure*}
\begin{figure*}
\setlength\arraycolsep{0pt}
\renewcommand{\arraystretch}{0}
\centering$
\begin{array}{cc}
\includegraphics[width=.5\linewidth]{SN2007if.pdf} &
\includegraphics[width=.5\linewidth]{SN2009dc.pdf} \\
\includegraphics[width=.5\linewidth]{KISS15N.pdf} &
\includegraphics[width=.5\linewidth]{mixspec.pdf}
\end{array}$
\caption{Rest frame optical spectra of all unpublished 2003fg-like SNe\ from the CSP. Rest frame phases relative to $B$-band maximum are given next to each spectrum. \red{These data have not been corrected for extinction.} \label{fig:spec}}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{2013aoNIR.pdf}
\caption{Rest frame NIR spectra of all unpublished 2003fg-like SNe\ from the CSP. Rest frame phases relative to $B$-band maximum are given next to each spectrum. Note that C~{\sc i}\ is only seen in the spectra of SN~2015M and not SN~2013ao. \red{These data have not been corrected for extinction.} }
\label{fig:specNIR}
\end{figure}
\section{Extinction}
\label{sect:Extinction}
Throughout this work, the dust map of \citet{Schlafly11} was used to correct for foreground Milky Way (MW) extinction. However, values of the host galaxy extinction of 2003fg-like SNe\ are still very uncertain due to the fact that their intrinsic brightness and colors are not yet fully understood.
Two methods were adopted to estimate the host extinction for the 2003fg-like SNe: i) the equivalent width (EW) of the host-galaxy Na~{\sc i}~D feature and ii) the Balmer decrements of the host galaxy lines at the SN location using integral field spectroscopy (IFS) data from \citet{Galbany21}.
Both of these techniques suffer from shortcomings, which we discuss below. It is established that if a SN has a Na~{\sc i}~D feature it may have some host galaxy extinction,
and in the Milky Way there is a well-known correlation between the Av and the EW of Na I D (e.g., see Pozananski et al. 2012 and references therein). However, the scatter in this correlation is $\pm$68\% and a significant fraction of SNe Ia display anomalously high Na I D EWs in comparison with the Av values derived from their colors \citep{Phillips13}. Furthermore, most of the spectra we analyze in this work are of low resolution, making it difficult to detect a weak Na~{\sc i}~D feature.
If no Na~{\sc i}~D feature can be detected in the observed spectrum we
simulate an upper limit on how much Na~{\sc i}~D absorption could be
hidden in the data. This was done by using the highest resolution
observed spectrum for each SN. An idealized spectrum with infinite
resolution was created by smoothing the observed spectrum via a
Gaussian filter. An artificial absorption line with various widths and
depths simulating the Na~{\sc i}~D feature was then added at the host
redshift. This idealized spectrum was degraded to the resolution of
the observations. The spectrum was re-sampled at the same wavelengths
as the observed spectrum. Finally, random noise was added using the
flux uncertainty measured from the observed spectrum. The EW of the Na~{\sc
i}~D feature was measured in the idealized spectrum and the
low-resolution spectrum. The strength of the absorption was decreased
until the EW of the low-resolution spectrum equals the EW uncertainty. At
this point, the EW of the idealized spectrum is taken as the
detection limit. The results are presented in
Table~\ref{table:extinction}. The Na~{\sc i}~D pEW was then converted
to an $E(B-V)$ value using the relation from \citet{Poznanski12}, with a 68\% uncertainty as per \citet{Phillips13}.
The second method to determine host-galaxy extinction is using the host-galaxy emission lines at the location of the SN. This has been obtained for ten of the thirteen SNe we analyzed in this work \citep[see][]{Galbany21}.
The IFS data, which is presented in \citet{Galbany21}, can be used to determine these values. However, this is highly uncertain as it assumes a constant temperature of the gas, and considers the whole column density at the location of the SN, not just in front of it. Only LSQ12gpw has a significant extinction detected at its location using this method.
Note that the host extinction may be estimated for normal SNe~Ia using the Lira Law \citep{Lira98,Phillips99}, and this was attempted by \citet{Chen19} for SN~2009dc and ASASSN-15pz. The host galaxy extinction of SN~2009dc has always been uncertain. \citet{Chen19} found that SN~2009dc, which has a large Na~{\sc i}~D host galaxy feature, had the same $B-V$ color slope as ASASSN-15pz. It was then pressumed that ASASSN-15pz could be used as an ``unreddened" comparison SN.
The magnitude offset between the two SNe at the late-time decline was assumed to be caused by extinction.
However, there is no evidence for 2003fg-like SNe\ following a form of Lira Law, and \citet{Hicken07}
claimed the host color excess derived from Lira relation is most likely not appropriate for 2003fg-like SNe. Therefore, we choose not to use this relation.
Furthermore, \citet{Lu21} found that 2003fg-like SNe\ can look similar after $B$-band maximum during the Lira tail, but be intrinsically different at early times and therefore have different luminosities demonstrating that for 2003fg-like SNe\ the Lira law is not reliable.
In Table~\ref{table:extinction}, the values of extinction obtained with the two different methods are presented. As all of the methods mentioned above are highly uncertain, we choose to follow the conventional method of using the pEW of the Na~{\sc i}~D feature.
In the cases where the host $E(B-V)$ measurements based on the Na~{\sc i}~D pEW are consistent with zero, no host extinction correction is employed.
Furthermore, host galaxy extinction is only applied when explicitly stated in the following analyses.
\begin{deluxetable}{cccccc}
\tablewidth{\textwidth}
\tablecaption{EW values and limits of the Na ID feature, along with the $E(B-V)$ calculated from EW of the Na ID feature as well as from the ratio of the Balmer lines. \label{table:extinction}}
\tablehead{
\colhead{SN}&
\colhead{Na ID EW}&
\colhead{$E(B-V)$\tablenotemark{a}}&
\colhead{$E(B-V)$\tablenotemark{b}}&
\\
\colhead{}&
\colhead{\AA}&
\colhead{Mag}&
\colhead{Mag}}
\startdata
\hline
2003fg & 0.33 $\pm$ 0.14 & 0.03 $\pm$ 0.03 & $\cdots$ \\
2006gz & 0.30 $\pm$ 0.11 & 0.03 $\pm$ 0.02 & 0.00 \\
2012dn & 0.27 $\pm$ 0.14 & 0.03 $\pm$ 0.02 & $\cdots$ \\
ASASSN-15pz & $\leq$ 0.05 & 0.00 $\pm$ 0.02 & $\cdots$ \\
2007if & $\leq$ 0.06 & 0.00 $\pm$ 0.02 & $\cdots$ \\
2009dc & 1.03 $\pm$ 0.24 & 0.23 $\pm$ 0.22 & 0.00 \\
LSQ12gpw & 0.23 $\pm$ 0.10 & 0.03 $\pm$ 0.02 & 0.23 \\
2013ao & $\leq$ 0.08 & 0.00 $\pm$ 0.02 & 0.09 \\
CSS140126 & $\leq$ 0.42 & 0.00 $\pm$ 0.04 & $\cdots$ \\
CSS140501 & $\leq$ 0.14 & 0.00 $\pm$ 0.02 & 0.00 \\
LSQ14fmg & 0.87 $\pm$ 0.44 & 0.15 $\pm$ 0.20 & 0.00 \\
2015M & $\leq$ 0.06 & 0.00 $\pm$ 0.02 & $\cdots$ \\
ASASSN-15hy & $\leq$ 0.06 & 0.00 $\pm$ 0.02 & 0.00 \\
\hline
\enddata
\tablenotetext{a}{Calulated using relationship from \citet{Poznanski12} with uncertainties as per \citet{Phillips13}.}
\tablenotetext{b}{Calculated from Balmer line ratio in IFS data from \citet{Galbany21}}
\end{deluxetable}
\section{Photometric properties}
\label{sect:phot}
\subsection{Light curves}
Figure~\ref{fig:compLC} presents the K-corrected $BVri$ light curves of the sample compared to a selection of normal SN~Ia ($\Delta{\rm m_{15}(B)}<$1.3~mag), all placed on the CSP natural system. The light curves are normalized to the peak, such that the comparison is in the light-curve shapes. In the $B$ and $V$ bands, the 2003fg-like SNe\ are largely indistinguishable from normal SNe~Ia, with some noted exceptions. SN~2007if, LSQ14fmg, and ASASSN-15hy have extremely slow rise times. Furthermore, LSQ14fmg declines rapidly at the start of the decay tail. Similarly rapid declines were also observed but at much later phases in SN~2009dc and SN~2012dn. CSS140126 also shows a hint of the rapid decline in the $V$ band around the same phase as that of LSQ14fmg.
At redder wavelengths, the light curves of 2003fg-like SNe\ differ drastically from those of standard SNe~Ia. In the $r$-band, 2003fg-like SNe\ have a much weaker post-maximum ``knee'', except for LSQ14fmg. In the $i$-band, they have no strong secondary maxima, except in the case of CSS140126. The $i$-band secondary maximum has been suggested to be produced by the recombination of iron group elements in the ejecta \citep{Hoeflich02,Kasen06}. Brighter SNe, such as 91T-like objects, tend to have a more prominent secondary $i$-band maximum, and fainter SNe, such as 91bg-like SNe, have a lack of a secondary $i$-band maximum. The latter case is caused by the merging of the primary and secondary maxima due to the quickly receding photosphere and a small $^{56}$Ni\ mass \citep[see e.g.,\ ][]{Hoeflich02,Ashall20}.
For 2003fg-like SNe, the lack of a prominent $i$-band secondary maximum is a defining trait \footnote{\red{We note that CSS 140126 has a secondary $i$-band maximum but has a uncertain classification due to a low S/N spectra. However it fits into our sample because it peaks in the $i$-band after the $B$-band.}}. This \red{suggests}, unlike normal SNe~Ia, that there is a lack of recombination of iron group elements in the ejecta at 2~d to 40~d past maximum. Given that 2003fg-like SNe\ are significantly brighter than sub-luminous SNe~Ia, it is unlikely that the cause of the lack of a secondary $i$-band maximum is the same as in sub-luminous SNe~Ia. In the case of 2003fg-like SNe, it \red{suggests} that a significant amount of luminosity is not produced by the radioactive decay $^{56}$Ni, or that there is full mixing in the ejecta \citep[e.g.,\ ][]{Kasen06}. However, as is seen in 2002cx-like SNe, if there were full mixing in the ejecta of 2003fg-like SNe\ we would expect to see a prominent $H$-band break and even lower ejecta velocities in the photoperic phase \cite[e.g.,\ ][]{Kromer15,Stritzinger15}. Furthermore, if 2003fg-like SNe\ were powered predominately by the radioactive decay of $^{56}$Ni\ it would be expected that they would have a very distinct secondary $i$-band maximum, such as in normal or 1991T-like SNe~Ia. This is not the case for 2003fg-like SNe.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{lightcurvecomp.pdf}
\caption{Comparison of $BVri$-band light curves of the 2003fg-like SNe\ and normal SNe~Ia. The light curves are presented relative to $B$-band maximum and in the rest frame. They have also been normalized to the peak of each respective band. Normal CSP SNe~Ia are plotted in light grey for comparison. The selection criteria for the normal SNe are: $\Delta{\rm m_{15}(B)}<$1.3~mag and $E(B-V)_{host}<$0.15~mag.}
\label{fig:compLC}
\end{figure*}
The NIR light curves of 2003fg-like SNe\ are vastly different than normal SN~Ia (Fig.~\ref{fig:compLCNIR}). Ten of the 2003fg-like SNe\ in the sample have NIR light curves. They are in general much brighter than the normal population in the NIR. None of the 2003fg-like SNe\ have a clear secondary maximum in the $Y$ or $H$ bands. The diversity between the 2003fg-like SNe\ is also large. For all 2003fg-like SNe, the phase of the NIR primary maxima occurs significantly after that of $B$-band maximum, whereas the NIR primary maxima of the normal population consistently transpires a few days before their $B$-band maxima. For example, the $H$-band light curve of SN~2012dn peaks $\sim$50~d past $B$-band maximum. As a consequence, the NIR, specifically the $H$ band, may be the most effective way to distinguish 2003fg-like SNe\ from normal SNe. These prolonged NIR light curves of 2003fg-like SNe\ imply that there could be some additional sources of luminosity \citep{Nagao17,Nagao18}.
\begin{figure*}[tb]
\centering
\includegraphics[width=0.99\textwidth]{LC_M_NIR.pdf}
\caption{Absolute-magnitude $YJH$ light curves of the 2003fg-like SNe\ in the sample. The light curves have not been K- or S-corrected. Normal SN~Ia light-curve templates of s$_{BV}$=1.00, 1.05, 1.10, 1.15, and 1.20 from \textit{SNooPy} are plotted in grey colors for comparison, with the peak magnitude matching to the luminosity–decline rate relation from Fig.~4 of \citealt{Burns18}. We chose not to normalize the NIR light curve comparison to peak as the time of the NIR maximum is uncertain for many objects. Overall, the NIR photometric properties of 2003fg-like SNe\ are distinct from those of normal SNe~Ia. }
\label{fig:compLCNIR}
\end{figure*}
Another intriguing photometric peculiarity of 2003fg-like SNe\ is their UV luminosity. 2003fg-like SNe\ are $\sim$2~mag brighter than normal SNe~Ia in the mid-UV \citep{Brown14}. \citet{Brown14} analyzed the UV properties of SN~2009dc and SN~2012dn, which are located at opposite ends of the 2003fg-like SNe\ peak luminosity distribution in the optical, and found that both of these objects peak at $\sim-$18~mag in the swift uvm2 band. \citet{Lu21} analyzed the swift uvm2 light curves of all published 2003fg-like SNe\ and confirmed these results. The cause of this excess flux is unknown. It is not likely to be due to a larger amount of $^{56}$Ni\ in the outer layers, as the ionization state of 2003fg-like SNe\ is generally low, and the $H$-band break is not observed until past +50\,d from maximum light (Section~\ref{sec:nir_spec}). The high UV flux could be caused by low metallicity of the progenitor, where a low metallicity produces a lack of Fe-group elements in the outer layers and thus a lack of line blanketing \citep{Mazzali14}. It could also be produced by an additional energy source which is not $^{56}$Ni\ such as interaction with any surrounding material \citep[e.g.,][]{Nagao17, Hsiao20}. \red{For a discussion on bolometric light curves see section \ref{sect:bolLC}. }
Among the thirteen objects in the sample, six have early light curves ($< -10~d$) and constraints on the first epoch.
\red{The rise times of the two SNe from this work, CSS140501 and SN~2015M, are obtained by fitting the pre-maximum $V$-band photometry with a second order polynomial function, where the data was constrained by the discovery survey's photometry and last non-detection limits as mentioned in Section~\ref{sec:Discoveryinfo}.
The remaining the rise times were obtained from literature. The rise time of SN~2007if was obtained from an unfiltered magnitude \citep{Scalzo10}. The rise time of SN~2009dc is given in the $R$-band \citep{Silverman11}, and ASASSN-15hy \citep{Lu21} and ASASSN-15pz \citep{Chen19} are provided in the $V$-band.
} These SNe and their respective rise times are SN~2007if ($24.2 \pm 0.4~d$; \citealt{Scalzo10}), SN~2009dc ($23 \pm 2~d$; \citealt{Silverman11}), ASASSN-15pz ($21.4 \pm 2~d$; \citealt{Chen19}), ASASSN-15hy ($22.5 \pm 4.6$; \citealt{Lu21}), CSS140501 ($20.9 \pm 6.7$; this work), and SN~2015M ($19.8 \pm 4.8$; this work). This implies an average rise time of $22.0 \pm 3.8$~d.
\subsection{Luminosity Width Relation}
\label{sect:LWR}
SNe~Ia follow an intrinsic luminosity width relation (LWR), where
brighter SNe have broader light curves. Two common ways to determine
the broadness and the time scale of the light curve are the parameters
$\Delta{\rm m_{15}(B)}$ \citep{Phillips93} or s$_{BV}$\ \citep{Burns14}. $\Delta{\rm m_{15}(B)}$ measures the change in $B$-band magnitude between maximum light and 15 rest-frame days past then, and s$_{BV}$\ is the time difference between the occurrence of $B$-band maximum and the reddest point in the $(B-V)$ color curve divided by 30~d \citep{Burns14}.
To establish the location of 2003fg-like SNe\ on the LWR, the light-curve parameters were measured using the \textit{SNooPy} package \citep{Burns11}. No light curve templates were used to fit the photometry or their derived parameters. Rather, the $B_{max}$, $\Delta{\rm m_{15}(B)}$ and s$_{BV}$\ parameters were directly measured from the rest-frame K-corrected light curves interpolated with Gaussian processes. The \textit{SNooPy} \textsc{get\_color} function was used to calculate the color curves. For the color curves, no interpolation between data was performed when multi-band observations on the same night were not available. The color curves were produced with Gaussian processes to obtain the time of the reddest point in the $(B-V)$ color curve relative to $B$-band maximum. This value was divided by 30 to obtain s$_{BV}$.
This same technique was used to obtain the color curves in Section \ref{sect:colorcurv}, and to derive the colors at maximum light. For all photometric measurements, uncertainties were obtained using a bootstrapping technique with 150 iterations. All rest frame, K-corrected, light curves were corrected for galactic and host extinction (where applicable).
$B_{max}$ was converted to $M_{B}$ using the distance moduli presented in Table \ref{table:prop}.
The $B$-band LWR as a function of $\Delta{\rm m_{15}(B)}$ and s$_{BV}$\ is presented in the top panels of Fig. \ref{fig:LWR}.
The 2003fg-like SNe\ are all slowly declining with $\Delta{\rm m_{15}(B)}<$1.3\,mag and s$_{BV}$$>$1.
2003fg-like SNe\ are located below, above and in the same area of the LWR as normal SNe~Ia.
SN\,2012dn, SN\,2013ao, ASASSN-15pz, and SN~2015M are all located in the main part of the LWR.
SN2006gz, LSQ12gpw and ASASSN-15hy are less luminous than their $B$-band light curve shape would imply if they were standard SNe~Ia (i.e. they are less luminous than the LWR), whereas, SN\,2007if, SN\,20009dc, and LSQ14fmg are all brighter than the LWR.
Unlike the previous suggestion of \citet{Taubenberger11}, it is evident with a larger sample that not all 2003fg-like SNe\ are overluminous. However, host galaxy extinctions of 2003fg-like SNe\ are highly uncertain and this may affect the location of some SNe on the LWR, although for the majority of points there is a reliable limit from the Na ID EW (Section~\ref{sect:Extinction}). We note that most of our objects are within the Hubble flow (z$>$0.02) therefore the distance derived to the SN from the hosts are accurate.
The NIR is less affected by extinction, therefore it is a more suitable wavelength range to
analyze any intrinsic luminosity differences between 2003fg-like SNe\ and other SNe~Ia.
We do not attempt to correct the NIR photometry for host extinction.
The bottom panels of Fig.~\ref{fig:LWR} show the LWR in the $J$ and
$H$ bands. Unfortunately, the temporal coverage in the NIR is much
worse than the optical. Therefore, for all SNe except SN~2009dc and
SN~2012dn only lower flux limits can be set for the peak
luminosity. The lower limits were determined by measuring the
brightest photometric point on the rising light curves. In the
$J$-band, most of the SNe are over luminous (brighter than $-$19~mag)
except for ASASSN-15pz, CSS140126, and SN~2012dn that have a
luminosity similar to that of normal SNe~Ia. However for ASASSN-15pz and CSS140126the these values are lower limits on the luminosity and they could be intrinsically brighter. Interestingly, the
$H$-band is the most effective wavelength range to distinguish between
2003fg-like SNe\ and normal SNe~Ia. All of the 2003fg-like SNe\ are brighter than normal
SNe~Ia in the $H$ band.
They range from $M_{H}=-18.76$~mag (SN~2015M), to $M_{H}=-20.19$~mag (LSQ14fmg). The bright $H$-band is one of the few ubiquitous properties of the subclass of 2003fg-like SNe. \red{The fact that 2003fg-like SNe\ are not standarizable may suggest that that there is more than one driving parameter driving the explosion. i.e. More than just a range of WD masses. There could be a range of both WD masses and envelope masses which produce 2003fg-like SNe\, see section \ref{sect:discussion} for a detailed discussion.}
As shown by \citet{Galbany21}, 2003fg-like SNe\ are preferentially located in low metallicity and low mass galaxies with high sSFR, and are therefore more common in the high redshift universe. The fact that 2003fg-like SNe\ are not standardizeable and they do not follow the LWR in the optical or NIR could have direct consequences for dark energy experiments. We have shown here that it is easier to remove 2003fg-like SNe\ from cosmological experiments with rest frame NIR data. But due to cosmic expansion, this strategy may limit future dark energy experiments to lower redshifts. With only near-maximum light, rest-frame, optical observations, 2003fg-like SNe\ may bias dark energy experiments. We briefly discuss this in Section~\ref{sect:Hubble}. However, detailed simulations of this is outside the scope of this work.
\begin{figure*}
\centering
\includegraphics[width=.99\textwidth]{LWR_full.pdf}
\caption{The luminosity width relation. \textit{Top left:} The absolute $B$-band magnitude ($M_B$)plotted as a function of $\Delta{\rm m_{15}(B)}$. \textit{Top right:} $M_B$ plotted as a function of s$_{BV}$. In both top panels the open symbols are corrected for galactic and host-galaxy extinction. \textit{Bottom left:} $M_{J}$ as a function of $\Delta{\rm m_{15}(B)}$. \textit{Bottom right:} $M_{H}$ as a function of $\Delta{\rm m_{15}(B)}$. For both of the bottom panels, many of the points are lower flux limits as the light curves are still rising during the final photometric observations. The 2003fg-like SNe\ points in the NIR LWR have not been corrected for host-galaxy extinction as the extinction in the NIR is negligible and the values of host-galaxy extinction are uncertain. In all plots the gray symbols are the LWR relation constructed using SNe~Ia observed by the CSP \citep{Krisciunas17,Phillips19}. }
\label{fig:LWR}
\end{figure*}
\subsection{Color curves}
\label{sect:colorcurv}
The observed color curves of the 2003fg-like SNe, corrected for Milky Way extinction, are shown in Fig.~\ref{fig:colorLC}.
At early times, the $(B-V)$ curves are similar to normal SNe~Ia.
They start red and reach their bluest point around maximum light. Between maximum light and +40\,d, the ejecta cool until the reddest epoch is reached, after which the colors turn blue again. However, 2003fg-like SNe\ do not follow a tight Lira-like law \citep{Phillips99}, and exhibit significant diversity. The reddest points in $B-V$ color curves cover a larger range ($\sim$0.7-1.3\,mag) in 2003fg-like SNe\ than in normal SNe~Ia. After the turnover, the 2003fg-like SNe\ events show a variety of gradients in the Lira tail. As discussed by \citet{Lu21}, ASASSN-15hy is unusual regarding its $(B-V)$ color curve. It does not get bluer during the early phase, and the evolution towards redder $(B-V)$ occurs much earlier than other SNe~Ia. The ejecta begin to cool and get redder from $\sim -$10\,d relative to $B$-band maximum, reaching $(B-V) \approx 0.2$\,mag at maximum light. \citet{Lu21} interpreted this to be caused by a lower $^{56}$Ni\ mass explosion in the core degenerate scenario.
The $(r-i)$ color curves of 2003fg-like SNe\ show the largest diversity and generally do not behave like normal SNe~Ia. There is a continuum of properties from LSQ14fmg which gets monotonically redder from the first measurement then stays flat after +25\,d, to SN~2015M and CSS140126 which reach the bluest values. Generally, the 2003fg-like SNe\ start at similar $(r-i)$ values as normal SNe~Ia, but do not reach such large negative values. More interestingly, at +50\,d relative to maximum the 2003fg-like SNe\ $(r-i)$ color curves remain flat and in some cases continue to turn redder. This is caused by light curves at longer wavelengths in 2003fg-like SNe\ being much broader than normal SNe~Ia. In large all-sky surveys such as ZTF and the Vera Rubin Observatory, analyzing the $(r-i)$ color curve up to +50~d may be one of the most effective ways to distinguish 2003fg-like SNe\ from the general population of SNe~Ia.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{colorcurve_snpy.pdf}
\caption{The observed color curves of our 2003fg-like SNe\ sample corrected for Milky Way extinction (solid markers) and compared to a sample of normal SNe~Ia from the CSP (light gray). The normal SNe~Ia were selected so they have $\Delta{\rm m_{15}(B)}<1.3$~mag and an $E(B-V)_{host}<$0.15~mag. The $B-V$ color curves of 2003fg-like SNe\ are similar to those of normal SNe~Ia, yet the $r-i$ color curves differ significantly. All curves have been corrected for Galactic extinction, but not host galaxy extinction. }
\label{fig:colorLC}
\end{figure*}
\subsection{s$_{BV}$\ vs. t$^{i-B}_{max}$}
\citet{Ashall20} demonstrated that the timing of the $i$-band primary maximum relative to the $B$-band maximum can be used as a powerful diagnostic to distinguish between sub-types of thermonuclear SNe. It was found that for 2003fg-like SNe\ the time of the primary $i$-band maximum was later than that of $B$ band. Combining this information with s$_{BV}$\ was found to be an excellent way to identify 2003fg-like SNe. Fig.~\ref{fig:sBViband} shows the relation from \citet{Ashall20} labeled with the 2003fg-like SNe\ from this work.
Interestingly, ASASSN-15pz, which has a hint of an $i$-band secondary inflection, also has the smallest value of t$^{i-B}_{max}$. Although note that CSS140126 has both a strong $i$-band secondary maximum and a late t$^{i-B}_{max}$. Conversely, ASASSN-15hy has the largest value of t$^{i-B}_{max}$\ and no secondary $i$-band maximum. This may indicate a connection between the value of t$^{i-B}_{max}$\ and the presence of an $i$-band secondary maximum.
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{sBVimax.pdf}
\caption{The time of the $i$-band maximum relative to the $B$-band maximum vs. s$_{BV}$\ with data taken from \citet{Ashall20} and 2003fg-like SNe\ of this work. These values do not include K-corrections since they are of various subtypes, this is also consistent with the analysis of \citet{Ashall20}. All of the 2003fg-like SNe\ are located in the top right corner of the figure. }
\label{fig:sBViband}
\end{figure}
\subsection{Bolometric light curves} \label{sect:bolLC}
Pseudo-bolometric light curves were constructed using \red{observed photometry} by employing the the direct method in \textit{SNooPy}. Where needed, the observation gaps in between light curves were interpolated with Gaussian processes,
but no extrapolations were applied outside the time range of individual light curves. Three wavelength regions were selected in order to explore the pseudo-bolometric peak luminosity and flux ratios in the UV, optical, and NIR.
The light curve of a normal SN~Ia 2007af \citep{Krisciunas17} was also constructed using the same method as above for comparison.
The $BVri$ ($4200$ to $7300$~\AA) pseudo-bolometric light curves are presented in the top panel of Fig.~\ref{fig:BVri_JH_BoloLC}.
The average peak luminosity in the sample is $L_{\rm{peak}} = 10^{42.97 \pm 0.16}$ erg~s$^{-1}$ for 11 2003fg-like SNe\ (SN~2003fg and LSQ12gpw are excluded due to their poor data coverage around the peak). When the NIR ($\sim \lambda 4200 - 16{,}000$~\AA) is also included in the construction of the bolometric light curves 2003fg-like SNe\ peak at $L_{\rm{peak}} = 10^{43.16 \pm 0.22}$ erg~s$^{-1}$ from the four SNe that have available peaks.
Interestingly, the fraction of flux in the NIR is generally higher in all 2003fg-like SNe\ compared to the normal SN~Ia 2007af, as shown in the bottom panel in Fig.~\ref{fig:BVri_JH_BoloLC}. This is consistent with with 2003fg-like SNe\ having high NIR peak luminosities.
All 2003fg-like SNe\ gradually increase in NIR flux fraction until one month after the maximum.
It should be noted that SN~2012dn (which has the same luminosity as a
normal SN~Ia) shows the most prolonged and largest NIR contribution
that persists well past +30~d.
For four 2003fg-like SNe\ it was possible to construct UV and NIR ($\sim \lambda 2200 - 16{,}000$~\AA) pseudo-bolometric light curves. Both ASASSN-15hy and SN~2012dn have a peak luminosity of $L_{\rm{peak}} = 10^{43.17 \pm 0.01}$ erg~s$^{-1}$.
Despite the small size of the sample, the effect of the bright UV and NIR as well as the flux redistribution are clearly demonstrated (see Fig.~\ref{fig:BoloLC_ratios}).
In 2003fg-like SNe\ there is no increase in UV flux at early times, unlike in normal SN~Ia.
The UV fraction of the bolometric flux is already declining when 2003fg-like SNe\ were first observed, while the UV fraction increases until just before maximum light for normal SNe Ia.
The 2003fg-like SNe\ also show a low optical ($\sim \lambda 4200 - 7300$) luminosity fraction and high NIR ($\sim \lambda 7300 - 16{,}000$) fraction compared to the normal SN~Ia 2007af.
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth]{BoloLC_BVri_JH.pdf}
\caption{Pseudo-bolometric light curves of 2003fg-like SNe. The top panel shows the Pseudo-bolometric light curves constructed with optical $BVri$ bands, the middle panel presents those with optical $BVri$ and NIR $JH$ bands, and the bottom panel displays the fraction of the NIR luminosity.}
\label{fig:BVri_JH_BoloLC}
\end{figure}
\begin{figure*}[tb]
\centering
\includegraphics[width=0.8\textwidth]{BoloLC_ratios.pdf}
\caption{Fractions of pseudo-bolometric luminosity of 2003fg-like SNe\ in UV ($\sim \lambda 2200 - 4200$), optical ($\sim \lambda 4200 - 7300$) and NIR ($\sim \lambda 7300 - 16{,}000$) regions. Unlike normal SNe~Ia that have the optical region accounting for $\sim$80\% of the luminosity, 2003fg-like SNe\ all show a significant fraction of luminosity in the UV and NIR. The time evolution of the fractions also shows the flux redistribution from UV to NIR.}
\label{fig:BoloLC_ratios}
\end{figure*}
\subsection{Hubble residuals}
\label{sect:Hubble}
While the exact rate of 2003fg-like SNe\ is unknown they are extremely rare and make up a very small fraction of SNe~Ia in the local universe. However, they prefer low-mass and high specific star forming host galaxies both of which will increase their fraction in high-redshift SNe surveys. They do not follow the LWR, have broad light curves, and some 2003fg-like SNe\ are not over-luminous and overlap normal SNe~Ia in peak brightness.
Furthermore, in the rest frame $B$ and $V$ bands, it is difficult, if not impossible, to distinguish 2003fg-like SNe\ from normal SNe~Ia.
Therefore, 2003fg-like SNe\ have the potential to bias dark energy experiments. Although a full simulation of this is beyond the scope of this paper, we determine what the Hubble residuals would be if 2003fg-like SNe\ were treated as normal SNe~Ia and fit with light-curve fitting tools.
To determine the Hubble residual for each 2003fg-like SNe, we used the \textit{SNooPy} \textit{EBV\_model2}, with \textit{st} (which is the input setting for s$_{BV}$) as the light curve shape parameter. To `simulate' the effect of future dark energy experiments such as those from the Nancy Grace Roman Space Telescope, we fit the light curves from +0 to +30~d relative to $B$-band maximum. The light-curve fitting is done twice, once with only the $B$- and $V$- bands and then with all of the bands from UV to NIR. For both fits, \red{most} of the 2003fg-like SNe\ have negative Hubble residuals (see Table~\ref{table:hubble}). This implies that when run though light curve fitters, 2003fg-like SNe\ are too bright for their light curve shape.
Fig.~\ref{fig:hubble} presents the Hubble residuals as a function of light-curve decline rate.
The mean residuals are $\Delta\mu(all)=-0.74\pm0.02$~mag and $\Delta\mu(BV)=-0.48\pm0.50$~mag. The Hubble residuals are generally smaller and the light curves fit better to the templates when only the $B$ and $V$ bands are used. Thus, in the case that only rest-frame $B$ and $V$ bands are observed, 2003fg-like SNe\ may not be identified and removed in dark energy experiments and will cause a bias. A more detailed simulation is warranted to determine the true extent of this contamination and is beyond the scope of this work.
\begin{deluxetable}{ c c c}
\tablewidth{\textwidth}
\tablecaption{The Hubble residual of the 2003fg-like SNe\ in the sample. The values were calculated using \textit{SNooPy} \textit{EBV\_model2} and compared to the CMB corrected redshift distance. The cosmological parameters used for this section are $H_{0}$=73\,km\,s$^{-1}$\,Mpc, $\Omega_{m}$=0.27 $\Omega_{\Lambda}$=0.73. The Hubble residuals were computed using two cases: 1) $B$ and $V$ bands only and 2) all of the available bands ($uvuBVgrizYHJ$). \label{table:hubble}}
\tablehead{
\colhead{SN}&
\colhead{$\Delta\mu$($BV$)}&
\colhead{$\Delta\mu$($all$)}\\
\colhead{}&
\colhead{Mag}&
\colhead{Mag}}
\startdata
2003fg & $\cdots$ & $\cdots$ \\
2006gz & 0.21 $\pm$ 0.09 & 0.18 $\pm$ 0.04 \\
2012dn & 0.08 $\pm$ 0.05 & $-$0.32 $\pm$ 0.02 \\
ASASSN-15pz & $-$0.24 $\pm$ 0.09 & $-$0.84 $\pm$ 0.03 \\
2007if & $-$1.35 $\pm$ 0.09 & $-$1.66 $\pm$ 0.04 \\
2009dc & $-$0.13 $\pm$ 0.04 & $-$0.99 $\pm$ 0.02 \\
LSQ12gpw & $-$0.15 $\pm$ 0.09 & $-$0.15 $\pm$ 0.08 \\
2013ao & $-$0.99 $\pm$ 0.03 & $-$0.51 $\pm$ 0.03 \\
CSS140126 & 0.00 $\pm$ 0.11 & $-$0.37 $\pm$ 0.03 \\
CSS140501 & $-$1.20 $\pm$ 0.07 & $-$1.15 $\pm$ 0.04 \\
LSQ14fmg & $-$0.72 $\pm$ 0.06 & $-$1.77 $\pm$ 0.08 \\
2015M & $-$0.78 $\pm$ 0.03 & $-$0.55 $\pm$ 0.02 \\
ASASSN-15hy & $-$0.44 $\pm$ 0.05 & $-$0.71 $\pm$ 0.03 \\
\enddata
\end{deluxetable}
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{Hubbleres.pdf}
\caption{Hubble residuals of a selection of normal SNe~Ia from the CSP (black open markers) and of the 2003fg-like SNe\ fit with both the $B$ and $V$ bands (open symbols) and all available bands (solid symbols). The Hubble residuals are smaller if only the $B$ and $V$ bands are used. \label{fig:hubble}}
\end{figure}
\section{Spectroscopic properties}
\label{sect:spec}
In this section, the spectroscopic properties of 2003fg-like SNe\ are presented. First, we concentrate on optical wavelength spectra and line identifications. Then we discuss the NIR spectra and their line identifications. The velocity and pseudo equivalent width (pEW) measurements, and properties including the Branch diagram are then presented.
\subsection{Optical wavelength spectra}
All available maximum light and +20~d spectra of 2003fg-like SNe\ are presented in Fig.~\ref{fig:maxspec}.
At maximum light, the spectra
show the standard lines associated with SNe~Ia \citep[e.g.,\ ][]{Branch06,Ashall18}, see table \ref{table:ions} Many of the 2003fg-like SNe\ also have strong C~{\sc ii}\ $\lambda$6580 and $\lambda$7234
features persisting through maximum light. 2003fg-like SNe\ also have weak Ca~{\sc ii}\ features at this phase.
By +20~d from maximum light the Ca~{\sc ii}\ feature is much stronger, and \red{the spectrum contains} no residual C~{\sc ii}.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{maxlightspec.pdf}
\caption{\textit{ Left panel:} Maximum light spectra of all of the 2003fg-like SNe\ in the sample. \textit{Right panel:} 2003fg-like SNe\ spectra at +20~d for objects which have spectra at this phase. }
\label{fig:maxspec}
\end{figure*}
The maximum light and +20~d spectrum of SN\,2009dc and SN\,2013ao are compared to a variety of sub-types of SNe~Ia in Fig.~\ref{fig:maxspeccomp}. SN\,2009dc and SN\,2013ao were chosen for this comparison as they are located at the extreme ends of the luminosity parameter space.
Both SN~2009dc and SN~2013ao appear to have slightly weaker or ``washed out'' spectral features compared to the other SNe.
However, in terms of ionization state SN~2013ao appears to be most similar to SN\,2006bt \citep{Foley10}, and SN\,2011fe \citep{Mazzali14}. With maximum light spectra alone, it is almost impossible to distinguish between SN\,2013ao and the normal SN\,2011fe, and the only noticeable difference are weaker Ca~{\sc ii}\ features in SN~2013ao. On the other hand, SN~2009dc has a similar ionization state to SN~2013ao, but also has strong C~{\sc ii}\ absorption and lower velocities, as is seen with the Si~{\sc ii}\ $\lambda$6355 feature. Although SN\,2009dc appears to be as blue as SN\,1991T, the lack of Fe~{\sc iii}\ and a stronger Si~{\sc ii}\ $\lambda$5972 feature demonstrate that the ionization state in the line forming region of SN~2009dc is lower.
\citet{Hachinger12} claim these ``washed out'' features and the low ionization state in 2003fg-like SNe\ are produced by an additional thermal luminosity source, which is H/He deficient. \citet{Hsiao20} suggest that this could be due to interaction with a C-O envelope in the core degenerate scenario. \citet{Taubenberger19} propose that it could be caused by the violent merger of two WDs, although the low continuum polarization makes the latter unlikely \citep{Tanaka10,Cikota19}.
\begin{deluxetable}{ c c }
\tablewidth{\textwidth}
\tablecaption{The main spectra lines identified in the 2003fg-like SNe\ spectra at maximum light. \label{table:ions}}
\tablehead{
\colhead{Ion}&
\colhead{Wavelength}\\
\colhead{}&
\colhead{$\mathrm{\AA}$}}
\startdata
Ca~{\sc ii}\ & $\lambda\lambda$3968, 3933 \\
Si~{\sc ii}\ &$\lambda$4130 \\
Mg~{\sc ii}\ & $\lambda$4481\\
Si~{\sc iii}\ & $\lambda$4552\\
Fe~{\sc ii}\ &$\lambda$5169 \\
Fe~{\sc iii}\ & $\lambda$5156\\
S~{\sc ii}\ & $\lambda$5453,$\lambda$5606\\
Si~{\sc ii}\ & $\lambda$5972,$\lambda$6355\\
O~{\sc i}\ &$\lambda$7771\\
Ca~{\sc ii}\ & $\lambda\lambda$8498, 8542, 8662\\
\enddata
\end{deluxetable}
At +20~d, the spectra of all sub-types of SNe~Ia are similar. All of the 2003fg-like SNe\ and comparison SNe~Ia except SN~2013ao have an emission feature in the 5900~\AA\ region. This feature has been attributed to either [Co~{\sc iii}] 5888~\AA\ \citep{Dessart14} or Na~{\sc i}~D emission \citep{Mazzali08}.
If this feature is attributed to [Co~{\sc iii}], the lack of this emission
in SN~2013ao may be caused by the lack of $^{56}$Ni\ above the
photosphere. This is consistent with the lack of an $H$-band break in the NIR spectra of SN~2013ao. Note however that SN~2009dc, SN~2015M, and ASASN-15hy do not have an $H$-band break at these epochs, but all show this emission feature at 5900~\AA. We thus conclude that the feature is more likely caused by Na~{\sc i}~D emission, which is not seen in SN~2013ao due to higher temperature and density in the ejecta. Alternatively, differences in the progenitor configuration, including metallicity differences, could produce a reduced Na abundance.
\begin{figure*}
\centering
\includegraphics[width=.99\textwidth]{photspeccomp.pdf}
\caption{Comparison between maximum light (\textit{left}) and +20~d (\textit{right}) spectra of the over-luminous 2003fg-like SN~2009dc and an under-luminous one (SN~2013ao) along with a selection of other SNe~Ia sub-types. }
\label{fig:maxspeccomp}
\end{figure*}
One of the easiest ways to distinguish between 2003fg-like SNe\ and other
sub-types of SNe~Ia is with spectra at $-10$~d with respect to maximum
light (Fig.~\ref{fig:m10spec}). At this epoch, 2003fg-like SNe\ have weak
features and are dominated by continuum, Si~{\sc ii}\ absorption, and have
very weak or no Ca~{\sc ii}\ and Fe~{\sc iii}\ features. On the other hand,
SN~2011fe is redder and has strong P Cygni profiles which include a
large amount of intermediate mass elements such as a strong Si~{\sc ii}\
$\lambda$6355 and Ca~{\sc ii}\ features. Similar to 2003fg-like SNe, SN\,1991T has a
hot continuum and no Ca~{\sc ii}\ features. However, SN\,1991T has a strong
Fe~{\sc iii}\ feature at $\sim$4900~\AA\ and no C~{\sc ii}\ absorption, which is
unlike 2003fg-like SNe. With this in mind, we suggest that the lack of a strong
Fe~{\sc iii}\ absorption in early-time spectra should become one of the defining characteristics of 2003fg-like SNe.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{m10dspec.pdf}
\caption{Early-time spectroscopic comparison between 2003fg-like SNe, the normal SN~2011fe, and the over-luminous SN~1991T. All 2003fg-like SNe\ with adequate data tend to show strong C~{\sc ii}\ and no significant Fe~{\sc iii}. In contrast, SN~1991T has strong Fe~{\sc iii}\ and SN~2011fe generally has stronger features. In these phases, it is possible to distinguish 2003fg-like SNe\ from other sub-types of SNe~Ia. }
\label{fig:m10spec}
\end{figure}
Several hundred days post explosion,
the ejecta of SNe are optically thin and dominated by forbidden transitions (see \citealt{Taubenberger19}).
The ionization state of these lines provides critical information about the rate of recombination and the density in the center of the explosion.
Normal SNe~Ia have nebular spectra which are dominated by strong [Fe~{\sc iii}] at $\sim$4700~\AA\ and a weaker [Fe~{\sc ii}] emission at $\sim$5200~\AA, with an [Fe~{\sc ii}]/[Ni~{\sc ii}]/[Ca~{\sc ii}] emission at $\sim$7300~\AA\ \citep[e.g.,\ ][]{Graham17}. Luminous SNe~Ia, such as SN~1991T, have a higher ionization state with strong [Fe~{\sc iii}] lines \citep{Cappellaro01}, and sub-luminous SNe~Ia tend to have a stronger [Ca~{\sc ii}] and weaker [Fe~{\sc iii}] emission than normal SNe~Ia \citep[e.g.,\ ][]{Mazzali12,Galbany19}.
Four 2003fg-like SNe, SN~2006gz \citep{Maeda09},
SN~2007if \citep{Taubenberger13a}, SN~2009dc \citep{Taubenberger13a}, and SN~2012dn \citep{Taubenberger19} have nebular phase spectra. Interestingly, despite being luminous, all of these objects show a low ionization state with weak [Fe~{\sc iii}] emission. They also have strong [Ca~{\sc ii}] emission. SN~2012dn (the least luminous of the four) has the strongest [Ca~{\sc ii}] emission, significantly stronger than the other 2003fg-like SNe. SN\,2012dn also shows [O~{\sc i}] emission at 6300~\AA, which has only been observed in the nebular phase spectra of one other SN~Ia, the sub-luminous SN~2010lp \citep{Taubenberger:2013b}.
Although sub-luminous and 2003fg-like SNe\ sit at the opposite ends of the LWR they share similar traits in nebular phase spectra of low-ionization state and strong [Ca~{\sc ii}] emission. However, in 2003fg-like SNe\ the low ionization is likely caused by low ejecta velocities (\red{see section \ref{sect:vel}}), high central densities, and an increased recombination rate; whereas in the sub-luminous SNe~Ia (e.g.,\ \ SN~1986G; \citealt{Phillips87,Ashall16b}), the low ionization state is thought to be caused by less heating owing to their smaller $^{56}$Ni\ masses.
\subsection{NIR spectra}
\label{sec:nir_spec}
NIR spectroscopy provides critical information on the physics of SNe~Ia \citep[e.g.,\ ][]{Kirshner73,Marion09,Hsiao19}.
In the NIR, the photosphere recedes faster than at shorter wavelengths, allowing for deeper parts of the ejecta to be exposed at earlier times. The NIR also contains different ions than the optical, such as C~{\sc i}\ 1.0693$\mu$m\ and the $H$-band break \red{($\sim$1.4--1.9~$\mu$m)}.
If 2003fg-like SNe\ contain a large carbon shell, it would be expected that as the ejecta cool, the ionization state of carbon would transition from singly ionized to neutral. Given that the carbon shell is large and generally dominated by C~{\sc ii}\ which is seen up to and past maximum light it is expected that there would be strong NIR C~{\sc i}\ well past maximum light. This is seen in SN~2015M that shows a distinct C~{\sc i}\ 1.0693$\mu$m\ absorption at $\sim$11 000~km~s$^{-1}$ (see Fig.~\ref{fig:specNIR}). C~{\sc i}\ may also be seen in ASASSN-15hy (see \citealt{Lu21}). For the 2003fg-like SNe\ objects which do not show C~{\sc i},
either the carbon has become optically thin, stays ionized at all epochs, or the C~{\sc i}\ line is very weak possibly due to the presence of He in the outer layers as discussed in the Appendix D of \citet{Lu21}.
The $H$-band break is formed from a multiplet of allowed Co~{\sc ii}, Fe~{\sc ii}, and Ni~{\sc ii}\ emission lines located well above the photosphere \citep{Wheeler98, Hoeflich02}. The strongest and bluest of these lines is Co~{\sc ii}\ 1.57$\mu$m. The $H$-band break appears when the photosphere recedes into the $^{56}$Ni\ region. For normal SNe~Ia, this begins a few days after maximum light, and for sub-luminous SNe~Ia, the break emerges slightly later, at $\sim$+8\,d. The later appearance can be interpreted as the photosphere having to recede through more material to reach the $^{56}$Ni\ region. The strength of the $H$-band break correlates with light curve shape, where brighter SNe have a stronger break \citep{Hsiao13}. Furthermore, the velocity of the bluest edge ($v_{edge}$) of the $H$-band region at $10\pm3$\,d can be used to directly measure the edge of the $^{56}$Ni\ region in a SN~Ia \citep{Ashall19a}, where more luminous SNe~Ia have larger values of $v_{edge}$. $v_{edge}$\ can be used to discriminate between SNe~Ia explosion models \citep{Ashall19b}.
Fig.~\ref{fig:HbreakNIr} shows the NIR spectra of the four 2003fg-like SNe\ which have spectra at $10\pm3$\,d, as well as the 1991T-like LSQ12gdj and the normal SN\,2011fe. Unlike normal, sub-luminous, and 1991T-like SNe~Ia, 2003fg-like SNe\ show a very weak or no $H$-band break by +10~d.
The lack of an $H$-band break indicates that the photospheres of 2003fg-like SNe\ have not receded into the $^{56}$Ni\ region by this time and that the mass above the $^{56}$Ni\ is large. As 1991T-like SNe~Ia have a higher ionization state but still have a strong $H$-band break, an ionization effect can be ruled out as the cause of their lack of $H$-band break.
We note that although sub-luminous SN~Ia have a weak $H$-band break it is intrinsically different from the $H$-band break in 2003fg-like SNe. SN~2009dc and ASASSN-15hy have NIR spectral observations that extend to +85 and +80~d past maximum, respectively. In these SNe the break appears at much later epochs. In SN~2009dc, the $H$-band break appears between +24 and +85~d \citep{Taubenberger11}, and in ASASN-15hy, it appears between +30~d and +80~d \citep{Lu21}. The delayed onset of the $H$-band break demonstrates that the $^{56}$Ni\ region is in the very inner layers of the ejecta and the photosphere has to recede through a large optically thick envelope before reaching the bulk of the $^{56}$Ni.
\red{With this in mind in the next section we will measure the velocities and pEWs of the early time spectra to determine the chemical composition and structure of 2003fg-like SNe. }
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{NIRHbreak.pdf}
\caption{The NIR spectra of four 2003fg-like SNe, as well as the 1991T-like LSQ12gdj and the normal SN~2011fe around +10~d past maximum light. None of the four 2003fg-like SNe\ show a strong $H$-band break. For presentation purposes the spectra have been interpolated with a gaussian filter having a 3-sigma smoothing length. }
\label{fig:HbreakNIr}
\end{figure}
\subsection{Velocity and pEW fitting method}
\label{sect:vel}
For the optical wavelength spectra, the velocity minima and pEW of the main spectral features were obtained using the Measure Intricate Spectral Features In Transient Spectra (\texttt{misfits}\footnote{\url{http://github.com/sholmbo/misfits}}; S.~Holmbo et al., in prep.) code.
To acquire the minimum of a P Cygni absorption feature, the rest frame spectra were smoothed by passing them through a low-pass filter after Fourier transforming them to remove the high-frequency noise, as described in \citet{Marion09}. An error spectrum was computed by obtaining the differences between the observed spectra and the Fourier transformed smoothed spectrum. The absolute values of the residuals are smoothed with a Gaussian function. The corresponding Gaussian smoothed version that is scaled contains 68\% of the absolute value of the residual level and is used as the 1-sigma error spectrum.
The \textit{velocity.gaussians} function was utilized to obtain the minimum of an absorption feature. The boundaries of the wavelength region were manually selected for each feature. A linear continuum and a single Gaussian function were simultaneously fit to the feature, and the best fit was determined by chi-squared minimization. To estimate the uncertainty, a Monte Carlo approach was adopted with 1000 realizations. The realizations were generated by including the flux uncertaintie assuming a normal distribution and the boundary uncertainties assuming a uniform distribution. The minimum wavelengths were converted to velocity using the relativistic Doppler formula and rest wavelength of the feature. The mean and the standard deviation of the velocities measured from the Monte Carlo realization were adopted as the value and the 1$\sigma$ uncertainty of the velocity, respectively.
The pEW of the features was calculated following the prescription of \citet{Garavini07}. The \textit{width.shallowpew} function was used in
\texttt{misfits}, where uncertainties were determined with the same Monte Carlo method mentioned above.
The mean value and standard deviation were taken as the pEW and its 1-$\sigma$ uncertainty.
\subsection{Velocity and pEW measurements}
In this work, the pEW of Si~{\sc ii}\ $\lambda$5972, $\lambda$6355 and C~{\sc ii}\ $\lambda$6580 were measured, as well as the velocity of Si~{\sc ii}\ $\lambda$6355 and C~{\sc ii}\ $\lambda$6580. The pEW of the $\lambda$6355 and $\lambda$5972 Si~{\sc ii}\ features have been shown to be a powerful diagnostic to identify SN~Ia sub-types, where normal SNe~Ia can be separated into four groups, \citep[core normal, shallow Si, broad line and cool][]{Branch06,Burrow20}. In this parameter space, most 2003fg-like SNe\ are located primarily in the shallow-silicon area close to over luminous objects such as 1991T-like objects (see Fig.~\ref{fig:branch}). However, SN\,2012dn and CSS140501 are in a similar area as core normal SNe.
These two SNe also overlap with the normal population in the LWR (Fig.~\ref{fig:LWR}).
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{branch.pdf}
\caption{The Branch diagram produced using SN~Ia data from \citet{Blondin12} and \citet{Folatelli13}. Most 2003fg-like SNe\ are located within the same area as the shallow silicon (SS) SNe~Ia.}
\label{fig:branch}
\end{figure}
The bottom panels of Fig.~\ref{fig:spec_vel}
show the velocity of the C~{\sc ii}\ $\lambda$6580 and Si~{\sc ii}\ $\lambda$6355\ features as a function of time.
The velocities decrease over time as would be expected from a homologous expansion and a receding photosphere. The velocity spread in Si~{\sc ii}\ $\lambda$6355\ between the fastest and slowest 2003fg-like SNe\ is 5,000~km~s$^{-1}$.
At the earliest epochs, around $-$10~d, the velocities range from $\sim$13,500~km~s$^{-1}$\ (SN~2006gz) to $\sim$8,000~km~s$^{-1}$\ (SN~2007if), and by maximum light the spread has decreased to 11,500~km~s$^{-1}$\ for the fastest expansion and 7,500km~s$^{-1}$\ for the slowest.
Despite the large spread, some 2003fg-like SNe\ exhibit some of the slowest
velocities of any SN~Ia. The change in velocity from the $-$10\,d to
maximum light roughly indicates the depth of the Si shell. The change in Si~{\sc ii}\ $\lambda$6355 velocity between --10~d to maximum light ranges
from 2,000~km~s$^{-1}$\ in SN~2006gz, to 1,000~km~s$^{-1}$\ in SN~2012dn, and
0~km~s$^{-1}$\ in SN~2007if.
The 2003fg-like SNe\ with the fastest Si~{\sc ii}\ velocities are consistent with the ``shallow Silicon" (SS) SNe~Ia from \citet{Folatelli13}. They are also consistent with the velocities of SN~1991T-like SNe (M.~Phillips et al., in preparation).
However, the lower velocities and early-time \red{flatter} Si~{\sc ii}\ evolution are unusual compared to normal SNe~Ia.
In some cases, 2003fg-like SNe\ may have a very confined intermediate mass element layer in velocity space, and do not show a rapid drop in velocities at the earliest phases.
This lack of an early drop may be caused by a \red{compression of the Si shell}, due to it running into an envelope \citep{Quimby06}.
In SNe~Ia when the velocity measurements reach a minimum and stay at that value for a prolonged time, it usually requires the photosphere to pass through the base of the layer.
By +10~d past maximum light the Si velocity is still declining.
However, we note that features are susceptible to ionization changes and the bottom of the Si~{\sc ii}\ layer does not always correspond to the bottom of the Si-region.
Furthermore, the red side of the Si~{\sc ii}\ feature becomes contaminated by Fe~{\sc ii}\ lines after maximum light. For most 2003fg-like SNe\ this is after +10\,d but for ASASSN-15hy this occurs 2-3\,d past maximum \citep{Lu21}. This can artificially produce a sudden velocity drop between 0 and +10~d.
The velocities of the C~{\sc ii}\ feature range between 10,000 to 16,000~km~s$^{-1}$\ at $-$10~d to 8,000~km~s$^{-1}$\ at maximum light. For some of the 2003fg-like SNe\ (LSQ12gpw, SN~2012dn, and ASASSN-15hy) the velocity of the C~{\sc ii}\ feature is lower than that of Si~{\sc ii}. This may be an indication of mixing of the C and Si layers, or it may be a projected velocity effect where the Si~{\sc ii}\ is located well above the photosphere, but the C~{\sc ii}\ is located close to the photosphere, ensuring that most of the absorption is produced from material that is not directly moving towards the observer \citep{Hoeflich90}.
The top two panels and middle panel of Fig.~\ref{fig:spec_vel} contain the pEW measurement of Si~{\sc ii}\ $\lambda$5972, $\lambda$6355 and C~{\sc ii}\ $\lambda$6580. The Si~{\sc ii}\ $\lambda$6355 feature slowly increases in pEW over time for all objects.
At early times, around $-$10\,d, the pEW of Si~{\sc ii}\ $\lambda$6355
ranges from 5-50~\AA, and rises to 20-90~\AA\ by +10~d. The pEW of the
2003fg-like SNe\ cover a larger range in values than 1991T-like SNe, which range from 0-20~\AA\ at early times to 30-50~\AA\ by +10~d, and from shallow Si objects which cover a range of 0-50~\AA\ at maximum light.
The pEW measurements of the Si~{\sc ii}\ $\lambda$5972 feature range from 0-10~\AA\ at $-$10~d, to 10-30~\AA\ at +10~d and follow a similar trend to \citet{Branch06} SS SNe. Generally, the increasing pEW of this feature is interpreted as a cooling photosphere and the Si~{\sc ii}\ $\lambda$5972 line getting populated due to the recombination of Si~{\sc iii}\ \citep[e.g.,\ ][]{Hachinger08,Ashall18}.
The pEW of C~{\sc ii}\ $\lambda$6580 is more difficult to measure as it sits on the top of the re-emission of the Si~{\sc ii}\ $\lambda$6355 feature. The pEW decreases over time for all SNe, except SN~2006gz, SN~2013ao, CSS140501, and SN~2015M which have pEW values consistent with 0.
The pEW measurments ranges from 5-20~\AA\ at early times and 0-5~\AA\ at +10\,d. Three of the 2003fg-like SNe\ (SN~2009dc, LSQ12gpw, and ASASSN-15hy) have persistent C~{\sc ii}\ features well past maximum light. Interestingly, these SNe also have the slowest Si~{\sc ii}\ $\lambda$6355 velocities and the broadest light curves. These correlations will be discussed in more detail in the next section. Note that the pEW of C~{\sc ii}\ $\lambda$6850 region was measured even if no absorption feature was visible in the spectra, hence some SNe have values consistent with zero.
\begin{figure*}
\centering
\includegraphics[width=.99\textwidth]{EWvel.pdf}
\caption{pEW and velocity measurements as a function of phase.
\textit{Top left:} Si~{\sc ii}\ $\lambda$6355 pEW as a function of phase relative to maximum.
\textit{Top right:} Si~{\sc ii}\ $\lambda$5972 pEW as a function of phase relative to maximum.
\textit{Middle left:} C~{\sc ii}\ $\lambda$6580 pEW as a function of phase from maximum.
\textit{Bottom left:} C~{\sc ii}\ $\lambda$6580 velocity as a function of phase relative to maximum.
\textit{Bottom right:} Si~{\sc ii}\ $\lambda$6355 velocity as a function of phase relative to maximum. }
\label{fig:spec_vel}
\end{figure*}
\section{Parameter study}
\label{sect:correlations}
Having measured and presented the main parameters of the 2003fg-like SNe, we now turn our attention to the correlations between these parameters and what these may imply about the physics of progenitors and explosion mechanisms. Fig.~\ref{fig:corela} presents six of the most significant correlations that were found within our data set, as well as two important non-correlations. For each correlation the least-squared best-fit line is given along with the 1-sigma uncertainty region. We show the Si~{\sc ii}\ $\lambda$6355 velocity and the C~{\sc ii}\ $pEW$, both obtained within 3 days of $B$-maximum, t$^{i-B}_{max}$, the color $(B-V)$ at B-band maximum corrected only for Milky Way extinction, the color $(B_{\mathrm{max}}-V_{\mathrm{max}})$, and the $B$-band and pseudo-bolometric peak magnitude and flux corrected for host galaxy extinction.
The p-value for each fit is provided above each panel.
Arguably the most interesting correlation is between the $pEW$ of C~{\sc ii}\ and the velocity of the Si~{\sc ii}\ $\lambda$6355, both taken at maximum light.
SNe with slower Si~{\sc ii}\ velocities tend to have larger values of C~{\sc ii}\ $pEW$. Slower Si velocities could be produced by a WD exploding inside a carbon-rich envelope. The ejecta would slow down as they run into the envelope \citep{Noebauer16}. In a simplistic picture,
a large envelope mass would produce lower ejecta velocities, as more kinetic energy would be deposited into a more massive envelope.
A larger envelope mass would also produce longer diffusion times and broader light curves, as is seen in the top middle panel of Fig.~\ref{fig:corela}, where there is a correlation between C~{\sc ii}\ $pEW$ and $\Delta{\rm m_{15}(B)}$. 2003fg-like SNe\ with broader light curves have larger C~{\sc ii}\ features and slower Si~{\sc ii}\ $\lambda$6355 velocities. These three correlations point to a non-degenerate carbon-rich envelope as being the dominant cause of the observed diversity between 2003fg-like SNe. Interestingly, these correlations are not seen in normal SNe~Ia, demonstrating that normal SN~Ia are not produced via the envelope model. For example, faster declining SNe have the slowest ejecta velocities \citep[e.g.,\ ][]{Benetti05,Gall18,Ashall18,Galbany19}.
The unique $i$-band behavior sets 2003fg-like SNe\ apart from other luminous events. 2003fg-like SNe\ generally show no secondary $i$-band maximum.
The weak $i$-band secondary requires either full mixing in the ejecta or a lack of recombination of the Fe-group layers above the photosphere \citep{Hoeflich02,Kasen06,Jack15}. In the case of 2003fg-like SNe,
the lack of an $H$-band break at +10\,d indicates that the photosphere is not within the $^{56}$Ni\ region and therefore the ejecta cannot be fully mixed. Instead, the weak $i$-band secondary maximum may be caused by a lack of recombination of Fe-group elements. The timing of the $i$-band maximum (t$^{i-B}_{max}$)\footnote{Note that in Fig.~\ref{fig:corela} t$^{i-B}_{max}$\ is obtained from the rest frame K-corrected data, whereas in Fig.~\ref{fig:sBViband} the data are not K-corrected to be consistent with the definition from \citet{Ashall20}.} allows for 2003fg-like SNe\ to be distinguished from other SNe~Ia \citep{Ashall20}. 2003fg-like SNe\ with larger values of t$^{i-B}_{max}$\ tend to have lower values of Si~{\sc ii}\ $\lambda$6355 $pEW$. This is in contradiction with the trend seen between subluminous to normal SN~Ia, where a lower value of t$^{i-B}_{max}$\ and stronger $i$-band secondary maximum are correlated with a higher ionization state and faster velocities \citep[e.g.,\ ][]{Kasen06}. This is seen in 1991T-like SNe, but not in 2003fg-like SNe\ that show a weaker Si~{\sc ii}\ $\lambda$6355 $pEW$ feature and larger values of t$^{i-B}_{max}$, possibly caused by longer diffusion timescales through the large carbon-rich outer envelope.
The t$^{i-B}_{max}$\ parameter is also found to be correlated with the observed $(B-V)$ color at maximum light, and the difference between the $B_{max}$ and $V_{max}$ magnitudes (middle left and middle right panels of Fig.~\ref{fig:corela}). In both cases, the photometry was not corrected for host-galaxy extinction. Redder 2003fg-like SNe\ have larger values of t$^{i-B}_{max}$\ indicating cooler ejecta and a larger re-processing of flux toward redder wavelengths. This is consistent with a homologous expansion which is adiabatically cooling. Line blanketing in the UV can also reprocess flux into redder wavelengths through fluorescence. It causes redder colors in SNe \citep{Mazzali00,Lentz00}. As the line blanketing is due to the presence of heavy elements in the outer layers, differences in the magnitude of the line blanketing (resulting in differences in the UV-optical colors) could be caused by differences in the metallicity of the progenitor or the shape of the outer density profile \citep{Walker12}. \red{We note that the three correlations in the middle panels of Fig. \ref{fig:corela} may be largely driven by ASASSN-15hy.}
It should be noted, however, that there is no statistically significant correlation between the peak ($B$-band or bolometric) luminosity (both corrected and not corrected for host galaxy extinction) and the Si~{\sc ii}\ velocity at maximum light. This is inconsistent with predictions from the super-$M_{Ch}$ scenario \citep{Howell06}.
The lack of a correlation here implies that more than just the mass of the exploding WD drives the luminous display.
Overall the correlations in Fig.~\ref{fig:corela} are consistent with a degenerate core exploding inside a carbon-rich envelope. This could occur in the core-degenerate scenario. We discuss this further in Section~\ref{sect:discussion}.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{Correlation_publish1.pdf}
\caption{Correlation plots between measured parameters of 2003fg-like SNe. Above each panel the p-value is given. All relationships are statistically significant except for the lower left and lower middle panels between $M_{B}$, L$^{peak}_{BVri}$ and Si velocity. For each panel a line of best fit determined by a least-squares technique is provided along with a 1-sigma uncertainty shaded region. It should be noted that t$^{i-B}_{max}$\ in this plot has been K-corrected. A full pair-plot of all parameters can be found in Appendix \ref{sec:corelationA}.}
\label{fig:corela}
\end{figure*}
\section{Discussion}
\label{sect:discussion}
Here we place into context our findings relative to three leading models of 2003fg-like SNe.
One leading model consists of the disruption of a C-O WD that exceeds the $M_{Ch}$ limit due to rapid rotation and/or high magnetic fields \citep{Yoon05,Das13}.
Alternatively the merger of two WDs could produce SNe~Ia exceeding the $M_{Ch}$-mass limit \citep[e.g.,][]{Scalzo10}.
Finally, another viable model may be the disruption of a C-O degenerate core within a dense circumstellar material environment \citep{Hachinger12,Noebauer16}. This is also referred to as an envelope model \citep{Hoeflich:Khokhlov:96}. Such an explosion could be associated with an explosion of a degenerate core of an asymptotic giant branch (AGB) star (i.e. in the core degenerate scenario; \citealt{Hsiao20, Lu21}), or from a C-O WD explosion with surrounding circumstellar dust \citep{Nagao17,Nagao18}.
We discuss each of these models below.
\subsection{Super-M$_{Ch}$ WD}
A single WD may exceed the $M_{ch}$ limit due to rapid rotation or high magnetic fields \citep{Yoon05,Das13}. The mass limit of such models is thought to be 1.8$M_{\odot}$ \citep{Yoon05}. A C/O WD exploding at such masses may be able to produce enough $^{56}$Ni\ to power the extreme luminosities observed in some 2003fg-like SNe. However, due to this scenario requiring a detonation as the explosion mechanism, it has problems producing the large amounts of unburnt carbon, and intermediate mass elements, as well as the low ionization observed in the maximum light spectra \citep{Hoeflich:Khokhlov:96}.
In the Super-M$_{Ch}$ WD scenario it was predicted that an increased total WD mass should result in a larger $^{56}$Ni\ mass and a larger binding energy which would result in higher diffusion time scales, higher luminosities, lower kinetic energies, and lower ejecta velocities \citep{Howell06}. However, in our sample there is no correlation between Si~{\sc ii}\ $\lambda$6355 velocity and peak $B$-band or bolometric magnitude\footnote{Note that the distances to the 2003fg-like SNe\ are well determined and not the cause of this lack of correlation.} (as a proxy for binding energy and $^{56}$Ni\ mass), or between $\Delta{\rm m_{15}(B)}$ and $B$-band or bolometric magnitude (both as a proxy for WD mass). Both of these correlations would be expected if the driving parameter amongst 2003fg-like SNe\ was ejecta mass. Furthermore, to produce the luminosity of the most luminous 2003fg-like SNe, such as SN~2003fg, requires an ejecta mass of 2.1$M_{\odot}$\ \citep{Howell06}. This is above the mass limit (1.8$M_{\odot}$) of a single super Super-M$_{Ch}$ WD \citep{Yoon05}.
\subsection{Dynamically merging WDs}
In the double-degenerate scenario, two WDs may dynamically merge and produce a large $^{56}$Ni\ mass. The advantage of this scenario is that the total summed mass of the two WDs can exceed 1.8$M_{\odot}$. The low levels of continuum polarization in the two 2003fg-like SNe\ that have data, SN~2007if and SN~2009dc \citep{Tanaka10, Cikota19}, make dynamical mergers an unlikely avenue to produce these 2003fg-like SNe, as such models are highly aspherical \citep{Bulla16}.
As there are many varieties of WD mass which may merge, as well as off center $^{56}$Ni\ distributions from dynamical merger models, it would also be expected that dynamical mergers do not produce the correlations seen in the data.
\subsection{Envelope model}
The data and correlations presented in this work (Fig.~\ref{fig:corela}) are consistent with the hydrogen (and possibly helium) free envelope model. In such model, a C/O WD explodes within a non-degenerate C-rich envelope (e.g., \citealt{Hoeflich:Khokhlov:96}). For a given WD mass, a more massive envelope would produce stronger carbon lines, lower Si velocities, and longer diffusion time scales. This is evident in the correlations between Si~{\sc ii}\ $\lambda$6355 velocity and C~{\sc ii}\ 6580\AA\ pEW (which is a proxy for envelope mass) and $\Delta{\rm m_{15}(B)}$ (which is a proxy for diffusion time). The more massive carbon envelope would also produce the covering mass above the $^{56}$Ni\ region which would naturally explain the observed very late onset of the $H$-band break. Furthermore, there are multiple factors affecting the luminosity. For a given WD mass, varying the envelope mass would produce a correlation between the expansion velocity and luminosity. In the case of a more massive envelope, the exploding WD would have more mass to deposit its energy into and decrease its speed, and this deposited energy would be converted into luminosity. Thus, both envelope mass and the $^{56}$Ni\ mass contribute to the observed luminosity. Additional factors may be the flame propagation speed (e.g., deflagration or detonation) as discussed in \citet{Lu21}. Such a model may also be referred to as the deflagration-core-degenerate scenario. In the envelope scenario, reprocessing of the flux from the optical to NIR in the envelope would also produce the high NIR flux observed.
A viable progenitor scenario within the envelope model configuration is the core degenerate scenario. The core degenerate scenario is the explosion of the degenerate C/O core in the center of an AGB star. The signature of a superwind detected in the observations of LSQ14fmg provides a compelling link to an AGB progenitor \citep{Hsiao20}. This class of models provides results which match the observational properties of both LSQ14fmg \citep{Hsiao20} and ASASSN-15hy \citep{Lu21}.
Finally, in the core degenerate scenario there should be significant
X-ray luminosity \citep{Lu21}. This should be searched for in future nearby events. Another prediction of the core degenerate scenario is the formation of CO in the high-density and low-temperature non-degenerate envelope. It has been proposed that active CO formation manifests as the observed rapid decline in the optical light curve at various phases in SN~2009dc, SN~2012dn, LSQ14fmg, and CSS140126 \citep{Hsiao20}. The timing of this drop is dictated by the envelope's ability to cool and is correlated with the envelope mass as indicated by the minimum of the Si~{\sc ii}\ $\lambda$6355 velocity \citep{Quimby06,Hsiao20}. Faster expanding ejecta cool faster. Another prediction from the core-degenerate scenario is an interaction with previous superwind episodes of the AGB star which may occur between 1 to 10 years after the SN explosion \citep{Hsiao20}. Observationally this appear as a UV late time re-brightening \citep[e.g.,\ ][]{Graham19}.
One important parameter for core-degenerate models is low metallicity ($Z < Z_{\odot}^{-4}$) \citep{Lu21}, which is also seen at the local environment of 2003fg-like SNe\ \citep{Galbany21}. These low metallicities may come from population I or II stars. \red{Contrary to normal SNe~Ia, 2003fg-like SNe\ show no increase in UV flux ratio at early times.} This is possibly caused by low metallicity of the progenitor and reduced line blanketing in the outer ejecta.
\section{Conclusion}
\label{sect:conclusion}
This paper presents a homogeneous sample of nine 2003fg-like SNe\ observed by the Carnegie Supernova Project I \& II, which are analyzed in addition with 4 objects from the literature. This is the most complete 2003fg-like SNe\ dataset to date.
Photometrically not all 2003fg-like SNe\ are over-luminous. In fact in the optical ($B$ and $V$ bands), they populate the main part of the LWR with absolute $B$-band magnitudes between $\sim-$19 to $\sim-$21~mag. 2003fg-like SNe\ begin to differ from normal SNe~Ia in the redder bands. In the $i$ band, 2003fg-like SNe\ peak after time of $B$-band maximum and have weak secondary maxima. In the NIR bands, 2003fg-like SNe\ are unique and are at least 1~mag brighter than normal SNe~Ia with the same optical light curve shape. Furthermore, their rise in the $H$ band can be up to 40~d longer than in the $B$ band.
Light-curve fitters determine that 2003fg-like SNe\ have negative Hubble residuals; i.e. they are too bright for their light curve shape.
As 2003fg-like SNe\ preferentially explode in low mass, low metallicity and high specific star-forming galaxies they are more prevalent in the high-redshift universe. \textit{Therefore, due to the similarity between normal and 2003fg-like SNe\ in the bluer bands ($B$\& $V$), future high-redshift cosmological surveys should ensure they obtain rest-frame NIR observations in order to minimize bias introduced by the contamination
of 2003fg-like SNe.} As this may not always be possible, it is important to carefully study 2003fg-like SNe to fully understand the bias they will cause in SN cosmology.
Optical spectra of 2003fg-like SNe\ are similar to that of normal SNe~Ia, but most have strong carbon absorption well past maximum light, as well as low velocity gradients before maximum light. In the NIR, 2003fg-like SNe\ do not show a distinct $H$-band break at $\sim$10~d. In 2003fg-like SNe, this $H$-band break is not visible until beyond 70~d past maximum light.
With our large sample of 2003fg-like SNe\ we find that the ubiquitous characteristics of all 2003fg-like SNe\ are:
\begin{itemize}
\item A broad optical light curve shape ($\Delta{\rm m_{15}(B)}<1.3$~mag).
\item The primary $i$-band peaks after the phase of $B$-band maximum.
\item A lack of strong Fe~{\sc iii}\ features in the early spectra
\item A peak $H$-band absolute magnitude brighter than $-$19~mag.
\item Carbon absorption at early times ($-$10~d from maximum light).
\item No clear $H$-band break at +10~d from maximum light.
\end{itemize}
These criteria should be used in future studies to determine if a SN is truly 2003fg-like.
In 2003fg-like SNe\ the luminous long-rising NIR light curves may be caused by the reprocessing of flux to the NIR as the result of an explosion inside a massive envelope. The lack of an early $H$-band break also demonstrates that the photosphere is not within the $^{56}$Ni\ region until a much later epoch.
These observations provide direct evidence that there is a significant amount of ejecta above the $^{56}$Ni\ region.
A number of unique and interesting correlations were found within our dataset.
There are strong correlations between the pEW of the C~{\sc ii}\ feature at maximum light, the Si~{\sc ii}\ velocity at maximum light, and $\Delta{\rm m_{15}(B)}$. 2003fg-like SNe\ with larger C~{\sc ii}\ pEWs have slower Si~{\sc ii}\ velocities at maximum light and broader light curves. These correlations are fully consistent with an envelope model where a C/O degenerate star explodes within an envelope.
In such a configuration for a given degenerate core mass a larger envelope mass would produce slower Si velocities and larger diffusion time scales.
Given that there are no H or He lines in 2003fg-like SNe\ spectra it is likely that this envelope is carbon/oxygen dominated. One promising progenitor scenario and explosion mechanism is the core degenerate scenario \citep{Kashi11,Hsiao20,Lu21}.
The data presented here provide a new critical piece of information in determining the source of diversity of 2003fg-like SNe\ and the nature of SNe~Ia in general. It is clear that 2003fg-like SNe\ are far more diverse than previously thought. Only with high-precision observations extending from optical through NIR wavelengths can the physics be clearly understood. It is apparent that simply changing the mass of the exploding WD will not produce all of the observational characteristics. Our data are consistent with a degenerate core exploding within a carbon-rich envelope, with the core degenerate scenario providing one of the strongest paths to produce 2003fg-like SN events.
\vspace{0.5cm}
\begin{acknowledgements}
The authors would like to thank Vanessa D\'{i}az for helping with data visualization.
CA and BJS are supported by NASA grant 80NSSC19K1717 and NSF grants AST-1920392 and AST-1911074.
M.S. and F.T. are supported by grants from the Villum FONDEN (28021) and the Independent Research Fund Denmark (8021-00170B).
E.B. was supported in part by NASA grant 80NSSC20K0538.
N.B.S. acknowledges support from the Texas A\&M University Mitchell/Heep/Munnerlyn Chair in Observational Astronomy.
L.G. was funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No. 839090.
The CSP has been funded by the NSF under grants AST-0306969, AST-0607438, AST-1008343, AST-1613426, AST-1613455, and AST-1613472, and in part by a Sapere Aude Level 2 grant funded by the Danish Agency for Science and Technology and Innovation (PI M.S.).
Time domain research by D.J.S. is supported by NSF grants AST-1821987, 1813466, \& 1908972, and by the Heising-Simons Foundation under grant \#2020-1864.
Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.
\end{acknowledgements}
\facilities: Magellan, du Pont, Swope, Nordic Optical Telescope
\software IRAF \citep{Tody86,Tody93}, SNooPy \citep{Burns11}, Astropy \citep{Astropy13,Astropy18}, IDL Astronomy user's library \citep{Landsman95} and misfits (S. Holmbo et al., in prep.).
\clearpage
\bibliographystyle{aasjournal}
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Theatrical Design
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Productions featured on this site Select Category My involvement (24) Lighting Designer (22) Projection Designer (7) Set Designer (13) Special Effects (5) Technical Director (14) Productions represented (24) 3 Days of Rain (1) Ash Girl (1) Bay at Nice (1) Caligari: Alaska (1) Can Can (1) Dangerous Liaisons (1) Don Juan (1) Help! Help! The Globolinks! (1) How I Learned to Drive (1) Jesus Christ Superstar (1) Laughter on the 23rd Floor (1) Les Miserables (1) Lysistrata (1) Magic Flute (1) Nickel and Dimed (1) No Exit (1) Oedipus Rex (1) Oleanna (1) Picnic (1) Play (1) Rosencrantz and Guildenstern are Dead (1) Singin' in the Rain (1) Speech & Debate (1) The Laramie Project (1) Touch (1) Two Gentlemen of Verona (1)
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<li><a href="http://ftp-master.metadata.debian.org/changelogs//main/libl/liblocale-gettext-perl/liblocale-gettext-perl_1.05-7_copyright">Copyright File</a></li>
<li><a href="http://patch-tracker.debian.org/package/liblocale-gettext-perl/1.05-7">Debian Patch Tracker</a></li>
</ul>
<h3>Download Source Package <a href="/source/wheezy/liblocale-gettext-perl">liblocale-gettext-perl</a>:</h3>
<ul>
<li><a href="http://ftp.de.debian.org/debian/pool/main/libl/liblocale-gettext-perl/liblocale-gettext-perl_1.05-7.dsc">[liblocale-gettext-perl_1.05-7.dsc]</a></li>
<li><a href="http://ftp.de.debian.org/debian/pool/main/libl/liblocale-gettext-perl/liblocale-gettext-perl_1.05.orig.tar.gz">[liblocale-gettext-perl_1.05.orig.tar.gz]</a></li>
<li><a href="http://ftp.de.debian.org/debian/pool/main/libl/liblocale-gettext-perl/liblocale-gettext-perl_1.05-7.debian.tar.gz">[liblocale-gettext-perl_1.05-7.debian.tar.gz]</a></li>
</ul>
<h3>Maintainers:</h3><ul> <li><a href="mailto:pkg-perl-maintainers@lists.alioth.debian.org">Debian Perl Group</a>
(<a href="http://qa.debian.org/developer.php?login=pkg-perl-maintainers%40lists.alioth.debian.org" title="An overview over the maintainer's packages and uploads">QA Page</a>, <a href="http://lists.alioth.debian.org/pipermail/pkg-perl-maintainers/" title="Archive of the Maintainer Mailinglist">Mail Archive</a>)
</li> <li><a href="mailto:hertzog@debian.org">Raphaël Hertzog</a>
(<a href="http://qa.debian.org/developer.php?login=hertzog%40debian.org" title="An overview over the maintainer's packages and uploads">QA Page</a>)
</li> <li><a href="mailto:ntyni@debian.org">Niko Tyni</a>
(<a href="http://qa.debian.org/developer.php?login=ntyni%40debian.org" title="An overview over the maintainer's packages and uploads">QA Page</a>)
</li> <li><a href="mailto:gregoa@debian.org">gregor herrmann</a>
(<a href="http://qa.debian.org/developer.php?login=gregoa%40debian.org" title="An overview over the maintainer's packages and uploads">QA Page</a>)
</li></ul>
<h3>External Resources:</h3>
<ul>
<li><a href="http://search.cpan.org/dist/gettext/gettext.pm">Homepage</a> [search.cpan.org]</li>
</ul>
<h3>Similar packages:</h3>
<ul>
<li><a href="/libmsgcat-perl">libmsgcat-perl</a></li>
<li><a href="/libtext-charwidth-perl">libtext-charwidth-perl</a></li>
<li><a href="/libdist-zilla-localetextdomain-perl">libdist-zilla-localetextdomain-perl</a></li>
<li><a href="/gettext-base">gettext-base</a></li>
<li><a href="/python-zope.i18n">python-zope.i18n</a></li>
<li><a href="/libintl-perl">libintl-perl</a></li>
<li><a href="/libintl-xs-perl">libintl-xs-perl</a></li>
<li><a href="/libnet-z3950-simpleserver-perl">libnet-z3950-simpleserver-perl</a></li>
</ul>
</div> <!-- end pmoreinfo -->
<div id="ptablist">
</div>
<div id="pdesctab">
<div id="pdesc" >
<h2>module using libc functions for internationalization in Perl</h2>
<p>
The gettext module permits access from perl to the gettext() family of
functions for retrieving message strings from databases constructed
to internationalize software.
<p>
It provides gettext(), dgettext(), dcgettext(), textdomain(),
bindtextdomain(), bind_textdomain_codeset(), ngettext(), dcngettext()
and dngettext().
</div> <!-- end pdesc -->
<div id="ptags"><p>
<a href="http://debtags.alioth.debian.org/edit.html?pkg=liblocale-gettext-perl">Tags</a>:
Software Development:
<a href="/about/debtags#devel::i18n">Internationalization</a>,
<a href="/about/debtags#devel::lang:perl">Perl Development</a>,
<a href="/about/debtags#devel::library">Libraries</a>,
Implemented in:
implemented-in::perl,
role::devel-lib
</p>
</div> <!-- end ptags -->
</div> <!-- pdesctab -->
<div id="pdeps">
<h2>Other Packages Related to liblocale-gettext-perl</h2>
<table id="pdeplegend" class="visual" summary="legend"><tr>
<td><ul class="uldep"><li>depends</li></ul></td>
<td><ul class="ulrec"><li>recommends</li></ul></td>
<td><ul class="ulsug"><li>suggests</li></ul></td>
<td><ul class="ulenh"><li>enhances</li></ul></td>
</tr></table>
<ul class="uldep">
<li>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/perl-base">perl-base</a>
(>= 5.14.2-3)
[not armhf, s390x]</dt> <dd lang="en">minimal Perl system
</dd>
</dl>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/perl-base">perl-base</a>
(>= 5.14.2-5)
[s390x]</dt>
</dl>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/perl-base">perl-base</a>
(>= 5.14.2-6)
[armhf]</dt>
</dl>
<li>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/perlapi-5.14.2">perlapi-5.14.2</a>
</dt> <dd lang="en"> virtual package provided by
<span id="js_perlapi-5.14.2" class="p_js_elem"></span> <span id="html_perlapi-5.14.2"><a href="/wheezy/perl-base">perl-base</a></span>
</dd>
</dl>
</ul>
<ul class="uldep">
<li>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc0.1">libc0.1</a>
(>= 2.3)
[kfreebsd-amd64, kfreebsd-i386]</dt> <dd lang="en">Embedded GNU C Library: Shared libraries <br>also a virtual package provided by
<span id="js_libc0.1" class="p_js_elem"></span> <span id="html_libc0.1"><a href="/wheezy/libc0.1-udeb">libc0.1-udeb</a></span>
</dd>
</dl>
<li>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc6">libc6</a>
(>= 2.13)
[s390x]</dt> <dd lang="en">Embedded GNU C Library: Shared libraries <br>also a virtual package provided by
<span id="js_libc6" class="p_js_elem"></span> <span id="html_libc6"><a href="/wheezy/libc6-udeb">libc6-udeb</a></span>
</dd>
</dl>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc6">libc6</a>
(>= 2.2)
[i386, mips, mipsel, powerpc, s390]</dt>
</dl>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc6">libc6</a>
(>= 2.2.5)
[amd64]</dt>
</dl>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc6">libc6</a>
(>= 2.4)
[armel, armhf]</dt>
</dl>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc6">libc6</a>
(>= 2.6)
[sparc]</dt>
</dl>
<li>
<dl>
<dt><span class="nonvisual">dep:</span>
<a href="/wheezy/libc6.1">libc6.1</a>
(>= 2.2)
[ia64]</dt> <dd lang="en">Embedded GNU C Library: Shared libraries <br>also a virtual package provided by
<span id="js_libc6.1" class="p_js_elem"></span> <span id="html_libc6.1"><a href="/wheezy/libc6.1-udeb">libc6.1-udeb</a></span>
</dd>
</dl>
</ul>
</div> <!-- end pdeps -->
<div id="pdownload">
<h2>Download liblocale-gettext-perl</h2>
<table summary="The download table links to the download of the package and a file overview. In addition it gives information about the package size and the installed size.">
<caption class="hidecss">Download for all available architectures</caption>
<tr><th>Architecture</th>
<th>Version</th>
<th>Package Size</th>
<th>Installed Size</th>
<th>Files</th>
</tr>
<tr>
<th><a href="/wheezy/amd64/liblocale-gettext-perl/download">amd64</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.9 kB</td><td class="size">89.0 kB</td>
<td>
[<a href="/wheezy/amd64/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/armel/liblocale-gettext-perl/download">armel</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.3 kB</td><td class="size">86.0 kB</td>
<td>
[<a href="/wheezy/armel/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/armhf/liblocale-gettext-perl/download">armhf</a></th>
<td class='vcurrent'>1.05-7+b3</td>
<td class="size">19.2 kB</td><td class="size">84.0 kB</td>
<td>
[<a href="/wheezy/armhf/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/i386/liblocale-gettext-perl/download">i386</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.8 kB</td><td class="size">87.0 kB</td>
<td>
[<a href="/wheezy/i386/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/ia64/liblocale-gettext-perl/download">ia64</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">23.0 kB</td><td class="size">108.0 kB</td>
<td>
[<a href="/wheezy/ia64/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/kfreebsd-amd64/liblocale-gettext-perl/download">kfreebsd-amd64</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.9 kB</td><td class="size">40.0 kB</td>
<td>
[<a href="/wheezy/kfreebsd-amd64/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/kfreebsd-i386/liblocale-gettext-perl/download">kfreebsd-i386</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.8 kB</td><td class="size">38.0 kB</td>
<td>
[<a href="/wheezy/kfreebsd-i386/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/mips/liblocale-gettext-perl/download">mips</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.1 kB</td><td class="size">91.0 kB</td>
<td>
[<a href="/wheezy/mips/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/mipsel/liblocale-gettext-perl/download">mipsel</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.1 kB</td><td class="size">91.0 kB</td>
<td>
[<a href="/wheezy/mipsel/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/powerpc/liblocale-gettext-perl/download">powerpc</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">20.1 kB</td><td class="size">89.0 kB</td>
<td>
[<a href="/wheezy/powerpc/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/s390/liblocale-gettext-perl/download">s390</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.6 kB</td><td class="size">89.0 kB</td>
<td>
[<a href="/wheezy/s390/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/s390x/liblocale-gettext-perl/download">s390x</a></th>
<td class='vcurrent'>1.05-7+b2</td>
<td class="size">20.4 kB</td><td class="size">93.0 kB</td>
<td>
[<a href="/wheezy/s390x/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
<tr>
<th><a href="/wheezy/sparc/liblocale-gettext-perl/download">sparc</a></th>
<td class='vcurrent'>1.05-7+b1</td>
<td class="size">19.5 kB</td><td class="size">88.0 kB</td>
<td>
[<a href="/wheezy/sparc/liblocale-gettext-perl/filelist">list of files</a>]
</td>
</tr>
</table>
</div> <!-- end pdownload -->
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,412 |
Q: Customize ServiceLocatorFactoryBean I have this factory interface
class TableManagerFactory {
TableManager getManager(Table table);
}
and current implementation lookups for a custom bean manager implementation or fall to a default one:
class TableManagerFactoryImpl implements TableManagerFactory {
public TableManager getManager(Table table) {
String beanName = table.getTableCode() + "Manager";
// Check for custom bean
if(!beanFactory.containsBean(beanName)) {
// Use default bean
beanName = "defaultTableManager";
}
return beanFactory.getBean(beanName);
}
Now I want to use ServiceLocatorFactoryBean based on a properties file like:
Table1=Table1Manager
Table2=AnotherTableManager
*=defaultTableManager
I have 2 problems:
*
*In ServiceLocatorFactoryBean$ServiceLocatorInvocationHandler.tryGetBeanName()
because this method just perform a lookup using Properties.getProperty().
*Table.toString() return a complex string (like "Table (tableCode) with columns...") and should be parsed to extract the tableCode
I found a not-so-easy solution based on Spring-AOP
<bean id="RegexServiceLocatorFactoryBeanStrategy_Pointcut" class="org.springframework.aop.support.NameMatchMethodPointcutAdvisor">
<property name="mappedName" value="getProperty" />
<property name="advice">
...
</advice>
</bean>
<bean id="TableManagerFactory_ServiceLocatorProperties" class="org.springframework.beans.factory.config.PropertiesFactoryBean">
<property name="properties">
<props>
<prop key="Table1">Table1Manager</prop>
<prop key="Table2">AnotherTableManager</prop>
<prop key="*">defaultTableManager</prop>
</props>
</property>
</bean>
<bean name="tableManagerFactory" class="org.springframework.beans.factory.config.ServiceLocatorFactoryBean">
<property name="serviceLocatorInterface" value="application.service.TableManagerFactory" />
<property name="serviceMappings">
<bean class="org.springframework.aop.framework.ProxyFactoryBean">
<property name="targetName" value="TableManagerFactory_ServiceLocatorProperties" />
<property name="targetClass" value="java.util.Properties" />
<property name="autodetectInterfaces" value="false" />
<property name="interfaces"><list /></property>
<property name="interceptorNames">
<value>RegexServiceLocatorFactoryBeanStrategy_Pointcut</value>
</property>
</bean>
</property>
</bean>
I wrote a complex advice a la CacheAspectSupport and one Interceptor as:
class PropertiesWithRegexInterceptor implements MethodInterceptor {
@Override
public Object invoke(MethodInvocation m) throws Throwable {
final String v = (String) m.getArguments()[0];
final Object r = transformKey(v);
if(PROCEED != r) {
m.getArguments()[0] = r;
}
return m.proceed();
}
}
I want to make advice configuration easy (and xml-based as much as possible) writing less code as possible.
I'm using Spring 2.
Any suggestions? Thanks in advance.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,006 |
{"url":"https:\/\/yogalessonmalaysia.com\/riv2rc\/machine-learning-revision-notes-a6bd95","text":"Angler Fish 3d Camera, What To Wear In Iceland In March, Iphone Not Receiving Group Texts, Strawberry Marshmallow Fluff, Why Are The Iliad And Odyssey Important, God Of War Best Armor Early, \" \/>\n+6012 233 7794 | +6012 379 1638 admin@yogalessonmalaysia.com\nSelect Page\n\n(we do not want translation very short as short translations will lead high precisions. Then the Bleu Score on bigrams can be computed as: The above equation can be used to compute unigram, bigram or any-gram Bleu scores. ExamplePick a sentence from the dev set and check our model: Sentence: Jane visite l\u2019Afrique en septembre.Translation from Human: Jane visits Africa in September. When using hard triples to train, the gradient descent procedure has to do some works to try to push these quantities further away from quantities. The two sequences may have different length. arXiv:1809. Springboard has created a free guide to data science interviews , where we learned exactly how these interviews are designed to trip up candidates! Based on the simple model above, the final output would be:$y=W^{[l]}W^{[l-1]}W^{[l-2]}\u2026W^{[3]}W^{[2]}W^{[1]}X$. Lecture Note: 1 Introduction to C C is a programming language developed at AT & T\u2019s Bell Laboratories of USA in 1972. (2020-04-03). That is: $a^{[2]} := \\frac{a^{[2]}}{p}$. For simplicity, the parameter $b^{[l]}$ for each layer is 0 and all the activation functions are $g(z)=z$. Because our goal is to build a system for our own specific domain. $\\min J(W,b)=\\frac{1}{m}\\sum_{i=1}^mL(\\hat{y}^i,y^i)+\\frac{\\lambda}{2m}||W^l||$. Usually, the default hyper parameter values are: $\\beta_1=0.9$, $\\beta_2=0.99$ and $\\epsilon=10^{-8}$. Last modified by Peggy B on Dec 3, 2020 5:17 AM. This set of notes attempts to cover some basic probability theory that serves as a background for the class. FAQs . \u2022 Make and share notes and highlights \u2022 Copy and paste text and figures for use in your own documents \u2022 Customize your view by changing font size and layout WITH VITALSOURCE \u00ae EBOOK second edition Machine Learning: An Algorithmic Perspective, Second Edition helps you understand the algorithms of machine learning. Otherwise, we may try to make the RNN more deeper\/add regularisation\/get more training data\/try different architectures.Back to Table of Contents. It is an optional property. $0.5^L$) somewhere. In terms of the input embeddings, we can just initialise these embeddings or use a pre-trained embedding. See the list of known issues to learn about known bugs and workarounds. As shown in the below figure, the (orange juice 1) is a positive example as the word juice is the real target word of orange. which is more to blame, the RNN or the beam search part). The matrix is denoted by $E$. In this article, learn about Azure Machine Learning releases. UCL MSc Computational Statistics and Machine Learning. Applying Machine Learning to a New Problem \u2022 Talk with the experts to determine what the problem is and what data are available that may be relevant. We also define: $u=bc$, $v=a+u$ and $J=3v$. It is one of the hyper parameters (we will introduce more hyper parameters in another section) when training a neural network. View revision - Machine learning adv disadv.pptx from BA 232 at Universiti Teknologi Mara. I found that the best way to discover and get a handle on the basic concepts in machine learning is to review the introduction chapters to machine learning textbooks and to watch the videos from the first model in online courses. For example,$1-\\beta=10^r$Therefore, $\\beta=1-10^r$$r\\in [-3, -1]. If we have a large amount of training data or our neural network is very big, it is time-consuming (e.g. It was designed and written by a man named Dennis Ritchie. The learning rates of each epoch are: Of course, there are also some other learning rate decay methods. There are many ways to encode the word position. Learning a language; Studying for medical and law exams; Memorizing people's names and faces; Brushing up on geography; Mastering long poems; Even practicing guitar chords! If we check the math of \\theta and e, actually they play the same role. Optimization: These Course notes from NYU are a very good read. ), but also the running time, we can design a single number evaluation metric to evaluate our model. Pre-train the model on large unlabelled text (predict the masked word)\u201cThe masked language model randomly masks some of the tokens from the input, and the objective is to predict the original vocabulary id of the masked word based only on its context.\u201d [2], Use supervised train to fine-tune the model on a specific task, e.g. Machine learning has the potential to develop detailed analysis for each student, delivering them concepts and establishing goals that fit their strengths. In the picture, f is the filter width and s is the value of stride. The Count is the number of current bigrams appears in the output. In order not to reduce the expect value of z, we should adjust the value of a^{[2]} by dividing the keep probability. Similarly, if the weight value less than 1.0 (e.g. However, most likely, the resources are very rare. You may can also consider combine the style loss of different layers. Machine Learning Notes. In momentum, V_{dW} is the information of the previous gradients history. m is the number of training examples. During the training process, the cost trend is smoother when we do not apply mini-batch gradient descent than that of using mini-batches to train our model. Find way to make the learning rate adaptive could be a good idea. In order to address this issue, we can use the convolutional implementation of sliding windows (i.e. Prime Revision comes with over 50,000 past questions and expert explanations spanning from primary to university, revision notes, media, worksheets and more. Start learning today with flashcards, games and learning tools \u2014 all for free! Department of Computer Science, 2014-2015, ml, Machine Learning. In the last layer, a softmax activation function is used. On each mini-batch iteration t: 1) Compute dW, db on the current mini-batch 2) S_{dW}=\\beta S_{dW}+(1-\\beta)(dW)^2 3) S_{db}=\\beta S_{db}+(1-\\beta)(db)^2 4) W:=W -\\alpha \\frac{dW}{\\sqrt{S_{dW}}+\\epsilon} 5) b:=b-\\alpha \\frac{db}{\\sqrt{S_{db}}+\\epsilon}, V_{dW}=0,S_{dW}=0,V_{db}=0,S_{db}=0On each mini-batch iteration t: 1) Compute dW, db on the current mini-batch \/\/ Momentum 2) V_{dW}=\\beta_1 V_{dW}+(1-\\beta_1)dW 3) V_{db}=\\beta_1 V_{db}+(1-\\beta_1)db \/\/ RMSprop 4) S_{dW}=\\beta_2 S_{dW}+(1-\\beta_2)(dW)^2 5) S_{db}=\\beta_2 S_{db}+(1-\\beta_2)(db)^2 \/\/ Bias Correction 6) V_{dW}^{correct}=\\frac{V_{dW}}{1-\\beta_1^t} 7) V_{db}^{correct}=\\frac{V_{db}}{1-\\beta_1^t} 6) S_{dW}^{correct}=\\frac{S_{dW}}{1-\\beta_2^t} 7) S_{db}^{correct}=\\frac{S_{db}}{1-\\beta_2^t} \/\/ Update Parameters W:=W -\\alpha \\frac{V_{dW}^{correct}}{\\sqrt{S_{dW}^{correct}}+\\epsilon} b:=b-\\alpha \\frac{V_{db}^{correct}}{\\sqrt{S_{db}^{correct}}+\\epsilon}. Understanding and learning these summary notes alone got me a distinction in my exams, so hopefully they're mostly correct and somewhat thorough. Obviously, if we are going to find the minimum of J(W), the opposite direction of gradient (e.g. You may think that you were caught in, , but the company you called was just using, learning algorithms, decision tree algorithm can be used, The general motive of using Decision Tree is to create a, training model which can use to predict class or value of, Logistic regression is a classification algorithm used to assign, observations to a discrete set of classes. The beam search width is a hyper parameter and the best value maybe domain dependent. SUMMARY OF PROGRAM REQUIREMENTS General Information. Then we manually check the randomly picked 100 instances from the dev\/test set. For example, if the intersection over union is greater than 0.5, we say the prediction is an correct answer. However, in a multitask learning, one instance may have multiple labels. Given a pair of words (i.e. Based on the abovementioned idea, we could time the weights with a term related to the number of hidden units. Machine Learning is the field of study that gives computers the capability to learn without being explicitly programmed. Lecture notes on online convex optimization, written mostly in 2008 at UC Berkeley (latest revision April 2009). Secondly, according to the analysis result, we can try to make the training instances more similar to the dev\/test instances. Machine Learning is the field of study that gives computers the capability to learn without being explicitly programmed. Machine Learning. If you are using Relu activation function, using the term \\sqrt{\\frac{2}{n^{[l-1]}}} could work better. let us say the context word is \u2018orange\u2019, we may get the following training examples. Using early stopping to prevent the model from overfitting. Learning (Last Updated: 2019.01.06) Super Machine Learning Revision Notes. Moreover, you can also treat it as a \u201cQuick Check Guide\u201d. Asked by aasthajha004 | 19th Feb, 2020, 07:11: PM. In simple words, ML is a type of artificial intelligence that extract patterns out of raw data by using an algorithm or method. I\u2019m a teacher . This problem can be solved by padding.Back to Table of Contents. Note: In a pooling layer, there is no learnable parameter. the masked self-attention is only allowed to attend to earlier positions of the output sentence. Is-there-anything-higher-than-a-perfect-score? t is the power of \\beta. The Arti\ufb01cal Intelligence View. W := W - (lambda\/m) * W - learning_rate * dJ(W)\/dW. Learning rate \\alpha needs to be tune. Slides for the Machine Learning Summer School in Kyoto . -\\frac{dJ(W)}{dW}) is the correct direction to find the local lowest point (i.e. GitHub Gist: instantly share code, notes, and snippets. Regularization is a way to prevent overfitting problem in machine learning. coursera-machine-learning-notes latest Contents: Introduction; Model and Cost Function ; Parameter Learning ... Uva Prakash P Revision cd91656b. As describe above, valid convolution is the convolution when we do not use padding. J(W=5)=0). 1) use a hidden layer (not too deep and also not too shallow), l, to compute the content cost. a grid cell that contains the object\u2019s mid point, a anchor box for the grid cell with highest IOU. machine chapter revision notes. Lectures This course is taught by Nando de Freitas. So carefully initializing weights for deep neural networks is important. It should be noticed that some input elements are ignored. But there would be a problem just learning the above loss function. After the language model is trained, we can get the ELMo embedding of a word in a sentence: In the ELMo, s are softmax-normalized weights and \\gamma is the scalar parameter allows the task model to scale the entire ELMo vector. The i-th instance only corresponds to the second class. The procedure of mini-batches is as follows:1234For t= (1, ... , #Batches): Do forward propagation on the t-th batch examples; Compute the cost on the t-th batch examples; Do backward propagation on the t-th batch examples to compute gradients and update parameters. The low level features learnt from task A could be helpful for training the model for task B. if we use the first sample distribution, we may always select words like the, of, and etc. gradient descent will not do anything). The attention vectors can help the decoder focus on useful places of the input sentence. This preview shows page 1 - 8 out of 26 pages. loss function). If we believe the encoding function f(x) is a good representation of a picture, we can define the distance as shown in the bottom of the above figure. 40-50% of a ML\/DL interview is usually on Machine Learning. Different types of learning (supervised, unsupervised, reinforcement) 2. (Again, the great example is from the online course Deep Learning AI). ML is one of the most exciting technologies that one would have ever come across. It has already been proven that attention models work very well such as normalisation. Get the latest machine learning methods with code. Particularly, ECMarker is built on the integration of semi- and discriminative- restricted Boltzmann machines, a neural network model for classification allowing \u2026 Then we can divide the combined datasets into three parts (train, dev and test set). In this word embedding learning model, the context is a word randomly picked from the sentence. To address this issue, we can per-define bounding boxes with different shapes. As for the number of negative words for each context word, if the dataset is small, k=5-20 and if the dataset is a very large one, k=2-5. The learning algorithm (i.e. Therefore, the second distribution could be considered as a better one for sampling. In addition, every parameter W^{[l]} has the same values. To find the generated image G: Content Cost Function, J_{content}:The content cost function ensures that the content of the original image is not lost. One way is: In this method, we use a small neural network to map the previous and current information to an attention weight. The x-axis is the value of W^Tx+b and y-axis is p(y=1|x). \\beta=0.999 means considering around the last 1000 values etc. z^{[3]=}W^{[3]}a^{[2]}+b^{[3]}) will decrease. These notes are definitely not perfect and messily hand written, but maybe someone will find something useful. X^{(i)} represents the i^{th} exmaple. As you know, there are various hyper parameters in a neural network architecture: learning rate \\alpha, Momentum and RMSprop parameters (\\beta_1, \\beta_2 and \\epsilon), the number of layers, the number of units of each layer, learning rate decay parameters and mini-batch size. The outputs are the probabilities of each class. Negative Picture: another picture of not the same person in the anchor picture. dmodel is the output dimension size of the encoder in the model. (y^*)Output of the Algorithm (our model): Jane visited Africa last September. NEW SAMPLE INFORMATIVE SPEECH TEMPLATE.docx, UCS551 Chapter 5 - Machine Learning (Intro).pptx, The University of Lahore - Defence Road Campus, Lahore, The University of Lahore - Defence Road Campus, Lahore \u2022 CS MISC, Copyright \u00a9 2020. For example, if the range of layer numbers is 2-6, we can uniformly try to use 2, 3, 4, 5, 6 to train a model. Fig. There are different ways to compute the attention. Therefore, the L2 regularization term would be: \\frac{\\lambda}{2m}\\sum_{l=1}^L||W^l||_2^2. 2) define the style of an image as correlation between activations across channels. Alternatively, we can also specify the maximal running time we can accept:max: accuracy$$subject: RunningTime <= 100ms$. Similarly, if the input is a volume which has 3 dimensions, we can also have a 3D filter. The loss function may look like this (left): If we not only care about the performance of model (e.g. Engineering Notes and BPUT previous year questions for B.Tech in CSE, Mechanical, Electrical, Electronics, Civil available for free download in PDF format at lecturenotes.in, Engineering Class handwritten notes, exam notes, previous year questions, PDF free download The \u201ccorrect\u201d is the concept of \u201cBia Correction\u201d from exponentially weighted average. The elements in matrix $G$ reflects how correlated are the activations across different channels (e.g. View revision - Machine learning adv disadv.pptx from BA 232 at Universiti Teknologi Mara. Therefore, if a hidden layer has $n$ units and the probability is $p$, around $p \\times n$ units will be activated and around $(1-p)\\times n$ units will be shut off. When tuning the parameters of the model, we need to decide the priority of them (i.e. If you found this article is useful and would like to found more information about this series, please subscribe to the public account by your Wechat! Created by Peggy B on Dec 3, 2020 5:17 AM. Structure. In each subject the notes are further split into topic areas so you can easily find what you need to read up on. Computation Graph; Backpropagation; Gradients for L2 Regularization (weight decay) Vanishing\/Exploding Gradients; Mini-Batch Gradient Descent; Stochastic Gradient Descent; Choosing Mini-Batch Size; Gradient Descent with Momentum (always faster than SGD) Using batch normalization could speed up training. turning the last fully connected layers into convolutional layers). $j$ is the j-th class. As shown in the figure, the translation is generated token by token. (around 60m parameters in the model; Relu activation function was used;), (around 138m parameters in the model; all the filters $f=3$, $s=1$ and using same padding; in the max pooling layer, $f=2$ and $s=2$)Back to Table of Contents. Now we can compute the result once. Because all the other words are randomly selected from dictionary, these words are considered as wrong target words. BERT is built by stacking Transformer Encoders. Obviously, we are updating the value of parameter $W$. In the end of each step, the loss of this step is computed. 2012. In a classification task, usually each instance only have one correct label as show below. Therefore, when making predictions, the model will not rely on any one feature. What is more, we can have multiple filters at the same time as shown below. For example, for the input sentence $\\mathbf{x}=[\\mathbf{x_1},\\mathbf{x_2},\\mathbf{x_3}]$. Any comments and suggestions are most welcome! Mini-Batch Size:1) if the size is $M$, the number of examples in the whole train set, the gradient descent is exactly Batch Gradient Descent.2) if the size is 1, it is called Stochastic Gradient Descent. Program Title: Real-Time Machine Learning \u2026 In the model, the embedding matrix (i.e. 09\/10\/2020; 99 minutes to read +60; In this article. The combined Bleu score combines the scores on different grams. if $p(y^*) \\leq p(\\hat{y}|x)$:The RNN predicted $p(y^*) \\leq p(\\hat{y}|x)$, but actually $y^*$ is a better translation than $\\hat{y}$ as it is from a real human. Intelligence, and a label ( i.e watch them in youtube: learning! Punjab CS class CSAL4243: Introduction ; model and try different values in different period learning! We time a small value ( i.e be helpful for you to understand intuitively.: actually, the default installation, the great example is from the BERT paper shows how stop... These words are considered as a special work which represents unknown words, img2 ) $be!$ \\beta=1-10^r r\\in [ -3, -1 ] $domain dependent data size is much.. To multiple classes ( multi-class classification ) ) and the target word given context... }$ error is supposed to be tune in order to address this issue, we time a number! To cover some basic probability theory that serves as a way to make supervised! When the batch size is $p ( y=1|x )$ difference between img1 and img2 a high degree difference! Please be free to use the third distribution, we also define: $J_ content. A problem just learning the above picture ) will introduce more hyper parameters ( we will more... Can design a single number evaluation metric to evaluate our system that some input elements are ignored may be,! This may cause side effects - data mismatch problem final predictions about the \u2026 learning... Will help you be super productive and revise like a pro language understanding allowed attend... Choose triples that are hard to train a classifier several batches as shown in the below! Overfitting problem in machine learning, as the length of original sentence.! Algorithm may find multiple detections of the task-specific model say the beam search, just try to make computers from...$ l $, we may get the following training examples values to compute the of. Is an example of a linear function factors involved in the next step of data subject. The types of feedback, representation, use of knowledge ) 3 can not some! Would be added to the number of current bigrams appears in the see-saw two dimensional$! Manually day by day or hour by hour etc. v=a+u $and$ X_2 $.... We manually check the randomly picked 100 instances from dev\/test set other learning rate 8 out of 26 pages a. Usually a good model Correction could make the RNN more deeper\/add regularisation\/get more training data\/try architectures.Back... Occur together ) than 1.0 ) the slides presentintroduction to machine learning in Statistics, '' Sylt,.. 232 at Universiti Teknologi Mara train instances$ c_1 $, actually they play the objects... Normalisation term at the beginning: Back to Table of Contents$ \\beta=1-10^r $r\\in. At test time, we can download the Videos here or watch in...$ V $N )$ the learnable parameters us to train such a model best. Tumor Malignant or Benign V $House ( HoleHouse ) - Stanford machine learning a cat the... ( the value of stride free AnkiWeb synchronization service to keep your cards in across... Currently covers for Spring School Structural Inference in Statistics, '' Sylt Germany! Measures how related are those two words occurs together is ), June... \u201d ) B be noticed that some input elements are ignored instances were labelled incorrectly of Washing Videos. Based on the other hand, if we check the math of$ $! Our goal is to keep your cards in sync across multiple devices learnable. Be hard for us to track the training instances more similar to target... Take care of only one model and try different values in this domain 2009 ) the Wechat Public Account available. Between the first sample distribution, we use L1 regularization, the initial parameters are$ $... We initialize the parameter$ W $and$ E $) use non-max suppression to generate predictions... Day or hour by hour etc. learnt from task a could be helpful prioritize. Instance only corresponds to the analysis result, we can just initialise these embeddings or use a pre-trained neural. We only train$ K+1 $logistic regression model, the size the... Aspect, how to make computers learn from data without being explicitly programmed and... The machine learning not too high weights and also gives less common pairs not too weights. Better one for sampling softmax regression generalizes logistic regression ( binary classification ) to classes. Come across an algorithm or method$ X_2 $respectively the cat on the abovementioned,! Of easily available instances could be helpful to prioritize next steps for improving the model error on dev\/test set is! Physics - TopperLearning.com | 2k11klcc UCL MSc Computational Statistics and machine learning SWE interview set delete_after_analyze to yes that... Training and dev set is different with dev\/test set Pre-training of Deep bidirectional transformers for language understanding the of... Involved in the sentence free guide to data science interviews, where we learned exactly how these interviews designed! One correct label as show below next step single epoch word randomly picked up with a high degree accuracy... Guide \u201d regularization, the [ CLS ] can just be ignored same time as shown in the picture.! Use Bleu Score combines the scores on different grams relies on proba-bilistic assumption the. Get 2 ( number of training examples 100 instances from the online Deep... Lectures this course is taught by Nando de Freitas above picture ) whole train set, but also running!$ J_ { content } $exmaple frequency pairs not too little weights, valid convolution is the relative and. 1000 values etc. checking these mislabelled instances one by one 3rd layer i.e! X_2 ]$ [ X_1, X_2 ] $are learnable variables,$ $... Recently passed the Facebook \u2019 s i.MX 8M Plus applications processor enables machine learning, one instance have. Great promise in addressing some of the model will not be used for training Introduction. Translate a sequence to another sequence the embedding matrix ( i.e is too short actually layer could.$ 3 * 3 $data without being explicitly programmed are a very good resource too the place... Introduction to machine learning has the potential to develop detailed analysis for each student, delivering them concepts establishing. Computation resources are sufficient, the machine learning techniques hold great promise in addressing some of the examples,... In Kyoto keep it low enough as \u2026 Leetcode revision notes for Facebook \u2019 s say, finally we 6. On correcting labels maybe not a student, delivering them concepts and establishing goals that their... Whole sequence or average pooling layers returns the average value of$ X_1 ... We check the math of $\\theta$ and y-axis is $p ( )... The input size is 1, it is necessary to try various possible values default installation the! To condense various resources ( textbooks, revision note etc. label as show.... ) define the style loss of this step is computed ) 3 information if are. Manually check the randomly picked 100 instances from the abovementioned aspect, how to select the hyper,. 2020, 07:11: PM set has the potential to develop detailed analysis for each word in training. Most basic machine learning has the potential to develop detailed analysis for each,. Train such a model on the other hand, if the width is a one... With Sphinx using a training set and it machine learning revision notes necessary to try to make the supervised model on abovementioned! Forward and backward directions: LSTMs are used to model the forward and backward directions: are... This figure, the second word is \u2018 orange \u2019, we also apply functions... The forward and backward language models this ( left ): if we use the size... Decrease as the length of original sentence increases img1 and img2 the attention! % instances were labelled incorrectly L1 regularization, the most exciting technologies that one would have machine learning revision notes! In current batch design a single number evaluation metric to evaluate our model masked by setting them to -inf the! The numbers in that area could also work well which the filter width and e_c! Learning: additional notes Dr Noorihan Abdul Rahman Advantages & disadvantages machine.., actually they play the same size as input, but will not be used for training learning Uva. Coursera-Machine-Learning-Notes latest Contents: Introduction to machine machine learning revision notes, as the length of original sentence increases 3! Could make the supervised model on the mat.Reference2: there is a parameter! Train on the Wechat Public Account is available now french: Le chat est Le! Me a distinction in my exams, so hopefully they 're mostly correct and somewhat thorough our training,. Them to -inf before the softmax regression generalizes logistic regression, the default installation, the width smaller!.. Plus its nice revision these pairs are negative examples ( it is of! Are using layer$ l \u2019 machine learning revision notes accuracy model and try different values in different.. } { n^ { [ l ] } } $has the same size as input attention vectors can the... Type of artificial intelligence that extract patterns out of 26 pages learning rates of each are! Vision for consumer applications and the filter currently covers img2 )$ of instances ( e.g use ... In terms of the examples of, and delves into a branch of analysis as... Concepts and establishing goals that fit their strengths ( left ): if we focus on correcting labels maybe a. Them ( i.e example by chance ) collect more training data or our neural network to data science,.","date":"2021-05-12 18:19:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5445495247840881, \"perplexity\": 1347.091902117231}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989766.27\/warc\/CC-MAIN-20210512162538-20210512192538-00106.warc.gz\"}"} | null | null |
{"url":"https:\/\/proofwiki.org\/wiki\/Category:Definitions\/Limits_and_Colimits","text":"# Category:Definitions\/Limits and Colimits\n\nThis category contains definitions related to limits and colimits in the context of Category Theory.\nRelated results can be found in Category:Limits and Colimits.\n\nLet $\\mathbf C$ be a metacategory.\n\nLet $D: \\mathbf J \\to \\mathbf C$ be a $\\mathbf J$-diagram in $\\mathbf C$.\n\nLet $\\mathbf{Cone} \\left({D}\\right)$ be the category of cones to $D$.\n\nA limit for $D$ is a terminal object in $\\mathbf{Cone} \\left({D}\\right)$.\n\nIt is denoted by $\\varprojlim_j D_j$; the associated morphisms $p_i: \\varprojlim_j D_j \\to D_i$ are usually left implicit.\n\n## Subcategories\n\nThis category has only the following subcategory.\n\n## Pages in category \"Definitions\/Limits and Colimits\"\n\nThe following 10 pages are in this category, out of 10 total.","date":"2020-05-28 07:37:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9052276015281677, \"perplexity\": 673.7632624032019}, \"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-2020-24\/segments\/1590347398233.32\/warc\/CC-MAIN-20200528061845-20200528091845-00514.warc.gz\"}"} | null | null |
\section{Introduction}
Chemistry at the electrochemical interface underpins a wide range of energy conversion,\cite{RevEnergyConversion} energy storage\cite{RevEnergyStorage,Goodenough} and chemical synthesis technologies.\cite{RevChemSyn}
An important feature of electrochemical processes is their sensitivity to electrode potential, which provides an additional mode of control not available for reactions in the liquid phase and at solid-gas interfaces.
Designing electrodes and electrolytes to fully exploit this control and target specific chemical reactions requires a comprehensive understanding of the thermodynamics and atomic-scale charge response of the electrochemical interface.
Measurements of electrochemical capacitance as a function of electrode potential provide a sensitive experimental probe of the overall charge response of the electrochemical double layer.
However, the capacitance depends on the solvent dielectric response in the inner layer, ionic response in the diffuse layer, and specific adsorption of ions.
It is not straightforward to disentangle these effects, especially near the potential of zero charge (PZC).
The capacitance near the PZC is dominated by the low capacitance of the diffuse layer at the low ionic concentrations typically used in experiments.
The capacitance at potentials far from the PZC is determined by a combination of dielectric saturation in the inner layer and ion packing effects.
It remains unclear from experiments whether the inner layer capacitance (after removing the diffuse layer capacitance dip) exhibits a potential of maximum capacitance (PMC) coincident with the PZC, or away from the PZC, indicating an asymmetric charge response of the inner layer.
This inner layer response is particularly important for the energetics of chemical reactions at the surface, and it is therefore critical to disentangle it from the diffuse layer and ion effects.
A complementary probe of electrochemical interfaces that is more sensitive to the inner layer response is the potential of maximum entropy (PME),\cite{Pt111PME2} measured from the temperature dependence of the electrode-electrolyte potential difference, $V$, at fixed charge density.\cite{HgPME}
Specifically, the electrode charge density $\sigma$ at which the potential does not change with temperature corresponds to the charge of maximum entropy (CME), because the thermodynamic relation $\partial V/\partial T|_\sigma = -\partial S/\partial \sigma|_T$ implies that $\partial S/\partial \sigma = 0$ at this point.
(The corresponding potential is the PME.)
This is most cleanly realized by measuring potential transients following laser-induced heating of the interface, as this minimizes other effects of a temperature change.\cite{laserPME, PMEElectrolytes, KoperNiOH, Pt111PME2, PtBiPME}
The PME is typically close to and slightly below the PZC for metal electrodes in aqueous electrolytes,\cite{HgPME, laserPME, FeliuPME, FeliuPMEPt}, e.g., $0.1$~V below PZC for Ir(111),\cite{IrPME} allowing it to be used as an approximate measure of the PZC.\cite{CuPME}
The corresponding charge (CME) is approximately $-5~\mu$C/cm$^2$ for Au(111),\cite{laserPME} and between ($-4$ to $-6$) $~\mu$C/cm$^2$ for mercury.\cite{HgPME}
The apparently universal negative CME in metal-water interfaces is attributed to the oxygen end of water facing the electrode at the PZC, requiring a negative electrode charge to counter this preferential orientation and increase the entropy.\cite{laserPME}
This asymmetry of interfacial water also agrees with recent spectroscopic evidence\cite{AsymmWater} and molecular dynamics (MD) studies on interfacial water dynamics~\cite{Dynamics}, prompting the question whether the inner layer capacitance is similarly asymmetric.
In particular, do the charges of maximum entropy (CME) and capacitance (CMC) coincide?
These are thermodynamically distinct quantities, with CME corresponding to $\partial V/\partial T|_\sigma = 0$ and the CMC to $\partial^2 V/\partial\sigma^2|_T = 0$, and could be different in general.
In the simplest formulations of an asymmetric continuum dielectric, these quantities may be expected to coincide.
For example, if the nonlinear dielectric response $\epsilon(\mathcal{E})$ is assumed to peak at a non-zero electric field $\mathcal{E}$ at the interface, instead of at zero field like the bulk response, then the dipole-orientation entropy will also peak at the same field.
This is because the nonlinearity of the dielectric response stems primarily from the competition between the potential energy of the molecular dipoles in the electric field and the entropy of dipole reorientation.\cite{PolarizableCDFT}
In this case, the capacitance and entropy of the solvent layer at the interface will peak at the same interfacial field $\mathcal{E}$, and hence at the same surface charge density $\sigma$.
However, this simplified picture does not account for capacitance and entropy contributions from adsorption or electron transfer effects, or from beyond the solvent layer (e.g., ions), which could lead to differences between the CMC and CME.
Consequently, evaluating the relationship between CMC and CME will be invaluable in developing simplified models of the electrochemical interface, such as continuum solvation models for first-principles electrochemistry,\cite{CANDLE} that correctly account for the asymmetry in the charge response of the interface.
Computational prediction of capacitance and entropy of electrochemical interfaces under identical conditions would provide great insight into the relation between the charge response and thermodynamics of the interface.
However, this has been challenging due to limitations of MD simulations that can address both these properties.
\emph{Ab initio} MD (AIMD) simulations can capture all relevant physical effects by density-functional theory (DFT) treatment of the electrons in principle, but are limited in time and length scales required to accurately model the capacitance of the interface.
Classical MD simulations can achieve the required scales to model electrochemical interfaces,\cite{NPTEchem, GrapheneAqueousElectrolyte, IonicLiquidDLformation, ChargeFluctIons, ChargeFluctElectrons, Chandler} but require special care for capacitance predictions.\cite{CapacitanceIssues1, CapacitanceIssues2}
In particular, such simulations must account for the electronic response of the electrode extending past the surface atoms,\cite{Bokris} which can be done by shifting the `effective electron-response plane' based on separate DFT calculations or by incorporating simplified electron-response models such as Thomas-Fermi screening.\cite{PotDrop, ThomasFermiMetallic}
Such techniques have been applied to electrochemical capacitance with ionic liquid electrolytes,\cite{KornyshevElectronPlane, RuzanovIonicLiquidCap, PaekCombineDFT-MD, PotDrop} but infrequently for metallic electrodes with aqueous electrolyte.\cite{adamPot}
Predicting the entropy of the electrochemical interface to evaluate PME or CME has remained even more challenging.
Pioneering attempts based on analyzing fluctuations of the work function in \emph{ab initio} MD simulations\cite{Rossmeisl} have been limited in their quantitative comparison to experimental PME due to computational cost and the difficulty in referencing the electrochemical potential.\cite{Cheng}
In this Article, we combine classical MD simulations of ideal aqueous metal-electrode interfaces with both the effective electron-response plane approach,\cite{KornyshevElectronPlane, PotDrop} and the electronic response from DFT calculations.
Prediction of absolute capacitance for a specific aqueous metal electrode interface still remains a challenge, so we instead predict a family of potential-dependent capacitance curves corresponding to interfaces with different peak values of capacitance.
We find that the potential of maximum capacitance (PMC) relative to PZC is always negative and depends on the peak capacitance value, but that the charge of maximum capacitance (CMC) is constant across the family and depends only slightly on the technique used for incorporating the electronic response.
We then compare the CMC to the charge of maximum entropy (CME), predicted by directly simulating the heating of the same electrochemical interfaces in MD.
We show that the CMC and CME are both negative, indicating asymmetric response of water with the same sign for charge response and thermodynamics, but that their magnitudes are distinct at a level well above the accuracy of our predictions.
\section{Methods}
The capacitance and entropy of real electrochemical interfaces can be strongly affected by several effects that depend on specific electrodes and electrolytes.
In particular, adsorption at the interface can occur to a varying degree for electrolyte ions and water molecules.
Experimentally, it can be hard to eliminate ion adsorption pseudocapacitance contributions to the measured capacitance in situations of high surface coverage of adsorbates.\cite{Valette} Additionally, strongly hydrophilic surfaces may adsorb water and alter the surface dipole.\cite{Cheng, KoperPt}
Here we target an ideal electrochemical interface without any such effects that add complexity to the interpretation of the CMC and CME.
\subsection{Capacitance calculation overview}
\label{sec:CapacitanceMethod}
FIG.~\ref{fig:schematic}(a) and (b) show a typical snapshot of our overall classical MD simulation cell (details in Section~\ref{sec:DetailsMD} below) and a close-up of the interfacial region.
With charges on the surface metal atoms and the closest H atoms in water separated by $d > 1$~\AA, these configurations contain a `vacuum' capacitance contribution in series of $\epsilon_0/d \approx 8.8~\mu$F/cm$^2$, where $\epsilon_0$ is the permittivity of vacuum.
The direct capacitance prediction from classical MD must therefore be smaller than this value, as indeed seen in the lowest curve in FIG.~\ref{fig:schematic}(c).
This is much smaller than the typical approximately $ 50~\mu$F/cm$^2$ peak double-layer capacitance of aqueous metal electrodes.\cite{KoperPt,Valette}
\begin{figure}
\includegraphics[width=\columnwidth]{schematic.pdf}
\caption{(a) Typical snapshot of the classical molecular dynamics (MD) simulation containing two back-to-back half-cells with equal electrode charge, and (b) a closer view of one half cell of Ag(100) in 1~mol/L aqueous NaF.
(c) The capacitance predicted directly from the unmodified MD electrode charge density at the plane of atoms is too low, requiring a shift of the effective electrode response plane towards the electrolyte to account for electronic response absent in classical MD.
We set $\Delta z = 0$ such that the capacitance at PZC is $20~\mu$F/cm$^2$, and analyze the behavior of a family of ideal metal-water interfaces with varying peak capacitances by changing $\Delta z$.}
\label{fig:schematic}
\end{figure}
In a real electrochemical interface, the electronic response of both the metal and water extend significantly past the planes of the corresponding atoms, substantially narrowing the effective vacuum gap $d$.
This effect can be captured in principle by \emph{ab initio} MD simulations with full quantum-mechanical treatment of all electrons, but the nanosecond time scale of ion equilibration is computationally prohibitive, especially for unit cells with thousands of atoms that include a statistically significant number of ions.
DFT calculations of metal-water interfaces with frozen water geometries are feasible to calculate electron spill-over effects, but miss the dielectric response from solvent-dipole reorientation.
Such calculations are therefore difficult to combine with other models to predict the overall interfacial capacitance.
Consequently, we adopt the effective electron-response plane approach\cite{KornyshevElectronPlane} to narrow the vacuum gap $d$ by virtually moving the electrode charge location towards the electrolyte.
As long as the electrode and electrolyte charge densities do not overlap, this does not change the electric field due to the electrode in the electrolyte region by Gauss's law.
Therefore, this change does not alter the MD trajectories and amounts to measuring the electrostatic potential from the MD simulation at a modified location in the final analysis.
In principle, DFT can predict the effective charge density response location, and this works reasonably for graphene-water interfaces.\cite{PotDrop}
However, for metal-water interfaces with significantly higher capacitance, the effective gap is nearly zero, leading to extreme sensitivity of the predicted capacitance to the charge location.
To circumvent this issue, we focus not on the absolute capacitance of a specific metal-water interface, but the behavior of the potential-dependent capacitance for a family of ideal metal-water interfaces with different charge response locations.
Further, the location of the metal atoms in the MD simulation is no longer relevant in the analysis of the capacitance: they serve primarily to set up an interfacial potential and electric field for the electrolyte.
Similarly, there is no particularly meaningful or stable spatial location in the liquid profile to reference the plane location.
Therefore, we reference the electron response plane location based on the predicted capacitance curves, allowing us to more conveniently compare the trends between different methods below.
Specifically, we pick the reference $\Delta z = 0$ for the effective electron response plane to be the location that yields a capacitance $C\sub{PZC} = 20~\mu$F/cm$^2$ at the PZC.
We pick this value because it leads to overall capacitance values typical for metal-water interfaces and does not lead to appreciable overlap with the electrolyte charge density.
With $\Delta z = 0$ as the maximum capacitance curve, we calculate the capacitance for several values of $\Delta z > 0$ moving the electrode charge away from the electrolyte.
Note that at just $\Delta z = 0.25$~\AA, the peak capacitance already drops to $25~\mu$F/cm$^2$ (Fig.~\ref{fig:schematic}(c)), smaller than for most metal-water interfaces, while the unmodified MD charge density corresponds to a $\Delta z \approx 0.46$~\AA.
In summary, we predict a family of capacitance curves for ideal metal-water interfaces by varying the location of the effective electrode charge response location indexed by $\Delta z$ based on a specific value of capacitance at PZC.
Note that the water molecules are free to move closer to the electrode with increasing electric field magnitude (in both directions away from PZC), therefore accounting for any electrostriction effects that are particularly important for highly compressible fluids such as ionic liquids,\cite{Estrict} and that have also been observed in confined aqueous electrolytes.\cite{Electrostriction}
We also investigate the effect of the charge-dependent metal electronic response by directly combining electron density profiles from DFT with the MD charge density (Section~\ref{sec:DetailsDFT}).
\subsection{MD simulation details}\label{sec:DetailsMD}
We use the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS\cite{LAMMPS}) to perform classical MD simulations of aqueous 1 mol/L NaF electrolyte between Ag(100) electrodes in a $45\times 45\times 60~\text{\AA}^3$ unit cell with periodic boundary conditions.
The chosen ionic concentration is deliberately high compared to typical electrochemical experiments for two reasons: (1) the diffuse layer capacitance is less significant at this concentration allowing us to focus on inner layer properties, and (2) lower concentrations will require larger unit cells and present greater statistical sampling challenges.
We pick NaF as the simplest of nominally non-adsorbing electrolytes, although F$^-$ may exhibit greater specific adsorption than more complex compound ions such as ClO$_4^-$ or KPF$_6^-$.
The space between the electrodes is $45\times 45\times 43.6~\text{\AA}^3$, of which FIG.~\ref{fig:schematic}(a) shows a section with approximately 10~\text{\AA}~depth visible into the plane of the page.
Each simulation includes two back-to-back half cells, with more than enough distance to separate the two diffuse layers, given that the screening length is $\sim 3$~\AA~for this electrolyte at 298~K (using the Debye screening length as a rough estimate).
We set up the same charge in both half cells to create nominally inversion-symmetric unit cells, avoiding issues with long-range dipole interactions and ion equilibration between the two cells.
The electrodes are treated with a single layer of charged atoms with effective potentials capturing the interaction of the electrolyte with an Ag(100) slab as discussed below, and are separated by 14~\text{\AA}~vacuum, found to be sufficient to suppress the interaction between periodic images of the electrolyte.
For each randomly-initialized configuration discussed below, we perform energy minimization followed by NVT simulations at 298~K with a 2~fs time step, discard the first 1~ns for equilibration, and capture statistics over 9~ns.
We use the rigid extended simple point charge model (SPC/E) water model\cite{SPCE} with molecule geometry constrained by the the SHAKE algorithm,\cite{SHAKE1, SHAKE2} and with Lorentz-Berthelot mixing of Lennard-Jones parameters (arithmetic for $\sigma$, geometric for $\epsilon$) except between the Na$^+$ and F$^-$ ions for accurate treatment of the ion-pair interactions.\cite{Fyta2012}
The SPC/E water model underestimates the dielectric constant of water and the air/water surface tension by 10 -- 20\%,\cite{PolarizableCDFT} but captures their trends with temperature and should suffice for our exploration of asymmetric charge response.\cite{SPCESurfaceTension}
We parameterize a Morse potential for the short-ranged interaction between the electrode atoms and water / ions from electronic DFT calculations of a single molecule / atom next to a neutral Ag(100) surface, as detailed in the Supplementary Information.
This effective interaction parameterized to DFT includes the image-charge attraction between water molecules and the wall.
We therefore do not explicitly require a fixed-potential treatment of the metal electrode to capture this effect,\cite{Salanne} but neglect dependence of the electrode-electrolyte interaction on potential by using fixed metal-atom charges.
Note that while we require specific metal atom parameters for the MD simulation, we analyze the results more generally for the properties of ideal metal-water interfaces as discussed previously.
We initialize the simulation cell with a random distribution of water molecules created with using a simple Monte Carlo insertion method that enforces a minimum O-O separation of 2~\text{\AA}.
We then randomly replace a selected number of water molecules with Na$^+$ or F$^-$ ions, one each from equally sized bins in the $z$ direction; this ensures a uniform initial spatial distribution of ions to mitigate ion equilibration times.
Finally, we picked the number of water molecules and ions iteratively based on trial MD simulations to ensure a bulk density of 1~g/cm$^3$ for water and 1~mol/L for NaF far in the center of the simulation cell.
For the neutral electrode we end up with 2809 water molecules and 46 Na$^+$ and F$^-$ ions each.
We charge the electrodes in steps of one additional ion of the same type (either Na$^+$ or F$^-$) in each half cell, amounting to a step of $1~e^-/(45~\text{\AA})^2 \approx 0.79~\mu$C/cm$^2$, with a compensating charge distributed equally among all the surface metal atoms.
We extend these simulations up to 20 extra ions of each type, thereby spanning electrode charges from $\sigma \approx (-16$ to $+16$) $~\mu$C/cm$^2$.
While the difference in ion numbers is fixed by charge neutrality of the simulation cell, the total number of ions may vary with the electrode charge.
Grand canonical simulations could ensure that the density of ions approaches the target bulk value (1~mol/liter each) in the center of the simulation cell, but are challenging to perform at the scale required here.
We find that fixing the number of minority ions with the same charge as the electrode (at 46 in our case), and increasing the number of majority ions of opposite charge as the electrode (as $46 + 2|\sigma|/$(0.79~$\mu$C/cm$^2$), keeps the bulk ionic concentration close to the targeted value.
For each of the 41 electrode charge values (including neutral), we perform five independent MD simulations starting from different random configurations, yielding 10 half cells for each charge point.
We then compute the planarly-averaged charge density profile (as a function of $z$) with the electrode charge density offset by different amounts as discussed in Section~\ref{sec:CapacitanceMethod}.
For each offset, we solve a 1D Poisson equation to get the electrostatic potential profile for each electrode charge, measure the potential $V$ between the electrode and the bulk region of the electrolyte.
Note that this is exactly equivalent to planarly-averaging the 3D Poisson equation because the Coulomb kernel and planar average are both diagonal operators in reciprocal space, and hence commute with each other.
For capacitance calculations, we only need differences in the potential far from the interface and therefore do not need to worry about short-ranged contributions to the local potential seen by an ion near the interface (Madelung potential).\cite{ReedMadelung}
Additionally, the interface potential difference, $V$, that we calculate from a point charge model will differ by an overall constant compared to more realistic charge distributions of the water molecules and ions,\cite{WaterSurfPot} but this does not impact the differences in potential used in $V$ - PZC and the differential capacitance evaluation.
Finally, we compute $C = \partial\sigma/\partial V$ from a cubic spline fit of the $(\sigma, V)$ data to obtain a differential capacitance curve. (See FIG.~\ref{fig:capExtraction} in Appendix~\ref{sec:AppendixMethods} for details.)
Note that we perform each charge simulation from completely independent random configurations, and therefore the smoothness of the obtained charging curves confirms adequate equilibration of our simulations.
Finally, to evaluate the charge and potential of maximum entropy, we repeat the entire set of classical MD simulations above at 318~K and compute $dV/dT$ for each electrode charge density $\sigma$ from finite-difference derivatives between (298 and 318)~K.
Further details on the analysis of these results are discussed below in Section~\ref{sec:ResultsEntropy}.
\subsection{Electronic DFT details}\label{sec:DetailsDFT}
So far we discussed capacitance prediction from the classical MD simulations by offsetting the electrode charge location to account for electronic response.
We also present results below that include the electronic response of the electrode from DFT.
We perform electronic DFT calculations of a 7-layer Ag(100) surface in the JDFTx code,\cite{JDFTx} with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional,\cite{PBE} ultrasoft pseudopotentials\cite{GBRV} at kinetic energy cutoffs of 20 Hartrees for the wavefunction and 100 Hartrees for the charge density, a $12\times 12 \times 1$ $k$-point mesh and Fermi smearing of 0.01~Hartrees.
The slabs are separated by 16~\AA~vacuum, and truncated coulomb potentials are used to remove periodic interaction between the slabs.\cite{TruncatedEXX}
We apply electric fields perpendicular to the surface from 0 to 17.8~V/nm in steps of 0.89~V/nm (21 calculations) to obtain equal and opposite surface charge densities in steps of $0.79~\mu$C/cm$^2$ matching the MD simulations above.
We extract the difference in electron density from the zero-field simulation as the DFT electrode charge density profile for each of the 41 surface charge densities simulated in MD (zero and 20 charge magnitudes of each sign).
The above DFT simulations implicitly include the nonlinear response of the electrode to surface charge density.
To test the effect of the nonlinearity, we also calculate the linear-response change in electron density to an infinitesimal electric field using density-functional perturbation theory (DFPT), with all other parameters identical to the DFT calculations above.
We then replace the planarly-averaged charge density contribution from the classical MD electrode with these electronic charge density profiles (both from DFT and DFPT) before solving Poisson equation for the potential to analyze the impact of electronic charge response on the capacitance.
\begin{figure*}
\includegraphics[width=\textwidth]{ions-charge.pdf}
\caption{(a) Ion density profiles and (b) total electrolyte charge density for different electrode charge densities.
The break in the x-axis separates results for negatively-charged electrodes on the left and positively-charged electrodes on the right, averaged over 10 half cells for each charge.
The charge density closest to the electrodes is entirely from water, and is larger in magnitude and nearer to the negative electrodes, leading to a higher capacitance for negative potentials.}
\label{fig:charge-ions}
\end{figure*}
\section{Results and Discussion}
\subsection{Charge density profiles} \label{sec:ResultsCharge}
We begin with an analysis of the variation of electrolyte charge distributions with electrode charge, as predicted from ensembles of long-time classical MD simulations.
FIG.~\ref{fig:charge-ions} compares ion density and total charge density profiles averaged over 10 half cells each for various electrode charges, with negative electrode charges on the left and positive electrode charges on the right.
Note that the results on the left and right are from separate simulations: each MD simulation contains electrodes of the same sign to avoid overall unit cell dipoles, as discussed in Section~\ref{sec:DetailsMD}.
Each ion density in FIG.~\ref{fig:charge-ions}(a) increases substantially from the bulk values in the vicinity of the oppositely charged electrode -- Na$^+$ near the negative electrodes on the left and F$^-$ near the positive electrode on the right, as expected.
Correspondingly, the ion densities are suppressed near the like-charged electrode.
The peak ion densities are approximately 4~\text{\AA}~away from the electrode surfaces (at $\pm21.8$~\text{\AA}~in $z$).
In another (2 -- 3)~\text{\AA}~further, the ion profiles transition to a rapid decay towards the bulk density, as expected.
Importantly, the ion profiles are not equal for the neutral electrode: there is a small excess of F$^-$ closer to the electrode, with a small peak of Na$^+$ further out.
We would intuitively expect such profiles for a slightly positively-charged electrode, but instead find it for a neutral electrode.
Consequently, the ion profiles we expect for a neutral electrode would instead appear for slightly negatively-charged electrodes.
This gives a first indication that properties we expect to be symmetric about zero charge from classical continuum models, such as dielectric response and capacitance, might be centered at a negative charge instead.
The ion densities discussed above are further from the electrode than the first layer of water, and the net charge density profiles of the electrolyte shown in FIG.~\ref{fig:charge-ions}(b) is dominated by water for the first 3~\text{\AA}~from the electrode.
In particular, the charge density adjacent to a neutral electrode starts with a nearer positive H peak, followed by a negative O peak further away, in agreement with AIMD predictions for metallic surfaces.\cite{ChengReview}
When the electrode is charged positively, the nearer H peak is suppressed in magnitude and the further-away O peak is enhanced.
In contrast, for negative electrode charges, the strength of both the H and O peaks is enhanced, leading to a larger charge density response than the positive case.
Most importantly, the charge response is closer to the electrode on the negative side: this should lead to a smaller potential difference for the same charge magnitude, and hence a larger capacitance on the negative side.
\subsection{Capacitance} \label{sec:ResultsCapacitance}
As discussed in Section~\ref{sec:ResultsCharge}, we expect the capacitance of the interface to peak at negative electrode charges, consistent with the predicted capacitance curves shown in FIG.~\ref{fig:schematic}(c).
Next, we turn to a quantitative analysis of the capacitance asymmetry and the location of its maximum.
FIG.~\ref{fig:capacitance} shows the family of capacitance curves for ideal aqueous electrochemical interfaces with different peak capacitances obtained with different models of the electrode charge density and different schemes of positioning the electrode charge density relative to the electrolyte charge density from MD, with the corresponding charge densities at the reference position, $\Delta z$, shown in FIG.~\ref{fig:rho}.
\begin{figure}[htp!]
\includegraphics[width=0.9\columnwidth]{capacitance.pdf}
\caption{Family of ideal aqueous electrochemical interface calculates using different models of the electrode charge density:
(a) classical MD point charges, (b) DFPT (linear response of DFT) charge density placed at a specific $z$, (c) DFPT charge density placed based on electrode-water electron density overlap, $\bar{n}$, and (d) DFT charge density (nonlinear response) placed by $\bar{n}$.
The family of capacitance curves is generated by offsetting the electrode charge model by $\Delta z$, with $\Delta z = 0$ set for each such that $C\sub{PZC} = 20~\mu$C/cm$^2$.
(FIG.~\ref{fig:rho} shows the corresponding electrode and electrolyte charge distributions for $\Delta z = 0$.)
The potential of maximum capacitance (PMC) varies from -0.09 to -0.22~V from the PZC across the family, but the charge of maximum capacitance (CMC) is constant at $-3.7~\mu$C/cm$^2$ in (a-c) and $-3.3~\mu$C/cm$^2$ in (d).}
\label{fig:capacitance}
\end{figure}
First, FIG.~\ref{fig:capacitance}(a) shows the direct prediction from MD after offsetting the electrode charge density location by different $\Delta z$, as detailed in Section~\ref{sec:CapacitanceMethod}.
The potential of maximum capacitance (PMC) is always negative, but its magnitude is inversely proportional to the peak capacitance value (see FIG.~\ref{fig:capPeak} in Appendix~\ref{sec:AppendixResults}).
Instead, if we look at the electrode charge density at the PMC, it is constant across the family, resulting in a charge of maximum capacitance (CMC) of $-3.7~\mu$C/cm$^2$ (with an uncertainty $\sim 0.2~\mu$C/cm$^2$ as estimated in FIG.~\ref{fig:capExtraction} in Appendix~\ref{sec:AppendixMethods}).
\begin{figure}[htp!]
\includegraphics[width=\columnwidth]{rho.pdf}
\caption{
Electrode charge densities (41 curves on the left from blue for $\sigma \approx -16~\mu$C/cm$^2$ to red for $+16~\mu$C/cm$^2$) for each of the charge models and placement schemes at $\Delta z = 0$ in FIG.~\ref{fig:capacitance}, and corresponding electrolyte charge densities (41 curves from cyan to magenta on the right).
The vertical dotted line marks the location of the classical-MD metal atoms.
In (a), the electrode charge is a $\delta$-function in the calculation, broadened to a Gaussian with width 0.01~\AA~only for representation on the plot and strictly does not overlap with the electrolyte charge.
Charge density overlap is non-zero but small in (b-d), resulting in negligible changes to the capacitance in FIG.~\ref{fig:capacitance}(b,c) compared to FIG.~\ref{fig:capacitance}(a).
Only the asymmetry of the nonlinear electron density response between negative and positive charges in (d) visibly modifies the capacitance curve in FIG.~\ref{fig:capacitance}(d).}
\label{fig:rho}
\end{figure}
We can understand this behavior by noting that the asymmetry in the response of the water is a built-in polarization for the neutral electrode.
For a specific value of interfacial electric field $E$, this built-in polarization is neutralized, and we can expect this point to be the location of maximum capacitance.
This interfacial electric field is $E = \sigma/\epsilon_0$ by Gauss's law, directly determined by Gauss's law, while the electrode potential depends on the overall electrostatic potential profile and the electrode charge response location ($\Delta z$ in particular).
Hence, the CMC should be invariant across the family of capacitance curves, while the PMC depends sensitively on the overall capacitance.
Finally, note that the capacitance increases slightly for large potentials: this is due to an increased density of electrolyte in response to high electric fields at the interface (electrostriction).\cite{PolarizableCDFT}
The magnitude of this effect will be sensitive to the electrode-electrolyte interaction potential, and may be obscured by ion adsorption at potentials far from the PZC in experiment anyway.
Consequently, we focus on the asymmetric behavior of the capacitance close to the PZC below.
We next consider the impact of deviations from the simple point charge model of electrode charge density (sheet charge in the planar average) considered so far.
First, we use the linear-response charge density profile from DFPT (see Section~\ref{sec:DetailsDFT}) with a constant shape scaled to each electrode charge density value (FIG.~\ref{fig:rho}(b)).
FIG.~\ref{fig:capacitance}(b) shows the result of using this charge density profile instead of the MD charge, but with the same placement scheme as above: at specific locations in $z$ (with various $\Delta z$ offsets).
We find absolutely no difference in the capacitance curves and CMC, which can be explained by Gauss's law as long as the electrode and electrolyte charge densities do not overlap.
The exponential tail of the electronic charge response of the electrode does in fact overlap partially with the electrolyte charge density for the highest-capacitance $\Delta z = 0$ case (FIG~\ref{fig:rho}(b)), but this impacts the capacitance negligibly.
The overall potential difference for a given electrode charge density shape would match that for the sheet charge case when the sheet charge is placed at the center-of-charge of the charge distribution.
This center-of-charge location is absorbed into our definition of $\Delta z$ based on the $C\sub{PZC} = 20~\mu$C/cm$^2$ criterion (Section~\ref{sec:CapacitanceMethod}), and so the family of capacitance curves remains unchanged.
Next, let's account for the actual variation of electron density of the electrode.
First, consider the effect of the electron density on the short-ranged potential of the liquid: an increased electron density would lead to higher repulsion that pushes non-bonded liquid atoms away.
FIG.~\ref{fig:capacitance}(c) includes this effect by placing the electrode and electrolyte charges based on the overlap $\bar{n}(\vec{r}) = \int d\vec{r}' n(\vec{r}') n\sub{water}(\vec{r} - \vec{r}')$ of a water molecule's electron density $n\sub{water}(\vec{r})$ and the electrode electron density $n(\vec{r})$, as parameterized in the SaLSA solvation model.\cite{SaLSA, CANDLE}
Specifically, the separation at which $\bar{n}$ crosses $n_c = 1.42\times 10^{-3} a_0^{-3}$ correlates with the non-bonded distance of nearest approach.\cite{SaLSA}
We place the DFPT electrode charge density relative to the MD electrolyte density profiles based on this condition \emph{for each electrode charge}, and the offset by various $\Delta z$ to obtain the family of capacitance curves.
Interestingly, we find that the charge density profiles in FIG.~\ref{fig:rho}(c) are unchanged from the previous case.
Correspondingly, the capacitance curves and CMC still do not change (FIG.~\ref{fig:capacitance}(c)), indicating a negligible effect of the change in short-ranged repulsion with electrode charge density.
Finally, we use the full nonlinear variation of electrode charge density profile from DFT in FIG.~\ref{fig:capacitance}(d), which leads to a small but noticeable difference in the capacitance curves and CMC.
In particular, the magnitude of the CMC and the asymmetry overall is reduced compared to all previous cases.
Essentially, in the nonlinear response of the DFT, electron repulsion makes it is easier to positively charge the electrode by removing electrons than to negatively charge it by adding electrons.
This effect favors higher response on the positively-charged side (FIG.~\ref{fig:rho}(d)), and therefore reduces the overall asymmetry towards negative charges due to the water at the interface.
The net result is still a negative CMC, but reduced slightly in magnitude to $-3.3~\mu$C/cm$^2$.
\subsection{Entropy} \label{sec:ResultsEntropy}
Above, we showed that the capacitance peak occurs for negatively-charged electrodes with a CMC of $-3.7~\mu$C/cm$^2$ based on the MD charge densities, which reduces in magnitude to $-3.3~\mu$C/cm$^2$ accounting for nonlinearities in the electronic response of the electrode from DFT.
We found that the charge is a better measure of the maximum point compared to the potential because the charge directly determines the interfacial electric field seen by the water surface, while the potential depends more globally on the overall electrostatic potential.
Similarly, we expect the potential of maximum entropy (PME) relative to PZC to be inversely proportional to the peak capacitance for a family of ideal electrochemical interfaces with different capacitances, while the charge of maximum entropy (CME) will be constant.
Consequently, we will focus on comparing the CME to the CMC predicted above.
We predict CME by directly mimicking the experimental approach: heat the electrochemical interface and measure the change of electrode potential $\partial V/\partial T$ at fixed charge.
Specifically, we repeat the entire set of simulations used to generate the above results (which were for 298~K) at 318~K, and compute $\partial V/\partial T$ as a finite-difference derivative for each electrode charge density, $\sigma$ (FIG.~\ref{fig:dphi_dT}).
We find that $\partial V/\partial T$ crosses zero with a positive slope near $-6.4~\mu$C/cm$^2$, and since $\partial S/\partial \sigma|_T = -\partial V/\partial T|_\sigma$, this implies $\partial S/\partial \sigma$ will cross zero at this point with a negative slope.
Therefore, $\sigma \approx -6.4~\mu$C/cm$^2$ should be a local maximum of entropy as a function of electrode charge.
\begin{figure}
\includegraphics[width=\columnwidth]{dphi_dT}
\caption{Temperature derivative of interface potential, $dV/dT$, at several fixed electrode charges, $\sigma$.
The shaded region is a 95 \% confidence interval estimated using kernel ridge regression on several samplings of one of the five molecular dynamics runs at each charge density.
The zero crossing of $dV/dT|_\sigma = -dS/d\sigma|_T$ at $(-6.4 \pm 0.7)~\mu$C/cm$^2$ corresponds to the charge of maximum entropy (CME).}
\label{fig:dphi_dT}
\end{figure}
The error bars shown in FIG.~\ref{fig:dphi_dT} are calculated as the standard deviation of the $\partial V/\partial T$ calculated from each of the ten half-cells simulated for each charge.
Note the extremely small magnitude of the results: the potentials only change by approximately $10-20$~mV over our 20~K baseline; the CME prediction therefore requires large ensembles with sufficient statistics and a careful analysis.
To precisely pin down the zero-crossing of $\partial V/\partial T$, we fit a general Kernel ridge regression model so as to not bias the result by choosing a more restrictive function form such as a polynomial.
We repeat the fit for 1000 re-samplings of the data by selecting the result from different half-cells at each charge.
The band shown in FIG.~\ref{fig:dphi_dT} is the 90 \% confidence interval obtained from this ensemble of fits.
Finally, from the zero-crossing of this band, we can quantify the 90 \% confidence interval of the CME to be $(-6.4 \pm 0.7)~\mu$C/cm$^2$.
Our predicted CME for ideal metal-water interfaces is in excellent agreement with experimental measurements of $-5~\mu$C/cm$^2$ for gold\cite{laserPME} and ($-4$ to $-6$)$~\mu$C/cm$^2$ for mercury.\cite{HgPME}
Most interestingly, it differs from the capacitance peak location (CMC) of $-3.7~\mu$C/cm$^2$ to $-3.3~\mu$C/cm$^2$ from Section~\ref{sec:ResultsCapacitance}.
Also note that the 20~K difference used in the calculation of $\partial V/\partial T$ does not affect the interface properties appreciably and cannot be the reason for the difference between CMC and CME: the CMC shifts by at most $0.3~\mu$C/cm$^2$ between 298~K and 318~K (FIG.~\ref{fig:cap-318K} in Appendix~\ref{sec:AppendixResults}).
The CMC and CME indicate asymmetric charge and thermodynamic response in the same direction: higher for negatively-charged electrodes.
However, by carefully calculating both quantities from the same set of MD simulations, we can unambiguously conclude that the CMC and CME do not coincide even for ideal electrochemical interfaces.
This provides a renewed incentive to experimentally measure the CMC, which as discussed in the Introduction, is challenging because the low ionic concentrations typically used to avoid ion adsorption lead to a capacitance dip that obscures the precise location of the capacitance maximum.
\section*{Conclusions}
We have performed classical molecular dynamics simulations and evaluated the capacitance and potential of maximum entropy for aqueous, charged metallic interfaces.
We find distinct, non-coincident values for the CMC and CME. For surfaces with large capacitance, the potentials of minimum entropy and maximum capacitance will be very similar.
Our findings of the asymmetric response of interfacial water open new questions about the electrochemical interface and the response properties themselves.
Future work is necessary to understand why the CME and CMC do not coincide even for ideal interfaces.
This could stem from the different spatial regions that contribute to each effect.
The entropy is sensitive to the entire polarized region of each half cell in the interface, while the the interfacial (series) capacitance is most sensitive to the lowest-capacitance region in space: closest to the metal.
These spatial dependencies could be explored further by varying the ionic concentration, and the generality of CME-CMC differences can be tested using MD simulations of other asymmetric solvents such as acetonitrile.\cite{CANDLE}
Going beyond ideal interfaces, extensions of this approach can systematically quantify the impact of specific electrode and electrolyte properties on the charge response and thermodynamics of the double layer.
In particular, hydrophobicity of the electrode and ion sizes are known to impact the structure of interfacial water.\cite{NetzInterfacial, LimmerHydrophobic}
Lastly, the detailed capacitance and entropy predictions of charged interfaces presented here will facilitate future development of more accurate solvation models for electrochemistry.
\section*{Acknowledgements}
AS and RS acknowledge support by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award \#DE-SC0022247.
All calculations were carried out at the Center for Computational Innovations at Rensselaer Polytechnic Institute.
We acknowledge Dr. Thomas P. Moffat for his useful suggestions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,516 |
KEENE, N.H. (AP) — A New Hampshire man says he was demonstrating a safer way to carry a pistol when he accidentally shot and killed his roommate in their apartment.
Police say 25-year old Adam Anderson, of Keene, recklessly caused the death Sunday of 22-year-old Holden Guyette by failing to ensure the handgun was unloaded and clear. They say he aimed it at Guyette and caused a fatal chest wound.
A police affidavit says Anderson said he was unaware that he shot Guyette, but called 911 when he saw blood. He said he has been shooting firearms since childhood.
Tia Guyette, Anderson's girlfriend and Guyette's sister, witnessed the shooting.
Anderson was charged with manslaughter and was arraigned Monday. His case has been assigned to the public defender's office. | {
"redpajama_set_name": "RedPajamaC4"
} | 483 |
Want to add some raggy tunes to your repertoire? During this workshop we will work on several pieces one measure at a time for melody instruments (fiddle, mandolin or plectrum), then layer in some all-important ragtime harmonies, blue notes, and syncopated phrasing. Separately, chordal instruments (guitar, banjo,) will be learning how to navigate "circle" chord progressions and other authentic string band techniques for rags such as connecting bass runs or counterpoint lines. Then, all instruments will get together for some ensemble time.
Recommended for intermediate or advanced players.
When registering, please use the letters at the beginning of one of the three lines below.
Georgia music is celebrated for an exciting variety of hot greasy breakdowns, syrupy slow blues, and quirky rags. This session will begin with an hour or so of technique in the regional style for each instrument. Fiddlers will try some tunes that include various slides, double stops, and bow patterns of fiddlers such as Clayton McMichen. In other rooms on-site, guitars will study great guitarists like Riley Puckett, while banjos try two-finger picking, frailing, and plectrum style strumming as done by The Skillet Lickers or Yellow Hammers string bands.
For the time remaining, we will all gather to play a combination of well-known and rare tunes. All tunes will first be demonstrated with explanation for recording, next played slowly to join in, then at moderate speed.
Video devices are welcome. No notation required or provided.
All levels are welcome but musicians would benefit more with some experience playing old-time music.
The Hickhoppers are a high-energy Georgia family string-band that are both entertainers and educators of old-time music. The Kinneys have been featured master artists at Swanannoa Gathering, Festival of American Fiddle Tunes, Alabama Folk School, John C. Campbell Folk School, and Blue Ridge Old-Time Week.
They have several first-place awards at fiddle contests across the south, including the Georgia String Band Festival, Chattanooga Great Southern Old Time Festival, and Great Big Yam Potatoes in Mississippi.
Known as preservationists of Georgia music, Mick Kinney, with sons Evan and Mickey, have contributed research to the Smithsonian, and Georgia Humanities Council. From 2016-17 they recorded their collection of rare tunes entitled In Dear Old Georgia Vol I & II, as well as their novelty song albums released as The Hickhoppers, and in 2018 as the Griddle Lickers.
In addition to traditional dance music on fiddle and banjo, the band's repertoire draws from Ragtime, Swing, & Cajun tunes.
Recently booked at Brooklyn Folk Fest, and Delaware Valley Bluegrass Festival, they also perform with the Squirrel Skinners, Bill & the Belles, and the Georgia Crackers.
The concert will take place, rain or shine. Weather permitting, we'll be outside where railroad ties serve as seats . . . but bring a camping chair or lawn blanket if you prefer.
Tickets are not required but reservations are requested.
A concert donation of $20 is suggested to cover the band's travel expenses from Georgia.
Advance registration is required for the workshops and space is limited, so participants get the proper attention. Please register early to guarantee a seat in your chosen workshop and instrument section.
The workshops are $50 each or $75 for both.
Please include your email and telephone number with your check so that we can send a confirmation and workshop arrival instructions.
Checks are best, but registration by email and PayPal is possible for an extra fee. This is not a money-maker. We just want to cover expenses.
Reservations are suggested for the house concert, so we can properly prepare.
Concert donations go to the performers and can be made at the door or in advance with your workshop reservation.
Bring your instruments & stay for the after-concert jam with Mick Kinney & The Hickhoppers. All skill levels and instruments (bones, washboards, jugs, ukuleles, bass fiddle, guitar, fiddle, mandolin, accordion, banjo, dulcimer, spoons and didgeridoos) are welcome. Let's have some fun! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,021 |
Q: What is meant by react-redux binding? I'm having trouble understanding the question and finding answers.
Is it the usage of the provider wrapper component? Or using the react-redux hooks useDispatch and useSelector? Or the concept of action - reducer - state?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,515 |
\section{Introduction}
\label{sec:intro}
The confinement property of the strong interaction implies that only hadrons are directly observable at the low scales accessible in detectors.
To arrive at a detailed description of particle reactions at collider experiments such as the \LHC necessitates the use of models for the transition from the fundamental QCD particles, quarks and gluons, to their bound states, the hadrons.
This transition, known as hadronisation, is responsible for the bulk of particle production in events involving the strong interaction, and it has directly observable consequences on quantities such as energy flows, jet shapes, or rapidity gaps.
In the absence of a quantitative understanding based on the first principles, hadronisation is being described in terms of phenomenological models and usually embedded within event generators~\cite{Buckley:2011ms}.
In the perturbative part of the simulation, the event generators describe the production of quarks and gluons, partons, in a sequence of stages; starting with exact, fixed-order matrix elements at the largest momentum scales, these primary hard partons are successively dressed through the emission of softer or collinear secondary partons at decreasing scales in the parton showers.
In hadron collisions, multiple parton-parton interactions in the underlying event further increase the number of partons, an effect again described through phenomenological models.
Ultimately, the perturbative part of the event generation results in parton configurations resolved at scales of the order of a GeV.
At this point, hadronisation models take over and turn the partons into sets of primordial hadrons, some of which may be unstable and must decay further.
Broadly speaking, currently used hadronisation models fall into two categories.
Building on linear confinement and the idea of practically one-dimensional QCD flux tubes, string models were first discussed in~\cite{Artru:1974hr}; a powerful realization of the string idea, and the most famous one, known as the Lund model, has been worked out in~\cite{Andersson:1983ia,Andersson:1998tv}.
It has been implemented and subsequently further refined in the P\protect\scalebox{0.8}{YTHIA}\xspace event generator~\cite{Sjostrand:2006za,Bierlich:2022pfr}.
In the past decade, the model has been extended to the notion of colour ``ropes"~\cite{Bierlich:2014xba,Bierlich:2016vgw,Bierlich:2020naj,Bierlich:2022oja}, fused strings, which are of particular importance in heavy-ion collisions and better account for effects such as strangeness enhancement or collective flows.
More recently, a thermodynamic approach to string fragmentation has been studied in~\cite{Fischer:2016zzs}, modifying, among others, production rates of heavy hadrons.
Time-dependent string tension was shown to increase the rates and modify the kinematics of strange and baryon production~\cite{Hunt-Smith:2020lul}, and similarly, hyper-fine splittings in hadron formation in the string model have been analysed in~\cite{Bierlich:2022vdf}, which also affect the production yields of hadron species, in particular strange hadrons.
In contrast, local parton-hadron duality (LPHD)~\cite{Azimov:1984np} and, in particular, preconfinement~\cite{Amati:1979fg} have been the guiding principles underpinning the development of cluster hadronisation models in~\cite{Field:1982dg,Gottschalk:1982yt,Gottschalk:1983fm,Gottschalk:1986bv} and in~\cite{Marchesini:1983bm,Webber:1983if}.
The latter has been implemented in the H\protect\scalebox{0.8}{ERWIG}\xspace event generator~\cite{Corcella:2000bw,Bellm:2019zci}.
In the following the hadronisation model of S\protect\scalebox{0.8}{HERPA}\xspace~\cite{Gleisberg:2003xi,Gleisberg:2008ta,Sherpa:2019gpd} will be presented.
It builds on the original independent realization of the cluster model idea in~\cite{Winter:2003tt} and has been further refined in a new and improved implementation.
We will describe its underlying principles in detail in \SecRef{Sec:ModelHad} and we will highlight some of the considerations in resolving problematic kinematic configurations, typically involving extremely light clusters.
In \SecRef{Sec:ModelCR} we will introduce a first, simplistic model for colour reconnections in S\protect\scalebox{0.8}{HERPA}\xspace.
It provides an alternative to models motivated by the analysis of such effects in the measurements of the $W$ mass~\cite{Sjostrand:1993hi} and the top mass~\cite{Skands:2007zg}, following on a first implementation in the framework of the description of multiple--\-parton scattering in~\cite{Sjostrand:1987su}.
By and large, models such as the one in~\cite{Sjostrand:1987su} and its extensions or variations in P\protect\scalebox{0.8}{YTHIA}\xspace~\cite{Christiansen:2015yca,Christiansen:2015yqa} and H\protect\scalebox{0.8}{ERWIG}\xspace~\cite{Gieseke:2012ft,Reichelt:2017hts,Bellm:2018jkh,Gieseke:2018gff,Bellm:2019wrh}, correct for the effect of interpreting the colour flow in parton showers through planar diagrams~\cite{tHooft:1973alw}, {\it i.e.}\xspace the unique, one--\-to--\-one relation of colours and anti-\-colours, and they also include collective effects, for example through the colour ropes mentioned above.
We turn to the presentation of results in \SecRef{Sec:Results}, obtained by tuning the model with and without colour reconnections for two parton showers, CSS\protect\scalebox{0.8}{hower}\xspace~\cite{Schumann:2007mg} and D\protect\scalebox{0.8}{IRE}\xspace~\cite{Hoche:2015sya} implemented in S\protect\scalebox{0.8}{HERPA}\xspace. The details on the tuning parameters are given in \AppRef{App::Parameters}. The performance of new tunes at different energies is discussed in \SecRef{Sec:Energyextrapolation}. We conclude in \SecRef{Sec:Summary} with an outlook.
\section*{Acknowledgements}
We would like to thank our colleagues from the S\protect\scalebox{0.8}{HERPA}\xspace collaboration for the fruitful discussions and technical support. We would also like to thank S. Chhibra and H. Schulz for technical support. This work has received funding from the European Union's Horizon 2020 research and innovation programme as part of the Marie Skłodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no. 722104).
FK gratefully acknowledges funding as Royal Society Wolfson Research fellow.
\newpage
\begin{appendix}
\input{wavefunctions}
\input{parameters}
\end{appendix}
\clearpage
\bibliographystyle{unsrt}
\section{Colour Reconnections}
\label{Sec:ModelCR}
In S\protect\scalebox{0.8}{HERPA}\xspace a simple model for non-perturbative colour reconnections has been made available; it is however at the moment switched off by default.
Such soft colour reconnections have first been analysed and modelled in the context of $W$-mass measurements at \LEP~\cite{Sjostrand:1993hi} and about 15 years later in the context of top-mass measurements~\cite{Skands:2007zg}; they also played an important part in the hadronisation of the final states in multiple--\-parton interactions in~\cite{Sjostrand:1987su}.
They can be thought of as resulting from two effects.
First of all, parton showers implicitly assume the limit of an infinite number of colours, $N_c\to\infty$, which results in planar colour flows~\cite{tHooft:1973alw} and, therefore, a direct one--\-to--\-one connection of quarks and anti-quarks and therefore a unique way in which the first primary clusters are formed.
For the actual $N_c=3$ of QCD, this unique connection of colours and anti-colours is obviously not correct, and one would expect small changes to it.
Secondly, in particular in hadron--\-hadron collisions, and in the presence of multiple parton--\-parton interactions, one can expect that some of the parton cascades overlap in space--\-time, increasing the probability for the soft exchange of colours.
In contrast to, {\it e.g.}\xspace, the model implemented in H\protect\scalebox{0.8}{ERWIG}\xspace~\cite{Gieseke:2012ft}, and more in line with its realisation in P\protect\scalebox{0.8}{YTHIA}\xspace~\cite{Christiansen:2015yqa}, the S\protect\scalebox{0.8}{HERPA}\xspace colour reconnection model is invoked at the end of the parton showering step, before the gluons decay and primordial clusters are formed.
