text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: SQL Server : how to get 2 different query results with 1 single query using case? The table I'm working on is a Employee log-in database. Amongst other things, it will be able to identify the time of entrance from an employee and to make a report on how his entrances have been during a set of dates.
The problem I am facing right now is: when the boss wants to see how a worker's log-in and log out work during a specific date. He wants to:
a) set a date and a name, and for the query to return the accesses from that employee on that specific date.
b) IF he inputs no name, then it will return a list with everyone for that date.
No | Name | ID | Date | Log-in | Log-out | Description
---+------+-----+------------+--------+---------+---------------------
1 | Joe1 | j1 | 2018/10/02 | 09:00 | 18:00 | Normal Exit
2 | Joe2 | j2 | 2018/10/02 | 10:00 | 17:00 | Meeting
.. | ... | . | . . . | . . | . . . | . . . .
N | JoeN | jN | yyyy/mm/dd | hh/mm | hh/mm | Whatever text
So for the first part I went with a simple
select *
from DB
where @name = '*the name of the person to look for*'
and @date = '*specific date*' (is a date variable yyyy-mm-dd)
all done. Now the next part seems weird and have not been able to figure it out yet.
I was trying with an IIF scenario as a subquery but doesn't do the part where it returns the complete list.
Next is the where case but cant make it work either. I am missing a variable for the program to check:
if there is no name to look for => show table with everyone for that day
if there is a name to look for => show table ONLY for that person
How can I make it so my program discriminates which table to show at the end?
Cheers!
EDIT:
ADDED (thanks for the recomendation) query for better assistance
CREATE TABLE DailyRegisters
(
No INTEGER NOT NULL PRIMARY KEY,
Name VARCHAR(50) NOT NULL,
ID VARCHAR(50) NOT NULL,
Date DATE NOT NULL,
Login DATETIME NOT NULL,
Logout DATETIME NOT NULL,
Description VARCHAR(50) NOT NULL
);
INSERT INTO DailyRegisters (No, Name, ID, Date, Login, Logout, Description)
VALUES (1,'Person 1','P1','10/2/2018','9:00','18:00','Normal'),
(2,'Person 2','P2','10/2/2018','9:01','18:00','Normal'),
(3,'Person 3','P3','10/2/2018','10:17','16:00','Plant'),
(4,'Person 4','P4','10/2/2018','10:10','18:00','Normal'),
(5,'Person 1','P1','10/3/2018','9:01','18:00','Normal'),
(6,'Person 2','P2','10/3/2018','9:02','18:00','Normal'),
(7,'Person 3','P3','10/3/2018','9:04','18:00','Normal'),
(8,'Person 1','P1','10/4/2018','9:00','18:00','Normal'),
(9,'Person 2','P2','10/4/2018','9:00','14:00','Meeting'),
(10,'Person 3','P3','10/4/2018','9:17','18:00','Normal'),
(11,'Person 4','P4','10/4/2018','9:00','18:00','Normal'),
(12,'Person 1','P1','10/5/2018','9:00','18:00','Normal'),
(13,'Person 3','P3','10/5/2018','9:00','20:00','Plant'),
(14,'Person 4','P4','10/5/2018','9:00','18:00','Normal');
A: You simply need to handle the case wherein no name is specified, like this:
select * from DB
where
((@name is not null AND name = @name) or (@name is null))
AND date = @date;
Basically the above query will not check for name if the @name parameter is null. And of course it goes without saying that @name and @date are parameters passed to the query.
I also suggest not to use select * but instead specify the columns: It is better for maintenance in case your table schema changes.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,357 |
Select a destination on the flight map to show all scheduled flights from Annaba Airport. Find out which airlines operate a specific flight route and what their flight schedule is.
Below you can find a list of alternative airports to Annaba (AAE). The nearest airports to Annaba (AAE) are Constantine (CZL) and Tébessa (TEE).
Annaba Airport (IATA: AAE, ICAO: DABB), also known as Rabah Bitat Airport (Les Salines , is a small airport in Algeria. It is an international airport and serves the area of Annaba, Algeria. Annaba Airport has passenger flights scheduled to 8 destinations in 2 countries.
At present, there are 3 domestic flights from Annaba.
The longest flight from Annaba AAE is a 1,162 mile (1,870 kilometers) non stop route to Istanbul IST. This direct flight takes around 2 hours and 45 minutes and is operated by Air Algerie. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,448 |
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<tr><td><div align="right"><a name="1"></a><a href="#1">1</a></div></td><td class="codeLine"><?php</td></tr>
<tr><td><div align="right"><a name="2"></a><a href="#2">2</a></div></td><td class="codeLine">namespace HealthCareAbroad\HelperBundle\Form;</td></tr>
<tr><td><div align="right"><a name="3"></a><a href="#3">3</a></div></td><td class="codeLine"></td></tr>
<tr><td><div align="right"><a name="4"></a><a href="#4">4</a></div></td><td class="codeLine">use Symfony\Component\Validator\Constraints\NotBlank;</td></tr>
<tr><td><div align="right"><a name="5"></a><a href="#5">5</a></div></td><td class="codeLine"></td></tr>
<tr><td><div align="right"><a name="6"></a><a href="#6">6</a></div></td><td class="codeLine">use HealthCareAbroad\AdminBundle\Entity\OfferedService;</td></tr>
<tr><td><div align="right"><a name="7"></a><a href="#7">7</a></div></td><td class="codeLine">use Symfony\Component\Form\AbstractType;</td></tr>
<tr><td><div align="right"><a name="8"></a><a href="#8">8</a></div></td><td class="codeLine">use Symfony\Component\Form\FormBuilderInterface;</td></tr>
<tr><td><div align="right"><a name="9"></a><a href="#9">9</a></div></td><td class="codeLine">use Symfony\Component\OptionsResolver\OptionsResolverInterface;</td></tr>
<tr><td><div align="right"><a name="10"></a><a href="#10">10</a></div></td><td class="codeLine"></td></tr>
<tr><td><div align="right"><a name="11"></a><a href="#11">11</a></div></td><td class="codeLine">class OfferedServiceFormType extends AbstractType</td></tr>
<tr><td><div align="right"><a name="12"></a><a href="#12">12</a></div></td><td class="codeLine">{</td></tr>
<tr class="danger"><td><div align="right"><a name="13"></a><a href="#13">13</a></div></td><td class="codeLine"> public function buildForm(FormBuilderInterface $builder, array $options)</td></tr>
<tr class="danger"><td><div align="right"><a name="14"></a><a href="#14">14</a></div></td><td class="codeLine"> {</td></tr>
<tr class="danger"><td><div align="right"><a name="15"></a><a href="#15">15</a></div></td><td class="codeLine"> $status = array(OfferedService::STATUS_ACTIVE => 'active', OfferedService::STATUS_INACTIVE => 'inactive');</td></tr>
<tr><td><div align="right"><a name="16"></a><a href="#16">16</a></div></td><td class="codeLine"></td></tr>
<tr class="danger"><td><div align="right"><a name="17"></a><a href="#17">17</a></div></td><td class="codeLine"> $builder->add('name', 'text', array('constraints'=>array(new NotBlank())));</td></tr>
<tr class="danger"><td><div align="right"><a name="18"></a><a href="#18">18</a></div></td><td class="codeLine"> $builder->add('status', 'choice', array('choices'=>$status));</td></tr>
<tr class="danger"><td><div align="right"><a name="19"></a><a href="#19">19</a></div></td><td class="codeLine"> }</td></tr>
<tr><td><div align="right"><a name="20"></a><a href="#20">20</a></div></td><td class="codeLine"></td></tr>
<tr class="danger"><td><div align="right"><a name="21"></a><a href="#21">21</a></div></td><td class="codeLine"> public function setDefaultOptions(OptionsResolverInterface $resolver)</td></tr>
<tr class="danger"><td><div align="right"><a name="22"></a><a href="#22">22</a></div></td><td class="codeLine"> {</td></tr>
<tr class="danger"><td><div align="right"><a name="23"></a><a href="#23">23</a></div></td><td class="codeLine"> $resolver->setDefaults(array(</td></tr>
<tr class="danger"><td><div align="right"><a name="24"></a><a href="#24">24</a></div></td><td class="codeLine"> 'data_class' => 'HealthCareAbroad\AdminBundle\Entity\OfferedService',</td></tr>
<tr class="danger"><td><div align="right"><a name="25"></a><a href="#25">25</a></div></td><td class="codeLine"> ));</td></tr>
<tr class="danger"><td><div align="right"><a name="26"></a><a href="#26">26</a></div></td><td class="codeLine"> }</td></tr>
<tr><td><div align="right"><a name="27"></a><a href="#27">27</a></div></td><td class="codeLine"></td></tr>
<tr class="danger"><td><div align="right"><a name="28"></a><a href="#28">28</a></div></td><td class="codeLine"> public function getName()</td></tr>
<tr class="danger"><td><div align="right"><a name="29"></a><a href="#29">29</a></div></td><td class="codeLine"> {</td></tr>
<tr class="danger"><td><div align="right"><a name="30"></a><a href="#30">30</a></div></td><td class="codeLine"> return 'offeredService';</td></tr>
<tr class="danger"><td><div align="right"><a name="31"></a><a href="#31">31</a></div></td><td class="codeLine"> }</td></tr>
<tr><td><div align="right"><a name="32"></a><a href="#32">32</a></div></td><td class="codeLine">}</td></tr>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,253 |
Who That Divines
A new collection of poetry from the important experimental writer of the "A Tonalist" lyric
Intergenre, Essay, Poetry
Who That Divines quantity
6.5 × .5 × 8 in
Who That Divines comprises short songs and puzzles and longer poems of memoir and history—all of which assert an unconventionally feminist sense of the possibility of locating the divine in language, politics, and daily life. Moriarty's position as one of the most important writers of a postmodern lyric is confirmed by this dynamic collection that also includes the text of her experimental memoir "An Air Force."
Who That Divines opens onto a vastness of inquiry wherein the political, the philosophical, and the autobiographical all commingle. Moriarty, our most adept chaosophist, marshals a "scattering/calibrated/ to assess the meant." Via these deep and decentering investigations, we are led to discover a "power not one of belief but of being."
-ELIZABETH ROBERTSON
paperback, 132 pages, 6 x 8 1/4 in
Moriarty (A Tonalist) calls upon a wide range of poetic styles to explore the connection between agency and the divine in her latest collection. Opening with a Luce Irigaray epigraph—"Divinity is what we need to become free, autonomous, sovereign"—Moriarty looks at the divine broadly, considering in her poems divinity arises from sources and structures of power of all kinds, not just gods, and focusing especially on acts of divination, both in the traditional sense and in more expansive acts of discovery, seeing, and knowing. Brief, tightly-wound lyrics examine the authorial position, asking "how not to go/ Hunted haunted// Divine undaunted/ With you as reader." Many of the poems have dedications or epigraphs, foregrounding a conceit of Moriarty's generative process but, more importantly, marking language that did not originate with the author herself. The book's many threads come together in "In the Air Force," a long autobiographical poem that tracks "1 body" coming of age amid the "Preexisting condition/ Eternall War." Through trenchant observations about knowing and being in lines like "Signs of impending rain/ Exist/ Rain itself/ Exits," Moriarty makes direct inquiries into the most basic structures of human experience, venturing that "All of creation/ Takes part/ In the same nothing."
-Publishers Weekly
So for me A Tonalism is a fiction, but at the same time it exists. I can recognize it. Brent Cunningham and I say to each other (not often anymore, but it happens) that people are A Tonalist. Standard Schaefer and I wrote an (unfinished) novel about it as a sort of cult. In the same way (as a whole other and much more recognizable discussion), the New York School dominates people from New York. They are in it. They write it. They exist through it. They are also apart from it, but it's a part of their DNA in a certain way.
-Andy Finch, Los Angeles Review of Books
Laura Moriarty was born in St. Paul, Minnesota, and grew up in Cape Cod, Massachusetts and Northern California. She attended the University of California at Berkeley. She was the Director …
More about Laura Moriarty
Essay, Poetry
Dear Kathleen
Stephen Motika
Susan Gevirtz
Camp Marmalade
Wayne Koestenbaum
Livingry
Gracie Leavitt
Field Theories | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,812 |
If you haven't done Chimaeron yet, the short version is that 10k health is a big effin deal. It's a huge break point. And Grid's low health status works on a percentage of total health, not an actual number of HP left. Setting "low health" at 10% is a sloppy approximation.
GridStatusChimaeron allows you to set a status for "below 10k health".
Now, you can show the indicator for below 10k health however you want.
You can use an indicator (box) somewhere in the frame.
You can color the health bar hot pink.
You can color the border around the Grid Frame.
If you're like me, you will set it up so that it shows health as a number somewhere on the frame (I used row 2 of center text).
Any of these options are fine. Just make sure that it's set at high enough priority so that it shows when necessary (mine is only superseded by death), and make sure it is noticeable.
Now, GridStatusRaidDebuff does have a sub-10k icon warning built in… but you might not have that plugin. You might want an extra reminder. This is just another option.
How hawt is a sound indicator right in your raid frames? Oh it's hawt. Much better than a list off to the side.
First, you set your sound thresholds. I don't even have the sound indicator showing before 40%. Then I have different colors to let me know when it goes over 60% and over (gulp) 80%.
You can color the health bar (I wouldn't recommend this… too distracting).
If you're like me, you will set it up so that it shows health as a number somewhere on the frame (Again, I used row 2 of center text).
Just make sure that it's set at high enough priority so that it shows when necessary, and that it's shown in a way that is useful to you.
GridStatusRaidDebuff does NOT have any debuffs for sound that I'm aware of.
Also, this addon can also be used for Cho'Gall… which I haven't gotten to yet. I'll let you know how that goes.
Also! If I were healing Chimaeron, I would set my grid to absolute health as an absolute numerical value.
Hrm, but will doing that highlight if a person drops below a certain threshold? I haven't done absolute health in a while, but maybe I should switch.
Hmm, I should see if Vuhdo can do sound levels like that. I do like Vuhdo for Chimaeron too; it's easy to set absolute health to display, and it puts a little tiny skull n crossbones icon on the frame of anyone who is under 10k health.
I am sure it is impossible with the inferior Vuhdo. Come to the Grid side.
I just have Grid show me whenever someone has the "Low Health" debuff, no joke. That'll pop up whenever someone's below 10k health, and I show it as the center icon. However… I am ASTOUNDED and totally in love with the idea of Grid showing me people's sound/corruption!!! I'm sad I've already killed Atramedes for the week!
I do use the "low health" debuff on gridstatusraiddebuff, and that's primarily how I tell the person is under 10k. However, my fellow healer (who is the tank healer) says that it's hard to tell if the MT is under 10k when the MT also has some other raid debuff (break?). This is a decent solution to that problem.
The sound thing is just AWESOME, especially since it's color-coded.
I used Grid for the longest time but new plugins and additional features Grid has now, I think I will reconsider using it again, currently I am using Vudho – its still good and I have no problems with it. but since I am mostly dpsing now as a shadow priest I rarely heal now a days, having any raid frames in my interface works fine.
My Grid over Vuhdo thing is just personal preference. It gets people riled up, which is fun. If you're comfortable with Vuhdo, there's no reason to switch. The two are about the same in functionality, and serious (and good) healers use both. See the poll on World of Matticus.
Zel how big are your raid frames anyway? If I was to put critical information in text on mine it would not be big enough to see… Mine are nice and long (so I can tell the difference between half-health and 2/3 health at a glance) but they are not tall at all, therefore rows of text is like… *squint* WUT?
I display everything I can in terms of colors, icons, and borders. I love that Vuhdo lets me change display of different debuffs too… some color the target's name (usually things I want to know about but can't remove, like poison), some color the whole frame (usually things that are really critical – like Mangle on the tank in Magmaw, or that I can remove – like curses). And then things that are encounter specific are usually icons, like low health on Chimaeron or Red-Beam-of-Death on the trons.
Let me dig up a screenie. (I'm eventually going to post my UI to my "ui" page but I haven't gotten to it yet.
Here's an old one, another old one but it shows the proportions (I am a creature of habit, so even though I've totally redone Grid since then, the frames are roughly the same size). I changed the aspect ratio of my screen using CT viewport, so i'm not obscuring anything in the "black bar" under my feet.
Hm, I think yours are slightly larger than mine, but not a whole lot.I think I could make mine bigger though… my UI was designed for allowing me to run the occasional 25 man (or with tweaking, a 40 man BG) so there's extra room in the raid frames department that I really don't need anymore. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,969 |
package com.justwayward.reader.ui.contract;
import com.justwayward.reader.base.BaseContract;
import com.justwayward.reader.bean.BookSource;
import java.util.List;
/**
* @author lfh.
* @date 2016/8/30.
*/
public interface BookSourceContract {
interface View extends BaseContract.BaseView {
void showBookSource(List<BookSource> list);
}
interface Presenter extends BaseContract.BasePresenter<View> {
void getBookSource(String view, String book);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,418 |
Fort Boyard 2014 is het derde seizoen van de Nederlandse versie van het televisieprogramma Fort Boyard. Het programma werd gestart op maandag 3 november 2014 en werd uitgezonden op Nederland 3. Freek Bartels, die Gerard Ekdom vervangt, en Lauren Verster verzorgden samen de presentatie. Daarnaast zijn er twee Franse dwergmensen "Passe Partout" (André Bouchet) en "Horloge" (Anthony Labourde) die als belangrijkste taak hebben het op verzoek van de presentatoren tonen van de tussenstand. Tevens begeleidden ze de presentatoren en kandidaten naar de juiste plekken in en rondom het fort.
Uitleg
Het programma bestaat uit een spel waarbij teams bestaande uit bekende Nederlanders tegen elkaar strijden door middel van een zoektocht naar sleutels en goudstukken. De kandidaten spelen in teams spellen in en rondom de cellen van het Fort Boyard. Hiermee kunnen zij sleutels verdienen. Als een kandidaat een spel niet binnen de tijd voltooit wint het team geen sleutel. Beide teams spelen twee of drie voorrondespellen waarbij de kandidaat op tijd op het startpunt terug moet zijn, anders volgt er per spel vijftig goudstukken aftrek na afloop van het finalespel. Ieder team speelt ook één voorrondespel waarbij twee sleutels kunnen worden verdiend. Een team dat met de voorrondespellen meer dan zes sleutels heeft verdiend krijgt per extra sleutel honderd goudstukken gratis na het finalespel.
Met behulp van de sleutels kan een finalespel worden gespeeld, dit spel duurt twee minuten en dertig seconden. Tijdens het finalespel dient ieder team zes sleutels te gebruiken om het eigen toegangshek tot een kooi met goudstukken te openen. Indien een team bij de start van het finalespel geen zes sleutels heeft dient aan een kolom gedraaid te worden die bovenaan extra sleutels geeft. Zodra een team zijn totaal op zes sleutels heeft gebracht mag begonnen worden met het verzamelen van goudstukken. Een team dat bij de start van het finalespel al zes of meer sleutels heeft verdiend mag direct met het verzamelen van goudstukken beginnen. Zij dienen wel te wachten tot de toegangspoort voldoende omhoog is gegaan. Hierdoor maakt het nagenoeg geen verschil of je met vijf of zes sleutels aan het finalespel begint.
Wanneer een deelnemer niet binnen de duur van het finalespel teruggekeerd is zit hij zogenaamd opgesloten in de kooi en worden tijgers los gelaten. In werkelijkheid mag deze persoon de kooi echter via een veiligheidsluik verlaten.
Poules
Afleveringen
Poulefase
Ma. 3 nov. 2014
Ma. 10 nov. 2014
Ma. 17 nov. 2014
Ma. 24 nov. 2014
Ma. 1 dec. 2014
Ma. 8 dec. 2014
Finales
Ma. 22 dec. 2014
Externe link
Informatie op AVRO website
Nederlands spelprogramma | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/3647987\/given-lambda-regular-cardinal-left-kappa-lambda-right-lambda-ka","text":"Given $\\lambda$ regular cardinal, $\\left(\\kappa^{<\\lambda}\\right)^{<\\lambda}=\\kappa^{<\\lambda}$?\n\nI'm studying forcing from Kunen's Set Theory (ed. 1983), and I came across this lemma\n\nLemma 6.10. Fn$$(I,J,\\lambda)$$ has the $$\\left(|J|^{<\\lambda}\\right)^+$$-cc.\n\nproof. [...] First assume $$\\lambda$$ regular. Then $$\\left(|J|^{<\\lambda}\\right)^{<\\lambda} = |J|^{<\\lambda}$$ [...]\n\nIn this lemma, we are not assuming GCH and no assumption is made on $$|J|$$. I tried to prove the cardinal arithmetic fact that appears in the proof, but I only succeeded in proving it in specific cases, not in generality (i.e. for all regular cardinals $$\\lambda$$).\n\nIn fact it is trivial in case $$\\lambda$$ is a successor cardinal, since then we have $$|J|^{<\\mu^+}=|J|^\\mu$$. It also follows if we assume that $$\\lambda$$ is limit (hence weakly inaccessible) and $$\\text{cof}\\left(|J|^{<\\lambda}\\right)\\neq \\lambda$$, since then we'd have that the the $$\\lambda$$-sequence $$\\left(|J|^\\kappa\\right)_{\\kappa < \\lambda}$$ cannot be cofinal in $$|J|^{<\\lambda}$$, hence it is eventually constant. But if we were to deal with a weakly inaccessible cardinal $$\\lambda$$ s.t. $$\\text{cof}\\left(|J|^{<\\lambda}\\right) = \\lambda$$, then my attempts fail.\n\nFor what I have seen afterward the problematic case does not appear since mostly we are dealing with successor cardinals or we are assuming some form of CH. But still, I wonder, how it can be proved in the general case?\n\nThanks\n\n\u2022 Use \\left(\\right) when you have some more complex terms, like fractions, powers, etc., so that all can fit inside the parentheses. (: \u2013\u00a0Invisible Apr 28 '20 at 11:06\n\u2022 @ms._VerkhovtsevaKatya Thanks! \u2013\u00a0Lorenzo Apr 28 '20 at 11:07\n\nNote that since $$\\lambda$$ is regular, for any $$\\mu<\\lambda$$, $$f\\colon\\mu\\to\\lambda$$ is bounded.\n\nNow think about $$g\\in\\left(\\kappa^{<\\lambda}\\right)^{<\\lambda}$$ as some $$g\\colon\\mu\\to\\kappa^{<\\lambda}$$. Then there is some $$\\nu<\\lambda$$ such that $$g\\colon\\mu\\to\\kappa^\\nu$$. So we get the wanted result, since clearly $$\\left(\\kappa^{<\\lambda}\\right)^\\mu=\\kappa^{<\\lambda}$$ for any $$\\mu<\\lambda$$.\n\n\u2022 Thank you! Why do you say that $(\\kappa^{<\\lambda})^\\mu = \\kappa^{<\\lambda}$? I would say, in order to finish, that $(\\kappa^\\nu)^{<\\lambda} = \\kappa^{<\\lambda}$... \u2013\u00a0Lorenzo Apr 28 '20 at 14:47\n\u2022 To quote Shelah, \"think!\" \u2013\u00a0Asaf Karagila Apr 28 '20 at 14:47\n\u2022 Let me give you a hint, $\\mu<\\lambda$, and $\\kappa^{<\\lambda}=\\sup\\{\\kappa^\\nu\\mid\\nu<\\lambda\\}$. \u2013\u00a0Asaf Karagila Apr 28 '20 at 16:06\n\u2022 Ok, so I'd like to say that $(\\sup\\{\\kappa^\\nu \\mid \\nu < \\lambda\\})^\\mu = \\sup\\{\\kappa^{\\nu \\ \\mu} \\mid \\nu < \\lambda\\} = \\sup\\{\\kappa^\\nu \\mid \\nu < \\lambda\\}$, right? So $\\sup\\{\\kappa^\\nu \\mid \\nu < \\lambda\\}$ should be a continuity point of $\\alpha \\mapsto \\alpha^\\mu$. The last question I have posted regards the continuity of such function, and what I got from the responses is that it is not obvious to check whether a certain cardinal is a continuity point or not. Probably I'm missing something trivial.. \u2013\u00a0Lorenzo Apr 28 '20 at 16:54\n\u2022 Yes, you can assume that $\\nu\\geq\\mu$. What's $\\nu\\mu$? \u2013\u00a0Asaf Karagila Apr 28 '20 at 17:33\n\nI want to add some details to Asaf's aswer and slightly modify its final argument:\n\nSuppose $$\\lambda$$ weakly inacessible and $$\\text{cof}(k^{<\\lambda})=\\lambda$$ (the other cases are dealt in the body of the question), then, if we have $$g \\in \\left(k^{<\\lambda}\\right)^{<\\lambda}$$ with $$g:\\mu \\longrightarrow k^{<\\lambda}$$, $$g$$ must be bounded in $$k^{<\\lambda}$$ (because of its cofinality),\nhence $$\\exists \\nu < \\lambda$$ s.t. $$g: \\mu \\longrightarrow k^\\nu$$\n\nSo we have $$\\left(k^{<\\lambda}\\right)^{<\\lambda} = \\left|\\bigcup_{\\mu,\\nu<\\lambda}\\left(\\kappa^\\nu\\right)^\\mu\\right| = \\kappa^{<\\lambda}$$","date":"2021-04-20 17:49:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 33, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9651957750320435, \"perplexity\": 270.5916887847233}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039476006.77\/warc\/CC-MAIN-20210420152755-20210420182755-00481.warc.gz\"}"} | null | null |
University of Cambridge debuts virtual talking head capable of expressing human emotions
By Chris Wood
University of Cambridge debuts...
Meet Zoe - a virtual talking head capable of expressing human emotions
The team recorded thousands of sentences from actress Zoe Lister
The University of Cambridge in the United Kingdom has unveiled a virtual "talking head" that is capable of expressing a range of human emotions. The lifelike face called Zoe, which the team believes is the most expressive controllable avatar ever created, could in the future be used as a digital personal assistant.
The voice controlled virtual assistant has been a staple of science fiction for decades, from Star Trek and Red Dwarf to Iron Man, it's up there with faster-than-light travel as one of the most mythologized technologies going. Though recent technologies like Apple's Siri and Samsung's S Voice have brought us a little closer to that dream by allowing us to have (fairly) life-like conversations with our smartphones, there's still a palpable sense that we're just talking to lifeless machines.
The University of Cambridge has been working to tackle this exact problem, and while the team's "Zoe" digital talking head may not be completely convincing, it's certainly a step in the right direction.
The team has developed a virtual, controllable avatar that is capable of expressing a range of emotions with what the team believes to be "unprecedented realism." It works by a user entering a line of text and selecting from a range of sliders that determine emotion. Hit the enter key and Zoe will read the message in however much of a happy, angry or sad tone you desire.
To create the system, the team spent days recording the face and voice of actress Zoe Lister while she recited more than 7000 lines of text in varying emotions. This data was then used to create six basic emotions for Zoe – happiness, sadness, fear, anger, tenderness and neutrality, as well as changeable pitch, speed and depth settings. Combinations of these levels allow for a huge range of emotions, something that has not been possible in other avatars of this type.
The resulting system was tested on volunteers who were able to recognize the emotion being conveyed with an impressive 77 percent success rate. Bizarrely, this was actually higher than the 73 percent recognition rate when the volunteers were shown the real-life Zoe Lister.
The program itself is impressively data light, coming in at just ten megabytes in size. This means that we could well see the technology in future mobile devices such as smartphones and tablets. The template surrounding the technology also has the potential to allow for people to upload their own faces and voices in just a few seconds, rather than the days it took the team to create Zoe.
"It took us days to create Zoe, because we had to start from scratch and teach the system to understand language and expression," Professor Roberto Cipolla from the University's Department of Engineering said. "Now that it already understands those things, it shouldn't be too hard to transfer the same blueprint to a different voice and face."
The team is working to improve the realism of the technology while exploring real world applications, such as sending friends a digital "face message" that conveys the emotion you're feeling, or helping autistic and deaf children to understand emotions and learn to lip-read.
The impressive (and occasionally terrifying) Zoe can be seen in action below.
Source: University of Cambridge
Face of the future rears its head
ScienceComputer Human InterfaceEmotionsVoiceVirtualCambridge UniversityResearch
Chris specializes in science and consumer technology for New Atlas, but also likes to dabble in the latest gaming gadgets. He has a degree in Politics and Ancient History from the University of Exeter, and lives in Gloucestershire, UK. In his spare time you might find him playing music, following a variety of sports or binge watching Game of Thrones.
Marcus Carr March 19, 2013 09:44 PM
Interesting, but getting information out of a computer is the easy side of the equation. The real gains are in getting it to understand our face and words, not the other way around.
Ong Chia Hooi March 20, 2013 06:40 AM
remind me of Zordon..
Rafael Kireyev March 20, 2013 07:20 AM
Wow! Every day more and more of a sci-fi becomes reality.
ringo the Baptist March 20, 2013 08:19 AM
For me the head movement is the next big bump to smooth over.
I find it too heavily mechanically derived from every individual syllable.
Individual syllables should contribute only a secondary element to the modulation of the head movement – the primary head movement modulation needs to come more from the points of emphasis in PHRASES, perhaps peaking on the adjectives, or the verbs when there are no adjectives
The average position that the head returns to must never be locked on dead centre. Mood should also modulate the average position - and the intensity of the modulation.
I appreciate the achievement and realize that these are early days yet.
Anybody else agree with me?
Easy to criticise when there is something there - hats off to you guys - keep going!
Patrick Young March 20, 2013 11:46 AM
LOL! It's "Hilly" the talking computer from Red Dwarf Series II, "Parallel Universe"
Sergio Pissanetzky March 20, 2013 01:00 PM
Neuroscientist Paul Frenger worked for many years in the simulation of emotions. He's developed an analog system and published many papers about it. Maybe his system can be combined with this one to simulate the emotions rather than just setting them.
Bob Tackett March 20, 2013 04:32 PM
It does not "express" emotion... it might fake, imitate, or emulate emotion, but since it does not actually have emotion, it can't express it... Not to get all OCD, but let's not give the computer too much credit until it actually gets to that point... and it might.
Mirmillion March 21, 2013 03:03 AM
Interesting but still bit too shape-shifting around the upper head for my tastes. The blinking eyes should also be just a touch slower and less alien-like. Otherwise very good. A bit scary but good.
machinephilosophy March 21, 2013 08:29 AM
Need some math and some empirical documentation of real humans speaking, to get rid of both the bobbing up and down of the back of the head and the general jerkiness. A few general approximation-transition algorithms that are readily available should resolve these problems. Hopefully these people aren't doing all this from scratch, since there's a lot of public domain knowledge of how to resolve both graphical and pronunciation jerkiness issues. In text to speech, check out the latest (and amazing) Ivona voices.
Tom Swift April 4, 2013 07:42 PM
Finally we can get rid of those overpaid talking heads on the TV news. | {
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César Enrique Román González (* 1930) ist ein ecuadorianischer Botschafter im Ruhestand.
Leben
César Enrique Román González trat in den auswärtigen Dienst und war 1972 Gesandtschaftsrat in Lima.
Von 24. Juli 1974 bis 26. Juli 1977 etablierte er als Geschäftsträger die ecuadorianische Botschaft in Warschau.
Er stand am Beginn einer Periode, in welcher die Regierung in Quito einen Botschafter in Warschau hatte, vorher war der Botschafter in Moskau auch in Warschau akkreditiert, so wie heute der Botschafter in Berlin auch in Warschau akkreditiert ist. Die polnische Regierung zeichnete ihn am 15. November 1977 mit dem 1974 gestifteten Order Zasługi Rzeczypospolitej Polskiej (Verdienstorden der Volksrepublik Polen) aus, welcher ihm vom polnischen Botschafter in Quito, Zygmunt Pietrusiński in einer Feierstunde angeheftet wurde.
Am 14. Dezember 1977 wurde er zum Leiter der Abteilung Wirtschaftspolitik ernannt und löste in dieser Funktion Luis Ortiz Terán ab.
1980 war er Botschafter in Bukarest.
Vom 7. September 1987 bis 1988 war er Rektor der Academia Diplomático de Ecuador und löste in dieser Funktion den Gründungsdirektor Mario Alemán Salvador ab
Von 1992 bis 1996 war er Botschafter in Peking.
Einzelnachweise
Ecuadorianischer Botschafter in Polen
Ecuadorianischer Botschafter in Rumänien
Ecuadorianischer Botschafter in China
Ecuadorianer
Geboren 1930
Mann | {
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from quicli import *
@run
@argument
def test_function(arg1):
pass
| {
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Topanga Aims to Cut Street, Stadium Lighting in Half
The company's fantasma plasma light is coming soon.
Michael Kanellos June 18, 2010
Michael Kanellos
LEDs have helped kick off the ongoing revolution in lighting, but they aren't perfect for every occasion, says Greg Barry, CEO of Topanga Technologies.
Topanga is in the midst of putting the final design changes on a plasma bulb that it says can roughly half the power and maintenance costs associated with high intensity discharge (HID) lights, those intense bulbs that illuminate tennis courts, streets, garages and big box retailers. Ordinarily, HID lighting costs $12.50 in electricity and maintenance per 1,000 lumens of light a year. Topanga will do the job for $6.50 per 1,000 lumens.
"And it's fully dimmable from 100 percent to 20 percent," said Barry. He brought a sample with him, and you can take our word for it: it's bright.
Like Luxim, Topanga's core technology revolves around using radio frequency waves to excite molecules trapped in a clear chamber. The chamber is about as big as a breath mint and looks like something from a Christmas tree. In about thirty seconds, the small bulb emits enough light to illuminate several rooms. In a sense, both Luxim's and Topanga's bulbs operate under a supersized version of the principle -- chemical excitation -- that animates fluorescent lights. LEDs, by contrast, are solid-state lights: light gets emitted by a chip.
A number of different solutions -- electronic ballasts from companies like HID Labs, LED streetlights, plasma lights -- have been proposed in recent years to cut power in the HID market. Government mandates and high power prices have driven these changes. To date, LED has made the most progress (check out the warehouse LED lights from Digital Lumens), but it remains something of an open market.
Luxim has marketed its bulbs for years and branched out into street lighting in 2008. What is Topanga going to offer that's different than Luxim? The resonator, which concentrates the radio frequency waves for the bulb, is hollow, he said, whereas Lumix has a solid resonator. Additionally, Topanga claims it will be cheaper to swap out its bulbs -- though a Luxim representative refuted the 'cheaper' part. (The photo shows a Luxim bulb and two of my fingers. Topanga showed me their bulb and electronics but pictures were not allowed.) More importantly, the question is what can these companies offer over LEDs?
Plasma lighting, Barry added, has an advantage over LEDs in HID applications because of the vast quantity of light that they can produce. To emit 10,000 lumens or more, LED fixture makers need to put 200 LEDs into a fixture, which adds weight and creates a challenge for dissipating the heat that emerges from the back of each LED. HID lights emit 15,000 to 30,000 lumens and can emit up to 90,000 lumens. (Note: LED makers will argue that point.)
"In LEDs, you can get hundreds of thousands of hours of light," Barry explained. "The driver only lasts 30,000 hours. Here you have the opposite problem. The lamp only lasts 30,000 hours but the drivers last longer."
Another problem with LEDs: the technology seems to have occupied all of the intellectual bandwidth at the Department of Energy. It has become tough for alternative light technologies to get funding.
Still, LEDs do deserve credit for something. These lights have grabbed the attention of customers and electronics manufacturers, which in turn has prompted lighting companies to respond with new products. Typically, the lighting industry has been somewhat slow moving. A company might come out with a new product, but then evolve it slowly because of apparent customer indifference.
"There hasn't been a lot of changes in the lighting business in the last 60 years. You got a little bit of change with CFLs in the '90s," he said. "But we've seen more changes in the last five years than in the previous 60. If LEDs hadn't come along, technologies like this might not have happened."
The company has landed five patents, has 25 pending, and is gearing up to file 23 more applications. Khosla Ventures and Nth Power have both invested in the company. Ford Tamer, who just left Khosla Ventures, will remain on the board.
07.15.19 Wind
Europe's Offshore Wind Market Continues Hammering Down Costs
07.09.19 Sponsored Media
The Challenge of Greening America's Cannabis Industry
07.05.19 Grid Edge | {
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\section{Introduction}
Thin films of $\beta$-FeSi$_2$ grown on a Si(001) substrate have generated considerable attention since the demonstration of intense infrared electroluminescence at $\sim$1.5 $\mu$m from a buried $\beta$-FeSi$_2$ epilayer on the Si(001) substrate was reported \cite{Leong1997}. In order to account for strong light emission from the epitaxial layer of $\beta$-FeSi$_2$, substantial efforts have been devoted to experimental and theoretical studies of the electronic structure of this semiconductor in both the form of a bulk single crystal and of a thin film, to address the issue of whether this semiconductor has a direct or indirect band gap \cite{Arushanov1995, Birdwell2002, Terai2011, Filonov1996, Clark1998, Moroni1999}. Recently, it has been accepted that a single crystal of $\beta$-FeSi$_2$ has a direct band gap of 0.80-0.95 eV, and an indirect band gap of 0.7-0.78 eV \cite{Udono2004}, and that the epitaxial film remains an indirect gap semiconductor, even though it is strained by the lattice mismatch between $\beta$-FeSi$_2$ and silicon \cite{Birdwell2008}.
On the other hand, the electronic structure of the $\beta$-FeSi$_2$ surface is still debated: an early scanning tunnelling spectroscopy (STS) study showed a semiconducting surface state with a band gap of $\sim$0.9 eV \cite{Raunau1993}, whereas another STM study combined with X-ray photoelectron diffraction and ultraviolet photoelectron spectroscopy (UPS) showed a finite density of states (DOS) at the Fermi level, indicating a metallic surface state \cite{Hajjar2003}. This latter work suggested the formation of an $\alpha$-FeSi$_2$ film on the Si(001) substrate because of the observed metallic surface state, despite the fact that the surface structure and morphology of the film appeared very similar to those reported in Ref. 10. Moreover, Romanyuk \textit{et al}. recently experimentally confirmed the presence of a single crystalline thin film of $\beta$-FeSi$_2$ when they grew iron silicide on a Si(001) substrate \cite{Romanyuk2014}. They also proposed the presence of a metallic surface, as well as a bulk-like band gap ($\sim$0.85 eV) inside the film, based on the results of density functional theory (DFT) calculations, but without experimental data. A general consensus on the electronic structure of this surface has not emerged despite two decades of study.
In this paper, we address the issue of the surface electronic structure of $\beta$-FeSi$_2$ ultrathin films grown on the Si(001) substrates from the results of precise STM and STS experiments, and DFT calculations. Our STS measurements detected a finite local density of states (LDOS) at the Fermi level, indicating that the reconstructed surface of $\beta$-FeSi$_2$ has a metallic surface state, which agrees well with our DFT calculations based on a $\beta$-FeSi$_2$/Si(001) slab model. Moreover, we observed negative differential conductance (NDC) at $\sim$0.45 eV above the Fermi level in the spectra. This phenomenon is reproduced well by a current-voltage (I-V) simulation using the surface electronic structure from our DFT calculations. The detection of NDC suggests the presence of a surface energy gap at the corresponding energy within the band gap of the $\beta$-FeSi$_2$ ultrathin film. Finally, we propose band diagrams for a $\beta$-FeSi$_2$ film grown on both n- and p-type Si(001) substrates estimated from the results of STS measurements. After discussing the methods used, we present STM and DFT results on the surface structure, before turning to the electronic structure. We present differential conductance measurements and DFT simulations, and explore the band alignment and source of the negative differential conductance before concluding.
\section{Methods}
\subsection{Experimental method}
The experiments were carried out in ultrahigh vacuum (UHV) chambers with an STM apparatus (UNISOKU, USM-1400). The base pressure in the UHV chambers was kept below 4.0$\times$10$^{-9}$ Pa. The substrates used were pieces of n-type (P-doped, 0.01 $\Omega$cm) or p-type (B-doped, 0.013 $\Omega$cm) Si(001) wafers. After the substrate was well degassed for over 12 hours at 870 K in UHV, it was flashed to 1420 -- 1470 K for 10 -- 20 s several times to obtain a clean surface. Subsequently, iron was deposited on the clean surface at room temperature (RT) with an e-beam evaporator (AVC, AEV-3) and the sample was annealed to the temperature of 730 K for 10 min. According to a cross-sectional analysis by transmission electron microscopy (TEM), this operation produced a $\beta$-FeSi$_2$ film with a thickness of 1.0 -- 2.0 nm \cite{Sagisaka20}. STM/S measurements were performed at the sample temperature of 78 K or 4.5 K. For the probe, chemically etched tungsten tips were used. All STM images were generated by the WSxM software \cite{Horcas07}.
\subsection{Computational method}
DFT calculations were performed with the VASP code \cite{kresse93, kresse96}. We employed the projector augmented wave (PAW) method with a plane wave cutoff of 500 eV, and used the generalized gradient approximation with the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional\cite{Perdew96}. The $\beta$-FeSi$_2$/Si(001) surface was modeled using a periodic slab consisting of a 2$\times$2 surface with 16 layers of $\beta$-FeSi$_2$ stacked on 21 layers of silicon and a vacuum thickness of 19 \AA\ (see figure \ref{figlabel_A1}). The crystal structure of bulk $\beta$-FeSi$_2$ is shown in Fig.~\ref{figlabel_A1}(a)\cite{Dusausoy71}. The top layer of the slab was terminated with silicon, according to a recent report on the structural determination of the $\beta$-FeSi$_2$ film by a low energy electron diffraction (LEED)-IV measurement \cite{Romanyuk2014}, and the bottom silicon layer was terminated by hydrogen atoms. The optimized PBE lattice constant for bulk Si in our calculation was 5.47 \AA\, leading to a surface unit cell length of 7.73 \AA. Experimentally, the $\beta$-FeSi$_2$ layer is observed to be pseudomorphically strained to the substrate; the interface structure between $\beta$-FeSi$_2$ and Si(001) was precisely determined by an analysis of the cross sectional TEM images \cite{Sagisaka20}. We also checked that the impact of the interface structure on the surface electronic structure is negligibly small for the thickness of the iron silicide film used in the present study.
To match the substrate, the lattice constant of $\beta$-FeSi$_2$ (a = 9.86 \AA, b = 7.79 \AA, and c = 7.83 \AA) [figure \ref{figlabel_A1}(a)]\cite{Dusausoy71} needed to be adjusted in the plane parallel to the surface (b and c axes). As a result, the lattice of $\beta$-FeSi$_2$ was compressed by 0.8 \% and 1.3 \% in the b and c axes, respectively.
Structural relaxations were performed with 8$\times$8$\times$1 Monkhorst-Pack mesh Brillouin zone (BZ) sampling centered at the $\Gamma$ point until the force on each atom was reduced less than 0.02 eV/\AA. Calculations of LDOS and charge density were carried out with 22$\times$22$\times$1 BZ sampling. After the electronic ground state was calculated by the VASP code, STM images and tunnelling spectra were generated with the Tersoff-Hamann method \cite{Tersoff83, Tersoff85} implemented in the bSKAN code \cite{Hofer03}.
\section{Results and discussion}
Figure \ref{figlabel_1}(a) shows a typical STM image of the ultrathin iron silicide film covering almost the entire surface of the substrate. The surface appears to be rumpled, with a pattern of patches each several nanometres wide, with different image contrast: the root mean square surface roughness in the image was estimated to be 0.129 nm [see the cross sectional profile in figure \ref{figlabel_1}(a)]. However, the surface lattice is continuous and the surface profile smoothly varies without steps across the boundaries of the patched areas [figure \ref{figlabel_1}(b)]. According to the result of our TEM observations \cite{Sagisaka20}, this film consisted of a single crystalline $\beta$-FeSi$_2$ whose surface was smooth, but rumpled; the underlying stepped structure of the Si (001) substrate was compensated by steps in the $\beta$-FeSi$_2$ film at the interface, leading to a variation of the thickness between three values: 1.0, 1.5 and 2.0 nm. These values are close to the unit cell size of $\beta$-FeSi$_2$ normal to the (100) plane ($a$ = 9.86 \AA), 3/2$a$ and 2$a$, respectively. The crystallinity of the surface over a wide area was also confirmed by LEED measurements, which show slightly diffuse (1/2, 1/2) and (1, 1) spots [inset of figure \ref{figlabel_1}(a)]. We note that this spot spread is derived from the patches on the surface observed by STM, and it suggests that the lattice constant along the surface fluctuates slightly from patch to patch, or along the boundaries between adjacent patches. Zoomed STM images clearly show protrusions with c(2 $\times$ 2) periodicity as shown in figures \ref{figlabel_1}(b) and \ref{figlabel_1}(c). These characteristics are consistent with the results of a recent study on the $\beta$-FeSi$_2$(001) film using LEED $I-V$ analysis and DFT calculations \cite{Romanyuk2014}.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{Fig1.pdf}
\end{center}
\caption{STM images of a $\beta$-FeSi$_2$ film grown on the Si(001) substrate. (a) Image size = 100 $\times$ 100 nm, Sample bias ($V_s$) = -1.5 V, tunnelling current ($I$) = 50 pA. Crystal orientations are denoted with respect to those of silicon substrate. The topographic profile was obtained from the white line crossing the $\beta$-FeSi$_2$ region and a small residual fraction of the Si(100) surface in the STM image. Inset: LEED pattern from the sample; incident electron energy = 67.5 eV, (b) Image size = 30 nm $\times$ 30 nm, $V_s$ = -1.5 V, $I$ = 10 pA, and (c) Image size = 5 $\times$ 5 nm, $V_s$ = +1.0 V, $I$ = 50 pA. Sample temperature ($T_s$) = 78 K.
}
\label{figlabel_1}
\end{figure}
The model proposed to explain the emergence of c(2 $\times$ 2) periodicity in STM images \cite{Romanyuk2014} proposed that four silicon atoms on the bulk-truncated surface [figure \ref{figlabel_2}(e)] cluster to form a tetramer [figure \ref{figlabel_2}(f)]. The Si tetramers are arranged with c(2 $\times$ 2) periodicity and are imaged as single protrusions by STM. Using this Si-tetramer model, our DFT-based simulation reproduced the experimental STM images in detail, at both positive and negative bias. Figure \ref{figlabel_2} displays a comparison of the experimental [(a) and (c)] and simulated [(b) and (d)] STM images of the $\beta$-FeSi$_2$ surface. Our DFT calculations found that the reconstruction of the surface reduces the energy by 0.275 eV per Si atom.
A careful inspection of both experimental and simulated images reveals that each Si cluster is actually elongated alternately along (100) and (010) crystal orientations at a positive sample bias (unoccupied state), while appearing almost circular at a negative sample bias (occupied state). This alternating elongation is probably derived from the atomic arrangement of Si atoms in the fourth layer [Si(2) in figure \ref{figlabel_A2}] and Fe atoms in the fifth layer [Fe(2) in figure \ref{figlabel_A2}] from the surface.
\begin{figure}
\begin{center}
\includegraphics{Fig2.pdf}
\end{center}
\caption{Experimental [(a) and (c)] and simulated [(b) and (d)] STM images of the $\beta$-FeSi$_2$ surface. $V_s$ = (a) -1.5 V, (b) -1.0 V, (c) 1.0 V, and (d) 1.0 V. The image contrast is enhanced in the inset in (c) to emphasise the diagonal feature. (e) Top view of the bulk truncated $\beta$-FeSi$_2$(100) surface terminated with silicon atoms and (f) reconstructed surface after structural optimization, with which STM simulation was conducted.}
\label{figlabel_2}
\end{figure}
Figure \ref{figlabel_3}(a) shows a typical normalized $dI/dV$ spectrum acquired from the $\beta$-FeSi$_2$ surface at 4.5 K. The flat and clean region of the surface did not have any distinct site dependence. A significant feature in the STS is the presence of finite conductance around the Fermi level, with three distinct peaks. Therefore, we conclude that the clean surface of the $\beta$-FeSi$_2$ has no energy gap, and so is metallic. This result is consistent with a previous report of a finite DOS of states at the Fermi level observed by UPS measurement \cite{Hajjar2003}, where the surface was assigned to be $\alpha$-FeSi$_2$, but actually it was $\beta$-FeSi$_2$. We also note that a semiconducting feature observed in the I-V curve reported by Raunau \textit{et al}. \cite{Raunau1993} was probably due to an experimental malfunction such as contamination of the STM tip. Another intriguing feature in our spectrum is a conductance drop to negative values above the sample bias of +0.45 V (NDC) which is discussed fully below.
To find the electronic structure responsible for the peaks detected in STS, we examined both the projected density of states (PDOS) for the top surface Si atoms, and the spatial distribution of electronic states in specific energy windows, using the slab model shown in figure \ref{figlabel_A1} (we note that the PDOS does not correspond directly to the normalised differential conductance, but gives more insight into the origin of different features; we compare experimental and theoretical differential conductance in figure~\ref{figlabel_5} below).
The peaks in the calculated PDOS plotted in figure \ref{figlabel_3}(b) match well with those in the experimental spectrum in figure \ref{figlabel_3}(a), with regard to the peak shape and energies; the strong peak at +1.3 V in the experimental spectrum is the exception, and is exaggerated relative to PDOS by the asymmetry of the tunneling process. While PDOS of surface atoms will give some insight into STS, the extended states through the slab are also important, and we examined a series of energy windows, notated from I to V in figure \ref{figlabel_3}(b), by plotting charge density across the slab [figure \ref{figlabel_3}(c)].
The peaks found in windows I, II and III correspond to bonding states in the Si tetramer, which produce the protrusion in the topographic STM images. In particular, the three peaks in range III, situated in the band gap of bulk silicon, consist of three bands crossing the Fermi energy (further detail of the band structure can be seen in figure \ref{figlabel_A3}). Therefore, electronic transport under small bias voltages ($\textless\pm$0.25 V) will occur only within the FeSi$_2$ film. In energy window IV, the surface states are completely absent and the charge density is depleted from the first to fifth atomic layers below the surface. We also note the presence of significant charge density at the interface between the FeSi$_2$ film and the Si(001) substrate in ranges III and IV (see also figure \ref{figlabel_A3}); a detailed characterization of the atomic and electronic structure of the interface between Si(001) and the FeSi$_{2}$ film will be presented in future work. The onset of empty (anti-bonding) states of the Si tetramer occurs in energy window V.
\begin{figure*}
\begin{center}
\includegraphics[width=16cm]{Fig3.pdf}
\end{center}
\caption{(a) Normalized $dI/dV$ spectrum recorded on the $\beta$-FeSi$_2$ surface at 4.5 K. $dI/dV$ measurement was performed by placing the STM tip over a Si tetramer under open-feedback through a lock-in technique with bias modulation of 10 mV at a frequency of 830 Hz. Set point : V$_s$ = 1.5 V and I = 100 pA. Ten spectra were averaged. (b) Projected density of states (PDOS) for the top Si atom of the $\beta$-FeSi$_2$ slab model, (c) Isosurface map of charge density across the entire slab integrated in energy windows of (I) -1.1 $\sim$ -0.70 eV, (II) -0.70 $\sim$ -0.23 eV, (III) -0.23 $\sim$ +0.30 eV, (IV) +0.30 $\sim$ +0.7 eV, and (V) +0.70 $\sim$ +1.0 eV. Isosurface value: 10$^{-2}$ e/\AA$^3$. Color scale represents the $z$ position in the slab.}
\label{figlabel_3}
\end{figure*}
To our knowledge, the band alignment between $\beta$-FeSi$_2$ and the Si substrate has not been assessed in previous works. In order to clarify this point, we have examined the band bending in the cases of a $\beta$-FeSi$_2$ film grown on n-type and p-type Si(001) substrates. Figure \ref{figlabel_4}(a) shows normalized $dI/dV$ spectra recorded on the $\beta$-FeSi$_2$ surface and Si dimers on both n-type and p-type substrates at 78 K. Although the spectral features of $\beta$-FeSi$_2$ grown both on n- (Spectrum A) and p-type (Spectrum B) Si(001) were thermally broadened compared to those observed at 4.5K in figure \ref{figlabel_3}(a), the shapes and positions of the peaks in both spectra were almost equivalent. This indicates that the Fermi level of the FeSi$_2$ films is pinned at the same energy in the surface state on both the n- and p-type substrates, as depicted in figures \ref{figlabel_4}(b) and \ref{figlabel_4}(c). To achieve this, the electrostatic potential in Si near the interface with $\beta$-FeSi$_2$ must be bent in different manners for n- or p-type substrates.
The evaluation of the band bending was performed by measuring the energetic positions of the Si dimer states in small windows of clean Si(001) in the $\beta$-FeSi$_2$ films [figure \ref{figlabel_4}(d)]. The Si dimer (Spectra C -- F) is characterized by a pronounced peak ($\pi$) in the occupied states and broad and intense peaks ($\pi_1$* and $\pi_2$*) in unoccupied states, separated by a surface band gap of 0.6$\sim$0.7 eV. \cite{Sagisaka05, Nakayama06}
The energetic locations of these states with respect to bulk valence and conduction bands have been recently reported. \cite{Sagisaka17} Based on that relationship, band diagrams for the clean Si surface are given for both n- and p-type substrates in figures \ref{figlabel_4}(b) and (c), respectively. \cite{Note1} By evaluating the peak positions, we found that the growth of FeSi$_2$ shifted the surface states of Si upward by 0.1 V on n-type Si [figure \ref{figlabel_4}(b)] and downward by 0.3 V on p-type Si [figure \ref{figlabel_4}(c)]. Since we did not observe contrast abnormality in STM images, or a shift of spectral features in STS of the $\beta$-FeSi$_2$ surface in the vicinity of the uncovered areas, we believe that the same amount of band bending occurred both in the silicon underneath the $\beta$-FeSi$_2$ and in areas uncovered by the FeSi$_2$ film. This is reflected in the band diagrams, which imply that electrons (holes) are depleted in the n- (p-) type Si substrate near the interface after the formation of $\beta$-FeSi$_2$.
\begin{figure*}
\begin{center}
\includegraphics[width=16cm]{Fig4.pdf}
\end{center}
\caption{(a) Normalized $dI/dV$ spectra recorded on $\beta$-FeSi$_2$ surface and Si dimers. Spectra A and B: $\beta$-FeSi$_2$ grown on p-type and n-type Si(001) substrates, Spectra C and D: Si dimers in a small domain not covered by $\beta$-FeSi$_2$ on p-type and n-type Si(001) substrates. Coverages of $\beta$-FeSi$_2$ on both samples were greater than 80\%. Spectra E and F: Si dimers on clean p-type and n-type Si(001) substrates without growth of $\beta$-FeSi$_2$. Set point: $V_s$= --2.0 V and $I$= 10 pA. Each spectrum was obtained by numerical differentiation after 16 I-V curves recorded in a 4 $\times$ 4 grid over an area of 1.5 $\times$ 1.5 nm$^2$ were averaged. (b) and (c) Band diagrams for clean Si(001) and $\beta$-FeSi$_2$/Si(001). (d) Representative STM image of a $\beta$-FeSi$_2$ film in the vicinity of a small Si dimer window.$V_s$ = +1.0 V, $I$ = 50 pA, $T_s$= 78 K.}
\label{figlabel_4}
\end{figure*}
The presence of a surface energy gap above the Fermi level affects the variation of the tunnelling conductance with bias voltage. To see this effect, we replotted the spectrum shown in figure \ref{figlabel_3}(a) in the form of I-V and dI/dV in figures \ref{figlabel_5}(a) and \ref{figlabel_5}(c), respectively. Above a sample bias ($V_S$) = +0.3, the current slightly drops, which is equivalent to the differential conductance becoming negative. Upon ramping up $V_S$, the current increases again (the differential conductance becomes positive). Similar NDC has been observed on various surfaces by STM \cite{Bedrossian89, Berthe08,Wang09,Kim13,Yin20}. For quantitative analysis, we computed an I-V curve from the DFT DOS based on the standard formula for the tunnel current:\cite{Hamers89}
\begin{equation}
I \propto \int^{eV}_{0}\rho_s(E)\rho_t(eV-E)T(E,V,z)dE
\end{equation}
where $\rho_s$ and $\rho_t$ are the DOS of the sample and tip, and $T$ is the tunnelling matrix element. In the WKB approximation for planar electrodes, $T$ is written as $T(E, V, z) = \exp(-2z\sqrt{m(2\phi+eV-2E)}/\hbar)$, where $\phi$ is the effective work function. We used $\phi = (\phi_{tip} + \phi_{sample})/2$ = 9.4 eV and the tip-sample distance $z$ = 6 \AA\ for the computation of the I-V curve. Furthermore, we treated $\rho_{t}$ as a constant with respect to the energy, which simplified eq. (1) to:
\begin{equation}
I \propto \int^{eV}_{0}\rho_s(E)T(E,V,z)dE
\end{equation}
For $\rho_s$, we used the Kohn-Sham eigenstates which also gave the PDOS shown in figure \ref{figlabel_3}(b). Computed I-V and dI/dV curves, shown in figures \ref{figlabel_5}(b) and \ref{figlabel_5}(d) respectively, successfully duplicate the overall peak positions and the feature of NDC of the experimental curves in figures \ref{figlabel_5}(a) and \ref{figlabel_5}(c). The NDC comes from the combined effect of the presence of an energy gap in the unoccupied state of the $\beta$-FeSi$_2$ surface and the behaviour of the tunnelling matrix $T$ with respect to energy, known as the tunnel-diode mechanism \cite{Bedrossian89} [figure \ref{figlabel_5}(e)].
The application of a small positive voltage to the sample induces tunnelling of electrons from the tip to the unoccupied state of the $\beta$-FeSi$_2$ surface [$V_S$ range I in figures \ref{figlabel_5}(b) and \ref{figlabel_5}(e)]. When the Fermi level of the tip ($E_{F, tip}$) matches the energy gap of the FeSi$_2$ surface, tunnelling of electrons from the tip to the surface is markedly suppressed, because $T$ is largest at $E_{F, tip}$ ($V_S$ range II). Consequently, the current decreases with increasing sample bias, while $E_{F, tip}$ is in the energy gap. As $E_{F, tip}$ exceeds the energy gap, the current increases again ($V_S$ range III). The excellent agreement between the experiment and simulation indicates that the NDC observed on the FeSi$_2$ surface is intrinsic to its electronic nature and is independent of that of the tip.
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{Fig5.pdf}
\end{center}
\caption{Experimental and calculated tunnelling spectra from $\beta$-FeSi$_2$ surface. Experimental I-V curve (a) and dI/dV curve (c), and calculated DFT I-V curve (b) and dI/dV curve (d). Set point: $V_s$= +1.5 V and $I$= 100 pA. The experimental dI/dV curve was acquired by a lock-in technique with bias modulation of 10 mV at 830 Hz. The sample temperature was 4.5 K. (e)-(h) Schematic band diagrams for tunnelling processes with $V_s$ = 0 V and three different ranges. $V_s$ range I, II, and III correspond to those in (b). The symbol X in (g) and (h) indicates that tunnelling of electrons is not allowed at the corresponding $V_s$ due to the lack of electronic states within the surface band gap.}
\label{figlabel_5}
\end{figure}
\section{Summary}
We have demonstrated the metallic nature of the surface of ultrathin $\beta$-FeSi$_2$ films grown on the Si(001) substrate. The excellent agreement of simulated STM, LDOS and I-V results with our experimental data confirms that the iron-silicide film in this study was $\beta$-FeSi$_2$, rather than another possible crystal structure. We also determined band alignment between $\beta$-FeSi$_2$ and n- or p-type Si(001) substrate from tunnelling spectroscopy. Finally, I-V curves show NDC, owing to the presence of an energy gap in the unoccupied state of the $\beta$-FeSi$_2$ surface.
The new data for the $\beta$-FeSi2 film grown on the Si(001) substrate presented in this paper will provide an important launch point for the development of optoelectronic devices using this silicide. Moreover, the identification of NDC arising from the gap in the silicide film may guide the design of new materials systems with this important characteristic. As we have shown, the origin of the metallic surface state is the Si tetramers, so the formation of Si clusters in or on $\beta$-FeSi2 may cause unwanted electronic states in the band gap. This implies that growth of a high quality film, with the surface state removed, will be necessary for device development. Finally, the band alignment between $\beta$-FeSi2 and silicon presented in figure~\ref{figlabel_4} will form a good starting point for designing and controlling the bands in this material by impurity doping.
\ack{Computational calculations were performed by using the Numerical Materials Simulator of
NIMS. This study was partly supported by JSPS KAKENHI Grant Number 22K04860.}
| {
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Уганда је учествовала на 14. Светском првенству у атлетици у дворани 2012. одржаном у Истанбулу од 9. до 11. марта дванаести пут. Репрезентацију Уганде представљала су двојица учесника, који су се такмичили у две дисциплине.
Такмичари Уганде нису освојили ниједну медаљу нити је постигнут неки рекорд.
У табели успешности према броју и пласману такмичара који су учествовали у финалним такмичењима (првих 8 такмичара) Уганда је са једним учесником у финалу делила 41 место са 2 бода.
Учесници
Мушкарци:
Џулијус Мутеканга — 800 м
Мозис Кипсиро — 3.000 м
Резултати
Мушкарци
Референце
Спољашње везе
Комплетни резултати на сајту ИААФ
Званични сајт ИААФ
Земље учеснице на Светском првенству у атлетици у дворани 2012.
Уганда на Светском првенству у атлетици у дворани | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,371 |
Q: Bash command stopping and not doing the third part I have this command that takes any video in a folder, it converts it in .ts segments and creates a m3u8 playlist.
I then need to concat them in a file playlist.txt, so this is what I'm doing:
cd /var/myfolder && find ./ -name "*.*" -exec ffmpeg -i {} -c:v h264 -crf 1 -flags +cgop -g 30 -hls_time 1 -hls_list_size 0 {}.m3u8 2>&1 > /dev/null \; && ls /var/myfolder/*.m3u8 | sed "s:.*/::" | xargs -i echo file {} > /var/myfolder/playlist.txt && echo done';
The first and second command work (cd in folder and ffmpeg conversion) then the command && ls /var/myfolder/*.m3u8 | sed "s:.*/::" | xargs -i echo file {} > /var/myfolder/playlist.txt && echo done'; is not executing.
I don't understand why is not executing the 3rd command.
A: You can't use shell redirects in find -exec.
xargs defaults to echo ... but since you already cd into the directory, we can substantially simplify the last line and get rid of both xargs and sed.
cd /var/myfolder &&
find ./ -name "*.*" -exec ffmpeg -i {} -c:v h264 -crf 1 -flags +cgop -g 30 -hls_time 1 -hls_list_size 0 {}.m3u8 \; 2>&1 > /dev/null &&
printf 'file %s\n' *.m3u8 > /var/myfolder/playlist.txt &&
echo done
| {
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Features Business Management Marketing
CannTrust celebrates local launch of We Care platform
By Andrew Snook
Diane Corkum, executive director of Project SHARE, talked about the impact of the organization's community gardens for families in Niagara Falls
Project Share's community garden at Our Lady of the Scapular Parish in Niagara Falls, Ont. was buzzing with members of the community and local media taking in the celebration of the local launch of CannTrust's We Care national social responsibility platform.
The event focused on a couple of the charities and non-profits that CannTrust is partnering with in the Niagara Region: Project SHARE and Hospice Niagara.
Morgan Cates, director of communication for CannTrust, says her company is very fortunate to have communities like Niagara Falls and Vaughan, Ont. (where they employ a total of more than 700 people combined) open to them setting up operations, and that CannTrust believes it is important to give back to these communities.
"We want to showcase how caring and giving the people in our company actually are," she said.
The We Care national platform was officially launched at the Lift & Co. Cannabis Business Conference earlier this month, and focuses on four key areas: lifting people out of poverty; health and wellness; safe consumption; and sustainability.
The first We Care partnerships announced were a $50,000 donation to the Canadian Alliance to End Homelessness (CAEH) for the non-profit's Built for Zero campaign; a donation of $25,000 to Project SHARE; and a three-year commitment of $25,000 in annual donations to Hospice Niagara for its music therapy program.
For Project SHARE's executive director Diane Corkum, who has worked with the organization off and on for the past 30 years, the donation from CannTrust was "a wonderful bolt of lightning from the sky".
Project SHARE provides emergency food items to 126 families living below the poverty line every day in the Niagara Falls area. The organization runs three community and allotment gardens that grow 4,000 lbs. of organic produce for local food banks. In addition to the $25,000 donation for the Project SHARE 2019 community garden season, CannTrust is donating volunteer hours and its knowledge as experts in cultivation.
Over the past three years, Project SHARE had been receiving its funding through the Ontario Trillium Foundation, but those funds had run out this past February; leaving the organization scrambling for new funding. CannTrust reached out to the organization to offer donations to cover the funding for materials and other needed funding for the 2019 growing season.
"It was a really nice surprise. They've been wonderful to deal with," Corkum said.
For Sam Piccirillo, the community gardens offer her family options for healthier living, both physically and mentally.
"We've been doing this for 13 years and it's been excellent for my family, we get to eat our own stuff," she said, adding that when she steps into the community garden it offers her a form of therapy. "All of the gardeners share information, they share plants; it's like a community. I've just really enjoyed doing this all these years."
Carol Nagy, director of Hospice Niagara, was also on hand for the event and thanked CannTrust for their support of their music therapy program, designed to help people through one of the most difficult times in their lives.
"One of our dreams was to start a music therapy program because we know how music can touch our lives," she said, adding that her organization spent years trying to search out corporate funding for the program without success. "CannTrust, we thank you."
Michael Caplin, general manager for CannTrust's Niagara operations, said that the giving and caring philosophies of his organization are something shared by all its employees.
"It has become our mantra that every CannTrust employee lives and breathes," he says.
For more information and updates on the We Care platform, visit www.Canntrust.com/WeCare.
CannTrust begins trading on New York Stock Exchange
LPs and research institutions are teaming up in pursuit of science
CannTrust to venture outdoors
CannTrust finalizes outdoor grow purchase
canntrust
Bruce Linton steps down as Canopy Growth co-CEO
Ontario to issue 50 new cannabis store licences | {
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{"url":"https:\/\/physics.stackexchange.com\/questions\/595841\/adjusting-the-rate-of-proton-decay-in-the-standard-rm-su5-grand-unified-the","text":"# Adjusting the rate of proton decay in the standard $\\rm SU(5)$ grand unified theory\n\nThe proton decay rate in the standard $$SU(5)$$ grand unified theory is given by $$\\Gamma \\sim \\left(\\frac{g^2}{M_x^2}\\right)^2 m_p^5 =\\frac{g^4}{M_x^4}m_p^5$$ Naively we could push up the bound for the decay rate $$\\Gamma$$ arbitrarily high by increasing the mass of the $$X$$ boson, $$M_x$$. However, it seems that we set $$M_x$$ to be the scale of\n\n(1). $$M_x \\sim M_{gut}$$ which is the energy scale when all three gauge couplings meet\n\nor\n\n(2). $$M_x$$ set to be the scale GUT's Higgs scale where the $$SU(5)$$ is broken down to the standard model group $$SU(3) \\times SU(2) \\times U(1)$$.\n\nMy questions are that:\n\n1. Are scales in point (1) and (2) are the same? Are they somehow related? It seems to me (1) and (2) can be different.\n\n2. Why can we arbitrarily push up the scale of $$M_x$$ to the scale where all three gauge couplings meet? (as long as it is below the Planck scale?)\n\n3. If possible, can we sketch other ways of varying the proton decay rate in these theories? (How exactly can SUSY help?) The current bound in experiment seems to suggest the half-life is about $$10^{34}$$ years..","date":"2021-10-18 18:09:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 12, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9206190705299377, \"perplexity\": 166.15645977208527}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585204.68\/warc\/CC-MAIN-20211018155442-20211018185442-00527.warc.gz\"}"} | null | null |
\section{Introduction}
It has been known for a long time that the description of ferroelectric oxides as insulators becomes less and less valid as their dimensions are reduced. The conductivity of thin films is often modelled using well established semiconductor theories and concepts for new devices such as resistive memories \cite{Contreras, MeyerKolstedt} are already emerging which exploit the semiconducting nature of thin ferroelectrics. The frequently observed deformation of hysteresis loops by leakage currents, which masks the true ferroelectric properties of the system, apart from being a nuissance to researchers, is not directly a problem for non-volatile memory applications. However, charge injection associated with finite conductivity is believed to be involved in accelerating fatigue as well as raising other reliability issues. The increased power consumption and heating due to leakage are also undesirable. Leakage measurements are therefore a crucial part of any electrical characterization of a ferroelectric device. These are often performed using automated systems designed for semiconductor characterization. However, as shown in this article, great care should be taken if one wants to obtain true steady state current-voltage (\emph{I-V}) data due to the the much slower dielectric relaxation in ferroelectrics compared with usual semiconductors as well as resistance degradation at high fields. Such artefacts in \emph{I-V} measurements can lead to incorrect interpretation of the conduction mechanism.
This paper reports a detailed study of the \emph{I-V} characteristics of Samsung's lead zirconium titanate (PZT) capacitors. Together with capacitance-voltage (\emph{C-V}) measurements and ferroelectric hystesis (\emph{P-V}) loops, the data are analyzed in the light of various conduction models with the aim of understanding the leakage mechanism.
\section{Experimental}
The 144nm PbZr$_{0.4}$Ti$_{0.6}$O$_3$ films were deposited by a sol-gel method at Samsung, using a high purity (99.9999\%) precursor, on a [111] oriented Pt bottom electrode, resulting, after annealing, in an almost epitaxial interface between the metal and the columnar PZT grains. For each capacitor the top Ir/IrO$_2$ electrode was patterned as a two-dimensional array of interconnected $\sim$13$\times 13\mu$m$^2$ squares with a total area $A=5.31\times 10^{-5}$cm$^2$. The capacitor structures were fully integrated and embedded in Si, with various metallic layers connecting the top and bottom electrodes to the external tungsten contact pads. Current-voltage-time (\emph{I-V-t}) measurements were performed using the Agilent 4155C semiconductor parameter analyser, dynamic capacitance measurements were obtained using the HP4192A impedance analyzer and the Radiant Technologies Precision Pro tester was used for ferroelectric hysteresis loops.
\section{Results}
\subsection{Current-voltage characteristics}
\begin{figure}[t]
\centering
\scalebox{0.75}{\includegraphics{SelectedF142C.eps}}
\caption{Typical current transients obtained at $42^{\circ}$C at various biases. Note the change of scale from logarithmic at low field to linear at higher biases.}
\label{Transients:OK}
\end{figure}
The \emph{I-t} response at $42^{\circ}$C of our capacitors upon a stepwise application of a constant bias is shown in figure~\ref{Transients:OK} and is typical of leakage behaviour of ferroelectric thin films. At short times the behaviour is dominated by dielectric relaxation which often follows the empirical Curie-von Schweidler law ($I\propto t^{-n}$ with $0<n<1$). This is then followed by a region of steady state leakage (in this case most clearly visible for the intermediate bias), which may or may not be observed depending on the point at which the current starts to increase again due to time-dependent dielectric breakdown or resistance degradation. It is common practice when measuring \emph{I-V} characteristics to choose a waiting time (between the application of the bias and the current measurement) which is longer than the dielectric relaxation time at intermediate biases. However, as is clear from figure~\ref{Transients:OK}, this can lead to an overestimate of the current at high biases due to resistance degradation and at low biases due to dielectric relaxation, sometimes leading to apparent negative differential resistivity or ohmic-like behaviour.
To obtain true steady state current readings and avoid unnecessary stress to the sample, we performed \emph{I-t} measurements at each voltage until a clear plateau in the current was reached. In a few cases at high biases and high temperatures where no clear plateau could be resolved before the onset of resistance degradation, the minimum value of the current was used. The tungsten pads were contacted using 10$\mu$m diameter tungsten probe tips, which were found to be more effective in achieving a good contact and penetrating through the thin tungsten oxide surface layer than thinner tips. Due to the high homogeneity of the films and the almost perfect reproducibility of all electrical measurements, the quality of contact could be checked by looking at the hysteresis loops, and in particular at the coercive voltages, which were found to be very sensitive to any contamination of the tungsten pad surfaces. Prior to taking any measurements the samples were poled at three times the coercive field (for $\sim$60s) and at the highest measurement bias (for 10-20s, depending on the temperature) to avoid any diplacement currents and internal field changes due to the reversal of ferroelectric polarization when measuring the \emph{I-V} characteristics. In this way the forward and reverse bias \emph{I-V} measurements were obtained for a range of temperatures; we will define forward bias as the case when the top IrO$_2$ electrode is at a higher potential than the bottom Pt electrode.
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics{FwdAndRvs.eps}}
\caption{Forward (open circles) and reverse (filled circles) bias \emph{I-V} at different temperatures.}
\label{FwdAndRvsIV:OK}
\end{figure}
Figure~\ref{FwdAndRvsIV:OK} shows that at intermediate temperatures the \emph{I-V} are quite symmetric apart from the very low field region below $\sim 1$V. At low and high temperatures, however, some asymmetry is seen. Although different capacitors were used to obtain the forward and reverse bias data, this cannot account for the observed asymmetry due to the high reproducibility of the measurements over many samples, especially in forward bias. The asymmetry is quite low, well below an order of magnitude for almost all biases. It is also interesting to note that whereas in low temperature measurements the reverse bias currents are higher, they are lower at high temperatures; this is not expected if the asymmetry is due to differences in the interfaces, such as different barrier heights, at the top and bottom electrodes. In general, reverse bias currents were somewhat less reproducible and less stable, with breakdown occuring at lower biases. For this reason we focused more on forward bias data for the quantitative analyses, althogh some reverse bias \emph{I-V} will also be discussed.
In the hope of understanding the mechanism responsible for conduction in these films we have looked at a number of models commonly used to describe carrier transport in ferroelectrics and tried to fit their predictions as quantitatively as possible to our own data. In the following subsections we therefore present the same set of results analyzed in terms of these different models.
\subsection*{Space charge limited (SCL) conduction}
When the internal field is dominated by space charge (either from free or trapped carriers) the current density $J$ is, in general, expected to have a power-law dependence on voltage, $J\propto V^n$. In the case of discrete traps a quadratic voltage and cubic thickness ($d$) dependence is expected \cite{HwangKao}
\begin{equation}
J=\frac{9}{8}\theta_f\epsilon\epsilon_0\mu \frac{V^2}{d^3}\nonumber,
\end{equation}
where $\mu$ is the carrier mobility, $\epsilon$ the relative dielectric constant and $\theta_f$ the fraction of injected carriers that are free (i.e. not trapped); in the trap-free limit $\theta_f=1$. Unfortunately, since these are commercial Samsung films of a fixed thickness, we have to rely purely on the voltage dependence when analyzing our data.
From figure~\ref{FwdAndRvsIV:OK} it is possible to identify, for reverse bias data in particular, regions which look linear on the log-log plot and could therefore satisfy the power-law dependence. At low negative biases the room temperature exponent $n$ was found to be 2.1, which is compatible with the trap-free or discrete trap case. However, an unexpected temperature dependence was found, with $n$ decreasing steadily to 1.65 at the highest temperature. The high-field exponents were in the range 3.0--4.8 for reverse bias and 3.5--6.9 for forward bias data respectively. Such high exponents are not common to SCL currents, and are certainly not possible for discrete traps. Values of $n>2$ are possible if traps are distributed within the bandgap and they are expected to be temperature dependent \cite{HwangKao}. In the case of an exponential trap distribution characterised by a temperature $T_1$
\[n-1=\frac{T_1}{T}\]
and for a Gaussian trap distibution with a standard deviation $\sigma_t$ we expect
\[(n-1)^2=1+\frac{2\pi\sigma_t^2}{16k^2T^2}.\]
However, neither of these relationships were well satisfied by our data. We therefore believe that SCL conduction is not likely to be the dominant leakage mechanism in our films, though other authors \cite{ Shin_SCLC, Cho_Jeon_SCLC} have tried to explain their data using this model.
\subsection*{Schottky thermionic emission}
The slight asymmetry between forward and reverse bias \emph{I-V} suggests that the interface properties might play a role and we therefore consider Schottky thermionic emission as a possible conduction mechanism. In this case the sample is considered as two back-to-back Schottky diodes, with the current controlled by the field-induced lowering of the orginal zero-field potential barrier height $\phi_{B0}$ at the reverse (or forward) biased electrode when electrons (or holes) are the majority carriers. The current density is given by \cite{Sze}
\begin{equation}
J=A^*T^2\exp(-q\phi_{B0}/kT)\exp(\beta\sqrt{E_m})
\label{Schottky:OK}
\end{equation}
with the Richardson constant $A^*=4\pi em^*k^2/h^3$ related to the effective carrier mass $m^*$, and $\beta=(e/kT)(e/4\pi\epsilon\epsilon_0)^{1/2}$; $k$ and $h$ are the Boltzmann and Planck constants and $q$ and $e$ are the carrier and electron charges respectively. The important parameter in equation~\ref{Schottky:OK} is the field $E_m$ at the potential maximum which determines the barrier lowering. We consider two cases.
\subsubsection*{Case 1: uniform field (full depletion)}
Bulk ferroelectrics are good insulators and even in thin films it is common to assume that $E=V/d$. If the films are semiconducting this corresponds to the case of full depletion, which is expected for low concentrations of ionised dopants.
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics{SchottkyF1F4.eps}}
\caption{Forward bias data represented on the standard Schottky plot. The data which deviate from the linear fit are shown in light grey.}
\label{SchottkyFullDepl:OK}
\end{figure}
Figure~\ref{SchottkyFullDepl:OK} shows that, especially at higher fields, the data satisfy the Schottky equation quite well; but we also have to see if the physical parameters that can be extracted from the data are sensible. The relative dielectric constant was calculated from slopes of the linear fits in figure~\ref{SchottkyFullDepl:OK} to be between 0.9 and 1.1 over the temperature range shown. The physical range for $\epsilon$ is limited from one side by the optical dielectric constant $\epsilon_\infty\approx 6$ \cite{Eps_Infty} and from the other side by the static dielectric constant $\epsilon_{dc}$, expected to be around 400 \cite{Haun} for this composition along [111]. The calculated value of around $\epsilon\sim 1$ is clearly outside this range. From the temperature dependence of the intercepts the Richardson constant and effective carrier mass were found to be 2.4~Acm$^{-2}$K$^{-1}$ and 0.02m$_e$ respectively, where m$_e$ is the free electron mass. These values also seem somewhat low compared to those previously reported for perovskites \cite{EffectiveMass}. A barrier height of 0.9eV was also calculated, which is plausible from band diagram considerations.
\subsubsection*{Case 2: partial depletion}
In more semiconducting films, the dopant concentration $N_D$ may be sufficiently high to allow the applied voltage to be dropped across a thin depletion layer near the reverse biased electrode (for an $n$-type or forward biased for a $p$-type material), resulting in a linear field profile in the sample and a modified value for $E_m$ which also depends on the built-in bias $V_{bi}$ \cite{Sze}
\begin{equation}
E_m=\sqrt{\frac{2qN_D}{\epsilon_{dc}\epsilon_0}\left(V+V_{bi}-\frac{kT}{q}\right)}.
\label{FieldPartDepl:OK}
\end{equation}
The $kT/q$ term originating from the free carrier contribution to the space charge within the depletion layer may usually be neglected.
Recently Pintilie \emph{et al.} \cite{Pintilie} have analysed their epitaxial single crystal PZT data using the partially depleted thermionic emission model, modified to account for the ferroelectric polarization, and have shown that they fit the $\log (J/T^2)\sim (V+V_{bi})^{1/4}$ law. We performed a similar analysis on our films, fitting the \emph{I-V} to the quarter-power law assuming various values of $V_{bi}$ and extracting the physical parameters from the slopes and intercepts of our fits.
We found that the quality of the fit could be adjusted by varying $V_{bi}$, with the high temperature data fitting best for low or negative values of $V_{bi}$ and the low temperature data giving the best fits assuming a high positive built-in bias. From the temperature dependence of the intercepts of $\log (J/T^2)$ vs. $(V+V_{bi})^{1/4}$ plots, the effective carrier masses and barrier heights were extracted and are shown in table~\ref{Parameters:OK} for different assumed values of $V_{bi}$.
\begin{table}[h]
\begin{tabularx}{\linewidth}{|
>{\setlength{\hsize}{1.0\hsize}}Y
>{\setlength{\hsize}{1.0\hsize}}Y
>{\setlength{\hsize}{1.0\hsize}}Y|}
\hline
$V_{bi}$ (V) & $m^*/m_e$ & $\phi_{B0}$ (eV) \\
\hline
-0.05 & 20 & 1.34 \\
0 & 2.1 & 1.28 \\
0.1 & 0.10 & 1.21 \\
0.2 & 0.010 & 1.16 \\
0.4 & 0.00026 & 1.09 \\
0.6 & 1.27$\times 10^{-5}$ & 1.02 \\
\hline
\end{tabularx}
\caption{Effective carrier masses are highly dependent on the $V_{bi}$ value assumed for the fit. It should also be noted that, due to the exponential dependence of $m^*$ values on the intercept, they are very sentitive to small errors.} \label{Parameters:OK}
\end{table}
The effective carrier masses in perovskites are expected to be of order a few $m_e$ so the most physically plausible value for $V_{bi}$ is close to zero. The $\log J$ vs. $(V+V_{bi})^{1/4}$ plot with $V_{bi}=0$ is shown in figure~\ref{PartDeplZeroVbi:OK} and, it is clear that only the high temperature data gives a good fit in this case.
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics{ZeroVbi_paper.eps}}
\caption{Forward bias \emph{I-V} fitted to Schottky thermionic emission model under partial depletion with zero $V_{bi}$.}
\label{PartDeplZeroVbi:OK}
\end{figure}
Provided the mobility of the carriers is not too low, so that they only spend a short time near the potential barrier maximum, $\epsilon$ in equation~(\ref{Schottky:OK}) should be replaced by $\epsilon_{\infty}$. Substituting expression~(\ref{FieldPartDepl:OK}) for $E_m$ into (\ref{Schottky:OK}) shows that the slopes in figure~\ref{PartDeplZeroVbi:OK} allow the product $N_D/\epsilon_\infty ^2\epsilon_{dc}$ to be calculated. Assuming $\epsilon_\infty=6$ and $\epsilon_{dc}\approx 400$ gives a rough estimate for the concentration of ionised impurities $N_D\approx 10^{20}$--$10^{22}$cm$^{-3}$.
\subsection*{Poole-Frenkel conduction}
In insulating samples, field-enhanced emission of carriers from traps can lead to Poole-Frenkel currents, which have a functional field dependence similar to Schottky emission currents with
\begin{equation}
J=\sigma_0E\exp\left(-\frac{E_t}{kT}\right)\exp(\beta_{PF}\sqrt{E}),
\label{PooleFrenkel:OK}
\end{equation}
where $E_t$ is the trap energy, $\beta_{PF}=2\beta$ and $\sigma_0$ is the sample dependent zero-field conductivity. Poole-Frenkel conduction is a bulk mechanism and therefore it is usual to assume $E=V/d$. Thickness dependence can be very useful in discriminating between bulk and interface effects, however, as already mentioned such a study was not possible in our case. Thus to test the Poole-Frenkel model we have to rely purely on the fitting of the \emph{I-V} curves. Figure~\ref{PooleFrenkelFig:OK} shows that the data fit equation~(\ref{PooleFrenkel:OK}) extremely well over the whole two decades in voltage at almost all temperatures. The relative dielectric constant extracted from the slopes was found to be 6.3 at room temperature and in the range of 6.3--7.4 for the six curves shown with no specific temperature dependence. These values are very close to the optical dielectric constant. The trap energy was found to be 1.0eV.
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics{PoolFrenk_.eps}}
\caption{Forward bias data showing a very good fit to the Poole-Frenkel model over two decades in bias.}
\label{PooleFrenkelFig:OK}
\end{figure}
\subsection{Capacitance-voltage}
The capacitance was measured as a function of dc bias using a 100kHz 25mV tickle voltage at high and low temperatures with typical curves shown in figure~\ref{C_V:OK}(a). The room temperature zero-bias dielectric constant extracted from the capacitance measurement via $\epsilon=Cd/A\epsilon_0$ was found to be around 460. The a-axis and c-axis dielectric constants for bulk PbZr$_{0.4}$Ti$_{0.6}$O$_3$ are 498 and 197 respectively \cite{Haun}. We believe our samples to be preferentially [111] oriented and therefore the extracted dielectric constant is not far from the value of 398 expected for perfectly [111] oriented bulk PZT. Figure~\ref{C_V:OK}(b) shows a Curie-Weiss plot for the inverse zero-bias dielectric constant obtained from the capacitance measurements. From the gradient of the plot and equation~(\ref{InvEps:OK}) a Curie constant $C=4.44\times 10^5$K$^{-1}$ was calculated.
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics{CV_Selected.eps}
\includegraphics{CurieWeiss.eps}}
\caption{(a) Capacitance-voltage curves at different temperatures. (b) Curie-Weiss plot for the inverse of the relative dielectric constant.}
\label{C_V:OK}
\end{figure}
If the samples were partially depleted, a change in bias would result in a change of the depletion widths and consequently of the depletion layer capacitance. The concentration profile of ionised impurities can therefore in principle be calculated from the \emph{C-V} measurements using
\begin{equation}
N_D(V)=\frac{2}{q\epsilon_{dc}\epsilon_0\frac{d}{dV}\left(\frac{1}{C^2}\right)},
\end{equation}
where each value of $V$ probes $N_D$ at a particular depth within the film. In practice, however, only the mobile carriers respond to the ac tickle voltage used for the \emph{C-V} measurements, thus only mobile carrier concentrations ($n$ or $p$) can be obtained. Using this method and again assuming $\epsilon_{dc}=400$ we obtained a value of $\sim 3\times 10^{18}$cm$^{-3}$ for the mobile carrier concentration.
\subsection{Ferroelectric properties}
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics{LandauFit_T1.eps}
\includegraphics{Pr2_vs_T.eps}}
\caption{(a) Room temperature \emph{P-V} loop; (b) Square of the remanent polarization $P_r$ as a function of temperature.}
\label{P_V:OK}
\end{figure}
A typical ferroelectric hysteresis loop is shown in figure~\ref{P_V:OK}a, illustrating the very well defined coercive voltages and a symmetric shape with very little imprint. The remanent polarization of 23$\mu$C/cm$^{2}$ is significantly smaller than the 33$\mu$C/cm$^{2}$ expected for bulk [111] PZT of the same composition.
The capacitors are known to be under compressive strain, so strain effects cannot be responsible for the lowering of the polarization as they would increase it. It was observed that $P_r$ progressively decreases after each etching step during the fabrication process and must therefore be associated with introduction of defects such as space charge or perhaps passive interfacial layers.
Within the framework of Landau-Ginzburg-Devonshire (LGD) theory, for ferroelectrics with a second order phase transition and with polarization low enough that the $P^6$ terms in the free energy expression can be neglected, the hysteresis loops are given by a cubic equation
\begin{equation}
E_3=\alpha P_3 + \beta P_3^3,
\label{cubic:OK}
\end{equation}
where the field and polarization are assumed to be along the c-axis (hence the subscript 3)and $\alpha$ and $\beta$ are the standard bulk LGD coefficients. Provided the coefficient coupling the polarizations along the different axes is smaller than $\beta$ or at high enough fields, the field $E_{[111]}$ and polarization $P_{[111]}$ along [111] are also expected to be related by equation~(\ref{cubic:OK}) but with modified coefficients. As was shown by in \cite{Pertsev} and \cite{Catalan} the effect of substrate strain is also to rescale the LGD coefficients. From equation~(\ref{cubic:OK}) the inverse susceptibility $\chi$, and the spontaneous polarization are given by
\begin{eqnarray}
P_s^2=-\frac{\alpha}{\beta}=\frac{(T_C-T)}{\beta\epsilon_0 C}, \;\;\;\;\label{Ps2:OK}\\
\frac{1}{\epsilon\epsilon_0}\approx \frac{1}{\chi}=-2\alpha=\frac{2(T_C-T)}{\epsilon_0 C}.
\label{InvEps:OK}
\end{eqnarray}
The plot of $P_r^2$ vs. $T$ is shown in figure~\ref{P_V:OK}(b) and was used to extract the values of $\beta '=1.52\times 10^9$ m$^5$C$^{-2}$F$^{-1}$ and $T_C '=620$~K; the latter together with the previously obtained Curie-Weiss constant $C$ give the parameter $\alpha '$ as a function of temperature; the primes are used to show that these are no longer the bulk LGD coefficients, but their effective values modified by strain, space charge and the fact that the samples are not [100] oriented.
In another paper \cite{Zubko} we have studied phenomenologically the effects of space charge on hysteresis loops for fully and partially depleted samples with various dopant concentrations. Our results showed that increasing the amount of space charge caused a decrease in polarization and coercive field values as well as an increase in the observable dielectric constant. We also found that in the case of fully depleted films the effect of space charge was to simply rescale the original $\alpha$ and $\beta$ coefficients; the \emph{P-E} loops and the zero-field dielectric constant are completely depermined by these rescaled coefficients via (\ref{cubic:OK}) and (\ref{InvEps:OK}). This was not the case for partially depleted films, where the hysteresis loops no longer fitted a cubic relationship at all.
The LGD fit using the experimentally extracted $\alpha '$ and $\beta '$ coefficients is shown in figure~\ref{P_V:OK}(a) and is almost perfect, which would be consistent with the idea of a fully depleted sample, as would be expected for an insulating film.
\section{Discussion}
We have presented a large amount of data analyzed in various ways to try and find some clues as to the nature of the leakage mechanism in these commercial PZT capacitors. We now try to summarise our findings and make some conclusions.
SCL conduction has been ruled out as the \emph{I-V} do not seem to satisfy the required power law or the possible temperature dependences. We also rule out Fowler-Nordheim tunnelling precisely because of the strong temperature dependence; the \emph{I-V} also did not fit the $\log I/V^2\propto 1/V$ law for any significant voltage range.
Schottky thermionic emission assuming a uniform internal field distribution yields an unphysical value for the dielectric constant, as does the modified Schottky model proposed by Simmons \cite{Simmons} and used by Zafar \emph{et al.} \cite{Zafar, JimsBook} to fit their data on (Ba,Sr)TiO$_3$ thin films. Thus both these mechanisms can be ruled out. Fow completeness, we should note that the ``standard'' Schottky equation (\ref{Schottky:OK}) and that due to Simmons are just limiting cases of the more general thermionic emission-diffusion model by Crowell and Sze (see \cite{Sze, Baniecki}) which predicts the same \emph{J-E} expression as (\ref{Schottky:OK}) but with $A^*$ replaced by $A^{**}(E)$. Due to the unknown field dependence of $A^{**}$ it is difficult to fit data to this generalized model, however, this field dependence is usually much weaker than exponential and thus any functional deviations from (\ref{Schottky:OK}) are significant mainly at low fields. Our data spans a large range of biases and we have already ruled out the two limiting cases of the thermionic emission-diffusion model, thus it is probably safe to rule it out completely for the case of uniform fields on the basis of the unphysical value it gives for $\epsilon$ upon fitting.
If the sample is assumed to be partially depleted with the field at the potential maximum given by (\ref{FieldPartDepl:OK}) it is possible to obtain a fairly good fit to the Schottky equation (\ref{Schottky:OK}) with physically plausible parameters, at least over a limited range of applied biases and temperatures. However, the fits yield a fairly high value for the concentration of ionised dopants that, according to our theoretical calculations \cite{Zubko}, is expected to produce a significant deformation of the hysteresis loop, which is not observed, and is also in fact expected to modify the internal field profile from a linear to a cubic, making equation~(\ref{FieldPartDepl:OK}) invalid and resulting in the loss of self-consistency in the analysis. Even without resorting to theoretical modelling, there seems to be a large discrepancy between the ionised dopant concentration of $10^{20}$--$10^{22}$cm$^{-3}$ extracted from \emph{I-V} data and the carrier concentration of $10^{18}$cm$^{-3}$ obtained from \emph{C-V} measurements. The mobile carrier density which can follow the ac signal used in \emph{C-V} measurements \emph{is} expected to be lower than $N_D$ \cite{Pintilie} but not by three or four orders of magnitude. The partial depletion model, although unlikely in this case, is very difficult to rule out definitively as it contains many adjustable parameters such as non-uniform dopant concentrations, unknown $m^*$, $V_{bi}$ and the field dependence of $A^{**}$ which cannot be independently measured.
The last possibility we have considered was Poole-Frenkel conduction which so far seems to fit all the experimental evidence available. The \emph{I-V} give very convincing straight lines on the $\log (I/V)$ vs $V^{1/2}$ plot for all temperatures and almost all two decades of applied bias. The extracted $\epsilon_\infty\approx 6.3$ shows good agreement with other independent measurements of the refractive index of PZT; and the assumption of a uniform field distribution is consistent with the shape of the \emph{P-E} loops according to phenomenological modelling. The \emph{I-V} also show little dependence on the polarity of the applied bias, again confirming that the leakage currents are probably bulk limited.
Poole-Frenkel conduction in PZT has been previously reported by Chen \emph{et al.} \cite{Chen_PF}. They observed highly symmetric \emph{I-V} despite having different top and bottom electrodes (Au and Pt respectively) with Poole-Frenkel conduction identified above 40kV/cm by the extracted dielectric constant value of 6.3, which is in perfect agreement with ours. Nagaraj \emph{et al.} \cite{Nagaraj} also observed this mechanism for 120--360nm thin films with epitaxial (La,Sr)CoO$_3$ electrodes. In their case, it was identified at fields above $\sim 100$kV/cm and yielded an activation energy of 0.5--0.6eV attributed by the authors to ionisation of Ti$^{4+}$ to Ti$^{3+}$.
Nagaraj \emph{et al.} refer the 0.5eV activation energy to a private communication. Previous papers by the referenced author \cite{Robertson}, however, state that the Ti$^{3+}$ centre lies at least 1eV below the conduction band edge; the value of 1eV for the electronic trap depth was reconfirmed in further publications \cite{Smyth, Smyth2}. The ionisation energies of Pb ions \cite{Robertson} and oxygen vacancies \cite{Smyth2} are believed to be much lower compared with our value of $E_t\approx 1eV$, so electron trapping at Ti$^{4+}$ sites may after all be responsible for the observed conductivity, as suggested in \cite{Nagaraj}. $E_t$ of 1eV is also similar to the activation energy for ionic transport of oxygen vacancies \cite{Smyth}, however, the Poole-Frenkel model explicitly assumes a Coulombic attraction between a singly charged mobile and an oppositely charged stationary species; doubly charged carriers such as oxygen vacancies would change the slope of the $\log (I/V)$ vs $V^{1/2}$ plot. It is thus believed that we are observing electron or hole but not ionic conduction in these integrated PZT capacitors.
Before concluding we would like to mention two more points. The first relates to distinguishing Poole-Frenkel and Schottky conduction on the basis of the extracted dielectric constant. If the value extracted from the Poole-Frenkel plot agrees with the optical dielectric constant, while that from the Schottky plot does not, then it is likely that the mechamism is indeed Poole-Frenkel conduction. However, the reverse is not necessarily true. As was discussed by Simmons \cite{Simmons_PF} and later by Mark and Hartman \cite{MarkHartman}, in some cases a factor of $\frac{1}{2}$ can enter the exponential term containing $\beta_{PF}$ in equation~(\ref{PooleFrenkel:OK}) giving equal slopes for both Schottky and Poole-Frenkel plots.
Secondly, we note that our phenomenological arguments regarding the fitting of the \emph{P-V} loop with coefficients obtained from measurements of $\epsilon$ and $P_r$ are only valid if the field is applied along the polar axis. If the field is along [111] (as in our case) and the polarization is along the c-axis, then the main contribution to the measured $\epsilon$ will be from the larger a-axis dielectric constant $\epsilon_a$, whereas the measured polarization will be the projection of $P_3$ onto the [111] direction. In this case the extracted Landau coefficients should not be able to reproduce the \emph{P-V} loop. Thus is we assume that the fit in figure~\ref{P_V:OK}(a) is not just fortuitous, then our polarization must be pointing along [111]. This may be possible, despite the tetragonal composition, through strain effects or through field induced change of symmetry to rhombohedral as described by Bell \cite{Bell}. However, so far we have not directly measured the direction of the polarization in our samples, nor have we explored in much detail the effects of different field orientations in our phenomenological modelling. Here we merely wish to point out that \emph{P-V} data may be of use in providing clues as to the extent of depletion in ferroelectric thin films.
\section{Conclusions}
The leakage mechanism in commercial PZT capacitors has been studied in detail. It was shown that standard automated \emph{I-V} tests are not always appropriate for ferroelectrics due to the large difference in relaxation times for current transients at different fields. We have thus used current-time measurements to ensure that reliable steady state \emph{I-V} were obtained. Together with \emph{C-V} these data were carefully analysed and it was concluded that Poole-Frenkel emission of carriers from traps was the most likely conduction mechanism. A phenomenological model was used to confirm that the assumed field distribution was consistent with the observed shape of the ferroelectric hysteresis loop.
\section{Acknowledgements}
The authors are very grateful to Lucian Pintilie and Marin Alexe for their help with the low temperature data, and to Finlay Morrison and to Gustau Catalan for interesting discussions and many useful suggestions.
| {
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{"url":"https:\/\/physics.stackexchange.com\/questions\/487758\/are-the-electron-and-positron-doublets-in-any-mathematical-sense","text":"# Are the electron and positron doublets in any mathematical sense?\n\nI know left handed electrons and electron neutrinos are isospin doublets. Is there any analogous way in which an electron and a positron are related? My first thought is some sort time-parity doublet.\n\n\u2022 In a Dirac spinor, two of the four components describe the particle and the other two describe the antiparticle. \u2013\u00a0G. Smith Jun 24 at 1:54\n\u2022 Yes, you can look at the outer automorphisms of the Lorentz group, they are \"doublets\" under CPT. One does not need Dirac spinors for that, already a Weyl fermion transforms under these transformations. Since this is a $\\mathbb{Z}_2$, not many would call this a \"doublet\", though. \u2013\u00a0user178876 Jun 24 at 1:58","date":"2019-10-18 01:08:39","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\": 0.8407052755355835, \"perplexity\": 557.722408069528}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986677412.35\/warc\/CC-MAIN-20191018005539-20191018033039-00050.warc.gz\"}"} | null | null |
{"url":"https:\/\/justindomke.wordpress.com\/2008\/10\/29\/linear-classifiers-and-loss-functions\/","text":"## Linear Classifiers and Loss\u00a0Functions\n\nA linear classifier is probably the simplest machine learning technique. Given some input vector $\\bf x$, predict some output $y$. One trains a vector of \u201cweights\u201d $\\bf w$, that determine the behavior of the classifier. Given some new input $\\bf x$, the prediction $f$ will be:\n\n$f({\\bf x}) = I[{\\bf w}^T {\\bf x}>0]$\n\nHere, $I[\\text{expr}]$ is the indicator function\u2013 $1$ if $\\text{expr}$ is true, and -1 otherwise. So, $y$ is one of two classes.\n\nOften the classifier will be written like $I[{\\bf w}^T {\\bf x}+b>0]$, i.e. including a \u201cbias\u201d term $b$. Here I\u2019ll ignore this\u2013 if you want a bias term, just append a $1$ to every vector $\\bf x$.\n\nNow, it seems like nothing could be simpler than a linear classifier, but there is considerable subtlety in how to train a linear classifier. Given some training data $\\{({\\hat {\\bf x}},{\\hat y})\\}$, how do we find the vector of weights $\\bf w$?\n\nThere are various popular ways of doing this. Three popular techniques are:\n\n\u2022 The Perceptron Algorithm\n\u2022 Logistic Regression\n\u2022 Support Vector Machines (Hinge loss)\n\nThe perceptron algorithm works, roughly speaking as follows: Go through the training data, element by element. If one datum is misclassified, slightly nudge the weights so that it is closer to being correct. This algorithm has the remarkable property that, if the training data is linearly separable, it will find a $\\bf w$ that correctly classifies it in a finite number of iterations. (The number of iterations that it takes depends on the \u201chow separable\u201d the data is, making fewer mistakes on data with a larger margin.)\n\nUnfortunately, the assumption that the training data is linearly separable is basically never true in practice. There is still a bound in terms of how much points would need to be moved in order to be linearly separable, but it isn\u2019t as strong as we might like. (I don\u2019t know what the folklore is for how this tends to work in practice.)\n\nLogistic regression and support vector machines take a different approach. There, a \u201closs function\u201d is defined in terms of the weights, and one just optimizes the loss to find the best weights. (I\u2019m ignoring regularization here for simplicity.) For logistic regression, the loss is\n\n$L = -\\sum_{({\\hat {\\bf x}},{\\hat y})} \\log p({\\hat y}|{\\hat {\\bf x}})$\n\nwhere $p(y=1|{\\bf x})=\\sigma({\\bf w}^T {\\bf x})$, for a sigmoid function $\\sigma$, and $p(y=-1|{\\bf x})=1-p(y=1|{\\bf x})$.\n\nFor a support vector machine, the loss is\n\n$L = \\sum_{({\\hat {\\bf x}},{\\hat y})} (1-{\\hat y} \\cdot {\\bf w}^T {\\hat {\\bf x}})_+$\n\nwhere $(a)_+$ is $a$ if $a>0$ and 0 otherwise. This is a hinge loss. Notice that it will be zero if ${\\hat y} \\cdot {\\bf w}^T {\\hat {\\bf x}} > 1$, or if that particular training element is comfortably on the correct side of the decision boundary. Otherwise, the \u201cpain\u201d is proportional to \u201chow wrong\u201d the classification is.\n\nBoth the hinge-loss and logistic regression are convex loss functions. This means that if we apply a nonlinear search procedure to find a local minima, that will also be the global minima.\n\nThese different loss functions are compared here.\n\nWHY NOT DO THE OBVIOUS THING?\n\nNow, the critical point about these three above methods is that none of them do the seemingly obvious thing: find the vector of weights ${\\bf w}$ that has the lowest classification error on the training data. Why not? Some would defend logistic regression or SVM for theoretical reasons (namely a meaningful probabilistic interpretation, and the reasonableness and theoretical guarantees of max-margin methods, respectively).\n\nHowever, probably the more significant hurdle is computational considerations. Namely, the problem of finding the weights with lowest classification error is (in order of increasing horribleness) non-convex, non-differentiable, and NP-hard.\n\nIn fact it is NP-hard even to find an approximate solution, with worst-case guarantees. (See citation 2 below, which, interestingly, gives an approximate algorithm in terms of a property of the data.)\n\nNevertheless, if classification error is what we want, I don\u2019t see how it makes sense to minimize some alternative loss function. As such, I decided today to try the following algorithm.\n\n1. Apply an SVM to get an initial solution.\n2. Apply heuristic search to minimize classification error, initialized to the solution of step 1.\n\nNow, I am well aware of the problems with non-convex optimization. However, simply the fact that logistic regression or hinge loss is convex is not an argument in their favor. If theoretical considerations dictate that we minimize classification error, just substituting a different loss, and then refusing to look at the classification rate is highly questionable. Sure, that can lead to a convex optimization problem, but at the that\u2019s because a different problem is being solved! The goal is accurate prediction of future data, not accurate minimization of a loss on the training data. If we use our best convex approximation to initialize the heuristic optimization of the loss we really want, we will never do worse.\n\nEXPERIMENT\n\nThere are some heuristics available for doing this (Reference 3), but they seem a little expensive. I decided to try something really simple.\n\n1. Fit a classifier (by hinge loss or logistic regression).\n2. Throw out a few of the most badly misclassified points.\n3. Repeat until all remaining points are correctly classified.\n\nThe idea is that, by \u201cgiving up\u201d on badly misclassified points, the boundary might be movable into a position where it correctly classifies other points. There is no guarantee in general that this will find a linear classifier with a lower misclassification rate, but it should not do worse.\n\nTo test this out, I got some data from the MNIST handwritten digit database. To make the problem harder, I took one class to be random images from either the set of 1\u2019s or 2\u2019s, while the other class was 3\u2019s or 4\u2019s. The images were downsampled to 7\u00d77, and a constant of one added. So we are looking for a linear classifier in a 50 dimensional space.\n\nThe data was trained on a database of 10,000 examples, with a test set of 10,000 examples.\n\nHere are the results where we throw out the worst 100 training points in step 2:\n\nHere are the results when we throw out 20 at a time:\n\nHere are the results when we throw out 5 at a time:\n\nThere seems to be a trade-off in step 2: Fewer models need to be fit if we throw out more points at a time. However, this seems to come with a small penalty in terms of terms of the classification error on the final model.\n\nSo, in summary\u2013 a drop in classification error on test data from .941 to .078. Thats a 17% drop. (Or a 21% drop, depending upon which rate you use as a base.) This from a method that you can implement in basically zero extra work if you already have a linear classifier. Seems worth a try.\n\nReferences:\n\n[2] Efficient Learning of Linear Perceptrons by Shai Ben-david, Hans Ulrich Simo\n\n[3] Optimizing 0\/1 Loss for Perceptrons by Random Coordinate Descent by L. Li and H.-T. Lin\n\nThis entry was posted in Uncategorized and tagged , , , . Bookmark the permalink.\n\n### 3 Responses to Linear Classifiers and Loss\u00a0Functions\n\n1. Mark Reid says:\n\nInteresting post. It\u2019s always good to question the prevailing thinking and try simple experiments like the one you\u2019ve proposed.\n\nOne part that is unclear in your experiment is exactly how you trained the linear classifier. Was it logistic regression or an SVM with hinge loss? If it was the latter, was it using a hard or soft margin?\n\nI ask because, at least conceptually, your throwing out of examples on the training set is similar to the use of slack variables to admit misclassifications in soft margin SVMs. With the nu-SVM you can even control the fraction of allowable errors on the training set.\n\nI\u2019d be curious to see whether the same reduction in test set error can be achieved by increasing the nu parameter on a soft-margin SVM.\n\n2. justindomke says:\n\nThanks for the comment. I couldn\u2019t remember how I fit the models, but looking at it today, it looks like it was the hinge loss. (I wouldn\u2019t call it an SVM, just a straightforward primal minimization of $\\sum_{({\\hat {\\bf x}},{\\hat y})} (1-{\\hat y} \\cdot {\\bf w}^T {\\hat {\\bf x}})_+$).\n\nThrowing out the examples is similar to slack variables, but I don\u2019t think it is quite the same. Slack variables still incur increasing cost the further away the misclassified points are from the decision boundary, while the misclassification error only cares which side of the decision boundary points are on. The picture I am thinking about is something like this, where \u201c+\u201d indicates an example of one class, and \u201c-\u201d one of another. (This is supposed to be a picture of some training data in 1-D)\n\n+++ + - - - - - ............................................ +\n\nClearly, the way to minimize misclassification error is to stick the decision boundary between the two groups on the left. A convex loss will be worried about the horrible margin for the \u201c+\u201d on the right. (Though it might still put the decision boundary in the right place, depending on the distances, and how many non-outlying points there are, etc.)\n\nI need to look at nu-SVMs in more detail. Thanks for the pointer.","date":"2017-06-26 03:37:45","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\": 28, \"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\": 0.7580884695053101, \"perplexity\": 547.9128879085405}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320669.83\/warc\/CC-MAIN-20170626032235-20170626052235-00480.warc.gz\"}"} | null | null |
This preview shows page 32 - 35 out of 68 pages.
2.8 Spatial Domain methods The value of a pixel with coordinates (x, y) in the enhanced image is the result of performing some operation on the pixels in the neighborhood of (x, y) in the input image, F. Neighborhoods can be any shape, but usually they are rectangular . Figure 2.7 Spatial domain of an image.
22 Image Processing in the spatial domain can be Expressed by: g (m, n) =T (f (m, n)). Where f (m, n) is the input image, g (m, n) is the processed Image and T is the operator defining the modification process. The operator ‗T' is typically a single -valued and monotone function that can operate on individual pixels or on selective value of the input Image is used to compute the corresponding pixel value for the output Image. Within the input image are used to compute the modified image at any given point. One can consider point processing as a special case of region processing where the region is composed of a single pixel. The point-processing operator can also be expressed by: S=T(r), Where r and s are variables denoting the intensity level of f( m , n) and g( m ,n) at any point ( m ,n). The following sections describe several point-and region-based image enhancement techniques. The second article will also discuss spatial smoothing, which is another example of region-based image processing, and compare its advantages and dies- advantages with transform domain methods. The smallest possible neighborhood is of size 1 X 1. In this case g depends only on the value of f at a single point (x, y) and T . 2.8.1 Creating Negative of an Image The most basic and simple operation in digital image processing is to compute the negative of an image. The pixel gray values are inverted to compute the negative of an image. For example, if an image of size R x C, where R represents number of rows and C represents number of columns, is represented by me (r, c). The negative N(r, c) of image I(r, c) can be computed as N(r, c) = 255- I(r, c) where 0≤ r ≥R and 0≤ c ≤C. It can be seen that every pixel value from the original image is subtracted from the 255. The resultant image becomes negative of the original image. Negative images are useful for enhancing white or grey detail embedded in dark regions of an image .
23 Figure 2.8 Creating Negative of an image . 2.8.2 Intensity Transformation Intensity transformations are among the simplest of all image processing techniques. The value of pixels, before and after processing, will be denoted by r and s, respectively. An intensity transformation function of the form s=T(r) . Where T is a transformation that maps a pixel value r into a pixel value s. because we are dealing with digital quantities, values of a transformation function typically are stored in a one-dimensional array and the mappings from r to s are implemented via table lookups. For an 8-bit environment, a lookup table containing the values of T will have 256 entries . | {
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Pierluigi Pairetto (nacido el 15 de julio de 1952) es un ex-árbitro de fútbol italiano. Nació en Torino. Entre 1981 y 1998 empezó a arbitrar partidos de la Serie B y la Serie A. Además, fue designado para pitar la final de la Eurocopa 1996, que enfrentó a República Checa con Alemania en Wembley.
Carrera
Arbitró su primer partido en Serie A en 1981, y arbitró las finales de Copa de Italia los años 1991 y 1996, y las finales de la supercopa de Italia en 1992 y 1994.
arbitró la final de la Recopa de Europa 1995-96, entre Paris Saint-Germain y Rapid Viena en Bruselas.
ha sido nombrado uno de los mejores árbitros de Italia, ha jugado un papel importante en el ámbito internacional, habiendo sido considerado en los años noventa en el Top Ten (Nº 2 del mundo en 1996) por los árbitros IFFHS.
Recientemente obtuvo un gran reconocimiento internacional siendo nombrado en el lugar 7 (junto al eslovaco Lubos Michel) de los mejores árbitros de la historia de IFFHS en el ranking al final de 2008.
Eurocopa 1992
3-1
Fue el único partido de primera ronda que dirigió Pairetto, perteneciente al grupo B. Pese a la tradicional rivalidad entre ambas selecciones, el italiano hizo un buen papel, y selección neerlandesa ganó el partido por 3 a 1.
Mundial de 1994
3-2
En octavos de final, Pairetto árbitro Rumania-Argentina. La selección argentina se disponía a jugar el partido por los octavos de final con un plantel afectado futbolística y anímicamente por la expulsión de Diego Armando Maradona del torneo. El equipo tenía delante un difícil partido ante , quien había clasificado en el primer lugar del Grupo A. El encuentro finalizaría con una derrota de los albicelestes por 3 a 2, lo que significó el fin de la participación argentina en esa Copa del Mundo.
Eurocopa 1996
0-2
En 1996 representa a la FIGC durante el campeonato europeo de fútbol en Inglaterra, están entre los doce árbitros seleccionados para la competencia. Fue designado para dirigir el partido entre Escocia y la Inglaterra, que tuvo lugar en Wembley el 15 de junio de 1996.
1-2 ; La Final
Además de un excelente rendimiento, la eliminación prematura en cómplice de la Nacional Italiana, la UEFA le considera el mejor del torneo y 24 de junio oficialmente designado para dirigir el final de la Eurocopa, que se celebra 30 de junio en Wembley, con la victoria de Alemania al primero Gol de oro de la historia de fútbol, segnado con Oliver Bierhoff.
Designación
Después de una carrera como árbitro en 1998 fue nombrado tribunal, el primero en la terzera división italiana (entre 1998 y 1999), entonces en la Serie A y B con Paolo Bergamo entre 1999 y 2005. Fue entre 2002 y 2006 vicepresidente del Comité de la UEFA Comisión de Árbitros (el primer italiano en la historia).
Calciopoli
En mayo del 2006 su nombre fue ligado al escando deportivo en Italia. Ocupó la posición de vicepresidente de la Comisión de Árbitros de la UEFA hasta el verano de 2006, cuando se descubrió que había estado en contacto telefónico regular con el entonces director general de la Juventus FC, Luciano Moggi sobre los árbitros que serían seleccionados para los partidos favoreciendo al mencionado club. Como resultado de su implicación en este escándalo, que inicialmente recibió una suspensión de dos años y seis meses, luego se incrementó a tres años y medio.
Referencias
Árbitros de fútbol de Italia
Árbitros de fútbol de la Copa Mundial de 1990
Árbitros de fútbol de la Copa Mundial de 1994
Nacidos en Turín | {
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Q: Bootstrap 4: Align dynamically added columns I am trying to use bootstrap 4 grid.
The number of cols vary in number and are added dynamically. It leads to adding more than 12 cols within a row(which is not ideal - but not sure how to deal with such scenarios when the cols are dynamically added).
The outcome: The grid works fine and rearranges the cols as expected. However, I think the gutter gets added twice for the inner cols. Refer to the image attached.
I searched the docs, used justify-space-between/around - but no luck
https://jsfiddle.net/8L3dt1xh/1/
<div class="row">
<div class="col-lg-2 col-md-3 col-sm-4 col-6"> .. image .. </div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6"> .. image .. </div>
... many such divs ..
<div class="col-lg-2 col-md-3 col-sm-4 col-6"> .. image .. </div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6"> .. image .. </div>
</div>
Any pointer on how to get uniform spacing between cols will be a great help.
A: Another way is to add a padding 15px to both sides of the container which will give a consistent result throughout the rows, no matter how many items
.row {
padding: 0 15px;
}
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.5.2/css/bootstrap.min.css">
<section class="">
<div class="container-fluid">
<div class="row">
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
<div class="col-lg-2 col-md-3 col-sm-4 col-6">
<a href="/">
<figure class="figure">
<img src="https://via.placeholder.com/850X150?text=X" style="height:150px" class="img-fluid rounded" alt="dummy">
<figcaption class="figure-caption text-center">Check</figcaption>
</figure>
</a>
</div>
</div>
</div>
</section>
A: I would personally go for building a flexbox myself, rather than using the Bootstrap's Grid system, because then I can add my own paddings on the items and calculate an uniform spacing between them.
<section class="items">
<a href="#" class="item">
<figure />
</a>
<a href="#" class="item">
<figure />
</a>
...
</section>
First of all, you want to make .items as a flexbox, displayed as a wrap-able row:
@import "../node_modules/bootstrap/scss/functions";
@import "../node_modules/bootstrap/scss/variables";
@import "../node_modules/bootstrap/scss/mixins";
.items {
display: flex;
flex-flow: row wrap;
}
I'm using SASS here but you can see the generated CSS in the demo.
I also saw you had .col-6, .col-sm-4, etc classes on each column. You can replicate that by setting the width of the .item on each breakpoint:
@import "../node_modules/bootstrap/scss/functions";
@import "../node_modules/bootstrap/scss/variables";
@import "../node_modules/bootstrap/scss/mixins";
.items {
display: flex;
flex-flow: row wrap;
.item {
width: calc(100%/2); // 2 per row
@include media-breakpoint-up(sm) {
width: calc(100%/3); // 3 per row
}
@include media-breakpoint-up(md) {
width: calc(100%/4); // 4 per row
}
@include media-breakpoint-up(lg) {
width: calc(100%/6); // 6 per row
}
}
}
Now it's time to calculate the gutter between two .items. Indeed, if you add padding on each .item, the spacing between 2 of them will be doubled. But you can easily accommodate that by adding half of the gutter/spacing on their parent flexbox, .items.
It would be even easier if you're using SASS, because you can declare a variable for the gutter spacing you want and you can do the calculations based on that:
@import "../node_modules/bootstrap/scss/functions";
@import "../node_modules/bootstrap/scss/variables";
@import "../node_modules/bootstrap/scss/mixins";
$items-gutter: 2rem;
.items {
display: flex;
flex-flow: row wrap;
padding: $items-gutter/2;
.item {
padding: $items-gutter/2;
width: calc(100%/2); // 2 per row
@include media-breakpoint-up(sm) {
width: calc(100%/3); // 3 per row
}
@include media-breakpoint-up(md) {
width: calc(100%/4); // 4 per row
}
@include media-breakpoint-up(lg) {
width: calc(100%/6); // 6 per row
}
}
}
That would add 1rem padding on the flexbox .items, and 1rem padding on each .item. That would make up total 2rem paddings around each item.
demo: https://jsfiddle.net/davidliang2008/rv0knh8b/15/
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Q: Direct (and simple!) AJAX upload to AWS S3 from (AngularJS) Single Page App I know there's been a lot of coverage on upload to AWS S3. However, I've been struggling with this for about 24 hours now and I have not found any answer that fits my situation.
What I'm trying to do
Upload a file to AWS S3 directly from my client to my S3 bucket. The situation is:
*
*It's a Single Page App, so upload request must be in AJAX
*My server and my client are not on the same domain
*The S3 bucket is of the newest sort (Frankfurt), for which some signature-generating libraries don't work (see below)
*Client is in AngularJS
*Server is in ExpressJS
What I've tried
*
*Heroku's article on direct upload to S3. Doesn't fit my client/server configuration (plus it really does not fit harmoniously with Angular)
*ready-made directives like ng-s3upload. Does not work because their signature-generating algorithm is not accepted by recent s3 buckets.
*Manually creating a file upload directive and logic on the client like in this article (using FormData and Angular's $http). It consisted of getting a signed URL from AWS on the server (and that part worked), then AJAX-uploading to that URL. It failed with some mysterious CORS-related message (although I did set a CORS config on Heroku)
It seems I'm facing 2 difficulties: having a file input that works in my Single Page App, and getting AWS's workflow right.
The kind of solution I'm looking for
If possible, I'd like to avoid 'all included' solutions that manage the whole process while hiding of all of the complexity, making it hard to adapt to special cases. I'd much rather have a simple explanation breaking down the flow of data between the various components involved, even if it requires some more plumbing from me.
A: You can use the ng-file-upload $upload.http method in conjunction with the aws-sdk getSignedUrl to accomplish this. After you get the signedUrl back from your server, this is the client code:
var fileReader = new FileReader();
fileReader.readAsArrayBuffer(file);
fileReader.onload = function(e) {
$upload.http({
method: 'PUT',
headers: {'Content-Type': file.type != '' ? file.type : 'application/octet-stream'},
url: signedUrl,
data: e.target.result
}).progress(function (evt) {
var progressPercentage = parseInt(100.0 * evt.loaded / evt.total);
console.log('progress: ' + progressPercentage + '% ' + file.name);
}).success(function (data, status, headers, config) {
console.log('file ' + file.name + 'uploaded. Response: ' + data);
});
A: I finally managed. The key points were:
*
*Let go of Angular's $http, and use native XMLHttpRequest instead.
*Use the getSignedUrl feature of AWS's SDK, instead on implementing my own signature-generating workflow like many libraries do.
*Set the AWS configuration to use the proper signature version (v4 at the time of writing) and region ('eu-central-1' in the case of Frankfurt).
Here below is a step-by-step guide of what I did; it uses AngularJS and NodeJS on the server, but should be rather easy to adapt to other stacks, especially because it deals with the most pathological cases (SPA on a different domain that the server, with a bucket in a recent - at the time of writing - region).
Workflow summary
*
*The user selects a file in the browser; your JavaScript keeps a reference to it.
*the client sends a request to your server to obtain a signed upload URL.
*Your server chooses a name for the object to put in the bucket (make sure to avoid name collisions!).
*The server obtains a signed URL for your object using the AWS SDK, and sends it back to the client. This involves the object's name and the AWS credentials.
*Given the file and the signed URL, the client sends a PUT request directly to your S3 Bucket.
Before you start
Make sure that:
*
*Your server has the AWS SDK
*Your server has AWS credentials with proper access rights to your bucket
*Your S3 bucket has a proper CORS configuration for your client.
Step 1: setup a SPA-friendly file upload form / widget.
All that matters is to have a workflow that eventually gives you programmatic access to a File object - without uploading it.
In my case, I used the ng-file-select and ng-file-drop directives of the excellent angular-file-upload library. But there are other ways of doing it (see this post for example.).
Note that you can access useful information in your file object such as file.name, file.type etc.
Step 2: Get a signed URL for the file on your server
On your server, you can use the AWS SDK to obtain a secure, temporary URL to PUT your file from someplace else (like your frontend).
In NodeJS, I did it this way:
// ---------------------------------
// some initial configuration
var aws = require('aws-sdk');
aws.config.update({
accessKeyId: process.env.AWS_ACCESS_KEY,
secretAccessKey: process.env.AWS_SECRET_KEY,
signatureVersion: 'v4',
region: 'eu-central-1'
});
// ---------------------------------
// now say you want fetch a URL for an object named `objectName`
var s3 = new aws.S3();
var s3_params = {
Bucket: MY_BUCKET_NAME,
Key: objectName,
Expires: 60,
ACL: 'public-read'
};
s3.getSignedUrl('putObject', s3_params, function (err, signedUrl) {
// send signedUrl back to client
// [...]
});
You'll probably want to know the URL to GET your object to (typically if it's an image). To do this, I simply removed the query string from the URL:
var url = require('url');
// ...
var parsedUrl = url.parse(signedUrl);
parsedUrl.search = null;
var objectUrl = url.format(parsedUrl);
Step 3: send the PUT request from the client
Now that your client has your File object and the signed URL, it can send the PUT request to S3. My advice in Angular's case is to just use XMLHttpRequest instead of the $http service:
var signedUrl, file;
// ...
var d_completed = $q.defer(); // since I'm working with Angular, I use $q for asynchronous control flow, but it's not mandatory
var xhr = new XMLHttpRequest();
xhr.file = file; // not necessary if you create scopes like this
xhr.onreadystatechange = function(e) {
if ( 4 == this.readyState ) {
// done uploading! HURRAY!
d_completed.resolve(true);
}
};
xhr.open('PUT', signedUrl, true);
xhr.setRequestHeader("Content-Type","application/octet-stream");
xhr.send(file);
Acknowledgements
I would like to thank emil10001 and Will Webberley, whose publications were very valuable to me for this issue.
A: To do multipart uploads, or those larger than 5 GB, this process gets a bit more complicated, as each part needs its own signature. Conveniently, there is a JS library for that:
https://github.com/TTLabs/EvaporateJS
via https://github.com/aws/aws-sdk-js/issues/468
A: Use s3-file-upload open source directive having dynamic data-binding and auto-callback functions - https://github.com/vinayvnvv/s3FileUpload
| {
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Music streaming services on the internet are at a very key point in their very short career to date; whoever gets in 'right' could expect to be the dominate player in the market till kingdom come - i.e the next Google. Leading the charge by a considerable distance is Spotify and today it announced deets of what to expect in the next OS update.
The changes in 0.4.3 Spotify have been described by them as a 'next generation' upgrade and focuses on integrating social networking sites such as Facebook as well as its users' personal iTunes music library – basically there's no better way to share what you're listening to with your friends.
If you haven't tried Spotify yet, then you must. It's a two tier model; the first being 100% free for you to listen to a gazillion songs with minor adverts every 20 mins and the second, for the price of a cigarette, is advert free. What's not to like? And no, we don't work for Spotify. | {
"redpajama_set_name": "RedPajamaC4"
} | 251 |
{"url":"https:\/\/manual.q-chem.com\/5.4\/Ch10.S3.SS1.html","text":"Searching....\n\n# 10.3.1 Overview\n\n(December 20, 2021)\n\nQ-Chem incorporates v.\u00a05 of the Natural Bond Orbital (NBO) package for molecular properties and wave function analysis, 374 Glendening E. D., Landis C. R., Weinhold F.\nWiley Interdiscip.\u00a0Rev.: Comput.\u00a0Mol.\u00a0Sci.\n(2012), 2, pp. 1.\n, 1192 Weinhold F., Landis C. R., Glendening E. D.\nInt.\u00a0Rev.\u00a0Phys.\u00a0Chem.\n(2016), 35, pp. 399.\nand can interface to v.\u00a06 374 Glendening E. D., Landis C. R., Weinhold F.\nWiley Interdiscip.\u00a0Rev.: Comput.\u00a0Mol.\u00a0Sci.\n(2012), 2, pp. 1.\nand v.\u00a07 375 Glendening E. D., Landis C. R., Weinhold F.\nJ.\u00a0Comput.\u00a0Chem.\n(2019), 40, pp. 2234.\nof the NBO program as well. The NBO5 package is invoked simply by setting NBO = TRUE in the $rem section and is initiated after the SCF wave function is obtained. If switched on for a geometry optimization, the NBO5 program will only be invoked at the end of the last optimization step. To use either NBO6 or NBO7, the desired version of the NBO program must be purchased, downloaded, and installed separately from nbo.chem.wisc.edu. Q-Chem can then interface to an external version of the NBO program by: (a) setting the NBOEXE environment variable appropriately, and (b) setting both NBO = TRUE and NBO_EXTERNAL = TRUE in the$rem section of the Q-Chem input file. Note that support for (and new downloads of) the NBO6 program were discontinued in 2020. Existing copies should still interface to Q-Chem, and the newer NBO7 program should interface in the same way. The older NBO5 program continues to be a part of Q-Chem and the latter is what runs if NBO = TRUE is set without specifying NBO_EXTERNAL.\n\nUsers should consult the NBO User\u2019s Manual for options and details relating to NBO calculations. NBO analysis is also available for excited states calculated using CIS or TDDFT. Excited-state NBO analysis is less well-developed, and users should be aware that the convergence of the NBO search procedure may be less well-behaved for excited states than it is for ground states. Excited state may require specification of additional NBO parameters in the $nbo section that is described below. Consult Ref. for details and suggestions. NBO NBO Controls the use of the NBO package. TYPE: INTEGER DEFAULT: 0 OPTIONS: 0 Do not invoke the NBO package. 1 Do invoke the NBO package, for the ground state. 2 Invoke the NBO package for the ground state, and also each CIS, RPA, or TDDFT excited state. RECOMMENDATION: None The general format for passing options from Q-Chem to the NBO program is shown below: $nbo\n{NBO program keywords, parameters and options}\n$end Note: 1.$rem variable NBO must be set to TRUE before the $nbo keyword is recognized. 2. Q-Chem does not support facets of the NBO package which require multiple job runs 3. Output of the NBOs can be triggered by the PRINT_ORBITALS and MOLDEN_FORMAT keywords. In this case two MOLDEN sections are written to outfile. The first one corresponds to the regular MOs, the second one to the NBOs. 4. Print-out of the full set of NAOs, NHOs, NBOs, and NLMOs can be triggered via the PLOT keyword in the$nbo section. The files of interest are FILE.31FILE.39 in $QCSCRATCH\/savename. These files can be opened by, e.g., the ChemCraft and Jmol programs. Example 10.6 Basic input for NBO computation on formaldehyde. The NBOs are printed to outfile in Molden format and the full set of files in native NBO format are written to$QCSCRATCH\/savename.\n\n$molecule 0 1 C 0.000000 0.000000 0.523383 O 0.000000 0.000000 -0.671856 H 0.931138 0.000000 1.117280 H -0.931138 0.000000 1.117280$end\n\n$rem METHOD pbe0 BASIS def2-sv(p) NBO 1 PRINT_ORBITALS true MOLDEN_FORMAT true$end\n\n$nbo PLOT$end\n\n\nView output","date":"2022-01-28 09:19:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 7, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4295012652873993, \"perplexity\": 7572.256875295125}, \"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-2022-05\/segments\/1642320305423.58\/warc\/CC-MAIN-20220128074016-20220128104016-00187.warc.gz\"}"} | null | null |
{"url":"https:\/\/datascience.stackexchange.com\/questions\/94776\/should-bag-of-words-in-training-set-include-test-set-data-when-doing-text-classi","text":"# Should bag of words in training set include test set data when doing text classification?\n\nI'm doing text classification to identify 'attacks' from Wikipedia comments using a simple bag of words model and a linear SVM classifier. Because of class imbalance, I'm using the F1 score as my error measure. I'm wondering if the tokens I have in the training data should also include words that exist only in the test data, or does it not matter? I was under the impression that it shouldn't matter since the counts for these features would be zero anyway in the training set. That should make them irrelevant to the model when training. Apparently, that's what some people on SO were saying as well (it didn't find any definitive answer though).\n\nIn order to test this, I decided to train my model both ways and see the difference: once with only features in the training data, and another with features that included test data. N folds for CV were set to 10. I got a very similar CV error for both of them, but when I generated predictions for my test data my F1 score was 0.06 higher for the model which included features from test data - 0.64 vs 0.58. Because this is a Kaggle assignment I cannot see the true labels for the test set. I'm inclined to believe that such a big difference can't simply be random. It seems like including all the features from test data did improve my model, but how could this be? Can anyone give me an explanation?\n\n\u2022 I wrote an answer but it's not clear to me what causes the difference in performance. What is the proportion of the majority class? Also what is the performance on the training set, if you can obtain it? May 22 '21 at 16:22\n\u2022 Thanks for the answer Erwan. I have 80% negative class. Training set performance was very similar at about 0.63 to 0.64 (I'm assuming you meant my CV error. If you mean predictions on training I will have to compute that separately). I was thinking whether it could be due to the elevation in dimensions by adding more features? SVM handles sparse matrices pretty well but I'm not sure whether having a larger, sparser matrix for training is known to improve performance May 23 '21 at 12:52\n\u2022 ok thanks. I also thought about the matrix dimension but I can't think of any reason why it would have this effect. at least I can say that no, it's not known to improve performance (if anything rather the opposite in general since it makes the model more likely to overfit). Maybe on thing you could try is to train the model with a random subset of the training instances, e.g. removing a few instances randomly. The goal would be to observe whether the model is stable: normally removing a few instances should not cause a big difference, so if the perf is very different it would be... May 23 '21 at 14:38\n\u2022 ... an indication that there's too much variance (probably due to overfitting) and that would be partial evidence that the difference with\/without these test features is just due to chance. I'm not sure but it might help understand what happens. May 23 '21 at 14:40\n\nThis is the problem of out of vocabulary (OOV) words.\n\nAs a rule, the training should not use anything from the test set for several reasons:\n\n\u2022 The risk of data leakage, which would cause an overestimated performance on the test set.\n\u2022 During training the model cannot use these words to distinguish between classes anyway, since they are not present. So it would be pointless to include them.\n\u2022 In principle, the model is meant to be used with any new input text, not only the documents in the current test set. Using the test set words as features in the model actually restricts the model, since it cannot be applied to any other text than the test set.\n\nThe correct way to deal with OOV words is either:\n\n\u2022 To simply ignore them completely, i.e. filter them out before applying the model.\n\u2022 To account for this possibility in the model from the start. Typically this can be done by having a special token UNKNOWN. This option is often combined with filtering out rare words from the training set: every occurrence of rare words can be replaced by the UNKNOWN token.\n\nThe cause of the higher performance when including these features is unclear. I suspect that the inclusion of these features causes the model to be different in a way that happens by chance to have a positive effect on the test set. It's quite unlikely in general, I'm not sure.","date":"2022-01-25 03:40:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5961386561393738, \"perplexity\": 398.76000990666586}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304749.63\/warc\/CC-MAIN-20220125005757-20220125035757-00310.warc.gz\"}"} | null | null |
<?php
declare(strict_types=1);
namespace Markup\JobQueueBundle\Entity;
class JobStatus
{
/** @var ?int */
private $id;
/** @var string */
private $command;
/** @var string */
private $arguments;
/** @var bool */
private $enabled;
public function __construct(
?int $id,
string $command,
string $arguments,
bool $enabled
) {
$this->id = $id;
$this->command = $command;
$this->arguments = $arguments;
$this->enabled = $enabled;
}
public function getCommand(): string
{
return $this->command;
}
public function getArguments(): string
{
return $this->arguments;
}
public function getEnabled(): bool
{
return $this->enabled;
}
public function setCommand(string $command): void
{
$this->command = $command;
}
public function setArguments(string $arguments): void
{
$this->arguments = $arguments;
}
public function setEnabled(bool $enabled): void
{
$this->enabled = $enabled;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,310 |
Camden est une ville australienne de l'agglomération de Sydney, située dans le conseil de Camden en Nouvelle-Galles du Sud.
Géographie
Camden est située dans la plaine de Cumberland, à au sud-ouest du centre-ville de Sydney. Elle est bordée par la rivière Nepean au nord et à l'est.
Camden possède un aéroport (code AITA : CDU).
Démographie
En 2016, la population s'élevait à habitants.
Politique
Camden appartient à la zone d'administration locale du même nom et relève de la circonscription de Hume pour les élections à la Chambre des représentants.
Références
Ville en Nouvelle-Galles du Sud | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,453 |
Sony Launches 'Music Unlimited' Streaming Service
Music-lovers can have access to more than six million songs with Sony's new cloud-based streaming music service, "Music Unlimited powered by Qriocity." However the service is only available to a limited population, released Wednesday to users in the U.K. and Ireland.
Music lovers can have access to more than six million songs with Sony's new cloud-based streaming music service, "Music Unlimited powered by Qriocity." However the service is currently only available to users in the U.K. and Ireland.
Sony said the Music Unlimited library includes songs from all four major labels: Universal Music Group, Sony Music Entertainment, Warner Music Group, and EMI Music. Throughout 2011, Sony will launch Music Unlimited in other countries including Australia, Canada, France, Germany, Italy, Spain, New Zealand, and the U.S.
In addition to devices like Sony's PlayStation 3, Internet-connected TVs, and Blu-ray players, Music Unlimited can be accessed via PCs and mobile devices. It is an expansion of "Video On Demand powered by Qriocity," another streaming service from Sony that was launched earlier this year.
"Seamless accessibility to content through these fresh user experiences will enrich Sony's network service offerings and add value to the unique aspects of Sony's network-enabled products," Kazuo Hirai, Sony's president of networked products and services group, said in a statement.
Music Unlimited features two subscription options: basic for four pounds a month and premium for 10 pounds a month. The basic service works similarly to streaming radio sites like Pandora or Slacker, except that Music Unlimited users can skip an unlimited number of tracks. There are pre-set stations organized by genre or era that users can personalize according to their tastes.
Sony Launches 'Uncharted 3' Site, Pre-Orders
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'Gran Turismo 5' Sales Top 5.5M Copies, Sony Says
Sony BDP-S770
With the premium service, subscribers get a 30-day trial in which they can listen to every song in the catalog at will and make custom playlists. These users can also listen to a premium list of top 100 that include current hit songs.
Sony has not specified what the service would cost when it is launched in other countries.
Music Unlimited forms stations based on users' tastes, similar to the previously mentioned services. Sony said that this will get more specific and accurate the longer a user is listening. Subscribers can also integrate their iTunes files and other previously downloaded music to their Music Unlimited account to access their favorite tunes at any time.
DDoS Attacks Continue to Plague Human Rights Sites
Top 10 Tech Figures of 2010 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,015 |
Rolf Gomes is an Australian cardiologist. He founded Heart of Australia, an organisation that delivers specialist medical services (currently cardiac and respiratory health) to people in rural Queensland using large trucks to carry the diagnostic and treatment equipment normally only available in cities.
References
Australian cardiologists
Living people
Year of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,000 |
<footer class="footer">
<div class="container">
<div>
<ul>
<li><a href="">Política de privacidad</a></li>
<li><a href="">Gerenciamiento de Riesgos</a></li>
<li><a href="">Términos y Condiciones</a></li>
</ul>
</div>
<small>Industrial and Commercial Bank of China (Argentina) S.A. Todos los Derechos Reservados</small>
</div>
</footer> | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,143 |
Q: Error in read.csv with colClasses: scan() expected 'a real' got 'NULL' I am reading in large csv file using read.csv. Several websites suggest using colClasses to define the classes for each column to make the import process faster.
t = read.csv("pca.csv",header=TRUE,colClasses = classes)
Error in scan(file, what, nmax, sep, dec, quote, skip, nlines, na.strings, :
scan() expected 'a real', got 'NULL'
classes = c("numeric","integer")
I obviously have nulls in some of my data. Is there a way to use colClasses where "numeric" or "integer" include nulls? Also, any other tips on importing large datasets faster into R would be very helpful. I have all the data in a SQL database and I've tried using RODBC which is surprisingly slower than read.csv().
A: Use na.strings='NULL' in your call to read.csv.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,472 |
O Colt New Service é um revólver de corpo grande, de ação dupla fabricado pela Colt's Patent Fire Arms de 1898 a 1941. Disponibilizado em vários calibres, a versão .45 Colt com cano de 5½ polegadas, foi adotada pelas Forças Armadas dos Estados Unidos, com a designação de Model 1909.
Ver também
Colt M1892
M1917 revolver
M1911
FitzGerald Special
Bibliografia
Ligações externas
The Colt Revolver in the American West—New Service
Guns and Ammo Magazine article on Colt New Service Revolver
Colt New Service Revolver (infographic tech. drawing)
Revólveres Colt
Revólveres de ação dupla
Revólveres militares
Armas de infantaria do Reino Unido na Primeira Guerra Mundial
Armas de infantaria dos Estados Unidos na Primeira Guerra Mundial
Armas de fogo dos Estados Unidos na Segunda Guerra Mundial
Armas da polícia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,035 |
Christine Bontemps (born 16 July 1969) is a French sports shooter. She competed in the women's 10 metre air rifle event at the 1992 Summer Olympics.
References
1969 births
Living people
French female sport shooters
Olympic shooters of France
Shooters at the 1992 Summer Olympics
People from Remiremont
Sportspeople from Vosges (department) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,281 |
create class aoo ( a nchar(10) primary key, b int, c int );
select attr_name, is_nullable from db_attribute where class_name = 'aoo' order by 1,2;
select * from db_index where class_name = 'aoo';
select * from aoo;
create class boo (b nchar(10) ,a nchar(10) );
ALTER TABLE boo ADD CONSTRAINT fk_a FOREIGN KEY (a) REFERENCES aoo(a);
select attr_name, is_nullable from db_attribute where class_name = 'boo';
select * from db_index where class_name = 'boo';
select * from boo;
drop boo;
drop aoo; | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,666 |
The ruling is the second big appellate win this year for Public.Resource and founder Carl Malamud. In July, the group won a major ruling at the D.C. Circuit in a similar challenge to copyrights covering federal regulations that are created by private industry groups. That case is now being retried at a district court.
See Also: Library Copyright Alliance Joins Amicus Brief: Georgia v. Public.Resource.Org, Inc. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,488 |
Q: can not use authentication with OAuth 2.0 with JavaMail 1.4 I want to use oauth with IMAP to authenticate to Microsoft Exchange Online in my project. I have log4j that has a dependency of javax.mail 1.4 so in all my project this version is used. To use oauth I have to update javax.mail to 1.6 and I was not successful to do that.
I tried to use javax.mail 1.6 and import it without log4j but it broke my project:
<dependency>
<groupId>com.sun.mail</groupId>
<artifactId>javax.mail</artifactId>
<version>1.6.2</version>
</dependency>
I tried to find a version of log4j that uses javax.mail that supports oauth, but I didn't find anything.
Javax.mail is used in other parts of my project, so I don't want to touch those parts if possible.
I'm working on a legacy code, so for me it's important backwards compatibility and to not break my base code.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,521 |
{"url":"https:\/\/www.vedantu.com\/maths\/cardioid","text":"# Cardioid\n\n## Cardioid Meaning\n\nThere are numerous two dimensional and three dimensional geometric figures in Mathematics. The two dimensional shapes include triangles, quadrilaterals, circles and other polygons. The three dimensional figures include prisms, pyramids, spheres etc. Cardioid is one of the most important two dimensional plane geometric shapes. A cardioid has a heart shaped curve. This forms a basis for its name. The word cardioid is derived from the Greek word which means \u2018heart\u2019. Cardioid can be understood by comparing its shape to that of the cross section of an apple without stalk. This shape is formed when a point on the boundary of the circle is traced and rolled on to another circle of the same radius.\n\n### How to Draw a Cardioid:\n\nA cardioid is a plane figure that has a heart shaped curve. The heart shaped curve is defined by the locus of a point lying on the circumference of a circle which rolles from outside without a slip on the boundary of another circle of radius. The figure above gives a clear idea about how to draw a cardioid. The cardioid curve is in the form of a spiral sinusoid. A parabola whose focus is at the center of inversion can be inverted\u00a0 to form a cardioid curve. A cardioid can have accurately 3 parallel tangents with a specific gradient. The two branches of the cardioid graph intersect to form a cusp. The arc of the cardioid passing through the cusp measures \u20184a\u2019 where \u2018a\u2019 is the radius of the curve.\n\n### Cardioid Equation:\n\nThe cardioid equation can be described in both polar and cartesian coordinate system as explained below.\n\n1. Polar Form of Representing Cardioid Equation:\n\nThe polar equation of a cardioid graph is represented with respect to \u2018r\u2019 and \u2018\u03b8\u2019 coordinates as follows:\n\n r = a (1 + Cos \u03b8)\n\nIn the above equation, \u2018a\u2019 is the radius of the circle that is traced and \u03b8 is the polar angle.\n\n1. Cartesian Form of Representing Cardioid Equation:\n\nThe equation of a cardioid in the cartesian coordinate system with respect to X and Y axes of a cartesian plane is given as follows.\n\n (x2 + y2 + ax)2 = a2 (x2 + y2)\n\nThe parametric equations of \u2018x\u2019 and \u2018y\u2019 in the above cardioid equation is given as:\n\nx = a Cos t (1 - Cos t)\n\ny = a Sin t ( 1 - Cos t)\n\nwhere \u2018a\u2019 is the radius of the circle which is traced.\n\n### Cardioid Graph:\n\nThe figure above represents a cardioid graph which has two perpendicular axes named X axis and Y axis.\u00a0 It has a heart shaped curve that has a cusp formed by the intersection of the two branches. This graph is obtained by tracing out the path of a circle rolling over another fixed circle. However, the radii of both the circles remains the same. A cardioid is most commonly represented in the polar coordinate system. However, the equation of cardioid obtained in polar form can be transformed into a cartesian form.\n\nThe region enclosed by a cardioid graph is called the area of a cardioid and is calculated using the formula:\n\n Area of Cardioid = 6 \u03c0 a2\n\n## Total Length of the Arc of a Cardioid is Given by the Formula:\n\n Length of the arc = 16 a\n\nIn both the above mentioned formula, \u2018a\u2019 stands for the radius of the circle whose path is traced to form the cardioid.\n\n### Cardioid Equation Example Problems:\n\n1. The equation of a cardioid is given as r = 3 (2 + 2Cos \u03b8). Find the total length of the arc and the area of the cardioid.\n\nSolution:\n\nThe equation of a cardioid in the given problem is\n\nr = 3 (2 + 2 Cos \u03b8)\n\nIf \u20182\u2019 is taken as common, the above equation becomes\n\nr = 6 (1 + Cos \u03b8)\n\nThe value of \u2018a\u2019 in the above equation is a = 6.\n\nArea of cardioid = 6 \u03c0 a2 = 6 x 3.14 x (6)2 = 678.24 square units\n\nLength of the arc = 16 a = 16 x 6 = 96 units\n\n1. What is the length of the arc and area of the cardioid whose equation is defined as r = 7 (1 + Cos \u03b8)?\n\nSolution:\n\nThe equation of a cardioid in the given problem is\n\nr = 7 (1 + Cos \u03b8)\n\nThe value of \u2018a\u2019 in the above equation is a = 7\n\nArea of cardioid = 6 \u03c0 a2 = 6 x $\\frac{22}{7}$ x (7)2 = 924 square units\n\nLength of the arc = 16 a = 16 x 7 = 114 units\n\n### Cardioid Graph Fun Facts:\n\n\u2022 A cardioid is a special case of limacon. It belongs to a class of curves studied by Etinne Pascal, the father of Blaise Pascal.\n\n\u2022 The word cardioid is also used to name the microphones and other sonic devices which are employed to extract the sound from front and sides only and not from the back side of the device.\n\n1. What is a Cardioid?\n\nA cardioid is a two dimensional plane geometric shape that is in the form of a heart shape. The shape of a cardioid can be understood better by analysing the shape of the cross section of an apple without stalk. The cardioid is a plane figure obtained as a path followed by a circle of certain radius over the circumference of a fixed circle of the same radius. A cardioid can be represented in both polar and cartesian coordinate systems. A critical analysis of cardioid is very helpful in analytical geometry and other branches of Mathematics.\n\n2. What are Limacons?\n\nAll the equations that can be represented in the form of any one of the following is represented as a limacon.\n\n\u2022 r = a + b Sin \u03b8\n\n\u2022 r = a - b Sin \u03b8\n\n\u2022 r = a + b Cos \u03b8\n\n\u2022 r = a - b Cos \u03b8\n\nThe different types of limacons are:\n\n1. Inner loop Limacon: The value of \u2018a\u2019 is less than that of \u2018b\u2019 in the above equations.\n\n2. Dimpled Limacon: The value of \u2018a\u2019 is greater than that of \u2018b\u2019\n\n3. Convex Limacon: The value of \u2018a\u2019 is greater than or equal to \u20182b\u2019\n\n4. Cardioid: The value of \u2018a\u2019 is equal to that of \u2018b\u2019.","date":"2020-07-04 21:41:30","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\": 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\": 0.8038100004196167, \"perplexity\": 518.4601663682573}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-29\/segments\/1593655886706.29\/warc\/CC-MAIN-20200704201650-20200704231650-00317.warc.gz\"}"} | null | null |
Q: Get the number index through boolean opearator in python-pandas Now I can get the expected value with boolean operator:
mask1 = df.val > 10
mask2 = df.val < 5
c1 = df[mask1]
c2 = df[mask2]
And I wish to get the slice between each point in c1 and c2, that is, given a data frame df as below:
val
0 9
1 12
2 5
3 2
4 11
5 9
6 9
7 3
The slices of df[1:3] and df[4:7] are what I want. First of all, I need to get the index 1, 3, 4, 7 through mask1 and mask2. Do you know how to do it?
A: I think you can use:
c1 = df[mask1].index
c2 = df[mask2].index
print (c1)
Int64Index([1, 4], dtype='int64')
print (c2)
Int64Index([3, 7], dtype='int64')
print (df[c1[0]:c2[0]])
val
1 12
2 5
print (df[c1[1]:c2[1]])
val
4 11
5 9
6 9
It is same as:
print (df[1:3])
val
1 12
2 5
print (df[4:7])
val
4 11
5 9
6 9
If you need slice by c1 or c2 use ix:
print (df.ix[c1])
val
1 12
4 11
print (df.ix[c2])
val
3 2
7 3
A: This suppose to work for you:
c1 = df[df['val'] > 10].index.tolist()
c2 = df[df['val'] < 5].index.tolist()
output: c1 = [1,4] c2 = [3,7]
A: Subset the bigger data-frame to smaller one by that mask condition
df_mask1 = df[df.val > 10]
#in df_mask1 you will have index 1 and 4 [Rows of main dataframe]
#similarly
df_mask2 = df[df.val < 5]
# you will have index 3 and 7 in df_mask2
or if you want to change some other column value by that mask condition you can use df.loc (just try serching on web).
| {
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{"url":"https:\/\/de.zxc.wiki\/wiki\/K%C3%BCrzester_Pfad","text":"# Shortest path\n\nIn graph theory, a shortest path is a path between two different nodes of a graph which has a minimum length with respect to an edge weight function . If the edges in the graph all have the weight 1, that is for all edges , then the shortest path is a - -path with the smallest possible number of edges between and . ${\\ displaystyle s, t \\ in V}$ ${\\ displaystyle c \\ colon E \\ to \\ mathbb {R}}$${\\ displaystyle c (e) = 1}$${\\ displaystyle e \\ in E}$${\\ displaystyle s}$${\\ displaystyle t}$${\\ displaystyle s}$${\\ displaystyle t}$\n\nIn the literature, the problem is often referred to as the Shortest Path Problem .\n\n## complexity\n\nThe complexity depends largely on the type of weight function and on whether paths or edge trains are being considered. In edge trains, nodes and edges can be repeated, while paths do not use any nodes twice. There are three types of weight functions:\n\n\u2022 Weight functions without negative weights;\n\u2022 Conservative weight function: A weight function is said to be conservative for the graph if for all cycles of ;${\\ displaystyle G}$${\\ displaystyle c (C) = \\ sum _ {e \\ in C} c (e) \\ geq 0}$ ${\\ displaystyle C}$${\\ displaystyle G}$\n\u2022 Weight functions with any weights.\n\nThe exact formulation of the problem is crucial in order to be able to answer the question of complexity.\n\nWithout negative weights\nWith Dijkstra's algorithm the problem can be solved in a running time of , where denotes the number of edges and the number of nodes in the graph. Note that the shortest paths are also the shortest edges. If all weights are genuinely positive, then the shortest paths match the shortest edges.${\\ displaystyle O (m + n \\ cdot \\ log (n))}$${\\ displaystyle m}$${\\ displaystyle n}$\nWith any weight and with edges\nIf the graph contains a cycle in which the sum over the weights is strictly negative, then there are nodes that do not have a shortest line of edges. If there is an edge from to this cycle and an edge from this cycle to , then you can create an arbitrarily short edge from to by running through the cycle only sufficiently often. The Bellman-Ford algorithm can find a shortest edge train (if it exists) in a running time of or prove that it does not exist by finding a negative cycle. The decision problem as to whether there is a path of length can thus be solved in polynomial time.${\\ displaystyle s, t}$${\\ displaystyle s}$${\\ displaystyle t}$${\\ displaystyle s}$${\\ displaystyle t}$${\\ displaystyle O (nm)}$${\\ displaystyle \\ leq C}$\nWith arbitrary weights and with paths\nThis variant of the problem is NP- hard. This can be proven, for example, by a reduction of the NP-heavy Hamiltonian path problem by setting all weights to \u22121 in the shortest path problem. Note that this construction contains negative cycles, and therefore the NP severity does not hold for conservative weight functions.\nConservative weight function and with paths\nThe Bellman-Ford algorithm can find a shortest path in a running time of .${\\ displaystyle O (nm)}$\n\nThe literature is mostly limited to non-negative weights or conservative weight function. With one of these additional requirements, every shortest path is automatically a shortest line of edges and that is why this distinction is often not made in the literature.\n\nIn contrast to the shortest path problem, the longest path problem is NP-hard even for unweighted graphs .\n\n## Variations of the problem\n\nAside from determining a shortest - path, there are a few other, but very similar problems: ${\\ displaystyle s}$${\\ displaystyle t}$\n\n### Single-source shortest path (SSSP)\n\nThis variant of the shortest paths problem deals with the problem of how to compute the shortest paths between a given starting node and all other nodes of a graph. For nonnegative weight functions, the Dijkstra algorithm or the A * algorithm can be adapted in order to calculate the shortest paths to all nodes of the graph. On the other hand, the Bellman-Ford algorithm always calculates the shortest paths to all other nodes for any conservative weight functions.\n\n### Single-destination shortest path (SDSP)\n\nThe aim here is to determine the shortest path between an end node and all other nodes of the graph. This problem can be described as SSSP by reversing the edge directions.\n\n### All-pairs shortest path (APSP)\n\nThis variant of the problem is about determining the shortest paths between all pairs of nodes in a graph. Depending on the weight function, it is more efficient to solve the SSSP for each node one after the other or to use specialized methods such as the Floyd-Warshall algorithm or the min-plus matrix multiplication algorithm , which determine the shortest paths for all pairs at the same time.\n\n## example\n\nSample graph\n\nIn the graph on the right there is a shortest path between the nodes and the path that starts in and goes over to . The path costs are here . However, if you want to find a path from to , the direct path with costs from is not the shortest possible path, since the path from via to only has costs from . ${\\ displaystyle D}$${\\ displaystyle C}$${\\ displaystyle D}$${\\ displaystyle B}$${\\ displaystyle C}$${\\ displaystyle 9 + 8 = 17}$${\\ displaystyle D}$${\\ displaystyle E}$${\\ displaystyle 15}$${\\ displaystyle D}$${\\ displaystyle F}$${\\ displaystyle E}$${\\ displaystyle 14 = 8 + 6}$\n\n## Formulation as a linear program\n\nA linear program can also be used to determine a shortest path . In this case, the path is interpreted as a flow with a flow value of 1 on the edges of the graph. The determination of the shortest path is then a special case of the min-cost-flow problem. The corresponding formulation is:\n\n{\\ displaystyle {\\ begin {aligned} \\ min & \\ sum _ {e \\ in E} c_ {e} x_ {e} \\\\ {\\ text {so that}} & \\ forall \\; v \\ in V \\ colon \\; \\ sum _ {e \\ in \\ operatorname {\\ delta ^ {-}} (v)} x_ {e} - \\ sum _ {e \\ in \\ operatorname {\\ delta ^ {+}} (v)} x_ {e} = {\\ begin {cases} -1, & {\\ text {if}} v = s \\\\ 1, & {\\ text {if}} v = t \\\\ 0, & {\\ text {otherwise} } \\ end {cases}} \\\\ & \\ forall \\; e \\ in E \\ colon \\; x_ {e} \\ geq 0 \\\\\\ end {aligned}}}\n\nIf there is a - path in the given graph, then the program has a feasible solution. However, the program is unlimited if the weight function is not conservative. In this case, the flow can be increased arbitrarily along a cycle with negative costs. Otherwise the problem has an optimal solution which corresponds to a vector with entries. The set then describes a shortest - path, the objective function value of the program corresponds to the length of the path. ${\\ displaystyle s}$${\\ displaystyle t}$${\\ displaystyle x}$${\\ displaystyle 0\/1}$${\\ displaystyle | E |}$${\\ displaystyle \\ {e \\ in E \\,: \\, x_ {e} = 1 \\}}$${\\ displaystyle s}$${\\ displaystyle t}$\n\n## Node potentials\n\nIt turns out that the dualization of the above linear program has an illustrative interpretation. The dual program is given by\n\n{\\ displaystyle {\\ begin {aligned} \\ max {} & y_ {t} -y_ {s} \\\\ {\\ text {so that}} & \\ forall \\; e = (u, v) \\ in E \\ colon \\ ; y_ {v} -y_ {u} \\ leq c_ {e} \\; \\; \\ end {aligned}}}\n\nA solution to the dual program is called a node potential . It is easy to see that the vector is also a solution for every solution , and one can choose at will. As a rule, the value of is set so that . The objective function is then given by . ${\\ displaystyle y}$${\\ displaystyle (y_ {v}) _ {v \\ in V}}$${\\ displaystyle (y_ {v} + \\ delta) _ {v \\ in V}}$${\\ displaystyle \\ delta \\ in \\ mathbb {R}}$${\\ displaystyle \\ delta}$${\\ displaystyle y_ {s} = 0}$${\\ displaystyle \\ max \\; y_ {t}}$\n\nIf there is any path between and a node , the length of the path can be estimated as follows: ${\\ displaystyle P}$${\\ displaystyle s}$${\\ displaystyle w \\ neq s}$\n\n${\\ displaystyle c (P) = \\ sum _ {e \\ in P} c_ {e} \\ geq \\ sum _ {e = (u, v) \\ in P} y_ {v} -y_ {u} = y_ { w}}$\n\nThe potential of each node is therefore a lower bound for the length of a path. An optimal solution of the dual program can be found if one sets the potential of a node as the length of the shortest - path with respect to the objective function . ${\\ displaystyle w \\ neq s}$${\\ displaystyle s}$${\\ displaystyle w}$${\\ displaystyle c}$\n\n## Applications\n\nAlgorithms that calculate a shortest path are often used in the calculation of travel routes. For example, the distance between two cities can be calculated. The cities are the nodes of the graph and the streets are the edges. Various algorithms are in the free Python - Library NetworkX implemented.\n\n## Shortest routes with constraints\n\nA generalization of the problem is obtained if one only considers - paths which obey the additional inequality . There is another weight function and a real number. ${\\ displaystyle s}$${\\ displaystyle t}$${\\ displaystyle P}$${\\ displaystyle \\ sum _ {e \\ in P} u_ {e} \\ leq U}$${\\ displaystyle u \\ colon E \\ to \\ mathbb {R} _ {+}}$${\\ displaystyle U}$\n\nThe resulting constrained shortest path problem is then also NP-difficult for conservative or nonnegative objective functions, see HC Joksch (1966).\n\n## Individual evidence\n\n1. ^ Bernhard Korte , Jens Vygen : Combinatorial Optimization. Theory and Algorithms. 4th edition. Springer, Berlin et al. 2008, ISBN 978-3-540-71844-4 ( Algorithms and Combinatorics 21)\n2. ^ Algorithms - Shortest Paths. In: NetworkX 2.2 documentation. Retrieved October 24, 2018 .\n3. HC Joksch (1966)","date":"2021-12-07 23:51:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 69, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.538478672504425, \"perplexity\": 1205.8661744336443}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363420.81\/warc\/CC-MAIN-20211207232140-20211208022140-00355.warc.gz\"}"} | null | null |
{"url":"http:\/\/ay17-chusic.blogspot.com\/2015\/12\/37-illustris-simulation.html","text":"## Sunday, December 6, 2015\n\n### 37 | The Illustris Simulation\n\nIn this post, we will explore the Illustris Simulation, a numerical cosmological simulation developed by a collaboration of scientists working to better understand the formation of large-scale structure in the universe. The online tools associated with Illustris allow us to explore the data generated from this simulation.\n\nWe will begin by looking closely at an over-dense region. By choosing such a region on the simulation explorer, the program will automatically select nearby subhalos, which are essentially regions with very high relative density contrast.\n\n a selection within the Illustris Explorer\n\nThe simulation explorer allows us to learn more about the nature of these halos. Let's start by generating a histogram of the masses of these halos. To make the data easier to visualize, we will consider the logarithm of the mass:\n\nFrom this plot, it is clear that the mass distribution seems to be shifted slightly more towards the low-mass halos, meaning that low-mass halos are more numerous. From the data provided, we can also learn about the total stellar mass within each halo. From the data set we collected, we find that about 93% of the halo mass comes from the stars contained within it.\n\nAt a large scale, the density of dark matter and gas seem to be quite similar, and we can see similar structural features in spatial plots of each component. The following plots show the density of dark matter and gas, respectively.\n\nNote how both appear to peak in similar regions, and exhibit \"filamentary\" structure elsewhere. However, it's interesting to note that the dark matter appears to be more closely defined to the filaments, whereas the gas is a bit more diffuse.\n\nOn a small scale, however, the gas density appears to be more defined than the dark matter density. Again these plots show dark matter and gas density, respectively--this time at a much smaller scale:\n\nIn some of the larger galaxies, we can see that the gas density is actually greatest in the disk of the galaxy and note in the nucleus. This is clearly visible in the image below where white represents the highest gas density. Note how the white appears in a ring around the center of galaxy, but the center of the galaxy is actually less dense.\n\nThe explorer also allows us to see that the largest galaxies tend to exist in clusters rather than on their own in the universe. This is evident in the screenshot below, where the most massive galaxies have been circled.\n\nNext, we can look at an animation which shows the development of dark matter and gas temperature over the evolution of our universe. This animation seems to show that structure formation is lead by dark matter, then baryons (gas) tend to develop along these filaments of dark matter. However, the gas is initially not very warm. In fact, researchers refer to the time before the gas heats up as the \"Dark Ages.\" Based on the animation, gas did not begin to warm up until a redshift of $$z \\approx 10$$, which corresponds to about 0.5 billion years after the Big Bang.\n\nThe model also provides us with the approximate rate of star formation as the universe forms. More specifically, the model predicts this by the rate of increase in stellar mass. There are several periods throughout the evolution of the universe when star formation greatly increases. In particular, between $$z = 1.5$$ and $$z=1.2$$, the rate of star formation is quite fast.\n\nAs time progresses, the animation shows large structures collapsing and generating \"outbursts\" of gas. This suggests that as time passes, large objects break up and this ultimately results in the formation of new, smaller structures throughout the universe. These small structures grow until they too collapse on themselves.\n\nFinally, it is curious why this structure forms in filaments. We can see that the dark matter occurs in filaments throughout the universe. The gravity of this dark matter pulls baryons towards it. As a result of this matter being pulled together, stars and galaxies can form. Consequently, this gravitational attraction from the dark matter generates structures along the filaments.\n\n#### 1 comment:\n\n1. you've gotta watch the Log(M) units. the ratio of the masses is more like 10-20%.","date":"2018-10-18 23:21:27","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\": 0.6652945876121521, \"perplexity\": 506.0966736200045}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583512015.74\/warc\/CC-MAIN-20181018214747-20181019000247-00369.warc.gz\"}"} | null | null |
Matthew Krousey received his BFA in ceramics from the University of Minnesota–Twin Cities. Krousey's ceramic vessels and murals have been exhibited locally and internationally, including an exhibition at Sanbao Ceramic Institute in Jingdezhen, China. He is a 2010 Fogelberg Fellowship recipient through Northern Clay Center. Krousey's work preserves memories and emotions evoked by the disappearing Minnesota landscape. "This region is being vastly altered by human hands, which is the reason I seek to preserve it upon the durable ceramic surface," he explains. | {
"redpajama_set_name": "RedPajamaC4"
} | 850 |
Q: Display message box from Task Scheduler on top of all other windows Is there a nice way to get a message box to pop up at a scheduled time on top of other windows? The default "Display a Message" action uselessly appears below everything else.
A: Also see https://www.howtogeek.com/136894/how-to-create-popup-reminders-with-no-additional-software/. In case of link death, I'll summarize here:
*
*Open Windows Task Scheduler (search 'Task Scheduler' in the Start menu)
*Under Actions, click 'Create Task'
*In the General tab, make sure 'Run only when user is logged on' is checked, and 'Hidden' is NOT checked
*In the Triggers tab, click 'New' to set a time for the message to trigger
*In the Actions tab, click 'New' and make sure the selected action at the top is 'Start a program'
*In 'Program/script', type CMD
*In 'Add arguments', copy and paste this:
/C TITLE [Your title here] &ECHO.&ECHO.&ECHO [Your message here] &ECHO.&ECHO.&TIMEOUT [timeout]
*Replace [Your title here] and [Your message here] with your preferred text (do not include the brackets)
*Replace [timeout] with the number of seconds you want the message to wait before timing out, or -1 if you don't want it to time out (either way, a key press while the console window is in focus will close it)
*Click 'OK'!
This will generate a console window with the given text, which will appear ON TOP of your current window, but won't automatically steal the focus (you can keep interacting with your current window as usual).
This was the best solution I found, hope it helps someone!
A: How about using Window's built-in msg command like so?
msg * "Message you would like to send"
You can add other parameters such as /TIME:x where x is the number of seconds you want the message to display. Of course, msg /? will show you all the options available.
This implies Windows XP and higher as the system where you want to display the message. If you have a Home Edition of the applicable OS, you're out of luck. See http://ss64.com/nt/msg.html for available parameters.
If you have a Home Edition, the following batch script will pop-up a message using VBSCript's PopUp method:
@echo off
::See http://msdn.microsoft.com/en-us/library/x83z1d9f(v=vs.84).aspx
::for an explanation of the PopUp method
::
::Use the directory from whence script was called as working directory
set CWD=%~dp0
::Use a random file name for the temporary VBScript.
set usrmsg=%CWD%%random%.vbs
::First parameter is the timeout in seconds. 0 = wait forever
set _timeout=%~1
::Second parameter is the message, enclosed in quotes.
set _Message=%~2
::Third parameter is the title of the window, enclosed in quotes.
set _Title=%~3
::This last variable is used to display a button/icon on the window.
::Setting this to 4096 sets the window to Modal (on top of everything else)
set _nType=4160
::Create the temp script using the provided information.
ECHO Set wshShell = CreateObject( "WScript.Shell" )>%usrmsg%
ECHO wshShell.Popup "%_Message%" ^& vbCrLf, %_Timeout%, "%_Title%", %_nType%>>%usrmsg%
::Run the script.
WSCRIPT.EXE %usrmsg%
::Delete the script.
DEL %usrmsg%
::Exit the batch file
exit /b
Hope this helps!
Added: Gregg mentions in the comments that in order for this to work in Windows 10, you must use "/time:0" if you want the message to stay on-screen longer than 60 seconds.
A: Use msgbox in vbs. like this:
Set filesys = CreateObject("Scripting.FileSystemObject")
Set shell = CreateObject("Shell.Application")
Set wshShell = WScript.CreateObject( "WScript.Shell" )
PCName = wshShell.ExpandEnvironmentStrings( "%COMPUTERNAME%" )
msgbox "Dear user on " & PCName & vbcrlf & " " & vbcrlf & "This is a message box on top of all other windows.", &h51000, "I am msgbox"
shell.Open "C:\Users"
Code &h51000 will make sure msgbox in central and on top of all other windows all the time.
If you only want to schedule a msgbox, you can simply use task scheduler, there is a built in function to schedule a message. see location of [start a program] in task scheduler.
A: Display a Message is deprecated, and msg command didn't work for me in Windows 10 1703. I'm using powershell to display the message box from a scheduled task, use this as arguments when defining the action:
-WindowStyle hidden -Command "& {[System.Reflection.Assembly]::LoadWithPartialName('System.Windows.Forms'); [System.Windows.Forms.MessageBox]::Show('<message body here>','<window title here>')}"
It's ugly AF, but it works.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,127 |
\section{Introduction and statement of results}
A \textit{covering system} is
a finite collection of arithmetic progressions that covers the integers.
It was first introduced by Erd\H{o}s \cite{Erdos1950}
in 1950 to answer a question of Romanoff. And in his collections of open problems \cite{Erdos1957, Erdos1963, AB1971, Erdos1973, EG1980}, Erd\H{o}s posed a number of problems concerning covering systems. One of them is the well-known minimum modulus problem: is there a uniform upper bound on the smallest modulus of all covering systems with distinct moduli? In 2007, Filaseta, Ford, Konyagin, Pomerance and Yu \cite{FFSPY2007} made a significant process towards this problem in the case when the reciprocal sum of the moduli of a covering system is
bounded. Building on their work, Hough \cite{Hough2015} made a remarkable breakthrough and gave an affirmative answer to Erd\H{o}s' minimum modulus problem in 2015, showing that the minimum modulus is at most $10^{16}$. In 2022, Balister, Bollob\'as, Morris, Sahasrabudhe and Tiba \cite{BBMRS2022} reduced this bound to $616,000$ by developing a general method for bounding the density of the uncovered set. This method is named by them \textit{the distortion method}; see \cite{BBMST2020_2} for more details about it. Recently, Cummings, Filaseta, and Trifonov \cite{CFT2022} reduced the bound to 118 for covering systems with squarefree moduli. They also proved in general the $j$-th smallest modulus in a minimal covering system with distinct moduli is bounded by some unspecified constant $C_j, j\ge1$. This result was improved by Klein, Koukoulopoulos and Lemieux \cite{KKL2022} later in the sense that they gave a specified bound for the $j$-th smallest modulus. In fact, they \cite{KKL2022} achieved the following generalization of Hough's theorem by using the distortion method. For a covering system $\mathcal{A}$, the \textit{multiplicity} of $\mathcal{A}$ is defined to be the largest number of times that a modulus appears in $\mathcal{A}$.
\begin{theorem}[\text{\cite[Theorem~2]{KKL2022}}]\label{KKL Thm2}
Let $\mathcal{A}$ be a covering system of multiplicity $s$. Then there exists an absolute constant $c > 0$ such that the smallest modulus of $\mathcal{A}$ is at most $\exp \big(c \log^2(s+1)/\log\log(s+2)\big)$.
\end{theorem}
In this paper, our goal is to establish the generalization of Theorem~\ref{KKL Thm2} to covering systems of the ring of integers of an arbitrary number field. In fact, the covering systems for number fields has already been actively studied in the literature; see for example \cite{Kim2009, Kim2012, JD2014, BB2020}. To explain our main result, we first introduce some notation in the number field setting.
Let $K/\mathbb{Q}$ be a number field. Let $\mathcal{O}_K$ be the ring of integers in $K$. For any ideal $I\subset \mathcal{O}_K$, we denote by $\norm{I}$ the norm of $I$, i.e., $\norm{I}=[\mathcal{O}_K\colon I]$. We say an ideal $I$ is \textit{distinguishable}, if there exists a unique prime ideal factor of $I$ obtaining the largest norm among prime ideal factors of $I$. If $I$ is distinguishable, then we denote by $P_{\min}(I)$ the prime factor of $I$ obtaining the largest norm. Let $\mathfrak{D}(K)$ be the set of distinguishable ideals of $\mathcal{O}_K$. Let $\mathcal{A}=\set{a_1+I_1, \dots, a_n+I_n}$ be a finite collection of congruence classes with $a_i\in\mathcal{O}_K$ and $I_i\in\mathfrak{D}(K)$, $1\le i \le n$. Let
\[
m(\mathcal{A})= \max_{I\subset \mathcal{O}_K} \#\set{1 \le j \le n: I_j=I}
\]
be the multiplicity of $\mathcal{A}$. We say $\mathcal{A}$ is a \textit{covering system} of $\mathcal{O}_K$ if $\mathcal{O}_K=\bigcup_{i=1}^n(a_i+I_i)$; see \cite[Page~124]{Kim2009} for an example. Now we are ready to state our main theorem.
\begin{theorem}\label{mainthm}
For a number field $K/\mathbb{Q}$, let $\mathcal{A}=\set{a_1+I_1, \dots, a_n+I_n}$ be a covering system of multiplicity $s$ with distinguishable ideals $I_1,\dots,I_n$ in $\mathcal{O}_K$. Then there exists a constant $c_K>0$ depending only on $K$ such that
\[
\min_{1\le i \le n}\norm{I_i} \le \exp\big(c_K\log^2(s+1)/\log\log(s+2)\big).
\]
\end{theorem}
\begin{remark} If one can generalize Crittenden and Vanden-Eynden's theorem \cite{CV1970} to any number field, then one may get an generalization of Theorem 1 in Klein, Koukoulopoulos and Lemieux's work \cite{KKL2022} to the setting of number fields. See \cite{BBMST2020_1} for a short proof of Crittenden and Vanden-Eynden's theorem.
\end{remark}
Theorem~\ref{mainthm} provides a solution to Erd\H{o}s' minimum modulus problem in number fields. That is, in a given number field, the smallest norm of the moduli of all covering systems with distinct distingushiable moduli is uniformly bounded. To prove Theorem~\ref{mainthm}, following the ideas in the proof of Theorem~2 in \cite{KKL2022}, we derive analogues of the necessary lemmas and theorems in \cite{BBMRS2022, KKL2022} in the number field setting. Most of the proofs are very similar to those of \cite{BBMRS2022, KKL2022}. First, in Section~\ref{sec_distortion_method} we present a notable generalization of the distortion method by introducing the theory of probability measures associated to an inverse system; see Theorem~\ref{thm_distortion_method}. Then we restrict it to the number fields setting for our purpose. One may apply Theorem~\ref{thm_distortion_method} to prove the analogue of Theorem~\ref{KKL Thm2} in other situations; say function fields. In Sections~\ref{boundind_moments}-\ref{sec_pf_mainthm}, we investigate the first and second moment appearing in Theorem~\ref{thm_distortion_method_nf} similarly as in \cite{KKL2022} and finally complete the proof of our main theorem.
\section{The distortion method}
\label{sec_distortion_method}
\subsection{Probability measures associated to an inverse system}
\label{sec_distortion_method_general}
In this section, we construct a sequence of probability measures on a finite set with an inverse system. Let $S$ be a finite set, $T$ be an arbitrary set, and $\pi:S\to T$ be a map. We say a function $f:S\to \mathbb{C}$ is \textit{$\pi$-measurable}, if $f(x) = f(x')$ whenever $\pi(x)=\pi(x')$ for any $x,x'\in S$. For any subset $B\subseteq S$, we say $B$ is \textit{$\pi$-measurable}, if the following property holds: for any $x\in B, x'\in S$, if $\pi(x')=\pi(x)$, then $x'\in B$.
Let $\P$ be a $\pi$-measurable probability measure on $S$. Let $J\ge1$ be an integer. Let $B_1, \dots, B_J$ be subsets of $S$, $T_1,\dots, T_J$ be arbitrary sets, and $\pi_j: S \to T_j$ be arbitrary maps for $1\le j \le J$. We say $\set{\pi_j}$ is a \textit{$\pi$-inverse system} (or \textit{inverse system}) if the following property holds: for any $1\le j\le J$ and $x,x'\in S$, if $\pi_j(x)=\pi_j(x')$ then $\pi_{j-1}(x)=\pi_{j-1}(x')$, where $\pi_0=\pi$. Let $\delta_1,\dots, \delta_J\in [0,1/2]$ be parameters. With these definitions in hand, following the steps in \cite{BBMRS2022,KKL2022} we can construct a sequence of $\pi_j$-measurable\footnote{In \cite{BBMRS2022,KKL2022}, it is called $Q_j$-measurable.} probability measures $\P_j (1\le j \le J)$ on $S$ with the triples $\set{(\pi_j, B_j, \delta_j): 1\le j\le J}$ and the initial probability measure $\P$ as follows.
First, we take $\P_0=\P$. Next, for $1\le j \le J$, suppose we have already constructed a $\pi_{j-1}$-measurable probability measure $\P_{j-1}$ on $S$. Then for all $x\in S$ we define
\begin{equation*}
\alpha_j(x) \colonequals \frac{|F_{j-1}(x)\cap B_j|}{|F_{j-1}(x)|} ,
\end{equation*}
where $F_{j-1}(x)=\set{x'\in S: \pi_{j-1}(x')=\pi_{j-1}(x)}$. Note that $\alpha_j$ is $\pi_{j-1}$-measurable, since $F_{j-1}(x')=F_{j-1}(x)$ holds for any $x', x\in S$ satisfying $\pi_{j-1}(x')=\pi_{j-1}(x)$. Now, we define $\P_j$ on $S$ as follows:
\[
\P_j(x)\colonequals \P_{j-1}(x) \cdot \left\{
\begin{aligned}
&\frac{1_{x\notin B_j}}{1-\alpha_j(x)}, & \text{if } \alpha_j(x)<\delta_j;\\
&\frac{\alpha_j(x)-1_{x\in B_j}\delta_j}{\alpha_j(x)(1-\delta_j)} , & \text{if } \alpha_j(x)\ge \delta_j.
\end{aligned}
\right.
\]
One may easily check that $\P_j$ is $\pi_j$-measurable, and we have
\begin{equation}\label{eqn_measurable}
\P_j(F_{j-1}(x)) = \P_{j-1}(F_{j-1}(x))
\end{equation}
for all $x\in S$. It follows that $\P_j(S)=\P_{j-1}(S)=1$, and hence it is a probability measure on $S$. Thus, by induction we obtain a sequence of probability measures $\set{\P_j}$.
Now, we define the expectations
\[
\mathbb{E}_j[f]\colonequals \sum_{x\in S} f(x)\P_j(x)
\]
for all functions $f\colon S\to \mathbb{C}$. For $j=1,\dots,J$, we set
\[
M_j^{(1)}\colonequals\mathbb{E}_{j-1}[\alpha_j]
\quad\text{and}\quad
M_j^{(2)}\colonequals\mathbb{E}_{j-1}[\alpha_j^2]
\]
as the first and second moments. Then the following is the key result of the distortion method in \cite[Theorem 3.1]{BBMRS2022} in this general setting. And we give a proof for the readers' convenience.
\begin{theorem}\label{thm_distortion_method}
Assume the above notation. Let $B_j$ be $\pi_j$-measurable subsets of $S$ for all $1\le j \le J$. If
\[
\sum_{j=1}^J
\min\left\{ M_j^{(1)},\frac{M_j^{(2)}}{4\delta_j(1-\delta_j)}\right\}<1 ,
\]
then
\[
\P_J\of{\bigcup_{1\le j \le J} B_j}<1.
\]
\end{theorem}
\begin{proof} Following the proof of \cite[Theorem 3.1]{BBMRS2022}, we first prove that
\begin{equation} \label{eqn_pjbj}
\P_j(B_j)\le \min\set{M_j^{(1)},\frac{M_j^{(2)}}{4\delta_j(1-\delta_j)}}
\end{equation}
for all $1\le j \le J$. Indeed, by the construction of $\P_j$, we have $\P_j(B_j) \le \P_{j-1}(B_j)=\mathbb{E}_{j-1}[\alpha_j]$. Moreover, by the inequality $\max\set{a-d,0}\le a^2/4d$ for all $a,d>0$ we have
\begin{align*}
\P_j(B_j) & = \sum_{x\in B_j} \max \set{0,\frac{\alpha_j(x)-\delta_j}{\alpha_j(x)(1-\delta_j)}}\P_{j-1}(x) \\
& = \frac{1}{1-\delta_j}\sum_{x\in S} \max \set{0,\alpha_j(x)-\delta_j}\P_{j-1}(x) \\
& \le \frac{1}{1-\delta_j}\sum_{x\in S} \frac{\alpha_j(x)^2}{4\delta_j}\cdot \P_{j-1}(x)\\
& = \frac{\mathbb{E}_{j-1}[\alpha_j^2]}{4\delta_j(1-\delta_j)}.
\end{align*}
Hence \eqref{eqn_pjbj} holds.
Now, we assume $B_j$ is $\pi_j$-measurable for all $1\le j \le J$. Then we have $F_j(x)\subset B_j$ for all $x\in B_j$. So $B_j$ is a disjoint union of some subsets of the form $F_j(x)$. By \eqref{eqn_measurable}, we see that $\P_{j+1}(B_j)=\P_j(B_j)$. Since $\set{\pi_j}$ is an inverse system, $B_j$ is $\pi_i$-measurable as well for any $i>j$. It follows by induction that we have $\P_j(B_j)=\P_{j+1}(B_j)=\cdots=\P_J(B_j)$. Thus, summing both sides of \eqref{eqn_pjbj} over $1\le j \le J$ we obtain the desired result.
\end{proof}
\subsection{The distortion method for number fields}
Let $K/\mathbb{Q}$ be a number field. Let $\mathcal{A}=\set{a_1+I_1, \dots, a_n+I_n}$ be a collection of congruence classes of multiplicity $s$ for $a_1,\dots, a_n\in \mathcal{O}_K$ and distinguishable ideals $I_1,\dots,I_n$ in $\mathcal{O}_K$ with $\norm{I_1}\le \cdots\le \norm{I_n}$. Let $Q=I_1\cap\cdots\cap I_n$. Then $\mathcal{O}_K/Q$ is a finite set. Suppose we have the prime ideal decomposition $Q=\prod_{i=1}^J\mathfrak{p}_i^{\nu_i}$ with $\norm{\mathfrak{p}_1} \le \cdots \le \norm{\mathfrak{p}_J}$. For $1\le j \le J$, we take $Q_j=\prod_{i=1}^j\mathfrak{p}_i^{\nu_i}$ and let $\pi_j:\mathcal{O}_K/Q\to \mathcal{O}_K/Q_j$ be the natural projection. Then $\set{\pi_j}$ is an inverse system. For $1\le j \le J$, let
\[
\mathcal{B}_j\colonequals\bigcup_{\substack{1\le i\le n\\ P_{\min}(I_i)=\mathfrak{p}_j } } \set{a+Q: a \equiv a_i \mod{I_i}}.
\]
Clearly, $\mathcal{B}_j$ is a $\pi_j$-measurable subset of $\mathcal{O}_K/Q$ for each $1\le j\le J$.
Now, take $\P_0$ to be the uniform probability measure on $\mathcal{O}_K/Q$. Then we can construct a sequence of probability measures $\P_1, \cdots, \P_J$ on $\mathcal{O}_K/Q$ with respect to the triples $\set{(\pi_j, \mathcal{B}_j, \delta_j): 1\le j\le J}$ and $\P_0$ by taking $S=\mathcal{O}_K/Q$ and $B_j=\mathcal{B}_j$ in Section~\ref{sec_distortion_method_general}. Here $\delta_1,\dots, \delta_J\in [0,1/2]$ are parameters. By Theorem~\ref{thm_distortion_method}, we have the following result of the distortion method in \cite{BBMRS2022, KKL2022} for number fields.
\begin{theorem}\label{thm_distortion_method_nf}
Let $K/\mathbb{Q}$ be a number field. Let $\mathcal{A}$ be a finite collection of congruence classes in $\mathcal{O}_K$. Assume the above construction. If
\[
\sum_{j=1}^J
\min\left\{M_j^{(1)},\frac{M_j^{(2)}}{4\delta_j(1-\delta_j)}\right\}<1 ,
\]
then $\mathcal{A}$ does not cover the algebraic integers of $K$.
\end{theorem}
To prove Theorem~\ref{mainthm} via Theorem~\ref{thm_distortion_method_nf}, we take $S=\mathcal{O}_K/Q$ and $B_j=\mathcal{B}_j$ in Section~\ref{sec_distortion_method_general} and assume the above notation in the following sections.
\section{Bounding the first and second moment}\label{boundind_moments}
We will estimate $M_j^{(1)}$ and $M_j^{(2)}$ in this section. Recall that
\[
\alpha_j(x)= \frac{|F_{j-1}(x)\cap \mathcal{B}_j|}{|F_{j-1}(x)|},
\]
where $F_{j-1}(x)=\set{a+Q: a\equiv c\mod{Q_{j-1}}}$ for any $x=c+Q\in\mathcal{O}_K/Q$ with $c\in\mathcal{O}_K$.
\begin{lemma} \label{lem:B in fibers}
For $x\in\mathcal{O}_K/Q$ and $1\le j \le J$, we have
\[
\alpha_j(x) \le \sum_{r=1}^{\nu_j} \sum_{H|Q_{j-1}} \sum_{\substack{1\le i\le n \\ I_i=H \mathfrak{p}_j^r}} \frac{1_{x \subseteq a_i+H}}{\norm{\mathfrak{p}_j}^r}.
\]
\end{lemma}
\begin{proof} First, we have $|F_{j-1}(x)|=\norm{Q}/\norm{Q_{j-1}}$. Write $x=c+Q$ for some $c\in\mathcal{O}_K$. Then for any $a\in F_{j-1}(x)\cap\mathcal{B}_j$, there exists some $1\le i \le n$ such that $a\equiv a_i\mod{I_i}$ and $P_{\min}(I_i)=\mathfrak{p}_j$. Moreover, we have $a\equiv c \mod{Q_{j-1}}$. Then it follows that
\begin{equation*}
\alpha_j(x)=\frac{|F_{j-1}(x)\cap\mathcal{B}_j|}{\norm{Q}/\norm{Q_{j-1}}}\leq \frac{\norm{Q_{j-1}}}{\norm{Q}}\sum_{\substack{1\le i\le n\\ P_{\min}(I_i)=\mathfrak{p}_j}}\sum_{\substack{a \mod{Q}\\ a\equiv c\mod{Q_{j-1}}\\ a\equiv a_i\mod{I_i}}}1.
\end{equation*}
For each $i$ with $P_{\min}(I_i)=\mathfrak{p}_j$, we may write uniquely $I_i=H\mathfrak{p}_j^r$ with $H|Q_{j-1}$ and $1\leq r\leq \nu_j$. In order that the congruences $a\equiv a_i\mod{I_i}$ and $a\equiv c\mod{Q_{j-1}}$ to be compatible, we need that $c\equiv a_i\mod{H}$ and hence $x\subseteq a_i+H$. Under this assumption, $a$ lies in some congruence class mod $Q_{j-1}\mathfrak{p}_j^r$. In particular, we see that there are $\norm{Q}/\norm{Q_{j-1}\mathfrak{p}_j^r}$ choices for $a \mod{Q}$. Therefore, the desired result follows immediately.
\end{proof}
\begin{lemma}\label{lemma applied in proof of Lemma 4.2}
For each $0\le j \le J$, $a\in\mathcal{O}_K$, and any ideal $I\subset \mathcal{O}_K$ such that $I|Q$, we have
\[
\P_j(a+I)\le \frac{1}{\norm{I}}\prod_{\mathfrak{p}_i|I, i\le j}\frac{1}{1-\delta_i}.
\]
\end{lemma}
\begin{proof} We follow the proof of \cite[Lemma 3.4]{BBMRS2022} by induction on $j$. Since $\P_0$ is the uniform probability measure, we have $\P_0(a+I)=1/\norm{I}$. So the estimate holds for the case $j=0$. Let $j\ge1$, and assume the estimate holds for $\P_{j-1}$. Suppose that $\mathfrak{p}_j | I$, then we have
\begin{align*}
\P_j(a+I)&\le \frac{1}{1-\delta_j}\P_{j-1}(a+I)\\
&\le \frac{1}{\norm{I}}\cdot \frac{1}{1-\delta_j}\prod_{\mathfrak{p}_i|I, i< j}\frac{1}{1-\delta_i}\\
&=\frac{1}{\norm{I}}\prod_{\mathfrak{p}_i|I, i\le j}\frac{1}{1-\delta_i}.
\end{align*}
If $\mathfrak{p}_j \nmid I$, then $I=ML$, where $M=\gcd(I,Q_j)=\gcd(I,Q_{j-1})$. It follows that
\begin{align*}
\P_j(a+I)&=\frac{\P_j(a+M)}{\norm{L}}=\frac{\P_{j-1}(a+M)}{\norm{L}}\\
&\le \frac{1}{\norm{L}\norm{M}} \prod_{\mathfrak{p}_i|M, i< j}\frac{1}{1-\delta_i}=\frac{1}{\norm{I}}\prod_{\mathfrak{p}_i|I, i\le j}\frac{1}{1-\delta_i},
\end{align*}
as required.
\end{proof}
Now, we are ready to prove the following bounds of $M_j^{(1)}$ and $M_j^{(2)}$.
\begin{lemma}\label{lem:moments}
Assume the above notation, let $s=m(\mathcal{A})$, and let $j\in\{1,2,\dots,J\}$.
\begin{enumerate}
\item If $\delta_i=0$ for $i\in\{1,\dots,j-1\}$, then
\[
M_j^{(1)} \le s \sum_{\substack{I\in \mathfrak{D}(K), \norm{I}\ge \norm{I_1} \\ P_{\min}(I)=\mathfrak{p}_j}} \frac{1}{\norm{I}} .
\]
\item We have
\[
M_j^{(2)} \ll \frac{s^2(\log \norm{\mathfrak{p}_j})^6}{\norm{\mathfrak{p}_j}^2}.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
Let $k$ be a positive integer. By Lemma~\ref{lem:B in fibers} we have
\begin{align}
\mathbb{E}_{j-1}(\alpha_j^k)&=\sum_{x\in \mathcal{O}_K/Q}\alpha_j^k(x)\P_{j-1}(x)\nonumber\\
&\le \sum_{1\le r_1,\dots, r_k\le \nu_j} \sum_{H_1,\dots, H_k | Q_{j-1}} \sum_{\substack{1\le i_1,\dots, i_k\le n\\ I_{i_\ell}=H_\ell \mathfrak{p}_j^{r_\ell}, \forall \ell}} \frac{\P_{j-1}\left(\bigcap_{\ell=1}^k(a_{i_\ell}+H_\ell)\right)}{\norm{\mathfrak{p}_j}^{r_1+\dots+r_k}}.\label{kth moment}
\end{align}
Observe that $\norm{H_\ell\mathfrak{p}_j^{r_\ell}}\geq \norm{I_1}$ for all $\ell$ since $\norm{I_i}\geq \norm{I_1}$ for all $i$. Moreover, given $r_1, \ldots, r_k$ and $H_1, \ldots, H_k$, there are at most $s^k$ choices for $i_1, \ldots, i_k$ with $I_{i_\ell}=H_\ell \mathfrak{p}_j^{r_\ell}$. It follows from the Chinese Remainder Theorem that for each such choice of $i_1, \ldots, i_k$ the set $\bigcap_{\ell=1}^k(a_{i_\ell}+H_\ell)$ is either empty, or a congruence class mod $\bigcap_{\ell=1}^kH_\ell$.
Next, using Lemma~\ref{lemma applied in proof of Lemma 4.2} we can see that
\begin{equation}\label{prob of intersection of APs}
\P_{j-1}\left(\bigcap_{\ell=1}^k(a_{i_\ell}+H_\ell)\right) \le \frac{1}{\norm{\bigcap_{\ell=1}^k H_\ell}}\prod_{\substack{i\leq j-1\\ \mathfrak{p}_i|\bigcap_{\ell=1}^k H_\ell}}(1-\delta_i)^{-1}.
\end{equation}
Then plugging \eqref{prob of intersection of APs} into \eqref{kth moment} we obtain
\[
\mathbb{E}_{j-1}(\alpha_j^k)\le s^k \sum_{1\le r_1,\dots, r_k\le \nu_j} \sum_{\substack{H_1,\dots, H_k | Q_{j-1}\\ \norm{H_\ell\mathfrak{p}_j^{r_\ell}}\ge \norm{I_1}, \forall \ell}} \frac{1}{\norm{\bigcap_{\ell=1}^k H_\ell}\norm{\mathfrak{p}_j}^{r_1+\dots+r_k}}\prod_{\substack{i\leq j-1\\ \mathfrak{p}_i|\bigcap_{\ell=1}^k H_\ell}}(1-\delta_i)^{-1}.
\]
In particular, let $k=1$ and $\delta_i=0$ for all $i\leq j-1$, it follows that
\begin{align*}
M_j^{(1)} & \le s\sum_{1\le r \le \nu_j} \sum_{\substack{H|Q_{j-1}\\ \norm{H\mathfrak{p}_j^r}\ge \norm{I_1}}} \frac{1}{\norm{H\mathfrak{p}_j^r}} \\
& \le s\sum_{\substack{I\in \mathfrak{D}(K), \norm{I}\ge \norm{I_1}\\ P_{\min}(I)=\mathfrak{p}_j}}\frac{1}{\norm{I}}.