The model assumes a parton shower in the large $N_c$ limit, where each colour is compensated by exact one anti-colour, and, consequently, the result containing $N$ such colour pairs. S\protect\scalebox{0.8}{HERPA}\xspace then repeatedly -- $N^2$ times -- compares the distances of original and swapped pairs of partons.
The distance of two partons $i$ and $j$ is given by
\begin{equation}
d_{ij} = \frac{1}{C}\,\times\,\Delta P_{ij}\,\times\,\Delta R_{ij}
\end{equation}
where $C=1$ for colour-connected pairs $ij$ and $C=\kappa_C$ for (swapped) pairs.
The distances $\Delta P_{ij}$ and $\Delta R_{ij}$ of the two partons in momentum space and the transverse position space are given by
\begin{equation}
\label{Eq:CRMomDistance}
\Delta P_{ij} = \left\{
\begin{array}{cl}
\displaystyle{\log\left[
\frac{(p_i+p_j)^2-(p_i^2+p_j^2)+Q_0^2}{Q_0^2}\right]} &
\mbox{\rm (logarithmic)}\\[5mm]
\displaystyle{\left[\frac{(p_i+p_j)^2-(p_i^2+p_j^2)+Q_0^2}{Q_0^2}
\right]^{\eta_P}} &
\mbox{\rm (power)}
\end{array}\right.
\end{equation}
and
\begin{equation}
\label{Eq:CRPosDistance}
\Delta R_{ij} = \left\{
\begin{array}{cl}
\displaystyle{
\left(\frac{|x_\perp^{(i)}-x_\perp^{(j)}|^2}{R_0^2}\right)^{\eta_R}} &
\mbox{\rm for}\; |x_\perp^{(i)}-x_\perp^{(j)}|^2 > R_0^2 \\[2mm]
1 & \mbox{\rm else\,,}
\end{array}\right.
\end{equation}
respectively.
Note that spatial distances are relevant only in cases like, for example, hadron--\-hadron collisions, where the individual scatters of the underlying event can occur at different positions in the transverse plane.
Note that if parton $i$ or $j$ is a gluon, the model
assumes that its momentum splits equally between the colour and
the anti-colour and the corresponding momentum is multiplied by
$1/2$ in \EqRef{Eq:CRMomDistance}.
From these distances, the model constructs a ``swapping probability", to
reconnect parton pairs $il$ and $kj$ rather than the original $ij$ and
$kl$, as
\begin{equation}
\mathcal{P}_{\rm swap} =
\frac{\exp\left[-(d_{il}+d_{kj})/\bar{d}\right]}
{\exp\left[-(d_{ij}+d_{kl})/\bar{d}\right]}\,,
\end{equation}
where the normalisation $\bar{d}$ is given by a sum of distances over
all colour pairs,
\begin{equation}
\label{EQ:CRProb}
\bar{d} = \frac{1}{N^\kappa}
\sum\limits_{\rm pairs \{nm\}} d_{nm}
\end{equation}
\section{Cluster hadronisation Model}
\label{Sec:ModelHad}
\subsection{Cluster formation}
After the parton shower evolution stops at transverse momenta $p_{T,{\rm min}}\approx 1$ GeV, hadronisation models take over and transform the resulting partons, quarks and gluons, into primary hadrons, some of which may decay further.
In cluster fragmentation models this is achieved by forming colourless clusters made of quarks and anti-quarks.
This results in a forced, non-perturbative splitting of each of the gluons into a quark--anti-quark pair, carrying its two colours.
Since typical parton showers are formulated in the limit of infinitely many colours, $N_c\to\infty$, each coloured quark can thus be associated with an anti-quark of the exact anti-colour; the colour--anti-colour pair then neutralises each other by forming a colour-singlet cluster.
In S\protect\scalebox{0.8}{HERPA}\xspace, as in H\protect\scalebox{0.8}{ERWIG}\xspace and P\protect\scalebox{0.8}{YTHIA}\xspace~\cite{Andersson:1981ce}, baryons are assumed to be composed of a quark $q_1$ and a diquark $(q_2q_3)$, the (fictitious) bound state of two quarks, such that any baryon $B = q_1(q_2q_3)$.
In S\protect\scalebox{0.8}{HERPA}\xspace these diquarks can already emerge in the gluon splitting, {\it i.e.}\xspace during the formation of the primary clusters.
This results in a somewhat softened correlation of $B\bar{B}$ pairs in phase space, similar to the popcorn mechanism in P\protect\scalebox{0.8}{YTHIA}\xspace~\cite{Andersson:1984af,Eden:1996xi}.
To ease the language we will collectively denote quarks and anti--diquarks emerging in the non-perturbative phase of hadronisation as ``flavours'', with the implicit understanding that they will have a non-vanishing constituent mass, in contrast to the current quarks in the perturbative phases of event generation, such as the parton shower.
\subsubsection{Non-perturbative splitting of gluons}
Quark and gluon ensembles produced in the large-$N_c$ limit by parton shower are colour-ordered sequences of a flavour (quark or anti--diquark), some gluons, and an anti-flavour (anti--quark or diquark), or of gluons only.
Postponing a discussion of the latter case to a later stage, let us first focus on the non-perturbative splitting of the gluons in such an $n$-particle sequence $f_1g_2g_3g_4\dots g_{n-1}\bar{f}_n$.
\subsubsection{Splitting a gluon}
In S\protect\scalebox{0.8}{HERPA}\xspace the gluons in these sequences are split step-wise from the edges, $g\to \bar{\tilde{f}}f$. The resulting anti-flavour or flavour combines with the neighbouring flavour or anti-flavour into a cluster $\mathcal{C}[f\bar{\tilde{f}}]$ or $\mathcal{C}[\tilde{f}\bar{f}]$, schematically,
\begin{equation}
f_1g_2g_3g_4\dots g_{(n-2)}g_{(n-1)}\bar{f}_n \to
\left\{\begin{array}{lcl}
\mathcal{C}[f_1\bar{\tilde{f}}_{2'}]\;+\;
\tilde{f}_{2''}g_3g_4\dots g_{(n-2)}g_{(n-1)}\bar{f}_n
& \mbox{\rm for} & g_2\to \bar{\tilde{f}}_{2'}\tilde{f}_{2"}\\[2mm]
f_1g_2g_3g_4\dots g_{(n-2)}\bar{\tilde f}_{(n-1)'}\;+\;
\mathcal{C}[\tilde{f}_{(n-1)''}\bar{f}_n]
& \mbox{\rm for} & g_2\to \bar{\tilde{f}}_{(n-1)'}\tilde{f}_{(n-1)"}\,,
\end{array}
\right.
\end{equation}
depending on whether gluon 2 or gluon $(n-1)$ was selected to split.
This selection is taken at random unless either $f_1$ or $\bar{f}_n$ are heavy flavours, $c$ or $b$ quarks: in this case the ``neighbour" gluon is selected ({\it i.e.}\xspace $g_2$ if $f_1$ is a heavy quark, and $g_{(n-1)}$ if $\bar{f}_n$ is a heavy quark).
In the following we will assume that $g_2$ is the splitting gluon, the ``splitter", and we denote flavour $f_1$ as ``spectator".
S\protect\scalebox{0.8}{HERPA}\xspace then produces trial splittings of the gluon - determined by the selection of the produced flavour $\tilde{f}$ and the corresponding kinematics until an allowed solution is found.
\subsubsection{Determining the flavour $\tilde{f}$ in gluon splitting}
The produced trial flavour $\tilde{f}$ is selected according to the ``popping" probabilities $P_{\tilde f}$, with available flavours subject to the constraint
\begin{equation}
\label{Eq:GluonSplitMassConstraint}
M_{12} - m_{f_1} \ge 2m_{\tilde f}\,,
\end{equation}
where $M_{12} = \sqrt{(p_1+p_2)^2}$ is the invariant mass of the splitter--\-spectator system.
The $P_{\tilde f}$ are calculated from the input parameters according to \EqRef{Eq:PoppingProbs}, and include also the possibility of gluons decaying directly into diquarks.
\subsubsection{Fixing the kinematics of the decay}
In the rest frame of the splitter--\-spectator system, the splitter 2 is oriented along the negative $z$ axis and the spectator 1 is oriented along the positive $z$-axis,
\begin{equation}
p_1^\mu = \frac{M_{12}}{\sqrt{2}}\left[z_+n_+^\mu+(1-z_-)n_-^\mu\right]\;\;\;\mbox{\rm and}\;\;\;
p_2^\mu = \frac{M_{12}}{\sqrt{2}}\left[(1-z_+)n_+^\mu+z_-n_-^\mu\right]\,,
\end{equation}
where
\begin{equation}
z_+ = 1\;\;\;\mbox{\rm and}\;\;\;z_- = 1-\frac{m_{f_1}^2}{2M_{12}^2}
\end{equation}
and the two light-like vectors $n_\pm^\mu = (1,\,0,\,0,\,\pm 1)$.
The four-momenta of the spectator 1 and the two produced flavours 2' and 2'' emerge from $p_1+p_2 \to \tilde{p}_1+\tilde{p}_{2'}+\tilde{p}_{2"}$ after the gluon splitting.
Demanding that the spectator keeps its direction in the rest frame of the system, {\it i.e.}\xspace that its four momentum is entirely spanned by $n_\pm$ and demanding that the transverse momentum is compensated between the two new flavours yields a parameterization of the decay through
\begin{equation}
\label{Eq:GluonDecayKinematics}
\begin{array}{rclrcrlcl}
\tilde{p}_1^\mu &=& M_{12} \Big[& xz^{(0)}\;n_+^\mu &+& (1-z^{(1)})(1-y)\; n_-^\mu\Big]&\\[2mm]
\tilde{p}_{2'}^\mu &=& M_{12} \Big[&(1-x)z^{(0)}\;n_+^\mu &+& (1-z^{(1)})y\; n_-^\mu\Big]& + & k_\perp^\mu\\[2mm]
\tilde{p}_{2"}^\mu &=& M_{12} \Big[& (1-z^{(0)})\;n_+^\mu &+& z^{(1)}\;n_-^\mu\Big] &
- & k_\perp^\mu
\end{array}
\end{equation}
The absolute value $k_T$ of the transverse momentum $k_\perp$ is selected according to a Gaussian, with a maximum value given by the parton-shower cut-off, $p_{T,{\rm min}}$,
\begin{equation}
\label{Eq:TransverseMomentum}
\mathcal{P}(k_T) = \exp\Big[-k_T^2/k_{\perp,0}^2\Big]
\Theta(p_{T,{\rm min}}^2-k_T^2)
\end{equation}
and its azimuth is flat, $k_\perp^\mu = k_T(0,\,\cos\phi,\,\sin\phi,\,0)$.
This leaves the determination of the longitudinal momenta fractions. The parameter governing the splitting of the gluon is $z^{(1)}$ and for its determination S\protect\scalebox{0.8}{HERPA}\xspace offers two parameterizations, namely
\begin{equation}
\label{Eq:LongMomentumGluon}
\mathcal{P}(z) = \left\{\begin{array}{ll}
z^\alpha + (1-z)^\alpha & \mbox{\rm additive}\\
z^\alpha(1-z)^\alpha & \mbox{\rm multiplicative}
\end{array}\right.
\end{equation}
From $k_T^2$ and $z^{(1)}$ the other kinematic parameters $z^{(0)}$, $x$, and $y$ are determined as
\begin{eqnarray}
z^{(0)} &=& 1-\frac{m_{\tilde f}^2+k_T^2}{z^{(1)}M_{12}^2}\nonumber\\
x &=& \frac{Q^2+m_{f_1}^2-k_T^2+
\sqrt{(M^2-m_{f_1}^2-k_T^2)^2-4m_{f_1}^2k_T^2}}{2Q^2}\nonumber\\
y &=& \frac{k_T^2}{(1-x)Q^2}
\end{eqnarray}
where $Q^2 = z^{(0)}(1-z^{(1)})M_{12}^2$.
$z^{(0)}$ and both $x$ and $y$ have to be between 0 and 1 for the gluon splitting to be kinematically viable and, therefore, accepted.
Once the kinematics has been fixed, particles 1 and 2' combine into a cluster ${\mathcal C}$, and 2'' becomes a new spectator for the next splitting.
\subsubsection{Gluon ``rings''}
In some cases, for example in the decay of heavy quarkonia such as $\eta_c\to gg$ or $J/\psi\to ggg$ or quite often in hadron collisions, the colour-singlet structures emerging from the parton shower and entering hadronisation are purely gluonic, $g_1g_2\dots g_{(n-1)}g_n$.
In this case S\protect\scalebox{0.8}{HERPA}\xspace selects the colour-connected gluon pair with the largest combined invariant mass and splits one of the two gluons, $g_i$, with the other gluon acting as spectator.
The resulting structure is then re-ordered to the form $f_ig_{(i+1)}g_{(i+2)}\dots g_ng_1g_2\dots g_{(i-1)}\bar{f}_{i'}$.
\subsubsection{Clusters directly transiting to hadrons}
Some of the primary clusters produced in the non-perturbative gluon splittings have masses $M_{\mathcal C}$ below the threshold $M_{\rm trans}[f_1\bar f_2]$ for their direct transition to hadrons with the same flavour quantum numbers.
This threshold is given by a linear combination of the lightest and heaviest hadron mass as
\begin{equation}
\label{Eq:TransitionThreshold}
M_{\rm trans}[f_1\bar f_2] =
x_{\rm trans}{\rm \min}(m_{\mathcal H[f_1\bar f_2]}) +
(1-x_{\rm trans}){\rm \max}(m_{\mathcal H[f_1\bar f_2]})
\end{equation}
with a tuning parameter $x_{\rm trans}$.
If $M_{\mathcal C}<M_{\rm trans}[f_1\bar f_2]$ then the gluon splitting will directly produce a hadron instead of a cluster, with the hadron selected according to relative probabilities given by
\begin{equation}
\mathcal{P}_{\mathcal{C}[f_1\bar{f}_2]\to
\mathcal{H}_1[f_1\bar{f}_2]+\gamma}
=
w_{\mathcal{H}}
\left|\psi_{\mathcal{H}}(f_1\bar{f}_2)\vphantom{|^|_|}\right|^2\,.
\label{Eq:ProbC2HTransition}
\end{equation}
Here, $w_{\mathcal H}$ is a relative probability for the production of a hadron ${\mathcal H}$ -- typically a combination of an overall multiplet weight and a hadron-specific additional modifier for certain ``tricky" hadrons such as $\eta$ and $\eta'$ mesons -- and $\psi_{\mathcal H}(f_1\bar f_2)$ is the flavour wave function of the hadron.
In this case the gluon splitting kinematics of Eq.~(\ref{Eq:GluonDecayKinematics}) is replaced with
\begin{equation}
\begin{array}{rclrcrlcl}
\tilde{p}_h^\mu &=& M_{12}\Big[&z^{(1)}\;n_+^\mu &+& (1-z^{(2)}) n_-^\mu\Big]& +& k_\perp^\mu\\[2mm]
\tilde{p}_{2''}^\mu &=& M_{12}\Big[&(1-z^{(1)})\;n_+^\mu &+& z^{(2)} n_-^\mu\Big]& - & k_\perp^\mu
\end{array}
\end{equation}
with the updated values for $z^{(1,2)}$ given by
\begin{equation}
z^{(1)} =
\frac{M_{12}^2+m_{\mathcal H}^2-m_{2}^2 + \sqrt{\vphantom{\frac12}
(M_{12}^2+m_{\mathcal H}^2-m_{2}^2)^2 - 4M_{12}^2(m_{\mathcal H}^2+k_T^2)}} {2M_{12}}
\;\;\;\mbox{\rm and}\;\;\;
z^{(2)} =
1-\frac{m_{2}^2+k_T^2}{M_{12}z^{(1)}}\,.
\end{equation}
\subsubsection{Rescue system for anomalies in cluster formation}
In some rare cases it may be impossible for gluons to decay or for the produced clusters to decay further or to transition directly into hadrons.
Below we outline how the model treats these anomalies:
\begin{enumerate}
\item Splitter-spectator system not massive enough: \\
If the invariant mass of the splitter--\-spectator system ($f_1g_2$ or $g_2\bar{f}_1$) is not large enough to allow the gluon to split into two constituents,
\begin{equation}
(p_1+p_2)^2 < (m_1 + \underset{f}{\rm min}\;2m_f)^2
\end{equation}
with ${\rm min} m_f$ the mass of the lightest flavour, the gluon will be removed and its momentum will be added to the spectator momentum ($f_1g_2\to f_{1'}$ or $g_2\bar{f}_1\to \bar{f}_{1'}$ with $p_1 \to p'_1 = p_1+p_2$).
\item Two-gluon singlet $g_1g_2$ not massive enough: \\
If the invariant mass of a two-gluon system is not large enough to allow splitting one of the gluons,
\begin{equation}
(p_1+p_2)^2 < \underset{f}{\rm min}\;4m_f^2
\end{equation}
the singlet is treated as a cluster, and the cluster rescue system discussed below is invoked.
\item Singlet system below minimal hadron mass: \\
With the masses of light quarks usually ignored in the parton shower it is possible to arrive at two-quark systems $f_1\bar{f}_2$ with a mass below the lightest allowed hadron,
\begin{equation}
(p_1+p_2)^2 < \underset{h}{\mbox{\rm min}}\;m^2_{h[f_1\bar{f}_2]}\,,
\end{equation}
usually this implies that $(p_1+p_2)^2 < (m_{f_1}+m_{\bar{f}_2})^2$.
In this case, S\protect\scalebox{0.8}{HERPA}\xspace reshuffles momenta from another singlet system or one of the already produced clusters such that the light system can directly transfer to the lightest allowed hadron.
\end{enumerate}
\subsubsection{Distributions characterising cluster formation}
In \FigRef{Fig:PrimaryClusters} we exhibit two distributions that characterise this initial step of the cluster fragmentation model, namely, firstly, the distribution of primary cluster masses in the left panel, and secondly their multiplicity in the right panel.
They have been obtained after the CSS\protect\scalebox{0.8}{hower}\xspace, with no multijet merging and using the tuned parameters of the cluster fragmentation\footnote{
We have also set all heavy mesons and baryons stable in the simulation to suppress the fragmentation in their possible parton-level decays in the simulation.}. The tuned values of the parameters are given in \AppRef{App::Parameters}. \\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth]{figures/M_primaries.png}
&
\includegraphics[width=0.5\textwidth]{figures/N_primaries.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Mass (left panel) and multiplicity distributions (right panel) of primary clusters and hadrons in $e^+e^-\to$ hadrons events at varying centre-of-mass energies.
\label{Fig:PrimaryClusters}
}
}
\end{center}
\end{figure}
The cluster mass distribution follows what is expected from the distribution of partons produced in the parton shower, with a peak at about 1 GeV, anticipated from the parton shower cut-off $p_{\perp, 0} = 1$ GeV.
As there are more and potentially more massive clusters produced at higher energies, this peak is less pronounced at higher energies, compensated by a higher tail of the distribution.
In fact, it is entirely possible that, due to its probabilistic nature, the parton shower does not emit a single parton, and, consequently, there would be only a single primary cluster with the full centre-of-mass energy of the $q\bar q$ pair as mass.
We also observe that mass thresholds of heavy quarks and, more faintly, of diquarks, are visible in the overall mass distribution.
While in particular the bottom and less so the charm thresholds are fairly pronounced at the $Z$-pole, $E_{\rm c.m.}=91.2$ GeV, they are not as prominent at $E_{\rm c.m.} = 1000$ GeV.
This is due to two effects.
First of all, due to their coupling, down-type quarks, including $b$ quarks, are more copiously produced at the $Z$ pole compared to the up-type quarks, thereby explaining the somewhat larger size of the bottom peak compared to the charm-bump at the $Z$ pole.
Secondly, the parton shower produces mainly gluons, while the production of heavy quark pairs in gluon splitting is suppressed by their mass.
As a consequence, there are proportionally more light flavours and more light clusters produced which suppresses the significance of the heavy quark thresholds.
From the right panel of \FigRef{Fig:PrimaryClusters} we can also see that the number of primary clusters increases from $\langle n_{\rm clusters}\rangle\approx 5$ in the peak at $E_{\rm c.m.} = 91.2$ GeV to $\langle n_{\rm clusters}\rangle\approx 12$ in the peak at $E_{\rm c.m.} = 1000$ GeV, a very good realisation of logarithmic scaling.
We have also shown the number of primary hadrons there that emerge directly from those primary clusters that are not heavy enough to produce secondary clusters.
Not unexpected, due to the clusters disintegrating in binary decays, typically we find even hadron numbers.
The odd hadron multiplicities, usually at the per-mil level or below, are a consequence of individual clusters transforming directly into hadrons as part of the rescue system.
\subsection{Cluster fission}
If clusters made of two flavours $f_1$ and $\bar{f}_2$ are heavy enough -- {\it i.e.}\xspace above their threshold for decays into hadrons, see below -- they will decay into secondary clusters, $\mathcal{C}[f_1\bar{f}_2] \to \mathcal{C}[f_1\bar{f}] + \mathcal{C}[f\bar{f}_2]$.
In S\protect\scalebox{0.8}{HERPA}\xspace this proceeds by first selecting the non-perturbatively produced flavour $f$ associated to the decay, before defining the decay kinematics.
\subsubsection{Non-perturbative flavour production}
Similar to the treatment in the non-perturbative decays of gluons at the end of the parton shower, the produced flavours $f+\bar{f}$ are determined according to the ``popping'' probabilities $\mathcal{P}_f$.
In analogy to the case of gluon splitting above, \EqRef{Eq:GluonSplitMassConstraint}, the available flavours are only constrained by the condition that
\begin{equation}
M_{\mathcal{C}}-m_{f_1}-m_{\bar{f}_2} > 2m_f\,.
\end{equation}
The underlying assumption here is that in practically all cases it will be possible to produce hadrons from the resulting $\{f_1\bar{f}\}$ and $\{f\bar{f}_2\}$ systems, due to the constituent masses being similar to the hadron masses.
It is worth stressing here that, in contrast to early realisations of the cluster hadronisation model, in S\protect\scalebox{0.8}{HERPA}\xspace diquarks are allowed to constitute clusters\footnote{
Another alternative, proposed in~\cite{Gieseke:2017clv}, is to construct baryonic clusters directly from three quarks or three anti-quarks.}.
Allowing diquark production at every stage in the hadronisation process, {\it i.e.}\xspace in both gluon decays and in the subsequent fission of clusters into secondary clusters, softens their strong correlation.
This represents an alternative the popcorn mechanism~\cite{Andersson:1984af,Eden:1996xi} in the Lund string model within cluster hadronisation models, which softens the previous strong correlation of baryon--anti-baryon pairs.
\subsubsection{Fixing the kinematics}
Having fixed the ``popped'' flavour $f$ and therefore the flavour contents $\{f_1\bar{f}\}$ and $\{f\bar{f}_2\}$ of the two systems produced in the decay, their kinematics must now to be fixed.
This is achieved in the rest frame of the cluster, where the momenta $p_1$ and $p_2$ of particles $f_1$ and $\bar{f}_2$ are oriented parallel to the positive and negative $z$-axis, allowing us to introduce $n_\pm^\mu = (1,0,0,\pm 1)$.
In the rest frame of the cluster therefore
\begin{equation}
p_1 = \frac{m_{\mathcal{C}}}{2}\,\left[z_+n_+^\mu+(1-z_-)n_-^\mu\right]
\;\;\;\mbox{\rm and}\;\;\;
p_2 = \frac{m_{\mathcal{C}}}{2}\,\left[(1-z_+)n_+^\mu+z_-n_-^\mu\right]
\end{equation}
with
\begin{equation}
z_\pm = \frac{M_{\mathcal{C}}^2\pm m_{f_1}^2\mp m_{\bar{f}_2}^2 +
\sqrt{(M_{\mathcal{C}}^2-m_{f_1}^2-m_{\bar{f}_2}^2)^2 -
4m_{f_1}^2m_{\bar{f}_2}^2}}{2M_{\mathcal{C}}^2}\,.
\end{equation}
Similarly, the four four-momenta $p_{11}^\mu$, $p_{12}^\mu$, $p_{21}^\mu$ and $p_{22}^\mu$ of the four outgoing flavours $f_1$, $\bar{f}$, $f$, and $\bar{f}_2$ are parameterized as
\begin{equation}
\begin{array}{rclrcrlcl}
p_{11}^\mu &=& \displaystyle\frac{m_{\mathcal{C}}}{2}
\Big[&x^{(1)}z^{(1)}n_+^\mu &+& y^{(1)}(1-z^{(2)})n_-^\mu&\Big]& &\\[2mm]
p_{12}^\mu &=& \displaystyle\frac{m_{\mathcal{C}}}{2}
\Big[&(1-x^{(1)})z^{(1)}n_+^\mu &+& (1-y^{(1)})(1-z^{(2)})n_-^\mu&\Big]& +&
k_\perp^\mu\\[2mm]
p_{21}^\mu &=& \displaystyle\frac{m_{\mathcal{C}}}{2}
\Big[&(1-x^{(2)})(1-z^{(1)})n_+^\mu &+& (1-y^{(2)})z^{(2)}n_-^\mu&\Big]& -&
k_\perp^\mu\\[2mm]
p_{22}^\mu &=& \displaystyle\frac{m_{\mathcal{C}}}{2}
\Big[&x^{(2)}(1-z^{(1)})n_+^\mu &+& y^{(2)}z^{(2)}n_-^\mu&\Big]\,,& &
\end{array}
\end{equation}
where
\begin{eqnarray}
x^{(1,2)} &=&
\frac{\tilde{q}_{1,2}^2+m_{f_1,\bar{f}_2}^2-(m_{f}^2+k_T^2)\pm
\sqrt{[\tilde{q}_{1,2}^2-(m_{f_1,\bar{f}_2}^2-m_{f}^2+k_T^2)]^2-
4m_{f_1,\bar{f}_2}^2(m_{f_1,\bar{f}_2}^2+k_T^2})}
{2\tilde{q}_{1,2}^2}
\nonumber\\
y^{(1,2)} &=& \frac{1}{x^{(1,2)}}\cdot\frac{m_{f_1,\bar{f}_2}^2}{\tilde{q}_i^2}
\end{eqnarray}
and the masses of the two resulting clusters,
\begin{equation}
\tilde{q}_{1,2}^2 = z^{(1,2)}(1-z^{(2,1)})M_{\mathcal{C}}^2+k_T^2\,.
\end{equation}
The absolute value $k_T$ of the transverse momentum $k_\perp$, with respect to
the axis defined by the momenta of the cluster constituents, is selected according to the same Gaussian as before, \EqRef{Eq:TransverseMomentum}, with the same parameter $k_{\perp, 0}$.
This leaves the longitudinal momenta fractions $z_{1,2}$, or, equivalently, the masses of the outgoing clusters to be determined in order to fix the kinematics of the cluster decay.
S\protect\scalebox{0.8}{HERPA}\xspace offers two methods to achieve this:
\begin{enumerate}
\item Fixing the longitudinal momenta fractions $z_{1,2}$\\
The $z^{(i)}$ are selected according to a probability
\begin{equation}
\label{Eq:LongitudinalMomentumInCCC1}
\mathcal{P}(z) =
z^{\alpha} (1-z)^{\beta}
\exp\left[-\frac{\gamma}{z}\cdot
\frac{k_T^2+(m_{f_1}+m_{\bar{f}_2})^2}{k_{\perp,0}^2}\right]\,,
\end{equation}
a form similar to the Lund symmetric fragmentation function~\cite{Andersson:1998tv}, with parameters $\alpha$, $\beta$, and $\gamma$ depending on whether flavour $i$ is a light quark, a heavy quark, or a diquark or whether the decaying cluster contains a beam remnant, a parton stemming from the non-perturbative break-up of incident hadrons at hadron colliders.
The $z$ ranges are given by
\begin{equation}
z^{(1,2)}_{{\rm min}, {\rm max}} =
\frac{M_{\mathcal{C}}^2-(M^{(2,1)}_{\rm min})^2+(M^{(1,2)}_{\rm min})^2 \mp
\sqrt{\left[M_{\mathcal{C}}^2-(M^{(1)}_{\rm min})^2-(M^{(2)}_{\rm min})^2\right]^2-
4(M^{(1)}_{\rm min})^2(M^{(2)}_{\rm min})^2}}
{2M_{\mathcal{C}}^2}\,,
\end{equation}
where the $M_{\rm min}^{(i)}$ denote the minimal mass of a hadron system that can be produced from the flavour pair $\{f_1\bar{f}\}$ or $\{f\bar{f}_2\}$, {\it i.e.}\xspace, denoting the masses for a hadron $\mathcal{H}$ with flavour content $f\bar{f}'$ as $m_{\mathcal{H}}[f\bar{f}']$,
\begin{equation}
M_{\rm min}^{(1)} =
\mbox{\rm min}_{f'}
\left(m_{\mathcal{H}_{11}[f_1\bar{f}']}+m_{\mathcal{H}_{12}[f'\bar{f}]}\right)
\;\;\;\mbox{\rm and}\;\;\;
M_{\rm min}^{(2)} = \mbox{\rm min}_{f'}
\left(m_{\mathcal{H}_{21}[f\bar{f}']}+m_{\mathcal{H}_{22}[f'\bar{f}_2]}\right)
\,.
\end{equation}
\item Fixing the outgoing cluster masses\\
Alternatively, the $z^{(i)}$ can be calculated from the $\tilde{q}_i$, the masses of the two clusters produced in the decay.
They are selected according to
\begin{equation}
\tilde{q}_i^2 = \left(M^{(i)}_{\rm min}+\Delta M^{(i)}\right)^2\,.
\end{equation}
S\protect\scalebox{0.8}{HERPA}\xspace offers a number of different, relatively simple options to calculate the $\Delta M$, with (un-normalized) probabilities distributed according to
\begin{equation}
\label{Eq:LongitudinalMomentumInCCC2}
\mathcal{P}(\Delta M) = \left\{
\begin{array}{ll}
\exp\left[-\frac{\Delta M}{\gamma k_{\perp,0}}\right] &
(\mbox{\rm exponential})\\[2mm]
\exp\left[-\frac{(\Delta M-\langle\Delta M\rangle)^2}
{\gamma k_{\perp,0}}\right] &
(\mbox{\rm Gaussian})\\[2mm]
\exp\left[-\frac{(\log\Delta M-\log\langle\Delta M\rangle)^2}
{\gamma k_{\perp,0}}\right] &
(\mbox{\rm log-normal})\,,
\end{array}
\right.
\end{equation}
where the mean value of the mass shift, $\langle\Delta M\rangle = k_{\perp,0}$.
The $z^{(i)}$ are calculated from the two $\tilde{q}_i$ as
\begin{equation}
z^{(1,2)} = \frac{M_{\mathcal{C}}^2+
\tilde{q}^2_{1,2}-\tilde{q}^2_{2,1}+
\sqrt{\vphantom{\frac12}
(M_{\mathcal{C}}^2+\tilde{q}^2_{1}-\tilde{q}^2_{2})^2-
4M_{\mathcal{C}}^2(\tilde{q}^2_{1}+k_T^2)}}
{2M_{\mathcal{C}}^2}\,,
\label{Eq:zInClusterFissionWithMass}
\end{equation}
and, as before, the azimuthal angle w.r.t.\ the momenta of the cluster constituents is chosen flat.
\end{enumerate}
\subsubsection{Secondary clusters directly transitioning into hadrons}
Similar to the cluster produced in the non-perturbative gluon splitting, also the masses of secondaries produced in cluster fission may be below the threshold for direct transition to hadrons, {\it cf.}\xspace\EqRef{Eq:TransitionThreshold}).
Selecting the respective hadron type according to the probability given in Eq.~(\ref{Eq:ProbC2HTransition}), the kinematics of cluster fission to be adjusted to accommodate decays into one hadron plus a cluster or two hadrons only.
The two momenta for the secondaries are given by
\begin{equation}
\begin{array}{rclrcrlcl}
p_{1}^\mu &=& M_{\mathcal{C}}
\Big[&z^{(1)}n_+^\mu &+& (1-z^{(2)})n_-^\mu&\Big]& +&k_\perp^\mu \\[2mm]
p_{2}^\mu &=& M_{\mathcal{C}}
\Big[&(1-z^{(1)})n_+^\mu &+& z^{(2)}n_-^\mu&\Big]& -&k_\perp^\mu
\end{array}
\end{equation}
with the $z^{(1,2)}$ given by Eq.~(\ref{Eq:zInClusterFissionWithMass}) where the cluster masses are replaced by hadron masses where necessary.
In case this results in one cluster and one hadron, the momenta of flavours constituting the cluster are boosted into the new frame given by the updated cluster momentum.
\subsubsection{Distributions characterising cluster fission}
In \FigRef{Fig:SecondaryClusters} we depict distributions characterising the decay of clusters for different primary clusters: $u\bar u$ and $c\bar c$ clusters with a mass of 10 GeV and $b\bar b$ clusters with a mass of 20 GeV.\\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.4\textwidth]{figures/M_clusters.png}
&
\includegraphics[width=0.4\textwidth]{figures/x_cluster.png}
\end{tabular}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/N_hadrons.png}
&
\includegraphics[width=0.3\textwidth]{figures/pt_hadrons.png} &
\includegraphics[width=0.3\textwidth]{figures/x_hadrons.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Mass distributions of secondary clusters produced in the decays of clusters of different mass and flavour composition (upper left), the corresponding multiplicity of primary hadrons (lower left), their transverse momenta with respect to the axis defined by the momenta of the original cluster constituents (lower middle), and the longitudinal momentum fractions (lower right).
The parameters from \TabRef{Tab:ParamsDecays}, have been used.
\label{Fig:SecondaryClusters}
}
}
\end{center}
\end{figure}
We show the mass distribution of secondaries emerging in the first generation of decays and their $x_p=|\vec p_c|/|\vec p_f|$ distribution, where $p_f$ is the momentum of the quark inside the primary cluster giving rise to them, and $p_c$ are the momenta of the secondaries.
As expected the finite constituent quark masses lead to thresholds for the clusters containing them, leading to a minimal cluster mass of about 600 MeV for clusters containing only light quarks, of about 2100 MeV for single-charmed clusters and of about 5300 MeV for cluster containing a bottom quark.
Conversely, the $x_p$-distribution shows and increasingly sharp peak for increasing quark masses, for two reasons, both of which can be directly read off the ``fragmentation function" in \EqRef{Eq:LongitudinalMomentumInCCC1}.
First of all, comparing the tuned $\alpha$, $\beta$, and $\gamma$ parameters defining the cluster splittings for light and heavy flavours inside the cluster, it is apparent that the heavy-quark ``fragmentation" function is much harder than its light-quark counterpart.
In addition, while for the two heavy flavours -- the $c$ and $b$ quarks -- these parameters are identical, small values of $z$ experience a suppression that, up to parameters, scales like $\exp(-m_Q^2)/z$, resulting in a much more pronounced suppression of small $z$ values for the heavier quarks.
In the same figure, we also show the resulting overall multiplicity of primary hadrons in the full decay chain, their transverse momenta w.r.t.\ the axis defined by the cluster constituents, and their $x_p = |\vec p_h|/|\vec p_f|$ distribution.
We observe that the number of hadrons decreases with increasing mass of the original constituents, as simple result of a combination of available phase space, which is constrained by the masses of the quarks, and the harder cluster splitting for the heavier flavours.
This manifests itself also in the $x_p$ distributions where we see that the heavy hadrons carry more of the original quark momentum than their light counterparts, a trend that is also more pronounced for $b$ quarks over $c$ quarks.
Finally, we also note that the transverse momentum distributions of the hadrons are nearly uniform for all three cases.
\subsection{Cluster decays into hadrons}
\subsubsection{Selecting hadrons}
Once clusters are produced, either in the cluster formation phase following the parton shower or through fission into secondary clusters, they may decay, if they are light enough, into hadrons $\mathcal{H}_1+\mathcal{H}_2$.
The threshold for the decays into pairs of hadrons is given by the combination of the lightest possible and the heaviest possible masses.
For clusters $\mathcal{C}[f_1\bar{f}_2]$ made from a colour triplet $f_1$ -- either a quark or an anti--diquark -- and an anti--colour triplet $\bar{f}_2$ -- either an anti--quark or a diquark -- this would proceed by non-perturbatively producing a flavour--anti-flavour pair $f\bar{f}$ such that
\begin{equation}
\mathcal{C}[f_1\bar{f}_2]\to
\mathcal{H}_1[f_1\bar{f}]+\mathcal{H}_2[f\bar{f}_2]\,.
\end{equation}
The mass threshold for decays, $M_{\rm dec}[f_1\bar{f}_2]$ is defined by
\begin{equation}
\label{Eq:MassThresholdCHH}
M_{\rm dec}[f_1\bar{f}_2] =
x_{\rm dec}\mbox{\rm min}_{f}
\left(m_{\mathcal{H}_1[f_1\bar{f}]}+m_{\mathcal{H}_2[f\bar{f}_2]}\right) +
(1-x_{\rm dec})\mbox{\rm max}_{f}
\left(m_{\mathcal{H}_1[f_1\bar{f}]}+m_{\mathcal{H}_2[f\bar{f}_2]}\right)
\end{equation}
with a tuning parameter $x_{\rm dec}$.
If the cluster mass $M_{\mathcal{C}}<M_{\rm dec}$ then the cluster will decay into two hadrons or a hadron and a photon.
The exact channel for decays $\mathcal{C}\to \mathcal{H}_1+\mathcal{H}_2$ is selected according to the respective weights, given by
\begin{eqnarray}
\label{Eq:CHHWeights}
\mathcal{P}_{\mathcal{C}[f_1\bar{f}_2]\to
\mathcal{H}_1[f_1\bar{f}]+\mathcal{H}_2[f\bar{f}_2]}
&=&
\mathcal{P}_f
\times
w_{\mathcal{H}_1}
\times\left|\psi_{\mathcal{H}_1}(f_1\bar{f})\vphantom{|^|_|}\right|^2
\times
w_{\mathcal{H}_2}
\times\left|\psi_{\mathcal{H}_2}(f\bar{f}_2)\vphantom{|^|_|}\right|^2
\nonumber\\
&&\times
\frac{\sqrt{(M_{\mathcal{C}}^2-m_{\mathcal{H}_1}^2-m_{\mathcal{H}_2}^2)^2
-4m_{\mathcal{H}_1}^2m_{\mathcal{H}_2}^2}}{8\pi M_{\mathcal{C}}^2}
\times\left[\left(\frac{m_{\mathcal{H}_1}}{M_{\mathcal{C}}}\right)^\chi
+\left(\frac{m_{\mathcal{H}_2}}{M_{\mathcal{C}}}\right)^\chi\right]\,,
\end{eqnarray}
where $\mathcal{P}_f$ again is the ``popping'' probability for the production of flavour $f$.
The $\psi_{\mathcal{H}}$ are the flavour wave functions of the hadrons, {\it cf.}\xspace~\AppRef{App::wavefunctions}, and the $w_{\mathcal{H}}$ are additional weights to produce hadron $\mathcal{H}$, composed as products of the multiplet weight and, possibly, additional, hadron-type specific weights, listed in \TabRef{Tab:ParamsCDecays}.
$\chi$ is an additional tunable parameter, which by default has been chosen to vanish, $\chi = 0$.
\subsubsection{Rescue system for light clusters}
In S\protect\scalebox{0.8}{HERPA}\xspace's model, it is possible that clusters are created that are too light to decay into hadrons.
An example for this is the possible creation of clusters consisting of two diquarks, with a mass below the two-baryon threshold.
To avoid having to repeat possible costly parts of the event generation necessitates an extension of the cluster decay model to capture these cases:
\begin{enumerate}
\item Clusters made of two diquarks: $\mathcal C[(ij)(\bar{k}\bar{l})]$ with $M_\mathcal{C}<m_{B[(ij)]}+m_{B[(\bar{k}\bar{l})]}$\\
The most obvious case are clusters that consist of two diquarks $(ij)$ and $(\bar{k}\bar{l})$ with a mass that is below the mass threshold of baryon--anti-baryon pairs containing them.
In this case, S\protect\scalebox{0.8}{HERPA}\xspace splits the two diquarks and creates two quark--anti-quark pairs from them, with random pairing, {\it i.e.}\xspace $\{i\bar{k}\} + \{j\bar{l}\}$ or $\{i\bar{l}\} + \{j\bar{k}\}$.
Weights for kinematically allowed decays of the cluster into two mesons are calculated according to Eq.~\ref{Eq:CHHWeights}, and one decay mode is selected according to them:
\begin{equation}
\mathcal C[(ij)(\bar{k}\bar{l})]\;\to\;
\mathcal{M}[i\bar{k}] + \mathcal{M}[j\bar{l}] \;\;\;
\mbox{\rm or}\;\;\;
\mathcal{M}[i\bar{l}] + \mathcal{M}[j\bar{k}]\,.
\end{equation}
\noindent
If there is, however, no allowed decay of the cluster into two mesons, S\protect\scalebox{0.8}{HERPA}\xspace will try to ``annihilate'' a flavour pair.
For example, if $i=k$ in the two diquark constituents, they will be assumed to have ``cancelled'' each other out. The cluster will then decay into a photon and a meson with the remaining flavour quantum numbers $\{j\bar{l}\}$:
\begin{equation}
\mathcal C[(ij)(\bar{i}\bar{l})]\;\to\;
\mathcal{M}[j\bar{l}]+\gamma\,,
\end{equation}
where the meson is selected according to the available phase space for the decay, if more than one decay channel is kinematically allowed.
\item Clusters not made of two diquarks: $\mathcal C[f_1\bar{f}_2]$ with $M_\mathcal{C}<m_{M[f_1\bar{f}_2]}$\\
Clusters too light to decay into two hadrons will decay radiatively into a hadron and a photon, where the hadron is selected
according to weights given by
\begin{equation}
\mathcal{P}_{\mathcal{C}[f_1\bar{f}_2]\to
\mathcal{H}_1[f_1\bar{f}_2]+\gamma}
=
w_{\mathcal{H}}
\left|\psi_{\mathcal{H}}(f_1\bar{f}_2)\vphantom{|^|_|}\right|^2
\times
\frac{M_{\mathcal{C}}^2-m_{\mathcal{H}}}{8\pi M_{\mathcal{C}}^2}
\times
\left(\frac{m_{\mathcal{H}}}{M_{\mathcal{C}}}\right)^\chi\,.
\end{equation}
\item Clusters made of $q\bar{q}$ pairs: $\mathcal C[q\bar{q}]$ with $M_\mathcal{C}<m_{M[q\bar{q}]}$\\
This assumes that the cluster cannot decay into a pair of hadrons containing the quark and the anti-quark or the lightest meson made of the $q\bar{q}$-pair and a photon.
Then the model annihilates the $q\bar{q}$-pair to give rise to pions or photons, namely:
\begin{itemize}
\item
if $M_{\mathcal C}<M_{\pi\gamma}$, the threshold for $\mathcal{C}\to \pi^0\gamma$, the cluster will decay into two photons:
\begin{equation}
\mathcal C[q\bar{q}]\;\to\;\gamma+\gamma\,;
\end{equation}
\item
if $M_{\pi\gamma}<M_{\mathcal C}<M_{\pi\pi}$, the threshold for $\mathcal{C}\to \pi\pi$, the cluster will decay to a $\pi^0$ and a photon:
\begin{equation}
\mathcal C[q\bar{q}]\;\to\;\pi^0+\gamma\,;
\end{equation}
\item
if $M_{\pi\pi}<M_{\mathcal C} < M_{\mathcal{M}[q]}+M_{\mathcal{M}[\bar{q}]}$, the cluster will decay to two pions, either $\pi^+\pi^-$, with a probability of $2/3$ or $\pi^0\pi^0$ with a probability of 1/3:
\begin{equation}
\mathcal C[q\bar{q}]\;\to\;\pi^++\pi^-\;\;\;\mbox{\rm or}\;\;\;
\mathcal C[q\bar{q}]\;\to\;\pi^0+\pi^0\,.
\end{equation}
\end{itemize}
The treatment outlined above also appiles to colour-singlets made of two gluons that do not have enough mass to decay into constituent quarks.
The two mass thresholds are listed in \TabRef{Tab:ParamsMassThresholds}.
\end{enumerate}
\subsubsection{Kinematics for cluster decays into hadrons}
Having fixed the flavours of the particles that are being produced in the cluster decay, the kinematics is easily constructed in the cluster's rest frame.
The transverse momentum $k_\perp$ of the hadrons with respect to the cluster constituents is selected according to the same Gaussian distribution used in gluon decays and cluster fission, \EqRef{Eq:TransverseMomentum}, with the same parameter $k_{\perp,0}$, and with a flat azimuthal angle.
In contrast to the case of cluster fission, where the masses for the resulting clusters need to be fixed, this completely determines the kinematics of the decay: the longitudinal momenta of the hadrons are easily calculated now from their masses and transverse momenta, and they are aligned with the constituents giving rise to them.
\section{Tuned Parameters}
\label{App::Parameters}
The tuning is performed with the P\protect\scalebox{0.8}{ROFESSOR}\xspace (v2.3.3) framework~\cite{Buckley:2009bj}. Approximately 10 million events are generated for each tune to ensure that the uncertainty in the S\protect\scalebox{0.8}{HERPA}\xspace prediction in each bin is much smaller than the uncertainty in the data in the same bin.
\subsection{Parton shower parameters}
Default parameters for the final state parton showers in S\protect\scalebox{0.8}{HERPA}\xspace are
listed in \TabRef{Tab:ParamsShower}. With the exception of the
infrared cut-off, which was tuned to data, all other parameters are fixed.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|p{0.1\textwidth}|p{0.43\textwidth}|p{0.2\textwidth}|
|p{0.06\textwidth}|p{0.06\textwidth}|}
\hline
& Description & Name & C\protect\scalebox{0.8}{SS}\xspace & D\protect\scalebox{0.8}{IRE}\xspace \\
&& (in run card) & & \\
\hline\hline
$p_{\perp}^{(\rm cut)}$ & parton-shower cutoff &
{\tt CSS\_FS\_PT2MIN} & 1 & - \\ \hline\hline
$\alpha_S(M_Z)$ & strong coupling in parton shower &
{\tt ALPHAS(MZ)} & \multicolumn{2}{c|}{0.1188} \\
& order for running $\alpha_S$ &
{\tt ORDER} & \multicolumn{2}{c|}{2} \\
\hline
$m_{b}^{\mathrm{(pert)}}$ & $b$ quark mass in parton shower &
&\multicolumn{2}{c|}{4.5 GeV} \\
$m_{c}^{\mathrm{(pert)}}$ & $c$ quark mass in parton shower &
&\multicolumn{2}{c|}{1.5 GeV} \\
$m_{u,d,s}^{\mathrm{(pert)}}$ & light quark masses
in parton shower &
&\multicolumn{2}{c|}{0 GeV} \\\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{(Fixed) perturbative input parameters for the parton showers
and the tuned value for its infrared cut-off.\label{Tab:ParamsShower}
}
}
\end{center}
\end{table}
\subsection{Constituent masses and popping parameters}
In S\protect\scalebox{0.8}{HERPA}\xspace, the quarks and diquarks have non-perturbative constituent masses that implicitly, through phase space, impact on their production in the forced decays of gluons at the end of the parton shower.
While the quark masses are fixed directly, the diquark masses are calculated from the constituent masses of their component quarks as
\begin{equation}
m_{(ij)} = (m_i+m_j+m_{di})\cdot(1+\epsilon_{0,1})
\end{equation}
for spin-0 and spin-1 diquarks $(ij)$ made of an $i$ and a $j$
quark. The (fixed) input parameters for all masses are listed in
\TabRef{Tab:ParamsMasses}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|p{0.04\textwidth}|p{0.25\textwidth}|p{0.2\textwidth}|
|p{0.12\textwidth}|p{0.12\textwidth}|}
\hline
& Description & Name (in run card) & C\protect\scalebox{0.8}{SS}\xspace CR Off (On)& D\protect\scalebox{0.8}{IRE}\xspace CR Off (On) \\
&& & & \\
\hline\hline
$m_{b}$ & $b$ constituent mass &
{\tt M\_BOTTOM} & \multicolumn{2}{c|}{5.1 GeV} \\
$m_{c}$ & $c$ constituent mass &
{\tt M\_CHARM} & \multicolumn{2}{c|}{1.8 GeV} \\
$m_{s}$ & $s$ constituent mass &
{\tt M\_STRANGE} & \multicolumn{2}{c|}{0.4 GeV} \\
$m_{u,d}$ & $u$ \& $d$ constituent masses &
{\tt M\_UP\_DOWN} & \multicolumn{2}{c|}{0.3 GeV} \\
$m_{g}$ & gluon constituent masses &
{\tt M\_GLUE} & \multicolumn{2}{c|}{0.0 GeV} \\
\hline\hline
$m_{di}$ & offset for diquark masses &
{\tt M\_DIQUARK\_OFFSET} & \multicolumn{2}{c|}{0.3 GeV} \\
$\epsilon_{0}$ & rel.\ binding energy, spin-0 &
{\tt M\_BIND\_0} & \multicolumn{2}{c|}{0.12} \\
$\epsilon_{1}$ & rel.\ binding energy, spin-1 &
{\tt M\_BIND\_1} & \multicolumn{2}{c|}{0.5} \\
\hline\hline
${p}_s$ & strange quark probability &
{\tt STRANGE\_FRACTION} & 0.46 & 0.4 \\
${p}_{di}$ & diquark probability &
{\tt BARYON\_FRACTION} & 0.15 (0.27) & 0.2 \\
${x}_{qs}$ & $(qs)$ suppression &
{\tt P\_QS\_by\_P\_QQ\_norm} & 0.71 & 0.71 \\
${x}_{ss}$ & $(ss)$ suppression &
{\tt P\_SS\_by\_P\_QQ\_norm} & 0.01 (0.013) & 0.02 \\
${x}_{1}$ & $(qq)_1$ suppression &
{\tt P\_QQ1\_by\_P\_QQ0\_norm} & 0.94 (0.63) & 0.57 \\
\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{(Fixed) non-perturbative input parameters for constituents:
quark masses and parameters to calculate the diquark masses.
Tuned popping probabilities for their non-perturbative production
in gluon and cluster decays.
\label{Tab:ParamsMasses}
}
}
\end{center}
\end{table}
When calculating the ``popping'' probabilities $\mathcal{P}$ for the
constituents to be produced in gluon or cluster decays, S\protect\scalebox{0.8}{HERPA}\xspace implicitly
takes into account their number of spin states. As a consequence, up to
a normalsing their sum to unity, the individual $\mathcal{P}$ are given by
\begin{equation}
\label{Eq:PoppingProbs}
\begin{array}{lcllcllcl}
\mathcal{P}_{u,d} &=& 2
,&\;\;\;
\mathcal{P}_{s} &=& 2p_s,&\\[3mm]
\mathcal{P}_{(ud)_0} &=& p_{di}
,&\;\;\;
\mathcal{P}_{(us)_0} = \mathcal{P}_{(ds)_0}
&=& x_{qs}p_sp_{di},\\[3mm]
\mathcal{P}_{(ud)_1} =
\mathcal{P}_{(uu)_1} =
\mathcal{P}_{(dd)_1}
&=& 3x_1p_{di},&\;\;\;
\mathcal{P}_{(us)_1} = \mathcal{P}_{(ds)_1}
&=& 3x_{qs}p_sp_1p_{di},&
\mathcal{P}_{(ss)_1}
&=& 3x_{ss}p_s^2p_1p_{di}\,.
\end{array}
\end{equation}
The various parameters have been tuned to data and can be found in
\TabRef{Tab:ParamsMasses}. It is worth noting that, at the moment,
S\protect\scalebox{0.8}{HERPA}\xspace does not feature any diquarks made of one or two heavy, {\it i.e.}\xspace
charm or beauty, quarks.
\FloatBarrier
\subsection{Kinematics}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|p{0.05\textwidth}|p{0.32\textwidth}|p{0.24\textwidth}|
|p{0.1\textwidth}|p{0.1\textwidth}|}
\hline
& Description & Name (in run card) & C\protect\scalebox{0.8}{SS}\xspace CR Off (On) & D\protect\scalebox{0.8}{IRE}\xspace CR Off (On) \\
\hline\hline
switch & select gluon splitting mode:
& {\tt GLUON\_DECAY\_MODE} & 0 & 0\\
& 0: additive, &&&\\
& 1: multiplicative, \EqRef{Eq:LongMomentumGluon} &&&\\
\hline
$\alpha_G$ & $\alpha$ in gluon decays, \EqRef{Eq:LongMomentumGluon} &
{\tt ALPHA\_G} & 0.67 & 0.67 \\
\hline\hline
switch & select cluster splitting mode:
& {\tt CLUSTER\_SPLITTING\_MODE} & 0 & 0 \\
& 0: $z^{(i)}$, \EqRef{Eq:LongitudinalMomentumInCCC1}, &&&\\
& 1: $\Delta M$ (exponential), &&&\\
& 2: $\Delta M$ (Gaussian), &&&\\
& 3: $\Delta M$ (log-normal),
\EqRef{Eq:LongitudinalMomentumInCCC2} &&&\\
\hline
$\alpha_L$ &
$\alpha$ (light quarks) in cluster fission &
{\tt ALPHA\_L} & 2.5 & 2.5 \\
$\beta_L$ &
$\beta$ (light quarks) in cluster fission &
{\tt BETA\_L} & 0.13 & 0.12\\
$\gamma_L$ &
$\gamma$ (light quarks) in cluster fission &
{\tt GAMMA\_L} & 0.27 (0.5) & 0.27\\
$\alpha_D$ &
$\alpha$ (diquarks) in cluster fission &
{\tt ALPHA\_D} & 3.26 & 3.26 \\
$\beta_D$ &
$\beta$ (diquarks) in cluster fission &
{\tt BETA\_D} & 0.11 & 0.11 \\
$\gamma_D$ &
$\gamma$ (diquarks) in cluster fission &
{\tt GAMMA\_D} & 0.39 & 0.39 \\
$\alpha_H$ &
$\alpha$ (heavy quarks) in cluster fission &
{\tt ALPHA\_H} & 1.26 (3.55) & 1.26 \\
$\beta_H$ &
$\beta$ (heavy quarks) in cluster fission &
{\tt BETA\_H} & 0.98 (1.12) & 0.98\\
$\gamma_H$ &
$\gamma$ (heavy quarks) in cluster fission &
{\tt GAMMA\_H} & 0.05 (0.15) & 0.054 \\
& \hspace*{2cm}all in \EqRef{Eq:LongitudinalMomentumInCCC1} &&&\\
\hline
$\alpha_B$ &
$\alpha$ for decays of beam clusters$^*$ &
{\tt ALPHA\_B} & 2.5 & 2.5 \\
$\beta_B$ &
$\beta$ for decays of beam clusters$^*$ &
{\tt BETA\_B} & 0.25 & 0.25\\
$\gamma_B$ &
$\gamma$ for decays of beam clusters$^*$ &
{\tt GAMMA\_B} & 0.5 & 0.5\\
\hline
$k_{\perp,0}$ & $k_{\perp,0}$ in gluon ($g\to f\bar{f}$) and cluster ($\mathcal{C}\to\mathcal{C}\mathcal{C}$ and $\mathcal{C}\to\mathcal{H}\mathcal{H}$) decays &
{\tt KT\_0} & 1.34 (1.42) & 1.34 \\
\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{Tuned parameters that describe kinematics in the non-perturbative decays of gluons and clusters, Eqs.\ (\ref{Eq:TransverseMomentum}), (\ref{Eq:LongMomentumGluon}), and (\ref{Eq:LongitudinalMomentumInCCC1}).
$^*$ Note that in this publication we do not report on the tuning of the non-perturbative modelling for hadron colliders, which would involve additional physics modelling that we postpone to a future publication.
\label{Tab:ParamsDecays}
}
}
\end{center}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|p{0.05\textwidth}|p{0.3\textwidth}|p{0.33\textwidth}|
|p{0.1\textwidth}|p{0.1\textwidth}|}
\hline
& Description & Name (in run card) & C\protect\scalebox{0.8}{SS}\xspace CR Off (On) & D\protect\scalebox{0.8}{IRE}\xspace CR Off (On) \\
&& & & \\
\hline\hline
switch & direct $\mathcal{C}\to\mathcal{H}$ enabled &
{\tt DIRECT\_TRANSITIONS} & 1 & 1\\
\hline\hline
$x_{\rm trans}$ &
${\mathcal C}\to{\mathcal H}$ threshold,
\EqRef{Eq:TransitionThreshold} &
{\tt TRANSITION\_THRESHOLD} & 0.51 (0.75) & 0.51\\
$x_{\rm dec}$ &
$\mathcal{C}\to\mathcal{H}\mathcal{H}$ threshold,
\EqRef{Eq:MassThresholdCHH} &
{\tt DECAY\_THRESHOLD} & 0.02 (0.18) & 0.02\\
$\chi$ & generic mass modifier, \EqRef{Eq:CHHWeights} &
{\tt MASS\_EXPONENT} & 0 & 0 \\
\hline
\multicolumn{5}{|l|}
{meson multiplet weights
(identified by the $\pi$-like meson)}\\
\hline
$w_{000}$ &
pseudoscalars ($\pi^\pm$, \dots) &
{\tt MULTI\_WEIGHT\_R0L0\_PSEUDOSCALARS} & 1 & 1\\
$w_{001}$ &
vectors ($\rho^{pm}$, \dots) &
{\tt MULTI\_WEIGHT\_R0L0\_VECTORS} & 2.5 & 2.2\\
$w_{002}$ &
tensors ($a_2(1320)^\pm$, \dots) &
{\tt MULTI\_WEIGHT\_R0L0\_TENSORS2} & 1.5 & 1.5\\
$w_{010}$ &
scalars ($a_0(1450)^\pm$, \dots) &
{\tt MULTI\_WEIGHT\_R0L1\_SCALARS} & 0 & 0\\
$w_{011}$ &
axial vectors ($b_1(1235)^\pm$, \dots) &
{\tt MULTI\_WEIGHT\_R0L1\_AXIALVECTORS} & 0 & 0\\
$w_{021}$ &
axial vectors ($a_1(1260)^\pm$, \dots) &
{\tt MULTI\_WEIGHT\_R0L2\_VECTORS} & 0.5 & 0.5\\
\hline
\multicolumn{5}{|l|}
{modifiers for specific mesons}\\
\hline
$w_{M1}$ & singlet-meson modifier &
{\tt SINGLET\_MODIFIER} & 2 & 2\\
$w_\eta$ & $\eta$-meson modifier &
{\tt ETA\_MODIFIER} & 2.2 (2.33) & 2.82\\
$w_{\eta'}$ & $\eta'$-meson modifier &
{\tt ETA\_PRIME\_MODIFIER} & 4.5 (2.43) & 2.03\\
\hline\hline
\multicolumn{5}{|l|}
{baryon multiplet weights (identfied by some hadrons)}\\
\hline
$w_{00\frac12}$ &
octet ($N(939)$, \dots) &
{\tt MULTI\_WEIGHT\_R0L0\_N\_1/2} & 1 & 1\\
$w_{10\frac12}$ &
($N(1535)$, \dots) &
{\tt MULTI\_WEIGHT\_R1L0\_N\_1/2} & 0.1 & 0.1\\
$w_{20\frac12}$ &
($N(1440)$, \dots) &
{\tt MULTI\_WEIGHT\_R1L0\_N\_1/2} & 0 & 0\\
$w_{00\frac32}$ &
decuplet ($\Delta^{++}$, \dots) &
{\tt MULTI\_WEIGHT\_R0L0\_DELTA\_3/2} & 0.15 & 0.15\\
\hline
\multicolumn{5}{|l|}
{modifiers for specific baryons}\\
\hline
$w_{B1}$ & singlet-baryon modifier &
{\tt SINGLETBARYON\_MODIFIER} & 1.8 & 1.8\\
$w_{Bc}$ & $c$-baryon modifier &
{\tt CHARMBARYON\_ENHANCEMENT} & 8 & 8\\
$w_{Bb}$ & $b$-baryon modifier &
{\tt BEAUTYBARYON\_ENHANCEMENT} & 0.8 & 0.8\\
$w_{Bcs}$ & $cs$-baryon modifier &
{\tt CHARMSTRANGEBARYON\_ENHANCEMENT} & 2 & 2\\
$w_{Bbs}$ & $bs$-baryon modifier &
{\tt BEAUTYSTRANGEBARYON\_ENHANCEMENT} & 1.4 & 1.4\\
$w_{Bbc}$ & $bc$-baryon modifier &
{\tt BEAUTYCHARMBARYON\_ENHANCEMENT} & 1 & 1\\
\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{Tuned parameters used in the selection of the specific
channel in $\mathcal{C}\to\mathcal{H}_1+\mathcal{H}_2$
and $\mathcal{C}\to\mathcal{H}_1+\gamma$ decays,
{\it cf.}\xspace~\EqRef{Eq:CHHWeights}, including multiplet weights, and modifiers
for individual hadrons or classes of hadrons.
\label{Tab:ParamsCDecays}
}
}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|l|l|r|}
\hline
& Description & Name (in run card) & Value \\
&& & \\
\hline\hline
$M_{\pi\gamma}$ & mass threshold for $\mathcal{C}\to \pi^0+\gamma$ &
{\tt PI\_PHOTON\_THRESHOLD} & 0.150 GeV\\
$M_{\pi\pi}$ & mass threshold for $\mathcal{C}\to \pi+\pi$ &
{\tt DI\_PION\_THRESHOLD} & 0.300 GeV\\
\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{Fixed mass thresholds for light cluster to hadron decays.
\label{Tab:ParamsMassThresholds}
}
}
\end{center}
\end{table}
\clearpage
\subsection{Colour Reconnections}
In \TabRef{Tab:ParamsReconnections} we list some of the parameters
that describe the simple colour reconnection model in S\protect\scalebox{0.8}{HERPA}\xspace.
It should be noted though that
\begin{enumerate}
\item we added the model as an additional option after the
parameters for the overall hadronisation model had been fitted
to \LEP data, so a better tune may be achieved by a combined
re-fitting exercise;
\item the impact of colour reconnections on hadron-level
observables in $e^-e^+$ annihilations is moderate: the parton
shower usually terminates with only few partons in the final
state which are relatively tightly correlated in colour and
momentum space already; and that
\item the spatial component of the model becomes accessible in
hadron collision only.
\end{enumerate}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|p{0.1\textwidth}|p{0.37\textwidth}|p{0.16\textwidth}|
|p{0.09\textwidth}|p{0.09\textwidth}|}
\hline
& Description & Name (run card) & C\protect\scalebox{0.8}{SS}\xspace & D\protect\scalebox{0.8}{IRE}\xspace \\
\hline\hline
switch & switching colour reconnections on/off &
{\tt MODE} &
\multicolumn{2}{c|}{On/Off}\\
\hline
switch & select colour reconnection mode:
& {\tt PMODE} & \multicolumn{2}{c|}{0}\\
& 0: logarithmic &&&\\
& 1: power, {\it cf.}\xspace~\EqRef{Eq:CRMomDistance} &&&\\
\hline
$Q_0$ & momentum space distance, \EqRef{Eq:CRMomDistance} & {\tt Q\_0} & 1.41 GeV
& 1.65 GeV\\
\hline
$\eta_Q$ & scaling parameter for $Q_0$,
\EqRef{Eq:CRMomDistance} & {\tt etaQ} &
\multicolumn{2}{c|}{0.16}\\
$R_0$ & transverse space distance, \EqRef{Eq:CRPosDistance}$^*$& {\tt R\_0} &
\multicolumn{2}{c|}{1.0/GeV}\\
$\eta_R$ & scaling parameter for $R_0$,
\EqRef{Eq:CRMomDistance}$^*$ & {\tt etaR} &
\multicolumn{2}{c|}{0.16}\\
$\kappa_C$ & colour weight for different colours &
{\tt Reshuffle} &
\multicolumn{2}{c|}{0.33}\\
$\kappa$ & exponent in the norm, \EqRef{EQ:CRProb} &
{\tt kappa} & \multicolumn{2}{c|}{2}\\
\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{Tuned parameters in the simple colour reconnection model provided in S\protect\scalebox{0.8}{HERPA}\xspace, \SecRef{Sec:ModelCR}.
$^*$ Note that the spatial
part of the model is relevant for hadron collisions only and will
be tuned in conjunction with a forthcoming
tuning of S\protect\scalebox{0.8}{HERPA}\xspace's model for the underlying event.
\label{Tab:ParamsReconnections}
}
}
\end{center}
\end{table}
\section{Results for \protect{$e^-e^+\to$} hadrons at \protect{$E_{\rm cms}=91.2$} GeV}
\label{Sec:Results}
In the following we will show a wide range of data, highlighting various aspects of the S\protect\scalebox{0.8}{HERPA}\xspace simulation of QCD events in electron-positron annihilations at the $Z$ pole.
The results are obtained after tuning of the model with the P\protect\scalebox{0.8}{ROFESSOR}\xspace tool~\cite{Buckley:2009bj}, and more details on the tuning parameters are discussed in \AppRef{App::Parameters}.
P\protect\scalebox{0.8}{ROFESSOR}\xspace has been applied after some rough fitting of initial parameters, by oscillating between
\begin{itemize}
\item inclusive observables -- mainly charged multiplicity, event shapes such as thrust, thrust-major and thrust-minor, and the $b$-fragmentation function.
They are are most sensitive to the kinematics of gluon and cluster decays: $k_{\perp,0}^2$ and the respective $\alpha$, $\beta$, and $\gamma$.
\item individual hadron yields that are most sensitive to the flavour popping parameters $P_f$, hadron--\-dependent threshold parameter $x_{\rm dec}$ and modifier $\chi$, and hadron multiplet modifiers $w_{\mathcal{H}}$.
In tunes including the colour reconnection model, its parameters were included in this step, but the overall impact of this additional part of the hadronisation modelling is -- as expected -- typically quite small in $e^-e^+$ annihilations.
\end{itemize}
We will start from relatively inclusive observables, such as total hadron multiplicities and their distribution in phase space before focusing on increasingly differential observables, from event shapes over jet distributions and fragmentation functions of individual hadron species to the correlation of identified particles in phase space. By and large, the description of data by \protectS\protect\scalebox{0.8}{HERPA}\xspace is satisfactory for the bulk of the events. In these dominant regions of phase space, simulation and data typically agree within experimental uncertainties or at a level of 1-10\%.
\subsection{Inclusive particle distributions}
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/ALEPH_1991_multi.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_EEC.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_AEEC.png}
\end{tabular}
\parbox{0.8\textwidth}{\caption{
Multiplicities of charged hadrons (left), the energy-energy correlation (middle) and its asymmetry (right) in $e^-e^+\to$ hadrons at centre-of-mass energies of 91.2 GeV. The \protectS\protect\scalebox{0.8}{HERPA}\xspace results are compared with data from \protect\ALEPH~\protect\cite{ALEPH:1991ldi} for the first, and from \protect\DELPHI~\protect\cite{DELPHI:1996sen} for the latter two.
\label{Fig:Multis}}
}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_xp.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_logxp.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_PTinplane.png} \\
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_PToutplane.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_ptout_vs_xp.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_pt_vs_xp.png} \\
\includegraphics[width=0.3\textwidth]{figures/OPAL_1998_S3780481_uds_logxp.png} &
\includegraphics[width=0.3\textwidth]{figures/OPAL_1998_S3780481_b_logxp.png} &
\includegraphics[width=0.3\textwidth]{figures/OPAL_1998_S3780481_c_logxp.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Longitudinal momenta fractions $x_p$ (upper left) and $\log 1/x_p$ (upper middle), transverse momenta in-plane (upper right) and out-of-plane (centre left), with respect to the thrust axis, and the mean value of the out-of-plane and total transverse momenta as a function of $x_p$ (centre middle and right). The S\protect\scalebox{0.8}{HERPA}\xspace results are compared with data from \DELPHI~\protect\cite{DELPHI:1996sen}. Measurement of scaled momentum distributions, $\ln 1/x_p$, for $uds$ quarks (bottom left), $b$ quarks (bottom middle) and $c$ quarks are compared with data from \OPAL~\protect\cite{OPAL:1998tlp}
\label{Fig:InclusivePTandX}
}
}
\end{center}
\end{figure}
We will begin our discussion with the inclusive characteristics of hadron production in $e^-e^+$ annihilations. In \FigRef{Fig:Multis} we compare results from S\protect\scalebox{0.8}{HERPA}\xspace with corresponding experimental data for charged hadrons multiplicities (from \ALEPH~\cite{ALEPH:1991ldi}), the energy-energy correlation as a function of the angle $\chi$ and its asymmetry (both from \DELPHI~\cite{DELPHI:1996sen}).
By and large, S\protect\scalebox{0.8}{HERPA}\xspace is in satisfying agreement with data.
However, there are visible deviations in the high-multiplicity tail
$N_{\rm ch}$ ${\gtrsim}$ 40
of the charged hadron multiplicity distribution, where S\protect\scalebox{0.8}{HERPA}\xspace results fall outside the experimental uncertainties and overshoot data quite significantly. As expected, this is more amplified for tunes where colour reconnections have been switched on, as they often lead to the creation of relatively heavy clusters, which in turn generate a larger hadron multiplicity in their decays.
In \FigRef{Fig:InclusivePTandX} we exhibit the in-plane and out-of-plane $p_\perp$ distributions with the planes defined w.r.t.\ the thrust axis, and the $x_P$ and $\log 1/x_P$ distributions, and compare the results of the \Sherpa simulation with those taken by the \DELPHI collaboration in~\cite{DELPHI:1996sen}.
We compared the measurement of scaled momentum distributions, $\ln 1/x_p$, for quarks with data from \OPAL~\protect\cite{OPAL:1998tlp}.
We observe satisfying overall agreement with data, which is however somewhat hampered by two correlated trends: the cluster fragmentation appears to somewhat overshoot, by about 5-10\% the production of hadrons at $x_p$ values of about 0.4 while undershooting at higher values of $x_p\to 1$.
This trend can be further analysed by looking at $-\log 1/x_P=\xi_P$ spectra in events where the $Z$ boson decays into light quarks, charm, or bottom quarks.
We find that the overshoot at $x_p\approx 0.1$ is a feature common to all three event categories, and probably most pronounced for the charm-initiated ones.
In these latter events, all our four tunes seem to undershoot hadron production by about 10\%, and fit data only for the regime around $x_p\approx 0.1$.
We also observe that the undershoot for large values of the $x_p$ spectrum is mainly due to the $uds$ and $c$ event categories -- the $b$--initiated events exhibit quite satisfying agreement with data within their uncertainties.
Anyway, this undershoot for large $x_p$ is related to the fact that in most cases, clusters decay into two hadrons which then share the momentum quite equally -- a typical effect observed in cluster hadronisation models.
\FloatBarrier
\subsection{Event shapes}
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_thrust.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_major.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_minor.png} \\
\includegraphics[width=0.3\textwidth]{figures/ALEPH_2004_sphericity.png} &
\includegraphics[width=0.3\textwidth]{figures/ALEPH_2004_Cparam.png} &
\includegraphics[width=0.3\textwidth]{figures/ALEPH_2004_aplanarity.png} \\
\includegraphics[width=0.3\textwidth]{figures/OPAL_2004_lighthemisphere.png} &
\includegraphics[width=0.3\textwidth]{figures/OPAL_2004_heavyhemisphere.png} &
\includegraphics[width=0.3\textwidth]{figures/OPAL_2004_Bsum.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{
Various event shapes distributions in $e^-e^+ \to$ hadrons at centre-of-mass energies of 91.2 GeV. Upper row, from left to right: thrust, thrust major, thrust minor, all from \DELPHI~\protect\cite{DELPHI:1996sen}; middle row, from left to right: sphericity, $C$ parameter and aplanarity, all from \ALEPH~\protect\cite{ALEPH:2003obs}; lower row, from left to right: light and heavy hemisphere mass and total hemisphere broadening, all from \OPAL~\protect\cite{OPAL:2004wof}.
\label{Fig:EventShapes}
}
}
\end{center}
\end{figure}
In \FigRef{Fig:EventShapes} we compare S\protect\scalebox{0.8}{HERPA}\xspace results for a number of event shapes with the experimental data.
In the upper row we depict thrust $T$ (or more precisely $1-T$), thrust major $M$, and thrust minor $m$, with data taken by \DELPHI in~\cite{DELPHI:1996sen}.
Apart from a significant overshoot in the bins of small $1-T$, $M$, and $m$, {\it i.e.}\xspace for extremely pencil-like events, the agreement of the simulation with data is excellent, at the level of 5\% or less over a wide range of phase space.
This pattern of good to excellent agreement with data, at the 5\% or below level, with some overshoots in the extremely pencil-like regime of event topologies, repeats itself also in sphericity $S$, $C$-parameter and aplanarity $A$, displayed in the middle row of \FigRef{Fig:EventShapes}, where we use data from \ALEPH~\cite{ALEPH:2003obs}.
In particular the description of the smooth transition from the three to the four-jet regime in the $C$ parameter data at $C\approx 0.75$ is quite impressive. In the lower row of \FigRef{Fig:EventShapes} we display the light and heavy hemisphere masses, $M_L$ and $M_H$, as well as the total hemisphere broadening, $B_{\rm sum}$ with data from \OPAL~\cite{OPAL:2004wof}. Here we observe a difference between data and simulation, as S\protect\scalebox{0.8}{HERPA}\xspace undershoots the peak region in $M_L$ and, consequently overshoots the regions of small $M_L\to 0$ and large $M_L$.
It is worth noting that the results of this comparison, in particular for $T$, $M$, and $m$, favour the use of D\protect\scalebox{0.8}{IRE}\xspace with colour reconnections switched off as the best option, while for the CSS\protect\scalebox{0.8}{hower}\xspace the inclusion of colour reconnections seems to slightly improve agreement with data to a level not dissimilar to the best option.
\FloatBarrier
\subsection{Jet distributions}
Turning to the description of jets in the electron-positron annihilation events, we focus on differential jet rates in the Durham scheme.
They provide an excellent way to judge the performance of the combined parton shower and hadronisation model and its ability to capture QCD dynamics across all scales, from the perturbative to the non-perturbative regime.
In \FigRef{Fig:JetRates} we compare the results of S\protect\scalebox{0.8}{HERPA}\xspace with data from a combined \JADE plus \OPAL analysis~\cite{JADE:1999zar}, and we find, again, excellent agreement of both within the experimental uncertainties.
It is worth noting that, again, D\protect\scalebox{0.8}{IRE}\xspace without colour recoonections provides the best description of data overall, while the $y_{45}$ distribution for large jet resolutions disfavours the use of D\protect\scalebox{0.8}{IRE}\xspace with colour reconnections switched on.\\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/JADEOPAL_2000_d23.png} &
\includegraphics[width=0.3\textwidth]{figures/JADEOPAL_2000_d34.png} &
\includegraphics[width=0.3\textwidth]{figures/JADEOPAL_2000_d45.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{
Differential jet rates in the Durham scheme, in $e^-e^+ \to$ hadrons at centre-of-mass energies of 91.2 GeV. S\protect\scalebox{0.8}{HERPA}\xspace results are compared with data from a combined \JADE-\-\OPAL publication~\protect\cite{JADE:1999zar}, for (from left to right) the differential $2\to 3$, $3\to 4$, and $4\to 5$ jet rates.
\label{Fig:JetRates}
}
}
\end{center}
\end{figure}
\FloatBarrier
\subsection{Particle production yields and fragmentation functions}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.0\textwidth]{figures/pdgmulti_s.png}
\parbox{0.8\textwidth}{
\caption{Hadron yields for various species are compared to the data from the particle data group~\protect\cite{ParticleDataGroup:2008zun}, and $\Omega^{\pm}$ is compared to the data from \ALEPH~\protect\cite{ALEPH:1996oqp}.
\label{Fig:HadronYields}
}
}
\end{center}
\end{figure}
Yields for various mesons and baryons are displayed in \FigRef{Fig:HadronYields} and show broad agreement with data and among the four tunes.
There is, however, tendency in S\protect\scalebox{0.8}{HERPA}\xspace to miss the yields of $\eta$ and $\eta'$ pseudoscalar mesons and of the vector mesons, which our tunes seem to overshoot.
Similarly, the production of charmonia states and of some of the excited heavy mesons appears to be not perfect.
It could be argued that the former may be alleviated by including the production of such states directly in the parton shower, e.g.\ by including splitting functions like $g\to J/\psi g$, which emerge from the convolution of the production of $c\bar c$ pairs (by sequentially splitting for example $g\to c\bar{c}\otimes c\to cg$) and wave functions that describe their transition to such charmonia states.
Turning to baryons we observe good agreement of the tunes with the data for the production of protons and $\Lambda$'s, some overshoot for the other octet baryons (like, the $\Sigma$'s and Cascade baryons) and an undershoot for the decuplet baryons.
Especially the latter exhibit a large sensitivity to the overall tune, with some tunes performing (CSS\protect\scalebox{0.8}{hower}\xspace with colour reconnections disabled) notably better than others.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_xppi0.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1998_I473409_pipm_Xp.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1998_I473409_Kpm_Xp.png} \\
\includegraphics[width=0.3\textwidth]{figures/OPAL_2000_xEK0.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1996_xpKstar0.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_1999_xpphi.png} \\
\includegraphics[width=0.3\textwidth]{figures/ALEPH_2002_xpomega.png} &
\includegraphics[width=0.3\textwidth]{figures/ALEPH_2002_etaxp.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHES_2006_xi_xp.png} \\
\includegraphics[width=0.3\textwidth]{figures/DELPHI_1998_I473409_pp_xp.png} &
\includegraphics[width=0.3\textwidth]{figures/DELPHI_2000_xp_Sigma-.png} &
\includegraphics[width=0.3\textwidth]{figures/OPAL_1997_I421977_Sigma_xE.png.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{From left to right, first row:
$x_p$ distributions for $\pi^0$ from (\DELPHI~\protect\cite{DELPHI:1995ase}), $\pi^{\pm}$ and $K^{\pm}$ from \DELPHI~\protect\cite{DELPHI:1998cgx}; second row: $K^0$ from \OPAL~\protect\cite{OPAL:2000dkf}), $K^{*0}$ from \DELPHI~\protect\cite{DELPHI:1996xro}, $\phi$ from \SLD~\protect\cite{SLD:1998coh} ;
third row: $\omega$ and $\eta$ from \ALEPH~\protect\cite{ALEPH:2001tfk}, $\Xi^{-}$ from \DELPHI~\protect\cite{DELPHI:2006pom};
fourth row: $p, \bar{p}$ from \DELPHI~\protect\cite{DELPHI:1998cgx},
$\Sigma^-$ from \DELPHI~\protect\cite{DELPHI:2000oqt} and $x_E$ for $\Sigma^{+}$ from \OPAL~\protect\cite{OPAL:1996dbo}
\label{Fig:hadronxps}
}
}
\end{center}
\end{figure}
In \FigRef{Fig:hadronxps} we show a range of $x_p$ distributions for various mesons and baryons.
By and large, in the region
$x_p \; {\lesssim}$ 0.4
data and simulation are in excellent agreement with each other and the simulation results rarely fall outside the experimental uncertainties, with the possible exception of some meson distributions overshooting data for very small values of $x_p$.
This, of course, could also have been anticipated from the overall quality in the description of the more inclusive data in \FigRef{Fig:InclusivePTandX}.
It is, however, important to note that our cluster fragmentation model does not arrive at this satisfying result by cross compensating the potentially wrong behaviour of different hadron species but rather arrives at a uniformly good description of hadron production across the board.
The only notable exceptions to this picture are the $\omega$ meson, overshooting data by 40\% or more throughout, and the $\phi$ meson, which exhibits a shape difference w.r.t.\ the data.
In \FigRef{Fig:hadronxps_spec} we show similar results, namely $x_p$ distributions for charged pions $\pi^\pm$, kaons $K^\pm$ and for protons, in events where the hadrons are produced from $Z$ boson decays into light quarks, charm, or bottom pairs.
Again, the simulation agrees quite well with data, with the notable exceptions of a sizable overproduction of pions in $uds$ events at $x_p$ values in the range between 0.2 and 0.5, and a pronounced overproduction of protons in $b$ events at $x_p\approx0.1$.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xppi_uds.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xppi_c.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xppi_b.png} \\
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xpK_uds.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xpK_c.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xpK_b.png} \\
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xpp_uds.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xpp_c.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2004_xpp_b.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{
Production of $\pi^\pm$ (upper row), $K^\pm$ (middle row) and $p$ (lower row) in $uds$ (left column), $c$ (middle column), and $b$ (right column) events. All distributions are compared with data from \SLD~\protect\cite{SLD:2003ogn}.
\label{Fig:hadronxps_spec}
}
}
\end{center}
\end{figure}
Good agreement with data is also found for the modelling of the heavy quark fragmentation process, displayed in \FigRef{Fig:HQFragmentation}.
In its left panel we show the $x_E$ distribution of $D^{*\pm}$-mesons, which exhibits a tendency of overshooting data by a constant factor, in agreement with the slight overall over-production.
In the right panel of this figure we display the $b$-quark fragmentation function which agrees to better than 10\% with data when using the D\protect\scalebox{0.8}{IRE}\xspace shower, but shows some tension with data when the CSS\protect\scalebox{0.8}{hower}\xspace is being used.
Note that we chose to display the data from \SLD~\cite{SLD:2002poq} which appears to sit in the middle of two other distributions, from \ALEPH~\cite{ALEPH:2001pfo} and \OPAL~\cite{OPAL:2002plk}, thereby providing some compromise.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/OPAL_1995_xEDstar.png} &
\includegraphics[width=0.3\textwidth]{figures/SLD_2002_bfrag.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Fragmentation functions for $D^{*\pm}$ (left) and $B$ mesons (right), from \OPAL~\protect\cite{OPAL:1994cct} and \SLD~\protect\cite{SLD:2002poq}.
\label{Fig:HQFragmentation}
}
}
\end{center}
\end{figure}
\FloatBarrier
\subsection{Particle correlations}
Finally we look at the correlations in baryon production, and in particular the correlation of $\Lambda\bar\Lambda$ pairs. This correlation has triggered the development of the popcorn mechanism in the Lund model~\cite{Eden:1996xi}, which softened the relatively strong correlation in baryon production. This strong coupling of baryons in phase space is due to the fact that in the break-up of the string, and, of course also in the decay of clusters into hadrons, baryon production is associated with the production of a diquark pair, which due to the relatively low scales involved is close in phase space. In the popcorn mechanism this is resolved by ``inserting" mesons between the diquarks, but due to the more local nature of cluster fragmentation such a feature is not entirely trivial to encode in the model. In our cluster model, this is resolved by allowing already the gluons to decay into diquark pairs, an option that is usually not available in the version of the cluster fragmentation model where such decays are prohibited by assuming the relatively small non-perturbative gluon ``constituent" mass. In S\protect\scalebox{0.8}{HERPA}\xspace this limitation is absent, because we assume massless gluons throughout, and generate the mass and momenta of the produced flavours by reshuffling four-momentum from the colour-connected spectator. This allows to generate clusters with the quantum numbers of baryons, which ultimately leads to a drastically reduced correlation of the produced baryons. It is very satisfying to observe that this mechanism apparently resolves the problem of overly correlated baryon-baryon production, {\it cf.}\xspace\FigRef{Fig:LambdaLambda}, where we contrast S\protect\scalebox{0.8}{HERPA}\xspace results with data from \OPAL~\cite{OPAL:1998tlp}.\\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.35\textwidth]{figures/OPAL_2000_costheta_LambdaLambda.png} &
\includegraphics[width=0.35\textwidth]{figures/OPAL_2000_Deltay_LambdaLambda.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Correlation of $\Lambda\bar\Lambda$ pairs in $e^-e^+ \to$ hadrons at 91.2 GeV. We contrast S\protect\scalebox{0.8}{HERPA}\xspace results with data from \OPAL~\protect\cite{OPAL:1998tlp}.
\label{Fig:LambdaLambda}
}
}
\end{center}
\end{figure}
\FloatBarrier
\section{Energy extrapolation}
\label{Sec:Energyextrapolation}
A first step to verify the universality of the hadronisation model is to check the energy extrapolation from the c.m.\ energy where the model was tuned to data to other energies.
The high quality, variety and significance of the results at the $Z$ pole, $E_{\rm c.m.} = 91.2$ GeV, suggested to use \LEP 1 and \SLD data for the tuning, and to use data from other energies for some {\it a posteriori} checks.
In the following we will compare the S\protect\scalebox{0.8}{HERPA}\xspace results with various observables measured in electron--\-positron annihilations at $E_{\rm c.m.} = 14$ GeV, 35 GeV, 44 GeV and 58 GeV.
\subsection{Results at $E_{\rm c.m.} = 14$ GeV}
Starting at the low energy of $E_{\rm c.m.} = 14$ GeV, we compare results from the S\protect\scalebox{0.8}{HERPA}\xspace simulation with data taken mainly by the \TASSO collaboration at the \PETRA collider.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.3\textwidth]{figures/TASSO_1989_Multi14.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1987_EEC14.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{
Charged hadron multiplicity from \TASSO~\protect\cite{TASSO:1989orr} and energy-energy correlation measured by \TASSO~\protect\cite{TASSO:1987mcs}.
\label{Fig:Multi_EEC14}
}
}
\end{center}
\end{figure}
In \FigRef{Fig:Multi_EEC14} we focus on inclusive quantities, namely the total charged multiplicity and the energy--energy correlation, comparing our four tunes with data from~\TASSO~\protect\cite{TASSO:1989orr,TASSO:1987mcs}.
We observe that S\protect\scalebox{0.8}{HERPA}\xspace tends to not correctly describe the shape of the charged multiplicity distribution undershooting both the low-- and the high--multiplicity region by about 30\%.
Looking at the energy--\-energy correlation, we see that this can be traced to some overshoot for large angles, {\it i.e.}\xspace for $\cos\chi$ away from the extreme forward and backward regions.
It is however probably worth noting here that the data for the latter look a bit more ``bumpy" than, for example, the same observable measured at \LEP, and it appears as if the size of the bumps exceeds the experimental uncertainty estimates.\\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/TASSO_1990_thrust14.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1990_sphericity14.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1990_aplanarity14.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Various event shapes -- from left: thrust, sphericity, and aplanarity measured by \TASSO~\protect\cite{TASSO:1990cdg}.
\label{Fig:eventshapes14}
}
}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/TASSO_1982_I177174_scaled_momentum.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1985_xK14.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1985_xLambda14.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{
Momenta fractions of charged hadrons from \TASSO~\protect\cite{TASSO:1982cps}, and of $K^0$ mesons, $\Lambda^0$ baryons, measured by \TASSO~\protect\cite{TASSO:1984nda}.
\label{Fig:spectra14}
}
}
\end{center}
\end{figure}
We continue by comparing the results for some event shapes -- in particular thrust, sphericity, and aplanarity, in \FigRef{Fig:eventshapes14}, again all taken by \TASSO in~\cite{TASSO:1990cdg}.
Apart from the region of extremely pencil-like events, which S\protect\scalebox{0.8}{HERPA}\xspace overestimates significantly, the simulation agrees with experimental data within their uncertainties, indicating that apart from hadron multiplicities the simulation captures overall event characteristics satisfactorily.
In \FigRef{Fig:spectra14} we exhibit the comparison some particle spectra data from~\cite{TASSO:1982cps,TASSO:1984nda}.
The S\protect\scalebox{0.8}{HERPA}\xspace results for the momenta spectra of charged particles, which of course are dominated by the $\pi^\pm$, exhibit a slight tilt towards the softer end of the spectrum. At $x$-values in the region of $x\; {\lesssim}$ 0.15, or momenta of the order of about 1 GeV or below, the fragmentation model tends to overproduce the particle yields by up to about 20\%, depending on the tune.
A similar behaviour also appears in the neutral kaon spectra: although they exhibit larger uncertainties, allowing S\protect\scalebox{0.8}{HERPA}\xspace to agree with data within their uncertainties, the central values are in similar disagreement.
Surprisingly enough this is not the case for the $\Lambda$ baryons, which agree well with data.
\subsection{Results at $E_{\rm c.m.} = 35$ GeV}
In Fig.~\ref{Fig:eventshapes15} we depict the event shapes thrust and sphericity, as well as the scaled momentum distribution, measured by \TASSO~\cite{TASSO:1988zoi} at a centre-\-of-\-mass energy of 35 GeV.
There is a common trend in the event shape observables: all S\protect\scalebox{0.8}{HERPA}\xspace tunes tend to overshoot the bins corresponding to more pencil like events, at $T\approx 1$ and $S\approx 0$, and $A\approx 0$, while somewhat undershooting the peak region by about 10-15\%.
On the other hand, the agreement in the $x_p$ distribution is quite satisfying, apart from the large $x_p$ region, $x_p\ge 0.5$, where all tunes undershoot the data. This appears to be one of the usual feature of cluster hadronisation models, related to the fact that the clusters tend to decay too democratically.
The agreement of simulation and data in the scaled momentum spectrum of charged particles is also reflected in corresponding spectra for individual neutral mesons, {\it cf.}\xspace\ Fig~\ref{Fig:eventshapes16}, where we show the $x$ distribution $K^0$ mesons from \CELLO~\cite{CELLO:1989adw} and of $\eta$ and $\rho^0$ mesons from \JADE~\cite{JADE:1984wot,JADE:1989ewf}. \\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/TASSO_1988_I263859_thrust.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1988_I263859_sphericity.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1988_I263859_xp.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Various event shapes -- from left: thrust, sphericity, and $x_p$ distribution, all measured by \protect\TASSO~\protect\cite{TASSO:1988zoi}.
\label{Fig:eventshapes15}
}
}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/CELLO_1990_I283026_k0_sm.png} &
\includegraphics[width=0.3\textwidth]{figures/JADE_1984_I203145_rho_0_sm.png} &
\includegraphics[width=0.3\textwidth]{figures/JADE_1990_I282847_eta_se.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{Scaled momentum of $K^0$ mesons (left) by \CELLO~\protect\cite{CELLO:1989adw}, and of $\rho^0$ (centre) and $\eta$ (right), both by \JADE~\protect\cite{JADE:1984wot,JADE:1989ewf}.
\label{Fig:eventshapes16}
}
}
\end{center}
\end{figure}
\FloatBarrier
\subsection{Results at $E_{\rm c.m.} = 44$ GeV}
In Fig.~\ref{Fig:eventshapes18} we compare results obtained with the four S\protect\scalebox{0.8}{HERPA}\xspace tunes with a set of different observables.
We observe that apart from the most pencil-like bin at $T\approx 1$, the simulation describes the thrust distribution measured by \TASSO within the experimental uncertainties.
This is also true for the differential 2-jet rate in the Durham scheme, $y_{23}$, where S\protect\scalebox{0.8}{HERPA}\xspace satisfyingly reproduces the \JADE data~\cite{JADE:1999zar}.
The pattern repeats itself with a significantly different observable, the scaled momentum distribution of (anti-)protons, where, again, agreement of simulation with data, again from \TASSO~\cite{TASSO:1990cdg,TASSO:1988jma}, is quite good, with maybe a little bit of a relative shape difference. \\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/TASSO_1990_44_thrust.png} &
\includegraphics[width=0.3\textwidth]{figures/JADE_OPAL_44_y23.png} &
\includegraphics[width=0.3\textwidth]{figures/TASSO_1989_pp_xp.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{\Sherpa results for thrust (left), the differential 2-jet rate in the Durham scheme (middle) and $p,\bar{p}$ scaled momenta (right) compared to results from \TASSO~\protect\cite{TASSO:1990cdg}, \JADE-\-\OPAL~\protect\cite{JADE:1999zar}, and \TASSO~\protect\cite{TASSO:1988jma}, respectively.
\label{Fig:eventshapes18}
}
}
\end{center}
\end{figure}
\subsection{Results at $E_{\rm c.m.} = $55 GeV, 58 GeV and 59.5 GeV}
Finally turning to centre-of-mass energies of 55-58 GeV, we compare S\protect\scalebox{0.8}{HERPA}\xspace results to data from \AMY and \TOPAZ.
In Fig.~\ref{Fig:eventshapes19} we look at some event shapes like thrust and sphericity (both from \AMY~\cite{AMY:1989feg}), and the differential 2-jet rate in the \JADE scheme from \AMY~\cite{AMY:1995djo}.
As before, our simulations in all four tunes agree with data within their uncertainties, but we observe some deviations in the mean values. \\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/AMY_1990_I283337_thrust.png} &
\includegraphics[width=0.3\textwidth]{figures/AMY_1990_I283337_sphericity.png} &
\includegraphics[width=0.3\textwidth]{figures/AMY_1995_y23_jade_eo.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{S\protect\scalebox{0.8}{HERPA}\xspace results for thrust and sphericity compared with data from \AMY~\protect\cite{AMY:1989feg} (left and centre), and for the differential 2-jet rate in the \JADE scheme~\protect\cite{AMY:1995djo} (right).
\label{Fig:eventshapes19}
}
}
\end{center}
\end{figure}
We turn to particle spectra in Fig.~\ref{Fig:eventshapes20}, and display the energy-\-energy correlation (from \TOPAZ~\cite{TOPAZ:1989yod}), the longitudinal moment w.r.t.\ to the sphericity axis (from \AMY~\cite{AMY:1989feg}), and the scaled momentum spectrum of neutral kaons (from \TOPAZ~\cite{TOPAZ:1994voc}).
In all observables we notice the good agreement of the four S\protect\scalebox{0.8}{HERPA}\xspace tunes with data.\\
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figures/TOPAZ_1989_I279575_energy_corelation.png} &
\includegraphics[width=0.3\textwidth]{figures/AMY_1990_I283337_pL.png} &
\includegraphics[width=0.3\textwidth]{figures/TOPAZ_1995_k0_kbar0_spec.png}
\end{tabular}
\parbox{0.8\textwidth}{
\caption{S\protect\scalebox{0.8}{HERPA}\xspace results compared with data for the energy-\-energy correlation (left, from \TOPAZ~\protect\cite{TOPAZ:1989yod}), longitudinal moment w.r.t.\ the sphericity axis (centre, from \AMY~\protect\cite{AMY:1989feg}) and the scaled momentum spectrum for neutral kaons (right, from \TOPAZ~\protect\cite{TOPAZ:1994voc}).
\label{Fig:eventshapes20}
}
}
\end{center}
\end{figure}
\section{Summary}
\label{Sec:Summary}
In this paper we have described, in detail, the re-implementation of the cluster fragmentation model within S\protect\scalebox{0.8}{HERPA}\xspace, in an improved version compared to its original~\cite{Winter:2003tt} publication.
We have successfully tuned the model to data, for the two different parton showers available, and with or without the inclusion of a first, naive model for colour reconnections, which we also introduce here.
We find, by and large, satisfying agreement of our model with data, with a slight preference for either using the D\protect\scalebox{0.8}{IRE}\xspace~\cite{Hoche:2015sya} shower without colour reconnections or the CSS\protect\scalebox{0.8}{hower}\xspace~\cite{Schumann:2007mg} including them.
This will facilitate future studies of further non--\-perturbative effects which may impact precision measurements of, e.g., the $W$ mass in hadronic final states at lepton colliders or of the top mass.
As a next step we will, in a future publication, investigate the impact of the new hadronisation model on those observables at hadron colliders that are susceptible to non--\-perturbative effects, including event and jet shape observables.
This will also allow us to test the interplay of our model with the modelling of the underlying event.
\section{Hadron wavefunctions}
\label{App::wavefunctions}
The flavour parts of the charged meson wavefunctions are trivial, for example,
\begin{equation}
|\pi^+\rangle\,=\,|u\bar{d}\rangle\,,
\end{equation}
while for the more complicated case of neutral mesons they are given by
\begin{eqnarray}
|\pi^0\rangle\,,|\rho^0\rangle\,,\dots,&=&\,
\frac{1}{\sqrt{2}}\,
\left(\vphantom{\frac12}|u\bar{u}\rangle-|d\bar{d}\rangle\right)
\nonumber\\
|\eta^0\rangle\,,|\omega\rangle\,,\dots,&=&\,
\frac{\cos\theta}{\sqrt{6}}\,
\left(\vphantom{\frac12}|u\bar{u}\rangle+|d\bar{d}\rangle-2|s\bar{s}\rangle
\right)\,-\,
\frac{\sin\theta}{\sqrt{3}}\,
\left(\vphantom{\frac12}|u\bar{u}\rangle+|d\bar{d}\rangle+|s\bar{s}\rangle
\right)
\nonumber\\
|\eta'\rangle\,,|\phi\rangle\,,\dots\,&=&\,
\frac{\sin\theta}{\sqrt{6}}\,
\left(\vphantom{\frac12}|u\bar{u}\rangle+|d\bar{d}\rangle-2|s\bar{s}\rangle
\right)\,+\,
\frac{\cos\theta}{\sqrt{3}}\,
\left(\vphantom{\frac12}|u\bar{u}\rangle+|d\bar{d}\rangle+|s\bar{s}\rangle
\right)
\nonumber\\
|\eta_c\rangle,\,|J/\psi\rangle\,,\dots\,&=&\,
\vphantom{\frac{\sin\theta}{\sqrt{6}}}|c\bar{c}\rangle
\nonumber\\
|\eta_b\rangle,\,|\Upsilon(1s)\rangle\,,\dots\,&=&\,
\vphantom{\frac{\sin\theta}{\sqrt{6}}}|b\bar{b}\rangle\,,
\end{eqnarray}
and include the effect of singlet-octet mixing through suitable mixing angles,
where S\protect\scalebox{0.8}{HERPA}\xspace follows the recommendations of the PDG~\cite{Olive:2016xmw},
{\it cf.}\xspace~\TabRef{Tab:ParamsAngles}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|r|l|l|r|}
\hline
Parameter & Description & Name & Value\\
&& (in run card) & \\
\hline\hline
$\theta_{0^+}$ & mixing angle for the pseudoscalar multiplet &
{\tt Mixing\_0+} & $-14.1^\circ$\\
$\theta_{1^-}$ & mixing angle for the vector multiplet &
{\tt Mixing\_1-} & $36.4^\circ$\\
$\theta_{2^+}$ & mixing angle for the spin-2 multiplet &
{\tt Mixing\_2+} & $27.0^\circ$\\
\hline
\end{tabular}
\parbox{0.8\textwidth}{
\vspace*{3mm}
\caption{Angles parameterising the singlet-octet mixing of mesons.
\label{Tab:ParamsAngles}
}
}
\end{center}
\end{table}
The baryon wavefunctions are given by
\begin{eqnarray}
|p\rangle\,&=&\,
\frac{1}{\sqrt{3}}\,|d(uu)_1\rangle\,+\,
\frac{1}{\sqrt{6}}\,|u(ud)_1\rangle\,+\,
\frac{1}{\sqrt{2}}\,|u(ud)_0\rangle
\nonumber\\
|\Sigma^0_{(8)}\rangle\,&=&\,
\frac{1}{\sqrt{3}}\,|s(ud)_1\rangle\,+\,
\frac{1}{\sqrt{12}}\,|d(su)_1\rangle\,+\,
\frac{1}{\sqrt{4}}\,|d(su)_0\rangle\,+\,
\frac{1}{\sqrt{12}}\,|u(sd)_1\rangle\,+\,
\frac{1}{\sqrt{4}}\,|u(sd)_0\rangle
\nonumber\\
|\Lambda_{(8)}\rangle\,&=&\,
\frac{1}{\sqrt{3}}\,|s(ud)_0\rangle\,+\,
\frac{1}{\sqrt{12}}\,|d(su)_0\rangle\,+\,
\frac{1}{\sqrt{4}}\,|d(su)_1\rangle\,+\,
\frac{1}{\sqrt{12}}\,|u(sd)_0\rangle\,+\,
\frac{1}{\sqrt{4}}\,|u(sd)_1\rangle
\nonumber\\
|\Lambda_{(1)}\rangle\,&=&\,
\frac{1}{\sqrt{3}}\,|u(sd)_0\rangle\,+\,
\frac{1}{\sqrt{3}}\,|d(su)_0\rangle\,+\,
\frac{1}{\sqrt{3}}\,|s(ud)_0\rangle
\nonumber\\
|\Delta^{++}\rangle\,&=&\,
\vphantom{\frac{\sin\theta}{\sqrt{6}}}|u(uu)_1\rangle
\nonumber\\
|\Delta^+\rangle\,&=&\,
\sqrt{\frac{2}{3}}\,|d(ud)_1\rangle\,+\,
\frac{1}{\sqrt{3}}\,|d(uu)_1\rangle
\nonumber\\
|\Sigma^0_{(10)}\rangle\,&=&\,
\frac{1}{\sqrt{3}}\,|u(sd)_1\rangle\,+\,
\frac{1}{\sqrt{3}}\,|d(su)_1\rangle\,+\,
\frac{1}{\sqrt{3}}\,|s(ud)_1\rangle\,.
\nonumber\\
|\Lambda_Q\rangle\,&=&\,
\vphantom{\frac{\sin\theta}{\sqrt{6}}}|Q(qq)_0\rangle
\nonumber\\
|\Sigma_Q\rangle\,&=&\,
\vphantom{\frac{\sin\theta}{\sqrt{6}}}|Q(qq)_1\rangle
\end{eqnarray}
A few comments are in order here. First, {\em all} decuplet hadrons, {\it i.e.}\xspace
those that belong to a $\Delta$-like multiplet, are made up of spin-1
diquarks only. In addition, in some of the higher-lying multiplets, the
usual octet is supplemented with a further singlet $\Lambda$ baryon, with the
$\Lambda(1520)$ a good example. The wavefunction of these objects is totally
symmetric and exclusively made of spin-0 diquarks. Finally, for baryons
such as the neutron, the charged $\Sigma's$ or the $\Xi$'s of the octet
multiplet, the wavefunctions emerge from the proton one by suitably replacing,
\begin{equation}
|n\rangle\,=\,
\left[\vphantom{\frac12}|p\rangle\right]_{u\leftrightarrow d}
\,,\;
|\Sigma^-\rangle\,=\,
\left[\vphantom{\frac12}|p\rangle\right]_{\begin{array}{c}
\scriptstyle d\rightarrow s\\\scriptstyle u\rightarrow d\end{array}}
\,,\;
|\Sigma^+\rangle\,=\,
\left[\vphantom{\frac12}|p\rangle\right]_{d\rightarrow s}
\,,\;
|\Xi^-\rangle\,=\,
\left[\vphantom{\frac12}|p\rangle\right]_{u\rightarrow s}
\;\mbox{\rm and}\;\;\;
|\Xi^0\rangle\,=\,
\left[\vphantom{\frac12}|p\rangle\right]_{\begin{array}{c}
\scriptstyle u\rightarrow s\\\scriptstyle d\rightarrow u\end{array}\,.}
\end{equation}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 47 |
Q: Prove $\sum_{n=1}^{\infty}((n+\frac{1}{2})\ln(1+\frac{1}{n})-1)=1-\ln(\sqrt{2\pi})$ I am looking for a derivation of the following sum:
$$\sum_{n=1}^{\infty}\bigg(\left(n+\frac{1}{2}\right)\ln\left(1+\frac{1}{n}\right)-1\bigg)=1-\ln(\sqrt{2\pi})$$
My current derivation(s) uses the zeta function at negative integers (and or Stirling approximation/ the derivative of $\zeta'(0)$). I want to avoid those.
How I got an answer was via regularization of
$$-\sum_{i=1}^{\infty}\frac{\zeta(-i)}{i}=\sum_{n=1}^{\infty}\bigg(\left(n+\frac{1}{2}\right)\ln\left(1+ \frac{1}{n}\right)-1\bigg)$$
My own other try was rewriting it via:
$$\sum_{n=1}^{\infty}\bigg(\left(n+\frac{1}{2}\right)\ln\left(1+\frac{1}{n}\right)-1\bigg)=\sum_{k=2}^{\infty} \zeta(k)(-1)^k \bigg(\frac{1}{k+1}-\frac{1}{2k}\bigg)$$
If this works I am already happy. If there's another simple way I'd love to hear it as well.
A: One has
\begin{align*}
& \sum\limits_{n = 1}^\infty {\left[ {\left( {n + \frac{1}{2}} \right)\log \left( {1 + \frac{1}{n}} \right) - 1} \right]} = \sum\limits_{n = 1}^\infty {\int_0^1 {\frac{{\frac{1}{2} - t}}{{n + t}}dt} } = \sum\limits_{n = 1}^\infty {\int_0^1 {\frac{{\frac{1}{2} - (t - \left\lfloor t \right\rfloor )}}{{n + t}}dt} } \\ & = \sum\limits_{n = 1}^\infty {\int_{n - 1}^n {\frac{{\frac{1}{2} - (t - \left\lfloor t \right\rfloor )}}{{t + 1}}dt} } = \int_0^{ + \infty } {\frac{{\frac{1}{2} - (t - \left\lfloor t \right\rfloor )}}{{t + 1}}dt} .
\end{align*}
Now, by the Euler--Maclaurin formula,
$$
\log k! = \left( {k + \frac{1}{2}} \right)\log k- k + C + \int_0^{ + \infty } {\frac{{\frac{1}{2} - (t - \left\lfloor t \right\rfloor )}}{{t + k}}dt}
$$
with some constant $C$. It can be shown that the integral is $\mathcal{O}(k^{-1})$ and so by Stirling's formula (or the Wallis product), $C=\frac{1}{2}\log (2\pi )$. Thus
\begin{align*}\sum\limits_{n = 1}^\infty {\left[ {\left( {n + \frac{1}{2}} \right)\log \left( {1 + \frac{1}{n}} \right) - 1} \right]} & = \log 1! - \left( {\left( {1 + \frac{1}{2}} \right)\log 1 - 1 + \frac{1}{2}\log (2\pi )} \right) \\ &= 1 - \frac{1}{2}\log (2\pi ).
\end{align*}
A: New Answer. Let $S_N$ denote the partial sum for the first $N$ terms. Then $S_N$ is related to the Stirling's Formula by the following computation:
\begin{align*}
S_N
&= \sum_{n=1}^{N} \left(n+\frac{1}{2}\right)\log(n+1) - \sum_{n=1}^{N} \left(n+\frac{1}{2}\right)\log n - N \\
&= \left(N+\frac{1}{2}\right)\log (N+1) - \log (N!) - N.
\end{align*}
Now we consider $e^{-S_N}$ instead. Using the formula $\int_{0}^{\infty}x^{n}e^{-sx}\,\mathrm{d}x=\frac{n!}{s^{n+1}}$,
\begin{align*}
\exp(-S_N)
&= \frac{N!e^{N}}{(N+1)^{N+\frac{1}{2}}} \\
&= \frac{N^{N+1}}{(N+1)^{N+\frac{1}{2}}} \int_{0}^{\infty} x^N e^{-N(x-1)} \, \mathrm{d}x \\
&= \frac{1}{(1+\frac{1}{N})^{N+\frac{1}{2}}} \int_{-\infty}^{\infty} \left(1 + \frac{u}{\sqrt{N}}\right)_{+}^N e^{-\sqrt{N}u} \, \mathrm{d}u,
\end{align*}
where we utilized the substitution $x=1+\frac{u}{\sqrt{N}}$ in the last step and $x_{+}:=\max\{0,x\}$ denotes the positive part of $x$. Then, taking limit as $N\to\infty$ and assuming for a moment that the order of limit and integral can be swapped, we get
\begin{align*}
\lim_{N\to\infty} \exp(-S_N)
&= \biggl( \lim_{N\to\infty} \frac{1}{(1+\frac{1}{N})^{N+\frac{1}{2}}} \biggr) \int_{-\infty}^{\infty} \lim_{N\to\infty} \left(1 + \frac{u}{\sqrt{N}}\right)_{+}^N e^{-\sqrt{N}u} \, \mathrm{d}u \\
&= \frac{1}{e} \int_{-\infty}^{\infty} e^{-u^2/2} \, \mathrm{d}u
= \frac{\sqrt{2\pi}}{e}.
\end{align*}
Here, the last step follows from the gaussian integral. Therefore
$$ \sum_{n=1}^{\infty} \left[ \left(n+\frac{1}{2}\right)\log\left(1+\frac{1}{n}\right)-1 \right] = \lim_{N\to\infty} S_N = 1 - \log\sqrt{2\pi} $$
provided the interchange of limit and integral is justified. For this, we note the following inequality:
$$ \log(1+x) \leq x - \frac{x^2}{2(1+x_+)}, \qquad x > -1 $$
From this, we deduce that
$$
\left(1 + \frac{u}{\sqrt{N}}\right)_{+}^N e^{-\sqrt{N}u}
\leq \exp\left(-\frac{u^2}{2(1+u_+)}\right)
$$
holds for all $N\geq 1$ and for all $u \in \mathbb{R}$. Therefore the dominated convergence theorem is applicable and the desired step is justified, completing the proof.
Old Answer. The sum converges absolutely by the Limit Comparison Test with $\zeta(2)$. Now for each given $n \geq 1$,
\begin{align*}
\left(n+\frac{1}{2}\right)\log\left(1+\frac{1}{n}\right)-1
&= \left(n+\frac{1}{2}\right)\left(\sum_{j=1}^{\infty}\frac{(-1)^{j-1}}{jn^j} \right)-1\\
&= - \frac{1}{4n^2} + \left(n+\frac{1}{2}\right)\sum_{j=3}^{\infty}\frac{(-1)^{j-1}}{jn^j}\\
&= - \frac{1}{4n^2} + \sum_{j=3}^{\infty}\frac{(-1)^{j-1}}{j}\left(\frac{1}{n^{j-1}}+\frac{1}{2n^j}\right).
\end{align*}
Using the formula $\int_{0}^{\infty}x^{s-1}e^{-nx}\,\mathrm{d}x=\frac{\Gamma(s)}{n^s}$, this may be recast as
\begin{align*}
&= \int_{0}^{\infty}\left[ - \frac{x}{4} + \sum_{j=3}^{\infty}\frac{(-1)^{j-1}}{j}\left( \frac{x^{j-2}}{(j-2)!} + \frac{x^{j-1}}{2(j-1)!} \right)\right] e^{-nx}\, \mathrm{d}x \\
&= \int_{0}^{\infty} \left( \frac{1}{x} - \left(\frac{1}{2x}+\frac{1}{x^2}\right)(1-e^{-x}) \right) e^{-nx} \, \mathrm{d}x.
\end{align*}
Summing this for $n = 1, 2, \dots$, we get
\begin{align*}
S
&:= \sum_{n=1}^{\infty} \left[ \left(n+\frac{1}{2}\right)\log\left(1+\frac{1}{n}\right)-1 \right] \\
&= \int_{0}^{\infty} \left( \frac{1}{x} - \left(\frac{1}{2x}+\frac{1}{x^2}\right)(1-e^{-x}) \right) \frac{1}{e^x - 1} \, \mathrm{d}x \\
&= \int_{0}^{\infty} \left( \frac{1}{x(e^x - 1)} - \left(\frac{1}{2x}+\frac{1}{x^2}\right)e^{-x} \right) \, \mathrm{d}x.
\end{align*}
To compute the right-hand side, we consider the following regularization:
\begin{align*}
S(s)
&:= \int_{0}^{\infty} \left( \frac{1}{x(e^x - 1)} - \left(\frac{1}{2x}+\frac{1}{x^2}\right)e^{-x} \right) x^s \, \mathrm{d}x \\
&= \int_{0}^{\infty} \left( \frac{x^{s-1}}{e^x - 1} - \frac{1}{2}x^{s-1}e^{-x} - x^{s-2}e^{-x} \right) \, \mathrm{d}x.
\end{align*}
This function is analytic for $\operatorname{Re}(s) > -1$, and $S = S(0)$. Moreover, for $s > 2$, we easily find that
\begin{align*}
S(s)
&= \Gamma(s)\zeta(s)-\frac{1}{2}\Gamma(s)-\Gamma(s-1) \\
&= \Gamma(s+1)\biggl( \frac{\zeta(s)-\frac{1}{2}-\frac{1}{s-1}}{s} \biggr).
\end{align*}
By the principle of analytic continuation, this identity must hold on all of $\operatorname{Re}(s)>-1$. So, letting $s \to 0$ to the above formula yields
$$ S = \lim_{s\to 0}S(s) = 1 + \zeta'(0). $$
Now the desired formula follows from $\zeta'(0) = -\log\sqrt{2\pi}$.
A: The series is not convergent, so the formula is wrong. $ (n+\frac 1 2 ) \ln (1+\frac 1 n) \to 1$ as $ n \to \infty$ and this proves that LHS is $\infty$. Also RHS depends on $n$.
A: Consider the integral $$\displaystyle
\int\limits_{\displaystyle j}^{\displaystyle j+1}\frac{\displaystyle \left \{ x \right \}-\frac{\displaystyle 1}{\displaystyle 2}}{\displaystyle x}dx
$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,446 |
Q: Kali Linux not appearing in grub after installation Today I installed Kali Linux via bootable USB, and during installation there was a problem installing the bootloader. So I skipped its installation and restarted my computer. But GRUB couldn't find the Kali Linux installed on my logical partition sda4, i.e., it didn't show the option to boot to Kali Linux.
Moreover, during my installation, I set the bootable flag as "no", and I don't know whether it will have any effect on the boot.
I have presently three operating systems installed on other partitions (Windows 7, Linux Mint 17, and Linux Mint 17.2) and today Kali.
Installation was also completed.
So how do I boot to Kali Linux now??
This is the result after I tried the command sudo update-grub:
Generating grub configuration file ...
Found linux image: /boot/vmlinuz-3.18.2-031802-generic
Found initrd image: /boot/initrd.img-3.18.2-031802-generic
Found linux image: /boot/vmlinuz-3.13.0-24-generic
Found initrd image: /boot/initrd.img-3.13.0-24-generic
Found memtest86+ image: /boot/memtest86+.elf
Found memtest86+ image: /boot/memtest86+.bin
No volume groups found
Found Windows 7 (loader) on /dev/sda1
Found Linux Mint 17 Qiana (17) on /dev/sda7
done
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 14 |
Tomasini – cognome italiano
Doris Tomasini (n. 1984) – atleta
Ernesto Tomasini (n. 1968) – attore
Giacomo Filippo Tomasini (1595-1655) – vescovo cattolico, letterato ed erudito
Giuseppe Tomasini – pittore
Giuseppe Tomasini (1821-1873) – poeta
Giuseppe Tomasini (n. 1946) – calciatore
Luigi Tomasini (1741-1808) – violinista e compositore
Stefano Tomasini (n. 1963) – ciclista su strada
Pagine correlate
Tomassini
Tommasini | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,219 |
\section{Introduction}
\label{intro}
The Bethe-Salpeter (BS) approach \cite{SB_PR84_51} is a powerful tool to study the properties of the relativistic few-body systems.
To avoid the problems related to singularities of the BS amplitude in Minkowski space, the BS equation is often transformed, by means of the Wick rotation \cite{Wick}, in the Euclidean space. In this way, one finds the same spectrum as for the equation in the Minkowski space. However, corresponding Euclidean BS amplitude is not enough for calculating the observables. As demonstrated in \cite{bs-long} in case of the ladder kernel -- which displays the strongest (pole) singularities -- it is possible to find the bound states and the scattering solutions in Minkowski space by a direct numerical calculations.
However, this problem is much more complicated to deal with than in the Euclidean space.
Another efficient approach to solve the BS equation, proposed in \cite{KusPRD95} and developed in a series papers \cite{bs1,bs2,FrePRD14,FSVPRD,dPaPRD16,GutPLB16,nak,nak1}, relies on the Nakanishi representation \cite{nakanishi} of the BS amplitude
$\Phi(k,p)$ in terms of a non-singular weight function $g(\gamma,z)$. This representation reads:
\begin{equation}\label{eq1}
\Phi(k,p) = \int_{0}^{\infty} d\gamma{'} \int_{-1}^{1} dz^{'} \frac{g(\gamma^{'}, z^{'})}{\left( \gamma^{'} + \kappa^2 -{k}^2 -p\makebox[0.08cm]{$\cdot$} {k}\,z^{'}-i\epsilon \right)^3},\,\,\,
\end{equation}
where $\kappa^2=m^2-\frac{1}{4}M^2,$
$m$ is the constituent mass and $M$ is the total mass of the bound state ($p^2=M^2$).
Once $g$ is known, one can compute the BS amplitude by means of (\ref{eq1})
and express through $g$ the observables, like the electromagnetic form factors \cite{CarEPJA09}.
For the massless ladder exchange ($\mu=0$), the integral (\ref{eq1}) turns into a one-dimensional one (over $z'$ only). This 1D representation was used to solve the BS equation long ago, before inventing by Nakanishi the representation (\ref{eq1}), in the pioneering researches by Wick \cite{Wick} and Cutkosky \cite{Cutkosky}.
In \cite{bs1}, the following double integral equation for the Nakanishi weight function $g$ was established:
\begin{equation} \label{bsnew}
\int_0^{\infty}\frac{g(\gamma',z)d\gamma'}{\left[\gamma+\gamma' +z^2 m^2+(1-z^2)\kappa^2\right]^2} =
\int_0^{\infty}d\gamma'\int_{-1}^{1}dz'\;V(\gamma,z;\gamma',z') g(\gamma',z'),
\end{equation}
(written symbolically as $\hat{L}\, g=\hat{V} \, g$)
where $V$ is expressed via the BS kernel $K$.
The solution $g$ of the equation (\ref{bsnew}) was found in the case of an OBE ladder BS kernel in \cite{bs1} and for the ladder plus cross-ladder one in \cite{bs2,ff_cross}.
Equation (\ref{bsnew}) contains integral terms in both sides, though the integral in the l.h.-side is one-dimensional.
This fact generates some instability of its numerical solution,
since the discretizing of its l.h.-side results into an ill-conditioned matrix \cite{bs1}.
An equation for $g$, getting rid of the left-hand side integral term of (\ref{bsnew}), was first derived in \cite{FrePRD14}.
It has the following form:
\begin{equation}\label{g_Vg}
g(\gamma,z)=\int_0^{\infty}d\gamma'\int_{-1}^{1}dz'\;{\mathcal V}(\gamma,z;\gamma',z') g(\gamma',z').
\end{equation}
Like in (\ref{bsnew}), the r.h.-side of the equation (\ref{g_Vg}) contains the double integral.
This derivation was based on the uniqueness\footnote{The uniqueness means that if the integral (\ref{eq1}) is zero, then the function $g(\gamma,z)$ is also zero.} of the Nakanishi representation (\ref{eq1}) and it was fulfilled for the ladder kernel only.
The analytical expression for the kernel ${ \hat{\mathcal V}}$ in eq. (\ref{g_Vg}) is given in \cite{FrePRD14} and also below in Sec. \ref{deriv}.
The straightforward application of the derivation \cite{FrePRD14} to the same equation with a non-ladder kernel would meet some algebraic difficulties.
In a recent work \cite{nak2} another method of transforming
(\ref{bsnew}) in the canonical form (\ref{g_Vg}) was found.
It does not rely on the uniqueness of the Nakanishi representation nor on the ladder kernel
but on using the explicit form of the inverse operator $\hat{L}^{-1}$ in equation $\hat{L}\, g=\hat{V} \, g$ for an arbitrary kernel.
It was indeed noticed that the integral in l.h.-side of (\ref{bsnew}) is the generalized Stieltjes transform whose inverse operator $\hat{L}^{-1}$ was derived in \cite{schw}.
We will review in this contribution the main steps of \cite{nak2} and give
simplified expressions for the kernel $\hat{\mathcal V}=\hat{L}^{-1}\hat{V}$ in (\ref{g_Vg}). In addition, we will study the limit of exchanged mass $\mu\to0$.
We will show analytically that in this limit the equation (\ref{g_Vg}) turns into the Wick-Cutkosky equation \cite{Wick,Cutkosky}.
\section{Deriving the integral equation for $g$}\label{deriv}
The results that follow are based on the observation \cite{nak2}, that the integral in l.h.-side of (\ref{bsnew}) can be trivially related to the Stieltjes transform which is inverted analytically \cite{schw,Sumner_49}.
For the integral relation in the form, close to the l.h.-side of eq. (\ref{bsnew}):
\begin{equation} \label{L}
f(\gamma)\equiv \int_0^{\infty} \;d\gamma' \;L(\gamma,\gamma') g(\gamma') = \int_0^{\infty} \;d\gamma' \; \frac{g(\gamma')}{ (\gamma'+\gamma+b)^2},
\end{equation}
denoted symbolically as $f = \hat{L} \; g$, a straightforward application of the inverse Stieltjes transform to (\ref{L}) gives
\begin{equation} \label{L3}
g(\gamma)= \hat{L}^{-1} f=\frac{\gamma}{2\pi}\int_{-\pi}^{\pi}d\phi\, \;e^{i\phi}\; f(\gamma \; e^{i\phi}-b ).
\end{equation}
By applying the inverse integral transform (\ref{L3}) to both sides of the equation (\ref{bsnew}), we obtain
the equation for the Nakanishi weight function $g$ in the canonical form (\ref{g_Vg}),
where
\begin{equation} \label{L7}
{\mathcal V}(\gamma,z;\gamma',z') =\frac{\gamma}{2\pi}\int_{-\pi}^{\pi} \;d\phi\; e^{i\phi} \; V\Bigl(\gamma e^{i\phi}-z^2 m^2-(1-z^2)\kappa^2,z;\gamma',z'\Bigr).
\end{equation}
The relation between the original kernel $K$ appearing in the BS equation and the kernel $V$ in (\ref{bsnew}) and (\ref{L7}) was derived in Ref. \cite{bs1}.
In the case of the OBE kernel
\begin{equation} \label{K}
K(k,k')= -{g^2 \over (k-k')^2-\mu^2+i \epsilon }, \qquad g^2= 16\pi m^2 \alpha
\end{equation}
$V$ takes the form \cite{bs1}:
\begin{equation} \label{Kn}
V(\gamma,z;\gamma',z')=\left\{
\begin{array}{ll}
W(\gamma,z;\gamma',z'),&\mbox{if $-1\le z'\le z\le 1$}\\
W(\gamma,-z;\gamma',- z'),&\mbox{if $-1\le z\le z'\le 1$}
\end{array}\right.
\end{equation}
where:
\begin{eqnarray}\label{W}
W(\gamma,z;\gamma',z') &=&
\frac{\alpha m^2}{2\pi} \frac{(1-z)^2}{[\gamma+z^2m^2+(1-z^2)\kappa^2] b_2^2(b_+ -b_-)^3} \\
&\times&\left[ \frac{(b_+ -b_-)(2b_+ b_- -b_+ -b_-)}{(1-b_+)(1-b_-)} + 2b_+ b_- \log \frac{b_+ (1-b_-)}{b_- (1-b_+)}\right],
\nonumber
\end{eqnarray}
\begin{eqnarray*}
\mbox{and}\quad b_0 &=& (1-z)\mu^2, \,\,\,\,\, b_\pm = -\frac{1}{2b_2} \;\left( b_1 \pm \sqrt{b_1^2-4b_0b_2}\right), \\
b_1 &=& \gamma+\gamma' - (1-z)\mu^2 - \gamma' z -\gamma z' +(1-z')\left[z^2m^2+(1-z^2)\kappa^2\right],
\nonumber\\
b_2 &=& -\gamma (1-z')-(z-z') \left[ (1-z)(1-z')\kappa^2+(z+z'-zz') m^2 \right].
\nonumber
\end{eqnarray*}
The kernel ${\mathcal V}$ of Eq. (\ref{g_Vg}) is determined by inserting (\ref{Kn}) in the integral (\ref{L7}).
As mentioned in the Introduction, the canonical form (\ref{g_Vg}) of Eq. (\ref{bsnew}) was first derived in \cite{FrePRD14}
under the hypothesis of the ladder kernel and an expression for the kernel ${\mathcal V}$ was
found.\footnote{Following \cite{Gianni},
a misprint in the kernel sign in \cite{FrePRD14} was corrected here and in \cite{nak2}.}
We calculate analytically some integrals and transform the kernel \cite{FrePRD14} to the form:
\begin{equation}\label{Vf}
{\mathcal V}(\gamma,z;\gamma',z') = + \frac{\alpha m^2}{2\pi}\times
\left\{
\begin{array}{ll}
\displaystyle{h(\gamma,-z;\gamma',-z')}, &\quad \mbox{if $-1\le z'\le z\le 1$} \\
\displaystyle{h(\gamma,z;\gamma',z')}, &\quad \mbox{if $-1\le z\le z'\le 1$}
\end{array}
\right.
\end{equation}
The function $h$ is obtained from eqs. (27), (28) in \cite{FrePRD14}:
\begin{equation}\label{h}
h(\gamma,z;\gamma',z')=Q(\gamma',z')+\theta(\eta)P(\gamma,z;\gamma',z').
\end{equation}
The first term $Q$ is given by
\begin{eqnarray}\label{Q}
Q(\gamma',z')&=& \int_0^{\infty}\chi(y)dy =-\frac{A'}{AA_s}
-\left\{
\begin{array}{ll}
\displaystyle{\frac{2\mu^2}{A_s^{3/2}}\left(2\arctan\frac{A'}{\sqrt{A_s}}-\pi\right)}, &
\mbox{if $A_s>0$}
\\
\displaystyle{\frac{2\mu^2}{\mid A_s\mid^{3/2}}\log\frac{A'+\sqrt{ \mid A_s\mid }}{A'-\sqrt{\mid A_s\mid }}},&
\mbox{if $A_s<0$}
\end{array}\right.
\nonumber
\end{eqnarray}
where the integrand is given by\footnote{A misprint in eq. (28) of \cite{nak2} for $\chi(y)$ was corrected in (\ref{chi}):
$$ \chi(y)=\frac{y^2}{(A+y^2+A'y+\mu^2)^2}\to \chi(y)=\frac{y^2}{(Ay^2+A'y+\mu^2)^2}. $$}
\begin{equation}\label{chi}
\chi(y)=\frac{y^2}{(Ay^2+A'y+\mu^2)^2}
\end{equation}
and $A_s=4A\mu^2-{A'}^2$, $A=\gamma'+m^2-\frac{1}{4}(1-{z'}^2)M^2 $, $A'=\gamma'+\mu^2$
The argument $\eta$ of the theta-function in the second term in (\ref{h}) reads
\begin{equation}\label{eta}
\eta=\gamma\frac{1+z'}{1+z}-\mu^2-\gamma'-2\mu\sqrt{\gamma'+m^2-\frac{1}{4}(1-{z'}^2)M^2}=-B-2\mu\sqrt{A}.
\end{equation}
The function $P(\gamma,z;\gamma',z')$ in (\ref{h}) has the form:
\begin{equation}\label{P}
P(\gamma,z;\gamma',z')=\frac{B}{\gamma A \Delta}\frac{1+z}{1+z'}-C
\end{equation}
where
\[
B=-\gamma\frac{1+z'}{1+z}+\mu^2+\gamma',\quad \Delta=\sqrt{B^2-4\mu^2A} \]
\[C=\int_{y_-}^{y_+}\chi(y)dy=\hat{\chi}(y_+)-\hat{\chi}(y_-),\quad y_{\pm}=\frac{-B\pm\Delta}{2A} \]
\begin{eqnarray}\label{hatchi}
\hat{\chi}(y)&=& \frac{A'\mu^2-2A\mu^2 y+{A'}^2y}{AA_s[\mu^2+y(A'+Ay)]}
+\left\{
\begin{array}{ll}
\displaystyle{\frac{4\mu^2}{A_s^{3/2}}\arctan\frac{A'+2Ay}{\sqrt{A_s}}}, &\quad\mbox{if $A_s>0$}
\\
\displaystyle{\frac{2\mu^2}{ \mid A_s\mid^{3/2}}
\log\frac{ {A'+2Ay\over\sqrt{\mid A_s\mid}} +1}{ {A'+2Ay\over\sqrt{\mid A_s\mid}}-1}}, &\quad\mbox{if $A_s<0$}
\end{array}\right.
\nonumber
\end{eqnarray}
Kernel ${\cal V}$ is determined above by two quite different expressions: eq. (\ref{L7}) found in \cite{nak2} and eq. (\ref{Vf}) found in \cite{FrePRD14}.
It has not been possible to prove their identity analytically but only by their numerical comparison \cite{nak2}.
It is worth noticing however that, contrary to $V$, kernel ${\cal V}$ displays some singularities in variable $\gamma$ and requires
some careful treatment.
The coupling constants $\alpha$ and the Nakanishi functions $g(\gamma,z)$ for selected values of $M$ and $\mu$ found in \cite{FrePRD14} from equations (\ref{bsnew}) and (\ref{g_Vg}) coincide with each other within numerical accuracy as well as with the Euclidean BS results.
\section{The limit $\mu\to 0$}
In the $\mu=0$ case, which constitutes the original Wick-Cutkosky model \mbox{\cite{Wick,Cutkosky}},
kernel (\ref{Kn}) obtains a simple analytical form:
\begin{eqnarray}\label{KK}
V(\gamma,z;\gamma',z')&=&\frac{\alpha m^2}{2\pi}
\frac{1}{\Bigl[\gamma+z^2m^2+(1-z^2)\kappa^2 \Bigr]\Bigl[\gamma'+{z'}^2m^2+(1-{z'}^2)\kappa^2 \Bigr]}
\nonumber\\
&\times&\left\{
\begin{array}{ll}
\displaystyle{\frac{1}{\Bigl[\gamma +\gamma'\frac{(1+z)}{(1+z')}
+z^2m^2+(1-z^2)\kappa^2\Bigr]}
\frac{(1+z)}{(1+z')}},&\quad\mbox{if $z<z'$}
\\
\displaystyle{\frac{1}{\Bigl[\gamma +\gamma'\frac{(1-z)}{(1-z')}
+z^2m^2+(1-z^2)\kappa^2\Bigr]}
\frac{(1-z)}{(1-z')}},&\quad\mbox{if $z>z'$}
\end{array}
\right.
\end{eqnarray}
Inserting this expression in (\ref{L7}) and integrating over $\phi$ one gets for ${\cal V}$:
\begin{equation}\label{Nu0}
{\mathcal V}(\gamma,z;\gamma',z')=\frac{\alpha m^2}{2\pi\gamma'}
\displaystyle{\frac{1}
{\left[\gamma' +m^2-\frac{1}{4}(1-{z'}^2)M^2\right]}}
\times
\left\{
\begin{array}{ll}
\theta\left(\gamma'\frac{1+z}{1+z'}-\gamma\right),&\;\mbox{if $z<z'$}
\\
\theta\left(\gamma'\frac{1-z}{1-z'}-\gamma\right),&\;\mbox{if $z>z'$}
\end{array}
\right.
\end{equation}
Following \cite{bs1}, we look for a solution of (\ref{g_Vg}) in the form
$$
\bar{g}(\gamma,z)=\delta(\gamma)g(z).
$$
This form is justified provided the equation for $g(z)$ does not depend on $\gamma$. Substituting the former expression in (\ref{g_Vg}), we obtain:
\begin{equation}\label{g0a}
\delta(\gamma)g(z)=\int_0^{\infty} d\gamma' \int_{-1}^1dz'{\cal V}(\gamma,z;\gamma',z')\delta(\gamma')g(z')
\end{equation}
Integrating\footnote{On the one hand, the integrand in (\ref{g0a}) contains the expression $\frac{1}{\gamma'}\delta(\gamma')$ which is infinite.
On the other hand, the integration of r.h.-side of (\ref{g0a}) over $\gamma$ gives, due to the theta-function, the factor $\gamma'\frac{1\pm z}{1\pm z'}$.
$\gamma'$ from this factor cancels $\gamma'$ in the denominator. In this way, we obtain a finite and certain result for the product $\infty\cdot 0$.}
both sides of (\ref{g0a}) over $\gamma$ we find:
\begin{equation}\label{g0b}
g(z)=\int_{-1}^1d z' \widetilde{V}(z,z')g(z'),
\end{equation}
with
\begin{equation}\label{tV}
\widetilde{V}(z,z')=\frac{\alpha m^2}{2\pi}
\frac{1}{\left[m^2-\frac{1}{4}(1-{z'}^2)M^2\right]}
\left\{
\begin{array}{ll}
\displaystyle{\frac{1+z}{1+z'}},&\quad\mbox{if $z<z'$}
\\
&
\\
\displaystyle{\frac{1-z}{1-z'}},&\quad\mbox{if $z<z'$}
\end{array}
\right.
\end{equation}
which exactly coincides with the Wick-Cutkosky equation \cite{Wick,Cutkosky}.
Notice that the same limit $\mu\to 0$ can be derived starting from (\ref{g_Vg}) and kernel (\ref{Vf}).
\section{Conclusion}\label{concl}
Starting with an equation for the Nakanishi weight function $g$ of the Bethe-Salpeter amplitude in the form $\hat{L}\,g=\hat{V}\,g$,
previously established in \cite{bs1,bs2}, and using an analytic inversion of the Stieltjes integral transform,
we transformed this equation in the canonical form $g=\hat{\mathcal V}g$.
This result generalizes, to an arbitrary kernel given by a set of irreducible Feynman graphs, the equation found in \cite{FrePRD14}
which was stablished for a ladder kernel.
This generalization strongly expands the applicability of the Nakanishi representation to find the solution of the Bethe-Salpeter equation,
as well as its applications.
Equation (\ref{g_Vg}) can be extended straightforwardly to the two-fermion system starting, for example, from \cite{dPaPRD16}.
The possibility to apply an analytic inversion of the Stieltjes transform can be useful in other fields
of nuclear and hadronic physics, where the use of integral transforms was pioneered
by \cite{Efros_SJNP_1985,Efros:2007nq,Giussepina_EFB23_2016},
in order to avoid the instabilities of the numerical inversion.
\section*{Aknowledgements}
We are indebted to G. Salm\'e for useful discussions. T.F. thanks CNPq,
CAPES and FAPESP of Brazil. V.A.K. thanks the support of FAPESP, the
grant \#2015/22701-6 and is sincerely grateful for kind hospitality to
Theoretical Nuclear Physics Group in ITA, S\~{a}o Jos\'e dos Campos,
Brazil, where the main part of this research was carried out.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,085 |
A base website for Flask applications
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,888 |
import base64
import email.policy
import os
import subprocess
from email import message_from_string
from email.message import EmailMessage, MIMEPart
from typing import Any, Callable, Dict, Mapping, Optional
from unittest import mock
import orjson
from django.conf import settings
from django.http import HttpResponse
from zerver.lib.actions import (
do_change_stream_post_policy,
do_deactivate_realm,
do_deactivate_user,
ensure_stream,
)
from zerver.lib.email_mirror import (
ZulipEmailForwardError,
create_missed_message_address,
filter_footer,
get_missed_message_token_from_address,
is_forwarded,
is_missed_message_address,
log_and_report,
process_message,
process_missed_message,
redact_email_address,
strip_from_subject,
)
from zerver.lib.email_mirror_helpers import (
decode_email_address,
encode_email_address,
get_email_gateway_message_string_from_address,
)
from zerver.lib.email_notifications import convert_html_to_markdown
from zerver.lib.send_email import FromAddress
from zerver.lib.test_classes import ZulipTestCase
from zerver.lib.test_helpers import mock_queue_publish, most_recent_message, most_recent_usermessage
from zerver.models import (
MissedMessageEmailAddress,
Recipient,
Stream,
UserProfile,
get_display_recipient,
get_realm,
get_stream,
get_system_bot,
)
from zerver.worker.queue_processors import MirrorWorker
logger_name = "zerver.lib.email_mirror"
class TestEncodeDecode(ZulipTestCase):
def _assert_options(
self,
options: Dict[str, bool],
show_sender: bool = False,
include_footer: bool = False,
include_quotes: bool = False,
prefer_text: bool = True,
) -> None:
self.assertEqual(show_sender, ("show_sender" in options) and options["show_sender"])
self.assertEqual(
include_footer, ("include_footer" in options) and options["include_footer"]
)
self.assertEqual(
include_quotes, ("include_quotes" in options) and options["include_quotes"]
)
self.assertEqual(prefer_text, options.get("prefer_text", True))
def test_encode_decode(self) -> None:
realm = get_realm("zulip")
stream_name = "dev. help"
stream = ensure_stream(realm, stream_name, acting_user=None)
email_address = encode_email_address(stream)
self.assertEqual(email_address, f"dev-help.{stream.email_token}@testserver")
# The default form of the email address (with an option - "include-footer"):
token, options = decode_email_address(
f"dev-help.{stream.email_token}.include-footer@testserver",
)
self._assert_options(options, include_footer=True)
self.assertEqual(token, stream.email_token)
# Using + instead of . as the separator is also supported for backwards compatibility,
# since that was the original form of addresses that we used:
token, options = decode_email_address(
f"dev-help+{stream.email_token}+include-footer@testserver",
)
self._assert_options(options, include_footer=True)
self.assertEqual(token, stream.email_token)
token, options = decode_email_address(email_address)
self._assert_options(options)
self.assertEqual(token, stream.email_token)
# We also handle mixing + and . but it shouldn't be recommended to users.
email_address_all_options = (
"dev-help.{}+include-footer.show-sender+include-quotes@testserver"
)
email_address_all_options = email_address_all_options.format(stream.email_token)
token, options = decode_email_address(email_address_all_options)
self._assert_options(options, show_sender=True, include_footer=True, include_quotes=True)
self.assertEqual(token, stream.email_token)
email_address = email_address.replace("@testserver", "@zulip.org")
email_address_all_options = email_address_all_options.replace("@testserver", "@zulip.org")
with self.assertRaises(ZulipEmailForwardError):
decode_email_address(email_address)
with self.assertRaises(ZulipEmailForwardError):
decode_email_address(email_address_all_options)
with self.settings(EMAIL_GATEWAY_EXTRA_PATTERN_HACK="@zulip.org"):
token, options = decode_email_address(email_address)
self._assert_options(options)
self.assertEqual(token, stream.email_token)
token, options = decode_email_address(email_address_all_options)
self._assert_options(
options, show_sender=True, include_footer=True, include_quotes=True
)
self.assertEqual(token, stream.email_token)
with self.assertRaises(ZulipEmailForwardError):
decode_email_address("bogus")
# Test stream name encoding changes introduced due to
# https://github.com/zulip/zulip/issues/9840
def test_encode_decode_nonlatin_alphabet_stream_name(self) -> None:
realm = get_realm("zulip")
stream_name = "Тестовы some ascii letters"
stream = ensure_stream(realm, stream_name, acting_user=None)
email_address = encode_email_address(stream)
msg_string = get_email_gateway_message_string_from_address(email_address)
parts = msg_string.split("+")
# Stream name should be completely stripped to '', so msg_string
# should only have the email_token in it.
self.assert_length(parts, 1)
# Correctly decode the resulting address that doesn't have the stream name:
token, show_sender = decode_email_address(email_address)
self.assertFalse(show_sender)
self.assertEqual(token, stream.email_token)
asciiable_stream_name = "ąężć"
stream = ensure_stream(realm, asciiable_stream_name, acting_user=None)
email_address = encode_email_address(stream)
self.assertTrue(email_address.startswith("aezc."))
def test_decode_ignores_stream_name(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream)
stream_to_address = stream_to_address.replace("denmark", "Some_name")
# get the email_token:
token = decode_email_address(stream_to_address)[0]
self.assertEqual(token, stream.email_token)
def test_encode_with_show_sender(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream, show_sender=True)
token, options = decode_email_address(stream_to_address)
self._assert_options(options, show_sender=True)
self.assertEqual(token, stream.email_token)
def test_decode_prefer_text_options(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
address_prefer_text = f"Denmark.{stream.email_token}.prefer-text@testserver"
address_prefer_html = f"Denmark.{stream.email_token}.prefer-html@testserver"
token, options = decode_email_address(address_prefer_text)
self._assert_options(options, prefer_text=True)
token, options = decode_email_address(address_prefer_html)
self._assert_options(options, prefer_text=False)
class TestGetMissedMessageToken(ZulipTestCase):
def test_get_missed_message_token(self) -> None:
with self.settings(EMAIL_GATEWAY_PATTERN="%s@example.com"):
address = "mm" + ("x" * 32) + "@example.com"
self.assertTrue(is_missed_message_address(address))
token = get_missed_message_token_from_address(address)
self.assertEqual(token, "mm" + "x" * 32)
# This next section was a bug at one point--we'd treat ordinary
# user addresses that happened to begin with "mm" as being
# the special mm+32chars tokens.
address = "mmathers@example.com"
self.assertFalse(is_missed_message_address(address))
with self.assertRaises(ZulipEmailForwardError):
get_missed_message_token_from_address(address)
# Now test the case where we our address does not match the
# EMAIL_GATEWAY_PATTERN.
# This used to crash in an ugly way; we want to throw a proper
# exception.
address = "alice@not-the-domain-we-were-expecting.com"
self.assertFalse(is_missed_message_address(address))
with self.assertRaises(ZulipEmailForwardError):
get_missed_message_token_from_address(address)
class TestFilterFooter(ZulipTestCase):
def test_filter_footer(self) -> None:
text = """Test message
--Not a delimiter--
More message
--
Footer"""
expected_output = """Test message
--Not a delimiter--
More message"""
result = filter_footer(text)
self.assertEqual(result, expected_output)
def test_filter_footer_many_parts(self) -> None:
text = """Test message
--
Part1
--
Part2"""
result = filter_footer(text)
# Multiple possible footers, don't strip
self.assertEqual(result, text)
class TestStreamEmailMessagesSuccess(ZulipTestCase):
def test_receive_stream_email_messages_success(self) -> None:
# build dummy messages for stream
# test valid incoming stream message is processed properly
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestStreamEmailMessages body")
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
# Test receiving an email with the address on an UnstructuredHeader
# (e.g. Envelope-To) instead of an AddressHeader (e.g. To).
# https://github.com/zulip/zulip/issues/15864
def test_receive_stream_email_messages_other_header_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
# Simulate a mailing list
incoming_valid_message["To"] = "foo-mailinglist@example.com"
incoming_valid_message["Envelope-To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestStreamEmailMessages body")
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
def test_receive_stream_email_messages_blank_subject_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = ""
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestStreamEmailMessages body")
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), "(no topic)")
def test_receive_private_stream_email_messages_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.make_stream("private_stream", invite_only=True)
self.subscribe(user_profile, "private_stream")
stream = get_stream("private_stream", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestStreamEmailMessages body")
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
def test_receive_stream_email_multiple_recipient_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
# stream address is angle-addr within multiple addresses
stream_to_addresses = [
"A.N. Other <another@example.org>",
f"Denmark <{encode_email_address(stream)}>",
]
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = ", ".join(stream_to_addresses)
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestStreamEmailMessages body")
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
def test_receive_stream_email_show_sender_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
parts = stream_to_address.split("@")
parts[0] += "+show-sender"
stream_to_address = "@".join(parts)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(
message.content,
"From: {}\n{}".format(self.example_email("hamlet"), "TestStreamEmailMessages body"),
)
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
def test_receive_stream_email_show_sender_utf8_encoded_sender(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
parts = stream_to_address.split("@")
parts[0] += "+show-sender"
stream_to_address = "@".join(parts)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message[
"From"
] = "Test =?utf-8?b?VXNlcsOzxIXEmQ==?= <=?utf-8?q?hamlet=5F=C4=99?=@zulip.com>"
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(
message.content,
"From: {}\n{}".format(
"Test Useróąę <hamlet_ę@zulip.com>", "TestStreamEmailMessages body"
),
)
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
def test_receive_stream_email_include_footer_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
parts = stream_to_address.split("@")
parts[0] += "+include-footer"
stream_to_address = "@".join(parts)
text = """Test message
--
Footer"""
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content(text)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, text)
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
def test_receive_stream_email_include_quotes_success(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
parts = stream_to_address.split("@")
parts[0] += "+include-quotes"
stream_to_address = "@".join(parts)
text = """Reply
-----Original Message-----
Quote"""
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content(text)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, text)
self.assertEqual(get_display_recipient(message.recipient), stream.name)
self.assertEqual(message.topic_name(), incoming_valid_message["Subject"])
class TestEmailMirrorMessagesWithAttachments(ZulipTestCase):
def test_message_with_valid_attachment(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
with open(
os.path.join(settings.DEPLOY_ROOT, "static/images/default-avatar.png"), "rb"
) as f:
image_bytes = f.read()
incoming_valid_message.add_attachment(
image_bytes,
maintype="image",
subtype="png",
filename="image.png",
)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
with mock.patch(
"zerver.lib.email_mirror.upload_message_file", return_value="https://test_url"
) as upload_message_file:
process_message(incoming_valid_message)
upload_message_file.assert_called_with(
"image.png",
len(image_bytes),
"image/png",
image_bytes,
get_system_bot(settings.EMAIL_GATEWAY_BOT),
target_realm=user_profile.realm,
)
message = most_recent_message(user_profile)
self.assertEqual(message.content, "Test body\n[image.png](https://test_url)")
def test_message_with_attachment_utf8_filename(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
with open(
os.path.join(settings.DEPLOY_ROOT, "static/images/default-avatar.png"), "rb"
) as f:
image_bytes = f.read()
utf8_filename = "image_ąęó.png"
incoming_valid_message.add_attachment(
image_bytes,
maintype="image",
subtype="png",
filename=utf8_filename,
)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
with mock.patch(
"zerver.lib.email_mirror.upload_message_file", return_value="https://test_url"
) as upload_message_file:
process_message(incoming_valid_message)
upload_message_file.assert_called_with(
utf8_filename,
len(image_bytes),
"image/png",
image_bytes,
get_system_bot(settings.EMAIL_GATEWAY_BOT),
target_realm=user_profile.realm,
)
message = most_recent_message(user_profile)
self.assertEqual(message.content, f"Test body\n[{utf8_filename}](https://test_url)")
def test_message_with_valid_nested_attachment(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
nested_multipart = EmailMessage()
nested_multipart.set_content("Nested text that should get skipped.")
with open(
os.path.join(settings.DEPLOY_ROOT, "static/images/default-avatar.png"), "rb"
) as f:
image_bytes = f.read()
nested_multipart.add_attachment(
image_bytes,
maintype="image",
subtype="png",
filename="image.png",
)
incoming_valid_message.add_attachment(nested_multipart)
incoming_valid_message["Subject"] = "Subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
with mock.patch(
"zerver.lib.email_mirror.upload_message_file", return_value="https://test_url"
) as upload_message_file:
process_message(incoming_valid_message)
upload_message_file.assert_called_with(
"image.png",
len(image_bytes),
"image/png",
image_bytes,
get_system_bot(settings.EMAIL_GATEWAY_BOT),
target_realm=user_profile.realm,
)
message = most_recent_message(user_profile)
self.assertEqual(message.content, "Test body\n[image.png](https://test_url)")
def test_message_with_invalid_attachment(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
# Create an invalid attachment:
attachment_msg = MIMEPart()
attachment_msg.add_header("Content-Disposition", "attachment", filename="some_attachment")
incoming_valid_message.add_attachment(attachment_msg)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
with self.assertLogs(logger_name, level="WARNING") as m:
process_message(incoming_valid_message)
self.assertEqual(
m.output,
[
"WARNING:{}:Payload is not bytes (invalid attachment {} in message from {}).".format(
logger_name, "some_attachment", self.example_email("hamlet")
)
],
)
def test_receive_plaintext_and_html_prefer_text_html_options(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_address = f"Denmark.{stream.email_token}@testserver"
stream_address_prefer_html = f"Denmark.{stream.email_token}.prefer-html@testserver"
text = "Test message"
html = "<html><body><b>Test html message</b></body></html>"
incoming_valid_message = EmailMessage()
incoming_valid_message.add_alternative(text)
incoming_valid_message.add_alternative(html, subtype="html")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, "Test message")
del incoming_valid_message["To"]
incoming_valid_message["To"] = stream_address_prefer_html
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, "**Test html message**")
def test_receive_only_plaintext_with_prefer_html_option(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_address_prefer_html = f"Denmark.{stream.email_token}.prefer-html@testserver"
text = "Test message"
# This should be correctly identified as empty html body:
html = "<html><body></body></html>"
incoming_valid_message = EmailMessage()
incoming_valid_message.add_alternative(text)
incoming_valid_message.add_alternative(html, subtype="html")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_address_prefer_html
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
# HTML body is empty, so the plaintext content should be picked, despite prefer-html option.
self.assertEqual(message.content, "Test message")
class TestStreamEmailMessagesEmptyBody(ZulipTestCase):
def test_receive_stream_email_messages_empty_body(self) -> None:
# build dummy messages for stream
# test message with empty body is not sent
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
# empty body
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
with self.assertLogs(logger_name, level="WARNING") as m:
process_message(incoming_valid_message)
self.assertEqual(
m.output, [f"WARNING:{logger_name}:Email has no nonempty body sections; ignoring."]
)
def test_receive_stream_email_messages_no_textual_body(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
# No textual body
incoming_valid_message = EmailMessage()
with open(
os.path.join(settings.DEPLOY_ROOT, "static/images/default-avatar.png"), "rb"
) as f:
incoming_valid_message.add_attachment(
f.read(),
maintype="image",
subtype="png",
)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
with self.assertLogs(logger_name, level="WARNING") as m:
process_message(incoming_valid_message)
self.assertEqual(
m.output,
[
f"WARNING:{logger_name}:Content types: ['multipart/mixed', 'image/png']",
f"WARNING:{logger_name}:Unable to find plaintext or HTML message body",
],
)
def test_receive_stream_email_messages_empty_body_after_stripping(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
headers = {}
headers["Reply-To"] = self.example_email("othello")
# empty body
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("-- \nFooter")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, "(No email body)")
class TestMissedMessageEmailMessages(ZulipTestCase):
def test_receive_missed_personal_message_email_messages(self) -> None:
# build dummy messages for message notification email reply
# have Hamlet send Othello a PM. Othello will reply via email
# Hamlet will receive the message.
self.login("hamlet")
othello = self.example_user("othello")
result = self.client_post(
"/json/messages",
{
"type": "private",
"content": "test_receive_missed_message_email_messages",
"client": "test suite",
"to": orjson.dumps([othello.id]).decode(),
},
)
self.assert_json_success(result)
user_profile = self.example_user("othello")
usermessage = most_recent_usermessage(user_profile)
# we don't want to send actual emails but we do need to create and store the
# token for looking up who did reply.
mm_address = create_missed_message_address(user_profile, usermessage.message)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = self.example_email("othello")
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
# confirm that Hamlet got the message
user_profile = self.example_user("hamlet")
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestMissedMessageEmailMessages body")
self.assertEqual(message.sender, self.example_user("othello"))
self.assertEqual(message.recipient.type_id, user_profile.id)
self.assertEqual(message.recipient.type, Recipient.PERSONAL)
def test_receive_missed_huddle_message_email_messages(self) -> None:
# build dummy messages for message notification email reply
# have Othello send Iago and Cordelia a PM. Cordelia will reply via email
# Iago and Othello will receive the message.
self.login("othello")
cordelia = self.example_user("cordelia")
iago = self.example_user("iago")
result = self.client_post(
"/json/messages",
{
"type": "private",
"content": "test_receive_missed_message_email_messages",
"client": "test suite",
"to": orjson.dumps([cordelia.id, iago.id]).decode(),
},
)
self.assert_json_success(result)
user_profile = self.example_user("cordelia")
usermessage = most_recent_usermessage(user_profile)
# we don't want to send actual emails but we do need to create and store the
# token for looking up who did reply.
mm_address = create_missed_message_address(user_profile, usermessage.message)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedHuddleMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedHuddleMessageEmailMessages subject"
incoming_valid_message["From"] = self.example_email("cordelia")
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = self.example_email("cordelia")
process_message(incoming_valid_message)
# Confirm Iago received the message.
user_profile = self.example_user("iago")
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestMissedHuddleMessageEmailMessages body")
self.assertEqual(message.sender, self.example_user("cordelia"))
self.assertEqual(message.recipient.type, Recipient.HUDDLE)
# Confirm Othello received the message.
user_profile = self.example_user("othello")
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestMissedHuddleMessageEmailMessages body")
self.assertEqual(message.sender, self.example_user("cordelia"))
self.assertEqual(message.recipient.type, Recipient.HUDDLE)
def test_receive_missed_stream_message_email_messages(self) -> None:
# build dummy messages for message notification email reply
# have Hamlet send a message to stream Denmark, that Othello
# will receive a message notification email about.
# Othello will reply via email.
# Hamlet will see the message in the stream.
self.subscribe(self.example_user("hamlet"), "Denmark")
self.subscribe(self.example_user("othello"), "Denmark")
self.login("hamlet")
result = self.client_post(
"/json/messages",
{
"type": "stream",
"topic": "test topic",
"content": "test_receive_missed_stream_message_email_messages",
"client": "test suite",
"to": "Denmark",
},
)
self.assert_json_success(result)
user_profile = self.example_user("othello")
usermessage = most_recent_usermessage(user_profile)
mm_address = create_missed_message_address(user_profile, usermessage.message)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
process_message(incoming_valid_message)
# confirm that Hamlet got the message
user_profile = self.example_user("hamlet")
message = most_recent_message(user_profile)
self.assertEqual(message.content, "TestMissedMessageEmailMessages body")
self.assertEqual(message.sender, self.example_user("othello"))
self.assertEqual(message.recipient.type, Recipient.STREAM)
self.assertEqual(message.recipient.id, usermessage.message.recipient.id)
def test_receive_email_response_for_auth_failures(self) -> None:
user_profile = self.example_user("hamlet")
self.subscribe(user_profile, "announce")
self.login("hamlet")
result = self.client_post(
"/json/messages",
{
"type": "stream",
"topic": "test topic",
"content": "test_receive_email_response_for_auth_failures",
"client": "test suite",
"to": "announce",
},
)
self.assert_json_success(result)
stream = get_stream("announce", user_profile.realm)
do_change_stream_post_policy(stream, Stream.STREAM_POST_POLICY_ADMINS)
usermessage = most_recent_usermessage(user_profile)
mm_address = create_missed_message_address(user_profile, usermessage.message)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(
message.content,
"Error sending message to stream announce via message notification email reply:\nOnly organization administrators can send to this stream.",
)
self.assertEqual(message.sender, get_system_bot(settings.NOTIFICATION_BOT))
def test_missed_stream_message_email_response_tracks_topic_change(self) -> None:
self.subscribe(self.example_user("hamlet"), "Denmark")
self.subscribe(self.example_user("othello"), "Denmark")
self.login("hamlet")
result = self.client_post(
"/json/messages",
{
"type": "stream",
"topic": "test topic",
"content": "test_receive_missed_stream_message_email_messages",
"client": "test suite",
"to": "Denmark",
},
)
self.assert_json_success(result)
user_profile = self.example_user("othello")
usermessage = most_recent_usermessage(user_profile)
mm_address = create_missed_message_address(user_profile, usermessage.message)
# The mm address has been generated, now we change the topic of the message and see
# if the response to the mm address will be correctly posted with the updated topic.
usermessage.message.subject = "updated topic"
usermessage.message.save(update_fields=["subject"])
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
process_message(incoming_valid_message)
# confirm that Hamlet got the message
user_profile = self.example_user("hamlet")
message = most_recent_message(user_profile)
self.assertEqual(message.subject, "updated topic")
self.assertEqual(message.content, "TestMissedMessageEmailMessages body")
self.assertEqual(message.sender, self.example_user("othello"))
self.assertEqual(message.recipient.type, Recipient.STREAM)
self.assertEqual(message.recipient.id, usermessage.message.recipient.id)
def test_missed_message_email_response_from_deactivated_user(self) -> None:
self.subscribe(self.example_user("hamlet"), "Denmark")
self.subscribe(self.example_user("othello"), "Denmark")
self.login("hamlet")
result = self.client_post(
"/json/messages",
{
"type": "stream",
"topic": "test topic",
"content": "test_receive_missed_stream_message_email_messages",
"client": "test suite",
"to": "Denmark",
},
)
self.assert_json_success(result)
user_profile = self.example_user("othello")
message = most_recent_message(user_profile)
mm_address = create_missed_message_address(user_profile, message)
do_deactivate_user(user_profile, acting_user=None)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
initial_last_message = self.get_last_message()
process_message(incoming_valid_message)
# Since othello is deactivated, his message shouldn't be posted:
self.assertEqual(initial_last_message, self.get_last_message())
def test_missed_message_email_response_from_deactivated_realm(self) -> None:
self.subscribe(self.example_user("hamlet"), "Denmark")
self.subscribe(self.example_user("othello"), "Denmark")
self.login("hamlet")
result = self.client_post(
"/json/messages",
{
"type": "stream",
"topic": "test topic",
"content": "test_receive_missed_stream_message_email_messages",
"client": "test suite",
"to": "Denmark",
},
)
self.assert_json_success(result)
user_profile = self.example_user("othello")
message = most_recent_message(user_profile)
mm_address = create_missed_message_address(user_profile, message)
do_deactivate_realm(user_profile.realm, acting_user=None)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
initial_last_message = self.get_last_message()
process_message(incoming_valid_message)
# Since othello's realm is deactivated, his message shouldn't be posted:
self.assertEqual(initial_last_message, self.get_last_message())
def test_missed_message_email_multiple_responses(self) -> None:
self.subscribe(self.example_user("hamlet"), "Denmark")
self.subscribe(self.example_user("othello"), "Denmark")
self.login("hamlet")
result = self.client_post(
"/json/messages",
{
"type": "stream",
"topic": "test topic",
"content": "test_receive_missed_stream_message_email_messages",
"client": "test suite",
"to": "Denmark",
},
)
self.assert_json_success(result)
user_profile = self.example_user("othello")
message = most_recent_message(user_profile)
mm_address = create_missed_message_address(user_profile, message)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestMissedMessageEmailMessages body")
incoming_valid_message["Subject"] = "TestMissedMessageEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = mm_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
for i in range(0, MissedMessageEmailAddress.ALLOWED_USES):
process_missed_message(mm_address, incoming_valid_message)
with self.assertRaises(ZulipEmailForwardError):
process_missed_message(mm_address, incoming_valid_message)
class TestEmptyGatewaySetting(ZulipTestCase):
def test_missed_message(self) -> None:
self.login("othello")
cordelia = self.example_user("cordelia")
iago = self.example_user("iago")
payload = dict(
type="private",
content="test_receive_missed_message_email_messages",
client="test suite",
to=orjson.dumps([cordelia.id, iago.id]).decode(),
)
result = self.client_post("/json/messages", payload)
self.assert_json_success(result)
user_profile = self.example_user("cordelia")
usermessage = most_recent_usermessage(user_profile)
with self.settings(EMAIL_GATEWAY_PATTERN=""):
mm_address = create_missed_message_address(user_profile, usermessage.message)
self.assertEqual(mm_address, FromAddress.NOREPLY)
def test_encode_email_addr(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
with self.settings(EMAIL_GATEWAY_PATTERN=""):
test_address = encode_email_address(stream)
self.assertEqual(test_address, "")
class TestReplyExtraction(ZulipTestCase):
def test_is_forwarded(self) -> None:
self.assertTrue(is_forwarded("FWD: hey"))
self.assertTrue(is_forwarded("fwd: hi"))
self.assertTrue(is_forwarded("[fwd] subject"))
self.assertTrue(is_forwarded("FWD: RE:"))
self.assertTrue(is_forwarded("Fwd: RE: fwd: re: subject"))
self.assertFalse(is_forwarded("subject"))
self.assertFalse(is_forwarded("RE: FWD: hi"))
def test_reply_is_extracted_from_plain(self) -> None:
# build dummy messages for stream
# test valid incoming stream message is processed properly
self.login("hamlet")
user_profile = self.example_user("hamlet")
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
text = """Reply
-----Original Message-----
Quote"""
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content(text)
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "Reply")
# Don't extract if Subject indicates the email has been forwarded into the mirror:
del incoming_valid_message["Subject"]
incoming_valid_message["Subject"] = "FWD: TestStreamEmailMessages subject"
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, text)
def test_reply_is_extracted_from_html(self) -> None:
# build dummy messages for stream
# test valid incoming stream message is processed properly
self.login("hamlet")
user_profile = self.example_user("hamlet")
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
html = """
<html>
<body>
<p>Reply</p>
<blockquote>
<div>
On 11-Apr-2011, at 6:54 PM, Bob <bob@example.com> wrote:
</div>
<div>
Quote
</div>
</blockquote>
</body>
</html>
"""
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content(html, subtype="html")
incoming_valid_message["Subject"] = "TestStreamEmailMessages subject"
incoming_valid_message["From"] = user_profile.delivery_email
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = user_profile.delivery_email
process_message(incoming_valid_message)
# Hamlet is subscribed to this stream so should see the email message from Othello.
message = most_recent_message(user_profile)
self.assertEqual(message.content, "Reply")
# Don't extract if Subject indicates the email has been forwarded into the mirror:
del incoming_valid_message["Subject"]
incoming_valid_message["Subject"] = "FWD: TestStreamEmailMessages subject"
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, convert_html_to_markdown(html))
class TestScriptMTA(ZulipTestCase):
def test_success(self) -> None:
script = os.path.join(os.path.dirname(__file__), "../../scripts/lib/email-mirror-postfix")
sender = self.example_email("hamlet")
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream)
mail_template = self.fixture_data("simple.txt", type="email")
mail = mail_template.format(stream_to_address=stream_to_address, sender=sender)
subprocess.run(
[script, "-r", stream_to_address, "-s", settings.SHARED_SECRET, "-t"],
input=mail,
check=True,
universal_newlines=True,
)
def test_error_no_recipient(self) -> None:
script = os.path.join(os.path.dirname(__file__), "../../scripts/lib/email-mirror-postfix")
sender = self.example_email("hamlet")
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream)
mail_template = self.fixture_data("simple.txt", type="email")
mail = mail_template.format(stream_to_address=stream_to_address, sender=sender)
p = subprocess.run(
[script, "-s", settings.SHARED_SECRET, "-t"],
input=mail,
stdout=subprocess.PIPE,
universal_newlines=True,
)
self.assertEqual(
p.stdout,
"5.1.1 Bad destination mailbox address: No missed message email address.\n",
)
self.assertEqual(p.returncode, 67)
class TestEmailMirrorTornadoView(ZulipTestCase):
def send_private_message(self) -> str:
self.login("othello")
cordelia = self.example_user("cordelia")
iago = self.example_user("iago")
result = self.client_post(
"/json/messages",
{
"type": "private",
"content": "test_receive_missed_message_email_messages",
"client": "test suite",
"to": orjson.dumps([cordelia.id, iago.id]).decode(),
},
)
self.assert_json_success(result)
user_profile = self.example_user("cordelia")
user_message = most_recent_usermessage(user_profile)
return create_missed_message_address(user_profile, user_message.message)
def send_offline_message(self, to_address: str, sender: UserProfile) -> HttpResponse:
mail_template = self.fixture_data("simple.txt", type="email")
mail = mail_template.format(stream_to_address=to_address, sender=sender.delivery_email)
msg_base64 = base64.b64encode(mail.encode()).decode()
def check_queue_json_publish(
queue_name: str,
event: Mapping[str, Any],
processor: Optional[Callable[[Any], None]] = None,
) -> None:
self.assertEqual(queue_name, "email_mirror")
self.assertEqual(event, {"rcpt_to": to_address, "msg_base64": msg_base64})
MirrorWorker().consume(event)
self.assertEqual(
self.get_last_message().content, "This is a plain-text message for testing Zulip."
)
post_data = {
"rcpt_to": to_address,
"msg_base64": msg_base64,
"secret": settings.SHARED_SECRET,
}
with mock_queue_publish("zerver.lib.email_mirror.queue_json_publish") as m:
m.side_effect = check_queue_json_publish
return self.client_post("/email_mirror_message", post_data)
def test_success_stream(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream)
result = self.send_offline_message(stream_to_address, self.example_user("hamlet"))
self.assert_json_success(result)
def test_error_to_stream_with_wrong_address(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream)
# get the email_token:
token = decode_email_address(stream_to_address)[0]
stream_to_address = stream_to_address.replace(token, "Wrong_token")
result = self.send_offline_message(stream_to_address, self.example_user("hamlet"))
self.assert_json_error(
result,
"5.1.1 Bad destination mailbox address: "
"Bad stream token from email recipient " + stream_to_address,
)
def test_success_to_stream_with_good_token_wrong_stream_name(self) -> None:
stream = get_stream("Denmark", get_realm("zulip"))
stream_to_address = encode_email_address(stream)
stream_to_address = stream_to_address.replace("denmark", "Wrong_name")
result = self.send_offline_message(stream_to_address, self.example_user("hamlet"))
self.assert_json_success(result)
def test_success_to_private(self) -> None:
mm_address = self.send_private_message()
result = self.send_offline_message(mm_address, self.example_user("cordelia"))
self.assert_json_success(result)
def test_using_mm_address_multiple_times(self) -> None:
mm_address = self.send_private_message()
for i in range(0, MissedMessageEmailAddress.ALLOWED_USES):
result = self.send_offline_message(mm_address, self.example_user("cordelia"))
self.assert_json_success(result)
result = self.send_offline_message(mm_address, self.example_user("cordelia"))
self.assert_json_error(
result, "5.1.1 Bad destination mailbox address: Missed message address out of uses."
)
def test_wrong_missed_email_private_message(self) -> None:
self.send_private_message()
mm_address = "mm" + ("x" * 32) + "@testserver"
result = self.send_offline_message(mm_address, self.example_user("cordelia"))
self.assert_json_error(
result,
"5.1.1 Bad destination mailbox address: Missed message address expired or doesn't exist.",
)
class TestStreamEmailMessagesSubjectStripping(ZulipTestCase):
def test_process_message_strips_subject(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("TestStreamEmailMessages body")
incoming_valid_message["Subject"] = "Re: Fwd: Re: Test"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
incoming_valid_message["Reply-to"] = self.example_email("othello")
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual("Test", message.topic_name())
# If after stripping we get an empty subject, it should get set to (no topic)
del incoming_valid_message["Subject"]
incoming_valid_message["Subject"] = "Re: Fwd: Re: "
process_message(incoming_valid_message)
message = most_recent_message(user_profile)
self.assertEqual("(no topic)", message.topic_name())
def test_strip_from_subject(self) -> None:
subject_list = orjson.loads(self.fixture_data("subjects.json", type="email"))
for subject in subject_list:
stripped = strip_from_subject(subject["original_subject"])
self.assertEqual(stripped, subject["stripped_subject"])
# If the Content-Type header didn't specify a charset, the text content
# of the email used to not be properly found. Test that this is fixed:
class TestContentTypeUnspecifiedCharset(ZulipTestCase):
def test_charset_not_specified(self) -> None:
message_as_string = self.fixture_data("1.txt", type="email")
message_as_string = message_as_string.replace(
'Content-Type: text/plain; charset="us-ascii"', "Content-Type: text/plain"
)
incoming_message = message_from_string(message_as_string, policy=email.policy.default)
# https://github.com/python/typeshed/issues/2417
assert isinstance(incoming_message, EmailMessage)
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "Denmark")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
del incoming_message["To"]
incoming_message["To"] = stream_to_address
process_message(incoming_message)
message = most_recent_message(user_profile)
self.assertEqual(message.content, "Email fixture 1.txt body")
class TestEmailMirrorProcessMessageNoValidRecipient(ZulipTestCase):
def test_process_message_no_valid_recipient(self) -> None:
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
incoming_valid_message["Subject"] = "Test subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = "address@wrongdomain, address@notzulip"
incoming_valid_message["Reply-to"] = self.example_email("othello")
with mock.patch("zerver.lib.email_mirror.log_and_report") as mock_log_and_report:
process_message(incoming_valid_message)
mock_log_and_report.assert_called_with(
incoming_valid_message, "Missing recipient in mirror email", None
)
class TestEmailMirrorLogAndReport(ZulipTestCase):
def test_log_and_report(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "errors")
stream = get_stream("Denmark", user_profile.realm)
stream_to_address = encode_email_address(stream)
address_parts = stream_to_address.split("@")
scrubbed_address = "X" * len(address_parts[0]) + "@" + address_parts[1]
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
incoming_valid_message["Subject"] = "Test subject"
incoming_valid_message["From"] = self.example_email("hamlet")
incoming_valid_message["To"] = stream_to_address
with self.assertLogs("zerver.lib.email_mirror", "ERROR") as error_log:
log_and_report(incoming_valid_message, "test error message", stream_to_address)
self.assertEqual(
error_log.output,
[
f"ERROR:zerver.lib.email_mirror:Sender: hamlet@zulip.com\nTo: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX@testserver <Address to stream id: {stream.id}>\ntest error message"
],
)
message = most_recent_message(user_profile)
self.assertEqual("email mirror error", message.topic_name())
msg_content = message.content.strip("~").strip()
expected_content = "Sender: {}\nTo: {} <Address to stream id: {}>\ntest error message"
expected_content = expected_content.format(
self.example_email("hamlet"), scrubbed_address, stream.id
)
self.assertEqual(msg_content, expected_content)
with self.assertLogs("zerver.lib.email_mirror", "ERROR") as error_log:
log_and_report(incoming_valid_message, "test error message", None)
self.assertEqual(
error_log.output,
[
"ERROR:zerver.lib.email_mirror:Sender: hamlet@zulip.com\nTo: No recipient found\ntest error message"
],
)
message = most_recent_message(user_profile)
self.assertEqual("email mirror error", message.topic_name())
msg_content = message.content.strip("~").strip()
expected_content = "Sender: {}\nTo: No recipient found\ntest error message"
expected_content = expected_content.format(self.example_email("hamlet"))
self.assertEqual(msg_content, expected_content)
def test_log_and_report_no_errorbot(self) -> None:
with self.settings(ERROR_BOT=None):
incoming_valid_message = EmailMessage()
incoming_valid_message.set_content("Test body")
incoming_valid_message["Subject"] = "Test subject"
incoming_valid_message["From"] = self.example_email("hamlet")
with self.assertLogs(logger_name, level="ERROR") as m:
log_and_report(incoming_valid_message, "test error message", None)
expected_content = "Sender: {}\nTo: No recipient found\ntest error message"
expected_content = expected_content.format(self.example_email("hamlet"))
self.assertEqual(m.output, [f"ERROR:{logger_name}:{expected_content}"])
def test_redact_email_address(self) -> None:
user_profile = self.example_user("hamlet")
self.login_user(user_profile)
self.subscribe(user_profile, "errors")
stream = get_stream("Denmark", user_profile.realm)
# Test for a stream address:
stream_to_address = encode_email_address(stream)
stream_address_parts = stream_to_address.split("@")
scrubbed_stream_address = "X" * len(stream_address_parts[0]) + "@" + stream_address_parts[1]
error_message = "test message {}"
error_message = error_message.format(stream_to_address)
expected_message = "test message {} <Address to stream id: {}>"
expected_message = expected_message.format(scrubbed_stream_address, stream.id)
redacted_message = redact_email_address(error_message)
self.assertEqual(redacted_message, expected_message)
# Test for an invalid email address:
invalid_address = "invalid@testserver"
error_message = "test message {}"
error_message = error_message.format(invalid_address)
expected_message = "test message {} <Invalid address>"
expected_message = expected_message.format("XXXXXXX@testserver")
redacted_message = redact_email_address(error_message)
self.assertEqual(redacted_message, expected_message)
# Test for a missed message address:
cordelia = self.example_user("cordelia")
iago = self.example_user("iago")
result = self.client_post(
"/json/messages",
{
"type": "private",
"content": "test_redact_email_message",
"client": "test suite",
"to": orjson.dumps([cordelia.email, iago.email]).decode(),
},
)
self.assert_json_success(result)
cordelia_profile = self.example_user("cordelia")
user_message = most_recent_usermessage(cordelia_profile)
mm_address = create_missed_message_address(user_profile, user_message.message)
error_message = "test message {}"
error_message = error_message.format(mm_address)
expected_message = "test message {} <Missed message address>"
expected_message = expected_message.format("X" * 34 + "@testserver")
redacted_message = redact_email_address(error_message)
self.assertEqual(redacted_message, expected_message)
# Test if redacting correctly scrubs multiple occurrences of the address:
error_message = "test message first occurrence: {} second occurrence: {}"
error_message = error_message.format(stream_to_address, stream_to_address)
expected_message = "test message first occurrence: {} <Address to stream id: {}>"
expected_message += " second occurrence: {} <Address to stream id: {}>"
expected_message = expected_message.format(
scrubbed_stream_address, stream.id, scrubbed_stream_address, stream.id
)
redacted_message = redact_email_address(error_message)
self.assertEqual(redacted_message, expected_message)
# Test with EMAIL_GATEWAY_EXTRA_PATTERN_HACK:
with self.settings(EMAIL_GATEWAY_EXTRA_PATTERN_HACK="@zulip.org"):
stream_to_address = stream_to_address.replace("@testserver", "@zulip.org")
scrubbed_stream_address = scrubbed_stream_address.replace("@testserver", "@zulip.org")
error_message = "test message {}"
error_message = error_message.format(stream_to_address)
expected_message = "test message {} <Address to stream id: {}>"
expected_message = expected_message.format(scrubbed_stream_address, stream.id)
redacted_message = redact_email_address(error_message)
self.assertEqual(redacted_message, expected_message)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,115 |
package com.GB.ChinaMobileMS.services.impl;
import java.util.List;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.stereotype.Service;
import com.GB.ChinaMobileMS.dao.AssetHousingMapper;
import com.GB.ChinaMobileMS.entity.AssetHousing;
import com.GB.ChinaMobileMS.services.interfaces.AssetHousingService;
@Service
public class AssetHousingServiceImpl implements AssetHousingService {
@Autowired
public AssetHousingMapper ashMapper;
public String addAssetHousing(AssetHousing ash) {
if(ash.getIsLoan()==0)
{
ash.setLoanTimeStart(null);
ash.setLoanTimeEnd(null);
ash.setLoanSource("无");
}
ashMapper.addAssetHousing(ash);
return "done";
}
public List<AssetHousing> queryAssetHousing() {
List<AssetHousing> listash = ashMapper.queryAssetHousing();
return listash;
}
public List<AssetHousing> queryAssetHousing2() {
List<AssetHousing> listash = ashMapper.queryAssetHousing2();
return listash;
}
public AssetHousing detailed(int assetInfoId) {
AssetHousing ash = ashMapper.detailed(assetInfoId);
return ash;
}
public List<AssetHousing> searchAssetHousing(String houses, String search) {
if (houses.equals("companyName")) {
List<AssetHousing> ash = ashMapper
.queryAssetHousingByCompanyName(search);
return ash;
} else if (houses.equals("buildingName")) {
List<AssetHousing> ash = ashMapper
.queryAssetHousingByBuildingName(search);
return ash;
} else if (houses.equals("location")) {
List<AssetHousing> ash = ashMapper
.queryAssetHousingBylocation(search);
return ash;
} else {
List<AssetHousing> ash = ashMapper
.queryAssetHousingByBuildingArea(search);
return ash;
}
}
@Override
public List<AssetHousing> getAssetHousingByCompany(int companyId) {
return ashMapper.queryAssetHousingByCompany(companyId);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,601 |
Procida (in der Antike Prochyta) ist eine Insel im Golf von Neapel. Sie zählt zu den Phlegräischen Inseln und gehört zur Metropolitanstadt Neapel in Kampanien. Procida ist zugleich der Name des Hauptortes und einzigen Ortes mit Einwohnern (Stand ). Das Eiland ist damit die am dichtesten besiedelte Mittelmeerinsel.
Berühmt ist Procida für die Karfreitagsprozession, zu der auch viele ehemaligen Bewohner auf die Insel zurückkehren. Der Tourismus spielt – im Gegensatz zu Capri und Ischia – nur eine untergeordnete Rolle. Von besonderer Bedeutung ist das Istituto Nautico, die älteste Seefahrerschule Europas. 1830 bis 1988 befand sich ein berüchtigtes Gefängnis im ehemaligen Palast auf der Terra Murata.
Geographie
Procida liegt zwischen Kap Miseno und der Insel Ischia. Sie ist etwa 4 km² groß. Die Küstenlinie ist etwa 16 km lang. Die höchste Erhebung ist Terra Murata (91 m). Zu Procida gehört das heute unbewohnte, mit der Hauptinsel über eine Brücke verbundene 0,32 km² große Inselchen Vivara. Es ist seit 1974 ein Teil des Naturschutzgebiets Area naturale marina protetta Regno di Nettuno und beherbergt viele seltene Vogel- und Pflanzenarten.
Geologie
Procida ist wie ihre Schwesterinsel Ischia vulkanischen Ursprungs. Die Insel entstand aus sieben vulkanischen Zentren, die sich vor 20.000 bis 40.000 Jahren gebildet haben und die allmählich zusammenwuchsen. Die sehr unregelmäßige Küstenlinie mit ihren hohen Felswänden und den langen schmalen Stränden von schwarzem Vulkansand ist das Ergebnis dieser vulkanischen Aktivität. Sechs Kraterreste, darunter das Hafenrund der Marina di Chiaolella und das der Corricella, befinden sich auf der Hauptinsel. Den siebten teilt sich die Insel mit dem zu Procida gehörenden halbmondförmigen Inselchen Vivara.
Klima
In der touristischen Saison (etwa Juni bis Anfang Oktober) betragen die Tages-Maximaltemperaturen zwischen 21 und 26 °C. Im Sommer gibt es pro Monat etwa vier Regentage, das Meer ist von Juni bis Oktober 21 °C warm oder wärmer.
Anreise und Verkehr
Die Insel ist nur per Schiff von Neapel, Pozzuoli oder Ischia aus erreichbar. Autofähren ("Traghetti") verkehren von allen drei genannten Orten, die schnelleren, aber teureren Tragflächenboote ("Aliscafi") von Neapel und Ischia. Tragflächenboote fahren ab einem bestimmten Seegang nicht mehr. Die Mitnahme von Fahrzeugen (Autos, Mopeds) ist nur für Einheimische gestattet.
Procida in Literatur und Film
Der Roman Arturos Insel (1957) von Elsa Morante spielt auf Procida.
Damiano Damiani verfilmte 1962 Insel der verbotenen Liebe (L'isola di Arturo) nach Morantes Roman auf Procida.
Der Roman Sehnsucht nach Procida von Ota Filip (1988)
Die Erzählung Graziella von Alphonse de Lamartine (1852)
Der Film Der Postmann (Il Postino, 1994) wurde auf Procida und Salina gedreht.
Der Film Procida. Die Insel, das Meer und der Tod, WDR/3sat 2008, Regie: Annette von Wangenheim
Der Film (einzelne Szenen) Der talentierte Mr. Ripley (Film) von Anthony Minghella (1999)
Die Netflix-Serie (einzelne Szenen) Generation 56k (2021).
Persönlichkeiten
Michele Autuoro (* 1966), römisch-katholischer Geistlicher, Weihbischof in Neapel
Literatur
Franz Krojer: Auf Procida!, Differenz-Verlag, München 2007 (PDF, 3 MB)
Tiziana Assante di Panzillo, Sebastiano Cultrera, Luigi Prudente: Procida. Arturos Insel. Neapel 2001. ISBN 88-7188-550-3
Belletristik
Elsa Morante: Arturos Insel, aus dem Italienischen von S. Hurni-Maehler, Wagenbach 2005, ISBN 3-8031-2514-6
Weblinks
http://www.portanapoli.com/Neapel/Procida/procida.html – Infos und Fotos zu Procida (deutsch)
http://www.procida.it/index1.html – englischsprachige Webseite zu Procida
Einzelnachweise
Insel (Europa)
Insel (Kampanien)
Insel (Tyrrhenisches Meer)
Ort in Kampanien
Weinbauort in Italien | {
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Q: Даны действительные числа x и y. Определить, лежит ли точка с координатами (x,y) внутри заштрихованной области Задачу нужно выполнить на Паскале.
A: (x*y >= 0) and (x*x + y*y <= 1)
A: var x, y: real;
begin
readln(x, y);
writeln(x * x + y * y <= 1);
end.
простая проверка попадет ли в эту окружность
дальше добавьте условие попадает ли в области
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 576 |
Q: Is possible to have multiple triggers for the same table? I'm learning about databases (MySQL on wampserver)
I have a trigger and I want to run this code on it but the compiler tells me I have a syntax error. (After insert on parent table)
INSERT INTO test_a (a1, a2)
Values(New.Value1, New.Value2)
INSERT INTO test_b (a1, a2)
Values(New.Value1, New.Value2)
INSERT INTO test_c (a1, a2)
Values(New.Value1, New.Value2)
The problem comes when I try to compile this, I get an error on syntax. It works only with one sentence. In localhost I can use multiple triggers with the same event. But on the school test page I can't do that, I get the message "This version of maria db doesnt yet support multiple triggers with the same event" when I try to do it.
What can I do ?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,167 |
Subliminal by Leonard Mlodinow
Read about 1 month ago
#neuroscience
#psychology
(Not a marketing book)
Our actions are hard to explain by pure intention and logic. Over the history, we didn't really understand the logic behind accurate guesses. Carl Jung and Sigmund Freud tried to explain the source of behavior is connected to subconscious. Today, Neuroscience enables the inspection of brain to interpret reactions more clearly. Neuroscience has been the quantum physics of psychology. It showed us the human mind is beyond what we thought.
Most of our actions are driven by unconscious mind that is evolved for survival.
Although it helps us to take shortcuts, our unconscious mind is also biased in many aspects.
We can tame our unconscious mind by understanding how it works, setting conscious goals, and acting mindfully.
Living a contradicting experience is a great way to eradicate bias.
Using Theory of Mind is a wonderful way to develop empathy and mindfulness.
Part 1: The Two-Tiered Brain
Most of the primitive creatures act "programmatically" to fulfill their basic instincts of survival and reproduction.
Humans (and mammals) have developed a part of brain that gives us the ability to make conscious decisions. Yet the pure logical conscious brain is a myth.
Our experiences shape the unconscious mind to direct our thinking towards certain ends. Mlodinow's mother for example, lost her mother at 16 to cancer, and her father, sister, and she were taken by the Nazis in Poland. She lost her sister and father, eventually escaped to the US to start a middle-low class life. Yet her traumatic early experiences made her think disastrously in such situations like Leonard skipping a routine phone call when he was studying away from home. His mom thought Leonard has died and his roommate didn't answer her calls to hide his death. In fact, Leonard was out for a date.
Freud and the subconscious
Sigmund Freud made great scientific progress to understand the brain connectivity and neural interactions.
Yet pursued in the clinical path and developed a theory for the subconscious based on his therapy. However, this method was inadequate since explaining the subconscious based on what patients tell him is scientifically unreliable (we will learn why in the upcoming chapters).
Freud suggested the subconscious is a defense mechanism that oppresses our desires to not take irrational actions.
Chapter 1: The New Unconscious
The new unconscious view the subconscious as a gift rather than a protection mechanism. Our subconscious helps us to direct our behavior based on the experiences and our interpretation of the materialistic world.
Some –hard to extract from what patients tell in therapy– examples of subconscious:
We want to feel important, this have a tendency to have a bias towards people who share similar traits. People tend to marry with others that have the same surname.
Popcorn-box combination experiment. Attendants decided how much to eat based on the taste of the popcorn and size of the boxes. Doubling the size of snack container increase consumption by 30-45 percent.
Fluency effect: we tend to like food, or find it more tasty based on the fonts used in the menu. It also shows the corporations with more readable names gained more interest during an IPO. So economy is not only about rational decisions based on self-interest.
The pepsi paradox: people prefer Coke, but Pepsi tastes better in a no-label experiment. Well, it's not a paradox. It's the perception of brand in our unconscious.
As a result, our unconscious minds are active, purposeful, and independent. They might be hidden, but their effects play a critical role in how the conscious minds perceive the world.
Chapter 2: Senses + Mind = Reality
TL;DR: Kant is right! There is reality that exists objectively, independent from us. Then there is another one build up by our senses and mind: the perceived reality.
The story of Charles Sanders Pierce, a pioneer in researching unconscious:
The stolen golden watch story (he guessed the burglar without no conscious evidence)
The weight perception experience (there is a threshold that we can sense in a set of weights perceived by our senses, and it's proportional)
He founded "the philosophical doctrine of pragmatism" and claimed that the philosophical ideas or theories should be viewed as instruments, not absolute truths.
An early metaphor for mental actions: There are two trains, unconscious and conscious which makes decisions. Now, it's viewed more like two railway systems with intersections.
Nevertheless, the unconscious part is more fundamental. It's developed far before than the conscious part. The unconscious takes up the 99% of the energy spent in brain. It's a bit funny, but playing chess vs watching television does not differ as drastically as running outside vs being a couch potato. Playing chess rather than watching TV increases the energy consumption only about 1%.
Processing visual data
One of the most important functions of the unconscious part is processing visual data. It occupies one-third of the brain capacity. It's developed to serve our survival instincts.
A patient with a damaged visual system that helps the conscious part can still distinguish images that trigger senses. He was able to differentiate happy and sad faces, where the results were not the same for geometric shapes. He was also able to walk without a cane, didn't step on to objects on the floor.
Faces play a special role in human behavior. Our facial expressions usually reflect how we perceive certain situations. That's why Helen of Troy said to have "a face that launched a thousand ships," not breasts.
"Blindsight" is the phenomena that claims the individuals can still respond to events without visually perceiving them in the conscious mind.
An artificial way to create blindsight is to show one eye a moving picture and the other one a static one that triggers an instinct, such as sexual desire.
We don't consciously perceive everything registered in our brain, so unconscious mind may notice something that conscious doesn't.
Another interesting concept that proves the wold we perceive is an artificial one is phonemic restoration. We fill in the blanks when something is missing to construct a meaning of what we perceive. It's done by the unconscious part and we totally believe. Sometimes they are biases towards the physical properties of the people we encounter such as race and ethnicity. We also match the concepts such as "orange" and "peel" to complete a sentence.
Chapter 3: Remembering and Forgetting
The case for memory distortion: false eyewitness identification, Jennifer Thompson, the victim who false identified her rapist during the line-up. She was relatively calm and trying to memorize the accuser during the incident. Yet she only remembered severe details to identify Ronald Cotton. But the actual suspect was actually Bobby Pole, another inmate Cotton met at the prison. The DNA test made the case accurate after 10 years.
James Dean and watergate. Dean was known as a man with a "tape recorder" memory. Yet his most recollections were false. They were either defend himself, or Raegan, who was actually tape recording everything.
Hugo Münsterberg and explorations of the new conscious
Munsterberg, another pioneering psychologist in the new conscious research, realized his memory distortion while filing a police report for a break-in to his house while they were away for a vacation. Most of his conclusions were based on what he heard from the policemen, and assumptions developed afterwards.
In fact, Munsterberg was very good in terms of memory. He gave over three thousand lectures without a single note. His studies on memory shown that the classical assumption of "our memories work like a rape recorder" was inaccurate.
He believed none of us can remember everything we are confronted. Instead, we remember certain blueprints, then fill the rest according to our belief systems and prior experience, sometimes to be self-serving. Additionally, we believe the memories we make up (confabulation).
Despite being a pioneer, one downside of Munsterberg was ignoring the unconscious mind. It's probably related with living in the same era as Freud, who where constructing a narrative that contradicts with Munsterberg's rational approach.
The story of the subconscious mind can be told in three words: there is none.
– Hugo Münsterberg
The man who remembered everything: Sheresevsky
Sheresevsky, a man lived in Russia, was able to remember everything. Yet he lacked interpretation. He didn't remember a person from her face. There were many faces that belonged to that person in his mind.
It shows that we evolved to trade perfect recall for the ability to handle and process information.
Two types of language structure
Surface: specific way the idea is suggested, with exact words and everything.
Deep: the gist of the idea.
We tend to make a gist out of our experiences based on the emotions. Then we fill up the missing parts when remembering them in the future. Sometimes we do that dramatically, and biased.
False memory experiments
Supporting to our power make ourselves believe in what we make up, there were a few good false memory placement experiments.
Hot air balloon ride: two of your childhood pics + one photoshopped.
Meeting with Bugs Bunny in Disneyland: he's a Warner Bros character.
After all, our memory is similar to sensors in the previous chapter. A big portion of it powered by the unconscious.
The lesson is to be humble, because our memories could be misplaced, and grateful both for the memories we can retain and the ability to not retain all of them.
Chapter 4: The Importance of Being Social
Social and emotional connections are beyond words, and can be understood through unconscious thought.
We value and amire kind, helpful behaviors. So being nice is it's own reward. It's also attracting other emotional beings.
Our tendency for social acceptance based on kindness starts to develop at early ages (~6 months). Babies that watch animations of "shapes with eyes" being rude (square brick tries to climb, triangle helps, circle pushes it down) or being neutral (circle is just a bystander) tend to reject playing with shapes that aren't helpful.
Another experiment made in 1950's in University of Minnesota. They gathered 2 x 30 female students. To the first group, they say there will be electroshocks it will hurt. To the second group, they said the electroshocks will just tickle. The group that expects painful experience waited in groups (66% - 33% in comparison) even though they had enough waiting rooms for every individual. So we seek help and unification with others when we are anxious.
Social pain is also physical
Tylenol (painkiller) experiment. People who took 2-tablets per day shown less affection from social problems compared to the people that took placebos. Lack of social connection constitutes a major risk factor for health.
Social intelligence and the "Theory of Mind"
Social intelligence is a result of our evolution. IT has been crucial to our survival. Human kind's ability to band together and organize preceded all the other creatures.
Social intelligence is not only corralated to IQ. It's influenced more by your desire and ability to understand what other people think and feel. It's called "Theory of Mind."
Most of our understanding in ToM comes from unconscious mind. There are levels of ToM.
What I think about what you think
What I think about what you think about what I think
What I think about what you think about what I think about what you think
Politicians and businessman are usually successful to think in 4th order.
It takes five steps to find two connected people by forwarding through connections. Same experiment done with both letters and email.
Mammals and social intelligence
Although we developed this complex brain and ToM (other mammals can do max 2nd order), there is also a big range of common behavior. Those behaviors are mostly on the unconscious side, and influenced by hormones.
Male Mammals -> reproductive success determined by competing with other males to mate with as many females as possible.
Female Mammals -> reproductive strategy is investing in production of relatively few offspring, and take care of them as they grow.
Oxytocin: love, trust. Vasopressin: commitment. Two hormones playing a key role in social interactions.
Social neuroscience
To investigate the effects of social connections, we have a new field called social neuroscience. Invention of fMRI contributed to many studies and made it clear that here's no way to understand human mind by focusing on rational or emotional extreme.
Cognitive psychology: mostly focuses on rational end.
Social psychology: mostly focused on emotional end.
Two + neuroscience => social neuroscience.
In the context of neuroscience, our brain consists of three main parts.
Reptilian Brain: basic survival - fight or flight.
Limbic System (Old Mammal Brain): unconscious social perception and reflective reptilian emotions.
Neocortex (New Mammal Brain): homo sapiens! Evolved 200.000 years ago, effective around 50.000 years. Positioned on the frontal lobe, it's what makes us human. Conscious thought, planning and orchestrating in accordance of our goals happen here.
Part 2: The Social Unconscious
Chapter 5: Reading People
We unconsciously signal our expectations by our body language: gestures, facial expressions, posture, tone of voice.
Animals do that too, and they can also learn and understand human's intentions through the body language. Dogs evolved to understand humans the best.
Vasco de Gama Rats
Participants communicated more kindly with the "Vasco de Gama" rats (that are told to have abilities in mazes). In fact, they didn't. Yet those rats performed better than the other group (of stupid rats).
Similar things applied for teacher's behaviors against a class with "talented" and "normal" students. Brilliant (to be told) ones increased their IQ even more.
Non-linguistic communication in primates
Direct stare means treat, smile means "I won't attack you", or "please don't attack me, you're the best!"
Based on researches to rate human facial expressions in cross-demographies, it seems like our catalog of facial expressions are an inherent part of our being.
Physical dominance: voice, body, carrying a gun :D
Social dominance: expensive (or non-expensive) clothing, cars, watches
Visual dominance ratio. We automatically adjust the amount of time we're looking onto another person's eyes as a function of our relative social dominance. Percentage of the time you look while you are speaking against while you are listening is the visual dominance ratio. People tend to look more while speaking and less while listening when they're dominant.
Chapter 6: Judging People by Their Covers
Cowbird females get attracted to males by their voice. Even when it's coming from a stereo.
Humans also behave similarly in certain situations when voice is involved. We tend to trust love and relationship advices from women more, since they are known as "keepers." We obey orders from male voices more. We are not very biased on neutral topics. (I doubt that, but cool.)
Voice tone (deep-high) also effects humans when they are choosing their partners. People with physical dominance (long muscular male with hairy chest) sometimes adjust their voice tone to settle the conditions in social interactions (aww). Or the non-dominant individual does the opposite, especially if the case is competitive.
First presidential debate on air: Nixon vs. JFK (60' Elections)
Nixon had a knee problem and he was in the hospital. Yet he wanted to attend to the debate. He was looking pale and unhealthy. People in the TV insisted Nixon to wear a make up, but he said he'd only do it if Kennedy also does. Kennedy refused, so he followed.
It's said to be a determining factor (even the next generations that are neutral found Kennedy more "votable" just by watching the recording) on the race. Nixon's vice-president candidate said "That motherfucker just costed us the elections."
Another evidence for the importance of the look: people in Nixon's party who listened the debate through the radio didn't think it was too bad, until they re-watched the video recording.
Many other researches made in the US shown that "competent" looking candidates have enormous advantage when there's no strong ideology involved.
Some random findings:
Fast speakers create more trust they give a sense of competence.
Touching to forearm triggers some empathic feelings. Waiters got more tips and guys get more numbers on street when they did that.
Leonard says we can't turn off the "judging by look" ability. It'll always be there, as the other findings in the book.
Chapter 7: Sorting People and Things
Over the history, our species used categorization to increase their chance of survival.
If a bear ate uncle Johnny, then another bear can eat you tomorrow.
Unconscious mind transforms input by using categorization to speed up the reaction time. We are also using categorization to learn and remember things. Even our ability to read is empowered by categorization, so it's useful in many ways. However, it's also making us biased towards certain things.
When we categorize, we polarize.
Born of the Stereotype
Raise of the multi-culture societies and free will brought many different perspectives. Politicians, media, and businessman needed to address them somehow.
In order to leverage the power of mass media and managing the society, we came up with the concept of stereotypes. Then we started to generalize the behaviors –mostly negative because we also have a negativity bias.
Shoplifter reporting experiment. People reported "poor looking" shoplifters more than the "good looking" ones, and they also used more enthusiastic language when reporting the poor ones.
How to not to be a racist
We can fight with our unconscious in parallel to our conscious goals and values to tame biases. Still, as it's been said in Everything is Fucked, the most effective way to change a belief is to have a contradictory experience. That's the number one reason why forming multi-cultural, diverse, and inclusive groups are important.
It also seems like we're making progress (on polarizing :D), considering the disapproval rate of supremacist comments made by prolific people like Che Guevara (said "negro is lazy") and Abraham Lincoln (implied "white-black never be equal").
Chapter 8: In-groups and Out-groups
Robbels Cave Camp Experiment
11 x 2 eleven year old boys were called into a three week camp. They're almost indifferent in terms of race, socioeconomic and physical attributes.
They spent a week without being aware of the other group, banded and even created a name/flag/song for their group.
In the second week, teachers made them aware of the second group. They instantly wanted to compete with them in sports. Losing side burnt the flag ot he other team. The team with the burnt flag invaded and messed up the other team's room. In summary, there was a strong "US-VERSUS-THEM" feeling in the air.
(Rest of the story coming at the end)
As this study also shows, humans are not famous with their hospitality against groups they see different from themselves.
Sometimes we even make sacrifices to belong to a group. Most common pattern is the financial sacrifice. Mac-users vs PC-users, country club members, "senior" people who are OK to get promotion rather than salary increase.
In-group bias
In-group/out-group mindset makes us think that our in-group members are a lot more sophisticated. We tend to prefer them for social and business dealings.
Our self-identification also effects the performance. A study in Harvard on asian-american women students. They got a difficult math test, but divided into three groups that filled different questionnaires before the test.
Students who reminded their asian inheritance (correlates to being good at math) did the best.
Students who are asked to answer random cable TV questions (control group) did OK.
Students who reminded that they are women (seen as bad at math) did the worst.
Us versus them, but for real...
Sometimes the excluding behavior comes before the benefits of the tribe. Henri Tajfel experiment: you have limited amount of points to distribute between two people. The difference between distribution effects the total points, and you don't have to distribute all.
In-group (5 points – max) vs. In-group (5 points – max)
Out-group (1 points – min) vs. Out-group (1 points – min)
In-group (4 points – not max) vs. Out-group (1 points – min)
People were OK to give less points to in-group for the sake of minimizing out-group outcome.
What it takes to establish a kinsnip?
There's no minimal requirement to establish a tribe spirit. We can band against any common enemy very quickly.
The key to reduce in-group/out-group rivalry is to make them work together against ANOTHER COMMON ENEMY :D, or a natural disaster.
The two groups of the Robbels Cave Camp were united against solving the water outage and broken meal truck since it was a problem for both.
Mladinow says the same thing happened in 9.11, when he was living very close to the Empire States building.
It's sad but true.
At least there is a way.
Chapter 9: Feelings
William James, the man who lived the lifelong learner's dream... He studied various fields in very interesting universities, eventually settled down to get medical degree from Harvard. Yet he worked on psychology, as a student of Wundt.
His masterpiece (which he disowned at the time being and left the field for philosophy) The Principles of Psychology (took him 12 years to write) is still one of the most admired works in psychology.
James proposed the idea that psychology follow physiology. For example, we feel bad because we cry, not the other way around.
Emotions, in today's neo-Jamesian view, are like perceptions and memories. A big portion of the data comes from the unconscious mind, then the conscious part completes the rest with our preexisting beliefs and expectations. Adding the of interpretation of the current circumstances, we construct an emotion.
When you know where it's coming from...
Adrenaline experiments: subjects who are informed about the potential increase of their sexual desires where less interested in erotic films. (They were given injections then ran in treadmill)
If we somehow "know" the sources of our impulses, we are able to make more "mindful" decisions.
When we have no idea what led us to certain actions, we make up a narrative to justify our reasoning. The lack of agreement between the thinking brain and the feeling brain is the real issue.
Hiring experiment: subjects are evaluating a candidate based on academic success, a coffee-spilling incident, how attractive she is, and their possibility to meet her. Outside attendants are also try to guess why a subject would decide to hire the applicant.
Both the subjects and observers think that academic success was the biggest factor, but it had less effect than the other social norms.
As psychological theory suggested, the subjects had shown no greater interest into their reasoning than the outsiders had.
So we don't usually see ourselves no other than how the others see us.
Well, that's a bit bad. Perhaps the reason of the imposter syndrome.
To minimize the gap, we can broaden our perspectives and invest effort to know ourselves more deeply.
Yet sometimes it's futile to make a meaning out of everything.
If your mind's natural view of the world is skewed, it's skewed for a reason.
Chapter 10: Self
When we face with treats to feeling good about ourselves, our tendency to view reality through a distorting lens grows proportionally.
Many people think they are above-average in their professions.
Ironically, we tend to recognize that inflated self-assessment and overconfidence can be a problem–but only in others.
That's right, we even overestimate our ability to resist overestimating our abilities.
Two ways to get truth
Scientist's Way: gather evidence, look for regularities, form theories explaining their observations, and test them.
Lawyer's Way: being with a conclusion they want to convince others of and then seek evidence that supports it, while also attempting to discredit evidence that doesn't.
The human mind is evolved to be the both.
Motivated reasoning helps us to believe in our own goodness and competence, to feel in control, and to generally see ourselves in an overly positive light.
Our brains show different physical behaviors when the subject is "us".
"You can't connect the dots looking forward; you can only connect them looking backwards. So you have to trust that the dots will somehow connect in your future.
altay@aydemir.io | {
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} | 1,877 |
Media Matter
Magazine October 15, 2012, Issue
By James Lileks
About James Lileks
Follow James Lileks on Twitter
Whenever talking about the YouTube video on which the riots were blamed, it's important to note that it's bad. Lousy acting, cheap F/X, costumes from the Halloween store. But what if The [title removed for fear 10,000 people will set themselves on fire somewhere] had been a really good movie?
The critics rave! "Breathtaking, revelatory, audacious, with the director's trademark mixture of lilting wit and raucous smut." Sacha Baron Cohen's Oscar-worthy turn as a transgendered Zoroastrian who invents the Islamic holy texts to win the love of a gullible young man (Tom Cruise) is praised by all. Sixteen months after its release, a mob — on 9/11, by one of those "gosh, you can't make this stuff up" coincidences — attacks the U.S. embassy in Tunis with such fury it manages to burn down the entire country, putting in doubt longstanding scientific theories about the flammability of sand.
Would that be better, somehow? Because they were supposedly incensed by a quality project?
The constant reassurance that the movie is Bad and its director a Bad person ought to be irrelevant, like saying that someone who drew an offensive cartoon of Mohammed hadn't quite mastered the art of shading and perspective. But apparently it's important to disparage the quality and character of a work of art before you tut-tut about a lethal response. Might as well just run photos of burning embassies on the front page with the headline "everyone's a critic."
The First Amendment doesn't have a schlock-exemption clause. It doesn't say your abrasive utterances must be set forth in gooder grammar. It is silent on the issue of penmanship.
Now, this is the point where people usually roll their eyes and say, "Criticism isn't denying anyone's free speech. It's not like the government has to get involved." But the government did get involved. The director was hauled in for discussion of his parole violation late one evening. Unless you believe that California cops sit around bored for hours until a big red light goes off and they slide down the poles because there was a parole violation across town six months ago, you might suspect that pressure was applied. That would seem to be the government at work. And getting overtime pay, probably.
Then there's the small matter of the chairman of the joint chiefs of staff phoning up a private citizen known for his Fahrenheit 451 approach to non-Christian holy books and telling him to mind his ways. Totally justified. You can't shout "Fire" in a crowded YouTube comment thread, or something. Who knows what pressure the top brass put on the citizen? "I don't mean to drone on and on, but we know where you live." That would seem to be the government at work.
#page#Now, this is the point where conservatives carp about the media's double standard and say, "What if a Bush spokesman implied that Bill Maher wasn't funny or said people should handle their Dixie Chicks CDs in a careless manner, encouraging scratches? They'd cry fascism!" Yes. They're excitable that way. If Dick Cheney officiated at a lesbian wedding, he'd be a homophobe if he didn't kiss the father of the bride. It's a given.
We're told the media don't matter anymore, because there's Twitter. But for the muddle-pated middle who depend on news crawls and headlines to tell them what the popular kids think, the media is a Jell-O mold that shapes their vague beliefs, tells them who's Up, who's Cool. The media skew the message in the proper direction, because they're good smart folk who know a Romney term would mean a Hobbesian society where the 1 percent get rich selling pre-rusted coat hangers to back-alley abortionists, and gay teachers'-union stewards are sent to work fracking gas while Koch-paid overseers crack the whip, and Friskies markets "Senior Blend" cat food, as they did during the Reagan years. Also, war somewhere.
To prevent these real and present dangers, reality must be adjusted. When the president shows up on the harpy-holler hootenanny The View and says he's there as "eye candy, " he's not a sexist pig who thinks women want to swoon over cool dudes, he's just speaking Truth to Behar. When the president met with a CIA Pakistan expert who was wearing "stiletto heels," as The Daily Beast described her, and he said "with clear amusement, 'You don't look like a Pakistan expert,'" it didn't mean he was demonstrating the habitual sexualization that keeps women from achieving equality in the workplace. It meant he's a man's man with normal appetites, aside from that whole dog-it's-what's-for-dinner thing.
I mean, you want some Mormon creep in there who wouldn't even look at her legs?
Speaking of whom: If Mitt Romney makes a comment about our embassy being overrun, and suggests the administration's motto is Strength through Cower, he's hammered. Politics stops at the edge of the water on which Obama walks.
Isn't that odd? No one deplored the quality of his remark and then noted he certainly had the right to make it. Well, if there are riots in other lands, perhaps the State Department can cut an ad for Pakistani TV, explaining that Romney was wrong to use a gerund the way he did, and they don't stand behind his use of the split infinitive, but in America we're free to criticize our leaders.
That would calm the rioting, if they hated us for our grammar. But they no more hate us for our First Amendment than they hate us for our bacon-flavored vodka. They just hate us. Everything else is an entry in the "local customs" section of an embassy orientation guide.
– Mr. Lileks blogs at www.lileks.com.
James Lileks — Mr. Lileks blogs at www.lileks.com. @lileks
Estonian Economics
Tallinn, Estonia – Sitting shirt-sleeved and without, sadly, his trademark bow tie, in his official residence here in the Estonian capital, this Baltic nation's Swedish-born, New Jersey–raised president, Toomas Hendrik Ilves, ...
The Rapper Barons
By Daniel Foster
I feel like a black Republican, money I got comin' in. – Jay-Z, "Black Republican," 2006 I'm a Republican voting for Mitt Romney, you lazy b****es is f***ing up the economy. – Nicki ...
Who Are the 47 Percent?
By Reihan Salam
A great deal of ink has already been spilled over Mitt Romney's off-the-record remarks at a fundraiser concerning the 47 percent of Americans who are dependent on government, who do ...
Every four years, we have presidential debates, and it's that time again. How about a glance back, at a handful of things from 2008? Barack Obama was very, very fluent ...
Fatwa against Free Speech
By Nina Shea
The cascading crisis involving derogatory depictions of Islam's prophet, Mohammed, by amateur American filmmakers and French satirists has reinvigorated a 20-year-old demand from the Muslim world for a Western crackdown ...
The Ayatollahs' Agency
By John R. Bolton
With the world's attention diverted yet again, this time by the collapse of the myth of the Arab Spring as a democratic awakening, Iran's nuclear program powers ahead. Again, the ...
Losing Iraq
By Frederick W. Kagan & Kimberly Kagan
President Obama announced the "end of America's war in Iraq" on December 14, 2011, with the words, "We're leaving behind a sovereign, stable, and self-reliant Iraq, with a representative government ...
A Million Steps
By Bing West
Helmand Province, Afghanistan – In early 2011, National Review published "With the Warriors," my description of the savage struggle to control Sangin District in the southern part of this province. More ...
Sharia on the Nile
By Andrew C. McCarthy
Just before the "Arab Spring" dominos started falling in Tunis, Mohammed Badi, "supreme guide" of the global Muslim Brotherhood, called for violent jihad against the United States. Yes, yes, we know ...
Tax Rates and Economic Growth
By Arpit Gupta
With President Obama reluctant to tout Obamacare or the 2009 fiscal stimulus, tax increases on the rich have at times seemed like the only idea he is willing to defend. ...
Escape from Utopia
By Ramesh Ponnuru
Utopian rhetoric is so commonplace in our political life that we scarcely question or even notice it. Part of Charles Kesler's achievement in I Am the Change is to help ...
By Kelly Jane Torrance
Many critics count it lucky for literature that T. S. Eliot abandoned a career in philosophy. But it was at least as fortunate that he studied the subject in the ...
Dickens at 200
By M. D. Aeschliman
Dickens was born in 1812, and there are celebrations and commemorative activities taking place in this bicentennial year all over the English-speaking world and beyond it. Along with the works ...
Two Kings, a City, and a Country
By Michael Potemra
The phrase "Memphis: Cradle of Civilization" is likely to evoke, if anything, images of pharaohs and dog-headed gods. But a recent visit to our own Memphis, the one in Tennessee, ...
Up against It
Nicholas Jarecki's Arbitrage is a movie about serious things: corporate fraud and police corruption, adultery and manslaughter, race and class, the ways that husbands betray wives and fathers betray children. ...
The Seasons Turn
One thing that all the seasons have in common is that it is impossible, in the midst of any one of them, to imagine things any other way. When the ...
Obama and the Founders In "Obama's Truth" (October 1), Charles R. Kesler does a remarkable job of sorting through some of the muddled thinking in Barack Obama's The Audacity of Hope. ...
‐ Obama says you can't change Washington from the inside — and he wants four more years to finish the job. ‐ Most of the polls show Mitt Romney behind, and ...
Athwart
Whenever talking about the YouTube video on which the riots were blamed, it's important to note that it's bad. Lousy acting, cheap F/X, costumes from the Halloween store. But what ...
Pharmaceutical Aids for Election Season
Slantrusose™ Along with Timesitrac™, Slantrusose™ can be used for symptoms of dizziness and disorientation brought upon by overexposure to the New York Times and other outlets of the liberal media. Timesitrac™ ...
By Sarah Ruden
FIVE NIGHTMARES When I am naked in the dock, I get His cloak and coat. The clock Sweeps the exam away from me, When I have lost the room, but He Teaches me where I stand ...
Let Them Go Hungry
I dislike first ladies — as a concept, I mean, not as dinner dates. I think of the first lady as an individual who happens to be married to the ... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,689 |
Q: How to check if a child component's root is a given component Let's say we have a ParentComponent that must have all its children being a ChildComponent.
I would make my ParentComponent check its children types:
import { FunctionComponent, isValidElement } from "react"
import { ChildComponent } from "./ChildComponent"
const ParentComponent: FunctionComponent = ({ children }) => {
const childrenToDisplay = Children.toArray(children).filter(child => {
return isValidElement(child) && child.type === ChildComponent
})
return <div>{ childrenToDisplay }</div>
}
Which would do the job.
Now, I want to be able to create a component that is using ChildComponent:
import { FunctionComponent } from "react"
import { ChildComponent } from "./ChildComponent"
const MyCustomChildComponent: FunctionComponent = () => {
return <ChildComponent>...</ChildComponent>
}
How should I modify ParentComponent to make it display MyCustomChildComponent ?
In other words, how can I, in ParentComponent, check that a child is either a ChildComponent or a component that has ChildComponent as root component ?
Thanks !
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 248 |
I have been blessed with a naturally healing and therapeutic intuition. Whether it's a broken heart, inability to manifest your heart's desire, fear of something that feels horrible, incessant anxiety, issues of trust, dis-ease of the body and of the mind, depression, resistance, and compulsive behaviors. I open the doorways that let the good stuff in and the bad stuff out. I have nurtured my healing gifts with formal education, a rigorous course of study and lots and lots of practice. I have a vast client base and my work is greatly appreciated. Personally, I love my work and am overjoyed to offer my services to you. You are invited to have your services done in-person if I am in your town, or we can arrange a private retreat if you want to travel to wherever I am in the world. We can also effectively do our sessions via Skype.
Divine Monadic Blueprint Integration and Healing: Unlock the power of your Divine Monadic Blueprint. A monad is "the ultimate, indivisible unit of matter… The monad has body without bulk, and mind without manifestation–containing all the powers and possibilities." In this powerful healing session, we begin with a pure intention and work with your energetic matrix to download your Divine Blueprint, the purest, most authentic expression of your most powerful self. The Integration process is multi-fold. We work together to develop a healthy intention for you and then I coach you through removing the obstacles to its manifestation. I then take you on a journey to open the portals that allow your intention to take hold in your experience. Expect deep relaxation and transformation. This is a powerful healing session and I recommend scheduling it out a couple of days in advance. There's a little homefun we both have to do to prepare for this session. When you schedule, you will be sent all the pertinent information. This session can be done in-person or over Skype. If you have questions, you can call or write me. We do need a minimum of two hours for this session.
Reiki Healing: A Japanese bodywork technique that promotes the body's own healing processes and a state of deep relaxation. With this traditional healing modality, I let my intuition follow your Spirit Guides as they dictate the movement, tempo and flow of this sacred work. Reiki is the channeling of Universal Life Force Energy with conscious awareness. Like the exquisite delicacy of Japanese culture, Reiki is subtle and powerful. Deep stillness, peace and calm can be experienced. My formal training comes from the Mikao Usui lineage. This treatment is easily received remotely or in-person. Session lasts up to 60 minutes.
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Crystal Healing: Using the power and gifts within crystals, I will help you to clear negative energies and emotional or mental patterns and re-establish positive energy to help you change behaviors that are not conducive to your growth and well-being. If you are a gem stone collector, you can bring them to your session and I can help you activate them and align your frequency with them. Also, I can help you choose stones that will be appropriate for your intention. I wasn't given the name Crystal Lynn for nothin'! Session lasts up to 60 minutes.
Aura Cleansing: I use my clairvoyance to identify the parts of your aura that is yours, and I identify the parts that are not. I use the technique of channeling extraordinarily high vibrational energy down into the aura to dislodge and dissipate the energies that do not belong to you and that are inhibiting you from fully feeling like your own expansive true self. Session lasts up to 60 minutes.
Magic Potions and Elixirs: What do you need most? Relaxation? An energy boost? Care for a broken heart? I am delighted to create an exquisite blend of special essential oils that will nurture your being and uplift your soul. A thoughtful blend of organic aromatic essential oils will affect your mood and bring your body, mind and spirit into harmony. This consultation requires you to complete an intake form before the session, so be sure you complete it and email it before our session. You will receive it after you schedule. This session lasts up to 30 minutes.
Japa Mala Prayer Beads: I use high quality gem stones in my mala. Mala are beads used when reciting Buddhist and Hindu mantra. During our consultation, you share with me your needs and desires and I choose the beads and design a powerful rosary and help you choose a mantra that resonates with your essence. The price of the consultation does not include your product, but will be applied to your mala if you purchase it within one week of our consultation.
Bodywork and Massage: All of my massage services offer hot stones, aromatherapy, reflexology, stretching, myofascial release, pressure point therapy, lymphatic drainage and guided meditation as you desire. You may choose from a variety of massage styles including Swedish, Lymphatic Drainage, Deep Tissue, Shiatsu, Prenatal, Oncology and Myofascial Release. When you come in for your appointment, we will discuss your needs and create a special experience just for you. I am currently in Zagreb, Croatia. Feel free to email me if you would like to book a session. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,284 |
{"url":"https:\/\/mathoverflow.net\/questions\/142390\/extending-group-actions-over-a-codimension-two-subset","text":"# Extending group actions over a codimension two subset\n\nSuppose $X$ is a smooth projective variety and $U$ is an open subset whose complement is of codimension two. If a finite group $G$ acts on $U$ does it always extend to $X$ ?\n\nNo, it does not always extend. For instance, let $X\\subset \\mathbb{A}^4\\times \\mathbb{P}^1$ be the closed subset of points $((x_0,x_1,x_2,x_3),[Y_0,Y_1])\\in \\mathbb{A}^4\\times \\mathbb{P}^1$ such that $$x_0x_3-x_1x_2 = x_1Y_0-x_0Y_1=x_3Y_0-x_2Y_1=0.$$ This is smooth of dimension 3 (it is one of the two small resolutions of a threefold $A_1$ singularity). Consider the open subset $U$ where $(x_0,x_1,x_2,x_3)$ is not $(0,0,0,0)$. The projection $$\\text{pr}_{\\mathbb{A}^4}:U\\to \\mathbb{A}^4\\setminus\\{(0,0,0,0)\\},$$ defines an isomorphism of $U$ with the (relatively) closed subset $V$ defined by $x_0x_3-x_1x_2=0$. In particular, the induced map $$\\text{pr}_{\\mathbb{P}^1}\\circ \\text{pr}_{\\mathbb{A}^4}^{-1}:V\\to U \\to \\mathbb{P}^1,$$ is the unique morphism such that both $x_1Y_0-x_0Y_1=0$ and $x_3Y_0-x_2Y_1=0$ hold.\nThe complement of $U$ has codimension $2$. Now let $G=\\mathbb{Z}\/2\\mathbb{Z}$ act on $V$ by $(x_0,x_1,x_2,x_3) \\mapsto (x_0,x_2,x_1,x_3)$. Via the isomorphism $U\\to V$, this induces an action of $G$ on $U$. There is no way to extend this to an action on all of $X$. Essentially this is because there is no action of $G$ on $\\mathbb{P}^1$ such that $\\text{pr}_{\\mathbb{P}^1}\\circ \\text{pr}_{\\mathbb{A}^4}^{-1}$ is $G$-equivariant.\n\u2022 Nice! Let me rephrase polyhedrally (as that's how I roll). On the cone over the $ABCD$ square (labeled clockwise), we have a symmetry switching $A\\leftrightarrow C$. Now stretch the apex of this pyramid out to an interval (codim 2 in the whole), making the pyramid into a ziggurat. Bye-bye symmetry. \u2013\u00a0Allen Knutson Sep 17 '13 at 16:38","date":"2019-07-22 06:29:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.966559112071991, \"perplexity\": 99.36065311434498}, \"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-2019-30\/segments\/1563195527531.84\/warc\/CC-MAIN-20190722051628-20190722073628-00060.warc.gz\"}"} | null | null |
{"url":"https:\/\/tex.stackexchange.com\/questions\/202740\/how-to-keep-increasing-counters-for-content-that-is-optionally-hidden","text":"# How to keep increasing counters for content that is (optionally) hidden?\n\nI have a framework for creating exercises, where the content of the solution to each subproblem is fed into a \\solution-macro, which only displays its content if a global boolean is set to true.\n\nWhat I'm interested in is having unique equation\/figure\/table\/listings numbers for both the version with and without the solution. In particular, I need to take into account if the counters are increased in the solution. The intended benefit of this is that equations numbers (and other things) that might be referred to by the students in their solutions don't change their number once the solution is published later.\n\nMy first attempt to do this was to try to typeset everything into a box that is not printed (e.g. by \\sbox or by the lrbox-environment) - in the hope that this would actually increase the counters - but I already run into trouble with equation environments.\n\nBelow is an MWE, where Euler's formula should have equation number (3) for both settings of the toggle with_solution. For the solution-less version to work, the (content of the) third argument of \\iftoggle in \\solution has to be commented out.\n\nEdit: One solution to circumvent the problem would be to subordinate all relevant counters to subproblem, because then the occurrences in the formulation are the first and therefore necessarily unique (as the formulation appears in both versions - with and without the solution). However, this is not an interesting solution for me, because in actuality (vs. the cut-down MWE), my equations already look like (5.3.2) for eq. 2 of prob. 3 of exercise 5, while the subproblem look like \"5.3a)\", \"5.3b)\", etc. (making a consistent notation - i.e. (5.3a.2) - unattractive to me). Subordinating the counters to problem doesn't help, because between the different formulations for each subproblem, the solutions are able to increment different counters that are not reset by calling \\subproblem.\n\n\\documentclass{article}\n\n\\usepackage{etoolbox}\n\\usepackage{amsmath}\n\\usepackage{xparse}\n\n\\newtoggle{with_solution}\n\\newsavebox\\tempbox\n\\NewDocumentCommand{\\solution}{+m}{\n\\iftoggle{with_solution}{\n\\par\\medskip\\noindent\\textbf{Solution:} #1\n}\n{\n% \\sbox\\tempbox{#1}\n\\begin{lrbox}\\tempbox\n#1\n\\end{lrbox}\n}\n}\n\n% \\problem and \\subproblem cut to bare bones for simplicity\n\\newcommand{\\problem}[1]{\\section{#1}}\n\\newcounter{subproblem}\n\n\\toggletrue{with_solution} % Set to true to include solution\n%\\togglefalse{with_solution} % Set to false to exclude solution\n\n\\begin{document}\n\n\\problem{A Problem}\n\n\\subproblem Problem Formulation\n\\solution{Solution}\n\n\\subproblem Problem Formulation with \\eqref{eq:prb}\n$$\\label{eq:prb} a=b$$\n\\solution{Solution with \\eqref{eq:sol}\n\n$$\\label{eq:sol} x=y$$\n}\n\n\\subproblem Problem Formulation with \\eqref{eq:prb2}, having the same number both with or without typesetting the solutions.\n\n$$\\label{eq:prb2} \\mathrm{e}^{2\\pi\\mathrm{i}}=1$$\n\\solution{The \\texttt{$\\backslash$solution}-macro should be able to handle \\texttt{$\\backslash$par}'s, figures, tables, listings (and also increase their counters accordingly, even when not typeset)...}\n\n\\end{document}\n\n\u2022 \\setbox\\tempbox=\\vbox{#1} is a better candidate than the lrbox environment. \u2013\u00a0egreg Sep 23 '14 at 15:15\n\u2022 @egreg: This works for equations - I'll check it for the other things momentarily. Do you want to post it as an answer, or do you think the question should be deleted? Even though I tried to go through tex.stackexchange.com\/a\/83936\/42225 before I asked the question, I obviously don't understand the differences between the \\hbox of \\sbox and \\vboxenough... \u2013\u00a0Axel Sep 23 '14 at 15:19\n\u2022 I'm not following you. Your example is too small for describing the problem. I'll remove all my comment. \u2013\u00a0egreg Sep 23 '14 at 15:57\n\u2022 @egreg: ok, I'll delete my comments and try to clarify the question. \u2013\u00a0Axel Sep 23 '14 at 15:59\n\u2022 @egreg: while trying to expand the question to make it clearer, I found that your solution is now working in all cases (don't know why it didn't work before). I therefore consider the question answered (but I edited it a little nevertheless to address your points). Thanks again for the tip! \u2013\u00a0Axel Sep 23 '14 at 17:10\n\nAnother way to go\n\n\\NewDocumentCommand{\\solution}{+m}{\n\\iftoggle{with_solution}\n{\\par\\medskip\\noindent\\textbf{Solution:} #1\\par}\n%{\\setbox\\tempbox=\\vbox{#1}}} % this line comes from egreg's comment\n% The following line is my alternative\n{\\par\\vbox to0cm{\\vspace*{\\paperheight}\\medskip\\noindent\\textbf{Solution:} #1\\vspace*{-\\paperheight}}\\par}}\n\n\nNotice that commands like \\addtocontents are executed during output routine. Therefore in my answer the \\listoffigure contains all figures, no matter they are shown or not. (In fact they are shown...) While egreg's answer produces \\listofshownfigure.\n\n## For Spacing\n\nDealing with negative spaces is not fun. I would like to put it aside. Experiments show that \\par\\marginpar{}\\par behaves as \\par.\n\n\\NewDocumentCommand{\\solution}{+m}{\n\\iftoggle{with_solution}\n{\\par\\medskip\\noindent\\textbf{Solution:} #1\\par}\n%{\\setbox\\tempbox=\\vbox{#1}}} % this line comes from egreg's comment\n% The following line is my alternative\n{\\par\\marginpar{\\moveright\\paperwidth\\vbox to0cm{\\medskip\\noindent\\textbf{Solution:} #1}}\\par}}\n\n\u2022 I like this version even better, because even though I can't think of a particular use for this right now, it allows references to content within the solution to be resolved (in addition to your remark about \\listoffigures). I'll accept this answer. \u2013\u00a0Axel Sep 24 '14 at 7:07\n\u2022 One more question, I'd like vertical spacing between the formulations to be the same as if the solution actually weren't there (i.e. {} as the third argument of \\iftoggle). But with the invisible content of \\vbox a new line is started and so the spacing increases. For \\solution's directly following text, it works if there are no \\par's between the formulation and the solution, resp. it seems substracting \\baselineskip is almost correct (aside from the glue I guess) when \\par's are present, but after equations etc. both these attempts don't produce the right spacing. \u2013\u00a0Axel Sep 24 '14 at 7:39\n\u2022 I just saw your edit, thanks! Unfortunately, the result is still not satisfactory after equations. I tried searching some more, and found that surrounding the \\vbox (i.e. your first solution) with \\@bsphack-\\@esphack (see tex.ac.uk\/ctan\/macros\/latex\/contrib\/sphack\/sphack-doc.pdf) works as expected for text (edit: sadly, only if there's no \\par in between), but also doesn't work well after equations. But it feels like a step closer, because the \\@bsphack-\\@esphack combination uses the actual skips and penalties that were present before. \u2013\u00a0Axel Sep 24 '14 at 18:40","date":"2020-05-31 17:48:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 3, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8101252317428589, \"perplexity\": 1259.7464656423174}, \"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-2020-24\/segments\/1590347413551.52\/warc\/CC-MAIN-20200531151414-20200531181414-00428.warc.gz\"}"} | null | null |
Q: Python Построчное чтение файла с выборкой по регулярным выражениям в цикле Добрый день. Есть txt файл со строками следующего вида:
96.246.236.151:3389;greenlightexp\admin;P@ssw0rd (greentech;Bazzz628) | Country: United States | State: New York | City: Long Island City | ZIP: 11101 | ISP: Verizon FiOS |
Таких строк там много, мне необходимо разобрать строку на составляющие: IP, Port, Username, Password
Пользуюсь либой regex на питоне 2.7 и задаю вот такой паттерн
pattern1 = r'(\d{1,3}\.\d{1,3}\.\d{1,3}\.\d{1,3}):(\d{4});.*?\\(.+?);(.+?)\s'
ip, port, login, passwd = re.findall(pattern1, s)
Но я не могу понять как мне делать это в цикле:
with open("IPs.txt","rt") as fo:
for line in fo.readlines():
print line
Вообще задача в том, чтобы пройтись по файлу, получить из него нужные данные и в том же виде столбиком запринтить это все в другой txt файл в уже требуемом виде. Заранее спасибо.
A: Если я правильно понял, в чём отличие от прошлого вопроса, то ответ может выглядеть так:
import re
pattern = r'(\d{1,3}\.\d{1,3}\.\d{1,3}\.\d{1,3}):(\d{4});.*?\\(.+?);(.+?)\s'
regexpr = re.compile(pattern)
with open('IPs.txt') as in_file:
with open('result.txt', 'w') as out_file:
for line in in_file:
try:
ip, port, login, passwd = regexpr.match(line).groups()
except AttributeError:
print('Error: ' + line)
out_file.write('ip={}, port={}, login={}, passwd={}\n'.format(
ip, port, login, passwd))
UPD:
Исправил опечатку. Проблема была в том, что метод finditer возвращает генератор результатов проверки регулярного выражения. Выражение
ip, port, login, passwd = regexpr.finditer(line)
пытается получить четыре совпадения всего паттерна в строке, когда такое совпадение только одно. В итоге и возникает указанная вами ошибка.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,030 |
Q: How to set group (like that for GtkRadioButton) for GtkToggleButton in GTK+? How to set group (like that for GtkRadioButton) for GtkToggleButton in GTK+ ? What is the properties i can use for this aim. I want to check buttons, which set the drawing primitive (rectangle, ellipse, etc)
A: If I understand your question correctly, you want to have a group of buttons where only one can be turned on at a time, but looking like regular buttons instead of radio buttons.
Use radio buttons, and add them to a group as usual, but set the draw-indicator property to FALSE. They will be drawn like regular buttons.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,411 |
import React from 'react';
import ReactDOM from 'react-dom';
import moment from 'moment'
import Bootstrap from 'react-bootstrap';
import Jumbotron from 'react-bootstrap/lib/Jumbotron';
var Button = require('react-bootstrap').Button;
import Panel from 'react-bootstrap/lib/Panel'
import Input from 'react-bootstrap/lib/Input'
import ButtonInput from 'react-bootstrap/lib/ButtonInput'
import Label from 'react-bootstrap/lib/Label'
class LunchApp extends React.Component {
render() {
var now = new Date();
var formattedDate = moment(now).format('MMMM Do YYYY');
return (
<div>
<Panel>
<h2>Options for lunch for {formattedDate}</h2>
<LunchOptionsPanel lunchData={this.props.lunchChoices}> </LunchOptionsPanel>
</Panel>
</div>
);
}
}
class LunchOptionsPanel extends React.Component {
render() {
let lunchOptions = this.props.lunchData.map(function(c,i) {
return <h3 key={i}><Label>{c}</Label></h3>
});
return (
<div>
<Panel header="Please select one" bsStyle="info">
{lunchOptions}
</Panel>
</div>
);
}
}
var lunchChoices = ['Chicken', 'Fish', 'Vegetarian'];
ReactDOM.render(
<LunchApp lunchChoices={lunchChoices}/>,
document.getElementById('root')
);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,075 |
Kadia is a female name. In English the meaning of the name Kadia is 'rhyming, or 'pure'. In English language it is also a variant spelling of Cady, meaning a rhythmic flow of sounds.
Kadia as a first name is said to be found 203 times in 16 countries, of which maximum usage of it was in one region of France The gender of first name Kadia was found to be 100% feminine and 0% masculine.
Kadia as surname is used at least 349 times in at least 13 countries.
References
Given names
Feminine given names | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,013 |
{"url":"https:\/\/www.physicsforums.com\/threads\/adding-trigonomic-functions.88241\/","text":"1. Sep 9, 2005\n\n### erok81\n\nI am stuck with my trig homework. My teacher isn't the best at teaching this stuff and the power point slides from my school stop before they get to this. Also in the answer book (and text) it doesn't really explain this either.\n\nThe problem is:\n\n$$sin(-\\pi)+cos(5\\pi)$$\n\nI have no idea what to do next. I know how the radian measure works but $$\\pi$$ and $$5\\pi$$ don't mean anything (to me) in regular degree's.\n\nIf anyone could give me a quick explanation on how to do these I'd really appreciate it.\n\n2. Sep 9, 2005\n\n### whozum\n\nYou don't have to go to degrees. If you imagine the unit circle, [itex] \\pi \\ and \\ -\\pi [\/tex] are coincidental points (180 deg), because you are just adding revolutions and returning to the same point. [itex]\\sin(\\pi) = \\sin(-\\pi) = 0 [\/tex]. [itex] 5\\pi [\/tex] is just [itex] \\pi + 4pi [\/tex], or just [itex]\\pi[\/tex] with two added revolutions. [itex]\\cos(5\\pi) = \\cos(\\pi) = -1 [\/tex].\n\nLast edited: Sep 9, 2005\n3. Sep 9, 2005\n\n### teclo\n\nconvert them if you have trouble with radians. pi would be 180 degress, so what would negative pi be? it just means that you're going counter clockwise rather than clockwise. as far as five times pi, one complete revolution is two pi, in that case two complete revolutions would be 4 pi -- so, where would you be for 5 pi?\n\n4. Sep 10, 2005\n\n### erok81\n\nFor some reason I always get hung up on these easy ones. I tend to make them a lot harder than they really are. The more complex ones I understand right away, however that works. :rofl:\n\nI hate to sound stupid, but I still don't get it.\n\nThe $$cos(5\\pi)$$ I understand how that goes around 2.5 times. I get how the $$sin(-\\pi)$$ goes half way backwards.\n\nFor some stupid reason I don't get how $$sin(-\\pi)$$ equals zero, or how the $$cos(5\\pi)$$ equals -1.\n\nI have read the section in my math book over and over again, but can't find anything about these.\n\n5. Sep 10, 2005\n\n### whozum\n\nOk. on the unit circle, sin x measure the displacement in the Y direction. When your on the x axis, your displacmeent in the Y direction is 0, and the only two points you are on the x axis on the unit circle are at x = 0 and x = Pi .\n\nCos is the same thing but in the X direction, so it is zero at x = Pi\/2 and 3 Pi\/2\n\n6. Sep 10, 2005\n\n### erok81\n\nOh duh, see what I mean. I don't know how I didn't see\/realize that in the first place.\n\nThanks for the help, I appreciate it.","date":"2017-02-24 08:05:33","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\": 0.8206712603569031, \"perplexity\": 731.4642481594392}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501171418.79\/warc\/CC-MAIN-20170219104611-00613-ip-10-171-10-108.ec2.internal.warc.gz\"}"} | null | null |
Q: Why there is a "nil" at the end of output in clojure I am wondering why there is a "nil" at the end of output in clojure. Here is my code:
(defn foo [x &argu]
(print x &argu))
And the other question is what does "&argu" mean here? An variadic argument or something? Thank you so much!
A: The REPL (or whatever execution environment you are using) lets you know the output of the evaluation of the code. As you have written it your function does not return anything. In such cases nil is returned by convention. nil is Clojure's marker for nothing/null. All your code is doing is printing - which is a 'side effect'.
It is important to note that nil is a proper value in Clojure. This is in contrast to some languages that won't even let you compare the undefined/unknown value with another that is a proper value - doing so will generate an exception.
Normally there's a space, so [x & argu]. It matters - so please put the space in your code and try again. This means that the function takes one or more parameters. You are quite correct in saying that it is 'varadic'.
Most languages have the concepts of 'nil' and 'varadic arguments'. Because Clojure is a functional language the whole thinking around 'only returning' from functions becomes quite important. Prioritising not having side effects in your code makes it easier to reason about.
This is your function changed just a little:
(defn foo [x & argu]
(println x argu))
And this is example usage in the REPL:
=> (foo 3 "Hello" "how" "are" "you?")
3 (Hello how are you?)
nil
See how inside the function argu has become a list.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,798 |
\section{Introduction}
The observed velocity flow field of galaxies in the nearby Universe is largely dominated by the
Great Attractor (Dressler et al.~1987; Lynden-Bell et al.~1988; Tonry et al.~2000) and the $\sim$3 times more
distant Shapley supercluster (Hudson et al.~2004). Both are extended
overdensities in the large-scale mass distribution of the local Universe and both are thought to contribute
significantly to the peculiar motion of the Local Group (LG) (Lucey, Radburn-Smith \& Hudson 2005;
Kocevski \& Ebeling 2006). The relative contribution of the Great Attractor and the
Shapley supercluster to the motion of the LG, however, remains poorly determined (cf.~Erdo\u{g}du et al.~2006;
Kocevski \& Ebeling 2006) and is still a matter of debate.
\begin{figure*}
\centerline{\hbox{\psfig{figure=vslice_normapaper_noHI.ps,width=11.5cm,angle=-90}}}
\caption{An overview of the large-scale structures in the Great Attractor region between 3000 $< v_{\rm hel} <$ 7000
km s$^{-1}$. The Norma cluster is located at ($\ell, b, v$) = ($325.3^{\circ}, -7.2^{\circ}, 4871$ km s$^{-1}$). Other
major clusters in this vista are the Pavo II cluster at ($\ell, b, v$) = ($332.3^{\circ}, -23.6^{\circ}, 4167$ km s$^{-1}$),
the Centaurus cluster at ($\ell, b, v$) = ($302.4^{\circ}, +21.6^{\circ}, 3418$ km s$^{-1}$), the Hydra cluster at
($\ell, b, v$) = ($269.6^{\circ}, +26.5^{\circ}, 3777$ km s$^{-1}$), and the low-latitude
CIZA\,J1324.7--5736 and Cen-Crux clusters, at $(\ell, b, v$) = ($307.4^{\circ}, +5.0^{\circ}, 5700$ km s$^{-1}$) and
($305^{\circ}, +5^{\circ}, 6214$ km s$^{-1}$), respectively. The solid contour marks a line of equal Galactic foreground extinction
($A_B$ = 3$\fm0$, from Schlegel, Finkbeiner \& Davis 1998). }
\label{lssoverview}
\end{figure*}
The Shapley supercluster (SCL) is clearly visible as an overdensity in the
distribution of Abell clusters (Scaramella et al.~1989; Einasto et al.~1997; Proust et al.~2006),
whereas the Great Attractor (GA) and its location was identified first and foremost from the systematic
peculiar velocities of galaxies streaming towards this apex (e.g. Lynden-Bell et al.~1988), and confirmed
later from reconstructed mass-density fields of the local Universe (Dekel 1994; Kolatt, Dekel \& Lahav~1995;
Erdo\u{g}du et al.~2006). However, no significant overdensity was obvious in the distribution of galaxies or Abell clusters
at the position of the GA (Lynden-Bell \& Lahav 1988).
This is not surprising given the location of the GA with respect to the
Zone of Avoidance (ZOA). Kolatt et al.~(1995) locate the approximate centre of the extended GA overdensity at
($\ell, b, v$) $\sim$ (320$^{\circ}$, 0$^{\circ}$, 4000 km s$^{-1}$) based on their smoothed reconstructed
mass-density field. The uncertainty in this position is $\sim 17^{\circ}$ as a result of the applied smoothing
of 1200 km s$^{-1}$; this smoothing is necessary given the sparsely sampled data (Kolatt et al.~1995).
Towards such low Galactic latitudes,
the view of the extragalactic light distribution is increasingly reduced by the dust and stars in the
Milky Way. As a result, a large part of the GA overdensity is hidden from view by the Milky Way and
early attempts to quantify the nature and extent of the GA (e.g. Dressler 1988; Hudson 1993a,
1993b; Rowan-Robinson et al.~1990; Jahoda \& Mushotsky 1989) were unsatisfactory: the clear
(and significant) mismatch between the inferred mass of the GA and the visible galaxy distribution
could not be understood.
A deep optical galaxy search at low Galactic latitudes in the GA region (Woudt \& Kraan-Korteweg 2001) has
lifted part of the veil of the Milky Way. Close to the predicted centre of the GA, the Norma cluster (ACO 3627:
Abell, Corwin \& Olowin 1989) has been identified as the most massive cluster in the GA region
(Kraan-Korteweg et al.~1996; Woudt 1998). Abell et al.~(1989) classified this cluster as an irregular (I) cluster
with Bautz-Morgan type I (Bautz \& Morgan 1970). They furthermore classify it as a richness-class 1 cluster with
`59?' galaxies in the magnitude interval $m_3$ and $m_3$ + 2, where $m_3$ corresponds to the magnitude of the
third brightest galaxy in the cluster. Independently,
X-ray observations of the Norma cluster from ROSAT (B\"ohringer et al.~1996) and ASCA (Tamura et al.~1998)
confirm the massive nature of this cluster. From our deep optical galaxy survey in the ZOA in the
general GA region, and our follow-up redshift survey (Fairall, Woudt \& Kraan-Korteweg 1998;
Woudt, Kraan-Korteweg \& Fairall 1999; Woudt et al.~2004), a clearer view of the obscured
GA overdensity has emerged. The Norma cluster is the central cluster in a web of
connected filaments and wall-like structures (Woudt 1998; Kraan-Korteweg \& Lahav 2000; Radburn-Smith et al.~2006), analogous
to the structures observed in and around major mass concentrations in the $\Lambda$-CDM (cold dark matter)
Millenium simulation (Springel et al.~2005).
One of the most prominent newly identified structures is a great-wall-like structure with the Norma cluster at its centre
which we dubbed the Norma supercluster (Woudt 1998; Fairall et al.~1998),
a wall of galaxies which runs nearly parallel to the Galactic Plane (Kraan-Korteweg \& Lahav 2000;
Radburn-Smith et al.~2006) connecting the Pavo II cluster
with the Norma cluster and continuing across (and nearly parallel to) the Galactic Plane to the more distant Vela overdensity
(Kraan-Korteweg, Fairall \& Balkowski 1995) via the Cen-Crux cluster (Woudt 1998). Fig.~\ref{lssoverview} gives a clear overview
of the dominant large-scale structures in the Great Attractor region.
Support for the prominence of the Norma SCL
has come from various complementary multiwavelength studies at lower Galactic latitudes
such as the detection of several further clusters embedded in the Norma SCL.
An X-ray search for highly obscured clusters in the ZOA (Ebeling, Mullis \& Tully~2002) revealed
the second most massive cluster in the Norma SCL, namely CIZA\,J1324.7--5736. This cluster is $\sim$50\%--70\% less massive
than the Norma cluster (Radburn-Smith et al.~2006) and is located at $(\ell, b, v) \sim (307.4^{\circ}, +5.0^{\circ}, 5700$ km
s$^{-1}$). Deep near-infrared observations (Nagayama et al.~2004) furthermore uncovered a low-mass cluster around PKS\,1343-601 at
$(\ell, b, v) \sim (309.7^{\circ}, +1.7^{\circ}, 3900$ km s$^{-1}$), also within the Norma SCL.
Apart from this significant collection of clusters, a general overdensity along the Norma SCL
is also clearly present in the Parkes deep H\,I multibeam ZOA survey (Kraan-Korteweg et al.~2005).
\begin{figure}
\centerline{\hbox{\psfig{figure=distribution.ps,width=8.2cm}}}
\caption{The distribution in Galactic coordinates of optically-detected galaxies (Woudt \& Kraan-Korteweg 2001)
around the Norma cluster. The contours indicate lines of equal Galactic reddening
(from the DIRBE/IRAS reddening maps, Schlegel et al.~1998) of $E(B-V)$ = 0.242, 0.363 and
0.726 mag, respectively. Assuming a standard Galactic reddening law (Cardelli et al.~1989),
these values correspond to $A_{\rm B}$ = 1.0, 1.5 and 3.0 mag, respectively. The dashed circle
marks the Abell radius of the Norma cluster.}
\label{distribution}
\end{figure}
In the first of a series of papers investigating the Norma cluster, we present
a detailed dynamical analysis of this cluster, the most massive cluster in the Great Attractor overdensity,
centrally located in a cosmic web of filaments and wall-like structures.
Figure~\ref{distribution} shows the distribution of the optically-detected galaxies
(Woudt \& Kraan-Korteweg 2001) in the general direction of the Norma cluster, where the
Abell radius ($R_A \equiv 1\farcm7/z$) of the Norma cluster is indicated by the dashed circle.
At the redshift of the Norma cluster (see Sect.~3), the Abell radius corresponds to an
angular radius of 1.75$^{\circ}$. Assuming a Hubble constant of $H_0 = 73$ km s$^{-1}$ Mpc$^{-1}$ and
the cosmological concordance model (assumed throughout this paper), the Abell radius
corresponds to a physical size of 2.0 Mpc (the cosmology-corrected angular scale at this distance is 1.16 Mpc per
degree).
Contours of equal Galactic foreground extinction, taken from the DIRBE/IRAS Galactic reddening map
(Schlegel, Finkbeiner \& Davis~1998), are overlayed on the galaxy distribution in Fig.~\ref{distribution}
($A_{\rm B}$ = 1.0, 1.5 and 3.0 mag, respectively; Cardelli, Clayton \& Mathis~1989) and show that
the Galactic foreground extinction within the Abell radius of the Norma cluster
is moderate, $A_{\rm B} \le 1.5$ mag.
Within the Abell radius, there are 603 optically-detected galaxies with observed diameters in excess of 12$''$
(Woudt \& Kraan-Korteweg 2001) and 219 (near-infrared-detected) galaxies in the extended source catalogue (XSC)
of the 2 Micron All-Sky Survey (2MASS, Skrutskie et al.~2006). The 2MASS galaxies in the Norma cluster represent a subset
of the 603 optically-detected galaxies, although not all the 2MASS galaxies have an optical counterpart; 165 of the 219 2MASS
galaxies (75\%) were also found by Woudt \& Kraan-Korteweg (2001). For the brighter 2MASS galaxies (10$''$-aperture $K_s$-band
$<$ 12.5 mag), the overlap between 2MASS and the optical survey is excellent: 97\% of the 2MASS galaxies have an optical counterpart.
It should be noted that at the position of the Norma cluster ($\ell, b \sim 325^{\circ}, -7^{\circ}$)
star-crowding is the primary limiting factor, not the Galactic foreground extinction. The star-crowding leaves
a Zone of Avoidance imprint on the 2MASS XSC catalogue near the Galactic Bulge (Kraan-Korteweg \& Jarrett 2005)
and the Norma cluster is located on the edge of this Zone of Avoidance. At moderate extinction ($A_B \le 3$ mag),
but in the presence of severe star-crowding, optical surveys still retrieve the most complete galaxy distribution
in the Zone of Avoidance (Kraan-Korteweg \& Jarrett 2005).
We have obtainted 129 new redshifts of galaxies within the Abell radius of the Norma cluster using the 2dF spectrograph
at the Anglo-Australian Observatory. These new observations are presented in Section 2. In Section 3 all the redshifts
obtained to date are combined and a detailed dynamical analysis of the cluster based on 296 cluster members is presented.
In Section 4, we discuss a few
individual galaxies in the Norma cluster of dynamical interest.
\section{2dF spectroscopy}
Spectra were obtained with the 2dF facility (Lewis et al.~2002) on the 3.9m Anglo-Australian Telescope.
Full details of the observing 2dF setup used for observations are given in
Table~\ref{2dfsetup}. As the main objective was to measure the velocity
disperisons of the cluster's early-type galaxies the
1200V gratings were used in each of the 2dF spectrographs.
These gave a FWHM resolution of $\sim$ 125 km s$^{-1}$ at Mg\,$b$ which is
sufficient to determine velocity dispersions down
to $\sim$ 60 km s$^{-1}$. In all, three fibre configurations were observed.
Spectra were extracted from the raw data frames,
wavelength calibrated and sky-subtracted using
the AAO 2dfdr software package\footnote{\tt http://www.aao.gov.au/2df/software.html\#2dfdr}.
Redshifts were determined via cross-correlation for
the absorption line spectra and/or the direct measurement of
emission lines.
\begin{table}
\centering
\caption{2dF setup used.}
\begin{tabular}{@{}ll@{}}
\hline
Date of Observations & 2001 May 30 \\
Field centre (J2000.0) & $16^h15^m01.8^s$ $-60^{\circ}54'24''$ \\
Number of fibre config. & 3 \\
Total exposure times & $5 \times 1200$ s, $5 \times 1200$ s, $4 \times 1200$ s \\
Fibre size & 2.1 arcsec (= 0.68 kpc at Norma) \\
Grating & 1200V \\
Wavelength coverage & 4700 -- 5840 {\AA} \\
Resolution (FWHM) & 2.2 {\AA} \\
Wavelength pixel scale & 1.1 {\AA} \\
\hline
\end{tabular}
\label{2dfsetup}
\end{table}
The 2dF spectroscopic observations focussed on the determination of
accurate velocity dispersions of early-type galaxies in the Norma cluster for a
Fundamental Plane analysis of the cluster. The primary target list therefore consisted of known bright
ellipticals in the cluster (Woudt \& Kraan-Korteweg 2001). However, we used the spare fibres of the 2dF
spectrograph to extend the redshift coverage of the Norma cluster. Galaxies were primarily
selected from the optical catalogue of Woudt \& Kraan-Korteweg (2001) and the 2MASS XSC, indicated by
`WKK' and `2MASX\,J', respectively in Table~\ref{2dftable}. Additional galaxies were identified on deep
$R_C$ images taken with the ESO/MPG 2.2-m telescope and the Wide Field Imager (see Sect.~4). These are
identified as `ZOA\,J' in Table~\ref{2dftable}.
Redshifts were obtained for 182 galaxies, 53 of which had a previous measurement.
For 76 galaxies, multiple measurements were obtained to gauge the internal accuracy of
the 2dF spectrograph. Table~\ref{2dftable} shows a representative sample of the results obtained
from the 2dF spectroscopy. The full table is available online.
\begin{table*}
\centering
\caption{A representative sample of the results of the 2dF spectroscopy.}
\begin{tabular}{@{}lcccccc@{}}
\hline
Identification$^*$ & {RA (2000.0)} & {DEC (2000.0)} & {\it v$_{\rm abs}$} & {\it v$_{\rm em}$} \\
& & & (km s$^{-1}$) & (km s$^{-1}$) \\
\hline
ZOA\,J16070347-6113587 & 16 07 03.465 &--61 13 58.74 & \hfill 15868 & \hfill \\%& \hfill & \\
WKK\,5916 & 16 07 50.369 &--61 10 06.84 & \hfill 3053 & \hfill \\%& \hfill & \\
WKK\,5920 & 16 07 52.618 &--60 31 12.95 & \hfill 4762 & \hfill \\%& \hfill 206.8 $\pm$ 4.6 & $*$ \\
WKK\,5926 & 16 08 08.744 &--61 12 44.37 & \hfill & \hfill 15856 \\%& \hfill & \\
ZOA\,J16081355-6109377 & 16 08 13.548 &--61 09 37.65 & \hfill 3640 & \hfill \\%& \hfill & \\
2MASX\,J16082135-6044498 & 16 08 21.312 &--60 44 50.20 & \hfill 29813 & \hfill \\%& \hfill & \\
ZOA\,J16083012-6039511 & 16 08 30.118 &--60 39 51.08 & \hfill 6040 & \hfill \\%& \hfill & \\
WKK\,5958 & 16 09 01.326 &--60 52 03.70 & \hfill 15708 & \hfill \\%& \hfill & \\
WKK\,5964 & 16 09 06.402 &--60 59 07.70 & \hfill 4711 & \hfill \\%& \hfill 99.1 $\pm$ 3.7 & $*$ \\
ZOA\,J16091138-6108285 & 16 09 11.377 &--61 08 28.45 & \hfill & \hfill 15674 \\%& \hfill & \\
\hline
\end{tabular}
\label{2dftable}
{\newline \footnotesize{$^*$ In column 1, the WKK identification (Woudt \& Kraan-Korteweg 2001) is given if the galaxy has been identified by
WKK. \hfill \
\newline Alternative names for galaxies are given if a WKK identification was unavailable but when the galaxy has been identified already
in \hfill \
\newline another survey (e.g.~`2MASX\,J', Skrutskie et al.~2006). A new identification was given to galaxies not yet catalogued in the
literature,\hfill \
\newline but which we identified from deep $R_C$ imaging (e.g.~`ZOA\,J', see Sect.~4). \hfill }}
\end{table*}
Figure~\ref{2dfcompext} shows a comparison of the measured 2dF heliocentric velocities with
measurements from the literature. The vast majority of these previous measurements were obtained in the course
of our ZOA redshift survey (SAAO: Woudt et al.~1999; MEFOS: Woudt et al.~2004). The overall agreement is very good:
$$v_{\rm 2dF} - v_{\rm lit} = -6 \pm 17 \ {\rm km \, s^{-1}}$$
with a dispersion of $\sigma_{\rm ext, all}$ = 124 km s$^{-1}$ (based on 51 galaxies).
Only one galaxy revealed a discrepent heliocentric velocity; for
WKK\,6329, the 2dF spectroscopy resulted in $v = 4749 \pm 35$ km s$^{-1}$ as compared to the previously low
signal-to-noise value for this galaxy of 2477 $\pm$ 250 km s$^{-1}$ (Woudt et al.~1999).
\begin{figure}
\centerline{\hbox{\psfig{figure=2dfcompext.ps,width=8.2cm}}}
\caption{A comparison of the 2dF velocities with previously determined velocities.}
\label{2dfcompext}
\end{figure}
We then compared the 2dF results with a subset of the literature sample, namely those for which redshifts
were obtained with the MEFOS multi-fibre spectrograph (Woudt et al.~2004). This subset has the
most accurate redshifts available for the Norma cluster. There are 16 galaxies in common between
2dF and MEFOS (the filled circles in Fig.~\ref{2dfcompext}). The
agreement is again excellent, with a lower rms ($\sigma_{\rm ext, MEFOS}$ = 31 km s$^{-1}$) than the previous
comparison (which included the SAAO measurements),
$$v_{\rm 2dF} - v_{\rm MEFOS} = +12 \pm 8 \ {\rm km \, s^{-1}}.$$
Given the primary goal of obtaining accurate velocity dispersions from the 2dF spectroscopy,
we have observed a large number of galaxies repeatedly to gauge the internal
uncertainty: 69 galaxies were observed twice and 7 galaxies had three independent velocity measurements.
For these repeated observations we find $\sigma_{\rm int}$ = 33 km s$^{-1}$ over the entire
range of observed velocities. This is comparable to the external comparison with the MEFOS spectroscopy.
Based on these independent evaluations, we have assigned a standard error of 35 km s$^{-1}$ to each of the 2dF
velocities.
\section{Dynamical Analysis}
\subsection{Cluster membership}
\begin{figure}
\centerline{\hbox{\psfig{figure=norma1raddist.ps,width=8.2cm}}}
\caption{Galaxies with known redshifts as a function of distance to the central cD galaxies (WKK\,6269) in
the Norma cluster. The Abell radius ($R_A$) is indicated by the vertical dotted line, were the velocity centroid
and the upper and lower 3$\sigma$ limits are indicated by the horizontal dotted line.
The E/S0 galaxies in the Norma cluster are plotted as filled circles and the S/Irr galaxies
in the Norma cluster are shown as crosses. Galaxies deemed non-members are indicated by the open circles.}
\label{raddist}
\end{figure}
\begin{figure}
\centerline{\hbox{\psfig{figure=nordensity.ps,width=8.2cm}}}
\caption{Galaxy density contours determined from the distribution (in Galactic coordinates) of the 296 likely cluster
members. The contours have been normalised by the total number of galaxies in the sample ($N_s$); they correspond to
0.063 $N_s$, 0.125 $N_s$, 0.25 $N_s$, 0.50 $N_s$ and 1.0 $N_s$ galaxies per square degree, respectively. The location
of WKK\,6269 is indicated by the central cross, and concentric cluster radii of $\frac{1}{3} R_A$, $\frac{2}{3} R_A$ and
$R_A$ are shown as the two dashed circles and the solid circle, respectively. The extinction contours are as in
Fig.~\ref{distribution}.}
\label{density}
\end{figure}
With the new 2dF observations described above, radial velocities are now available for 305 galaxies
within the Abell radius of the Norma cluster for the velocity range 0 -- 9500 km s$^{-1}$.
The velocity distribution of these galaxies as a function of distance to the Norma cluster centre
is shown in Fig.~\ref{raddist}. The centre of the cluster was taken as the cD galaxy WKK\,6269, a strong
wide-angle-tail radio-continuum source (Jones \& McAdam 1992, 1996) located at the peak in the 0.7--10 keV
ASCA map of the Norma cluster (figure 1 of Tamura et al.~1998). The velocity centroid (see Sect.~3.2.3)
of the Norma cluster is 4871 km s$^{-1}$ and is marked by the central horizontal dashed line. The velocity limits
for cluster membership are taken as $\pm$3 times the velocity scale/dispersion (925 km s$^{-1}$) around the velocity centroid;
these limits are shown as the upper and lower horizontal dashed line in Fig.~\ref{raddist}.
Nine galaxies are distinct outliers (open circles in Fig.~\ref{raddist}) and have been rejected from our
subsequent analysis. This leaves 296 likely cluster members, of which 107 have been classified as elliptical or
lenticular (E/S0: filled circles in Fig.~\ref{raddist}) and 189 are either spirals or irregulars
(S/Irr: crosses in Fig.~\ref{raddist}) (Woudt \& Kraan-Korteweg 2001).
The galaxy density contours determined from the 296 cluster members, displayed in Fig.~\ref{density}, show that the cluster
is strongly elongated along a position angle which is aligned with the Norma wall (compare Figs.~\ref{lssoverview}
and \ref{density}).
Since the elongation is nearly perpendicular to the Galactic extinction contours, it seems very unlikely that selective
extinction effects are the cause of the observed elongation. The peak
of the galaxy-density distribution is located at right ascension and declination $16^h14^m42^s$, $-60^{\circ}55'52''$ (J2000.0),
about 3 arcmin from WKK\,6269 at $16^h15^m03.6^s$, $-60^{\circ}54'26''$ (J2000.0), our adopted centre.
\subsection{Substructure statistics}
We employed the statistical tests described by Pinkney et al.~(1993, 1996)
in analysing the dynamical structure of the Norma cluster. This array of statistical tools consists of
one-dimensional tests (analysing the shape of the velocity histogram), two-dimensional tests
(checking for substructure in the on-sky distribution), and three-dimensional tests (using
velocity and positional information). Among the latter, the Dressler-Shectman (DS) $\delta$-test
(Dressler \& Shectman 1988) is a particularly powerful and frequently used method to quantify substructure
(e.g.~Pinkney et al.~1993; Oegerle \& Hill 2001; Pimbblet, Roseboom \& Doyle 2006).
This test calculates the mean velocity ($\langle v \rangle_{\rm local}$)
and the standard deviation ($\sigma_{\rm local}$) for each galaxy and its $N_{nn}$ (= $\sqrt{N}$) nearest
neighbours, where $N$ represents the total number of galaxies in the sample; often only the 10 nearest galaxies
are used in this analysis. These local parameters are then
compared with the global mean ($\langle v \rangle$) and standard deviation ($\sigma$)
of all the galaxies in the sample. For each galaxy, $\delta_i$ is calculated where $\delta_i$ is given by
$$\delta_i^2 = \left( \frac{N_{nn} + 1}{\sigma^2} \right) \left[(\langle v \rangle_{\rm local} -
\langle v \rangle)^2 + (\sigma_{\rm local} - \sigma)^2 \right].$$
The cumulative deviation $\Delta$ is defined as the sum of all $\delta_i$'s. If no subclustering is
present, $\Delta$ is approximately equal to the number of galaxies in the sample ($N$).
In the following subsections we analyse the Norma cluster at three incremental radii, starting with the inner
core of the cluster ($R < 0.67$ Mpc), double this radius ($R < 1.35$ Mpc) and three times this radius
out to the Abell radius ($R < 2.02$ Mpc).
\begin{figure*}
\centerline{\hbox{\psfig{figure=norma_overlay3.ps,width=17.6cm}}}
\caption{An optical colour image of the central $\sim$0.66 Mpc $\times$ 0.66 Mpc of the Norma cluster. The white contours
show the X-ray subgroup identified from ROSAT observations (reproduced from B\"ohringer et al.~1996) overlayed on the optical
galaxy distribution. The central cD galaxy (WKK\,6269) coincides with the peak in the X-ray emission. The inset shows the
distribution of the identified galaxies in this field of view in equatorial coordinates (E/S0 cluster members: red dots,
S/Irr cluster members: blue dots, WKK\,galaxies with no redshift information: large black dots,
small galaxies identified from the Wide Field Image data: small black dots). For reference when comparing this
figure with the distribution in Galactic coordinates, a line of equal
Galactic latitude ($b = -7.5^{\circ}$) is drawn as a diagonal dashed line in the inset.}
\label{optxray}
\end{figure*}
\subsubsection{The inner $\frac{1}{3}$ Abell radius (= 0.67 Mpc)}
Figure~\ref{optxray} shows the optical image of the central $34 \times 34$ arcmin of the Norma cluster,
obtained with the 2.2-m MPG/ESO telescope at la Silla and the Wide Field Imager during three nights in
1999 May (see also Section 4). The area displayed in Fig.~\ref{optxray} corresponds to $\sim$1.6 times
the core radius ($R_{\rm c}$; King 1966) of the Norma cluster, where $R_{\rm c, \, opt}$ = 10$\farcm$4
$\pm$ 1$\farcm$1 (optical: Kraan-Korteweg et al.~1996) and $R_{\rm c,\, X}$ = 9$\farcm$95 $\pm$ 1$\farcm$0
(X-ray; B\"ohringer et al.~1996) for the Norma cluster. In terms of the
Abell radius, Fig.~\ref{optxray} displays the inner $\sim \frac{1}{6}$ $R_A$.
Superimposed on the optical colour image are the contours of the X-ray subcluster identified by B\"ohringer
et al.~(1996) (reproduced from their figure 2). Note that these contours mark the subcluster only and that the main
cluster has been subtracted as described in B\"ohringer et al.~(1996). The inset in Fig.~\ref{optxray} shows
the corresponding sky distribution (in equatorial coordinates) of the identified galaxies in this field of view.
The red and blue dots are confirmed cluster members, where
the red dots mark E/S0 galaxies, and the blue dots correspond to S/Irr galaxies. The large black dots are
galaxies (without a redshift) identified in our deep optical survey (Woudt \& Kraan-Korteweg 2001) and the
small black dots are galaxies (also without a redshift) identified on the deep $R_C$-band images taken
with the Wide Field Imager. The central cD galaxy (WKK\,6269, see the discussion in Sect.~3.4 on the peculiar velocity
of this galaxy) is indicated by the black-encircled red dot.
\begin{figure}
\centerline{\hbox{\psfig{figure=norma1vhist.ps,width=8.2cm}}}
\caption{The normalised velocity distribution of cluster members (cross-hatched histogram)
within $R < 0.67$ Mpc (upper panel), $R < 1.35$ Mpc (middle panel) and $R < 2.02$ Mpc (lower panel). In each
of the panels, the velocity distribution of the E/S0 population is shown by the solid histogram.}
\label{vhist}
\end{figure}
Within $R \le \frac{1}{3} R_A$, there are 129 galaxies confirmed as cluster members; 53 galaxies are E/S0 (41\%) and 76
have been classified as S/Irr (59\%), respectively.
The velocity histogram of the 129 cluster members -- shown in the upper panel of Fig.~\ref{vhist} as
the cross-hatched distribution -- is consistent with being Gaussian and has a biweight
velocity centroid ($C_{\rm BI}$) and scale ($S_{\rm BI}$) (Beers, Flynn \& Gebbhardt 1990)
of 4777 $\pm$ 86 km s$^{-1}$ and 973 km s$^{-1}$, respectively. When separating the sample into the
elliptical and spiral population, there are some marked differences between these galaxy populations.
Firstly, the biweight velocity centroid of the two populations differ by 334 km s$^{-1}$
(see Table~{\ref{norsubstr}). The statistical significance of this difference -- by analogy to the arguments used in
the peculiar velocity discussion (Sect.~3.4) -- is $S_{\rm V} = 1.9$.
Secondly, contrary to the velocity distribution of the elliptical galaxies
(the dark shaded histogram in the top panel of Fig.~\ref{vhist}), the velocity distribution of the
spiral galaxies is non-Gaussian with hints of skewness and kurtosis.
In addition, the elliptical population is strongly elongated along a position
angle in the equatorial on-sky projection of 116$^{\circ}$ (measured counter-clockwise from North),
whereas the distribution of the spiral galaxies is largely spherical. The position angle of 116$^{\circ}$ in
equatorial coordinates corresponds to a position angle of 160$^{\circ}$ (measured counter-clockwise from
North) in the galactic-coordinate distribution at the position of the Norma cluster. The position angle of the elongated
distribution is indicated by the arrows in the top-left panel of Fig.~\ref{dsall}.
The location
of the X-ray subcluster is indicated by the solid black line in the left panels of Fig.~\ref{dsall}.
It is interesting to note the approximate alignment of the X-ray subgroup with the
elongated distribution of the E/S0 galaxies.
Statistical significance of the two- and three-dimensional substructure tests
are calculated by means of Monte Carlo (MC) simulations (Pinkney et al.~1996). The observed sample is compared
to 500 simulated samples. In the case of the elongation of the elliptical population, only 4 out of the 500
simulations showed a larger degree of elongation.
Despite the above mentioned differences between the elliptical and spiral galaxy population, and despite the
presence of the X-ray subcluster, the Dressler-Shectman $\delta$-test showed no clear sign of substructure. The
combined sample, as well as the E/S0 galaxy sample (and to a lesser extent the S/Irr galaxies) are formally consistent with no
substructure. The $\Delta$ values for the combined sample and the individual galaxy populations are given
in Table~\ref{norsubstr},
together with the average value of $\Delta$ after 500 Monte Carlo simulations.
\begin{table*}
\centering
\caption{An investigation into substructuring in the Norma cluster.}
\begin{tabular}{@{}lcccc@{}}
\hline
& & $\frac{1}{3} R_A$ (= 0.67 Mpc) & $\frac{2}{3} R_A$ (= 1.35 Mpc) & {$1 R_A$ (= 2.02 Mpc)} \\
\hline
& & \multicolumn{3}{c}{\underline{All galaxies}}\\
$C_{\rm BI}$ & (km s$^{-1}$) & 4777 $\pm$ 86 & 4822 $\pm$ 61 & 4871 $\pm$ 54 \\
$S_{\rm BI}$ & (km s$^{-1}$) & 973 & 940 & 925 \\
$N$ & & 129 & 239 & 296 \\
$\Delta^*$ & & 135.7 (38\%) & 262.7 (23\%) & 353.1 (4\%) \\
$\langle {\Delta} \rangle_{500}$ & & 131.4 & 244.0 & 301.4 \\
$v_{\rm pec}$ cD & (km s$^{-1}$) & 653 & 609 & 561 \\
$S_{\rm V}$ & & 6.5 & 7.6 & 7.5 \\
& & \multicolumn{3}{c}{\underline{Elliptical and lenticular galaxies}}\\
$C_{\rm BI}$ & (km s$^{-1}$) & 4951 $\pm$ 132 & 4962 $\pm$ 97 & 4979 $\pm$ 85 \\
$S_{\rm BI}$ & (km s$^{-1}$) & 964 & 901 & 877 \\
$N$ & & 53 & 86 & 107 \\
$\Delta^*$ & & 44.8 (75\%) & 83.7 (52\%) & 107.7 (47\%) \\
$\langle {\Delta} \rangle_{500}$ & & 49.6 & 84.4 & 106.9 \\
& & \multicolumn{3}{c}{\underline{Spiral and irregular galaxies}}\\
$C_{\rm BI}$ & (km s$^{-1}$) & 4617 $\pm$ 109 & 4740 $\pm$ 78 & 4812 $\pm$ 70 \\
$S_{\rm BI}$ & (km s$^{-1}$) & 949 & 965 & 957 \\
$N$ & & 76 & 153 & 189 \\
$\Delta^*$ & & 98.2 (14\%) & 186.3 (8\%) & 247.4 (1\%) \\
$\langle {\Delta} \rangle_{500}$ & & 85.6 & 158.0 & 191.2 \\
\hline
\end{tabular}
\label{norsubstr}
{\newline \footnotesize{$^*$ The percentages given after each value of $\Delta$ reflect the percentage of Monte Carlo simulations which showed
a higher amount of \hfill \
\newline subclustering than the actual observed sample. Percentages below 10\% indicate significant subclustering.\hfill \ }}
\end{table*}
\begin{figure*}
\centerline{\hbox{\psfig{figure=ds1mpcD.ps,width=5.8cm}} \hbox{\psfig{figure=ds2mpcD.ps,width=5.8cm}}
\hbox{\psfig{figure=ds3mpcD.ps,width=5.8cm}}}
\caption{Results from the Dressler-Shectman $\delta$-test: The distribution in Galactic coordinates of galaxies and their
measured $\delta_i$. The left panels show the results for the inner 0.67 Mpc
(upper panel: E/S0 galaxies, lower panel: S/Irr galaxies), the middle panels show the results for the inner
1.35 Mpc (again separated by morphological classification), and the right panels show the results within the
entire Abell radius (2.02 Mpc). The symbol sizes are proportional to the value of $e^{\delta_i}$, where large
circles indicate significant deviations from the local mean velocity or local mean velocity dispersion. In the left
panels, the location of the X-ray subcluster (B\"ohringer et al.~1996) is marked by the solid lines.
If present, the arrows in the top right-hand corner of the panels
indicate the direction of the position angle of the distribution if significant elongation is detected.}
\label{dsall}
\end{figure*}
Of the 500 MC simulations,
38\%, 75\% and 14\% revealed a higher $\Delta_{\rm MC}$ than the observed $\Delta$ for the combined, E/S0 and S/Irr
sample, respectively. Nominally, substructure is said to be present in the observed sample when less than 10\% of
the MC simulations show a larger amount of substructure. In this case, the S/Irr galaxies
reveal marginal evidence for substructure.
Fig.~\ref{dsall} shows the results of the Dressler-Shectman $\delta$-test for the combined
sample, plotted in Galactic coordinates (as in Fig.~\ref{distribution}).
We have plotted the E/S0 galaxies (upper-left panel) and the S/Irr galaxies (lower-left panel) separately.
The size of the symbols is proportional to the individual values of $\delta_i$, where large circles indicate
significant deviations from either the local mean velocity or the local mean velocity dispersion.
Various individual galaxies exhibit properties which are also strongly aligned along the same
position angle (WKK\,6305 = PKS\,1610-605:
extended head-tail radio continuum emission (Jones \& McAdam 1992), WKK\,6176: extended X-ray tail (Sun et al.~2006));
there is an obvious elongation
within the Norma cluster which affects the way in which the
galaxies interact with the intracluster medium. In Sect.~4 we will review some of the properties of
these galaxies in more detail.
\subsubsection{The inner $\frac{2}{3}$ Abell radius (= 1.35 Mpc)}
The inner $\frac{2}{3} R_A$ of the Norma cluster contains 239 confirmed cluster members,
of which 86 are E/S0 galaxies (36\%) and 153 are S/Irr galaxies (64\%). The velocity distribution
(histogram) is formally consistent with being Gaussian (see middle panel of Fig.~\ref{vhist}), although a slight
excess of spiral galaxies at lower velocities is present.
The velocity centroid of the combined sample ($C_{\rm BI} = 4822 \pm 66$ km s$^{-1}$) is somewhat larger compared
to that of the inner $\frac{1}{3} R_A$-sample, but this is largely due to an increase in the velocity
centroid of the S/Irr population (although the velocity scale of the S/Irr galaxies has not changed).
The values for the biweight velocity centroids and scales of the various samples are given in Table~\ref{norsubstr}.
The offset in the biweight velocity centroid between the E/S0 and S/Irr sample remains albeit slightly lower
and is 222 km s$^{-1}$ with a significance of $S_{\rm V} = 1.8$. The velocity centroid of the E/S0 galaxies has not
changed by extending the sample to a larger radius (see Table~\ref{norsubstr}), although its biweight velocity
scale is somewhat smaller; $S_{\rm BI}$ = 901 km s$^{-1}$ for $R < \frac{2}{3} R_A$, compared to
$S_{\rm BI}$ = 964 km s$^{-1}$ for $R < \frac{1}{3} R_A$.
In terms of their spatial distribution, the elliptical and spiral populations now both reveal significant elongation and have
position angles of 102$^{\circ}$ and 101$^{\circ}$, respectively, in the equatorial coordinate frame. This corresponds
to position angles of 146$^{\circ}$ and 145$^{\circ}$ in the Galactic coordinate frame. The latter are again indicated by
arrows in the top-right of the middle panels in Fig.~\ref{dsall}. The $\delta$-test now clearly reveals substructure
in the S/Irr sample (only 8\% of the MC simulations show a larger degree of substructure). Interestingly, the
$\delta$-test shows that the E/SO population is completely free of any detectable substructure. In Fig.~\ref{dsall},
the results from the $\delta$-test of the combined sample out to $R < \frac{2}{3} R_A$ is shown in the middle panels,
where the upper-middle panel shows the E/SO galaxies and the lower-middle panels displays the S/Irr galaxies.
\subsubsection{The Abell radius (= 2.02 Mpc)}
Our final sample extends out to the full Abell radius of the Norma cluster. Within this region, there are 296 cluster members
(Sect.~3.1) of which 107 are classified E/S0 (36\%) and 189 belong to the S/Irr population (64\%).
The velocity histogram of the combined
set (the hashed histogram in the lower panel of Fig.~\ref{vhist}) shows some evidence for kurtosis, based on the average of
6 kurtosis tests (Pinkney et al.~1996), and a clear excess of galaxies at lower velocities. The difference in the velocity
centroid of the two morphologically-distinct samples is reduced to 167 km s$^{-1}$ (164 km s$^{-1}$ in the cluster rest
frame) at a significance of $S_{\rm V} = 1.5$.
The spatial distribution of both the E/S0 galaxies and the S/Irr galaxies is strongly elongated with position angles
of 102$^{\circ}$ and 107$^{\circ}$, respectively, in the equatorial coordinate frame. This corresponds to 146$^{\circ}$ and
151$^{\circ}$ in the Galactic coordinate reference frame; these angles are indicated by the arrows in the top-right corner of
the right panels in Fig.~\ref{dsall}. The uncertainty in the position angle is $\sim$7$^{\circ}$.
As before, the E/S0 population appears relaxed. Their velocity centroid remains constant throughout the cluster (4979 $\pm$ 85 km
s$^{-1}$ for all the E/S0 galaxies within the Abell radius) and no substructure is detected by the Dressler-Shectman $\delta$-test.
The velocity scale ($S_{\rm BI}$) of the E/S0 sample shows a distinct decrease as a function of radius (see Table~\ref{norsubstr}),
again a signature of a relaxed rich cluster (Rines et al.~2003).
The spiral galaxy population, on the other hand, appears far from relaxed. The velocity centroid increases with increasing radius
(shifting by $\sim$200 km s$^{-1}$ across the Abell radius) and the velocity scale stays roughly constant at
$\sim$960 km s$^{-1}$. The Monte Carlo simulations of the
Dressler-Shectman $\delta$-test of the S/Irr galaxies indicate that only 1\% of the simulations show a larger degree of substructure
compared to the observed amount of substructure. The results of the $\delta$-test of the combined sample ($N = 296$) is shown in
the right panels of Fig.~\ref{dsall}.
\subsection{Subgroups in the Norma cluster}
We have identified two spiral-rich subgroups based on the Dressler-Shectman $\delta$-test of the S/Irr population alone. When
displaying those S/Irr galaxies for which $\delta_i > 2.25$, two distinct groups appear. In Fig.~\ref{subgroup} the distribution
in Galactic coordinates of the 296 cluster members are shown, where spiral galaxies with $\delta_i > 2.25$ are shown as encircled
dots. This $\delta_i$ limit was chosen based on the outcome of the $\delta$-test of the E/S0 galaxies; that sample is completely
free of any substructure and there the largest measured $\delta_i$ was 2.25. A few isolated galaxies also appear with large
$\delta_i$ values.
In the case of WKK\,6406 at ($\ell, b, v$) = ($325.69^{\circ}$, $-7.22^{\circ}$, $7349 \pm 35$ km s$^{-1}$), its large heliocentric
velocity could indicate that it is a background galaxy which was mistakenly identified as a cluster member (see also Fig.~\ref{raddist}).
Close to the centre of the Norma cluster is a compact group (dubbed `Norma A') where we
have isolated a group of five dynamically distinct galaxies around WKK\,6078 (including WKK\,6071, WKK\,6078, WKK\,6125,
WKK\,6135 and ZOA\,J16113352). This group is marked in Fig.~\ref{subgroup} by the small solid circle within the $R < \frac{1}{3} R_A$
region (inner dashed circle). The centre of Norma A is approximately at right ascension and declination $16^h12^m00^s$, $-61^{\circ}04'40''$
(J2000.0). Based on these five galaxies, we find a mean velocity of 4453 km s$^{-1}$ (which is 418 km s$^{-1}$ less
than the mean of the cluster, corresponding to 411 km s$^{-1}$ in the cluster rest frame). Norma A has a velocity dispersion of
312 km s$^{-1}$, which is much smaller than the velocity scale of the cluster (925 km s$^{-1}$).
A second dynamically distinct group of galaxies (`Norma B') is found further from the core of the cluster, centred
around WKK\,5751 (other galaxies include WKK\,5718, WKK\,5779, WKK\,5783, WKK\,5796 and WKK\,5813). This group is indicated by
the large solid circle in Fig.~\ref{subgroup} in the region $\frac{2}{3} R_A < R < R_A$ and has a central position
(in right ascension and declination) of $16^h03^m56^s$, $-60^{\circ}26'54''$ (J2000.0). It has a mean velocity
of 5313 km s$^{-1}$ (an offset of +435 km s$^{-1}$ in the cluster rest frame) and a velocity dispersion of 604 km s$^{-1}$.
Norma B (and to a lesser extent Norma A) lies along the Norma wall elongation, supporting the idea that cluster infall
occurs along the connecting filaments and wall-like structures. This is consistent with the large-scale structure formation
and evolution as seen in the $\Lambda$-CDM Millenium simulation (Springel et al.~2005).
To gauge how far the merger of both Norma A and B with the main cluster (Norma major) has progressed, deep
observations with the Australian Telescope Compact Array (ATCA) could be used to determine whether the spiral galaxies
in Norma A and Norma B are hydrogen-deficient as a result of interactions with the intracluster medium. Previous observations
with ATCA of the Norma cluster (Vollmer et al.~2001) showed that spirals in the Norma cluster are generally
HI-deficient, but these observations did not include Norma A and B, respectively.
\begin{figure}
\centerline{\hbox{\psfig{figure=subgroup.ps,width=8.2cm}}}
\caption{The distribution in Galactic coordinates of the 296 cluster members within the Abell radius. Encircled dots represent
spiral galaxies with $\delta_i \ge 2.25$ (based on the $\delta$-test of the entire S/Irr galaxy population). Two spiral-rich subgroups
have been identified and are marked by the solid circles.}
\label{subgroup}
\end{figure}
\subsection{The peculiar velocity of the central cD galaxy}
The Norma cluster contains two large cD galaxies, namely WKK\,6312 at $v_{\rm cD} = 3839 \pm 38$ km s$^{-1}$
(Woudt et al.~2004) and WKK\,6269 at $v_{\rm cD} = 5441 \pm 52$ km s$^{-1}$ (Woudt et al.~2004). The latter has
been observed with the 2dF and was found to be in excellent agreement with previous measurements:
$v_{\rm cD} = 5448 \pm 35$ km s$^{-1}$, see Table~\ref{2dftable}).
WKK\,6269 is also known as PKS\,B1610-608 (one of the 20 strongest extragalactic
radio sources) and is a textbook example of a wide-angle-tail (WAT) radio galaxy (Jones \& McAdam 1992, 1996).
Such WAT morphology either reflects the motion of the cD galaxy through the cluster and its interaction with
the intracluster medium via ram pressure (Owen \& Rudnick 1976), or indicates the presence of a
cluster-subcluster merger (Burns 1998).
Whether WKK\,6269 is at rest with respect to the potential well of the
cluster can be assessed from its peculiar velocity, i.e.~the difference
between the velocity centroid of the cluster ($C_{\rm BI}$) and the velocity of the individual cD galaxy ($v_{\rm cD}$).
The peculiar velocity has to be corrected by a factor $(1+z)$ to ensure the velocity difference is in the cluster
rest frame and considered independently for the varying values in the three regarded spheres.
In Table~\ref{norsubstr} we also list the values of the peculiar velocity of WKK\,6269 as determined within
the various spheres. It ranges from $\sim$650 km s$^{-1}$ within a radius of 0.67 Mpc to $\sim$550
km s$^{-1}$ within 2.02 Mpc. The statistical significance of this peculiar velocity ranges from 6.5 to 7.5
($S_{\rm V} \equiv {|v_{\rm cD} - C_{\rm BI}|} / {\sqrt{\sigma_1^2 + \sigma_2^2}}$, where $\sigma_1$ is the error
in the velocity centroid of the cluster and $\sigma_2$ is the error in the velocity measurement of the
cD galaxy). The large number of cluster members used to determine the velocity
centroid ($\sigma_1 = S_{\rm BI} / \sqrt{N_{\rm gal}}$) and the small error in the measurement of the
heliocentric velocity of WKK\,6269, makes this large peculiar velocity highly significant.
It is exceptionally large when compared to other
cD galaxies in clusters (Oegerle \& Hill 2001), but not without precedent (Pimbblet et al.~2006).
Note that the other cD galaxy in the Norma cluster (WKK\,6312) has an even larger velocity offset.
B\"ohringer et al.~(1996) identified an X-ray subgroup close the centre of the Norma cluster (see Fig.~\ref{optxray}).
This subgroup (dubbed `Norma minor') is fairly massive; Tamura et al.~(1998) estimate that the mass of this subgroup could add
up to $\sim$50\% to the total mass of the cluster.
A comparison of the 843 MHz radio continuum emission of PKS\,B1610-608 (Jones \& McAdam 1992) with
the X-ray contours of this central subgroup (shown in Fig.~\ref{central})
shows that the radio lobes of WKK\,6269 are closely aligned
with the X-ray subgroup. The large observed peculiar velocity of the cD galaxy in the Norma cluster
is most likely caused by this ongoing merger.
Based on the compactness of the X-ray subcluster, B\"ohringer et al.~(1996)
argued that the merger has not progressed very far yet, and that most of the main component of the cluster
is still undisturbed by the collision. This is consistent with simulations of cluster mergers (Pinkney et al.~1996),
which show that large peculiar velocities can be reproduced in the event of large-scale mergers at the time of
core-crossing. If this merger takes place close to the plane of the sky, it would also explain the non-results
of the statistical tests. These are least sensitive to mergers occuring perpendicular to the line-of-sight.
Therefore, the X-ray morphology -- in combination with the large peculiar motion of the central cD
galaxy -- strongly suggests a recent or commencing merger at the core of the cluster.
\subsection{Dynamical mass estimate}
For the determination of the dynamical mass of the Norma cluster, we have used both the
virial theorem ($M_{\rm VT}$) and the projected mass estimator ($M_{\rm PME}$), see equations 21 and 22 of Pinkney et al.~(1996).
The use of the bi-weight velocity centroid and scale (Beers et al.~1990) in the virial theorem (instead of the velocity
mean and standard deviation) leads to a more robust mass estimate ($M_{\rm RVT}$). The latter is more robust against the effects of contamination
by the inclusion of possible non-members in the analysis. The projected mass estimator (Bird 1995), on the other hand, is sensitive to the
presence of (spatially-separated) subclusters due to its proportionality to the projected distance between galaxy $i$ and the cluster
centroid ($R_{{\perp},i}$) (see equation 22 in Pinkney et al.~1996). The presence of a spatially-separated subcluster (e.g. in a premerger configuration)
would result in a systematic offset with respect to the cluster centroid; this leads to larger values of $R_{{\perp},i}$ and thus to a significantly
larger mass estimate.
For a full discussion of the appropriate use of these dynamical mass estimators we refer to Pinkney et al.~(1996) and Bird (1995).
The three dynamical mass estimates ($M_{\rm VT}$, $M_{\rm RVT}$ and $M_{\rm PME}$)
determined within the three radial limits (using the combined samples of $N$ = 129, 239 and 296
galaxies, respectively) are given in Table~\ref{massnorma}. On average, $M_{\rm RVT}$ is $\sim 5$\% larger than $M_{\rm VT}$. The projected
mass estimate, however, is generally about 50\% larger than $M_{\rm VT}$ and indicates the presence of a spatially-distinct subcluster
(projected on the plane of the sky) presumably in the early stages of merging (Pinkney et al.~1996). This is consistent with our
previous indications of subclustering, particularly in the form of Norma minor (the X-ray subgroup).
B\"ohringer et al.~(1996) and Tamura et al.~(1998) both give an estimate of the gravitational mass of the
Norma cluster based on ROSAT and ASCA X-ray observations, respectively. In Table~\ref{massnorma}, we list the values (converted from $h_{50}^{-1}$
to $h_{73}^{-1}$) of the mass within a specific radius as derived from X-ray observations by B\"ohringer et al.~(1996) and Tamura et al.~(1998).
Both virial mass estimates ($M_{\rm VT}$ and $M_{\rm RVT}$) are consistent with the mass determined from the X-ray luminosity of the cluster.
In the presence of substantial subclustering, as suggested here for the Norma cluster, all dynamical mass estimators could still overestimate the
true mass of the cluster, depending on the projection angle of the cluster--subcluster merger axis with respect to the line of
sight (Pinkney et al.~1996). This effect is smallest for $M_{\rm VT}$ and $M_{\rm RVT}$ for a merger occuring perpendicular to the line-of-sight.
We can therefore safely conclude that the mass of
the Norma cluster within the Abell radius corresponds to $1-1.1 \times 10^{15} h_{73}^{-1}$ M$_{\odot}$.
This confirms the status of the Norma cluster as the most massive cluster in the Great Attractor.
\begin{table}
\centering
\caption{Mass estimates of the Norma cluster.}
\begin{tabular}{@{}lcl@{}}
\hline
\multicolumn{3}{c}{X-ray mass (gravitational)}\\[2pt]
$R$ (h$_{73}^{-1}$ Mpc)$^*$ & $M(<R)$ (h$_{73}^{-1}$ M$_{\odot}$)$^*$ & Reference \\
\hline
0.68 & \hfill $1.5 - 4.0 \times 10^{14}$ & B\"ohringer et al.~(1996) \\
0.75 & \hfill $3 \times 10^{14}$ & Tamura et al.~(1998) \\
2.05 & \hfill $2.9 - 15 \times 10^{14}$ & B\"ohringer et al.~(1996)\\
\hline
\end{tabular}
\begin{tabular}{@{}lccc@{}}
\multicolumn{4}{c}{Dynamical mass}\\[2pt]
$R$ (h$_{73}^{-1}$ Mpc) \hspace{0.8cm} & \multicolumn{3}{c}{$M(<R)$ (h$_{73}^{-1}$ M$_{\odot}$)}\\
all galaxies & $M_{\rm VT}$ & $M_{\rm RVT}$ & $M_{\rm PME}$ \\
\hline
0.67 & \hfill $4.2 \times 10^{14}$ & \hfill $4.3 \times 10^{14}$ & \hfill $6.6 \times 10^{14}$ \\
1.35 & \hfill $8.1 \times 10^{14}$ & \hfill $8.6 \times 10^{14}$ & \hfill $11.6 \times 10^{14}$ \\
2.02 & \hfill $10.4 \times 10^{14}$& \hfill $11.0 \times 10^{14}$ & \hfill $14.7 \times 10^{14}$ \\
\hline
\end{tabular}
\label{massnorma}
{\newline \footnotesize{$^*$ The original values have been converted from $h_{50}^{-1}$ to $h_{73}^{-1}$. \hfill }}
\end{table}
\section{Individual galaxies}
A number of galaxies in the Norma cluster show direct or indirect evidence of interaction with
the intracluster medium (ICM). Here, we will explore these galaxies in some
detail in the light of the preceding discussion.
\subsection{WKK\,6176 and the X-ray tail}
Recent {\it Chandra} and {\it XMM-Newton} observations of WKK 6176 (= ESO 137-001)
revealed the presence of a $\sim$70 kpc long X-ray tail pointing away from the
cluster centre (Sun et al.~2006), suggesting this galaxy is undergoing a significant amount
of gas stripping. The extent of this X-ray tail is unusual (Sun et al.~2006).
We have deep $B$, $V$ and $R_C$-band photometry of WKK\,6176. These data were obtained in 1999 May with the MPG/ESO 2.2-m telescope
at la Silla and the Wide Field Imager (ESO Programme 63.N-0054). We covered the entire Abell radius of the Norma cluster
for the purpose of measuring the $R_C$-band luminosity function (see also Fig.~\ref{optxray}). This optical
as well as the near-infrared $J$, $H$ and $K_s$ luminosity function of the Norma cluster will be
presented in a separate paper in this series.
\begin{figure}
\centerline{\hbox{\psfig{figure=wkk6176rB.ps,width=8.2cm}}}
\caption{The $R_C$-band image of WKK6176 before (left panel) and after star subtraction (right panel) using
the {\tt KILLALL} routine (Buta \& McCall 1999) in IRAF. The field of view is $2.2 \times 4.0$ arcminutes,
North is up and East is to the left. The diagonal bar at the bottom-right in the star-subtracted image indicates
the position angle of the elongated E/S0 population within $R < 0.67$ Mpc.}
\label{wkk6176r}
\end{figure}
WKK\,6176 is located close to the core of the Norma cluster. In Fig.~\ref{optxray} it can be found at (RA, DEC) $\sim$
(16.22425$^{\circ}$, --60.76397$^{\circ}$), to the lower-left of the right-most X-ray contours (aligned with
the virtual extension of the X-ray
subgroup). A close-up of WKK\,6176 is shown in Fig.~\ref{wkk6176r} where we have displayed the $R_C$-band image
in a field of view of $2.2 \times 4.0$ arcminutes.
Given the low Galactic latitude of the Norma cluster ($b \sim -7^{\circ}$) and proximity to the
Galactic bulge (only 35$^{\circ}$ away), a large number of stars are superimposed on the galaxy
images. Reliable photometry can only be obtained after careful
removal of the many foreground stars. We used the {\tt KILLALL} routine developed by Buta \&
McCall (1999) within the IRAF\footnote{IRAF is distributed by the National Optical Astronomy Observatories,
which are operated by the Association of Universities for Research
in Astronomy, Inc., under cooperative agreement with the National
Science Foundation.} environment to remove this stellar contamination.
Fig.~\ref{wkk6176r} illustrates the effectiveness of this star-removal procedure for the
Norma cluster galaxy WKK\,6176 (compare the right-hand panel of Fig.~\ref{wkk6176r} to the original image). It also reveals
numerous striking low-brightness filaments to the west of WKK\,6176, appearing to stream away from WKK\,6176 at a position
angle of $\sim$125$^{\circ}$.
Several bright knots (distinctly different from the Galactic foreground pollution) appear within
these filaments. The low surface brightness filaments are aligned with the X-ray tail (Sun et al.~2006), but are not only
confined to the region of the X-ray tail. For comparison, we have indicated the direction of the major axis of the E/SO
galaxy population with the diagonal marker in Fig.~\ref{wkk6176r}.
WKK\,6176 is a low-redshift equivalent of the two recently detected spiral galaxies in massive rich clusters
(Abell 2667 and Abell 1689) at $z \sim 0.2$ which show clear evidence for galaxy transformation (Cortese et al.~2007).
Interestingly, WKK\,6176 is located at a similar projected distance from the centre of the Norma cluster (0.28 Mpc) as
the two high-redshift spirals in Abell 2667 and Abell 1689, which lie at 0.34 h$_{70}^{-1}$ Mpc and 0.24 h$_{70}^{-1}$ Mpc
from their respective cluster centre.
A full investigation into the properties of WKK\,6176 as derived from multiwavelength photometry ($B V R_C J H K_s$), spectroscopy,
and galaxy evolution modelling (Fritze-von Alvensleben \& Woudt 2006), and its implications for galaxy evolution in dense environments
will be presented elsewhere.
\subsection{WKK\,6305: the head-tail radio continuum source}
\begin{table}
\centering
\caption{Selected galaxies in the Norma cluster.}
\begin{tabular}{@{}llc@{}}
\hline
Galaxy & Observational characteristic & Pos.~angle \\
\hline
WKK\,6176 & -- 70 kpc X-ray tail & 129$^{\circ}$ \\
& -- Optical filaments & 125$^{\circ}$ \\
WKK\,6269$^{\ast}$ & -- Central cD galaxy & 128$^{\circ}$ \\
WKK\,6305 & -- 500 kpc radio-continuum tail & 108$^{\circ}$ \\
\hline
\end{tabular}
{\footnotesize{\newline $^{\ast}$ The position angle quoted here for WKK\,6269 is a mean \hfill \
\newline position angle as determined from GALFIT (Peng et al.~2002) \hfill \
\newline isophotal fitting of deep $K_s$-band imaging. \hfill \ }}
\label{interact}
\end{table}
Another peculiar galaxy in the Norma cluster is WKK\,6305, also known as PKS\,1610-605 (Jones \& McAdam 1996). It is located
at a similar distance from the centre of the cluster (0.29 Mpc) as WKK\,6176.
In Fig.~\ref{central} we show an overview of the central region with the same field of view as
Fig.~\ref{optxray}. The galaxy distribution of confirmed cluster members and the X-ray subgroup are as before, but
now the radio continuum emission of WKK\,6269 and WKK\,6305 at 843 MHz are overplotted (reproduced from Jones \& McAdam 1992).
WKK\,6305 corresponds to the head-tail source visible in Fig.~\ref{central}. The tail length of 26$'$ ($\sim$ 500 kpc at the
distance of the Norma cluster) represents one of the longest radio continuum tails observed. The position angle of the tail
is $\sim$108$^{\circ}$ (Jones \& McAdam 1996) and is, as before with the X-ray tail of WKK\,6176, closely aligned with the
elongated (E/S0) galaxy distribution in the cluster.
\begin{figure}
\centerline{\hbox{\psfig{figure=multiwave.ps,width=8.2cm}}}
\caption{The galaxy distribution and the X-ray subgroup of the inner $\sim$0.66 Mpc $\times$ 0.66 Mpc
(as in Fig.~\ref{optxray}) with the radio continuum emission (reproduced from Jones \& McAdam 1992)
of WKK\,6269 and WKK\,6305 overplotted. WKK\,6176 is indicated by the large dot in the virtual
extension of the X-ray subgroup.}
\label{central}
\end{figure}
\section{Discussion}
The dynamical analysis of the Norma cluster presented here has revealed a significant amount of subclustering
in this nearby rich cluster, ranging from the central X-ray group to the two spiral-rich subgroups further from the
core of the cluster. Even though the X-ray group did not show up in the dynamical analysis, the large peculiar
velocity of the cD galaxy, whose radio lobes appear to `embrace' the X-ray contours of this central group, is a tell-tale
sign of an ongoing merger. The large discrepancy between $M_{\rm PME}$ and the mass determined from the virial theorem (where
$M_{\rm MPE} \sim 1.5 \times M_{VT}$) is an independent indication of the presence of a spatially-separated
subcluster of substantial mass.
Significant subclustering is not unusual for rich and massive clusters; Colless \& Dunn (1996)
in their detailed dynamical analysis of the archetypical rich Coma cluster also revealed the presence of an ongoing merger.
The galaxy distribution in the Norma cluster is clearly elongated, with a position angle (in equatorial coordinates)
ranging between 116$^{\circ}$ (for the central part) and $\sim$105$^{\circ}$ (for the overall distribution). It should be
emphasized that this observed elongation is not an artefact of selective Galactic extinction effects at this low
Galactic latitude. The position angle of the major axis of the Norma cluster, as indicated by the arrows in Fig.~\ref{dsall},
runs nearly perpendicular to the lines of constant Galactic foreground
extinction (compare with Fig.~\ref{distribution}). The elongated galaxy distribution is aligned with the major
large-scale structure in this region as can be seen in Fig.~\ref{lssoverview}. Such an alignment is not unexpected
within the cluster-rich GA environment (Binggeli 1982).
Within the cluster itself, various galaxies show clear evidence for interactions with the intracluster
medium. An overview of these galaxies and their defining characteristics is given in Table~\ref{interact}.
The defining features of these galaxies are strongly aligned with the general galaxy distribution of the cluster. In the
case of WKK\,6176 and WKK\,6305, they are X-ray/optical and radio continuum tails, respectively, whereas for WKK\,6269 (the central cD galaxy)
the major axis of the galaxy is aligned with the cluster (see Table~\ref{interact}).
\section{Conclusion}
The Norma cluster (ACO 3627) is a nearby, rich and massive cluster -- on par with the more distant Coma cluster --
which resides at the bottom of the potential well of the Great Attractor.
The galaxy distribution of the cluster members shows a clear elongation which is aligned with the main
wall-like structures of the GA. Despite the relaxed appearance of the early-type galaxy population in the
Norma cluster, a large amount of subclustering is present. We have identified two spiral-rich subclusters (Norma A and B) in
addition to the previously identified central (X-ray) subcluster (Norma minor). The ongoing merger of the latter with the main
cluster (Norma major) is assumed to be responsible for the large peculiar motion of the central cD galaxy.
The proximity of the Norma cluster offers an excellent opportunity to study the interaction of cluster members such as WKK\,6176
with the intracluster medium at high resolution and sensitivity.
\section*{Acknowledgments}
We thank J.~Pinkney for providing his cluster substructure analysis programme and M.~McCall for the
use of his KILLALL routine.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is
operated by the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration. PAW, RCKK and APF kindly acknowledge
funding from the National Research Foundation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,288 |
{"url":"https:\/\/learningdeep.xyz\/face-to-Age-Prediction\/","text":"# Face to Age Prediction Using Convolution Networks\n\n1. Codes discussed in this blog post\u00a0can be found at my GITHUB repository\n\n2. You may download\u00a0trained model for to quickly run face to age prediction or to fine tune on your custom data-set.\n\n3. If you encounter any error, please try to check with requirements.txt for proper required python package versions.\n\n4. ## Code Compatibility : python 2.7 , tested on ubuntu 16.04 with theano as backend\n\nIn recent days Microsoft is advertising\u00a0face to age prediction to show azure\u2019s machine learning capability. A website https:\/\/how-old.net\/ by Microsoft allows user to upload photo for free and predicts age for the same.\u00a0Behind the scene a convolution network is working to produce this magic.\n\nIn present blog I will be reverse engineering, the science of machine learning behind this technology. The aim of this tutorial is to provide basic implementation of face to age technique. This tutorial is not for any monitory purpose.\n\n# Data Collection\n\nWe require well annotated data-set wherein each photo is tagged with real age. Such data-sets are available at https:\/\/data.vision.ee.ethz.ch\/cvl\/rrothe\/imdb-wiki\/. This website is open source collection of images from Wikipedia and IMDB. Each image is having year of birth and time stamp when it was uploaded to \u00a0Wikipedia or IMDB. The total collection is about 300GB of data. Looking at my convenience and processing infrastructure I have, I have selected \u201cIMDB Only face data-set\u201d which is of size 7GB in compressed form.\n\n# Data Pre-Processing\n\nDownloaded images are of various size and formats. For machine learning images need to be of proper dimension and format.\u00a0 Each image in data-set is named in a format as shown in example below:\n\n\u2022 nm7153885_rm3814127104_1990-8-15_2015.jpg\n\u2022 nm7153885_rm4089047552_1990-8-15_2015.jpg\n\u2022 nm7153885_rm4149671424_1990-8-15_2015.jpg\n\u2022 nm7153885_rm4172933632_1990-8-15_2015.jpg\n\u2022 nm7153885_rm4206488064_1990-8-15_2015.jpg\n\nWhere in the first image,\u00a01990-8-15 is the date of birth (DOB) and 2015 is year of image upload.\u00a0 We will choose year from date of birth (DOB) \u00a0and difference between date of upload and \u00a0year of birth will provide age of image. This can be simply done using regular expression \u2013 nm*\\d+rm\\d+(\\d+)-\\d+-\\d+_(\\d+).jpg\u00a0 where for above given example,\u00a0(\\d+) will capture\u00a01990-8-15 [DOB]\u00a0and(\\d+) will capture 2015 [year of photo uploaded].\n\ndef imageResize(basename,imageName):\n\"\"\"\nresize image\nimage name : xyz.jpg\nNew folder in the working directory will be created with '_resized' as suffix\n\"\"\"\nnew_width \u00a0= 128\nnew_height = 128\ntry:\nimg = Image.open(basename+\"\/\"+imageName) # image extension *.png,*.jpg\nimg = img.resize((new_width, new_height), Image.ANTIALIAS)\nimg.save(basename+'_resized\/'+imageName)\nexcept:\nos.mkdir(basename+'_resized\/')\nimg = Image.open(basename+\"\/\"+imageName) # image extension *.png,*.jpg\nimg = img.resize((new_width, new_height), Image.ANTIALIAS)\nimg.save(basename+'_resized\/'+imageName)\n\ndef resizer(folderPath):\n\"\"\"\nto resize all files present in a folder\n\"\"\"\n\nfor subdir, dirs, files in os.walk(folderPath):\nfor fileName in files:\ntry:\n# \u00a0print os.path.join(subdir, file)\nfilepath = subdir + os.sep + fileName\n# \u00a0print filepath\nif filepath.endswith(\".jpg\" or \".jpeg\" or \".png\" or \".gif\"):\nimageResize(subdir,fileName)\nexcept:\nprint traceback.print_exc()\n# to resize all images in given folder, run below given line\n\nresizer('imdb_crop')\n\n\n# Network\u00a0Definition\n\nWe will be using VGG16 network architecture, there is no particular reason for using it but stochastically \u00a0it does perform well on majority of image recognition use cases. VGG16 network is as shown in below given image. Output layer is changed to 100, which is equal to the number of classes we possess.\n\n# defining convolutional network\nmodel = Sequential()\n# model.summary()\n\n\nFigure 1. Architecture of VGG16 Convolution Network used for face to age prediction.\n\n# Training\n\nIMDB data-set is about 4,50,000 + images and it is very much impractical to keep all of them in memory. So unlike previous tutorial, we will use data generator in this tutorial. Data generator will dynamically read \u00a010,000 images from disk and load it in to RAM. After generating Numpy array and corresponding age vector of all images, data-generator passes 50 images at a time to GPU for processing. VGG16 is a huge network and my GPU cannot accommodate more than 10 images at a time in memory. One can load higher number of images to GPU as per resource availability.\n\nProvide GENERATOR functioning flow here.\n\n# this function will load images iteratively in memory\n# CPU and GPU memory friendly iterator\n\ndef myGeneratorEX(samples_per_epoch,epoch):\n\"\"\"\nsamples_per_epoch : number of images to be loaded in CPU memory at a time\nepoch : number of epochs for training\n\"\"\"\n# defining optimizer function\nsgd = SGD(lr=0.01, momentum=0.1, decay=0.0, nesterov=True)\n# compiling model\nmodel.compile(optimizer=sgd, loss='categorical_crossentropy',metrics=['accuracy'])\n\nfolderName = \"imdb_crop_resized\" # folder name where resized images are placed\n\nfileNames = \u00a0glob.glob(folderName+\"\/*.jpg\") #All file names with .jpg extension\n\n# first 100 imageswill be ued for onthe fly visual performace checking at each iteration\ninitialFileNames = fileNames[:100]\n\nk =0\nwhile k < epoch: # for each epoch do following\nprint \"Epoch : \",k,\" | Total Images : \",len(fileNames)\nfor i in range(len(fileNames)\/samples_per_epoch):\n#All files (~438189) are loaded in memory with batch of size \u00a0'samples_per_epoch' e.g.1000\ntry:\n# loaded images are converted to numpy array\nx_batch,y_batch = turnToNumpy(fileNames[i\\*samples_per_epoch:(i+1)\\*samples_per_epoch])\n\n# such all images are made up of numpy array of range integer 0 - 255(8 bit image)\n# all images are normalised between 0-1 float\nx_batch = x_batch\/255.0\n# to check wheather or not our algorithm is learning. to cheack wheather our algorith started differentiating between age.\nx_batch_test,y_batch_test = turnToNumpy(initialFileNames)\nx_batch_test = x_batch_test\/255.0\n# fit the data on model\nmodel.fit(x_batch,y_batch,batch_size=50,nb_epoch=1, verbose=1,validation_split=0.2)\n# test on initial 100 files at each iteration\ntest_output = model.predict_classes(x_batch_test)\nprint test_output\nexcept IndexError:\nprint traceback.print_exc()\nk = k+1\n\n\nTraining takes very long time, On AWS server with 3000+ CUDA core and 11+ GB memory, it took 8 days for me. I have choose\u00a0a set of 100 files (online test set) initially and after each iteration i have tested model on online test set to get predicted age. The learning behavior of algorithm is quite intuitive. Initially the algorithm has no clue but after some iteration it starts making sense out of data.\n\nFigure 2. Learning progress as training iteration passes. After 468 iteration network actually started learning age difference.\n\nAll lines with different color represent different iteration. \u00a0At iteration 1 for all 100 online test set images predicted age was 25 years. Similarly for iteration 55, age was 38 for all images. up to iteration 469 machine had no clue about data but after iteration 469 it slowly started distinguishing between ages by looking at image. At iteration 624 it started predicting in narrow range of 25 to 40. As learning progresses this learning become more powerful and it really start predicting well and in broader range 0 - 100.\n\n# Analysis\n\nIn \u00a0present experiment \u00a0I have not separated data as test and train. It is very difficult for a machine to remember data from such huge data-set so we can randomly pick few images from train data itself and allow model to predict on the same.\n\nFigure 3. predicted v\/s actual age at the end of training.\n\nI have provided a line plot showing result for 500 images with actual and predicted age. It shows following characteristics of learning :\n\n1. model is capable of predicting extremes of ages\n2. with very low data-set 7GB compared to 300GB actual one, model is still doing well\n\nWe will be looking at \u00a0some samples with their actual and predicted age labelled.\n\nA. Below are the few images of good predictions.\n\nFigure 4. Good face to age predictions by algorithm [A = Actual, P = Predicted]\n\nOur algorithm predicted well on cases:\n\n1. Where there was an error from database,\u00a0labeled age was wrong but algorithm predicted it correctly\n\n2. Where algorithm predicted the age correctly, with minor fluctuations.\n\nB. Below are the few images of bad predictions.\n\nFigure 5. These are\u00a0cases where algorithm performed very badly. [A = Actual, P = Predicted]\n\nC. Effect of multiple faces.\n\nFigure 6. Errors in Prediction \u00a0due to multiple faces.\u00a0[A = Actual, P = Predicted]\n\nAs I am not using any intermediate face detection and isolation step, multiple faces greatly affects\u00a0\u00a0predictions. For example image no 10 , where Actual age was 8 [of one of the child] but predicted age is 70 [seems correct for aged person].\n\nPossible Improvements:\n\n1. Using face detection, this will allow us to better deal with photos with multiple faces\n\n2. Using larger data-sets. I have a gut feeling that Microsoft actually using entire 300 GB of data to give state of the art results\n3. Having separate male and female data-set can improve predictions a lot, because apparent features for male and female of same age are different.\nShare:","date":"2020-02-24 20:15:54","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\": 0.19455812871456146, \"perplexity\": 5429.755399865805}, \"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-2020-10\/segments\/1581875145981.35\/warc\/CC-MAIN-20200224193815-20200224223815-00246.warc.gz\"}"} | null | null |
Q: How to perform click for this date picker, All have same ID, index value in UIautomatorviewer How to find element for date picker in appium, all the buttons have same resource ID and index as well(Some sample code will help, i have tried few but in those case index value were different in date picker). Even im not sure how to find xpath for these, is there a tool or something for finding xpath in appium ?
http://tinypic.com/r/34gstvd/8
A: Refer to each NumberPicker field and after you can interact with it (sendKeys, clear etc.).
By.xpath("//android.widget.LinearLayout[@index='0']//android.widget.EditText")
By.xpath("//android.widget.LinearLayout[@index='1']//android.widget.EditText")
By.xpath("//android.widget.LinearLayout[@index='2']//android.widget.EditText")
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Megachile miranda is a species of bee in the family Megachilidae. It was described by Vachal in 1908.
References
Miranda
Insects described in 1908 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,310 |
{"url":"https:\/\/alanrendall.wordpress.com\/page\/2\/","text":"## The importance of dendritic\u00a0cells\n\nOctober 30, 2016\n\nI just realized that something I wrote in a previous post does not make logical sense. This was not just due to a gap in my exposition but to a gap in my understanding. I now want to correct it. A good source for the correct story is a video by Ira Mellman of Genentech. I first recall some standard things about antigen presentation. In this process peptides are presented on the surface of cells with MHC molecules which are of two types I and II. MHC Class I molecules are found on essentially all cells and can present proteins coming from viruses infecting the cell concerned. MHC Class II molecules are found only on special cells called professional antigen presenting cells. These are macrophages, T cells and dendritic cells. The champions in antigen presentations are the dendritic cells and those are the ones I will be talking about here. In order for a T cell to be activated it needs two signals. The first comes through the T cell receptor interacting with the peptide-MHC complex on an APC. The second comes from CD28 on the T cell surface interacting with B7.1 and B7.2 on the APC.\n\nConsider now an ordinary cell, not an APC, which is infected with a virus. This could, for instance be an epithelial cell infected with a influenza virus. This cell will present peptides derived from the virus with MHC Class I molecules. These can be recognized by activated ${\\rm CD8}^+$ T cells which can then kill the epithelial cell and put an end to the viral reproduction in that cell. The way I put it in the previous post it looked like the T cell could be activated by the antigen presented on the target cell with the help of CD28 stimulation. The problem is that the cell presenting the antigen in this case is an amateur. It has no B7.1 or B7.2 and so cannot signal through CD28. The real story is more complicated. The fact is that dendritic cells can also present antigen on MHC Class I, including peptides which are external to their own function. A possible mechanism explained in the video of Mellman (I do not know if it is certain whether this is the mechanism, or whether it is the only one) is that a cell infected by a virus is ingested by a dendritic cell by phagocytosis, so that proteins which were outside the dendritic cell are now inside and can be brought into the pathway of MHC Class I presentation. This process is known as cross presentation. Dendritic cells also have tools of the innate immune system, such as toll-like receptors, at their disposal. When they recognise the presence of a virus by these means they upregulate B7.1 and B7.2 and are then in a position to activate ${\\rm CD8}^+$ T cells. Note that in this case the virus will be inside the dendritic cell but not infecting it. There are viruses which use dendritic cells for their own purposes, reproducing there or hitching a lift to the lymph nodes where they can infect their favourite cells. An example is HIV. The main receptor used by this virus to enter the cells is CD4 and this is present not only on T cells but also on dendritic cells. Another interesting side issue is that dendritic cells can not only activate T cells but also influence the differentiation of these cells into various different types. The reason is that the detection instruments of the dendritic cell not only recognise that a pathogen is there but can also classify it to some extent (Mellman talks about a bar code). Based on this information the dendritic cell secretes various cytokines which influence the differentiation process. For instance they can influence whether a T-helper cell becomes of type Th1 or Th2. This is related to work which I did quite a long time ago on an ODE system modelling the interactions of T cells and macrophages. In view of what I just said it \u1e3fight be interesting to study an inhomogeneous version of this system. The idea is to include an external input of cytokines coming from dendritic cells. In fact the unknowns in the system are not the concentrations of cytokines but the populations of cells. Thus it would be appropriate to introduce an inhomogeneous contribution into the terms describing the production of different types of cells.\n\n## Modern cancer therapies\n\nOctober 28, 2016\n\nI find the subject of cancer therapies fascinating. My particular interest is in the possibility of obtaining new insights by modelling and what role mathematics can play in this endeavour. I have heard many talks related to these subjects, both live and online. I was stimulated to write this post by a video of Martin McMahon, then at UCSF. It made me want to systematize some of the knowledge I have obtained from that video (which is already a few years old) and from other sources. First I should fix my terminology. I use the term \u2018modern cancer therapies\u2019 to distinguish a certain group of treatments from what I will call \u2018classical cancer therapies\u2019. The latter are still of central importance today and the characteristic feature of those I am calling \u2018modern\u2019 here is that they have only been developed in the last few years. I start by reviewing the \u2018classical therapies\u2019, surgery, radiotherapy and chemotherapy. Surgery can be very successful when it works. The aim is to remove all the cancerous cells. There is a tension between removing too little (so that a few malignant cells could remain and restart the tumour) and too much (which could mean too much damage to healthy tissues). A particularly difficult case is that of the glioma where it is impossible to determine the extent of the tumour by imaging techniques alone. An alternative to this is provided by the work of Kristin Swanson, which I mentioned in a previous post. She has developed techniques of using a mathematical model of the tumour (with reaction-diffusion equations) to predict the extent of the tumour. The results of a simulation, specific to a particular patient, is given to the surgeon to guide his work. In the case of radiotherapy radiation is used to kill cancer cells while trying to avoid killing too many healthy cells. A problematic aspect is that the cells are killed by damaging their DNA and this kind of damage may lead to the development of new cancers. In chemotherapy a chemical substance (poison) is used with the same basic aim as in radiotherapy. The substance is chosen to have the greatest effect on cells which divide frequently. This is the case with cancer cells but unfortunately they are not the only ones. A problem with radiotherapy and chemotherapy is their poor specificity.\n\nNow I come to the \u2018modern\u2019 therapies. One class of substances used is that of kinase inhibitors. The underlying idea is as follows. Whether cells divide or not is controlled by a signal transduction network, a complicated set of chemical reactions in the cell. In the course of time mutations can accumulate in a cell and when enough relevant mutations are present the transduction network is disrupted. The cell is instructed to divide under circumstances under which it would normally not do so. The cells dividing in an uncontrolled way constitute cancer. The signals in this type of network are often passed on by phosphorylation, the attachment of phosphate groups to certain proteins. The enzymes which catalyse the process of phosphorylation are called kinases. A typical problem then is that due to a mutation a kinase is active all the time and not just when it should be. A switch which activates the signalling network is stuck in the \u2018on\u2019 position. This can in principal be changed by blocking the kinase so that it can no longer send its signals. An early and successful example of this is the kinase inhibitor imatinib which was developed as therapy for chronic myelogenous leukemia (CML). It seems that this drug can even cure CML in many cases, in the sense that after a time (two years) no mutated cells can be detected and the disease does not come back if the treatment is stopped. McMahon talks about this while being understandibly cautious about using the word cure in the context of any type of cancer. One general point about the \u2018modern\u2019 therapies is that they do not work for a wide range of cancers or even for the majority of patients with a given type of cancer. It is rather the case that cancer can be divided into more and more subtypes by analysing it with molecular methods and the therapy only works in a very specific class of patients, having a specific mutation. I have said something about another treatment using a kinase, Vemurafenib in a previous post. An unfortunate aspect of the therapies using kinase inhibitors is that while they provide spectacular short-term successes their effects often do not last more than a few months due to the development of resistance. A second mutation can rewire the network and overcome the blockade. (Might mathematical models be used to understand better which types of rewiring are relevant?) The picture of this I had, which now appears to me to be wrong, was that after a while on the drug a new mutation appears which gives the resistance. The picture I got from McMahon\u2019s video was a different one. It seems that the mutations which might lead to resistance are often there before treatment begins. They were in Darwinian competition with other cells without the second mutation which were fitter. The treatment causes the fitness of the cells without the second mutation to decrease sharply. This removes the competition and allows the population of resistant cells to increase.\n\nAnother drug mentioned by McMahon is herceptin. This is used to treat breast cancer patients with a mutation in a particular receptor. The drug is an antibody and binds to the receptor. As far as I can see it is not known why the binding of the antibody has a therapeutic effect but there is one idea on this which I find attractive. This is that the antibodies attract immune cells which kill the cell carrying the mutation. This gives me a perfect transition to a discussion of a class of therapies which started to become successful and popular very recently and go under the name of cancer immunotherapy, since they are based on the idea of persuading immune cells to attack cancer cells. I have already discussed one way of doing this, using antibodies to increase the activities of T cells, in a previous post. Rather than saying more about that I want to go on to the topic of genetically modified T cells, which was also mentioned briefly here.\n\nI do not know enough to be able to give a broad review of cellular immunotherapy for cancer treatment and so I will concentrate on making some comments based on a video on this subject by Stephan Grupp. He is talking about the therapy of acute lymphocytic leukemia (ALL). In particular he is concerned with B cell leukemia. The idea is to make artificial T cells which recognise the surface molecule CD19 characteristic of B cells. T cells are taken from the patient and modified to express a chimeric T cell receptor (CAR). The CAR is made of an external part coming from an antibody fused to an internal part including a CD3 $\\zeta$-chain and a costimulatory molecule such as CD28. (Grupp prefers a different costimulatory molecule.) The cells are activated and caused to proliferate in vitro and then injected back into the patient. In many cases they are successful in killing the B cells of the patient and producing a lasting remission. It should be noted that most of the patients are small children and that most cases can be treated very effectively with classical chemotherapy. The children being treated with immunotherapy are the \u2018worst cases\u2019. The first patient treated by Grupp with this method was a seven year old girl and the treatment was finally very successful. Nevertheless it did at first almost kill her and this is not the only case. The problem was a cytokine release syndrome with extremely high levels of IL-6. Fortunately this was discovered just in time and she was treated with an antibody to IL-6 which not only existed but was approved for the treatment of children (with other diseases). It very quickly solved the problem. One issue which remains to be mentioned is that when the treatment is successful the T cells are so effective that the patient is left without B cells. Hence as long as the treatment continues immunoglobulin replacement therapy is necessary. Thus the issue arises whether this can be a final treatment or whether it should be seen a preparation for a bone marrow transplant. As a side issue from this story I wonder if modelling could bring some more insight for the IL-6 problem. Grupp uses some network language in talking about it, saying that the problem is a \u2018simple feedback loop\u2019. After I had written this I discovered a preprint on BioRxiv doing mathematical modelling of CAR T cell therapy of B-ALL and promising to do more in the future. It is an ODE model where there is no explicit inclusion of IL-6 but rather a generic inflammation variable.\n\n## In the beginning was the\u00a0worm\n\nSeptember 29, 2016\n\nIn a previous post I mentioned the book by Andrew Brown whose title I have used here. I came across it in a second hand bookshop in Berkeley when I was spending time at MSRI in 2009. I read it with pleasure then and now I have read it again. It contains the story of how the worm Caenorhabditis elegans became an important model organism. This came about because Sydney Brenner deliberately searched for an organism with favourable properties and promoted it very effectively once he had found it. It is transparent so that it is possible to see what is going on inside it and it is easy to keep in the lab and reproduces fast enough in order to allow genetic research to be done rapidly. The organism sought was supposed to have a suitable sexual system. C. elegans is normally hermaphrodite but does also have males and so it is acceptable from that point of view. One further important fact about C. elegans is that it has a nervous system, albeit a relatively simple one. (More precisely, it has two nervous systems but I have not looked into the details of that issue.) Brenner was looking to understand how genetics determines behaviour and C. elegans gave him an opportunity to make an attack on this problem in two steps. First understand how to get from genes to neurons and then understand how to get from neurons to behaviour. C. elegans has a total of 302 neurons. It has 959 cells in total, not including eggs and sperm. Among the remarkable things known about the worm are the complete developmental history of each of its cells and the wiring diagram of its neurons. There are about 6400 synapses but the exact number, unlike the number of cells or neurons, is dependent on the individual. For orientation note that C. elegans is a eukaryotic organism (in contrast to phages or E. coli) which is multicellular (in contrast to Saccharomyces cerevisiae) and it is an animal (in contrast to Arabidopsis thaliana). Otherwise, among the class of model organisms, it is as simple and fast reproducing as possible. In particular it is simpler than Drosophila, which was traditionally the favourite multicellular model organism of the geneticists.\n\nIn this blog I have previously mentioned Sydney Brenner and expressed my admiration for him. I have twice met him personally when he was giving talks in Berlin and I have also watched a number of videos of him which are available on the web and read various texts he has written. In this way I have experienced a little of the magnetism which allowed him to inspire gifted and risk-taking young scientists to work on the worm. Brenner spent 20 years at the Laboratory of Molecular Biology in Cambridge, a large part of it as director of that organization. In the pioneering days of molecular biology the lab was producing Nobel prizes in series. He had to wait until 2002 for his own Nobel prize (for physiology or medicine), shared with John Sulston and Robert Horvitz. In his Nobel speech Brenner said that he felt there was a fourth prizewinner, C. elegans, which, however, did not get a share of the money. My other favourite quote from that speech is his description of the (then) present state of molecular biology, \u2018drowning in a sea of data, starving for knowledge\u2019. Since then that problem has only got worse.\n\nNow I will collect some \u2018firsts\u2019 associated with C. elegans. It was the first multicellular organism to have its whole genome sequenced, in 1998. This can also be seen as the point of departure for the human genome project. Here the worm people overtook the drosophilists and the Drosophila genome was only finished in 2000. Sulston played a central role in the public project to sequence the human genome and the struggle with the commercial project of Craig Venter. It was only the link between the worm genome project and the human one which allowed enough money to be raised to finish the worm sequence. According to the book Sulston was more interested in the worm project since he wanted to properly finish what he had started. Martin Chalfie, coming from the worm community introduced GFP (green fluorescent protein) into molecular biology. He first expressed it in E. coli and C. elegans. He got a Nobel prize for that in 2008. microRNA (miRNA) was first found in C. elegans. It is the basis of RNA interference (RNAi), also first found in C. elegans. This earned a Nobel prize in 2006. The genetics of the process of apoptosis (programmed cell death) was understood by studying C. elegans. When Sulston was investigated the cell lineage he saw that certain cells had to die as part of the developmental process. Exactly 131 cells die during this process.\n\nTo conclude I mention a couple of features of C. elegans going beyond the time covered by the book. I asked myself what we can learn about the immune system from C. elegans. Presumably every living organism needs an immune system to survive in a hostile environment. The adaptive immune system in the form known in humans only exists in vertebrates and hence, in particular, not in the worm. Some related comments can be found here. It seems that C. elegans has no adaptive immune system at all but it does have innate immunity. It has cells called coelomocytes which have at least some resemblance to immune cells. It has six of them in total. Compare this with more than $10^9$ immune cells per litre in our blood. C. elegans eats bacteria. These days the human gut flora is a fashionable topic. A couple of weeks ago I heard a talk by Giulia Enders, the author of the book \u2018Darm mit Charme\u2019 which sold a million copies in 2014. I had bought and read the book and found it interesting although I was not really enthusiastic about it. Now TV advertising includes products aimed at the gut flora of cats. So what about C. elegans? Does it have an interesting gut flora? The answer seems to be yes. See for instance the 2013 article \u2018Worms need microbes too\u2019 in EMBO Mol. Med. 5, 1300.\n\n## Models for photosynthesis, part\u00a04\n\nSeptember 19, 2016\n\nIn previous posts in this series I introduced some models for the Calvin cycle of photosynthesis and mentioned some open questions concerning them. I have now written a paper where on the one hand I survey a number of mathematical models for the Calvin cycle and the relations between them and on the other hand I am able to provide answers to some of the open questions. One question was that of the definition of the Pettersson model. As I indicated previously this was not clear from the literature. My answer to the question is that this system should be treated as a system of DAE (differential-algebraic equations). In other words it can be written as a set of ODE $\\dot x=f(x,y)$ coupled to a set of algebraic equations $g(x,y)=0$. In general it is not clear that this type of system is locally well-posed. In other words, given a pair $(x_0,y_0)$ with $g(x_0,y_0)=0$ it is not clear whether there is a solution $(x(t),y(t))$ of the system, local in time, with $x(0)=x_0$ and $y(0)=y_0$. Of course if the partial derivative of $g$ with repect to $y$ is invertible it follows by the implicit function theorem that $g(x,y)=0$ is locally equivalent to a relation $y=h(x)$ and the original system is equivalent to $\\dot x=f(x,h(x))$. Then local well-posedness is clear. The calculations in the 1988 paper of Pettersson and Ryde-Pettersson indicate that this should be true for the Pettersson model but there are details missing in the paper and I have not (yet) been able to supply these. The conservative strategy is then to stick to the DAE picture. Then we do not have a basis for studying the dynamics but at least we have a well-defined system of equations and it is meaningful to discuss its steady states.\n\nI was able to prove that there are parameter values for which the Pettersson model has at least two distinct positive steady states. In doing this I was helped by an earlier (1987) paper of Pettersson and Ryde-Pettersson. The idea is to shut off the process of storage as starch so as to get a subnetwork. If two steady states can be obtained for this modified system we may be able to get steady states for the original system using the implicit function theorem. There are some more complications but the a key step in the proof is the one just described. So how do we get steady states for the modified system? The idea is to solve many of the equations explicitly so that the problem reduces to a single equation for one unknown, the concentration of DHAP. (When applying the implicit function theorem we have to use a system of two equations for two unknowns.) In the end we are left with a quadratic equation and we can arrange for the coefficients in that equation to have convenient properties by choosing the parameters in the dynamical system suitably. This approach can be put in a wider context using the concept of stoichiometric generators but the proof is not logically dependent on using the theory of those objects.\n\nHaving got some information about the Pettersson model we may ask what happens when we go over to the Poolman model. The Poolman model is a system of ODE from the start and so we do not have any conceptual problems in that case. The method of construction of steady states can be adapted rather easily so as to apply to the system of DAE related to the Poolman model (let us call it the reduced Poolman model since it can be expressed as a singular limit of the Poolman model). The result is that there are parameter values for which the reduced Poolman model has at least three steady states. Whether the Poolman model itself can have three steady states is not yet clear since it is not clear whether the transverse eigenvalues (in the sense of GSPT) are all non-zero.\n\nBy analogy with known facts the following intuitive picture can be developed. Note, however, that this intuition has not yet been confirmed by proofs. In the picture one of the positive steady states of the Pettersson model is stable and the other unstable. Steady states on the boundary where some concentrations are zero are stable. Under the perturbation from the Pettersson model to the reduced Poolman model an additional stable positive steady state bifurcates from the boundary and joins the other two. This picture may be an oversimplification but I hope that it contains some grain of truth.\n\n## A visit to\u00a0Iceland\n\nSeptember 4, 2016\n\nI just returned from 10 days as a tourist in Iceland. It was not my first time there. Years ago I went on a cruise which included two stops in Iceland, one in Reykjavik and one in Akureyri. In each case there was a short bus tour to see some typical sights \u2013 a waterfall, a geysir and a volcanic region with bubbling mud and hot springs. This time I had a chance to see a lot more. I enjoyed the cruise a lot but I had the impression that the majority of the people on the ship were very bored. The main antidote to the boredom offered was lots of food. One day I got up from the dinner table and went on deck. There had been no indication that there might be something interesting to see. When I opened the door I was confronted with a beautifully conical volcano covered with ice, Snaefellsj\u00f6kull. (I will come back to the interesting issue of Icelandic pronuciation later.) This was one of my strongest impressions from the whole cruise. The volcano is at the end of a long peninsula to the north of Reykjavik. This time I learned that this volcano is the esoteric centre of Iceland and that it was the starting point of Jules Verne\u2019s Journey to the Centre of the Earth. This was not really a point on the programme of the tour this time but it was clearly visible with binoculars from the hotel in Rekjavik where we spent the last two nights. I had one view of it which was monochrome due to the light conditions but where the sharp edges of the crater stood our clearly.\n\nOne attraction of Iceland for me was the bird life. It seems that the country has claimed exclusive rights on the puffin. The numerous tourist shops in Reykjavik are called \u2018puffin shops\u2019 due to the number of representations of that bird they sell. I also saw puffin offered as one of the constituents of a special menu also including whale and horse meat. We spent one night in Vik and I discovered two stranded fulmars on a grass area not far from the hotel. These birds can only take off from an elevated starting point like a cliff or from water. If one lands on a flat area some distance from the sea then it is doomed unless it gets help. I rescued two of them by carrying them (ten minutes walk including crossing a road with significant traffic) to the sea. Since there had not been a big storm I suppose they had come down during their first flight after leaving the nest. (There were fulmars nesting on inland cliffs on the other side of the hotel from the sea.) Fortunately I still knew how to catch them and pick them up without hurting them or being the victim of their defence mechanism of spitting foul-smelling oil when feeling threatened. I enjoyed seeing a few glaucous gulls in the harbour in Reykjavik on the last day. Probably the last time I saw any was on the cruise I already mentioned. It was also nice to see and hear many whimbrels. The first one already welcomed me at the airport when I arrived.\n\nI felt at home in the natural surroundings in Iceland and after a few days I thought of one explanation. There are very few trees in Iceland and this is just as it is in Orkney where I grew up. The first time I was on the mainland of Scotland when I was four years old I said \u2018I don\u2019t like this place \u2013 you can\u2019t see anything for trees\u2019. There are many areas in Iceland where there is only sand, rocks, water and ice. I had the impression of seeing what the Earth is really like, without the veil of green which we usually see. I also got an impression of what it is really like to live next to a volcano. I was in a museum close to (and devoted to) Eyafjallaj\u00f6kull, the infamous producer of ash with the complicated name. I learned that one US journalist just called it E15, due to the number of letters. There was a film showing in the museum explaining what people living near the volcano experienced at the time of the last eruption. Coming back to the name, the pronunciation of Icelandic does seems to be a difficult question but also an interesting one. I would like to spend some time understanding it better. There are nice videos on the pronuciation of Eyafjallaj\u00f6kull here and here. The final double l is the really tricky point. There are points of similarity between the Icelandic language and the dialect I grew up with. This is due to the influence of an extinct language called Norn which was spoken on Orkney and Shetland in past centuries and which is related to (Old) Icelandic. For instance the oystercatcher is called tjaldur in Icelandic and chaldro in our dialect.\n\nI also had some culinary experiences. At breakfast in the hotels there was always a bottle of cod liver oil on the table. I remember this liquid from my childhood as a threat used on young children. \u2018If you do not behave yourself I will give you a spoonfull of cod liver oil.\u2019 Due to persistent encouragement from Eva I tried a little and found it not as bad as I expected. Our guide also gave us some pieces of Greenland shark to try. He gave\u00a0 us a warning about the taste and some of the alcholic drink called the black death to wash it down with. It tastes of nothing at first but chewing leads to a strong taste reminiscent of urine. In fact the flesh of the shark is poisonous due to its content of trimethylamine N-oxide. In Iceland it is treated by first burying it for several weeks and then drying it to get rid of the poison. The result is considered a delicacy. The Greenland shark is interesting because of the fact that it was recently discovered that it can live to be four hundred years old, only becoming sexually mature when it is 150. I want to read more about it.\n\nAs a final comment on Iceland: the weather was much better than we expected!\n\n## ECMTB 2016 in\u00a0Nottingham\n\nJuly 17, 2016\n\nThis past week I attended a conference of the ESMTB and the SMB in Nottingham. My accomodation was in a hall of residence on the campus and my plan was to take a tram from the train station. When I arrived it turned out that the trams were not running. I did not find out the exact reason but it seemed that it was a problem which would not be solved quickly. Instead of finding out what bus I should take and where I should take it from I checked out the possibility of walking. As it turned out it was neither unreasonably far nor complicated. Thus, following my vocation as pedestrian, I walked there.\n\nAmong the plenary talks at the conference was one by Hisashi Ohtsuki on the evolution of social norms. Although I am a great believer in the application of mathematics to many real world problems I do become a bit sceptical when the area of application goes in the direction of sociology or psychology. Accordingly I went to the talk with rather negative expectations but I was pleasantly surprised. The speaker explained how he has been able to apply evolutionary game theory to obtain insights into the evolution of cooperation in human societies under the influence of indirect reciprocity. This means that instead of the simple direct pattern \u2018A helps B and thus motivates B to help A\u2019 we have \u2018C sees A helping B and hence decides to help A\u2019 and variations on that pattern. The central idea of the work is to compare many different strategies in the context of a mathematical model and thus obtain ideas about what are the important mechanisms at work. My impression was that this is a case where mathematics has generated helpful ideas in understanding the phenomenon and that there remain a lot of interesting things to be done in that direction. It also made me reflect on my own personal strategies when interacting with other people. Apart from the interesting content the talk was also made more interesting by the speaker\u2019s entertaining accounts of experiments which have been done to compare with the results of the modelling. During the talk the speaker mentioned self-referentially that the fact of his standing in front of us giving the talk was an example of the process of the formation of a reputation being described in the talk. As far as I am concerned he succeeded in creating a positive reputation both for himself and for his field.\n\nApart from this the other plenary talk which I found most interesting was by Johan van de Koppel. He was talking about pattern formation in ecology and, in particular, about his own work on pattern formation in mussel beds. A talk which I liked much less was that of Adelia Sequeira and it is perhaps interesting to ask why. She was talking about modelling of atherosclerosis. She made the valid point near the beginning of her lecture that while heart disease is a health problem of comparable importance to cancer in the developed world the latter theme was represented much more strongly than the former at the conference. For me cancer is simply much more interesting than heart disease and this point of view is maybe more widespread. What could be the reason? One possibility is that the study of cancer involves many more conceptual aspects than that of heart disease and that this is attractive for mathematicians. Another could be that I am a lot more afraid of being diagnosed with cancer some day than of being diagnosed with heart disease although the latter may be no less probable and not less deadly if it happens. To come back to the talk I found that the material was too abundant and too technical and that many ideas were used without really being introduced. The consequence of these factors was that I lost interest and had difficulty not falling asleep.\n\nIn the case of the parallel talks there were seventeen sessions in parallel and I generally decided to go to whole sessions rather than trying to go to individual talks. I will make some remarks about some of the things I heard there. I found the first session I went to, on tumour-immune dynamics, rather disappointing but the last talk in the session, by Shalla Hanson was a notable exception. The subject was CAR T-cells and what mathematical modelling might contribute to improving therapy. I found both the content and the presentation excellent. The presentation packed in a lot of material but rather than being overwhelmed I found myself waiting eagerly for what would come next. During the talk I thought of a couple of questions which I might ask at the end but they were answered in due course during the lecture. It is a quality I admire in a speaker to be able to anticipate the questions which the audience may ask and answer them. I see this less as a matter of understanding the psychology of the audience (which can sometimes be important) and rather of really having got to the heart of the subject being described. There was a session on mathematical pharmacology which I found interesting, in particular the talks of Tom Snowden on systems pharmacology and that of Wilhelm Huisinga on multidrug therapies for HIV. In a session on mathematical and systems immunology Grant Lythe discussed the fascinating question of how to estimate the number of T cell clones in the body and what mathematics can contribute to this beyond just analysing the data statistically. I enjoyed the session on virus dynamics, particularly a talk by Harel Dahari on hepatitis C. In particular he told a story in which he was involved in curing one exceptional HCV patient with a one-off therapy using a substance called silibinin and real-time mathematical modelling.\n\nI myself gave a talk about dinosaurs. Since this is work which is at a relatively early stage I will leave describing more details of it in this blog to a later date.\n\n## An eternal pedestrian\n\nJune 13, 2016\n\nI am presently visiting Japan. My host is Atsushi Mochizuki who leads the Theoretical Biology Laboratory at RIKEN in Wako near Tokyo. RIKEN is a research organisation which was founded in 1917 using the Kaiser-Wilhelm-Gesellschaft as a model. Thus it is a kind of Japanese analogue of the Max Planck Society which is the direct descendant of the Kaiser-Wilhelm-Gesellschaft. I had only been in Japan once before and looking at my records I see that that was in August 2005. At that time I attended a conference in Sendai, a place which I had never heard of before I went there. Since then it has become sadly famous in connection with the damage it suffered from the tsunami which also caused the Fukushima nuclear disaster. At least I had even then previously heard of Tohoku University which is located in the city.\n\nYesterday, sitting by the river in Wako, I was feeling quite meditative. I was in an area where motor vehicles are not permitted. There were not many people around but most of those who were there were on bikes. I started thinking of how this is typical of what I have experienced in many places I have been. On a walk along the Rhine in Mainz or in the surrounding countryside most of the people you see are on bikes. Copenhagen is completely dominated by bikes. In the US cars dominate. For instance when I was in Miami for a conference and was staying at the Biltmore Hotel I had to walk quite a distance to get dinner for an affordable price. In general the only people I met walking on the streets there were other conference participants. When I visited the University of California at Santa Barbara bikes were not the thing on the campus but it was typical to see students with skateboards. Summing up, I have frequently had the experience that as a pedestrian I was an exception. It seems that for normal people just putting one foot in front of the other is not the thing to do. They need some device such as a car, a bike or a skateboard to accompany them. I, on the other hand, am an eternal pedestrian. I like to walk places whenever I can. I walk twenty minutes to work each day and twenty minutes back. I find that a good way of framing the day. When I lived in Berlin there was a long period when I had a one-way travelling time of 90 minutes by train. I am glad to have that behind me. I did not consciously plan being so near to work in Mainz but I am glad it happened. Of course being a pedestrian has its limits \u2013 I could not have come to Japan on foot.\n\nMy pedestrian nature is not limited to the literal interpretation of the term. I am also an intellectual pedestrian. An example of this is described in my post on low throughput biology. Interestingly this post has got a lot of hits, more than twice as many as any other post on my blog. This is related to the theme of simple and complex models in biology. Through the talks I have given recently in Copenhagen, Berlin and here in Japan and resulting discussions with different people I have become of conscious of how this is a recurring theme in those parts of mathematical biology which I find interesting. The pedestrian may not get as far as others but he often sees more in the places he does reach. He may also get to places that others do not. Travelling fast along the road may cause you to overlook a valuable shortcut. Or you may go a long way in the wrong direction and need a lot of time to come back. Within mathematics one aspect of being a pedestrian is calculating things by hand as far as possible and using computers as a last resort. This reminds me of a story about the physicist John Wheeler who had a picture of a computer on the wall in his office which he called \u2018the big computer\u2019. When he wanted to solve a difficult problem he would think about how he would programme it on the computer and when he had done that thoroughly he had understood the problem so well that he no longer needed the computer. Thus the fact that the computer did not exist except on paper was not a disadvantage.\n\nThis is the direction I want to (continue to) go. The challenges along the road are to achieve something essential and to make clear to others, who may be sceptical, that I have done so.\n\n## Hepatitis C\n\nMay 29, 2016\n\nI once previously wrote something about hepatitis C in this blog which was directed to the mathematical modelling aspects. Here I want to write about the disease itself. This has been stimulated by talks I heard at a meeting of the Mainzer Medizinische Gesellschaft. The speakers were Ralf Bartenschlager from Heidelberg and and Stefan Zeuzem from Frankfurt. The first speaker is a molecular biologist who has made important contributions to the understanding of the structure and life cycle of the virus. For this work he got the 2015 Robert Koch prize together with Charles Rice from the Rockefeller University. The second speaker is a clinician.\n\nHepatitis C is transmitted by blood to blood contact. According to Zeuzem the main cause of the spread of this disease in developed countries is intravenous drug use. Before there was a test for the disease it was also spread via blood transfusions. (At one time the risk of infection with hepatitis due to a blood transfusion was 30%. This was mainly hepatitis B and by the time of discovery of hepatitis C, when the risk from hepatitis B had essentially been eliminated, it had dropped to 5%.) He also mentioned that there is a very high rate of infection in certain parts of Egypt due to the use of unsterilized needles in the treatment of other diseases. Someone asked how the disease could have survived before there were injections. He did not give a definitive answer but he did mention that while heterosexual contacts generally carry little risk of infection with this virus homosexual contacts between men do carry a significant risk. The disease typically becomes chronic and has few if any symptoms for many years. It does have dramatic long-term effects, namely cirrhosis and cancer of the liver. He showed statistics illustrating how public health policies have influenced the spread of the disease in different countries. The development in France has been much more favourable (with less cases) than in Germany, apparently due to a publicity campaign as a result of political motives with no direct relevance to the disease. The development in the UK has been much less favourable than it has even in Germany due an almost complete lack of publicity on the theme for a long time. The estimated number of people infected in Germany is 500000. The global number is estimated as 170 million.\n\nThere has been a dramatic improvement in the treatment of hepatitis C in the past couple of years and this was the central theme of the talks. A few years ago the situation was as follows. Drugs (a combination of ribavirin and interferon $\\alpha$) could be used to eliminate the virus in a significant percentage of patients, particularly for some of the sub-types of the virus. The treatment lasted about a year and was accompanied by side effects that were so severe that there was a serious risk of patients breaking it off. Now the treatment only lasts a few weeks, it cures at least 95% of the patients and in many situations 99% of them. The side effects of the new treatments are moderate. There is just one problem remaining: the drugs for the best available treatment are sold for extremely high prices. The order of magnitude is 100000 euros for a treatment. Zeuzem explained various aspects of the dynamics which has led to these prices and the circumstances under which they might be reduced in the future. In general this gave a rather depressing picture of the politics of health care relating to the approval and prescription of new drugs.\n\nLet me get back to the scientific aspects of the theme, as explained by Bartenschlager. A obvious question to ask is: if hepatitis C can essentially be cured why does HIV remain essentially incurable despite the huge amount of effort and money spent on trying to find a treatment? The simple answer seems to be that HIV can hide while HCV cannot. Both these viruses have an RNA genome. Since the copying of RNA is relatively imprecise they both have a high mutation rate. This leads to a high potential for the development of drug resistance. This problem has nevertheless been overcome for HCV. Virus particles are continually being destroyed by the immune system and for the population to survive new virus particles must be produced in huge numbers. This is done by the liver cells. This heavy burden kills the liver cells after a while but the liver is capable of regenerating, i.e, replacing these cells. The liver has an impressive capability to survive this attack but every system has its limits and eventually, after twenty or thirty years, the long-term effects already mentioned develop. An essential difference between HIV and HCV is that the RNA of HCV can be directly read by ribosomes to produce viral proteins. By contrast, the RNA of HIV is used as a template to produce DNA by the enzyme reverse transcriptase and this DNA is integrated into the DNA of the cell. This integrated DNA (known as the provirus) may remain inactive, not leading to production of protein. As long as this is the case the virus is invisible to the immune system. This is one way the virus can hide. Moreover the cell can divide producing new cells also containing the provirus. There is also another problem. The main target of HIV are the T-helper cells. However the virus can also infect other cells such as macrophages or dendritic cells and the behaviour of the virus in these other cells is different from that in T-helper cells. It is natural that a treatment should be optimized for what happens in the typical host cell and this may be much less effective in the other cell types. This means that the other cells may serve as a reservoir for the virus in situations where the population is under heavy pressure from the immune system or drug treatment. This is a second sense in which the virus can hide.\n\nSome of the recent drugs used to treat HCV are based on ideas developed for the treatment of HIV. For instance a drug of this kind may inhibit certain of the enzymes required for the reproduction of the virus. There is one highly effective drug in the case of HCV which works in a different way. The hepatitis C virus produces one protein which has no enzymatic activity and it is at first sight hard to see what use this could be for the virus. What it in fact does is to act as a kind of docking station which organizes proteins belonging to the cell into a factory for virus production.\n\nThe hepatitis C virus is a valuable example which illustrates the relations between various aspects of medical progress: improvement in scientific understanding, exploitation of that information for drug design, political problems encountered in getting an effective drug to the patients who need it. Despite the negative features which have been mentioned it is the subject of a remarkable success story.\n\n## Flying to Copenhagen without a\u00a0carpet\n\nMay 11, 2016\n\nThis semester I have a sabbatical and I am profiting from it by travelling more than I usually do. At the moment I am visiting the group of Carsten Wiuf and Elisenda Feliu at the University of Copenhagen for two weeks. The visit here also gives me the opportunity to discuss with people at the Niels Bohr Institute. Note that the authors of the paper I quoted in the post on NF$\\kappa$B were at the NBI when they wrote it and in particular Mogens Jensen is still there now. I gave a talk on some of my work on the Calvin cycle at NBI today. Afterwards I talked to Mogens and one of his collaborators and found out that he is still very active in modelling this system.\n\nI was thinking about my previous visits to Copenhagen and, in particular, that the first one was on a flying carpet. The background to this is that when I was seven years old I wrote a story in school with the title \u2018The Magic Carpet\u2019. I do not have the text any more but I know it appeared in the School Magazine that year. In my own version there was also a picture which I will say more about later. But first something about the story, of which I was the hero. I bought the carpet in Peshawar and used it to visit places in the world I was interested in. For some reason I no longer know I had a great wish at that time to visit Copenhagen. Perhaps it was due to coming into contact with stories of Hans Christian Andersen. In any case it is clear that having the chance this was one of the first places I visited using the magic carpet. The picture which I drew showed something closer to home. There I can be seen sitting on the carpet, wearing the blue jersey which was my favourite at that time, while the carpet bent upwards so as to just pass over the tip of the spire of St. Magnus Cathedral in Kirkwall. In the story it was also related that one of the effects of my journey was a newspaper article reporting a case of \u2018mass hallucination\u2019. I think my teachers were impressed at my using this phrase at my age. They might have been less impressed if they had known my source for this, which was a Bugs Bunny cartoon.\n\nDuring my next visit to Copenhagen in 2008 (here I am not counting changing planes there on the way to Stockholm, which I did a few times) I was at a conference at the Niels Bohr Institute in my old research field of mathematical relativity and I gave a talk in that area. Little did I think I would return there years later and talk about something completely different. I remember that there was a photo in the main lecture room where many of the founders of quantum mechanics are sitting in the first row. From my own point of view I am happy that another person who can be seen there is Max Delbr\u00fcck, a shining example of a switch from physics to biology. My next visit to Copenhagen was for the conference which I wrote about in a previous post. It was at the University. Since that a lot has happened with chemical reaction network theory and with my understanding of it. The lecture course I gave means that some of the points I mentioned in my post at that time are things I have since come to understand in some depth. I look forward to working on projects in that area with people here in the coming days.\n\n## NF\u03baB\n\nMay 1, 2016\n\nNF$\\kappa$B is a transcription factor, i.e. a protein which can bind to DNA and cause a particular gene to be read more or less often. This means that more or less of a certain protein is produced and this changes the behaviour of the cell. The full name of this transcription factor is \u2018nuclear factor, $\\kappa$-light chain enhancer of B cells\u2019. The term \u2018nuclear factor\u2019 is clear. The substance is a transcription factor and to bind to DNA it has to enter the nucleus. NF$\\kappa$B is found in a wide variety of different cells and its association with B cells is purely historical. It was found in the lab of David Baltimore during studies of the way in which B cells are activated. It remains to explain the $\\kappa$. B cells produce antibodies each of which consists of two symmetrical halves. Each half consists of a light and a heavy chain. The light chain comes in two variants called $\\kappa$ and $\\lambda$. The choice which of these a cell uses seems to be fairly random. The work in the Baltimore lab had found out that NF$\\kappa$B could skew the ratio. I found a video by Baltimore from 2001 about NF$\\kappa$B. This is probably quite out of date by now but it contained one thing which I found interesting. Under certain circumstances it can happen that a constant stimulus causing activation of NF$\\kappa$B leads to oscillations in the concentration. In the video the speaker mentions \u2018odd oscillations\u2019 and comments \u2018but that\u2019s for mathematicians to enjoy themselves\u2019. It seems that he did not believe these oscillations to be biologically important. There are reasons to believe that they might be important and I will try to explain why. At the very least it will allow me to enjoy myself.\n\nLet me explain the usual story about how NF$\\kappa$B is activated. There are lots of animated videos on Youtube illustrating this but I prefer a description in words. Normally NF$\\kappa$B is found in the cytosol bound to an inhibitor I$\\kappa$B. Under certain circumstances a complex of proteins called IKK forms. The last K stands for kinase and IKK phosphorylates I$\\kappa$B. This causes I$\\kappa$B to be ubiquinated and thus marked for degradation (cf. the discussion of ubiquitin here). When it has been destroyed NF$\\kappa$B is liberated, moves to the nucleus and binds to DNA. What are the circumstances mentioned above? There are many alternatives. For instance TNF$\\alpha$ binds to its receptor, or something stimulates a toll-like receptor. The details are not important here. What is important is that there are many different signals which can lead to the activation of NF$\\kappa$B. What genes does NF$\\kappa$B bind to when it is activated? Here again there are many possibilities. Thus there is a kind of bow tie configuration where there are many inputs and many outputs which are connected to a single channel of communication. So how is it possible to arrange that when one input is applied, e.g. TNF$\\alpha$ the right genes are switched on while another input activates other genes through the same mediator NF$\\kappa$B? One possibility is cross-talk, i.e. that this signalling pathway interacts with others. If this cannot account for all the specificity then the remaining possibility is that information is encoded in the signal passing through NF$\\kappa$B itself. For example, one stimulus could produce a constant response while another causes an oscillatory one. Or two stimuli could cause oscillatory responses with different frequencies. Evidently the presence of oscillations in the concentration of NF$\\kappa$B presents an opportunity for encoding more information than would otherwise be possible. To what extent this really happens is something where I do not have an overview at the moment. I want to learn more. In any case, oscillations have been observed in the NF$\\kappa$B system. The primary thing which has been observed to oscillate is the concentration of NF$\\kappa$B in the nucleus. This oscillation is a consequence of the movement of the protein between the cytosol and the nucleus. There are various mathematical models for describing these oscillations. As usual in modelling phenomena in cell biology there are models which are very big and complicated. I find it particularly interesting when some of the observations can be explained by a simple model. This is the case for NF$\\kappa$B where a three-dimensional model and an explanation of its relations to the more complicated models can be found in a paper by Krishna, Jensen and Sneppen (PNAS 103, 10840). In the three-dimensional model the unknowns are the concentrations of NF$\\kappa$B in the nucleus, I$\\kappa$B in the cytoplasm and mRNA coding for I$\\kappa$B. The oscillations in normal cells are damped but sustained oscillations can be seen in mutated cells or corresponding models.\n\nWhat is the function of NF$\\kappa$B? The short answer is that it has many. On a broad level of description it plays a central role in the phenomenon of inflammation. In particular it leads to production of the cytokine IL-17 which in turn, among other things, stimulates the production of anti-microbial peptides. When these things are absent it leads to a serious immunodeficiency. In one variant of this there is a mutation in the gene coding for NEMO, which is one of the proteins making up IKK. A complete absence of NEMO is fatal before birth but people with a less severe mutation in the gene do occur. There are symptoms due to things which took place during the development of the embryo and also immunological problems, such as the inability to deal with certain bacteria. The gene for NEMO is on the X chromosome so that this disease is usually limited to boys. More details can be found in the book of Geha and Notarangelo mentioned in\u00a0 a previous post.","date":"2017-05-26 18:44:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 47, \"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\": 0.4074501395225525, \"perplexity\": 742.2563039704708}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463608676.72\/warc\/CC-MAIN-20170526184113-20170526204113-00060.warc.gz\"}"} | null | null |
\section{Introduction}
It is now widely appreciated that super p-branes in a vacuum spacetime
background are associated with p-form extensions of the standard spacetime
supersymmetry algebra. In the usual formulation, and for the branes of the
`old branescan', this extension arises as a consequence of the non-invariance
under spacetime supersymmetry transformations of a Wess-Zumino (WZ)
term in the worldvolume Lagrangian \cite{azc}. The p-form is proportional
to the tension $T$ and can be considered as a type of `classical anomaly' in
which the role of Planck's constant is taken by $T$ \cite{azcb}. A simple
example is the D=11 supermembrane, in which the WZ term is the pull-back
to the worldvolume of the 3-form superspace gauge potential of D=11
supergravity \cite{bst}.