\end{align*}
This proves part (1) of the lemma.
Next, let $k=2$ and $\delta_i\in[0,1/2]$ for all $i$. It is clear that
\begin{equation*}
\prod_{\mathfrak{p}_i| H_1\cap H_2}(1-\delta_i)^{-1}\leq 2^{\omega(H_1\cap H_2)},
\end{equation*}
where $\omega(H_1\cap H_2)$ is the number of distinct prime ideals of $H_1\cap H_2$. Hence, we have
\begin{align*}
M_j^{(2)} & \le s^2\sum_{1\leq r_1, r_2\leq \nu_j}\sum_{H_1, H_2\mid Q_{j-1}}\frac{2^{\omega(H_1\cap H_2)}}{\norm{H_1\cap H_2}\norm{\mathfrak{p}_j}^{r_1+r_2}}\\
&\le \frac{s^2}{(\norm{\mathfrak{p}_j}-1)^2}\sum_{H_1, H_2\mid Q_{j-1}}\frac{2^{\omega(H_1\cap H_2)}}{\norm{H_1\cap H_2}}\\
&= \frac{s^2}{(\norm{\mathfrak{p}_j}-1)^2}\prod_{i<j}\left(1+\sum_{\nu\geq 1}\frac{2}{\norm{\mathfrak{p}_i}^\nu}\cdot (2\nu+1)\right)\\
&\le \frac{s^2}{\norm{\mathfrak{p}_j}^2}\prod_{i<j}\left(1+\frac{6}{\norm{\mathfrak{p}_i}}+O\left(\frac{1}{\norm{\mathfrak{p}_i}^2}\right)\right)\\
&\leq \frac{s^2}{\norm{\mathfrak{p}_j}^2}\,\mathrm{exp}\left\lbrace \sum_{i<j}\frac{6}{\norm{\mathfrak{p}_i}}+O\left(\frac{1}{\norm{\mathfrak{p}_i}^2}\right)\right\rbrace\\
&\ll \frac{s^2(\log\norm{\mathfrak{p}_j})^6}{\norm{\mathfrak{p}_j}^2}.
\end{align*}
Note that the last inequality in above calculations follows from Mertens' estimate for number fields, i.e., $\sum_{\norm{\mathfrak{p}}\leq x}1/\norm{\mathfrak{p}}=\log\log x+O(1)$, which follows from the prime number theorem for number fields (see \cite[p.~670]{Landau1903}) and partial summation. In conclusion, this completes the proof of part (2) of the lemma.
\end{proof}
\section{Proof of Theorem \ref{mainthm}}\label{sec_pf_mainthm}
In this section, let $\mathcal{A}=\set{a_1+I_1, \dots, a_n+I_n}$ be a collection of congruence classes of multiplicity $s$ for $a_1,\dots, a_n\in \mathcal{O}_K$ and distinguishable ideals $I_1,\dots,I_n$ in $\mathcal{O}_K$ with $\norm{I_1}\le \cdots\le \norm{I_n}$. To prove Theorem \ref{mainthm}, it suffices to show that there exists a constant $c_K>0$ depending only on $K$ such that if
\[
\norm{I_1} > \exp\big(c_K\log^2(s+1)/\log\log(s+2)\big)
\]
then $\mathcal{A}$ does not cover $\mathcal{O}_K$. Moreover, using the notation in Section~\ref{sec_distortion_method}, by Theorem~\ref{thm_distortion_method_nf} it reduces to show that
\[
\eta\colonequals \sum_{j=1}^J
\min\left\{ M_j^{(1)},\frac{M_j^{(2)}}{4\delta_j(1-\delta_j)}\right\}<1.
\]
Let $y = C s^3$, where $C$ is a large constant to be chosen later. Let $k = \max\{j \in [1, J] \cap \mathbb{Z}: \norm{\mathfrak{p}_j} \leq y\}$ and let
\[
\delta_i =
\begin{cases}
0 \text{ for } i \leq k \\
\frac{1}{2} \text{ for } i > k.
\end{cases}
\]
Then
\[
\eta \leq \sum_{1 \leq j \leq k} M_j^{(1)} + \sum_{k < j \leq J} M_j^{(2)} =: \eta_1 + \eta_2.
\]
Then it follows from Lemma~\ref{lem:moments} (2) that
\[
\eta_2 \ll \sum_{\norm{\mathfrak{p}_j} > y} \frac{s^2 (\log \norm{\mathfrak{p}_j})^6}{\norm{\mathfrak{p}_j}^2}
\ll \sum_{\norm{\mathfrak{p}_j} > y} \frac{s^2}{\norm{\mathfrak{p}_j}^{1.9}}
\ll \frac{s^2}{y^{0.9}}
= \frac{1}{C^{0.9} s^{0.7}}.
\]
Now we choose and fix $C$ to be large enough such that $\eta_2 < \frac{1}{2}$.
To bound $\eta_1$, we apply Lemma~\ref{lem:moments} (1) and our assumption that
\[
\norm{I_1} > x \colonequals \exp \{ c_K (\log(s + 1))^2 / \log \log (s + 2)\}
\]
and obtain
\[
\eta_1
\le s \sum_{1 \leq j \leq k}
\sum_{\substack{I\in \mathfrak{D}(K), \norm{I}\ge \norm{I_1} \\ P_{\min}(I)=\mathfrak{p}_j}} \frac{1}{\norm{I}}
\le s
\sum_{\substack{\norm{I}\ge x \\ P(I)\le y}} \frac{1}{\norm{I}}.
\]
Now, we use Rankin's trick (cf. \cite[(16.11)]{Koukoulopoulos2019}) to estimate the right summation as follows:
\[
\sum_{\substack{\norm{I}\ge x\\ P(I)\le y}}\frac{1}{\norm{I}} \le \sum_{P(I)\le y}\frac{(\norm{I}/x)^{1/2}}{\norm{I}} =x^{-1/2}\prod_{\norm{\mathfrak{p}}\le y} \of{1+\sum_{m=1}^\infty \frac{1}{\norm{\mathfrak{p}}^{m/2}}} =Wx^{-1/2},
\]
where $W\colonequals\prod_{\norm{\mathfrak{p}}\le y} (1-\norm{\mathfrak{p}}^{-1/2})^{-1}$ is fixed as $y$ is fixed. Hence we get that
\[
\eta_1 \le Wsx^{-1/2},
\]
which tends to $0$, as $x\to\infty$. Therefore, we have $\eta < 1$ for sufficiently large $c_K$, which completes the proof.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,129 |
Death of Pedestrian in Austin Ruled a Vehicular Homicide
By John Greenfield
The alley where the victim was found.
Following an autopsy, the death of a woman who was struck and killed by an SUV driver and found in an Austin alley early Sunday morning has been ruled a homicide, according to the Cook County Medical Examiner's office.
LaTasha Amos, 37, was found unresponsive in the alley in the 1100 block of South Mason at 3 a.m. and was taken to Loyola University Medical Center, according to Officer Michael Sullivan of Police News Affairs. Amos, who lived on the 4700 block of West Adams Street, was pronounced dead at 3:23 a.m., according to the morgue.
The autopsy found that Amos died of multiple injuries after the motorist struck her and then fled the scene, authorities said. The police are conducting a death investigation, but no one is in custody, and no further information is currently available, Sullivan said.
Fatality Tracker: 2013 Chicago pedestrian and bicyclist deaths
Pedestrian: 19 (15 were from hit-and-run crashes, 2 in semi- and box truck crashes)
Bicyclist: 3
Filed Under: Streetsblog Chicago, Fatality Tracker
Adam Herstein
While this is a terrible tragedy, I'm glad that the CPD is treating this incident seriously.
This wasn't an accident this was a murder. The person will be charged and the amos and Wilcox family will get justice. The person who did this will get just what they deserve. God bless and keep the Amos and Wilcox block family.
Man Killed in Little Village Was 3rd Vehicular Homicide Victim of 2014
By John Greenfield | Sep 3, 2014
View Larger Map The 4000 block of West 26th Street. A Brighton Park man who was beaten and then run over by a motorist early Sunday morning was the third victim of a vehicular homicide this year. On Monday, April 28, Darrell Cooper, 41, intentionally struck and killed Artez McBride, 20, in South Austin following […]
Police: Teen Intentionally Struck 3 Relatives With a Car, Killing His Father
By John Greenfield | Feb 17, 2016
A 39-year-old man is dead after his teenage son intentionally struck him and two other relatives with a car in Austin, according to police. On Saturday at around 11:15 p.m., a domestic dispute took place outside in the 5400 block of West Kamerling, according to the police. After the fight, a 19-year-old man intentionally struck […]
Jessie Rosales, 45, Fatally Struck in Lawndale Alley by Hit-and-Run Driver
By John Greenfield | Aug 29, 2016
45-year-old Jessie Rosales was struck and killed in the North Lawndale neighborhood by a motorist who fled the scene last Friday, according to police. At about 10 p.m. the driver of a dark-colored vehicle was heading south on the 1500 block of South Karlov Avenue when they made a left turn, eastbound, into the north […]
Hit-and-Run Drivers Killed Pedestrians in Englewood and Bridgeport on Friday
By John Greenfield | Jun 18, 2013
Last week two different people lost their lives to hit-and-run drivers, in two different South Side neighborhoods, on the same day. Both of the drivers are still at large. On Friday at about 3:40 a.m., a southbound motorist fatally struck Timothy Jones, 41, of the 200 block of South Sacramento, in the 6000 block of […]
Drivers Killed Four Pedestrians in Chicago This Month
August has been one of the deadliest months for pedestrians in Chicago this year, with four fatalities so far. However, with 16 pedestrian deaths so far this year, the fatality rate is down from last year, when there were 20 deaths by this time, according to the Chicaog Department of Transportation. Here's a summary of […]
Man Who Tried to Stop DUI Dragged and Fatally Struck; Driver Arrested
By John Greenfield | Mar 24, 2014
View Larger Map The victim was fatally struck just south of the Aberdeen Tap. Shane Stokowski, 33, was killed Saturday while trying to prevent a fellow bar patron from driving drunk, according to Officer Michael Sullivan from Police News Affairs. Stokowski, of the 3500 block of West Altgeld in Logan Square, had been hanging out […] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,715 |
\section{Conclusion}\label{sec:conclusion}
In this article, we have studied the implications of the twisted Riemann period relations to the study of accordiohedra. We made use of the recent analysis by the author \cite{Kalyanapuram:2019nnf,Kalyanapuram:2020tsr} which revealed an interpretation of scattering amplitudes at tree level in scalar theories are patterns of intersection between accordiohedra. The fact that these amplitudes could be written as intersection numbers naturally suggested that the twisted Riemann period relations may be an interesting direction of study, which is what we have set out to carry out in the note.
The twisted Riemann period relations have already been employed in the study of theories with cubic interactions, where they were shown to give rise to the KLT relations between gravity and Yang-Mills amplitudes at tree level. This allows us to make some comments on the relationship borne by our analysis to these previous studies as well as lay out some directions for further study.
In Section \ref{sec:scattering}, we have presented a synthesis of the data derived so far in the literature that describe the combinatorial structures of accordiohedra for a number of examples. We treated examples of accordiohedra in $\phi^4$ theory as well as the theory $\phi^3+\phi^4$ with mixed vertices. After providing details on how these can be understood as hyperplane arrangements in $\mathbb{CP}^{n}$, we used the methods of twisted cohomology theory to derive the analogue of the scattering equations due to Cachazo, He and Yuan for these more generic quantum field theories. A salient feature throughout was the fact that since for a given scattering process accordiohedra are generically not unique, we obtain a \emph{family} of scattering equations, rather than a single one as in the case of the CHY formalism.
We then made use of the fact that the number of equations of the scattering equations coincides with the dimension of the cohomology group to compute the dimensions of all the relevant groups in our analysis. Applying now the fact that scattering amplitudes of the relevant scalar theories are simply intersection numbers of the accordiohedra, in Section \ref{sec:relations} we found the KLT relations for these theories by studying the twisted Riemann period relations, which amount to simply an insertion of two complete sets of cohomology bases into the intersection number. Again, an important feature of the analysis was the lack of uniqueness; one KLT relation is present for every accordiohedron contributing to the scattering.
There are a few key differences that distinguish the KLT relations in this work from the original relations between gravity and Yang-Mills amplitudes. In the original KLT relations, the KLT matrix was a very special object from the point of view of quantum field theory - it is the inverse of the matrix of partial amplitudes in the biadjoint scalar theory. No such simple picture seems to arise in our analysis. Indeed, certain bases give rise to elements of the KLT matrix containing poles which are spurious and cannot arise out of any scattering channel. Another important difference is more practical in bent. In this work, we have focused on the simplest examples, dealing only with one and two dimensional examples. Other than the obvious ease of computation this facilitates, we have saved ourselves of having to deal with the complicated geometry of hyperplanes in higher dimensions. We have been able to specify cohomology bases essentially by inspection. It would be quite unsatisfactory if one were forced to do this at higher dimensions as well. Accordingly, one definite line of future research could be to study the possibility of finding a canonical presentation of cohomology classes for accordiohedron hyperplane arrangements \footnote{A possible answer to the question posed here might be to refine the basis due to Aomoto described in \cite{Abe:2015ucn,Abe:2018bgl}. It doesn't seem to always apply in the case by accordiohedra, due in part to the specific nature of the arrangement.}.
Field theory amplitudes are not alone in admitting a representation as intersection numbers. String amplitudes in the closed and open sectors as well as more exotic amplitude with $\alpha'$ corrections such as $Z$-theory \cite{Mafra:2011nv,Mafra:2011nw,Carrasco:2016ldy,Carrasco:2016ygv,Mafra:2016mcc} can also be represented as intersection numbers. There also exist deformations of the Parke-Taylor factor \cite{Stieberger:2013hza,Stieberger:2013nha,Mizera:2017sen} which recast string amplitudes as essentially CHY type formulae. There is a natural generalization of the Koba-Nielsen factor in the accordiohedron context, which was touched upon briefly in \cite{Kalyanapuram:2020vil} as well as in more conventional string-like contexts \cite{Baadsgaard:2015hia,Baadsgaard:2015ifa,Baadsgaard:2015voa,Baadsgaard:2016fel}. Exploring the analogue of $\alpha'$ corrections in generic scalar theories and the possibilities of double copy relations in terms of string-like generalizations of scalar amplitudes studied in \cite{Mizera:2016jhj,Kalyanapuram:2019nnf} might be interesting.
Another very popular avatar of the double copy is due to Bern, Carrasco and Johansson (see \cite{Bern:2008qj,Bern:2010tq,Bern:2010ue,Bern:2019prr} and references thereof and therein), in which the double copy is carried out at the level of individual Feynman diagrms, where kinematical and colour factors in the numerators obery identities that enable an essential 'squaring' of a gluon amplitude into a gravity amplitude. It has transpired of late that string amplitudes and intersection theory \cite{Mizera:2019gea} may play a role in providing some insight into their origins. It may prove worthwhile to pursue this connection for theories with higher point interactions using the tools developed here.
In this work and work on accordiohedra more generally, we have not yet tried to understand the implications of the theory of polytopes to the study of quantum field theory with numerators and possible double copy structures therein. One may consider extending the analysis carried out in the work to the study of theories such as $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$ supergravity, which admit geometric representations \cite{ArkaniHamed:2008gz,ArkaniHamed:2009dn,ArkaniHamed:2009si,ArkaniHamed:2009vw,ArkaniHamed:2010gg,ArkaniHamed:2010gh,ArkaniHamed:2010kv,ArkaniHamed:2012nw,Arkani-Hamed:2013jha,Arkani-Hamed:2014dca} as well. We note here parenthetically that the construction of double copy relations for polynomial vertices may also make possible future analyses of double copy relations without having recourse to string theory, thereby possibily enabling future applications of double copy relations to realistic theories.
Finally, we note that the intersection theory of accordiohedra has been developed thus far with no recourse being had to string theory methods. However, higher point interactions have been studied using string based methods in the past \cite{Baadsgaard:2016fel}. Modern developments in superstring field theory \cite{Moosavian:2017fta,Moosavian:2017qsp,Moosavian:2017sev,Moosavian:2019ydz}, especially in light of the use made of hyperbolic geometry in the closed sector, may consequently prove to be valuable in decoding the relationship between higher loop interactions and polytopes and in shedding light on the nature of the double copy more generally.
\section{Introduction}\label{sec:intro}
A double copy relation broadly defined is a method by which the scattering amplitudes of a given theory - such as Einstein gravity - can be defined equivalently as a convolution of two scattering amplitudes of some other quantum field theory, such as Yang-Mills. These relations often generate considerable interest due to the perspective they provide on the structure of gravitational amplitudes. The most famous double copy relation is that of Kawai, Lewellen and Tye \cite{KAWAI19861}, which is a statement about the relationship borne by gravity amplitudes to gluon amplitudes at tree level. The relation is expressed by the formula,
\begin{equation}
\mathcal{A}_{GR}(1,..,n) = \sum_{\alpha,\beta}\mathcal{A}_{YM}(\alpha)S[\alpha|\beta]\A_{YM}(\beta).
\end{equation}
The notation is by now quite standard, but we will review it for completeness. $\A_{GR}$ and $\A_{YM}$ are the scattering amplitudes for gravity and Yang-Mills at tree level, here evaluated for $n$ particles. The Yang-Mills amplitudes are colour ordered, denoted by $\alpha$ and $\beta$. The sum is over $(n-3)!$ colour orderings, known now as a Bern-Carrasco-Johansson basis \cite{Bern:2005hs}. The matrix $S[\alpha|\beta]$ is the KLT kernel, which fuses together the gauge theory amplitudes to return the gravity amplitude.
One of the most interesting lines of research in the modern study of scattering amplitudes deals with the subtleties of the KLT kernel, which enjoys several interesting mathematical properties. The kernel can be computed as the inverse of a matrix whose entries are $m(\alpha|\beta)$, the double-partial ordered amplitudes of a theory known as biadjoint scalar field theory. This fact was originally discovered by Cachazo, He and Yuan \cite{Cachazo:2013gna}, and is most easily derived in their scattering equations formalism \cite{Cachazo:2013hca,Cachazo:2013iaa,Cachazo:2013iea,Cachazo:2014nsa,Cachazo:2014xea,Cachazo:2015aol}. Indeed, a particular upshot of this formalism is the trivialization of the KLT relations as a simple fact of linear algebra. This formalism, now known as the CHY formalism, inspired a growing understanding of scattering amplitudes in quantum field theories as essentially geometric and topological objects. Two key aspects in this study are the theory of positive geometries and polytopes and the theory of twisted intersection numbers.
The positive geometry program, which found its inception in the work of Arkani-Hamed \emph{et al.} \cite{Arkani-Hamed:2017mur,Arkani-Hamed:2017tmz} provided a new understanding of scattering amplitudes in biadjoint theories as the volume of a fairly special geometric object, known as the associahedron. In more technical terms, the associahedron $\mathcal{A}_{n}$ of dimension $(n-3)$ admits a convex embedding in the space $\mathcal{K}_{n}$ of Mandelstam invariants for $n$ massless particles. A differential form $\Omega$ of top degree, known in the literature as a canonical form, is fixed uniquely upto a prefactor by simply demanding a logarithmic divergence as the boundaries of the associahedron are approached. The residue of this form on the associahedron can be shown to be equal to the scattering amplitude of the $\phi^3$ at tree level,
\begin{equation}
\mathrm{Res}_{\mathcal{A}_{n}}\Omega_{n} = m_{\phi^3}(1,...,n).
\end{equation}
The geometric structure of the associahedron and the analytic structure of the canonical form trivialise quantitative features of the scattering amplitude, such as locality and unitarity. The local structure of the amplitude is fixed by the logarithmic divergence condition, while unitarity is assured at tree level by the boundary structure of the associahedron. Any boundary $\mathcal{B}$ of the associahedron factors into two lower point associahedra $\mathcal{A}_{n_1}\times \mathcal{A}_{n_2}$. Approaching the boundary amounts to a unitarity cut, and the residue of the canonical form factorizes accordingly, encoding the factorization of the scattering amplitude, implying unitarity. A polytope equipped with a canonical form is known as a \emph{positive geometry}.
The associahedron encodes the amplitudes of theories with cubic vertices due to its simple combinatorial definition. The vertices of an associahedron $\mathcal{A}_{n}$ are in one to one correspondence with the triangulations of an $n$-gon. A given triangulation of an $n$-gon can be dualized into a trivalent tree, which can be interpreted as a Feynman diagram for a cubic theory. Accordingly, it becomes intuitively clear why the associahedron encodes so naturally the analytic structure of scattering amplitudes for such theories.
This approach to understanding scattering amplitudes has enjoyed quick progress over the last few years. In particular, much work has gone into trying to extend these ideas to theories which have more complicated interactions. The first in this line of work was due to Banerjee, Laddha and Raman \cite{Banerjee:2018tun}, in which it was found that a class of polytopes known as Stokes polytopes furnish the positive geometries of quartic scalar theories at tree level in the planar limit. Stokes polytopes are convex polytopes whose vertices are in one to one correspondence with \emph{quadrangulations} of an $n$-gon. They differ qualitatively from associahedra in that they are not unique for a given number of particles interacting. Even for $8$ particles, there are two different types of Stokes polytopes, which give rise to partial amplitudes. The total amplitude is obtained by a sum over these partial amplitudes weighted by factors determined entirely by the combinatorics of the Stokes polytopes. This picture was extended to $\phi^p$ theories and to theories with generic interactions (see respectively \cite{Raman:2019utu} and \cite{Jagadale:2019byr}), and it was found that for all these theories, a large variety of polytopes collectively known as \emph{accordiohedra} played the role of the positive geometry. The accordiohedra for a large class of scalar theories at tree level have now been determined, studies of which have led to a renewed interest in their geometric and analytic structures \cite{Jagadale:2020qfa,Kojima:2020tox,John:2020jww}.
Accompanying this line of work nearly simultaneously was the application of a novel class of mathematical objects known as twisted intersection numbers to the study of scattering amplitudes. Originally, the techniques of twisted intersection theory were used to study the KLT kernel in string theory \cite{Mizera:2017cqs}, but have since been applied to general quantum field theories \cite{Mizera:2017rqa}, Feynman integrals \cite{Mastrolia:2018uzb,Frellesvig:2019kgj,Frellesvig:2019uqt,Frellesvig:2020qot} and other aspects of scattering amplitudes, including recursion \cite{Mizera:2019gea} and the double copy \cite{Mizera:2019blq}.
The essential data characterizing a twisted intersection theory is a one-form $\omega$ on $\mathbb{CP}^{n}$ with a collection of hyperplanes $\lbrace{H_{1},...,H_{N}\rbrace}$ removed, with the property that it diverges logarithmically as the hyperplanes are approached. The residue prescriptions near these singularities encode kinematical information. Defining a quantum field theory scattering amplitude now requires the definition of two forms $\phi_L$ and $\phi_R$, which have middle dimension (real dimension $n$). These forms however must be defined up to a cohomology prescription, where the cohomology is now \emph{twisted} by redefining the exterior derivative as $\nabla_{\omega} = d + \omega \wedge$. Now it can be shown that a pairing can be defined between two such forms. For suitable choices of the forms and hyperplane arrangement, the scattering amplitudes for an enormous class of quantum field theories can be obtained. Indeed, the CHY formula is a special case of a twisted intersection number, in which the hyperplane arrangement yields the moduli space of marked Riemann spheres. For an extensive review of methods and formalism, the reader should consult \cite{Mizera:2019gea}.
One of the advantages of the method of twisted intersection numbers is the trivialization of identities such as the KLT relations, recasting them as statements of linear algebra. These relations, known as the twisted Riemann period relations \cite{cho1995}, can be stated as follows. Suppose we have two forms $\varphi_1$ and $\varphi_2$ belonging to the $n$th cohomology of $\nabla_{\omega}$ and bases $\lbrace{\phi_{i}\rbrace}$ and $\lbrace{\phi'_{i}\rbrace}$ of this cohomology group. The twisted intersection number $\langle{\varphi_1,\varphi_2\rangle}$ can be shown to satisfy,
\begin{equation}\label{eq1.3}
\langle{\varphi_1,\varphi_2\rangle} = \langle{\varphi_1,\phi_i\rangle}(\mathbf{C}^{-1})^{T}_{ij}\langle{\phi_j,\varphi_2\rangle},
\end{equation}
where the matrix $\mathbf{C}_{ij}$ is the intersection matrix $\langle{\phi_i,\phi'_j\rangle}$. Indeed, when the cohomology is defined on $\mathcal{M}_{0,n}$ and the basis of the cohomology class is chosen to be a Parke-Taylor basis, the intersection matrix simply becomes the KLT kernel in field theory.
Now, given these two geometric approaches to scattering amplitudes, a natural question to ask is whether or not the advantages of twisted intersection theory can be exploited by the technology of polytopes. This question was addressed by the author for the case of $\phi^4$ theory and Stokes polytopes in \cite{Kalyanapuram:2019nnf}, where the planar amplitudes in this theory were written as intersection numbers, enabling an extension to generic kinematics as well. The extension to all scalar theories was then carried out in \cite{Kalyanapuram:2020tsr}, with Jha. Using this framework, the goal of the present work will be to carry out a systematic study of the scattering equations for generic scalar theories and provide a concrete realization of the twisted period relations for the relevant polytopes as well.
In order to study the twisted period relations for the polytopes controlling generic scalar theories, known as accordiohedra, some relevant analytic data must first be secured. In particular, the specific convex realizations of accordiohedra in $\mathbb{CP}^{n}$ have to be understood. The twist for accordiohedra, encoding kinematical information now play the role of the scattering equations, the number of solutions of which determines the size of the KLT matrix. The task of finding the KLT relations then devolves upon the definition of a suitable basis of the cohomology and the computation of the intersection matrix of these elements. To ensure that the context and motivations for these calculations are clear, we carry out the analysis for most of the examples worked out in the literature so far, reviewing most of the background material needed for the study.
Throughout the paper, we will only be dealing with scalar theories and won't concern ourselves with theories containing nontrivial kinematical numerators. It is important to note then that what we are really doing is \emph{expanding} scalar amplitudes into a so-called Bern-Carrasco-Johansson basis. For the purposes of simplicity, we will continue to refer to the expansions we will derive as KLT relations, but it should be kept in mind that we are not yet dealing with theories with numerators.
\section{The KLT Relations for Accordiohedra}\label{sec:relations}
The double copy relations due to Kawai, Lewellen and Tye were first developed in the study of string amplitudes. They observed that the scattering of $n$ massless closed string states could be rearranged as a product of two open string scattering amplitudes of $n$ massless states each. This product however cannot be carried out blindly. An object known as the KLT kernel must be used to convolve $(n-3)!$ open string amplitudes with another copy of $(n-3)!$ amplitudes, where the permutations are carried out over two choices of $n-3$ particles for each of the open string amplitudes.
The original derivation of the KLT relations used contour deformation arguments, which combined with the splitting into holomorphic and antiholomorphic parts of the closed string integrand breaks the integral over $\mathcal{M}_{0,n}$ into two integrals over the $n$-simplex $\Delta_{n}$. The trade-off is the introduction of the KLT kernel, which fuses the two open string integrals together. Schematically we have,
\begin{equation}
\int_{\mathcal{M}_{0,n}} |\mathcal{I}|^{2} = \sum_{\alpha,\beta}\left(\int_{\Delta_{n}}\mathcal{I}(\alpha)\right)K[\alpha|\beta]\left(\int_{\Delta_{n}}\mathcal{I}(\beta)\right).
\end{equation}
Here, the closed string integrand has been factorized into two open string integrals. Doing this requires the introduction of two bases $\alpha$ and $\beta$, which run over two arbitrary permutations of the $n$ particles, playing the role of a colour ordering. In the field theory limit, this expression reduces to a relationship between the scattering of gravitons and Yang-Mills amplitudes at tree level. We will focus on this avatar of the KLT kernel henceforth.
While the KLT relation in field theory can be derived in a number of ways, the approach that we are most interested in is that of twisted intersection theory. To review the basic details involved, consider the moduli space of marked Riemann spheres $\mathcal{M}_{0,n}$. Let us equip this space with the following differential form,
\begin{equation}
\omega = \sum_{i<j}s_{ij}d\ln(z_{i}-z_{j}),
\end{equation}
where $s_{ij} = (p_i+p_j)^2$, and all particles are considered massless. This form acts as a \emph{twist}, which modifies the exterior derivative to
\begin{equation}
\nabla_{\omega} = d + \omega\wedge .
\end{equation}
This twisted differential operator can be verified to be nilpotent, which leads to a natural generalization of the de-Rham complex into the twisted de Rham complex and a corresponding cohomology theory, which can be defined by,
\begin{equation}\label{eq3.4}
H^{k}\left(\mathcal{M}_{0,n},\nabla_{\omega}\right) = \frac{\lbrace{h\in \Omega^{k}\left(\mathcal{M}_{0,n}\right)|\nabla_{\omega}h = 0\rbrace}}{\lbrace{h\in \Omega^{k}\left(\mathcal{M}_{0,n}\right)|h = \nabla_{\omega }f\rbrace}}.
\end{equation}
An important theorem due to Aomoto established that this cohomology theory is trivial for all $k\neq (n-3)$\footnote{Note that we have $n \geq 3$.}, which is middle dimensional.
A basis of this cohomology class which is particularly valuable for the study of scattering amplitudes is known as the Parke-Taylor basis. For a given permutation $\sigma\in S_{n-3}$ the following form may be defined,
\begin{equation}
\mathrm{PT}(\sigma) = \frac{dz_1\wedge ... \wedge dz_{n}}{(z_{1}-z_{2})(z_{2}-z_{\sigma(3)})(z_{\sigma(3)}-z_{\sigma{4}})...(z_{n}-z_{\sigma(n-1)})(z_{1}-z_{1})}.
\end{equation}
It can be shown (a key result of \cite{Mizera:2017cqs}) that this collection of $(n-3)!$ forms furnishes a complete basis for $H^{(n-3)}\left(\mathcal{M}_{0,n},\nabla_{\omega}\right)$.
Given two representatives $\varphi_1$ and $\varphi_2$ of the cohomology group in \ref{eq3.4}, it was observed in a number of studies \cite{cho1995,Mimachi_2003,yoshida1,yoshida2,yoshida3,AOMOTO1997119,AOMOTO-KITA,YOSHIDA} that a pairing $\langle{\varphi_1,\varphi_2\rangle}$ may be defined between the two, which acts as a twisted generalization of the traditional notion of an intersection number. These ideas found their first application to field theory when Mizera proved in \cite{Mizera:2017rqa} that this pairing between two Parke-Taylor forms $\mathrm{PT(\alpha)}$ and $\mathrm{PT(\beta)}$ gives precisely the tree level planar scattering amplitude for the biadjoint scalar theory:
\begin{equation}
\langle{\mathrm{PT}(\alpha),\mathrm{PT}(\beta)\rangle} = m(\alpha|\beta).
\end{equation}
To relate this fact to the KLT relations, one simply notes that the Pfaffian $\mathrm{Pf}'\Psi$, which arises in the CHY formalism can be recast in a Parke-Taylor basis. The pairing between a Pfaffian and Parke-Taylor factor then becomes precisely the CHY formula for Yang-Mills amplitudes, while the pairing between two Pfaffians becomes the CHY formula for gravity amplitudes. An application of the twisted period relations \ref{eq1.3} with $\varphi_1 = \varphi_2 = \mathrm{Pf}'\Psi$ with a Parke-Taylor basis for the cohomology group yields the KLT relations.
Our main goal in this section will be to repeat this analysis when we have hyperplane arrangements in $\mathbb{CP}^{n}$ that produce accordiohohedra rather than $\mathcal{M}_{0,n}$. In order to do this, we have to use the formalism developed in \cite{Kalyanapuram:2019nnf} and \cite{Kalyanapuram:2020tsr}. We will now recall some of the principal aspects of the techniques developed therein.
We start with a collection of hyperplanes, say $f_{1},...,f_{N}$ in $\mathbb{CP}^{n}$, which realizes an $n$-dimensional polytope $\mathcal{A}_{n}$\footnote{We will also use this symbol to denote the complement of the hyperplane arrangement; there should be no confusion on this point} as the interior of the arrangement. Given this, we can now define a twist $\omega_{\A_n}$, encoding kinematical data. The main result of the recent work of the author was that the form
\begin{equation}
\varphi_{\mathcal{A}_{n}} = \sum_{f_{i_1}\cap f_{i_2}\cap...\cap f_{i_n}\in \mathrm{Vertex}}d\ln f_{i_{1}}\wedge ... \wedge d\ln f_{i_{n}}
\end{equation}
when viewed as an element of $H^{n}_{\omega_{\mathcal{A}_{n}}}\left(\mathcal{A}_{n},\nabla_{\omega_{\A_n}}\right)$ gives rise to the pairing $\langle{\varphi_{\A_n},\varphi_{\A_n}\rangle}$. The pairing was then shown to equal the scattering amplitude defined by the accordiohedron using the localization formula proved in \cite{Mizera:2017rqa}.
In this section, our aim will be to use this fact to develop a suitable generalization of the KLT relations for amplitudes derived from accordiohedra. Our basic technique will mirror the one used for associahedra, namely that we will find a basis for the twisted cohomology group and employ the twisted period relations. There are a couple of qualitative differences here that are worth noting however. An important fact that differentiates accordiohedra from associahedra is that we can have several accordiohedra for a given process, while associahedra are unique. Accordingly, we will obtain a family of KLT relations when working with these polytopes. Additionally, it seems that the elements of the KLT matrix in the case of accordiohedra are generically nonplanar amplitudes. There does not seem to be any canonical choice like the Parke-Taylor basis in the case of accordiohedra. Thus, for the time being, we are left with having to do brute force calculations for the time being. Clarifying the implications of these observations should be fertile ground for future research.
\subsection{One-Dimensional KLT Relations}
We start with the simplest example of KLT relations for accordiohedra - the case of one dimensional accordiohedra, which are basically associahedra. Since the essential aspects of these relations have already been worked out in \cite{Mizera:2017cqs}, the main point of this section will be to provide the correct analogue for those processes which admit geometric representations in terms of intersection numbers of one dimensional accordiohedra.
A one dimensional accordiohedron is simply a line, so we have the following hyperplane arrangement,
\begin{equation}
\begin{aligned}
f_{1} &= 0 \\
f_{2} &= 1
\end{aligned}
\end{equation}
in $\mathbb{CP}^{1}$. Thus, we are led to working on the complement
\begin{equation}
\mathcal{A}_{1} = \mathbb{CP}^{1} - \lbrace{0,1,\infty\rbrace}.
\end{equation}
On this space, we now define a twist, which encodes kinematical information. To keep the discussion as general as possible, we will use the following twist,
\begin{equation}
\omega_{\mathcal{A}_{1}} = \alpha_{1} d\ln(x) + \alpha_{2}d\ln(x-1).
\end{equation}
We have now the form,
\begin{equation}\label{eq3.12}
\varphi_{\mathcal{A}_{1}} = d\ln\left(\frac{x}{x-1}\right)
\end{equation}
and the intersection number
\begin{equation}
\langle{ \varphi_{\mathcal{A}_{1}}, \varphi_{\mathcal{A}_{1}}\rangle} = \frac{1}{\alpha_1} + \frac{1}{\alpha_2}.
\end{equation}
To find the KLT relations, we first note that the dimension is
\begin{equation}
\mathrm{dim} H^{1}_{\omega_{\A_1}}\left(\A_1, \nabla_{\omega_{\A_1}}\right) = 1.
\end{equation}
Accordingly, use of the twisted Riemann period relations, which requires inserting two copies of bases of the, can be made by specifying a two basis elements, each of which separately furnishes a basis of the twisted cohomology group. We will look at three such examples.
\paragraph{Example 3.1}
Consider a basis for the twisted cohomology group given by the element $\phi_1 = \varphi_{\mathcal{A}_{1}}$. In this case, the KLT matrix becomes,
\begin{equation}
\langle{\phi_1,\phi_1\rangle}^{-1} = \frac{\alpha_1\alpha_2}{\alpha_1+\alpha_2}.
\end{equation}
Applying the twisted Riemann period relations is now a trivial multiplication,
\begin{equation}
\left(\frac{1}{\alpha_1} + \frac{1}{\alpha_2}\right)\frac{\alpha_1\alpha_2}{\alpha_1+\alpha_2}\left(\frac{1}{\alpha_1} + \frac{1}{\alpha_2}\right) = \frac{1}{\alpha_1} + \frac{1}{\alpha_2}.
\end{equation}
Let's now consider a slightly more complicated example.
\paragraph{Example 3.2}
We now consider a basis given by the element,
\begin{equation}
\phi_1 = d\ln\left(\frac{x-1}{x^{-1}}\right)
\end{equation}
where we have indicated the hyperplane at infinity by $x^{-1}$. The self intersection number of this form gives
\begin{equation}\label{eq3.17}
\langle{\phi_1,\phi_1\rangle}^{-1} = \left(\frac{1}{\alpha_2}-\frac{1}{\alpha_1+\alpha_2}\right)^{-1} = \frac{(\alpha_1+\alpha_2)\alpha_2}{\alpha_1}.
\end{equation}
We have the partial intersection number
\begin{equation}\label{eq3.18}
\langle{\varphi_{\A_1},\phi_1\rangle} = \frac{1}{\alpha_2}.
\end{equation}
using which the twisted Riemann period relation becomes
\begin{equation}
\langle{\varphi_{\A_1},\phi_1\rangle} \langle{\phi_1,\phi_1\rangle}^{-1} \langle{\varphi_{\A_1},\phi_1\rangle} = \frac{1}{\alpha_1} + \frac{1}{\alpha_2}.
\end{equation}
Before considering one final example, we note that it is quite easy to see how the KLT matrix in equation \ref{eq3.17} fails to be planar - indeed, it isn't really even the inverse of an amplitude most of the time! Consider the case of $6$-particle scattering in $\phi^4$ theory, in which we choose $\alpha_1 = X_{14}$ and $\alpha_2 = X_{36}$. Clearly, $X_{14}+X_{36}$ is not a Mandelstam variable; it doesn't even correspond to a scattering channel.
\paragraph{Example 3.3}
Let's consider a final example, now using a mixed basis. On the left, we consider the basis given by $\phi_1 = \varphi_{\A_1}$ and on the right we consider the basis given by $\phi'_2=\varphi^{(2)}_{\A_1}$. This leads to the twisted period relation,
\begin{equation}
\langle{\varphi_{\A_1},\varphi_{\A_1}\rangle} = \langle{\varphi_{\A_1},\phi_1\rangle}\langle{\phi_1',\phi_1\rangle}^{-1}\langle{\phi_1',\varphi_{\A_1}\rangle},
\end{equation}
which can be seen to be true essentially trivially, but can be verified readily by making use of equations \ref{eq3.12} and \ref{eq3.18}.
\subsection{Two-Dimensional KLT Relations}
Let us now discuss the case of KLT relations that arise out of the twisted Riemann period relations for two dimensional accordiohedra intersecting. We have two possibilities, the square and pentagon accordiohedra.
A square is defined by the hyperplane arrangement
\begin{equation}
\begin{aligned}
f_{1} &= 1-y \\
f_{2} &= 1-x \\
f_{3} &= 1+y \\
f_{4} &= 1+x.
\end{aligned}
\end{equation}
With these we have the complement and twist as
\begin{equation}
\A_{2}^{(1)} = \mathbb{CP}^{2}-\lbrace{f_1,f_2,f_3,f_4\rbrace}
\end{equation}
\begin{equation}
\omega_{\A^{(1)}_{2}} = \beta_{1}d\ln f_{1} + \beta_{2}d\ln f_{2} + \beta_{3}d\ln f_{3} + \beta_{4}d\ln f_{4}.
\end{equation}
Similarly for the case of the pentagon we use the data
\begin{equation}
\begin{aligned}
g_{1} &= 1-x\\
g_{2} &= 1+y\\
g_{3} &= 2+x\\
g_{4} &= 2+x-y\\
g_{5} &= 2-y,
\end{aligned}
\end{equation}
\begin{equation}
\A_{2}^{(2)} = \mathbb{CP}^{2}-\lbrace{g_1,g_2,g_3,g_4,g_{5}\rbrace},
\end{equation}
and
\begin{equation}
\omega_{\A^{(2)}_{2}} = \gamma_{1}d\ln g_{1} + \gamma_{2}d\ln g_{2} + \gamma_{3}d\ln g_{3} + \gamma_{4}d\ln g_{4} + \gamma_{5}d\ln g_{5}.
\end{equation}
Consider first the case of the square. Take the form
\begin{equation}
\varphi_{\A^{(1)}_{2}} = d\ln\left(\frac{f_{1}}{f_{3}}\right)\wedge d\ln\left(\frac{f_{2}}{f_{4}}\right).
\end{equation}
The self intersection number of this form on $A^{(1)}_{2}$ is computed as
\begin{equation}
\langle{\varphi_{\A^{(1)}_{2}},\varphi_{\A^{(1)}_{2}}\rangle} = \left(\frac{1}{\beta_{1}}+\frac{1}{\beta_3}\right)\left(\frac{1}{\beta_{2}}+\frac{1}{\beta_4}\right).
\end{equation}
Indeed, this is just the square of the amplitude we obtained from the one dimensional case. Accordingly, we have the following.
\paragraph{Example 3.4}
Since the twisted cohomology group in this case is one dimensional, we need to specify again only one form in order to derive the KLT relations. The KLT relations are determined here entirely by the KLT relation for the line. For purposes of clarity, suppose we consider the basis given by
\begin{equation}
\phi_{1} = d\ln\left(\frac{f_{1}}{y^{-1}}\right)\wedge d\ln\left(\frac{f_{2}}{x^{-1}}\right).
\end{equation}
The KLT relation
\begin{equation}
\langle{\varphi_{\A^{(1)}_{2}},\varphi_{\A^{(1)}_{2}}\rangle} = \langle{\varphi_{\A^{(1)}_{2}},\phi_{1}\rangle} \langle{\phi_{1},\phi_{1}\rangle}^{-1} \langle{\phi_{1},\varphi_{\A^{(1)}_{2}}\rangle}
\end{equation}
can be established by noting that
\begin{equation}
\langle{\varphi_{\A^{(1)}_{2}},\phi_{1}\rangle} = \frac{1}{\beta_{1}\beta_{2}}
\end{equation}
and
\begin{equation}
\langle{\phi_{1},\phi_{1}\rangle} = \left(\frac{1}{\beta_{1}}-\frac{1}{\beta_1 + \beta_3}\right)\left(\frac{1}{\beta_{2}}-\frac{1}{\beta_2 + \beta_4}\right).
\end{equation}
Moving on, we now consider three examples for the pentagon. Defining the form
\begin{equation}
\varphi_{\A^{(2)}_{2}} = d\ln\left(\frac{f_{1}}{f_{2}}\right)\wedge d\ln\left(\frac{f_{2}}{f_{3}}\right) + d\ln\left(\frac{f_{3}}{f_{4}}\right)\wedge d\ln\left(\frac{f_{4}}{f_{5}}\right) + d\ln\left(\frac{f_{5}}{f_{1}}\right)\wedge d\ln\left(\frac{f_{1}}{f_{3}}\right)
\end{equation}
the self intersection number is
\begin{equation}\label{3.41}
\langle{\varphi_{\A^{(2)}_{2}},\varphi_{\A^{(2)}_{2}}\rangle} = \frac{1}{\gamma_{1}\gamma_{2}} + \frac{1}{\gamma_{2}\gamma_{3}} + \frac{1}{\gamma_{3}\gamma_{4}} + \frac{1}{\gamma_{4}\gamma_{5}} + \frac{1}{\gamma_{5}\gamma_{1}}.
\end{equation}
To determine a basis for the cohomology group, we note that the dimension of the group is $4$, meaning that we need to specify four basis elements. We can glean a natural way of doing this by inspecting a real slice of the hyperplane arrangement, as shown in figure \ref{fig3.1}\cite{Kalyanapuram:2020tsr}.
\begin{figure}[H]
\centering
\includegraphics[width=0.59\textwidth]{Pentagon_Hyperplanes.pdf}
\caption{Real Slice of Arrangement $\A^{(2)}_{1}$}\label{fig3.1}
\end{figure}
One can infer simply by inspection that there are four bounded chambers, which is a simple consistency check of the dimension of the cohomology group. An element of the cohomology group is furnished by a chamber; this can be seen by noting the Poincar\'e duality - a form with singularities on the boundaries of a chamber will be dual to the chamber, a cycle on the complement space.
\paragraph{Example 3.5}
Consider a basis for the cohomology group given by the four \emph{distinct} bounded chambers. These are given by
\begin{equation}
\phi_{1} = d\ln\left(\frac{f_{1}}{f_{2}}\right)\wedge d\ln\left(\frac{f_{2}}{f_{3}}\right) + d\ln\left(\frac{f_{3}}{f_{4}}\right)\wedge d\ln\left(\frac{f_{4}}{f_{5}}\right) + d\ln\left(\frac{f_{5}}{f_{1}}\right)\wedge d\ln\left(\frac{f_{1}}{f_{3}}\right),
\end{equation}
\begin{equation}
\phi_{2} = d\ln\left(\frac{f_{2}}{f_{3}}\right)\wedge d\ln\left(\frac{f_{3}}{f_{4}}\right),
\end{equation}
\begin{equation}
\phi_{3} = d\ln\left(\frac{f_{3}}{f_{4}}\right)\wedge d\ln\left(\frac{f_{4}}{f_{5}}\right)
\end{equation}
and
\begin{equation}
\phi_{4} = d\ln\left(\frac{f_{4}}{f_{5}}\right)\wedge d\ln\left(\frac{f_{5}}{f_{1}}\right).
\end{equation}
To determine the KLT relations, we have to compute the partial amplitudes $\langle{\varphi_{\A^{(2)}_{1}},\phi_{i}\rangle}$ and the matrix $\langle{\phi_i,\phi_j\rangle}$. We have
\begin{equation}\label{eq3.39}
\begin{split}
&\langle{\varphi_{\A^{(2)}_{1}},\phi_{1}\rangle} = \frac{1}{\gamma_{1}\gamma_{2}} + \frac{1}{\gamma_{2}\gamma_{3}} + \frac{1}{\gamma_{3}\gamma_{4}} + \frac{1}{\gamma_{4}\gamma_{5}} + \frac{1}{\gamma_{5}\gamma_{1}} \\
&\langle{\varphi_{\A^{(2)}_{1}},\phi_{3}\rangle} = \frac{1}{\gamma_{3}\gamma_{4}}+\frac{1}{\gamma_{4}\gamma_{5}}
\end{split}\qquad
\begin{split}
&\langle{\varphi_{\A^{(2)}_{1}},\phi_{2}\rangle} = \frac{1}{\gamma_{2}\gamma_{3}} + \frac{1}{\gamma_{3}\gamma_{4}} \\
&\langle{\varphi_{\A^{(2)}_{1}},\phi_{4}\rangle} = \frac{1}{\gamma_4\gamma_5}+\frac{1}{\gamma_5\gamma_1}.
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
&\langle{\phi_{1},\phi_{1}\rangle} = \frac{1}{\g_{1}\g_{2}} + \frac{1}{\g_{2}\g_{3}} + \frac{1}{\g_{3}\g_{4}} + \frac{1}{\g_{4}\g_{5}} + \frac{1}{\g_{5}\g_{1}} \\
&\langle{\phi_{1},\phi_{2}\rangle} = \frac{1}{{\g_{4}\g_{5}}}+\frac{1}{{\g_{5}\gamma_{1}}}\\
&\langle{\phi_{1},\phi_{3}\rangle} = \frac{1}{{\g_{3}\g_{4}}}+\frac{1}{{\g_{4}\g_{5}}} \\
&\langle{\phi_{1},\phi_{4}\rangle} = \frac{1}{{\g_{2}\g_{3}}}+\frac{1}{{\g_{3}\g_{4}}}\\
&\langle{\phi_{2},\phi_{2}\rangle} = \frac{1}{\g_{2}\g_{3}} + \frac{1}{\g_{3}\g_{4}} + \frac{1}{\g_{4}\g_{2}}\\
\end{split}\qquad
\begin{split}
&\langle{\phi_{2},\phi_{3}\rangle} = \frac{1}{{\g_{4}\g_{5}}} \\
&\langle{\phi_{2},\phi_{4}\rangle} = 0\\
&\langle{\phi_{3},\phi_{3}\rangle} = \frac{1}{\g_{3}\g_{4}} + \frac{1}{\g_{4}\g_{5}} + \frac{1}{\g_{5}\g_{3}}\\
&\langle{\phi_{3},\phi_{4}\rangle} = \frac{1}{{\g_{3}\g_{4}}} \\
&\langle{\phi_{4},\phi_{4}\rangle} = \frac{1}{\g_{4}\g_{5}} + \frac{1}{\g_{5}\g_{1}} + \frac{1}{\g_{1}\g_{4}}\\
\end{split}
\end{equation}
With these, the KLT relation is given by the resolution,
\begin{equation}
\langle{\varphi_{\A^{(2)}_{1}},\varphi_{\A^{(2)}_{1}}\rangle} = \langle{\varphi_{\A^{(2)}_{1}},\phi_{i}\rangle}\langle{\phi_{j},\phi_{i}\rangle}^{-1}\langle{\phi_{j},\varphi_{\A^{(2)}_{1}}\rangle}
\end{equation}
which has been verified using {\scshape Mathematica}.
The expressions for the inverse of the KLT matrix $\langle{\phi_i,\phi_j\rangle}$ computed in {\scshape Mathematica} were extremely unilluminating and baroque. As a result of this, we have not attempted to reproduce the results herein. It may be of value to carry out a systematic study of this issue in future work.
\paragraph{Example 3.6}
Consider now the following basis for the cohomology group of the pentagon
\begin{equation}
\phi'_{1} = d\ln\left(\frac{f_{2}}{f_{3}}\right)\wedge d\ln\left(\frac{f_{3}}{f_{4}}\right),
\end{equation}
\begin{equation}
\phi'_{2} = d\ln\left(\frac{f_{2}}{f_{1}}\right)\wedge d\ln\left(\frac{f_{1}}{f_{4}}\right),
\end{equation}
\begin{equation}
\phi'_{3} = d\ln\left(\frac{f_{4}}{f_{5}}\right)\wedge d\ln\left(\frac{f_{5}}{f_{1}}\right)
\end{equation}
and
\begin{equation}
\phi'_{4} = d\ln\left(\frac{f_{3}}{f_{4}}\right)\wedge d\ln\left(\frac{f_{4}}{f_{5}}\right).
\end{equation}
Now, we have to compute $\langle{\varphi_{\A^{(2)}_{1}},\phi'_{i}\rangle}$ and the matrix $\langle{\phi_i',\phi'_j\rangle}$. We have,
\begin{equation}\label{3.46}
\begin{split}
&\langle{\varphi_{\A^{(2)}_{1}},\phi'_{1}\rangle} = \frac{1}{\gamma_{2}\gamma_{3}} + \frac{1}{\gamma_{3}\gamma_{4}}\\
&\langle{\varphi_{\A^{(2)}_{1}},\phi'_{3}\rangle} = \frac{1}{\gamma_4\gamma_5}+\frac{1}{\gamma_5\gamma_1}
\end{split}\qquad
\begin{split}
&\langle{\varphi_{\A^{(2)}_{1}},\phi'_{2}\rangle} = \frac{1}{\gamma_{1}\gamma_{2}}\\
&\langle{\varphi_{\A^{(2)}_{1}},\phi'_{4}\rangle} = \frac{1}{\gamma_{3}\gamma_{4}}+\frac{1}{\gamma_{4}\gamma_{5}},
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
&\langle{\phi'_{1},\phi'_{1}\rangle} = \frac{1}{\g_{2}\g_{3}} + \frac{1}{\g_{3}\g_{4}} + \frac{1}{\g_{2}\g_{4}}\\
&\langle{\phi'_{1},\phi'_{2}\rangle} = \frac{1}{\g_{2}\g_{4}}\\
&\langle{\phi'_{1},\phi'_{3}\rangle} = 0\\
&\langle{\phi'_{1},\phi'_{4}\rangle} = \frac{1}{\g_{3}\g_{4}}\\
&\langle{\phi'_{2},\phi'_{2}\rangle} = \frac{1}{\g_{1}\g_{2}} + \frac{1}{\g_{1}\g_{4}} + \frac{1}{\g_{2}\g_{4}}\\
\end{split}\qquad
\begin{split}
&\langle{\phi'_{2},\phi'_{3}\rangle} = \frac{1}{\g_{1}\g_{4}} \\
&\langle{\phi'_{2},\phi'_{4}\rangle} = 0\\
&\langle{\phi'_{3},\phi'_{3}\rangle} = \frac{1}{\g_{4}\g_{5}} + \frac{1}{\g_{5}\g_{1}} + \frac{1}{\g_{1}\g_{4}}\\
&\langle{\phi'_{3},\phi'_{4}\rangle} = \frac{1}{\g_{4}\g_{5}}\\
&\langle{\phi'_{4},\phi'_{4}\rangle} = \frac{1}{\g_{3}\g_{4}} + \frac{1}{\g_{4}\g_{5}} + \frac{1}{\g_{3}\g_{5}}.\\
\end{split}
\end{equation}
Using {\scshape Mathematica}, we have verified that the KLT relation
\begin{equation}
\langle{\varphi_{\A^{(2)}_{1}},\varphi_{\A^{(2)}_{1}}\rangle} = \langle{\varphi_{\A^{(2)}_{1}},\phi'_{i}\rangle}\langle{\phi'_{j},\phi'_{i}\rangle}^{-1}\langle{\phi'_{j},\varphi_{\A^{(2)}_{1}}\rangle}
\end{equation}
holds.
\paragraph{Example 3.7}
Finally, in this example, we consider a case of the mixed KLT relation for the pentagon, by inserting two different bases in the intersection number \ref{3.41}. We use the bases $\lbrace{\phi_{i}\rbrace}$ and $\lbrace{\phi'_{i}\rbrace}$ and the resolution,
\begin{equation}\label{eq3.49}
\langle{\varphi_{\A^{(2)}_{1}},\phi'_{j}\rangle}\langle{\phi'_{i},\phi_{j}\rangle}^{-1}\langle{\phi_{i},\varphi_{\A^{(2)}_{1}}\rangle}.
\end{equation}
Proving the KLT relation in this case amounts to proving that this quantity is equal to \ref{3.41}. To do this, we have the quantities $\langle{\phi_{i},\varphi_{\A^{(2)}_{1}}\rangle}$ and $\langle{\varphi_{\A^{(2)}_{1}},\phi'_{j}\rangle}$ already computed. We are now left with the task of finding the elements of the matrix $\langle{\phi_{i},\phi'_{j}\rangle}$. We have
\begin{equation}
\begin{split}
&\langle{\phi_{1},\phi'_{1}\rangle} = \frac{1}{\g_{2}\g_{3}}+\frac{1}{\g_{3}\g_{4}} \\
&\langle{\phi_{1},\phi'_{2}\rangle} = -\frac{1}{\g_{1}\g_{2}}\\
&\langle{\phi_{1},\phi'_{3}\rangle} = \frac{1}{\g_{4}\g_{5}}+ \frac{1}{\g_{5}\g_{1}}\\
&\langle{\phi_{1},\phi'_{4}\rangle} = \frac{1}{\g_{3}\g_{4}}+\frac{1}{\g_{4}\g_{5}}\\
&\langle{\phi_{2},\phi'_{1}\rangle} = \frac{1}{\g_{2}\g_{3}} + \frac{1}{\g_{2}\g_{4}}\\
&\langle{\phi_{2},\phi'_{2}\rangle} = \frac{1}{\g_{2}\g_{4}} \\
&\langle{\phi_{2},\phi'_{3}\rangle} = 0\\
&\langle{\phi_{2},\phi'_{4}\rangle} = \frac{1}{\g_{3}\g_{d}}\\
\end{split}\qquad
\begin{split}
&\langle{\phi_{3},\phi'_{1}\rangle} = \frac{1}{\g_{3}\g_{4}}\\
&\langle{\phi_{3},\phi'_{2}\rangle} = 0\\
&\langle{\phi_{3},\phi'_{3}\rangle} = \frac{1}{\g_{4}\g_{5}}\\
&\langle{\phi_{3},\phi'_{4}\rangle} = \frac{1}{\g_{3}\g_{4}}+\frac{1}{\g_{4}\g_{5}}+\frac{1}{\g_{5}\g_{3}}\\
&\langle{\phi_{4},\phi'_{1}\rangle} = 0\\
&\langle{\phi_{4},\phi'_{2}\rangle} = \frac{1}{\g_{1}\g_{4}}\\
&\langle{\phi_{4},\phi'_{3}\rangle} = \frac{1}{\g_{4}\g_{5}}+\frac{1}{\g_{5}\g_{1}}+\frac{1}{\g_{1}\g_{4}}\\
&\langle{\phi_{4},\phi'_{4}\rangle} = \frac{1}{\g_{4}\g_{5}}.\\
\end{split}
\end{equation}
Using this, \ref{eq3.39} and \ref{3.46}, we have verified using {\scshape Mathematica} that the KLT relation \ref{eq3.49} holds.
As a final note, we point out the challenge involved in extending this framework to higher points. Take the case of a $10$-point amplitude in $\phi^4$. Neglecting the trivial cases of the box and product polytope, we saw that the dimension of the cohomology class of the associahedron type polytope was $24$ dimensional while the cohomology class of the Lucas type polytope was $12$ dimensional. The three dimensional nature of these polytopes makes a direct counting and enumeration of the bounded chambers more complicated. One possible approach to writing out cohomology classes would perhaps be to find the intersection lattice of the arrangement and use that data to extract information about chambers, which could serve as an interesting direction of future research.
\section{Accordiohedra and Scattering Equations}\label{sec:scattering}
In this section, we will present a generalization of the scattering equation formalism relevant to the study of the amplitudes defined by accordiohedra. Much of the technical material of this section has already appeared in previous works by the author (see \cite{Kalyanapuram:2019nnf} for applications to Stokes polytopes and \cite{Kalyanapuram:2020vil,Kalyanapuram:2020tsr} for elementary applications to accordiohedra). Accordingly, the main goal of this section will be to provide a number of examples which have not yet been considered or only received treatment in passing, providing a coherent synthesis of existing techniques and setting the context for later sections.
\subsection{Accordiohedra as Hyperplane Arrangements}
Accordiohedra are polytopes whose vertices are defined in one to one correspondence with given dissections of an $n$-gon. Such dissections can be mapped directly into Feynman diagrams of a corresponding quantum field theory. The canonical example considered in the literature is the case of a hexagon dissected into two triangulations and one quadrangulation. In the language of quantum field theory, such dissections are exactly dual to tree level Feynman diagrams proportional to $\lambda^2_3\lambda_4$ in a theory with interactions $\lambda_3\phi^3 + \lambda_4\phi^4$. In figure \ref{fig2.1}, this has been illustrated for the dissection $(13,36)$ (vertices are labelled starting at the top left clockwise).
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{ACC_13_36.pdf}
\caption{Dissection $(13,36)$ of a hexagon and the dual Feynman diagram}\label{fig2.1}
\end{figure}
Given a collection of dissections of a given type, accordiohedra are defined for each dissection separately, with the vertices being labelled by those dissections satisfying a compatibility condition with the reference dissection. For a review of the details involved and a number of examples, the reader may consult \cite{Jagadale:2019byr}. This procedure usually results in several different types of accordiohedra for a given class of scattering amplitudes, which must be appropriately summed over in order to produce the correct scattering amplitude.
One of the more mundane reasons why one would expect this to be the case can be realized by a simple counting experiment. Consider the $6$-particle scattering amplitude in $\phi^4$ theory in the planar limit. Three possible scattering channels contribute to this process, namely $X_{14}$, $X_{25}$ and $X_{36}$\footnote{The notation used here is $X_{ij} = (p_{i}+...+p_{j-1})^2$.}. Since the process involves a single propagator, the positive geometry is expected to be one dimensional. There exists no one dimensional polytope with three vertices. Accordingly, it is more likely that specific subsets of the diagrams can be arranged into positive geometries, the sum over which would be expected to yield the amplitude. The three scattering channels each contribute one Stokes polytope. Each of these is a line. These are illustrated in the figure below.
\begin{figure}[h]
\centering
\includegraphics[width=1.1\textwidth]{ACC_1D-PHI4.pdf}
\caption{Accordiohedra for $6$-particle scattering in $\phi^4$. Reference dissections are on the left.}\label{fig2.2}
\end{figure}
While the foregoing case furnishes the simplest of examples, accordiohedra can be computed for arbitrarily complicated scattering processes. More to the point, since they are convex polytopes (see \cite{Thibault:2017nnf} for more details), they admit realizations as hyperplane arrangements. Correspondingly, twisted intersection theory may naturally be applied to their study. In order to see how this works out for the $6$-particle case, we note that each of the accordiohedra can be realized as a line in $\mathbb{CP}^{1}$ by removing two distinct points $p_1$ and $p_2$. Indeed, this is a hyperplane arrangement in one complex dimension.
Given a hyperplane arrangement $\lbrace{f_1,...,f_{N}\rbrace}$ in $\mathbb{CP}^n$ realizing an accordiohedron, kinematical data is introduced into the discussion by defining a twist,
\begin{equation}
\omega = X_{f_{1}}d\ln f_{1}+\dots+X_{f_{N}}d\ln f_{N}.
\end{equation}
The coefficients $X_{f_{i}}$ are the planar $X_{ij}$ variables associated to the facet described by the hyperplanes $f_{i}$.We note parenthetically that the twisted differential $\nabla_{\omega} = d + \omega\wedge$ gives rise to twisted cohomology classes, the homology duals of which can be recovered by the Poincare duality. We will have occasion to use this fact soon.
The twist gives rise to a natural generalization of the scattering equations. Given the foregoing twist, we have,
\begin{equation}
\omega = \omega_{x_1}dx_{1}+...+\omega_{x_n}dx_{n}
\end{equation}
where $dx_{i}$ are the the inhomogeneous coordinates on $\mathbb{CP}^{n}$. The twist equations $\omega_{x_i}$ furnish the analogue of the scattering equations for accordiohedra. To make contact with the original idea of the scattering equations, we note that the scattering equations are essentially the twist equations of associahedra, a fact proven in \cite{Mizera:2017cqs,Mizera:2017rqa}.
With these basic points noted, our purpose in the rest of this section will be to write down the scattering equations for a number of accordiohedra and compute the number of solutions for each system. The data provided in this section will then be used to generalize the KLT relations to accordiohedra in the next section.
\subsection{One Dimensional Polytopes}
\paragraph{$6$ Particles in $\phi^4$ Theory}
For the case of $6$-particle scattering in $\phi^4$ theory, there is a single reference dissection, namely $(14)$, meaning all other dissections can be obtained by cyclic permutations thereof. Accordingly, we have a single accordiohedron to think of - the line with vertices labelled by $(14)$ and $(36)$. This can be realized as the arrangement $\mathcal{A}_{(14)} = \mathbb{CP}^{1}-\lbrace{0,1\rbrace}$ with the twist,
\begin{equation}
\omega_{14} = \left(\frac{X_{14}}{x}-\frac{X_{36}}{1-x}\right)dx.
\end{equation}
The scattering equation,
\begin{equation}
\frac{X_{14}}{x}-\frac{X_{36}}{1-x} = 0
\end{equation}
has a single solution. This may be seen from the fact that this arrangement has one bounded chamber, namely the line between $0$ and $1$. We remark here that there is a hyperplane at $x=\infty$ as well, with residue $-X_{14}-X_{36}$. This point will come up when we discuss the KLT relations in later sections. Finally, we note,
\begin{equation}
\mathrm{dim}H^{1}_{\omega_{(14)}}(\mathcal{A}_{14}) = 1.
\end{equation}
Indeed, we have the same result for the dissections $(25)$ and $(36)$.
\paragraph{$5$ Particles in $\phi^3+\phi^4$ Theory}
We now consider $5$-particle scattering in $\phi^3+\phi^4$ theory with amplitudes having one cubic and one quartic vertex. At the planar level, there are five diagrams, as shown below
\begin{figure}[H]
\centering
\includegraphics[width=1\textwidth]{ACC-5-PHI34.pdf}
\caption{Planar Feynman Diagrams for $5$ Particles in $\phi^3 + \phi^4$ Theory}\label{fig2.3}
\end{figure}
In the figure, the diagram to the left is $(13)$, with the rest obtained by cyclic rotations. As a result, we have here only one accordiohedron, corresponding to the reference $(13)$, from which the others may be easily inferred. This turns out to be the line, with the compatible dissections being $(13)$ and $(26)$, which can be realized again by $\mathcal{A}_{(13)} = \mathbb{CP}^{1}-\lbrace{0,1\rbrace}$ with the twist,
\begin{equation}
\omega_{(13)} = \left(\frac{X_{13}}{x}-\frac{X_{26}}{1-x}\right)dx.
\end{equation}
Once again, we see that the scattering equation has a single solution. Thus we have,
\begin{equation}
\mathrm{dim}H^{1}_{\omega_{(13)}}(\mathcal{A}_{13}) = 1.
\end{equation}
It goes without saying that this result holds for the remaining dissections as well.
\subsection{Two Dimensional Polytopes}
\paragraph{$8$ Particles in $\phi^4$ Theory}
We turn now to our first nontrivial example. In $\phi^4$, for $8$-particle scattering, we need to consider quadrangulated octagons. There are two inequivalent classes of such quadrangulations, given in the diagrams below.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{ACC-8-PHI4.pdf}
\caption{Inequivalent Diagrams for $8$-Particle Scattering in $\phi^4$}\label{fig2.4}
\end{figure}
The diagram on the left is dissection $(14,58)$ and the diagram on the right is $(14,47)$. It can be shown that all other planar diagrams for this process can be obtained by cyclic permutations of one of these two diagrams. Now, we need the accordiohedra corresponding to these two different dissections. Let us start with $(14,58)$. The accordiohedron for this reference dissection is a square, as there are four quadrangulations of the $8$-gon which are compatible with this dissection. These are $(14,58)$, $(14,47)$, $(38,58)$ and $(38,47)$.
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{ACC-8-14_58.pdf}
\caption{Accordiohedron for Dissection $(14,58)$}\label{fig2.5}
\end{figure}
In figure \ref{fig2.5}, the accordiohedron has been represented in terms of the compatible dissections. The upper left vertex in the diagram is the dissection $(14,58)$. As a convex polytope, this can be described by the following collection of hyperplanes,
\begin{equation}\label{eq2.8}
\begin{aligned}
f_{1} &= 1-y\\
f_{2} &= 1-x\\
f_{3} &= 1+y\\
f_{4} &= 1+x
\end{aligned}
\end{equation}
in $\mathbb{CP}^{2}$. Now we come across the first nontrivial case in which the twist has to be properly defined. The facets have to be brought in correspondence with the correct factorization channels. From diagram \ref{fig2.5}, we can identify facets $f_{1}$ through $f_{4}$ with channels $X_{14}$, $X_{47}$, $X_{38}$ and $X_{58}$ respectively, with the twist defined accordingly,
\begin{equation}
\omega_{(14,58)} = X_{14}d\ln f_{1} + X_{47}d\ln f_{2} + X_{38}d\ln f_{3} + X_{58}d\ln f_{4}.
\end{equation}
The scattering equations can now be read off from the twist. We obtain
\begin{equation}
\frac{X_{14}}{1-y}-\frac{X_{38}}{1+y} = 0
\end{equation}
and
\begin{equation}
\frac{X_{47}}{1-x}-\frac{X_{58}}{1+x} = 0.
\end{equation}
From the above equations, we can infer that again, we have a single solution to the scattering equations. We can record this observation according to the formula
\begin{equation}
\mathrm{dim}H^{2}_{\omega_{14,58}}(\mathcal{A}_{(14,58)}) = 1.
\end{equation}
This is not especially surprising; a square is after all the product of two lines, each of which has one solution. We remark parenthetically that this phenomenon more generally can be stated as follows. Given a $d = d_1+d_2$ dimensional accordiohedron $\mathcal{A}_{D}$ factorizing into two accordiohedra $\mathcal{A}_{D_1}$ and $\mathcal{A}_{D_2}$ of dimensions $d_1$ and $d_2$ respectively, we have the formula,
\begin{equation}
\mathrm{dim}H^{d}_{\omega_{D}}(\mathcal{A}_{D}) = \mathrm{dim}H^{d_1}_{\omega_{D_1}}(\mathcal{A}_{D_1})\times \mathrm{dim}H^{d_2}_{\omega_{D_2}}(\mathcal{A}_{D_2})
\end{equation}
which is simply a statement of the K\"unneth theorem combined with the fact that twisted cohomology is supported on the middle dimension.
Now we deal with the second case - that of the dissection $(14,47)$. Here, the accordiohedron is a pentagon, with the vertices labelled by the quadrangulations $(14,47)$, $(38,47)$, $(36,38)$, $(16,36)$ and $(14,16)$. This is illustrated in figure \ref{fig2.6}.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{ACC-8-14_47.pdf}
\caption{Accordiohedron for Dissection $(14,47)$}\label{fig2.6}
\end{figure}
The hyperplane arrangement furnishing this polytope in $\mathbb{CP}^{2}$ is,
\begin{equation}\label{eq2.14}
\begin{aligned}
f_{1} &= 1-x\\
f_{2} &= 1+y\\
f_{3} &= 2+x\\
f_{4} &= 2+x-y\\
f_{5} &= 2-y.
\end{aligned}
\end{equation}
The twist capturing the correct kinematics is
\begin{equation}
\omega_{(14,47)} = X_{36}d\ln f_{1} + X_{16}d\ln f_{2} + X_{14}d\ln f_{3} + X_{47}d\ln f_{4} + X_{38}d\ln f_{5},
\end{equation}
which can be verified by inspection of figure \ref{fig2.6}. We now encounter the first genuinely nontrivial scattering equations. They are as follows,
\begin{equation}
\frac{X_{36}}{1-x} - \frac{X_{14}}{2+x} - \frac{X_{47}}{2+x-y} = 0
\end{equation}
and
\begin{equation}
\frac{X_{36}}{1+y} - \frac{X_{47}}{2+x-y} - \frac{X_{38}}{2-y} = 0.
\end{equation}
Our task now is to determine the number of solutions for these equations. As it turns out, the problem is actually computationally tedious. The \texttt{Solve} routine in {\scshape Mathematica} may be employed, and this determines the existence of four solutions for generic kinematics. A simpler method to do this however is to count the number of bounded chambers of the arrangement. This can be done far more readily, and the answer agrees with the former method. Thus, we note
\begin{equation}
\mathrm{dim} H^{2}_{\omega_{(14,47)}}(\mathcal{A}_{(14,47)}) = 4.
\end{equation}
We note here that the \texttt{euler} routine in \texttt{Macaulay2} may be used to similar effect. This will prove useful when we consider three dimensional cases, where both of the previous methods - the counting of bounded chambers and brute force solving prove to be prohibitive.
Finally, before moving on to the next example, we point out that there are a total of $12$ quadrangulations of the $8$-gon. $4$ of these give rise to square type accordiohedra and $8$ give rise to pentagons.
\paragraph{$6$ Particles in $\phi^3$ + $\phi^4$}
The case of $6$-particle scattering in $\phi^3+\phi^4$ gives rise to a two dimensional accordiohedron when the process involves two trivalent and one quartic vertex. In this case, which was first studied in \cite{Jagadale:2019byr}, it was found that there were four primitive dissections, from which all of the others could be obtained by permutation. These are $(13,14)$, $(13,15)$, $(13,36)$ and $(13,46)$.
The first three of these give rise to pentagons and the last one gives rise to a square. Since the combinatorics are essentially identical to that of the previous section, we simply list the accordiohedra in the images below.
\begin{figure}[H]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{ACC-6-13_14.pdf}
\captionof{figure}{Accordiohedron for Dissection $(13,14)$}
\label{fig2.7}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{ACC-6-13_15.pdf}
\captionof{figure}{Accordiohedron for Dissection $(13,15)$}
\label{fig2.8}
\end{minipage}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{ACC-6-13_36.pdf}
\captionof{figure}{Accordiohedron for Dissection $(13,36)$}
\label{fig2.9}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{ACC-6-13_46.pdf}
\captionof{figure}{Accordiohedron for Dissection $(13,46)$}
\label{fig2.10}
\end{minipage}
\end{figure}
It should be clear from this that the hyperplane arrangements effecting these polytopes are identical to the arrangements considered in the previous section. For the polytopes in figures \ref{fig2.7}, \ref{fig2.8} and \ref{fig2.9}, we can use the arrangement in \ref{eq2.14} and for the polytope in figure \ref{fig2.10}, we use arrangement \ref{eq2.8}. For the sake of completeness, let us list all the scattering equations involved. For figure \ref{fig2.7} we have,
\begin{equation}
\begin{aligned}
\omega_{(13,14),x}&:= \frac{X_{26}}{1-x} - \frac{X_{13}}{2+x} - \frac{X_{14}}{2+x-y} = 0\\
\omega_{(13,14),y}&:=\frac{X_{35}}{1+y} - \frac{X_{14}}{2+x-y} - \frac{X_{24}}{2-y} = 0.
\end{aligned}
\end{equation}
We have similarly for figure \ref{fig2.8},
\begin{equation}
\begin{aligned}
\omega_{(13,15),x}&:= \frac{X_{26}}{1-x} - \frac{X_{13}}{2+x} - \frac{X_{15}}{2+x-y} = 0\\
\omega_{(13,15),y}&:=\frac{X_{46}}{1+y} - \frac{X_{15}}{2+x-y} - \frac{X_{24}}{2-y} = 0,
\end{aligned}
\end{equation}
and for figure \ref{fig2.9}
\begin{equation}
\begin{aligned}
\omega_{(13,36),x}&:= \frac{X_{24}}{1-x} - \frac{X_{13}}{2+x} - \frac{X_{36}}{2+x-y} = 0\\
\omega_{(13,36),y}&:=\frac{X_{15}}{1+y} - \frac{X_{36}}{2+x-y} - \frac{X_{26}}{2-y} = 0.
\end{aligned}
\end{equation}
Finally for figure \ref{fig2.10} we have,
\begin{equation}
\begin{aligned}
\omega_{(13,46),x}&:= \frac{X_{26}}{1-x} - \frac{X_{13}}{2+x}=0\\
\omega_{(13,46),y}&:=\frac{X_{35}}{1+y} - \frac{X_{46}}{2-y} = 0.
\end{aligned}
\end{equation}
From these we infer the dimensions
\begin{equation}
\mathrm{dim}H_{\omega_{(13,14)}}^{2}(\mathcal{A}_{(13,14)}) = \mathrm{dim}H_{\omega_{(13,15)}}^{2}(\mathcal{A}_{(13,15)}) = \mathrm{dim}H_{\omega_{(13,36)}}^{2}(\mathcal{A}_{(13,36)}) = 4
\end{equation}
and
\begin{equation}
\mathrm{dim}H_{\omega_{(13,46)}}^{2}(\mathcal{A}_{(13,46)}) = 1.
\end{equation}
\subsection{Three Dimensional Polytopes}
\paragraph{$10$ Particles in $\phi^4$}
The last case we consider is the scattering of $10$-particles with quartic interactions. This case is far more nontrivial than those considered till this point. For the first time, we encounter three-dimensional accordiohedra, and the counting of the number of solutions of the scattering equations becomes more involved. This case was originally considered in \cite{Banerjee:2018tun} when Stokes polytopes were applied for the first time to scattering amplitudes. In this section, we review the combinatorial details and provide precise convex realizations of these polytopes and the relevant scattering equations. In this section, we will omit the onerous details involving the vertices of the accordiohedra and only present data about the facets.
For this scattering process, there are four classes of accordiohedra that appear. The simplest, known as cube type, arises out of the quadrangulation $(14,510,69)$, which has been illustrated in figure \ref{fig2.11}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{ACC-10-14_510_69.pdf}
\caption{Dissection $(14,510,69)$}\label{fig2.11}
\end{figure}
The accordiohedron for this dissection has six facets, labelled by partial quadrangulations $(14)$, $(510)$, $(69)$, $(310)$, $(49)$ and $(58)$, realized as a convex polytopes by the following hyperplanes $f_{1}$ through $f_{6}$ in $\mathbb{CP}^{3}$
\begin{equation}
\begin{aligned}
&f_{1} = x - 3 \hspace{1cm} &f_{2} = y - 3 \\
&f_{3} = z - 3 \hspace{1cm} &f_{4} = x + 2 \\
&f_{5} = y + 2 \hspace{1cm} &f_{6} = z +2.
\end{aligned}
\end{equation}
This arrangement is presented in figure \ref{fig2.12}
\begin{figure}[H]
\centering
\includegraphics[width=0.29\textwidth]{Stokes_14_510_69.pdf}
\caption{Accordiohedron for Dissection $(14,510,69)$}\label{fig2.12}
\end{figure}
The scattering equations are now just three copies of the scattering equation for a line,
\begin{equation}
\omega_{(14,510,69),x} := \frac{X_{14}}{x-3} - \frac{X_{310}}{x+2} = 0,
\end{equation}
\begin{equation}
\omega_{(14,510,69),y} := \frac{X_{510}}{y-3} - \frac{X_{49}}{y+2} = 0
\end{equation}
and
\begin{equation}
\omega_{(14,510,69),z} := \frac{X_{510}}{z-3} - \frac{X_{58}}{z+2} = 0.
\end{equation}
The K\"unneth formula may be readily applied to see (trivially so) that,
\begin{equation}
\mathrm{dim}H^{3}_{\omega_{(14,510,69)}}(\mathcal{A}_{(14,510,69)}) = 1
\end{equation}
corresponding to one solution of the scattering equations.
The second type of accordiohedron is a three dimensional associahedron, arising from three types of primitive quadrangulations, namely $(14,16,18)$, $(14,16,69)$ and $(14,49,69)$. The latter two are entirely analogous to the first case. Accordingly, we focus on the former in this discussion. Diagramatically we have the quadrangulation shown in \ref{fig2.13}.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{ACC-10-14_16_18.pdf}
\caption{Dissection $(14,16,18)$}\label{fig2.13}
\end{figure}
The facets of the accordiohedron generated by this dissection are $9$ in number, and are labelled by the partial dissections $\lbrace{14,16,18,36,58,710,38,510,310\rbrace}$. The corresponding hyperplane arrangement $\lbrace{f_{1},...,f_{9}\rbrace}$ is as follows,
\begin{equation}
\begin{split}
&f_{1} = x - 4 \\
&f_{3} = z-6\\
&f_{5} = z-y-6\\
&f_{7} = z-x-7 \\
&f_{9} = x+6\\
\end{split}\qquad
\begin{split}
&f_{2} = y - 6 \\
&f_{4} = y - x - 6 \\
&f_{6} = z+3 \\
&f_{8} = y+6 \\
\end{split}
\end{equation}
which yields figure \ref{fig2.14}.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{Stokes_14_16_18.pdf}
\caption{Accordiohedron for Dissection $(14,16,18)$}\label{fig2.14}
\end{figure}
The scattering equations for this arrangement are given as follows
\begin{equation}
\begin{aligned}
\omega_{(14,16,18),x} := \frac{X_{14}}{x-4} -\frac{X_{310}}{x+6} -\frac{X_{36}}{x+6-y} -\frac{X_{38}}{x-z+7} = 0,
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\omega_{(14,16,18),y} := \frac{X_{16}}{y-5}- \frac{X_{510}}{y+6} +\frac{X_{36}}{y-6+x} -\frac{X_{58}}{y-z+6} = 0
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\omega_{(14,16,18),z} := \frac{X_{18}}{z-6} -\frac{X_{710}}{z+3} + \frac{X_{58}}{z-y-6} + \frac{X_{38}}{z-x-7} = 0.
\end{aligned}
\end{equation}
In order to find the number of solutions of this system of equations, two approaches may be employed. One possible method is direct computation on \texttt{Solve} in {\scshape Mathematica}. This method however was prohibitive due to the CPU time required. A simpler approach involves using the \texttt{euler} routine in the \texttt{HyperplaneArrangements} package in \texttt{Macaulay2}. The Euler characteristic of this hyperplane arrangement computed in this manner is $24$. We thus have,
\begin{equation}
\mathrm{dim}H^{3}_{\omega_{(14,16,18)}}\left(\mathcal{A}_{(14,16,18)}\right) = 24.
\end{equation}
This implies that there are $24$ bounded chambers for this hyperplane arrangement.
We consider next the case of the dissection $(14,47,710)$, which gives rise to an accordiohedron of so-called Lucas type, which is a genuine accordiohedron in the sense that it cannot be understood as an associahedron.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{ACC-10-14_47_710.pdf}
\caption{Dissection $(14,47,710)$}\label{fig2.15}
\end{figure}
This polytope, obtained by finding all dissections of a $10$-gon with three quadrangulations compatible with $(14,47,710)$ has $12$ vertices, with $8$ facets, labelled by $ \lbrace{14,47,710,16,49,310,36,69\rbrace}$. The corresponding hyperplane arrangement in $\mathbb{CP}^{3}$ is
\begin{equation}
\begin{split}
&f_{1} = x - 5 \\
&f_{3} = y-4\\
&f_{5} = y-z-5\\
&f_{7} = y+6 \\
\end{split}\qquad
\begin{split}
&f_{2} = y - 5 \\
&f_{4} = x-y-6 \\
&f_{6} = x+4 \\
&f_{8} = z+4. \\
\end{split}
\end{equation}
Visually this is represented in figure \ref{fig2.16}.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{Stokes_14_47_710.pdf}
\caption{Accordiohedron for Dissection $(14,47,710)$}\label{fig2.16}
\end{figure}
The scattering equations can be inferred from the twist, which is determined by the kinematic assignements. They are
\begin{equation}
\omega_{(14,47,710),x}:= \frac{X_{14}}{x-5} -\frac{X_{16}}{x-y-6} -\frac{X_{310}}{x+4} = 0,
\end{equation}
\begin{equation}
\omega_{(14,47,710),y}:= \frac{X_{47}}{y-5} +\frac{X_{16}}{y-x+6} -\frac{X_{36}}{x+4} +\frac{X_{49}}{y-z-5} = 0,
\end{equation}
and
\begin{equation}
\omega_{(14,47,710),z}:= \frac{X_{710}}{z-4} -\frac{X_{49}}{z-y+5} -\frac{X_{69}}{z+4} = 0.
\end{equation}
The number of solutions to this system was determined in \texttt{Macaulay2} to be $12$. Hence we have,
\begin{equation}
\mathrm{dim}H^{3}_{\omega_{(14,47,710)}}\left(\mathcal{A}_{(14,47,710)}\right) = 12
\end{equation}
following which we may state that there are $12$ bounded chambers in this arrangement.
The final case we have to consider is known as the mixed accordiohedron, which is generated by the quadrangulations $(14,510,710)$ and $(14,16,17)$. Again, we focus on the former case for convenience.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{ACC-10-14_510_710.pdf}
\caption{Dissection $(14,510,710)$}\label{fig2.17}
\end{figure}
The accordiohedron in this case actually ends up being a product polytope, formed out of a pentagon times a line. Accordingly, we have $7$ facets, labelled by the dissections $\lbrace{14,510,710,310,57,69,49\rbrace}$. This convex polytope is afforded by the arrangement
\begin{equation}
\begin{split}
&f_{1} = x - 4 \\
&f_{3} = z-2\\
&f_{5} = z-y-4\\
&f_{7} = y+4 \\
\end{split}\qquad
\begin{split}
&f_{2} = y - 3 \\
&f_{4} = x+3 \\
&f_{6} = z+4 \\
\end{split}
\end{equation}
which produces the polytope shown in figure \ref{fig2.18}.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{Stokes_14_510_710.pdf}
\caption{Accordiohedron from Dissection $(14,510,710)$}\label{fig2.18}
\end{figure}
One can observe from the diagram that this is indeed a pentagon times a line. Accordingly, the K\"unneth formula is applied readily to obtain,
\begin{equation}
\mathrm{dim}H^{3}_{\omega_{(14,510,710)}}\left(\mathcal{A}_{(14,510,710)}\right) = 4\times 1 = 4.
\end{equation}
For the sake of completeness, we include the scattering equations,
\begin{equation}
\omega_{(14,510,710),x}:= \frac{X_{14}}{x-4} - \frac{X_{310}}{x+3} = 0,
\end{equation}
\begin{equation}
\omega_{(14,510,710),y}:= \frac{X_{510}}{y-3} - \frac{X_{49}}{y+4} - \frac{X_{57}}{y-z+4} = 0,
\end{equation}
and
\begin{equation}
\omega_{(14,510,710),z}:= \frac{X_{710}}{z-2} - \frac{X_{69}}{z+4} + \frac{X_{57}}{z-y-4} = 0.
\end{equation}
These cases exhaust all the accordiohedra that arise for this scattering process, as all of the others can be obtained by permuting the indices of the primitives considered.
Let us summarize the salient results of this section before moving on. What we have attended to so far is a class of convex realizations for a number of accordiohedra considered in the literature in one, two and three dimensions. Rather than treating them in the sense of the traditional positive geometry program, we have examined these constructions in the general context of twisted intersection theory. While we have not yet evaluated any amplitudes explicitly, we have provided the data needed to do so, namely the hyperplane arrangements supplying the polytopes, as well as the twist differential forms that encode the kinematical structure of the scattering processes. The components of the twist have been interpreted as the analogue of the scattering equations of Cachazo, He and Yuan for the more complicated interactions that we want to study.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,703 |
Lantlôs (middelhøjtysk for "hjemløs" eller "uden hjemland") er et tysk black metal/post-rock-band, dannet i 2005. Frontmanden Herbst valgte i sin tid bandnavnet for at indikere at han ikke følte sig hjemme i Tyskland eller noget andet sted. Medstifter Angrrau forlod bandet efter udgivelsen af deres debutalbum. En overgang var Stéphane "Neige" Paut - kendt fra Alcest og det tidlige Peste Noire - forsanger i bandet, men han forlod Lantlôs i 2013. Samme år blev de to livemusikere Cedric Holler og Felix Wylezik indlemmet som fuldgyldige medlemmer.
Medlemmer
Markus "Herbst" Siegenhor – guitar, bas, keyboard, vokal (tidligere også trommer)
Cedric Holler - guitar, vokal
Felix Wylezik - trommer
Tidligere medlemmer
Angrrau – vokal, bas, piano (2005-2008)
Tom Innocenti – vokal, guitar, keyboard (2008)
Stéphane "Neige" Paut – vokal (2010-2013)
Diskografi
Studiealbum
2008: Lantlôs
2010: .neon
2011: Agape
2014: Melting Sun
Singler
2014: "Melting Sun I: Azure Chimes"
Demoer
2008: Îsern Himel
Fodnoter
Eksterne henvisninger
Officielle hjemmeside
Black metal-grupper
Post-rock-grupper
Metalgrupper fra Tyskland
Musikgrupper fra 2005 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,304 |
yea zhar, we should have "party" only chars to avoid others getting up too high.
i could remake a char for party play. normal and NM should be fine, but if we are going to venture into hell and inferno we need a concrete strategy and build order.
I've started solo'ing hell – smoked skele king alone, they hit *hard* but it's not as bad as you'd think. I imagine inferno is a little different 😀 Hell is basically meant as a place to farm for inferno-ready gear me thinks. Build order changes depending on the engagements, some bosses need much different builds than say, hordes of wee mobs which need ae mowing down! All in all happy with the game so far..
I wouldn't mind starting my monk over again. I geared him wrong, and am still trying to figure out how to do combos correctly. Actually, I could start any of my characters again.
I'm cool making a new character, that'll work. What I heard the other day that is really cool is that different characters on the same account can share gold.
Zhar can't you just move around the runes to suit your needs? Its not like you allocate skill points in D2? how is your monk ruined?
think thats a good protoype inferno party, but it will be some time until we get there!
Oh, he is not ruined… I just geared him up wrong. For some reason I was thinking he was a tank, so started beefing up his strength instead of his dexterity. ::facepalm:: Anyways, he can use the gear that's on my demon hunter. No big deal for such a lowbie character.
Playing for 10 minutes then heading to the pug for some tacos.
EDIT: Make that 5 minutes, pug time!!
You might know this already, but it's an interesting addition of Blizzard.
nice article, kept me entertained at work for a bit yesterday. like anything in life, moderation is key. just make a study schedule and a gold/exp schedule. learning chinese is like a RPG, just set some goals.
I noticed the same thing a few days ago. It's not worth the trouble of messing with traditional characters for me either, although simplified would be fun (and difficult). I played DOTA in Chinese, really difficult to understand a lot of what's going on since I don't really play that game in English either.
One thing I also discovered the other day is that you can play D3 on Asian and US servers simultaneously. My roommate was wanting to play D3 and had it installed but didn't have an account to login with. I let him use my account and now he's farther than I am. I tried to install it on my desktop yesterday but the installer got stuck at 1%. Not sure if I have to download the entire 7gb package again or what.
I finally got D3 installed last night on my desktop. I have a level 13 witch doctor. Holla at me in game.
Nice pun. My witch doctor is named Dhalsim.
my 45 WD is "FelaQT" Unfortunately WD kind of gest rocked in later NM and HELL unless you do one specific build which takes the spontaneity out of it.
until then, have fun experimenting with zombies, frogs, spiders, poison and voodoo!
I get facemelted in hell/hell without a partner as a wiz, unless yer madly geared up we're a squishy class, need to build specifically for survival – and later on it's hard to keep distance from a lot of mob types….. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,664 |
{"url":"https:\/\/worddisk.com\/wiki\/Outline_of_logic\/","text":"# Outline of logic\n\nLogic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments.\n\n## Foundations of logic\n\nPhilosophy of logic\n\n## Philosophical logic\n\n#### Fallacies\n\n\u2022 Fallacy \u00a0(list) incorrect argumentation in reasoning resulting in a misconception or presumption. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor (appeal to emotion), or take advantage of social relationships between people (e.g. argument from authority). Fallacious arguments are often structured using rhetorical patterns that obscure any logical argument. Fallacies can be used to win arguments regardless of the merits. There are dozens of types of fallacies.\n\n## Formal logic\n\n#### Symbols and strings of symbols\n\n##### Logical symbols\n###### Logical connectives\n\nLogical connective\n\nProposition\n\nObject language\n\nMetalanguage\n\n#### Propositional and boolean logic\n\n##### Propositional logic\n\nPropositional logic\n\n#### Predicate logic and relations\n\nPredicate logic\n\n##### Relations\n\nMathematical relation\n\n## Mathematical logic\n\nMathematical logic\n\n#### Metalogic\n\nMetalogic The study of the metatheory of logic.\n\n##### Proof theory\n\nProof theory The study of deductive apparatus.\n\n##### Model theory\n\nModel theory The study of interpretation of formal systems.\n\n#### Computability theory\n\nComputability theory branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. The basic questions addressed by recursion theory are \"What does it mean for a function from the natural numbers to themselves to be computable?\" and \"How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?\". The answers to these questions have led to a rich theory that is still being actively researched.\n\n## Semantics of natural language\n\nFormal semantics (natural language)\n\n\u2022 Formal systems\n\u2022 Concepts\n\nClassical logic\n\nModal logic\n\n## Non-classical logic\n\nNon-classical logic\n\n## Concepts of logic\n\nMathematical logic\n\nHistory of logic","date":"2021-07-31 09:43:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 13, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5442944169044495, \"perplexity\": 2873.209770820178}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046154085.58\/warc\/CC-MAIN-20210731074335-20210731104335-00178.warc.gz\"}"} | null | null |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace Telerik.UI.Xaml.Controls.Primitives.SideDrawer.Commands
{
/// <summary>
/// Encapsulates the command-related routine within a <see cref="RadSideDrawer"/> instance.
/// </summary>
public class CommandService : CommandServiceBase<RadSideDrawer>
{
internal CommandService(RadSideDrawer owner)
: base(owner)
{
this.InitKnownCommands();
}
internal CommandCollection<RadSideDrawer> UserCommands
{
get
{
return this.userCommands;
}
}
/// <summary>
/// Attempts to find the command, associated with the specified Id and to perform its Execute routine, using the provided parameter.
/// </summary>
/// <param name="id">The <see cref="CommandId"/> value to look for.</param>
/// <param name="parameter">The parameter that is passed to the CanExecute and Execute methods of the command.</param>
/// <returns>True if the command is successfully executed, false otherwise.</returns>
public bool ExecuteCommand(CommandId id, object parameter)
{
return this.ExecuteCommandCore((int)id, parameter, true);
}
/// <summary>
/// Executes the default (built-in) command (without looking for user-defined commands), associated with the specified Id.
/// </summary>
/// <param name="id">The <see cref="CommandId"/> value to look for.</param>
/// <param name="parameter">The parameter that is passed to the CanExecute and Execute methods of the command.</param>
/// <returns>True if the command is successfully executed, false otherwise.</returns>
public bool ExecuteDefaultCommand(CommandId id, object parameter)
{
return this.ExecuteCommandCore((int)id, parameter, false);
}
/// <summary>
/// Determines whether the default command, associated with the specified Id can be executed given the parameter provided.
/// </summary>
/// <param name="id">The <see cref="CommandId"/> value to look for.</param>
/// <param name="parameter">The parameter that is passed to the CanExecute and Execute methods of the command.</param>
/// <returns>True if the command can be executed, false otherwise.</returns>
public bool CanExecuteDefaultCommand(CommandId id, object parameter)
{
var command = this.GetCommandById((int)id, false);
if (command != null)
{
return command.CanExecute(parameter);
}
return false;
}
private void InitKnownCommands()
{
this.defaultCommands[(int)CommandId.DrawerStateChanged] = this.CreateKnownCommand(CommandId.DrawerStateChanged);
this.defaultCommands[(int)CommandId.GenerateAnimations] = this.CreateKnownCommand(CommandId.GenerateAnimations);
}
private SideDrawerCommand CreateKnownCommand(CommandId id)
{
SideDrawerCommand command = null;
switch (id)
{
case CommandId.DrawerStateChanged:
command = new DrawerStateCommand();
break;
case CommandId.GenerateAnimations:
command = new GenerateAnimationsCommand();
break;
default:
throw new ArgumentException("Unknown command id!", "id");
}
command.Id = id;
command.Owner = this.Owner;
return command;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,686 |
(Main)
DC Movies Wiki talk
File talk
MediaWiki talk
Help talk
User blog
User blog comment
Blog talk
Module talk
Message Wall Greeting
Board Thread
Arthur Fleck
Arthur Fleck was an amateur stand-up comedian and a professional clown. He lost his job as a clown after being framed by a co-worker. He was later repeatedly assaulted by various people and
https://dcmovies.fandom.com/wiki/Arthur_Fleck
Welcome to DC Movies Wiki DC Movies Wiki is an online encyclopedia created by fans, for fans, dedicated to documenting all films related to DC comics. Browse released movies • Browse upcoming movies
https://dcmovies.fandom.com/wiki/DC_Movies_Wiki
Damian Wayne (DC Animated Film Universe)
Damian Wayne is the son of Batman and Talia al Ghul who became the fourth Robin. Damian Wayne was born from a drug-induced affair between Bruce Wayne and Talia al Ghul. Talia gave birth
https://dcmovies.fandom.com/wiki/Damian_Wayne_(DC_Animated_Film_Universe)
Catwoman: Hunted is an upcoming animated anime-style film centered around Catwoman. Catwoman: Hunted, picks up with Catwoman as she attempts to steal a priceless jewel. The heist puts her squarely in the crosshairs of
https://dcmovies.fandom.com/wiki/Catwoman:_Hunted
Talon (DC Animated Film Universe)
Talon was the latest in a long line of master assassins employed by the mysterious Court of Owls, the Talon came to Gotham hoping to recruit Damian Wayne to be his successor. There once was
https://dcmovies.fandom.com/wiki/Talon_(DC_Animated_Film_Universe)
Justice League: Mortal
Justice League: Mortal is a film that was shelved and is no longer in production. Was going to be loosely based off the comic book storylines JLA: Tower of Babel (2000) and Superman: Sacrifice (2006
https://dcmovies.fandom.com/wiki/Justice_League:_Mortal
Samantha Vanaver
Samantha Vanaver was a wealthy woman whose family, like the Waynes, had positions of influence in Gotham for hundreds of years. She was secretly the Grandmaster of the Court of Owls and the lover of
https://dcmovies.fandom.com/wiki/Samantha_Vanaver
The Flash (film) rumors
Rumors about The Flash. Casting[] CONFIRMED: Ezra Miller has been cast as Barry Allen/The Flash., Jai Courtney will return as Captain Boomerang., CONFIRMED:Kiersey Clemons, Rita Ora, Lucy Boynton, Kathryn Prescott, and Stefanie Scott
https://dcmovies.fandom.com/wiki/The_Flash_(film)_rumors
The Titans (DC Animated Film Universe)
The Titans (also called the Teen Titans) were a group of powered teenagers led by Starfire. The purpose of this team was for members to help each other control their gifts and help one another
https://dcmovies.fandom.com/wiki/The_Titans_(DC_Animated_Film_Universe)
John Constantine (DC Animated Film Universe)
John Constantine is a purveyor of the dark arts and lifeline of last resort. While he wasn't born with a natural ability to create magic, Constantine has an affinity for it and the supernatural
https://dcmovies.fandom.com/wiki/John_Constantine_(DC_Animated_Film_Universe)
The Flash is an upcoming movie and the film is the twelfth film in the DC Extended Universe about Barry Allen in an alternative reality after traveling back in time to save his mother and
https://dcmovies.fandom.com/wiki/The_Flash_(film)
Richard Grayson (DC Animated Film Universe)
Richard "Dick" Grayson was the first Robin, partner of Batman, from when he was a boy. In his adult years, Grayson took on a new costumed identity Nightwing. Nightwing stopped Damian Wayne from killing the
https://dcmovies.fandom.com/wiki/Richard_Grayson_(DC_Animated_Film_Universe)
Batman vs. Robin is the fourteenth animated Batman and the fifth feature length film in the DC Animated Film Universe that's partially based on the Batman: The Court of Owls storyline which features the
https://dcmovies.fandom.com/wiki/Batman_vs._Robin
The DC Animated Film Universe is the cohesive universe in which a selection of DC Universe Animated Original Movies are a part of following Justice League: War. Timeline[] Alternate Timeline Events of Justice League: The
https://dcmovies.fandom.com/wiki/DC_Animated_Film_Universe
Barbara Gordon (Batman: The Killing Joke)
Barbara Gordon was the superhero known as Batgirl, later after an incident with he Joker she later works as the information broker and hacker known as Oracle. During most meetings between Batman and Commissioner Gordon
https://dcmovies.fandom.com/wiki/Barbara_Gordon_(Batman:_The_Killing_Joke)
Injustice is an animated Justice League movie that's based on the video game of the same name. The story takes place on an alternate Earth in which the Joker has tricked Superman into killing
https://dcmovies.fandom.com/wiki/Injustice
Joker (film)
Joker is a 2019 film that serves as an origin story of The Joker. In 1981, party clown and aspiring stand-up comedian Arthur Fleck lives with his mother, Penny, in Gotham City. Gotham is
https://dcmovies.fandom.com/wiki/Joker_(film)
Teen Titans: The Judas Contract is the second animated film in the Teen Titans series and the ninth feature length film in the DC Animated Film Universe. It is based loosely on the storyline of
https://dcmovies.fandom.com/wiki/Teen_Titans:_The_Judas_Contract
Justice League vs. The Fatal Five is a 2019 animated film set in the DC Animated Universe and the eleventh animated Justice League film. The fate of the earth hangs in the balance when the
https://dcmovies.fandom.com/wiki/Justice_League_vs._The_Fatal_Five
Smallville: Absolute Justice
Smallville: Absolute Justice is a two-hour movie episode for Season Nine of Smallville. The plot revolves around Justice League teaming up with Justice Society of America to catch a villain who's killing members
https://dcmovies.fandom.com/wiki/Smallville:_Absolute_Justice
Batgirl (film)
Batgirl is an upcoming 2022 movie set in the DC Extended Universe. To be added
https://dcmovies.fandom.com/wiki/Batgirl_(film)
Jason Todd (Batman: Under the Red Hood)
Jason Todd was the second Robin. He was killed by the Joker and revived by Ra's al Ghul's Lazarus Pit. After this, he became the Red Hood. Sometime after Dick Grayson left Batman
https://dcmovies.fandom.com/wiki/Jason_Todd_(Batman:_Under_the_Red_Hood)
Justice League Dark: Apokolips War is the tenth Justice League animated film, the sequel to Justice League Dark, and the fifteenth and current latest feature length film in the DC Animated Film Universe. The world
https://dcmovies.fandom.com/wiki/Justice_League_Dark:_Apokolips_War
Nightwing and Robin
Nightwing and Robin is a 45-second short set during the events of Justice League: Throne of Atlantis. Damian and Dick are called upon by Batman to find and capture the Scarecrow since he had
https://dcmovies.fandom.com/wiki/Nightwing_and_Robin
DC Showcase Original Shorts Collection
DC Showcase Original Shorts Collection (full cover title: DC Showcase Animated Original Shorts: Superman/Shazam! The Return of Black Adam) is a collection of animated short films which featured for the first time Superman/Shazam
https://dcmovies.fandom.com/wiki/DC_Showcase_Original_Shorts_Collection
Related Community
View More Wikis | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,769 |
\section*{\refname}}
\fancypagestyle{newstyle}{
\fancyhf{}
\fancyfoot[R]{\thepage\centering}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}}
\pagestyle{newstyle}
\renewcommand\thesection{\arabic{section}}
\renewcommand\thesubsection{\thesection.\arabic{subsection}}
\renewcommand{\thesubsubsection}{\thesubsection.\arabic{subsubsection}}
\newcommand{\ubar}[1]{\underaccent{\bar}{#1}}
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{claim}[thm]{Claim}
\newtheorem{obs}[thm]{Observation}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defin}{Definition}
\newtheorem{cond}{Condition}
\newtheorem{hyp}{Hypothesis}
\newtheorem{exmp}{Example}
\newtheorem{rem}[thm]{Remark}
\newtheorem*{lem*}{Lemma}
\newcommand\ve{\varepsilon}
\newcommand\vf{\varphi}
\newcommand\nn{\nonumber}
\newcommand{J_{\mathrm{b}}}{J_{\mathrm{b}}}
\newcommand{\chi^+_{\ve}}{\chi^+_{\ve}}
\newcommand{[-\ve^{-{1\over 2}},\ve^{-{1\over 2}}]}{[-\ve^{-{1\over 2}},\ve^{-{1\over 2}}]}
\newcommand{[-\ve^{-1},\ve^{-1}]}{[-\ve^{-1},\ve^{-1}]}
\newcommand{[0,\ve^{-1}]}{[0,\ve^{-1}]}
\newcommand{\int_{\mathbb{R}}}{\int_{\mathbb{R}}}
\newcommand{-\ve^{-1}\le x\le\ve^{-1}}{-\ve^{-1}\le x\le\ve^{-1}}
\newcommand{\ve^{-1}}{\ve^{-1}}
\newcommand{\ve^{-{1\over 2}}}{\ve^{-{1\over 2}}}
\newcommand{\mathrm{d}}{\mathrm{d}}
\newcommand{\mathrm{arc}\tanh}{\mathrm{arc}\tanh}
\newcommand{\sqrt{\ve}}{\sqrt{\ve}}
\newcommand{m^{(\kappa)}}{m^{(\kappa)}}
\newcommand{m_{\beta}}{m_{\beta}}
\newcommand{m^{*}\left(\beta\right)}{m^{*}\left(\beta\right)}
\newcommand{\chi_{\beta}}{\chi_{\beta}}
\newcommand{\hat{m}_{\ve}^{\mathrm{macro}}}{\hat{m}_{\ve}^{\mathrm{macro}}}
\newcommand{\hspace*{0.4mm}}{\hspace*{0.4mm}}
\linespread{1.125}
\begin{document}
\title{\normalfont{\Large Stationary currents in long-range interacting magnetic systems}}
\author{Roberto Boccagna}
\affiliation{{Universit{\`a} dell'Aquila, Via Vetoio, Loc. Coppito, 67010 L'Aquila, Italia.}}
\begin{abstract}
\noindent
We construct a solution for the $1d$ integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in \cite{GIAC}. This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials. The microscopic system is in contact with reservoirs of fixed magnetization and infinite volume, so that their density is not affected by any exchange with the bulk in the original Kawasaki dynamics. At the mesoscopic level, this condition is mimicked by the adoption Dirichlet boundary conditions. We derive the stationary equation of the model starting from the Lebowitz-Penrose free energy functional defined on the interval $[-\ve^{-1},\ve^{-1}]$, $\ve>0$. For $\ve$ small, we prove that below the critical temperature there exists a solution that carries positive current provided boundary values are opposite in sign and lie in the \textit{metastable} region. Such profile is no longer monotone, connecting the two phases through an antisymmetric interface localized around the origin. This represents an analytic proof of the existence of diffusion along the concentration gradient in one-component systems undergoing a phase transition, a phenomenon generally known as \textit{uphill diffusion}. However uniqueness is lacking, and we have a clue that the stationary solution obtained is not unique, as suggested by numerical simulations.
\end{abstract}
\maketitle
\noindent
\textbf{Keywords}: uphill diffusion, Kac potentials, Fick's law, phase transitions
\section{Introduction}
The aim of this paper is to study Fick's law of transport in one-component systems undergoing a second order phase transition. In this context, it represents a step forward towards the establishment of a well posed theory for diffusion along the gradient (\textit{uphill diffusion}). Fick's law relates the flux $J$ of a given substance to the gradient of its concentration $\rho$, which we suppose to be a differentiable function of the position in $\left[0,L\right]$:
\begin{equation}\label{fick}
J=-D\,\frac{\mathrm{d}}{\mathrm{d} x}\rho,
\end{equation}
at fixed boundary conditions $\rho\left(0\right)=\rho_-$, $\rho\left(L\right)=\rho_+$, with $\rho_-<\rho_+$ WLOG. Here, $D>0$ is the diffusion coefficient. According to \eqref{fick}, the flux is always in the direction of decreasing gradient, i.e. from the region at higher concentration to the region at lower concentration. Thus, the solution of \eqref{fick} connects monotonically $\rho_-$ to $\rho_+$, as represented in a sketchy way in Figure \ref{diff1}. Indeed, \eqref{fick} should be modified when considering systems that consist of many components, since diffusion may be also affected by possible microscopic, chemical interactions among different substances. Evidences of surprising behaviors have already been reported by Nernst \cite{NERNST}, Onsager \cite{ONS} and especially Darken \cite{DARKEN1,DARKEN2,LSD}, who performed an acknowledged experiment in the late 40's. His setup consisted of pairs of doped steels (Fe-Si with a different wt. $\%$ of silicon, Fe-Si and Fe-Mn or Fe-Si and Fe-Mo) containing a small difference in the carbon concentration at the edges. The steels were welded together and eventually held in a furnace in order to let diffusion occur. In fact, it was observed that carbon diffused following the gradient in the mixtures with slightly differences in carbon concentration. This is shown in Figure \ref{diffu}, which refers to the Fe-Si-Mn compound after two weeks the experiment started.
\begin{figure}[htbp!]
\centering
\begin{tikzpicture}[trim axis left, trim axis right]
\begin{axis}[
width=10cm,
height=4.9cm,
scale only axis,
axis lines=middle,
inner axis line style={=>},
ylabel=$$,
xlabel=$x$,
xmin=-1,
xmax=4,
ymin=0,
ymax=1,
domain=-0.2:4.1,
xtick={10},
ytick={5},
extra x ticks={0,3},
extra x tick labels={$0$,$L$},
]
\draw[thick,color=black] (axis cs:-1,0.2) -- (axis cs:0,0.2);
\draw[color=black] (axis cs:3,0) -- (axis cs:3,1);
\draw[dashed,color=black] (axis cs:0,0.8) -- (axis cs:3,0.8);
\draw[thick,color=black] (axis cs:3,0.8) -- (axis cs:4,0.8);
\draw[thick,color=black] (axis cs:0,0.2) -- (axis cs:3,0.8);
\node[] at (80,80) (a) {$\rho_L$};
\node[] at (80,13) (a) {$\rho_0$};
\node[] at (245,60) (a) {$J<0$};
\end{axis}
\end{tikzpicture}
\caption{Sketchy representation of the solution of \eqref{fick}.}
\end{figure}\label{diff1}
\begin{figure}[htbp!]
\centering
\begin{tikzpicture}[trim axis left, trim axis right]
\begin{axis}[
grid=both,
width=10cm,
height=5.9cm,
scale only axis,
ylabel=$\text{weight-percent C}$,
xlabel=$\text{distance from weld (inches)}$,
ymin=0.30,
ymax=0.80,
xmin=-1,
xmax=1,
xtick={-1.0,-0.5,0,0.5,1.0},
ytick={0.3,0.4,0.5,0.6,0.7,0.8},
]
\addplot[only marks,mark=*,mark size=0.6mm,mark options={color=black,line width=0.9pt,fill=white},color=black]
table{SiMn.dat};
\addplot[domain=0.001:1]{0.561987+0.166301*exp(-3.03827*x)};
\addplot[domain=-1:0.001]{0.506756-0.182751*exp(2.83615*x)};
\draw[<-,thin] (axis cs:-1,0.55) -- (axis cs:-0.675,0.55);
\draw[->,thin] (axis cs:-0.325,0.55) -- (axis cs:0,0.55);
\draw[<-,thin] (axis cs:0,0.55) -- (axis cs:0.3,0.55);
\draw[->,thin] (axis cs:0.7,0.55) -- (axis cs:1,0.55);
\draw[black,semithick] (axis cs:0,0.3) -- (axis cs:0,0.8);
\node[] at (axis cs: -0.5,0.55) {3.60$\%$ Si};
\node[] at (axis cs: 0.5,0.55) {6.45$\%$ Mn};
\node[] at (axis cs: -0.2,0.73) {0.717$\%$ C};
\node[] at (axis cs: 0.2,0.324) {0.324$\%$ C};
\node[] at (axis cs: -0.875,0.46) {0.493};
\node[] at (axis cs: 0.8775,0.605) {0.566};
\end{axis}
\end{tikzpicture}
\caption{Carbon concentration in the Fe-Si-Mn compound after 10 days at 1050 $\degree$C (figure extrapolated from \cite{LSD}).}
\end{figure}\label{diffu}
This counterintuitive behavior has a microscopic origin: in fact, silicon decreases the chemical affinity of carbon, while manganese increases it. This results in a driving force that acts in the opposite direction with respect to the concentration gradient and it might beat the gradient, if the difference in the carbon concentration at the edges is small. Such mechanism, which actually sustains uphill diffusion, works until dopants penetrate the weld; then, ``standard'' diffusion regime is restored. In formulae, this can be modeled by replacing \eqref{fick} with a vectorial relation:
\begin{equation}\label{fick2}
J_i=-c_i \sum_j L_{ij}\nabla \mu_j, \qquad i=1,\ldots,n
\end{equation}
where $n>2$ is the number of components of the system, $i$ refers to a given component, $c_i$ is the molar concentration of $i$, $L_{ij}$ represents the Fick diffusivity of $i$ given the presence of $j$, while the $\mu_i$'s are chemical potentials. \eqref{fick2} describes a system of $n-1$ linearly independent equations, because of the Gibbs-Duhem relation for chemical potentials \cite{KR1,KR2}.
\newline
Very surprisingly, numerical simulations suggest that uphill diffusion may also occur also in one-component systems undergoing a phase separation. Colangeli et al. considered in \cite{CDMP} a $1d$ stochastic automaton describing a dissipative system of particles interacting at large distances. After a transient, a stationary state with non zero current emerges and, moreover, a region in which diffusion follows the concentration gradient can be spotted tuning the characteristic parameters of the system. Similar results have been obtained by running a Kawasaki dynamics for an Ising spin chain with Kac potentials below the critical temperature, in which particles located at the edges may flip according to assigned rates, in order to mimic interactions with reservoirs of infinite volume and opposite magnetization \cite{RB2}. When the magnetizations of the reservoirs are suitably chosen, the flux follows the ``magnetization gradient''. The resulting steady profile, called \textit{bump}, is no longer monotone and connects the two boundary values through an interface that is localized in the nearby of one of the edges, randomly selected by dynamics. Colangeli et al. \cite{CGGV} obtained analogue numerical results for the $2d$ nearest neighbors Ising model.
\newline
The microscopic mechanism underlying uphill diffusion in one-component systems has not a chemical origin. We speculate that the ``force'' that counteracts the gradient is provided in this case by the separation of phases; however, we believe that such state is in fact metastable, meaning that bumps are local minima for the corresponding Gibbs free energy, but not global ones. Hence, after a transient, the flux should reverse to be directed from the state at higher magnetization to the state at lower magnetization. Nevertheless, such inversion does not take place in the time considered for simulations.
\newline
\newline
\indent
Here we prove analytically the occurrence of uphill diffusion considering the model that is the continuous limit of the Ising chain with Kac potentials and Dirichlet boundary conditions. Our starting point is the Lebowitz-Penrose free energy functional, that is a non-local version of the scalar Ginzburg-Landau functional and that we postulate to describe the Physics of the system at the \textit{mesoscopic} level. This represents the intermediate scale between the microscopic, discrete chain and the macroscopic model, which is obtained letting the size of the system diverge. We know that in this case the phase diagram (i.e. the free energy density vs. magnetization diagram) has a global minimum for $\beta<1$, $\beta$ the inverse temperature in our units, while for $\beta>1$ the graph is flat in $\left[-m_{\beta},m_{\beta}\right]$, $m_{\beta}$ the positive solution of the mean field equation $m=\tanh\left(\beta m\right)$. This indicates the occurrence of a phase transition. Any value in $\left(-m_{\beta},m_{\beta}\right)$, the so called \textit{spinodal region}, is then forbidden, so that any stationary profile containing values smaller of $-m_{\beta}$ and larger than $m_{\beta}$ must be discontinuous. At the mesoscopic level, the spinodal region is actually available and the discontinuity is replaced by a smooth interface, as proved by De Masi et al. for the free-boundary Stefan problem \cite{EP1,EP2,EP3,EP4}. However, the discontinuity is recovered when the hydrodynamic limit is performed.
\newline
We then fix $\beta>1$ ($\beta=1$ the critical inverse temperature in the mean-field model) and look for stationary solutions of $\dot{m}=-\frac{\partial }{\partial x} I$ in the space of bounded antisymmetric functions, $I$ being the local current which is supposed to be proportional to the functional derivative of the free energy. Thus, we reduce to the problem $\frac{\partial }{\partial x} I=0$ in the finite interval $[-\ve^{-1},\ve^{-1}]$, $\ve>0$ fixed, which turns out to be an integro-differential equation \cite{GIAC}. The corresponding Dirichlet problem has been already studied in \cite{RB2} and \cite{RB}, although in the presence of an external, antisymmetric magnetic field. In that case it has been proved that, whatever the intensity of this field, the provided ``external force'' cannot reverse the flux when the positive boundary condition is in the interval $\left(m^{*}\left(\beta\right),m_{\beta}\right)$, $m^{*}\left(\beta\right)=\sqrt{1-1/\beta}$ the positive saddle point of the mean field free energy (Figure \ref{dyy}).
\begin{figure}[htbp!]
\centering
\begin{tikzpicture}[trim axis left, trim axis right]
\begin{axis}[
white,
every x tick/.style={black},
width=10cm,
height=4.9cm,
scale only axis,
axis lines=middle,
inner axis line style={=>},
xlabel=$m$,
xmin=-1.25,
xmax=1.25,
ymin=0.638,
ymax=0.71,
domain=-1.1:1.1,
xtick={5},
ytick={-1},
extra x ticks={-1,-0.710412,-0.4472,0.4472,0.710412,1},
extra x tick labels=
{$\textcolor{black}{-1}^{\textcolor{white}{*}}$,
$\textcolor{black}{-m_{\beta}}^{\textcolor{white}{*}}$,
$\textcolor{black}{-m^*\left(\beta\right)}$,
$\textcolor{black}{m^*\left(\beta\right)}$,
$\textcolor{black}{m_{\beta}}^{\textcolor{white}{*}}$,
$\textcolor{black}{1}^{\textcolor{white}{*}}$},
]
\draw[dashed,color=black] (axis cs:0.4472,0.63) -- (axis cs:0.4472,0.65);
\draw[dashed,color=black] (axis cs:-0.4472,0.63) -- (axis cs:-0.4472,0.65);
\draw[dashed,color=black] (axis cs:1,0.63) -- (axis cs:1,0.8);
\draw[dashed,color=black] (axis cs:-1,0.63) -- (axis cs:-1,0.8);
\draw[thick,color=black] (axis cs:-0.4472,0.638) -- (axis cs:0.4472,0.638);
\draw[thick,color=black] (axis cs:-0.71,0.638) -- (axis cs:-0.4472,0.638);
\draw[thick,color=black] (axis cs:0.71,0.638) -- (axis cs:0.4472,0.638);
\draw[thick,color=black] (axis cs:0.71,0.638) -- (axis cs:1,0.638);
\draw[thick,color=black] (axis cs:-0.71,0.638) -- (axis cs:-1,0.638);
\draw[black,->] (axis cs:0,0.638) -- (axis cs:0,0.71);
\node[] at (141,652) {$\textcolor{black}{\phi_{\beta}(m)}$};
\addplot[thick,color=black,samples=500]
{-(x^2/2)+0.8*((1-x)/2)*ln((1-x)/2)+0.8*((1+x)/2)*ln((1+x)/2)+1.2214};
\end{axis}
\end{tikzpicture}
\caption{The mean field free energy at $\beta>1$.}
\end{figure}\label{dyy}
We solve the stationary equation and prove that our resulting profile actually carries positive current. For $x\ge 0$ the solution firstly increases, jumping ``instantly'' from zero to $m_{\beta}$, then decreases to the metastable value at the right boundary; however, the region in which the current flows in the ``wrong'' direction reduces to a set of zero Lebesgue measure in the hydrodinamic limit, so that Fick's Law actually holds almost everywhere (w.r.t. the Lebesgue measure). The weak spot of the analysis is that our solution is supposedly unstable; in fact, numerical simulations suggest that bumps should be stable points for the gradient dynamics.
\newline
\indent
The stationary Stefan problem in bounded domains has been already considered by De Masi, Presutti and Tsagkarogiannis in \cite{DMPT}, despite Neumann conditions have been adopted there. Apart from technicalities in the proof, the two approaches are quite different, since in the Neumann setting the magnetization profile naturally selects the boundary values imposed by the choice of current. However, as clarified in that work, there are solutions of the mesoscopic Neumann version of problem that converge to any solution of the Dirichlet problem as $\ve\to 0$.
\section{Background}
\subsection{The Lebowitz-Penrose functional}
\noindent
Indicate for notational convenience $\Lambda\coloneqq[-\ve^{-1},\ve^{-1}]$ and $\Lambda^c\coloneqq\mathbb{R}\setminus[-\ve^{-1},\ve^{-1}]$. Consider $m_{\Lambda}\in L^{\infty}\left(\Lambda,\left[-1,1\right]\right)$, $m_{\Lambda^c}\in L^{\infty}\left(\Lambda^c,\left[-1,1\right]\right)$, $m_{\Lambda}$ being the magnetization density of the bulk and $m_{\Lambda^c}$ the magnetization of the reservoirs. Our starting point is the mesoscopic Lebowitz-Penrose free energy functional at zero external magnetic field, that is
\begin{eqnarray}
&&\mathcal{F}_{\beta}\left[m_{\Lambda}\mid m_{\Lambda^c}\right] = \mathcal{F}_{\beta}\left[m_{\Lambda}\right] +{1\over 2} \int_{\Lambda}\int_{\Lambda^c}\tilde{J}\left(x,y\right)\left[m_{\Lambda}\left(x\right)-m_{\Lambda^c}\left(y\right)\right]^2\mathrm{d} x\,\mathrm{d} y, \label{freeen}\\
&&\mathcal{F}_{\beta}\left[m_{\Lambda}\right] = -{1\over\beta}\int_{\Lambda}\phi_{\beta}\left(m_{\Lambda}\left(x\right)\right)\mathrm{d} x + {1\over 4} \int_{\Lambda}\int_{\Lambda}\tilde{J}\left(x,y\right)\left[m_{\Lambda}\left(x\right)-m_{\Lambda}\left(y\right)\right]^2\mathrm{d} x\,\mathrm{d} y \label{freeen2}
\end{eqnarray}
where
\begin{equation}
\phi_{\beta}\left(m\right)=-\frac{1}{2}m^2-\frac{1}{\beta}S\left(m\right),
\end{equation}
and $S\left(m\right)$ is the standard binary entropy for an Ising spin system:
\begin{equation}\label{entr}
S\left(m\right)=-{1+m\over 2}\log\left({1+m\over 2}\right)-{1-m\over 2}\log\left({1-m\over 2}\right).
\end{equation}
$\tilde{J}$ is a probability kernel that actually depends on the distance between two points. The assumptions made on $\tilde{J}$ are precisely listed below:
\begin{itemize}
\item[-] $\tilde{J}$ is translational invariant, that is $\tilde{J}\left(x,y\right)=\tilde{J}\left(0,\left|x-y\right|\right)$ for any $x,y\in\mathbb{R}$;
\item[-] $\tilde{J}$ is twice differentiable, compactly supported: $\tilde{J}\in C_K^{2}\left(\left[-1,1\right],\left[0,1\right]\right)$;
\item[-] $\tilde{J}$ is normalized, $\int_{\mathbb{R}} \tilde{J}\left(\,\cdot\,,y\right)\mathrm{d} y=1$;
\item[-] $\tilde{J}\left(0,x\right)$ is strictly decreasing in $\left[0,1\right]$.
\end{itemize}
We treat expressions \eqref{freeen}, \eqref{freeen2} as primitive quantities, by postulating them to define our model at a mesoscopic level. Indeed, it can be shown that this is precisely what one obtains taking the continuous limit of the underlying microscopic Ising chain with Kac potentials after a feasible scaling. We indicate reference \cite{GIAC} for details on this procedure, that is known as the \textit{Lebowitz-Penrose limit}.
We drop hereafter the suffix $\Lambda$, also indicating $\mathcal{F}_{\beta}\left[m\mid \mu\right]\equiv\mathcal{F}_{\beta}\left[m_{\Lambda}\mid m_{\Lambda^c}\right]$ for the sake of simplicity.
\newline
The axiomatic theory provides that the magnetization evolves in time according to a gradient dynamics
\begin{equation}\label{motion}
\dot{m}\left(x,t\right)=-\frac{\partial I}{\partial x}\left(x,t\right), \qquad t\ge 0
\end{equation}
where $I$ represents the local current
\begin{equation}\label{stat}
I\left(x,t\right)\coloneqq-\chi_{\beta}\left(m\left(x,t\right)\right)\frac{\partial}{\partial x}\frac{\delta\mathcal{F}_{\beta,\mu}\left[m\mid \mu\right]
}{\delta m\left(x,t\right)},
\end{equation}
in which $\chi_{\beta}\left(m\right)=\beta\left(1-m^2\right)$ is the mobility coefficient for an Ising spin system. Hence, the stationary problem $\dot{m}=0$ reads
\begin{equation}\label{current}
I=-\chi_{\beta}\left(m\left(x\right)\right)\frac{\partial}{\partial x}\frac{\delta\mathcal{F}_{\beta}\left[m\mid \mu\right]
}{\delta m\left(x\right)},
\end{equation}
that is an integro-differential equation in the unknown function $m$ at constant $I$ and given boundary conditions. We call $I=j\ve$, $j\in\mathbb{R}$ a constant, as we expect the current to be of order $\ve$.
\subsection{Notation}
\noindent
It is worth redefining the convolution kernel in the distributional sense as follows
\begin{equation}
J\left(x,y\right)\stackrel{{\mathcal{D}'}}{=}\tilde{J}\left(x,y\right)\mathbf{1}_{\left|y\right|<\ve^{-1}}+b_{\ve}\left(\left|x\right|\right)\left[\delta(\ve^{-1}-y)+\delta(\ve^{-1}+y)\right]\mathbf{1}_{\left|y\right|\ge\ve^{-1}},
\end{equation}
where
\begin{equation}
b_{\ve}\left(x\right)\coloneqq\int_{\ve^{-1}}^{\ve^{-1}+1}\tilde{J}\left(x,y\right)\mathrm{d} y.
\end{equation}
This way we act on functions defined in the interior of the bulk. For any bounded function in $[-\ve^{-1},\ve^{-1}]$, we indicate $\left\|\,\cdot\,\right\|_{\ve} \equiv \left\|\,\cdot\,\right\|_{L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right)}$ as the sup norm in that interval. For any $m\in L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right)$, we define
\begin{equation}
\left(J\ast m\right)\left(x\right)\coloneqq \int J\left(x,y\right)m\left(y\right) \mathrm{d} y.
\end{equation}
If not specified, integrals are intended to be performed on $[-\ve^{-1},\ve^{-1}]$. For any $m,h\in L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right)$ we call
\begin{equation}
p_{m,h}\left(x\right) \coloneqq \beta \cosh^{-2}\Big\{\beta\Big[\left(J\ast m\right)\left(x\right)+h\left(x\right)
\Big]\Big\}
\end{equation}
and
\begin{equation}
p'_{m,h}\left(x\right)\coloneqq p_{m,h}\left(x\right) \tanh\Big\{\beta\Big[\left(J\ast m\right)\left(x\right)+h\left(x\right)
\Big]\Big\}.
\end{equation}
Let $\mathsf{A}_{m,h}$ the linear operator acting on a bounded function $f$ as follows
\begin{equation}
\mathsf{A}_{m,h}f\left(x\right)= \int p_{m,h}\left(x\right) J\left(x,y\right)f\left(y\right) \mathrm{d} y.
\end{equation}
$\mathsf{A}_{m,h}$ is the linear operator with kernel $\mathsf{A}_{m,h}\left(x,y\right)=p_{m,h}\left(x\right)J\left(x,y\right)$. The action of the $n$-th power of $\mathsf{A}_{m,h}$ on $f$ is explicitly given by
\begin{equation}
\mathsf{A}_{m,h}^n f\left(x_0\right)=\int \prod_{i=1}^n p_{m,h}\left(x_{i-1}\right)J\left(x_{i-1},x_i\right)f\left(x_n\right)\mathrm{d} x_1\ldots\mathrm{d} x_n.
\end{equation}
\subsection{Instantons}
\noindent
We briefly recall a fundamental result obtained by De Masi et al. \cite{EP1,EP2,EP3,EP4}. This regards the free boundary version of the problem. Let
\begin{equation}
\mathcal{F}_{\beta}\left[m\right] = -{1\over\beta}\int_{\mathbb{R}}\phi_{\beta}\left(m\left(x\right)\right)\mathrm{d} x + {1\over 4} \int_{\mathbb{R}} \int_{\mathbb{R}} J\left(x,y\right)\left[m\left(x\right)-m\left(y\right)\right]^2\mathrm{d} x\,\mathrm{d} y
\end{equation}
be the free energy functional on $\mathbb{R}$ defined for functions that belong to the Banach space
\begin{equation}
\mathcal{N}\coloneqq\Big\{m\in L^{\infty}\left(\mathbb{R},\left[-1,1\right]\right) \mid \liminf_{x\to -\infty}m\left(x\right)<0, \; \limsup_{x\to +\infty}m\left(x\right)>0 \Big\}.
\end{equation}
In this case the flux is null, so that the stationary problem reduces to
\begin{equation}\label{evo}
\dot{m}\left(x,t\right)=-\frac{\delta\mathcal{F}_{\beta}\left[m\right]
}{\delta m\left(x,t\right)}, \qquad t\ge 0.
\end{equation}
Let the \textit{instanton} $\bar{m}$ the set of minimizers of $\mathcal{F}_{\beta}\left[m\right]$ in $\mathcal{N}$ solution of
\begin{equation}\label{inst}
\bar{m}\left(x\right)=\tanh\Big\{\beta\Big[\left(J\ast \bar{m}\right)\left(x\right)\Big]\Big\}, \qquad x\in\mathbb{R}
\end{equation}
that satisfies $\lim_{x\to\pm\infty}m\left(x\right)=\pmm_{\beta}$.
\newpage
\begin{thm}
For any $\beta>1$ the following holds:
\begin{itemize}
\item[-] the variational problem $\delta\mathcal{F}_{\beta}\left[m\right]=0$ has a minimizer which is unique up to translations;
\item[-] the mean field equation \eqref{inst} has a solution which is unique in $\mathcal{N}$, up to translations;
\item[-] for any $m\in\mathcal{N}$, there is $\xi\in\mathbb{R}$ such that $\lim_{t\to\infty} \left\|m\left(\,\cdot\,,t\right) -m_{\xi}\right\|_{\infty}=0$, where $\bar{m}_{\xi}\left(x\right)\coloneqq \bar{m}\left(x-\xi\right)$.
\end{itemize}
$\bar{m}\in C^{\infty}\left(\mathbb{R},\left[-1,1\right]\right)$ is a strictly increasing, antisymmetric function which converges exponentially fast to $\pmm_{\beta}$ as $x\to\pm\infty$.
\end{thm}
\noindent
Hereafter, we call
\begin{equation}
\bar{p}\left(x\right)\coloneqq \beta\left[1-\bar{m}^2\left(x\right)\right], \qquad {\mathsf{A}_{\bar{m}}}\left(x,y\right)\coloneqq \bar{p}\left(x\right)J\left(x,y\right)
\end{equation}
for any $x$ and $y$ in $\mathbb{R}$. We refer as $\bar{m}'$ to the derivative of $\bar{m}$ with respect to $x$.
\section{Uphill Diffusion}
\noindent
Our main result is
\begin{thm}\label{teo}
At fixed $\beta>1$ and $\mu\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$, there is ${\ve}_{\beta}>0$ such that for any $\ve<{\ve}_{\beta}$ there are an antisymmetric, continuous function $m$ and a positive constant $j$ that solve
\begin{equation}\label{pbm}
\begin{dcases}
j\ve=-\chi_{\beta}\left(m\left(x\right)\right)\frac{\partial}{\partial x}\frac{\delta\mathcal{F}_{\beta}\left[m\mid \mu\right]
}{\delta m\left(x\right)},\qquad &x\in [-\ve^{-1},\ve^{-1}] \\
m\hspace*{0.4mm}(-\ve^{-1}) =-\mu, \qquad m\hspace*{0.4mm}(\ve^{-1}) =\mu.
\end{dcases}
\end{equation}
\end{thm}
\subsection{Outline of the Proof}
\noindent
For our purposes, it is worth performing the following change of variables:
\begin{equation}
h\left(x\right)\coloneqq \frac{\delta\mathcal{F}_{\beta}\left[m\mid \mu\right]
}{\delta m\left(x\right)}
\end{equation}
where, explicitly
\begin{equation}\label{deltaf}
\frac{\delta\mathcal{F}_{\beta}\left[m\mid \mu\right]}{\delta m\left(x\right)}=
-\frac{1}{\beta}\,\mathrm{arc}\tanh\left(m\left(x\right)\right)+\int J\left(x,y\right)m\left(y\right) \mathrm{d} y.
\end{equation}
In this position \eqref{current} becomes, after a straight integration
\begin{equation}\label{chvar}
h\left(x\right)=h\left(x_0\right)-j\ve\int_{x_0}^x \frac{\mathrm{d} y}{\chi_{\beta}\left(m\left(y\right)\right)}.
\end{equation}
Observe that $h\left(0\right)=0$ if $m$ is odd, so we eventually formulate problem \eqref{pbm} as a system of coupled equations:
\begin{equation}\label{pbmh}
\begin{dcases}
m\left(x\right)= \tanh\Big\{\beta\Big[\left(J\ast m\right)\left(x\right)+h\left(x\right)
\Big]\Big\} \\
h\left(x\right)=-j\ve\int_0^x \frac{\mathrm{d} y}{\chi_{\beta}\left(m\left(y\right)\right)}
\end{dcases}
\qquad x\in [-\ve^{-1},\ve^{-1}],
\end{equation}
with $m\hspace*{0.4mm}(-\ve^{-1})=-\mu$, $m\hspace*{0.4mm}(\ve^{-1})=\mu$. Notice that the first equation in \eqref{pbmh} is just \eqref{deltaf} expressed in the new variables. We will often indicate
\begin{equation}
\left[\mathsf{H}\left(m\right)\right]\left(x\right) \coloneqq -j\ve\int_0^x \frac{\mathrm{d} y}{\chi_{\beta}\left(m\left(y\right)\right)}\qquad \forall x\in[-\ve^{-1},\ve^{-1}].
\end{equation}
The existence of a solution of problem \eqref{pbmh} is proved by iteration (Newton's method): we start from a couple $\left(m_0,h_0\right)$ and fixed $\mu$ and $j$ and define a feasible map $\left(m_{n},h_{n}\right)\mapsto\left(m_{n+1},h_{n+1}\right)$ that converges uniformly to a couple $\left(m,h\right)$ that solves \eqref{pbmh} and satisfies certain boundary conditions $m\hspace*{0.4mm}(\pm\ve^{-1})=\pm\nu$, $\nu\neq\mu$ in general. Afterwards, we prove that $j$ can be actually tuned in order to cover the whole metastable region, that is for any $\nu\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$ there exists at least one $j>0$ such that $\lim_{n\to\infty} m_n\left(x\right)$ solves \eqref{pbmh} with $m\,(\ve^{-1})=\nu$. In this scheme, the choice of $m_0$ (and $h_0$ as a function of $m_0$) turns out to be crucial, as we would like to start with a profile that is ``almost'' a fixed point.
\newline
\newline
The technical part of the paper is organized as follows: after having established the recursive method and chosen $m_0$, we perform in Section \ref{quattro} some estimates that are needed in the course of the proof; in particular, we prove the invertibility of $\mathsf{I}-\mathsf{A}_{m,h}$. In Section \ref{cinque} we construct the sequence $\left(m_{n},h_{n}\right)_{n=0}^{\infty}$ and prove convergence to a certain solution of \eqref{pbmh} with $j>0$. In Section \ref{sei} we deal with the invertibility issue mentioned above.
\subsection{Choice of $m_0$}
\begin{prop}
The ``macroscopic'' problem at $\beta>1$
\begin{equation}\label{pbmmacro}
\begin{dcases}
j_{M} = -\big[1-\chi_{\beta}\left(M\left(x\right)\right)\big]M'\left(x\right), \qquad x\in \left[0,1\right]\\
M\left(0\right)= \mu_-, \qquad M\left(1\right)=\mu_+
\end{dcases}
\end{equation}
with $0<\mu_+<\mu_-<1$, admits a unique solution in $C^{\infty}\left(\left[0,1\right]\right)$. Such solution is decreasing in $\left[0,1\right]$.
\end{prop}
\noindent
\textbf{Proof.} We refer to \eqref{pbmmacro} as the macroscopic equation because it comes from the variational problem that one obtains after performing the macroscopic limit (see \cite{RB2,RB}). A straight integration gives
\begin{equation}\label{terzogrado}
x=-\frac{1}{j_{M}}\Big[\left(\beta-1\right)\left(M\left(x\right)-\mu_-\right)-\frac{\beta^3}{3}\left(M^3\left(x\right)-\mu_-^3\right)\Big],
\end{equation}
with $j_M$ fixed by the choice of $\mu_-$ and $\mu_+$:
\begin{equation}
j_{M}=\left(\beta-1\right)\big(\mu_- -\mu_+\big)-\frac{\beta^3}{3}\big(\mu_-^3-\mu_+^3\big).
\end{equation}
As a function of $M$, $x$ is infinitely times differentiable and moreover, $x\left(M\right)$ is invertible since $M'$ is negative. $M$ can be obtained as the unique real solution of the cubic equation \eqref{terzogrado}.
\qed
\newline
\newline
Notice that problem \eqref{pbmmacro} can be formulated as a system of coupled equations as well
\begin{equation}\label{pbmhm}
\begin{dcases}
M\left(x\right)= \tanh\Big\{\beta\Big[M\left(x\right)+H\left(x\right)
\Big]\Big\} \\
H\left(x\right)=\widetilde{H}-j_{M}\int_0^x \frac{\mathrm{d} y}{\chi_{\beta}\left(M\left(y\right)\right)}
\end{dcases}
,\qquad x\in [0,1]
\end{equation}
with $\widetilde{H} =\beta^{-1}\mathrm{arc}\tanh\left(\mu_-\right)-\mu_-$. We define
\begin{equation}\label{m0}
m_0\left(x\right)\coloneqq
\begin{dcases}
\bar{m}\left(x\right) \qquad &0\le x\le\ve^{-{1\over 2}}\\
M_{\mu}\left(\frac{\ve x-\sqrt{\ve}}{1-\sqrt{\ve}}\right)\eqqcolon \hat{M}_{\mu}\left(x\right) \qquad &\ve^{-{1\over 2}}\le x\le\ve^{-1},
\end{dcases}
\end{equation}
with $m_0\left(x\right)=-m_0\left(-x\right)$ for $x<0,$ where $M_{\mu}$ is the solution of \eqref{pbmmacro} that satisfies $M_{\mu}\left(0\right)=\bar{m}\hspace*{0.4mm}(\ve^{-{1\over 2}})$ and $M_{\mu}\left(1\right)=\mu$. For technical reasons, we choose $\ve^{-1}$ so large that $\bar{m}\hspace*{0.4mm}(\ve^{-{1\over 2}}/2)=m_{\beta}-\delta$, $\delta>0$ a small parameter specified further on. We speculate that if $\ve^{-1}$ is large enough, the solution should not differ so much from the instanton in the nearby of the origin. Once reached the value $m_0\hspace*{0.4mm}(\ve^{-{1\over 2}})\approxm_{\beta}$, we suppose the solution to be monotone decreasing and ``close'' to the (rescaled) macroscopic profile. This will be very clear a posteriori, as we will show that in fact the distance between $m_0$ and the stationary solution $m$ is of order $\ve$ in the sup norm.
\subsection{Iterative scheme}
\noindent
The following results explicitly defines the method.
\begin{prop}\label{p3}
Let $m_0$ as in \eqref{m0} satisfying $m_0\,(\ve^{-1})=\mu_0$, $\mu_0\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$ fixed and $h_0 = \mathsf{H}\left(m_0\right)$. For any $n$, there is $m_n\in C\left([-\ve^{-1},\ve^{-1}],\left[-1,1\right]\right)$ that solves
\begin{equation}\label{mn}
\begin{dcases}
m_{n}\left(x\right)=\tanh\Big\{\beta\Big[\left(J\ast m_n\right)\left(x\right)+h_{n-1}\left(x\right)
\Big]\Big\}\\
h_{n-1}\left(x\right)=\left[\mathsf{H}\left(m_{n-1}\right)\right]\left(x\right)
\end{dcases}
\end{equation}
with $m_n\hspace*{0.4mm}(-\ve^{-1})=-\mu_n$, $m_n\hspace*{0.4mm}(\ve^{-1})=\mu_n$, $\mu_n\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$, provided $\ve$ is small enough. The sequence $\left(m_n,h_n\right)_{n=0}^{\infty}$ converges uniformly to a pair $\left(m,h\right)$, where $h=\mathsf{H}\left(m\right)$, which is a solution of problem \eqref{pbmh} with boundary conditions $m\hspace*{0.4mm}(-\ve^{-1})=-\mu$, $m\hspace*{0.4mm}(\ve^{-1})=\mu$, $\mu$ in the metastable region. Then, $m$ also solves \eqref{pbm}.
\end{prop}
\begin{prop}\label{p4}
In the same hypothesis of Proposition \ref{p3}, for any $\mu\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$ there is at least one $j>0$ such that the iterative scheme defined above converges to a solution of \eqref{pbm} with boundary conditions $m\hspace*{0.4mm}(-\ve^{-1})=-\mu$, $m\hspace*{0.4mm}(\ve^{-1})=\mu$.
\end{prop}
\noindent
In the iterations, $j=j\left(\mu_0\right)$ is fixed parameter, whose value is actually specified by the boundary value $m_0\,(\ve^{-1})=\mu_0$ (and $\bar{m}\,(\ve^{-{1\over 2}})$ that depends on $\ve$ only):
\begin{equation}
j=\left(\beta-1\right)\big(\bar{m}\hspace*{0.4mm}(\ve^{-{1\over 2}}) -\mu\big)-\frac{\beta^3}{3}\big(\bar{m}^3\hspace*{0.4mm}(\ve^{-{1\over 2}})-\mu^3\big).
\end{equation}
Every time an iteration is performed, the boundary value changes, and therefore we cannot rule out the possibility that our constructive method defines a map $j\mapsto \left(m^{*}\left(\beta\right),m_{\beta}\right)$ which is not surjective. Hence, Proposition \ref{p4} is needed in order to close the proof of Theorem \ref{teo}.
\section{Some properties of $\mathsf{A}_{m,h}$}\label{quattro}
\subsection{A preliminary result}
\noindent
We recall here a result proved in \cite{DMOP} and that can be even found in \cite{EP}. Define the scalar product on $\mathbb{R}$
\begin{equation}
\left\langle f\right\rangle_{\infty}\coloneqq\int_{\mathbb{R}} f\left(x\right)\frac{\mathrm{d} x}{\bar{p}\left(x\right)}
\end{equation}
and indicate $\widetilde{m}'\coloneqq\bar{m}'/\sqrt{\left\langle (\bar{m}')^2\right\rangle_{\infty}}$. We have the following
\begin{prop}
There are positive constants $a$ and $c$ such that for any $f\in L^{\infty}\left(\mathbb{R}\right)$ and any integer $n$:
\begin{equation}
\left|\,\int_{\mathbb{R}^n}{\mathsf{A}^n_{\bar{m}}}\left(x,y\right)\widetilde{f}\left(y\right)\mathrm{d} y\,\right| \le
c\hspace*{0.4mm}\|\widetilde{f} \|_{\infty}\mathrm{e}^{-an}, \qquad \widetilde{f}\coloneqq f-\left\langle f\widetilde{{m}}'\right\rangle_{\infty}\widetilde{m}'.
\end{equation}
\end{prop}
\noindent
There is a very straight consequence of this result, which is however essential for our purposes.
\begin{cor}
For any bounded, antisymmetric function $\psi$ on $\mathbb{R}$ and any integer $n$:
\begin{equation}\label{est}
\left|\,\int_{\mathbb{R}^n}{\mathsf{A}^n_{\bar{m}}}\left(x,y\right)\psi\left(y\right)\mathrm{d} y\,\right| \le c\hspace*{0.4mm}\mathrm{e}^{-an} \left\|\psi\right\|_{\infty}.
\end{equation}
\end{cor}
\noindent
\textbf{Proof.} Since $\widetilde{m}'$ is symmetric, $\left\langle \psi\widetilde{{m}}'\right\rangle_{\infty}=0$ and then $\widetilde{\psi}\equiv\psi$.
\qed
\subsection{Invertibility of $\mathsf{I}-\mathsf{A}_{m,h}$}
\noindent
Define the set:
\begin{equation}
\Sigma_{\delta,\ve}\coloneqq\bigg\{m\in L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right) \mid \left\|p_{m,h}-p_{m_0,h_0}\right\|_{\ve}<\delta\bigg\}.
\end{equation}
\begin{lem}\label{lem11}
Let $m^{(1)},m^{(2)}\in\Sigma_{\delta,\ve}$. Then
\begin{equation}
\|\mathsf{H}(m^{(2)})-\mathsf{H}(m^{(1)})\|_{\ve}\le \tilde{c}\hspace*{0.4mm} \|m^{(2)}-m^{(1)}\|_{\ve}, \qquad \tilde{c}= \frac{2\beta j}{\chi_{\beta}^2\left(m_{\beta}\right)}.
\end{equation}
\end{lem}
\noindent
\textbf{Proof.} We have, for any $x\in[-\ve^{-1},\ve^{-1}]$:
\begin{eqnarray}
\big|[\mathsf{H}(m^{(2)})]\left(x\right)-[\mathsf{H}(m^{(1)})]\left(x\right)\big|&\le&
\beta j\ve \int_{0}^x \frac{\left|(m^{(2)})^2\left(y\right)-(m^{(1)})^2\left(y\right)\right|}{\chi_{\beta}(m^{(1)}\left(y\right))\chi_{\beta}(m^{(2)}\left(y\right))}\mathrm{d} y\nn\\
&\le& \frac{2\beta j}{\chi_{\beta}^2\left(m_{\beta}\right)}\|m^{(1)}-m^{(2)}\|_{\ve}.
\end{eqnarray}
\qed
\begin{comment}
\begin{lem}
Let $m\in\Sigma_{\delta,\ve}$, $h=\mathsf{H}\left(m\right)$. Then,
\begin{equation}\label{wil}
\left\|p_{m,h}-p_{m_0,h_0}\right\| \le 4\beta^2\left(1+\tilde{c}\right)\delta.
\end{equation}
\end{lem}
\noindent
\textbf{Proof.} For any $x_1$ and $x_2$ in $\mathbb{R}$
\begin{equation}\label{th}
\left|\tanh^2\left(x_2\right)-\tanh^2\left(x_1\right)\right|\le 4 \tanh\left|x_2-x_1\right|,
\end{equation}
therefore
\begin{equation}
\left|p_{m,h}\left(x\right)-p_{m_0,h_0}\left(x\right)\right|\le4\beta^2\left(\left|m\left(x\right)-m_0\left(x\right)\right|+\left|h\left(x\right)-h_0\left(x\right)\right|\right).
\end{equation}
Using Lemma \ref{lem11} we get \eqref{wil}.
\qed
\end{comment}
\begin{prop}\label{pe}
Let $m\in\Sigma_{\delta,\ve}$, $h=\mathsf{H}\left(m\right)$. There exists a constant $\theta=\theta\left(\delta,\ve\right)$ such that for any bounded, antisymmetric function $\psi$ on $\mathbb{R}$ and any integer $n$:
\begin{equation}\label{est1}
\left|\,\int_{|x_i|\le\ve^{-{1\over 2}},\, i=1,\ldots,n}\prod_{i=1}^n
\mathsf{A}_{m,h}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n\,\right| \le c\hspace*{0.4mm}\mathrm{e}^{-an}\theta^n\sup_{|x_0|\le\ve^{-{1\over 2}}}\left|\psi\left(x_0\right)\right|
\end{equation}
for any $\ve$ small enough.
\end{prop}
\noindent
\textbf{Proof.} Call, at fixed $x_0\in[-\ve^{-{1\over 2}},\ve^{-{1\over 2}}]$:
\begin{equation}
\Delta_n^{(\psi)}\left(x_0\right)\coloneqq \int_{|x_i|\le\ve^{-{1\over 2}},\, i=1,\ldots,n}
\left|\prod_{i=1}^n \mathsf{A}_{m,h}\left(x_{i-1},x_i\right)-\prod_{i=1}^n {\mathsf{A}_{\bar{m}}}\left(x_{i-1},x_i\right)\right|\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n.
\end{equation}
We have
\begin{eqnarray}
\Delta_n^{(\psi)}\left(x_0\right)&\le&\int_{|x_i|\le\ve^{-{1\over 2}},\, i=1,\ldots,n}\hspace*{-0.5mm}
\left|\prod_{i=1}^n \hspace*{-0.5mm}p_{m,h}\left(x_{i-1}\right)\hspace*{-0.75mm}J\left(x_{i-1},x_i\right)\hspace*{-0.5mm}-\hspace*{-0.5mm}
\prod_{i=1}^n \hspace*{-0.5mm} \bar{p}\left(x_{i-1}\right)\hspace*{-0.75mm}J\left(x_{i-1},x_i\right)\right|\hspace*{-0.75mm}\psi\left(x_n\right)\hspace*{-0.5mm}\mathrm{d} x_1\ldots \mathrm{d} x_n\nn\\
&\le& \int_{|x_i|\le\ve^{-{1\over 2}},\, i=1,\ldots,n}\hspace*{-0.5mm}
\left|\prod_{i=1}^n\hspace*{-0.5mm}\frac{p_{m,h}\left(x_{i-1}\right)}{\bar{p}\left(x_{i-1}\right)}-1\right|\prod_{i=1}^n {\mathsf{A}_{\bar{m}}}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots\mathrm{d} x_n.
\end{eqnarray}
Write
\begin{equation}
\frac{p_{m,h}\left(x\right)}{\bar{p}\left(x\right)}=1+\frac{\left|p_{m,h}\left(x\right)-p_{m_0,h_0}\left(x\right)\right|}{\bar{p}\left(x\right)}+\frac{\left|p_{m_0,h_0}\left(x\right)-\bar{p}\left(x\right)\right|}{\bar{p}\left(x\right)}
\end{equation}
and observe that
\begin{equation}
\left|p_{m,h}\left(x\right)-p_{m_0,h_0}\left(x\right)\right|\le 4\beta^2\tilde{c}\sqrt{\ve} \qquad \mathrm{if}\, \left|x\right|\le\ve^{-{1\over 2}},
\end{equation}
where we used the fact that $\left|\tanh^2\left(x_2\right)-\tanh^2\left(x_1\right)\right|\le 4 \tanh\left|x_2-x_1\right|$ for any real numbers $x_1$ and $x_2$. Hence,
\begin{equation}
\sup_{\left|x\right|\le\ve^{-{1\over 2}}}\frac{p_{m,h}\left(x\right)}{\bar{p}\left(x\right)}\le 1+\chi_{\beta}^{-1}\left(m_{\beta}\right)\left[\delta+4\beta^2\tilde{c}\sqrt{\ve}\right]\eqqcolon \theta.
\end{equation}
Therefore,
\begin{eqnarray}
\Delta_n^{(\psi)}\left(x_0\right)&\le&\left(\theta^n-1\right)\int_{|x_i|\le\ve^{-{1\over 2}},\, i=1,\ldots,n}
\prod_{i=1}^n {\mathsf{A}_{\bar{m}}}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots\mathrm{d} x_n\nn\\
&\le&c\hspace*{0.4mm}\mathrm{e}^{-an}\left(\theta^n-1\right)\sup_{|x_0|\le\ve^{-{1\over 2}}}\left|\psi\left(x_0\right)\right|.
\end{eqnarray}
Combining the trivial inequality $|\mathsf{A}^n_{m,h}\psi|\le |{\mathsf{A}^n_{\bar{m}}}\psi|+|({\mathsf{A}^n_{m,h}}-{\mathsf{A}^n_{\bar{m}}})\psi|$ with \eqref{est}, we get the result.
\qed
\begin{prop}\label{45}
Let $n_{\ve}\coloneqq \lfloor \ve^{-{1\over 2}} -\frac{\ve^{-{1\over 2}}}{2} \rfloor$ and $m\in\Sigma_{\delta,\ve}$, $h=\mathsf{H}\left(m\right)$. If $\ve$ is small enough, there is $0<\gamma<1$, $\gamma=\gamma\left(\delta,\ve\right)$, such that for any bounded, antisymmetric function $\psi$ on $[-\ve^{-1},\ve^{-1}]$ and any $n\le n_{\ve}$:
\begin{equation}\label{crst}
\left|\,\int \prod_{i=1}^{n}
\mathsf{A}_{m,h}\left(x_{i-1},x_i\right)\psi\left(x_{n}\right)\mathrm{d} x_1\ldots \mathrm{d} x_{n}\,\right| \le
\gamma^{n}\left\|\psi\right\|_{\ve}\qquad \forall x_0\in[-\ve^{-1},\ve^{-1}].
\end{equation}
\end{prop}
\noindent
\textbf{Proof.} If $x_0\in[0,{\ve^{-{1\over 2}}}/{2}]$, $\left|x_i\right|\le\ve^{-{1\over 2}}$ for any $i=1,\ldots,n_{\ve}$ because $J$ has range $1$, thus estimate \eqref{est1} applies. If $x_0\in[{\ve^{-{1\over 2}}}/{2},\ve^{-1}]$, we distinguish two cases; if $\left|x_i\right|<\ve^{-{1\over 2}}$, $\left|p_{m_0,h_0}\left(x_i\right)-\bar{p}\left(x_i\right)\right|\le 4\beta\tilde{c}\sqrt{\ve}$, while $\left|p_{m,h}\left(x_i\right)-p_{m_0,h_0}\left(x_i\right)\right|\le\delta$ by hypothesis. Since $\bar{p}\left(x_i\right)\le
\bar{p}\hspace*{0.4mm}(\ve^{-{1\over 2}}/2)=\chi_{\beta}\left(m_{\beta}\right)+\left(2m_{\beta}-\ve\right)\ve$, $\left|p_{m,h}\left(x_i\right)\right|\le\chi_{\beta}\left(m_{\beta}\right)+\delta+O\left(\ve\right)$, that is less than 1 for $\ve$ small enough. If $\left|x_i\right|>\ve^{-{1\over 2}}$, we claim that $\left|p_{m_0,h_0}\left(x_i\right)-\chi_{\beta}\left(m_0\left(x_i\right)\right)\right|=O\left(\ve\right)$. Indeed,
\begin{eqnarray}\label{oggi}
\left|p_{m_0,h_0}\left(x_i\right)-\chi_{\beta}\left(m_0\left(x_i\right)\right)\right|&=\beta&\hspace*{0.4mm}\left|\tanh^2\Big\{\beta\Big[\big(J\ast \hat{M}_{\mu}\big)\left(x\right)+h_0\left(x\right)\Big]\Big\} -\hat{M}^2_{\mu}\left(x\right)\right|\nn\\
&\le&4\beta^2 \left|\big(J\ast \hat{M}_{\mu}\big)\left(x\right)-\hat{M}_{\mu}\left(x\right)\right|\nn\\
&\le&4\beta^2 \max_{y:\hspace*{0.4mm}\left|y-x\right|\le 1}\left|M_{\mu}\left(y\right)- M_{\mu}\left(x\right)\right|\nn\\
&\le&4\beta^2\left\|\frac{\mathrm{d} M_{\mu}}{\mathrm{d} x}\right\|_{\ve} = \mathrm{const}\,\cdot\,\ve,
\end{eqnarray}
then $\left|p_{m,h}\left(x_i\right)\right|\le\chi_{\beta}\left(\mu\right)+\delta+O\left(\ve\right)$. Therefore, if $\ve$ is small enough there exists a constant $\theta'<1$ that bounds $p_{m,h}\left(x_i\right)$ for any $\left|x_i\right|>\ve^{-1}/2$. Define $\gamma$ as the maximum between $\theta$ and $\theta'$ to obtain \eqref{crst}.
\qed
\newline
\newline
Proposition \ref{45} induces the following
\begin{prop}\label{mj}
Let $m\in\Sigma_{\delta,\ve}$, $h=\mathsf{H}\left(m\right)$. For any bounded, antisymmetric function $\psi$ on $ [-\ve^{-1},\ve^{-1}]$:
\begin{equation}\label{9-1}
\left|\sum_{n=0}^{\infty}\int\prod_{i=0}^n\mathsf{A}_{m,h}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n\right| \le \frac{\left\|\psi\right\|_{\ve}}{1-\gamma}
\end{equation}
if $\ve$ is small enough.
\end{prop}
\noindent
\textbf{Proof.} It suffices to show that for any integer $n$, $\big\|\mathsf{A}^n_{m,h}\psi\big\|_{\ve}\le\gamma^n \left\|\psi\right\|_{\ve}$. If $n\le n_{\ve}$ this is true because of \eqref{crst}. If $n>n_{\ve}$, we write $n=k_n n_{\ve} + m_n$, with $m< n_{\ve}$ so that
\begin{eqnarray}
\big\|\mathsf{A}^n_{m,h}\psi\big\|_{\ve}&\le&\sup_{\left|x_0\right|\le\ve^{-1}}\bigg|\,
\int\prod_{i_1=1}^{n_{\ve}}\mathsf{A}_{m,h}\left(x_{i_1-1},x_{i_1}\right)\mathrm{d} x_1\ldots \mathrm{d} x_{n_{\ve}}\nn\\
&\times&\int\prod_{i_2=n_{\ve}+1}^{2n_{\ve}}\mathsf{A}_{m,h}\left(x_{i_2-1},x_{i_2}\right)\mathrm{d} x_{n_{\ve}+1}\ldots \mathrm{d} x_{2n_{\ve}}\nn\\
&\vdots&\nn\\
&\times&\int\prod_{i_k=\left(k-1\right)n_{\ve}+1}^{k n_{\ve}}\mathsf{A}_{m,h}\left(x_{i_k-1},x_{i_k}\right)
\mathrm{d} x_{\left(k-1\right)n_{\ve}+1}\ldots \mathrm{d} x_{k n_{\ve}}\nn\\
&\times&\int\prod_{j=k n_{\ve}+1}^{n}\mathsf{A}_{m,h}\left(x_{j-1},x_{j}\right)\psi\left(x_n\right)\mathrm{d} x_{k n_{\ve}+1}\ldots \mathrm{d} x_n\bigg|.
\end{eqnarray}
Call
\begin{equation}
\Psi^{(k)}\left(x_{k n_{\ve}}\right)\coloneqq\int\prod_{j=k n_{\ve}+1}^{n}\mathsf{A}_{m,h}\left(x_{j-1},x_{j}\right)\psi\left(x_n\right)\mathrm{d} x_{k n_{\ve}+1}\ldots \mathrm{d} x_n
\end{equation}
and notice that $\Psi^{(k)}$ is antisymmetric and satisfies $\|\Psi^{(k)}\|_{\ve}\le \gamma^{m_n}\left\|\psi\right\|_{\ve}$ by virtue of \eqref{crst}. Thus, for any $i<k$:
\begin{equation}
\Psi^{(i)}\left(x_{i n_{\ve}}\right)\coloneqq
\int\prod_{j=i n_{\ve}+1}^{\left(i+1\right)n_{\ve}}\mathsf{A}_{m,h}\left(x_{j-1},x_{j}\right)\Psi^{(i+1)}\left(x_n\right)\mathrm{d} x_{i n_{\ve}+1}\ldots \mathrm{d} x_{\left(i+1\right)n_{\ve}}
\end{equation}
is antisymmetric and therefore by iteration
\begin{eqnarray}
\big\|\mathsf{A}^n_{m_0,h_0}\psi\big\|_{\ve}&\le&\sup_{\left|x_0\right|\le\ve^{-1}}\bigg|\,
\int\prod_{i_1=1}^{n_{\ve}}\mathsf{A}_{m,h}\left(x_{i_1-1},x_{i_1}\right)\Psi^{(1)}\left(x_{n_{\ve}}\right)\mathrm{d} x_1\ldots \mathrm{d} x_{n_{\ve}}\nn\\
&\le& \gamma^{n_{\ve}}\|\Psi^{(1)}\|_{\ve}\le \gamma^{2n_{\ve}}\|\Psi^{(2)}\|_{\ve}\le\ldots\le\gamma^{kn_{\ve}}\|\Psi^{(k)}\|_{\ve}\le \gamma^n\left\|\psi\right\|_{\ve}.
\end{eqnarray}
Summing on $k$ we get \eqref{9-1}.
\qed
\newline
\newline
The previous bound induces the existence of the inverse of $\mathsf{I}-\mathsf{A}_{m,h}$. Explicitly
\begin{equation}
\left(\mathsf{I}-\mathsf{A}_{m,h}\right)^{-1}\coloneqq\sum_{k=0}^{\infty}\mathsf{A}^k_{m,h}, \qquad
\|(\mathsf{I}-\mathsf{A}_{m,h})^{-1}\|_{\ve}\le \frac{1}{1-\gamma}.
\end{equation}
\begin{lem}\label{fund}
Let $m\in\Sigma_{\delta,\ve}$, $h=\mathsf{H}\left(m\right)$. For any bounded, antisymmetric function $F$ on $[-\ve^{-1},\ve^{-1}]$ the equation
\begin{equation}
\vf\left(x\right)-p_{m,h}\left(x\right)\left(J\ast\vf\right)\left(x\right)=F\left(x\right)
\end{equation}
can be solved in the unknown function $\vf$. Furthermore, $\left\|\vf\right\|_{\ve}\le \left(1-\gamma\right)^{-1}\left\|F\right\|_{\ve}$.
\end{lem}
\noindent
\textbf{Proof.} It is a straightforward consequence of Proposition \ref{mj}.
\begin{comment}
The previous Proposition induces the existence of the inverse of $\mathsf{I}-\mathsf{A}_{m_0,h_0}$.
\begin{prop}\label{nd}
There exists $\delta'>0$ such that for any $\delta<\delta'$, if $\left\|p_{m,h}-p_{m_0,h_0}\right\|_{\ve}<\delta$, there is a constant $\gamma_{\delta}<1$ such that
\begin{equation}\label{9-1}
\left|\sum_{n=0}^{\infty}\int\prod_{i=0}^n\mathsf{A}_{m,h}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n\right| \le \gamma_{\delta}\left\|\psi\right\|_{\ve}
\end{equation}
for any bounded antisymmetric function $\psi$. Moreover,
\begin{equation}\label{9-2}
\sum_{n=0}^{\infty}\int\prod_{i=0}^n\mathsf{A}_{m,h}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n \equiv \left(\mathsf{I}-\mathsf{A}_{m,h}\right)^{-1}\psi\left(x_0\right).
\end{equation}
\end{prop}
\noindent
\textbf{Proof.}
It works as follows: we shall prove that \eqref{9-1} and \eqref{9-2} hold true for $m=m_0$ and $h=h_0$. This is a matter of combining properly estimates \eqref{est1} with the ``complementary'' inequality
\begin{equation}\label{est2}
\left|\,\int_{|x_i|>\ve^{-{1\over 2}},\, i=1,\ldots,n}\prod_{i=1}^n
\mathsf{A}_{m_0,h_0}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n\,\right|\le \zeta^n \hspace{-1mm}\sup_{|x_0|>\ve^{-{1\over 2}}}\hspace{-1mm}\left|\psi\left(x_0\right)\right|,
\end{equation}
where $\zeta$ is a positive constant which can be chosen equal to $p_{m_0,h_0}\,(\ve^{-1})$, that is the maximum of $p_{m_0,h_0}$ in $[\ve^{-{1\over 2}},\ve^{-1}]$ (which is less than 1). The bound over $[-\ve^{-1},\ve^{-1}]$ can be then obtained through
an integral decomposition.
\newline
Call
\begin{equation}
\mathcal{I}_n\left(x_0\right)\coloneqq\int\prod_{i=0}^n\mathsf{A}_{m,h}\left(x_{i-1},x_i\right)\psi\left(x_n\right)\mathrm{d} x_1\ldots \mathrm{d} x_n
\end{equation}
and, for the sake of notational simplicity:
\begin{eqnarray}
\Omega_{1}&\coloneqq& \bigg[-\frac{\ve^{-{1\over 2}}}{2},\frac{\ve^{-{1\over 2}}}{2}\bigg]\nn\\
\Omega_{2}&\coloneqq& \bigg[-\ve^{-{1\over 2}},-\frac{\ve^{-{1\over 2}}}{2}\bigg] \,\cup \, \bigg[\frac{\ve^{-{1\over 2}}}{2},\ve^{-{1\over 2}}\bigg] \nn\\
\Omega_{3}&\coloneqq& \bigg[-\ve^{-1},-\frac{\ve^{-{1\over 2}}}{2}\bigg] \,\cup \, \bigg[\frac{\ve^{-{1\over 2}}}{2},\ve^{-1}\bigg].
\end{eqnarray}
Suppose that $\left|x_0\right|<\ve^{-{1\over 2}}/2$ WLOG. We have
\begin{eqnarray}
&\left|\mathcal{I}_n\left(x_0\right)\right|&\le
\sum_{k=1}^n \sum_{\substack{t_1,\ldots,t_k\\ t_1+\ldots+t_k= n}}\sum_{\sigma\in S\left(k\right)}
\int_{\Omega^{t_1-1}_{\sigma(t_1)}} \mathsf{A}_{m_0,h_0}\left(x_{0},x_{1}\right)\ldots\mathsf{A}_{m_0,h_0}\left(x_{t_1-1},x_{t_1}\right)\mathrm{d} x_1\ldots \mathrm{d} x_{t_1-1}\nn\\
&\times&\int_{\Omega^{t_2}_{\sigma(t_2)}} \mathsf{A}_{m_0,h_0}\left(x_{t_1},x_{t_1+1}\right)\ldots\mathsf{A}_{m_0,h_0}\left(x_{t_2-1},x_{t_2}\right)\mathrm{d} x_{t_1}\ldots \mathrm{d} x_{t_2-1}\nn\\
\nn\\
&\vdots&\nn\\
\nn\\
&\times&
\int_{\Omega^{t_k}_{\sigma(t_k)}} \mathsf{A}_{m,h}\left(x_{t_{k-1}},x_{k_{k-1}+1}\right)\ldots\mathsf{A}_{m,h}\left(x_{t_k-1},x_{t_k}\right)\left|\psi\left(x_{t_k}\right)\right|\mathrm{d} x_{t_{k-1}}\ldots \mathrm{d} x_{t_k}\nn\\
\end{eqnarray}
where $S_{x_0}\left(k\right)$ is the set of all possible sequences $\left\{\sigma\left(1\right),\ldots,\sigma\left(k\right)\right\}$ with $\sigma\left(i\right)\in\left\{1,2,3\right\}$ and startig point $x_0$ allowed. Considering that we bound $\sup_{\frac{\ve^{-{1\over 2}}}{2}\le\left|x\right|\le\ve^{-{1\over 2}}}p_{m_0,h_0}\left(x\right)<1$ by definition of $m_0$, the worst case is when
\begin{eqnarray}
&\left|\mathcal{I}_n\left(x_0\right)\right|&\le
\end{eqnarray}
However, a small variation of $p_{m_0,h_0}$ produces a small variation of $g_\ve$ in estimate \eqref{est1} that can be controlled. We omit the details of the proof, which can be found in \cite{RB2}.
\qed
\end{comment}
\section{Newton's Method}\label{cinque}
\subsection{Small perturbations to $m_0$}
\noindent
In this section we construct $m_1$ as a series
\begin{equation}
m_1\left(x\right)=m_0\left(x\right)+\sum_{n=0}^{\infty}\vf_n\left(x\right)
\end{equation}
in which each correction $\vf_n$ depends on the previous ones
\begin{equation}
\vf_n\left(x\right)=\vf_n\left(x; m_0\left(x\right),\vf_1\left(x\right),\ldots,\vf_{n-1}\left(x\right)\right), \qquad n\ge1\\
\end{equation}
with $\vf_0\equiv 0$.
For notational convenience, we will often indicate
\begin{equation}
\phi_n\left(x\right)\coloneqq\sum_{m=1}^{n-1}\vf_m\left(x\right), \qquad n\ge 1.
\end{equation}
\begin{prop}
For any $\ve$ small enough, there exists an antisymmetric function $\vf_1\in L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right)$ that solves
\begin{equation}\label{vf1}
\begin{dcases}
\vf_1\left(x\right)-p_{m_0,h_0}\left(x\right)\left(J\ast\vf_1\right)\left(x\right)=\tanh\Big\{\beta\Big[\left(J\ast m_0\right)\left(x\right)+h_0\left(x\right)\Big]\Big\} -m_0\left(x\right)\\
\vf_1\hspace*{0.4mm}(-\ve^{-1})=- t_1, \qquad \vf_1\hspace*{0.4mm}(\ve^{-1})=t_1
\end{dcases}
\end{equation}
for some $t_1\in\mathbb{R}$. Moreover, there is $c_0>0$ such that $\left\|\vf_1\right\|_{\ve}\le c_0\ve$.
\end{prop}
\noindent
\textbf{Proof.} Call
\begin{equation}
S_0\left(x\right)\coloneqq \tanh\Big\{\beta\Big[\left(J\ast m_0\right)\left(x\right)+h_0\left(x\right)\Big]\Big\} -m_0\left(x\right).
\end{equation}
We restrict to the positive semiline. We have, for any $0\le x\le\ve^{-{1\over 2}}-1$:
\begin{equation}
\left|S_0\left(x\right)\right|=\left|\tanh\Big\{\beta\Big[\left(J\ast \bar{m}\right)\left(x\right)+h_0\left(x\right)\Big]\Big\} -\bar{m}\left(x\right)\right|\le\beta j\ve \int_0^x \frac{\mathrm{d} y}{\chi_{\beta}\left(\bar{m}\left(y\right)\right)}\le\mathrm{const}\,\cdot\,\sqrt{\ve}.
\end{equation}
In the interval $\ve^{-{1\over 2}}+1\le x \le\ve^{-1}$, similarly to estimate \eqref{oggi}:
\begin{eqnarray}
\left|S_0\left(x\right)\right|&=&\left|\tanh\Big\{\beta\Big[\big(J\ast \hat{M}_{\mu}\big)\left(x\right)+h_0\left(x\right)\Big]\Big\} -\hat{M}_{\mu}\left(x\right)\right|\nn\\
&=&\left|\tanh\Big\{\beta\Big[\big(J\ast \hat{M}_{\mu}\big)\left(x\right)+h_0\left(x\right)\Big]\Big\} -
\tanh\Big\{\beta\Big[\hat{M}_{\mu}\left(x\right)+h_0\left(x\right)\Big]\Big\}
\right|\nn\\
&\le&\beta \left|\big(J\ast \hat{M}_{\mu}\big)\left(x\right)-\hat{M}_{\mu}\left(x\right)\right|\nn\\
&\le&\beta \max_{y:\hspace*{0.4mm}\left|y-x\right|\le 1}\left|M_{\mu}\left(y\right)- M_{\mu}\left(x\right)\right|\nn\\
&\le&\beta\left\|\frac{\mathrm{d} M_{\mu}}{\mathrm{d} x}\right\|_{\ve} = \mathrm{const}\,\cdot\,\ve
\end{eqnarray}
where we used Lagrange's Theorem. The remaining case is when $\ve^{-{1\over 2}}-1\le x\le\ve^{-{1\over 2}}$. We consider at first the sub-case $\ve^{-{1\over 2}}-1\le x\le\ve^{-{1\over 2}}$. We get
\begin{eqnarray}
\left|S_0\left(x\right)\right|&=&\Bigg|
\tanh\Big\{\beta\Big[\int_{x-1}^{\ve^{-{1\over 2}}}J\left(x,y\right)\bar{m}\left(y\right)\mathrm{d} y
+\int_{\ve^{-{1\over 2}}}^{x+1}J\left(x,y\right)\hat{M}_{\mu}\left(y\right)\mathrm{d} y+h_0\left(x\right)\Big]\Big\}
-\bar{m}\left(x\right)\Bigg|\nn\\
&\le&\beta\left| \int_{\ve^{-{1\over 2}}}^{x+1}J\left(x,y\right)\left(\hat{M}_{\mu}\left(y\right)-\bar{m}\left(y\right)\right)\mathrm{d} y +h_0\left(x\right)\right|\nn\\
&\le& \beta\left|\hat{M}_{\mu}\left(x+1\right)-\bar{m}\left(x+1\right)\right| + \mathrm{const}\,\cdot\,\sqrt{\ve}
\end{eqnarray}
because $\hat{M}_{\mu}$ is non increasing while $\bar{m}$ is increasing. Since by definition $\hat{M}_{\mu}\,(\ve^{-{1\over 2}})=\bar{m}\,(\ve^{-{1\over 2}})$:
\begin{equation}
\left|\hat{M}_{\mu}\left(x+1\right)-\bar{m}\left(x+1\right)\right|\le \left|\hat{M}_{\mu}\left(x+1\right)-\hat{M}_{\mu}\,(\ve^{-{1\over 2}})\right|+\left|\bar{m}\left(x+1\right)-\bar{m}\,(\ve^{-{1\over 2}})\right|=O\left(\ve\right)
\end{equation}
where we bounded the first term with the sup norm of the derivative of $\hat{M}_{\mu}$, while the second term is exponentially small in $\ve^{-{1\over 2}}$. At last, if $\ve^{-{1\over 2}} \le x\le\ve^{-{1\over 2}}$:
\begin{eqnarray}
\left|S_0\left(x\right)\right|&=&\Bigg|
\tanh\Big\{\beta\Big[\int_{x-1}^{\ve^{-{1\over 2}}}J\left(x,y\right)\bar{m}\left(y\right)\mathrm{d} y
+\int_{\ve^{-{1\over 2}}}^{x+1}J\left(x,y\right)\hat{M}_{\mu}\left(y\right)\mathrm{d} y+h_0\left(x\right)\Big]\Big\}
-\hat{M}_{\mu}\left(x\right)\Bigg|\nn\\
&\le&\beta\left|
\int_{x-1}^{\ve^{-{1\over 2}}}J\left(x,y\right)\left[\bar{m}\left(y\right)-\hat{M}_{\mu}\left(y\right)\right]\mathrm{d} y
+\int_{x-1}^{x+1}J\left(x,y\right)\hat{M}_{\mu}\left(y\right)\mathrm{d} y -\hat{M}_{\mu}\left(x\right)
\right|\nn\\
&\le&\beta\left|\bar{m}\left(x-1\right)-\hat{M}_{\mu}\left(x-1\right)\right|=O\left(\ve\right)
\end{eqnarray}
by virtue of the previous estimates. This proves that $\left\|S_0\right\|_{\ve}=O\left(\ve\right)$. Indeed, the existence of $\vf_1$ follows from the invertibility of $\mathsf{I}-\mathsf{A}_{m_0,h_0}$, and explicitly:
\begin{equation}
\vf_1\left(x\right)= \left(\mathsf{I}-\mathsf{A}_{m_0,h_0}\right)^{-1}S_0\left(x\right).
\end{equation}
By Lemma \ref{fund}, since $S\left(x\right)$ is antisymmetric, we get $\left\|\vf_1\right\|_{\ve}\le\left(1-\gamma\right)^{-1}\left\|S_0\right\|_{\ve}$.
\qed
\begin{prop}\label{5s}
For any $\ve$ small enough, we have
\begin{itemize}
\item[$\mathrm{(i)}$] for any $n\ge 2$ there exists an antisymmetric, bounded function $\vf_{n}\in L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right)$ that solves
\begin{equation}
\begin{dcases}
\vf_n\left(x\right)-p_{m_0+\phi_{n},h_0}\left(x\right)\left(J\ast\vf_n\right)\left(x\right)\hspace{-3mm}&=\tanh\Big\{\beta\Big[\left(J\ast m_0+\phi_n\right)\left(x\right)+h_0\left(x\right)\Big]\Big\}\nn\\
&-m_0\left(x\right)-\phi_n\left(x\right)\\
\vf_n\hspace*{0.4mm}(-\ve^{-1})=-t_n, \qquad \vf_n\hspace*{0.4mm}(\ve^{-1})=t_n
\end{dcases}
\end{equation}
for some $t_n\in\mathbb{R}$;
\item[$\mathrm{(ii)}$] there is a constant $\tau$ such that $\left\|\vf_n\right\|_{\ve}\le \tau \left\|\vf_{n-1}\right\|_{\ve}^2$ for any $n\ge 2$;
\item[$\mathrm{(iii)}$] $\lim_{n\to\infty}\left\|m_1-m_0-\phi_n\right\|_{\ve}=0$,
where $m_1$ is a solution of \eqref{mn} at $n=1$;
\item[$\mathrm{(iv)}$] $\left\|m_1-m_0\right\|_{\ve}=O\left(\ve\right)$.
\end{itemize}
\end{prop}
\noindent
\textbf{Proof.} It works by induction. In particular, suppose that (i) and (ii) hold for any integer less or equal to a certain $k$. Since $\left\|\vf_1\right\|_{\ve}\le c_0\ve$, iterating (ii) we get
\begin{equation}
\left\|\vf_k\right\|_{\ve}\le \tau^{2^{k-1}-1}\left(c_0\ve\right)^{2^{k-1}}
\end{equation}
and then
\begin{equation}\label{bd}
\left\|\phi_k\right\|_{\ve}\le 2c_0\ve
\end{equation}
provided $\ve\le\left(2c_0\tau\right)^{-2}$. Moreover if $\ve$ is so small that $m_0+\phi_k\in\Sigma_{\delta,\ve}$, and then $\mathsf{A}_{m_0+\phi_k,h_0}$ is invertible, and $\vf_{k+1}$ exists. We prove (ii); expand the hyperbolic tangent in Taylor series:
\begin{eqnarray}\label{expan}
\tanh\Big\{\beta\Big[\left(J\ast m_0+\phi_k\right)\left(x\right)+h_0\left(x\right)\Big]\Big\}&=&
\tanh\Big\{\beta\Big[\left(J\ast m_0+\phi_{k-1}\right)\left(x\right)+h_0\left(x\right)\Big]\Big\}\nn\\
&+&p_{m_0+\phi_{k-1},h_0}\left(x\right)\left(J\ast\vf_{k-1}\right)\left(x\right)\nn\\
&+&p'_{m_0+\phi_{k-1},h_0}\left(x\right)\left(J\ast\vf_{k-1}\right)^2\left(x\right)+\ldots .
\end{eqnarray}
Combining \eqref{expan} with the definition of $\vf_n$ we get
\begin{equation}
\vf_k\left(x\right)=\left(\mathsf{I}-\mathsf{A}_{m_0,\phi_{k-1},h_0}\right)^{-1}\left(
p'_{m_0,\phi_{k-1},h_0}\left(x\right)\left(J\ast\vf_{k-1}\right)\left(x\right)+\ldots\right),
\end{equation}
hence
\begin{equation}
\left\|\vf_k\right\|_{\ve}\le \left(1-\gamma\right)^{-1}\sup_{0\le x'\le \ve^{-{1\over 2}}}p'_{m_0,\phi_{k-1},h_0}(x')\sup_{0\le x''\le \ve^{-{1\over 2}}}\vf_{k-1}^2(x'')\le \frac{\beta}{1-\gamma}\left\|\vf_{k-1}\right\|^2_{\ve},
\end{equation}
so we can identify $\tau\equiv\beta/1-\gamma$. This proves that for any integer $n$:
\begin{equation}
m_0\left(x\right)+\phi_{n}\left(x\right)=\tanh\Big\{\beta\Big[\left(J\ast m_0+\phi_n\right)\left(x\right)+h_0\left(x\right)\Big]\Big\}+ O\left(\vf_n\left(x\right)\right).
\end{equation}
Taking the limit $n\to\infty$, by continuity of the hyperbolic tangent, we get the uniform convergence to $m_1$. (iv) follows from \eqref{bd}.
\qed
\subsection{Small perturbations to $h_0$}
\begin{prop}\label{55}
There is $\delta'>0$ such that for any $h\in C\left([-\ve^{-1},\ve^{-1}]\right)$, if $\left\|h-h_0\right\|_{\ve}\le\delta'$:
\begin{itemize}
\item[$\mathrm{(i)}$] there exists a continuous, antisymmetric function $m$ solution of
\begin{equation}
m\left(x\right)= \tanh\Big\{\beta\Big[\left(J\ast m\right)\left(x\right)+h\left(x\right)\Big]\Big\};
\end{equation}
\item[$\mathrm{(ii)}$] $\left\|m-m_0\right\|_{\ve}=O\left(\delta'\right)$.
\end{itemize}
\end{prop}
\noindent
\textbf{Proof.} We construct $m$ using again Newton's method with starting point $\left(m_0,h\right)$. Observe that
\begin{equation}
\left\|p_{m_0,h}-p_{m_0,h_0}\right\|_{\ve}\le 4\beta\tanh\Big\{\beta \left\|h-h_0\right\|_{\ve}\Big\}\le 4\beta^2\delta';
\end{equation}
hence, if $\delta'<\delta/4\beta^2$ we are in the hypothesis of Lemma \ref{fund}, so we conclude that there exists an antisymmetric function $\psi_1$ that solves
\begin{equation}\label{vf1}
\begin{dcases}
\psi_1\left(x\right)-p_{m_0,h}\left(x\right)\left(J\ast\psi_1\right)\left(x\right)=\tanh\Big\{\beta\Big[\left(J\ast m_0\right)\left(x\right)+h\left(x\right)\Big]\Big\} -m_0\left(x\right)\\
\psi_1\hspace*{0.4mm}(-\ve^{-1})=-s_1, \qquad \psi_1\hspace*{0.4mm}(\ve^{-1})=s_1
\end{dcases}
\end{equation}
for some $s_1\in\mathbb{R}$, and moreover $\left\|\psi_1\right\|_{\ve}\le \tau\left\|h-h_0\right\|_{\ve}$. The rest of the proof is the same as that of Proposition \ref{5s}, provided $\delta'\le\delta/\left(2\tau+1\right)$, which is the condition needed in order to apply Lemma \ref{fund} recursively.
\qed
\subsection{Further corrections}
\noindent
As we shall see, it is worth emphasizing scaling properties of the magnetization profile by introducing the following weighted norm at fixed $\alpha>0$:
\begin{equation}\label{alpha}
\left\|m\right\|_{\ve,\alpha}\coloneqq \sup_{\left|x\right|\le \ve^{-1}}\mathrm{e}^{-\alpha\ve \left|x\right|}\left|m\left(x\right)\right|, \qquad
m\in L^{\infty}\left([-\ve^{-1},\ve^{-1}]\right).
\end{equation}
Notice that convergence in the $\alpha$-norm implies uniform convergence as the inclusion $\left\|\,\cdot\,\right\|_{\ve,\alpha}\le\left\|\,\cdot\,\right\|_{\ve}\le\mathrm{e}^{\alpha}\left\|\,\cdot\,\right\|_{\ve,\alpha}$ holds. The iterability of our method directly follows from the fact that $\left(m_n,h_n\right)\mapsto \left(m_{n+1},h_{n+1}\right)$ is a contraction in the $\alpha$-norm (for a feasible choice of parameters).
\begin{lem}\label{sera}
Let $h^{(i)}$, $i=1,2$ such that $\|h^{(i)}-h_0\|_{\ve}\le \delta'$, and $m^{(i)}$ the corresponding profiles constructed via Newton's method that solve
\begin{equation}
m^{(i)}\left(x\right)= \tanh\Big\{\beta\Big[\big(J\ast m^{(i)}\big)\left(x\right)+h^{(i)}\left(x\right)\Big]\Big\}.
\end{equation}
Then,
\begin{equation}
\|m^{(2)}-m^{(1)}\|_{\ve,\alpha}\le \tau \|h^{(2)}-h^{(1)}\|_{\ve,\alpha}
\end{equation}
for $\ve$ small enough.
\end{lem}
\noindent
\textbf{Proof.} $m^{(1)}$ and $m^{(2)}$ exist by Proposition \ref{55}. By Taylor's Theorem
\begin{equation}\label{6s}
m^{(2)}\left(x\right)= \tanh\Big\{\beta\Big[\big(J\ast m^{(1)}\big)\left(x\right)+h^{(1)}\left(x\right)\Big]\Big\}+R_{1,2}\left(x\right),
\end{equation}
where
\begin{equation}
R_{1,2}\left(x\right)=p_{1,2}\left(x\right)\left[\big(J\ast \big(m^{(2)}-m^{(1)}\big)\big)\left(x\right)+h^{(2)}\left(x\right)-h^{(1)}\left(x\right)\right]
\end{equation}
$p_{1,2}$ being some interpolating function between $p_{m^{(2)},h^{(2)}}$ and $p_{m^{(1)},h^{(1)}}$. We multiply by $\mathrm{e}^{-\alpha\ve \left|x\right|}$ equation \eqref{6s} and take absolute values to get
\begin{equation}\label{eqineq}
\mathrm{e}^{-\alpha\ve \left|x\right|}\big|m^{(2)}\left(x\right)-m^{(1)}\left(x\right)\big|-p_{1,2}\left(x\right)\mathrm{e}^{-\alpha\ve}\big(J\ast \big|m^{(2)}-m^{(1)}\big|\big)\left(x\right)\nn\\
\le p_{1,2}\left(x\right)\mathrm{e}^{-\alpha\ve \left|x\right|}\big|h^{(2)}\left(x\right)-h^{(1)}\left(x\right)\big|.
\end{equation}
\eqref{eqineq} as an equality can be solved in the unknown function $\mathrm{e}^{-\alpha\ve \left|x\right|}|m^{(2)}\left(x\right)-m^{(1)}\left(x\right)|$ provided $\|p_{1,2}\,\mathrm{e}^{\alpha\ve}-p_{m_0,h_0}\|_{\ve}\le\delta$. In this case, i.e. if $\ve$ is small enough:
\begin{equation}
\|m^{(2)}-m^{(1)}\|_{\ve,\alpha}\le\frac{\left\|p_{1,2}\right\|_{\ve}}{1-\gamma}\|h^{(2)}-h^{(1)}\|_{\ve,\alpha}\le \tau\|h^{(2)}-h^{(1)}\|_{\ve,\alpha}.
\end{equation}
\qed
\subsection{Convergence to $\left(m,h\right)$}
\noindent
We now prove Proposition \ref{p3}.
\newline
\newline
\textbf{Proof.} Suppose that for any $k<n$, $n$ a fixed integer, the following hypothesis hold true:
\begin{itemize}
\item[(H1)] there is a continuous, antisymmetric function $m_k$ which solves
\begin{equation}
m_k\left(x\right)= \tanh\Big\{\beta\Big[\left(J\ast m_k\right)\left(x\right)+h_{k-1}\left(x\right)\Big]\Big\}
\end{equation}
with boundary conditions $m_k\hspace*{0.4mm}(-\ve^{-1})=-mu_k$, $m_k\hspace*{0.4mm}(\ve^{-1})=mu_k$, $\mu_k\in(m^{*}\left(\beta\right),m_{\beta})$, where $h_{k}=\mathsf{H}\left(m_k\right)$.
\item[(H2)] there is a constant $\rho\in\left(0,1\right)$ independent of $k$ such that
\begin{equation}
\left\|h_k-h_{k-1}\right\|_{\ve,\alpha}\le\rho \left\|h_{k-1}-h_{k-2}\right\|_{\ve,\alpha}, \qquad 2\le k<n.
\end{equation}
\end{itemize}
Notice that (H1) and (H2) imply:
\begin{eqnarray}
&&\left\|h_k-h_{k-1}\right\|_{\ve,\alpha}\le \frac{\rho}{1-\rho} \left\|h_1-h_{0}\right\|_{\ve,\alpha}\\
\nn\\
&&\left\|h_k-h_{0}\right\|_{\ve,\alpha}\le \frac{\rho}{1-\rho} \left\|h_1-h_{0}\right\|_{\ve,\alpha}. \label{h4}
\end{eqnarray}
According to \eqref{h4}, $\alpha$ and $\ve$ can be suitably tuned in order to apply Lemma \ref{sera}; in particular, this is true uniformly in $k$ provided $\left\|h_1-h_0\right\|_{\ve}\le \left(1-\rho\right)/\mathrm{e}^{\alpha}\rho$. In this hypothesis there exists $m_n$ solution of
\begin{equation}
m_n\left(x\right)= \tanh\Big\{\beta\Big[\left(J\ast m_n\right)\left(x\right)+h_{n-1}\left(x\right)\Big]\Big\}
\end{equation}
satisfying a certain boundary condition. It remains to prove that (H2) holds for $k=n$. We have, for any $x\in[-\ve^{-1},\ve^{-1}]$:
\begin{eqnarray}
\left|h_n\left(x\right)-h_{n-1}\left(x\right)\right|&\le&2\beta\tilde{c}\,\ve
\int_0^x \mathrm{e}^{-\alpha\ve y} \left|m_n\left(x\right)-m_{n-1}\left(x\right)\right|\mathrm{e}^{\alpha\ve y}\,\mathrm{d} y\nn\\
&\le&2\beta\tilde{c}\,\ve\left\|m_n-m_{n-1}\right\|_{\ve,\alpha}
\int_0^x \mathrm{e}^{\alpha\ve y} \mathrm{d} y\nn\\
&\le&\frac{2\beta\tilde{c}}{\alpha}\left\|m_n-m_{n-1}\right\|_{\ve,\alpha}.
\end{eqnarray}
Hence, in the $\alpha$-norm:
\begin{equation}
\left\|h_n-h_{n-1}\right\|_{\ve,\alpha}\le \frac{2\beta\tilde{c}}{\alpha}\left\|m_n-m_{n-1}\right\|_{\ve,\alpha}\le\frac{2\beta\tilde{c}\,\tau }{\alpha}\left\|h_{n-1}-h_{n-2}\right\|_{\ve,\alpha}.
\end{equation}
If $\rho\le 2\beta\tilde{c}\,\tau/\alpha$ and $\ve$ is accordingly small, $\left(m_n,h_n\right)\mapsto\left(m_{n+1},h_{n+1}\right)$ is a contraction in the $\alpha$-norm. This implies uniform convergence to a solution of \eqref{pbmh}.
\qed
\section{Invertibility of the method}\label{sei}
\noindent
We indicate in this section $m_0\left(\,\cdot\,,j\right)$ as the starting magnetization profile carrying current $j$. This also fix the boundary value $m_0\hspace*{0.4mm}(\ve^{-1})=\mu_0$. We use the same notation for the corresponding auxiliary magnetic field.
\subsection{Lipschitz continuity in $j$}
\begin{prop}
$m_0$ and $h_0$ are Lipschitz continuous in the current $j$.