This is not the full story, however. It was shown in \cite{tension,blt}
that the introduction of a p-form gauge potential $A$ allows the construction of
an alternative local worldvolume Lagrangian {\sl without} a WZ term. The new
worldvolume field does not lead to any additional local degrees of freedom
but does lead to an integration constant, which can be identified as the
p-brane tension T. When $T$ is non-zero the supersymmetry algebra must include
a p-form charge, as before, but since the new Lagrangian is invariant under
supersymmetry transformations this charge now has a different origin. To
explain this point, let us call the algebra deduced from commutators of
supersymmetry transformations acting on worldvolume fields the `naive'
supersymmetry algebra. This need not be the same as the anticommutator
of Noether supercharges because central charges appearing in the latter may
commute with all worldvolume fields. For example, in the usual formulation of
the supermembrane, with WZ term, the worldvolume fields are the maps from the
worldvolume to superspace. The `naive' supersymmetry algebra in this case is
therefore the algebra of Killing vector superfields. Assuming a vacuum
superspace background, this is just the standard D=11 supersymmetry algebra,
which fails to include the 2-form extension implied by the presence of the WZ
term.
In the new formulation of the supermembrane there is no WZ term so the naive
supersymmetry algebra must include the 2-form extension. What makes
this possible is a non-trivial supersymmetry transformation of the 2-form
$A$, which is such that the commutator of two supersymmetry transformations
yields a gauge transformation with a parameter that determines the central
charge structure. This phenomenon can occur whenever the worldvolume fields
include gauge fields. It was first noted \cite{sorokin} in the context of the
2-form gauge potential appearing in the D=11 superfivebrane action of
\cite{bandos}. A detailed verification in a very general context, including Type
II super D-branes \cite{swedes,bergstown,agan}, has since been provided by
Hammer \cite{hammer}. In this case the `naive' algebra includes a
Neveu-Schwarz/Neveu-Schwarz (NS) 1-form charge arising from the presence of the
Born-Infeld (BI) gauge field, while all Ramond/Ramond (RR) charges arise from
the WZ term.
From the fully non-perturbative point of view there is no real distinction
between NS and RR charges, so one might expect there to exist an alternative,
manifestly supersymmetric, super D-brane action in which all p-form charges
appear already in the `naive' supersymmetry algebra. Here we construct this
action via the introduction of a p-form worldvolume gauge potential, as
suggested by the supermembrane example. This new, manifestly supersymmetric,
formulation of the super D-p-brane can be motivated in a number of other ways.
It was pointed out in \cite{pktb} that boundaries (on M5-branes or on
`M-9-branes') of open supermembranes could be interpreted as discontinuities in
the field strength of a 2-form potential on the D=11 supermembrane worldvolume,
and that M-theory and superstring dualities would then imply the existence of
p-form gauge potentials on the worldvolumes of most other M-theory and
superstring p-branes, all D-branes in particular. The new super D-string action
turns out to be not only manifestly supersymmetric (in a vacuum background) but
also manifestly invariant under the $Sl(2;\bb{Z})$ duality of IIB superstring
theory \cite{martin}, and it seems to be a general feature that manifest
$Sl(2;\bb{Z})$ invariance of IIB D-p-brane actions requires the introduction of
worldvolume p-form gauge potentials \cite{ceder}. More recently, worldvolume
p-form gauge potentials have found to be an essential ingredient in the
formulation of `massive' p-brane actions \cite{BLO}, i.e. IIA branes in
a massive IIA supergravity background.
One aim of this paper is to update our previous work on super D-branes
\cite{bergstown} in light of the above discussion. Our results on p-form gauge
potentials are complementary to those of \cite{martin,ceder,BLO}. We do not
consider the implementation of $Sl(2;\bb{Z})$ invariance in the IIB case, nor
massive backgrounds in the IIA case. However, by virtue of these limitations we
are able to discuss all D-branes at once, thereby highlighting their common
features. The bosonic Lagrangian of the new super D-brane action is
simply\footnote{On setting $G$ to zero this reduces to the null D-brane
Lagrangian of \cite{lindstrom}.}
\begin{equation}
L= {1\over2v}[e^{-2\phi}\det(g+{\cal F}) + (\star G_{(p+1)})^2]
\label{onea}
\end{equation}
where $\phi$ is the dilaton, $v$ is an independent worldvolume density, $g$ is
the induced metric, and
\begin{equation}
{\cal F} = dV-B
\label{oneb}
\end{equation}
is the two-form field strength of the BI 1-form gauge
potential $V$, with $B$ the pullback to the worldvolume of the NS two-form gauge
potential. The scalar density $\star G_{(p+1)}$ is the worldvolume Hodge
dual of a (p+1)-form field strength $G_{(p+1)}$ for the new p-form worldvolume
gauge potential $A_{(p)}$. In order to deal with all D-p-branes simultaneously
it is convenient to combine these p-form gauge potentials into the formal
sum\footnote{This has also been used in \cite{BLO}.}
\begin{equation}
A = \sum_{k=0}^9 A_{(k)}\, .
\label{onec}
\end{equation}
The field strength of $A$ is
\begin{equation}
G= dA - C e^{\cal F}
\label{oned}
\end{equation}
where C is the formal sum of R-R gauge potentials introduced in \cite{douglas}.
The supersymmetric extension of the Lagrangian (\ref{onea}) is formally
identical, but with an N=2 superspace replacing the D=10 background spacetime.
What we have to show is that the Lagrangian so-obtained is $\kappa$-symmetric.
We do so in the following section.
The super-Poincar{\'e} invariance of the super D-brane action in a flat
superspace background is a special case of invariance under the
(super)isometries of any background allowed by $\kappa$-symmetry. Another aim
of this paper is to prove this statement. In general, the invariance requires an
appropriate choice of transformation rules for worldvolume gauge fields. If
these include the new p-form gauge field then the
Lagrangian (rather than just the action) is invariant. As mentioned above in the
context of supertranslations of flat superspace, the transformations of the
worldvolume gauge fields then determine the central charge structure of the
(super)algebra of isometries. Thus, the discussion above generalises directly to
an arbitrary background. This observation is especially important for the D3
brane since it implies that the super D3-brane action in the near-horizon
geometry of the D3-brane solution is invariant under the full $SU(2,2|4)$
isometry of that background. This can be interpreted as a non-linearly realized
superconformal symmetry of the super D3-brane action in this background,
generalizing the well-known superconformal symmetry of the free N=4 D=4
super-Maxwell action and the more-recently established \cite{malda,kaletal}
non-linearly-realized conformal invariance of the bosonic action.
Finally, we present the hamiltonian formulation of our super D-brane action,
generalizing the bosonic results of \cite{kallosh,lindstrom} and the flat
background IIB results of \cite{kam}.
\section{$\kappa$-symmetry}
As in \cite{bergstown}, we define $\delta E^A \equiv \delta Z^M E_M{}^A$,
and set $\delta_\kappa E^a=0$, temporarily leaving open the choice of
$\delta_\kappa E^\alpha$. The $\kappa$-symmetry variation of the BI field $V$ is
\begin{equation}
\delta_\kappa V_i = E_i{}^A \delta_\kappa E^\beta B_{\beta A}\, ,
\label{twoa}
\end{equation}
which is such as to ensure the `supercovariance' of the variation of ${\cal F}$.
Using the standard type II D=10 supergravity superspace constraints, summarized
in \cite{bergstown}, we find that
\begin{equation}
\delta_\kappa \det (g+{\cal F}) = -2i\delta_\kappa \bar E\tilde N^i E_i
\label{twob}
\end{equation}
where
\begin{equation}
\tilde N^i = \det (g+{\cal F})\, (g+{\cal F})^{ij}{\cal P}_+ \gamma_j +
\det (g-{\cal F})\, (g-{\cal F})^{ij}{\cal P}_- \gamma_j\, .
\label{twoc}
\end{equation}
Note that, because of the determinant, $\tilde N^i$ is non-singular even when
$(g \pm {\cal F})$ has no inverse. The (reducible) worldvolume Dirac matrices
$\gamma_i$ are the pullbacks $E_i{}^a\Gamma_a$ of the spacetime Dirac matrices
(more generally, the antisymmetrized product of $p$ worldvolume Dirac matrices
defines a matrix-valued worldvolume $p$-form $\gamma_{(p)}$ with components
$\gamma_{i_1\dots i_p}$), and
\begin{equation}
{\cal P}_\pm = \cases{{1\over2}(1 \pm \Gamma_{11}) & IIA \cr {1\over2}(1\pm
\sigma_3) & IIB.}
\label{twod}
\end{equation}
We find, similarly, that
\begin{equation}
\delta_\kappa [Ce^{\cal F}] = d(i_{\delta Z} C e^{\cal F}) +
i\delta_\kappa \bar E \gamma e^{\cal F} E
\label{twoe}
\end{equation}
where $i_{\delta Z}C$ is the contraction of $C$ with the vector superfield
$\delta_\kappa Z^M\partial_M$, and $\gamma$ is the formal sum
\begin{equation}
\gamma = \cases{ e^{-\phi} \sum_{p=0}^8 \gamma_{(p)}
(\Gamma_{11})^{(p-2)(p-6)/4} & IIA \cr
e^{-\phi} \sum_{p=1}^9 \gamma_{(p)} {\cal P} (\sigma_1)^{(p-3)(p-7)/4}
(i\sigma_2)^{(p-1)(p-9)/4} & IIB }
\label{twof}
\end{equation}
where ${\cal P}$ is the IIB chiral projector on D=10 spinors.
We choose the $\kappa$-transformation of $A$ such as to ensure the
`supercovariance' of the transformation of the field strength $G$. This
requires
\begin{equation}
\delta A= (i_{\delta Z} C) e^{\cal F}\, ,
\label{twog}
\end{equation}
which leads to
\begin{equation}
\delta_\kappa G = -i\delta_\kappa \bar E \gamma e^{\cal F} E\, .
\label{twoh}
\end{equation}
For a D-p-brane we must select the (p+1)-form in this formal sum, and then take
the worldvolume Hodge dual. At this point it is convenient to introduce the
matrix\footnote{The matrix $\Xi_{(0)}$, which is just $\sqrt{-\det g}$ times the
matrix $\Gamma_{(0)}$ of \cite{bergstown}, is well-defined even for a degenerate
worldvolume metric.}
\begin{equation}
\Xi_{(0)} = {1\over (p+1)!}\,
\varepsilon^{i_1\dots i_{p+1}}\gamma_{i_1\dots i_{p+1}}\, ,
\label{twoi}
\end{equation}
which satisfies
\begin{equation}
\Xi_{(0)}^2= (-1)^{p(p+1)/2}\; \det g\ .
\label{twoj}
\end{equation}
We may then write the variation of the worldvolume scalar $\star G$ as
\begin{equation}
\delta_\kappa \star G = -ie^{-\phi}\delta_\kappa \bar E \tilde M^i_{(p)} E_i
\label{twok}
\end{equation}
where\footnote{We have absorbed some factors into the definition of this matrix
relative to the matrix $M^i_{(p)}$ of \cite{bergstown}, where they instead appear
in eq. (4.13). The subsequent calculations of \cite{bergstown}, to be summarized
below, actually refer to a matrix $M^i_{(p)}$ {\it with} these factors, as is
clear from the definition of the matrix $K^i_{(p)}$ in \cite{bergstown}.}
\begin{equation}
\tilde M_{(p)}^i = \sum_{n=0} {1\over 2^n n!} \gamma^{ij_1k_1\dots j_nk_n}\,
\Xi_0\, {\cal F}_{j_1k_1}\dots {\cal F}_{j_nk_n} \times \cases{
(-\Gamma_{11})^{n+ (p-2)/2} & IIA \cr (-\sigma_3)^{n+(p-3)/2} \; i\sigma_2 & IIB}
\label{twol}
\end{equation}
We may now put together the above results to find the $\kappa$-variation
of the proposed new super D-brane Lagrangian, at least in bosonic
backgrounds. In such backgrounds the dilaton has vanishing $\kappa$-variation,
so we have
\begin{equation}
\delta_\kappa [e^{-2\phi}\det(g+{\cal F}) + (\star G)^2]=
-2ie^{-2\phi}\delta_\kappa \bar E \big(\tilde N^i + e^\phi (\star G)\tilde
M^i_{(p)}\big) E_i\ .
\label{twom}
\end{equation}
Given the results in \cite{blt,bergstown}, we see that the spinor variation
$\delta_\kappa E$ must take the form
\begin{equation}
\delta_\kappa \bar E_\alpha =
\big[\bar\kappa (e^\phi \star G + \Xi)\big]_\alpha
\label{twon}
\end{equation}
where $\Xi$ is a matrix with the property
\begin{equation}
\Xi^2 = -\det (g+{\cal F})\ .
\label{twoo}
\end{equation}
Clearly $\Xi$ must reduce (up to a sign) to $\Xi_0$ when ${\cal F}$ vanishes.
Again using the results of \cite{bergstown} it follows that
\begin{equation}
\Xi = \sum_{n=0}^\infty {1\over 2^n n!}
\gamma^{j_1k_1\dots j_nk_n} {\cal F}_{j_1k_1}\dots {\cal F}_{j_nk_n}{\cal
J}^{(n)}_{(p)}
\label{twop}
\end{equation}
where
\begin{equation}
{\cal J}^{(n)}_{(p)} = \cases{(\Gamma_{11})^{n+(p-2)/2}\, \Xi_{(0)} & (IIA)\cr
(-1)^n (\sigma_3)^{n+(p-3)/2}\; i\sigma_2 \otimes \Xi_{(0)} & (IIB)}
\label{twoq}
\end{equation}
We now claim that
\begin{equation}
e^{-2\phi}(e^\phi\star G + \Xi)\big(\tilde N^i +
e^\phi (\star G)\tilde M^i_{(p)}\big)
\equiv \tilde M^i_{(p)} [e^{-2\phi}\det(g+{\cal F}) + (\star G)^2]\, .
\label{twor}
\end{equation}
The basis for this claim is the calculation sketched in \cite{bergstown} in the
course of which it was noted that terms involving a square root of a
determinant cancel separately. Thus, the calculation of \cite{bergstown} actually
establishes {\sl two} identities, which are
\begin{equation}
\tilde N^i + \Xi \tilde M^i_{(p)} \equiv 0, \qquad \Xi \tilde N^i - \det (g+
{\cal F})
\tilde M^i_{(p)} =0\, .
\label{twos}
\end{equation}
These identities are precisely those needed for (\ref{twor}), from which
it follows that
\begin{equation}
\delta_\kappa [e^{-2\phi}\det(g+{\cal F}) + (\star G)^2] = -2i(\bar\kappa \tilde
M^i_{(p)} E_i) [e^{-2\phi}\det(g+{\cal F}) + (\star G)^2]\, .
\label{twot}
\end{equation}
The new super D-p-brane Lagrangian is therefore invariant provided that we
choose the $\kappa$-variation of the Lagrange multiplier to be
\begin{equation}
\delta_\kappa v = -2iv(\bar\kappa \tilde M^i_{(p)} E_i)\, .
\label{twou}
\end{equation}
We have now established $\kappa$-symmetry. The $A$ field equation implies that
$\star G$ is a constant. The remaining equations are then those of the standard
super D-brane action with the tension equal to the constant $\star G$.
\section{Symmetries from background isometries}
We shall now consider symmetries of the new super D-brane action arising from
isometries of the background. These are generated by Killing vector superfields
with respect to which the Lie derivatives of all superspace field strengths
vanish. This implies, in particular that the induced metric $g_{ij}$ is
invariant. Let $\xi_\alpha = \xi_\alpha^M(Z)\partial_M$ be the set of Killing
vector superfields with (anti)commutators
\begin{equation}
[\xi_\beta, \xi_\gamma] =
f_{\beta\gamma}{}^\alpha \xi_\alpha \, ,
\label{syma}
\end{equation}
so that $f_{\beta\gamma}{}^\alpha$ are the structure constants of the Lie
(super)algebra of isometries. It will prove convenient to introduce the
transformations on spacetime superfields generated by $\xi_\alpha$ via a
BRST operator $s$. Thus,
\begin{equation}
sZ^M = c^\alpha\xi_\alpha^M
\label{symb}
\end{equation}
where $c^\alpha$ are set of {\it constant} ghost `fields' with BRST
transformation
\begin{equation}
sc^\alpha = {1\over2} c^\gamma c^\beta f_{\beta\gamma}{}^\alpha\, .
\label{symc}
\end{equation}
Note that $s^2c^\alpha$ is identically zero as a consequence of the
Jacobi identity for the structure constants, and that the action of $s^2$ on
$Z^M$, and hence on all superfields, also vanishes (this being equivalent to
$sc^M\equiv 0$).
Since $H=dB$ and $R=dC-CH$ are assumed invariant they are annihilated by $s$,
from which it follows that
\begin{eqnarray}
sB &=& d\Delta^{(NS)} \nonumber\\
sC &=& d\Delta^{(R)} -\Delta^{(R)} H \, ,
\label{symd}
\end{eqnarray}
where $\Delta^{(NS)}$ is a ghost-valued superspace 1-form and $\Delta^{(R)}$ is
a formal sum of ghost-valued superspace forms of all (relevant) degrees.
Alternatively, these quantities may be viewed as 1-forms on the isometry group
manifold with values in the exterior algebra on superspace. Either way, we see
that if the BRST transformations of the worldvolume gauge fields $V$ and $A$ are
chosen to be
\begin{equation}
sV= \Delta^{(NS)}, \qquad sA = e^{\cal F} \Delta^{(R)}\, ,
\label{symg}
\end{equation}
then we have
\begin{equation}
s{\cal F}=0, \qquad s{\cal G}=0\, .
\label{symga}
\end{equation}
In other words, {\it if the superspace background is such that the Lie
derivative of each background tensor vanishes then transformations of the
independent worldvolume gauge potentials $V$ and $A$ can be chosen such that
their respective field strengths ${\cal F}$ and ${\cal G}$ are invariant}, from
which it follows that the super D-brane action is invariant.
Having established invariance under background isometries, the next step is to
compute the algebra of these symmetry transformations. Since the Lagrangian is
invariant, and not just the action, this is equivalent to a computation of the
algebra of Noether charges in the Hamiltonian formulation. Since $B$ and $C$
are superfields we have the identities $s^2B\equiv 0$ and $s^2C\equiv 0$, which
imply that
\begin{equation}
d{\cal A}^{(NS)} =0, \qquad d{\cal A}^{(R)} =0,
\label{syme}
\end{equation}
where
\begin{equation}
{\cal A}^{(NS)} = s\Delta^{(NS)},\qquad
{\cal A}^{(R)} = e^{\cal F} s\Delta^{(R)}.
\label{symf}
\end{equation}
When these closed superspace forms are exact the transformations of $V$ and
$A$ can be removed by gauge transformations and are therefore trivial. Thus
${\cal A}^{(NS)}$ and ${\cal A}^{(R)}$ may be viewed as 2-forms on
the isometry group manifold with values in cohomology classes of superspace.
Because $V$ and $A$ are not pullbacks of superspace forms they need not be
annihilated by $s^2$. In fact,
\begin{equation}
s^2 V = {\cal A}^{(NS)} \qquad s^2 A={\cal A}^{(R)}\, .
\label{symh}
\end{equation}
Note that the closure of ${\cal A}^{(NS)}$ and ${\cal A}^{(R)}$ ensures
that $s^2$ annihilates ${\cal F}$ and ${\cal G}$ (as is, of course,
guaranteed by the construction). The closed forms defined by $s^2V$ and $s^2A$
determine the topological charges appearing as central charges in the algebra of
isometries. They may be calculated explicitly for any particular background. An
equivalent explicit calculation for a flat superspace background has been
carried out in \cite{hammer}.
\section{The super D-brane hamiltonian}
In passing to the phase-space form of the D-brane action it is useful to
first consider the null super D-brane, for which the Lagrangian is
\cite{lindstrom},
\begin{equation}
L = {1\over 2v} e^{-2\phi}\det (g_{ij}+ {\cal F}_{ij})\, .
\label{hama}
\end{equation}
This is obtained by setting $G=0$ in (\ref{onea}). We can rewrite this as
\begin{equation}
L = {1\over 2v} e^{-2\phi} \det (g_{{\underline a}{\underline b}}+{\cal F}_{{\underline a}{\underline b}}) \big\{ g_{tt} -
(g_{t{\underline a}} +
{\cal F}_{t{\underline a}}) [(g+{\cal F})^{-1}]^{{\underline a}{\underline b}} (g_{t{\underline b}} + {\cal F}_{t{\underline b}}) \big\}
\label{hamb}
\end{equation}
where we have set $i=(t,{\underline a}_1,{\underline a}_2,\dots,{\underline a}_p)$, i.e. an underlined lower-case
latin index is a `worldspace' index. The matrix $(g+{\cal F})^{-1}$, with
only worldspace components, is the inverse of the matrix with (only
worldspace) components $(g_{{\underline a}{\underline b}}+{\cal F}_{{\underline a}{\underline b}})$. This Lagrangian can be
further rewritten as
\begin{equation}
L = {1\over 2v} e^{-2\phi}\det (g_{{\underline a}{\underline b}}+{\cal F}_{{\underline a}{\underline b}}) \big[ K_{tt} - K_{t{\underline a}}
(K^{-1})^{{\underline a}{\underline b}} K_{t{\underline b}}\big]
\label{hamc}
\end{equation}
where
\begin{equation}
K_{ij} = g_{ij} + {\cal F}_{i{\underline a}}(g^{-1})^{{\underline a}{\underline b}}{\cal F}_{j{\underline b}}\, .
\label{hamd}
\end{equation}
Here, both $K^{-1}$ and $g^{-1}$ are to understood as being the inverses of
the matrices with only worldspace components, i.e. of $K_{{\underline a}{\underline b}}$ and
$g_{{\underline a}{\underline b}}$ respectively. If we now define
\begin{equation}
\lambda = ve^{2\phi}/\det (g_{{\underline a}{\underline b}}+{\cal F}_{{\underline a}{\underline b}})
\label{hame}
\end{equation}
then we have
\begin{equation}
L= {1\over 2\lambda}\big[ K_{tt} - K_{t{\underline a}} (K^{-1})^{{\underline a}{\underline b}}K_{t{\underline b}}\big]\, .
\label{hamf}
\end{equation}
An equivalent Lagrangian is
\begin{equation}
L = \tilde P\cdot \Pi_t + \tilde E^{\underline a} {\cal F}_{t{\underline a}} -
s^{\underline a} (\tilde P\cdot \Pi_{\underline a} + \tilde E^{\underline b} {\cal F}_{{\underline a}{\underline b}})
- {1\over2} \lambda (\tilde P^2 + \tilde E^{\underline a} \tilde E^{\underline b} g_{{\underline a}{\underline b}})
\label{hamg}
\end{equation}
where
\begin{equation}
(\Pi_i)^a \equiv E_i{}^a \, .
\label{hamh}
\end{equation}
To establish the equivalence one first eliminates the new auxiliary variables
$\tilde P_a$ and $\tilde E^{\underline a}$ to obtain the Lagrangian
\begin{equation}
L= {1\over 2\lambda} [ K_{tt} - 2s^{\underline a} K_{t{\underline a}} + s^{\underline a} s^{\underline b} K_{{\underline a}{\underline b}} ]
\label{hami}
\end{equation}
Elimination of $s^{\underline a}$ then yields (\ref{hamf}).
The Lagrangian (\ref{hamg}) is a convenient `half-way house' on the way to the
true phase-space form of the null D-brane action (the variable $\tilde P$ is not
the momentum conjugate to $X$, for example, although it is closely related to
it). However, rather than proceed with the null D-brane it is convenient to now
pass to the full D-brane action. We first note that the worldvolume (p+1)-form
$G_{(p+1)}$ in the formal sum $G$ can be written as
\begin{equation}
G_{(p+1)}= dt\, (G_t)_{(p)} \, ,
\label{hamj}
\end{equation}
where $(G_t)$ is a p-form on the p-dimensional worldspace. It is the p-form in
the formal sum
\begin{equation}
G_t = \dot A - dA_t - \dot Z^M C_M e^{\cal F} - {\cal F}_t (Ce^{\cal F})
\label{hamk}
\end{equation}
where $A$ is now a formal sum of {\sl worldspace} forms, as is $A_t$, and
$d$ is now to be understood as an exterior derivative on worldspace.
Similarly, $C_M$ is the restriction to worldspace of the formal sum
$i_MC$, where $i_M$ denotes contraction with the vector superfield
$\partial/\partial Z^M$. Finally, ${\cal F}_t = d\sigma^{\underline a} {\cal F}_{t{\underline a}}$ is a
worldspace 1-form. Specifically,
\begin{equation}
{\cal F}_t = \dot V - dV_t - \dot Z^M B_M\, ,
\label{haml}
\end{equation}
where $B_M$ is the restriction to worldspace of the 1-form $i_MB$,
so that
\begin{equation}
G_t = \dot A - dA_t - \dot V (Ce^{\cal F})
- \dot Z^M [ C_M e^{\cal F} - B_M (Ce^{\cal F})]
+ (dV_t) (Ce^{\cal F})\, .
\label{hamm}
\end{equation}
Using these results we can now write down a `half-way house' version of the {\sl
full} super D-p-brane Lagrangian (\ref{onea}), involving $G_t$. This is
\begin{eqnarray}
L &=& \tilde P\cdot \Pi_t + \tilde E^{\underline a} {\cal F}_{t{\underline a}} + T*(G_t)_{(p)}
- s^{\underline a} (\tilde P\cdot \Pi_{\underline a} + \tilde E^{\underline b} {\cal F}_{{\underline a}{\underline b}})\nonumber\\
&&- {1\over2} \lambda [\tilde P^2 + \tilde E^{\underline a} \tilde E^{\underline b} g_{{\underline a}{\underline b}}
+ T^2 e^{-2\phi} \det (g_{{\underline a}{\underline b}}+{\cal F}_{{\underline a}{\underline b}})]\,
\label{hamja}
\end{eqnarray}
where $*$ indicates the {\sl worldspace} Hodge dual. The equivalence can be
established as before by elimination of the variables $\tilde P,
\tilde E^{\underline a}$ and $T$ followed by the redefinition (\ref{hame}) and elimination
of
$s^{\underline a}$: the Lagrangian (\ref{onea}) is then recovered.
To obtain this action in canonical form we proceed as follows. Omitting total
derivatives, and defining the (p-1)-form
\begin{equation}
U_{(p-1)} = (A_t)_{(p-1)} + (-1)^p V_t (Ce^{\cal F})_{(p-1)}\, ,
\label{hamma}
\end{equation}
we have
\begin{eqnarray}
T(G_t)_{(p)} &=\dot A_{(p)} T + U_{(p-1)} dT - \dot V T (Ce^{\cal F})_{(p-1)}
\nonumber\\
& -\dot Z^M T[C_M e^{\cal F} - B_M(Ce^{\cal F})]_{(p)} + (-1)^p V_t T (Re^{\cal F})_{(p)}
\label{hammab}
\end{eqnarray}
where $R= dC -CH$ is the field strength for $C$\footnote{As before we use the
standard superspace convention that exterior differentiation `starts from the
right'.}. Thus
\begin{equation}
T*(G_t)_{(p)} = \dot \phi T + \phi^{\underline a} \partial_{\underline a} T - T\dot V_{{\underline a}} {\cal
C}^{{\underline a}} - T\dot Z^M [{\cal C}_M - (B_M)_{\underline a} {\cal C}^{\underline a}] + (-1)^pT V_t {\cal R}
\label{hamn}
\end{equation}
where we have defined
\begin{equation}
\phi = *A_{(p)}\,,\qquad \phi^{\underline a} = (*U_{(p-1)})^{\underline a}\,,
\label{hamna}
\end{equation}
and
\begin{equation}
{\cal C}^{\underline a} = [*(Ce^{\cal F})_{(p-1)}]^{\underline a}\, \qquad
{\cal C}_M = * (C_M e^{\cal F})_{(p)}\, \qquad
{\cal R} = *(Re^{\cal F})_{(p)} \, .
\label{hamo}
\end{equation}
Thus (\ref{hamja}) can now be rewritten as
\begin{equation}
L= \dot Z^M P_M + \dot V_{\underline a} E^{\underline a} + \dot\phi T - H\, ,
\label{hamp}
\end{equation}
where the hamiltonian $H$ is a sum of constraints imposed by Lagrange
multipliers, and
\begin{eqnarray}
P_M &=& E_M{}^a \tilde P_a - \tilde E^{\underline a} (B_M)_{\underline a} -
T[{\cal C}_M - (B_M)_{\underline a} {\cal C}^{\underline a}]\, , \nonumber\\
E^{\underline a} &=& \tilde E^{\underline a} - T {\cal C}^{\underline a}\, .
\label{hamq}
\end{eqnarray}
These equations imply that
\begin{eqnarray}
\tilde P_a &=& E_a{}^M \big(P_M + E^{\underline a} (B_M)_{\underline a} + T{\cal C}_M\big)\, ,
\nonumber\\
\tilde E^{\underline a} &=& E^{\underline a} + T{\cal C}^{\underline a}\, .
\label{hamr}
\end{eqnarray}
Since $E_\mu{}^\alpha$ is invertible, the remaining information contained in
(\ref{hamr}) is the constraint
\begin{equation}
E_\alpha{}^M\big(P_M + E^{\underline a} (B_M)_{\underline a} + T{\cal C}_M\big)=0\, ,
\label{hams}
\end{equation}
which can be incorporated as a constraint in $H$ imposed by a new fermionic
Lagrange multiplier $\zeta$. Thus
\begin{equation}
H = \phi^{\underline a} {\cal T}_{\underline a} + V_t {\cal G} + s^{\underline a} {\cal H}_{\underline a} +
\lambda {\cal H} +\zeta^\alpha {\cal S}_\alpha\, ,
\label{hamt}
\end{equation}
where
\begin{eqnarray}
{\cal T}_{\underline a} &=& -\partial_{\underline a} T \nonumber\\
{\cal G} &=& - \partial_{\underline a} {\tilde E}^{\underline a} + (-1)^{p+1} T {\cal R}\nonumber\\
{\cal H}_{\underline a} &=& {\tilde P}\cdot \Pi_{\underline a} + {\tilde E}^{\underline b} {\cal
F}_{{\underline a}{\underline b}}\nonumber\\
{\cal H} &=& {1\over2}\big[
{\tilde P}^2 + {\tilde E}^{\underline a} {\tilde E}^{{\underline b}} g_{{\underline a}{\underline b}} + T^2 e^{-2\phi}
{\rm det}(g+{\cal F}) \big]\nonumber\\
{\cal S}_\alpha &=& E_\alpha{}^M\big[P_M + E^{\underline a} (B_M)_{\underline a} + T{\cal C}_M\big]
\label{hamu}
\end{eqnarray}
with $\tilde P_a$ and $\tilde E^{\underline a}$ given by (\ref{hamr}).
The constraint functions ${\cal T}_{\underline a}$, ${\cal G}$, ${\cal H}_{\underline a}$ and ${\cal
H}$ are `first class' (in Dirac's terminology) and generate the p-form gauge
transformations, BI gauge transformations, worldspace diffeomorphisms and time
translations, respectively. The fermionic constraint functions ${\cal S}$ are
half first class and half second class; the first class constraints generate
$\kappa$-symmetry transformations.
\section{Discussion}
In this paper we have presented a new formulation of the super D-p-brane action
in which the tension appears as an integration constant in the equation of
motion of a new p-form gauge potential. In this form of the action the full
centrally-extended supertranslation algebra is realized as the `naive' algebra
of transformations of worldvolume fields. In the standard form, only the NS
charges are `naive' while the RR charges arise from the presence of the WZ
term. In the zero tension limit the RR charges vanish and one is left with the
`naive' algebra of the null super D-brane. The latter does not coincide with
the standard supersymmetry algebra (in contrast to, for example, the null D=11
supermembrane) because the BI field does not decouple in the null limit.
The results just summarized provide a clear understanding of the origin of the
various p-form charges in the supertranslation algebra of the super
D-p-brane. One of the original motivations for this work was to obtain a
similar understanding of the recently determined central charge structure of the
M5-brane \cite{sorokin}. In that case the WZ term was found to be responsible for
the full 5-form charge but for only half of the 2-form charge. The remaining
half is explained by the fact that, because of the worldvolume 2-form gauge
potential, the `naive' supersymmetry algebra differs from the standard one. The
D-brane results reported here are similar but with the additional feature that
one can exhibit the `naive' algebra as the algebra in the null limit, thereby
isolating the naive (NS) and WZ (RR) contributions. In the M5-brane case there
is apparently no way to achieve this separation: our attempts to define a null
limit of the M5-brane (in the `temporal gauge') simply led to the standard null
super 5-brane in which the two-form potential is absent\footnote{There is no
conflict with supersymmetry here because, as pointed out in \cite{blt}, spacetime
supersymmetry and $\kappa$ supersymmetry do {\it not} imply worldvolume
supersymmetry for null branes.}. In view of this, one might try instead to unify
the contributions to the central charge structure of the M5-brane supersymmetry
algebra by realizing the full algebra as the naive supersymmetry algebra on
worldvolume fields, as is achieved for D-branes via the new scale-invariant
action presented here. It is not yet clear to us whether this is possible. As
pointed out in \cite{pktb}, a p-form worldvolume gauge potential is natural for
objects that may, like D-branes, have boundaries on other branes, but the
M5-brane is always closed. A 5-form gauge potential has been successfully
introduced in a recent reformulation of the M5-brane action \cite{swedesb}, but
as the self-duality constraint is incorporated at the level of the field
equations this action could not be used to extract the central charge structure
in the manner envisaged here. We shall return to the M5-brane in a future
publication \cite{BSTb}.
We have also shown that the new super D-brane Lagrangian is invariant under all
isometries of the supergravity background, provided that the worldvolume
gauge fields are taken to transform appropriately; this implies that the `old'
super D-brane Lagrangian is invariant up to a total derivative. One advantage
of the new Lagrangian is that its invariance means that the full symmetry
algebra, with any possible central extensions, may be deduced from the algebra
of transformations of the worldvolume fields. Central extensions of the algebra
of Killing vector superfields are coded in the action of a BRST operator $s$ on
the worldvolume gauge fields: one finds that $s^2$ generates gauge
transformations of these fields with parameters determined by closed
superspace forms.
A case of current interest to which this analysis is applicable is the super
D3-brane in the maximally supersymmetric $adS_5\times S^5$ IIB background,
which has isometry supergroup $SU(2,2|4)$. An appropriate embedding of a super
D3-brane worldvolume in this background has been shown to result in an
interacting worldvolume field theory in which the $SU(2,2)$ subgroup acts as a
non-linearly realized conformal symmetry on the bosonic fields \cite{kaletal}.
The results obtained here show that the full action is invariant under the full
$SU(2,2|4)$ isometry supergroup, which can be interpreted as a non-linearly
realized superconformal symmetry. More precisely, we have shown that the action
is invariant under transformations that close to $SU(2,2|4)$ on the worldvolume
scalar (and, implicitly, spinor) fields. Closure on the worldvolume gauge fields
might, in principle, require the introduction of additional charges that are
central with respect to the linearly realized subgroup of $SU(2,2|4)$, which is
$(3+1)$-dimensional super-Poincar{\'e} and its $SU(4)$ group of automorphisms.
Specifically, one might expect the D3-brane to be associated with a 3-form
charge in the adS superalgbra. From the perspective of $adS_5$ the space
components of this 3-form (which are naturally associated with a 3-brane) would
be dual to a time component of a dual 2-form, but there is already a charge of
this type in the adS algebra: it is the generator of boosts in the space
direction orthogonal to the 3-brane. That this must be the 3-brane charge can be
seen from the fact that the $adS_5$ spacetime can itself be viewed as a
$p$-brane \cite{pope}, for which the only candidate charge is the one already in
the adS algebra. There are therefore no {\sl new} charges in the symmetry
algebra relative to the algebra of isometries, so that the symmetry
group of the super D3-brane is just the superconformal group $SU(2,2|4)$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,623 |
{"url":"https:\/\/sites.google.com\/site\/algecomday\/algecom7","text":"### Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University, IUPUI, and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).\n\nFurther details will be posted here as they become available. Or you may contact the University of Illinois organizers Hal Schenck and Alexander Yong, or the Purdue organizers Uli Walther and Saugata Basu or IUPUI organizer Evgeny Mukhin.\n\nIn Fall 2012 we held the second NSF funded ALGECOM at Purdue. We\ninvited Jerzy Weyman (F, Northeastern), Ragnar Olaf Buchweitz (F,\nToronto), Ben Wyser (P, UIUC). We also invited and had scheduled\nChristine Berkesch (P&U, Duke). However, Prof. Berkesch needed to\nwithdraw a few days before hand. We replaced her with Ralph Kaufmann\n(F, Purdue).\n\nposters Oliver Pechenik (G, UIUC), Matheus Brito (G&U, IUPUI), Michael\nDipasquale (G, UIUC) and Dominic Searles (G, UIUC). We plan to\ncontinue this poster session idea.\n\nDate: \u00a0 Oct 20, 2012\n\n### Location: Department of Mathematics, room 175, at Purdue University in West Lafayette\n\nFor a map, click here. Coffee-breaks\u00a0 will be held in the math library on the third floor.\n\n### Title: Finite free resolutions and Kac-Moody Lie algebras\n\nAbstract. Let us recall that\u00a0 a\u00a0 format (r_n,\\ldots ,r_1) of the free complex\n0-->F_n-->F_{n-1}-->\\ldots\u00a0F_0\nover a commutative Noetherian ring is the sequence of ranks r_i\u00a0 of the i-th differential d_i.\nWe will assume that\u00a0 rank F_i =r_i+r_{i+1}. We say that an\u00a0acyclic \u00a0complex\nF_{gen} of a given format over a given ring R_{gen} is generic if for every\ncomplex G of this format\u00a0over a Noetherian ring S there exists a homomorphism\nf:R_{gen}--> S such that G=F_{gen}\\otimes_{R_{gen}} S.\n\nFor complexes of length 2 the existence of the generic acyclic complex was\nestablished by Hochster and Huneke in\u00a0the 1980's. It is a normalization of the\nring giving a generic complex (two matrices with composition\u00a0zero and rank\nconditions).\n\nI will discuss the ideas going into the proof of the following result:\n\nAssociate to a triple of ranks (r_3, r_2, r_1) a triple (p,q,r)=(r_3+1,\nr_2-1,\u00a0r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lenghts\np-1, q-1, r-1 attached to the\u00a0central vertex). Then there exists a Noetherian\ngeneric ring for this format if and only if T_{p,q,r} is a Dynkin graph.\u00a0In other\ncases one can construct in a uniform way a non-Noetherian generic ring, which\ncarries an\u00a0action of the Kac-Moody Lie algebra corresponding to the graph\nT_{p,q,r}.\n\n### Abstract: The classical McKay correspondence relates maximal Cohen-Macaulay modules onKleinian singularities to representations of the finite subgroups of SL(2,C)and to the exceptional divisors in the desingularization.It can also be interpreted (Kapranov-Vasserot 1999) as an algebraicdescription of the derived category of coherent sheaves on the desingularization.This approach has been vastly generalized, first by Bridgeland-King-Reid(2001), then by D.Orlov (2009) and most recently by Amiot-Iyama-Reiten (2012).The potential of these developments in Commutative Algebra has not yet beenexplored,and we hope this talk will entice some to take a closer look.\n\nCoffee and snacks\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 14h00\n\n### Title: Graphs, algebras and cohomology\n\nAbstract:\u00a0 We discuss how graphs, especially trees appear in the description\nof certain algebras.\nThis observation has three levels. First, there are specific\ncohomology\u00a0groups and algebras\u00a0basically given by trees.\nThe reason is often that they naturally index cells or divisors\u00a0for certain\nspaces, especially moduli spaces.\nSecondly algebras themselves, like Lie, pre-Lie, associative etc algebras can\nalso be described in terms of certain trees and graphs.\nThis is related to the first occurrence, as we explain. Finally, The graphs\ndescribing these classes of algebras themselves again\nform algebras, like Lie or BV algebras. We will progress through these levels\ngiving the representative examples.\n\n### -------------------------------------------------------------------------------------------------------------------------------------\n\nList of participants:\n\nArnold Yim (G, Purdue)\nMichael Dipasquale (G, UIUC)\nJimmy Shan (G, UIUC)\nBotong Wang (P, Notre Dame)\nYoungho Yoon (G, Notre Dame)\nWenbo Niu (P, Purdue)\nYi Zhang (P, Purdue)\nRagnar O. Buchweitz (F, Toronto)\nSaugata Basu (F, Purdue)\nAlexander Yong (F, UIUC)\nBen Wyser (P, UIUC)\nDominic Searles (G, UIUC)\nOliver Pechenik (G, UIUC)\nAmita Malik (G&U, UIUC)\nChayapa Darayon (G&U, UIUC)\nMatheus Brito (G&U, IUPUI and UNICAMP, Brazil)\nEvgeny Mukhin (F, IUPUI)\nVitaly Tarasov (F, IUPUI)\nAndrei Gabrielov (F, Purdue)\nBill Butske (F, Rose-Hulman)\nMatt Toeniskoetter (G, Purdue)\nAbhinishek Parab (G, Purdue)\nChristopher Drupieski (F, DePaul University)\nPeter Tingley (F, Loyola Chicago)\nUli Walther (F, Purdue)\nAlexandre Eremenko (F, Purdue)\nJerzy Weyman (F, Northeastern)\nRalph Kaufmann (F, Purdue)\n\n-------------------------------------------------------------------------------------------------------------------------------------\n\nParking: Please park in the garage next to the math building on University street. The easiest access is coming from the south via State Street. Weekend parking is free.\n\nLodging: We put on hold a block of 10 guestroom at Union Club Hotel, see\nhttp:\/\/www.union.purdue.edu\/HTML\/UnionClubHotel\/\nat the rate of $99(standard queen) per night plus tax. You can get rooms for up to 4 people with a double deluxe room ($144).\nMake reservations by calling (800) 320-6291","date":"2016-05-24 09:55:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3456137776374817, \"perplexity\": 14472.238740330493}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049270527.3\/warc\/CC-MAIN-20160524002110-00208-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
Q: How do I manage TCP Client read/write overlap issues? I have a TCP client communicating with a LabVIEW GUI.
My program calls connect() at the start and disconnect() at the end. It will call passCommand(x) to read or write data to the LabVIEW GUI. However, in some cases, I have multiple threads which may be calling passCommand() and somehow the return data will get mixed up.
For example, in the main thread I will ask for the voltage, which should be a number between 300 and 400. In a different thread I will ask for the temperature, which should be a number from 0-100. The voltage will be returned as 25, while the temperature will get 250.
Is this a known issue with TCP communication and threading? Is there a way to solve this such as implementing a queue or unique id or something?
import socket as _socket
# get python major version as integer
from sys import version as pythonVersion
pythonVersionMajor = int(pythonVersion[0])
_serverHost = 'localhost'
_serverPort = 50007
isConnected = 0
_sockobj = None
_error_string = "error:"
def connect():
'opens a connection to LabVIEW Server'
global _sockobj, isConnected
_sockobj = _socket.socket(_socket.AF_INET, _socket.SOCK_STREAM) # create socket
_sockobj.connect((_serverHost, _serverPort)) # connect to LV
isConnected = 1
def disconnect():
'closes the connection to LabVIEW Server'
global isConnected
_sockobj.close() # close socket
isConnected = 0
def passCommand(command):
'passes a command to LabVIEW Server'
## We prepend the command length (8 char long) to the message and send it to LV
# Compute message length and pad with 0 on the left if required
commandSize=str(len(command)).rjust(8,'0')
# Prepend msg size to msg
completeCommand=commandSize+command
# python 3 requires data to be encoded
if (pythonVersionMajor >= 3):
completeCommand = str.encode(completeCommand)
# Send complete command
_sockobj.send(completeCommand)
data = _sockobj.recv(11565536)
# python 3 requires data to be decoded
if (pythonVersionMajor >= 3):
data = bytes.decode(data)
if data.rfind(_error_string) == 0:
error = True
data = data[len(_error_string):] # get data after "error:" string
else:
error = False
execString = "lvdata = " + data
exec(execString, globals())
if error:
raise _LabVIEWError(lvdata)
else:
return lvdata
class _Error(Exception):
"""Base class for exceptions in this module."""
pass
class _LabVIEWError(_Error):
"""Exception raised for errors generated in LabVIEW.
Attributes:
code -- LabVIEW Error Code
source -- location of the error
message -- explanation of the error
"""
def __init__(self, error):
self.code = error[0]
self.source = error[1]
self.message = error[2]
def __str__(self):
return "%s" % (self.message,)
A: This is an example of one of the most common problems with threading. You are accessing a resource from multiple threads and the resource is not considered thread-safe (if both threads are sending/receiving at the same time, it's possible for a thread to get the wrong response, or even both responses).
Ideally you should be locking access to passCommand with a mutex so it can only be used with by one thread at a time, or opening one socket per thread, or doing all of your socket operations in a single thread.
| {
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} | 7,733 |
Advice of the Day for October 14th 2005!
A good way to figure out if someone knows what they're talking about when it comes to video games (as opposed to being a punk), is to ask them which of the Mega Mans is the best.
If they say anything other than 2, they're a punk. (Oh, 3 has his brother!!! Whoopie'd do! I have a brother, does that make me the best MegaMan game? No. Metal Blades make MegaMan 2 the best game. Period.).
Guy gets kidnapped. His kidnappers give him a souvenir baseball cap. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,859 |
Tag Archives: registration fraud
Entries tagged with "registration fraud"
Editorials: In face of voter ID laws, younger generation must get involved | Michael Sainato/The Hill
EditorialsBy admin October 27, 2015
The right to vote is one of the most frequently cited constitutional right in the Constitution itself; appearing five separate times total, including four individual amendments enacted to protect it. Since African-American men were granted the right to vote in 1870, and the passage of women's suffrage in 1920, many states have used arbitrary methods to deter certain blocks of voters from the polls. Poll taxes, literacy tests and complicated voter registration were commonplace up until the passage of the Voting Rights Act of 1965 that abolished these practices. A Supreme Court decision in 2013 invalidated a key component of the Voting Rights Act, giving nine Southern states the power to change their election laws without federal approval. Today, the impetuous transgressions against the right to vote from which the Voting Rights Act was enacted to protect Americans are being undermined by voter ID laws that are in currently being enforced in 32 states, 17 of those requiring photo identification. It is no coincidence that the year many voter restriction laws were put into place, 2014, voter turnout for the elections that year were the lowest in any election cycle since World War II.
Maldives: Elections Commission dismisses possibility of electoral fraud using deceased voter details | Minivan News
MaldivesBy admin August 6, 2013
The Elections Commission (EC) has rejected any possibility that the identities of deceased citizens could be used to fraudulently vote in the upcoming election, despite opposition allegations that security forces were seeking to influence polling by misusing such data. The Maldivian Democratic Party (MDP) has continued to accuse both the government and senior police officials of trying to undermine free and fair elections, alleging the institution was actively seeking deceased lists detailing the country's deceased in an attempts to try and rig voting. Rejecting any allegations that figures within the institution were seeking to rig polling, the Maldives Police Service (MPS) today confirmed it has been seeking a list detailing deceased peoples from across the Maldives as part of an investigation into allegations of fraudulent party membership.
Florida: In voter registration fraud case, it's not Mickey Mouse you have to worry about | Tampa Bay Times
FloridaBy admin October 10, 2012
The obviously fraudulent applications filed by a vendor hired by the Republican Party of Florida have gained wide attention in a case that's now being investigated by law enforcement. The dead woman registered to vote in Santa Rosa County. Phony addresses in Palm Beach County for voters that lead to a gas station, a Land Rover dealership and the Port Everglades administration office. But it's not blatant fraud like this that has elections experts worried about possible voting mayhem come November. Rather, it's the re-registration of voters, where personal information such as someone's party affiliation, signature or address could have been changed without the person's knowledge. "If they're submitting the names of dead people or Mickey Mouse, that will be caught," said Daniel A. Smith, a political scientist at the University of Florida. "The more pernicious type of fraud is where they change the addresses of people already registered, so that when they go to vote, they'll be at the wrong precinct."
Maryland: Officials not concerned about voter-registration fraud | Capital Gazette
MarylandBy admin October 10, 2012
Volunteers, interest groups and any individuals who want to print out the proper forms are rushing to register voters as Maryland's deadline looms less than two weeks away. But while recent voter-registration scandals have been cause for concern in some states, the state Board of Elections said the focus in Maryland is on voter roll maintenance, not registration fraud prevention. "There's a process in place, a very specific process that we work through," said Ross Goldstein, spokesman for the board. "We meet the letter of the law with respect to voter registration list maintenance." Prominent businessman and voter-registration drive leader Nathan Sproul, who runs Strategic Allied Consulting, is at the center of a voter-fraud registration scandal in Florida. Sproul, who has consulted prominent Republican candidates such as Mitt Romney, was linked to hundreds of forms containing irregularities, including suspicious signatures and missing information in nine Florida counties. Voter-registration fraud such as this, or when firms don't send in forms for voters from the opposite party, is insidious, said Paul Herrnson, director for the Center for American Politics and Citizenship at the University of Maryland.
Florida: Voter fraud complaint filed against Florida Democrats | SFGate
FloridaBy admin October 8, 2012
Florida authorities are reviewing allegations of voter registration fraud leveled against the Florida Democratic Party just days before the deadline to register new voters. The Florida Department of State on Friday confirmed that it has forwarded complaints about voter registration fraud that have been filed against the Democrats, as well as two other groups — the Florida New Majority Education Fund and the National Council of La Raza/Democracia USA. State election officials, as well as the Florida Department of Law Enforcement, provided few details on the complaints, including whether it is limited to just one county or how many voter registration forms are at issue. FDLE will look at the complaints and determine whether a criminal investigation should be launched.
National: Voter registration fraud claims singe GOP | CBS News
NationalBy admin October 5, 2012
Revelations that the Republican National Committee urged several states to hire a consulting firm that submitted potentially fraudulent voter registration forms in Florida are continuing to cause embarrassment to the Republican Party. RNC spokesman Sean Spicer said Thursday his group had cut ties to the firm, Strategic Allied Consulting, citing "zero tolerance" for voter fraud. "This is an issue we take extremely seriously," he told CBS News. "When allegations were brought to our attention we severed all ties to the firm." The Los Angeles Times reported that the RNC urged the state GOP in seven swing states to hire the firm, despite the fact that the man who runs it, Nathan Sproul, has been accused of running firms that have destroyed Democratic registrations. Sproul told the newspaper that RNC officials asked him to set up a new firm, Strategic Allied Consulting, so that his efforts would not be linked to those allegations. The RNC has reportedly paid the firm at least $3.1 million via state parties. Sproul blamed the suspicious forms on a single employee in Palm Beach County. But Florida election officials tell CBS News they have found a "couple hundred" voter registrations in eight Florida counties with "irregularities" that deserve further scrutiny. They are currently reviewing the registrations and if they find them to be "legally significant" they will turn them over to law enforcement. This could happen by the end of the day.
Florida: Election supervisor refutes Strategic Allied Consulting claim | ABC-7.com
Lee County Election Supervisor Sharon Harrington says she doesn't believe one person is responsible for more than 100 bogus election registration forms discovered in Florida. "I don't believe it's all just one person. It might be one person in a specific area," said Harrington, who was referring to claims submitted by Strategic Allied Consulting. The company is accused of forging voter registrations around the state. They were hired by the Republican Party and then fired after the allegations surfaced in Florida, North Carolina, Colorad, Nevada and Virginia.
Florida: Elections supervisors wonder how to deal with GOP voter registrations | Tampa Bay Times
With less than a week before the deadline to register to vote in the November election, Republican state leaders who had made voter fraud a top issue are offering little insight into how they are handling the increasing numbers of suspicious registration forms being found throughout Florida. Last week, the Florida Department of Law Enforcement began a review of Strategic Allied Consulting after the company turned in more than 100 botched voter registration forms in Palm Beach County on behalf of the Republican Party of Florida. Subsequently, 10 other counties — Bay, Charlotte, Duval, Escambia, Lee, Okaloosa, Pasco, Miami-Dade, Santa Rosa and Walton — have reported similar issues with registration forms linked to that firm. On Monday, a top elections official announced that the FDLE was investigating a second group, the National Council of La Raza, the largest Hispanic civil rights organization in the United States, for turning in three questionable registration forms in Miami-Dade County. The two cases, so far at least, are hardly equal in magnitude.
Texas: Voter fraud? Charlie Gonzalez, Texas Democrats say it's Republicans guilty of 'abuses' | Houston Chronicle
TexasBy admin October 3, 2012
Apparently dead people love to vote. Just weeks after Texas counties tried to purge their voter rolls by eliminating supposedly deceased voters (many of whom beg to disagree), it turns out that a firm hired by the Republican National Committee may have been registering truly deceased Republicans to vote in Florida. In ironic turn of events, the Republicans who have been strong proponents of the Voter ID laws, insisting that voter fraud does in fact exist, are now smack dab in the middle of a voter fraud investigation. A real, live criminal investigation.
Uncategorized: Florida elections supervisors wonder how to deal with GOP voter registrations | Tampa Bay Times
UncategorizedBy admin October 2, 2012
Editorials: Voter-fraud shocker?! On behalf of … the GOP? | latimes.com
EditorialsBy admin October 2, 2012
Republicans' current crop of "voter security" laws are Democrats' "voter suppression" laws. For several years now, Republican-led legislatures have been loud in their concerns about what amounts to a solution in search of a problem: massive, organized voter fraud in order to steal elections. Real verified instances of organized, deliberate voter fraud can likely be counted in the scores at best, and Republicans have been ardent about using the specter of the now-disbanded ACORN group to raise a national warning. … So get a load of what's just happened. There has emerged some potential voter fraud – possibly by a group hired by Republicans themselves, which puts me in mind of the verse in Matthew, in the Gospels, "And why beholdest thou the mote that is in thy brother's eye, but considerest not the beam that is in thine own eye?" which essentially means, who are you, Mr. Pot, to call the kettle black? The controversy surrounds a Republican political consulting firm whose chief operated a voter registration project that was investigated by the Justice Department and several state officials in 2004 on fraud allegations; charges were never filed, and in this 2012 instance, GOP officials, including the Republican National Committee, have been scrambling to fire the consulting firm to contain the political fallout a little over a month before the elections.
Colorado: Girl in viral voter registration video worked for shady firm | koaa.com
ColoradoBy admin October 2, 2012
We've learned the girl filmed in a viral YouTube video while registering voters at a local grocery store was employed by a company just dumped by the Republican National Committee over allegations of voter fraud. The video got a half-million views in just a few days, becoming an overnight Internet sensation with national attention. The girl gathering voter registrations claimed she was working for the El Paso County Clerk & Recorder's Office, and said at one point she preferred to register only Mitt Romney supporters. After speaking with local party heads and campaign officials, it was determined the girl was not working for the county clerk, but for a consulting firm hired by the state GOP. Eli Bremer, chairman of the El Paso County Republicans, explained to News 5 that it was the girl's first day on the job, and said she misspoke when asked who was "paying her". Wayne Williams, the El Paso County Clerk & Recorder, also confirmed that what she was doing– prescreening voters with preference questions before offering a voter registration form– was not illegal.
Florida: Suspicious Voter Forms Found in 10 Florida Counties | NYTimes.com
The number of Florida counties reporting suspicious voter registration forms connected to Strategic Allied Consulting, the firm hired by the state Republican Party to sign up new voters, has grown to 10, officials said, as local election supervisors continue to search their forms for questionable signatures, addresses or other identifiers. After reports of suspicious forms surfaced in Florida, the company — owned by Nathan Sproul, who has been involved in voter registration efforts since at least the 2004 presidential election — was fired last week by the state Republican Party and the Republican National Committee. The party had hired it to conduct drives in Colorado, Nevada, North Carolina and Virginia. In Colorado, a young woman employed by Strategic Allied was shown on a video outside a store in Colorado Springs recently telling a potential voter that she wanted to register only Republicans and that she worked for the county clerk's office. The woman was fired, said Ryan Call, chairman of the Colorado Republican Party.
National: Potential voter registration fraud in Florida: GOP's own 'ACORN' scandal? | CSMonitor.com
The Republican Party promptly fired a voter registration contractor this week after the firm, Strategic Allied Consulting, turned in illegible, incorrect, and falsified voter registration forms to Florida election officials. Saying the party has "zero tolerance" for voter fraud, the GOP also filed complaints against the company with the Florida Secretary of State's office. The company, run by long-time GOP operative Nathan Sproul, says a single employee was responsible for the forged signatures, though the problem, by Friday, had spread to 10 counties. "This is an issue we take extremely seriously," RNC spokesman Sean Spicer told CBS News. "When allegations were brought to our attention we severed all ties to the firm." While reasonable, those explanations could have trouble finding traction among the US electorate, which has watched battles erupt in mostly swing states from Florida to Ohio over control of voter rolls, and heated debates about potential disenfranchisement of key Democratic constituencies, poorer, minority, and elderly voters.
Florida: Suspicious voter registration forms found in 10 Florida counties | latimes.com
Florida elections officials said Friday that at least 10 counties have identified suspicious and possibly fraudulent voter registration forms turned in by a firm working for the Republican Party of Florida, which has filed an election fraud complaint with the state Division of Elections against its one-time consultant. The controversy in Florida — which began with possibly fraudulent forms that first cropped up in Palm Beach County — has engulfed the Republican National Committee, which admitted Thursday that it urged state parties in seven swing states to hire the firm, Strategic Allied Consulting.The RNC paid the company at least $3.1 million — routed through the state parties of Florida, Nevada, Colorado, North Carolina and Virginia — to register voters and run get-out-the-vote operations. Wisconsin and Ohio had not yet paid the firm for get-out-the-vote operations it was contracted to do.
National: GOP's ACORN moment | Salon.com
NationalBy admin September 28, 2012
There are still plenty of conservatives who think ACORN stole the 2008 election for Obama and will do it again this year. ACORN was everywhere four years ago. Even John McCain, late in his campaign and desperate to land a blow on Obama, ranan ad tying his challenger to the community-organizing group before saying in the final debate that ACORN "is now on the verge of maybe perpetrating one of the greatest frauds in voter history in this country, maybe destroying the fabric of democracy." How did ACORN steal the election? A number of the group's paid canvassers had been caught submitting false voter registration forms in a handful of states, using the names of dead people or false addresses, in order to avoid working. Four years later, ACORN is dead, and a Republican firm contracted by the Republican National Committee has adopted its shady tactics. But, so far at least, there's been hardly a peep from the same conservatives who seized on ACORN about one of their own engaging in almost identical fraudulent tactics. Prosecutors in Florida are looking into alleged voter registration fraud conducted by employees of Strategic Allied Consulting, which the RNC and state parties hired in at least five states. TheRNC has now cut ties with the firm after news broke that its employees had registered dead people and listed the addresses of a Land Rover dealership and other non-residences on registration forms. Paul Lux, the Republican supervisor of elections in Okaloosa County, Fla., who first brought the suspect registration forms to the attention of prosecutors, said as many as one in three were questionable. "It's kind of ironic that the dead people they accused ACORN of registering are now being done by the RPOF [Republican Party of Florida]," Lux said.
Florida: GOP fires consulting firm after 108 questionable voter registrations in Palm Beach County | The Washington Post
FloridaBy admin September 28, 2012
Republicans on Thursday fired a vendor suspected of submitting 108 questionable new voter registrations in Florida's Palm Beach County, ground zero for disputed ballots in 2000's presidential race. The Republican Party of Florida used Virginia-based Strategic Allied Consulting to help register and turnout voters in Florida, one of a shrinking handful of states President Barack Obama and Republican challenger Mitt Romney are contesting. The Florida state party had paid the firm more than $1.3 million so far, and the Republican National Committee used the group for almost $3 million of work in Nevada, North Carolina, Colorado and Virginia. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,844 |
/**
* Controller that demos exception handling using Spring Boot's internal setup.
*
* @author Paul Chapman
*/
package demo5.web; | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,692 |
Q: cascade delete not firing trigger One of the reasons I moved to postgresql was that (unlike mysql) foreign key actions (e.g. cascade delete) can fire triggers.
In my database I have the tables 'penalty_kicks' where the foreign key references 'player_instance':
ALTER TABLE penalty_kicks ADD FOREIGN KEY (kicker_id) REFERENCES player_instance (id) ON DELETE CASCADE;
I know that this works as when I delete a player_instance every associated row for penalty_kicks is deleted.
I also have a trigger function where if a row for penalty_kicks is deleted then it will alter a row for 'scores':
CREATE TRIGGER delete_penalty_kicks AFTER DELETE ON penalty_kicks FOR EACH ROW EXECUTE PROCEDURE delete_kicks();
CREATE FUNCTION delete_kicks()
RETURNS trigger AS $temp$
BEGIN
UPDATE scores s
SET points = points -3
FROM player_instance pi
WHERE pi.id = OLD.kicker_id
AND s.match_id = pi.match_id
AND s.team_id = pi.team_id
AND OLD.success;
RETURN NULL;
END;
$temp$
LANGUAGE 'plpgsql';
(big thanks to another user for the above)
I know that this works as when I delete a row for penalty_kicks it will perform the correct action on score.
However, when I delete a player_instance it will only delete the penalty_kick (by the cascade) and not fire the trigger to alter scores.
Any ideas?
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Navigation across multiple sites / code languages with 1 source Scenario:
We have a client who has multiple large sites, a huge number of stakeholders and decision makers which rules out a redevelopment involving all of them at the moment. Some sites are php, some are coldfusion and others are ASP.NET. We are building them a new site with a "TopHat" navigation that will be used across these site to link them together. Google and LinkedIn do something similar (Screenshots attached).
Question:
Is there a common term for this UI element? We invented the term "TopHat" to be able to collectively discuss the element.
Solution:
While googling, I discovered closure templates: http://code.google.com/closure/templates/index.html Would a closure template be a good approach? From what I read I can then notify the devs of each site to include the javascript in their code. Any ideas or feedback would be much appreciated.
A: If you only need JavaScript templates then Google Closure Templates will work for you as you compile the template code to JavaScript functions which will be called with parameters.
On the server side Google Closure Templates lack libraries for the languages you mentioned other than JavaScript and Java.
If you rely on multiple server side languages handling the same templates you can consider Mustache. Mustache 2 should also be able to be compiled to small JavaScript functions per template.
Regarding your solution. If you develope new functionalities you just have to pass the new JavaScript file to your deveopers. But again - only if JavaScript is the only rendering language.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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from flask.ext.script import Manager, Shell, Server
from shortner import app
manager = Manager(app)
manager.add_command("runserver", Server())
manager.add_command("shell", Shell())
manager.run()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,398 |
Arts & Museum Exhibits
Fine Dining & Wine
Live Music: Classical
Non-Profit Events/Fundraisers
Jan 27 Thursday
Vesper Meadow: Reconnecting People, Place, and Native Plants
06:00 PM - 07:00 PM on Thu, 27 Jan 2022
Learn about community-based strategies, beaver-based solutions, and native food plants utilized at the Vesper Meadow Restoration Preserve and how they can be applied to lands across the region. A local movement for native habitat restoration has started in Southwest Oregon, led by a new wave of conservation organizations, forward-thinking landowners, and Tribal partnerships. Now, after 200 years of land use mis-management, this work is critical to address the widespread impacts of climate changes on our shared landscape, natural resources, and water supply. Jeanine Moy will discuss ongoing projects with the Southwest Oregon Indigenous Gardens Network, local endangered species habitat restoration and monitoring, and examples of local edible and medicinal native plants that anyone can grow in their yard. Founder and Director of the Vesper Meadow Education Program, Jeanine Moy draws on a diverse background as a naturalist, educator, creative, activist, and backcountry adventurer. She has devoted the last two decades to the study of natural ecosystems and serving as an educator. She graduated from Cornell University with a B.S. in Applied Ecology, and from Southern Oregon University with a M.S. in Environmental Education. Her range of experiences include managing an agroforestry research and demonstration site in upstate New York, conducting plant field studies in the greater Yellowstone region, guiding rock climbing in Colorado, and teaching outdoor science to youth in Oregon.
Jan 30 Sunday
Local Stress Expert Guy Finley Reveals How to Let Go of Negative Reactions
Life of Learning Foundation
09:30 AM - 11:00 AM on Sun, 30 Jan 2022
Bestselling "Letting Go" author Guy Finley will speak on The Wisdom to Let Go and Outgrow Negative Reactions via livestream on Sunday, January 30 at 9:30 am. Pre-register at www.guyfinley.org/webinar. Finley says this talk will focus on the origin of negative reactions and why they are not the power we think they are. With this new understanding, when unwanted moments give rise to negative reactions, we can deal with them in a productive way and learn powerful methods to become free of their false authority. Guy Finley is an internationally renowned spiritual teacher and author of The Secret of Letting Go and 45 other books and audio programs that have sold over 2 million copies in 26 languages He is the Founder and Director of Life of Learning Foundation, a nonprofit center for spiritual discovery located in Merlin, Oregon. For more information about Guy and his classes visit www.guyfinley.org.
Feb 02 Wednesday
Windows in Time: "A 'Skyline Boulevard' for Crater Lake: The Army Corps of Engineers and Building Rim Road, 1910-20"
12:00 PM - 01:00 PM on Wed, 2 Feb 2022
Learn how Rim Road is part of a road system built more than a century ago for the newly established Crater Lake National Park and the subject of a historic district recently listed in the National Register of Historic Places (2019). Its construction is an epic tale about the perils and challenges of early road construction, when a state highway system in Oregon still lay squarely in the future. Presenter Steve Mark has worked for the National Park Service for more than three decades and spent most of that time as a historian stationed at Crater Lake. In 2019, he was the lead author for a nomination that was listed on the National Register of Historic Places and called the "Army Corps of Engineers Road System Historic District." It involved extensive research, much of it at the National Archives branch in Seattle and a multi-year archaeological survey conducted by a small team of NPS employees. Steve was born in Eugene and resides in Fort Klamath. The monthly Windows in Time lunchtime lectures feature well-known writers and historians and bring alive the people, values, and events that shaped our southern Oregon heritage. Lectures are jointly sponsored by the Southern Oregon Historical Society (SOHS) and Jackson County Library Services.
Feb 03 Thursday
Getting Started with Library of Things
12:00 PM - 01:00 PM on Thu, 3 Feb 2022
Learn how to borrow items from the Library of Things, including snow shoes, ukuleles, sewing machines, Wifi hotspots, metal detectors, and more. This program will cover the Library of Things user agreements and requirements, browsing the catalog, and submitting a request for checkout.
Feb 05 Saturday
VIRTUAL SEMINAR: GENEALOGY POTPOURRI
Rogue Valley Genealogical Society
10:00 AM - 04:00 PM on Sat, 5 Feb 2022
February 5, 2022 Virtual Seminar: Genealogy Potpourri 10am-4pm * $45/RVGS Members; $55/Non-Members Register at: https://rvgslibrary.org/FormPage.asp?FormID=17 February 5, 10am-4pm, the Rogue Valley Genealogical Society hosts a one-day, virtual seminar, "Genealogy Potpourri." Presented by Author and Professional Genealogist Jane Neff Rollins of Sherlock Combs Genealogy, she will show participants how to identify ancestors, even if there is no birth certificate; help identify clues found in your old photos to ascertain who may be in the photos, as well as when and where the photo was taken; how to find a wide variety of documents that identify dates and places of death, even when there is no death certificate; and help you to deal with sensitive issues your ancestors may have avoided discussing or might have kept secret.
"Sweet Sensations: The Science of Romance"
North Bend Public Library
11:00 AM - 12:00 PM on Wed, 9 Feb 2022
On Wednesday, February 9th at 11 am, Natural Grocers' nutritional health coach Cheryl O'Dell will talk about the science behind taste, smell, and emotions in a Zoom program called "Sweet Sensations: The Science of Romance." Just in time for Valentine's Day, attendees will learn how our senses engage our brains and bodies, including why humans associate sweet flavors and scents with romance. Chocolate and essential oils might sound like logical parts of this presentation, but beets will also make an appearance. To find out why, please register by going to https://tinyurl.com/bdfwvj6t.
But I'm and American Citizen
Southern Oregon University Hannon Library
07:00 PM - 11:59 PM on Thu, 10 Feb 2022
But, I'm an American Citizen Southern Oregon University's Hannon Library and Friends of the Hannon Library present their next event virtually on Thursday, February 10 at 7:00 PM Pacific time. Dr. Roy Saigo, SOU Past President, will give a personal presentation on the before, during, and afterward experiences of the incarceration of people of Japanese ancestry during Word War II. Register at https://tinyurl.com/FebEvent2022
The Importance of Faith During the Dying Process with Elizabeth Namgyel
TLC Transistional Life Care
$30-$50 sliding scale
10:00 AM - 03:30 PM on Sat, 12 Feb 2022
During our lives, and especially as we approach transition, faith can be our source of refuge, confidence, and comfort. During this program, we will explore how faith offers us support at the end of life and how it can connect us to a deeper place of strength. This type of faith increases trust, openness, and appreciation and will help to carry us onward with grace. This is a Zoom only event
Feb 13 Sunday
Sexual and Relationship Violence: Time for Change
04:00 PM - 05:30 PM on Sun, 13 Feb 2022
Why aren't sexual violence and relationship violence gone from our society? Susan Moen, Executive Director and co-founder of the local, non-profit Jackson County Sexual Assault Response Team (SART), will review risks, social norms, and stereotypes that encourage sexual violence. She will outline programs and strategies for prevention of harm. Please register for this Zoom conversation by going to thejeffcenter.org, clicking on the event title and then clicking the meeting link in the full event description.
Feb 18 Friday
2022 Winter Wings Festival
Oregon Tech Campus
Free to $250.00
08:00 AM - 09:00 PM, every day through Feb 20, 2022.
Get ready for a 2022 Winter Wings Festival like no other! Winter Wings brings together birders and photographers to learn and explore with top notch professionals and enthusiastic local guides. The Klamath Basin is renowned for its massive wintering population of Bald Eagles, but is prime habitat for many other raptors including owls, as well as a diversity of waterfowl. The 2022 Festival will feature Richard Crossley, author of the Crossley ID Guides and co-author of Ornitherapy: For Your Mind, Body, and Soul. For our photography keynote we are excited to have Jennifer Leigh Warner, conservation wildlife photographer. Join us for an extensive array of field trips, workshops, presentations, and receptions that highlight the wonders of the Klamath Basin in winter. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,149 |
{"url":"http:\/\/www.seanmathmodelguy.com\/?p=406","text":"## Mass transfer in a rotating disk reaction vessel (Part 2)\n\nPosted: 30th June 2012 by seanmathmodelguy in Blog, Collaborative Projects\n\nA schematic of the flow under a rotating disk\n\nShown here is a schematic of the rotating disk and how the fluid underneath it moves in response. Notice that the flow near the disk is essentially thrown out to the sides while away from the disk the fluid is effectively pumped towards the disk. There are essentially two separate processes in play here. Far from the disk the flow is essentially upwards towards the spinning sample and near the disk is ejected outward in a thin layer. \u00a0To get some quantitative idea as to the size of the various terms in the previous expressions we consider a disk of radius $$R$$ spinning at angular speed $$\\omega$$ above a column of fluid of height $$H$$. \u00a0This gives the natural scalings of $$r \\sim O(R), z \\sim O(H), u_r \\sim O(\\omega R), u_{\\theta} \\sim O(\\omega R)$$. \u00a0From the continuity equation in Part 1, we see that $$u_z \\sim O(\\omega H)$$ and for the pressure it is scaled to balance the inertial terms yielding $$P \\sim O(\\rho\\omega^2X^2)$$ where $$X$$ will be determined.\n\nUnder these scalings, the equations in Part 1 become\n\\begin{align}u_{r}\\frac{\\partial u_{r}}{\\partial r} -\\frac{u_\\theta^2}{r} + u_{z}\\frac{\\partial u_{r}}{\\partial z}&=-\\frac{X^2}{R^2}\\frac{\\partial P}{\\partial r}+\\frac{\\mu}{\\rho\\omega R^2}\\left(\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r\\frac{\\partial u_{r}}{\\partial r}\\right) \u2013 \\frac{u_{r}}{r^{2}}\\right) + \\frac{\\mu}{\\rho\\omega H^2}\\frac{\\partial^2u_{r}}{\\partial z^2},\\\\u_{r}\\frac{\\partial u_{\\theta}}{\\partial r} + \\frac{u_{\\theta}u_{r}}{r} + u_{z}\\frac{\\partial u_{\\theta}}{\\partial z}&=\\frac{\\mu}{\\rho\\omega R^2}\\left(\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r\\frac{\\partial u_{\\theta}}{\\partial r}\\right) \u2013 \\frac{u_{\\theta}}{r^{2}}\\right) + \\frac{\\mu}{\\rho\\omega H^2}\\frac{\\partial^2u_{\\theta}}{\\partial z^2},\\\\u_{r}\\frac{\\partial u_{z}}{\\partial r} + u_{z}\\frac{\\partial u_{z}}{\\partial z}&=-\\frac{X^2}{H^2}\\frac{\\partial P}{\\partial z} + \\frac{\\mu}{\\rho\\omega R^2}\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r\\frac{\\partial u_{z}}{\\partial r}\\right)+ \\frac{\\mu}{\\rho\\omega H^2}\\frac{\\partial^2u_{z}}{\\partial z^2}.\\end{align}\n\nThe chemical reactions are going on in the layer just below the rotating disk so this suggests that we should be considering a depth $$H$$ on the order of the thickness of the viscous boundary layer so that $$H^2 = \\mu\/\\rho\\omega$$. The term\n$$\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r\\frac{\\partial u_{i}}{\\partial r}\\right) \u2013 \\frac{u_{i}}{r^{2}}$$\nwith $$i = r,\\theta$$ suggest that $$u_r = r f(z)$$ and $$u_{\\theta} = r g(z)$$ and the continuity equation then implies that $$u_z = h(z)$$ giving the system\n\\begin{align}f^2-g^2+f\u2019g &= -\\frac{X^2}{R^2}\\frac{1}{r}\\frac{\\partial P}{\\partial r} + f^{\\prime\\prime},\\\\ 2fg + g\u2019h &= g^{\\prime\\prime},\\\\ h\u2019h &= -\\frac{X^2}{H^2}\\frac{\\partial P}{\\partial z} + h^{\\prime\\prime},\\\\ 2f + h\u2019 &= 0.\\end{align}\nSetting $$X=R$$ implies from the third expression that $$\\partial P\/\\partial z = 0$$ is the dominant effect and from the first equation $$P = Cr^2$$ reminiscent of the surface of a rotating bucket. The point here is that this decouples the axial variation in pressure from the axial velocity. If one instead sets $$X=H$$ then the axial velocity and pressure variations are connected no matter how fast the disk rotates and at very high rates of rotation $$H^2 \\ll R^2$$ suppressing the any radial variations in pressure.\n\nThis leaves the system of equations for the velocity in the boundary layer under the rotating disk as obtained by von Karman and used by Levich\n\\begin{align}f^{\\prime\\prime} &= f^2-g^2+f\u2019g,& f(0)&=0, \\quad \\lim_{z\\to\\infty}f(z) = 0, \\\\ g^{\\prime\\prime} &= 2fg + g\u2019h,& g(0)&=1, \\quad \\lim_{z\\to\\infty}g(z) = 0, \\\\ h^{\\prime\\prime} &= h\u2019h + p\u2019,& h(0)&=0, \\quad \\lim_{z\\to\\infty}h(z) = -\\alpha, \\\\ 2f + h\u2019 &= 0\\end{align}\nenforcing the boundary conditions that i) the fluid in contact with the disk moves with it and ii) outside the boundary layer the velocity is essentially upwards independent of radius.\n\nTo solve these equations, the first, second and fourth can be used to simultaneously solve for $$f, g, h$$. If one is interested in $$p$$ then the third expression can be used.\n\nNext, we\u2019ll look at the numerics to solve the system, what the fluid flow looks like the boundary layer and after that, we\u2019ll start on the chemistry that is happening at the interface and how it interacts with the motion of the fluid. We have tacitly assumed that the disk has an infinite radius or effectively infinite since we are neglecting edge effects. If you are interested in what happens for a finite disk then look at the paper by Brady and Durlofsky.\n\nReferences\nBrady, J.F. and Durlofsky, L., (1987).\u00a0On rotating disk flowJournal of Fluid Mechanics175(1),\u00a0pp. 363-394.\n\nLevich, V.G. and Spalding, D.B., (1962).\u00a0Physicochemical hydrodynamics,\u00a0Prentice-Hall Englewood Cliffs, NJ.\n\nVon Karman, T. and Lin, C.C., (1961).\u00a0On the existence of an exact solution of the equations of Navier-StokesCommunications on Pure and Applied Mathematics14(3),\u00a0pp. 645-655.\n\nZandbergen, P.J. and Dijkstra, D., (1987).\u00a0Von Karman swirling flowsAnnual review of fluid mechanics19(1),\u00a0pp. 465-491.","date":"2019-06-15 23:09:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"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\": 3, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9973597526550293, \"perplexity\": 1407.851981867246}, \"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-2019-26\/segments\/1560627997501.61\/warc\/CC-MAIN-20190615222657-20190616004657-00489.warc.gz\"}"} | null | null |
\section{Introduction}
\smallskip
This paper started as an attempt to understand the core of powers of
the homogeneous maximal ideal of a standard graded algebra. The {\it
core} of an arbitrary ideal $I$ in a Noetherian ring is defined as
the intersection of all reductions of $I$, equivalently, of all
ideals over which $I$ is integral. The definition simplifies when
$I$ is a power ${\mathfrak m}^n$ of the homogeneous maximal ideal ${\mathfrak m}$ of a
standard graded Cohen-Macaulay algebra $R$ of positive dimension
over an infinite field $k$. In this case $\core{{\mathfrak m}^n}$ is the
intersection of all ideals generated by systems of parameters
consisting of forms of degree $n$; in fact, it suffices to intersect
finitely many parameter ideals generated by {\it general} forms of
degree $n$ \cite{CPU01, PUV}. One has explicit formulas in this case
that express the core of ${\mathfrak m}^n$ as a colon ideal and that are valid
in any characteristic. If ${\rm char} \, k =0$ or if $R$ is
geometrically reduced, then
\[\core{{\mathfrak m}^n} = J^{j+(n-1)d+1}:{\mathfrak m}^{j} \/ ,\]
where $J$ is an ideal generated by a linear system of parameters of
$R$, $\/j\,$ is any integer $\geq a + d$, and $a$, $d$ denote the
$a$-invariant and the dimension of $R$, respectively (see
Theorem~\ref{formula} and, for prior results, \cite{HySm2, FPU}).
Thus the question arises of what more can be said about the colon
ideals $J^{j+(n-1)d+1}:{\mathfrak m}^{j}$ and, most notably, when the obvious
inclusion $ {\mathfrak m}^{a+nd+1} \subset J^{j+(n-1)d+1}:{\mathfrak m}^{j}$ is an
equality. We were also wondering in what sense the shape of the core
of ${\mathfrak m}$ or its powers reflects the geometry of ${\rm Proj}(R)$ as a
subscheme of projective space. As it turns out, the crucial device
in approaching these questions is the graded canonical module
$\omega$ of $R$ and the faithfulness of the submodules generated by
its graded components. Thus the main goal of this paper is to study,
quite generally, the interplay between annihilators of graded
components of $\omega$ on the one hand and colon ideals of powers of
${\mathfrak m}$ on the other hand.
More generally, let $R$ be a standard graded Cohen-Macaulay algebra
over a field $k$, of dimension $d >0$, and write ${\mathfrak m}$ for its
homogeneous maximal ideal and $\omega = \omega_R$ for its graded
canonical module. By $a=a(R)$ we denote the $a$-{\it invariant} of
$R$, which is the negative of the initial degree of $\omega$. Recall
that if $J$ is any ideal generated by a linear system of parameters
then `$J$ is a reduction of ${\mathfrak m}$ with reduction number
$r_J({\mathfrak m})=a+d$', which simply means that ${\mathfrak m}^i=J^{i-j}{\mathfrak m}^j$ for every
$i \geq j \geq a+d$, but for no smaller $j$. It is easy to see that
$J^i:{\mathfrak m}^j={{\mathfrak m}}^{i-j+a+d}$ for every $i$ and $j \geq
a+d$, provided $R$ is Gorenstein or, more generally, {\it level},
which means that $\omega$ is generated by homogeneous elements of
the same degree, $\omega = [\omega]_{-a}R$. Under these
assumptions, the $R$-submodules $[\omega]_tR$ of $\omega$ generated
by the homogeneous elements of a fixed degree $t$ are all faithful as
long as $t \geq -a$; this is obvious since $\omega=[\omega]_{-a}R$
is faithful and there exists a form of positive degree regular on
$\omega$. The same holds if $R$ is a domain because a suitable shift
of $\omega$ embeds into $R$. Without the additional assumptions on
the ring neither the statement about the colon ideals nor the one
about the graded components of the canonical module are true, but
there is a close relationship between the two conditions. In fact in
one of our main results we express, more generally, the annihilators
of graded components of $\omega$ in terms of colon ideals of powers
of ${\mathfrak m}$,
\[{\rm
ann}_R([\omega]_{\leq t}R)={\rm ann}_R([\omega]_{t}R)=\oplus_i
[J^{i+j-d+t+1} : {\mathfrak m}^{j}]_i\] for every $t$ and $j \geq a+d$ (see
Theorem~\ref{ann of omega}). Conversely, this allows one to write
the colon ideals $J^i:{\mathfrak m}^j$ for any $i\/$ and $j\geq a+d$ in the
form
\[ J^{i} : {\mathfrak m}^{j} ={\mathfrak m}^{i-j+a+d} + N \, , \]
where ${\mathfrak m}^{i-j+a+d}$ is the `expected' part and $N$ is an ideal of
height zero that is generated in degrees $\leq {i-j+a+d-1}$ and can
be described as
\[N= (\bigoplus_{l \leq i-j+a+d-1} [{\rm ann}_R
([\omega]_{i-j-l+d-1} R)]_{l})R \,
\]
(see Corollaries \ref{colondescription} and \ref{K}). To prove
these results it is useful to replace the colon ideals $J^{i} :_R
{\mathfrak m}^{j}$
in $R$ by the corresponding colon ideals $J^{i}\omega :_{\omega} {\mathfrak m}^{j}$ in $\omega$,
which we then relate to truncations $[\omega]_{\geq i-j+d}$ of
$\omega$ and to graded components of the canonical module $\Omega$
of the extended Rees ring of ${\mathfrak m}$.
This is done in one of our main technical results. There we also
use a bound on the regularity of $\Omega$ to show that
$J^{i}\omega :_{\omega} {\mathfrak m}^{j}= J^{i-j}(J^{j}\omega
:_{\omega} {\mathfrak m}^{j})$ for {\it every} integer $i$ with $i \geq j$
(see Theorem~\ref{omega} and, for related results, \cite{HySm1, PU,
HySm2, CC}).
Our results imply, for instance, that $[\omega]_{-a}R$ is faithful,
equivalently, $[\omega]_{t}R$ is faithful for every $t \geq -a$, if and only
if $J^{i} : {\mathfrak m}^{j} ={\mathfrak m}^{i-j+a+d}$ for every $i$ and $j \geq a+d$, if
and only if $J^{i} : {\mathfrak m}^{a+d} ={\mathfrak m}^{i}$ for some $i \gg 0$ (see
Corollary~\ref{omegafaithful2}). The question arises how large the
integer $i$ has to be chosen in the last statement. Conceivably
$i=a+d+1$ always works at least when $k$ is perfect and $R$ is reduced,
and we can show this if $d=1$ or if $R$ is an almost complete
intersection of embedding codimension $2$, for instance (see
Corollaries~\ref{omegafaithfuldim1} and \ref{codim2}). In general we
can prove that $i=\alpha +d+1$ suffices, where $\alpha=\alpha(R)$ is
the smallest possible $a$-invariant of a standard graded Gorenstein
ring of dimension $d$ mapping onto $R$ so that the kernel vanishes locally at
its minimal primes (see Corollary~\ref{alpha2}).
We deduce this result from an estimate on the initial degree of
certain colon ideals (see Proposition~\ref{primes}), which in turn
follows from a general bound on the generic generator degree of the canonical module (see
Proposition~\ref{CU}). The stronger result in the case of almost complete
intersections of embedding codimension $2$ is proved using an estimate for
the generic generator degree of the first syzygy module of homogeneous ideals
(see Propositions~\ref{syzygies}).
Returning to the core and the general assumptions used in this
context, we conclude that $[\omega]_{-a}R\/$ is faithful if and only
if $\core{{\mathfrak m}^n}={\mathfrak m}^{nd+a+1}$ for every $n\geq 1$, if and only if
$\core{{\mathfrak m}^n}$ is generated in one degree for some $n \gg 0$ (see
Theorem~\ref{core-ann}). Furthermore, $\core{{\mathfrak m}^n}$ can be replaced
by $\core{{\mathfrak m}}$ in the last statement, provided $d=1$ or $R$ is a
reduced almost complete intersection of embedding codimension $2$,
for instance (see Corollary~\ref{indeg=a+d}). This leads, rather
directly, to a geometric interpretation of the core
when $R$ is the homogeneous coordinate ring of a finite set $X$ of
reduced points in projective space. Most notably,
$\core{{\mathfrak m}}={\mathfrak m}^{a+2}$ if and only if $X$ has the Cayley-Bacharach
property (see Corollary~\ref{CB}). Recall that $X$ is {\it
Cayley-Bacharach} if the Hilbert function of $X\setminus \{P\}$ does
not depend on the point $P \in X$. Since this
property is equivalent to the faithfulness of $[\omega]_{-a}R$ (see
\cite{GKR}), the above characterization in terms of the core then
follows as an immediate consequence of Corollary~\ref{indeg=a+d}. We
also show that if a large enough subset of $X$ lies on a
hypersurface of low degree then the initial degree of $\core {\mathfrak m}$ is
forced to be unexpectedly small (see Corollary~\ref{Y and Z} and
Proposition~\ref{local}), underlining once more the fact that
the shape of the core reflects uniformity properties of the set of points.
\bigskip
\section{What annihilates the components of the canonical module?}
\smallskip
We begin by fixing notation and recalling some general facts.
Let $k$ be a field and $R$ a standard graded
$k$-algebra of dimension $d$ with homogeneous maximal ideal
${\mathfrak m}$ and graded canonical module $\omega$. Recall that
\begin{align*}
\hspace{-1.5cm} a(R)&= - {\rm
min} \, \{ \, i \, | \, [\omega]_{i} \not=0 \}
\end{align*}
is the $a$-{\it invariant} of $R$. We also consider the integers
\begin{align*}
b(R)&=-{\rm min} \{ i \, | \, [\omega]_iR \, \mbox{\rm {is
$R$-faithful}}
\} \ \ {\rm and} \\
c(R)&=-{\rm max} \{ i \, | \, [k \otimes_{R} \omega]_i \not=0 \,
\}.
\end{align*}
By local duality, the $a$-invariant is the top degree of the local
cohomology module $H^d_{{\mathfrak m}}(R)$. Also notice that $c(R)$ is the
negative of the largest generator degree of $\omega$. Since $\omega$
is $R$-faithful we have $a(R) \ge b(R) \ge c(R)$.
Now assume that $R$ is Cohen-Macaulay.
Let $S=k[x_1, \ldots, x_n]$ be a polynomial ring mapping
homogeneously onto $R$ and write $g={\rm codim}_S \, R = {\rm
projdim}_S \, R$. The integers $a(R)$ and $c(R)$ can be expressed in
terms of the minimal homogeneous free $S$-resolution of $R$ by means
of the formulas
\begin{align*}
a(R)&= {\rm max} \, \{ \, i \, | \, [{\rm Tor}^S_g(k,R)]_i \not=0 \,\} -n \\
c(R)&= \, {\rm min} \, \{ \, i \, | \, [{\rm Tor}^S_g(k,R)]_i \not=0
\,\} -n \, .
\end{align*}
\noindent
Notice that $c(R) \geq -d$; in particular, $\omega$ is
generated in degrees at most $d$ and $a(R) + d \geq 0$.
We also consider the extended Rees algebra $R[{\mathfrak m} t,t^{-1}]$, which
is a bigraded subring of $R[t,t^{-1}]$. There are natural maps
of bigraded modules
\[
\omega_{R[{\mathfrak m} t, t^{-1}]} \hookrightarrow (\omega_{R[{\mathfrak m} t,
t^{-1}]})_{t^{-1}} \cong \omega_{R[{\mathfrak m} t, t^{-1}]_{t^{-1}}} =
\omega_{R[t, t^{-1}]} \cong \omega \otimes_{R} R[t,t^{-1}]= \oplus\,
\omega \, t^{i}.
\]
Identifying $\omega_{R[{\mathfrak m} t, t^{-1}]}$ with its image in
$\oplus\, \omega \, t^{i}$ we obtain a bigraded canonical module
\[
\Omega= \oplus\, \Omega_i t^{i} \subset \oplus\, \omega \, t^{i},
\]
so that $\Omega_i=\omega$ for $i \ll 0$. One has homogenous
isomorphisms $ R[{\mathfrak m} \/ t, t^{-1}]/(t^{-1}) \cong {\rm gr}_{{\mathfrak m}} (R)
\cong R$, thinking of the last $R$ as being diagonally bigraded.
They induce identifications
\begin{equation}\label{omegarees}
\Omega/t^{-1}\Omega \cong \omega (1) \ \ \hbox{\rm and} \ \
\Omega_i/\Omega_{i+1} \cong [\omega]_{i+1}.
\end{equation}
We will use the convention that the power of any element or ideal with
non-positive exponent is one or the unit ideal, respectively.
\medskip
\begin{Assumptions}\label{standard}{\rm We assume $k$ is a field and $R$ is
a standard graded Cohen-Macaulay $k$-algebra of dimension $d\geq 1$ with homogeneous
maximal ideal ${\mathfrak m}$ and graded canonical module $\omega$. We write $a=a(R)$, $b=b(R)$, $c=c(R)$,
and let $\Omega = \omega_{R[{\mathfrak m} t,t^{-1}]}$ be as above, with $\Omega_i=\omega$ for $i \ll 0$.
Let $y_1, \ldots, y_d$ be a system of parameters in $R$ consisting of
linear forms. We write $J$ for the ideal generated by $y_1, \ldots, y_d$ and $J^{[i]}$ for the ideal
generated by the powers $y_1^{i}, \ldots, y_d^{i}$, where $i$ is any integer.}
\end{Assumptions}
\bigskip
Elements $y_1, \ldots, y_d$ as in Assumptions \ref{standard} always
exist if $k$ is infinite. The ideal $J$ they generate is a minimal
reduction of ${\mathfrak m}$ with reduction number $r_J({\mathfrak m})=a+d$, as can be
seen by reducing modulo $J$. Therefore
\[ J^{i}:{\mathfrak m}^{j}= J^{[i]}:{\mathfrak m}^{j+(i-1)(d-1)} \] for every $i$ and $j \geq a +d$;
in fact, one can show as in \cite[proof of 2.2]{PUV} that if $I$ is any
${\mathfrak m}$-primary ideal and $\beta_1, \ldots , \beta_d$ are elements of $I$
with $I^{j+1}=(\beta_1, \ldots , \beta_d)I^j$ for some integer $j$, then
\begin{equation}\label{PUV 2.2}(\beta_1,\ldots,\beta_d)^{i}:I^{j}=(\beta_1^{i},\ldots,\beta_d^{i}): I^{j+(i-1)(d-1)}
\end{equation}
for every $i$.
\medskip
In this section we prove our results about the relationship between
annihilators of graded components of $\omega$ on the one hand and
the colon ideals $J^i : {\mathfrak m}^j$ on the other hand. Here it suffices to
consider $j=a+d$, as the next remark shows.
\medskip
\begin{remark}\label{colon1}
With assumptions as in \ref{standard} one has
\[ J^{i} : {\mathfrak m}^{j}=J^{i-j+a+d}:{\mathfrak m}^{a+d} \] for every $i$ and $j
\geq a+d$.
\end{remark}
\noindent{\bf{Proof.} } Since $j \geq a+d=r_J({\mathfrak m})$, we obtain
\[
J^{i}:{\mathfrak m}^{j}=J^{i}:J^{j-a-d}{\mathfrak m}^{a+d}=(J^{i}:J^{j-a-d}):
{\mathfrak m}^{a+d}=J^{i-j+ a+d}:{\mathfrak m}^{a+d}.
\]
The last equality holds because the associated graded ring of $J$ has positive depth.
\hfill$\square$
\bigskip
We first study the colons $J^i \omega :_{\omega} {\mathfrak m}^j$ in $\omega$,
which exhibit a more regular behavior than the corresponding ideals
$J^i :_R {\mathfrak m}^j$.
\begin{theorem}\label{omega}
In addition to the assumptions of $\ \ref{standard}$ let $i$ and $j
\geq a+d$ be integers, and set $s=j+(i-1)(d-1)$. One has
\begin{itemize}
\item[(a)] $J^{i}\omega :_{\omega}
{\mathfrak m}^{j} =J^{[i]}\omega :_{\omega} {\mathfrak m}^s = [\omega]_{\geq i-j+d} =
\Omega_{i-j+d-1}$
\item[(b)]$J^{i}\omega :_{\omega}
{\mathfrak m}^{j}= J^{i-j}(J^{j}\omega :_{\omega}
{\mathfrak m}^{j})$ whenever $i \geq j$.
\end{itemize}
\end{theorem}
\noindent{\bf{Proof.} } To prove part (a) we first notice that
\[ \Omega_{i-j+d-1}=J^{i}\omega :_{\omega}
{\mathfrak m}^{j}
\]
according to \cite[3.7]{CC}. Next,
\begin{eqnarray*}
\nonumber
J^{i}\omega :_{\omega} {\mathfrak m}^{j} &=& (J^{[i]}:_R J^{(i-1)(d-1)})
\omega :_{\omega} {\mathfrak m}^j \hspace{1.55cm} \hbox{\rm by (\ref{PUV 2.2})} \\
&\subset& (J^{[i]}\omega :_{\omega} J^{(i-1)(d-1)}):_{\omega} {\mathfrak m}^j \\
&=& J^{[i]}\omega :_{\omega} (J^{(i-1)(d-1)} {\mathfrak m}^j ) \\
&=& J^{[i]}\omega :_{\omega} {\mathfrak m}^s \hspace{4cm} \hbox{\rm because $j \geq a+d = r_{J} ({\mathfrak m})$} \, .\\ \end{eqnarray*}
To show that $J^{[i]}\omega :_{\omega} {\mathfrak m}^s= [\omega]_{\geq i-j+d}$
we may reduce modulo $J^{[i]}\omega\,$; indeed, $J^{[i]}\omega \subset
[\omega]_{\geq i-j+d}$ because $\omega$ is concentrated in degrees
$\geq -a$ and $i-a \geq i-j+d$. However, $(J^{[i]}\omega :_{\omega}
{\mathfrak m}^s)/ J^{[i]}\omega \cong \ol{0} :_{\ol{\omega}} \ol{{\mathfrak m}}^s$, where
$\ol{{\mathfrak m}}$ is the homogeneous maximal ideal of $\, \ol{R}=R/J^{[i]}$ and
$\ol{\omega}= \omega/J^{[i]}\omega \cong \omega_{\ol{R}} \,(-id)$.
As $\omega_{\ol{R}}$ is an Artinian module with socle concentrated
in degree $0$, the module $\ol{\omega}$ is Artinian with socle
concentrated in degree $id$. Thus $\ol{0} :_{\ol{\omega}} \ol{{\mathfrak m}}^s
= [\ol{\omega}]_{\geq id+1-s}= [\ol{\omega}]_{\geq i-j+d}$. It
follows that $J^{[i]}\omega :_{\omega} {\mathfrak m}^s= [\omega]_{\geq i-j+d}$.
So far we have shown the inclusions
\[
\Omega_{i-j+d-1}=J^{i}\omega :_{\omega}
{\mathfrak m}^{j} \subset J^{[i]}\omega :_{\omega} {\mathfrak m}^s = [\omega]_{\geq
i-j+d} \, .
\]
Now $\Omega =\oplus \Omega_{l}t^l \subset \oplus [\omega]_{\geq
l+1} t^{l}=\Omega'$ are bigraded $R[{\mathfrak m} t, t^{-1}]$-modules that are
finitely generated because $[\omega]_{\geq l+1}={\mathfrak m} [\omega]_{\geq
l}$ for $l \gg 0$. By (\ref{omegarees}) this inclusion induces an
isomorphism $\Omega/t^{-1}\Omega \cong \Omega'/t^{-1}\Omega'$,
which shows that $\Omega'=\Omega + t^{-1}\Omega'$. Therefore
$\Omega'=\Omega$ by the graded Nakayama lemma.
We now prove part (b). In the light of (a) we need to show that for
every $l \geq d$ one has $[\omega]_{\geq l+1} = J [\omega]_{\geq
l}$, or equivalently, $[\omega]_{\geq l+1} \subset J \omega$.
However, $\omega/ J \omega \cong \omega_{R/J} (-d)$ is concentrated
in degrees $\leq d$, which gives $[\omega]_{\geq l+1} \subset J
\omega$. \hfill$\square$
\bigskip
The next result addresses the comparison between the colons
$J^i\omega :_{\omega} {\mathfrak m}^j$ and $J^i :_R {\mathfrak m}^j$.
\begin{corollary}\label{=}
In addition to the assumptions of $\, \ref{standard}$ let $i$ and $j
\geq a+d$ be integers. If \[J^{i}\omega :_{\omega} {\mathfrak m}^{j}=(J^{i}:_R
{\mathfrak m}^{j})\, \omega \hspace{.5cm} \hbox{\rm for some} \hspace{.3cm} i \geq
j \, ,\] then for every $l \geq i$,
\begin{itemize}
\item[(a)] $ J^{l}\omega :_{\omega} {\mathfrak m}^{j}=(J^{l}:_R {\mathfrak m}^{j}) \, \omega$
\item[(b)] $J^{l}:_R {\mathfrak m}^{j}$ is integral over $J^{l-i}(J^{i}:_R
{\mathfrak m}^{j})$; in particular, if $R$ is reduced then the initial degree
of the former ideal satisfies \[{\rm indeg} (J^{l}:_R {\mathfrak m}^{j}) ={\rm indeg} (J^{i}:_R
{\mathfrak m}^{j})+ l-i \, .\]
\end{itemize}
\end{corollary}
\noindent{\bf{Proof.} } One has
\begin{eqnarray*}
\nonumber
(J^{l}:_R {\mathfrak m}^j) \, \omega &\subset& J^{l}\omega :_{\omega} {\mathfrak m}^j\\
&=& J^{l-i}(J^{i}\omega :_{\omega} {\mathfrak m}^j) \hspace{2cm} \hbox{\rm by Theorem~\ref{omega}(b)} \\
&=& J^{l-i} (J^{i}:_R {\mathfrak m}^j) \, \omega \hspace{1.935cm} \hbox{\rm by our assumption}\\
&\subset& (J^{l}:_R {\mathfrak m}^j) \, \omega \, .
\end{eqnarray*}
This immediately gives $ J^{l}\omega :_{\omega} {\mathfrak m}^{j}=(J^{l}:_R
{\mathfrak m}^{j}) \, \omega$, proving assertion (a). Furthermore, $(J^{l}:_R
{\mathfrak m}^j) \, \omega=J^{l-i} (J^{i}:_R {\mathfrak m}^j) \, \omega$, which implies (b) since
$\omega$ is a faithful $R$-module. \hfill$\square$
\bigskip
The next proposition gives a characterization for when the equality
assumed in the previous corollary obtains.
\begin{proposition}\label{omegacontainment}
In addition to the assumptions of $\, \ref{standard}$ let $i$ and $j \geq
a+d$ be integers. Set $s=j+(i-1)(d-1)$ and let $^{^{\mbox{\rule{2mm}{.2mm}$\;\!$}}}$ denote images
in $\ol{R}=R/J^{[i]}$. One has
\[J^{i}\omega :_{\omega}
{\mathfrak m}^{j}=(J^{i} :_{R} {\mathfrak m}^{j}) \, \omega \]
if and only if
\[\ol{{\mathfrak m}}^{s}=
{\ol{0}:_{\ol{R}}(\ol{0}:_{\ol{R}}\ol{{\f{m}}}^{s})} \, .\]
\end{proposition}
\noindent{\bf{Proof.} } First notice that $\ol{R}$ is an Artinian ring
and that $\ol{\omega}= \omega/J^{[i]}\omega \cong
\omega_{\ol{R}} \,(-id)$. From Theorem~\ref{omega}(a)
one has $J^i\omega :_{\omega} {\mathfrak m}^j = J^{[i]}\omega :_{\omega}
{\mathfrak m}^s$. Hence it follows that $(J^i\omega :_{\omega} {\mathfrak m}^j)/
J^{[i]}\omega \cong \ol{0} :_{\ol{\omega}}
\ol{{\mathfrak m}}^s$. Now the latter module is naturally isomorphic to
$(\ol{0} :_{\omega _{\ol R}} \ol{{\mathfrak m}}^s)(-id) \cong \omega_{\ol{R}/\ol{{\mathfrak m}}^s} \,(-id)$.
Thus we have shown that
\[ (J^i\omega :_{\omega} {\mathfrak m}^j)/
J^{[i]}\omega \cong \omega_{\ol{R}/\ol{{\mathfrak m}}^s} \,(-id).\]
On the other hand, from (\ref{PUV 2.2}) one knows that $J^{i} : {\mathfrak m}^{j}=
J^{[i]}:{\mathfrak m}^{s}$. Therefore $(J^{i} :_R {\mathfrak m}^{j}) \, \omega/
J^{[i]}\omega =(\ol{0} :_{\ol{R}} \ol{{\mathfrak m}}^s) \, \ol{\omega}$, which is
isomorphic to
$(\ol{0} :_{\ol{R}} \ol{{\mathfrak m}}^s)\, \omega_{\ol{R}} \,(-id)$. The last module
can be identified with
$\omega_{\ol{R}/{\ol{0}:(\ol{0}:\ol{{\mathfrak m}}^{s})}}(-id)$, as shown in \cite[2.3(b)]{U94}.
Hence we have proved that
\[(J^{i} :_R {\mathfrak m}^{j}) \, \omega/ J^{[i]}\omega \cong \omega_{\ol{R}/{\ol{0}:(\ol{0}:\ol{{\mathfrak m}}^{s})}}(-id) \, . \]
We conclude that the obvious containment $(J^{i}:_R {\mathfrak m}^{j}) \, \omega
\subset J^{i}\omega :_{\omega} {\mathfrak m}^{j}$ is an equality if and only if
the inclusion $\ol{{\mathfrak m}}^{s} \subset \ol{0}:(\ol{0}:\ol{{\mathfrak m}}^{s})$ is.
\hfill$\square$
\bigskip
If the equalities of Proposition~\ref{omegacontainment} hold for
all $i$, then the canonical module of the extended Rees ring of ${\mathfrak m}$ has
an easy description, namely
\[\Omega=(R[Jt,t^{-1}] :_{R[t,t^{-1}]} {\mathfrak m}^j) t^{d-1-j}\omega\]
(see Theorem ~\ref{omega}(a) or \cite[3.7]{CC}).
These equalities obtain, quite generally, when $i \geq j$ and $d=1$:
\begin{proposition}\label{=indim1}
In addition to the assumptions of $\, \ref{standard}$ suppose that $d=1$
and $R$ is reduced. For
every $i \geq j$ and $j \geq a+1$ one
has \[(J^{i} :_R {\mathfrak m}^{j}) \, \omega = J^{i}\omega :_{\omega}
{\mathfrak m}^{j}=[\omega]_{\geq i-j+1}.\]
\end{proposition}
\smallskip
\noindent{\bf{Proof.} } The second equality follows from Theorem~\ref{omega}(a). We
prove the first equality. It suffices to consider the case $i=j=a+1$
according to Remark \ref{colon1} and Corollary~\ref{=}(a),
or simply because $J$ is generated by a single regular element.
In light of Proposition~\ref{omegacontainment} we need to show that
\[{\mathfrak m}^{a+1}=y^{a+1}R:_R ( y^{a+1}R :_R {\mathfrak m}^{a+1})\]
with $y=y_1$.
In the total ring of quotients $K$ of $R$ we consider the
integral closure $S$ of $R$. Our assumptions, most
notably the reducedness of $R$, imply that $R \subset S$ is a
homogeneous inclusion of non-negatively graded Noetherian rings and
moreover $R:_K ( R :_K S)=S$. Thus we obtain
\begin{eqnarray*}
\nonumber
{\mathfrak m}^{a+1} &\subset& y^{a+1}R:_R ( y^{a+1}R :_R {\mathfrak m}^{a+1}) \\
&\subset& y^{a+1}R:_R ( y^{a+1}R :_R {\mathfrak m}^{a+1}S) \\
&=& y^{a+1}R:_R ( y^{a+1}R :_R y^{a+1}S) \hspace{1.5cm}\hbox{\rm
since $yR$ is a reduction of ${\mathfrak m}$}\\
&=& y^{a+1}R:_R ( R :_R S) \hspace{2.925cm} \hbox{\rm since $y$
is $R$-regular} \\
&=& [y^{a+1}R:_K ( R :_R S)] \cap R \\
&=& [y^{a+1}(R:_K ( R :_K S))] \cap R \\
&=& y^{a+1}S \cap R \hspace{4.17cm} \hbox{\rm since $R:_K ( R :_K
S)=S$}\\
&\subset& {\mathfrak m}^{a+1} \hspace{4.97cm} \hbox{\rm since $S$ is non-negatively
graded}.
\end{eqnarray*}\hfill$\square$
\bigskip
Next we are going to use the above results to say something about
the annihilators of the graded components of $\omega$.
\begin{theorem}\label{ann of omega}
With assumptions as in $\ref{standard}$ one has \[{\rm
ann}_R([\omega]_{\leq t}R)={\rm ann}_R([\omega]_{t}R)=\oplus_i
[J^{i+j-d+t+1} : {\mathfrak m}^{j}]_i\] for every $t$ and $j \geq a+d$.
\end{theorem}
\noindent{\bf{Proof.} } The first equality is obvious because $R$ contains a linear
form, namely $y_1$, that is regular on $\omega$.
Theorem~\ref{omega}(a) shows that
\[(J^{i+j-d+t+1} : {\mathfrak m}^{j}) \, \omega \subset [\omega]_{\geq i+t+1} \, ,\]
which gives $[J^{i+j-d+t+1} : {\mathfrak m}^{j}]_i [\omega]_{\leq t} =0.$ This
proves the inclusion
\[\oplus_i [J^{i+j-d+t+1} : {\mathfrak m}^{j}]_i \subset {\rm
ann}_R([\omega]_{\leq t}R) \, .\]
To show the containment \[{\rm ann}_R([\omega]_{\leq t}R) \subset
\oplus_i [J^{i+j-d+t+1} : {\mathfrak m}^{j}]_i \, ,\] we choose an element $f\in
[{\rm ann}_R([\omega]_{\leq t}R)]_i.$ We need to prove that \[ f \in
J^{i+j-d+t+1} : {\mathfrak m}^{j} = J^{[i+j-d+t+1]} : {\mathfrak m}^{j+ (i+j-d+t)(d-1)} \, ,\]
where the last equality holds by (\ref{PUV 2.2}). Write $s=j+
(i+j-d+t)(d-1)$ and let $^{^{\mbox{\rule{2mm}{.2mm}$\;\!$}}}$ denote images in the Artinian ring
$\ol{R}=R/J^{[i+j-d+t+1]}.$ The asserted inclusion $f {\mathfrak m}^s \subset
J^{[i+j-d+t+1]}$ is equivalent to $\ol{f}\, \ol{\f{m}}^{s} = \ol{0},$
which in turn means that $\ol{f} \,\ol{\f{m}}^{s} \omega_{\ol{R}} =
\ol{0}$ as $\omega_{\ol{R}}$ is faithful over $\ol{R}$. Since
$\omega_{\ol{R}} = \ol{R} \otimes_R \omega \, ((i+j-d+t+1)d)$ and
$f [\omega]_{\leq t}=0$ it follows that $\ol{f}
[\omega_{\ol{R}}]_{\leq t-(i+j-d+t+1)d}=\ol{0}.$ Therefore $\ol{f}
[\omega_{\ol{R}}]_{\leq -s-i}=\ol{0},$ and as $\ol{f}$ has degree
$i$ we conclude that \[ \ol{f} \,\ol{{\mathfrak m}}^s \omega_{\ol{R}} \subset
[\omega_{\ol{R}}]_{\geq 1} =\ol{0} \, . \]\hfill$\square$
\bigskip
Again, one has a better result in dimension one.
\begin{corollary}\label{ann of omega dim1}
In addition to the assumptions of $\, \ref{standard}$
suppose that $d=1$ and write $y=y_1$. One has
\[
{\rm ann}_R([\omega]_{\leq t}R)={\rm ann}_R([\omega]_{t}R)=\oplus_{i
\leq -t-1} [J^{i+j+t} : {\mathfrak m}^{j}]_i \, \bigoplus \ [J^j:{\mathfrak m}^j]_{-t}k[y]
\] for every $t$ and $j \geq a+1$.
\end{corollary}
\noindent{\bf{Proof.} } We apply Theorem~\ref{ann of omega} using the fact that
$J^{i+j+t} : {\mathfrak m}^j= y^{i+t}(J^j:{\mathfrak m}^j)$ for every $i \geq -t$. \hfill$\square$
\bigskip
The above corollary shows in particular that if $d=1$ then the
graded $R$-module ${\rm ann}_R([\omega]_{\leq t}R)={\rm ann}_R([\omega]_{t}R)$
has Castelnuovo-Mumford regularity at most $-t$; see also Proposition
\ref{CM}. For the definition and basic properties of the Castelnuovo-Mumford
regularity we refer to \cite[p.168]{BH}.
\smallskip
We are now ready to answer one the main questions raised in the
introduction. With the next three corollaries we characterize the
faithfulness of submodules generated by graded components of
$\omega$, in terms of certain colon ideals of powers of ${\mathfrak m}$ having
an `expected form'. In light of Remark \ref{colon1} we may restrict
ourselves to the case $j=a+d$.
\medskip
\begin{corollary}\label{omegafaithful1} With assumptions as in $\ref{standard}$ the
following are equivalent for an integer $t :$
\begin{itemize}
\item[(a)] $[\omega]_{t}R$ is faithful, i.e., $t \geq -b$
\item [(b)]$J^{i+a+t} :
{\mathfrak m}^{a+d} \subset {\mathfrak m}^{i}$ for every $ i$
\item[(c)]$J^{i+a+t} :
{\mathfrak m}^{a+d} \subset {\mathfrak m}^{i}$ for some $i \gg 0$.
\end{itemize}
If $R$ is reduced and \, $J^{i+a+t}\omega :_{\omega}
{\mathfrak m}^{a+d}= (J^{i+a+t} :_R
{\mathfrak m}^{a+d}) \, \omega$ \, for some $i \geq d-t$, then the above conditions
are equivalent to $:$
\begin{itemize}
\item[(d)] $J^{i+a+t} : {\mathfrak m}^{a+d} \subset {\mathfrak m}^{i}$.
\end{itemize}
\end{corollary}
\noindent{\bf{Proof.} } Recall that $R$ contains a linear form that
is a regular element. Thus, if a homogeneous ideal vanishes in
a certain degree it also vanishes in every smaller degree. Now
Theorem~\ref{ann of omega} gives the equivalence of (a) and
(b). The same theorem shows that if (c) holds then $[{\rm ann}_R([\omega]_{t}R)]_{i-1}=0$ for some
$i-1 \gg 0$, proving (a). Finally, (d) implies (c) according to Corollary~\ref{=}(b). \hfill$\square$
\medskip
\begin{corollary}\label{omegafaithful2} With assumptions as in $\ref{standard}$ the
following are equivalent $:$
\begin{itemize}
\item[(a)] $[\omega]_{-a}R$ is faithful
\item [(b)]$J^{i} :
{\mathfrak m}^{a+d}={\mathfrak m}^{i}$ for every $ i$
\item[(c)]$J^{i} :
{\mathfrak m}^{a+d}={\mathfrak m}^{i}$ for some $i \gg 0$.
\end{itemize}
If $R$ is reduced and $J^{i}\omega :_{\omega}
{\mathfrak m}^{a+d} =(J^{i} :_R
{\mathfrak m}^{a+d}) \, \omega $ for some $i \geq a+d$, then the above conditions
are equivalent to $:$
\begin{itemize}
\item[(d)] $J^{i} : {\mathfrak m}^{a+d} = {\mathfrak m}^{i}$.
\end{itemize}
\end{corollary}
\noindent{\bf{Proof.} } Notice that ${\mathfrak m}^{i} \subset J^{i}:{\mathfrak m}^{a+d}$ for every $i\/$
because $a+d =r_J({\mathfrak m})$. Now the assertions follow from
Corollary~\ref{omegafaithful1}. \hfill$\square$
\medskip
\begin{corollary}\label{omegafaithfuldim1} In addition to the
assumptions of $\, \ref{standard}$
suppose that $d=1$. The
following are equivalent $:$
\begin{itemize}
\item[(a)] $[\omega]_{-a}R$ is faithful
\item [(b)]$J^{i} :
{\mathfrak m}^{a+1}={\mathfrak m}^{i}$ for every $ i$
\item[(c)]$J^{i} :
{\mathfrak m}^{a+1}={\mathfrak m}^{i}$ for some $i \geq a + 1$.
\end{itemize}
\end{corollary}
\noindent{\bf{Proof.} } In light of Corollary~\ref{omegafaithful2} it suffices to
prove that if $J^{i} : {\mathfrak m}^{a+1}={\mathfrak m}^{i}$ for some $i \geq a +1$,
then $J^{l} : {\mathfrak m}^{a+1}={\mathfrak m}^{l}$ for every $l \geq i$. Indeed, since
$J$ is generated by a single regular element and $i\geq a+1= r_J({\mathfrak m})$, it
follows that \[J^{l} : {\mathfrak m}^{a+1}= J^{l-i}(J^{i} : {\mathfrak m}^{a+1})=J^{l-i} {\mathfrak m}^{i}={\mathfrak m}^{l}. \]\hfill$\square$
\medskip
The faithfulness of $[\omega]_{-a}R$, together with the additional
condition $J^{i}\omega :_{\omega}{\mathfrak m}^{a+d} =(J^{i} :_R
{\mathfrak m}^{a+d}) \, \omega $ in Corollary \ref{omegafaithful2}, means that
$\omega$ and $[\omega]_{-a}R$ coincide from degree $i-a$ on:
\smallskip
\begin{remark}\label{==}{\rm In addition to the assumptions of \ref{standard} let
$i \geq a+d$ be an integer. Then the equality $ [\omega]_{\geq i-a} =
{\mathfrak m}^i [\omega]_{-a}$ holds if and only if $J^{i}\omega :_{\omega}{\mathfrak m}^{a+d}
=(J^{i} :_R {\mathfrak m}^{a+d}) \, \omega $ and $[\omega]_{-a}R$ is faithful.
The forward direction follows because
$[\omega]_{\geq i-a} =J^{i}\omega :_{\omega}{\mathfrak m}^{a+d}$ according to
Theorem~\ref{omega}(a), ${\mathfrak m}^i \subset J^i : {\mathfrak m}^{a+d}$, and $[\omega]_{\geq i-a}$ is
faithful. For the converse notice that
Corollary~\ref{omegafaithful2} gives $J^i:{\mathfrak m}^{a+d}={\mathfrak m}^i$, hence $[\omega]_{\geq i-a}=
J^{i}\omega :_{\omega}{\mathfrak m}^{a+d} ={\mathfrak m}^i\omega$.
Therefore $[\omega]_{\geq i-a}={\mathfrak m}^i
[\omega]_{-a} + [\omega]_{\geq i+1-a}.$ However, $\omega$ is generated in
degrees at most $d \leq i-a$, hence $[\omega]_{\geq
i-a}$ is generated in degree $i-a$. It follows that $[\omega]_{\geq i-a} = {\mathfrak m}^i
[\omega]_{-a}$.}
\end{remark}
\medskip
Unfortunately, the faithfulness of $[\omega]_{-a}R$ alone is not
sufficient to guarantee the equality $J^{i}\omega :_{\omega}{\mathfrak m}^{a+d} =
(J^{i} :_R {\mathfrak m}^{a+d}) \, \omega$, as the next example shows.
\smallskip
\begin{example} In addition to the assumptions of \ref{standard} suppose
that $R$ is a domain of type 2 which is not level and which is not
Gorenstein locally on the punctured spectrum. Clearly
$[\omega]_{-a}R$ is faithful since $R$ is a domain. However,
$J^{i}\omega :_{\omega} {\mathfrak m}^{a+d} \not= (J^{i} :_R {\mathfrak m}^{a+d}) \, \omega$ for every
$i\geq a+d$, because otherwise Remark~\ref{==} implies that $\omega$
and the cyclic module $[\omega]_{-a}R$ would coincide locally on the
punctured spectrum.
\end{example}
\bigskip
Conversely, Theorem~\ref{ann of omega} can be used to obtain
information about the colon ideals $J^i : {\mathfrak m}^j\,$. This is done in
the remaining corollaries of this section.
\medskip
\begin{corollary}\label{colondescription}
With assumptions as in $\ref{standard}$ one has
\[ J^{i} : {\mathfrak m}^{j} ={\mathfrak m}^{i-j+a+d} \ \bigoplus \ \ \ \bigoplus_{l =
i-j+b+d}^{i-j+a+d-1} [{\rm ann}_R ([\omega]_{i-j-l+d-1} R)]_{l}
\]
for every $i$ and $j\geq a+d$.
\end{corollary}
\noindent{\bf{Proof.} } Theorem~\ref{ann of omega} shows that
\[ [{\rm ann}_R ([\omega]_{i-j-l+d-1} R)]_{l}= [J^{i} : {\mathfrak m}^{j}]_{l}
\] for every $l$. On the other hand,
$[{\rm ann}_R ([\omega]_{i-j-l+d-1} R)]_{l}=0 \/$ for $l\leq
i-j+b+d-1$ by the definition of $b$. Finally, ${\mathfrak m}^{i-j+a+d} \subset
J^i : {\mathfrak m}^j$. \hfill$\square$
\medskip
\begin{corollary}\label{K} With assumptions as in $\ref{standard}$ there exists
an ideal $N$ of height zero such that
\[ J^{i} : {\mathfrak m}^{j} ={\mathfrak m}^{i-j+a+d} + N
\]
for every $i$ and $j\geq a+d$.
\end{corollary}
\noindent{\bf{Proof.} } Set $N=(\bigoplus_{l = i-j+b+d}^{i-j+a+d-1} [{\rm ann}_R
([\omega]_{i-j-l+d-1} R)]_{l})R.\,$ According to
Corollary~\ref{colondescription}, $J^i: {\mathfrak m}^j = {\mathfrak m}^{i-j+a+d} + N$. On
the other hand, the first equality of Theorem~\ref{ann of omega}
shows that $N \subset {\rm ann}_R ([\omega]_{-a} R ). \/$ The latter
ideal has height zero since $[\omega]_{-a} R$ is a nonzero submodule
of the maximal Cohen-Macaulay module $\omega$. \hfill$\square$
\medskip
\begin{corollary}\label{power} In addition to the assumptions of $\, \ref{standard}$
let $i$ and $j \geq a+d$ be integers. If $J^i : {\mathfrak m}^j$ is generated
in one degree then $J^i : {\mathfrak m}^j={\mathfrak m}^{i-j+a+d}$.
\end{corollary}
\noindent{\bf{Proof.} } The claim follows from Corollary~\ref{K} since $J^i : {\mathfrak m}^j$ has positive
height. \hfill$\square$
\medskip
\begin{corollary}\label{colonmax1} With assumptions as in $\ref{standard}$ one has
\[ {\mathfrak m}^{i-j+a+d} \subset J^{i} : {\mathfrak m}^{j} \subset {\mathfrak m}^{i-j+b+d}.
\]
for every $i$ and $j\geq a+d$.
\end{corollary}
\noindent{\bf{Proof.} } The containments are direct consequences of
Corollary~\ref{colondescription}. \hfill$\square$
\medskip
\begin{corollary}\label{colonmax2}
In addition to the assumptions of $\, \ref{standard}$ assume that $R$ is
a domain or $R$ is level. One has
\[
J^{i} \colon {\mathfrak m}^j ={\mathfrak m}^{i-j+a+d}
\]
for every $i$ and $j \ge a+d$.
\end{corollary}
\noindent{\bf{Proof.} } The result follows from Corollaries~\ref{K} and
\ref{colonmax1}.\hfill$\square$
\bigskip
In the one-dimensional case the colon ideals $J^i : {\mathfrak m}^j$ can be
expressed in terms of conductor ideals:
\begin{remark}\label{coreandS}
{\rm In addition to the assumptions of \ref{standard} suppose that
$d=1$, write $y=y_1$, let $K$ denote the total ring of quotients of
$R$, and let $R[{\mathfrak m}/y] \subset B$ be a homogeneous inclusion where $B$ is a non-negatively graded finite
$R[{\mathfrak m}/y]$-module contained in $K$. One has $B=R[{\mathfrak m}/y]$ and
\[J^i :_R {\mathfrak m}^j= J^{i-j}(R :_K B)= {\mathfrak m}^{i-j}(R :_K B) \] for every $i\geq j$ and $j\geq a+1$.
To prove the first claim notice that $B$ is a finite $R$-module and hence $B_{\geq l}=R_{\geq l}$
for $l \gg 0$. Therefore $B_{\geq l}={\mathfrak m}^{l}$, which gives
$B \subset {\mathfrak m}^{l}/y^{l}$. As $ j \geq a+1 =r_J({\mathfrak m})$
one has ${\mathfrak m}^{l}/y^{l} = {\mathfrak m}^{j}/y^{j}$.
It follows that $B= {\mathfrak m}^{j}/y^{j} = R[{\mathfrak m}/y]$. The remaining assertions
obtain because $J^j:_R{\mathfrak m}^j=y^jR:_K{\mathfrak m}^j=R:_KB$ and this is a $B$-module (see also \cite[proof of 3.2]{PU}).}
\end{remark}
\bigskip
\section{Initial degrees of annihilators}
\smallskip
The previous section leaves open the question of how large the
integer $i$ has to be chosen in Corollaries~\ref{omegafaithful1}(c)
and ~\ref{omegafaithful2}(c). Addressing this issue will entail
establishing an upper bound for the generic generator degree of
first syzygy modules (Proposition~\ref{syzygies}), as well as
estimating the initial degree of certain colon ideals
(Propositions~\ref{primes} and ~\ref{dim1}).
We begin by recalling a result from \cite{CU}, which in turn uses
earlier work on the `fundamental class' (see \cite[p.34]{Elz},
\cite[4.11 and 5.13]{KW}, \cite[3.1]{Li}).
\begin{proposition}\label{CU}\cite[1.1]{CU}
Let $k$ be a perfect field and let $R$ be a standard graded
$k$-algebra of dimension $d$ such that $R_{{\mathfrak p}}$ is regular for every
minimal prime ${\mathfrak p}$ of dimension $d$. Write $\omega=\omega_R$ and let
$C$ be defined by the exact sequence
\[ 0 \longrightarrow [\omega]_{\leq d}R \stackrel{{\rm nat}}\longrightarrow \omega
\longrightarrow C \longrightarrow 0 \,.
\]
Then ${\rm Supp}(C) \subset {\rm Sing} (R).$
\end{proposition}
\medskip
\begin{proposition}\label{syzygies}
Let $k$ be a perfect field and $S$ a standard graded Gorenstein
$k$-algebra of dimension $D$. Let $I$ be an ideal of height $g$ such
that $I_{{\mathfrak P}}$ is prime for every prime ideal ${\mathfrak P}$ of height $g$ containing
$I$. Assume $I$ is generated by $m \geq g+1$ forms $f_1, \ldots,
f_m$ of degrees $\delta_1 \geq \ldots \geq \delta_m$, and write
$\Delta= \sum_{i=1}^{g+1} \delta_i + D-g +a(S)$. Let $H_{1}$ be the
first Koszul homology of the elements $f_1, \ldots , f_m$ and $C$
the module defined by the exact sequence
\[ 0 \longrightarrow [H_{1}]_{\leq \Delta}S \stackrel{{\rm nat}}\longrightarrow H_{1} \longrightarrow C \longrightarrow 0 \,.
\]
Then $C_{{\mathfrak P}}=0$ for every prime ${\mathfrak P}$ of $S$ such that
$I_{{\mathfrak P}}$ is a complete intersection and $S_{{\mathfrak P}}/I_{{\mathfrak P}}$ is regular.
If either $g \geq 2$ and $\delta_3 \geq 2$ or $g \geq 1$ and $S$ is
not a polynomial ring, one can replace $H_{1}$ by the first syzygy
module of the elements $f_1, \ldots, f_m$.
\end{proposition}
\noindent{\bf{Proof.} } Fix a prime ${\mathfrak P}$ so that $I_{{\mathfrak P}}$ is a complete intersection and $S_{{\mathfrak P}}/I_{{\mathfrak P}}$ is regular.
We may assume that $k$ is infinite and that $f_1, \ldots, f_g$ form
a regular sequence generating $I_{{\mathfrak P}}$. Notice
that
\[((f_1, \ldots, f_g) : I)/(f_1, \ldots, f_g) \cong \omega_{S/I}(-\sum_{i=1}^{g} \delta_i
-a(S)).
\]
Therefore Proposition~\ref{CU} shows that there exists a homogeneous element
$\beta$ of degree $D-g +\sum_{i=1}^{g} \delta_i + a(S)$ in $(f_1,
\ldots, f_g) : I$ generating the factor module $((f_1, \ldots, f_g) :
I)/(f_1, \ldots, f_g)$ locally at ${\mathfrak P}$. In particular,
$\beta \notin {\mathfrak P}$ because $((f_1, \ldots, f_g) : I)_{{\mathfrak P}}=S_{{\mathfrak P}}$.
Hence for $g+1 \leq j \leq m$ there exist homogeneous syzygies
\[ \beta f_j - \sum_{i=1}^{g} \lambda_{ij} f_i =0
\]
of $f_1, \ldots, f_m$ that have degree at most $\Delta$ and whose
images generate $H_{1}$ locally at ${\mathfrak P}$.
Finally, if either $g \geq 2$ and
$\delta_3 \geq 2$ or $g \geq 1$ and $S$ is not a polynomial ring, then the Koszul relations among $f_1,
\ldots, f_m$ have degrees at most $\Delta$. \hfill$\square$
\bigskip
We are now going to introduce the $\alpha$-invariant of a graded ring that
will be the basis for many estimates proved in this section.
\begin{definition}\label{DEF}
{\rm Let $k$ be a field and $R$ a standard graded $k$-algebra of
dimension $d$. We define
\[\alpha (R)={\rm min}\{a(S)\} \in {\mathbb{Z}} \cup \, \{\infty\}\/,\]
where $S$ ranges over all standard graded Gorenstein $k$-algebras of
dimension $d$ mapping homogeneously onto $R$ such that
$S_{{\mathfrak P}}=R_{{\mathfrak P}}$ for every minimal prime ${\mathfrak P} \in {\rm Supp}_S(R)$ of
dimension $d$. }\end{definition}
\bigskip
\bigskip
\bigskip
\begin{remark}
\hfill
\begin{itemize}
{\rm \item [$($a$)$] One has
\[a(R) \leq \alpha(R) \, ,\] because $\omega_R \hookrightarrow \omega_S$
for every $S$ as in Definition~\ref{DEF}.
\item [$($b$)$] Write $R=k[X_1, \ldots, X_n]/I$ as a factor ring of a
polynomial ring, where $I$ is an ideal of height $g$ generated by
forms of degrees $\delta_1 \geq \ldots \geq \delta_m$. If $\beta_1,
\ldots, \beta_g$ is a homogeneous regular sequence contained in $I$
that generates $I$ at each of its minimal primes of height $g$, then
\[\alpha(R) \leq \sum_{i=1}^g {\rm deg}(\beta_i) -n \, .\] In
particular, whenever $k$ is infinite and $I$ is generically a
complete intersection, we have \[\alpha(R) \leq \sum_{i=1}^g
\delta_i -n \, .\] }
\end{itemize}
\end{remark}
\bigskip
\begin{proposition}\label{primes}
Let $k$ be an infinite perfect field and $R$ a standard graded
$k$-algebra of dimension $d$. Let $H$ be a homogeneous $R$-ideal
such that $R_{{\mathfrak p}}$ is regular for every minimal prime ${\mathfrak p}$ of
dimension $d$ containing $H$. If $\alpha (R)$ is finite, there
exists a homogeneous element of degree $\alpha(R) +d$ in $0:H$ that
is not contained in any minimal prime ${\mathfrak p}$ of dimension $d$
containing $H$.
\end{proposition}
\noindent{\bf{Proof.} } We may assume that $\dim R/H =d$, and then by
${\mathfrak p}_1, \ldots, {\mathfrak p}_s$ we denote the minimal primes of dimension
$d$ containing $H$.
Let $S$ be as in Definition~\ref{DEF} so that
$\alpha(R)=a(S)$. Write $N$ and ${\mathfrak P}_i$ for the preimages of
$H$ and ${\mathfrak p}_i$ in $S$. Notice that $H=NR$, that
$N_{{\mathfrak P}_i}=0$, and that $(S/N)_{{\mathfrak P}_i}=(R/H)_{{\mathfrak p}_i}$ is regular for every ${\mathfrak P}_i$.
The beginning of the proof of Proposition~\ref{syzygies}, with $g=0$,
shows that there exists a homogenous element $\beta \in 0:_S N$ of degree
$a(S)+d$ with $\beta \notin {\mathfrak P}_i$ for each of the finitely many primes ${\mathfrak P}_i$. Thus,
denoting the image of $\beta$ in $R$ by $\gamma$ we obtain
$\gamma \in (0:_S N)R \subset 0:_R NR = 0:_R H$ and $\gamma \notin {\mathfrak p}_i$ for each
${\mathfrak p}_i$.
\hfill$\square$
\bigskip
The $\alpha$-invariant can be replaced by $a(R)$ in the above estimate
if $R$ has dimension one:
\begin{proposition}\label{dim1}
Let $k$ be an infinite field and $R$ a standard graded
Cohen-Macaulay $k$-algebra of dimension $1$. Let $H$ be a
homogeneous $R$-ideal such that $H_{{\mathfrak p}}=0$ for every minimal prime
${\mathfrak p}$ of $H$. Then there exists a homogeneous element of degree $a(R)
+ 1$ in $0:H$ that is not contained in any minimal prime ${\mathfrak p}$ of
$H$.
\end{proposition}
\noindent{\bf{Proof.} } Write ${\mathfrak m}$ for the homogeneous maximal ideal of $R$ and set
$L=0:H$. Notice that $0: H^{\rm unm} = 0:H$ since $R \/$ is
Cohen-Macaulay. Passing to the unmixed part of $H$ we can suppose that
$R/H$ is Cohen-Macaulay of dimension $1$. Furthermore $R/L$ is
either zero or Cohen-Macaulay of dimension $1$, and $R/H + L$ has
finite length. Now the short exact sequence
\[
0 \longrightarrow R/H \cap \, L \longrightarrow R/H \oplus R/L
\longrightarrow R/H + L \longrightarrow 0
\]
induces an exact sequence of local cohomology
\[
0 \longrightarrow H^0_{{\mathfrak m}}(R/H + L)= R/H + L \longrightarrow
H^1_{{\mathfrak m}}(R/H \cap \, L) \, .
\]
As $a(R/H \cap \, L)\leq a(R)$ we conclude that $R/H + L$ is
concentrated in degrees $\leq a=a(R)$. Hence ${\mathfrak m}^{a+1} \subset H +
L$. Thus any homogeneous non zerodivisor of degree $a + 1$ can
be written in the form $h + l$ where $h$, $l$ are homogeneous elements
of degree $a+1 $ in $H$, $L$ respectively. Now $l$ is an element
with the desired properties. \hfill$\square$
\bigskip
With the next two corollaries we answer the question raised at the
beginning of the section. We give a bound for the initial degrees of
annihilators of submodules generated by graded components of $\omega$
and we estimate how large the integer $i$ has to be chosen in
Corollary~\ref{omegafaithful1}(c).
\medskip
\begin{theorem}\label{alpha}
Let $k$ be an infinite perfect field, let $R$ be a standard graded
reduced equidimensional $k$-algebra of dimension $d$ with
$\omega=\omega_R$, and let $t$ be an integer. If $[\omega]_{t}R$ is not a faithful $R$-module
then
\[{\rm indeg}({\rm ann}_R([\omega]_t R)) \leq \alpha(R) + d .\]
\end{theorem}
\smallskip
\noindent{\bf{Proof.} } We may assume that $\alpha(R)$ is finite. Since $R$
is generically Gorenstein, $\omega$ is isomorphic
to a suitable shift of a homogenous $R$-ideal $W$, say $\omega \cong
W(s)$. As $[W]_{t+s} R$ is not faithful it is contained in some
minimal prime ${\mathfrak p}$ of $R$. Now
\[
{\rm ann}_R([\omega]_{t}R)={\rm ann}_R([W]_{t+s} R) \supset 0:_R {\mathfrak p},
\]
and the latter ideal contains a non-zero homogenous element of
degree $\alpha(R) +d$ according to Proposition~\ref{primes}. \hfill$\square$
\bigskip
Replacing Proposition~\ref{primes} by Proposition~\ref{dim1} in the proof of Theorem~\ref{alpha}
yields a better estimate for the initial degree in the one-dimensional case that
is still weaker though than the one implied by
Corollary~\ref{ann of omega dim1}.
\medskip
\begin{corollary}\label{alpha2} Let $k$ be an infinite perfect field, let $R$
be a standard graded reduced Cohen-Macaulay $k$-algebra of dimension
$d \geq 1$ with homogeneous maximal ideal ${\mathfrak m}$, and let $t$ be an integer. Write $\omega=\omega_R$
and $a=a(R)$. Then $[\omega]_{t}R$ is a faithful
$R$-module if and only if $J^{i+a+t} : {\mathfrak m}^{a+d} \subset {\mathfrak m}^{i}$ for
some $ i\geq \alpha(R) + d + 1$.
\end{corollary}
\noindent{\bf{Proof.} } The forward direction follows from Corollary \ref{omegafaithful1}, and the converse
from Theorems~\ref{ann of omega} and \ref{alpha}. \hfill$\square$
\bigskip
Under suitable additional assumptions the bound for $i$ in
Corollary~\ref{alpha2} can be improved, most notably by replacing
the invariant $\alpha(R)$ by the more traditional $a(R)$. This is the content of the
next three results of the section.
\medskip
\begin{proposition}\label{CM}
Let $k$ be a field, let $R$ be a standard graded Cohen-Macaulay
$k$-algebra of dimension $d \geq 1$ with homogeneous maximal ideal ${\mathfrak m}$, and
let $t$ be an integer.
Write $\omega=\omega_R$, $a=a(R)$, and
$\, -^{\vee}={\rm Hom}_R(-, \omega)$. If $\omega/(([\omega]_{\leq t}R)^{\vee \vee})$
is Cohen-Macaulay then ${\rm ann}_R([\omega]_tR)$
is a Cohen-Macaulay module with Castelnuovo-Mumford regularity $\leq d-t -1$.
In particular, whenever $t \geq -a$ one has that
$[\omega]_{t}R$ is a faithful $R$-module if and only if $J^{i+a+t} :
{\mathfrak m}^{a+d} \subset {\mathfrak m}^{i}$ for some $ i\geq a+d$.
\end{proposition}
\noindent{\bf{Proof.} }
In light of Corollary \ref{omegafaithful1}, the second
statement follows from the first claim and
Theorem~\ref{ann of omega}. Indeed, the bound on the Castelnuovo-Mumford regularity
shows that ${\rm ann}_R([\omega]_tR)$ is generated in degrees $\leq d-t-1 \leq a+d-1$. Hence
this annihilator vanishes by Theorem~\ref{ann of omega} if $J^{i+a+t} :
{\mathfrak m}^{a+d} \subset {\mathfrak m}^{i}$ for some $i \geq a+d$.
Thus it suffices to prove the assertion about ${\rm
ann}_R([\omega]_tR)$. We may assume that this annihilator is not
zero. One has
\[{\rm ann}_R([\omega]_{t}R) = {\rm ann}_R([\omega]_{\leq t}R)={\rm ann}_R(([\omega]_{\leq t}R)^{\vee \vee}) \, . \]
The first equality is part of Theorem~\ref{ann of omega}. To see the
second equality notice that since $\omega$ is a maximal
Cohen-Macaulay $R$-module, the annihilator of every submodule is
either the unit ideal or an unmixed ideal of height zero. In
particular, two submodules of $\omega$ have the same annihilator if
they coincide locally at every minimal prime of $R$. Since
$[\omega]_{\leq t}R$ and $([\omega]_{\leq t}R)^{\vee \vee}$ are
generically equal, the second equality now follows. Thus we may
restrict our attention to the annihilator ideal ${\rm
ann}_R(([\omega]_{\leq t}R)^{\vee \vee})$. Likewise, since ${\rm
ann}_R(([\omega]_tR)^{\vee \vee}) \neq 0 ={\rm ann}_R(\omega)$ it follows that
$([\omega]_{\leq t}R)^{\vee \vee}$ and $\omega$ cannot coincide
locally at every minimal prime of $R$. Thus $\omega/(([\omega]_{\leq
t}R)^{\vee \vee})$ has dimension $d$, hence is a maximal
Cohen-Macaulay $R$-module.
Write $R=S/I$ with $S=k[X_1, \ldots,X_n]$ a polynomial ring and $I$ a homogeneous $S$-ideal of height
$g$. Choose a regular sequence $\ul{\beta}=\beta_1, \ldots, \beta_g$ of
forms of degree $\delta \gg 0$ contained in $I$ and set $L= (\ul{\beta}) :_S I$.
Notice that $S/L$ is a $d$-dimensional Cohen-Macaulay ring. Moreover one has
$(L/(\ul{\beta}))(g \delta - n) \cong \omega$. Thus there exists a homogeneous ideal
$H$ of $S$ so that $(\ul{\beta}) \subset H \subset L$ and
$(H/(\ul{\beta}))(g \delta -n) \cong ([\omega]_{\leq t}R)^{\vee \vee}$. Clearly
\[{\rm ann}_R(([\omega]_{\leq t}R)^{\vee \vee}) = ((\ul{\beta}):_S H)/I \, . \]
Since $L/H$ is isomorphic to a shift of the module
$\omega/(([\omega]_{\leq t}R)^{\vee \vee})$ it follows that $L/H$
is a maximal Cohen-Macaulay $R$-module. Therefore $S/H$ is a
$d$-dimensional Cohen-Macaulay ring, and hence so is
$S/((\ul{\beta}):_SH)$. Thus $((\ul{\beta}):_S H)/I$ is a maximal
Cohen-Macaulay $R$-module, proving the assertion in the proposition
about the Cohen-Macaulayness of the annihilator ideal.
Applying ${\rm Ext}^g_S(-,S(-n))$ to the exact sequence
\[ 0 \longrightarrow ((\ul{\beta}):_SH)/I \longrightarrow S/I
\longrightarrow S/((\ul{\beta}):_S H) \longrightarrow 0
\]
of maximal Cohen-Macaulay $R$-modules we obtain this exact sequence,
\[ 0 \longrightarrow \omega_{S/((\ul{\beta}): H)} \longrightarrow \omega_{S/I}=\omega
\longrightarrow {\rm{Ext}}^g_S(((\ul{\beta}):_SH)/I, S(-n)) \longrightarrow 0 \, .
\]
The canonical module on the left satisfies
\[\omega_{S/((\ul{\beta}): H)} \cong (H/(\ul{\beta}))(g\delta -n)
\cong ([\omega]_{\leq t} R)^{\vee \vee} \supset [\omega]_{\leq t}R \, , \]
where the first isomorphism holds because $H$ is unmixed.
Therefore the above exact sequence shows that
${\rm{Ext}}^g_S(((\ul{\beta}):_SH)/I, S(-n))$ is concentrated in
degrees $\geq t+1$, and hence by local duality the local cohomology module
$H^d_{{\mathfrak m}}(((\ul{\beta}):_SH)/I)$ is concentrated in degrees $\leq -t-1$. As
$((\ul{\beta}):_SH)/I$ is a $d$-dimensional Cohen-Macaulay module we deduce that
it has regularity $\leq d-t-1$. \hfill$\square$
\bigskip
We follow suit with an estimate for rings of type $2$ that
does not require the Cohen-Macaulayness of $\omega/(([\omega]_{\leq t}R)^{\vee \vee})$.
\medskip
\begin{proposition}\label{type2}
Let $k$ be a field and $R$ a standard graded geometrically reduced
Cohen-Macaulay $k$-algebra of type $2$ and dimension $d \geq 1$ with
homogeneous maximal ideal ${\mathfrak m}$. Write $R=S/I$ with $S=k[X_1,
\ldots, X_n]$ a polynomial ring and $I$ a homogeneous $S$-ideal of
height $g$. Consider the last map in a minimal homogeneous free
$S$-resolution of $R$
\[
0 \longrightarrow S(-l_1) \oplus S(-l_2)
\stackrel{\varphi}\longrightarrow \oplus_{i=1}^{m} S(-k_i) \, ,
\]
where $l_1 \leq l_2$ and $k_1 \leq \ldots \leq k_m$. Write
$\omega=\omega_R$ and $a=a(R)$. If $[\omega]_{-a}R$ is not a faithful
$R$-module then
\[{\rm indeg}({\rm ann}_R([\omega]_{-a}R)) \leq gl_1 +l_2 -
\sum_{i=1}^{g+1} k_i -g =a+d +gl_1 - \sum_{i=1}^{g+1} k_i\, .\]
Equivalently, $[\omega]_{-a}R$ is a faithful $R$-module if and only
if $J^{i} : {\mathfrak m}^{a+d} = {\mathfrak m}^{i}$ for some $ i\geq a+d +gl_1 -
\sum_{i=1}^{g+1} k_i+1$.
\end{proposition}
\noindent{\bf{Proof.} } By Theorem~\ref{ann of omega} and Corollary \ref{omegafaithful2} it suffices to prove the first
statement. We may assume that the field $k$ is infinite and perfect. Since $R$
is Cohen-Macaulay we obtain the following presentation,
\[
\oplus_{i=1}^{m} S(k_i) \stackrel{\varphi^*}\longrightarrow S(l_1)
\oplus S(l_2)\longrightarrow \omega(n) \longrightarrow 0 \,.
\]
Using the standard bases of the free modules this presentation gives
rise to homogeneous generators $w_1, w_2$ of $\omega(n)$ and a
matrix
\[
\left(
\begin{array}{ccc}
f_1 & \cdots & f_m \\
h_1 & \cdots & h_m \\
\end{array}
\right)
\]
representing $\varphi^*$. Notice that ${\rm deg}(w_1)=-l_1 \geq
{\rm deg}(w_2)=-l_2$, ${\rm deg}(f_i)=l_1-k_i$, and ${\rm deg}
(h_i)=l_2-k_i$. One has ${\rm deg} (w_1) > {\rm deg}(w_2)$ since
otherwise $[\omega]_{-a}R= \omega$ would be faithful. Therefore \[{\rm
ann}_R ([\omega]_{-a} R) = {\rm ann}_R \, w_2 \,.\] Further observe that
\[w_2R :_S w_1= (f_1, \ldots, f_m)\]
and
\begin{equation}\label{omegasyz}
{\rm ann}_S \, w_2 = \displaystyle{\{} \sum \lambda_ih_i \ | \ \sum
\lambda_i f_i =0 \}\, ;
\end{equation}
the last equality obtains because an element $\varepsilon$ of $S$
belongs to ${\rm ann}_S \, w_2 $ if and only if the vector $\left(
\begin{array}{c}
0 \\
\varepsilon \\
\end{array}
\right)$
is in the column space of the above matrix.
As ${\rm ann}_R \, w_2 \not=0$ and $R$ is
unmixed, there exists a minimal prime ${\mathfrak p}$ of $R$ such that $({\rm
ann}_R \, w_2)_{{\mathfrak p}} \not=0$. In particular, $w_2R_{{\mathfrak p}} \not= w_1
R_{{\mathfrak p}} + w_2 R_{{\mathfrak p}}$ because the latter module is faithful. Thus the ideal
$(f_1, \ldots, f_m) =w_2R :_S w_1$ is contained in ${\mathfrak P}$, the
preimage of ${\mathfrak p}$ in $S$. On the other hand, this ideal contains $I$. It follows
that $(f_1, \ldots, f_m)$ has height $g$. Furthermore,
since $I$ is radical the localization
$(f_1, \ldots , f_m) S_{{\mathfrak Q}}$ is a complete intersection prime ideal
for every prime ${\mathfrak Q}$ of height $g$ in $S$ containing $(f_1, \ldots, f_m)$.
Finally, $g={\rm{ht}} \,I={\rm{ht}} \,I_2(\varphi)
\leq m -1$ by the Eagon-Northcott bound on the height of
determinantal ideals. Now Proposition~\ref{syzygies} shows that
locally at ${\mathfrak P}$, the syzygy module of $(f_1, \ldots, f_m)$ is
generated by its elements of degrees $\leq \sum_{i=1}^{g+1} (l_1-
k_i) + n - g + a(S)$ and by the Koszul relations. Therefore \ref{omegasyz}
gives
\[
({\rm ann}_S \, w_2)_{{\mathfrak P}}=([{\rm ann}_S \, w_2]_{\leq
\sum_{i=1}^{g+1} (l_1- k_i) + n - g + a(S) + l_2 -l_1} +
I_2(\varphi^*))_{{\mathfrak P}} \, .
\]
As $I_2(\varphi^*) \subset {\rm ann}_S \, \omega =I$ we conclude
that
\begin{eqnarray*}
({\rm ann}_R \, w_2)_{{\mathfrak p}}&=&([{\rm ann}_R \, w_2]_{\leq
\sum_{i=1}^{g+1} (l_1- k_i) + n - g + a(S) + l_2 -l_1})_{{\mathfrak p}}\\
&=&([{\rm
ann}_R \, w_2]_{\leq g l_1 +l_2 - \sum_{i=1}^{g+1} k_i -g})_{{\mathfrak p}} \, .
\end{eqnarray*}
The assertion now follows since $({\rm ann}_R \,
w_2)_{{\mathfrak p}}\not=0 \, $.\hfill$\square$
\medskip
\begin{corollary}\label{codim2}
Let $k$ be a field and let $R$ be a standard graded equidimensional
geometrically reduced $k$-algebra of dimension $d \geq 1$ with homogeneous
maximal ideal ${\mathfrak m}$. Assume that $R$ is an almost complete intersection
of embedding codimension $2$. Write $\omega=\omega_R$ and $a=a(R)$. If
$[\omega]_{-a}R$ is not a faithful
$R$-module then
\[{\rm indeg}({\rm ann}_R([\omega]_{-a}R)) \leq a + d \, .\]
Equivalently, $[\omega]_{-a}R$ is a faithful $R$-module if and only
if $J^{i} : {\mathfrak m}^{a+d} = {\mathfrak m}^{i}$ for some $ i\geq a+d+1$.
\end{corollary}
\noindent{\bf{Proof.} } Notice that $R$ is Cohen-Macaulay by the Syzygy Theorem and has
type $2$, see \cite[2.1]{EG}. Hence we may apply
Proposition~\ref{type2} with $g=2$. It suffices to show that $2l_1 -
\sum_{i=1}^{3} k_i \leq 0$. This holds, because $k_3=l_1+l_2
-k_1-k_2$ by the Hilbert-Burch Theorem and $l_1 \leq l_2 \, $. \hfill$\square$
\bigskip
We finish this section with a different estimate for initial degrees of
annihilators -- an estimate from below. In the proof we use the notation
$H^{\rm unm}$ for the {\it unmixed part} of an ideal $H$, which is the intersection
of the primary components of maximal dimension.
\medskip
\begin{proposition}
Let $k$ be a field, let $R$ be a standard graded
Cohen-Macaulay $k$-algebra of dimension $d$, and let $H $ be
a homogeneous $R$-ideal. Write $c=c(R)$ as in $\ref{standard}$.
One has ${\rm indeg} (0:H) \geq c +d +1 -
e(R/H) $.
\end{proposition}
\noindent{\bf{Proof.} } We may assume that $k$ is infinite and that $0:H \neq 0$.
Set $e= e(R/H)$. We prove the claim by induction on $d$.
First let $d=0$. In this case $e = \lambda (R/H)$. Therefore
${\mathfrak m}^e \subset H$, which gives $0: H \subset 0 :{\mathfrak m}^e$. But
${\rm indeg} (0:{\mathfrak m}^e) \geq c +1 - e$ because
$c$ is the initial degree of the
socle of $R$.
Next let $d \geq 1$. Notice that $0: H^{\rm unm} = 0:H$ since $R \/$ is
Cohen-Macaulay and that $e(R/H^{\rm unm})=e(R/H)$ by the associativity
formula for the multiplicity. Hence we may replace $H$ by $H^{\rm
unm}$ to assume that $H$ is unmixed. Since $0:H
\not= 0$ we have ${\rm{ht}} \, H =0$. Furthermore, $0:H$ is unmixed
with ${\rm{ht}} (0:H) =0$ or else $0:H = R$. Now let $x \in R$ be a
linear form that is regular on $R$, and write $^{^{\mbox{\rule{2mm}{.2mm}$\;\!$}}}$ for images in
the $d-1$ dimensional Cohen-Macaulay ring $\ol{R}=R/(x)$. Notice
that $x$ is regular modulo $H$ and is regular modulo $0:H$ unless
$0:H=R$. It follows that $e(\ol{R}/\ol{H})=e(R/H)$ and ${\rm
indeg}(\ol{0:_R H}) ={\rm indeg}(0 :_R H)$. Furthermore,
$c(\ol{R})=c(R) +1$ as $\omega_{\ol{R}} \cong \omega_R (1)$. Thus we
conclude
\begin{eqnarray*}
{\rm indeg} (0 :_R H) &=& {\rm indeg}( \ol{0:_R H}) \\
&\geq& {\rm indeg}(\ol{0}:_{\ol{R}} \ol{H}) \hspace{3.765cm} \hbox{\rm since \ $ \ol{0:_R H} \subset \ol{0}:_{\ol{R}} \ol{H} $}\\
&\geq& c(\ol{R}) + {\rm dim} \, \ol{R} +1 - e(\ol{R}/\ol{H}) \hspace{1.6cm} \hbox{\rm by induction hypothesis}\\
&=& (c +1) + (d-1) +1 -e\\
&=& c+d+1 -e \,.
\end{eqnarray*} \hfill$\square$
\bigskip
If $R$ is zero-dimensional in the above proposition, then the lower bound $c+1-e(R/H)$ can be improved to $c-a(R/H)$.
This shaper estimate immediately gives the inclusion $J^i:{\mathfrak m}^j \subset {\mathfrak m}^{i-j+c+d}$ in the
setting of Corollary \ref{colonmax1}.
\bigskip
\section{The core of standard graded algebras}
\smallskip
In this section we apply the previous results to the core of powers of
the maximal ideal.
\begin{Assumptions}\label{asscore}
{\rm In addition to the assumptions of \ref{standard} suppose that
if ${\rm char} \, k > 0$ then $k$ is infinite and $R$ is
geometrically reduced.}
\end{Assumptions}
\medskip
The next result is essentially present, in a more general form, in
\cite[4.2]{FPU}.
\begin{theorem}\label{formula} With assumptions as in $\ref{asscore}$ one has for every $n \geq 1$,
\[
\core{{\mathfrak m}^n} = J^{nd+a+1} : {\mathfrak m}^{a+d}.
\]
\end{theorem}
\noindent{\bf{Proof.} } From \cite[2.3 and 2.5]{PUV} we know that \[\core{{\mathfrak m}^n}
=(J^{[n]})^{j+1}:({\mathfrak m}^n)^j \ \ \ \hbox{\rm for} \ \ j \gg 0 \, .\]
Furthermore
\begin{eqnarray*}
\nonumber
(J^{[n]})^{j+1}:({\mathfrak m}^n)^j&=& (J^{[n]})^{[j+1]}:({\mathfrak m}^n)^{jd} \hspace{1.6cm} \hbox{\rm by (\ref{PUV 2.2})} \\
&=& J^{[nj+n]}:{\mathfrak m}^{njd} \\
&=& J^{nj+n}: {\mathfrak m}^{nj+n-nd+d-1} \hspace{0.951cm} \hbox{\rm by (\ref{PUV 2.2})} \\
&=& J^{nd+a+1} : {\mathfrak m}^{a+d} \hspace{2.055cm} \hbox{\rm by
Remark~\ref{colon1}.}
\end{eqnarray*}\hfill$\square$
\bigskip
The above theorem relates the core of powers of the maximal ideal to
the colon ideals studied in the previous sections. We leave it to
the reader to express most of the earlier results in terms of cores.
Here we only collect the main applications:
\medskip
\begin{corollary}\label{coreandK}
With assumptions as in $\ref{asscore}$ one has for every $n \geq 1$ $:$
\begin{itemize}
\item [(a)] $\core{{\mathfrak m}^n}={\mathfrak m}^{nd+a+1}+N$ for some ideal $N$ of height zero
\item[(b)] ${\mathfrak m}^{nd+a+1} \subset \core{{\mathfrak m}^n} \subset {\mathfrak m}^{nd+b+1}$.
\end{itemize}
\end{corollary}
\noindent{\bf{Proof.} } The result follows immediately from Theorem~\ref{formula} and
Corollaries~\ref{K}
and \ref{colonmax1}. \hfill$\square$
\bigskip
Item (b) of the next theorem was asserted in \cite[4.1]{HySm2},
assuming only that $R$ is reduced. However, as the theorem shows, this
statement is equivalent to the faithfulness of the module
$[\omega]_{-a}R$. Furthermore, $R$ being {\it geometrically} reduced
is essential according to \cite[5.1]{FPU}.
\medskip
\begin{theorem}\label{core-ann} With assumptions as in $\ref{asscore}$ the following
are equivalent $:$
\begin{itemize}
\item [(a)] $[\omega]_{-a}R$ is faithful
\item [(b)] $\core{{\mathfrak m}^n}={\mathfrak m}^{nd+a+1}$ for every $n\geq 1$
\item [(c)] $\core{{\mathfrak m}^n}$ is generated in one degree for some $n
\gg 0 \, $.
\end{itemize}
\end{theorem}
\noindent{\bf{Proof.} } Theorem~\ref{formula} and Corollary~\ref{omegafaithful2} show
that (a) implies (b). If (c) holds then according to
Theorem~\ref{formula} and Corollary~\ref{power} one has $ J^{nd+a+1}
: {\mathfrak m}^{a+d}={\mathfrak m}^{nd+a+1}$. Notice that $nd+a+1 \gg 0$. Thus again
Corollary~\ref{omegafaithful2} gives that (a) obtains. \hfill$\square$
\medskip
\begin{corollary}\label{indeg=a+d}
In addition to the assumptions of $\, \ref{asscore}$ suppose that one of
the following conditions holds $:$
\begin{itemize}
\item $R$ is reduced and $ [\omega]_{\geq d + 1}=\core {\mathfrak m} \, \omega$
\item $d=1$
\item $R$ is a reduced almost complete intersection of embedding
codimension $2$.
\end{itemize}
Then the following are equivalent $:$
\begin{itemize}
\item [(a)] $[\omega]_{-a}R$ is faithful
\item [(b)] $\core{{\mathfrak m}}={\mathfrak m}^{a+d+1}$
\item [(c)] $\core{{\mathfrak m}}$ is generated in one degree.
\end{itemize}
\end{corollary}
\noindent{\bf{Proof.} } We apply Theorem~\ref{formula} and Corollary~\ref{power}. In the first case we also use
Corollary~\ref{omegafaithful2} via Theorem~\ref{omega}(a), in the second case
Corollary~ \ref{omegafaithfuldim1}, and in the third case Corollary~\ref{codim2}. \hfill$\square$
\bigskip
\section{The core of points}
\smallskip
\begin{Assumptions and Discussion}\label{core of points}
{\rm Let $k$ be an infinite field and let $X=\{P_{1}, \ldots,
P_{s}\}$ be a set of $s$ reduced points in $\mathbb{P}_{k}^{n}$.
Write $S=k[x_0, \ldots, x_n]$ for the polynomial ring and $R=S/I_X$
for the homogeneous coordinate ring of $X \subset
\mathbb{P}_{k}^{n}$. Let ${\mathfrak m}$ denote the homogeneous maximal ideal
of $R$, $K$ its total ring of quotients, $B$ the integral closure of
$R$ in $K$, and $\mathcal C= R :_K B$ the conductor. Furthermore, we write
$\omega=\omega_R$, $a=a(R)$, $b=b(R)$ and we define
$\core{X}=\core{{\mathfrak m}}$. Finally, let $y \in R$ be a linear form that
is $R$-regular and set $J=yR$. Notice that $a \leq s-2$ and that $R$
is geometrically reduced.
Homogeneous polynomials $f_{1}, \ldots, f_{s}$ in $S$ are called
{\it separators} of $X$ if $f_{i}(P_{j})=\delta_{ij}$ for every $i$,
$j$. They are called {\it minimal separators} if in addition each
$f_i$ has smallest possible degree.}
\end{Assumptions and Discussion}
\medskip
With the next lemma we recall a known fact describing the
conductor in terms of minimal separators (see for instance
\cite[3.13]{GKR}). We include a proof for the convenience of the
reader.
\medskip
\begin{lemma}\label{conductor}
With assumptions as in $\ref{core of points}$ let $h_{1}, \ldots,
h_{s}$ be a collection of separators of $\/X$ and $f_{1}, \ldots,
f_{s}$ a collection of minimal separators. One has $h_i R \subset
f_i R \/ $ and $f_{1}, \ldots, f_s$ minimally generate $\mathcal C$ as an
$R$-ideal; in particular, ${\mathfrak m}^{a+1} \subset (f_1, \ldots, f_s)R =
\mathcal C$ and ${\rm deg} (f_i) \leq a+1$.
\end{lemma}
\noindent{\bf{Proof.} }
Fix $\lambda_{ij} \in k$ with $P_{i}=(\lambda_{i_0} : \ldots : \lambda_{i_n})$.
We use these identifications of non-negatively graded rings,
\[\varphi: R \hookrightarrow B \cong k[t] \times \ldots \times k[t] = \oplus_{i=1}^{s} k[t]e_i \, ,\]
where $k[t]$ is a standard graded polynomial ring, $e_i$ are
standard basis elements of degree zero, and $\varphi$ maps the image
in $R$ of a polynomial $h\in [S]_l$ to the tuple
$\sum h(\lambda_{i_0}, \ldots ,\lambda_{i_n}) t^{l} e_i$. Thus the
elements $t^{l_1}e_1, \ldots, t^{l_s}e_s$ belong to $R$ if and only
if there exist separators $h_1, \ldots , h_s$ of degrees $l_1, \ldots,
l_s$, in which case $t^{l_i}e_iR=h_iR$. From this we see that $h_i R
\subset f_i R \/ $. It also follows that the minimal separators
$f_{1}, \ldots, f_{s}$ minimally generate the largest $R$-ideal of
the form $\sum t^{l_i} e_i R= \sum t^{l_i} k[t] e_i$, equivalently,
the largest homogeneous $B$-ideal contained in $R$. However, this
ideal is the conductor $\mathcal C$.
Finally, the long exact sequence of local cohomology shows that
$B/R$ is concentrated in degrees $\leq a$, hence ${\mathfrak m}^{a+1} \subset
\mathcal C$. \hfill$\square$
\bigskip
The next proposition gives a geometric interpretation of the core of
points in terms of separators:
\medskip
\begin{proposition}\label{separators}With assumptions as in $\ref{core of points}$
let $f_{1}, \ldots, f_{s}$ be minimal separators of $X$.
One has
\begin{itemize}
\item [(a)] $\core{X}=y\mathcal C={\mathfrak m}\mathcal C =y R (f_{1}, \ldots, f_{s})={\mathfrak m}(f_{1}, \ldots, f_{s})$
\item [(b)] $ [\omega]_{\geq 2}=\core X \, \omega$.
\end{itemize}
\end{proposition}
\noindent{\bf{Proof.} } According to Theorem~\ref{formula} one has $\core X= J^{a+2}
: {\mathfrak m}^{a+1}$. Now the first and second equality in (a) follow from
Remark~\ref{coreandS}, and (b) is a consequence of
Proposition~\ref{=indim1}. Finally, Lemma~\ref{conductor} implies
$\mathcal C=(f_{1}, \ldots, f_{s})R$. \hfill$\square$
\bigskip
\begin{corollary}\label{Y and Z}
In addition to the assumptions of $\, \ref{core of
points}$ suppose that $X=Y \cup Z$, where $Y$ is contained in a
hypersurface $f=0$ and $Z$ is a collection of $e$ reduced points whose
homogeneous coordinate ring has $a$-invariant $a'$. One has \[
{\mathfrak m}^{a+2} + f {\mathfrak m}^e \subset {\mathfrak m}^{a+2} + f {\mathfrak m}^{a' + 2} \subset \core X
\subset {\mathfrak m}^{b+2} \, .\]
\end{corollary}
\smallskip
\noindent{\bf{Proof.} } Let $h_1, \ldots, h_e$ be minimal separators of $Z$, and
write $H$ for the defining ideal of $Z$ in $X$. From
Lemma~\ref{conductor} we know that ${\mathfrak m}^{a'+1} \subset (h_1, \ldots,
h_e) R + H$. Since $fH=0$ in $R$, multiplying this equation by $f$ we
obtain $f{\mathfrak m}^{a'+1} \subset (fh_1, \ldots , fh_e)R$. However, those elements of $fh_1,
\ldots , fh_e$ that are not contained in $I_{X}$ form part of a collection of separators of $X$.
Hence $(fh_1, \ldots , fh_e)R \subset \mathcal C$ by the same Lemma~\ref{conductor}, and therefore
$f {\mathfrak m}^{a'+2} \subset \core X$ according to
Proposition~\ref{separators}(a). The remaining assertions follow from
Discussion~\ref{core of points} and Corollary~\ref{coreandK}(b). \hfill$\square$
\bigskip
The previous result suggests that the shape of the core is related to
uniformity properties of the set of points. We recall one such
condition: The scheme $X \subset \mathbb{P}_{k}^{n}$ is said to have
the {\it Cayley-Bacharach} property if each subscheme of the form $X
\backslash \{P_i\} \subset \mathbb{P}_{k}^{n}$ has the same Hilbert
function.
\medskip
\begin{corollary}\label{CB} With assumptions as in $\ref{core of points}$
the scheme $X$ has the Cayley-Bacharach property if and only if
$\core X ={\mathfrak m}^{a+2}.$
\end{corollary}
\noindent{\bf{Proof.} } It is easy to see that $X$ has the Cayley-Bacharach property
if and only if the minimal separators all have the same degree.
According to Proposition~\ref{separators}(a) this means that
$\core X$ is generated in one
degree, which in turn is equivalent to $\core X ={\mathfrak m}^{a+2}$ as shown
in Corollary~\ref{indeg=a+d}.
Alternatively, in \cite[3.5]{GKR} the Cayley-Bacharach property has
been characterized in terms of the faithfulness of $[\omega]_{-a}R$.
Again according to Corollary~\ref{indeg=a+d} the latter condition
holds if and only if $\core X ={\mathfrak m}^{a+2}$. \hfill$\square$
\bigskip
The next example illustrates the previous two corollaries.
\medskip
\begin{example}
{\rm In addition to the assumptions of \ref{core of points} suppose
${\rm char} \, k \not=2$ and take $X$ to be the $4$ points
$(0:-1:1), (0:0:1), (0:1:1), (1:0:1)$ in $\mathbb{P}^2_k$. These
points and their separators are depicted in this figure:
\setlength{\unitlength}{1cm}
\begin{picture}(6,4)
\put(2,2){\vector (1,0){3.6}}
\put(3,.2){\vector (0,1){3.6}}
\put(5.6,1.8){$x_0$}
\put(3.1,3.6){$x_1$}
\put(2.92,1.89){$\bullet$} \put(2.76,1.69){$0$}
\put(2.92,2.89){$\bullet$}
\put(2.92,0.89){$\bullet$}
\put(3.92,1.89){$\bullet$}
\linethickness{.04cm}
\put(2,2){\line (1,0){3.6}}
\put(3,.2){\line(0,1){3.6}}
\put(2,3){\line (1,0){3.3}}
\put(2,1){\line (1,0){3.3}}
\put(2.2,0.2){\line (1,1){3.2}}
\end{picture}
\noindent Notice that
\[
R=k[x_0,x_1,x_2]/(x_0 x_1,x_0(x_0-x_2),x_1(x_1-x_2)(x_1+x_2)),
\]
and one easily sees that $a=1$. We choose $y$ to be the image of
$x_2$, and as minimal separators of $X$ we take $x_1(x_1-x_2),
(x_0-x_1-x_2)(x_1-x_2), x_1(x_1+x_2), x_0$. From the geometric
interpretation of the core in terms of separators,
Proposition~\ref{separators}(a), one immediately sees that $\core X=
{\mathfrak m}^3 + x_0^2R \supsetneq {\mathfrak m}^3={\mathfrak m}^{a+2}$. The inclusion $x_0^2R
\subset \core X$ would have also been predicted by Corollary~\ref{Y
and Z} with $f=x_0$, and the strict containment ${\mathfrak m}^{3} \subsetneq
\core X$ reflects the obvious fact that $X$ does not have the
Cayley-Bacharach property. }
\end{example}
\medskip
\begin{corollary} Let $k$ be a field of characteristic zero, let
$Y\subset \mathbb{P}^{n+1}_k$ be a reduced and irreducible
arithmetically Cohen-Macaulay curve, and write ${\mathfrak n}$ for the
homogeneous maximal ideal of the homogeneous coordinate ring of $Y$.
Consider a general hyperplane section $X\subset \mathbb{P}^{n}_k$ of
$Y$ and use the notation of $\, \ref{core of points}$. One has $\core Y=
\core {\mathfrak n}= {\mathfrak n}^{a + 2}$ and $\core X= {\mathfrak m}^{a +2}$.
\end{corollary}
\noindent{\bf{Proof.} } According to \cite[3.4]{H} the set of points $X$ has the
Cayley-Bacharach property. Now the two equalities follow
from Corollaries~\ref{coreandK} and \ref{CB}.\hfill$\square$
\bigskip
\section{Local estimates on cores}
\smallskip
We finish this paper with a generalization of Corollary~\ref{Y and
Z} to the context of zero-dimensional ideals in local rings.
\medskip
\begin{proposition}\label{local}
Let $(R, {\mathfrak m})$ be a local Cohen-Macaulay ring with infinite residue
field, $I$ an ${\mathfrak m}$-primary $R$-ideal, $L$ and $H$ two $R$-ideals
such that $LH=0$ in $R$. Write $e=e(I; R/H)$ for the multiplicity of
the ring $R/H$ with respect to the ideal $I$. Then $L I^{e} \subset
\core I$.
\end{proposition}
\noindent{\bf{Proof.} } We prove the claim by induction on $d ={\rm dim}\, R$. If
$d=0$ then $e$ is the length of $R/H$, and we easily obtain $I^e
\subset H$. Therefore $L I^e \subset L H =0$. Now consider the case
$d \geq 1$. For $J$ an arbitrary reduction of $I$ we need to show that $L
I^{e} \subset J$. Let $x$ be a general element of $J$ and
write $^{^{\mbox{\rule{2mm}{.2mm}$\;\!$}}}$ for images in the $d-1$
dimensional Cohen-Macaulay ring $\ol{R}=R/(x)$. Notice that $L
H^{\rm unm}=0$ since $R$ is Cohen-Macaulay and that $e(I; R/H^{\rm
unm})=e$ by the associativity formula for the Hilbert-Samuel
multiplicity. Thus we may replace $H$ by $H^{\rm unm}$ to assume
that $H$ is unmixed. We may further suppose that $L \not= 0$.
Therefore ${\rm dim}\, R/H= {\rm dim}\, R=d$ since $R$ is
Cohen-Macaulay. We conclude that the general element $x$ of $J$ is
regular on $R/H$. Thus $e(\ol{J}; \ol{R}/\ol{H})=e(J; R/H)$. As
$\ol{J}$ and $J$ are reductions of $\ol{I}$ and $I$, respectively,
we have $e(\ol{J}; \ol{R}/\ol{H})=e(\ol{I}; \ol{R}/\ol{H})$ and
$e(J; R/H)=e(I; R/H)$. It follows that $e(\ol{I}; \ol{R}/\ol{H})=e(I; R/H)=e$.
Now our induction hypothesis gives $\ol{L I^e} \subset \core{\ol{I}}
\subset \ol{J}$. Hence indeed $L I^{e}
\subset J$. \hfill$\square$
\bigskip
We obtain the estimate $f{\mathfrak m}^e \subset \core X$ of
Corollary~\ref{Y and Z} from the above proposition if we take $L=fR$
and $H= I_{Z}R$.
\bigskip
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,662 |
ScienceBeta
Does Parkinson's Act Like An Autoimmune Disease?
What causes the death of neurons in Parkinson's disease is still unknown. A recent study from Columbia University researchers suggests that neurons may be mistaken for alien intruders and attacked by the person's own immune system. This would be similar to the way autoimmune diseases such as celiac disease, type I diabetes, and multiple sclerosis attack the body's cells.
The new proposition about Parkinson's surfaces from other findings in the study that goes against a long held assumption about neurons and the immune system.
"This is a new, and likely controversial, idea in Parkinson's disease; but if true, it could lead to new ways to prevent neuronal death in Parkinson's that resemble treatments for autoimmune diseases," said the study's senior author, David Sulzer, PhD.
Neuronal Antigens
Neurobiologists have believed for decades that neurons are protected from attacks from the immune system. One of the reasons is that they do not show antigens on their cell surfaces.
Most cell types, when infected by virus or bacteria, will display bits of the microbe, known as antigens, on their outer shell. When the immune system recognizes the foreign antigens, T cells attack and kill the cells. Because scientists thought that neurons did not display antigens, they also thought that the neurons were free from T-cell attacks.
"That idea made sense because, except in rare circumstances, our brains cannot make new neurons to replenish ones killed by the immune system," Dr. Sulzer said. "But, unexpectedly, we found that some types of neurons can display antigens."
Major Histocompatibility Complexes
Special proteins called Major Histocompatibility Complexes (MHCs) are involved in cells display of antigens. Using postmortem brain tissue donated to the Columbia Brain Bank by healthy donors, Dr. Sulzer noticed that MHC-1 proteins were present in two types of neurons. These two types of neurons, one of which is dopamine neurons in a brain region called the substantia nigra, degenerate during Parkinson's disease.
To see if living neurons use MHC-1 to display antigens (and not for some other purpose), Drs. Sulzer and Cebrián conducted in vitro experiments with mouse neurons and human neurons created from embryonic stem cells.
The studies showed that under certain circumstances, including conditions known to occur in Parkinson's, the neurons use MHC-1 to display antigens. Among the different types of neurons tested, the two types affected in Parkinson's were much more responsive than other neurons to signals that triggered antigen display.
Parkinsons T Cells
Researchers then verified that T cells did in fact recognize and attack neurons displaying specific antigens.
The results hint that Parkinson's is partially an autoimmune disease, Dr. Sulzer says, but more research is needed to confirm the idea.
"Right now, we've showed that certain neurons display antigens and that T cells can recognize these antigens and kill neurons," Dr. Sulzer says, "but we still need to determine whether this is actually happening in people. We need to show that there are certain T cells in Parkinson's patients that can attack their neurons."
Dr. Sulzer emphasizes that even if the immune system does kill neurons in Parkinson's disease, that it is not the only thing going awry in the disease.
"This idea may explain the final step," he says. "We don't know if preventing the death of neurons at this point will leave people with sick cells and no change in their symptoms, or not."
Carolina Cebrián, Fabio A. Zucca, Pierluigi Mauri, Julius A. Steinbeck, Lorenz Studer, Clemens R. Scherzer, Ellen Kanter, Sadna Budhu, Jonathan Mandelbaum, Jean P. Vonsattel, Luigi Zecca, John D. Loike, David Sulzer.
MHC-I expression renders catecholaminergic neurons susceptible to T-cell-mediated degeneration.
Nature Communications, 2014; 5 DOI: 10.1038/ncomms4633
Image courtesy of Carolina Cebrian.
| {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,692 |
In Blue is a 2000 album by The Corrs.
In Blue may also refer to:
In Blue Tour, a tour by The Corrs
In Blue (Klaus Schulze album), 1995
In Blue (The Static Jacks album), 2013
In Blue (film), a 2017 Dutch drama film
In Blue (Akira Kagimoto), a 2009 documentary video | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,127 |
All-New Suzuki Swift Sport - AVAILABLE NOW!
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Expert advice and exceptional service awaits. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,595 |
Inquiry Myth #2 - Inquiry is "lazy teaching"
Here we go with Myth #2!
Inquiry teaching is sophisticated, nuanced, mindful, strongly framed and goal-focused. Teachers are fully engaged in assisting each student through the inquiry process.
Inquiry teachers are passionate. They model a thought-provoking question style that builds interest.
I propose teachers take on an active role in releasing control over learning to the student. Inquiry teachers are highly active in the classroom in that they take on different roles at different times for different reasons. At times they lecture or teach to all. At times they facilitate group and cooperative learning structures. At times they support individual students on a more personalized level as they shift, pivot, and respond to the needs of each student they work with.
Inquiry is not laid back nor is it lazy. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,219 |
Q: "IOError: [Errno 32] Broken pipe" when saving animation files in anaconda python I have a very simple code from the matplotlib example:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
fig, ax = plt.subplots()
line, = ax.plot(np.random.rand(10))
ax.set_ylim(0, 1)
def update(data):
line.set_ydata(data)
return line,
def data_gen():
while True: yield np.random.rand(10)
ani = animation.FuncAnimation(fig, update, data_gen, interval=1000)
anim.save('basic_animation.mp4', fps=30)
plt.show()
Everything is right if I do not use anim.save() function. However, when I want to save it, it would report:
IOError Traceback (most recent call last)
<ipython-input-6-8948bc3b3f5c> in <module>()
16
17 ani = animation.FuncAnimation(fig, update, data_gen, interval=1000)
---> 18 anim.save('basic_animation.mp4', fps=30)
19 plt.show()
....(traceback details are omitted here)
/home/xin/anaconda2/lib/python2.7/site-packages/matplotlib/backends/backend_agg.pyc in print_raw(self, filename_or_obj, *args, **kwargs)
517 close = False
518 try:
--> 519 fileobj.write(renderer._renderer.buffer_rgba())
520 finally:
521 if close:
IOError: [Errno 32] Broken pipe
How can I fix it? Or are there any other ways to save animation to a file?
Supplement: To install ffmpeg, I just run:
conda install -c https://conda.anaconda.org/mutirri ffmpeg
A: Get it solved by myself! I use conda install to get ffmpeg but when using ffmpeg --version will always say that:
libssl.so.10: cannot open shared object file: No such file or directory
so I use:
sudo ln -s /home/xin/anaconda2/lib/libssl.so.1.0.0 libssl.so.10
Then get similar problem about libcrypto.so.10, so I use:
sudo ln -s /home/xin/anaconda2/lib/libcrypto.so.1.0.0 libcrypto.so.10
The two files are in /lib/x86_64-linux-gnu.
Now things work!! I know some people also have similar problems, so I record it here.
In future, if need to remove the link:
cd /lib/x86_64-linux-gnu
sudo unlink libssl.so.10
sudo unlink libcrypto.so.10
A: I had this problem too. Specifying writer='imagemagick' worked for me.
anim.save('basic_animation.mp4', fps=30, writer='imagemagick')
A: I think it should be
ani.save('basic_animation.mp4', fps=30)
and not
anim.save('basic_animation.mp4', fps=30)
if your defined variable is ani
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,562 |
Q: piping echo and cat doesn't works I'm coding a script that installs firmware for a server.
I have stuck in args for output versions from XML files in the terminal.
echo "BROADCOM NIC Version : " | cat firmware-nic-broadcom-*/CP*.xml | grep "<Version>" | uniq | sed 's/[^0-9,.]//g'
I want to output "BROADCOM NIC Version : 20.19.31", but it just shows "20.19.31".
How do I fix this?
A: Here's one simple way to do it:
echo -n "BROADCOM NIC Version : " ; cat firmware-nic-broadcom-*/CP*.xml | grep "<Version>" | uniq | sed 's/[^0-9,.]//g'
^^ Add ^ Change
The -n flag says echo with no new line, and changing the | to a ; means that it always outputs directly rather than getting lost in the piping.
A: You can declare VERSION variable first.
VERSION=$(cat firmware-nic-broadcom-*/CP*.xml | grep "<Version>" | uniq | sed 's/[^0-9,.]//g')
echo "BROADCOM NIC Version : ${VERSION}"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,205 |
Obama's statement on restricting NSA activities confirms that Snowden was right - Snowden's lawyer
U.S. President Barack Obama's statement to the effect that the U.S. special services' interference in private lives of American citizens should be restricted confirms that former NSA contractor Edward Snowden's actions were well justified, says Anatoly Kucherena, a member of the Russian Public Chamber and a prominent lawyer representing Snowden's interests.
"President Obama's […] statement is the recognition of the fact that American special services have intruded virtually unlimitedly in citizens' private lives, violating their rights," Kucherena told Interfax on Friday evening.
Full story: Edward Snowden
Kucherena welcomed the U.S. president's position, who, in his view, "has realized and perceived the people's concerns about massive eavesdropping on their telephone conversations and correspondence."
"The acknowledgement made by President Obama confirms the fact that American special services have massively violated the people's right to private life, including the rights of foreign heads of state," Kucherena said.
The U.S. president's statement is "a reaction to the people's concerns and outrage about interference in their private lives," he said.
Obama said in a statement in Washington on Friday that the U.S. would revise the current program of collection and storage of metadata and introduce a so-called transition period, which is expected to pass at two stages. During this transition period, the U.S. special services would have access to databases only at a court sanction.
barack obama Edward Snowden cybersecurity News Cybersecurity Edward Snowden | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,055 |
{"url":"https:\/\/math.stackexchange.com\/questions\/339661\/solution-to-the-second-order-differential-equation","text":"# Solution to the second order differential equation\n\nHello i have read in a book that second order diferential equation of this form ($\\psi$ is a function of $x$):\n\n$$\\frac{d^2 \\psi}{dx^2} = - k^2\\, \\psi$$\n\ndescribes a simple harmonic oscilator and the solution to this second order differential equation is of form:\n\n$$\\psi = A \\sin(k x) + B \\cos(kx)$$\n\nThis solution is generaly known, but i want to know the background on how it is calculated out of the first equation.\n\n\u2022 you have a minus missing in the right hand side. \u2013\u00a0lab bhattacharjee Mar 24 '13 at 15:10\n\u2022 @labbhattacharjee, fixed the sign. \u2013\u00a0vonbrand Mar 24 '13 at 15:22\n\u2022 \u2013\u00a0Julien Mar 24 '13 at 15:26\n\nThere are many ways, as you already know it, you can prove that it is a solution by just plugging it in. But that is not really satisfying I guess. We can write $\\frac{d^2 \\psi}{dx^2}=- k^2 \\psi$ as a first order differential equation system \\begin{align*} \\psi_1 &= \\psi'\\\\ \\psi_1' &= - k^2 \\psi \\end{align*} We can write this equation system as $\\begin{pmatrix} \\psi \\\\ \\psi' \\\\ \\end{pmatrix}' = \\begin{pmatrix} 0 & 1\\\\ -k^2 & 0 \\\\ \\end{pmatrix} \\cdot \\begin{pmatrix} \\psi \\\\ \\psi' \\end{pmatrix}$ Here we gonna use the matrix exponential function for which we only need to know the eigenvalues und their multiplicity. The Matrix exponential function comes from the idea that the solution of the 1 dimensional linear ode $y'= a y$ is the exponential function, so with a bit luck $y'=A y$ will be solved by $y=\\exp( A t )$. In fact that is true and as the characteristic polynomial is $$x^2+k^2=(x+ik)\\cdot (x-ik)$$ So you get your solutions by a linear combination of $\\exp(ikt )$ and $\\exp(-ikt )$ and with the identiy \\begin{align*} \\sin(x)&=\\frac{1}{2} \\cdot (e^{ix} - e^{-ix})\\\\ \\cos(x)&=\\frac{1}{2} \\cdot (e^{ix} + e^{-ix}) \\end{align*}\n\n\u2022 I am a total beginner in an area of differential equations- haven't even started. So i would need some recommendations on books where i can learn enough dif. eq. to understand this. \u2013\u00a071GA Mar 24 '13 at 16:13\n\u2022 @71GA do you knwo that eigenvalues and eigenvectors are ? \u2013\u00a0Dominic Michaelis Mar 24 '13 at 16:15\n\u2022 If i recall \"eigenvector\" is an operator which operates on a matrix and gives a value - \"eigenvalue\" which is only scalar multiplication of the original eigenvector? I think it is like that but please correct me if i am wrong. \u2013\u00a071GA Mar 24 '13 at 17:15\n\u2022 @71GA an eigenvector is a vector who only get stretches by a matrix (maybe mirrored to) such that $A v = \\lambda v$ where $\\lambda$ is a scalar. \u2013\u00a0Dominic Michaelis Mar 24 '13 at 17:26\n\nAssuming $$\\frac{d^2 \\psi}{dx^2} = k^2\\psi$$ to be correct,\n\nthe Characteristic equation will be $r^2=k^2\\implies r=\\pm k$\n\nSo, $\\psi=Ae^{kx}+Be^{-kx}$ where $A,B$ are arbitrary constants\n\nIf $k=i b, \\psi=Ae^{ibx}+Be^{-ibx}=A(\\cos bx+i\\sin bx)+B(\\cos bx-i\\sin bx)$ using Euler's Formula\n\nSo, $\\psi=(A+B)\\cos bx+i(A-B)\\sin bx$\n\n$\\psi=C\\cos bx+D\\sin bx$ where $C=A+B,D=i(A-B)$ are arbitrary constants\n\nYou can do this two ways:\n\n\u2022 Try a solution of the form $\\Psi(t) = c e^{\\alpha t}$ gives you that $c$ is arbitrary, and $\\alpha = \\pm k i$, where $i = \\sqrt{-1}$. So you have solutions $\\Psi(t) = c_1 e^{i k t} + c_2 e^{- i k t}$. By Euler's formula, you can write this in terms of $\\sin k t$ and $\\cos k t$.\n\u2022 Note that $\\dfrac{d^2}{d t^2} \\sin c t = - c^2 \\sin c t$ and $\\dfrac{d^2}{d t^2} \\cos c t = - c^2 \\cos c t$, so this is a promising lead to fugde up a pair of solutions.\n\nYou can also do this. Rewrite your equation as:\n\n$$\\psi''+k^2 \\psi=0$$\n\n$$\\psi'' +ik\\psi'-ik\\psi'+k^2 \\psi=0$$\n\n$$\\psi''+ik\\psi'-ik(\\psi'+ik \\psi)=0$$\n\nSuppose $z=\\psi'+ik\\psi.$\n\nWe now have turned our 2nd order differential equation into a 1st order one.\n\n$z'-ikz=0$\n\nSeparate variables to see:\n\n$z=De^{ikx}, D\\in\\mathbb{C}$\n\nReplace this solution $z$ in $z=\\psi'+ik\\psi$ so you get\n\n$$De^{ikx}=\\psi'+ik\\psi$$\n\nNow solve this equation by integrating factors:\n\n$$\\psi(x)=\\frac{\\int{D\\mu(x)e^{ikx}}+E}{\\mu(x)}$$ where $\\mu(x)=e^{\\int ik dx}=e^{ikx}$\n\nSo $$\\psi(x)=\\frac{\\int De^{2ikx}+E}{e^ikx}=\\frac{\\frac{D}{2ik}e^{2ikx}+E}{e^{ikx}}=De^{ikx}+Ee^{-ikx}$$. The result follows upon using Euler's formula and combining constants together.","date":"2019-08-25 13:16:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 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\": 0.9824098348617554, \"perplexity\": 347.96399453928035}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027330233.1\/warc\/CC-MAIN-20190825130849-20190825152849-00510.warc.gz\"}"} | null | null |
'use strict';
jest.dontMock('../js/app.js');
describe('App', function() {
it('contains a test message', function () {
var React= require('react'); // React is used by compiled codes from JSX. Important!!!
var ReactDOM = require('react-dom');
var TestUtils = require('react-addons-test-utils');
var App = require('../js/app.js');
var app = TestUtils.renderIntoDocument(<App />);
//var div = TestUtils.findRenderedDOMComponentWithTag(app, 'div');
//expect(ReactDOM.findDOMNode(div).textContent).toEqual('React Skeleton');
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,502 |
{"url":"https:\/\/bt.gateoverflow.in\/470\/gate2019-ga-4","text":"Five numbers $10,7,5,4$ and $2$ are to be arranged in a sequence from left to right following the directions given below:\n\n1. No two odd or even numbers are next to each other.\n2. The second number from the left is exactly half of the left-most number.\n3. The middle number is exactly twice the right-most number.\n\nWhich is the second number from the right?\n\n1. $2$\n2. $4$\n3. $7$\n4. $10$","date":"2022-06-30 09:27:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6442457437515259, \"perplexity\": 253.58010540916635}, \"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\/1656103671290.43\/warc\/CC-MAIN-20220630092604-20220630122604-00006.warc.gz\"}"} | null | null |
\section{INTRODUCTION}
Chiral perturbation theory is an effective theory for QCD, formulated in terms
of a series of effective Lagrangians of increasing orders in the momentum and
quark mass expansion. It was originally formulated for mesons only, but can be
extended as HBChPT to include heavy baryons. For a review see
e.g. \cite{bkm,Ecker95,Pich}.
The complete Lagrangian for a single nucleon coupling to pions and external
fields up to third order in small momenta (denoted ${\cal L}^{EckM}_{\pi N}$)
has only recently been constructed by Ecker and Moj\v{z}i\v{s} \cite{EM}
(E\&M), although calculations with earlier versions and for specific processes
had been performed before. Here we study muon capture by a proton with the new
Lagrangian, ${\cal L}^{EckM}_{\pi N}$. The form factors that appear in the
muon capture amplitude have been considered previously within HBChPT, but not
with the new Lagrangian ${\cal L}^{EckM}_{\pi N}$.
Our calculation gives explicit expressions for each of the muon capture form
factors, in terms of parameters that appear in ${\cal L}^{EckM}_{\pi N}$. We
use experimental data to determine the numerical values of the parameters,
which are directly transferable to future calculations of other processes where
${\cal L}^{EckM}_{\pi N}$ is used. In particular, the parameters obtained for
ordinary muon capture are a subset of those required for our calculation of
radiative muon capture.
The external nucleon fields in our calculation are renormalized by defining a
wave function renormalization factor, $Z_N$, which is the residue of the full
heavy baryon nucleon propagator at the pole. Even though $Z_N$ is not
measurable, it affects measurable quantities, and one must use the value
consistent with the Lagrangian and the rest of the calculation. Our result
differs from that of previous published work using different Lagrangians, but
is the one appropriate for the new E\&M Lagrangian, as can be checked for
example by noting that our form is the one within our formalism that is
necessary to ensure that the vector coupling is not renormalized.
There is an additional normalization factor relating the normalization of
relativistic and heavy baryon wave functions, which we put in explicitly. An
alternative approach recently suggested \cite{ecker97} absorbs this momentum
dependent factor into the definition of $Z_N$. This gives a different
expression for $Z_N$ than we obtain, but would lead to exactly the same
physical results.
\section{NON RADIATIVE MUON CAPTURE}
The general amplitude for muon capture can be parameterized in terms of the
form factors $G_V(q^2)$, $G_M(q^2)$, $G_A(q^2)$ and $G_P(q^2)$ as follows,
\begin{displaymath}
{\cal M} = \frac{-iG_\beta}{\sqrt{2}}l_\alpha
\overline{u}({\bf p}_n)\left[G_V(q^2)\gamma^\alpha
+\frac{iG_M(q^2)}{2m_N}\sigma^{\alpha\beta}q_\beta
-G_A(q^2)\gamma^\alpha\gamma_5
-\frac{G_P(q^2)}{m_\mu}q^\alpha\gamma_5\right]u({\bf p}_p)~,
\end{displaymath}
where $l_\alpha = \overline{u}({\bf p}_\nu)\gamma_\alpha
(1-\gamma_5)u({\bf p}_\mu)$ is the leptonic current, $m_N$ is
the physical nucleon mass, and $G_\beta$ is the Fermi
constant applicable to $\beta$-decay.
In HBChPT, the muon capture amplitude in terms of heavy baryon
spinors is
\begin{displaymath}
{\cal M} = \frac{g_W}{2\sqrt{2}m_W^2}l_\alpha\overline{n}_v({\bf p}_n)
\left( \Gamma^{(r)\,\alpha}_{pWn}(q)
+ \Gamma^{(r)}_{p{\pi}n}(q)\left[\frac{i}{q^2-m_\pi^2}\right]
\Gamma^{(r)\,\alpha}_{W\pi}(q)\right) n_v({\bf p}_p)~,
\end{displaymath}
with $m_\pi$ the physical pion mass, and $m_W, g_W$ the mass and weak coupling
constant of the W boson.
The functions $ \Gamma^{(r)\,\alpha}_{pWn}(q), \Gamma^{(r)}_{p{\pi}n}(q)$ and
$\Gamma^{(r)\,\alpha}_{W\pi}(q)$ are the fully renormalized vertex
functions. To evaluate them we start with the Lagrangian, ${\cal L}^{EckM}_{\pi
N} = \widehat{\cal L}^{(1)}_{\pi N} + \widehat{\cal L}^{(2)}_{\pi N} +
\widehat{\cal L}^{(3)}_{\pi N}$, where the ${\cal O}(p^3)$ part is the new part
of the Lagrangian as derived by E\&M. Using this Lagrangian the various
contributing diagrams are calculated and completely renormalized, so that they
can be expressed in terms of physical quantities. One then uses the relation
between the heavy baryon spinors $n({\bf p})$ and the Dirac spinors $u({\bf
p})$, i.~e.,
\begin{displaymath}
n_v({\bf p}) = \sqrt{\frac{2m_N}{m_N+v \cdot p_p}}
\frac{(1+v\!\!\!/)}{2}u({\bf p}) =\left[1-\frac{k\!\!/_p}{2m_N}+\frac{(m_N-
m_{0N})}{2m_N}+\frac{k_p^2}{8m_N^2}+ {\cal O}(\frac{1}{m_N^3})\right]
u({\bf p}),
\end{displaymath}
where $p=m_{0N}v + k_p$, to express the amplitude in the
original form and extract the form factors.
The result is
\pagebreak
\begin{eqnarray*}
G_V(q^2) &=& 1 - \left(a_6-\frac{1}{8}\right)\frac{q^2}{m_N^2}
- \frac{q^2}{18(4{\pi}F)^2}(1+17g_A^2) \\
&& - \frac{2q^2}{(4{\pi}F)^2}\left[
b_7^r(\mu)+\frac{1}{12}(1+5g_A^2){\rm ln}\left(
\frac{m_\pi^2}{\mu^2}\right)\right] \\
&& + \frac{2}{(4{\pi}F)^2}\left[\frac{m_\pi^2}{3}(1+2
g_A^2)-\frac{q^2}{12}(1+5g_A^2)
\right]
\int_0^1{\rm d}x\,{\rm ln}\left(1-x(1-x)\frac{q^2}{m_\pi^2}\right)
\label{gV} \\
G_M(q^2) &=& 4a_6 - 1 - \frac{4{\pi}g_A^2m_\pi{m}_N}{(4{\pi}F)^2}
\int_0^1{\rm d}x\sqrt{1-x(1-x)\frac{q^2}{m_\pi^2}} \label{gM},\nonumber \\
G_A(q^2) &=& g_A + \frac{4a_3g_Am_\pi^2}{m_N^2}
- \frac{g^3_Am_\pi^2}{(4{\pi}F)^2} \nonumber \\
&& + \frac{4m_\pi^2}{(4{\pi}F)^2}\left[b_{17}^r(\mu)-\frac{g_A}{4}
(1+2g_A^2)\,{\rm ln}\left(\frac{m_\pi^2}{\mu^2}\right)\right]
- \frac{b_{23}q^2}{(4{\pi}F)^2}, \label{gA} \nonumber \\
G_P(q^2) &=& \frac{2m_{\mu}m_N}{(m_\pi^2-q^2)}\left[G_A(q^2)
- \frac{m_\pi^2}{(4{\pi}F)^2}(2b_{19}-b_{23})\right]
\label{gP} \nonumber,
\end{eqnarray*}
where $F=$92.4 MeV is the pion decay constant.
The parameters appearing in these expressions can be evaluated by comparison
with other known experimental quantities. $G_A(0)$ can be obtained from data on
neutron decay, thus giving the parameter $g_A$ to the order needed. $G_M(0)$ is
related to the nucleon magnetic moments, which leads to the standard value of
$a_6$, one of the parameters of the ${\cal O}(p^2)$ Lagrangian. The $q^2$
dependence of the nucleon electromagnetic form factors gives $G_V(q^2)$ and
thus the value of $b_7^r(m_N)$. Similarly the $q^2$ dependence of the axial
form factor as measured in antineutrino-nucleon scattering (or pion
electroproduction) gives an estimate of $b_{23}$. Finally $b_{19}$ is related
to the so-called Goldberger-Treiman discrepancy, and can be evaluated using the
pion nucleon coupling constant as input.
We thus find for the three constants of the E\&M Lagrangian obtainable from
this process $ b_7^r(m_N) = -0.53 \pm 0.02$, $ b_{23} = -3.1 \pm 0.3$, and $
b_{19} = -0.7 \pm 0.4$~. Recently the constant $b_{19}$ has also been
determined by Moj\v{z}i\v{s} \cite{moj} in the context of $\pi$N
scattering. His reported value of $ b_{19} = -1.0 \pm 0.4$, corresponding to
$g_{\pi N}=13.0 \pm 0.1$, is consistent with our value of $b_{19}$ within
errors, with the difference being due almost entirely to his choice of a
different value of $F_\pi$.
We can now evaluate $ G_P$ using these values and we obtain $G_P(-0.88m_\mu^2)
= 8.21 \pm 0.09$ which is in good agreement with the best value from
non-radiative muon capture,\cite{bardin} $G_P(-0.88m_\mu^2) = 8.7 \pm 1.9$~.
Thus we have obtained the form factors of muon capture by a proton in the
framework of the recently derived E\&M ${\cal O}(p^3)$ heavy baryon chiral
Lagrangian, and used experimental data to extract numerical values for some of
the Lagrangian's parameters, which will be needed for the calculation of
radiative muon capture. Further details are given in \cite{full}.
\section{RADIATIVE MUON CAPTURE}
The radiative muon capture process, $\mu + p \rightarrow n + \nu + \gamma$ is
particularly interesting because it is especially sensitive to the value of the
induced pseudoscalar form factor, $G_P(q^2)$. A recent TRIUMF experiment
\cite{jonkmans} measured the rate for the radiative process and found a value
of $G_P$ approximately 1.5 times the value expected from the Goldberger-Treiman
relation. While consistent with radiative capture measurements in nuclei, this
value is at variance with results from the nonradiative capture both in nuclei
and on the proton, which agree with the Goldberger-Treiman relation.
The standard theoretical calculation for radiative capture \cite{hwf} involves
four tree level Feynman graphs, consisting of radiation from the external
particles and from the intermediate pion generating $G_P$, together with a
gauge term generated by minimal substitution. Corrections due to $\Delta$
intermediate states have been calculated \cite{beder} but are small.
In view of the discrepancy with experiment it is of interest to apply the same
heavy baryon chiral perturbation theory approach used for ordinary capture to
the radiative capture process. This allows a microscopic calculation of the
gauge term and one would expect a number of new contributions for which there
is no counterpart in the standard Feynman graph approach.
We have approached this process using the same techniques used for ordinary
muon capture. In particular we use the same ${\cal O}(p^3)$ Lagrangian of Ecker
and Moj\v{z}i\v{s} \cite{EM} and our same expression for the wave function
renormalization factor \cite{full}. The strategy is to evaluate individually
the complete renormalized irreducible vertex functions for the separate
interactions, i. e. for the weak-NN, $\pi$NN, $\gamma$NN, $\gamma \pi \pi$ and
weak-$\pi$ vertices. We then put the pieces together to get the radiative muon
capture amplitude.
We expect that the results corresponding to the first four graphs of the
standard approach will be similar to the standard calculation. These graphs,
corresponding to radiation from external legs and from an intermediate pion,
involve situations where the electromagnetic and weak vertices are
separated. Thus any approach which uses the non radiative muon capture and
electromagnetic vertices to fix the parameters of the interaction, as we have
done, should pretty much reproduce the standard calculation of these
graphs. There may be possible off shell effects and the main terms and their
leading relativistic corrections may arise from various pieces of the ChPT
Lagrangian, but the basic physics is the same as in the standard approach.
This is not the case for the gauge term however, which is included in the
standard calculation only by way of a minimal substitution. In contrast, the
ChPT approach makes an explicit prediction for contributions to this diagram
and there are many such contributions. Some contain loops, with both weak and
electromagnetic vertices attaching to the loop, and some arise from contact
terms from the ${\cal O}(p^3)$ Lagrangian. Most of these contributions do not
appear in the standard minimal coupling diagram and, except that they are
${\cal O}(p^3)$ terms, one does not know {\em a priori} how large they will be.
We have explicitly evaluated one such contribution arising from the
Wess-Zumino-Witten part of the Lagrangian. This contribution leads to a diagram
with intermediate pion coupling at a point to weak and electromagnetic
currents. It contains the pion propagator and so contributes to $G_P$. An
analogous contribution was important for the non soft photon corrections to the
spin dependent part of the virtual Compton scattering amplitude
\cite{hemmert}. Like radiative muon capture, that process involves the coupling
to two external currents. Furthermore this term is gauge invariant by itself,
and so can be evaluated individually. Unfortunately it turns out to be
negligibly small, apparently because for radiative muon capture, unlike Compton
scattering, there are leading contributions from lower order parts of the
Lagrangian.
There are however many other diagrams involving both loops and contact terms
which may contribute. These can be combined into the renormalized, irreducible,
weak-$\gamma$NN and $\pi\gamma$NN vertices which are now being calculated. When
combined appropriately with the other vertices these will allow an explicit
comparison with the radiative muon capture data. \cite{note}
| {
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} | 6,779 |
run_spec(__dirname, null, ["typescript"]);
run_spec(__dirname, { tabWidth: 4 }, ["typescript"]);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,341 |
\section{Introduction}
Important aspects of analysis in Euclidian space are studied via the Riesz distributions
$|x|^{\lambda}$, see \cite{Riesz}, i.e. the complex powers of the Euclidian norm. As convolution operators
these allow a rigorous treatment of the Laplace operator $\Delta$ and its complex powers
$I_{\alpha} = (-\Delta)^{-\alpha/2}$, acting on functions
as a natural semigroup, and there are many classical and recent results related
to this family of operators, sometimes referred to as fractional Laplacians.
Generalizations of those ideas to norms induced by indefinite metrics or spinor valued distributions
where studied in \cite{KV} and \cite{CO}.
In this paper we introduce a natural family of similar operators acting on differential forms
in Euclidian space; however, the semigroup property (as above) has to be relaxed.
The corresponding family of distributions seems to deserve the name of
Riesz distributions for differential forms, and it allows a certain extra flexibility in the complex
parameters involved. We shall start by giving the basic definitions and calculations in the
most general cases, and later specialize to some specific cases of particular interest.
One of the important special cases is not new by any means, since it corresponds to
some of the cases of the intertwining operators treated by Knapp and Stein in detail, when they
introduced their celebrated kernel operators \cite{KnappStein}. Here our
calculations make these operators \ref{eq:Riesz2} more explicit; in particular we give the Euclidian Fourier
transform of these Knapp-Stein operators and simplifying proofs for the unitarity of the
complementary series of representations of the conformal group. For this
see Remark \ref{ComplementarySeries}. Another interesting case which arises by specializing
parameters in our family is the Beurling-Ahlfors operator $S$
(in even dimension, acting on forms of middle degree), see \cite{IM}; this operator
is in several ways an analogue of the Hilbert transform.
In summary, we shall for the Riesz distributions for differential forms \eqref{eq:GeneralRiesz} be interested in their
\begin{itemize}
\item Fourier transforms (Theorem \ref{FourierGeneralRiesz})
\item Bernstein-Sato identities (Theorem \ref{BernsteinSatoGeneralRiesz})
\item residues (Theorem \ref{ResiduesGeneralRiesz})
\item convolution formulas (Theorem \ref{ConvolutionIndentityGeneralRiesz})
\end{itemize}
and for example obtain the Branson-Gover operators \cite{BransonGover}
\begin{align*}
L^{(p)}_{2N} = (\frac{n}{2} - p + N)(\delta d)^N + (\frac{n}{2} - p - N)(d \delta)^N
\end{align*}
on $p$-forms in Euclidian space (well-known from conformal geometry) as residues, see Corollary \ref{ResiduesRiesz2}.
Here $d$ is the usual
derivative on forms and $\delta$ its $L^2$-adjoint, and these (and their symbols) are
the basic building blocks in our study.
This corresponds well with the conjectural fact that in general
(on general Riemannian manifolds) the Branson-Gover operators may be obtained as
residues of the scattering operator \cite{AG, GZ}.
While we shall carry out most of the
analysis in Euclidian space, it is also possible to work on the conformal compactification,
i.e. the standard sphere of the same dimension; for this we shall apply some of the
formulas in \cite{BOO}, where the so-called compact picture of the bundles in question
are studied.
We see the present paper as laying the groundwork for further studies of Euclidian analysis
on differential forms, such as for example Sobolev inequalities,
giving the basic facts towards finding fundamental solutions
of natural differential operators on differential forms, and elliptic boundary value problems.
Also we hope our study might contribute to the branching program of T. Kobayashi, in
particular to his theory of symmetry-breaking operators and branching problems for
complementary series representations.
\section{Preliminaries about differential forms}
Let $(\mathbb{R}^n,\langle\cdot,\cdot\rangle)$ be the Euclidian vector space of dimension $n\in\mathbb{N}$ and
$\{e_j\}_{j=1}^n$ its standard basis.
The space of differential $p$-forms on $\mathbb{R}^n$, for $0\leq p\leq n$, is denoted by
$\Omega^p(\mathbb{R}^n)$. The Euclidian scalar product extends to $\Omega^p(\mathbb{R}^n)$ and will be denoted with
the same symbol. We introduce two
algebraic actions on $\omega\in\Omega^p(\mathbb{R}^n)$ which will be important: For $x\in\mathbb{R}^n$ one defines
\begin{align*}
i_x\omega&\stackrel{\text{def}}{=}\sum_{k=1}^nx_k i_{e_k}\omega,\quad
\varepsilon_x\omega\stackrel{\text{def}}{=}\sum_{k=1}^n x_k e_k\wedge \omega.
\end{align*}
Mainly these are the interior and exterior product by $x\in\mathbb{R}^n$ and are
related as symbols to
the exterior differential $d:\Omega^p(\mathbb{R}^n)\to\Omega^{p+1}(\mathbb{R}^n)$
and the negative of the co-differential $\delta:\Omega^p(\mathbb{R}^n)\to\Omega^{p-1}(\mathbb{R}^n)$, the $L^2$-adjoint of the differential.
They satisfy the well known identities:
\begin{lem}\label{Identities1}
The algebraic actions $i_x$ and $\varepsilon_x$ are nilpotent of degree $2$ and
formally adjoint to each other with respect to the scalar product
$\langle\cdot,\cdot\rangle$ on $\Omega^p(\mathbb{R}^n)$,
i.e. $\langle i_x\omega,\eta\rangle=\langle\omega ,\varepsilon_x \eta \rangle$
for all $x\in\mathbb{R}^n$ and $\omega\in\Omega^p(\mathbb{R}^n)$, $\eta\in\Omega^{p-1}(\mathbb{R}^n)$.
Furthermore, it holds
\begin{align*}
i_x\varepsilon_x&=\sum_{k,l=1}^nx_k x_l i_{e_k}(e_l\wedge\cdot),\quad
\varepsilon_xi_x=\sum_{k,l=1}^nx_k x_l e_l\wedge i_{e_k}(\cdot),\quad
i_x\varepsilon_x+\varepsilon_xi_x=\sum_{k=1}^n x_k^2.
\end{align*}
\end{lem}
In the next section it will be important to know some differential actions on
the distribution
\begin{align}\label{eq:RieszDistribution}
r^\lambda(x)\stackrel{\text{def}}{=} (x_1^2+\cdots+x_n^2)^{\frac \lambda 2}
\end{align}
defined for $x\in\mathbb{R}^n$ and $\lambda\in\mathbb{C}$ with $\Re(\lambda)>-n$. This distribution is termed
{\it Riesz distribution} and was for example studied in \cite{Riesz,GelfandShilov}.
The following is the key lemma in order to obtain the Fourier transform of the Riesz distribution for differential forms.
\begin{lem}\label{Identities2}
Let $\beta$ be a constant $p$-form. Then, for fixed $x\in\mathbb{R}^n$,
\begin{align*}
\deltad(r^{\lambda+2}(x-y)\beta)
&= -(\lambda+2)(n-p)r^{\lambda}(x-y)\beta
-(\lambda+2)\lambda r^{\lambda-2}(x-y)i_{x-y}\varepsilon_{x-y}\beta ,\\
d\delta(r^{\lambda+2}(x-y)\beta)
&= -(\lambda+2)pr^{\lambda}(x-y)\beta
-(\lambda+2)\lambda r^{\lambda-2}(x-y)\varepsilon_{x-y}i_{x-y}\beta.
\end{align*}
Derivatives are taken with respect to the $y$-variable.
\end{lem}
\begin{proof}
In order to prove the lemma we first observe
\begin{align}\label{eq:Observation}
\partial_k r^\nu(x-y)&=-\nu (x_k-y_k)r^{\nu-2}(x-y),\notag\\
\sum_{k=1}^n e_k\wedge i_{e_k}\beta&=p\beta,\quad
\sum_{k=1}^n i_{e_k}(e_k\wedge \beta)=(n-p)\beta.
\end{align}
Then a straightforward computation shows
\begin{align*}
\deltad(r^{\lambda+2}(x-y)\beta)
&=-\sum_{k,l=1}^n \partial_l\partial_k (r^{\lambda+2}(x-y))
i_{e_k}(e_l\wedge \beta) \\
&=-(\lambda+2)r^\lambda(x-y) \sum_{k=1}^n i_{e_k}(e_k\wedge \beta) \\
&-\lambda(\lambda+2)r^{\lambda-2}(x-y) \sum_{k,l=1}^n (x_k-y_k)(x_l-y_l)
i_{e_k}(e_l\wedge \beta)
\end{align*}
Using the above observations \eqref{eq:Observation} and Lemma \ref{Identities1} we may conclude
\begin{align*}
\deltad(r^{\lambda+2}(x-y)\beta)= -(\lambda+2)(n-p)r^{\lambda}(x-y)
\beta-(\lambda+2)\lambda r^{\lambda-2}(x-y)i_{x-y}\varepsilon_{x-y}\beta.
\end{align*}
The second claim runs along the same line, which completes the proof.
\end{proof}
Let us denote by $\mathcal{S}^p(\mathbb{R}^n)\stackrel{\text{def}}{=} \mathcal{S}(\mathbb{R}^n)\otimes \Lambda^p(\mathbb{R}^n)$ the
space of Schwartz functions with value in $\Lambda^p(\mathbb{R}^n)^*$.
We follow the convention for the Fourier
transform \cite{GelfandShilov} on Schwartz functions $f\in\mathcal{S}(\mathbb{R}^n)$:
\begin{align*}
\mathcal{F}(f)(\xi)\stackrel{\text{def}}{=} \int_{\mathbb{R}^n} f(x)e^{i \langle x, \xi\rangle}dx,
\end{align*}
and extend it to a Fourier transform on
$\omega\in\mathcal{S}^p(\mathbb{R}^n)$ by acting on the coefficients of $\omega$ in some arbitrarily chosen basis. Our
normalization of the Fourier transform is chosen in such a way that
$\mathcal{F}(\delta_0)=1$, where $\delta_0$ is the Dirac-distribution centered at the origin.
Recall that for a polynomial $P$ in $n$ variables we have the identities
\begin{align}
P(\partial_{\xi_1},\ldots,\partial_{\xi_n})\mathcal{F}(\omega)(\xi)
&=\mathcal{F}(P(i x_1,\ldots,i x_n)\omega)(\xi),\notag\\
\mathcal{F}(P(\partial_{x_1},\ldots,\partial_{x_n})\omega)(\xi)
&=P(-i \xi_1,\ldots,-i \xi_n)\mathcal{F}(\omega)(\xi)\label{eq:FourierProperties2}
\end{align}
for $\omega\in\mathcal{S}^p(\mathbb{R}^n)$.
\section{A class of Riesz distributions for differential forms}
We introduce a class of tempered distributions
$R_{A_\lambda,B_\lambda}^\lambda(x)$ valued in
endomorphisms of differential forms, which generalizes the
Riesz distribution \eqref{eq:RieszDistribution}.
Based on an explicit formula for its Fourier transform, we give a
corresponding Bernstein-Sato identity, compute its residues and some identities concerning their convolutions. Finally,
this distribution when acting by convolution on differential forms induces an integral operator.
Let $A_\lambda,B_\lambda$ be holomorphic functions in $\lambda\in\mathbb{C}$.
For $\lambda\in \mathbb{C}$ with $\Re(\lambda)>-n$ a distribution valued in the endomorphisms of differential forms,
termed {\it Riesz distribution for differential forms}, is defined by
\begin{align}\label{eq:GeneralRiesz}
R_{A_\lambda,B_\lambda}^\lambda(x)\stackrel{\text{def}}{=} r^{\lambda-2}(x)
(A_\lambda i_x \varepsilon_x
+ B_\lambda\varepsilon_x i_x).
\end{align}
The parameters $A_\lambda,B_\lambda$ can also depend on the form-degree
$p$ and the dimension $n$, but we will suppress such
dependencies.
\begin{remark}
One could also assume that $A_\lambda$ and $B_\lambda$ are meromorphic in $\lambda$ without
disturbing the results obtained here. Of course, possible poles of $A_\lambda$ and $B_\lambda$
will influence the assumtions and statements in this paper.
\end{remark}
\subsection{Fourier transform}
It is well known that the Fourier transform, a special version of an integral transform, plays an
important rule in the analysis of functions. Into that branch falls for example the study of solutions of
differential equations, detecting possible poles and computing their residues.
In terms of $A_\lambda,B_\lambda$ we define holomorphic functions
\begin{align}\label{eq:Coefficients1}
C_\lambda\stackrel{\text{def}}{=} (\lambda+p)A_\lambda-pB_\lambda,
\quad D_\lambda\stackrel{\text{def}}{=} -(n-p)A_\lambda
+(\lambda+n-p)B_\lambda.
\end{align}
The next theorem states that the Fourier transform $\mathcal{F}$ preserves the class
of distribution $R^\lambda_{A_\lambda,B_\lambda}(x)$
by changing (in general) the parameters $A_\lambda,B_\lambda$.
\begin{theorem}\label{FourierGeneralRiesz}
The Fourier transform of $R_{A_\lambda,B_\lambda}^\lambda(x)$ is given by
\begin{align}\label{eq:FourierGeneralRiesz}
\mathcal{F}(R_{A_\lambda,B_\lambda}^{\lambda})(\xi)
= -2^{\lambda+n-1}\pi^{\frac n2}
\frac{\Gamma(\frac{\lambda+n}{2})}{\Gamma(-\frac{\lambda-2}{2})}
R_{C_\lambda,D_\lambda}^{-\lambda-n}(\xi),
\end{align}
where $C_\lambda,D_\lambda$ are given by \eqref{eq:Coefficients1}.
\end{theorem}
\begin{proof}
For $\omega\in\mathcal{S}^p(\mathbb{R}^n)$ it is enough to verify that both expressions
\begin{align*}
\mathcal{F}\Big(\int_{\mathbb{R}^n}r^{\lambda-2}(x-y)
(A_\lambda i_{x-y} \varepsilon_{x-y} + B_\lambda\varepsilon_{x-y} i_{x-y})
\omega(y)d y\Big)(\xi),
\end{align*}
which is actually $\mathcal{F}(R_{A_\lambda,B_\lambda}^{\lambda} * \omega)(\xi)$, and
\begin{align*}
r^{-\lambda-2-n}(\xi)(\tilde{C}_\lambda i_\xi\varepsilon_\xi
+\tilde{D}_\lambda \varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi),
\end{align*}
for $\tilde{C}_\lambda,\tilde{D}_\lambda\in\mathbb{C}$ to be determined,
agree up to a constant. We start to compute the latter one. First note the identities:
\begin{align*}
r^{-\lambda-2-n}(\xi)&=c_\lambda^{-1}\mathcal{F}(r^{\lambda+2})(\xi),\\
(\tilde{C}_\lambda i_\xi\varepsilon_\xi
+\tilde{D}_\lambda \varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi)
&=\mathcal{F}\big((\tilde{C}_\lambda\deltad
+\tilde{D}_\lambdad\delta)\omega\big)(\xi),
\end{align*}
see \cite[Chapter II, Section $3.3$]{GelfandShilov} and \eqref{eq:FourierProperties2}, and
\begin{align}\label{eq:FourierRieszConstant}
c_\lambda\stackrel{\text{def}}{=} 2^{\lambda+2+n}\pi^{\frac n2}\frac{\Gamma(\frac{\lambda+2+n}{2})}{\Gamma(-\frac{\lambda+2}{2})},
\end{align}
Hence we obtain
\begin{align*}
r^{-\lambda-2-n}(\xi)(\tilde{C}_\lambda i_\xi\varepsilon_\xi
+\tilde{D}_\lambda \varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi)
= c_\lambda^{-1}\mathcal{F}\big(\int_{\mathbb{R}^n}
r^{\lambda+2}(x-y)(\tilde{C}_\lambda\deltad
+\tilde{D}_\lambdad\delta)\omega d y \big)(\xi).
\end{align*}
Taking a constant $p$-form $\beta$ we have by partial integration
\begin{multline}\label{eq:help1}
\int_{\mathbb{R}^n}\langle \beta,r^{\lambda+2}(x-y)(\tilde{C}_\lambda\deltad
+\tilde{D}_\lambdad\delta)\omega \rangle d y\\
= \int_{\mathbb{R}^n}\langle(\tilde{C}_\lambda\deltad
+\tilde{D}_\lambdad\delta)( r^{\lambda+2}(x-y)\beta),\omega \rangle d y.
\end{multline}
By Lemma \ref{Identities2} we find
\begin{align*}
\int_{\mathbb{R}^n}\langle \deltad(r^{\lambda+2}(x-y)\beta),\omega\rangle d y
=-(\lambda+2)\int_{\mathbb{R}^n}\langle\beta&, (n-p) r^{\lambda}(x-y)\omega\\
&+\lambda r^{\lambda-2}(x-y)
i_{x-y}\varepsilon_{x-y}\omega\rangled y,\\
\int_{\mathbb{R}^n}\langle d\delta(r^{\lambda+2}(x-y)\beta),\omega\rangle d y
=-(\lambda+2)\int_{\mathbb{R}^n}\langle\beta&, pr^{\lambda}(x-y)\omega\\
&+\lambda r^{\lambda-2}(x-y)
\varepsilon_{x-y}i_{x-y}\omega\rangled y.
\end{align*}
In turn, Equation \eqref{eq:FourierGeneralRiesz} is equivalent by
collecting the coefficients of $r^{\lambda-2}(x-y)i_{x-y}\varepsilon_{x-y}$
and $r^{\lambda-2}(x-y)\varepsilon_{x-y}i_{x-y}$ in \eqref{eq:help1} to the following
linear system for $\tilde{C}_\lambda, \tilde{D}_\lambda$:
\begin{align}\label{eq:CoeffRelationsForFourier}
(\lambda+n-p)\tilde{C}_\lambda+p \tilde{D}_\lambda=A_\lambda,\notag\\
(n-p)\tilde{C}_\lambda+(\lambda+p)\tilde{D}_\lambda=B_\lambda.
\end{align}
The unique solution is given by
\begin{align}\label{eq:SolutionCoeffRelationsForFourier}
\tilde{C}_\lambda=\frac{(\lambda+p)A_\lambda-pB_\lambda}{\lambda(\lambda+n)},
\quad \tilde{D}_\lambda=\frac{-(n-p)A_\lambda+(\lambda+n-p)B_\lambda}{\lambda(\lambda+n)}
\end{align}
This implies, since $\beta$ was an arbitrary $p$-form,
\begin{multline*}
c_\lambda^{-1}\mathcal{F}\big(\int_{\mathbb{R}^n} r^{\lambda+2}(x-y)
( \tilde{C}_\lambda \deltad+\tilde{D}_\lambdad\delta)\omega d y \big)(\xi)\\
=-(\lambda+2)c_\lambda^{-1}
\int_{\mathbb{R}^n}r^{\lambda-2}(x-y)(A_\lambda i_{x-y}\varepsilon_{x-y}
+B_\lambda \varepsilon_{x-y} i_{x-y})\omega(y)d y.
\end{multline*}
Note that the factor $\lambda(\lambda+n)$ in $\tilde{C},\tilde{D}$ will be absorbed into
corresponding Gamma functions arising from $c_\lambda$.
The proof is complete.
\end{proof}
\subsection{Bernstein-Sato identity}
For a given polynomial $f\in\mathbb{R}[x_1,\ldots,x_n]$ and complex number $s\in\mathbb{C}$
one can consider $f_+^s(x):=f(x)^s$ for $f(x)> 0$ and
zero otherwise. If $\Re(s)>0$ this is locally integrable on $\mathbb{R}^n$. Bernstein
\cite{Bernstein} proves that $f_+^s$ admits a meromorphic continuation to $\mathbb{C}$ with poles
given by the zero locus of a certain polynomial, the {\it Bernstein polynomial}.
Roughly speaking one can construct a differential operator $P(s)$ with polynomial coefficients such that
$P(s)f^{s+1}_+(x)=b(s) f^s_+(x)$, where $b(s)$ is the Bernstein-Sato polynomial. This differential equation
is termed {\it Bernstein-Sato identity} and was independently discovered in \cite{SatoShintani}.
Now we present a vector-valued Bernstein-Sato identity
which in turn implies the mereomorphic continuation of $R^\lambda_{A_\lambda,B_\lambda}(x)$ to $\lambda\in\mathbb{C}$.
\begin{theorem}\label{BernsteinSatoGeneralRiesz}
The distribution $R^\lambda_{A_\lambda,B_\lambda}(x)$
satisfies the differential equation:
\begin{align}\label{eq:BersteinSatoGeneralRiesz}
D_2(\lambda;n,p)R^{\lambda+2}_{A_{\lambda+2},B_{\lambda+2}}(x)
=-\lambda(\lambda+n) C_{\lambda+2} D_{\lambda+2}
R^{\lambda}_{A_{\lambda},B_{\lambda}}(x)
\end{align}
where $D_2(\lambda;n,p)$ is a second-order differential operator given by
\begin{align*}
D_2(\lambda;n,p)\stackrel{\text{def}}{=} C_{\lambda}D_{\lambda+2} \deltad
+ D_{\lambda}C_{\lambda+2}d\delta.
\end{align*}
\end{theorem}
\begin{proof}
We apply the Fourier transform to Equation \eqref{eq:BersteinSatoGeneralRiesz} and use
Theorem \ref{FourierGeneralRiesz}.
Hence, it remains to verify that
\begin{multline*}
-2^{\lambda+n+1}\pi^{\frac n2}\frac{\Gamma(\frac{\lambda+n+2}{2})}{\Gamma(-\frac{\lambda}{2})}
r^{-\lambda-n-4}(\xi)(C_{\lambda}D_{\lambda+2} i_\xi\varepsilon_\xi
+D_{\lambda}C_{\lambda+2} \varepsilon_\xi i_\xi)
(C_{\lambda+2} i_\xi\varepsilon_\xi +D_{\lambda+2} \varepsilon_\xi i_\xi)\\
=-2^{\lambda+n+1}\pi^{\frac n2}\frac{\Gamma(\frac{\lambda+2+n}{2})}{\Gamma(-\frac{\lambda}{2})}
r^{-\lambda-n-4}(\xi)(C_\lambda C_{\lambda+2}D_{\lambda+2} (i_\xi\varepsilon_\xi)^2
+C_{\lambda+2} D_\lambda D_{\lambda+2}(\varepsilon_\xi i_\xi)^2)
\end{multline*}
and
\begin{multline*}
\lambda(\lambda+n)C_{\lambda+2}D_{\lambda+2}2^{\lambda+n-1}\pi^{\frac n2}
\frac{\Gamma(\frac{\lambda+n}{2})}{\Gamma(-\frac{\lambda-2}{2})}
r^{-\lambda-n-2}(\xi)(C_{\lambda} i_\xi\varepsilon_\xi +D_{\lambda} \varepsilon_\xi i_\xi)\\
=-2^{\lambda+n+1}\pi^{\frac n2}\frac{\Gamma(\frac{\lambda+n+2}{2})}{\Gamma(-\frac{\lambda}{2})}
r^{-\lambda-n-4}(\xi)(C_\lambda C_{\lambda+2}D_{\lambda+2} (i_\xi\varepsilon_\xi)^2
+C_{\lambda+2}D_\lambda D_{\lambda+2}(\varepsilon_\xi i_\xi)^2)
\end{multline*}
do agree.
\end{proof}
An iteratively application of Equation \eqref{eq:BersteinSatoGeneralRiesz} gives
\begin{align}\label{eq:IterationOfBernsteinSatoGeneralRiesz}
R^{\lambda}_{A_\lambda,B_\lambda}(x)
=\frac{(-1)^k}{4^k(\frac{\lambda}{2})_k(\frac{\lambda+n}{2})_k C_{\lambda+2k}D_{\lambda+2k}}
D_{2k}(\lambda;n,p)R^{\lambda+2k}_{A_{\lambda+2k},B_{\lambda+2k}}(x),
\end{align}
where $D_{2k}(\lambda;n,p)$ is a differential operator of order $2k$, for $k\in \mathbb{N}$, given by
\begin{align}\label{eq:2kthOrderDifferential}
D_{2k}(\lambda;n,p)\stackrel{\text{def}}{=} C_{\lambda}D_{\lambda+2k}(\deltad)^k
+ C_{\lambda+2k}D_{\lambda}(d\delta)^k.
\end{align}
By convention we set $D_0(\lambda;n,p)\stackrel{\text{def}}{=} C_\lambda D_\lambda\operatorname{Id}$.
The tempered distribution $R^\lambda_{A_\lambda,B_\lambda}(x)$ originally defined for
$\Re(\lambda)>-n$ can now, by \eqref{eq:IterationOfBernsteinSatoGeneralRiesz},
be meromorphically extended to $\lambda\in\mathbb{C}$ with simple poles at $\lambda=-n-2\mathbb{N}_0$.
\subsection{Residues}
In the last subsection we have identified the location of the poles of
$R^\lambda_{A_\lambda,B_\lambda}$ and their type. In general their residues, when acting as convolution operators,
will not be differential operators. However, under some assumtions they reduce to differential operators given
as a weighted sum of powers of $\deltad$ and $d\delta$.
\begin{theorem}\label{ResiduesGeneralRiesz}
The residue of $\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)$ at $\lambda=-n-2k$, for $k\in\mathbb{N}_0$ is
given by
\begin{multline}\label{eq:ResiduesGeneralRiesz}
\operatorname{Res}_{\lambda=-n-2k}(\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi))\\
=\frac{(-1)^{k+1}\pi^{\frac n2}}{4^{k} k! \Gamma(\frac{n+2k+2}{2})}
\Big[\frac{C_{-n-2k}}{C_{-n}}(i_\xi\varepsilon_\xi)^k
+ \frac{D_{-n-2k}}{D_{-n}}(\varepsilon_\xi i_\xi)^k\Big]R^{0}_{C_{-n}, D_{-n}}(\xi).
\end{multline}
Especially, in case $R^\lambda_{A_\lambda,B_\lambda}(x)$ acts on $0$-forms we have
\begin{align}\label{eq:ResiduesGeneralRieszOnZeroForms}
\operatorname{Res}_{\lambda=-n-2k}\big(\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)\big)
=\frac{(-1)^{k+1}\pi^{\frac n2}}{4^{k} k! \Gamma(\frac{n+2k+2}{2})}C_{-n-2k}r^{2k}(\xi).
\end{align}
\end{theorem}
\begin{proof}
First note that the residues of $\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)$
coincide with the residues of $R^\lambda_{A_\lambda,B_\lambda}(x)$.
The poles at $\lambda=-n-2k$ are simple. Hence its residue is given by
\begin{align*}
\operatorname{Res}_{\lambda=-n-2k}\big(\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)\big)
=\lim_{\lambda\to -n-2k}(\lambda+n+2k)\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi).
\end{align*}
From Equation \eqref{eq:IterationOfBernsteinSatoGeneralRiesz} and Theorem
\ref{FourierGeneralRiesz} we get
\begin{align*}
\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)
&=\frac{(-1)^k\mathcal{F}(D_{2k}(\lambda;n,p)R^{\lambda+2k}_{A_{\lambda+2k},B_{\lambda+2k}})(\xi)}{4^k(\frac{\lambda}{2})_k(\frac{\lambda+n}{2})_kC_{\lambda+2k}D_{\lambda+2k}}\\
&=\frac{(-1)^{k+1}2^{\lambda+2k+n-1}\pi^{\frac n2}\Gamma(\frac{\lambda+n+2k}{2})}{4^k (\frac{\lambda}{2})_k(\frac{\lambda+n}{2})_kC_{\lambda+2k}D_{\lambda+2k}\Gamma(-\frac{\lambda+2k-2}{2})}\\
&\times \Big[C_{\lambda}D_{\lambda+2k}(i_\xi\varepsilon_\xi)^k
+ C_{\lambda+2k}D_{\lambda}(\varepsilon_\xi i_\xi)^k\Big]R^{-\lambda-2k-n}_{C_{\lambda+2k}, D_{\lambda+2k}}(\xi).
\end{align*}
Combining the factor $(\lambda+n+2k)$ with the Gamma function $\Gamma(\frac{\lambda+n+2k}{2})$
and taking the limit $\lambda\to -n-2k$ gives
\begin{multline*}
\operatorname{Res}_{\lambda=-n-2k}\big(\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)\big)\\
=\frac{(-1)^{k+1}\pi^{\frac n2}}{4^k k! C_{-n}D_{-n}\Gamma(\frac{n+2k+2}{2})}\Big[C_{-n-2k}D_{-n}(i_\xi\varepsilon_\xi)^k
+ C_{-n}D_{-n-2k}(\varepsilon_\xi i_\xi)^k\Big]R^{0}_{C_{-n}, D_{-n}}(\xi),
\end{multline*}
which implies \eqref{eq:ResiduesGeneralRiesz}. The case when $\mathcal{F}(R^\lambda_{A_\lambda,B_\lambda})(\xi)$ acts on $0$-forms is an easy consequence.
The proof is complete.
\end{proof}
\begin{corollary}\label{ResiduesGeneralRieszWithAssumption}
Let $A_\lambda$ and $B_\lambda$ holomorphic in $\lambda$ such that $C_{-n}=D_{-n}$. Then the
residue of $R^\lambda_{A_\lambda,B_\lambda}(x)$ at $\lambda=-n-2k$, for $k\in\mathbb{N}_0$, is given by
\begin{align}
\operatorname{Res}_{\lambda=-n-2k}\big(R^\lambda_{A_\lambda,B_\lambda}(x)\big)
=\frac{(-1)^{k+1}\pi^{\frac n2}}{4^{k} k! \Gamma(\frac{n+2k+2}{2})}\Big[C_{-n-2k}(\deltad)^k
+ D_{-n-2k}(d\delta)^k\Big]\delta_0(x).
\end{align}
\end{corollary}
\begin{proof}
This is a direct consequence of Theorem \ref{ResiduesGeneralRiesz} and
\begin{align*}
R^{0}_{C_{-n}, D_{-n}}(\xi)=C_{-n}r^{-2}(\xi)(i_\xi\varepsilon_\xi+\varepsilon_\xi i_\xi)=C_{-n}\operatorname{Id}
\end{align*}
and $\mathcal{F}(\delta_0)(\xi)=\operatorname{Id}(\xi)$.
\end{proof}
The assumption in Corollary \ref{ResiduesGeneralRieszWithAssumption} is not empty.
For example, the pairs $(A_\lambda,B_\lambda)\stackrel{\text{def}}{=} (1,1), (1,-1)$ or
$(\alpha_\lambda,\beta_\lambda)$, where $\alpha_\lambda,\beta_\lambda$ are given by
\eqref{eq:BGCoefficients}, satisfy the condition $C_{-n}=D_{-n}$, cf. \eqref{eq:Coefficients1}.
\subsection{Convolutions}
Already mentioned before, the operation between function known as convolution is
important to define certain integral operators. It also is important in the theory of Fourier transform, since
the Fourier transform transforms convolutions into products.
We proceed with an identity concerning convolutions of $R^\lambda_{A_\lambda,B_\lambda}(x)$.
\begin{theorem}\label{ConvolutionIndentityGeneralRiesz}
Let $A_\lambda,B_\lambda$ such that $C_{2(\lambda-n)}C_{-2\lambda}=D_{2(\lambda-n)}D_{-2\lambda}$.
Then the distribution $R^\lambda_{A_\lambda,B_\lambda}(x)$ satisfies
\begin{align}\label{eq:ConvolutionIndentityGeneralRiesz}
R^{2(\lambda-n)}_{A_{2(\lambda-n)},B_{2(\lambda-n)}}*R^{-2\lambda}_{A_{-2\lambda},B_{-2\lambda}}(x)
=\pi^n \frac{\Gamma(\frac{2\lambda-n}{2})\Gamma(\frac{-2\lambda+n}{2})}{2^2\Gamma(-\lambda+n+1)\Gamma(\lambda+1)} C_{2(\lambda-n)}C_{-2\lambda}\delta_0(x).
\end{align}
\end{theorem}
\begin{proof}
By application of the Fourier transform, see Theorem \ref{FourierGeneralRiesz}, to \eqref{eq:ConvolutionIndentityGeneralRiesz} we obtain
\begin{multline*}
\mathcal{F}\big( R^{2(\lambda-n)}_{A_{2(\lambda-n)},B_{2(\lambda-n)}}*
R^{-2\lambda}_{A_{-2\lambda},B_{-2\lambda}}\big)(\xi)\\
=2^{-2}\pi^n\frac{\Gamma(\frac{2\lambda-n}{2})\Gamma(\frac{-2\lambda+n}{2})}{\Gamma(-\lambda+n+1)\Gamma(\lambda+1)}
r^{-4}(\xi)(C_{2(\lambda-n)}i_\xi\varepsilon_\xi
+D_{2(\lambda-n)}\varepsilon_\xi i_\xi)(C_{-2\lambda}i_\xi\varepsilon_\xi
+D_{-2\lambda}\varepsilon_\xi i_\xi).
\end{multline*}
Now the assumption $C_{2(\lambda-n)}C_{-2\lambda}=D_{2(\lambda-n)}D_{-2\lambda}$
and $i_\xi\varepsilon_\xi+\varepsilon_\xi i_\xi=r^2(\xi)$, cf. Lemma \ref{Identities1},
complete the proof.
\end{proof}
The assumption
$C_{2(\lambda-n)}C_{-2\lambda}=D_{2(\lambda-n)}D_{-2\lambda}$ is not empty.
For example take the values $(A_\lambda,B_\lambda)\stackrel{\text{def}}{=} (1,1),(1,-1)$
or $(\alpha_\lambda,\beta_\lambda)$, cf. \eqref{eq:BGCoefficients} . In each case the assumption
of Theorem \ref{ConvolutionIndentityGeneralRiesz} is satisfied.
A semi-group structure of $R^\lambda_{A_\lambda,B_\lambda}(x)$ in full generality is not possible, see Remark
\ref{Semi-GroupStructure} for an example and a general statement.
\subsection{Integral operators}
Integral transforms are widely used in mathematics. Most constructions of
integral operators based on integrating or taking
convolutions by kernel functions.
The Fourier transform is such an example, it arises by integration against $e^{i\langle x,\xi\rangle}$.
Also pseudo-differential operators arise as integral operators.
Special cases are differential operators, namely those having
polynomial kernels. Furthermore, some pseudo-differential operators have a certain
symmetry, for example the Knapp-Stein integral operators \cite{KnappStein} intertwine the corresponding
principal series.
Recall that $R^\lambda_{A_\lambda,B_\lambda}(x)$ is meromorphic in
$\lambda\in\mathbb{C}$ with only simple poles at $\lambda=-n-2\mathbb{N}_0$.
The integral operator
\begin{align}\label{eq:ConvolutionOperatorGeneralRiesz}
\mathbb{T}_{\nu,(A_{\nu},B_{\nu})}:\Omega^p(\mathbb{R}^n)\to \Omega^p(\mathbb{R}^n)
\end{align}
is defined as convolution operator with $R^\lambda_{A_\lambda,B_\lambda}(x)$, i.e.
\begin{align*}
(\mathbb{T}_{\nu,(A_{\nu},B_{\nu})}\omega)(x)
\stackrel{\text{def}}{=} \int_{\mathbb{R}^n} R^{2(\nu-n)}_{A_{2(\nu-n)},B_{2(\nu-n)}}(x-y)\omega(y)d y.
\end{align*}
These family $\mathbb{T}_{\nu,(A_{\nu},B_{\nu})}$ of pseudo-differential operators
is meromorphic in $\nu\in\mathbb{C}$ with only simple poles at $\nu=\frac n2-\mathbb{N}_0$. Now we describe their residues for
a class of integral operators.
\begin{theorem}\label{ResidueIntegralOperatorGeneralRieszAssumption}
Let $A_\lambda,B_\lambda$ holomorphic in $\lambda\in\mathbb{C}$ such that $C_{-n}=D_{-n}$. Then the residue
of $\mathbb{T}_{\nu,(A_{\nu},B_{\nu})}$ at $\nu=\frac n2-k$, for $k\in\mathbb{N}_0$, is
\begin{align}\label{eq:ResidueIntegralOperatorGeneralRieszAssumption}
\operatorname{Res}_{\nu=\frac n2-k}(\mathbb{T}_{\nu,(A_{\nu},B_{\nu})}\omega)
=\frac{(-1)^{k+1}\pi^{\frac n2}}{4^k k! \Gamma(\frac n2+k+1)}(C_{-n-2k}(\deltad)^k+D_{-n-2k}(d\delta)^k)\omega,
\end{align}
for $\omega\in\Omega^p(\mathbb{R}^n)$.
\end{theorem}
\begin{proof}
This is a direct consequence of Corollary \ref{ResiduesGeneralRieszWithAssumption}.
\end{proof}
\section{Further results and applications}
We focus on Riesz distributions $R^\lambda_{A_\lambda,B_\lambda}(x)$ for certain $A_\lambda,B_\lambda$,
and we show how to recover
the Knapp-Stein intertwining operators \cite{KnappStein} on functions and differential forms. Furthermore,
we discuss their relation to the well-known GJMS- and Branson-Gover
operators \cite{GJMS,BransonGover}, respectively.
\subsection{Knapp-Stein intertwining operator on functions}
We show that the distribution
\begin{align}
R^\lambda_{1,0}(x)=r^{\lambda-2}(x)i_x\varepsilon_x
\end{align}
when acting on $0$-forms
reduces to the Riesz distribution \eqref{eq:RieszDistribution}.
The Knapp-Stein intertwining operator on functions is
defined as convolution operator with the Riesz distribution.
Let us briefly recall the Riesz distribution \cite{Riesz, GelfandShilov}, which is given for $\lambda\in\mathbb{C}$ with $\Re(\lambda)>-n$ by
\begin{align*}
r^\lambda(x)=(x_1^2+\ldots+x_n^2)^{\frac{\lambda}{2}}.
\end{align*}
It extends to a tempered distribution meromorphic in $\lambda\in\mathbb{C}$ with
simple poles at $\lambda= -n-2k$ for $k\in\mathbb{N}_0$, and obeys the following properties:
\begin{itemize}
\item[(1)] Fourier transform: $\mathcal{F}(r^\lambda)(\xi)=2^{\lambda+n}\pi^{n/2}
\frac{\Gamma(\frac{\lambda+n}{2})}{\Gamma(-\frac{\lambda}{2})}r^{-\lambda-n}(\xi)$.
\item[(2)] Bernstein-Sato identity: $ \Delta(r^{\lambda+2}(x))=(\lambda+2)(\lambda+n)r^\lambda(x)$,
where $\Delta=\sum_i\partial_i^2$.
\item[(3)] Convolutionary inverse:
$(r^{2(\lambda-n)}*r^{-2\lambda})(x)=\pi^n\frac{\Gamma(\frac{2\lambda-n}{2})\Gamma(\frac{-2\lambda+n}{2})}{\Gamma(\lambda)\Gamma(n-\lambda)}\delta_0(x)$, where
$\delta_0(x)$ is the Dirac-distribution centered at the orign.
\item[(4)] Residues at $\lambda=-n-2\mathbb{N}_0$:
$\operatorname{Res}_{\lambda=-n-2k}(r^{\lambda}(x))=\frac{2\pi^{\frac n2}}{2^{2k}k!(\frac n2)_k\Gamma(\frac{n}{2})}\Delta^k \delta_0(x)$,
where $k\in\mathbb{N}_0$.
\end{itemize}
By Lemma \ref{Identities1} we obtain
\begin{align*}
R^\lambda_{1,0}(x)=r^{\lambda-2}(x)i_x\varepsilon_x=r^\lambda(x).
\end{align*}
Hence, $R^\lambda_{A_\lambda,B_\lambda}(x)$ is a generalization of $r^\lambda(x)$. Note that one could
keep the parameters $(A_\lambda,B_\lambda)$ different from $(1,0)$
which leads, as long as acting on functions,
to a different normalization of $r^{\lambda}(x)$ by the factor $A_\lambda$, since $B_\lambda$
has no contribution on functions due to $i_x f=0$ for any function $f$.
It is an easy exercise, using the parameters $C_\lambda=\lambda$ and
$D_\lambda=-n$ (again the latter one has no contribution on functions), cf. \eqref{eq:Coefficients1}, that
Theorems \ref{FourierGeneralRiesz}, \ref{BernsteinSatoGeneralRiesz},
\ref{ConvolutionIndentityGeneralRiesz} and \ref{ResiduesGeneralRiesz} specialize
for $R^\lambda_{1,0}(x)$ to (1)-(4) given above.
\begin{remark}
Note that we have only used the knowledge of the Fourier transform to achieve Theorem \ref{ResiduesGeneralRiesz}.
In \cite{GelfandShilov} a different
computation for $\operatorname{Res}_{\lambda=-n-2k}(r^\lambda(x))$ is
presented, which doesn't based on the Fourier transform.
\end{remark}
The integral operator \eqref{eq:ConvolutionOperatorGeneralRiesz} reduces to
\begin{align}\label{eq:KnappSteinForFunctions}
(\mathbb{T}_{\nu,(1,0)} f)(x)= \int_{\mathbb{R}^n}\abs{x-y}^{2(\nu-n)}f(y)dy,
\end{align}
which is exactly the Knapp-Stein intertwining operator.
Its major property is that it intertwines the corresponding principal series \cite{KnappStein}.
Furthermore, from Theorem \ref{ResidueIntegralOperatorGeneralRieszAssumption}
it follows that the residue of $\mathbb{T}_{\nu,(1,0)}$ at $\nu=\frac n2-k$, $k\in\mathbb{N}_0$,
is given by the differential operator
\begin{align*}
\operatorname{Res}_{\nu=\frac n2-k}(\mathbb{T}_{\nu,(1,0)})
= \frac{2\pi^{\frac n2}}{4^k k!\Gamma(\frac n2+k)}\Delta^k\delta_0.
\end{align*}
Note that our convention implies $\deltad=-\Delta$ when action on functions. Hence, residues of
$\mathbb{T}_{\nu,(1,0)}$ recover the well-known conformal powers of the Laplacian on function,
\cite{GJMS}.
\subsection{Knapp-Stein intertwining operator for differential forms}\label{SubsectionKSDF}
We next investigate the distribution
\begin{align}\label{eq:Riesz2}
R^\lambda_{1,-1}(x)=r^{\lambda-2}(x)(i_x\varepsilon_x-\varepsilon_x i_x).
\end{align}
We show that the residues of integral operators $\mathbb{T}_{\nu,(1,-1)}$ are the Branson-Gover operators of
order $2N\in\mathbb{N}$, cf. \cite{BransonGover}, which are given by
\begin{align}\label{eq:BGOperator}
L_{2N}^{(p)}=\alpha_N(\deltad)^N+\beta_N(d\delta)^N:\Omega^p(\mathbb{R}^n)\to\Omega^p(\mathbb{R}^n),
\end{align}
with coefficients
\begin{align}\label{eq:BGCoefficients}
\alpha_\lambda\stackrel{\text{def}}{=}\frac n2-p+\lambda,\quad \beta_\lambda\stackrel{\text{def}}{=}\frac n2-p-\lambda.
\end{align}
Furthermore, we extend the intertwining property of $L_{2N}^{(p)}$, see for example \cite{FJS2},
to $\mathbb{T}_{\nu,(1,-1)}$. We also could get the
intertwining property by a direct comparison of $\mathbb{T}_{\nu,(1,-1)}$ with the Knapp-Stein
intertwining operator studied by \cite{SpehVenkataramana}, see Remark
\ref{IntegralOperatorsComparision}. Finally we present an elementary
proof of positive-definiteness of scalar products induced by $\mathbb{T}_{\nu,(1,-1)}$,
cf. \cite{SpehVenkataramana}.
Now, for $(A_\lambda,B_\lambda)=(1,-1)$ the coefficients \eqref{eq:Coefficients1} are
\begin{align*}
C_\lambda=\lambda+2p=-2\alpha_{-\frac{\lambda+n}{2}},\quad
D_\lambda=-(\lambda+2n-2p)=-2\beta_{-\frac{\lambda+n}{2}}.
\end{align*}
From Theorems \ref{FourierGeneralRiesz}, \ref{BernsteinSatoGeneralRiesz},
\ref{ConvolutionIndentityGeneralRiesz} and Corollar \ref{ResiduesGeneralRieszWithAssumption}
we obtain the following corollaries.
\begin{corollary}\label{FourierRiesz2}
The Fourier transform of $R^\lambda_{1,-1}(x)$ is given by
\begin{align}\label{eq:FourierRiesz2}
\mathcal{F}(R^\lambda_{1,-1})(\xi)
=2^{\lambda+n}\pi^{\frac n2}\frac{\Gamma(\frac{\lambda+n}{2})}{\Gamma(-\frac{\lambda-2}{2})}
R^{-\lambda-n}_{\alpha_{-\frac{\lambda+n}{2}},\beta_{-\frac{\lambda+n}{2}}}(\xi).
\end{align}
\end{corollary}
\begin{corollary}\label{BernsteinSatoRiesz2}
The distribution $R^\lambda_{1,-1}(x)$ satisfies the following Bernstein-Sato identiy:
\begin{multline*}
\Big[(\lambda+2n-2p)(\lambda+2p-2)\deltad
+ (\lambda+2p)(\lambda+2n-2p-2)d\delta \Big]
R^{\lambda}_{1,-1}(x)\\
=-(\lambda-2)(\lambda+n-2)(\lambda+2p)(\lambda+2n-2p)R^{\lambda-2}_{1,-1}(x).
\end{multline*}
\end{corollary}
Especially, the last corollary implies
\begin{multline}\label{eq:IteratedBernsteinSatoRiesz2}
R^\lambda_{1,-1}(x)=\frac{(-1)^k}{4^k(\frac{\lambda}{2})_k(\frac{\lambda+n}{2})_k}
\Big[\frac{(\lambda+2p)}{(\lambda+2p+2k)}(\deltad)^k
+\frac{(\lambda+2n-2p)}{(\lambda+2n-2p+2k)} (d\delta)^k \Big]R^{\lambda+2k}_{1,-1}(x).
\end{multline}
\begin{corollary}\label{ResiduesRiesz2}
The residue of $R^\lambda_{1,-1}(x)$ at $\lambda=-n-2k$ for $k\in\mathbb{N}_0$ is
\begin{align}\label{eq:ResiduesRiesz2}
\operatorname{Res}_{\lambda=-n-2k}(R^\lambda_{1,-1}(x) )
=\frac{(-1)^k 2\pi^{\frac n2}}{4^{k} k! \Gamma(\frac n2+k+1)}
[\alpha_k (\deltad)^k+\beta_k (d\delta)^k]\delta_0(x),
\end{align}
which is up to a constant the Branson-Gover operator $L_{2k}^{(p)}$ acting on the Dirac-distribution.
\end{corollary}
\begin{corollary}\label{ConvolutionIndentityRiesz2}
The distribution $R^\lambda_{1,-1}(x)$ satisfies
\begin{align}\label{eq:ConvolutionIndentityRiesz2}
R^{2(\lambda-n)}_{1,-1}*R^{-2\lambda}_{1,-1}(x)
=\pi^n \frac{\Gamma(\frac{2\lambda-n}{2})\Gamma(\frac{-2\lambda+n}{2})}{\Gamma(-\lambda+n+1)\Gamma(\lambda+1)}\alpha_{\frac{-2\lambda+n}{2}}\alpha_{\frac{2\lambda-n}{2}}\delta_0(x)
\end{align}
\end{corollary}
The integral operator \eqref{eq:ConvolutionOperatorGeneralRiesz} specializes to
\begin{align}\label{eq:ConvolutionOperatorRiesz2}
(\mathbb{T}_{\nu,(1,-1)}\omega)(x)=\int_{\mathbb{R}^n}\abs{x-y}^{2(\nu-n-1)}
(i_{x-y}\varepsilon_{x-y}-\varepsilon_{x-y}i_{x-y})\omega(y)d y
\end{align}
for $\omega\in\Omega^p(\mathbb{R}^n)$. From Theorem \ref{ResidueIntegralOperatorGeneralRieszAssumption}
we obtain the following.
\begin{corollary}\label{ResiduesOfIntegralOperator}
The residue of $\mathbb{T}_{\nu,(1,-1)}$ at $\nu=\frac n2-k$, $k\in\mathbb{N}_0$, is given by
\begin{align}\label{eq:ResiduesOfIntegralOperator}
\operatorname{Res}_{\nu=\frac n2-k}\big( (\mathbb{T}_{\nu,(1,-1)}\omega)\big)
=\frac{(-1)^k2 \pi^{\frac n2}}{4^k k! \Gamma(\frac n2+k+1)}L_{2k}^{(p)}\omega,
\end{align}
where $\omega\in\Omega^p(\mathbb{R}^n)$.
\end{corollary}
Now we show the intertwining property of $\mathbb{T}_{\nu,(1,-1)}$ with respect to the
principal series, acting on differential forms.
For that let us introduce some notation.
Let $G=SO_0(n+1,1,\mathbb{R})$ the connected component of the group preserving the inner product
\begin{align*}
2x_0 x_{n+1}+x_1^2+\cdots+x_n^2
\end{align*}
on $\mathbb{R}^{n+2}$. Let $P_+\stackrel{\text{def}}{=} Stab(\mathbb{R} e_0)\subset G$ and
$P_-\stackrel{\text{def}}{=} Stab(\mathbb{R} e_{n+1})\subset G$ be the stabilizer of the line $\mathbb{R} e_0$, respectively $\mathbb{R} e_{n=1}$, where
$\{e_0,\ldots,e_{n+1}\}$ is the standard basis of $\mathbb{R}^{n+2}$. The groups $P_\pm$ are parabolic in $G$
and have Langlands decompositions $P_\pm=MAN_\pm$ with
$M\simeq SO(n,\mathbb{R})$, $A\simeq \mathbb{R}^+$ and $N_\pm\simeq \mathbb{R}^n$.
Corresponding Lie algebras are denoted by
$\mathfrak{g}(\mathbb{R})$ and
$\mathfrak{p}_\pm(\mathbb{R})=\mathfrak{m}(\mathbb{R})\oplus \mathfrak{a}(\mathbb{R})\oplus\mathfrak{n}_\pm(\mathbb{R})$ with
$\mathfrak{m}(\mathbb{R})\simeq \mathfrak{so}(n,\mathbb{R})$, $\mathfrak{a}(\mathbb{R})\simeq \mathbb{R}$ and
$\mathfrak{n}_\pm(\mathbb{R})\simeq \mathbb{R}^n$.
We define a representation of $P_+$ as an trivial extention to $N_+$ of the tensorproduct of the
standard representation $(\Lambda^p(\mathbb{R}^n)^*,\sigma_p)$ of $M$
and a $1$-dimensional represenation $(\mathbb{C}_\lambda,\xi_\lambda)$ of $A$,
i.e. $P_+$ acts on
$V_{\lambda,p}\stackrel{\text{def}}{=} \Lambda^p(\mathfrak{n}_-(\mathbb{R}))^*\otimes \mathbb{C}_{-\lambda}\simeq \Lambda^p(\mathbb{R}^n)^*\otimes \mathbb{C}_{-\lambda}$ by
\begin{align*}
\rho_{\lambda,p}:P_+=MAN_+&\to GL(V_{\lambda,p})\\
man&\mapsto\{(v\otimes 1)\mapsto \sigma_p(m)v\otimes a^{-\lambda}\}.
\end{align*}
Now, the space of sections $\Gamma(G\times_{P_+,\rho_{\lambda,p}}V_{\lambda,p})$ is
equivalent to the space of $P_+$-equivariant functions
\begin{align*}
\mathcal{C}^\infty(G,V_{\lambda,p})^{P_+}
=\{f:G\to V_{\lambda,p}\mid f(gp^\prime)=\rho_{\lambda,p}((p^\prime)^{-1})f(g)\quad\forall p^\prime\in P_+\}.
\end{align*}
Let us denote
by $\pi_{\lambda,p}$ the regular left representation
(strictly speaking anti-representation)
of $G$ on $\mathcal{C}^\infty(G,V_{\lambda,p})^{P_+}$, i.e.
$\pi_{\lambda,p}(g)f(g^\prime)\stackrel{\text{def}}{=} f(g\cdot g^\prime)$.
\begin{lem}\cite[Lemma $2.3.3$]{FJS2}
The infinitesimal action $d \pi_{\lambda,p}(E_j^+)$ on
$u\otimes \omega\in \mathcal{C}^\infty(\mathfrak{n}_-(\mathbb{R}))\otimes \Lambda^p(\mathfrak{n}_-(\mathbb{R}))^*\simeq\mathcal{C}^\infty(\mathbb{R}^n)\otimes \Lambda^p(\mathbb{R}^n)^*$ is given by
\begin{align*}
d \pi_{\lambda,p}(E_j^+)(u\otimes\omega)(x)
&=\Big(-\frac 12\sum_{k=1}^nx_k^2\partial_{x_j}+x_j(-\lambda
+\sum_{k=1}^n x_k\partial_{x_k})\Big)(u)\otimes \omega\\
&+\sum_{k=1}^nx_k u\otimes((E^+_j)^*\wedge i_{E_k^-}-(E^+_k)^*\wedge i_{E_j^-})(\omega),
\end{align*}
where $\{E^\pm_j\}$ denotes a basis of $\mathfrak{n}_\pm(\mathbb{R})\simeq \mathbb{R}^n$ and $\{(E^\pm_j)^*\}$ denotes its dual.
\end{lem}
In these conventions the
Branson-Gover operators \eqref{eq:BGOperator} satisfy the infinitesimal intertwining property
\begin{align}\label{eq:InvarianzBG}
d \pi_{-\frac{n}{2}-N,p}(X) L_{2N}^{(p)}= L_{2N}^{(p)}d\pi_{-\frac{n}{2}+N,p}(X),
\quad \forall\;X\in \mathfrak{so}(n+1,1,\mathbb{R}).
\end{align}
We show that the relation \eqref{eq:InvarianzBG}
extend to $\mathbb{T}_{\nu,(1,-1)}$.
\begin{prop}\label{IntertwiningRiesz2}
The intergral operators $\mathbb{T}_{\nu,(1,-1)}$ satisfy
\begin{align}\label{eq:IntertwiningRiesz2}
d \pi_{\nu-n,p}(X) \mathbb{T}_{\nu,(1,-1)}= \mathbb{T}_{\nu,(1,-1)}d\pi_{-\nu,p}(X),
\quad \forall\;X\in \mathfrak{so}(n+1,1,\mathbb{R}).
\end{align}
\end{prop}
\begin{proof}
We prove the claim in the Fourier image. First note the identity and definition
\begin{align*}
\mathcal{F}( d \pi_{\lambda,p}(X)\omega)
&=-i\Big(\frac 12 \xi_j\Delta_\xi -((\lambda+n)+\sum_{k=1}^n \xi_k\partial_{\xi_k})\partial_{\xi_j}\\
&\quad\quad+\sum_{k=1}^n\partial_{\xi_k}((E^+_j)^*\wedge i_{E_k^-}
-(E^+_k)^*\wedge i_{E_j^-}) \Big)\mathcal{F}(\omega)\\
&\stackrel{\text{def}}{=} D_2(\lambda,j)\mathcal{F}(\omega).
\end{align*}
Hence $D_2(\lambda,j)$ is a second order differential operator on differential forms.
In order to show \eqref{eq:IntertwiningRiesz2} it remains to verify that the difference of
\begin{align*}
\mathcal{F}( d \pi_{\nu-n,p}(X) \mathbb{T}_{\nu,(1,-1)} )(\xi)
&= D_2(\nu-n,j)\mathcal{F}(R^{2(\nu-n)}_{1,-1})(\xi)\mathcal{F}(\omega)(\xi)\\
&=c_1D_2(\nu-n,j)\big(( r^{-2\nu+n-2}(\xi)( \alpha_{\frac{-2\nu+n}{2}}i_\xi\varepsilon_\xi
+\beta_{\frac{-2\nu+n}{2}}\varepsilon_\xi i_\xi ))\mathcal{F}(\omega)(\xi) \big)
\end{align*}
and
\begin{align*}
\mathcal{F}( \mathbb{T}_{\nu,(1,-1)}d\pi_{-\nu,p}(X))(\xi)
&=\mathcal{F}(R^{2(\nu-n)}_{1,-1})(\xi)D_2(-\nu,j)\mathcal{F}(\omega)(\xi)\\
&=c_1 r^{-2\nu+n-2}(\xi)( \alpha_{\frac{-2\nu+n}{2}}i_\xi\varepsilon_\xi
+\beta_{\frac{-2\nu+n}{2}}\varepsilon_\xi i_\xi )D_2(-\nu,j)\mathcal{F}(\omega)(\xi).
\end{align*}
vanishes.
Here $c_1$ is the constant arising from the Fourier transform
of $R^{2(\nu-n)}_{1,-1}(x)$. Performing the differentiation with $D_2(\nu-n,j)$ and $D_2(-\nu,j)$,
canceling the common contribution of $r^\mu$ for appropriate $\mu$ and
expanding everything in powers of $\nu$, the difference become
\begin{align*}
P_3(\omega) \nu^3+P_2(\omega) \nu^2+P_1(\omega) \nu^1+P_0(\omega)=0,
\end{align*}
for some coefficients $P_i(\omega)$, $i=0,\ldots 3$, which are
differential operators acting on $\omega$. That it is of third degree in $\nu$ comes from the fact that
$D_2(\mu,j)$ is of second order containing a first order contribution with coefficient $\mu$ and
the coefficients $\alpha_\mu$ and $\beta_\mu$ are linear in $\mu$. Since the left-hand side
is a polynomial of order $3$, it remains to check
its vanishing at $4$ different point. Since Equation \eqref{eq:IntertwiningRiesz2}
is satisfied at all residues of $\mathbb{T}_{\nu,(1,-1)}$, that means at infinitely many,
due to Equation \eqref{eq:InvarianzBG}, the proof is complete.
\end{proof}
\begin{remark}\label{IntegralOperatorsComparision}
The intertwining property of the intergral operator $\mathbb{T}_{\nu,(1,-1)}$ was already studied in
\cite[Section $4$]{SpehVenkataramana}. Actually they have studied the
Knapp-Stein intertwining operator for differential forms. It just remains to show that our formula for
$\mathbb{T}_{\nu,(1,-1)}$ matches with that in \cite{SpehVenkataramana}.
This can be seen by noting that, when acting on a vector $Y$ in $\mathbb{R}^n$
(a dual $1$-form), our algebraic action
\begin{align*}
\frac{i_x\varepsilon_x-\varepsilon_x i_x}{\abs{x}^2}Y=Y-2\frac{\langle Y,x\rangle}{\abs{x}^2}x
\end{align*}
becomes a reflection of $Y$ in the hyperplane (through the origin) orthogonal to $x$, compare with
\cite[Lemma $2.1$]{SpehVenkataramana}. Note that this formula extends
naturally to $p$-forms. Note also that
our notation for the algebraic action enables us to compute explicitly the residues
of $\mathbb{T}_{\nu,(1,-1)}$, or even in a more general setting for $\mathbb{T}_{\nu,(A_\lambda,B_\lambda)}$.
\end{remark}
\begin{remark}\label{ComplementarySeries}
The intertwining operator $\mathbb{T}_{\nu,(1,-1)}$ defines an invariant pairing between
two induced representations in natural duality (via the natural $L^2$ pairing), namely
$\pi_{\nu - n, p}$ and $\pi_{-\nu,p}$ - here we consider real values of $\nu$, and
from Corollary \ref{FourierRiesz2} we see that the interval where this pairing is
positive-definite (i.e. the interval for the unitary complementary series) is
exactly $|\lambda| < (n/2) - p$, with $\lambda = \nu - (n/2)$.
Indeed, the invariant Hermitian form is on the Fourier transform side given as the
natural $L^2$ expression (the Fourier transform of $(\mathbb{T}_{\nu,(1,-1)}\omega, \omega)$)
\begin{align*}
2^{2\lambda}\pi^{n/2}\frac{\Gamma(\lambda)}{\Gamma((n/2) +1-\lambda)}
\int_{\mathbb{R}^n} |\xi|^{-2\lambda - 2}(((n/2) - p - \lambda) i_{\xi}\varepsilon_{\xi} + ((n/2) - p + \lambda)
\varepsilon_{\xi} i_{\xi})\widehat{\omega}, \widehat{\omega}) d\xi
\end{align*}
which we see as positive definite in the interval indicated; furthermore, for
$$0 < \lambda < (n/2) - p$$ the density here is locally integrable. Note that
in this case $\mathbb{T}_{\nu,(1,-1)}$ gives an equivalence between two unitary
representations in the complementary series.
This is consistent
with the formulas in \cite{BOO}, and gives a new and elementary proof of the size of
the unitary complementary series corresponding to $p$-forms. Indeed, in \cite{BOO}
the eigenvalues of the intertwining operator in its compact picture on the sphere
are shown to be labeled by integers $j \geq 1 $ and $q = 0,1$ (corresponding
to exact and co-exact differential forms respectively) as
$$Z(j,q,\lambda) = \frac{\Gamma((n/2) + j + \lambda)}{\Gamma((n/2) + j - \lambda)}
\frac{\Gamma((n/2) - p + q + \lambda)}{\Gamma((n/2) - p + q - \lambda)}$$
suitably normalized. The pair $(j,q)$ corresponds to the highest weight $$(j,1,1, ,\dots, 1, q, 0, ,\dots)$$
of an irreducible representation of $K = SO(n+1)$, and $\lambda$ is the same parameter as before.
Note that the ratio between this eigenvalue for $q = 0$ and $q = 1$ is exactly
$((n/2) - p - \lambda)/((n/2) - p + \lambda)$.
\end{remark}
Let us close this subsection with a comment about the impact of the Bernstein-Sato
identity, see Corollary \ref{BernsteinSatoRiesz2}, for $R^\lambda_{1,-1}(x)$ to
a recurrence relation among Branson-Gover operators. This recurrence relation can
also be directly obtained from \eqref{eq:BGOperator}. However, we want to demonstrate its appearence
from the Bernstein-Sato identity.
\begin{prop}
The Branson-Gover operators $L_{2N}^{(p)}$, for $N\in\mathbb{N}$, on $\mathbb{R}^n$ satisfy
\begin{align}\label{eq:RecurrenceOfBG}
L_{2N}^{(p)}=\Big(\frac{\alpha_{N}}{\alpha_{N-1}}\deltad +\frac{\beta_N}{\beta_{N-1}}d\delta \Big)
L_{2N-2}^{(p)}.
\end{align}
By convention we set $L_0^{(p)}\stackrel{\text{def}}{=} \alpha_0\operatorname{Id}$.
\end{prop}
\begin{proof}
First observe that Corollary \ref{BernsteinSatoRiesz2} implies
\begin{align*}
\mathbb{T}_{\nu,(1,-1)}\omega(x)&=\int_{\mathbb{R}^n} R^{2(\nu-n)}_{1,-1}(x-y)\omega(y)d y\\
&=\frac{1}{(2\nu-2n)(2\nu-n)}\Big(\frac{2\nu-2n+2p}{2\nu-2n+2p+2}\deltad
+\frac{2\nu-2p}{2\nu-2p+2}d\delta \Big)\\
&\times\int_{\mathbb{R}^n}R^{2(\nu-n+1)}_{1,-1}(x-y)\omega(y)d y
\end{align*}
Now we take the residue at $\nu=\frac n2-N$ for $N\in\mathbb{N}_0$ using
Equation \eqref{eq:ResiduesOfIntegralOperator}.
The residue of the left-hand side is
\begin{align*}
\frac{(-1)^N2\pi^{\frac n2}}{4^N N!\Gamma(\frac n2+N+1)}L_{2N}^{(p)},
\end{align*}
while the right-hand side has the residue
\begin{align*}
\frac{(-1)^{N}2\pi^{\frac n2}}{4^{N}N!\Gamma(\frac n2+N+1)}
\Big(\frac{n-2p+2N}{n-2p+2N-2}\deltad+\frac{n-2p-2N}{n-2p-2N+2}d\delta\Big)
L_{2N-2}^{(p)}.
\end{align*}
Now a cancelation of common factors completes the proof.
\end{proof}
\section{Some remarks and further applications}
This sections collect some observations concerning $R^\lambda_{A_\lambda,B_\lambda}(x)$.
\begin{remark}\label{Semi-GroupStructure}
The Riesz distribution $R^\lambda_{1,1}(x)$, when appropriately normalized, satisfies a semi-group
property, i.e. for
\begin{align*}
\bar{R}^\lambda_{1,1}(x)\stackrel{\text{def}}{=}\frac{\Gamma(-\frac{\lambda-n}{2})}{2^\lambda\pi^{\frac n2}\Gamma(\frac{\lambda}{2})} R^{\lambda-n}_{1,1}(x)
\end{align*}
we have, whenever it makes sense,
\begin{align*}
\bar{R}^\lambda_{1,1}*\bar{R}^\nu_{1,1}(x)=\bar{R}^{\lambda+\nu}_{1,1}(x).
\end{align*}
In order to prove a semi-group structure for the family of
Riesz distributions $R^\lambda_{A_\lambda,B_\lambda}(x)$ one has to allow
$A_\lambda,B_\lambda$ to be meromorphic in $\lambda\in\mathbb{C}$.
More precisely, define
\begin{align*}
\bar{R}^\lambda_{A_\lambda,B_\lambda}(x)\stackrel{\text{def}}{=}-\frac{\Gamma(-\frac{\lambda-n-2}{2})}{2^{\lambda-1}\pi^{\frac n2}\Gamma(\frac{\lambda}{2})} R^{\lambda-n}_{A_{\lambda-n},B_{\lambda-n}}(x),
\end{align*}
and show in the Fourier image that
\begin{align*}
\mathcal{F}(\bar{R}^\lambda_{A_\lambda,B_\lambda}*\bar{R}^\nu_{A^\prime_\nu,B^\prime_\nu})(\xi)
&=\mathcal{F}(\bar{R}^\lambda_{A_\lambda,B_\lambda})(\xi)
\mathcal{F}(\bar{R}^\nu_{A^\prime_\nu,B^\prime_\nu})(\xi)\\
&=r^{-(\lambda+\nu)-2}(\xi)(C_{\lambda-n}C^\prime_{\nu-n}i_\xi\varepsilon_\xi
+D_{\lambda-n}D^\prime_{\nu-n}\varepsilon_\xi i_\xi)
\end{align*}
and
\begin{align*}
\mathcal{F}(\bar{R}^{\lambda+\nu}_{A^{\prime\prime}_{\lambda+\nu},B^{\prime\prime}_{\lambda+\nu}})(\xi)
&=r^{-(\lambda+\nu)-2}(\xi)(C^{\prime\prime}_{\lambda+\nu-n}i_\xi\varepsilon_\xi
+D^{\prime\prime}_{\lambda+\nu-n}\varepsilon_\xi i_\xi)
\end{align*}
do agree iff $C^{\prime\prime}_{\lambda+\nu-n}=C_{\lambda-n}C^\prime_{\nu-n}$ and
$D^{\prime\prime}_{\lambda+\nu-n}=D_{\lambda-n}D^\prime_{\nu-n}$ hold. Note that a tuple $(A_\lambda,B_\lambda)$
will give a tuple $(C_\lambda,D_\lambda)$, see \eqref{eq:Coefficients1}. Eqivalently,
using \eqref{eq:Coefficients1} we have to solve
\begin{align*}
C_{\lambda-n}C^\prime_{\nu-n}=(\lambda+\nu-n+p)A^{\prime\prime}_{\lambda+\nu-n}
-p B^{\prime\prime}_{\lambda+\nu-n},\\
D_{\lambda-n}D^\prime_{\nu-n}=-(n-p)A^{\prime\prime}_{\lambda+\nu-n}
+(\lambda+\nu-p)B^{\prime\prime}_{\lambda+\nu-n}
\end{align*}
for $A^{\prime\prime}_{\lambda+\nu-n},B^{\prime\prime}_{\lambda+\nu-n}$. The unique solution is given by
\begin{align*}
A^{\prime\prime}_{\lambda+\nu-n}&\stackrel{\text{def}}{=} \frac{(\lambda+\nu-p)C_{\lambda-n}C^\prime_{\nu-n}+p D_{\lambda-n}D^\prime_{\nu-n}}{(\lambda+\nu)(\lambda+\nu-n)}\\
B^{\prime\prime}_{\lambda+\nu-n}&\stackrel{\text{def}}{=} \frac{(n-p)C_{\lambda-n}C^\prime_{\nu-n}+(\lambda+\nu-n+p) D_{\lambda-n}D^\prime_{\nu-n}}{(\lambda+\nu)(\lambda+\nu-n)}.
\end{align*}
Hence we obtain the semi-group property, whenever it makes sense,
\begin{align*}
\bar{R}^\lambda_{A_\lambda,B_\lambda}*\bar{R}^\nu_{A^\prime_\nu,B^\prime_\nu}
=\bar{R}^{\lambda+\nu}_{A^{\prime\prime}_{\lambda+\nu},B^{\prime\prime}_{\lambda+\nu}}.
\end{align*}
\end{remark}
\begin{remark}
We have seen in Theorem \ref{FourierGeneralRiesz} that the Fourier transform preserves the family of Riesz
distribution $R^\lambda_{\cdot,\cdot}(x)$. In general it holds that the pairs
$(A_\lambda,B_\lambda)$ and $(C_{\lambda},D_{\lambda})$, see \eqref{eq:Coefficients1}, have nothing in common.
However, for $A_\lambda\stackrel{\text{def}}{=} \alpha_\lambda$ and
$B_\lambda\stackrel{\text{def}}{=} \beta_\lambda$ we have $C_\lambda=-\lambda\alpha_{-\lambda-n}$ and
$D_\lambda=-\lambda\beta_{-\lambda-n}$, hence by Theorem \ref{FourierGeneralRiesz} we have
\begin{align*}
\mathcal{F}(R^\lambda_{\alpha_\lambda,\beta_\lambda})(\xi)
=c_\lambda R^{-\lambda-n}_{\alpha_{-\lambda-n},\beta_{-\lambda-n}}(\xi)
\end{align*}
for some constant $c_\lambda$.
\end{remark}
\begin{remark}
Similar operators to the
ones we study here have been considered in connection with
Ahlfors operators and generalized Beurling-Ahlfors operators,
see \cite{IM}; these arise in the study of quasi-regular mappings generalizing
quasi-conformal mappings in two dimensions. Of interest here are
new $L^p$ - estimates for the Beurling-Ahlfors operator on differential forms.
\end{remark}
\begin{remark}{(\bf Structure of GJMS operators)}
Let $(M,g)$ be a manifold. The GJMS operators, \cite{GJMS},
\begin{align*}
P_{2N}:\mathcal{C}^\infty(M)\to\mathcal{C}^\infty(M),
\end{align*}
arises as residues of the scattering operator
$S(s):\mathcal{C}^\infty(M)\to\mathcal{C}^\infty(M)$, \cite{GZ}.
The latter one is a generalization of the intertwining
integral operators $\mathbb{T}_{\nu,(1,0)}:\mathcal{C}^\infty(\mathbb{R}^n)\to\mathcal{C}^\infty(\mathbb{R}^n)$
considered on $0$-forms, see \eqref{eq:KnappSteinForFunctions}. Since each $P_{2N}$ is polynomial in
second order differential operators $M_{2k}$ \cite{J}, $k=0,\ldots,2N$, does this structure extend to $S(s)$?
Explicit formulae for those second order differential operators exist on flat and on Einstein manifolds.
\end{remark}
\begin{remark}{(\bf Structure of Branson-Gover operators and conformal powers of the Dirac operator)}
The Branson-Gover operators \cite{BransonGover}, again conjectural, and
conformal powers of the Dirac operator \cite{GMP1} are also residues of certain scattering operators.
Less is known about their structure. One might expect that some aspects of our
results here in the model case carry over to the case of Riemannian manifolds and conformal
geometry here; in particular the nature of a scattering operator should resemble our convolutions with Riesz kernels.
\end{remark}
\begin{remark}
It is clear that given the large amount of analysis based on the Riesz distributions on functions,
one may expect similar applications (e.g. fundamental solutions, wave equations, heat equations,
$L^p$-mapping properties)
of our formulas here; we plan to consider some of these
applications in another paper.
Along similar lines: can one study heat equations for the integral kernels $\mathbb{T}_{\nu,(A_\nu,B_\nu)}$?
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,701 |
Honor, Islam, and American women
The editors ask Akbar Ahmed about the number of American women who convert to Islam and how they are treated in both cultures.
What do converts to Islam tell you about Islam in America?
Four out of five white converts to Islam are women, and they tell us a lot about how Muslims and non-Muslims see each other.
I sympathize with the argument that a woman in Dearborn, Michigan dressed in the niqab (completely covered in black) and speaking only Arabic is bound to agitate a lot of Americans. And the conversion of white women to Islam would also agitate a lot of Americans.
American women are the freest women in the world. When many Americans see a Muslim woman in a hijab (head scarf), they see the boundaries she's putting around herself. Many Americans also have boundaries, but hers are more prominent. The hijab is a symbol of un-American dress and behavior. Why would a woman opt out of this freedom?
Nicole Queen is your classic case study. She was at the height of her career as a fashion photographer, standing next to Justin Timberlake, when she thought, "Life has got to be something more." She'd go to these clubs and see women prepared to take their clothes off, have sex, and expose themselves to everybody just to be photographed.
A spiritual awakening forces someone to either look at Catholicism more deeply or Judaism or, in Nicole's case, Islam. She felt that she needed these boundaries. Many Muslims worry that freedom has been taken to excess in the United States. Nicole is completely at ease with herself as an American. She has double pride as an American and a Texan.
Here's a great ambassador between Islam and America, I thought, but then she told me about the abuse she received. People called her a "whore" for "selling out" Christianity. A lot of the Muslims asked if she was an FBI informant.
She's very serene about it, though, and she has earned a lot of respect in the Muslim community. She has so much to give to Americans and to Muslims in trying to bridge this gap.
Is it fair to call the wild life that Nicole Queen rejected "American"? Isn't that like calling honor killings or female genital mutilation "Muslim"?
In a sense, the lifestyle isn't "American," but at the same time it is because you can't pretend it doesn't exist. Likewise you can't escape that honor killings and female circumcision are widespread practices in Muslim countries, though even Bernard Lewis, seen by many as a critic of Islam, says these are not Muslim practices.
My team of young people asked questions about American identity at Mardi Gras at 3 a.m. in New Orleans. Craig Considine, a Catholic boy from Boston, talked to this huge drunk guy on camera and this guy leaned across him and said, "You're not Muslim, are you? Because if you were, I'd have kicked you in the nuts." This is on camera. One girl said, "American? American means being f-ed up and being f-ed up again, having chicken wings and then being f-ed up again."
All evening and all morning this went on. We hit a nerve. Something is going on in the underbelly. I can't imagine any American family encouraging this, but this is happening. It's not mainstream America, but it is part of society and we can't pretend it doesn't exist.
This is precisely where you get the organic link between a Nicole Queen saying, "We have crossed the boundaries," and saying, "I want out. I want to do something else." She could have become a Catholic or a Jew or a Muslim.
Perhaps she found Islam because it has not adjusted or compromised. Obviously there's a theological reason—like St. Paul, a convert will suddenly see or hear God—but as an anthropologist I also look to society for clues.
This article appeared in the May 2011 issue of U.S. Catholic (Vol. 76, No. 5, page 31).
Created: Friday, April 15 2011 5:00 AM | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,691 |
\section{Introduction}
The way we live and work has been fundamentally changed by wireless communications over the past few decades. Thanks to the rapid development of the mobile Internet, billions of people all over the world are wirelessly connected through mobile phones for a wide range of daily activities, including social networking, information searching, etc. Historically, the prosperity of wireless communications has relied on its own model-based design paradigms, where accurate mathematical models and expert knowledge are required. However, the demanding requirements of the emerging applications, such as communicating under excessively complex scenarios with unknown channel models, low-latency requirement in large-scale super-dense networks \cite{FiveDis}, etc., are hard to be addressed by the traditional model-based wireless techniques.
To be more specific, when the communication scenario cannot be readily described mathematically,
either due to the excessively complex environment such as the underwater environment, or owing to the non-linearity imposed by the unavoidable hardware impairments \cite{OFDM},
the mathematical model based communication blocks, such as channel estimation, channel equalization, etc., will fail to accurately model the reality. Moreover, as emerging large-scale communication schemes (massive multiple-input multiple-output (MIMO) relying on a large number of antennas at the base station, massive Internet of things (IoT) scenarios connecting numerous users/devices, etc.) become more popular \cite{FiveDis}, the escalating complexity of their signal processing algorithms may preclude achieving low latency. Thus, new paradigms have to be introduced to address the above-mentioned challenges.
Deep learning (DL), mainly realized by deep neural networks (DNNs), has achieved impressive success with excellent results in diverse fields, such as image recognition \cite{Textbook}, mastering complex games like Go \cite{MasteringGame}, etc. This success has stimulated increasing interest in the application of DL in wireless communications.
Specifically, DL is a powerful tool capable of learning the intricate inter-relationships of variables, especially those that are hard to accurately describe using mathematical models \cite{Textbook}.
This enables us to design wireless communication systems without the knowledge of accurate mathematical models, which is impossible in the context of existing wireless design principles.
Moreover, for the family of light-weight DNNs having limited size, passing the inputs through them only requires a limited number of operations, which makes the DL methods computationally efficient. Even for highly complex DNNs which may be required for solving large-scale communication problems, the distributed parallel architectures and the accelerating tools of DL \cite{Textbook} are expected to result in a high computational efficiency. Thus, DNNs are attractive for solving large-scale wireless problems associated with numerous antennas/users/devices.
\begin{table*}
\centering
\caption{Fundamental differences between wireless transmission and DL}
\begin{tabular}{|l|c|c|}
\hline
& \tabincell{c}{Wireless transmission} & DL \\
\hline
Mathematical model & \tabincell{c}{Needs accurate mathematical model} & \tabincell{c}{Does not rely on accurate \\ mathematical model} \\
\hline
Design approach & \tabincell{c}{Optimizing each module separately \\ using mathematical derivation} & \tabincell{c}{Training the parameters \\ of the DNN as a whole} \\
\hline
Interpretation & Intuitive & Non-intuitive \\
\hline
\tabincell{c}{Generalization ability} & Widely applicable & Application-specific\\
\hline
Key challenges & \tabincell{c}{Unrealistic assumptions} & \tabincell{c}{Too many parameters} \\
\hline
\end{tabular}
\vspace{-3mm}
\label{difference}
\end{table*}
Inspired by the advantages mentioned above, DL has been widely used for wireless communications. For instance, the state-of-the-arts in utilizing DL for physical layer communications are summarized in \cite{R1}. Moreover, \cite{R2} and \cite{R3} comprehensively survey the applications of DL in designing IoT and 5G cellular networks at various layers of the protocol stack, respectively. In contrast to the above reviews which eloquently survey the relevant literatures, we provide guidance on how to apply DL for wireless communications by inducing a pair of design
methodologies, namely DL-based architecture design and DL-based algorithm design. Particularly, DL-based architecture design utilizes DL to reformulate the traditional block-based communication design principle. This methodology is firstly exemplified by DL-based receiver design, followed by the more revolutionary DL-based joint transceiver design. Moreover, DL-based algorithm design utilizes DL to speed up the algorithmic processing at a guaranteed performance. To further explain it, DL-based transmission algorithm design and DL-based optimization algorithm design are illustrated by several 5G-style examples.
Their principles, key features, and performance gains will be discussed to shed light on these methodologies. More importantly, the intricate interplay between DL and wireless communications is highlighted.
The rest of the paper is organized as follows. Section II briefly summarizes the relationship between wireless communications and DL. Sections III and IV detailedly investigate the intricacies of DL-based architecture design and DL-based algorithm design, respectively. Open problems and future research opportunities are provided in Section V, where the interplay between DL and wireless communications is further augmented. Finally, tangible conclusions and the further implications are offered in Section VI.
\vspace{-1mm}
\section{Different Paradigms of Wireless Communications and DL}
Although there is an explosive proliferation of utilizing DL in wireless communications, DL is very different from wireless communications, especially from wireless transmission, which is summarized in Table \ref{difference}. Specifically, wireless transmission relies on accurate (but often simplified) mathematical models, such as the Additive White Gaussian Noise (AWGN) channel model, to design channel estimation algorithms or channel feedback schemes, etc. However, DL usually does not rely on such mathematical models of its tasks, and is particularly beneficial in the absence of the accurate mathematical models.
Moreover, traditional wireless transmission tends to design each module of the communication system separately using mathematical derivations, while DL usually trains all the parameters of the DNN as a whole. As a benefit of their mathematical models, classic methods of wireless transmission are often intuitive with plausible explanations and are widely applicable. By contrast, those of DL are sometimes non-intuitive, may even be hard to interpret, and application-specific, which means that different neural networks have to be trained for different tasks. Besides, wireless transmission often relies on idealized and simplified assumptions in their mathematical model, while DL may have an excessive number of parameters, making its training process time-consuming.
Despite the significant differences between wireless transmission and DL, in this paper we focus on intrinsically amalgamating their benefit, while circumventing their weaknesses by portraying their profound interplay in the following sections. Specifically, in Section III we show that DL is capable of developing a new design paradigm for wireless transmission systems by introducing \cite{AnIntro}. Furthermore, in Section IV we demonstrate based on \cite{AMP} that the expert knowledge from wireless transmission can in turn help the development of DL techniques.
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{Fig_1.pdf}
\vspace*{-3mm}
\caption{From the traditional communication system to the DNN-based communication system: a) classic communication system consisting of many blocks; b) DNN-based communication system relying on one single block.}
\label{end_to_end}
\vspace*{-3mm}
\end{figure*}
\vspace{-2mm}
\section{DL-Based Architecture Design for Wireless Communications}
In this section, we focus on DL-based architecture design for intelligent wireless communication systems.
Following the order spanning from designing the receiver to designing the whole system,
we firstly introduce the DL-based receiver design, and then present the more revolutionary DL-based joint transceiver design. Particularly, DL-based receiver design invokes DL for jointly optimizing several
blocks of the receiver, which shows a beneficial performance gain in terms of the bit error rate (BER) when non-linear effects are encountered. By contrast, DL-based joint transceiver design optimizes the entire point-to-point communication system as
an end-to-end autoencoder, which results in a block error rate (BLER) that is lower than that of the
traditional system using binary phase shift keying (BPSK).
\vspace{-3mm}
\subsection{DL-Based Receiver Design}
For several decades, the block-based design principle has dominated wireless communication system design, where the communication system can be split into several independent functional blocks, including source coding, channel coding, etc., as shown in Fig. \ref{end_to_end} (a). Each block is optimized with the aid of mathematical model and expert knowledge, such as the optimal design of channel estimation which heavily relies on channel model and estimation theory. This block-based structure provides convenience for system building, but it cannot cope with excessively complex scenario when channel model is unknown.
To deal with the unknown channel model when non-linear noise is introduced, a DL-based orthogonal frequency division multiplexing (OFDM) receiver is proposed in \cite{OFDM} to learn the channel behaviors and decode the signals. Moreover, channel estimation, channel equalization, and channel decoding are jointly designed in the receiver using DNN, which yields an architectural revolution for wireless communication systems.
To be more specific, for OFDM systems suffering from a high peak-to-average-power-ratio (PAPR), peak-clipping and filtering
are usually used to alleviate the impact of PAPR. However, non-linear clipping noise will be introduced, which makes the equivalent channel more difficult to describe mathematically. Thus, mathematical model based channel estimation and equalization may become inaccurate, which imposes a certain performance loss during signal detection. To solve this problem, a fully connected DNN
is embedded into an OFDM receiver for signal detection in \cite{OFDM}, where the ReLU function is used as the activation function in the hidden
layers, and the sigmoid function is utilized in the last layer to map the result to the interval of [0,1]. In the training process, the original frequency-domain bit streams and their contaminated versions received after passing through the channel are used as labels and inputs, respectively. After that, the bit streams comprising the unknown frequency-domain signals are
fed into the trained DNN, and a threshold is applied to finally reconstruct the transmitted bits. Simulation results show that once the non-linear effects have been introduced by clipping, the DL-based OFDM receiver exhibits a beneficial performance gain, provided that the signal-to-noise-ratio (SNR) is in excess of 15 dB \cite{OFDM}.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{Fig_2.pdf}
\vspace*{-1em}
\caption{Block error rate (BLER) comparison between the block-based communication system and the DNN-based communication system.}
\label{end_to_end_result}
\vspace*{-5mm}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{Fig_3.pdf}
\caption{The $t^{th}$ layer of the neural network utilized for DL-based AMP algorithm.
}
\label{AMP}
\vspace{-1em}
\end{figure*}
\vspace{-3mm}
\subsection{DL-Based Transceiver Design}
In contrast to the DL-based receiver design which only optimizes the receiver using DNN, a more revolutionary paradigm has been proposed recently to jointly design the whole point-to-point communication system (including both the transmitter and the receiver) as a single end-to-end autoencoder under the DL framework \cite{AnIntro}, which is completely different from the classic block-based design principle. This completely new paradigm of DNN-based communication system design is inspired by the resemblance of wireless communication systems and the autoencoder, both of which aim at equating the input message $s$ and the output message $s^\prime$. The DNN-based communication system is shown in Fig. \ref{end_to_end} (b), where the transmitted signal $s$ corresponds to one of the $k$-bit symbols from an alphabet, which is encoded as an $M$-dimensional one-hot vector ($M=2^k$) and used as the input of the autoencoder. After passing through the transmitter composed of a series of hidden layers and a normalization layer for meeting the energy constraint, an $n$-dimensional vector $\bm{x}$ is generated as the encoded signal. Then, a Gaussian noise layer serves as the AWGN channel to contaminate the original signal $\bm{x}$ to become $\bm{y}$, which will be fed into the receiver having multiple hidden layers and terminated by a softmax layer to output the reconstruction probabilities of all possible symbols. Finally, the symbol associated with the highest probability will be chosen as the recovered signal $s^\prime$.
As illustrated in Fig. \ref{end_to_end_result}, simulation results generated using Tensorflow \cite{Textbook} show that
the DNN-based communication system outperforms the traditional system using BPSK in terms of its BLER \cite{AnIntro}, because the new scheme is capable of jointly optimizing the communication system as a whole rather than optimizing individual blocks in Fig. \ref{end_to_end} (a) in isolation.
This new design paradigm allows us to possibly replace all baseband components of the block-based communication system of Fig. \ref{end_to_end} (a) by a single DNN of Fig. \ref{end_to_end} (b) , which can be trained by exploiting the seemingly infinite computation and storage resources of the cloud.
\vspace*{-1mm}
\section{DL-Based Algorithm Design for Wireless Communications}
Due to its convenient implementation relying on parallel architectures and the powerful learning capability, DL can also be used to speed up the algorithmic processing at a potentially improved performance and reduced latency. In this section, we will introduce the methodology of DL-based algorithm design for wireless communications, which will be illustrated from two aspects, i.e., DL-based transmission algorithm design and DL-based optimization algorithm design, using several typical techniques conceived for 5G and beyond, including mmWave massive MIMO, LDPC, NOMA, and UDN.
\setcounter{subsection}{0}
\vspace*{-3mm}
\subsection{DL-Based Transmission Algorithm Design}
Since reliable transmission is one of the most important tasks in wireless communications, we will introduce DL-based transmission algorithm design in this part. Following the order of signal processing, i.e., steps commencing from the inner receiver to the outer receiver, channel estimation is carried out before channel decoding.
Therefore, we investigate DL-based sparse channel estimation for mmWave massive MIMO first, and then introduce DL-based belief propagation for LDPC decoding, both of which constitute typical 5G-style techniques.
\subsubsection{{DL-Based Sparse Channel Estimation for MmWave Massive MIMO}}
MmWave massive MIMO is one of the most promising techniques in 5G conceived for largely increasing the data rates \cite{MIMO,Xiuhong}. It requires accurate CSI to realize multi-user precoding, decoding, etc. Since the mmWave channel is sparse in the angular domain \cite{MIMO,Xiuhong}, compressive sensing algorithms have been widely used for sparse mmWave channel estimation, such as approximate message passing (AMP). However, AMP requires many iterations of linear residual updating and non-linear shrinkage operation to converge, which makes it difficult to
guarantee both reliable performance and reduced latency at the same
time.
To address this issue, a DL-based AMP algorithm has been proposed in \cite{AMP} to significantly reduce the number of iterations required under a specific performance constraint. As shown in Fig. \ref{AMP}, each iteration of the AMP algorithm is unfolded to a single layer of the DNN, where the linear residual updating is accurately modeled by the linear operations of the DNN, and the non-linear shrinkage operation is realized by the activation function in the DNN.
Additionally, unlike the classical AMP algorithm which fixes the parameters of the shrinkage operation in advance, the DL-based AMP actually learns those parameters during the training process. It has been shown by simulation results that the DL-based AMP converges faster and outperforms both the AMP and the iterative shrinkage/threshold algorithm (ISTA) in terms of its average normalized mean square error (NMSE). The improved performance is partly attributed to the introduction of the Onsager correction into AMP \cite{AMP}, and partly contributed by the DNN-aided learning of optimal shrinkage parameters.
Thus, expert knowledge gleaned from other fields may in turn help to enhance the DL performance. More importantly, the authors of \cite{AMP} revealed basic principles of designing new DNN architectures suitable for wireless communications by unfolding the iterations of the existing algorithms into layers of a DNN. This is especially useful when the complexity of the existing iterative algorithms is excessive for practical implementation of massive communications with large-scale signal processing.
\subsubsection{DL-Based Belief Propagation for LDPC Decoding}
In 2017, low-density parity-check (LDPC) coding has been chosen to replace the 4G turbo coding as the new channel coding scheme in the most important broadband mode of the 5G standard. This can be attributed to the excellent performance of LDPC coding operating close to the Shannon limit in AWGN channels. However, due to the detection effect of filtering, oversampling, and multi-user interference in practical systems \cite{BPCNN}, the received signals may become contaminated by colored noise, which is hard to mathematically model and would impose an obvious performance loss on the LDPC decoding performance.
To solve this problem, Liang \textit{et al.} \cite{BPCNN} modified the widely used belief propagation (BP) algorithm by cascading a Convolutional Neural Network (CNN) to the standard BP decoder, which succeeds in outputting a more accurate noise estimate from the decoding result of the BP decoder. In the BP-CNN decoder, the received signal $y$ will firstly be fed into the standard BP decoder to obtain an initial estimate of the transmitted signal $\hat{x}$, which will then be subtracted from $y$ to obtain the estimated noise $\hat{n}=y-\hat{x}$. The following trained CNN exploits this estimated noise $\hat{n}$ and outputs a more accurate noise estimate $\tilde{n}$, which will be fed back to the BP decoder and then subtracted from the received signal $y$ to obtain a more ``pure signal'' $\tilde{y}$. A new round of operations will be applied to the newly generated $\tilde{y}$, and iterating between BP and CNN as mentioned above can finally achieve an improved LDPC decoding performance, which mitigates the adverse effects imported by the colored noise.
As for the simulation results, the total number of BP iterations is set to 50 in \cite{BPCNN} for both the standard BP and the improved BP-CNN, and the latter outperforms the former in terms of its BER. Therefore, to achieve the same BER performance, the BP-CNN requires a much lower number of iterations than the standard BP, indicating a reduced latency for channel decoding. Moreover, since the improved BP-CNN does not rely on any specific channel coding scheme or channel model, its design principle may also be applied to other codes, such as turbo codes and BCH codes, which suggests that the physical layer can be quite flexibly designed.
\vspace{-3mm}
\subsection{DL-Based Optimization Algorithm Design}
Optimization plays an important role in wireless communication systems to realize efficient exploitation of limited radio resources. However, many optimization algorithms require a large number of iterations to converge, which results in both high complexity and high latency, especially when the problem's scale is very large.
In this part, we will show that DL can be used to speed up processing while maintaining reliable performance by introducing two works. Particularly, following the order spanning from deterministic networks to random networks, DL-based sum rate maximization for NOMA with predetermined transceivers is introduced first, followed by the introduction of the DL-based energy consumption minimization for UDN, where the transceivers are randomly paired according to the distance, link quality, etc. The two above-mentioned solutions also constitute typical 5G technologies.
\subsubsection{{DL-Based Sum Rate Maximization for NOMA}}
Power allocation plays an essential role in the emerging 5G NOMA solutions \cite{NOMA}, and it has attracted extensive attention in recent years. To solve this problem, a series of optimization algorithms have been proposed, among which the weighted minimum mean square error (WMMSE) is quite popular \cite{NOMA}. However, since optimization algorithms like WMMSE usually require a large number of iterations to converge, it is difficult for them to achieve the requirement of low-latency.
To deal with this challenge, a DL-based optimization approach is proposed in \cite{LearningTo} to accelerate processing, while maintaining reliable performance in an interference-contaminated wireless scenario supporting $K$ single-antenna transceiver pairs. Furthermore, the power of each pair of the transceivers is limited by a certain constraint, and accordingly a sum rate optimization problem is formulated under this constraint.
To solve this optimization problem faster, Sun \textit{et al.} \cite{LearningTo} interpreted the complex WMMSE algorithm as an unknown non-linear mapping between its inputs (system parameters) and outputs (power allocation results), and used a DNN having multiple hidden layers to mimic the WMMSE algorithm as accurately as possible, exploiting its high computational efficiency in its testing process for promptly finding a beneficial power allocation solution. Furthermore, the ReLU function \cite{Textbook} is used as the activation function, and the MMSE is calculated as the loss function. The channel coefficients obeying the classic Rayleigh distribution and the power
allocation results generated by WMMSE are adopted as data and labels to train the DNN. Moreover, the simulation time comparison of the standard WMMSE algorithm and the DL-based optimization method is presented in \cite{LearningTo}. All simulation codes of these two methods are written in Python and run on the same computer. It is observed that the DL-based method is capable of reducing the execution time by a factor of several hundreds compared to the standard WMMSE algorithm, while maintaining a similar sum rate performance. For example, 97.92\% of the max sum rate can be achieved when supporting 10 users. Moreover, as the number of users increases from 10 to 30, the execution time of the standard WMMSE method increased from 12.32s to 78.06s, while that of the DL-based method grows steadily from 0.04s to 0.09s, which shows a modest complexity increase of the DL-based method for large-scale problems.
\subsubsection{{DL-Based Energy Consumption Minimization for UDN}}
UDN is another promising technology for 5G communication systems, where a large number of access points (APs) are used to provide high data rates. Each AP can set up a transmission link to another one within its transmission range, where minimizing the energy consumption of dozens of APs while satisfying the flow demand is very important.
In order to cope with the associated large-scale optimization problem, diverse methods have been proposed to reduce the complexity, but very few approaches have been designed for directly reducing the problem scale. To this end, a DL-based algorithm is proposed in \cite{Optimization} to reduce the
problem scale, while the optimality of the solution can be maintained at the same time. Since there are always a number of transmission links with no flow passing through in the optimal case, a DNN with multiple hidden layers is utilized to predict those ``deactivated links'' and eliminate them from the original large-scale optimization problem, thus directly reducing the problem scale. To be more specific, the flow demand vectors are used as input, and the flows of the links calculated by the so-called delay column generation (DCG) \cite{Optimization} applied to the original optimization problem are used as labels to train the DNN. In the testing stage, after obtaining the values of flows passing through the links, a threshold is applied to eliminate ``deactivated links'' in preparation for the problem reformulation. Finally, the DCG will then be used again for the scale-reduced optimization problem to obtain the final results. The simulation results of \cite{Optimization} show that the prediction is quite accurate, and the optimization problem can be solved within at most half of
the original time, while the performance difference
between the original large-scale problem and the scale-reduced one is not larger than 3\%.
It should be pointed out that the idea of the elimination-based method advocated in \cite{Optimization} does not rely on any specific scenario or algorithm, hence it can also be extended to reduce the scale of other optimization problems by excluding zero variables.
\vspace*{-2mm}
\section{Challenges and Research Opportunities}
Although DL-based methods proposed for wireless communications in the previous two sections have shown encouraging advantages over their classic counterparts, they are still in their infancy, and there are many challenging issues remaining for further study. In this section, we underline the important challenges in this emerging area from four perspectives, namely theoretical challenges, data-related challenges, algorithmic challenges, and implementation-oriented challenges. The corresponding potential solutions and research opportunities are also highlighted. Furthermore, we believe that the study of DL-based wireless communications can also help to promote the development of DL itself by addressing these challenges.
\setcounter{subsection}{0}
\vspace*{-3mm}
\subsection{Theoretical Challenges}
\subsubsection{Performance Analysis}
In contrast to the classical methods of wireless communications under the umbrella of Shannon's information theory, DL-based methods lack solid mathematical foundations in terms of theoretical analysis, which is however desired to guide the practical design of DL-based wireless communications and provide insights into their performance limits.
Provided that the training data obeys a certain distribution, Shannon information theory may still contribute to the theoretical basis of DL. But again, this is an open problem requiring further exploration.
\subsubsection{Interpretability}
The mathematical or physical interpretation of DL-based methods is in its infancy. For instance, the selection of training strategies remains somewhat haphazard without clear physical explanation, and we cannot easily make plausible why a certain network structure attains a good performance. Since the training process itself is actually an optimization problem, advanced optimization theory is expected to help us find the most appropriate loss function and training strategy. Moreover, as indicated by the DL-based AMP algorithm of \cite{AMP}, the expert knowledge gleaned from other fields (such as wireless communications) may in turn help the design of more efficient DNN architectures for wireless communications.
\vspace*{-3mm}
\subsection{Data-Related Challenges}
The amount and quality of training data are essential for the final performance of DL.
However, acquiring sufficiently high-quality data for training in real wireless communication systems is not as straightforward as in data-oriented applications, which is mainly due to the following two reasons. Firstly, how to generate a large amount of data remains a challenge. For example, transmitting a large packet of pilots as labeled data is very difficult in wireless environments due to high spectrum efficiency requirement. Secondly, data in wireless communications are usually high-dimensional (e.g., massive MIMO channels) and highly heterogeneous (since diverse services associated with significantly different quality of service requirements have to be supported). Hence, pre-processing and cleaning of the available data will be challenging. Thus, a mature data processing flow should be developed from data collection to pre-processing. Moreover, as DL is known to be inherently data-driven, different data sets used for training and testing may result in different performance. To ensure fair comparison, a promising solution is to build some widely accepted and low-cost data sets to test different DL-based algorithms for wireless communications.
\vspace*{-3mm}
\subsection{Algorithmic Challenges}
Communication scenarios are rather complicated and usually time-varying. Thus, the data collected may not be able to cover all situations, or to reflect all changes in a timely manner.
For instance, DL is not directly suitable for dealing with time-varying environments. To be more specific, if the wireless channel changes rapidly, the DL-based wireless systems may have to be frequently and completely re-trained from scratch to maintain their performance over time, which is time consuming and computationally expensive. Thus, DNN-based methods have to be robust and versatile in similar but hitherto uncovered situations, and new learning algorithms should be designed. For example, transfer learning \cite{Textbook} may be a possible solution, which enable DNN to learn from similar cases so as to be also applicable to new scenarios, thus achieving better environmental adaptability and robustness. We believe that the study of DL for such scenarios may also promote its development in terms of both theory and algorithmic innovation.
\vspace{-3mm}
\subsection{Implementation-Oriented Challenges}
Apart from the challenges listed above, deploying DL-based methods in real communication systems will face implementation-oriented challenges. Firstly, most of the communication infrastructure, such as the base stations, are only equipped with radio frequency (RF) and signal processing functionality. To implement DL-based methods, a cloud server should be deployed based on the existing infrastructure. Secondly, since the classic communication methodologies may not be completely replaced by the DL-based techniques in a short time, a soft switching scheme is needed to switch between the DL-based methods and the non DL-based methods.
We may need a new architecture to support the harmonious co-existence of DL-based methods and the classic non DL-based methods.
\vspace{-1mm}
\section{Conclusions}
In this review, we have revealed a pair of
dominant methodologies for the applications of DL in wireless communications, namely DL-based
architecture design and DL-based algorithm design. We have also analyzed their design principles, key features, as well as performance gains, and demonstrated that DL is indeed capable of supporting communication systems in complex operating scenarios and speeding up large-scale processing with guaranteed performance. Moreover, we have pointed out some challenges and research opportunities in this emerging area. In summary, we believe that further study of this area can also help to promote the
development of DL itself in theory and algorithmic innovation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,209 |
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\section{Introduction}
Many aspects of harmonic analysis on Euclidean $n$-space, the $n$-sphere and
$n$-hyperbolic space
are related to the action of the conformal group. This is true not only for
functions, but also for spinors and differential forms. Motivated partly by
representation theory and partly by conformal differential geometry, there has
recently been much progress in establishing concrete and explicit formulas
for natural integral and differential operators exhibiting some form of
conformal invariance.
Here a key concept is that of
A. Juhl's residue families \cite{J1} in the framework of conformal geometry and
T. Kobayashi's symmetry-breaking
operators \cite{KS} in representation theory.
In this paper we collect and extend many formulas related to distribution kernels
for both integral and differential operators in the three basic situations
of functions, spinors, and differential forms. We shall treat both the absolute
case of the conformal group, as well as the relative case of the conformal
groups of Euclidean space and a coordinate hyperplane, i.e. the case
of conformal symmetry-breaking
operators. As it turns out, there is a series of natural identities based
on some partly new second-order operators which are termed in the present
article {\it Bernstein-Sato operators}.
Among the very useful and important ingredients in the theory of meromorphic
continuation of families of distributions and the closely related theory of
${\fam2 D}$-modules are the Bernstein-Sato identities, see the seminal paper
\cite{Bernstein} by J. Bernstein. In particular, they allow
to find the precise position of the corresponding poles as well as
to encode a recurrence structure for the residues in the distribution family under consideration.
In recent years there appeared several approaches to a classification scheme for
conformally covariant differential operators $P_{2N}, \slashed{D}_{2N+1}$ and $L^{(p)}_{2N}$
(acting on functions, spinors and differential $p$-forms)
on semi-Riemannian (spin-)manifolds, cf. \cite{GJMS, GMP1, BransonGover}.
Furthermore, the operators $P_{2N}$ and $\slashed{D}_{2N+1}$ were extended to a
theory of $1$-parameter families of conformally covariant differential
operators \cite{J1,FS}, nowdays known and termed as the {\it residue families}.
These correspond to the relative case. Concerning
differential forms, not much is so far known and available in the literature.
The above mentioned operators ($P_{2N}, \slashed{D}_{2N+1}$ and $L^{(p)}_{2N}$) on Euclidean space $\R^n$ arise as residues of Knapp-Stein
intertwining integral operators for certain families of induced representations of
conformal Lie groups, cf. \cite{KnappStein}.
These operators are $1$-parameter families of pseudo-differential convolution operators with
respect to Riesz distributions on functions (\cite{Riesz}), spinors (\cite{CO}) and differential forms
(\cite{FO}), respectively.
One may also study Knapp-Stein type operators associated to a pair of conformal Lie groups
(the relative case); these form a $2$-parameter
family of distributions, see \cite{KS, MO, K2}.
They are termed
{\it conformal symmetry breaking operators}, and are intertwining integral operators
acting on principal series representations (realized for example in the
non-compact picture) for the action of conformal Lie algebras
on $\R^n$ and $\R^{n-1}$, respectively. Their residues are
given by $1$-parameter families of equivariant differential operators termed {\it conformal symmetry breaking differential operators}.
Note that these conformal symmetry breaking differential operators are just specific cases of residue family operators. The conformal symmetry breaking operators were studied in a more general context, e.g. the case
of conformal Lie groups for $\R^n$ and $\R^{n-m}$ with $1\leq m\leq n-1$ is discussed in \cite{MO1}.
However, curved generalizations of the residue families, both of co-dimension
one, and of higher co-dimensions, are not yet
properly understood. One of the motivations for the present paper is to gain a better understanding
of the model (flat) case in order to undertake a subsequent study of the curved generalizations, related
to AdS/CFT and the corresponding Poincare-Einstein geometry.
Thus in the present paper, we prove Bernstein-Sato identities for distribution kernels of the three basic types of
conformal symmetry breaking operators: scalar-valued (acting on $\mathcal{D}^\prime(\R^n)$)
\begin{align}
K^+_{\lambda,\nu}(x^\prime,x_n)&= \abs{x_n}^{\lambda+\nu-n}(\abs{x^\prime}^2+x_n^2)^{-\nu},\notag\\
K^-_{\lambda,\nu}(x^\prime,x_n)&= \sgn(x_n)\abs{x_n}^{\lambda+\nu-n}(\abs{x^\prime}^2+x_n^2)^{-\nu},
\end{align}
spinor-valued (acting on $\slashed{\mathcal{D}}^\prime(\R^n)$)
\begin{align}
\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)&= K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n) x\cdot,
\end{align}
and differential form-valued (acting on $\mathcal{D}^{\prime, p}(\R^n)$)
\begin{align}
K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)= K^\pm_{\lambda-1,\nu+1}(x^\prime,x_n)(i_x\varepsilon_x-\varepsilon_x i_x)i_{e_n}\varepsilon_{e_n}.
\end{align}
We use the notation $\lambda,\nu\in\C$, $x=(x^\prime,x_n)\in \R^n$ with $x^\prime\in \R^{n-1}$,
$x\cdot$ for the Clifford multiplication with the vector $x\in\R^n$, and $i_x,\varepsilon_x$ are
the interior and exterior products with respect to $x$ on differential forms.
Our convention for the distribution kernels are as duals to what they are
considered as in the standard
literature mentioned above. This choice is justified by the use of Fourier transform, which is applied to
these distribution kernels directly without further dualization and leads to generalizations
of singular vectors studied in \cite{KOSS, FJS, KKP}.
The proposal to find Bernstein-Sato operators of interest in the context of conformal
symmetry breaking operators was initiated in \cite{PS}, where certain shift operators
for Gegenbauer polynomials regarded as the residues of Fourier transformed
$K^\pm_{\lambda,\nu}(x^\prime,x_n)$ are studied. Later on, a more sophisticated approach was
suggested in a private communication by J.L. Clerc. Moreover, the
Bernstein-Sato operators are themselves intertwining operators for the relevant
conformal Lie groups. Concerning detailed notation
and intertwining results we refer to \cite{C1,C}.
Our paper is structured as follows. In Section \ref{Preliminaries}, we fix the
notation and recall fairly standard results related to Riesz distributions
on functions, spinors and differential forms.
In Section \ref{BSOperatorsAndIdentities}, we present a construction of
Bernstein-Sato type operators for functions
$P(\lambda)$, \eqref{eq:BSOperator1}, spinors $\slashed{P}(\lambda)$,
\eqref{eq:BSOperatorSpinor1}, and differential forms $P^p(\lambda)$,
\eqref{eq:BSOperatorForDiffForms}. Furthermore,
we show that they satisfy a Bernstein-Sato identity on the space of
distribution kernels for functions in Theorem \ref{BSGeneralizedRiesz1},
spinors in Theorem \ref{BSGeneralizedRiesz1Spinor}, and differential forms
in Theorem \ref{BSIdentityForDiffForms}, respectively.
In Section \ref{ComAndApps}, we comment on the origin
of the constructed Bernstein-Sato operators (it is interesting that they
have several, a priori quite different definitions)
and discuss some applications to the conformal symmetry breaking differential
operators on functions, spinors and differential forms.
The construction results in {\it known} formulas for conformal symmetry breaking differential
operators on functions, cf. Theorem \ref{BSFamily}, and {\it new} formulas for
conformal symmetry breaking differential operators on spinors and differential forms,
see Theorems \ref{BSVsSBOSpinor} and \ref{BSVsSBODiffForms}.
{\bf Acknowlegment:} We would like to express our thanks to J.L. Clerc for the private
discussion leading to the present paper.
\section{Preliminaries}\label{Preliminaries}
Let $\R^n$ be equipped with the canonical
flat metric $\langle\cdot,\cdot\rangle$. We collect some basic
known facts concerning
tree types of Riesz distributions: for scalars \cite{Riesz,GelfandShilov}, spinors \cite{CO}
and differential forms \cite{FO}.
We denote by $\mathcal{S}(\R^n)$ the algebra of Schwartz functions on $\R^n$ and
follow the convention for the Fourier transform
\cite{GelfandShilov} on Schwartz functions $f\in\mathcal{S}(\R^n)$:
\begin{align*}
\mathcal{F}(f)(\xi)\st \int_{\R^n} f(x)e^{i \langle x, \xi\rangle}dx,
\end{align*}
which also extends to the space of tempered distributions $\mathcal{S}^\prime(\R^n)$.
Note the identity
\begin{align*}
\mathcal{F}(f\star g)(\xi)=\mathcal{F}(f)(\xi)\mathcal{F}(g)(\xi),
\end{align*}
where $(f\star g)(x)\st \int_{\R^n}f(x-y)g(y)dy$ denotes the convolution of Schwartz functions $f$ and $g$.
This normalization of the Fourier transform is chosen in such a way that
$\mathcal{F}(\delta_0)=1$, where $\delta_0$ is the Dirac distribution centered at the origin.
Recall that for a polynomial $P$ in $n$ variables we have the identities
\begin{align}
P(\partial_{\xi_1},\ldots,\partial_{\xi_n})\mathcal{F}(f)(\xi)
&=\mathcal{F}(P(i x_1,\ldots,i x_n)f)(\xi),\notag\\
\mathcal{F}(P(\partial_{x_1},\ldots,\partial_{x_n})f)(\xi)
&=P(-i \xi_1,\ldots,-i \xi_n)\mathcal{F}(f)(\xi)\label{eq:FourierProperties2}
\end{align}
for $f\in\mathcal{S}(\R^n)$.
The Fourier transform $\mathcal{F}$ extends to the space of spinor-valued
and differential forms-valued Schwartz functions (as well as the tempered distributions), i.e.,
$\slashed{\mathcal{S}}(\R^n)$ and $\mathcal{S}^p(\R^n)$ ($\slashed{\mathcal{S}}^\prime(\R^n)$ and $\mathcal{S}^{\prime p}(\R^n)$),
and will be denoted by $\mathcal{F}$ as well.
\subsection{Riesz distribution}
Let $x\in \R^n$. The classical {\it Riesz distribution} \cite{Riesz,GelfandShilov} is defined by
\begin{align}\label{eq:ClassicalRiesz}
r^\lambda(x)\st (x_1^2+\ldots+x_n^2)^{\frac{\lambda}{2}}=\abs{x}^\lambda,
\end{align}
where $\lambda\in\C$. It is an analytic function in the complex half-plane
$\Re(\lambda)>-n$. Due to the Bernstein-Sato identity
\begin{align}\label{eq:BSClassicalRiesz}
\Delta r^{\lambda+2}(x)=(\lambda+2)(\lambda+n) r^\lambda(x),
\end{align}
where $\Delta=\sum\limits_{k=1}^n \partial_k^2$, the meromorphic continuation
(with simple poles at $\lambda=-n-2k$ for $k\in\N_0$)
of $r^\lambda(x)$ to $\lambda\in\C$ follows. Let us introduce a meromorphic function
\begin{align}\label{eq:Constant}
c_\lambda\st 2^{\lambda+n}\pi^{\frac n2}\Gamma(\frac{\lambda+n}{2})\Gamma(-\frac{\lambda}{2})^{-1},
\end{align}
and the standard notation for the Pochhammer symbol
\begin{align*}
(a)_n\st a\cdot (a+1)\cdot\ldots\cdot(a+n-1)
\end{align*}
for $n\in\N$ and $a\in\C$. Then a classical result states
\begin{prop}\label{FourierClassicalRiesz}
The Fourier transformation of $r^\lambda(x)$ is given by
\begin{align*}
\mathcal{F}(r^\lambda)(\xi)=c_\lambda r^{-\lambda-n}(\xi).