\end{prop}
\noindent
\textbf{Proof.} We differentiate $m_0$ and $h_0$ with respect to $j$:
\begin{eqnarray}
\frac{\partial m_0}{\partial j}\left(x,j\right)&=&
\begin{dcases}
0 &0\le x\le\ve^{-{1\over 2}} \\
-\frac{1}{1-\chi_{\beta}\left(m_0\left(x,j\right)\right)}\frac{\ve x-\sqrt{\ve}}{1-\sqrt{\ve}} &\ve^{-{1\over 2}}\le x\le\ve^{-1},
\end{dcases}
\\ \nn \\ \nn \\
\frac{\partial h_0}{\partial j}\left(x,j\right)&=&
\begin{dcases}
0 &0\le x \le\ve^{-{1\over 2}} \\
-\frac{1}{\chi_{\beta}\left(m_0\left(x,j\right)\right)}\frac{\ve x-\sqrt{\ve}}{1-\sqrt{\ve}} &\ve^{-{1\over 2}} \le x\le\ve^{-1}
\end{dcases}
\end{eqnarray}
and similarly for $x<0$. Then, for any $j_1$ and $j_2$, using Lagrange's Theorem:
\begin{eqnarray}
&&\left\|m_0\left(\,\cdot\,,j_2\right)-m_0\left(\,\cdot\,,j_1\right)\right\|_{\ve}\le \left(1-\chi_{\beta}\left(m^{*}\left(\beta\right)\right)\right)^{-1}\left|j_2-j_1\right|\\
\nn \\
&&\left\|h_0\left(\,\cdot\,,j_2\right)-h_0\left(\,\cdot\,,j_1\right)\right\|_{\ve}\le \chi_{\beta}^{-1}\left(m_{\beta}\right)\left|j_2-j_1\right|.
\end{eqnarray}
\qed
\begin{prop}
The sequence $\left(m_n\left(\,\cdot\,,j\right)\right)_{n=0}^{\infty}$ is Lipschitz continuous in $j$. The limit profile $m\left(\,\cdot\,,j\right)$ is Lipschitz in $j$ as well.
\end{prop}
\noindent
\textbf{Proof.} Suppose that
\begin{itemize}
\item[(H)] There is $\rho'\in\left(0,1\right)$ such that, for any $k<n$, $n$ fixed:
\begin{equation}
\left\|h_k\left(\,\cdot\,,j_2\right)-h_k\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}\le \rho'
\left\|h_{k-1}\left(\,\cdot\,,j_2\right)-h_{k-1}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}.
\end{equation}
\end{itemize}
It can be proved (in a way similar at all to that used to prove Lemma \ref{sera}) that (H) implies
\begin{equation}\label{7set}
\left\|m_{k+1}\left(\,\cdot\,,j_2\right)-m_{k+1}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}\le \rho'
\left\|h_k\left(\,\cdot\,,j_2\right)-h_k\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}
\end{equation}
and therefore, there exists a constant $L>0$ such that for any $x\in[-\ve^{-1},\ve^{-1}]$:
\begin{equation}
\left|h_{k+1}\left(x,j_2\right)-h_{k+1}\left(x,j_1\right)\right|\le\frac{L\mathrm{e}^{\alpha\ve \left|x\right|}}{\alpha}
\left\|m_{k+1}\left(\,\cdot\,,j_2\right)-m_{k+1}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}.
\end{equation}
Multiplying both members by $\mathrm{e}^{-\alpha\ve \left|x\right|}$ and taking the supremum with respect to $x$ we get the inequality in the $\alpha$-norm:
\begin{eqnarray}\label{rec}
\left\|h_{k+1}\left(\,\cdot\,,j_2\right)-h_{k+1}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}&\le& \frac{L}{\alpha}\left\|m_{k+1}\left(\,\cdot\,,j_2\right)-m_{k+1}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}\nn\\
&\le&\frac{L\rho'}{\alpha}\left\|h_{k+1}\left(\,\cdot\,,j_2\right)-h_{k+1}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}.
\end{eqnarray}
Choose $\alpha>L\rho'$ and let $\rho'\equiv L\rho'/\alpha<1$; then, (H) also holds for $n+1$. Moreover we get, applying recursively \eqref{rec}:
\begin{eqnarray}
\left\|h_{k}\left(\,\cdot\,,j_2\right)-h_{k}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}&\le&\frac{\rho'}{1-\rho'}\left\|h_{0}\left(\,\cdot\,,j_2\right)-h_{0}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}\nn\\
&\le& \frac{\rho'}{\left(1-\rho'\right)\chi_{\beta}\left(m_{\beta}\right)}\left|j_2-j_1\right|
\end{eqnarray}
and then, by \eqref{7set}:
\begin{equation}
\left\|m_{k}\left(\,\cdot\,,j_2\right)-m_{k}\left(\,\cdot\,,j_1\right)\right\|_{\ve,\alpha}\le
\frac{\rho'}{\left(1-\rho'\right)\chi_{\beta}\left(m_{\beta}\right)}\left|j_2-j_1\right|.
\end{equation}
The fact that the function $m$ that is obtained as limit of Newton's method is Lipschitz is a consequence of the uniform continuity of the sequence.
\qed
\subsection{Proof of Theorem \ref{teo}}
\noindent
What remains to prove is the surjectivity of $j$ as a function of $\mu_0$.
\begin{prop}
At fixed $\mu_0\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$, let $\mu =m\,(\ve^{-1},j\left(\mu_0\right))$ the right boundary condition satisfied by the limit profile of the sequence starting with $m_0\left(\,\cdot\,,j\left(\mu_0\right)\right)$. For any $\mu_0\in\left(m^{*}\left(\beta\right),m_{\beta}\right)$ there are $j^-\left(\mu_0\right)$ and $j^+\left(\mu_0\right)$ such that
\begin{equation}
m\,(\ve^{-1},j^-\left(\mu_0\right))=\mu^-, \qquad m\,(\ve^{-1},j^+\left(\mu_0\right))=\mu^+
\end{equation}
with $\mu^-<\mu<\mu^+$.
\end{prop}
\noindent
\textbf{Proof.} This result closes the proof of theorem \ref{teo} as it provides sufficient conditions in order to apply the intermediate value theorem. Indeed, set
\begin{equation}
j^{\pm}_{\eta}\left(\mu_0\right)\coloneqq j\left(\mu_0\right)\mp\left[\left(\chi_{\beta}\left(\mu_0\right)-1\right)\eta + \beta\mu_0\eta^2 -\frac{\beta}{3}\eta^3\right],
\end{equation}
which are the currents corresponding to the starting magnetization value $\mu_0\pm\eta$ (notice that $j^{-}_{\eta}\left(\mu_0\right)>j^{+}_{\eta}\left(\mu_0\right)$). We have proved that $\left|\mu-\mu_0\right|=O\left(\ve\right)$, thus we can choose $\ve$ so small (depending on $\eta$ fixed) such that $\mu^-$ is smaller than $\mu$ while $\mu^+$ is larger than $\mu$. Since $m\left(\,\cdot\,,j\right)$ is continuous in $j$, the intermediate value theorem can be applied. Then, $m\,(\ve^{-1},j)$ takes any value between $m\,(\ve^{-1},j_{\eta}^+\left(\mu_0\right))$ and $m\,(\ve^{-1},j_{\eta}^-\left(\mu_0\right))$.
\qed
\section*{Acknowledgements}
\noindent
We thank Anna De Masi, Errico Presutti and Dimitrios Tsagkarogiannis for enlightening discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,190 |
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/the-atomic-masses-three-elements-x-y-z-having-similar-chemical-properties-are-7-23-39-respectively-a-calculate-average-atomic-mass-elements-x-z-b-how-does-average-atomic-mass-elements-x-z-compare-dobereiner-s-triads_29262","text":"# The Atomic Masses of Three Elements X, Y and Z Having Similar Chemical Properties Are 7, 23 and 39 Respectively. (A) Calculate the Average Atomic Mass of Elements X and Z. (B) How Does the Average Atomic Mass of Elements X and Z Compare with the Atomic Mass of Element Y? (C) Which Law of Classification of Elements is Illustrated by this Example? (D) What Could the Elements X, Y and Z Be? (E) Give Another Example of a Set of Elements Which Can Be Classified According to this Law. - Science\n\nThe atomic masses of three elements X, Y and Z having similar chemical properties are 7, 23 and 39 respectively.\n\n(a) Calculate the average atomic mass of elements X and Z.\n(b) How does the average atomic mass of elements X and Z compare with the atomic mass of element Y?\n(c) Which law of classification of elements is illustrated by this example?\n(d) What could the elements X, Y and Z be?\n(e) Give another example of a set of elements which can be classified according to this law.\n\n#### Solution\n\n(a) The atomic masses of three elements X, Y and Z are 7, 23 and 39 respectively. The average atomic mass of X and Z is nothing but the arithmetic mean of the atomic masses of X and Z.\n\nTherefore, the average atomic mass of \u00a0X and Z =(7+39)\/2=23\n\n(b) From the above calculation, we observe that the average atomic mass of the elements X and Z is equal to the atomic mass of the element Y.\n\n(c) Dobereiner's law of triads is illustrated by\u00a0the above\u00a0example.\nAccording to Dobereiner's law, when\u00a0elements are arranged in an increasing order of atomic masses, a three element group (triads), having the same chemical properties is formed. The atomic mass of the middle element is equal to the average of the atomic masses of the other two elements.\n\n(d) The elements X,Y and Z\u00a0, having atomic masses 7, 23, 39 are lithium (Li), sodium (Na) and potassium (K). They belong to the alkali metal group\u00a0(group 1),\u00a0with the same valency of 1 and\u00a0exhibit similar chemical properties.\n\n(e) The elements calcium (Ca), strontium (Sr) and barium (Ba) are a set of alkali earth metals (group 2 elements), with atomic masses 40,88 and 137 respectively, which are classified according to Dobereiner's law of triads.\n\nIs there an error in this question or solution?\n\n#### APPEARS IN\n\nLakhmir Singh Class 10 Chemistry (Science)\nChapter 5 Periodic Classification Of Elements\nQ 42 | Page 284","date":"2021-03-08 04:05:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5106702446937561, \"perplexity\": 639.381554420628}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178381803.98\/warc\/CC-MAIN-20210308021603-20210308051603-00344.warc.gz\"}"} | null | null |
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Stand-in Captain Rohit Sharma scored 208 runs remaining unbeaten for the entire inning.
After the appalling defeat at the first ODI, India batsmen did come back with a massive attack in the second venue at Mohali. In the match, sent in to bat first, India blitz a score of 392 for 4, with Sri Lanka bowlers remaining largely futile in their attacks. Stand in captain Rohit Sharma alone bombarded 208 runs as the opener in the match. It was the third double century knock of the batsman in the format. His impressive knock was unbeaten throughout the inning.
Meanwhile, bowlers Suranga Lakmal, spearhead of the first match, as well as Nuwan Pradeep, were plundered for massive boundaries. The opening partnership between Rohit Sharma and Shikhar Dhawan hauled India's score to 115 runs. Sri Lanka could not made a breakthrough in the inning in the first 21 overs. It was only Sacthith Pathirana who removed Dhawan in the first ball of the 22nd over. Dhawan was batting at 68 off 67 balls, hammering a half-century. India's run rate was moderate until half the inning. In the first 10 overs, the team scored only 33 runs.
On the other hand, Sri Lanka bowlers maintained a perfect length in their attack in the stretch. When the visitors introduced the new bowler into the attack, India openers got the pace back adding some momentum to the score. The players began their knocks with boundaries. The next stretch of 10 overs had impressive boundaries of both Dhawan and Sharma. The latter hammered 9 boundaries for his score.
Rohit Sharma remained unbeaten in the inning batting for 50 overs full. He played 153 balls at the strike rate of 135.94 to score 208 runs. After Dhawan, the player forged a partnership of 213 runs with Shreyas. The players were steady in the inning until 45.3 overs when Thisara Perera removed Iyer. The young batsman scored 88 off 70 balls in the inning.
MS Dhoni, the only salvager of first ODI, came in for a short stretch to bat. He was trapped by Perera 12 balls Iyer. It was the only consecutive wicket removed in a short gap. Sri Lanka bowlers were able to dismiss another wicket of Hardik Pandya in the last ball of the inning. But, by them, India's scored soared up to 392. | {
"redpajama_set_name": "RedPajamaC4"
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In my last post (The Word of the Day is "Transition"), I shared our magical cure for the rough mornings. We added a few minutes to the transition time and it was enough to reduce the separation anxiety in the morning. It worked great on Monday through Thursday…and then there was Friday.
As I was writing that post, I was concerned that those very words might come back to bite me. Boy did they!
Friday morning went so poorly, that he was sent home right away for his behavior. I won't go into the gory details because it was even too much for me to repeat. Let's just say it was not his best day. Once home, there was lots of studying and very little fun. There was an apology note written, privileges taken away, and a lecture that was repeated probably more times than necessary. Even the "Friday Family Movie Night" selection had a theme of boy-making-bad-decisions-learns-to-make-good-ones.
Unfortunately, his actions were bad…but not completely a surprise.
The night before we had a reoccurrence of Operation Eyelashes! My TBP's long eyelashes were clumped under his eyelid causing great discomfort and anxiety. Refusing to let anyone touch them or help him remove the lashes, this episode went on for HOURS! His eye was very irritated (actually we were ALL feeling pretty irritated) when he finally fell asleep at 10:oo pm. After he was asleep, we were able to flip the lashes out in a matter of seconds. Because his anxiety and emotions were on overdrive, he woke up in the middle of the night worried about his eye. When it was morning and time to get up for school, he was very tired and grumpy. The morning was rushed and chaotic which unfortunately set him up for a rough beginning. Now I'm not saying that this completely makes up for his actions, but it does help explain the change in attitude.
I have decided that I'm not going to gloat about new parenting strategies that are working. Just when I think that we can cross a stressful situation off the list, it reappears like a bad case of strep! What I can celebrate is talking about trends. At this point, we have more success with classroom and recess behavior than ever before! I can also say that when there is an outburst at home, it tends to be much shorter and easier to resolve. Again, this is a move in the right direction!
Because Mine Craft had been taken away as a punishment, there was lots of time to talk and do other activities. As we drew together, we talked about the consequences of our behaviors and the need for heartfelt apologies. Today, my son spent part of his tooth fairy money on a gourmet chocolate bar to go with the apology note. I would like to say that it was his idea and that he felt good about the purchase…but he did not. He didn't want to spend his "hard-earned money that I was saving to buy something for me" but once he did, he was able to explain why it was necessary.
I won't pretend that this is the last of the poor choices…I know that it isn't. What I hope is that these "learning opportunities" come frequent enough when he is little, that he learns to be less impulsive and more responsible in situations that are not dangerous and/or life changing.
What am I learning? That he makes mistakes! We all make mistakes! The important part is what we do to fix our mistakes and to learn from them. How do you help your child learn from his/her mistakes?
Let me know when you figure it out!!! One thing that works in a situation such as yours is telling him that, "Other parents will not like you being an example to their child." He really prides himself on being a good example. We say sometimes kids look at the anyone older as a way they should act. Although not always, usually in school he "over" follows the rules. Such as letting all the other kids know that their behavior is illegal or he would get in huge trouble for that and asking are you aware that that is against the rules or makes you not smart. (I know!! I know!!!) This might not work, but has helped in a lot of situations for us. He loves knowing that someone appreciates when he does good. He is able to restrain when he knows that a little one is looking.
Great suggestions, but so far mine doesn't buy into the role model thing. I wished that he did! Like yours, my boy likes to tell when others aren't following the rules, but doesn't appreciate the favor toward him.
Oh this doesn't always work, but I will take sometimes. Mine is so quick to react and will spend forever making sure you know that was his only choice and he had to do it instead of moving on. It is so frustration!! GRRRR!! I'm glad you are able to be home with him more since he needs you right now!
It's true…nothing works all the time does it?! I am feeling lucky about being home now. For me, he needs me more now than ever! Thanks for your continued support and comments! | {
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\section{Introduction}
In the last decade most electricity providers replaced their old meters (that had to be read by a technician) with smart connected devices, which are configured remotely and upload readings at predefined intervals without human assistance. Smart meters have numerous advantages beyond saving costs of manual readings: high granularity load monitoring, better anomaly and fraud detection, preventive maintenance or the ability to reduce billing frequency, just to name a few.
The configuration and data upload formats for electricity smart meter communications are focused on flexibility, resulting in highly configurable but verbose protocols, collectively known as \textit{Device Language Message Specification} (DLMS\footnote{https://www.dlms.com/}). Data compression options have been part of this de-facto standard for a long time, but the supported algorithms no longer provide adequate size reductions considering recent service demands. As utility providers want to receive consumption readings more frequently (e.g. every 15-60 minutes instead of once a day), the basic assumptions of the built-in DLMS compressors about payload size do not hold any more, and compression rates decline significantly \cite{feher2020smartmeter}. When a single reading is uploaded at a time, the existing compressors do not compress at all.
In this paper we continue our efforts to find better data compression methods for DLMS-based smart meters, leveraging a technique called \textit{Generalized Deduplication}\cite{vestergaard2019bounds}. We propose a new algorithm and companion data format that achieves the same level of compression when readings are uploaded in real time as the commonly used DLMS Null compressor uploading readings once a day. We evaluate our method on real life data captured over a 9 months from 95 households, provided by a major danish smart meter manufacturer, Kamstrup A/S.
The paper is organized as follows. After reviewing existing literature about smart meter reading compression in Section \ref{sec:relatedwork}, we describe the important properties of the DLMS data format and built-in compressors, as well as the concept of generalized deduplication in Section \ref{sec:background}. Next, we introduce our proposed compression scheme in Section \ref{sec:contribution}, and evaluate it in Section \ref{sec:evaluation}. In Section \ref{sec:conclusions} we summarize our findings.
\section{Related Work}\label{sec:relatedwork}
Compressing smart meter data is a relatively new field, as the need for more frequent uploads is a recent trend. In \cite{feher2020smartmeter} we introduced generalized deduplication-based compressors using a single GD transformation function and configuration, and processing the encoded APDU bytes sequentially. We now extend this method with positioning GD chunks to the underlying structure of the data, and add support for different settings in a single compressed packet.
Other methods in this field typically require a model of power consumption to compress.
Lee \textit{et al.}~\cite{lee2019unified} proposes compressive sensing with empirical modeling of the electricity consumption of different devices in homes of residential customers.
In \cite{kraus2012optimal} the authors describe a method using the LZMA algorithm and predictive modeling, but their technique requires a dictionary and additional preprocessing before compression can start.
A different approach in \cite{abuadbba2017gaussian} aims to minimize the parameters required to represent meter readings. They use a method based on Gaussian approximation.
Our proposed compressor does not require any preprocessing or modeling step.
The authors of \cite{blalock2018sprintz} propose a forecasting-based stream compressor that encodes the difference between the prediction and the actual measurements.
\section{Background} \label{sec:background}
In order to describe the context of the proposed compression method, we start by a brief introduction of the DLMS data format of smart meter readings, as well as the built-in compressors that our results will be compared to. We also give a high level introduction to the underlying technique of our algorithms: generalized deduplication.
\begin{figure*}[tbp]
\centerline{\includegraphics[width=\textwidth]{figures/chunk_alignment.png}}
\caption{ When GD chunks are not aligned with the similar regions of data, compression potential is poor(above). Pattern-based GD moves chunks to where similarity is expected (below). }
\label{fig:chunk_alignment}
\vspace{-2mm}
\end{figure*}
\subsection{DLMS Data Format}
The DLMS protocol specifies the communication protocol between a smart meter backend and the meters both for configuration (server to meter) and reporting (meter to server). Meters typically record active and reactive energy consumption every 15 minutes, and upload them once or multiple times a day, depending on local regulations and utility provider preferences.
Meters can be configured with multiple so-called \textit{Load Profiles} (LP), which determine what signals to record, how frequently, and when to upload the readings. Even though we only focus on compressing uploaded data packets from the Load Profile that records power consumption, there are other profiles as well, for example for anomaly detection or power quality measurements.
When a Load Profile trigger condition is met, the meter collects the corresponding readings since the last upload and constructs a data packet (called \textit{Application Data Unit} or APDU) by encoding the data values and adding a header that describes the LP. The header contents and internal structure can be heavily customized by the meter manufacturer and the utility provider, but the data buffer is only based on the included signals, therefore identical across providers. For this reason, our compression algorithm only processes the APDU data buffer, and assumes a different compressor handles the header.
\subsection{DLMS Compressors}
The DLMS protocol includes a few (optional) compression methods to decrease the APDU data buffer size, which we will compare our algorithms with.
\textbf{Null Data} is useful when repeating or strictly predictable values are present in the APDU. It works by replacing the actual value by a single NULL byte if it is identical to the previous one (for example the same reactive energy was measured twice in a row), or the current value was expected from the previous one (e.g., the timestamp was increased by the amount defined in the Load Profile).
\textbf{Delta Array} aims to encode the difference of consecutive values on fewer bytes instead of the values themselves. It can efficiently compress the energy consumption columns, since it typically requires only 1 or 2 bytes to represent the delta, instead of 4 bytes for the original full values. Both of these methods use the previous reading as a basis for compression, therefore only effective when at least two readings are included in an APDU.
\textbf{V.44} is a general purpose statistical compressor that is included in the DLMS protocol suite. Unlike the previous methods, it processes the APDU as a continuous byte array, e.g. without any regards to the internal structure of the encoded readings.
\subsection{Generalized Deduplication}
Deduplication is a well known technique that decreases size by storing repeating data chunks only once. While it is great for data with identical segments (e.g file servers), it fails to compress even small changes between chunks. Vestergaard \textit{et al.} ~\cite{vestergaard2019generalized} extended this method with the possibility of deduplicating similar chunks. The resulting algorithm, called \textit{Generalized Deduplication} (GD) works by first transforming each data chunk with a user-supplied function to a pair of basis and deviation. Ideally, the basis stores the repeating information across chunks, while the difference is captured in the smaller deviation. As a result, similar data chunks produce the same basis, which can be deduplicated in the compressed representation. The decompressor applies the inverse transformation to basis-deviation pairs to reconstruct the original data.
This compression method is highly efficient in scenarios where the data has high correlation and there exists a reversible transformation that can separate the variance from the constant part of each chunk. IoT data in general and smart meter readings specifically are very good fits for this technique.
\begin{figure*}[tpb]
\centerline{\includegraphics[width=0.9\textwidth]{figures/data_format.png}}
\caption{ Data format of the pattern-based GD stream compressor. }
\label{fig:data_format}
\vspace{-2mm}
\end{figure*}
\section{GD Pattern-Based Stream Compression}\label{sec:contribution}
Due to the fact that the APDU header is customizable by the utility provider and differs significantly between companies, we focus our efforts to compressing only the Data sections, e.g. the encoded list of power consumption readings. Furthermore, since the header is constant for a Load Profile, it can be represented by a predefined ID or a hash digest, as we proposed in \cite{feher2020smartmeter}. Consequently, the best strategy is to split the APDU and compress the two sections with different approaches.
Similarly to general-purpose statistical compressors like V.44, in our previous works we used GD-based techniques on APDUs as continuous byte sequences, without leveraging their internal structure. This led to good compression rates and straightforward applicability over a wide range of APDU configurations. However, a lot of compression potential was left unexploited, as the deduplicated chunks were not aligned with the similar data sections in the APDU. Figure \ref{fig:chunk_alignment} illustrates this issue, as the 4-byte GD chunks are not aligned with the data columns. This skew does not eliminate compression completely, but GD basis duplications are rather accidental and rare than intentional and exploited every time.
Since the data section of an APDU follows a fixed, table-like structure where column widths are predictable\footnote{ The DLMS protocol guarantees that values are always encoded to a constant number of bytes, depending on the data type. }, it is possible to align the GD chunks to the similar byte regions (data columns).
\subsection{Pattern-based Generalized Deduplication}
To exploit this predictability, our new GD-based compressor allows the user to define a pattern of transformation function configurations that are applied to the input byte array (encoded APDU) continuously. This way it is possible to position the deduplicated chunks precisely to the data columns (see lower half of Figure \ref{fig:chunk_alignment})
The pattern is given as a compact string that consists of GD algorithms and their configurations. Each pair directly or implicitly defines the chunk size and the size of GD basis and deviation. For example, the pattern \textbf{\texttt{[H5 H6]}} prescribes using Hamming code with 5 parity bits, then Hamming code with 6 parity bits. This transformation function only has one parameter (parity bits), which determines the basis and deviation sizes (3+1 bytes for H5, 7+1 bytes for H6), therefore the chunk size as well (4 bytes for H5, 8 bytes for H6). During compression, the pattern is repeated until the end of the input data stream.
To define the pattern used on Figure \ref{fig:chunk_alignment}, we cannot use Hamming code as the transformation function, since it does not have a setting for 3, 5 and 14 byte chunks. Furthermore, the highly variable bytes are always at constant positions in an APDU, something that Hamming code does not exploit very well, and a different transformation function is needed.
\begin{figure*}[t]
\centerline{\includegraphics[width=0.95\textwidth]{figures/patterns_wide.png}}
\caption{ The evaluated GD patterns (deviation indicated by underline). }
\label{fig:patterns_list}
\vspace{-2mm}
\end{figure*}
\subsection{LastByte GD Transformation Function}
We propose a new transformation function for generalized deduplication that is applicable when the highly variable bytes of a chunk are at the end, and their length is known.
\input{lastbyte_formula}
\vspace{2mm}
During compression, the LastByte function takes an $n+k$ byte chunk and splits it: the first $n$ bytes become the basis and the subsequent $k$ bytes form the deviation. The inverse function is used in the decompressor, which glues the basis and deviation back together to reconstruct the original data chunk. This simple transformation very efficiently produces identical bases and small deviations when applied to unsigned integers in big endian encoding, when the range of values, and therefore the number of highly varying bytes can be estimated well. All of these conditions are true for DLMS APDUs, and the computational simplicity also makes this function ideal for low powered embedded systems like a smart meter.
A LastByte-transformed chunk is encoded in a pattern string as $Ln,k$, or $Lnk$ when both parameters are single digits. A possible pattern for the coloring of Figure \ref{fig:chunk_alignment} is the following \textit{(note that the colored regions do not indicate where the basis ends and the deviation starts)}:
\vspace{1mm}
\textbf{\texttt{[L20 L41 L14,0 L21 L32 L41 L50 L41 L41]}}
\vspace{1mm}
Pattern-based generalized deduplication with the LastByte transformation function on APDUs has numerous advantages over our previously proposed compressors, which used Hamming code and a single chunk size:
\begin{itemize}
\item Deduplicated data chunks are positioned exactly to the data columns, not skewed any more.
\item LastByte enables extracting the bytes with high variance as deviation, instead of the bits dictated by Hamming code.
\item The encoded readings can be covered without any gaps, therefore no data padding is necessary.
\end{itemize}
In Section \ref{sec:evaluation} we evaluate different patterns to see how to best separate the encoded readings to chunks, and where to position the deviations to maximize compression gain.
\subsection{Serialization Format}
The compression process and data format of the compressed data format is illustrated on Figure \ref{fig:data_format}, where for two APDUs, both containing two readings of only the A14 and R12 signals are processed. First, DLMS-encoded input data is converted to a list of basis-deviation pairs according to the pattern. Next, new unique bases are added to in-memory arrays. These arrays form the compressor state that needs to be stored between transmissions. To transmit the new APDU, only the new GD bases and the (basis index,deviation) pairs are necessary. Since the decompressor also knows the pattern (it can either be sent in the first APDU, or inferred from the Load Profile), both the index and deviation lists can be transmitted without any separators. For example, if the \textbf{\texttt{[L41 L32]}} pattern is used, the decompressor picks bases from the 4-byte and the 3-byte arrays in an alternating manner. Furthermore, since both the compressor and decompressor knows how many unique bases are in each array, the number of bits to address them can be minimized, and packed tightly into bytes. In our example, the whole Base Reconstruction List (that holds indexes to the unique base arrays) fits into 4 bits per APDU. Finally, the list of deviations are also serialized without delimiters, because the decompressor knows from the pattern how many bytes to read at a time.
This data format heavily leverages the information present in the pattern itself, and achieves very low overhead for serializing the new information of the next compressed APDU.
\section{Performance Evaluation}\label{sec:evaluation}
We measure the compression of the pattern-based generalized deduplication method using a real life dataset provided by Kamstrup A/S, a major smart meter manufacturer in Denmark. It includes readings data from 95 consumers over 9 months between 2020 January and September. Every 15 minutes the smart meters recorded a reading with 4 quantities, listed in Table \ref{table:apdu_contents}. A single reading is encoded to a total of 49 bytes, including DLMS type codes that precede each value and additional overheads.
\begin{table}[htbp]
\caption{Contents of a reading}
\centering
\begin{tabular}{c l l}
\textbf{Order} & \textbf{Name} & \textbf{Data type} \\
1 & Log Id & 4 byte unsigned \\
2 & Timestamp & 12 bytes \\
3 & Log Status & 2 byte bitmap \\
4 & Data Quality & 4 byte bitmap \\
5 & Active energy consumption (A14) & 4 byte unsigned \\
6 & Active energy generation (A23) & 4 byte unsigned \\
7 & \makecell[l]{Reactive energy consumption\\phases 1 and 2 (R12)} & 4 byte unsigned \\
8 & \makecell[l]{Reactive energy consumption\\phases 3 and 4 (R34)} & 4 byte unsigned \\
\end{tabular}
\label{table:apdu_contents}
\end{table}
As the motivation of our research is the desire of utility providers to receive consumption data more frequently, we focus the compressor evaluations to smaller APDUs. Namely, we will calculate compression gains at the sizes listed in Table \ref{table:apdu_sizes_horizontal}. Typical data upload periods have been decreasing from 24-12 hours to 2-6 hours. The ultimate goal is to lower this to 1 hour and under, the most valuable being real time upload of every single reading.
\begin{table}[htbp]
\caption{Evaluated APDU sizes}
\centering
\begin{tabular}{r c c c c c c c }
\textbf{Readings/APDU:} & 1 & 2 & 4 & 12 & 24 & 48 & 96 \vspace{1mm} \\
\textbf{Upload period:} & 15min & 30min & 1h & 3h & 6h & 12h & 24h \\
\end{tabular}
\label{table:apdu_sizes_horizontal}
\end{table}
To evaluate our compressor against the built-in methods and compare different patterns, we calculate the compression gain over the whole duration of the dataset, separately for each consumer, and report the average across all households. For the GD pattern-based stream method, we report the average after the first transient APDU. The compression gain is expressed as $1 - compression\ rate$ in percentage, where higher values are better (0\%: no compression, 100\%: data is reduced to zero bytes). We have replaced the V.44 compressor, which is patented and no implementation is freely available, with the LZMA algorithm. It is a fundamentally very similar statistical compressor, built into Python.
We will evaluate the GD pattern-based compressor in Stream Mode (as described in \cite{feher2020smartmeter}), where consecutive APDUs of a single meter are considered a data stream. With this technique the unique GD bases that repeat across APDUs are sent only once, the first time they appear. Data savings are realized two ways: when there are similar values within a data packet, and across packets from the same meter. Figure \ref{fig:stream_convergence} shows how the compression rate changes with successive APDUs. Our pattern-based stream compressor achieves very high compression already from the second APDU, even if every reading is uploaded immediately. The downside of this approach is the introduction of state in the compressor and decompressor that needs to be stored between data uploads.
\newcommand\Tstrut{\rule{0pt}{2.6ex}}
\newcommand\Bstrut{\rule[-0.9ex]{0pt}{0pt}}
\begin{figure*}[htpb]
\centerline{\includegraphics[width=0.9\textwidth]{figures/stream_convergence_flat_wide.png}}
\caption{ The GD-based stream compressor achieves its eventual compression rate already at the second APDU. }
\label{fig:stream_convergence}
\end{figure*}
\subsection{Different Patterns}
Since the pattern-based compressor is highly flexible, we can easily evaluate variations that correspond to different assumptions about the dataset, and see which compresses better.
In the following, we describe five handcrafted and a generic pattern, and evaluate their performace. Figure \ref{fig:patterns_list} shows the pattern strings and illustrates how they map to a few encoded readings.
\textbf{Pattern \#1} exploits the constant \texttt{0208} type code (meaning: structure of 8 elements) in the beginning of each encoded reading, and merges it with the LogID in the first chunk. The last two LogID bytes form the deviation, which allows 256 consecutive readings to produce the same GD basis. The DLMS standard encodes the timestamp in a way that the highly variable hour and minute bytes are at offsets 8 and 9, so we define a 7+2 LastByte that covers the first half of the timestamp and captures these bytes as deviation. The rest of timestamp, together with the Status and Data Quality bitmaps are expected to be constant throughout the dataset. For this reason, the following chunk goes all the way to the end of A14, where the next highly varying bytes are expected. In this pattern we assume higher range of active energy readings and allow two bytes for the deviation. The last three data columns are processed as separate 5-byte chunks, where the one byte is reserved for deviation. This equals to the assumption of lower variance in the produced active energy and reactive energy quantities. \textbf{Pattern \#2} only differs in second rule: it assigns two additional bytes of the timestamp to the deviation, the ones that encode the day of the reading. \textbf{Pattern \#3} goes back to having only the hours and minutes as deviation, and assumes A23 to be constant across readings by merging it into the basis of the next chunk. Since very few households are feeding electricity back to the power grid, this change is likely to increase compression. \textbf{Pattern \#4} combines shorter LogID deviation, assumption of constant A23 and higher A14 variance, while \textbf{Pattern \#5} shortens the A14 deviation to a single byte. \textbf{Pattern \#6} is fundamentally different from the previous ones. It explores whether a fixed GD pattern can be assigned to each quantity, and locality is not exploited within the encoded reading.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.95\columnwidth]{figures/patterns_compression.png}}
\caption{ Average compression rates of the patterns, over the whole time period }
\label{fig:patterns_results}
\end{figure}
\begin{figure*}[htpb]
\centerline{\includegraphics[width=0.95\textwidth]{figures/compression_rates_sizes_all.png}}
\caption{ Compression gains (a) and total uploaded data (b) of the evaluated compressors. }
\label{fig:compression_rates_sizes_all}
\end{figure*}
The results show that some of our assumptions yield slightly better compression than others. Since patterns \#3 and \#4 are the best, the better choice about timestamp is to maximize the deduplicated basis and keep only the hours and minutes in the deviation. Also, it is best to separate only the last byte of the LogID, but A14 varies so much that two bytes deviation is the better choice. Since pattern \#6 achieves only 3\% lower compression rates than the best hand-optimized ones, it is safe to declare that automatic generation of patterns is a viable option. This proves the real life applicability of this compression method, since the GD pattern can be generated from the Load Profile definition, and does not require a specialist to construct the optimal pattern string on a case-by-case basis.
Since the stream compressor achieves low transmission size by introducing a state of GD bases sent in previous APDUs, it is important to see how large this state is for each pattern. Figure \ref{fig:compressor_state_size} shows how the state grows over the 9 months time period. Note, that the state size is independent of the upload period, since the same amount of information is being transferred over time, the same GD bases form the state regardless of how frequently a new APDU is uploaded. The relative growth rate is constant with each pattern, and the memory requirements correspond to the compression gains: patterns with higher compression require larger state.
Note, that our proposed compressor keeps old GD bases in the unique basis arrays indefinitely, therefore the memory requirement constantly increases over time. An interesting future research direction is to remove the unused bases from the state over time, either by reference counting or simply defining a sliding window and evicting old bases automatically. Both improvements can be implemented easily with the existing compressor architecture, and will be explored later.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.95\columnwidth]{figures/compressor_state_size.png}}
\caption{ Size of the compressor state over time, per meter. }
\label{fig:compressor_state_size}
\end{figure}
\subsection{Comparison with DLMS Compressors}
The existing DLMS compressors fall short when APDU size is drastically reduced. Null and Delta methods use the first reading as the reference, and only process the consecutive ones. Therefore, their performance with only two and four readings per APDU is not satisfactory, and they do not compress when a single reading is uploaded.
Another issue with the built-in compressors, including V.44, is they only compress one APDU at a time, and not leveraging information from the previous data packets.
The left side of Figure \ref{fig:compression_rates_sizes_all} shows the compression gains of the compressors over all 95 meters and 9 months (using pattern \#4 in the GD-Pattern method). The DLMS methods and general purpose statistical algorithms achieve satisfactory compression rates when the upload period is multiple hours, since they need a larger data buffer to be efficient. The statistics-based LZMA method is noticably good at 12 and 24 hour reporting periods, but due to its computational complexity, it is rarely used in real life. None of the methods currently available in the DLMS standard provide compression of a single reading. Our proposed compressor is able to reduce APDU size to 64\% when every reading is uploaded separately, and up to 74\% with larger upload periods.
In right side of Figure \ref{fig:compression_rates_sizes_all} shows the total amount of data uploaded by one meter over 9 months, and added the total uncompressed APDU data size as a reference. We report the complete APDU size (header and data together), where the header is represented by a 4-byte ID. Our GD-P method can process the header together with the APDU data by simply extending the pattern to the following: \textbf{\texttt{L40 [ L52 L72 L16,2 L91 L41 ]}}. The first 4 bytes (header ID) are treated as a GD basis and referenced/deduplicated together with all other 4-byte bases. Then, the rest of the pattern is repeated over the encoded readings, just like before. As a result, the header occupies 4 bytes in the first GD-P compressed APDU, and only a few bits in all subsequent ones, depending on the number of unique 4-byte bases. In the other compressors the header ID is repeated in each APDU.
The data shows that our proposed compressor is able to report each reading immediately with similar amount of data transfer (413 kB) than the best DLMS method with only one upload per day (DLMS Null, 414 kB), or a heavy statistical compressor reporting every 3 hours (LZMA, 398 kB). This compression efficiency allows utility providers to access near real time information with no additional data transfer cost.
\section{Conclusions} \label{sec:conclusions}
In this paper we have presented a new GD transformation function and a flexible, pattern-based extension of a previously introduced strem compression technique for smart meter readings. We have shown that the pattern can be automatically inferred from the Load Profile definition, making the method easily applicable at scale. The proposed compressor is highly efficient in reducing transfer sizes when data is uploaded frequently, and allows utility providers to upload every reading immediately.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,593 |
\section{Introduction}
Dynamical systems are powerful objects which can be found in numerous
applications. However, this versatility does not come without cost: dynamical
systems are objects which are inherently very hard to study.
Recently, digital computers have been successfully used to analyze dynamical
systems, through the use of numerical simulations. However, with the known
existence of phenomena like the \textquotedblleft butterfly
effect\textquotedblright\ -- a small perturbation on initial conditions can be
exponentially amplified along time -- and the fact that numerical simulations
always involve some truncation errors, the reliability of such simulations for
providing information about the long-term evolution of the systems is questionable.
For instance, numerical simulations suggested the existence of a
\textquotedblleft strange\textquotedblright\ attractor for the Lorenz system
\cite{Lor63}, and it was widely believed that such an attractor existed.
However, the formal proof of its existence remained elusive, being at the
heart of the 14th from the list of 18 problems that the Fields medalist
S.\ Smale presented for the new millennium \cite{Sma98}. Finally, after a 35
years hiatus, a computer-aided formal proof was achieved in \cite{Tuc98},
\cite{Tuc99}, but this example painstakingly illustrates the difference
between numerical evidence and a full formal proof, even if computer-based.
For the above reasons, it is important to understand which properties can be
accurately computed with a computer, and those which cannot. We have
previously dwelled on this subject on our paper \cite{GZ10} which focused on
\textquotedblleft stable\textquotedblright\ dynamical systems having
hyperbolic attractors. The latter systems were extensively studied in the XXth
century and were thought to correspond to the class of \textquotedblleft
meaningful\textquotedblright\ dynamical systems. This happened for several
reasons: the \textquotedblleft stability\textquotedblright\ -- structural
stability -- which implies robustness of behavior to small perturbations was
believed to be of uttermost importance, since it was thought that only such
systems could exist in nature; there were also results (Peixoto's Theorem
\cite{Pei62}) which showed that, in the plane, these systems are dense and
their invariant sets (which include all attractors) can be fully characterized
(they can only be points or periodic orbits).There was hope that such results
would generalize for spaces of higher dimensions, but it was shattered by S.
Smale, who showed that there exist \textquotedblleft chaotic\textquotedblrigh
\ hyperbolic invariant sets which are neither a point nor a periodic orbit
(Smale's \ horseshoe \cite{Sma67}) and that for dimensions $\geq3$
structurally stable systems are not dense \cite{Sma66}.
Moreover, the notion of structural stability was shown to be too strong a
requirement. In particular, the Lorenz attractor, which is embedded in a
system modeling weather evolution, is not structurally stable, although it is
stable to perturbations in the parameters defining the system \cite{Via00} (in
other words, robustness may not be needed for \emph{all} mathematical
properties of the system).
Despite the fact that systems with hyperbolic attractors are no longer
regarded as \textquotedblleft the\textquotedblright\ meaningful class of
dynamical systems for spaces of dimension $n\geq3$, they still play a central
role in the study of dynamical systems. In essence, what characterizes a
hyperbolic set is the existence at each point of an invariant splitting of the
tangent space into stable and unstable directions, which generate the local
stable and unstable manifolds (see Section \ref{Sec:Dynamical_systems}).
In this paper we will study the computability of the stable and
unstable manifolds for hyperbolic equilibrium points. The stable and
unstable manifolds are constructs which derive from the stable
manifold theorem. This theorem states that for each hyperbolic
equilibrium point $x_{0}$ (see Section \ref{Sec:Dynamical_systems}
for a definition), there exists a manifold $S$ such that a
trajectory on $S$ will converge to $x_{0}$ at an exponential rate as
$t\rightarrow+\infty$. Moreover every trajectory converging to
$x_{0}$ lies entirely in $S$. $S$ is call a stable manifold.
Similarly one can obtain the unstable manifold $U$, using similar
conditions when $t\rightarrow-\infty$. (The stable and unstable
manifolds of a hyperbolic equilibrium point are depicted in Fig.\
\ref{fig:hyperbolic}.) It is important to note that classical proofs
on existence of $S$ and $U$ are non-constructive. Thus computability (or, for the matter, semi-computability) of the stable/unstable manifold does not follow straightforwardly from these proofs.
We address the following basic question: given a dynamical system
and some hyperbolic equilibrium point, can we compute its stable and
unstable manifolds? We will show that the answer is twofold: (i) we
can compute the stable and unstable manifolds $S$ and $U$ given by
the stable manifold theorem which are, in some sense, local, since
there are (in general) trajectories starting in points which do not
lie in $S$ that will converge to $x_{0}$ (the set of all these point
united with $S$ would yield the global stable manifold -- see
Section \ref{Sec:Dynamical_systems}), but (ii) the global stable and
unstable manifolds are not, in general, computable from the
description of the system and the hyperbolic point. Since classical
proofs on existence of $S$ and $U$ are non-constructive, a different
approach is needed if one wishes to construct an algorithm that
computes $S$ and $U$. Our approach aiming at a constructive proof
makes use of function-theoretical treatment of the resolvents (see
the first paragraph of Section 4 for more details).
A most likely interpretation for these results is that, since the definition of
hyperbolicity is local, computability also applies locally. However global
homoclinic tangles can occur \cite{GH83}, leading to chaotic behavior for such
systems. So it should not be expected that the global behavior of stable and
unstable manifolds be globally computable in general, as indeed our results show.
In the end of the paper we also show that the prototypical example of an
hyperbolic invariant set which is neither a fixed point nor a periodic orbit
-- the Smale horseshoe -- is computable.
The computability of simple attractors -- hyperbolic periodic orbits and
equilibrium points -- was studied in our previous papers \cite{Zho09},
\cite{GZ10} for planar dynamics, as well as the computability of their
respective domains of attraction.
A number of papers study dynamical systems, while not exactly in the context
used here. For example the papers\ \cite{Moo90}, \cite{BBKT01}, \cite{Col05},
\cite{BY06}, \cite{Hoy07} provide interesting results about the long-term
behavior of dynamical systems using, among others, symbolic dynamics or a
statistical approach. See \cite{BGZ10} for a more detailed review of the
literature
\begin{center}
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=5cm]{stable_unstable}
\end{center}
\caption{Dynamics near a hyperbolic equilibrium point.}
\label{fig:hyperbolic}
\end{figure}
\end{center
\section{Dynamical systems\label{Sec:Dynamical_systems}}
In short, a dynamical system is a pair consisting of a state space where the
action occurs and a function $f$ which defines the evolution of the system
along time. See \cite{HS74} for a precise definition. In general one can
consider two kinds of dynamical systems: discrete ones, where time is discrete
and one obtains the evolution of the system by iterating the map $f$, and
continuous ones, where the evolution of the system along time is governed by a
differential equation of the typ
\begin{equation}
\dot{x}=f(x) \label{Eq:ODE
\end{equation}
where $t$ is the independent variable (the \textquotedblleft
time\textquotedblright) and $\dot{x}$ denotes the derivative $dx(t)/dt$.
Continuous-time systems can be translated to discrete-time using time-one maps
and, in some cases, the Poincar\'{e} map, and vice-versa (using the suspension
method). \ Therefore to study dynamical systems one can focus on
continuous-time ones.
Although the essential feature of hyperbolic attractors is the existence of an
invariant splitting of the tangent space into stable and unstable directions
generating the local stable and unstable manifolds, for a hyperbolic
equilibrium point $x_{0}$, it can be described equivalently in terms of the
linearization of the flow around $x_{0}.$ We recall that $x_{0}$ is an
equilibrium point of (\ref{Eq:ODE}) iff $f(x_{0})=0$.
\begin{definition}
An equilibrium point $x_{0}$ of (\ref{Eq:ODE}) is hyperbolic if none of the
eigenvalues of $Df(x_{0})$ has zero real part.
\end{definition}
According to the value of the real part of these eigenvalues, one can
determine the behavior of the linearized flow near $x_{0}$: if the eigenvalue
has negative real part, then the flow will converge to $x_{0}$ when it follows
the direction given by the eigenvector associated to this eigenvalue. A
similar behavior will happen when the eigenvalue has positive real part, with
the difference that convergence happens when $t\rightarrow-\infty$. See Fig.
\ref{fig:hyperbolic}. The space generated by the eigenvectors associated to
eigenvalues of $Df(x_{0})$ with negative real part is called the stable
subspace $E^{s}_{Df(x_{0})}$, and the space generated by the eigenvectors
associated to eigenvalues with positive real part is called the unstable
subspace $E^{u}_{Df(x_{0})}$.
We now state the Stable Manifold Theorem (as it appears in \cite{Per01}).
\begin{proposition}
[Stable Manifold Theorem]Let $E$ be an open subset of $\mathbb{R}^{n}$
containing the origin, let $f\in C^{1}(E)$, and let $\phi_{t}$ be the flow of
the system (\ref{Eq:ODE}). Suppose that $f(0)=0$ and that $Df(0)$ has $k$
eigenvalues with negative real part and $n-k$ eigenvalues with positive real
part (\emph{i.e.} 0 is a hyperbolic equilibrium point). Then there exists a
$k$-dimensional differentiable manifold $S$ tangent to the stable subspace
$E^{s}_{Df(0)}$ such that for all $t\geq0$, $\phi_{t}(S)\subseteq S$ and for
all $x_{0}\in S
\[
\lim_{t\rightarrow+\infty}\phi_{t}(x_{0})=0;
\]
and there exists an $n-k$ dimensional differentiable manifold $U$ tangent to
the unstable subspace $E^{u}_{Df(0)}$ such that for all $t\leq0$, $\phi
_{t}(S)\subseteq S$ and for all $x_{0}\in U
\[
\lim_{t\rightarrow-\infty}\phi_{t}(x_{0})=0.
\]
\end{proposition}
From the local stable and unstable manifolds given by the Stable Manifold
Theorem, one can define the \emph{global stable and unstable manifolds} of
(\ref{Eq:ODE}) at a hyperbolic equilibrium point $x_{0}$ b
\begin{align}
W_{f}^{s}(x_{0}) & = \bigcup_{j=0}^{\infty}
\phi_{-j}(S)\label{Eq:global}\\
W_{f}^{u}(x_{0}) & =
\bigcup_{j=0}^{\infty}
\phi_{j}(U),\nonumber
\end{align}
respectively, where
\[
\phi_{t}(A)=\{x(t)|x\text{ is a solution of (\ref{Eq:ODE}) with }x(0)\in A\}.
\]
We note that $W^{s}(x_{0})$ and $W^{u}(x_{0})$ are $F_{\sigma}$-sets of
$\mathbb{R}^{n}$ (recall that a subset $F\subseteq\mathbb{R}^{n}$ is called an
$F_{\sigma}$-set if $F=\bigcup_{j=0}^{\infty}A_{j}$, where $A_{j}$,
$j\in\mathbb{N}$, is a closed subset of $\mathbb{R}^{n}$).
We end this section by introducing Smale's horseshoe \cite{Sma67}. We refer
the reader to \cite{GH83} for a more thorough discussion of this set. In
essence, Smale's horseshoe appears when we consider a map $f$ defined over
$S=[0,1]^{2}$. This map is bijective and performs a linear vertical expansion
of $S$, and a linear horizontal contraction of $S$, by factors $\mu>1$ and
$0<\lambda<1$, respectively, followed by a folding.
\begin{definition}
In the conditions defined above (see \cite{GH83} or \cite{HSD04} for precise
definitions), the Smale horseshoe is the set $\Lambda$ given b
\[
\Lambda
{\displaystyle\bigcap\limits_{j=-\infty}^{+\infty}}
f^{j}(S).
\]
This set is invariant for the function $f$ (i.e.\ $f(\Lambda)=\Lambda$).
\end{definition}
\section{Computable analysis}
Now that we have introduced the main concepts of dynamical systems theory we
will use, we need to introduce concepts related to computability. The theory
of computation can be rooted in the seminal work of Turing, Church, and
others, which provided a framework in which to achieve computation over
discrete identities or, equivalently, over the integers.
However, this definition was not enough to cover computability over continuous
structures, and was then developed by other authors such as Turing himself
\cite{Tur36}, Grzegorczyk \cite{Grz57}, or Lacombe \cite{Lac55} to originate
\emph{computable analysis}.
The idea underlying computable analysis to compute over a set $A$ is to encode
each element $a$ of $A$ by a countable sequence of symbols from a finite
alphabet (called a $\rho$-\emph{name} for $a$). Each sequence can encode at
most one element of $A$. The more elements we have from a sequence encoding
$a$, the more precisely we can pinpoint $a$. From this point of view, it
suffices to work only with names when performing a computation over $A$. To
compute with names, we use Type-2 machines, which are similar to Turing
machines, but (i) have a read-only tape, where the input (i.e.\ the sequence
encoding it) is written; (ii) have a write-only output tape, where the head
cannot move back and the sequence encoding the output is written. For more
details the reader is referred to \cite{PR89}, \cite{Ko91}, \cite{Wei00}.
At any finite amount of time we can halt the computation, and we will have a
partial result on the output tape. The more time we wait, the more accurate
this result will be. We now introduce notions of computability over
$\mathbb{R}^{n}$.
\begin{definition}
\begin{enumerate}
\item A sequence $\{r_{k}\}$ of rational numbers is called a $\rho$-name of a
real number $x$ if there are three functions $a,b$ and $c$ from $\mathbb{N}$
to $\mathbb{N}$ such that for all $k\in\mathbb{N}$, $r_{k}=(-1)^{a(k)
\frac{b(k)}{c(k)+1}$ and
\begin{equation}
\left\vert r_{k}-x\right\vert \leq\frac{1}{2^{k}}. \label{PourElReal
\end{equation}
\item A double sequence $\{r_{l,k}\}_{l,k\in\mathbb{N}}$ of rational numbers
is called a $\rho$-name for a sequence $\{x_{l}\}_{l\in\mathbb{N}}$ of real
numbers if there are three functions $a,b,c$ from $\mathbb{N}^{2}$ to
$\mathbb{N}$ such that, for all $k, l\in\mathbb{N}$, $r_{l,k}=(-1)^{a(l,k)
\frac{b(l,k)}{c(l,k)+1}$ and
\[
\left\vert r_{l,k}-x_{l}\right\vert \leq\frac{1}{2^{k}}.
\]
\item A real number $x$ (a sequence $\{x_{l}\}_{l\in\mathbb{N}}$ of real
numbers) is called computable if it has a computable $\rho$-name, i.e. the
functions $a$, $b$, and $c$ are computable or, equivalently, there is a Type-2
machine that computes a $\rho$-name of $x$ ($\{x_{l}\}_{l\in\mathbb{N}}$,
respectively) without any input.
\end{enumerate}
\end{definition}
In general, $\rho$-names for real numbers do not need to be exactly those
described above: different definitions can provide the same set of computable
points (e.g.\ the $\rho$-name could be a sequence of intervals containing
$x_{0}$ with diameter $1/k$ for $k\geq1$, etc.). A notable exception is the
decimal expansion of $x_{0}$, which cannot be used since it can lead to
undesirable behavior \cite{Tur37} because this representation does not respect
the topology of the space $\mathbb{R}$.
The notion of the $\rho$-name can be extended to points in $\mathbb{R}^{n}$ as
follows: a sequence $\{(r_{1k},r_{2k},\ldots,r_{nk})\}_{k\in\mathbb{N}}$ of
rational vectors is called a $\rho$-name of $x=(x_{1},x_{2},\ldots,x_{n
)\in\mathbb{R}^{n}$ if $\{r_{jk}\}_{k\in\mathbb{N}}$ is a $\rho$-name of
$x_{j}$, $1\leq j\leq n$. Using $\rho$-names, we can define computable functions.
\begin{definition}
Let $X$ and $Y$ be two sets, where $\rho$-names can be defined for elements of
$X$ and $Y$. A function $f:X\rightarrow Y$ is computable if there is a Type-2
machine such that on any $\rho$-name of $x\in X$, the machine computes as
output a $\rho$-name of $f(x)\in Y$.
\end{definition}
Next we present a notion of computability for open and closed subsets of
$\mathbb{R}^{n}$ (cf. \cite{Wei00}, Definition 5.1.15). We implicitly use
$\rho$-names. For instance, to obtain names of open subsets of $\mathbb{R
^{n}$, we note that the set of rational balls $B(a,r)=\{x\in\mathbb{R
^{n}:\left\vert x-a\right\vert <r\}$, where $a\in\mathbb{Q}^{n}$ and
$r\in\mathbb{Q}$, is a basis for the standard topology over $\mathbb{R}^{n}$.
Thus a sequence $\{(a_{k},r_{k})\}_{k\in\mathbb{N}}$ such that $E=\cup
_{k=0}^{\infty}B(a_{k},r_{k})$ gives a $\rho$-name for the open
set $E$.
\begin{definition}
\label{def_r_e_open}
\begin{enumerate}
\item An open set $E\subseteq\mathbb{R}^{n}$ is called recursively enumerable
(r.e.\ for short) open if there are computable sequences $\{a_{k}\}$ and
$\{r_{k}\}$, $a_{k}\in\mathbb{Q}^{n}$ and $r_{k}\in\mathbb{Q}$, such that
\[
E=\cup_{k=0}^{\infty}B(a_{k},r_{k}).
\]
Without loss of generality one can also assume that for any $k\in\mathbb{N}$,
the closure of $B(a_{k},r_{k})$, denoted as $\overline{B(a_{k},r_{k})}$, is
contained in $E$.
\item A closed subset $A\subseteq\mathbb{R}^{n}$ is called r.e.\ closed if
there exists a computable sequence $\{b_{k}\}$, $b_{k}\in\mathbb{Q}^{n}$, such
that $\{b_{k}\}$ is dense in $A$. $A$ is called co-r.e. \ closed if its
complement $A^{c}$ is r.e. open. $A$ is called computable (or recursive) if it
is both r.e.\ and co-r.e.
\item An open set $E\subseteq\mathbb{R}^{n}$ is called computable (or
recursive) if $E$ is r.e.\ open and its complement $E^{c}$ is r.e.\ closed.
\item A compact set $K\subseteq\mathbb{R}^{n}$ is called computable if it is
computable as a closed set and, in addition, there is a rational number $b$
such that $||x||\leq b$ for all $x\in K$.
\end{enumerate}
\end{definition}
In the rest of the paper, we will work exclusively with $C^{1}$ functions $f:
E\to\mathbb{R}^{n}$, where $E$ is an open subset of $\mathbb{R}^{n}$. Thus it
is desirable to present an explicit $\rho$-name for such functions.
\begin{definition}
\label{Def:rho_C1_function} Let $E=\bigcup_{k=0}^{\infty}B(a_{k},r_{k})$,
$a_{k}\in\mathbb{Q}^{n}$ and $r_{k}\in\mathbb{Q}$, be an open subset of
$\mathbb{R}^{n}$ (assuming that the closure of $B(a_{k},r_{k})$ is contained
in $E$) and let $f:E\rightarrow\mathbb{R}^{n}$ be a continuously
differentiable function. Then a ($C^{1}$) $\rho$-name of $f$ is a sequence
$\{P_{l}\}$ of polynomials ($P_{l}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$)
with rational coefficients such that
\[
d_{C^{1}(E)}(f,P_{l})\leq2^{-l}\quad\mbox{for all $l\in \mathbb{N}$}
\]
where
\[
d_{C^{1}(E)}(f,P_{l})=\sum_{k=0}^{\infty}2^{-k}\left( \frac{||f-P_{l}||_{k
}{1+||f-P_{l}||_{k}}+\frac{||Df-DP_{l}||_{k}}{1+||Df-DP_{l}||_{k}}\right)
\]
and
\[
||g||_{k}=\max_{x\in\overline{B(a_{k},r_{k})}}|g(x)|.
\]
\end{definition}
We observe that this $\rho$-name of $f$ contains information on both $f$ and
$Df$ in the sense that $(P_{1}, P_{2}, \ldots)$ is a $\rho$-name of $f$ while
$(DP_{1}, DP_{2}, \ldots)$ is a $\rho$-name of $Df$. See \cite{ZW03} for
further details.
Throughout the rest of this paper, unless otherwise mentioned, we will assume
that, in (\ref{Eq:ODE}), $f$ is continuously differentiable on an open subset
of $\mathbb{R}^{n}$ and we will use the above $\rho$-name for $f$.
\section{Computable stable manifold theorem}
\label{sec_computable}
The stable manifold theorem states that near a hyperbolic equilibrium point
$x_{0}$, the nonlinear system
\begin{equation}
\dot{x}=f(x(t)) \label{e1
\end{equation}
has stable and unstable manifolds $S$ and $U$ tangent to the stable and
unstable subspaces ${\mathbb{E}}_{A}^{s}$ and ${\mathbb{E}}_{A}^{u}$ of the
linear system
\begin{equation}
\dot{x}=Ax
\end{equation}
where $A=Df(x_{0})$ is the gradient matrix of $f$ at $x_{0}$. The classical
proof of the theorem relies on the Jordan canonical form of $A$. To reduce $A$
to its Jordan form, one needs to find a basis of generalized eigenvectors.
Since the process of finding eigenvectors from corresponding eigenvalues is
not continuous in general, it is a non-computable process. Thus if one wishes
to construct an algorithm that computes some $S$ and $U$ of (\ref{e1}) at
$x_{0}$, a different method is needed. We will make use of an analytic, rather
than algebraic, approach to the eigenvalue problem that allows us to compute
$S$ and $U$ without calling for eigenvectors. The analytic approach is based
on function-theoretical treatment of the resolvents (see, \emph{e.g.},
\cite{Nag42}, \cite{Kat49}, \cite{Kat50}, and \cite{Rob95}). \newline
Let us first show that the stable and unstable subspaces are computable from
$A$ for the linear hyperbolic systems $\dot{x}=Ax$. We begin with several
definitions. Let $\mathfrak{A}_{H}$ denote the set of all $n\times n$ matrices
such that the linear differential equation $\dot{x}=Ax$, $x\in{\mathbb{R}
^{n}$, is hyperbolic, where a linear differential equation $\dot{x}=Ax$ is
defined to be hyperbolic if all the eigenvalues of $A$ have nonzero real part.
The Hilbert-Schmidt norm is used for $A\in\mathfrak{A}_{H}$: $\norm{A}=\sqrt
{\sum_{i=1}^{n}\sum_{j=1}^{n}|a_{ij}|^{2}}$, where $a_{ij}$ is the
$ij^{\mbox{th}}$ entry of $A$. For each $A\in\mathfrak{A}_{H}$, define the
stable subspace $\mathbb{E}^{s}_{A}$ and unstable subspace $\mathbb{E}^{u
_{A}$ to be
\begin{align*}
{\mathbb{E}}^{s}_{A} & = \mbox{span}\left\{ v\in{\mathbb{R}}^{n}:
\mbox{$v$ is a generalized
eigenvector for an eigenvalue $\lambda$ of $A$} \right. \\
& \qquad\quad\left. \mbox{with $Re(\lambda)<0$}\right\}
\end{align*}
\begin{align*}
{\mathbb{E}}^{u}_{A} & = \mbox{span}\left\{ v\in{\mathbb{R}}^{n}:
\mbox{$v$ is a generalized
eigenvector for an eigenvalue $\lambda$ of $A$} \right. \\
& \qquad\quad\left. \mbox{with $Re(\lambda)>0$}\right\}
\end{align*}
Then ${\mathbb{R}}^{n}={\mathbb{E}}^{s}_{A}\bigoplus{\mathbb{E}}^{u}_{A}$. The
stable subspace $\mathbb{E}^{s}_{A}$ is the set of all vectors which contract
exponentially forward in time while the unstable subspace $\mathbb{E}^{u}_{A}$
is the set of all vectors which contract backward in time.
As mentioned above, the process of finding eigenvectors from corresponding
eigenvalues is not computable; thus the algebraic approach to ${\mathbb{E
}^{s}_{A}$ and ${\mathbb{E}}^{u}_{A}$ is a non-computable process. Of course,
this doesn't necessarily imply that it is impossible to compute ${\mathbb{E
}^{s}_{A}$ and ${\mathbb{E}}^{u}_{A}$ from $A$, but rather this particular
classical approach fails to be computable. So even for the linear hyperbolic
system $\dot{x}=Ax$, one needs a different approach to treat the
stable/unstable subspace when computability concerned. The approach used below
to treat the stable/unstable subspace is analytic, rather than algebraic. Let
$p_{A}(\lambda)$ be the characteristic polynomial for $A$, $\gamma_{1}$ be any
closed curve in the left half of the complex plane that surrounds (in its
interior) all eigenvalues of $A$ with negative real part and is oriented
counterclockwise, and $\gamma_{2}$ any closed curve in the right half of the
complex plane that surrounds all eigenvalues of $A$ with positive real part,
again with counterclockwise orientation. Then
\[
P^{1}_{A}{\mathbb{R}}^{n}={\mathbb{E}}^{s}_{A}, \qquad P^{2}_{A}{\mathbb{R
}^{n}={\mathbb{E}}^{u}_{A
\]
where
\[
P^{1}_{A}v=\frac{1}{2\pi i}\int_{\gamma_{1}}(\xi I-A)^{-1}vd\xi, \quad
P^{2}_{A}v=\frac{1}{2\pi i}\int_{\gamma_{2}}(\xi I-A)^{-1}vd\xi
\]
(See Section 4.6 of \cite{Rob95}.)
\begin{theorem}
\label{t0} The map $H^{s}:{\mathfrak{A}}_{H}\rightarrow\mathcal{A
({\mathbb{R}}^{n
)=\{X|\mbox{$X\subseteq\mathbb{R}^n$ is a closed subset of $\mathbb{R}^n$}\}$,
$A\mapsto{\mathbb{E}}_{A}^{s}$, is computable, where ${\mathfrak{A}}_{H}$ is
represented \textquotedblleft entrywise": $\rho=(\rho_{ij})_{i,j=1}^{n}$ is a
name of $A$ if $\rho_{ij}$ is a $\rho$-name of $a_{ij}$, the $ij$th entry of
$A$.
\end{theorem}
\begin{proof}
First we observe that the map $A\mapsto$ the eigenvalues of $A$,
$A\in{\mathfrak{A}}_{H}$, is computable. Assume that $\lambda_{1
,\ldots,\lambda_{k},\mu_{k+1},\ldots,\mu_{n}$ (counting multiplicity) are
eigenvalues of $A$ with $Re(\lambda_{j})<0$ for $1\leq j\leq k$ and
$Re(\mu_{j})>0$ for $k+1\leq j\leq n$. Then from the $\rho$-names of the
eigenvalues, one can compute two rectangular closed curves $\gamma_{A}^{1}$
and $\gamma_{A}^{2}$, where $\gamma_{A}^{1}$ is in the left half of the
complex plane that surrounds all eigenvalues $\lambda_{j}$ with $1\leq j\leq
k$ and $\gamma_{A}^{2}$ is in the right half of the complex plane that
surrounds all eigenvalues $\mu_{j}$ with $k+1\leq j\leq n$, and both
$\gamma_{A}^{1}$ and $\gamma_{A}^{2}$ are oriented counterclockwise. From
$\gamma_{A}^{1}$ and $\gamma_{2}^{A}$ one can further computes the maps
$P_{A}^{1},P_{A}^{2}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{n}$ , where
$P_{A}^{1}v=\frac{1}{2\pi i}\int_{\gamma_{A}^{1}}(\xi I-A)^{-1}vd\xi$ and
$P_{A}^{2}v=\frac{1}{2\pi i}\int_{\gamma_{A}^{2}}(\xi I-A)^{-1}vd\xi$. We note
that, on the one hand, $\mathbb{E}^{s}_{A}=(P^{2}_{A})^{-1}(\{0\})$, and on
the other hand, $\mathbb{E}^{s}_{A}=\overline{\mathbb{E}^{s}_{A}
=\overline{P^{1}_{A}\mathbb{R}^{n}}$, where $\overline{K}$ denotes the closure
of the set $K$. Then by Theorem 6.2.4 of \cite{Wei00}, $\mathbb{E}^{s}_{A}$ is
both r.e. and co-r.e. closed, thus computable. The same argument shows that
${\mathbb{E}}_{A}^{u}$ is computable from $A$.
\end{proof}
Next we present an effective version of the stable manifold theorem. Consider
the nonlinear system
\begin{equation}
\label{e0}\dot{x}=f(x(t))
\end{equation}
Assume that (\ref{e0}) defines a dynamical system, that is, the solution $x(t,
x_{0})$ to (\ref{e0}) with the initial condition $x(0)=x_{0}$ is defined for
all $t\in\mathbb{R}$. Since the system (\ref{e0}) is autonomous, if $p$ is a
hyperbolic equilibrium point, without loss of generality, we may assume that
$p$ is the origin $0$. \newline
\begin{theorem}
\label{t1} Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a $C^{1
$-computable function (meaning that both $f$ and $Df$ are computable). Assume
that the origin $0$ is a hyperbolic equilibrium point of (\ref{e0}) such that
$Df(0)$ has $k$ eigenvalues with negative real part and $n-k$ eigenvalues with
positive real part (counting multiplicity), $0<k\leq n$. Let $x(t,x_{0})$
denote the solution of (\ref{e0}) with the initial value $x_{0}$ at $t=0$.
Then there is a (Turing) algorithm that computes a $k$-dimensional manifold
$S\subset{\mathbb{R}}^{n}$ containing $0$ such that
\begin{itemize}
\item[(i)] For all $x_{0}\in S$, $\lim_{t\rightarrow+\infty}x(t,x_{0})=0$;
\item[(ii)] There are three positive rational numbers $\gamma$, $\epsilon$,
and $\delta$ such that $|x(t,x_{0})|\leq\gamma2^{-\epsilon t}$ for all
$t\geq0$ whenever $x_{0}\in S$ and $|x_{0}|\leq\delta$.
\end{itemize}
Moreover, if $k<n$, then a rational number $\eta$ and a ball $D$ with center
at the origin can be computed from $f$ such that for any solution $x(t,x_{0})$
to the equation (\ref{e0}) with $x_{0}\in D\setminus S$, $\{ x(t, x_{0}):
\ t\geq0\}\not \subset B(0, \eta)$ no matter how close the initial value
$x_{0}$ is to the origin.
\end{theorem}
\begin{proof}
First we note that under the assumption that $f$ is $C^{1}$-computable, the
solution map $x:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$,
$(t,a)\mapsto x(t,a)$, is computable (\cite{GZB07}).
Let $f(x)=Ax+F(x)$, where $A=Df(0)$ and $F(x)=f(x)-Ax$. Then the equation
(\ref{e0}) can be written in the form of
\begin{equation}
\dot{x}=Ax+F(x) \label{e2
\end{equation}
The first step in constructing the desired algorithm is to break the flow
$e^{At}$ governed by the linear equation $\dot{x}=Ax$ into the stable and
unstable components, denoted as $I_{\Gamma_{1}}(t)$ and $I_{\Gamma_{2}}(t)$
respectively. By making use of an integral formula, we are able to show that
the breaking process is computable from $A$. The details for the first step:
Since $F(x)=f(x)-Ax$, it follows that $F(0)=0$, $DF(0)=0$, $F$ and $DF$ are
both computable because $f$ and $Df$ are computable functions by assumption.
Thus there is a computable modulus of continuity $d:\mathbb{N}\rightarrow
\mathbb{N}$ such that
\begin{equation}
|F(x)-F(y)|\leq2^{-m}|x-y|\ \ \mbox{whenever}\ \ |x|\leq2^{-d(m)
\ \&\ |y|\leq2^{-d(m)} \label{ine-0
\end{equation}
Since $Df$ is computable, all entries in the matrix $A$ are computable;
consequently, the coefficients of the characteristic polynomial
$\mbox{det}(A-\lambda I_{n})$ of $A$ are computable, where
$\mbox{det}(A-\lambda I_{n})$ denotes the determinant of $A-\lambda I_{n}$ and
$I_{n}$ is the $n\times n$ unit matrix. Thus all eigenvalues of $A$ are
computable, for they are zeros of the computable polynomial
$\mbox{det}(A-\lambda I_{n})$. Assume that $\lambda_{1},\ldots,\lambda_{k
,\mu_{k+1},\ldots,\mu_{n}$ (counting multiplicity) are eigenvalues of $A$ with
$Re(\lambda_{j})<0$ for $1\leq j\leq k$ and $Re(\mu_{j})>0$ for $k+1\leq j\leq
n$, where $Re(z)$ denotes the real part of the complex number $z$. Then two
rational numbers $\sigma>0$ and $\alpha>0$ can be computed from the
eigenvalues of $A$ such that $Re(\lambda_{j})<-(\alpha+\sigma)$ for $1\leq
j\leq k$ and $Re(\mu_{j})>\sigma$ for $k+1\leq j\leq n$. We break $\alpha$
into two parts for later use: Let $\alpha_{1}$ and $\alpha_{2}$ be two
rational numbers such that
\begin{equation}
0<\alpha_{1}<\alpha\text{ \ \ \ and \ \ \ }\alpha_{1}+\alpha_{2}=\alpha.
\label{Eq:alphas
\end{equation}
Let $M$ be a natural number such that $M>\max\{\alpha+\sigma,1\}$ and
$\max\{|\lambda_{j}|,|\mu_{l}|:1\leq j\leq k,k+1\leq l\leq n\}\leq M-1$. We
now construct two simple piecewise-smooth close curves $\Gamma_{1}$ and
$\Gamma_{2}$ in $\mathbb{R}^{2}$: $\Gamma_{1}$ is the boundary of the
rectangle with the vertices $(-\alpha-\sigma,M)$, $(-M,M)$, $(-M,-M)$, and
$(-\alpha-\sigma,-M)$, while $\Gamma_{2}$ is the boundary of the rectangle
with the vertices $(\sigma,M)$, $(M,M)$, $(M,-M)$, and $(\sigma,-M)$. Then
$\Gamma_{1}$ with positive direction (counterclockwise) encloses in its
interior all the $\lambda_{j}$ for $1\leq j\leq k$ and $\Gamma_{2}$ with
positive direction encloses all the $\mu_{j}$ for $k+1\leq j\leq n$ in its
interior. We observe that for any $\xi\in\Gamma_{1}\bigcup\Gamma_{2}$, the
matrix $A-\xi I_{n}$ is invertible. Since the function $g:\Gamma_{1
\bigcup\Gamma_{2}\rightarrow\mathbb{R}$, $g(\xi)=||(A-\xi I_{n})^{-1}||$, is
computable (see for example \cite{Zho09}), where $(A-\xi I_{n})^{-1}$ is the
inverse of the matrix $A-\xi I_{n}$, the maximum of $g$ on $\Gamma_{1
\bigcup\Gamma_{2}$ is computable. Let $K_{1}\in\mathbb{N}$ be an upper bound
of this computable maximum. Now for any $t\in\mathbb{R}$, from (5.47) of
\cite{Kat95},
\begin{align}
e^{At} & =-\frac{1}{2\pi i}\int_{\Gamma_{1}}e^{\xi t}(A-\xi I_{n})^{-1
d\xi-\frac{1}{2\pi i}\int_{\Gamma_{2}}e^{\xi t}(A-\xi I_{n})^{-1
d\xi\label{e-lin}\\
& =I_{\Gamma_{1}}(t)+I_{\Gamma_{2}}(t)\nonumber
\end{align}
We recall that $e^{At}$ is the solution to the linear equation $\dot{x}=Ax$.
Since $A$ is computable and integration is a computable operator, it follows
that $I_{\Gamma_{1}}$ and $I_{\Gamma_{2}}$ are computable. A simple
calculation shows that $||-\frac{1}{2\pi i}\int_{\Gamma_{1}}e^{t\xi}(A-\xi
I_{n})^{-1}d\xi||\leq4K_{1}Me^{-(\alpha+\sigma)t}/\pi$ for $t\geq0$ and
$||-\frac{1}{2\pi i}\int_{\Gamma_{2}}e^{t\xi}(A-\xi I_{n})^{-1}d\xi
||\leq4K_{1}Me^{\sigma t}/\pi$ for $t\leq0$. Let
\begin{equation}
K=4MK_{1} \label{eK
\end{equation}
Then
\begin{equation}
\mbox{$||I_{\Gamma_1}(t)||\leq Ke^{-(\alpha +\sigma) t}$ for
$t\geq 0$ \ and \ $||I_{\Gamma_2}(t)||\leq Ke^{\sigma t}$ for
$t\leq 0$} \label{eI
\end{equation}
The two estimates in (\ref{eI}) show that $I_{\Gamma_{1}}(t)$ and
$I_{\Gamma_{2}}(t)$ are stable component and unstable component of $e^{A}$.
The first step is now complete.
The second step in the construction is to compute a ball, $B(0, r)$,
surrounding the hyperbolic equilibrium point 0 such that $B(0, r)$ contains a
set of potential solutions $x(t, x_{0})$ to (\ref{e2}) satisfying the
conditions (i) and (ii) described in Theorem \ref{t1}. The details for step 2:
\newline
\noindent\textbf{Claim 1.} Consider the integral equation
\begin{equation}
u(t,a)=I_{\Gamma_{1}}(t)a+\int_{0}^{t}I_{\Gamma_{1}}(t-s)F(u(s,a))ds-\int
_{t}^{\infty}I_{\Gamma_{2}}(t-s)F(u(s,a))ds \label{e3
\end{equation}
where the constant vector $a$ is a parameter. If $u(t,a)$ is a continuous
solution to the integral equation, then it satisfies the differential equation
(\ref{e2}) with the initial condition $u(0,a)=I_{\Gamma_{1}}(0)a-\int
_{0}^{\infty}I_{\Gamma_{2}}(-s)F(u(s,a))ds$. \newline
\noindent\textbf{Proof of Claim 1.} See Appendix 1. \newline
We observe that the solution to the integral equation is the fixed point of
the operator defined on the right hand side of the equation (\ref{e3}). Next
we compute an integer $m_{0}$ such that $2^{-m_{0}}\leq\frac{\sigma}{4K}$, and
then set
\begin{equation}
r=2^{-d(m_{0})}/2K,\qquad B=B(0,r)=\{a\in\mathbb{R}^{n}:|a|<r\} \label{er
\end{equation}
where the computable function $d$ is as in (\ref{ine-0}). \newline
\noindent\textbf{Claim 2.} For any $a\in B$ and $t\geq0$, define the
successive approximations as follows:
\begin{align}
\label{e4}u^{(0)}(t, a) & = 0\\
u^{(j)}(t, a) & = I_{\Gamma_{1}}(t)a+\int_{0}^{t}I_{\Gamma_{1
}(t-s)F(u^{(j-1)}(s, a))ds\nonumber\\
& -\int_{t}^{\infty}I_{\Gamma_{2}}(t-s)F(u^{(j-1)}(s,a))ds, \ \ j\geq
1\nonumber
\end{align}
then the following three inequalities hold for all $j\in\mathbb{N}$:
\begin{equation}
\label{ine-1}|u^{(j)}(t,a)-u^{(j-1)}(t,a)|\leq K|a|e^{-\alpha_{1}t}/2^{j-1
\end{equation}
\begin{equation}
\label{ine-2}|u^{(j)}(t, a)|\leq2^{-d(m_{0})}e^{-\alpha_{1}t
\end{equation}
and for any $\tilde{a}\in B$,
\begin{equation}
\label{ine-3}|u^{(j)}(t, a)-u^{(j)}(t, \tilde{a})|\leq3K|a-\tilde{a}|\\
\end{equation}
\noindent\textbf{Proof of Claim 2.} See Appendix 2. \newline
It then follows from (\ref{e4}) and (\ref{ine-1}) that $\{u^{(j)
(t,a)\}_{j=1}^{\infty}$ is a computable Cauchy sequence effectively convergent
to the solution $u(t,a)$ of the integral equation (\ref{e3}), uniformly in
$t\geq0$ and $a\in B$. Consequently the solution, $t,a\mapsto u(t,a)$, is
computable. Furthermore, (\ref{ine-2}) and (\ref{ine-3}) imply that for all
$t\geq0$ and $a,\tilde{a}\in B$,
\begin{equation}
|u(t,a)|\leq2^{-d(m_{0})}e^{-\alpha_{1}t},\qquad|u(t,a)-u(t,\tilde
{a}|<3K|a-\tilde{a}| \label{ine-2-3-c
\end{equation}
Thus $\lim_{t\rightarrow\infty}u(t,a)=0\quad\mbox{for all $a\in
B$}$. Moreover, the first inequality in (\ref{ine-2-3-c}) shows that $u(t, a)$
satisfies condition (ii) of Theorem \ref{t1} with $\gamma=2^{-d(m_{0})}$,
$\epsilon=\alpha_{1}$, and $\delta=2^{-d(m_{0})}/2K$.\newline
Although Claim 2 shows that for any $a\in B$, the integral equation (\ref{e3})
has a computable solution $u(t, a)$ and this solution satisfies the conditions
(i) and (ii) of Theorem \ref{t1}, we may not take $B$ as a desired stable
manifold $S$ because if $u(0, a)\neq a$, then $u(t, a)$ is not a solution to
the equation (\ref{e0}) with the initial value $a$ at $t=0$. Nevertheless, the
set $B$ provides a pool of potential solutions to (\ref{e0}) on a stable
manifold. This leads us to the next step of the proof. \newline
The last step in the construction of the desired algorithm is to extract a
$k$-dimensional manifold $S$ from $B$, using a computable process, such that
for any $a\in S$, $u(0, a)=a$. Then if we set $x(t, a)=u(t,a)$ for $a\in S$,
by Claims 1 and 2, $x(t, a)$ is the solution to the differential equation
(\ref{e2}) with the initial value $a$ at $t=0$ and satisfies the conditions
(i) and (ii) of Theorem \ref{t1}. Now for the details.
Let $P_{1}=I_{\Gamma_{1}}(0)=-\frac{1}{2\pi i}\int_{\Gamma_{1}}(A-\xi
I_{n})^{-1}d\xi$ and $P_{2}=I_{\Gamma_{2}}(0)=-\frac{1}{2\pi i}\int
_{\Gamma_{2}}(A-\xi I_{n})^{-1}d\xi$. Then $P_{1}\mathbb{R}^{n}={\mathbb{E
}_{A}^{s}$, $P_{2}\mathbb{R}^{n}={\mathbb{E}}_{A}^{u}$, $\mathbb{R
^{n}={\mathbb{E}}_{A}^{s}\bigoplus{\mathbb{E}}_{A}^{u}$ with
$\mbox{dim}{\mathbb{E}}_{A}^{s}=k$ and $\mbox{dim}{\mathbb{E}}_{A}^{u}=n-k$,
$P_{j}P_{k}=\delta_{jk}P_{j}$ ($\delta_{jk}=1$ if $j=k$ and $\delta_{jk}=0$ if
$j\neq k$), and $P_{1}+P_{2}=I$ is the identity map on $\mathbb{R}^{n}$
(\emph{c.f.} $\S 1.5.3$ and $\S 1.5.4$, \cite{Kat95}; $\S 4.6$ of
\cite{Rob95}). Since $A$ is computable, so are $P_{1}$ and $P_{2}$. Moreover,
$I_{\Gamma_{1}}(t)P_{2}=0$ for any $t\in\mathbb{R}$ as the following
calculation shows: Let $R(\xi)$ denote $(A-\xi I_{n})^{-1}$. Then we hav
\begin{align*}
R(\xi_{1})-R(\xi_{2}) & =R(\xi_{1})(A-\xi_{2}I_{n})R(\xi_{2})-R(\xi
_{1})(A-\xi_{1}I_{n})R(\xi_{2})\\
& =R(\xi_{1})[(A-\xi_{2}I_{n})-(A-\xi_{1}I_{n})]R(\xi_{2})\\
& =R(\xi_{1})(\xi_{1}-\xi_{2})I_{n}R(\xi_{2})\\
& =(\xi_{1}-\xi_{2})R(\xi_{1})R(\xi_{2}).
\end{align*}
Using the last equation we obtain that for any $v\in\mathbb{R}^{n}$,
\begin{align}
I_{\Gamma_{1}}(t)P_{2}v & =\left( \frac{1}{2\pi i}\right) ^{2}\int
_{\Gamma_{1}}e^{\xi t}R(\xi)d\xi\int_{\Gamma_{2}}R(\xi^{\prime})vd\xi^{\prime
}\label{e5}\\
& =\left( \frac{1}{2\pi i}\right) ^{2}\int_{\Gamma_{1}}\int_{\Gamma_{2
}e^{\xi t}R(\xi)R(\xi^{\prime})vd\xi d\xi^{\prime}\nonumber\\
& =\left( \frac{1}{2\pi i}\right) ^{2}\int_{\Gamma_{1}}\int_{\Gamma_{2
}e^{\xi t}\frac{R(\xi)-R(\xi^{\prime})}{\xi-\xi^{\prime}}vd\xi d\xi^{\prime
}\nonumber\\
& =\left( \frac{1}{2\pi i}\right) ^{2}\int_{\Gamma_{1}}e^{\xi t}\left(
\int_{\Gamma_{2}}\frac{R(\xi)}{\xi-\xi^{\prime}}d\xi^{\prime}-\int_{\Gamma
_{2}}\frac{R(\xi^{\prime})}{\xi-\xi^{\prime}}d\xi^{\prime}\right)
vd\xi\nonumber\\
& =\left( \frac{1}{2\pi i}\right) ^{2}\int_{\Gamma_{1}}e^{\xi t}\left(
-\int_{\Gamma_{2}}\frac{R(\xi^{\prime})}{\xi-\xi^{\prime}}d\xi^{\prime
}\right) vd\xi\nonumber\\
& =\left( \frac{1}{2\pi i}\right) ^{2}\int_{\Gamma_{2}}R(\xi^{\prime
})\left( \int_{\Gamma_{1}}\frac{e^{\xi t}}{\xi^{\prime}-\xi}d\xi\right)
vd\xi^{\prime}=0\nonumber
\end{align}
A similar computation shows that for any $t\in\mathbb{R}$,
\begin{equation}
I_{\Gamma_{2}}(t)P_{1}v=0 \quad\mbox{and} \quad P_{2}I_{\Gamma_{2
}(t)v=I_{\Gamma_{2}}(t)v, \quad v\in\mathbb{R}^{n} \label{e6
\end{equation}
Now let us use these results to compute the projections of $u(0,a)$ in
${\mathbb{E}}_{A}^{s}$ and ${\mathbb{E}}_{A}^{u}$: for any $a\in\mathbb{R
^{n}$,
\begin{align}
P_{1}u(0,a) & =P_{1}\left( I_{\Gamma_{1}}(0)a-\int_{0}^{\infty
I_{\Gamma_{2}}(-s)F(u(s,a))ds\right) \nonumber\\
& =P_{1}P_{1}a-\int_{0}^{\infty}P_{1}I_{\Gamma_{2}}(-s)F(u(s,a))ds\nonumber\\
& =P_{1}a \label{Eq:P1_a
\end{align}
and
\begin{align*}
P_{2}u(0,a) & =P_{2}\left( I_{\Gamma_{1}}(0)a-\int_{0}^{\infty
I_{\Gamma_{2}}(-s)F(u(s,a))ds\right) \\
& =P_{2}P_{1}a-P_{2}\left( \int_{0}^{\infty}I_{\Gamma_{2}
(-s)F(u(s,a))ds\right) \\
& =-\int_{0}^{\infty}I_{\Gamma_{2}}(-s)F(u(s,a))ds
\end{align*}
We note that for any $a\in\mathbb{R}^{n}$,
\begin{align*}
I_{\Gamma_{1}}(t)a & =I_{\Gamma_{1}}(t)(P_{1}a+P_{2}a)\\
& =I_{\Gamma_{1}}(t)P_{1}a+I_{\Gamma_{1}}(t)P_{2}a\\
& =I_{\Gamma_{1}}(t)P_{1}a
\end{align*}
which implies that if the solution $u(t,a)$ of the integral equation
(\ref{e3}) is constructed by successive approximations (\ref{e4}), then
$P_{2}a$ does not enter the computation for $u(t,a)$ and thus may be taken as
zero. Therefore the projection of $u(0,a)$ in ${\mathbb{E}}_{A}^{u}$ satisfies
the equation
\begin{equation}
\label{e-P2}P_{2}u(0,a)=-\int_{0}^{\infty}I_{\Gamma_{2}}(-s)F(u(s,P_{1}a))ds
\end{equation}
Next we define a map $\phi_{A}:{\mathbb{E}}_{A}^{s}(r)\rightarrow{\mathbb{E
}_{A}^{u}$, $b\mapsto-\int_{0}^{\infty}I_{\Gamma_{2}}(-s)F(u(s,b))ds$ for
$b\in{\mathbb{E}}_{A}^{s}(r)$, where $r$ is the rational number defined in
(\ref{er}) and ${\mathbb{E}}_{A}^{s}(r)=\{b\in{\mathbb{E}}_{A}^{s}:|b|\leq
r/2\}$. We observe that the compact set ${\mathbb{E}}_{A}^{s}(r)$ is
computable since the closed set ${\mathbb{E}}_{A}^{s}$ is computable (proved
in Theorem \ref{t0}).
Obviously the map $\phi_{A}$ is computable. By Theorem 6.2.4 \cite{Wei00} the
compact set $\phi_{A}[{\mathbb{E}}_{A}^{s}(r)]$ is computable.
Now we are ready to define $S$:
\[
S=\{b+\phi_{A}(b):b\in{\mathbb{E}}_{A}^{s}(r)\}
\]
Then $S$ is a manifold of dimension $k$. For any $a\in S$, $a=b+\phi_{A}(b)$
for some $b\in\mathbb{E}_{A}^{s}(r)$. Since ${\mathbb{R}}^{n}={\mathbb{E}
_{A}^{s}\bigoplus{\mathbb{E}}_{A}^{u}$, $b\in{\mathbb{E}}_{A}^{s}$ and
$\phi_{A}(b)\in{\mathbb{E}}_{A}^{u}$, it follows that
\begin{equation}
\label{e-S}b=P_{1}a \quad\mbox{and} \quad\phi_{A}(b)=\phi_{A}(P_{1}a)=P_{2}a
\end{equation}
Combining (\ref{Eq:P1_a}), (\ref{e-P2}), and (\ref{e-S}) we obtain that for
any $a\in S$, $u(0,a)=P_{1}u(0,a)+P_{2}u(0,a)=P_{1}a+\phi_{A}(P_{1
a)=P_{1}a+P_{2}a=a$. Thus for any $a\in S$, $u(0,a)=a$; that is, $a$ is the
initial condition of $u(t,a)$ at $t=0$. \newline
\noindent\textbf{Claim 3.} $S$ is a computable closed subset of ${\mathbb{R
}^{n}$. \newline
\noindent\textbf{Proof.} Since ${\mathbb{E}}_{A}^{s}(r)$ is computable, there
is a computable sequence $\{b_{j}\}\subseteq{\mathbb{E}}_{A}^{s}(r)$ that is
effectively dense in ${\mathbb{E}}_{A}^{s}(r)$; that is, there is a computable
function $\psi:{\mathbb{N}}\rightarrow{\mathbb{N}}$ such that ${\mathbb{E
}_{A}^{s}(r)\subseteq\bigcup_{j=1}^{\psi(k)}B(b_{j},2^{-k})$ for all
$k\in{\mathbb{N}}$ (\emph{c.f.} \cite{Zho96}). The following estimate shows
that the computable sequence $\{b_{j}+\phi_{A}(b_{j})\}$ is effectively dense
in $S$, thus the closed manifold $S$ is computable. For any $b+\phi
_{A}(b),\tilde{b}+\phi_{A}(\tilde{b})\in S$, we have
\begin{align*}
& |b+\phi_{A}(b)-(\tilde{b}+\phi_{A}(\tilde{b}))|\\
& \leq|b-\tilde{b}|+|\phi_{A}(b)-\phi_{A}(\tilde{b})|\\
& =|b-\tilde{b}|+\left\vert -\int_{0}^{\infty}I_{\Gamma_{2}
(-s)F(u(s,b))ds+\int_{0}^{\infty}I_{\Gamma_{2}}(-s)F(u(s,\tilde{b
))ds\right\vert \\
& \leq|b-\tilde{b}|+\int_{0}^{\infty}||I_{\Gamma_{2}}(-s)||\cdot
|F(u(s,b))-F(u(s,\tilde{b}))|ds\\
& \leq|b-\tilde{b}|+2^{-m_{0}}|u(s,b)-u(s,\tilde{b})|\int_{0}^{\infty
}Ke^{-\sigma s}ds\\
& =|b-\tilde{b}|+\frac{K}{\sigma2^{m_{0}}}|u(s,b)-u(s,\tilde{b})|\\
& =|b-\tilde{b}|+\frac{K}{\sigma2^{m_{0}}}\cdot3K|b-\tilde{b}|
\end{align*}
The estimates (\ref{eI}), (\ref{ine-0}), (\ref{er}), and (\ref{ine-3}) are
used in the above calculation. The proof of claim 3 is complete.
For every $a\in S$, set $x(t, a)=u(t,a)$; then by Claims 1, 2, and 3, $x(t,a)$
is the solution to the differential equation (\ref{e2}) with initial value $a$
at $t=0$ and $x(t, a)$ satisfies the conditions $(i)$ and $(ii)$ of Theorem
\ref{t1}.
Finally we show that if $x(t,x_{0})$ is a solution to the differential
equation (\ref{e2}) satisfying $0<|x_{0}|<2^{-d(m_{0})}/4K^{2}$ but
$x_{0}\not \in S$, then there exists $t^{\prime}>0$ such that $|x(t^{\prime
},x_{0})|> 2^{-d(m_{0})}$. This proves the last part of the theorem if we set
$\eta=2^{-d(m_{0})}$ and $D=\{ x\in\mathbb{R}^{n}: |x|<2^{-d(m_{0})}/4K^{2
\}$. Indeed, if otherwise $|x(t,x_{0})|\leq2^{-d(m_{0})}$ for all $t\geq0$. We
show in the following that this condition implies $x_{0}\in S$, which is a contradiction.
Since $x(t,x_{0})$ is the solution to $\dot{x}=Ax+F(x)$ with the initial value
$x_{0}$, it follows that (\emph{c.f.} Theorem 4.8.2 \cite{Rob95})
\[
x(t,x_{0})=e^{At}x_{0}+\int_{0}^{t}e^{(t-s)A}F(x(s,x_{0}))ds
\]
which, using (\ref{e-lin}), can be rewritten as
\begin{align}
\label{e-rep} & x(t,x_{0})\\
& = I_{\Gamma_{1}}(t)x_{0}+I_{\Gamma_{2}}(t)x_{0}+\int_{0}^{t}I_{\Gamma_{1
}(t-s)F(x(s,x_{0}))ds+\int_{0}^{t}I_{\Gamma_{2}}(t-s)F(x(s,x_{0
))ds\nonumber\\
& = I_{\Gamma_{1}}(t)x_{0}+\int_{0}^{t}I_{\Gamma_{1}}(t-s)F(x(s,x_{0
))ds-\int_{t}^{\infty}I_{\Gamma_{2}}(t-s)F(x(s,x_{0}))ds+I_{\Gamma_{2
}(t)b\nonumber
\end{align}
where $b=x_{0}+\int_{0}^{\infty}I_{\Gamma_{2}}(-s)F(x(s,x_{0}))ds$
(\emph{c.f.} \S 1.5.3 \cite{Kat95}). Note that $b$ is well defined since we
assume that $|x(t,x_{0})|\leq2^{-d(m_{0})}$ for all $t\geq0$, then, from
(\ref{ine-0}), $|F(x(t,x_{0}))|$ is bounded for all $t\geq0$; consequently,
by\ (\ref{eI}), the integral $\int_{0}^{\infty}I_{\Gamma_{2}}(-s)F(x(s,x_{0
))ds$ converges. We also note that the first three terms in the above
representation for $x(t,x_{0})$ are bounded. Moreover, $I_{\Gamma_{2
}(t)b=I_{\Gamma_{2}}(t)P_{1}b+I_{\Gamma_{2}}(t)P_{2}b=I_{\Gamma_{2}}(t)P_{2}b$
since $I_{\Gamma_{2}}(t)P_{1}b=0$ by (\ref{e6}). We claim that if
$P_{2}b\not \equiv 0$, then $I_{\Gamma_{2}}(t)b$ is unbounded as
$t\rightarrow\infty$. We make use of the residue formula to prove the claim.
Recall that $\Gamma_{2}$ is a closed curve in the right-hand side of the
complex plane with counterclockwise orientation that contains in its interior
$\mu_{j}$, $k+1\leq j\leq n$, where $\mu_{j}$ are the eigenvalues of $A$ with
$Re(\mu_{j})>0$, which are the exact singularities of $R(\xi)=(A-\xi
I_{n})^{-1}$ in the right complex plane. Then by the residue formula,
\begin{equation}
\label{e-res}\frac{1}{2\pi i}\int_{\Gamma_{2}}e^{t\xi}R(\xi)d\xi=\sum
_{l=k+1}^{n}e^{\mu_{j}t}\mbox{res}_{\mu_{j}}R
\end{equation}
where $\mbox{res}_{\mu_{j}}R$ is the residue of $R$ at $\mu_{j}$. Since
$Re(\mu_{j})>0$, if $P_{2}b\not \equiv 0$, then
\begin{align*}
& I_{\Gamma_{2}}(t)b = I_{\Gamma_{2}}(t)P_{2}b\\
& =-\frac{1}{2\pi i}\int_{\Gamma_{2}}e^{t\xi}(A-\xi I_{n})^{-1}P_{2}bd\xi\\
& =-\sum_{l=k+1}^{n}e^{\mu_{j}t}res_{\mu_{j}}RP_{2}b
\end{align*}
is unbounded as $t\to\infty$. This is however impossible because the first
three terms in (\ref{e-rep}) are bounded and we have assumed that
$|x(t,x_{0})|\leq2^{-d(m_{0})}$ for all $t\geq0$. Therefore, $P_{2}b\equiv0$;
consequently, $I_{\Gamma_{2}}(t)b=I_{\Gamma_{2}}(t)P_{2}b=0$. This last
equation together with (\ref{e-rep}) shows that $x(t,x_{0})$ satisfies the
integral equation (\ref{e3}). Now let $x^{\prime}(t, x_{0})$ be the solution
to the integral equation (\ref{e3}) with parameter $x_{0}$ and constructed by
the successive approximations (\ref{e4}). Then
\[
x^{\prime}(t, x_{0})=I_{\Gamma_{1}}(t)x_{0}+\int_{0}^{t}I_{\Gamma_{1
}(t-s)F(x^{\prime}(s, x_{0}))ds-\int_{t}^{\infty}I_{\Gamma_{2}
(t-s)F(x^{\prime}(s, P_{1}x_{0}))ds
\]
By the uniqueness of the solution, $x(t, x_{0})=x^{\prime}(t, x_{0})$; in
particular, $x_{0}=x(0, x_{0})=x^{\prime}(0, x_{0})=P_{1}x_{0}-\int
_{0}^{\infty}I_{\Gamma_{2}}(-s)F(x^{\prime}(s, P_{1}x_{0}))ds$. Since
$P_{1}x_{0}\in\mathbb{E}^{s}_{A}$ and $\norm{P_1x_0}\leq
\norm{P_1}\norm{x_0}\leq K\cdot2^{d(m_{0})}/4K^{2}=r/2$ (recall that
$\norm{P_1}\leq K$ by (\ref{eI})), it follows that $P_{1}x_{0}\in
\mathbb{E}^{s}_{A}(r)$, which further implies that $x_{0}=P_{1}x_{0}-\int
_{0}^{\infty}I_{\Gamma_{2}}(-s)F(x^{\prime}(s, P_{1}x_{0}))ds=P_{1}x_{0
+\phi_{A}(P_{1}x_{0})\in S$. This contradicts the fact that $x_{0}$ is not on
$S$. The proof is complete.
\end{proof}
\begin{theorem}
Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a $C^{1}$-computable
function. Assume that the origin $0$ is a hyperbolic equilibrium point of
(\ref{e0}) such that $Df(0)$ has $k$ eigenvalues with negative real part and
$n-k$ eigenvalues with positive real part (counting multiplicity), $0\leq
k<n$. Let $x(t,x_{0})$ denote the solution to (\ref{e0}) with the initial
value $x_{0}$ at $t=0$. Then there is a (Turing) algorithm that computes a
$(n-k)$-dimensional manifold $U\subset{\mathbb{R}}^{n}$ containing $0$ such that
\begin{enumerate}
\item[(i)] For all $x_{0}\in U$, $\lim_{t\rightarrow-\infty}x(t,x_{0})=0$;
\item[(ii)] There are three positive rational numbers $\gamma$, $\epsilon$,
and $\delta$ such that $|x(t,x_{0})|\leq\gamma2^{\epsilon t}$ for all $t\leq0$
whenever $x_{0}\in U$ and $|x_{0}|\leq\delta$.
\end{enumerate}
Moreover, if $k>0$, then a rational number $\eta$ and a ball $D$ can be
computed from $f$ such that for any solution $x(t,x_{0})$ to the equation
(\ref{e0}) with $x_{0}\in D\setminus U$, $\{ x(t,x_{0}): \ t\leq
0\}\not \subset B(0, \eta)$ no matter how close the initial value $x_{0}$ is
to the origin.
\end{theorem}
\begin{proof}
The unstable manifold $U$ can be computed by the same procedure as the
construction of $S$ by considering the equation
\[
\dot{x}=-Ax-F(x(t))
\]
\end{proof}
The proof can be easily extended to show that the map: ${\mathfrak{F}
_{H}\rightarrow{\mathfrak{K}}\times{\mathfrak{K}}$, $f\mapsto(S_{f},U_{f})$,
is computable, where ${\mathfrak{F}}_{H}$ is the set of all $C^{1}$ functions
having the origin as a hyperbolic equilibrium point, ${\mathfrak{K}}$ is the
set of all compact subsets of ${\mathbb{R}}^{n}$, and $S_{f}$ and $U_{f}$ are
some local stable and unstable manifolds of $f$ at the origin respectively.
\section{Computability and non-computability of global stable/unstable
manifolds\label{Sec:NotComputable}}
Although the stable and unstable manifolds can be computed locally as shown in
the previous section, globally they may not be
necessarily computable.
In \cite{Zho09} it is shown that there exists a $C^{\infty}$ and
polynomial-time computable function $f: \mathbb{R}^{2}\to\mathbb{R}^{2}$ such
that the equation $\dot{x}=f(x)$ has a sink at the origin \textbf{0} and the
basin of attraction (also called the domain of attraction), $B_{f
(\mathbf{0})$, of $f$ at \textbf{0} is a non-computable open subset of
$\mathbb{R}^{2}$. Since a sink is a hyperbolic equilibrium point (with all
eigenvalues of the gradient matrix $Df(\mathbf{0})$ having negative real
parts) and the basin of attraction at a sink is exactly the global stable
manifold at this equilibrium point, we conclude immediately that the global
stable manifold is not necessarily computable.
One the other hand, it is also shown in \cite{Zho09} that, although not
necessarily computable, the basin of attraction $B_{f}(x_{0})$ is r.e. open
for a computable sink $x_{0}$ of a computable system $f$. In other words, the
open subset $B_{f}(x_{0})$ can be plotted from inside by a computable sequence
$\{B_{n}\}$ of rational open balls, $B_{f}(x_{0})=\bigcup B_{n}$, but one may
not know the rate at which $B_{f}(x_{0})$ is being filled up by $B_{n}$'s if
$B_{f}(x_{0})$ is non-computable. When an equilibrium point $x_{0}$ of the
system $\dot{x}=f(x)$ is a saddle (not a sink nor a source), then the global
stable (unstable) manifold $W_{f}^{s}(x_{0})$ ($W_{f}^{u}(x_{0})$) is in
general an $F_{\sigma}$-set of $\mathbb{R}^{n}$. As in the special case of the
basin of attraction, $W_{f}^{s}(x_{0})$ can also be plotted from inside by a
computable process. To make the concept precise, we introduce the following
definition. Let $\mathcal{F}(\mathbb{R}^{n})$ denote the set of all
$F_{\sigma}$-subsets of $\mathbb{R}^{n}$.
\begin{definition}
A function $f:X\rightarrow\mathcal{F}(\mathbb{R}^{n})$ is called
semi-computable if there is a Type-2 machine such that on any $\rho$-name of
$x\in X$, the machine computes as output a sequence $\{a_{j,k}\}$, $a_{j,k
\in\mathbb{Q}^{n}$, such that
\[
f(x)=\bigcup_{j=0}^{\infty}\overline{\{ a_{j,k} : k\in\mathbb{N}\}}
\]
where $\overline{A}$ denotes the closure of the set $A$.
\end{definition}
We call this function semi-computable because we can plot a dense subset of
the set $f(x)$ by a computable process, but we cannot tell in a finite amount
of time what the ``density" is.
\begin{theorem}
\label{t-5-1} The map $\psi: C^{1}(\mathbb{R}^{n})\times\mathbb{R}^{n
\to\mathcal{F}(\mathbb{R}^{n})$, $(f, x_{0})\to W^{s}_{f}(x_{0})$, is
semi-computable, where $(f, x_{0})\in\mbox{dom}(\psi)$ if $x_{0}$ is a
hyperbolic equilibrium point of $f$, and $W^{s}_{f}(x_{0})$ is the global
stable manifold of $f$ at $x_{0}$.
\end{theorem}
\begin{proof}
From Theorem \ref{t1} one can compute from $f$ and $x_{0}$ a compact subset
$S$ of $\mathbb{R}^{n}$, which is a local stable manifold of $f$ at $x_{0}$.
Note that the global stable manifold of $f$ at $x_{0}$ is the union of the
backward flows of $S$, $\emph{i.e.}$,
\[
W_{f}^{s}(x_{0})=\bigcup_{j=0}^{\infty}\phi_{-j}(S)
\]
where $\phi_{t}(a)$ is the flow induced by the equation $\dot{x}=f(x)$ at time
$t$ with the initial data $x(0)=a$. Since the sequence $\{\psi_{-j}(a)\}$ is
computable from $f$ and $a$ \cite{GZB07}, it follows that the sequence
$\{\phi_{-j}(S)\}$ of compact sets are computable from $f$ and $x_{0}$
(Theorem 6.2.4 \cite{Wei00}). In particular, a sequence $\{a_{j,k
\}_{j,k\in\mathbb{N}}\subset\mathbb{R}^{n}$ can be computed such that
$W_{f}^{s}(x_{0})=\bigcup_{j=0}^{\infty}\overline{\{a_{j,k}:k\in\mathbb{N}\}}$.
\end{proof}
The function $f$ in the counterexample mentioned at the beginning of this
section is $C^{\infty}$ but not analytic. It is an open problem whether or not
the stable/unstable manifold(s) of a computable analytic hyperbolic system
$\dot{x}=f(x)$ (\emph{i.e.} $f$ is computable and analytic) is computable. In
the following we present a negative answer to a weaker version of the open
problem; we show that it is impossible to compute uniformly the closure of the
global stable/unstable manifold from $(f, x_{0})$, where $f$ is analytic and
$x_{0}$ is a hyperbolic equilibrium point of $f$. Let's denote by
$\omega(\mathbb{R}^{n})$ the set of real analytic functions $f:\mathbb{R
^{n}\to\mathbb{R}^{n}$ and $\mathcal{A}(\mathbb{R}^{n})$ the set of all closed
subsets of $\mathbb{R}^{n}$. With Wijsman topology $\mathcal{W}$ on
$\mathcal{A}(\mathbb{R}^{n})$, $(\mathcal{A}(\mathbb{R}^{n}), \mathcal{W})$ is
a topological space that is separable and metrizable with a complete metric.
It can be shown that the Wijsman topology on $\mathcal{A}(\mathbb{R}^{n})$ is
the same as the topology induced by the $\rho$-names giving rise to the notion
of computable close subsets of $\mathbb{R}^{n}$ as defined in Def.
\ref{def_r_e_open} (\cite{BW99}). Thus if a map $F:X\to\mathcal{A
(\mathbb{R}^{n})$ is computable (with respect to above $\rho$-names for
elements of $\mathcal{A}(\mathbb{R}^{n})$), then $F$ is continuous (with
respect to Wijsman topology on $\mathcal{A}(\mathbb{R}^{n})$) (Corollary
3.2.12 of \cite{Wei00}).
\begin{theorem}
The map $\psi:\omega(\mathbb{R}^{n})\times\mathbb{R}^{n}\rightarrow
\mathcal{A}(\mathbb{R}^{n})$, $(f,x_{0})\rightarrow\overline{W_{f}^{u}(x_{0
)}$ (the closure of $W_{f}^{u}(x_{0})$), is not computable, where
$(f,x_{0})\in\mbox{dom}(\psi)$ if $x_{0}$ is a hyperbolic equilibrium point of
$f$, and $W_{f}^{u}(x_{0})$ is the global unstable manifold of $f$ at $x_{0}$.
\end{theorem}
\begin{proof}
Consider the following system $\dot{x}=f_{\mu}(x)$ taken from \cite{HSD04}
\begin{align}
x^{\prime} & =x^{2}-1\label{Eq:big_system}\\
y^{\prime} & =-xy+\mu(x^{2}-1)\nonumber
\end{align}
The system (\ref{Eq:big_system}) has two equilibria: $z_{1}=(-1,0)$ and
$z_{2}=(1,0)$. $Df_{\mu}(z_{1})$ has eigenvalues $-2$ and $1$, associated to
the eigenvectors $(-1,-2\mu/3)$ and $(0,1)$, respectively, and $Df_{\mu
(z_{2})$ has eigenvalues $2$ and $-1$, associated to the eigenvectors
$(-1,-2\mu/3)$ and $(0,1)$, respectively. Therefore both points $z_{1}$ and
$z_{2}$ are saddles (the behavior of the system is sketched in
Fig.\ \ref{fig:dynamics}). From this information and the fact that any point
$(-1,y)$ or $(1,y)$ can only move along the $y$ axis, one concludes that the
line $x=-1$ is the unstable manifold of $z_{1}$ and the line $x=1$ gives the
stable manifold of $z_{2}$. Note that $z_{1},z_{2}$ and the above manifolds do
not depend on $\mu.$
Let us now study the unstable manifold of $z_{2}$. It is split by the stable
manifold $x=1$, and hence there is a \textquotedblleft right\textquotedblrigh
\ as well as a \textquotedblleft left\textquotedblright\ portion of the
unstable manifold$.$ Let us focus our attention to the \textquotedblleft
left\textquotedblright\ portion. From the system (\ref{Eq:big_system}) one
concludes that any point $z$ near $z_{2}$, located to its left (i.e.\ $z<z_{2
$) will be pushed to the line $x=-1$ at a rate that is independent of the
$y$-coordinate of $z$.
When $\mu=0$, the eigenvector of $Df_{\mu}(z_{2})$ associated to the unstable
manifold is $(-1,0)$. Looking at (\ref{Eq:big_system}), one concludes that the
\textquotedblleft left\textquotedblright\ portion of the unstable manifold of
$z_{2}$ can only be the segment of line
\[
U_{l,0}=\{(x,0)\in\mathbb{R}^{2}|-1<x<1\}.
\]
Now let us analyze the case where $\mu<0$. Since $(-1,-2\mu/3)$ is the
eigenvector of $Df_{\mu}(z_{2})$ associated to the unstable manifold, as the
unstable manifold of $z_{2}$ moves to the left, its $y$-coordinate starts to
grow. The \textquotedblleft left\textquotedblright\ portion of the unstable
manifold is always above the line $y=0$ (if it could be $y=0$, the dynamics of
(\ref{Eq:big_system}) would push the trajectory upwards), and as soon as its
$x$-coordinate is less than 0 (this will eventually happen), the trajectory is
pushed upwards with $y$-coordinate converging to $+\infty$. Notice that the
closer $\mu$ is to $0$, the lesser the $y$-component of the trajectory grows,
and the closer to $z_{1}$ the unstable manifold will be.
If $U_{l,\mu}$ represents the left portion of the unstable manifold of $z_{2}$
and
\[
A=U_{l,0}\cup\{(-1,y)\in\mathbb{R}^{2}|0\leq y\}
\]
then one concludes that
\[
\lim_{\mu\rightarrow0^{-}}d(A,U_{l,\mu})=0
\]
where $d$ is the Hausdorff distance on $\mathbb{R}^{n}$.
Suppose that $\psi$ is computable. Then, in particular, the map $\chi
:\mathbb{R}\rightarrow\mathcal{A}(\mathbb{R}^{n})$ defined by $\chi
(\mu)=\overline{W_{f_{\mu}}^{u}(z_{2})}$, is also computable.
Since computable maps must be continuous (Corollary 3.2.12 of \cite{Wei00}),
this implies that $\chi$ should be a continuous map. But the point $\mu=0$ is
a point of discontinuity for $\chi$ since (without loss of generality, we
restrict ourselves to the semi-plane $x<1$
\[
\lim_{\mu\rightarrow0^{-}}\chi(\mu)=\lim_{\mu\rightarrow0^{-}}\overline
{U}_{l,\mu}=\overline{A}\neq\overline{U}_{l,0}=\chi(0).
\]
and hence this map cannot be computable, which implies that $\psi$ is not
computable either.
\end{proof}
\begin{center}
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=13cm]{dynamics}
\end{center}
\caption{Dynamics of the system \eqref{Eq:big_system}.
\label{fig:dynamics
\end{figure}
\end{center}
\section{The Smale horseshoe is computable}
In this section we show that Smale's horseshoe is computable.
\begin{theorem}
\label{thm_smale} The Smale Horseshoe $\Lambda$ is a computable (recursive)
closed subset in $I=[0,1]\times[0,1]$.
\end{theorem}
\begin{proof}
We show that $\Omega=I\setminus\Lambda$ is a computable open subset in $I$ by
making use of the fact \cite{Zho96}: An open subset $U\subseteq I$ is
computable if and only if there is a computable sequence of rational open
rectangles (having rational corner points) in $I$, $\{ J_{k}\}_{k=0}^{\infty
$, such that
\begin{itemize}
\item[(a)] $J_{k}\subset U$ for all $k\in{\mathbb{N}}$,
\item[(b)] the closure of $J_{k}$, $\bar{J}_{k}$, is contained in $U$ for all
$k\in{\mathbb{N}}$, and
\item[(c)] there is a recursive function $e:{\mathbb{N}}\to{\mathbb{N}}$ such
that the Hausdorff distance $d(I\setminus\cup_{k=0}^{e(n)}J_{k}, I\setminus
U)\leq2^{-n}$ for all $n\in{\mathbb{N}}$. \newline
\end{itemize}
Let $f:I\to{\mathbb{R}}^{2}$ be a map such that $\Lambda=\bigcap_{n=-\infty
}^{\infty}(f^{n}(I)\cap I)$ is the Smale horseshoe. Without loss of generality
assume that $f$ performs a linear vertical expansion by a factor of $\mu=4$
and a linear horizontal contraction by a factor of $\lambda=\frac{1}{4}$. For
each $n\in{\mathbb{N}}$, let $U_{n}=I\setminus\bigcap_{k=-n}^{n}(f^{k}(I)\cap
I)$. Then $I\setminus\Lambda=\bigcup_{n=0}^{\infty}U_{n}$. Moreover,
\begin{itemize}
\item[(1)] there exists a computable function $\alpha: {\mathbb{N}
\to{\mathbb{N}}$ such that $U_{n}$ is a union of $\alpha(n)$ rational open
rectangles in $I$,
\item[(2)] $d(\Lambda, I\setminus\bigcup_{k=0}^{n}U_{n})\leq(1/4)^{n}$.
\end{itemize}
Define $e:{\mathbb{N}}\to{\mathbb{N}}$, $e(n)=\sum_{k=0}^{n}\alpha(k)$,
$n\in{\mathbb{N}}$. Then it follows from the lemma that $I\setminus\Lambda$ is
indeed a computable open subset of $I$.
\end{proof}
\section{Conclusions}
We have shown that, locally, one can compute the stable and unstable manifolds
of some given hyperbolic equilibrium point, though globally these manifolds
are, in general, only semi-computable. It would be interesting to know if
these results are only valid to equilibrium points or can be extended e.g.\ to
hyperbolic periodic orbits. In \cite{GZ10} we provide an example which shows
that the global stable/unstable manifold cannot be computed for hyperbolic
periodic orbits. However, the question whether locally these manifolds can be
computed remains open.
\noindent\textbf{Appendix 1.} Proof of Claim 1. Assume that $u(t,a)$ is a
continuous solution to the integral equation (\ref{e3}). We show that it
satisfies the differential equation (\ref{e2}). To see this, we first
establish a relation between $I_{\Gamma_{j}}$, $j=1,2$, and the Jordan
canonical form of $A$:
\[
A=C\left(
\begin{array}
[c]{ll
P & 0\\
0 & Q
\end{array}
\right) C^{-1
\]
where $C$ is an $n\times n$ invertible matrix, $P$ is a $k\times k$ matrix
with eigenvalues $\lambda_{j}$, $1\leq j\leq k$, and $Q$ is a $(n-k)\times
(n-k)$ matrix with eigenvalues $\mu_{j}$, $k+1\leq j\leq n$. By (\ref{e-lin}),
we have
\[
e^{At}=I_{\Gamma_{1}}(t)+I_{\Gamma_{2}}(t)
\]
On the other hand, it is straightforward to show that
\[
e^{At}=C\left(
\begin{array}
[c]{cc
e^{Pt} & 0\\
0 & 0
\end{array}
\right) C^{-1}+C\left(
\begin{array}
[c]{ll
0 & 0\\
0 & e^{Qt
\end{array}
\right) C^{-1
\]
Since the first $k$ columns of $C$ consists of a basis of the stable subspace
$P_{1}\mathbb{R}^{n}$ and the last $(n-k)$ columns of $C$ is a basis of the
unstable subspace $P_{2}\mathbb{R}^{n}$, it then follows from (\ref{e5}) and
(\ref{e6}) that
\[
I_{\Gamma_{2}}(t)C\left(
\begin{array}
[c]{cc
e^{Pt} & 0\\
0 & 0
\end{array}
\right) C^{-1}=0,\quad\mbox{and}
\
\[
I_{\Gamma_{1}}(t)C\left(
\begin{array}
[c]{cc
0 & 0\\
0 & e^{Qt
\end{array}
\right) C^{-1}=0
\]
Thus
\[
I_{\Gamma_{1}}(t)=C\left(
\begin{array}
[c]{cc
e^{Pt} & 0\\
0 & 0
\end{array}
\right) C^{-1
\]
and
\[
I_{\Gamma_{2}}(t)=C\left(
\begin{array}
[c]{ll
0 & 0\\
0 & e^{Qt
\end{array}
\right) C^{-1
\]
(recall that $\mathbb{R}^{n}=P_{1}\mathbb{R}^{n}\bigoplus P_{2}\mathbb{R}^{n
$). Consequently the integral equation (\ref{e3}) can be written in the
following form:
\begin{align}
u(t,a) & =C\left(
\begin{array}
[c]{cc
e^{Pt} & 0\\
0 & 0
\end{array}
\right) C^{-1}a+\int_{0}^{t}C\left(
\begin{array}
[c]{cc
e^{P(t-s)} & 0\\
0 & 0
\end{array}
\right) C^{-1}F(u(s,a))ds\label{e-jordan}\\
& -\int_{t}^{\infty}C\left(
\begin{array}
[c]{ll
0 & 0\\
0 & e^{Q(t-s)
\end{array}
\right) C^{-1}F(u(s,a))ds\nonumber
\end{align}
It is known that if $u(t,a)$ is a continuous solution to (\ref{e-jordan}),
then it is the solution to (\ref{e2}) (\emph{c.f.} \S 2.7 \cite{Per01}). The
proof is complete. \newline
\noindent\textbf{Appendix 2.} Proof of Claim 2. For $a,\tilde{a}\in B$ and
$t\geq0$, where $B=B(0,r)=\{x\in\mathbb{R}^{n}:|x|<r\}$, $r=2^{-d(m_{0})}/2K$
($d$ is the computable function defined in (\ref{ine-0}), and $m_{0}$ is a
positive integer such that $2^{-m_{0}}\leq\frac{\sigma}{4K}$, define
\begin{align*}
u^{(0)}(t,a) & =0\\
u^{(j)}(t,a) & =I_{\Gamma_{1}}(t)a+\int_{0}^{t}I_{\Gamma_{1}
(t-s)F(u^{(j-1)}(s,a))ds\\
& -\int_{t}^{\infty}I_{\Gamma_{2}}(t-s)F(u^{(j-1)}(s,a))ds,\ \ j\geq1
\end{align*}
We show that the following three inequalities hold for all $j\in\mathbb{N}$:
\begin{align*}
|u^{(j)}(t,a)-u^{(j-1)}(t,a)| & \leq K|a|e^{-\alpha_{1}t}/2^{j-1}\\
|u^{(j)}(t,a)| & \leq2^{-d(m_{0})}e^{-\alpha_{1}t}\\
|u^{(j)}(t,a)-u^{(j)}(t,\tilde{a})| & \leq3K|a-\tilde{a}|
\end{align*}
We argue by induction on $j$. Since $u^{(0)}(t,a)=0$ for any $a$, by
(\ref{Eq:alphas}) and (\ref{eI}) we get $|u^{(1)}(t,a)|=||I_{\Gamma_{1
}(t)a||\leq Ke^{-(\alpha+\sigma)t}\cdot2^{-d(m_{0})}/2K<2^{-d(m_{0
)}e^{-\alpha_{1}t}$, and $|u^{(1)}(t,a)-u^{(1)}(t,\tilde{a})|=|I_{\Gamma_{1
}(t)(a-\tilde{a})|\leq Ke^{-(\alpha+\sigma)t}|a-\tilde{a}|\leq K|a-\tilde{a
|$, the three inequalities hold for $j=1$. The estimate (\ref{eI}) is used in
calculations here.
Assume that the three inequalities hold for all $k\leq j$. Then for $k=j+1$,
\begin{align*}
& |u^{(j+1)}(t,a)-u^{(j)}(t,a)|\\
& \leq\int_{0}^{t}||I_{\Gamma_{1}}(t-s)||\cdot|F(u^{(j)}(s,a))-F(u^{(j-1)
(s,a))|ds\\
& +\int_{t}^{\infty}||I_{\Gamma_{2}}(t-s)||\cdot|F(u^{(j)}(s,a))-F(u^{(j-1)
(s,a))|ds\\
& \leq\int_{0}^{t}||I_{\Gamma_{1}}(t-s)||\cdot2^{-m_{0}}|u^{(j)
(s,a)-u^{(j-1)}(s,a)|ds\\
& +\int_{t}^{\infty}||I_{\Gamma_{2}}(t-s)||\cdot2^{-m_{0}}|u^{(j)
(s,a)-u^{(j-1)}(s,a)|ds\\
& \leq2^{-m_{0}}\int_{0}^{t}Ke^{-(\alpha_{1}+\alpha_{2}+\sigma)(t-s)
\frac{K|a|e^{-\alpha_{1}s}}{2^{j-1}}ds+2^{-m_{0}}\int_{t}^{\infty
Ke^{\sigma(t-s)}\frac{K|a|e^{-\alpha_{1}s}}{2^{j-1}}ds\\
& =2^{-m_{0}}\frac{K^{2}|a|}{2^{j-1}}e^{-(\alpha_{1}+\alpha_{2}+\sigma)t
\int_{0}^{t}e^{(\alpha_{2}+\sigma)s}ds+2^{-m_{0}}\frac{K^{2}|a|}{2^{j-1
}e^{\sigma t}\int_{t}^{\infty}e^{(-(\alpha_{1}+\sigma)s}ds\\
& =2^{-m_{0}}\frac{K^{2}|a|}{2^{j-1}}e^{-(\alpha_{1}+\alpha_{2}+\sigma
)t}\frac{e^{(\alpha_{2}+\sigma)t}-1}{\alpha_{2}+\sigma}+2^{-m_{0}}\frac
{K^{2}|a|}{2^{j-1}}e^{\sigma t}\frac{0-e^{-(\alpha_{1}+\sigma)t}}{-(\alpha
_{1}+\sigma)}\\
& \leq2^{-m_{0}}\frac{K^{2}|a|}{2^{j-1}}\frac{e^{-\alpha_{1}t}}{\alpha
_{2}+\sigma}+2^{-m_{0}}\frac{K^{2}|a|}{2^{j-1}}\frac{e^{-\alpha_{1}t}
{\alpha_{1}+\sigma}\\
& <\frac{K|a|}{2^{j}}e^{-\alpha_{1}t}\quad(\mbox{recall that
$2^{-m_0}\leq \frac{\sigma}{4K}$})
\end{align*}
and furthermore,
\begin{align*}
& |u^{(j+1)}(t,a)|\\
& \leq|u^{(j)}(t,a)|+|u^{(j+1)}(t,a)-u^{(j)}(t,a)|\\
& \leq|u^{(j)}(t,a)|+\frac{K|a|}{2^{j}}e^{-\alpha_{1}t}\\
& \leq\sum_{k=1}^{j}\frac{K|a|}{2^{k}}e^{-\alpha_{1}t}\qquad
(\mbox{induction hypothesis on $u^{(k)}(t, a)$ for $k\leq j$})\\
& \leq2K|a|e^{-\alpha_{1}t}\leq2Ke^{-\alpha_{1}t}\cdot2^{-d(m_{0
)}/2K=2^{-d(m_{0})}e^{-\alpha_{1}t
\end{align*}
Lastly we show that if $|u^{(k)}(t,a)-u^{(k)}(t,\tilde{a})|\leq3K|a-\tilde
{a}|$ holds for all $k\leq j$, then it holds for $j+1$.
\begin{align*}
& |u^{(j+1)}(t,a)-u^{(j+1)}(t,\tilde{a})|\\
& =\left\vert I_{\Gamma_{1}}(t)(a-\tilde{a})+\int_{0}^{t}I_{\Gamma_{1
}(t-s)\left( F(u^{(j)}(s,a))-F(u^{(j)}(s,\tilde{a}))\right) ds-\right. \\
& -\left. \int_{t}^{\infty}I_{\Gamma_{2}}(t-s)\left( F(u^{(j)
(s,a))-F(u^{(j)}(s,\tilde{a}))\right) ds\right\vert \\
& \leq\left\vert I_{\Gamma_{1}}(t)(a-\tilde{a})\right\vert +\int_{0
^{t}||I_{\Gamma_{1}}(t-s)||\cdot\left\vert F(u^{(j)}(s,a))-F(u^{(j)
(s,\tilde{a}))\right\vert ds+\\
& \int_{t}^{\infty}||I_{\Gamma_{2}}(t-s)||\cdot\left\vert F(u^{(j)
(s,a))-F(u^{(j)}(s,\tilde{a}))\right\vert ds\\
& \leq K|a-\tilde{a}|+2^{-m_{0}}\left\vert u^{(j)}(s,a)-u^{(j)}(s,\tilde
{a})\right\vert \left( \int_{0}^{t}Ke^{-(\alpha+\sigma)(t-s)}ds+\int
_{t}^{\infty}Ke^{\sigma(t-s)}ds\right) \\
& \leq K|a-\tilde{a}|+2^{-m_{0}}\cdot3K|a-\tilde{a}|\cdot\left( \frac
{K}{\alpha+\sigma}+\frac{K}{\sigma}\right) \\
& =K|a-\tilde{a}|\left( 1+2^{-m_{0}}\frac{3K}{\alpha+\sigma}+2^{-m_{0}
\frac{3K}{\sigma}\right) \\
& \leq3K|a-\tilde{a}|\quad
(\mbox{Recall that $2^{-m_0}\leq \frac{\sigma}{4K}$})
\end{align*}
The proof is complete.\smallskip
\textbf{Acknowledgments.}
D.\ Gra\c{c}a was
partially supported by \emph{Funda\c{c}\~{a}o para a Ci\^{e}ncia e a
Tecnologia} and EU FEDER POCTI/POCI via SQIG - Instituto de
Telecomunica\c{c}\~{o}es. N. Zhong was partially supported by the Charles
Phelps Taft Memorial Fund of the University of Cincinnati.
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,840 |
class Group < ActiveRecord::Base
has_and_belongs_to_many :users
# TODO remove events_groups table
# has_and_belongs_to_many :events
validates :name, presence: true, uniqueness: true
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,898 |
suji had the most liked content!
new demo band #zekkooo_lalala_ will change to new band "ballad"
ゲルニカ (guernica) 1st digital single "Blast" has been made available since 4/17 https://linkco.re/bNfchd73 btw Ba.LEO is ex-lll-Ligro- (as a roadie), Dr.Kiro is ex-ナルコレプシィ (Narcolepsy) and has also supported bands like Re:lia and ballad. 2 announcements with English subs (!!!) They will be putting up new songs on their Youtube channel.
I understand, that's why we're here to help ^^ Don't consider it an attack when others put up where you left off; like I always say, newsposting is a team effort. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,129 |
Saj Huq
Creating a world class cyber ecosystem in the UK
As we close out our virtual cyber conference LORCA director, Saj Huq, reflects on the week in his closing remarks covering what we've learnt through delivering LORCA and why it's essential, now more than ever, to continue to support the UK's startup community.
This week was a bit of experiment for LORCA. We were originally supposed to run this event on 18 March – sadly, due to the impending lockdown, plans had to be changed. Undeterred by the pandemic, we set out to create the largest virtual event possible to showcase the achievements of British cyber startups, celebrate the growth of the UK cyber ecosystem and hear about the innovative cyber issues set to define our digital future.
Over the last five days, we've had Lords, Ministers, unicorns, professors, the biggest VCs in Europe, companies with the biggest technology budgets for cyber and fast-growth business founders, all convening to debate, engage and collaborate.
Why did we want to be more ambitious than ever before?
Because UK cyber startups have stepped up during the pandemic, providing resilience and security to the technology services that have allowed society to continue operating.
Because cyber startups have never attracted more investment than during the pandemic, and never experienced such potential for growth and success.
And because this is a coming of age moment for the UK cyber sector, the moment when it could stand alongside the biggest international cyber hubs as a truly world-class cyber ecosystem.
Consider this. In the worst economic crisis of our lifetimes, none of the 72 cohort companies within the LORCA programme have failed. Of course they've had challenges over the last six months, but every single cyber startup in our programme is alive and well and many have actually closed funding rounds and continued to grow during this period.
They are the future. And they need our continued support.
So anything less than a virtual experience that was bigger, better and further reaching than what we'd done before would have been a disservice to the entrepreneurs driving cyber innovation in 2020.
We've also tried to capture the breadth and influence of the cybersecurity industry through our content. We wanted to reflect the extent to which cyber is now a critical business issue and a theme which is increasingly woven through technology, business and policy trends now and into the future.
To work in cyber in 2020 is to be at the forefront of some of the biggest ethical, regulatory and political questions facing society.
Like all of those individuals and organisations who are a part of LORCA, what fascinates me just as much as the growth potential and commercialisation of our cohort companies is the opportunity to work alongside people who are developing technologies that will shape the future in years to come.
And to work alongside people who are trying to address some of the unanswered questions that technology has posed. Questions like how can we combat the wave of disinformation flooding social media? Who is responsible, when a cyber event renders a driverless car inoperable at 70 mph on the motorway? How can we create secure and accessible digital identities, to help society bridge a widening digital divide?
The digital world is the new Wild West, and it is cyber experts, working with those in adjacent fields, including ethics and public policy as we have heard this week, who are introducing rules and principles to govern it. If nothing else, cyber in 2020 means predicting and asking the right questions, in advance of them needing to be asked.
LORCA Live is our salute to everyone working in cybersecurity across startups, corporates, investors, Government and academia. We are with you every step of the way.
Why is the UK cyber ecosystem so important?
Throughout the week, we've talked about the importance of the UK cyber startup ecosystem.
I wanted to clarify the two key reasons why the success of cyber startups up and down the country is so critical to the future of the whole country.
The first is economic growth. We've spent so much time discussing the record levels of investment going into British cyber startups because it is the best indicator for the market's confidence and optimism in the growth potential of these businesses.
This means that the international investment market is looking to the UK to produce world-class cyber companies and create products that will be sold around the world.
Considering the economic crisis in which we find ourselves, this is such good news because it means cyber companies will be the ones driving economic growth, creating jobs and taking British IP worldwide.
In the same way that fintech emerged out of the financial crash as a new and growing sector of our economy, I believe that cyber can and will define the post-pandemic economic recovery, both directly, through the continued growth of its sectoral economic contribution, and indirectly, as an enabler for our wider digital economy, which has expanded and leapt forward by at least a decade in the space of a few short months. Cyber security is now equally as critical for the multinational corporation, as it is for the local baker who has pivoted their business during the last 6 months, and now accepts online orders and payments for the first time.
The second key reason for the importance of a robust and sustainable UK cyber ecosystem is because we're stepping into new and unknown territory.
The Covid-19 pandemic means we're walking into a perfect storm of an expanded attack surface, unknown vulnerabilities and low levels of cyber literacy and security hygiene. The use of data has never been more important, more people are working from home in insecure environments and we're spending more and more time online – buying food, watching content and connecting to families and friends.
Even before the pandemic, we knew there were challenges ahead. From driverless cars to digital identities, cyber-attacks launched by hostile states to an increasing dependence on tech-enabled national infrastructure. The problem is, as we've heard today, there is a realism gap; between the perceived risk of being online at an individual level, and the ultimate consequence when things go wrong.
Or, to frame this another way; we are in the midst of a global health pandemic. Concurrently, we have heard this week about the increase in attacks on the health sector. Today it is being reported that a murder investigation has been opened by the German police, after a patient died following a ransomware attack on a hospital.
Cyber resilience is a matter of national security, and the consequences of a concerted cyberattack on the UK means we can't afford to do anything other than maximise our investment, further accelerate innovation and create our best possible digital future.
The only way to do this is to support our cyber startup community, which is driving the new technologies and solutions that we don't even know we need yet.
In an increasingly digital world, where the physical and the digital is increasingly converging, and it becoming ever harder to envisage the collateral damage of an attack; we all have a responsibility, at some level, for our country's cybersecurity. And we need the tools and the knowledge to protect ourselves, our businesses and our economy.
We wouldn't be able to deliver LORCA alone.
I want to thank again the Department for Digital, Culture, Media & Sport for your continued support for LORCA; our wonderful corporate partners and our LORCA Live event sponsors – AWS and Splunk.
I'd also like to thank everyone who has supported LORCA over the last two years – and continues to support us. Our Industry Advisory Board and the cyber leaders who join our needs accelerators, workshops and give their time to work with and mentor our cohort companies.
We've learnt a lot through LORCA
What I've learnt from two years running LORCA is that we have plenty of reasons to be optimistic.
Despite the pandemic and the economic challenges which we may face in the coming months, the UK has a long-term track record of producing world-class cyber companies and innovations.
We have world-class academic institutions, we have a critical mass of global investors, we have a Government fully committed to supporting the cyber industry, we have a vibrant community of entrepreneurs and a wealth of expertise within industry collaborating with cyber startups day in, day out.
Of course, there are still challenges that need to be overcome.
The funding gap for early stage cyber businesses needs to be addressed to ensure the sustainability of the sector. Too few rounds are being raised by first-time entrepreneurs and those raising capital for the first time. Too much money continues to go to too few, later-stage startups, who have already survived the famous 'valley of death.'
Without more intervention and support for our early-stage startups, we risk limiting the potential of a generation of talent; talent which we will rely on to grow our way back to a prosperous economic future.
But through the thousands of hours that we've spent with our cohort companies, and the willingness from the wider industry to come together and support the next generation of cyber businesses in the UK, there is a clear appetite from entrepreneurs, business leaders and Government to learn, collaborate and champion the ecosystem as a whole.
Most importantly, I want to recognise the startup founders and the startup teams whose determination, grit and courage is the future of our country. Speaking directly to those entrepreneurs – I want to say that LORCA is here to support you, we admire what you're achieving, and we want to make your success a reality.
What LORCA could achieve
So what role does LORCA have to play in all of this?
In our live broadcast on the first day of LORCA Live, Andrew Roughan, managing director of Plexal, gave an update on what LORCA has achieved over the last two years.
And we're incredibly proud of the numbers. £153m raised by cohort companies, £26m in revenue generated and 844 contracts secured.
But I believe we're just getting started.
We've created an ecosystem to commercialise and scale cyber startups and the success of the 72 cohort companies we have supported shows what a difference Government backing combined with private sector support can make.
Our cohort companies are set to create 800 jobs by 2022. This could be a vital lifeline during the economic recovery as the employment landscape re-balances.
We've supported many digital skills initiatives and campaign to increase diversity and inclusion in cyber. We want to continue this work and stimulate startup growth in a way that makes cyber careers accessible to as many people as possible.
We're very proud that 58% of our cohort companies are based outside of London, with 36% based across regional areas of the UK, and 22% having moved to the UK through LORCA, to engage and grow into the UK market.
LORCA is an access point to growth capital, global industry and capital markets in London for startups around the country, and a critical springboard to international markets around the world.
We want to continue this and make our programme available to startups in the cyber clusters that are emerging nationwide.
We can't do this without the support of Government. And specifically, the support of DCMS has been essential to stimulating cyber startup success through our programmes. We are confident Government stands shoulder to shoulder with us in our ongoing commitment to creating a world-class cyber ecosystem in the UK.
These are exciting times to be a cyber entrepreneur in the UK. My hope is that LORCA continues to sit at the heart of the UK cyber ecosystem, helping our most promising start-ups and scale-ups realise their full potential, and helping to create a world-class cyber ecosystem for many years to come.
The conversation doesn't end here.
Our website and all of our content will be online and available on-demand. And LORCA Live will be returning in March 2021. Crucially, the debates this week are actually the start of an important narrative; of LORCA raising, exploring and debating the big issues in cyber security long beyond today.
Thank you to all of our speakers, sponsors, partners for making the week so successful and for enabling us to do this.
But I want to finish this where I started, by thanking our LORCA startup community. Thank you for your ambition, vision and determination, in creating new value at a time when it has never been more important to us all.
Article / The COVID-19 risk landscape
Article / Education and skills
Article / Growing UK cyber
Article / Emerging challenges
Day one live event with DCMS, Deloitte and Dell Technologies: watch playback
Should you be concerned about the security of your video conferencing platform?
We ask Darktrace: is AI a threat or an opportunity? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,306 |
Q: Ajax request sends url rather than data I have a very weird error running on my plugin. I am currently developing a plugin that will allow the user to create routes and specific dates for the route. The dates are more or less tickets that can be purchased by the visitor.
I am running ajax request for the data creation and removal of a date.
The return of these ajax requests are HTML snippets that are used to update the table that shows the available dates for the route.
When I create a route I get the proper HTML back and put it into the table via jQquery.
However, when I run pretty much the same request to delete a row from the table I do not get the same result back. Instead of getting the HTML back I am getting the URL back of my website. I have looked everywhere and cannot find anything why this could happen. The codex says, that in term of an error I get -1 or 0 back. The ajax request works fine and the row gets deleted just as I request.
I hope somebody could find the little devil who is preventing my code from being executed properly.
Here are the following files:
PHP:
witchroute/index.php
witchroute/wit_witchroute.class.php
witchroute/wit_date.class.php
HTML:
witchroute/views/date-display.html
witchroute/views/date-display-table.html
witchroute/views/date-insert-form.html
What you should know:
The ajax remove request is located in the wit_witchroute.class.php in the constructor
add_action('wp_ajax_remove_date_from_route', function()
{
$wit_date = wit_date::getInstance();
$wit_date->removeDate($_POST);
});
the actual method is in the wit_date.class.php
The jQuery request is started in the date-display.html with the following testing code:
jQuery(function() {
jQuery(".witchroute_delete_btn").click(function(){
var id = jQuery(this).attr('date-id');
var deleteData = { id: id,
action: "remove_date_from_route",
security: '<?php print wp_create_nonce("adsfadsfasdfadsfd"); ?>'
};
var r2 = jQuery.ajax({
url: ajaxurl,
type: "POST",
data: deleteData,
dataType: "html"
});
r2.done(function(msg) {
//jQuery("#witchroute_date_display_body").html(msg);
alert(msg);
});
});
});
Please not, that I am fairly new to the plugin development. I havent done any form validation yet as I am still testing, so please dont comment on that. However, if you had some tips for structuring or so I would appreciate any help- that might even help solving my problem ?
I am looking forward to your help and hope somebody can see what I cannot see :)
All the best,
Richard
A: Some Notes
*
*You shouldn't use anonymous/lambda fn, closures for hooks - Those can't get unregistered.
*Create your nonces, when localizing the script - See for ex. this answer.
*The ajaxurl isn't set by default. You'll have to define it (and let it point to admin-ajax.php) when localizing the script. Then you access it via the localized object (the 2nd argument inside wp_localize_script()) like this: your_obj_name.ajaxurl.
You should also take a look at the Codex article about Ajax in Plugins.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,090 |
\section{Introduction}
\label{sec:intro}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/mobile-former-overview1.png}
\end{center}
\vspace{-3mm}
\caption{\textbf{Overview of Mobile-Former}, which parallelizes MobileNet \cite{sandler2018mobilenetv2} (on the left side) and Transformer \cite{NIPS2017_transformer} (on the right side). Different with vision transformer \cite{dosovitskiy2021vit} that uses image patches to form tokens, the transformer in Mobile-Former takes \textit{very few learnable tokens} as input that are randomly initialized. \textit{Mobile} (refers to MobileNet) and \textit{Former} (refers to transformer) are communicated by a bidirectional bridge, which is modeled by the proposed light-weight across attention. Best viewed in color.}
\label{fig:overview}
\vspace{-2mm}
\end{figure}
Recently, Vision Transformer (ViT) \cite{dosovitskiy2021vit, touvron2020deit} demonstrates the advantage of global processing and achieves significant performance boost over CNNs. However, when constraining the computational budget within 1G FLOPs, the gain of ViT diminishes. If we further challenge the computational cost, MobileNet \cite{howard2017mobilenets, sandler2018mobilenetv2, Howard_2019_ICCV_mbnetv3} and its extensions \cite{Han_2020_CVPR_ghostnet, li2021micronet} still dominate their backyard (e.g. less than 300M FLOPs for ImageNet classification) due to their efficiency in local processing filters via decompostion of depthwise and pointwise convolution. This in turn naturally raises a question:
\begin{quote}
\textit{How to design \textbf{efficient} networks to \textbf{effectively} encode both local processing and global interaction?}
\end{quote}
A straightforward idea is to combine convolution and vision transformer. Recent works \cite{wu2021cvt, graham2021levit, Xiao-2021-early-cnns-help-transformers} show the benefit of combining convolution and vision transformer in \textit{series}, either using convolution at the beginning or intertwining convolution into each transformer block.
In this paper, we shift the design paradigm from \textit{series} to \textit{parallel}, and propose a new network that parallelizes MobileNet and transformer with a two-way bridge in between (see \Fref{fig:overview}). We name it \textit{Mobile-Former}, where \textit{Mobile} refers to MobileNet and \textit{Former} stands for transformer.
\textit{Mobile} takes the image as input and stacks mobile (or inverted bottleneck) blocks \cite{sandler2018mobilenetv2}. It leverages the efficient depthwise and pointwise convolution to extract local features at pixel level. \textit{Former} takes a few learnable tokens as input and stacks multi-head attention and feed-forward networks (FFN). These tokens are used to encode global features of the image.
\textit{Mobile} and \textit{Former} communicate over a two-way bridge to fuse local and global features. This is crucial since it feeds local features to \textit{Former's} tokens as well as introduces global views to every pixel of featuremap in \textit{Mobile}. We propose a light-weight cross attention to model this bidirectional bridge by (a) performing the cross attention at the bottleneck of \textit{Mobile} where the number of channel is low, and (b) removing projections on query, key and value ($\bm{W}^Q$, $\bm{W}^K$, $\bm{W}^V$) from \textit{Mobile} side.
This parallel structure with a bidirectional bridge leverages the advantage of both MobileNet and transformer. The decoupling of local and global features in parallel leverage MobileNet's efficiency in extracting local features as well as transformer's power in modeling global interaction. More important, this is achieved in an efficient way via a thin transformer (with very few tokens) and a light-weight bridge to exchange local and global features between \textit{Mobile} and \textit{Former}. The bridge and \textit{Former} consumes less than 20\% of the total computational cost, but significantly improve the representation capability. This showcases an efficient and effective implementation of part-whole hierarchy proposed in \cite{Hinton2021-part-whole}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/top-1.png}
\end{center}
\vspace{-3mm}
\caption{\textbf{Comparison among Mobile-Former, efficient CNNs and Vision Transformers}, in terms of accuracy-flop tradeoff. The comparison is performed on ImageNet classification. Mobile-Former consistently outperforms both efficient CNNs and vision transformers in low FLOP regime (from 25M to 500M MAdds). Note that we implement Swin \cite{liu2021Swin} and DeiT \cite{touvron2020deit} at low computational budget from 100M to 2G FLOPs. Best viewed in color.}
\label{fig:top-1}
\vspace{-2mm}
\end{figure}
Mobile-Former achieves solid performance on both image classification and object detection. For example, it achieves 77.9\% top-1 ImageNet accuracy at 294M FLOPs, outperforming MobileNetV3 \cite{Howard_2019_ICCV_mbnetv3} and LeViT \cite{graham2021levit} by a clear margin (see \Fref{fig:overview}). More important, Mobile-Former consistently outperforming both efficient CNNs and vision transformers from 25M to 500M FLOPs (shown in \Fref{fig:top-1}), showcasing the usage of transformer at the low FLOP regime where efficient CNNs dominate. Furthermore, when transferring from Image classification to object detection, Mobile-Former significantly outperforms MobileNetV3, e.g. gaining 8.6 AP (35.8 vs. 27.2) with even less computational cost.
Finally we note that \textit{exploring the optimal network parameters (e.g. width, height) in Mobile-Former is not a goal of this work}, rather we demonstrate that the parallel design of Mobile-Former provides an efficient and effective network architecture.
\section{Related Work}
\label{sec:related-work}
\noindent \textbf{Light-weight Convolutional Neural Networks (CNNs):} MobileNets \cite{howard2017mobilenets, sandler2018mobilenetv2, Howard_2019_ICCV_mbnetv3} proposes an efficient way to model local filter processing by using depthwise and pointwise convolution in an inverted bottleneck structure. \cite{Zhang_2018_CVPR, ma_2018_ECCV} proposes ShuffleNet that uses group convolution and channel shuffle to simplify pointwise convolution. Furthermore, MicroNet \cite{li2021micronet} presents micro-factorized convolution that optimizes the combination of inverted bottleneck and group convolution and achieves solid performance at extremely low FLOPs. Dynamic operators \cite{Hu_2018_CVPR, Yang2019CondConvCP, Chen2019DynamicCA, Chen2020DynamicReLU} have been studied to boost performance for MobileNet with negligible computational cost.
Other efficient operators include butterfly transform \cite{vahid_2020_CVPR}, cheap linear transformations in GhostNet \cite{Han_2020_CVPR_ghostnet}, and using cheap additions to trade massive multiplications in AdderNet \cite{Chen_2020_CVPR_addernet}. In addition, different architectures and compound scaling method have been studied. MixConv \cite{Tan-bmvc2019-mixconv} explores mixing up multiple kernel sizes, and Sandglass \cite{Daquan_2020_ECCV_RethinkingBS} flips the structure of inverted residual block. EfficientNet \cite{tan-ICML19-efficientnet, Tan_2020_CVPR} and TinyNet \cite{NEURIPS2020_e069ea4c} study the compound scaling of depth, width and resolution.
\vspace{2mm}
\noindent \textbf{Vision Transformers (ViT):} Recently, ViT \cite{dosovitskiy2021vit} and its follow-ups \cite{touvron2020deit, yuan2021tokens, liu2021Swin, dong2021cswin, Vaswani_2021_CVPR_halo} achieve impressive performance on multiple vision tasks. The original ViT requires training on large dataset such as JFT-300M to perform well. Later, DeiT \cite{touvron2020deit} demonstrates that good performance can be achieved on the smaller ImageNet-1K dataset by introducing several important training strategies. To enable ViT for high resolution images, several hierarchical transformers are proposed. For example, Swin \cite{liu2021Swin}
presents shifted window approach for computing self-attention within a local window and CSWin \cite{dong2021cswin} further improves it by introducing cross-shaped window self-attention. T2T-ViT \cite{yuan2021tokens} progressively converts the image to tokens by recursively aggregating neighboring tokens, such that the local structure can be well modeled. HaloNet \cite{Vaswani_2021_CVPR_halo} develops two attention extensions (blocked local attention and attention downsampling) that result in improvement of speed, memory usage and accuracy.
\vspace{2mm}
\noindent \textbf{Combination of CNNs and ViT:} Recent works \cite{Srinivas_2021_CVPR_bot, wu2021cvt, d2021convit, Xiao-2021-early-cnns-help-transformers, graham2021levit} show that combining convolution and transformer achieves improvement in prediction accuracy as well as training stability. BoTNet \cite{Srinivas_2021_CVPR_bot} shows significant improvement in instance segmentation and object detection by just replacing the spatial convolutions with global self-attention in the final three bottleneck blocks of a ResNet \cite{he2016deep}. ConViT \cite{d2021convit} improves ViT with soft convolutional inductive biases by introducing a gated positional self-attention (GPSA).
CvT \cite{wu2021cvt} introduces depthwise/pointwise convolution before each multi-head attention. LeViT \cite{graham2021levit} and ViT$_C$ \cite{Xiao-2021-early-cnns-help-transformers} use convolutional stem (stacking 3$\times$3 convolutions) to replace the patchify stem \cite{Xiao-2021-early-cnns-help-transformers}. LeViT and ViT$_C$ show clear improvement at the low FLOP regime. In this paper, we propose a different design that parallelizes MobileNet and transformer with bidirectional cross attention in between. Our approach is both efficient and effective, outperforming both efficient CNNs and ViT variants at low FLOP regime.
\section{Our Method: Mobile-Former}
\label{sec:method}
In this section, we describe the design process of Mobile-Former and its building blocks. The architecture is summarized in \Fref{fig:overview} and \Fref{fig:MF-block}.
\subsection{Overview}
Mobile-Former parallelizes MobileNet and transformer, and connects them by bidirectional cross attention (see \Fref{fig:overview}). In Mobile-Former, \textit{Mobile} (refers to MobileNet) takes an image as input ($\bm{X}_0 \in \mathbb{R}^{H \times W \times 3}$) and applies inverted bottleneck blocks \cite{sandler2018mobilenetv2} to extract local features. \textit{Former} (refers to transformer) takes learnable parameters (or tokens) as input, denoted as $\bm{Z}_0 \in \mathbb{R}^{M \times d}$ where $d$ and $M$ are the dimension and number of tokens, respectively. These tokens are randomly initialized and each represents a global prior of the image. This is different with Vision Transformer (ViT) \cite{dosovitskiy2021vit}, where tokens project the local image patch linearly. This difference is important as it significantly reduces the number of tokens ($M \leq 6$ in this paper), resulting in an efficient \textit{Former}.
\textit{Mobile} and \textit{Former} are connected by a two-way bridge where local and global features are fused bidirectionally. Here, we denote two directions of the bridge as \textit{Mobile$\rightarrow$Former} and \textit{Mobile$\leftarrow$Former}, respectively. We propose a light-weight cross attention to model this bidirectional bridge, which will be discussed next.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/overview.png}
\end{center}
\vspace{-3mm}
\caption{\textbf{Mobile-Former Block} that includes four modules: \textbf{\textit{Mobile}} sub-block modifies inverted bottleneck block in \cite{sandler2018mobilenetv2} by replacing ReLU with dynamic ReLU \cite{Chen2020DynamicReLU}. \textbf{\textit{Mobile$\rightarrow$Former}} uses light-weight cross attention to fuse local features into global features. \textbf{\textit{Former}} sub-block is a standard transformer block including multi-head attention and FFN. Note that the output of \textit{Former} is used to generate parameters for dynamic ReLU in \textit{Mobile} sub-block. \textbf{\textit{Mobile$\leftarrow$Former}} bridges from global to local features.}
\label{fig:MF-block}
\vspace{-2mm}
\end{figure}
\subsection{Low Cost Two-Way Bridge}
We leverage the advantage of cross attention to fuse the local features (from \textit{Mobile}) and global tokens (from \textit{Former}). Here, two changes are introduced to the standard cross attention for the sake of low computational cost: (a) computing the cross attention at the bottleneck of \textit{Mobile} where the number of channels is low, and (b) removing the projections ($\bm{W}_i^Q$, $\bm{W}_i^K$, $\bm{W}_i^V$) from \textit{Mobile} side where the number of positions is large, but keeping them at \textit{Former} side.
Let us denote the local feature map as $\bm{x}$, and the global tokens as $\bm{z}$. They are split as $\bm{x}=[\bm{x}_{h}]$ and $\bm{z}=[\bm{z}_{h}]$ ($1\leq h \leq H$) for multi-head attention with $H$ heads. The light-weight cross attention from local to global is defined as follows:
\begin{align}
\bm{z}^{out}&=\bm{z}+\left[Attention(\bm{z}_h\bm{W}_h^Q, \bm{x}_h, \bm{x}_h)\right]_{h=1:H}\bm{W}^O,
\label{eq:mobile2former}
\end{align}
where $\bm{W}_h^Q$ is the query projection matrix in the $h^{th}$ head, $\bm{W}^O$ is used to combine multiple heads together, $Attention(Q, K, V)$ is the standard attention function (in \cite{NIPS2017_transformer}) over query $Q$, key $K$, and value $V$ as follows:
\begin{align}
Attention(Q, K, V) = softmax(\frac{QK^T}{\sqrt{d_k}})V.
\label{eq:attn}
\end{align}
Note that the global feature $\bm{z}$ is the query and the local feature $\bm{x}$ is the key and value. $\bm{W}_h^Q$ and $\bm{W}^O$ are applied on global tokens $\bm{z}$. The diagram of this cross attention is shown in \Fref{fig:MF-block} (\textit{Mobile$\rightarrow$Former}).
In a similar manner, the cross attention from global to local is computed as:
\begin{align}
\bm{x}^{out}&=\bm{x}+ \left[ Attention(\bm{x}_h, \bm{z}_h\bm{W}_h^K, \bm{z}_h\bm{W}_h^V) \right]_{h=1:H},
\label{eq:Former2Mobile}
\end{align}
where $\bm{W}_h^K$ and $\bm{W}_h^V$ are the projection matrices for key and value. Here, the local feature $\bm{x}$ is the query and the global feature $\bm{z}$ is the key and value. The diagram of this cross attention is shown in \Fref{fig:MF-block} (\textit{Mobile$\leftarrow$Former}).
\subsection{Mobile-Former Block}
Mobile-Former can be decoupled as a stack of \textit{Mobile-Former blocks} (see \Fref{fig:overview}). Each block includes a \textit{Mobile} sub-block, a \textit{Former} sub-block, and a bidirectional bridge (\textit{Mobile$\leftarrow$Former} and \textit{Mobile$\rightarrow$Former}). The details of Mobile-Former block are shown in \Fref{fig:MF-block}.
\vspace{2mm}
\noindent \textbf{Input and Output:}
Mobile-Former block has two inputs: (a) local feature map $\bm{X}_i \in \mathbb{R}^{L \times C}$, which has $C$ channels and $L$ spatial positions ($L=hw$ where $h$ and $w$ are the height and width of the feature map), and (b) global tokens $\bm{Z}_i \in \mathbb{R}^{M \times d}$, where $M$ and $d$ are the number and dimension of tokens, respectively. Mobile-Former block outputs the updated local feature map $\bm{X}_{i+1}$ and global tokens $\bm{Z}_{i+1}$, which are used as input for the next ($i+1^{th}$) block. Note the number and dimension of global tokens are identical across all blocks.
\vspace{2mm}
\noindent \textbf{\textit{Mobile} sub-block:} \textit{Mobile} sub-block takes feature map $\bm{X}_i$ as input. It is slightly different to the inverted bottleneck block in \cite{sandler2018mobilenetv2} by replacing ReLU with dynamic ReLU \cite{Chen2020DynamicReLU} as the activation function after the first pointwise convolution and the 3$\times$3 depthwise convolution. Different with the original dynamic ReLU, in which the parameters are generated by applying two MLP layers on the average pooled feature, we save the average pooling by applying the two MLP layers ($\theta$ in \Fref{fig:MF-block}) on the first global token output from \textit{Former}. Note that the kernel size of depthwise convolution is 3$\times$3 for all blocks. The output of \textit{Mobile} sub-block, denoted as $\bm{X}_i^{hidden}$, is taken as the input for \textit{Mobile$\leftarrow$Former} (see \Fref{fig:MF-block}). Its computational complexity is $O(2LEC^2+9LEC)$, where $L$ is the number of spatial positions, $E$ is the channel expansion ratio, and $C$ is the number of channels before the expansion.
\noindent \textbf{\textit{Former} sub-block:} \textit{Former} sub-block is a standard transform block with a multi-head attention (MHA) and a feed-forward network (FFN). Here, we follow \cite{NIPS2017_transformer} to use post layer normalization. To save computations, we use expansion ratio 2 instead of 4 in FFN. Note that \textit{Former} sub-block is processed between the two-way cross attention, i.e. after \textit{Mobile$\rightarrow$Former} and before \textit{Mobile$\leftarrow$Former} (see \Fref{fig:MF-block}). Its complexity is $O(M^2d+Md^2)$. The first item relates to computing dot product between query and key, and aggregating values based on attention values, while the second item covers linear projections and FFN. Since \textit{Former} only has a few tokens ($M\leq6$), the first item $M^2d$ is ignorable.
\vspace{2mm}
\noindent \textbf{\textit{Mobile$\rightarrow$Former}:} The proposed light-weight cross attention (\Eref{eq:mobile2former}) is used to fuse local features $\bm{X}_i$ to global tokens $\bm{Z}_i$. Compared to the standard attention, the projection matrices for key $\bm{W}^K$ and value $\bm{W^V}$ (on the local features) are removed to save computations (shown in \Fref{fig:MF-block}). Its computational complexity is $O(LMC+MdC)$, where the first item relates to computing cross attention between local and global features and aggregating local features for each global token, and the second item is the complexity to project global features to the same dimension of local features $C$ and back to dimension $d$ after aggregation.
\vspace{2mm}
\noindent \textbf{\textit{Mobile$\leftarrow$Former}:} Here, the cross attention (\Eref{eq:Former2Mobile}) is on the opposite direction to \textit{Mobile$\rightarrow$Former}. It fuses global tokens $\bm{Z}_i$ to local features $\bm{X}_i$. The local features are the query and global tokens are key and value. Therefore, we keep the projection matrices for key $\bm{W}^K$ and value $\bm{W}^V$, but remove the projection matrix for query $\bm{W}^Q$ to save computations, as shown in \Fref{fig:MF-block}. The computational complexity is $O(LMC+MdC)$.
\vspace{2mm}
\noindent \textbf{Computational Complexity:} The four pillars of Mobile-Former block have different computational costs. \textit{Mobile} sub-block consumes the most computations ($O(2LEC^2+9LEC)$), as it grows linearly with the number of spatial positions $L$ and grows quadratically with the number of channels in local features $C$. \textit{Former} sub-block and the two-way bridge are computational efficient, consuming less than 20\% of total computation for all Mobile-Former models.
\subsection{Network Specification}
\noindent \textbf{Architecture:}
\Tref{table:mobile-former-specification} shows a Mobile-Former architecture at 294M FLOPs, which stacks 11 Mobile-Former blocks at different input resolutions. All Mobile-Former blocks have 6 global tokens with dimension 192. It starts with a 3$\times$3 convolution as stem which is followed by a lite bottleneck block at stage 1. The lite bottleneck block is proposed in \cite{li2021micronet}, which uses a 3$\times$3 depthwise convolution to expand the number of channels and uses a pointwise convolution to squeeze the number of channels. The classification head applies average pooling on the local features, concatenates with the first global token, and then passes through two fully connected layers with h-swish \cite{Howard_2019_ICCV_mbnetv3} in between.
\vspace{2mm}
\noindent \textbf{Downsample Mobile-Former Block:}
Note that stage 2--5 has a downsample variant of Mobile-Former block (denoted as Mobile-Former$^{\downarrow}$) to handle the spatial downsampling. In Mobile-Former$^{\downarrow}$, only the convolution layers in \textit{Mobile} sub-block are changed from three layers (pointwise$\rightarrow$depthwise$\rightarrow$pointwise) to four layers (depthwise$\rightarrow$pointwise$\rightarrow$depthwise$\rightarrow$pointwise), where the first depthwise convolution layer has stride 2. The number of channels expands in each depthwise convolution, and squeezes in the following pointwise convolution. This saves computations as the two costly pointwise convolutions are performed at the lower resolution after downsampling.
\vspace{2mm}
\noindent \textbf{Mobile-Former Variants:} Mobile-Former has 7 models of different computational costs from 26M to 508M FLOPs. They share the similar architecture, but have different widths and heights. We follow \cite{Xiao-2021-early-cnns-help-transformers} to refer our models by their flops, e.g. Mobile-Former-294M, Mobile-Former-96M. The details of network architecture for these Mobile-Former models are listed in the appendix (see \Tref{table:mf-achs}).
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{1.7mm}{
\begin{tabular}{c|c|c|c|c|c}
\specialrule{.1em}{.05em}{.05em}
Stage & Input & Operator & exp size & \#out & Stride \\
\specialrule{.1em}{.05em}{.05em}
tokens & 6$\times$192 & -- & -- & -- & -- \\
\hline
stem & 224$^2\times$3 & conv2d, 3$\times$3 & -- & 16 & 2 \\
\hline
1 & 112$^2\times$16 & bneck-lite & 32 & 16 & 1 \\
\hline
\multirow{2}{*}{2} & 112$^2\times$16 & Mobile-Former$^{\downarrow}$ & 96 & 24 & 2 \\
&56$^2\times$24 & Mobile-Former & 96 & 24 & 1 \\
\hline
\multirow{2}{*}{3} & 56$^2\times$24 & Mobile-Former$^{\downarrow}$ & 144 & 48 & 2 \\
& 28$^2\times$48 & Mobile-Former & 192 & 48 & 1 \\
\hline
\multirow{4}{*}{4} & 28$^2\times$48 & Mobile-Former$^{\downarrow}$ & 288 & 96 & 2 \\
& 14$^2\times$96 & Mobile-Former & 384 & 96 & 1 \\
&14$^2\times$96 & Mobile-Former & 576 & 128 & 1 \\
&14$^2\times$128 & Mobile-Former & 768 & 128 & 1 \\
\hline
\multirow{4}{*}{5} &14$^2\times$128 & Mobile-Former$^{\downarrow}$ & 768 & 192 & 2 \\
& 7$^2\times$192 & Mobile-Former & 1152 & 192 & 1 \\
&7$^2\times$192 & Mobile-Former & 1152 & 192 & 1 \\
&7$^2\times$192 & conv2d, 1$\times$1 & -- & 1152 & 1 \\
\hline
\multirow{4}{*}{head} & 7$^2\times$1152 & pool, 7$\times$7 & -- & -- & 1 \\
& 1$^2\times$1152 & concat w/ cls token & -- & 1344 & 1 \\
&1$^2\times$1344 & FC & -- & 1920 & 1 \\
&1$^2\times$1920 & FC & -- & 1000 & 1 \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Specification for Mobile-Former-294M}. ``bneck-lite" denotes light bottleneck block. ``Mobile-Former$^{\downarrow}$" denotes the Mobile-Former block for downsampling.}
\label{table:mobile-former-specification}
\end{table}
\section{Experimental Results}
We conduct experiments on ImageNet classification \cite{deng2009imagenet}, and COCO object detection \cite{lin2014microsoft} to evaluate the proposed Mobile-Former.
\subsection{ImageNet Classification}
We now evaluate our Mobile-Former models on ImageNet \cite{deng2009imagenet} classification. ImageNet has 1000 classes, including 1,281,167 images for training and 50,000 images for validation.
\vspace{2mm}
\noindent \textbf{Training Setup:} The image resolution is 224$\times$224. All models are trained from scratch using AdamW \cite{loshchilov2018decoupled} optimizer for 450 epochs with cosine learning rate decay. A batch size of 1024 is used. Data augmentation includes Mixup \cite{zhang2018mixup}, auto-augmentation \cite{Cubuk_2019_CVPR}, and random erasing \cite{zhong2020random}. Different combinations of initial learning rate, weight decay and dropout are used for models with different complexities, which are listed in the appendix (see \Tref{table:imagenet-hyper}).
\vspace{2mm}
\noindent \textbf{Main Results:}
\Tref{table:imagenet-cls-cnn} shows the comparison between Mobile-Former and classic efficient CNNs: (a) MobileNetV3 \cite{Howard_2019_ICCV_mbnetv3}, (b) EfficientNet \cite{tan-ICML19-efficientnet}, and (c) ShuffleNetV2 \cite{ma_2018_ECCV} and its extension WeightNet \cite{Ma_2020_eccv_WeightNetRT}. The comparison covers the FLOP range from 26M to 508M, organized in seven groups based on similar FLOPs. Mobile-Former consistently outperforms efficient CNNs with even less computational cost except the group around 150M FLOPs, where Mobile-Former costs slightly more FLOPs than ShuffleNet/WeightNet (151M vs. 138M/141M), but achieves significantly higher top-1 accuracy (75.2\% vs. 69.1\%/72.4\%). This demonstrates that our parallel design improves the representation capability efficiently.
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{0.7mm}{
\begin{tabular}{l|crr|c}
\specialrule{.1em}{.05em}{.05em}
Model & Input & \#Param & MAdds &Top-1 \\%& Top-5 \\
\specialrule{.1em}{.05em}{.05em}
MobileNetV3 Small 1.0$\times$ \cite{Howard_2019_ICCV_mbnetv3} & 160$^2$ & 2.5M & 30M & 62.8 \\
\textbf{Mobile-Former-26M} & 224$^2$ & 3.2M & \textbf{26M} & \textbf{64.0} \\%& \textbf{84.4} \\
\hline
MobileNetV3 Small 1.0$\times$ \cite{Howard_2019_ICCV_mbnetv3} & 224$^2$ & 2.5M & 57M & 67.5 \\
\textbf{Mobile-Former-52M} & 224$^2$ & 3.5M &\textbf{52M} & \textbf{68.7} \\%& \textbf{88.1} \\
\hline
MobileNetV3 1.0$\times$ \cite{Howard_2019_ICCV_mbnetv3} & 160$^2$ & 5.4M & 112M & 71.7 \\
\textbf{Mobile-Former-96M} & 224$^2$ & 4.6M & \textbf{96M} & \textbf{72.8} \\%& \textbf{90.5} \\
\hline
ShuffleNetV2 1.0$\times$ \cite{ma_2018_ECCV} & 224$^2$ & 2.2M & \textbf{138M} & 69.1 \\
ShuffleNetV2 1.0$\times$+WeightNet 4$\times$ \cite{Ma_2020_eccv_WeightNetRT} & 224$^2$ & 5.1M & 141M & 72.4 \\
MobileNetV3 0.75$\times$ \cite{Howard_2019_ICCV_mbnetv3} & 224$^2$ & 4.0M & 155M & 73.3 \\
\textbf{Mobile-Former-151M} & 224$^2$ & 7.6M & 151M & \textbf{75.2} \\
\hline
MobileNetV3 1.0$\times$ \cite{Howard_2019_ICCV_mbnetv3} & 224$^2$ & 5.4M & 217M & 75.2 \\
\textbf{Mobile-Former-214M} & 224$^2$ & 9.4M & \textbf{214M} & \textbf{76.7} \\%& \textbf{93.1} \\
\hline
ShuffleNetV2 1.5$\times$ \cite{ma_2018_ECCV} & 224$^2$ & 3.5M & 299M & 72.6 \\
ShuffleNetV2 1.5$\times$+WeightNet 4$\times$ \cite{Ma_2020_eccv_WeightNetRT} & 224$^2$ & 9.6M & 307M & 75.0 \\
MobileNetV3 1.25$\times$ \cite{Howard_2019_ICCV_mbnetv3} & 224$^2$ & 7.5M & 356M & 76.6 \\
EfficientNet-B0 \cite{tan-ICML19-efficientnet} & 224$^2$ & 5.3M & 390M & 77.1 \\%& 93.3 \\
\textbf{Mobile-Former-294M} & 224$^2$ & 11.4M & \textbf{294M} & \textbf{77.9} \\%& \textbf{93.7} \\
\hline
ShuffleNetV2 2$\times$ \cite{ma_2018_ECCV} & 224$^2$ & 5.5M & 557M & 74.5 \\
ShuffleNetV2 2$\times$+WeightNet 4$\times$ \cite{Ma_2020_eccv_WeightNetRT} & 224$^2$ & 18.1M & 573M & 76.5 \\
\textbf{Mobile-Former-508M} & 224$^2$ & 14.0M & \textbf{508M} & \textbf{79.3} \\% & \textbf{94.4} \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Comparing Mobile-Former with efficient CNNs} evaluated on ImageNet \cite{deng2009imagenet} classification. }
\label{table:imagenet-cls-cnn}
\end{table}
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{2.6mm}{
\begin{tabular}{l|crr|c}
\specialrule{.1em}{.05em}{.05em}
Model & Input & \#Param & MAdds &Top-1 \\%& Top-5 \\
\specialrule{.1em}{.05em}{.05em}
T2T-ViT-7 \cite{yuan2021tokens} & 224$^2$ & 4.3M & 1.2G & 71.7 \\
DeiT-Tiny \cite{touvron2020deit} & 224$^2$ & 5.7M & 1.2G & 72.2 \\
ConViT-Tiny \cite{d2021convit} & 224$^2$ & 6.0M & 1.0G & 73.1 \\
ConT-Ti \cite{yan2104contnet} & 224$^2$ & 5.8M & 0.8G & 74.9 \\
ViT$_C$ \cite{Xiao-2021-early-cnns-help-transformers} & 224$^2$ & 4.6M & 1.1G & 75.3 \\
ConT-S \cite{yan2104contnet} & 224$^2$ & 10.1M & 1.5G & 76.5 \\
Swin-1G \cite{liu2021Swin} $^{\ddag}$ & 224$^2$ & 7.3M & 1.0G & 77.3 \\
\textbf{Mobile-Former-294M} & 224$^2$ & 11.4M & \textbf{294M} & \textbf{77.9} \\%& \textbf{93.7} \\
\hline
PVT-Tiny \cite{wang2021pvtv1} & 224$^2$ & 13.2M & 1.9G & 75.1 \\
T2T-ViT-12 \cite{yuan2021tokens} & 224$^2$ & 6.9M & 2.2G & 76.5 \\
CoaT-Lite Tiny \cite{xu2021coscale} & 224$^2$ & 5.7M & 1.6G & 76.6 \\
ConViT-Tiny+ \cite{d2021convit} & 224$^2$ & 10.0M & 2G & 76.7 \\
DeiT-2G \cite{touvron2020deit} $^{\ddag}$ & 224$^2$ & 9.5M & 2.0G & 77.6 \\
CoaT-Lite Mini \cite{xu2021coscale} & 224$^2$ & 11.0M & 2.0G & 78.9 \\
BoT-S1-50 \cite{Srinivas_2021_CVPR_bot} & 224$^2$ & 20.8M & 4.3G & 79.1 \\
Swin-2G \cite{liu2021Swin} $^{\ddag}$ & 224$^2$ & 12.8M & 2.0G & 79.2 \\
\textbf{Mobile-Former-508M} & 224$^2$ & 14.0M & \textbf{508M} & \textbf{79.3} \\% & \textbf{94.4} \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Comparing Mobile-Former with Vision Transformer variants} evaluated on ImageNet \cite{deng2009imagenet} classification. Here, we choose ViT variants that use image resolution 224$\times$224 and are trained \textit{without} distillation from a teacher network. We split ViT models based on FLOPs (using 1.5G as threshold) and rank them based on top-1 accuracy. $^{\ddag}$ indicates our implementation.
}
\label{table:imagenet-cls-vit}
\end{table}
In \Tref{table:imagenet-cls-vit}, we compare Mobile-Former with multiple variants (DeiT \cite{touvron2020deit}, T2T-ViT \cite{yuan2021tokens}, PVT \cite{wang2021pvtv1}, ConViT \cite{d2021convit}, CoaT \cite{xu2021coscale}, ViT$_C$ \cite{Xiao-2021-early-cnns-help-transformers}, Swin \cite{liu2021Swin}) of vision transformer. All variants use image resolution 224$\times$224 and are trained \textit{without} distillation from a teacher network. Mobile-Former achieves higher accuracy but uses 3$\sim$4 times less computational cost. This is because that Mobile-Former uses significantly fewer tokens to model global interaction and leverages MobileNet to extract local features efficiently.
Note that our Mobile-Former (trained in 450 epochs without distillation) even outperforms LeViT \cite{graham2021levit} which leverages the distillation of a teacher network and much longer training (1000 epochs). Our method achieves higher top-1 accuracy (77.9\% vs. 76.6\%) but uses less computation (294M vs. 305M FLOPs) than LeViT.
\Fref{fig:top-1} compares Mobile-Former with more efficient CNNs (e.g. GhostNet \cite{Han_2020_CVPR_ghostnet}) and vision transformer variants with lower FLOPs (e.g. Swin \cite{liu2021Swin} and DeiT \cite{touvron2020deit} from 100M to 2G FLOPs). Note that we implement Swin and DeiT for the low computational budget from 100M to 2G FLOPs, by carefully reducing the network width and height. Mobile-Former clearly outperforms both CNNs and ViT variants, demonstrating the advantage of the parallel design to integrate MobileNet and transformer. Although vision transformers are inferior to efficient CNNs by a large margin, our work showcases that the transformer can also contribute to the low FLOP regime with proper architecture design.
\subsection{Object Detection}
\begin{table*}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{3.0mm}{
\begin{tabular}{l|ccc|ccc|cc|cc}
\specialrule{.1em}{.05em}{.05em}
\multirow{2}{*}{Model} & \multirow{2}{*}{AP} & \multirow{2}{*}{AP$_{50}$} & \multirow{2}{*}{AP$_{75}$} & \multirow{2}{*}{AP$_S$} & \multirow{2}{*}{AP$_M$} & \multirow{2}{*}{AP$_L$} & \multicolumn{2}{c|}{MAdds (G)} & \multicolumn{2}{c}{\#Params (M)} \\
& & & & & & & backbone & all & backbone & all \\
\specialrule{.1em}{.05em}{.05em}
ShuffleNetV2 \cite{ma_2018_ECCV} & 25.9 & 41.9 & 26.9 & 12.4 & 28.0 & 36.4 & 2.6 & 161 & 0.8 & 10.4\\
\textbf{Mobile-Former-151M} & \textbf{34.2} & 53.4 & 36.0 & 19.9 & 36.8 & 45.3 & \textbf{2.4} & 161 & 4.9 & 14.4\\
\hline
MobileNetV3 \cite{Howard_2019_ICCV_mbnetv3} & 27.2 & 43.9 & 28.3 & 13.5 & 30.2 & 37.2 & 4.7 & 162 & 2.8 & 12.3\\
\textbf{Mobile-Former-214M} & \textbf{35.8} & 55.4 & 38.0 & 21.8 & 38.5 & 46.8 & \textbf{3.6} & 162 & 5.7 & 15.2 \\%8.3 & 17.9\\
\hline
ResNet18 \cite{he2016deep} & 31.8 & 49.6 & 33.6 & 16.3 & 34.3 & 43.2 & 29 & 181 & 11.2 & 21.3 \\
\textbf{Mobile-Former-294M} & \textbf{36.6} & 56.6 & 38.6 & 21.9 & 39.5 & 47.9 & \textbf{5.2} & 164 & 6.5 & 16.1\\%9.3 & 18.8 \\
\hline
ResNet50 \cite{he2016deep} & 36.5 & 55.4 & 39.1 & 20.4 & 40.3 & 48.1 & 84 & 239 & 23.3 & 37.7 \\
PVT-Tiny \cite{wang2021pvtv1} & 36.7 & 56.9 & 38.9 & 22.6 & 38.8 & 50.0 & 70 & 221 & 12.3 & 23.0 \\
ConT-M \cite{yan2104contnet} & 37.9 & 58.1 & 40.2 & 23.0 & 40.6 & 50.4 & 65 & 217 & 16.8 & 27.0 \\
\textbf{Mobile-Former-508M} & \textbf{38.0} & 58.3 & 40.3 & 22.9 & 41.2 & 49.7 & \textbf{9.4} & 168 & 8.4 & 17.9\\%11.4 & 20.9\\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{COCO object detection results}. All models are trained on \texttt{train2017} and tested on \texttt{val2017}. All models are trained for 12 epochs (1$\times$) from ImageNet pretrained weights.}
\label{table:coco-det-results}
\end{table*}
\iffalse
\begin{table*}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{1.9mm}{
\begin{tabular}{l|c|c|ccc|ccc|cc}
\specialrule{.1em}{.05em}{.05em}
\multirow{2}{*}{Model} & \multirow{2}{*}{Framework} & \multirow{2}{*}{Iteration} & \multirow{2}{*}{AP} & \multirow{2}{*}{AP$_{50}$} & \multirow{2}{*}{AP$_{75}$} & \multirow{2}{*}{AP$_S$} & \multirow{2}{*}{AP$_M$} & \multirow{2}{*}{AP$_L$} & \multicolumn{2}{c}{MAdds} \\
& & & & & & & & & backbone & all \\
\specialrule{.1em}{.05em}{.05em}
ResNet50 & & & 36.5 & 55.4 & 39.1 & 20.4 & 40.3 & 48.1 & 84G & 239G \\
MobileNetV3 & & & & & & & & & & \\
ShuffleNetV2 & & & & & & & & & & \\
\hline
ViP-Mo & & & 36.5 & 56.7 & 38.6 & 23.4 & 39.7 & 48.4 & 15G & 166G \\
ViP-Ti & & & 39.7 & 60.6 & 42.2 & 23.9 & 42.9 & 53.0 & 29G & 181G \\
PVT-Tiny \cite{wang2021pvtv1} & & & 36.7 & 56.9 & 38.9 & 22.6 & 38.8 & 50.0 & 70G & 221G \\
PVTv2-B0 \cite{wang2021pvtv2}& & & 37.2 & 57.2 & 39.5 & 23.1 & 40.4 & 49.7 & -- & -- \\
PVTv2-B1 \cite{wang2021pvtv2}& & & 41.2 & 61.9 & 43.9 & 25.4 & 44.5 & 54.3 & -- & -- \\
ConT-m \cite{yan2104contnet} & & & 37.9 & 58.1 & 40.2 & 23.0 & 40.6 & 50.4 & & 217G \\
Mobile-Former-151M & & & 34.2 & 53.4 & 36.0 & 19.9 & 36.8 & 45.3 & 2.4G & 160.8G \\
Mobile-Former-213M & RetinaNet & 1$\times$ & 35.8 & 55.4 & 38.0 & 21.8 & 38.5 & 46.8 & 3.6G & 162.0G \\
Mobile-Former-294M & & & 36.6 & 56.6 & 38.6 & 21.9 & 39.5 & 47.9 & 5.2G & 163.6G \\
Mobile-Former-508M & & & 38.0 & 58.3 & 40.3 & 22.9 & 41.2 & 49.7 & 9.4G & 168.0G \\
\hline
ViP-Mo & & & 39.2 & 59.7 & 41.4 & 25.5 & 42.3 & 51.7 & 15G & 166G \\
ViP-Ti & & & 41.6 & 62.6 & 44.0 & 27.2 & 45.1 & 54.2 & 29G & 181G \\
ResNet50 & & & 39.0 & 58.4 & 41.8 & 22.4 & 42.8 & 51.6 & 84G & 239G \\
PVT-Tiny & & & 39.4 & 59.8 & 42.0 & 25.5 & 42.0 & 52.1 & 70G & 221G \\
Mobile-Former-151M & & & 36.5 & 55.9 & 38.6 & 24.2 & 38.9 & 48.1 & 2.4G & 160.8G \\
Mobile-Former-213M & RetinaNet & 3$\times$ & 37.9 & 57.6 & 40.1 & 23.3 & 40.9 & 48.5 & 3.6G & 162.0G \\
Mobile-Former-294M & & & 38.0 & 57.7 & 40.2 & 23.3 & 40.7 & 49.1 & 5.2G & 163.6G \\
Mobile-Former-508M & & & 39.2 & 59.2 & 42.0 & 24.5 & 41.7 & 51.2 & 9.4G & 168.0G \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{COCO object detection results}. All models are trained on \texttt{train2017} and tested on \texttt{val2017}. All models are trained for 12 epochs (1$\times$) from ImageNet pretrained weights.}
\label{table:coco-det-results}
\end{table*}
\fi
Object detection experiments are conducted on COCO 2017 \cite{lin2014microsoft}, which contains 118K training and 5K validation images. We use RetinaNet \cite{Lin_2017_ICCV_retinanet_focal} (one-stage) as the detection framework and follow the standard settings to use our Mobile-Former as backbone to generate feature map at multiple scales. All models are trained for 12 epochs (1×) from ImageNet pretrained weights.
In \Tref{table:coco-det-results}, we compare Mobile-Former with both CNNs (ResNet \cite{he2016deep}, MobileNetV3 \cite{Howard_2019_ICCV_mbnetv3}, ShuffleNetV2 \cite{ma_2018_ECCV}) and vision transformers (PVT \cite{wang2021pvtv1} and ConT \cite{yan2104contnet}). Mobile-Former significantly outperforms MobileNetV3 and ShuffleNet by 8.3+ AP under similar computational cost. Compared to ResNet and vision transform variants, our Mobile-Former achieves higher AP with significantly less FLOPs in the backbone. Specifically, our Mobile-Former-508M only takes 9.4G FLOPs in backbone but achieves 38.0 AP, outperforming ResNet-50, PVT-Tiny, and ConT-M which consume 7 times more computation (65G to 84G FLOPs) in the backbone. This demonstrates that Mobile-Former is also effective and efficient in the object detection task.
\section{Ablations and Discussion}
In this section, we show Mobile-Former is effective and efficient via several ablations performed on ImageNet classification. Here, Mobile-Former-294M is used and all models are trained for 300 epochs. Moreover, we visualize the two-way cross attention to understand the communication between \textit{Mobile} and \textit{Former}. Finally, the limitations of Mobile-Former are discussed.
\subsection{Mobile-Former is Effective}
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{1.2mm}{
\begin{tabular}{l|cc|ll}
\specialrule{.1em}{.05em}{.05em}
Model & \#Param & MAdds &Top-1 & Top-5 \\
\specialrule{.1em}{.05em}{.05em}
\textit{Mobile} (using ReLU) & 6.1M & 259M & 74.2 & 91.8 \\
+ \textit{Former} and Bridge & 10.1M & 290M & 76.8$_{(+2.6)}$ & 93.2$_{(+1.4)}$ \\
+ DY-ReLU in \textit{Mobile} & 11.4M & 294M & 77.8$_{(+1.0)}$ & 93.7$_{(+0.5)}$ \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Ablation of Former, Bridge and DY-ReLU} evaluated on ImageNet classification. Mobile-Former-294M is used.}
\label{table:ablation-former-bridge}
\end{table}
Mobile-Former is more effective than MobileNet as it encodes global interaction via \textit{Former}, resulting in more accurate prediction. As shown in \Tref{table:ablation-former-bridge}, adding \textit{Former} and bridge (\textit{Mobile$\rightarrow$Former} and \textit{Mobile$\leftarrow$Former}) only costs 10.6\% of the computational cost, but gains 2.6\% top-1 accuracy over the baseline that uses \textit{Mobile} alone. In addition, using the first global token to generate parameters for dynamic ReLU \cite{Chen2020DynamicReLU} in \textit{Mobile} sub-block (see \Fref{fig:MF-block}) achieves additional 1.0\% top-1 accuracy.
This validates our parallel design in Mobile-Former.
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{2.3mm}{
\begin{tabular}{c|cc|cc}
\specialrule{.1em}{.05em}{.05em}
Kernel Size in Mobile & \#Param & MAdds &Top-1 & Top-5 \\
\specialrule{.1em}{.05em}{.05em}
3$\times$3 & 11.4M & 294M & 77.8 & 93.7 \\
5$\times$5 & 11.5M & 332M & 77.9 & 93.9 \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Ablation of kernel size of depthwise convolution in Mobile} evaluated on ImageNet classification. Mobile-Former-294M is used.}
\label{table:ablation-kernel-size}
\end{table}
We also perform another ablation on the kernel size of the depthwise convolution in \textit{Mobile}, to validate the contribution of \textit{Former} and bridge on global interaction. \Tref{table:ablation-kernel-size} shows that the gain of increasing kernel size (from 3$\times$3 to 5$\times$5) is negligible. We believe this is because \textit{Former} and the bridge enlarge the reception field for \textit{Mobile} via fusing global features. Therefore, using larger kernel size is not necessary in Mobile-Former.
\subsection{Mobile-Former is Efficient}
Mobile-Former is not only effective in encoding both local processing and global interaction, but achieves this \textit{efficiently}. The key finding is that \textit{Former only requires a small number of global tokens}. Here, we first show Mobile-Former is efficient in terms of both the number of tokens and dimension. Then, we show the efficient parallel design of Mobile-Former is stable when removing FFN in \textit{Former} and replacing multi-head attention (MHA) with MLP.
\iffalse
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{1.9mm}{
\begin{tabular}{r|c|c|c}
\specialrule{.1em}{.05em}{.05em}
& Former & Bridge & Mobile \\
\specialrule{.1em}{.05em}{.05em}
\# Tokens & \cmark & & \\
Token Dimension & \cmark & & \\
FFN & \cmark & & \\
Attention & \cmark & \cmark & \\
Kernel Size & & & \cmark \\
Activation Function & & & \cmark \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Key components} for ablations.}
\label{table:ablation-components}
\end{table}
\fi
\vspace{2mm}
\noindent \textbf{Number of tokens in \textit{Former}:} \Tref{table:ablation-number-of-tokens} shows the ImageNet classification results for using different number of global tokens in \textit{Former}. The token dimension is 192. Interestingly, even a single global token achieves a good performance (77.1\% top-1 accuracy).
Additional improvement (0.5\% and 0.7\% top-1 accuracy) is achieved when using 3 and 6 tokens. But the improvement stops when more than 6 tokens are used. This ablation shows the compactness of global tokens which is important to the efficiency of Mobile-Former.
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{3.9mm}{
\begin{tabular}{c|cc|cc}
\specialrule{.1em}{.05em}{.05em}
\#tokens & \#Param & MAdds &Top-1 & Top-5 \\
\specialrule{.1em}{.05em}{.05em}
1 & 11.4M & 269M & 77.1 & 93.2 \\
3 & 11.4M & 279M & 77.6 & 93.6 \\
6 & 11.4M & 294M & 77.8 & 93.7 \\
9 & 11.4M & 309M & 77.7 & 93.8 \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Ablation of number of tokens} on ImageNet classification. Mobile-Former-294M is used.}
\label{table:ablation-number-of-tokens}
\end{table}
\vspace{2mm}
\noindent \textbf{Token Dimension:} \Tref{table:ablation-tokens-dimension} shows the results for different token lengths (or dimension). Here, six global tokens are used in \textit{Former}. The performance keeps improving from 76.8\% to 77.8\% when token dimension increases from 64 to 192, but converges when higher dimension is used. This further supports the efficiency of \textit{Former}. When using six tokens with dimension 192, the total computational cost of \textit{Former} and the bridge only consumes 12\% of the overall budget (35M/294M).
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{2.9mm}{
\begin{tabular}{c|cc|cc}
\specialrule{.1em}{.05em}{.05em}
Token Dimension & \#Param & MAdds &Top-1 & Top-5 \\
\specialrule{.1em}{.05em}{.05em}
64 & 7.3M & 277M & 76.8 & 93.1 \\
128 & 9.1M & 284M & 77.3 & 93.5 \\
192 & 11.4M & 294M & 77.8 & 93.7 \\
256 & 14.3M & 308M & 77.8 & 93.7 \\
320 & 17.9M & 325M & 77.6 & 93.6 \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Ablation of token dimension} on ImageNet classification. Mobile-Former-294M is used.}
\label{table:ablation-tokens-dimension}
\end{table}
\vspace{2mm}
\noindent \textbf{FFN in \textit{Former}:}
As shown in \Tref{table:ablation-ffn-mlp}, removing FFN introduces a small drop in top-1 accuracy (-0.3\%). Compared to the important role of FFN in the original vision transformer, FFN has limited contribution in Mobile-Former. We believe this is because FFN is not the only module for channel fusion in Mobile-Former. The 1$\times$1 convolution in \textit{Mobile} helps the channel fusion of local features, while the projection matrix $\bm{W}^O$ in Mobile$\rightarrow$Former (see \Eref{eq:mobile2former}) contributes to the fusion between local and global features.
\begin{table}[t!]
\begin{center}
\footnotesize
\setlength{\tabcolsep}{2.8mm}{
\begin{tabular}{cc|cc|cc}
\specialrule{.1em}{.05em}{.05em}
Attention & FFN & \#Param & MAdds &Top-1 & Top-5 \\
\specialrule{.1em}{.05em}{.05em}
MHA & \cmark & 11.4M & 294M & 77.8 & 93.7 \\
MHA & \xmark & 9.8M & 284M & 77.5 & 93.6 \\
MLP & \cmark & 10.5M & 284M & 77.3 & 93.5 \\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
}
\end{center}
\vspace{-2mm}
\caption{\textbf{Ablation of multi-head attention (MHA) and FFN} evaluated on ImageNet classification. Mobile-Former-294M is used.}
\label{table:ablation-ffn-mlp}
\end{table}
\vspace{2mm}
\noindent \textbf{Multi-head Attention (MHA) vs. MLP:} \Tref{table:ablation-ffn-mlp} shows the result of replacing multi-head attention (MHA) with MLP in both \textit{Former} and bridge (\textit{Mobile$\rightarrow$Former} and \textit{Mobile$\leftarrow$Former}). The top-1 accuracy drops from 77.8\% to 77.3\%. The implementation of MLP is more efficient by a single matrix multiplication, but it is static (i.e. not adaptive to different input images).
\subsection{Mobile-Former is Explainable}
\begin{figure*
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/vis-1-tokens.jpg}
\end{center}
\vspace{-4mm}
\caption{\textbf{Visualization of cross attention} on the two-way bridge: \textit{Mobile$\rightarrow$Former} and \textit{Mobile$\leftarrow$Former}. Mobile-Former-294M is used, which includes 6 tokens (each corresponds to a column). Four blocks with different input resolutions are selected and each has two attention heads that are visualized in two rows. Attention in \textit{Mobile$\rightarrow$Former} (left half) is normalized over pixels, showing the focused region per token. Attention in \textit{Mobile$\leftarrow$Former} (right half) is normalized over tokens showing the contribution per token at each pixel. The cross attention has more diversity across tokens at lower levels than higher levels. At the last block, token 2--5 have very similar cross attention.}
\label{fig:vis}
\end{figure*}
\begin{figure
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/vis-M2F.jpg}
\end{center}
\vspace{-4mm}
\caption{\textbf{Cross attention over featuremap for the first token in \textit{Mobile$\rightarrow$Former}} across all Mobile-Former blocks. Attention is normalized over pixels, showing focused regions. The focused regions change from low to high levels. The token starts paying more attention to edges/corners at block 2--4. Then it focused more on bigger region rather than scattered small pieces at block 5--12. The focused region shifts between the foreground (person and horse) and background (grass). Finally, it locks the most discriminative part (horse body and head) for classification.}
\label{fig:vis-m2f}
\end{figure}
\begin{figure
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/vis-F2M.jpg}
\end{center}
\vspace{-4mm}
\caption{\textbf{Cross attention in \textit{Mobile$\leftarrow$Former} separates foreground and background at middle layers}. Attention is normalized over tokens showing the contribution of different tokens at each pixel. Block 8 is chosen where the background pixels pay more attention to the first token and the foreground pixels pay more attention to the last token.
}
\label{fig:vis-f2m}
\end{figure}
To understand the collaboration between \textit{Mobile} and \textit{Former}, we visualize the cross attention on the two-way bridge (\textit{Mobile$\rightarrow$Former} and \textit{Mobile$\leftarrow$Former}) in \Fref{fig:vis}, \Fref{fig:vis-m2f} and \Fref{fig:vis-f2m}. The ImageNet pretrained Mobile-Former-294M is used, which includes 6 global tokens and 11 Mobile-Former blocks. We observe three interesting patterns.
First, the attention has more diversity across tokens at lower levels than higher levels. As shown in \Fref{fig:vis}, each column corresponds to a token, and each row corresponds to a head in the corresponding multi-head cross attention. Note that the attention is normalized over pixels in \textit{Mobile$\rightarrow$Former} (left half), showing the focused region per token. In contrast, the attention in \textit{Mobile$\leftarrow$Former} is normalized over tokens, comparing the contribution of different tokens at each pixel.
Clearly, the six tokens at block 3 and 5 have different cross attention patterns in both \textit{Mobile$\rightarrow$Former} and \textit{Mobile$\leftarrow$Former}. Similar attention patterns among tokens is clearly observed at block 8. At block 12, the last five tokens share very similar attention pattern. Note that the first token is class token feed into the classifier head. The similar observation has been identified in recent studies on ViT \cite{zhou2021refiner, zhou2021deepvit, touvron2021going}.
Second, the focused regions of global tokens change progressively from low to high levels. \Fref{fig:vis-m2f} shows the cross attention over pixels for the first token in \textit{Mobile$\rightarrow$Former}. This token begins focusing on local features, e.g. edges/corners (at block 2-4). Then it pays more attention to regions with connected pixels. Interestingly, the focused region shifts between foreground (person and horse) and background (grass) across blocks. Finally, it locates the most discriminative region (horse body and head) for classification.
Thirdly, the separation between foreground and background is surprisingly found at middle layers (e.g. block 8) of \textit{Mobile$\leftarrow$Former}. \Fref{fig:vis-f2m} shows the cross attention over 6 tokens for each pixel at featuremap. Clearly, the foreground and background are separated by the first and last tokens. This shows that some global tokens learn meaningful prototypes that cluster similar pixels.
\subsection{Limitations}
The major limitation of Mobile-Former is the model size. This is due to two reasons. Firstly, the parallel design is not efficient in terms of parameter sharing as \textit{Mobile}, \textit{Former} and bridge have their own parameters. Although \textit{Former} is efficient in computation due to the small amount of tokens, but it does not save the numer of parameters. Second, Mobile-Former consumes many parameters in the classification head (two fully connected layers) when performing ImageNet classification task. For instance, Mobile-Former-294M spends 40\% (4.6M of 11.4M) parameters in the classification head. The model size problem mitigates when switching from image classification to object detection task, as the classification head is removed. We will explore the parameter efficiency in the future work.
\section{Conclusion}
This paper presents Mobile-Former, a new parallel design of MobileNet and Transformer with two-way bridge in between to communicate. It leverages the efficiency of MobileNet in local processing and the advantage of Transformer in encoding global interaction. This design is not only effective to boost accuracy, but also efficient to save computational cost. It outperforms both efficient CNNs and vision transformer variants on image classification and object detection in the low FLOP regime with a clear margin. We hope Mobile-Former encourage new design of efficient CNNs and transformers.
{\small
\bibliographystyle{ieee_fullname}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,922 |
{"url":"https:\/\/codereview.stackexchange.com\/questions\/260012\/i-am-trying-to-filter-a-queryset-based-on-dates-i-keep-getting-a-duplicate-erro\/260024","text":"# I am trying to filter a queryset based on dates, I keep getting a duplicate error on the for loops,How do I refactor these code to remove duplicates? [closed]\n\nI am trying to filter the events table to get ongoing events and complete events, but pycharm keeps underlining my code because of duplicated code. How do I prevent for loop code duplicates on the two functions below? Better yet, how do I optimally refactor these two functions? Thanks\n\n\ndef get_ongoing_events():\nongoing_events = Events.objects.filter(\nQ(event_begin_datetime__lte=current_time),\nQ(event_end_datetime__gt=current_time),\n)\nfor event in ongoing_events:\nevent.event_status = 'ongoing'\nevent.save()\n\nevent.venue.status = 'booked'\nevent.venue.save()\n\nreserve_data = dict()\nreserve_data[\"sensor_id\"] = event.venue.sensor_id\nreserve_data[\"status\"] = event.venue.status\n\nreturn reserve_data\n\ndef get_complete_reservation():\n\"\"\"\nUpdate reservations and sensors\n:return:\n\"\"\"\ncompleted_events = Events.objects.filter(\nQ(reservation_begin_datetime__lt=current_time),\nQ(reservation_end_datetime__lte=current_time),\n)\n\nfor event in completed_events:\nevent.reservation_status ='complete'\nevent.save()\n\nevent.venue.status = 'free'\nevent.venue.save()\nreserve_data = dict()\nreserve_data[\"sensor_id\"] = event.venue.sensor_id\nreserve_data[\"status\"] = event.venue.status\nreturn reserve_data\n$$$$\n\n\u2022 The current question title, which states your concerns about the code, applies to too many questions on this site to be useful. The site standard is for the title to simply state the task accomplished by the code. Please see How do I ask a good question?. Apr 26, 2021 at 7:30\n\u2022 Apr 26, 2021 at 7:30\n\n# Factoring duplication out\n\nWhenever I have (near-)duplicate code, what I do is take a look at the repeated code. The parts that change ever so slightly are going to be controlled with arguments to functions and etc, whereas the parts that remain the same will just be left more-or-less as-is.\n\nHaving said that, here is what I see:\n\n\u2022 the queries being used change, so those are function arguments;\n\u2022 the attributes of the event being updated change, so those are function arguments;\n\u2022 the final dict being generated is the same, so we leave it the same.\n\nFrom your code alone, here is a suggested modification:\n\ndef filter_and_update_events(Qs, event_changes, venue_changes):\nfor event in Events.objects.filter(*Qs)\nfor attr, new_value in event_changes.items():\nsetattr(event, attr, new_value)\nevent.save()\nfor attr, new_value in venue_changes.items():\nsetattr(event.venue, attr, new_value)\nevent.venue.save()\n\nreserve_data = dict()\nreserve_data[\"sensor_id\"] = event.venue.sensor_id\nreserve_data[\"status\"] = event.venue.status\nreturn reserve_data\n\ndef get_ongoing_events():\nreturn filter_and_update_events(\n[\nQ(event_begin_datetime__lte=current_time),\nQ(event_end_datetime__gt=current_time),\n],\n{\"event_status\": \"ongoing\"},\n{\"status\": \"booked\"},\n)\n\ndef get_complete_reservation():\n\"\"\"\nUpdate reservations and sensors\n:return:\n\"\"\"\nreturn filter_and_update_events(\n[\nQ(reservation_begin_datetime__lt=current_time),\nQ(reservation_end_datetime__lte=current_time),\n],\n{\"reservation_status\": \"complete\"},\n{\"status\": \"free\"},\n)\n\n\nNotice that I used setattr docs to set the attributes of the event and the venue. Also notice that my new function has a bit of duplication in the loops, but that is the easiest way to deal with the fact that venue is inside event and there is no obvious way to use setattr to deal with the nesting. If you need to expand your function to update even more things inside event, then I would also recommend creating a helper function that takes an event and a \"setting name\" and sets it, so that filter_and_update_events doesn't need to take one dictionary per object inside event.\n\nDoes this make sense?\n\n\u2022 If there is already an answer, it might be better not to edit the question, especially the code in the question since everyone needs to be able to see the code as the first reviewer saw it. Apr 26, 2021 at 12:18\n\u2022 This makes sense and works like a charm, thankyou @RGS, just add .index() during filter so as to avoid the unpacking elements error. for attr, new_value in event_changes.index(): Apr 27, 2021 at 8:04\n\u2022 @pythonista woops that slipped, I meant for attr, new_value in event_changes.items():, and similarly for the venues, sorry for that. Answer has been fixed.\n\u2013\u00a0RGS\nApr 27, 2021 at 8:37\n\u2022 oops meant .items(), thanks for the fix Apr 29, 2021 at 11:30\n\nDuplication could be a non-problem, but is often an alarm that something is misplaced.\n\nThis repeated piece of code\n\n for event in completed_events:\nevent.reservation_status = 'complete'\nevent.save()\n\nevent.venue.status = 'free'\nevent.venue.save()\nreserve_data = dict()\nreserve_data[\"sensor_id\"] = event.venue.sensor_id\nreserve_data[\"status\"] = event.venue.status\nreturn reserve_data\n\n\nIs problematic because the action to be taken when an even is ongoing\/reserved\/etc should be performed by the event itself; that's why the status and the venue objects are inside the event, because they are in its domain. I see that you already have a Events class, maybe there is also an Event class?. I suggest you to have a Event class that would do something like this\n\n# Perhaps this constants could be stored in the Events class?\nONGOING = 'ongoing'\nCOMPLETE = 'complete'\nVENUES_STATUS = {\nONGOING: 'booked',\nCOMPLETE: 'free'\n}\n\nclass Event:\ndef trigger(self, new_status, save=True):\nif new_status == ONGOING:\nself.event_status = ONGOING\nelif new_status == COMPLETE:\nself.reservation_status = COMPLETE\nelse:\npass # you may have other statuses here?\nif new_status in VENUES_STATUS:\nself.venue.status = VENUES_STATUS[new_status]\nif save:\nself.save()\nself.venue.save()\n\ndef data(self): # this could be moved to the Venue class, also a better name can be definitely found.\nreturn {\n\"sensor_id\": self.venue.sensor_id,\n\"status\": self.venue.status\n}\n\n\nHaving that the loop becomes (Here I'm guessing what you're trying to collect, because in your code you have a return inside the for loop; that would always return the first event data.)\n\ndef get_complete_reservation():\ncompleted_events = Events.objects.filter(\nQ(reservation_begin_datetime__lt=current_time),\nQ(reservation_end_datetime__lte=current_time),\n)\ndata = []\n\nfor event in completed_events:\nevent.trigger(ONGOING)\ndata.append(event.data())\nreturn data\n\n\n\nThe name of the function doesn't really say what you're doing, your not just \"getting\" the events, you are triggering a transaction of status from one to another, so I would rewrite that like this:\n\ndef trigger_new_status(selected_events, new_status):\nfor event in selected_events:\nevent.trigger(new_status)\nreturn [event.data() for event in selected_events]\n\ndef main():\ncompleted_events = Events.objects.filter(\nQ(reservation_begin_datetime__lt=current_time),\nQ(reservation_end_datetime__lte=current_time),\n)\nongoing_events = Events.objects.filter(\nQ(event_begin_datetime__lte=current_time),\nQ(event_end_datetime__gt=current_time),\n)\ncompleted_events_data = trigger_new_status(completed_events, COMPLETE)\nongoing_events_data = trigger_new_status(ongoing_events, ONGOING)\n\n\n\nAnother thing: if you have the possibility to modify the filter method, it would be better to use a functional approach:\n\ndef filter(self, filter_function):\nreturn [event for event in self.events if filter_function(event)]\n\n\nand that would be used like\n\nevents.filter(lambda e: e.reservation_end < today and e.reservation_begin> today)\n\n\n\nalthough your code has duplication, but it is also modular. Depending on code functionality, you can trade-off this modularity with less duplication.\n\nhow about merging the 2 functions and returning two lists like as follows? this will remove code duplication, however, the trade-off would be less code-modularity.\n\nAlso, rather than using a dictionary to store booked and free sensor ids, you can just use lists with descriptive names as I have done.\n\ndef get_booked_and_free_events():\nongoing_events = Events.objects.filter(\nQ(event_begin_datetime__lte=current_time),\nQ(event_end_datetime__gt=current_time),\n)\ncompleted_events = Events.objects.filter(\nQ(reservation_begin_datetime__lt=current_time),\nQ(reservation_end_datetime__lte=current_time),\n)\nfor event in ongoing_events:\nevent.event_status = 'ongoing'\nevent.save()\n\nevent.venue.status = 'booked'\nevent.venue.save()\n\nbooked_sensors = []\nbooked_sensors.append(event.venue.sensor_id)\n\nfor event in completed_events:\nevent.reservation_status ='complete'\nevent.save()\n\nevent.venue.status = 'free'\nevent.venue.save()\n\nfree_sensors = []\nfree_sensors.append(event.venue.sensor_id)\n\nreturn booked_sensors, free_sensors\n`","date":"2022-08-19 11:31:16","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\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2867142856121063, \"perplexity\": 3733.0642663227036}, \"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-2022-33\/segments\/1659882573667.83\/warc\/CC-MAIN-20220819100644-20220819130644-00463.warc.gz\"}"} | null | null |
Графство Сайпрес () — муніципальний район в Канаді, у провінції Альберта.
Населення
За даними перепису 2016 року, муніципальний район нараховував 7662 жителів, показавши зростання на 6,2%, порівняно з 2011-м роком. Середня густина населення становила 0,6 осіб/км².
З офіційних мов обидвома одночасно володіли 210 жителів, тільки англійською — 7 405, тільки французькою — 5, а 25 — жодною з них. Усього 805 осіб вважали рідною мовою не одну з офіційних, з них 5 — українську.
Працездатне населення становило 73,8% усього населення, рівень безробіття — 7,2% (9% серед чоловіків та 5,1% серед жінок). 69,5% були найманими працівниками, 30% — самозайнятими.
Середній дохід на особу становив $66 213 (медіана $41 920), при цьому для чоловіків — $83 898, а для жінок $47 264 (медіани — $55 488 та $30 816 відповідно).
31,2% мешканців мали закінчену шкільну освіту, не мали закінченої шкільної освіти — 17,7%, 51,2% мали післяшкільну освіту, з яких 24,7% мали диплом бакалавра, або вищий, 10 осіб мали вчений ступінь.
Населені пункти
До складу муніципального району входять місто Медисин-Гет, містечко Редкліфф, а також хутори, інші малі населені пункти та розосереджені поселення.
Клімат
Середня річна температура становить 5,4°C, середня максимальна – 25,1°C, а середня мінімальна – -17,9°C. Середня річна кількість опадів – 332 мм.
Примітки
Муніципалітети Альберти
Засновані в Канаді 1985 | {
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This page, Delays may soon be over in scheduling breath test cases , is offered by
Berkshire District Attorney's Office
Press Release Delays may soon be over in scheduling breath test cases
Berkshire District Attorney Paul J. Caccaviello
Media Contact for Delays may soon be over in scheduling breath test cases
Frederick A. Lantz
Call Frederick A. Lantz at (413) 443-5951
Email Frederick A. Lantz at fred.a.lantz@state.ma.us
PITTSFIELD — Since 2014 there have been delays in criminal trials in Massachusetts involving defendants charged with OUI who submitted to a breath test using the Draeger 9510 breath testing instrument. A group of defendants challenged the admissibility of these test results.
Today all Massachusetts District Attorney's Offices and legal counsel for the defendants filed in the Concord District Court a Joint Stipulation of Facts and Recommended Resolution to the Defendants' Motion for Sanctions. This is a mutual effort to resolve the litigation that has delayed the criminal trials of numerous OUI defendants throughout the Commonwealth.
The Office of Alcohol Testing (OAT), a unit within the Massachusetts State Police Crime Laboratory, initially, and in accordance with a court order, provided annual calibration and certification data for the widely-used Draeger 9510 breath testing instrument. A Concord District Court judge reviewed the documents and found that some of the testing devices were not properly certified. He ruled in January 2017 that the Draeger 9510 data results could not be automatically used by the Commonwealth at OUI trials. This affected all OUI cases in the Commonwealth where a breath test result was obtained from a Draeger 9510 between June of 2011 and September 14, 2014.
A few months later, the Concord District Court judge subsequently learned that OAT had withheld from both prosecutors and defense counsel a number of documents that recorded failed calibration tests. These documents were provided to prosecutors and defense counsel on August 17, 2017. The late disclosure of these documents resulted in additional delay in resolving the OUI cases and caused the defendants to file suit and seek monetary sanctions against OAT for withholding this exculpatory evidence.
The Joint Stipulation of Facts and the Recommended Resolution filed today in Concord District Court, includes an agreement by the prosecutors and defense counsel to expand the period of time, to be determined by the Court, when the breath test results are not automatically admissible as evidence by the Commonwealth. The Commonwealth further agrees not to admit these breath test results at trials in OUI cases within this period, except for cases alleging motor vehicle homicide by OUI, OUI resulting in serious bodily injury, manslaughter by motor vehicle, and OUI 5th Offense or greater. The parties also agree that OAT will apply for national accreditation for the laboratory by August of 2019.
The state-wide delay in resolving OUI cases with a Draeger 9510 breath test result has affected 75 cases in Berkshire County as of August 2017.
The Concord District Court has the matter under advisement at this time.
Thanks, your message has been sent to Berkshire District Attorney's Office! | {
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Featured / Sports / The Prairie Fire / March 5, 2014
Winter weather Knox's first hurdle
With the goal of another MWC South championship in mind, the Prairie Fire have only been able to vicariously take the diamond. Subzero temperatures and consistent accumulation of snow has resigned the baseball team to the confines of the T. Fleming Fieldhouse, where the Prairie Fire have been forced to split the four tennis courts with the softball team, who are also inside due to the inclement weather.
Senior Peter Marquardt, who anchored the Knox pitching staff last year with a team-leading 49 innings pitched and 36 strikeouts, was quick to point out defense as the predominant difficulty of the situation, saying, "The biggest problem with indoor practice is the fact that our outfielders aren't able to get any real fly balls. They've been practicing their reads with tennis balls, which is nothing like a game situation. They're just not ready quite yet."
Senior Peter Marquardt pitches against St. Norbert in the first round of the MWC Tournament. Marquardt, one of the anchors of the pitching staff last season, has remained optimistic on the team's season outlook despite some reservation of the in-game readiness of the Prairie Fire squad. (Courtesy of the Office of Communications)
It would be easy for the novelty and fresh confidence associated with the nascent season to dwindle, but Marquardt remains optimistic. "Despite all we've had to put up with, I feel like our team is still very well prepared for our upcoming game as well as our Florida trip," he said. "We've been really getting after it in practice every day, getting the necessary work in to assure we're ready. Although we can't go outside, we've still been coming into practice with the mindset of getting better every single day, and I definitely think we're succeeding at that."
While indoor and outdoor practices share a common structure, one of the larger challenges of indoor practice is the inability to experience the game-like feel that outdoor practices have, an issue that has required some serious innovation on the part of the coaching staff. To compensate, the team has been running more scrimmage-oriented drills and games, such as a nine-man bunt drill, having pitchers throw to hitters off the mound in a cage so as hitters can get looks and swings off live pitching, or even full-blown scrimmages (which prove very difficult given space constraints and shared time of the fieldhouse with the softball team).
Junior Andrew Sparrow delivers a pitch to junior outfielder Michael Bohnhoff during a bunting drill in T. Fleming Fieldhouse on Tuesday. The inclement weather has forced the Prairie Fire indoors for longer than usual, forcing them to run more innovative drills like the one above. (Casey Mendoza/TKS)
Despite the challenges of indoor work, senior Michael O'Connell maintains that the coaching staff has kept the Prairie Fire with the right mindset, saying "We know that we can't do anything about [indoor practice], but we also know that it can't be a distraction and can't stop us from working hard … as a team, we have a number of players who take more time out of their own day to go lift or hit on their own to prepare themselves for games, which acts as a huge motivator for the entire team."
The pressure has ramped up on Knox after their record-setting 22-win season, earning the program a stigma of success and high expectations for the upcoming year.
The Prairie Fire were predicted to win the MWC South in a coach's poll, and they're certainly preparing like they're going to take home some hardware. Their first game of the year comes on Tuesday at MacMurray College, and given the forecast it seems unlikely they'll be able to hit the field too much before then.
They're lucky enough, however, to bring back a wealth of veteran experience to guide the young guys through the first few games. While the first game that matters for conference play isn't until early April, it's imperative that Knox gets off to a strong start to silence the snapping at their ankles from the four teams below them in the South, all of whom are surely anxious to take down the team on top.
Tags: baseball Jami Isaacson michael o'connell MWC peter marquardt
Nailing down theme houses
Tracks to have on loop for weeks 9 and 10
Gavin Crowell
Gavin Crowell is a junior with interests in neuroscience and psychology. He has been playing baseball ever since he could walk, playing throughout his childhood and winning two IHSA regional titles in his three years of varsity baseball at Walter Payton College Prep. He currently plays on the Knox College Ultimate team. Gavin is an Illinois State Scholar and has been involved with writing throughout high school. This is his third year working with TKS. Over the summer after his sophomore year, he had a sports internship at the Chicago Sun-Times, Chicago's second largest paper.
Baseball, softball, predicted last in MWC March 4, 2015
George Moore knows baseball October 29, 2014
White says farewell to his Knox career May 28, 2014
Senior Day provides hope for future May 7, 2014
Baseball stuck in reverse April 16, 2014
Spring roundup: Baseball drops four April 10, 2014
Baseball at top of MWC South poll February 12, 2014
The best of 2012-2013: Gavin Crowell's picks May 29, 2013
Baseball places five on All-MWC first team May 22, 2013
Explicit definitions of permanent theme houses and cultural centers will be established by the end of the year, according... | {
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Q: Converting "3H 30m" into 3.5 I am creating a SQL report for someone in which exists a column called Labour Hours of Engineer. In the column, the answers are shown as "3H 30M".
I want to change it to decimal. For example, "3H 30M" becomes 3.5.
Or another example is if an engineer works for 23 minutes, in the column, the answer should be 0.38 (rounded to 2 decimal places). 0.38 is the answer when you divide 23 mins by 60.
My current Formula is:
CONVERT(VARCHAR(40), CAST(Labour AS INT)%(24*60)/60) + '.' + CONVERT(VARCHAR(40), CAST(Labour AS INT)%60)
Any way to achieve this, please share.
A: Assuming that the 'H' and 'M' are always there, one method would be some use of CHARINDEX to find the 'H' and return the "first" characters and also strip those first characters. For minutes, you can just use simple division to get the decimal value
DECLARE @Time varchar(7) = '3H 30M';
SELECT @Time,
CONVERT(int,LEFT(@Time,CHARINDEX('H',@Time)-1)) + (CONVERT(decimal(2,0),REPLACE(STUFF(@Time,1,CHARINDEX('H',@Time),''),'M',''))/60);
A: This code will cover Hour minute or only minute
select Convert(decimal(26,2),
IIF(Labour like '%H%',(Convert(int, substring(Labour,0,(CHARINDEX('H',Labour)))) * 60 )+IIF(Labour like '%M%',Convert(int, substring(Labour,CHARINDEX('H',Labour)+1,CHARINDEX('M',Labour)-CHARINDEX('H',Labour)-1)),0)
,IIF(Labour like '%M%',Convert(int, substring(Labour,0,CHARINDEX('M',Labour))),0)) /60.00)
| {
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\section{Introduction}
Infinite-state systems such as distributed protocols remain challenging to verify despite decades of work developing interactive and automated proof techniques.
Such proofs rely on the fundamental notion of an \emph{inductive invariant}.
Unfortunately, specifying inductive invariants is difficult for users, who must often repeatedly iterate through candidate invariants before achieving an inductive invariant.
For example, the Verdi project's proof of the Raft consensus protocol used an inductive invariant with 90 conjuncts and relied on significant manual proof effort~\cite{verdi-pldi,verdi-cpp}.
The dream of \emph{invariant inference} is that users would instead be assisted by automatic procedures that could infer the required invariants.
While other domains have seen successful applications of invariant inference,
using techniques such as abstract interpretation~\cite{DBLP:conf/popl/CousotC77} and property-directed reachability~\cite{ic3,pdr},
existing inference techniques fall short for interesting distributed protocols, and often diverge while searching for an invariant.
These limitations have hindered adoption of invariant inference.
\para{Our Approach}
The idea of this paper is that invariant inference can be made drastically more effective by utilizing \emph{user-guidance} in the form of \emph{phase structures}.
We propose user-guided invariant inference, in which the user provides some additional information to guide the tool towards an invariant.
An effective guidance method must (1) match users' high-level intuition of the proof, and
(2) convey information in a way that an automatic inference tool can readily utilize to direct the search.
\ifsketch, and
(3) allow varying degrees of control over the inference procedure by providing detail in specifying the proof by the user, from a full proof to a mere sketch, so that the user will always be able to find an invariant using the tool by providing more details until the tool converges.
In this paper we introduce \emph{phase invariants} as the basis of user-guided invariant inference that enjoys these properties.
\fi
In this setting invariant inference turns a partial, high-level argument accessible to the user into a full, formal correctness proof, overcoming scenarios where procuring the proof completely automatically is unsuccessful.
Our approach places \emph{phase invariants} at the heart of both user interaction and algorithmic inference.
Phase invariants have an automaton-based form that is well-suited to the domain of distributed protocols.
They allow the user to convey a high-level temporal intuition of why the protocol is correct in the form of a \emph{phase structure}. The phase structure provides hints that direct the search and allow a more targeted generalization of states to invariants, which can facilitate inference where it is otherwise impossible.
\OMIT{
\yotam{Perhaps stop here and move to the next paragraph?}
Phase invariants capture an innate structure of protocol's correctness in a way that can be utilized to direct the search and improve the applicability of inference.
\yotam{omitting:
\yotam{consider moving these points to a discussion of phase \emph{sketches}}
Phase invariants are useful as an interface for guiding the search for an invariant, as they
(1)~are easier to specify manually than classical inductive invariants,
(2)~naturally allow presenting the correctness proof at different levels of detail, ranging
from the full proof to only the essential structure while omitting the low-level details,
(3)~capture the temporal aspects of the algorithm, and
(4)~support automatic inference for filling in missing details in the proof, guided by the automaton structure.
Our approach places phase invariants at the heart of both user interaction and algorithmic inference, leveraging these benefits.}
}
\OMIT{
\paragraph{Phase Invariants}
Phase invariants are so named because they describe executions of the protocol
as transitioning between different logical stages, or phases.
These phases form the states of an automaton, whose edges are labeled by actions of the protocol that trigger transitions between phases.
The phases themselves are labeled
by \emph{phase characterizations}: assertions that hold whenever the protocol is in that phase.
Phase invariants closely match the way domain experts already think about the correctness of distributed protocols by means of state-machine refinement \`a la \citet[e.g.][]{DBLP:books/aw/Lamport2002}.
In essence, phase invariants reveal a \emph{logical} control structure in the protocol, guided by the correctness property.
\yotam{R1: ``it is also possible that most of the benefit of the technique is simply in the structuring mechanism of phase automaton for specifying the inductive invariant. In other words, once the programmer has come up with the intuition for the phase structure and transition relations on the edges it is not much more work to write down the assertions in each state and have them be checked by standard VC checking without the more unpredictable step of invariant inference.''}
\OMIT{
Often, the latent phase structure of a protocol becomes clear only after focusing on a particular entity in the system (e.g.\ a single node or a single piece of data).
Phase invariants are thus parameterized by a \emph{view}, a finite set of elements (implicitly universally quantified)
whose evolution is described by the phase structure.
The elements in the view may be threads or processes (as is common when dealing with parameterized systems), but may also be different objects, e.g. locks, as demonstrated by our examples (see \Cref{sec:overview,sec:evaluation}).
}
\yotam{Sharon, how the technical section looks is relevant here}
A full correctness full arises from an \emph{inductive phase automaton}, in which
the phase characterizations are sufficiently strong to show that the automaton includes transitions for all actions possible at the phase,
and that the characterizations are preserved along each automaton edge.
An inductive phase invariant thus establishes a simulation relation between the protocol and the phase automaton,
parameterized by the view,
ensuring that the automaton overapproximates all executions of the protocol \jrw{merged:} for every view.
\paragraph{User-guided inference of phase invariants}
\yotam{omitting: The automaton structure of a phase invariant is more closely aligned with the intuition of protocol developers, but specifying all the details necessary for an inductive phase invariant is still challenging and tedious.
\yotam{added:} Thus in this paper we are interested in \emph{invariant inference}.}
Our approach utilizes the additional structure of phase invariants to facilitate effective \emph{invariant inference} which is \emph{guided} by the user's phase-based understanding of correctness. In our approach, users convey their intuition of the proof by specifying an automaton structure.
\ifsketch , possibly with some \emph{partial} (i.e.\ not necessarily inductive) phase characterizations.
An automaton with partial characterizations (if any) provided for the sake of inference is called a \emph{phase sketch}.
\yotam{added:} A phase sketch is still far from a full correctness proof.
\fi
Our inference procedure automatically finds phase characterizations to achieve an inductive phase invariant out of the provided phase structure.
The inference algorithm, which is parameterized by a language of possible phase characterizations,
and may return either a completed inductive phase invariant
or a witness showing that the user's phase structure is inappropriate, in the sense that no choice of phase characterizations in the given language can form an inductive phase invariant to establish safety. \yotam{Sharon, lingo also here}
From the perspective of automatic invariant inference, the given phase structure,
the localized view it induces,
and disabled transitions---actions of the system that are impossible at some phase---are utilized by our algorithm to guide the search for an inductive invariant.
The decomposition to phases guides inference towards finding an invariant.
Ultimately, the phase structure facilitates the gradual construction of a disjunctive invariant corresponding to the different phases under the scope of universal quantification over the view,
by combining manual insight with automatic inference.
\TODO{put results much earlier}
\OMIT{
Perhaps surprisingly, we show that applying the automaton's structure itself yields a speedup in inference. \TODO{move elsewhere}
One reason is that the phase structure can also specify
\emph{disabled transitions}, actions of the system that are impossible at some phase. \yotam{TODO: modify this explanation after we understand the reasons well}
Inference uses these disabled transitions as additional safety properties to guide the search.
}
\OMIT{
\jrw{cut the following paragraph?} \yotam{cutting:
From the user's perspective, where inductive invariants attempt to summarize all possible states of the system at once, phase automata instead allow the user to refer to distinct phases in the computation, each of which is summarized separately and semi-automatically. Furthermore, the witness provided in case no inductive strengthening exists may assist the user in diagnosing the failure and modifying the phase sketch in order to achieve an inductive phase invariant.
}
}
\OMIT{
From the user's perspective, phase structure express a temporal intuition of the protocol's behavior, including when certain transitions are impossible.
\ifsketch
The specification of partial characterizations builds on the local view of phases, as the provided characterizations need not hold globally but only in the specific phase.
\fi
\jrw{Okay, decided to edit and merge with previous paragraph. Is it still correct?}
Ultimately, the phase structure facilitates the gradual construction of a disjunctive invariant corresponding to the different phases under the scope of universal quantification over the view,
by combining manual insight with automatic inference.
}
\paragraph{Results}
We instantiate our user-guided phase-based inference to the language of universally quantified invariants for EPR programs,
which previous work has used to model distributed protocols~\cite{pldi/PadonMPSS16,DBLP:journals/pacmpl/PadonLSS17,DBLP:conf/pldi/TaubeLMPSSWW18}.
Using a variant of $\text{PDR}^\forall$~\cite{DBLP:journals/jacm/KarbyshevBIRS17}, we infer universally quantified phase characterizations, in addition to the universal quantifiers that arise from the automaton's view.
We use our approach to infer phase invariants from phase structures for several interesting distributed protocols.
We show that inference guided by a phase structure can infer proofs for distributed protocols that are currently beyond reach for inference, and can also achieve convergence faster than unguided state-of-the-art inference.
}
This paper makes the following contributions:
(1) We present \emph{phase invariants}, an automaton-based form of safety proofs, based on the distinct logical phases of a certain view of the system.
Phase invariants closely match the way domain experts already think about the correctness of distributed protocols by state-machine refinement \`a la Lamport~\cite[e.g.][]{DBLP:books/aw/Lamport2002}.
(2) We describe an algorithm for inferring \emph{inductive phase invariants} from \emph{phase structures}.
The decomposition to phases through the phase structure guides inference towards finding an invariant.
The algorithm finds a proof over the phase structure or explains why no such proof exists. In this way, phase invariants facilitate user interaction with the algorithm.
(3) Our algorithm reduces the problem of inferring inductive phase invariants from phase structures to the problem of solving a linear system of Constrained Horn Clauses (CHC), irrespective of the inference technique and the logic used. In the case of universally quantified phase inductive invariants for protocols modeled in EPR (motivated by previous deductive approaches~\cite{pldi/PadonMPSS16,DBLP:journals/pacmpl/PadonLSS17,DBLP:conf/pldi/TaubeLMPSSWW18}), we show how to solve the resulting CHC using a variant of $\text{PDR}^\forall$~\cite{DBLP:journals/jacm/KarbyshevBIRS17}.
(4) We apply this approach to the inference of invariants for several interesting distributed protocols. (This is the first time invariant inference is applied to distributed protocols modeled in EPR.) In the examples considered by our evaluation, transforming our high-level intuition about the protocol into a phase structure was relatively straightforward.
The phase structures allowed our algorithm to outperform in most cases an implementation of $\text{PDR}^\forall$ that does not exploit such structure, facilitating invariant inference on examples beyond the state of the art and attaining faster convergence.
\iflong
It is surprising that invariant inference---operating in the realm of logical clauses and implications---can so effectively benefit from guidance by phase structures, which exhibit a much higher level of abstraction.
While there remain significant challenges to applying invariant inference on complex distributed protocols---notably, inference of invariants with quantifier alternations, necessary, e.g.\ for Paxos~\cite{DBLP:journals/pacmpl/PadonLSS17}---our approach demonstrates that the seemingly inherent intractability of sifting through a vast space of candidate invariants can be mitigated by leveraging users' high-level intuition.
\else
Overall, our approach demonstrates that the seemingly inherent intractability of sifting through a vast space of candidate invariants can be mitigated by leveraging users' high-level intuition. \OMIT{make sure to mention Paxos somewhere}
\fi
\OMIT{We should find a natural place somewhere in the intro to mention Ivy in passing and cite,
so that the overview does not have to explain what Ivy is.}
\OMIT{
We seek to improve on these results by leveraging two further types of automation.
First, we model systems using decidable fragments of first-order logic, leading to fully automated checking of inductive invariants.
Second, we also semi-automate the process finding a state-machine abstraction of the protocol, by allowing the user to sketch a \emph{phase automaton}.
The details of the abstraction are then filled out automatically by solving a system of second-order constrained Horn clauses using a variant of $\text{PDR}^\forall$.
\jrw{I was just trying to sound fancy with the previous sentence. Please check that I used all the words properly.}
Building on existing literature from the distributed systems community, our key insight is that state-machine refinement techniques such as our phase automata
more naturally capture the structure of distributed protocols.
Where inductive invariants attempt to summarize all possible states of the system at once, phase automata instead allow the user to refer to distinct phases in the computation, each of which is summarized separately and semi-automatically.
Such a summary describes the mechanisms used by the protocol during execution, which provide the essential reason for its correctness to be checked in the proof.
\TODO{explain what is a phase invariant}
We demonstrate the benefits of phase invariants by using them to verify to sophisticated protocol from the distributed systems literature.
We also compare our phase automaton characterizations to traditional inductive invariants used in state-of-the-art deductive verification tools.
We find that phase invariants improve proof automation by localizing the search, yielding a speedup of \TODO{insert results} over baseline inductive invariant inference.
Phase invariants also facilitate user interaction since users can provide hints about the meaning of each phase, which the system either completes to a proof or explains why no such proof exists.
When given such guidance, our results further improve to \TODO{insert results}, showing that users can trade off machine time for their own effort.
}
\OMIT{
\yotam{The use of phase invariants for infinite-state systems relies on the use of a
\emph{view}. The view is a finite list of elements, fixed throughout the execution, to which the phases refer. Intuitively, each set of elements from the domain induce a phase automaton, and they all execute in parallel. The proofs we consider are \emph{view-modular}, meaning the proof of a phase invariant refers only to single arbitrary choice of the elements for the view, and not to the automata of other choices of view. \TODO{probably not clear at all...} This is sound because the view is in essence universally quantified.
One observation of this work is that often the safety property calls for a view on elements other than nodes or threads. For example, the view can consist of a shared resource or two different values in a consensus protocol.
}
}
\OMIT{
\paragraph{Inferring phase invariants}
We note that providing fully inductive (phase) invariants in distributed systems can be tricky. Typically, the user has to account for the exact inductive argument for correctness, including all corner cases. Therefore, we also describe and implement an algorithm for inferring phase invariants. This is an adaptation of IC3/PDR~\cite{} and $\text{PDR}^\forall$~\cite{} for inferring quantified invariants.
\TODO{insert results here}
\paragraph{Why is deductive verification hard?}
\paragraph{Why is deductive verification of distributed protocols hard?}
-> Inductive invariants are properties of states rather than traces.
More restricted than temporal logic. (Is it contained in LTL?)
\paragraph{Why is inference hard?}
\begin{itemize}
\item CNF good for PDR, DNF for AI (?). Natural invariants of distributed protocols are a disjunction of clauses, like we find here
\item Problem of disjunction
\end{itemize}
Phases:
Better fit.
Easy to partially characterize phases, to be completed by inference.
Even just the structure provides valuable insight that can improve inference.
\subsection{Some Potential Phrases}
\begin{itemize}
\item Distributed protocols are often viewed and described as progressing between distinct phases of the computation~\cite{}.
Our key insight is that \emph{correctness proofs} of distributed protocols enjoy a similar structure.
\item Providing control to unstructured protocols
\item While an inductive phase invariant induces a standard inductive invariant, we view the phase invariant as the fundamental building block of the proof, as it more closely captures the intuition users have of the protocol's correctness. \yotam{This one is even more dubious than the rest :)}
\sharon{maybe the point is that if we want to convert it to CNF it will become exponentially larger and much less clear?}
\item When the phase structure of the proof deviates from that of the protocol, this elucidates the subtlety in the proof in the form of a refinement of the protocol's straightforward, operational, description.
\sharon{Here I assume you refer to the one round vs two rounds in paxos, but I am not sure that this is a good thing. }
\item For the purpose of automatic verification, we harness the phase structure in a PDR-based invariant inference procedure, which aims to complete a self-sufficient phase invariant out of a phase sketch.
\sharon{more generally: given a phase sketch, we formulate/reduce the problem of computing a phase invariant using CHCs. Talk about linear vs. non linear?}
\item Impossible transitions are valuable in directing the search for invariant, as they constitute additional safety constraints to be satisfied by phase characterizations.
\sharon{vague thought: can the guards on transitions be used to give us the ``bounded occurrences of arbitrary relations'' that were used in POPL16 for decidability?}
\item (Phase specification can be useful also bug finding, restricting the possible transitions in phases)
\item Making the distinction between phases is useful when there is a property which holds in one phase but not in the other, and when this fact is important to the proof.
Without separating the phases, the invariant would guard the property by some other formula expressing indirectly which phase the execution is currently in (as the lockservice example does).
\item The phases are parameterized by a certain view of the system, e.g., the view of a single thread/round or the view of a pair of threads/rounds.
\sharon{this is related to the point about control: in distributed protocols there is no clear control partly because the players are not fixed, but when we fix a view, the control does become clear.}
\item (?) Phase invariants combine intuition from abstraction (simulation as in existential abstraction) and deductive verification (inductiveness).
\item (?) Phase invariants provide a convenient way for a user to convey intuition to a tool, with a spectrum of how much information can be provided, and with "focused" counterexamples from the tool (since they refer to specific phases).
\end{itemize}
}
\iflong
\else
\vspace{-0.4cm}
\fi
\section{Preliminaries}
\iflong
\else
\vspace{-0.4cm}
\fi
\iflong
In this section we provide background on modeling and verifying systems using first-order logic.
\else
In this section we provide background on first-order transition systems.
\fi
\iflong
We assume familiarity with first-order logic.
\fi \iflong
We use many-sorted first order logic, but omit sorts here to simplify the presentation.
\else
Sorts are omitted for simplicity.
\fi
\iflong
Although in this paper we will mostly deal with uninterpreted first-order logic, our definitions and results extend to logics with a background theory.
\else
Our results extend also to logics with a background theory.
\fi
\para{Notation}
$\fv{\varphi}$ denotes the set of free variables of $\varphi$.
$\form{\vocabulary}{V}$ denotes the set of first-order \iflong well-formed \fi formulas \iflong $\varphi$\fi over vocabulary $\vocabulary$ with $\fv{\varphi} \subseteq V$.
We
\iflong
extend the notation $\implies$ to quantified formulas and
\fi
write $\forall \mathcal{V}.\ \varphi \implies \psi$ to denote that the formula $\forall \mathcal{V}.\ \varphi \to \psi$ is valid.
We sometimes use $f_a$ as a shorthand for $f(a)$.
\para{Transition systems}
We represent transition systems symbolically, via formulas in first-order logic. The definitions are standard.
A vocabulary $\Sigma$ consisting of constant, function, and relation symbols is used to represent states\iflong (each function and relation symbol is associated with its arity)\fi.
Post-states of transitions are represented by a copy of $\Sigma$ denoted $\Sigma' = \{ a' \mid a \in \Sigma\}$\iflong (where the arity of each function and relation symbol is inherited from $\Sigma$)\fi.
A \emph{first-order transition system} over \iflong vocabulary\fi $\Sigma$ is a tuple ${\textit{TS}} = (\textit{Init}, {\textit{TR}})$, where $\textit{Init} \in \form{\vocabulary}{\emptyset}$ describes the initial states, and ${\textit{TR}} \in \form{\hat{\vocabulary}}{\emptyset}$ with $\hat{\vocabulary} = \Sigma \uplus \Sigma'$ describes the transition relation.The states of ${\textit{TS}}$ are first-order structures over $\vocabulary$\iflong, denoted $\structOfVoc{\vocabulary}$\fi.
\iflong
Each state ${s} \in \structOfVoc{\vocabulary}$ is a pair ${s} = ({D},{\mathcal{I}})$ where ${D}$ is the \emph{domain} and ${\mathcal{I}}$ is the \emph{interpretation function} mapping each symbol in $\vocabulary$ to its interpretation over ${D}$. We denote by $\structdom{\vocabulary}{{D}}$ the set of structures with domain ${D}$.
In this way, every closed formula over $\vocabulary$ represents the set of states (first-order structures) that satisfy it.
\fi
\iflong
In particular, a state ${s}$ is initial if ${s} \models \textit{Init}$. A transition of ${\textit{TS}}$ is a pair of states ${s}_1 = ({D}, {\mathcal{I}}_1),{s}_2 = ({D}, {\mathcal{I}}_2)$ with a shared domain such that $({s}_1,{s}_2) \models {\textit{TR}}$, where $({s}_1,{s}_2)$ is a shorthand for the structure ${s} = ({D}, {\mathcal{I}})$ obtained by defining ${\mathcal{I}}(a) = {\mathcal{I}}_1(a)$ if $a \in \vocabulary$, and ${\mathcal{I}}(a) = {\mathcal{I}}_2(a)$ if $a \in \vocabulary'$.
\else
A state ${s}$ is initial if ${s} \models \textit{Init}$. A transition of ${\textit{TS}}$ is a pair of states ${s}_1,{s}_2$ over a shared domain such that $({s}_1,{s}_2) \models {\textit{TR}}$, $(s_1,s_2)$ being the structure over that domain in which $\vocabulary$ in interpreted as in ${s}_1$ and $\vocabulary'$ as in ${s}_2$.
\fi
${s}_1$ is also called the \emph{pre-state} and ${s}_2$ the \emph{post-state}.
Traces are finite
sequences of states $\sigma_1, \sigma_2, \ldots$ starting from an initial state
such that there is a transition between each pair of consecutive states.
The \emph{reachable states} \iflong of ${\textit{TS}}$\fi are those that reside on traces starting from an initial state.
\para{Safety}
\iflong
A safety property $P$ is a formula in $\form{\vocabulary}{\emptyset}$. We say that ${\textit{TS}}$ is \emph{safe} if all the reachable states satisfy $P$, in which case we also say that $P$ is an \emph{invariant} of ${\textit{TS}}$.
A prominent way to prove safety is via \emph{inductive invariants}. An inductive invariant $\textit{Inv}$ is a closed first-order formula over $\vocabulary$ such that the following requirements hold:
\begin{inparaenum}[(i)]
\item $\textit{Init} \implies \textit{Inv}$ (initiation), and
\item $\textit{Init} \wedge {\textit{TR}} \implies \textit{Inv}'$ (consecution), \end{inparaenum}
where $\textit{Inv}'$ is obtained from $\textit{Inv}$ by replacing each symbol from $\vocabulary$ with its primed counterpart.Initiation and consecution ensure that all the reachable states satisfy $\textit{Inv}$. If, in addition, $\textit{Inv}$ satisfies:
\begin{inparaenum}[(iii)]
\item $\textit{Inv} \implies P$ (safety),
\end{inparaenum}
it follows that all the reachable states satisfy $P$, and
${\textit{TS}}$ is safe.
\else
A safety property $P$ is a formula in $\form{\vocabulary}{\emptyset}$. We say that ${\textit{TS}}$ is \emph{safe}, and that $P$ is an \emph{invariant}, if all the reachable states satisfy $P$.
$\textit{Inv} \in \form{\vocabulary}{\emptyset}$ is an \emph{inductive invariant} if
\begin{inparaenum}[(i)]
\item $\textit{Init} \implies \textit{Inv}$ (initiation), and
\item $\textit{Init} \wedge {\textit{TR}} \implies \textit{Inv}'$ (consecution),
\end{inparaenum}
where $\textit{Inv}'$ is obtained from $\textit{Inv}$ by replacing each symbol from $\vocabulary$ with its primed counterpart.
If also
\begin{inparaenum}[(iii)]
\item $\textit{Inv} \implies P$ (safety),
\end{inparaenum}
then it follows that
${\textit{TS}}$ is safe.
\fi
\iflockserv
\input{running-lockserv}
\else
\section{Running Example: Distributed Key-Value Store}
\label{sec:running}
\iflong
\begin{figure*}[t]
\centering
\begin{minipage}{1.1\textwidth}
\begin{minipage}{0.53\textwidth}
\begin{lstlisting}[numbers=left, numberstyle=\tiny, numbersep=5pt, name=lockserv, escapeinside={(*}{*)}]
type key(*\label{kv-code:types-start}*)
type value
type node
type sequnum(*\label{kv-code:types-end}*)
relation owner: node, key(*\label{kv-code:state-start}*)
relation table: node, key, value
relation transfer_msg: node, node,
key, value, seqnum
relation ack_msg: node, node, seqnum
relation seqnum_sent: node, seqnum
relation unacked: node, node,
key, value, seqnum
relation seqnum_recvd: node, node, seqnum(*\label{kv-code:state-end}*)
init $\forall n_1,n_2,k.$(*\label{kv-code:init-start}*) owner($n_1$,$k$)$\land$owner($n_2$,$k$)
$\rightarrow n_1=n_2$(*\label{kv-code:init-start}*)
init // all other relations are empty(*\label{kv-code:init-end}*)
action reshard(n_old:node, n_new:node,(*\label{kv-code:action-start}*)(*\label{kv-code:reshard-start}*)
k:key, value:sequnum)
require table(n_old, k, v)
$\land$$\neg$seqnum_sent(n_old, s)
seqnum_sent(n_old, s) := true
table(n_old, k, v) := false
owner(n_old, k) := false(*\label{kv-code:cede-own}*)
transfer_msg(n_old, n_new, k, v, s) := true(*\label{kv-code:inserting-tuple}*)
unacked(n_old, n_new, k, v, s) := true(*\label{kv-code:reshard-end}*)
action drop_transfer_msg(src:node, dst:node,(*\label{kv-code:transfer-drop-start}*)
k:key, v:value, s:seqnum)
require transfer_msg(src, dst, k, v, s)(*\label{kv-code:requiring-tuple}*)
transfer_msg(src, dst, k, v, s) := false(*\label{kv-code:removing-tuple}*)(*\label{kv-code:transfer-drop-end}*)
action retransmit(src:node, dst:node,(*\label{kv-code:transfer-retransmit-start}*)
k:key, v:value, s:seqnum)
require unacked(src, dst, k, v, s)
transfer_msg(src, dst, k, v, s) := true(*\label{kv-code:transfer-retransmit-end}*)
\end{lstlisting}
\end{minipage}
\hspace{0.0\textwidth}\begin{minipage}{0.4\textwidth}
\begin{lstlisting}[numbers=left, numberstyle=\tiny, numbersep=5pt, name=lockserv, escapeinside={(*}{*)}]
action recv_transfer_msg(src:node, n:node,(*\label{kv-code:receive-transfer-start}*)
k:key, v:value, s:seqnum)
require transfer_msg(src, n, k, v, s)
$\land$$\neg$seqnum_recvd(n, src, s)(*\label{kv-code:ignore-dup}*)
seqnum_recvd(n, src, s) := true
table(n, k, v) := true
owner(n, k) := true(*\label{kv-code:receive-transfer-end}*)
action send_ack(src:node, n:node,(*\label{kv-code:ack-start}*)
k:key, v:value, s:seqnum)
require transfer_msg(src, n, k, v, s)
$\land$seqnum_recvd(n, src, s)
ack_msg(src, n, s) := true(*\label{kv-code:ack-end}*)
action drop_ack_msg(src:node, dst:node,(*\label{kv-code:ack-drop-start}*)
k:key, s:seqnum)
require ack_msg(src, dst, s)
ack_msg(src, dst, s) := false(*\label{kv-code:ack-drop-end}*)
action recv_ack_msg(src:node, dst:node,
k:key, s:seqnum)
require ack_msg(src, dst, s)
unacked(src, dst, *, *, s) := false(*\label{kv-code:qf-update:recv_ack_msg}*)
action put(n:node, k:key, v:value)
require owner(n, k)
table(n, k, *) := false(*\label{kv-code:qf-update:put}*)
table(n, k, v) := true(*\label{kv-code:action-end}*)
safety $\forall k, n_1, n_2, v_1, v_2.$(*\label{kv-code:safety-start}*)
table($n_1$,$k$,$v_1$) $\land$
table($n_2$,$k$,$v_2$) $\to$
$n_1 = n_2 \land v_1 = v_2$(*\label{kv-code:safety-end}*)
\end{lstlisting}
\end{minipage}
\captionof{figure}{\footnotesize Sharded key-value store with retransmissions (KV-R) in a first-order relational modeling.}
\label{fig:kv-ivy}
\end{minipage}
\end{figure*}
\else
\begin{figure*}[t]
\centering
\begin{minipage}{1.1\textwidth}
\begin{minipage}{0.53\textwidth}
\begin{lstlisting}[numbers=left, numberstyle=\tiny, numbersep=5pt, name=lockserv, escapeinside={(*}{*)}]
type key(*\label{kv-code:types-start}*)
type value
type node
type sequnum(*\label{kv-code:types-end}*)
relation owner: node, key(*\label{kv-code:state-start}*)
relation table: node, key, value
relation transfer_msg: node, node,
key, value, seqnum
relation ack_msg: node, node, seqnum
relation seqnum_sent: node, seqnum
relation unacked: node, node,
key, value, seqnum
relation seqnum_recvd: node, node, seqnum(*\label{kv-code:state-end}*)
init $\forall n_1,n_2,k.$(*\label{kv-code:init-start}*) owner($n_1$,$k$)$\land$owner($n_2$,$k$)
$\rightarrow n_1=n_2$(*\label{kv-code:init-start}*)
init // all other relations are empty(*\label{kv-code:init-end}*)
action reshard(n_old:node, n_new:node,(*\label{kv-code:action-start}*)(*\label{kv-code:reshard-start}*)
k:key, value:sequnum)
require table(n_old, k, v)
$\land$$\neg$seqnum_sent(n_old, s)
seqnum_sent(n_old, s) := true
table(n_old, k, v) := false
owner(n_old, k) := false(*\label{kv-code:cede-own}*)
transfer_msg(n_old, n_new, k, v, s) := true(*\label{kv-code:inserting-tuple}*)
unacked(n_old, n_new, k, v, s) := true(*\label{kv-code:reshard-end}*)
action drop_transfer_msg(src:node, dst:node,(*\label{kv-code:transfer-drop-start}*)
k:key, v:value, s:seqnum)
require transfer_msg(src, dst, k, v, s)(*\label{kv-code:requiring-tuple}*)
transfer_msg(src, dst, k, v, s) := false(*\label{kv-code:removing-tuple}*)(*\label{kv-code:transfer-drop-end}*)
action retransmit(src:node, dst:node,(*\label{kv-code:transfer-retransmit-start}*)
k:key, v:value, s:seqnum)
require unacked(src, dst, k, v, s)
transfer_msg(src, dst, k, v, s) := true(*\label{kv-code:transfer-retransmit-end}*)
\end{lstlisting}
\end{minipage}
\hspace{0.0\textwidth}\begin{minipage}{0.4\textwidth}
\begin{lstlisting}[numbers=left, numberstyle=\tiny, numbersep=5pt, name=lockserv, escapeinside={(*}{*)}]
action recv_transfer_msg(src:node, n:node,(*\label{kv-code:receive-transfer-start}*)
k:key, v:value, s:seqnum)
require transfer_msg(src, n, k, v, s)
$\land$$\neg$seqnum_recvd(n, src, s)(*\label{kv-code:ignore-dup}*)
seqnum_recvd(n, src, s) := true
table(n, k, v) := true
owner(n, k) := true(*\label{kv-code:receive-transfer-end}*)
action send_ack(src:node, n:node,(*\label{kv-code:ack-start}*)
k:key, v:value, s:seqnum)
require transfer_msg(src, n, k, v, s)
$\land$seqnum_recvd(n, src, s)
ack_msg(src, n, s) := true(*\label{kv-code:ack-end}*)
action drop_ack_msg(src:node, dst:node,(*\label{kv-code:ack-drop-start}*)
k:key, s:seqnum)
require ack_msg(src, dst, s)
ack_msg(src, dst, s) := false(*\label{kv-code:ack-drop-end}*)
action recv_ack_msg(src:node, dst:node,
k:key, s:seqnum)
require ack_msg(src, dst, s)
unacked(src, dst, *, *, s) := false(*\label{kv-code:qf-update:recv_ack_msg}*)
action put(n:node, k:key, v:value)
require owner(n, k)
table(n, k, *) := false(*\label{kv-code:qf-update:put}*)
table(n, k, v) := true(*\label{kv-code:action-end}*)
safety $\forall k, n_1, n_2, v_1, v_2.$(*\label{kv-code:safety-start}*)
table($n_1$,$k$,$v_1$) $\land$
table($n_2$,$k$,$v_2$) $\to$
$n_1 = n_2 \land v_1 = v_2$(*\label{kv-code:safety-end}*)
\end{lstlisting}
\end{minipage}
\captionof{figure}{\footnotesize Sharded key-value store with retransmissions (KV-R) in a first-order relational modeling.}
\label{fig:kv-ivy}
\end{minipage}
\end{figure*}
\fi
We begin with a description of the running example we refer to throughout the paper.
The \emph{sharded key-value store with retransmissions (KV-R)}, adapted
from IronFleet~\cite[\S 5.2.1]{IronFleet}, is a distributed hash table where each node owns a subset of the keys, and keys can be dynamically transferred among nodes to balance load.
The safety property ensures that each key is globally associated with one value, even in the presence of key transfers.
Messages might be dropped by the network, and the protocol uses retransmissions and sequence numbers to maintain availability and safety.
\Cref{fig:kv-ivy} shows code modeling the protocol in a relational first-order language akin to Ivy~\cite{DBLP:conf/sas/McMillanP18}, which compiles to EPR transition systems.
The state of nodes and the network is modeled by global relations.
\Crefrange{kv-code:types-start}{kv-code:types-end} declare uninterpreted
sorts for keys, values, clients, and sequence numbers.
\Crefrange{kv-code:state-start}{kv-code:state-end} describe the
state, consisting of:
\begin{inparaenum}[(i)]
\item local state of clients pertaining to the table (which nodes are \codefont{owner}{}s of which keys, and the local shard of the \codefont{table}{} mapping keys to values);
\item local state of clients pertaining to sent and received messages (\codefont{seqnum\_sent}{}, \codefont{unacked}{}, \codefont{seqnum\_recvd}{}); and
\item the state of the network, comprised of two kinds of messages (\codefont{transfer\_msg}{}, \codefont{ack\_msg}{}).
Each message kind is modeled as a relation whose first two arguments indicate the source and destination of the message, and the rest carry the message's payload.
For example, \codefont{ack\_msg}{} is a relation over two nodes and a sequence number, with the intended meaning that a tuple $(c_1, c_2, s)$ is in
\codefont{ack\_msg}{} exactly when there is a message in the network from $c_1$ to $c_2$ acknowledging a message with sequence number $s$.
\end{inparaenum}
The initial states are specified in \crefrange{kv-code:init-start}{kv-code:init-end}.
Transitions are specified by the actions declared in \crefrange{kv-code:action-start}{kv-code:action-end}.
Actions can fire nondeterministically at any time when their precondition (\codefont{require} statements) holds.
Hence, the transition relation comprises of the disjunction of the transition relations induced by the actions.
The state is mutated by modifying the relations. For example,
message sends are modeled by inserting a tuple into the corresponding relation
(e.g. \cref{kv-code:inserting-tuple}),
while message receives are modeled by requiring a tuple to be in the relation
(e.g. \cref{kv-code:requiring-tuple}),
and then removing it
(e.g. \cref{kv-code:removing-tuple}).
The updates in \cref{kv-code:qf-update:recv_ack_msg,kv-code:qf-update:put} remove a set of tuples matching the pattern.
\iflong
\para{KV-R protocol}
\fi
Transferring keys between nodes begins by sending a \codefont{transfer\_msg}{} from the owner to a new node (\cref{kv-code:reshard-start}), which stores the key-value pair when it receives the message (\cref{kv-code:receive-transfer-start}).
Upon sending a transfer message the original node cedes ownership (\cref{kv-code:cede-own}) and does not send new transfer messages.
Transfer messages may be dropped (\cref{kv-code:transfer-drop-start}). To ensure that the key-value pair is not lost, retransmissions are performed (\cref{kv-code:transfer-retransmit-start}) with the same sequence number until the target node acknowledges (which occurs in \cref{kv-code:ack-start}). Acknowledge messages themselves may be dropped (\cref{kv-code:ack-drop-start}).
Sequence numbers protect from delayed transfer messages, which might contain old values (\cref{kv-code:ignore-dup}).
\iflong
\para{KV-R safety property}
\fi
\Crefrange{kv-code:safety-start}{kv-code:safety-end}
specify the key safety property: at most one value is associated with any key, anywhere in the network.
Intuitively, the protocol satisfies this because each key $k$ is either currently
\begin{inparaenum}[(1)]
\item \emph{owned} by a node, in which case this node is unique, or
\item it is in the process of \emph{transferring} between nodes, in which case the careful use of sequence numbers ensures that the destination of the key is unique.
\end{inparaenum}
As is typical, it is not straightforward to translate this intuition into a full correctness proof. In particular, it is necessary to relate all the different components of the state, including clients' local state and pending messages.
\OMIT{This is a more elaborate version that makes it seem easy to write by hand:
When $k$ is \emph{owned}, it is owned by a single node, and there are no pending messages concerning this key. When $k$ is transferring \emph{transferring}, the key is not owned by any node, and pending non-duplicate transfer messages are unique.
}
\OMIT{In this flow I don't have a version of this:
As is typical, the lock service safety property is not inductive by itself,
because it is not strong enough to be preserved by the transitions of the system.
In particular, since the safety property only mentions \codefont{holds\_lock}{}, it does not
constrain the other components of the state (including all network messages and the server state).}
\emph{Invariant inference} strives to automatically find an inductive invariant establishing safety.
This example is challenging for existing inference techniques (\Cref{sec:evaluation}).
This paper proposes \emph{user-guided invariant inference} based on \emph{phase-invariants} to overcome this challenge. The rest of the paper describes our approach, in which inference is provided with the phase structure in
\iflockserv
\Cref{fig:lockserv-automaton},
\else
\Cref{fig:kv-automaton},
\fi
matching the high level intuitive explanation above. The algorithm then automatically infers facts about each phase to obtain an inductive invariant. \Cref{sec:phase-invariants} describes phase structures and inductive phase invariants, and \Cref{sec:inference} explains how these are used in user-guided invariant inference.
\commentout{
\subsection{Inductive Invariants for Distributed Protocols}
The standard approach to formally prove safety for a system such as the lock service is to find an
\emph{inductive invariant}, which is a property that
(1) implies safety,
(2) is true in the initial state, and
(3) is preserved by all transitions of the system.
}
\commentout{
Unfortunately, inductive invariants are notoriously hard and laborious to specify, making \emph{invariant inference} an appealing direction.
In this setting an automatic algorithm finds an inductive invariant for a given system and safety property, relieving the user from specifying it by hand.
Unfortunately, state-of-the-art invariant inference techniques typically fall short for interesting distributed protocols, and often diverge in their search.
\OMIT{
\yotam{Include only if we can do better:
Furthermore, these techniques suffer from sensitivity to minor details in the program---for example, insubstantial details in the data representation might throw them off course---and even to the details of the underlying solver they use.}
These problems hinder the applicability of invariant inference techniques to distributed protocols.
}
}
\commentout{
\paragraph{Inference implementation}
Our algorithm reduces the inference of inductive phase invariants from a given phase structure to a set of linear Constrained Horn Clauses.
Such systems can be solved in various ways, including abstract interpretation and property-directed reachability (PDR). We implement a PDR-based inference procedure for universally quantified characterizations over uninterpreted first-order logic, based on $\text{PDR}^\forall$~\cite{DBLP:journals/jacm/KarbyshevBIRS17}, and apply it to infer inductive phase invariants for various distributed protocols modeled in EPR. The implementation and evaluation are described in \Cref{sec:evaluation}.
}
\section{Phase Structures and Invariants}
\label{sec:phase-invariants}
\iflong
Phase invariants describe the
protocol as transitioning
between different logical stages.
\fi
In this section we introduce \emph{phase structures} and \emph{inductive phase invariants}\iflong
and explain their role in verifying safety properties.
\else. \fi
\iflong
In \Cref{sec:inference} we explain how we use these in guiding automatic invariant inference.
\else
These are used for guiding automatic invariant inference in \Cref{sec:inference}.
\fi
\ifapp
For brevity, proofs are deferred to \refappendix{sec:proofs}.
\else
Proofs appear in~\cite{extendedVersion}. \fi
\subsection{Phase Invariants}
\begin{definition}[Quantified Phase Automaton]
A \emph{quantified phase automaton} (\emph{phase automaton} for short) over $\vocabulary$ is a tuple
$\mathcal{A} = (\mathcal{Q}, \iota, \mathcal{V}, \edgelabel{}, \statelabel{})$ where:
\iflong
\begin{itemize}
\item $\mathcal{Q}$ is a finite set of \emph{phases}.
\item $\iota \in \mathcal{Q}$ is the initial phase.
\item $\mathcal{V}$ is a set of variables, called the \emph{automaton's quantifiers} \sharon{or view, or view quantifiers?}.
\item $\edgelabel{}: \mathcal{Q} \times \mathcal{Q} \to \form{\hat{\vocabulary}}{\mathcal{V}}$ is a function labeling every pair of phases by a transition relation formula, such that $\fv{\edgelabel{(q,p)}} \subseteq \mathcal{V}$ for every $(q,p) \in \mathcal{Q} \times \mathcal{Q}$. \item $\statelabel{}: \mathcal{Q} \to \form{\vocabulary}{\mathcal{V}}$ is a function labeling every phase by a \emph{phase characterization} formula, such that $\fv{\statelabel{q}} \subseteq \mathcal{V}$ for every phase $q \in \mathcal{Q}$.
\end{itemize}
\else
$\mathcal{Q}$ is a finite set of \emph{phases}.
$\iota \in \mathcal{Q}$ is the initial phase.
$\mathcal{V}$ is a set of variables, called the \emph{automaton's quantifiers}.
$\edgelabel{}: \mathcal{Q} \times \mathcal{Q} \to \form{\hat{\vocabulary}}{\mathcal{V}}$ is a function labeling every pair of phases by a transition relation formula, such that $\fv{\edgelabel{(q,p)}} \subseteq \mathcal{V}$ for every $(q,p) \in \mathcal{Q} \times \mathcal{Q}$. $\statelabel{}: \mathcal{Q} \to \form{\vocabulary}{\mathcal{V}}$ is a function labeling every phase by a \emph{phase characterization} formula, s.t.\ $\fv{\statelabel{q}} \subseteq \mathcal{V}$ for every $q \in \mathcal{Q}$.
\fi
\end{definition}
Intuitively, $\mathcal{V}$ should be understood as free variables that are implicitly universally quantified outside of the automaton's scope. For each assignment to these variables, the automaton represents the progress along the phases from the point of view of this assignment, and thus $\mathcal{V}$ is also called the \emph{view} (or \emph{view quantifiers}).
We refer to $(\mathcal{Q}, \iota, \mathcal{V}, \edgelabel{})$, where $\statelabel{}$ is omitted, as the \emph{phase structure} (or the \emph{automaton structure}) of $\mathcal{A}$.
\iflong
\fi
We refer by the \emph{edges} of $\mathcal{A}$ to
$
\mathcal{R} = \{(q,p) \in \mathcal{Q} \times \mathcal{Q} \mid \edgelabel{(q,p)} \not\equiv {\textit{false}} \}.
$
\iflong
\fi
A \emph{trace} of $\mathcal{A}$ is a sequence of phases $q_0,\ldots,q_n$ such that $q_0 = \iota$ and $(q_i,q_{i+1}) \in \mathcal{R}$ for every $0 \leq i < n$.
\iflong
\fi
We say that $\mathcal{A}$ is \emph{deterministic} if for every $(q,p_1), (q,p_2) \in \mathcal{R}$ s.t. $p_1 \neq p_2$,
the formula $\edgelabel{(q,p_1)} \wedge \edgelabel{(q,p_2)}$ is unsatisfiable.
\iflockserv
\input{figs/lockserv-phase-automaton-w-char}
\else
\iflong
\begin{figure*}[t]
\begin{subfigure}{\textwidth}
\centering
\begin{tikzpicture}[
state/.style={
draw,
circle,
font=\footnotesize\ttfamily,
minimum size=9mm,
inner sep=0pt
},
action/.style={
font=\ttfamily\scriptsize }
]
\node[state] (O) at (180:1.5cm) {{\small O[$k$]}};
\node[state] (T) at (0:1.5cm) {{\small T[$k$]}};
\draw[->] (O) edge[bend left] node[action,above] {\footnotesize reshard(*,*,$k$,*)} (T);
\draw[->] (T) edge[bend left] node[action,below] {\footnotesize recv\_transfer\_msg(*,*,$k$,*,*)} (O);
\draw[->] (O) edge[loop left] node[action,left,align=center]
{drop\_transfer\_msg(*,*,$k$,*,*) \\
retransmit(*,*,$k$,*,*) \\
send\_ack(*,*,$k$,*,*) \\
drop\_ack\_msg(*,*,$k$,*) \\
recv\_ack\_msg(*,*,$k$,*) \\
put(*,$k$,*)} (O);
\draw[->] (T) edge[loop right] node[action,right,align=center]
{drop\_transfer\_msg(*,*,$k$,*,*) \\
retransmit(*,*,$k$,*,*) \\
send\_ack(*,*,$k$,*,*) \\
drop\_ack\_msg(*,*,$k$,*) \\
recv\_ack\_msg(*,*,$k$,*) \\
put(*,$k$,*)} (T);
\end{tikzpicture}
\end{subfigure}
\\
\begin{subfigure}{\textwidth}
\centering
\begin{lstlisting}[numbers=left, numberstyle=\tiny, numbersep=5pt, name=lockserv, morekeywords={phase}]
phase O[$k$]:
invariant $\forall n_1,n_2.$ owner($n_1$,$k$)$\land$owner($n_2$,$k$)$\rightarrow n_1=n_2$
invariant $\forall n,v.$ table($n$,$k$,$v$)$\rightarrow$owner($n$,$k$)
invariant $\forall {\normalfont \textrm{src}},{\normalfont \textrm{dst}},v,s.$ $\neg$(transfer_msg(${\normalfont \textrm{src}}$,${\normalfont \textrm{dst}}$,$k$,$v$,$s$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}$,${\normalfont \textrm{src}}$,$s$))$\label{kv-char:disallow-recv-transfer}$
invariant $\forall n_1,n_2,v_1,v_2.$ table($n_1$,$k$,$v_1$)$\land$table($n_2$,$k$,$v_2$)$\rightarrow n_1 = n_2 \land v_1 = v_2$
invariant $\forall {\normalfont \textrm{src}},{\normalfont \textrm{dst}},v,s.$ $\neg$(unacked(${\normalfont \textrm{src}}$,${\normalfont \textrm{dst}}$,$k$,$v$,$s$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}$,${\normalfont \textrm{src}}$,$s$))
phase T[$k$]:
invariant $\forall n.$ $\neg$owner($n$,$k$)
invariant $\forall n,v.$ table($n$,$k$,$v$)$\rightarrow$owner($n$,$k$)
invariant $\label{kv-char:unique-transfer-unreceived}$$\forall {\normalfont \textrm{src}}_1,{\normalfont \textrm{src}}_2,{\normalfont \textrm{dst}}_1,{\normalfont \textrm{dst}}_2,v_1,v_2,s_1,s_2.$ transfer_msg(${\normalfont \textrm{src}}_1$,${\normalfont \textrm{dst}}_1$,$k$,$v_1$,$s_1$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_1$,${\normalfont \textrm{src}}_1$,$s_1$)
$\land$transfer_msg(${\normalfont \textrm{src}}_2$,${\normalfont \textrm{dst}}_2$,$k$,$v_2$,$s_2$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_2$,${\normalfont \textrm{src}}_2$,$s_2$)$\rightarrow {\normalfont \textrm{src}}_1 = {\normalfont \textrm{src}}_2 \land {\normalfont \textrm{dst}}_1 = {\normalfont \textrm{dst}}_2 \land v_1=v_2 \land s_1=s_2$
invariant $\forall {\normalfont \textrm{src}}_1,{\normalfont \textrm{src}}_2,{\normalfont \textrm{dst}}_1,{\normalfont \textrm{dst}}_2,v_1,v_2,s_1,s_2.$transfer_msg(${\normalfont \textrm{src}}_1$,${\normalfont \textrm{dst}}_1$,$k$,$v_1$,$s_1$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_1$,${\normalfont \textrm{dst}}_1$,$s_1$)
$\land$unacked(${\normalfont \textrm{src}}_2$,${\normalfont \textrm{dst}}_2$,$k$,$v_2$,$s_2$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_2$,${\normalfont \textrm{src}}_2$,$s_2$)$\rightarrow {\normalfont \textrm{src}}_1 = {\normalfont \textrm{src}}_2 \land {\normalfont \textrm{dst}}_1 \land {\normalfont \textrm{dst}}_2 \land v_1 = v_2 \land s_1 = s_2$
invariant $\forall {\normalfont \textrm{src}}_1,{\normalfont \textrm{src}}_2,{\normalfont \textrm{dst}}_1,{\normalfont \textrm{dst}}_2,v_1,v_2,s_1,s_2.$ unacked(${\normalfont \textrm{src}}_1$,${\normalfont \textrm{dst}}_1$,$k$,$v_1$,$s_1$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_1$,${\normalfont \textrm{src}}_1$,$s_1$)
$\land$unacked(${\normalfont \textrm{src}}_2$,${\normalfont \textrm{dst}}_2$,$k$,$v_2$,$s_2$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_2$,${\normalfont \textrm{src}}_2$,$s_2$)$\rightarrow {\normalfont \textrm{src}}_1 = {\normalfont \textrm{src}}_2 \land {\normalfont \textrm{dst}}_1 = {\normalfont \textrm{dst}}_2 \land v_1 = v_2 \land s_1 = s_2$
\end{lstlisting}
\end{subfigure}
\caption{\footnotesize Phase structure for key-value store (top) and phase characterizations (bottom). The user provides the phase structure, and inference automatically produces the phase characterizations, forming a safe inductive phase automaton.
}
\label{fig:kv-automaton}
\label{fig:kv-invariants}
\label{fig:kv-phase}
\end{figure*} \else
\begin{figure*}[t]
\begin{subfigure}{\textwidth}
\centering
\begin{tikzpicture}[
state/.style={
draw,
circle,
font=\footnotesize\ttfamily,
minimum size=9mm,
inner sep=0pt
},
action/.style={
font=\ttfamily\scriptsize }
]
\node[state] (O) at (180:1.5cm) {{\small O[$k$]}};
\node[state] (T) at (0:1.5cm) {{\small T[$k$]}};
\draw[->] (O) edge[bend left] node[action,above] {\footnotesize reshard(*,*,$k$,*)} (T);
\draw[->] (T) edge[bend left] node[action,below] {\footnotesize recv\_transfer\_msg(*,*,$k$,*,*)} (O);
\draw[->] (O) edge[loop left] node[action,left,align=center]
{drop\_transfer\_msg(*,*,$k$,*,*) \\
retransmit(*,*,$k$,*,*) \\
send\_ack(*,*,$k$,*,*) \\
drop\_ack\_msg(*,*,$k$,*) \\
recv\_ack\_msg(*,*,$k$,*) \\
put(*,$k$,*)} (O);
\draw[->] (T) edge[loop right] node[action,right,align=center]
{drop\_transfer\_msg(*,*,$k$,*,*) \\
retransmit(*,*,$k$,*,*) \\
send\_ack(*,*,$k$,*,*) \\
drop\_ack\_msg(*,*,$k$,*) \\
recv\_ack\_msg(*,*,$k$,*) \\
put(*,$k$,*)} (T);
\end{tikzpicture}
\end{subfigure}
\\
\begin{subfigure}{\textwidth}
\centering
\begin{lstlisting}[numbers=left, numberstyle=\tiny, numbersep=5pt, name=lockserv, morekeywords={phase}]
phase O[$k$]:
invariant $\forall n_1,n_2.$ owner($n_1$,$k$)$\land$owner($n_2$,$k$)$\rightarrow n_1=n_2$
invariant $\forall n,v.$ table($n$,$k$,$v$)$\rightarrow$owner($n$,$k$)
invariant $\forall {\normalfont \textrm{src}},{\normalfont \textrm{dst}},v,s.$ $\neg$(transfer_msg(${\normalfont \textrm{src}}$,${\normalfont \textrm{dst}}$,$k$,$v$,$s$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}$,${\normalfont \textrm{src}}$,$s$))$\label{kv-char:disallow-recv-transfer}$
invariant $\forall n_1,n_2,v_1,v_2.$ table($n_1$,$k$,$v_1$)$\land$table($n_2$,$k$,$v_2$)$\rightarrow n_1 = n_2 \land v_1 = v_2$
invariant $\forall {\normalfont \textrm{src}},{\normalfont \textrm{dst}},v,s.$ $\neg$(unacked(${\normalfont \textrm{src}}$,${\normalfont \textrm{dst}}$,$k$,$v$,$s$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}$,${\normalfont \textrm{src}}$,$s$))
phase T[$k$]:
invariant $\forall n.$ $\neg$owner($n$,$k$)
invariant $\forall n,v.$ table($n$,$k$,$v$)$\rightarrow$owner($n$,$k$)
invariant $\label{kv-char:unique-transfer-unreceived}$$\forall {\normalfont \textrm{src}}_1,{\normalfont \textrm{src}}_2,{\normalfont \textrm{dst}}_1,{\normalfont \textrm{dst}}_2,v_1,v_2,s_1,s_2.$ transfer_msg(${\normalfont \textrm{src}}_1$,${\normalfont \textrm{dst}}_1$,$k$,$v_1$,$s_1$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_1$,${\normalfont \textrm{src}}_1$,$s_1$)
$\land$transfer_msg(${\normalfont \textrm{src}}_2$,${\normalfont \textrm{dst}}_2$,$k$,$v_2$,$s_2$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_2$,${\normalfont \textrm{src}}_2$,$s_2$)$\rightarrow ({\normalfont \textrm{src}}_1, {\normalfont \textrm{dst}}_1, v_1, s_1)=({\normalfont \textrm{src}}_2, {\normalfont \textrm{dst}}_2, v_2, s_2)$
invariant $\forall {\normalfont \textrm{src}}_1,{\normalfont \textrm{src}}_2,{\normalfont \textrm{dst}}_1,{\normalfont \textrm{dst}}_2,v_1,v_2,s_1,s_2.$transfer_msg(${\normalfont \textrm{src}}_1$,${\normalfont \textrm{dst}}_1$,$k$,$v_1$,$s_1$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_1$,${\normalfont \textrm{dst}}_1$,$s_1$)
$\land$unacked(${\normalfont \textrm{src}}_2$,${\normalfont \textrm{dst}}_2$,$k$,$v_2$,$s_2$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_2$,${\normalfont \textrm{src}}_2$,$s_2$)$\rightarrow ({\normalfont \textrm{src}}_1, {\normalfont \textrm{dst}}_1, v_1, s_1) = ({\normalfont \textrm{src}}_2, {\normalfont \textrm{dst}}_2, v_2, s_2)$
invariant $\forall {\normalfont \textrm{src}}_1,{\normalfont \textrm{src}}_2,{\normalfont \textrm{dst}}_1,{\normalfont \textrm{dst}}_2,v_1,v_2,s_1,s_2.$ unacked(${\normalfont \textrm{src}}_1$,${\normalfont \textrm{dst}}_1$,$k$,$v_1$,$s_1$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_1$,${\normalfont \textrm{src}}_1$,$s_1$)
$\land$unacked(${\normalfont \textrm{src}}_2$,${\normalfont \textrm{dst}}_2$,$k$,$v_2$,$s_2$)$\land$$\neg$seqnum_recvd(${\normalfont \textrm{dst}}_2$,${\normalfont \textrm{src}}_2$,$s_2$)$\rightarrow ({\normalfont \textrm{src}}_1, {\normalfont \textrm{dst}}_1, v_1, s_1) = ({\normalfont \textrm{src}}_2, {\normalfont \textrm{dst}}_2, v_2, s_2)$
\end{lstlisting}
\end{subfigure}
\caption{\footnotesize Phase structure for key-value store (top) and phase characterizations (bottom). The user provides the phase structure, and inference automatically produces the phase characterizations, forming a safe inductive phase automaton.
}
\label{fig:kv-automaton}
\label{fig:kv-invariants}
\label{fig:kv-phase}
\end{figure*} \fi
\fi
\iflockserv
\begin{example}[Quantified Phase Automaton and Structure] \label{ex:phase-structure}
\Cref{fig:lockservice-phase} shows a phase structure of the running example.
The phase invariant is structured as an automaton parameterized by a lock $\ell$.
The automaton, displayed in \Cref{fig:lockserv-automaton}, describes the protocol as transitioning between 4 distinct (logical) phases of $\ell$.
Clockwise from top, the phases for the lock $\ell$ are: Server Holds Lock, Grant message destined for some client, lock held by some client, and Unlock message sent from some client and pending to be received by the server. The transitions are labeled by actions of the system; a wildcard \texttt{*} means that the action is executed by some arbitrary node.
Each phase has
a self loop for \texttt{send\_lock}, which can happen at any time. Actions related to locks other than $\ell$ are considered as self-loops, and omitted here for brevity. There
are four non-self-loop edges, forming the core of the automaton and
capturing the lifecycle of the lock service.
The automaton transitions between phases when the protocol executes an action that matches the automaton's edge between the phases.
For example, a \codefont{recv\_grant} for some client and the lock $\ell$ is possible only when starting from phase \texttt{G}($\ell$), in which case the automaton transitions to phase \texttt{HL}($\ell$); in all other phases, the transition is disallowed.
Our algorithm exploits missing edges as additional safety properties to guide the search.
Each phase in the automaton is \emph{characterized} by a set of logical formulas, depicted in \Cref{fig:lockserv-invariants}.
When these are omitted \Cref{fig:lockserv-automaton} represents a \emph{phase structure}, which is the input to our inference algorithm.
\end{example}
\else
\iflong
\begin{example}[Quantified Phase Automaton and Structure]
\else
\begin{example}
\fi
\label{ex:phase-structure}
\Cref{fig:kv-phase} shows a phase automaton for the running example, with the view of a single key $k$.
It describes the protocol as transitioning between two distinct (logical) phases of $k$: \emph{owned} ($\texttt{O}[k]$) and \emph{transferring} ($\texttt{T}[k]$).
The edges are labeled by actions of the system. A wildcard \texttt{*} means that the action is executed with an arbitrary argument.
The two central actions are (i) {\small \texttt{reshard}}, which transitions from $\texttt{O}[k]$ to $\texttt{T}[k]$, but cannot execute in $\texttt{T}[k]$, and (ii) {\small \texttt{recv\_transfer\_message}}, which does the opposite.
The rest of the actions do not cause a phase change and appear on a self loop in each phase.
Actions related to keys other than $k$ are considered as self-loops, and omitted here for brevity.
\iflong
Intuitively, the automaton transitions between phases when the protocol executes an action that matches the automaton's edge between the phases.
\fi
Some actions are \emph{disallowed} in certain phases, namely, do not label \emph{any} outgoing edge from a phase, such as {\small \texttt{recv\_transfer\_msg($k$)}} in $\texttt{O}[k]$.
\emph{Characterizations} for each phase are depicted in \Cref{fig:kv-invariants} (bottom).
Without them, \Cref{fig:kv-automaton} represents a \emph{phase structure}, which serves as the input to our inference algorithm.
\iflong
\begin{remark}
\fi
We remark that the choice of automaton aims to reflect the safety property of interest.
\iflockserv
In the running example, one might instead imagine taking the view of a single client
as it interacts with multiple locks.
\else
In our example, one might instead imagine taking the view of a single node
as it interacts with multiple keys, which might seem intuitive
\fi
from the standpoint of implementing
the system.
However, it is not appropriate for the proof of
\iflockserv
mutual exclusion,
since locks pass in and out of the view of a single client over their lifetime.
\else
value uniqueness, since keys pass in and out of the view of a single client\iflong over their lifetime\fi.
\fi
\iflong
The phase automaton should not aim to capture the phase structure of the implementation but that arising from the correctness intuition.
\end{remark}
\fi
\fi
\end{example}
\fi
\commentout{
\begin{remark}[Phases under a view]
\TODO{move? explain somewhere earlier about the view?}
The different phases are understood with respect to a \emph{view},
\iflockserv
a single lock in the case of the lock service.
\else
a single key in the case of the running example.
\fi
This encodes another part of the intuitive argument: the focus on a
\iflockserv
single lock at a time,
which exploits the fact that different locks
\else
single key at a time,
which exploits the fact that different keys
\fi
do not interact in this protocol. \TODO{improve} \TODO{footnote}
\end{remark}
}
We now formally define \emph{phase invariants} as phase automata that overapproximate the behaviors of the original system.
\begin{definition}[Language of Phase Automaton]
\label{def:language-phase-automaton}
Let $\mathcal{A}$ be a quantified phase automaton over $\vocabulary$, and $\overline{\sigma} = \sigma_0,\ldots,\sigma_n$ a finite sequence of states \iflong (first-order structures)\fi \iflong in $\structOfVoc{\vocabulary}$\else over $\vocabulary$\fi, all with domain ${D}$.
Let $v: \mathcal{V} \to {D}$ be a valuation of the automaton quantifiers.
We say that:
\begin{itemize}
\item $\overline{\sigma}, v \models \mathcal{A}$ if there exists a trace of phases $q_0,\ldots,q_n$ such that $(\sigma_i, \sigma_{i+1}), v \models \edgelabel{(q_i, q_{i+1})}$ for every $0 \leq i < n$ \iflong (in particular, $q_0,\ldots,q_n$ is a trace of $\mathcal{A}$)\fi and $\sigma_i, v \models \statelabel{q_i}$ for every $0 \leq i \leq n$.
\item $\overline{\sigma} \models \mathcal{A}$ if $\overline{\sigma}, v \models \mathcal{A}$ for every valuation $v$.
\end{itemize}
The language of $\mathcal{A}$ is $\Lang{\mathcal{A}} = \{ \overline{\sigma} \mid \overline{\sigma} \models \mathcal{A}\}$.
\end{definition}
\iflong
\subsection{Phase Invariants}
\fi
\OMIT{omitting: We use phase automata to over-approximate the set of traces of a transition system, in which case we refer to them as phase invariants:}
\begin{definition}[Phase Invariant]
A phase automaton $\mathcal{A}$ is a \emph{phase invariant} for a transition system ${\textit{TS}}$ \iflong(both over $\vocabulary$)\fi if $\Lang{{\textit{TS}}} \subseteq \Lang{\mathcal{A}}$, where $\Lang{{\textit{TS}}}$ denotes the set of \iflong all\fi finite traces of ${\textit{TS}}$.
\end{definition}
\commentout{
That is, $\mathcal{A}$ is a phase invariant of ${\textit{TS}}$ if the language of the latter is included in the language of the former (when viewing the set of traces of ${\textit{TS}}$ as its language).
In other words, $\mathcal{A}$ over-approximates the set of traces of ${\textit{TS}}$.}\iflong{
Trace inclusion ensures that a phase invariant $\mathcal{A}$ can be soundly used to verify safety properties of ${\textit{TS}}$.
\fi
\iflockserv
\iflong
\begin{example}[Phase Invariant]
\else
\begin{example}
\fi
The phase automaton of \Cref{fig:lockserv-automaton} is a \emph{phase invariant} for the lock service protocol: the phase characterizations of each phase hold whenever an execution trace of the original system reaches the phase.
The phases and characterizations correspond exactly to the properties from the intuitive correctness argument in \Cref{sec:running}; one out of the four properties is true in each phase, while all others are false.
\end{example}
\else
\iflong
\begin{example}[Phase Invariant]
\else
\begin{example}
\fi
The phase automaton of \Cref{fig:kv-automaton} is a \emph{phase invariant} for the protocol: intuitively, whenever an execution of the protocol reaches a phase, its characterizations hold.
This fact may not be straightforward to establish. To this end we develop the notion of \emph{inductive} phase invariants.
\end{example}
\iflong
\subsection{Establishing Phase Invariants with Inductive Phase Invariants}
\else
\subsection{Establishing Safety and Phase Invariants with Inductive Phase Invariants}
\fi
To establish phase invariants, we use inductiveness:
\begin{definition}[Inductive Phase Invariant] \label{def:ind}
$\mathcal{A}$ is \emph{inductive w.r.t.\ ${\textit{TS}} = (\textit{Init}, {\textit{TR}})$} if\iflong the following conditions hold\fi:
\begin{description}
\item[{\bf Initiation:}] $\textit{Init} \implies \left(\forall \mathcal{V}. \ \statelabel{\iota}\right)$ .
\item[{\bf Inductiveness}:] for all $(q,p) \in \mathcal{R}$, \quad $\forall \mathcal{V}. \ \left(\statelabel{q} \land \edgelabel{(q,p)} \implies \statelabel{p}'\right)$.
\item[{\bf Edge Covering}:] for every $q \in \mathcal{Q}$, \quad $\forall \mathcal{V}. \ \left(\statelabel{q} \land {\textit{TR}} \implies \bigvee_{(q,p) \in \mathcal{R}}{\edgelabel{(q,p)}}\right)$ .
\end{description}
\end{definition}
\iflong
\begin{example}[Inductive Phase Invariant]
\else
\begin{example}
\fi
The phase automaton in
\iflockserv
\Cref{fig:lockservice-phase}
\else
\Cref{fig:kv-phase}
\fi
is an inductive phase invariant\iflong: the characterizations are such that in every possible phase, when the protocol executes a valid action starting from a state satisfying the current phase characterizations, there is an outgoing edge that matches this action (covering), and the resulting program state satisfies the characterization of the target phase (inductiveness)\fi.
\iflockserv
For example, it can be seen that every transition of the protocol starting from a state that satisfies the characterizations of \texttt{G}($\ell$) matches the labeling of at least one automaton edge outgoing from \texttt{G}($\ell$), and if, for example, the transition matches the labeling of the edge to \texttt{HL}($\ell$) (i.e.\ a \codefont{recv\_grant}(\texttt{*},$\ell$) action), the protocol necessarily ends in a state satisfying the characterizations of \texttt{HL}($\ell$).
\else
For example, the only disallowed transition in \texttt{O[$k$]} is {\small \texttt{recv\_transfer\_message}}, which indeed cannot execute in \texttt{O[$k$]} according to the characterization in \cref{kv-char:disallow-recv-transfer}.
Further, if, for example, a protocol's transition from \texttt{O[$k$]} matches the labeling of the edge to \texttt{T[$k$]} (i.e.\ a {\small \texttt{reshard}} action on $k$), the post-state necessarily satisfies the characterizations of \texttt{T[$k$]}: for instance, the post-state satisfies the uniqueness of unreceived transfer messages (\cref{kv-char:unique-transfer-unreceived}) because in the pre-state there are none (\cref{kv-char:disallow-recv-transfer}).
\fi
\end{example}
\commentout{
The requirements listed in \Cref{eq:ind-auto-init,eq:ind-auto-consec,eq:ind-edge-covering} define \emph{verification conditions} for verifying that a given phase automaton is inductive w.r.t.\ ${\textit{TS}}$, \sharon{rephrase as follws: Even though the focus of this paper is on inference of inductive invariants, this already gives rise to deductive verification based on user-provided phased automata, as justified by the following lemma} and form the basis for \emph{deductive verification} based on phase automata, as justified by the following lemma.
}
\begin{lemma}
\label{lemma:phase-ind-to-phase-inv}
If $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ then it is a phase invariant for ${\textit{TS}}$.
\end{lemma}
\commentout{
To prove the lemma we first define a notion of \emph{simulation} which provides a sufficient condition for $\mathcal{A}$ being a phase invariant of ${\textit{TS}}$. We then show that for an inductive phase invariant, the phase characterizations induce a simulation relation.
\begin{definition}[Simulation]
\label{def:simulation}
We say that $\mathcal{A}$ \emph{simulates} ${\textit{TS}} = (\textit{Init}, {\textit{TR}})$ (both over $\vocabulary$) if for every domain ${D}$ and every valuation $v$ over ${D}$ there is a relation $H \subseteq \structdom{\vocabulary}{{D}} \times \mathcal{Q}$ such that
\begin{enumerate}
\item \label{it:simulation-labeling-agree} $(\sigma, q) \in H$ implies $\sigma,v \models \statelabel{q}$.
\item \label{it:simulation-init} For every $\sigma_0 \models \textit{Init}$, $(\sigma_0, \iota) \in H$.
\item \label{it:simulation-next} For every transition $(\sigma, \sigma') \in {\textit{TR}}$, if $(\sigma, q) \in H$ then there exists $q' \in \mathcal{Q}$ such that $(\sigma', q') \in H$ and
$(\sigma, \sigma'), v \models \edgelabel{(q,q')}$.
\end{enumerate}
\end{definition}
\begin{lemma}\label{lem:sim-to-phase-inv}
If $\mathcal{A}$ simulates ${\textit{TS}}$ then it is a phase invariant for ${\textit{TS}}$.
\end{lemma}
\begin{proof}
Let $\overline{\sigma} = \sigma_0, \ldots, \sigma_n$ be a finite trace of ${\textit{TS}}$, and let $v$ be a valuation of $\mathcal{A}$'s quantifiers to the domain of $\overline{\sigma}$.
As is standard, by induction on $n$ we can show that there exists a trace $q_0, \ldots, q_n$ of the phase automaton such that
$\sigma_i, v \models \statelabel{q_i}$ and $(\sigma_i, \sigma_{i+1}), v \models \edgelabel{(q_i, q_{i+1})}$.
This implies that $\overline{\sigma}, v \models \mathcal{A}$, as required.
\qed
\end{proof}
Equipped with the notion of simulation, we now return to the proof of \Cref{lemma:phase-ind-to-phase-inv}, and show that the definition of an inductive phase invariant ensures that
$\statelabel{}$ induces a simulation relation, and therefore ensures that the automaton is a phase invariant.
\begin{proof}[Proof of \Cref{lemma:phase-ind-to-phase-inv}]
We show that if $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ then $\mathcal{A}$ simulates ${\textit{TS}}$ (and hence by \Cref{lem:sim-to-phase-inv} it is a phase invariant).
Fix a domain ${D}$ and a valuation $v$.
Take the simulation relation to be
\begin{equation}
H = \{(\sigma, q) \, | \, \sigma, v \models \statelabel{q}\}.
\end{equation}
To see that this is a simulation relation as required in \Cref{def:simulation}, note that requirement \ref{it:simulation-labeling-agree} holds by construction.
The condition of \Cref{eq:ind-auto-init} implies requirement \ref{it:simulation-init}.
For requirement \ref{it:simulation-next}, let $(\sigma, q) \in H$ and $(\sigma, \sigma') \in {\textit{TR}}$.
We have that $\sigma, v \models \statelabel{q}$, so, from edge covering (\Cref{eq:ind-edge-covering}), there exists $p \in \mathcal{Q}$ such that $(\sigma, \sigma'), v \models \edgelabel{(q,p)}$.
From the condition of \Cref{eq:ind-auto-consec} necessarily $\sigma', v \models \statelabel{p}'$, and thus $(\sigma', p) \in H$, as required.
\end{proof}
}
\OMIT{
\begin{corollary}
\label{cor:ind-safe-implies-safe}
If $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ and safe w.r.t\ a safety property $\forall \mathcal{V}. \ \mathcal{P}} \newcommand{\Bad}{Bad$ then $\forall \mathcal{V}. \, \mathcal{P}} \newcommand{\Bad}{Bad$ is an invariant of ${\textit{TS}}$.
\end{corollary}
}
\begin{remark} \label{rem:weaker-ind}
The careful reader may notice that the inductiveness requirement is stronger than needed to ensure that the characterizations form a phase invariant. It could be weakened to require for every $q \in \mathcal{Q}$:
$
\forall \mathcal{V}. \ \statelabel{q} \wedge {\textit{TR}} \implies \bigvee_{(q,p) \in \mathcal{R}}{\edgelabel{(q,p)} \wedge \statelabel{p}'}.
$
However, as we explain in \Cref{sec:inference}, our notion of inductiveness is crucial for \emph{inferring} inductive phase automata, which is the goal of this paper.
Furthermore, for deterministic phase automata, the two requirements coincide.
\end{remark}
\para{Inductive invariants vs. inductive phase invariants}
Inductive invariants and inductive phase invariants are closely related:
\iflong
an inductive phase invariant induces a ``standard'' inductive invariant, and vice versa: \fi
\begin{lemma}
\label{lemma:phase-to-ind}
If $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ then
$
\forall \mathcal{V}. \ \bigvee_{q \in \mathcal{Q}}{\statelabel{q}}
$
is an inductive invariant for ${\textit{TS}}$. \iflong
Conversely, if
\else
If
\fi
$\textit{Inv}$ is an inductive invariant for ${\textit{TS}}$, then the phase automaton $\mathcal{A}_{\textit{Inv}} = (\{q\}, \{q\}, \emptyset, \edgelabel{}, \statelabel{})$, where $\edgelabel{(q,q)} = {\textit{TR}}$ and $\statelabel{q} = \textit{Inv}$ is an inductive phase automaton w.r.t. ${\textit{TS}}$.
\end{lemma}
In this sense, phase inductive invariants are as expressive as inductive invariants.
However, as we show in this paper, their structure can be used by a user as an intuitive way to guide an automatic invariant inference algorithm.
\iflong
\begin{remark}
It is straightforward to add more flexibility to a phase automaton by allowing a set of initial states, $\mathcal{Q}_0$.
In this case, for the automaton to over-approximate all the reachable states of ${\textit{TS}}$, it suffices that every initial state corresponds to some initial phase of ${\textit{TS}}$, possibly depending on the valuation of $\mathcal{V}$.
Therefore, the initiation constraint in the definition of an inductive phase invariant for ${\textit{TS}}$ may be relaxed into:
\[
\textit{Init} \implies \left(\forall \mathcal{V}. \ \bigvee_{q_0 \in \mathcal{Q}_0}\statelabel{q_0}\right).
\]
\end{remark}
\fi
\iflong
\subsection{Safe Inductive Phase Invariants}
\else
\para{Safe Inductive Phase Invariants}
\fi
Next we show that an inductive phase invariant can be used to establish safety.
\begin{definition}[Safe Phase Automaton]
\label{def:safe-automaton}
Let $\mathcal{A}$ be a phase automaton over $\vocabulary$ with quantifiers $\mathcal{V}$
\iflong, and let $\forall \mathcal{V}.\ \mathcal{P}} \newcommand{\Bad}{Bad$ be a safety property\fi. Then $\mathcal{A}$ is \emph{safe} w.r.t.\ $\forall \mathcal{V}. \ \mathcal{P}} \newcommand{\Bad}{Bad$ if
\iflong
\begin{equation*}
\label{eq:safe-automaton}
\forall \mathcal{V}. \ \left(\statelabel{q} \implies \mathcal{P}} \newcommand{\Bad}{Bad\right)
\end{equation*}
\else
\, $\forall \mathcal{V}. \ \left(\statelabel{q} \implies \mathcal{P}} \newcommand{\Bad}{Bad\right)$
\fi
holds for every $q \in \mathcal{Q}$.
\end{definition}
\OMIT{
\iflong
The following lemma shows that if $\mathcal{A}$ is a phase invariant of ${\textit{TS}}$ then the safety of $\mathcal{A}$ implies safety of ${\textit{TS}}$.
\begin{lemma} \label{lem:phase-inv-to-safety}
If $\mathcal{A}$ is a phase invariant for ${\textit{TS}}$ and safe w.r.t.\ $\forall \mathcal{V}.\ \mathcal{P}} \newcommand{\Bad}{Bad$, then $\forall \mathcal{V}. \, \mathcal{P}} \newcommand{\Bad}{Bad$ is an invariant of ${\textit{TS}}$.
\end{lemma}
\begin{proof}
Let $\sigma_1,\ldots,\sigma_n$ be a finite trace of ${\textit{TS}}$.
Let $v$ be a valuation for $\mathcal{V}$.
$\overline{\sigma} \models \mathcal{A}$, so there exists a trace of phases $q_0,\ldots,q_n$ such that $\sigma_i, v \models \statelabel{q_i}$.
Since $v \models \statelabel{q_i} \to \mathcal{P}} \newcommand{\Bad}{Bad$, for every $i$ it holds that $\sigma_i, v \models \mathcal{P}} \newcommand{\Bad}{Bad$.
The claim follows.
\end{proof}
\begin{corollary}
\label{cor:ind-safe-implies-safe}
\iflong
If $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ and safe w.r.t.\ a \TODO{omit next 2 words if it helps} safety property $\forall \mathcal{V}. \ \mathcal{P}} \newcommand{\Bad}{Bad$ then $\forall \mathcal{V}. \, \mathcal{P}} \newcommand{\Bad}{Bad$ is an invariant of ${\textit{TS}}$.
\else
If $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ and safe w.r.t.\ $\forall \mathcal{V}. \ \mathcal{P}} \newcommand{\Bad}{Bad$ then $\forall \mathcal{V}. \, \mathcal{P}} \newcommand{\Bad}{Bad$ is an invariant of ${\textit{TS}}$.
\fi
\end{corollary}
}
\begin{lemma}
\label{cor:ind-safe-implies-safe}
If $\mathcal{A}$ is inductive w.r.t.\ ${\textit{TS}}$ and safe w.r.t.\ $\forall \mathcal{V}. \ \mathcal{P}} \newcommand{\Bad}{Bad$ then $\forall \mathcal{V}. \, \mathcal{P}} \newcommand{\Bad}{Bad$ is an invariant of ${\textit{TS}}$.
\end{lemma}
\fi
\commentout{
\subsection{Obtaining Inductive Phase Invariants by Strengthening} \label{sec:stregthening}
Given a phase invariant that is not inductive w.r.t.\ $TS$, we may establish that it is an invariant by \emph{strengthening} it into one that is inductive:
\begin{definition}[Strengthening]
\label{def:strengthening}
A phase automaton $\mathcal{A}_2$ is a \emph{strengthening} of a phase automaton $\mathcal{A}_1$ if $\mathcal{A}_1, \mathcal{A}_2$ have the same automaton structure, and the characterization of every phase is stronger in $\mathcal{A}_2$ than it is in $\mathcal{A}_1$, namely, for every $q \in \mathcal{Q}^1 = \mathcal{Q}^2$,
\begin{equation}
\forall \mathcal{V}. \ \left(\statelabel{q}^2 \implies \statelabel{q}^1\right)
\end{equation}
\end{definition}
\begin{lemma}
\label{lem:ind-strengthening-to-phase-inv}
If $\mathcal{A}$ has a strengthening $\mathcal{A}'$ that is inductive for ${\textit{TS}}$ then $\mathcal{A}$ is a phase invariant for ${\textit{TS}}$.
\end{lemma}
\iflong
\begin{proof}
Let $\overline{\sigma} = \sigma_0,\ldots,\sigma_n$ be a trace of ${\textit{TS}}$ starting from an initial state.
We show that $\overline{\sigma} \in \Lang{\mathcal{A}}$.
Let $v$ be a valuation of $\mathcal{V}$.
By \Cref{lemma:phase-ind-to-phase-inv}, $\mathcal{A}'$ is a phase invariant for ${\textit{TS}}$. Hence, there exists a trace $q_0,\ldots,q_n$ of phases that corresponds to $(\overline{\sigma},v)$ in $\mathcal{A}'$. We show that the same trace is also a witness for $\mathcal{A}$.
Since $\mathcal{A}$ has the same structure as $\mathcal{A}'$, we know that $q_0 = \iota$ and $(\sigma_i, \sigma_{i+1}), v \models \edgelabel{(q_i, q_{i+1})}$ for every $0 \leq i < n$.
It remains to show that $\sigma_i, v \models \statelabel{q_i}$ for every $0 \leq i \leq n$, but since $\mathcal{A}'$ is a strengthening of $\mathcal{A}$, this follows immediately from $\sigma_i, v \models \statelabel{q_i}'$. \TODO{the prime here is confusing!}
\end{proof}
\fi
We note that $\mathcal{A}$ may be a phase invariant for ${\textit{TS}}$ but not have any strengthening that is inductive.
This may happen for two reasons.
First, as with standard inductive invariants, it is possible that the necessary strengthening of the phase characterizations is not expressible in the logic available to us.
Second, even if we assume an unrestricted language of phase characterizations, it is possible that the edge labeling is too permissive, thus adding transitions that are not necessary for the edge covering requirement.
Such ``redundant'' transitions may sometimes be harmless, but they may also violate preservation along some edge.
Namely, if no state that has such an outgoing transition can reach the corresponding phase $q$ from previous phases, such violations
can be overcome by strengthening $\statelabel{q}$ to exclude all states that have such an outgoing transition (assuming an unrestricted language of phase characterizations), thus disabling the problematic transition along the edge.
In these cases, the automaton can be strengthened into one that is inductive.
However, in other cases, such strengthening
would exclude states that \emph{can} reach phase $q$ and as such would damage the inductiveness property along incoming edges of $q$.
In such cases, the only way to disable problematic transitions along automaton edges is by strengthening the transition relation formulas (i.e., updating the automaton structure). Hence, no inductive strengthening exists.
The second reason for the absence of a strengthening has no counterpart in standard inductive invariants; it reflects the additional structure expressed by a phase automaton, which is enforced by our stronger definition of inductiveness (as opposed to the weaker definition mentioned in \Cref{rem:weaker-ind}).
Fortunately, this reason can be avoided by considering deterministic phase automata:
\begin{lemma} \label{lem:strengthening-exists}
Let $\mathcal{A}$ be a deterministic phase automaton, and assume an unrestricted language of phase characterizations. Then $\mathcal{A}$ is a phase invariant for ${\textit{TS}}$ if and only if it has a strengthening $\mathcal{A}'$ that is inductive w.r.t.\ ${\textit{TS}}$.
\end{lemma}
\iflong
\begin{proof}[Proof (sketch)]
The implication from right to left is clear. Consider the other direction.
For a deterministic phase automaton, $\Lang{{\textit{TS}}} \subseteq \Lang{\mathcal{A}}$ if and only if there exists a simulation relation between ${\textit{TS}}$ and $\mathcal{A}$.
Furthermore, in this case, the inductiveness requirement coincides with the requirement that the phase characterizations induce a simulation relation. Hence, in this case, by defining the characterization of $q$ to include all the states simulated by it, we obtain an inductive strengthening of $\mathcal{A}$.
\end{proof}
\fi
We point out that restricting our attention to deterministic phase automata does not lose generality in the context of safety verification since every inductive phase invariant, which are the ones we seek, can be translated into a deterministic one:
\begin{lemma}
Let $\mathcal{A} =(\mathcal{Q}, \iota, \mathcal{V}, \edgelabel{}, \statelabel{}) $ be an inductive phase invariant w.r.t.\ ${\textit{TS}}$. Define an arbitrary total order, $<$, on $\mathcal{Q}$, and define $\mathcal{A}' = (\mathcal{Q}, \iota, \mathcal{V}, \edgelabel{}', \statelabel{})$ where
\[
\edgelabel{(q,p)}' = \edgelabel{(q,p)} \wedge \bigwedge_{p' < p} \neg \edgelabel{(q,p')}
\]
Then $\mathcal{A}'$ is a deterministic inductive phase invariant w.r.t.\ ${\textit{TS}}$.
\end{lemma}
\iflong
\begin{proof}
The definition of $\edgelabel{}'$ ensures that it is deterministic, and inherits the edge covering property from $\edgelabel{}$.
Initiation is not affected by the edge labeling, and inductiveness cannot be damaged by strengthening $\edgelabel{(q,p)}'$.
\end{proof}
\fi
We note that when the language of phase characterizations is restricted and an inductive strengthening does not exist for this reason, it may be possible to obtain an inductive strengthening in the given language by changing the automaton structure.
}
\section{Inference of Inductive Phase Invariants}
\label{sec:inference}
\iflockserv
\else
\fi
In this section we turn to the \emph{inference} of safe inductive phase invariants over a given phase structure, which guides the search\iflong for invariant\fi.
\commentout{
Providing a full specification of an inductive phase invariant may still be difficult.
We therefore consider the problem of inferring a safe inductive phase invariant from a phase structure. }
\iflong
This section defines the problem, shows that it can be reduced to a set of Constrained Horn Clauses, discusses the aspects by which a phase structure guides inference, and considers witnesses for the case that no solution exists.
\fi
Formally, the problem we target is:
\begin{definition}[Inductive Phase Invariant Inference] Given a transition system ${\textit{TS}} = (\textit{Init}, {\textit{TR}})$, a phase structure $\mathcal{S} = (\mathcal{Q}, \iota, \mathcal{V}, \edgelabel{})$ and a safety property $\forall \mathcal{V}. \ \mathcal{P}} \newcommand{\Bad}{Bad$, all over $\vocabulary$, \emph{find} a safe inductive phase invariant $\mathcal{A}$ for ${\textit{TS}}$ over the phase structure $S$, if one exists.
\end{definition}
\iflockserv
\TODO{}
\else
\begin{example} \label{ex:inference}
Inference of an inductive phase invariant is provided with the phase structure in \Cref{fig:kv-automaton}, which embodies an intuitive understanding of the different phases the protocol undergoes (see \Cref{ex:phase-structure}).
The algorithm automatically finds phase characterizations forming a safe inductive phase invariant over the user-provided structure.
We note that inference is valuable even after a phase structure is provided: in the running example, finding an inductive phase invariant is not easy; in particular, the characterizations in \Cref{fig:kv-invariants} relate different parts of the state and involve multiple quantifiers.
\end{example}
\fi
\subsection{Reduction to Constrained Horn Clauses}
\label{sec:chc-reduction}
We view each unknown phase characterization, $\statelabel{q}$, which we aim to infer for every $q \in \mathcal{Q}$, as a predicate $I_q$.
The definition of a safe inductive phase invariant induces a set of second-order Constrained Horn Clauses (CHC) over $I_q$: \begin{align}
\label{eq:chc-init}
&\text{\bf Initiation.} && \textit{Init} \implies \left(\forall \mathcal{V}. \ I_{\iota}\right) && \\
&\text{\bf Inductiveness.} \text{ \ For every } (q,p) \in \mathcal{R}:
&& \forall \mathcal{V}. \ \left(I_q \land \edgelabel{(q,p)} \implies I'_p\right) \label{eq:chc-ind} && \\
&\text {\bf Edge Covering.} \text{ \ For every } q \in \mathcal{Q}:
&& \forall \mathcal{V}. \ \bigg(I_q \land {\textit{TR}} \implies \bigvee_{(q,p) \in \mathcal{R}}{\edgelabel{(q,p)}}\bigg) \label{eq:chc-edge-cover} && \\
&\text{\bf Safety.} \ \text{For every } q \in \mathcal{Q}: && \forall \mathcal{V}. \ \left(I_q \implies \mathcal{P}} \newcommand{\Bad}{Bad\right) \label{eq:chc-safe} &&
\end{align}
where $\mathcal{V}$ denotes the quantifiers of $\mathcal{A}$.
\iflong
The number of constraints is $1 + |\mathcal{R}| + 2|\mathcal{Q}|$.
\ifsketch
\TODO{}
If $\mathcal{A}$ is nothing more than a phase structure, constraint \ref{eq:chc-strengthen} can be omitted, and the number of constraints is
$1 + |\mathcal{R}| + 2|\mathcal{Q}|$.
\fi
\fi
All the constraints are \emph{linear}, namely at most one unknown predicate appears at the lefthand side of each implication.
\commentout{Using CHC terminology, \Cref{eq:chc-init} defines ``facts'' (since no unknown predicate appears in the body);
\Cref{eq:chc-ind} defines ``rules''; and
\Cref{eq:chc-edge-cover,eq:chc-safe,eq:chc-strengthen} define ``queries'' (since no unknown predicates appear in the righthand side of the implications).}
Constraint (\ref{eq:chc-safe}) captures the original safety requirement, whereas (\ref{eq:chc-edge-cover}) can be understood as additional safety properties that are specified by the phase automaton (since no unknown predicates appear in the righthand side of the implications).
\OMIT{Omitting: It requires that a system trace that reaches phase $q$ with state
$\sigma$ has a matching outgoing edge of $q$ for every outgoing transition of $\sigma$.}
A \emph{solution} $\mathbf{I}$ to the CHC system associates each predicate $I_q$ with a formula $\psi_q$ over $\Sigma$ (with $\fv{\psi_q} \subseteq \mathcal{V}$) such that when $\psi_q$ is substituted for $I_q$, all the constraints are satisfied (i.e., the corresponding first-order formulas are valid).
A solution to the system induces a safe inductive phase automaton through characterizing each phase $q$ by the interpretation of $I_q$, and vice versa. Formally:
\OMIT{
\iflong
\begin{lemma}
Let $\mathbf{I}_q$ be a solution to the CHC system.
Then $\mathcal{A}' = (\mathcal{Q}, \mathcal{R}, \iota, \mathcal{V}, \edgelabel{}, \statelabel{}')$ with $\statelabel{q}' = \mathbf{I}_q$ is a safe inductive phase invariant wrt.\ ${\textit{TS}}$ and $\mathcal{P}} \newcommand{\Bad}{Bad$ and strengthens $\mathcal{A}$.
\end{lemma}
\begin{proof}
$\mathcal{A}'$ is safe due to constraint \ref{eq:chc-safe}. Constraint \ref{eq:chc-edge-cover} implies that $\mathcal{A}'$ covers ${\textit{TR}}$, constraint \ref{eq:chc-ind} implies that it is inductive, and with constraint \ref{eq:chc-init} this implies that $\mathcal{A}'$ is inductive wrt.\ ${\textit{TS}}$. Constraint \ref{eq:chc-strengthen} implies that $\mathcal{A}'$ is a strengthening of $\mathcal{A}$.
\end{proof}
The converse also holds:
\begin{lemma}
A safe inductive phase invariant which is a strengthening of $\mathcal{A}$ induces a solution to the CHC system. \end{lemma}
\else
}
\begin{lemma}\label{lem:inference}
Let $\mathcal{A} = (\mathcal{Q}, \mathcal{R}, \iota, \mathcal{V}, \edgelabel{}, \statelabel{})$ with $\statelabel{q} = {\mathbf{I}}_q$. Then $\mathcal{A}$ is a safe inductive phase invariant wrt.\ ${\textit{TS}}$ and $\forall \mathcal{V}.\ \mathcal{P}} \newcommand{\Bad}{Bad$
if and only if $\mathbf{I}$ is a solution to the CHC system.
\end{lemma}
Therefore, to infer a safe inductive phase invariant over a given phase structure, we need to solve the corresponding CHC system.
In \Cref{sec:updr-phase} we explain our approach for doing so for the class of universally quantified phase characterizations.
Note that the weaker definition of inductiveness discussed in \Cref{rem:weaker-ind} would prevent the reduction to CHC as it would result in clauses that are \emph{not} Horn clauses.
\para{Completeness of inductive phase invariants}
There are cases where a given phase structure induces a safe phase invariant ${\cal A}} \newcommand{\abs}[1]{ \vert #1 \vert $, but not an inductive one, making the CHC system unsatisfiable. However, a strengthening into an inductive phase invariant can always be used to prove that $\mathcal{A}$ is an invariant if
\begin{inparaenum}[(i)]
\item the language of invariants is unrestricted, and
\item the phase structure is deterministic, namely, does not cover the same transition in two outgoing edges. Determinism of the automaton does not lose generality in the context of safety verification since every inductive phase automaton can be converted to a deterministic one; non-determinism is in fact unbeneficial as it mandates the same state to be characterized by multiple phases (see also \Cref{rem:weaker-ind}).
\end{inparaenum}
These topics are discussed in detail in \refappendix{sec:completeness}.
\begin{remark}
\label{rem:phase-decomposition}
Each phase is associated with a set of states that can reach it, where a state $\sigma$ can reach phase $q$ if there is a sequence of program transitions that results in $\sigma$ and can lead to $q$ according to the automaton's transitions. This makes a phase structure different from a simple syntactical disjunctive template for inference, in which such semantic meaning is unavailable.
\end{remark}
\iflong
\begin{remark} \TODO{remove? (space)}
When the safety property is of the form $\forall \mathcal{V}. \ \textit{Grd} \to \psi$ where $\textit{Grd}, \psi \in \form{\vocabulary}{\mathcal{V}}$, we sometime seek an inductive phase invariant where $\textit{Grd}$ guards the entire phase structure of the automaton.
This may be represented by splitting the initial phase into two: one whose characterization includes $\textit{Grd}$, and another ``dummy'' initial phase whose characterization is $\neg \textit{Grd}$. The dummy initial phase has a single self loop labeled ${\textit{TR}}$, whereas the other maintains the actual phase structure. Moreover, $\textit{Grd}$ is added to the characterization of all other phases.
\end{remark}
\fi
\subsection{Phase Structures as a Means to Guide Inference}
\label{sec:inference-benefit}
The search space of invariants over a phase structure is in fact \emph{larger} than that of standard inductive invariants, because each phase can be associated with different characterizations.
Sometimes the disjunctive structure of the phases (\Cref{lemma:phase-to-ind}) uncovers a significantly simpler invariant than exists in the syntactical class of standard inductive invariants explored by the algorithm, but this is not always the case.\footnote{
As an illustration, \refappendix{sec:kv-ind} includes an inductive invariant for the running example which is comparable in complexity to the inductive phase invariant in \Cref{fig:kv-invariants}.
}
Nonetheless, the search for an invariant over the structure is \emph{guided}, through the following aspects: \\
(1) \emph{Phase decomposition.}
Inference of an inductive phase invariant aims to find characterizations that overapproximate the set of states reachable in each phase (\Cref{rem:phase-decomposition}).
The distinction between phases is most beneficial when there is a considerable \emph{difference} between the sets associated with different phases and their characterizations. For instance, in the running example, all states without unreceived transfer messages are associated with $\texttt{O[$k$]}$, whereas all states in which such messages exist are associated with $\texttt{T[$k$]}$---a distinction captured by the characterizations in \cref{kv-char:disallow-recv-transfer,kv-char:unique-transfer-unreceived} in \Cref{fig:kv-invariants}.
Differences between phases would have two consequences. First, since each phase corresponds to fewer states than all reachable states, generalization---the key ingredient in inference procedures---is more focused. The second consequence stems from the fact that inductive characterizations of different phases are correlated. It is expected that a certain property is more readily learnable in one phase, while related facts in other phases are more complex. For instance, the characterization in \cref{kv-char:disallow-recv-transfer} in \Cref{fig:kv-invariants} is more straightforward than the one in \cref{kv-char:unique-transfer-unreceived}. Simpler facts in one phase can help characterize an adjacent phase when the algorithm analyzes how that property evolves along the edge. Thus utilizing the phase structure can improve the gradual construction of overapproximations of the sets of states reachable in each phase.
\\
(2) \emph{Disabled transitions.} A phase automaton explicitly states which transitions of the system are enabled in each phase, while the rest are disabled. Such impossible transitions induce additional safety properties to be established by the inferred phase characterizations.
\iflockserv
For example, the phase structure in \Cref{fig:lockservice-phase} forbids a $\texttt{recv\_grant}(\texttt{c}, \ell)$ action in all phases but $\texttt{G}(\ell)$.
\else
For example, the phase invariant in \Cref{fig:kv-invariants} forbids a {\small \texttt{recv\_transfer\_message($k$)}} in $\texttt{O[$k$]}$, a fact that can trigger the inference of the characterization in \cref{kv-char:disallow-recv-transfer}.
\fi
These additional safety properties direct the search for characterizations that are likely to be important for the proof.
\\
(3) \emph{Phase-awareness.} Finally, while a phase structure can be encoded in several ways (such as ghost code), a key aspect of our approach is that the phase decomposition and disabled transitions are \emph{explicitly} encoded in the CHC system in \Cref{sec:chc-reduction}, ensuring that they guide the otherwise heuristic search.
\\
In \Cref{sec:eval-dissection} we demonstrate the effects of aspects (1)--(3) on guidance.
\newcommand{16}{16}
\newcommand{1 hour}{1 hour}
\newcommand{a 3.4GHz AMD Ryzen Threadripper 1950X with 16 physical cores, running Linux 4.15.0, using Z3 version 4.7.1}{a 3.4GHz AMD Ryzen Threadripper 1950X with 16 physical cores, running Linux 4.15.0, using Z3 version 4.7.1}
\section{Implementation and Evaluation}
\label{sec:evaluation}
In this section we apply invariant inference guided by phase structures to distributed protocols modeled in EPR, motivated by previous deductive approaches\iflong to safety of distributed protocols\fi~\cite{pldi/PadonMPSS16,DBLP:journals/pacmpl/PadonLSS17,DBLP:conf/pldi/TaubeLMPSSWW18}.
\iflong
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/workflow.jpg}
\caption{The work-flow of user-guided invariant inference through phase structures.
The input to the tool is a (1) program, (2) safety property, and (3) phase structure.
The output is either (4) inductive phase characterizations over the given phase structure, or (5) proof that universally quantified inductive phase characterizations do not exist, in the form of an abstract trace.
}
\label{fig:workflow}
\end{figure}
The work-flow for our approach is illustrated in \Cref{fig:workflow}.
\fi
\subsection{Phase-$\text{PDR}^\forall$ for Inferring Universally Quantified Characterizations}
\label{sec:updr-phase}
We now describe our procedure for solving the CHCs system of \Cref{sec:chc-reduction}.
It either (i) returns universally quantified phase characterizations that induce a safe inductive phase invariant, (ii) returns an abstract counterexample trace demonstrating that this is not possible, or (iii) diverges.
\para{EPR}
Our procedure handles transition systems expressed using the extended {\bf E}ffectively {\bf PR}opositional fragment (EPR) of first order logic~\cite{epr,pldi/PadonMPSS16}, and infers universally quantified phase characterizations.
Satisfiability of (extended) EPR formulas is decidable, enjoys the finite-model property, and supported by
\iflong
existing SMT solvers such as Z3~\cite{de2008z3} and first order logic provers such as iProver~\cite{DBLP:conf/cade/Korovin08}.
\else
solvers such as Z3~\cite{de2008z3} and iProver~\cite{DBLP:conf/cade/Korovin08}.
\fi
\para{Phase-$\text{PDR}^\forall$}
Our procedure is based on $\text{PDR}^\forall$~\cite{DBLP:journals/jacm/KarbyshevBIRS17}, a variant of PDR~\cite{ic3,pdr} that infers universally quantified inductive invariants.
PDR computes a sequence of \emph{frames} $\mathcal{F}_0, \ldots, \mathcal{F}_n$
such that $\mathcal{F}_i$ overapproximates the set of states reachable in $i$ steps.
In our case, each frame $\mathcal{F}_i$ is a mapping from a phase $q$ to characterizations.
The details of the algorithm are standard for PDR; we describe the gist of the procedure in \refappendix{sec:algorithm-long}. We only stress the following:
Counterexamples to safety take into account the safety property as well as disabled transitions. Search for predecessors is performed by going backwards on automaton edges, blocking counterexamples from preceding phases to prove an obligation in the current phase. Generalization is performed w.r.t.\ all incoming edges. As in $\text{PDR}^\forall$, proof obligations are constructed via diagrams~\cite{chang1990model}; in our setting these include the interpretation for the view quantifiers (see
\ifapp
\refappendix{sec:algorithm-long}
\else
\cite{extended-version}
\fi
for details).
\para{Edge covering check in EPR}
\iflong
When the transition relation formula is in EPR and the phase characterizations are universally quantified, the checks induced by \Cref{eq:chc-init,eq:chc-ind,eq:chc-safe} translate to checking (un)satisfiability of EPR formulas, potentially causing divergence of the solver.
\Cref{eq:chc-edge-cover} is trickier as checking implication between two EPR transition relations falls outside of EPR.
To use a decidable edge covering check, we exploit the typical structure of transition relations in our setting, which is a disjunction between the transition relation of exported actions (the different {\small \texttt{action}}s in \Cref{fig:kv-ivy}).
In the phase automaton we label an edge $(q,p)$ by a set of exported actions, each action $a$ with a guard $g^{(q,p)}_a$ which is an alternation-free formula (a Boolean combination of universal and existential closed formulas). The edge covering check (\Cref{eq:chc-edge-cover}) can now be written
\begin{equation}
\forall \mathcal{V}. \ \bigg(I_q \land {\textit{TR}}_a \implies \bigvee_{(q,p) \in \mathcal{R}}{g^{(q,p)}_a}\bigg).
\end{equation}
The righthand side of the implication is alternation-free and thus the check falls into the decidable EPR class.
\else
In our setting, \Cref{eq:chc-init,eq:chc-ind,eq:chc-safe} fall in EPR, but not \Cref{eq:chc-edge-cover}.
Thus, we restrict edge labeling so that each edge is labeled with a ${\textit{TR}}$ of an {\small \texttt{action}},
together with an alternation-free precondition. It then suffices to check implications between the preconditions and the entire ${\textit{TR}}$ (see the extended version~\cite{extendedVersion}).
\fi
Such edge labeling is sufficiently expressive for \iflong the phase structures of\fi all our examples.
Alternatively, sound but incomplete bounded quantifier instantiation~\cite{DBLP:conf/tacas/FeldmanPISS17} could be used, potentially allowing more complex decompositions of ${\textit{TR}}$.
\para{Absence of Inductive Phase Characterizations}
\label{sec:infer-absence}
What happens when the user gets the automaton wrong?
One case is when there does not exist an inductive phase invariant with universal phase characterizations over the given structure. When this occurs, our tool can return an \emph{abstract counterexample trace}---a sequence of program transitions and transitions of the automaton (inspired by~\cite{DBLP:journals/jacm/KarbyshevBIRS17,DBLP:conf/popl/PadonISKS16})---which constitutes a proof
of that fact
(see \refappendix{sec:abstract-traces}). The counterexample trace can assist the user in debugging the automaton or the program and modifying them.
For instance, missing edges occurred frequently when we wrote the automata of \Cref{sec:evaluation}, and we used the generated counterexample traces to correct them.
Another type of failure is when an inductive phase invariant exists but the automaton does not direct the search well towards it. In this case the user may decide to terminate the analysis and articulate a different intuition via a different phase structure. In standard inference procedures, the only way to affect the search is by modifying the transition system; instead, phase structures equip the user with an ability to guide the search.
\subsection{Evaluation}
We evaluate our approach for user-guided invariant inference
\iflong
based on phase structures
\fi
by comparing Phase-$\text{PDR}^\forall$ to standard $\text{PDR}^\forall$\iflong, its inductive invariant inference counterpart\fi.
We implemented $\text{PDR}^\forall$ and Phase-$\text{PDR}^\forall$ in \Tool~\cite{mypyvy}, a new system for invariant inference inspired by Ivy~\cite{DBLP:conf/sas/McMillanP18}, over Z3~\cite{de2008z3}.
We study\iflong the following effects of guidance by phase structures\fi:
\begin{enumerate}
\item Can Phase-$\text{PDR}^\forall$ \empheval{converge} to a proof when $\text{PDR}^\forall$ does not (in reasonable time)?
\item Is Phase-$\text{PDR}^\forall$ \empheval{faster} than $\text{PDR}^\forall$? \ifsketch
Do additional phase characterizations in a phase sketch yield further speedup?
\fi
\OMIT{
\item How do Phase-$\text{PDR}^\forall$ and $\text{PDR}^\forall$ \empheval{scale} with the number of phases in the protocol? (\Cref{sec:eval-scalability})
}
\item Which aspects of Phase-$\text{PDR}^\forall$ contribute to its performance benefits? \end{enumerate}
\ifsketch
In each protocol, we compare Phase-$\text{PDR}^\forall$ with a phase structure and
Phase-$\text{PDR}^\forall$ with a phase sketch which includes partial characterizations,
to the baseline $\text{PDR}^\forall$ inferring a standard inductive invariant.
\fi
\para{Protocols}
We applied $\text{PDR}^\forall$ and Phase-$\text{PDR}^\forall$ to the most challenging examples admitting universally-quantified invariants, which previous works verified using deductive techniques.
The protocols we analyzed are listed below and in \Cref{tab:examples-evaluation-table}. The full models appear in~\cite{mypyvy-examples}.
The KV-R protocol analyzed is taken from one of the two realistic systems studied by the IronFleet paper~\cite{IronFleet} using deductive verification.
\para{Phase structures}
The phase structures we used appear in~\cite{mypyvy-examples}.
In all our examples, it was straightforward to translate the existing high-level intuition of important and relevant distinctions between phases in the protocol into the phase structures we report. For example, it took us less than an hour to finalize an automaton for KV-R.
We emphasize that phase structures do not include phase characterizations; the user need not supply them, nor has to understand the inference procedure. Our exposition of the phase structures below refers to an intuitive meaning of each phase, but this is not part of the phase structure provided to the tool.
\OMIT{
\paragraph{Specifying phase structures} In our experience, producing phase structures exercises existing intuition about the protocol's correctness.
Abstract counterexample traces aided us in articulating the phase structures, in most cases indicating an omitted edge. \TODO{is this a good place for this?}
}
\iflong
\newcolumntype{S}{>{\scriptsize}c}
\begin{table}[t]
\centering
\begin{threeparttable}\centering
\footnotesize
\begin{tabular}{|l|c|c||c|c||c|c||cccc|cccc|}
\hline
\multirow{2}{*}{Program} & \multirow{2}{*}{$\text{PDR}^\forall$} & \multirow{2}{*}{Phase-$\text{PDR}^\forall$} & \multirow{2}{*}{\#p} & \multirow{2}{*}{\#v} & \multirow{2}{*}{\#r} & \multirow{2}{*}{|a|}
& \multicolumn{4}{c|}{Inductive} & \multicolumn{4}{c|}{Phase-Inductive}
\\
& & & & & & & |f| & \#c & \#q & \#l & |f| & \#c & \#q & \#l
\\
\hline
\begin{tabular}{l}
Lock service \\
(single lock)\end{tabular} & 2.21 (00.03) & 0.67 (0.01) & 4 & 1 & 5 & 1
& 11 & 9 & 15 & 21
& 6 & 3--4 & 3--4 & 3--6
\\
\begin{tabular}{l}
Lock service\\(multiple locks)\end{tabular} & 2.73 (00.02) & 1.06 (0.01) & 4 & 1 & 5 & 2
& 11 & 9 & 24 & 21
& 6 & 4 & 3--4 & 4--6
\\
Consensus & 60.54 (2.95) & 1355 (570)$^*$ & 3 & 1 & 7 & 2
& 9 & 6 & 15 & 15
& 12 & 5--6 & 10--14 & 9--15
\\
KV (basic) & 1.79 (0.02) & 1.59 (0.02) & 2 & 1 & 3 & 3
& 5 & 7 & 27 & 19
& 5 & 4 & 9--10 & 8--9
\\
Ring leader & 152.44 (39.41) & 2.53 (0.04) & 2 & 2 & 4 & 3
& 6--7 & 6 & 11 & 16
& 5 & 1--2 & 0--1 & 1--4
\\
\hline
KV-R & 2070 (370)$^*$ & 372.5 (35.9) & 2 & 1 & 7 & 5
& 12--15 & 24 & 156 & 106
& 11--13 & 5--11 & 15--67 & 12--52
\\
Cache coherence & > 1 hour & 90.1 (0.82) & 10 & 1 & 11 & 2
& n/a & n/a & n/a & n/a
& 13 & 10--15 & 12--27 & 14--39
\\
\hline
\end{tabular}
\end{threeparttable}
\caption{
\label{tab:examples-evaluation-table}
\footnotesize Running times in seconds of $\text{PDR}^\forall$ and Phase-$\text{PDR}^\forall$, presented as
the mean and standard deviation (in parentheses) over 16 different Z3 random seeds.
``${}^*$'' indicates that some runs did not converge after 1 hour and were not included in the summary statistics.
``> 1 hour'' means that no runs of the algorithm converged in 1 hour.
\#p refers to the number of phases and \#v to the number of view quantifiers in the phase structure.
\#r refers to the number of relations and |a| to the maximal arity.
The remaining columns describe the inductive invariant/phase invariant obtained in inference. |f| is the maximal frame reached. \#c, \#q, \#l are the mean number of clauses, quantifiers (excluding view quantifiers) and literals per phase, ranging across the different phases.
}
\end{table}
\else
\newcolumntype{S}{>{\scriptsize}c}
\begin{table}[t]
\centering
\begin{threeparttable}\centering
\footnotesize
\begin{tabular}{|l|c|c||c|c||c|c||ccc|ccc|}
\hline
\multirow{2}{*}{Program} & \multirow{2}{*}{$\text{PDR}^\forall$} & \multirow{2}{*}{Phase-$\text{PDR}^\forall$} & \multirow{2}{*}{\#p} & \multirow{2}{*}{\#v} & \multirow{2}{*}{\#r} & \multirow{2}{*}{|a|}
& \multicolumn{3}{c|}{Inductive} & \multicolumn{3}{c|}{Phase-Inductive}
\\
& & & & & & & |f| & \#c & \#q & |f| & \#c & \#q
\\
\hline
\begin{tabular}{l}
Lock service \\
(single lock)\end{tabular} & 2.21 (00.03) & 0.67 (0.01) & 4 & 1 & 5 & 1
& 11 & 9 & 15
& 6 & 3--4 & 3--4
\\
\begin{tabular}{l}
Lock service\\(multiple locks)\end{tabular} & 2.73 (00.02) & 1.06 (0.01) & 4 & 1 & 5 & 2
& 11 & 9 & 24
& 6 & 4 & 3--4
\\
Consensus & 60.54 (2.95) & 1355 (570)$^*$ & 3 & 1 & 7 & 2
& 9 & 6 & 15
& 12 & 5--6 & 10--14
\\
KV (basic) & 1.79 (0.02) & 1.59 (0.02) & 2 & 1 & 3 & 3
& 5 & 7 & 27
& 5 & 4 & 9--10
\\
Ring leader & 152.44 (39.41) & 2.53 (0.04) & 2 & 2 & 4 & 3
& 6--7 & 6 & 11
& 5 & 1--2 & 0--1
\\
\hline
KV-R & 2070 (370)$^*$ & 372.5 (35.9) & 2 & 1 & 7 & 5
& 12--15 & 24 & 156
& 11--13 & 5--11 & 15--67
\\
Cache coherence & > 1 hour & 90.1 (0.82) & 10 & 1 & 11 & 2
& n/a & n/a & n/a
& 13 & 10--15 & 12--27
\\
\hline
\end{tabular}
\end{threeparttable}
\caption{
\label{tab:examples-evaluation-table}
\footnotesize Running times in seconds of $\text{PDR}^\forall$ and Phase-$\text{PDR}^\forall$, presented as
the mean and standard deviation (in parentheses) over 16 different Z3 random seeds.
``${}^*$'' indicates that some runs did not converge after 1 hour and were not included in the summary statistics.
``> 1 hour'' means that no runs of the algorithm converged in 1 hour.
\#p refers to the number of phases and \#v to the number of view quantifiers in the phase structure.
\#r refers to the number of relations and |a| to the maximal arity.
The remaining columns describe the inductive invariant/phase invariant obtained in inference. |f| is the maximal frame reached. \#c, \#q are the mean number of clauses and quantifiers (excluding view quantifiers) per phase, ranging across the different phases.
}
\vspace{-0.5cm}
\end{table}
\fi
\subsubsection{(1) Achieving Convergence Through Phases}
\label{sec:eval-convergence}
In this section we consider the effect of phases on inference for examples on which standard $\text{PDR}^\forall$ does not converge in 1 hour{}.
\para{Examples}
\iflockserv
\emphexample{Sharded key-value store with retransmissions (KV-R)}, based on IronKV from IronFleet~\cite[\S 5.2.1]{IronFleet}.
This is a distributed hash table where each node owns some subset of the keys,
and keys can be dynamically transferred among nodes to balance load.
The safety property ensures that each key is globally associated with one value, even in the presence of key transfers.
Messages might be dropped by the network, and the protocol uses retransmissions and sequence numbers to maintain availability and safety.
This example is especially challenging for invariant inference in part due to the high arity of relations in its model (two relations have arity 5).
The \empheval{phase structure} takes the view of a single key in the store, and has two phases: \emph{owned} and \emph{transferring}, intuitively meaning that the key is owned by a node or that an ownership transfer is in progress, respectively.
The automaton transitions from \emph{owned} to \emph{transferring} upon the system's action in which a node cedes ownership of a key and sends a message (with a sequence number) to the new owner-to-be. The automaton transitions from \emph{transferring} to \emph{owned} when the target node receives the message (and it is not a duplicate).
\ifsketch
The \empheval{phase sketch} we consider states, in the form of partial characterizations, that
in \emph{owned} the key is owned by at most one node and there are no pending messages concerning this key; in phase \emph{transferring}, the key is not owned by any node, and pending non-duplicate transfer messages are unique.
\fi
\else
\emphexample{Sharded key-value store with retransmissions (KV-R)}: see \Cref{sec:running} and \Cref{ex:phase-structure}. This protocol has not been modeled in decidable logic before.
\fi
\emphexample{Cache coherence}. This example implements the classic MESI protocol for maintaining cache coherence in a shared-memory multiprocessor~\cite{hennessy-patterson-arch-6ed}, modeled in decidable logic for the first time.
Cores perform reads and writes to memory, and caches snoop on each other's requests using a shared bus and maintain the invariant that there is at most one writer of a particular cache line.
For simplicity, we consider only a single cache line, and yet the example is still challenging for $\text{PDR}^\forall$.
Standard explanations of this protocol in the literature already use automata to describe this invariant, and we directly exploit this structure in our phase automaton.
\empheval{Phase structure}:
There are 10 phases in total, grouped into three parts corresponding to the modified, exclusive, and shared states in the classical description.
Within each group, there are additional phases for when a request is being processed by the bus.
For example, in the shared group, there are phases for handling reads by cores without a copy of the cache line, writes by such cores, and also writes by cores that \emph{do} have a copy.
Overall, the phase structure is directly derived from textbook descriptions, taking into account that use of the shared bus is not atomic.
\OMIT{
\item \emphexample{Single-decree Paxos}, based on the model from~\cite{DBLP:journals/pacmpl/PadonLSS17}; see~\cite{DBLP:journals/pacmpl/PadonLSS17} for an exposition to the protocol in EPR. The ``choosable'' relations is provided as a derived relation to reduce the invariant to a universal form---without this, Phase-$\text{PDR}^\forall$ returns an abstract counterexample trace. \TODO{also true for $\text{PDR}^\forall$}
The \empeval{phase structure} takes the view of two values $v_1,v_2$ as they compete to become proposed in the highest possible round. The automaton transitions between phases in which the highest round (of rounds proposing either $v_1$ or $v_2$) has proposed $v_1$, and a symmetric one in which the highest round has proposed $v_2$.
The \empheval{phase structure} we consider states exactly these facts about the proposal of the highest round.
Phase-$\text{PDR}^\forall$ automatically infers additional required characterizations, stating that when the proposal from the highest round belongs to $v_1$ then (1) $v_2$ is not choosable in lower rounds, and that (2) $v_1$ itself it not choosable in rounds even lower than the highest round proposing $v_2$.
We note that the automaton's structure conveys a slightly different intuition Paxos's safety proof than the standard standpoint of inductive invariant based-proof in previous works.
\item
}
\para{Results and discussion}
Measurements for these examples appear in \Cref{tab:examples-evaluation-table}.
Standard $\text{PDR}^\forall$ fails to converge in less than an hour on 13 out of 16 seeds for KV-R and all 16 seeds for the cache\iflong coherence protocol\fi.
In contrast, Phase-$\text{PDR}^\forall$ converges to a proof in a few minutes in all cases.
\ifsketch
(Discussion of the improvement gained from partial characterizations in these examples is deferred to the next section.)
\fi
These results demonstrate that phase structures can effectively guide the search and obtain an invariant quickly where standard inductive invariant inference does not.
\subsubsection{(2) Enhancing Performance Through Phases}
\label{sec:eval-performance}
In this section we consider the use of phase structures to improve the speed of convergence to a proof.
\para{Examples}
\iflockserv
\emphexample{Distributed lock service}, which is described in detail in \Cref{sec:running} and \Cref{ex:phase-structure}.
\ifsketch
The phase sketch includes in each phase its main invariant: the first invariant in each phase in \Cref{fig:lockserv-invariants}.
\fi
We also consider a simpler variant with only a single lock,
which reduces the arity of all relations and removes the need for a (nontrivial) automaton view. Its \empheval{phase structure} is the same, only for a single lock.
\else
\emphexample{Distributed lock service}, adapted from~\cite{verdi-pldi},
allows clients to acquire and release locks by sending requests to
a central server, which guarantees that only one client holds each lock at a
time.
\empheval{Phase structure}: for each lock, the phases follow the 4 steps by which a client completes a cycle of acquire and release.
We also consider a simpler variant with only a single lock,
reducing the arity of all relations and removing the need for an automaton view. Its \empheval{phase structure} is the same, only for a single lock.
\fi
\emphexample{Simple quorum-based consensus}, based on the example in~\cite{DBLP:conf/pldi/TaubeLMPSSWW18}.
In this protocol, nodes propose themselves and then receive votes from other nodes.
When a quorum of votes for a node is obtained, it becomes the leader and decides on a value.
Safety requires that decided values are unique.
The \empheval{phase structure} distinguishes between the phases before any node is elected leader, once a node is elected, and when values are decided.
Note that the automaton structure is unquantified.\ifsketch
The \empheval{phase sketch} states that first there are no leaders and no decisions, then unique leaders and no decisions, and then unique leaders and unique decisions. The rest of the inductive phase invariant relates quorums, leaders and decisions, and votes and vote messages.
\fi
\emphexample{Leader election in a ring}~\cite{chang1979improved,pldi/PadonMPSS16},
in which nodes are organized in a directional ring topology with unique IDs,
and the safety property is that an elected leader is a node with the highest ID.
\empheval{Phase structure}: for a view of two nodes $n_1,n_2$,
in the first phase, messages with the ID of $n_1$ are yet to advance in the ring past $n_2$, while
in the second phase, a message advertising $n_1$ has advanced past $n_2$.
\ifsketch
We did not consider a phase sketch in this example due to the small number of clauses in the inductive phase invariant. \yotam{does it sound bad?}
Phase-$\text{PDR}^\forall$ automatically infers the characterizations that
in the first phase $n_1$ cannot be a leader and all messages with the ID of $n_1$ are between $n_1$ and $n_2$ in the ring,
and that in the second phase necessarily the ID of $n_1$ is greater than that of $n_2$ (because otherwise $n_2$ would have dropped the message).
This latter characterization includes another quantifier on nodes, which constrains interference (see \Cref{sec:related}).
\else
The inferred characterizations include another quantifier on nodes, constraining interference (see \Cref{sec:related}).
\fi
\emphexample{Sharded key-value store (KV)} is a simplified version of KV-R above, without message drops and the retransmission mechanism.
The \empheval{phase structure} is exactly as in \mbox{KV-R}, omitting transitions related to sequence numbers and acknowledgment.
This protocol has not been modeled in decidable logic before.
\para{Results and discussion}
We compare the performance of standard $\text{PDR}^\forall$ and Phase-$\text{PDR}^\forall$ on the above examples,
with results shown in~\Cref{tab:examples-evaluation-table}.
For each example, we ran the two algorithms on 16 different Z3 random seeds.
Measurements were performed on a 3.4GHz AMD Ryzen Threadripper 1950X with 16 physical cores, running Linux 4.15.0, using Z3 version 4.7.1{}.
By disabling hyperthreading and frequency scaling and pinning tasks to dedicated cores,
variability across runs of a single
seed was negligible.
In all but one example, Phase-$\text{PDR}^\forall$ improves performance, sometimes drastically; for example, performance for leader election in a ring is improved by a factor of 60.
Phase-$\text{PDR}^\forall$ also improves the \emph{robustness} of inference~\cite{DBLP:conf/cav/0001LMN14} on this example, as the standard deviation falls from 39 in $\text{PDR}^\forall$ to 0.04 in Phase-$\text{PDR}^\forall$.
The only example in which a phase structure actually diminishes inference effectiveness is simple consensus.
We attribute this to an automaton structure that does not capture the essence of the correctness argument very well, overlooking votes and quorums.
This demonstrates that a phase structure might guide the search towards counterproductive directions if the user guidance is ``misleading''.
This suggests that better resiliency of interactive inference framework could be achieved by combining phase-based inference with standard inductive invariant-based reasoning. We are not aware of a single ``good'' automaton for this example.
The correctness argument of this example is better captured by the conjunction of two automata (one for votes and one for accumulating a quorum) with different views, but the problem of inferring phase invariants for mutually-dependent automata is a subject for future work.
\ifsketch
We now turn to the effect of adding partial characterizations in a phase sketch on inference.
In almost all examples, additional phase characterizations provided by the user improve or perform as well as when just a phase structure is provided.
The phase sketch improve over a phase structure especially in the more challenging examples: KV-R and cache coherence, in which the additional characterizations reduce mean time from \textasciitilde 6 minutes to 3.5 and from 1.5 minutes to 44 seconds (respectively).
Broadly, the speedup obtained from adding additional phase characterizations to the sketch is weaker than the benefit of a phase structure over standard inductive invariant inference.
This suggests that in most cases the characterizations the user provides are in fact easy for the automatic algorithm to infer by itself from the phase structure, and thus do not offer much additional benefit. Speedup from partial characterizations (as apparent in KV-R and cache coherence) can occur when the user provides characterizations which are not readily apparent to the algorithm.
\fi
\OMIT{
\input{figs/microbenchmark}
\para{Phase structures and scalability} In the following benchmark we study the effect of different phase structures and scalability.
\emphexample{$n$-phase commit} is an artificial protocol, based on 2-phase commit, achieving agreement between all nodes in $n$ rounds.
In each round, the coordinator sends a message to all participants and waits for unanimous acknowledgment
before proceeding to the next round.
The safety property is that if the final round has completed, then all nodes acknowledged in
the first round.
This property requires to reason about the entire protocol to associate all different rounds, so the complexity of the proof is in some sense proportional to $n$.
We compared two \empheval{phase structures}: (long) an automaton structure with one phase per round,
and (short) one with two phases, the first corresponding to round one and the second to the rest.
Both structures achieve a performance improvement as seen in \Cref{fig:microbenchmark}, with a preference to the long automaton.
The additional structure of phase invariants uncovers a disjunctive invariant that is not easily approximated by a conjunction of lemmas by $\text{PDR}^\forall$.
Somewhat surprisingly, the long automaton, which we find more intuitive, improves inference even further in spite of the overheard incurred by an increasing number of phases that must be handled.
}
\OMIT{
Overall, the results suggest that harnessing the phase structure can improve the performance of invariant inference.
\ifsketch
, and that additional phase characterizations in the phase sketch potentially offer some additional improvement in certain cases.
\fi
}
\OMIT{
almost all examples improved by phase structure
comparing sketch versus structure: sometimes the sketch doesn't provide what algorithm needs,
and adds some overhead (because it's an extra safety obligation)
when we're worse, it's because the algorithm can already infer what the user gave.
in other words, the difference between structure and sketch is explained by saying that
the user provided something already inferrable from the phase structure
(but if we're better than updr, then not by updr).
this also suggests that it would be fruitful to spend additional engineering effort
to reduce the overhead of additional safety checks.
about toy consensus: not many disabled transitions,
the phase characterizations aren't that different among phases
calls for some combination of phase-based and inductive invariant based search
}
\subsubsection{(3) Anatomy of the Benefit of Phases}
\label{sec:eval-dissection}
We now demonstrate that each of the beneficial aspects of phases discussed in \Cref{sec:inference-benefit} is important for the benefits reported above.
\para{Phase decomposition}
Is there a benefit from a phase structure even without disabled transitions? An example to a positive answer to this question is leader election in a ring, which demonstrates a huge performance benefit even without disabled transitions.
\para{Disabled transitions}
Is there a substantial gain from exploiting disabled transitions?
We compare Phase-$\text{PDR}^\forall$ on the structure with disabled transitions and a structure obtained by (artificially) adding self loops labeled with the originally impossible transitions, on the example of lock service with multiple locks (\Cref{sec:eval-performance}), seeing that it demonstrates a performance benefit using Phase-$\text{PDR}^\forall$ and showcases several disabled transitions in each phase.
The result is that without disabled transitions, the mean running time of Phase-$\text{PDR}^\forall$ on this example jumps from 2.73 seconds to 6.24 seconds.
This demonstrates the utility of the additional safety properties encompassed in disabled transitions.
\para{Phase-awareness}
Is it important to treat phases explicitly in the inference algorithm, as we do in Phase-$\text{PDR}^\forall$ (\Cref{sec:updr-phase})?
We compare our result on convergence of KV-R with an alternative in which standard $\text{PDR}^\forall$is applied to an encoding of the phase decomposition and disabled transition by \emph{ghost state}: each phase is modeled by a relation over possible view assignments, and the model is augmented with update code mimicking phase changes; the additional safety properties derived from disabled transitions are provided; and the view and the appropriate modification of the safety property are introduced.
This translation expresses all information present in the phase structure, but does not explicitly guide the inference algorithm to use this information.
The result is that with this ghost-based modeling the phase-oblivious $\text{PDR}^\forall$ does not converge in 1 hour on KV-R in any of the 16 runs, whereas it converges when Phase-$\text{PDR}^\forall$ explicitly directs the search using the phase structure.
\section{Related Work} \label{sec:related}
\iflong
\else
\vspace{-0.4cm}
\fi
\para{Phases in distributed protocols}
Distributed protocols are frequently described in informal descriptions as transitioning between different phases.
Recently, PSync~\cite{DBLP:conf/popl/DragoiHZ16} used the Heard-Of model~\cite{DBLP:journals/dc/Charron-BostS09}, which describes protocols as operating in rounds, as a basis for the implementation and verification of fault-tolerant distributed protocols.
Typestates~\cite[e.g.][]{DBLP:journals/tse/StromY86,DBLP:journals/scp/FieldGRY05} also bear some similarity to the temporal aspect of phases.
State machine refinement~\cite{DBLP:journals/tcs/AbadiL91,Garland:2000:UIA:336431.336455} is used extensively in the design and verification of distributed systems (see e.g.~\cite{Newcombe:2015:AWS:2749359.2699417,IronFleet}).
The automaton structure of a phase invariant is also a form of state machine; our focus is on inference of characterizations establishing this.
\para{Interaction in verification}
Interactive proof assistants such as Coq~\cite{DBLP:series/txtcs/BertotC04} and Isabelle/HOL~\cite{DBLP:books/sp/NipkowPW02} interact with users to aid them as they attempt to prove candidate inductive invariants. This differs from interaction through phase structures and counterexample traces.
Ivy uses interaction for invariant inference by interactive generalization from counterexamples~\cite{pldi/PadonMPSS16}. This approach is less automatic as it requires interaction for every clause of the inductive invariant.
In terminology from synthesis~\cite{DBLP:conf/synasc/Gulwani12}, the use of counterexamples is \emph{synthesizer-driven} interaction with the tool, while interaction via phase structures is mainly \emph{user-driven}. Abstract counterexample traces returned by the tool augment this kind of interaction.
As~\cite{DBLP:journals/acta/JhaS17} has shown, interactive invariant inference, when considered as a synthesis problem (see also~\cite{DBLP:conf/cav/0001LMN14,DBLP:journals/fmsd/SharmaA16}) is related to inductive learning.
\para{Template-based invariant inference} Many works employ syntactical templates for invariants, used to constrain the search~\cite[e.g.][]{DBLP:conf/cav/ColonSS03,DBLP:conf/sas/SankaranarayananSM04,DBLP:conf/pldi/SrivastavaG09,DBLP:journals/sttt/SrivastavaGF13,DBLP:series/natosec/AlurBDF0JKMMRSSSSTU15}. The different phases in a phase structure induce a disjunctive form, but crucially each disjunct also has a distinct semantic meaning, which inference overapproximates, as explained in \Cref{sec:inference-benefit}.
\para{Automata in safety verification}
Safety verification through an automaton-like refinement of the program's control has been studied in a number of works. We focus on related techniques for proof automation.
The \emph{Automizer} approach to the verification of sequential programs~\cite{DBLP:conf/sas/HeizmannHP09,DBLP:conf/cav/HeizmannHP13} is founded on the notion of a \emph{Floyd-Hoare automaton}, which is an unquantified inductive phase automaton; an extension to parallel programs~\cite{DBLP:conf/popl/FarzanKP15} uses thread identifiers closed under the symmetry rule,
which are related to view quantifiers.
Their focus is on the automatic, incremental construction of such automata as a union of simpler automata, where each automaton is obtained from generalizing the proof/infeasibility of a single trace.
In our approach the structure of the automaton is provided by the user as a means of conveying their intuition of the proof, while the annotations are computed automatically.
A notable difference is that in Automizer, the generation of characterizations in an automaton constructed from a single trace does not utilize the phase structure (beyond that of the trace), whereas in our approach the phase structure is central in generalization from states to characterizations.
\iflong
\fi
In \emph{trace partitioning}~\cite{DBLP:journals/toplas/RivalM07,DBLP:conf/esop/MauborgneR05}, abstract domains based on transition systems partitioning the program's control are introduced. The observation is that recording historical information forms a basis for case-splitting, as an alternative to fully-disjunctive abstractions. This differs from our motivation of distinguishing between different protocol phases.
The phase structure of the domain is determined by the analyser, and can also be dynamic. In our work the phase structure is provided by the user as guidance. We use a variant of $\text{PDR}^\forall$, rather than abstract interpretation~\cite{POPL:CC79}, to compute universally quantified phase characterizations.
\iflong
\fi
Techniques such as \emph{predicate abstraction}~\cite{DBLP:conf/cav/GrafS97,DBLP:conf/popl/FlanaganQ02} and \emph{existential abstraction}~\cite{DBLP:books/daglib/0007403}, as well as the safety part of \emph{predicate diagrams}~\cite{DBLP:conf/ifm/CansellMM00}, use finite languages for the set of possible characterizations and lack the notion of views, both essential for handling unbounded numbers of processes and resources.
\iflong
\fi
Finally, \emph{phase splitter predicates}~\cite{DBLP:conf/cav/SharmaDDA11} share our motivation of simplifying invariant inference by exposing the different phases the loop undergoes. Splitter predicates correspond to inductive phase characterizations~\cite[Theorem~1]{DBLP:conf/cav/SharmaDDA11}, and are automatically constructed according to program conditionals. In our approach, decomposition is performed by the user using potentially non-inductive conditions, and the inductive phase characterizations are computed by invariant inference. Successive loop splitting results in a sequence of phases, whereas our approach utilizes arbitrary automaton structures.
Borralleras et al.~\cite{DBLP:conf/tacas/BorrallerasBLOR17} also refine the control-flow graph throughout the analysis by splitting on conditions, which here are discovered as preconditions for termination (the motivation is to expose termination proof goals to be established): in a sense, the phase structure is grown from candidate characterizations implying termination. This differs from our approach in which the phase structure is used to guide the inference of characterizations.
\para{Quantified invariant inference}
We focus here on the works on quantifiers in automatic verification most closely related to our work.
In \emph{predicate abstraction}, quantifiers can be used internally as part of the definitions of predicates, and also externally through predicates with free variables~\cite{DBLP:conf/popl/FlanaganQ02,DBLP:journals/tocl/LahiriB07}.
Our work uses quantifiers both internally in phases characterizations and externally in view quantifiers.
The view is also related to the bounded number of quantifiers used in \emph{view abstraction}~\cite{DBLP:journals/sttt/AbdullaHH16,DBLP:conf/vmcai/AbdullaHH13}.
In this work we observe that it is useful to consider views of entities beyond processes or threads,
\iflockserv
such as a single lock in the running example.
\else
such as a single key in the store.
\fi
\iflong
\fi
Quantifiers are often used to their full extent in verification conditions, namely checking implication between two quantified formulas, but they are sometimes employed in weaker checks as part of thread-modular proofs~\cite{DBLP:journals/toplas/AbadiL95,Jones:1983:TST:69575.69577}.
This amounts to searching for invariants provable using specific instantiations of the quantifiers in the verification conditions~\cite{DBLP:conf/sigsoft/GurfinkelSM16,DBLP:conf/popl/HoenickeMP17}.
In our verification conditions, the view quantifiers are localized, in effect performing a single instantiation.
This is essential for exploiting the disjunctive structure under the quantifiers, allowing inference to consider a single automaton edge in each step, and reflecting an intuition of correctness.
When necessary to constrain interference, quantifiers in phase characterizations can be used to establish necessary facts about interfering views.
Finally, there exist algorithms other than $\text{PDR}^\forall$ for solving CHC by predicates with universal invariants~\cite[e.g.][]{DBLP:conf/atva/GurfinkelSV18,DBLP:conf/cav/DrewsA16}.
\OMIT{
Also Coq?
Houdini
ACL/2
Rahul Sharma
Dynamo
}
\section{Conclusion}
Invariant inference techniques aiming to verify intricate distributed protocols must adjust to the diverse correctness arguments on which protocols are based. In this paper we have proposed to use phase structures as means of conveying users' intuition of the proof, to be used by an automatic inference tool as a basis for a full formal proof. We found that inference guided by a phase structure can infer proofs for distributed protocols that are beyond reach for state of the art inductive invariant inference methods, and can also improve the speed of convergence.
The phase decomposition induced by the automaton, the use of disabled transitions, and the explicit treatment of phases in inference, all combine to direct the search for the invariant.
We are encouraged
by our experience of specifying phase structures for different protocols.
It would be interesting to integrate the interaction via phase structures with other verification methods and proof logics, as well as interaction schemes based on different, complementary, concepts.
Another important direction for future work is inference beyond universal invariants, required for example for the proof of Paxos~\cite{DBLP:journals/pacmpl/PadonLSS17}.
\subsubsection*{Acknowledgements}
{\footnotesize
We thank
Kalev Alpernas, Javier Esparza, Neil Immerman, Oded Padon, Andreas Podelski, Tom Reps,
and
the anonymous referees
for insightful comments which improved this paper.
This publication is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No [759102-SVIS]).
The research was partially supported by
Len Blavatnik and the Blavatnik Family foundation,
the Blavatnik Interdisciplinary Cyber Research Center, Tel Aviv University,
the Israel Science Foundation (ISF) under grant No. 1810/18,
the United States-Israel Binational Science Foundation (BSF) grant No. 2016260,
and the National Science Foundation under Grant No. 1749570.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
}
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 290 |
We get a lot of requests for love magic here at the apothecary; people need help finding a new lover or reconciling with one from the past. Here is an old spell from the turn of the twentieth century using dragon's blood to fetch the love partner from a distance when you are separated.
But 's heart I wish to turn.
Till he returns to me to me to speak.
This spell, published in Folk-lore : A Quarterly Review, Volume 20 in 1909, continues an old tradition of love spells that border on cursing. Christopher Farone's "Ancient Greek Love Magic" contains many fascinating examples of this type of spell from the ancient world. The directive to neither rest nor sleep means the target of the spell will not be able to get you out of their mind, going over the argument you may have had, longing to clear the air and make things right.
The spell is performed on a Friday night, the day of Venus, mistress of the matters of the heart. I suspect the mention of the first friday of the month does not refer to the calendar month, but instead the lunar month, in which case the moon is waxing. As the moon swells and grows, so will your lover's thoughts of you, until the pull of your spell, like the moon on the tides of the ocean, cannot be resisted.
We sell two different kinds of dragon's blood at the apothecary. One is the glassy dark red resin from the fruit of a palm (mostly growing in Indonesia) which smells sharp, piney, and slightly acrid, much like frankincense. The other is the dried sap of a tropical tree and looks like a dull, dark red myrrh. When burned, its scent is woody and spicy and kind of metallic. We have no idea which resin you would have gotten from a chemist in Britain in 1909, but for the purposes of this spell, we'd recommend using the former Indonesian Dragon's Blood.
I would like to try this. My boyfriend left me. I want him back in my life, but i do not know when he will come back to my life. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,104 |
Table of Contents
Title Page
Copyright Page
Dedication
Acknowledgements
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-one
Chapter Twenty-two
Chapter Twenty-three
Chapter Twenty-four
Chapter Twenty-five
Chapter Twenty-six
Chapter Twenty-seven
Chapter Twenty-eight
Chapter Twenty-nine
Chapter Thirty
Chapter Thirty-one
**Praise for the novels of Moira J. Moore**
**Heroes Adrift**
"A lovely installment in a consistently entertaining series."
— _Locus_
"Good entertainment in the romantic fantasy subgenre . . . _Heroes Adrift_ is well paced, never bogging down . . . a pleasant way to spend the time."— _Grasping for the Wind_
**Resenting the Hero**
"This incredible romantic fantasy will appeal equally to fans of both genres. The sexual tension between the two protagonists is so strong that readers will feel sparks fly off the pages."— _The Best Reviews_
"An enchanting fantasy that introduces two interesting and complex protagonists and a fascinating world . . . The tale has everything—magic, mayhem, a hint of romance and a thread of wry humor."— _Romance Reviews Today_
"[A] fast-paced plot . . . The various threads come together in a satisfying way."— _SFRevu_
"A wry twist on classic fantasy . . . _Resenting the Hero_ is a funny book with occasional dramatic spans . . . a good choice for a rainy afternoon."— _Infinity Plus_
"An entertaining read . . . sure to be a hit with romance as well as fantasy readers."— _Fresh Fiction_
_Ace titles by Moira J. Moore_
RESENTING THE HERO
THE HERO STRIKES BACK
HEROES ADRIFT
HEROES AT RISK
**THE BERKLEY PUBLISHING GROUP**
**Published by the Penguin Group**
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Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental. The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.
HEROES AT RISK
An Ace Book / published by arrangement with the author
PRINTING HISTORY
Ace mass-market edition / September 2009
Copyright © 2009 by Moira J. Moore.
All rights reserved.
No part of this book may be reproduced, scanned, or distributed in any printed or electronic form
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ACE and the "A" design are trademarks of Penguin Group (USA) Inc.
<http://us.penguingroup.com>
_For Seán and Cate, who surprised me_
Acknowledgments
Thank you to my mother, who is my most dedicated PR agent, to Jack Byrne, my agent, and to Anne Sowards, my editor.
Chapter One
The residence of the Source and Shield Service was unimpressive in appearance, a plain, large square of a building. Simple in style, drab in material, anyone would look right past it. It was boring and forgettable.
But to me, it was beautiful. I had been away from it for over a year. I couldn't wait to get settled back within it. And it seemed to me that the carriage we had rented was taking its own sweet time to draw up before the building and come to a complete stop. Once it had, Taro and I leapt out with our bags, and I thought about kissing the ground.
"All we need are fires!" Taro announced as he kicked open the front door to the residence.
I looked at him and couldn't help grinning. He was healthy and relaxed, his black eyes practically glowing, his black hair mussed by a hard day of travel. It was good to see him finally back to himself. While he had been steadily improving in body and disposition since we'd returned to the mainland a few months earlier, I realized just then that he still hadn't regained his usual buoyancy. Not until we'd gotten home.
"Aye, we certainly do." I, of course, had no idea what his words meant. Shintaro Karish was, in many ways, the most stereotypical Source I'd ever met, which meant that at times he was compelled to make incomprehensible statements. I was a Shield, the fairly average Dunleavy Mallorough, which meant I was very sensible, and knew when not to bother twisting my brain trying to figure out the meaning of my Source's words.
Besides, all I needed to know right then was that his words were expressing his joy in being home. A joy I shared. The Triple S residence in High Scape and the Shield Academy in Shidonee's Gap were the only homes I'd ever known. Well, the only ones I remembered. I had spent seventeen years in the Academy, and then had been sent straight to High Scape. In the three years that my official address had been the Triple S residence of High Scape, I'd spent more time away from it than in it. Still, it felt like a proper home, a place I belonged, and a place I had the right to bar others from entering. That was all a home was, really.
"It's probably too much to expect that any of the others are in," Taro commented, and he carried both of our bags into the foyer.
He wouldn't let me carry my own bag, unless I wanted to get into an argument over it, which I rarely did. Sometimes I just grabbed my bag and we got into a tugging match, which made me feel ridiculous, so I would let go. Taro, of course, didn't look ridiculous. Just patient and lordly. He was good at that, damn him.
Seven Pairs were needed to keep High Scape stable. It was one of the hottest sites in the world, constantly barraged with earthquakes and tornadoes and other natural events that would normally tear down the buildings, bury the crops, and decimate the population. Sources were born with the ability to channel the power of these natural disasters and keep the land stable. Shields were born with the ability to make sure Sources weren't killed by the forces that swirled through and around them.
I was a Shield; Taro was my Source. All we were supposed to do was keep High Scape calm. But when Taro had been abducted by Stevan Creol, I'd found myself playing amateur Runner trying to find him, with an incompetence that made me cringe every time I thought of it. Stevan Creol had been a Source with a lot of anger he didn't keep nearly repressed enough. He had been taught how to prevent disasters while in the Source Academy, just like every other Source, but once he was released from the Academy he discovered how to create natural disasters, something no one had ever thought of doing, because why would they? He had been using that perversion of a skill to try to destroy High Scape as a means of expressing his frustration with his lot in life. During his captivity, Taro had picked up the same skill, because Creol had liked to show off and Taro has eyes. It turned out to be a handy talent, one Taro had already used more times than I liked.
I was the one who had figured out how to kill Creol while he was attacking High Scape. I did it by manipulating his Shields while he channeled. Words couldn't describe how very uncomfortable I was with that. I didn't know whether the fact that no one other than Taro knew about it made me feel better or worse. I had no desire to admit to my actions and face the consequences. We wouldn't be executed or placed in prison—Pairs were considered too valuable to be unrecoverably destroyed—but we could be sent to a cold site for the rest of our lives, where no one needed or respected our talents. It would be a waste, and it would be hell.
Yet shouldn't I be punished for killing someone? How could something like that just pass away, like it had never happened? Was there anything I could do to make up for that?
I thought about it a lot. Good ideas never came to me.
After that mess, we had very little time in High Scape before Empress Constia, unhappy with the quality of her son and heir, had sent Taro and me to the remote southern island of Flatwell to search for the descendants of her exiled sister. It was Taro who was wanted for the job, for my illustrious Source had first managed to catch Her Majesty's eye and then earn her trust. I was just dragged along to make travel more comfortable for Taro, whatever that was supposed to mean.
I was fully justified in resenting the Empress for this folly. We were a Pair, and we had responsibilities. To pull us off our roster for personal use was an abuse of her position. I didn't care why she had felt Taro was the only person she could trust with such a delicate task. She had unlimited wealth and ultimate power; she could have sent anyone. She should have sent someone better suited to the task. She could have paid them enough that they would have done the job and kept her secrets.
What made the whole situation even worse was that the one descendant we found, Aryne, was not to the royal taste. Not properly educated, Her Majesty declared, and lacking that certain quality that every ruler needed. Apparently, being clever and resilient didn't mean much.
Fortunately, we hadn't told Aryne she had the potential to be the future ruler of our world. Even more fortunately, she was a Source. So she'd had a place to go when her great-aunt hadn't wanted her. The Source Academy.
That was one good thing that had come out of that ridiculous trip. Aryne had gotten out of a situation of ignorance and abuse, and was on her way to a much better life. Provided she hadn't run away from the Academy yet.
Now we were finally home. Perhaps, after disappointing just about everyone who had ever expected more from us, we would be left alone to do what we were supposed to do. Be a Pair. Channel and Shield.
Please.
"Do you think our rooms are still ours?" Taro asked.
"Why wouldn't they be?"
"They might have brought in another Pair to replace us."
I stared at him, shocked. I had never thought of that. "They wouldn't do that."
"We've been gone a long time, Lee," he said. "And we've been gone a lot. A volatile place like High Scape, maybe they need a Pair that's not unreliable."
We weren't unreliable. Things outside of our control kept pulling us off the roster. "They'd better not."
That didn't come out quite the way I'd meant.
Taro snickered.
Ben Veritas stepped into the corridor from the kitchen. "Source Karish, Shield Mallorough, welcome home." He reached out to take the bags from Taro, who eased them out of reach. Ben was a regular, neither a Shield nor a Source, of late middle age, retained by the Triple S to clean up after us and make our meals. I found him disturbing, though I wasn't sure why. There had been people at the Academy who prepared our meals and cleaned up after us, usually Shields who had been through the Matching ceremonies for decades and hadn't managed to bond to a Source. They hadn't disturbed me at all. There was just something in Ben's manner, like he was watching us more closely than he should. "You should have sent word ahead of your arrival. I would have had a hot meal ready for you."
"Optimism is despised by nature," said Taro. "Besides, we're going out to eat."
"We are?" I asked. It was the first I'd heard of it.
"We have to celebrate our return. Delicacies excellently prepared and accompanied by chilled goblets of the best wine. Brought to you on trays carried by handsome young men and women. Sometimes there's music."
I was starving. And one of the good things about eating in a tavern was the variety of possible dishes. I wasn't sure I wanted to just sit at home, tired though I was. I wanted to really soak in High Scape again, walk down the streets, hear the familiar accents and turns of phrase, really feel that I was back. I'd missed it.
"I thought I heard you two." Source Kyna Riley came clomping down the stairs. "Shintaro, Dunleavy, welcome home."
"Kyna!" Taro crowed, dropping our bags and throwing his arms open and hugging Riley whether she liked it or not. Fortunately, Riley remembered Taro's exuberance, and she accepted the embrace with only a roll of her eyes. "Tell me all the exciting things that have happened while we were gone."
Ben silently picked up our bags and carried them upstairs.
"How can anything exciting happen in your absence, Shintaro?" she asked, and there was the slightest bitter edge in her tone that informed me that all was not calm on her sea.
"What's going on?" I asked.
"Giles and I have been transferred to Ice Ridge."
"Ice Ridge," I echoed. That had been a cold site, the last time I'd looked.
"A volcano erupted a few months ago and has been giving off little spurts ever since."
"And they're just sending a Pair now?" I used to have high respect for the Triple S council. I now found myself becoming distressingly disenchanted. It wasn't my fault. They kept indulging in behavior that smacked of ludicrous incompetence.
Riley shrugged. "Ice Ridge isn't important to anyone."
Which may have been the cause of her bitterness. A transfer from High Scape to Ice Ridge was taking a big step down in prestige.
"They have the best oranges!" Taro protested.
They did, actually. "Was anyone hurt?" I asked.
"No lives lost, but a lot of structural damage, and they can't rebuild with the ongoing events."
I wondered why we hadn't heard of this during our travels from Erstwhile. I would have thought that was the kind of news that would get around.
"We're not the only ones being transferred," Riley added. "Vera and Lauren left for Blue Rock a few weeks ago."
This was disappointing. Triple S Pairs never stayed in one place for too long, but assignments usually lasted at least a few years. "Do you know who's coming to replace them?"
"No one," said Riley, "as far as I know."
"Why not?"
"High Scape's been a lot calmer for the last year. Not quite cold, but, I don't know, it might be on its way there. I haven't channeled in weeks. So we don't need as many Pairs."
High Scape was known for its frequent turbulence. The population practically prided themselves on it. Certainly, hot spots and cold sites could switch designations, but usually it took decades. What could have caused such an abrupt change? Creol couldn't have been responsible for all of the natural events in High Scape before his death. And I refused to believe that the half year the Reanists had spent sacrificing aristocrats to their gods had actually accomplished the stability they had claimed to seek. That was just ridiculous. "Do you know how many Pairs are going to be transferred?"
"Not yet."
And if they had to send Pairs away, why couldn't they send the annoying ones? Like Beatrice and Benedict? Or Wilberforce and Ladin?
I couldn't help feeling oddly deflated. I'd had an image of what to expect from home. And that had included all the Pairs. Even the ones I didn't like. I hated change.
Riley was looking at me with an expression of puzzlement. "Are you all right, Dunleavy?"
"Of course."
"You look different."
That was probably the remnants of the southern sun. While I never got as brown as Taro, I'd turned a kind of golden beige. It made my hair even more blazingly red than usual, too. Those effects had faded a great deal during our journey home, but there was still a hint of additional color here and there. However, Riley wasn't supposed to know where we'd gone, so I couldn't tell her any of that. "Oh," I said, because I was a witty person.
"There's something different about your voice, too," she added.
"I see," I said. My voice sounded the same to me.
"We don't go until next week," said Riley, clearly giving up that line of conversation as a nonstarter. "And the night before, those of us not on watch are going out to, I don't know, celebrate or say farewell. Will you two go?"
Taro gently took her hand and bowed over it, kissing the back of it. "I am devastated that we will be losing you. If it does not conflict with my duties, I will surely be there to mourn your departure."
I must have been smirking, because the look Riley sent me was slightly hostile. I straightened my mouth. "My apologies, Riley." I had been laughing at Taro and his melodrama, not Riley. "Of course we will be there. I am very sorry to lose you."
"Thank you." Riley nodded. "I'm pleased you were able to get back before we left. I was worried we'd miss each other." She headed off toward the kitchen.
Taro grinned at me then, a particularly wicked grin that I had seen many times before. He grabbed my hand, and the next thing I knew, I was running up the stairs behind him. I ran behind him to his door, through his sitting room, and into his bedroom.
His bedroom. I had always avoided it. I never wanted anyone to expect to find me in his bedroom. I had never had any interest in knowing what was in it.
Well, I had never _wanted_ to have any interest in it.
Now I was being dragged into it and I was smiling at the feeling of wickedness it engendered. I was going to have sex with the Stallion of the Triple S in the Stallion's bedroom.
He hated that nickname, so I didn't use it unless I wanted to tease him. But it wasn't one I'd come up with; it was too lurid for my taste. But he was known for sleeping with everything on two legs, though he claimed the numbers of rumor were highly exaggerated.
Taro closed the door and immediately curled his arms around my waist. "You have no idea how many times I imagined you in here," he said, just before he kissed me.
I had lost all common sense, or sense of self-preservation, when it came to Taro. I'd fallen in love with him. With my own Source. One of the most stupid things I could ever have done.
The regulars liked to think of a Pair feeling nothing but everlasting love for each other. That assumption seemed to be supported by the fact that when one partner died, the other followed into death. But in truth, for partners to fall in love with each other was a complication that could have disastrous repercussions. No one had figured out exactly how the bond really worked, but it did affect the emotions of the partners. Sometimes it put them into a form of harmony with each other, sometimes it brought out the worst in each other, and sometimes it created weird possessive and obsessive behavior. Falling in love made the latter all the more likely.
Which was why the instructors at the Shield Academy—and no doubt the Source Academy as well _—_ drilled the lesson into our thick skulls again and again and again. Don't sleep with your Source. Certainly, don't fall in love with your Source. If you become unstable, you won't be trusted to guard any of the prestigious sites, and you don't want to ruin your chances over something so fleeting and so stupid.
It had all sounded very logical to me. It had seemed even more logical after I was bonded to Lord Shintaro Karish, the Stallion of the Triple S, handsome and engaging and full of life. Aggravatingly handsome. Annoyingly engaging. Exhaustingly full of life. It would be dangerous for someone like me to get sexually involved with someone like him, because I was of a nature to take things more seriously, and he was of a nature to cast a wide net over the world in his search for pleasure. I wasn't sure I could remain mature and professional once he moved on to new partners.
I still didn't know if I could remain professional and mature once he was no longer interested in me. But it hadn't taken me that long to come to want him, and he had seemed to want me, too. And although I knew I would end up paying for it in the long run, the damage I had been doing to our relationship by sticking to my principles, principles Taro didn't agree with, could have lasted a lifetime. He had thought I was refusing him because I thought there was something wrong with him. If he had come to believe I thought him in any way inferior, we would have never recovered. It would have poisoned everything.
Besides, I wanted him. In a moment of clear self-indulgence, I had asked myself why I shouldn't have him. And while I was leery of the turmoil I would feel once he made his waning interest clear to me, I didn't at all regret the decision to enjoy him while I could. Maybe not the most sensible decision, but the right one.
So we indulged in a thorough homecoming, bathed together, and then headed out for supper in the most expensive tavern Taro knew of, to his surprised pleasure. "I'll have you thoroughly corrupted yet," he gloated.
"Corrupted?" I wouldn't go that far. Perhaps he'd ruined me for other lovers, but that wasn't his fault. He would be a hard act to follow.
"Before we were banished off to that damned island, you never would have consented to eating in a tavern like this."
Shields and Sources weren't required to pay for anything, as a sort of compensation for the fact that we weren't paid for our services. That didn't mean I felt we should always seek the most expensive of everything. We didn't need it, and I couldn't help feeling regulars would feel ill used if it became a habit. Still: "This is a special occasion."
"Aye, sure. And the reason you agreed to stay at the Imperial when we first got back to Erstwhile?"
I shrugged. I had been very tired of living rough for a year, and not in the mood for an argument.
"Face it: I'm corrupting you," he said. "Soon, you'll be just like me."
"Zaire forfend," I muttered.
I ordered the buffalo broth, a dish unique to High Scape that I hadn't realized I'd missed until I saw it written up on the menu board. And how could I have forgotten it? It teased the nose with heated spice and positively drowned the tongue in thick, savory flavor, the meat so perfectly tender it practically melted in the mouth.
Gods, I'd missed home. I almost couldn't believe I was actually here.
The food was wonderful, but the meal itself was not restful. Taro was well-known in High Scape, despite the frequency and length of his absences. Every few moments, someone would stop to exchange greetings and chat for a bit. Taro would try to draw me into the conversations, but neither I nor his devotee of the moment was interested. I watched him laugh and smile, and I wondered which of these people he had already slept with, and which would succeed me.
I would not act foolish when I was replaced. I would not. I had some pride. In fact, if I had my way, I would know it was coming before he did.
After enjoying a light, creamy dessert _—_ oh, I had missed dessert! _—_ we left the tavern and decided to stroll about High Scape to reacquaint ourselves with our post. It wasn't necessary in order for us to do our jobs well, but I was anxious to see the city.
High Scape was unlike any city or settlement I had ever seen, and so far I'd seen more settlements than I'd expected to by my age. It was divided by three rivers into six sections, called quads by the residents who apparently lacked a clear understanding of basic mathematics. Each quad had its own hospital and Runner Headquarters and government buildings. And each quad had its own character, largely influenced by the level of wealth enjoyed by its residents. The North Quad had the wealthiest residents of High Scape, and that was where we were headed. I rarely went there, having little reason to, and it was interesting to see the perfectly cobbled streets, the large, clean buildings with so much space between them, and the many large and leafy trees. As it got dark, lighters lit street lamps, adding a nice glow to everything.
A loud hissing sound, followed by a sharp pop that seemed to shake the night air, jolted out of nowhere. The shock of it pushed my heart right into my throat. "What the hell was that?" Taro demanded, and he leapt over the short iron fence beside him. He didn't immediately start running, as I had expected him to. "Ech!" he uttered. "Know what this is?"
I took a quick glance over the small field, seeing through the darkening air the scattering of small stone, iron and wooden posts. "It looks like an ash grove." Where the ashes of the dead were buried in copper urns.
Taro shed his hesitancy and started running, presumably in the direction from which he thought the noise had come. To me, it seemed as though it had come from everywhere. I climbed over the fence with the plan of running after him.
But I took only two steps before having to halt. I felt something odd. Some kind of resonance. Or vibration. A quick look around gave no evidence of anything that could create such an uncomfortable sensation. It seemed to skitter underneath my skin. It was horrible.
"Hey!" I heard Taro shout, and I looked up to see him sharply change direction.
"What the hell are you doing?" I called, and I got no answer. Why was my Source racing about in the dark? Chasing around after people was rarely a good idea under any circumstances. To do so in the dark was just begging for a disaster. One of us was sure to trip over one of the grove markers and break a leg.
"Hey!" Taro shouted again. "Stop running! That's an order!"
An order? Taro was no one to be giving orders. But maybe regulars didn't know that.
In the midst of running in the dark and trying not to fall and kill myself, I felt Taro's mental protections, the personal inner shields that guarded his mind when he wasn't channeling, drop down. That meant I had to raise mine around him. Which meant I had to stop running, because I wasn't sure I could run and Shield at the same time, and I wasn't insane enough to try. But really, what the hell was he doing? There were Pairs on duty. It was their job to settle any disasters, and they would resent us for stepping on their toes.
Almost immediately, though, I realized Taro wasn't channeling a natural event. When he channeled normally, he opened himself to the forces causing the disaster, and they rushed through him. In this manner, a Source drained a disaster of its power and diverted the forces in a harmless direction. But this time, he was pulling in selected forces, much fewer than usual, and he was directing them far more precisely than the raw power of a natural disaster normally allowed.
Taro was creating an event, not eliminating it. Damn it. I wished he would keep in mind that that was a skill he'd learned from a madman. It was unnatural to create disasters, and I didn't know why he was doing it right then. There was no good reason for it.
There was nothing I could do, though, except Shield him and keep him from killing himself. I'd yell at him after.
The ground trembled beneath my feet. Another sign that this was an artificial event. If it were real, I wouldn't be able to feel any of the physical manifestations. Taro would be stopping it before it got that far.
I hated earthquakes. I hated any kind of natural event, really. I was never supposed to have to feel any. That was part of what being bonded meant.
At least the tremors weren't very strong. No serious upheaval in the ground. None of the markers fell over. Nor did it last very long. The tremors faded away with nothing, as far as I could tell, being accomplished. I waited, ready to Shield again. Maybe he would try something else.
Moments burned by with nothing happening. It was eerily quiet, and I couldn't see anything. Zaire, this had been a stupid idea.
Then I heard, "Lee?"
And just like that, I was furious. "What the hell was that?" I demanded.
"There were people messing about," he answered as he appeared from the darkness and trotted up beside me.
"Doing what?"
"I don't know, but it probably had something to do with that racket we heard."
"So you decided to run after them? To what purpose? What would you do with them once you caught them?" Other than get beaten black and blue, if they were doing something illegal. Though I had no reason to think they were doing anything illegal. Just because I wouldn't imagine what anyone would be doing in an ash grove at night didn't mean their behavior had to be nefarious. Maybe they were visiting the remains of a family member. I could imagine plenty of scenarios out of novels and plays where some exiled member of a family would feel compelled to visit the ashes of a loved one under the cover of darkness.
It was possible.
"I couldn't do nothing," he insisted.
"Were they hurting another person?"
"I couldn't tell. I couldn't see."
He'd be able to tell. Any such victim would have been left behind, or would have run off in another direction. "I can see you risking our lives to help another person, but who cares about markers in an ash grove?"
"The families of those whose ashes are buried under the markers," he chided, trying to shame me.
He was unsuccessful. It was stupid to risk one's life for stone, no matter what the significance. "What was the point of the earthquake?"
"They weren't stopping."
"And did the earthquake stop them?" I knew damn well it hadn't because, clearly, they weren't there. "Wait, you didn't bury them, did you?" Because he could do that, too, and that disturbed me as much as his other unnatural skills.
"No, I did not," he snapped. "And no, it didn't stop them. Whichever Pair was on duty was trying to channel the earthquake. I didn't want to get into a battle for control over it. That could get messy."
It could get stupid, was what it could get. What was he thinking?
"Did you see what they were doing?" Taro asked.
"I haven't seen anything." I was only taking his word that there had been people in the grove at all. For all I knew, there'd been nothing going on and we'd been running around like idiots for no reason.
"Let's see if we can find what they were doing." Taro seemed enthusiastic about the possibility of finding something weird.
I really didn't care. I didn't like this place. It felt very strange, and I just wanted to leave.
But something small and white, almost glowing in the darkness of the night, caught my eye. Frowning, I stared at it, and as my focus cleared, other white somethings appeared. As I drew closer I was able to see that they were set in a sort of pattern. They were candles, unlit, and they made a circle. There was a line of a very pale powdery substance drawn from candle to candle, barely perceptible in the grass. The circle surrounded a marker. Dead center in the circle, a hole had been dug into the ground, and the marker had been uprooted.
I thought about stepping into the circle for about half an instant. Something about the whole scene made me very, very uncomfortable. Not uncomfortable in the "someone dug up someone's ashes" sort of way, though that was disgusting enough. I felt jittery, like something inside me was screaming to run while something else within me was keeping me rooted to the ground against my will. My heart was pounding, my breath was short, I thought I might have been sweating, and I could tell I was trembling.
It was probably time to give up on that ridiculous dream of mine, that I could ever be the cool, calm, unflappable Shield I had been trained to be.
I sighed.
It was our first damn day back.
Chapter Two
Risa Demaris was a Runner. It was her task to hunt and catch criminals. It was a task demanding good health and an iron spine, but no family connections or education of distinction. From what I had observed _—_ and Risa's bitter comments _—_ it didn't pay very well, and I now had the experience to understand just what that meant.
Risa was a glorious woman. She was tall and beautifully shaped, strong and lean. Her skin was a gorgeous warm brown and her eyes were stunning. Really, if I was going to envy any woman for her appearance, it would be Risa.
She was wearing the uniform of a Runner right then, the tall black boots, the black trousers, the black tunic, and a black cape. She looked stern and imposing. When she was not on duty, she wore flowing garments of bright orange and yellow and half a dozen pairs of earrings. In either scenario, she stood out in a crowd, and she liked it.
Risa was my friend, one of the few I'd been able to make in High Scape. She had been put to the task of finding Taro when he'd been abducted by Stevan Creol. She hadn't been successful with it. I had, but only because I'd had a follower of Creol leading me every step of the way.
She wasn't visiting us for a social call. The evening before, Taro and I had reported the disturbance in the ash grove to the first Runner we could find, but Risa had come to do a follow-up interview. She said people had been digging up ashes all over the city, which was apparently what Taro and I had stumbled into the night before, and her superiors thought we would be more comfortable being questioned by her, as we knew her.
I didn't think I could possibly be comfortable with any conversation concerning digging up human ashes, regardless of the participants. Why would anyone want to mess around with that sort of thing?
"To start off," Risa said, leaning back into the settee in my suite. "Do either of you recall any new details about last night? Something you've remembered since you spoke with Runner Elliot?"
"I have nothing to remember," I told her. "I didn't see anything."
"Aside from the defiled marker."
"Aye."
"But, Karish, you saw people."
"Yes, but not faces or hair color or anything like that. Except . . ." He trailed off, his eyes narrowing.
Risa leapt on him. Verbally, of course. "You're remembering something."
"Nothing useful. There was just something familiar about the way one of them moved."
"But you have no idea why he seemed familiar."
Taro shook his head and shrugged.
"Well, keep thinking on it." She looked at me. "What about the grass immediately around the marker? Did you notice anything unusual?"
"I didn't get close enough to see."
"Why not?"
"I wasn't going to step inside the circle."
"Why in the world not?"
"It felt strange." I didn't like admitting that. It was fantasti cal and childish. But it was what had happened.
"It felt strange?" Risa asked with a snicker.
"Yes," I snapped.
She narrowed her eyes at me. "You're different," she accused me. "So are you," she said to Taro. "Why are you so quiet?"
"Silence enhances my beauty," he announced solemnly.
Risa clearly didn't know how to respond to that, so she didn't. She was smart that way. "Did you notice anything unusual about the grass around the marker?" she asked him.
"I didn't step into the circle, either."
"Why not?"
"Lee said it felt strange."
Risa rolled her eyes. "So neither of you touched anything?"
"Correct."
"And you didn't see or hear or smell anything other than what you've already mentioned?"
"No."
"You're sure."
"Yes."
"Then I guess all I can ask is that you let us know if you think of anything else."
Hopefully, we would never have any reason to ever think of it again. It was weird.
"You said this has been going on a lot?" Taro asked.
_Shut up, Taro. This is none of our business. Oh, except—_ "Were any of the other markers for the ashes of aristocrats?" Just in case. Taro had been born into an aristocratic family, and that caused him a stupid amount of problems, even though he'd given up all rights to the family title.
"Aye, but not all of them. There's everything you could think of. Merchants. Gamblers. Actors. Farmers. All sorts of different people."
An interesting selection of victims. If one thought the dead could be victims. Which I didn't, as they were dead.
"Do they have any idea who's doing it?" Taro asked.
"Not yet," Risa said in a curt voice. "Why? Are the two of you going to be supplying desperately needed assistance?"
That wasn't fair. We didn't try to get involved in these things. It just happened. "No," I answered loudly, before Taro could say anything. "It is none of our business. We are a Pair. That's what we do. That's all we're going to do."
Risa looked amused by my words.
Taro raised an eyebrow. "I would think after the last year you'd have a different opinion about that sort of thing."
Flatwell had been a rude shock for us both. It had been terrifying to be in a place where Pairs were neither respected nor supported. We had been expected to pay coin, which we didn't have, for everything we needed. It had been a harsh lesson in the dangers of having only one skill and relying entirely on it.
On the other hand, it was wise to know one's limitations. I was not a Runner. Who was I to get involved in a Runner's business?
"Where did you two go, anyway?" Risa demanded. "And why?"
No one was supposed to know where we'd gone, or why. Showing up after more than a year of absence, it was hard to lie, and I wasn't good at lying. "Triple S business." Which wasn't entirely a lie, as taking Aryne to Shidonee's Gap was Triple S business, and it was also a catchall phrase used to tell regulars to stop asking questions. Handy.
"Huh," said Risa, unimpressed.
"Did you bring your costume back with you?" Taro asked, sounding innocent when he was acting anything but.
I couldn't kill him. I wasn't suicidal. But I could hurt him a lot without experiencing any ill effects myself. Testing the limits of that ability might prove educational.
"Costume?" Risa demanded, eyes alight.
"It's not as interesting as it sounds, Risa."
"I beg to differ," said Taro.
Why was he doing this to me? I couldn't remember aggravating him recently. "Part of the task I was required to perform demanded particular clothing."
"Task, eh?" said Risa. "Some task other than Shielding, I suppose?"
"Bloody barbarians," Taro muttered, suddenly losing his good mood. "Wasn't enough that we were a Pair. They expected us to work."
Risa gaped at him for a moment, then started cackling with laughter, nearly falling off her chair in her enthusiasm.
"It's not funny," Taro objected, and no, he didn't sound at all petulant.
"It's beautiful!" Risa exclaimed. "Finally! You two had to work to earn an honest coin?"
That was all it took to bring Lord Shintaro Karish back. He sat up even straighter in his chair, his shoulders squaring back and the haughty mask slipping over his face. "We spend our lives risking our lives settling natural disasters, and have never gotten a damned coin for it."
I loved the way his _r_ 's rolled whenever he was particularly annoyed.
But Risa was not the sort to be intimidated. If pushed, she could tell dozens of stories about lordlings whom she'd picked up in various drunken positions of embarrassment or destruction. "So what did you do?"
"Nothing we can tell you about," I told her.
"You can't tease me with a mention of a costume and then just drop it."
"Just watch us."
"At least tell me why you had to wear it."
"I'm sorry, Risa," I said. "Really. It's just that our time away was difficult, and to come home and come across _—_ " More weird ritual trash. The people we'd known on Flatwell had taken their belief in ritual and superstition to fatal extremes. I hated rituals. They were never a good thing. I sighed. "It's disappointing, really." And frightening. And frustrating.
Risa tried to stare us into confessing. We stared back, silently. Then she shrugged. "All right. For now. But I'm getting some of that horrible white wine you like and pulling the story out of you."
Like hell. I was making a note of it. No more drinking with Risa.
We were able to tell her some of the more ridiculous things that had been going on in the Imperial court, claiming we had heard the stories during our travels. Risa told us of some of the goings-on in High Scape, which included a rash of home fires, a mayor who'd been caught spending too much government money on personal pleasure, and a series of successful jewel thefts. Risa's world was largely shaped by criminal activity, which was perfectly natural given her occupation, but it could make for a depressing conversation.
Risa was always, for some reason, a little less fun when Taro was with us. She seemed less relaxed and more likely to be offended. So it was a bit of a relief when she decided it was time to leave. But that wasn't necessarily an improvement of my overall circumstances, because Taro declared that it was time to go shopping. "For what?" I demanded.
"For whatever takes our interests. We work damn hard at our jobs and deserve whatever compensation we desire."
I looked at him with concern, discomforted by the tone of bitterness in his voice. "I don't need anything," I said.
He rolled his eyes. "That's subject to debate."
And that quickly, the bitterness was gone. Perhaps it had been a momentary aberration. "You can stop it right there." I found myself pointing at him, which was rude. "Only my mother is allowed to nag at me about my clothes." And only because I'd failed to find an effective means of gagging her.
"The point of shopping, my love, is not to go hunting for things you need, but allowing yourself to stumble across things you like and delighting in the discovery."
"How like a flighty lordling you sound."
"Hush, you, and be a good girl. It's not like you have anything better to do."
He was unfortunately right, but that didn't mean I was going to obey him unequivocally. "We are not shopping for clothes."
"Aye, aye. There's plenty else to find, you know. We need to get you some trinkets. You almost never wear any."
I shrugged. It wasn't as though I didn't like jewelry; I just never thought of it much. And when I did, I was uncomfortable with the idea of taking it from merchants. It was expensive, and I didn't need it at all. What few pieces I did have had been gifts from my family, and those would have been properly bought.