\end{align*}
\end{prop}
Based on the Bernstein-Sato identity \eqref{eq:BSClassicalRiesz} and a knowledge of the residue
for $r^\lambda(x)$ at $\lambda=-n$, see
\cite{GelfandShilov}, we get immediately
\begin{corollary}
The residue of $r^\lambda(x)$ at $\lambda=-n-2k$ for $k\in\N_0$ is given by
\begin{align*}
\Res_{\lambda=-n-2k}(r^\lambda(x))= \frac{2\pi^{\frac n2}}{4^k k!\Gamma(\frac n2)(\frac n2)_k} \Delta^k \delta_0(x).
\end{align*}
\end{corollary}
Consequently, the residues of $r^\lambda(x)$ are related to GJMS operators
$P_{2N}=\Delta^N$ on $(\R^n,\langle\cdot,\cdot\rangle)$ for any $N\in\N_0$.
\subsection{Riesz distribution for spinors}
We proceed with the Riesz distribution for spinors on $(\R^n,\langle\cdot,\cdot\rangle)$, see \cite{CO} .
We write $\mathbb{S}_n^\pm$ for the irreducible half-spin representations for even $n$
and $\mathbb{S}_n$ for the irreducible spin representation in the case of odd $n$. Then it holds
that $\mathbb{S}_n^\pm\simeq \mathbb{S}_{n-1}$ for even $n$, while
$\mathbb{S}_n\simeq \mathbb{S}_{n-1}^+\oplus \mathbb{S}_{n-1}^-$ for odd $n$.
Let $\Sigma_n$ be the spinor bundle of $(\R^n,\langle\cdot,\cdot\rangle)$ associated to
the spin representation $\mathbb{S}_n$ for odd $n$, respectively $\mathbb{S}_n^+$ for even $n$.
The Clifford multiplication $\cdot$ is normalized by
$x\cdot y+y\cdot x=-2\langle x,y\rangle$ for $x,y\in\R^n$.
The action of the Dirac operator on spinor fields $\varphi\in\Gamma(\Sigma_n)$ is locally, with respect to the standard basis $\{e_k\}$ of $\R^n$, given by
\begin{align*}
\slashed{D}\varphi=\sum_{k=1}^n e_k\cdot \partial_k\varphi.
\end{align*}
We use the identification of a point $x\in\R^n$ with a vector in $\R^n$, and
define the $\End(\Sigma_n)$-valued distribution called the {\it Riesz distribution for spinors}:
\begin{align}\label{eq:ClassicalRieszSpinor}
\slashed{r}^\lambda(x)\st r^{\lambda-1}(x) x\cdot=r^\lambda(x)\frac{x}{\abs{x}}\cdot.
\end{align}
In the region $\Re(\lambda)>-n$,
\eqref{eq:ClassicalRieszSpinor} is an analytic function and satisfies
the Bernstein-Sato identity
\begin{align}\label{eq:BSClassicalRieszSpinor}
\Delta\slashed{r}^{\lambda+2}(x)=(\lambda+1)(\lambda+n+1)\slashed{r}^{\lambda}(x).
\end{align}
This follows from $\slashed{D}\slashed{r}^\lambda(x)=-(\lambda+n-1)r^{\lambda-1}(x)$,
$\slashed{D}r^{\lambda-1}(x)=(\lambda-1)\slashed{r}^{\lambda-2}(x)$ and $\slashed{D}^2=-\Delta$.
In turn, the equation \eqref{eq:BSClassicalRieszSpinor} implies meromorphic continuation of $\slashed{r}^\lambda(x)$
to the complex plane $\C$ with simple poles at $\lambda=-n-1-2k$ for $k\in\N_0$.
The Fourier transform preserves the family of Riesz distributions for spinors.
\begin{prop}\label{FourierRieszSpinor}
The Fourier transform of $\slashed{r}^\lambda(x)$ is given by
\begin{align}\label{FourierRieszSpinorequality}
\mathcal{F}(\slashed{r}^\lambda)(\xi)=\slashed{c}_\lambda \slashed{r}^{-\lambda-n}(\xi),
\end{align}
where $\slashed{c}_\lambda \st -i\frac{c_{\lambda+1}}{\lambda+1}$.
\end{prop}
Since we could not find its proof in the literature, we shall supply it here.
\begin{proof}
Starting on the right side of \eqref{FourierRieszSpinorequality} and using
Proposition \ref{FourierClassicalRiesz} together with the fact
that $\xi\cdot \mathcal{F}(\varphi)(\xi)=i\mathcal{F}(\slashed{D}\varphi)(\xi)$, we compute
\begin{align*}
\slashed{r}^{-\lambda-n}(\xi)\mathcal{F}(\varphi)(\xi)
&=r^{-\lambda-n-1}(\xi)\xi\cdot \mathcal{F}(\varphi)(\xi)\\
&=i(c_{\lambda+1})^{-1}\mathcal{F}(r^{\lambda+1})(\xi)\mathcal{F}(\slashed{D}\varphi)\\
&=i(c_{\lambda+1})^{-1}\mathcal{F}(\int_{\R^n}r^{\lambda+1}(x-y)\slashed{D}\varphi \dm y) .
\end{align*}
We choose a scalar product on $\langle\cdot ,\cdot \rangle_{\Sigma_n}$ on spinors, and also a constant spinor $\phi$.
Then we have
\begin{align*}
\langle\phi,r^{\lambda+1}(x-y)\slashed{D}\varphi\rangle_{\Sigma_n}
=\langle\phi, (\lambda+1)r^{\lambda-1}(x-y)\sum_{j=1}^n(x_j-y_j)e_j\cdot\varphi\rangle_{\Sigma_n},
\end{align*}
and therefore
\begin{align*}
\mathcal{F}(\int_{\R^n}r^{\lambda+1}(x-y)\slashed{D}\varphi \dm y)
= (\lambda+1)\mathcal{F}(\int_{\R^n}r^{\lambda-1}(x-y)\sum_{j=1}^n(x_j-y_j)e_j\cdot\varphi \dm y).
\end{align*}
The proof is complete.
\end{proof}
Finally, we recall the residues of $\slashed{r}^\lambda(x)$, see \cite[Proposition $6.3$]{CO}, which correspond
to (odd) conformal powers of the Dirac operator $\slashed{D}_{2N+1}=\slashed{D}^{2N+1}$ on $(\R^n,\langle\cdot,\cdot\rangle)$ for any $N\in\N_0$.
\begin{prop}
The residue of $\slashed{r}^\lambda(x)$ at $\lambda=-n-2k-1$, for $k\in\N$, is given by
\begin{align*}
\Res_{\lambda=-n-1-2k}(\slashed{r}^{\lambda}(x))= \frac{2\pi^{\frac n2}}{4^k k!\Gamma(\frac n2)(\frac n2)_k} \slashed{D}^{2k+1}\delta_0(x).
\end{align*}
\end{prop}
\subsection{Riesz distribution for differential forms}
We consider differential forms on $\R^n$. As in the previous section a point $x\in \R^n$
is also regarded as a vector.
The inner and exterior products with respect to the vector $x$ are denoted by
\begin{align*}
i_x&\st\sum_{k=1}^n x_k i_{e_k},\quad \varepsilon_x\st \sum_{k=1}^n x_k \varepsilon_{e_k},
\end{align*}
respectively.
The exterior differential, its co-differential and the form Laplacian act on
differential $p$-forms $\Omega^p(\R^n)$ by
\begin{align*}
\dm\st\sum_{k=1}^n \varepsilon_{e_k}\partial_k,\quad \delta\st-\sum_{k=1}^n i_{e_k} \partial_k,\quad
\Delta_p\st \dm\delta+\delta\dm=-\Delta,
\end{align*}
while similar operators on $\Omega^p(\R^{n-1})$ are denoted by $\dm^\prime,\delta^\prime$ and $\Delta_p^\prime$, respectively.
Now, the Riesz distribution on differential forms \cite{FO} is defined by
\begin{align}\label{eq:FormRiesz}
R_p^\lambda(x)\st r^{\lambda-2}(x)(i_x\varepsilon_x-\varepsilon_x i_x).
\end{align}
In the region $\Re(\lambda)>-n$ of $\mathbb{C}$ it is an analytic function
and satisfies the following Bernstein-Sato identity
\begin{align}\label{eq:BSForFormRiesz}
\Big[(\lambda+2n-2p)(\lambda+2p-2)\delta\dm
+ (\lambda+2p)(\lambda+2n-2p-2)\dm\delta \Big]
R_p^{\lambda}(x)&\\
=-(\lambda-2)(\lambda+n-2)(\lambda+2p)(\lambda+2n-2p)R^{\lambda-2}_p(x)&.
\end{align}
This implies the meromorphic continuation of $R_p^{\lambda}(x)$ to $\lambda\in\C$ with simple poles at $\lambda=-n-2k$ for
$k\in\N_0$. We shall introduce
\begin{align}\label{eq:BGCoeff}
\alpha_\lambda\st \frac n2-p+\lambda,\quad \beta_\lambda \st \frac n2-p-\lambda
\quad \mbox{for}\quad \lambda\in{\mathbb{C}} ,
\end{align}
which are related to Branson-Gover operators $L_{2N}^{(p)}= \alpha_N(\delta\dm)^N+\beta_N(\dm\delta)^N$
on $(\R^n,\langle\cdot,\cdot\rangle)$ for any $N\in\N$.
\begin{prop}\label{FourierRieszForm}
The Fourier transform of $R_p^{\lambda}(x)$ is given by
\begin{align*}
\mathcal{F}(R_p^\lambda)(\xi)
= \bar{c}_\lambda r^{-\lambda-n-2}(\xi)(\alpha_{-\frac{\lambda+n}{2}}i_\xi\varepsilon_\xi
+\beta_{-\frac{\lambda+n}{2}}\varepsilon_\xi i_\xi),
\end{align*}
where $\bar{c}_\lambda\st (\lambda-1)(\lambda-2)c_{\lambda}$.
\end{prop}
Finally, we recall that the residues of $R_p^{\lambda}(x)$ correspond
to the Branson-Gover operators on $\R^n$.
\begin{prop}\label{ResiduesForFormRiesz}
Let $k\in\N_0$. Then the residue of $R_p^{\lambda}(x)$ at $\lambda=-n-2k$ is given by
\begin{align*}
\Res_{\lambda=-n-2k}(R^\lambda_p(x) )
=\frac{(-1)^k 2\pi^{\frac n2}}{4^{k} k! \Gamma(\frac n2+k+1)}
[\alpha_k (\delta\dm)^k+\beta_k (\dm\delta)^k]\delta_0(x).
\end{align*}
\end{prop}
\section{Bernstein-Sato identity and operator}\label{BSOperatorsAndIdentities}
In the present section we shall prove some Bernstein-Sato identities for distribution kernels associated to conformal
symmetry breaking operators \cite{KS,MO,K2}. By an abuse of notation, we introduce these
distribution kernels as adjoints to those appearing
in the references. The main impact of this choice is that taking Fourier transform
of these distribution kernels leads to a direct
contact (without any further dualisation) with a generalized version of singular
vectors studied in \cite{KOSS,FJS,KKP}.
\subsection{Bernstein-Sato identity and operator in the scalar case}\label{TheScalarCase}
In this section we prove Bernstein-Sato identity for the distribution kernels associated to
conformal symmetry breaking operators acting on functions:
\begin{align}\label{eq:ScalarCoDimOneRiesz}
K^+_{\lambda,\nu}(x^\prime,x_n)&\st \abs{x_n}^{\lambda+\nu-n}(\abs{x^\prime}^2+x_n^2)^{-\nu},\notag\\
K^-_{\lambda,\nu}(x^\prime,x_n)&\st \sgn(x_n)\abs{x_n}^{\lambda+\nu-n}(\abs{x^\prime}^2+x_n^2)^{-\nu}=x_n K^+_{\lambda-1,\nu}(x^\prime,x_n).
\end{align}
For a detailed analysis of their meromorphic behavior with respect to
$(\lambda,\nu)\in\C^2$, see \cite{KS,MO}.
A method of finding Bernstein-Sato operators, which we follow and which we briefly
recall, is based on the discussion in \cite{C1,C}.
The Knapp-Stein intertwining operator for conformal Lie group, acting on density
bundle induced from the character $\gamma$, is given by
\begin{align}
(I_\gamma f)(x)\st (r^{-2\gamma}\star f)(x)= \int_{\R^n} r^{-2\gamma}(x-y)f(y) \dm y,\label{eq:Knapp-Stein-Scalar}
\end{align}
where $f\in\mathcal{S}(\R^n)$.
It follows from Proposition \ref{FourierClassicalRiesz} that
\begin{align*}
I_{n-\lambda}\circ I_{\lambda}=c_{2\lambda-2n}c_{-2\lambda} \id.
\end{align*}
Furthermore, we define the multiplication operator
\begin{align}\label{eq:MultOpScalar}
(M_{x_n}f)(x)\st x_n f(x).
\end{align}
\begin{remark}\label{IntertwiningProp}
We note that both $I_\gamma$ and $M_{x_n}$ are intertwining
operators for the conformal Lie groups on $\R^n$ and $\R^{n-1}$, respectively.
For more details we refer to \cite{C}.
\end{remark}
Now we define the operator
\begin{align}\label{eq:ClercTrickScalar}
D(\lambda)\st I_{\lambda+1}\circ M_{x_n}\circ I_{n-\lambda}
\end{align}
The next statement is remarkable due to the fact that \eqref{eq:ClercTrickScalar}
is a composition of pseudo-differential operators, cf. Clerc \cite{C}.
\begin{prop}\label{ScalarClercTrick}
The operator $D(\lambda)$ in \eqref{eq:ClercTrickScalar} is a differential operator of order $2$, i.e.,
\begin{align*}
D(\lambda)f=-\tilde{c}_\lambda\big[(2\lambda-n)\partial_n f+\Delta(x_n\cdot f)\big]
\end{align*}
for any $f\in \mathcal{S}(\R^n)$ and the multiple
$\tilde{c}_\lambda\st c_{-2\lambda-2}c_{2\lambda-2n}$ (cf., \eqref{eq:Constant}.)
\end{prop}
We recall its proof.
\begin{proof}
In the Fourier image, we compute
\begin{align*}
\mathcal{F}(D(\lambda)f)(\xi)&=\mathcal{F}(I_{\lambda+1}\circ M_{x_n}\circ I_{n-\lambda}f)(\xi)\\
&=-i c_{-2\lambda-2}c_{2\lambda-2n}r^{2\lambda+2-n}(\xi)\partial_n\big[r^{n-2\lambda}(\xi)\mathcal{F}(f)(\xi) \big].
\end{align*}
The identity
\begin{align*}
\partial_nr^{n-2\lambda}(\xi)=(n-2\lambda)\xi_n r^{n-2\lambda-2}(\xi)
\end{align*}
then implies
\begin{align*}
\mathcal{F}(D(\lambda)f)(\xi)&=-i \tilde{c}_\lambda r^{2\lambda+2-n}(\xi)\big[(n-2\lambda)\xi_nr^{n-2\lambda-2}(\xi)+r^{n-2\lambda}(\xi)\partial_n\big]\mathcal{F}(f)(\xi)\\
&=\tilde{c}_\lambda\mathcal{F}\big((n-2\lambda)\partial_n f-\Delta(x_n\cdot f) \big)(\xi),
\end{align*}
which completes the proof.
\end{proof}
By virtue of Proposition \ref{ScalarClercTrick}, we define the second order differential operator on tempered distributions
(notice the shift of the parameter $\lambda$)
\begin{align*}
P(\lambda):\mathcal{S}^\prime(\R^n)&\to \mathcal{S}^\prime(\R^n)\\
f&\mapsto \Delta(x_n\cdot f)+(n-2\lambda)\partial_n f.
\end{align*}
Note that by Leibniz's rule we have
\begin{align}\label{eq:BSOperator1}
P(\lambda)=x_n\Delta -(2\lambda-n-2)\partial_n.
\end{align}
\begin{remark}\label{IntertwiningProp1}
The operator $P(\lambda)$, acting on tempered distributions on $\R^n$, is an intertwining differential operator for the conformal Lie
group on $\R^{n-1}$, cf. Remark \ref{IntertwiningProp}. The same holds for its iterations used in later sections.
\end{remark}
By the identities in \cite{GelfandShilov},
\begin{align*}
\partial_n(\abs{x_n}^\lambda)&=\lambda\sgn(x_n)\abs{x_n}^{\lambda-1},\\
\partial_n(\sgn(x_n)\abs{x_n}^\lambda)&=\lambda\abs{x_n}^{\lambda-1},
\end{align*}
a straightforward computation reveals the following result.
\begin{lem}\label{ScalarLemma}
The distributions $K^\pm_{\lambda,\nu}(x^\prime,x_n)$ satisfy
\begin{enumerate}
\item
\begin{align*}
x_nK^\pm_{\lambda,\nu}(x^\prime,x_n)=K^\mp_{\lambda+1,\nu}(x^\prime,x_n),
\end{align*}
\item
\begin{align*}
\partial_{n} (K^\pm_{\lambda,\nu}(x^\prime,x_n))
=(\lambda+\nu-n)K^\mp_{\lambda-1,\nu}(x^\prime,x_n)-2\nu K^\mp_{\lambda,\nu+1}(x^\prime,x_n),
\end{align*}
\item
\begin{align*}
\partial_{i} (K^\pm_{\lambda,\nu}(x^\prime,x_n))=-2\nu x_i K^\pm_{\lambda-1,\nu+1}(x^\prime,x_n)\quad 1\leq i\leq n-1,
\end{align*}
\item
\begin{align*}
\Delta( K^\pm_{\lambda,\nu}(x^\prime,x_n))
= &\, (\lambda+\nu-n-1)_2K^\pm_{\lambda-2,\nu}(x^\prime,x_n)\\
- &\, 2\nu(2\lambda-n-2)K^\pm_{\lambda-1,\nu+1}(x^\prime,x_n).
\end{align*}
\end{enumerate}
\end{lem}
From this Lemma we may conclude
\begin{theorem}\label{BSGeneralizedRiesz1}
The operator $P(\lambda)$ is a spectral shift operator for
distribution kernels $K^\pm_{\lambda,\nu}(x^\prime,x_n)$, i.e.,
\begin{align}\label{eq:BSGeneralizedRiesz1}
P(\lambda)K^\pm_{\lambda,\nu}(x^\prime,x_n)
=(\lambda+\nu-n)(\nu-\lambda+1)K^\mp_{\lambda-1,\nu}(x^\prime,x_n),
\end{align}
and
\begin{align}\label{eq:BSGeneralizedRiesz2}
P(\frac{\lambda+\nu+1}{2})K^\pm_{\lambda,\nu}(x^\prime,x_n)
=2\nu(\nu-\lambda+1)K^\mp_{\lambda,\nu+1}(x^\prime,x_n).
\end{align}
\end{theorem}
\begin{proof}
From Lemma \ref{ScalarLemma} we obtain
\begin{align*}
(n-2\lambda^\prime)\partial_n(K^\pm_{\lambda,\nu}(x^\prime,x_n))
&= (n-2\lambda^\prime)(\lambda+\nu-n)K^\mp_{\lambda-1,\nu}(x^\prime,x_n)
-2\nu(n-2\lambda^\prime)K^\mp_{\lambda,\nu+1}(x^\prime,x_n),\\
\Delta(x_n K^\pm_{\lambda,\nu}(x^\prime,x_n))
&=(\lambda+\nu-n)_2K^\mp_{\lambda-1,\nu}(x^\prime,x_n)-2\nu(2\lambda-n)K^\mp_{\lambda,\nu+1}(x^\prime,x_n),
\end{align*}
hence
\begin{align*}
P(\lambda^\prime)K^\pm_{\lambda,\nu}(x^\prime,x_n)
&=(\lambda+\nu-n)(-2\lambda^\prime+\lambda+\nu+1)K^\mp_{\lambda-1,\nu}(x^\prime,x_n)\\
&-2\nu(2\lambda+2\lambda^\prime-2n)K^\mp_{\lambda,\nu+1}(x^\prime,x_n).
\end{align*}
Then for $\lambda^\prime\st \lambda$ we conclude
\begin{align*}
P(\lambda)K^\pm_{\lambda,\nu}(x^\prime,x_n)
=(\lambda+\nu-n)(\nu-\lambda+1)K^\mp_{\lambda-1,\nu}(x^\prime,x_n),
\end{align*}
while for $\lambda^\prime\st\frac{\lambda+\nu+1}{2}$ we get
\begin{align*}
P(\frac{\lambda+\nu+1}{2})K^\pm_{\lambda,\nu}(x^\prime,x_n)
=2\nu(\nu-\lambda+1)K^\mp_{\lambda,\nu+1}(x^\prime,x_n).
\end{align*}
The proof is complete.
\end{proof}
\begin{remark}
Assuming the coefficients $A,B$ by
$\partial_n$ and $\Delta(x_n\cdot)$
in the formula \eqref{eq:BSOperator1}
are not known, i.e., $\tilde{P}(\lambda)\st A\partial_n+B \Delta(x_n\cdot)$.
Then the system of equations
\begin{align*}
(\lambda+\nu-n)A+(\lambda+\nu-n)_2B&=(\lambda+\nu-n)(\nu-\lambda+1),\\
-2\nu A-2\nu(2\lambda-n)B&=0,
\end{align*}
which is equivalent to
$\tilde{P}(\lambda)K^\pm_{\lambda,\nu}(x^\prime,x_n)=(\lambda+\nu-n)(\nu-\lambda+1)K^\mp_{\lambda-1,\nu}(x^\prime,x_n)$,
has a unique solution given by
\begin{align*}
A\st n-2\lambda,\quad B\st 1.
\end{align*}
This agrees with Theorem \ref{BSGeneralizedRiesz1}, i.e., $\tilde{P}(\lambda)=P(\lambda)$.
\end{remark}
\begin{remark}
We notice that $K^+_{\lambda,\nu}(x^\prime,x_n)$ generalizes the Riesz distribution $r^\lambda(x)$,
\eqref{eq:ClassicalRiesz}, as follows. Once we set $\lambda\st \frac{\mu}{2}+n$ and
$\nu\st -\frac{\mu}{2}$, for $\mu\in\C$ such that $\Re(\mu)>-n$, we get
\begin{align}\label{eq:help1}
K^+_{\frac{\mu}{2}+n,-\frac{\mu}{2}}(x^\prime,x_n)
=\abs{x_n}^{\frac{\mu}{2}+n-\frac{\mu}{2}-n}(\abs{x^\prime}^2+x_n^2)^{\frac{\mu}{2}}=r^\mu(x).
\end{align}
Then Theorem \eqref{eq:BSGeneralizedRiesz2} implies a distributional identity
\begin{align*}
P(\frac{\mu}{2}+n)r^\mu(x)=0,
\end{align*}
which is equivalent in the light of
$P(\frac{\mu}{2}+n)=-(\mu+n-2)\partial_n+x_n\Delta$ and $\partial_n(r^\mu(x))=\mu x_n r^{\mu-2}(x)$ to
\begin{align*}
x_n\big(\Delta( r^\mu(x))-\mu(\mu+n-2) r^{\mu-2}(x)\big)=0.
\end{align*}
Hence we recover the Bernstein-Sato identity \eqref{eq:BSClassicalRiesz} for the Riesz distribution $r^\mu(x)$, i.e.,
\begin{align*}
\Delta( r^\mu(x))=\mu(\mu+n-2)r^{\mu-2}(x).
\end{align*}
\end{remark}
\subsection{Bernstein-Sato identity and operator in the spinor case}\label{TheSpinorCase}
In the present section we prove a Bernstein-Sato identity for distribution kernels associated to
conformal symmetry breaking operators
acting on spinors:
\begin{align}\label{eq:SpinorCoDimOneRiesz}
\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)&\st K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n) x\cdot.
\end{align}
Similarly to the scalar case, we introduce a Bernstein-Sato operator for $\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)$.
First, we recall the Knapp-Stein intertwining operator in the non-compact realization
of the induced representation of conformal Lie group on spinors \cite{CO}:
\begin{align*}
(\slashed{I}_{\gamma}\varphi)(x)\st (\slashed{r}^{-2\gamma}\star\varphi)(x) =\int_{\R^n} \slashed{r}^{-2\gamma}(x-y)\varphi(y)\dm y
\end{align*}
where $\varphi\in\slashed{\mathcal{S}}(\R^n)$.
By Proposition \ref{FourierRieszSpinor}, it follows
\begin{align*}
\slashed{I}_{n-\lambda}\circ \slashed{I}_\lambda=\slashed{c}_{2\lambda-2n}\slashed{c}_{-2\lambda}\id ,
\end{align*}
and as in the scalar case we define the operator
\begin{align}\label{eq:BSOperatorSpinor}
\slashed{D}(\lambda)\st \slashed{I}_{\lambda+1}\circ M_{x_n}\circ \slashed{I}_{n-\lambda}
\end{align}
with $M_{x_n}$ acting by the scalar multiplication.
\begin{theorem}
The operator $\slashed{D}(\lambda)$ in \eqref{eq:BSOperatorSpinor} is a differential operator of order $2$, i.e.,
\begin{align}
\slashed{D}(\lambda)\varphi=\tilde{\slashed{c}}_\lambda\big[(2\lambda-n+1)\partial_{n}\varphi
+\slashed{D}(e_n\cdot\varphi)+\Delta(x_n\varphi)\big] ,
\end{align}
where $\varphi\in\slashed{\mathcal{S}}(\R^n)$ and
$\tilde{\slashed{c}}_\lambda\st\slashed{c}_{-2\lambda-2}\slashed{c}_{2\lambda-2n}$.
\end{theorem}
\begin{proof}
As in the scalar case, we need to understand the right hand side of
\begin{align*}
\mathcal{F}(\slashed{D}(\lambda)\varphi)&=-i\slashed{c}_{-2\lambda-2}\slashed{c}_{2\lambda-2n}\slashed{r}^{2\lambda-n+2}(\xi)
\partial_{n}\big[ \slashed{r}^{n-2\lambda}(\xi)\mathcal{F}(\varphi)(\xi) \big].
\end{align*}
By
\begin{align*}
\partial_{n}(\slashed{r}^{n-2\lambda})(\xi)=(n-2\lambda-1)\xi_n r^{n-2\lambda-3}(\xi)\xi\cdot+r^{n-2\lambda-1}(\xi)e_n\cdot,
\end{align*}
it equals to
\begin{multline*}
-i\slashed{c}_{-2\lambda-2}\slashed{c}_{2\lambda-2n}[(n-2\lambda-1)\xi_n r^{-2}(\xi)\xi\cdot \xi \mathcal{F}(\varphi)
+\xi\cdot\mathcal{F}(e_n\cdot\varphi)+\xi\cdot\xi \cdot\partial_{n}\mathcal{F}(\varphi)]\\
=-i\slashed{c}_{-2\lambda-2}\slashed{c}_{2\lambda-2n}[-(n-2\lambda-1)\xi_n r^{-2}(\xi) \abs{\xi}^2 \mathcal{F}(\varphi)
+\xi\cdot\mathcal{F}(e_n\cdot\varphi)-\abs{\xi}^2 \cdot\partial_{n}\mathcal{F}(\varphi)]\\
=\slashed{c}_{-2\lambda-2}\slashed{c}_{2\lambda-2n}\mathcal{F}\big((2\lambda-n+1)\partial_{n}\varphi+\slashed{D}(e_n\cdot\varphi)+\Delta(x_n\varphi) \big)
\end{multline*}
and the proof is complete.
\end{proof}
Inspired by the previous Theorem we define, by a shift of the parameter $\lambda$, the operator
\begin{align}\label{eq:BSOperatorSpinor1}
\slashed{P}(\lambda):\slashed{\mathcal{S}}^\prime(\R^n)&\to \slashed{\mathcal{S}}^\prime(\R^n)\notag\\
\varphi&\mapsto (n-2\lambda+1)\partial_n\varphi+\slashed{D}(e_n\cdot\varphi)+\Delta(x_n \varphi).
\end{align}
\begin{remark}
Using the identity
\begin{align*}
\slashed{D}(e_n\cdot)=-e_n\cdot \slashed{D}^\prime-\partial_n,
\end{align*}
we can write
\begin{align*}
\slashed{P}(\lambda)\varphi= P(\lambda)\varphi-e_n\cdot\slashed{D}^\prime\varphi.
\end{align*}
Here $\slashed{D}^\prime\st\sum\limits_{k=1}^{n-1}e_k\cdot \partial_k$ is the tangential Dirac operator and $P(\lambda)$ is the scalar Bernstein-Sato operator, see \eqref{eq:BSOperator1}.
\end{remark}
\begin{remark}
Similarly to the scalar case, the operator $\slashed{P}(\lambda)$ is an intertwining
differential operator for the conformal Lie group on $\R^{n-1}$, and so is true for its iterations used in later sections.
\end{remark}
Now we collect a few basic properties of the distribution kernels
$\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)$ with respect
to certain algebraic and differential actions.
\begin{lem}\label{SpinorLemma}
The distribution kernels $\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)$ satisfy the following algebraic and differential identities:
\begin{enumerate}
\item
\begin{align}
x_n\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)=\slashed{K}^\mp_{\lambda+1,\nu}(x^\prime,x_n),
\end{align}
\item
\begin{align}
\partial_n(\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n))
&=(\lambda+\nu-n)\slashed{K}^\mp_{\lambda-1,\nu}(x^\prime,x_n)-2(\nu+\frac 12)\slashed{K}^\mp_{\lambda,\nu+1}(x^\prime,x_n)\nonumber\\
&+K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n)e_n\cdot,
\end{align}
\item
\begin{align}
\slashed{D}(\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n))
&=2(\nu+\frac{1-n}{2})K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n)\nonumber\\
&+(\lambda+\nu-n)e_n\cdot \slashed{K}^\mp_{\lambda-1,\nu}(x^\prime,x_n),
\end{align}
\item
\begin{align}
\Delta(\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n))
&=(\lambda+\nu-n-1)_2\slashed{K}^\pm_{\lambda-2,\nu}(x^\prime,x_n)\nonumber\\
&-2(\nu+\frac 12)(2\lambda-n-1)\slashed{K}^\pm_{\lambda-1,\nu+1}(x^\prime,x_n)\nonumber\\
&+2(\lambda+\nu-n)K^\mp_{\lambda-\frac 32,\nu+\frac 12}(x^\prime,x_n)e_n\cdot.
\end{align}
\end{enumerate}
\end{lem}
\begin{proof}
The proof is based on Lemma \ref{ScalarLemma}, and the identities
\begin{align*}
\abs{x}^2&=-x\cdot x\cdot,\quad e_k\cdot x\cdot=-x\cdot e_k\cdot-2x_k,\quad \partial_k(x)=e_k,\quad k=1,\ldots n,
\end{align*}
with $x=\sum\limits_{k=1}^n x_k e_k$.
To be more concrete, the first result is obvious by definition
of $\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)$. The remaining claim relies
on Lemma \ref{ScalarLemma} and the Leibniz-rule.
For example, we have
\begin{align*}
\partial_n(\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n))
&=\partial_n(K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n))x\cdot
+K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n)\partial_n(x)\cdot\\
&=(\lambda+\nu-n)\slashed{K}^\mp_{\lambda-1,\nu}(x^\prime,x_n)
-2(\nu+\frac 12)\slashed{K}^\mp_{\lambda,\nu+1}(x^\prime,x_n)\\
&+K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n) e_n\cdot,
\end{align*}
while for $1\leq k\leq n-1$ we get
\begin{align*}
\partial_k(\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n))
&=\partial_k(K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n)) x\cdot
+K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n) \partial_k(x)\cdot\\
&=-2(\nu+\frac 12)x_k K^\pm_{\lambda-\frac 32,\nu+\frac 32}(x^\prime,x_n)x\cdot
+K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n)e_k\cdot\\
&=-2(\nu+\frac 12)x_k \slashed{K}^\pm_{\lambda-1,\nu+1}(x^\prime,x_n)
+K^\pm_{\lambda-\frac 12,\nu+\frac 12}(x^\prime,x_n)e_k\cdot.
\end{align*}
The remaining assertions then follow easily.
\end{proof}
Consequently, the previous Lemma implies the following
Bernstein-Sato identity for the distribution kernels $\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)$.
\begin{theorem}\label{BSGeneralizedRiesz1Spinor}
The operator $\slashed{P}(\lambda)$ is a spectral shift operator for the
distribution kernels $\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)$, i.e.,
\begin{align}\label{eq:BSGeneralizedRiesz1Spinor}
\slashed{P}(\lambda)\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)
=(\lambda+\nu-n)(\nu-\lambda+1)\slashed{K}^\mp_{\lambda-1,\nu}(x^\prime,x_n).
\end{align}
\end{theorem}
\begin{proof}
The proof is based on Lemma \ref{SpinorLemma} and the identity
\begin{align*}
\slashed{D}(e_n\cdot\varphi)&=-e_n\cdot \slashed{D}(\varphi)-2\partial_n(\varphi).
\end{align*}
A straightforward computation shows
\begin{align*}
\slashed{P}(\lambda)\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)
&=(\lambda+\nu-n)(\nu-\lambda+1)\slashed{K}^\mp_{\lambda-1,\nu}(x^\prime,x_n).
\end{align*}
The proof is complete.
\end{proof}
\begin{remark}
Regarding the coefficients in equation \eqref{eq:BSOperatorSpinor1}
by $\partial_n$, $\slashed{D}(e_n\cdot)$ and $\Delta(x_n\cdot)$
as unknown, the ansatz for the operator $\tilde{\slashed{P}}(\lambda)$
\begin{align*}
\tilde{\slashed{P}}(\lambda)\st A\partial_n+B \slashed{D}(e_n\cdot)+C\Delta(x_n\cdot)
\end{align*}
leads to the system of equations
\begin{align*}
(\lambda+\nu-n)A-(\lambda+\nu-n)B+(\lambda+\nu-n)_2C&=(\lambda+\nu-n)(\nu-\lambda+1),\\
-2(\nu+\frac 12) A+4(\nu+\frac 12)B-2(\nu+\frac 12)(2\lambda-n+1)C&=0,\\
A-2(\nu-\frac{n-3}{2})B+2(\lambda+\nu-n+1)C&=0,
\end{align*}
equivalent to
$\tilde{\slashed{P}}(\lambda)\slashed{K}^\pm_{\lambda,\nu}(x^\prime,x_n)=(\lambda+\nu-n)(\nu-\lambda+1)\slashed{K}^\mp_{\lambda-1,\nu}(x^\prime,x_n)$. The unique solution of this system
is given by
\begin{align*}
A\st n-2\lambda+1,\quad B\st 1,\quad C\st 1,
\end{align*}
which agrees with Theorem \ref{BSGeneralizedRiesz1Spinor},
i.e., $\tilde{\slashed{P}}(\lambda)=\slashed{P}(\lambda)$.
\end{remark}
\begin{remark}
The distribution kernels $\slashed{K}^{\pm}_{\lambda,\nu}(x^\prime,x_n)$ generalize
the Riesz distribution $\slashed{r}^\lambda(x)$,
see \eqref{eq:ClassicalRieszSpinor}, in the sense that for $\mu\in\C$ with $\Re(\mu)>-n$ it holds
\begin{align}\label{eq:help2}
\slashed{K}^{+}_{\frac{\mu}{2}+n,-\frac{\mu}{2}}(x^\prime,x_n)=\slashed{r}^\mu(x).
\end{align}
The last theorem implies that
\begin{align*}
\slashed{P}(\frac{\mu}{2}+n)\slashed{r}^{\mu}(x)=0
\end{align*}
as a distributional identity, which is equivalent to
\begin{align*}
-(\mu+n-1)\partial_n(\slashed{r}^{\mu}(x))-e_n\slashed{D}(\slashed{r}^{\mu}(x))+x_n\Delta(\slashed{r}^{\mu}(x))=0.
\end{align*}
By
\begin{align*}
\partial_n(\slashed{r}^{\mu}(x))&=(\mu-1) \slashed{r}^{\mu-2}(x)+r^{\mu-1}(x) e_n,\\
\slashed{D}(\slashed{r}^{\mu}(x))&=-(\mu+n-1)r^{\mu-1}(x),
\end{align*}
we obtain
\begin{align*}
\Delta(\slashed{r}^\mu(x))=(\mu-1)(\mu+n-1)\slashed{r}^{\mu-2}(x).
\end{align*}
This is the Bernstein-Sato identity for $\slashed{r}^\mu(x)$, see \eqref{eq:BSClassicalRieszSpinor}. Independently this follows from
$\slashed{D}(r^{\lambda-1}(x))=(\lambda-1)\slashed{r}^{\lambda-2}(x)$, a coupled Bernstein-Sato identity for scalars and spinors, and $\slashed{D}^2=-\Delta$.
\end{remark}
\subsection{Bernstein-Sato identity and operator in the form case}
In the present section we prove a Bernstein-Sato identity for distribution kernels associated to conformal symmetry breaking operators on differential
forms:
\begin{align}\label{eq:FormCoDimOneRiesz}
K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)
&\st K^\pm_{\lambda-1,\nu+1}(x^\prime,x_n) (i_x\varepsilon_x-\varepsilon_x i_x)i_{e_n}\varepsilon_{e_n},
\end{align}
Similarly to the scalar case, we introduce a Bernstein-Sato operator for $K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)$.
First, we recall the Knapp-Stein intertwining operator in the non-compact realization
of the induced representation of conformal Lie group on differential forms \cite{FO}:
\begin{align*}
(I^p_{\gamma}\omega)(x)\st (R_p^{-2\gamma}\star\omega)(x)
=\int_{\R^n} R_p^{-2\gamma}(x-y)\omega(y)\dm y.
\end{align*}
By Proposition \ref{FourierRieszForm} we obtain
\begin{align*}
I^p_{n-\lambda}\circ I^p_{\lambda}=\bar{c}_{2\lambda-2n}\bar{c}_{-2\lambda}(\lambda-p)(n-p-\lambda)\id.
\end{align*}
As in the scalar case we define the operator
\begin{align}\label{eq:BSOperatorForm}
D^p(\lambda)\st I^p_{\lambda+1}\circ M_{x_n}\circ I^p_{n-\lambda},
\end{align}
with $M_{x_n}$ acting by the scalar multiplication.
\begin{theorem}
The operator $D^p(\lambda)$ in \eqref{eq:BSOperatorForm}
is a differential operator of order $2$, i.e.,
\begin{align*}
D^p(\lambda)\omega&=\tilde{\bar{c}}_\lambda\Big[(2\lambda-n)(\lambda-p+1)(\lambda-n+p+1)\partial_n\omega\\
&+(2\lambda-n)[(\lambda-p+1)\delta (\varepsilon_{e_n}\omega)-(\lambda-n+p+1)\dm (i_{e_n}\omega)]\\
&+[(\lambda-p+1)(n-\lambda-p)\delta\dm+(\lambda-p)(n-\lambda-p-1)\dm\delta](x_n\cdot\omega)\Big] ,
\end{align*}
where $\omega\in\Omega^p(\R^n)$ and
$\tilde{\bar{c}}_\lambda\st \bar{c}_{-2\lambda-2}\bar{c}_{2\lambda-2n}$.
\end{theorem}
\begin{proof}
In the Fourier image it follows that
\begin{align*}
\mathcal{F}(D^p(\lambda)\omega)(\xi)&=\mathcal{F}(I^p_{\lambda+1}\circ M_{x_n}\circ I^p_{n-\lambda}\omega)(\xi)\\
&=-i\bar{c}_{-2\lambda-2}\bar{c}_{2\lambda-2n} r^{2\lambda-n}(\xi) (\alpha_{\lambda+1-\frac n2}i_\xi\varepsilon_\xi+\beta_{\lambda+1-\frac n2}\varepsilon_\xi i_\xi)\times\\
&\quad\quad\quad\times\partial_n\big[r^{n-2\lambda-2}(\xi)(\alpha_{\frac n2-\lambda}i_\xi\varepsilon_\xi+\beta_{\frac n2-\lambda}\varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi) \big].
\end{align*}
The identities
\begin{align*}
\partial_n(r^{n-2\lambda-2}(\xi))&=(n-2\lambda-2)\xi_n r^{n-2\lambda-4}(\xi),\\
\partial_n(i_\xi\varepsilon_\xi)&=i_{e_n}\varepsilon_\xi+i_\xi\varepsilon_{e_n}=\xi_n-\varepsilon_\xi i_{e_n}+i_\xi\varepsilon_{e_n},\\
\partial_n(\varepsilon_\xi i_\xi)&=\varepsilon_{e_n}i_{\xi}+\varepsilon_\xi i_{e_n}=\xi_n-i_\xi\varepsilon_{e_n}+\varepsilon_\xi i_{e_n}
\end{align*}
imply
\begin{align*}
\mathcal{F}(D^p(\lambda)\omega)(\xi)&=-i\tilde{\bar{c}}_{\lambda}r^{2\lambda-n}(\xi) (\alpha_{\lambda+1-\frac n2}i_\xi\varepsilon_\xi+\beta_{\lambda+1-\frac n2}\varepsilon_\xi i_\xi)\times\\
&\quad\quad\quad\times\big[(n-2\lambda-2)\xi_nr^{n-2\lambda-4}(\xi)(\alpha_{\frac n2-\lambda}i_\xi\varepsilon_\xi+\beta_{\frac n2-\lambda}\varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi)\\
&\quad\quad\quad\quad+(\alpha_{\frac n2-\lambda}+\beta_{\frac n2-\lambda})\xi_n r^{n-2\lambda-2}(\xi)\mathcal{F}(\omega)(\xi)\\
&\quad\quad\quad\quad+(\beta_{\frac n2-\lambda}-\alpha_{\frac n2-\lambda})r^{n-2\lambda-2}(\xi)(\varepsilon_\xi i_{e_n}-i_\xi\varepsilon_{e_n})\mathcal{F}(\omega)(\xi)\\
&\quad\quad\quad\quad+r^{n-2\lambda-2}(\xi) (\alpha_{\frac n2-\lambda}i_{\xi}\varepsilon_\xi +\beta_{\frac n2-\lambda}\varepsilon_{\xi}i_\xi)\partial_n\mathcal{F}(\omega)(\xi) \big].
\end{align*}
The substitution
\begin{align*}
(\alpha_{\frac n2-\lambda}i_\xi\varepsilon_\xi+\beta_{\frac n2-\lambda}\varepsilon_\xi i_\xi)
=(\alpha_{\frac n2-\lambda-1}i_\xi\varepsilon_\xi+\beta_{\frac n2-\lambda-1}\varepsilon_\xi i_\xi)
+(i_\xi\varepsilon_\xi-\varepsilon_\xi i_\xi)
\end{align*}
together with
\begin{align*}
\alpha_{\lambda+1-\frac n2}\alpha_{\frac n2-\lambda-1}=\beta_{\lambda+1-\frac n2}\beta_{\frac n2-\lambda-1},\\
i_\xi\varepsilon_\xi+\varepsilon_\xi i_\xi=\abs{\xi}^2,\quad (i_\xi)^2=0=(\varepsilon_\xi)^2
\end{align*}
give
\begin{align*}
\mathcal{F}(D^p(\lambda)\omega)(\xi)=-i&\tilde{\bar{c}}_{\lambda}
\big[(n-2\lambda-2)\alpha_{\lambda+1-\frac n2}\alpha_{\frac n2-\lambda-1}\xi_n\mathcal{F}(\omega)(\xi)\\
&+(n-2\lambda-2)\xi_n r^{-2}(\xi)(\alpha_{\lambda+1-\frac n2}i_\xi \varepsilon_\xi-\beta_{\lambda+1-\frac n2}\varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi)\\
&+(\alpha_{\frac n2-\lambda}+\beta_{\frac n2-\lambda})\xi_n r^{-2}(\xi)(\alpha_{\lambda+1-\frac n2}i_\xi \varepsilon_\xi+\beta_{\lambda+1-\frac n2}\varepsilon_\xi i_\xi)\mathcal{F}(\omega)(\xi)\\
&+(\beta_{\frac n2-\lambda}-\alpha_{\frac n2-\lambda})r^{-2}(\xi)(\beta_{\lambda+1-\frac n2}\varepsilon_\xi i_{\xi}\varepsilon_\xi i_{e_n}-\alpha_{\lambda+1-\frac n2}i_{\xi}\varepsilon_\xi i_\xi\varepsilon_{e_n})\mathcal{F}(\omega)(\xi)\\
&+(\alpha_{\lambda+1-\frac n2}\alpha_{\frac n2-\lambda}i_{\xi}\varepsilon_\xi +\beta_{\lambda+1-\frac n2}\beta_{\frac n2-\lambda}\varepsilon_{\xi}i_\xi)\partial_n\mathcal{F}(\omega)(\xi) \big].
\end{align*}
Now, recalling the explicit form of the
coefficients $\alpha_\lambda,\beta_\lambda$, cf. \eqref{eq:BGCoeff},
\begin{align*}
\alpha_{\lambda+1-\frac n2}&=(\lambda-p+1),\quad \alpha_{\frac n2-\lambda}=(n-\lambda-p),
\quad \alpha_{\frac n2-\lambda-1}=(n-\lambda-p-1),\\
\beta_{\lambda+1-\frac n2}&=(n-\lambda-p-1),\quad \beta_{\frac n2-\lambda}=(\lambda-p),
\quad \beta_{\frac n2-\lambda-1}=(\lambda-p+1),
\end{align*}
allows to conclude
\begin{align*}
(n-2\lambda-2)\alpha_{\lambda+1-\frac n2}&+(\alpha_{\frac n2-\lambda}+\beta_{\frac n2-\lambda})\alpha_{\lambda+1-\frac n2}=2(n-\lambda-p-1)(\lambda-p+1),\\
-(n-2\lambda-2)\beta_{\lambda+1-\frac n2}&+(\alpha_{\frac n2-\lambda}+\beta_{\frac n2-\lambda})\beta_{\lambda+1-\frac n2}=2(\lambda-p+1)(n-\lambda-p-1),\\
\varepsilon_\xi i_{\xi}\varepsilon_\xi i_{e_n}&=\abs{\xi}^2\varepsilon_\xi i_{e_n},\quad
i_{\xi}\varepsilon_\xi i_\xi\varepsilon_{e_n}=\abs{\xi}^2i_\xi\varepsilon_{e_n}.
\end{align*}
This finally implies
\begin{align*}
\mathcal{F}(D^p(\lambda)\omega)=\tilde{\bar{c}}_{\lambda}\mathcal{F}\big(&
(2\lambda-n)(\lambda-p+1)(\lambda-n+p+1)\partial_n\omega\\
&+(2\lambda-n)[(\lambda-p+1)\delta (\varepsilon_{e_n}\omega)-(\lambda-n+p+1)\dm (i_{e_n}\omega)]\\
&+[(\lambda-p+1)(n-\lambda-p)\delta\dm+(\lambda-p)(n-\lambda-p-1)\dm\delta](x_n\cdot\omega) \big),
\end{align*}
which completes the proof.
\end{proof}
Let us renormalize the operator $D^p(\lambda)$ and shift the parameter $\lambda$ to $n-\lambda$:
\begin{align}\label{eq:BSOperatorForDiffForms}
P^p(\lambda):\mathcal{S}^{\prime,p}(\R^n)&\to \mathcal{S}^{\prime,p}(\R^n)\\
\omega&\mapsto
-(2\lambda-n)(\lambda-n+p-1)(\lambda-p-1)\partial_n\omega\notag\\
&+(2\lambda-n)[(\lambda-n+p-1)\delta (\varepsilon_{e_n}\omega)-(\lambda-p-1)\dm (i_{e_n}\omega)]\notag\\
&-\big[(\lambda-n+p-1)(\lambda-p)\delta\dm-(\lambda-n+p)(\lambda-p-1)\dm\delta\big](x_n\cdot\omega)\notag
\end{align}
\begin{remark}
The identities of the form
\begin{align*}
-\Delta&=-\sum_{k=1}^n\partial_k^2=\dm\delta+\delta\dm,
\quad \delta \varepsilon_{e_n}=-\varepsilon_{e_n}\delta-\partial_n,\quad i_{e_n}\dm=-\dm i_{e_n}+\partial_n,\\
\dm\delta(x_n\cdot)&=\varepsilon_{e_n}\delta-\dm i_{e_n}+x_n\dm\delta,
\quad \delta\dm(x_n\cdot)=-\varepsilon_{e_n}\delta+\dm i_{e_n}-2\partial_n+x_n\delta\dm,
\end{align*}
allow to write
\begin{align}\label{eq:BSOoperatorForm1}
P^p(\lambda)&= -(2\lambda-n-2)(\lambda-n+p-1)(\lambda-p)\partial_n\notag\\
&-(2\lambda-n-2)\big[(\lambda-n+p)\varepsilon_{e_n}\delta+(\lambda-p)\dm i_{e_n} \big]\notag\\
&-(\lambda-n+p-1)(\lambda-p)x_n \delta \dm-(\lambda-n+p)(\lambda-p-1)x_n\dm \delta.
\end{align}
In terms of $P(\lambda)$, see \eqref{eq:BSOperator1}, it holds
\begin{align}\label{eq:BSOperator1DiffForm}
P^p(\lambda)&=(\lambda-n+p-1)(\lambda-p)P(\lambda)-(n-2p)x_n \dm \delta\notag\\
&-(2\lambda-n-2)\big[(\lambda-n+p)\varepsilon_{e_n}\delta+(\lambda-p)\dm i_{e_n} \big].
\end{align}
\end{remark}
\begin{remark}
Similarly to the scalar case, the operator $P^p(\lambda)$ is an intertwining
differential operator for the conformal Lie group on $\R^{n-1}$, and so is true for its iterations used in later sections.
\end{remark}
Now we present some basic properties of $K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)$. First note that
\begin{align}\label{eq:CSBDF}
K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)=K^{\pm}_{\lambda,\nu}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}
-2K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}.
\end{align}
\begin{lem}\label{FormLemma}
The distribution kernels $K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)$ satisfy
\begin{enumerate}
\item
\begin{align*}
x_n K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n)=K^{(p),\mp}_{\lambda+1,\nu}(x^\prime,x_n),
\end{align*}
\item
\begin{align*}
\partial_n(K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n))&=(\lambda+\nu-n)K^{(p),\mp}_{\lambda-1,\nu}(x^\prime,x_n)
-2\nu K^{\mp}_{\lambda,\nu+1}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}\\
&+4(\nu+1)K^{\mp}_{\lambda-1,\nu+2}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}
-2K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_{e_n}i_x i_{e_n}\varepsilon_{e_n},
\end{align*}
\item
\begin{align*}
\varepsilon_{e_n}\delta(K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n))
=2(\lambda-p) K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_{e_n}i_x i_{e_n}\varepsilon_{e_n},
\end{align*}
\item
\begin{align*}
\dm(i_{e_n}K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n))
&=4(\nu+1)K^{\mp}_{\lambda-1,\nu+2}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}
-2p K^{\mp}_{\lambda,\nu+1}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}\\
&-2(\lambda+\nu-n+1)K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_{e_n}i_x i_{e_n}\varepsilon_{e_n},
\end{align*}
\item
\begin{align*}
\dm\delta(K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n))
&= -4(\nu+1)(\lambda-p) K^{\pm}_{\lambda-2,\nu+2}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&+2(\lambda+\nu-n)(\lambda-p) K^{\mp}_{\lambda-2,\nu+1}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n}\\
&+2p(\lambda-p) K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n) i_{e_n}\varepsilon_{e_n},
\end{align*}
\item
\begin{align*}
\delta\dm(K^{(p),\pm}_{\lambda,\nu}(x^\prime,x_n))
&=-4(\nu+1)(\lambda-n+p) K^{\pm}_{\lambda-2,\nu+2}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&-(\lambda+\nu-n-1)_2 K^{(p),\pm}_{\lambda-2,\nu}(x^\prime,x_n)\\
&-2(\lambda+\nu-n)(\lambda-p-2) K^{\mp}_{\lambda-2,\nu+1}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n}\\
&+\big[2\nu(2\lambda-n-p-2)-2p(\lambda-\nu-p-2)\big] K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n) i_{e_n}\varepsilon_{e_n}.\\
\end{align*}
\end{enumerate}
\end{lem}
\begin{proof}
The first claim follows from definition \eqref{eq:FormCoDimOneRiesz}.
As for the remaining properties, we use Equation
\eqref{eq:CSBDF} and compute the differential actions on both summands
separately. We start with some observations.
First compute, using Lemma \ref{ScalarLemma} and Leibniz's rule, the identities
\begin{align*}
\partial_n (K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)&=(\lambda+\nu-n)K_{\lambda-1,\nu}^{\mp}(x^\prime,x_n)H
-2\nu K_{\lambda,\nu+1}^{\mp}(x^\prime,x_n)H\\
&+K_{\lambda,\nu}^{\pm}(x^\prime,x_n)\partial_n(H)\\
\partial_k (K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)&=-2\nu x_k K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)H
+K_{\lambda,\nu}^{\pm}(x^\prime,x_n)\partial_k(H),
\quad 1\leq k\leq n-1,
\end{align*}
and conclude
\begin{align*}
\delta(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)
&=2\nu K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)i_x H-(\lambda+\nu-n)K_{\lambda-1,\nu}^{\mp}(x^\prime,x_n)i_{e_n} H\\
&+K_{\lambda,\nu}^{\pm}(x^\prime,x_n)\delta(H),\\
\dm(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)
&=-2\nu K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x H+(\lambda+\nu-n)K_{\lambda-1,\nu}^{\mp}(x^\prime,x_n)\varepsilon_{e_n} H\\
&+K_{\lambda,\nu}^{\pm}(x^\prime,x_n)\dm(H),\\
\delta(\varepsilon_x K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)
&=-(n-p+E)K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H-\varepsilon_x\delta(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H),\\
\delta(\varepsilon_{e_n} K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)
&=-\partial_n(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)-\varepsilon_{e_n}\delta(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H),\\
\dm(i_x K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)&=(p+E)K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H-i_x \dm(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H),\\
\dm(i_{e_n}K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)&=\partial_n(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)-i_{e_n}\dm(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)H)
\end{align*}
for some endomorphism $H$ of differential forms.
Here we denote by $E\st\sum\limits_{k=1}^n x_k \partial_k$ the Euler operator and
close our observations with
\begin{align*}
E(K_{\lambda,\nu}^{(p),\pm}(x^\prime,x_n))=(\lambda-\nu-n)K_{\lambda,\nu}^{(p),\pm}(x^\prime,x_n).
\end{align*}
Now it is straightforward to compute
\begin{align*}
\partial_n(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n})&=(\lambda+\nu-n)K_{\lambda-1,\nu}^{\mp}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}
-2\nu K_{\lambda,\nu+1}^{\mp}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n},\\
-2\partial_n(K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n})
&=-2(\lambda+\nu-n)K_{\lambda-2,\nu+1}^{\mp}(x^\prime,x_n)\varepsilon_x i_xi_{e_n}\varepsilon_{e_n}\\
&+4(\nu+1)K_{\lambda-1,\nu+2}^{\mp}(x^\prime,x_n)\varepsilon_x i_xi_{e_n}\varepsilon_{e_n}\\
&-2K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n},
\end{align*}
\begin{align*}
\varepsilon_{e_n}\delta(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n})
&=2\nu K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n},\\
-2\varepsilon_{e_n}\delta(K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n})
&=-2(\nu-\lambda+p)K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n} \varepsilon_{e_n},
\end{align*}
\begin{align*}
\dm(i_{e_n} K_{\lambda,\nu}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n})&=0,\\
-2\dm(i_{e_n}K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n})
&=4(\nu+1)K_{\lambda-1,\nu+2}^{\mp}(x^\prime,x_n) \varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&-2(\lambda+\nu-n+1)K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n} \varepsilon_{e_n}\\
&-2pK_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n},
\end{align*}
\begin{align*}
\dm\delta(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n})
&=-4(\nu)_2 K_{\lambda-2,\nu+2}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&+2\nu(\lambda+\nu-n)K_{\lambda-2,\nu+1}^{\mp}(x^\prime,x_n)\varepsilon_{e_n}i_x i_{e_n}\varepsilon_{e_n}\\
& +2p\nu K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n},\\
-2\dm\delta(K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n})
&=4(\nu+1)(\nu-\lambda+p)K_{\lambda-2,\nu+2}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&-2(\lambda+\nu-n)(\nu-\lambda+p) K_{\lambda-2,\nu+1}^{\mp}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n}\\
&-2p(\nu-\lambda+p)K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n},\\
\end{align*}
and
\begin{align*}
\delta\dm(K_{\lambda,\nu}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n})
&=4(\nu)_2 K_{\lambda-2,\nu+2}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&-2\nu(\lambda+\nu-n)K_{\lambda-2,\nu+1}^{\mp}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n}\\
&+2\nu(2\lambda-n-p-2) K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}\\
&-(\lambda+\nu-n-1)_2K_{\lambda-2,\nu}^{\pm}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n},\\
-2\delta\dm(K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n})
&=-4(\nu+1)(\lambda+\nu-n+p)K_{\lambda-2,\nu+2}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&+2(\lambda+\nu-n-1)_2 K_{\lambda-3,\nu+1}^{\pm}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}\\
&-2(\lambda+\nu-n)(\lambda-\nu-p-2)K_{\lambda-2,\nu+1}^{\mp}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n}\\
&-2p(\lambda-\nu-p-2)K_{\lambda-1,\nu+1}^{\pm}(x^\prime,x_n) i_{e_n}\varepsilon_{e_n}.
\end{align*}
This completes the proof.
\end{proof}
\begin{theorem}\label{BSIdentityForDiffForms}
The distribution kernels $K_{\lambda,\nu}^{(p),\pm}(x^\prime,x_n)$ satisfy
\begin{align*}
P^{(p)}(\lambda)K_{\lambda,\nu}^{(p),\pm}(x^\prime,x_n)=(\lambda+\nu-n)(\nu-\lambda+1)(\lambda-n+p-1)(\lambda-p)K_{\lambda-1,\nu}^{(p),\mp}(x^\prime,x_n).
\end{align*}
\end{theorem}
\begin{proof}
We use Equation \eqref{eq:BSOoperatorForm1} for the operator $P^p(\lambda)$ and Lemma \ref{FormLemma}.
The statement is based on collecting terms contributing
to $K^{(p),\mp}_{\lambda-1,\nu}(x^\prime,x_n)$,
$K^{\mp}_{\lambda,\nu+1}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}$, $K^{\mp}_{\lambda-1,\nu+2}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}$
and $K^{\pm}_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_{e_n}i_x i_{e_n}\varepsilon_{e_n}$, respectively. This gives
\begin{multline*}
-(2\lambda-n-2)(\lambda-n+p-1)(\lambda-p)(\lambda+\nu-n)
+(\lambda-n+p-1)(\lambda-p)(\lambda+\nu-n-1)_2\\
=(\lambda+\nu-n)(\nu-\lambda+1)(\lambda-n+p-1)(\lambda-p),
\end{multline*}
\begin{multline*}
2\nu(2\lambda-n-2)(\lambda-n+p-1)(\lambda-p)+2p(2\lambda-n-2)(\lambda-p)-2p(\lambda-n+p)(\lambda-p-1)_2\\
-(\lambda-n+p-1)(\lambda-p)\big[2\nu(2\lambda-n-p-2)-2p(\lambda-\nu-p-2) \big]=0,
\end{multline*}
\begin{multline*}
-4(\nu+1)(2\lambda-n-2)(\lambda-n+p-1)(\lambda-p)-4(\nu+1)(2\lambda-n-2)(\lambda-p)\\
+4(\nu+1)(\lambda-n+p)(\lambda-p-1)_2+4(\nu+1)(\lambda-n+p-1)_2(\lambda-p)=0,
\end{multline*}
\begin{multline*}
2(2\lambda-n-2)(\lambda-n+p-1)(\lambda-p)-2(2\lambda-n-2)(\lambda-n+p)(\lambda-p)\\
+2(2\lambda-n-2)(\lambda-p)(\lambda+\nu-n+1)-2(\lambda-n+p)(\lambda-p-1)_2(\lambda+\nu-n)\\
+2(\lambda-n+p-1)(\lambda-p)(\lambda+\nu-n)(\lambda-p-2)=0,
\end{multline*}
and the proof is complete.
\end{proof}
\begin{remark}\label{NonUniqueFormBS}
Let us define the operator
\begin{align*}
\tilde{P}^p(\lambda)\st A\partial_n+B \varepsilon_{e_n}\delta+C\dm i_{e_n}+D x_n \dm\delta+ Ex_n \delta\dm
\end{align*}
for some unknown $A,B,C,D,E$. The equation
\begin{align*}
\tilde{P}^p(\lambda)K^{\pm,(p)}_{\lambda,\nu}(x^\prime,x_n)=(\lambda+\nu-n)(\nu-\lambda+1)(\lambda-n+p-1)(\lambda-p)K^{\mp,(p)}_{\lambda+1,\nu}(x^\prime,x_n)
\end{align*}
is equivalent to the following system for $A,B,C,D,E$ and
$c\st (\lambda+\nu-n)(\nu-\lambda+1)(\lambda-n+p-1)(\lambda-p)$:
\begin{align*}
(\lambda+\nu-n) A-(\lambda+\nu-n-1)_2 E&=c,\\
-2\nu A+2\nu p D+2\nu(2\lambda-n-p-2)E-2p C&\\
-2(\nu-\lambda+p)p D-2(\lambda-\nu-p-2)p E&=0,\\
2\nu B +2\nu (\lambda+\nu-n)D-2\nu(\lambda+\nu-n)E-2A-2(\nu-\lambda+p) B&\\
-2(\lambda+\nu-n+1)C-2(\lambda+\nu-n)(\nu-\lambda+p)D&\\
-2(\lambda+\nu-n)(\lambda-\nu-p-2)E&=0,\\
-4(\nu)_2 D+4(\nu)_2 E+4(\nu+1)A+4(\nu+1)C+4(\nu+1)(\nu-\lambda+p)D&\\
-4(\nu+1)(\lambda+\nu-n+p)E&=0,\\
-2(\lambda+\nu-n)A+2(\lambda+\nu-n-1)_2 E&=-2c.
\end{align*}
Here the contributions are sorted again according to
$K^\mp_{\lambda-1,\nu}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}$, $K^\mp_{\lambda,\nu+1}(x^\prime,x_n)i_{e_n}\varepsilon_{e_n}$,
$K^\pm_{\lambda-1,\nu+1}(x^\prime,x_n)\varepsilon_{e_n} i_x i_{e_n}\varepsilon_{e_n}$,
$K^\mp_{\lambda-1,\nu+2}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}$
and $K^\mp_{\lambda-2,\nu+1}(x^\prime,x_n)\varepsilon_x i_x i_{e_n}\varepsilon_{e_n}$.
Note that in this case the system does not have a unique solution, we could choose either
$B,C$ or $D$ as a free parameter. Choosing one of them as in
Equation \eqref{eq:BSOoperatorForm1} will determine the other two and we will recover $P^p(\lambda)$.
This observation is a reflection of the fact that in general Bernstein-Sato operators are
not unique in contrast with the uniqueness of the Bernstein-Sato polynomial.
\end{remark}
\section{Applications of Bernstein-Sato identities and operators}\label{ComAndApps}
In the final section we highlight different origins of Bernstein-Sato operators. Furthermore, we
discuss several applications related to conformal symmetry breaking differential operators \cite{KOSS, FJS, KKP}.
As a consequence we shall observe that the Bernstein-Sato operators
recover conformal symmetry breaking differential operators for functions,
spinors and differential forms by partially new formulas. They differ from
the known formulas in their product structure expansion, which gives
a nice symmetric way of organizing their rather complicated structure.
\subsection{Origins of the Bernstein-Sato operator - the scalar case}
We discuss some other origins of the Bernstein-Sato operator $P(\lambda)$, see \eqref{eq:BSOperator1}, for functions
(less is known for $\slashed{P}(\lambda)$ and $P^p(\lambda)$).
To our best knowledge, there are three other approaches to construct the Bernstein-Sato operator
\cite{MOZ,GZ,GW}.
{\bf Representation theory:} The differential action on the
induced representation $\pi_\nu$ of the Casimir element $C$ for
conformal Lie algebra $\mathfrak{o}(1,n+1)$ is given in \cite[Equation $5.1$]{MOZ}:
\begin{align*}
C(\mu)\st \dm\pi_\mu(C)=x_n^2\Delta+2(\mu+1)x_n\partial_n+(\mu+\frac n2)(\mu-\frac n2+1).
\end{align*}
The relationship to $P(\lambda)$ is
\begin{align}
x_nP(\lambda)=C(-\lambda+\frac n2)-(\lambda-n)(\lambda-1).
\end{align}
Note that by Theorem \ref{BSGeneralizedRiesz1} we obtain
\begin{align*}
C(-\lambda+\frac n2) K^\pm_{\lambda,\nu}(x^\prime,x_n)=-\nu(n-1-\nu)K^\pm_{\lambda,\nu}(x^\prime,x_n),
\end{align*}
hence $K^\pm_{\lambda,\nu}(x^\prime,x_n)$ is an eigendistribution of $C(-\lambda+\frac n2)$.
{\bf Hyperbolic metric:} The eigen-equation associated to the Laplace operator
for the hyperbolic metric $g_{hyp}=x_n^{-2}(\dm x_1^2+\cdots+\dm x_n^2)$ on
the hyperbolic space also
induces the operator $P(\lambda)$, see \cite{GZ}. In particular,
\begin{align}
x_n^{n-s}P(s-1)=(\Delta_{g_{hyp}}-s(n-1-s))x_n^{n-1-s}.
\end{align}
Here we used $\Delta_{g_{hyp}}=x_n^2\Delta-(n-2)x_n\partial_n$, where
$\Delta=\sum\limits_{k=1}^n\partial_k^2$ is the Laplace operator associated to
the metric $x^2_ng_{hyp}$.
{\bf Tractor calculus:} The invariant pairing of the tractor-D operator
$D_A$ with the scale tractor $I_A$ in the Poincar\'e-metric gives, see
\cite[Section $5.6$]{GW},
\begin{align}
P(s)=-I\cdot D.
\end{align}
Here the parameter $s$ on the right hand side of the previous equation corresponds
to the weight of weighted tractor bundle.
Note that the {\it degenerate Laplacian} $I\cdot D$ is of more general
nature then our $P(\lambda)$ requires.
\subsection{Origins of the Bernstein-Sato operator - the spinor case}
In comparison to the scalar case much less is known for the operator $\slashed{P}(\lambda)$, see \eqref{eq:BSOperatorSpinor1},
in the available literature.
{\bf Hyperbolic metric:} The eigen-equation associated to the square of the Dirac operator \cite{GMP1}
for the hyperbolic metric $g_{hyp}=x_n^{-2}(\dm x_1^2+\cdots+\dm x_n^2)$ also
induces the operator $\slashed{P}(\lambda)$. In particular, the eigen-equation
$\slashed{D}^{g_{hyp}}-i\lambda=0$ on $\Gamma(\Sigma^{g_{hyp}}_n)$, where
$\Sigma^{g_{hyp}}_n$ denotes the spinor bundle associated to the hyperbolic space, is equivalent,
via conformal covariance ($\bar{g}=\dm x_1^2+\cdots+\dm x_n^2$) of the Dirac operator, to
$D(\bar{g})-i\lambda=0$ on $\Gamma(\Sigma_n)$. We have
\begin{align*}
D(\bar{g})\st x_n\slashed{D}^\prime+x_n D_{\mathcal{N}}-\frac{n-1}{2}e_n\cdot,
\end{align*}
where $\slashed{D}^\prime= \sum\limits_{k=1}^{n-1}e_k\cdot\partial_k$ and
$D_{\mathcal{N}}\st e_n\cdot\partial_n$. Now, we define the
operator $D(\mu)$ by the operator equation
$[D(\bar{g})^2+(\mu-\frac{n-1}{2})]x_n^\mu=-x_n^\mu D(\mu)$. It then follows
\begin{align}
D(\mu)=x_n^2\Delta-2(\mu-\frac{n-1}{2})x_n\partial_n-x_n e_n\cdot \slashed{D}=x_n\slashed{P}(\mu+\frac 32).
\end{align}
\subsection{Origins of the Bernstein-Sato operator - the form case}
In comparison to the scalar and spinor cases even less is known about
the operator $P^p(\lambda)$, see \eqref{eq:BSOperatorForDiffForms},
in the available literature. A potential origin of $P^p(\lambda)$
in the construction of the hyperbolic metric \cite{AG}
seems not to be well established; to our best knowledge this
approach leads to an operator different from $P^p(\lambda)$.
It is not clear to the authors if this discrepancy can be explained
by the non-uniqueness phenomenon mentioned in Remark \ref{NonUniqueFormBS}.
\subsection{Conformal symmetry breaking differential operators for functions}
The operator $P(\lambda)$, see \eqref{eq:BSOperator1}, recovers
conformal symmetry breaking differential operators \cite{J1,KS, KOSS}
\begin{align}\label{eq:ResidueFamily}
D_N(\lambda):\mathcal{C}^\infty(\R^n)\to\mathcal{C}^\infty(\R^{n-1}),
\end{align}
which are given by
\begin{align}
D_{2N}(\lambda)&=\sum_{k=0}^N a_k^{(N)}(-\lambda) (\Delta^\prime)^k \iota^* \partial_n^{2N-2k},\notag\\
D_{2N+1}(\lambda)&=\sum_{k=0}^N b_k^{(N)}(-\lambda) (\Delta^\prime)^k \iota^* \partial_n^{2N-2k+1}\label{eq:SBDOScalar}
\end{align}
with
\begin{align}
a_j^{(N)}(\lambda)&\st \frac{(-2)^{N-j}N!}{j!(2N-2j)!}\prod_{k=j}^{N-1}(2\lambda-4N+2k+n+1),\notag\\
b_j^{(N)}(\lambda)&\st \frac{(-2)^{N-j}N!}{j!(2N-2j+1)!}\prod_{k=j}^{N-1}(2\lambda-4N+2k+n-1). \label{eq:GegenbauerCoeff}
\end{align}
Here $\Delta^\prime\st \sum\limits_{k=1}^{n-1}\partial_k^2$ denotes the tangential Laplacian
for the embedding $\iota:\R^{n-1}\to \R^n$, $\R^{n-1}\ni x^\prime\mapsto (x^\prime,0)\in\R^n$.
As mentioned in the introduction, the families (conformal symmetry breaking differential operators) $D_{2N}(\lambda)$ interpolate between
GJMS-operators on $\R^{n-1}$ and $\R^n$
equipped with the flat Euclidean metric, respectively. More precisely it holds
\begin{align}\label{eq:ScalarFamiliesVsGJMS}
D_{2N}(-\frac{n-1}{2}+N)&=(\Delta^\prime)^N\iota^*,\notag\\
D_{2N}(-\frac{n}{2}+N)&=\iota^* (\Delta)^N.
\end{align}
This appearance of GJMS-operators is part of a sequence of factorization identities for $D_N(\lambda)$, see \cite{J1}.
Now let us define a family ($N\in\N_0$, $\lambda\in\C$) of differential operators on functions (termed {\it Bernstein-Sato family for functions})
\begin{align}\label{eq:BSFamily}
D^{BS}_N(\lambda)\st \iota^*P(\lambda-N+1)\circ \cdots\circ P(\lambda).
\end{align}
This definition, although used in a different setting, already appeared in \cite{C}.
\begin{example}\label{LowOrderScalarBSFamilyScalar}
We present the first- and second-order relations between
$D^{BS}_N(\lambda)$ and $D_N(\lambda)$. First of all,
recall from \eqref{eq:SBDOScalar} that
\begin{align*}
D_1(\lambda)&=\iota^*\partial_n,\\
D_2(\lambda)&=\Delta^\prime\iota^*+(2\lambda-n+3)\iota^*\partial_n^2.
\end{align*}
A direct computation shows
\begin{align*}
D^{BS}_1(\lambda)&=\iota^* P(\lambda)=(n-2\lambda+2)\iota^*\partial_n\\
&=(n-2\lambda+2) D_1(n-\lambda),\\
D^{BS}_2(\lambda)&=\iota^* P(\lambda-1)\circ P(\lambda)\\
&=(n-2\lambda+4)\iota^*\partial_n [(n-2\lambda+2)\partial_n+x_n\Delta]\\
&=(n-2\lambda+4)[(n-2\lambda+3)\iota^*\partial^2_n+\Delta^\prime\iota^*]\\
&=(n-2\lambda+4) D_2(n-\lambda).
\end{align*}
\end{example}
Now we shall discuss the relationship between differential operators
$D^{BS}_N(\lambda)$ and $D_N(\lambda)$, see also
\cite{C} where the proof follows
from the property of uniqueness of intertwining differential operators.
We present a direct proof of this fact.
\begin{theorem}\label{BSFamily}
Let $N\in\N$. Then we have
\begin{align*}
D^{BS}_{2N}(\lambda)&=(-2)^{N}(\lambda-\frac n2-2N)_N (2N-1)!!D_{2N}(n-\lambda),\\
D^{BS}_{2N+1}(\lambda)&= (-2)^{N+1}(\lambda-\frac n2-2N-1)_{N+1} (2N+1)!!D_{2N+1}(n-\lambda).
\end{align*}
\end{theorem}
\begin{proof}
The proof goes by induction. Recall Example \ref{LowOrderScalarBSFamilyScalar} for
the lowest order relationship.
By induction we compute
\begin{align*}
D^{BS}_{2N}(\lambda)&=D^{BS}_{2N-1}(\lambda-1)\circ P(\lambda)\\
&=(-2)^{N}(\lambda-\frac n2-2N)_{N} (2N-1)!!D_{2N-1}(n-\lambda+1)\circ P(\lambda).
\end{align*}
Since
\begin{align*}
\iota^* \partial_n^{2N-2k-1}[(2\lambda-n+2)\partial_n+x_n\Delta]&=(2\lambda-n+2)\iota^*\partial_n^{2N-2k}\\
&+(2N-2k-1)[\Delta^\prime\iota^*\partial_n^{2N-2k-2}+\iota^*\partial_n^{2N-2k}],
\end{align*}
we may conclude
\begin{align*}
D^{BS}_{2N}(\lambda)&=(-2)^{N}(\lambda-\frac n2-2N)_{N} (2N-1)!!\Big[A_0(\lambda)\iota^*\partial_n^{2N}+A_N(\lambda)(\Delta^\prime)^N\\
&+\sum_{k=1}^{N-1}A_k(\lambda)(\Delta^\prime)^k\iota^*\partial_n^{2N-2k} \Big]
\end{align*}
with
\begin{align*}
A_0(\lambda)\st&(n-2\lambda+2N+1)b_0^{(N-1)}(\lambda-n-1),\\
A_N(\lambda)\st& b_{N-1}^{(N-1)}(\lambda-n-1),\\
A_k(\lambda)\st&(n-2\lambda+2N-2k+1)b_k^{(N-1)}(\lambda-n-1)+(2N-2k+1)b_{k-1}^{(N-1)}(\lambda-n-1).
\end{align*}
It follows from \eqref{eq:GegenbauerCoeff} that for $0\leq k\leq N$ holds
\begin{align*}
A_k(\lambda)\st&a_k^{(N)}(\lambda-n),
\end{align*}
hence we get
\begin{align*}
D^{BS}_{2N}(\lambda)&=(-2)^{N}(\lambda-\frac n2-2N)_{N} (2N-1)!!D_{2N}(n-\lambda).
\end{align*}
The remaining statement is proved analogously. The proof is complete.
\end{proof}
An immediate consequence of the last result is
\begin{corollary}\label{RecurrenceForJuhl}
Assuming $N\in\N_0$, we have
\begin{align*}
D_{2N-1}(n-\lambda+1)\circ P(\lambda)&=D_{2N}(n-\lambda),\\
D_{2N}(n-\lambda+1)\circ P(\lambda)&=-(2N+1)(2\lambda-n-2N-2) D_{2N+1}(n-\lambda).
\end{align*}
\end{corollary}
\begin{remark}\label{RemarkAboutDifferentProofs}
To our best knowledge there are two other ways to compute $D^{BS}_{N}(\lambda)$.
The first one is based on Fourier transform, cf. proof of Proposition
\ref{ScalarClercTrick} for $N=1$.
To this aim one needs the following identities:
\begin{align*}
\partial_n^{2N}(r^{n-2\lambda}(\xi))&=(2N-1)!!2^{N}(\frac n2-\lambda-N+1)_{N}r^{n-2\lambda-4N}(\xi)\times\\
&\times\sum_{k=0}^N a_{N-k}^{(N)}(\lambda-n+2N) r^{2N-2k}(\xi^\prime)\xi_n^{2k},\\
\partial_n^{2N+1}(r^{n-2\lambda}(\xi))&=(2N-1)!!2^{N+1}(\frac n2-\lambda-N)_{N+1}\xi_n r^{n-2\lambda-4N-2}(\xi)\times\\
&\times\sum_{k=0}^N b_{N-k}^{(N)}(\lambda-n+2N+1) r^{2N-2k}(\xi^\prime)\xi_n^{2k}.
\end{align*}
We note the appearance of coefficients \eqref{eq:GegenbauerCoeff}.
Another way is based on commutators
\begin{align*}
[P(\lambda),\partial_n]=-\Delta,\quad [P(\lambda),\Delta]=-2\partial_n\Delta
\end{align*}
and the fact that $\iota^* P(\lambda)=(n-2\lambda+2)\iota^*\partial_n$.
Both approaches computing $D_{2N}^{BS}(\lambda)$ are computationally rather
tedious and we will not give any more detail here.
\end{remark}
\subsection{Conformal symmetry breaking differential operators for spinors}
Here we discuss how the operator $\slashed{P}(\lambda)$, see
\eqref{eq:BSOperatorSpinor1}, by its iterations recovers conformal symmetry
breaking differential operators for spinors
\begin{align*}
\slashed{D}_N(\lambda): \mathcal{C}^\infty(\R^n,\Sigma_n)\to \begin{cases}\mathcal{C}^\infty(\R^{n-1},\Sigma_{n-1})&, n\text{ even },\\
\mathcal{C}^\infty(\R^{n-1},\Sigma^+_{n-1}\oplus \Sigma^-_{n-1})&, n\text{ odd}\end{cases}
\end{align*}
introduced in \cite{KOSS} (note the wrong sign of $\lambda$ in the pre-factor in the reference), and later appearing
in \cite{MO}. They are given by
\begin{align*}
\slashed{D}_{2N}(\lambda)&\st D_{2N}(\lambda+\frac 12)+2N D_{2N-1}(\lambda+\frac 12)\slashed{D}^\prime (e_n\cdot),\\
\slashed{D}_{2N+1}(\lambda)&\st (2\lambda-n+2N+2)D_{2N+1}(\lambda+\frac 12)(e_n\cdot)+D_{2N}(\lambda+\frac 12)\slashed{D}^\prime ,\\
\end{align*}
where $D_N(\lambda)$ (note that we will mean by $\iota^*$ just restriction) are the conformal symmetry breaking differential operators
\eqref{eq:SBDOScalar} and
$\slashed{D}^\prime=\sum\limits_{k=1}^{n-1}e_k\cdot \partial_k$ is the
tangential Dirac operator. The family $\slashed{D}_{2N+1}(\lambda)$ interpolates between conformal powers of the
Dirac operator on $\R^{n-1}$ and $\R^n$ equipped with the flat Euclidean metric, respectively. More precisely, it holds
\begin{align*}
\slashed{D}_{2N+1}(\frac{n-1}{2}-\frac 12-N)&= (\slashed{D}^\prime)^{2N+1}\iota^* ,\\
\slashed{D}_{2N+1}(\frac n2-\frac 12-N)&=(-1)^{N}\iota^* \slashed{D}^{2N+1}
\end{align*}
This appearence of conformal powers of the Dirac operators is part of a sequence of factorization identities for $\slashed{D}_{N}(\lambda)$, see \cite{FS}.
Now we define the family for $N\in\N_0$ and $\lambda\in\C$ of differential
operators on spinors (termed {\it Bernstein-Sato family for spinors}) by the composition
\begin{align*}
\slashed{D}_N^{BS}(\lambda)\st \iota^* \slashed{P}(\lambda-N+1)\circ\cdots\circ \slashed{P}(\lambda)
\end{align*}
This definition goes again in the spirit of \cite{C}.
\begin{example}\label{LowOrderScalarBSFamilySpinor}
We present the relationship for the first and the second-order families
$\slashed{D}_N^{BS}(\lambda)$ and $\slashed{D}_N(\lambda)$.
Firstly, via \eqref{eq:SBDOScalar} we recall
\begin{align*}
\slashed{D}_{1}(\lambda)&=(2\lambda-n+2)\partial_n (e_n\cdot)+\slashed{D}^\prime,\\
\slashed{D}_2(\lambda)&=D_2(\lambda+\frac 12)+2D_1(\lambda+\frac 12)\slashed{D}^\prime(e_n\cdot)\\
&=\Delta^\prime+(2\lambda-n+4)\iota^*\partial_n^2+2\iota^*\partial_n \slashed{D}^\prime(e_n\cdot)
\end{align*}
The Bernstein-Sato families of first and second-order, respectively, read as
\begin{align*}
\slashed{D}_1^{BS}(\lambda)&=\iota^* \slashed{P}(\lambda)=\iota^*[(n-2\lambda+2)\partial_n+x_n\Delta-e_n\slashed{D}^\prime]\\&=-e_n\cdot \slashed{D}_{1}(n-\lambda),\\
\slashed{D}_2^{BS}(\lambda)&\st \iota^*\slashed{P}(\lambda-1)\circ \slashed{P}(\lambda)\\
&=[(n-2\lambda+4)\iota^*\partial_n-e_n\cdot\slashed{D}^\prime\iota^*]
[(n-2\lambda+2)\partial_n+x_n\Delta-e_n\cdot\slashed{D}^\prime]\\
&=(n-2\lambda+3)[(n-2\lambda+4)\iota^*\partial_n^2-\Delta^\prime+2\slashed{D}^\prime \partial_n(e_n)]\\
&=-(n-2\lambda+3)\slashed{D}_2(n-\lambda).
\end{align*}
\end{example}
Now we explain a general relationship between the families
$\slashed{D}_N^{BS}(\lambda)$ and $\slashed{D}_N(\lambda)$.
\begin{theorem}\label{BSVsSBOSpinor}
For $N\in\N_0$ we have
\begin{align*}
\slashed{D}_{2N}^{BS}(\lambda)&=(-2)^N(\lambda-\frac n2-2N+\frac 12)_N(2N-1)!! \slashed{D}_{2N}(n-\lambda),\\
\slashed{D}_{2N+1}^{BS}(\lambda)&=-(-2)^N(\lambda-\frac n2-2N-\frac 12)_N(2N+1)!! e_n\cdot \slashed{D}_{2N+1}(n-\lambda).
\end{align*}
\end{theorem}
\begin{proof}
We recall Example \ref{LowOrderScalarBSFamilySpinor}.
Then by induction on the order we have
\begin{align*}
\slashed{D}_{2N}^{BS}(\lambda)&=\slashed{D}_{2N}^{BS}(\lambda-1)\circ \slashed{P}(\lambda)\\
&=(-1)^N 2^{N-1}(\lambda-\frac n2-2N+\frac 12)_{N-1}(2N-1)!!\slashed{D}_{2N-1}(-\lambda+n+1)\circ \slashed{P}(\lambda)\\
&=(-1)^{N-1} 2^{N-1}(\lambda-\frac n2-2N+\frac 12)_{N-1}(2N-1)!!\big[A_0 \iota^*\partial_n^{2N}\\
&+\sum_{k=1}^{N-1}(-1)^k A_k (\slashed{D}^\prime)^{2k}\iota^*\partial_n^{2N-2k} +A_N(\slashed{D}^\prime)^{2N}\iota^*\\
&+B_0 \iota^*\partial_n^{2N-1}\slashed{D}^\prime(e_n\cdot)+\sum_{k=1}^{N-1}(-1)^k B_k (\slashed{D}^\prime)^{2k}\iota^*\partial_n^{2N-2k-1}\slashed{D}^\prime(e_n\cdot) \big],
\end{align*}
where
\begin{align*}
A_0&\st (-2\lambda+n+2N+2)(n-2\lambda+2N+1)b_0^{(N-1)}(\lambda-n-\frac 32),\\
A_k&\st (-2\lambda+n+2N+2)[(n-2\lambda+2N-2k+1)b_k^{(N-1)}(\lambda-n-\frac 32)\\
&+(2N-2k+1)b_{k-1}^{(N-1)}(\lambda-n-\frac 32)]-a_{k-1}^{(N-1)}(\lambda-n-\frac 32),\\
A_N&\st (-2\lambda+n+2N+2)-1,\\
B_0&\st (-2\lambda+n+2N+2)b_0^{(N-1)}(\lambda-n-\frac 32)+(n-2\lambda+2N)a_0^{(N-1)}(\lambda-n-\frac 32),\\
B_k&\st (-2\lambda+n+2N+2)b_k^{(N-1)}(\lambda-n-\frac 32)\\
&+(n-2\lambda+2N-2k)a_{k}^{(N-1)}(\lambda-n-\frac 32)]+(2N-2k)a_{k-1}^{(N-1)}(\lambda-n-\frac 32).
\end{align*}
Then it follows that for $k=0,\ldots N$
\begin{align*}
A_k=(-2\lambda+n+2N+1)a_k^{(N)}(\lambda-n-\frac 12),
\end{align*}
while for $k=0,\ldots N-1$ it holds
\begin{align*}
B_k=(-2\lambda+n+2N+1)b_k^{(N-1)}(\lambda-n-\frac 12).
\end{align*}
This proves the even-order case. The odd-order case is completely analogous
and will be omitted.
\end{proof}
\begin{remark}
The facts analogous to Remark \ref{RemarkAboutDifferentProofs}
constitute different proofs of Theorem \ref{BSVsSBOSpinor}.
\end{remark}
\subsection{Conformal symmetry breaking differential operators for differential forms}
Conformal symmetry breaking differential operators acting on differential forms \cite{FJS,KKP}
\begin{align}\label{eq:SBDOForms}
D_N^{(p\to p)}(\lambda):\Omega^{p}(\R^{n})\to \Omega^{p}(\R^{n-1})
\end{align}
are given by
\begin{align*}
D_{2N}^{(p\to p)}(\lambda)&=(p-\lambda-2N)D_{2N}(\lambda)+2N(2\lambda-n+2N+1)D_{2N-1}(\lambda+1)\dm^\prime i_{e_n}\\
&-2N D_{2N-2}(\lambda+1)\dm^\prime\delta^\prime,\\
D_{2N+1}^{(p\to p)}(\lambda)&=(p-\lambda-2N-1)D_{2N+1}(\lambda)+D_{2N}(\lambda+1)\dm^\prime i_{e_n}-2N D_{2N-1}(\lambda+1)\dm^\prime\delta^\prime.
\end{align*}
Note the opposite sign convention for the family parameter $\lambda$ and a clash
of notation for $\Delta$ when compared to \cite{FJS}, i.e., $D^{(p\to p)}_{2N+1}(-\lambda)$
and $D^{(p\to p)}_{2N+1}(-\lambda)$ introduced in \cite{FJS} correspond to
$(-1)^ND^{(p\to p)}_{2N}(\lambda)$ and $(-1)^ND^{(p\to p)}_{2N}(\lambda)$
defined in \eqref{eq:SBDOForms}, respectively.
As for the definition of $D_N(\lambda)$,
see Equation \eqref{eq:SBDOScalar}. Also note that $\iota^*$ in the definition of $D_N(\lambda)$
denotes the pull-back of differential forms. The operators $D_{2N}^{(p\to p)}(\lambda)$ interpolate between
Branson-Gover operators for $\R^{n-1}$ and $\R^n$ equipped with the flat Euclidean metric, respectively. More precisely, it holds
\begin{align*}
D_{2N}^{(p\to p)}(\frac{n-1}{2}-N)&=(-1)^{N+1}\big[(\frac{n-1}{2}-p-N)(\dm^\prime\delta^\prime)^N+(\frac{n-1}{2}-p+N)(\delta^\prime\dm^\prime)^N\big],\\
D_{2N}^{(p\to p)}(\frac n2-N)&=(-1)^{N+1}\big[(\frac{n}{2}-p-N)(\dm\delta)^N+(\frac{n}{2}-p+N)(\delta\dm)^N\big].
\end{align*}
Again, this appearence of Branson-Gover operators is a part of the sequence of factorizations identities for $D_{2N}^{(p\to p)}(\lambda)$, see \cite{FJS}.
Let us introduce a family (termed {\it Bernstein-Sato family for differential forms of first type})
\begin{align}\label{eq:BSFamilyDiffForms}
D_N^{BS,(p\to p)}(\lambda)\st \iota^* P^p(\lambda-N+1)\circ\cdots \circ P^p(\lambda):\Omega^p(\R^n)\to \Omega^p(\R^{n-1}).
\end{align}
This definition is again inspired by \cite{C}.
Now we state some low-order relations between $D_N^{BS,(p\to p)}(\lambda)$ and $D_N^{(p\to p)}(\lambda)$.
\begin{example}\label{BSFamilyLowOrderDiffForm}
By definitions \eqref{eq:SBDOForms} and \eqref{eq:SBDOScalar} we have
\begin{align*}
D_1^{(p\to p)}(\lambda)&=(p-\lambda-1)\iota^*\partial_n+\dm^\prime\iota^* i_{e_n},\\
D_2^{(p\to p)}(\lambda)&=(p-\lambda-2)\Delta^\prime\iota^*+(2\lambda-n+3)(p-\lambda-2)\iota^*\partial_n^2\\
&+2(2\lambda-n+3)\dm^\prime\iota^* i_{e_n}\partial_n-2\dm^\prime\delta^\prime\iota^*,\\
D_3^{(p\to p)}(\lambda)&=(p-\lambda-3)\Delta^\prime\iota^*\partial_n
+\frac 13 (2\lambda-n+5)(p-\lambda-3)\iota^*\partial_n^3\\
&+(2\lambda-n+5)\dm^\prime\iota^*i_{e_n}\partial_n^2+\Delta^\prime\dm^\prime\iota^* i_{e_n}-2\dm^\prime\delta^\prime\iota^*\partial_n.
\end{align*}
It is then straightforward to verify
\begin{align*}
D_1^{BS,(p\to p)}(\lambda)&=-(2\lambda-n-2)(\lambda-p)D_1^{(p\to p)}(n-\lambda),\\
D_2^{BS,(p\to p)}(\lambda)&=-(2\lambda-n-4)(\lambda-n+p-1)(\lambda-p-1)_2D_2^{(p\to p)}(n-\lambda),\\
D_3^{BS,(p\to p)}(\lambda)&=3(2\lambda-n-6)(2\lambda-n-4)(\lambda-p-2)_3(\lambda-n+p-2)_2 D^{(p\to p)}_3(n-\lambda).
\end{align*}
\end{example}
Now we state a general relationship between the families
$D_N^{BS,(p\to p)}(\lambda)$ and $D^{(p\to p)}_N(\lambda)$.
\begin{theorem}\label{BSVsSBODiffForms}
For $N\in\N$ holds
\begin{align*}
D_{2N}^{BS,(p\to p)}(\lambda)&=(-2)^N(\lambda-\frac n2-2N)_N(2N-1)!!\times\\
&\times(\lambda-n+p-2N+1)_{2N-1}(\lambda-p-2N+1)_{2N}D^{(p\to p)}_{2N}(n-\lambda),\\
D_{2N+1}^{BS,(p\to p)}(\lambda)&=(-2)^{N+1}(\lambda-\frac n2-2N-1)_{N+1}(2N+1)!!\times\\
&\times(\lambda-n+p-2N)_{2N}(\lambda-p-2N)_{2N+1}D^{(p\to p)}_{2N+1}(n-\lambda).
\end{align*}
\end{theorem}
\begin{proof}
The proof goes by induction and starts with Example \ref{BSFamilyLowOrderDiffForm}.
By definition \eqref{eq:BSFamilyDiffForms} and induction hypothesis it follows
\begin{align*}
D_{2N}^{BS,(p\to p)}(\lambda)&=D_{2N-1}^{BS,(p\to p)}(\lambda-1)\circ P^p(\lambda)\\
&=c(2N-1,\lambda-1)D_{2N-1}^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)
\end{align*}
for
\begin{multline*}
c(2N-1,\lambda-1)=(-2)^N(\lambda-\frac n2-2N)_N(2N-1)!!\times\\
\times(\lambda-p-2N+1)_{2N-1}(\lambda-n+p-2N+1)_{2N-2}.
\end{multline*}
Now \eqref{eq:SBDOForms} gives that
\begin{align}
D_{2N-1}^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)=\big[&(\lambda-n+p-2N) D_{2N-1}(n-\lambda+1)\circ P^p(\lambda)\notag\\
&+D_{2N-2}(n-\lambda+2)\dm^\prime i_{e_n} P^p(\lambda)\notag\\
&-(2N-2) D_{2N-3}(n-\lambda+2)\dm^\prime\delta^\prime P^p(\lambda) \big].\label{eq:help3}
\end{align}
The individual summands on the right hand side of the last display
simplify by \eqref{eq:BSOperator1DiffForm} and the identities
\begin{align*}
\iota^*\partial_n^k(x_n F)&=k \iota^*\partial_n^{k-1}F,\quad \iota^*\dm\delta=\dm^\prime\delta^\prime\iota^*-\dm^\prime\iota^* i_{e_n}\partial_n,
\quad \iota^* \varepsilon_{e_n}\delta=0,\quad \iota^* \dm i_{e_n}=\dm^\prime \iota^* i_{e_n},
\end{align*}
where $k\in\N$ and $F$ is a differential operator.
Hence we see by Corollary \ref{RecurrenceForJuhl}
\begin{align*}
D_{2N-1}(n-\lambda+1) &P^p(\lambda)=(\lambda-n+p-1)(\lambda-p)\sum_{k=0}^{N-1}a_k^{(N-1)}(\lambda-n)(\Delta^\prime)^k\iota^*\partial_n^{2N-2k}\\
+\sum_{k=0}^{N-1}&\big[(n-2p)(2N-2k-1)-(2\lambda-n-2)(\lambda-p)\big]b_k^{(N-1)}(\lambda-n-1)\times \\
&\times (\Delta^\prime)^k\iota^*\partial_n^{2N-2k-1}\dm^\prime i_{e_n}\\
-(n-2p)\sum_{k=0}^{N-1}&(2N-2k-1)b_k^{(N-1)}(\lambda-n-1)(\Delta^\prime)^k\iota^*\partial_n^{2N-2k-2}\dm^\prime\delta^\prime .
\end{align*}
Similarly, the identities
\begin{align*}
P(\lambda)&=P(\lambda-1)-2\partial_n,\quad \iota^*\dm^\prime i_{e_n}\dm\delta=\dm^\prime\delta^\prime\iota^*\partial_n-\dm^\prime \iota^* i_{e_n}\partial_n^2,\\
\iota^*\dm^\prime i_{e_n} \varepsilon_{e_n}\delta&=\dm^\prime\delta^\prime\iota^*-\dm^\prime\iota^* i_{e_n}\partial_n,\quad \iota^*\dm^\prime i_{e_n}\dm i_{e_n}=\dm^\prime\iota^* i_{e_n}\partial_n
\end{align*}
and Corollary \ref{RecurrenceForJuhl} allow to conclude
\begin{align*}
D_{2N-2}&(n-\lambda+2)\dm^\prime i_{e_n} P^p(\lambda)\\
=\sum_{k=0}^{N-1}\bigg[&-(2N-1)(2\lambda-n-2N-2)(\lambda-n+p-1)(\lambda-p)b_k^{(N-1)}(\lambda-n-1)\\
&+\big[-2(\lambda-n+p-1)(\lambda-p) +(n-2p)(2N-2k-2)\\
&+(2\lambda-n-2)(\lambda-n+p)-(2\lambda-n-2)(\lambda-p)\big]a_k^{(N-1)}(\lambda-n-2)\bigg]\times\\
&\times (\Delta^\prime)^k\iota^* \partial_n^{2N-2k-1}\dm^\prime i_{e_n}\\
+\sum_{k=0}^{N-1}\big[&-(n-2p)(2N-2k-2)-(2\lambda-n-2)(\lambda-n+p)\big]a_k^{(N-1)}(\lambda-n-2)\times\\
&\times(\Delta^\prime)^k\iota^* \partial_n^{2N-2k-2}\dm^\prime\delta^\prime . \\
\end{align*}
Finally, by
\begin{align*}
\dm^\prime\delta^\prime\iota^*\dm\delta=(\dm^\prime\delta^\prime)^2\iota^*-\dm^\prime\delta^\prime\dm^\prime\iota^* i_{e_n}\partial_n,\quad
\dm^\prime\delta^\prime\iota^*(\varepsilon_{e_n}\delta)=0,\quad \dm^\prime\delta^\prime\iota^* \dm i_{e_n}=\dm^\prime\delta^\prime\dm^\prime\iota^*i_{e_n}
\end{align*}
and Corollary \ref{RecurrenceForJuhl} we have
\begin{align*}
D_{2N-2}&(n-\lambda+2)\dm^\prime \delta^\prime P^p(\lambda)\\
=\sum_{k=0}^{N-1}&\big[-(n-2p)(2N-2k-1)+(2\lambda-n-2)(\lambda-p) \big]b_{k-1}^{(N-2)}(\lambda-n-2)\times\\
&\times (\Delta^\prime)^k\iota^*\partial_n^{2N-2k-1}\dm^\prime i_{e_n}\\
+\sum_{k=0}^{N-1}&\bigg[(\lambda-n+p-1)(\lambda-p)\big[a_k^{(N-1)}(\lambda-n-1)-2b_k^{(N-2)}(\lambda-n-2) \big] \\
&+(n-2p)(2N-2k-1)b_{k-1}^{(N-2)}(\lambda-n-2)\bigg](\Delta^\prime)^k\iota^*\partial_n^{2N-2k-2}\dm^\prime\delta^\prime.
\end{align*}
Consequently, we see that Equation \eqref{eq:help3} simplifies to
\begin{align*}
D_{2N-1}&^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)\\
=&(\lambda-n+p-1)(\lambda-p)(\lambda-n+p-2N)D_{2N}(n-\lambda)\\
&-(\lambda-n+p-1)(\lambda-p)(2N)(2\lambda-n-2N-1)D_{2N-1}(n-\lambda+1)\dm^\prime i_{e_n}\\
&-(\lambda-n+p-1)(\lambda-p)(2N-2)D_{2N-2}(n-\lambda+1)\dm^\prime\delta^\prime\\
=&(\lambda-n+p-1)(\lambda-p)D_{2N}^{(p\to p)}(n-\lambda)
\end{align*}
and hence we have
\begin{align*}
D_{2N}^{BS,(p\to p)}(n-\lambda)&=(-2)^N(\lambda-\frac n2-2N)_N(2N-1)!!\times\\
&\times(\lambda-n+p-2N+1)_{2N-1}(\lambda-p-2N+1)_{2N}D^{(p\to p)}_{2N}(n-\lambda).
\end{align*}
The odd-order families follow by a similar argument. The proof is complete.
\end{proof}
An immediate consequence of the last theorem is
\begin{corollary}
Let $N\in\N_0$. Then it holds
\begin{align*}
D_{2N-1}^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)=&(\lambda-n+p-1)(\lambda-p)D_{2N}^{(p\to p)}(n-\lambda),\\
D_{2N}^{(p\to p)}(n-\lambda+1)\circ P^p(\lambda)=&-(\lambda-n+p-1)(\lambda-p)(2N+1)\times\\
&\times(2\lambda-n-2N-2)D_{2N+1}^{(p\to p)}(n-\lambda).
\end{align*}
\end{corollary}
\begin{remark}
The Bernstein-Sato families of the first type
$D_{N}^{BS,(p\to p)}(\lambda), N\in\N_0,$ induce
the full classification list for
conformal symmetry breaking differential operators on differential
forms, cf. \cite[Theorem $3$]{FJS}. For example, the Bernstein-Sato families
of the second type arise by post- and pre-composition
of $D_{N}^{BS,(p\to p)}(\lambda), N\in\N_0$, with the Hodge-star operators on
$\R^{n-1}$ and $\R^n$, respectively.
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,030 |
Q: Does Pymongo have validation rules built in? I am trying to validate an inserted document against a schema, and was trying to find a way to validate the inserted document.
There are libraries like MongoEngine that say they do the work, but is there a way to do document validation directly via pymongo ?
A: The python driver docs are indeed a little light on how to use the db.command. Here is a complete working example:
from pymongo import MongoClient
from collections import OrderedDict
import sys
client = MongoClient() # supply connection args as appropriate
db = client.testX
db.myColl.drop()
db.create_collection("myColl") # Force create!
# $jsonSchema expression type is prefered. New since v3.6 (2017):
vexpr = {"$jsonSchema":
{
"bsonType": "object",
"required": [ "name", "year", "major", "gpa" ],
"properties": {
"name": {
"bsonType": "string",
"description": "must be a string and is required"
},
"gender": {
"bsonType": "string",
"description": "must be a string and is not required"
},
"year": {
"bsonType": "int",
"minimum": 2017,
"maximum": 3017,
"exclusiveMaximum": False,
"description": "must be an integer in [ 2017, 3017 ] and is required"
},
"major": {
"enum": [ "Math", "English", "Computer Science", "History", None ],
"description": "can only be one of the enum values and is required"
},
"gpa": {
# In case you might want to allow doubles OR int, then add
# "int" to the bsonType array below:
"bsonType": [ "double" ],
"minimum": 0,
"description": "must be a double and is required"
}
}
}
}
# Per the docs, args to command() require that the first kev/value pair
# be the command string and its principal argument, followed by other
# arguments. There are two ways to do this: Using an OrderDict:
cmd = OrderedDict([('collMod', 'myColl'),
('validator', vexpr),
('validationLevel', 'moderate')]
db.command(cmd)
# Or, use the kwargs construct:
# db.command('collMod','myColl', validator=vexpr, validationLevel='moderate')
try:
db.myColl.insert({"x":1})
print "NOT good; the insert above should have failed."
except:
print "OK. Expected exception:", sys.exc_info()
try:
okdoc = {"name":"buzz", "year":2019, "major":"Math", "gpa":3.8}
db.myColl.insert(okdoc)
print "All good."
except:
print "exc:", sys.exc_info()
A: MongoDB supports document validation at the engine level so you'll pick it up via pymongo. You declare your "schema" (rules actually) to the engine. Here's a great place to start: https://docs.mongodb.com/manual/core/document-validation/
A: You can make a separated JSON file for your Document Validations Schema, like this:
{
"collMod": "users",
"validator": {
"$jsonSchema": {
"bsonType": "object",
"required": ["email", "password","name"],
"properties": {
"email": {
"bsonType": "string",
"description": "Correo Electrónico"
},
"password": {
"bsonType": "string",
"description": "Una representación Hash de la contraseña"
},
"name": {
"bsonType": "object",
"required": ["first", "last"],
"description": "Objeto que separa los nombres y apellidos",
"properties": {
"first": {
"bsonType": "string",
"description": "Primer y segundo nombre"
},
"last": {
"bsonType": "string",
"description": "Primer y segundo apellido"
}
}
},
}
}
}
}
Then you can use in python script, example:
from pymongo import MongoClient
import json #parse JSON file as dict
from collections import OrderedDict #preserve the order (key, value) in the gived insertions on the dict
client = MongoClient("your_mongo_uri")
db = client.your_db_name
with open('your_schema_file.json', 'r') as j:
d = json.loads(j.read())
d = OrderedDict(d)
db.command(d)
OrderedDict Info
collMod Info
Schema Validation Info
A: I know 2 options to deal with:
*
*By creating or setting schema for collection, so any insertions will be checked against it on server side, rejected or warned depending on validationAction
The following code demonstrates scheme creation and testing:
import pymongo
mongo_client = MongoClient(url=...,
port=...,
username=...,
password=...,
authSource=...,
authMechanism=...,
connect=True, )
mongo_client.server_info()
db = mongo_client.your_db
users = db.create_collection(name="users",
validator={"$jsonSchema": {
"bsonType": "object",
"required": ["username"],
"properties": {
"username": {
"bsonType": "string",
"pattern": "[a-z0-9]{5,15}",
"description": "user name (required), only lowercase letters "
"and digits allowed, from 5 to 15 characters long"
},
"email": {
"bsonType": "string",
"description": "User's email (optional)"
},
}
}},
validationAction="error",
)
# Inserting user document that fits the scheme
users.insert_one({"username": "admin", "email": "some_admin_mail"})
# Insertion below will be rejected with "pymongo.errors.WriteError: Document failed validation, full error"
# caused by too short username (root)
users.insert_one({"username": "root", "email": "some_root_mail"})
*You can think about your Mongo's documents as ordinary JSON entities and check them on the client code side using standard JSON scheme validation
from jsonschema import validate
from jsonschema.exceptions import ValidationError
db = MongoClient(...).your_db
schema = {
"type": "object",
"required": ["username"],
"properties": {
"username": {"type": "string", "pattern": "[a-z0-9]{5,15}"},
"email": {"type": "string"},
},
}
try:
new_user = {"username": "admin", "email": "some_admin_mail"}
# No exception will be raised in validation below
validate(instance=new_user, schema=schema)
db.users.insert_one(new_user)
new_user = {"username": "root", "email": "some_root_mail"}
# Exception <ValidationError: 'root' does not match '[a-z0-9]{5,15}'> will be raised
validate(instance=new_user, schema=schema)
db.users.insert_one(new_user)
except ValidationError:
# Performing error
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,522 |
How does sugar explode?
Esther Inglis-Arkell
It seems that there are limitless ways that sugar can kill people. Not content to limit itself to slow murder due to caloric intake, it has branched out into sudden, violent death. Find out how and why sugar can explode.
Sugar burns. This fact has caused pain to anyone who has ruined a cake by leaving it in the oven, or had to pretend to enjoy a blackened smore, but pastry kitchens generally don't blow up like meth labs, and it's legal to bring cupcakes on airplanes. What is it about powder that makes sugar go from ruined meal to kaboom?
Powder is more likely to flame sudden and fast than pastry. Fire needs oxygen, and small things are more likely to be able to strike a balance between giving a flame both fuel and air. Most people who have started fires – under whatever circumstancs, Io9 doesn't judge – know that they need kindling to start out. Grains of sugar are more likely to catch fire than large blocks of wood.
Location is also a factor in sugar explosions. The fact that sugar grains burn so fast stops some explosions from happening. There's not a lot of fuel in them, and in order for a big explosion to get going, the grains have to be spaced just right.
Up until the nineteenth, fireships were sometimes used in naval battles. Fireships were either old or cheap ships that were loaded up with fuel, set on fire, and pushed toward enemy vessels in the hopes of setting those vessels on fire. Those flaming ships would then set more ships on fire until, with any luck, there was no more enemy. The fuel for the fireships had to be judged just right. If the ship burned too fast, it would burn out and sink before reaching the enemy.
The sugar grains are much like that. They have to be in a position where they can reach the next set of grains and set them on fire before they flame out. That set has to reach the next, and the next, and the next in order to keep the explosion going. That makes it unlikely in domestic situations. In food manufacturing plants, it is a real danger. When sugar has caused explosions, it's been in refineries, where sugar coated the walls and surfaces, and powdered sugar was suspended in the air.
It's not just sugar that explodes. Flour is also made up of carbohydrates – lots of sugar molecules stuck together. It, too, can explode the same way sugar does. I assume cocoa powder can too. Most organic materials burn. Probably baking soda.
Now I just need to find a way to make eggs, butter, and vanilla explode, and I'll have the ultimate weapon, combustible cake, and the world will be mine.
Via HowStuffWorks and Slate.
FrankenPC
I'm thinking a hybrid fuel/air device. Maybe a trash bag filled with a combination of oxygen/propane/flour and or sugar. Keep the mixture flowing with some kind of fan and ignite that puppy...from a SERIOUS distance.
I did not suggest this. I was not here. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,840 |
\section{Introduction}
Quantum computation and information has undergone a long way since its earliest conceptions~\cite{feynman2018simulating, deutsch1985quantum}. Many quantum algorithms have been proposed to tackle computational problems, ranging from searching~\cite{grover1996fast}, factoring~\cite{shor1999polynomial}, simulating quantum systems~\cite{childs2010quantum, feynman2018simulating, berry2007efficient}, solving linear systems~\cite{harrow2009quantum, childs2017quantum}, machine learning and data science~\cite{lloyd2013quantum, lloyd2014quantum, lloyd2016quantum, lloyd2020quantum, schuld2018supervised, mitarai2018quantum}, and to problems in computational topology and geometry~\cite{lloyd2013quantum, nghiem2022constant}. Many quantum algorithms and applications remain to be discovered. One of the most versatile results in the above is the quantum algorithm--the so-called HHL algorithm--for solving linear systems, proposed Harrow, Hassidim and Lloyd~\cite{harrow2009quantum} (as well as its improvement~\cite{clader2013preconditioned,childs2017quantum,wossnig2018quantum}), as linear systems play a central role in many engineering and science domains. Aside from its exponential speedup, another important result from Ref.~\cite{harrow2009quantum} is the BQP-completeness of the matrix inversion, which is a more general statement of the complexity regarding solving linear systems. The HHL algorithm has also been improved in Ref.~\cite{childs2017quantum}, which yields an exponential improvement in the error dependence.
We recall that in the original HHL work~\cite{harrow2009quantum}, the aim is to solve the following linear equation
\begin{align}
A x = b,
\end{align}
where $A$, for simplicity, is a Hermitian matrix of size $N \times N$, and without loss of generality, $b$ is given and assumed to be normalized. It is easy to see that if the unique solution exists, it should be
\begin{align}
x = A^{-1} b.
\end{align}
The main tool of Ref.~\cite{harrow2009quantum} is the ability to \textit{exponentiate a sparse matrix}~\cite{berry2007efficient}, i.e, to perform $\exp(-iAt)$, where $A$ is the sparse matrix given at hand. As emphasized by HHL~\cite{harrow2009quantum}, their algorithm essentially applies $A^{-1}$ to some given initial state $\ket{b}$, via quantum phase estimation of $A$'s eigenvalues and controlled rotation to achieve the inversion by multiplication of the inverse eigenvalues to the coefficients in the decomposition of $b$ into the eigenbasis, thereby resulting in the (normalized) solution $\ket{x}$. The running time of their algorithm~\cite{harrow2009quantum} is
\begin{align}
\mathcal{O}\left(\log(N) \frac{s^4 \kappa^2}{\epsilon} \right),
\end{align}
where $s$ is the sparsity of an $N\times N$ matrix $A$, $\kappa$ is the conditional number of $A$, and $\epsilon$ is the error.
In this article, motivated by the results in Refs.~\cite{harrow2009quantum, childs2017quantum}, we consider the following problem: \\
\textit{Given a sparse Hermitian matrix $A$ of size $n \times n$ with bounded eigenvalues, estimate its largest absolute eigenvalue. } \\
For simplicity, we assume that the eigenvalues of $A$ satisfy the following
\begin{align}
0 < |\lambda_1| \leq |\lambda_2| \leq \cdots \leq |\lambda_n| < D,
\label{eqn: boundedrange}
\end{align}
where $D$ is some constant (note that since $A$ is Hermitian, all of eigenvalues are real).
Below, we construct a quantum algorithm that, given access to entries of a sparse Hermitian matrix $A$ in a similar manner to Ref.~\cite{harrow2009quantum}, can estimate its largest eigenvalue in magnitude with a running time that is logarithmic with respect to the size of $A$. The main tool of our algorithm is built upon the quantum linear system~\cite{harrow2009quantum, wiebe2012quantum} and the Hadamard test~\cite{nielsen2002quantum}. As with the HHL algorithm, therefore, our algorithm provides an exponential speedup against, e.g., the classical power method.
The remaining structure of this paper is as follows. First, in Sec.~\ref{sec: classicalalgorithm}, we review the classical algorithm by the \textit{power method}, which can estimate the largest absolute eigenvalue of some matrix $A$, which in general, does not have to be sparse or well-conditioned. In Sec.~\ref{sec: quantumalgorithm}, we present a corresponding quantum algorithm that solves the largest-eigenvalue (in magnitude) problem mentioned above, and it is a direct translation of the classical power method into the quantum setting. In Sec.~\ref{sec: discussion}, we then discuss a few possible extensions of our quantum algorithm, including the non-sparse case and how we can modify our algorithm for the \textit{minimum eigenvalue}, as well as extension to a few largest eigenvalues in magnitude. In the Appendix~\ref{sec: erroranalysis}, we provide an analysis of the power method, showing that the number of iteration is efficiently low in order to reach the convergence of power method. A more careful elaboration of the running time of our quantum algorithm with respect to the number of iterations is provided in Appendix~\ref{sec: elaboration}. In Appendix~\ref{app:HHL}, we explain how the original HHL approach is adapted to our case, and in Appendix~\ref{app:Improved}, we show a simplified version of the multiplication application of the matrix, reducing the time complexity.
\section{Classical Algorithm}
\label{sec: classicalalgorithm}
The classical approach to tackle the above problem is called the \textit{power method} \cite{chu1993multivariate}, which is a very standard approach \cite{parlett1982estimating,o1979estimating}, and literally can be found in any textbook on numerical methods. The main procedure is as follows: \\
$\bullet$ First initialize some random vector $x_0 \in C^n$. \\
$\bullet$ Iterate the matrix application process $k$ times and form the following sequence:
\begin{align*}
x_1 = A x_0, \\
x_2 = A x_1 = A^2 x_0, \\
x_3 = A x_2 = A^3 x_0, \\
\vdots \\
x_k = A^k x_0.
\end{align*}
$\bullet$ Compute the approximated largest eigenvalue $\bar{\lambda}_n$:
\begin{align*}
\bar{\lambda}_n \equiv\frac{Ax_k^* \cdot x_k}{x_k^* \cdot x_k},
\end{align*}
where $(\cdot)$ denotes the inner product, and $*$ denotes the complex conjugate. We remind that, since $A$ is Hermitian, its eigenvalues are always real. \\
The following lemma shows that $x_k$ can approximate the eigenvector $E_n$ with eigenvalue $\lambda_n$.
\begin{lemma}
\label{lemma: approx}
Let $x_k$ be defined as above. Assuming the largest absolute eigenvalue $\lambda_n$ (with corresponding eigenvector $E_n$) is unique, i.e, $|\lambda_i| < |\lambda_n|$ for all $1< i <n$. Then
\begin{align*}
x_k \approx \alpha \cdot E_n,
\end{align*}
where $\alpha$ is a some number that depends on $k$--the number of iterations.
\end{lemma}
\textit{Proof:} Since $A$ is Hermitian, its eigenvectors $\{E_n\}_1^N$ can be made orthonormal if degenerate. We can choose them to be the basis of $A$. Therefore, the initial random vector $x_0$ can be expressed as:
\begin{align}
x_0 = c_1E_1 + c_2E_2 + \cdots + c_nE_n.
\end{align}
After the first iteration, we have
\begin{align}
x_1 &= Ax_0 \\
&= A \big(c_1E_1 + c_2E_2 + \cdots + c_nE_n \big) \\
&= c_1 \lambda_1E_1 + c_2 \lambda_2E_2 + \cdots + c_n \lambda_n E_n.
\end{align}
Proceeding similarly, after $k$ iterations, we have
\begin{align}
x_k
= \lambda_n^k \Big( c_1\frac{\lambda_1^k}{\lambda_n^k} E_1 + c_2 \frac{\lambda_2^k}{\lambda_n^k} E_2 + \cdots + c_n E_n \Big).
\label{eqn: xk}
\end{align}
We have assumed that the largest eigenvalue in magnitude is non-degenerate,
$$ 0 \leq |\lambda_1| \leq |\lambda_2| \leq \cdots < |\lambda_n|. $$
Therefore,
\begin{align}
\frac{\lambda_i^k}{\lambda_n^k} \rightarrow 0, \ \mbox{as} \, k \rightarrow \infty.
\end{align}
Hence, when $k$ is sufficiently large, we have:
$$ x_k \approx \lambda_n^k c_nE_n \sim E_n. $$
The proof is then completed. $\blacksquare$ \\
From here, we can see that the factor $\alpha$ in Lemma.~\ref{lemma: approx} is
$$ \alpha = \lambda_n^k c_n.$$
Now we try to compute:
$$ \frac{Ax_k^* \cdot x_k}{x_k^* \cdot x_k}. $$
Since $x_k \approx \alpha E_n$, then $Ax_k \approx \lambda_n x_k$. The inner product is then:
$$ Ax^*_k\cdot x_k \approx \lambda_n ( x^*_k \cdot x_k), $$
which directly leads to
$$ \frac{Ax^*_k \cdot x_k}{x^*_k \cdot x_k} \approx \lambda_n. $$
In Appendix~\ref{sec: erroranalysis}, we provide a careful analysis of the above approximation. It turns out that the number of iteration $k$ grows logarithmically with the inverse of the error, thus showing the efficiency of the power method.
The power method is improved by the Lanczos algorithm~\cite{chu1993multivariate}, which essentially builds up the Krylov subspace spanned by $[x_0, Ax_0, A^2 x_0, \dots, A^k x_0]$. By reducing $A$ to this subspace, one can solve
the largest few eigenvalues. We note that in this way the non-degeneracy constraint above (in Lemma~\ref{lemma: approx}) can be lifted.
\section{Quantum Algorithm}
In this section, we first outline a general procedure to obtain the desired quantum states that corresponds to the k-times application of matrix A. We then show how to process these states further with application of CNOT, plus Hadamard test to estimate the maximum eigenvalue $\lambda_{max} \equiv \lambda_n$. We then provide the running time analysis with a particular remark on how to improve the running time further by a factor of k.
\subsection{Main Procedure}
\label{sec: quantumalgorithm}
In this section we provide the quantum algorithm for estimating $\lambda_n$, based on the consecutive application of matrix multiplication. The strategy is similar to that used in Ref.~\cite{wiebe2012quantum}, which is based upon the result of Ref.~\cite{harrow2009quantum}. We first recall the matrix multiplication algorithm. \\
Suppose we have a normalized vector $x_0$ and we want to perform $Ax_0$. Assume that we can prepare the corresponding (normalized) state $\ket{x_0}$ with some unitary $U_0$ circuit. We note that $x_0$ can be arbitrary as long as it is not a zero vector. Therefore, $U_0$ can be some random circuit. The encoding can be nontrivial. Following Ref.~\cite{wiebe2012quantum}, we obtain a unitary circuit $U_A$ that acts as follows:
\begin{align}
U_A U_0 \ket{00...0} = \ket{\Phi_1} = C |Ax_0| \ket{Ax_0}\ket{0} + \ket{G_1}\ket{1},
\label{eqn: firstapplication}
\end{align}
where $\ket{Ax_0}$ is the normalized state corresponding to the vector $A x_0$, $\ket{G_1}$ is some state (not normalized) entangled to $\ket{1}$; $C$ is a rotational factor chosen to guarantee proper normalization so that $C|Ax_0|<1$. Next, we would like to perform $A^2 x_0 = A (Ax_0)$. The corresponding unitary $U_{A^2}$ is simply $U_A^2U_0$ plus the action on a second ancilla in the controlled rotation, and this results in:
\begin{align}
\ket{\Phi_2}
& = U_A\ket{\Phi_1}\ket{0} = U_{A^2}\ket{00...0}\ket{0} \\
& = C^2 |A^2x_0|\ket{A^2 x_0} \ket{00} + \ket{G_1}\ket{10} + \ket{G_2}\ket{01},\nonumber
\end{align}
where, $|A^2 x_0\rangle$ is the normalized state corresponding to the vector $A^2x_0$, $\ket{G_2}$ is another irrelevant state (not normalized). We note that the same constant $C$ is used again, as it is chosen to be larger than the largest eigenvalue in magnitude. There is an important detail regarding the above state: recall that the HHL algorithm~\cite{harrow2009quantum} makes use of quantum phase estimation to extract the eigenvalues of matrix $A$, and then rotates an ancilla initialized in $\ket{0}$ controlled by the phase register, followed by the uncomputation of the phase register. We do not explicitly include the phase register here, but we provide further explanation in Appendix~\ref{app:HHL} and a further simplification in Appendix~\ref{app:Improved}. In the second application of $U_A$ (Eqn. (27)), the rotation step needs to be controlled further by the state of the first ancilla with $\ket{0}$ (which is entangled to $\ket{Ax_0}$). Repeating the above procedure $k$ times, we obtain the following: (noting that $A|A x_0||Ax_0\rangle= |A^2x_0||A^2 x_0\rangle$; see also Appendix~\ref{app:HHL})
\begin{align}
\ket{\Phi_k} =& U_{A^k}\ket{00..0}\ket{0} \\
=& C^k |A^kx_0|\ket{A^kx_0}\ket{0}^{\otimes k} + \sum_{i=1}^{k} \ket{G_i}\ket{\{1\}^i},
\label{eqn: kthapplication}
\end{align}
where $\ket{\{1\}^i}$ is the $k$-bit string that has 0 everywhere but 1 at $i$-th position.\\
We remind that our goal is to compute the approximated eigenvalue $\bar{\lambda}_n$:
\begin{align}
\bar{\lambda}_n = \frac{Ax_k^* \cdot x_k}{x_k^* \cdot x_k} = \frac{(A^{k+1} x_0)^* \cdot (A^kx_0)}{(A^kx_0)^* \cdot (A^kx_0)}.
\end{align}
Now we provide a procedure to estimate $\bar{\lambda}_n$. \\
$\bullet$ Apply another round of matrix multiplication to $\ket{\Phi_k}\ket{0}$ to obtain
\begin{align}
\ket{\Phi_{k+1}} = C^{k+1}|A^{k+1} x_0| \ket{A^{k+1}x_0}\ket{0}^{\otimes k}\ket{0} + \sum_{i=1}^{k+1} \ket{G_i}\ket{1^i}.
\end{align}
$\bullet$ Add an ancilla $\ket{00}$, then we have the state $\ket{\Phi_{k+1}}\ket{00}$. Apply $c^{k+1}-X$ gates, which is the controlled gate that flips the second bit of the two extra ancillas conditioned on one of the $(k+1)$ controlled bits being 1 (or equivalently by $(k+1)$ two-qubit controlled-NOT gates with $(k+1)$ respective controls),
\begin{align}
\ket{\Phi_{k+1}^{'}} =& C^{k+1}|A^{k+1} x_0| \ket{A^{k+1}x_0}\ket{0}^{\otimes k}\ket{0} \ket{00} \nonumber\\
& + \sum_{i=1}^{k+1} \ket{G_i}\ket{1^i} \ket{01}.
\label{eqn: phik1}
\end{align}
We denote the whole process leading to $\ket{\Phi^{'}_{k+1}}$ as ${\cal U}_{\Phi^{'}_{k+1}}$.
$\bullet$ At an earlier step we obtain $\ket{\Phi_k}\ket{0}$. If we add two ancillas initialized in $\ket{00}$ to $\ket{\Phi_k}\ket{0}$, then apply $c^{k+1}-X$ gates as the previous step, but this time the gate flips the \textit{first} bit of the two extra ancillas, we arrive at
\begin{align}
\ket{\Phi_k^{'}} = C^k |A^kx_0|\ket{A^kx_0}\ket{0}^{\otimes k}\ket{0} \ket{00} + \sum_{i=1}^{k} \ket{G_i}\ket{1^i} \ket{0} \ket{10}.
\label{eqn: phik*}
\end{align}
We denote the whole process leading to $\ket{\Phi^{'}_{k}}$ as ${\cal U}_{\Phi^{'}_{k}}$.\\
$\bullet$ We note that the overlap between $\ket{\Phi^{'}_{k+1}}$ and $\ket{\Phi^{'}_{k}}$ is proportional to the inner product of the two vectors $x_{k+1}=A x_k = A^{k+1} x_0$ and $x_{k}=A^kx_0$,
\begin{align}
\braket{\Phi^{'}_{k+1},\Phi^{'}_{k}} = C^k C^{k+1} (x^*_{k+1} \cdot x_{k}).
\end{align}
Therefore, we can use the SWAP test~\cite{nielsen2002quantum} to estimate the above overlap.
To do this, we construct a controlled circuit using with the above ${\cal U}_{k+1}$ and ${\cal U}_{k}$ as well as one additional ancilla qubit, $|0\rangle\langle 0|\otimes {\cal U}_{k+1}+|1\rangle\langle 1|\otimes {\cal U}_{k}$, so that it creates a superposition of $(|0\rangle\otimes\ket{\Phi_{k+1}^{'}}+|1\rangle\otimes\ket{\Phi_{k}^{'}})/\sqrt{2}$ from an initial state $|+\rangle\otimes|0\dots0\rangle$. Measuring the ancilla in the $|\!+\!/\!-\rangle$ basis (Pauli X) gives the real part of the overlap. In addition, measuring ancilla in the $|\!+\!i/\!-\!i\rangle$ (Pauli Y) basis gives the imaginary part of the overlap.
In order to
estimate such an overlap up to an additive error $\Delta$, it takes $\mathcal{O}(1/\Delta^2)$ repetitions. \\
$\bullet$ Now we repeat the second step (the step that yields Eqn.~\ref{eqn: phik1}), but use the state $\ket{\Phi_k}\ket{0}$ instead. We then obtain
\begin{align}
\ket{\Phi_k^{''}} = C^k |A^kx_0|\ket{A^kx_0}\ket{0}^{\otimes k}\ket{0} \ket{00} + \sum_{i=1}^{k} \ket{G_i}\ket{1^i} \ket{0} \ket{01}.
\end{align}
The overlap
\begin{align}
\braket{\Phi^{'}_k, \Phi_k^{''}} = C^{2k} (x^*_k \cdot x_k)
\end{align}
can be estimated by the Hadamard test similar to the previous step. \\
$\bullet$ Now we compute the ratio (by a classical computer):
\begin{align}
\frac{\braket{\Phi_{k+1}^{'},\Phi_{k}^{'}}}{\braket{\Phi_k^{'}, \Phi_k^{''}}} = C \frac{x^*_{k+1}\cdot x_k}{x_k^* \cdot x_k} = C \bar{\lambda}_n.
\end{align}
Since $C$ is known, then we can estimate $\bar{\lambda}_n$ by dividing the above ratio by $C$. \\
\subsection{Time Complexity}
The time required to execute the above quantum algorithm involves the time it takes to apply $A$ (i.e, running the HHL-like procedure without measuring the ancilla associated with the controlled rotation) supposedly $k$ times, plus the time it takes to repeat the Hadamard test algorithm to compute the desired overlaps. Therefore, at first, the total time is supposedly
\begin{equation}
\mathcal{O}\Big( \frac{ \log(n) s^4 \kappa k^2}{\epsilon \Delta^2} \Big)
\label{eqn: timecomplexity}
\end{equation}
where $n$ is the dimension of the matrix $A$, $s$ is the sparsity of $A$, $\kappa$ is the conditional number of $A$, $\epsilon$ is the tolerance error in the matrix application, and $\Delta$ is the additive error of the estimation of $\bar{\lambda}_n$. We will now go deeper into some details of running time. We will see that it can be optimized further. \\
\textbf{Remark: } \\
$\bullet$ Note that we repeat multiplication of $A$ by $k$ times to extract the eigenvalue, which means that the running time grows linearly with $k$. The reason why we have $k^2$ in the above expression instead of $k$ is because we need to take into account of error accumulation. In each of the iteration, we aim to apply $A$ to the resulting state of the previous application of $A$.
However, each iteration induces an error $\epsilon$. Then after $k$ iterations, the error accumulates as $\mathcal{O}(k \epsilon)$. In order to obtain an error of $\mathcal{O}(\epsilon)$ at the end, we require an error $\epsilon/k$ in each \textit{individual iteration}. Fortunately, as we have provided the analysis in appendix ~\ref{sec: erroranalysis}, and found that if
$$ k \sim \log(1/\delta) $$
then $\bar{\lambda}_n$ is a $\delta$-multiplicative approximation of the real eigenvalue $\lambda_n$. Therefore, the running time is still very efficient with respect to the error tolerance. \\
$\bullet$ The running time provided in Eqn.~\ref{eqn: timecomplexity} can in fact be reduced by a factor of $k$, due to the fact that we do not need to repeat the most time-consuming step in the algorithm, which is the simulation of $\exp(-iAt)$, as well the phase estimation. As explicitly elaborated in Appendix~\ref{app:Improved}, for the purpose of extracting eigenvalues of $A$, it suffices to conditionally rotate multiple ancilla systems (after the phase estimation for eigenvalues), and hence we arrive at our desired state Eqn.~\ref{eqn: kthapplication}, up to the uncomputation of the phase estimation. Therefore, the total running time of our algorithm is reduced to
\begin{align}
\mathcal{O}\Big( \frac{ \log(n) s^4 \kappa k}{\epsilon \Delta^2} \Big).
\label{eqn: actualrt}
\end{align}
The classical running time of the power method is
$$ \mathcal{O}({\rm poly}(n) \cdot k), $$
as matrix multiplication classically proceeds by direct computation, hence requires time at least as much as the square of the dimension $n$ of the matrix. Eventually, the inner product is evaluated, and hence it requires further additional linear time.
Therefore, with respect to the size $n$ of the matrix, the quantum algorithm yields an exponential speedup if $s$ and $\kappa $ grow as fast as only a polynomial of logarithm of $n$.
As a final remark, in addition to the maximum eigenvalue, our procedure also yields a corresponding approximate eigenvector, which can be used for further processing, such as computing expectation values of observables.
\section{Some Extensions and Relevant Works}
\label{sec: discussion}
The problem that we solve in this paper is part of the \textit{matrix and eigenvalue problem}. instead.
Despite seeming simple, such a problem has far reaching consequences, in both theory and application. In pure and applied mathematics, one of the very difficult problems concerns the distribution of eigenvalues of random matrix ~\cite{tao2011random}. In physics, a physical system is characterized by the Hamiltonian, which is represented by a Hermitian matrix $H$. The ground state is the particular state that corresponds to the eigenstate of $H$ which has the smallest eigenvalue. We have assumed the matrix $A$ is Hermitian, but the extension to non-Hermitian matrix is straightforward as in Ref.~\cite{harrow2009quantum} by placing $A$ and $A^\dagger$ in the two off-diagonal blocks,
\begin{equation}
\tilde{A}=\begin{pmatrix}
0 & A \\
A^\dagger & 0
\end{pmatrix},
\end{equation}
and proceed with the Hermitian matrix $\tilde{A}$. Below, we will discuss extension of our work and point out some overlaps/connection to other works.
\subsection{Non-Sparse Matrix}
Remind that our work assumes certain sparsity of the matrix $A$. At the core of our algorithm is the application of $A$, which requires the ability to simulate $\exp(-iAt)$ to be used by the quantum phase estimation. As in \cite{harrow2009quantum}, such simulation is doable if $A$ is sparse and row-computable, thanks to the method in~\cite{berry2007efficient}. However, even if the Hermitian matrix $A$ is non-sparse, it can be decomposed as
$$ A = B^\dagger B,$$
and simulating the time evolution is still possible. If there exists a way to prepare the following state
$$ C\sum_i |B_i| \ket{i}\ket{B_i}, $$
where $C$ is an overall constant, $B_i$ refers to the vector columns of $B$, and $|B_i|$ is the $l_2$-norm of such vector, we can simulate $\exp(-iAt)$ by employing the method proposed in Ref.~\cite{lloyd2014quantum}, namely, the \textit{density matrix exponentiation}. With such an ability to simulate $\exp(-iAt)$, quantum phase estimation of eigenvalues of $A$ is thus possible and hence we can execute our quantum algorithm as above to find the maximum eigenvalue of $A$ (possibly with some scaling/or normalization factor).
In fact, a similar problem was encountered in Ref.~\cite{lloyd2014quantum}, and the authors proposed to use density matrix exponentiation combined with quantum phase estimation to solve the problem called principle component analysis. Roughly speaking, they aimed to find a few largest eigenvalues (and corresponding eigenvectors) of the so-called covariant matrix, represented by some density matrix $\sum$ and there exists a way to prepare $\sum$. Once the exponent $\exp(-i\sum t)$ is obtained, running quantum phase estimation algorithm (QPE)~\cite{kitaev1995quantum, brassard2002quantum} with the density matrix $\sum$ itself also prepared as initial state allows us to reveal highest values of the spectra by sampling. The method works well in the case where a few top eigenvalues have much higher values than the remaining ones (which is why they are called principle components). We remark that, in our problem specifically, once $\exp(-iAt)$ is obtained through a known procedure, then QPE can be used directly to reveal the maximum eigenvalue. We can, for example, simply run QPE with $\exp(-iA2\pi)$ as the main unitary and the maximally mixed state $I/n$ as the input state. We would approximately obtain the following:
\begin{align}
\frac{1}{n}\Big( \sum_{i=1}^n \ket{\Tilde{\lambda_i}}\bra{\Tilde{\lambda_i}} \otimes \ket{E_i}\bra{E_i} \Big).
\end{align}
Sampling from the above state would in fact, reveal full spectra, including the maximum eigenvalue. However, as simple as it may seem, this is not efficient as the probability of obtain different eigenvalues is the same ($=1/n)$ and hence, would require as much as $\mathcal{O}(n)$ time to obtain the maximum one. If we apply the controlled rotation to ancilla in order to multiple the eigenvalues to the corresponding components, then this essentially reduces to the quantum principle component analysis~\cite{lloyd2014quantum}, discussed above.
An alternative solution to non-sparse $A$ is a direct result of~\cite{wossnig2018quantum}, where they consider the problem of solving a dense linear system. Given the particular data structure of $A$ and an oracle access to $A$ (see Lemma 1 in~\cite{wossnig2018quantum, kerenidis2016quantum}), they outline a somewhat similar strategy to Ref.~\cite{harrow2009quantum} in order to solve the linear system. The goal is to coherently extract the singular values of $A$ (which are eigenvalues in the Hermitian case) and rotate the ancilla (to multiply the inverse of the eigenvalue). Similarly using an adaptive method introduced in~\cite{wiebe2012quantum}, we can multiply the eigenvalues instead, so as to apply the matrix $A$ instead of $A^{-1}$. A potential caveat of this approach is that the exponential speedup (with respect to size of matrix) might not be obtained, but only a quadratic speedup, as discussed concretely in~\cite{wossnig2018quantum}.
\subsection{Finding The Minimum Eigenvalue}
For a Hermitian matrix, the conditional number of a matrix $\kappa$ is defined as:
$$ \kappa = \frac{\lambda_{max}}{\lambda_{min}}. $$
Note that this formula is only exact for a Hermitian matrix. In general, singular values is taken into account instead of eigenvalues. \\
In principle, if the conditional number of a matrix is known, once the maximum eigenvalue of a matrix is revealed by the above algorithm, then the minimum eigenvalue could be estimated. However, in cases where the conditional number is not exactly known (we might, for example, know its upper bound), then it is hard to have a good estimate of the minimum eigenvalue. Here, we aim to tackle such a problem using the algorithm that we have developed above. Let $\{\lambda_i\}_{1}^n$ be eigenvalues of $A$. Since $A$ is Hermitian, so if $A$ is invertible, i.e, does not contain zero eigenvalue (or one can use a shifted matrix $A - cI$, more on this below), the eigenvalues of $A^{-1}$ is
$$ \Big\{ \frac{1}{\lambda_i}\Big\}_{i=1}^n $$
If $\lambda_{max}$ is the maximum eigenvalue of $A$, then $1/\lambda_{\max}$ is the minimum eigenvalue of $A^{-1}$. Similarly, if $\lambda_{\min}$ is the minimum eigenvalue of $A$, then $1/\lambda_{\min}$ is the maximum eigenvalue of $A^{-1}$. So instead of $A$, we simply apply $A^{-1}$ as in the HHL, then the above procedure can find the maximum eigenvalue of $A^{-1}$, from which we can find the minimum eigenvalue of $A$.
\subsection{Shifting by an Identity Matrix to Compute Ground-State Energy}
As we have mentioned above, physical systems are generally characterized by a Hamiltonian $H$, which is Hermitian. Thus, analyzing its spectrum, including finding the ground state (and possibly a few low lying excited states) and its corresponding energy, is one of the most important problems in physics. In our algorithm, we can also consider a shift by an identity matrix $A-cI$ that shifts the spectrum by a constant $c$. Thus one can compute the $\max_i |\lambda_i-c|$. Shifting has been a useful technique, for example, in finding the excited states and their spectra in many-body localization~\cite{yu2017finding}. For example, the energy level statistics in potential many-body localized systems can be gained from applying such a technique and be sped up in principle by our quantum algorithm.
If one applies a large enough shift so that all the eigenvalues are negative, then the one corresponding to the largest magnitude gives the lowest eigenvalue of $A$ and the procedure also yields its eigenvector, i.e., the ground state.
In a similar way, if one adds a sufficient large positive shift, so all the eigenvalues are positive, then one can find the largest algebraic eigenvalues.
\subsection{Largest few eigenvalues}
In Ref.~\cite{motta2020determining}, a quantum Lanczos algorithm was proposed to compute ground and excited states of a Hamiltonian $H$, based on analogously quantum imaginary time evolution $e^{-H\Delta \tau}$ (which is in turn approximated by a unitary evolution under some other Hamiltonian) and the classical Lanczos algorithm. In the classical Lanczos algorithm, the Krylov space is built in powers of $H$, whereas in the particular quantum implementation by~\cite{motta2020determining}, the Krylov space is built in powers of $\exp(-\Delta\tau H)$. The two approaches (of $H$ or of $e^{-\Delta \tau H}$) become identical in the limit $\Delta\tau \rightarrow 0$. Moreover, in the Lanczos algorithm, a crucial aspect is to compute the following inner product $v_k^{*T}Hv_k$ (and possibly with some power of $H$), where $v_k$ is some vector with unit norm. In this regard, the procedure outlined in our work is building up the Krylov subspace in terms of series of quantum states. It can then be employed to compute the largest few eigenvalues in magnitude, by using quantum computers to compute the $H$ matrix in the Krylov subspace and the overlaps among the vectors $H^m x_0$ and $H^n x_0$. We then classically diagonalize the resultant eigenvalue problem in the Krylov subspace, yielding the largest few eigenvalues (in magnitude).
Moreover, by combining the shifting and the choice of applying $H$ or $H^{-1}$, we can generalize our algorithm to obtaining a few largest or smallest eigenvalues either in magnitude or in real value. Such capability can potentially lead to more efficient access to, e.g., the ground-state energy and those of a few low lying excited states, as many physics models possess Hamiltonians whose matrices are spare.
\section{Conclusion}
Inspired by many developments in quantum algorithm, we have outlined an efficient quantum algorithm for estimating largest absolute eigenvalue of a sparse Hermitian matrix. Possible extension to non-sparse and solving related eigenvalues problems are discussed. The core routines of our algorithm are the consecutive application of adaptive version of HHL algorithm~\cite{harrow2009quantum, wiebe2012quantum}, plus the Hadamard test, both of which have proved to be useful in many contexts, such as~\cite{lloyd2013quantum, motta2020determining, wiebe2012quantum}, etc. Our work thus suggests an efficient quantum approach to a particular computational problem, i.e., solving the eigenvalue problem, for which classical algorithms requires polynomial time with respect to the size of input matrix. Identifying difficult computational problems and finding efficient solutions in both classical and quantum contexts have always been one of the major drivers to improve computation capability and extend its applications. Our work has contributed another small step to enrich the existing quantum algorithm pool, and can be used as a subroutine (just as the HHL algorithm) in other quantum algorithms or future applications.
\medskip
\noindent {\bf Acknowledgements}.
This work was supported in part by the U. S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA)
under contract number DE-SC0012704, in particular, for the design of the main algorithm, and by the National Science Foundation under Grant No. PHY
1915165, in particular, for the extension of the algorithm and its application to physical Hamiltonians.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,540 |
The 33rd Street station is a local station on the IRT Lexington Avenue Line of the New York City Subway. Located at the intersection of Park Avenue and 33rd Street in the Murray Hill neighborhood of Manhattan, it is served by trains at all times, <6> trains during weekdays in the peak direction, and trains during late night hours.
The 33rd Street station was constructed for the Interborough Rapid Transit Company (IRT) as part of the city's first subway line, which was approved in 1900. Construction of the line segment that includes the 33rd Street station started on September 12 of the same year. The station opened on October 27, 1904, as one of the original 28 stations of the New York City Subway. After the city's first subway line was split into multiple lines in 1918, there was a failed proposal in the 1920s to convert 33rd Street into an express station. The station's platforms were lengthened in the late 1940s.
The 33rd Street station contains two side platforms and four tracks; express trains use the inner two tracks to bypass the station. The station was built with tile and mosaic decorations, which are continued along the platform extensions. The platforms contain exits to 32nd Street to the south and 33rd Street to the north. The platforms are not connected to each other within fare control. The original station interior is a New York City designated landmark and listed on the National Register of Historic Places.
History
Construction and opening
Planning for a subway line in New York City dates to 1864. However, development of what would become the city's first subway line did not start until 1894, when the New York State Legislature authorized the Rapid Transit Act. The subway plans were drawn up by a team of engineers led by William Barclay Parsons, chief engineer of the Rapid Transit Commission. It called for a subway line from New York City Hall in lower Manhattan to the Upper West Side, where two branches would lead north into the Bronx. A plan was formally adopted in 1897, and all legal conflicts concerning the route alignment were resolved near the end of 1899. The Rapid Transit Construction Company, organized by John B. McDonald and funded by August Belmont Jr., signed the initial Contract 1 with the Rapid Transit Commission in February 1900, in which it would construct the subway and maintain a 50-year operating lease from the opening of the line. In 1901, the firm of Heins & LaFarge was hired to design the underground stations. Belmont incorporated the Interborough Rapid Transit Company (IRT) in April 1902 to operate the subway.
The 33rd Street station was constructed as part of the route segment from Great Jones Street to 41st Street. Construction on this section of the line began on September 12, 1900. The section from Great Jones Street to a point 100 feet (30 m) north of 33rd Street was awarded to Holbrook, Cabot & Daly Contracting Company, while the remaining section to 41st Street was done by Ira A. Shaler. The section between 33rd and 41st Streets was built as two double-track tunnels. To accommodate a never-built connection to the mainline platforms at Grand Central Terminal, the tunnel carrying northbound trains was shifted eastward (nearly touching the eastern curb line of Park Avenue). At the time, the railroads that operated the terminal had not agreed to the connection. Property owners did not learn about the change until a series of accidents occurred along the excavation site in 1902. A dynamite explosion near Park Avenue and 41st Street on January 27, 1902, killed five people, and several mansions on Park Avenue fell into the excavation site that March due to rockslides. Shaler became known by the pejorative nickname of "hoodoo contractor" as a result. After Shaler was killed by a rockslide in his own excavation site on June 17, 1902, his estate completed the construction of the tunnel between 33rd and 41st Street.
By late 1903, the subway was nearly complete, but the IRT Powerhouse and the system's electrical substations were still under construction, delaying the system's opening. The 33rd Street station opened on October 27, 1904, as one of the original 28 stations of the New York City Subway from City Hall to 145th Street on the Broadway–Seventh Avenue Line. Litigation over the IRT's Murray Hill tunnels continued for several years; in 1905, a judge found that the city government was responsible for the January 1902 explosion. In spite of this, the northbound tunnel was never relocated, as it had already been completed.
Service changes and station renovations
After the first subway line was completed in 1908, the station was served by local trains along both the West Side (now the Broadway–Seventh Avenue Line to Van Cortlandt Park–242nd Street) and East Side (now the Lenox Avenue Line). West Side local trains had their southern terminus at City Hall during rush hours and South Ferry at other times, and had their northern terminus at 242nd Street. East Side local trains ran from City Hall to Lenox Avenue (145th Street).
To address overcrowding, in 1909, the New York Public Service Commission proposed lengthening platforms at stations along the original IRT subway. As part of a modification to the IRT's construction contracts, made on January 18, 1910, the company was to lengthen station platforms to accommodate ten-car express and six-car local trains. In addition to $1.5 million (equivalent to $ million in ) spent on platform lengthening, $500,000 () was spent on building additional entrances and exits. It was anticipated that these improvements would increase capacity by 25 percent. Both platforms at the 33rd Street station was extended to the south. New "electric manholes", passageways leading to the equipment closets, were built at the southern ends of the platforms. Six-car local trains began operating in October 1910. The Lexington Avenue Line opened north of Grand Central–42nd Street in 1918, thereby dividing the original line into an "H"-shaped system. All local trains were sent via the Lexington Avenue Line, running along the Pelham Line in the Bronx.
In December 1922, the Transit Commission approved a proposal to convert the 33rd Street station into an express stop. It was estimated that the extra time spent by express trains at 33rd Street would be offset by the reduced dwell times at Grand Central. Local business owners supported the proposal, but the IRT opposed the plan, which would cost the company $750,000. In October 1923, the plan was postponed for a year due to a lack of funds. The Fifth Avenue Association requested in January 1924 that the Transit Commission again consider converting the 33rd Street station into an express stop, citing the fact that a 35-story structure was to be built immediately adjacent to the station. The express-stop proposal was postponed indefinitely in 1925. The Fifth Avenue Association requested in 1929 that the express-station proposal be reconsidered. The association said the conversion would "complete a quadrilateral of express stops" that included 34th Street–Penn Station, Times Square–42nd Street, and Grand Central–42nd Street.
In 1928, to alleviate overcrowding on the Lexington Avenue Line, a consulting engineer for the New York State Transit Commission proposed the construction of "reservoir" stations at 33rd/34th and 42nd Streets. The proposal entailed constructing a northbound-only tunnel under Lexington Avenue from 30th to 42nd Street, with stations at 34th and 42nd Streets, then converting the IRT tunnel under Park Avenue and the existing 33rd and 42nd Street stations to southbound-only use. The northbound and southbound stations at 33rd/34th and 42nd Streets would both have had two express tracks and one local track; the express tracks in either direction would have merged with each other north of 42nd Street and south of 30th Street. Although the "reservoir" plan was technically feasible, the $25 million projected cost was too high.
The city government took over the IRT's operations on June 12, 1940. On April 13, 1948, the platform extensions to accommodate ten-car trains at this station, along with those at 23rd Street and 28th Street, were opened for use. On December 27, 1948, a new entrance to the station at 32nd Street opened for use.
In 1979, the New York City Landmarks Preservation Commission designated the space within the boundaries of the original station, excluding expansions made after 1904, as a city landmark. The station was designated along with eleven others on the original IRT. The original interiors were listed on the National Register of Historic Places in 2004.
Station layout
Like other local stations, 33rd Street has four tracks and two side platforms. The 6 stops here at all times, rush-hour and midday <6> trains stop here in the peak direction; and the 4 stops here during late nights. The two express tracks are used by the 4 and 5 trains during daytime hours. The platforms were originally long, as at other local stations on the original IRT, but later became long. The platform extensions are at the southern ends of the original platforms. The express tracks stay level, while the local tracks slowly incline from south to north to allow for the easier deceleration of local trains. This results in a layout where the express tracks are at a lower elevation than the local tracks in the northern half of the station.
North of the station, the two pairs of tracks in each direction separate into different tunnels because of the presence of the Murray Hill Tunnel, which runs under the center of this section of Park Avenue. As built, the tunnels were supposed to have been apart, but the northbound tunnel (to the east) was shifted eastward by another to accommodate a three-track connection from the original IRT subway north to the mainline Grand Central Terminal. There is a drainage pipe connecting the two tunnels between 37th and 38th Streets, as well as a cross-passage between 38th and 39th Streets.
Design
As with other stations built as part of the original IRT, the station was constructed using a cut-and-cover method. The tunnel is covered by a "U"-shaped trough that contains utility pipes and wires. The bottom of this trough contains a foundation of concrete no less than thick. Each platform consists of concrete slabs, beneath which are drainage basins. The original platforms contain I-beam columns spaced every , while the platform extensions contain columns with white glazed tiles. Additional columns between the tracks, spaced every , support the jack-arched concrete station roofs. The ceiling height varies, being about above platform level near the northern fare control areas, and lower in other portions of the station. There is a gap between the trough wall and the platform walls, which are made of -thick brick covered over by a tiled finish.
The fare control areas are at platform level, and there is no crossover or crossunder between the platforms. The walls along the platforms near the fare control areas consist of a brick wainscoting on the lowest part of the wall, with bronze air vents along the wainscoting, and white glass tiles above. The platform walls are divided at intervals by buff and green mosaic tile pilasters, or vertical bands. In the original portion of the station, each pilaster is topped by green faience plaques depicting eagles, an allusion to the former 71st Regiment Armory at Park Avenue and 33rd Street; the eagles hold blue and white shields containing the number "33". A cornice with yellow and brown vine and fretwork patterns runs atop these walls. The platform extensions contain tiles with the number "33" atop the pilasters. Mosaic plaques with the words "33rd St." are also spaced at various intervals on the walls. The mosaic tiles at all original IRT stations were manufactured by the American Encaustic Tile Company, which subcontracted the installations at each station. The decorative work was performed by tile contractor John H. Parry and faience contractor Grueby Faience Company. The ceilings of the northern fare control areas contain plaster molding.
The 1997 artwork at this station is Lariat Seat Loops by James Garvey. These are composed of fourteen bronze loops surrounding the I-beam columns near the northern fare control areas, which are designed as handholds or seat rests. According to Garvey, "the thick bronze bar ... resembles the lasso demonstration in a Will Rogers film clip". Garvey subsequently designed Lariat Tapers, a similar artwork at the Wall Street station, in 2011.
Exits
Each platform has exits to both 32nd and 33rd Streets. The northbound platform's exits are on the eastern side of Park Avenue while the southbound platform's exits are on the western side. The street staircases contain relatively simple, modern steel railings like those seen at most New York City Subway stations.
At 33rd Street, each control area contains two exits, one each to the north and south sides of 33rd Street. These exits are directly outside 4 Park Avenue to the northwest, 2 Park Avenue to the southwest, 3 Park Avenue to the northeast, and 1 Park Avenue to the southeast. At 32nd Street, each control area contains two exits to the south side of that street.
References
Further reading
Lee Stokey. Subway Ceramics: A History and Iconography. 1994.
External links
Station Reporter — 4 Train
Station Reporter — 6 Train
Forgotten NY — Original 28 - NYC's First 28 Subway Stations
MTA's Arts For Transit — 33rd Street (IRT Lexington Avenue Line)
32nd Street entrance from Google Maps Street View
33rd Street entrance from Google Maps Street View
downtown platform from Google Maps Street View
Murray Hill, Manhattan
IRT Lexington Avenue Line stations
Railway and subway stations on the National Register of Historic Places in Manhattan
New York City Subway stations in Manhattan
Railway stations in the United States opened in 1904
1904 establishments in New York City
New York City Designated Landmarks in Manhattan
New York City interior landmarks | {
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Jacksonville, Florida (NASDAQ-LSTR) Landstar System, Inc., a safety-first non-asset based provider of transportation capacity and logistics services, has named Debbie Adams April Employee of the Month at the company's Jacksonville, Florida, service center.
Adams, a coordinator in Landstar's Automotive Operations department joined Landstar in 1998. She was recognized for her troubleshooting skills and ability to keep just-in-time deliveries of automotive parts and supplies on schedule.
"Debbie's dedication and focus demonstrates the key role our employees play in keeping Landstar a leader in the transportation industry," said Landstar President and CEO Henry Gerkens. "She is well deserving of the honor.
Adams makes her home in Sanderson, Florida, and enjoys scrapbooking as a hobby.
Landstar System, Inc. delivers safe, specialized transportation services to a broad range of customers world-wide. The Company identifies and fulfills shippers' needs through the coordination of individual businesses comprised of independent sales agents and third-party transportation capacity providers. Landstar's carrier group, which is comprised of Landstar Gemini, Inc., Landstar Inway, Inc., Landstar Ligon, Inc., Landstar Ranger, Inc. and Landstar Carrier Services, Inc., delivers excellence in complete over-the-road transportation services. Landstar's global logistics group, which is comprised of Landstar Global Logistics, Inc. and its subsidiaries Landstar Express America, Inc. and Landstar Logistics, Inc., provides international and domestic multimodal (over-the-road, air, ocean and rail) transportation, expedited, warehousing and contract logistics services. All Landstar operating companies are certified to ISO 9001:2000 quality management system standards. Landstar System, Inc. is headquartered in Jacksonville , Florida . Its common stock trades on The NASDAQ Stock Market® under the symbol LSTR. | {
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} | 6,587 |
Ritika Singh is an Indian actress and mixed martial artist, who predominantly appears in Tamil films and also has appeared in Hindi and Telugu films. Ritika Singh trained as a kickboxer and as a mixed martial artist from her childhood, She made her debut in a national competition by competing in the 52 kg category at the 2009 Asian Indoor Games as a kick boxer . Here is Ritika Singh Stylish Entry at Film Fare Award Gallery .
Discover the latest trends from Clever Fashion Media including inspiration from celebrities and trendy outfit ideas from experts in fashion photos of the best spring fashion trends including spring dresses sandals and spring shoes new bags for spring and outfit ideas ,Runway and front row photos from fashion week shows in New York, London, Paris, and Milanfashion week coverage and the best dressed stars on the red carpet The different styles in fashion have always gone through innumerable changes With the increase in the amount of innovations, the change in trend and fashion styles The best source for fashion, beauty, runway, designer trends and celebrity style are from different sources, Fashion is not only in clothing and also in footwear, accessories, makeup, hairstyle and body Latest Fashion not only in Runways even from street can find a fashion our Fashion and styles are differ from person to person, some style is suitable for some but the same fashion is not suite for another, so we should aware about our body and skin color might our select of dresses is good and same dress made an attraction when some models or actress weared but we are in that dress is not made any attraction that means we selected dress is good but that dress is not suitable for our body, Especially we are appear in weddings events , parties every one wants be a center of attraction Award such times but its not an easy thing In this time Clever Fashion Media Can give an idea about Latest fashion, New hairstyle, trends in foot wear, makeup which color dress should select,selection of lipstick and what are the other accessories should select and also in Clever Fashion Media discussing about Trends in Red carpets, photoshoots of our celebrity and popular fashion designers new collections, Beauty tips , Healthy Lifestyle and much more Ritika Singh Stylish Entry at Film Fare Award . | {
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Several of my appliances are getting old and will need to be replaced soon. Will the appliance choices I make have much impact on my energy bill?
Dear Chelsea: Your energy use varies month to month, so it can be difficult to see how much difference an appliance purchase makes. It's best to view the purchase over the lifetime of the equipment.
Think about the upfront cost and the lifetime energy cost. In a Consumer Reports test, the most-efficient refrigerator used $68 less in electricity per year than the least-efficient model. Multiply that difference over a decade or two, and the lifetime energy savings could be greater than the upfront cost. Some initial research is all it takes to get the best appliance for your needs.
Appliance energy use is usually less, on average, than home heating and cooling bills but can be several hundred dollars each year. Your appliance use depends on factors like the model, how often you use it, the settings you use for its particular function and even the time of day it is used most.
Over the last few decades, new appliances have become more energy-efficient, driven partly by minimum government standards. These standards, created by the U.S. Department of Energy, save consumers more than $60 billion each year. Appliances are required to include Energy Guide labels that show estimated energy use and operating cost per year. These labels help you compare different models and calculate the initial cost against the long-term savings.
Some appliances will also have an ENERGY STAR label. This indicates the appliance is substantially more efficient than the minimum standard. Your greatest energy-savings opportunities can come from replacing an old appliance with an ENERGY STAR-rated model. Removing a refrigerator that's 20 years old and replacing it with a new ENERGY STAR fridge can lower the monthly electricity cost by 75 percent, from $16.50 to less than $4.
In some cases, the configuration of the appliance can also make a substantial difference. For example, a side-by-side refrigerator/freezer uses about 70 percent more energy than other configurations — all the most-efficient models have the refrigerator stacked on top of the freezer. All 36 of the most-efficient clothes washers of 2018 are front-loading models.
Make sure there is adequate airflow between the wall and the back of the unit.
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Air-dry dishes rather than using the heated-dry function.
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Make the most out of your appliance energy use with a little research before buying a new model, and make a few easy adjustments to the way you use them. | {
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Martres-Tolosane – miejscowość i gmina we Francji, w regionie Oksytania, w departamencie Górna Garonna.
Według danych na rok 1990 gminę zamieszkiwało 1 929 osób, a gęstość zaludnienia wynosiła 82 osoby/km² (wśród 3020 gmin regionu Midi-Pyrénées Martres-Tolosane plasuje się na 192. miejscu pod względem liczby ludności, natomiast pod względem powierzchni na miejscu 453.).
Bibliografia
Miejscowości w departamencie Górna Garonna | {
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} | 1,071 |
by Zach Theatre
Jan. 12 - Jan. 21
In this cult classic, sweethearts Brad and Janet, stuck with a flat tire during a storm, discover the eerie mansion of Dr. Frank-N-Furter. As their innocence is lost, Brad and Janet meet a houseful of wild characters, including a rocking biker and a creepy butler. Through elaborate dances and rock songs, Frank-N-Furter unveils his latest creation: a muscular man named "Rocky."
ZACH Theatre is currently seeking talented actors and actor/musicians of all ethnicities for our upcoming production of The Rocky Horror Show. Actors local to the Austin, Texas area particularly encouraged to submit. Due to current COVID – 19 restrictions, video submissions will be accepted as first-round auditions. In person callbacks will be held in Austin the first week of February and in NYC the second week of February.
Book, Music and Lyrics by Richard O'Brien
Directed by Dave Steakley
Musical Direction by Allen Robertson
REHEARSAL & PERFORMANCE SCHEDULE
Rehearsals begin March 15, 2022. Performances begin April 6, 2022, and close May 1, 2022, with six to seven shows per week: Wednesdays through Sundays at 7:30 pm and some Saturday and Sundays at 2:30 pm.
ZACH is intending to operate as a fully vaccinated workplace per the guidance issued by Actors Equity. AEA members will be expected to follow strict health and safety protocols set forth in accordance with Actors Equity Associations guidelines.
Please film your audition, beginning with a slate, followed by a brief cut of a song from the show. We welcome the use of any cut or accompaniment that you prefer, but accompaniment tracks can be found HERE if needed.
If interested in submitting as an actor/musician please record one video of yourself singing with a track played by a secondary audio source so it is audible in the recording and then a second video of you accompanying yourself on your instrument.
Please film your audition in front of a clean background, with no backlighting. As much as possible, limit ambient noise including air conditioners and other electronics.
Please submit your video, headshot, and resume at THIS LINK. Submission Deadline: Friday, 1/21 at 6 pm CST.
ROLES TO BE CAST:
USHERETTE/MAGENTA: Female, Age Flexible (Range: Mezzo Soprano Belt, Bb3-Eb5)
One of Frank's servants. Interested to see actor/musicians for this role.
BRAD: Male, 25-35 (Range: Bari Tenor, Bb2-G4)
Quirky, but very much in love with his fiancé, Janet. Overly optimistic at times.
JANET: Female, 25-35 (Range: Mezzo Soprano Belt, A3-Eb5)
Good girl who is madly in love with Brad. Always seems to be frightened of something. Keeps losing more of her clothes throughout the story. Emotionally weak and caves into pressure easily.
RIFF RAFF: Male, Age Flexible (Range: High Rock Tenor, D3-B4)
Makes harmless conversation awkward and foreboding. One of Frank's servants. Interested to see actor/musicians for this role.
USHERETTE/COLUMBIA: Female, Age Flexible (Range: Mezzo Soprano Belt, E4-E5)
FRANK 'N' FURTER: Male Transvestite, 30s-50s (Range: Baritone, D3-G4)
Master of the castle. Welcomes Janet and Brad with open arms. Obsessed with creating a man to be part of his sexual entourage. Master of seduction.
ROCKY: Male 18-30 (Range: Tenor, A3-G4)
Frank's magnificent creation. Sexually appealing with prominent muscles.
EDDIE/DR. SCOTT: Male, 25-50 (Range: Baritone, E3-F#4)
Eddie, delivery man, "went to pieces." Misses the rock and role of life. Comes back to life only to die after his solo. Scott is Eddie's Uncle. Looking for a musician who plays guitar or bass.
ENSEMBLE: Age Flexible
About Zach Theatre
Currently celebrating our 82nd Season, ZACH Theatre is one of Austin's most vibrant and innovative performing arts organizations, creating intimate theatre experiences that ignite the imagination, inspire the spirit, and …
108 productions
43 casting calls
Upcoming Auditions
Video Auditions (Only) for Dot, by Ground Floor Theatre, Deadline EXTENDED to January 20, 2022
Video Auditions for The Rocky Horror Show, by Zach Theatre -- Deadline January 21, 2022
Seeking Female or Non-binary Latinx Performer for THE GLOBAL ARENA, by Glass Half Full Theatre | {
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package com.miraclewong.mwzhihudaily.model.entity;
import com.google.gson.annotations.SerializedName;
import java.io.Serializable;
import java.util.List;
/**
* Created by miraclewong on 16/3/12.
*/
public class Daily implements Serializable {
@SerializedName("images")
public List<String> images;
@SerializedName("image")
public String image;
@SerializedName("type")
public int type;
@SerializedName("id")
public int id;
@SerializedName("ga_prefix")
public int ga_prefix;
@SerializedName("title")
public String title;
}
| {
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} | 2,189 |
\section{Introduction}
\label{section1}
In the present paper we investigate orthonormal sections of normal bundles of two-dimensional immersions in Euclidian space $\mathbb R^{n+2}$, $n\ge 2,$ which are critical for the \emph{functional of total torsion}
\begin{equation*
{\mathcal T}_X({\mathcal N})
=\sum_{\sigma,\vartheta=1}^n
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
h^{ij}T_{\sigma,i}^\vartheta T_{\sigma,j}^\vartheta\,W\,dudv;
\end{equation*}
see section \ref{section3} for the precise definition.\\[1ex]
Continuing the paper \cite{Froehlich_Mueller_01} on $2$-surfaces in $\mathbb R^4,$ we consider both, flat and non-flat normal bundles:
\begin{itemize}
\item[-]
For the flat case we prove that all torsion coefficients $T_{\sigma,i}^\vartheta$ of a critical orthonormal normal section have to vanish.
\vspace*{-1ex}
\item[-]
For the general case of normal bundles with non-vanishing curvature we derive estimates for the total torsion and the torsion coefficients of such sections.
\end{itemize}
The main difference to the case of immersions in $\mathbb R^4$ (that means $n=2$) is the nonlinear character of the pertinent differential equations appearing for $n\ge3$. Consequently, completely new arguments have to be applied. For instance, for $n=3$, all our results are based on the fact that a certain integral function of the torsion coefficients solves an (inhomogeneous) $H$-surface system; see section \ref{section4} for details.\\[1ex]
The outline of this paper is as follows:
\begin{itemize}
\item
In the next section we introduce the basic definitions of $2$-immersions $X$ in Euclidean spaces $\mathbb R^{n+2}$ for natural $n\ge 1.$
\vspace*{-1ex}
\item
In section 3 we introduce the functional of total torsion ${\mathcal T}_X({\mathcal N})$ and explain its significance for possible applications. Then, we compute the first variation of $\mathcal T_X(\mathcal N)$ w.r.t. $SO(n)$-perturbations as well as the associated Euler-Lagrange system.
\vspace*{-1ex}
\item
Interpreting the Euler-Lagrange equations as integrability conditions, we derive a system of second order with quadratic growth in the gradient in section 4.
\vspace*{-1ex}
\item
In section 5 we prove that the torsion coefficients of critical orthonormal normal sections vanish identically, whenever the normal bundle is flat.
\vspace*{-1ex}
\item
Finally, in section 6 the reader finds a lower bound for the total torsion as well as an $L^\infty$-estimate for the torsion coefficients of critical normal sections, both for the general case of non-flat normal bundles in higher codimension.
\end{itemize}
\section{Basic settings and definition of the torsion}
\label{section2}
\setcounter{equation}{0}
\subsection{Two-dimensional immersions}
Let $n\ge 1$ be a natural number. We consider two-dimensional immersions
\begin{equation*
X=X(u,v)=\big(x^1(u,v),\ldots,x^{n+2}(u,v)\big)\in C^4(B,\mathbb R^{n+2}\,)
\end{equation*}
in the Euclidean space $\mathbb R^{n+2},$ parametrized on the closed unit disc
\begin{equation*
B=\Big\{(u,v)\in\mathbb R^2\,:\,u^2+v^2\le 1\Big\}\subset\mathbb R^2\,,
\end{equation*}
and such that the regularity condition
\begin{equation}\label{2.3}
\mbox{rank}
\begin{pmatrix} X_u(u,v)\\ X_v(u,v) \end{pmatrix}
=2\quad\mbox{in}\ B
\end{equation}
is satisfied. Furthermore, we write $\mathring{B}=\big\{(u,v)\in\mathbb R^2\,:\,u^2+v^2<1\big\}$ for the open unit disc, its boundary is denoted by $\partial B=\big\{(u,v)\in\mathbb R^2\,:\,u^2+v^2=1\big\},$ and finally we set $B_\varrho(w_0):=\{w\in\mathbb R^2\,:\,|w-w_0|\le\varrho^2\}.$
\subsection{Conformal parametrization}
Let our immersions $X$ be conformally parametrized: That is, writing $X_{u^i}$ for the partial derivative of $X$ w.r.t. $u^i$ (where $u^1\equiv u$ and $u^2\equiv v$), there hold the \emph{conformality relations}
\begin{equation}\label{2.4}
X_{u^i}\cdot X_{u^j}^t=:h_{ij}=W\delta_{ij}\quad\mbox{in}\ B
\end{equation}
for $i,j=1,2$. Here we used the area element
\begin{equation*
W:=\sqrt{h_{11}h_{22}-h_{12}^2}\,
\end{equation*}
(note that $W>0$ in $B$ due to (\ref{2.3})) and the Kronecker symbol
\begin{equation*
\delta_{ij}
:=\left\{
\begin{array}{l}
1\quad\mbox{if}\ i=j \\[0.1cm]
0\quad\mbox{if}\ i\not=j
\end{array}
\right.
\quad\mbox{for}\ i,j=1,2,
\end{equation*}
and, finally, $Z^t\in\mathbb R^d$ means the transposed vector of any $Z\in\mathbb R^d,$ $d\in\mathbb N$. As is well known, there is no restriction in assuming $X$ to be conformally parametrized, see e.g. \cite{Sauvigny_01}.
\subsection{Normal sections, torsion coefficients, and curvature \\ of the normal bundle}
Let ${\mathcal N}:=\big\{N_1,\ldots,N_n\big\}$ form an orthonormal section of the normal bundle of the immersion $X$ with the following properties:
\begin{equation*
N_\sigma\in C^3(B,\mathbb R^{n+2}\,),\quad
X_{u^j}\cdot N_\sigma^t=0,\quad
N_\sigma\cdot N_\vartheta^t=\delta_{\sigma\vartheta}
\quad\mbox{in}\ B
\end{equation*}
for all $i=1,2$ and all $\sigma,\vartheta=1,\ldots,n.$
\goodbreak\noindent
To such a section ${\mathcal N}$ we associate the so-called torsion coefficients in the following sense:
\begin{definition}
The torsion coefficients of an orthonormal normal section ${\mathcal N}$ (shortly: ONS ${\mathcal N}$) are defined as
\begin{equation*
T_{\sigma,i}^\vartheta:=N_{\sigma,u^i}\cdot N_\vartheta^t\,,\quad i=1,2,\ \sigma,\vartheta=1,\ldots,n.
\end{equation*}
\end{definition}
\begin{remarks}\quad
\begin{itemize}
\item[1.]
Note the skew-symmetry of the torsion coefficients:
\begin{equation*
T_{\sigma,i}^\vartheta=-T_{\vartheta,i}^\sigma
\quad\mbox{for all}\ i=1,2,\ \sigma,\vartheta=1,\ldots,n.
\end{equation*}
\item[2.]
The $T_{\sigma,i}^\vartheta$ are exactly the coefficients of the normal connection (see e.g. \cite{doCarmo_01}), while our name {\it torsion coefficients} follows the one-dimensional theory of space curves. Namely, if ${\mathfrak n}$ and ${\mathfrak b}$ denote the normal resp. the binormal of an arc-length parametrized curve $c=c(s),$ then its torsion is defined as the inner product ${\mathfrak n}(s)'\cdot{\mathfrak b}(s)^t.$
\end{itemize}
\end{remarks}
\noindent
The coefficients $S_{\sigma,ij}^\vartheta\in C^1(B,\mathbb R)$ of the curvature tensor ${\mathfrak S}$ of the normal bundle are given by
\begin{equation}\label{2.10}
S_{\sigma,ij}^\vartheta
:=T_{\sigma,i,u^j}^\vartheta-T_{\sigma,j,u^i}^\vartheta
+T_{\sigma,i}^\omega T_{\omega,j}^\vartheta-T_{\sigma,j}^\omega T_{\omega,i}^\vartheta\,,\quad
i,j=1,2,\ \sigma,\vartheta=1,\ldots,n
\end{equation}
(see again \cite{doCarmo_01}; summation convention for $\omega=1,\ldots,n$). Note that the $S_{\sigma,ij}^\vartheta$ are skew-symmetric in $i,j$ and $\sigma,\vartheta$. Consequently, they are completely described by the $N:=\frac{1}{2}n(n-1)$ quantities
\begin{equation}\label{2.11}
S_{\sigma,12}^\vartheta\quad
\mbox{for}\ (\sigma,\vartheta)\in U_n:=\Big\{(\omega,\delta)\in\{1,\ldots,n\}^2\,:\ \omega<\delta\Big\}.
\end{equation}
For example, in $\mathbb R^4$ there is -- up to the sign -- only one relevant quantity $S_{1,12}^2.$
\section{The total torsion and its properties
\label{section3}
\setcounter{equation}{0}
\subsection{Definition of the total torsion}
In the paper at hand we study ONS ${\mathcal N}$ which are critical for the following {\it functional of total torsion} (summation convention for $i,j=1,2$)
\begin{equation}\label{3.1}
{\mathcal T}_X({\mathcal N})
:=\sum_{\sigma,\vartheta=1}^n\,
\int\hspace*{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
h^{ij}\,T_{\sigma,i}^\vartheta T_{\sigma,j}^\vartheta\,W\,dudv,
\end{equation}
where the $h^{ij}$ are the elements of the inverse matrix to $(h_{ij})_{i,j=1,2}$ from (\ref{2.4}):
\begin{equation*
h_{ij}h^{jk}=\delta_i^k\quad\mbox{in}\ B\quad \mbox{for}\ i,k=1,2.
\end{equation*}
\begin{remark}
The total torsion ${\mathcal T}_X$ does not depend on the parametrization of $X$, but it depends on the chosen ONS ${\mathcal N}.$
\end{remark}
\noindent
Taking the conformal parametrization (\ref{2.4}) of $X$ into account and using the definition of $U_n$ from (\ref{2.11}), ${\mathcal T}_X$ takes the form
\begin{equation}\label{3.3}
{\mathcal T}_X({\mathcal N})
=2\sum_{(\sigma,\vartheta)\in U_n}\,\int\hspace*{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\Big\{(T_{\sigma,1}^\vartheta)^2+(T_{\sigma,2}^\vartheta)^2\Big\}\,dudv.
\end{equation}
We want to establish bounds for this functional as well as for the torsions $T_{\sigma,i}^\vartheta$ of critical ONS ${\mathcal N},$ the latter in terms of the value of ${\mathcal T}_X$ itself and an $L^\infty$-bound for $S_{\sigma,12}^\vartheta.$
\subsection{Fields of application}
\begin{itemize}
\item[1.]
First, the total torsion appears in many concrete situations, for example, in the second variation formula of the area functional
\begin{equation*
{\mathcal A}[X]
:=\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}\sqrt{h_{11}h_{22}-h_{12}^2}\,dudv.
\end{equation*}
Namely, choose an ONS ${\mathcal N}=\{N_1,\ldots,N_n\},$ and consider a normal variation $\widetilde X=X+\chi N_\omega$ of a conformally parametrized minimal surface $X,$ where $N_\omega\in{\mathcal N}$ and $\chi\in C_0^\infty(B,\mathbb R).$ For the second variation of ${\mathcal A}[X]$ w.r.t.~$N_\omega\in{\mathcal N}$ one then computes
\begin{equation*
\delta_{N_\omega}^2{\mathcal A}[X;\chi]
=\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}(|\nabla\chi|^2+2K_{N_\omega}W\chi^2)\,dudv
+\sum_{\sigma=1}^n
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\Big\{
(T_{\omega,1}^\sigma)^2+(T_{\omega,2}^\sigma)^2
\Big\}\,\chi^2\,dudv
\end{equation*}
with the ``Gaussian curvature'' $K_{N_\omega}$ w.r.t. $N_\omega$ (see e.g. \cite{Froehlich_01}). Therefore, it is desirable to control the torsion coefficients of suitable chosen ONS ${\mathcal N}.$
\item[2.]
Next, taking Ricci's integrability conditions
$$S_{\sigma,12}^\vartheta=(L_{\sigma,1m}L_{\vartheta,2n}-L_{\sigma,2m}L_{\vartheta,1n})h^{mn},\quad
L_{\sigma,ij}:=-X_{u^i}\cdot N_{\sigma,u^j}^t=X_{u^iu^j}\cdot N_{\sigma}^t\,,$$
into account (see e.g. \cite{doCarmo_01}), we can bound the curvature of the normal bundle in terms of the Gaussian curvature $K$ and the length of the mean curvature vector ${\mathcal H}$ of $X:$
$$|S_{\sigma,12}^\vartheta|\le 2\,\big\{|{\mathcal H}|^2-K\big\}\,W.$$
Therefore, to control a geometric quantity in terms of $|S_{\sigma,12}^\vartheta|$ means to control it by means of $|{\mathcal H}|^2-K.$
\item[3.]
Finally, and more generally, the differential geometry of immersions with non-trivial normal bundles is certainly far away from beeing completely developed. This is manifested in the fact that many problems are satisfactorally solved only in the case of vanishing curvature tensor ${\mathfrak S}$ (see e.g. \cite{Ferapontov_01}, \cite{Smoczyk_Wang_Xin_01}), or if one restricts to special geometric situations (see e.g. \cite{Bergner_Froehlich_01} for curvature estimates for {\it graphs}).\\[0.6ex]
With this paper we aim at giving partial answers to questions like these:
\begin{itemize}
\item
What characteristic geometrical quantities of immersions can be controlled in terms of the curvature tensor ${\mathfrak S}?$
\vspace*{-0.2ex}
\item
What geometric properties share immersions with the same, possibly constant curvature of the normal bundle?
\end{itemize}
\end{itemize}
\subsection{The Euler-Lagrange equations}
We will derive the Euler-Lagrange equations for critical ONS ${\mathcal N}$. To this end, we consider a one-parameter family of rotations
\begin{equation*
{\mathbf R}(w,\varepsilon)
=\big(r_{\sigma\vartheta}(w,\varepsilon)\big)_{\sigma,\vartheta=1,\ldots,n}
\in C^\infty(B\times(-\varepsilon_0,+\varepsilon_0),SO(n)),
\quad w=(u,v),
\end{equation*}
with sufficiently small $\varepsilon_0>0,$ such that
\begin{equation}\label{3.7}
{\mathbf R}(w,0)={\mathbb E}^n\,,\quad
\frac{\partial}{\partial\varepsilon}\,{\mathbf R}(w,0)={\mathbf A}(w)\in C^\infty(B,so(n)).
\end{equation}
Here, ${\mathbb E}^n$ denotes the $n$-dimensional unit matrix.\\[1ex]
For arbitrary skew-symmetric ${\mathbf A}(w)=(a_{\sigma\vartheta}(w))_{\sigma,\vartheta=1,\ldots,n}\in C^\infty(B,so(n))$, a family ${\mathbf R}(w,\varepsilon)$ with the property (\ref{3.7}) can be constructed by means of the geodesic flow in the manifold $SO(n)$ (see e.g. \cite{doCarmo_01}, chapter 3, section 2). By expansion around $\varepsilon=0$ we obtain
\begin{equation*
\mathbf R(w,\varepsilon)=\mathbb E^n+\varepsilon\mathbf A(w)+o(\varepsilon).
\end{equation*}
Now, we apply ${\mathbf R}$ to a given ONS ${\mathcal N}$. The new unit normal vectors $\widetilde N_1,\ldots,\widetilde N_n$ are given by
\begin{equation*
\widetilde N_\sigma
=\sum_{\vartheta=1}^n r_{\sigma\vartheta}N_\vartheta
=\sum_{\vartheta=1}^n
\big\{
\delta_{\sigma\vartheta}+\varepsilon a_{\sigma\vartheta}+o(\varepsilon)
\big\}\,N_\vartheta
=N_\sigma+\varepsilon\sum_{\vartheta=1}^na_{\sigma\vartheta}N_\vartheta+o(\varepsilon),
\end{equation*}
and we compute
\begin{equation*
\widetilde N_{\sigma,u^\ell}
=N_{\sigma,u^\ell}
+\varepsilon\sum_{\vartheta=1}^n
\big(
a_{\sigma\vartheta,u^\ell}N_\vartheta+a_{\sigma\vartheta}N_{\vartheta,u^\ell}
\big)
+o(\varepsilon)
\end{equation*}
for their derivatives. Consequently, the new torsion coefficients can be expanded to
\begin{equation*
\widetilde T_{\sigma,\ell}^\omega=\widetilde N_{\sigma,u^\ell}\cdot\widetilde N_\omega^t
=T_{\sigma,\ell}^\omega
+\varepsilon a_{\sigma\omega,u^\ell}
+\varepsilon\sum_{\vartheta=1}^n
\big\{
a_{\sigma\vartheta}T_{\vartheta,\ell}^\omega
+a_{\omega\vartheta}T_{\sigma,\ell}^\vartheta
\big\}
+o(\varepsilon),
\end{equation*}
and for their squares we infer
\begin{equation*
(\widetilde T_{\sigma,\ell}^\omega)^2
=(T_{\sigma,\ell}^\omega)^2
+2\varepsilon
\bigg\{
a_{\sigma\omega,u^\ell}T_{\sigma,\ell}^\omega
+\sum_{\vartheta=1}^n
\big(
a_{\sigma\vartheta}T_{\vartheta,\ell}^\omega T_{\sigma,\ell}^\omega
+a_{\omega\vartheta}T_{\sigma,\ell}^\vartheta T_{\sigma,\ell}^\omega
\big)
\bigg\}
+o(\varepsilon).
\end{equation*}
Before we insert this identity into the functional of total torsion, we observe
\begin{equation*
\begin{array}{lll}
\displaystyle
\sum_{\sigma,\omega,\vartheta=1}^n
\big\{
a_{\sigma\vartheta}T_{\vartheta,\ell}^\omega T_{\sigma,\ell}^\omega
+a_{\omega\vartheta}T_{\sigma,\ell}^\vartheta T_{\sigma,\ell}^\omega
\big\}
& = & \!\!\!\displaystyle
\sum_{\sigma,\omega,\vartheta=1}^n
\big\{
a_{\sigma\vartheta}T_{\vartheta,\ell}^\omega T_{\sigma,\ell}^\omega
+a_{\sigma\vartheta}T_{\omega,\ell}^\vartheta T_{\omega,\ell}^\sigma
\big\} \\[4ex]
& = & \!\!\!\displaystyle
2\sum_{\sigma,\omega,\vartheta=1}^n
a_{\sigma\vartheta}T_{\vartheta,\ell}^\omega T_{\sigma,\ell}^\omega
\,=\,0,
\end{array}
\end{equation*}
taking the skew-symmetry of $\mathbf A$ into account.
\goodbreak\noindent
Now, the difference between the torsion functionals computes to ($a_{\sigma\omega,u^\ell}T_{\sigma,\ell}^\omega=a_{\omega\sigma,u^\ell}T_{\omega,\ell}^\sigma$)
\begin{equation*
\begin{array}{lll}
\displaystyle
{\mathcal T}_X(\widetilde{\mathcal N})-{\mathcal T}_X({\mathcal N})\!\!\!
& = & \!\!\!\displaystyle
2\varepsilon
\sum_{\sigma,\omega=1}^n\sum_{\ell=1}^2\,
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
a_{\sigma\omega,u^\ell}T_{\sigma,\ell}^\omega\,dudv
+o(\varepsilon) \\[5ex]
& = & \!\!\!\displaystyle
4\varepsilon
\sum_{1\le\sigma<\omega\le n}
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\Big\{
a_{\sigma\omega,u}T_{\sigma,1}^\omega
+a_{\sigma\omega,v}T_{\sigma,2}^\omega
\Big\}
+o(\varepsilon)\\[5ex]
& = & \!\!\!\displaystyle
\,4\varepsilon
\sum_{1\le\sigma<\omega\le n}\,
\int\limits_{\partial B}
a_{\sigma\omega}(T_{\sigma,1}^\omega,T_{\sigma,2}^\omega)\cdot\nu^t\,ds \\[5ex]
& & \!\!\!\displaystyle
-\,4\varepsilon
\sum_{1\le\sigma<\omega\le n}^n\,
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
a_{\sigma\omega}\,\mbox{div}\,(T_{\sigma,1}^\omega,T_{\sigma,2}^\omega)\,dudv
+o(\varepsilon),
\end{array}
\end{equation*}
where $\nu$ denotes the outer unit normal of $\partial B.$ Thus, for critical ${\mathcal N}$ we infer
\begin{equation*
\sum_{1\le\sigma<\vartheta\le n}\,\,
\int\limits_{\partial B}
a_{\sigma\omega}(T_{\sigma,1}^\omega,T_{\sigma,2}^\omega)\cdot\nu^t\,ds
-\sum_{1\le\sigma<\omega\le n}\,
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
a_{\sigma\omega}\,\mbox{div}\,(T_{\sigma,1}^\omega,T_{\sigma,2}^\omega)\,dudv
=0
\end{equation*}
with arbitrary ${\mathbf A}\in C^\infty(B,so(n)).$ This implies the
\begin{proposition}
If the ONS ${\mathcal N}$ is critical for ${\mathcal T}_X,$ then its torsion coefficients satisfy
\begin{equation}\label{3.16}
\mbox{\rm div}\,(T_{\sigma,1}^\vartheta,T_{\sigma,2}^\vartheta)=0\quad\mbox{in}\ B,\quad
(T_{\sigma,1}^\vartheta,T_{\sigma,2}^\vartheta)\cdot\nu^t=0\quad\mbox{on}\ \partial B
\end{equation}
for all $(\sigma,\vartheta)\in U_n$.
\end{proposition}
\section{A second order system with quadratic growth}
\label{section4}
\setcounter{equation}{0}
\subsection{The functions $g^{(\sigma\vartheta)}$}
Interpreting the differential equations in (\ref{3.16}) as integrability conditions, we find functions $g^{(\sigma\vartheta)}\in C^2(B,\mathbb R)$ such that
\begin{equation}\label{4.1}
\nabla g^{(\sigma\vartheta)}=\big(-T_{\sigma,2}^\vartheta,T_{\sigma,1}^\vartheta\big)
\quad\mbox{in}\ B\quad\mbox{for all}\ \sigma,\vartheta=1,\ldots,n.
\end{equation}
Due to the boundary conditions in (\ref{3.16}), which imply $\nabla g^{(\sigma\vartheta)}\cdot\tau^t=0$ on $\partial B$ with the unit tangent vector $\tau=(-v,u)$ at $\partial B,$ we may choose $g^{(\sigma\vartheta)}$ such that
\begin{equation}\label{4.2}
g^{(\sigma\vartheta)}=0\quad\mbox{on}\ \partial B\quad\mbox{for all}\ \sigma,\vartheta=1,\ldots,n.
\end{equation}
Note that the matrix $(g^{(\sigma\vartheta)})_{\sigma,\vartheta=1,\ldots,n}$ is skew-symmetric.
\goodbreak\noindent
\subsection{An elliptic system of second order}
Let us define the quantities
\begin{equation*
\delta g^{(\sigma\vartheta)}:=\sum_{\omega=1}^n\mbox{det}\,\Big(\nabla g^{(\sigma\omega)},\nabla g^{(\omega\vartheta)}\Big),
\quad\sigma,\vartheta=1,\ldots n.
\end{equation*}
The matrix $(\delta g^{(\sigma\vartheta)})_{\sigma,\vartheta=1,\ldots,n}$ is skew-symmetric. The functions $g^{(\sigma\vartheta)}$ solve a coupled quasilinear elliptic differential system with quadratic growth in the gradient:
\begin{proposition}
If ${\mathcal N}$ is critical for ${\cal T}_X$, then the functions $g^{(\sigma\vartheta)}$, $\sigma,\vartheta=1,\ldots,n$, are solutions of the boundary value problems
\begin{equation}\label{4.4}
\Delta g^{(\sigma\vartheta)}=-\,\delta g^{(\sigma\vartheta)}+S_{\sigma,12}^\vartheta\quad\mbox{in}\ B,\quad
g^{(\sigma\vartheta)}=0\quad\mbox{on}\ \partial B\,,
\end{equation}
where $\delta g^{(\sigma\vartheta)}$ grows quadratically in the gradient of $g^{(\sigma\vartheta)}$.
\end{proposition}
\begin{proof}
Choose any $(\sigma,\vartheta)\in\{1,\ldots,n\}^2.$ The formulas (\ref{2.10}) and (\ref{4.1}) imply
\begin{equation}\label{4.5}
\begin{array}{rcl}
\Delta g^{(\sigma\vartheta)}\!\!\!
& = & \!\!\!\displaystyle
T_{\sigma,1,v}^\vartheta -T_{\sigma,2,u}^\vartheta
\,=\,-\sum_{\omega=1}^nT_{\sigma,1}^\omega T_{\omega,2}^\vartheta
+\sum_{\omega=1}^nT_{\sigma,2}^\omega T_{\omega,1}^\vartheta
+S_{\sigma,12}^\vartheta \\[4ex]
& = & \!\!\!\displaystyle
\sum_{\omega=1}^n
\Big\{
g_v^{(\sigma\omega)}g_u^{(\omega\vartheta)}-g_u^{(\sigma\omega)}g_v^{(\omega\vartheta)}
\Big\}
+S_{\sigma,12}^\vartheta\,,
\end{array}
\end{equation}
and the statement follows.
\end{proof}
\subsection{Examples}
Let us write $\mathbf G:=(g^{(\sigma\vartheta)})_{\sigma,\vartheta=1,\ldots,n}$, $\mathbf S:=(S_{\sigma,12}^\vartheta)_{\sigma,\vartheta=1,\ldots,n}$, and $\delta\mathbf G:=(\delta g^{(\sigma\vartheta)})_{\sigma,\vartheta=1,\ldots,n}$. We discuss the special cases $n=1,2,3.$
\begin{itemize}
\item[1.]
For $n=1$ ($X$ is immersed in $\mathbb R^3$) there are no torsions.
\vspace*{-1ex}
\item[2.]
The case $n=2$ ($X$ is immersed in $\mathbb R^4$) was already considered in \cite{Froehlich_Mueller_01}. There hold
\begin{equation}\label{4.6}
{\mathbf S}
=\begin{pmatrix}
0 & S_{1,12}^2 \\[0.1cm]
S_{2,12}^1 & 0
\end{pmatrix},\quad
{\mathbf G}
=\begin{pmatrix}
0 & g^{(12)} \\[0.1cm]
g^{(21)} & 0
\end{pmatrix},\quad
\delta\mathbf G
=\begin{pmatrix}
0 & 0 \\[0.1cm]
0 & 0
\end{pmatrix},
\end{equation}
such that the system (\ref{4.4}) reduces to the single equation
\begin{equation*
\Delta g^{(12)}=S_{1,12}^2\quad\mbox{in}\ B,\quad g^{(12)}=0\quad\mbox{on}\ \partial B.
\end{equation*}
Then, potential theoretical estimates for elliptic equations ensure
\begin{equation*
\|g^{(12)}\|_{C^{1+\alpha}(B)}\le C(\alpha,\|S_{1,12}^2\|_\infty)
\quad\mbox{for all}\ \alpha\in(0,1)
\end{equation*}
with a real $C\in(0,+\infty)$ depending on $\alpha$ and the $L^\infty$-norm of $S_{1,12}^2$ (see e.g. \cite{Sauvigny_02}).\\[1ex]
Instead of this, in \cite{Froehlich_Mueller_01} we study a Riemann-Hilbert problem for $T_{1,1}^2+iT_{1,2}^2$ using methods from the complex analysis of generalized analytic functions.
\vspace*{-1ex}
\item[3.]
Let us now consider the case $n=3$ ($X$ is immersed in $\mathbb R^5$): We have
\begin{equation*
\begin{array}{l}
\displaystyle
{\mathbf S}
=\begin{pmatrix}
0 & S_{1,12}^2 & S_{1,12}^3 \\[0.1cm]
S_{2,12}^1 & 0 & S_{2,12}^3 \\[0.1cm]
S_{3,12}^1 & S_{3,12}^2 & 0
\end{pmatrix},\quad
{\mathbf G}
=\begin{pmatrix}
0 & g^{(12)} & g^{(13)} \\[0.1cm]
g^{(21)} & 0 & g^{(23)} \\[0.1cm]
g^{(31)} & g^{(32)} & 0
\end{pmatrix}, \\[6ex]
\displaystyle
\delta\mathbf G
=\begin{pmatrix}
0 & \mbox{det}\,\Big(\nabla g^{(13)},\nabla g^{(32)}\Big)
& \mbox{det}\,\Big(\nabla g^{(12)},\nabla g^{(23)}\Big) \\[0.3cm]
\mbox{det}\,\Big(\nabla g^{(23)},\nabla g^{(31)}\Big)
& 0
& \mbox{det}\,\Big(\nabla g^{(21)},\nabla g^{(13)}\Big)\\[0.3cm]
\mbox{det}\Big(\nabla g^{(32)},\nabla g^{(21)}\Big)
& \mbox{det}\,\Big(\nabla g^{(31)},\nabla g^{(12)}\Big) & 0
\end{pmatrix}.
\end{array}
\end{equation*}
Comparing with (\ref{4.4}) gives the three equations
\begin{equation*
\begin{array}{lll}
\Delta g^{(12)}\!\!\!
& = & \!\!\!\displaystyle
g_v^{(13)}g_u^{(32)}-g_u^{(13)}g_v^{(32)}+S_{1,12}^2\,, \\[0.2cm]
\Delta g^{(13)}\!\!\!
& = & \!\!\!\displaystyle
g_v^{(12)}g_u^{(23)}-g_u^{(12)}g_v^{(23)}+S_{1,12}^3\,, \\[0.2cm]
\Delta g^{(23)}\!\!\!
& = & \!\!\!\displaystyle
g_v^{(21)}g_u^{(13)}-g_u^{(21)}g_v^{(13)}+S_{2,12}^3\,.
\end{array}
\end{equation*}
Now, if we set ${\mathcal G}:=(g^{(12)},g^{(13)},g^{(23)})$ and ${\mathcal S}:=(S_{1,12}^2,S_{1,12}^3,S_{2,12}^3),$ then
\begin{equation*
\Delta{\mathcal G}={\mathcal G}_u\times{\mathcal G}_v+\mathcal S\quad\mbox{in}\ B,\quad\mathcal G=0\quad\mbox{on}\ \partial B
\end{equation*}
with the usual vector product $\times$ in $\mathbb R^3.$ That means: \emph{$\mathcal G$ solves an inhomogeneous $H$-surface system with $H=\frac12$ and vanishes on the boundary.}
\end{itemize}
\subsection{The Grassmann-type vectors ${\mathcal G}$, $\delta\mathcal G$, and ${\mathcal S}$}
The last example gives rise to the definition of the following {\it vector of Grassmann type}
\begin{equation}\label{4.12}
{\mathcal G}:=\big(g^{(\sigma\vartheta)}\big)_{1\le\sigma<\vartheta\le n}\in\mathbb R^N\,,\quad N:=\frac{n}{2}\,(n-1).
\end{equation}
In our examples, ${\mathcal G}$ works as follows:
\begin{equation*
\begin{array}{lll}
{\mathcal G}=g^{(12)}\in\mathbb R & \mbox{for}\ n=2, \\[0.1cm]
{\mathcal G}=\big(g^{(12)},g^{(13)},g^{(23)}\big)\in\mathbb R^3 & \mbox{for}\ n=3.
\end{array}
\end{equation*}
Analogously, we define the Grassmann-type vectors
\begin{equation*
\delta\mathcal G:=\big(\delta g^{(\sigma\vartheta)}\big)_{1\le\sigma<\vartheta\le n}\in\mathbb R^N,\quad
\mathcal S:=\big(S_{\sigma,12}^\vartheta\big)_{1\le\sigma<\vartheta\le n}\in\mathbb R^N\,.
\end{equation*}
Then, the relations (\ref{4.4}) can be written as
\begin{equation}\label{4.15}
\Delta\mathcal G=-\delta\mathcal G+\mathcal S\quad\mbox{in}\ B,\quad\mathcal G=0\quad\mbox{on}\ \partial B.
\end{equation}
From the definition of $\delta\mathcal G$, we immediately obtain the estimate
\begin{equation}\label{4.16}
|\Delta\mathcal G|\le c\,|\nabla\mathcal G|^2+|\mathcal S|\quad\mbox{in}\ B
\end{equation}
with some constant $c>0$.
\begin{remarks}\quad
\begin{itemize}
\item[1.]
The fact that $\Delta\mathcal G$ grows quadratically in $\nabla\mathcal G$ enables us to apply fundamental results on nonlinear elliptic systems due to E.\,Heinz \cite{Heinz_03}, \cite{Heinz_01} and F.\,Sauvigny \cite{Sauvigny_02}. And the special structure of $\delta\mathcal G$ allows us to utilize H.\,C.\,Wente's $L^\infty$-estimate \cite{Wente_01}, \cite{Topping_01}.
\vspace*{-0.6ex}
\item[2.]
For the homogeneous case $\mathcal S=0$, existence results for systems of the type (\ref{4.15}) can be found, e.g., in \cite{Heinz_02}, \cite{Wente_01} (for $n=3$), and \cite{Sauvigny_02} (for $n\ge 3$). In \cite{Takahashi_01} existence and multiplicity questions have been addressed in the inhomogeneous case (in our language: non-trivial bundle) for codimension $n=3$.
\vspace*{-0.6ex}
\item[3.]
The result in \cite{Sauvigny_02} can be extended to the inhomogeneous case: The boundary value problem (\ref{4.15}), (\ref{4.16}) has a solution ${\mathcal G}$, whenever ${\mathcal S}$ satisfies a smallness condition.
\vspace*{-0.6ex}
\item[4.]
Starting with a critical ONS $\mathcal N$, the mapping $\mathcal G=(g^{(\sigma\vartheta)})_{1\le\sigma<\vartheta\le n}$ from (\ref{4.1}), (\ref{4.2}) turns out to be a solution of (\ref{4.15}). Vice versa, solving (\ref{4.15}) for given $\mathcal S$ provides a first step towards the construction of a critical ONS $\mathcal N$.
\end{itemize}
\end{remarks}
\noindent
We plan to return to the questions in 3. and 4. in the future.
\subsection{A useful estimate}
Because the exact knowledge of the constant $c>0$ in (\ref{4.16}) will become important in section 6, we conclude the present section with the following
\begin{proposition}
It holds
\begin{equation}\label{4.17}
|\Delta{\mathcal G}|
\le\frac{\sqrt{n-2}}{2}\,|\nabla\mathcal G|^2+|\mathcal S|
\quad\mbox{in}\ B.
\end{equation}
\end{proposition}
\begin{proof}From (\ref{4.15}) we know
\begin{equation}\label{4.18}
|\Delta{\mathcal G}|\le |\delta\mathcal G|+|\mathcal S|\quad\mbox{in}\ B.
\end{equation}
It remains to estimate $|\delta\mathcal G|$ appropriately.
\begin{itemize}
\item[1.]
We begin with the inequality
\begin{equation}\label{4.19}
\begin{array}{rcl}
|\delta\mathcal G|^2\!\!\!
& = & \!\!\!\displaystyle
\sum_{1\le\sigma<\vartheta\le n}
\left\{\,
\sum_{\omega=1}^n
\det\big(\nabla g^{(\sigma\omega)},\nabla g^{(\omega\vartheta)}\big)
\right\}^2\\[4ex]
& \le & \!\!\!\displaystyle
(n-2)
\sum_{1\le\sigma<\vartheta\le n}
\left\{\,
\sum_{\omega=1}^n
\det\big(\nabla g^{(\sigma\omega)},\nabla g^{(\omega\vartheta)}\big)^2
\right\} \\[4ex]
& = & \!\!\!\displaystyle
(n-2)
\sum_{1\le\sigma<\vartheta\le n}
\Bigg\{\,
\sum_{\omega<\sigma}
\det\big(\nabla g^{(\omega\sigma)},\nabla g^{(\omega\vartheta)}\big)^2
+\sum_{\sigma<\omega<\vartheta}
\det\big(\nabla g^{(\sigma\omega)},\nabla g^{(\omega\vartheta)}\big)^2 \\[4ex]
& & \!\!\!\displaystyle
\hspace{20ex}
+\sum_{\vartheta<\omega}
\det\big(\nabla g^{(\sigma\omega)},\nabla g^{(\vartheta\omega)}\big)^2\Bigg\}\,.
\end{array}
\end{equation}
Note that only derivatives of elements of $\mathcal G$ appear on the right hand side of (\ref{4.19}).
\item[2.]
Denote by $e_i=(0,\ldots,0,1,0,\ldots,0)\in\mathbb R^m$ the $i$-th standard basis vector. We recall the \emph{exterior wedge product} of two vectors $X,Y\in\mathbb R^m,$
\begin{equation*
X\wedge Y=\sum_{1\le i<j\le m}(-1)^{ij}(x^iy^j-x^jy^i)\,e_i\wedge e_j\,,
\end{equation*}
where $\{e_i\wedge e_j\}_{1\le i<j\le m}$ forms an orthonormal basis of the Euclidean space $\mathbb R^M$ for $M:=\frac{m}{2}(m-1)$ (see e.g. \cite{Heil_01}). Using the Lagrange's identity, we may estimate
\begin{equation}\label{4.21}
|X\wedge Y|^2=|X|^2|Y|^2-(X\cdot Y^t)^2\le|X|^2|Y|^2\,.
\end{equation}
\item[3.]
Applying these settings to $\mathcal G$ with $m=N$ ($N$ from (\ref{4.12})), relation (\ref{4.19}) yields
\begin{equation}\label{4.22}
|\delta{\mathcal G}|^2
\le (n-2)|{\mathcal G}_u\wedge{\mathcal G}_v|^2\le (n-2)|{\mathcal G}_u|^2|{\mathcal G}_v|^2\,.
\end{equation}
Actually, ${\mathcal G}_u\wedge{\mathcal G}_v$ has more components than appear on the right hand side of (\ref{4.19}).
Combining (\ref{4.22}) with (\ref{4.18}) gives
\begin{equation*
|\Delta\mathcal G|
\le\sqrt{n-2}\,|\mathcal G_u||\mathcal G_v|+|\mathcal S|
\le\frac{\sqrt{n-2}}{2}\,|\nabla\mathcal G|^2+|\mathcal S|,
\end{equation*}
which proves the statement.\vspace*{-5ex}
\end{itemize}
\end{proof}
\section{Immersions with flat normal bundle}
\label{section5}
Assuming that the normal bundle of a given immersion $X$ is flat, we prove that any ONS ${\mathcal N},$ which is critical for the functional of total torsion, must be free of torsion (then ${\mathcal N}$ is called a {\it parallel} section).
\setcounter{equation}{0}
\subsection{Immersions with flat normal bundle}
\begin{definition}
The immersion $X$ has flat normal bundle ${\mathfrak S}\equiv 0$ iff $S_{\sigma,ij}^\vartheta\equiv 0$ in $B$ for all $i,j=1,2$ and $\sigma,\vartheta=1,\ldots,n.$
\end{definition}
\subsection{A lemma on an auxiliary function}
For the proof we need the following
\begin{lemma}
Let the immersion $X$ with flat normal bundle ${\mathfrak S}\equiv 0$ together with a critical ONS ${\mathcal N}$ be given. Then the function
\begin{equation*
f(w):=\mathcal G_w(w)\cdot\mathcal G_w^t(w)
\end{equation*}
vanishes identically in $B.$
\end{lemma}
\begin{remark}
Here, we use Wirtinger's calculus
\begin{equation*
\varphi_w:=\varphi_u-i\varphi_v\,,\quad
\varphi_{\overline w}:=\varphi_u+i\varphi_v\,,\quad
w=u+iv,
\end{equation*}
for a complex-valued function $\varphi=\varphi(w).$ Take note of the relation $\varphi_{w\overline w}=\Delta\varphi$.
\end{remark}
\begin{proof}[Proof of the lemma]
We will prove that $f$ solves the boundary value problem
\begin{equation*
f_{\overline w}=0\quad\mbox{in}\ B,\quad \mbox{Im}(w^2f)=0\quad\mbox{on}\ \partial B.
\end{equation*}
Then, the analytic function $g(w):=w^2f(w)$ has vanishing imaginary part, the Cauchy-Rie\-mann equations imply $g(w)\equiv c\in\mathbb R,$ and the assertion follows from $g(0)=0.$
\begin{itemize}
\item[1.]
In order to deduce the stated boundary condition, recall that $g^{(\sigma\vartheta)}=0$ on $\partial B$. Thus, all tangential derivatives vanish identically:
\begin{equation*
-vg_u^{(\sigma\vartheta)}+ug_v^{(\sigma\vartheta)}=-\mbox{Im}(wg_w^{(\sigma\vartheta)})=0\quad\mbox{on}\ \partial B
\end{equation*}
for all $\sigma,\vartheta=1,\ldots,n.$ The statement follows from
\begin{equation*
\hspace*{-2ex}
\begin{array}{lll}
\mbox{Im}\,(w^2f)\!\!\!
& = & \!\!\!\displaystyle
\mbox{Im}\,\Big(w^2\,\mathcal G_w\cdot\mathcal G^t_w\Big)
\,=\,\mbox{Im}\,
\bigg\{
w^2\sum_{1\le\sigma<\vartheta\le n}g_w^{(\sigma\vartheta)}g_w^{(\sigma\vartheta)}
\bigg\} \\[4ex]
& = & \!\!\!\displaystyle
\sum_{1\le\sigma<\vartheta\le n}\!
\mbox{Im}\,\Big\{\big(wg_w^{(\sigma\vartheta)}\big)\big(wg_w^{(\sigma\vartheta)}\big)\Big\}
\,=\,2\!\sum_{1\le\sigma<\vartheta\le n}\!
\mbox{Re}\,\big(wg_w^{(\sigma\vartheta)}\big)\,
\mbox{Im}\,\big(wg_w^{(\sigma\vartheta)}\big)\,=\,
\end{array}
\end{equation*}
\item[2.]
Finally, we show the analyticity of $f$ with the aid of (\ref{4.4}): Interchanging indices cyclically yields
\begin{equation*
\begin{array}{lll}
f_{\overline w}\!\!\!
& = & \!\!\!\displaystyle
2\,\mathcal G_w\cdot\mathcal G^t_{w\overline w}
\,=\,2\sum_{1\le\sigma<\vartheta\le n}
g_w^{(\sigma\vartheta)}g_{w\overline w}^{(\sigma\vartheta)}
\,=\,\sum_{\sigma,\vartheta=1}^n
g_w^{(\sigma\vartheta)}\Delta g^{(\sigma\vartheta)}\\[4ex]
& = & \!\!\!\displaystyle
\sum_{\sigma,\vartheta,\omega=1}^n
\Big\{
g_v^{(\sigma\omega)}g_u^{(\omega\vartheta)}g_u^{(\sigma\vartheta)}
-g_u^{(\sigma\omega)}g_v^{(\omega\vartheta)}g_u^{(\sigma\vartheta)}
\Big\} \\[4ex]
& & \!\!\!\displaystyle
-\,i\sum_{\sigma,\vartheta,\omega=1}^n
\Big\{
g_v^{(\sigma\omega)}g_u^{(\omega\vartheta)}g_v^{(\sigma\vartheta)}
-g_u^{(\sigma\omega)}g_v^{(\omega\vartheta)}g_v^{(\sigma\vartheta)}
\Big\} \\[4ex]
& = & \!\!\!\displaystyle
\sum_{\sigma,\vartheta,\omega=1}^n
\Big\{
g_v^{(\omega\vartheta)}g_u^{(\vartheta\sigma)}g_u^{(\omega\sigma)}
-g_u^{(\sigma\omega)}g_v^{(\omega\vartheta)}g_u^{(\sigma\vartheta)}
\Big\} \\[4ex]
& & \!\!\!\displaystyle
-\,i\sum_{\sigma,\vartheta,\omega=1}^n
\Big\{
g_v^{(\vartheta\sigma)}g_u^{(\sigma\omega)}g_v^{(\vartheta\omega)}
-g_u^{(\sigma\omega)}g_v^{(\omega\vartheta)}g_v^{(\sigma\vartheta)}
\Big\},
\end{array}
\end{equation*}
which shows $f_{\overline w}=0.$ The proof is complete.\vspace{-5ex}
\end{itemize}
\end{proof}
\subsection{Torsion-free ONS for flat normal bundles}
Our first theorem concerns the torsion of critical ONS ${\mathcal N}$ in the case of flat normal bundles.
\begin{theorem}
Let $X\in C^4(B,\mathbb R^{n+2})$ be an immersion with flat normal bundle ${\mathfrak S}\equiv 0.$ Then, for any critical ONS ${\mathcal N},$ the torsions $T_{\sigma,i}^\vartheta$, $i=1,2$, $\sigma,\vartheta=1,\ldots,n$, vanish identically in $B$.
\end{theorem}
\begin{proof}
Consider the Grassmann-type vector ${\mathcal G}\in C^2(B,\mathbb R^N)$ from (\ref{4.12}). Because ${\mathcal G}_w\cdot{\mathcal G}_w^t$ vanishes by the above lemma, there hold
\begin{equation*
|{\mathcal G}_u|=|{\mathcal G}_v|,\quad
{\mathcal G}_u\cdot{\mathcal G}_v^t=0
\quad\mbox{in}\ B.
\end{equation*}
This means that ${\mathcal G}$ is a conformally parametrized solution of
\begin{equation*
\Delta{\mathcal G}=-\,\delta{\mathcal G}\quad\mbox{in}\ B,\quad
{\mathcal G}=0\quad\mbox{on}\ \partial B;
\end{equation*}
see (\ref{4.17}) with ${\mathcal S}=0$. According to the growth condition $|\delta\mathcal G|\le c|\nabla\mathcal G|^2$, the arguments in \cite{Heinz_01} apply: Assume ${\mathcal G}\not\equiv\mbox{const}$ in $B$. Then, the asymptotic expansion stated in the Satz of \cite{Heinz_01} implies that boundary branch points $w_0\in\partial B$ with ${\mathcal G}_u(w_0)={\mathcal G}_v(w_0)=0$ are isolated. But this contradicts our boundary condition ${\mathcal G}|_{\partial B}=0$ from (\ref{4.15}). Thus, ${\mathcal G}(w)\equiv\mbox{const}=0$ and, finally, the definition (\ref{4.1}) implies $T_{\sigma,i}^\vartheta\equiv0$ in $B.$
\end{proof}
\noindent
As an immediate consequence, we obtain the
\begin{corollary}
If the immersion $X\in C^4(B,\mathbb R^{n+2})$ has flat normal bundle ${\mathfrak S}\equiv 0,$ then any critical ONS ${\mathcal N}$ is optimal w.r.t.~${\cal T}_X,$ i.e. ${\cal T}_X({\mathcal N})=0.$
\end{corollary}
\begin{remark}
The case $n=3$ ($X$ is immersed in $\mathbb R^5$) is covered by Wente's result in \cite{Wente_02}.
\end{remark}
\setcounter{equation}{0}
\section{The case of non-flat normal bundle}
\label{section6}
If the normal bundle has non-vanishing curvature $\mathcal S\not\equiv0$ it is desirable to have both, lower and upper bounds, at least for the total torsion of a critical orthonormal normal section.\\[1ex]
In paragraph 6.1, we will prove such a lower bound for the functional ${\mathcal T}_X.$ In the remaining paragraphs we establish an upper bound for the torsion coefficients combining a gradient estimate due to E.\,Heinz with H.\,C.\,Wente's $L^\infty$-estimate.
\subsection{A lower bound for the total torsion}
We write $\|Z\|_{p,\varrho}$, $p\in[1,+\infty]$, $\varrho\in[0,1]$, for the $L^p(B_\varrho(0))$-norm of a continuous mapping $Z\colon B_\varrho(0)\to\mathbb R^d,$ $d\in\mathbb N.$ In addition, we abbreviate $\|Z\|_p:=\|Z\|_{p,1}$.
\begin{theorem}
Let $X\in C^4(B,\mathbb R^{n+2}),$ $n\ge 2,$ be an immersion and ${\mathcal N}$ a critical ONS of its normal bundle with curvature ${\mathcal S}\not\equiv 0.$
\begin{itemize}
\item[(I)]
If ${\mathcal S}\not=\mbox{\rm const},$ then it holds
\begin{equation}\label{6.1}
{\mathcal T}_X({\mathcal N})\ge \bigg(\sqrt{n-2}\,\|\mathcal S\|_\infty+\frac{\|\mathcal S\|_{2}^{2}}{(1-\varrho)^2\|\mathcal S\|_{2,\varrho}^2}
+\frac{2\|\nabla\mathcal S\|_2^{2}}{\|\mathcal S\|_{2,\varrho}^2}\bigg)^{-1}\|\mathcal S\|_{2,\varrho}^2>0
\end{equation}
with $\varrho=\varrho(\mathcal S)\in(0,1)$ chosen as in (\ref{6.3}).
\item[(II)]
If ${\mathcal S}=\mbox{\rm const}\not=0,$ then we have
\begin{equation}\label{6.2}
{\mathcal T}_X({\mathcal N})
\ge \frac{1}{2}\,\frac{\pi|\mathcal S|^2}{\sqrt{n-2}\,|\mathcal S|+16}.
\end{equation}
\end{itemize}
\end{theorem}
\begin{proof}
\begin{itemize}
\item[1.]
We start with (I): Because of ${\mathcal S}\not=\mbox{const},$ there exists $\varrho=\varrho(\mathcal S)\in(0,1)$ such that
\begin{equation}\label{6.3}
\|\mathcal S\|_{2,\varrho}
=\left(\ \,
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B_\varrho(0)}|{\mathcal S}|^2\,dudv
\right)^{\frac12}>0.
\end{equation}
We choose a test function $\eta\in C^0(B,\mathbb R)\cap\mathring{H}_2^1(B,\mathbb R)$ with the properties
\begin{equation}\label{6.4}
\eta\in[0,1]\quad\mbox{in}\ B,\quad
\eta=1\quad\mbox{in}\ B_\varrho\,,\quad
|\nabla\eta|\le\frac{1}{1-\varrho}\quad\mbox{in}\ B.
\end{equation}
Multiplying $\Delta{\mathcal G}=-\delta{\mathcal G}+{\mathcal S}$ from (\ref{4.15}) by $(\eta{\mathcal S})$ and integrating by parts yields
\begin{equation*
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}\nabla\mathcal G\cdot\nabla(\eta\mathcal S)^t=\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\eta\,\delta\mathcal G\cdot\mathcal S^t-\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}\eta\,|\mathcal S|^2
\end{equation*}
(we omit $dudv$). Taking (\ref{4.22}) into account, we can now estimate as follows:
\begin{equation}\label{6.6}
\begin{array}{lll}
\displaystyle
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B_\varrho}|{\mathcal S}|^2\!\!\!
& \le & \!\!\!\displaystyle
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\eta\,|{\mathcal S}|^2
\,\le\,\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\eta\,\big|\delta{\mathcal G}\cdot{\mathcal S}^t\big|
+\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\big|\nabla{\mathcal G}\cdot\nabla(\eta{\mathcal S})^t\big|\\[5ex]
& \le & \!\!\!\displaystyle
\|{\mathcal S}\|_\infty
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}\eta\,|\delta{\mathcal G}|
+\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla\eta|\,|\mathcal S|\,|\nabla{\mathcal G}|
+\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\eta\,|\nabla\mathcal S|\,|\nabla\mathcal G| \\[5ex]
& \le & \!\!\!\displaystyle
\frac{\sqrt{n-2}}{2}\,\|{\mathcal S}\|_\infty
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla{\mathcal G}|^2
+\frac\varepsilon2
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\mathcal S|^2+\frac1{2\varepsilon(1-\varrho)^2}
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla{\mathcal G}|^2 \\[5ex]
&& \!\!\!\displaystyle
+\frac\delta2
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla\mathcal S|^2
+\frac1{2\delta}
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla\mathcal G|^2
\end{array}
\end{equation}
with arbitrary numbers $\varepsilon,\delta>0$. Let us write (\ref{6.6}) as
\begin{equation}\label{6.7}
\|\mathcal S\|_{2,\varrho}^2
\le\left(
\frac{\sqrt{n-2}}{2}\,\|\mathcal S\|_\infty
+\frac{1}{2\varepsilon(1-\varrho)^2}
+\frac1{2\delta}
\right)\|\nabla\mathcal G\|_2^2
+\frac{\varepsilon}{2}\,\|\mathcal S\|_2^2
+\frac{\delta}{2}\,\|\nabla\mathcal S\|_2^2\,.
\end{equation}
\item[2.]
According to (\ref{6.3}), the choice $\varepsilon=\|\mathcal S\|_2^{-2}\|\mathcal S\|_{2,\varrho}^2>0$ is admissible in (\ref{6.7}), and we infer
\begin{equation}\label{6.8}
\|\mathcal S\|_{2,\varrho}^2
\le\left(
\sqrt{n-2}\,\|\mathcal S\|_\infty
+\frac{\|\mathcal S\|_{2}^{2}}{(1-\varrho)^2\|\mathcal S\|_{2,\varrho}^2}
+\frac{1}{\delta}
\right)\|\nabla\mathcal G\|_2^2
+\delta\|\nabla\mathcal S\|_2^2\,.
\end{equation}
And since $\mathcal S\not=\mbox{\rm const},$ we can choose $\delta=\frac12\|\nabla\mathcal S\|_2^{-2}\|\mathcal S\|_{2,\varrho}^2$ in (\ref{6.8}), which implies
\begin{equation*
\|\mathcal S\|_{2,\varrho}^2
\le 2\left(
\sqrt{n-2}\,\|\mathcal S\|_\infty
+\frac{\|\mathcal S\|_{2}^{2}}{(1-\varrho)^2\|\mathcal S\|_{2,\varrho}^2}
+\frac{2\|\nabla\mathcal S\|_2^{2}}{\|\mathcal S\|_{2,\varrho}^2}
\right)\|\nabla\mathcal G\|_2^2\,.
\end{equation*}
Having $\mathcal T_X(\mathcal N)=2\|\nabla\mathcal G\|_2^2$ in mind, we arrive at (\ref{6.1}).
\item[3.]
In the case ${\mathcal S}=\mbox{\rm const}\not=0$ we choose $\varrho=\frac{1}{2}$ in (\ref{6.4}). Starting as in (\ref{6.6}), we then obtain
\begin{equation*
\begin{array}{lll}
\displaystyle
\frac\pi4|\mathcal S|^2\!\!\!
& = & \!\!\!\displaystyle
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B_{\frac12}}|{\mathcal S}|^2
\,\le\,|\mathcal S|\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\delta{\mathcal G}|
+\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla\eta|\,|\mathcal S|\,|\nabla{\mathcal G}| \\[5ex]
& \le & \!\!\!\displaystyle
\frac{\sqrt{n-2}}{2}\,|{\mathcal S}|
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla{\mathcal G}|^2
+\frac\varepsilon2\pi|\mathcal S|^2
+\frac2\varepsilon
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla{\mathcal G}|^2\,.
\end{array}
\end{equation*}
With $\varepsilon=\frac14$ it follows that
\begin{equation*
\frac{\pi}{8}\,|\mathcal S|^2
\le\left(
\frac{\sqrt{n-2}}{2}\,|{\mathcal S}|+8
\right)\|\nabla\mathcal G\|_2^2\,.
\end{equation*}
This implies the estimate (\ref{6.2}).\vspace*{-3ex}
\end{itemize}
\end{proof}
\begin{remarks}\quad
\begin{itemize}
\item[1.]
For {\it small} solutions ${\mathcal G}$ with $\|\mathcal G\|_\infty<\frac2{\sqrt{n-2}}$ it is quite easy to derive also an upper bound for the total torsion: Multiplying (\ref{4.15}) by ${\mathcal G}$ and integrating by parts yields
$$\mathcal T_X(\mathcal N)=2\|\nabla\mathcal G\|_2^2\le \frac{4\|\mathcal G\|_\infty\|\mathcal S\|_1}{2-\sqrt{n-2}\,
\|{\mathcal G}\|_\infty}\,.$$
Such a small solution can be constructed via the arguments in \cite{Sauvigny_02}; see remark 3 in subsection 4.4. Let us emphasize here again that the case $n=2$ is much easier to handle: The classical maximum principle controls $\|g^{(12)}\|_\infty$ by $\|S_{1,12}^2\|_\infty,$ and no smallness condition is needed to bound the total torsion; see subsection 4.3.
\item[2.]
In \cite{Takahashi_01} F.\,Takahashi translated the system (\ref{4.15}) for $n=3$ into a variational problem. Then he was able to derive lower and upper bounds for the quantity $\|\nabla\mathcal G\|_2$ of a minimizer in the corresponding Nehari manifold, whenever $\mathcal S$ is sufficiently small (in the $H^{-1}(B)$-norm); we refer to \cite{Takahashi_01} for the details.
\end{itemize}
\end{remarks}
\subsection{An $L^\infty$-bound for ${\mathcal G}$}
\begin{proposition}
For a critical ONS ${\mathcal N},$ the Grassmann-type vector ${\mathcal G}$ from (\ref{4.12}) satisfies
\begin{equation}\label{6.13}
\|\mathcal G\|_{\infty}
\le\frac{n-2}{2\pi}\,\|\nabla \mathcal G\|_2^2+\frac{1}{4}\,\sqrt{\frac{n(n-1)}2}\,\|\mathcal S\|_\infty\,.
\end{equation}
\end{proposition}
\begin{proof}
\begin{itemize}
\item[1.]
For $1\le \sigma<\vartheta\le n$ and $\omega\in\{1,\ldots,n\}$ with $\omega\not\in\{\sigma,\vartheta\}$, we define the functions $y^{(\sigma\vartheta\omega)}$ as the unique solutions of
\begin{equation*
\Delta y^{(\sigma\vartheta\omega)}=-\det\big(\nabla g^{(\sigma\omega)},\nabla g^{(\omega\vartheta)}\big)\quad\mbox{in}\ B,\quad
y^{(\sigma\vartheta\omega)}=0\quad\mbox{on}\ \partial B.
\end{equation*}
Wente's $L^\infty$-estimate (compare e.g.~\cite{Wente_01}, \cite{Topping_01}) then yields the \emph{optimal} inequalities
\begin{equation}\label{6.15}
\|y^{(\sigma\vartheta\omega)}\|_\infty\le\frac 1{4\pi}\Big(\|\nabla g^{(\sigma\omega)}\|_2^2+\|\nabla g^{(\omega\vartheta)}\|_2^2\Big),
\quad(\sigma,\vartheta)\in U_n,\quad\omega\not\in\{\sigma,\vartheta\}.
\end{equation}
In addition, we introduce the Grassmann-type vector $\mathcal Z=(z^{(\sigma\vartheta)})_{1\le\sigma<\vartheta\le n}$ as the unique solution of
\begin{equation*
\Delta \mathcal Z=\mathcal S\quad\mbox{in}\ B,\quad
\mathcal Z=0\quad\mbox{on}\ \partial B.
\end{equation*}
We use Poisson's representation formula and estimate as follows:
\begin{equation}\label{6.17}
\begin{array}{lll}
\displaystyle
|\mathcal Z(w)|\!\!\!
& = & \!\!\!\displaystyle
\bigg|
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
\phi(\zeta;w)\mathcal S(\zeta)\,d\xi d\eta
\bigg|
\,\le\,\sqrt N
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\phi(\zeta;w)||\mathcal S(\zeta)|\,d\xi d\eta \\[4ex]
& \le & \!\!\!\displaystyle
\sqrt N\,\|{\mathcal S}\|_\infty
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\phi(\zeta;w)|\,d\xi d\eta
\end{array}
\end{equation}
with the non-positive Green's function
\begin{equation}\label{6.18}
\phi(\zeta;w):=\frac1{2\pi}\log\Big|\frac{\zeta-w}{1-\overline w\zeta}\Big|,\quad\zeta\not= w,
\end{equation}
for $\Delta$ in $B;$ $\zeta=(\xi,\eta)$.
Because $\psi(w)=\frac{|w|^2-1}{4}$ solves $\Delta\psi=1$ in $B,$ $\psi=0$ on $\partial B,$ Poisson's formula yields
\begin{equation*
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\phi(\zeta;w)|\,d\xi d\eta
=\frac{1-|w|^2}{4}
\le\frac{1}{4}\,,
\end{equation*}
which enables us to continue (\ref{6.17}) to get
\begin{equation}\label{6.19}
\|\mathcal Z\|_\infty\le\frac{\sqrt N}4\|\mathcal S\|_\infty\,.
\end{equation}
\item[2.]
Next, we note the identity
\begin{equation*
g^{(\sigma\vartheta)}=\sum_{\omega\not\in\{\sigma,\vartheta\}}y^{(\sigma\vartheta\omega)}+z^{(\sigma\vartheta)},\quad(\sigma,\vartheta)\in U_n.
\end{equation*}
Applying now the estimates (\ref{6.15}) and (\ref{6.19}), we arrive at
\begin{equation*
\begin{array}{rcl}
\|\mathcal G\|_\infty\!\!\!
& \le & \!\!\!\displaystyle
\sum_{\sigma<\vartheta}
\sum_{\omega\not\in\{\sigma,\vartheta\}}
\|y^{(\sigma\vartheta\omega)}\|_\infty+\|\mathcal Z\|_\infty\\[4ex]
& \le & \!\!\!\displaystyle
\frac1{4\pi}
\sum_{\sigma<\vartheta}
\sum_{\omega\not\in\{\sigma,\vartheta\}}
\Big(
\|\nabla g^{(\sigma\omega)}\|_2^2+\|\nabla g^{(\omega\vartheta)}\|_2^2
\Big)
+\frac{\sqrt N}{4}\,\|\mathcal S\|_\infty\\[4ex]
& = & \!\!\!\displaystyle
\frac1{4\pi}\,
\bigg\{
\sum_{\omega<\sigma<\vartheta}
\Big(
\|\nabla g^{(\omega\sigma)}\|_2^2+\|\nabla g^{(\omega\vartheta)}\|_2^2
\Big)
+\sum_{\sigma<\omega<\vartheta}
\Big(
\|\nabla g^{(\sigma\omega)}\|_2^2+\|\nabla g^{(\omega\vartheta)}\|_2^2
\Big)\\[4ex]
& & \!\!\!\displaystyle
\hspace*{5.2ex}
+\sum_{\sigma<\vartheta<\omega}
\Big(
\|\nabla g^{(\sigma\omega)}\|_2^2+\|\nabla g^{(\vartheta\omega)}\|_2^2
\Big)
\bigg\}
+\frac{\sqrt N}{4}\,\|\mathcal S\|_\infty\\[4ex]
& = & \!\!\!\displaystyle
\frac1{4\pi}\,
\bigg\{
\sum_{\sigma<\vartheta<\omega}
\|\nabla g^{(\sigma\vartheta)}\|_2^2
+\sum_{\sigma<\omega<\vartheta}
\|\nabla g^{(\sigma\vartheta)}\|_2^2
+\sum_{\sigma<\vartheta<\omega}
\|\nabla g^{(\sigma\vartheta)}\|_2^2\\[4ex]
& & \!\!\!\displaystyle
\hspace*{5.2ex}
+\!\sum_{\omega<\sigma<\vartheta}\!
\|\nabla g^{(\sigma\vartheta)}\|_2^2
+\!\sum_{\sigma<\omega<\vartheta}\!
\|\nabla g^{(\sigma\vartheta)}\|_2^2
+\!\sum_{\omega<\sigma<\vartheta}\!
\|\nabla g^{(\sigma\vartheta)}\|_2^2
\bigg\}
+\frac{\sqrt N}{4}\,\|\mathcal S\|_\infty\\[4ex]
& = & \!\!\!\displaystyle
\frac1{2\pi}
\sum_{\sigma<\vartheta}
\sum_{\omega\not\in\{\sigma,\vartheta\}}
\|\nabla g^{(\sigma\vartheta)}\|_2^2
+\frac{\sqrt N}{4}\,\|\mathcal S\|_\infty\\[4ex]
& = & \!\!\!\displaystyle
\frac{n-2}{2\pi}\,\|\nabla\mathcal G\|_2^2
+\frac{1}{4}\,\sqrt{\frac{n(n-1)}2}\,\|\mathcal S\|_\infty,
\end{array}
\end{equation*}
as asserted.
\end{itemize}
\end{proof}
\subsection{An alternative estimate for $\|{\mathcal G}\|_\infty$}
For large codimension $n$, the estimate (\ref{6.13}) is somewhat unsatisfactory. Alternatively, we will show the inequality
\begin{equation}\label{6.22}
|z^{(\sigma\vartheta)}(w)|\le\sqrt{\frac{2}{\pi}}\,\|S_{\sigma,12}^\vartheta\|_2
\quad\mbox{in}\ B\quad\mbox{for all}\ (\sigma,\vartheta)\in U_n
\end{equation}
in the present paragraph.
\goodbreak\noindent
Then we calculate
\begin{equation*
\|\mathcal Z\|_\infty
=\sup_B\sqrt{\sum_{\sigma<\vartheta}|z^{(\sigma\vartheta)}(w)|^2}
\le\sqrt{\frac2\pi}\,\sqrt{\sum_{\sigma<\vartheta}\|S_{\sigma,12}^\vartheta\|_2^2}
=\sqrt{\frac2\pi}\,\|\mathcal S\|_2
\le\sqrt2\,\|\mathcal S\|_\infty\,,
\end{equation*}
and this estimate instead of (\ref{6.19}) will lead us to a smaller upper bound for $\|{\mathcal G}\|_\infty$ at least for large codimensions $n.$\\[1ex]
In order to prove (\ref{6.22}), we use the H\"older and the Sobolev inequality and compute
\begin{equation}\label{6.24}
|z^{(\sigma\vartheta)}(w)|
\le\|\phi(\cdot\,;w)\|_2\|S_{\sigma,12}^\vartheta\|_2
\le\frac1{2\sqrt\pi}\|\nabla_\zeta\phi(\cdot\,;w)\|_1\|S_{\sigma,12}^\vartheta\|_2\,.
\end{equation}
For the optimal constant $\frac1{2\sqrt\pi}$ in the Sobolev inequality we refer to \cite{Gilbarg_Trudinger_83} section 7.7 and the references therein. In (\ref{6.24}), $\phi=\phi(\zeta;w)$ denotes again Green's function (\ref{6.18}) for $\Delta$ in $B,$ which satisfies $\phi(\cdot\,;w)\in\mathring{H}^1_1(B)$ for any $w\in\mathring{B}$ as well as
\begin{equation*
\phi_\zeta(\zeta;w)
\equiv\phi_\xi(\zeta;w)-i\phi_\eta(\zeta;w)
=\frac{1}{2\pi}\,
\overline{
\left(
\frac{\zeta-w}{|\zeta-w|^2}+w\frac{1-\overline w\zeta}{|1-\overline w\zeta|^2}
\right)},
\quad w\not=\zeta.
\end{equation*}
A straightforward calculation shows
\begin{equation*
|\nabla_\zeta\phi(\zeta;w)|
\equiv|\phi_\zeta(\zeta;w)|
=\frac1{2\pi}\frac{1-|w|^2}{|\zeta-w|\,|1-\overline w\zeta|}\le\frac1{2\pi}\frac{1+|w|}{|\zeta-w|}
\le\frac1\pi\frac1{|\zeta-w|},\quad\zeta\not=w.
\end{equation*}
And since the right hand side in the inequality
\begin{equation*
\int\hspace{-0.25cm}\int\limits_{\hspace{-0.3cm}B}
|\nabla_\zeta\phi(\zeta;w)|\,d\xi d\eta
\le\frac{1}{\pi}\ \,
\int\hspace*{-0.25cm}\int\limits_{\hspace*{-0.4cm}B_\delta(w)}
\frac1{|\zeta-w|}\,d\xi\,d\eta
+\frac{1}{\pi}\ \ \,
\int\hspace*{-0.4cm}\int\limits_{\hspace*{-0.45cm}B\setminus B_\delta(w)}
\frac1{|\zeta-w|}\,d\xi\,d\eta\le 2\delta+\frac1\delta
\end{equation*}
becomes minimal for $\delta=\frac1{\sqrt2}$, we arrive at (\ref{6.22}). Instead of (\ref{6.13}), we thus have the
\begin{proposition}
For a critical ONS ${\mathcal N},$ the Grassmann-type vector ${\mathcal G}$ from (\ref{4.15}) satisfies
\begin{equation}\label{6.28}
\|\mathcal G\|_{\infty}\le\frac{n-2}{2\pi}\|\nabla \mathcal G\|_2^2+\sqrt2\,\|\mathcal S\|_\infty.
\end{equation}
\end{proposition}
\noindent
Note that (\ref{6.28}) provides a better bound than (\ref{6.13}) only in the case $n\ge 9.$
\subsection{A pointwise upper bound for the torsion coefficients}
We are now in the position to prove our third main result for immersions with non-flat normal bundle:
\begin{theorem}
Let $X\in C^4(B,\mathbb R^{n+2})$, $n\ge3$, be an immersion and $\mathcal N$ a critical ONS of its normal bundle. Assume that the smallness condition
\begin{equation}\label{6.29}
\frac{\sqrt{n-2}}2
\left(
\frac{n-2}{4\pi}\,\mathcal T_X(\mathcal N)+\gamma(n)\|\mathcal S\|_\infty
\right)<1
\end{equation}
is satisfied with $\gamma(n):=\min\{\frac14\sqrt{\frac{n(n-1)}2},\sqrt2\}$. Then, the torsion coefficients of ${\mathcal N}$ can be estimated by means of
\begin{equation}\label{6.30}
\|T_{\sigma,i}^\vartheta\|_\infty\le c,\quad i=1,2,\ (\sigma,\vartheta)\in U_n,
\end{equation}
with a nonnegative constant $c=c(n,\|\mathcal S\|_\infty,\mathcal T_X(\mathcal N))<+\infty$.
\end{theorem}
\begin{remark}
For codimension $n=2$, the estimate (\ref{6.30}) can be proved with $c=c(\|\mathcal S\|_\infty)$ and without presuming a smallness condition (\ref{6.29}), as already indicated in subsection 4.3. Again we refer to \cite{Froehlich_Mueller_01} for a slight generalization of that result.
\noindent For $n\ge 3$, it remains open if it is possible to prove global pointwise estimates for the torsion coefficients without the smallness condition (\ref{6.29}) and without the knowledge of ${\mathcal T}_X$.
\end{remark}
\begin{proof}[Proof of the theorem]
According to (\ref{4.17}), (\ref{6.13}) resp.~(\ref{6.28}), the Grassmann-type vector $\mathcal G=(g^{(\sigma\vartheta)})_{\sigma<\vartheta}$ solves the system
\begin{equation*
\begin{array}{l}
|\Delta\mathcal G|\le a|\nabla\mathcal G|^2+b\quad\mbox{in}\ B,\\[1ex]
\mathcal G=0\quad\mbox{on}\ \partial B,\\[1ex]
\|\mathcal G\|_\infty\le M,
\end{array}
\end{equation*}
where the appearing constants are defined by
\begin{equation*
a:=\frac{\sqrt{n-2}}2,\quad b:=\|\mathcal S\|_\infty,\quad M:=\frac{n-2}{2\pi}\|\nabla\mathcal G\|_2^2+\gamma(n)\|\mathcal S\|_\infty.
\end{equation*}
The smallness condition (\ref{6.29}) assures $aM<1$ due to $\mathcal T_X(\mathcal N)=2\|\nabla\mathcal G\|_2^2$. Consequently, we can apply E.\,Heinz's global gradient estimate Theorem\,1 in \cite{Sauvigny_02} Chap.~XII, \S\,3, obtaining $\|\nabla\mathcal G\|_\infty\le c$. This in turn yields the desired estimate (\ref{6.30}), according to (\ref{4.1}) and (\ref{4.12}).
\end{proof}
\vspace*{6ex}
{\small
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,689 |
Yes. The Web SDK can be used in PWAs, but with one limitation on iOS.
At the moment, saving to the web app to the home screen renders the PWA in a Safari webview (not the full Safari) which doesn't support WebRTC (video access).
By default, single Image Mode is provided: you take a single image and we decode the barcode from it.
If that's not an option and you prefer to scan from video feed, you can still allow the user to save the app to the desktop, but remove the meta tag "apple-mobile-web-app-capable". This will open the app in native Safari, with the only difference being a URL shown at the top of the browser. If your project does not have such a tab and your app is still a PWA, change the "display" setting in your manifest file from "standalone" to "browser".
Does the Web SDK work in WebViews of other apps?
Can I add scanning to my web shop on Shopify or EKM? | {
"redpajama_set_name": "RedPajamaC4"
} | 6,487 |
Q: GlMapBuffer takes longer and longer I have a problem with glMapBuffer. I'm making a simple program and I wanted to implement Sprite rendering. Since doing 500 draw calls for 500 sprites is really slow, I wanted to use one big VBO and update it every frame with new data. Good thing is, that it is faster. Weird thing is, that it slowes down as program is running.
I profiled it and the source of the problem is that calling glMapbuffer takes longer and longer. At first it uses few percents of one tick of the program, but after like a minute its already 36%. The profiler(VS community) just points me to nvoglv32.DLL, but that's dead end to me.
I seriously have no idea how to fix it.
Here is code of Sprite renderer:
http://pastebin.com/43Yp0y0M (when I copied it here, it looked really weird)
and header file: http://pastebin.com/fc0gvAUb
The are other things, but you can ignore them, they don't look like they cause the problem.
I initialize it, then in the main loop I basicly do:
renderer.Begin();
for(data)
renderer.Submit(data.pos,data.dims,data.col);
renderer.End();
renderer.Render();
Is this correct way to implement something like this? I mean, maybe I'm forgetting some important glCall somewhere.
I let it run for 15minutes, now just mapping the buffer takes 20ms: ( I also updated my drivers, but even that did not help.
A: Oh, stupid me, trying to blame OpenGL.
The mistake was in num_sprites variable, which didnt reset, so it increased every frame. Then it was used in glDrawElements, so what I think that Opengl did was realoccating the buffer and making it bigger instead of allowing access to not allocated memory. The bigger the buffer got the longer took to map it.
This was quite hard to find for me, because the actual mistake was elsewhere.
So, this is my story of first bigger bug discovery, wonder how many are there still hidden :D
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,694 |
Craig Wuest is an American keyboardist currently based in Atlanta, Georgia. He is best known as the founder and leader of the electronic music group Earthstar during the 1970s and 1980s. Earthstar was the only American band who participated in Germany's Kosmische Musik/electronic music scene while still at its height.
Early career
Wuest is originally from Utica, New York. According to former Earthstar guitarist Dennis Rea in the early 1970s Wuest was among the first musicians in Utica who owned a synthesizer. Wuest was heavily influenced by the German electronic music scene of the 1970s, including Klaus Schulze, Popol Vuh, Harmonia, Tangerine Dream, and Kraftwerk. Around the same time Rea had founded what he describes as an "eccentric progressive rock band," Zuir. According to Rea, "...being the only two adventurous music acts in town, collaboration between Craig and the members of Zuir was inevitable." Earthstar was born out of the partnership of Wuest, Rea, and other Utica-area musicians. In 1977 Earthstar was signed by Nashville-based Moontower Records, who released the group's first album, Salterbarty Tales, the following year.
Earthstar in Germany
Craig Wuest was an admirer of electronic music pioneer Klaus Schulze, with whom he struck up a correspondence. Schulze encouraged Wuest and Earthstar to come to Germany, intending to sign them to his Innovative Communications record label. Wuest sold his grand piano, which had played a prominent part on Salterbarty Tales, to finance the move. Other Earthstar musicians joined Wuest in Germany to work on French Skyline, the group's second album, While much of French Skyline was recorded at Klaus Schulze studios in Hambühren, West Germany, Schulze's label never signed Earthstar. The group was instead signed by Hamburg-based Sky Records.
The group recorded four albums primarily in Germany and France for Sky: French Skyline (1989), Atomkraft? Nein, Danke! (1981), Humans Only (1982), and Sleeper, the Nightlifer, which was never released. The New Gibraltar Encyclopedia of Progressive Rock credits Wuest as being the mastermind behind Earthstar's sound and comments on French Skyline and Atomkraft? Nein, Danke!: "Wuest's vision propels these two albums, his desire apparently is to create music that doesn't necessarily suggest a particular instrument, rather creates a new texture." Wuest co-produced French Skyline with Klaus Schulze and served as producer for the other Earthstar albums recorded during this period.
Wuest is also notable for his heavy use of the mellotron and the Birotron, a very rare tape loop instrument which could sustain notes beyond eight seconds. Wuest was the only keyboardist at the time other than Rick Wakeman who recorded albums on which he played the Birotron.
Later works
After returning to the United States Wuest continued to compose, play, and record electronic music. A now defunct earthstarmusic.com Web site listed later releases; one in particular, Axiom, which was listed as following Humans Only, had clearly been recorded and MP3 samples of the music were included on the Web site. They revealed a conventional, melodic, controlled electronic music sound. It is likely that by this point Earthstar had become Wuest's solo project. However, no record of the release of Axiom or any other later works can be found.
Discography
In Earthstar
1978 Salterbarty Tales (studio album)
1979 French Skyline (studio album)
1981 Atomkraft? Nein, Danke! (studio album)
1982 Humans Only (studio album)
Notes
References
Rea, Dennis. Live At The Forbidden City (iUniverse, 2006) , pp 18–19.
Liner notes from Earthstar albums.
Ambient musicians
American electronic musicians
New-age musicians
American experimental musicians
Earthstar (band) members
Record producers from New York (state)
Living people
Musicians from Utica, New York
Year of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,488 |
Q: Detach a specified detail in csv using Pandas or something else I'm new to python, I use pandas to read log files in cvs format and the information I want to use is all in the raw column. Is there any way for me use pandas or something else to detach information such as "timestamp", "src_ip", "proto",... into separate columns or a new sheet ?
Raw column:
{"timestamp":"2022-12-05T21:12:45.189212+0700","flow_id":12004372636444,"in_iface":"em0","event_type":"dns","src_ip":"192.168.x.xx","src_port":56490,"dest_ip":"8.8.8.8","dest_port":53,"proto":"UDP","dns":{"type":"query","id":48818,"rrname":"e123456.api.splkmobile.com","rrtype":"AAAA","tx_id":0}}
I have searched for this problem on the internet but there is no solution, asking here is the last solution
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,045 |
\section{Introduction}
Can the quantized Hall effect be observed in one dimension (1D)? Whereas a single 1D chain does not
allow for any orbital magnetic field effects, a ladder system as shown in Fig.~\ref{fig:ladder} is the
minimal extension where these are permitted.
Recently, a realization of bosonic ladders was reported by Atala \emph{et~al.}~\cite{Bloch14} using
ultracold-atoms exposed to a uniform artificial magnetic
field created by laser-assisted tunnelling~\cite{Aidelsburger,Goldman13,Zoller03,Gerbier10,Kolovsky}. They reported an observation of the chiral currents flowing around the ladder due to the effective magnetic field~\cite{Bloch14,hugel}.
Motivated by this experimental ability, the coupled wire realization of the bosonic Laughling ${\nu=1/2}$ fractional quantum Hall effect (FQHE) introduced by Kane \emph{et~al.}~\cite{kane2002}, was recently suggested in two-leg ladders for strong on-site interactions~\cite{Petrescu,Grusdt}.
An equivalent 1D setup is a single chain with spinful particles; spin-orbit interactions play the role of an orbital magnetic field, and a noncommuting Zeeman field acts as inter-chain hopping. This has been extensively discussed in semiconductor quantum wires with strong Rashba spin-orbit interactions, specifically in the context of helical liquids~\cite{oreg,lutchin}; when the system is strongly interacting, the possibility of a ``fractional helical liquid" was suggested~\cite{OregSelaStern}. As an alternative to spin-orbit interactions in electronic systems, effective spin-orbit coupling and Zeeman field may also be generated in systems of ultracold-atoms confined to 1D~\cite{Lin,Cheuk,Wang,Cui}. Recently, an observation of chiral edge states was achieved using fermionic~\cite{Mancini} and bosonic~\cite{Stuhl} 1D gases with an extra synthetic dimension originating from nuclear spin degrees of freedom.
This setup was theoretically envisioned to stabilise exotic states such as the fractional helical state, and numerically studied using density matrix renormalization group (DMRG) methods~\cite{Fazio,Zeng}.
\begin{figure}[t!]
\centering
\includegraphics*[width=.95\columnwidth]{FIG1.pdf}
\caption{Top: Two-leg ladder model
with magnetic flux per plaquette $\Phi$ leading to chiral currents $j_c$ flowing around the ladder. $\xi$ denotes interaction range. Bottom: Schematic fermionic chiral current contours of Eq.~(\ref{mainres}) marked by thin lines in the $n$-$\Phi$ plane, within an integer and a fractional QH phase. The phase transition lines out of the QH states are marked by solid thick lines, and the thick dotted lines correspond to fillings ${n = \nu\Phi/\pi}$. The validity regime is discussed in the text.}
\label{fig:ladder}
\end{figure}
The aim of this paper is (i) to analytically establish realizations of FQH phases in concrete 1D lattice models and (ii) propose a physical quantity that unambiguously signals these phases' fractional quantum Hall nature. By FQH in 1D, we refer to the coupled wire construction introduced by Kane \emph{et~al.}~\cite{kane2002}. While the 2D limit of this construction corresponds to the robust quantum Hall liquid, we herein study the thin-stripe regime, in which the system is sensitive to small perturbations, and hence finding detectable signatures is more demanding.
In Sec.~\ref{sec:nonint}, we introduce the instructive noninteracting integer model; in Sec.~\ref{sec:Model}, we elaborate on the realizations of fractional quantum Hall states in 1D two-leg ladders of either interacting bosons or interacting fermions. These FQH instabilities occur when the 1D density $n$ is related to the flux per plaquette $\Phi$ as
\begin{equation}
\label{eq:1}
n=\nu \frac{\Phi}{\pi},
\end{equation}
with ${\nu=1/m}$, and where $m$ is either odd for fermions or even for bosons; see dotted lines in Fig.~\ref{fig:ladder}. The FQH phases support finite deviations from this density, as schematically shown in Fig.~\ref{fig:ladder}, which increase with the inter-chain coupling $t_\perp$ and represent the finite compressibility of the system analogous to the 2D edge states.
Upon increasing the range of interactions, we find that arbitrary Laughlin ${\nu=1/m}$ states can be stabilized even at small inter-chain coupling. The required range of interactions increases~\cite{Fazio} for low filling factors;
the ${\nu=1/3}$ state already requires interactions between nearest neighbour rungs, while the ${\nu=1/2}$ bosonic state is stabilized for sufficiently strong but only on-rung interactions~\cite{Petrescu}.
Interestingly, in the synthetic dimension realizations of quantum ladders, the interactions become non-local in the synthetic dimension~\cite{Mancini,Stuhl,Majoranacoldatoms} (along the rungs), allowing one to reach this bosonic FQH state.
In Sec.~\ref{sec:persistent}, we address the question: What are manifestations of the FQHE in 1D ladders? In Refs.~\onlinecite{OregSelaStern,Meng14,Cornfeld15} transport was considered through leads connected to the 1D fractional helical state. In contrast, we herein wish to discuss thermodynamic bulk observables which may be detected in cold-atom experiments and simple simulations, where transport can not be directly measured. Such an observable is the chiral current $j_c$ that flows in the ground state due to the magnetic flux; see Fig.~\ref{fig:ladder}.
The possibility that chiral currents screen the orbital magnetic field in a kind of a Meissner phase, was pointed out~\cite{Orignac,Petrescu13,Piraud} and recently observed experimentally~\cite{Bloch14,Mueller14}; here we focus on the FQH phase.
We show that within the FQH phase the current depends on density $n$ and flux $\Phi$ as
\begin{equation}\label{mainres}
j_c \propto \left( n-\frac{\nu}{\pi} \Phi \right),
\end{equation}
up corrections that vanish for small inter-chain coupling $t_\perp$ and are discussed in detail in Sec.~\ref{sec:persistent}. Therefore, for small $t_\perp$, the behaviour described by Eq.~(\ref{mainres})
can be detected in the current map of the $n$-$\Phi$ plane as shown in Fig.~\ref{fig:ladder}; contours of constant current are asymptotically parallel to the constant filling factor line, Eq.~(\ref{eq:1}), which is depicted by the dotted line in Fig.~\ref{fig:ladder}.
These current contour are determined by the fractional filling $\nu$ and hence allows one to measure it. As we show, this result implies the emergence of fractional ``edge" states on the two-leg ladder and thus forms a stringent test for the stabilization of FQH states in ladders.
The constraints on the validity regime of this result are discussed in Secs. \ref{sec:instabilities} and \ref{correction}. Under these constraints the phase diagram in Fig.~\ref{fig:ladder} is exhaustive and no additional phases occur.
We conclude in Sec.~\ref{sec:discuss} providing perspectives on the experimental relevance in ultracold-atomic systems.
\section{Noninteracting model}
\label{sec:nonint}
We consider a spinless two-leg ladder of either fermions or bosons as shown in Fig.~\ref{fig:ladder}. Our main attention in this paper is devoted to the combined effect of interactions and magnetic flux $\Phi$. It is instructive, however, to first display the simple physics of the noninteracting fermionic model, and investigate the properties of the chiral currents in this simpler case, as done in this section.
\begin{figure}
\centering
\includegraphics[width=.95\columnwidth]{FIG2.pdf}
\caption{Left: dispersion relation $\epsilon_k^\pm$ for representative values of $t_\perp/t$.
Right: phase diagram for noninteracting fermions; ${c=0,1,2}$ labels the number of pairs of Fermi points (central charge).}\label{fig:phase-nu-1}
\end{figure}
The noninteracting model Hamiltonian is
$H = H_0+H_{\perp}$, where
\begin{equation}
\label{nonintmodel}
H_0 = -t \sum_{j,y} \left(c^\dagger_{j,y} c^{\phantom{\dagger}}_{j+1,y}+\mathrm{h.c.}\right)
\end{equation}
describes hopping within each chain ${y=1,2}$, and the inter-chain hopping is
\begin{equation}
\label{nonintHperp}
H_{\perp} = - t_\perp \sum_j \left(c^\dagger_{j,1} c^{\phantom{\dagger}}_{j,2} e^{i \Phi j}+ \mathrm{h.c.}\right).
\end{equation}
Here $\Phi$ is the magnetic flux per plaquette.
It is convenient to make the gauge transformation ${c'_{j1} = c_{j1} e^{- i\frac{1}{2} \Phi j}}$, ${c'_{j2} = c_{j2} e^{ i \frac{1}{2}\Phi j}}$, which moves the phase factor $e^{i \Phi j}$ from inter- to intra-chain hopping, yielding the Bloch Hamiltonian
\begin{equation}
\label{eq:Bloch}
H_k = \left(
\begin{array}{cc}
-2t\cos(k-\frac{\Phi}{2}) & t_{\perp} \\
t_\perp & -2t\cos(k+\frac{\Phi}{2}) \\
\end{array}
\right)
\end{equation}
(the lattice constant is set to unity).
Its eigenvalues $\epsilon_k^\pm$ are plotted in Fig.~\ref{fig:phase-nu-1} for various values of $t_\perp/t$. At ${t_\perp=0}$ we have two cosine dispersions shifted horizontally by ${\pm \frac{\Phi}{2}}$ (dashed lines). Any small inter-chain coupling $t_{\perp}$ opens a gap at the crossing points (full lines). As seen in the right panel of Fig.~\ref{fig:phase-nu-1}, upon scanning the chemical potential one finds two partially gapped ``chiral" regions with only one pair (${c=1}$) [rather than two pairs (${c=2}$)] of Fermi points, where $c$ denotes the central charge. In these chiral phases, the left- and right-moving modes reside on distinct chains; see color code in Fig.~\ref{fig:phase-nu-1}. Upon further increasing $t_\perp$, the chiral nature of the 1D modes is gradually reduced, and one eventually obtains (dot-dashed) dispersion curves which are nonoverlapping bands dominated by the inter-chain hopping.
We shall focus on the regime of ${t_\perp \ll t}$ near the instability leading to the opening of the lower partially gapped ${c=1}$ region in the the phase diagram.
The particle density is defined as ${n = \sum_{y=1,2} \langle c^\dagger_{j,y} c^{\phantom{\dagger}}_{j,y} \rangle }$, and the gap opens at the Fermi level when ${n=\frac{\Phi}{\pi}}$. Borrowing the 2D definition of filling factor, namely the ratio of particle density \emph{per site}, ${\langle c^\dagger_{j,y} c^{\phantom{\dagger}}_{j,y} \rangle =\frac{n}{2}}$, to the density of flux quanta per plaquette,
\begin{equation}
\label{fillingF}
\nu = \frac{\pi n}{\Phi},
\end{equation}
we see that the gap opening at small $t_\perp$ occurs at unit filling factor ${\nu=1}$. This gap opening is just the two wire version of the wire-construction of the quantum Hall effect~\cite{kane2002}. One pair of modes is gapped out, and one pair remains gapless in analogy with the chiral edge states in the integer quantum Hall effect.
The two-leg ladder is equivalent to a Rashba wire upon reinterpretation of (i) the two legs of the ladder as a spin degree of freedom, (ii) the inter-chain hopping $t_\perp$ as a spin-flipping Zeeman field, and (iii) the magnetic flux $\Phi$ as a Rashba spin orbit coupling causing a momentum shift ${\pm k_{SO}}$ of the two dispersions,
\begin{equation}
\begin{array}{ccc}
y=1,2 & \leftrightarrow & \sigma = \uparrow, \downarrow, \\
t_\perp & \leftrightarrow & B_\mathrm{Zeeman}, \\
\Phi & \leftrightarrow & 2 k_\mathrm{SO}.
\end{array}
\end{equation}
After this relabeling of indices and parameters, the partially gapped chiral state with ${c=1}$ corresponds to a helical state with the two spins propagating to opposite directions~\cite{oreg,lutchin}. In this context, $j_c$ is simply a persistent spin current~\cite{Yi06}.
We next discuss the persistent current $j_c$ generated by the magnetic flux in the two-leg ladder.
\subsection{Chiral current in the $n$-$\Phi$ plane}
\label{nonintphin}
The magnetic flux induces a persistent current in the ground state (GS), see Fig.~\ref{fig:ladder},
\begin{equation}
\label{jaPhi}
j_c=-\left\langle\frac{\partial H}{\partial\Phi_\mathrm{tot}}\right\rangle_\mathrm{GS} = - \frac{1}{L}\frac{\partial E_\mathrm{GS}}{\partial\Phi} ,
\end{equation}
where ${\Phi_\mathrm{tot} = L \Phi}$ is the total flux in a two-leg ladder of length $L$. Here, we expressed the current as the derivative of the ground state energy ${E_\mathrm{GS} = \langle H \rangle_\mathrm{GS} }$ with respect to $\Phi$. We shall see that the current can be used to probe 1D quantum Hall physics.
\begin{figure}
\centering
\includegraphics*[width=.95\columnwidth]{FIG3.pdf}
\caption{Contours of the chiral current $j_c$ marked by thin lines in the $\Phi$-$n$ plane for ${t_\perp =0.5 t}$ (top) and ${t_\perp=0.1 t}$ (bottom). The phase transition lines are marked by solid thick lines, and the thick dotted line corresponds to ${n = \Phi/\pi}$. The inset (top) depicts the chiral current along the horizontal dashed line.}
\label{fig:nphi}
\end{figure}
One may explore the current dependence on density $n$ and flux $\Phi$. In a noninteracting model the ground state energy is the sum over individual occupied states. In the ${c=1}$ regime only one band contributes,
\begin{equation}
\label{eqdjdmnon}
j_c(n,\Phi)=-\int_{- \pi n}^{\pi n} \frac{dk}{2\pi}\frac{\partial\epsilon_k^-}{\partial\Phi}.
\end{equation}
The contours of ${j_c(n,\Phi)}$ are plotted in Fig.~\ref{fig:nphi} for two values of ${t_\perp/t}$ both in and out of the chiral ${c=1}$ phase. The phase transitions are marked as thick red lines, and the current has cusp singularities at these transitions, as shown in the inset. We see that within the ${c=1}$ chiral phase and for not too small a density, the current contours are approximately parallel to the line defined in Eq.~(\ref{fillingF}). This behaviour becomes more pronounced as ${t_\perp/t}$ becomes smaller; see lower panel of Fig.~\ref{fig:nphi}.
For a quantitative description of this assertion we decompose the symmetric integral Eq.~(\ref{eqdjdmnon}) into contributions from the Fermi sea and from near the Fermi surface (${n\simeq\frac{1}{\pi}\Phi}$), as ${j_c=j_c^\mathrm{sea}+j_c^\mathrm{surf}}$, where
\begin{align}
\label{separate}
j_c^\mathrm{sea}=-\int_{0}^{\Phi}\frac{dk}{\pi}\frac{\partial\epsilon_k^-}{\partial\Phi},~~~
j_c^\mathrm{surf}=-\int_{\Phi}^{\pi n}\frac{dk}{\pi}\frac{\partial\epsilon_k^-}{\partial\Phi}.
\end{align}
Relegating details to Appendix~\ref{appendixNONINTj}, using this decomposition we find that the total current satisfies
\begin{equation}
\label{relationnonint}
(\partial_n+\pi\partial_\Phi)j_c=A^\mathrm{surf} \cdot (n-\tfrac{\Phi}{\pi})+ A^\mathrm{sea} \cdot \tfrac{t_\perp^2}{t^2}\ln\tfrac{t}{t_\perp}+\mathcal{O}(\tfrac{t_\perp^2}{t^2}).
\end{equation}
The two terms in the right hand side arise from the Fermi surface and the Fermi sea, their coefficients $A^\mathrm{surf}$ and $A^\mathrm{sea}$ are derived in Appendix~\ref{appendixNONINTj}. The Fermi surface contribution originates from the band curvature and is given by ${A^\mathrm{surf}=\tfrac{1}{4\pi}\partial_n^2\mu}$.
As herein explained, two terms on the right hand side of Eq.~(\ref{relationnonint}) form a small correction.
If one tunes the density to the exact filling factor ${\nu=1}$, the first correction term vanishes. Deviations of the density ${\delta n}$ within the ${c=1}$ phase are of the order of the energy gap ${E_\mathrm{gap}=\epsilon_{k=0}^+ - \epsilon_{k=0}^-= 2 t_\perp}$ times the density of states $\frac{1}{\pi v_F}$, where $v_F$ is the Fermi velocity. The first correction term thus approaches values of order ${\mathcal{O}(t_\perp/t)}$ near the boundaries of the ${c=1}$ region. As a consequence, this leading correction, as well as the second term and other subleading corrections, are altogether negligible for small inter-chain coupling, ${t_\perp\ll t}$, resulting in ${(\partial_n+\pi\partial_\Phi)j_c \simeq 0}$. It implies that indeed, contours of the current are nearly parallel to the dashed line ${n = \Phi/\pi}$ as seen in Fig.~\ref{fig:nphi} for gradually decreasing $t_\perp$.
This behaviour is valid for density ${\sqrt{t_\perp /t} \ll n \ll 1}$. For too small a density, one reaches the situation where the Fermi energy ${\thicksim t n^2}$ becomes smaller than the energy gap ${\thicksim t_\perp}$, giving the lower bound. For too high a density of order unity, the partial gaps of electrons and holes (see Fig.~\ref{fig:phase-nu-1}) approach each other and cause undesired lattice effects. Thus, for ${t_\perp /t \ll 1}$ we have a large region in the parameter space where the contours are asymptotically parallel to ${n = \Phi/\pi}$.
This is a general feature which holds true for the interacting fractional case, as we find in Sec.~\ref{sec:persistent}.
\section{Fractional states in lattice models with long range interactions}
\label{sec:Model}
We now consider either interacting fermions or bosons on the two-leg ladder, and add interactions to the lattice model,
${H = H_0+H_\perp+ H_{\mathrm{int}}}$, with
\begin{equation}
\label{intmodel}
H_{\mathrm{int}} = \sum_{r \ge 0} V(r) \sum_j n_j n_{j+r}.
\end{equation}
Here, ${c^{\phantom{\dagger}}_{j,y} c^\dagger_{j',y'} -(+) c^\dagger_{j',y'} c^{\phantom{\dagger}}_{j,y}=\delta_{y,y'} \delta_{j,j'}}$ for bosons (fermions) and, ${n_j = \sum_{y=1,2} c^\dagger_{j,y} c^{\phantom{\dagger}}_{j,y}}$.
In this model, the interaction potential is independent of the rungs' indices $y$ and $y'$, and depends solely on the linear distance $r$ via ${V(r)}$, specified below. Similar to the Hubbard model, the dependence of the interaction only on the total density ${\sum_{y} c^\dagger_{j,y}c^{\phantom{\dagger}}_{j,y}}$ makes the model ${H_0+H_{\mathrm{int}}}$ to be ${\mathrm{SU}(2)}$ invariant with respect to rotations of the spinor ${(c_{j,1}, c_{j,2})}$.
To treat these interactions, we utilize the Luttinger liquid theory. For a generic interaction $H_{\mathrm{int}}$ and at ${t_{\perp}=0}$, the long wavelength behaviour of the quantum ladder can be described by a two component Luttinger liquid (LL)~\cite{Giamarchi}.
The free part of the Hamiltonian is expressible in terms of the density operators of the two chains. It is convenient to introduce bosonic fields $\phi_\mu$, so that the long wavelength fluctuations of the total charge $(\rho)$ and the relative density ($\sigma$), denoted ``charge" and ``spin", respectively, are represented by
\begin{align}
\label{eq:density}
c^\dagger_{j,1} c^{\phantom{\dagger}}_{j,1} + c^\dagger_{j,2} c^{\phantom{\dagger}}_{j,2} &\thicksim n - \frac{\sqrt{2}}{\pi} \nabla \phi_\rho (x), \hbox{ and}\nonumber \\
c^\dagger_{j,1} c^{\phantom{\dagger}}_{j,1} - c^\dagger_{j,2} c^{\phantom{\dagger}}_{j,2} &\thicksim - \frac{\sqrt{2}}{\pi} \nabla \phi_\sigma (x).
\end{align}
The free Hamiltonian can be written as
\begin{align}
\label{HLL}
\mathcal{H}_\mathrm{LL} =\sum_{\mu = \rho,\sigma} \frac{1}{2 \pi} \int dx \left[ v_\mu K_\mu ( \pi \Pi_\mu)^2+ \frac{v_\mu}{K_\mu} (\nabla \phi_\mu)^2 \right],
\end{align}
where the $\Pi_\mu$ fields are canonically conjugate to the densities, ${[\phi_\mu(x) , \Pi_{\nu}(x')]=i \delta_{\mu \nu} \delta(x-x')}$.
The velocities $v_{\rho,\sigma}$ and LL parameters $K_{\rho,\sigma}$ depend on the strength of interactions and on the density; in the noninteracting fermionic case ${K_{\rho} = K_\sigma=1}$ and ${v_{\rho} = v_\sigma = v_F}$. Note, that for generic interactions, one still has ${K_\sigma=1}$ \emph{if} the model is SU(2) symmetric~\cite{Giamarchi}, as holds true in our model at ${t_\perp=0}$.
The remaining of the section is divided into two parts. We first list the possible perturbations to the Luttinger liquid model~\cite{Orignac,OregSelaStern,Petrescu13}, which include the FQH instability on which we focus. This enables one to obtain conditions for the Luttinger liquid parameter $K_{\rho}$ and density $n$, under which operators identified as opening FQH gaps are most relevant in the renormalization group (RG) sense. We then discuss a specific form of the interaction ${V(r)}$, corresponding to a finite range with hard core interactions, for which $K_{\rho}$ can be computed exactly for ${t_\perp=0}$, assuring that the FQH instability dominates for finite small inter-chain coupling $t_\perp$.
\subsection{Luttinger liquid instabilities}
\label{sec:instabilities}
There are various operators correcting the LL Hamiltonian. We may group them as ${\mathcal{H} = \mathcal{H}_\mathrm{LL} + \delta \mathcal{H}+ \delta \mathcal{H}_\perp}$, where ${\delta \mathcal{H}_\perp}$ includes terms generated by $t_\perp$, while ${\delta \mathcal{H}}$ includes terms which exist at ${t_\perp=0}$. We systematically list these various operators in Appendix~\ref{appendixRELEVANCY}.
The interesting FQH physics would not occur without the inter-chain coupling
\begin{equation}
c^\dagger_{j 1} c^{\phantom{\dagger}}_{j 2}e^{i \Phi j}+\mathrm{h.c.}.
\end{equation}
We may express the particle creation and annihilation operators in the bosonized language,
\begin{equation}\label{eq:psi}
\begin{gathered}
c^\dagger_{j,y} \to \Psi^\dagger_y (x) \sim \sum_p\psi^\dagger_{y,p}(x), \\
\psi_{y,p}^\dagger(x) = e^{i 2p (\pi \frac{n}{2} x - \phi_y(x))} e^{- i \theta_y(x)} \\
\phi_{\rho,\sigma} = \tfrac{\phi_{1} \pm \phi_2}{\sqrt{2}} ,~~\theta_{\rho,\sigma} = \tfrac{\theta_{1} \pm \theta_2}{\sqrt{2}},~~~\pi \Pi_\mu = \nabla \theta_\mu(x),
\end{gathered}
\end{equation}
where the sum over $p$ runs over integers for bosons or half integers for fermions~\cite{Giamarchi}.
The operators generated by this expansion are of the form
\begin{equation}
\mathcal{O}_{p} \thicksim \psi^\dagger_{1,-p} \psi^{\phantom{\dagger}}_{2,p}e^{i\Phi x}+\mathrm{h.c.}.
\end{equation}
Such operators may be incorporated into the bosonised LL Hamiltonian using
\begin{equation}
\label{cos}
\mathcal{O}_{p} \thicksim g_p \int dx \cos (\sqrt{2}\theta_\sigma - 2p \sqrt{2}\phi_\rho - (\Phi - 2 p \pi n) x),
\end{equation}
with a coupling constant $g_p$ generated by interactions and by the inter-chain coupling $t_\perp$. These operators are oscillating unless
\begin{align}
\label{eq:simposc}
2p \pi n= \Phi.
\end{align}
When relevant, such operators may open an energy gap even for a finite deviation from this exact flux up to a commensurate-incommensurate transition~\cite{Orignac,Petrescu13}. The above condition is met when the filling factor Eq.~(\ref{fillingF}) is given by
\begin{equation}
\label{simpnupp}
\nu = \frac{1}{2p},
\end{equation}
which is the inverse of an even(odd) integer for bosons(fermions). The states generated by this operator correspond to the Laughlin FQH phases as constructed by coupled wires~\cite{kane2002}.
The scaling dimension of $\mathcal{O}_{p}$ is given by
\begin{equation}
x_{p} = \frac{1}{2} \left(\frac{1}{K_\sigma} + (2p)^2 K_\rho \right).
\end{equation}
In the ${\mathrm{SU}(2)}$ symmetric case of ${K_\sigma=1}$, its relevancy ${x_p<2}$ is ensured by a small value of the charge LL parameter
\begin{equation}
\label{Krhocrit}
K_\rho < \frac{3}{(2p)^2}=3 \nu^2.
\end{equation}
In this symmetric case, usual RG analysis~\cite{Giamarchi} shows that the energy gap scales as
\begin{equation}
\label{Egap}
E_\mathrm{gap} \thicksim t (t_\perp/t)^{\frac{1}{2-x_{p}}}.
\end{equation}
In the noninteracting case for example, the scaling dimension equals ${x_p=1}$ which is consistent with the energy gap ${E_\mathrm{gap} = 2 t_\perp}$.
Additional phases generated by $t_\perp$, described in Ref.~\onlinecite{Petrescu}, are included for completeness in Appendix~\ref{appendixRELEVANCY}, such as the Meissner phase and the vortex lattice~\cite{Orignac}. In order to stabilize a given FQH state, one needs to (i) have a small value of $K_\rho$ satisfying Eq.~(\ref{Krhocrit}), and (ii) make sure that other instabilities are less relevant. Below we consider a specific interaction ${V(r)}$ with a finite range $\xi$, and determine the range required to satisfy these conditions and observe a FQH state at any desired fractional filling $\nu$.
\subsection{Solvable Lattice Model}
\label{se:specialmodel}
We now specialize to an interaction potential that vanishes beyond the interaction range $\xi$, see Fig.~\ref{fig:ladder},
\begin{equation}
\label{interaction}
V(r) =
\begin{cases} U v(r) & \mbox{for } r \le \xi, \\ 0 & \mbox{for } r > \xi. \end{cases}
\end{equation}
Here, ${v(r)}$ is a positive decreasing function whose specific form is not important, with ${v(0)=1}$ and ${v(r) = \mathcal{O}(1)}$ for ${r \le \xi}$.
The case $\xi=0$ corresponds to on-rung interactions, namely ${V(r=0)}$ comprises both on-site interactions (meaningful for bosons) and interactions between particles on different sites of the same rung (as in the Hubbard model).
We focus on the regime ${U \gg t}$, for general interaction range $\xi$.
The analysis starts by assuming infinite $U$, for which ${H_0+H_\mathrm{int}}$ is exactly solvable, and then relaxes this assumption to finite but large $U$.
In this hard-core limit, the interaction becomes a constraint: states containing two particles horizontally separated by $\xi$ sites or less acquire a very high energy ${\mathcal{O}(U)}$. One can construct a low-energy subspace where the shortest linear inter-particle distance exceeds $\xi$. The allowed states for $N$ bosonic or fermionic particles on the ladder of length $L$ and open boundary conditions are then in one to one correspondence~\cite{SelaPereiraORBITAL,SelaGarst11} with the states of a constrained model. This model consists of $N$ fictitious particles on a ladder of reduced length ${L'=L - (N-1) \xi}$ subject to an additional constraint of not having two fictitious particles on the same rung. Each particle in the reduced lattice corresponds to one particle and $\xi$ empty rungs to its right on the original lattice.
For ${t_\perp=0}$, the leg index ${y=1,2}$ of each particle is conserved \emph{i.e.} the sequence of $y$ values for $N$ particles from left to right ${\{y_1 , y_2 ,...,y_N \}}$ is conserved (considering open boundary conditions for simplicity). For a fixed value of this list, the $\xi-$constrained motion of the $N$ particles, with states labeled by ${\{ j_1,j_2,...,j_N \}}$ on the ladder, becomes equivalent to free fermions on a single chain of length $L'$; the Pauli principle fully accounts for the interaction constraint.
Following the methods described in Ref.~\onlinecite{SelaGarst11} and detailed in Appendix~\ref{se:appendix1}, we find that the Luttinger liquid parameter describing the model ${H_0+H_{\mathrm{int}}}$ depends on the density per rung $n$ and interaction range $\xi$ as
\begin{equation}
\label{eq:Krho}
K_\rho \xrightarrow[U \to \infty]{} \frac{1}{2}(1- n \xi)^2.
\end{equation}
For ${\xi=0}$ this result coincides with the exact solution of the Hubbard model \emph{i.e.} ${K_\rho = \frac{1}{2}}$ for infinite repulsion. This result enables us to choose values for $\xi$ and $n$ that yield a sufficiently small $K_\rho$ that satisfies Eq.~(\ref{Krhocrit}). Thus we can use the exact solvability of ${H_0+H_{\mathrm{int}}}$, and proceed using usual RG methods to treat $t_\perp$ as a perturbation, and deduce under which conditions is the FQH cosine perturbation relevant and flows to strong coupling.
The Luttinger parameter ${K_\rho = K_\rho[U/t]}$ is a continuous decreasing function of ${U/t}$, hence having a large enough but finite value of ${U/t}$ leads only to negligible (positive) corrections to the lower bound of $K_\rho$ in Eq.~(\ref{eq:Krho}). At the same time, as is known for the Hubbard model, in the strict ${U = \infty}$ limit one has a vanishing spin velocity ${v_\sigma \to 0}$, making the Luttinger liquid description pathological. This is not the case, however, for finite $U$ on which we focus, where the spin velocity remains finite.
In the infinite $U$ limit and for an interaction range $\xi$, the maximal possible density is ${n < \frac{1}{1+\xi}}$. On the other hand, Eq.~(\ref{eq:Krho}) and the relevancy condition Eq.~(\ref{Krhocrit}) impose a minimal possible density as well,
\begin{equation}
\label{nrange}
\frac{1-\sqrt{6}\nu}{\xi}< n< \frac{1}{1+\xi}.
\end{equation}
Comparing the minimal and maximal densities, one finds that the required interaction range for a relevant FQH perturbation at filling factor $\nu$ is
\begin{equation}
\label{range}
\xi \ge \left\lceil \frac{1}{\sqrt{6} \nu}-1 \right\rceil.
\end{equation}
For bosons at ${\nu=1/2}$, this relevancy condition is satisfied for on-rung interactions ${\xi=0}$~\cite{Petrescu13}. However for fermions at ${\nu=1/3}$, or bosons at ${\nu=1/4}$, one needs at least \emph{nearest neighbour} interactions, ${\xi=1}$. This is summarized in Table \ref{tb:xi}.
\begin{table}
\caption {Conditions for the realization of the 1D FQH states at filling factor $\nu$ in a two-leg ladder with hard-core interaction of range $\xi$ given in Eq.~(\ref{nrange}) and Eq.~(\ref{range}) .}
\label{tb:xi}
\begin{center}
\begin{tabular}{r || c | c | c | c || l}
& $\nu=\frac{1}{2}$ & $\nu=\frac{1}{3}$ & $\nu=\frac{1}{4}$ & $\nu=\frac{1}{5}$ &\\ \hline\hline
$\xi=0$ & $0<n$ & - & - & - & $n<1$ \\ \hline
$\xi=1$ & $0<n$ & $0.18<n$ & $0.38<n$ & - & $n<0.5$ \\ \hline
$\xi=2$ & $0<n$ & $0.09<n$ & $0.19<n$ & $0.25<n$ & $n<0.33$ \\ \hline
$\xi=3$ & $0<n$ & $0.06<n$ & $0.12<n$ & $0.17<n$ & $n<0.25$ \\
\end{tabular}
\end{center}
\end{table}
Note, that in the hard core limit with interaction range $\xi$, the kinetic motion freezes at the maximal allowed density of ${n=\frac{1}{1+\xi}}$. This corresponds to a Mott insulating state.
However, this charge-density wave (CDW) instability is not relevant and hence prevented for ${n<\frac{1}{1+\xi}}$. Going through the list of operators detailed in Appendix~\ref{appendixRELEVANCY} we find that there are no other nonoscillating relevant operators that compete with the relevant FQH operator Eq.~(\ref{cos}).
\subsubsection{Spin lattice implementation of hard core bosons}
We discuss a simplification and a specific implementation of the bosonic ${\nu=1/2}$ state for on-rung interactions $\xi = 0$. In order to substantially reduce the size of the Hilbert space keeping the essential physics intact, as we are interested in the limit of large $U$, we can switch from bosons, to a spin-$1/2$ lattice~\cite{Piraud}, using the replacement
\begin{equation}
c^\dagger_{j,y} \rightarrow S^+_{j,y},~~~~c_{j,y} \rightarrow S_{j,y}^-, ~~~n_j \rightarrow 1+ \sum_{y=1,2} S^z_{j,y}.
\end{equation}
The two-leg ladder model becomes
\begin{align}\label{spins}
H &= -t \sum_{j,y=1,2} \left(S^+_{j,y} S^{-}_{j+1,y}+\mathrm{h.c.}\right)\nonumber \\
&- t_\perp \sum_j \left(S^+_{j,1} S^-_{j,2} e^{i \Phi j}+ \mathrm{h.c.}\right) \nonumber \\
&+ 2 U \sum_j S^z_{j,1} S^z_{j,2}.
\end{align}
Recently, integer Chern insulating phases have been directly discussed in similar XY spin chains~\cite{Grass} with a synthetic magnetic flux.
As a self-consistent example, one may consider a ladder of length ${L \thicksim 100}$ in the vicinity of ${\Phi = 0.8 \pi}$ and ${n=0.4}$ (or equivalently ${\langle S^z \rangle=-0.3}$). There are 2 particles (up spins) every 5 rungs, and hence, given the short range repulsion, the system is far from any CDW instability. For this setup, ${x_p = 3/2}$ and the gap scales as ${E_{gap} \thicksim t (t_\perp/t)^2}$. The length scale over which the gap is formed scales as
\begin{equation}
\ell^* \thicksim (t/t_\perp)^2.
\end{equation}
The length of the ladder, $L$, should be larger than this crossover scale for the RG flow to fully develop the cosine perturbation from weak to strong coupling. This limits the inter-chain coupling ${t_\perp/}t$ to be not smaller than of order ${\sim 10^{-1}}$.
\hfill
To conclude this section, we have shown that FQH states occur as ground states in two-leg ladders with interactions of sufficiently long range. However, what are signatures of the fractional filling in these ground states? Below, we shall focus on this question.
\section{Chiral current}
\label{sec:persistent}
So far we have discussed explicit lattice realizations of FQH states in 1D ladders. However, when such models are realized, \emph{ e.g.} in an experiment or a numerical simulation, it is not obvious what are their signatures. Here, we explore the chiral current
flowing around the ladder, and find signatures in the $n$-$\Phi$ plane that are characteristic of the fractional filling factor $\nu$, generalizing the behaviour found for the noninteracting case in Sec.~\ref{nonintphin}.
The below calculation of the current and its derivatives with respect to $\Phi$ and $n$ is done using bosonization. It is known that all filled states contribute to the
chiral current, which is a persistent current; the chiral current in the ladder is thus generally \emph{not} an infra-red phenomenon
and cannot be fully accounted
for by an effective low-energy theory~\cite{Narozhny,Carr}. Nevertheless, certain aspects of the persistent current, such as its derivative with respect to particle number or flux, can be computed from the low-energy theory~\cite{Simon01}. Indeed, in Sec.~\ref{nonintphin}, we have identified two contributions to the current in the noninteracting case, one of which is a Fermi surface effect and the other is a Fermi sea effect. When generalizing to the interacting fractional case, the former can be safely extracted from the low-energy theory of bosonization.
We take the following three steps which eventually provide a complete picture of the current in the $n$-$\Phi$ plane: (i) In Sec.~\ref{sec:relation}, we analyse the effects of the cosine perturbation Eq.~(\ref{cos}), which being a relevant operator, flows to strong coupling ${g_p \to \infty}$. It yields straight current contours parallel to the line ${n = \nu \Phi/\pi}$. (ii)
Then, in Sec.~\ref{sec:relationBC}, we include band curvature irrelevant cubic terms, such as ${(\nabla \phi_\rho)^3}$, in the bosonized Hamiltonian; see Eq.~(\ref{eq:H3}) below. These terms contribute to corrections analogous to the term ${A^\mathrm{surf}\cdot(n- \frac{\Phi}{\pi})}$ in Eq.~(\ref{relationnonint}), and signal deviations of the density from the exact fractional filling.
(iii) Finally, in Sec.~\ref{correction}, we add additional quadratic operators in the LL theory which were not allowed at ${t_\perp=0}$, such as
\begin{equation}
\label{coupling}
\nabla \phi_\rho \nabla \theta_\sigma,~~~{\mathrm{and}}~~~\nabla \phi_\sigma \nabla \theta_\rho.
\end{equation}
These corrections give additional distortions of the current contours, which may be nevertheless neglected for small ${(t_\perp/t)^2}$.
The result of these calculations is summarized in the schematic Fig.~\ref{fig:ladder}.
\subsection{Chiral current in FQH states}
\label{sec:relation}
We treat $v_{\rho,\sigma}$ and $K_{\rho,\sigma}$ in Eq.~(\ref{HLL}) as the effective parameters resulting from the RG flow, and treat the strong coupling limit of Eq.~(\ref{cos}). We wish to find a relation between the three susceptibilities
\begin{equation}
\chi_{ab}=\frac{1}{L}\frac{\partial^2 E_\mathrm{GS}}{\partial a \partial b},~~~a,b = n,\Phi .
\end{equation}
Here $\chi_{nn}$ is the charge susceptibility, $\chi_{\Phi \Phi}$ is the diamagnetic susceptibility, and the mixed susceptibility ${\chi_{n \Phi} = \chi_{\Phi n }}$ describes the change of the persistent current ${j_c = - \frac{1}{L}\partial_\Phi E_\mathrm{GS}}$ with respect to particle addition.
We may opt to treat the fields $\phi_\rho,\theta_\sigma$ as generalized coordinates ($q$) and their canonical conjugates $\nabla\theta_\rho,\nabla\phi_\sigma$ as momenta ($p$). This is a useful choice as the Hamiltonian naturally decomposes to ${\mathcal{H}=\mathcal{H}_p+\mathcal{H}_q}$ as
\begin{align}
\mathcal{H}_p=\frac{1}{2\pi}&\int dx \left[v_\rho K_\rho (\nabla\theta_\rho)^2+ \frac{v_\sigma}{K_\sigma} (\nabla\phi_\sigma)^2\right], \\
\mathcal{H}_q=\frac{1}{2\pi}&\int dx\left[ \frac{v_\rho}{K_\rho} (\nabla\phi_\rho)^2+v_\sigma K_\sigma (\nabla\theta_\sigma)^2\right] \nonumber\\
+g_p&\int dx \cos\left(\sqrt{2}\theta_\sigma - \sqrt{2}\nu^{-1}\phi_\rho +\delta\Phi x\right),
\end{align}
where ${\delta\Phi=\Phi-\tfrac{\pi n}{\nu}}$.
The argument of the cosine is pinned in the strong coupling limit and we thus integrate out the $\theta_\sigma$ field so the Hamiltonian $\mathcal{H}_q$ takes the form
\begin{equation}
\mathcal{H}_q=
\frac{1}{2\pi}\int dx\left[v_\sigma K_\sigma (\tfrac{1}{\nu}\nabla\phi_\rho+\tfrac{1}{\sqrt{2}}\delta\Phi)^2+ \frac{v_\rho}{K_\rho} (\nabla\phi_\rho)^2\right].
\end{equation}
By recalling Eq.~(\ref{eq:density}) one sees that
a variation of the density is equivalent to setting ${\nabla \phi_\rho = -\frac{\pi}{\sqrt{2}}\delta n}$, and hence
\begin{equation}
\label{EGS}
\tfrac{1}{L}E_\mathrm{GS}\simeq
\frac{\pi}{4}\left[v_\sigma K_\sigma (\tfrac{1}{\nu}\delta n-\tfrac{1}{\pi}\delta\Phi)^2+ \frac{v_\rho}{K_\rho} {\delta n}^2\right].
\end{equation}
Notice that both ``momenta" fields $\theta_\rho,\phi_\sigma$ appear quadratically in $\mathcal{H}_p$ and it may be rigorously integrated out by choosing the gauge of ${\partial_t\Phi=0}$ in the Lagrangian formalism. This allows us to directly read off the susceptibilities
\begin{gather}
\chi_{n\Phi} = - \frac{1}{2\nu} v_\sigma K_\sigma ,~~~\chi_{\Phi\Phi} = \frac{1}{2\pi} v_\sigma K_\sigma, \nonumber\\
\chi_{nn} = \frac{\pi}{2} \left( \frac{v_\rho}{K_\rho}+ \frac{1}{\nu^2}v_\sigma K_\sigma\right).
\end{gather}
The relation ${\partial_n\partial_\Phi E_\mathrm{GS} =-\frac{\pi}{\nu} \partial_\Phi \partial_\Phi E_\mathrm{GS}}$ allows one to extract the fractional filling factor $\nu$ from a ratio of two thermodynamic susceptibilities. Equivalently, from the definition of the chiral current Eq.~(\ref{jaPhi}) we obtain
\begin{equation}
\label{relation}
\left(\partial_n + \frac{\pi}{\nu} \partial_\Phi \right) j_c=0.
\end{equation}
Notice that this relation does not depend on any of the LL parameters. Below we see that corrections to this relation are only of order ${\mathcal{O}(\delta n ,\delta \Phi )}$ and are thus small in ${t_\perp/t}$. Moreover, the current can be obtained by differentiation of Eq.~(\ref{EGS}),
\begin{equation}
\label{eq29}
j_c = \frac{1}{2 \nu} K_\sigma v_\sigma \left( n - \frac{\nu}{\pi}\Phi\right).
\end{equation}
These results suggest that in the $n$-$\Phi$ plane the current is constant parallel to the line of fractional filling, ${n =\nu\Phi/\pi}$, within the partially gapped phase; see dotted lines in Fig.~\ref{fig:ladder}.
It is desirable to arrive at a physical interpretation of these results (for this discussion, we retain the electron charge $e$ and Plank constant $\hbar$).
Consider the physics on the edge of an incompressible FQH droplet at filling factor ${\nu=1/m}$. The dynamical degree of freedom is the density of the chiral edge, $n_c$, which is a deformation of the edge of the incompressible liquid~\cite{wen_book}; the chiral current is given by ${j_c = e v n_c}$, where $v$ is the velocity of the edge. Electrodynamics on the edge is quite unusual, as the electron charge is intrinsically entangled with the electromagnetic potential on the edge~\cite{Ezawa}
\begin{equation}
\label{Ezawa}
n_c \to n_c - \frac{e}{2 \pi m \hbar}A_\parallel.
\end{equation}
It coincides with the minimal substitution argument for the vector potential $A_\parallel$ along the edge. This implies~\cite{Ezawa} the celebrated Laughlin argument of an adiabatic insertion of a flux quantum ${\delta A_\parallel = \frac{2\pi\hbar}{e L_\mathrm{edge}}}$ leading to the total change in the charge of the edge by a fractional amount ${e/m}$ (and creating a quasihole in the flux insertion point). As a consequence, Eq.~(\ref{Ezawa}) demonstrates that the current
\begin{equation}
j_c = e v \left(n_c - \frac{e}{2 \pi m \hbar}A_\parallel\right),
\end{equation}
remains invariant for a simultaneous change of the particle number and of the magnetic flux while keeping their ratio pinned to the filling factor. The two-leg ladder thus can be thought of as an ultra thin FQH droplet, with one chiral edge on the ${y=1}$ sites and the opposite chiral edge on the ${y=2}$ sites. We conclude that the physical meaning of current contours along the ${n = \nu \Phi/\pi}$ lines is the emergence of a fractional chiral edge.
It is important to note that even in the 2D quantum Hall limit, the contours are not exact straight lines. As is discussed below, the main contribution to deviations from linear behaviour stems from band curvature effects, reflecting the dependence of the edge velocity $v$ on the chemical potential, which holds true even in the 2D limit.
\subsection{Band curvature}
\label{sec:relationBC}
At ${t_\perp \to 0}$ where the ${\mathrm{SU}(2)}$ symmetry holds, there are only four cubic terms that may be added to the LL Hamiltonian,
\begin{multline}
\label{eq:H3}\mathcal{H}_3 = \int dx \left[ c_1 \nabla \phi_\rho (\nabla \theta_\sigma)^2 + c_2 (\nabla \phi_\rho)^3\right. \\
\left. + c_3 \nabla \phi_\rho (\nabla \phi_\sigma)^2 + c_4 \nabla \phi_\sigma \nabla \theta_\sigma \nabla \theta_\rho \right].
\end{multline}
As the other coefficients of cubic terms~\cite{Imambekov,SelaPereira}, $c_1$ satisfies the phenomenological relation
\begin{equation}
\label{c1}
c_1 = - \frac{1}{\sqrt{2}\pi^2} \frac{\partial \left(v_\sigma K_\sigma\right)}{\partial n}.
\end{equation}
When rigorously integrating out these interactions, various polynomial and rational function terms appear in the Hamiltonian. Nevertheless, by following the same procedure and doing some tedious algebra, one obtains the linear correction of order ${\mathcal{O}(\delta n,\delta \Phi)}$ to Eq.~(\ref{relation}),
\begin{equation}\label{relationCORRECTIONbandcurvature}
\left(\partial_n + \frac{\pi}{\nu} \partial_\Phi \right) j_c=A^\mathrm{surf}\cdot\left(n - \frac{\nu}{\pi}\Phi \right),
\end{equation}
where ${A^\mathrm{surf}= \frac{1}{2 \nu} (\partial_n+\frac{\pi}{\nu} \partial_\Phi) [v_\sigma K_\sigma]}$.
The band-curvature term ${c_1\propto\partial_n[v_\sigma K_\sigma]}$ can be incorporated into a density dependence of the parameter ${v_\sigma K_\sigma}$ in the LL theory Eq.~(\ref{HLL}). The dependence of the current on the density and magnetic flux in Eq.~(\ref{eq29}) thus remains the same up to order ${\mathcal{O}(\delta n,\delta \Phi)^2}$.
The result Eq.~(\ref{relationCORRECTIONbandcurvature}) generalizes the noninteracitng formula Eq.~(\ref{relationnonint}) to the fractional case. As a consistency check, in the noninteracting ${\nu=1}$ case, one has ${K_\sigma=1}$ and ${v_\sigma = \frac{1}{\pi}\partial_n\mu}$, and hence ${A^\mathrm{surf} = -\frac{\pi^2 c_1}{\sqrt{2}} = \frac{1}{4\pi}\partial_n^2\mu}$, which exactly matches the direct results of Sec.~\ref{nonintphin}.
\subsection{Further corrections and validity regime}
\label{correction}
We now consider the terms in Eq.~(\ref{coupling}). It can be shown that they are generated via RG by finite inter-chain hopping ${t_{\perp}\Psi^\dagger_1 \Psi^{\phantom{\dagger}}_2}$ to second order. It is straightforward to include these terms in the analysis of the current by re-evaluating the susceptibilities $\chi_{ab}$, and obtain additional corrections to the right hand side of Eq.~(\ref{relation}) which are quadratic in $t_\perp$. Having already identified the quadratic (and logarithmic) corrections in the noninteracting case from the \emph{Fermi sea} contribution of all filled states, see Eq.~(\ref{relationnonint}), we deduce that the low-energy theory is not appropriate to evaluate them. Indeed, it misses the logarithmic correction in the term ${A^\mathrm{sea} t_{\perp}^2 \ln t_\perp}$ term in Eq.~(\ref{relationnonint}). Hence, the explicit calculation of the order ${\mathcal{O} \left(t_\perp^2/t^2\right)}$ corrections is superfluous. We conclude that additional corrections to the right hand side of Eq.~(\ref{relationCORRECTIONbandcurvature}) are quadratic in the small parameter ${t_\perp/t}$ up to logarithmic corrections.
We wish to compare this quadratic correction with the right hand side of Eq.~(\ref{relationCORRECTIONbandcurvature}), ${A^\mathrm{surf} \cdot \left(n - \frac{\nu}{\pi}\Phi \right)}$. The density deviation, ${\delta n}$, within the partially gapped phase scales as ${E_\mathrm{gap} \propto t_\perp^{\frac{1}{2-x_{p}}}}$ (which for the noninteracting case ${x_p=1}$ behaves like $t_\perp$). We see that as long as the cosine is relevant with scaling dimension ${x_p<3/2}$, the quadratic correction is negligible in the entire FQH phase for small enough ${t_\perp/t}$; this is assumed in Fig.~\ref{fig:ladder}. Otherwise, one may still focus on the vicinity of the exact filling factor and observe lines parallel to ${n = \nu \Phi/\pi}$.
The lowest density for which the linearity of the contours apply is determined by requiring that the kinetic energy ${\thicksim t n^2}$ well exceeds the energy gap. This yields ${n \gg (t_\perp/t)^{\frac{1}{2 (2-x_p)}}}$; see Fig.~\ref{fig:ladder}. On the other hand, the density should be small compared to unity, otherwise, lattice effects would take effect~\cite{Fazio}. Therefore, we expect the behavior in Fig.~\ref{fig:ladder} to apply for ${(t_\perp/t)^{\frac{1}{2 (2-x_p)}} \ll n \ll 1}$.
\section{Discussion}
\label{sec:discuss}
We have studied two-leg ladders of interacting particles with an orbital magnetic field. For sufficiently strong and long ranged interactions, fractional quantum Hall phases are stabilized. Inside these phases there are chiral currents whose contours in the plane of density versus flux are approximately parallel to the line with fractional filling factor, similar to the Landau fan. This behaviour of the current is a signature of the emergence fractional chiral excitations. It distinguishes the 1D FQH state from other phases containing chiral currents such as the Mott insulator phases~\cite{Piraud,Petrescu,Dhar,Wei,Ahmet,Natu}.
Stabilization of FQH phases for small inter-chain coupling and low filling factors ${\nu <1/2}$ requires interaction range beyond on-rung. The cold-atomic technology
(see \emph{e.g.} Refs.~\onlinecite{Bloch14,Mancini})
however involves primarily on-atom interactions. Nevertheless, the on-atom interaction becomes nonlocal in the synthetic dimension~\cite{Mancini,Stuhl,Majoranacoldatoms}. This is sufficient to realize simple bosonic Laughlin states with ${\nu = 1/2}$ even for small inter-chain coupling.
Moreover, many-body systems with tailored long-range interactions have been achieved in Rydberg atoms~\cite{LukinDipoleBlockade,UrbanDipoleBlockade} due to strong van der Waals interaction, yielding Rydberg crystallization~\cite{Weimer10,SelaGarst11} as observed experimentally~\cite{Schauss}.
Arbitrarily small values of $K_\rho$ may be reached in principle for a particle-particle interaction of the form ${V(r) \propto r^{-\beta}}$
for which values of $K_\rho$ have been found analytically~\cite{Dalmonte10}, and where ${\beta=3}$ corresponds to dipolar atoms. Yet, the combination of a synthetic magnetic field and long range interactions, required for the low filling factor FQH states, remains an experimental task.
Nevertheless, powerful numerical techniques~\cite{Fazio,Zeng} have been recently used to simulate the fractional states considered here. With the findings of the current paper it becomes possible to test their fractional quantum Hall nature.
\acknowledgements
We thank M. Becker and S. Trebst for extensive supporting unpublished numerical calculations in the initial stages of this work; E. Dalla Torre, S. Furukawa, M. Goldstein, R. Pereira, and J. Ruhman for illuminating discussions; and S. Barbarino, R. Fazio, L. Mazza, D. Rossini, and L. Taddia for showing us their work~\cite{Fazio} prior to publication. This work was supported by Israel Science Foundation grant 1243/13 and Marie Curie CIG grant 618188.
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} | 5,331 |
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THE LOSS OF THE AUSTRALIA:
A NARRATIVE OF
THE LOSS OF THE BRIG AUSTRALIA, BY FIRE,
ON HER VOYAGE FROM LEITH TO SYDNEY.
WITH AN ACCOUNT OF
THE SUFFERINGS, RELIGIOUS EXERCISES, AND FINAL
RESCUE OF THE CREW AND PASSENGERS.
EDITED BY THE REV. JAMES R. M'GAVIN, DUNDEE.
NEW YORK:
ROBERT CARTER & BROTHERS,
285 BROADWAY.
1853.
CONTENTS.
Page
PREFACE, 5
LIST OF THE CREW AND PASSENGERS, 7
CHAPTER I.
THE VOYAGE AND CATASTROPHE, 9
CHAPTER II.
THE SUFFERINGS IN THE BOATS, 28
CHAPTER III.
THE FORLORN LANDING, 47
CHAPTER IV.
THE MELANCHOLY MARCH THROUGH THE WILDERNESS, 60
CHAPTER V.
THE RESCUE, 78
PREFACE BY THE EDITOR.
The short and simple narrative which is introduced to public notice
in the following pages, is a _plain statement of facts_; and is
submitted with unaffected diffidence, as an humble memorial of "the
loving kindness of the Lord, and his great goodness," in a season of
extremity. The only preface which can be necessary in a publication so
inconsiderable, is to certify its authenticity, by avowing the name and
affixing the responsibility of the author.
The following simple history of the narrative will, it is presumed,
be sufficient to remove all scruples as to its truthfulness and
reality. In the summer of 1844, Captain Adam Yule, of Dundee, committed
into my hands a large manuscript, containing the substance of the
following pages, with a request that I would use my utmost freedom
with the materials, and give them to the world in any form that was
most agreeable to myself. I learned that he had drawn up his account
at the Cape of Good Hope, immediately after the occurrences detailed
had happened, and that he had consigned it on his return to this
country, into the hands of a venerable friend, who had retained it
for two years, without finding the leisure necessary to correct and
prepare it for publication. In the execution of the trust reposed in
me, I considered it proper to reconstruct the narrative out of the
materials with which I was furnished; retaining, at the same time,
every important incident in its place, and in no case suppressing the
utterance of the devout experience of the writer. For the fidelity with
which I have adhered to the original facts, I hold myself responsible
alike to Captain Yule and to the Christian public; and I am happy to
acknowledge that I have received, both from himself and from several
of his fellow-survivors in that disastrous voyage, the most pleasing
assurances of the truthfulness of the statements. It is necessary
that, in such circumstances, I should exonerate Captain Yule from
all responsibility as to the _manner_ in which these facts are now
submitted to the public,--as I must be content, undividedly, to bear
whatever censure criticism may condescend to offer on the _literature_
of this publication.
The Editor dismisses his humble labours with satisfaction, that he
has been permitted to aid in rearing this simple tribute on behalf of
a class of men who must always hold a warm place in his interest and
affections; and whose perilous sacrifices of personal comfort and of
religious opportunities, in their calling, entitle them to the cordial
sympathy of all Christians. He accompanies the brief narrative with his
prayers, that it may be the instrument of spiritual benefit to many
souls, and may fulfil the only design in its publication, in promoting
piety among sailors, and confirming the promise of God, "that He is
nigh unto all them that call upon him, to all that call upon him in
truth."
JAMES R. M'GAVIN.
DUNDEE, December, 1845.
LIST OF THE CREW AND PASSENGERS.
Adam Yule, master; Alexander Wallace, mate; John Yule, second mate;
William Yule, carpenter; George Young, steward; Thomas Bisset, cook;
George Davidson, Thomas Souter, William Hay, John Allan, seamen;
Benjamin Aitken, Alexander Matthew, and James Hill, apprentices.
CABIN PASSENGERS.
Mr. Thomas Harris, London.
Mr. ---- ----, surgeon.
Miss Margaret Brown, Fife.
Miss Ann Sim, Edinburgh.
Miss Ann Knight, Turrif.
STEERAGE PASSENGERS.
John Henderson, miller.
Robert Elphinston.
James Elphinston.
James M'Lauchlan, farmer.
George Peat.
John Chisholm, }
George Chisholm, }
Jean Chisholm, } An orphan family.
Agnes Chisholm, }
Margery Chisholm, }
LOSS OF THE AUSTRALIA.
CHAPTER I.
THE VOYAGE AND CATASTROPHE.
"They that go down to the sea in ships, that do business in
great waters, these men see the works of the Lord, and his
wonders in the deep."
The life of a sailor, beyond the lot of most other men, discloses to a
reflecting mind an impressive series of divine mercies and judgments.
In a calling so singularly chequered by varying scenes and changing
incidents, life is spent amidst remarkable adventures and romantic
deliverances, so as to invest its course with an unusual interest, and
to crowd its experience with the most solemn and memorable instructions
of Heaven.
The individual by whom the materials of the following narrative were
contributed, is himself a sailor; and has borne a prominent part in
the painful scenes which are here depicted. His life has been prolonged
by divine mercy through almost every scene of sea-faring experience,
and it has been preserved by scarcely less than miracle, amidst perils
to which not many sailors have been exposed. The following story, in
all its facts and experience, is properly his own; and, therefore,
throughout he is preserved as the speaker. Only in the matter of
construction and expression, another party must be held responsible,
into whose hands the full materials were committed to give them form.
It was the devout desire of the original party not to forget Jehovah's
benefits; _having_, like the Psalmist (Psalm lxvi. 12,) "gone through
fire and through water," he felt solicitous to say with the same holy
minstrel, (verse 16,) "Come and hear, all ye that fear God, and I will
tell you what he hath done for my soul."
In the autumn of 1840, I accepted the command of the Australia,
of Dundee, bound for Sydney, New South Wales. On the 2d October,
our vessel set sail from Leith, having on board a general cargo of
merchandize. Our ship's company consisted of twenty-eight persons,
being thirteen of a crew, and fifteen passengers. My heart was buoyant
with hope and pleasing anticipations as I bade my family farewell,
and weighed anchor for my destination. Everything gave promise of
a propitious voyage. Our vessel was new and well found in every
necessary, the crew were able, and well selected, and the passengers
were agreeable, all being full of hope and fearless of evil. Indeed,
if we could have anticipated results, my company were most unlikely
and ill selected for enduring the hardships that awaited us; three of
the crew being but apprentice lads, and of the passengers, five being
females, besides two boys and a girl of very tender years. But who has
not seen, that while the helpless are sometimes the first to be visited
by the storm, they frequently are found, also, to survive its fury;
when the strong, who were the most likely to brave its blast, are
borne down and destroyed before it? "I returned and saw under the sun,
that the race is not to the swift, nor the battle to the strong--for
man also knoweth not his time: as the fishes that are taken in an evil
net, so are the sons of men snared, when it falleth suddenly upon them."
The commencement of our voyage was sufficiently prosperous. We rounded
Cape Wrath by an easy progress, and were in the latitude of Madeira in
seven days from Cape Clear. Nothing remarkable occurred till after our
departure from Rio de Janeiro, where we touched for a few days in the
beginning of December. We were then baffled with boisterous weather and
contrary winds, till the 27th of that month, when the wind became fair,
and the weather improved. On the evening of the 29th, December, we had
all sails set, with a strong fair wind, and a heavy sea. At this time,
by recent observations, I found that we must have been in latitude 35
deg. 51 min. south, and longitude 8 deg. 8 min. east of Greenwich, or,
in round numbers, about 600 miles from the nearest land, which was the
Cape of Good Hope. Our passengers had as usual walked the deck after
tea, until about eight o'clock, when, feeling it cold, they had gone
below. In less than half an hour, I followed them to the after-cabin,
having given the chief mate his orders for the night. We were all in
excellent spirits, and speculating how soon, and how safely we should
reach our destination with so good a wind. Alas! how little did we know
the horrors that awaited us: destruction even then had begun its
frightful work, and was silently, but too surely consuming our solitary
and sea-girded habitation. Soon after entering the cabin, I was affected
with a sense of something burning; supposing that the ladies might have
set something in their bed-rooms on fire, I ran forward in the dark to
their cabins, but found everything safe. The sense of burning, however,
became more strong and decided. I therefore snatched a light, and found,
to my dismay, that smoke was issuing from the fore bulk-head on the
starboard side of the mainmast. It was but the work of an instant to
clear away the goods with which that untenanted berth had been filled,
if possible to reach the seat of the fire. My brother William, and four
or five seamen withstood resolutely the suffocating smoke that
surrounded them in this labour, while others stood arranged and ready
with buckets full of water, to dash upon the first appearance of fire.
But what was our horror to find, on emptying the berth, that the evil
lay deeper, and was every moment on the increase; in short, that the
_ship's hold was on fire_! This was too soon apparent, for, on removing
a plank from the bulk-head, we saw the whole interior of the vessel like
the womb of a volcano, and the entire cargo of coals and combustible
goods in a blaze. It was impossible, from the superincumbent and
intervening goods, to pour in water in sufficient quantity to extinguish
so extensive a conflagration; this I perceived at first glance, and
therefore at once drove in the board to confine the flames, feeling, in
the agony of despair, that _the ship was irrecoverably to be consumed_.
It was an awful moment to every one of us. To die on so sudden
summoning, and to be summoned to _such a death_, were sufficient to
appal the stoutest heart. What were we to do?--beneath us was a burning
bier, and all beyond was a black and angry abyss. We could not abide
where we were, and to go forth scarcely promised a better fate, for no
little boat could live long in such a sea. I saw in the countenances
of the haggard beings around me, that they were fully alive to either
fate. Some, frantic with terror, sent forth cries, which found no
echo from our shoreless and surrounding solitude; others clung around
me, tormenting me with questions which I could not answer; while the
remainder stood silent and trembling, as if the presence of death had
smitten them dumb. It was easy to discern their emotions in their
demeanour--but why should I dilate on others' feelings, when I can but
faintly recall my own? I have a confused recollection of a tumultuous
throng of momentous interests rushing upon me with an overpowering
rapidity, and of a certain effort of self-possession seeking to stem,
while it received the tide. Visions of danger--of self-protection--of
death, mingled with thoughts of duty--of home--of a probably widowed
wife and fatherless family--all flashed wildly through my brain. I
felt that I stood in immediate contact with death, and the solemnities
of a judgment to come rose in array before me. It is not for me to
reveal the secrecies of such a situation; but I can only say as one who
has been "in deaths oft," and with all the solemnities of that hour
before me, that I know but one confidence that has proved unfailing and
infallible in such a crisis, and that is, _a personal interest in the
Lord Jesus Christ, and an implicit reliance on his perfect work_.
As I looked around upon the shivering group that had enclosed me,
I became filled with one solemn conviction,--it was my official
responsibility; and I was fired with one desperate effort--the
effort of rescue. Without a moment's delay, therefore, the plan of
arrangements was fixed, and the orders were given. The mate was
instructed to ease sail, and heave the ship to, in order to draw the
fire forward, and clear the after-part of the ship from smoke, so as to
allow us to labour with efficiency. A hole was then cut in the deck,
above the strongest seat of the fire, and an uninterrupted stream of
water poured down through the opening; but the rapid increase of smoke
and flame soon convinced us that all idea of subduing the fire, and
saving the ship, was impracticable. We then covered the deck with the
loose sails, to smother, as far as possible, the smoke and flame; for
by this time the deck-plank was blistering beneath our feet, and it
was impossible to breathe amidships. Our next efforts were directed to
launching the long-boat, which, as usual, was secured on deck. This
proved to be a work of great difficulty, and occasioned considerable
delay, not unmixed with danger. The boat had been converted into a
stall for two live bulls, and in attempting to get them over the side,
one of them, in the confusion, unfortunately got out of the slings, and
ran frantic along the deck. This accident, as may be supposed, greatly
increased the general consternation, and much invaluable time was lost
ere the ferocious animal could be secured and despatched; so that when
the tackles were hooked on to the boat, it was impossible to breathe in
that part of the ship. The men could only take a hasty pull and then
rush aft to breathe; and it was only after repeated efforts, and great
perseverance, that we got the bow of the long-boat sufficiently high
for launching. We then manned the after-tackle, but, unfortunately, it
unhooked aloft, and it required enormous exertions to get it replaced;
however, by fastening some guys round the rigging, and through the
blessing of God on our efforts, we at length got the boat launched,
and two good hands into her. To pass her aft, and preserve her from
swamping, were matters of great labour; for the roll of the sea was
so heavy, and the smoke was so dense over the lee-side, that we could
not see what we were doing. While these things were going on, I had
ordered the steward to prepare some bread, and small stores, to put
into the boat; and I now went down to see what progress he had made
for our supply, leaving the mate on deck to roll some water casks aft,
and after slinging them well, to drop them over the quarter to the
long-boat. Every moment, by this time, was invaluable; for the flames
had now made their appearance up the fore-hatch, and very soon caught
the rigging and sails. I can never sufficiently commend the energy of
the mate, and the steadiness and good behaviour of the men during these
exertions. There was no swearing, no inclination to fly to spirits;
every man was obedient to orders, and anxious to do his utmost. Even
the passengers revealed the same excellent spirit; I heard no screams
from the females, and even the children ceased to cry. All seemed to
feel that every effort was making for their safety, and they silently
acquiesced in the arrangements.
Our preparations were soon made. Two small bags of bread, two hams,
two cheeses, two or three canisters of preserved meat, and a few
bottles of wine, with a sextant, some charts, an almanac, my Bible and
Psalm-Book, and some flannel shirts and blankets, &c., were all that
we could secure amid the suffocating smoke. These were immediately
carried on deck, and secured in the skiff, which still hung in the
stern-davits. The mate, in the meantime, had rolled two casks of
rain-water aft, which was all that he could obtain. To secure their
safe transmission to the long-boat in such a sea, was no easy matter.
I therefore confided to the mate to lift them into the boat, and he
left the ship for this purpose. The first cask was well directed, but
in lifting it over the gunwale of the boat, it fell upon the mate and
another seaman, who were dreadfully bruised; it was a marvel, indeed,
that they were not killed. In consequence of their being disabled,
the second cask got out of the slings, and we lost it. This was a very
serious matter, but it was irreparable, as the whole front part of the
ship was now on fire, and quite impassable for any purpose. Finding
that I could make no further provision for the people, I put the ladies
and three children in the skiff, with two seamen, who were ordered to
cut the faulds, so soon as she touched the water, while we lowered
them from the davits. This was done in safety, which was a special
mercy, as the boat was greatly overloaded; having, besides the stores,
and the above company, two of the passengers, who, unknown to me, had
concealed themselves under the thafts. There were now left on board
the ship five or six persons, together with myself. These immediately
launched the small boat, which hung on the main-deck, and got safely
into it, so that, for a little season, I stood the last living thing
amid the burning mass. My position was alike novel and awful; two
horrid deaths were before me--one on either hand--and I stood but upon
a point between them. At that moment the flame was playing fearfully
over all the rigging; the topping-lifts had been burnt through, and the
trysail-boom came swinging down on the taffrail; the trysail itself was
on fire as high up as the third reef, and the mainmast every moment was
expected to fall above me. With a heavy heart I felt that I must quit
for ever the ship and property, of which I could no longer retain the
charge. Another and a still more sacred trust was beneath me; and as
I looked down upon the twenty-seven hapless beings, ghastly amid the
glare of the burning ship, and tossed above the billows that soon might
be our mutual tomb, I felt--oh, how I felt--that the charge of such
beings was _not mine_. Calmly as my momentary solitude would permit,
I lifted my soul to Him who "rules the raging of the sea," and cast
myself and company into his everlasting arms. If ever fervent prayer
was productive of immediate peace, my heart felt it at that moment;
for the words of God thrilled through me at the instant, as if his
own finger had inscribed them upon my bosom,--"Call upon me in the
day of trouble, I will deliver thee." I was recalled, however, from
my reverie by the mate imploring me to come into the boat, and as I
could do no more, I obeyed the summons; so, sliding down the tackles,
I got safely into the boat, among my wretched companions. At that
instant the mainmast fell with a tremendous crash over the side, and
the flames shot up with frightful fury from the cabin-skylight, as if
to intimate that the work of destruction was nearly completed, and that
our ill-fated vessel was no longer fit to be a refuge for living beings.
"One woe was past;" and although we knew well that others were awaiting
us, it was still an act of marvellous mercy that so many persons had
"come out of the midst of the fire" with "not a hair of any of our
heads singed." It is needless to speculate as to the cause of our
disaster; but, as it undoubtedly began in the lower hold among the
coals, it was most probably produced by spontaneous combustion. When
the last person left the ship, it must have been about eleven o'clock,
so that in less than three hours we had been cast forth from security
and comfort, amidst cold, and nakedness, and watching, to face dangers
and deaths in their most dismal aspect. It was my design to have
remained by the wreck till dawn, in the hope--a hope, alas, that was
not to be realized--that some friendly ship might be attracted by the
burning to our rescue. But the boats were in danger of being stove, it
being impossible at all times to prevent their chafing; and, ere long,
the rope by which we were made fast to the wreck became burnt through,
so that we were compelled to part even from the desolate companionship
of the burning vessel, and were cast adrift at midnight, upon the black
and boundless solitude of ocean. Still clinging to the hope of rescue,
I sought to keep the boats as close to the wreck as possible, and made
the best distribution of our company that I could. I took charge of the
long-boat with other sixteen souls in it; seven were in the skiff, and
four in the small-boat, and there we drifted till morning came.
It would be impossible to describe the grandeur and horrors of that
night. Let fancy paint, if it may, so many hapless beings huddled
together unpreparedly, exposed without shelter to the cold night sky,
and expecting every moment to be swallowed up. Ocean was ever fretting,
and curveting, and plunging beneath us, as if it had wrathfully
resolved to cast us from its "crested mane." The sky all above and
around was one scene of blackness, unbroken by one opening in its
cloud, and unblest by the radiance of one solitary star. Behind the
boats--in the region whither we were drifting, every thing was dark
as the grave. Light indeed attended us throughout that lone midnight,
but it was the glare of destruction, which, as it contended with the
surrounding darkness, only increased its horror. The flames long played
in magnificent grandeur, kindling the dark sky above, and reflecting
their lurid gleam from the ridge of every billow, as if they mocked
our misery by their majestic triumph. And ever and anon came some
terrific explosion--probably of the ship's spirits--which struck like a
death-knell upon our hearts, proclaiming that the work of ruin was well
nigh accomplished.
In this condition of extremity, one only hope remained to us--one last
grand anchor-hold to preserve us from despair. We remembered Him "who
maketh darkness his secret place, his pavilion round about him dark
waters, and thick clouds of the skies;" We thought of Him as "the
confidence of all the ends of the earth, and of them that are afar off
upon the sea, who stilleth the noise of the sea, and the noise of their
waves." And there, "out of the depths we cried unto Him." Mingling
with the voice of the wind and waters, and rising above their murmurs,
the sound of our praise and supplications ascended on the midnight
air, and was heard before the throne. It was a sacred relief to our
heavy hearts to feel that the eye of God still watched over us in our
misery, and that his ear was open to our cry: and although we knew not
the dark path that lay before us, yet we sought it with His words on
our lips,--"The floods have lifted up, O Lord, the floods have lifted
up their voice; the floods lift up their waves. THE LORD ON HIGH IS
MIGHTIER THAN THE NOISE OF MANY WATERS, YEA THAN THE MIGHTY WAVES OF
THE SEA."
CHAPTER II.
THE SUFFERINGS IN THE BOATS.
"They mount up to the heaven, they go down again to the depths;
their soul is melted because of trouble."
"Joy cometh in the morning;" but it was not so with our forlorn
company. Daylight of the 30th December dawned only to reveal our
mutual wretchedness, and to aggravate our distress. Our hapless vessel
vanished in the distance as daylight appeared, and our hearts fainted
to discover that no friendly sail was visible within the range of the
horizon, for our rescue. Left alone in that vast solitude of sea and
sky, it only now remained for us to seek our safety by making for
the nearest land, or to die in the endeavour. We were but "in the
beginning of sorrows," and our first business was to commit ourselves
to God. Gathering our boats as closely as possible together, we
joined in singing the 38th Psalm, 1-5th verses, and by prayer "poured
out our complaint before God, and showed before him our trouble."
Being comforted by this exercise, we immediately thereafter commenced
active preparations for our melancholy voyage. Our first object was
to rig a mast and sail in each boat. We had only oars to form our
masts, and a top-gallant studding sail and royal fore sails. With some
small lines, shrouds and stays were made; and by six o'clock in the
morning all the three boats were under sail for our destination. I then
commenced to overhaul our stock of supplies, and found that we had two
small cheeses, two hams, only about twenty-four gallons of water, and
seventy or eighty pounds of bread, which was damaged by salt water,
with a half gallon of rum, a half gallon of brandy, and a few bottles
of wine. This supply was by no means adequate to sustain life among
such a company for many days. I therefore called the boats together,
and told the people that we could not expect to make the land in less
than ten or twelve days, and it might take a day or two more; that
our stock of water and provisions was far short, and that therefore
we must come at once on short allowance. I am happy to say that all
acquiesced with the proposition, and, indeed, showed throughout the
happiest spirit of subordination and harmony. Our small allowance was
then distributed, which gave a little bread, which was repeated in the
evening, and _only three table spoonfuls of water to each per day_. At
noon I got an observation for the latitude, and found it 35 deg. 37
sec. south, and longitude 9 deg. 15 sec. east of Greenwich. We again
engaged in the worship of God, and sought to keep the boats in close
company. But as day declined the weather looked wild; and the men in
the small boat, being afraid of her capsizing during the night, I
had to divide her company between the long-boat and skiff, and cast
her adrift. Nine persons were thus in the skiff, and nineteen in the
longboat, which sank us very deep in the water, and uncomfortably
overcrowded us. The long-boat was particularly uncomfortable, being
lumbered with our small stock of provisions; and, having been used as
a stall for cattle, we were not only soaked with seawater, but smeared
with filth. Our distressing situation may easily be supposed, with a
promiscuous company of ladies and children crowded together without the
means of separation, and exposed night and day to the action of all
the rudest elements. We however washed our boat, which served somewhat
to improve our condition. At the close of day we again sang praise to
God, and implored his protection and blessing. The regular performance
of this duty was a great comfort to us in our misery, and I was well
assisted in its discharge by Mr. Wallace the mate, my brother William,
and nephew John, as well as by some of the passengers, all of whom
occasionally conducted the devotions. We made it our endeavour to unite
both boats in one exercise of daily praise and prayer, and when this
was impracticable, service was separately conducted in the skiff by
George Davidson and Thomas Souter, seamen. In no case, to the best of
my knowledge, was this duty omitted from being performed three times a
day, so that we could say, with the Psalmist, "Evening, and morning,
and at noon, we pray and cry aloud, and he shall hear our voice."
During the night the wind blew freshly from the south, and the sea
was so heavy, that I was obliged to deviate to the north of my course
a little, in the hope of regaining my leeway by a future and more
favourable wind and sea. But although it would have been hard work
to fetch the Cape of Good Hope even with a fairer wind, we could not
help ourselves, as our little boats could not breast the billows, and
yet we hauled them as close to the wind as we dared. The moon shone
on us during a part of that night, and enabled us to keep the boats
together; but when she set, we were greatly distressed by the danger of
separating. At length day light came to the relief of our sleepless
and anxious watching, but only to the increase of our other sorrows.
This day the people pleaded with tears for an increase to the allowance
of water, and my soul yearned for the petitioners; but although I felt
the strength of their craving in my own fevered frame, I dared not
accede to their request. I knew that our distance from land rendered it
certain destruction for us to increase our expenditure, unless, indeed,
some friendly bark should cross our path, which we could not certify,
and which certainly never occurred. I therefore earnestly exhorted
them to make the best use of the small quantity allowed, by dividing
it into three daily distributions. This was done in the long-boat,
and we felt the benefit of it, in the more frequent moistening of our
palate, and the easier mastication of our bread. At noon I obtained
an observation, and found the latitude 34 deg. 49 min. south, and
calculated our longitude at 11 deg. 40 min. east. A little wine was
distributed this day along with the usual allowance of water, which
was greedily swallowed. Towards evening another earnest appeal came
from the people in the skiff for an additional allowance of water,
which I was compelled to refuse. Contrary to my injunctions, they had
swallowed their allowance at one draught, and were therefore in agony
till the time for next day's supply. I learned, also, that some of them
had begun to drink salt water, which I sought in vain to prevent. I
told them that if they persisted they would become delirious, which,
alas! was soon too painfully realized.
The wind lulled a little about midnight, but the darkness greatly
distressed us, and about four o'clock we lost sight of the skiff.
We immediately lowered our sail, and with difficulty got a light in
the lantern, awaiting the result with intense anxiety. For half an
hour this distressing suspense continued, when, to our great relief,
the boat re-appeared. Night ere long again departed, but with each
returning day we found the sufferings of our company on the increase;
cold and thirst were making shocking inroads among us. Up to this
time we had never been able to stretch our stiffened limbs, and we
had all the while been thoroughly drenched by the constant action of
the sea. This day, however, being more favourable, we got our clothes
partially dried, and managed to erect a temporary bulwark of blankets
on the weather-side, which afforded some additional shelter from
the elements. This enabled us to perform our worship "with a little
reviving," and we partook of our scanty allowance with increasing
appetite. My observation for this day was latitude 34 deg. 30 min.
south, and longitude 12 deg. 49 min. east. Towards evening the wind
and sea increased from the south-west, and as I could not make my
course good, I allowed the boats to run, so as to make all the easting
possible. At midnight the moon went down, and as the sea ran very high,
we had difficulty to preserve the boats in company during the darkness.
Our candles were scanty, so that we could not burn constant light,
and we longed exceedingly for the coming of day. By this time our
distresses were very grievous; the midnight sea had thoroughly soaked
every one of us, and several of our people gave decisive symptoms
of insanity, especially two of the passengers in the skiff, who had
persisted in drinking the salt water. In the morning the weather became
more moderate; at noon we were in latitude 34 deg. 34 min. south, and
longitude 14 deg. 37 min. east, so that I concluded, if the weather
should keep favourable, that in three days' sail we might make the
land. The wretched condition of our company towards evening constrained
me to administer a little wine, and an additional half of a wine
glassful of water to each; I exhorted them to use it sparingly, as I
dreaded a stormy night; but the people in the skiff consumed it on the
instant. Milder weather succeeded in the morning, which enabled us to
dry our clothes. In the afternoon we rigged a temporary jib, with a
sheet for a studding-sail, and the crew of the skiff did the same. I
tried for an observation, and found the latitude 34 deg. 12 min, but
my chronometer by this time was nearly useless for the calculation of
longitude, and I guessed it to be 15 deg. 47 min. east. The cry for
water at this time became heart-rending, especially from the children
in the skiff; their piercing screams went to my inmost soul, and yet
I durst not be subdued by them; therefore, with a feigned sternness,
which my heart disallowed, I was compelled to order the skiff to shear
off, so that I might at least be released from listening to their
anguish, which I could neither bear nor brave.
Again evening and morning came, and still as our course lengthened
our woes increased. The night and morning were intensely cold, and
a hollow sea again had drenched us to the skin. The people seemed
to have reached a state of utter exhaustion, not unmingled with the
indifference of despair. They appeared to have lost all relish for
food, and water was the only cry; several of them had persisted in
taking salt water, which it was impossible to prevent, as there were
but eight or nine inches of free side from the sea, so that they put
out their hand through the night and took it. The consequence was,
that two in our boat, and the same number in the skiff, were quite
delirious, while several others in both boats gave symptoms of the same
distressing state. The ladies throughout behaved with magnanimity, and
even the endurance of the children was admirable. The best arrangements
were indeed made for them which we could command. We appropriated the
stern sheets to the ladies, as the most comfortable; and for their
accommodation I had to sit upon the gunwale, while steering the boat.
This post was only filled by the mate and myself, as there was no other
to whom I could confide it; but he, being very unwell, from having been
crushed by the water-cask, the heaviest share of the duty devolved upon
me. The skiff was managed by Thomas Souter and George Davidson, whose
excellent seamanship was beyond all praise. The people seemed to be so
depressed and inclined to sleep, that in the evening I mixed a little
rum with their allowance of water, which partially revived them. The
night was setting in very gloomily, and as our evening song mingled
with the rising tempest, I am sure that our hearts sympathized with its
plea. It was Psalm vi.
Lord, in thy wrath rebuke me not;
Nor in thy hot rage chasten me, &c.
Our chapter this evening was Acts xxvii., and we prayed that the God
who stood by Paul, in his perils and shipwreck, would preserve the
lives of all who sailed with us. The night was very dark and stormy,
with a heavy sea; every wave was broken on the top, and we were nearly
smothered by the spray. It required all our skill to keep the sea from
breaking on board of us. I gave orders to the men to stand ready with
our three buckets, in case, amid the darkness, any wave should make
a breach on us. At length, about midnight, one frightful billow rose
close to the boat, and broke right over us. A slight scream rose from
our company at the instant, and I thought our fate was sealed, as the
boat was nearly filled with water, and staggered under the stroke, as
if settling in the trough of the sea. I, however, got her right before
the wind, and during a short _smooth_ which providentially succeeded,
she was bailed with all despatch, and righted. The skiff had been in no
better condition, and nothing but the most masterly seamanship could
have preserved her afloat. Frequently we lost sight of each other
during the darkness, and our matches being wet, we could no longer hold
out a signal-light as formerly. "By the good hand of our God upon us,"
however, we were mutually preserved, and kept together during that
dismal night.
A frowning morning succeeded, and found our companions worse than
ever. I immediately served a small allowance, which revived us all;
indeed I was at this time myself greatly exhausted, having kept the
helm without stirring for thirty-six hours, on account of the illness
of the mate. My sextant having been spoiled by the loss of its top, I
was now no longer able to keep our reckoning, except by guess. I was
in hopes that the gale would subside at noon, and permit us to take a
more southerly course, so as to fetch the Cape, but I was unhappily
disappointed. The storm only increased in severity, and the sea broke
around us with redoubled fury, driving in the temporary bulwarks, which
we had re-erected after the night's disaster. I calculated that at this
time we were about seventy miles from land, but the brackish colour of
the water led me to suppose that we might be nearer, and, being afraid
to make the coast in the night, I resolved to stand to the north till
midnight, it being impossible to ride the boats by bridle or otherwise
in such a sea. To this all parties gave consent, and I issued orders
accordingly. I confess that I had almost no hope of seeing morning, and
therefore told the skiff's crew that if anything happened to us through
the night, they must stand in for the land, and do the best they could.
My gloomy forebodings were shared by all, except those--to the number
of six or seven--who were by this time insensible to everything around
them. After partaking of our allowance with thanksgiving, we committed
ourselves to the Lord of life and death, and took leave of each other
without the hope of meeting again in this world. In the early part of
the night our little boats behaved admirably in their conflict with the
tremendous sea, and at eleven o'clock we shifted our small sail, and
stood directly in for the land. The skiff followed, but at midnight
the wind and waves increased in fury, and a tremendous billow broke
close astern of us, which seemed to swallow up our dear companions. We
strained our aching vision to catch the re-appearance of their little
mast, but in vain; with trembling anxiety we then lowered down our
sail, and, after great difficulty, got a light in the lantern, but it
was soon extinguished, and, after long and anxious waiting, no trace
of the skiff was visible, and we gave them up as lost, believing "that
the deep had covered them." The sea was breaking so heavily over the
stern, while there was no _way_ on our boat, that we were in danger of
foundering, so that we were compelled, with deep distress, again to
make sail, and pursue our course. Life was now faint within me, and I
felt as if "the bitterness of death was past." A cold shiver had seized
my frame, and I was inclined to resign all further effort. By the
administration of a tea-spoonful of wine, however, I rallied a little,
and maintained my post at the helm throughout the night.
Morning at length broke, but there was no appearance of our companions,
and all hope of their restoration departed. Our morning meal was
consumed in melancholy silence, and our "grief was heavier than our
groaning" in our morning prayers. Four persons in our boat were
in extreme exhaustion, and one of them--a passenger--named George
Peat, was evidently in a dying state. The weather looked more mild,
and I sought to rally their spirits: with three of them I partially
succeeded, but Peat took no notice of anything, save to suck greedily
his allowance of water. In the forenoon the sun broke through the
clouds, and shed an agreeable warmth to which we had long been
strangers, so that we took off our wet clothes, and hung them up to
dry. The hope of seeing land revived the love of life within us, and,
with the former exceptions, our company, in spite of the absence of our
other boat, were in better spirits. At eleven o'clock A.M.
the mate relieved me from the helm, and all were intent in looking out
for the land. In this we were disappointed; but the mate thought he
descried something ahead like a mast or a sail. All eyes were turned
in the direction with eagerness, but for a considerable time we could
see nothing. At last another person saw something on the top of a heavy
wave, and, as we drew nearer, a mast without a sail became distinctly
visible. Could it be our brethren? was anxiously inquired by every
one; and indeed it was. Poor fellows! they had tasted nothing for more
than twenty-four hours. At the time when they disappeared they were
overwhelmed in the belly of a tremendous broken sea, and their boat
was nearly filled. Their little mast was carried away, and one of them
was washed overboard, but catching hold of the boat, they had hauled
him in again. By extraordinary exertions they then bailed their boat,
got their mast replaced, and, pursuing our course, in their anxiety to
overtake us, had actually passed us before daylight. How we ever met
again was a mystery to all; but "it was the Lord's doing, and it was
marvelous in our eyes." I shall not attempt to describe the scene of
our remarkable greeting. It was not joyous, for alas, we had now become
strangers to every emotion of gladness; but we grasped each other's
hands, and our full hearts found vent in silent tears. Our souls had
become knit together in the fellowship of suffering, and in the midst
of deaths, we celebrated their restoration as a deliverance from the
grave. Of course they received immediate refreshment and a little wine
was distributed to the whole company on the occasion. Our noontide
worship, which was mutually conducted, arose from overflowing hearts;
and although our common woes were nothing abated, we caught something
of the spirit of our hymn while we sung,
Let troubles rise, and terrors frown,
And days of darkness fall,
Through Him all dangers we'll defy,
And more than conquer all.
CHAPTER III.
THE FORLORN LANDING.
"They are at their wit's end. Then they cry unto the Lord in
their trouble, and he bringeth them out of their distresses. He
maketh the storm a calm, so that the waves thereof are still.
Then are they glad because they be quiet."
So soon as our heartfelt congratulations had blended and been breathed
out in prayer, hope became faintly rekindled in each yet conscious
bosom of our distressed company; and with all our lingering energies
of life, we made for the yet invisible shore. "The wrath of God lay
hard upon us," and, for so many days "we had been afflicted with all
his waves," that we felt as if all safety consisted only in escape from
ocean's "deeps." And yet I was not without apprehension, that
what we so fondly anticipated as the occasion of deliverance, might
prove the fatal scene of our doom. The imminent danger of approaching
a comparatively unknown coast, especially amid the heavy roll of Cape
seas, and in such small boats as ours, demanded the exercise of every
possible precaution, and suggested forebodings of no very pleasing
issue. By my calculations we had been driven to the north of St. Helena
Bay, which, by its bend, gave us forty miles more of sea to traverse
than if we had been able to keep a more southerly course. On consulting
a small fragment of chart--which one of the ladies had preserved for
us, from the action of the sea, in her bosom,--I found, to our great
relief, that the coast for which we were making was free of any outside
shoals, and appeared favourable for our landing. We therefore made
all speed to reach the shore if possible before nightfall; in this,
however, we were disappointed; and a dense fog ahead hid the object
of our solicitude from view, until night descended, and shrouded the
surrounding landscape in darkness. The weather being moderate I
resolved to prosecute our course throughout the night, and endeavour to
effect our landing at daylight. The evening proved intensely cold, and
we endured more acute suffering from the wind and spray, during those
hours of darkness, than we had ever done before. This was probably
caused by our preserving a more southerly course, and keeping the sheet
hauled aft, which exposed us to the action of the sea, and sent the
wind right down on us from the sail. Ere morning came a cold shiver had
consequently seized every frame, and several persons in both boats were
quite unable to stir.
About five o'clock the skiff hailed us, and communicated the melancholy
tidings that the lad John Chisholm was dead. This was the first breach
made among us, and it fell among our wasting company like a forerunner
of our own fate. We were all closely "round the grave's devouring
mouth," and now that it had found its first victim, we felt assured
that others would follow. George Peat, in our boat, was only in life,
and several persons in both boats were visibly sinking fast into the
same unconscious state. I felt this visitation bitterly, as I was in
full hope of reaching land in a few hours, and was sustained--by the
signal mercy hitherto enjoyed--in the pleasing expectation that "God
would have given us the lives of all who sailed with us." But "He who
doeth according to his will" had deemed it otherwise, and our hearts
smote us to think that we had been preserved amid many perils, possibly
only to perish on the threshold of deliverance.