But there was no harm in looking.
In my experience, no city did markets as High Scape did them. It wasn't merely a matter of size, though High Scape did have a much greater number of merchants than any other city or settlement I had ever been to. It was also the fact that it was pretty much spread out all over the city, with a concentration in each quad and tendrils of stalls winding through the surrounding streets.
And then there was the sheer variety of goods. Clothing, of course. Clothing already made to fit general standard sizes, at which Taro always turned his nose. Fabric, to be purchased by those forced to make their own clothes. Tailors for those who could afford to have clothes made for them. Consumables, such as chocolate _—_ Taro picked up three bars _—_ and other luxuries, as well as vegetables and fruits and fish and meats and breads and cheeses and ales and wines and other liquors. Paintings and rugs and wall hangings and trinkets and toys. Dyes and cosmetics and perfumes and hair combs. Playing cards and name cards and news circulars and books. Just everything imaginable.
The noise of the market was often deafening, the scents in the air a battle between the sublime and the disgusting, and the streets completely crammed with people, animals and stalls. Sometimes I found it overwhelming. As well, fingering items I didn't need and listening to people bicker just wasn't terribly interesting to me.
Still, shopping with Taro could be fun, for he obviously enjoyed it, and his enthusiasm was a pleasure to observe.
Besides, there was a part of me that still hadn't quite recovered from worrying about every coin and whether we'd have enough to buy food, which had been a constant preoccupation of mine while on Flatwell. I couldn't say I wasn't thinking about how much things cost—I could never return to such a state of perfect ignorance—but knowing I could have it if I wanted it, regardless of price, was an almost dizzying relief.
I was disgusted that I felt that way. I was ashamed of how much I'd hated my time as a regular, how stressful I'd found it, how eager I was to return to the ease of being a Shield. I would ignore such self-assessment, if I could.
"That," he said, nodding at a bolt of blue cloth, "would look brilliant on you."
Aye, it would. "No clothes," I reminded him.
"Blue brings out the green in your eyes and makes your hair look exceptionally red."
And that was a good thing? "No clothes." Especially clothes that had to be made for me. That meant fittings, and fittings were time-consuming, irksome things.
"But you agreed to jewelry."
"I agreed to look at it." And I hadn't even done that. I just hadn't verbally opposed it.
Jewelry itself covered a lot of variety. There were plenty of rings, bracelets, anklets and chokers woven from leather or a variety of fabrics, and some of it was quite pretty. But that, of course, wasn't what Taro had in mind. "Gold," he declared.
"Copper would do, too," I suggested, but only because it made him give me that look of affronted disdain.
"Gold," he repeated.
"You know, if copper were more expensive, you'd be insisting on that over gold."
"There's a reason gold is more expensive."
"The merchants flipped a coin?"
"Because it's more rare, and it's prettier."
"Pretty being so much more valuable than useful."
"That's something that damn island taught us, didn't it?"
Taro never called Flatwell by its name if he could avoid it.
The stalls with the valuable jewelry were easy to spot. They had solid wooden walls on three sides and a guard or two standing at the front. The guards gave us a glance before returning to scanning the rest of the crowd, recognizing us as a Pair by the braids we wore on our left shoulders, and therefore not a threat.
Ah, and there was that disappointed look from the merchant when he realized some of his precious stock might be walking out with no coin in return. I'd almost missed that look.
"And don't even think of hiding your best stock," Taro warned him.
I looked up at him, startled. As the first words out of his mouth, they were a little harsh.
"I wouldn't dream of it," was the merchant's snappish reply.
"In fact, why don't we start with your strong box and move from there?"
The merchant sighed and ducked behind his display. He pulled out one black box, about a foot square, and then a second, unlocking both with keys secured to his belt. He raised the lids and pulled out the drawers.
I knew immediately that this was not the stuff for me. Gem-encrusted necklaces, rings, earrings and bracelets, thick and heavy and glittering. My skin hurt just looking at it.
Taro seemed to be reading my mind, for he tugged on one of my naked earlobes. "The holes are fairly small. You might have to work your way up to earrings such as these."
"Don't be ridiculous, Taro. I've no occasion for earrings such as these." I didn't expect any repetitions of the one party I'd attended in honor of the Crown Prince. Thank Zaire. I enjoyed music and drinking with friends as much as the next person, but there was something rather stiff about the gatherings of the High Landed. Plus I knew they were looking down at me, which was never a comfortable sensation.
The look of relief on the merchant's face was hilarious.
"You have in the past."
"And if I have any luck at all, I never will again." I looked at the merchant. "You can put these away."
He was pushing in the drawers before I'd finished speaking.
Taro growled but didn't press, instead choosing to glance over the rest of the man's inventory.
It wasn't a matter of modesty with me. I truly didn't care for jewelry that was too fussy, no more than I liked fussy clothing or fussy furnishings.
"So, what are these things?" Taro asked, holding up what looked like a gold lump on a gold chain. The chain was too short to be a bracelet, too long to be an earring. "I've seen people wearing them."
So had I, now that I thought about it. A faint memory teased at my mind and faded away. I took the item from him. I found a small pinning loop attached to the chain. The lump, upon closer inspection, was beautiful golden knot work.
"Not like that one, sir," the merchant objected. "Few have the taste required to _—_ "
"Yes, yes," Taro interrupted him with a wave of his hand. "But what is it?"
"Are you visitors here?" the merchant asked in return.
Taro raised a haughty eyebrow. "We are Shield Mallorough and Source Karish."
And before the merchant would make the mistake of admitting the names meant nothing to him, I added, "We returned to the roster only yesterday. We've been at another post."
The merchant nodded in a quick, abrupt gesture, because it wasn't as though he actually cared. "These are called harmony bobs. They have become highly fashionable. They are said to bring good luck."
"Bring good luck?" Taro picked up another bob from the display. "How do they do that?"
The merchant shrugged. "The spells cast on them."
Taro chuckled. "Spells?" he asked incredulously.
"Casting has become popular this season."
How could belief in casting become popular? Like a style of boot. Who believed in casting outside of plays, poetry and novels? And, of course, Flatwell.
"You are to wear it pinned to your clothing, over your heart," the merchant explained. "The beating of your heart is supposed to wake the power in the metal. The bob itself must be given enough length to hang freely, to give it direct exposure to the forces it is supposed to affect."
What a ridiculous idea. "What has brought this on?" I asked.
"I suspect it has something to do with the Riverfront Ravage," said the merchant, and I could hear him capitalizing the words.
The what?
"Excuse me?" Taro asked.
"Some kind of illness. It started a few months ago. No one knows how it started or where it came from. Some people think it's from the trade boats. Some think it's because of all those little flies that seemed to be springing up down there."
How had we not heard of this during our travels?
"What's being done about it?"
The merchant shrugged. "There are healers looking into it, I expect. It's nothing too serious. No one's died or anything. It's pretty common for these kinds of illnesses to flare up in larger cities. It's nothing to worry about."
"Yet they are developing this interest in casting because of it?" Taro gestured at the merchant's display.
"The ignorant get scared," the merchant answered. "I believe people started feeling uncertain about things during the Harsh Summer."
Though I had not heard the title before, I imagined he was referring to the summer before last, when unseasonable weather had a devastating impact on crops and stocks, on all number of livelihoods and lives.
"Many have still not recovered. I imagine there are those who never will. And then this illness comes, and the healers can do nothing. The Pairs can do nothing, the healers can do nothing, the mayor can do nothing. People start looking for other solutions."
How incredibly sad. I couldn't imagine what was worse than feeling utterly helpless, having terrible things happen and knowing there was nothing to be done about it. "But no one has died from this illness?"
"Of course not."
All right, then. It was unpleasant but not fatal. I was really tired of fatal things.
"Does the design have special meaning?" Taro asked of the bobs.
"Yes. A design will determine the nature of the luck you're seeking. In wealth, in health, in love and so on."
"And is there a limit on the number of designs?"
The merchant frowned. "Not that I know of."
"You have several repetitions here."
"Ah, those are partner bobs. Two people, wishing for the same thing, means twice the luck for both."
"Really?" Taro grinned, and I began to feel nervous. "What are these for?" He held up the bob.
"Luck with cards."
"Don't need it." He pointed at a design still on the display. "And that one?"
"Many children."
Taro apparently desired no luck in that area, either, for he moved on to the other bobs. He found none of the harmony bobs appealing, in meaning or style. So we left the merchant stall empty-handed. Which was a relief to the merchant and fine with me. I didn't want matching jewelry.
Except he dragged me directly to another jewelry stall. "You've got to be kidding me, Taro." It was a fashion. Taro in the past had sneered at those who felt compelled to follow fashion. He himself was always stylish, but never fashionable.
"What? They're funny."
"They're childish."
"Too much maturity makes you old before your time."
"No matter how old I get, I'll always be younger than you."
His response was to pinch my ear.
"I'm not wearing one of these harmony bobs, Taro. They're ridiculous." And while I wasn't overly concerned with what I wore or what others thought of it, that didn't mean I went out of my way to attract ridicule.
Besides, wearing a piece of jewelry identical to what Taro wore, it was too suggestive. People who were promised to each other in marriage wore matching jewelry, or those who belonged to some kind of organization. Either way, I'd look a complete fool once Taro went back to his licentious ways.
"So don't wear it if you don't want to. But you're getting one with me."
I sighed. "Yes, milord."
We went through three more stalls like the first one, and they didn't have anything that appealed to Taro, either. The design, the metal, or the meaning put him off.
It was the fifth stall that had what he was looking for. And I should have predicted that such would be the case, because this stall looked different from the others. There were no guards, and the only solid wall was the one at the back. Apparently this merchant wasn't overly concerned about thieves, for the wares were pinned to slanted boards that faced outward, rather than inward like at the other stalls.
This merchant, unlike the others we'd so far encountered that day, didn't seem worried about the loss of revenue to us. He greeted us with a smile that appeared genuine. He looked different, too, now that I bothered to really look at him. His brown hair, surprisingly curly, was worn longer than was the fashion for men, and his blue eyes fairly beamed out of a face that was masculine but sensual, his full lips curled as though he were gently amused at something. Perhaps us. His posture was relaxed, his clothes loose and colorful and designed for comfort rather than style. Something about him felt odd. Not dangerous or unpleasant. Just odd.
He was extremely handsome, an embodiment of all that was lush. There was something about his face that made me want to reach out and touch, starting at those amazing cheekbones with a side route to the enticing mouth and along that beautifully defined jawline. Really, everything about him was just so inviting. I folded my hands together to keep them still.
I looked to Taro, wondering what he thought of this toothsome young man. My Source didn't seem to have noticed him yet, focusing instead on the jewelry.
He found a bob he liked almost immediately. A simple silver chain supporting a symbol that looked like a sidewise number eight, only slimmer and elongated.
"It's the wrong metal," Taro said regretfully. "Silver doesn't suit you."
"It suits you, though. And you're the one who'll be wearing it." I liked the look of it myself. It was nonfussy, but elegant, and something about the figure implied balance. I liked the idea of balance. It was calming.
"What does this one mean?" Taro asked the merchant.
This was the first time Taro had addressed the merchant directly. I watched his face, waiting for the coy smile, the spark in his eyes. There was nothing, no recognition of the merchant's beauty. What was going on?
"Eternity."
"Eternity? An eternity of what?"
"Maybe it means something like immortality," I suggested.
Taro frowned. "Never understood the appeal of living forever."
"But eternity is not limited to immortality," said the merchant. "It can symbolize the desire for everlasting youth, or love, or the success of one's family, through every generation. It can refer to a search for knowledge that spans all existence, or an understanding of one's connection to everything else. Its meaning can adapt to the desires of the wearer."
That didn't make sense. Objects didn't adapt to their users. They were what they were.
"We'll take these," Taro told the merchant, and he moved to pin the thing on my chest before I could move to stop him.
The merchant was faster. "No, no, my lord," said the merchant, surprising us both with the use of Taro's former title. "There is a way to be doing such things."
That couldn't be good.
"Please." From beneath one display the merchant pulled out a very short stool, a second one from beneath another. "Sit."
One couldn't properly sit, the stools were so low. It was more like kneeling on the mat, with the stool supporting the buttocks. Taro and I ended up facing each other a short distance away from one another.
He wasn't going to sacrifice a chicken or anything, was he?
He got our names first, my two and Taro's string of them. He lit two candles, setting them down on the mat, one before Taro, one before me. Beside each candle he placed a black feather, a silver ring, a triangle of smoky incense and a bowl of clean water.
"Think of what you wish would last for an eternity," the merchant said. "If both of you desire the same, the power of the symbol will be brought to bear. If you do not, the bobs shall be nothing more than pretty pieces of silver."
That was all they were anyway. "We don't need to put you through all this trouble," I said to him. I didn't add that I had noticed a few curious onlookers pausing outside the stall, wondering what was going on. Taro and I no doubt looked like a couple of right fools.
"If you're not to give me the proper coin," the merchant said coolly, "you might at least observe the appropriate rituals."
That told me.
To make us even more uncomfortable, the merchant started to walk, circling us. "Think of what you wish to be eternal. Don't speak of it, for you will feel guarded in what you say, and may shape each other's wishes. Only if you are identical and sincere in your desire can the casting work."
Ah, so when the casting didn't work, it would be our fault.
But just as a mental exercise, I thought about what I would wish to be eternal. Not my life. I couldn't imagine living forever, continuously watching everyone I knew die and die and die. To have an unlimited thirst for knowledge might be useful, but it might be discomforting, too, to want more and more information and never be satisfied. Like an addiction. I didn't care about having an unlimited amount of possessions, though that might be useful for any children I might have.
Actually, I did know of one thing I would wish to last forever or, at least, as long as I lived. It was a weak and childish kind of wish. I couldn't even form the words in my mind.
The merchant stood beside Taro, a bowl in one hand and a knife in the other. "If you would take the bowl, my lord."
"What are you planning?" I asked sharply.
"Just a few drops of his blood, to be mingled with yours."
"No." I didn't know why that idea offended me so much. It just did.
"Relax, Lee," said Taro. "It's harmless. Unless you think I've got some fearsome disease I've been hiding from you."
"Of course not." If he did, I'd no doubt already gotten it. "It's just barbaric."
"Spoken like someone who has no understanding of the process," said the merchant. He wasn't sarcastic or snide or condescending. Just gently amused. I didn't know that I liked appearing amusing to other people.
Taro took the bowl, in his right hand as the merchant hinted, and held his left over it. The merchant carefully sliced into the fleshy part of Taro's palm. Taro hissed in reaction, but shifted his hand over so the first drops of blood landed in the copper bowl. The merchant pressed the bob into the bloodied palm, curling the hand into a fist.
Then the merchant was beside me, and I rolled my eyes at the childishness of it all. I took the bowl and let the knife cut into my palm. I felt it, but it didn't hurt. One of the benefits of being a Shield, I didn't feel things as much as other people did. I held my hand over the bowl and watched the blood drop down.
I actually felt a little odd. Almost a little faint. Which was stupid. I'd never before been squeamish over the sight of blood. But this was causing some kind of buzzing sensation under my skin, and I didn't like it.
The merchant put the other bob in my hand and balled my hand into a fist. He took the bowl from me, mixing the blood within it with the tip of his knife. He dribbled a drop of blood into the wax of each of the candles, right by the flame, then set the bowl on the floor, an equal distance between Taro and me.
"To the river from the north," the merchant said, much more loudly than he needed to. "To the river from the south. To the river from the east. As they join together, let Dunleavy and Shintaro join together in their will and their desire." Lovely. It had only needed that. He walked around us, continuing to speak in what seemed like a blending of a poem and a speech. "As the waters flow together to a single end, so will the blood of Shintaro and Dunleavy." At times, he would pause beside one of us, picking up the feather, waving it in the smoke of the incense and returning it to the ground. After he had done that with each feather three times, he slid the rings onto them and placed them both on the ground, never lifting them again.
I looked at Taro. He was smirking.
And then the merchant stopped. "My lord, you may pin the bob on Shield Mallorough. Go directly to her; don't step outside the circle. Wash the bob off in the bowl beside her feet. Then dry the bob on the inner wrist of your left hand before pinning it directly over Shield Mallorough's heart."
Taro did as instructed, and with the lightest touch pinned the bob over my left breast. No inappropriate fondling.
"Please return to your post, my lord. Shield Mallorough, if you would do the same for your lord."
That made Taro snicker, again. He was enjoying this far too much. With a straight face I repeated the actions required by the merchant, ending by pinning the bob on Taro's chest. It suited him, I had to admit. Smooth and elegant, just like him.
"And now," the merchant said once I'd returned to my stool, "each of you remove your candle from its setting and stand. Each of you approach the center bowl. Now douse the flame in the blood." We did so, and I, for one, was feeling silly. "Put the candles in the bowl, and stand straight." And we did. "It is done."
The applause startled me. An audience had formed. Lovely. I had no doubt they totally misunderstood the significance of what we had done. Which was to humor the merchant so Taro could get his meaningless matching bobs.
Taro had his hand shaken by many. I was given flowers. And someone pressed a wrapped package into my hand, which I would later discover held a loaf of spice bread, gushing best wishes for a happy future.
Oh, aye, totally misunderstood.
And no, there would be no embarrassing repercussions from this.
Chapter Three
"All right, that's enough for me," I said to Taro, my voice low because we were still surrounded by well-wishers. "Let's head back to the residence."
"What? Why? We just got out."
"We have what we came for. You wanted to get some jewelry, and we got some jewelry. Let's go home." They were still watching us. What did they expect us to do now? _The show is over, people._
"Lee, we've been away for over a year. I don't want to be shut up in the residence."
"People are going to be watching us because of that spectacle."
"So? That spectacle wasn't my fault."
"I didn't say it was." Though it was, sort of. If he hadn't insisted on requisitioning harmony bobs, we wouldn't have had to endure that ridiculous ritual. But then, I could have refused, so it was sort of my fault, too.
"And you had no problem being a spectacle on that damned island."
I glared at him. That wasn't fair. "I wasn't a spectacle."
"Oh, no? Leavy the Flame Dancer. You couldn't be anything but."
"My making a spectacle of myself put food on our table," I hissed.
"Which was more than I could do, right?"
"Stop putting words in my mouth." What was wrong with him?
"All I'm saying is that you're letting other people's interest control what you do. Why do you care whether other people stare at you?"
"How can you not care?" Who enjoyed being stared at by strangers, every move made and word spoken judged and possibly communicated to someone else? "Maybe you're used to having strangers speculate about you, but I'm not, and I don't plan to get used to it."
"You're overreacting. I just want to go shopping."
"And I don't. I've done enough shopping."
"Fine." He sketched out a little bow that somehow managed to appear sarcastic. "I need clothes, so I'll be off. I hope you enjoy your afternoon." He turned on his heel and strode off. Most of the crowd either followed him or dispersed.
Having an argument in front of an audience was humiliating. What had gotten Taro so upset? So what if I wanted to go home? Taro knew hundreds of people he could go shopping with, if that was what he wanted. Why did he have to turn it into an argument?
There were still a few people watching me, wondering what entertaining thing I would do next. Repressing the embarrassing urge to smile or wave, I turned away to head back to the residence. I felt like I could sense their eyes boring into the back of my head.
As I passed the jewelry stalls we had examined earlier on, I looked for signs of other such rituals being performed. There was nothing. I even watched a couple purchase a pair of harmony bobs and they were required to do nothing more than hand over the coin and pin the bobs on each other. So that blue-eyed merchant had done all that to Taro and me just for the hell of it. I felt even more of a fool. No wonder so many people had been watching. The display had no doubt appeared positively freakish.
I caught a carriage for the rest of the way back to the Triple S residence. I could have walked the distance easily enough, but the streets just felt too crowded, and I needed some space. That was one positive aspect of Flatwell; there was always space and it had almost always been quiet.
I hated being stared at. I really, really did.
I entered the residence with a sense of relief. Standing in the foyer were Source Claire Firth and her Shield, Dee Stone. They were ladies in their fifties who loved to drink and dance and have sex with a lot of people. They were a delight and a hoot and I was happy to see them. I'd managed to miss them since returning the day before. "Firth, Stone, how are you?"
"I am well," Stone answered.
And Firth said, her words treading on Stone's, "Is it true that you slept with Shintaro last night?"
I stared at her, shocked speechless by the question.
"Hush, Claire," said Stone. She was blushing.
Firth wasn't. "Answer the question, Dunleavy."
I finally found my voice. "I will not." How dare she interrogate me? She had no authority over me.
"Oh, so you're ashamed of it, are you?"
"I've done nothing to be ashamed of." But I did feel odd. Yes, it was suggested one shouldn't sleep with one's partner. This reaction, though, was unexpected, especially from one I considered a friend.
"Haven't you?"
"It is simply none of your business."
"What goes on in our house is our business."
"Now's not the time, Claire," said Stone, grabbing Firth's arm. "We're going to be late for our watch."
"We'll be discussing this, Dunleavy," said Claire. "All of us will. And we have every right to. Because it's wrong." And then she let herself be dragged out by her Shield.
It wasn't wrong. It was merely foolish. And while, if I had thought about it, I would have expected some ribbing, outright disapproval was a shock.
Oh, what a day.
I went up to my room. My luggage, still packed, was on the floor in the center of the bedchamber. No time like the present to get that done. I pulled out the trousers and shirts I'd worn for most of my travels, the two gowns I'd worn again and again before the Empress in Erstwhile, and the sandals I'd purchased with money I had earned on Flatwell. My one skill, besides Shielding, was dancing the benches. That had been a skill of value to the troupe with whom we had traveled. But bench dancing couldn't be done in ordinary, loose-fitting clothes, not on Flatwell. Oh, no. There had to be drama. There had to be flair.
The costume consisted of two pieces, a brown halter that left the midriff bare, and a scandalously short skirt. Copper beads had been sewn onto the brown material. Not by me—I couldn't sew to save my life—but by the woman whose assistance was the only reason Taro and I had managed to survive at all.
The clothing wasn't the entire costume. There had been cosmetics, too. Shadowing my face, bringing out the color of my eyes, coiling down my arms and legs in temporary tattoos. All of it applied, before every performance, by the same woman who had sewn the beads.
The first time I'd been in that costume, I'd felt exotic and beautiful. Then Taro had seen it. He'd thought it looked ridiculous.
He was right. I had made a spectacle of myself. But it had been necessary to get the coins we needed in order to eat.
I put the costume in my wardrobe, far in the back. I should get rid of it—I certainly wouldn't be wearing it again—but I wasn't ready to do that quite yet.
I craved a bath, but lacked the patience to wait for water to be heated and brought up. I poured some water into the wash basin and disrobed entirely. With the cloth, I scrubbed the grime of the market from my skin.
I was so happy to be home.
Once I was bathed and I had put on a fresh dress, I picked up the bundle of discarded clothes. At the moment before putting them in the laundry bag, I remembered the bob. I took it off and, after depositing the clothes in the bag, I put the bob in my jewelry box. I went to the door of my suite, preparing to go down to the kitchen for some coffee.
And I couldn't make myself go through the door. I didn't know what it was; I just felt restless and uncomfortable. I felt as though I was forgetting something important.
I felt better when I took a step back, and better again when I went another step back. I felt best of all when I went back to my bedchamber and stood over my jewelry box.
It was stupid. It was all the power of suggestion. That merchant had performed that ritual, and now I had spells and casting on my mind. It was making me act all ridiculous.
Still, I opened the jewelry box and picked up the harmony bob, holding it at eye level. It was a pretty little thing.
And I had to admit, I wanted to wear it. I would have had no problem wearing it if Taro weren't wearing the same thing. Why did we have to get matching bobs? What was Taro thinking?
I was so adverse to the idea of putting the bob in the jewelry box that it shocked me. But I was not going to pin it on my dress for all the world to see.
Feeling like a complete idiot, I stripped down to my chemise and pinned the bob to that. I felt better. I dressed again, and when I went to the door of my suite again, I was able to pass through it with no difficulties.
It made absolutely no sense, but I wasn't going to think about it. I would wear the bob and I wouldn't think about it. And when I went to bed that night, the merchant's influence would have dissipated and everything would be normal. I really, really wanted everything to be normal.
Chapter Four
Stomp, stomp, stomp, stomp, stomp. Stop. Grumble, growl or sigh. Turn. Stomp, stomp, stomp, stomp, stomp. Stop. Grumble, growl or sigh. Turn. Do it again.
The Stall was a very small building. One small room, in fact. Built just outside the general sprawl of High Scape, its official name, according to the Triple S, was the Observation Post of High Scape, where the Pair on duty waited while they watched for natural disasters to channel. It had been nicknamed the paranoia stall by one of our predecessors, so everyone called it the Stall.
It was a very boring place to be. That was the intention. The Pair on duty was not to be distracted by anything interesting.
High Scape was unusual in having multiple Pairs. Most sites that required Pairs had only one, and that Pair was not set to any kind of schedule. A Source and Shield didn't have to be together when they channeled, though it made it easier when they were. In most cases a Source and Shield went about their separate business. The Source would feel the onset of an event and lower his or her natural protections in preparation for channeling. The Shield would feel the Source's protection going down and would erect his or her Shields. This could be accomplished over a distance.
But High Scape was the largest city in the world and, until recently, had been one of the most turbulent, with events at least once a day, and often once a shift. It was filled with constant distractions, both day and night. The Triple S had deemed it wise to arrange to have a Pair on official duty at all times, and while on duty, to have them in the Stall.
One of the consequences of having fewer than the usual seven Pairs was that our watches in the Stall were much longer. I had been told that the shifts had lengthened from seven hours to nine when Taro and I were sent to the Southern Islands, and then to eleven when the circuit Pair had left. They had stayed at eleven once Taro and I returned, as Riley and Sabatos would soon be leaving.
For some reason, eleven hours felt so much longer than seven. It felt as if it were twice as long, not just four hours more. The books we'd brought had all been read; the games had been played too many times. All we had to keep ourselves from going mad with boredom was each other. There were times when that wasn't enough.
Don't get me wrong; Taro could be a very entertaining fellow. I really thought the flaw was with me. I wasn't the best conversationalist, and I couldn't listen to someone else for more than a few hours before getting tired of reacting to everything he said. I was used to having my attention absorbed by reading and writing, not people.
And if I felt that way, I could only imagine how Taro felt. He wasn't a person who read, or could sit still in quiet contemplation. He needed people and activity, and the long, eventless shifts tested what little patience he had.
And he tested mine. "Stop pacing," I snapped.
"I'm bored!"
"So? Is that any reason to torment me?"
"It's ridiculous for them to expect us to just sit here and do nothing for eleven hours. There is no reason why we couldn't just stay together in more interesting surroundings. This is a stupid waste of time."
That was true. Why didn't they trust us to stay together at the residence? "We could play cards."
"I'm sick of cards."
Taro loved playing cards, even when I was his only opponent and there were no real stakes. We had whiled away many a slow hour losing our firstborn to each other. He was just being difficult.
When I thought about it, I couldn't believe Taro had never developed any solitary pursuits, with the upbringing he had. What did he do all those hours he was locked alone in his room as a child? "Then please find yourself something to do that isn't so distracting." Because I couldn't settle down as long as he didn't settle down.
"Let's practice the weather," Taro announced, suddenly seating himself at the table.
"Practice the weather?"
"Aye, see what you can do with it."
I was shocked as I realized he wanted to experiment with my uncertain ability to tamper with the weather, an ability we'd discovered during the Harsh Summer. The regulars hadn't been able to understand why Pairs who could calm tsunami and earthquakes and volcanoes were helpless against things like blizzards and temperature. I'd thought they'd had a point, at the time, so I'd bullied Taro into experimenting, seeing if he could do anything. After all, he could create natural events when he wished, not just channel them. There had been reason to think he might be able to tamper with the weather, too.
Except he couldn't. He would open himself up to the forces, let them swarm through him, but was unable to glean the sensations that were caught up in the movement of the weather. It was as though all of his attention was absorbed in the grander sweeps of the forces, making him blind to the more subtle shifts of weather.
I wasn't. I could see them, hear them, feel them. More astounding, I could give those sensations a nudge, which resulted in a change in the weather.
Unfortunately, the changes were largely unpredictable, and ended up being negative at least as often as they were positive. So I was leery of the whole thing. There was no one to act as a mentor for me, and I had no way of knowing what the long-term implications of changing the weather might be.
So, of course, I was caught in a circle. I didn't want to experiment because I couldn't predict the results, but I would never be able to learn how to predict the results if I didn't experiment.
There were worse conundrums. I didn't like the idea of doing it unless it was absolutely necessary, anyway. I didn't want to do it just to see what I could do. That seemed disrespectful somehow.
"I'm not playing with that sort of thing just so you have something to do."
"I'm bored."
I rolled my eyes.
"Don't do that!" he snapped. "I hate it when you do that."
"Don't do what?" What was wrong now?
"Dismiss me," he interrupted. "I hate it when you dismiss me."
His leaping about in tangents was not a good sign, but if I stayed calm, I could possibly steer him back into making sense. "I'm not dismissing you. But it's not my fault you're bored. You knew what it would be like. Why didn't you bring something to entertain you?"
"I brought you."
Cute. "I was thinking along the lines of racing circulars. Or letters to write. Or read." I knew that the man was practically bombarded by letters from various admirers.
"You're acting like you did before," he accused me.
How bizarre. People had been telling me I looked different, that I was acting differently. Here was Taro, telling me I was acting as I had before. But perhaps I was misunderstanding him. "Before what?"
"Before we went to that damn island."
Why was that a bad thing? What was wrong with the way I had been then? "What do you mean?"
"Why did you leave your things in your room?"
Sometimes his conversational leaps just took too much energy to follow. I wished he'd pick a topic and stick with it for a while. "Because we have separate rooms. We've always had separate rooms."
"We always shared a room on that damn island."
"That's not entirely true. And when we did share, it was often a matter of saving space or money."
His eyes widened in shock before narrowing into a glare. "I see," he said coolly.
Clearly, it had been a bad idea to mention that. I should have said I'd wanted to share space for pure sentimentality. But damn it, it was the truth. "I assumed that once life got back to normal _—_ "
"What does back to normal mean to you?" he demanded.
What, he was going to make me say it? "Well, coming back here, getting back on the roster." I wasn't going to mention the fact that now we were home, Taro would wish to return to his more philandering ways. Because I wasn't stupid. Comments to that effect had always infuriated him.
Taro wasn't stupid, either. Sometimes he could hear what wasn't being said. "You are assuming a great deal," were the chilly controlled words that came out of his mouth. "But then, you always have."
The anger in him seemed to spark off some anger of my own. "Seems to me I'm not the only one making assumptions."
"Really?"
When Taro got angry, he had a whole repertoire of responses. He snapped and snarled. He shouted. He became rigidly polite. But I really hated that soft voice with the almost hidden edge.
"And what assumptions am I making?" he asked.
"Well, you obviously assumed we'd be sharing quarters once we got back here." Why should he be making all the decisions about how we did things? He was going to be the one to decide when it was over. It was only fair that I would be able to have some influence over the rest of the relationship.
"And you think that expectation presumptuous, given the circumstances."
Yes, but I knew better than to say so. "Just illogical. Why settle for less space when we can have more?" It made perfect sense to me. We'd spent so long in cramped quarters. Didn't he enjoy being able to stretch out more?
Didn't he enjoy having his own space again? Somewhere that was purely his when he wanted to be away from me, when he wanted privacy to do whatever kinds of things he did that needed privacy?
Sure, it meant a little more inconvenience on the sex side of things. He could no longer just roll over and find me there. But the trip down the hall wasn't too taxing. And it would make things so much easier when the relationship was over. No awkward moving-out scene.
Taro opened his mouth, no doubt ready to say something cutting. Then he frowned. "Something's happening."
Wonderful. He'd shifted moods again. I was still stuck in anger, because I was a normal person, and once I felt something, I felt it for a while. "No need to go overboard on the specifics, Taro, my love." No, that wasn't sarcastic at all.
He scowled at me briefly. "An earthquake, but not in High Scape."
"Oh." It was assumed that Sources could only feel events that were in their vicinity, but no one really knew what "vicinity" really meant, when it came to actual measurements. "Can you tell if there's anyone doing anything about it?"
"There doesn't seem to be."
"So, do you want to do something about it?" It was possible that the reason no one was doing anything was that the earthquake was hitting an empty area, but it was also possible that someone wasn't doing their job, or that this was an area that didn't normally experience events and so didn't have a Pair to protect it.
"I'd really like to."
That, I thought, was an odd thing for him to say, and it made me curious. Were there events that Sources preferred to channel over others? What made a particular event more appealing? I couldn't believe I'd never thought to ask.
But it wasn't time to ask right then.
I felt that little spurt of excitement I always felt when I knew I was going to Shield. This was what I was good at. It was what I was born to do. And there was something so cleansing about it. It was almost like a good, challenging, physically exhausting but satisfying bench-dancing competition, except it was for the mind.
His barriers lowered, I raised my Shields around him. I felt his mind open to the forces, only not to the forces around him, which was what he usually did. It was more like he was reaching out, trying to find and pull in the forces that were setting off an earthquake, somewhere far away. This was the moment, when it was too late, that I wondered if his channeling an event so far away was such a good idea. What if it drained him too quickly?
There was a bit of a wait before he found the forces he was looking for, and then he drew them in, channeling them through his own body. My task was not only to prevent all the other forces that were swirling about from rushing in and crushing him, but also to make sure the blood didn't burst from his veins, his heart didn't beat itself to explosion, and his brain didn't tear itself apart, literally, under the strain of world forces being funneled through a vessel that really hadn't been designed to do the job it was doing.
This took great concentration on my part, as well as a firm knowledge of how Taro's mind worked, how thoughts traveled over his mind, how his blood moved. That, as far as we knew, was the reason for the bond, to enable me to feel all of that information within him. I could do it with other Sources, but not as well as I did with him.
The flickering of color behind my eyes was my next hint that this was going to be an unusual channeling. Sometimes, I got images when I Shielded. This was not common. In fact, I'd never heard or read of another Shield seeing images when they Shielded. That didn't necessarily mean they never had. I hadn't gone out of my way to tell anyone, and perhaps all other Shields who had seen images had been similarly embarrassed. If I was the only one to see such things, I was clearly a freak. Both of these possibilities were a reason, as a rule, to keep the experience to myself.
The images never really made any particular sense, and this time was no different. What was different was that the sensations were not limited to the visual. A dark, clouded sky. High, rocky cliffs. Some kind of eagle or hawk screaming. Fast, rushing gray water laced with whitecaps that made me feel cold. The smell of rotting greens, and a taste of salt.
The assault on my senses was confusing, distracting. And the forces Taro was channeling seemed to be rushing through him particularly fast and particularly hard. I felt like I was scrambling to keep everything under control. It wasn't the first time that had happened to me, nor was it the first time it had frightened me. I didn't know why it was happening. Was it because the source of the event was so far away?
At least there wasn't any pain this time.
As the channeling continued, the screaming of the birds became continuous, and louder, pressing against the insides of my ears. My nose and mouth felt coated with salt. And the waves kept reaching higher and higher until thoughts of drowning trickled through my mind.
The rushing of the water nearly obscured the rushing of Taro's blood, which strained against the sides of its carriers in an unusual bid to break free. His heart was racing, pulling against the confinement of its own form. And the fluids moving over his brain seemed to be, what, curdling? Something vile. Something not good. I pushed against the entrancing images in my mind in an almost desperate bid to prevent Taro killing himself.
I had lost awareness of everything around me. I couldn't see the table in front of me, couldn't feel the floor beneath my feet. Usually I didn't so thoroughly lose all sense of what was going on around me. Should this worry me?
And then, finally, it was over. Taro was re-erecting his own protections and I breathed in a deep sigh of relief. "What the hell was that?" I demanded. Without thinking, I put my hands on my legs to see if my dress was wet.
"What do you mean? What is wrong?"
"Far too hard and fast."
He grinned. "Not to your taste, then?"
I didn't roll my eyes. Men. "What happened?"
"Nothing happened. Why?"
It had all seemed normal to him? Really? "I saw strange things," I admitted.
"I see," he said, appearing serious. At least he believed me. Probably because I'd told him of the other times I'd seen images while channeling, and it had always been a bad sign. "What did you see?"
"Cold water and high cliffs. Lots of screaming white birds. The air tasted like salt. It was very gray."
"Hm," he said. "That sounds familiar."
"Really? I can remember no place like it."
"I can't remember if I've actually seen such a place. It just seems familiar." He shrugged. "Oh well."
In my mind, he was dismissing it too easily. "Don't do it again."
He arched a brow. "Excuse me?"
I was reluctant to talk about what I had seen within him as he channeled. The images, the potential for danger, had been too gruesome. "Your heart was going far too fast. For a moment, I thought it was going to rip right out of your chest." That was a nightmare of mine, watching his body literally burst apart because I'd failed to Shield him properly.
"You can handle it," he said with utmost flattering and foolish confidence.
"I'm serious, Taro. I almost lost control there." It was hard enough admitting my failings without having him disregard my warnings so lightly.
"Almost means nothing. You did your job and that's all that counts. You always do your job."
It seemed to me that as time went by, Taro was becoming a little careless, a little too quick to believe that I could follow wherever he went, with no warning. I wasn't sure what to do about that. A part of me believed that it was only natural for him to become more confident in his abilities as he acquired more experience. The other part wondered why I wasn't acquiring the same confidence.
Our watch dragged by. The episode of channeling seemed to have calmed Taro down considerably, and it had washed my anger away. Still, we bickered as we usually did when we were stuck somewhere with nothing to do. It occurred to me that Taro and I never really talked about anything of any substance, except for our work. We either bickered or bantered. I wondered if that was a problem.
Eventually Source Chris LaMonte and Shield Fehvor Hammad showed up for their watch. A little more eventually than I'd been expecting. Source LaMonte was a pompous, aggravating stiff who thought too much of his advanced experience, but his sense of duty was infallible. The very idea of him being late for anything was shocking.
"My apologies," was his greeting, which shocked me even more. "But there's madness in the streets. Empress Constia has died."
My mouth fell open as I stared at him. "She what?"
"Apparently the word reached the deputy mayor last night. The Empress died nearly three weeks ago. Some sort of wasting sickness."
Taro, who had risen upon the opening of the door, fell back into his seat. "Zaire," he muttered.
It shouldn't have been a surprise. Rumors of Her Majesty's poor health had been circulating for the past couple of years, and she hadn't looked all that well the last time I saw her. And even if she weren't ill, well, she'd been in her sixties. At that age, people started dying for no reason.
It was just that I wasn't ready for her to die. I didn't like who would be taking her place on the throne. Even though she was married, no one assumed that her husband, Prince Albert, would be taking over her title when she died. Spouses usually didn't if there were children to inherit, and given Prince Albert's complete absence from public life, I, for one, wasn't sure he was still alive.
The only surviving child of the Empress was Crown Prince Gifford, and his mother had been so thrilled with him she'd sent us out to Flatwell to look for possible options for heirs. Prince Gifford, who'd conspired with Taro's mother, for some bizarre unknown reason, to get Taro's title back after he'd abjured it. Which was illegal, but it wasn't like the Prince was the one who would have suffered the consequences. That would have been Taro. And me, by extension. That little plot had been easy enough to circumvent, but it had caused a lot of sleepless nights.
Gifford's interest had resulted in my enduring a lot of exposure to Taro's mother, which I could have lived without. I'd never met someone I'd so very badly wanted to hit, and considering she had usually ignored me, her ability to raise my ire so quickly and so thoroughly revealed real talent.
I didn't know a lot about Gifford, but my limited exposure had shown me a man of exceptional arrogance, and little concern for the law. And he was to be our new ruler. Zaire save us.
Chapter Five
"So, Dunleavy, how's married life treating you?"
Oh, no, that wasn't the first time I'd heard that in the week since that ridiculous harmony bob ceremony. Instead of answering, because it was a stupid question, I took a sip of wine. As I let the wine roll over my tongue, I wondered why wine always tasted so much better in taverns than it did at home, even when it was the same kind of wine.
"Ya done it now," said Shield Giles Sabatos. "She's fuming."
I was not fuming. I was calm. I was serene. It was merely that the comment was so lacking in originality, it deserved no reaction at all.
The misunderstanding of the bob ceremony was far worse than I'd predicted. Those who had seen us indulge in it had thought we were getting married, of all things. Married! Why in the world would we be getting married? We had neither titles nor property. But the gossip about the married Pair had spread everywhere. Including to the other Pairs of High Scape. Who abused us mercilessly.
Harshly, actually. I was surprised by some of the things they'd been saying, about a lack of proper control and a weak understanding of professionalism and ethics. Especially Source Claire Firth and her partner, Shield Dee Stone. I was a little relieved they weren't able to join us for Riley and Sabatos's farewell dinner.
Taro's only comment about it was that he hoped the gossip reached his mother. She'd have an apoplectic fit.
I hated it. It was stupid, irritating, humiliating and no one's business. Of all the things to be right about, all the things I'd been worried about when I'd first experienced the dizzying disappointment of being bonded to Taro, this had to be it. People who didn't even know me, talking about me, about my private life. It was awful.
It was baffling. Where were people's priorities? The Empress had just died, but while those who could afford it wore the dark purple of mourning, few were speaking about her, and it seemed to me there was too little being said about the man who was going to be ruling our lives. More than enough was being said about me, though, and about Taro. I couldn't believe it.
Really, we weren't that interesting.
"This is not a matter for levity," LaMonte scolded, and I had to silently rebuke myself for being on the same side as he on any issue. "Really, what were the two of you thinking, to make such a spectacle of yourselves?"
"But being a spectacle is what I do best, Chris," said Taro.
I didn't answer. I couldn't think of anything to say that wasn't embarrassing. I hadn't objected to the little ritual because Taro had wanted to do it, and the merchant had been gorgeous. I wasn't admitting that.
"At least Dunleavy has the good sense not to wear the trinket," LaMonte continued, ignoring Taro's response. I didn't blame him. "I've yet to see you without it, Shintaro. I imagine you enjoy the attention it brings you."
It was true. Taro did wear the thing all the time. Even I'd noticed it. I hated seeing it on him. The symbol of eternity seemed to be mocking me, a grotesque reminder of how fragile the more emotional side of our relationship was, how it would be ended at any time, with no warning. Despite this, my only words to him on the subject had been that I thought he was too much of a gambler to believe in luck. He'd said a little good luck never hurt anyone, and I guess I had to agree with that.
Besides, I tried to avoid being a hypocrite. I was wearing mine, too. It was stupid but I just couldn't leave it off. I wore it every day. But I didn't let Taro know. Every night I took it off in my bedchamber before joining him in his.
I had to recognize in myself the unworthy trait of cowardice.
"This is our last night in High Scape," Riley announced. "And yet we are speaking of Dunleavy and Shintaro. How unusual."
"Not at all," said Taro. "We are the most interesting people here."
"Here" was the rather reclusive Silver Penny, one of the few taverns that hadn't become a dangerous place to eat during the Harsh Summer. At least, the servers had continued to treat us well, so I chose to believe they hadn't spit in the food, or done anything even more disgusting that I didn't want to contemplate. We were supposed to be eating and drinking and talking of old times. Except we didn't have many old times in common. Taro and I had been bonded for only around three years, and were the Pair who had been in High Scape the longest. Of those three years, over half of our time had been spent either absent or otherwise off the roster. And the other Pairs had never worked together before coming to High Scape.
Besides, we all lived together. We talked to one another all the time.
"One would think," Sabatos said in response to Taro. "But you've been characteristically closemouthed about what you were up to while you were away."
See? It wasn't our fault that the conversation stayed on us. They kept asking questions they should have known better than to ask. "Why don't you tell us what you've learned about Ice Ridge?" I asked.
Sabatos shrugged. "The name is the result of someone's twisted sense of humor. It's hot and humid and far from anywhere civilized."
It sounded like Flatwell.
"The only ridges are those belonging to the volcano. And every spring they sacrifice a virgin to it."
Hammad's eyes widened. "You're not serious."
"Well, not literally, because it's been totally inactive until recently. But aye, they dress up some old spinster crone and carry her up and around the top in a litter, and cut off some of her hair and throw it around. Then they bring her down and there's a big feast and dancing. It's supposed to bring good luck."
Of all the idiocy. Were there really so many superstitious people in the world? How had I missed them?
"At least it sounds interesting," Riley muttered. "The volcano may keep things active enough to keep me from falling asleep on watch. Or getting pregnant."
Hell, I hadn't thought of that. Channeling prevented pregnancy. I had gone about a year with very few events, and had managed to avoid getting pregnant. If High Scape was settling down to the point where I rarely channeled, or didn't channel at all, I did risk quickening. Not that I didn't want children, I did. Just not yet. I didn't know anything about kids yet.
I'd have to see how regular women avoided having children.
"Something strange is going on, though," said Shield Elata Benedict.
"Strange how?" Riley asked.
Benedict shrugged. "I don't know. It's just a feeling."
Source Derek Beatrice, Benedict's partner, snorted in derision.
Benedict ignored him. "All this new dabbling in superstition and casting. I know Karish is wearing that bob as a joke, but there are people who are really starting to believe in this stuff."
"A fad," LaMonte said in a dismissive tone. "I doubt it will last the winter."
"I'm not talking about those ridiculous metal cages some women are weaving into their hair," Benedict objected. "Aye, those'll go out of style, and none too soon. This is different. People are really starting to do things out of the belief it will bring them good luck. And not just the peasants, either. Lord Noirden sacrificed a cow on his front lawn this morning."
My wine almost went down the wrong way at that. Riley sputtered and coughed. "He what?"
"He did it personally, and of course he didn't know what he was doing. The poor animal was moaning and practically squealing as the idiot was sawing at its neck" _—_ Sabatos grimaced and dropped his knife and fork on his plate before pushing the remains of his meal away _—_ "and there was blood everywhere. Once the poor thing was finally dead, he poured oil all over it and set it on fire. There was a merry little blaze going before the Runners showed up, and once they got there, they didn't really know what to do. Apparently no one knows whether it's illegal to sacrifice an animal on your own property, because no one's done it before."
"I thought the casting was being attempted only by people with that riverfront illness," I said.
"Who told you that?"
What did that matter? "So it's more widespread than that?"
"Clearly, if Noirden was trying it. His son has a bleeding disease and probably won't live to be a man. It's said the healers can't do anything."
"Did he end up getting arrested?" Hammad asked.
"I don't think so. And if he did, he'd just buy his way out of it."
"I wonder if that'll become illegal after this," Taro mused, whirling his wine in his goblet. "Sacrificing animals, I mean."
"The Runners have a hard enough time trying to catch people who murder other people. I don't think they can afford to add murdering animals to the list of crimes," said Sabatos.
"Who was he sacrificing the cow to, anyway?" Beatrice demanded.
"No one seems to know."
"And where did he get the ritual? He didn't just make it up, did he?"
"He got it out of a book."
"A book about what?" I asked.
"Now you've got her started," Taro groaned.
"How could anyone not be interested in a book about how to murder animals for the purposes of healing someone? It's insane." Who would write such a book? Who would print it? How would Noirden even learn about it?
"I don't know anything about the book," Benedict told me. "The Runners confiscated it."
Maybe Risa would be able to tell me. She wasn't supposed to give me details about investigations, especially those that had nothing to do with me or Taro, but surely it wouldn't hurt to tell me the title of a book. I wanted to read the book that taught people to kill animals as some kind of ritual. I thought the writer was brilliant to be able to pander to this fashion so quickly. Cruel, and possibly lacking any normal sense of responsibility, but definitely brilliant.
None of this made any sense. A whole bunch of people couldn't just start believing in spell casting, all out of nowhere, no matter what difficulties they were experiencing. How did all of this start? The lunacy of it kept me distracted through the rest of the evening.
Taro and I were the first to leave. We had an early watch in the Stall the next morning, and it would take us a good hour and a half to walk to the Triple S residence from the tavern. Not that the distance was so great. It was just that Taro would need the time to chat with the dozens of people who would want a word with him when they saw him.
However, the first person to stop us was not one of Taro's many admirers. It was someone far worse. Someone much more able to create the sharpest discomfort within me.
It was Doran Laidley. Doran was a lord who, he said, had no money. There was an estate, which he claimed was really nothing more than an ugly stone house on a miserable plot of land with a handful of tenant families barely eking out an existence on the barren soil. Doran joked that the family kept a roof over their heads by marrying money, though he confessed himself unable to understand why anyone would be willing to pay for such a tired old title.
Doran Laidley. With whom I had been shamelessly flirting before Taro and I had been sent to Flatwell. With whom I'd been contemplating doing considerably more than flirt, because he had a pleasing face and shining green eyes and a comfortable manner. To whom I had given a too polite brush-off because I wasn't sure how long I'd be gone or if I'd ever really be coming back.
Doran, whom I'd forgotten all about almost as soon as I'd stepped beyond the city borders of High Scape and hadn't thought of once since our return. I couldn't believe it.
He looked startled to see us, and then he grinned with genuine cheer. "Lee! Taro! How wonderful to see you! When did you get back?"
Taro slung an arm around my shoulders. "A few weeks ago," he answered in an attempt at a casual tone that he didn't quite pull off.
"Just one week, actually," I added quickly. I would not have Doran believing I'd been back for ages without seeing him. "I do apologize for not contacting you immediately, Doran. Things have been strange here, since we got back."
"Yes, there have been a lot of changes since you've been away," Doran agreed. "Have you heard about _—_ "
"Our apologies, Laidley," Taro interrupted with civility so cool it was almost rude. "But we're on our way home. We have an early watch tomorrow."
"Oh," Doran said, clearly taken aback by Taro's manner. And I couldn't remember Taro ever calling Doran by his family name, not once since they'd first been introduced. "Yes, of course. Will you have time to meet me tomorrow, Lee?"
"No," Taro answered, curtly, before I could.
I gouged him in the ribs with my elbow. "Take your lordly temperament on ahead. I'll catch up." And I stepped out from under his arm when he seemed disinclined to remove it.
"It is inappropriate to leave you alone on the street at this time of night," Taro objected.
"I didn't turn into a child over the last half hour." Really, he was acting like a prat. "I won't be long."
He glared at me.
I raised an eyebrow at him. What was his problem?
He stormed away, muttering under his breath.
I turned from watching him to see a pensive look on Doran's face. "Your relationship with him changed while you were away," he commented.
I didn't know what to say to that. On the one hand, it was no one's business. On the other hand, one might say it was Doran's business, due to the nature of the relationship we'd had before I'd left. And if I had another hand, I could worry about how I was going to explain to Doran that the current stage in my relationship with Taro was only temporary without sounding addled or desperate for reassurance. So I said nothing.
"When can we meet?" he asked.
"I'm not sure," I answered. "It's not the best time. We just got back, and we're scrambling hard to catch up." Oh, such lies. Could he tell I was lying? He appeared only politely interested. "May I contact you in a few days?"
"Of course." He kissed my cheek. "I'm looking forward to it."
I wasn't. It was going to be awkward. If he suspected there had been a change between Taro and me, why couldn't he just accept that and fade away?
Zaire. How self-centered was that?
I caught up to Taro, expecting him to be in the same bad mood in which he had left. He didn't appear to be, though. He grinned when he saw me, slung his arm over my shoulder and strode on with a relaxed gait. I didn't know why he had gotten so upset; I didn't know why he was now so calm. It was frustrating to be unable to read him.
Chapter Six
High Scape had a lot of celebrations. Days where government and most businesses closed and people watched performances and played games and drank too much and ended up getting arrested for being really stupid in public. The reasons High Scape found for closing down varied from the profound to the trivial. There would be something massive for Crown Prince Gifford's coronation, Zaire help us all. Every year there was a celebration to mark the day we guessed our ancestors first arrived on the world from who knew where. There was a marking for the changing of the seasons. There were annual celebrations for the existence of the sun and the moon each, a festival in praise of balance, and days marking battles that had occurred back when cities still fought one another.
And there were others that I hadn't had a chance to learn about yet, due to my many absences from High Scape. For the last few days, I had seen evidence of another citywide occasion, extra scrubbing of public statues, wreaths and colorful drapes decorating the buildings. I ran through my mental list of holidays and couldn't think of anything that fell at that precise time of year. I had meant to ask someone what they were celebrating this time, but I kept forgetting.
So I got up late on a day on which I had no watch at all in the Stall, and the only other person in the Triple S residence was Ben. He was in the kitchen, which, as usual, was my first destination in the morning, and he poured me a hot cup of coffee. I breathed in the delicious aroma. My plan for the morning was to drink coffee and read, and maybe write a letter to Aryne.
"How are you settling back in, Shield Mallorough?"
Damn it. He'd asked a question of me. That meant I couldn't take the coffee to the parlor and consume it in sweet silence. "It's nice to be back," I said.
"You must have enjoyed your trip away," he said, wiping at a countertop that looked perfectly clean to me. "When you returned, you seemed much more relaxed than I remember seeing you before."
What was he talking about? I was always relaxed.
"Pairs are so very fortunate, that they may travel so much." He sounded wistful.
Recognizing I was going to be there for a while, I perched on one of the stools at the table and sipped at the coffee. "Is traveling so difficult for people to arrange?"
"Most can't spare the coin. Or the time away from making their living."
Yes, that made sense. I felt stupid for not thinking of that. "I would have rather stayed here." I didn't care how much he thought the trip to Flatwell had relaxed me: it had been awful.
We sat in silence a short while, both of us watching him wipe his cloth over the counter. Then he bit his lip and asked, "What do you think makes a Source or a Shield?"
That was odd. From what I understood, Ben had been working around members of the Triple S for most of his life. He should know the answer to such questions as well as anyone did. "They channel forces."
"No, no, excuse me. I mean, what decides a person is going to be a Source or a Shield?"
I frowned, still not sure I understood quite what he meant.
"Before they're born," he added.
Oh lords. Philosophy. "Circumstances," I said. "Life. That was just the way things fell together."
"Really? That's it?"
It was far too soon after waking _—_ I'd had only one cup of coffee _—_ to be having a discussion like this. "What are you suggesting?"
He smiled sheepishly. "Nothing, Shield Mallorough. Sometimes it just strikes me as odd, how different life can be for different people."
Yes, life was unfair. I was lucky enough to have been born a Shield, which was, I knew, a privileged position. If I hadn't been a Shield, I still would have been the daughter of wealthy merchants, also a fairly privileged position. I didn't know why I was so fortunate. I only knew I was uncomfortable when I was reminded how much harder other people had it. People like Ben. "Aye, it is strange."
"Hm," he said. Then he folded the cloth in a neat little square. "Are you going to the parade today?"
Ah, a parade. So that was the big event. I wasn't a great admirer of parades. Waiting around for hours while various people drove by. They tended to be crowded and extremely noisy. "I'm not sure. What's it for?"
He looked shocked. "To celebrate the appointment of a new mayor."
"When did that happen?"
"Word of it was announced a few days ago."
Oh. I supposed I should have known that. I had to get back in the habit of reading the news circulars. "I'm surprised they were able to throw together a parade so quickly." And surely it was hugely inappropriate? The Empress had just died.
"Aye, it does take a while to organize. The parade has been planned for weeks, since the corruption of the last mayor was announced. I am told the city had been assured the name of the new mayor would be given in time for the parade."
I drained my mug and Ben immediately reached for the coffee warming on the stove. I found it bizarre that a parade was planned without knowing the identity of the new mayor, with the Empress _—_ or whoever _—_ making sure the name was provided in time. Almost as though the parade were more important than the mayor it was designed to celebrate. And what if the people didn't like the person appointed?
Actually, maybe that was why they did it. It would be too late to cancel the parade if the new mayor was despised by the populace, and that was probably a good thing. It would be embarrassing for everyone if a mayor were appointed and everyone refused to celebrate. By picking the day in advance, all the participants and merchants would already be committed.
"So, are you going to the parade?" Ben asked again.
I guess I'd better be. Ben certainly seemed enthusiastic about it. If I didn't know better, I would think he was trying to get me out of the house. "Certainly. I love a good parade."
"I'll wrap something up for you to eat."
"I can get something while I'm out," I said quickly.
"Festival stall food," he sniffed.
Hey, I'd always liked it. "Have you seen Taro?"
"He left with a friend very early this morning."
Taro and his bizarre early-morning habits. I refused to ask how good-looking said friend was. I really didn't want to know.
I left the residence carrying a small bundle of fresh bread, cold cooked bacon, and cheese—good, wholesome food that wouldn't stand a chance against the more savory fare that would be available closer to the parade route. Of course, it was a bit of a hike to Center Street, the popular choice for parades, and most of the carriages usually available for hire would either be somehow involved with the parade itself or sticking to the streets with the residences of those who could afford rates that had been tripled in honor of the day.
So that meant walking, which meant I'd be starving by the time I got to any stalls. I dug into the little package Ben had made. Cold bacon and cheese could be really tasty when one was starving.
I'd never seen the appointment of a mayor before. I had seen parades. With this in mind, I asked for directions and made my way to Ivy Point, more popularly known as Confusion Square, where the parade would end. That was the first place where royal proclamations and other political speeches were given in High Scape, and it was apparently where the mayor would be appointed. It was a large empty space, basically, where Center Street ended and dozens of other, smaller streets branched off in crazy directions. Instead of the usual cobblestones, the street of the Square had large slabs of stones in circling shades of light gray. There was a huge memorial in the middle of the northern half of the Square, a ridiculously tall statue of Emperor Benik in a ridiculously heroic pose. And there was ivy everywhere, crawling up the walls of the buildings lining the Square. That was why it was named Ivy Point, but people still called it Confusion Square back from the days when a handful of merchants had fought for control over who could sell in the area, with the result that on any given day no one knew what would be available for sale, if they would be risking their lives trying to buy it or if they'd even be allowed in the Square at all.
When I reached the Square, I saw that many others had had the same idea as I, and a small selection of merchants had set up stalls. I'd been uncertain as to whether it was appropriate to wear mourning purple to the celebration of a new mayor, but I saw many others were also wearing the color to some degree or another. So I wouldn't stand out.
It looked like everyone was wearing harmony bobs.
A platform had been constructed beside the memorial, and on the platform, a canopy with cushioned chairs. That was where the important people would sit, when it was time to swear in the new mayor. Runners kept a huge swath of Center Street clear where it entered the Square and stretched up right to the memorial. That was where the parade would end, the participants splitting off before the platform and disappearing into the crowd or down the smaller streets as they chose.
I was aware, from the chatter going on around me, that the parade had started before I'd risen for the day, but none of the entrants had reached the Square yet. I browsed through the stalls. Most of the fare seemed of a cheap variety, the kind of things a merchant would be less devastated to lose if the Runners chose to enforce the laws and confiscate their goods. I found a stall selling books, thin with jagged paper and loose bindings.
After perusing the titles for a while, I asked the merchant, "Have you any books about spell casting?"
Her eyebrows shot up. "Of course not," she said quickly.
That was a strange response. "What do you mean by that?"
"Look, I've answered your question. Do you see something you want?"
"No."
"Then please leave."
Well, that was weird.
The first participants in the parade showed up, a troop of Imperial soldiers wearing copper breastplates and blue capes, marching to the beat of the troop drummers and horn blowers coming in behind them. They were impressive. They were dangerous. At least, to me.
Marching music was bad. It made me want to march somewhere and do something martial. I was such an idiot. Of course I knew parades had music. I'd just, somehow, forgotten the impact such music would have on me. Because I was an idiot.
Music could have a really bad effect on Shields, worse than alcohol, and probably down there with drugs if it were the right kind of music and the wrong kind of Shield. I'd nearly started a tavern brawl once because of the music being played in it. One entry in my endless list of embarrassments was how particularly sensitive to music I appeared to be.
I noticed people giving me uneasy looks. The white braid sewn into my left shoulder practically glowed against the dark purple material of my dress. And a Source, who would be known by a black braid over the left shoulder, was nowhere in sight. My Source was supposed to keep me under control. That was aggravating, but necessary.
I had a method for getting through musical assault. Not a great method, and it didn't always work, but it was better than nothing. I made my way to the platform, to the side away from everyone else and, an added plus, the farthest from the marching band, and I wrapped my arms around one of the support beams. Hold on to something and don't let go. And when the music stopped, I would go home.
But the music didn't stop. It seemed that the military musicians had taken up a position on the other side of the Square and planned on serenading each entrant in the parade. Just lovely.
At least there were plenty of Runners around. They'd stop me if I tried to attack anyone, which was the sort of behavior this kind of music could inspire in me. Violence was far less embarrassing than the other possibilities.
I pressed my forehead against the wooden beam, trying to focus on the grains of wood digging into my skin. They didn't hurt, and normally I wouldn't even feel them. But I concentrated on the rough grains biting so slightly into my flesh. It almost felt good.
I closed my eyes. Images popped into my head. Images of running. With a sword. Charging toward the enemy. Rising up to the crest of a hill and ready to leap into the fray below. It didn't matter that I wouldn't know what to do with a sword if I had one, and leaping into the fray would no doubt result in my tripping and falling on my face.
When someone tapped me on the shoulder, I swear I almost jumped right out of my skin.
"My apologies." The man, middle-aged and almost rigidly slim, bowed low. He wore the plain tunic and trousers of a servant. "But Trader Fines has learned of your difficulties, and offers a place on the platform as a means of collecting yourself."
I stared at him, my mind blank. He was going to make me talk, damn it. I tried to think about something other than the music. "I thank him." Hey, I could still form words. "But it would be better if I stayed out of everyone's way." The top of the platform was just as open to noise as under it, with the added disadvantage of exposing me to everyone's easy view.
"There is no one to look after you here," said the man. "We may be able to assist you and allow you to preserve your dignity."
There was no dignity in losing control in front of an audience. I would know.
"Trader Fines will be able to attend to you, give you something on which to put your attention and prevent you from harming yourself or anyone else. Surely that is better than cowering down here."
I wasn't cowering, prat. I was being discreet.
"Please. Trader Fines fears what havoc your parents might wreak on his trade relations, should they learn he left their daughter in distress and without her Source out on the street."
Trader Fines. I knew no Trader Fines. That didn't mean my parents didn't know him. And if he lived in High Scape, it wouldn't be ridiculous to believe he knew who Trader and Holder Mallorough's daughter was.
It was better than accepting aid from someone who didn't know me at all. But would accepting his assistance put my parents into some kind of debt? Surely not.
Was it normal for complete strangers to assist each other in this way? We had been assisted on Flatwell, but with the expectation that we would bring our benefactors money. Would Fines expect something in return from me? But there was nothing I, as a Shield, could do for him.
It was really hard to think. "I can't let go," I confessed.
"Your pardon?"
"I have to hold on and not let go." I closed my eyes again.
"Then perhaps I can be of assistance." I opened my eyes when the man touched my wrist. He pulled gently, but my fingernails were dug as deeply into the wood as I could force them. "Please," he said softly.
I looked at the hand in question, and with great effort forced myself to unclench my fingers. Once my hand was free, the man took it, tsked at the blood under my nails and wrapped it around his forearm. It was hard work to avoid sinking my nails into him, but it gave me focus and a distraction from the music bombarding me.
He slowly escorted me around to the front of the platform, and then up the steps. At the top, I realized that people had arrived and taken some of the seats. A quick glance at the Square showed me that it was much more full, and that there were a lot more parade participants mingling about. How long had I been standing under the platform, anyway?
I recognized two of the people on the platform as attendees at the disastrous party held by Lord Yellows, who had been using the party as a means to get all of the local aristocracy, as well as the visiting Prince Gifford, into one place so his fellow Reanists could sacrifice them all at once. The two, along with everyone else on the platform, turned to look at who was joining them, and the two sacrifice survivors surprised me by nodding greetings at me. I nodded back.
The servant led me to a man who remained seated as he watched me approach. His black hair had a single wide stretch of white so perfectly linear and white that I wondered if he'd dyed it so for effect. His eyes were dark brown and his skin was the color of copper coins. His face seemed free of lines, but there was an edge to his features that made me think he was much older than he looked.
I really didn't know much about fashion, certainly not as it pertained to men, but even I could tell the cloth was of the finest quality. He was wearing the dark purple for mourning, and a cloak that looked similar in cut to what I'd seen many other men wear.
He was wearing a harmony bob. That made me respect him less, even though I knew this made me a hypocrite. I had a sentimental reason for wearing mine. What was his excuse?
He rose when I was about a foot away from him. "Trader Fines," said my escort. "May I present Shield Dunleavy Mallorough. Shield Mallorough, this is Trader Richard Fines."
He bowed. I always tried to avoid bowing, because it seemed too subservient. It was bad enough that I had to do it for royalty. But when the other person did it first, well, I had to hunker into a curtsy. So I did. "A joy to meet you," I said.
"A joy shared is twice blessed," he responded, which was a greeting I had never heard before. "I understand you are experiencing some difficulty."
"Yes, I hadn't expected the music to be so . . . loud." I realized my nails were digging into the servant's arm after all. The poor man hadn't made a sound.
I was afraid to let go of him.
"Please be seated."
"No. Thank you. I need to go." But I couldn't move. Not by myself. I was surprised I was able to string words into coherent sentences. So stupid, to have come to the parade alone. Or at all.
"Nonsense. I will take care of you. To my understanding, it is merely a matter of having you give your attention to something else, correct?"
Ah, not entirely. It depended on the Shield, and how badly music affected him or her. Music affected me very badly.
But I couldn't move on my own, and for a moment the music welled up, sweeping through my mind.
Perhaps Fines continued to speak. I didn't hear him. I found myself pressed into his chair, my hands transferred from the poor servant to the arms of the chair. And once I was seated, I felt frozen in place. This had become my new anchor.
I was able to notice that Fines had ousted the person from the chair next to mine and seated himself.
A few _—_ or a great many, I couldn't be sure _—_ moments later, the music changed to something a little softer. Still a little rambunctious for my taste, but it was easier to keep still, and I could actually listen to what this Fines person was telling me.
"Your father outbid me on the shipment of buttersilk," said Fines, and it sounded like he was at the end of a story.
"Oh." What did one say in response to that? Good for him? I'm sorry? I wasn't really listening?
Fines appeared pleased by my response. Perhaps he hadn't been expecting one, which wouldn't be surprising, considering my behavior. "What with the frost destroying so many trees in the Beatrum Triangle the year before last, buttersilk is scarce."
I didn't know the location of the Beatrum Triangle, I didn't know there had been frost there, and I didn't know what buttersilk was. Should I tell him that?
"Your mother will make an absolute fortune."
"I'm pleased to hear it." Because I had to say something. And I was always happy to know my family was doing well.
"Your family has quite able traders and holders. It's rare to see so much talent in one family. Usually that sort of ambition skips a few."
"Does it?" The servant offered me a goblet of something. I shook my head.
"Do you have any skill with trade?"
"None at all."
"Well, it would be wasted in a Shield, wouldn't it?"
It would. Completely. And yet I found his comment insulting nonetheless. As if a Shield could do nothing but Shield. It was true that I could do nothing but Shield, but that didn't mean no Shields had any other skills. I was offended on their behalf.
Zaire, I wasn't even making sense to myself. It had to be the music.
"And I understand that you are an exceptionally skilled Shield."
I wouldn't have said "exceptionally." I was good, I knew that, but there were a whole lot of Shields out there that I had never met. "That sort of thing is really very difficult to quantify. It's more a matter of the talent of the Source than that of the Shield." After all, if the Source couldn't channel well, the skill of the Shield didn't really matter much.
"And Source Karish is a highly skilled Source."
"Yes." I felt I should do more in this conversation than answer questions, even though I hadn't asked to be part of it. "I gather from your place on the platform that you have a part in the appointment ceremony?"
"Actually, Mayor Izen has already been appointed. There was a ceremony held at city hall last week. I am merely introducing him to the people. He is a good friend of mine, and he is well suited to his new role. I am honored and pleased to be part of his introduction."
I wanted to ask why he was given what I assumed was a significant honor, introducing the mayor to High Scape, but I had a feeling I should already know. The thing was, he was just a merchant. With all the aristocrats who lived in High Scape, I was surprised to find a merchant so singled out. "Have you done this before?"
"Oh, no," he chuckled. "I think I was chosen because I have a seat on the Imperial Council. I just received it, because of the bill. You know the one, introducing a quota for merchant seats on the council?"
Ah, yes. I'd even met one of the people who wrote it.
Fines must, I thought, have an awful lot of money.
I was getting cynical in my old age.
"As well, I was part of the campaign for Mayor Izen to get this post."
"Really?" I said. "What does such a campaign involve?" I had no idea how a monarch chose a mayor.
He winked. "Now, that would be telling."
That would be telling me what? What couldn't he tell me? Had there been something nefarious about the campaign?
I was being completely ridiculous. Of course there had been nothing nefarious going on. Fines was just amusing himself by playing with me.
It was so hard to think.
The Square was becoming uncomfortably packed as more and more jugglers, dignitaries, acrobats and musicians decided to wait around for the introduction of the mayor. At least the musicians stopped playing once they entered the Square. The cacophony of competing musicians would have driven me insane.
Fines talked on. About local politics, of which I knew little, about trade, of which I knew nothing, and about scandal, of which I knew more than I liked. And time passed. The rest of the chairs on the platform filled. More and more people crammed into Confusion Square.
And then, finally, a carriage was drawing up before the platform, a heavyset man with a chain draped over his shoulders and chest stepping down from the carriage and lumbering up to the platform. The new mayor, I assumed. The music, thank Zaire, finally stopped. I breathed a sigh of relief, unclenched my fingernails and worried at the sliver that had worked itself into my left index finger.
And I realized, really realized, that I was on the platform upon which the mayor would be introduced to everyone. The platform everyone was watching. What the hell was I doing on the platform?
My hair clashed horribly with my dress.
Fines rose to his feet and assisted the man I presumed to be the new mayor up the last few stairs. The two men walked to the center of the platform, near the front, and waited for the crowd to quiet down. It took a while, but the silence, when it finally came, was soothing to my abused ears.
Everyone heard the single person who, hidden by the crowd, booed.
I pressed my lips together to keep from smiling.
Ignoring the malcontent, Fines called out, "Good citizens of High Scape! Welcome to this glorious day of renewal." He paused, and a lot of people obligingly applauded. "Although it is a time when many of us carry sorrow in our hearts _—_ " Did we really? How could people mourn the death of someone they had never met? "We must always seek solace in new triumphs, and recognize that change brings with it opportunity as well as loss."
I hated change. Change brought anxiety and uncertainty. I didn't think I knew anyone who actually liked change. At best, they accepted it with a sort of grim resolution. At worst, they denied the reality of that change to the detriment of themselves and everyone around them.
Did Fines like change?
"Today is our opportunity to welcome to the helm of our great city an able man of discipline, compassion and wisdom."
I could swear I heard someone snicker.
"There were many who could have done the job of acting as mayor of High Scape. There were even more who wanted the position."
Really? I wondered why. There was a goodly amount of money to it, I supposed, and prestige. But the mayor probably learned a whole lot more about the inner workings of the city than I would ever want to know. I imagined there were a lot of long meetings while people debated endlessly about building bridges or water ducts. It certainly wasn't how I'd want to spend my time.
"But few would be able to bring to the role the dedication and honor of the man it is my privilege to introduce to you." Fines put a hand on Izen's shoulder, squeezing it slightly. "I give you Yuri Izen, mayor of High Scape."
The applause was thunderous. I was made aware once more that I had no idea who Izen was and whether anyone in High Scape, aside from the person who had booed, really knew anything about him. Maybe the only thing they liked about him was that his introduction provided an excuse to take a day off and drink beer.
My studies at the Shield Academy had never given much attention to local politics. We had been encouraged to meet regulars, of course, and to understand that it was important to protect them, but not to get too attached, and certainly not to get involved in anything that could make the blood boil as politics could. It was not our place to interfere. We would always be moving on soon.
"Dear people of High Scape!" Mayor Izen shouted. He didn't have a voice for public speaking. It cracked on every word. "I stand before you fully aware of my responsibilities as the mayor of the greatest city in the world." There was more loud applause at that, and I wondered if Izen had visited every city in the world, that he could make that claim. "We have been put through some difficult times in the past, but we made it through. We are facing some challenges now, and we'll face and defeat those, too. Because we are the strongest, smartest, bravest people in the world." More applause, and I wondered if the spectators believed what they were hearing. How could they think they were better than other people in other cities? It was silly. "I was born in this city. I grew up here. My parents were both on the city council and they instilled in me the knowledge that service to others is the greatest employment a person can have. I bring to my role as mayor this knowledge, as well as great pride in this city, and the willingness to do whatever it takes to assist in the prosperity of High Scape."
The new mayor droned on with more rhetoric about what a wonderful leader he was going to be. The audience applauded dutifully where the mayor paused. I wondered how much longer he would be.
With the music silenced, my mind cleared. I looked at Fines, who was standing behind Izen. I couldn't see his face, but he was standing very straight. I imagined he looked proud. I wondered why he had wanted Izen to be the mayor. How did he benefit from it? While I didn't know much about such things, I did know that little was done in political circles without favors changing hands.
Ah, it had nothing to do with me. None of my business.
Izen finally stopped talking. I noticed wagons with barrels of what I assumed was beer being pulled into the Square. So did a great many of the spectators, it looked like, for many didn't bother applauding Izen's last words as they headed for the beer.
Despite the defection, Fines spoke again. "Please remain with us," he called, "as we celebrate this great man's rise to our highest office."
Really, people didn't need the encouragement.
I rose from my seat. I wanted to get out of there in case any of the music started up again.
Fines was suddenly by my side. "Shield Mallorough, will you not remain with us for the afternoon? There are people I would like you to meet."
Why would he want me to meet anyone? "I apologize, but being exposed to such rousing music for as long as I was can be very draining. It would be irresponsible for me to remain here without my Source. Truly, I shouldn't have come at all."
"I am glad you did come, though I am, of course, regretful that you experienced such difficulty. It gave us a chance to meet."
I didn't know how to address that. "Thank you for your kindness." And I made my escape.
I was tired from my interaction with the music, I had been telling the truth about that. All the muscles of my body felt loose and watery from being held so tensely for so long. There were still no carriages available for rent. I was thoroughly exhausted by the time I dragged myself over the threshold of the Triple S entrance. I went straight to bed.
The next day, I learned that Izen had been murdered in his sleep.
Chapter Seven
A bench-dancing competition that didn't conflict with one of my lengthened watches at the Stall, one that I was free to enter. For the first time since we'd gotten back. Finally.
And it was a proper bench-dancing competition. Never mind that it was one of the smaller ones, one that no one of exceptional skill would enter. It was enough for me that it would be a regulation match, with two benches and four bars, and the only musical accompaniment would be the drums. The kind of match I'd been trained to expect when I was in the Academy, the kind I hadn't been able to enjoy since before Taro and I were sent to Flatwell.
There were no costumes, because no one cared what anyone looked like. My hair was tightly bound, though much of it was likely to escape at some point. My trousers and shirt were loose and comfortable and covered me from throat to wrist to ankle. I wore no cosmetics, no flashy baubles. I'd used nothing more than my plain name, no adjectives included, when I'd signed up.
There was a Runner among the gathering spectators, his solid-black uniform standing out against the varying clothing of the others. I thought it odd that one would waste time watching a bench-dancing competition while he was on duty. But perhaps he was looking for suspicious behavior. Since the mayor had been killed, Runners seemed to have multiplied, and they interfered with the most innocuous of activities.
I wouldn't let his presence distract me. Finally, my muscles would be used as they were meant to be used. A good, proper working out. I was going to enjoy it.
And no one would be tossing coins on the ground for me to scoop up after my performance. That had been so demeaning.
I was excited, to my disgust. I tried to calm down. Not just look calm, but be calm. Deep breaths, sedate thoughts. It didn't help. I was excited. I stretched a little harder than was good for me, ground my bare feet into the chalk box a little too enthusiastically, and I found it difficult to stay still once I was standing on the bench. I practically shivered once the drums started rolling.
But the drums weren't the only instrument that sounded through the air. I heard the winding tones of a double-reed recorder, felt it low in the pit of my stomach. That wasn't traditional. Why were they playing that? I shook my head in an attempt to shake off its influence.
And then the bars started moving. I nearly lost right then as a strange battle developed between my mind and my body. My mind expected me to dance the proper way, the way I had always been trained. My body, however, remembered a different way of dancing, the way that had been drilled into me over endless practices and performances on Flatwell. A way with a slower pace and lower bars, where legs and arms curled and coiled unnecessarily, for the show of it rather than the need of it.
The first couple of steps were taken in such confusion that I almost slipped from the benches for no reason at all.
I was not going to do that again. I struggled to get myself under control.
_Remember what you were taught._
_Get your arms back where they belong._
There were two benches. Two dancers stood facing each other, one on either end of the benches, with a foot on each bench. Four bars were held and moved by people called stalkers, two at each end of the benches. They brought the bars up and over the benches, banging them together, and it was the task of the dancers to leap over the bars without getting their feet or ankles caught, or resting two feet on a bench at the same time, or falling off. It was dangerous, with a real possibility of permanent injury.
Ignore the flashes of gold, the memory of the slithering tones of the sandpipe, the dark, hot, tropical air. This was not a performance before the ignorant; I was not striving for the coins needed to keep Taro and me in clothing and food. This was a competition _—_ that was all _—_ and what was at stake was my pride.
Listen to the drums. The drums that are actually present, not the ones you remember. Let the solid beats mingle with that of the heart, warming it and the blood it sent racing. Let it fill your chest and mind and drive your thighs and feet.
Ignore the recorder. The recorder shouldn't be there. It was fighting the drums. The drums were a steady beat. The recorder was sinuous and entrancing. Why did they have a recorder? Did they know what that could do to a Shield? Were they trying to drive Shields out of the competitions?
It was insidious. It coiled around my spin and slid into my mind, clouding my eyes. And suddenly, I was transported. The hot, moist air, the flickering torches, the desperation of knowing my survival relied on how well I danced.
I was there. My arms curling. My hips swaying. My heart pounding so hard I could feel it in my throat. I lost sight of my opponent. I forgot he was even there.
I didn't know I had missed this.
The recorder sounded so good with the drums. Why had no one thought to mix those two before? Why hadn't bench dancing changed at all during my life? It was glorious to add the recorder, to add the hips and the arms, to add color. Why had it taken people so long to realize that?
And then, just as suddenly as it had started, it ended. My brain full of the echoes of the music, I half stumbled to the ground, panting. As my eyes cleared, I realized people were staring at me. My opponent, the stalkers handling the bars, the other competitors, the adjudicators, all of them looking at me like I'd lost my mind. As I clearly had.
Oh my gods. What the hell had I been doing?
One of the adjudicators cleared her throat. "Shield Mallorough is disqualified for . . ." There was a pause as she clearly strove for appropriate words. ". . . unsanctioned maneuvers."
I was pretty sure that wasn't anywhere in the rules, but I was too humiliated to object.
I slunk to the competitors' bench to wipe my feet and strap on my boots.
"You're in fine form," a voice said from behind me, "but I'm still happy I didn't wager on you."
Of course I didn't need to look up to know who it was. "You're making a habit of this, Doran." And wasn't that annoying? I'd told him I would contact him when I had time, I didn't contact him at all, and he caught me at a bench-dancing competition. He could be forgiven for wondering why I had time to dance but no time to send him a note.
Plus he'd seen me act like an idiot.
"I like to watch you dance," he said, sitting beside me on the bench, but with his back to the competition. "Was there something different about it today?"
He was being kind, or he was subtly teasing me. Either way, it just made me more embarrassed. "I thought bench dancing wasn't your sport."
"It isn't. I like to watch you dance."
I shot him a glance. "I hardly look my best on such occasions."
"It's not about your hair being perfect, or wearing a gown of the latest fashion. It's like you let the real you come shining out when you dance."
What the hell did that mean? "I'm always the real me." I didn't play games.
"No. You're always so worried about keeping your face blank and your voice moderate and your eyes . . ."
Dead? I'd been accused of having dead eyes before.
". . . unrevealing. It's admirable, of course, and I assume it's necessary for your work. But when you dance, it's like you forget all that. You become what you really are, at the core. Full of fire and drive."
"I hate to break it to you, Doran, but the reserve you so generously described" _—_ the prat _—_ "is the real me."
"It was imposed upon you by your training," he insisted. "Just like healers are supposed to give only good news and not react when things go wrong. Just like barristers are always supposed to look like they expect everything that happens even when they've been shocked six ways from rest day."
I cocked my head to one side. I didn't know anything about healers or barristers but I could ask, "Do you assume all Shields are alike?"
"Well, I _—_ " He cut off his words, abruptly.
He did. He really did. How very disappointing. How could any sensible person think that every member of a profession had the same personality? That just defied logic.
"You're the only Shield I've ever met," was how he tried to save himself.
That was easily fixed. I could introduce him to the other Shields of High Scape. I wouldn't, though. I wouldn't encourage his entrance into that part of my life. The fact that he had nothing to do with the Triple S was one of the things I liked about him. I needed people that kept me in touch with the world outside of the Triple S.
On the other hand, ignorance could be tiresome. "Don't assume you know the whole from meeting the one," I said.
"Yes, ma'am," he answered, not chastened at all. "Do you have a watch today?"
"Not until late this evening."
"So you have time for a picnic with me in Gray's Park?"
I had time. I just didn't know if I had the inclination. I liked him. A lot. And if Taro and I hadn't started sleeping together on Flatwell, I would have no problem accepting his invitation. But things were different, and I had to tell him that, and that was something I would like to put off for a while. "I need to wash up and change," I warned him, hoping it would discourage him, for the day, at least.
"That's perfect," he said. "It will give me time to gather our meal. I'll pick you up at your boardinghouse, shall I?"
Oh, aye, that was a brilliant idea. Someone else might be there, and I didn't know which would be worse, Taro or someone who would fill Taro's head with all the wrong ideas.
And the fact that I was worried irritated the hell out of me. I had every right to see a friend anytime I wished to, damn it.
I wouldn't hide that I was seeing Doran. In fact, I would make sure to tell Taro about it, because I knew Taro wouldn't see Doran as just another friend of mine, and sneaking around would only make it worse.
Of course, maybe I was being presumptuous. It was possible Taro really wouldn't care. It was possible he would even be relieved.
Since returning to High Scape, I'd felt a little buried in Taro. He was my Source, the most important relationship I would ever have. But that shouldn't make him everything, and it felt to me that that was what he had become. We spent most nights sleeping together, me changing in my bedchamber before joining him in his, we usually ate together, and then there were the endless watches. It wasn't healthy for me, and it would leave me at a total loss once Taro tired of me.
I wanted Doran to remain a part of my life. That would depend on how he handled my news regarding Taro, and my decision that anything between Doran and me would have to be nothing more than friendship. But only in my nightmares would any part of that discussion take place in Taro's presence. "I'd prefer to meet you at the park. I'm sure to finish before you do, and I'd rather wait at the park, where there's something to look at, than wait around at the residence looking like I'm being stood up. You have no idea how the others would torment me."
Doran probably didn't believe my reasons, because he wasn't stupid, but being a proper gentleman, he agreed to the lady's wishes. He bowed and left.
It turned out that none of the others was at the Triple S residence, not unusual at that time of day. So I quickly bathed and dressed in the same kind of clothes I normally wore when I wasn't doing anything in particular, a comfortable long skirt of sturdy cotton, a loose cotton shirt of a creamy color, and my hair left free. I was going to be comfortable, physically if not emotionally.
If I had any discipline left at all, I wouldn't worry about being emotionally comfortable.
I practiced what I was going to say to Doran as I walked the streets toward the park. I lectured myself on the importance of being calm and not falling into any behavior that may seem leading or suggestive with him. I told myself that I was an idiot for meeting him at all. I then argued with myself that I was allowed to have friends, whether Taro approved of them or not. And I wasn't betraying Taro because I wasn't sleeping with Doran.
If we had never gone to Flatwell, Doran and I would have continued on the path we'd started, into something steady and possibly permanent. I knew, without a doubt, that nothing would have happened between Taro and me had we not been so far from home and so very miserable about it. This thing I had with Taro, it was just a fancy, and like all fancies, it was insubstantial and would fade in the hard light of reality.
When that happened, I would need as many friends as I could find, people to be with so I could be away from Taro when I needed to be, when he was off somewhere with his new lovers. I knew my limitations. I wouldn't be able to maintain an emotional balance after Taro had tired of me without Risa to listen to me complain about him, and Doran to distract me from it all. Provided, of course, he was willing to fulfill that function.
Of course, if I had any self-respect at all, I would break it off with Taro immediately, instead of waiting for him to do it.
But I didn't want to do it. I couldn't. I wanted to enjoy this wonderful thing I had with him, for as long as it lasted. Once it was over, I knew I'd have nothing like it again.
At the same time, I couldn't dive into it wholeheartedly. It would end. If I put all my belief into it, once it ended, I wouldn't be able to bear it.
I had to be careful.
Engrossed in my internal argument, I could perhaps be forgiven for missing the early signs that something unusual was going on around me. The purple smoke, the hissing sound, and the windows cracking in the building I was passing all escaped my notice.
But then the roof blew off. I noticed that.
My internal argument halted. I stared at the trembling building. Made of wood to allow give in case of natural events. The slats of the roof blown high and dropping down, deadly little missiles falling down on the heads of people not quick enough to dodge out of the way.
Then the windows blew out, and the purple smoke became mixed with black. It was only then that I heard the screaming.
I happened to be closest to the building. I could feel the painful heat of fire. I was stupid enough to run to the front door, screaming as my palm was scalded by the handle.
I was pulled away by firm hands on my shoulders, and a long leg kicked out and forced the door open. Smoke poured out and the heat was searing.
"Hell," muttered the stranger beside me. He took a few steps back. "You're a Shield, right? Can you do anything about this?"
"No, because I'm a Shield, not a member of the fire brigade." Normally, I would have stopped my response after the first word, but my hand hurt, and it was bloody hot. I was not at my best.
There were screams coming from within the house. Damn it.
I was not a member of the fire brigade. I'd never aspired to be. But the fire brigade was not there. My eyes blinking with tears, I ducked low and through the door, because I was a complete idiot. I crouched. Near the floor the smoke was lighter and it was easier to see.
I didn't waste much time looking at the furniture. My eyes were drawn to a woman dragging two young children toward a set of stairs that was across the room from the door. In other words, away from the way out. Stupid woman.
I looked up. There was a hole in the ceiling, clean through the second floor. I could see the sky through what remained of the roof. What could do this?
Gods, the heat was unbelievable. I felt as though my skin was blistering off my face, my eyes boiling in their sockets. But I couldn't leave while those people were in there.
The older child was close to the woman in size. She pulled away to make for the door. The woman astonished me by grabbing the collar of the child's dress and yanking her back. I wanted to shout at her for that bit of stupidity, but the air felt too dry to talk.
This caused the woman to look up and notice me. She released the smaller child to throw a hand out at me. "No!" she shouted. "Stop!"
What the _—_? I looked to the floor at my feet, searching for the gaping hole of fire I'd obviously missed the first time around.
What I saw was a circle of white powder on the bare wooden floor, symbols that I didn't recognize drawn within the circle, a knife with a white handle, a knife with a black handle, and a small silver bowl with a red liquid within. Oddly enough, nothing within the circle had been disturbed.
It was nearly the size of the floor, and the room couldn't be crossed without crossing the circle as well.
I stared at the woman, stunned. She was trying to cast a spell? And when her house caught on fire, she was more concerned about preserving her precious little circle than the lives of her children? Look at her, herding them up the stairs, with flames reaching down from above. Heartless bitch.
Some people should not be allowed to breed.
I swept at the circle with the length of my foot, the first step to crossing the room.
And immediately, the flames and the smoke, both the purple and the black, were gone. The noise dove into thick silence. Even the smell disappeared.
The damage remained, though. The shattered windows, the holes in the ceiling, the scorches on the walls. I couldn't quite understand what was going on, but before there was chaos, and suddenly the only sound was the crying of the two girls.
I felt so annoyed, so restless and jittery, that I wanted to charge across the room and slap that woman.
The man who'd kicked in the door stepped inside. "You've been casting!" he gasped, pointing at the floor.
It didn't make sense. There were no candles within the circle. If there had been, that would have explained how the fire had started. Though not how it had ended so abruptly.
There had to be some explanation other than my breaking the circle. There absolutely had to be. Just because I couldn't think of one didn't mean one didn't exist.
Damn, my hand was killing me.
"Get out of my house!" the woman shrieked. She was a pretty thing, I noticed almost absently. Very fair skin, wide blue eyes, true blond hair. "You have no right to be here!"
"Your house was on fire," I pointed out, just in case she had failed to notice it herself.
"And now it's not."
All right, so not unobservant. Just really, really stupid.
"Get out!" she shouted, and there were tears in her eyes. "This is my house! Get out! All of you, get out!"
All of us? Oh. Others were looking in through the broken windows and the door.
I looked at the two girls, about six and twelve, I guessed, with the white blond hair and blue eyes of the woman I assumed was their mother. They wore shapeless white dresses, possibly pristine prior to the fire, and their feet were bare. They looked confused and frightened, and although their mother clutched them close, they didn't appear reassured by her presence. At least they weren't idiots.
"Get out!" the woman screamed again, her voice cracking.
I didn't know what else to do. It was her house. I didn't really want to have to help her clean up the place. And now that the fire was out, however creepily that had been accomplished, whatever was going on in the house was none of my business.
The man who'd come in with me had a different opinion. "I'm getting the Runners!" he threatened before charging back out of the house.
"Go ahead!" the woman shouted after him. "You!" I found her staring at me. "Out!"
Why was I standing around here? If anyone needed saving, and I had a feeling all three of them did for different reasons, they didn't want to be relying on me.
Well, maybe the girls did. They didn't look like they were comfortable with their mother. But there was nothing I could do about that, either.
I walked out of the house. There was a crowd outside, waiting for the Runners to show up, no doubt. There was still no sign of the fire brigade. I wondered if they would be showing up at all.
An older man tsked and touched the back of my right hand. "Best have that looked at, lass," he said.
There was a truly ugly burn in the palm of my hand and along the insides of my fingers. It stung, and I didn't want it getting infected.
Still, I couldn't help lingering a little, looking at the mess that had been made of the house. Really, how had she managed that? And then the scowling woman slammed the door shut.
I did leave once I heard the Runners approaching. I didn't want to get caught up in their investigation. I made my way to the closest hospital, waited forever and met with a healer who slathered my palm and fingers with salve and wrapped my hand to uselessness. I was instructed to wash my hand daily and change the bandages every time I washed.
Life would be just too unusual if there wasn't something wrong with me. And it gave me something to think about other than the fact that that woman appeared to have created a fire by casting a spell. Spells weren't real. There had to be some other explanation.
Chapter Eight
I'd forgotten about Doran and our picnic. The whole thing with the fire had driven all such mundane ideas out of my head.
I was reminded of them with painful immediacy when I returned to the Triple S residence. Because Doran was in the kitchen, which was, as usual, the first place I went. I needed some water to clear out my throat. Doran was seated at the long wooden table, picnic paraphernalia piled on the chair beside him.
Taro was at the other end of the kitchen, leaning back against one of the counters, a mug of something in his hand. Oh, lovely.
Not that they appeared to be arguing, or sitting in stony silence, or anything else so adolescent. I seemed to have interrupted a polite conversation, Doran with his own mug of what smelled like tea resting on the table. The two men looked comfortable and casual, and would have fooled any stranger.
I wasn't a stranger. Doran was wary, but hiding it well. Taro was furious, and the only way to tell was by looking at his eyes. I was pretty sure it wasn't Doran that he was angry with. I probably held that honor.
Yes, this was going to be a treat.
But Doran noticed my hand and jumped to his feet, crossing to the door. "What happened?" he demanded, and I backed a step away before he could touch my hand.
"I passed a building that was on fire, and there were people inside screaming, so I _—_ " Did something really stupid.
"Decided to join the fire brigade?" Taro asked coolly. He didn't move.
"It's just a burn. Not a bad one. No harm done."
"Your face is very red."
"That will fade."
"So you're all right?"
"Aye."
"And up for talking, are you?"
Well, I'd rather skip the conversation I feared was coming. I really didn't feel like dealing with the two of them at once. Should I take the coward's way out and claim I was shocked and tired and needed to lie down?
And suddenly, I was irritated. An ambush, that was what it was. How dare Taro try to call me to the carpet? I was doing nothing wrong. "Certainly."
"I was concerned when you didn't meet me at the park," said Doran.
I'd already explained that. "I was at the hospital for a while." I waved my hand. "I do apologize for making you wait. And I don't know that I'm up for a picnic right now. I'm not really hungry."
"Of course," said Doran. "I understand. Perhaps tomorrow?"
I wanted to put him off, tell him I'd discuss it with him another time. But that was cowardly, too. And the firm line of Doran's mouth indicated he knew of the temptation, and he probably wasn't going to let me give into it. Not quietly, anyway. Well, fine, then. "Yes, I can do that."
He smiled only slightly. There was something almost smug in the expression, something I had never seen in him before. I didn't care for it. "The third hour of the decline?"
"The third hour."
"Excellent. I'll pick you up here." Another challenge.
"That would be lovely." Challenge accepted.
He nodded. "I'll see you tomorrow, then." He looked at Taro. "It was a pleasure speaking with you again, Shintaro."
"Aye, it was, Doran," Taro responded.
Never had I seen the men so chilly with each other.
I saw Doran out, well aware that the more difficult man was the one still in the kitchen. He was going to get all emotional. It was important that I stayed calm. Because there was no doubt he was misunderstanding all of this.
Doran was no rival for Taro's affections. Doran had never inspired in me the depth of feeling I had for Taro, and I didn't believe he ever would. It wouldn't be fair of me to pretend otherwise, and I didn't want to play with people.
I needed to make sure both men understood that. It would have been so much easier if I could have met with Doran as expected, had been given the opportunity to make everything clear. Now everything I said was going to sound weak or insincere, because Doran had shown up with clear expectations, and Taro had known nothing about it. Damn it.
Taro was pacing in the kitchen, all pretense of courtesy gone. "What the hell are you playing at?" he demanded.
_Oh, let's dive right into the irrational part of the discussion, shall we?_ "I think we should go somewhere more private." I did not want to be providing the evening's entertainment for the other Pairs.
"As loud as I'll be getting, there'll be no privacy no matter where we go."
Wonderful. "I see."
"Don't you dare!" he snapped, pointing a finger at me. "Don't you even think of pretending you're the mature, stable one when you're the one stepping out!"
"I am not stepping out."
"Picnicking with Doran? I beg to differ."
"Differ all you like. I have every right to see my friends."
"He's not a friend! He's in love with you!"
I nearly laughed. "Don't be ridiculous. You don't believe that any more than I do."
He didn't deny that, but he glowered at me, and I waited for another explosion.
But there was no explosion. He pulled in a deep breath through his nose. Then he closed his eyes and pinched the bridge of his nose. He let the breath out through his mouth. He dropped his hand and opened his eyes.
The glower was still there, but his shoulders were a little lower.
"You can't claim he is a friend like any other."
His voice was almost calm, but the 'risto in his accent was coming on thick. Never a good sign. At least, not when it was directed at me. I enjoyed it when he used it on others.
"He is now." Or he would be, once I was able to explain everything to him.
"You would go on a picnic with Risa?"
"Sure." Though I couldn't see Risa enjoying a picnic. It seemed a little tame for her.
He snorted in derision. No doubt he agreed.
"Have you slept with him?"
My mouth dropped open. It had been doing that a lot lately. "Of course not!"
"Before we went to that damned island?"
"That's none of your business!" So much for staying calm.
"So you did sleep with him!"
"No, I did not!" I snapped. "I don't _—_ " I cut myself off before I said something really, really stupid.
The way Taro's eyes narrowed, though, told me I was too late. "You. Don't. What."
It was interesting how much words relied on tone. Technically, his words should have constituted a question. The flat tone made it a dare. He was daring me to finish.
Sleep with people at the drop of a hat. That had been the end of my aborted sentence. That would have sounded like an accusation, which wouldn't have been my intention. And I knew Taro didn't sleep with absolutely everyone he met. I really did. But sometimes the habit of speaking as though I thought he did snuck back up on me. "I don't see how that matters now, anyway."
"It matters a lot. It matters that you were slithering off to see him!"
"I wasn't slithering."
"What do you call going off to see him without telling me?"
"He asked me to go to the picnic this morning at the bench-dancing match. I came here to change, and you weren't here. If you had been, I would have told you." I would have. "If we'd had a picnic and I came back, I would have told you then."
"Why are you seeing him, anyway?"
"I like his company."
His hard-fought-for calm was slipping. His hands rose to clutch at his hair. "I can't believe you! You say it like there's nothing wrong with it!"
"There is nothing wrong with it!" I searched my mind for an explanation as to why there was nothing wrong with it. It was sometimes difficult to explain the obvious. "Where were you today, when I came back from the match?"
"Racing track!" he growled.
"All alone, I assume," I said sarcastically.
"With friends."
"So what's the difference?"
"None of them want to sleep with me!"
I snickered. I couldn't help it.
"Oh, my mistake," he said with heavy sarcasm. "I'd forgotten. That's the only reason anyone would want to spend any time with me, right? To see if they have a chance."
"No!" I was horrified. "That's not what I meant _—_ "
"Because all I am is a face and a penis, right?"
After all the times I had told him he was smart and talented and generous and kind, and he had the nerve to say that to me? "I didn't say that and I don't believe that."
"Doran is not just a friend and he's interested in more than just sleeping with you."
Damn. I'd kind of thought we'd gotten away from Doran. "It doesn't matter what Doran's interested in. I'm only spending time with him because I enjoy his company."
"Does he know that?"
"He would if I'd had a chance to meet with him today."
"You were going to tell him you never wanted to see him again at a nice romantic picnic?" Taro asked skeptically.
"I have no intention of telling him I don't want to see him again. I will tell him that any expectations he might have had before I went to Flatwell will have to be put aside."
"Oh, I'm sure he'll be delighted to accommodate you." He rolled his eyes. "You're such a child."
How very obnoxious of him. "Then, if he can't accept that, yes, I won't be seeing him anymore."
"I can tell you right now that he won't agree to _—_ whatever you're expecting to do. Not if he's any kind of man at all. So what you will do is forget all these unnecessary steps you're talking about and you will send him a note canceling your plans for tomorrow and that will be the end of it."
"You will go to hell!" How dare he try to tell me what to do? This was between Doran and me.
"Or do you like having him dangling? Hm? Maybe you like having two men acting like idiots over you."
"Of course not!"
Well, maybe a little.
No, of course not.
"After all, it can't be a common thing for someone like you."
"Someone like me?" I echoed without thought. Definitely without thought. It wasn't as though I actually wanted him to elaborate.
"Someone so rigid. With your blank eyes and your flat voice. So determined to show the world that she doesn't give a damn what we think and she doesn't need us and we can all go to hell!"
"This is interesting," I mused. "You perceive that I am insulting you, so you have permission to verbally abuse me."
"I notice you're not actually denying what I'm saying."
"Would there be a point?"
He made another grab for his hair. What was his problem? I was, for the most part, refusing to rise to all of the insults he was throwing my way. I was answering all of his questions honestly. I was doing my best to prevent this discussion from degenerating into something really nasty.
And he was being ridiculous. I would wager my last coin while stuck on Flatwell that he hadn't acted this way with any of his other lovers. And, aye, I could see why I was different, in that I was his Shield and the Pair bond always made things a little weird. Still, he neither expected nor wanted the relationships with his other lovers to be exclusive. So why did he have to act like a domineering lord of the manor from a bad melodrama with me?
Perhaps I could end all the posturing if I pointed out what should be obvious, even to him. We simply were not suited for the long term. He was Karish, and as much as he tried to deny it, he was unique. He was dazzling and exciting, such characteristics unusually coupled with generosity and genuine kindness. He deserved someone who matched him.
That was not me. I was quiet and steady and serious, interesting only in my connection to him.
But I dared not say that to him. Much as he accused me of purposely misunderstanding him, he just as often did the same with me. If I gave him my description of him, he would accuse me of thinking him nothing more than a pretty shell. If I gave him my description of me, he'd either agree with me, or think he needed to reassure me. Neither reaction would assist in this situation.
"Please make yourself comfortable," I heard Veritas say, because he was speaking much more loudly than he usually did. "I will see if they are in."
I wondered how many arguments Veritas had overheard in his years of working at the Triple S residence. He'd probably heard things much stranger and more damaging than what Taro and I had been shouting at each other. That didn't make me feel any better about the fact that I knew he'd no doubt been listening to us.
Ben showed up at the kitchen door. "That Runner woman is here to see you, Shield Mallorough."
Interruptions were wonderful things. "Thank you, Ben."
"What happened to your hand, Shield?"
I'd forgotten about it. "A burn."
"What kind of burn, if I may know?"
What difference did it make? "Metal. Heated metal."
"A healer has seen to it?"
"Aye."
"And has given you a poultice for it?"
"He spread something on it."
"But gave you no more for the following days?"
"Ah, no." Should he have?
Ben tsked. "Come see me tomorrow morning, after you have washed but before you leave for the day. I will refresh the treatment."
"You have healing skills? I didn't know that."
"There is much you don't know of me, if you'll beg my pardon, Shield Mallorough."
I thought that was an odd thing for him to say. It was almost as though he was accusing me of something. I didn't think that was like him.
His words did make me wonder that Taro hadn't touched me once since I'd come to the residence. Not only was that not like him, in general, but he knew if he touched me, it would ease any discomfort I might be feeling. That was sometimes one of the characteristics of the bond between a Source and Shield. We were lucky enough to have it.
I could ask Taro to touch me, I supposed. After that ridiculous argument, though, I wasn't going to ask him for any favors.
"Thank you, Ben." I headed to the door.
"I shall bring you some wine, shall I?" he offered.
"Thank you. That would be lovely." I said nothing to Taro. I had no desire to spark off the argument again. I was aware, however, that he wasn't following me from the kitchen.
Risa was lounging on one of the settees, examining a small statuette that had been left by some long-gone resident of the house. It was the naked form of a youth, stretching one hand up over his head as though reaching for something. It was a pretty piece, but I'd never understood what it was supposed to mean.
"Good evening, Risa," I said. "I am pleased to see you here."
"Well, I'm not here for pleasure, I'm afraid," she answered. "Business."
Oh, that was never good. "What kind of business?"
She gestured at my wrapped hand. "Of course, the fire isn't in my docket. But it seems everyone at Headquarters thinks I'm the one to look into things when you're involved. And you do end up involved in an awful lot of crimes."
"I don't commit crimes," I huffed. Then I froze. Because I'd killed Creol. That was murder. I couldn't believe I'd actually forgotten that.
Risa didn't seem to notice that part of my reaction. "No, but you do always seem to be in the vicinity when crimes are committed," she said.
That was an exaggeration. All sorts of crimes were committed in High Scape and I was nowhere nearby. "What do you want to know?"
"Just tell me everything."
"Oh, is that all?" I did as she ordered, describing my knowledge of the fire with as much detail as I could. During that time, Taro left the house without pausing to say where he was going, and Ben served us red wine. "But is that really a crime? Accidentally setting fire to your house?" Criminally stupid, maybe.
"No, that's not what the strike is. Pretending to cast spells is a crime."
"I don't think she thought she was pretending."
Risa laughed. "The only alternative is that she was actually casting a spell. A spell that worked. Are you saying it worked? Do you want me to quote you on that?"
"No, thanks all the same, but just out of curiosity, what is the difference in the punishment?"
"The difference?"
"Between pretending to cast a spell and actually casting a spell."
Risa's expression suggested she thought I was suffering some form of delusion. "There's no such thing as truly casting spells."
"Well, no, of course not, but it doesn't seem to make sense that pretending to cast is illegal while actually casting isn't."
"Of course it makes sense. One is possible and the other isn't."
I understood her point, but there was a fundamental lack of logic involved that disturbed me. "So why is pretending to cast illegal?"
"People use it to steal."
"How?"
"You know, selling love potions and things like that."
"How is that different from medicine men who sell phys icks that don't work?"
Risa scowled. "I don't write the laws, Dunleavy."
Did that mean we couldn't discuss the inconsistencies? Apparently so.
"Speaking of fake spells, have you heard what those freaks digging up ash groves are doing with the ashes?"
"No, and I don't really care to." I didn't want to imagine doing anything with the ashes. Just the act of digging them up was disgusting.
She gave me a toothy grin. "They mix them into a bowl of the liquid of their choice and rub the mixture on their skin in preparation for casting a spell."
My throat closed up in protest and I put down my glass of red wine. "You can't be serious."
"We've learned that the victims are all people who were considered particularly fortunate by those who knew them. These spell freaks dig up the ashes and try to absorb the luck of the dead person through their skin."
"That's revolting." My stomach twinged just at the thought of it.
"Or they sell them."
"They sell the ashes?"
"There's quite a market developing. People dig up the ashes and sell them to a slew of customers who are desperate for luck. There's even talk that they're running out of people to dig up."
"This is insane." And also none of my business. "Are you supposed to be telling me this?"
"I'm not giving you any specifics of any particular investigation." Risa shrugged. "But if it disturbs you, we can talk of other things. Why did Shintaro go storming out of here?"
I could be so stupid. I kind of set myself up for that. "Storming?"
"He didn't tell you where he was going."
"He doesn't always."
"Of course he does."
How in the world could she claim to know that? "You don't live here, Risa."
"So? Seeing habits isn't hard for someone with my job."
I sighed, because she was right. "We had an argument."
"About what?"
"Do you honestly expect me to tell you?"
"That's what friends do."
Not since I left the Academy. I'd never found talking about such things helped me feel any better, nor did they improve the situation in general. Besides, Taro would kill me. "So if I were really your friend, I would feel obligated to tell you, is that it?"
"No, that's not it. Sometimes you can be terribly stupid, Dunleavy."
Oh. So if I were truly her friend, I would want to tell her. Really? She wanted to play that game? "How much debt are you in right now?" See? I could ask highly inappropriate questions, too.
"You have a nasty habit of attacking people when they ask questions you don't want to answer."
"Isn't that what you're doing now?"
"How about this, then? I'll answer your question after you answer mine."
Perfectly fair, yet that wasn't going to happen. Really, Taro would see it as a betrayal, were I to talk about our affairs with Risa or anyone else, and I wouldn't blame him. I assumed he didn't talk about us with others and if I was wrong and I found out he did, I'd have to kill him.
Of course, once he was done with me, things would be different.
I felt like I'd offended every person I'd encountered that day. It was all catching up with me, the craziness and the people I'd infuriated for no good reason. I didn't want to add Risa to the list, not that day. "So both of us will have to survive having our curiosity unsatisfied," I said. "Want more wine?"
She grinned and I felt relieved. "Sounds great," she said. "But if you're not going to tell me about Karish, the least you can do is show me the costume you had to wear while you were away."
I rolled my eyes. "Are you crazy? Believe it or not, I don't go out of my way to humiliate myself." That always happened by accident.
After Taro's reaction to the costume, I wasn't going to show it to any other Northerner. Ever.
"So you still have it."
"No," I lied.
"I could go to your room and find it."
She wasn't seriously suggesting she would go rifling through my things. "You don't know which room is mine."
She actually pouted. "You won't wear it for me?"
"No."
"Do you wear it for Taro?"
What an odd question. "No."
She grinned. "You've just given me another reason to get you drunk."
And she'd given me another reason to kill Taro. He was the one who'd told her about the costume in the first place. It should have been _me_ who stormed out on _him_.
Chapter Nine
Alone in my sitting room, wearing a light night shift, I sat cross-legged on the floor. I had pushed the furniture close to the walls to give myself plenty of space. I had lit two candles and set them on the floor before me. It was late, everyone in the residence asleep. It was quiet, not a sound coming in from the streets. I stared at the taller candle, letting the golden glow fill my eyes and wrap around my mind.
I didn't usually rely on such artificial relaxation techniques. I couldn't remember doing it even once since leaving the Shield Academy. It seemed to me that it must be unhealthy to try to put the mind asleep while it still had thoughts to work over. But I couldn't sleep, and I really, really wanted to. My mind was spinning so hard with so many thoughts, it was stiffening my muscles, shallowing my breathing and creating imaginary explosions in my ears. The thoughts in my mind were not resolving into any conclusions, just circling uselessly. It made sleep impossible, and it made me anxious.
The shambles of my mind made it clear how very sloppy I had gotten since leaving the Academy. As time went by, I seemed to have less and less control over myself and my emotions and how I expressed them. That was dangerous. A Shield without self-control was useless.
So, out of desperation, I had dragged out some old half-forgotten lessons about finding peace, and the tools to be used when simple deep breathing wasn't enough. So, two candles. One tall and unobstructed. The second low, to accommodate the small metal arch that covered it. On that arch rested a small bowl filled with water and a few drops of dark bark essence. Staring at the flame of the tall candle, breathing in the soothing scent and blowing it out slowly, I followed the thoughts of serenity.
_The candle brings light._
_Lights shows truth._
_Truth brings knowledge._
_Knowledge crafts balance._
_Balance brings peace._
_Peace crafts clarity._
_Clarity is light._
_The candle brings light._
A simple sequence of words, yet I couldn't keep them straight. I kept remembering other words that didn't belong. Colors flashed behind my eyes, soiling the golden glow of the candle. The explosions increased within my ears, and my heart wouldn't slow down.
So, eventually, I gave up and stared at the candle and got angry at the fact that everyone else was able to sleep.
I heard the door to the corridor open with just the faintest of creaks. I looked up.
It was Taro, of course. His hair was loose, his earlobe was bare, and he was wearing a nightshirt that covered more skin than mine. He slipped into the room and leaned back against the door after closing it. "Are you all right?"
"Of course."
"You're sitting on the floor in the middle of the night."
"Is there a better time to sit on the floor?"
He tsked and crossed the room so he could sit on the floor on the other side of the candles. He had in one hand some crumpled paper, and I wondered if he'd received another letter from his mother. That always put him in a precarious mood. "What's wrong?"
"Nothing serious." I didn't want to talk about it. It was all so childish and embarrassing.
"I will decide what is serious," he declared.
"Will you?"
"Indeed."
That made me smile, for a moment. "Do you ever get the feeling that the Academies don't really prepare us for the real world?" Though perhaps I was speaking too broadly. The Source Academy might be a very different experience, as Sources spent a great deal of time outside the limits of their Academy in order to practice channeling different kinds of events in different environments. While still closely guarded to avoid spontaneous bonding, it was a lot more experience and exposure than that granted to Shields.
"Yes," was his answer. And that surprised me. I'd really thought it was just me. "I think they think they're being thorough. They tell us of things like illness and crime, the dangers of Shields in the grip of music, the friction that can accidentally happen with regulars."
"Aye," I said. "That's pretty much what they taught us. I guess there isn't a lot more that we can expect from them. But they don't tell us much about what it feels like when other Shields and Sources, people you live and work with, die. Or when the city you're living in suffers some kind of devastation that Sources and Shields can do nothing about."
"Those aren't trivial concerns," said Taro.
But those were concerns from a while ago. "The Empress dies. People are digging up ashes because they believe in luck. And spells. A woman burned her house down because she thought she could cast a spell and have it actually accomplish something. It's like people are going crazy. Again."
"Or still," Taro suggested.
"Still?"
He shrugged. "It's not like they were sane while we were gone and only got crazy after we came back."
Now, there was an idea. Maybe we had brought the craziness with us. "I think I like that idea better than the possibility that they're just getting progressively crazier with no end in sight."
"There's always an end," he said.
"Aye, but some ends are better than others."
"We're not going to get into a philosophical discussion, are we?"
"I certainly hope not."
"Air is better blown to water," Taro added.
"Exactly," I said, though I didn't really know what that meant. I assumed it was something relevant.
"How can a school prepare you for murderers and mad-men?" Taro asked. "Friends who would betray you to your death. People in authority who use their power to perform the most unnatural acts. People hating you because you can't do things you aren't supposed to be doing anyway. That's a lot to expect of a school."
"Maybe if we hadn't gone to the Academies, a more normal upbringing would have better prepared us for the real world."
He snickered. "Depends what you call normal, love."
True enough. If some smart and plucky servant hadn't recognized him as a Source _—_ and then promptly gotten fired _—_ Taro might have spent the rest of his life locked up somewhere, by parents who thought he was embarrassingly mad. And that was assuming some event didn't occur and he didn't kill himself trying to channel it without a Shield.
If I'd been a regular, I would have been sent to a boarding school that, from what my brothers told me, was far more rigid than the Shield Academy. That probably wouldn't be considered too normal an upbringing by the average regular, either.
"Speaking of which," said Taro. "I received today a letter."
That was never good.
"From Her Gracious Duchess of Westsea."
It took me a moment to remember who that was. Not Taro's mother, the woman I always associated with that title. The current Duchess of Westsea was a cousin of Taro's, and the woman to whom he'd given the means to acquire the title so he could avoid taking it himself. "Does she write to you often?" I didn't really like the idea of his maintaining contact with that world. It might manage to lure him back.
"This is only the second letter, the first being to thank me for helping her get the title and offering her services should I need anything. This one is giving me details about how things are going. Did you have any idea that there are all sorts of fishers belonging to the estate?"
I'd never thought about it, but it made sense. "It is on the water."
"Aye, but how can there be both fishers and farmers on one estate?"
"What can I say? It's a crazy world." I found it more surprising that he hadn't known that about his own land, though perhaps I shouldn't have. He'd rarely been allowed out of his room before he was sent to the Academy. His parents had never visited him at the Academy. Perhaps they hadn't written much, either, or hadn't bothered to give much detail when they had.
He tapped me on the forehead with the paper as a mild reproof. Then he unfolded it. "It was a lengthy missive, but I'll just read the interesting bits." He squinted his eyes in the dim light, and I almost scolded him for it. He'd wrinkle his lovely skin. "'I have to confess, cousin,'" he read. "'You have a reputation in our family for being a lack wit, or at least half mad.'"
I clenched my teeth to keep from voicing my outrage. How dare they? And what the hell did they know, the thick-skulled provincials?
"'But have you met your mother? She's truly insane. And sometimes I think I mean that literally. She will not stay in the dowager house. She invades the manor at any time of day or night, rousing the servants for mulled wine or chilled ale, waking everyone. She tried to give orders to my gardener about the ash on the east corner of the main plot. They have begun to rot, you see, and will collapse if they aren't cut down. And yes, they are beautiful and have stood there for generations, but are virtually dead and will be a danger if they're left to collapse on their own. So I ordered them cut down, and your mother saw one of the gardeners beginning to work at the first tree, because apparently she has nothing to do all day but watch what's going on on my land, and she descended upon the poor man, denigrating his intelligence and his appearance and his lineage, and then she tried to fire him. My gardener. Fortunately, I'd been pretty close myself, seeing if there were any other changes that needed to be made to the garden, and one of the under gardeners fetched me over. When her imperious Dowager Duchess saw me coming, she changed her target and told me my lineage was nothing more than a graft onto hers and my upbringing had clearly been deficient if I thought I could stride into a position that was never meant to be mine and make changes to a place that would be mine only temporarily.' "
"'Temporarily'?" I interrupted. "What does that mean?"
"Well, when you think about it, every titleholder has the title and the seat only temporarily. They don't die with the holder. That's the nature of it. The title's been in my direct line for a few generations, though. Perhaps Her Grace feels that the recent transference to another line is only an anomaly."
"But there is no one else in your line. Unless you've got some other siblings you haven't told me about." Or maybe some children.
He shrugged and resumed reading the letter. "'While she was talking, I took the axe from poor Forrester and swung at the trunk of the nearest tree. The blade kind of bounced off the bark, which was really embarrassing. But it did shut the woman up, at least for a few moments.' "
"I like this person," I commented.
Taro smiled and continued to read. "'She started up again, though, once I gave the axe back to Forrester and he started chopping the proper way, but we all kept working. Or watching Forrester work, as a more accurate depiction of events. And then your mother, overcome by our brutal attack on her delicate nerves, fainted. It appeared a painful collapse. I think she was expecting someone to catch her.'"
I snickered.
"'Her useless little companion pulled out a handkerchief and fluttered it around her mistress's face, and then she looked at the gardeners, outraged. She demanded they carry your mother back to her home. I told her we were all busy and she would have to fetch one of the dowager's servants to take her home. The little companion actually tried to throw some of her own orders around, warning us that the mistress would be most displeased. When that didn't work, she walked off with a loud huff, and it was clear that your mother was miming her entire episode, with the sour expression she pulled at being left there to lie on the ground.'" Taro paused to laugh. "'We kept cutting, and eventually a couple of handsome young men _—_ whose functions within your mother's household I've never been able to ascertain _—_ came and carried her away.
"'The fun wasn't over, though. A few weeks later, my housekeeper came to me and told me she had just had a lengthy audience with your mother, who had given her instructions for the menu, the music and the decorations for the ball I was apparently having.
"'I confronted your mother, who had not only sent out dozens of invitations to this ball in my home and in my name, but had also received all the responses. Which were all positive. And all humor aside, it was a little disquieting to talk to her about it. She really thought it was perfectly acceptable to plan this huge undertaking in my name without telling me about it. Because she believed I was too stupid or too badly raised to know what was required of me as the Duchess of Westsea, or how to go about it. She didn't care that she was insulting me grossly, or that she was engaging in behavior to which she flatly had no right. It was baffling. And frustrating. I've never so badly wanted to hit a woman in my life.
"'So now I wonder if I should revoke the gratitude I expressed to you in my last letter. It seems to me you've enjoyed a lucky escape, and I wonder if I've been had.'"
"Zaire," I breathed. "She's blunt, isn't she?"
"She has to be, if she's dealing with Her Grace."
Actually, it seemed she did have the knack for handling Taro's mother, though it was difficult to be sure from a letter. Maybe she could give Taro lessons. Though, truly, I was hoping we could spend the rest of our lives without seeing the Dowager Duchess ever again.
"She has something to say about our favorite royal, too." Taro flipped to another part of the letter. "'Our Emperor-to-be has sent notice of a new tax against everyone with servants. Can you believe it? I am to pay three sovereigns a year for every adult in my employ. It is sheer robbery, in my opinion, and I have informed him that as he has not yet ascended to the throne, he lacks the authority to tax me, and he can just wait until he has the right before his gets his grubby little hands on my money.' "
I put my hand over my mouth, as it had gaped open in shock. "And she says your mother is mad?" I mumbled through my fingers.
"She certainly doesn't seem worried about making enemies." He folded the letter up. "I think she will be good for Flown Raven. I'm pleased I chose her."
I really wished I had been there when the Dowager Duchess had feigned a faint and no one had bothered to catch her.
Taro reached around the candles and caught the fingers of my right hand, turning it over. "I never really let you explain what this was about."
I didn't want to go back to our discussion of earlier that day. I didn't want another argument. I didn't think I could bear it well, not right then. "This woman set her house on fire trying to cast some kind of spell. She had two young children with her."
"And you rushed in to rescue them," he commented in a soft voice, one corner of his mouth curled into a small smile.
He was just gorgeous in the candlelight.
I shrugged. I didn't know if I was thinking of rescuing anyone. I didn't think I was thinking at all, really.
"And just out of curiosity, what would you have done if I had gone rushing into a burning building?" he asked.
The very idea made my chest tight. "Slapped you up the back of the head," I admitted.
He leaned forward so he could reach around with his free hand and give me a gentle tap on the back of the head.
"But you're different," I said solemnly. "You're so much more fragile than I." I was joking, but only a little. It was sort of true. As a Shield, I was resistant to things like extreme temperature, and pain was much less acute to me than it would be to Taro.
"Fragile!" he roared. He placed the candles to one side and grabbed one of my ankles, jerking me to him so I collapsed onto my back. My admonishment to stay quiet was obscured by giggles as he pulled me beneath him. "I'll show you fragile."
He felt so good. Strange and safe and sure and exciting and so many other things. I loved pushing my hands through his hair. I loved the low tones of his voice whispering against my ear. I loved the taste of his skin.
I loved being forgiven when I acted like an idiot.
Chapter Ten
The next morning, Ben had prepared a white, milky glop to put on my palm, and he appeared to be waiting in the kitchen for me to show up so he could apply it. It did take a bit of the sting out of the burn, and Ben seemed so pleased about being able to do it. I was happy to be able to do something that made him feel appreciated.
Taro was already out, helping a friend change residences. Apparently, the fellow was falling behind on his rent and skipping out. While I couldn't approve of anyone slithering out on their financial obligations, it wasn't unlike an action Taro and I had been considering not too long ago back on Flatwell, and the rent Taro's friend was expected to pay for that hovel was criminal.
Taro's absence meant I got to go book hunting, something that would have bored Taro into incoherence.
All this casting craziness, it had nothing to do with me, and there was nothing I could do about any of it. Screaming at people to start making sense was surprisingly ineffective. But I felt driven to find out where these people were getting their crazy ideas. If nothing else, it had to make for interesting reading.
So I headed out to the nearest print shop, and I asked for materials about spells, and the printer looked appalled. "Of course we don't sell such filth here!" she exclaimed.
"Filth?" I echoed. "How is it filth?"
"Encouraging people to believe in casting. It's unnatural."
I didn't disagree. Believing in casting was unnatural. Still, that didn't mean books about it could be classified as filth. "You print a circular with a mystery series. The descriptions of the deaths are horrible. Really graphic." I didn't like serials, as a rule, preferring to read a full book in a sitting or two. I'd started the serial to which I was referring because I'd thought the writing was unusually good, but the gruesome nature of the murders had put me off.
"That's fiction."
"So's casting."
She looked irritated by the reminder. "That may be, but selling materials purporting to instruct the reader in how to perform spells is illegal."
"Really? Since when?"
"Since always."
"Really?" That was surprising. On the one hand, it kind of made sense, in that if it were illegal to pretend to cast spells, it had to be illegal to print materials teaching someone how to pretend to cast spells. On the other hand, really, where was the harm? "But people are getting it from somewhere."
"Not from this shop, they aren't," the woman declared. "And you should be ashamed of yourself for looking for it. You should set a better example."
"A better example?" I frowned. "Of what? To whom?"
She stared at me as though she thought I'd said something outrageous. "You're a Shield! You are to be a good example of honor and decency for all."
"I see." That was the first I'd heard of it. "Thank you for your time."
I left the shop feeling puzzled and a little disturbed. I didn't like the idea that printed material about casting was illegal. I could sort of see the sense of it, if idiots tried whatever was in a book and ended up setting their houses on fire. But I wasn't an idiot. And I wasn't going to try any spells. I just wanted to know what was being written and read. And I didn't like being told I couldn't see it.
Of course, I knew that just because something was deemed illegal didn't mean it wasn't being done. People were getting the books from somewhere, and book stalls and printing shops were the logical places to look. My search required a lot of stops, a lot of walking and a few carriage rides. I would wait until the shops were empty of customers, which sometimes took some time, and then I'd have to convince the printer that I was no kind of authority and that I had neither the ability nor the desire to entrap anyone. Only once that was accomplished _—_ and it wasn't always _—_ would I be allowed to see the hidden cache of banned goods.
The end result was a collection of pamphlets and small, cheaply bound books with no indication of the contents embossed on the covers. I was given a cloth bag to carry my bounty, so no one could guess that what I was carrying wasn't anything other than respectable literature. I supposed I should have been feeling guilty or nervous or something, since I was doing something illicit. But it didn't feel illicit, and in my opinion it shouldn't have been illicit. I was tempted to sit down on a bench by the road and start reading one of the pamphlets, just to see what would happen.
But I didn't have time to sit and read and scandalize everyone who walked by. It had taken me much longer than I'd expected to find the books. I had to go home and bathe and change for my meeting, my picnic, with Doran.
I was glad that Taro knew about the picnic in advance. And he was apparently fine with it, having gotten over his initial reactionary snit. And so he should be. Nothing Doran did would have any impact on my relationship with Taro, which was more than could be said of Taro's friends.
When Doran came to the residence, he was not carrying the same sort of supplies he had had the previous day. He said he'd left everything at the park, and I wondered if that was a wise decision. Surely there was a good chance of something being stolen.
"I confess, I half expected you to find yourself suddenly unavailable today," he said as he insisted in helping me into the carriage he had borrowed for the occasion.
"Really," I said coolly. "And why is that?"
"Your Taro seemed a little upset with you last night."
He had better not expect me to complain to him about Taro, because that was never, ever going to happen. "He can get nervous if I'm away too long and he has no idea where I am. Things have happened in the past." Though they had usually happened to Taro.
"I hope there was no unpleasantness after I left."
I studied him for a moment. If he had expressed any hope that there was unpleasantness, I would have been unimpressed with him. But his face showed only polite concern. "No, no unpleasantness." And it wasn't really a lie. Not when one took the long view. We had had a little, insignificant spat, and made up for it gloriously.
"That's good," he said, and he seemed sincere.
I wondered if he would feel the same way after I told him what I had to say.
Stepping down from the carriage, I saw a vision that put Taro right out of my mind. Under a large tree with a wide, solid trunk and leafy branches stretching out into a glorious sun-spattered umbrella, the ubiquitous picnic blanket stretched out on the ground in loose white waves. Where it sheered off from my experience was the small white table set up on the blanket, with two white cane chairs. There were two servants, one male, one female, and they either ruined or complemented the look of the scene by being dressed entirely in black; I couldn't tell which. They were moving a little before they noticed our arrival, at which point they stilled into silent attendance.
"Where did you get them?" I asked, nodding toward the servants.
"Borrowed them from my mother."
We sat at the table, the servants waving serviettes into our laps. From side bars they produced covered plates, revealing slices of cheese and skinned grapes.
"Is it really that bad?" Doran asked.
"Excuse me?"
"You look tense, like you're expecting something to be dropped on your head without notice."
I forced my shoulders down, and I knew I was blushing. I could feel it, damn it. "I'm sorry. I just wasn't expecting anything like this."
"Don't you like it?"
"It's lovely." It was conspicuous. It made sure that everyone who entered the park or passed on the road would get a good look at us. But it was definitely lovely.
The cheese was great. Strong and chewy.
"I had some difficulty choosing the menu for today," Doran commented.
"Oh?"
"I was thinking of a light spinach salad," he said.
"Oh." Not my favorite but it would have been fine.
"Followed by a light serving of sliced roast goat."
"Goat meat?" I couldn't help wrinkling my nose at that. The only time I'd ever had goat meat had been at that horrible dinner party at the Yellows estate. "Oh my gods. Are you serious?" That would have been in such bad taste.
He chuckled. "The meal we were enjoying the night we met."
"Or not enjoying, to be precise." It had been a ritualistic meal meant to prepare us to meet whatever gods the Reanists believed in. Meant to make us slow and stupid and easier to sacrifice. And Doran had thought that meal might be something appropriate to serve to me? "You can be a twisted sort of fellow, can't you?"
He grinned. "My dear, sometimes the best results can come from the worst circumstances."
"I suppose." I knew what he was implying, but I didn't want to encourage that line of thinking.
"But given how unpleasant those circumstances ultimately ended up being, I decided to spare you."
"Thank you. I appreciate it." Ugh. Could this conversation be any less witty? What was wrong with me? What was wrong with us? We were usually so much more relaxed than this.
"Dunleavy!" I heard a voice call out, and I looked to the right. The woman I saw, leading two young children by the hand, was a prostitute I'd met through Risa. Doran rose to his feet.
"Zeva." I smiled. "Good afternoon." The woman was gorgeous, curvaceous and blond, her hair tightly pinned back and coiled at the back of her head. "Who are your friends?"
The young woman smiled down at her charges, wearing a warm, gentle expression that I couldn't recall seeing on her before. "This is my daughter, Amber." The little girl dipped into a cute little curtsy. "And my son, Viker." The little boy, he was maybe four, bowed. "Say good afternoon to Shield Mallorough."
"Good afternoon, Shield Mallorough," they said in perfect unison, and it would have been a little creepy if it weren't for the fact that Viker had trouble pronouncing my name.
"And who's your friend?" Zeva asked, an eyebrow raised.
"This is Lord Doran Laidley," I said, avoiding his proper title in favor of his actual name. He could correct me if he didn't like it. "Doran, this is Zeva Smith."
Doran took her hand and bowed over it. "It is a pleasure to meet you," he said.
"It certainly could be," she smirked. Then her eyes widened in alarm and she looked at me. "Dunleavy, I'm _—_ "
I waved away any impending apology. For some reason, the idea that Zeva might flirt with Doran didn't disturb me. He deserved more fun than I could give him.
"I'll wager everyone wishes they had husbands as understanding as yours," Zeva said impishly, quickly over her discomfort. "Allowing you romantic picnics with other handsome men."
Oh lords, spare me.
Doran narrowed his eyes at me. "Really, Lee? You've gotten married in the last couple of days?"
I wasn't sure what he meant by those words, or by the odd tone in his voice. He seemed annoyed. Did he think I had gotten married and just hadn't bothered to tell him? "You'll have to ask Zeva, I suppose. I didn't have a husband the last time I looked."
Zeva pretended to pout. "You're not wearing it."
"Not wearing what?" Doran demanded.
I sipped at the apple wine that had been poured into the goblet before me. It was nice. Sweet.
"I'm surprised you haven't heard, my lord," said Zeva. "Dozens saw it."
I looked at her more closely. Was she trying to cause trouble? She appeared to be teasing, but I didn't know her very well.
I decided to stop any further insinuation or misunderstanding. "Taro had his eye on a matched set of those harmony bobs, and the merchant made us go through a ritual that was supposed to trigger their power. It was in no way a wedding. We are not married and won't ever be married. And" _—_ just to be thorough _—_ "Taro was there when Doran invited me to this picnic."
Zeva cackled. "You've got him well under heel, then. Good work, Dunleavy."
Strangely enough, I didn't find the suggestion that I was somehow oppressing my Source at all flattering.
Really, this picnic wasn't turning out to be nearly as pleasant as I'd expected.
The boy tugged on Zeva's hand, and she leaned down so he could whisper into her ear. "I'm afraid we must be going," she said. "It was pleasant to meet you, Lord Laidley. Nice seeing you again, Dunleavy."
"You, too," I lied.
"She likes to stir things up, doesn't she?" Doran commented once she was out of earshot.
"Apparently." Not a trait I'd noticed in her before.
"I don't think I've once seen you wear a harmony bob."
I shrugged. "It was a whim of Taro's."
"Hm," said Doran.
All right. It was time to get the unpleasantness started. "You were right when you said things had changed between Taro and me," I confessed. "We do have a different sort of relationship now."
"Do you plan to marry him?" Doran asked.
"What? No!" Why would I do that?
"Have children with him?"
"Definitely not." It wasn't likely, within a Pair, both of us channeling. Channeling and Shielding tended to prevent conception. And even if it were possible, Zaire, what a bad combination. With our luck any children would have my looks and his mood swings.
"So it's not a permanent liaison," he said.
I knew the answer to that, but I wasn't prepared to say it out loud. To anyone.
Doran nodded, though, as if I had spoken. "I understand that Taro was almost grown before he was sent to the Source Academy."
"He was eleven." And what did that have to do with anything?
"No doubt he was taught something of the courtship rites."
As far as I knew, Taro didn't learn anything from his family except how to sit excrutiatingly straight and use his cutlery. Apparently, he could barely read when he'd left his family. And at the age of eleven, that was just shocking. "The what?"
"The way we do things."
"Who's we? What things?" It suddenly felt like we were speaking in a different language. And maybe we were. The language of the aristocrats, which I had never learned.
"Nothing is settled between you. There has been no promise made, correct?"
I wasn't sure where this was going, but it didn't seem a positive place. "That's really none of your business." Aye, yes, blunt, but if he had any manners, he wouldn't be pursuing this line of questioning.
"Sometimes being open and honest can really save a lot of time," said Doran.
And when had I started appearing closed and dishonest?
It wasn't about being honest, anyway. It was just that I didn't want to talk to anyone about this thing between Taro and me. I didn't want to add to the gossip. What rule said I had to talk about it, anyway?
Was Doran entitled to know about this? Did the relationship we had before I left for Flatwell give him that right? Really, I had no idea.
"I like Taro very much," said Doran. "I mean no disrespect to him in what I am about to say. But he is not the sort to make that sort of commitment, is he?"
I was suddenly amused. Doran's aristocratic upbringing was showing. What normal person talked about commitment in reference to sexual relationships? And even if normal people did think that way, Taro and I were bonded. An expression of commitment was redundant. Where could he go?
Doran's eyebrows rose. "My dear, where did you pick up that smirk?"
Smirk? I did not smirk. Smirking was something children and obnoxious men did. I was far too mature for such an expression.
"He hasn't spoken of commitments, has he?" Doran persisted.
"Of course not." I wanted to add that I wouldn't welcome any such discussion, but I had a feeling Doran would interpret such words as evidence that I really did want a commitment, I was merely too proud to admit it.
Which was stupid. If I wanted a commitment, I would ask for one.
"I really don't care for this subject of discussion," I said coolly.
And suddenly, his posture relaxed, and he smiled. "Can't blame a man for trying."
I certainly could, even when I didn't know exactly what he had been trying.
"Lydia would like to see you," he said, mentioning a friend of his, whom I had met and liked.
She was another person I hadn't thought of since leaving for Flatwell. What was wrong with me? "How did the wedding go?"
He grinned. "It was a total disaster. Her fiancé's father forced himself into the planning and turned it into an overly elaborate nightmare." He took a sip of wine. He didn't appear about to expound.
"You can't just leave it there," I chided. "Details."
Details there were many, from people being given the wrong location for the ceremony in the invitations, to the celebrant showing up dead drunk, to no fewer than seven chairs collapsing under guests during dinner. It was clearly a day of endless calamity and Doran described it with heartless glee. I couldn't help laughing, badly though I felt for Lydia.
It felt good to laugh.
The list of problems was so long that Doran wasn't able to reach the end of it before a Runner appeared beside the table, without warning. I'd forgotten we were in a public place. I felt embarrassed for having laughed so much while being observed. Was that why the Runner was approaching us? Were we not supposed to laugh so much in public?
"My apologies, my lord, Shield," the Runner said. "Due to the illness in the riverfront spreading to other parts of the city, the deputy mayor has outlawed the use of the parks. Any unnecessary public gatherings are prohibited."
I was shocked. "I had no idea the illness was so severe."
"Well," said the Runner, "there is no reason why you would, is there?"
What was that supposed to mean? Was that an insult? "So this illness is contagious?"
"No, not at all," the Runner answered rather quickly.
"If it's not contagious, what does it matter whether people gather in public or not?"
"It is a preventative measure."
"In case it might be contagious?"
He looked impatient. "It is not contagious, ma'am. I know nothing more about it."
That sounded cagey, to me. Maybe I was always too suspicious, but this seemed odd.
"How long is this ban expected to last?" Doran asked.
"I don't have that information," the Runner said in a rather snippy tone. "Please gather your things. I have other areas to clear."
I wanted to object, or to think about it, or ask more questions. This had all come out of nowhere. But I didn't know what to say.
"I'm sorry, Lee," said Doran, which was the cue for the servants to start packing up.
"Not at all," I answered without thought. What was the nature of this illness? Where had it spread? What were the healers doing about it?
Why hadn't I learned more about it? How could I be so ignorant of what was going on in the city in which I lived?
"Well, that was disappointing," Doran commented once we were back in his carriage.
"It was very pleasant, aside from the interruptions."
"So you wouldn't mind doing it again?"
We couldn't do it again, because they were closing all of the parks, but that wasn't what he meant. "Not at all."
"We can remain friends?" he asked.
Relief seemed to loosen every muscle in my body. He understood. I smiled. "I would like that." I couldn't believe he would be willing to do that. And yet, why not? I'd always known he was an exceptional person.
"We can still see each other?"
"Of course."
"Shintaro won't mind?"
"He had better not, not after I explain things to him." Taro spent all sorts of time with all sorts of people, and although he had neatly avoided admitting it during our last argument, he knew as well as I did that a lot of people he ran with were in lust, if not love, with him. I'd never suggested he stop associating with his friends, and he had never offered.
It didn't seem right that he should be able to spend time with people who admired him while I wasn't to have the same right. At the same time, it didn't really disturb me that Taro was meeting with his friends, while it clearly disturbed Taro, a lot, that I met with Doran.
Did my insisting on doing something that I knew made Taro upset make me mean-spirited?
Would pandering to his feelings, while not insisting he observe the same behavior, make me weak?
Would insisting that if I couldn't see Doran, he couldn't keep his menagerie be just plain childish?
I would have to think about that.
But in the meantime, I got to enjoy the company of a good and entertaining friend. There was no harm in that. Taro would just have to accept it.
Chapter Eleven
Taro was waiting for me when I returned to the residence. He was lingering in the parlor and jumped out as soon as I was through the front door. "What did you tell him?" he demanded.
I was annoyed to be assaulted so quickly. I looked through the correspondence deposited on the table in the foyer, surprised to find a letter from Trader Fines addressed to me. I was tempted to read it right then, an indication that I didn't appreciate his behavior, but that was just a little too rude.
"I made it clear that you and I were _—_ " What? I hated all the descriptors that immediately leapt to mind. They were either too saccharine or too coarse. "Together."
"And?" he prodded.
"That any plans he might have had for the two of us were to be put out of his head, because they weren't going to happen."
"And?"
I just looked at him. And what? I had had nothing more to say.
"And you're not going to see him again?"
"I didn't say that." And I wouldn't say that. "I'm not going to talk about this anymore, Taro. I'm not going to be told who my friends are."
"He's not a friend, Lee! He loves you!"
I grimaced at the melodrama. "That's an exaggeration, Taro." Doran had never said so, and had given me no reason to believe that was the way he felt. And I respected him for it. He really didn't know me well enough to fall in love with me. "I told him that any kind of romantic relationship between us was over. He argued about it for a bit, and then he accepted it and asked if we could be friends. And I was happy to agree. I'm not like you, Taro. I don't have a lot of friends."
"You can't be this naive! No one wants to go from lovers to friends."
"We were never lovers," I reminded him in a sharp tone. "And are you saying you don't stay friends with the people you've had as lovers?"
"Not usually," he muttered.
"Really?" I said, my surprise evident in my voice. That was a bad sign for us.
We were still standing in the front foyer, which meant we were in the way when Stone and Firth came clattering in. "Hells, not another argument," were the first words out of Firth's mouth.
"We don't argue that much," Taro objected.
"You argue constantly," Firth snapped. "And it is much more bitter than it was before you left." She poked me in the chest. Hard. "This is one of the many reasons why partners should never sleep with each other. It was incredibly foolish for you to let things go that far."
I stared at her. Why was I, of the two of us, responsible for letting things go anywhere? And where was this censure coming from? All the Pairs had thought we were sleeping together, I was sure, from the moment they'd come to High Scape, and there hadn't been a hint of disapproval. Annoyance and a lot of eye rolling, certainly, but not actual disapproval.
And Firth was always going on about sleeping with everything on two legs in her long, adventure-filled life. She tormented Taro every time she saw him, about how delicious she found him. What had all that been about, if it didn't demonstrate a certain flexibility in her morals?
And yes, sexual relations between partners was discouraged by the official policy of the Triple S. We were lectured about it the whole time we were in school. But while it was considered an extremely bad idea, it wasn't illegal or even really immoral. I'd met a Pair who were married to each other, for Zaire's sake, so how wrong could it really be?
"Ignore her," Stone said to me. "She's just upset that Prince Albert died."
Hold on, the Empress's husband had died? When?
"Don't tell her to ignore me," Firth snapped. "What they're doing is disgusting."
Disgusting. _Disgusting?_ Seriously, where the hell was this coming from?
"We've talked about this," Stone said to Firth.
"No, you lectured me about it. And where did you get the idea that you could lecture me, I wonder? You've been spending too much time with this one." Firth nodded at me with a sniff, before turning on her heel and striding away from us, ascending the stairs with an uncharacteristic heavy tread.
"She's actually disgusted by us?" I asked Stone.
Stone shrugged. "I believe so," she said. Which surprised me. I had been expecting her to discount Firth's accusations as an aberration.
"But why is she only getting upset about it now?" Unless she had always disapproved of us and had been able to hide it better before.
Stone frowned, looking puzzled. "She's disapproved of it since she realized the nature of your relationship," she said. "This is fairly new, is it not? At least, you weren't sleeping together before you went on your trip."
There were too many surprises happening in a single conversation. I had really thought the other Pairs had thought Taro and I had been sleeping together all along. They'd certainly acted as though that was what they believed.
"I'd felt like she'd been avoiding us," Taro commented. "But I'd thought it was just me."
"But she's always talked as though she enjoyed . . ." Sleeping with a lot of different people. I wasn't comfortable saying that about someone so much older than I. I wasn't sure why.
Stone rescued me. "She likes to talk," she said. "And she likes to tease. But her morals are actually very firm."
"Meaning what, exactly?" I asked. "She really thinks what we're doing is immoral?" That was a stupid question. I knew it as soon as it left my mouth. I just couldn't believe she actually thought what we were doing was wrong.
"Yes," Stone answered, with no hesitation.
"And do you think we're immoral?"
It took her a little longer to respond that time. "Not immoral, precisely," she said finally. "Just incredibly foolish. I have to say I expected better of you."
And again, I was fairly sure the use of the word "you" was meant to refer only to me, not to Taro and me collectively. And seriously, what was that about? Why was I the only one to blame?
"If you will excuse me," said Stone, moving away.
"Dee," said Taro.
Stone kept moving. "I don't care to continue this conversation."
"No, I just want to know how Prince Albert died."
Stone hesitated and turned back. "They say he died in his sleep. Apparently, he has been bedridden for close to fifteen years, and demented for most of them."
Demented? I'd heard no such rumors, and I would have, wouldn't I? If this had been going on for fifteen years? I had spent time at Erstwhile, staying right in the palace. People living in the palace, or spending their days at the court, they would have been aware of something like that, and spoken of it. Wouldn't they?
"Thank you," said Taro. Stone nodded and went upstairs, presumably following her Source.
I couldn't believe they had the gall to think there was something immoral about us. "Why am I to blame?" I muttered.
"Oh?" Taro arched an eyebrow. "So you feel what we're doing is blameworthy?"
"No, no. Bad choice of words. But why am I the only one considered responsible?"
"Because you're the only one considered able to be responsible," Taro said bitterly. "I'm just a Source, after all. Little better than a child."
"That can't be it. Firth is a Source and she holds me responsible, too."
"She wouldn't be the only Source to think that way. Sometimes we're our own worst enemies."
That didn't make sense, to think one's own kind was inferior to another. And what did that mean, in Firth's case? That Taro was incapable of making his own decisions, and that I was abusing a position of authority in the pursuit of sex?
I felt a little sick. I'd never been despised before. Not by someone who knew me. I'd never been thought of as something vile and reprehensible. It was a disquieting position to be in.
"Who is your letter from?" Taro asked in a welcome change of subject.
I had forgotten about the folded paper I held in my left hand. "Trader Fines."
"The man who guided you during the parade?" Taro's expression was flat. I had told him of my experiences at the parade, and he had read me a fine lecture about my idiocy in attending alone.
He'd been right, but it was so aggravating to be limited that way. Talk about feeling like a child. "Aye."
"What does he want?"
We moved to the parlor as I opened the letter. It was addressed only to me, though the invitation within was for both Taro and me. It spoke at length of Fines's laxity in not inviting me sooner _—_ revealing that he hadn't been aware of my absence from High Scape, though there was no reason why he should have been _—_ but his holdings had enjoyed a sudden expansion that had left him and his staff scrambling. He owed me no explanation. I hadn't known he existed before the day of the parade. "Trader Fines wants us to come to dinner."
"You're not going to refuse," he said, because he knew me quite well.
"Why shouldn't I? It's likely to be deadly boring."
"You said he said he knew your parents."
"Aye, as business rivals. Not friends."
"That you know of. And I wager that you haven't even written to your parents to find out."
I didn't respond to that challenge, because he was right. I hadn't even thought to do that. Why would I? "It doesn't matter what kind of relationship he has with my parents. That has nothing to do with me."
"They're your family."
"Those ties are severed once you enter the Academy." He raised his eyebrows at me to inform me that my opinion was stupid. "That's what the rules say," I insisted. While any child sent to a Triple S Academy could retain his place in his family, all obligations that would normally bind him were considered eliminated.
"I know for a fact that doesn't actually happen."
"Your case is different. No one expected your highly titled brother to die without producing an heir. And most families wouldn't have expected a Triple S member to take on the obligation of a title. No one's going to expect me to play nice with merchants because of my family." They'd better not. I didn't know the first thing about trade, and it wasn't fair for anyone to expect me to play the game.
"All right, then go to thank him for the kindness he showed you during the parade."
"Won't that be just acquiring another debt to him?"
"Not if you play it right."
"I don't play things, Taro."
He winked. "That's what you have me for."
"Fine. I'll send an acceptance." Clearly, I had no willpower at all.
And then I felt Taro's inner protections fall.
Really, I should stop expecting not to have to Shield merely because we weren't on duty. Time was, Taro would warn me when he was going to channel in unusual circumstances. It was odd for him to desert that courtesy. I'd have to talk to him about this, after he finished channeling whatever he was channeling.
It wasn't long before I knew what he was doing. It was an event from that place of cliffs and water. Or at least, it was the place that inspired those images in my mind, and filled my nose with the scent of salt, and my mouth with the impression of water. As with the first time, Taro let the forces rush through him too quickly, pushing the workings of his body and his mind too hard. The anger that behavior inspired in me made it a little bit easier to Shield him in the slightly chaotic circumstances.
And when the forces dissipated and Taro's inner protections were resumed, I glared at my idiot of a Source. "I asked you not to channel like that again," I snapped.
"No, you ordered me not to," he responded in a cool tone. "When did you develop the delusion that you can give me orders?"
When I saw him acting like an idiot. "We are both put at risk when you _—_ "
But he turned on his heel and walked out of the room, his lack of interest in any further conversation clear. Arrogant prat.
Chapter Twelve
I curled up on the settee in my sitting room with the bag of books and pamphlets I'd gotten from the printers. Taro and I didn't have a watch at the Stall that day, and I was finding it more and more important to learn something about this spell casting everyone was talking about. Really, where did it come from? How long would it take to go away?
I picked up a book at random and upon reading the very first page noticed with shock and discomfort that it had first been written a good seventy years before. So much for the theory that this belief in spells had been created by the effects of the Harsh Summer or the Riverfront Ravage. People had believed in magic for at least seventy years? What had happened seventy years ago, to get that belief started?
I found it amusing that the first chapter of the book was dedicated to the discipline and hard work needed to achieve results. Because wasn't casting all about trying to gain things one couldn't acquire because one lacked the discipline or the will to do the hard work? If casting required all of this effort, wouldn't it be easier just to get things the natural way?
I found it less amusing that many of the methods described in the book for getting into the proper frame of mind to cast spells were similar to the methods taught to Shields to stay focused and calm. It was enough to make me wonder if a Shield had written the book.
Of course, that was ridiculous. Being a Shield had nothing to do with casting spells. It had nothing to do with magic of any kind.
Being a Shield, or a Source, was a talent one was born with, like being able to sing or having skills with one's hands. There was no magic. We didn't create forces that weren't already there. Even Taro's ability to heal and my ability to influence the weather _—_ both skills that were not, as far as I knew, the usual skills associated with Sources and Shields _—_ were still just the manipulation of forces that were already there. From what I could determine, casting spells was about creating something from nothing. Which was what made it impossible.
We had encountered people who claimed what Taro and I did was magic. It was not. I knew that. So, that there were similarities between what a spell caster did to prepare and what a Shield did to remain calm was disquieting, but not particularly significant.
I almost got drawn into a description of the history of the use and development of spells. The book I was reading claimed the tradition of using spells was brought to the world by our ancestors, when they'd relocated here from some other world several hundred years ago. That the knowledge had been lost and then rediscovered. But that was ridiculous. From all we had learned of the Landing, the ancestors had had enormous and complicated machinery at their disposal, machines that let them fly between worlds and communicate over great distances. Such people would have had no use for spells, no reason to dream up such beliefs.
Of course, the machinery hadn't worked on our world, for some reason. But that had merely caused most of the ancestors to leave this world. The failure of the machines was no motive to suddenly start believing in spells. It didn't make sense.
It was something I wanted to look into further, but later. Right then, I wanted to read more about what kinds of spells people were going to attempt, killing cows and setting houses on fire. Actually, spells about the ashes, those were what I wanted to know about first. The bit with the ashes was what I found most disturbing.
I didn't know exactly what I was looking for, though. Was there going to be a spell mixed in with all the others that dealt with ashes? Was there a separate section dealing strictly with supplies, of which ashes would be one? I didn't know, and the book I was examining didn't have any kind of index, so I skimmed through the pages. Within the one book, there was an astounding array of subjects, or goals, for the spells. Love spells, of course, a popular subject for plays. Raising people from the dead, which I kind of expected, grief being what it was. I wasn't surprised to find spells that gave one a talent, or attempted to direct another person's will. I was shocked, however, to find spells allowing the caster to inflict an illness or death on another person, or to render them unable to have children. That was horrible. What kind of twisted mind came up with such things?
I was almost tempted to wonder if having such books criminalized wasn't such an awful thing, after all. I didn't want anyone encouraged to try such terrible things, even knowing they wouldn't work. When the spells failed, having sparked a person's imagination, said person might then move on to more reliable methods of accomplishing the same things.
As I continued to read, I was surprised by the degree of complexity some of the spells required. While some needed nothing more than facing the right direction, closing one's eyes and reciting a few words, others took months in preparation, lists of ingredients several pages long, multiple participants and several sonnets' worth of incantations.
Who had the time to make this stuff up?
What made people think any of it worked? Even though some of the rhetoric describing how spells functioned was intelligently written and appeared almost logical, it was still casting. When had a spell ever worked for anyone? Even the books didn't point to any example for whom a single spell had worked.
It was in the next book, a thicker volume with heavy gray paper, that I found what I was looking for, a lengthy chapter dedicated to the procurement and use of human ashes. And even the procurement was much more complicated than I would have expected. The ashes were to be harvested _—_ and wasn't that a horrible word for it? _—_ during the night of the new moon. I couldn't remember if the night we'd interrupted a harvesting had had a new moon. I never noticed such things. A copper bowl was considered best for holding the ashes, an ivory spade the best for digging them out. That particular item had to cost more than most could afford. The list of instructions for the making of the candles that circled the ash grove marker was more than three pages long.
The book confirmed that the best source of ashes was someone who had enjoyed good fortune during their life. There was no definition of good fortune. I supposed that could be a subjective determination, but I had no doubt Taro would be considered fortunate. If he were dead, I'd have to be guarding his ashes, if I were able, which I wouldn't be, because I would be dead, too.
There were a lot of uses for the ashes. Often they were an ingredient in other spells. Sometimes they were put in a vial and worn around the neck _—_ which was just disgusting _—_ to increase the user's general good fortune or the talent of the user when they lacked the skill to perform one of the more specific spells. There was even a recommendation to wear the ashes on a daily basis to improve health and mind. And then there was, of course, the practice of mixing the ashes into a paste and rubbing it into the skin.
Just reading about it made my stomach gurgle and created the sensation of something thick and slimy coating my mouth and throat.
A quick, loud knock on my door made me jump. The door opened and Taro stuck his head in. "Come to my suite," he ordered and he closed the door.
Sir, yes, sir. I tossed the book aside and went to Taro's suite. He gestured at me kind of frantically until I closed the door behind me.
A young man I'd never before seen was standing in Taro's sitting room, leaning against a settee. He was pale and sweaty, and his dark and ragged clothes were dirty and rank. He looked at me with more intensity than I found comfortable. "Good day," I said, hoping that would be enough to prompt someone to give me an explanation.
"Good day, Shield," he responded in a low voice that cracked unpleasantly.
"Lee, this is Lan Kafar," Taro said. "Lan, please go through there." He pointed at the door to his bedchamber. "Lie down. We'll join you in a moment." And Kafar did as directed.
We were going to join him? What the hell? "What the _—_ "
"He's ill," said Taro. "He's from the riverfront."
"And you brought him here?" I demanded in a low voice.
"He just showed up demanding to see me. He remembers those rumors about me healing people during the Harsh Summer. He says the healers won't go to the riverfront anymore, and the hospitals turned him away."
"Because it's contagious!" No matter what that Runner had said, I was sure it was contagious. Why else would they be closing the parks? Why would the hospitals refuse to treat people? And were they really? Were they allowed to do that? I couldn't believe that. It was appalling if it was true.
"He kept saying I was a healer, over and over, and getting louder and louder. People on the street were noticing."
"So you close the door."
"Dunleavy!" he said, clearly appalled. I didn't think he had ever addressed me by my formal name before. "He's ill and he can't get treatment."
"No one's dying from this illness." And if every single member of the Triple S residence caught it, that could leave everyone in High Scape vulnerable to natural disasters. A whole lot of people would be dying then.
"That's not what he's saying."
"So he's claiming people are dying from this and you've brought him into the residence and exposed us all to it?" Seriously, what was he thinking?
"The more time we stand here talking about it, the more of a chance the illness has to spread."
Was that true? Was that how it worked? I had no idea. "You don't even know if you can do anything for him."
"And I won't know until I try."
I couldn't believe how careless and selfish he was being. Aye, it was horrible to be ill, and if it was true that the healers were refusing to help, I was appalled and felt nothing but sympathy for the man in Taro's bedchamber. But to expose us all to this illness to perform a task he had no duty to perform would bring us the ire of all the other Pairs, and so it should.
"I'm going to try," said Taro, having clearly lost patience with trying to persuade me. "You can join me if you choose." He headed to the bedchamber.
I was tempted to threaten him with a refusal to Shield him _—_ really, this was something that required the both of us, and he shouldn't be making unilateral decisions _—_ but I always tried to avoid making empty threats. As if I would ever refuse to Shield him, even when he was being ridiculous.
When I entered the bedchamber, Taro was instructing the man to lie out on the bed, and I winced as the man did so. All of that bedding would have to be burned. Taro sat on the edge of the bed. "I'm not a healer," he said to Kafar. "I can't be sure I can do anything."
"Neither can anyone else," the man croaked.
"I'll do my best. That's all I can promise."
The man nodded. Taro put a hand on his shoulder.
Taro had learned he could ease the pain of others while he was at the Source Academy. Since we met, I'd had reason to believe he could actually heal some people. These were not abilities I'd heard of a Source having before meeting Taro, but I was coming to suspect there were elements to Source abilities that were largely unexplored. It made me wonder what else Sources might be able to do.
Taro had always kept his additional abilities as much a secret as possible. He claimed that if the Triple S council suspected he could do odd things, they would order him back to the Source Academy for tests and he would never be seen again. He refused to say more than that. And while I couldn't believe the Triple S would do something nefarious with Sources who fell outside the mold, I also didn't believe Taro was lying. It was frustrating, though, that he wouldn't say more.
He couldn't heal everyone; we knew that much. He seemed able to ease everyone's pain, but he hadn't been able to save a woman enduring a difficult childbirth, and he couldn't mend broken bones. He could heal frostbite for some reason, which was what he was doing during the Harsh Summer. That was what created the rumors that caused his secret to grow legs and skitter away from him.
Really, I was surprised there hadn't been any repercussions before this.
Shielding while Taro attempted to heal someone was a much more gentle experience than regular channeling. The forces were weaker, and Taro used a light touch, so light it almost felt like he wasn't doing anything at all. His blood and mind worked at their usual pace, and he was in almost no danger. Sometimes I suspected I was superfluous during such occasions.
"The illness is not contagious," Taro announced.
I saw Kafar's eyes widen. I wondered if he was feeling something in this process. I'd never thought of that before. "Everyone says it's contagious," he said. "Often every person in a household is struck."
"That may be," Taro answered as he continued to channel, "and I can't explain why I know this, but this doesn't feel like a proper illness."
"What is a proper illness supposed to feel like?" I asked.
"Something other than what I'm feeling here," he retorted.
"Can you fix it?" Kafar demanded.
"I'm not finished yet."
So we let him continue. Shielding while he healed was a little boring, but it was a nice change from what we'd been doing recently.
In time, Taro ceased channeling and I withdrew my Shields. He removed his hand from Kafar's shoulder. "I'm sorry," he said with soft sincerity. "I can't help you."
"Keep trying," the man ordered. "You barely tried."
"I'm not a healer," Taro reminded him. "I told you when I let you in that there would likely be little I could do."
"Why don't you want to help me?" It was more a complaint than a question. "You're a Source. You're supposed to want to help people."
Why, that ungrateful little plonker. He comes to our house, a stranger demanding services that we have no obligation to provide, and after an attempt to provide those services is made, he starts throwing around accusations? How dare he? "He has no training in this and he's doing this to be kind," I said, making no attempt to dull the sharpness in my tone. "He is an honorable man and a Source, and you accuse him of lying."
To my shock, the indignation seemed to drain right out of him, and his eyes lowered. "I am sorry, Source," he mumbled as he sat up on the bed. "This was my last hope."
Really? He didn't look to be in that bad a shape. He was of a decent weight and he seemed able to move easily. Of course, I wasn't a healer. I didn't know.
"You are certain this isn't contagious?" Kafar asked Taro.
"Completely."
I was very uncomfortable with Taro's claim, but no one was asking me, and it wasn't like I actually knew anything about any of this.
"My sister lives on Gray Fields Row," Kafar said, naming an area outside of the riverfront area. "She said I can move in with her for a while and help her with her shop if I can prove this thing doesn't carry. I can say you said it's not contagious?"
Sounded to me like his sister really didn't want him anywhere near her. How could anyone prove the absence of a contagion?
"Aye," said Taro. "I can't promise she won't get ill, but she won't get it from you."
"And I'll get better?"
"I can't answer that. I really don't know."
"Have you tried all of the hospitals?" I asked.
Kafar glared at me. "Now you are calling me a liar?"
For all I knew, he'd merely been told the hospitals were refusing to treat people, and he hadn't investigated it personally. Or he'd tried only one hospital. Perhaps the others weren't following the same policy. But I didn't try to defend myself. Clearly he wasn't in the mood for logic.
"We'll see you down," Taro said as a means to hint the man out of the room.
Kafar didn't object to being herded down the stairs and out the front door. I fought the urge to order the man not to tell anyone else about Taro's talent. I supposed Taro's failure would take care of that.
"How do you know this thing isn't contagious?" I asked Taro.
He shrugged. "The illness doesn't feel natural to the body."
"I would think illness is never natural."
"Aye, it is. It is unpleasant. It involves a breakdown of the natural workings of the body. But a natural illness is a natural reaction to a breakdown within the body."
"I don't understand," I confessed.
He scratched the back of his head. "To be honest, neither do I. All I can tell you is that there is something off about what that man is carrying. It's not natural, so it can't spread from person to person."
Huh. I still didn't know what that meant. However, Taro wasn't in the habit of claiming abilities or accomplishments he didn't actually have. If he said Kafar wasn't contagious, then he wasn't contagious.
But I still thought Taro should burn his bedding.
Chapter Thirteen
Like most of the wealthy, Trader Fines lived in the North Quad on the outer edge of the city limits. He surprised me by having a house made of wood, more sensible than stone in a place where earthquakes could be prevalent. The house wasn't particularly large, either, being only two stories and squarish in frame. It was set far back from the road and was shielded from public view by trees as opposed to wall and gate.
It was kind of ugly, from the outside at least. But most buildings in High Scape were, in my opinion.
I had no interest in being there. I'd felt terrible all day, my stomach gurgling unpleasantly. Our watch at the Stall had been nothing more than a long test of endurance with my head in my arms on the table. Not sleeping, of course, just tired and queasy. There were no events, which in a way was a good thing, because it might have given me a nasty headache to try Shielding while feeling under the weather, but it was also a bad thing, because there was little to distract me from my discomfort.
I'd worried that I'd picked up something from our uninvited guest from the riverfront. Taro was certain that wasn't the problem. That didn't reassure me at all. He wouldn't know. He wasn't able to do with me whatever he did with the people he healed, as I was his Shield. I couldn't Shield him while he tried to heal me; it just didn't work.
After our watch, I had gone to a hospital to see what was wrong. I had no difficulty getting a healer to see me, though perhaps that was because I was a Shield and didn't live in the riverfront area. She said I was suffering from too much bile and that would wear away in time if I drank enough water. It was nothing serious.
Still, the last thing I wanted to do after a long, uncomfortable watch at the Stall was eat a formal dinner with strangers. Taro had insisted that we go. It would be terribly rude, he said, after having accepted, not to show up for any reason other than a genuine calamity. I wondered what it said about me that I didn't care at all about being rude, especially to someone like Fines, who was probably too used to having people fawning all over him. But I had to admit Taro had a point.
I dressed nicely, but not too nicely, because this was a merchant's dinner, and not an aristocrat's ball. I was a little pale, but that was easily obscured with face powder and a light-colored lip ointment. I pinned up some of my hair with a wooden hair comb and let the rest hang loose, and in my ears I put small hoop earrings of yellow gold. These were both items that Taro had picked up in the market, on a whim, and their understated elegance appealed to me. My gown was of a simple cut, with long loose sleeves and a relatively modest neckline, light green in color with cream-colored piping, so that my white Shield braid had a chance of blending in with the ensemble.
Taro was dressed in his most severe style. All in black, his shirt buttoned up to the throat and down to the wrist, his hair tied tightly back at the nape of his neck. The only brightness about him was the harmony bob he wore on his chest and the small emerald stud he wore in his ear.
His choice of outfit told me he viewed the evening with extreme discomfort. He felt he was going to be out of his element, and perhaps an object of ridicule or abuse. Which just made me wonder all the more why the hell we were even going.
We took a carriage. Even were I in top form, the walk would have been a bit of a challenge, and I wanted to arrive at the home feeling as fresh as possible. We were met at the merchant's door by an older woman who curtsied before taking our wraps, which custom required we wear though the weather didn't demand. This process distracted me, at first, from the mural in the small room that acted as the foyer. The entire wall facing the entrance was painted dark green, and rising from the floor was some kind of sun, painted in silver, with its rays spreading out in stark lines and stopping only when it reached a corner of the wall.
It was quite ugly. And it gave me bad memories of eating dinner in the house of another rich man who also had atrocious taste in decorating and had served a positively vile meal, and had then tried to sacrifice us to his bloodthirsty gods. "Trader Fines doesn't believe in anything, does he?" I whispered to Taro.
"Not that I've heard of," Taro whispered back.
Not that that really meant anything. Neither of us had known Lord Yellows was a Reanist.
The servant led us around a corner to the left, through a short series of three doors and into a parlor. "Sir, Source Karish and Shield Mallorough have arrived," she announced with another curtsy.
The parlor, thankfully, was much more tastefully decorated than the foyer, not too fussy but with enough color and enough knickknacks to appeal to the eye without being overwhelming. I did, however, notice small and less vibrant replicas of the silver and green sunrise above each window. They clearly weren't meant to be as obvious as the image in the foyer, but they seemed of serious significance to Fines.
That didn't mean there was anything sinister about them. I knew that. Yet their existence made me uncomfortable, and I wondered if I really wanted to know what they represented.
"Dunleavy, Shintaro." Fines, who had been standing near the unlit fireplace, saluted us with a drink. "Delighted you could join us. What can I get you to drink?" Taro was given a glass of wine, and I took advantage of the lovely light sweet juice chilling near the liquor stand. "Please sit, and I'll introduce you to Dean Gamut, the finest actor in High Scape, maybe in the civilized world."
That actor was a middle-aged man, a head of abundant black hair with the gray dyed out of it, his skin ruddy and rough, his neck thick along with his body. I'd seen him on the stage a few times, and he was brilliant. He usually looked more handsome on the stage, but for me it was really his deep voice that took my breath away, and he was an exceptional actor, sliding from laughter to lust to rage and every subtle shade in between with dignity and grace.
"Ayana Cree, the foremost healer in High Scape. She has her own practice. She doesn't work through any of the hospitals."
That surprised me, and it must have shown when I looked at the older lady, sharply thin with beautifully styled iron gray hair and dark brown eyes that appeared strangely warm in her stern face. "It allows me more control," she said in a low voice. "My choice of treatments, medicines, suppliers and staff."
"I would think that an expensive undertaking, Healer Cree," said Taro.
"It is, but my patients are willing to pay for the superior quality of their care."
Ah. Medicine for the wealthy. I shouldn't have been surprised that such a thing existed. I wondered if she would treat people from the riverfront if they could scrape together enough money to pay the fees.
The woman was staring at me in a way I did not like. "Are you quite well, Shield Mallorough?"
The actor laughed then, and the explosive sound was too big for the room. "You don't want her as a patient, healer," he boomed out. "Her sort are notoriously bad payers." He laughed again. The laughter seemed to physically bounce off the walls.
From the look the healer was giving the actor, she found him about as hilarious as I did. "What injury have you done to your hand, Shield Mallorough?"
Oh, that. "I burned it on heated metal." Zaire, did that make me sound stupid.
"Is that dressing being changed daily?"
"Yes, Healer Cree." Ben was insistent about it.
"With a vinegar and soda salt compound?"
"I have no idea." Well, I was pretty sure there was no vinegar involved, and for that I was properly thankful. That would hurt.
"Find out."
I was tempted to salute.
Fines resumed the introductions. "This is Grace Ahmad, high master of the Construction Guild of High Scape."
High Master Ahmad was an elderly lady, her brown skin heavily lined, hands gnarled by arthritis and resting on the head of a sturdy cane. There was something solid and settled about her, and she merely nodded in return to our greeting.
"Is Morgan going to be here?" the actor demanded. "Or is he only late?"
"He promised he was coming," Fines said with a look of amusement. "You know as well as I how much that is worth." He then looked a little alarmed. "Not that he is not a man of his word," he said to me, quickly. "It is only that he gets caught up in things"—the actor snickered—"and sometimes it makes him forgetful."
There was an exchange of glances that made me curious. It led me to believe there might be something off about this Morgan. I wondered how rude it would be to ask.
"So, Dean," said Fines. "That play of yours."
The actor grimaced and groaned. "I am not, alas, in charge of the repertoire."
"Or the script," Ahmad added. Her voice was crinkly. "But surely you have some influence over the selection?"
"One would think," Gamut sniffed. "I have warned Beezly—"
"The owner of the theater," Fines clarified for the benefit of Taro and me.
"That contracts were made to be broken and I would have no difficulty leaving a house so ready to debase my talents. There are other theaters."
So perhaps I should forgo attending his play. I had been planning to go. I was likely to see any play that had Gamut in it. But now, even if the play would be something I would normally enjoy, knowing that Gamut despised it would make it impossible for me to see him in character.
"Have you ever thought to put your hand to writing yourself, Dean?" Fines asked.
"Writers," Gamut snorted. "Such temperamental creatures where they have no right to be. Always flying into hysterics if you change so much as a single precious word."
"I believe I have heard some expression," said Healer Cree, "that if it hasn't been written on the page, it cannot be acted on the stage."
"Aye, aye, I am not saying playwrights are not without their uses, but it is the players who breathe life into their words, and give shape to their characters. Players are what give a play-wright's work meaning, for if scripts are not made into plays, they may as well never be written at all. It's not as though anyone reads scripts."
I did.
The same servant who had greeted us entered the room, a middle-aged man, very lean and with a shocking white mane of hair, trailing in behind her. "Morgan Williams," she announced before curtsying and vacating.
Interesting. Apparently he had no honorific of any kind. I couldn't remember ever meeting an adult with no title. It implied that he had no occupation at all.
"Morgan!" Fines greeted him. "What can I get you to drink?"
I found it interesting that Fines was serving everyone, and that there were no servants remaining in the room. I had never spent much time among people of my family's or Taro's family's class, but what little exposure I had taught me they used servants if they could afford it. Fines could clearly afford it.
We were introduced to Morgan Williams, and as he was introduced to us we were told, "He owns the Mercury Brothels."
"Really?" Taro asked with sincere interest. I raised an eyebrow at him and resisted the urge to ask him if he'd ever visited one of those brothels. These people were strangers and I wasn't comfortable teasing Taro in front of them.
"You've heard of them, then?" Williams asked with a slight emphasis on the word "heard."
"Aye. I've been told it's unusual for one person to own more than one brothel."
"I should hope that's not all you've heard about them," Williams said a little huffily. "We also accommodate a wider variety of tastes. Except, of course, anything involving animals or children. We don't deal in perversion," he sniffed.
I bit back a smile. There were those who believed that paying for sex was a perversion all its own.
Another servant stepped into the room and then stepped out without saying anything.
"Supper is ready," Trader Fines announced.
We followed Fines through a corridor to the dining room, which was considerably longer than the table it held. It appeared to me that most of the leaves of the table had been removed to allow our small number to sit around the table in comfortable proximity to one another. The table was almost completely obscured by an array of dishes, most of them covered, and by each chair there was a wine stand holding a dark bottle. "I would beg your indulgence in the informality," Fines said as he closed the door. "We prefer to entertain without servants always getting in our way."
"Ears are too big," Gamut muttered, and the silence that followed seemed oddly tense.
"So, everyone, please take a seat," Fines invited, surprising me by seating himself at the side of the table instead of one of the ends. "We don't follow any rules for seating. I find that a little pompous in a private home."
Well, all right, then. I took an end seat, just to make a point. There was a small brown clay pot at my place, and one at each place of the others. Everyone except Taro and I took the lid off the pot, dipped out some of the substance, and rubbed it vigorously on their hands. "Please try it," Fines prompted with a smile. "Ayana created it, to cleanse the hands before eating, and it's relaxing."
I lifted the lid from the pot. The contents, an off-white translucent paste, had a pleasant, clean scent. It spread smoothly onto the skin with no noticeable grease. It was nice.
"Ayana," Fines said to the healer, who had sat at the head of the table. "I believe you have the main course before you, but I propose that everyone dip into whatever dish is placed in front of you and pass it to the right until everyone has had access to everything."
The next several moments were spent in silence, following Fines's suggestion. It was an odd way to have a meal served, but it had to be fashionable or otherwise common, because the other guests easily handled the platters that were small and light enough to be held with one hand, with utensils and portions easily manipulated from platter to plate.
The soup wasn't even in a tureen, but in a silver jug for pouring into the small bowls unusually placed to the upper left of the dining plates.
All of the food was piping hot, though, and delicious. Unusually delicious. I couldn't remember ever eating anything that tasted quite that good. It was really astonishing. And soothing on my stomach.
"Please try the wine," Fines prompted, though Gamut and Ahmad were already filling their goblets. I joined them. The wine was red, not my favorite, but it was lighter than most reds I'd had before, and I enjoyed it.
"This is excellent wine, Trader Fines," said Taro.
"Richard, please. And I'm pleased you enjoy the wine. It's one of my best sellers."
"The best wine on the market," Williams drawled, sounding impatient about it, like it was something he had heard too many times before. But Fines saluted him with his goblet as though it were a genuine compliment.
"And you're my most reliable customer," he said.
Williams shrugged. "My clientele like it, and they can afford it."
I was curious about his bordellos, and I had a thousand questions to ask. I'd heard of bordellos, of course—some of the best plays took place in them—but I'd never given the reality of them much thought. But then, I'd never sat down with someone who owned one. Would it be rude to ask him how he got into the business in the first place? That had to be a fascinating story.
"So tell me, Dunleavy," said the healer, who had the food on her plate in discrete little piles, no one item touching another. "Which theory do you prefer for explaining how Shields and Sources become Shields and Sources?"
Damn it, why did I always get that question? Taro was right there. Why didn't she ask him? "I feel it's just one of those things. Just like being born with any other kind of talent."
"There are those who believe talent is inherited."
"Yes," I said, because that was one I'd heard often enough.
"So that is a theory you agree with?"
What did it matter? Shields and Sources were born, and that was good, and why not leave it at that? I was no expert on the subject. What would I know about it? She was the doctor. "I don't agree with it, necessarily. I don't disagree with it, either. I really don't know." And I never really thought about it.
"Are there others in your family who have this talent?" she asked.
"A cousin on my mother's side is a Shield," I told her. "He is the only one I know of."
"Taro?"
He plastered on one of the fakest smiles I'd ever seen him use. "Just me," he said. "For I, Madam Healer, am unique and far outside the ken of the Karish ranks."
"Hm," said Gamut, perhaps in recognition of the only other person at the table who was as melodramatic as he.
"I'm serious," said the healer. "Where does this ability come from, if it is not a matter of breeding?"
"It's just happenstance," I said.
"Or luck?" Ahmad suggested.
I would have felt fine about agreeing that it was merely a matter of luck a few months earlier, before people were buying harmony bobs and killing cows with their bare hands in a futile quest for luck. Now I was afraid to use the word, afraid of the meaning listeners might take from it. "Happenstance," I repeated.
"Do you think there's any way to influence happenstance?"
Wasn't happenstance beyond the reach of any influence? Wasn't that the nature of happenstance? "I can't imagine how."
"Well, let's take that Yellows's fiasco a couple of years ago," Ahmad said. "You two were there, were you not?"
"We were so honored," Taro answered with a bitter twist to his words.
"And are the rumors true? That the dinner was just an elaborate ritual for their gods?"
"More like a big trap for aristocrats," said Taro. "With very bad wine."
"Do you believe what the Reanists believe? That killing aristocrats calms the world?"
"Of course not," Taro scoffed.
"So there were still events happening even while aristocrats were being abducted and killed?"
I was glad that I wasn't being asked that question. Because the truth was, no events threatened over the months that the Reanists were abducting and killing aristocrats. That didn't mean one had anything to do with the other, but try convincing some regulars of that.
"If the Pairs are doing their jobs properly, there never are events," Taro answered smoothly.
"And you, Dunleavy? Do you believe there might be any magical potential in killing aristocrats?"
Aside from ridding ourselves of some deadwood? "I don't believe in magic," I said. I wanted to say there was no such thing as magic, but I had the feeling that in doing so I might offend our hosts, and possibly some of their other guests. "And I've been given no reason to believe that if magic does exist, the blood of aristocrats is any different from the blood of anyone else."
Gamut was snickering.
"I find it entertaining," said Ahmad, "when a Shield claims not to believe in magic."
Claims? "I don't understand," I said, my voice nice and flat.
"What do you think you do, if not magic?"
Ah. We'd encountered this argument before. "It is a talent we are born with. From what I have learned of the magic people are currently exploring, it involves the casting of spells; it is something one studies in order to become proficient at it. We don't cast spells. There are no specific ingredients or words that we use. We just do it. It's natural to us."
"But perhaps there are many forms of magic," said Fines. "People who are born magical, and people who must study to acquire magical skills. Just think of it, we're sitting down with two magical beings."
What a ridiculous thing to say. I almost winced.
"Have you noticed," Cree asked, "that the incidents of spells that have worked have all occurred in specific locations about the city?"
"Spells have worked?" I retorted, and I knew I sounded sarcastic, but I couldn't help it. She was a healer. Wasn't she supposed to be smart?
"There has been some dispute about it," Cree admitted. "But the people performing the spells have claimed they work."
Huh. "And they all live in the same place?"
"You should read the news circulars more often."
Aye, I knew that.
"Speaking about rituals and their supernatural results," Williams interjected. "Ahmad, it took only eight months to have that crater at Center and Dove streets fixed. How was that accomplished?"
Ahmad shot him a crusty look.
The conversation moved on to other things, and after dinner we moved back to the parlor and drank more alcohol. That was where Gamut started telling stories about his early days in the theater, about humiliating costumes and missed lines and sets that collapsed midperformance. He had a brilliant knack for description and a way with accents, and he had me laughing so hard it made my back hurt.
It was a pleasant evening, but I didn't know why Taro and I had been invited. These were all tradespeople of one sort or another. And they were all older than us. What interest could they have in us?
I never liked to be the first to leave a gathering—I was always afraid the host would think I was anxious to get away—but all of a sudden my stomach started to slosh, and no one else was making any signs of moving, so Taro and I were the first to say our thanks and farewells. Fines sent out a servant to flag down a carriage for us.
"None of them had their partners there," Taro said once we were one our way.
He was right. I hadn't really noticed or thought about that. But, "I've been to a lot of parties where people didn't bring their partners." Though, granted, it wasn't the rule.
"People in positions like that always have partners, usually even marrying them. To handle things while they work, like buyers or other contacts. Except Dean, I guess, but even he would bring someone. They all would, to a social event. But they didn't even talk about having spouses or partners, or offer excuses as to why they weren't there tonight."
"Are you saying it wasn't a social gathering?" I asked. "What else could it be? Why else would they meet? They could be a group of friends who had grown up knowing each other, which would explain why partners and spouses weren't welcome."
Taro shrugged. "Maybe it was. I don't know. I'm just saying it felt strange."
I decided I didn't care. It wasn't likely that I'd ever encounter Fines or any of the others again.
Chapter Fourteen
I was taking Cree's advice and reading news circulars. I used to read them regularly, but I had fallen out of the habit while on Flatwell. Which was unfortunate, because there were all sorts of interesting things in news circulars. Like a long editorial ranting about the transfer of so many Triple S Pairs with no replacements, accusing the Triple S council of negligence in its duty to an important city like High Scape. I wondered whether it would be appropriate to tell the writer that High Scape hadn't been threatened with a single event for weeks, if not longer. For all I knew, such information would cause that same writer to complain that the remaining Pairs were a waste of resources.
"Shield Mallorough."
The sudden voice made me jerk in my chair, and I glared at Ben, who was standing in the doorway of the parlor.
He didn't appear to notice. "You're up early."
"I slept poorly." Lately, I was always sleeping poorly. It was getting annoying. "Have you been away? I feel like I haven't seen you for a few days."
"Is that why you've taken your wrap off your hand?" he chided, carefully taking my right hand in his.
The burns were still discolored and tacky. I'd had no idea burns took so long to heal. "That seemed safer than using something harmful out of ignorance."
"It is safer to leave the wrap on. The plaster remains viable for several days. I'll use a stronger plaster today. Hopefully that will address any damage you might have done."
Hey, he was the one who left without telling me how to take care of my hand.
"You're not having coffee?" Ben asked.
"I didn't feel like making any when I got up." Which was bizarre, but so was getting up early.
"I'll put some on for you."
"Oh, no. That's not necessary. But thank you."
He nodded and left the parlor. I continued to read and came across the information that the ashes of the mayor for a day had been stolen. It was ridiculous how many ashes were going missing. People were getting obsessed.
As Ben rewrapped my hand, Taro came in for some breakfast, and then we headed out to the Stall for our watch. Almost as soon as we were on the streets, we saw the new notices pasted on flat surfaces all over the place. The paste was so fresh and smelled so strong I couldn't stand to get close enough to read the notice. "What are they about?" I asked Taro.
"New punishments for people pretending to engage in the practice of casting spells," he said.
"Really?" I didn't even know what the old punishments were. "What are they?"
"Lashes," he said with distaste.
"Excuse me?"
"A public whipping. A single lash for each book of spells owned." He shot me a look. "Five for performing any alleged spell or ritual. An additional lash if it's a love spell. An additional ten if the spell harms someone else's property. Twenty lashes for each instance of collecting, possessing, selling or consuming human ashes."
I didn't know why the idea of digging up ashes for the purposes of selling them disgusted me more than digging them up for personal use. It just did. Not as much, though, as the idea that someone would get tortured with twenty lashes just for doing it. After all, ashes were merely the remains of dead people. Dead people weren't capable of caring. Yes, the practice was disgusting and stupid, but so was torture, and torture was perpetrated against live people.
"I didn't know there were standard punishments for crimes." I'd always thought it was up to each individual judge. "Who decided these?"
"His Imperial Majesty Emperor Gifford."
"Can he do that?" He wasn't an emperor yet.
Taro shrugged. "Who's going to tell him he can't? You might think about getting rid of those books."
I stared at him. "What did you say?"
He tapped the notice with the tip of his finger. "A lash for each book. How many books would you say you have? About twenty-five?"
"No one is going to tell me what I can and cannot read."
"Would you be saying that if you weren't fairly confident that as a Shield you are excluded from the force of this decree?"
"Yes." But I would also be getting rid of the books, coward that I was. I wasn't going to be tortured for reading about something I wasn't all that interested in.
Taro seemed to stiffen, and I felt his inner shields shift. "We're not on duty yet, Taro," I said sharply.
"No one's doing anything. It's not here." The words were rushed out of his mouth, and then his shields were down. I had to put up mine and there was no point in talking any further because he was committed and there was nothing more to be done but follow along.
But damn it, I was tired. Besides, the natural events had to be allowed to happen somewhere. If nothing was allowed to happen anywhere, well, the world would just explode, wouldn't it?
Taro was just showing off, damn him.
Again, it was an event from the place of crashing waves and jagged cliffs. What was it about this place that Taro kept feeling the events taking place there? He wasn't feeling events from anywhere else.
Then something strange happened. An unpleasant sliding sensation that hit me right in the stomach. I felt myself stumble. "Taro."
Water heaving and swirling brought a stinging to my nostrils and dizzying tension to the bridge of my nose.
And Taro was channeling too much, all at once. It frightened me. The racing of his heart was mixing in with the cacophony exploding in my ears. "Taro, slow down."
I was vaguely aware of falling against a wall. A stench assaulted my nose, which didn't help with the nausea.
And the forces rushed through Taro, much like the gushing water pressing against the backs of my eyes.
I felt a rippling in my Shields. Were they slipping?
My gods. That couldn't happen. That didn't happen. "Taro!"
Couldn't he hear me? The panic in my voice was obvious to anyone who cared to listen, I was sure. Why wouldn't he answer me?
But maybe he couldn't hear. He was letting too much rush through him, and I had no idea how aware he might be of what was going on around him. Or of me. Maybe when his blood was rushing about that hard, he couldn't really hear anything.
Yet I would have thought he would notice that there was something happening to my Shields.
The rippling became more intense. "Taro!" I shrieked. "Slow down!"
He couldn't stop once he'd started. I knew that. We were in trouble.
But the forces lessened in their volume. By slowing the channeling of the forces, Taro lessened the demands on his blood and his organs. That released some of the pressure from my Shields. Unfortunately, the more the forces lessened, the more my Shields seemed to weaken. That made no sense.
I couldn't let my Shields fall. It would kill Taro. It would kill me. But the sliding turned into spinning, and there were moments when I wasn't sure whether I was standing up or falling or lying down. I curled my arms around my stomach, because letting them hang loose made them whip around.
I couldn't believe my Shields were failing. They never failed. I'd been in far more trying situations without a doubt that my Shields would hold. What the hell was going on?
The forces smoothed out; the flow lessened and lessened again. Still, I had to push hard, my arms tight against my stomach, my knees hard against my forehead. The images faded from my mind, the screaming birds were silenced, and all I could feel was the tension in my muscles as I fought to keep my Shields up.
I managed to say out loud, "We need another Shield here!"
"No, we don't!" Taro snapped. "We're all right!"
"We're not."
"It's almost over. You can hold on."
"I can't. We need another Shield."
"No one's here. And another Shield can't Shield me. Hang on. We're almost done."
But he lied, and the channeling went on and on and on. The pain in my nose tightened and spread over my forehead and into my temples. I ground my fingernails into my arms and just held on.
It took too long. It should never take so long. Our bodies weren't made for doing this for such an extended period. Why was Taro doing this?
The forces lessened a little more, and in time, a little more, and then, finally, they disappeared. Taro's own protections drew up around his mind, and I let my Shields fall. Finally. Thank gods.
Taro knelt beside me, his face unusually serious. "What happened?"
"I couldn't keep my Shields up." And that was so hard to admit—that I'd almost failed my Source.
"You did keep them up," he reminded me.
"Only because you slowed down," I said. He didn't respond to that. "This is too dangerous. I needed you to slow down because I couldn't keep my Shields up. That's a serious problem." I had never, ever thought I would be in danger of letting my Shields drop. There were Shields who had failed in that way. It almost always meant the death of the Pair. Once a Source began channeling, it was supposed to be impossible for him or her to stop, even when the Shield's protections failed. The forces would kill the Source, and the Source's death would kill the Shield.
That was why Taro and I had to be brutally honest about what was happening.
"So what are you saying?" Taro asked. "That you can't Shield? Do you want us taken off the roster?"
"Of course not," I snapped. Nothing so drastic was needed. "You're channeling events you're not supposed to channel. Stop doing that."
"I'm not going to stop doing that," he snapped back. "I'm supposed to channel these events."
"The only events you're supposed to channel are those that are occurring within High Scape."
"I know, I know, but something feels familiar about these events. I feel I should channel them. That something terrible will happen if I don't."
Now, that was alarming. "Familiar how?" I demanded. "Like when Creol was sending events to High Scape?" Channeling those events had been painful, and I had gotten images while Shielding. But there couldn't possibly be another bitter, crazy Source out there.
"No, nothing like that."
"Then how is it familiar?" I persisted.
"I don't know! There's just something about it that I've felt before."
I resisted the urge to ask, again, what he meant. There was no point in going around and around about it. "All right, fine. But these events aren't threatening High Scape, right?"
"Correct."
"So I need you to stop channeling them."
"There might be people there," he protested.
"We don't know that there are, and I could barely hold out this time."
"It's my fault, Lee," Taro said in a soothing, yet annoying, tone. "I channeled when you weren't expecting it, in a manner you've asked me in the past not to channel. There's no need to panic."
"I am not panicking," I said through my teeth. "I am being responsible."
"Aren't you always," he mocked. He kissed my forehead. "Come on, we're going to be late."
Why wouldn't he listen to me? He was risking our lives to channel events that, for all we knew, were taking place out in the middle of nowhere. Why was he being so stubborn about this?
Because he was always stubborn, especially when it came to the use of his abilities. But it wasn't his abilities at issue. He was apparently having no difficulty channeling. I was the one having problems. He needed to listen to me when it came to how my Shielding was functioning.
How was I going to convince him of that?
Chapter Fifteen
"I'm not going to let this go, Dunleavy," Risa said just before taking another long swallow from her roofer's black, a heavy dark beer that I thought tasted terrible. She'd brought it with her, and the windows in the room would all have to be left open for hours to get rid of the heavy yeasty odor. It was making me nauseous.
"Let what go?" I knew exactly what she was talking about. I wasn't going to help her along.
"Costume," she drawled.
I rolled my eyes.
"Dunleavy, I will nag at you and nag at you until you tell me," she warned me.
Fine. "It really wasn't that interesting, Risa. I danced the benches against spectators who paid for the privilege." It was the truth, if not the whole truth.
"And the costume?" she prodded.
All right. Something less truthful than the short-skirted, midriff-baring embarrassment in scraps that I'd been forced to wear, but strange enough that Risa would believe I'd be made uncomfortable by it. "These strange trousers that came only to my knees. And a blouse that wasn't too bad, except it was covered in all this golden glitter, and they put glitter in my hair and on my face."
"Oh," said Risa, and she sounded disappointed. "That doesn't sound too bad."
Really? I'd like to see her wearing something like that. "It was bad enough."
"And you had me thinking there was something interesting going on."
"Come now, Risa. When have you ever known me to be involved in anything interesting?"
And I knew right then that I'd overplayed my hand, because that was a stupid thing for me to say, and the look Risa gave me told me she thought the same.
"I'll just get Shintaro alone and ask him," she promised. Or threatened.
I shrugged as though I didn't care. And I didn't. While Taro was happy enough to tease me when he had an audience—and often when we didn't—he knew how much I'd hated the whole Leavy the Flame Dancer experience, and I knew he wouldn't betray me.
"You look like hell," Risa said, because she could be ridiculously blunt if she wanted to be.
"Thank you so much," I said without heat. I'd been hearing similar comments, though less blunt, so often that I couldn't get angry at every one of them. I'd be exhausted. "Did I invite you over? I really can't remember."
I sipped on the revolting tea Ben had brewed for me. It tasted foul, but it did ease the ache in my head a little, which was the point of it.
"I came over to see why you're begging off of drinks all the time. Can see for myself, now. What's wrong with you?"
"Just tired and whatnot."
She eased back on the chair. "Nothing catching, I hope?"
"No. It's too much bile or something like that."
"Are you sure that's all it is? You look really bad."
"The healer said it was nothing to worry about." And I was beyond being bored with talking about my health.
"Where do you get your water?"
Wasn't that a bizarre leap? And one I wasn't able to follow. "What?"
"The water that's used in this house. Where does it come from?"
"There's a well out back."
"Is it connected to one of the rivers?"
"Aren't they all?"
"No, though a lot of them are."
"Well, I have no idea. I've never thought about it."
"They think that's where the Riverfront Ravage is coming from. It's something in the water. That's why the illness is showing up in other parts of the city. Because their wells are fed by the rivers."
"Holy hell." That would be a nightmare. What would people do without water? "What's wrong with the water?"
"They haven't been able to figure that out yet."
"Brilliant."
"Is anyone else here ill?"
"I'm not ill," I said. "I'm just tired. And I have too much bile. I think if I had the symptoms, the healer would have at least mentioned the riverfront illness."
"You look really pale."
"I'm always pale."
"Really, you don't look well."
"Risa." Her insistence on talking about it was making me think about it, and that just made me feel worse. "Leave it alone. Please."
"All right, all right."
"Thank you."
"Anyway, I didn't come to talk about any of that. We found out who killed that mayor."
"Oh?" I didn't really care. I was upset about the fact that he'd been murdered, but I didn't really care about who'd done it. There was no reason for Risa to think I should know anything about it.
"It was one of his servants. A woman named Sara Copper."
"I see." Still had nothing to do with me.
"She's your Ben's daughter."
That shocked the hell out of me. "Ben Veritas's daughter?"
"Aye."
"My gods. Does he know?" He hadn't said anything while he wrapped my hand early this morning, and I hadn't seen him since. If he did know, it would explain his absence today. Though none of his earlier absences. He'd been gone a lot recently.
"Probably not. She was arrested just this morning."
Oh. "And you're telling me first? Why?"
"She killed him," Risa continued in a lowered voice, "so she could sell his ashes."
"She killed him for his ashes?" I echoed dumbly. Yes, he'd been murdered. Yes, his ashes had been stolen. I hadn't thought both crimes had been committed by the same person.
"Aye."
"I thought they were digging the ashes out of groves."
"Maybe they're running out of likely sources in the ash groves," said Risa. "Now that more and more people are doing it."
"She didn't want to wait for someone to die? She murdered him? That's insane."
"Murderers aren't known for being rational."