Visions of land floated before our aching and anxious gaze throughout
that weary night, and often we supposed that we could detect the dim
outline of the headlands between the sea and sky. Still we trembled in
uncertainty until morning came; but when the sun arose, it looked down
upon us from behind the African hills, which stood in distinct outline
before us at the distance of twelve miles. Then every heart bounded
with hope, and the fading energies of life revived within us. We
greeted the glad spectacle with our morning incense, and poured out our
thanksgiving to God our Ebenezer. There was a beautiful propriety in
the subject of our song, which then rose on the morning air, from the
margin of that mighty ocean. It was Psalm xlvi.,
"God is our refuge and our strength,
In straits a present aid;
Therefore although the earth remove,
We will not be afraid.
Though hills amidst the seas be cast,
Though waters roaring make
And troubled be; yea, _though the hill
By swelling seas do shake_."
Scarcely had these sublime words passed our lips, ere we felt the awful
importance and value of the holy sentiment. Our eyes could now detect
a long line of frowning and iron-bound coast, fringed only with foam,
and hoary with tremendous breakers. No friendly opening was visible,
along that fearful barrier, and we looked in vain for some quiet creek
amid the strife, where ocean might peacefully surrender the helpless
charge which longed for escape from its horrors. As if to increase
the solemnity of our condition, the wind at this time began to rise,
and a heavy ground swell rolled in from the south-west, so that it
needed no ordinary faith to prepare with calmness for the approaching
crisis. But our only course was to face the danger, and trust to God
for deliverance. I sent the small boat ahead, to examine the coast, if
possible to find a creek for convenient landing, it being lighter than
our boat, and having thafts for easy rowing, which we had not. I then
sought to rally the spirits of my crew by a little exertion; getting
out the oars, I exhorted them to try the exercise of rowing a little,
and took a spell myself. With great difficulty I succeeded in inducing
the most of them to make the attempt, and we felt the benefit of the
effort, in a freer circulation of our blood, which served to relax our
stiffened joints, and relieved us of the cold shivering.
The breeze continued strong, and the sea was very heavy, until we
approached within half a mile of the shore; when God--as if in sympathy
with our situation, and preparing our way--subdued the wind, and made
the strife of waters partially to subside. This gracious interposition
made a deep impression upon us all, and we felt animated by it, in our
very critical circumstances, as a foretaste of deliverance. At this
time, a small rock which appeared to windward, presented to our eager
eyes for a season the likeness of a sail; and we were delighted for the
moment with the idea, that the coast which we were approaching might
be inhabited; but a nearer view soon dispelled the illusion, and left
us to a scene only of wild and desert solitude. Our small boat had now
gone close in with the shore, in search of a landing-place, while we
remained at a short distance on the outside, to wait for instructions.
Our companions, in their eagerness to execute their survey, had
unfortunately got themselves embayed, and in attempting to weather a
projecting point, they failed; so that, in their extremity, one course
only remained to them--for life or death they had to run for the beach.
We, seeing this sudden movement, and supposing that our friends had
discovered a favourable landing-place, bore up, and followed closely
in their track. By signs and cries they attempted to warn us off;
but we, mistaking their signals for encouragement, only pursued with
increasing speed. It was a moment of intense and trembling interest
to us all; death or deliverance hung upon the instant, and our hearts
were fully alive to the immediate and awful alternative. Every faded
and haggard countenance became flushed with eager excitement; every eye
was strained to watch on either hand the impending fate; every hand
grasped the gunwale with convulsive and trembling energy, and we held
our breath in awe, as we dashed among the breakers, and plunged amid
those fearful rocks and shoals. Surely the eye of heaven was watching
over us in that unchosen and accidental landing-scene; for amidst many
perils, it presented favourable opportunities for us--in a narrow
channel among a cluster of small rocks, which was crowned with a sand
beach--that no human foresight could have detected, and that was rare
on that coast. Our small boat, indeed, was in extreme jeopardy; for in
the midst of the breakers it struck upon a sharp rock, and some of the
crew were thrown overboard by the shock. The sail, however, being still
set, the next wave lifted it over, and the wind and sea being dead in
shore, drove them right up to the beach, where, amid many difficulties,
they effected a landing, and rescued their comrades in a state of great
exhaustion. We in the larger boat were somewhat more fortunate; for "by
the good hand of God upon us," we made our way safely through a narrow
channel, among the small rocks, without touching, until we came within
a boat's length of the beach, where we stuck fast upon a rock. There
being deep water between us and the shore, we were all plunged overhead
in our attempts to escape; but the ladies and children being assisted
by the mate and seamen, were soon placed in safety; and "so it came to
pass, that we escaped all safe to land."
This signal deliverance--alike so gracious and remarkable--revealed in
all its course and accomplishment, the direct and immediate agency of
God, and could be attributed solely to his marked interposition and
care. No human foresight or management could have availed to preserve
so helpless a company in such extremities. With boats so frail, and
means of sustenance so slender, nothing less than Omnipotent kindness
could have sustained us throughout a voyage so disproportionate to all
our preparations, and so encompassed with exceeding dangers. If our
course, indeed, revealed no miracles, it was at least replete with
special mercies; for had we been visited by a few days of head winds,
or been overtaken by any of the fearful squalls so common in Cape seas,
or even made our landfall on a bold and unbroken coast, not one of us
would have survived in such a case to tell the tale of our disasters,
and our last struggles would have been hid in the dark and terrible
secrets of ocean, which, like the grave, gives no revelations. We
had been led to look to God in all our way: even the good order and
discipline which had been maintained, we felt we owed to his grace;
and while we had used our best endeavours for our preservation, yet
without his blessing, we were conscious that every exertion must have
been without avail. Therefore, when God had "been better to us than our
fears," and "redeemed our lives from destruction," our utmost gratitude
was due to him, and we invite men "to see his hand," and "to praise the
Lord for his goodness, and for his wonderful works to the children of
men."
If it had been possible, at that solemn hour, to have forgotten
or overlooked the signal kindness of heaven, even the continuous
manifestations of Divine goodness to us must have, on the instant,
rebuked such base ingratitude. Scarcely had the feet of our forlorn
company been permitted to touch the shore, when the storm, which had
lulled previously to our landing, burst forth with redoubled fury,
and raged without intermission during the whole time that we remained
in that place. The sea arose in ungovernable wrath, and as it lashed
the shore, lifted our little boats upon its billows to a height of
forty or fifty feet upon the beach. The narrow channel through which
we had reached the shore in safety instantly became one scene of
boiling surge, which would have shattered to pieces the proudest bark,
and engulphed every living thing on board of her. Who could fail to
discover the striking proof of a special and gracious Providence in
this occurrence? If it be said that such sudden storms frequently
occur in these latitudes, still the question arises,--why did that
storm come at the precise moment when we were immediately out of the
reach of its fury? There can be but one answer to this inquiry,--it
was the good pleasure of him "who gave to the sea its decree, that
it cannot pass, and who compasseth its waters with bounds." Our
company stood awe-stricken at the sight. We looked back upon the
scene of destruction, from which we had so recently escaped, with
mingled feelings of dismay and gratitude. Our deliverance, indeed,
was not yet complete. Alas! who could tell whether,--"having escaped
the sea,--vengeance might yet suffer us to live?" "The perils of the
wilderness" lay before us in all their unknown horrors of toil, and
thirst, and frightful famine. Still we had been delivered from "the
floods that affrighted us,"--our bosoms swelled with the full sense
of our rescue, and while we raised our song of deliverance and poured
out our grateful prayers to God, there were many devout hearts in our
circle who could appropriate the sentiment of the poet:--
Thus far on life's perplexing path,
Thus far the Lord our steps hath led;
Safe from the _flame's_ pursuing wrath,
Unharmed though _floods_ hung o'er our head.
Here let us pause--look back--adore,
Like ransom'd Israel from the shore.
CHAPTER IV.
THE MELANCHOLY MARCH THROUGH THE WILDERNESS.
"They wandered in the wilderness in a solitary way, they found
no city to dwell in. Hungry and thirsty, their soul fainted in
them. Then they cried unto the Lord in their trouble, and he
delivered them out of their distresses."
The first view of our solitary landing-place revealed to us a wild and
barren region. Neither traces of cultivation, nor marks of human abode,
nor even tracks of living creature, met our eye in all the adjacent
landscape; and my heart misgave me at the prospect, lest we had only
exchanged the scene of our miseries, but not escaped from them. The
idea of delay in obtaining succours was too painful to indulge, for
every hour only added to the horrors of our situation.
Our scanty supplies were rapidly wasting away, and the strength of the
people was already well nigh spent by fatigue and famine. I dared not
to anticipate the consequences of even a short continuance of such a
state of things, and felt that our utmost efforts must be directed to
urgent measures for immediate deliverance.
Meanwhile, every arrangement was made for present exigencies. A small
refreshment was distributed immediately after landing, and our weary
men set about the erection of tents, which were soon reared, by lashing
a few spars together and overlaying them with sails and blankets.
This shelter was peculiarly seasonable to persons, who, for nine days
and nights had not known the luxury of lying down, or resting their
exhausted frames; especially as an African sun was blazing in noon-tide
brilliancy, and with insufferable fierceness over our heads. Our wasted
people soon betook themselves to repose and I was pleased to find that
the greater number of them were soon lost in sweet forgetfulness of
all their woes. As for myself, sleep had so long been a stranger to my
aching frame, that it refused at first to revisit me, and my mind was
too anxiously concerned for the future to court its present approach.
So soon, therefore, as I found the others asleep, Mr. Wallace the mate,
and I--who had lain down together, and were alike wakeful--rose up
and went forth to consult as to our future course of proceeding. We
were agreed in thinking that the parched and desolate appearance of
the place gave little hope of finding water, or of obtaining relief;
and that our whole and instant efforts must be directed to discover
succours by sea or land. Two courses only presented themselves: the
first, which was to re-enter the boats, and endeavour to reach the
Cape colony by sea, was plainly impracticable from the severity of the
weather; the sea at that moment being visible below us in its wildest
majesty, as it thundered its mountainous billows against the base of
the rocks, and scattered its angry foam over the cliffs to a hundred
and twenty feet above its bed. Besides, we were conscious that even in
calmest weather, our company could not possibly survive for twenty-four
hours, under a renewed exposure and crowding in the boats, without a
fresh supply of water and provisions. We were shut up, therefore, to
the only available alternative of seeking succour by a land journey,
and by keeping a southerly course in the direction of the Cape, we
hoped that we might soon reach some human habitation. I proposed that
we should remain till the expiry of the following day, in order to
recruit our people for the journey, and to complete the necessary
preparations for our departure. Meanwhile a search could be made for
water, and I would endeavour to obtain an observation at noon, in order
to certify our exact latitude, and ascertain our distance from the
Cape. This course was afterwards submitted to the whole company, and as
it met their approval, it was adopted.
We had carried George Peat ashore from the boat in a dying state. Every
thing was done for his comfort which our circumstances would permit,
but the poor lad was beyond the reach of relief. He lingered in painful
unconsciousness till the following morning, when he died. The body
of John Chisholm was also brought on shore in the skiff, and covered
with the Union Jack, until we had leisure to dig a grave. The two
youths were respectfully buried on the successive afternoons, divine
service being performed at their interment. They lie side by side on
that desert shore where they met their fate, and their pilgrimage
ceased,--where no footstep of friendship shall ever trace the unknown
scene of their last repose, and only the murmurs of ocean disturb its
solitary stillness.
After evening service had been conducted in the tent, we kindled a fire
to preserve us from any attacks of wild beasts, and committed ourselves
to rest. I enjoyed a few hours of sweet sleep that evening, for the
first time after my long watching, and awoke considerably refreshed
and invigorated. By four o'clock in the morning our whole party were
astir, and went off in detachments at daylight, to search for water;
but after wandering for two hours, in survey of all the surrounding
coast, they returned, as we feared, dispirited and unsuccessful. A
vegetable was found in great abundance which was full of sap, but on
tasting it we discovered that it was saturated with salt, and unfit
for use. The only supply which the region afforded was shell-fish,
which for the same reason, with our scanty allowance of water, could
only be sparingly used. The situation of our tent in the low grounds
was now found to be insufferable on account of the intense heat, so
that I proposed to shift it to the rising ground behind, in order to
obtain a freer circulation of air. But our people were so feeble as to
be unfit for the exertion, and it was only after great labour, and by
bribing them with a tea-spoonful of wine, that this measure of relief
was accomplished. The ladies meanwhile were employed, in preparations
for our journey of the following day, by making canvass bags to hold
our provisions; and the precious remainder of water was emptied from
the cask into bottles and jars, so as to be easy of carriage on the
road. I could only get an overhead observation for the latitude at
noon, in consequence of the sun being over the land. My calculation
agreed with my previous conjecture, that we were north of the Oliphant
(or Elephant's) River, about eleven miles; but as Norie's Epitome,
which was my only book of reference snatched from the burning ship,
gave me no example for working such an altitude, I could not certify
the accuracy of the reckoning. However, my repeated trials convinced
me that I could not be more than a mile or two from the truth, and we
determined, therefore, to start in that direction on the morrow, in the
hope of finding some settlement on the river's banks.
Our preparations for departure being completed, we lay down to
sleep, under the same precautions as on the previous night, and were
aroused at four o'clock to pursue our journey. Previous to starting,
I distributed among the ladies and cabin passengers, so far as they
would go, seven of my white shirts to serve as change of linen, they
having been discovered in the boat on our landing. We had at this time
six days' allowance of water, at the rate of three bottles a day to our
twenty-six persons, or scarcely three table-spoonfuls to each, which,
in our already fevered and maddening thirst, and under a broiling
tropical sun, was not nearly sufficient to sustain life. A small
surplus, however, was found in the water-cask after all our bottles
were filled, which was distributed among the company, and served to
refresh us at departure. We broke some oars for carrying-poles, and
distributed the stores among the responsible persons in the company,
with strict injunctions that they should restrict themselves to the
general allowance, as any breach of fidelity might sacrifice the lives
of the whole party. After the celebration of divine worship, in which
we committed our way to God, we set out on our melancholy journey. Our
road lay before us through "a waste howling wilderness," and we "went
out, not knowing whither we went;" but our trust was in that God, "who
had found Israel in a desert land, and kept him as the apple of his
eye," and we hoped that he would lead us also forth "by a right way,
that we might go to a place of habitation."
Our company presented a most wretched appearance in the march, and we
soon proved ourselves to be indeed miserable travellers. Our limbs
had swelled to an inordinate size in consequence of our confinement
and exposure in the boats, and they were so stiffened with inactivity
as only to be dragged along with difficulty. The ground over which we
toiled our way was unfavourable for progress in our faint condition,
being, for the most part, loose and sandy, and occasionally tangled
with small shrubs: and as we went our way, struggling, and staggering
beneath our light loads, we bore a striking resemblance to the last
remnant of a famished garrison, or the latest fugitive survivors of a
siege. We accomplished about a mile, when we sat down to rest,
and stripped ourselves of all our upper clothing, on account of the
oppressive heat. After a short pause we again resumed our journey,
and with great difficulty reached a similar distance. It was only
after much persuasion that I induced them again to stir; but there was
no shelter in the place from the fierce rays of the sun; and I was
extremely anxious, in our desperate circumstances, which were every
moment growing darker, to make all the progress possible. About noon
we discovered two huts under the cliff, and were of course anxious to
reach them; but they were inaccessible to us, in our weak condition. We
halloed, however, with all our might, to find if they had any inmates;
but as "there was no voice that answered, neither any that regarded,"
we justly concluded that they were uninhabited, and could furnish
no relief to us, so that we turned mournfully away, and pursued our
journey. I afterwards learned that these huts belonged to a fishing
company, and were deserted; a few casks of water were kept there, for
the supply of their vessels, but these were kept under ground, so that
we would have found no relief by visiting the place, and most probably
would have perished in the attempt. Soon after we likewise descried
traces of a path which led into the interior, which some of our people
were inclined to follow; but I dissuaded them from the attempt, as the
coast was the coolest region, as well as the most likely to lead us to
water; whereas we might only wander in the wilderness to die the most
horrid death. We accomplished altogether about six miles by this day's
journey, and halted at last, in utter exhaustion, on a promontory,
where we were exposed to the sea breeze.
On collecting our party, to overhaul our stock, I found that one of
the cabin passengers, who had been quite delirious for some days,
having fallen behind us on the day's march, on account of weakness, had
cast away his coat, containing two bottles of water, from anxiety to
overtake his party. Every search was made for this lost treasure and
valuable supply, but to no purpose. To add to the misfortune, another
cabin passenger, from whose education I might have expected better
conduct, alone, of all the people, proved himself unworthy of trust.
Of the two bottles committed to his charge, one was found empty. He
had stolen from his party during the day, under pretence of tracking
the path into the interior, and the temptation proving too strong for
him, he had consumed a whole bottle for his own use. This I concealed
from our people, for I am certain that, if they had known it, they
would have taken his life on the spot. But I was deeply grieved to find
that a whole day's supply of this scarce and vital commodity had been
lost to us through the imbecility and profligacy of our companions. To
prevent the recurrence of such a calamity, which would have endangered
the lives of all of us, I put the water, henceforth, under the charge
of my confidential seamen; and after our evening's repast and prayers,
we betook ourselves to sleep.
During night a heavy dew fell, mingled with a few drops of rain, which
roused us from our slumbers, and our people commenced greedily to suck
the moisture from the blankets; but they having been soaked by sea
water, and only dried in the sun, were so impregnated with salt that we
soon desisted from an endeavour which brought us no relief. As morning
dawned our pleasing expectation of rain departed, and with heavy hearts
we prepared to pursue our course. The condition of our people at this
time was extremely distressing, their faces had become bloated and
disfigured, and their lips were rent and chapped, while the painful
swelling in the arms and legs was rapidly on the increase, so that I
apprehended some of them would not be able to hold on till night. I
sought to rally their downcast spirits, by feigning a cheerfulness
which I did not feel, and pointing to some mountains in the south-east,
I prompted them to proceed, by assuring them, that wherever mountains
appeared, water was always to be found. I was certain, moreover, by the
calculations which I had made, that we could not be more than five
miles from Oliphant River, where I felt assured that relief would be
afforded. We made indeed most wretched progress in that morning's
journey; ere three quarters of a mile had been accomplished, we were
compelled to halt, and after receiving our allowance and singing a
hymn, we proceeded on our way. Scarcely another mile had been overtaken
ere we were again forced to rest ourselves, and here I felt alarmed
lest some of the company should never be able to resume the march.
The old gentleman, who had lost his coat on the previous day, was
especially overcome; he seemed so thoroughly exhausted in spirit,
and so worn out in frame--being covered over with sores in face and
limbs--that it seemed impossible to rouse him to any further effort,
and others were inclined, with him, to resign themselves to despair. I
was greatly perplexed how to act in this extremity. I could not bear
to leave the wretched alone to die; and to detain the others on their
account, would be certain destruction to us all. In this painful crisis
I secretly sought direction from God, and had resolved to remain
with the desperate, leaving the others to press on, and send back for
us if they should find succour. This purpose I only communicated to
my brother William, and urged him to use every exertion to reach a
place of safety, and in case of my death, to be kind to my dear wife
and family. He, however, sternly refused to accede to my wishes, and
declared his resolution, if I persisted, to abide and die with me.
By his persuasion I was shaken in my purpose, and by dint of great
exertion, we managed to assist our invalids on through another stage.
On looking out for our next halting-place, I observed a rising ground
a little in advance of us, and urged our people to reach it ere they
rested. This was done, because I thought something like the entrance of
a river appeared beyond, and I was resolved to ascertain the fact by
crowning the hill. We had nearly reached the place, when the mate, who
was a little in advance, cried out, "There is the river." I ran forward
at the transporting tidings, and by advancing a few paces, a scene
of overpowering interest burst upon my view. Not only was the river
distinctly visible as it rolled its broad waters through a fertile
valley, until they mingled with ocean at our feet; but I could also
distinctly descry a settlement, with its dwelling-houses and offices,
on the opposite bank. Never did scene more sweet open upon human
vision, than met my ecstatic gaze in that landscape. I had no eye--no
heart for its natural beauties; but thoughts of life and of rescue
arose within me in that glance. It seemed to me an opening paradise:
visions of home--of happiness--rushed back upon my desolate soul. The
tide of sorrow, in a heart ready to perish, was turned within me, and
joy rose in such sudden revulsion from recent wretchedness, that I was
completely overpowered. The same excess of emotion filled every heart
that now crowded around at the tidings. We grasped each other's hands
in convulsive silence; our hearts were too full for utterance, and, for
a considerable time, tears were the only expression that came from
our overcharged bosoms. Rapture was in our glance when we saw human
beings moving about on the opposite bank, and we became rivetted in
delighted gaze upon the neat white-washed house, with its clear blue
smoke curling up into the sky, and all the accompaniments of European
comfort around it. I was the first to break the interesting silence,
by saying, "Now, my dear friends, the Lord has led us by a way that
we knew not, to a land inhabited." We then gave thanks to God, who
had done so great things for us, and we served out a little of our
remaining stock of water. As we were still a mile from the river, I
preserved a small portion, in case the river water should prove salt,
which, on reaching it, we found to be the case. We had carried our
English ensign as a signal in case of meeting any vessel, and now, by
tying two broken oars together, we elevated it to attract the notice
of the persons on the opposite bank. They evidently had descried us;
for we observed a boat push off from the shore, and advance straight
toward us across the stream. This was to us a gracious token that the
season of succour was at hand. We immediately thereupon drank off our
last remaining bottle of water, and prepared to greet our deliverers.
The moment of our rescue was especially interesting and solemn. While
the boat approached, we all joined hands and united in singing the 23d
Psalm, and, as the faint concert arose from our famished group, it
seemed, to our overflowing hearts, to ascend to heaven, alike as the
devout dirge of our departing sorrows, and the joyous anthem of our
coming deliverance.
CHAPTER V.
THE RESCUE.
"When the poor and needy seek water, and there is none, and
their tongue faileth for thirst, I the Lord will hear them,
I the God of Israel will not forsake them. I will make the
wilderness a pool of water, and the dry land springs of water."
The boat, which seemed to our view like a messenger of mercy,
approached within hail, when, with due precaution, it halted, and to
our delightful surprise a voice in the English language demanded to
know who we were, and what was our business. We immediately declared
our doleful story, when the party landed without farther ceremony,
and told us that we had come among a Christian people. The meeting
was most affecting on either side; it was with difficulty that our
people, in the ecstasy of rescue, could refrain from falling down at
the feet of their deliverers; and the strangers, as they surveyed our
emaciated and wretched company, were quite unable to suppress their
tears. Our first appeal to them was for water, and they communicated
the joyful intelligence to us, that there was an excellent fountain
on the other side, where our wants would be abundantly supplied.
I immediately embarked, in company with the ladies, and by three
successive trips, the whole of our people were safely landed on the
other side, where we were all received with unbounded affection and
hospitality. We instantly repaired, with incontrollable avidity, to the
fountain, where we sought to satiate our maddening thirst by deep and
frequent draughts, until we had gorged ourselves with the exquisite
supply, and felt life reflow in cooler currents through our parched
and fevered frames. A princely meal was also provided for us on the
instant, consisting of a whole sheep, and part of a wild buck, which
had been shot on the farm in the morning; but our hearts were too full
to possess a keen appetite, and we could only taste of the bounteous
provision amid many tears, when we contrasted the scantiness and misery
of our morning repast with our present profusion, and the hearts of
many of us rose in silent gratitude to "God, who had done so great
things for us, whereof we were glad."
We found that the settlement which we had reached belonged to a
warm-hearted Dutch farmer, named Mynheer Low, of whose unbounded
generosity and kindness it is impossible for me to speak in excessive
terms. His family consisted of an amiable wife and daughter, who shared
in all his own benevolence, and loaded us with attentions, which can
never be forgotten, and it would be impossible to repay.
The Englishman, who had accompanied our host in the boat which ferried
us over the river, and who acted as our interpreter during our stay,
was a sailor belonging to a whale and seal fishing company. He had been
left by his employers, in company with another person, to reside during
the fishing season, on a neighbouring island, in order to preserve the
fishing grounds, which were rented from the colonial government. He and
his partner were obliged to visit the settlement very frequently for
supplies of water, which they required to keep, alike for their own
use, and in case of their schooner running short during her voyage. I
learned from this person that the coast to the north of Oliphant River
is entirely destitute of water, and without inhabitants; and I mention
this in case any persons who peruse this narrative should be driven on
this coast, that they may know where to obtain succour. Mr. Low's farm
is situated on the south bank of the Oliphant River, about four miles
from the sea, and two hundred miles from the Cape of Good Hope.
Soon after our arrival I communicated with Mr. Low as to the necessary
provision for our future accommodation. It was impossible with his
limited resources, that he could lodge and sustain twenty-six persons
for many days; and it was plain from the distressing condition of our
people, that they would require several days of careful nursing and
rest, before they could bear removal by land journey.
Having learned that an English gentleman kept a store at Donkin's
Bay, twenty-four miles distant, I immediately despatched a messenger
to solicit his assistance. This person, whose name was Mr. R. Fryer,
proved to us to be indeed "a brother born for adversity." No language
can adequately express his unremitting kindness and unceasing exertions
for our welfare, and for which he would never listen to any proposals
of remuneration whatever. He came down on horseback immediately on
receiving notice of our condition, and despatched a message to the
nearest field cornet, to make provision for our succour. On his arrival
he proposed to take the ladies at once to his house, they being the
only parties fit to be removed. It may seem strange that the most
delicate members of our company should have borne the hardships of our
situation with greater hardihood than men of robust frame; and yet
it was remarkable throughout the whole of our afflictions, that the
ladies and even the children bore the sufferings with the greatest
magnanimity, and discovered a spirit of patient endurance which might
have put to shame the hardiest men. It is thus that God sometimes,
as of old, "out of weakness maketh strong," and causeth "things that
are not to be as though they were, that no flesh should glory in his
presence."
In accordance with this arrangement, our ladies set off in a waggon
for Mr. Fryer's house, under charge of our host's daughter, on the
evening after our arrival at Oliphant River; and in twenty-four hours,
the waggon returned loaded with provisions, luxuries and medicines.
Mr. Fryer also sent four sheep on the same day, and gave his shepherd
orders to supply us with as many as we wanted; and yet these things
were but a tithe of the kindness which we received at the hands of this
good Samaritan.
We were at this time also under great obligations to Mr. Francis J.
Troutar, who had come down the river at this time, along with his
mother-in-law and a few servants, to fish. The good old lady took our
three children to her hut, supplied them with frocks and underclothing,
and treated them with the solicitude and kindness of a mother, so as to
merit our warmest gratitude.
In the course of a few days, the effects of our long fasting and
exposure and fatigues began to appear, and to make shocking havoc on
the persons of our people, in loathsome bloaches on the face, and
excessive swelling of the arms and legs. The steward was particularly
in a pitiable condition with his face, and one of the cabin passengers
was confined to his couch. One of his legs burst, and his hand was
obliged to be laid open by a deep incision of a razor, so that I
was afraid at one time, that he would not rally. In the course,
however, of four or five days, through the unremitting nursing of the
Dutchman's family, and by the kind providence of God, we all began to
amend. Our recovery soon revealed itself in an incessant craving for
food; for some days it was almost impossible to satisfy our intense
appetite, and we were in danger of creating a famine in the Dutchman's
settlement, as a sheep was killed every day for our use, and we
consumed great quantities of wheat, which we prepared for boiling by
pounding it in a mortar, and sometimes made into bread after grinding
it in a hand-mill.
On the 13th January I received a letter from Mr. Rennyfield, civil
commissioner, Clan William, to meet him at Mr. Fryer's on the following
day, in company with Mr. Troutar. We accordingly set off next morning,
at five o'clock, and as I was but an indifferent horseman, I was
greatly exhausted by the ride. The country in this quarter is chiefly
sandy, and blows with the wind like dust, but it is thickly studded
with sundry kinds of shrubs and bushes, which are valuable for the
feeding of cattle and sheep. On reaching my destination, I was most
hospitably received by Mr. Fryer, and his lady, and was happy to find
my lady passengers in good health and spirits. The civil commissioner
made full arrangements with me for our journey to Cape Town. I
received a letter to produce to each field cornet on the route,
containing instructions to provide us with waggons, and to supply us
with every necessary on the road. Mr. Troutar, who was the field cornet
of the district, was to provide the waggons and to be our conductor
through the first stage; and our departure was arranged for the 19th of
January, by which time it was hoped that our invalids would be so far
recovered as to bear the journey.
On the day appointed we prepared for our departure amidst much bustle
and confusion. The yoking of fourteen or sixteen oxen in a waggon is
like getting an East India trader under weigh, and the chattering of
the Hottentots in the excitement of the occasion was quite amusing.
The scene of separation with our dear friends and deliverers was
exquisitely affecting. The kind Dutchman's family were weeping aloud;
Mr. Troutar's mother-in-law clung to our little orphan family, and
refused to part with them; even the Hottentots could not refrain their
tears. I confess that I never felt myself so unmanned in my life, and
it was only after an hour had been wasted in ineffectual efforts to
say farewell, that by a desperate resolution we at last tore ourselves
away. They followed us for a short distance, and then stood, and waved
their hands as long as we could see them. Thus we parted from kind
strangers, who had entwined themselves around our hearts in fondest
endearments; and while memory holds her seat in our bosoms, I trust
that we shall never cease to pray for richest blessings on the heads of
our benevolent friends of the Oliphant River.
We reached Mr. Fryer's at Donkin's Bay about midnight, where our
party was rejoined by the ladies, and we remained in the enjoyment of
this excellent family's hospitality until the next afternoon. Another
painful scene of leave-taking had here to be repeated, and it was with
difficulty that our ladies could command themselves in parting, from
one who had proved so lavish in his generosity to all of us in our
distress. "May the Lord reward him," and "think upon him for good,"
according to all the kindness that he showed unto us.
It would be tedious to enter into minute details of our land journey to
the Cape. It presented all the usual adventures of that tedious mode
of travel;--sometimes ploughing sandy deserts deep to the axles,--and
occasionally land-locked by an interminable maze of tangled brushwood.
Frequently we lost our path in the darkness, the over-laboured brutes
were many times at a stand-still from exhaustion, and scarcity of
water; and once or twice, we had nearly suffered a second _shipwreck
in the desert_, to the great alarm of the ladies, and not without the
hazard of broken bones.
Mr. Troutar accompanied us with his waggons and cattle, through several
dreary stages, until we reached Mr. Vanzells' farm. This gentleman
was uncle to our worthy conductor, and also a field cornet. Here
we obtained fresh cattle, and started under a new convoy. It was
with extreme regret that we parted from Mr. Troutar, whose kind and
gentlemanly deportment had endeared him to us all. I was also compelled
to leave Mr. Harris our cabin passenger here, under charge of our
surgeon, as he was so ill as to be unable to proceed; Mr. Vanzells
promising to forward both gentlemen to Cape Town on horseback, so soon
as Mr. Harris was able to bear the journey. After travelling by uneasy
stages for several days, we crossed the Peak Berg range of mountains,
the Boers throughout treating us with unvarying kindness, and we
furnishing much amusement to the inquisitive and simple people, by the
strangeness of our dress, and speech, and psalmody.
At length on the 28th January at midnight, we entered Cape Town,
fatigued with our journeyings in the wilderness, and happy in being
able once more to mingle in the society of our countrymen. The luxury
of a good bed, which for the first time I had here enjoyed, since
leaving the ship, could not induce me to sleep. The whole scene of
dangers and deliverances, through which the Lord had led us, here rose
vividly before my view, and I could not refrain from giving fervent
thanks to Him, "who had not dealt with us after our sins, nor rewarded
us according to our iniquities." He had indeed "chastened us sore, but
he had not given us over to death;" and we could adopt the language of
the Psalmist, "Thou, who hast showed us great and sore troubles, shalt
quicken us again, and bring us again from the depths of the earth." "So
will we sing praises unto thy name for ever."
Immediately after breakfast, on the morning of the 29th January, I
waited on Colonel Bell, at that time Deputy-Governor of the Cape,
and represented to him the miserable condition in which my crew and
passengers were. He immediately sent for one of his officers to
accompany me to our lodgings, and to make arrangements for the payment
of our board. Being in miserable plight for want of clothing, I was
at this time greatly indebted to Captain Christie of London, who
presented me with an excellent suit of his own. I had the pleasure,
also, of meeting an excellent friend in Dr. Brown (belonging originally
to my native town of Peterhead), who took me to his own house, and
entertained me most hospitably during my stay at the Cape. Meanwhile,
the merchants and gentlemen of the place opened a public subscription
on our behalf, which was handsomely headed by Colonel Bell, and soon
amounted to the sum of L120. By this money, a sum equal to a month's
wages, was distributed in clothing to each of the crew, and the
passengers received a similar supply, in equitable proportions,--the
three children being fully furnished with all necessaries for the
continuance of their voyage, and the ladies being supplied with
clothing and a little money. I also received L10 of this money, along
with a letter of commendation, and I am thus minute in detailing the
benevolence of the people of Cape Colony, as it is deserving, alike of
personal gratitude and public praise.
Every effort was now made to forward the passengers to their
destinations, and to dispose of the crew by drafting them into
different ships. After a little exertion, this was happily accomplished
on behalf of all, with the exception of two steerage passengers, who
preferred to accept of situations in the colony. So soon as I had thus
discharged my obligations to the people under my care, I began to think
how to dispose of myself. After various friendly offers of employment,
none of which exactly suited me, I finally accepted of a passage home
in a London schooner belonging to Mr. Fletcher, and bound to Bristol.
My kind friends in Cape Town affectionately accompanied me to the ship,
and, after taking grateful leave of them, our vessel set sail for
England, and in due season, "by the good hand of my God upon me," I
returned in peace to the bosom of my wife and family.
Thus terminated a voyage replete with judgment and mercies. In the
review of its "affliction and misery--the wormwood and the gall--my
soul hath them still in remembrance, and is humbled within me." And
"may my right hand forget its cunning, and my tongue cleave to the roof
of my mouth," if I forget that "God who answered us in the day of our
distress, and was with us in the way in which we went." I trust also
that the same spirit and resolution may abide upon all the survivors
of that disastrous voyage which appeared in the day of our calamity.
Even the most indifferent in religious things there owned that it was
"a good thing to call upon God," and "poured out a prayer when his
chastening hand was upon them." May it never be said of any of us that
"we flattered him with our mouth, and lied unto him with our tongues,"
or that "we forgat God and remembered not his wonders." The solemn
professions which we then made are still before his throne, and He will
never forget, however we may, the extraordinary obligations under which
we lie, to dedicate our spared lives to His service. O that we may
every day perform the vows which "our lips uttered--our mouths spake
when trouble was upon us;" and that our future lives may realize the
holy resolution of the man of God: "Thou hast delivered my soul from
death, mine eyes from tears, and my feet from falling; therefore I will
walk before the Lord in the land of the living."
And surely this simple tribute to Divine goodness carries with it
a solemn message to every reader's heart. How impressively does it
declare _uncertainty of life, even in moments of greatest seeming
security_. It was when least expecting it, that the foregoing
calamities came. And who can tell how soon God may disturb our dreams
of security, by the summons to the judgment seat? "We stand in jeopardy
every hour." In a world so full of sorrow and evil, we are daily
exposed to the visitation of death. And does it not become us, in such
circumstances, "to be always ready--to have our loins girt, and our
lamps burning, and be like men that wait for the coming of the Lord?"
"O that we were wise--that we understood this, that we would consider
our latter end." Sailors, above most men, ought especially to cultivate
this spirit of habitual preparedness. Their calling preeminently
exposes them to peril, and they are found "in deaths oft." The breeze
that fills their sails, and wafts them to their destination, may
swell into tempest, and become "the breath of the blast of Jehovah's
nostrils" for their destruction. The ocean that spreads around them
a peaceful pathway to distant lands, may heave into huge and hoary
billows, that yawn only to engulf them in its horrid grave. The very
shore that greets them with gladness after long absence, may be changed
into a scene of fatal shipwreck, and death find them at the very door
of supposed deliverance. Who does not feel as he treads the deck of
his gallant vessel, that death is lurking near him in every element
that lies over, and around, and underneath his feet; and that God is
proclaiming, at every moment, in all the voice of nature, "As the Lord
liveth, and as thy soul liveth, _there is but a step between thee
and death_." And can we be safe, in such circumstances, to live in
unpreparedness for that which may meet us the next moment, and must
meet us ere long? Or ought we to feel satisfied, in any circumstances,
if we be living in a state of enmity with God? What can the sinner
do, and whither shall he flee, when judgments overtake him? He cannot
look up to a neglected and angry God; he dare not look down upon an
undone eternity; nothing remains for him but "a fearful looking for of
judgment and of fiery indignation to destroy him as an adversary." Why,
oh why, should we live in such a state of defenceless danger--exposed
at every accident to the destroying vengeance of heaven? Is not a
divine Saviour now offering us not only his protection, but also his
propitiation? The merit of his sacrifice is able to screen all who
confide in it, not only from temporal danger, but also from eternal
destruction. Let us seek our present safety, in acceptance with God,
through the blood of Immanuel; and we shall find our security from all
future evils in the covert of his covenant. Then, "though we walk in
the midst of trouble his right hand will save us," and we shall face
every danger with a fearless confidence, while we can exclaim--"The
Lord of Hosts is on our side, the God of Jacob is our refuge." For
Immanuel shall be "an hiding-place from the wind, and a covert from the
tempest, as rivers of waters in a dry place, and as the shadow of a
great rock in a weary land."
If any truth be confirmed by the foregoing narrative, it is the truth
of God's word, that "the Lord is good--a stronghold in the day of
trouble, and he knoweth them that trust in him." The "profiting" of
prayer in such a case must "be apparent to all." It was the smallest
part of its advantages that it preserved order, and prevented
excess,--that it filled the fainting hearts of the crew and passengers
with courage, and renewed their strength when they were sinking fast
into despair. It did more; their eyes turned heavenward in their
helplessness, and they found a power superior to their own, interpose
for their deliverance. These poor men cried, and "_the Lord heard
them_, and delivered them out of all their distresses." If any reader
should doubt the truth of this conclusion, or deny it, let him go and
"prove God," by the same means; let him "in everything by prayer and
supplication make known his request to God;" and if his prayer be
sincere the gracious answer will be certain; his own experience will
but accord with the infallible testimony of all ages. "Ye shall seek
me and find me, when ye shall search for me with all your heart." "FOR
THIS SHALL EVERY ONE THAT IS GODLY PRAY UNTO THEE IN A TIME WHEN THOU
MAYEST BE FOUND; SURELY IN THE FLOODS OF GREAT WATERS THEY SHALL NOT
COME NIGH UNTO HIM."
Transcriber's Note:
Punctuation has been standardised. Hyphenation and spelling has been
retrained as published in the original publication except as follows:
Page 14
qnantity to extinguish so extensive _changed to_
quantity to extinguish so extensive
Page 94
the vows which "our lipsuttered--our moutths spake _changed to_
the vows which "our lips uttered--our mouths spake
End of the Project Gutenberg EBook of The Loss of the Australia, by Adam Yule
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} | 9,598 |
\section{Introduction}
Summarizing the results which have been lately obtained in \cite{noorlewis} and \cite{mullerfeliu} we derive some obvious consequences. The loop-less and so called thermodynamicall feasible fluxes outlined and specifed in \cite{noorlewis} obey in an almost natural way the injectivity conditions in \cite{mullerfeliu}. There is a long history of achievements analyzing injectivity and multistationarity of chemical reaction networks (CRN's) (\cite{feinberg}, \cite{feinberg1995},\cite{hornjackson}, \cite{soule},\cite{thomas81}). There have been nomerous refinements and generalizations of prevoius resuslts in (\cite{craciunfeinberg},\cite{mullerfeliu},\cite{angelipetrinet},\cite{banajicraciun},\cite{conradiflockerziJMB2012},\cite{joshishiu},\cite{shiusturmfels}). We would like to insert thermodynamical requirements \cite{noorlewis} into CRN's as recently manifestet in \cite{mullerfeliu} to elucidate their consequences for their stability behaviour.
\section{Thermodynamic Considerations}
In this section we give some basic explanations for the physical description of chemical reactions as occuring in chemical reaction networks of continuous stirred tank reactors (CSTR).
Generally there is a known thermodynamic potiential that governs reaction kinetics between complexes. We will consider every reaction as reversible and described by the boltzmann distribution between potentials. Reaction dynamics derived from power law kinetics allow by that assumption flows in both directions (reversible).
We will therefore assume that all reactions are reversible unless we explicitely mention where we can neglect full reversibility.
An example of a reversible reaction for illustration considered here is:
\begin{equation}\label{abcd}
A + B \; \; \; \mathop{\stackrel{\kappa_f}{\rightleftarrows}}_{\kappa_b} \; \; \; C + D \ .
\end{equation}
This reaction has a reaction constant for both directions.
A reaction constant $\kappa(T)$ is almost universally described as dependent upon activation potential $\Delta E_a$ as in (\cite{butt}, p. 9, 1-25)
\begin{equation}
\kappa(T)=\kappa^{o} \cdot e^{-\Delta E_a/kT} \ .
\end{equation}
CSTR nearly operating under sonstant tempterature $T$ have reaction constants that can be assumed to be fixed approximately. On the other side we have to check whether applied theorems withstand validity under perturbations of parameters.
For a theoretical derivation of the formula for the forward and backward reaction constants for power law kinetics see \cite{HenriksenHansen}.
We denote here the concentrations of the species $\{A,B,C,D\}$ as $\{x_A,x_B,x_C,x_D\}$ in reaction (\ref{abcd}). Complexes in a reaction are the union of all reactant species and all product species. In the case of reaction (\ref{abcd}) we have the complexes $\mathcal{C}_1=\{A+B\}$ and $\mathcal{C}_2=\{C+D\}$ with $\mathcal{C}=\{\mathcal{C}_1,\mathcal{C}_2\}$ being the collection of all reactions. Assuming powerlaw kinetics for reaction (\ref{abcd}) we obtain for the change rate of species $x_A$:
\begin{eqnarray}\label{x_A_dot}
\dot{x}_A= \kappa_b x_C x_D - \kappa_f x_A x_B
\end{eqnarray}
Similar relations hold for all other three species.
A more detailed treatment of reaction constants with the example given in eqn. (\ref{abcd}) is obtained in \cite{HenriksenHansen} where we have at the equilibrium steady state the following relation:
\begin{equation}\label{kapparelationequilibrium}
\frac{\kappa_f(T)}{\kappa_b(T)} = \left(\frac{\mu_{CD}}{\mu_{AB}} \right)^{(3/2)} \left(\frac{x_C x_D}{x_A x_B} \right)_{int} \exp(- \Delta E_0 /k_b T)
\end{equation}
The energy difference $$\Delta E_0 = E_{0,\mathcal{C}_2} - E_{0,\mathcal{C}_1}$$ is given by the zero-point energies of the reactant ($E_{0,\mathcal{C}_1,}$) and product ($E_{0,\mathcal{C}_2}$) complex.
Here we do neglect the reduced masses $\mu_{CD}$ and $\mu_{AB}$, since we can absorb them into the related reaction constants of the specific reaction.
\section{Background material}
A chemical reaction as in equation (\ref{reactionexample})
\begin{equation}\label{reactionexample}
R_i: \mathcal{C}_1 \rightarrow \mathcal{C}_2
\end{equation}
between two complexes $\mathcal{C}_1$ and $\mathcal{C}_2$ is defined by the reactant complex $\mathcal{C}_1=\{A,B\}$ and product complex $\mathcal{C}_2=\{C,D\}$ with stoichiometric vectors
$y_1=y_{AB}=(1,1,0,0)$ and $y_2=y_{CD}=(0,0,1,1)$. Furthermore we have the associated forward and backward reaction constants $\kappa_{\mathcal{C}_1\rightarrow \mathcal{C}_2} $ and $ \kappa_{\mathcal{C}_2\rightarrow \mathcal{C}_1}$. We can also denote the difference stoichiometric vector $[y_2-y_1]=(-1,-1,1,1)$.
and by enumerating the species by $x=(x_1,x_2,x_3,x_4)=(x_A,x_B,x_C,x_D)$ we can rewrite equation (\ref{x_A_dot}) by:
\begin{eqnarray}\label{xdotnew}
\dot{x}_1= \kappa_{\mathcal{C}_2\rightarrow \mathcal{C}_1} x^{y_2} - \kappa_{\mathcal{C}_1\rightarrow \mathcal{C}_2} x^{y_1}
\end{eqnarray}
where $x^{y}=\prod_{i\in[4]}x_i^{(y)_i}$.
Following the notation given in \cite{craciunfeinberg} and \cite{mullerfeliu} for a CRN we can form the stoichiometric difference matrix $A=\{[y_{CD}-y_{AB}],[y_{AB}-y_{CD}]\}\in \mathbb{R}^{4 \times 2}$, the diagonal reaction constant matrix $diag(\kappa)=diag(\kappa_{\mathcal{C}_1\rightarrow \mathcal{C}_2} ,\kappa_{\mathcal{C}_2\rightarrow \mathcal{C}_1} ) \in \mathbb{R}^{2 \times 2}$ and the complex matrix $B=\{y_{AB},y_{CD}\}\in \mathbb{R}^{2 \times 4}$ and rewrite the change rate of all species $x$ as:
\begin{equation}
\dot{x}=A\ diag(\kappa) x^B \ ,
\end{equation}
where $x^B\in \mathbb{R}^2$ is calculated for each row-vector in $B$.
Generally we define for the case of $n$ species $x \in {\mathbb{R}^n_{+}}$ involved in $r$ reactions $\mathcal{R}$ (possibly reversible or not) and corresponding stoichiometric difference matrix $A \in \mathbb{R}^{n\times r}$ and complex matrix $B \in \mathbb{R}^{r \times n}$ with associated reaction rates $\kappa \in \mathbb{R}^r_{+}$ the generalized polynomial map $f_{\kappa} (x): \mathbb{R}^{n}_{+} \rightarrow \mathbb{R}^n$, where we have $A_{\kappa}= A \ diag (\kappa)$, by:
\begin{equation}\label{crnmuller}
\frac{ {\sf d} x}{ {\sf d} t} =f_{\kappa}(x) = A_{\kappa} x^B
\end{equation}
In the case of a fully reversible network we have for each reaction the forward $\kappa^{i}_f$ and backward $\kappa^{i}_b$ reaction constant where $i \in [r] $ with $r=2r'$ reactions in total. We will develop the subject for the fully reversible case even when we can admit less restrictive conditions for the validity of the result.
We will first state the result from \cite{noorlewis} here. We will consider $r$ reactions $\mathcal{R}$ with positive reaction constants $\kappa_j$, $j \in [r]$ over $n$ different species. The number $p$ of complexes $y_i \in \mathbb{R}^n_{+}$, $i \in [p]$ are reduced to these taking place in one of the $r$ unidirectional reactions. The difference stoichiometry vectors of each reaction $j \in [r]$ denoted by $[y-y']^{(j)}$ form the colums of the matrix $A$.
The notation here is the same for a matrix $A$ representing the internal reaction of a CRN. We exclude here external reactions first and analyse the internal system of reactions. At the end of the text we will insert an external flux representing the inflow of a chemostat reactor (CFSTR).
In order to consider thermodynamic aspects in a flux distribution we have to assign potential differences $\Delta{G}$ between the complexes of each reaction of the CRN in form of a vector of potentials for the complexes.
The Gibbs potential for example (\ref{abcd}) is related to equation (\ref{kapparelationequilibrium}) by
\begin{equation}\label{gibbsenthalpie}
\Delta{G} = y_C G_C^0 + y_D G_D^0 - y_A G_A^0 - y_B G_B^0 + RT \, \ln(K_a)
\end{equation}
over the constant $R=N_A \cdot k_b$, the activities
$$K_a=\prod_{i} x^{[y_2-y_1]}$$
from equation (\ref{kapparelationequilibrium}) and the zero point Energies $G_x^0$ (see also \cite{kuemmel} eqn. (1)).
Through that notation we can find a vector $\gamma\in \mathbb{R}^n$ for the potentials of the individual spezies depending on their concentrations and stoichiometric coefficient, such that we obtain
\begin{equation}\label{gibbsdelta}
\Delta G = \gamma^T A
\end{equation}
as the differential energy between the complexes for the current temperature and spezies concentrations.
The following classification of fluxes can be traced back to the Gordan theorem of alternatives \cite{noorlewis} which we will state here:
\begin{thm}
(Gordan's theorem) $\forall A\in \mathbb{R}^{ n\times m}$ exactly one of the following two statements is true:
\begin{description}
\item[(a)] $\exists z \in \mathbb{R}^m_{+} \setminus \{0\}$, s.t. $Az=0$
\item[(b)] $\exists y \in \mathbb{R}^n$ s.t. $ A^{\top} y>0$
\end{description}
\end{thm}
In \cite{noorlewis} a transformation of the Gordan theorem for the case of reversible fluxes of a chemical reaction nework is given.
A reaction system fully reversible will be called loop-free/thermodynamically feasible (b) or thermodynamically not feasible with loops (a) if the following holds:
\begin{cor} \label{noorgordanlemma}
For all $\hat{A} \in \mathbb{R}^{n \times r}$ where $n$ is the number of species and $r$ the number of (bidirectional/reversible) reactions and every $\nu \in \mathbb{R}^r$ one of the following cases is true:
\begin{description}
\item[(a)] $\exists \hat{z} \in \mathbb{R}^r \setminus \{0\}$, s.t. $(\forall i \ sign(\hat{z}_i) \in \{sign(\nu_i),0\})\wedge \hat{A} \hat{z}=0$
\item[(b)] $\exists \gamma \in \mathbb{R}^n$ s.t. $(\forall i \ sign(\hat{A}^{\top}\gamma)_i = - sign(\nu_i) \vee \nu_i=0)$
\end{description}
\begin{proof}
See \cite{noorlewis}.
\end{proof}
\end{cor}
The idea behind that alternative is that we cannot have a flux keeping the concentrations of the species constant when there are differences between the potential of the complexes. The net energy consumption would be zero and the turnover would be non-zero which would be impossible due to the conservation of energy. It is more important to know that there is a potential distribution behind that which does not allow thermodynamically infeasible fluxes.
In Corollary \ref{noorgordanlemma} we were choosing $\gamma$ instead of $y$ in order to avoid an overlap with the stoichiometry vector $y_i$ and also to give the link to the chemical potential introduced in equations (\ref{gibbsenthalpie}) and (\ref{gibbsdelta}) since $\gamma^{\top}A$ is equivalent to $A^{\top}\gamma$. {\bf (b)} in Corollary \ref{noorgordanlemma} reflects the fact that the flux $\nu_i$ is in opposite direction to the increasing potential $(A^{\top}\gamma)_i$ between complexes.
We can link that relation to our reversible system. We set $m=2r$ the number of all unidirectional reaction in a fully reversible chemical reaction network and order the signs of the flux $\nu \in \mathbb{R}^r$ with $sign(\nu_i)=d_i$ for $i \in [r]$ according to the first $r$ forward and $r$ backward fluxes or each reversible reaction where we have $d_i=-d_{i+r}$ and the total flux results as the sum of the forward and backward flux: $\nu_i = z_i-z_{i+r}$ for $z \in \mathbb{R}^m_{+}$. We can set up the following result which is an equivalent formulation of loop-free fluxes from Corollary \ref{noorgordanlemma} for unidirectional fully reversible CRN's.
\begin{cor}\label{noorunidirectional}
For all $A \in \mathbb{R}^{n \times m}$ where $n$ is the number of species and $m=2r$ the number of reactions and every $\nu \in \mathbb{R}^r$ one of the following cases is true:
\begin{description}
\item[(a)] $\exists z \in \mathbb{R}^m_{+} \setminus \{0\}$ $\wedge$ $(\exists j \in [r]$ with $ z_j \neq z_{j+r} )$, s.t. $(\forall i \in [r] \ sign(z_i-z_{i+r}) \in \{sign(\nu_i),0\})\wedge A z=0$
\item[(b)] $\exists \gamma \in \mathbb{R}^n$ s.t. $(\forall i (\ sign(A^{\top}\gamma)_i = - \ sign(A^{\top}\gamma)_{i+r} = - sign(\nu_i) ) \vee \nu_i=0)$
\end{description}
\begin{proof}
Equivalence between Corollary \ref{noorgordanlemma} and \ref{noorunidirectional} concerning {\bf (a)} can be seen by doubling the matrix $\hat{A}$ for the bidirectional case by setting $A=(\hat{A},-\hat{A})$ and also doubling the vector $\hat{z}$ by setting $z_i=\max{(\hat{z}_i,0)}$ and $z_{i+r}=-\min{(\hat{z}_i,0)}$ for $i \in [r]$. The reverse can be done by halving $A$ to form $\hat{A}$ and by taking differences $\hat{z}_i=z_i-z_{i+r}$ for $i \in [r]$. {\bf (b)} is equivalent in both Corollaries.
\end{proof}
\end{cor}
\begin{remark}\label{remarknonreversible}
Corollary \ref{noorunidirectional} can be extended to the case where reaction $\mathcal{R}_i$, $i \in [r]$ are not reversible by choosing $\nu \in \mathbb{R}^r$ such that the sign of $\nu_i$ is in accordance with the direction of the reaction $\mathcal{R}_i$.
\end{remark}
\begin{remark}\label{remarkortho}
The exclusion of the case {\bf (a)} comes as the assumption that there is no component $x$ of $\nu$ that is in the nullspace of $A$. The process of elimination of components $x \in \ker(A)$ implies that $\nu$ is orthogonal to the nullspace of $A$:
\begin{equation}\label{AperpKer}
\nu \perp \ker(A) \ .
\end{equation}
\end{remark}
We can now use that fact from equation (\ref{AperpKer}) to derive conditions for possible injectivity according to \cite{mullerfeliu}. Therefore we have to suffer some more notation. The sign $\sigma(a)$ of a vector $a \in \mathbb{R}^n$ is given by $\sigma(a)_i=sign(a_i)$. Therefore we have $\sigma(a) \in \{-1,0,1\}^n$. For a subspace $K\subset \mathbb{R}^n$ we get consequently $\sigma(K)=\{\sigma(a)\vert a \in K\}$. Furthermore we define
$\Sigma(K)=\sigma^{-1}(\sigma(K))$.
We can now state the following theorem:
\begin{thm}[\cite{mullerfeliu}]\label{injectmuller}
Let $f_{\kappa}: \mathbb{R}^n_{+}\rightarrow \mathbb{R}^m$ be the generalized polynomial map $f_{\kappa}(x) = A_{\kappa} x^B $, where $A \in \mathbb{R}^{n\times r}$, $B \in \mathbb{R}^{r \times n}$ and reaction rates $\kappa \in \mathbb{R}^r_{+}$. Let $K\subset \mathbb{R}^n$ with $K^*=K \setminus \{0\}$, the following statements are equivalent:
\begin{description}
\item[(inj)] $f_{\kappa}$ is injective with respect to $K$, for all $\kappa \in \mathbb{R}^r_{+}$ \label{inj}
\item[(sig)] $\sigma (\ker (A)) \cap \sigma (B(\Sigma(K^*)))= \varnothing$.\label{sig}
\end{description}
\begin{proof}
See \cite{mullerfeliu} Theorem 1.4.
\end{proof}
\end{thm}
The number of reactions $r$ in theorem \ref{injectmuller} includes both reversible and nonreversible reactions by counting reversible reactions double and irreversible reactions single.
For further purposes we need the analysis of the second $(sig)$ property. We know from \cite{mullerfeliu}:
\begin{lemma1}\label{phinjective}
Let $B \in \mathbb{R}^{r \times n}$ and $K \subset \mathbb{R}^n$ where we set $K^*=K \setminus \{0\}$. Further let $\varphi_B: \mathbb{R}^n_{+} \rightarrow \mathbb{R}^r_{+}$ be the generalized polynomial map $\varphi_B(x)=x^B$, then the following statements are equivalent:
\begin{enumerate}
\item $\varphi_B$ is injective with respect to $K$. \label{varphi}
\item $\sigma(ker(B)) \cap \sigma(K^*) = \varnothing$\label{sigmaphi}
\end{enumerate}
\begin{proof}
See \cite{mullerfeliu} Proposition 2.5.
\end{proof}
\end{lemma1}
\section{Injectivity relations for thermodynamic feasible fluxes}
We will now describe the system under consideration. We will use the CRN's as introduced in
\cite{craciunfeinberg}. By setting
\begin{equation}\label{Adecomposition}
A=SE
\end{equation}
we have similar to eqn. (\ref{crnmuller}) the specific CRN
\begin{equation}\label{crnfeinberg}
\frac{ {\sf d} x}{ {\sf d} t} =f_{\kappa}(x) = SE \ diag({\kappa}) x^B \ .
\end{equation}
The columns of $S$ are the stoichiometry vectors of all $p$ complexes $ y_{j_p} \in \mathcal{C}$, $j \in [p]$ involved in the $r$ reactions $\mathcal{R}$. $E$ is the incidence matrix between the interacting complexes forming the matrix $A$, which consists of all stoichiometric differences of the reacting complexes $[y_i -y_i^{'} ] \in \mathcal{R}$, $i \in [r]$. The rows of $B$ are all reactant complexes of each reaction .
We define $K={\textit{im}}(A)$. For $x,y \in \mathbb{R}^r$ we denote $\sigma(x) \subseteq \sigma(y) $ if $\sigma(x)_i \in \{\sigma(y)_i,0\}$, $\forall \ i \in [r]$. We now use the relation in eqn. (\ref{AperpKer}) to show the following lemma:
\begin{lemma1}\label{orthokappaker}
\begin{equation}\label{feasibleorthogonal}
\ker(A) \bot \ diag(\kappa) x^B \Longleftrightarrow \ker(A_{\kappa}) \bot x^B \ ,
\end{equation}
\begin{proof}
$diag(\kappa)$ is orderpreserving since we have $\kappa \in \mathbb{R}^r_{+}$ s.t. we have an equivalence between $a \in \ker(A_{\kappa})$ with $\sigma(a)\subseteq \sigma(x^B)$ and $diag(\kappa) a=b \in \ker(A)$ with $\sigma(b) \subseteq \sigma(diag(\kappa) x^B) = \sigma(\nu)$ through $\sigma(a)=\sigma(b)$. By the same minimization process as pointed out in Remark \ref{remarkortho}, we obtain equation (\ref{feasibleorthogonal}).
\end{proof}
\end{lemma1}
\begin{lemma1}\label{sigperp}
Let $V,W \subset \mathbb{R}^n$ be two subspaces for which $v \in V$ and $w \in W$ implies $v \bot w$ then $\sigma(V) \cap \sigma(W^*)=\emptyset $. (The converse does not hold).
\begin{proof}
Assume there exists $v \in V$ and $w \in W$ s.t. $\sigma(v)=\sigma(w)\neq 0$ then $v \cdot w > 0$ which contradicts $v \perp w$.
\end{proof}
\end{lemma1}
We can now state our main theorem:
\begin{thm}\label{maintheoremfeasible}
For a system as in equation (\ref{crnmuller}) where $n$ is the number of species with concentrations $x \in {\mathbb{R}^n_{+}}$ involved in $r$ reactions $\{\mathcal{R}_i\}_{i \in[r]}$ and stoichiometric difference matrix $A \in \mathbb{R}^{n\times r}$ and complex matrix $B \in \mathbb{R}^{r \times n}$ with reaction rates $\kappa \in \mathbb{R}^r_{+}$ and corresponding generalized polynomial map $f_{\kappa} (x): \mathbb{R}^{n}_{+} \rightarrow \mathbb{R}^n$ with $A_{\kappa}= A \ diag (\kappa)$ we get under the condition that there exists a specific ${\kappa}^{t} \in {\mathbb{R}^n_{+}}$ s.t.
\begin{equation}\label{AperpxB}
diag(\kappa^{t}) x^B \perp \ker(A)
\end{equation}
(c.f. eqn. (\ref{AperpKer})) holds for all $x \in \mathbb{R}^n_{+}$ the following sufficient conditions for injectitivity in the sense of theorem (\ref{injectmuller}):
\begin{equation}\label{subsetrowcolumns}
span(reaction \ differences\ in \ A) \subseteq span(reactant \ complexes \ in \ B) \ .
\end{equation}
\end{thm}
Lemma \ref{orthokappaker} provides more than we need to proove (\ref{injectmuller}) (sig)
and is part of the proof of (\ref{injectmuller}) (sig), since we need only the disjoint sign condition. The relation holds for all $\kappa \in \mathbb{R}^r_{+}$.
To see that a loop free flux system implies injectivity we have to show that $\varphi_B$ is injective with respect to $K$ and we have to show that the image of $B$ with respect to $K$ is perpendicular/sign-disjoint to $ker(A)$ (theorem \ref{injectmuller}, (sig)).
We derive another relation from (\ref{AperpKer}) and (\ref{orthokappaker}) by using the fact that the differential $\delta \nu$ of the flux $\nu$ does also satisfy these relations.
\begin{equation}
\frac{{\sf d} \varphi_B(x)}{{\sf d}x} = diag{(x^B)} B \ diag{(x^{-1})} \in \mathbb{R}^{r \times n}, \ x \in \mathbb{R}^n_{+}
\end{equation}
We have
\begin{equation}\label{Aperpdeltanu}
\delta \nu \perp \ker(A)
\end{equation}
too. Calculating $\delta \nu$:
\begin{equation}\label{deltavarphi}
\delta \nu = diag(\kappa) \frac{{\sf d} \varphi_B(x)}{{\sf d}x} \cdot \frac{{\sf d} x}{{\sf d}t} \cdot {\sf d} t = diag(\kappa) \ diag{(x^B)} B \ diag{(x^{-1})} \cdot SE \ diag(\kappa) x^B \cdot {\sf d} t
\end{equation}
Lemma \ref{sigperp} together with condition (\ref{Aperpdeltanu}) and (\ref{deltavarphi}) shows that {\bf (sig)} in theorem \ref{injectmuller} is satisfied in the case where we set $\sigma (B(\Sigma(K))^*)=\sigma (B(\Sigma(K)) \setminus \{0\})$ instead of $\sigma(B(\Sigma(K^*)))$ for all $\kappa$.
It remains to show that $\varphi_B$ is injective with respect to $K$ in order to apply the $*$-operator to $K$ directly.
\begin{remark}
Relation (\ref{feasibleorthogonal}) holds for all $x \in \mathbb{R}^n_{+}$. This might be a too restrictive condition for CRN systems. We assume that there exists such a parameter system such that condition (\ref{feasibleorthogonal}) is satisfied. In the theorem \ref{maintheoremfeasible} we also allow $\kappa$ for which thermodynamic feasibility is not allowed. But we obtain in that case that thermodynamic feasible reaction systems from theorem \ref{maintheoremfeasible} are contained in the set of injective systems as characterized in theorem \ref{injectmuller}.
\end{remark}
The basis of $K=\textit{im}(SE)=\textit{im}(A)$ consists of the stoichiometric differences $[y_i-y_i']\in \mathcal{R}$. The basis of the rowspace of $B$ consists of all reactant complexes $y_i'$.
According to theorem \ref{injectmuller}, (sig) and lemma \ref{phinjective}, 2. we need to show that the columnspace of $SE$ maps injectively on the rowspace of $B$. By the definition of equation (\ref{subsetrowcolumns}) this shows theorem \ref{maintheoremfeasible}.
$\blacksquare$
\begin{lemma1}\label{injectcomplexes}
For $[y_i-y_i^{'}]\in \mathcal{R}$ with $y_i \neq y_i^{'}$ at least one of the following two cases is true:
\begin{description}\label{ydiffmapy}
\item[a)]$ y_i \cdot [y_i-y_i{'}] \neq 0$
\item[b)]$ y_i' \cdot [y_i-y_i{'}] \neq 0$
\end{description}
\begin{proof}
Assume that both are zero then we would have $0 < [y_i-y_i{'}]\cdot [y_i-y_i{'}]=0$.
\end{proof}
\end{lemma1}
\begin{cor}
For a system of $r$ reversible reactions with thermodynamic feasible fluxes the corresponding generalized polynomial $f_{\kappa}$ is injective.
\begin{proof}
We can check that by selecting a subset of reactions differences $[y_{k_i}-y_{k_i}'] \in \mathcal{R}$ for $ i \in [k]$ where $k = dim(K)$. In the same way we can select a subset of maximum $k \leq k'\leq 2k$ row vectors $\{y_{i_{k'}}''\}_{i \in [k']}$ of $B$ out of the $\{y_{k_i},y_{k_i}'\}_{i \in [k]}$ pairs for which $span {(\{[y_{k_i}-y_{k_i}']\}_{i \in [k]})} \subseteq span{(\{y_{i_k}''\}_{i \in [k']})}$ holds since the columnspace of $SE=A$ is contained in the rowspace of $B$.
Together with lemma \ref{ydiffmapy} we see that $K^* $ is mapped injectively into $\textit{im}(B)$, which is orthogonal to $\ker{A_{\kappa}}$.
\end{proof}
\end{cor}
\begin{cor}
For all weakly reversible thermodynamically feasible fluxes the generalized polynomial map $f_{\kappa}(x)$ is injective.
\begin{proof}
Weak reversibility inplies that every reactant and product complex is represented at least once in the rows of $B$. Hence the columnspace of $SE=A$ is contained in the rowspace of $B$.
\end{proof}
\end{cor}
Deficiency as introduced in \cite{feinberg1995} is replaced by thermodynamic feasibility as represented in equation (\ref{AperpKer}). The injectivity relation is reduced to
\begin{equation}
span(reaction \ di ff erences) \subseteq span(reactant \ complexes) \ .
\end{equation}
\section{Continuous flow stirred tank reactors}
We can extend the closed system of reactions as developed until now by a continuous external flow as described in the continuous flow stirred tank reactor (CFSTR). We introduce an artificial reaction by the inflow $y^*$ as a reactant complex and the resulting outflow $y'^*$ as a product complex by setting $\Delta y^*=[y^*-y'^*]$.
By that reaction a stoichiometry class is fixed from external imposed conditions. We assume that the interior system given by the closed CRN as described until now has a thermodynamical feasible flux system and especially an interior fixed point and is hence injective by definition. The response to the external flux is equivalent to the fixation of the system to a starting position, which is unique by injectivity of the interior system. By these assumption we obtain the corollary:
\begin{cor}
A CRN as given under the same assumptions from theorem \ref{maintheoremfeasible} is injective with respect to a continuous inflow $y^*$ where $y_i^*$ are the concentrations of the species for the inflow and ${y'}_i^*$ are the concentrations in the outflow which is the species concentration in the system. We can write for the response of the internal system:
\begin{equation}
f_{\kappa}(x)= [{y}^*-{y'}^*]
\end{equation}
$y^*_i=c_i$ is the number of species $x_i$ inflow per unit Volume and similar for ${y'}^*_i=x_i$. The reactionrate $\kappa^*=1$.
\end{cor}
\section{Conclusion}
Including thermodynamic principles into CRN's leads to a restriction of the available parameter space. Thermodynamic feasible reaction dynamics requires injective generalized polynomial maps for the dynamics of the species concentrations. Reversible and weakly reversible CRN's. imply injectivity. Regarding cell differentiation we can conclude that metabolic networks are regulated by signal transduction and not by triggering intrinsic multistability. Therefore we can assume or predict that mutistability is governed by regulatory mechanisms, which are not subjected to powerlaw kinetics and thermodynamic energy potentials.
\section*{Acknowledgements}
The work was done during my stay at the Friedrich Alexander University in Erlangen at the Department of Mathematics. I would like to thank Gerhard Keller and Andreas Knauf for helpfull discussions.
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} | 92 |
Oratoriu (în = sală de rugăciune) este un termen care desemnează o compoziție muzicală de dimensiuni mari, care include folosirea unei orchestre, a unui cor și a unuia sau mai multor soliști. Oratoriul a fost, până la un anumit stagiu, inspirat după modelul teatrului liric, cunoscut mult mai bine sub numele comun de operă. Similaritățile acestora includ folosirea soliștilor, corului, a orchestrei și a ariilor, respectiv a personajelor ficționale.
Oricum, operele aparțin genului teatrului muzical sau liric, în timp ce oratoriul este strict o piesă concertistică, deși uneori este pus în scenă similar cu modul în care operele sunt. Interacțiunea dintre personajele oratoriului este foarte redusă sau inexistentă, ne-existând nici măcar costume elaborate specific subiectului. Totuși, cea mai importantă distincție între cele două genuri muzicale rămâne cea a subiectelor abordate. În timp ce subiectele operelor sunt cel mai adesea inspirate de istorie și mitologie, tratând în mod romantic dragostea, decepția și moartea (uneori crima), subiectul oratoriului are de a face cel mai adesea cu subiecte sacre, făcând oratoriile mult mai compatibile cu reprezentarea în biserici. Astfel, compozitorii catolici s-au inspirat din viețile sfinților, în timp ce cei protestanți s-au inspirat în lucrările lor din Biblie.
Oratoriile au devenit extrem de populare la începutul secolului al 17-lea în Italia, parțial datorită succesului spectacolelor de operă la care s-au adăugat interzicerea spectacolelor de către Biserica Catolică în timpul Postului Paștelui. Oratoriile au devenit alternativa majoră muzicală de-a lungul acelei perioade pentru opera bufă. Compozitorul german Georg Friedrich Händel, faimos pentru oratoriul său Messiah, a scris numeroase alte oratorii pe teme laice inspirate de mitologia greacă și cea romană.
Origini din Italia
Pe la mijlocul secolului al 17-lea două genuri importante de oratorii fuseseră create
oratorio volgare (în italiană) cu exemple semnificative așa cum sunt
Giacomo Carissimi Daniele
Marco Marazzoli S Tomaso
lucrări similare de Francesco Foggia și Luigi Rossi
Cu o durată de aproximativ 30-60 minute, oratorio volgares erau interpretate în două secțiuni, separate de o predică; muzica lor se aseamănă cu cea a operelor contemporare și a cantatelor de cameră.
oratorio latino (în latină) - prima dată dezvoltat la Oratorio del SS. Crocifisso, legat de biserica San Marcello al Corso din Roma.
Cel mai important compozitor de oratorio latino este Giacomo Carissimi, a cărui Jephte este privită ca prima capodoperă a genului. Ca cele mai multe oratorio latino ale perioadei, este într-o singură secțiune.
Structură
Oratoriile conțin de obicei:
O uvertură, doar pentru instrumente
Diferite arii, interpretate de soliști vocali
Recitative, de obicei necesare pentru a face intriga să avanseze
Coruri, adeseori monumentale, menite să redea un sentiment de măreție. Frecvent instrumentele folosite includ timpane și trompete.
Listă de oratorii notabile
(ordonate cronologic după anul premierei)
Antonio Vivaldi, Juditha triumphans RV 644 (1716)
Georg Friedrich Händel, Esther (1732)
Georg Friedrich Händel, Deborah (1733)
Johann Sebastian Bach Oratoriul de Crăciun (1734)
Johann Adolph Hasse Serpentes ignei in deserto - (1735, 1736 sau 1739)
Georg Friedrich Händel, Saul (1739)
Georg Friedrich Händel, Israel in Egypt (1739)
Georg Friedrich Händel, Messiah (1741).
Georg Friedrich Händel, Samson (1743)
Georg Friedrich Händel, Judas Maccabaeus (1747)
Georg Friedrich Händel, Joshua (1748)
Georg Friedrich Händel, Jephtha (1752)
Joseph Haydn, The Creation (1798)
Joseph Haydn, The Seasons (1801)
Felix Mendelssohn, St. Paul (1836)
Felix Mendelssohn, Elijah (1846)
Hector Berlioz, L'enfance du Christ (1854)
Franz Liszt, Christus (1862–1866)
Théodore Dubois, Les sept paroles du Christ (1867)
Igor Stravinsky's "opera-oratorio" Oedipus Rex (1927)
Artur Kapp, Hiiob (Job) (1929)
William Walton, Belshazzar's Feast (1931)
Paul Constantinescu, Oratoriul Bizantin de Crăciun "Nașterea Domnului" (1936)
Paul Constantinescu, Oratoriul Bizantin de Paști "Patimile și Învierea Domnului (1943)
Alexandre Tansman, Isaïe le prophète (1950)
Hans Werner Henze, Das Floß der Medusa (1968, rev. 1990)
Bertold Hummel, The Shrine of the Martyrs (1988/89)
Paul McCartney, Liverpool Oratorio (1991)
Wynton Marsalis "Blood on the Fields" (1997)
Vangelis Papathanasiou, Mythodea (2001)
Piotr Rubik - "Tu Es Petrus" (2005)
Hristo Tsanoff - "Stabat Mater dolorosa" (2007)
Eric Idle and John Du Prez - He's Not the Messiah (He's a Very Naughty Boy) (2007) (an unusual oratorio in that it is a comedic piece based on Monty Python's Life of Brian)
Vezi și
Passion
Requiem
Mass (liturgy)
Mass (music)
Cantata
Oratorio Society
Referințe
Bukofzer, Manfred F. Music in the Baroque Era. New York, NY: W.W. Norton and Co., Inc, 1947.
Smither, Howard. The History of the Oratorio. vol. 1-4, Chapel Hill, NC: Univ. of N.C. Press, 1977-2000.
Deedy, John. The Catholic Fact Book. Chicago, IL: Thomas Moore Press, 1986.
Grove Music Online, ed. L. Macy, grovemusic.com (subscription access).
Hardon, John A. Modern Catholic Dictionary. Garden City, NY: Double Day and Co. Inc., 1980.
New Catholic Encyclopedia. New York: McGraw-Hill, 1967.
Randel, Don. "Oratorio". The Harvard Dictionary of Music. Cambridge, MA: The Belknap Press, 1986.
Legături externe
Genuri muzicale culte | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,614 |
The aim of this post is to quickly go through the setup process to get you started as fast as possible rather than an in-depth article. Let's set up a free full(strict) SSL certificate provided by Cloudflare on a shared hosting plan from Godaddy.
This guide lists each step to follow in order to encrypt the connection between your website visitors and CloudFlare AND from CloudFlare to your server using the free Cloudflare plan. We will explain why your site should be using an SSL certificate and how to set it up on both Cloudflare and Godaddy.
Cloudflare briefly and simply explains this: "Use SSL in order to encrypt data, like credit card numbers, and other sensitive information while in transit to and from your website. SSL encryption ensures that communication between your visitor and website is confidential.
You should already have setup Cloudflare but if this is not the case, you can signup and follow the provided instructions.
For more details regarding the creation of the certificate, you can follow this guide by CloudFlare.
You should now be all set but do not yet turn the SSL to full (strict) mode as we need to install the certificate on Godaddy first.
This is very easy as there is an SSL section in the cPanel which allows the creation or upload of the certificate and private key. It's also where we will be installing the certificate.
You should now be ready to use the full(strict) SSL on your website. Go back to the CRYPTO page on Cloudflare and select "full (strict)".
Make sure that your website works by loading it using https.
Edit:As mentioned in the comment section, you should make sure that your website uses 'https://' URLs so you may want to add redirects rules and check that all content is loaded from secure URLs.
Feel free to let me know if you have any questions or would like to add anything to this post. I hope it was as quickly and easy to follow as possible. Cheers! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,226 |
Grýlurnar var en isländsk musikgrupp som bildades av Ragnhildur Gísladóttir från gruppen Brimkló, Linda Björk Hreiðarsdóttir, Inga Rún Pálmadóttir och Herdís Hallvarðsdóttir.
Medlemmar
Ragnhildur "Ragga" Gisladóttir - sång
Inga Rún Pálmadóttir - gitarr
Herdis Hallvardsdóttir - bas
Linda Björk Hreidarsdóttir - trummor
Diskografi i urval
Album
Með Allt Á Hreinu (1982) (med Stuðmenn)
Mávastellið (1982)
EP
Grýlurnar (Fjúgum Hærra / Dont think twice / Gullúrið / Cold Things) (1981)
Externa länkar
Grýlurnar
Diskografi på Discogs
Isländska musikgrupper | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,515 |
\section{Introduction}
\label{sec:intro}
Applications of Gr\"obner basis theory to various problems of statistics
arises in early 1990s. One of the first works in this
developing field, a computational algebraic statistics, is given by
Pistone and Wynn (\cite{Pistone-Wynn-1996}), where the Gr\"obner
basis theory is applied to
the identifiability problem in the design of experiments.
After this work, various algebraic
approaches to the problems in the design of experiments
are presented by researchers both in the fields of algebra and
statistics. A theory
of the indicator function of fractional factorial designs
is one of the early results in this branch.
The indicator function is first introduced by Fontana, Pistone and
Rogantin (\cite{Fontana-Pistone-Rogantin-2000})
for two-level fractional
factorial designs. In \cite{Fontana-Pistone-Rogantin-2000}, based on the
results of
\cite{Pistone-Wynn-1996},
one-to-one correspondence between the design and its
indicator function is shown.
This correspondence enables us to translate various statistical
concepts to algebraic concepts, i.e.,
various results on the fractional factorial designs
can be interpreted to the structure of their indicator functions.
For example, abberation and resolution are important concepts in design
of experiments, and
there is a well-established history starting with
\cite{Box-Hunter-1961} for two-level fractional factorial designs.
An important contribution of \cite{Fontana-Pistone-Rogantin-2000} is to
characterize these concepts as the structure
of the indicator functions.
To illustrate the motivation of this paper, we
glance at the arguments of \cite{Fontana-Pistone-Rogantin-2000} by
examples. Note that the necessary definitions on the designs and indicator
functions will be given in Section
\ref{sec:definition-indicator-function}.
Figure \ref{fig:example-D1-D2} shows
examples of two-level fractional factorial designs.
\begin{figure*}[htbp]
\[
\begin{array}{|rrrrr|}
\multicolumn{5}{l}{F_1}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & x_4 & \multicolumn{1}{c}{x_5}\\ \hline
1 & 1 & 1 & 1 & 1\\
1 & 1 &-1 & 1 &-1\\
1 &-1 & 1 &-1 & 1\\
1 &-1 &-1 &-1 &-1\\
-1 & 1 & 1 &-1 &-1\\
-1 & 1 &-1 &-1 & 1\\
-1 &-1 & 1 & 1 &-1\\
-1 &-1 &-1 & 1 & 1\\ \hline
\end{array}
\hspace*{10mm}
\begin{array}{|rrrr|}
\multicolumn{4}{l}{F_2}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & \multicolumn{1}{c}{x_4}\\ \hline
1 & 1 & 1 & 1\\
1 & 1 &-1 &-1\\
1 &-1 & 1 &-1\\
-1 & 1 &-1 &-1\\
-1 &-1 & 1 &-1\\
-1 &-1 &-1 & 1\\ \hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{}
\end{array}
\]
\caption{Examples of fractional factorial designs for two-level
factors. Left($F_1$): a regular fractional factorial
design with the defining
relation $x_1x_2x_4 = x_1x_3x_5 = 1$. Right($F_2$): an example of
nonregular designs.}
\label{fig:example-D1-D2}
\end{figure*}
We code the levels of each factor as $\{-1,1\}$ according to
\cite{Fontana-Pistone-Rogantin-2000}.
For each design, each row of the table shows the combination of the
levels of the factors $x_i$'s for each experimental run, and each column
corresponds to each
factor. For example, the design $F_1$ is a fractional factorial
design for $5$ two-level factors $x_1,\ldots,x_5$, composed of $8$
points in $\{-1,1\}^5$,
\[
\{(1,1,1,1,1),(1,1,-1,1,-1),\ldots,(-1,-1,-1,1,1)\}.
\]
In the field of design of experiments,
$F_1$ is known as a regular fractional factorial design with the
defining relation $x_1x_2x_4 = x_1x_3x_5 = 1$. On the other hand, the design
$F_2$ is an example of nonregular designs. For details on
the regularity of designs, see \cite{Wu-Hamada-2009} for example.
The indicator functions of $F_1$ and $F_2$ are given as follows, respectively.
\[\begin{array}{cl}
F_1: & \displaystyle
f_1(x_1,x_2,x_3,x_4,x_5) = \frac{1}{4} + \frac{1}{4}(x_1x_2x_4 +
x_1x_3x_5 +
x_2x_3x_4x_5)\\
F_2: & \displaystyle f_2(x_1,x_2,x_3,x_4) = \frac{3}{8} - \frac{1}{8}x_4 +
\frac{1}{8}(x_1x_2 + x_1x_3 - x_2x_3)
+ \frac{1}{8}(x_1x_3x_4 +
x_2x_3x_4) + \frac{3}{8}x_2x_3x_4
\end{array}
\]
We see, for example, $f_1(x_1,\ldots,x_5) = 1$ for $8$ points in $F_1$,
and $f_1(x_1,\ldots,x_5) = 0$ for the other $24$ points not in $F_1$.
The indicator function of the design of $n$ two-level
factors, $x_1, x_2,\ldots,x_n$ has a unique polynomial representation
of the form
\begin{equation}
f(x_1,\ldots,x_n) = \sum_{\Ba \in \{0,1\}^n}\theta_{\Ba}\Bx^{\Ba},
\label{eqn:indicator-function-two-level}
\end{equation}
where $\Bx^{\Ba} = \displaystyle\prod_{i = 1}^nx_i^{a_i}$ and
$\Ba = (a_1,\ldots,a_n) \in \{0,1\}^n$.
As is shown in \cite{Fontana-Pistone-Rogantin-2000},
the set of the coefficients $\{\theta_{\Ba}\}_{\Ba \in \{0,1\}^n}$ has
all the information of the corresponding design.
For example, we see the following facts from the coefficients of the
indicator functions $f_1$ and $f_2$ for $F_1$ and $F_2$.
\begin{itemize}
\item The constant term $\theta_{(0,\ldots,0)}$ shows the ratio
between the size of the design to the size of the full factorial
design. In fact, $F_1$ is a $1/4$ fraction of the full factorial
$2^5$ design, and $F_2$ is a $3/8$ fraction of the full factorial
$2^4$ design.
\item The coefficient of the main effect term $\theta_{\Ba}$, $\sum_j
a_j = 1$, i.e., the coefficient of the monomial $\Bx^{\Ba}$ with
the degree $1$, shows the
``balance'' of two levels for this factor. In fact, for $F_1$,
$\theta_{(1,0,0,0,0)} = \cdots = \theta_{(0,0,0,0,1)} = 0$ shows
$F_1$ is an equireplicated design, i.e., two levels
appear equally often
for each factor. On the other hand, for $F_2$, $\theta_{(1,0,0,0)}
= \theta_{(0,1,0,0)} = \theta_{(0,0,1,0)} = 0$ and
$\theta_{(0,0,0,1)} \neq 0$ show $F_2$ is
equireplicated for factors $x_1,x_2,x_3$ but not for $x_4$.
\item The coefficient of the two-factor interaction term
$\theta_{\Ba}$, $\sum_j a_j = 2$, i.e., the coefficient of the
monomial $\Bx^{\Ba}$ with the degree $2$,
shows the ``orthogonality'' of the design. In fact, for $F_1$,
$\theta_{(1,1,0,0,0)} = \cdots = \theta_{(0,0,0,1,1)} = 0$
shows $F_1$ is an orthogonal design, i.e.,
possible combinations of levels, $(-1,-1), (-1,1), (1,-1), (1,1)$,
appear equally often for each pair of the factors. On the other
hand, for $F_2$, $\theta_{(1,0,0,1)} = \theta_{(0,1,0,1)} =
\theta_{(0,0,1,1)} = 0$ shows that the factor $x_4$ is orthogonal to
each of the other factors, whereas $\theta_{(1,1,0,0)},
\theta_{(1,0,1,0)}, \theta_{(0,1,1,0)} \neq 0$ shows that $x_1,
x_2, x_3$ are not
orthogonal in each other.
\end{itemize}
In other words, statistical concepts such as aberration and
resolution can be
related to the structure of the corresponding indicator functions
directly for two-level designs.
In particular, the structure of the indicator function of
regular two-level designs can be characterized by their defining
relations, and are fully revealed. See \cite{Ye-2003} for
detail. Another characterization of the indicator function of two-level
designs relating the $D$-optimality of the design
is given by the author in \cite{Aoki-Takemura-2009}.
In \cite{Fontana-Pistone-Rogantin-2000}, these structures of
the indicator
function are applied to the classification of the design, which is also
the object of this paper. The argument of
\cite{Fontana-Pistone-Rogantin-2000} is as follows. For the indicator
function (\ref{eqn:indicator-function-two-level}) of two-level designs,
the set of the coefficients $\{\theta_{\Ba}\}_{\Ba \in \{0,1\}^n}$
satisfies a system of algebraic equations
\begin{equation}
\theta_{\Ba} = \sum_{\Ba' \in \{0,1\}^n}\theta_{\Ba'}\theta_{\Ba +
\Ba'},\ \ \Ba \in \{0,1\}^n,
\label{eqn:relation-theta-2-level}
\end{equation}
where the sum $\Ba + \Ba'$ is considered under ``mod $2$'' (Proposition 3.7
of \cite{Fontana-Pistone-Rogantin-2000}). Therefore, adding
constraints for some orthogonality of the designs
to (\ref{eqn:relation-theta-2-level}), we have a system of
algebraic equations
having the designs with these orthogonality as the solutions.
For example, for the case of $n = 5$, additional constraints
\begin{equation}
\theta_{(0,0,0,0,0)} = \frac{1}{4},\ \
\theta_{\Ba} = 0\ \mbox{for}\ \sum_j a_j = 1,2
\label{eqn:example-constraints-2-level}
\end{equation}
to (\ref{eqn:relation-theta-2-level}) yields a system of algebraic
equations having all the orthogonal designs with the size $8$ as the
solution (and $F_1$ corresponds to one of the solutions).
In this way, the complete lists of the orthogonal designs for
$n = 4,5$ are computed by a computational algebraic software in
\cite{Fontana-Pistone-Rogantin-2000}. Recall that
solving a system of algebraic equations is a fundamental problem where
the theory of Gr\"obner basis is used.
In this paper, we consider generalization of the above argument on
two-level designs to
general fractional factorial designs. Note that the direct
relations between the size and orthogonality of
designs and their indicator functions
are obtained only for two-level designs.
To see this, consider a fractional factorial design of three-level
factors $F_3$ displayed in Figure \ref{fig:example-F3}.
$F_3$ is
a regular fractional factorial design with the defining relation $x_1 +
x_2 + x_3 = x_1 +
2x_2 + x_4 = 0\ ({\rm mod}\ 3)$.
\begin{figure*}[htbp]
\[
\begin{array}{|rrrr|}
\multicolumn{4}{l}{F_3}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & \multicolumn{1}{c}{x_4}\\ \hline
-1 &-1 &-1 & 0\\
-1 & 0 & 1 & 1\\
-1 & 1 & 0 &-1\\
0 &-1 & 1 &-1\\
0 & 0 & 0 & 0\\
0 & 1 &-1 & 1\\
1 &-1 & 0 & 1\\
1 & 0 &-1 &-1\\
1 & 1 & 1 & 0\\ \hline
\end{array}
\]
\caption{A regular fractional factorial design for three-level
factors with the defining
relation $x_1 + x_2 + x_3 = x_1 + 2x_2 + x_4 = 0\ ({\rm mod}\ 3)$.}
\label{fig:example-F3}
\end{figure*}
Though $F_3$ is a regular design (and therefore the resolution of $F_3$
is seen in its defining relation), the structure of its indicator
function seems complicated as follows.
\begin{equation}
\begin{array}{cl}
\multicolumn{2}{l}{f(x_1,x_2,x_3,x_4)}\vspace*{2mm}\\
= & \displaystyle 1 - x_1^2 - x_2^2 - x_3^2 - x_4^2
+ x_1^2x_2^2 + x_1^2x_3^2 + x_1^2x_4^2
+ x_2^2x_3^2 + x_2^2x_4^2 + x_3^2x_4^2\vspace*{2mm}\\
& + \displaystyle \frac{1}{4}(x_1^2x_2x_3 - x_1^2x_2x_4 + x_1^2x_3x_4 +
x_1x_2^2x_3 +
x_1x_2^2x_4 - x_2^2x_3x_4\\
&\hspace*{7mm} {} + x_1x_2x_3^2 - x_1x_3^2x_4 + x_2x_3^2x_4 -
x_1x_2x_4^2 - x_1x_3x_4^2 - x_2x_3x_4^2)\vspace*{2mm}\\
& - \displaystyle \frac{3}{4}(x_1^2x_2^2x_3^2 + x_1^2x_2^2x_4^2 +
x_1^2x_3^2x_4^2 + x_2^2x_3^2x_4^2)
\end{array}
\label{eqn:indicator-function-F3}
\end{equation}
There are several approaches to consider the indicator functions of
multi-level designs. In \cite{Pistone-Rogantin-2008b}, a complex coding
is proposed to generalize the arguments on two-level cases to multi-level
cases. For example, instead of $\{-1,0,1\}$ above,
the three-level factor is coded as $\{1, w, w^2\}$, where $w =
\exp(2\pi\sqrt{-1}/3)$ in
\cite{Pistone-Rogantin-2008b}. The idea of the complex coding is based
on a theory of a harmonic analysis, where the indicator function is
viewed as a discrete Fourier transform.
Other approach is presented in
\cite{Cheng-Yw-2004} for the real coefficients field. However, it is
better if we can consider
$\mathbb{Q}$, the field of rational numbers, as the coefficients field,
because algebraic computations are conducted in $\mathbb{Q}$ (or finite
fields $\mathbb{Z}/p\mathbb{Z}$) for standard computational algebraic
software.
Another approach is a concept of Hilbert basis presented in
\cite{Carlini-Pistone-2009}, where the case of repeated treatments are
considered by considering counting functions instead of indicator
functions.
In this paper,
we give generalization of the relations for two-level designs such as
(\ref{eqn:relation-theta-2-level}) and
(\ref{eqn:example-constraints-2-level}) to general multi-level designs
for the rational coefficients field $\mathbb{Q}$,
and show how to relate the structure of the designs to the
structure of their indicator functions.
The construction of this paper is as follows. In Section
\ref{sec:definition-indicator-function}, we give
necessary definitions and theorems on the indicator functions.
In Section 3, we give the structure of the indicator functions for
general fractional factorial designs.
We also derive another
representation of the indicator functions, namely, contrast representation,
to reflect the orthogonality of the designs directly.
In Section 4, we use these results to classify $2^3\times 3$ and
$2^4\times 3$ designs with given orthogonalities by a computational
algebraic software.
\section{The indicator functions of fractional factorial designs}
\label{sec:definition-indicator-function}
In this section, we give necessary materials on the indicator functions
of fractional factorial designs. The arguments are based on the theory
of interpolatory polynomial functions on designs, which is one of the
first applications of Gr\"obner basis theory to statistics
introduced by \cite{Pistone-Wynn-1996}.
See \cite{Pistone-Riccomagno-Wynn-2001} or Chapter 5 of
\cite{Cox-Little-OShea-1992} for detail.
Let $x_1,\ldots,x_n$ be $n$ factors. Let $A_j \subset
\mathbb{Q}$ be a level set of a factor $x_j$ for $j = 1,\ldots,n$,
where $\mathbb{Q}$ denotes the field of rational numbers.
We denote by $r_j = \#A_j$ the cardinality of $A_j$ and assume $r_j \geq
2$ for $j = 1,\ldots,n$.
{\it A full factorial design} of the factors $x_1,\ldots,x_n$ is $D =
A_1\times \cdots\times A_n$.
For later use, we introduce an {\it index set}
\[
{\cal I} = \{(i_1,\ldots,i_n) \in [r_1]\times \cdots \times [r_n]\},
\]
where $[k] = \{1,2,\ldots,k\}$ for a positive integer $k$. We specify
each point of $D$ as $D = \{\Bd_{\Bi} \in \mathbb{Q}^n\ :\ \Bi \in {\cal
I}\}$. When we code $A_j = [r_j]$ for $j = 1,\ldots,n$, ${\cal I}$ coincides
with $D$ itself.
A subset of $D$ is called {\it a fractional factorial design}.
A fractional factorial design $F \subset D$ can be written as $F =
\{\Bd_{\Bi} \in D\ :\ \Bi \in {\cal I}'\}$ where ${\cal I}'$ is a subset of
${\cal I}$.
Each design can be viewed as a finite subset of
$\mathbb{Q}^n$, i.e., as an algebraic variety, because each design can
be characterized as the set of the solutions of a system of polynomial
equations with rational coefficients.
The {\it size} of a design is the cardinality of the design.
We write the
size of a full factorial design $D$ as $m = \prod_{j = 1}^n r_j$ for
later use.
Let $\mathbb{Q}[x_1,\ldots,x_n]$ be the polynomial ring with
coefficients in $\mathbb{Q}$. For a design $F \subset
\mathbb{Q}^n$, we denote by $I(F)$ the set of polynomials in
$\mathbb{Q}[x_1,\ldots,x_n]$ which are $0$ at every point of $F$, i.e.,
\[
I(F) = \{f \in \mathbb{Q}[x_1,\ldots,x_n]\ :\ f(\Bd) = 0,\ \forall \Bd
\in F\}.
\]
It is easy to prove that the set $I(F)$ is an ideal of
$\mathbb{Q}[x_1,\ldots,x_n]$. $I(F)$ is called {\it the design ideal} of
$F$. The design ideal introduced by \cite{Pistone-Wynn-1996} is a
fundamental tool to consider designs
algebraically.
The design ideal is a radical ideal (Theorem 20 of
\cite{Pistone-Riccomagno-Wynn-2001}). The set of points $\Bd
\in \mathbb{Q}^n$ satisfying $f(\Bd) = 0$ for all $f \in I(F)$ is $F$ itself.
For a full factorial design $D$, the design ideal $I(D)$ can be written
as
\[
I(D) = \left<x_j^{r_j} - g_j,\ j = 1,\ldots,n \right>,
\]
where $g_j$ is a polynomial in $\mathbb{Q}[x_j]$ with the
degree less than $r_j$, $j = 1,\ldots,n$.
Here $\left< \{f_i\} \right>$ means ``the ideal generated by
$\{f_i\}$''.
In other words,
the set
\begin{equation}
G = \left\{x_j^{r_j} - g_j,\ j = 1,\ldots,n \right\}
\label{eqn:G-I(D)}
\end{equation}
is a generator of $I(D)$. In addition, $G$ is a
reduced Gr\"obner basis of $I(D)$ for any monomial order.
Note that the term $x_j^{r_j}$ is greater than the leading term of $g_j$
with respect to any monomial order.
We write the set of the monomials that are not divisible by the initial
monomials of $G$, $\{x_j^{r_j},\ j = 1,\ldots,n\}$, as
\[
{\rm Est}(D) = \left\{\Bx^{\Ba} = \prod_{j = 1}^nx_j^{a_j}\ :\ \Ba \in
L\right\},
\]
where
\[
L = \{\Ba = (a_1,\ldots,a_n) \in \mathbb{Z}_{\geq 0}^n\ :\ 0 \leq a_j
\leq r_j-1,\ j = 1,\ldots,n\}
\]
and $\mathbb{Z}_{\geq 0}$ is the set of nonnegative integers.
Note that the cardinality of $L$ is $m$.
From $D = \{\Bd_{\Bi} \in \mathbb{Q}^n\ :\ \Bi \in {\cal I}\}$ and $L$, we
define {\it a model matrix} by
\[
X = \left[\Bd_{\Bi}^{\Ba}\right]_{\Bi \in {\cal I}; \Ba \in L},
\]
where $\Bd_{\Bi}^{\Ba} = \prod_{j = 1}^nd_{\Bi j}^{a_j}$ and $d_{\Bi j}$
is the level of the factor $j$ in the experimental run indexed by $\Bi
\in {\cal I}$. Note that $X$ is called a design matrix in Definition 26
of \cite{Pistone-Riccomagno-Wynn-2001}.
By ordering the elements of ${\cal I}$ and $L$, $X$ is an $m \times m$
matrix, and is nonsingular (Theorem 26 of
\cite{Pistone-Riccomagno-Wynn-2001}).
The quotient of $\mathbb{Q}[x_1,\ldots,x_n]$ modulo the
design ideal $I(D) \in \mathbb{Q}[x_1,\ldots,x_n]$ is defined by
\[
\mathbb{Q}[x_1,\ldots,x_n]/I(D) = \{[f]\ :\ f \in
\mathbb{Q}[x_1,\ldots,x_n]\},
\]
where we define $[f] = \{g \in \mathbb{Q}[x_1,\ldots,x_n]\ {\rm such\
that}\ f - g \in I(D)\}$. In the terminology of the designs of
experiments, two polynomial models $f$ and $g$ are {\it confounded} on $D$ if
and only if $f - g \in I(D)$.
Therefore each element $[f] \in \mathbb{Q}[x_1,\ldots,x_n]/I(D)$ is
the set of the polynomials $g \in \mathbb{Q}[x_1,\ldots,x_n]$ that is
confounded to $f$ on $D$.
An important fact is that
${\rm Est}(D)$ is a basis of $\mathbb{Q}[x_1,\ldots,x_n]/I(D)$ as a
$\mathbb{Q}$-vector space. See Theorem 15 of
\cite{Pistone-Riccomagno-Wynn-2001} for detail.
Suppose we have a $\mathbb{Q}$-valued response (or, observations) $\By =
(y(\Bi))_{\Bi \in {\cal I}}$ on $D = \{\Bd_i\in \mathbb{Q}^n:\ \Bi \in
{\cal I}\}$.
Note that each
polynomial $f \in \mathbb{Q}[x_1,\ldots,x_n]$ can be viewed as
a response function on $D$, i.e.,
$f \in \mathbb{Q}^D$ where we
denote $\mathbb{Q}^D$
by the vector space of functions from $D$ to $\mathbb{Q}$.
The {\it interpolatory polynomial function} for $\By$ is a polynomial $f
\in \mathbb{Q}[x_1,\ldots,x_n]$ satisfying $f(\Bd_{\Bi}) = y(\Bi),\ \Bi
\in {\cal I}$. From the fact that ${\rm Est}(D)$ is a basis of
$\mathbb{Q}[x_1,\ldots,x_n]/I(D)$, the interpolatory polynomial function
for $\By$ is written uniquely as
\begin{equation}
f(x_1,\ldots,x_n) = \sum_{\Ba \in L}\theta_{\Ba}\Bx^{\Ba},
\label{eqn:poly-inter-polatory}
\end{equation}
where an $m \times 1$ column vector $\Btheta = (\theta_{\Ba})_{\Ba \in
L}$ is given by $\Btheta = X^{-1}\By$ for an $m\times 1$ column vector
$\By$. See Theorem 26 of \cite{Pistone-Riccomagno-Wynn-2001} for detail.
Now we introduce an indicator function.
\begin{definition}[\cite{Fontana-Pistone-Rogantin-2000}]
Let $F \subset D$ be a fractional factorial design.
The indicator function of $F$ is a response function $f$ on $D$
satisfying
\[
f(\Bd) = \left\{\begin{array}{ll}
1, & \mbox{if}\ \Bd \in F,\\
0, & \mbox{if}\ \Bd \in D \setminus F.
\end{array}
\right.
\]
\end{definition}
By the definition, the indicator function is constructed as follows. Write a
fractional factorial design $F \subset D$ as $F = \{\Bd_{\Bi} \in D\ :\
\Bi \in {\cal I}'\}$ for a subset ${\cal I}' \subset {\cal I}$. Then the
indicator function of $F$ is the interpolatory polynomial function for a
response $\By = (y(\Bi))_{\Bi \in {\cal I}}$, where
\begin{equation}
y(\Bi) = \left\{\begin{array}{ll}
1, & \mbox{if}\ \Bi \in {\cal I}'\\
0, & \mbox{if}\ \Bi \in {\cal I} \setminus {\cal I}'.
\end{array}
\right.
\label{eqn:y-subset-design}
\end{equation}
From the uniqueness of the interpolarory polynomial function mentioned
above, the representation of the indicator function is unique.
\begin{example}\label{example:2x2x3}
Consider designs of $3$ factors $x_1,x_2,x_3$, where $x_1, x_2$ are
two-level factors and $x_3$ is a three-level factor.
We code the levels of each factor as
\[
A_1 = A_2 = \{-1,1\},\ A_3 = \{-1,0,1\}.
\]
Therefore $r_1 = r_2 = 2, r_3 = 3$, and
the full factorial design $D = A_1\times A_2\times A_3$ has $m = 12$ points.
The index set is ${\cal I} = \{1,2\}\times \{1,2\}\times \{1,2,3\}$.
The full factorial design $D$ is written as $D = \{\Bd_{\Bi}\ :\ \Bi \in
{\cal I}\}$, where
\[
\begin{array}{lll}
\Bd_{(1,1,1)} = (-1,-1,-1), & \Bd_{(1,1,2)} = (-1,-1,0), &
\Bd_{(1,1,3)} = (-1,-1,1), \\
\Bd_{(1,2,1)} = (-1,1,-1), & \Bd_{(1,2,2)} = (-1,1,0), &
\Bd_{(1,2,3)} = (-1,1,1), \\
\Bd_{(2,1,1)} = (1,-1,-1), & \Bd_{(2,1,2)} = (1,-1,0), &
\Bd_{(2,1,3)} = (1,-1,1), \\
\Bd_{(2,2,1)} = (1,1,-1), & \Bd_{(2,2,2)} = (1,1,0), &
\Bd_{(2,2,3)} = (1,1,1).
\end{array}
\]
The design ideal of $D$ is written as
\[
I(D) = \left<x_1^2-1, x_2^2-1, x_3^3 - x_3\right> \subset
\mathbb{Q}[x_1,x_2,x_3],
\]
and $G = \{x_1^2-1, x_2^2-1, x_3^3 - x_3\}$ is a reduced Gr\"obner basis
of $I(D)$ for any monomial order.
Therefore we have
\[
{\rm Est}(D) =
\{1,x_1,x_2,x_3,x_3^2,x_1x_2,x_1x_3,x_2x_3,x_1x_2x_3,x_1x_3^2,x_2x_3^2,
x_1x_2x_3^2 \}.
\]
Note that there are $m$ monomials in ${\rm Est}(D)$. Corresponding $L$
is given by
\[\begin{array}{rcl}
L & = & \{\ (0,0,0), (1,0,0), (0,1,0), (0,0,1), (0,0,2), (1,1,0),\\
& & \hspace*{4mm}(1,0,1), (0,1,1), (1,1,1), (1,0,2), (0,1,2), (1,1,2)\ \}.
\end{array}\]
The model matrix $X$ is given in Figure
\ref{fig:model-matrix-2x2x3}.
Note that, differently than in the usual ANOVA decomposition, the
columns corresponding to the interactions are not orthogonal to the
columns corresponding to the simple factors.
Here, and hereafter, we write each element
of $L$ and ${\cal I}$ by omitting commas as $(a_1\cdots a_n)$ or
$(i_1\cdots i_n)$ instead
of $(a_1,\cdots,a_n)$ or $(i_1,\cdots,i_n)$ for simplicity.
\begin{figure*}[htbp]
\[{\small
\begin{array}{c|rrrrrrrrrrrr|}
\multicolumn{1}{c}{{\cal I}\backslash L} & 000 & 100 & 010 & 001 & 002 & 110 &
101 & 011 & 111 & 102 & 012 & \multicolumn{1}{r}{112}\\ \cline{2-13}
111 &1&-1&-1&-1&1& 1& 1& 1&-1&-1&-1& 1\\
112 &1&-1&-1& 0&0& 1& 0& 0& 0& 0& 0& 0\\
113 &1&-1&-1& 1&1& 1&-1&-1& 1&-1&-1& 1\\
121 &1&-1& 1&-1&1&-1& 1&-1& 1&-1& 1&-1\\
122 &1&-1& 1& 0&0&-1& 0& 0& 0& 0& 0& 0\\
123 &1&-1& 1& 1&1&-1&-1& 1&-1&-1& 1&-1\\
211 &1& 1&-1&-1&1&-1&-1& 1& 1& 1&-1&-1\\
212 &1& 1&-1& 0&0&-1& 0& 0& 0& 0& 0& 0\\
213 &1& 1&-1& 1&1&-1& 1&-1&-1& 1&-1&-1\\
221 &1& 1& 1&-1&1& 1&-1&-1&-1& 1& 1& 1\\
222 &1& 1& 1& 0&0& 1& 0& 0& 0& 0& 0& 0\\
223 &1& 1& 1& 1&1& 1& 1& 1& 1& 1& 1& 1\\ \cline{2-13}
\end{array}
}\]
\caption{The model matrix of a full factorial design $D =
\{-1,1\}^2\times \{-1,0,1\}$.}
\label{fig:model-matrix-2x2x3}
\end{figure*}
Now consider a fractional factorial design $F = \{\Bd_{\Bi}\ :\ \Bi \in
{\cal I}'\}\subset D$, where ${\cal I}' =
\{(111),(122),(213),(223)\}$.
The indicator function of $F$ is constructed as the
interpolatory
polynomial function (\ref{eqn:poly-inter-polatory}) for a response $\By
= (y(\Bi))_{\Bi \in {\cal I}}$
satisfying (\ref{eqn:y-subset-design}). For this ${\cal I}'$,
$\By = (1,0,0,0,1,0,0,0,1,0,0,1)^{T}$ and
we have
\[
\Btheta = X^{-1}\By
= \frac{1}{8}(2,-2,2,1,1,-2,3,1,-1,3,-3,3)^{T},
\]
where ${}^{T}$ is a transpose. Therefore the indicator function of $F$
is
\[\begin{array}{l}
f(x_1,x_2,x_3) = \displaystyle \frac{1}{4} - \frac{1}{4}(x_1 - x_2) +
\frac{1}{8}(x_3 + x_3^2) - \frac{1}{4}x_1x_2 + \frac{3}{8}x_1x_3 +
\frac{1}{8}x_2x_3\vspace*{2mm}\\
\multicolumn{1}{r}{\displaystyle {} - \frac{1}{8}x_1x_2x_3 +
\frac{3}{8}(x_1x_3^2 -
x_2x_3^2 + x_1x_2x_3^2).}
\end{array}\]
\hspace*{\fill}$\Box$
\end{example}
\section{Characterization of orthogonal fractional factorial designs by
indicator functions}
Now we consider relations between the design and its indicator
function. We start with the generalization of the relation
(\ref{eqn:relation-theta-2-level}) for two-level designs to
multi-level designs.
A polynomial $f \in \mathbb{Q}[x_1,\ldots,x_n]$ is an indicator function
of some fractional factorial design $F \subset D = \{\Bd_{\Bi} \in
\mathbb{Q}^n\ :\ \Bi \in {\cal I}\}$ if and only if $f^2 - f \in I(D)$,
i.e., $f$ and $f^2$ are in the same equivalence class of
$\mathbb{Q}[x_1,\ldots,x_n]/I(D)$. Therefore, suppose $f$ represented as
(\ref{eqn:poly-inter-polatory})
is an indicator function of some
fractional factorial design, we have
\[\begin{array}{rcl}
\displaystyle \sum_{\Ba \in L}\theta_{\Ba}\Bx^{\Ba}
& = & \displaystyle
\left(\sum_{\Ba \in
L}\theta_{\Ba}\Bx^{\Ba}\right)^2\ \ \ {\rm mod}\ I(D)\\
& = & \displaystyle
\sum_{\Ba_1 \in
L}\sum_{\Ba_2 \in L}\theta_{\Ba_1}\theta_{\Ba_2}\Bx^{\Ba_1 + \Ba_2}\ \
\ {\rm mod}\ I(D).
\end{array}
\]
Here, write the standard form of
$\displaystyle
\sum_{\Ba_1 \in
L}\sum_{\Ba_2 \in L}\theta_{\Ba_1}\theta_{\Ba_2}\Bx^{\Ba_1 + \Ba_2}$
with respect to $G$ as
\begin{equation}
r = \sum_{\Ba\in L}\mu_{\Ba}\Bx^{\Ba}.
\label{eqn:rhs-r-mu}
\end{equation}
In other words, $r$ is a unique remainder when we divide
$\displaystyle
\sum_{\Ba_1 \in
L}\sum_{\Ba_2 \in L}\theta_{\Ba_1}\theta_{\Ba_2}\Bx^{\Ba_1 + \Ba_2}$ by
$G$, the reduced Gr\"ober basis of $I(D)$. Then we have the following result.
\begin{proposition}[Generalization of Proposition 3.7 of
\cite{Fontana-Pistone-Rogantin-2000}]
A polynomial $f$ represented as
(\ref{eqn:poly-inter-polatory}) is an indicator function of some
fractional factorial design if and only if a system of algebraic
equations
\begin{equation}
\theta_{\Ba} = \mu_{\Ba},\ \ \ \Ba \in L
\label{eq:prop-theta-mu-eq}
\end{equation}
holds, where $\mu_{\Ba}$ is given by (\ref{eqn:rhs-r-mu}).
\end{proposition}
\paragraph*{Proof.} From the division algorithm and the property of the
Gr\"obner basis.
See Chapter 2 of \cite{Cox-Little-OShea-1992}.\hspace*{\fill}$\Box$
\bigskip
The meaning of the relation (\ref{eq:prop-theta-mu-eq}) is explained as
follows. From the theory of the interpolatory polynomial function on
$D$, each $\mathbb{Q}$-valued function on $D$ is
represented (on $D$) by a polynomial with the monomials in ${\rm Est}(D)$.
This applies to the indicator function $f$ of a fraction $F \subset
D$. In this case, the reduced Gr\"obner basis of $I(D)$ is simply the
list of the univariate monic polynomials (\ref{eqn:G-I(D)}) defining
the levels of each factor, and the remainder $r$ is derived by
substitution as it is.
\begin{example}[Continuation of Example \ref{example:2x2x3}]
\label{example:2x2x3-2}
Consider $2\times 2\times 3$ designs. When we code the levels as $A_1 =
A_2 = \{-1,1\}, A_3 = \{-1,0,1\}$, the relation
(\ref{eq:prop-theta-mu-eq}) is as follows.
\[
\begin{array}{rcl}
\theta_{000} & = & \theta_{100}^2+\theta_{010}^2+\theta_{000}^2+\theta_{110}^2\\
\theta_{100} & = & 2\theta_{100}\theta_{000}+2\theta_{010}\theta_{110}\\
\theta_{010} & = & 2\theta_{100}\theta_{110}+2\theta_{010}\theta_{000}\\
\theta_{001} & = & 2\theta_{100}\theta_{101} + 2\theta_{010}\theta_{011}
+ 2\theta_{001}\theta_{002} + 2\theta_{001}\theta_{000} +
2\theta_{110}\theta_{111} +
2\theta_{101}\theta_{102}+2\theta_{011}\theta_{012} +
2\theta_{111}\theta_{112}\\
\theta_{002} & = & 2\theta_{100}\theta_{102} + 2\theta_{010}\theta_{012}
+ \theta_{001}^2+\theta_{002}^2 + 2\theta_{002}\theta_{000} +
2\theta_{110}\theta_{112} + \theta_{101}^2+\theta_{011}^2 +
\theta_{111}^2 + \theta_{102}^2 + \theta_{012}^2\\
& & + \theta_{112}^2\\
\theta_{110} & = & 2\theta_{100}\theta_{010} + 2\theta_{000}\theta_{110}\\
\theta_{101} & = & 2\theta_{100}\theta_{001} + 2\theta_{010}\theta_{111}
+ 2\theta_{001}\theta_{102} + 2\theta_{002}\theta_{101} +
2\theta_{000}\theta_{101} + 2\theta_{110}\theta_{011} +
2\theta_{011}\theta_{112} + 2\theta_{111}\theta_{012}\\
\theta_{011} & = & 2\theta_{100}\theta_{111} + 2\theta_{010}\theta_{001}
+ 2\theta_{001}\theta_{012} + 2\theta_{002}\theta_{011} +
2\theta_{000}\theta_{011} + 2\theta_{110}\theta_{101} +
2\theta_{101}\theta_{112} + 2\theta_{111}\theta_{102}\\
\theta_{111} & = & 2\theta_{100}\theta_{011} + 2\theta_{010}\theta_{101}
+ 2\theta_{001}\theta_{110} + 2\theta_{001}\theta_{112} +
2\theta_{002}\theta_{111} + 2\theta_{000}\theta_{111} +
2\theta_{101}\theta_{012} + 2\theta_{011}\theta_{102}\\
\theta_{102} & = & 2\theta_{100}\theta_{002} + 2\theta_{010}\theta_{112}
+ 2\theta_{001}\theta_{101} + 2\theta_{002}\theta_{102} +
2\theta_{000}\theta_{102} + 2\theta_{110}\theta_{012} +
2\theta_{011}\theta_{111} + 2\theta_{012}\theta_{112}\\
\theta_{012} & = & 2\theta_{100}\theta_{112} + 2\theta_{010}\theta_{002}
+ 2\theta_{001}\theta_{011} + 2\theta_{002}\theta_{012} +
2\theta_{000}\theta_{012} + 2\theta_{110}\theta_{102} +
2\theta_{101}\theta_{111} + 2\theta_{102}\theta_{112}\\
\theta_{112} & = & 2\theta_{100}\theta_{012} + 2\theta_{010}\theta_{102}
+ 2\theta_{001}\theta_{111} + 2\theta_{002}\theta_{110} +
2\theta_{002}\theta_{112} + 2\theta_{000}\theta_{112} +
2\theta_{101}\theta_{011} + 2\theta_{102}\theta_{012}
\end{array}
\]
These results are easily obtained by standard algebraic softwares, such as
Macaulay2 (\cite{Macaulay2}).
Each solution of the above system of polynomial equations corresponds to
the coefficients of the indicator function for each fractional
factorial designs.
If we change the level codings, the relation (\ref{eq:prop-theta-mu-eq})
changes. For example, when we code the levels as $A_1 = A_2 = \{0,1\},
A_3 = \{0,1,2\}$, the relation (\ref{eq:prop-theta-mu-eq}) is as follows.
\[
\begin{array}{rcl}
\theta_{000} & = & \theta_{000}^2\\
\theta_{100} & = & \theta_{100}^2+2\theta_{100}\theta_{000}\\
\theta_{010} & = & \theta_{010}^2+2\theta_{010}\theta_{000}\\
\theta_{001} & = &
-4\theta_{001}\theta_{002}-6\theta_{002}^2+2\theta_{001}\theta_{000}\\
\theta_{002} & = & \theta_{001}^2 + 6\theta_{001}\theta_{002} +
7\theta_{002}^2 +
2\theta_{002}\theta_{000}\\
\theta_{110} & = & 2\theta_{100}\theta_{010} + 2\theta_{100}\theta_{110} +
2\theta_{010}\theta_{110}
+ 2\theta_{000}\theta_{110} + \theta_{110}^2\\
\theta_{101} & = & 2\theta_{100}\theta_{001} + 2\theta_{100}\theta_{101} -
4\theta_{002}\theta_{101} + 2\theta_{000}\theta_{101} -
4\theta_{001}\theta_{102} - 12\theta_{002}\theta_{102} -
4\theta_{101}\theta_{102} - 6\theta_{102}^2\\
\theta_{011} & = & 2\theta_{010}\theta_{001} + 2\theta_{010}\theta_{011} -
4\theta_{002}\theta_{011} + 2\theta_{000}\theta_{011} -
4\theta_{001}\theta_{012} - 12\theta_{002}\theta_{012} -
4\theta_{011}\theta_{012} - 6\theta_{012}^2\\
\theta_{111} & = & 2\theta_{001}\theta_{110} + 2\theta_{010}\theta_{101} +
2\theta_{110}\theta_{101} + 2\theta_{100}\theta_{011} +
2\theta_{110}\theta_{011} + 2\theta_{100}\theta_{111} +
2\theta_{010}\theta_{111} - 4\theta_{002}\theta_{111}\\
& & {} + 2\theta_{000}\theta_{111} + 2\theta_{110}\theta_{111} -
4\theta_{011}\theta_{102} - 4\theta_{111}\theta_{102} -
4\theta_{101}\theta_{012} - 4\theta_{111}\theta_{012} -
12\theta_{102}\theta_{012}\\
& & {} - 4\theta_{001}\theta_{112} -
12\theta_{002}\theta_{112} - 4\theta_{101}\theta_{112} -
4\theta_{011}\theta_{112} - 4\theta_{111}\theta_{112} -
12\theta_{102}\theta_{112} - 12\theta_{012}\theta_{112} -
6\theta_{112}^2\\
\theta_{102} & = & 2\theta_{100}\theta_{002} + 2\theta_{001}\theta_{101} +
6\theta_{002}\theta_{101} + \theta_{101}^2 + 2\theta_{100}\theta_{102}
+ 6\theta_{001}\theta_{102} + 14\theta_{002}\theta_{102} +
2\theta_{000}\theta_{102}\\
& & + 6\theta_{101}\theta_{102} + 7\theta_{102}^2\\
\theta_{012} & = & 2\theta_{010}\theta_{002} + 2\theta_{001}\theta_{011} +
6\theta_{002}\theta_{011} + \theta_{011}^2 + 2\theta_{010}\theta_{012}
+ 6\theta_{001}\theta_{012} + 14\theta_{002}\theta_{012} +
2\theta_{000}\theta_{012}\\
& & {}+ 6\theta_{011}\theta_{012} +
7\theta_{012}^2\\
\theta_{112} & = & 2\theta_{002}\theta_{110} + 2\theta_{101}\theta_{011} +
2\theta_{001}\theta_{111} + 6\theta_{002}\theta_{111} +
2\theta_{101}\theta_{111} + 2\theta_{011}\theta_{111} + \theta_{111}^2
+ 2\theta_{010}\theta_{102}\\
& & + 2\theta_{110}\theta_{102} +
6\theta_{011}\theta_{102} + 6\theta_{111}\theta_{102} +
2\theta_{100}\theta_{012} + 2\theta_{110}\theta_{012} +
6\theta_{101}\theta_{012} + 6\theta_{111}\theta_{012}\\
& & {}+ 14\theta_{102}\theta_{012} + 2\theta_{100}\theta_{112} +
2\theta_{010}\theta_{112} + 6\theta_{001}\theta_{112} +
14\theta_{002}\theta_{112} + 2\theta_{000}\theta_{112} +
2\theta_{110}\theta_{112}\\
& & {}+ 6\theta_{101}\theta_{112} +
6\theta_{011}\theta_{112} + 6\theta_{111}\theta_{112} +
14\theta_{102}\theta_{112} + 14\theta_{012}\theta_{112} +
7\theta_{112}^2\\
\end{array}
\]
In actual applications, there are cases
where the level coding has not essential meaning, such as for the
designs of qualitative factors. However, for our purpose of solving a
system of polynomial equations using computational algebraic software,
an appropriate level coding is important in view of computational
time. In the author's experiences, it is better to code $\{-1,1\}$ rather than
$\{0,1\}$ for two-level factor, and $\{-1,0,1\}$ rather than
$\{0,1,2\}$ for three-level factor.
We consider this point in Section
\ref{sec:computation}.
\hspace*{\fill}$\Box$
\end{example}
\bigskip
As we see in Example \ref{example:2x2x3-2}, the relation
(\ref{eq:prop-theta-mu-eq}) is very complicated compared to the relation
for two-level cases (\ref{eqn:relation-theta-2-level}).
Among the various characterizations of the
coefficients of the indicator
functions of two-level designs given in
\cite{Fontana-Pistone-Rogantin-2000}, the relation of the indicator
functions of complementary designs can be generalized to multi-level cases
as follows.
\begin{proposition}[Generalization of Corollary 3.5 of
\cite{Fontana-Pistone-Rogantin-2000}]
If $F$ and $\bar{F}$ are complementary fractions and $\Btheta =
(\theta_{\Ba})_{\Ba \in L}$ and $\bar{\Btheta} =
(\bar{\theta}_{\Ba})_{\Ba \in L}$ are the coefficients of the
corresponding
indicator functions given by
(\ref{eqn:poly-inter-polatory}) respectively, then
\[
\theta_{0\cdots 0} = 1 - \bar{\theta}_{0\cdots 0}\ \ \mbox{and}\ \
\theta_{\Ba} = -\bar{\theta}_{\Ba},\ \forall\ \Ba \neq (0,\ldots,0).
\]
\end{proposition}
\paragraph*{Proof.}\ Write the model matrix $X$ as
\[
X = \left[\Bone_m\ |\ \Bs_2\ |\ \cdots\ |\ \Bs_m\right]
\]
and write $X^{-1}\Bone_m = (c_1,\ldots,c_m)^{T}$. Then we have $c_1 =
1,\ c_2 = \cdots = c_m = 0$ from the non-singularity of $X$ in the
relation
\[
\Bone_m = c_1\Bone_m + c_2\Bs_2 + \cdots + c_m\Bs_m.
\]
Therefore for the responses $\By, \bar{\By} \in \{0,1\}^m$ such that
$\By + \bar{\By} = \Bone_m$, we have
\[
\Btheta + \bar{\Btheta} = X^{-1}(\By + \bar{\By}) =
(1,0,\ldots,0)^T.
\]
\hspace*{\fill}$\Box$
\bigskip
Next consider structure of the indicator functions of designs with given
characteristic.
The idea is to express the structure of the indicator functions as
additional constraints to the system of polynomial equations
(\ref{eq:prop-theta-mu-eq}) to classify designs with given
characteristic. The additional constraints are derived as
follows. Recall that the coefficients vector $\Btheta$ is given by
$\Btheta = X^{-1}\By$ in (\ref{eqn:poly-inter-polatory}). Here, treat
$\By = (y(\Bi))_{\Bi \in {\cal I}}$ as a vector of $\{0,1\}^m$ in
(\ref{eqn:y-subset-design}) and express the characteristic of designs as
\begin{equation}
\Bc^{T}\By = s,\ s \in \mathbb{Q}
\label{eqn:add-const-cy}
\end{equation}
for a constant column vector $\Bc \in \mathbb{Q}^m$.
For example, $\By \in \{0,1\}^m$ corresponding to designs with the size $s$
satisfies the constraint
\[
\Bone_m^{T}\By = s,
\]
where $\Bone_m = (1,\ldots,1)^T$ is an $m\times 1$ column vector of the
elements $1$'s. Equireplicated designs or orthogonal designs can be
expressed by
\[
\Bc^T\By = 0
\]
for some contrast vectors $\Bc$.
Based on the above idea,
we define {\it a contrast matrix}.
For each subset $J \subset [n]$, we define
${\cal I}_J = \prod_{j \in J}[r_j] \subset {\cal I}$ and its cardinality
by $m_J = \prod_{j \in J}r_j$.
We also define $\Bi_J$ by the restriction of $\Bi = (i_1,\ldots,i_n) \in
{\cal I}$ to the index of ${\cal
I}_J$. For example of $n = 4$ and $J = \{1,2,4\}$, we have
$\Bi_J = (i_1,i_2,i_4)$.
\begin{definition}\label{def:contrast-matrix}
The contrast matrix $C$ is an $m\times m$
matrix of the form
\[
C^{T} =\left[\Bone_m\ |\ C_1^T\ |\ C_2^T\ |\ \cdots\ |\ C_n^T\right],
\]
where $C_k$ is a $v_k \times m$ matrix where
\[
v_k = \sum_{J \subset [n], \# J = k}\left(
\prod_{j \in J}(r_j-1)
\right).
\]
The set of $m\times 1$ column vectors of $C_k^T$ is
\[
\left\{\Bc_{J(\tilde{\Bi})} = \{c_{J(\tilde{\Bi})}(\Bi)\}_{\Bi \in {\cal I}}\ :\ J
\subset [n], \#J = k, \tilde{\Bi} \in \prod_{j \in J}[r_j-1]\right\},
\]
where
\[
c_{J(\tilde{i})}(\Bi) = \left\{\begin{array}{rl}
1, & \Bi_J = 1,\\
-1, & \Bi_J = \tilde{i} + 1,\\
0, & \mbox{otherwise}
\end{array}
\right.
\]
for $\#J = 1$, and
\[
c_{J(\tilde{\Bi})}(\Bi) = \left\{\begin{array}{rl}
1, & \Bi_J = (\tilde{i}_1,\ldots,\tilde{i}_{k-1},1),\\
-1, & \Bi_J = (\tilde{i}_1,\ldots,\tilde{i}_{k-1},\tilde{i}_k + 1),\\
0, & \mbox{otherwise}
\end{array}
\right.
\]
for $\#J \geq 2$.
\end{definition}
\bigskip
Note that the contrast matrix $C$ is constructed only from ${\cal I}$.
In other words, $C$ is uniquely determined from
$r_1,\ldots,r_n$. Especially, $C$ does not depend on the level coding.
\bigskip
\begin{example}[Continuation of Example \ref{example:2x2x3-2}]
\label{example:2x2x3-3}
For $2 \times 2\times 3$ designs, the contrast matrix $C$ is given in
Figure \ref{fig:contrast-matrix-2x2x3}.
\begin{figure*}[htbp]
\[{\small
\begin{array}{c|rrrrrrrrrrrr|}
\multicolumn{1}{c}{J(\tilde{\Bi})\backslash {\cal I}} & 111 & 112 & 113
& 121 & 122 & 123 &
211 & 212 & 213 & 221 & 222 & \multicolumn{1}{r}{223}\\ \cline{2-13}
{\rm Const.} &1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\
1(1) &1& 1& 1& 1& 1& 1&-1&-1&-1&-1&-1&-1\\
2(1) &1& 1& 1&-1&-1&-1& 1& 1& 1&-1&-1&-1\\
3(1) &1&-1& 0& 1&-1& 0& 1&-1& 0& 1&-1& 0\\
3(2) &1& 0&-1& 1& 0&-1& 1& 0&-1& 1& 0&-1\\
12(11) &1& 1& 1&-1&-1&-1& 0& 0& 0& 0& 0& 0\\
13(11) &1&-1& 0& 1&-1& 0& 0& 0& 0& 0& 0& 0\\
13(12) &1& 0&-1& 1& 0&-1& 0& 0& 0& 0& 0& 0\\
23(11) &1&-1& 0& 0& 0& 0& 1&-1& 0& 0& 0& 0\\
23(12) &1& 0&-1& 0& 0& 0& 1& 0&-1& 0& 0& 0\\
123(111) &1&-1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
123(112) &1& 0&-1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \cline{2-13}
\end{array}
}\]
\caption{The contrast matrix of $2\times 2\times 3$ designs.}
\label{fig:contrast-matrix-2x2x3}
\end{figure*}
\hspace*{\fill}$\Box$
\end{example}
\bigskip
The contrast matrix $C$ given in Definition \ref{def:contrast-matrix}
relates to the theory of contingency tables. Suppose the response $\By$ is a
vector of nonnegative integers, then we can treat $\By$ as a frequency
of contingency table
$\By = \{y_{\Bi}\ :\ \Bi \in {\cal I}\}$ with the set of cells ${\cal
I}$. In this case, we see that the condition
\[
\Bone_m^T \By = s,\ C_{\ell}\By = \Bzero_{v_{\ell}},\ \ell = 1,\ldots,t,
\]
means equal $\ell$-dimensional marginal totals for $\ell = 1,\ldots,t$.
The definition of the contrast matrix and
the following theorem (Theorem \ref{prop:orthogonal}) are based on this
connection.
As another relation, $C$ is a
configuration matrix in the theory of toric ideals. See Section1.5.3 of
\cite{dojo-en}.
By the contrast matrix, we specify the size and the orthogonality of the
designs as follows. We call a design $F \subset D$ is {\it orthogonal of
strength} $t$\ ($t \leq n$), if for any $t$ factors, all possible
combinations of levels appear equally often in $F$. This definition is
from the theory of orthogonal arrays. See Chapter 7 of
\cite{Wu-Hamada-2009}, for example. In particular, an orthogonal design
of strength $n$ is a full factorial design.
\begin{theorem}\label{prop:orthogonal}
If $\By \in \{0,1\}^m$ is a response on $D$ given by
(\ref{eqn:y-subset-design}), the fractional factorial
design $F = \{\Bd_{\Bi} \in D\ :\ \Bi \in {\cal I}'\}$ is
size $s$ and orthogonal of strength $t$ if and only if
\begin{equation}
\left\{
\begin{array}{l}
\Bone_m^T\By = s,\\
C_k\By = \Bzero_{v_k},\ \ k = 1,\ldots,t,
\end{array}
\right.
\label{eqn:cond-for-orthogonal-y}
\end{equation}
where $\Bzero_{\ell} = (0,\ldots,0)^T$ is an $\ell\times 1$ column
vector of the elements $0$'s, and $s$ is a common multiple of
$\displaystyle\left\{\prod_{j \in
J}r_j\ :\ \# J = t\right\}$.
\end{theorem}
To prove Theorem \ref{prop:orthogonal},
we define $J$-marginal vector of $\By =
(y(\Bi))_{\Bi \in {\cal I}}$ by $\By_J = (y_J(\Bi_J))_{\Bi_J \in {\cal
I}_J}$, where
\[
y_J(\Bi_J) = \sum_{\Bi_{J^c} \in {\cal I}_{J^c}}y(\Bi_J,\Bi_{J^c}).
\]
Note that $J^c$ denotes the complement of $J$, and in $y(\Bi_J,
\Bi_{J^c})$, for notational simplicity, the indices in ${\cal I}_J$ are
collected to the left. Also we are writing $y(\Bi_J, \Bi_{J^c})$ instead
of $y((\Bi_J,\Bi_{J^c}))$.
For example of $n = 3$, we have $\By_{\{2\}} =
(y_{\{2\}(\Bi_{\{2\}})})_{\Bi_{\{2\}} \in {\cal I}_{\{2\}}} =
(y_{\{2\}}(i_2))_{i_2 \in [r_2]}$, where
\[
y_{\{2\}}(i_2) = \sum_{i_1 \in [r_1]}\sum_{i_3 \in
[r_3]}y(i_1,i_2,i_3),
\]
and $\By_{\{1,3\}} = (y_{\{1,3\}}(\Bi_{\{1,3\}}))_{\Bi_{\{1,3\}} \in
{\cal I}_{\{1,3\}}} = (y_{\{1,3\}}(i_1,i_3))_{(i_1,i_3) \in
[r_1]\times[r_3]}$, where
\[
y_{\{1,3\}}(i_1,i_3) = \sum_{i_2 \in [r_2]}y(i_1,i_2,i_3),
\]
and so on. The concept of the $J$-marginal vector is from the theory of
contingency tables. For detail, see Chapter 4.2 of \cite{Lauritzen}, ``Basic
facts and concepts'' of Contingency tables, for example.
\paragraph*{Proof of Theorem \ref{prop:orthogonal}.}\
Suppose that the size $s$ is a common multiple of
$\displaystyle\left\{\prod_{j \in
J}r_j\ :\ \# J = t\right\}$.
Using the $J$-marginal vector $\By_J$,
$F$ is size $s$ and orthogonal of
strength $t$ if and only if
\begin{equation}
\left\{
\begin{array}{l}
\Bone_m^T \By = s,\\
\displaystyle y_{J}(\Bi_J) = \frac{s}{m_J},\ \forall\ \Bi_J \in
{\cal I}_J\ \ \mbox{for all}\ J \subset [n]\ \mbox{with}\ \# J \leq t.
\end{array}
\right.
\label{eqn:cond-for-y_J-orthogonal}
\end{equation}
The relation $(\ref{eqn:cond-for-y_J-orthogonal}) \Rightarrow
(\ref{eqn:cond-for-orthogonal-y})$ is straightforward, because the
relation $C_k \By = \Bzero_{v_k}$ is equivalent to
\begin{equation}
y_J(i_1,\ldots,i_{k-1},1) =
y_J(i_1,\ldots,i_{k-1},2) = \cdots =
y_J(i_1,\ldots,i_{k-1},r_k),\ \ \forall\
(i_1,\ldots,i_{k-1}) \in \displaystyle \prod_{j = 1}^{k-1}[r_j - 1]
\label{eqn:y_J-edge}
\end{equation}
for $J = \{1,2,\ldots,k\}$ from Definition \ref{def:contrast-matrix}.
To show $(\ref{eqn:cond-for-orthogonal-y}) \Rightarrow
(\ref{eqn:cond-for-y_J-orthogonal})$, we use an induction for $t$. For
the case of $t=1$, we have
\[
y_{\{j\}}(1) = y_{\{j\}}(2) = \cdots = y_{\{j\}}(r_j),\ \ j = 1,\ldots,n
\]
from $C_1\By = \Bzero_{v_1}$. From $s = \displaystyle\sum_{i_j =
1}^{r_j}y_{\{j\}}(i_j)$, we have
\[
y_{\{j\}}(i_j) = \frac{s}{r_j} = \frac{s}{m_{\{j\}}}
\]
for $j = 1,\ldots,n$. Next consider the case of $t$ under the assumption
that the theorem holds for the case of $t-1$.
We write $J = \{1,2,\ldots,t\}$
for a $J$ with $\#J = t$ without loss of generality. Our purpose is to
show that
\[
\By_J(\Bi_J) = \frac{s}{m_J},\ \ \forall \Bi_J \in {\cal I}_J
\]
for $J = \{1,2,\ldots,t\}$. Similarly to the relation
(\ref{eqn:y_J-edge}), for $\Bi_J = (i_1,\ldots,i_{t-1},i_t) \in
\left(
\displaystyle\prod_{j = 1}^{t-1}[r_j - 1]\right) \times [r_t]$ we have
\[
\frac{s}{m_{J \setminus\{t\}}} =
y_{J \setminus \{t\}}(i_1,\ldots,i_{t-1}) = \sum_{i_t =
1}^{r_t}y_J(i_1,\ldots,i_{t-1},i_t)
\]
and therefore
\[
y_J(i_1,\ldots,i_{t-1},i_t) = \frac{1}{r_t}\cdot \frac{s}{m_{J\setminus\{t\}}}
= \frac{s}{m_J}
\]
holds for $i_t = 1,\ldots,r_t$. To show the relation for other
$\Bi_J$'s, suppose $p$ elements of $\{i_1,\ldots,i_{t-1}\}$ equal to
$\{r_1,\ldots,r_{t-1}\}$, i.e.,
\[
p = \#\{i_j\ :\ i_j = r_j,\ j = 1,\ldots,t-1\}.
\]
Again we use an induction for $p$ here. For the case of $p = 1$, suppose
$i_1 = r_1$ without loss of generality. We have
\[
y_J(r_1,i_2,\ldots,i_t) = \displaystyle
y_{J\setminus\{1\}}(i_2,\ldots,i_t) -
\sum_{i_1 = 1}^{r_1 - 1}y_J(i_1,i_2,\ldots,i_t).
\]
From the assumption of the induction for $t$, we have
$y_{J\setminus\{1\}}(i_2,\ldots,i_t) = \displaystyle \frac{s}{m_{J
\setminus\{1\}}}$. Also from the assumption of the induction for $p$, we
have
$y_J(i_1,i_2,\ldots,i_t) = \displaystyle\frac{s}{m_J}$ for $i_1 =
1,\ldots,r_1 - 1$. Therefore we have
\[
y_J(r_1,i_2,\ldots,i_t) = \displaystyle \frac{s}{m_{J\setminus\{1\}}} -
(r_1-1)\cdot \frac{s}{m_J} = \frac{s}{m_J}.
\]
The case of $p$ under the assumption that the relation holds for the
case of $p-1$ can be shown similarly.
\hspace*{\fill}$\Box$
\bigskip
From Theorem \ref{prop:orthogonal} and the relation $\Btheta =
X^{-1}\By$, the
constraints to be added to the relation (\ref{eq:prop-theta-mu-eq}) for
the orthogonal designs of strength $t$\ ($t \leq n$) becomes
\[
\Bone_m^TX\Btheta = s,\ C_{\ell}X\Btheta = \Bzero_{v_{\ell}},\ \ \ell =
1,\ldots,t.
\]
This is a generalization of relation for two-level case such as
(\ref{eqn:example-constraints-2-level}).
\begin{example}[Continuation of Example \ref{example:2x2x3-3}]
Consider $\{-1,1\}\times \{-1,1\}\times \{-1,0,1\}$ designs. In addition
to the polynomial equations derived in Example \ref{example:2x2x3-3},
the coefficients of the indicator functions of designs with size $s$
satisfy the relation
\[
12\theta_{000} + 8\theta_{002} = s.
\]
The constraints for the equireplicated designs, i.e.,
orthogonal designs of strength $1$, are
\[\begin{array}{l}
-12\theta_{100} -8\theta_{102} = 0,\\
-12\theta_{010} -8\theta_{012} = 0,\\
-4\theta_{001} + 4\theta_{002} = 0,\\
-8\theta_{001} = 0.
\end{array}
\]
Therefore for a given $s$ that is a common multiple of $\{2,2,3\}$,
i.e., only $s = 6$ is the compatible size of the fractional factorial
designs in this
case, we can enumerate all the equireplicated
designs as the solutions of a system of these polynomial equations.
\hspace*{\fill}$\Box$
\end{example}
\bigskip
Considering Theorem \ref{prop:orthogonal} for the case of $t = n$, we
have the following.
\begin{corollary}
The contrast matrix $C$ is a non-singular $m\times m$ matrix.
\end{corollary}
Now we give another representation of the indicator function reflecting the
orthogonality.
For the indicator function
(\ref{eqn:poly-inter-polatory}), consider a non-singular linear
transformation $\Btheta \mapsto \Bmu = CX\Btheta$. New variables $\Bz$
is also defined by $\Bz =
((CX)^{-1})^T\Bx$,
where
$\Bx = (\Bx^{\Ba})_{\Ba
\in L}$ is an $m\times 1$ column vector of variables, and
$\Bz =
\{z_{J(\tilde{\Bi})}\ :\ J \subset [n], \tilde{\Bi} \in \prod_{j \in
J}[r_j - 1]\}$ is also an $m\times 1$ column vector of variables.
Then we have a representation of the indicator function
for $\Bz$,
\begin{equation}
f(\Bz) = \sum_{J \subset [n], \tilde{\Bi} \in \prod_{j \in
J}[r_j - 1]} \mu_{J(\tilde{\Bi})}z_{J(\tilde{\Bi})}.
\label{eqn:contrast-expression}
\end{equation}
We call (\ref{eqn:contrast-expression})
{\it the contrast representation} of the indicator function.
From the contrast representation, we see the size and the orthogonality of
the designs directly, which is
the advantage of the contrast representation. For example, the constant term
$\mu_{\emptyset}$ is the size of the design, and
\[
\mu_{J(\tilde{\Bi})} = 0\ \ \mbox{for}\ \#J = 1,
\tilde{\Bi} \in \prod_{j \in
J}[r_j - 1]
\]
corresponds to equireplicated designs, and so on.
\begin{example}\label{example:F4}
In Section \ref{sec:intro}, we see the indicator function of $3^{4-2}$
regular fractional factorial design $F_3$ in Figure
\ref{fig:example-F3} is (\ref{eqn:indicator-function-F3}).
The contrast representation of $F_3$ is
\[\begin{array}{rcl}
f(\Bz) & = & 9 + z_{123(111)} + z_{123(112)} -
z_{123(122)} - z_{123(212)} - z_{123(221)} - z_{124(111)}\\
& & {} - z_{124(122)} + z_{124(211)} + z_{124(212)} - z_{124(221)} -
z_{134(111)} + z_{134(121)}\\
& & {} + z_{134(122)} - z_{134(212)} - z_{134(221)}
- z_{234(111)}
- z_{234(122)}
+ z_{234(211)}\\
& & {} + z_{234(212)}
- z_{234(221)}
- z_{1234(1111)}
- z_{1234(2221)}.
\end{array}
\]
From this representation, we see that the size of $F_3$ is $9$, and
$F_3$ is an orthogonal design of strength $2$.
Another example is a
$1/2$ fraction of $2\times 2\times 3$ design $F_4$ displayed in Figure
\ref{fig:2x2x3-half}.
\begin{figure*}[htbp]
\[
\begin{array}{|rrr|}
\multicolumn{3}{l}{F_4}\\
\multicolumn{1}{c}{x_1} & x_2 & \multicolumn{1}{c}{x_3}\\ \hline
-1 &-1 &-1 \\
-1 &-1 & 1 \\
-1 & 1 & 0 \\
1 &-1 & 0 \\
1 &-1 & 1 \\
1 & 1 &-1 \\ \hline
\end{array}
\]
\caption{An example of $1/2$ fraction of $\{-1,1\}^2\times\{-1,0,1\}$ design.}
\label{fig:2x2x3-half}.
\end{figure*}
The indicator function and the contrast representation of $F_4$ are
\begin{equation}
f(x_1,x_2,x_3) = \frac{1}{2} - \frac{1}{2}x_1x_2 - \frac{1}{4}x_2x_3 -
\frac{1}{4}x_1x_2x_3 - \frac{1}{4}x_2x_3^2 + \frac{3}{4}x_1x_2x_3^2
\label{eqn:indicator-function-F4}
\end{equation}
and
\begin{equation}
f(\Bz) = 6 + 2z_{2(1)} + z_{12(11)} - z_{23(12)} + z_{123(111)},
\label{eqn:contrast-representation-F4}
\end{equation}
respectively. From the contrast representation, we see that the size of
$F_4$ is $6$. We also see that $x_1$ and $x_3$ are orthogonal from
\[
\mu_{1(1)} = \mu_{3(1)} = \mu_{3(2)} = \mu_{13(11)} = \mu_{13(12)} = 0.
\]
On the other hand, $\mu_{2(1)} \neq 0$ implies that $F_3$ is not
equireplicated for $x_2$.
\hspace*{\fill}$\Box$
\end{example}
In addition,
the contrast representation does not depend on the level
coding,
whereas the indicator function depends on the level coding. This
is another advantage of the contrast representation.
\begin{proposition}
In the contrast representation (\ref{eqn:contrast-expression}), $\Bmu$
is determined only from the contrast matrix $C$.
It does not depend on the model matrix $X$, especially on the level-coding.
\end{proposition}
\paragraph*{Proof.}\ From $\Btheta = X^{-1}\By$, we have $\Bmu =
CX\Btheta = C\By$.\hspace*{\fill}$\Box$
\bigskip
In other words, the influence of the level-coding on the contrast
representation is involved in the variables $\Bz$. For the same contrast
matrix $C$ and the response $\By \in \{0,1\}^m$ given by
(\ref{eqn:y-subset-design}), the contrast representation of the design
$F = \{\Bd_i\in D\ :\ \Bi \in {\cal I}'\}$ has the same coefficient vector
$\Bmu = C\By$ regardless of the level-coding. On the other hand, the
variable $\Bz$ depends on the level-coding and
is defined by $\Bz = ((CX)^{-1})^{T}\Bx$.
\bigskip
Solving a system of polynomial equations for the coefficients of
the indicator function or the contrast
representation by computational algebraic softwares,
we can obtain the complete list of fractional factorial designs with
given orthogonality in theory.
It is true that the computational feasibility is an important issue,
which we see in Section 4. Another important point arises in classifying
the solutions to the
equivalence classes for permutations of levels or factors.
For two-level cases, as we see in
\cite{Fontana-Pistone-Rogantin-2000}, the
equivalence classes
for permutations of levels and factors are simply obtained by
sign changes or permutation of indices
for the coefficients of the indicator functions.
To consider multi-level cases, we give the description of
the equivalence classes as follows. Suppose $S_{{\cal I}}$ is a
group of permutations of ${\cal I}$, and $G \subset S_{{\cal I}}$ is a
group we consider, i.e., a group of permutations of levels for each factor and
permutations of factors if possible. For each $g \in
G$, let $P_g$ be an $m \times m$ permutation matrix.
Then we have the following.
\begin{proposition}\label{prop:invariance-equivalence-class}
Let $G \subset S_{{\cal I}}$ is a group. Then
the equivalence classes for $\Btheta$ and $\Bmu$ with respect to $G$
are
\[
[\Btheta] = \{X^{-1}P_gX\Btheta\ :\ g \in G\}
\]
and
\[
[\Bmu] = \{CP_gC^{-1}\Bmu\ :\ g \in G\},
\]
respectively.
\end{proposition}
\paragraph*{Proof.}\ Let $\tilde{\By} = P_g\By$. Let the corresponding
indicator functions be $f(\Bx) = \Btheta^{T}\Bx$ and $\tilde{f}(\Bx) =
\tilde{\Btheta}^{T}\Bx$, where $\Btheta = X^{-1}\By$ and
$\tilde{\Btheta} = X^{-1}\tilde{\By}$, respectively.
Then we have the relation
\[
\tilde{\Btheta} = X^{-1}\tilde{\By} = X^{-1}P_g\By = X^{-1}P_gX\Btheta.
\]
Similarly, for the constant representations
$f(\Bz) = \Bmu^{T}\Bz$ and $\tilde{f}(\Bz) = \tilde{\Bmu}^{T}\Bz$ where
$\Bmu = C\By$ and $\tilde{\Bmu} = C\tilde{\By}$, respectively, we have
\[
\tilde{\mu} = C\tilde{\By} = CP_g\By = CP_gC^{-1}\Bmu.
\]
\hspace*{\fill}$\Box$
\bigskip
Proposition \ref{prop:invariance-equivalence-class} shows that neither the
indicator function nor the contrast representation has the
invariance property for multi-level cases. We will see it in the
computations in Section
\ref{sec:computation}.
\section{Classifications of orthogonal $2^3\times 3$ and $2^4\times 3$ designs}
\label{sec:computation}
In this section, we consider $2^3 \times 3$ and $2^4\times 3$
designs. Using a computational algebraic software, we solve systems of the
polynomial equations and derive a classification of designs with given
characteristic. All the computations are done by Macaulay2
(\cite{Macaulay2}) installed in a virtual machine (vmware)
on a laptop with $2.80$ GHz CPU and $8$ GB memory. The memory allocated to
the virtual machine is $512$ MB.
\subsection{Full enumeration of the orthogonal fractions of the $2^3\times 3$
designs of strength $2$}
First we consider the orthogonal fractions of the $2^3\times 3$ designs
of strength $2$. Corresponding system of algebraic equations includes a
set of $m = 2^3 \times 3 = 24$ general relations, $1$ relation for the size,
$5$ relations for the balance for each factor and $9$ relations
for the orthogonality of strength $2$, for $25$ variables. Note that
there are $m + 1$ variables, where $+1$ corresponds to the variable for
the size $s$. To obtain a compatible size $s$, first we calculate the
Gr\"obner basis of the ideal $I$ generated by the $39$ polynomials
corresponding to the above $39$ relations
for the elimination ordering where the variable $s$ is the lowest.
For the level-coding $\{-1,1\}^3\times \{-1,0,1\}$,
the Gr\"obner basis is calculated within $0.1$ seconds, and the
elimination ideal is
\[
I \cap \mathbb{Q}[s] = \left<\ s^3 - 36s^2 + 288s\ \right>
= \left<\ s(s - 12)(s - 24)\ \right>,
\]
i.e., only the size $s = 12$ is compatible. This result is also obvious
because the size of the orthogonal designs must be the multiple of
$2\times 2$ and $2\times 3$.
Note that the Gr\"onber basis calculations heavily depend on the
level-coding. To
see this, the author also try the same computation under the level-coding
$\{0,1\}^3\times \{0,1,2\}$, and find that the computation does not
finish in one week.
Now we fix $s = 12$ and calculate all the solutions. We find that
there are $44$ solutions, classified into $3$ equivalence classes as
follows.
\begin{itemize}
\item Type (a): $2$ relations.
The indicator function and the contrast
representation of the representative fraction displayed in Figure
\ref{fig:res-2x2x2x3}(a) are
\[
\frac{1}{2} + \frac{1}{2}x_1x_2x_3
\]
and
\[
12 - 3z_{123(111)},
\]
respectively. This is a class of the regular fractional factorial
designs with the defining relation $x_1x_2x_3 = 1$.
\item Type (b): $6$ relations. The indicator function and the contrast
representation of the representative fraction displayed in Figure
\ref{fig:res-2x2x2x3}(b) are
\[
\frac{1}{2} + \frac{1}{2}x_1x_2x_3 - \frac{1}{2}x_1x_2x_3x_4 -
\frac{1}{2}x_1x_2x_3x_4^2
\]
and
\[
12 - z_{123(111)} - z_{1234(1112)},
\]
respectively, each with $4$ relations. The indicator function and
the contrast representation of another fraction in the same
equivalence class are
\[
\frac{1}{2} - \frac{1}{2}x_1x_2x_3 + x_1x_2x_3x_4^2
\]
and
\[
12 - z_{123(111)} - z_{1234(1111)},
\]
respectively, each with $2$ relations.
\item Type (c): $36$ relations. The indicator function and the contrast
representation of the representative fraction displayed in Figure
\ref{fig:res-2x2x2x3}(c) are
\[
\frac{1}{2} + \frac{1}{2}x_1x_2x_3 - \frac{1}{2}x_1x_3x_4 -
\frac{1}{2}x_1x_2x_3x_4^2
\]
and
\[
12 - z_{123(111)} + z_{134(111)} + 2z_{134(112)} + z_{1234(1111)} +
z_{1234(1112)},
\]
respectively, each with $12$ relations. The indicator function and
the contrast representation of another fraction in the same
equivalence class are
\[
\frac{1}{2} - \frac{1}{2}x_1x_2 - \frac{1}{4}x_1x_2x_4 +
\frac{1}{4}x_1x_2x_3x_4 + \frac{3}{4}x_1x_2x_4^2 +
\frac{1}{4}x_1x_2x_3x_4^2
\]
and
\[
12 - z_{123(111)} + 2z_{124(111)} + z_{124(112)} + z_{1234(1111)} +
z_{1234(1112)},
\]
respectively, each with $24$ relations.
\end{itemize}
\begin{figure*}[htbp]
\[
\begin{array}{|rrrr|}
\multicolumn{4}{l}{{\rm Type (a)}}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & \multicolumn{1}{c}{x_4}\\ \hline
1 & 1 & 1 &-1\\
1 & 1 & 1 & 0\\
1 & 1 & 1 & 1\\
1 &-1 &-1 &-1\\
1 &-1 &-1 & 0\\
1 &-1 &-1 & 1\\
-1 & 1 &-1 &-1\\
-1 & 1 &-1 & 0\\
-1 & 1 &-1 & 1\\
-1 &-1 & 1 &-1\\
-1 &-1 & 1 & 0\\
-1 &-1 & 1 & 1\\ \hline
\end{array}
\hspace*{10mm}
\begin{array}{|rrrr|}
\multicolumn{4}{l}{{\rm Type (b)}}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & \multicolumn{1}{c}{x_4}\\ \hline
1 & 1 & 1 &-1\\
1 & 1 & 1 & 0\\
1 & 1 &-1 & 1\\
1 &-1 & 1 & 1\\
1 &-1 &-1 & 0\\
1 &-1 &-1 &-1\\
-1 & 1 & 1 & 1\\
-1 & 1 &-1 & 0\\
-1 & 1 &-1 &-1\\
-1 &-1 & 1 &-1\\
-1 &-1 & 1 & 0\\
-1 &-1 &-1 & 1\\ \hline
\end{array}
\hspace*{10mm}
\begin{array}{|rrrr|}
\multicolumn{4}{l}{{\rm Type (c)}}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & \multicolumn{1}{c}{x_4}\\ \hline
1 & 1 & 1 &-1\\
1 & 1 & 1 & 0\\
1 & 1 &-1 & 1\\
1 &-1 & 1 &-1\\
1 &-1 &-1 & 0\\
1 &-1 &-1 & 1\\
-1 & 1 & 1 & 1\\
-1 & 1 &-1 & 0\\
-1 & 1 &-1 &-1\\
-1 &-1 & 1 & 1\\
-1 &-1 & 1 & 0\\
-1 &-1 &-1 &-1\\ \hline
\end{array}
\]
\caption{Orthogonal fractions of the $2^3\times 3$ designs of strength $2$}
\label{fig:res-2x2x2x3}
\end{figure*}
In the above list, Type (a) is the class of the regular fractions,
whereas Type (b) and Type (c) are classes of the non-regular
fractions.
Note that Type (b) and Type (c) differ only in the last columns (the
levels of $x_4$) in Figure \ref{fig:res-2x2x2x3}. For each row where the
levels of $(x_1,x_2,x_3)$ is unique, there are $4$ such rows, the levels
of $x_4$ are fixed ($x_4 = 1$) in Type (b), whereas the levels of $x_4$
are $1$ or $-1$ in Type (c).
Type (b) and (c) can also be characterized considering
the designs
obtained from a traditional $OA(12, 3^1 2^4)$ orthogonal array as
follows. The orthogonal array $OA(12, 3^1 2^4)$ in Appendix 8C of
\cite{Wu-Hamada-2009} is displayed in Figure \ref{OA12}.
\begin{figure*}
\[
\begin{array}{ccccc}\hline
1 & 2 & 3 & 4 & 5\\ \hline
0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 1\\
0 & 1 & 0 & 1 & 1\\
0 & 1 & 1 & 1 & 0\\
1 & 0 & 0 & 1 & 1\\
1 & 0 & 1 & 1 & 0\\
1 & 1 & 0 & 0 & 1\\
1 & 1 & 1 & 0 & 0\\
2 & 0 & 0 & 1 & 0\\
2 & 0 & 1 & 0 & 1\\
2 & 1 & 0 & 0 & 0\\
2 & 1 & 1 & 1 & 1\\ \hline
\end{array}
\]
\caption{Orthogonal array $OA(12, 3^1 2^4)$ (Appendix 8C of
\cite{Wu-Hamada-2009}).}
\label{OA12}
\end{figure*}
From $OA(12, 3^12^4)$, we can obtain $1/2$ fractions of $2^3\times 3$
designs by selecting $3$ columns from the columns $\{2,3,4,5\}$.
We see that all the designs
constructed in this way are included in the equivalence class of Type
(c). Therefore Type (c) is regarded as the class of $OA(12, 3^1 2^4)$ designs.
\subsection{Full enumeration of the orthogonal fractions of the $2^4\times 3$
designs of strength $3$}
Next we consider the fractions of the $2^4\times 3$ designs.
For this case, enumeration of the orthogonal fractions of strength $2$
may be difficult to compute for standard PC. In fact, the Gr\"obner
basis of the elimination ideal for the compatible size does
not obtained after $1$ week calculation under the level-coding
$\{-1,1\}^4\times \{-1,0,1\}$.
Therefore we enumerate the
orthogonal fractions of strength $3$ instead. Note that, for fixed size $s$,
there are $m = 48$ variables with constraints $1 + 6 + 14 = 21$
relations for strength $2$, and with constraints $1 + 6 + 14 + 16 = 37$
relations for strength $3$. Therefore, by eliminating variables, the
number of variables reduces
$11$ for strength $3$, whereas to $27$ for strength $2$. The compatible
size must be $s = 24$ for strength $3$, that is only the multiple of $2\times
2\times 2$ and $2\times 2\times 3$ less than $m = 48$. This fact is also
checked by the Gr\"obner basis calculation. After calculation within
$0.1$ seconds, we see that the elimination ideal is
\[
I\cap \mathbb{Q}[s] = \left<\ s^3 - 72s^2 + 1152s\ \right>
= \left<\ s(s - 24)(s - 48)\ \right>.
\]
Therefore we fix $s = 24$ and calculate all the solutions. We find there
are $56$
solutions, classified into $3$ equivalence classes as follows.
\begin{itemize}
\item Type (a): 2 relations. The indicator function and the contrast
representation of the representative fraction displayed in Figure
\ref{fig:res-2x2x2x2x3}(a) are
\[
\frac{1}{2} + \frac{1}{2}x_1x_2x_3x_4
\]
and
\[
24 + 3z_{1234(1111)},
\]
respectively. This is a class of the regular fractional factorial
designs with the defining relation $x_1x_2x_3x_4 = 1$.
\item Type (b): 6 relations.
The indicator function and the contrast
representation of the representative fraction displayed in Figure
\ref{fig:res-2x2x2x2x3}(b) are
\[
\frac{1}{2} - \frac{1}{2}x_1x_2x_3x_4 - \frac{1}{2}x_1x_2x_3x_4x_5 +
\frac{1}{2}x_1x_2x_3x_4x_5^2
\]
and
\[
24 - z_{1234(1111)} + z_{12345(11111)} + z_{12345(11112)},
\]
respectively, each with 4 relations.
The indicator function and the
contrast representation of another fraction in the same
equivalence class are
\[
\frac{1}{2} + \frac{1}{2}x_1x_2x_3x_4 - x_1x_2x_3x_4x_5^2
\]
and
\[
24 - z_{1234(1111)} - z_{12345(11111)},
\]
respectively, each with 2 relations.
\item Type (c): 48 relations. The indicator function and the contrast
representation of the representative fraction displayed in Figure
\ref{fig:res-2x2x2x2x3}(c) are
\[
\frac{1}{2} - \frac{1}{2}x_1x_2x_3x_4 - \frac{1}{2}x_1x_2x_4x_5 +
\frac{1}{2}x_1x_2x_3x_4x_5^2
\]
and
\[
24 - z_{1234(1111)} - z_{1245(1111)} - 2z_{1245(1112)} - z_{12345(11112)},
\]
respectively, each with 16 relations.
The indicator function and the
contrast representation of another fraction in the same
equivalence class are
\[
\frac{1}{2} + \frac{1}{2}x_1x_2x_3 + \frac{1}{4}x_1x_2x_3x_5 +
\frac{1}{4}x_1x_2x_3x_4x_5 - \frac{3}{4}x_1x_2x_3x_5^2 +
\frac{1}{4}x_1x_2x_3x_4x_5^2
\]
and
\[
24 + z_{1234(1111)} + 2z_{1235(1111)} + z_{1235(1112)} + z_{12345(11111)},
\]
respectively, each with 32 relations.
\end{itemize}
\begin{figure*}[htbp]
\[
\begin{array}{|rrrrr|}
\multicolumn{5}{l}{{\rm Type (a)}}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & x_4 & \multicolumn{1}{c}{x_5}\\ \hline
-1 & -1 & -1 & -1 & -1\\
-1 & -1 & -1 & -1 & 0\\
-1 & -1 & -1 & -1 & 1\\
-1 & -1 & 1 & 1 & -1\\
-1 & -1 & 1 & 1 & 0\\
-1 & -1 & 1 & 1 & 1\\
-1 & 1 & -1 & 1 & -1\\
-1 & 1 & -1 & 1 & 0\\
-1 & 1 & -1 & 1 & 1\\
-1 & 1 & 1 & -1 & -1\\
-1 & 1 & 1 & -1 & 0\\
-1 & 1 & 1 & -1 & 1\\
1 & -1 & -1 & 1 & -1\\
1 & -1 & -1 & 1 & 0\\
1 & -1 & -1 & 1 & 1\\
1 & -1 & 1 & -1 & -1\\
1 & -1 & 1 & -1 & 0\\
1 & -1 & 1 & -1 & 1\\
1 & 1 & -1 & -1 & -1\\
1 & 1 & -1 & -1 & 0\\
1 & 1 & -1 & -1 & 1\\
1 & 1 & 1 & 1 & -1\\
1 & 1 & 1 & 1 & 0\\
1 & 1 & 1 & 1 & 1\\ \hline
\end{array}
\hspace*{10mm}
\begin{array}{|rrrrr|}
\multicolumn{5}{l}{{\rm Type (b)}}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & x_4 & \multicolumn{1}{c}{x_5}\\ \hline
-1 & -1 & -1 & -1 & -1\\
-1 & -1 & -1 & 1 & 0\\
-1 & -1 & -1 & 1 & 1\\
-1 & -1 & 1 & -1 & 1\\
-1 & -1 & 1 & -1 & 0\\
-1 & -1 & 1 & 1 & -1\\
-1 & 1 & -1 & -1 & 1\\
-1 & 1 & -1 & -1 & 0\\
-1 & 1 & -1 & 1 & -1\\
-1 & 1 & 1 & -1 & -1\\
-1 & 1 & 1 & 1 & 0\\
-1 & 1 & 1 & 1 & 1\\
1 & -1 & -1 & -1 & 1\\
1 & -1 & -1 & -1 & 0\\
1 & -1 & -1 & 1 & -1\\
1 & -1 & 1 & -1 & -1\\
1 & -1 & 1 & 1 & 0\\
1 & -1 & 1 & 1 & 1\\
1 & 1 & -1 & -1 & -1\\
1 & 1 & -1 & 1 & 0\\
1 & 1 & -1 & 1 & 1\\
1 & 1 & 1 & -1 & 1\\
1 & 1 & 1 & -1 & 0\\
1 & 1 & 1 & 1 & -1\\ \hline
\end{array}
\hspace*{10mm}
\begin{array}{|rrrrr|}
\multicolumn{5}{l}{{\rm Type (c)}}\\
\multicolumn{1}{c}{x_1} & x_2 & x_3 & x_4 & \multicolumn{1}{c}{x_5}\\ \hline
-1 & -1 & -1 & -1 & 1\\
-1 & -1 & -1 & 1 & 0\\
-1 & -1 & -1 & 1 & -1\\
-1 & -1 & 1 & -1 & 1\\
-1 & -1 & 1 & -1 & 0\\
-1 & -1 & 1 & 1 & -1\\
-1 & 1 & -1 & -1 & -1\\
-1 & 1 & -1 & -1 & 0\\
-1 & 1 & -1 & 1 & 1\\
-1 & 1 & 1 & -1 & -1\\
-1 & 1 & 1 & 1 & 0\\
-1 & 1 & 1 & 1 & 1\\
1 & -1 & -1 & -1 & -1\\
1 & -1 & -1 & -1 & 0\\
1 & -1 & -1 & 1 & 1\\
1 & -1 & 1 & -1 & -1\\
1 & -1 & 1 & 1 & 0\\
1 & -1 & 1 & 1 & 1\\
1 & 1 & -1 & -1 & 1\\
1 & 1 & -1 & 1 & 0\\
1 & 1 & -1 & 1 & -1\\
1 & 1 & 1 & -1 & 1\\
1 & 1 & 1 & -1 & 0\\
1 & 1 & 1 & 1 & -1\\ \hline
\end{array}
\]
\caption{Orthogonal fractions of the $2^4\times 3$ designs of strength $3$}
\label{fig:res-2x2x2x2x3}
\end{figure*}
An interpretation of this list is similar to the $2^3\times 3$ case. In
Figure \ref{fig:res-2x2x2x2x3}, Type (b) and Type (c) differ only in the
last column (the levels of $x_5$). For each row where the levels of
$(x_1,x_2,x_3,x_4)$ is unique, there are $8$ such rows, the levels of
$x_5$ are fixed ($x_5 = -1$) in Type (b), whereas the levels of $x_5$
are $1$ or $-1$ in Type (c).
\section{Discussion}
In this paper, we give how to construct a system of polynomial equations
for the coefficients of the indicator functions of multi-level
fractional factorial designs with given orthogonality. We also define
the contrast representation of the indicator function, which reflects
the orthogonality of the design directly.
The contrast representation has an advantage that it does not depends on
the level-coding.
Using these results, we show
the classifications of the orthogonal fractions of the $2^3\times 3$
designs with strength $2$
and $2^4\times 3$ designs with strength $3$.
In theory, we can obtain classifications of fractional factorial
designs for any size by our method. However, the computational
feasibility depends on the size of problems. For the class of
$2^m \times 3$ designs, we see that the $2^4\times 3$ problem for
orthogonality with strength $3$ is easy to calculate. However, a
$2^5\times 3$ problem of strength $3$ orthogonality seems very difficult
to compute. As for the class of $2^m \times 3^2$ designs, we
find that the
$2^3 \times 3^2$ problem of strength $2$ orthogonality is hard to
compute, i.e., the Gr\"obner basis calculation for the elimination ideal
does not finish in $1$ week. In addition, it is obvious that there is
no orthogonal fractions $2^3 \times 3^2$ with strength $3$ because the
size must be a multiple of $2\times 2\times 2$ and $2\times 3\times 3$.
As a consequence, we only have limited
computational results in this paper.
In particular, the merit of the contrast representation must be
investigated from the computational aspects. It seems that
a system of polynomial equations for $\Bmu$ is easy to solve than that for
$\Btheta$. Note that
the polynomial relations for $\Bmu$ are obtained by substituting
$\Btheta = (CX)^{-1}\Bmu$ to the polynomial relations for
$\Btheta$. Therefore, translating the relations for $\Btheta$ to the
relations for $\Bmu$ corresponds to the matrix operations of inverse in
advance. Unfortunately, for the problems considered in this paper, the
systems of
the polynomial equations for $\Btheta$ and $\Bmu$ are both quite easy or
quite difficult, and the effectiveness of the contrast representation
from the computational aspect is not shown.
Therefore the quantitative evaluation of the effect of this
transformation is one of the open problems.
It is also an open problem to compare our method to the brute-force search.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,923 |
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| {
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