"I'd think that would depend on the murderer." Was it possible? Were people really getting that crazy? Because this was a whole different issue. Digging up ashes, while disgusting and disrespectful, didn't really hurt anyone.
But killing people. Killing people deemed to be particularly lucky. Just so their ashes could be consumed by or sold to de lusional layabouts who wanted spells to fix whatever they thought was wrong with their lives. "She said she killed him for his ashes?" Zaire, this was Ben's daughter. Poor man.
"Zaire, no. She's not admitting anything. But she was one of the servants who went missing from the mayor's house."
"That's hardly conclusive evidence," I objected. "There are all sorts of reasons why she would run. Fear of being blamed for his murder would be a big one." I would do the same, in her shoes.
"There is more evidence than that. I can't tell you what it is. I shouldn't really be telling you this, but I thought it was important you people knew the kind of man who's living with you and working for you," said Risa.
"What kind of man? You mean Ben?"
"Of course I mean Ben. His daughter is a murderer."
"So? That doesn't mean he's a threat to us. It's not like it's the sort of thing that runs in a family."
"You'd be amazed how much crime does run in the family, Dunleavy."
That was, I thought, a horrible and dangerous assumption to make. "How old is this woman?"
"Twenty-five or so."
"Was she raised by Ben?"
Risa shrugged.
"He lives here. He's worked for the Triple S for years. Decades, maybe." I realized, to my shame, that I didn't know for sure. "I'm pretty sure if his daughter is into this sort of thing, it's off her own bat."
"Oh, she did it all right."
"You can't know that unless she confessed."
"No, there will have to be a trial. But she did it. There's no doubt. And after she's convicted, she'll get as many lashes as the charges demand. If she survives that, she'll be hanged." I grimaced. "She's a murderer, Dunleavy. Don't waste your sympathies on her."
"It's barbaric," I insisted. "And it says something about us, all of us, that we'll inflict that kind of pain on her before killing her. And that there are those of us who'll enjoy watching it."
"Not you, of course," Risa sniffed. "You're above that sort of thing."
Well, yes. Did that make me arrogant? "It is not to my taste." And I didn't understand how it could be to anyone else's. "So you are here to tell Ben about his daughter?"
"No, that's not my place."
It wasn't her place to tell me, either. "Who will tell him?"
"Some member of the family, I guess."
Wonderful. I knew something that was none of my business, and no one was going to tell Ben? That wasn't right. "I'm out of tea," I said, carefully rising to my feet. "Please excuse me while I get some more." I walked from the room before she could utter more than a couple of words in objection. But Ben wasn't in the kitchen, or in the hall, and I didn't want to alert Risa to the fact that I was looking for him by calling out for him.
And perhaps it wasn't kind to spring Ben on Risa and force her to tell him what she'd told me. Who was I to tell her what to do? Damn it. I filled my cup from the pot of tea brewing on the stove, wrinkling my nose at the strong smell and returning to the parlor.
"What was that about?" Risa demanded.
"I just felt a little unwell," I lied. "The tea really helps."
"Ah," said the Runner, swirling her ale in her mug. "You don't need to be delicate about such things around me."
I did when I was lying. "Wait a moment," I said as a few things clicked together in a mind that had apparently gone to sleep at some time. "You think she killed him to get his ashes?"
"That's what I said."
"And the ashes need to come from someone lucky."
"Apparently."
Damn it to hell. Why was it that every time there were weird killers about, the characteristics of the potential victims matched Taro? It was getting ridiculous.
And Taro didn't know. He was out with friends, doing who knew what. One woman had killed a man for his ashes, and had gotten caught, but if one person would try something like that, someone else was sure to try it, too. People were stupid like that.
My stomach clenched with dread and sharp panic. I had to find him and tell him. I didn't know where he was, damn it.
"What's wrong now?" Risa asked.
Damn it, why was my face so easy to read now? "Nothing," I said. "Just worried about Taro."
She rolled her eyes. "Why?"
"It's what I do." I didn't want to tell her why. She would tell me I was overreacting, which would be annoying, or she would agree I had something to worry about, which I really didn't need to hear.
I wasn't much of a hostess after that. I became increasingly worried about Taro. Because what if the mayor had been murdered because he was lucky and they wanted his ashes, but Ben's daughter wasn't the actual murderer? There might be some homicidal idiot wandering about, cocky because he or she had gotten away with murder.
And the people Taro banged around with, crowds of people. He probably didn't know them all well. He probably wouldn't notice or care if someone new joined the throng. Someone who could lure him away to somewhere more remote. He wouldn't know enough to beware of strangers.
Risa finally, finally left. I was tired and nauseous, but my growing panic gave me the fortitude necessary to leave the residence. I had to find Taro.
Chapter Sixteen
My first step out of the residence was an unpleasant education for me. The sunlight speared into my eyes and the odor churned a stomach already too delicate. And the noise, it set my ears to ringing.
I hadn't been out since the day before, for the watch at the Stall. And I hadn't expected to need to be out that day until our late-evening watch, something I had looked forward to, getting some solid, uninterrupted sleep. I would be able to get a handle on this illness and start getting better if I could just be still and quiet for a while. That was why I had picked at Taro until he had stormed away in a huff, else he would have hung around the residence asking me if I needed anything every half hour or so.
But then Risa had arrived, refusing to be put off. So that was the end of my peaceful day.
This was also the first time I'd been out of the residence without Taro in a good long while. I'd had no idea I'd come to rely so completely on his supporting arm. After only a few steps I was unusually tired. But there was nothing to be done about it. Taro had to know someone had killed someone they'd deemed lucky, just so they could have his ashes after he'd been cremated. He needed to know he had to be on guard.
Realizing I wouldn't get far by foot, I tried to flag down a carriage. The first three carriages that approached me, all without passengers, refused to stop, the nasty buggars. My hand went to my left shoulder, and yes, the braid was still there.
The fourth carriage stopped when I raised my hand, but the driver narrowed his eyes at me. "Here, now, you've not got some plague, have you?"
"No, just a headache." Was that why the others had refused to stop? Did I really look that bad? I knew I was a little pale, and my difficulty sleeping had darkened the shadows under my eyes, and I hadn't bothered with cosmetics that morning, but I didn't think my fatigue was beaming out to strangers seeing me from a distance.
The driver studied me for a bit, chewing on something hidden in his mouth with slow, long slides of the jaw. He grunted. "Right, then. Where to?"
"The Lyre Loft," I said, naming Taro's current favorite watering hole.
I was gasping for breath after the ordeal of climbing into the carriage, and sweating unpleasantly. The shocks and swaying of the carriage were causing my nausea to bubble up, almost enough to make me disgrace myself right then. But I had to find Taro. He had no idea he was wandering around out there with someone looking to kill him.
It wasn't a long ride, and though I could have used some more time sitting down, I was relieved to exit. I almost tripped getting out, and the driver whistled and whipped the carriage off barely after I had my two feet on the street. Prat.
The Lyre Loft was one of the nicer drinking and gambling establishments in High Scape, though it avoided ascending into exclusivity. Anyone could go there, but anyone who appeared too obnoxious was encouraged, usually successfully, to leave. It was a little larger than the usual tavern, and cleaner, with furnishings of light red wood and small alcoves along the walls that allowed enough privacy for intimate conversation but not so much privacy that people were tempted to try activities less appropriate.
There weren't that many people there, it being early in the afternoon. Perhaps that was why the two fiddle players in the corner were so nerve-piercingly bad. They were practicing. No one else seemed to notice the flat notes or that the two players kept falling out of tempo with each other. I wouldn't have thought it possible for only two players to fall out of unison so regularly, as though each thought he was playing alone.
Taro was not among the patrons. Damn it.
Jek, one of the regular workers at the tavern, came in from the back room carrying a box of bottles. I headed toward him, and when he noticed me, he immediately called out, "Tasa! Vinori! Cut the music!"
The music wasn't having any kind of impact on my Shield sensibilities. Because it was so bad? Was that a possibility? It didn't make sense that only well-played music could influence my behavior, did it?
The grating wails halted, and when the fiddlers looked up and saw me, they lowered their instruments and started talking to each other. It wasn't necessary for them to stop, but the silence was blissful, so I didn't correct their misapprehension. "Thank you, Jek."
He nodded. "Are you all right, Dunleavy?"
I didn't know if he was referring to the music or the fact that I apparently looked awful. It didn't matter. "Yes, quite. Has Taro been in today?"
"Afraid not."
I sighed. Damn it. This was his favorite place. Aside from the racetrack. The chances of finding him at the track, even if he was there, were slim. And the thought of going there, with the noise and the crowds and, oh my gods, the smell, made me queasy, pressure building around my eyes as the bar in front of me seemed to slide back a few feet.
"Are you all right?" Jek asked again.
I nodded by means of lowering my head, slightly and just once, and swallowed so I could say in a low voice, "Thank you for your help." Though I wasn't sure whether he'd really done anything helpful. "If you see Taro, please let him know I'm looking for him. Please tell him it's important." And what was so important? Oh. Aye. Someone was trying to kill him.
And suddenly, Jek was standing right in front of me. How had he managed to walk right through the bar like that? It was made of wood, wasn't it? I tapped it with my knuckles just to make sure.
"You're looking really bad," Jek was saying, and then he went on to talk about carriage and home and I didn't really listen.
"Thank you," I said, and I made my way carefully to the entrance, keeping myself tall and upright, walking with dignity.
Jek said something else from behind me, but I wasn't sure he was speaking to me, and it didn't matter anyway. I needed to find Taro.
Why wouldn't the door stay still?
Without quite knowing how I'd accomplished it, I was back outside. All right, that was good. Now I just needed to get to the racetrack. If Taro wasn't at the Lyre Loft, then he must be at the racetrack.
I called for a carriage, and a sound must have come out, because suddenly there was a carriage standing there in front of me. It took two tries to get my foot on the footplate and whoever was standing directly behind me was surely Zaire-sent, for I wasn't sure I could have climbed into the carriage without his help.
It was only once I was seated in the carriage—if one could call it seated, my head down at one end of the cushion and my rear end pressed against the opposite wall of the carriage—that I realized I couldn't remember giving the driver directions. But I must have, because the carriage was moving.
A lot, damn it. I pressed a hand to my stomach, trying to prevent anything in it from escaping. I was not going to throw up in a carriage.
I wondered if there was any kind of spell to make me feel better. I'd have to look into it. Not that I actually believed in any of that stuff, but hey, it couldn't hurt to try. As long as I didn't have to eat or drink anything disgusting.
Oh, I shouldn't have thought about drinking disgusting concoctions. I groaned and curled more tightly around my stomach.
The carriage came to such a jolting halt that I landed on my knees on the floor. I experienced no surprise at this, because that was just the sort of day I was having. I tried to crawl back onto the cushion. I hadn't accomplished that by the time the door behind me had opened. Then there seemed to be hands all over me.
"Hey," I said in protest. And that was all I said. I couldn't really think of anything else to say as I was pulled out of the carriage and set unsteadily on my feet.
I stared up at the ugly gray building before me, stretching for the full length of a long block and with almost no windows. "This isn't the racetrack," I said stupidly.
"No, ma'am. If you'd please lie down here."
"I need to go to the racetrack." Why were people trying to keep me from finding Taro? I turned against the hands on my arms and shoulders to get back into the carriage. But the carriage was no longer there. "Hey!"
"Please come in, Shield Mallorough. We're going to take care of you."
"No!" I shook at the hands grabbing me, slapping at everything that connected. "I have to find Taro! They're going to kill him!"
"I'm sure Source Karish is quite all right. Just lie down here and we'll get you comfortable."
"No!" I pushed and shoved and shimmied with all my strength. Unfortunately, I didn't have much. "Let me go, gods damn it! Help!"
"Hush," a stern voice said. "You're making a spectacle of yourself." And despite my best efforts, I felt myself being lifted off my feet and laid across some surface.
And then straps were tightened over my chest, my feet and my waist, tying my arms, tying everything down.
I just started screaming. What were these people doing? Who were they?
There was the sharp incline of stairs, and then there was darkness. My screams seemed to echo. Something soft and sweet smelling covered my mouth and nose, and that really alarmed me. I tried to jerk my head to the side, but the softness just followed, and a palm on my forehead forced me to stillness.
With the second breath, warmth spread from my face down my arms, torso and legs, forcing muscles to relax whether I wanted them to or not. With the third breath, the whirling thoughts in my brain slowed right down. Whether I took a fourth breath or not would always be a little unclear to me.
The next thing I knew was silence, and it took me a few moments to determine whether I was asleep or not. Then my stomach gurgled into almost unmanageable nausea, and I knew I was awake. I put a hand over my eyes as I fought not to throw up.
"You had me worried there," I heard Taro say, and I uncovered my eyes. The ceiling above me was stone, as was the wall out of the right corners of my eyes. I looked to my left, realizing I was in the only bed of a stone cell, and that Taro was seated in a chair beside the bed, a racing circular on his knee. "I don't think they meant for you to sleep this long."
They. Hospital. The taste in my mouth was foul beyond the telling of it. I looked at the hand I'd put on my face. At least they'd untied me.
I was so tired.
"Would you like some water?"
There was a small table on the left side of the bed. On it rested the candle that was the only source of illumination in the room, a mug and a single flower—a blue lily—in a slim glass vase. Water sounded divine, so I shifted up on the bed to sit up. It was hard work.
"No, I'll get it," Taro said quickly.
I was sure he would, but I hated the idea of him propping up my head while I tried to drink lying down, so I continued to press myself upright. By the time my back was against the wall, I was shaking and sweating and dreading the effort it would take to lie back down again. Taro, having sighed loudly over my stubbornness, waited until I held out a hand for the mug.
I didn't object when he continued to hold on to the mug after I'd gripped the handle and brought it to my lips. My hand was shaking badly, and I might have dropped the mug without his assistance.
The water tasted awful, though that may have been due to the taste already in my mouth. It did feel good going down, clearing out the sticky sensation in my throat. There were a few tense moments, however, where it was in danger of coming back up.
"What happened?" Taro asked after I'd sipped down most of the mug. "You look terrible."
I would have stuck my tongue out at him if it wouldn't have taken so much energy.
What did happen? I had to think about that.
"They said you were in hysterics," Taro continued.
I was not. Not until they started grabbing me and yanking me and tying me down, and wouldn't that make anyone upset? "Where is this?" I demanded.
"Just a private room at the hospital," he said. "I had you moved from the public ward when I got here."
I wasn't entirely reassured. "Why's it made of stone?" Cells were made of stone, weren't they?
"It's quieter. They also prefer these rooms if they feel someone needs to be quarantined. But they don't seem to think you're contagious."
Of course I wasn't contagious. I was just . . . tired. Gods, I was so tired. But my mind was slowly clearing. "Someone's trying to kill you," I said, but the words seemed to come from nowhere. I remembered believing that someone was trying to kill him, but I wasn't quite sure why.
His eyebrows rose. "Again?"
All right. Try to remember. That morning, I woke up too early, an annoying habit I'd developed recently. Spent the morning not eating the breakfast I had put together. Ben freshened the poultice on my hand. Risa had shown up and told me about Ben's daughter. Ah, there it was. "Ben's daughter was arrested for killing the mayor," I told him. And while the shock widened his eyes, I added, "Risa thinks she killed him for the ashes."
"That's kind of extreme, isn't it?"
I shrugged. "That's what she said. That the mayor was killed because he was thought to be lucky."
Taro whistled. "Poor Ben."
"So I was worried that you might be considered a good target for the same kind of attack. Because, you know . . ." I trailed off to allow him to come to the proper conclusion on his own.
He frowned. "Because what? Because I'm lucky? You think I'm lucky?"
From his tone, it appeared that he had taken some sort of offense. I hadn't expected that. "You don't?"
He stiffened. "Do you know me at all?"
I was an idiot. Having been with him for the past few years, I could understand why he wouldn't think himself a particularly fortunate person. "Look at your life from the point of view of someone who is desperate enough to use a spell to fix whatever problems they have in their lives. To them, you lead a charmed life. The son and brother of a wealthy duke, and all that means to people who don't know any better. You're a Source. You're terribly handsome."
"You always make beauty seem such a bad thing," he murmured.
"I didn't mean 'terribly' in the negative sense of the word."
"There's a positive sense?"
There was no point in turning that into an argument. "Do you admit that people who don't know you well might think you're lucky?"
He shrugged.
"I just didn't feel comfortable having you out and about without knowing someone had been killed because they were thought lucky."
"But you couldn't wait until I got home?"
"Apparently not." Thinking back, I couldn't remember why I was so sure he had to know immediately. Even if he were a potential target, he'd been out with friends, in daylight, and he rarely spent the entire day out. Why didn't I think it could wait until he got home?
"That doesn't explain why you were hysterical."
"I wasn't hysterical," I objected. "I was upset." And why was that, again? Oh, right. "I got in the carriage expecting to go to the racetrack and ended up here. I didn't know where I was at the time." I didn't think I'd ever been to this hospital before. "People pulled me out of the carriage and were pulling me this way and that. They tied me down—"
"They tied you?"
"Aye. I was angry." But I remembered screaming, and not making much sense. How humiliating. Why had I been unable to express, calmly, rationally, why I couldn't stay? All I'd had to say was that yes, I was a little unwell, but it was nothing serious, and it was vital that I find Taro. My Source. They would have left me alone. Instead, I'd shrieked and struck out like a mad-woman. What was wrong with me?
After a brisk knock, the door opened and a healer walked into the tiny room, his bald head scraping the ceiling. He had oddly sunken eyes, and while I usually enjoyed prominent cheekbones, his appeared more like squarish blocks of bone pressing right up through his pale skin. His shoulders were freakishly broad, too broad for his body, so his long arms hung down as though they were attached just to the underside of his shoulders, and not the rest of his body. His hands were huge, with prominent knuckles. His loose trousers hid the shape of his legs, but they were long in proportion to his body.
I couldn't recall ever being so disturbed by someone's appearance before. It was as though he'd been too carelessly thrown together and no one had taken the time to sand down all the edges.
"You're looking better," he said. His voice was as deep as his height would suggest, and oddly slurred, as though he'd been drinking. He'd better not have been. "Sort of."
"And you are?" Taro asked coolly.
"Healer Pearson," he answered in an absent tone as he unwrapped my right hand. "This isn't healing as it should be. Is this dressing changed daily?"
"Aye," I said.
"With what kind of compound?"
"I have no idea," I admitted.
"That seems careless." He left my hand unwrapped. "All right, now." He fixed me with a stern eye that seemed too small for its socket. "Time for some honesty. Were you drunk when you first arrived here?" He pressed two thick fingers against my throat as he waited for my answer.
"Of course not."
"Had you taken any nonmedicinal drugs?"
"No."
"Any medicinal drugs?"
"Just this." I raised my right hand. "But I have been under the weather lately. Nausea and headache, mostly."
"That wouldn't explain your hysteria when you got here."
"I wasn't hysterical," I objected snappishly. "People were yanking me this way and that and tying me down and they wouldn't listen to me."
"Because you were incoherent. Are you sure you weren't drunk?"
Prat, calling me a liar. "Yes." I let the sibilant stretch out.
From the bag he'd carried in with him, he picked out some short strips of cream-colored cloth. One he briefly placed over my temple, another on my tongue, and a third over the burn on my palm. "You can rest here while I test these," he said, placing each strip into its own small bowl.
"How long will that take?" I asked. "We have watch tonight."
For the first time, the healer appeared surprised. "You're still on the roster?"
"Of course." Why shouldn't we be? Whatever this illness was, it didn't seem to be too serious.
I remembered then the difficulty I'd had Shielding Taro recently. Was that because of me, rather than because of the nature of the events or how he was channeling? Hell, I could be so stupid.
"How disappointingly irresponsible of you," he said. "You're off the roster now."
"You may not be aware of how few Pairs we have in High Scape right now," said Taro. "We're stretched quite thin. I don't think we can afford to lose another Pair."
"You will take yourselves off the roster, or I will take you off myself. And if I do it, it will take you much longer to get back on it."
I found this healer off-putting, and I felt inclined to do the opposite of whatever he was ordering regardless of how much sense he made. It really irked that this stranger, a regular, could have any impact on my ability to perform my duties, regardless of how right he might be. But it alarmed me that he thought an upset stomach and some headaches were serious enough to necessitate such a drastic step.
"All right," I said.
"Good decision. I'll be a couple of hours with these."
The room felt much larger once he left.
Chapter Seventeen
Taro folded the racing circular, then he unfolded it and flattened it on his thigh. Then he folded it again, along different lines, and then he unfolded it again.
I watched him for a little while in silence. Now that the creepy healer had left, I was content to lie still and try to will my stomach into settling down. There was something almost soothing about Taro's movements, and I could watch him for a while without getting bored.
In time, though, it flittered into my sluggish mind that Taro had to be agitated if he was fiddling like that. "What's wrong?"
He looked up at me with an expression of shock. "What's wrong?" he demanded. "Are you serious?"
"Oh." I was so stupid. "Are you worried that this is something fatal?" Because if it were, he'd be dying, too, and I remembered what it was like to be in that position. Walking around, feeling fine, no way of knowing whether I'd suddenly die from one breath to the next. It nearly made me crazy.
"Aren't you?"
"I don't think it's anything that serious." Maybe it was ridiculous to be of that opinion, but I couldn't help believing that if I were dying, I would know it, or at least be worrying about it somewhere in the back of my mind. But I wasn't. I'd never heard of anyone dying just because they were tired all the time.
"Jek saw you at the Loft. He said you were a complete wreck."
"I was not." My manners had been perfectly correct. And Jek could mind his own damned business.
"Tell me the truth!" he practically shouted.
I blinked in shock. What had just happened? "Excuse me?"
"You've been trying to hide how ill you've been," he accused me.
"I have not. I'm not going to complain every time I get a headache. You'd find that pretty tiresome after a while."
"You collapsed today!"
"I did not." I took a deep breath and continued in my most soothing tone. "For some reason, today I felt particularly bad. But this has been the first time I've felt light-headed. And I'm here now, and they'll—"
"It should have never gotten this far. You should have told me what was going on immediately. Why are you always hiding things from me?"
"I'm not hiding things from you," I said in my most soothing tone. "Please calm down."
"Stop pretending you're the rational one!" he shouted.
I stared at him. What was going on with him?
"Why are you doing this?" he demanded.
"Doing what?" Getting ill? Totally wasn't deliberate.
"Playing these games."
"What games?" Why did he have to pick this moment to go crazy?
Oh, and there went the hands, right into the hair. "What games?" he bellowed. "You're making time with Laidley. And you don't want to go about town with me, and you won't let me go out with you."
Why in the world was he dragging all this up again?
I wasn't making time with Doran. I hadn't seen him since our picnic, which Taro knew all about. "I haven't been with Doran that you haven't known about it." And I resented having to tell him that. "What is wrong with you?"
"On Flatwell and the whole trip back here you were all over me."
I wouldn't have described myself that way.
"And then we're back in High Scape, and suddenly it's a whole new set of rules. And you're the only one who gets to decide what the rules are."
Rules? There were no rules. Certainly none that I had made up.
"Are you embarrassed by me?"
There was a long, stunned silence after that question. I felt like I was scrambling to come up with a response. "No, of course not." He didn't really think that I could possibly find him embarrassing, did he?
"Then what? What is it?"
I sighed. "There is no 'it.'" But that wasn't true. Zaire, I didn't want this conversation now. Or ever, really, but especially not right now. I was too tired and my head was still cloudy. Yet I couldn't deny that my behavior had changed, and I couldn't let Taro continue to think there was something wrong with him. That just wasn't fair. "I believe that now we are back . . . home . . . you will no longer wish me to have the role I had during our travels." There. It was out. Something I'd hoped I'd never have to say. It was humiliating to say. That should be enough.
But, of course, it was not. "Role?" he said, and from his tone, I had somehow made things worse. "Role? That was all just some act?"
"No!" All right. I was going to have to tell him the whole thing, flat out, and it would end things. I wasn't ready for this. "You're used to variety, Taro."
"What the hell do you know what I'm used to?" he demanded. "You've never bothered to find out."
What could he mean by that? I'd been living with him for three years. What further investigation was required? "There's nothing wrong with enjoying multiple partners," I said, and I meant it. It wasn't for me, but I could understand why it would be appealing for others. "As long as everyone knows the circumstances and they are willing. But I'm not willing to be part of a crowd. It's just not who I am."
He twisted the racing circular into a crumpled stick and tossed it aside, resuming his preferred manner of expressing anger and frustration: pacing. The room really didn't have the space for it, allowing him only three strides each way. He looked a little silly, really, constantly stepping and turning. "I'm to tell you everything while you tell me nothing, is that it?"
"I'm not asking you to tell me anything." To be honest, I'd rather not hear him be brutally honest. I wouldn't be able to keep my expression serene in the face of his true opinion, I was sure.
"So you can so nobly claim," he scoffed, "but then you can make all these ridiculous assumptions, and you can be all smug about how reasonable you are."
"Why are you yelling at me?" I asked plaintively.
"It's the only way I can ever get you to listen to me."
"That's not true," I muttered.
"I've not slept with anyone since I met you."
I was sure my shock was written clearly all over my face. Shock that he had just said that, so bluntly. Shock that he could have endured such abstinence, and for no reason.
He swore suddenly. "No, that's not exactly true," he amended.
Hah. I knew it. Though if he'd slept with someone else since we'd started sleeping together, I'd have to kill him.
Or just curl up and die myself.
"A couple of times, in our first few months in High Scape," he admitted through clenched teeth. Then he kicked the door.
I wondered if he'd hurt himself. "There's nothing wrong with that." But, only two? I couldn't believe it. Not that I thought he was lying. Perhaps, as at first he hadn't remembered those first two incidents, he wasn't remembering others. Though I really couldn't imagine being unable to remember everyone I'd slept with.
I had no intention of interrogating him about it. It was none of my business. And I really didn't need to hear a list of his conquests.
"I know there's nothing wrong with that!" he snapped. "Listen to me! Not dozens! Not hundreds!"
Not even I had been thinking in terms of hundreds.
"Two!"
"I understand," I said.
"And that doesn't change the way you think at all?" he pressed.
Oh. That was what he wanted.
Well, to be honest, even if he were forgetting some encounters, that was a lot fewer than I ever would have guessed. "How about before we bonded?"
"You really think you have any right to ask about what I did before we bonded?"
"No, not really. I guess I'm just trying to understand."
"What's to understand? I just told you!"
Gods, he was confusing. "Are you trying to tell me that you did not enjoy a variety of lovers before we met?"
He opened his mouth several times to give me an answer that he clearly changed his mind about, because he finally pressed his lips together without saying anything.
"Are you trying to claim that somehow meeting me changed all that?"
"No!" was his immediate answer, much to my private disappointment. "Damn it, Lee, that was three years ago!"
"What does that have to do with anything?" Three years wasn't that long a time, once one was an adult.
"When people are bonded, it is time to put aside childish things," he said.
"I don't know that many would call sex a childish pursuit."
"Will you stop with the word games!" he roared.
"I'm not playing games," I insisted. "I'm just saying that no one expects you to change your personality once you bond."
Unexpectedly, he laughed. It wasn't a happy sound.
Just as unexpectedly, the door swung open. It was not Healer Pearson but a young woman wearing the colors of the cleaning staff, poking her head and shoulders just beyond the door without actually stepping into the room. "Will you be quiet?" she hissed at Taro.
He sniffed. "I was assured we could not be heard beyond the walls of this room," he said with a pronounced roll to his _r_ 's.
"Not when you make such efforts to shout at the top of your voice." Without waiting for any further response, she was back out of the room, closing the door behind her with a soft, but firm, rebuking snick.
Taro scooped the circular up from the floor and sat back on the chair, untwisting the paper and flattening it on his thigh. He stared down at it as though he hadn't already read the whole thing while waiting for me to wake up. So I guessed the argument was over, without having really accomplished anything. But that was the usual nature of arguments, wasn't it?
Two people? That couldn't really be possible, could it? And why would he restrict himself that way? He'd made it clear that it had nothing to do with me, and I believed him. There had been no reason to change his habits so completely from before we'd bonded to after we'd bonded. Unless he was claiming that he'd barely slept with anyone before we'd bonded, a question he'd avoided.
So perhaps he didn't require the kind of variety I'd suspected. At least, not all the time. Then again, it had been a chaotic three years. Maybe things just needed to settle down into a regular schedule for more than a week or so in order for him to resume his former activities.
I didn't know what to think and my head still hurt. I had a feeling things would get worse if I continued to talk, so I let myself fall asleep instead. I woke when Healer Pearson came back in, jerking into consciousness at the sound of his abrupt knock. "Well, it's nothing contagious," he announced.
We knew that. "But what is it?" I asked.
"Niyacin powder. It's a common means of fighting infection, but there are those who have a strong reaction to it. Reactions can be anything from light nausea to death."
There had been too many bizarre surprises that day. "Death?"
"Aye."
"But I wouldn't have died."
"I wouldn't say that. It looks like the effects have been building up over time. It's possible that you would have died if this had continued."
My gods. This was unbelievable.
And he was so cavalier about it. The tone he'd used was appropriate for announcing it was raining. What was it about healers that they could be so cold? "I see."
It didn't seem possible. A few headaches, some fatigue, nausea now and then. How could such innocuous symptoms add up to possible death? It was just too melodramatic. "Are you sure?"
He stiffened. "I would thank you not to question my competence."
I was not going to apologize for asking legitimate questions. "I'm not doing that."
"Would you like to hear the treatment or is there a traveling man you'd like to consult?"
"I'd like to hear your treatment for comparison purposes," I snapped back.
He glared at me. "Drink as much mint and anise tea as you can," he said. "For the next two weeks, the only reason you should be without a cup of tea in your hand is because you're asleep."
"And what will that do?"
The question seemed to irritate him. "It will flush the adulteration out."
"Really." It sounded a little fantastic to me. I was on my way to dying, and tea was going to fix it?
"My, Shield Mallorough, I wasn't aware you were a healer as well as everything else you do," he commented with crude sarcasm. "How do you find the time?"
I didn't bother to hide the rolling of my eyes, as he was being a prat and I didn't have to worry about managing his feelings. I was sick. I could get away with it.
"You need to eat as much beef as you can, as rare as you can tolerate, to build up the healthy elements in your blood. And I want you to bathe in hot water at least twice a day, to clear away the toxins as they rise from your skin."
The path to recovery sounded exhausting.
"And while there appears to be no infection about your hand, you're to have your bandage changed at least three times a day to make sure it stays completely clean. I don't want it complicating matters. Make sure there is salt and mercury in the poultice."
Salt and mercury. That wasn't what Healer Cree had said. But then, she hadn't actually examined my hand.
"You're to have lots of sleep and no excitement. And don't expect to be back to normal anytime soon. Something like this can take months to clean out of the body."
"Months!" I said, shocked. I was going to feel like this for months longer? I would go mad. "What about bench dancing? Can I do that?"
"Do you feel like bench dancing?"
"No." The very idea of it just made me want to curl up and sleep.
"There's your answer."
"But wouldn't it be good for me? To sweat out the illness or something?"
He snorted derisively. "How positively barbaric."
All right, then.
"Take her home," Pearson said to Taro. "And be quick about it. We need this room."
He left then. Really, he should assume a more sedate pace. Watching him move about could make a person dizzy.
Chapter Eighteen
We took a carriage home, and Taro was all for bundling me up to bed, but there were a few things I felt I had to do before shutting myself away for the day.
Ben was back from his errands, cooking soup in the kitchen. I watched him from the doorway for a moment. He was so involved in what he was doing, chopping vegetables with fluid precision. He had no idea what was coming.
I cleared my throat. "Ben."
He looked up at me. "Shield," he said. Then he seemed to take a closer look at me. "You don't look well, Shield Mallorough."
"That's one of the things I have to talk to you about."
He put down his knife. "Is there something wrong?"
"Can you leave this for a while? I'd like you to come up to my room for a bit."
"Of course." He covered the pot and followed me up to my sitting room.
Once we were settled, I asked, "Have you been using niyacin powder in the poultices for my hand?"
He blinked in surprise. "Yes, of course. It's a common medicine to prevent infection."
"Well, it turns out I have a negative reaction to it. I turns out it could kill me." I hated saying that. It sounded so ridiculous. And it sounded like I was accusing him of something. I wasn't. I didn't blame him.
"My lords, I had no idea." He leaned back from his proper position at the edge of the settee. "I will, of course, stop using it immediately."
"Thank you. I'm looking forward to feeling better soon." And quickly getting back on the roster.
"Please accept my apologies, Shield. I had no idea—"
I raised a hand to stop him. "The healer said the powder is often used. You had no way to know I'd react that badly to it."
"That's very gracious of you."
"No, no, not at all." No stalling now. It was time to get it over with. "I'm afraid I have some bad news for you."
"I see," he said calmly.
An odd reaction to a warning of bad news, I thought. Was it possible he already knew? It was his daughter, after all. "I have been told that Sara has been arrested for the murder of Yuri Izen."
His eyes widened, and then he stood abruptly, turning his back on me.
The silence stretched, and I had to resist the stupid urge to ask him if he understood. Of course he understood. He wasn't an idiot.
So instead I said, "I am very sorry."
He turned back around. It was hard to read his expression. He seemed annoyed more than anything else. "I have to admit I was worried she might have had something to do with it."
"You what?" I squeaked.
"She was a servant of his. She left his house immediately after his death was discovered. And while we didn't see each other often, I was always able to find her when I wanted to. I haven't been able to contact her since she left the mayor's home. I'd hoped it was only a coincidence."
"I see." He hadn't told me that. Had he told anyone? Was it realistic to think he should? She was family, and he would have had no way to confirm whether she had anything to do with Izen's death if he hadn't been able to speak with her. For all he knew, she really had nothing to do with the mayor's death but had feared suspicion and had run in an ill-advised attempt to avoid it.
Except they thought she'd stolen his ashes, too. There had to be a reason for that belief. "To your knowledge, did she believe that using the ashes of lucky people would bring her good fortune?"
He appeared even more surprised than when I'd told him his daughter had been arrested. That was odd.
"You know of such things?" he asked in a voice lowered nearly to a whisper.
"Of course. Everyone knows." Didn't they?
"People of your," Ben stammered, "of your stature shouldn't be bothered with such things."
"What things?" I asked, deciding that to question him on what he meant about my stature would take us off on a tangent.
"Foolish things," he muttered.
"You feel believing in casting is foolish?" Thank Zaire. I thought absolutely everyone in the city was going mad.
He wiped his mouth and avoided answering that question. "Has she confessed?"
"I've been told she hasn't." I found it curious that he immediately assumed she was guilty. And was prepared to admit that to me.
"So there will be a trial?"
"There damn well better be." With a possibility of a finding of not guilty. To hear Risa speak of a trial followed by a sentence, as though the trial were nothing more than a show for the sake of procedure, had chilled me.
"Good." He nodded, and he kept nodding. "Good. Good. Thank you for your time, Shield Mallorough."
I didn't try to keep the poor man with me. I doubted he would want to discuss such a personal tragedy with me any more than he had already. And there was nothing I could say to comfort him. Soothing lies never did anyone any good, and it wasn't as though I actually knew anything about how a trial or anything else in this matter would work.
I had another distasteful task to perform. One I wasn't sure I should be performing. Risa had told me about Ben's daughter because she felt we, the people who lived with him, had a right to know. I didn't know if I agreed with that. This information fell purely in the category of Ben's personal business.
On the other hand, this did involve murder. Perhaps something this serious transcended personal business. And Ben might need more time to himself, might want to spend more time with his daughter. If everyone knew why, they would be more understanding. I hoped.
If there was anyone who was the arbiter of correct behavior, at least in his own mind, it was Chris LaMonte. He was the oldest of all the Sources and Shields in High Scape, and he thought that gave him some sort of authority. There was no real hierarchy among Pairs, and if there were, Taro and I would be considered the senior Pair in High Scape because we had been there the longest.
I wanted someone else's opinion on what I should do with the information I had, and it seemed to me that LaMonte would be the best person to ask. He thought a lot about what kind of conduct was appropriate in what kind of circumstances.
I went to his suite, considering it the first logical place to look for him. He surprised me by opening the door. That was unexpectedly easy.
"Dunleavy," he said in greeting. Then he squinted at me. "You look like the moon."
I was pretty sure that was a Source way of saying I looked terrible. "Thanks. Can I talk to you?"
His eyebrows flew up in surprise. I didn't blame him. I never really wanted to talk to LaMonte. He was so arrogant.
"Please come in," he said, standing aside and letting me into a sitting room that was surprisingly dark. Dark brown walls, black furniture, deep red carpeting. Very depressing. "Please have a seat."
"Thank you." Once we were seated, I went straight to the point. "I learned today that Ben's daughter has been arrested for the murder of the new mayor. They think she stole his ashes, as well."
LaMonte was clearly shocked. "Who told you that?"
"Risa. The Runner."
"She shouldn't be telling you things like that."
I kind of agreed with him, but when he said so in that chiding, pompous tone, it made me want to find a good reason to disagree. "She felt we had a right to know because we live with him."
"That may have been so had Ben been the one suspected of murder, but his daughter has nothing to do with us. His family business is his own."
"So you don't feel we should tell the others?"
"Certainly not. I am ashamed of you for telling me."
All right, all right, I got the point. I was completely in the wrong.
"I hope I can rely on you not to treat Ben any differently because of this."
"Of course not." It was nice to know he thought so highly of me.
"Then we need say no more on the subject."
Fine with me. Gods, he was so annoying. I was so glad I'd asked. Damned waste of time. And it was past time I was in bed.
Chapter Nineteen
Abrupt though his manner had been, Healer Pearson appeared to know what he was talking about, as after a day or so I began to feel a great deal better. I hadn't realized how tired I had felt, how constant the nausea had gotten, how much my head had hurt.
I didn't suggest we return to the roster. I wanted to, now that I was feeling a little better, but I didn't want Pearson to feel forced to take us off. If he did, he would then send a report to the Triple S council to inform them why the removal was necessary. While the removal was quick, I had no doubt that resuming our duties would be a much longer process, because we would have to convince Pearson I was well again, wait for him to write another report and send it to the council, then wait for the council to actually read it, make a decision and send word back that we could work. That could take ages.
I sipped on a cup of tea the healer had recommended. I had to give him this, he recommended good tea. It smelled nice and was soothing on the stomach.
"You have a letter from Morgan Williams," Taro called from the foyer.
I was in the kitchen. It was so childish to shout from the kitchen to the foyer. I did it anyway. "Who's Williams?"
"The chap who owns the bordellos."
Oh, that was right.
Firth, who had been silently buttering some toast as a prelude to escaping my presence, raised her eyebrows at me in an expression of disapproval. "You are keeping company with the owner of a bordello now, are you?"
My first impulse was to exult over the quality of his services. Since that would be a lie in which I could be too easily caught, I said, "I am."
Taro entered the kitchen at this time, handing me the correspondence.
"Do you really think that's appropriate?" Firth demanded.
"Why wouldn't it be?" Taro asked.
"He lives off the profits of people selling their bodies for money!"
"It's perfectly legal."
"Unfortunately, a great many immoral acts are perfectly legal. That does not mean those acts should be condoned or executed."
This was not a discussion worth having. I opened the envelope and found within it an invitation to dinner. This created a sort of conflict within me. The dinner at Fines's home had been interesting enough, but I'd felt very out of place there. However, it would clearly irritate Firth if Taro and I accepted the invitation, and given how much Firth was aggravating me, I thought it was time to return the favor. I handed the letter to Taro. "I'd like to go."
He read it quickly and smirked. "Should be interesting."
Firth huffed. "If you aren't going to have any respect for yourselves, you might show some consideration for the rest of us. What you two do reflects badly on all of us."
"What the two of us do also reflects very well on all of you," Taro retorted, and I thought it timely that Firth be reminded of all the exceptional things Taro had done.
Firth clearly disagreed. "Arrogance does not become you."
"On the contrary, it's one of my most endearing qualities. Ask anyone."
"Dunleavy," Firth said to me, attempting to cut Taro out of the conversation, going so far as to put her back to him. Which was really odd, as he was standing right there. "You have never been foolish. You know this is an inappropriate course of action. Why are you pursuing this?"
Really, I had to frown at her, she was overreacting so very much. "It's dinner," I said. "I was introduced to this man by respectable people and found him amiable. I'm not going to let people who have never met him shape my opinion of him."
Firth stared at me. "What happened to you while you were away? You were so decent before you left."
"And you weren't nearly so judgmental." I'd really liked her before we'd left. She'd made me laugh. She'd been able to make Taro blush. Anyone who could do that was someone to admire. This new—or old but previously unrevealed—side to her nature was really disappointing.
"I'm only saying what all the others are thinking."
Was she? That was even more disappointing. It still wasn't going to change my mind. "I'll send an acceptance when we get back," I told Taro, for Firth's benefit.
Firth left. Success.
Taro took the mug from my hand, sniffed at its contents and gave it back.
Then I smelled the mug. The tea smelled normal.
"Aside from annoying Firth," he said "which is a motive I can commend given her behavior, you don't really want to accept Williams's invitation, do you?"
"There's no reason not to."
"Really," he said with flat skepticism.
"What are you trying to say?"
"Just that you aren't the sort to approve of someone like Williams."
I found that insulting. "And you're the sort who would approve?"
"I'm not saying I approve, just that I don't disapprove."
"But you feel I do disapprove." Because I, apparently, was judgmental and rigid and would therefore look down on people because of issues that were none of my business.
"I'm just saying I thought you'd be uncomfortable with the idea."
"All out of nowhere, without my having said anything about anything to make you think that way."
"I'm not trying to start an argument."
"Is there some reason why you don't want me to accept this invitation?"
He held up his hands in a mocking gesture of surrender. "Far be it from me to have reasons." He backed out of the kitchen.
I wasn't sure what that meant, but it was probably insulting to someone.
Really, I didn't know what was wrong with him lately. He was so touchy. He was almost never in a good mood. I hated to even think this, it was so shallow, but he hadn't been much fun lately.
"Not one more step!" I heard LaMonte roar, and I ran from the kitchen to the front door, from where I'd heard the shout. LaMonte shouting was such an unanticipated event that I envisioned something catastrophic occurring.
At first, all I saw was LaMonte glaring at a strange woman standing in the threshold of the residence. Then I saw the crowd of strangers behind her. I thought of the time a furious, frustrated crowd had nearly torn down the residence during the Harsh Summer. Then, they had felt the Pairs weren't doing enough to calm the violent, unseasonable weather. What had them so enraged this time?
"Kafar told us," the woman claimed with the air of someone who was repeating herself. She was wearing a dark dress that hung close to her arms and legs, and a dark leather apron. I could smell her from where I stood. Not necessarily a bad smell, just kind of astringent. "Why won't you heal us?"
Oh, hell. That was right. Kafar. The man Taro had completely failed to heal, because Taro couldn't do that sort of thing. I should have expected that to come back on us in some way.
"Sources are not healers, you ridiculous woman," LaMonte said, and I had a feeling he was repeating himself, too. He held his head high so it was easier to peer down his nose at the interloper, who was about his height. "Clear off before the Runners come and arrest you."
"You have to heal us! It's your duty!"
"It's our duty to protect the city. That is all."
That was, I thought, a rather warped way to interpret our obligations. What was the point of preserving a city if one didn't preserve the people who lived there? Not that we could do anything for individuals.
It was quite a crowd, several dozens of men, women and even children, all wearing dark, practical clothing. They covered the driveway and were obstructing traffic on the street. This couldn't end well. I stood beside LaMonte. "Source Karish wasn't able to help Kafar," I told the woman.
"Hush, Dunleavy," LaMonte hissed. "I'm handling this."
Don't hush me. "Source Karish can't help you."
The woman squinted at me. She was pale and sweating, and the lines about her eyes made me wonder if she had a headache. "You his Shield?"
"Aye."
"Fetch him."
Really, how dare she give me orders?
But I could see that all the people behind her were as pale and shiny as she, and at least a handful of them looked an alarming shade of green. One man was holding a young boy, about five years old, I guessed, in his arms. The child was exhausted, his head lying against the man's shoulder, too tired to fuss about his discomfort. The sight of him made me ashamed of my impatience.
It did not, unfortunately, change reality. "I am truly sorry, but Source Karish wasn't able to do anything for Kafar."
"Kafar is healed," she announced. "He told us. He sent us here."
There had to be a saying somewhere about how being kind could swing back to punch you in the face.
"He was reckless to make such a claim," LaMonte interjected. "You have no right to be here. Leave."
Not the best way to get the goal accomplished, I thought.
And as if in confirmation of that opinion, the woman looked back at her supporters, and almost as one they all took a step forward. I could almost feel the air they pushed before them. Their rigid determination sent a shiver down my spine. "All right!" I said quickly. "I'll find Source Karish. Please stay calm."
"What are you doing?" LaMonte demanded in a sharp whisper.
"Preventing a riot," I whispered back.
I found Taro in his suite, reading a letter. It appeared to be a good letter. He was smiling and his eyes were sparkling. I wished I didn't have to be the one to drive that expression off his face. "We've got a problem."
"I love my cousin," he declared with a grin. "She's driving Her Grace insane."
I was in full support of anything that tormented Taro's mother, and I wanted to hear all about it, but it would have to be later. "I'm serious. Kafar told a whole bunch of people you healed him, and they're all here." I was surprised he hadn't heard the noise in the street, even if his suite was on the other side of the building.
He folded up the letter. "I didn't heal him. I told him that."
"Apparently he's better and he's telling people you're responsible. That crowd expects you to treat them. And there was a point when I thought they were going to push their way in whether we liked it or not."
"We?"
"LaMonte and I."
"Chris. Oh, that's just lovely." He sighed as he rubbed his face. "What should we do?"
"We?" I squeaked. "I have no idea."
"Of course you do."
"I really, really don't." All the options were bad. I didn't know if we could get them to leave if we refused to try to help them, and they might turn into a mob. We could wait until Runners could be fetched to send them away, but the crowd would only return and we'd find ourselves in the same spot. Or Taro could pretend to try to heal them, and if they, like Kafar, thought they had actually been healed, we could find ourselves with a new, even bigger crowd. There was just no winning.
Taro swore and jumped up. I followed him downstairs, where Firth and Beatrice had joined LaMonte at the front door. Lovely. All my favorite people, all in the same place.
"Explain this, Dunleavy," Firth ordered.
I contemplated the various hand gestures that might be appropriate in the circumstances. "LaMonte knows."
"No, LaMonte doesn't know," the man in question stated. "LaMonte hasn't been graced with adequate explanations himself."
Taro ignored all of us. He could be good at that. "Who's in charge here?"
It was almost entertaining the way the crowd eased back from the woman who'd been doing all the talking. She didn't seem surprised by their actions. "Kafar said you healed him," she said.
"He's mistaken."
"He's better."
"That's wonderful, but it's not because of anything I did. However," he added quickly, for she had opened her mouth to object. "I will do what I did with him. I will do my best for you, and for everyone here, on the condition that not one of you mentions this to anyone else."
"Pure idiocy," LaMonte muttered.
I kind of agreed with him. I hated it when that happened.
"Do you accept those conditions?" Taro asked.
"Aye," the woman answered.
"And the others?"
"They'll hold to it. I'll make sure."
He suddenly seemed to gain in height as he said, "You'd better. You'll be held responsible if they don't."
I was impressed by his delivery. He sounded sure, serious and authoritative. One would almost assume he had the means to punish people who failed to do as he wished.
"Get everyone lined up in an orderly fashion. For gods' sake, get them out of the street. Send them in one at a time."
"So you can do what, exactly?" LaMonte demanded.
"Demonstrate that I'm unable to heal them."
"One at a time?" Firth felt compelled to toss in a coin, too. "That'll take all day."
At least.
"It will cause great disruption for all of us," Firth complained. "Can you never think of anyone other than yourselves?"
I thought the fact that Taro wasn't telling everyone to clear off demonstrated great thought for others.
"I take it, then, Claire, that you won't be assisting in serving refreshments while they wait?" Taro asked.
Firth glared at him and stomped off. Beatrice wandered off after her. I had the feeling he really didn't care one way or the other.
"You can't use our residence for this purpose," said LaMonte. "I forbid it."
Taro laughed.
I wondered whether LaMonte's deep red flush was due to anger or to humiliation.
"I really don't think we have the option of saying no, LaMonte." For what I believed was the first time in our acquaintance, I was trying to soothe LaMonte. The world had to be ending. "They don't look like they're prepared to leave quietly. It could have gotten messy."
LaMonte snorted, but he didn't actually dispute my common sense. He wasn't prepared to help, though, either, and I was disappointed when he walked off. It would have been nice to have some assistance in dealing with all these people.
"Who first?" Taro asked the woman.
And thus began what turned into a very long, hard day. Taro took them to the private dining room, as we hoped that would be less disturbing to the other Pairs than to have dozens and dozens of strangers traipsing up and down the stairs all day. Not that everyone in the house didn't find a reason and a chance to complain anyway.
Every person was asked to sit in a chair. Taro put a palm on his or her shoulder. He didn't spend nearly as long with any of them as he had spent with Kafar, but he did make an honest effort at channeling. After, of course, explaining that he couldn't heal them, and he was merely humoring them.
No one seemed to believe him, which I found odd. They were trusting him to heal them, yet they thought he was a liar.
As a whole, they were a quiet, timid group. Even the bold spokeswoman seemed to lower her voice and her eyes once she was in the residence, moving from place to place only as instructed and asking no further questions. The one exception was a pretty young man who flirted with Taro with disturbing skill. He was also clearly terrified. I would have thought less of Taro had he not flirted back, a manner of making him feel better.
It was difficult working with these people. They were so desperate, so hopeful that Taro was providing some miracle cure. But he wasn't. He wasn't able to do anything.
I wanted to ask them about this illness, about which I was receiving so many conflicting reports. Were people dying of it, or were they not? Was it contagious or was it not? Did they know of the suspicions that it was a problem with the water? And even if it were, what could they do about it? People had to use water. But even I wasn't so tactless as to treat them to an inquisition while they were pursuing a fool's path to a fake cure.
Why weren't the healers handling this, anyway?
Dare I ask Cree?
There were so many of them—that was the frightening thing. And I was under no illusion that these were all the ill people in the riverfront areas. These were just the most desperate of what was probably a much larger group.
The sun was dipping down by the time we escorted the last visitor to the front door. I had a blistering headache. It was an easy form of Shielding, but I'd never done it for hours at a stretch.
It was a relief to finally close the front door of the residence. I was tired and desperately wanted to lie down. But the sight that greeted us in the foyer stopped me cold.
On the table was a vase, an odd-looking vase of an earthy reddish brown color and a rectangular shape. In the vase were flowers with which I was unfamiliar, their blossoms an eye-piercing yellow. There was a small, dark, round cake balanced right in there among the blossoms, which was just bizarre. And hanging from the rim of the vase, a pair of earrings, short lengths of gold with small pearls at the end.
There was a small card leaning against the base of the vase, the script informing us that this floral thing was for me and from Doran. There was no other information on the card. "What the hell is this?"
"An expression of intent," Taro said with a scowl.
"Intent to what?"
"Don't be stupid," he snapped.
"I've never heard of anything like this."
"An intent to engage in negotiations for a contract of marriage," he explained, almost spitting out the words.
Marriage? Was Doran insane? Why would he do that? Even if we had ever had a relationship close enough to suggest marriage, I had no property and no powerful connections. It was unnecessary and ridiculous.
"Flowers signify personal affection," Taro continued. " The color of the vase represents stability, the earrings represent financial security and the cake represents fertility."
There was so much wrong with all of that. "Why would he send me something like this?"
"Excellent question," Taro said shortly.
"Don't you dare," I warned him. "I told him that I wasn't interested in anything like that."
"Are you sure?"
"Of course I'm damn well sure."
Taro didn't answer. He just stormed off. I had no interest in calling him back, as he was acting like an idiot.
So was Doran. Really, the flower arrangement was quite hideous. Why did he send it? I didn't believe for a moment that he was suggesting marriage or anything like it, because that was just insane. So what was he playing at?
"If you don't want to start negotiations, you send it back."
I looked up. Benedict was leaning against the wall near the entrance to the kitchen, a cup of what smelled like tea in her hand. "Throwing it away isn't sufficient?" I'd planned on sending the vase and the earrings back, of course, but that seemed troublesome for the cake and the flowers.
She shook her head. "You have to send it back in its entirety. Anything less denotes acceptance of the gesture."
That struck me as a deceitful way to handle things, and I was disappointed in Doran for engaging in that kind of behavior. He knew many of these little customs were outside my circle of knowledge. What if I had been alone when the flowers had arrived, and I'd thought the arrangement was nothing significant? I would have taken the arrangement to my suite, and no one would have seen it to warn me of its meaning. I would have sent back the earrings, possibly the vase, and kept the flowers, thereby accidentally committing myself to something.
I sighed and grabbed the ugly arrangement, leaving the residence to find the nearest messenger port. Really, men did the stupidest things sometimes.
Chapter Twenty
I slept in my own bed that night, as Taro had been acting like a plonker. And I had a vicious headache. And while it was sometimes nice to wake up snuggled up to someone, it was also nice to stretch out and have the whole bed to oneself.
I slept late for the first time in a long while. I woke with no nausea. That was lovely. And I had an appetite; I was really hungry. I went down to the kitchen and enjoyed a hearty breakfast of eggs and sausage. It was wonderful.
Then Taro strode in. He was scowling. "Where's Laidley's gesture of affection?"
I rolled my eyes. "I sent it all back. Thanks for telling me what would happen had I failed to do so."
"It's not my place to interfere."
"It's not your place to cause trouble."
"No, that would be Laidley."
Was he really willing to have me promised to Doran, or whatever would have happened, just because he was annoyed with me? Why was he annoyed with me, anyway? Doran was the one who was playing games.
"I don't want to talk about Laidley."
"Then why'd you bring him up?"
He sort of growled, then took a deep breath before saying, "There's something strange about this illness."
"So you've said."
"Kafar was ill when he came here. I could feel it. There was nothing I could do for him. He was still ill when he left."
"Are you sure?"
He nodded. "I could feel it."
"Sometimes you're not sure how much you've done for someone."
"Not this time. There really was nothing I could do for him."
"All right." What was this about?
"But he moved in with his sister. And he got better."
"Maybe he just got over it. If they don't die, they eventually have to get better. It's possible he's not even truly better. He might just feel better for some reason and think he's healed."
"Perhaps, but the most logical explanation is that he actually is better."
I had to give him that. "Aye, that's true."
"So maybe there's something in the riverfront that's making people sick."
"Risa thinks it's the water."
"It can't be the water. The water goes everywhere. The illness would be more widespread."
"It has spread beyond the riverfront area."
"But not everywhere the water goes. And most of the illness is restricted to the riverfront."
"If it's not the water, what else could it be?"
"I have no idea. But the logical assumption is that there is some unnatural element in the riverfront that's causing this. Maybe it's something only they are eating. Maybe it's something only they use to build their houses. But it's something that's in the riverfront and nowhere else."
All right, I could go with that theory for the purposes of discussion. "Do you want to tell Risa about this?"
He shook his head. "I don't want to have to explain to her about why I think this is unnatural. She won't believe me."
"Why not? You're a Source. She'll believe you can do anything you say you can do."
"I'm not worried about her believing me. I just don't want someone else learning about this little trick I have. Besides, this sort of thing isn't really the responsibility of the Runners. It's an illness, not a crime."
"But they must be investigating it, if Risa thinks it's the water."
"I don't know where she got that idea. The rumors are that the Runners aren't really doing much about this. They can't. They're too busy with other things, and the riverfront illness doesn't fall within their sphere of authority."
"Whose authority does it fall into?"
He shrugged. "I have no idea. No one's, it seems like. Or maybe the healers, but they're not going down there."
"That's alarming." Was that possible? How could something so serious not have a group of people meant to deal with it?
"I think we should go to the riverfront and see what we can find out."
I nearly choked on my tea. "What in the world makes you think we could find something the others can't?"
"Why wouldn't we be able to see something? We're reasonably intelligent people."
"Why would you even want to do this? You're the man who got so furious when people expected us to do something about the weather during the Harsh Summer. You resented the fact that they expected us to do something beyond our responsibilities."
He sniffed. "I resented the way they expressed their expectations. They were obnoxious."
That was true. Obnoxious and violent. But this was bizarre. This wasn't our task. I'd made a promise to myself not to get involved in things that were not my business. It rarely worked out.
"We're not doing anything else, Lee. We're off the roster."
That was true, too. If we couldn't be useful one way, we might as well try to be useful in another. Not that I thought we would find anything, but it would do no harm to humor Taro. "When did you want to go?"
"Might as well go now."
I sighed and finished off my tea.
We left the residence. It was a bright, mild day. A market day, which meant most people would be haunting the merchant stalls.
The riverfront technically referred to the banks of all three rivers, along their entire length through the city. In reality, it was concentrated in the center of the city, where the three rivers met and crossed. The riverfront of the wealthy quad was kept empty, but the other five quads were filled with dye makers, trash collectors and manure mongers. It was a very fragrant area. And loud. And dirty. The buildings were narrow and dark and a lot of the windows were broken or gone altogether.
"So what are we looking for?" I asked Taro.
"I have no idea. Just something that jumps out as being strange."
I wasn't familiar with the area. How would I know what was strange? It seemed to me we were going to waste an afternoon just wandering around. "This all looks unusual to me."
"You're such a snob, Lee."
"So you spend a lot of time here, do you, Taro?"
"There's nothing here that I want."
"How convenient for you."
People were staring at us. I imagined they didn't see Sources and Shields around there very often.
They looked awful. Pale with dark circles under their eyes, green and greasy. Those who moved did so slowly. Many were just sitting or lying in doorways.
"Let's ask someone," Taro suggested, leading me to a thin woman with lanky brown hair, sitting in a rocking chair in front of her hovel. "Excuse me, ma'am," Taro said politely. "I'm Source Karish. This is—"
"You're the Source what's been healing people," she said.
So much for them keeping that quiet. I'd known that was too much to ask.
"Actually, I haven't been. I can't heal people."
"Healed Kafar, didn't you?"
"No, actually, I didn't. I believe he got better because he moved away from the riverfront."
I suddenly wondered if it was the best idea in the world to imply that the place in which she lived was somehow causing the illness.
But she didn't seem to leap to that conclusion. "So you're not going to heal any more of us?"
How many times did he have to say it? He couldn't heal people of this illness. At least, he didn't think so. I still wasn't sure.
He chose not to answer that question. "Have you noticed anything unusual about? Something other than the illness itself?"
She positively cackled with laughter. "Everything about the riverfront is unusual."
Aye, she was going to be a lot of help.
But Taro gave it one more try. "There's nothing you've seen that you can tell me?"
"What are you, a Runner?"
"Have Runners been here?"
"Runners never come here."
"Really?" I said. That surprised me.
"Nothing here they think is worth protecting."
How odd.
We worked our way down the riverbank, stopping to talk to whoever looked open to conversation. No one knew of anything strange happening. A few of them asked Taro to heal them, and were quite bitter when Taro said no. I was glad I didn't have to convince him to refuse. I could just imagine everyone in the riverfront hearing that he was there and the two of us getting mired in a futile attempt to see to everyone.
A little boy, about eight, I guessed, ran up to us. He was too skinny, he was barefoot, and his clothes were too tight. "You a Source?" he demanded of Taro.
"Aye, I am."
"I want to be a Source," the boy announced. "What do I have to do?"
"I'm sorry, son. You have to be born a Source. There's nothing you can do to make yourself a Source."
The boy scowled.
He was only eight. In theory, he could be a Source, and no one had figured it out yet. Taro had been eleven before his family realized he was a Source. We couldn't know whether this boy was a Source unless we spent more time with him. It was usually the families who discovered it.
"You been asking all sorts of questions," the boy said. "I can answer them, if you take me with you and make me a Source."
"We can't take you with us, and we can't make you a Source," Taro told him. "But I can give you this." He took the emerald stud out of his ear and held it out for the boy's inspection.
I thought he was being careless to offer a bribe, of any kind. The boy could easily make something up, and we wouldn't know the difference. I didn't know how to communicate that to Taro without outright calling the boy a liar to his face.
The boy grabbed for the earring. Taro held it out of reach. "What have you got to say?"
"A woman comes at night sometimes. Real late. She's dressed too nice to be someone from around here."
"What does she look like?"
"I don't know. She wears a hood, and it's dark."
"What does she do?"
The boy shrugged. "Don't know. But she does it at the hub."
"What's the hub?"
The boy snickered. He thought the question stupid. "Where the rivers cross."
"When was the last time you saw her?"
"I don't know. A week, maybe."
Taro gave him the earring. "Thank you."
The boy crowed in delight and dashed off.
"A woman who comes in the middle of the night," I said. "That could be for anything. And would he even be up that late?"
"No harm in looking around the hub, though."
"No, if you want to."
There were bridges all over the city, spanning the rivers at various points. The bridge spanning where the rivers met was massive, wide enough to allow two wagons to cross at a time. The bridge was made of wood, and there were chunks rotted out of it.
Walls about shoulder high were built along the rivers, for about twenty cubits each way. Open drains were built low into the walls to allow some of the water to escape for residential use. The grass in the area was beaten down and dead.
And I felt jittery. It reminded me forcibly of how I'd felt in the ash grove. "Ashes."
"What?"
"I think someone's been using ashes here. I think I have a reaction against human ashes."
"That seems odd. Why would you have a reaction?"
"I react to niyacin powder. That's not common."
"That's true. So you're saying someone was trying to cast spells here."
"Or dumping something in the water. At the drains, not the rivers themselves, which is why only some people are getting ill."
"You're saying someone is making them ill on purpose."
As soon as he said that, I wanted to scramble away from the idea. "I don't know. Maybe it's just a side effect. Because why would anyone want to make a bunch of strangers ill?"
Taro nodded, though he didn't look entirely convinced. "You know what the next logical step is."
"Tell the Runners."
"No." Now he looked impatient. "We come back tonight and see if the woman comes."
I sighed. "Really, why are you so enthusiastic about this?"
"I told you. We have nothing else to do. Why don't you want to do this?"
"Because I'm really bad at this sort of thing. I'll make a fool of myself."
"You just don't want to go through the discomfort of waiting around here all night."
There was a kernel of truth to that, but Taro was right. I had nothing better to do.
"We need to get long black cloaks. We want to stay hidden."
That meant shopping.
So we found nice long black cloaks, and then we went back to the residence for a nap. Luckily our watches at the Stall had gotten us used to strange sleeping patterns. Once the sun was down, we headed out with our melodramatic black cloaks draped over our arms.
The riverfront was quieter at night, which surprised me. I was ashamed to realize I had expected people to be walking around drunk and fighting and causing chaos. Simply because they were poor and lived in a grim environment. What was wrong with me?
We put on our cloaks, though really, who were we kidding? If a lone woman had been noticed simply because she was dressed too nicely, we would stand out by a league. But I pulled the hood as far over my face as I could and held my hands within the sleeves. We huddled into the shadowed corner of the walls, and we waited.
And waited.
And waited.
If spending a watch in the Stall was boring, crouching in silence all night was brutal. Despite my nap, I found myself nodding off, my eyes heavy and blurred.
This was so stupid. If someone did come, what were we going to do? Stop her or merely watch? Go back with the news to the Runners? The latter was the smart thing, but while Risa didn't seem disturbed by my previous foray into the responsibilities of Runners, it seemed to annoy the others. They probably wouldn't take us seriously.
But it turned out that I needn't have worried about that. No one showed up, and when the sun started to rise we stood with stiff joints. "That was horrible," I complained.
"Aye, and unfortunately, we'll have to do it again."
"You can't be serious."
"We didn't catch her."
"How long do you expect to do this?"
"Until we do catch her."
I suppressed a groan. There was nothing I could reasonably do but follow along. But in all the rumors I had heard about Lord Shintaro Karish before I met him, no one mentioned an insane dedication to the duties of others.
Chapter Twenty-one
To my surprise, the location of Williams's home was in the Upper Eastern Quad. It was a good enough area in High Scape, home to midlevel merchants and minor politicians. It was not, however, where I would expect someone as successful as Williams to live. Though perhaps it was an unreasonable assumption, on my part, to think a man who owned bordellos would live in the same area as a merchant as successful as Fines.
The house was nice enough, large, made of a decent wood, with a lot of windows. It was hidden from the road with trees and hedges. The driveway didn't take us to the front of the house, as most driveways did, but instead curved around to the back. We didn't exit the carriage right away, unsure as to whether being delivered to the back of the house wasn't a mistake.
"Are you getting out or what?" a voice demanded from above us. The carriage driver, sounding irritable. So we left the carriage, which was jolted into movement as soon as we cleared it.
Not getting paid for services could certainly put people in a sour mood.
The door in the back of the house opened and a young man stepped out, smiling in greeting. He was extremely handsome, slim with a strong jaw, golden blond hair and bright blue eyes. He was dressed in blue, a strange style of clothing that was loose in cut but of a material that seemed to cling to his very attractive form.
"Source Karish? Shield Mallorough?" he asked, his voice deep and smooth. He probably had a gorgeous singing voice.
"Yes," Taro answered.
"My name is Akira," he said. "Please come in. The others are waiting."
He gestured at the back door. We were actually expected to enter the house through the back door. Had anyone asked me whether I'd be offended to enter a residence through the back, I would have said no, of course not, I wasn't so petty as to be disturbed by such things. But I was. I was ashamed of myself. It was ridiculous. Yet to be asked to enter through the back was bizarre.
I was surprised, upon stepping through the door, to be greeted by a huge foyer, with a grand staircase curving up to the second floor. To my knowledge, the back entrances to residences were usually small and utilitarian, as guests never saw them. There seemed to be a lot of furniture scattered about, lots of comfortable-looking chairs and settees. There were chaise lounges, which I could never recall seeing in the public areas of a residence before. And up against the wall, was that a stage? What would anyone want with a stage in their home?
Akira relieved us of our wraps and, shockingly enough, our footwear. While I had been required to remove my footwear while indoors on Flatwell, it had never stopped feeling unnatural and I'd never heard of anyone doing it anywhere else. For a moment, I wondered if Akira was playing a game with us, but then he removed his own footwear. So I guessed it was just a weird custom of the house.
It was odd to see a grown man in a formal situation with bare feet. He had nice feet.
I did not have a great deal of familiarity with private homes. I wasn't sure whether there were rules as to how a house should be organized. But something about the huge space, with the furniture that merely lined the walls and created no conversation areas, struck me as unhomelike.
The smell of perfume was overwhelming. And if I found it so, poor Taro had to be almost gagging.
We were taken to the largest and strangest dining room I'd ever seen. The table with its chairs was set up on one side of the room, by the windows, which were covered by heavy drapes. The other half of the room was left virtually empty except for more chairs and settees and chaise lounges pushed against the wall.
The other guests were identical to those when we had dined at Fines's, and again, there were no spouses or other partners. Williams was serving drinks, as there were no servants in the room. "How strange you should arrive at just this time," Fines said once Taro and I were settled. "We were just wondering what our dear Prince Gifford has been doing to show his authority over the Triple S."
I felt that question came out of nowhere, and I was a little confused by it. I looked to Taro, who grinned. "Our beloved monarch is causing some difficulty, is he?"
Gamut snorted. "It has been suggested to my theater manager that it might be in the best interests of the community if our program were examined by some government agent before being performed for the public."
"I thought that was happening already," I said. "Some plays are outlawed."
"Aye, and I agree with some of their choices. Some pieces are nothing but excrement from curtain rise to curtain fall, with no technical merit whatsoever. What the Prince seems to be proposing now is different. Everything we would seek to put onstage would first need the approval of the Emperor's agent, and that is insulting. To us and the audience. Not to mention tiresome, unnecessary and expensive."
"Why would it be expensive?" I asked.
"We'd need to give the agent coin to get the plays approved, of course."
I frowned. "Surely the agent would be paid by the government, not private citizens."
Healer Cree managed to convey, with her placid expression and unfathomable eyes, her belief that I was an idiot. "He is speaking of bribes, Dunleavy."
"Bribes," I echoed, because sometimes I was an idiot.
"One needs bribes to encourage government agents to act in one's favor."
"You mean you need to give extra coin to get these people to do their jobs?" I demanded, appalled. Sometimes I really just didn't want to learn anything more about people. Or the world. It was so often disappointing.
Gamut chuckled. "She's so cute."
I was not cute.
"She has no reason to know such things, Dean," Ahmad chided him before she addressed me. "Whenever the services of a government agent are needed, it is expected that the agent will receive a gift. That is the way things are done."
"And this is legal?"
"Oh, no. Quite the opposite. But there would be no point in attempting to fight it. After all, the only people to report it to are other government agents."
"Everyone's in on it," Williams added. "Including anyone who enforces the laws. The Runners. The judges."
"Not the Runners," I objected. Not Risa. Nothing could make me believe that she required bribes to do her job. She was an honorable person.
"How about you, Grace?" Fines asked. "Has the Crown Prince pulled any of his stunts with you?"
There was a curious tone to his voice, and something seemed forced about the question. I felt I was missing something.
Ahmad snorted. "It has been suggested that all members of the guild should hire services and buy supplies only from sources approved by a government overseer."
"That doesn't seem wholly unreasonable," said Fines. "Especially when you're doing work on government projects."
"None of it's reasonable," she snapped. "But yes, it would be tolerable if it were limited to such circumstances. However, I understand the plan is to restrict us in this way for all works performed by the guild."
"At least they don't plan on eliminating your guild altogether." Williams poked at a chunk of fruit floating in his wine, then licked the wine off his finger. "There have been rumors of plans to close down all bordellos and make them illegal. Which is pure stupidity. Without the protection of the bordellos, the prostitutes are in more physical danger from their clients and the patrons are at greater risk of becoming diseased."
"Why in the world would he outlaw bordellos?" Taro asked.
"Apparently His Royal Highness finds the practice of prostitution distasteful."
"He can't seriously believe," Taro said, "it's equitable to destroy the livelihoods of so many just because he finds it distasteful?"
"Ah, but he will be the Emperor, and what he considers distasteful is well within his power to change. And he will."
Well, aye, that was disturbing, that the Prince would have the ability to destroy something so harmless for no good reason. What would be the point beyond demonstrating his power merely for the sake of it? Not that the Empress hadn't thrown her weight around just because she could, but I'd had more respect for her judgment. The Prince had yet to do a single thing that I hadn't thought was stupid.
"Ayana?" Fines prompted, and it did feel like a prompt.
Cree shrugged. "There has been a suggestion there should be regular inspections of my medications and accounts. The justification is to make sure my methods are sound and my rates are fair."
To be honest, I didn't think that was a horrible idea. The idea of healers being able to sell and do anything without being accountable to anyone was chilling. Those crazy elixirs could do a lot of damage, and the idea of someone without competence setting bones and cutting into delicate tissue made me squirm.
"And I am to be told who I may employ and who I may not," said Fines. "If any of my people are not considered appropriate according to age and training, I will be fined."
I wondered if my family were facing similar restrictions. They hadn't written to me about it. They didn't tend to get into that level of detail with me when it came to trade.
"Many are already suffering new restrictions or fines at the hands of the Prince," Fines added.
"But the Prince hasn't the authority to do those things yet," I said. It took time to make all the preparations needed to have a title change hands, and even more time when it involved the throne.
"He will when he ascends the throne, and he'll remember those who have opposed him. It would be idiotic for anyone to refuse his edicts merely because he temporarily lacks the authority to make them."
I wondered what that would mean for Taro's cousin and her refusal to pay additional taxes.
"Have you not received any such regulations from the Prince?" Williams asked.
Of course not. That was a stupid question. "The Triple S is a self-regulating institution, and it always has been. The monarch has no authority over us."
"And what will the Triple S do should a monarch choose to exercise some authority?"
"I doubt it would ever come up." The Empress had sent Taro and me to Flatwell, which was an unusual exercise of a monarch's authority, but that wasn't the day-to-day interference the others were discussing.
The door opened and in streamed about half a dozen of the handsomest people I'd ever seen in one place, pushing carts and carrying decanters. Three men and three women, all young, their brushed and shining hair left loose, which was unusual for staff who were serving food. They were dressed in a fashion similar to Akira, with loose trousers and the shirts that flowed closely under breasts and against flat stomachs, in colors that perfectly suited the wearer. And they all had bare feet.
This was a weird place.
The food smelled good, though.
"Please seat yourselves," Williams invited once the servants had left. Once again, Taro and I found ourselves sitting across the table from each other. Though there was no reason why we should always sit together. We weren't children in need of proximity for moral support.
There were the same little clay pots on the table, and we all rubbed the translucent paste within on our hands. I rather liked that little ritual. Maybe it was something I could do at home.
The food was delivered in the same manner as it had been at Fines's. Everyone served themselves from the dishes and then passed them along. Each of us had our own decanters of wine. And as had been the case at Fines's house, everything tasted absolutely delicious.
"An appalling waste of life," Ahmad was saying, and I realized I had missed some conversation while contemplating the quality of the food.
"His actions did result in the death of two jockeys and five horses," said Fines.
What was this? I looked at Taro.
"A ditch appeared in the middle of the track during a race," he told me. "No one could stop in time. Five horses broke their legs and had to be put down right there. Two jockeys were trampled after they fell off their horses. There were many non-fatal injuries as well."
"The ditch just appeared out of nowhere?"
"That's what it looked like."
"How could that happen?"
"It's most likely that someone dug it, early in the morning before there were any staff at the track, and camouflaged it in some way, and no one noticed it until one of the horses actually stepped in it."
"Now, Shintaro," Fines chided him. "I expected better from you."
"Excuse me?" was Taro's frosty response.
"I take it you weren't at the track that day."
"I was, actually."
He'd witnessed this event and hadn't told me? Why not? It had clearly been a horrible experience. I'd need to talk about it if I'd seen all that. "When was this?"
My question was ignored.
"You weren't questioned by the Runners?"
"I was."
"But they didn't mention to you that they found the paraphernalia for casting near the track."
"They did."
I could tell that Taro's short answers were annoying Fines. I could sympathize with the trader, because they were annoying me, too. It wasn't like Taro to be so reticent. Nor so impolite. "They think the ditch was created by a spell?" I asked. Why would anyone do that?
Ahmad snorted. "They don't know what to think. They claim a spell couldn't have created the ditch, yet they have convicted and hanged a man for the creation of a ditch resulting in multiple deaths, as well as for possession of implements designed for the purpose of casting spells."
"So they believe he created the ditch, and they believe he was trying to use a spell, or pretending to use a spell, but not that he created the ditch with a spell." I didn't believe it, either. I just thought there was a piece of information missing in what I was being told.
"Isn't it ridiculous?" Ahmad said in apparent agreement.
"But you don't believe a spell was involved, Shintaro?" Fines asked.
Taro was committing the sin of playing with his food, pushing morsels of meat about his plate as he thought about what to say. "It seems an odd thing to do, if one had the ability to do such things."
"Oh? What would you do if you had the power to do anything you wanted?"
"I'm afraid I'd have to give that matter more thought."
"Oh, come now. Don't tell me you haven't been contemplating the possibilities since people have begun exploring casting so openly?"
Taro swirled the wine in his goblet. "The sky is obscured by boundaries."
Heh. No one understood that.
"This appears to be a similar wine to what Trader Fines served," Taro said, in an attempt to divert everyone's attention. And it worked, to my surprise. Williams complained, in a good-natured way, about how much the wine cost, especially considering how many wealthy clients he exposed the vintage to, which was why, Fines explained, he sold the wine to Williams at such a reduced rate. This somehow led to a lengthy discussion between Gamut and Ahmad about someone Ahmad knew who would best be able to make repairs to the theater in which Gamut worked. And then a further discussion of services Cree supplied to Williams's prostitutes.
I liked those topics of conversation much better, even though I couldn't contribute to them.
After dessert had been consumed and cleared away, Williams took his wine and sat in one of the chairs in the other half of the room. The others followed, so Taro and I did, too. Conversation felt awkward due to the organization of the seating.
And then the door opened and seven servants walked in. Five carried basins and towels, one was pushing another cart, and one had a flute of some kind. A light scent of citrus filled the air, carried on a thin waft of cedar, and it relaxed me in spite of myself.
"Dunleavy, Shintaro, you may not be aware of this," said Williams, "but many of my patrons enjoy having their feet bathed. It is a service we provide."
Oh, damn it to hell. This wasn't Williams's home. It was one of his bordellos. Could I be any more stupid?
Was this why we had bare feet?
Suddenly, it felt as though my mind were spinning with questions. Was this appropriate? Were they prostitutes? Not that I had anything against prostitutes, but should I be accepting services from them? Was this what anyone had in mind when they talked about Sources and Shields being allowed to requisition goods and services for free? What should I be doing? They never talked about this at the Academy.
I looked at Taro. He was looking back at me, wearing the tiniest smile, obviously aware of my discomfort and entertained by it. He had no intention of indicating which way we should go with this. Bastard.
Everyone else, I could tell, was observing me as well, with various degrees of subtlety. What was this, a test?
Fine. This was something being provided by my host. Something that was not harmful. It was bizarre, but to refuse might offend my host. So, fine, I was going to have my feet bathed. I could accept that attention with a little grace.
The man with the flute sat in a corner of the room with fluid confidence. The piece he played was unfamiliar to me, but it was lovely, the lower sliding notes curling deep in my belly. Moving, but not dangerous.
Akira knelt at my feet, setting a basin on the floor. With gentle hands he placed my feet in the basin. Then he poured warm, scented water over my feet, and when he judged that enough water was in the basin, he poured oil into the palms of his hands. He rubbed his hands together and grasped my shin to rest my wet foot against his thigh, unconcerned with the soaking the thin material he was wearing was suffering.
And then he started rubbing the bottom of my foot.
I had had my feet rubbed before, primarily by Taro, back on Flatwell when the constant bench dancing wrenched my ankles and made the soles of my feet sting. That had felt nice. But this was different. More skilled, for one thing. And the results were different. Especially at one moment when he pressed his thumb hard against the sole of my foot. It sent an oddly pleasant jolt right through me, making me sit up straighter in my seat.
Akira looked up at me and smirked. Clearly, my reaction had been anticipated. He was playing a game with me. I looked at the others, who were all enjoying their foot baths with varying displays of enjoyment. No one appeared shocked.
They were all playing a game.
Except for Taro. His attendant was a broad-shouldered youth with closely cropped hair, and his hands looked big on Taro's slender feet. Taro appeared relaxed, and there was a slight smile playing about his lips.
The fact that that disappointed me was entirely illogical. Of course he would enjoy the skills of other people. Just as I would enjoy the skills of Akira, if he weren't being paid to do this, and I didn't suspect this was all some ploy to make me feel foolish, or something.
It was all just bizarre, and I worried about what the next step would be.
"Have you ever enjoyed the services of a bordello, Dunleavy?"
There it was.
"No, I haven't," I answered Williams.
"Why not?"
I didn't know. Was it something most people contemplated? "I've just never thought of it."
"You should think of it. You're welcome at any of my establishments at any time."
Um, thank you? I glanced at Taro before I could help it.
Williams caught the look, of course. "Certainly, the two of you can come together. We provide excellent services for couples."
"Thank you." Seriously? A couple would come and share a prostitute? Or would they each hire one? I really had no idea.
Well, I was certainly getting an education from these people.
"Would the two of you like to spend the night and savor some of our services?"
"No, thank you," I said quickly.
Taro chuckled, no doubt delighting in my discomfort.
"My people can make you feel quite fine. Anyone here can attest to that."
Seriously, that was information I neither needed nor wanted. "I am sure that's true, but I'm really not interested. Sorry."
"Perhaps another time."
"Perhaps."
No one asked Taro anything. He was always the lucky one.
Once everyone's feet were determined sufficiently clean, they were gently dried, and Ahmad left. That meant we could leave, too, and we did. I was relieved. Because that had been odd, and I felt there was some kind of agenda going on. And that thought made me feel ridiculous. Because what would such a group want with us?
Chapter Twenty-two
"Where do you and Source Karish go so long at night?" Ben asked me as I enjoyed tea and a huge chunk of fresh bread. "If you don't mind me asking."
"I'm afraid I can't answer that, Ben." I had slept late that day after another fruitless wait at the hub. We weren't telling anyone what we were doing, mostly because they would think we were insane.
"I suppose I shouldn't have asked."
"There's nothing wrong with curiosity." But his curiosity made me uncomfortable, so I searched for a change of subject. "This bread is incredible. I don't think you've made it before." It was heavy and dark and rich with nuts.
"I don't make it often. It is a laborious recipe."
"Oh." I felt guilty about the fact that I was on my third piece. I was always so hungry after a night at the hub, and the bread had been the only readily available thing to eat.
Ben smiled. "I made it to be eaten, Shield. Please have as much as you like."
I watched Ben clean up some of the crumbs I had spilled on the counter. I wanted to tell him to stop. I could clean up my own crumbs. "How is Sara, Ben?"
He stilled for a moment, but quickly resumed his wiping. "She is hopeful that she will be vindicated at the trial."
"Has new evidence come to light?"
"She isn't telling me much about it. She's afraid what might happen if the information got into the wrong hands."
"I see. I understand." I didn't really, as I didn't know what kind of information and whose were the wrong hands, but he clearly didn't want to talk about it, and I could respect that.
"But you're kind to ask."
"No, not at all." But I was very uncomfortable. I shoved another piece of bread into my mouth.
I went back up to my room to read the last of the spell books. I would be getting rid of them soon. I didn't like how obsessed people were getting with the idea of casting. While I really didn't think I'd be punished for having the books, I couldn't help feeling leery about the sheer lack of logic people were displaying about spells, and I didn't want to get caught up in anything crazy. So the books would be gone.
After I read them. Because some of them were entertaining. In fact, some of them were hysterical. There were points where I wondered if the writer was insane. Or had been smoking something creative.
There was just so much about the practice of casting that was ridiculous. Wearing wool when it was hot and silk when it was cold. Honey gathered under a full moon—neat trick, that—or blood gathered from a sheep slaughtered at the mark of noon. Powerful locations included the peak of a mountain, the crossing of two or more rivers and the east end of a ravine. Some spells needed a particular timbre of voice, soprano or alto or tenor or bass. And the tools needed, all made out of materials that, if not expensive, needed to be constructed in ridiculously complex fashions.
Seriously, who had the money to cast spells? Who had the time?
But aye, an entertaining read.
Then Ben knocked on my door, telling me Doran had come to see me.
Damn it, I hadn't seen Doran since before he'd sent me that yellow-flowered monstrosity. I had sent it back to him. I'd also sent a note telling him not to be an idiot.
I knew I needed to have it out with him, but I really didn't want to. I just wanted him to accept the words in my messages and just go away. That was cowardly of me, but I felt Doran was being too pushy.
I didn't want what was sure to be an unpleasant conversation overheard by everyone in the house. I asked Ben to show Doran to my sitting room, and I quickly hid the book I'd been reading in my bedroom.
"Lee, dearest, I'm so glad to finally be able to see you," Doran said upon his entrance. He made no effort to make the emphasis on the word "finally" at all subtle.
I had to admit, there was something that irked me about the belief that a person who showed up unexpectedly should be able to see you, just because you happened to be there when they arrived. "I am very sorry," I said, as an apology was expected. "Things have been busy here."
"But you're up for anything now, aren't you?"
"I wouldn't go that far." I didn't know that I would ever be up for the conversation we had to have.
He grinned. "Want to go dancing?"
What was he doing? He had sent me all that stuff, and I had sent it all back. Why was he pretending nothing had changed?
He was going to make me say it. That was ungentlemanly of him. "Doran, we have to talk."
"We are talking."
"Doran, I'm being serious."
He sobered. "You look a little pale."
"I'm always pale."
"I don't think you should make final decisions when you're not at your best."
"This is a decision I've made. Are you really going to make me say it?"
"You don't have to say anything. I understand what's going on. Shintaro is handsome and a Source and he's full of flair. But he's not the sort to last. And if I have to wait, I will."
"You can't mean that." No man with any pride would.
"Don't tell me what I feel."
"I'm sorry, but you can't mean you're prepared to wait until Taro's done with me and you're going to step in. That doesn't make any sense."
"I don't think of you in such base terms, but yes, I'm prepared to wait."
This was unbelievable. There was no way I could inspire such insanity in a man. What was I supposed to say to this?
There was a knock on the door. "Lee?"
Ah, hell, Taro. What was the best thing to do? I wanted to get this conversation out of the way before I talked to Taro. On the other hand, had Ben told Taro Doran was there? "I can't talk to you right now, Taro."
"Why not?"
"Because I'm here," Doran answered.
Taro, of course, charged in. "What are you doing here?" he demanded.
"You're not suggesting you have a right to control who Lee sees, are you?" Doran asked, attempting to sound innocent.
"Her name is Dunleavy!" Taro snapped back. "And if you had any honor, you would have taken the hints she's given to you to leave her alone."
"And yet, she agreed to see me today."
Only because he'd left me no choice.
"Lee has manners. You apparently have no sense of what's proper."
"What do you know about what's proper?" Doran scoffed.
"I know you come sniffing around like a vagrant hoping for a spare crumb instead of declaring your true intentions as an honorable gentleman would."
I couldn't quite figure out whom Taro was insulting more with that simile, Doran or me. And hadn't that yellow-flower monstrosity been all about declaring unwanted intentions?
Doran was laughing with disbelief. "You're not seriously expecting me to announce a challenge."
A challenge?
"Not at all. Such requires a certain elevation of character that you clearly lack."
Doran flushed. "You are hardly in a position to criticize someone else's character."
"And you are? Who are you? What do you do? Spend all your days playing and gambling while your mother and your sisters handle all of your responsibilities. You do nothing for anyone else. Just another useless lordling whose existence is irrelevant."
Doran emitted another false laugh. "You mean what you would have been, if you hadn't been born a Source?"
Ouch. That was a telling blow. Because it was true. I adored Taro, and he took his responsibilities as a Source seriously, but I had a hard time believing that if those responsibilities hadn't been imposed upon him by an accident of birth, he would have chosen to be industrious in some other way. Would he have ever developed the gumption to break out of the seclusion in which his family had held him? Though, if he weren't a Source, and didn't express himself in the nonsensical manner Sources sometimes employed, his parents wouldn't have thought him mad, and probably wouldn't have locked him in his room until he was eleven. Instead, they would have raised him to be the second son he was, with an older brother who died from his philandering, and apparently a father and a mother who had much the same habits. Would Taro have been able to fashion himself into something different? I couldn't help but doubt it.
"And everyone knows it's the Shield who does all the work in any Pair." Just like that, Doran had swung from forced humor to unreasonable anger. "Shields make plans and study maps and learn about things and write reports. Sources just flit around and open themselves to whatever disasters might occur. And from what I've been hearing, there's been little of that around here lately."
"That's not fair," I objected, wondering how he'd heard that. The regulars should never know whether events were happening or not, not if the Pairs were doing their jobs properly. Someone had to have been telling him that, and it hadn't been me.
"No, it's not," Doran agreed, taking a different meaning from my words than I'd intended. "But it is correct, isn't it?"
"No," I asserted, my voice weaker than I liked. All the tension in the air was giving me a headache.
"As though your opinion mattered," Taro sneered. "What would you know of Sources and Shields? Nothing but what we choose to let you know."
"I know you're not lovers," said Doran. "Or, you're not supposed to be. That's considered something of a perversion among your kind, isn't it?"
Nausea erupted in my stomach. It was breathtakingly painful.
"It happens all the time," said Taro, bending the truth a little. "And you regulars love all those romance novels and plays that drone on about the love between Source and Shield being unlike any other."
When had Taro been reading romance novels?
"That's fiction," Doran jeered. "Though I can understand if the distinction eludes you. There are other sources of information, you know, books that speak of what the bond is truly like, and what it's supposed to be, and that bringing any sexual element into it is rare, because it is so very wrong. Base. A perversion of something meant to be pure."
"Perversion." That word again. I wanted to object to its use, but I couldn't spare the air to speak. Horrible cramps were developing in my stomach, and it was all I could do not to curl into a ball and moan.
"I have to admit, I'm a little disappointed that Lee would allow herself to be lured into such perversion."
Hey! Who the hell gave him the right to be disappointed in me?
"But she can't be blamed. She was so young when she met you."
I wasn't that young. Twenty-one. Most were hard at labor and parenthood long before that age.
"I can imagine how someone like you would appear to someone so sheltered and immature. The noble bloodline and the flashy looks. And no doubt she was taught to worship Sources at that school of hers."
These words sounded so familiar. Who had said them? Or was it just a common assumption made by all who met me? Did I really appear that weak? Was I that weak?
Apparently. I was sleeping with the man, after all.
"I am confident," Doran continued, "that exposure to someone normal will assist her in seeing how wrong her infatuation with you is, and she will end it. She is, at her core, a sensible woman."
I was seeing something, all right. A whole new side to Doran that I really didn't like. I didn't understand it. I thought Doran really liked Taro. He saw him as a sort of rival, and he couldn't be thrilled with that, but everything Doran had ever said about Taro had implied admiration and respect. Where had all this disdain been hiding?
"And you think you are the someone normal she'll be using as a comparison?" Taro asked with an unpleasant smile. "She is interested in you only for your similarities to me, because she considers you a safer version of me."
That was not true. The two men were hugely different. Doran was less moody, less complicated, and I could be fairly confident that with him, I wasn't merely one among many.
"Please," Doran sneered. "What she appreciates about me is how I am different from you. I'm stable and reliable."
"How romantic." Taro snickered.
"Deride it all you like, but it doesn't change this basic, fundamental truth. You're the sort a woman plays around with. I'm the sort she marries."
What a pile of bull. How dare he say something like that? How dare he? And what did he know about it, anyway? And when did he turn into a bastard?
I wasn't marrying anyone. The mere mention of marriage was stupid, and I didn't believe for an instant that Doran was actually interested in marrying me. This was a game he was playing, only I couldn't decipher the point of it.
Taro didn't seem at all moved by Doran's insult. There was no real reason why he should be, I supposed. He knew Doran was no threat to him. Yet he didn't seem anxious to end the argument. "If you are the sort one marries, surely you have an understanding of procedure, duty and tradition."
"I'm not going to issue a challenge over Lee," said Doran. "It's barbaric and stupid."
"No more than trying to slither your way in between us without properly declaring your intentions."
"I did declare my intentions," said Doran. "To Lee."
Taro looked at me, and I would have crawled under the settee to avoid his gaze if it wouldn't have taken too much effort. "I said properly declaring your intentions."
"To you?" Doran snorted. "Why do you keep trying to draw me into a lineage challenge? Do you think yourself at an advantage, because you're higher born than I? From what I've heard, your parents kept you in a box in a cellar until they shipped you off to the Academy."
"Get out." The words were out of my mouth before I'd formed any intention of saying them. They were spoken too softly, though, for either man to hear me.
"You would be the sort to listen to rumor," Taro sneered.
"No rumor," said Doran. "Straight from a member of your own family, who was only too willing to tell me how everyone thought you a mad idiot, and that some still weren't convinced they weren't at least half right."
"Get out!" This time I'd pulled in a deep breath before I spoke, the words coming out in a shout that broke the air of the genteel restraint the men had been exhibiting.
Other than silence, I couldn't say what Taro's reaction was. I was looking at Doran, who was watching me in shock.
"Get out now," I ordered.
He seemed surprised by my words. How was that possible? How could he think I would just sit by and watch him denigrate my Source in such a foul manner? "Lee?"
"That's Shield Mallorough to you. Leave." I was infuriated when he didn't obey. This was my room in my home.
"Lee, I know it's appalling of us to argue in front of you, but you mustn't overreact."
"Get out now!" I shouted again, and the effort it took caused my stomach to crush in on itself so hard I couldn't help a little yip of pain, pressing a hand to my stomach as I curled up on the settee.
Taro knelt on the floor beside the settee, putting a hand on my arm. The pain eased immediately, and I could breathe again. "You may leave now," he said to Doran.
"I can help."
"Stop being selfish," Taro snapped back. "We will take care of her."
"As you've done so well so far."
"Get out. And send Ben up on your way out. Tell him she has a fever."
I was aware of Doran hesitating before I closed my eyes and concentrated on trying to breathe without causing more pain. I heard the door open and close, and I assumed Doran left.
"You're on fire," Taro said. "I'm taking your gown off, and I'm going to put you in some cool water in the tub, all right?"
It sounded good to me, especially as, now that he'd mentioned it, my skin felt hot and blistery. Cool water sounded wonderful.
Taro unlaced my gown with light hands. I couldn't help him, for once he'd removed his touch from my skin, the crippling pain jolted back into my stomach.
The door burst open. Ben carried in two buckets. "Why isn't she in the tub yet?" he demanded with uncharacteristic heat. "Put her in the tub."
Choosing not to bother with removing my chemise, Taro lifted me from the settee and carried me to my water closet, where Ben was pouring the water into the tub. Showing off his upper body strength, Taro lowered me into the tub with no apparent difficulty.
I admit it, I squeaked at the first touch of the water. "That's freezing!"
"I assure you it's not, Shield Mallorough. It merely feels that way, because of your fever. Please have a care of your hand. The poultice can't get wet."
Aye, aye, aye, the useless poultice, mustn't get it wet. Made bathing difficult. Made everything difficult. And right then, I was pretty sure my burned hand was the least of my worries.
"No, no, no, have to get out," I groaned.
"We need to get the fever down, Lee."
"Going to vomit."
The next thing I knew, Taro was propping me up and I was throwing up into a basin Ben thrust beneath my face, and it was so humiliating. Painful, too, twisted around so I could aim for the basin, my knees and ankles smacking against the sides of the tub with the force of my retching.
"You have to fetch a healer," Taro ordered Ben. "Tell them who you work for. Tell them they're in danger of losing a Pair."
I thought that was overstating the case a little, but I was too busy throwing up to say so.
"Aye. Don't let her get too cold. And make sure she drinks something."
"Send in one of the others, if they're about. I'm going to need some help."
Lovely. I was going to have an audience.
Then I was throwing up again, and dignity lost its priority.
Chapter Twenty-three
Hell. I was in some version of hell. I could swear I could believe in it. Was there anything worse than sitting in a freezing tub, so close to naked the distinction was irrelevant, throwing up the lining of an empty stomach with my head trying to explode with every heave?
I supposed it would have been worse if Taro weren't right there, if having him touching me didn't ease the pain somewhat. I couldn't imagine how bad it would feel if he weren't there. Or if I'd been a regular, without the nice buffer from sensation that being a Shield offered.
It wasn't long before I just couldn't bear to be in the tub anymore. It was too confining and it was making me crazy. With the help of Benedict, who was apparently the only other member in the residence, Taro got me out of the tub and out of my wet chemise and into my bed. Naked, because cloth felt unbearably abrasive against my skin. That meant I couldn't bear to have the covers over me, either.
It was a good thing I had no pride at all.
Instead of soaking me, they tried draping clothes drenched in ice water over my ankles, wrists, throat and forehead. That felt awful. And I just disrupted the whole setup every time I threw up, which seemed to happen every other heartbeat or so.
Taro, taking Ben's advice, forced water down my throat. I threw that up, too. That time, there was blood.
That threw Taro, who had been staying surprisingly calm, into some kind of panic. "Tell me what to do!" he shouted at Benedict.
"I know nothing of healing!" she shouted back.
"You have to know something!"
"She's your Shield!"
"And you're far older than us! Surely you've learned something other than how to bait Derek all those years!"
"You're asking for a tanning, boy."
I hoped they weren't going to start some stupid loud argument. My head wouldn't be able to stand it. I curled onto my side, wishing my stomach into steadiness. I didn't know how much longer my throat could tolerate the constant throwing up. It felt like it had been torn bloody.
"Come on, Lee. Don't do this!"
My head hurt so much, but I opened my eyes just a little. Taro was sitting beside me on the bed, tilting my world crazily, and I lifted my left hand to rest it against his arm. "Be all right," I tried to say, though it came out in a croak.
"Damn it, Lee." He squeezed my hand.
"You say that a lot."
"You deserve it. You're always doing stupid things."
I was sure there was something wrong with that, I was sure I was usually a sensible person, but I couldn't remember why I thought so.
"That stupid healer," Taro muttered. "I knew he was an idiot. I should have insisted that someone else see you, when all the huge brainless wonder thought you should do was drink tea. Wouldn't admit he hadn't a clue. Ignorance and arrogance is such a bad combination. I've never heard of such quackery. Except for that lug on that damned island. Remember him, Lee? That monster who was using Aryne. Surely one of his vile potions would have been better than nothing at all."
I felt so bad for Taro. He had to be so worried. I had no doubt I looked awful, and he was thinking that I was about to die and take him along. I knew what that felt like, that uncertainty, that helpless fear. "S'all right," I said again.
"It's not all right, damn it. Why didn't you tell me you were still wearing that damned bob?"
What a bizarre question. He must have seen the bob when he undressed me. It didn't have anything to do with what was going on.
I was so, so tired. Apparently too tired for the cramps to work as painfully as they had been before, which was a glorious relief. The headache was there, but no longer so piercing that my skull was in danger of flying apart from the force of it. That was nice. And hey, it had been, what, thirty heartbeats since I'd last thrown up? Paradise.
And then I lost track of things for a while. Taro was a constant, of course, draping careful fingertips over my temples and down the side of my face. There seemed to be a lot of slamming doors, and at times I became more aware of myself when a fresh strip of ice water was draped over my skin. I didn't understand what was being said around me for a while. But then Ben was there, and there seemed to be shouting.
I hated shouting. It was nasty and totally unnecessary. "Taro?" I said.
"I'm sorry, Lee." He squeezed my hand, too hard this time, and it hurt. "Ben couldn't get a healer to come to the residence. Apparently a bridge collapsed over the Silver River and there are a lot of victims. We're supposed to bring you to them, but you won't be considered a priority once we get there."
Oh, Zaire, lying around in a hospital like this, waiting for hours until someone could look in on someone with something as insignificant as a fever? What was the point?
Silver River. Because of the color of the water, much lighter than that of the other rivers, and because it was favored by merchants, wide and just deep enough to manage boats and barges but not deep enough to encourage currents that might capsize vessels and destroy precious cargo. I'd never really looked at it before, though there were poems that painted beautiful images of moonlight dappling the little ripples. It was too bad I'd never gone to look at it.
My parents would like that river.
"I'm going to find you a healer, Lee."
And that was when I realized he planned to leave me. "No," I said, and I tried to hold on to him. But my grip was weak to the point of uselessness.
"You have to have a healer, Lee. I don't know what else to do for you, and I've got to get a healer who's not working at the hospitals. I think I have the best chance of finding one."
That was true. Taro could convince anyone of anything.
"Ben and Elata are here. They'll look after you."
But they couldn't ease my pain as he could. "Don't go." I couldn't bear to have him go.
I felt a light touch on my temple. "I love you," he whispered. And then he was rising from the bed, shaking off my weak grip. The door slammed again.
The pain in my stomach leapt into something unbearable, and I screamed, twisting against the hands that held me to the bed. It was too much. I couldn't breathe. I couldn't find peace. I just screamed and screamed until I could scream no more, could bear the pain no more.
When I opened my eyes, I could tell from the light in the room that I had slept. I felt damp and grimy, my whole body heavy in a way that was almost pleasant. I felt numb, and my tongue was thick and unwieldy in my mouth.
Ben was sitting in a chair beside my bed. Benedict wasn't anywhere in sight.
"She had to go for her watch," Ben told me.
I swallowed a few times, and even then I could barely whisper out, "Taro?"
Ben shook his head. "He is still looking for a healer. I fear he won't find one, though. That bridge killed and injured dozens upon dozens of people. And it's caused disruption and stoppage along many streets. Even if there were a healer to spare, one who had chosen not to go to a hospital during such a disaster but was willing to come here, it would be extremely difficult for Taro to get him here."
Oh. That was too bad.
Wait. Wait a moment. Wasn't Ben supposed to be trying to reassure me, and telling me that, of course, Taro would be able to convince a healer to come to the residence, even if he knew it wasn't true?
But maybe he didn't know the part he was supposed to play. He wasn't part of a Pair. If he had a companion of any kind, I had never heard of them. So maybe he didn't know how these events went. Poor man. He was supposed to tell me I was overreacting.
"It won't be long now," Ben was saying. "I know this has been hard on you, but the end is supposed to be painless."
The end? What end? What was he talking about?
"It's really an honor to be chosen, you know."
Why did that sound familiar? Who'd said that to me? So many people I couldn't properly count. I just knew that whenever someone told me something was an honor, it usually really wasn't. "Reanist," I said.
He sniffed. "Of course not. Gods. What nonsense."
That was true. And I wasn't an aristocrat. Pure merchant class, all the way back to my great-grandmother, at least.
Damn it. Why couldn't I think? "Water?" I asked.
"I don't think that would be a good idea, Shield Mallorough. I've given you some blue root to ease your pain." He had? Really? I had no memory of that at all. "Mixing water with it, that's likely to start the stomach cramps again. You don't want that, do you? Not with Source Karish gone. You'd feel them fully, without him here to blunt them for you."
Aye, that made sense. I let my eyes drift closed.
Ben kept talking, though. "It's really not that common, you know, for partners to be able to ease each other's pain the way you and Source Karish are able. It happens, of course, obviously, but it's rather rare. Only truly fortunate Pairs are able to enjoy such benefits from their partnerships."
Wait, wait, wait. How did he know we were able to ease each other's pain? It wasn't something we really talked about.
"Source Karish is a rather remarkable person, isn't he?" said Ben. "Of the finest family, so handsome, so personable, and so very talented. One would have expected him to have an equally talented partner."
Where was Taro? Why was he taking so long? If he would just come back, I could curl around him, and just touching him would make me feel so much better. Perhaps better enough to sleep. I just needed some sleep.
"It's so strange how things turn out sometimes," Ben said. "I'd hoped, so much, that one of my children would be a Source or a Shield and have the kind of life you enjoy."
Would he please just shut up? I'd never be able to get back to sleep with him nattering on.
"Sources and Shields are better off even than the aristocracy, in a way. No matter how wealthy or how highly placed an aristocrat is, a change in fortune can have him disgraced and digging out his dinner from someone else's land. But a member of the Triple S, no matter what they do, they'll always have a roof over their head, clothes on their back, food on their table."
Sleep, or the potential for it, was slipping farther away as the pain in my head grew more prominent.
"But none of them had any talent. In anything, really."
Who never had any talent? If he insisted on going on and on and on about nothing, he could at least make sense as he went about it.
My skin was heating up. Those strips of cloth dipped in ice now seemed a wonderful idea. I could see the bowl they'd been using earlier, sitting on the vanity. It probably wasn't so cold anymore, but even warm water would be soothing. "Water?" I asked.
"I said no, dear. It'll make you feel worse."
"No." He didn't understand. Why couldn't I speak properly?
"It won't be long now, I promise." He put a hand on my left arm, brushing my skin. It burned.
"Don't," I said.
There was a clatter of hooves in the street, which happened often enough, but it made Ben rise to his feet and leave the suite for a few moments. When he came back, he announced, "Source Karish is back."
He sounded surprised.
Thank Zaire. He was back. I would be able to sleep.
Chapter Twenty-four
Karish came into the room carrying a huge trunk, a heavy one from the way he was using a leg to balance it as he walked. And behind him was Healer Cree.
She halted when she saw Ben. "Who is this?" she demanded in a frosty tone.
"He runs the house." Taro grunted as he set the trunk on the floor.
"I want him out of the house."
"He lives here," Taro objected.
"It is a condition of my assistance."
"It is of no moment, Source Karish," said Ben, backing toward the door.
"Tell everyone else to stay out of the house as well," said Cree. "Their presence will disturb my work."
"Of course."
Cree watched him leave. I heard the door to the suite being closed, and Cree waited a few moments beyond that before stepping to the trunk and opening it. The first thing she took out of it was a bundle of wooden sticks that unfolded into a small, short table. I hadn't seen folding furniture since we'd left Flatwell.
Taro sat beside me on the bed, taking my left hand in his, and my headache eased. "Healer Cree is going to take care of you, Lee," he said. "But you have to remember that she's a different sort of healer from what you've seen before. She's going to do things differently."
Sure. Fine. Whatever.
Only Cree put immediate lie to that statement by pulling out strips similar to what Healer Pearson used. Like him, she put a strip each to my tongue, my temple, the burn on my right hand. With a small knife, she made a tiny cut in my arm, and she laid a strip against that, too.
"It took Pearson hours to examine his strips," said Taro. "She may not have hours."
Really, if he had to exaggerate about such a thing, couldn't he do it out of my hearing?
Cree glared at him. "We discussed this before I agreed to come," she reminded him. "You are to trust me, yes?"
Taro nodded.
"Besides, Pearson is incompetent, and too arrogant to know it. Don't insult me by comparing me to him." From a small pocket in the lid of the trunk she pulled out four glass vials with flat bottoms. From within the trunk she pulled out a bottle. She put a strip palm in each vial, then uncorked the bottle and poured clear yellow liquid into the vials. After corking the bottle and putting it on the table, she picked up each vial and swirled the contents at the bottom. In each case, the liquid clouded from yellow to a dull, smoky brown.
Cree sucked in a quick breath. She put down the last vial and stepped closer to the bed. "Move," she ordered Taro, and Taro moved. Cree sat on the bed and stared at me, putting a careful fingertip below each eye and gently pulling the skin down slightly. Then she had me open my mouth and move my tongue around, which under the current circumstances felt as easy as pushing a tree over with my pinky. She raised my left hand from the bed, sniffed it, then licked it, which was just disgusting. She peered at the fingernails of both hands. "Who wraps your hand for you?" she asked me, glancing at Taro.
"Ben," I whispered.
"Where does he get the poultice?"
"Makes it."
"Where does he get the ingredients?"
"Don't know."
"How long has he worked here?"
"Years," said Taro. "Years before we came here."
"Hm." Cree went back to her table, picking up the knife she'd used to cut my arm. "She's reacting to the niyacin powder."
"No, no, Ben stopped using it," I said. "I felt better."
"I can't say anything to that. Reactions aren't always predictable, and they aren't always progressive. You can feel fine one day and awful the next. But there are huge amounts on your palm and in your mouth. I'd say you've consumed it, very recently. What have you eaten today?"
"Just bread."
"Where did you get the bread?"
"Ben made it. But there was no way he would have put niyacin powder in it." Even I knew that wasn't an ingredient for bread.
"Are you sure?"
"You're saying Ben poisoned her deliberately." Taro looked appalled. And furious. "That's a disgusting thing to say! He's been nothing but good to us."
"I am not making accusations. It is not my job to determine who is responsible. It's my job to identify the problem and fix it."
"Can you?"
"We'll see."
I frowned at that. "I might die?"
Her eyebrows rose. "You're not terribly intelligent, are you?"
"Stop that," Taro snapped. "Just make her better."
"Of course, it's as simple as that," Cree murmured sarcastically. "I must see how far the poison has traveled." Apparently that meant nicking the skin low on my stomach, the bottoms of both of my feet and on my forehead near the hairline, collecting samples of the blood on more strips. It also meant being examined more intimately than I'd ever been examined by anyone before, and that included anyone I'd ever slept with. It should have hurt. It should have been humiliating. But I really didn't feel anything. I was too tired and too numb.
"This is very serious," Cree said once she had fiddled with more vials to her satisfaction. "It must have been a massive dose."
I couldn't bear to watch Taro. He was pacing and it was making me dizzy. "But there is something you can do, yes?" he asked.
There was no answer.
I was going to die. Poor Taro. It really wasn't fair. "Dunleavy," said Cree.
"Mm."
"Open your eyes."
Did I have to? I was so tired.
"Open your eyes!" Cree ordered.
I sighed and looked at her. She was standing beside my bed, staring down at me with eyes that seemed darker and larger than I remembered. "If nothing is done for you, you won't live through the night," she said.
"All right," I said.
She huffed. "All right. Stupid girl. As though I were offering you a cup of coffee you're too polite to refuse. I might be able to help you, but medicine won't be enough."
"All right," I said, because she had paused and clearly expected me to say something.
"What do you mean, no medicine?" Taro barked out.
"We discussed this before we came here, Source Karish."
"But I didn't think you weren't going to use any medicine at all."
"Adding any kind of medicine to the mix will nullify the spell."
The spell. Oh lords. I tried to giggle, but all that came out was a wheezing sound.
"Dunleavy, you'll have to trust me. I know your beliefs, but you're beyond the help of common medicine."
Spells. Sure. Why not? Go ahead.
"Your life is in my hands. Just as, when we are finished, my life will be in yours. I will expect you to honor that."
"All right." It seemed the thing to say.
"Source Karish?"
"I have no interest in reporting you to the Runners. I think those laws are ridiculous."
Oh. That was what she was worried about. Perhaps I should show her my books. They were around somewhere.
"That is because you don't believe spells truly work," said Cree. "We shall see how you feel after."
"If you heal her, all I'll feel is ecstatic."
"And if I fail to heal her?"
"I won't be around to feel anything."
She nodded. "Point taken." She removed from the trunk a square black bag and filled it with bottles and jars, candles and what looked like alarmingly long needles. "Does this place have a cellar?"
"Yes," said Taro.
"How big is it?"
"A bit larger than this room here."
"Is there much in it that would obstruct movement?"
"A big table in the middle of the room. Very solid."
She swore. "Can you drag it out?"
"I don't think so. The stairs would be far too narrow. I'm pretty sure it was built in the cellar."
"I don't suppose you know how to wield an axe?" she asked without much hope.
"I do."
"Really?" she asked with surprise.
"You'd be astonished at the breadth of my knowledge."
Once upon a time, that comment would have been made flirtatiously. The dignified healer's reaction would have been interesting to see.
"Then chop that table into pieces, as small as possible. Wait, what is the cellar floor made from?"
"It's just packed dirt."
"Perfect. Chop the table into kindling. I want to be able to light it easily. Make four piles of wood, one in each corner of the cellar. I'm assuming the cellar has four corners?"
"There's no place for smoke to escape," Taro said dubiously.
"You mentioned a stairway. That will have to do. Start chopping. Press anything else you find down there close to the walls, and take down anything that's hanging from the ceiling. And be quick about it. We haven't much time. Oh, take this with you." She gave him the black bag. "Come back up when you're done, but leave the bag down there."
Taro stared at me a moment, clutching the black bag, indecision clear upon his face.
"Move, man!" Cree ordered.
Taro left.
"All right, my dear," said Cree, who was, of all the strangest things, taking off her clothes. "I need something of yours that's of great sentimental value. I imagine that's the closest thing to magic that you have."
Things of sentimental value? I wasn't really the sentimental sort. A leather hair tie I'd made as a child, about the only thing I'd ever made with my hands that wasn't hideous. The history treatise given to me by my favorite professor. There was the bookmarker Lamer made for me, braided out of incredibly soft blue fabric, just before we left the Academy. The gold hoops Taro had given me. Would one of those do?
She wasn't going to destroy it in one of her spells, was she?
Though right then, I couldn't say exactly where any of those items were.
"Wait a moment," Cree said, stepping out of her skirt. And yes, it looked like she was getting completely naked. That couldn't be good. "I feel something in here."
Possibly a draft.
"Here it is." She scooped up my chemise, which had been left in a puddle on the floor beside the wardrobe.
I thought I didn't have much time left. Why was she wasting it?
"No, no, no. Ah! This." She unpinned the harmony bob and let it hang from her thumb and forefinger. "Why, Shield Mallorough, I didn't know you had it in you."
"Jus' jewelry," I slurred.
"Not at all. It's so much more than that. Expertly crafted, expertly enspelled. And worn next to your heart ever since. This might be the saving of you."
Wonderful. My life depended on a trinket. This would end well.
"Here, hold this." She put the bob in my right hand. "Don't change hands," she ordered as she saw that I was about to do just that.
Cree stripped down to her skin, then pulled from her trunk a garment of dark green with white designs embroidered down both lapels. It looked like some sort of robe, but made only of linen, the sleeves deep and wide, the bodice of the garment falling freely from the shoulders except where Cree fastened it down her front with hooks and eyes. It fell just to the top of her bare feet. Next she unpinned her hair, shaking it out so it hung loosely about her shoulders. Finally, she took off her rings, earrings and necklace.
Then she bent over her trunk again, throwing into another black bag another collection of bottles and vials and some sticklike things. Once she had everything she needed, she closed and locked the trunk.
I could hear the irregular thuds of Taro destroying the table in the basement. He'd done his share of chopping wood while we'd been on Flatwell, but that had been a while ago, and chopping something like a table had to be trickier than chopping logs. I just hoped he didn't lose a foot or something.
There was something kind of comforting about hearing the thud and the clatter. Homey.
And then: "What the hell is going on here?"
Ah. LaMonte. Always a pleasure.
"Who is that?" Cree asked, going into the suite to lock the door to the hall.
"LaMonte," I answered. "Source. Old. Thinks he's in charge."
"And he's not in charge?"
"No. No one is." That didn't mean he couldn't cause problems for us if he wanted.
The sound of Taro chopping wood halted. Clearly, LaMonte had gone down to the cellar. After only a few moments, the sound of chopping resumed, and a few moments after that I could hear him coming up the stairs to the second story. Then a pounding on the door to my suite. "Dunleavy! What's going on?"
"You can't come in," said Cree. "Dunleavy has become extremely contagious. You must leave this residence."
The snort I heard even through a door and across two rooms made me smile. "Nonsense. She was fine yesterday."
"She is now extremely ill. And contagious."
"I've never heard such tripe. Open up at once."
"Please, Source LaMonte," I said, and it took enormous effort to make my voice loud enough to reach him. "It came on suddenly. Healer Cree was the only healer willing to come to see me, and she really knows what she's doing. She had to send Ben away, too."
"And what about Shintaro? How is he immune?"
"He's not," said Cree. "I have no doubt he's already infected, so I need his help for as long as he can give it, before he succumbs to his symptoms. I would ask you to leave before you risk infection, and make sure none of the others come into the house. Consider it under quarantine."
There was a long pause. "This seems highly irregular," he said eventually.
And LaMonte did hate the irregular. Not that I blamed him. I hated the irregular, too.
"So is her illness," Cree responded tartly. "Are you going to force me to risk infecting you and who knows how many others just so I can find a Runner to enforce my quarantine on this residence?"
"You can't have me ejected from my own home."
"I am a healer. Believe me, I can."
I had no idea whether that was true or not. If LaMonte's silence meant anything, neither did he.
I did think it would be hysterical if the two imposing figures met under more normal circumstances. I would have to introduce them. If I survived.
"I will be looking into this," LaMonte threatened.
"You do that." In a much lower voice, too low to be heard by LaMonte, she added, "As long as you look into it somewhere else."
We listened to LaMonte retreat from the door and go down the stairs. I didn't hear the outer door open and close, but that didn't mean LaMonte hadn't actually exited the building. I wouldn't hear that door unless it was wrenched open and slammed shut.
So there was no way to know, one way or the other.
In time, the thud and clatter from the basement ceased, and we heard footsteps on the stairs again. There was a knock on the door. "It's me," said Taro.
"Has that other person left?" Cree asked.
"Chris? Aye."
"Are you sure?"
"There's no reason for him to hide from me," Taro said impatiently. "Aren't we running short on time?"
Cree sighed with impatience, and she went to the sitting room to unlock the door. "Please assist Dunleavy down to the cellar."
"Need to get dressed," I said.
"No, you need to remain as you are."
Wonderful.
Taro came into the room, strands of hair flying free of the tie at the base of his skull, sweat plastering his cream-colored shirt against his chest and back. I wished I had an artist's skill, that I could make renderings of him in all his states of beauty. He would never want to look at them, or even know about them. I would just like them for myself. Maybe he would want to see them when he was much older, and beautiful in a different way.
Chapter Twenty-five
Helping me down to the cellar meant carrying me down to the cellar, as my legs were useless. It was horribly embarrassing, though I couldn't say why. Both Taro and Cree had seen me naked, and no one else was there. That there was a serious danger of being dropped only increased my discomfort. Taro had exercised a lot of stamina in destroying the table, and I wasn't light.
I knew where the cellar was. I'd been there once or twice. There was something about the cellar that made me uneasy. It was completely underground. The air smelled heavy and unpleasantly musty. And I never felt I could move about freely.
Right then, lit only by a lantern set on the floor by the door, casting distorted shadows everywhere, the cellar looked unfamiliar and bizarre, the table gone and replaced by four piles of wood, one in each corner. Taro had dug a sort of trench around each pile in an attempt to prevent any spreading of fire once the piles were lit.
It struck me just how incredibly bad an idea this was.
"Lay her on the floor," Cree said. "Head facing this way, feet facing there."
"Directly on the dirt?" Taro objected.
"Now."
Wonderful. So I lay down on the floor, and despite the weird calm that had descended upon my mind, I had images of all the insects crawling in the dirt and possibly into exposed parts of my body.
Cree lit each of the bundles of wood, having poured some kind of oil over them. The warmth was immediately apparent. From her bag she took a black-handled knife, which she dug into the dirt near my feet. Shuffling backward, she cut a wide circle around me, ending back at my feet. Small sticks—rods, I supposed—were thrust into the ground, one at my head, one at the tip of each hand and each foot. From a bottle she poured a clear liquid from one rod to the next, connecting them all. This line was then frosted with salt.
"Take off your clothes, Shintaro," Cree ordered. "And step inside the circle, but stay outside the salt line." In a few moments, Taro was seated on the dirt beside me on my left.
Cree took my left hand and sliced a small cut into the palm. She took Taro's left hand and cut his palm, too. She took the bob from my right hand and put it in my left. She clasped my left hand and Taro's left hand together. "Don't let go until I give you leave."
Cree opened more bottles, and at various places within the circle poured out the contents. Sometimes the contents were a liquid, sometimes a powder. They were being poured to create patterns, but I couldn't see the patterns against the dirt.
A strange scent was developing in the air. It smelled like wet hay.
Once Cree was finished dumping stuff on the floor, she stepped outside the circle, standing near my feet. She plunged the knife deep into the soil. Then from her bag she took out another knife, this one with a white handle. "It is time for me to begin. Once I begin, I can't stop, or the fires will break loose and kill us all."
I was very displeased to be learning that death was a possibility this late in the proceedings.
"Dunleavy, you must not move from within the circle of salt. Shintaro, you must not cross the salt line nor leave the circle. Dunleavy, this will hurt. A lot. Shintaro, you must keep her from moving over the salt line, but you can't touch her with anything other than your left hand. Neither of you must try to get my attention. This is very important. I can't lose my concentration." She looked at me then. "Trust me, Dunleavy, as I am trusting you."
"Aye," I said. Apparently, I had no choice.
I was going to die. I wasn't ready to die. There were things I hadn't done. Gone to my parents' home. I'd never been. Gone back to talk to Professor McAuley. And I'd really wanted to see how Aryne was going to turn out. I had no doubt she was terrorizing the Source Academy, and more power to her.
Poor Taro. It wasn't fair.
Cree held the knife in both hands, at chest level, the tip of the blade pointed down. She looked ridiculous. What the hell was I doing?
"I call on the power of the earth," Cree intoned. "Source of all which heals. Source of the poison which defiles the blood of this subject. I call on you to draw this poison out, and take back that which should have never been drawn from you. Reclaim your essence."
The chants I'd read in the book had rhymed.
"I give to you blood. Blood of the half, who is strong and clean. Blood of the caller, who is wise and clean." She sliced her own right palm, then held her hand over the floor, palm facing down so the blood could drip from it. "Reclaim your essence."
Nothing happened, of course. Except the room was warming up and the smell of wet hay grew heavier, almost suffocating. Was this what her patients paid good money for?
"Reclaim your essence."
It was going to be so embarrassing. When I died, and Taro died with me, Cree would clear out her props and leave. She wasn't stupid enough to be found with a dead Pair and tools Runners would recognize as being connected to casting. And what would people think, what would my family think, of how we would be discovered? Naked in the cellar. And how people would talk, how the Stallion of the Triple S and his Shield had ended up. I hated the thought that I cared; it was stupid to worry about what people would think after I was dead. Still, what would my parents hear? What would my father think?
"Reclaim your essence!"
The circle flared into flame, low and blue, and Taro jerked in surprise. Damn it, the fire was spreading. How stupid were we, lighting fires in this small space? The temperature rose uncomfortably, while the scent of wet hay seemed to coat my nostrils and throat. It was foul.
I felt that jittery sensation again, piercing through all my discomforts and the general assault on my senses. And I realized something. It wasn't the ashes that gave me that feeling. It was the power of the spell itself. I could feel it when a spell was being used. My gods.
"Reclaim your essence!"
And the fire leapt closer, and this time the flames were dark green, tracing along the patterns on the floor.
That was it. I wasn't going to let myself be burned alive. But when I shifted to move, Taro's grip on my hand became painfully tight. I looked up at him, and he was glaring down at me. I opened my mouth to tell him not to be ridiculous, and he squeezed my hand so hard I thought a bone was going to break.
I was stunned. Despite appearances to the contrary, Taro was not a foolish person. How could he think we were doing something sane?
It was more likely that he thought there was nothing else we could do. And maybe he was right. I was hot and nauseous and so tired. No one else knew what the hell to do.
I figured if Cree was still in the cellar, and she was, we couldn't be in imminent danger of being fried.
Maybe Ben was out looking for a real healer.
"Reclaim your essence!"
Pain flared through every vein I had, sharp and icy and splintering, so severe I couldn't even scream. Gods.
"Reclaim your essence!"
I couldn't move. I wanted to. I needed to try to curl away from the pain, which didn't make sense, because the pain was everywhere. Except one spot that I couldn't really define, a location that didn't have the ice but instead some odd sucking sensation. It wasn't pleasant, but in comparison to the rest of my body it was a haven of comfort. I wanted more of it. But I couldn't move.
"Reclaim your essence!"
The shards of ice grew larger, tearing at my skin from the inside. For Zaire's sake.
"Reclaim your essence!"
I must have screamed then. How could I not? But I couldn't hear anything and I couldn't feel anything other than the icy, shredding pain tearing through every part of me, slicing at the bottom of my feet, ripping jagged holes into my stomach and chest, sending jolts of ice through my limbs. My left hand was hot, burning hot, and the focus of the strange sucking sensation. Gods, gods, gods.
"Reclaim your essence!"
The words pierced my ears and struck my brain. And then I felt nothing, a sublime relief.
When I opened my eyes, the fires within the circle were all gone, though the scent of wet hay lingered. It was dark and smokey and I felt filthy.
Taro was shouting. "There's blood all over her!"
Was there? It took far too much effort to move either head or limb to look.
"The poison is bleeding out of her." Cree was chillingly calm.
"Or maybe she's bleeding to death!"
"Keep your hold on her hand. She is taking blood from you."
There was something wrong with mixing blood from different people. I knew that. I just couldn't remember what the danger was, or what it meant.
"Don't touch anything," Cree said. "Don't move from your positions. This isn't finished yet." She went to each corner of the cellar, dousing the flames with a handful of powder. I didn't understand how that worked, a dash of powder putting out fire.
"Thank you," Cree said, and I knew she wasn't talking to either of us.
The salt she had lined around me she dug up with one of the rods she had sunk into the ground. She mixed it up in the soil, then poured more salt over it. "Thank you."
She plucked up the remaining rods, and when she had the last one in her hands, she said again, "Thank you."
"For what?" Taro asked her.
She didn't answer. From her bag she took out shorter, slimmer rods and she stuck them into the dirt in roughly the same locations as the first set. She lit them, and though none of them held a flame, they immediately began to smoke, thin white lines curving up from the tips. The smoke smelled nice, clean and refreshing, a little like mint.
All this smoke. I wondered if any of the food in the cellar was still fit for eating.
Cree knelt beside Taro, her hands clasped on her knees, her head bowed.
"Can't I—"
Cree shushed Taro sharply. He shut up.
So we just waited, in silence. I didn't know for what. I wished I could sleep. I was too tired to sleep. And too filthy.
And then, in time, she reached out and separated my hand from Taro's. The fact that both of our hands were smeared with unexpected amounts of blood should have probably been alarming to me, but I was beyond strong emotions. Cree plucked out the bob and placed it in a handkerchief, which she tied around Taro's neck.
Another bottle appeared from her bag, and in this was a liquid, clear and just a little thicker than water. This Cree slowly poured over me, over my forehead, temple to temple, hairline to eyebrows, and then my face, throat, torso and limbs, all very thorough, with the same intimacy she'd shown during my earlier examination.
That should have bothered me more than it did. I just didn't care. I was just happy the pain was gone, and the nausea.
"Thank you," Cree said once she'd finished with my feet. She untied the handkerchief from Taro's neck and tied it around my own. It was awkward, because I couldn't shift my head to help her.
Once the new rods had burned down to small stubs, she removed them from the dirt and threw them into her bag. Then she put her hands flat on the dirt, bowing until her forehead touched the floor. "Thank you," she said, and the difference in her tone told me this whole thing was over.
So. That was casting. It was real.
I would panic about that later. When I wasn't so exhausted.
"You can get dressed, Shintaro," she said, rising to her feet and shaking dirt from her robe.
Taro tried to stand, only to flop over on the floor.
"Don't worry. It's natural to be weak. Dunleavy has taken a great deal from you."
"How am I going to get her back upstairs?"
I laughed. For some reason that question struck me as hilarious.
"I will assist you."
And, oh, wasn't that an exercise in graceless danger? Carrying me was beyond Taro at that point, which meant he and Cree sort of dragged me up the stairs between them, each with one of my arms around their shoulders. There were several moments when someone nearly lost their balance. It was hysterical.
And of course, I was still naked. Later I would be pathetically relieved that there was no one else there to see me.
Cree directed us to Taro's suite rather than my own, though I balked at being laid on his clean sheets. "Filthy," I mumbled as they sat me down on the side of the bed, and I let myself fall back. I didn't think I could sit up unassisted if I tried.
"Yes, and you'll remain filthy until I come back to see you tomorrow. Neither of you will wash anything, not even your hands or face."
"You can't be serious." Taro's lip curled in disgust.
"Lie down." She left the room without waiting to see if she was obeyed.
Taro shifted me so I was properly stretched out on the bed. "How do you feel?"
"Don't know."
"Think any of what she did down there did any good?"
"Don't know."
And I didn't want to think about it. At least, not yet. The very idea of casting spells, that something so powerful and limitless existed for anyone who could read a book, was appalling. People were too stupid to be trusted with that sort of thing. People were too avaricious. And too vengeful. The possibilities were terrifying, and I was too tired to be terrified. I would be terrified later.
Taro sat on the other side of the bed. He was shaking, and his eyes seemed a little unfocused. "You all right?" I asked.
"Don't know," he answered with the tiniest smile.
In time, Cree came back into the room, carrying a tray with a jug and two mugs. "This is water," she explained, filling the mugs. "This is all you are to consume between now and my return. Neither of you will be able to tolerate anything else in your stomach, and vomiting when you're this drained can be dangerous." She gave Taro one mug and put a strong arm under my shoulders to lift me up and have me drink the water, whether I wanted it or not.
"You both need to sleep," said Cree. "Shintaro, I want you to lock the door behind me. I'll be leaving a note for your colleagues, instructing them not to disturb you. Should you somehow end up speaking with any of them before I return, you are to tell them I took you downstairs to sweat the illness out. No mention of casting to anyone. Understood?"
She gave each of us a hard look, and I was sure it would have been most effective had I not been so desperate for sleep.
"Sure," I muttered.
"Of course," said Taro.
She nodded and moved away from the bed, and I had no idea whether Taro followed her to lock the door, because I fell asleep.
I woke to pounding on the door, my heart leaping into triple time, because pounding on the door was never a good thing. I heard Taro swearing, and there was movement, and then I heard him shout out an inquiry, but I didn't hear an answer.
It turned out to be Cree. "What's wrong?" I asked, feeling confused and dizzy.
She was clean again, her hair tightly coiled at the back, her dress of simple clean lines and a delicate fabric unsuited to any kind of labor. She looked completely respectable. No one would think she'd been chanting in the basement with multicolored fire dancing around her.
"Nothing," she said. "I told you I would be back."
It was morning already? I rubbed at my eyes, which felt gritty. So did my fingers, and I looked at my hand, which, to my surprise, was streaked with some kind of orange reddish substance. And suddenly I felt just disgusting.
Cree put me through another examination, involving more little strips, and at the end of it declared all the niyacin had been cleared from my system.
And, really, I couldn't believe it. It was all so unlikely, to be so throughly sick so quickly, and to be cured so quickly.
Though I said nothing, Cree seemed to know what I was thinking. "How is your stomach?"
I'd been trying not to think about it. Now that I did, it cramped painfully. "Empty."
"You're hungry."
"That adjective seems somehow inadequate."
"Shintaro, how does she feel?"
"Excuse me?"
"Is she fevered or clammy?"
He put a hand to my forehead, my cheek, and then my throat. "No. She feels good. Normal."
"And her eyes?"
"They've lost their glaze."
My eyes were glazed? "But I'm so tired," I complained. "Of course you are." You idiot. "You were poisoned, or as good as, and you lost a lot of blood last night. You won't be back to full health for weeks. But you will be back to full health. Provided you don't do anything stupid." She looked at Taro, who was leaning against the wall looking awful, the disarray of his hair crazed as opposed to sexy, his white shirt and black trousers wrinkled and encrusted with various stains. His face, pale with wrinkles about the eyes, had dabs of the same orange red that was streaked all over my hands and arms. "She needs lots of broth, and tea with high amounts of sugar and cream. No coffee, nothing with alcohol. Tomorrow, add fruit and greens, fresh. Nothing dried. The next day, you can add cheese and small amounts of meat. Don't touch that bread. I want you to bring it to me. If there's as much niyacin in it as I suspect, it isn't safe for anyone, whether they're sensitive to the powder or not."
It couldn't be the bread. I refused to believe Ben had baked niyacin powder, which he knew I reacted badly to, into bread that he had then calmly watched me eat. That was unbelievably cold-blooded. And why would he do it? He had never shown anything but reverence for Sources and Shields.
There had to be another explanation.
Cree was packing the last of her things. "I am having a dinner," she said. "All of the guests are people you know. I expect your attendance as payment for my services."
Really, we didn't have to pay her, but I tried not to be an ungrateful wench when I could help it. "Be honored," I said.
"Thank you so much for all you've done," Taro added. "We're aware of the significance."
"Are you?" she asked.
"We are."
She had either cast a spell, or had pretended to cast a spell. Which it was wouldn't matter to the law. She had put an enormous amount of trust in us, trust we hadn't earned.
"Good. I will write to you about the details. And Shintaro, while I understand your desire to attend to Dunleavy, your body experienced something traumatic last night. You need rest, too. Share the burden. Understood?"
"Yes, ma'am."
"Good. Let me know if there are any further problems."
"Thank you," I called out as she left, not sure if she heard me, my voice was so weak.
I pushed the blankets aside and dragged my legs over the side of the bed. They were heavy and barely felt like they were attached to me, just dead loads I needed to shift around with other parts of my body.
"Where do you think you're going?" Taro demanded.
"Bath."
"You need sleep."
"Bath first." And then something to eat, and then sleep. I was disgusting, filthy, in every crack and crevice. If I weren't so tired, my state of filthiness would have been making me crazy. I felt like I hadn't been clean in months.
"Lee."
"Bath," I insisted.
"Fine, then." He took me by the shoulders to pull me from the bed.
It was Taro's bedroom. I wouldn't call Taro a vain lad, unless I was trying to wind him up, but he did have more mirrors than I did. So I got a good look at myself that I could have done without.
Those orange streaks were all over my face and throat, all over my body bizarre patterns with no regularity. There were cleaner bits around my eyes, cleaner strips down my cheeks. And the substance, whatever it was, had gotten into my hair, creating clumps and patches all around my face. I looked demonic. "What is it?"
"You were bleeding, Lee. It was coming out through your skin." Taro shuddered. "I'd never seen anything like it. I thought you were bleeding to death right in front of me." He took a few slow breaths. "Then Cree poured that stuff all over you, and I guess it turned the blood orange."
How thoroughly disgusting. I could have done just fine without knowing any of that.
I hated mirrors.
Chapter Twenty-six
Bathing turned out to be an unanticipated exercise in brute force, the garbage on my skin proving to be somewhat resistant to soap and water. The water, being cold, was little help until Firth, bless her, brought up two kettles full of hot water. That made shifting the orange sludge a little easier. And then Taro washed my hair, which felt almost luxurious. All the effort was worth it—it was wonderful to finally be clean—but it was exhausting, and by the time we were done I was too tired to eat.
When we moved back to Taro's bed, the linens had been changed, and Taro curled around me. I felt comfortable and safe. I slept.
It was night again when I woke, and I was ravenous. Taro was still asleep beside me, and I was tempted to rejoin him, but at this point my hunger was almost overpowering.
My exit from the bed was not graceful, but Taro didn't stir. Lords, I was so tired, feeling weighed down just everywhere. But hunger could provide great motivation.
The residence wasn't as quiet as it usually was. When I reached the bottom of the stairs, I heard a gaggle of voices from the parlor, and it was so rare to have so many of the Pairs in the house at the same time that I was drawn away from the kitchen.
Beatrice and Benedict weren't there. On watch at the Stall, no doubt. But all the others were, talking about the disaster that had happened the day before. Apparently more people had died than the collapse of the bridge had really warranted, simply because of the lack of organization in the rescue and relief efforts.
Maybe if Cree had been there, she'd have been able to do some real good.
I wasn't going to think about that yet.
"Dunleavy!" Firth cried out, putting aside a cup of tea. "You look awful."
Her Shield gave an amused eye roll at that. "Have a seat, Mallorough. We have some soup on the stove. I'll get you a bowl."
What was going on with those two? Wasn't I still a sexual deviant? Or had my trials of the day before cleansed away the perversion from my activities?
Manners prodded me to object. I didn't. The walk to the kitchen, after the trek down the stairs, seemed an impossible journey to make right then, so I collapsed in the chair closest to the entrance to the room.
"Where is Shintaro?" Wilberforce asked.
"Asleep."
Wilberforce, on the other hand, hadn't changed at all. It was refreshing. "Is he all right?" Wilberforce stood. "Maybe I should check on him."
"Don't you dare wake him," I ordered with less heat than I liked. "He needs sleep."
"But really, I should just make sure he—"
"Sit down, Franklin!" LaMonte barked.
Wilberforce sat.
I pressed my hand against my lips to hold down the insane urge to giggle.
"Now that you're up and about, Dunleavy," said LaMonte, leaning back on the settee with legs crossed and hands laced together, "you can tell us what went on here yesterday." The fact that I didn't know what he was talking about showed on my face. "When that woman was here."
I still hadn't had a chance to properly think things through. I didn't want to talk about it yet. Or ever, really, but certainly not yet.
But LaMonte had been there that day, had been there long enough to know something odd was going on, and if I didn't keep him happy, he had the power to make things really difficult for Taro and me. Possibly Cree as well. And really, if someone was casting spells in the residence, didn't LaMonte and the others have a right to know? Even though I resented being am-bushed as soon as I stuck my head in the door?
So, start at the beginning. "I became very ill. The only healer we could get, because of the bridge collapsing, was someone Taro and I met through a friend. Soon after getting here, she realized I was having a bad reaction to the niyacin powder Ben had been putting in the poultices for my hand." I didn't want to reveal that Cree had also thought it was in the bread. That was a serious allegation, and I wasn't going to speak about it without some kind of proof. "Where is Ben?" I wanted to talk to him and get this cleared up.
LaMonte frowned. "I haven't seen him since the other day. He left a note saying he needed to be with his daughter."
All right, that looked bad, but I still couldn't start throwing accusations about. "Does the note say when he's coming back?"
"No," said Wilberforce.
"He's been treating your hand for weeks," said LaMonte. "You just suddenly got ill with no warning?"
"You were taken off the roster ages ago," said Stone, handing me a mug of broth. "I thought that was because of your reaction to niyacin."
I'd forgotten I'd told her that. "It was."
"Did you not tell Ben you couldn't have niyacin powder?" LaMonte demanded.
"I did." This was going to come out whether I wanted it to or not.
"And he kept giving it to you?"
"We don't know what happened. That's why I want to talk to Ben." My conscience prodded me on, even though it was a little late. "Has anyone, over the past couple of days, eaten bread that Ben baked that had a lot of nuts in it?"
"What a bizarre question," LaMonte said dryly.
"The healer thinks I actually consumed the powder, and the bread was all I'd had to eat that day. She said it wouldn't be safe for anyone to eat."
"Why the hell would Ben put niyacin in bread?" Firth asked.
"I have no idea."
"For Zaire's sake, Dunleavy!" LaMonte threw up his hands. "Don't be so naive. He was trying to harm you."
"But why would he? It doesn't make sense. And it's a stupid way to kill someone. Everyone would suspect him."
"They would have a hard time determining what had killed you once you were dead," said Stone. "And even if someone did, Ben might be able to get away with claiming it was some kind of accident."
"I haven't seen the bread you're speaking of," said Firth. "I went out myself to get some because we were out." The others echoed her.
"So he must have disposed of it," said LaMonte. "And given this latest development, I think it's time we revealed the other piece of information we have about Ben."
I knew what he was talking about. "I don't think it's any more appropriate now than it was then."
"Then you're a fool," he snapped. "Everyone, Ben's daughter has been arrested for the murder of Mayor Izen."
And cue the outrage, the demands as to why we hadn't told them, and the assertions that they knew there had been something strange about Ben. I let LaMonte handle the bulk of the questions. He had the energy for it.
"All right, that's enough," LaMonte ordered. "We did what we did, right or wrong. What's important now is how we proceed. Franklin, I think you need to find a Runner. They need to know what's going on."
"We're jumping to conclusions," I objected.
"That's for a Runner to decide. Please, Franklin."
Wilberforce was torn. I could see it on his face. He was delighted to be singled out by LaMonte. He was offended to be sent on an errand. I guessed which impulse would win and was proven right when Wilberforce went scurrying out of the room.
I thought calling in a Runner was too hasty, and I couldn't believe Ben had been poisoning me deliberately. It just didn't make sense. I knew I wasn't beloved by all, but surely I'd never aggravated anyone to the point that they would seriously go to the effort of killing me. Aside from Creol, that is, and he'd been crazy. And I'd always treated Ben well, hadn't I? And he knew as well as anyone that killing me would kill Taro. Surely Ben didn't hate me enough to kill Taro.
I couldn't believe anyone could hate me that much.
"Why did that Cree woman say you were contagious?" LaMonte asked.
Ah, hell. I'd forgotten that. It didn't work with the truth that I'd been poisoned. "She said the treatment was . . . unorthodox. It would be dangerous to have it interrupted in the middle, and I guess she figured telling you I was contagious was the best way to avoid interference."
"Unorthodox?" That was Hammad, LaMonte's Shield, who rarely spoke, so it always surprised me a little whenever he did. "What does that mean?"
"That I sat in the cellar for hours while the table burned." Naked, but no one needed to know that. "It was quite disgusting."
"I heard—" said LaMonte, and he cut himself off.
What the hell did he hear? He was supposed to have been out of the damned house before anything was done. If he got Cree arrested, oh what a bastard.
"You just seemed in considerable distress," said LaMonte, his expression virtually unreadable.
That wasn't what he'd planned on saying.
"I was. She needed to do something kind of extreme, to get all the niyacin out."
"This is why we no longer have a dicing table?" Stone asked. "And the cellar smells something like what I imagined a bordello smells like?"
Williams's bordello hadn't smelled anything like the cellar did once Cree was done with it. Maybe cheaper bordellos did.
"Not to mention the cuts on your arms and on your face," Firth added.
"It was all about getting the niyacin out," I said. "The cuts to let the niyacin out, the burning stuff to draw the niyacin out."
"And this all worked?" Hammad asked, looking dubious.
I shrugged. "It seems to have. I'm dead tired, but I feel cleaner, somehow. She says, though, that it'll be weeks before I'm completely back to normal."
"Whatever that means," Firth said with a smirk.
It was close to an hour before Wilberforce returned with a Runner, an extremely young man who didn't normally work in our neighborhood. He introduced himself as Runner Calvin and explained that most of the Runners were still dealing with the many repercussions of the collapsed bridge.
LaMonte wanted to oversee my interview with the Runner. The Runner politely refused to allow it. I almost liked him for that. We went to the private dining room and the young man—really, did he even shave?—sat across the table from me.
"The excitable fellow who brought me here," he began, and I was even more impressed with him, "said you'd been poisoned. You look well for it."
"The healer said I'd been given niyacin, which I reacted to badly."
"Who is this healer?"
I almost answered, realizing just in time that Cree was unlikely to appreciate having a Runner showing up at her premises demanding answers that may lead to further uncomfortable questions. "I will suggest to her that she should speak with you."
"How about you just give me her name?"
"How about I just have her contact you?"
He scowled. "Are you wasting my time?"
"I might be," I admitted. "I don't really know what happened. The healer said I was given niyacin. I have no reason to doubt her, but I don't know anything about it."
"The excitable fellow said your servant poisoned you on purpose."
"He is not our servant." I knew my precision irritated him, but saying Ben was our servant had a host of inappropriate implications. He was a servant of the Triple S, not its individual members.
"Do you think he gave this niyacin to you on purpose, knowing it could kill you?"
It looked like he had. I didn't want to believe it. Or admit it. "I don't know."
"What has he said about all this?"
"I haven't seen him since I got ill." I told him about the note. I told him I hadn't seen it myself, and I didn't know whether Ben had more than one daughter, though I assumed he was with Sara. Calvin continued to press me for Cree's name, and I refused to tell him. Then he insisted on going into Ben's room, ignoring my objections.
"He's done a runner," he announced from inside the room.
I was waiting by the door, torn between respecting Ben's privacy and wanting to make sure the Runner didn't do anything to his stuff. Torn between my duty to watch this stranger in our home and my desperation to sit down. Leaning against the threshold wasn't going to be good enough for long. "He's what?"
And out of nowhere, LaMonte appeared. "What's going on?"
"Your servant has packed everything up," said Calvin, and I could hear him opening and closing drawers in quick succession. "Unless someone else has been in here and cleared everything out."
"We wouldn't dream of intruding into Ben's personal space," LaMonte chided.
"Clearly you should have," Calvin retorted. "Where's the note he left?"
The note had been thrown away. No one was able to find it.
Calvin felt all this needed to be reported. He ordered us to stay out of Ben's room and to look for anything that might be Ben's throughout the residence, though not to touch anything we found. Then he left.
Taro had risen by this time and was demanding to know what was going on. I was ready to crawl back into bed, my brain clouding with exhaustion. I asked LaMonte to fill Taro in and went back to bed.
It was dark when I woke again. A candle was lit on the table by the bed, and in that light I could see the steam rising from the mug of broth, also on the table and beside a glass of water. Taro was seated by the bed, watching me.
I drained down the water first, then started on the broth. Its heat was soothing and it filled my stomach pleasantly.
"I could demand an explanation as to why you came to this bed instead of mine," Taro commented.
A quick glance about told me, yes, I was in my own bed. I was surprised. I'd made no conscious decision about where to sleep; I'd just gone where my feet took me.
"I imagine, though," Taro continued, "that you're not up to much of an argument right now, so how about I do all the talking?"
Uh, all right. And it was the least he could do, since he had evidently decided to start an uncomfortable conversation while I was trapped in bed.
"I've been trying to understand why you were wearing the harmony bob on your underclothes the other day."
Oh lords. I didn't want to talk about that. But then, he'd just said I didn't have to, didn't he? So maybe the timing was actually excellent. Maybe.
"You would have no expectation that I would see it. Not that I think you would wear it just to please me. You don't do anything just to please me."
Prat. I did, too.
"So that means you have been wearing it all along, and I have to wonder why. And why you would wear the bob in secret, obviously with some effort of hiding that fact from me, as I haven't seen it on you during prior opportunities. It's made me very curious."
I took another sip of soup.
"Why do you think I've always worn the harmony bob?" he asked.
Hey, he'd just said I didn't have to talk in this conversation. I just raised my eyebrows and drank my soup.
"It did amuse me, of course. People wrapping up so many expectations in these little trinkets. But I was also making a public gesture. Except, clearly, you never perceived that. You know"—he smiled with just one corner of his mouth—"you can be quite clever in some ways but astoundingly oblivious in others."
People could stop calling me stupid anytime now.
"I shouldn't have to do this, I know," he went on. "It's embarrassing, and unmanly, and the very idea of it threatens to send me into a full-body cringe. But you'll never clue into it on your own, and sometimes you just have to do something flat out humiliating in order to get what you want. You taught me that, on that damned island."
Bastard.
"So here it is. It was a public gesture of my fealty, if you will, to you. An announcement that I love you, and will do so forever."
I choked on my soup.
"And so you laugh," he said with disgust. "That's what you do. Strip all the finer attributes out of everything."
"I wasn't laughing," I protested. "I was surprised."
"Why should you be surprised? What have I done to make you doubt that I love you?"
Nothing. Nothing at all. Nor had he done anything to make me believe he loved me. Not that way.
Unless he had been expecting me to take seriously all those off-the-cuff comments he'd made about what would be happening in the future.
This was a horribly uncomfortable conversation, even without having to contribute to it.
"I take it back," said Taro. "You're not clever at all." He leaned forward, so far forward that he could rest his forehead against mine. "I love you. I plan to continue loving you for the foreseeable future. And I would really appreciate it if you sent Doran packing."
I suddenly had trouble breathing. This was nothing like what I'd been expecting.
"Poor girl," he said. "I've thrown you into a panic."
Not a panic. Of course not. There was nothing to panic about. But the very idea of the Stallion of the Triple S telling me he loved me, and that he wanted to have it demonstrated as something more than a passing fancy, it was something out of a farce. The handsome lord did not fall in love with the merchant's daughter, not in real life.
Not that I doubted his sincerity. Taro was never cruel, and he didn't take things as lightly as many would believe. But this was just unnatural. And really, no one swore lifelong love outside of poetry and music. Because people couldn't know how they would feel about things ten or more years in the future.
"Do you love me?" he asked.
"Of course."
A small quirk of a smile. "Of course," he mocked. "And do you wish to do so permanently?"
Permanently? That seemed like an awful long time. Unrealistic, too. But Taro and I did ridiculous things all the time, usually forced into it by circumstances. Why not do something crazy purely for the hell of it? "Yes."
"And you'll send Doran packing."
"Yes." That wasn't working anyway. And I felt like a skeevy wench for ever entertaining the idea of keeping Doran hanging on until things with Taro settled down.
This was not what I'd ever contemplated for myself whenever I thought about who a long-term partner might be. Taro was not calm. He drew everyone's eye whenever he walked down the street. We liked none of the same things.
And maybe none of that mattered. I'd have to think about that. When I wasn't so tired.
Taro kissed me on the forehead, the chin, and each cheek. It felt strangely ritualistic, and although I'd never been fond of ritual, there was something calming about his gestures.
"Drink the rest of your soup," he said. "Then I'll let you get back to sleep."
Seriously, this whole thing deserved a joke or three, but I was too tired to come up with anything good. I sipped at the broth, until the mug was empty. Taro took the mug and blew out the candle before kissing me on the forehead again and leaving.
I snuggled back into bed, feeling relaxed and full and content.
Chapter Twenty-seven
The next morning, I woke aching and tired, filthy and thirsty. Taro ordered me to stay in bed and of course I ignored him. I was tired of lying down. And I wanted to be in a public part of the house. I felt like I'd been hidden away from the other Pairs for far too long. They had all seemed genuinely concerned about my illness, and I felt I hadn't given them enough credit for the basic decency they all habitually displayed despite their annoying quirks. So I decided to go to the kitchen, easily the most frequented room in the residence, with Taro trailing me anxiously.
And, of course, the only person there was LaMonte.
"Has anyone seen Ben?" I asked him.
"No," he answered bluntly. "I'd like to speak to you, Dunleavy." He glanced at Taro. "Alone."
Taro didn't leave. He did cross his arms and glare at LaMonte, though.
And LaMonte smiled, which was not an expression I was used to seeing on his face. "Has the definition of 'alone' changed and no one told me?"
Taro bristled, and I said, "Whatever you want to say to me can be said in front of Taro."
"Really? My dear, I had no idea you'd developed the ability to read minds."
All right. I deserved that.
"The misunderstanding is my fault, of course. People do have the unfortunate tendency to say 'talking to' when they mean 'talking with.' I'd like you to participate in this conversation, and whenever you and Shintaro are together, you have the distressing habit of letting him do all the talking. You are the one I want to speak with."
Well, what if I didn't want a conversation with him?
LaMonte sighed with impatience when Taro didn't move. "Do you really think you need to protect her from me?" he asked the younger Source.
"She has had a very hard time. She does not need a lecture."
"I'm not going to lecture her. Don't be ridiculous."
"You never think you're lecturing," Taro retorted. "Talk about ridiculous."
We were all getting ridiculous. We'd be here all day. "I don't feel as bad as I look," I said to Taro.
Taro scowled. I waved a hand toward the door. Taro huffed as he left.
LaMonte looked at the closed door for a moment. "Might he be listening at the door?"
Obnoxious creature. "Taro does not lurk about eavesdropping on people."
He held up his hands in a gesture meant to be placating. "He seems to be crowding you at times."
Not that I'd noticed, but we could continue trading insults or I could hear what he had to say and move on to something more restful. "What's on your mind?"
LaMonte sat on the stool next to the one I was occupying. I noticed with a shock that his hair was rapidly thinning. When had that started happening? "I understand it's been confirmed that Ben had been trying to . . . well, to kill you."
"Nothing's been confirmed," I objected.
"He moved out without telling us."
I wasn't denying that it looked bad, only that it actually proved anything.
"And his daughter has been arrested for killing Izen for his ashes, for the use in casting spells."
"Apparently."
"It is possible that he might have been killing you for the same reason."
"No one has suggested that." It never even occurred to me.
"It is a logical assumption, given Ben's circumstances."
I didn't want to agree. There was a certain logic to it, I supposed, but why would anyone want my ashes? I was privileged, but not particularly lucky. No more so than the other Shields in the residence, who had never been put in the position of having to kill someone, or sent off to a place where Shields and Sources weren't respected.
"And that woman who was here, that healer, she cast a spell to heal you."
Now I was panicking. "Oh, no. Nothing like that. As if I'd—"
He put up a hand and I stopped talking. My automatic response to his gesture annoyed me, and I wanted to go on talking just to demonstrate that I knew he had no authority over me, but I had no convincing lies to say about Cree's use of spells anyway.
"There is something going on," he said. "Something monumental. This belief in casting, it's not just in High Scape. I've been writing to Sources at other sites, Shidonee's Gap, Seldom Go By, everywhere that I know someone. It's not as prevalent everywhere else as it is here, but there are incidents happening all over, indicating that there are more and more people believing in casting. That is the alarming thing to me, that it is happening in places outside of High Scape. There are reasons why people would want something to believe in here. It has been explained to me that the Harsh Summer shattered people's understanding of how the world works, and they need something they feel they have some control over."
I wondered who had been talking to LaMonte, because it was clearly someone who had made a considerable impression on him. But perhaps that person wasn't as knowledgeable as LaMonte thought. Because people had been believing in casting for a long time, for generations. How did anyone explain that?
I wasn't going to tell him that, though. He'd ask how I knew. I'd have to tell him about the illegal books I had. I didn't really think he'd report me to any authorities for it, but there was no telling who else he'd tell, and that could result in nasty repercussions. If I told no one, no one would ever know.
"Many are embracing these beliefs," LaMonte continued. "Others, including the Crown Prince, oppose it, loudly and publicly. Yet the Triple S is doing nothing about any of it. The council doesn't react to questions and hasn't given any public statements. One gets the feeling they are trying to ignore the whole issue."
Well, why wouldn't they? It had nothing to do with them, as far as I knew.
"I was thinking this was just a temporary foolishness. Like that harmony bob Shintaro insists on wearing. I thought people would become bored with it all, and move on to some other fashion. But this isn't like other fashions I've seen. So pervasive, with followers in every class. And there are equal numbers, including the Crown Prince, who are violently opposed to the idea of casting spells. It's becoming a serious criminal offense. And I've been hearing of episodes in schools, like in Far Flung, where a teacher was teaching spells from a book. She was released from her position and dragged out of the school for an immediate flogging that nearly killed her."
Good hell, what was wrong with people?
"Everyone, everyone, is dealing with this in some way, to support it or deny it or criminalize it. Except the Triple S. I've sent them a whole series of letters. They refuse to answer in any way. There has been no guidance as to how we, as individual members of the Triple S, are to react to all this."
Why did he have to be told how to react to the trend? He was so quick to behave as though he were the authority in all things, yet he was equally quick to subjugate his opinion to that of the Triple S. Such an interesting contradiction.
"It isn't wise to ignore something so powerful."
I would have never expected to hear from LaMonte any criticism of the Triple S, no matter how oblique.
"Whether casting is actually real or not, the belief in it and the reactions to it are having a significant impact on people, and this impact shows no sign of abating. On the contrary, the influence is growing."
He had a point. People did seem to be going crazy over this thing.
"So I am going to ask you a question. And I want your honest answer. I want the truth. I believe you owe me that."
I wondered where he got the idea that I owed him anything in particular.
"Do you believe that people can cast actual, effective spells?"
Ah, hell. The last question I'd expected, and the question I'd least wanted to answer. I'd been happy enough continuing to avoid thinking about it.
I could ignore what was going on right in front of my face with the best of them, but really, I wasn't stupid. And I had been thinking about things, remembering things, whether I liked it or not. The whole time the Reanists were killing aristocrats in a bid to stop natural disasters from striking High Scape, there were no natural events, and since the Reanists were stopped, the natural events had returned, though at a greatly reduced rate.
The people of Flatwell had been great believers in ritual. I'd thought it all superstitious nonsense, the bad luck the troupe had believed in greatly assisted by a dangerous lifestyle and the deliberate interference of Atara's murderous son. But the fact was that a member of the troupe died every time they lingered at a place for too long, and unless Yesit had spent all of his life trailing the troupe with no one noticing, it seemed the belief that Yesit's curse was the cause of all their difficulties might have had some merit.
And then there were more recent events, the bizarre nature of the fire during which I'd burned my hand, and Cree's ritual in the cellar. I was better. That hadn't been a medical treatment. At least, no kind of medical treatment I had ever heard of.
No one had ever told me spells were nothing more than a figment of imagination. No one had ever talked to me about spells at all, not while I was growing up. When I'd seen them in plays or read about them in books, I had dismissed them as fantasy. It was an assumption I'd made. Most of my assumptions were disastrous.
I didn't want to say it. I hated the very idea of it. The possibility that spells could influence the natural order undermined everything I had learned about everything. How did I know what to believe in?
"Dunleavy." LaMonte put the tips of his fingers on the back of my uninjured hand. "Stop."
I was breathing too fast. And hard. I was close to hyperven tilating. I forced myself to pull in a long, deep breath and released it slowly. I was humiliated over losing control in front of LaMonte.
How dare he ask me, anyway? There was no good reason for him to do so, except to find yet another reason to sneer at me.
"I suppose that's my answer, then," said LaMonte.
"It's not that simple," I responded. "I don't feel comfortable giving a solid affirmative. I don't know enough about it, and part of me is waiting for the announcement that this is all a big hoax. But I've seen enough to know that it's definitely possible that there is real power behind it all."
Let the derision come.
LaMonte sighed. "I was afraid of that."
Why? Because it was now his duty to report me to the Triple S for being insane?
"I was hoping I could continue to dismiss this all as some form of temporary madness."
"You and me both," I muttered.
He actually smiled again. That had to be his quota for the year. "You will be well again?" he asked.
"Yes, I believe I will."
"Good," he said, and he cleared his throat before leaning forward to tap the back of my good hand. "I'll go tell Shintaro he can come back in."
"Wait."
He paused.
Really, I was so stupid. "Stone and Firth, they've said Taro and I are perverted, that the relationship we have is wrong." I wasn't going to say "sex" to LaMonte. I'd eat my boots first.
"So they've said."
I really didn't need the confirmation that they'd been talking about us when we weren't there. "What do you think?" Why the hell was I asking him? Why was I asking for more abuse? What was wrong with me?
"Why, Dunleavy, you've never cared whether I approved of you or not before. Why the change now?"
The weevil was playing with me, damn it. So I didn't answer. I was very good at not answering.
Finally, he got tired of waiting. "Like everything else, there will be those who believe it is a vile defilement of the natural order, those who think it is the most natural and the purest relationship in life, and those who fall somewhere between the two points."
"Where do you fall?"
"Dunleavy, it has nothing to do with me."
I was clenching my teeth to prevent myself from bursting out with something inappropriate. Was this really LaMonte? Where had this LaMonte been all the other times he appeared to be commenting on matters that had nothing to do with him?
"This is about you and Shintaro. It's not about a Shield and a Source. Only the two of you can decide what's right for you."
Well, as answers go, that was about as useful as wheels on a horse, but I supposed it was a little reassuring. I'd really expected him to castigate me brutally.
"Now, can I get Shintaro?"
"Yes, thank you very much."
I should have expected Risa to show up later that day. Whenever I was involved with Runners in any way, she came around. I couldn't believe that her superiors really felt it necessary to soothe my feelings by having her act as some kind of liaison for me. I thought it far more likely that she learned of my involvement and came around just to make sure I was all right and that the information the Runners were getting was correct. It was the sort of thing she would do.
My interpretation of her actions was confirmed when she showed up bearing a bundle of items meant to soothe me during my recovery. A berry wine meant to quiet the mind, a cream soap to soften the skin, a tea smelling of sandalwood to sooth the stomach—and chocolate for, well, chocolate.
All of these items were presented in beautiful jars and boxes. All of them cost more money than Risa should be spending. In my current state of exhaustion I was almost moved to tears that Risa would be so generous with me when she couldn't afford it. It infuriated me that there was nothing I could do to ease Risa's financial straits.
Well, nothing ethical, anyway.
After hearing about my poisoning and Ben's possible involvement in it, Risa sent Taro away. "Business is done. I want to enjoy myself now."
Taro scowled. It was cute.
"Stop hovering like a crow. I'll see to anything she needs."
"What is with everyone today?" he grumbled. He kissed me on the cheek before leaving.
Risa waited until she was sure Taro was gone before saying, "He's been acting strange since you two got back."
"I hadn't noticed." Of course I had, but I wasn't going to talk about that to outsiders.
"That's not surprising," she smirked. "You don't seem to notice a lot."
I sighed. I was getting very tired of having my flaws thrust in my face. What was it about me that invited people to speak to me that way?
Risa sobered. "All right, fine. What I have to say now is important, and I want you to promise not to tell anyone else about it, all right? Not even Shintaro."
"I can't promise not to tell until I know what it is." I tried to avoid making blind promises. That way led to idiocy and melodrama.
"Damn it, Dunleavy, this is serious. You don't have any idea how visible you are. As a Shield, I mean. People watch you, they talk about you, and sometimes information gets to the wrong ears."
I stared at her. People watched me? What the hell for? Didn't they have things to do? "You're not clarifying things."
She rolled her eyes. "Fine. You were seen buying—or requisitioning, whatever you people call it—a book of spells in the market. And the person who saw you reported you to Headquarters." What a dirty thing to do, watching and eavesdropping and tattling. "I said there was no way you'd have done anything like that, that you thought it was all bunk and you were above such lunacy, but I would talk to you about it to make sure." She held up a hand when she saw me opening my mouth to speak. "I am sure, having been away so long and then so busy what with fewer Pairs in High Scape, you would not recognize a spell book for what it was had you inadvertently picked one up. Shields and Sources pick up so many things without thought."
Hey, that made us sound like thieves.
"And then, when you actually did read it, you would probably think it was fiction. Badly written fiction. So you would throw it out. Or even burn it." She put heavy emphasis on the word "burn." "And you would never dream of picking up anything so trashy again."
_All right, all right, you don't have to club me in the face with it._ "I promise not to say anything. To anyone."
She nodded. "Good. And it would be great if your Source didn't go so far out of his way to start rumors about having an unnatural ability to heal."
I rubbed my face with my left hand. So much for their promise to keep it quiet. There really was no point in doing anything for anyone anymore. "He can't heal people. He told them that."
"Too bad he can't. There are too many idiots claiming the riverfront areas are cursed, and that's why they're getting sick."
"You said you thought the water was making them sick." I was so tempted to tell her what Taro and I thought we knew, but we had nothing to back it up. In all of our nights of surveillance of the hub, we hadn't once seen a woman come to do anything to the drains with ashes.
"Aye, but no one can figure out how. We can't really prove it. And now people are trying to move into the other areas. Only no one else will take them in. Because they're sick. It's getting nasty."
Really, sometimes it felt like this city was really falling apart. Had it always been so chaotic, and that never made it into the textbooks? Or was I merely living in interesting times?
"You don't want them all landing on your doorstep looking for a miracle."
Too late. "No, I don't." I held the package of tea to my nose and breathed in the scent. It was calming.
"Do you understand what I'm telling you?" Risa prodded.
"Yes." I understood that she suspected we'd been engaging in illegal activities, and she was warning me to stop it and get rid of the evidence. That had to be against the rules of being a Runner. "Thank you."
Risa left shortly thereafter, once more warning me that Ben had not yet been apprehended. I promptly went up to my suite, lit my fireplace and threw all the books and pamphlets in. It irked me to do it. I should never have to destroy books just because I was afraid someone would learn I had them. But people were acting crazy. And anything I did, Taro would share the repercussions.
Besides, I'd already read all the books.
Taro wandered in while I was poking the burning books into ashes. "Am I allowed to be around you now?"
I slanted a look up at him. "I've often been the one sent out of the room."
"Not that often. What are you doing?"
"Taking Risa's advice."
"She advised you to burn your possessions?"
I told him of the warnings Risa had given me.
He sighed as he sat beside me on the floor. "I'm sorry."
"What for? You've done nothing wrong."
"I shouldn't have tried to heal those people. It brought too much attention to us."
Taro, unfortunately, couldn't help bringing attention on himself, the poor lad. "Aye, but what if it had worked? That would have been wonderful."
He seemed to be squirming a little. "That wasn't the real reason I tried."
I waited.
"I just feel so bloody useless here," he confessed heatedly. "No events all this time. There's no point to me being here."
"We're all in the same position when it comes to that," I reminded him. "All the Sources and Shields. That's why they've been transferring Pairs to other sites."
"Aye, but unlike me, channeling isn't the only tool in your kit bag, is it?" he said bitterly. "That damned island proved that."
Ah, that. Damned Flatwell, convincing Taro he was inferior. I wished we'd never gone. I wished he could forget about it. "I'm sorry you feel that way. I don't feel that way about you. I doubt anyone who matters feels that way about you. But you feel that way, and I wish I had the words that would prove to you that you're wrong."
He smirked. "I think that's the longest string of words you've given me in months."
I slapped him up the back of the head. But gently.
And then his inner protections went up.
"No," I said, knowing it was too late. "I'm not strong enough."
"I can't help it!"
He couldn't help it? Of course he could help it. Nothing could force a trained Source to begin to channel.
But he was channeling, so I had to Shield. And the images and tastes and sounds of cliffs and sea tore through me. They flooded my mind and swirled behind my eyes, filling my throat and mouth until I felt I couldn't breathe.
And I couldn't do it. I just wasn't strong enough. I could practically see my Shields shaking from the pounding of the forces. I was going to get us both killed. "I need a Shield!" I shouted as loudly as I could. "Help! Stone! Ladin! Benedict! Hammad! Help!" Please, please, let there be someone within earshot. Someone with the skill to Shield someone else's Source. Please.
"No!" Taro roared. And there was a huge wrenching sensation, painful in its violence. The rise of his protections scraped like a serrated blade across my mind.
What the hell was that? I looked at Taro to ask just that. He was pale and sweaty and shaking. "Are you all right? What happened?"
"I stopped channeling before the disaster was finished."
That was supposed to be impossible. When had he started being able to do that?
I didn't know how he could claim to feel useless when he was developing a new ability every time he turned around.
Then he fainted, and I had other things to worry about.
Chapter Twenty-eight
Several days later, Taro had gone in search of chocolate, having eaten all of mine. He was pale and complained of a piercing headache. He was also impatient and restless and seemed unable to remain in the residence. So he left. To find chocolate.
After several hours, he still hadn't returned. And now I was the one who was impatient and restless.
He was late, but that meant nothing, I was sure. He had met up with some friends, that was all. He was having a few drinks at a tavern. He had been dancing attendance on me for a while, and then had been laid low with a headache that had taxed him ever since he collapsed from wrenching himself out of a channeling. Of course he needed some time away.
But he had left late in the morning, and now the sun was setting. I was starting to worry.
As I wouldn't have, had we not been sleeping together. Did that mean I had been a heartless, thoughtless wench before, or that I was a nagging, paranoid wench now?
Really, it only made sense that he was lengthening the leash a little. I slept like the dead for long hours at a stretch. When I was awake, I was usually no good for anything more than dragging my sorry posterior from the bed to a chair in the kitchen to a settee in the parlor. I was pretty much confined to the residence.
Maybe that was why I was so restless about Taro. I wasn't worried about him; I envied him. Maybe it was time I ventured out a little, put some muscles back on my legs.
Or maybe not. Just because he said he loved me didn't mean he wanted to spend every waking moment with me. It would be so humiliating if I were to show up and all he felt was exasperation because I would never leave him alone. I would hate to turn into a clinging weed, wrapping around everything in his life.
This was why it was so bad to fall in love. It turned a person into an idiot.
I was in the kitchen brewing coffee—doing it myself as Ben had not yet returned, and no, that didn't prove anything—when I felt Taro's protections lowering. I raised my Shields, of course—I would never do otherwise—but I was furious. He was getting way too cocky with his off-duty channeling, and we'd be having a nice little argument about that when I saw him next.
I quickly realized, however, that Taro wasn't channeling an event that was not in our jurisdiction, or attempting to soothe pain or heal an injury. He was creating an event. He only did that when something was wrong.
I didn't know what to do. I had no idea what kind of difficulty Taro might be in. He might be merely trying to stop someone else from doing something he considered wrong, or he might be in danger. The mere possibility that he might be in jeopardy meant I had to try to find him.
Only there was no way I could channel and search for Taro simultaneously, not properly. I needed help. Risa would have been my first choice, but I didn't know where to find her. I couldn't go out and find her while channeling, either.
I would need to get help from someone in the house. I hated that idea. I would have to ask for a favor from one of my colleagues. Worse, I would have to tell someone how I knew Taro was in trouble. We didn't want people to know he could create disasters. Telling another member of the Triple S would all too likely lead to questions from the Triple S council, something Taro was desperate to avoid.
But it couldn't be avoided. I needed help. So whom should I ask?
The name that first leapt to mind surprised me. Moving as carefully as I could while Shielding, I left the parlor and checked the kitchen and the private dining room before heading up the stairs to LaMonte's suite.
Taro was still channeling. What was going on with him? It was difficult, holding up the Shields so long while climbing up stairs.
I didn't know what I was going to do if LaMonte wasn't in. Panic, I supposed.
I pounded on LaMonte's door. I was so relieved when it opened that I almost couldn't breathe for a moment. I couldn't see his expression all that well—I wasn't really able to focus on that kind of detail—but he was probably annoyed at being interrupted. I didn't care.
"Taro's in trouble," I blurted out.
"What's wrong?"
"I don't know."
"Well, where is he?"
"I don't know."
He probably still looked annoyed.
"Are you Shielding right now?"
"Aye."
"Come in." He took me by the elbow, and the next thing I knew, I was sitting on a settee.
LaMonte surprised me by kneeling before me as opposed to sitting beside me. "What's happening?"
"Taro is in trouble. I need to find him." Why was he making me repeat myself?
"Taro is channeling right now?"
"Aye."
"You're not on duty. Why is he channeling?"
"I don't know. Can you find him?"
"How would I be able to find him?"
"You can feel where the event is." He would have felt the event, like all the other Sources in High Scape, but would have ignored it as he wasn't on duty. Apparently, only my Source felt the need to channel absolutely everything.
"I can't determine its exact location."
Oh, how disappointing. Now what was I going to do?
"Derek might be able to, depending on the nature of the event. He has a knack that way."
Oh, thank gods. Where was Beatrice?
"But why would that information assist you?"
"Does it really matter?"
"If you're going to ask me to do something unorthodox, yes, it does."
I supposed that was fair. Annoying, but fair. "That's where Taro will be."
"And you know this how?"
"He's creating the event. It means he's in trouble."
And that was when I learned that LaMonte was truly a beautiful person. He didn't imply I was lying. He didn't demand to know how Taro could possibly create an event. He merely said, "Wait here. I'll find Derek. Oh." He paused before leaving. "Never tell anyone else what you just told me."
Later, I would need to think more carefully on his words.
Then I had to wait while LaMonte tracked down Beatrice, convinced him we weren't crazy and brought him back. The whole time, Taro was channeling and I was Shielding. Taro was keeping all of the forces moving slowly, which meant he was causing tremors instead of a full-out earthquake. So we had some time. No one was killing him yet.
It was difficult to hold on so long. Luckily, I'd had some practice with that recently.
It was Beatrice who came back to me. "Chris is flagging down a carriage," he told me. It was the first time in my memory that he didn't sound at least slightly petulant. "We're going to find Shintaro."
Oh, thank gods. "You know where Taro is?"
"I can't just pinpoint the location from here."
Then what good was he?
"We'll have to track him down. I can do that."
I supposed it was better than nothing. It was certainly more that I could do. Would it be enough?
"Come with me. Let me guide you. I'll see you get to Shintaro. You just concentrate on Shielding. Don't worry about anything else." He took my wrists and gently urged me to my feet.
It was sweet of him to suggest I just put myself in his hands that way. It wasn't something I could do with anyone other than Taro. It wasn't as though I thought Beatrice would do me any harm. I just didn't know him. Which was a sad thing to admit after living in the same residence so long.
LaMonte had a carriage waiting by the time Beatrice and I reached the front door of the residence. LaMonte climbed into the carriage with me and Beatrice joined the driver up top. The carriage jolted into movement, pressing me hard against the back of my seat, the hooves of the horses clattering loudly against the surface of the street.
I couldn't recall traveling so quickly in a carriage before.
"Get out of the way!" I heard Beatrice bellow. "Move, move, move!"
It appeared Beatrice was taking this seriously. Who would have thought?
People out in the street shouted back. Some of the words were foul.
We took a sharp corner, and I was thrown against LaMonte. We seemed to take another quick corner, and I was forced back against the wall of the carriage. I clenched my fists in an effort to hold on to my Shields.
It was only a few moments later that I realized my left hand was actually digging my nails into LaMonte's thigh. That was embarrassing.
I could swear the next corner the carriage took, a couple of wheels left the ground.
"Don't kill us before we get there!" LaMonte shouted.
"Are we in a hurry or not?"
"All right, then, how about you try not to kill anyone else?"
Another sharp swerve had me tumbling to the floor.
"Don't let go of Shintaro," LaMonte ordered. "Maybe you should stay down there."
That actually seemed like a good idea, what little thought I could spare to it. So I stayed on the dirty floor.
There were more curves and corners and people swearing at us. Then, finally, we began to slow down.
"We're entering the riverfront district," LaMonte commented shortly thereafter.
Huh. There was a reason that was significant. I couldn't recall it right then.
The carriage slowed down. "Can you see him?" I asked LaMonte, meaning Taro.
"No. We're not stopping. I believe we're slowing down because Derek has narrowed the search to a particular area. He has to slow down to determine Shintaro's exact location."
I really hadn't heard much past the word "no." I hated that we had slowed down, even if that meant no longer getting thrown about the carriage. We had already taken too long.
And then the forces flowing through Taro intensified. "Something's happening," I hissed at LaMonte. "We have to go faster."
"He's going as fast as he can."
That wasn't good enough. I pounded on the side of the carriage. "Hey!" I shouted. "You have to go faster!" I reached for the window so I could shout through it.
LaMonte grabbed me by a handful of hair and pulled me back into my seat. "Derek knows how to do this. You do not. Let him work. Your sole task is to Shield Shintaro. You can't properly do that if you're worrying about what everyone else is doing. Settle down."
Bastard. Damn him for being right.
Street after street after street. And despite the fact that we were going slower, the ride was much bumpier. A couple of times I was sure we'd lost a wheel and I'd have to swallow down a spurt of panic.
Finally, we stopped. Finally. "Are we there?" Wherever there was. LaMonte was between me and the carriage door. Why wasn't he moving? Was he really going to make me climb over him?
The door opened, and LaMonte still didn't move. Beatrice was standing in the way. "There's something strange going on," he announced. "There's a group, about six, I'd guess. They've got fires lit, and that's never a good thing. Shintaro's there, and I think he's alive. I couldn't see too well, but he seems to be tied up."
"They're doing this out in the open?" LaMonte asked in disbelief.
Beatrice shrugged. "They don't seem to be worried about being seen."
"Good," I said, frustrated with the chatter. "So let's go."
"That's a bad idea," Beatrice objected. "We should bring the Runners here. These people obviously mean harm and they outnumber us. We could just end up tied like Shintaro."
"I don't care," I said. "I'm not wasting more time looking for Runners. They'll kill him before we can get back. Or move him." And if I had to bite and pull hair to get out of that carriage, I would.
LaMonte spared me that humiliation by saying, "I agree with Dunleavy, Derek. We should at least keep watch while you get the Runners."
Beatrice sighed gustily. "Fine. We circled around after we spotted them so we wouldn't be noticed. You take this street up two blocks and then you go left. Follow that street until it ends. They're all gathered up against where the rivers meet, down by the wall. It shields them from the river traffic."
They were at the hub.
And we were, at long last, getting out of the carriage. I let LaMonte lead me down the proper streets. The ground was shaking, but not enough to do significant damage. Taro was developing a delicate touch with that.
There were no hiding places to allow us to stand and watch. The buildings just stopped suddenly and there were no trees, just dead, flattened grass between the last rickety building and the solid stone wall that stood where the three rivers met. Moving any farther would bring us out in the open.
I didn't care. Nor was I concerned about the lack of cover. I didn't need to hide. I knew what Taro could do.
So I started running toward the group. That was surprisingly difficult, running while Shielding Taro, and I couldn't run as fast as I normally could.
"Dunleavy!" LaMonte hissed in horror.
I ignored him. He'd have to run and tackle me if he wanted to stop me, and that would get everyone's attention as easily as anything else.
I didn't want to take the chance he'd try, though. "We're here, Taro!" I shouted. "Bury 'em!" I could barely see him, between the thickening darkness and my focus on my Shielding, but there was his form huddled against the wall.
The forces grew stronger, just for a few moments. There were varying shouts of alarm. But there were also people running around. Whatever Taro was doing, he wasn't burying people. That was something he had done before, and it was the easiest way, I thought, of dealing with these people. But maybe he couldn't bury more than one at a time.
I was hit from behind and struck the ground with a painful pounding. That hurt a hell of a lot, and I got a mouthful of dirt out of it. The weight was heavy on my back. I kicked out and connected with nothing.
"Stop channeling," I called out. I couldn't think while I was Shielding.
The tremors stopped.
That seemed to surprise the person on top of me, who gasped. I pushed at him, and his shock enabled me to wiggle out from under him. He grabbed at my legs and I kicked him in the face. I thought I heard a crunch.
I scrambled free, but I was merely tackled again, by two people. And when we were all on the ground, I heard a voice say, "How fortunate. You were the one we wanted anyway."
That made no sense.
The two pulled me to my feet and then dragged me to the wall next to Taro. And then I saw the problem. Taro was blindfolded. He couldn't see to bury people.
I had little strength left, but I saw a rope in someone's grip. I kicked and I scratched and I bit, and I was able to reach out and pull the cloth from Taro's eyes.
His protections went down, my Shields went up, and the ground began to heave, sharply. Everyone fell, and that was all Taro needed to sink each person far enough into the ground that they couldn't easily escape.
"What the hell was that?"
Heh. We made LaMonte swear.
Six people. Three men, three women. One of them was Ben Veritas.
He really had been trying to kill me.
There were two bonfires going. A table had been knocked over, dumping books and rods and bowls of powder on the ground. Taro was naked except for the ropes that bound him. That wasn't as shocking as the bruises on his face and body. Had he acquired those in the course of his abduction or after his capture?
I thought about suggesting he bury his captors all the way.
Then LaMonte was beside us, and apparently he was one of those people who was always prepared, because he was draping a cloak over Taro's lower half while I worked on the bloodied ropes binding his wrists.
"They wanted to kill me for my ashes," he said. "You were the preferred target, and when that didn't work they went after me."
How could they be so stupid as to try to murder a Pair? Was the alleged power they received from the ashes supposed to prevent them from getting caught? Had that worked for anyone yet? My attack could be explained away as a freak illness, maybe, but how could they disguise this much more blunt assault?
I looked at Ben, who was sunk to the ground up to his knees. He appeared to be wearing some kind of robelike garment of a deep blue. I would have expected him to be angry, glaring at us and shouting. At the very least, he should be trying to free himself as his collaborators were, digging up the soil with fingernails. Instead, he stared down at the ground silently, his shoulders hunched, sort of collapsing in on himself. He seemed pale. I noticed he was trembling.
"It was for my daughter," he said suddenly. "For Sara. She is going to trial. She is going to be hanged."
Only if she was found guilty. And how the hell were our ashes going to influence that? Was there honestly some spell out there that could determine the outcome of a trial?
"Are you claiming she didn't kill the mayor?" Taro demanded.
"What difference does that make?" Ben demanded in return with a flash of heated anger. "She's nothing but a servant. She has no money and no one to speak for her."
In another flash, the anger disappeared, and to my horror, Ben started crying. Loud, hacking sobs that had to hurt. It was painful to watch. It was difficult to believe that he was acting purely to garner our sympathy.
"It's not right," he blubbered. "You have everything, easy lives and food and clothes and everything you ever wanted, just because you're born what you are. You don't even have to work, if you don't want to. And you're not even needed here. High Scape's gone cold and isn't getting any more events. You've all said it. So where's the harm?"
"For gods' sake, man, shut up!" one of the women hissed. Ben didn't even glance at her.
Where's the harm? Was he crazy? "So you used me to get to Taro."
Gasping, Ben raised wet eyes to me. "I didn't need Source Karish," he said. "His death would merely be an unfortunate consequence of yours."
"I thought you needed the ashes from someone lucky."
"A woman of no birth, beauty or exceptional talent," he said, "Paired with a highly skilled, noble Source. Who is more fortunate than you?"
It was unfathomable that a man who had been trying to kill me had the power to make me feel insulted.
It wasn't that what he said didn't make sense. I was the lucky one, wasn't I? Maybe not for the reasons so many people thought. Family connections didn't matter to me. I didn't care what family Taro came from. In fact, I could really do without his mother, and I thought he could, too. And while I admired his abilities, I didn't think skill in anything made anyone a superior or inferior person. But he was a good man. He was someone I could trust and rely on and feel comfortable with. And those were the qualities that made him a superior Source, partner, friend and lover.
So maybe I should just appreciate all that, instead of worrying about the fact that I didn't deserve it. So I didn't deserve it. I was lucky. There were worse things.
It was made clear to me how very stupid I'd been by believing in Ben's innocence for so long. But Ben had worked for the Triple S for decades. He had always treated me so kindly. So what if he'd made me uncomfortable? That had just been my own foolishness. So what if his daughter was suspected of murder? Why did that mean I was supposed to suspect him, too? There was no sense in that.
Every time I'd assumed the worst of someone, I'd been embarrassingly wrong. And a lot of times when I'd trusted someone from the start, I'd been dangerously wrong. There was just no winning. Understanding people should have been part of the formal curriculum at the Academy.
People had planned to kill me before, and I'd never imagined having to say that about myself. But with Ben it felt so much worse than the others. Because we lived in the same house, and he acted like he thought well of me. Really, what was wrong with me that people were prepared to do such things to me?
LaMonte called out when he saw the Runners approaching, and Taro unearthed all the crazy people. They immediately began gathering up all their casting gadgetry in the hopes of running away. All except Ben, who sat on the ground and stared into space. There were a solid twelve Runners galloping through the shoddy streets, because Taro had always been considered an important person. Ben's group was easily rounded up.
They claimed they had been buried by Taro. Taro claimed he saw them all snorting something. Taro was believed.
We took Taro home, where he was pampered and spoiled. Everyone was outraged that a man of criminal tendencies had been living among us, going so far as to attack two of us. The fact that no one else had suspected Ben didn't make my oblivion any easier to accept. Would I ever reach a point where I could see what was right in front of my face?
Chapter Twenty-nine
The Cree residence was located in the North Quad, on a long winding road with very few other houses. It was small, but the wood used for its construction had a deep dark red tint that I hadn't seen before, and I assumed it was rare and expensive. The door was opened by a very pretty young man, and the first thing I saw beyond him was a silver sun set into the opposing wall, its rays stretching out to touch every edge of the wall. And like the house of Trader Fines, there were similar suns over every window we passed as we were led straight to the dining room.
Again, everyone else was already there. Gamut, Ahmad, Fines and Williams. They were all enjoying some comment made just before Taro and I entered the room. I was pretty sure they weren't laughing at us. Still, I felt awkward. I didn't like being the last to arrive at a party any more than I liked being the first.
I felt the muscles in Taro's forearm tighten under my hand.
"Dunleavy, Shintaro," said Cree, rising gracefully from her chair. "I'm so pleased you could join us. Dunleavy, you are looking hale."
"Thanks to you," I replied. Cree took my hand and led me to the empty seat beside Fines. That left the seat beside Gamut, at the opposite end of the table, for Taro. The table was set in the same manner as Fines's had been, but the food had not yet been placed. The clay pots were there, though.
"We understand Ayana has made a convert of you," Fines commented.
"Convert." I'd never liked that word. To me it had always suggested coercion and a weakness of mind. "Even I can't deny the evidence of my own eyes."
For some reason Taro found that funny, if his snicker was any indication.
Fines smirked. "Perhaps it is our definition of evidence which differs."
Smug bastard.
We were still rubbing the paste into our hands when the door to the dining room opened and several servants entered the room, two of them pushing trays piled with covered dishes. A bottle was placed before Taro while I was given a tall, narrow pot. "It is not wise for you to drink alcohol so soon after your treatment," Cree explained.
Well, I had been drinking wine with no noticeable effects, and it was a little annoying to be dictated to like a child, but the woman had saved my life, and she was my hostess. Besides, the tea smelled wonderful, fresh and clean, and was light in color as it was poured from the pot.
The other servants placed covered plates all around us and withdrew. Clearly, we were once more to rotate the dishes about the table. It kind of reminded me of the Academy, only much, much quieter. And the food was better.
"We heard that distressing news about Ben Veritas," said Ahmad.
It was, I thought, an odd sentence to speak and just leave out there, hanging entirely on its own. The logical assumption was that she was commiserating with us on the fact that Ben Veritas had tried to kill us. But it could be an expression of sorrow over the fact that he had been caught, for all I knew.
"It is certain, then, that he gave you the niyacin powder after learning of its effect on you?" Cree asked. "That he was doing it deliberately?"
"He's all but admitted it," Taro answered, rather shortly, I thought.
Trader Fines made a sound of disapproval in his throat. "To engage in such behavior toward a member of one's own household is highly inappropriate."
I put a spoonful of soup in my mouth to make sure I didn't ask if he thought it was less inappropriate to engage in such behavior with a person not a member of one's household. I was in a snarky mood that evening, for some reason.
"Is he being charged with attempted murder or purveying ashes or both?"
"I haven't been told," I said. "Wait a moment, a purveyor of ashes? We know that was his plan, ultimately, but I don't know that they could accuse him of selling ashes. He didn't get a chance to do that."
"Actually," said Fines, "he's well-known as a supplier of ashes."
There was no way to respond to that without sounding like an idiot. He sold ashes? He was well-known? Who was this person who had lived in our home and seen to our every need? How blind was I?
"How could he be well-known?" Taro asked. "People haven't been doing it for so very long."
"People have always done it."
"I thought it was only for the last half year," I said.
"It only became so very popular in the last year, but trust me, it's always been done."
How did he know that? "Always since when?"
"Always since the Landing. Before that, I have no way of knowing."
"But more and more people started doing it recently," Cree added. "And that enabled the Runners to decipher the pattern."
"It doesn't make sense that Ben has been doing this for years," said Taro. "He doesn't have any money. He wouldn't still be working for the Triple S if he was making a lot of money selling ashes."
"I don't know how much the ashes garner for the diggers," said Fines.
"They're the ones who do the digging, who risk getting caught." And here I was arguing the economic unfairness of a criminal activity that nearly resulted in my death. It was ridiculous.
Fines shrugged. "Suppliers often get the least of any profit. It is the way of things."
I thought about what had been said as I, following the example of the others, placed my empty soup bowl in a sort of repository in the center of the table. "You knew Ben Veritas before he was arrested?"
"Knew of him. I've never met him."
"But you knew him as a source of ashes."
"Yes. He was well-known as being able to supply good ashes."
Good ashes. From the luckiest people? Was that what that meant?
"I don't know that anyone realized he was stepping into killing anyone, though," said Williams.
"Of course not," said Gamut. "It's ridiculous to think effective ashes can be harvested from a murder victim. Being murdered eliminates any luck the person enjoyed previously."
I kept my eyes on my plate, which had to be less rude than staring at Gamut. What kind of language was that to be using about the lives and the remains of human beings?
Maybe it was easier to speak that way when one wasn't a potential victim. Though why the others at the table were not potential victims was a bit of a puzzle. They all appeared to be enormously successful in their respective fields.
"How do you know about all this?" Taro demanded.
"Ashes are an important part of casting, Shintaro," said Fines. "This is what we do."
I didn't know why that surprised me. Cree obviously used spells. I supposed I just never considered whether the others did, too.
"You're a trader," said Taro.
"That is merely how I make money," Fines responded. "It is not what I am."
"And your interest in spells, that is what you are?"
"Not quite, but it is an important part." Fines smiled. "Being a Source is an important part of what you are, wouldn't you say?"
I didn't understand what Fines was suggesting. Surely he wasn't claiming his interest in casting was as important to him as being a Source was to Taro. Fines playing with spells was just a hobby.
"It is not all that I am," said Taro. "I wouldn't even say it's the most important part of what I am."
I looked up at him, surprised. Was that what he really thought?
"You don't agree, Dunleavy?" Cree asked me.
Damn it, no one was supposed to be looking at me. "I wouldn't presume to tell Taro what he is."
Her eyes did that slight shift that suggested she really wanted to roll her eyes but felt herself too well mannered. "I am asking for your opinion of yourself."
I liked that question even less. Because I did think being a Shield was the largest part of what I was. From birth it had made me different from everyone in my family, though they didn't recognize my true nature until I was four. It shaped everything about me, not just my education and my occupation. My very personality was influenced—perhaps a better word would be "controlled"—by the fact that I was a Shield.
Yet I didn't want to admit that, not before these people. I wasn't sure of the reason for my reluctance. It wasn't as though I was ashamed of being a Shield and all it meant. But these people were disturbing me right then, and I didn't like the idea of thinking like they did. "I doubt there are many who would enjoy being reduced to nothing more than a personification of a single talent."
"Then you don't know a great many people all that well," said Cree. "There are many who consider a talent they deem worthy to be their defining characteristic, and many more who wish for a talent by which to be defined. Perhaps, surrounded as you have been all your life by unusually gifted people, you have difficulty understanding what it is like to lack such gifts."
"You're a healer," Taro said to Cree. "Why would you need to use spells to feel special?"
The silence that followed was tense, and Cree's face stiffened.
"None of us need spells for anything in particular," Fines said quickly. "Understanding magic is much greater than any one particular activity or desired outcome. It gives us a connection to the ground and the air and to the animal world and each other in a way nothing else can. It is something anyone can learn and embrace, with diligence and discipline. It's bigger than any one person, spanning any distance, and it has the power to last forever. Why else do you think the Crown Prince is outlawing the use of it?"
I thought it was because people were dabbling in things they couldn't control and burning their houses down. "Why would people want to be bound to everyone else for an eternity?" I asked. It seemed suffocating to me.
"To belong to something greater than themselves."
"You don't need to use spells for that."
"You belong to the Triple S," said Fines. "You have a Source who is bound to you for the rest of your life. Most of us don't have such things."
"But you have families, friends, your community."
"And what if you have none of those things?"
"How could anyone have none of those things?"
"It is possible. There are orphans. There are those who are isolated and outcast for one reason or another."
"And casting will somehow make them feel better about all that?"
"It gives them a connection."
"Enough of a connection to replace all positive human contact?" I'd be worried about someone so thoroughly unfortunate having access to magic. What would they do with it?
"Mere human contact is too fragile, too easily disrupted by inconsequential things. An understanding of magic and how it connects us to each other and the world, that is stronger and more powerful than any human relationship."
"And nothing about human relationships will tell you how the natural world works," Williams said. "The power of fire. Air. Water."
He put a heavy emphasis on "water" that made me look at him. He was trying to tell us something.
Gamut thought so, too, and he didn't like it. "Shut it, Morgan."
"No. I'm sick of this dancing around. If they're going to join us—"
Join them? For what?
"It's too soon, Morgan," Fines chided.
"It became too late once Ayana used a spell to heal the Shield. They already know what we are."
"It's certainly too late now, Morgan, thanks to you," Fines snapped. Williams communicated his lack of concern with a shrug. "Our Dunleavy has a reputation for not leaving things alone."
I did? When did that happen? Why were people talking about me so much? And why did they get everything wrong?
"Good." Williams crossed his arms. "So lets get things moving."
The tension in the air became stronger, which was all we needed. I really didn't want to hear what they had to say. I didn't want to be part of whatever weird club they had running. But I also didn't have the spine to interrupt them all with a "No, thanks" and "Where's my carriage?"
"This little group formed through our businesses," said Fines with a wave of the hand that took in everyone at the table. "In many ways our interests and our clients intersect. We began to work together, and we learned that each of us is more successful working together than as an individual. I learned of Cree's ability to use spells in a way similar to yours, Dunleavy. At one point, while overseeing some loading of inventory at the docks, I was badly cut. The vein was one of those that bleed out in less than a third of an hour. One of my servants fetched Ayana, and she used a spell that saved my life. In time, she introduced all of us to the practice. We owe her a great deal."
I didn't want to hear about it. I really, really didn't.
"How do you expect us to assist you?" Taro asked. "We can't help you with your businesses."
"It is true that you have no business interests to complement ours. But that isn't your most valuable asset."
"Our most valuable assets are our skills as a Source and Shield," I said sharply. And they had better not be thinking about asserting any control over those. We didn't channel, or not, according to anyone else's wishes.
"It is not those assets, strictly speaking, that appeal to us."
"Quit dragging this out, Richard," Gamut grumbled.
"As you are no doubt aware," said Fines, "ashes are a powerful component in many spells. And ashes are relatively easy to procure. But blood is even more potent."
I stared at him, appalled. "You want our blood?" I squeaked.
"As far as we know, no one's done any real experimentation with the blood of Sources and Shields, but it is only logical to assume the blood of magical creatures has all sorts of useful properties. Ayana has told us the spell she used to heal Dunleavy was unusually quick."
That had been quick? I guess I wouldn't have survived the usual spell.
"We are not magical creatures," Taro declared with noticeable impatience.
"My dear Shintaro," said Williams, "what do you call what you do when you protect the city?"
"Not magic," Taro retorted. "And, by the way, not illegal."
"Only the latter is true, and only because the lawmakers made a random distinction about what magic is. What do you think those same lawmakers would call what you do when you heal people?"
Damn it, could no one in this city keep their mouths shut?
"It's not a talent traditional to a Source," Williams continued. "Would not most people think it magic?"
"We are not giving you our blood!" I said. "That's disgusting. And ridiculous." What were they thinking? That every week or so we'd let them bleed us?
"You can't make the usual contributions to our group," said Fines. "Blood would be an acceptable substitute."
Taro was shaking his head. "We have no reason to want to be part of your group."
"I saved Dunleavy's life," Cree reminded him bluntly. "There is no other healer in this city who could have done so. You would not have known to come to me, had it not been for this group."
She didn't add that she probably wouldn't have risked using a spell on me if she hadn't known me. It was still alarming to me that if it weren't for her, I would be dead. I hated feeling indebted to her.
"You would have access to everything we have, now and in the future," Ahmad said.
"There is no predicting how we could all benefit in the long term," said Fines. "And remember, you don't have just the two of you to consider. A merchant's fortune is a very fragile thing. Things can be done to assist it, or to destroy it. All it takes is a ship to sink and there begins a downward spiral to destitution."
I glared at him. "Are you threatening my family?" I demanded.
"Not at all. I am merely pointing out that people's affairs are more entwined than you might imagine."
Of course he was threatening them. Did he really have the ability to hurt my family? Was he somehow able to sink a ship?
"And, of course, most estates rely heavily on the goods they produce. Were an estate to fall into disfavor, it could quickly fall to ruin. Your family's estate, Shintaro, don't they sell a great deal of whale oil?"
Taro forced a small laugh. "You're aiming for air if you try to use my family against me."
"Lord Shintaro, you abjured your title but kept your name. Do not try to convince us you have no family feeling."
"Why are you trying to blackmail us into joining you?" I asked. "From the looks of it, you are enormously successful. You don't need us."
"All things must move forward or they die," Cree told us. "Now that more and more people are using spells, more people will become proficient at casting, and our advantages will diminish. We must always be ready to act on new opportunities."
"Or be ready to create them," Williams added.
"And it represents a brilliant opportunity to acquire knowledge," Ahmad added. "Knowledge of how the world works."
"How the mind works," Gamut said.
"How water works." Williams stared at me.
I had no idea what he was trying to tell me.
"Shut up, Morgan!" Gamut ordered.
"My gods," Taro breathed, his eyes widening. "You're the ones who have been poisoning the water in the riverfront areas."
Certain pieces of my memory slotted into a new pattern. By gods, he was right. I looked at Cree. She must have been the one who had gone to the hub to perform some spell.
"Ben didn't guard his tongue much when they captured me," Taro explained. "He was very enthusiastic about all the people leaving the riverfront areas. He was going to use our ashes to create for himself ownership of that area where the rivers meet, because it's supposed to be so powerful. Or something like that. He was rambling a bit."
"You're killing people to have control of the riverfront?" I demanded, horrified.
"Don't be ridiculous," Cree answered sharply. "None of them have died."
"That's not what they're saying."
"People have a higher mortality rate in the riverfront area than those who live in other parts of the city. Those that died while ill from our tampering died of other causes."
She couldn't know that for sure. She was just making assumptions.
"It sounds far worse than it is, Dunleavy," Fines said in a soft voice, as though by mere tone he could somehow soothe me. Wasn't going to happen, friend. "All we are doing is encouraging these people to leave."
"But they have nowhere to go. If they did, they probably wouldn't be living there in the first place."
"This is a big city. And there are other cities."
So heartless.
"The whole area is a cesspool," Gamut muttered. "It's needed to be shaken up for decades. It could have used an earthquake or two if you lot would have just let it happen."
I covered my mouth with my hand, the only way I could hold in my shock.
"What do you plan to do with this riverfront area?" Taro asked.
No one jumped in to answer that question. I noticed a lot of glances being exchanged.
"I think we've said enough for now, Shintaro," said Fines. "I think it's time for the two of you to do some talking."
We were not spilling any of our secrets. I hadn't wanted to know their secrets. I shouldn't have to pay for things I had neither wanted nor asked for. I said nothing. Neither did Taro. There was a bit of a waiting game.
Which we won. They spoke first.
"All we ask is that you continue to join us for our meals and honestly participate in our discussions on a committed basis."
Oh, and our blood. Don't forget that.
"Thank you for your kind invitation," Taro said in delightfully chilly tones. "But I fear we'll have to decline."
The others ate in silence for a few moments. I'd lost my appetite. So had Taro, from the looks of it.
"To your knowledge, has any member of the Triple S been found violating the prohibition against casting?" Williams asked.
Taro glanced at me for confirmation before indicating he knew of no such instances.
"I imagine that Sources and Shields will be protected from things such as incarceration and flogging, should any of them be found to be engaging in magical practices, but I do believe members of the Triple S are punished if the crime is considered serious enough."
"Aye," said Taro.
"Pairs who are found guilty of serious crimes are sent to remote areas that have no need for their skills, is that not so?"
"Aye," said Taro.
"It's a shameful thing. Something Pairs don't recover from. Such a Pair would never be assigned to somewhere they considered reputable, is that not so?"
"Correct."
By the exaggerated roll of his _r_ 's, I could tell Taro was getting as irritated by this line of questioning as I.
So after threatening our families, they were threatening us. That was infuriating. "You were the ones to cast spells," I said.
"There are five of us."
And only two of us. "People believe Sources don't lie."
"But we have impeccable reputations. Yours, you must admit, are a little dodgy. Long, unexplained absences from your duties. Healing people. You were both seen acting strangely at the hub in the middle of the night."
Damn it, they had seen us. That must have been why nothing had happened during our long nights of surveillance. Cree saw us or something, and stayed away.
"And then there are those strange stories about how the Reanists were brought down at Lord Yellows's dinner. Definitely sounds like there were spells involved in that."
"And then, of course, there's this." Fines pointed at one of the clay pots. "Made from the best ashes."
My knife and fork fell to the plate with a harsh clatter as I gagged. We had been spreading human ashes on our hands?
My gods, my gods. I shook my hands, not that I thought that would accomplish anything.
"It's perfectly sanitary," Cree asserted.
"Are you crazy?" I demanded.
"I believe using this paste helped you survive the first use of niyacin powder on your burn."
"There is no way you can know that," I objected.
"I am a healer and I am proficient at using spells," she pointed out. "What expertise do you have to bear?"
Taro was scrubbing at his hands. He poured wine into his palm and rubbed it into his skin.
"You buy ashes." I couldn't believe it. Risa and her colleagues were running around trying to stop the trade in ashes, and we'd been eating with people who bought them. Who used them. We'd used them.
"Yes," said Cree.
"You buy ashes from Ben Veritas."
"Not directly, of course," said Fines, though I didn't understand what was so "of course" about it. "And we had no idea he was killing people."
I was curious about whether his killing people was a problem for Fines because killing people was wrong, or because murder tainted the ashes.
"We had not planned to introduce you to our practices in this way," said Fines. "However, Cree put herself, and by extension all of us, in great danger by her actions at your home. We needed to be sure of your silence. And your loyalty."
"You've made it impossible for us to give you either," I said.
"Don't answer tonight," Fines suggested. "Think about it for a few days."
"We don't need to think about it."
"Think of your families," Fines persisted.
They were really irritating me. "My family wouldn't want me being bled for spells to spare them trouble."
"And my family can go hang," Taro added in a sharp tone.
"We'll see if you feel the same way when you start getting their letters of misfortune."
Bastard. "Do you really expect us to join people who would blackmail us?"
"You're not approaching this with an open mind."
"I'm content in my close-mindedness." They had been collecting information on us. They had induced us to use illegal human ashes. Cree had saved my life, and I owed her for that. My blood was too much to ask. "If you do anything to our families, we will tell everyone, including Prince Gifford, what you are doing, and let things happen as they will."
Fines's frown was pinched. "It is disappointing to learn you have so little family loyalty."
"Blame it on my upbringing. We're done here."
No one said anything as Taro and I left the table and left the room. My heart was pounding in my throat. The people I had eaten with tonight had not been the people I had met before. There had been no playfulness, no humor. They had been grim and ruthless. Which was the truth?
I felt tainted. I had no idea what it would take to feel clean again. For some reason I felt water and soap wouldn't be enough.
Could they really hurt my family?
I didn't know what was the best thing to do. To submit to blackmail rankled. But when it came to my family's well-being, maybe blood and time weren't so much to lose.
Then again, that might not be all that they asked for. Once successful at getting blood from us, they might demand more. And they were hardly honorable, getting us to use human ashes without our knowledge. Something within me screamed that entering into any kind of agreement with those people would be a horrible idea.
"We're not going to do what they want, are we?" Taro asked.
"Certainly not."
He nodded. "Good."
I would just have to think of a way to warn my parents of possible problems without giving them any facts.
Chapter Thirty
I had an unpleasant duty to perform. I had to end things with Doran. No more delay. No more games. The thought of the conversation we would have to have made my stomach clench, but it had to be done. So I went to his boardinghouse wearing the plainest gown I owned, my hair loosely tied at the nape of my neck, a particularly unattractive style on me.
He was in. He looked pleased to see me. I wondered why. Every interaction we'd had recently had been negative. Why did he think this would be a positive meeting?
He escorted me to the boardinghouse's parlor and kissed my cheek before settling into the chair next to mine. "You look wonderful," he said without the slightest trace of sarcasm in his voice.
Because of that, I merely said, "Thank you. You are very kind."
"It's good to see you looking so hale. I have to admit, I was worried the last time I saw you."
Oh, that made sense. I'd probably looked half dead the last time we saw each other. Anything would be an improvement on that.
"I'm so sorry to have caused you concern." Well, that was chilling and polite. I felt uncomfortable parroting meaningless phrases in the face of his sincerity. "Thank you for agreeing to see me on such short notice."
He frowned then. "Why wouldn't I?"
Um, well, yes, that had been a really stupid thing to say, hadn't it? As though he were some sort of stranger. "Well, it wouldn't be unexpected for you to have already made plans."
He shrugged. "Are you up to much yet? Because Weller is holding a small concert in his home the day after tomorrow, and from what I've heard of the repertoire, it's perfectly safe for you to hear it."
I didn't usually trust a regular's definition of safe. They often had no idea. And while I was tempted to delay things by talking about music and what kind I usually liked to listen to, I knew I had to get things over with. "I'm not good at saying important things delicately," I confessed. "So I'm just going to say it. You are truly a wonderful person, smart and witty and kind. But I can't see you anymore. In any capacity. I am sorry." There. That wasn't bad, was it?
His eyebrows rose. He was shocked. He hadn't considered this possibility.
How could he not, after our last meeting?
He rose abruptly, circling his chair, rubbing his nose. I got the feeling he didn't know how to react.
I felt compelled to say something more, but I had no idea what would be the appropriate words. Repeating sentiments with different wording would only weaken my original statement, and ran the risk of stumbling into a logistical frailty or emotional quagmire.
"I mean no disrespect to Shintaro," Doran said finally. "But he is not the sort of person to which one should attach aspirations of longevity."
I was offended even though I agreed with him. I had the right to think that way. He didn't have the right to say it. And it annoyed me that the first response he had about something that was relevant to the two of us, was a negative comment about Taro. "I really don't have a choice about that," I said. Doran once more looked surprised. "I'm bonded to him," I reminded him. "I hope that will enjoy a certain amount of longevity." As the alternative was an early death.
"That's different," he said with impatience. "You're being difficult."
I was being difficult? I wasn't the one trying to create a debate. There had been plenty of warnings that this was my mind-set. He was behaving as though it was coming out of nowhere.
I wished he would sit down. I didn't like having to look up at him.
"You are looking for someone to settle down with. For permanence. Aren't you?"
I frowned. "I wouldn't say that. I haven't been looking for anyone." For one thing, my life had been too chaotic since I left the Academy. For another, well, the logical side of my nature kept telling me that forever was a long time, and it was a little unrealistic for people to fall in love with the expectation that it would last for the rest of their lives.
Certainly, I had spoken of permanence with Taro, but I knew that really just meant for as long as we were both interested. That wasn't the same thing as forever.
"Shintaro will never be the sort of person to remain faithful to one person."
Really, how the hell would he know? He'd met Taro only a handful of times. "He has so far."
"You can't know that."
"Taro doesn't lie." Not to me.
"You can't know that," Doran insisted.
"Maybe not, but I can believe it."
"You'd be a fool to do so."
I found it fascinating how many people thought insults were an effective means of persuasion.
"I mean it when I say I admire him. I do. But in some ways he's like a child."
I opened my mouth to deny that and found nothing sprang immediately to mind, I was that shocked. A child? That was a new one. There was nothing childlike about Taro.
"I've known many like him, Lee. They're spoiled. Not mean-spirited, but thoughtless, because all their lives they've been given everything they've ever wanted, with little effort expended. And people like that are too easily bored, because they don't understand the effort it takes to build something, and the value there is in creating something with their own labor."
I wasn't going to explain that Taro had spent much of his life being anything but spoiled. That was Taro's story to tell, and none of Doran's business. "I knew someone else who said something similar about Taro. He was crazy." And I'd killed him, though not because he hadn't liked Taro.
Doran snorted. "Only one?"
"Crazy person?" Unfortunately, no. The Academy hadn't warned me about all the crazy people I would encounter out in the real world.
"Person who agreed Taro's a—"
I cut him off. "I don't want to hear it!" I wasn't going to sit there and listen to him denigrate Taro. If he was going to say such things, he could say them to Taro's face.
"Don't want to hear the truth?"
"The truth? Rumors spread by a bunch of people who don't even know him?"
"Oh, and you do?"
Was he serious? "I'd wager that I've spent more time in Taro's company over the last few years than everyone else in High Scape combined."
"And you think that means you know him better?"
Of course. What else could it mean? "Yes."
"You're being naive."
Certainly. If I didn't agree with him, it had to be because I didn't know what I was talking about. "And you're being obnoxious. What do you think you're going to accomplish here?"
"I don't know," he admitted. "I wasn't expecting this, and everything is coming out all wrong."
Well, I knew how that could happen. "I'm sorry, Doran." And I really was. I liked Doran. I wanted to want him. It would make so much more sense. "But nothing you say now can possibly come out right."
He sighed. "I'm worried about you. I can't see this ending well."
Neither could I, but I was going to give it my best effort. And I couldn't give it my best effort if I kept Doran waiting in the wings. Besides, Doran should be marrying someone with money, as he had none of his own. And that reminded me of something. "That bundle of yellow flowers with the cake and the earrings, someone told me that was a token indicating an intention to open marital negotiations." I took his silence for assent. "What was that about? You don't truly wish to marry me."
He flushed and said nothing.
"Please tell me," I prompted. I wanted to be able to tell Taro the answer.
"Can I assume Shintaro saw it?" Doran asked.
"He told me what it meant."
"I assume the delivery created a great deal of"—he hesitated in his word selection—"discussion between you."
I felt my eyes narrow. He'd sent a token he probably knew I'd send back in order to cause arguments between Taro and me. "I underestimated you Doran," I said flatly. "I had no idea you were so devious."
"It's for your own good, Lee. It's all going to fall apart. You know that. And I can be there to help you through it if you want."
Unbelievable. Doran, not Taro, was the man I didn't know, and I was so sick of being manipulated by people. "My name is Shield Mallorough," I reminded him. "And it's past time I left."
"I'll be here when Shintaro grows tired of you," Doran said spitefully.
"I won't."
Well, I had predicted that one correctly. That had been really unpleasant. Not to mention disappointing. Just because I had to stop seeing Doran didn't mean I wanted to think less of him. I headed home, feeling more weary than the slight physical exertion warranted.
Taro was waiting for me. As soon as I closed the door at the entrance of the Triple S residence, I heard clattering down the stairs. "Lee!" Taro called, a certain urgency to his voice. "Come up to my suite. Right now."
That didn't sound good. And he had called the suite his. He had been calling it ours. Something had rattled him. I wasn't up to jogging up the stairs right then, but I pushed myself a little harder than usual. I didn't know why I was trying to be quick. The bad news would be there whenever I got there.
Taro's hair was a tangled mess. That was another bad sign. "Close the door," he ordered when I reached his suite. So I closed the door. "We've been ordered to attend the coronation."
"Gifford's coronation? Ordered?"
"Technically it's an invitation, but it's not like we could refuse. It was dropped off by members of the Imperial Guard. They've offered to escort us to Erstwhile."
To make sure we actually went, no doubt. And sure, that was annoying. We'd just gotten back on the roster. I had no interest in going to Erstwhile and even less, if that were possible, in seeing Gifford crowned. It wasn't a disaster, though. "Why the panic?"
"He doesn't like us," Taro said. "Why is he summoning us?"
I took the letter and looked at it. "The only thing personal about the summons is our names. It's a copy of thousands of others, I'm sure. He calls us because he can. It's an inconvenience, but nothing to get excited about."
"Gifford isn't like the Empress," Taro said. "He's more likely to let his feelings rule him, and he feels he has something to prove."
"What do you think he's trying to prove with this?"
"I don't know, but it can't be good."
"It could be nothing more than that he can snap his fingers and we have to come running."
"It could be."
"But you don't think so."
"No."
"But we're of no use to him."
"I would have said the same about his mother, and look what she had us do."
I couldn't tolerate another journey like the one to Flatwell. I would go insane. But the Empress had had particular reasons to choose Taro for that task. Weird reasons, but she had clearly liked Taro. As Taro had said, Gifford didn't.
Maybe there were more family secrets that we were going to be exposed to.
Or maybe, upon the Empress's death, Gifford had learned about Aryne, and that Taro and I had been the ones to find her. He would be furious that his mother had sought another heir. And he would be furious with us for assisting her, whether we had had the option of refusing or not.
Now I was starting to panic. I went down to the kitchen for some wine to help me stop thinking. There was no point worrying about it until we were there.
Chapter Thirty-one
So Taro and I were in Erstwhile yet again. Erstwhile was a cold site, which meant we should have never had to visit the city at all, as far as I was concerned. Most Pairs never did. It was just because the monarchs liked playing games that we were ever there at all.
At least this time we weren't expected to stay at the palace. For the past few days we had been enjoying the relative privacy of an expensive boardinghouse. It gave me hope that we would escape the personal attention of Gifford altogether.
The city had been enjoying a week of celebration. There was a variety of competitions, running and jumping and fighting and bench dancing. There was free food and drink. There were plays and musical performances. There had been a series of breakfasts, dinners and balls at the palace, to none of which Taro and I, thank gods, had been invited.
This was to be the final day, marked by a massive parade leading to Gifford finally having the crown placed on his head. I was curious as to how the Crown Prince felt about that moment. Was he excited or merely impatient?
The only contact we received from the palace once we were in Erstwhile was the personal invitation to witness the coronation. That meant we had reserved seating immediately in front of the platform on which the Prince would be crowned. This was a highly privileged position, as thousands would be forced to stand. I would have preferred to be lost in a crowd.
All the other people milling about the collection of seats were the highest of titleholders, politicians and guild masters. No one like us. I'd met a couple of them before, during my other visits to Erstwhile, and we exchanged distant greetings. And through it all, I was aware of a woman, tall and buxom and blond and wearing a duchess's coronet, watching us closely.
When I finally looked directly at her, she approached us. "Excuse me," she said to Taro. "Are you Source Shintaro Karish?"
He smiled at her. "I am."
"My gods," she said. "You look so much like your mother."
His expression chilled. It was true—he and his mother looked freakishly alike—but he didn't like to be reminded of it. "How may I help you?" he asked coolly.
"I'm Fiona," she announced.
Taro grinned widely. "Cousin!"
"How are you?" She threw her arms around him.
I realized who she was. The current Duchess of Westsea. The woman who wrote the hilarious letters and stood up to the grasping Prince.
Taro seemed comfortable hugging her back. "It's wonderful to finally meet you," he said before drawing away. "This is Dunleavy Mallorough, my Shield."
The Duchess of Westsea turned her warm smile to me and offered her forearm for me to take. "A joy to meet you," I said.
"Shintaro has written about you," she told me.
"That's unfortunate," I responded.
"So what do you think of this idiot getting the Crown?"
I almost choked on that. "You might want to keep your voice down," Taro advised her in a mild tone.
She snickered. "I'm no politician."
I didn't think the titleholder of an estate as large and powerful as Westsea could afford not to be a politician. I agreed with her sentiment, though. I was very uneasy with the idea of Gifford being our ruler. I didn't trust him.
"Maybe you can give me some information, though," Fiona was saying. "I've applied to the Source and Shield Service for a Pair."
Taro frowned. "Flown Raven doesn't have natural disasters."
"It does now. There was a horrific earthquake a couple of months ago, and a few smaller ones since."
Taro gasped, his eyes wide.
"If everyone would take their seats!" some official called out.
"It's a delight to meet you both," Fiona said, squeezing our hands. "Maybe we can talk some more after the deed is done." She headed to the front row.
"What's wrong?" I asked Taro as we took our seats at the back.
"That's where those events were coming from," he said. "The ones from a distance. That's why they felt like something familiar."
I slapped his arm. "You're not supposed to channel events at your birthplace!" Weird things happened when Sources tried to channel events from the place where they had been born. In the past, the Pairs had died. More often, the events got away from them. There was a theory that the familiarity of their home made it difficult for a Source to decipher which forces needed to be channeled and which should be left alone. That was why Sources were never posted in the area in which they were born. Or raised.
It was why, probably, Taro had let so many forces flow through him, and why I'd had such difficulty Shielding him.
"I didn't know," he protested.
"Don't do it anymore."
"I haven't been."
And soon there would be a Pair at Flown Raven who would channel the events, and Taro wouldn't feel tempted to do so. That was one danger avoided, so I could stop thinking about it.
I could feel the racket of the parade approaching. When I learned of the parade that was to introduce the Prince's coronation, I'd thought nothing more of it than another event on a day that was already too long. And yet I had never seen a parade quite like this. The sheer number of participants, for one thing, all in matching and highly tailored clothing that wasn't quite warm enough in the autumn air. There was dancing and tumbling, all in time to expertly played music that really had me wanting to jump and dance around, too. I had to sink my nails hard into Taro's arm to resist that impulse.
The floats were unbelievable. One was a huge horse, about four men high, black with blazing red eyes, pulled by four enormous live black horses. Visually stunning and a little menacing. I wondered what kind of message the Prince was trying to give with that float. Maybe I was reading too much into it. It wasn't likely that the Prince had made any of those decisions.
The next float was of the sun rising, a huge ball of metallic yellow—that wasn't gold, was it?—rising over waves of purple and dark blue, surrounded by creeping white smoke released by the various people walking around the float, dressed in frothy gowns of light blue.
It reminded me of the tacky decorations in the homes of Fines and Cree. And lords, I didn't need to be reminded of them. Taro and I had received several more invitations from them and the others of their group. We had turned all of them down. I hoped they would lose interest in us soon. I didn't know what I was going to do if they didn't.
At least I hadn't yet received news from my family describing business difficulties.
The next float was a large three-dimensional replica of the royal seal, a gaudy thing of red, purple and blue ribbons surrounding a round black shield, topped by a thick book, on which stood a white horse wearing a gold crown at a jaunty angle. I'd always thought that a hideous emblem, and learning what it all meant in history class didn't improve my opinion. The ribbons represented royal blood and the seas, the two being mastered by the monarch. The shield represented military might; the horse represented wealth. The crown represented wisdom, and the book represented law and, therefore, in theory, justice.
Ben Veritas had had his trial. He had been convicted of attempted murder and a string of magic-related offenses. That pleased me, at least as far as attempted murder was concerned. He was sentenced to one hundred and twenty lashes followed by death by hanging, which disgusted me. There had to be a better way to punish people. As a victim, I'd been offered a reserved seat to witness the execution of the sentence, but I'd turned it down. A large part of me felt morally obligated to watch the results of my actions, but I knew I wouldn't last through the flogging. It was such a cruel form of punishment, and totally unnecessary.
Veritas hadn't survived the flogging, but his body had been hanged anyway, because that was what the sentence had dictated.
Ben Veritas was dead, as was his daughter, Sara Copper, who had also been found guilty at her trial. And from what Risa told me, people were still digging up ashes. I assumed people were still buying them. I certainly had no reason to believe Fines and his group weren't still using them.
A loud crash shocked me from my thoughts. Cymbals. Cymbals were too popular in parades as far as I was concerned.
The musicians were followed by a troupe of acrobats determined to kill themselves, if the way they flung themselves off high surfaces was any indication. And then another float followed, this one an elaborate throne propped high on a dais of gold. I was surprised the Prince wasn't sitting on it. For him to do so before being crowned would be a horrible violation of etiquette, but I'd never known that to stop him before. The float was pulled by dozens of men, marching in time to the music, which brought a disquieting uniformity to the display.
And following the float was the man himself, the Crown Prince, a space maintained between him and everything else by a perfect square of his personal guard marching around him, marred only by the two pretty young men who carried the trail of his long red cape. Aside from the cape, he was dressed entirely in black. He had small brooches or the like pinned to his chest. I knew they were symbols related to the military and the justice system, but I didn't know exactly what they signified.
The parade was progressing up the center street of Erstwhile, which was wide and white and led straight up to the steps of the palace. I thought it a rather stupid setup, allowing invaders a direct path to a vulnerable part of the palace despite the tall iron gates that had been pushed open for the occasion. But I didn't know anything about such things, except that there hadn't been a large-scale attack on the palace in centuries.
The participants of the parade didn't cross into the palace grounds, instead splitting off into the crowds. The Prince and his attendants did cross over, heading toward the small stage set up in front of the steps to the palace. On the stage was an elderly man who I'd been informed was the solicitor to the royal family. The solicitor formally held the title of ruler between monarchs and kept the secret code that identified the next sovereign. A series of spectacularly brutal executions had made clear the fate of solicitors who attempted to abuse the trust placed on them by trying to claim the title for the long term.
The solicitor stood beside a table. From where I sat, I couldn't see the articles on them, but I knew what they were.
Once the Prince stood before the stage, the music stopped and all the murmured conversations were cut short. In silence, the Prince ascended the steps to the stage. The solicitor bowed to him, the Prince offering no gesture of his own, which I thought rude.
I could hear Taro's teeth chattering beside me, and I glanced at him. He was trying to keep his teeth still, but his shivering was noticeable. His stiff black garments stood out among the colored finery of the others, and they weren't suitable for the weather. I wasn't going to remind him that I'd warned him to wear a cloak. I didn't want to turn into a nag.
The Prince faced the masses.
"State your name," the solicitor ordered.
The Prince didn't seem to react to the tone. "Gifford William Hiroshi Madas," the Prince announced, loudly enough to be heard by everyone seated and, I was sure, a good many who were standing. I wondered if he'd practiced.
"State your lineage."
"Son of Constia Alexandra Fiona Madas, who was daughter of Kemmeth Elisia Yuuki Madas, who was daughter of Aneck Randolf Emery Madas, who was son of Benik Chapry James Madas, and stretching back into the mists of time."
It was Benik who stole the throne from Emperor Koitchi through means both legitimate and not. I supposed it wasn't regal to admit to that.
"How have you proven your right to the position of high commander of the legions of the empire?"
"Through tests of strength and strategy."
Test of strength could mean anything. Lifting a weighty object. And perhaps he'd won a game of chess or two. I was fairly sure the Prince had never been in any kind of combat.
"How have you proven your right to the position of high justice of the empire?"
"Through study and reflection."
Fortunately, the Emperor didn't actually hear cases. The monarch used to, ages ago, and apparently it was a nightmare. The title, like that of high commander, was purely ceremonial. In theory, people who actually knew what they were doing ran the military and the judiciary.
"Who will attest to your character?"
A man in the front row rose. "I do," he said in a voice I could hear even though the man was facing in the other direction. These people must have been chosen for their ability to bellow.
"And you are?"
"Lord Gray, the Duke of Conrad."
"And what say you, my lord?"
"I attest that the Crown Prince is a man of honor, of honesty, of valor and of sound mind."
Well, to be strictly honest, I couldn't dispute any of that, not really. Just because I didn't like what he did didn't mean it wasn't all aboveboard. His mother was just sneakier about what she did. I wondered why I had liked her better.
"Why should you be our Emperor?" the solicitor asked.
"I have been born and bred in the knowledge of my responsibility to the people, and tutored in the honor to be found in the service of others. I am prepared to spend my life in the service of others. Always will I know that every decision I make must be made with the best interests of the people in mind, that their needs come before my own, that I am merely the most visible of servants to the people."
I wondered how hard that last phrase made him want to choke.
"I have been privileged to have had before me, my entire life, the best example of what a ruler should be. A ruler who never took a single action without first thinking how that action would affect her people, who put the needs of her people before her own, whose every action was one of compassion, of mercy, of justice. She was the model of all that was good and strong and right."
It alarmed me that he could lie so convincingly. At least, his praise of his mother sharply contradicted all the rumors I'd heard about how much he'd despised her. Maybe he'd really admired her and no one knew about it. Or maybe he just wished he'd been able to admire her.
"With this understanding of the needs of the people, I will dedicate myself to the peace and prosperity of all. This shall be my guiding principle for the rest of my life."
From the table beside him, the solicitor picked up the monarch's sword, said to have been carried by the first emperor and never used again. I wasn't sure if that story was true or not. Certainly it was a very shiny sword, thinner than the ones I sometimes saw the Runners wear, and it was reputed to be stronger than any sword any contemporary blacksmith could make. "Do you affirm," the solicitor asked, "on your name, your life and your lineage, that you will keep the position of high commander of the empire with all honor, putting aside personal connection and favor where it may conflict with duty, for as long as you shall live?"
"I so affirm," said the Prince.
The solicitor presented the sword to the Prince over his forearm, hilt first. "I bestow upon you the rank of high commander."
The Prince took the sword in his right hand, holding it rather awkwardly and making sure no part of it touched anything else.
The solicitor then picked up the staff, standing as high as the average man's shoulder and as wide as a thumb, a slim, plain rod made of gold that was, for some reason, the symbol of the justice system. "Do you affirm on your name, your life and your lineage that you will keep the position of high justice with all honor, putting aside personal connection and favor where it may conflict with duty, for as long as you shall live?"
"I so affirm."
Like the sword, the solicitor presented the staff over his forearm. "I bestow upon you the rank of high justice."
Due to the length of the staff and the lack of natural grace in either man, the transfer of the staff looked cumbersome, and the Prince held it at an unwieldy angle.
Finally, the solicitor held up the crown, a band of gold with no decoration or finesse. I'd never seen it before. It looked bulky and heavy and was probably uncomfortable to wear.
"Do you affirm on your name, your life and your lineage that you will keep the position of emperor with all honor, putting aside personal connection and favor when it may conflict with duty, for as long as you shall live?"
"I so affirm."
And standing behind him, the solicitor carefully placed the crown on the Prince's head. I imagined if he got the angle wrong, the thing would have slid off and hit the floor with a clang. That would have been fun to see. "I bestow upon you the rank of emperor."
May the gods save us all.
The crowd roared. I wondered how they could. Perhaps they had no idea what the Prince was like and what his new position would mean for them. Perhaps his ascension really did mean nothing to them, one ruler being pretty much the same as another. Perhaps they were just happy they hadn't had to work for the past few days.
The Prince, now Emperor, wearing the crown and holding the sword and the staff, looked unfortunately impressive, awkwardness notwithstanding. Clothing could lie so well. Anyone would look regal with such costumes and props.
"Troubling influences have been creeping into our great cities," he announced once the cheering faded away. "Our beloved Empress stood firm against all that was divisive and destructive, yet over the course of her failing health a darkness has begun to poison the land, attacking our traditions and our laws, endangering the peace and prosperity of all. To restore the world to its proper path will be Our first and most important task. To return glory to the people will be Our just reward."
The crowd cheered again. I hoped they understood what the Emperor had just said, because it had sounded like nothing more than gibberish to me. Ominous gibberish.
Lord Gray stood and climbed the few steps to the stage, lowering with some difficulty to one knee and bowing his head. "I, Lord Gray, Duke of Conrad, offer to the Emperor and his heirs my eternal faithfulness, and that of my descendants. I offer the best portion of my land and stock, the might of my servants, and the enforcement of the laws of the land."
"We, Emperor Gifford, accept the offers of Lord Gray and give in return protection against danger and want." That, I knew, was the traditional oath of the monarch to one of his vassals, meaning not a whole lot since the powers of the monarch were given boundaries by the Council. The following words were not traditional. "In honor of your recent services to the Crown, it is Our duty and Our honor to grant to you the lands of Vast Greens, yours and your heirs' in perpetuity."
That caused some gasps. I'd never heard of the place myself. Gray's expression as he returned to his seat was blank, so I couldn't tell if the boon was expected or not, an advantage or not.
Gray was only the first of many titled landowners to swear their fealty. The second was the new Duchess of Westsea, Taro's cousin, Fiona. She swore her oath in the flattest voice I'd ever heard in a regular, which made me think I might really like her.
The titleholders were followed by the heads of the guilds, a judge, a representative from the Runners of Erstwhile, and various other important people. Most of them merely gave their oaths and returned to their seats. A few were granted boons as Gray had been. It was all excruciatingly boring.
And as time wore on, I felt that distinctive jittery sensation. Someone was casting a spell. Right then and there. Who was it? I looked around, but no one and nothing seemed out of place. Certainly no one was waving any symbols around.
Except for the Emperor. Holy hell.
Aye, the Emperor's lips moved at times other than when he was responding to the oaths. And he was looking around a lot. My gods. The Emperor was performing a spell. He who had increased the sanctions against spell books and the tools of casting. What a hypocrite.
I couldn't figure out what the spell was meant to do. I couldn't see anything happening. Was it a spell to bind people, the people swearing the oaths, to him? That would be horrible. It had to be wrong.
So what was all that about increasing the sanctions against people casting spells? Or, excuse me, pretending to cast spells. Did he really find it a dark poison? Or did he just want to keep all that potential to himself?
What else had he accomplished by casting spells?
Hell.
And then I was horrified to hear the Emperor say, "Source Karish may approach."
I looked at Taro. His shock was clear to see. And the Emperor was looking right at him, perhaps having been aware of his location throughout the entire ceremony.
Taro recovered himself, easing out from the chairs before striding up to the stage, looking all dark and gorgeous. If he was feeling frantic and out of his depth, no one would know it by looking at him.
I really never wanted to be him.
He bowed, which I supposed was correct, instead of kneeling. I couldn't imagine Taro ever being incorrect. Yet the Emperor frowned. "You may kneel."
I clenched my teeth. It was unreasonable, I knew, to be offended on Taro's behalf, that the Emperor would make him kneel when he had no oath to give. We were all required to show deference to the Emperor. It was just that Taro was by far the worthier man.
Taro knelt, as demanded.
"There is an oath We would have from you," the Emperor announced.
Taro couldn't speak for the Triple S, who were not required to give fealty to any monarch. Nor could he speak for the duchy of Westsea. He could speak only for himself.
"We are aware you have not come prepared to speak," said the Emperor. "And indeed, why would you? A mere Source, at an occasion of this magnitude. But We are aware of your loyalty to Our Honored Mother, and We would enjoy the comfort of sharing such a worthy gift. Solicitor Tarin, if you will."
The solicitor stepped forward, and if Taro wasn't panicking, I was. He couldn't swear an oath of loyalty as Source Shintaro Karish. It was essential that a Source remained above and beyond influence. If people thought a Source's skills could be controlled by anyone outside the Triple S, the results would be . . . well, unimaginable by me right then, but definitely negative. One of the reasons everyone was expected to help support members of the Triple S was to avoid the perception that anyone owned us.
"If you would repeat after me, Source Karish . . ."
And yet, Taro could hardly refuse to give whatever oath the Emperor might demand, especially in front of all these people. Those repercussions would be highly negative as well. No doubt that was one of the reasons the Emperor chose to do it at this time. And damn him for doing this. What was he playing at?
"I, Shintaro Ivor Cear Karish"—and the solicitor thought Taro was an idiot, because he paused there so Taro could repeat his own name—"do swear personal loyalty to you, the Emperor." Taro hesitated just long enough to be noticeable to everyone, and when he did speak the words, it was in a voice so low I couldn't hear them. But I didn't doubt they had been said. I was sure the Emperor wouldn't feel too ashamed to berate Taro, loudly and at length, should my Source fail to give the oath.
"Putting my services to the use of the Emperor . . ."
That got a reaction from the crowd, those that understood the significance of what was going on. Damn it, damn it, damn it.
". . . and his heirs in perpetuity . . ."
Once Taro was allowed to stop talking, the Emperor gave no responding oath of protection. Wonderful. So Taro was expected to serve the Emperor in some way and would be left hanging when things went wrong. He did say this: "As a reward for the loyalty you have demonstrated before these great people, We grant you the comfort of your true home. Flown Raven has been enduring earthquakes recently and is in need of a Pair. Henceforth, your post shall be Flown Raven. You are dismissed."
Son of a bitch!
| {
"redpajama_set_name": "RedPajamaBook"
} | 3,944 |
\section{Introduction}
The constants of Landau and Lebesgue are defined for all integers $n\ge 0$, respectively, by
\begin{align}
G(n)=\sum_{k=0}^{n}\frac{1}{16^k}\binom {2k}{k}^2\quad \mbox{and}\quad
L_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\frac{\sin\left((n+\frac 12)t\right)}{\sin(\frac t2)}\right|dt.
\end{align}
The constants $G(n)$ are important in complex analysis.
In 1913, Landau~\cite{La} proved that if $f(z)=\sum_{k=0}^{\infty}a_kz^k$ is an analytic function in the unit disc satisfying
$|f(z)|<1$ for $|z|<1$, then $\left|\sum_{k=0}^{n}a_k\right|\le G(n)$, and that this bound is optimal. Furthermore, Landau~\cite{La} showed that
\begin{align}
G(n)\sim \frac{1}{\pi}\ln n, (n\rightarrow \infty).
\label{Landau-C-1}
\end{align}
Let $\gamma$ denote the Euler-Mascheroni constant. In 1930, Watson~\cite{Wat} obtained a more precise asymptotic
formula than~\eqref{Landau-C-1}
\begin{align}
G(n)\sim \frac{1}{\pi}\ln (n+1)+c_0-
\frac{1}{4\pi (n+1)}+O\left(\frac{1}{n^2}\right),
(n\rightarrow \infty),\label{Landau-C-2}
\end{align}
where
\begin{align}
c_0=\frac{1}{\pi}(\gamma+4\ln 2)=1.0662758532089143543\cdots.
\label{c0-definition}
\end{align}
In fact, the work of Watson opened up a novel insight into the asymptotic behavior of the Landau sequences $(G(n))_{n\ge0}$. Inspired by~\eqref{Landau-C-2}, many authors investigated the
upper and lower bounds of $G(n)$. We list some main results as follows:
\begin{eqnarray}
~~&&\frac{1}{\pi}\ln (n+1)+1\le G(n)
<\frac{1}{\pi}\ln (n+1)+c_0\quad (n\ge 0),\quad\mbox{(Brutman~\cite{Br},1982)}\\
~~&&\frac{1}{\pi}\ln \left(n+\frac 34\right)+c_0<G(n)
\le\frac{1}{\pi}\ln\left(n+\frac 34\right)+1.0976\quad (n\ge 0),\quad\mbox{(Falaleev~\cite{Fal}, 1991)}\label{Landau-Falaleev}\\
~~&&\frac{1}{\pi}\ln \left(n+\frac 34\right)+c_0<G(n)
<\frac{1}{\pi}\ln\left(n+\frac 34+\frac{11}{192n}\right)+c_0\quad (n\ge 1),\quad\mbox{(Mortici~\cite{Mor4}. 2011)}\label{Landau-Mortici}
\end{eqnarray}
Recently, Chen~\cite{Ch2} found the following better
approximation for $G(n)$: as $n\rightarrow\infty$,
\begin{align}
G(n)
=&c_0+\frac{1}{\pi}\ln\left(n+\frac 34+\frac{11}{192(n+\frac 34)}-\frac{2009}{184320(n+\frac 34)^3}
+\frac{2599153}{371589(n+\frac 34)^5}\right)\\
&+O\left(\frac{1}{(n+\frac 34)^8}\right),\nonumber
\end{align}
and the better upper bound:
\begin{align}
G(n)
<c_0+\frac{1}{\pi}\ln\left(n+\frac 34+\frac{11}{192(n+\frac 34)}\right), (n\ge 0).\label{chen-1}
\end{align}
More recently, Cao, Xu and You~\cite{CXY} improved the rate of convergence to $n^{-14}$, and attained the following tight double-sides inequalities
\begin{align}
\frac{C_1}{(n+\frac 32)^6}
<G(n)-\frac{1}{\pi}\ln(n+\frac 34)-c_0-\frac{\frac{11}{192\pi}}{(n+\frac 34)^2+\frac{1541}{7040}}
<\frac{C_1}{(n+\frac 12)^6},(n\ge 0),
\end{align}
where $C_1=\frac{89684299}{18166579200\pi}$.
Another direction for developing the approximation to $G(n)$ was
initiated by Cvijovi\'c and Klinowski~\cite{CK}, who established the following estimates
of $G(n)$ in terms of the Psi(or Digamma) function $\psi(z):=\frac{\Gamma'(z)}{\Gamma(z)}$:
\begin{align}
&\frac{1}{\pi}\psi\left(n+\frac 54\right)+c_0<G(n)
<\frac{1}{\pi}\psi\left(n+\frac 54\right)+1.0725,\quad(n\ge 0),\\
&\frac{1}{\pi}\psi\left(n+\frac 32\right)+0.9883<G(n)
<\frac{1}{\pi}\psi\left(n+\frac 32\right)+c_0,\quad(n\ge 0).
\end{align}
Since then, many authors have made significant contributions to sharper
the inequalities and the asymptotic expansions for $G(n)$, see e.g.
Alzer~\cite{Al}, Chen~\cite{Ch1}, Cvijovi\'c and Srivastava~\cite{CS}, Granath~\cite{Gra}, Mortici~\cite{Mor4}, Nemes~\cite{Ne1,Ne2}, Popa~\cite{Po}, Popa and Secelean~\cite{PS}, Zhao~\cite{Zhao}, Gavrea and M. Ivan~\cite{GI}, Chen and Choi~\cite{CC1,Ch2,CC2}, etc. To the best knowledge of the authors, the latest lower and upper bounds of $G(n)$ along this research direction are due to Chen and Choi~\cite{CC2}.
In 1906, Lebesgue~\cite{Le} showed that if a function $f$ is integrable on the interval $[-\pi,\pi]$
and $S_n(f;x)$ is the $n$-th partial sum of the Fourier series of $f$, then, we have
\begin{align*}
&a_k=\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(kt)dt\quad\mbox{($k\ge 0$) and}
\quad b_k=\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin(kt)dt,\\
&S_n(f;x)=\frac{a_0}{2}+\sum_{k=1}^{n}\left(a_k \cos(kx)
+b_k \sin(kx)
\right),
\end{align*}
where the sum for $n=0$ is usually stipulated to be zero. If $|f(x)|\le 1$ for all
$x\in [-\pi,\pi]$, then
\begin{align}
\max_{x\in [-\pi,\pi]}|S_n(f;x)|\le L_n\quad (n\ge 0).
\label{Lebesgue-definition-1}
\end{align}
It is noted that $L_n$ is the smallest possible constant for which
the inequality~\eqref{Lebesgue-definition-1} holds for all
integrable functions $f$ on $[-\pi,\pi]$.
The Lebesgue constants play an important role in the theory of Fourier
series. Therefore, they have attracted much attention of several well-known mathematicians such as Fej\'er~\cite{Fe}, Gronwall~\cite{Gro}, Hardy~\cite{Ha}, Szeg\"o~\cite{Sz}, Watson~\cite{Wat}, who established some remarkable properties of these
numbers including monotonicity theorems, and various series and integral representations for $L_n$. Watson~\cite{Wat} showed
\begin{align}
L_{n/2}=\frac{4}{\pi^2}\ln (n+1)+c_1+O\left(\frac{1}{n^2}\right)\quad (n\rightarrow\infty),
\label{Lebesgue-watson}
\end{align}
where
\begin{align}
c_1=\frac{8}{\pi^2}\sum_{k=1}^{\infty}
\frac{\ln k}{4k^2-1}+\frac{4}{\pi^2}(\gamma+2\ln 2)=0.98943127383114695174\cdots.\label{c1-definition}
\end{align}
Since then, many authors have made important contributions to this research topic, see e.g. Galkin~\cite{Gal}, Wong~\cite{Wo}, Alzer~\cite{Al}, Zhao~\cite{Zhao}, Chen and Choi~\cite{CM}, etc.
Let
\begin{align}
u_n=&L_{n/2}-\left(c_1+\frac{4}{\pi^2}\psi\left(
n+\frac 32+\frac{\frac 18-\frac{\pi^2}{72}}{n+1}\right)\right),\\
v_n=&L_{n/2}-\left(c_1+\frac{4}{\pi^2}\ln\left(
n+1+\frac{a}{n+1}+\frac{c}{(n+1)^3}
\right)\right),
\end{align}
where
\begin{align*}
a=\frac{1}{6}-\frac{\pi^2}{72}\quad \mbox{and}\quad
c=-\frac{37}{360}+\frac{\pi^2}{135}+\frac{67\pi^4}{259200}.
\end{align*}
Recently, Chen and Choi~\cite{CC1} obtained
\begin{align}
\lim_{n\rightarrow\infty}n^4u_n=&\frac{-23625+
1770\pi^2+67\pi^4}{64800\pi^2},\\
\lim_{n\rightarrow\infty}n^6v_n=&-\frac{-188637120+
15135120\pi^2+308196\pi^4+7537\pi^6}{97977600\pi^2}.
\end{align}
\noindent {\bf Notation.} Throughout the paper, the notation
$P_k(x)$(or $Q_k(x)$) as usual denotes a polynomial of degree $k$
in terms of $x$. The notation $\Psi(k;x)$
means a polynomial of degree $k$ in terms of $x$ with all of its non-zero coefficients being positive, which may be different at each occurrence.
Notation $\Phi(k;x)$ denotes a polynomial of degree $k$ in terms of $x$ with
the leading coefficient being equal to one, which may be different at different subsections.
This paper is organized as follows. In Section 2, we prepare
some preliminary lemmas. In Section 3, we explain how to find
a finite continued fraction approximation by using the
\emph{multiple-correction method}. In Section 4 and Section 5,
we discuss the constants of Landau and Lebesgue, respectively.
In the last section, we consider to refine the works of Lu~\cite{Lu}
and Xu and You~\cite{XY} for the Euler-Mascheroni constant.
\section{Some Preliminary Lemmas}
The following lemma gives a method for measuring the rate of convergence, whose proof can be found in~\cite{Mor1,Mor2}.
\begin{lem}
If the sequence $(x_n)_{n\in \mathbb{N}}$ is convergent to zero and there exists the limit
\begin{align}
\lim_{n\rightarrow +\infty}n^s(x_n-x_{n+1})=l\in [-\infty,+\infty]
\label{LEM1-1}
\end{align}
with $s>1$, then
\begin{align}
\lim_{n\rightarrow +\infty}n^{s-1}x_n=\frac{l}{s-1}.\label{LEM1-2}
\end{align}
\end{lem}
In the study of Landau constants, we need to apply a so-called
Brouncker's continued fraction formula.
\begin{lem}
For all integer $n\ge 0$, we have
\begin{align}
q(n):=\left(\frac{\Gamma(n+\frac 12)}{\Gamma(n+1)}\right)^2=
\frac{4}{1+4n+\frac{1^2}{2+8n+\frac{3^2}{2+8n
+\frac{5^2}{2+8n+\ddots}}}}.\label{LEM2-1}
\end{align}
\end{lem}
In 1654 when Brouncker and Wallis collaborated on the problem of squaring the circle, Lord William Brouncker found this remarkable fraction formula. Formula~\eqref{LEM2-1} was not published by Brouncker himself, and first
appeared in~\cite{Wal}. For a general $n$, it actually follows from Entry 25 in Chapter 12 in Ramanujan's notebook~\cite{Ber}, which gives a more
general continued fraction formula for quotients of gamma functions.
Writing continued fractions in the way of~\eqref{LEM2-1} takes a lot of space and thus, we use the following shorthand notation
\begin{align}
q(n)=\frac{4}{1+4n+}\frac{1^2}{2+8n+}\frac{3^2}{2+8n+}
\frac{5^2}{2+8n+}\cdots
=\frac{4}{1+4n+}\K_{k=0}^{\infty}
\frac{(2k+1)^2}{2+8n},
\end{align}
and its $k$-th approximation $q_k(n)$ is defined by
\begin{align}
q_1(n)&=\frac{4}{1+4n},\\
q_k(n)&=\frac{4}{1+4n+}\frac{1^2}{2+8n+}\frac{3^2}{2+8n+}
\frac{(2k-3)^2}{2+8n}
=\frac{4}{1+4n+}\K_{j=0}^{k-1}
\frac{(2j-1)^2}{2+8n}
,\quad (k\ge 2).\label{qk-definition}
\end{align}
\begin{lem}
Let $c_1$ be defined by~\eqref{c1-definition}. Then,
for $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, we have
\begin{align}
&\frac{4}{\pi^2}\ln (n+1)+c_1+
\sum_{j=1}^{2N}
\frac{a_j}
{(n+1)^{2j}}\\
&<L_{n/2}
<\frac{4}{\pi^2}\ln (n+1)+c_1+
\sum_{j=1}^{2N+1}
\frac{a_j}{(n+1)^{2j}},\nonumber
\end{align}
where
\begin{align*}
a_j:=\frac{8}{\pi^2}\frac{B_{2j}}{2j}\left(2^{2j-1}-1\right)
\left(1+\sum_{k=1}^{j}\frac{(-1)^k}{(2k)!}B_{2k}\pi^{2k}\right),
\end{align*}
and the Bornoulli numbers $B_k$
is defined by
\begin{align*}
\frac{z}{e^z-1}=\sum_{k=0}^{\infty}B_k\frac{z^k}{k!}\quad (|z|<2\pi).
\end{align*}
\end{lem}
\proof This is Theorem 3.1 of Chen and Choi~\cite{CC2}, and also see~(3.8) in Chen and Choi~\cite{CC1}.\hfill$\Box$
In the proof of our inequalities for the constants of Landau and
Lebesgue, we also need to use the following simple inequality,
which is a consequence of Hermite-Hadamard inequality.
\begin{lem} Let $f$ be twice derivable, with $f''$ continuous. If $f''(x)>0$, then
\begin{align}
\int_{a}^{a+1}f(x) dx > f(a+1/2).\label{LEM3}
\end{align}
\end{lem}
\section{The multiple-correction method}
First, let us briefly review a so-called \emph{multiple-correction method} presented in our previous paper~\cite{CXY}.
Let $(v(n))_{n\ge 1}$ be a sequence to be approximated. Throughout
the paper, we always assume that the following three
conditions hold.
\noindent {{\textbf{Condition} (i)}.} The initial-correction function $\eta_0(n)$ satisfies
\begin{align*}
&\lim_{n\rightarrow\infty}\left(v(n)-\eta_0(n)\right)=0,\\
&\lim_{n\rightarrow\infty}n^{l_0}\left(v(n)-v(n+1)-\eta_0(n)
+\eta_0(n+1)
\right)=C_0\neq 0,
\end{align*}
for some positive integer $l\ge 2$.
\noindent {{\textbf{Condition} (ii)}.} The $k$-th correction function $\eta_k(n)$ has the form of $-\frac{C_{k-1}}{\Phi_k(l_{k-1};n)}$, where
\begin{align*}
\lim_{n\rightarrow\infty}n^{l_{k-1}}\left(v(n)-v(n+1)-
\sum_{j=0}^{k-1}\left(\eta_j(n)-\eta_j(n+1)\right)
\right)=C_{k-1}\neq 0,
\end{align*}
\noindent {{\textbf{Condition} (iii)}.} The difference $\left(v(1/x)-v(1/x+1)-\eta_0(1/x)
+\eta_0(1/x+1)
\right)$ is an analytic function in a neighborhood of point $x=0$.
Actually, the \emph{multiple-correction method} is a recursive algorithm. If the assertion
\begin{align*}
\lim_{n\rightarrow\infty}n^{l_{k-1}}\left(v(n)-v(n+1)-
\sum_{j=0}^{k-1}\left(\eta_j(n)-\eta_j(n+1)\right)
\right)=C_{k-1}\neq 0
\end{align*}
is true, then, it is not difficult to observe
\begin{align*}
\lim_{n\rightarrow\infty}\left(\left(v(n)-v(n+1)-
\sum_{j=0}^{k-1}\left(\eta_j(n)-\eta_j(n+1)\right)
\right)-\frac{C_{k-1}}{n^{l_{k-1}}}\right)=0.
\end{align*}
Roughly speaking, the idea of the \emph{multiple-correction method} is to use the polynomial $\Phi_k(l_{k-1};n)$ of degree $l_{k-1}$ instead of $n^{l_{k-1}}$ for improving the convergence rate. In other words, we view $n^{l_{k-1}}$ as a special polynomial of degree $l_{k-1}$ in terms of $n$.
Here, we note that the polynomial $\Phi_k(l_{k-1};n)$ contains $l_{k-1}$ undetermined
parameters $a_j$ $(0\le j\le l_{k-1}-1)$, and hence, in some cases we hope that we can attain ``more gains".
The initial-correction is a very important step. With this, we hope to further develop the above method starting from the second-correction. To find
the proper structure of finite continued fraction, we must try many times
by using $-\frac{C_0}{\Phi_1(l_{1};n)+\frac{b_j}{n^j}}$ instead of $-\frac{C_0}{\Phi_1(l_{1};n)}$, where $j$ is a positive integer.
To do that, we need to begin from $j=1$ and try step by step. Once we have found that the convergence rate can be improved for the first positive integer, say $j_0$, we use $\Phi(j_0; n)$ to replace $n^{j_0}$ immediately, and then, determine all the corresponding coefficients of the polynomial $\Phi(j_0; n)$. We continue this process until the desired structure of finite continued fraction is found.
It is for this reason that we call it as the \emph{multiple-correction method}.
In addition, to determine all the related coefficients, we often use an appropriate symbolic computation software, which needs a huge of computations. On the other hand, the exact expressions at each occurrence also takes a lot of space. Hence, in this paper we omit some related details for space limitation. For interesting readers, see our previous paper~\cite{CXY}.
It is a natural question whether or not \emph{multiple-correction method} can be used to accelerate convergence in some BBP-type or Ramanujan-type series, we hope to return to this topic elsewhere.
\section{The Landau Constants}
\begin{thm} Let sequences ${\rm MC}_k(n)$ be defined as follows:
\begin{align}
\mathrm{MC}_0(n): =&\frac{1}{\pi}\ln\left(n+\frac 34\right)+c_0,\\
\mathrm{MC}_1(n):=&\frac{1}{\pi}\frac{\kappa_1}{(n+\frac 34)^2+\lambda_1},
\\
\mathrm{MC}_k(n):=&\frac{1}{\pi}\frac{\kappa_1}{(n+\frac 34)^2+\lambda_1+}
\K_{j=2}^{k}
\frac{\kappa_j}{(n+\frac 34)^2+\lambda_j},
(k\ge 2),\label{LMC-definition}
\end{align}
where $c_0$ is determined by~\eqref{c0-definition} and
\begin{align*}
\kappa_1=&\frac{11}{192},&\lambda_1=\frac{1541}{7040},\\
\kappa_2=&-\frac{89684299}{1040793600},&\lambda_2=\frac{815593360691}
{631377464960},\\
\kappa_3=&-\frac{791896453750695892475}{691850212268234428416},
&\lambda_3=\frac{79124827964452580408836456738931}
{23635681749960244849264556808320}.
\end{align*}
If we let the $k$-th correction error term $E_k(n)$ be denoted by
\begin{align}
E_k(n):=G(n)-\mathrm{MC}_0(n)-\mathrm{MC}_k(n),\label{LE-definition}
\end{align}
then, for all positive integer $k$, we have
\begin{align}
&\lim_{n\rightarrow\infty}n^{4k+3}\left(E_k(n)-E_k(n+1)\right)
=(4k+2)C_k,\\
&\lim_{n\rightarrow\infty}n^{4k+2}E_k(n)=C_k,\label{LMC-result}
\end{align}
where
\begin{eqnarray*}
C_1&=&\frac{89684299}{18166579200\pi},\\
C_2&=&\frac{31675858150027835699}{5605686531912433139712\pi},\\
C_3&=&\frac{9662255454831353335643376823083291821}
{310776980771128296411407710029663436800\pi}.
\end{eqnarray*}
\end{thm}
\begin{rem}
Theorem 1 tells us that it could be possible for us to find a simpler asymptotic expansion than Theorem 2.1 of Chen and Choi~\cite{CC1} for the Landau constants.
\end{rem}
\proof Let us consider the initial-correction.
\noindent{\bf (Step 1) The initial-correction.}
Motivated by inequalities \eqref{Landau-Falaleev} and \eqref{Landau-Mortici},
we choose $\mathrm{MC}_0(n)=\frac{1}{\pi}\ln(n+\frac 34)+c_0$, and define
\begin{align}
E_0(n)=G(n)-\mathrm{MC}_0(n)=G(n)-\frac{1}{\pi}\ln(n+\frac 34)-c_0.
\label{L-E0-definition}
\end{align}
Then, it follows
immediately from \eqref{L-E0-definition}
\begin{align}
E_0(n)-E_0(n+1)=G(n)-G(n+1)-\frac{1}{\pi}\ln(n+\frac 34)+
\frac{1}{\pi}\ln(n+\frac 74).\label{L-E0-difference}
\end{align}
Now, by using the duplication formula (Legendre, 1809)
\begin{align}
2^{2z-1}\Gamma(z)\Gamma(z+\frac 12)=\sqrt\pi\Gamma(2z),
\label{duplication formula}
\end{align}
one can prove
\begin{align}
G(n)-G(n-1)=\frac{(\Gamma(2n+1))^2}{16^2(\Gamma(n+1))^4}
=\left(\frac{(2n)!}{4^n(n!)^2}\right)^2=\frac{1}{\pi}q(n),
\label{G-difference-def}
\end{align}
where $q(n)$ is defined by~\eqref{LEM2-1}. Also see Page 739 in Granath~\cite{Gra} or Page 306 in Chen~\cite{Ch2}.
Combining~\eqref{L-E0-difference} with \eqref{G-difference-def} yields
\begin{align}
E_0(n)-E_0(n+1)=-\frac{1}{\pi}q(n+1)-\frac{1}{\pi}\ln(n+\frac 34)+
\frac{1}{\pi}\ln(n+\frac 74).\label{L-E0-difference-1}
\end{align}
On one hand, by utilizing Lemma 2 and~\eqref{qk-definition}, we can obtain that for all positive integer $j$,
\begin{align}
q_2(n)<q_4(n)<\cdots<q_{2j}(n)<q(n)
<q_{2j+1}(n)<\cdots<q_3(n)<q_1(n).\label{Landau-1}
\end{align}
On the other hand, by using \emph{Mathematica} software, we can attain
\begin{align}
q_9(n)-q_8(n)=O\left(\frac{1}{n^{17}}\right).\label{Landau-1-2}
\end{align}
Now, combining \eqref{Landau-1} and \eqref{Landau-1-2} gives us
\begin{align}
q(n+1)=q_{8}(n+1)+O\left(\frac{1}{n^{16}}\right).
\label{Landau-1-3}
\end{align}
Again, by making use of the \emph{Mathematica} software, we can
expand $q_{8}(n+1)$ into a power series in terms of $n^{-1}$ so that
\begin{align}
q(n+1)=&q_{8}(n+1)+O\left(\frac{1}{n^{16}}
\right)\label{q8-approximation}\\
=&\frac{1}{n}-\frac 54\frac{1}{n^2}+\frac{49}{32}\frac{1}{n^3}
-\frac{235}{128}\frac{1}{n^4}+\frac{4411}{2048}\frac{1}{n^5}
-\frac{20275}{8192}\frac{1}{n^6}
+\frac{183077}{65536}\frac{1}{n^7}\nonumber\\
&-\frac{815195}{262144}\frac{1}{n^8}+
\frac{28754131}{8388608}\frac{1}{n^9}
-\frac{125799895}{33554432}\frac{1}{n^{10}}
+\frac{1091975567}{268435456}\frac{1}{n^{11}}\nonumber\\
&-\frac{4702048685}{1073741824}\frac{1}{n^{12}}
+\frac{80679143663}{17179869184}\frac{1}{n^{13}}-
\frac{346250976095}{68719476736}\frac{1}{n^{14}}\nonumber\\
&+\frac{2947620308941}{549755813888}\frac{1}{n^{15}}
+O\left(\frac{1}{n^{16}}\right).\nonumber
\end{align}
In addition, it is not difficult to obtain
\begin{align}
-\ln(n+\frac 34)+
\ln(n+\frac 74)=\frac{1}{n}-\frac{5}{4}\frac{1}{n^2}
+\frac{79}{48}\frac{1}{n^3}+O\left(\frac{1}{n^{4}}\right).
\label{Landau-1-4}
\end{align}
Inserting \eqref{q8-approximation} and \eqref{Landau-1-4} into
\eqref{L-E0-difference-1} results in
\begin{align}
E_0(n)-E_0(n+1)=\frac{11}{96\pi}\frac{1}{n^3}
+O\left(\frac{1}{n^{4}}\right).\label{Landau-1-5}
\end{align}
Note that the inequalities \eqref{Landau-Mortici} implies $E_0(\infty)=0$. By Lemma 1 again, we obtain
\begin{align}
\lim_{n\rightarrow\infty}n^2E_0(n)=\frac{11}{192\pi}=C_0=\kappa_1.
\label{Landau-1-6}
\end{align}
\noindent{\bf (Step 2) The first-correction.} For simplicity, let
\begin{align}
\mathrm{MC}_1(n)=\frac{1}{\pi}\frac{\kappa_1}{\Phi_1(2;n)}
=\frac{1}{\pi}\frac{\kappa_1}{(n+\frac 34)^2+\lambda_1},\label{Landau2-1}
\end{align}
and define
\begin{align}
E_1(n):=G(n)-\mathrm{MC}_0(n)-\mathrm{MC}_1(n)
=E_0(n)-\mathrm{MC}_1(n).\label{Landau2-2}
\end{align}
Combining~\eqref{L-E0-difference-1}, \eqref{Landau-1-3} and~\eqref{Landau2-2}, we can obtain
\begin{align}
E_1(n)-E_1(n+1)=&\left(E_0(n)-E_0(n+1)\right)
-\left(\mathrm{MC}_1(n)-\mathrm{MC}_1(n+1)\right)\label{Landau2-3}\\
=&-\frac{1}{\pi}q_8(n+1)-\frac{1}{\pi}\ln(n+\frac 34)+
\frac{1}{\pi}\ln(n+\frac 74)\nonumber\\
&-\mathrm{MC}_1(n)+\mathrm{MC}_1(n+1)
+O\left(\frac{1}{n^{16}}\right).\nonumber
\end{align}
By taking advantage of formulae~\eqref{q8-approximation}
and \eqref{Landau2-1}, and~\emph{Mathematica} software, we expand $E_1(n)-E_1(n+1)$ into power series in terms of $n^{-1}$:
\begin{align}
\pi\left(E_1(n)-E_1(n+1)\right)=
&\frac{-\frac{1541}{30720}+\frac{11\lambda_1}{48}}{n^5}
+\frac{\frac{7705}{24576}-\frac{275 \lambda_1}{192}}{n^6}
\label{Landau2-4}\\
&+\frac{-\frac{183077}{65536}+\frac{275463+973280\lambda_1-59136 \lambda_1^2}{172032}}{n^7}+O\left(\frac{1}{n^{8}}\right).
\nonumber
\end{align}
By Lemma 1, the fastest sequence $(E_1(n))_{n\ge 1}$ is obtained when the first
coefficient of this power series vanish. In this case
\begin{align}
\lambda_1=\frac{1541}{7040},\label{Landau2-5}
\end{align}
and thus,
\begin{align*}
E_1(n)-E_1(n+1)=\frac{89684299}{3027763200\pi}\frac{1}{n^7}+
O\left(\frac{1}{n^{8}}\right).
\end{align*}
Now, by Lemma 1 again, we attain
\begin{align}
\lim_{n\rightarrow\infty}n^6E_1(n)=
\frac{89684299}{18166579200\pi}=C_1.
\label{Landau2-8}
\end{align}
\noindent{\bf (Step 3) The second-correction.} Let
\begin{align}
\mathrm{MC}_2(n)=\frac{1}{\pi}\frac{\kappa_1}{(n+\frac 34)^2+\lambda_1+}\frac{\kappa_2}{(n+\frac 34)^2+\lambda_2},\label{Landau3-1}
\end{align}
and define
\begin{align}
E_2(n)=G(n)-\mathrm{MC}_0(n)-\mathrm{MC}_2(n).\label{Landau3-2}
\end{align}
Following the way similar to the proof of~\eqref{Landau2-3}, we can obtain
\begin{align}
E_2(n)-E_2(n+1)=&-\frac{1}{\pi}q_8(n+1)-\frac{1}{\pi}\ln(n+\frac 34)+
\frac{1}{\pi}\ln(n+\frac 74)\label{Landau3-4}\\
&-\mathrm{MC}_2(n)+\mathrm{MC}_2(n+1)
+O\left(\frac{1}{n^{16}}\right).\nonumber
\end{align}
By utilizing~\emph{Mathematica} software, $E_2(n)-E_2(n+1)$ can be expanded into power series in terms of $n^{-1}$
\begin{align}
\pi\left(E_2(n)-E_2(n+1)\right)=&\frac{\frac{89684299}{3027763200}+
\frac{11\kappa_2}{32}}{n^7}
-\frac{\frac{89684299}{346030080}+\frac{385\kappa_2}{128}}{n^8}
\label{Landau3-5}\\
&+\frac{\frac{2961426180353}{2283798528000}+\frac{120119\kappa_2}{7680}
-\frac{11\kappa_2\lambda_2}{24}}{n^9}\nonumber\\
&+\frac{-\frac{988371602353}{203004313600}
-\frac{129357\kappa_2}{2048}
+\frac{165\kappa_2\lambda_2}{
32}}{n^{10}}+O\left(\frac{1}{n^{12}}\right)\nonumber\\
&+\frac{\frac{65353200785578639}{4287451103232000}+\frac{6308113241
\kappa_2}{28835840}-\frac{207019\kappa_2\lambda_2}{6144}+
\frac{55\kappa_2\lambda_2^2}{96}-\frac{55\kappa_2^2}{96}}
{n^{11}}.\nonumber
\end{align}
The fastest sequence $(E_2(n))_{n\ge 1}$ is obtained by enforcing the first
four coefficients of this power series to be zeros. In this case
\begin{align}
\kappa_2=-\frac{89684299}{1040793600}\quad\mbox{and}\quad
\lambda_2=\frac{815593360691}{631377464960},\label{Landau3-6}
\end{align}
and hence,
\begin{align*}
E_2(n)-E_2(n+1)=\frac{158379290750139178495}
{2802843265956216569856\pi}\frac{1}{n^{11}}+
O\left(\frac{1}{n^{12}}\right).
\end{align*}
Combining this with Lemma 1 leads to
\begin{align}
\lim_{n\rightarrow\infty}n^{10}E_2(n)=
\frac{31675858150027835699}{5605686531912433139712\pi}:=C_2.
\label{Landau3-7}
\end{align}
\noindent{\bf (Step 4) The third-correction.} Let
\begin{align}
\mathrm{MC}_3(n):=\frac{1}{\pi}\frac{\kappa_1}{(n+\frac 34)^2+\lambda_1+}
\frac{\kappa_2}{(n+\frac 34)^2+\lambda_2+}
\frac{\kappa_3}{(n+\frac 34)^2+\lambda_3},
\label{Landau4-1}
\end{align}
If we define
\begin{align}
E_3(n)=G(n)-\mathrm{MC}_0(n)-\mathrm{MC}_3(n).\label{Landau4-2}
\end{align}
then, by using the same approach as Step~3, we can prove
\begin{align}
E_3(n)-E_3(n+1)=&-\frac{1}{\pi}q_8(n+1)-\frac{1}{\pi}\ln(n+\frac 34)+
\frac{1}{\pi}\ln(n+\frac 74)\label{Landau4-3}\\
&-\mathrm{MC}_3(n)+\mathrm{MC}_3(n+1)
+O\left(\frac{1}{n^{16}}\right),\nonumber
\end{align}
and thus, find
\begin{align*}
\kappa_3=-\frac{791896453750695892475}{691850212268234428416},
\lambda_3=\frac{79124827964452580408836456738931}
{23635681749960244849264556808320}.
\end{align*}
Similarly, using the~\emph{Mathematica} software can produce
\begin{align*}
E_3(n)-E_3(n+1)=\frac{67635788183819473349503637761583042747}
{155388490385564148205703855014831718400}\frac{1}{n^{15}}
+O\left(\frac{1}{n^{16}}\right).
\end{align*}
Finally, by Lemma 1 we have
\begin{align}
\lim_{n\rightarrow\infty}n^{14}E_3(n)=
\frac{9662255454831353335643376823083291821}{
310776980771128296411407710029663436800\pi}:=C_3.
\end{align}
This completes the proof of Theorem 1.~\hfill\qed
\begin{thm} Let $\mathrm{MC}_2(n)$ be defined in Theorem~1. Then, for all integer $n\ge 0$, we have
\begin{align}
\frac{C_2}{(n+\frac 74)^{10}}
<G(n)-\frac{1}{\pi}\ln\left(n+\frac 34\right)-c_0-\mathrm{MC}_2(n)
<\frac{C_2}{(n+\frac 34)^{10}},\label{theorem2}
\end{align}
where $C_2=\frac{31675858150027835699}{5605686531912433139712\pi}
$.
\end{thm}
\begin{rem}
In fact, Theorem 2 implies that
$E_2(n)$ is a strictly decreasing function of $n$. In addition, it should be possible to establish many these types of inequalities by using the same method of Theorem 2.
\end{rem}
\proof
First, it is not difficult to verify that~\eqref{theorem2}
is true for $n=0$. Hence, in the following we only need to prove that~\eqref{theorem2} holds for $n\ge 1$. For notational simplicity, we let
$D_2=\frac{158379290750139178495}{254803933268746960896}$, and
\begin{align}
E_2(n)=G(n)-\frac{1}{\pi}\ln(n+\frac 34)-c_0-\mathrm{MC}_2(n).\label{Landau42-1}
\end{align}
Then, it follows from \eqref{G-difference-def}
\begin{align}
E_2(n)-E_2(n+1)=&-\frac{1}{\pi}q(n+1)
-\frac{1}{\pi}\ln(n+\frac 34)-\mathrm{MC}_2(n)\label{Landau42-2}\\
&+\frac{1}{\pi}\ln(n+\frac 74)
+\mathrm{MC}_2(n+1).\nonumber
\end{align}
If we let
\begin{align}
U(x)=&-\frac{1}{\pi}q_8(x+1)-\frac{1}{\pi}\ln(x+\frac 34)-\mathrm{MC}_2(x)
+\frac{1}{\pi}\ln(x+\frac 74)
+\mathrm{MC}_2(x+1),\label{Landau42-3}\\
V(x)=&-\frac{1}{\pi}q_7(x+1)-\frac{1}{\pi}\ln(x+\frac 34)-\mathrm{MC}_2(x)
+\frac{1}{\pi}\ln(x+\frac 74)
+\mathrm{MC}_2(x+1),\label{Landau42-4}
\end{align}
then, combining~\eqref{Landau-1} and~\eqref{Landau42-2}-\eqref{Landau42-4} yields
\begin{align}
V(n)<E_2(n)-E_2(n+1)<U(n).\label{Landau42-5}
\end{align}
In the following, we establish the lower bound of $V(n)$ and
the upper bound of $U(n)$. First,
by using the \emph{Mathematica} software, we can obtain
\begin{align}
-U'(x)-\frac{D_2}{\pi(x+\frac 54)^{12}}=-\frac{1}{\pi}
\frac{\Psi_1(32;n)}{75937489649280 (3+4n)(5+4n)^{12}(7+4 n)\Psi_2(32;n)}<0.
\label{Landau42-6}
\end{align}
Noticing $U(+\infty)=0$, and utilizing \eqref{Landau42-6} and Lemma 4, we have
\begin{align}
U(n)&=\int_{n}^{\infty}-U'(x)dx<\int_{n}^{\infty}\frac{D_2}
{\pi(x+\frac 54)^{12}}dx
=\frac{D_2}{11\pi}\frac{1}{(n+\frac 54)^{11}}\label{Landau42-7}\\
&<\frac{D_2}{11\pi}\int_{n+\frac 34}^{n+\frac 74}\frac{1}{x^{11}}dx.\nonumber
\end{align}
Similarly, we can attain
\begin{align}
-V'(x)-\frac{D_2}{\pi(x+\frac 74)^{12}}=\frac{1}{\pi}\frac{\Psi_3(30;n)}{75937489649280(3+4 n)(5+4 n)^2(7+4n)^{12}\Psi_4(28;n)}>0.
\label{Landau42-8}
\end{align}
Therefore, integrating~\eqref{Landau42-8} with $V(+\infty)=0$ results in
\begin{align}
V(n)&=\int_{n}^{\infty}-V'(x)dx>\int_{n}^{\infty}
\frac{D_2}{\pi(x+\frac 74)^{12}}dx
=\frac{D_2}{11\pi}\frac{1}{(n+\frac 74)^{11}}\label{Landau42-9}\\
&>\frac{D_2}{11\pi}\int_{n+\frac {7}{4}}^{n+\frac {11}{4}}\frac{1}{x^{11}}dx,\nonumber
\end{align}
which, along with $E_2(\infty)=0$ and~\eqref{Landau42-5} gives us
\begin{align}
E_2(n)=&\sum_{m=n}^{\infty}\left(E_2(m)
-E_2(m+1)\right)
>\sum_{m=n}^{\infty}\frac{D_2}{11\pi}\int_{m+\frac 74}^{m+\frac {11}{4}}\frac{1}{x^{11}}dx\label{Landau42-11}\\
=&\frac{D_2}{11\pi}\int_{n+\frac 74}^{\infty}\frac{1}{x^{11}}dx=\frac{D_2}{110\pi}\frac{1}{(n+\frac 74)^{10}}.
\nonumber
\end{align}
Similarly, combining~\eqref{Landau42-7} with~\eqref{Landau42-5} yields
\begin{align}
E_2(n)=&\sum_{m=n}^{\infty}\left(E_2(m)
-E_2(m+1)\right)
<\sum_{m=n}^{\infty}\frac{D_2}{11\pi}\int_{m+\frac 34}^{m+\frac 74}\frac{1}{x^{11}}dx\label{Landau42-10}\\
=&\frac{D_2}{11\pi}\int_{n+\frac 34}^{\infty}\frac{1}{x^{11}}dx
=\frac{D_2}{110\pi}\frac{1}{(n+\frac 34)^{10}}.\nonumber
\end{align}
This finishes the proof of Theorem 2.~\hfill\qed
\section{The Lebesgue constants}
For the Lebesgue constants, we will prove the following hybrid-type finite continued fraction approximations, which has a structure similar to that of the Landau constants.
\begin{thm} Let the initial-correction function be given by $\mathrm{MC}_0(n)=\frac{4}{\pi^2}\ln (n+1)+c_1$, where $c_1$ is defined by~\eqref{c1-definition}. If we let the $k$-th correction
function $\mathrm{MC}_k(n)$ for $k\ge 1$ be defined by
\begin{align}
\mathrm{MC}_1(n):=&\frac{\rho_1}{(n+1)^2+\varrho_1},\\
\mathrm{MC}_k(n):=&\frac{\rho_1}{(n+1)^2+\varrho_1+}
\K_{j=2}^{k}\frac{\rho_j}{(n+1)^2+\varrho_j},
(k\ge 2).\label{LMC-definition}
\end{align}
where
\begin{align*}
\rho_1=&\frac{12-\pi^2}{18 \pi^2},\\
\varrho_1=&\frac{7(-720+60\pi^2+\pi^4)}
{600(-12+\pi^2)},\\
\rho_2=&-\frac{7515244800-1252540800\pi^2
+46937520\pi^4+65640 \pi^6
+23797\pi^8}{52920000(-12+\pi^2)^2},\\
\varrho_2=&7(-36262162944000+9065540736000\pi^2
-720128102400\pi^4+16206350400\pi^6\\
&+117169920\pi^8+288540\pi^{10}+230953\pi^{12})/
(600(-90182937600+22545734400\pi^2\\
&-1815791040\pi^4+46149840\pi^6-219924\pi^8
+23797 \pi^{10}))
\end{align*}
and the corresponding $k$-th correction error term $E_k(n)$ be defined by
\begin{align}
E_k(n):=L_{n/2}-\mathrm{MC}_0(n)-\mathrm{MC}_k(n),
\label{Lebesgue-E-definition}
\end{align}
then, for all positive integer $k$, we have
\begin{align}
&\lim_{n\rightarrow\infty}n^{4k+3}\left(E_k(n)
-E_k(n+1)\right)
=(4k+2)C_k,\\
&\lim_{n\rightarrow\infty}n^{4k+2}E_k(n)=C_k,
\label{LMC-result}
\end{align}
where
\begin{align*}
C_1=&\frac{-7515244800 + 1252540800\pi^2-46937520\pi^4-65640\pi^6-23797\pi^8}
{952560000\pi^2(-12+\pi^2)},\\
C_2=&(7633889107527073628160000
- 1908472276881768407040000\pi^2
+146687085183488661504000\pi^4\\
&-3184401328004768256000\pi^6+50811629937851059200\pi^8
-5860796365392595200\pi^{10}\\
&+73433337261096960\pi^{12}-2698623258901920\pi^{14}
-13989723377364\pi^{16}-552278517605\pi^{18})\\
&/(97592743987200\pi^2(7515244800-1252540800\pi^2
+46937520\pi^4+65640\pi^6+23797\pi^8)).
\end{align*}
\end{thm}
\proof Since the proof of Theorem 3 is very similar to that of Theorem 1, we only outline the idea of the proof here. First, we recall that
\begin{align}
E_k(n):=L_{n/2}-\mathrm{MC}_0(n)-\mathrm{MC}_k(n).
\end{align}
For every positive integer $M$, we let
\begin{align}
W_M(n):=\sum_{j=1}^{M}\frac{a_j}{(n+1)^{2j}},
\end{align}
where $a_j$ is given in Lemma 3. It is not hard to see that $a_j>0$ for odd $j=1,3,\cdots$, and $a_j<0$ for even $j=2,4,\cdots$. It follows easily from Lemma 3 and (5.6) that
\begin{align}
E_k(n)=&W_{2k+1}(n)-\mathrm{MC}_k(n)
+O\left(n^{4k+4}\right),\\
E_k(n)-E_k(n+1)=&W_{2k+1}(n)-\mathrm{MC}_k(n)-W_{2k+1}(n+1)+
\mathrm{MC}_k(n+1)
+O\left(n^{4k+4}\right).
\end{align}
Hence, it suffices for us to approximate $W_{2k+1}(n)$. Similar to the proof of Theorem~1, we expand $ W_{2k+1}(n)-\mathrm{MC}_k(n)-W_{2k+1}(n+1)+
\mathrm{MC}_k(n+1)$ into a power series in terms of $n^{-1}$, and then check \eqref{LMC-result} holds. \qed
The main purpose of this section is to prove the following theorem,
which corresponds to inequalities (1.10) in the case of Landau constants.
\begin{thm} Let $\mathrm{MC}_1(n)$ be defined in Theorem 3. Then, for all integer $n\ge 0$, we have
\begin{align}
\frac{C_1}{(n+\frac {13}{8})^6}<L_{n/2}-\frac{4}{\pi^2}\ln (n+1)-c_1-\mathrm{MC}_1(n)<\frac{C_1}{(n+\frac 58)^6},
\label{Theorem 4}
\end{align}
where $C_1=
\frac{-7515244800+1252540800\pi^2-46937520\pi^4-
65640\pi^6-23797\pi^8}{952560000\pi^2(-12+\pi^2)}>0.$
\end{thm}
\begin{rem}
In fact, Theorem 4 implies that
$E_1(n)$ is a strictly decreasing function of $n$.
\end{rem}
\proof First, since $L_{n/2}=1$ for $n=0$, one may verify that \eqref{Theorem 4}
is true for $n=0$. When $n=1$, it follows from Lemma 3 and (5.7)
that
\begin{align*}
\mathrm{MC}_0(n)+W_4(n)<L_{n/2}<\mathrm{MC}_0(n)+W_3(n),
\end{align*}
it is not difficult to verify that \eqref{Theorem 4}
is also true for $n=1$. Hence, in the following we only need to prove that \eqref{Theorem 4} holds for $n\ge 2$. By \eqref{Lebesgue-E-definition} and Lemma 3 we have
\begin{align}
&E_1(n)-E_1(n+1)\label{L-E-Difference-ub}\\
=&\left(L_{n/2}-\mathrm{MC}_0(n)\right)
-\mathrm{MC}_1(n)-
\left(L_{(n+1)/2}-\mathrm{MC}_0(n+1)\right)+\mathrm{MC}_1(n+1)
\nonumber\\
<& W_3(n)-\mathrm{MC}_1(n)-W_4(n+1)+\mathrm{MC}_1(n+1)\nonumber\\
=&\left(W_3(n)-\mathrm{MC}_1(n)-W_3(n+1)+\mathrm{MC}_1(n+1)\right)
-\frac{a_4}{(n+2)^8}.\nonumber
\end{align}
Similarly, we also have
\begin{align}
E_1(n)-E_1(n+1)>\left(W_3(n)-\mathrm{MC}_1(n)-W_3(n+1)
+\mathrm{MC}_1(n+1)\right)
+\frac{a_4}{(n+1)^8}.\label{L-E-Difference-lb}
\end{align}
For notational simplicity, we let $D_1=42C_1$. Now we define for $x\ge 1$
\begin{align}
F(x):=&W_3(x)-\mathrm{MC}_1(x)-W_3(x+1)
+\mathrm{MC}_1(x+1),\label{F(x)-Def}\\
U(x):=&\frac{D_1}{(x+\frac 54)^8},\quad V(x):=\frac{D_1}{(x+\frac 32)^8}.
\end{align}
In the following, we establish the upper bound and lower bounds of $F(x)$, respectively. By using \emph{Mathematica} software, one can check
\begin{align}
-F'(x)-U(x)=\frac{P_1(21;x)}
{19845000\pi^2(-12+\pi^2)(1+x)^7(2+x)^7(5+4x)^8\Psi_1(4;x)\Psi_2(4;x)},
\end{align}
where polynomial $P_1(21;x)$ may be expressed as
\begin{align}
P_1(21;x)=(x-1)\left(\frac{b_{-1}}{x-1}+b_0+b_1x
+\cdots+b_{20}x^{20}\right).
\end{align}
By using \emph{Mathematica} software again, it is not hard to verify that all coefficients $b_j(-1\le j\le 20)$ are positive. Thus, the inequality $P_1(21;x)>0$ holds for $x\ge 1$. Noticing that $-12+\pi^2<0$, one obtains by Lemma 4
\begin{align}
-F'(x)<&U(x),\quad x\ge 1,\\
F(n)=&\int_{n}^{+\infty}-F'(x)dx\le \int_{n}^{+\infty}U(x)dx
=\frac{D_1}{7}\frac{1}{(n+\frac 54)^7}\le
\frac{D_1}{7}\int_{n+\frac 34}^{n+\frac 74}\frac{dx}{x^{7}}.
\label{F(n)-upper bound}
\end{align}
On the other hand, we can prove by using \emph{Mathematica} software \begin{align}
-F'(x)-V(x)=\frac{P_2(20;x)}{19845000\pi^2(-12+\pi^2)(1+x)^7)(2+x)^7(3+
2x)^8 \Psi_1(4;x)\Psi_2(4;x)},
\end{align}
where
\begin{align}
P_2(20;x)=d_0+d_1x
+\cdots+d_{20}x^{20},
\end{align}
and all coefficients $d_j(0\le j\le 20)$ are negative. Thus, this yields
\begin{align}
-F'(x)>& V(x),\quad x\ge 1,\\
F(n)=&\int_{n}^{+\infty}-F'(x)dx\ge \int_{n}^{+\infty}V(x)dx
=\frac{D_1}{7}\frac{1}{(n+\frac 32)^7}\ge
\frac{D_1}{7}\int_{n+\frac 32}^{n+\frac 52}\frac{dx}{x^{7}}.
\label{F(n)-lower bound}
\end{align}
Combining \eqref{L-E-Difference-ub}, \eqref{L-E-Difference-lb}, \eqref{F(x)-Def}, \eqref{F(n)-upper bound} and \eqref{F(n)-lower bound} gives
\begin{align}
\frac{D_1}{7}\int_{n+\frac 32}^{n+\frac 52}\frac{dx}{x^{7}}+\frac{a_4}{(n+1)^8}
<E_1(n)-E_1(n+1)<\frac{D_1}{7}\int_{n+\frac 34}^{n+\frac 74}\frac{dx}{x^{7}}-\frac{a_4}{(n+2)^8}.
\end{align}
By adding the estimates from $n$ to $\infty$ and noticing $E_1(\infty)=0$, we attain
\begin{align}
\frac{D_1}{7}\int_{n+\frac 32}^{\infty}\frac{dx}{x^{7}}
+a_4\sum_{m=n}^{\infty}\frac{1}{(m+1)^8}<E_1(n)<
\frac{D_1}{7}\int_{n+\frac 34}^{\infty}\frac{dx}{x^{7}}
-a_4\sum_{m=n}^{\infty}\frac{1}{(m+2)^8}.
\label{L-E_1(n)-double bounds}
\end{align}
By Lemma 4 again, one has
\begin{align}
\sum_{m=n}^{\infty}\frac{1}{(m+2)^8}\le\sum_{m=n}^{\infty}
\int_{m+\frac 32}^{m+\frac 52}\frac{dx}{x^8}=\frac{1}{7(n+\frac 32)^7}.
\end{align}
On the other hand, one has the following trivial estimate
\begin{align}
\sum_{m=n}^{\infty}\frac{1}{(m+1)^8}\ge \int_{n+1}^{\infty}\frac{dx}{x^8}=\frac{1}{7(n+1)^7}.
\end{align}
Substituting the above two estimates into \eqref{L-E_1(n)-double bounds} produces
\begin{align}
\frac{C_1}{(n+\frac 32)^6}+\frac{a_4}{7(n+1)^7}
<E_1(n)<
\frac{C_1}{(n+\frac 34)^6}
-\frac{a_4}{7(n+\frac 32)^7}.\label{L-E1(n)-double bounds-1}
\end{align}
By using \emph{Mathematica} software, it is not difficult to check
\begin{align}
&\frac{C_1}{(n+\frac 58)^6}-\left(\frac{C_1}{(n+\frac 34)^6}
-\frac{a_4}{7(n+\frac 32)^7}\right)\\
=&\frac{P_3(12;n)}{29767500\pi^2(-12+\pi^2)(3+2n)^7 (3+4n)^6(5+8n)^6},\nonumber
\end{align}
where
\begin{align}
P_3(12;n)=(n-1)\left(\frac{\theta_{-1}}{n-1}+\theta_0+\theta_1n+
\cdots+\theta_{11}n^{11}\right).
\end{align}
By utilizing \emph{Mathematica} software again, we observe that all coefficients
$\theta_j (-1\le j\le 11)$ are negative. Hence
\begin{align}
\frac{C_1}{(n+\frac 58)^6}-\left(\frac{C_1}{(n+\frac 34)^6}
-\frac{a_4}{7(n+\frac 32)^7}\right)\ge 0,\quad (n\ge 1).
\end{align}
Similarly, one may check
\begin{align}
&\left(\frac{C_1}{(n+\frac 32)^6}+\frac{a_4}{7(n+1)^7}\right)-
\frac{C_1}{(n+\frac {13}{8})^6}\\
=&\frac{(n-2)\left(\frac{\vartheta_{-1}}{n-2}+\vartheta_0
+\vartheta_1n+\cdots+\vartheta_{11}n^{11}
\right)}
{3810240000\pi^2(-12+\pi^2)(1+n)^7(3+2n)^6(13+8n)^6},\nonumber
\end{align}
and all coefficients $\vartheta_j(-1\le j\le 11)$ are negative.
Thus, we obtain
\begin{align}
\left(\frac{C_1}{(n+\frac 32)^6}+\frac{a_4}{7(n+1)^7}\right)-
\frac{C_1}{(n+\frac {13}{8})^6}>0,\quad (n\ge 2).
\end{align}
Finally, Theorem 4 follows from \eqref{L-E1(n)-double bounds-1}
, (5.30) and (5.32) immediately.\qed
\section{The Euler-Mascheroni constant}
The Euler constant was first introduced by Leonhard Euler (1707-1783) in 1734 as the limit of the sequence
\begin{align}
\gamma(n):=\sum_{m=1}^{n}\frac 1m -\ln n.
\end{align}
It is also known as the Euler-Mascheroni constant. There are many famous unsolved problems about the nature of this constant. See e.g. the survey papers or books of R.P. Brent and P. Zimmermann~\cite{BZ}, Dence and Dence~\cite{DD}, Havil~\cite{Hav} and Lagarias~\cite{Lag}. For example, a long-standing open problem is whether or not it is a rational number.
In fact, the sequence $\left(\gamma(n)\right)_{n\in \mathbb{N}}$ converges very slowly toward $\gamma$, like $(2n)^{-1}$. Up to now, many authors are
preoccupied to improve its rate of convergence. See e.g.~\cite{CM,DD,De,GI,GS,Lu,Lu1,Mor1,MC} and references therein.
Let $R_1(n)=\frac{a_1}{n}$ and for $k\ge 2$
\begin{align}
R_k(n):=\frac{a_1}{n+\frac{a_2 n}{n+\frac{a_3n}{n+\frac{a_4n}{\frac{\ddots}{n+a_k}}}}
}\label{Lu-CF},
\end{align}
where
$(a_1,a_2,a_4,a_6,a_8,a_{10},a_{12})=\left(
\frac{1}{2},\frac{1}{6},\frac{3}{5},
\frac{79}{126},\frac{7230}{6241},
\frac{4146631}{3833346},
\frac{306232774533}{179081182865}\right)$,
$a_{2k+1}=-a_{2k}$ for $1\le k\le 6$, and
\begin{align}
r_k(n):=\sum_{m=1}^{n}\frac 1m -\ln n-R_k(n).
\end{align}
Lu~\cite{Lu} introduced the continued fraction method to investigate
this problem, and showed
\begin{align}
\frac{1}{120(n+1)^{4}}< r_{3}(n)-\gamma<\frac{1}{120(n-1)^{4}}.
\label{Lu-inequalities}
\end{align}
Xu and You~\cite{XY} continued Lu's work
to find $a_5,\cdots, a_{13}$ with the help of \emph{Mathematica} software, and obtained
\begin{align}
\lim_{n\rightarrow\infty}n^{k+1}\left(r_k(n)-\gamma\right)=C_k',
\label{XY-results}
\end{align}
where
$(C_1',\cdots,C_{13}')=\left(-\frac{1}{12},-\frac{1}{72},\frac{1}{120},
\frac{1}{200},-\frac{79}{25200},-\frac{6241}{3175200},
\frac{241}{105840},
\frac{58081}{22018248},-\frac{262445}{91974960},\right.$
$\left. -\frac{2755095121}{892586949408},\frac{20169451}{3821257440},
\frac{406806753641401}{45071152103463200},
-\frac{71521421431}{5152068292800}\right)$.
Hence the rate of the convergence of the sequence $\left(r_k(n)\right)_{n\in \mathbb{N}}$ is $n^{-(k+1)}$. Moreover, they improved \eqref{Lu-inequalities} to
\begin{align}
&C_{10}'\frac{1}{(n+1)^{11}}<\gamma -r_{10}(n)<C_{10}'\frac{1}{n^{11}},\\
&C_{11}'\frac{1}{(n+1)^{12}}< r_{11}(n)-\gamma<C_{11}'\frac{1}{n^{12}}.
\end{align}
The purpose of this section is to further refine the works of Lu~\cite{Lu} and Xu and You~\cite{XY} by using the \emph{multiple-correction method}, and prove the following theorem.
\begin{thm}\label{th_euler}
For every positive integer $k$, let the $k$-th correction
function $\mathrm{MC}_k(n)$ be defined by
\begin{align}
\mathrm{MC}_1(n):=&\frac{a_1}{n+b_1},\\
\mathrm{MC}_k(n):=&\frac{a_1}{n+b_1+}
\K_{j=2}^{k}\frac{a_j}{n+b_j}
,(k\ge 2),
\end{align}
where
\begin{align*}
&a_1=\frac{1}{2}, \quad &b_1=\frac{1}{6},\\
&a_2=\frac{1}{36}, \quad &b_2=\frac{13}{30},\\
&a_3=\frac{9}{25}, \quad &b_3=\frac{17}{630},\\
&a_4=\frac{6241}{15876}, \quad &b_4=\frac{417941}{786366},\\
&a_5=\frac{52272900}{38950081}, \quad &b_5=-\frac{1835967509}{23923912386},\\
&a_6=\frac{17194548650161}{14694541555716}, \quad &b_6=\frac{431312596940299603}{686480136010816290},\\
&a_7=\frac{93778512198179213368089}{32070070056327569608225}, \quad &b_7=-\frac{75178865368857369613934863 }{437108607837436422694763190},\\
&a_8=\frac{14093175882028689333655328914081}
{5957702453097198927838844740836}, \quad &b_8=\frac{152838545298199920648591716358691154137
}{212256305311307139071033336233757302422}.
\end{align*}
If we let the $k$-th correction error term $E_k(n)$ be defined by
\begin{align}
E_k(n):=\sum_{m=1}^{n}\frac 1m-\ln n-\gamma -\mathrm{MC}_k(n),
\end{align}
then, for all positive integer $k$, we have
\begin{align}
&\lim_{n\rightarrow\infty}n^{2k+2}\left(E_k(n)-E_k(n+1)\right)
=(2k+1)C_k,\\
&\lim_{n\rightarrow\infty}n^{2k+1}E_k(n)=C_k,
\end{align}
where $(C_1,C_2,C_3,\cdots,C_{6})=(-\frac{1}{72},\frac{1}{200},
-\frac{6241}{3175200},\frac{58081}{22018248},
-\frac{2755095121}{892586949408},\frac{406806753641401}
{45071152103463200})$, $C_7=-\frac{5115313723510706087761}{239581189590134660611200}$ and $C_8=\frac{26329150006913625404731665769}
{241842252367746831359300280968}$.
\end{thm}
\begin{rem}
For comparison with~\eqref{Lu-CF}, Theorem~\ref{th_euler} is more convenient for us to
find $a_k$ and $b_k$, since the parameter $a_k$ and the variable $n$ in Lu's continued faction
need to be iterated more times than ours in the recursive algorithm.
\end{rem}
\begin{rem}
It is interesting to note that we have $|C_k|<1$ for all positive integers $k~(1\le k\le 8)$, and that the correction function $\mathrm{MC}_k(n)$ for all positive integers $k$ is a rational function in the form of $\frac{P_{k-1}(n)}{Q_k(n)}$ with $P_{k-1}(x),Q_k(x)\in \mathbb{Q}[x]$. Therefore, the rate of convergence for the $(E_k(n))_{k\ge 1}$ is much faster than the geometric series if we can repeat the multiple-correction infinite times. Hence, Theorem 5 predicts that it may be possible for us to find a more rapidly or
BBP-type series expansion for the Euler-Mascheroni constant in the future.
\end{rem}
\noindent{\emph{Proof of Theorem~\ref{th_euler}}}:
This proof consists of the following steps.
\noindent{\bf (Step 1) The initial-correction.}
We choose $\mathrm{MC}_0(n)=0$, in other words, we don't need the initial-correction, and let
\begin{align}
E_0(n)=\sum_{m=1}^{n}\frac 1m -\ln n -\gamma -\mathrm{MC}_0(n)=\sum_{m=1}^{n}\frac 1m -\ln n.\label{section6-1}
\end{align}
It is not difficult to check that
\begin{align}
\lim_{n\rightarrow \infty}n^2\left(E_0(n)-E_0(n+1)\right)
=\lim_{n\rightarrow \infty}n^2\left(\ln (1+\frac 1n)-\frac{1}{n+1}\right)=\frac 12.\label{section6-2}
\end{align}
Using Lemma~1 and Noting that $E_0(\infty)=0$, we have
\begin{align}
\lim_{n\rightarrow \infty}n E_0(n)=\frac{1}{2}=:a_1=C_0.
\label{section6-3}
\end{align}
\noindent{\bf (Step 2) The first-correction.} We let
\begin{align}
\mathrm{MC}_1(n)=\frac{a_1}{\Phi_1(1;n)}
=\frac{a_1}{n+b_1},\label{section6-4}
\end{align}
and define
\begin{align}
E_1(n)=\sum_{m=1}^{n}\frac 1m -\ln n -\gamma -\mathrm{MC}_1(n).\label{section6-5}
\end{align}
By making use of \emph{Mathemaica} software, we expand the difference $E_1(n)-E_1(n+1)$ into a power series in terms of $n^{-1}$:
\begin{align}
E_1(n)-E_1(n+1)=&\ln (1+\frac 1n)-\frac{1}{n+1}-\mathrm{MC}_1(n)+\mathrm{MC}_1(n+1)\\
=&\frac{-\frac 16+b_1}{n^3}+\frac{1-6b_1-6b_1^2}{4 n^4}+
O\left(\frac{1}{n^5}\right).\nonumber
\end{align}
By Lemma 1, the fastest sequence $E_1(n)_{n\ge 1}$ is obtained by enforcing the first
coefficient of this power series to be zero. In this case, $b_1=\frac 16$ and thus,
\begin{align}
\lim_{n\rightarrow \infty}n^4\left(E_1(n)-E_1(n+1)\right)=-\frac{1}{24}.
\label{section6-6}
\end{align}
Applying Lemma 1 again yields
\begin{align}
\lim_{n\rightarrow \infty}n^{3}E_1(n)=-\frac{1}{72}:=C_1.
\label{section6-7}
\end{align}
\noindent{\bf (Step 3) The second-correction.}
We choose
\begin{align}
\mathrm{MC}_2(n)=\frac{a_1}{n+b_1+}\frac{a_2}{n+b_2}
\end{align}\label{section6-8}
and define
\begin{align}
E_2(n)=\sum_{m=1}^{n}\frac 1m -\ln n -\gamma -\mathrm{MC}_2(n).\label{section6-9}
\end{align}
Similar to the first-correction, \emph{Mathemaica} software helps us find the power series of $E_2(n)-E_2(n+1)$ in terms of $n^{-1}$:
\begin{align}
E_2(n)-E_2(n+1)=&\ln (1+\frac 1n)-\frac{1}{n+1}-\mathrm{MC}_2(n)+\mathrm{MC}_2(n+1)
\label{section6-10}\\
=&\frac{-\frac{1}{24}+\frac{3a_2}{2}}{n^4}+\frac{\frac{17}{135}-
\frac{11a_2}{3}-2a_2 b_2}{n^5}\nonumber\\
&+\frac{-641+17820a_2-6480a_2^2+15120 a_2 b_2+6480 a_2 b_2^2}{
2592 n^6}+
O\left(\frac{1}{n^7}\right).\nonumber
\end{align}
In order to obtain the fastest convergence of the sequence from~\eqref{section6-10}, we enforce
\begin{align*}
\begin{cases}
-\frac{1}{24}+\frac{3a_2}{2}=0,\\
\frac{17}{135}-
\frac{11a_2}{3}-2a_2 b_2=0,
\end{cases}
\end{align*}
which is equivalent to
\begin{align}
a_2=\frac{1}{36} \quad \mbox{and}\quad b_2=\frac{13}{30}.
\end{align}
Therefore, we attain
\begin{align}
E_2(n)-E_2(n+1)=\frac{1}{40}\frac{1}{n^6}
+O\left(\frac{1}{n^7}\right).\label{section6-11}
\end{align}
Now by Lemma 1 again, we have
\begin{align}
\lim_{n\rightarrow \infty}n^{6}E_2(n)=\frac{1}{200}:=C_2.
\label{section6-12}
\end{align}
Since the derivations from the third-correction to the eighth-correction are very similar, here we only give the proof of the eighth-correction.
\noindent{\bf (Step 9) The eighth-correction.} We let
\begin{align}
\mathrm{MC}_8(n)=\frac{a_1}{n+b_1+}
\K_{j=2}^{8}\frac{a_j}{n+b_j},
\end{align}\label{section6-13}
and define
\begin{align}
E_8(n)=\sum_{m=1}^{n}\frac 1m -\ln n -\gamma -\mathrm{MC}_8(n).\label{section6-14}
\end{align}
Again, we resort to \emph{Mathemaica} software to expand the difference $E_8(n)-E_8(n+1)$ into a power series in terms of $n^{-1}$:
\begin{align}
&E_8(n)-E_8(n+1)\label{section6-15}\\
=&\ln (1+\frac 1n)-\frac{1}{n+1}-\mathrm{MC}_8(n)+\mathrm{MC}_8(n+1)\nonumber
\\
=&\frac{-\frac{5115313723510706087761}{15972079306008977374080}+
\frac{1220420260924203 a_8}{9014230420692640}}{n^{16}}
+O\left(\frac{1}{n^{19}}\right)
\nonumber\\
&+\frac{\phi_1}{427954568312084496528235420668724754292663281100
n^{17}}\nonumber\\
&+\frac{\phi_2}{393249450106833933773320732143901514534088027635
9961361938547200
n^{18}},\nonumber
\end{align}
where
\begin{align*}
\phi_1=&1651455193723916359597152051576300956283985319207\\
&-653628703213874127417970758864749834347105473339 b_8\\
&-61802656649700006308879054758037254288686256518 a_8 b_8\\
\phi_2=&-840200304617857764114169221449745246233826647190833037
98933256161\\
&+34897177003226720866547593123620552491030169390762369825972005484 a_8\\
&+6683314557607540897917106944093661063444345868437190779381411152 a_8b_8\\
&+603401714833886490440852089976344096105815206107927672101641232
b_8^2 a_8\\
&- 603401714833886490440852089976344096105815206107927672101641232 a_8^2.
\end{align*}
By enforcing $a$ and $b$ in~\eqref{section6-15} to satisfy the following condition:
\begin{align*}
\begin{cases}
-\frac{5115313723510706087761}{15972079306008977374080}+
\frac{1220420260924203 a_8}{9014230420692640}=0,\\
\phi_1=0,
\end{cases}
\end{align*}
i.e.,
\begin{align*}
a_8=\frac{14093175882028689333655328914081}
{5957702453097198927838844740836},
b_8=\frac{152838545298199920648591716358691154137
}{212256305311307139071033336233757302422},
\end{align*}
we obtain
\begin{align}
E_8(n)-E_8(n+1)=\frac{26329150006913625404731665769}
{14226014845161578315252957704}\frac{1}{n^{18}}
+O\left(\frac{1}{n^{19}}\right).\label{section6-16}
\end{align}
Now by Lemma 1 again, we finally attain
\begin{align}
\lim_{n\rightarrow \infty}n^{17}E_2(n)=\frac{26329150006913625404731665769}
{241842252367746831359300280968}:=C_8.
\end{align}
This completes the proof of Theorem 5.~\hfill\qed
\noindent{\bf Appendix.} For the reader's convenience, here we give some more constants in our theorems. In fact, one can use \emph{Mathematica command} ``Together" and ``Coefficient" to find
more constants.
\begin{align*}
\kappa_4=&-\frac{382149699786434954423663192287642772100258949239}{
69454986539981103777883874787703791756725862400},\\
\kappa_5=&-\frac{6793276653107510374395619138801559957929538247747
9617083158574631741
016483865590781675}{
400160623295066935399490057280725508008329690840570325893115626008
0111590006888051712},\\
\lambda_4=&\frac{3047183642643398321446537081211433153790774725
879204120678621187}{
476180753216552458418280167270798333222960626510492964456863360}.\\
\rho_3=&-25(91606669290324883537920000- 30535556430108294512640000 \pi^2 \\
&+3668717299083632345088000 \pi^4-184899901119545880576000\pi^6\\
&+3794140887258980966400\pi^8-121141186322562201600\pi^{10}\\
&+6741996412525758720\pi^{12}-105816816367920000\pi^{14}\\
&+2530746578373552\pi^{16}+7362381166104 \pi^{18}+552278517605\pi^{20})/\\
&(2561328(7515244800-1252540800\pi^2+46937520\pi^4+65640\pi^6+
23797 \pi^8)^2)\\
a_9=&\frac{38559153745620009525389781729558359566448528400
}{7562099567591782725341311886983340261624011969}, \\ b_9=&-\frac{4311810252990337765692084981855831368824641381949822699}
{16443847302827668255907904514549005300064801885045499646}\\
a_{10}=&\frac{1424408165569510820157486371128758386293642530
67475021984438416009
}{35757280329598209749962500807452853821298673049007961786630549764},
\\
b_{10}=&\frac{106368952896545249534816650756049857087954719240036868
2729425454967212
48718541
}{13163895580746317309555747820198647160355807805927403216735894475
7539476827550}.
\end{align*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,012 |
Q: Use imagemagic to put several JPG on one page I have a folder full of jpg files. When I use the following command,
convert *.jpg assembly.pdf
I get a pdf file with one image per page of the pdf. I would like to put 9 images on each page. Is there a way to do this with convert or mogrify?
Alternatively, I'd like to combine 9 of the jpg images into one jpg image so that I can convert that into a PDF.
Thanks,
z.
A: You're looking for montage from ImageMagick :
NAME
montage - create a composite image by combining several sepa‐
rate images. The images are tiled on the composite image
optionally adorned with a border, frame, image name, and more.
If you already have convert then you almost certainly also have montage. To get a grid of 3x3 images per page on your pdf try this:
montage *.jpg -tile 3x3 assembly.pdf
This assumes your images all have roughly the same size. Have a read through the documentation for more complex options.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,023 |
{"url":"https:\/\/keplerlounge.com\/number-theory\/2021\/06\/09\/PNT-error.html","text":"Given that the density of the primes in $$[1,N]$$ is on the order of $$\\frac{1}{\\ln N}$$ we\u2019ll note that:\n\n$$\\text{Li}(N) = \\int_{2}^N \\frac{1}{\\ln x}dx = \\text{li}(N)-\\text{li}(2)$$\n\nwhere\n\n$$\\text{li}(N) \\sim \\frac{N}{\\ln N} \\sum_{k=0}^\\infty \\frac{k!}{(\\ln N)^k} \\sim \\frac{N}{\\ln N} + \\frac{2N}{(\\ln N)^2} + \u2026$$\n\nand therefore we obtain the asymptotic relation:\n\n$$\\text{li}(N) \\sim \\frac{N}{\\ln N} + \\mathcal{O}\\big(\\frac{N}{(\\ln N)^2}\\big)$$\n\nwhich yields the error term:\n\n$$\\lvert \\pi(N) - \\frac{N}{\\ln N} \\rvert = \\mathcal{O}\\big(\\frac{N}{(\\ln N)^2}\\big)$$\n\nwhere $$\\pi(N)$$ is the prime counting function.\n\nNote: The reflections considered here are those which have probably occurred to Gauss, before the Prime Number Theorem was proven.","date":"2021-11-28 05:27:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 4, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9550868272781372, \"perplexity\": 302.3792277743253}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358469.34\/warc\/CC-MAIN-20211128043743-20211128073743-00226.warc.gz\"}"} | null | null |
Mannschaft und Erfolge des BMC Racing Team in der Saison 2009.
Erfolge
Erfolge in der UCI America Tour
In der Saison 2009 gelangen dem Team nachstehende Erfolge in der UCI America Tour.
Erfolge in der UCI Europe Tour
In der Saison 2009 gelangen dem Team nachstehende Erfolge in der UCI Europe Tour.
Mannschaft
Kader
Zugänge – Abgänge
Weblinks
Radsportteam 2009 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 315 |
Address : Post Master, Garasni, Asop Taluk, Jodhpur District, RAJASTHAN, India.
What is Pin code of Garasni?
Garasni Pin code or Postal zip code of Garasni is an unique Post index number provided to the post office of Garasni, Jodhpur Division, Jodhpur District of Rajasthan state, India. These Pin codes are provided by India Post. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,226 |
\section{Introduction}
\label{sec1}
For the numerical simulation of solid dynamic applications undergoing large strain,
as an alternative approach to the conventional finite element method (FEM),
the meshless methods have attracted more and more interests in recent years \cite{belytschko1996meshless,liu2005introduction, liu2010smoothed}.
One notable Lagrangian meshless method is smoothed particle hydrodynamics (SPH),
which was first introduced by Lucy \cite{lucy1977numerical} and Gingold and Monaghan \cite {gingold1977smoothed} for astrophysics applications.
In SPH method, discretization operators are approximated
by using the particles within a neighborhood
with the help of a weight or smoothing kernel function.
This distinct non-local feature makes SPH method can handle
large deformations straightforwardly.
After its inception, SPH method has been extensively studied and applied
to a broad range of problems including fluid dynamics \cite{monaghan1994simulating,hu2006multi,shao2006simulation,zhang2019weakly},
solid dynamics \cite{benz1995simulations,randles1996smoothed,monaghan2000sph},
fluid-structure interactions (FSI) \cite{antoci2007,zhang2020multi}
and multi-physics applications \cite{zhang2020integrative}.
In addition,
to further improve the numerical accuracy and address
the problems of tensile instability and inconsistency
\cite{swegle1995smoothed,morris1996analysis},
many variants of SPH method have been developed.
These modifications include
the symmetric and normalization formulations \cite{monaghan1992smoothed,johnson1996artificial,randles1996smoothed},
the corrective smoothed particle method (CSPH) \cite{chen1999corrective,chen2000generalized,bonet2002simplified},
the finite particle method (FPM) \cite{liu2005modeling,liu2006restoring}
and the stress point algorithm \cite{dyka1995approach,dyka1997stress,randles2000normalized}.
More applications and their corresponding algorithms can be found
in comprehensive reviews as Refs. \cite{liu2010smoothed,monaghan2012smoothed,ye2019smoothed}.
Although the accuracy and stability are achieved by SPH methods
for the simulation of solid dynamics problems,
one has to note that obtaining static solution is very time-consuming
because SPH is a typical method with explicit time-stepping.
Therefore, it is necessary to find an efficient method to approximate
the converged static or quasi-static displacement.
Besides resorting to traditional implicit methods,
which solve the entire system with iterative solvers,
it is attractive to apply dynamic relaxation,
which was proposed for FEM method originally
\cite{otter1966dynamic,belytschko2013nonlinear,jung2018dynamic},
because its explicit iterative algorithm is more efficient
for large problems, especially in highly nonlinear cases.
Generally,
the dynamic relaxation technique can be classified into two groups,
i.e. viscous dynamic relaxation (VDR) and kinetic dynamic relaxation (KDR) \cite{rodriguez2011numerical,alamatian2012new,rezaiee2014fictitious},
according to the manner of applied damping.
In VDR, an artificial viscous damping term is added into
the equation of motion to accelerate the convergence.
One difficulty of VDR is that the solutions for nonlinear problems may be path dependent
and vary with different damping ratio as noted in Ref. \cite{lin2014efficient}.
Also, the relaxation is slow if insufficient damping ratio is applied,
on the other hand, too much damping would also hinder the system
from achieving final steady state because the damped velocity can be very small.
In KDR, damping is imposed by resetting the velocity to zero at every time step
or when kinetic energy reaches maximum.
Although
the procedure is simple and no damping factor is required,
the momentum conservation property is not satisfied in the process of KDR
and the convergent rate is not optimal \cite{alamatian2012new,zardi2020new}.
The main objective of this paper is to propose a dynamic relaxation method,
where efficient damping is introduced to SPH for obtaining the static solution.
Following VDR, an artificial viscous term is added into the momentum conservation equation
to dissipate the velocity gradient of the system.
To address the issue of computational inefficiency
when applying explicit scheme to integrate the viscous term,
we propose two operator splitting schemes,
one is inspired by a method for highly dissipative system \cite{litvinov2010splitting}.
Since such methods treat the viscous term implicitly,
they can be unconditional stable and relax the viscous time-step limit,
and allow very large damping ratio.
To provide a remedy to the issues of path dependency
and large damping ratio encountered by the original VDR and KDR,
and further accelerate the convergence,
we propose a random-choice strategy by which
the artificial damping is only carried out at a small fraction of time steps randomly.
For large scale SPH simulations,
two parallelization strategies,
viz. shared and distributed memories,
are implemented for accelerating computational performance.
For shared memory parallelization strategy,
thread-conflict occurs when implicit scheme is applied.
To address this issue,
a thread-safe splitting cell-linked list (CLL) scheme is proposed for implementing shared memory parallelization to accelerate implicit scheme.
In our code,
the open-source Threading Building Blocks (TBB) library \cite{contreras2008characterizing} developed by INTEL is applied for providing the share-memory parallel programming environment.
The remainder of this paper is organized as follows.
Section 2 briefly reviews the governing equations and
total Lagrangian SPH (TLSPH) formulations for solid dynamics.
The details of the present artificial damping method and
the splitting CLL scheme are described in Section 3.
Numerical examples are presented and discussed in Section 4 and
then concluding remarks are given in Section 5.
The corresponding code of this work are available on GitHub at \url{https://github.com/Xiangyu-Hu/SPHinXsys} \cite{zhang2020sphinxsys}.
\section{Preliminary}
\label{preliminary}
\subsection{Total Lagrangian solid dynamics}\label{subsec:governing equations}
In total Lagrangian formalism,
the conservation of mass and momentum for solid dynamics can be expressed as
\begin{equation}
\begin{cases}
\rho =\rho^0 J^{-1} \\
\rho^0 \frac{\text{d} \mathbf v}{\text{d} t} = \nabla^0 \cdot \mathbb P^T + \rho^0 \mathbf g \\
\end{cases},
\label{governing-equations}
\end{equation}
where $\rho$ is the density, Jacobian $J = \text{det} (\mathbb F)$, $\mathbf v$ the velocity, $\mathbf g$ body-force and $\mathbb P$ the first Piola-Kirchhoff stress tensor.
$\mathbb F$ is the deformation gradient and is defined as
\begin{equation}
\mathbb F = \nabla^0 \mathbf u + \mathbb I,
\label{deformation-gradient}
\end{equation}
where $\mathbb I$ denotes the identity matrix and $\mathbf u$ the displacement vector.
Note that the superscript $\left( \cdot \right)^0 $ denotes the quantities in the reference configuration thereafter.
For ideal elastic or Kirchhoff material, the first Piola-Kirchhoff stress tensor $\mathbb P$,
can be evaluated by
\begin{equation}
\mathbb P = \mathbb F \mathbb S.
\label{piola}
\end{equation}
Here, $\mathbb S$ denotes the second Piola-Kirchhoff stress tensor
and it can be evaluated via constitutive equation.
In particular, for isotropic and linear elastic materials, the constitutive equation can be simplified to
\begin{equation}
\begin{split}
\mathbb S &= K \text{tr}\left( \mathbb E\right) \mathbb I + 2G\left( \mathbb E-\frac{1}{3} \text{tr}\left( \mathbb E\right) \mathbb I \right) \\
&= \lambda \text{tr} \left( \mathbb E\right) \mathbb I + 2\mu \mathbb E,
\end{split}
\label{constitutive-equation}
\end{equation}
where $\lambda$ and $\mu$ are Lam$\acute{e}$ coefficients and
$\mathbb E$ represents the Green-Lagrangian strain tensor defined by
\begin{equation}
\mathbb E = \frac{1}{2} \left( \mathbb F^T \mathbb F - \mathbb I \right).
\label{strain-tensor-e}
\end{equation}
Also, $K = \lambda + 2\mu/3$ and $G = \mu$ denote the bulk and the shear modulus, respectively.
They have the following relationship
\begin{equation}
E = 2 G \left( 1 + 2 \nu \right) = 3 K \left( 1 - 2 \nu \right),
\label{modulus-relation}
\end{equation}
with $E$ representing the Young's modulus and $\nu$ the Poisson's ratio.
In this work,
the neo-Hookean material with the stain energy density function
\begin{equation}
{\mathcal W} = \mu \text{tr} \left( \mathbb E \right) - \mu \text{ln} J + \frac{\lambda}{2} \left(\text{ln} J \right) ^2
\label{strain-density}
\end{equation}
is also considered to predict the nonlinear stress-strain behavior of materials undergoing large deformations.
The corresponding second Piola-Kirchhoff stress $\mathbb S$ can be evaluated by
\begin{equation}
\mathbb S = \frac{\partial {\mathcal W}}{\partial \mathbb E}.
\label{PK-stress}
\end{equation}
\subsection{Total Lagrangian SPH method}\label{subsec:tl-sph}
In TLSPH method,
the kernel correction matrix $\mathbb B^0$ is defined as \cite{vignjevic2006sph}
\begin{equation}
\mathbb B^0_i = \left(- \sum_{j}^{N} V^0_j \mathbf r^0_{ij} \otimes \nabla^0_i W_{ij} \right) ^{-1},
\label{correction-matrix}
\end{equation}
where $ \mathbf r^0_{ij} = \mathbf r^0_i - \mathbf r^0_j $ and
\begin{equation}
\nabla^0_i W_{ij} = \left( \frac{\partial W_{ij}}{\partial r_{ij}}\right) ^0 \mathbf e^0_{ij}
\label{correction-matrix1}
\end{equation}
represents the gradient of the kernel function calculated at the initial reference configuration and $V^0_j$ denotes the particle volume.
Here, $\mathbf e^0_{ij} = \frac{\mathbf r^0_{ij}}{\left| \mathbf r^0_{ij}\right|}$.
According to the equation of mass conservation Eq. \eqref{governing-equations},
the density can be evaluated directly as
\begin{equation}
\rho_i = \rho^0_i J^{-1}.
\label{evaluate-mass}
\end{equation}
Then, the equation of momentum conservation in Eq. \eqref{governing-equations} is discretized as
\begin{equation}
\frac{\text{d} \mathbf v_i}{\text{d} t}= \frac{2}{m_i}\sum_{j}^{N} V^0_i V^0_j \tilde{\mathbb P}_{ij} \nabla^0_i W_{ij} + \mathbf g,
\label{evaluate-momentum}
\end{equation}
where
\begin{equation}
\tilde{\mathbb P}_{ij}=\frac{1}{2}\left( \mathbb P_i \mathbb B^0_i + \mathbb P_j \mathbb B^0_j\right)
\label{averaged-Piola-Kirchhoff}
\end{equation}
denotes the inter-particle averaged first Piola-Kirchhoff stress and $m_i$ is the particle mass.
Here,
the first Piola-Kirchhoff stress tensor is computed with the constitutive law where the deformation tensor $\mathbb{F}$ is updated by
\begin{equation}
\frac{ d \mathbb{F}_i}{dt} = -\left( \sum_j V^0_j \mathbf{v}_{ij} \otimes \nabla^0_i W_{ij} \right) \mathbb{B}^0_i,
\end{equation}
with $ \mathbf v_{ij} = \mathbf v_i - \mathbf v_j $ the relative velocity between particle $i$ and $j$.
\subsection{Time integration}\label{subsec:time integration}
Following the work of Zhang et al. \cite{zhang2020multi},
the position-based Verlet scheme is adopted for time integration.
First, update the deformation tensor, density and particles position to a midpoint stage as
\begin{equation}
\begin{cases}
\mathbb F^{n+\frac{1}{2}}_i = \mathbb F^n_i + \frac{1}{2} \Delta t \left( \frac{\text d \mathbb F_i}{\text d t} \right)^n\\
\rho^{n+\frac{1}{2}}_i = \rho^{0}_i \left( J^n_i \right) ^{-1} \\
\mathbf r^{n+\frac{1}{2}}_i = \mathbf r^{n}_i + \frac{1}{2} \Delta t \mathbf v_i^{n} \\
\end{cases}.
\label{timestep-1}
\end{equation}
Then, particles velocity at new time step is update by
\begin{equation}
\mathbf v^{n+1}_i = \mathbf v^{n}_i +\Delta t \left( \frac{\text{d} \mathbf v_i}{\text{d}t} \right) ^ {n}.
\end{equation}
Finally, the deformation tensor, density as well as positions of particles at the new stage are evaluated by
\begin{equation}
\begin{cases}
\mathbb F^{n+1}_i = \mathbb F^{n+\frac{1}{2}}_i + \frac{1}{2} \Delta t \left( \frac{\text d \mathbb F_i}{\text d t} \right)^{n+\frac{1}{2}}\\
\rho^{n+1}_i = \rho^{0}_i \left( J^{n+\frac{1}{2}}_i \right) ^{-1} \\
\mathbf r^{n+1}_i = \mathbf r^{n+\frac{1}{2}}_i + \frac{1}{2} \Delta t \mathbf v_i^{n+\frac{1}{2}} \\
\end{cases}.
\label{timestep-3}
\end{equation}
For numerical stability, the acoustic and body-force time-step criterion are considered, i.e.
\begin{equation}
\Delta t = 0.6 \text{min} \left( \frac{h}{c+\left| \mathbf v \right|_{max} }, \sqrt{\frac{h}{\left|\frac{\text d \mathbf v}{\text d t} \right|_{max}}} \right),
\label{time-step-criterion}
\end{equation}
where $h$ denotes the smoothing length and $c$ is the sound speed of solid structure given by $c=\sqrt{K/\rho}$.
\section{Dynamic relaxation method}\label{dynamic relaxation}
To damp the velocity gradient of a system,
an artificial viscous force can be introduced into the momentum conservation equation Eq. \eqref{governing-equations}.
Subsequently,
the momentum conservation equation for solid dynamics can be rewritten as
\begin{equation}
\rho^0 \frac{\text{d} \mathbf v}{\text{d} t} = \nabla^0 \cdot \mathbb P^{T} + \rho^0 \mathbf g + \eta (\nabla^0)^2 \mathbf v,
\label{momentum-conservation-m}
\end{equation}
where $\eta = \rho^{0} \nu$ denotes the artificial dynamic viscosity.
Following the total Lagrangian formulation,
the viscous damping term can be discretized as
\begin{equation}
\left( \frac{\text{d} \mathbf v_i}{\text{d} t}\right) ^{\nu} = \eta (\nabla^0)^2 \mathbf v = \frac{2\eta}{m_i}\sum_{j}^{N} V^0_i V^0_j \frac{\mathbf v_{ij}}{r_{ij}} \left( \frac{\partial W_{ij}}{\partial r_{ij}}\right) ^0.
\label{damping-term}
\end{equation}
Note that if the added damping term is solved by an explicit scheme,
the time-step size, along with the acoustic and body-force time-step size criterion
of Eq. (\ref{time-step-criterion}),
would be also constrained by
\begin{equation}
\Delta t \leq 0.5\frac{h^2}{\nu D},
\label{viscous-criteria-o}
\end{equation}
where $D= \left\lbrace 1, 2, 3 \right\rbrace$ for one-, two- or three-dimensional cases, respectively.
To achieve fast convergence,
a suitable large damping ratio is preferred.
Therefore,
the overall time-step size may be limited by the above viscous criteria,
in particular when high spatial resolution is applied,
and this leads to excessive computational efforts.
In this work,
we first apply an operator splitting method following Ref. \cite{wang2019split} to decouple the momentum conservation equation Eq.\eqref{momentum-conservation-m} into two procedures,
i.e. the original momentum part
\begin{equation}
S_m \quad :\quad \rho^0 \frac{\text{d} \mathbf v}{\text{d} t} = \nabla^0 \cdot \mathbb P^{T} + \rho^0 \mathbf g,
\label{fist-step-m}
\end{equation}
and the damping part
\begin{equation}
S_d \quad :\quad \frac{\text{d} \mathbf v}{\text{d} t} = \eta (\nabla^0)^2 \mathbf v.
\label{second-step-m}
\end{equation}
Here,
two operators $S_m$ and $S_d$ are introduced to represent the original momentum equation without viscosity and the damping process, respectively.
Subsequently,
the first order accuracy Lie-Trotter splitting scheme \cite{mclachlan2002splitting} is applied to approximate the solution from time $t$ to $t+\Delta t$ by
\begin{equation}
\mathbf v \left( t+\Delta t \right) = S_d^{(\Delta t)} \circ S_m^{(\Delta t)}\mathbf v (t),
\label{operator-splitting}
\end{equation}
where the symbol $\circ$ denotes the separation of each operator and
indicates that $S_d^{(\Delta t)}$ is applied after $S_m^{(\Delta t)}$.
If implicit methods are adopted to discretize the viscous term,
the damping step can be unconditionally stable and thus a larger time-step size is allowed.
One should note that
large scale matrix operations or iterations over the entire system
for traditional implicit methods lead to large memory requirement
and difficulties in paralleization.
To avoid this problem,
we continue to split the entire-domain-related damping step into particle-by-particle operators, e.g. by second-order Strang splitting \cite{strang1968}, as
\begin{equation}
S_d^{(\Delta t)} = D_{1}^{(\frac{\Delta t}{2})} \circ D_{2}^{(\frac{\Delta t}{2})} \circ \dots \circ D_{N_t-1}^{(\frac{\Delta t}{2})} \circ D_{N_t}^{(\frac{\Delta t}{2})} \circ D_{N_t}^{(\frac{\Delta t}{2})} \circ D_{N_t-1}^{(\frac{\Delta t}{2})} \circ \dots \circ D_{2}^{(\frac{\Delta t}{2})} \circ D_{1}^{(\frac{\Delta t}{2})},
\label{particle-by-particle splitting}
\end{equation}
where $N_t$ denotes the total number of particles and $D_{i}$ the split damping operator corresponding to particle $i$.
In the following, we propose two efficient schemes to the local damping operator, where velocities are updated implicitly and locally.
Then,
the new time-step velocity updating for the entire field can be achieved
by carrying out the local split operator to all particles for half a time step and then backwards for another half time step \cite{nguyen2009mass} as shown in Eq.\eqref{particle-by-particle splitting}.
\subsection{Particle-by-particle splitting scheme}\label{subsec:semi-implicit}
In an implicit formulation,
the local damping term in Eq. \eqref{damping-term} can be rewritten as
\begin{equation}
\left( \frac{\text{d} \mathbf v_i}{\text{d} t}\right) ^{\nu} = \frac{2\eta}{m_i}\sum_{j}^{N} V^0_i V^0_j \frac{\mathbf v^{n+1}_{ij} }{r_{ij}} \left( \frac{\partial W_{ij}}{\partial r_{ij}}\right) ^0
\label{damping-term-implicit}
\end{equation}
with $\mathbf v^{n+1}_{ij} = \mathbf v^{n}_{ij} + \text d \mathbf v_i - \text d \mathbf v_j$.
We denote
\begin{equation}
B_j = 2 \eta V^0_i V^0_j \frac{1}{r_{ij}} \left( \frac{\partial W_{ij}}{\partial r_{ij}}\right) ^0 \text{d} t
\label{parameters1}
\end{equation}
and
\begin{equation}
\mathbf E_i = - 2\eta \sum_{j}^{N} V^0_i V^0_j \frac{ \mathbf v^{n}_{ij} }{r_{ij}} \left( \frac{\partial W_{ij}}{\partial r_{ij}}\right) ^0 \text{d} t= - \sum_{j}^{N} B_j \mathbf v^{n}_{ij}.
\label{parameters2}
\end{equation}
The implicit formulation Eq.\eqref{damping-term-implicit} can be further transformed as
\begin{equation}
\mathbf E_i = \left( \sum_{j}^{N} B_j - m_i\right) \text{d} \mathbf v_i - \sum_{j}^{N} B_j \text{d} \mathbf v_j,
\label{implicit-equation}
\end{equation}
where $\text{d} \mathbf v_i$ and $\text{d} \mathbf v_j$ represent the incremental change for velocity of particle $i$ and its neighboring particles $j$ induced by viscous acceleration.
Here, we adopt the gradient descent method \cite{nielsen2015neural} to evaluate $\text{d} \mathbf v_i$ and $\text{d} \mathbf v_j$.
Specifically, the left hand side (LHS) of Eq. \eqref{implicit-equation}
can be decreased by following its gradient.
With respect to variables $\left( \text{d} \mathbf v_i, \text{d} \mathbf v_1, \text{d} \mathbf v_2, \cdots, \text{d} \mathbf v_N \right)^T$ ,
the gradient $\nabla \mathbf E_i$ gives
\begin{equation}
\nabla \mathbf E_i = \left( \sum_{j}^{N} B_j - m_i, -B_1, -B_2, \cdots, -B_N \right)^T.
\label{gradient-E}
\end{equation}
We choose
\begin{equation}
\left( \text{d} \mathbf v_i, \text{d} \mathbf v_1, \text{d} \mathbf v_2, \cdots, \text{d} \mathbf v_N \right)^T = \mathbf k \nabla \mathbf E_i,
\label{gradient-learning-rate}
\end{equation}
where $\mathbf k$ is known as the learning rate \cite{nielsen2015neural}.
By substituting Eqs. \eqref{gradient-E} and \eqref{gradient-learning-rate} into Eq. \eqref{implicit-equation},
the learning rate can be obtained, i.e.
\begin{equation}
\mathbf k=\left(\left( \sum_{j}^{N} B_j - m_i\right)^2+ \sum_{j}^{N} \left( B_j\right)^2 \right)^{-1} \mathbf E_i.
\label{learning-rate-solve}
\end{equation}
According to Eqs. \eqref{gradient-E} and \eqref{gradient-learning-rate},
the incremental change for velocity by viscous damping can be thus achieved.
In order to ensure momentum conservation,
the velocities of neighboring particles are then modified by the above predicted incremental change.
In summary,
the local update of velocities includes two steps as follows.
The first step calculates the incremental change for velocity by gradient descent method, i.e.
\begin{equation}
\begin{cases}
\mathbf v_i^{n+1}&=\mathbf v_i^{n} + \text{d} \mathbf v_i = \mathbf v_i^{n} + \left( \sum_{j}^{N} B_j - m_i\right) \mathbf k\\
\mathbf v_1^{p}&=\mathbf v_1^{n} + \text{d} \mathbf v_1 = \mathbf v_1^{n} -B_1 \mathbf k \\
\mathbf v_2^{p}&=\mathbf v_2^{n} + \text{d} \mathbf v_2 = \mathbf v_2^{n} -B_2 \mathbf k \\
&\cdots \\
\mathbf v_N^{p}&=\mathbf v_N^{n} + \text{d} \mathbf v_N = \mathbf v_N^{n} -B_N \mathbf k \\
\end{cases},
\label{first-step}
\end{equation}
where the superscript $p$ denotes the predicted value.
The second step ensures momentum conservation, which yields
\begin{equation}
\begin{cases}
\mathbf v_1^{n+1}&= \mathbf v_1^{n} -B_1 \left( \mathbf v_i^{n+1} -\mathbf v_1^{p} \right) / m_1 \\
\mathbf v_2^{n+1}&= \mathbf v_2^{n} -B_2 \left( \mathbf v_i^{n+1} -\mathbf v_2^{p} \right) / m_2 \\
&\cdots \\
\mathbf v_N^{n+1}&= \mathbf v_N^{n} -B_N \left( \mathbf v_i^{n+1} -\mathbf v_N^{p} \right) / m_N \\
\end{cases}.
\label{second-step}
\end{equation}
As the velocities are updated implicitly and much larger time-step size is allowed.
In this work, we choose the following viscous criterion
\begin{equation}
\Delta t \leq 50\frac{h^2}{\nu D}.
\label{viscous-criteria}
\end{equation}
Note that this criterion is about 100 times larger than the corresponding explicit method as presented in Eq. \eqref{viscous-criteria-o}.
Besides, the artificial dynamic viscosity $\eta = \rho^{0} \nu$ as shown in Eq. \eqref{momentum-conservation-m}
is defined by
\begin{equation}
\eta = \frac{1}{4} \beta \rho^{0} \sqrt{\frac{E}{\rho}}L=\frac{\beta}{4} \sqrt{\rho^{0} E}L,
\label{viscous-parameter}
\end{equation}
where $L$ is the characteristic length scale of the problem and $\beta$ denotes a parameter relating to the body shape. Note that choose different value for the parameter $\beta$
may variate, though not much, the convergence rate.
\subsection{Pairwise splitting scheme}\label{subsec:pairwise-splitting}
Inspired by the work of Ref. \cite{litvinov2010splitting},
we present herein the pairwise splitting scheme where particle velocity is updated implicitly and locally in a pairwise fashion.
By adopting the second-order Strang splitting \cite{strang1968},
the damping operator corresponding to each particle $i$ as given in Eq. \eqref{particle-by-particle splitting}
is further split based on its neighbors, i.e.
\begin{equation}
D_i^{(\Delta t)} = D_{i,1}^{(\frac{\Delta t}{2})} \circ D_{i,2}^{(\frac{\Delta t}{2})} \circ \dots \circ D_{i,N-1}^{(\frac{\Delta t}{2})} \circ D_{i,N}^{(\frac{\Delta t}{2})} \circ D_{i,N}^{(\frac{\Delta t}{2})} \circ D_{i,N-1}^{(\frac{\Delta t}{2})} \circ \dots \circ D_{i,2}^{(\frac{\Delta t}{2})} \circ D_{i,1}^{(\frac{\Delta t}{2})},
\label{pairwise-splitting}
\end{equation}
where $D_{i,j}^{(\frac{\Delta t}{2})}$ denotes the interaction between particle $i$ and its neighbors.
Specifically,
the incremental changes for velocity of a specific particle pair induced by viscosity can be written in implicit form as
\begin{equation}
\begin{cases}
m_i \text{d} \mathbf v_i = B_{j} \left( \mathbf v_{ij} + \text d \mathbf v_i - \text d \mathbf v_j\right) \\
m_j \text{d} \mathbf v_j = -B_{j} \left( \mathbf v_{ij} + \text d \mathbf v_i - \text d \mathbf v_j\right) \\
\end{cases}.
\label{pairwise-eq}
\end{equation}
Here,
$B_j$ is defined in Eq. \eqref{parameters1} and it is obvious that this process does not change the conservation of momentum.
Then, $\text{d} \mathbf v_i$ and $\text{d} \mathbf v_j$ can be obtained straightforwardly by solving Eq. \eqref{pairwise-eq},
which yields
\begin{equation}
\begin{cases}
\text{d} \mathbf v_i = m_j\frac{ B_{j} \mathbf v_{ij}}{ m_i m_j - \left( m_i + m_j\right) B_j } \\
\text{d} \mathbf v_j = - m_i\frac{ B_{j} \mathbf v_{ij}}{ m_i m_j - \left( m_i + m_j\right) B_j } \\
\end{cases}.
\label{pairwise-solve}
\end{equation}
By sweeping over all neighboring particle pairs for half a time step and then backwards for another half time step,
the incremental changes for velocity of particle $i$ and all its neighbors can be thus achieved.
Compared to the particle-by-particle splitting method in Section \ref{subsec:semi-implicit},
this scheme leads more errors in solving viscosity due to the further splitting in pairwise fashion.
However, it is unconditional stable and thus more suitable for problems with high spatial resolution and high damping ratio.
\subsection{Random-choice strategy}\label{subsec:random-chosen}
As noted in Ref.\cite{lin2014efficient},
the solutions for nonlinear problems are path dependent
and may vary with different damping ratio.
The added viscous force would hinder the system achieving correct static state
especially when the damping radio is large.
Thus, a suitable damping ratio has to be selected \cite{lin2014efficient,crisfield1997non}
for faster convergence of the system with the aforementioned viscous damping methods.
To avoid this problem and relax the limitation on the choice of large damping ratio,
we present herein a random-choice strategy
in which the viscosity term is imposed randomly rather than at every time step.
For this, the artificial dynamic viscosity $\eta$ is modified as
\begin{equation}
\tilde{\eta} =
\begin{cases}
\eta / \alpha \quad \quad & {\rm if} \quad \alpha < \phi\\
0 & {\rm otherswise} \\
\end{cases},
\label{viscosity-reset}
\end{equation}
where $\phi$ is a random number uniformly distributed between $0$ to $1$.
$\alpha$ is a parameter determining the probability,
and we set $\alpha= 0.2$ in this work.
Therefore, the resistance on displacement
induced by the large artificial viscosity can be released randomly,
as will shown the numerical example (Section \ref{bending-cantilever}),
which eliminates the path dependency and accelerates the convergence.
Note that this strategy also helps to save much of computational cost
since the computation of damping is only carried out
at a small fraction of time steps.
\subsection{Splitting cell-linked list }\label{subsec:splitting cell-linked list}
Shared-memory parallel technique is widely used in SPH simulation due to its easy-implementation nature.
However,
for the above two proposed implicit approaches,
implementing shared-memory approach taking advantage of multicore performance, is problematic.
Since the velocities of particle $i$ and the neighbors are updated simultaneously in above implicit method,
conflicts may happen when the values of one particle pair are updated at the same time by multiple threads.
In order to avoid this problem, a splitting CLL method is proposed.
Figure \ref{split-cell-linked} presents a sketch of the splitting CLL method in two-dimensional space.
Similar to the conventional CLL method,
the computational domain is divided into cells with the same size to reduce the searching space \cite{liu2003smoothed}
and the cell spacing $r_c$ is equal to the cut-off radius of the kernel function.
Then,
the cell domain is divided into $9$ blocks ($27$ blocks for three-dimensional space) and all the cells are assigned to blocks.
To implement the share-memory parallelization between blocks,
each block satisfies the condition that all its cells are not neighboring cells to each other.
As shown in Figure \ref{split-cell-linked},
cells are assigned to block $1$ with condition of $\left( \text{Cell}_i \subset \text{Block}1 \middle| \left\lbrace i = 1,4,7,28,31,34 \right\rbrace\right) $
and also cells are assigned to block $9$ with condition of $\left( \text{Cell}_i \subset \text{Block}9 \middle| \left\lbrace i = 21,24,27,48,51,54 \right\rbrace\right) $.
When the tasks are distributed to threads based on blocks,
the particles executed by different threads are at least two cell space away from each other at any time.
The conflicts can be thus avoid, since the neighboring particles only locate in the same cell or other adjacent 8 cells.
\begin{figure}[htb!]
\centering
\includegraphics[trim = 2cm 4cm 2cm 3.8cm, clip,width=0.90\textwidth]{split-cell-link}
\caption{Sketch of the splitting CLL method in two-dimensional space.
}
\label{split-cell-linked}
\end{figure}
\section{Numerical examples}\label{validation}
In this section,
various tests for static equilibrium simulation including
cantilever bending and twisting, ball free falling and a fluid-structure interaction problem
are simulated to demonstrate the effectiveness and efficiency of the present method.
In the following cases, the 5th-order C2 Wendland kernel \cite{wendland1995} is adopted and the smoothing length $h=1.3dp$ with $dp$ denoting the initial particle spacing.
Term "No damping" denotes the conventional TLSPH method without damping imposed,
term "Reference" is the static solution obtained by the conventional TLSPH method and
term "Present" represents the particle-by-particle splitting damping method presented in Section \ref{subsec:semi-implicit} if not mentioned otherwise.
\subsection{Bending cantilever}\label{bending-cantilever}
We first consider a three dimensional cantilever bending under gravity.
As shown in Figure \ref{figs:bending-cantilever-setup},
one end of the cantilever is clamped to the wall and the body bends freely under the gravity $g=9.8 m/s^2$.
The neo-Hookean material with density $\rho=1265 kg/m^3$, Young's modulus $E=5\times 10^4 Pa$ and Poisson's ratio $\nu=0.45$ is applied.
This case is studied with three different spatial resolutions, i.e. $h/dp = \left\lbrace 6, 12, 24 \right\rbrace $.
The artificial viscosity $\eta_0=\frac{\beta}{4}\sqrt{\rho E} d = 32 kg/(m\cdot s)$ with $\beta=d/l=0.4$ is applied.
Figure \ref{figs:bending-cantilever-configuration} presents the deformed configuration in equilibrium state with three different spatial resolutions.
The time histories of the displacement in $y-$direction of the point $S$ are given in Figure \ref{figs:bending-cantilever-three-resolution}.
It is clear that the conventional TLSPH method exhibits oscillations and takes long time to reach the steady state.
As for the present method, only a few oscillations are exhibited and the steady state is reached quickly which reduces much computational time for static solution.
Besides, with the conventional TLSPH method,
the final displacements of point $S$ in $y-$direction with $h/dp = 6$, $h/dp = 12$ and $h/dp = 24$
are $-9.90 \times 10^{-3} m$, $-6.89 \times 10^{-3} m$ and $-5.86 \times 10^{-3} m$, respectively.
We can observe that the same static results are obtained by the present method and
a converge solution is achieved with increasing resolutions.
To test the performance of the adopted random-choice strategy,
the setup with parameter $\alpha=1$ of Eq. \eqref{viscosity-reset} and artificial viscosity $\eta=5\eta_0$ and $\eta=10\eta_0$ are also considered.
The top panel of Figure \ref{figs:bending-cantilever-compare} shows the comparison of the present method with different parameters at the resolution of $h/dp=12$.
We can observe that the results of $\alpha=1$, in which the viscous damping is imposed at every time-step,
cannot achieve the final state even at $t=1.5s$ especially for higher artificial viscosity.
Also note that lager damping leads larger distance between the solution to that of the final state and the velocity approaching to final state becomes very small due to too much damping.
However, with the help of the random-choice strategy, e.g. $\alpha=0.2$,
the system is released randomly from large damping and the same final state can be reached efficiently for both $\eta=5\eta_0$ and $\eta=10\eta_0$.
Besides, the pairwise splitting scheme as presented in Section \ref{subsec:pairwise-splitting} is also introduced to discretize the viscous term in this case.
The bottom panel of Figure \ref{figs:bending-cantilever-compare} gives the corresponding results with different parameters at the resolution of $h/dp=12$.
Compared to that of the top panel,
it can be observed that the viscosity achieved by the pairwise splitting scheme
is less than the splitting scheme presented in Section \ref{subsec:semi-implicit}.
However, with higher artificial viscosity, the steady state of the cantilever can also be achieved efficiently.
Meanwhile,
this pairwise splitting scheme is more stable and free from the limitation of viscous criterion, which is more suitable for the problem with high resolution and high damping ratio.
The computational performance of the present method by implementing the shared-memory parallelization and random-choice strategies is then tested.
The computation is conducted with spatial resolution of $h/dp=12$ and the total physical time of $t = 2s$ on an AMD Ryzen 5 3600 6-Core CPU Desktop computer with 16GB RAM and Windows system (10).
Table \ref{bending-cantilever-time} reports the CPU wall-clock time for the process of viscous damping.
We can see that the shared-memory parallelization strategy demonstrates about 2.8 speedup compared to that of serial computing.
Besides, the random-choice strategy also saves much computational efforts and only takes $20\%$ ($\alpha=0.2$) CPU time compared to the one without applying this strategy.
The viscous damping process only takes $2.38s$ during the whole computation and
optimized computational performance is achieved.
\begin{figure}[htb!]
\centering
\includegraphics[trim = 10cm 10cm 10cm 7cm, clip,width=0.6\textwidth]{bending-cantilever}
\caption{Bending cantilever: Initial configuration. Probe $S$ is located at the free-end of the cantilever. }
\label{figs:bending-cantilever-setup}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 1cm 2cm 1cm 2cm, clip,width=.95\textwidth]{bending-cantilever-all}
\caption{Bending cantilever: Deformed configuration in steady state at resolution of $h/dp = 6$ (left panel), $h/dp = 12$ (middle panel)
and $h/dp = 24$ (right panel).
}
\label{figs:bending-cantilever-configuration}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 0.2cm 0cm 0cm, clip,width=.8\textwidth]{bending-no-damping-all} \\
\includegraphics[trim = 0cm 0.2cm 0cm 0cm, clip,width=.8\textwidth]{bending-damping-all}
\caption{Bending cantilever: The time history displacement of the end point $S$ in vertical direction obtained by conventional method (top panel) and the present method (bottom panel). }
\label{figs:bending-cantilever-three-resolution}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 0.2cm 0cm 0cm, clip,width=.8\textwidth]{bending-12-compare} \\
\includegraphics[trim = 0cm 0.2cm 0cm 0cm, clip,width=.8\textwidth]{bending-12-compare-pairwise}
\caption{Bending cantilever: Comparisons of the time history displacement of point $S$ at the resolution of $h/dp = 12$ by the operator splitting method (top panel) and the pairwise splitting method (bottom panel).
}
\label{figs:bending-cantilever-compare}
\end{figure}
\begin{table}
\scriptsize
\centering
\caption{CPU time for viscous damping process in the simulation of Bending cantilever with different parameters.}
\begin{tabularx}{8.5cm}{@{\extracolsep{\fill}}lcc}
\hline
CPU time (s) & serial computation & parallel computation\\
\hline
$\alpha=1.0$ & $32.82$ & $11.61$ \\
\hline
$\alpha=0.2$ & $6.66$ & $2.38$ \\
\hline
\end{tabularx}
\label{bending-cantilever-time}
\end{table}
\subsection{Twisting cantilever}
\label{twisting-cantilever}
Under a rotational body-force, the cantilever as presented in Section \ref{bending-cantilever} can be twisted from the initial configuration.
As shown in Figure \ref{figs:twisting-cantilever-setup},
the bottom of the cantilever is also clamped and a sinusoidal rotational body-force is imposed.
The body-force relates to the origin and is defined by
\begin{equation}
g = \left[0,y \gamma, z \gamma \right] ^{T},
\label{rotation-body-force}
\end{equation}
where $\gamma = \frac{20g}{h} \text{sin}(\pi x /2L)$ and $g=9.8 m/s^2$.
The similar neo-Hookean material property as in bending case is adopted and this case is performed at the resolution of $h/dp=12$.
Figure \ref{figs:twisting-cantilever-contour} and \ref{figs:twisting-cantilever-compare}
present several snapshots of the deformation and the time history displacement in $z-$direction of the point $S$, respectively.
With the present viscous damping imposed,
the final steady state of the system is reached much faster and the same result with that of the conventional TLSPH is achieved
implying the efficiency and accuracy of the present method.
\begin{figure}[htb!]
\centering
\includegraphics[trim = 9cm 7cm 9cm 5cm, clip,width=0.6\textwidth]{twisting-cantilever}
\caption{Twisting cantilever: Initial configuration. Probe $S$ locates at the free-end of the cantilever. }
\label{figs:twisting-cantilever-setup}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 0.2cm 0cm 0cm, clip,width=.8\textwidth]{twisting-cantilever-contour}
\caption{Twisting cantilever: Four time instants as labeled by blue dots in Figure \ref{figs:twisting-cantilever-compare} of deformation and von Mises stress.
}
\label{figs:twisting-cantilever-contour}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 0.2cm 0cm 0cm, clip,width=.8\textwidth]{twisting-cantilever-compare}
\caption{Twisting cantilever: The time history of displacement at the end point $S$ in $z-$ direction. }
\label{figs:twisting-cantilever-compare}
\end{figure}
\subsection{Ball free falling}\label{subsec0}
A two dimensional ball free falling is considered herein.
Figure \ref{figs:ball-collision-setup} gives the initial configuration and the body-fitted particle distribution of the ball.
In this case, a ball falls freely accelerated by gravity $g=9.8 m/s^2$ and collides with the floor.
A linear elastic material with density $\rho=1000 kg/m^3$, Young's modulus $E=5\times 10^5 Pa$ and Poisson's ratio $\nu=0.45$ is adopted
and the initial particle spacing $dp = 0.02d$.
For the present method,
the artificial viscosity is given by $\eta=\frac{\beta}{4}\sqrt{\rho E} d = 5590 kg/(m\cdot s)$ with $\beta = 1$.
Figure \ref{figs:ball-collision-instant} shows four snapshots of the ball deformation and the corresponding von Mises stress.
After several collisions, the equilibrium of this system is reached and the ball stands still on the wall.
Figure \ref{figs:ball-collision-compare} gives the time history of the displacement of the ball center in vertical direction.
It is obvious that achieving steady state of the system for the conventional TLSPH method is time-consuming and more collisions are exhibited in the simulation.
The amplitude reduction is mainly caused by the numerical dissipation of the scheme.
However, with the help of the present viscous damping method,
the system achieves steady state faster and only few collisions is exhibited.
Term "Reference" in Figure \ref{figs:ball-collision-compare} denotes the final displacement obtained by the conventional TLSPH method and
it is clear that the present method achieves the same final results with that of the classic method.
\begin{figure}[htb!]
\centering
\includegraphics[trim = 11cm 7cm 11cm 6cm, clip,width=0.45\textwidth]{ball-collision}
\includegraphics[trim = 1.0cm 1.0cm 1.0cm 1.5cm, clip,width=0.45\textwidth]{particle-distribution}
\caption{Ball free falling: Initial configuration (left panel) and body-fitted particle distribution of the ball (right panel). }
\label{figs:ball-collision-setup}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 2cm 0cm 2cm, clip,width=.95\textwidth]{ball-collision-contour-all}
\caption{Ball free falling: Four time instants as labeled by blue dots in Figure \ref{figs:ball-collision-compare} of the ball deformation and von Mises stress.
}
\label{figs:ball-collision-instant}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 0.2cm 0cm 0.2cm, clip,width=0.8\textwidth]{ball-collision-compare}
\caption{Ball free falling: The time history of the displacement at ball center in vertical direction. }
\label{figs:ball-collision-compare}
\end{figure}
\subsection{Hydrostatic water column on an elastic plate}
\label{subsec4}
This case is taken from Ref.\cite{zhang2020multi} and relates to the deformation of an elastic plate induced by the hydrostatic pressure of a water column.
The geometric parameters and basic setups are given in Figure \ref{figs:hydrostatic-setup}.
An aluminium plate with thick $bh=0.05m$ is suddenly exposed to the hydrostatic pressure loading by a $2m$ height water column and
an equilibrium state can be reached after initial oscillations.
Following Refs. \cite{zhang2020multi, fourey2017efficient, khayyer2018enhanced},
the material with density of $\rho=2700 kg/m^3$, Young's modulus $E=67.5GPa$ and Poisson's ratio $0.34$ is applied for the aluminium plate.
As for the water,
the density is $\rho=1000 kg/m^3$ and the weakly-compressible model is adopted with
the sound speed of $c=10 U_{max}$,
where $U_{max}= 2\sqrt{gH} =8.86 m/s$ is the maximum anticipated flow speed.
In fluid-structure coupling, multiple time steps are applied.
Similar to Refs. \cite{zhang2020multi, fourey2017efficient, khayyer2018enhanced},
a constant fluid time-step size $\Delta t^{F}=2.0\times 10^{-5} s$ is adopted.
More details of fluid structure interaction formulation with SPH method are referred to Ref. \cite{zhang2020multi}.
In this case, the viscous damping is also introduced to water part
and the corresponding artificial dynamic viscosity is given by $\eta_f= \rho U_{max} L_f = 8858.9 kg/(m\cdot s)$
with $L_f$ denoting the characteristic length.
According to theoretical solution,
the magnitude of the static deformation at the mid-span of the plate is $d=-6.85 \times 10 ^{-5} m$.
The time history of mid-span displacement for spatial resolution $bh/dp = 4$ is presented in the top panel of Figure \ref{figs:hydrostatic}.
Term "Present-plate" denotes the viscous damping only imposed on the elastic plate,
"Present-fluid" is the solution with viscous damping introduced to water part and "Exact" represents the theoretical solution.
It can be observed that the solution with plate damping is similar to that of without artificial damping imposed and oscillations are not repressed effectively.
In fact, the oscillation of the plate is mainly induced by the water.
When the present viscous damping is introduced to water, the solution reaches the equilibrium state rapidly as shown in Figure \ref{figs:hydrostatic}.
Moreover, a convergence study with the present viscous damping imposed on water part is carried out by increasing the spatial resolution.
The results are given in bottom panel of Figure\ref{figs:hydrostatic} and it is clear that a convergent displacement of the mid-span is achieved implying the accuracy of the present method.
The CPU wall-clock time of the viscous damping process is recorded in Table \ref{hydrostatic-time} until the physical time of $t=2s$.
It can be observed that the damping process only takes $26.89s$ during the computing and almost $5$ times speedup is achieved by implementing share-memory parallelization.
Besides, if an explicit scheme instead of the present implicit schemes is implemented for viscous damping,
the fluid time-step size would be limited to $\Delta t^F \leq 0.25\frac{h^2}{\nu}=4.4 \times 10^{-6}s$ according to Eq.\eqref{viscous-criteria-o},
which is much smaller than the adopted constant fluid time-step size $\Delta t^F = 2.0 \times 10^{-5}s$ demonstrating the efficiency of the present method.
\begin{figure}[htb!]
\centering
\includegraphics[trim = 4cm 4cm 4cm 4cm, clip,width=0.6\textwidth]{hydrostatic-fsi-setup}
\caption{Hydrostatic water column on an elastic plate: Initial configuration. }
\label{figs:hydrostatic-setup}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[trim = 0cm 0.5cm 0cm 0cm, clip,width=.8\textwidth]{hydrostatic-04} \\
\includegraphics[trim = 0cm 0.5cm 0cm 0cm, clip,width=.8\textwidth]{hydrostatic-convergence}
\caption{Hydrostatic water column on an elastic plate: Mid-span displacement at resolution of $dp = bh/4$ (top panel) and a convergence study with increasing spatial resolutions (bottom panel). }
\label{figs:hydrostatic}
\end{figure}
\begin{table}
\scriptsize
\centering
\caption{CPU time for viscous damping process in the simulation of hydrostatic water column on an elastic plate.}
\begin{tabularx}{8.5cm}{@{\extracolsep{\fill}}lcc}
\hline
\quad & serial computation & parallel computation \\
\hline
CPU time (s) & $133.57$ & $26.89$ \\
\hline
\end{tabularx}
\label{hydrostatic-time}
\end{table}
\section{Conclusions}
\label{conclusions}
In this paper, an efficient dynamic relaxation method for SPH method is proposed by imposing viscous damping to a dynamic system.
This method mainly includes the following aspects:
(a) a viscous damping term is added to the momentum conservation equation and two operator splitting schemes are introduced to discretize this term;
(b) a random-choice strategy is introduced to release the system from large damping;
(c) a splitting CLL method is proposed to address the conflicts between different threads in shared-memory parallelization.
The implicit nature of the splitting scheme makes the discretization stable and large time-step size is allowable.
The random-choice strategy helps to eliminate the path dependency, improve the convergence rate and also reduce much computational cost.
A various of test cases demonstrate the improved efficiency and performance of the
present method for achieving the final converged state of a system compared to conventional SPH method.
Besides,
the proposed splitting CLL is thread-conflict free and makes it possible for implicit methods adopting share-memory parallel techniques to further improve computational performance.
Moreover,
other implicit methods,
e.g. the splitting scheme \cite{litvinov2010splitting} for highly dissipative problem and
compact finite different scheme \cite{lele1992compact} for hyperbolic problems,
encounter the same problem of thread-conflict in shared-memory parallelization.
The present splitting CLL scheme also provides a practicable solution to these problems
by dividing computational domain into several blocks, which is the subject of our ongoing work.
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgement}
The first author is partially supported by Xidian University (China) and
the project of National Natural Science Foundation of China (NSFC) (Grant No:91952110).
C. Zhang and X.Y. Hu would like to express their gratitude to Deutsche Forschungsgemeinschaft for their sponsorship
of this research under grant number DFG HU1572/10-1 and DFG HU1527/12-1.
\bibliographystyle{elsarticle-num}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,882 |
Mužská čtyřhra Erste Bank Open 2020 probíhala na přelomu října a listopadu 2020. Do deblové soutěže vídeňského tenisového turnaje hraného na tvrdém povrchu Wiener Stadthalle nastoupilo šestnáct dvojic. Obhájcem titulu byl americko-britský pár Rajeev Ram a Joe Salisbury, který odstoupil před čtvrtfinálem s Murraym a Skupskim.
Vítězem se stal třetí nasazený, polsko-brazilský pár Łukasz Kubot a Marcelo Melo, který ve finále za 1.55 hodiny zdolal dvojici Britů Jamieho Murrayho a Neala Skupského po zvládnutých koncovkách obou setů 7–6 a 7–5. Oba šampioni a obhájci finálové účasti si do žebříčku ATP připsali 500 bodů. Na Vienna Open získali po triumfech v letech 2015 a 2016 třetí trofej a celkově patnáctou společnou, respektive druhou v probíhající sezóně. Kubot vybojoval dvacátý sedmý deblový titul na okruhu ATP Tour. Pro Mela to bylo třicáté páté takové turnajové vítězství.
Nasazení párů
Mate Pavić / Bruno Soares (čtvrtfinále)
Rajeev Ram / Joe Salisbury (čtvrtfinále, odstoupili)
Łukasz Kubot / Marcelo Melo (vítězové)
Pierre-Hugues Herbert / Nicolas Mahut (1. kolo)
Pavouk
Odkazy
Reference
Externí odkazy
Vienna Open
Tenis v Rakousku v roce 2020
ATP Tour 2020 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,470 |
1993 was a banner year for breakthroughs.
The World Wide Web opened to the general public, forever changing the way we gather and distribute information around the globe.
NASA put a pair of "glasses" on the Hubble Space Telescope, bringing its pictures into focus and showing us the wonders that exist beyond our atmosphere.
And ETFs (exchange-traded funds) debuted in the United States, making an indelible impression on how we invest. In the 25+ years since, ETFs have become an increasingly attractive way for investors of all types to harness the powerful combination of diversification, tax efficiency, and low costs that comes with indexing.
But let's go back a little further than that. In 1976, Vanguard took what was traditionally an institutional investment strategy—indexing—and released it to the masses so we could all benefit from low-cost, diversified investing. Why not do it again with ETFs?
VTI now sits alongside dozens more Vanguard ETFs®, which have combined assets exceeding $850 billion,** making us one of the leading providers of ETFs in the U.S.
Best new U.S. equity ETF: Vanguard ESG U.S. Stock ETF (ESGV).
Best new international/global fixed-income ETF: Vanguard Total World Bond ETF (BNDW).
Best new active ETF: Vanguard U.S. Multifactor ETF (VFMF).
In all, we were nominated in 10 of 30 categories—more than ever in a single year. I left that room proud to represent a company that's continuously aiming to improve investor outcomes.
The popularity of ETFs has been nothing short of explosive. ETFs present a fresh alternative that combines the well-known benefits of indexing (diversification, tax efficiency, and lower costs) with real-time pricing (instead of having to wait until the end of the day) and the more manageable investment minimums of an individual stock.
It didn't take long for ETF popularity to expand beyond big institutions and grab the attention of individual investors.
ETFs originally focused on the U.S. stock market. Then came the first international ETF (1996), followed by the first bond ETF (2002). In 2010, total ETF assets surpassed $1 trillion.† By year-end 2018, assets topped $3.4 trillion.† And the number of ETFs available for investment in the U.S. reached nearly 2,000, offering essentially complete coverage of U.S. and international stock and bond markets.
Are ETFs here to stay?
The surge toward ETFs doesn't seem at all interested in slowing down. More than $311 billion flowed into ETFs industry-wide in 2018, with nearly $85 billion into Vanguard alone.** During that same time, nearly 250 new ETFs were launched.† And the menu is getting longer and more diverse, with new alternatives in the pipeline like ESG (environmental, social, and governance), factor-based, and more variations of actively managed ETFs.
At Vanguard, we're always exploring new opportunities—but we stop short of experimenting with ETFs that reflect short-term fads or narrow market niches. Instead, we make thoughtful, intentional decisions and only introduce new ETFs that we believe will have enduring value for our investors.
With the increased expansion of the ETF market, there are more options—and more companies to choose from—than ever before. As with any industry, this heightened competition presents more opportunities for you to save money on expense ratios and commissions. Or, more simply put, while the number of ETFs (and the companies that offer them) is going up, costs are coming down.
For example, in addition to offering all Vanguard ETFs commission-free (an offer that's been around for nearly a decade), last summer we expanded that to include nearly all ETFs, including around 1,800 ETFs from about 100 other companies.
Wider acceptance of ETFs, with more alternatives and more firms offering them, can make it harder to sift through everything that's out there. That's why I was excited about our narrowed list of Vanguard Select ETFs™ and how it might help make the selection process feel less overwhelming.
Expansion of the industry has also attracted greater scrutiny and increased regulation, so it's important to stick with a company you can trust. For example, while hundreds of new ETFs made their debut in 2018, 80 were completely liquidated as a result of low assets and/or low trading volume.† So it's become increasingly important to be watchful for very narrowly focused "trial and error" ETFs and recognize the ones that are carefully crafted to have longer staying power.
In the end let's not forget that, whether you prefer ETFs or mutual funds, Vanguard's investment philosophy remains the same: Focus first on your investment timeline and finding an appropriate balance between stocks and bonds. Next, look for well-diversified options that could help reduce your investment risk. Then make your specific selections.
You just might find the best choice is an ETF.
***ETF.com Award winners were selected in a 3-step process designed to leverage the insights and opinions of leaders throughout the ETF industry. The awards process began with an open nomination period that ran from December 3, 2018, through January 3, 2019. Following the open nominations, the ETF.com Awards Nominating Committee—made up of senior leaders at ETF.com, Inside ETFs, and FactSet—voted to select up to 5 finalists in each category. Winners from these finalists were selected by a majority vote of the ETF.com Awards Selection Committee, a group of independent ETF experts. Committee members recused themselves from voting in any category in which they or their firms appeared as finalists. Ties were decided where possible with head-to-head runoff votes.
Factor funds are subject to investment style risk, which is the chance that returns from the types of stocks in which the fund invests will trail returns from the stock market. Factor funds are also subject to manager risk, which is the chance that poor security selection will cause the fund to underperform relevant benchmarks or other funds with a similar investment objective.
Rich Powers is the head of ETF Product Management in Vanguard Portfolio Review Department since 2015.
He and his team are responsible for conducting surveillance of competitor products and positioning, meeting with clients and prospects to discuss Vanguard's ETF lineup, publishing on noteworthy developments in the ETF marketplace and Vanguard lineup, and supporting ETF education initiatives.
For the majority of his tenure in Portfolio Review, which began in 2003, Mr. Powers was a senior member of the Fund Oversight and Manager Search team, which is responsible for identifying subadvisory partners for Vanguard's active fund lineup and monitoring the firm, people, process, portfolio, and performance of Vanguard funds on behalf of the firm's senior leadership team and board of directors.
Mr. Powers earned a B.S.B.A. in finance from Shippensburg University and an M.B.A. in investment management from Drexel University.
ETFs can only be bought in whole shares. This makes investing an entire amount, say for a Roth IRA not possible. For instance if VT (total world ETF) costs $74.50 (market price at Vanguard's website as of 4/4/19) then if I invest $6000 I can only buy 80 shares, not 81. 80 shares will buy me $5960, 81 shares would buy $6034.50 which would be over the IRA limit. Therefore, I can only invest $5960, not $6000 in the stockf fund of my choice and $40 sits in money market account not fully invested.
When will ETFs be able to be purchased with fractional shares like mutual funds? Once ETFs can be purchased with fractional shares, this will even the playing field even more, though spreads may always be an issue with ETFs and not so with mutual funds.
Any thoughts on when ETFs will be able to be purchased in fractional shares rather than whole shares as at present?
Third time to post today-4/5/2019, first was this morning 745am, then later this morning at 1130am, still NOTHING, what's going on vanguard?! One more time at 225pm, 4/5/2019!
Does anyone else have problems posting, and ACTUALLy seeing your response posted?
packaged as " INFO ".
or so ago, talking about fixations, and how some folks have 15-20 funds, again I ask-WHY?
USELESS, and their past history pretty much proves this.
If you like etf's-GREAT, if you prefer index funds-GREAT! It doesn't really matter-my opinion.
Because one can trade them "all day long" (as I recall John Bogle put it), is it possible that ETFs are attractive to an investor with a more speculative orientation? Would the prices of ETFs then be more volatile?
There is likely something I'm not understanding here. Still, wouldn't the opportunity to get in and out often encourage acting on greater sensitivity to cycles of fear and greed?
Rich, when the market declined in December, ETF issuance increased by $50 billion in US equities alone, and when the market soared in Jan, it decreased by $5 billion. Explain that? ETFs are not pooled investments. Why do you say "ETF assets under management?" That's misleading. ETFs manage no one's money but the ETF sponsor's. When I buy Vanguard ETFs, Vanguard doesn't get my money (unless I use your brokerage) and doesn't give me a customer account. The person commenting immediately above me has no idea that he owns derivatives that have no redeemable value within the meaning of the Investment Company Act of 1940, thanks to SEC exemptive orders. It's up to the INDUSTRY to tell people the truth about ETFs — the wholesale market, retail market, that APs have no obligation to trade them and you have no obligation to redeem them. The inviolable investment principle of the prudent is KNOW WHAT YOU OWN. The industry is inhibiting that directive.
I have been buying the S&P500 for a long time.
Was Warren Buffet said–do not look for the needle in the hay stack–just buy the whole hay stack–S&P500.
I have only bought–not sell at any time.
Excellent post! I have invested almost exclusively in ETFs since they became available in the market place years ago. I have been pleased with how they perform, ever lower expense ratios and, obviously, diversification. Same day trade executions enhance their overall ability to fit into a well researched portfolio. Add commission-free trades to the mix and I don't understand why anyone would object to them. I switched from mutual funds years ago and I have no regrets. Do some research and you will be pleased with the results. Best regards to everyone. | {
"redpajama_set_name": "RedPajamaC4"
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Michael Louis Vespoli (born December 14, 1946) is a former American rower and rowing coach. He is the founder and chief executive officer of Vespoli USA, Inc., a boat manufacturer in New Haven, Connecticut, that makes shells for rowing teams and individual rowers. Vespoli was born in New Haven, Connecticut.
Career
His career in rowing started in 1964 when he joined the Georgetown University crew as a walk-on. He went on to row with teams that won several national championships, placed in the 1972 Munich Olympics, and in 1974, was part of a team of eight American rowers that won a World Championship in Switzerland. He later coached universities and was a sculling coach for the 1980 Olympics in Moscow, which the U.S. boycotted.
As boat technology developed rapidly in England with the advent of carbon fiber and other materials used by the aerospace industry, Vespoli started his company with the help of his retired machinist father in 1980 in the U.S. His wife Nancy Vespoli, who was on the women's crew at Dartmouth College, a member of the 1980 U.S. Olympic rowing crew and has a master's degree in chemical engineering from M.I.T., is also involved in the business. The company has become a well-known boat maker with models that have won many elite races.
Coaching career
1968 to 1972, Saint Joseph's Preparatory School
1972 to 1974, University of Massachusetts Amherst
1974 to 1977, Wichita State University
1977 to 1980, Yale University
St. Joseph Prep School
National Schoolboy Champions
University of Massachusetts
Two consecutive Dad Vail Men's 8 Champions
Wichita State University
Vespoli was given the goal to develop a nationally recognized rowing program in three years.
1976, Novice Men's 8, Wichita State defeats Yale.
1977, Novice HW Men's 8, Wichita State loses to University of Wisconsin by 2.57 seconds
Yale University
1979, Freshmen HW Men's 8 places first at the Eastern Sprints.
The Tony Johnson-Mike Vespoli coaching combination proves to be formidable. 1979, Yale sweeps in in HW Men's V and JV the Eastern Sprints. Yale's Heavyweight Men's 8 places first again at the 1981 Eastern Sprints. In the four years (1981–1984) following Mike Vespoli's departure as Yale's Frosh Men's Heavyweight coach, Yale's Varsity Men's Heavyweight Crew defeated Harvard in the annual Harvard-Yale race. Yale had not defeated Harvard in 18 years.
From an interview with Rowing News' Jeff Moag, November 2009, Mike Vespoli said, "We won the Eastern Sprints, we beat Harvard, I made enemies with Harry Parker. I think my proudest coaching moment is that when I left Yale and my three classes of freshmen filled the sophomore, junior, and senior years at Yale, those were classes that didn't lose to Harvard."
U.S. National Team
1979, Head Coach, U.S. Lightweight Men; Sılver Medal
1980, Assistant Coach, U.S. Heavyweight Men Scullıng Coach. Dıd not compete due to US boycott.
Mike Vespoli's career has been closely associated with Coach Tony Johnson. Johnson coached Vespoli at Georgetown from 1967 to 1969. Johnson was the assistant coach and Vespoli an oarsman on the 1972 U.S. Olympic Crew. Johnson recommended Vespoli for the Wichita State University position, and Johnson hired Vespoli as Yale's Frosh Men's Coach in 1977. Vespoli is Committee Co-Chair of the Georgetown University Boathouse project where Johnson has coached since 1989.
Vespoli is a former member of the U.S. Olympic Committee for Men's Rowing, a consultant to the President's Commission on Olympic Sports and was an NBC commentator for rowing in Barcelona in 1992.
A member of both the National Rowing Foundation Hall of Fame and the Georgetown Athletic Hall of Fame, Mike has maintained a strong connection to the Georgetown crew program that launched his professionally and personally rewarding career. He served as the chair of the rowing association's board of directors for 16 years and has been a member of the board of regents for another 16. In 2000, Georgetown honored Mike with the Outstanding Service to Athletics award. Mike and his wife, Nancy, are members of the Georgetown 1789 Society.
Mike and Nancy Vespoli established the Vespoli Family Crew Scholarship at Georgetown University in 2003. The scholarship provides financial aid to one or more intercollegiate rowers who demonstrate financial need. The Vespolis have also donated money in support of the Georgetown University Boathouse project. In April 2006, Nancy Vespoli donated a shell to the Dartmouth College Women's Crew. The crew christened the new shell, the "Fast Eddie." The shell was named after Nancy's father, Eddie Parssinen.
In 2000, Vespoli was honored with the "Power Ten" award. The award is given to those who are committed to supporting the sport of rowing, both nationally and internationally.
References
External links
Interview on May 28, 2009
1946 births
Living people
American male rowers
Georgetown Hoyas rowers
UMass Minutemen and Minutewomen rowing coaches
Wichita State Shockers rowing coaches
Yale Bulldogs rowing coaches
World Rowing Championships medalists for the United States
Pan American Games medalists in rowing
Pan American Games gold medalists for the United States
Congressional Gold Medal recipients
Olympic rowers of the United States
Rowers at the 1972 Summer Olympics
Rowers at the 1975 Pan American Games | {
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Сэмюэл Каньон Доу (; 6 мая 1951, дер. Тузон, Гранд-Геде — 9 сентября 1990) — фактический руководитель Либерии с 1980 по 1990 год. Президент Либерии с 1986 по 1990 год.
Биография
Происходил из племени кран из группы народов кру, его отец был солдатом либерийской армии. С 17 лет на службе в вооружённых силах Либерии, дослужился до звания старшего сержанта (с мая 1979 года). В 1979 году прошёл курс подготовки в подразделении коммандос. В армии сблизился с оппозицией.
12 апреля 1980 года он возглавил военный переворот, в ходе которого президент Республики Уильям Толберт был убит. Вслед за этим были убиты или казнены и многие члены правительства Уильяма Толберта. Доу стал главой государства и председателем Совета народного спасения, присвоив себе звание генерала. С 1981 года — верховный главнокомандующий вооружённых сил.
В 1984 году упразднил Совет народного спасения, заменив его Временной народной ассамблеей. В июле 1984 года в стране прошёл референдум, одобривший проект новой конституции. Возглавил вновь созданную Национально-демократическую партию, которая выдвинула его на пост президента на всеобщих выборах 15 октября 1985 года (получил 50,9 % голосов и официально стал президентом страны с января 1986 года).
Первоначально его режим пользовался относительно широкой поддержкой, но затем он постепенно установил в стране этническую диктатуру племени кран. Однако всё громче звучали призывы к переходу к гражданскому правлению, и в 1986 году с многочисленными нарушениями он был избран президентом Республики. Чтобы соответствовать Конституции, по которой президентом Республики мог стать только гражданин, достигший возраста 35 лет, в документах приписал себе год.
С его приходом прекратила существование старейшая партия Либерии «Партия истинных вигов». а преемница первой легальной оппозиционной ей силы (Прогрессивного альянса Либерии), Объединённая народная партия, была поставлена вне закона за свой «социалистический уклон». Во внешней политике полностью следовал курсу США и 18 июля 1985 года разорвал дипломатические отношения с СССР.
Пережил несколько попыток переворота. В частности, 1 апреля и 12 ноября 1985 года: в первом случае в него стрелял из пулемёта заместитель командира батальона президентской охраны, во втором — попытка переворота во главе с активным участником переворота 1980 года и главкомом сухопутных сил в 1980—1983 годах генералом Куиуонкпа (погиб в ходе полуторасуточных боёв в столице), уже организовывавшим в 1983 году заговор. Ситуация была использована президентом как повод для разгрома всех несогласных. В ходе последовавших репрессий погибло около 1,5 тысячи человек, главным образом, гражданских лиц.
После попытки государственного переворота Доу объявил по радио и телевидению, что любой, кого найдут на улице после комендантского часа в 18:00, будет считаться мятежником и немедленно казнен.
6 января 1986 года, в день его инаугурации в качестве двадцать первого президента, на стадионе состоялось шоу с несколькими либерийскими девушками, которые артистично танцевали в его честь с различными обручами, позже танцоры танцевали с маракасами, наконец, армия выстроилась в строй и в первую очередь, ряды трубили в толстые трубы.
3 мая 1986 года в выступлении по телевидению С. Доу заявил, что экономика страны на грани краха. Для «совместного принятия решений» в ключевые министерства и финансовые организации были приглашены представители США.
В декабре 1989 года на северо-востоке страны был образован Национальный патриотический фронт Либерии — НПФЛ во главе с Чарльзом Тейлором. В Либерии началась первая гражданская война. Национальный патриотический фронт быстро рос, и вскоре его бойцы контролировали 90 % территории Либерии. Однако от НПФЛ откололась группировка во главе с Принсом Джонсоном, который воевал как против Сэмюэла Доу, так и против Чарльза Тейлора, возглавлявшего НПФЛ. В сентябре 1990 года отряды Джонсона окружили Монровию. Под видом переговоров Джонсон вызвал С. Доу в миссию ООН, там Доу был похищен, а затем жестоко убит (для демонстрации отсутствия его магической защиты) — перед смертью ему сначала сломали руки, позже ошеломленного Доу допрашивают, приковывая кандалы к его ногам, в записях видно, как половой орган Доу был перевязан некой веревкой. После убийства Доу в стране установилась диктатура Чарльза Тейлора, а страна погрузилась в длительную гражданскую войну, унёсшую жизни 150 тысяч человек.
Личная жизнь
Был баптистом. Одно время принадлежал к Первой Баптистской Церкви в городе Зведру.
Память
В Монровии есть футбольный стадион имени Сэмюэла Доу.
Примечания
Фельдмаршалы
Баптисты Либерии
Казнённые президенты
Президенты Либерии
Убитые президенты | {
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At Hoburne, we know what makes a holiday special. It's not just the places you visit, the things you do, or the people you are with - it's how all of this makes you feel that really makes the difference. That deeprooted, away from the hum-drum, all-out total fun and relaxation that all our guests experience when spending time with us. With this in mind, we invest everything we have learned in our 100 years as a family run company in creating those wonderful feelings. Our unique park is set in a wonderful countryside location. Here, you'll find our extensive choice of holiday homes are designed to suit every pocket. Then there's our continued investment in facilities. Pools, playgrounds, cafés and restaurants are refreshed regularly (just like you will be!) And, with our people there to welcome you with a warm smile and help all the way, you're guaranteed that unmistakable Hoburne holiday feeling every time you visit. No wonder many of our customers return year after year.
A forest setting holiday is wholly enchanting. Especially when the setting is Hoburne Bashley. Where the beauty of the Hampshire and Dorset coastlines can be explored on foot, on horseback or by bus. A place where you're never far from a cosy country pub or the freeroaming New Forest ponies. Somewhere your children are safe to cycle or be happy hide-and-seekers on park. And where four-legged family members are warmly welcomed. You'll find Bashley is home to much more besides all this. Because this is the New Forest. Where you'll want to return to before you've even left.
Hoburne Bashley. From the great outdoors… to the great indoors.
At Hoburne Bashley, the outdoors is just as wonderful as the indoors. Our nature trail and 9 hole, par 3 golf course mean you can actively enjoy the forest surroundings; while our beauty h treatment rooms, sauna and steam rooms and entertainment complex mean you can relax on the inside too. If you are bringing the children with you, they will discover a range of fun and exciting activities. Sammy's Squad children's club, an indoor soft play area and outdoor adventure playground, an amusement arcade and two pools will keep them busy day until night. And, with Woodland and Bushcraft courses available for all ages, they can learn something new that they'll remember for years to come. All your dining options are taken care of on-park too. You'll find a choice of food and drink all day at Café h, delicious evening meals at Brasserie 1912 and our bars are the ideal spot to sip a drink or two while you decide what you want to eat.
Whatever you want to do, you'll find it ready and waiting.
*As a guide, prices are per room per night for hotels, guest houses, B&Bs and serviced apartments, per pitch per night for campsites and per week for self-catering accommodation and holiday parks.
From Lyndhurst take A35 signposted Christchurch for approx 10 miles. After passing East Close on left take second turning left onto B3055 signposted to Sway. after approx. 2 1/2 miles go strainght across at roundaboutand park entrance is 1/4 mile on left. | {
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Augusto Chaves Batista (1916 - 1967) fue un micólogo brasileño.
Publicó más de 600 trabajos de investigación, ya sean solos o en colaboración con otros. En el momento de su muerte a la edad de 51 años, Batista era el director del Instituto de Micologia de la Universidad Federal de Pernambuco.
Honores
Epónimos
Varios taxones han sido nombrados en su honor, incluyendo Batistia, publicado por Raffaele Ciferri, colaborador frecuente.
Véase también
Lista de micólogos
Referencias
Enlaces externos
Botánicos de Brasil
Micólogos de Brasil | {
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The 2001–02 Eastern Michigan Eagles men's basketball team represented Eastern Michigan University during the 2001–02 NCAA Division I men's basketball season. The Eagles, led by 2nd year head coach Jim Boone. The Eagles played their home games at the Eastern Michigan University Convocation Center and were members of the West Division of the Mid-American Conference. They finished the season 6–24, 2–16 in MAC play. They finished 6th in the MAC West. They were knocked out in the 1st round of the MAC Tournament by the Toledo Rockets.
Roster
Source:
The team captains were Ricky Cottrill, Steve Pettyjohn and Ryan Prillman.
Schedule
|-
!colspan=9| Regular Season
|-
!colspan=9| 2002 MAC men's basketball tournament
Season Highlights
11/26 vs Maryland Eastern Shore
Ryan Prillman records a career-high 14 rebounds.
12/01 vs Tennessee Tech
Ricky Cottrill sets individual three-point shooting record at EMU Convocation Center with 8 three-point field goal.
EMU sets a team record with 14 three-point field goals.
EMU sets a Convocation Center record with 14 three-point field goals.
12/03 vs Detroit
Ryan Prillman scores a career-high 24 points.
01/07 at Akron
Ricky Cottrill scores a career-high 31 points.
EMU wins its first MAC road game under head coach Jim Boone.
References
Eastern Michigan Eagles men's basketball seasons
Eastern Michigan
2001 in sports in Michigan
2002 in sports in Michigan | {
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Román Fernando Martínez (Morón (Buenos Aires), Argentina, 27 de març de 1983) és un futbolista argentí que juga de centrecampista al Reial Club Deportiu Espanyol de Barcelona.
Va iniciar la seva carrera professional com a futbolista en el Deportivo Morón l'any 2000. Tres anys més tard, va traslladar-se a l'Arsenal de Sarandí. No obstant això, a causa que no podia demostrar les seves facultats com a futbolista, va optar per passar-se al Tigre l'any 2006. En aquest equip, el jugador va començar-se a destacar i va esdevenir una de les màximes figures de l'equio i del campionat. Al maig de 2008, el director tècnic de la selecció de futbol de l'Argentina, Alfio Basile, va elogiar el seu estil de joc i va dir que el tindria en compte per a les properes convocatòries. El juliol de 2008, fitxa pel RCD Espanyol, començant així una nova etapa com a jugador fora del continent americà.
Futbolistes de la província de Buenos Aires
Futbolistes del RCD Espanyol de la dècada de 2000
Futbolistes del Club Atlético Tigre
Futbolistes del CD Tenerife
Futbolistes de l'Arsenal de Sarandí
Futbolistes del Club Atlético Lanús
Futbolistes del Club Estudiantes de La Plata
Futbolistes del San Lorenzo de Almagro | {
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{"url":"https:\/\/analyzethedatanotthedrivel.org\/tag\/latex2e\/","text":"# Pandoc based J Syntax\u00a0Highlighting\n\nJohn MacFarlane\u2019s excellent command line utility Pandoc is a Haskell program that converts to and from various text markup languages. Pandoc\u2019s help option lists its supported input and output formats.\n\nThe following examples are Linux bash shell commands. Windows shell commands are identical.\n\n$pandoc --help pandoc [OPTIONS] [FILES] Input formats: native, json, markdown, markdown+lhs, rst, rst+lhs, docbook, textile, html, latex, latex+lhs Output formats: native, json, html, html5, html+lhs, html5+lhs, s5, slidy, slideous, dzslides, docbook, opendocument, latex, latex+lhs, beamer, beamer+lhs, context, texinfo, man, markdown, markdown+lhs, plain, rst, rst+lhs, mediawiki, textile, rtf, org, asciidoc, odt, docx, epub Some Pandoc conversions are better than others. Pandoc does a better job of turning markdown into LaTeX than LaTeX into markdown. It\u2019s also better at converting HTML into LaTeX than LaTeX into HTML. Pandoc works best when converting markdown, the simplest of its inputs, to other formats. In fact Pandoc does such a good job of converting markdown to HTML, HTML+MathJax, LaTeX or PDF that many writers are now saving their source documents as markdown text knowing they can easily produce other formats as needed. As handy as Pandoc\u2019s markup conversions are this nifty tool also supports syntax highlighting for over a hundred programming languages. Unfortunately, my favorite language J is not on Pandoc\u2019s list of highlighted languages. [1] Where have I run into this problem before? Luckily for me Pandoc is an open source tool and Pandoc\u2019s author has made it easy to add new highlight languages. Pandoc is a Haskell program. I\u2019ve been aware of Haskell\u2019s existence for years but until I decided to take on this specialized Pandoc hack I had never studied or used the language. Usually when you set out to modify a large program in an unfamiliar programming language you\u2019re in for what can only be described as an f\u2019ing educational experience. It\u2019s a testament to the quality of the Haskell\u2019s global libraries and standard tools that a complete Haskell novice can effectively tweak large Haskell programs. Here\u2019s what you have to do. 1. Install the Haskell Platform. The Haskell Platform is available for all the usual suspects. I\u2019ve used both the Windows and Linux versions. I almost installed the Mac version on my wife\u2019s Mac but resisted the urge. 2. Get with the Cabal. Cabal is the main Haskell package distribution and build utility. Cabal comes with the Haskell Platform and is easily accessed from the command line. Type cabal --help in your favorite shell to view the program\u2019s options. 3. Spend sometime playing with Hackage. Hackage contains a large set of Haskell packages including all the source code required to build Pandoc. After installing the Haskell Platform and familiarizing yourself with Cabal try building Pandoc. This will thoroughly exercise your Haskell system. Instructions for building Haskell packages are here. After reading the package build instructions run the following in your command shell: $ cabal update\n$cabal install pandoc This will download, compile and install a number of Haskell packages. Where Cabal puts the packages depends on your operating system. Cabal saves Linux packages in a hidden local directory. On my machine they ended up in: \/home\/john\/.cabal\/lib If you managed to build Pandoc you\u2019re now ready to add a new highlighting language. Pandoc uses the highlighting-kate package for highlighting. highlighting-kate works by reading a directory of Kate editor xml language regex based definition files and generating custom language parsers. We want to generate a custom J parser so we need to download highlighting-kate source and add a Kate xml definition file for J. You can find such a J Kate file on the J Wiki here. Download this file by cutting and pasting and save it as j.xml. Now do the following. 1. Run the Pandoc version command pandoc --version of the Pandoc you just built to determine the version of the highlighting-kate package you need. 2. Use Cabal to unpack the required highlighting-kate package. This downloads the required package and creates a temporary subdirectory in your current directory that contains package source code. $ cabal unpack highlighting-kate-0.5.3.2\nUnpacking to highlighting-kate-0.5.3.2\/\n3. Move into the temporary subdirectory and copy the Kate j.xml\u00a0file to the package\u2019s xml subdirectory.\n$cd highlighting-kate-0.5.3.2$ cp ~\/pd\/blog\/j.xml ~\/temp\/highlighting-kate-0.5.3.2\/xml\/j.xml\n4. Configure the package.\n$cabal configure Resolving dependencies... Configuring highlighting-kate-0.5.3.2... 5. Build the highlighting-kate package. $ cabal build\nResolving dependencies...\n... (omitted) ...\n6. If highlighting-kate\u00a0builds without problems run the command.\n$runhaskell ParseSyntaxFiles.hs xml Writing Text\/Highlighting\/Kate\/Syntax\/SqlPostgresql.hs Writing Text\/Highlighting\/Kate\/Syntax\/Scala.hs ... (omitted) ... ParseSyntaxFiles scans the package\u2019s xml subdirectory and generates language specific parsers. If all goes well you will find J.hs in this directory. ~\/temp\/highlighting-kate-0.5.3.2\/Text\/Highlighting\/Kate\/Syntax J.hs, like all the files referred to in this post, are available in the files sidebar in the Haskell\/Pandoc subdirectory. 7. Now rebuild the highlighting-kate package. This compiles your new J.hs parser file. $ cabal build\nResolving dependencies...\n... (omitted) ...\n8. After rebuilding the package run the Cabal copy command to put the modified package in the expected library location.\n$cabal copy Installing library in \/home\/john\/.cabal\/lib\/highlighting-kate-0.5.3.2\/ghc-7.4.1 Now that the highlighting library is up to date we have to rebuild Pandoc. To do this mirror the steps taken to download and build the highlighting package. 1. Use Cabal to unpack the Pandoc package. $ cd ~\/temp\n$cabal unpack pandoc-1.9.4.2 Unpacking to pandoc-1.9.4.2\/ 2. Switch to the Pandoc subdirectory and configure the package. $ cabal configure\nResolving dependencies...\n[1 of 1] Compiling Main ( Setup.hs, dist\/setup\/Main.o )\n... (omitted) ...\n3. Rebuild Pandoc.\n$cabal build Building pandoc-1.9.4.2... Preprocessing executable 'pandoc' for pandoc-1.9.4.2... ... (omitted) ... If all goes well a Pandoc executable will be written to this subdirectory. ~\/temp\/pandoc-1.9.4.2\/dist\/build\/pandoc 4. You can check the new executable by running pandoc --version. The result should display J in the list of supported languages. Now that we have a Pandoc that can highlight J we\u2019re almost ready to blog gaudy J code. However before doing this we need to install some custom CSS. Custom CSS is not available on free WordPress.com blogs. To apply custom coloring schemes get the custom package and learn how to use WordPress\u2019s custom CSS editor. As daunting as this sounds it\u2019s no problemo for my limited purposes. To enable tango style Pandoc syntax highlighting on your WordPress blog paste tango.css into the custom CSS editor, check the \u201cAdd my CSS to CSS stylesheet\u201d button and then press the \u201cSave Stylesheet\u201d button. Now your WordPress blog will be sensitive to the HTML span tags generated by Pandoc. To show that all this hacking works as intended you can check out the Pandoc generated versions of this blog post. I\u2019ve posted the original markdown source with PDF, LaTeX and HTML versions. All these files are available via the files sidebar. You can generate the HTML version with the command: $ pandoc -s --highlight-style=tango PJHighlight.markdown -o PJHighlight.html\n\nTo get other versions simply change the file extension of the output -o file.\n\nBonebridge puzzle in MYST IV\nClick for \u201cHaven Age\u201d Walkthrough\n\nFinally we are ready to post syntax highlighted J code. The following J verb bonebridge generates all \u201clikely\u201d lock combinations for the\u00a0MYST IV Bonebridge puzzle in Pandoc\u2019s tango style.\u00a0At one time I was a big fan of MYST computer games. I always enjoyed being lost in a beautiful puzzle which, if you discard the beautiful bit, is a pretty accurate description of my programmer day job.\n\nbonebridge=:3 : 0\n\nNB.*bonebridge v-- lists totem symbol permutations for bone\nNB. bridge.\nNB.\nNB. The solution to this MYST IV puzzle is similiar to the book\nNB. shelf puzzle in Tomanha but requires far more exploration of\nNB. the age.\nNB.\nNB. You are confronted with 5 bones on the lock. All the bones\nNB. move independently. You can see the settings for 4 bones. One\nNB. bone has a broken display. The four visible bones have 8\nNB. symbols on them in the same order. The 5th bone also has 8\nNB. symbols and you can \"safely\" infer they are in the same order\nNB. as the visible bones.\nNB.\nNB. Four bone symbols match symbols found on totem poles\nNB. distributed around the age. There is a 5th totem pole but\nNB. fruit eating mangrees obscure the totem symbol and I have\nNB. never seen it. The totem poles are associated with age\nNB. animals. In addition to the totem poles there is a chart in\nNB. the mangree observation hut that displays a triangular\nNB. pattern of paw prints. The paw prints define an animal\nNB. ordering. The order seems to be how dangerous a particular\nNB. animal is; big scary animals are at the top and vegetarians\nNB. are at the bottom.\nNB.\nNB. Putting the clues together you infer:\nNB.\nNB. a) the bridge combination is some permutation of five\nNB. different symbols\nNB.\nNB. b) two possible symbol orders are given by the paw chart\nNB.\nNB. c) you know 5 symbols and the 4th is one of the remaining 4\nNB.\nNB. If this is the case the number of possible lock settings\nNB. shrinks from 32768 to the ones listed by this verb.\nNB.\nNB.\nNB. bonebridge 0\n\nNB. known in paw order\nknown=. s: ' square triangle hourglass yingyang'\nunknown=. s: ' clover cross xx yy'\n\nNB. all possible lock permutations\nsettings=. ~. 5 {.\"1 tapl known,unknown\nassert. ((!8)%!8-5) = #settings\n\nNB. possible ordering - we don't know\nNB. what the fifth symbol is but it\nNB. occurs in the 3rd slot\norder=. 8#s:<''\norder=. known (0 1 6 7)} order\norder=. unknown (2 3 4 5)} order\n\nNB. keep unknown only in 3rd slot\nsettings=. settings #~ -. +.\/\"1 (0 1 3 4{\"1 settings) e. unknown\nsettings=. settings #~ (2 {\"1 settings) e. unknown\n\nsrsm=. 1 : '*.\/\"1 u\/&> 2 <\\\"1 y'\n\nNB. retain strictly increasing and strictly decreasing rows\n)\n\n[1] J has its own syntax highlighting tools but they are not part of a document generation system. Pandoc\u2019s highlighters elegantly feed into many output formats making them far more useful.\n\n# WordPress to LaTeX with Pandoc and J: Using TeXfrWpxml.ijs (Part\u00a03)\n\nWordPress to LaTeX\n\nIn this post I will describe how to use the J script TeXfrWpxml.ijs to generate LaTeX source from WordPress export XML.\u00a0 I am assuming you have worked through (Part 1) and (Part 2) and have:\n\n1. Successfully installed and tested Pandoc.\n2. Installed and tested a version of J.\n3. Set up appropriate directories (Part 2).\n4. Know how to use LaTeX.\n\nItem #4 is a big if.\u00a0 Inexperienced LaTeX users will probably not enjoy a lot of success with this procedure as the source generated by TeXfrWpxml.ijs requires manual edits to produce good results. \u00a0However, if you\u2019re not a LaTeX guru, do not get discouraged. It\u2019s not difficult to create blog documents like bm.pdf.\n\nHow to download WordPress export XML is described here. \u00a0Basically you go to your blog\u2019s dashboard, select Tools, choose Export \u00a0and select the All content option.\n\nTools > Export > All Content\n\nc:\/pd\/blog\/wordpress\/analyzethedatanotthedrivel.wordpress.xml\n\n#### Step 2: download TeXfrWpxml.ijs\n\nDownload TeXfrWpxml.ijs and remember where you save it. \u00a0I put this script here.\n\nc:\/pd\/blog\/TeXfrWpxml.ijs\n\n#### Step 3: start J and load TeXfrWpxml.ijs\n\nTeXfrWpxml.ijs was generated from JOD dictionaries. With JOD it\u2019s easy to capture root word dependencies and produce complete standalone scripts. TeXfrWpxml.ijs needs only the standard J load profile to run. \u00a0It does not require any libraries or external references and should run on all Windows and Linux versions of J after 6.01. \u00a0Loading this script is a simple matter of executing:\n\nload 'c:\/pd\/blog\/TeXfrWpxml.ijs'\n\nThe following shows this script running in a J 7.01 console. The console is the most stripped down J runtime.\n\n#### Step 4: review directories and necessary LaTeX files\n\nThe conversion script assumes proper directories are available\u00a0up: see Part 2. The first time you run TeXfrWpxml.ijs it\u2019s a good idea to check that the directories and files the script is expecting are the ones you want to process.\u00a0 You can verify the settings by displaying TEXFRWPDIR, TEXINCLUSIONS, TEXROOTFILE and TEXPREAMBLE.\n\n TEXPREAMBLE\nbmamble.tex\nTEXFRWPDIR\nc:\/pd\/blog\/wp2latex\/\nTEXINCLUSIONS\ninclusions\nTEXROOTFILE\nbm.tex\nTEXPREAMBLE\nbmamble.tex\n\n\nIf all these directories and files exist go to step (5).\n\n#### Step 5: make sure you are online\n\nThe first time you run the converter it will attempt to download all the images referenced in your blog. This is where wget.exe gets executed.\u00a0 Obviously to download anything you must be connected to the Internet.\n\n#### Step 6: run LatexFrWordpress\n\nRun the verb LatexFrWordpress.\u00a0 The monadic version of this verb takes a single argument: the complete path and file name of the export XML file you downloaded in step (1).\n\nxml=: 'c:\/pd\/blog\/wordpress\/analyzethedatanotthedrivel.wordpress.xml'\n\nLatexFrWordpress xml\n\nAs the verb runs you will see output like:\n\n LatexFrWordpress xml\nFake Programming\nLaws or Suggestions\nLens Lust\n\n... many lines omitted ...\n\nWordPress to LaTeX with Pandoc and J: LaTeX Directories (Part 2)\n+-++\n|1||\n+-++\n\n\nWhen the verb terminates you should have a directory c:\/pd\/blog\/wp2latex full of *.tex files:\u00a0 one file for each blog post. Now the hard work starts.\n\n#### Step 7: editing LaTeX posts\n\nThe conversion from WordPress XML to LaTeX produces files that require manual edits. The more images, video, tables and other elements in your posts the more demanding these edits will become.\u00a0 My blog has about one image per post.\u00a0 Most of these images are wrapped by text. LaTeX has a mind of its own when it comes to floating figures\u00a0and getting illustrations to behave requires far more parameter tweaking than it should. This is a longstanding weakness of LaTeX that pretty much everyone bitches about. My advice is start at the front of your document and work through it post by post. The files generated by LatexFrWordpress do not attempt to place figures for you but they do bolt in ready-made figure templates as comments that you can experiment with.\u00a0 Each post file is also set up for separate LaTeX compilation. You don\u2019t have to compile your entire blog to tweak one post. The one good thing about this edit step is once you have sorted out your old posts you do not have to revisit them unless you make major global document changes. The next time you run LatexFrWordpress it will only bring down new posts and images.\n\n#### Step 8: compile your LaTeX blog\n\nI use batch files and shell scripts to drive LaTeX compilations. \u00a0I processed my blog with this batch file.\n\necho off\nrem process blog posting (bm.tex) root file\ntitle Running Blog Master\/LaTeX ...\n\nrem first pass for aux file needed by bibtex\nlualatex bm\n\nrem generate\/reset bbl file\nbibtex bm\nmakeindex bm\n\nrem resolve all internal references - may\nrem comment out when debugging entire document\nlualatex bm\nlualatex bm\n\nrem display pdf - point to prefered PDF reader\ntitle Blog Master\/LaTeX complete displaying PDF ...\n\"C:\\Program Files\\SumatraPDF\\SumatraPDF.exe\" bm.pdf\n\n\nThe presence of Unicode APL, see this post, forced me to use lualatex. I needed some very nonstandard APL fonts. \u00a0See bm.pdf\u00a0\u2014\u00a0also available on the Download this Blog page \u2014 to judge the effectiveness of my edits.\u00a0Producing nice figure laden typeset blog documents is work but, as I will describe in the next post, producing image free eBooks is a simple and far less laborious variation on this process.\n\n# WordPress to LaTeX with Pandoc and J: LaTeX Directories (Part\u00a02)\n\nWordPress to LaTeX\n\nIn this post I will describe the LaTeX directory structure the J script TeXfrWpxml.ijs is expecting. To convert WordPress export XML to LaTeX with this script you will have to set up similar directories.\n\nLaTeX documents are built from *.tex[1] files. This makes LaTeX more like a compiled programming language than a word processing program. There are advantages and disadvantages to the LaTeX way. In LaTeX\u2019s favor, the system is enormously adaptable, versatile and powerful. There is very little that LaTeX\/TeX and associates cannot do. \u00a0Unfortunately, \u201cwith great power comes great responsibility.\u201d LaTeX is demanding! You have to study LaTeX like any other programming language. It\u2019s not for everyone but for experienced users it\u2019s the best way to produce documents with the highest typographic standards.\n\n#### LaTeX directory structure\n\nTo use LaTeX efficiently it\u2019s wise to pick a document directory structure and stick with it. I use a simple directory layout. Each document has a root directory. The root directory used by TeXfrWpxml.ijs\u00a0is:\n\n Windows c:\/pd\/blog\/wp2latex Linux \/home\/john\/pd\/blog\/wp2latex\n\nI put my document specific *.tex, *.bib, *.sty and other LaTeX\/TeX files in the root. To handle graphics I create an immediate subdirectory called inclusions.\n\nc:\/pd\/blog\/wp2latex\/inclusions\n\nThe inclusions directory holds the document\u2019s *.png, *.jpg, *.pdf, *.eps and other graphics files. \u00a0To reference files in the inclusions directory with the standard LaTeX graphicx package insert\n\n\\usepackage{color,graphicx,subfigure,sidecap}\n\\graphicspath{{.\/inclusions\/}}\n\nin your preamble. Finally, to track document changes I create a GIT repository in the root directory.\n\nc:\/pd\/blog\/wp2latex\/.git\n\n#### Self contained directories\n\nI take care to keep my document directories self-contained. Zipping up the root and inclusions directory collects all the document\u2019s files. This means that I sometimes have to copy files that are used in more than one document. Many LaTeX users maintain a common directory for such files but I\u2019ve found that common directories complicate moving documents around. You\u2019re always forgetting something in the damn common directory or you are copying a buttload of mostly irrelevant files from one big confusing common directory to another.\n\n#### TeXfrWpxml.ijs files\n\nThe TeXfrWpxml.ijs\u00a0script searches for these files in the root directory.\n\n bm.tex Main LaTeX root file bmamble.tex LaTeX preamble\n\nbm.tex references bmtitlepage.tex. \u00a0I prefer a separate title page file; simply comment out this file if you create titles in other ways. The zip file wp2latex.zip contains a test directory in the format expected by TeXfrWpxml.ijs.\u00a0 It also has a subset of my blog posts already converted to LaTeX. To get ready for WordPress to LaTeX with Pandoc and J: Using TeXfrWpxml.ijs (Part 3) download wp2latex.zip and attempt to compile bm.tex. \u00a0You might have to download a number of LaTeX packages.\u00a0 Once you have successfully compiled bm.tex\u00a0you are ready for the next step.\n\n[1] LaTeX uses many other file types but key files are usually *.tex files.\n\n# WordPress to LaTeX with Pandoc and J: Prerequisites (Part\u00a01)\n\n#### There are no quick WordPress to LaTeX fixes\n\nWordPress to LaTeX\n\nOver the next three posts I will describe how to convert WordPress\u2019s export XML to LaTeX source code. \u00a0I know that many of you are looking for a quick WordPress to LaTeX fix; unfortunately there are no quick fixes. The two formats come from different worlds and are used in different ways. \u00a0Producing useful LaTeX source from WordPress export XML will require manual edits. \u00a0My goal here is to minimize manual edits, produce high quality LaTeX source and to outline what you will have to contend with. To get an idea of what you can expect download the LaTeX compiled version of this post.\n\n#### Visual and Logical composition\n\nWordPress and LaTeX are examples of the two basic approaches, visual\u00a0and\u00a0logical,\u00a0taken by writing software.\u00a0 Visual systems value appearance. It matters what things look like and no effort is spared to get the right look. Logical systems value content.\u00a0What\u2019s said is far more important than what it looks like. Logical systems impose order and structure and typically defer visual elements.\u00a0 As you might expect there is no such thing as a pure visual or logical writing system.\u00a0Successful systems use both approaches to a greater or lesser degree. Composing WordPress blog posts is roughly 35% visual and 65% logical.[1]\u00a0 LaTeX composition is about 10% visual and 90% logical. The numbers do not line up; there is a basic mismatch here.\n\nMany format X to LaTeX converters tackle this mismatch by attempting to maintain visual fidelity.\u00a0This is a catastrophic error that renders the entire conversion useless. \u00a0Here\u2019s a hint. If you\u2019re using a predominantly logical system like LaTeX you don\u2019t give a rodent\u2019s posterior about visual fidelity. This method dispenses with all but the most basic of visual elements.\u00a0No attempt is made to preserve fonts, type sizes, image scale, justification, hyphenation, text color and so forth.\u00a0 The goal is to produce working LaTeX source that can be transformed to whatever final layout the author desires.\n\n#### Prerequisite\u00a0Software\n\nI use two programs to transform WordPress export XML to LaTeX: \u00a0the J programming language and John MacFarlane\u2019s Pandoc. \u00a0Pandoc is an excellent text mark-up to mark-up converter.\u00a0 It wisely avoids attempting to convert entire complex documents and focuses on getting parts of documents right. \u00a0It does a particularly good job of converting HTML to LaTeX which is a crucial part of this process. \u00a0I use Pandoc to transform the HTML embedded in WordPress export XML CDATA elements to *.tex files and I use J to preprocess and post process Pandoc inputs and outputs and to stitch everything together into a set of LaTeX ready files.\n\nDownload Pandoc from here. I use the Windows command line version. There are Linux and Mac versions as well.\u00a0Download J from here.\u00a0 The easiest J install is the 32 bit Windows J 6.02 version.\u00a0Other versions require additional steps to configure and deploy. If you are already a J user there is no need to install a particular system but you will need:\n\n1. The task library require 'task'\n2. The utility program wget.exe\n\nBoth of these components are typically part of the J distribution.\n\n#### Install and check prerequisites\n\nTo continue download and install Pandoc and J and run the following tests; if you succeed you\u2019re system is ready for WordPress to LaTeX with Pandoc and J: LaTeX Directories (Part 2).\n\nPandoc Test:\n\nDownload the test file: cdata.html and run Pandoc from the command line:\n\npandoc \u2013o cdata.tex cdata.html\n\ncdata.html is an example of the HTML code you find in WordPress export XML CDATA elements. \u00a0Note: required files are also\u00a0available\u00a0in the files sidebar in the WordPress to LaTeX directory.\n\nJ Test:\n\nStart a J session and enter the following commands:\n\nrequire 'task'\n\nshell 'wget \u2013help'\n\nshell 'wget http:\/\/conceptcontrol.smugmug.com\/photos\/i-mNK4RHL\/0\/L\/i-mNK4RHL-L.png'\n\nIf the shell command is properly loaded and wget.exe is found you will see help text. The second shell command downloads an image file.\u00a0 Downloading post images is part of the overall conversion process.\n\n[1] Actually this is not bad. Page layout systems are far worse. A typical layout system might be 90% visual and 10% logical making layout systems polar opposites of LaTeX.\n\n# Typesetting UTF8 APL code with the LaTeX lstlisting\u00a0package\n\nUTF8 APL characters within a LaTeX lstlisting environment. Click for *.tex source code\n\nTypesetting APL source code has always been a pain in the ass! In the dark ages, (the 1970\u2019s), you had to fiddle with APL type-balls and live without\u00a0luxuries\u00a0like lower case letters. With the advent of general outline fonts it became technically possible to render APL\u00a0glyphs on standard display devices provided you:\n\n1. Designed your own APL font.\n2. Mapped the atomic vector of your APL to whatever encoding your font demanded.\n3. Wrote WSFULL\u2018s of junk transliteration functions to dump your APL objects as font encoded text.\n\nIt\u2019s a testament to either the talent, or pig headedness of APL programmers, that many actually did this. We all hated it! We still hate it! But, like an abused spouse, we kept going back for more. \u00a0It\u2019s our fault; if we loved APL more it would stop hitting us!\n\nWhen Unicode appeared APL\u2019ers cheered \u2014 our long ASCII nightmare was ending. The more politically\u00a0astute\u00a0worked to include the APL characters in the Unicode standard.\u00a0Hey if Klingon is there why not APL? Everyone thought it was just a matter of time until APL vendors abandoned their nonstandard atomic vectors and fully embraced Unicode. With a few notable exceptions we are still waiting. While we wait the problem of typesetting APL source code festers.\n\nMy preferred source code listing tool is the $\\LaTeX$ lstlisting package. lstlisting works well for standard ANSI source code. \u00a0I use it for J, C#, SQL, C, XML,\u00a0Ocaml, Mathematica, F#, shell scripts and $\\LaTeX$ source code, i.e. everything except APL! lstlisting is an eight bit package; it will not handle arbitrary Unicode out of the box. \u00a0I didn\u2019t know how to get around this so I handled APL by enclosing UTF8 APL text in plain \\begin{verbatim} \u2026 \\end{verbatim} environments. This works for XeLaTeX and LuaLaTeX but you lose all the lstlisting goodies. Then I saw an interesting tex.stackexchange.com\u00a0posting about The \u2018listings\u2019 package and UTF-8. One solution to the post\u2019s \u201cFrench ligature problem\u201d showed how to force Unicode down\u00a0lstlisting\u2018s throat.\u00a0I wondered if the same method would work for APL. It turns out that it does!\n\nIf you insert the following snippet of TeX code in your document preamble LuaLaTeX and XeLaTeX\u00a0will properly process UTF8 APL text in lstlisting environments. You will need to download and install the\u00a0APL385 Unicode font if it\u2019s not on your system. \u00a0A test $\\LaTeX$ document illustrating this hack is available here. The compiled PDF is available here. As always these files can be accessed in the files sidebar.\n\n% set lstlisting to accept UTF8 APL text\n\\makeatletter\n\\lst@InputCatcodes\n\\def\\lst@DefEC{%\n\\lst@CCECUse \\lst@ProcessLetter\n^^80^^81^^82^^83^^84^^85^^86^^87^^88^^89^^8a^^8b^^8c^^8d^^8e^^8f%\n^^90^^91^^92^^93^^94^^95^^96^^97^^98^^99^^9a^^9b^^9c^^9d^^9e^^9f%\n^^b0^^b1^^b2^^b3^^b4^^b5^^b6^^b7^^b8^^b9^^ba^^bb^^bc^^bd^^be^^bf%\n^^c0^^c1^^c2^^c3^^c4^^c5^^c6^^c7^^c8^^c9^^ca^^cb^^cc^^cd^^ce^^cf%\n^^d0^^d1^^d2^^d3^^d4^^d5^^d6^^d7^^d8^^d9^^da^^db^^dc^^dd^^de^^df%\n^^e0^^e1^^e2^^e3^^e4^^e5^^e6^^e7^^e8^^e9^^ea^^eb^^ec^^ed^^ee^^ef%\n^^f0^^f1^^f2^^f3^^f4^^f5^^f6^^f7^^f8^^f9^^fa^^fb^^fc^^fd^^fe^^ff%\n^^^^20ac^^^^0153^^^^0152%\n^^^^20a7^^^^2190^^^^2191^^^^2192^^^^2193^^^^2206^^^^2207^^^^220a%\n^^^^2218^^^^2228^^^^2229^^^^222a^^^^2235^^^^223c^^^^2260^^^^2261%\n^^^^2262^^^^2264^^^^2265^^^^2282^^^^2283^^^^2296^^^^22a2^^^^22a3%\n^^^^22a4^^^^22a5^^^^22c4^^^^2308^^^^230a^^^^2336^^^^2337^^^^2339%\n^^^^233b^^^^233d^^^^233f^^^^2340^^^^2342^^^^2347^^^^2348^^^^2349%\n^^^^234b^^^^234e^^^^2350^^^^2352^^^^2355^^^^2357^^^^2359^^^^235d%\n^^^^235e^^^^235f^^^^2361^^^^2362^^^^2363^^^^2364^^^^2365^^^^2368%\n^^^^236a^^^^236b^^^^236c^^^^2371^^^^2372^^^^2373^^^^2374^^^^2375%\n^^^^2377^^^^2378^^^^237a^^^^2395^^^^25af^^^^25ca^^^^25cb%\n^^00}\n\\lst@RestoreCatcodes\n\\makeatother\n\n\n# More on Kindle Oriented\u00a0LaTeX\n\nI\u2019ve been compiling $\\LaTeX$ PDFs for the Kindle. If you like $\\LaTeX$ typefaces,\u00a0especially\u00a0mathematical fonts, you\u2019ll love how they render on the Kindle. It\u2019s a good thing because you won\u2019t like the Kindle\u2019s cramped page dimensions. For simple\u00a0flow-able\u00a0text this isn\u2019t a big deal but for complex $\\LaTeX$ documents it is!\n\nThere are two basic $\\LaTeX$ $\\Longrightarrow Kindle$ \u00a0workflows.\n\n1. Convert your $\\LaTeX$ to HTML and then convert the HTML to\u00a0mobi.\n2. Compile your $\\LaTeX$ for Kindle page dimensions.\n\nFor simple math and figure free documents\u00a0mobi is the best choice because it\u2019s a native Kindle format. You will be able to re-flow text and change font sizes on the fly. There are many $\\LaTeX$ to HTML converters. This is a good summary of your options. You can also find a variety of HTML to mobi converters. I\u2019ve used Auto Kindle; it\u2019s slow but produces decent results.\n\nCompiling $\\LaTeX$ for Kindle page dimensions is more work. First decide what works best for your document: landscape or portrait. Portrait is the Kindle default but I\u2019ve found that landscape is better for math and figure rich documents. You can flip back and forth between landscape and portrait on the Kindle but it will not\u00a0re-paginate\u00a0PDFs. Of course with mobi this is no problemo!\n\nAfter\u00a0choosing\u00a0a basic layout expunge all hard-coded lengths from your source\u00a0*.tex\u00a0files. Replace all fixed lengths with\u00a0relative page\u00a0lengths. For example,\u00a04in\u00a0might become 0.75\\textwidth. If you have hundreds of figures and images to adjust write a little program to replace fixed lengths. I did this while\u00a0preparing a Kindle version\u00a0of Hilbert\u2019s\u00a0Foundations of Geometry.\n\nThe next hurdle to overcome is the Kindle\u2019s blase attitude about length units. $\\LaTeX$ is extremely precise: an inch is an inch to six\u00a0decimals. This is not the case on the Kindle! You will have to load your PDFs on the Kindle and inspect margins for text overflows. Be prepared for a few rounds of page dimension tweaking! For more details about preparing $\\LaTeX$ source check out\u00a0LaTeX Options for Kindle.\n\nFinally, after you have compiled your PDF and loaded it on your Kindle, there are some Kindle options you should set to optimize your PDF reading experience. My next post will walk you through setting these options.\n\nThe following *.tex file loads packages that are useful for Kindle sizing. It also shows how to print out $\\LaTeX$ dimensions with the\u00a0printlen\u00a0package.\n\n% A simple test document that displays some packages and settings\n% that are useful when compiling LaTeXe documents for the Kindle.\n% Compile with pdflatex or xelatex.\n%\n% Tested on MikTeX 2.9\n% July 22, 2011\n\n\\documentclass[12pt]{article}\n\n% included graphics in immediate subdirectory\n\\usepackage{graphicx}\n\\graphicspath{{.\/image\/}}\n\n% extended coloring\n\\usepackage[usenames,dvipsnames]{color}\n\n% hyperref link colors are chosen to display\n% well on Kindle monochrome devices\npdfcreator={MikTeX+LaTeXe with hyperref package},\npdfkeywords={your,key,words},\n]{hyperref}\n\n\\usepackage{breqn} % automatic equation breaking\n\\usepackage{microtype} % microtypography, reduces hyphenation\n\n% kindle page geometry (no page numbers)\n%\\usepackage[papersize={3.6in,4.8in},hmargin=0.1in,vmargin={0.1in,0.1in}]{geometry}\n\n% portrait kindle page geometry space reserved for page numbers\n\\usepackage[papersize={3.6in,4.8in},hmargin=0.1in,vmargin={0.1in,0.255in}]{geometry}\n\n% landscape geometry\n%\\usepackage[papersize={4.8in,3.6in},hmargin={0.1in,0.18},vmargin={0.1in,0.255in}]{geometry}\n\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyfoot{} % clear page footer\n\n\\setlength{\\abovecaptionskip}{2pt} % space above captions\n\\setlength{\\belowcaptionskip}{0pt} % space below captions\n\\setlength{\\textfloatsep}{2pt} % space between last top float or first bottom float and the text\n\\setlength{\\floatsep}{2pt} % space left between floats\n\\setlength{\\intextsep}{2pt} % space left on top and bottom of an in-text float\n\n% print LaTeX dimensions\n\\usepackage{printlen}\n\n% reduces footer text separation adjusted for page numbers\n\\setlength{\\footskip}{14pt}\n\n% scales down page number font size if document is at 12pt -> page numbers 10 pt\n\\renewcommand*{\\thepage}{\\footnotesize\\arabic{page}}\n\n\\begin{document}\n\nThe \\verb|\\textwidth| is \\printlength{\\textwidth} which is also\n\\uselengthunit{in}\\printlength{\\textwidth} and\n\\uselengthunit{mm}\\printlength{\\textwidth}.\n\n\\uselengthunit{pt}\nThe \\verb|\\textheight| is \\printlength{\\textheight} which is also\n\\uselengthunit{in}\\printlength{\\textheight} and\n\\uselengthunit{mm}\\printlength{\\textheight}.\n\n\\end{document}\n\n\n# Open Source Hilbert for the\u00a0Kindle\n\nDavid Hilbert\n\nWhile searching for free Kindle books I found Project Gutenberg. Project Gutenberg offers free Kindle books but they also have something better! Would you believe $\\LaTeX$ source code for some mathematical classics.\n\nThe best book I\u2019ve found so far is an English translation of David Hilbert\u2019s Foundations of Geometry. Hilbert\u2019s Foundations exposed some flaws in the ancient treatment of Euclidean geometry and recast the subject with modern axioms. Because it is\u00a0relatively easy to follow, compared to Hilbert\u2019s more recondite publications, this little book exercised disproportionate influence on 20th century mathematics. We still see its style aped, but rarely matched, in mathematics texts today.\n\nI couldn\u2019t resist the temptation of compiling a mathematical classic so I eagerly downloaded the source and ran it through $\\LaTeX$. \u00a0Foundations compiled without problems and generated a nice letter-sized PDF. Letter-size is fine but I was looking for free Kindle books! I decided to invest a little energy modifying the source to produce a Kindle version. Project Gutenberg makes it clear that we are free to modify the source. Isn\u2019t open source wonderful!\n\nConverting Foundations was simple. The main $\\LaTeX$ file included 52 *.png illustrations with hard-coded widths in \\includegraphics commands. I wrote a J script that converted all these fixed\u00a0widths to relative \\textwidth\u2018s. This lets $\\LaTeX$ automatically resize\u00a0images for arbitrary page geometries. When compiled with Kindle page dimensions this fixed most of the illustrations.\u00a0I had to tweak a few wragfig\u2018s to better typeset images surrounded by text. The result is a very readable Kindle oriented PDF version of Hilbert\u2019s book. There are still a few problems. The Table of Contents is\u00a0a plain tabular that does not wrap well and one table rolls\u00a0off the right Kindle margin. Neither of these deficiencies seriously impair the readability of the text. \u00a0If these defects annoy you download the Project Gutenberg source with my modifications and build your own version.\n\nThis little experiment convinced me that providing free classic books, in source code form, is a service to mankind. \u00a0Not only does it allow you to \u201cpublish\u201d classics on new media it also fundamentally\u00a0changes your attitude toward books. Hilbert was one of the great mathematical\u00a0geniuses\u00a0of the 19th and 20th century. It\u2019s hard to suppress we are not worthy moments and maintain a sharp critical eye when reading his \u201cprinted\u201d works. \u00a0You don\u2019t get the same vibe when reading raw $\\LaTeX$. \u00a0Source code puts you in a, it\u2019s just another bug infested program, frame of mind. You expect errors in code and you typically find them. This is exactly the hard-nosed attitude you need when reading mathematics.","date":"2019-01-21 10:04:26","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\": 20, \"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\": 0.47850343585014343, \"perplexity\": 5637.040533304208}, \"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-2019-04\/segments\/1547583771929.47\/warc\/CC-MAIN-20190121090642-20190121112642-00505.warc.gz\"}"} | null | null |
A type of paper folding in which each fold runs in the opposite direction to the previous fold, creating a pleated or accordion effect.
The non-colors... black, white, and gray.
Paper that is patterned by sizing, and then coated with powders of wool or cotton, (flock). | {
"redpajama_set_name": "RedPajamaC4"
} | 2,973 |
Q: rank of extended matrix Show that the linear system $AX=B$ has a solution if and only if the rank of $A$ equals the rank of the extended matrix $(AB)$.
I started by saying that $AX=a_1 x_1+a_2 x_2+\cdots+a_n x_n=B$, where $a_{i}$ represent the columns of $A$ and $x_i$ represent the coefficients of the system. It then follows that $B$ is a linear combination of the columns of $A$, so that $B \in C(A)$. Now I don't know how to go on further...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,552 |
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**Mark Dodgson and David Gann**
# **INNOVATION**
**A Very Short Introduction**
**Innovation: A Very Short Introduction**
'Despite the differences in surname, Mark Dodgson and I are brothers. I have known him and his faults all his life. How he wrote a book like this with David Gann I have no idea, but here it is, and a very good book too.
It tells a fascinating story, and one of growing importance. The ability to innovate is both expected and valued in the worlds of science and the arts: here we read about its importance in the field of business, and about how vastly our lives have changed – and continue to change – because of the innovation talents of individuals, and the innovation strategies of forward-thinking companies. There is a great deal here to fascinate not only those who are professionally engaged in business, but everyone who takes an intelligent interest in how the world is managed.'
Philip Pullman
## **Contents**
Preface
List of illustrations
**1** Josiah Wedgwood: the world's greatest innovator
**2** Joseph Schumpeter's gales of creative destruction
**3** London's wobbly bridge: learning from failure
**4** Stephanie Kwolek's new polymer: from labs to riches
**5** Thomas Edison's organizational genius
**6** Building a smarter planet?
References
Further reading
Index
_**For Yo and Anne**_
## **Preface**
When we were born, not so very long ago, there were no information technologies or television companies, and airline travel was rare and luxurious. Our parents were born into a world even more different than today's, where television had yet to be invented, and there was no penicillin or frozen food. When our grandparents were born, there were no internal combustion engines, aeroplanes, cinemas, or radios. Our great grandparents lived in a world with no light bulbs, cars, telephones, bicycles, refrigerators, or typewriters, and their lives probably had more in common with a Roman peasant than with ours. In the relatively short period of 150 years, our lives at home and work have been completely transformed by new products and services. The reason why the world has changed so much can be explained in large part by innovation.
This _Very Short Introduction_ defines innovation as ideas, successfully applied, and explains why it has the ability to affect us so profoundly. It will describe how innovation occurs, what and who stimulates it, how it is pursued and organized, and what its outcomes are, both positive and negative. It will argue that innovation is essential to social and economic progress, and yet that it is hugely challenging and besotted with failure. It describes how innovation has many contributors and takes different forms, adding to its complexity. It provides an analysis of the innovation process; the ways organizations marshal their resources to innovate, and the eventual outcomes of innovation, which can take a number of forms.
Innovations are found not only in the activities organizations do, but how they do them. The innovation process is presently going through a period of change, stimulated in large part by the opportunities of using new internet and visualization technologies to access ideas distributed from around the world. The potential sources of innovation are growing rapidly. There are, for example, more scientists and engineers alive today than in past history combined. Furthermore, the locus of innovation is changing as economies become dominated by service sectors and the ownership of, or access to, knowledge is ever more valuable compared to physical assets. Innovation is becoming more internationalized, with important new sources emerging in China, India, and elsewhere outside of the industrial powers of Europe, North America, and Japan. We explore the extent to which our understanding of innovation, developed over the past century or more, might be applied to deal with the restless transformations and turbulence we will witness in the global economy in the future.
The first three chapters explain what innovation is, its importance, and its outcomes. The subsequent chapters examine the contributors to innovation and how it is organized, and speculate on its future.
Our understanding of innovation is based on our research into countless innovative organizations around the world and our learning from the accumulated efforts of numerous scholars in the international innovation research community. Our grateful thanks are extended to all those innovators, and students of innovation, who make our journey so exciting and rewarding. We especially acknowledge Irving Wladawsky-Berger and Gerard Fairtlough, two great innovators who have had profound influences on our thinking.
## **List of illustrations**
1 Josiah Wedgwood
© Hulton Archive/Getty Images
2 Joseph Schumpeter
© Bettmann/Corbis
3 Millennium Bridge, London
© 2004 UPP/TopFoto
4 IBM System/360 computer
Courtesy of IBM Corporate Archives
5 Stephanie Kwolek
© Michael Branscom
6 Pasteur's Quadrant
From D. Stokes, _Pasteur's Quadrant_ (1997). By permission of Brookings Institution Press
7 Letter to _Nature_
Reprinted by permission from Macmillan Publishers Ltd. _Nature_ 171, Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid, by J. D. Watson and F. H. Crick, 25 April 1953 © 1953
8 Thomas Edison factory employees attending a 'sing'
US Department of the Interior, National Park Service, Edison National Historic Site
9 'Put out the light' cartoon, by Clifford K. Berryman, 1931
Courtesy of the Library of Congress
10 Touchlight
Courtesy of EON Reality, Inc.
11 Innovation cartoon
© A. Bacall/Cartoonstock.com
## [**Chapter 1
Josiah Wedgwood: the world's greatest innovator**](contents.html#ch01)
We begin with a study of an exemplary innovator, a person who tells us a great deal about the innovator's agenda. He established an enduring, high-profile company creating substantial innovations in the products made, the ways they were produced, and the manner in which they created value for himself and his customers. He made significant contributions to building national infrastructure, helped create a dynamic regional industry, pioneered new export markets, and positively influenced government policies. His outstanding scientific contribution was recognized by election as a Fellow of the Royal Society. He was a marketing genius, and bridged the scientific and artistic communities by a wholly new approach to industrial design. His most important contribution lay in the way he improved the quality of life and work in the society in which he lived. He is the potter Josiah Wedgwood (1730–95).
Born in modest circumstances into a family of Staffordshire potters, Wedgwood was the youngest of 13 children, and his father died when he was young. He was put to work as a potter when he was 11. He suffered badly from smallpox as a child and this had a big impact on his life. As William Gladstone put it, his disease 'sent his mind inwards, it drove him to meditate upon the laws and secrets of his art... and made for him... an oracle of his own inquiring, searching, meditative, fruitful mind'. For the first part of his career, he worked in a number of partnerships, studying every branch of the manufacture and sale of pottery. By the time Wedgwood began his own business, aged 29, he had mastered every aspect of the pottery industry.
**1. The world's greatest innovator**
In his mid-30s, the lameness resulting from smallpox proved too much of a constraint, so he had his leg amputated, without, of course, the aid of antiseptic or anaesthetic. As testament to his energy and drive, he was writing letters within a couple of days. A few weeks later, he suffered the tragic loss of one of his children, but he was back at work within a month of the operation.
By the mid-18th century, the European ceramics industry had been dominated by Chinese imports for around 200 years. Chinese porcelain, invented nearly a thousand years before, achieved a quality in material and glaze that could not be matched. It was much prized by the wealthy, but was too expensive for the expanding industrial classes whose incomes and aspirations were growing during this period of the Industrial Revolution. Trade restrictions on Chinese manufactures further increased the price of imports into Britain. The situation was ripe for innovation to provide attractive, affordable ceramics for a mass market.
Wedgwood was a product innovator, constantly searching for innovation in the materials he used, and in glazes, colours, and design forms of his wares. He applied extensive trial-and-error experiments to continually improve quality by removing impurities and making results more predictable. His favourite motto was 'Everything yields to experiment'. Some innovations resulted from incremental improvements to existing products. He refined a new cream-coloured earthenware being developed in the industry at the time, transforming it into a high-quality ceramic that was very versatile in that it could be thrown on a wheel, turned on a lathe, or cast. After producing a dinner service for Queen Charlotte, wife of George III, and receiving her approval, he named this innovation 'Queen's Ware'. Other innovations were more radical. In 1775, after around 5,000 recorded experiments that were often difficult and expensive, he produced Jasper, a fine ceramic, commonly blue in colour. This was one of the most significant innovations since the invention of porcelain. His major product innovations were still being produced by the Wedgwood company more than 200 years later.
He collaborated with numerous artists and architects in the design of his products, including George Hepplewhite, the furniture maker; Robert Adam, the architect; and George Stubbs, the artist. One of his great achievements was the application of design to the everyday. The renowned sculptor John Flaxman, for example, produced inkstands, candlesticks, seals, cups, and teapots. Products that were previously unattractive were made elegant.
Wedgwood searched everywhere for ideas for designs, from customers, friends, and rivals. He looked in museums and great houses, and trawled antique shops. One valuable source of designs was a coterie of amateur artists amongst well-bred women. Part of Wedgwood's successful approach to working with artists, according to Llewellyn Jewitt, his 19th-century biographer, lay in his effort 'to sharpen the fancy and skill of the artist by a collision with the talents of others'.
In a speech by William Gladstone, a generation after Wedgwood's death, he says of the potter:
His most signal and characteristic merit lay... in the firmness and fullness of his perception of the true law of what we term industrial art, or in other words, the application of the higher art to industry: the law which teaches us to aim first at giving to every object the greatest possible degree of fitness and convenience for its purpose, and next making it the vehicle for the highest degree of beauty, which compatibility with the fitness and convenience it will bear: which does not substitute the secondary for the primary end, but recognises as part of the business the study to harmonize the two.
In his manufacturing process innovations, Wedgwood introduced steam power into his factory, and as a result the Staffordshire pottery industry was the earliest adopter of this new technology. Steam power brought many changes to production processes. Previously the potteries were distant from the mills that provided power for mixing and grinding raw materials. Having power on-site significantly reduced transportation costs. It also mechanized the processes of throwing and turning pots, previously driven by foot or hand wheels. Technology enhanced efficiency in the way the use of lathes to trim, flute, and checker products improved production throughput.
He was preoccupied with quality, and spent vast amounts on pulling down and rebuilding kilns to improve their performance. Famously intolerant of poor product quality, legend has him prowling the factory smashing substandard pots and writing in chalk 'this won't do for Josiah Wedgwood' on offending workbenches.
One of the perennial challenges of making ceramics was measuring high temperatures in kilns in order to control the production process. Wedgwood invented a pyrometer, or thermometer, that recorded these temperatures, and for this achievement he was elected a Fellow of the Royal Society in 1783.
Many of Wedgwood's most popular products were produced in large numbers in plain shapes, which were then embellished by designers to reflect current trends. Other, more specialist products were produced in short, highly varied batches, quickly changing colour, fashion, style, and price as the market dictated. He subcontracted the manufacture of some products and their engraving to reduce his own inventory. When orders exceeded his production capacity, he outsourced from other potters. Wedgwood's innovative production system aimed to minimize proprietary risk and reduce fixed costs. He was highly aware of costs, having at one time complained that his sales were at an all-time high, yet profits were minimal. He studied cost structures and came to value economies of scale, trying to avoid producing one-off vases 'at least till we are got into a more methodicall way of making the same sorts over again'.
Wedgwood was an innovator in the way work was organized. His organizational innovations were introduced into an essentially peasant industry, with primitive work practices. When Wedgwood founded his main Staffordshire factory, Etruria, he applied the principles of the division of labour espoused by his contemporary, Adam Smith. Replacing previous craft production techniques, where one worker produced entire products, specialists concentrated on one specific element of the production process to enhance efficiency. Craftsmanship improved, allowing artists, for example, to improve the quality of designs, and innovation flourished. One of his proudest boasts was that he had 'made artists of mere men'.
Wedgwood paid slightly higher wages than the local average and invested extensively in training and skills development. In return, he demanded punctuality, introducing a bell to summon workers and a primitive clocking-in system, fixed hours, and constant attendance; high standards of care and cleanliness; avoidance of waste; and a ban on drinking. Wedgwood was conscious about health and safety, especially in relation to the ever-present dangers of lead poisoning. He insisted on proper cleaning methods, work attire, and washing facilities.
As a business innovator, Wedgwood created value by engaging with external parties in a number of ways. He innovated in sources of supply and distribution, astutely used personal and business partnerships to advantage, and introduced a remarkable number of marketing and retailing innovations.
Wedgwood sought the best-quality raw materials from wherever he could find them. In what today would be called 'global sourcing', he purchased clay from America in a deal struck with the Cherokee nation, from China, and the new colony in Australia.
He had a wide range of friends with very diverse interests upon whom he drew in his business dealings. Wedgwood belonged to a group of similarly minded polymaths who became known as the Lunar Men, because of their meeting during the full moon. Along with Wedgwood, they comprised a core of Erasmus Darwin, Matthew Boulton, James Watt, and Joseph Priestley. The friendship and business partnership with Boulton was particularly influential on Wedgwood's thinking about work organization, as he observed the efficiency, productivity, and profitability of the Boulton and Watt factory making steam engines in Birmingham. Jenny Uglow's book on the Lunar Men argues that they were at the leading edge of almost every movement of their time, in science, in industry, and in the arts. She evocatively suggests that: 'In the time of the Lunar men, science and art were not separated, you could be an inventor and designer, an experimenter and a poet, a dreamer and an entrepreneur all at once.'
Although Wedgwood had somewhat contradictory views on the ownership of intellectual property, he encouraged collaborative research and was a proponent of what today would be called 'open innovation'. In 1775, he proposed a cooperative programme with fellow Staffordshire potters to solve a common technical problem. It was a plan for what was the world's first collaborative industrial research project. The scheme failed to get off the ground, but it illustrates a desire to use a form of organization that was not again explored for over a century.
Wedgwood was the first in his industry to mark his name on his wares, denoting ownership of the design, but he disliked patents, and only ever owned one. Speaking of himself, he explains his approach:
When Mr. Wedgwood discovered the art of making Queen's ware... he did not ask for a patent for this important discovery. A patent would greatly have limited its public utility. Instead of one hundred manufacturies of Queen's ware there would have been one; and instead of an exportation to all quarters of the world, a few pretty things would have been made for the amusement of the people of fashion in England.
The period of the Industrial Revolution was one of great optimism as well as social upheaval. Consumption and lifestyle patterns changed as industrial wages were paid and new businesses created novel sources of wealth. The population of England doubled from around 5 million in 1700 to 10 million in 1800. Until the 18th century, English pottery had been functional; mainly crude vessels for storing and carrying. Pots were crudely made, ornamented in an elementary way, and glazed imperfectly. The size and sophistication of the market developed throughout the 18th century. Stylish table accessories were in huge demand in the burgeoning industrial cities and increasingly wealthy colonies. Drinking tea, and more fashionable coffee and hot chocolate, joined the traditional British pastime of imbibing beer as a national characteristic.
Wedgwood sought to meet and shape this burgeoning demand in a number of ways. Initially he sold his completed wares to merchants for resale, but he opened a warehouse in London, followed by a showroom that took direct orders. Browsing customers commented on the wares on display, and Wedgwood took particular note of criticisms of uneven quality, explaining his devotion to researching how to achieve better consistency. The showroom, run by Wedgwood's close friend, Thomas Bentley, became a place for the fashionable to be seen, and major new collections were visited by royalty and aristocracy. Bentley expertly interpreted new trends and tastes, informing design and production plans back in Staffordshire.
Wedgwood assiduously sought patronage from politicians and aristocrats: what he called his 'lines, channels, and connections'. He produced a 952-piece dinner service for Catherine the Great, Empress of Russia, shamelessly using her patronage in his advertising. His view was that if the great and the good bought his products, the new middle classes, merchants and professionals, and even some of the wealthier lower classes, artisans and tradespeople, would aspire to emulate them.
An astonishing number of retailing innovations were introduced by Wedgwood and Bentley, including the display of wares set out in full dinner service, self-service, catalogues, pattern books, free carriage of goods, money-back guarantees, travelling salesmen, and regular sales, all aiming 'to amuse, and divert, and please, and astonish, nay, and even to ravish the Ladies'. Jane Austen wrote of the pleasure of the safe delivery of a Wedgwood order.
Wedgwood's international marketing efforts were pioneering. When he started his business, it was rare for Staffordshire pottery to reach London, let alone overseas. To sell in international markets, he again used the strategy of courting royalty by using his English aristocratic connections as ambassadors. By the mid-1780s, 80% of his total production was exported.
Products were not sold on the basis of low prices. Wedgwood's products could be two or three times as expensive as his competitors'. As he put it, 'it has always been my aim to improve the quality of the articles of my manufacture, rather than to lower their prices'. He was contemptuous of price cutting in the pottery industry, writing to Bentley in 1771:
the General Trade seems to me to be going to ruin on the gallop... low prices must beget a low quality in their manufacture, which will beget contempt, which will beget neglect, and disuse, and there is an end of the trade.
Wedgwood's innovations extended into many other areas. He expended substantial efforts in building the infrastructure supporting the manufacture and distribution of his products and those in his industry. He devoted significant amounts of time and money to improving communications and transportation, especially with the ports that supplied raw materials and provided his routes to market. He promoted the development of turnpike roads and became centrally engaged in the construction of major canals. He actively lobbied the government on trade and industry policy and helped form the first British Chamber of Manufacturers.
Wedgwood's legacy extended well beyond his company. He had an enormous impact on the Staffordshire Potteries more generally, in what today might be called an innovative 'industrial cluster'. Pottery production in Staffordshire developed rapidly due to the efforts of numbers of firms, such as Spode and Turner, but Wedgwood was the acknowledged leader of the industry.
His 19th-century biographer, Samuel Smiles, wrote of the change from the 'poor and mean villages' brought about by Wedgwood's innovations:
From a half-savage, thinly peopled district of some 7000 persons in 1760, partially employed and ill remunerated, we find them increased, in the course of some twenty-five years, to about treble the population, abundantly employed, prosperous, and comfortable.
Wedgwood's contributions to public life included improving the education, health, diet, and housing of his employees. Etruria's 76 homes were, in their time, considered a model village.
Wedgwood built a dynasty. He inherited £20 from his father, and when he died he left one of the finest industrial concerns in England with a personal worth of £500,000 (around £50 million at present prices). Wedgwood's children used their good fortune well. One son created the Royal Horticultural Society and another contributed centrally to the development of photography. Wedgwood's wealth was used to great effect to fund the studies of his grandson, Charles Darwin.
The Wedgwood case raises a number of core issues that we shall be examining in this Very Short Introduction and reveals the approach to innovation we shall be taking. We focus on the organization, the mechanism for creating and delivering innovation. The individuals and their personal connections whose importance is so clearly shown in the Wedgwood case will be discussed here only to the extent to which they contribute to organizational outcomes. We do not discuss the meanings of innovation for us individually. Nor do we adopt the perspective of the user of innovation, although we shall argue that innovative organizations need to try to understand how innovations are consumed and for what purpose. With this observation in mind, Wedgwood shows us that innovation occurs in many forms and ways. It occurs in what organizations produce: their products and services. It is found in the ways in which organizations produce: in their production processes and systems, work structures and practices, supply arrangements, collaboration with partners, and very importantly how they engage with and reach customers. Innovation also happens in the context within which organizations operate: for example, in regional networks, supportive infrastructure, and government policies.
Wedgwood illustrates an enduring truth about innovation: it involves new combinations of ideas, knowledge, skills, and resources. He was a master at combining the dramatic scientific, technological, and artistic advances of his age with rapidly changing consumer demand. Gladstone said: 'He was the greatest man who ever, in any age or in any country, applied himself to the important work of mixing art with industry.' The way Wedgwood merged technological and market opportunities, art and manufacturing, creativity and commerce, is, perhaps, his most profound lesson for us.
## [**Chapter 2
Joseph Schumpeter's gales of creative destruction**](contents.html#ch02)
All economic and social progress ultimately depends on new ideas that contest the introspection and inertia of the status quo with possibilities for change and improvement. Innovation is what happens when new thinking is successfully introduced in and valued by organizations. It is the arena where the creation and application of new ideas are formally organized and managed. Innovation involves deliberate preparations, objectives, and planned benefits for new ideas that have to be realized and implemented in practice. It is the theatre where the excitement of experimentation and learning meets the organizational realities of limited budgets, established routines, disputed priorities, and constrained imagination.
There are a great many ways of understanding innovation that provide a wide range of rich insights and perspectives. The variety of different analytical lenses used depends on the particular innovation issues being studied. Some analyse the extent and nature of innovation: whether any change is incremental or radical, how it sustains or disrupts existing ways of doing things, and if it occurs in whole systems or their components. Other analyses are concerned with how the focus of innovation changes over time, that is, from the development of new products to their manufacture, their patterns of diffusion, how particular design configurations, such as video recorders and music players, come to dominate in the market, and how organizations appropriate value from innovation.
### Defining
The English language allows broad definitions of innovation, which can be both helpful and confusing. This is helpful in as much as it can usefully cover a wide range of activities and is confusing for the same reason – the word can be used promiscuously. Even the relatively simple definition of innovation we use – ideas, successfully applied – raises questions. What is 'success'? Time is influential, and innovations may be initially successful and eventually fail, or vice versa. What does 'applied' imply? Is it applied within a single part of an organization, or diffused internationally amongst a massive group of users? What and who are the sources of 'ideas'? Can anyone lay claim to them, especially as they inevitably combine new and existing thinking?
Typologies of innovation also face difficulties because of blurred boundaries and overlaps between categories. Innovation occurs in products, for example in new cars or pharmaceuticals; and services, for example in new insurance policies or in health monitoring. But many service firms describe their offerings as products, such as new financial products. Innovation occurs in operational processes, in the way new products and services are delivered. These processes may take the form of equipment and machinery, which are the providers' products, and logistics in the form of transportation, which are providers' services.
There are some similar definitional problems when thinking about levels of innovation. A minor innovation for one organization may be substantial for another. It is difficult in practice to develop anything but a nominal scale of the differences between levels of innovation, and categorization is best thought of as ideal types along a continuum. Most innovations are incremental improvements – ideas used in new models of existing products and services, or adjustments to organizational processes. They might include the latest versions of particular software packages, or decisions to add more representatives from the marketing department to development teams. Radical innovations change the nature of products, services, and processes. Examples include the development of synthetic materials, such as nylon, and decisions to use open-source software to encourage community development of new services, rather than doing it proprietarily. At the highest level, there are rarer, periodic transformational innovations, which are revolutionary in their impact and affect the whole economy. Examples would be the development of oil as an energy source, or the computer or internet.
We think about innovation as ideas, successfully applied in organizational outcomes and processes. Innovation can be thought of as practical and functional: the outcomes of innovation are new products and services, or they are the organizational processes supporting innovation that occur in departments such as R(esearch)&D(evelopment), Engineering, Design, and Marketing. Innovation can also be thought of more conceptually: the outcomes of innovation are enhanced knowledge and judgement, or they are the processes that support the capacity of organizations to learn.
We have chosen to focus on innovations other than those described as 'continuous improvement' that tend to be routine and highly incremental in nature. Although these small improvements are cumulatively important, our concern lies rather with ideas that stretch and challenge organizations as they attempt to survive and thrive. By concentrating on innovations beyond the ordinary that occur in both the outcomes of organizational efforts and the processes that produce them, we capture a great degree of what is generally understood to be innovation.
### Importance
The reason why innovation is so important has to be seen in the context of the relentless demands made of contemporary organizations as they face the challenges of a complex and turbulent world. Innovation is crucial for their continuing existence as they struggle to adapt and evolve to deal with constantly changing markets and technologies.
In the private sector, the threat of new competitors in globalized markets is ever present. In the public sector, the demand for efficiencies and enhanced performance is continual, as governments attempt to manage demands for expenditure to improve the quality of life that exceed their incomes. The motivation to innovate in all organizations is stimulated by the knowledge that if they are not capable of innovation, others are: new players that may threaten their very existence. Simply, if organizations are to progress – to develop and grow, become more profitable, efficient, and sustainable – they need to successfully implement new ideas. They have to be continually innovative. As the economist Joseph Schumpeter put it, at its most blunt, innovation 'offers the carrot of spectacular reward or the stick of destitution'.
One of the features of innovation is that it can be found in every organization. Although the cost of innovation can be very high – it can, for example, cost up to $800 million to develop a new pharmaceutical – new ideas can be successfully implemented cheaply. It is not only high-tech firms making semiconductors or working with biotechnology that rely on innovation in their businesses, it is all parts of the economy. Insurance firms and banks continually search for new ideas for services for customers; shops use computer-controlled ordering and stock management; farms use new seeds, fertilizers, and irrigation technologies, satellites can assist the optimization of their planting and harvesting, and new uses are being made of their products, such as biofuels and health-promoting functional foods. Innovation is found in construction, in new materials and building techniques; in packaging that keeps food fresher; and in clothing firms introducing new designs more quickly and cheaply. Innovation is sought by public services, in health, transportation, and education. While one might not wish for too much innovation in some areas, such as amongst the firms investing our pension funds or designing the aeroplanes in which we fly, generally the business or organization that doesn't benefit from using new ideas is rare indeed.
### Challenges
The challenges of innovation are immense. Many people are uncomfortable with the change brought about by innovation. Especially when it is broad-ranging, innovation can have negative effects on employees, inducing uncertainty, fear, and frustration. Organizations have social contracts by which their members develop loyalty, commitment, and trust. Innovation can disrupt this contract by redistributing resources, altering the relationships between groups, and emphasizing the ascendancy of one part of the organization to the disadvantage of others. It can disturb the technical and professional skills people acquire over many years, and with which they strongly identify themselves. Its organizational context means it is inseparable from the exertion of power and resistance to it.
Most attempts at innovation fail. History is littered with unsuccessful attempts to apply the – often very good – new ideas of individuals and organizations. The ill-fated development of a costeffective battery electric car with significant environmental benefits in the USA in the 1990s is illustrative of the way innovation can provide a serious threat to established interests. A coalition of political and business interests combined to prevent this new idea reaching the market. Although the product was popular with consumers, it had to compete with the interests of the established energy infrastructure, oil companies, and petrol distribution networks, and massive existing automotive industry investments in petrol engine car manufacture and maintenance.
Organizations simultaneously need to do things that allow them to operate in the short term, exploiting their existing know-how and skills, and explore new things that will develop capacities to support their continued long-term existence in a changing world. These demand different and sometimes conflicting behaviours and practices. Indeed, organizations are occasionally confronted with the paradox of needing to apply new ideas threatening to the practices that have created their current successes. If generals are said to fight the last rather than the current war, managers rely on ways of doing things that contributed to their organizations', and their own, past progress, rather than ways that will deal more effectively with the future. Since Edison established the first organization dedicated to producing innovations at the turn of the 19th century, many different ways of organizing the creation and use of ideas have periodically been favoured. As the business environment has changed, the large, centralized corporate Research and Development (R&D) laboratory, and the distinctly separate innovation team (sometimes called the 'skunkworks'), are no longer used as often as in the past. The search for ways of balancing routines with innovation is constant.
Organizations rarely innovate alone: they do so in association with others, including their suppliers and customers. They innovate in particular regional and national contexts. Access to innovation-supporting skills and university research, for example, often has a local dimension, as seen in the case of Silicon Valley in California and other international centres of innovation. Government policies and regulations affect innovation, as do national financial and legal systems that influence issues such as the availability of risk-taking investment capital, the creation of technical standards, and protection of intellectual property rights. Availability and cost of infrastructure for communications and transportation matter greatly. These factors add to the complexity, and hence unpredictability, of innovation, as innovators are never completely masters or mistresses of their destiny. They also point to the essentially idiosyncratic nature of innovation: each innovation occurs in its own particular set of circumstances.
In all the major elements of contemporary economies – in the services, manufacturing, and resources industries, and in the public sector – organizational progress depends upon owning or accessing and using knowledge and information. Being competitive and efficient relies on being innovative with all the resources organizations possess: their people, capital, and technology, and the ways they connect with those contributing to and using what they do.
### Innovation thinking
The American economist William Baumol argues that virtually all of the economic growth that has occurred since the 18th century is ultimately attributable to innovation. The successful application of ideas has been recognized within industry as the primary source of its development since this time.
The 18th century also saw the beginning of the study and recognition of the importance of the relationships between organization, technology, and productivity with the publication of Adam Smith's _Wealth of Nations_ in 1767. Smith produced his now famous analysis of the importance of the division of labour in a pin factory, which was so influential on Wedgwood's factory organization. Smith showed how specialization in specific manufacturing processes in pin production significantly increased the productivity of the workforce compared to when individuals produced each pin themselves. A man alone, even with the 'utmost industry', could produce 1 to a maximum of 20 pins a day, yet with a division of labour, 'very poor' labour 'indifferently accommodated with the necessary machinery', could produce 4,800 'when they exerted themselves'.
A century later, Karl Marx was highly aware of the significance of innovation, but more concerned with its negative consequences. In the first volume of _Capital_ , he declared:
Modern industry never views or treats the existing form of a production process as the definitive one... By means of machinery, chemical processes and other methods, it is continually transforming not only the technical basis of production, but also the functions of the worker and the social combinations of the labour process.
The possibilities of technological change, Marx argued, were contradicted by its use under capitalism, which inevitably led to the suppression of workers. Capitalism, he contended, subordinated workers to machines, but he believed technology held the possibility of liberating them from the burden of mechanical and repetitive work and enriching social relations.
Marx's emphasis on the strong social dimensions to technological development and use is a recurrent theme in research into the history of innovation. Study of the development of automated machine tools in the USA, for example, illustrates how often technology is shaped by dominant social forces. The automated, or numerical, control of machine tools, such as lathes, could have been configured in various ways to give the machine's operator more or less discretion over how it was used. The technology was constructed in such a way that control resided in engineering planning offices, not with their operators. This was less economically efficient, but complied with the expectations of the major customer for the new technology, the US Air Force, and hence reflected existing power structures.
At a more aggregate level, all the past revolutions in technology – in steam power, electricity, automobiles, information and communications technology – have required enormous adjustment and adaptation in industry and society. The economists Christopher Freeman and Carlota Perez show how in history the diffusion of new technologies since the Industrial Revolution required massive structural adjustments in industry and society, and also in the legal and financial framework, education and training systems for new skills and professions, new management systems, and new national and international technical standards.
The importance of clever 'human capital' has long been recognized. Observing the development of German industry in the mid-19th century, the political scientist Friedrich List declared that national wealth is created by intellectual capital: the power of people with ideas. In 1890, the British economist Alfred Marshall noted that knowledge is the most powerful engine of production available to economies. An economic theorist who unusually kept his feet on the ground by regularly visiting companies, Marshall celebrated the importance of innovation and is especially remembered for his analysis of the benefits of the 'clustering' of progressive firms in 'industrial districts'.
If any economist lays claim to be the first to include innovation centrally within their theory of development, it is Joseph Schumpeter (1883–1950), who remains today as one of the most influential thinkers on the subject. A complex man with a rich history, including being one time Finance Minister in Austria, director of a failed bank, and Harvard professor, Schumpeter argued that innovation unleashed the 'gales of creative destruction'. It arrives in a great storm of revolutionary technologies, such as oil and steel, that fundamentally change and develop the economy. Innovation is creative and beneficial, bringing new industries, wealth, and employment, and at the same time is destructive of some established firms, many products and jobs, and the dreams of failed entrepreneurs. For Schumpeter, innovation is essential for competitive survival:
**2. Schumpeter placed innovation centrally in his theory of economic development**
the competition from the new commodity, the new technology, the new source of supply, the new type of organization... competition which commands a decisive cost or quality advantage and which strikes not at the margins of the profits and the outputs of the existing firms but at their foundations and their very lives...
Schumpeter's views on the primary sources of innovation altered during his lifetime, reflecting changes in the practices in industry. His early 'Mark I' model, published in 1912, celebrated the importance of individual, heroic, risk-taking entrepreneurs. His 'Mark II' model, by contrast, published 30 years later, advanced the role of the formal, organized innovative efforts in large companies. It was during this period that the modern research laboratory became firmly established, initially in the chemical and electrical industries in Germany and the USA. By 1921, there were more than 500 industrial research laboratories in the USA.
### Five models
One of the first and most influential studies of the relationship between scientific progress and industrial innovation was conducted immediately after the Second World War by Vannevar Bush, the USA's first presidential science advisor. In his report _Science: The Endless Frontier_ , Bush advocated a national policy for open-ended research on a massive scale. The book proved popular; it was serialized in _Fortune_ magazine, and Bush appeared on the front page of _Time._ The view that investments in research held the solutions to most seemingly intractable problems was a legacy of Bush's association with the Manhattan Project to develop the atom bomb, which, to the minds of many, successfully curtailed the war in the Pacific. Although it took a simplistic interpretation of Bush's report, the view that all product and process innovations are founded in painstaking basic research became the fundamental precept of the 'science push' model of innovation, a perspective that remains popular with many in the scientific research community to this day.
An alternative view, which emphasized the importance of market demand as the primary source of innovation, emerged in the 1950s and 1960s. This resulted from a number of factors, including studies that showed that in sectors such as the military, technological outcomes resulted more from the demands of its users rather than any scientifically predetermined configurations. At the same time, there was a growth of large, corporate planning offices with belief in the conceit that sufficient market research could identify what was required of new science and technology to meet consumer needs. This mirrored the rise of social science at the time with its claims to predictive powers. Counter to the post-war enthusiastic embrace of science and technology, social movements – such as Ralph Nader's car safety campaign in the 1960s, developed in response to dangerous car designs – began to question the use to which they were being put and demand greater attention to consumer needs. In housing, research into the demographics of the baby-boomer generation led to 'predict and provide' strategies internationally, where innovation was sought to help satisfy growing demand. This view became known as the 'demand-pull' model of innovation.
Both these models of innovation are linear in their progression: research leads to new products and processes introduced into the market, or market demand for new products and processes leads to research to develop them. But increasing volumes of research conducted in the 1970s questioned the assumption of linearity. Pioneering studies, such as Project SAPPHO at the University of Sussex, UK, found differences between sectors: for example, the chemical industry innovated differently from the scientific instruments industry. And, furthermore, the pattern of innovation changed over time. Abernathy and Utterback at MIT developed the theory of product life cycles, with high levels of innovation in the development of products occurring initially, then reducing in scale and being replaced by high levels of innovation focusing on their application and their processes of production. Innovation was seen not to be unidirectional, but more iterative, with feedback loops.
The organizational and skills issues underlying this 'coupling' model of innovation came to the fore in the 1980s, driven primarily by the remarkable success of Japanese industry. A study of the car industry at the time showed Japanese auto manufacturers were twice as efficient as their international competitors in every measure of innovative performance, such as how long it takes to design and make a car. The explanation for this was an approach described as 'lean production', which contrasted with the mass-production techniques used in other countries. Mass production, typified by Henry Ford, is based on assembly lines producing standardized products. 'You can have any colour Model T car you want, as long as it is black', as Ford is reputed to have said. Lean production introduced greater flexibility into the assembly line, allowing a broader range of products to be made. It includes a system of relationships with suppliers of components that allow them to deliver 'just-in-time' to be assembled, thereby reducing the cost of holding inventory, and increasing the speed of response to market changes. Lean production also entailed an obsessive concern for quality control, which in many areas became the responsibility of shop-floor workers.
When analysing the differences between the way Japanese and Western firms organized themselves to innovate, the metaphors were used at the time of the former playing rugby (although netball is similarly suitable) and the latter running a relay race. In the West, innovation entailed one part of the organization, say R&D, beginning the process, running with it for a while, then handing it over to another, say Engineering, which similarly worked on it before passing it over to Manufacturing and then on to Marketing. This linear process was considered hugely wasteful to Japanese firms, with the likelihood of significant miscommunication as projects moved from one part of the organization to another. A rugby or netball player metaphor can be used, as their games involve the simultaneous combination of different kinds of players, with various skills and abilities, some big and tall, but generally slow, and some smaller, skilful, and fast, all working to the same objective. All parts of the organization were combined in innovation activities.
Collaboration between, as well as within, innovative Japanese companies was a feature of their 1980s' success story. As well as the extensive collaboration between customers and suppliers in the same industrial groups – Keiretsu – the Japanese government also encouraged collaboration between competing firms. The Fifth Generation Computer programme, for example, attempted to encourage the usually highly competitive computer manufacturers to cooperate around shared research agendas. This 'collaborative' model of innovation strategies and public innovation policies was also enthusiastically pursued in Europe in information technology and the USA in semiconductors.
By the 1990s, Roy Rothwell, one of the founders of innovation research, began to identify a number of changes occurring in the strategies firms were using to innovate and the technologies they were using to support it. He argued firms were developing innovation strategies that were highly integrated with its partners, including 'lead customers', demanding users, and co-developers of innovation. Also important, he contended, was the use of new digital technologies, like computer-aided design and computer-aided manufacturing, that brought different parts of the firm together when developing innovations, and helped link external parties into internal development efforts. Rothwell called this the 'strategic integration and networking' model of innovation. The trend towards greater strategic and technological integration in support of innovation is one that continues with the use of massive computing power, the internet, and new visualization and virtual-reality technologies.
These models of the innovation process have their antecedents in an industrialized economy where innovation predominantly occurs in the manufacturing industry. We are now in economies where services dominate, accounting for around 80% of Gross Domestic Product in most developed nations. Economies based on tangible, physical objects that can be measured and seen have been transformed into ones where outputs are weightless and invisible. Furthermore, as the global financial crisis that emerged in 2008 shows, we live in an era of extraordinary turbulence and uncertainty where any established formulae and prescriptions are likely to be tested by emerging and unforeseen circumstances. The models of innovation in the future will be far more organic and evolutionary where the sources of innovation are unclear, the organizations involved initially unknown, and the outcomes highly constrained by unpredictabilities. In these circumstances, it will be valuable to assess whether or not anything we know of the past might be a guide to the future. It will also be useful to understand how theory of innovation might help.
### Theory
There is no single, unified theory of innovation. There are partial explanations from, for example, economics, political science, sociology, geography, organizational studies, psychology, business strategy, and from within 'innovation studies', which draws on all these disciplines. This is to be expected given innovation's multiple influences, pathways, and outcomes. The utility of various theories will depend on the particular issues being explored. Theories from psychology may be more useful when the subject is an individual innovator, business strategy when it is organizational innovation, and economics when it is national innovation performance. It is important to consider theories of innovation not only to explain contemporary issues, significant as they are, but also to enlighten its future use in helping to deal with major social, economic, and environmental concerns.
Over recent years, there has been an emergence of a number of perspectives that share common ground in their theories of innovation. These include evolutionary economics and 'dynamic capabilities' frameworks for business strategy.
The challenge for any theory of innovation is that it has to explain an empirical phenomenon that takes many guises. It also has to encompass its complexity, dynamism, and uncertainty, often compounded by the way innovation results from the contribution of many parties with occasionally divergent and not fully established agendas. In this way, innovation has emergent properties: it results from a collective process whose outcomes may not be known or expected when it begins.
Evolutionary economics – with a Schumpeterian legacy – sees capitalism as a system that produces continuous variety in the new ideas, firms, and technologies created by entrepreneurs and the innovative activities of research groups. Decisions by organizations, consumers and governments make selections from within this variety. Some selections are successfully propagated and are fully developed into new organizations, businesses and technologies that provide the basis and resources for future investments in creating variety. Much of the variety and selections that occur are disruptive or fail to be propagated, so the evolutionary development of the economy is typified by significant uncertainty and failure.
Dynamic capabilities theory includes the ways firms search for, select, configure, deploy, and learn about innovations. Its focus is on the skills, processes, and organizational structures that create, use, and protect intangible and difficult to replicate assets, such as knowledge. This approach to strategy reflects the continual dynamism of technology, markets, and organizations where the capacity to sense threats and realize opportunities – when information is constrained and circumstances unpredictable – is the key to sustainable corporate advantage.
These theoretical explanations for innovation embrace complexity and emergent circumstances. They incorporate the messy organizational realities of innovation found in economies where there is constant change and adaptation and when the strategies of firms are often experimental.
### Time
There has to be a time dimension to any understanding of innovation. Whether considering outcomes, what happened, or processes of innovation, how it came about, it is necessary to know the period in which they occurred. Comparisons to what existed before the innovation determine the extent of novelty.
If an innovation is ahead of its time, as perhaps might be argued to have happened in the case of the battery electric car, no matter how much effort is expended, it will not gain the momentum needed for its wide diffusion and sustained growth. If an innovation takes too long to be developed, it may fail because a superior or cheaper idea emerges. Sometimes markets and technologies shift quickly, and rapidly move on from what at one point seemed like a good idea. Innovative organizations therefore have to think about timescales of new ideas. They can do this by considering their position based on past innovation diffusion patterns, and use tools and techniques to speed up innovation through formal project management techniques that progressively decide on the levels of resources needed. Returns on innovation investment are planned over periods of years, and decisions are made to invest if they pay back suitably over an acceptable time period. Risk is managed by attempts to reduce how long it takes to develop and introduce innovation. Speed is usually, but not always, seen to be a benefit. Compressing time reduces the chance of being caught up by competitors or wastefully squandering resources. Moving too quickly, however, leads to mistakes and failures to learn from them.
Short-term horizons are appropriate for incremental innovation, but long-term perspectives are needed to provide a wider view on where, why, and how radical innovation has occurred or failed. Understanding the relationships between scientific discovery, innovation, and societal changes requires deep historical interpretation.
Innovative organizations – as we shall see in the case of Edison in Chapter 5 – can improve their future chances of success by creating options that allow different potential paths to be followed, delaying decisions that do not need to be made until a later date when their consequences may be clearer. By organizing and equipping themselves for unforeseen eventualities, innovators can change course, or re-calibrate timetables. As Louis Pasteur observed about scientific discovery through experimentation, 'chance favours the prepared mind'.
Rates of innovation and diffusion vary considerably between different business sectors. In pharmaceuticals, for example, it usually takes between 12 and 15 years to bring a new drug to market, but new digital services can grow large within months. Organizations can make strategic choices on whether they should attempt to lead innovation in their sector, or follow others. Sometimes leaders have the best opportunity for reaping the greatest rewards from their ideas. The chemical company DuPont, for example, has consistently led other firms in bringing new products to market for over a century. But 'first-mover advantage' may be difficult to capture and sustain. It often brings greater risks, as the market may not be fully formulated, and higher costs may be accrued to stimulate demand.
Other organizations choose to learn from leaders, emulating innovations that appear to work well, and avoiding any pitfalls they have observed. Fast-followers can receive huge rewards, as Microsoft has done for consistently reacting quickly to the innovations of others who have borne initial risks. Many organizations do not have the skills or resources to be first-movers or fast-followers. They may, nevertheless, benefit from innovation that improves upon, adapts, or extends products, processes, or services. Whatever its position as an innovator, and whichever strategy an organization may wish to pursue, its ability to appreciate the time dimension is likely to have a significant bearing on its performance.
## [**Chapter 3
London's wobbly bridge: learning from failure**](contents.html#ch03)
Schumpeter's analysis of innovation being a process of creative destruction implies the outcomes of innovation can simultaneously be positive and negative. It both creates and destroys wealth and jobs. Innovation profoundly affects us all by creating new industries, firms, and products, as seen in the new industry established by Wedgwood. It is seen in services such as discount airlines, and infrastructure such as airports. It improves productivity, and quality of life in the form, for example, of new pharmaceuticals, means of transportation, communication, entertainment, and greater variety in and accessibility of food. It has helped lift millions out of poverty, especially over recent decades in Asia. Jobs can be more creative, interesting, and challenging as a result of innovation. But the successful application of ideas can also have profoundly adverse consequences. Nations and regions get left behind when they are not as innovative as their competitors, and increasing disparities in wealth result. Jobs can be deskilled, job satisfaction decreased, and unemployment increased, because of innovation. Innovation has given us the environmental consequences of the internal combustion engine and chlorofluorocarbons, and the toxic results from the complex financial instruments behind 2008's global financial crisis.
Predicting the adverse consequences of innovation can be as challenging as foreseeing its positive effects: these are unpredictable and can be mixed. On the positive side, the internal combustion engine democratized travel, chlorofluorocarbons in refrigerators improved nutrition, financial innovations gave us the security of better life insurance and pensions. But the occasionally ambiguous nature of the results of innovation is seen in the case of failure. Most attempts at innovation fail, and there is a highly skewed distribution in its returns, but failure is itself an important outcome, and it is to this that we turn.
### Failure
Innovation is risky, as, for example, innovators have to consider:
• Demand risk – how big will the market be for a new product or service? Will new competitors emerge?
• Business risk – is appropriate finance available to meet the costs of innovation? What effect will an innovation have on organizational reputation and brands?
• Technology risk – will a technology work, is it safe, and how does it complement other technologies? Will better competing technologies emerge?
• Organization risk – are the right management and organizational structures being used? Are the necessary skills and teams available?
• Network risk – are the right collaborative partners and supply chains in place? Are there important gaps?
• Contextual risks – how volatile are government policies, regulations, and taxation, and finance markets?
In theory, risk can be measured and managed by making assumptions on probabilities, although there are always dangers in assuming the past can predict the future. Uncertainty, on the other hand, has a truly unknown outcome and cannot be measured, and its management depends on decisions based on deep experience and intuition. It is because of risks and uncertainties that there is so much failure in innovation, but at the same time, they provide an incentive. If there were no risk and uncertainty, and therefore everyone could innovate easily, then innovation would provide little advantage over competitors.
Failures also provide valuable opportunities for future improvements, as seen in the highly embarrassing case of London's Millennium Bridge. Linking the Tate Gallery and St Paul's Cathedral, this was the first footbridge to be built across the River Thames for more than 100 years. It is an extraordinary engineering, architectural, and sculptural achievement – a design of such beauty it has been described as a 'blade of light' across the river. The bridge opened on the 10 June 2000, when between 80,000 and 100,000 people walked across it. When large groups of people were crossing, however, it became noticeably and increasingly unsteady, and it quickly gained notoriety as the 'wobbly bridge'. The bridge was closed after two days, causing immense discomfiture to all concerned.
After an intensive international effort, the cause was found and rectified. The problem, apparently, was the way men tend to walk with splayed feet, like ducks. When many of them walk in unison, unusual 'lateral excitation' occurs. Had it been a women-only bridge, there would not have been a problem. As a result of this debacle, new knowledge about bridge design was developed, and future projects will allow large numbers of men to waddle happily together over rivers.
The Millennium Bridge is an example of the way in which much progress in science, engineering, and innovation is built upon failure. As the chemist Humphry Davy said: 'The most important of my discoveries has been suggested to me by my failures.' And as Henry Ford put it: 'Failure is only the opportunity to begin again more intelligently.' Empirical evidence shows how returns to new ideas are highly skewed – there is what physicists and economists call a 'power law distribution'. Only a few academic papers, patents, products, and company start-ups are successes. In most cases, the majority of returns come from 10% of innovative investments. In some areas it is even more skewed. At any one time there may be up to 8,000 potential new pharmaceuticals being researched around the world, but maybe only one or two will prove successful.
**3. The Millennium Bridge: a great success after a wobbly start**
There is a strongly temporal element to failure: things deemed failures can become successful, such as the Millennium Bridge, and successes can over time turn into failures. Following its introduction in 1949, the de Havilland Comet aircraft was instrumental in creating the international commercial airline industry. The Comet was considered a highly successful product innovation until the period in the mid-1950s when the aircraft started falling out of the sky with alarming regularity. Aircraft engineers at the time knew little about metal fatigue, which was the cause of the crashes, but aircraft design improved as a result of the lessons learned from these failures.
Products may succeed technologically but fail in the market. The Sony Betamax was technically a better video recorder than its competitor, Matsushita's VHS system, but it lost the competitive battle to become the dominant design in the market. Concorde was a technological marvel in its time, but it sold only to the British and French governments, its joint manufacturer.
It is not always possible to judge what is going to be valuable in the future. Apple's Newton – an early personal digital assistant – is a notorious product failure. It cost more than a computer, and one memorable technical review said it was so big and heavy it could only be carried around by kangaroos. Its failure cost Apple's CEO his job. Yet ten years later and its operating system was found in the iPod, and several Newton-like features were incorporated in the iPhone.
Failure has a personal cost, and innovators have to develop strategies to deal with failure that entail personal recognition of its value for learning, reflection, and self-awareness. Similarly, organizations need to appreciate the value, and learn the lessons, of failure.
#### Learning
Innovation is manifested in new products, services, and processes. Less material, but no less real, are the options for the future it provides and the organizational and personal learning it encourages.
Organizations learn to do better the things they already do, learn to do new things, and learn about the need to learn. Organizations inevitably learn by doing familiar things; the more you do something, generally the better at it you become. But radical and disruptive innovations – those that involve significant breakthroughs and fracture past ways of doing things – pose great difficulties for organizations and the ways they learn. Established routines and ways of doing things actually restrain learning about these forms of innovation. Focus on the status quo produces returns that are positive, proximate, and predictable; focus on the novel produces returns that are uncertain, distant, and often negative. This produces a tendency to substitute exploitation of known alternatives for the exploration of unknown ones. Radical innovation involves technologies that are destabilizing of existing capabilities, and disruptive innovations entail disengagement with existing customers and secure income streams. There are compelling reasons why organizations try to avoid them.
This is where leadership comes in, providing the encouragement and resources to do things organizations find hard, but which are necessary for their continuing viability. Positive affirmation of the outcomes of innovation, through reviews and post-project assessments, and their wide communication throughout the organization builds support for new forms of learning. When positive results from innovation become remembered as organizational stories and corporate myths, they assist efforts to break with routine and institutionalized practices, and stimulate learning in all its forms.
#### Employment and work
There is a continuing debate on the impact of innovation on employment and its effect on the quantity and quality of jobs. Innovation has contributed to the massive historical shift in aggregate employment from agriculture to industry to the service sectors, but its impact on industries and organizations depends on their particular circumstances and choices.
The debate itself has a long history. Adam Smith would argue that increases in market size lead to greater opportunities for the division of labour, the replacement of people with machines, and potential deskilling. For Marx, automation inevitably led to labour replacement, wage reductions, and greater oppression of the workers. Schumpeter would argue that, as innovation both creates and destroys jobs, there will be mismatches between jobs and skills in declining industries and regions and emerging innovative new sectors, and painful adjustments will be needed with periods of skill shortages and unemployment.
One view has product and service innovation producing positive effects on jobs and skills, and process and operations innovation producing negative effects. As we shall see in Chapter 5, Edison created highly skilled jobs in his 'invention factory' and large numbers of unskilled jobs in his production factory. The skilled jobs were allied to product innovation, where thinking was at a premium, the unskilled jobs were linked to process innovation, where machinery reduced the need for thinking. But there is value in having skilled workers on production lines, and organizations often make choices about how they use innovations. The way machinery is designed and tasks configured affects the use of skills. Because of these choices, and as a result of the adjustments necessary as industries evolve in response to innovation, there are major incentives for individuals, employers, and governments to invest in education and training.
Organizations need to understand how innovation can be personally rewarding and stressful, stimulating, and frightening. It can provide incentives and motivation and fear of change and loss of status. It can be divisive, with one part of the organization undertaking well-remunerated and satisfying work and others poorly paid and dissatisfied. It may be exclusive, denying people without a particular type of education, or in some cases women, access to jobs.
#### Economic returns
Productivity, the index of outputs to inputs, increases when resources are used more efficiently. Improvements in the use of labour and capital increase productivity. It also increases when innovation, and improvements in technology and organization, contribute to what is known as multi-factor productivity (MFP). Ultimately, economic wealth depends upon improved productivity, and this is frequently driven by innovation. MFP growth in the USA in the 1990s, for example, was linked to the information and communications industry and the use of its products in other sectors of the economy. Much recent growth in MFP has occurred in the service industries, such as retailing and wholesaling, and this can partly be attributed to the use of digital technologies.
Profitability is driven by a wide number of factors such as how much better and efficient organizations are compared to competitors at designing, making, and delivering things, and the preferences of customers for particular brand names and their preparedness to pay prices that provide the required return to innovators. Innovation contributes to profits by providing distinctive advantages in the sale of products and services; in their features, prices, delivery times, upgrade opportunities, or maintenance. Intellectual property can be sold and licensed, and new start-up businesses can be created, to produce profit from innovation. Large-scale innovative activity, in investments in R&D or plant and equipment, can deter competition and hence improve opportunities for profit.
For organizations to benefit financially from investments in innovation, they have to appropriate returns. In some circumstances, innovation can be protected by using intellectual property law for patents, copyrights, and trademarks. In others, protection derives from difficult to replicate skills and behaviours, such as the capacity to quickly stay ahead of competitors, being able to maintain secrecy, or retaining important staff. In all cases, the contributions of innovations to profits are frequently skewed, with most returns from a few innovations.
Technical standards that allow inter-operability between components and systems confer economic advantage. Those organizations that own standards, or whose offerings comply with them, have advantages over those that do not. Battles over technical standards can become particularly heated, as we shall see in the case of Edison in Chapter 5.
## [**Chapter 4
Stephanie Kwolek's new polymer: from labs to riches**](contents.html#ch04)
Many people and organizations contribute to innovation. Large-scale surveys of innovative firms, such as the European Union's Community Innovation Survey, for example, show a wide range of contributors. These surveys also rank the importance of various sources, showing the most important of them lies within the organization. Innovation derives primarily from the energy, imagination, and local knowledge of employees identifying and solving problems. It is stimulated by innovative individuals and workplaces, and by formal organization structures and practices, such as R&D departments and management tools for developing new products.
Second in importance as sources of innovation, according to these surveys, are customers and clients, followed by suppliers of goods and services. Fairs and exhibitions, professional conferences and meetings, and academic and trade journals are reported to be important by a minority of firms. The least important sources, these surveys show, are universities and government research laboratories.
These rankings hide a much more complicated picture. Reliance on internally derived innovation, for example, makes organizations introspective and perhaps unprepared to deal with changes occurring externally in markets and technologies. Depending on customers for innovative ideas will likely produce conservative 'don't rock the boat' approaches. Universities are critically important contributors to invention for science-based sectors, and for innovative products and services at early stages of gestation, and they also educate and train employees with the skills to innovate.
As Josiah Wedgwood showed us, innovation usually involves the combination of ideas derived from many different starting places. The great scientist Linus Pauling said the best way to have a good idea is to have lots of them, and the same sentiment can apply to the pursuit of innovation from multiple contributors. Schumpeter's contention that innovation requires 'new combinations' amongst markets, technologies, and knowledge often entails integrating ideas from many different parts of the organization and with various external parties. The stimulus to innovate may not result from particular sources, with hierarchical contributions, but from multiple sources of ideas that intersect and blur in circumstances of intrinsic necessity and the impulsive quest for survival in volatile times.
Innovation is also affected by wider social, cultural, political, and economic factors. These include the contributions made by cities and regions, government policies, and the 'systems of innovation' organizations belong and contribute to.
### Continual pursuit: the case of IBM
The continual, broad-ranging, and challenging pursuit of innovation is seen throughout the history of the IBM Corporation. IBM is widely recognized as one of the most innovative companies in the world, playing a central role in the discovery and development of, amongst other things, supercomputers, semiconductors, and superconductivity. It has invested enormous resources in innovation. It spends billions of dollars on R&D each year, produces more patents than any other company, regularly creates iconic products and services, and its staff have won five Nobel Prizes. It has huge advantages in innovation compared to almost any other company in the world, yet its pursuit of innovation holds lessons for other organizations.
IBM was incorporated in 1924, but its history can be traced back to Herman Hollerith's foundation of the Tabulating Machine Company in 1896. Hollerith (1860-1929) developed a machine using electricity and card processors to mechanize the data-processing of US Census data. He called the machine 'hardware' and the cards 'software'. Hollerith worked for a time at the US Census Bureau and was acutely aware of its need for improved efficiency in processing data. The 1880 Census had taken seven years to compile, and there were fears that the 1890 version was going to take longer. Hollerith's tabulating machine met the Census Bureau's requirement for fast and efficient data collection and management. By using it, the 1890 data were analysed in six months, saving millions of dollars, and it was subsequently used in censuses in Canada and Europe. By 1912, Hollerith had sold his business and, although he remained as chief consulting engineer, he had less and less association with the company. For many years, he had refused to respond to requests and ideas from the Census Bureau for improvements to his machines. When Hollerith's main patents expired in mid-1906, the Bureau developed a tabulator of its own, which it used in the 1910 Census. It took the arrival of Thomas Watson in 1914 to improve the technical performance of tabulating machines and improve the company's relationships with its customers.
As Chairman of IBM, Thomas Watson (1874-1956) was instrumental in developing the company's use of electronics. He underwrote Harvard scientist Howard Aiken's 1930s research into developing a digital calculating machine. In 1945, in collaboration with Columbia University, he opened the first Watson Scientific Computing Laboratory, in New York. The IBM Thomas Watson laboratory remains today as one of the largest industrial research laboratories in the world. During the Second World War, the company developed very close relationships with the US government, especially in military ordnance and planning wartime logistics. It limited its profit margins for military work to contribute to the war effort.
During his 42 years at IBM, Watson built the company into a major international corporation. His son, Thomas Watson Jr, succeeded him as Chairman. From the late 1950s to the 1980s, following massive investments in R&D, IBM became the world leader in mainframe computers, especially with its System 360, launched in 1964. The System 360 remains in real terms one of the largest private investments ever made in R&D. The company, which was valued at $1 billion at the time, committed $5 billion to its development. Tom Watson Jr had 'bet the company' on its development. By 1985, IBM had 70% of the world mainframe market. It had unmatched expertise in hardware and software, and its business skills made it one of the world's most admired companies.
**4. The IBM System/360 computer. IBM 'bet the company' on its development**
By the mid-1970s, the company began the pursuit for smaller computers. The IBM Personal Computer (PC), launched in 1981, was along with the System 360 one of the most iconic products of the last century; it essentially created the mass market for PCs. It emerged from an IBM development group in Boca Raton in Florida that had failed in three previous attempts to create a PC. The successful development of the PC required the rejection of IBM's past strategy of self-reliance and developing everything itself internally. It decided to buy major components such as integrated circuits and operating software from small suppliers. Initially, the product was a huge success, capturing 40% of the market.
By the late 1980s and early 1990s, however, IBM was in serious trouble and nearly went bankrupt. The IBM PC had helped sow the seeds of its own demise. IBM did not control the intellectual property rights to its components, and the small suppliers – Intel and Microsoft – grew quickly to be larger and more powerful than IBM and supplied its competitors with their technology. Furthermore, the overall culture of IBM remained focused on historically profitable mainframes at the same time as price competition from Japanese manufacturers led profit margins to collapse. _The New York Times_ of 16 December 1992 was led to the opinion in an editorial that 'The IBM era is over... what was once one of the world's most vaunted high-tech companies has been reduced to the role of a follower, frequently responding slowly and ineffectively to the major technological forces shaping the industry.' The story of Herman Hollerith's rise and fall resonated again.
One response to IBM's 'near-death' experience was to appoint a new CEO, Lou Gerstner, the first appointed from outside IBM.
The company went through a massive restructuring and fundamental shift in business strategy. It changed from being a supplier of technology to being a provider of solutions to customer problems. Its objective was to provide the best possible service for customers, even if this meant using competitors' technologies. At the same time, despite its financial difficulties, the decision was made that as the company's strength in the past derived from its 'science and engineering mindset', its research investment should continue in the future. The search was on to find increasing innovations from within the company's technological community and R&D centres. These internal sources essentially reinvented the mainframe using microprocessors and parallel architectures. IBM also became much more open to ideas from outside, attempting to break away from its past introspection and 'not-invented-here' syndrome. It began to use open technical standards and software rather than those it owned proprietarily and started to collaborate more in its technological developments, embarking annually on scores of collaborations with other organizations. Its new 'market-facing' innovations included supercomputing, e-business, social networks, and Web 2.0 technologies.
The company presently makes extensive use of its intranet and social networking technologies to access and share ideas amongst its staff. With around 400,000 employees, half of whom are scientists and engineers, and 75 research centres globally, the company has massive technological skills on which to draw. The ways it uses these skills to support its current and emerging innovation process will be discussed in Chapter 6.
IBM illustrates the extensive pursuit of innovation throughout its history, involving inventor-entrepreneurs, customers, suppliers, universities, R&D departments, relationships with government, collaborative partners, and the broad community of its own staff and contacts. We now turn to these various contributors.
### Entrepreneurs and venture capitalists
In contrast with the large-scale activities of companies such as IBM, innovation also results from individual entrepreneurs using it to build new businesses. The term 'entrepreneur' began to be used in the early 18th century and is applied to individuals who discover, recognize, or create opportunities and then manage resources and bear risks to take advantage of them. Wedgwood demonstrates the substantial contribution entrepreneurs can make to innovation and economic development.
From Matthew Boulton in the 18th century, to Thomas Edison in the 19th, to Bill Gates in the 20th, and Sergey Brin and Larry Page in the 21st, entrepreneurs are commonly associated with the creation of technology-based companies. These companies grow rapidly on the basis of novel technologies that create new industries and transform old ones. Some entrepreneurs transform entire economies and societies. Boulton and his partner, James Watt, developed the steam engine and the world's first mechanized factory and helped usher in the Industrial Revolution. Edison, amongst many other contributions, developed electric power-generation technology and created the General Electric Company. Software from Gates' Microsoft popularized the personal computer; Brin and Page's Google transformed the use of the internet; and both companies have changed the nature of work and leisure.
These are highly exceptional examples. Around half a million new companies are formed annually in the USA alone, and very few, if any, will be as successful as Microsoft and Google. Yet the creation of new firms, and the challenges they present to incumbent firms, is an essential element and a major contribution of capitalism. In Schumpeter's Mark I model, creative destruction is driven by the entrepreneurial task of 'breaking up old, and creating new tradition'.
### Schumpeter on entrepreneurs: some selected quotations
Some of the many motivations and characteristics of the entrepreneur identified by Schumpeter have resonance today.
Motivation:
'the dream and will to found a private kingdom, usually, though not necessarily, also a dynasty';
'the will to conquer: the impulse to fight, to prove oneself superior to others, to succeed for the sake, not of the fruits of success, but of success itself';
'the joy of creating, of getting things done, or simply exercising one's energy and ingenuity';
'private property... pecuniary gain... (and) other social arrangements not involving private gain'.
Character:
The entrepreneur...
'seeks out difficulties, changes in order to change, delights in ventures';
needs: 'extraordinary physical and nervous energy';
possesses: 'that special kind of ''vision''... concentration on business to the exclusion of other interests, cool and hard-headed shrewdness – traits by no means irreconcilable with passion';
knows how to: 'woo support' among colleagues, 'handle men (sic) with consummate skills,' and 'give others ample credit for the organization's achievements'.
Schumpeter's Mark II model recognized that entrepreneurship occurs in large firms as well as newly created firms, reflecting the changing industrial realities as formally organized, large-scale R&D activities grew in scale from the 1920s. Entrepreneurship is therefore the organizational process by which opportunities are sought, developed, and exploited in many different kinds of company.
In some circumstances, entrepreneurial start-up firms receive investments from venture capitalists that are prepared to assume higher risks than high street and investment banks. Many of the US success stories of entrepreneurial information technology and biotechnology companies, such as Google and Genentech, received venture capital. Different international models of venture capital exist, but the US is often considered exemplary. US venture capital may include funds from private investors or corporations, and their managers may possess deep experience or knowledge of particular technological sectors and become engaged in the governance of start-up companies. The objective of venture capitalists is usually to acquire shares in companies in their early years that then reap extraordinary returns after they exit when the firms have reached sufficient maturity to attract a purchaser or to be floated on a stock market. Amongst their portfolio of investments, venture capitalists recognize that the majority of returns will come from a limited number of cases. Generally, venture capitalists tend to invest in better-established, rather than new and speculative ventures, when technological and market opportunities have been clearly identified.
### Research and Development (R&D)
R&D is a significant, but not always essential, source of innovation. Investments in R&D help organizations search for and find new ideas and improve their capacities to absorb knowledge from external sources.
R&D ranges from basic research driven by curiosity and little concern for its application, to highly practical problem-solving (see the definitions in the Frascati Manual below). Its expenditure reflects highly varied national, sectoral, and corporate commitments to its use in pursuit of innovation. Internationally, around $800 billion is spent annually on R&D. At an aggregate level, it is concentrated in a few main industries, including information and communication technology and pharmaceuticals. The USA is the major spender on absolute amounts of R&D. When relative expenditures on R&D are assessed – usually measured as a portion of a nation's Gross Domestic Product – smaller European nations, such as Finland, Sweden, and Switzerland head the list, committing over 3% of GDP annually. A marked trend in recent years has been the rapid growth in R&D expenditures in Asian nations, such as Korea, Taiwan, and China. Over 95% of R&D globally is spent in the USA, Europe, and Asia (primarily northeast Asia), so many nations, especially in the Southern hemisphere, cannot compete in this important source of wealth creation and growth.
There are wide differences among nations in the breakdown in R&D expenditures between that spent in business and government. In some countries, such as Korea and Japan, business expenditure predominates. In others, such as Poland and Portugal, government is the major source of R&D spending.
### The Frascati Manual
In 1963, the Organization for Economic Cooperation and Development (OECD) decided it would be useful for policy-making to have consistent international data on R&D statistics. Following a meeting in Frascati, Italy, the Proposed Standard Practice for Surveys of Research and Development was produced. This became known as the Frascati Manual. The sixth edition of the Manual was produced in 2004.
R&D is defined as comprising creative work undertaken on a systematic basis in order to increase the stock of knowledge, including knowledge of man, culture, and society, and the use of this stock of knowledge to devise new applications.
R&D is described as covering three activities:
• Basic research is experimental or theoretical work undertaken primarily to acquire new knowledge of the underlying foundation of phenomena and observable facts, without any particular application or use in view.
• Applied research is also original investigation undertaken in order to acquire new knowledge. It is, however, directed primarily towards a specific practical aim or objective.
• Experimental development is systematic work, drawing on existing knowledge gained from research and/or practical experience, which is directed to producing new materials, products, or devices, to installing new processes, systems, and services, or to improving substantially those already produced or installed.
The Frascati Manual has been useful in building consistent data sets on R&D expenditures internationally. It has continually evolved and improved. Nonetheless, significant problems remain in measuring collaborative R&D and activities undertaken in services.
The OECD has also developed the Oslo Manual to guide national innovation surveys, the Canberra Manual for measuring human resources in science and technology, and a Patent Manual on the use of patent statistics.
### Stephanie Kwolek's new polymer
Stephanie Kwolek (b. 1923) has saved thousands of police officers and military personnel from death or disability. As a result of a traditional R&D process, she invented Kevlar, a fibre used in body armour. The product, which is one of the strongest fibres ever made, has over 200 applications, including in brake pads, spacecraft, sporting goods, fibreoptic cables, fireproof mattresses, storm protectors, and wind turbines. It has produced several hundreds of million dollars annually for the chemical company DuPont. It is best known, however, for its use in bullet-proof vests. In 1987, the International Association of Chiefs of Police and DuPont began a Kevlar Survivors' Club for those saved by the product from death or serious injury. Its 3,000th member was inducted in 2006. Kevlar's protective properties have also been used extensively in the military.
Kwolek was born in New Kensington, Pennsylvania. Her steelworker father died when she was young, but she retained his curiosity: he was a keen amateur naturalist. She remembers spending hours designing and making clothes for her dolls and being very interested in fashion. She studied at a college that became part of Carnegie-Mellon University, and, unable to afford to study medicine, she majored in chemistry.
**5. Stephanie Kwolek, inventor of Kevlar**
She decided she wanted to work for DuPont. DuPont was and remains one of the world's leading and most innovative companies. In the 1920s, it was one of the first companies to invest in basic research with the 'object of establishing or discovering new scientific facts'. It developed neoprene synthetic rubber in 1933 and nylon in 1938. With the shortages of male chemists resulting from the Second World War, women were being attracted into the chemical industry. During her interview, Kwolek forcefully demanded to know when she would hear about the job as she had another offer. The offer was made that evening.
Kwolek began work for DuPont in 1946. She worked at the DuPont Research Laboratory in Delaware for 36 years, having previously worked for the same group for 4 years in Buffalo, New York. Her job was to develop new polymers and ways of making them. Shortly after her arrival, she was given the job of looking for a breakthrough fibre to be used to make tyres lighter and stiffer. There was a concern at the time to improve vehicle performance to address a petrol shortage. Others had been offered the task, but weren't interested. Kwolek felt that, while her competence was recognized by male colleagues, she was often overlooked.
She liked the working atmosphere, however, and the challenges that arose, and as one of the few women scientists at the time, she worked exceptionally hard to retain her job after the men returned from the war. She was given a high degree of independence and freedom to do what she wanted. (She complains about modern research being too rushed and short term with not enough time for thinking.)
Kwolek's specialization lay in low-temperature processes for the preparation of condensation polymers. In 1964, she discovered that the molecules of extended-chain aromatic polyamides formed under certain conditions a liquid-crystalline solution that could be spun into a strong fibre. She took her polymer, which was unpromisingly cloudy and thin, to a machine to be spun. She says the polymer had such strange features that anyone not thinking or being unaware would have thrown it out. The technician in charge of the spinner was deeply sceptical, thinking his machine would become bunged up by this contaminated substance, but was eventually persuaded to try. It spun successfully a product that was so strong that Kwolek had to repeat tests many times before she was convinced about her discovery. She did not tell anyone about her polymer until she was sure of its properties. Kevlar is heat-resistant, five times stronger than steel, and about half as light as fibreglass.
DuPont was immediately convinced of the value of Kwolek's new crystalline polymers, and the Pioneering Laboratory was given the job of finding commercial applications. She provided a small amount of fibre to a colleague experimenting with bullet-proof armour. Kevlar was introduced for this purpose in 1971. One of the reasons that it is used in such a broad range of applications is its versatility: it can be converted into yarn or thread, continuous filament yarn, fibrillated pulp, and sheeting. The new chemistry that Kwolek developed helped DuPont develop a range of other fibres, such as spandex Lycra and the heat-resistant Nomex.
Kwolek attributes her success to the way she can see things that others cannot. And she says:
To invent, I draw upon my knowledge, intuition, creativity, experience, common sense, perseverance, flexibility, and hard work. I try to visualize that desired product, its properties, and means of achieving it... Some inventions result from unexpected events and the ability to recognize these and use them to advantage.
Kwolek has 17 patents, including 5 for the Kevlar prototype. She has won numerous prestigious awards and has spoken about the great need for recognition of scientists and other people who do things that benefit mankind. She admits to being very pleased when a police officer asked for her autograph on the jacket that saved his life.
The case of Kwolek and Kevlar epitomizes the corporate R&D department's contribution to innovation. It also points to some of its shortcomings. The polymer was 18 years in development, and its commercialization took 7 years. Few, if any, organizations have the capacity to take such a long-term approach nowadays.
### Customers and suppliers
Innovations do not succeed unless customers or clients use them, and if the users of new products and services are involved in designing what they need, there is generally a better chance of success than if something is being designed for them. Demands and needs can never be articulated fully and communicated completely across organizational boundaries between producers of innovation and their customers and suppliers, and active engagement between them overcomes these barriers.
In some fields, such as medical instruments, the innovator is commonly the user of the innovation. Surgeons and medical practitioners are regular contributors of ideas for new tools and techniques that help them do their jobs better. The world's largest producer of implanted hearing devices, Cochlear, originated with Professor Graeme Clark, a medical researcher whose father was profoundly deaf. Clark was highly sensitized to the suffering of people whose deafness could not be helped by hearing aids, and was driven to improve their lives.
According to one estimate, up to one-quarter of all men over 30 suffer from sleep apnoea, a condition that causes potentially dangerous breathing irregularities when they are asleep. Respiratory medical devices can help manage the problem. The origins of the world's largest manufacturer of respiratory devices – ResMed – lay with Professor Colin Sullivan, a medical researcher working in a hospital sleep clinic. He overcame the problem by regularly blowing puffs of air up the nasal passage. Fortunately, for sufferers and their partners, as a result of continuous design improvements the present discreet and quiet devices are a big improvement from the original version built from a gas mask and vacuum cleaner.
Some companies go to great lengths to engage customers in designing new products. When Boeing developed the 777 aircraft, it involved its major customers, United, British Airways, Singapore Airlines, and Qantas, in trying to understand the demands of the market. It needed to know about optimum passenger loadings for airlines' favoured routes. But it also worked to understand the demands of the users of the aircraft: the pilots and aircrew, maintenance engineers, and cleaners. It aimed to be sympathetic to flight attendants brewing coffee in turbulence and maintenance engineers fixing an external component in Alaska at midnight in –40 degrees centigrade or Jeddah at midday in 50+ degrees. As Boeing developed the 787, it built a website to gain immediate input into the design process from interested parties around the world. Around 500,000 people voted on the choice of the aircraft's name: Dreamliner.
Software companies sometimes release their products in 'beta' form, that is, in prototype, to allow users to play with the software and suggest improvements. Essentially, customers do much of the final polishing of the product. This strategy is pursued for products that are proprietary, when the company aims to profit from them. This is different from open-source software, such as the web browser Mozilla Firefox and the operating system Linux, which is built, maintained, and continually improved by networks of volunteer programmers.
Excluding customers from the process of product improvement can be very shortsighted. When Sony developed its robot dog -the Aibo – it kept its software code secret. A community of hackers grew up, developing a much more extensive range of moves for the robot, including a number of amusing dances that made it a much more attractive product for customers. Sony sued the hackers and closed down their community, but soon after recognized its mistake and realized that the company could learn from the software developed externally. Sony no longer produces the Aibo, but its subsequent products have benefited from the technology developed for the robot dog in areas such as visualization.
Customers can also inhibit innovation. They can be conservative and complacent and locked into ways of doing things that preclude novelty and risk. Clayton Christensen identified the 'innovator's dilemma'; the problem of listening too closely to customers. If innovators only respond to the immediate demands of customers, they often miss big changes occurring in technologies and markets that may eventually put them out of business. Here there is advantage in working with 'lead customers', governments, firms, or individuals that are prepared to take risks to promote innovation in the belief that greater benefits will accrue than pursuing the safer short-term option of not innovating. In the 1980s, Roy Rothwell described the relationship between Boeing and Rolls Royce as 'tough customers; good designs', implying that Boeing's very demanding requirements of its aero engine suppliers made Rolls Royce design and produce better products.
Innovative suppliers are also major stimulants to new ideas. In the automotive industry, a high percentage of the value of a car is bought from suppliers of components, accounting in Toyota's case for up to 70% of the car's total cost. Toyota enjoys very close relationships with Nippondenso, a very large components supplier of innovative products such as lighting and braking systems. The automotive supplier Robert Bosch plays a similar role in the European car industry. Large automotive companies use numerous methods, including websites and technical conferences and fairs, to encourage their suppliers to provide innovative solutions to the problems they face. Innovative automobiles are based on suppliers of innovative components to car companies. The task of the car manufacturer – or the organization responsible for integrating any system of different elements – is to encourage innovation in suppliers of modules or components while ensuring the compatibility of components with overall design architectures or systems.
The encouragement of innovative suppliers is also a key objective of many governments. In the USA, the Small Business Innovation Research scheme uses the government's enormous procurement budget to support small companies by purchasing innovative products and services. This particular government scheme invests more in innovation in start-up companies than the US venture capital industry, and does so in more early stages of their development.
### Collaborators
Innovation rarely results from the activities of single organizations and more commonly occurs when two or more organizations collaborate. For many organizations, the benefits of using collaboration to contribute to innovation outweigh the costs of sharing the returns to that innovation. Collaborations take the form of joint ventures and various types of partnerships, alliances, and contracts that involve joint commitments to mutually agreed aims. They can be with customers and suppliers, organizations in other industries, and even with competitors. They are a feature of the world's industrialized economies, with some collaborations operating for many decades.
Organizations collaborate to reduce the costs of developing innovation, to access different knowledge sets and skills to the ones they possess, and use it as an opportunity to learn from partners about new technologies, organizational practices, and strategies. In uncertain and evolving circumstances, innovating collaboratively provides a greater chance of success than going it alone. Information, communications, and other technologies have made collaboration cheaper and easier. Governments around the world have actively promoted collaboration as a source of innovation. And organizations have become less self-reliant and more open to collaboration in their strategies for innovation.
Different types of collaboration work better in different situations. When the objectives of collaboration are clear, or the focus is on quickly reducing costs, it works better when organizations are similar. The opportunities for misunderstandings and miscommunication are fewer. When objectives are emergent, and the objective is exploration and learning, collaboration benefits from dissimilar organizations working together. More is learned from variety than uniformity. Larger numbers of partners increase the scale of effort; fewer partners improve speed.
Collaboration can be difficult to manage. Partners may have different priorities and organizational cultures. There are many opportunities for misunderstanding, as the following, perhaps apocryphal, anecdote reveals. Some years ago, a collaboration was proposed between a group of IBM and Apple staff. Prior to the first joint meeting, IBM staff discussed their approach.
Conscious of their reputation for formality – blue suits were the uniform of the day – they decided to put the Apple staff – who were usually casually dressed – at ease by wearing their weekend clothes to the meeting. They arrived in jeans and sweatshirts to find the Apple staff sitting uncomfortably in newly purchased blue suits. That this could occur amongst organizations in the same industry and country highlights the potential problems that might occur in collaborations in different sectors and nations.
### Universities
Clark Kerr, renowned social scientist and president of the University of California, was highly prescient in identifying the importance of universities for economic development when he wrote in 1963 that:
... the university's invisible product, knowledge, may be the most powerful single element in our culture... the university is being called on to produce knowledge as never before... And it is also being called upon to transmit knowledge to an unprecedented proportion of the population.
He argued new knowledge is the most important factor in economic growth and highlighted the role of the university in developing new industries and generating regional growth, and emphasized the contribution of the entrepreneurial professor, consulting and working closely with business. In the intervening decades, universities have been increasingly encouraged by governments and business to devote their energies to actively translating their knowledge into economic activity, a policy they have often enthusiastically endorsed. This activity is now so elevated that it has emerged, according to some, to be as important a university function as research and teaching. The ways knowledge is transferred to industry, and universities contribute to innovation, are often conceived too simply, however, and the paths to market are commonly complex, multifaceted, and subtle. The idea that ideas and knowledge are something that universities produce and 'transmit' to industry has also been replaced by the notion that it is co-created and exchanged.
### Teaching
By educating skilled undergraduates, graduates, and post-doctoral students, universities prepare a labour force's capacity to create and apply new ideas. The history of the successful development of new industries – such as electrical, chemical, aeronautical, and information technology – is in major part explained by the provision by universities of sufficient numbers of graduates with the requisite new skills, especially in engineering and management. It is said that the best form of knowledge exchange between university and industry is carried on two legs and by the movement of problem solvers from university to industry.
It is not only science and engineering graduates that contribute to innovation. At various times, philosophers and anthropologists have been in demand in Silicon Valley, and the creative industries provide a home for many Humanities students. Business schools increasingly provide courses in innovation management and entrepreneurship for students of all disciplines. The management literature discusses how successful firms are said to require a combination of 'I'-shaped people, with deep knowledge in a particular field, and those that are 'T'-shaped, having a breadth of knowledge with a particular specialism. The capacity to see connections 'across the T' between various disciplines is a major stimulus to innovation, but poses significant challenges for educators (see MIT and the education of engineers).
Technical colleges also play an important role in innovation, for example in training technicians to produce instrumentation which they themselves occasionally commercialize.
### Massachusetts Institute of Technology and the continuing challenge of educating engineers
Engineering is problem-solving, and to encourage this ability MIT has traditionally attempted to encourage interdisciplinarity in its educational approach. In the MIT Bulletin of 1954-5, it was said that the aim of its School of Humanities and Social Studies was to develop 'the first-rate human and social values which must accompany technical competence if an individual is to make his (sic) maximum contribution as a citizen'. The curriculum aimed to reflect these values. All students in the first two years of their four-year degrees undertook core courses encompassing history, philosophy, and literature. Their focus was on problems rather than solutions, and the development of an attitude that education was something they have to use continuously and develop, rather than something they have had.
The contemporary challenges of educating 'rounded' engineers are described by Rosalind Williams, MIT's former Dean of Students. She says that engineers nowadays need to understand how things are designed and brought to market, and how organizations work and innovations become successful. Indeed, she quotes a colleague saying how MIT had long ago given up training professional engineers and were really training technological innovators. She claims they need to understand science, on the one hand, and humanities, arts, social sciences, and management, on the other, and as a result the amount of information needed to be crammed into the head of a student doubles every 18 months. To accommodate this breadth, the trend, she reports, is the separation of engineers with different psychological and sociological profiles into 'systems integrators' and 'designers'; the former more intent on managing large technological systems in established firms, the latter more interested in entrepreneurially creating new products and services.
### Science and research
Science, from the Latin _scientia_ – knowledge – has been a feature of human development since the first civilizations. The application of science in industrial innovation, however, only began in earnest during the Industrial Revolution and has been most particularly a feature of the last 150 years or so.
One of the traditional distinctions in research, seen in the Frascati Manual, is between that which is 'basic' and that which is 'applied'. The former is thought to be curiosity-driven, with no consideration of its application, and the particular concern of universities. The latter is believed to be directed towards an identified use, usually in industry. Yet some businesses invest substantially in basic research and universities conduct extensive applied research, especially in professional departments such as medicine and engineering.
Furthermore, as Donald Stokes argued, the classic distinction between 'pure' or basic research, driven by a desire to understand, and applied research, with the purpose to be used, fails to capture a third category which aims to do both by improving understanding and being useful. He calls this 'Pasteur's Quadrant' of use-inspired basic research (see Figure 6). Pasteur's microbiology research was always concerned with useful applications, but created a new field of scientific understanding. Stokes contrasts this with Bohr's research in physics, in which his understanding of atomic structure provided a basis for developing the theory of quantum mechanics, and Edison's research that was driven by a concern for use and profit, although he was also influenced by theory. There is a direct and obvious connection between research and innovation in Edison's and Pasteur's Quadrants: the connection in Bohr's may or may not occur, and if it materializes, it might be in unexpected or unimagined areas. Bohr, one imagines, would have had little appreciation of how quantum theory is used to explain lasers and potentially provide the basis of future quantum computers.
**6. Pasteur's Quadrant: From Donald Stokes _Pasteur's Quadrant_ (Washington, DC: 1997)**
Similarly, in their short letter to _Nature_ on 25 April 1953, Watson and Crick modestly stated: 'We wish to suggest a structure for the salt of deoxyribose nucleic acid (D.N.A.) This structure has novel features which are of considerable biological interest.' They did not imagine the considerable commercial interest that was to emerge more than 20 years later, or the way in which their discovery has transformed old and created new businesses with the development of biotechnology.
In reality, basic and applied research are elements of a continuum, with many interconnections. Applied research may result from basic research findings, and basic research may be undertaken to explain how an existing technology works. One of the most useful outcomes from pure basic research is the instrumentation developed to assist experimentation. The computer, laser, and Worldwide Web were developed to this end, with little appreciation of potential industrial use as now ubiquitous innovations.
**7. Letter to _Nature_ announcing the discovery of DNA**
When we consider the world's most complex scientific and social questions, including global warming, sustainable energy, food security, and genetic engineering, the answers will depend on fundamental understanding developed by universities and its practical use in industry.
### Engagement
Dr Jonas Salk was reputedly once asked who owned the polio vaccine he developed. His answer was, 'why the people, I would say'. Such a response is unlikely today. Since the passing of the Bayh-Dole Act in the USA in 1980, which allowed research institutions to own the results of publicly funded research, universities in developed economies have become preoccupied with making money from their research. This has usually taken the form of patent-protected intellectual property, licensed to businesses, or through start-up companies, spun out of and part-owned by the university. Evidence suggests, however, that the number of successful instances of this model of commercialization is limited. There are some impressive success stories, such as the biotechnology firm Genentech. This company was formed in 1976 to help commercialize the discovery of recombinant DNA at Stanford University, and was sold to a Swiss pharmaceutical company in 2009 for just under $50 billion. Such companies, however, are a tiny fraction of the total amount of entrepreneurial activity encouraged by universities.
The focus of the majority of attention by governments, and indeed by many universities themselves, has been on issues such as patents and licensing, contract and cooperative research, and incubation and entrepreneurship centres. These activities are important for innovation in industries that are based on science and technology, but not for all industry sectors. They are generally less relevant for services, resources, and traditional industries, such as clothing and textiles. Furthermore, they ignore the importance of the social and networking activities that are crucial to the 'conversations' between universities and business about new developments and their potential applications. Although for many businesses, particularly smaller ones, the purpose of collaborating with universities is immediate problem-solving, larger firms will engage in broader dialogue with universities to learn about the directions of future research. Businesses claim that the attraction of working with universities is that they have different cultures to their own. University staff have more time to think about and test new ideas.
As contributors to inventive ideas and knowledge creation and diffusion, universities and research institutes need continually to communicate their capabilities and assess how they should best engage with external parties. Government and business cannot be expected to invest in universities and research institutions as providers of innovation without them fully articulating their broad contributing roles.
### Regions and cities
Innovation agglomerates by localizing within particular geographies, such as in the Staffordshire Potteries. It does so for economic reasons, as proximity reduces the costs of transactions and transportation, and firms in close association stimulate the creation and diffusion of innovation through improved awareness and knowledge of each other. Innovation clusters for social and cultural reasons, including the advantages derived from shared identity and higher trust in affiliated and cohesive groups. Communications are assisted by propinquity because knowledge is sticky and travels badly from its source, especially when it is complex or tacit and cannot be written down.
The best-known innovative region is Silicon Valley, near San Francisco, an area of high technology business concentration and employment, and one that has stimulated countless and often fruitless attempts at replication around the world. A number of factors contributed to the development and growth of Silicon Valley. Government played a central role, from the gift of land to local universities to stimulate industrial development, to being a large-scale customer of high-tech goods in defence markets. Universities have contributed their research and the education and training of scientists, technologists, and entrepreneurs. Institutions such as Stanford University have proactively developed policies to encourage academic engagement with business in fields such as electronics and information technology. Numerous high-tech businesses have started up and some have grown rapidly into major corporations, such as Hewlett-Packard, Apple, and Intel, assisted by a highly skilled and mobile labour market attractive to talented employees, links to university research, and ready access to professional services such as venture capitalists and patent lawyers. These factors contribute to a local culture, or 'buzz', that is technology-focused, risk-taking, and highly competitive, and creates a virtuous circle of initiative and reward. It has created enormous wealth and extensive experience of innovation and entrepreneurship to be reinvested back into new initiatives.
It is often not regions but cities that provide the locus of innovation. Throughout history, cities have at various stages been associated with creativity and innovation, from Athens in the 5th century BC to Florence in the 14th century, to _fin de siècle_ 19th-century Paris.
Cities are major contributors to the supply and demand of innovation. Most patents emanate and R&D is conducted in cities, and their higher disposable incomes ensure greater consumption of innovation. Some cities are renowned as centres of learning, such as Oxford or Heidelberg, others for their engineering ingenuity, such as Stuttgart or Birmingham, financial and services innovation, such as London and New York, and creativity and design, such as Copenhagen and Milan. Some cities are known for their particular technology expertise, such as Bangalore and Hyderabad in India, or their support for technological entrepreneurship, such the Hsinchu area in Taiwan and the Zhongguancun area in Beijing, China. The efforts of many city governments have been directed towards policies to identify and harness innovation that provides comparative advantage over other cities internationally. Although many have been blinded by the attraction of the Silicon Valley, technology-led, model, it is important that others have different approaches by, for example, addressing health, fashion, or the media. The issues of innovation in cities will be discussed further in Chapter 6.
### Government
The debate about the role of the government in supporting innovation commonly reflects political ideology. State intervention in innovation is considered essential in many nations, including most Asian countries, but in more 'free-market' economies, such as the USA, rhetorically at least, government intervention is regarded sceptically and avoided, usually with reference to government's inability to 'pick winners'. Nonetheless, the past polarities of the views that argue, on the one hand, that interventionist innovation policies distort markets and promote inefficiencies or, on the other, are essential components of sound economic planning and effective industry policies are tending nowadays towards a pragmatic middle ground. Here it is recognized that government has an important role to play in innovation, but policies have to be selective.
Governments contribute to innovation in many ways apart from their innovation policies. A stable, growing economy enhances the preparedness of firms and individuals to invest in innovation and take risks. Effective monetary and fiscal policies are crucial in providing confidence in the future. A nation with more wealthy firms and individuals is better placed to be innovative. Good educational policies produce employees and entrepreneurs with skills to create, assess, and realize opportunities for innovation. Well-educated citizens are more capable of contributing to national debates on innovation, and determining which sciences and technologies are acceptable, and what form new products and services should take. Government investments in research – which in developed nations account on average for about one-third of total expenditures on R&D – provide many of the opportunities for innovation. These investments can often take a longer perspective than those made in the private sector. Competition policies prevent monopolies erecting barriers to innovation; trade policies increase the size of markets for innovative products and services; intellectual property laws can provide incentives to innovate; regulations in areas such as environmental protection stimulate the pursuit of innovation. Free and open access to information stored by government increases opportunities for innovation. Innovation in a highly digitally connected world is inhibited unless government acts to ensure personal privacy and encourage ethical codes of practice when it comes to the collection and use of data. Open immigration policies allow the flow of talent from overseas and are the source of diversity, which is so important for innovative thinking. Industrial relations laws can help provide equitable, secure, and participative workplaces encouraging of innovation.
Governments can encourage innovation through their procurement power: they are the primary purchaser of innovation in any nation. Public expenditures on information technology, infrastructure, pharmaceuticals, and many other areas, exceeds that of the private sector, so government purchasing is a major stimulus to innovation.
Leadership by governments can set a tone or atmosphere in which innovation is encouraged. When the political discourse is future orientated and ambitious – think of John Kennedy's plan to have man on the moon, or Harold Wilson's 'white heat' revolution of science and technology – it is more supportive of innovation than when it is relaxed and comfortable with the status quo. Public servants are more likely to support innovation when they are not fearful of censure for the slightest mistake or risk-taking behaviour.
Apart from these forms of support, many governments develop specific innovation policies. These have tended in the past, especially in scale of expenditures, to focus on R&D, usually in the form of tax credits: by spending on R&D, firms can reduce their tax bills. There have been a plethora of other types of policy designed to encourage innovation. These include demonstration schemes, highlighting the benefits of particular innovations; consultancy schemes, helping organizations to improve their ability to innovate; investment schemes offering subsidies or increasing the amount of available venture capital for innovation; and creating new intermediary organizations that help build connections between research and business.
Many justifications for government innovation policy have been championed. These include, at their most practical, fear of international competition. The US government's response to the growing dominance of Japanese competition in semiconductors in the 1980s, for example, led it to create a well-funded consortia of US manufacturers, Sematech, directed to produce competitive technologies. Many pan-European schemes in the IT industry during the same period were designed to build the capability in Europe to resist US and Japanese competition. Some policies purported to encourage innovation are simple forms of industry support – or corporate welfare, to take a less charitable perspective. Schemes around the world to continually support the ailing car manufacturing industry in marginal electoral constituencies would provide a case in point.
Much of the justification for government intervention is presented in the form of an argument about 'market failure'. R&D, it is argued, produces knowledge that can be cheaply accessed by the competitors of those who undertake the risk of investing in it. The 'public' returns to investments thereby exceed the 'private' returns, and therefore there is a tendency towards under-investment. To address this market failure, government justifies the financial support of R&D in firms.
This form of support, which assumes the bulk of government investment in innovation policy, has several limitations. First, it is concerned with R&D, which is but one input into innovation, and in many industries and circumstances not the most important one. What is construed as 'R&D' might also be limited, and exclude important inputs to innovation such as software development and prototyping. Second, it misunderstands the investments required for public returns. The capacity of firms to access the R&D conducted by others is not costless. It requires investments to allow recipients to absorb the new ideas. Third, if market failure leads to sub-optimal investments in R&D, then there must be an optimal level, but there is little evidence about what this might be. Fourth, the delivery mechanisms of R&D support are highly generic, usually taking the form of tax credits for spending on R&D, rather than its performance. There is rarely provision for support of additional R&D to that which would be invested in without government money. The tax allowances are broadly available across industry, without the capacity to select strategic targets. Furthermore, application and compliance costs are usually resource-intensive, favouring larger, wealthier applicants rather than their generally more deserving smaller counterparts.
An additional case for government innovation policy can be made from the perspective of systems failure. Despite reservations about the dangers of the mechanical and predictable ways the national innovation systems we discuss below are viewed by government, as opposed to their commonly fluid and unpredictable reality, there is value in conceiving them from a government perspective. Government is the only actor capable of taking an overall national innovation systems view, and the only one able to influence its whole construction and function. It can assess performance, identify gaps and weaknesses, and support institutions and policies that build connections. The challenge for policy-making related to national innovation systems is that much attention is directed towards describing the components of the system, rather than what the system does, or perhaps even more importantly, what it should do.
The essential criterion for innovation policy is the extent to which it encourages and facilitates the flow of ideas across the economy and within national innovation systems and enhances the chance of their being successfully combined together and implemented. These flows of ideas occur in many, frequently unpredictable, directions: between the manufacturing, service, and resource industries; the public and private sectors; science, research, and business; and internationally within research networks or production supply chains. Innovation policy should therefore be concerned with the encouragement of the flow of ideas, the capacity of organizations to receive and use them, and the impediments to preventing effective connections between the various contributors to innovation.
Encouragement of the flow of ideas comes from open access to information and publicly funded research results, institutions that 'broker' connections between users and suppliers of knowledge, regulations that stimulate or at least fail to impede innovative investments, and judicious intellectual property laws that deal with the profound challenge of providing the confidence in their ownership to encourage trading, without providing the disincentives that occur from the award of monopoly positions. The receptivity to innovation in organizations depends upon the skills, organization, and quality of management of the recipients. Blunt policy initiatives such as R&D tax concessions are valuable only to the extent that they increase the quality and quantity of organization's capacity to select and use new ideas.
### Systems
The incredible success of Japanese industry in the 1970s and 1980s led to a search for its explanation. One analysis argued that it resulted from Japan's ability to organize the various elements of its economy into a national innovation system. In this view, the Japanese government played a central role coordinating large corporations' investments in important and emerging areas of industrial technology. Japan's strength in consumer electronics, for example, was believed to have resulted from the country's highly effective Ministry of International Trade and Industry (MITI) collecting information from around the world on new technologies and organizing the efforts of large electronics firms, such as Toshiba and Matsushita, to take advantage of new opportunities. The ability of the Japanese government to do this was exaggerated, but it did play an influential role and researchers began to think about the contributions to innovation made by national institutions and characteristics and the ways they combined into a system. The search was on to try and understand the role of the major players and the most important of their interactions and provide some capacity to encourage innovation at national level.
Early research into national innovation systems took two forms. One, primarily US in focus, took an economic and legal perspective and concentrated on the nation's key institutions, including those of research, education, finance, and law. The characteristics of effective national innovation systems were seen to be high-quality research, providing new options for business; education systems that produced well-qualified graduates and technicians; the availability of capital for investments in risky projects and new and growing ventures; and strong legal protection of intellectual property. The other approach, primarily Scandinavian in focus, was concerned more with the quality of business relationships in a society. The characteristics of effective national innovation systems were seen to be close ties between customers and suppliers of innovation, influenced by the amount of trust between people and organizations in a society and the learning this engenders.
These approaches were initially developed by academics interested in analysing and understanding the reasons why innovation occurs, and why it takes particular forms. The question emerged, for example, of why some nations, such as the USA, are particularly strong in radical innovation – explained by its strength in basic research – and why others, such as Japan, are very strong at incremental innovation – explained by efficient coordination of information exchange between customers and suppliers. The idea of national innovation systems, however, quickly took hold in government and public policy circles, as a way of prescribing and planning how institutions and their relationships could be configured. International organizations, such as the OECD, have produced numerous reports on the institutions of various nations, but these tend to be highly descriptive and static, failing to explain how national systems evolve over time. They do make the valuable observation, however, that what matters is not only the institutions that exist in a nation, but how effectively they work together.
At the same time that research on national innovation systems was blossoming, some began to ask whether the nation was the most useful level of analysis. The question was raised of why nations are often successfully innovative in some industries and regions, but not in others. The USA has Silicon Valley in California, but it also has the Rust Belt of declining heavy engineering and steel industries in the northeast. Researchers have argued the importance of regional, sectoral, and technological innovation systems. They examine the characteristics of successful regions, such as Route 128 around Boston and Cambridge in Massachusetts, Cambridge in the UK, Grenoble in France, and Daejon in Korea. They examine differences in patterns of innovation in the machine tool and textile industries. And they explore why innovation in biotechnology occurs differently from that in nanotechnology. Given the high levels of investment in innovation made by large multinational companies, operating across borders, researchers have also argued the role of global innovation systems.
The notion of systems of innovation is a helpful framework, but social systems are not engineering systems for which components and their interactions are known, planned, and constructed. Unpredictable events occur, and the systems evolve and change in unexpected ways. The early lead in biotechnology research in Harvard University, for example, was lost to Stanford University because of the election of a populist mayor in Boston who built on people's fear of unknown consequences of genetics research. What matters is thinking about the ways in which all the innovation-supporting institutions interrelate and evolve over time along with business practices and relationships. And whatever the level of analysis – global, national, regional, sectoral, technological – what is important is to understand how they relate to one another and co-evolve. The interplay between the many contributors to innovation systems is shown in the following examples of the social, cultural, political, and economic factors affecting the housing industry in Japan and research institutes in China.
### Japanese housing
Japan's industrial development is rooted in a long history of deep craft traditions. These continue to pervade Japanese society, from the traditional Japanese tea ceremony and the way food is prepared, to the design of ceramics. Examining the relationship between craft skills and innovation in Japan shows the influence of social and cultural factors on innovation systems.
For centuries, all Japanese housing was produced by craftsmen using local timber. This continued until the Meiji period (1868-1912), when architectural influences and construction techniques from other countries were introduced into Japan. Japanese housing design, with its simple layout and sliding screens, also influenced Western architects such as Gropius and Corbusier.
Housing production was historically based on tens of thousands of small carpentry and building firms, each producing a few hand-crafted houses a year in conventional post and beam constructions. Tradition in housing design remains very strong and is seen in a continuing Japanese preference for the elegant and intricate timber joints that have long been a hallmark of the craftsman's art. As well as their aesthetic appeal, these joints provide rigidity in earthquakes. Yet the world's most advanced, factory-made housing industry emerged from this very conservative background. Housing innovation resulted from changing demand and new sources of supply and, while the new industry is highly automated, craft skills have been retained.
A combination of factors spurred innovation in Japanese housing following the Second World War. There were severe materials and skills shortages. There was also a large increase in demand following a wave of mass urbanization in the 1950s. Hundreds of thousands of people moved each year from agrarian communities to rapidly growing conurbations in Tokyo, Nagoya, and Osaka, stimulated by employment in new manufacturing firms and the lifestyle attractions of urban living. Mass urbanization continued throughout the 1960s and 1970s. Western styles of living became more popular, and some consumers became prepared to put their faith in large-scale manufactured products from the swiftly growing companies many of them worked for.
The drive to industrialize housing was pushed by manufacturing companies in materials and components industries, in particular steel, chemicals, plastics, and plywood. Firms turned their attention to developing new markets and several began industrial production of housing for their own workforce. Toyota opened a housing division led by the son of the company's founder. Toyota's main purpose was to create high-quality, mass-produced housing for its own workforce, and its first housing production line ran next to its car production line. In 2009, it had 6 dedicated housing factories and recently took a 50% equity stake in the country's second largest factory-made housing company -a business that has won Japan's Good Design Awards for the last 19 years consecutively.
Large industrial companies marketed homes to the new Japanese middle classes, drawing on traditions of craft in design and the benefits of quality control and reliability of industrialized production. They formed R&D centres to study technologies for housing and to assess lifestyle demands and patterns of use. Lack of space in Japanese cities focused attention as much on design and functionality in use as on the development of new materials and production processes. Yet these firms continue to offer a traditional hand-crafted tatami room even in the most contemporary of house designs, reflecting lifestyles and housing preferences that combine the convenience of modernity with the traditions of craft.
The need to rapidly produce accommodation for the 1964 Tokyo Olympic Games triggered innovation in the design and manufacture of modular bathrooms. This created an industry in which several factories each produce more than 10,000 high-quality, fully-fitted bathroom modules to individual customer specifications each month.
The housing industry invests extensively in R&D, ranging from the development of new materials, including nanotechnology coatings for building façades, to designs for multiple generations of occupants. Modular designs allow single properties to be reconfigured for young people, with space to party; young parents with close bedrooms for infants; parents with teenagers with more distant bedrooms; empty nesters, with space for guests; and the elderly, with an emphasis on ease of access.
Investment in R&D has moved from process technologies to product improvements with an emphasis on environmental and energy management. Research focuses on zero-carbon homes, safety and housing performance, and 'smart' homes with electronic sensors and controls. Firms such as Toyota have invested in the development of fuel cells and renewable sources of energy for housing. As well as the home providing electricity for the Toyota car, systems are designed so that, when required, the car can provide energy for the home. All the major producers conduct research on waste reduction and the re-use and recycling of components. Once their foundations have been laid, customized homes can be delivered, installed, and fitted out in a few weeks.
The new industry posed challenges for rural, craft-based construction by exposing its inefficiencies, high costs, and lack of innovation. Demand for craft remained high, despite most people being unable to afford a traditionally built home. Carpenters and small builders did not have the resources to invest in modern production techniques, and large industrial housing manufacturers were not interested in the fragmented rural market. The craft-based housing industry was dying and housing standards started to fall.
The problem was resolved when the forestry industry that supplied the timber for traditional craft-built homes, led by Sumitomo Forestry, took the lead in innovation. It focused effort on automating the time-consuming and expensive process of cutting traditional timber joints. Computer numerically controlled timber-cutting machines were developed and installed in some 600 micro-factories across rural Japan. Local carpenters were able to take their designs to these factories and have their timber frames produced in a fraction of the time it would have taken to cut by hand. This resulted in huge improvements in productivity, and the survival of what remained of the traditional craft industry alongside modern industrialized business.
### China's science and technology institutes
The industrialization of Asia in recent decades has led to the extraordinary social and economic development of the region. Korea, for example, has been transformed from the second poorest country on Earth in the 1950s to a member of the OECD, the group of the world's 30 richest nations. Asian industrialization has required rapid developments in research, education, finance, and law to encourage the dynamic corporate and technological changes required for contemporary competitiveness. Countries such as Korea, Taiwan, and Singapore are developing coherent national innovation systems and becoming important international contributors to innovation. The models of development have varied. Korea, for example, has depended on large conglomerate companies, Taiwan on networks of small firms, Singapore on direct foreign investment by large multinationals, and China has pragmatically used all of these approaches. China is therefore a particularly valuable example of evolving innovation systems and the role of institutions in them. In East Asia, the process of development has been strongly directed by the state, and this is, of course, especially so in China.
China has experienced the most rapid and remarkable industrial development in history. From the devastation of the Second World War, civil war, and cultural revolution, it has emerged as a global manufacturing powerhouse, investing massively in science, technology, and education, and potentially challenges Western hegemony in innovation. The evolution of China's national innovation system, its features, past successes, and future challenges, can be seen in the changes affecting its science and technology research institutes. It shows the influence of political and economic factors on innovation systems and the continual challenges of change.
These institutes – which employ around 1 million people – have experienced over 20 years of major organizational reform, and substantially expanded investments over recent years. National R&D spending has increased by around 20% annually since 1999. Since China's economic reforms in the mid-1980s, these institutes have completely transformed their previous 1950s Soviet-style practice of conducting research disengaged from industry. They were encouraged in the 1980s by some severe government cuts to their budgets to focus their attention on working with business. In addition, in the 1990s a process of reforming government industrial departments led to the corporatization of around 2,000 industrial research institutes.
The reforms have been successful, but challenges still remain. The increasing nexus between research institutes and business has led to the creation of some of China's most successful start-up companies, such as Lenovo. It has redirected the focus of China's research capacity towards the market. But research institutes struggle to retain basic science investments while conducting research attractive to industry. There are complaints that commercialization of their research has distracted them from their main mission. Concern remains about the effectiveness of connections with industry. While there has been a cultural change as researchers recognize the benefits of being market-facing, establishing new forms of engagement that industry finds attractive remains elusive.
This is partly a problem of the lack of receptivity in Chinese firms. There are shortages of innovation skills in areas such as risk assessment, limiting investments in R&D and entrepreneurial start-ups. Venture capital investment tends to be concentrated in established firms and innovation-supporting investments by banks are generally made in large State-Owned Enterprises rather than entrepreneurial start-ups. Much of the focus on innovation has focused on manufacturing rather than services, and on high-technology sectors.
There is growing recognition on the part of government that innovation policy involves more than directing the research sector, and attention is being paid to improving the innovation performance of business. State direction of the former is proving much easier than the latter. Innovation systems researchers, such as Shulin Gu and Bengt-Åke Lundvall, have also questioned whether there is the extent of the social capital and trust required for the deep engagements required by researchers and businesspeople to work together and be successfully innovative.
The transformation of innovation in China experienced over the past decade has resulted from strong political leadership. It was recognized at the top levels of government that the export-led, manufacturing-based pattern of economic development behind its stunning economic growth since the 1980s would not maintain the level of growth necessary to fund China's social expectations. President Hu Jintao has called for an innovation oriented country, pursuing a path of innovation with Chinese characteristics. The political discourse in China refers to 'harmonious growth', and the imperative for inclusive development is the most important challenge confronting innovation in China. This encompasses the need to use innovation as a means of reducing income disparities between the poor and wealthy, and the economic disparities between coastal regions and inner China. The evolution of China's national innovation system to one that allows it to compete with the West in innovation is incomplete and continuing.
## [**Chapter 5
Thomas Edison's organizational genius**](contents.html#ch05)
Organizations have choices on how they organize themselves for the continually evolving challenges in innovation; the structures and procedures they adopt, the staffing and incentives they use. These reflect the strategy of the organization and its innovation objectives.
### Edison
Thomas Edison (1847–1931) is remembered for his inventiveness and the wide range of innovations he introduced. He held over 1,000 patents, and, amongst other remarkable achievements, he developed the phonograph, electric light bulb, and electrical power distribution, and improved the telephone, telegraph, and motion picture technology. He founded numerous companies, including General Electric. He was also responsible for pioneering a highly structured way of organizing innovation, and it is this that is our concern here.
Like Josiah Wedgwood, Edison was the youngest of a large family of modest circumstances, received little formal education, started work early, aged 12, and was afflicted by a disability, deafness, that influenced his life and work. He was similarly driven and hard-working, and shared with Wedgwood an appreciation of Thomas Paine, which also influenced his democratic worldview. Edison could be blunt, irascible, and impatient, but he could also be personable, kind, and generous.
Edison began his working life as a telegraph operator, and started to experiment during the night shift, when he could not be observed. His first patent, an electric vote recorder, was awarded when he was 22. The notoriety of his inventions moved him from his humble beginnings into high circles. He demonstrated the phonograph to President Hayes in the White House in 1878, and was close friends with Henry Ford. He is reputed to have influenced Ford over the potential of petrol engines. His business partners included the leading capitalists of the day, such as JP Morgan and the Vanderbilts.
Edison's approach to business was relentless and ruthless. He demanded continual improvement in innovations from his employees and energetically denigrated opposition. His campaign against Alternating Current (AC) and promotion of Direct Current (DC), his preferred option for electricity transmission, sank to the unsavoury level of a publicity war over their relative merits for the electric chair. Edison did not shrink from demonstrations of electrocuting animals with AC to reveal its dangers. These included the unfortunate, if ill-tempered, Topsy the elephant, whose demise at Lunar Park was filmed by Edison for further publicity value. AC, the superior system, eventually became dominant, and the cut-throat nature of the battle between these competing technical standards clearly shows the value of owning the dominant version.
While Edison enjoyed great commercial successes, he had his fair share of failures. There were comparatively expensive and unproductive diversions into mining and the manufacture of concrete. He failed to recognize public interest in the celebrity of musicians, when for years he refused to name them on recordings. With characteristic aplomb he claimed never to have failed, but to have discovered 10,000 ways that didn't work.
Possession of intellectual property was crucial for Edison. Patents emerging from research in his laboratories were attributed to Edison irrespective of his contribution. One of his long-term assistants said: 'Edison is in reality a collective noun and refers to the work of many men.' Fiercely protective of his own patents, he occasionally disregarded the intellectual property of others. He and his business partners regularly used patents to block the development of competitors.
Fêted during his lifetime, and called a 'wizard' by the press, he faced hostile criticism from rivals. Critics included Nikola Tesla, who had every reason to be bitter. Tesla was working for Edison when he developed AC, prior to commercializing it with the Westinghouse Corporation. Tesla claimed he was not paid what he was promised. In later life, Edison regretted the way he treated him. It is speculated that the reason why Edison did not proceed with AC himself, despite having many opportunities to do so, was that he did not develop it himself; a case of the 'not-invented-here' syndrome. After Edison's death, Tesla reported for posterity his ex-boss's utter disregard of the most elemental rules of hygiene.
The way Edison organized his inventive efforts derived from his overall approach to innovation. He always pursued several lines of research, wishing to keep options open until the strongest contender emerged, when resources and effort were concentrated. By working on numerous projects simultaneously, Edison hedged his bets so future income streams did not depend upon one development. He was well aware of how when pursuing one problem it would lead to others, often completely unexpected, and understood the value of chance, serendipity and 'accident'.
He explored how ideas from different areas of research could potentially be combined and had a strategy of re-using proven components of other machines and applying them as building blocks in new designs. Edison said he readily absorbed ideas from every source, frequently starting where others left off. The development and commercialization of the light bulb, for example, combined ideas by drawing on a network of researchers, financiers, suppliers, and distributors. Although the idea of the light bulb had existed for decades, Edison, by using low-current electricity, a carbonized filament, and a high-quality vacuum, developed a product that was relatively long-lasting. His principles were to experiment and prototype as much as possible on a small scale, and to make designs as simple as possible. Once a breakthrough had occurred, he appreciated it would take a great amount of continuing research and experimentation to turn it into a successful product. He said it usually took him 5 to 7 years to perfect a thing, and some things remained unresolved after 25 years. As he said: 'genius is one percent inspiration, ninety-nine percent perspiration'.
Edison understood that most value returned to the controller of the technical system, not the producer of its individual components, who was dependent upon the system configuration. His systems thinking was most apparent in the development of the electricity distribution industry that began operating in New York in 1882. Recognizing people's apprehension of the unfamiliar, Edison cleverly mixed the new and the existing in his electricity system. He used recognizable infrastructure to deliver electricity, including putting wires underground like gas pipes and utilizing existing gas fittings in homes.
Like many of his innovations, Edison's approach to organizing his research laboratories built on the experience of others. The telegraph industry in which Edison began his career had a number of small research shops with ranges of experimental equipment. Edison had conducted experiments in one such shop in Boston, and upon his arrival in New York in 1869, he used another before setting his own laboratory in Newark to make his design of stock ticker machines.
Edison's organizational innovation lay in the range and scale of the research activities undertaken. He invested more financial and technological resources in the search for innovation than any other organization had done previously.
Edison established the Menlo Park laboratory in 1876 so he could devote himself entirely to the 'invention business'. He brought with him key workers, including a draftsman, machinist, accountant, mathematician, assayist, chemist, glass blower, and bookkeeper. Located 25 miles from Manhatten, in what was then a small village, by 1880, 75 of the 200 Menlo Park residents worked for Edison. Menlo Park began with an office, laboratory, and machine shop. Over the years, Edison added a glass house, photographer's studio, carpenter's shop, carbon production shed, blacksmiths', and additional machine shop. He also added a library.
At this time, only a few of the best universities in the USA had laboratories, and these were ill-equipped and focused mainly on teaching. Yet Edison had fine scientific equipment, including an expensive reflecting galvanometer, electrometer, and photometrical devices. Within a couple of years, his stock of tools was worth $40,000 ($890,000 at 2008 prices).
Edison's objective was to have all tools, machines, materials, and skills needed for invention and innovation in one place. The combination of the diverse skills in Menlo Park was assisted by their close social integration in the local community.
At its peak, Edison had more than 200 machinists, scientists, craftsmen, and labourers assisting with inventions. Work was organized into teams of 10 to 20, each working simultaneously in seeing ideas transformed into working prototypes. As everyone in the team had the same objective, communications and mutual understanding were at a premium. In 6 years at Menlo Park, Edison registered 400 patents. He aimed for a minor invention every 10 days and a big thing every 6 months or so.
In 1886, Edison moved his main laboratory to West Orange, New Jersey, to increase the scale of his research and manufacturing capacity. West Orange was 10 times bigger than Menlo Park. Edison's biographer, Josephson, describes the reason behind the move:
I will have the best equipment & largest Laboratory extant, and the facilities superior to any other for rapid & cheap development of an invention & working it up into Commercial shape with models patterns & special machines... Inventions that formally took months & cost large sums can now be done two or three days with very small expense, as I shall carry a stack of almost every conceivable material.
Edison's factory made the parts necessary for research, and research developed and made the machines for large-scale production in the factory. During the development of the phonograph over 40 years, the cylinders developed by research were first made of tinfoil, followed by a wax compound, followed by plastic. The eventual primary use of the phonograph was not the one originally imagined. As this technological and market learning occurred, the capacity to quickly scale up production of new configurations helped Edison gain substantial market share. Edison's New York factories at one stage employed over 2,000 people, one of the largest industrial concerns of the time. In contrast to the high-performance workplace laboratories, these mass-production factories operated with extensive division of labour, and the repetitive, unskilled work led to many industrial disputes.
The scale of activities in West Orange inevitably led to greater departmentalization and administration, which took up more of Edison's time. Although it was highly productive, it never matched the extraordinary output of the Menlo Park period.
Edison is quoted as saying: 'From his neck down a man is worth a couple of dollars a day, from his neck up he is worth anything that his brain can produce.' He excoriated 'pinheads' and 'lunkheads', and said: 'A man who doesn't make up his mind to cultivate the habit of thinking misses the greatest pleasure in life.' He hired graduates, but generally preferred generalists over specialists, and this is argued by some to have limited the future development of his research organization. His recruitment methods were idiosyncratic. In the early years, he would point applicants at a pile of junk and tell them to put it together and tell him when they had done so. The junk was a dynamo and those who succeeded in assembling it passed the test for employment. In later years, he compiled lengthy general knowledge questionnaires that prospective inspectors needed to pass before being promoted.
**8. Edison encouraged play as well as hard work. Here workers attend a 'sing' session**
Edison's style was to provide staff with a general outline of what he wanted, and then leave them to decide the best ways of achieving the objectives. He is reputed to have said: 'Hell, there are no rules here – we're trying to accomplish something.' One of Edison's staff said: 'Nothing here is private. Everyone is at liberty to see all he can, and the boss will tell him all the rest.'
He 'managed by walking about', advising and encouraging the teams. Edison worked about 18 hours a day, and the exercise he received walking from one laboratory table to another gave him 'more benefit and entertainment... than some of my friends and competitors get from playing games like golf'. Edison's biographer, Baldwin, has him 'compelled' to 'wander democratically and visibly up and down the aisles, ubiquitous, endlessly snooping, his sleeves rolled up and his unattended cigar ash dropping onto the shoulders of welders and diecutters'.
Staff worked exceptionally long hours. Tesla complained that in his first two weeks he only managed 48 hours' sleep. Legend has Edison working for five straight days and nights, but it was probably three, and it was known that the best time to contact him in the factory was after midnight. Another of his biographers, Miller, suggests: 'The capital crime of the Edison laboratory was to go to sleep. This was a source of disgrace, unless the boss could be caught napping, and then they all followed in line.' Various methods were used to dissuade the somnambulant, including the 'corpse reviver', a terrifying noise released beside the ear and the 'resurrector of the dead', which apparently involved setting sleepers alight with a small exploding substance.
It could be dangerous to work for Edison. His chief assistant, Clarence Dally, lost an arm and most of a hand during experiments with fluoroscopy during which Edison nearly lost his own eyesight. The local press reported Edison generously saying that, although Dally was unable to do any work, he would keep him on the payroll.
Josephson revealingly records the reflections of two of Edison's staff. The first, a young job applicant, was told 'Everyone applying for a job wants to know two things: how much we pay and how long we work. Well, we don't pay anything and we work all the time.' The applicant took the job. The second, a man reflecting on working for Edison for 50 years, told of his sacrifices resulting from long hours at work, including not seeing his children grow up. When asked why he did it, he responded: 'Because Edison made your work interesting. He made me feel I was making something for him. I wasn't just a workman.'
Despite these practices, which seem draconian today, Edison encouraged a creative and productive workforce. Key employees were paid bonuses from profits on inventions, although this incentive did not extend to Nikola Tesla. He socialized with staff with snacks, cigars, jokes, tales, dancing, and singing. He organized a popular midnight lunch. There was an electric toy railroad to play with and a pet bear. According to management academic Andrew Hargadon:
The muckers [engineers] would work for days straight in pursuit of a solution, then punctuate their work with late-night breaks of pie, tobacco and bawdy songs around their giant organ that dominated one end of the laboratory.
One of Edison's assistants, quoted in Millard, said that there was 'a little community of kindred spirits, all in young manhood, enthusiastic about their work, expectant of great results', for whom work and play were indistinguishable.
Tesla complained of Edison's concentration on instinct and intuition over theory and calculation, and practices at the laboratory did occasionally seem haphazard. When searching for the best material for the light bulb filament, he experimented with unlikely materials ranging from horsehair to cork to the beards of his workers. When the breakthrough came in the carbon filament incandescent lamp, Edison's staff did not realize the extent of their discovery for several months after the event.
Nonetheless, there was focus and discipline. Edison claimed never to have perfected an invention he did not think about in terms of the service it provided, saying he found out what the world needed, then proceeded to invent. Projects had to have a practical commercial application. Famed for his 'guesswork', Edison insisted laboratory assistants kept detailed records of their experiments in over 1,000 notebooks, although this also helped with patent registration and dispute. Experimentation was extensive. 6,000 distinct species of plants, mainly bamboos, were used for carbonized filaments. 50,000 separate experiments were undertaken in developing Edison's nickel-iron battery. One of Edison's assistants, working closely with his employer, recorded 15,000 experiments into a particular problem.
West Orange possessed an extensive library of some 10,000 volumes, and Edison constantly read about biology, astronomy, mechanics, metaphysics, music, physics, and political economy. Although criticized for his disparagement of formal education, he employed two eminent mathematicians, one of whom went on to be a professor at Harvard and MIT. One of his key chemists was known as 'Basic Lawson' because of his adherence to basic scientific principles. Edison met and admired Pasteur and the German physicist and physician Helmholtz. Somewhat incongruously, George Bernard Shaw worked for Edison for a time in London.
Artefacts and drawings were important sources of creativity and communications. Edison is quoted as saying: 'Inspiration can be found in a pile of junk. Sometimes, you can put it together with a good imagination and invent something.' In 1887, his laboratory was reputed to have contained 8,000 kinds of chemical, every kind of screw, cord, wire, and needle, animals from camels to minx, feathers from peacocks and ostriches, hooves, horns, shells, and shark's teeth. Edison found it easier to think in pictures rather than words. When he was contracted by the Western Union Telegraph Company in 1877 to improve upon the telephone invented by Alexander Graham Bell, he produced over 500 sketches leading to his improved design.
As well as his internal efforts, Edison assiduously cultivated his business and research networks. He was a broker of technology, transferring research between industries. As well as his own experiments, he undertook contract research for the telegraph, electric light, railroad, and mining industries. As Hargadon puts it:
Edison quietly blurred the line between the experiments he did for others and those he did for himself. Who was to know if a result from contract research was applied to another project or if experimental equipment built for one customer was used in work for another.
His ability to continually innovate, according to Hargadon, lay in the way he knew how to exploit the networked landscape of the time.
Edison's approach was one of trial and error, hard work, and persistence, being methodical, rigorous, and purposeful, and using prepared minds and careful monitoring. He believed innovation arose not from individual genius but collaboration, and this capacity to work together and across boundaries resulted from a supportive culture, environment, and social and industrial relations.
**9. President Hoover called for electric lights to be switched off for a 'minute of darkness' in memory of Edison's achievements**
Edison worked on the cusp of the transition between the era of the great individual inventor and systematic, corporate organization of innovation. He created a form of organization for the emerging, modern technological society that was rapidly emulated by large corporations such as Bell and General Electric. In an article in the _New York Times_ on 24 June 1928, it was estimated that Edison's inventions had built industries valued at $15 billion ($188 billion at 2008 prices). His fame was universal. President Hoover called Edison a 'benefactor of all mankind', and when he died asked people to switch off their lights for a 'minute of darkness' in his memory. His obituary in the _New York Times_ on 18 October 1931 began: 'Thomas Alva Edison made the world a better place in which to live and brought comparative luxury into the life of the workingman.' An innovator can make no greater contribution.
### Workplaces
As Edison so clearly showed, innovation is more likely to occur in organizations that are forward-looking, accepting of risk, and tolerant of diversity and failure. A workplace that is playful and fun and where conversation and laughter is common is more likely to be innovative than one that is highly formal, bureaucratized, and impersonal. Where expression of opinion is welcomed, ideas are not only generated more regularly, but are also implemented faster. Opposition is voiced when it has a chance to be productive rather than in the subsequent subversion of decisions.
IDEO is a company with a highly innovative workplace that emulates some of the lessons from Edison. It is a successful provider of design and innovation services, employing over 550 people in offices around the world. It has built a reputation for helping other firms to innovate in their products and services by applying creative techniques learnt in the design studio and design school environments. The company combines 'human factors' and aesthetic design with product engineering knowledge to produce products for firms from Apple to Nike to Prada. Its designs include the computer mouse, Palm Pilot, and a range of cameras and toothbrushes. It designed the whale that starred in the film _Free Willy._ IDEO has contributed to the design of over 3,000 products, and works on 60 to 80 products at any one time. IDEO has been described by _Fast Company_ magazine as the 'world's most celebrated design firm'; by the _Wall Street Journal_ as 'imagination's playground'; and _Fortune_ described its visit to IDEO as 'a Day at Innovation U'.
For the company to deal with many diverse projects, it recruits a wide range of talent, and also enjoys special links with the Stanford University Institute of Design. It employs graduates from psychology, anthropology, and biomechanics as well as design engineering.
The leaders of IDEO have a very high profile in the international design community. They claim to have an innovative culture -low on hierarchy, big on communications and requiring a minimum of ego' – that uses:
a collaborative methodology that simultaneously examines user desirability, technical feasibility, and business viability, and employs a range of techniques to visualize, evaluate, and refine opportunities for design and development, such as observation, brainstorming, rapid prototyping, and implementation.
IDEO sells its design methodologies to other companies in the form of courses and training materials. It possesses a large repository – a 'toy box' – of devices and designs from a wide range of products which staff play with when seeking solutions to new problems. It is particularly skilled at playing with creative ideas developed for one industry or project to explore their innovative application in others. Playfulness in this environment enables the cross-fertilization and the serendipitous connection and combination of unrelated ideas.
### Structures
Edison pioneered a way of organizing, but organizations have wide choices in how they structure opportunities for innovation. Some choose to be highly formal and bureaucratized, others to be informal and unrestricted. Some try to do both, with parts of the organization encouraged to behave very differently from others.
One of the earliest studies of innovation in organizations, Burns and Stalker in 1961 distinguished between mechanistic and organic forms of organizing. They argued the former is appropriate to stable, predictable conditions and the latter to changing conditions and unpredictable situations. The basic general principle still applies; the way things are organized should be appropriate to the particular circumstances and aims of innovation. When technologies and markets are rapidly evolving and their future uncertain, the need – as in the case of Menlo Park – is to encourage experiment and creativity without constraining it with bureaucracy. When some of these uncertainties diminish, a more planned approach is needed to the development of projects, with highly prescribed budgets and operations geared up to deliver the innovation. Furthermore, the form of organization used often changes over time as different issues of innovation emerge. As the process of developing the innovation progresses, the supportive organizational structures move from being 'loose' to 'tight'.
### R&D
R&D can be structured in very diverse ways. Many leading firms in the past relied exclusively on large corporate laboratories to undertake their research: their own large-scale Menlo Park. The archetype of this form of 'centralized' R&D was Bell Labs, which employed 25,000 at its peak and has been awarded 30,000 patents. It has received six Nobel Prizes for Physics and, amongst other things, discovered the transistor, digital switching, communication satellites, cellular mobile radio, sound motion pictures, and stereo recording. One of its basic science discoveries led to the development of radio astronomy. Founded in 1925, and based in New Jersey, it was the research group for AT&T before that company was acquired by Alcatel-Lucent. Renowned for its past strength in basic research, like many corporate laboratories, it has progressively moved towards more applied research.
The criticism of this way of organizing from a business perspective is the way research tends to be too divorced from the needs of customers and is generally too long term in orientation. In contrast, rather than having a central laboratory, other firms 'decentralize' their R&D organization structures, with laboratories located close to particular businesses or customers.
The problem with this form of structure is research tends to focus on short-term issues and miss opportunities for more radical or disruptive innovations. To try and gain the benefits of both forms, some firms combine a central laboratory with numbers of decentralized R&D laboratories, but this is a choice open only to a handful of the wealthiest.
Other organizations dispense with formal R&D organizational structures altogether. Intel, the semiconductor company, despite having a billion dollar R&D budget, has never had an internal R&D structure. It relies on networks in universities and the technology community in Silicon Valley to provide research inputs. The challenge for this 'networked' form of R&D organization is that to be receptive to knowledge from external research, organizations need to have the internal capacity to absorb it. They need skills to understand, interpret and utilize externally sourced knowledge, and often require their own in-depth expertise to attract high-quality research partners.
The organizational challenge in R&D is finding the balance between longer-term research that provides new options and insights into potentially disruptive technologies and research that deals with short-term or immediate well-defined problems. Firms often appear dissatisfied with whatever R&D structures they have. With centralized structures they feel customer needs are relegated in importance, and decentralized structures might miss potentially valuable innovations. When both forms are used, continual tensions exist over relative funding levels and ownership of projects. The problems of networked R&D lie in managing and combining inputs from multiple parties and disputes over ownership of intellectual property rights.
One strategy firms pursue to improve returns to internal R&D and access external collaborators for innovation has in recent years been described by Henry Chesbrough as 'open innovation'. The household products company, Procter and Gamble, is an example of an open innovator. It is a science-based company with a strong internal commitment to research. Its strategy is described as 'Connect and Develop' and rather than being 90% reliant on its own research investments as in the past it aims to source half its innovations from outside the company. The way it combines its own internal research with external connections is indicative of a strategy that attempts to benefit from complementary ways of organizing innovation in the same company.
The rapid growth of research capacity in China and India in recent years has the potential to change the ways many multinational companies organize their R&D. Firms create laboratories overseas to adopt their products and services to local markets, take advantage of particular local research expertise, and to create international networks of research collaboration. Many US and European firms have established substantial R&D organizations in India and China, especially in information and communications technology. The strategy these firms use can change over time. Ericsson, the Swedish telecommunications company, for example, began to invest in R&D in China in the 1980s because it helped win government contracts and was evidence of goodwill and commitment. R&D expenditure was ramped up in the early 1990s to take advantage of cheap research staff and to help adopt Ericsson products to the rapidly growing local market. Recognizing the quality and potential of Chinese researchers, both in the company and in local universities, in the late 1990s Ericsson began to locate R&D for its world markets in China. By the early 2000s, some of its R&D groups around the world were closed down and moved to China, and Ericsson's Chinese research groups became core components of the company's global R&D efforts.
### New developments
R&D is one of the ways organizations create options for the future. The ways they organize their new product and service development is crucial for how successful they will be at realizing the future options they have. While R&D is generally the organizational space for scientists and technical specialists, new product and service development usually encompasses a broader range of people, including from design, marketing, and operations. These specialists help address the questions of why and how things are bought, and whether and at what cost can they be made and delivered.
There are many tools and techniques – such as 'stage-gate' systems, which operate a number of stop/go decision points in the development process – available to help plan new products and services. These are designed to help decide between competing projects and to ensure those that progress are appropriately resourced. These tools have limitations: they might be very helpful in managing the process of developing products, but they do not tell you whether they are the right products in the first place. They can also become very procedural and kill initiative.
To overcome the rigidities of bureaucracy, some organizations sanction 'bootlegging', or allowing staff to spend time working on their own projects. By giving people time – which can extend to one or two days a week – outside their formal job commitments, highly innovative companies such as Google and 3M encourage personal motivation to innovate and new ideas to emerge and blossom.
Another method used to circumvent organizational constraints to innovation is the so-called 'skunkworks'. Used initially by the Lockheed Corporation to quickly and secretly develop aircraft during the Cold War, the term is used to describe a small, tightly bound group working on a special project with considerable operational discretion within a larger organization.
### Operations and production
The ways new products and services are made and delivered have been the focus of considerable innovation themselves. Production, for example, is automated, and operations – the processes for turning inputs into outputs – have seen major innovations in the ways work is organized. Innovation in production and operations has helped create mass markets for affordable, high-quality products and services, such as automobiles, consumer goods and electronics, supermarkets, and hotel chains.
One of the key principles in organizing operations and production is Adam Smith's analysis of the division of labour. It was after reading Smith that Josiah Wedgwood saw how combining specialists in particular tasks with the new technology of steam power would improve productivity in his factory. Smith argued the division of labour is limited by the extent of the market. When markets grow large enough, benefits are gained by subdividing work and employing specialists dedicated to specific tasks rather than more expensive, broadly skilled craftsmen. He also noted specialization is a function of the division of labour, so the more work can be subdivided into discrete elements, the greater the potential to employ specialists.
Smith explained the economic benefits derived from the efficiency of the division of labour. By concentrating on a smaller range of tasks, individuals can improve their dexterity and undertake tasks more accurately and swiftly. Time is saved as there is no need to move from one task to another. When tasks are visible and discrete, machinery can more easily be devised to automate or enhance them to increase productivity.
Henry Ford used the principles of specialization and automation in developing his assembly line producing automobiles for the emerging mass market in the early 20th century. Ford's objective was tighter managerial control over production processes than previous craft forms of production allowed. His solution was the development of the mass-production line with high volumes of standardized products made from interchangeable parts. Ford learned the value of interchangeable parts from the manufacture of guns by the Colt Armory and mass production from breweries, canneries and meat packers. He combined, refined and simplified these approaches to speed up production and standardize quality on an assembly line.
His system allowed the subdivision and specialization of labour, employing unskilled or semiskilled workers on high-cost machinery dedicated to the manufacture of specific parts. Management and design were the responsibility of narrowly skilled professionals. The control of work by craftsmen was replaced by management, and the pace of work was dictated by the need to maximize use of equipment. Because machinery was so expensive, firms could not afford to allow the assembly line to grind to a halt. Buffers of extra supplies of materials and labour were added to the system to assure smooth production. Standard designs were kept in production for as long as possible because changing machinery was expensive, resulting in consumers benefiting from lower costs but at the expense of variety and choice.
Ford's friend Edison had already experienced the problem that unskilled, repetitive work caused with the industrial disputes he faced. General Motors showed Ford the limitations of his marketing approach and the benefits of producing varieties of vehicles. Alfred Sloan's approach at General Motors aimed to produce 'a car for every purse and purpose'. But the real innovation allowing both efficiency in production, wide customer choice, and better use of skills came from Japan.
After the Second World War, Toyota recognized that to realize its ambition of becoming an international carmaker it needed to harness the efficiency of American mass-production techniques and craft quality of Japanese work practices. At that time local Japanese automobile markets were small and demanded a wide variety of vehicles, production techniques were primitive in comparison to those in the USA, and investment capital was scarce. Unionized Japanese factory workers insisted they retain their skills and were unwilling to be treated as variable costs, like the interchangeable parts in Ford's and Edison's factories. Toyota understood the dangers of repetitive and boring tasks resulting in fatigue or injury to workers, with diminishing returns to efficiency.
In 1950, Toyota's President, Eiji Toyoda, spent three months at Ford's Rouge factory in the USA. He was amazed at the total output of the plant, which in one year produced over 2.5 times the number of cars made by Toyota in the previous 13 years. But while total output was impressive, Toyoda thought the system was wasteful in effort, materials and time. Toyota could not afford to produce cars with such narrowly skilled professionals and unskilled workers tending expensive, single purpose machines with their buffers of extra stocks and re-work areas. Toyoda's objectives were to simplify Toyota's production system combining some advantages of skilled craft working with those of mass production, but avoiding the high costs of craft and rigidities of factory systems. The result was the evolution of Toyota's lean production system employing teams of multi-skilled workers at all levels of the organization and highly flexible, automated machines producing large volumes of highly varied products. Rather than having buffer stocks of inventory, wasting resources, Toyota's system delivers components just-in-time to be used.
Teams of Toyota workers are given time to suggest improvements to production processes in 'quality circles': Toyota has several thousand quality circles that complete tens of thousands of small improvement projects every year. Quality circles are linked to efforts for continuous improvement (kaizen) in collaboration with industrial engineers. Emphasis on problem solving is an important part of everyone's job and on-the-job training, collective education, and self-development are all encouraged.
The success of lean production improved the whole system of designing and making cars and it made Toyota the car producer against which other manufacturing firms compared themselves. The combination of technical and organizational innovation in Toyota's production system has produced both economies of scale and scope: volume and variety.
The search for innovations that help combine economies of scale through standardization with economies of scope to satisfy diverse consumer choices is a continuing challenge. The ultimate objective in many cases is producing economically for markets of one. Toyota continues to invest in automation and new technologies such as advanced materials and techniques integrating computer-aided design with flexible computerized manufacturing systems. Unmanned supply vehicles are used to transport components and parts, and vertical computer controlled warehousing is used for storage. Despite the company's concern with skills and encouragement of quality circles, critics of the Toyota system point to the very demanding work pace required which as well as adversely affecting the health of the workforce, may inhibit innovation. The further development of its production system will depend on its accordance with what is acceptable to its employees.
Services organizations similarly look for innovation in their operations. The discount airline, easyJet, provides an example of innovative 'mass customization', or providing large-scale personalized service. The company began in 1995 with two leased aircraft and a telephone booking system. It launched a website in 1997, and by 1999, it sold its millionth ticket online. By 2005, it sold its 100 millionth. The use of the internet has been crucial to this growth and has underpinned the business model of time-dependent pricing – where prices vary with the length of time prior to booking and with demand – and highly customized customer requirements such as boarding priority, and baggage handling. It also allows the optimization of aeroplane utilization and cost reduction by not issuing tickets. It is one of Europe's largest internet retailers with 95% of its flights sold online, and also offers hotels and has a car hire partnership. All the company's documentation is stored on globally accessible servers. It has launched a desktop gadget to personalize flight information and bookings.
Another example of innovative use of operations is provided by the supermarket, Tesco, and the use of data on its 13 million regular customers. By individually classifying 25,000 products, data mining of purchasing behaviour, and the use of loyalty cards, the company creates a 'lifestyle DNA profile' of each customer. These are grouped together for particular targeted promotions. The 13 million Tesco Club Card holders are mailed four times annually with details of their rewards and vouchers on offers tailored to their profiles. Seven million variations of offers are made, and customer take up is between 10 and 25 times higher than the average 2% for direct marketing. The data is used to make sure goods available in existing and future stores are tailored to fit the profiles of local customers.
### Networks and communities
Edison's development of the electrical lighting industry was an example of innovation in a technical system brought about within a network of innovators. Most innovation involves the participation of numbers of collaborating organizations and from the perspective of the individual organization this brings benefits and difficulties. The benefits lie in being able to access knowledge, skills, and other resources it does not itself possess. The difficulties lie in the absence of organizational sanction in getting others to do as you wish.
The key to effective networking is building partnerships with high degrees of trust. Trust is needed in the technical competence of collaborators, their ability to deliver what is expected of them, and their overall integrity in protecting proprietary knowledge and being prepared to admit when things go wrong. Collaborations usually start as a result of personal connections. These can break down as people move to other jobs or organizations. Effective trust between partners therefore involves the extension of inter-personal trust to inter-organization trust, with the value of the collaboration becoming institutionally engrained: legally, administratively, and culturally.
In some fields, such as open-source software, the community of users is the innovator. Here it is users of the product or service that provide new content and improvements. Despite the rhetoric of unconstrained engagement in many of these communities, a degree of organization is required. Wikipedia, for example, recognizes the efforts of its contributors to its online encyclopedia by creating a hierarchy, with significant community status accorded to wikipedians whose contributions have reached high levels of quality and quantity.
Organizations are becoming more adept at using Web 2.0 social networking sites, wikis, and blogs in their innovation activities. They are using social networking analyses by, for example, surveys or tracking email correspondence, to understand the key personal and organizational nodes in the organization and help improve decision-making. To assist communications in what are called 'massively multiparty activities' they are using virtual worlds such as Second Life, where people assume a representation of themselves as avatars. These new forms of organization pose questions about their legitimization at work – given their frequent association with 'games' – and the appropriate incentive and reward systems and skills profiles of their users.
### Projects
A large part of modern economies is comprised of large, complex infrastructural projects, such as telecommunications networks, energy production and distribution, and transportation systems of airports, railways, and motorways. These projects, which commonly cost billions of dollars, entail the coordination of large numbers of firms that assemble to contribute their various skills and resources during different stages of the project's progression. They are notorious for cost over-runs and delays. The Channel Tunnel, between England and France, for example, was 80% over budget.
London Heathrow Airport's Terminal 5 (T5) was a large and highly complex project, with a budget of £4.3 billion and involving over 20,000 contracting organizations. Overseen by the British Airport Authority (BAA), the project client, airport owner and operator, it entailed the construction of major buildings, a transit system, and road, rail, and subway links, alongside the world's busiest airport working at overcapacity. T5 is the size of London's Hyde Park, and has an annual capacity of 30 million passengers. Although often remembered for the disastrous first few days of operation where British Airways misplaced 20,000 bags and cancelled 500 flights, the design and construction of the project itself was a success and was delivered to budget and on time. This success resulted from an innovative approach to managing large, complex projects.
BAA took care to learn lessons from previous projects, made sure any technologies used were already proven elsewhere, and trialled new approaches on smaller projects before applying them to T5. Use was made of digital simulation, modelling and visualization technologies to help integrate designs and construction. Underpinning the success of the T5 project was a contract between the client, BAA, and its major suppliers that differed considerably from industry norms – which were commonly adversarial – and encouraged collaboration, trust, and supplier responsibility. The risk in the project was assumed by BAA, work was conducted in integrated project teams with first-tier suppliers, and incentives were designed to reward high performing teams. Although the processes and procedures to be followed were highly specified, the project was formulated in a way that allowed managers to confront the unforeseen problems that inevitably arise in complex projects flexibly and on the basis of their previous experience.
The lessons from T5 are that success in large, complex projects involves standardized, repetitive and carefully prepared routines, processes, and technologies and the capacity to be innovative to be able to deal with unexpected events and problems. Organizing projects involves a judicious balance between performing routines and promoting innovation.
### Creative people and teams
As Edison showed in Menlo Park, innovation involves a team effort, bringing together different ideas and expertise. The construction of teams involves decisions about the most appropriate balance of skills given the problems faced. It also entails deciding on the comparative value of organizational memory – keeping people in teams together – and refreshment -bringing in new skills. Teams that work together for long periods tend to become introspective and become immune to innovative ideas from outside. Teams that are newly constructed or contain many new members have to learn to work together effectively and develop a modus operandi. Harmony in teams holds many virtues, but sometimes it is important for innovation to have disruptive elements – the grit in the oyster – asking difficult questions and shaking things up.
The structure of teams has to reflect their objectives. Those devoted to more radical innovation need more creativity and flexibility in objectives, with the freedom to respond to emerging and potentially unforeseen opportunities. They often need heavyweight support from higher levels in the organization as their objectives do not quickly add to the bottom line and as a result they are vulnerable to criticism and cost-saving exercises. A balance has to be found between incentives for individuals and teams. The factors that encourage innovation team effectiveness are often subjective – to do with professional satisfaction and recognition. Those that inhibit performance are more instrumental – to do with project objectives and resource limitations. As Edison found out, employees will work extraordinarily hard when given the incentive of interesting, rewarding, and appreciated jobs.
Creativity is not only important to design companies such as IDEO. Innovation in all organizations relies on creative people and teams to produce new ideas, and creativity is an issue that infuses the whole of the world of work. By its stimulation of innovation, many contemporary organizations see the encouragement of creativity as core to their development and competitiveness. Creativity provides a means of making work more attractive, improving the engagement and commitment of existing staff, and a winning strategy in the 'war for talent' amongst highly skilled and mobile employees.
Creativity has an individual and a group component. Psychologists tell us about the characteristics of creative people, and how imaginative ideas emerge from individuals with the capacity to think differently and see connections and possibilities. Creative individuals are said to have a tolerance for ambiguity, contradiction, and complexity. Cognitive scientists, such as Margaret Boden, argue that creativity is something that can be learnt by everyone and is based in ordinary abilities that we all share, and in practiced expertise to which we can all aspire.
Organizations expend large amounts of time and resources on creativity training, and constructing incentives and rewards for individual creativity. They are also concerned with the promotion of creativity in groups and with formulating the most conducive team structures and organizational processes and practices. Groups bring together the disparate perspectives and knowledge that are valuable for creativity and are essential for the new combinations in innovation. Recent research into creativity has focused more on the organizational and business circumstances that encourage creativity and the systems and strategies that shape its manifestation.
Creative ideas become useful innovations when they are successfully applied. Creativity in itself may be inspiring, stimulating, and beautiful, but it has no value economically until it is manifested as an innovation. It takes different forms in incremental and radical innovations. Incremental innovations usually involve a form of creativity that is more structured, managed, and deliberate. Radical innovation requires creativity that may be unbounded by existing practices and ways of doing things.
### People
#### Leaders
Innovation rarely occurs in organizations without the commitment and visible support of their leaders, although those leaders may have little idea of the specific nature of new developments. One of the key aspects of leadership is encouragement for the creation of new ideas and their implementation. Leaders find resources for support and offer protection from the opponents of innovation. When new ideas threaten the status quo, established interests will inevitably oppose them. As Machiavelli says in _The Prince:_
There is nothing more difficult to plan, nor more dangerous to manage than the creation of a new order of things... Whenever his enemies have the ability to attack the innovator they do so with the passion of partisans, while the others defend him sluggishly, so that the innovator and his party alike are vulnerable.
One of the lessons of renowned leaders of innovative organizations, such as Edison, is they create a supportive culture where staff are encouraged to try new things and are not discouraged when they fail. In 1948, the chairman of 3M, William McKnight, summarized his approach that characterized the company's strategy for decades to follow...
As our business grows, it becomes increasingly necessary to delegate responsibility and encourage men and women to exercise their initiative. This requires considerable tolerance. Those men and women to whom we delegate authority and responsibility, if they are good people, are going to want to do their jobs in their own way.
Mistakes will be made... Management that is destructively critical when mistakes are made kills initiative. And it's essential that we have many people with initiative if we are to continue to grow...
A nervous young manager who had led a failed project once offered Henry Ford a resignation letter. Ford's response was that he was not about to let someone go and work for a competitor after learning a valuable lesson with his money.
### Managers
As well as supportive leadership at the top of organizations, particular innovations need enthusiastic and powerful managerial 'champions' or sponsors with significant decision-making responsibility. As well as being good at managing teams, coordinating technical/design issues, and implementing processes and decisions, innovation managers also have to be skilful at advocating the virtues of the innovation, lobbying for its support, and creating a vision of what it will do and contribute.
### Boundary spanners
One of the most important individual roles in innovation is the boundary spanner, the person capable of communicating and building bridges between and within organizations. In manufacturing firms, this person used to be known as the technological gatekeeper. These people are avaricious acquirers of information – attained through reading and attending conferences and trade shows – and skilled at communicating useful information to the part of the organization that needs it. Organizations sometimes find it difficult to justify appointing boundary spanners. Their remit to travel, go to conferences, and talk to lots of people is sometimes unappreciated by those bound to a desk or workbench. But their role is highly beneficial to innovation.
#### Everyone
One of 3M's most successful innovations was the Post-It Note. Due recognition has been made of the developers of the technical core of the innovation – its non-sticky glue. And sufficient opprobrium has been dealt to the company's marketing department that claimed no one would buy it. But too little credit has been accorded to the people in the organization that recognized the product's potential and encouraged its development. Following the rejection of the idea of the Post-It from the marketing department, the product's developers sent samples to the secretaries of the company's General Managers. The secretaries immediately saw the value of the product and elicited the support of their bosses in getting the idea developed.
Innovation affects everyone in an organization and to a greater or lesser extent is everyone's responsibility. The computerization of many traditional craft skills, such as toolmaking, provided opportunities for de-skilling or re-skilling jobs. Many employers took the de-skilling path – as in the case of numerical control machine tools- but subsequently learned the advantages of re-skilling and giving the shop-floor worker discretion over the tasks they perform. This reflects the capacity of people to change and productively and creatively respond to innovation, if given the chance. It gives them, in Edison's words, the pleasure of cultivating the capacity to think. The potential of innovation derived from the factory floor has led some to describe them as laboratories and places for experiment.
An important tool used to encourage innovation is the use of reward and recognition programmes. Many organizations have suggestion schemes, and companies such as IBM and Toyota elicit hundreds of thousands of ideas from employees. These can be rewarded financially or by peer recognition. Often the most effective form of recognition is the implementation of the idea by the organization. The capacity of individuals throughout the organization to have innovative ideas and pursue their implementation shows that innovation leadership is not only the responsibility of those with high hierarchical positions.
Innovators in all forms are best supported in organizations whose commitment to human resource development and training attracts, rewards and retains talented managers and staff unafraid of change, and placates those who are fearful of it. Innovative organizations have the appointment procedures, payment and incentive systems, and career progression paths, to ensure appropriate staffing for innovation. While some people thrive on creating innovation, and need to be encouraged and rewarded, others are better at developing the procedures for its application, requiring different forms of recognition. Others still are temperamentally fearful of innovation, or at least too much change, seeing it as threatening and can suffer stress and poor performance as a result. A reputation as an innovative organization is very attractive to potential recruits who want to be innovative, and selection mechanisms should vet unsuitable appointments. Employees who find it disturbing need to be supported and guided through the introduction of innovation.
#### Technology
During the 1960s, the research of Joan Woodward into factory organization in the southeast of England began to explain the relationship between technology and organization. She showed how organization varied according to core underlying technology, whether production took the form of small or large batches, mass production, or continuous flow processes. The view that organization results from the technology used – technological determinism – has been discounted by research showing the extent to which choices can be made, a view to which Joan Woodward subscribed. Nonetheless, technology is highly influential and there is a relationship between the ways industries are organized and the extent to which they can benefit from innovation through the division of labour. The products and services of industries vary considerably and production and operations techniques vary accordingly.
#### Innovation technologies
Edison knew the value of high-quality scientific instrumentation, on the one hand, and 'junk', odd pieces of machines, and a vast array of unusual materials, on the other. These machines and artefacts stimulate innovation. Just as Edison's many sketches aided his thinking and improved the communication of his ideas with others, the creation of tangible designs and prototypes focuses efforts and builds connections between people with different skills and perspectives. In many cases, ideas for innovation grow organically and iteratively around emerging and increasingly focused designs.
Information and communications technologies move design and connections across boundaries into a digital world where Edison's aim for 'rapid and cheap development of an invention & working it up into Commercial shape', occurs in ways he could not imagine.
Digital technologies combine design and manufacture in Computer-Aided Design/Computer-Aided Manufacturing systems. Digital design information on new products is transferred to the equipment used to make them. The designs are guided by the system as to what is possible to manufacture. The internet, local area networks and enterprise resource planning systems help organizations combine different inputs from people with very different skills.
Developments of massive computing power, software that allows the merger of different data sets, and new visualization technologies used extensively in the computer games industry, has led to a new kind of technology supporting innovation. 'Innovation technology' (IvT) is so called because it helps combine various components of the innovation process. It is being used to improve the speed and efficiency of innovation by pulling together different inputs within and between organizations. IvT includes: virtual reality suites used to help customers design new products and services; simulation and modelling tools used to substantially improve the speed of new designs; E-science, or Grid computing, building new communities of scientists and researchers, and helping them to manage collaborative projects; sophisticated data-mining technology used to help understand customers and manage suppliers; and virtual and rapid prototyping technology used to improve the speed of innovation. Together, these technologies are being used to make customers on the one hand, and scientific researchers on the other, combine more effectively in decisions about innovation.
By moving experiments and prototyping into the digital world, IvT allows firms to experiment cheaply and 'fail often and early'. IvT is also very important in the design of large, complex systems, such as utilities, airport infrastructures, communications systems, where it is not usually feasible to test full-scale prototypes.
One of the most important aspects of IvT is how it assists the representation and visualization of knowledge and its communication across different domains, disciplines, professions and 'communities of practice'. By way of illustration, compare the design of a new building using traditional methods and IvT. The use of IvT makes complex data, information, perspectives, and preferences from diverse groups visible and comprehensible. Virtual representation assists architects to visualize their eventual designs and help to clarify clients' expectations by giving them a good understanding of what a building will look and feel like before work begins. Clients can 'walk through' their virtual building getting a sense of its layout and 'feel' prior to a brick being laid. IvT informs contractors and builders of specifications and requirements, and allows regulators, such as fire inspectors, to confidently assess whether buildings are likely to meet regulatory requirements. IvT can allow various players in the innovation process, suppliers and users, contractors and subcontractors, systems integrators and component producers, to collaborate more effectively in the delivery of new products and services.
**10. Engineering and design increasingly use computerized visualization and virtual reality tools**
Using IvT can produce some quite dramatic innovations. A great many died in the World Trade Center in 2001 because occupants trying to get down the fire escape stairs were trapped with firefighters going up them. New ways of getting people out of tall buildings in extreme events were considered for the replacement Freedom Tower in New York. Computer simulations and visualizations of the behaviour of buildings and people in emergencies led fire engineers to believe that the safest form of egress was by means of the lift. Changing entrenched views on safety to one where the message is: 'In case of fire, use the lift' requires a great deal of persuasion of building owners and occupants, engineers and architects, firefighters and fire regulators, and insurers. Mutual, shared comprehension of this radical change was assisted by moving detail from complex drawings and data sets to readily understandable computerized images. Fire engineers have used a range of new simulation and visualization technologies to help transform these diverse parties' understanding of safety in tall buildings and encourage the exploration of innovative approaches to rapid evacuation.
## [**Chapter 6
Building a smarter planet?**](contents.html#ch06)
We began this book with an illustration of innovation at the beginning of the Industrial Revolution. We end it with a speculative glimpse at what the future may hold. The challenges and opportunities for innovation are immense. As well as creating new sources of wealth from ideas, innovation is essential if we are to cope with climate change, provide better water and food, improve health and education, and produce energy sustainably. It will be essential for our continued co-existence on an increasingly crowded planet.
The innovation processes that will be used have become increasingly more complex. They have evolved from the activities of 18th-century entrepreneurs such as Josiah Wedgwood, to the formal organization of research in the 19th century and large corporate R&D departments of the mid- and late 20th century, to the present-day involvement of multiple contributors in distributed networks of innovators supported by new technologies.
The keys to future innovativeness will lie in an organization's ability to foster creativity and to make decisions and choices on the basis of being well prepared, informed and connected. The many sources of ideas – employees, entrepreneurs, R&D, customers, suppliers, and universities – will continually produce opportunities to innovate. The challenge lies in encouraging, selecting and configuring the best ideas from them. To explore how organizations in the future might respond to these challenges we turn to the example of IBM, a company that is using a distributed innovation process, and is preparing the ground for its further development in the future. We choose IBM because, in contrast to organizations of a more recent vintage, such as Microsoft and Toyota, or smaller, entrepreneurial firms, such as IDEO, IBM has had a long history of transforming itself. IBM has continually faced challenges, some of its own making, others imposed, and its survival in the future as much as the past has depended upon innovation. IBM has shown how it can change its innovation processes as it adapts to and shapes demands for new products, services, and technologies, and responds to emerging challenges. Although we cannot predict whether it will succeed in the future, the company shows us some of the aspects of a contemporary networked innovator; in the ways it supports and encourages expertise, internal and external connections, and decision-making in uncertain situations.
### Future thinking: the case of IBM
In 2006, IBM conducted a 'massively parallel conference' or 'Innovation Jam', which involved creating a web portal and inviting its staff to post ideas about four carefully identified future fields of potential development. The results were remarkable: in two three-day phases, over 40,000 suggestions were made by 150,000 IBM employees, family members, business partners, clients and university researchers from 104 countries. The interactive 'Jam' process saw ideas being discussed and refined, rated and scored as they were reduced in number to 36 and eventually 12. Nick Donofrio, IBM's head of innovation and technology at the time, says the process saw ideas getting momentum, evolving and turning into something completely different as those involved took over and changed them. Over $70 million was allocated in funding 10 new businesses as a result of the 2006 Jam, generating around $300 million in revenues in two years. In this way, IBM uses the internet to draw on the creativity of a massive community of potential innovators. It also uses a web portal, ThinkPlace, where employees can identify, share, and be rewarded for innovative suggestions. By these means, IBM has developed a way of systematically attracting ideas for, and making choices about, innovation on a huge scale.
Many of these proposals derive from the company's 200,000 scientists and engineers. To encourage creative science and engineering, and to ensure strong external connections amongst its technological leaders, IBM has created the positions of Distinguished Engineers and IBM Fellows. Around 650 staff currently hold these high-status positions within the organization. To become a Distinguished Engineer, individuals need to demonstrate a sustained record for invention and innovation, and recognition amongst internal and external peers. The position of IBM Fellow is the highest accolade for technical achievement in the company, and Fellows are given considerable latitude to pursue their own areas of research. Appointments to these prestigious positions are important motivators for career development and signals of achievement.
As part of its efforts to continually build its technological expertise, IBM has an Academy of Technology, founded in 1989, modelled on the US National Academies of Science and Engineering. Its purpose is to advise IBM's executives on technical trends, directions, and issues, and to develop and connect the IBM technical community across the world. The Academy produces reports, has an annual conference, and plays an important role in raising awareness about emerging trends and building and sharing knowledge.
The company's search for innovation extends well beyond its boundaries and it considers itself part of an 'innovation ecology' with a myriad of external relationships and connections. This ecology includes independent software providers, technical standards bodies, universities, government agencies, and customers. A regular series of publications – Innovation Outlooks – is produced to demonstrate thought leadership and help engage with communities that contribute to IBM's development and use of innovation.
IBM filed 4,186 new US patents in 2008, more than any other company. At the same time, as an indicator of the extent of its efforts to engage with the innovation community, it announced plans to increase by 50%, to more than 3,000 annually, the publication of its inventions and technical contributions, making this research freely available. By being 'open' in its innovation, and providing intellectual property for use by others, it demonstrates its expertise and helps build the scale of its technologies and markets as other parties develop complementary new products and services.
As an example of the way IBM engages with its customers, it has a system of Client Technology Advisers (CTAs) who develop long-term relationships with key clients, offering advice and developing roles as trusted guides to strategic developments in their industries. IBM benefits from this arrangement by helping shape major new investments at their early stages.
An example of IBM's attempts to lead in the development of new ideas is seen in the concept of 'services science'. This emerged in 2004 as a description of the shift towards complex services and systems in many industrial sectors and markets. The concept was adopted by a number of IBM's clients and collaborators such as BAE Systems and HP. IBM also works with the academic community, sponsoring research projects and university symposia aimed at exploring what it hopes will become a new discipline. IBM's encouragement of courses in Services Science, Management and Engineering (SSME), has seen their adoption worldwide in around 400 universities.
One method IBM uses to choose amongst its many potential opportunities for innovation is its Emerging Business Opportunities (EBO) process. EBO was created in 2000 to improve the company's ability to explore new technologies, such as virtual worlds, and quickly respond to new business opportunities. EBOs are usually managed in decentralized businesses, comprising small teams to encourage focus, and are expected to quickly generate financial returns, demonstrating there is a market for its products and services. Some EBOs, such as cloud computing – which provides immense computing power over the internet on a needs basis – are such major initiatives they are incubated centrally in the organization. The greatest incentive for encouraging new ideas is often the reward of seeing them put into practice, and the EBO process, which enjoys top-level management support, signals IBM's intent to systematically pursue the best of them.
IBM's organizational efforts to build and support its innovation process are complemented by its approach to using technology to build connections and make decisions. This is one of the key features of the company's Smart Planet strategy.
### IBM's Smart Planet strategy
Launched in 2008, this strategy recognizes that dealing with complex and emerging problems – in areas such as energy, health and the environment – requires understanding of the relationships within and between systems. This depends on the ability to monitor performance and make sense of massive amounts of data.
The strategy is based in part on the potential for using data from large numbers of sensing and monitoring instruments, including mobile devices. An indicator of the ubiquity of these instruments is IBM's estimate that there are around 1 billion transistors for every human on the planet. An example of this form of instrumentation is radio frequency identification (RFID) devices. RFID offers innovative approaches to managing supply chains and logistical systems, for example by tracking meat from farm to shop, helping to ensure fresh food is readily available for consumption.
IBM's strategy is to connect these sensors, to allow systems and objects to communicate together, creating what is known as the 'internet of things'. To create intelligent ways of finding and delivering solutions to problems using the Internet, decisions about how to design, configure, and operate systems will need to be made, using the powerful analytical capabilities of supercomputers and cloud computing. This allows data to be mined and patterns to be recognized, such as when insurance companies spot patterns in millions of claims, or police make correlations in forensic evidence to identify patterns of crime. Their analyses and diagnoses can lead to new understanding of the ways systems perform and evolve, and better management of resources. The use of innovation technologies, such as simulation techniques and the visualization of their results, can also offer the promise of decision-makers engaging with a wide range of stakeholders when making choices.
Examples of the potential value of the Smart Planet approach can be found in energy, transportation, and health. The scale of the challenge in energy provision is seen in IBM's estimate that the current inability to manage and balance electricity supply and demand in the US results in enough electricity being lost annually to provide the power for India, Germany and Canada. By utilizing an extraordinary density of instruments for measurement and monitoring, innovation can optimize energy supply and demand. The innovation opportunity is to develop new systems in which everything can be instrumented and analysed in real time: from the meter in the home, through the distribution network, to power plants. These 'smart grids' have the potential to allow better decisions and make energy supply more efficient, reliable, and adaptive to changing needs.
Similar issues are seen in the case of transport systems. According to IBM, congestion on US roads costs around $80 billion annually, with over 4 billion lost working hours and wastage of nearly 3 billion gallons of fuel, which produces huge carbon dioxide emissions. Innovation in public policy using instrumentation of traffic systems has resulted in improvements in many cities. Traffic congestion charges in Milan, for example, have a scale depending upon the level of pollution emitted by individual vehicles. When a vehicle enters the city, cameras almost instantaneously signal to a database identifying the model and relevant charge band. Stockholm's smart traffic system, which uses cameras and lasers to identify and charge vehicles according to the time of day, has reduced congestion by nearly 25% and emissions by about 12%. The complexity of the problems being confronted is seen in the way IBM worked with 300 different organizations in developing the Stockholm system.
Another example is seen in a means for addressing the notorious traffic problems in Chinese cities. Siemens, the German electronics and engineering company, is working with Chinese researchers to optimize traffic flow based on positional data provided automatically by drivers' mobile telephones.
An example of the Smart Planet approach in healthcare is provided by IBM's project with Google Health and Continua Health Alliance aiming to create telemedicine systems that allow individuals and families to track health information by streaming data from medical devices. RFID is also used to confirm the authenticity of medical supplies, reducing errors and improving compliance with medical regulations and procedures. It is being used by Siemens to track the number of sterile pads used in surgical operations to prevent any being left behind in the patient, and to monitor the temperatures of blood supplies throughout the process of donation, cell concentration, storage and use.
Many of the challenges of energy, transportation, and health will be addressed in cities. In 1900, 13% of the world's population lived in cities. By 2007, the majority were urban dwellers, and this is expected to increase to 70% by 2050, by which time the world's population will have risen from 6 to 9 billion. The difficulties for city planners and authorities are seen in the way cities are responsible for 75% of global energy consumption and 80% of greenhouse gas emissions.
An indication of the way IBM intends to develop services that address problems in urban systems is seen in the example of the creation of its Business Analytics Research Centre, announced by Sam Palmisano – IBM's President, Chairman and CEO – at the company's Smarter Cities Summit in Berlin in 2009. The German Centre will employ 100 scientists to research urban systems. The Centre's international, interdisciplinary, and inter-organizational research is an example of the new distributed models needed to tackle pressing contemporary problems.
IBM describes itself as a globally integrated enterprise, with global scale in many core activities, but whose delivery of services focuses on local market requirements. The company has the advantage of drawing on the huge amount of its science and technology expertise found around the world. At the same time, rather than concentrating R&D in the USA or a few major global hubs, the choice of Berlin for the new Centre shows how IBM aims to distribute its expertise closer to markets, especially in this case those of Eastern Europe.
The Centre therefore operates as a hub in a distributed research network, collaborating with more than 300 German mathematicians, consultants and software specialists. It links with IBM's global research centres, drawing on the broad expertise that exists on the complex interactions within and between systems. The Centre's work is collaborative, builds on IBM's services science approach, and uses innovation technologies to create and analyse new mathematical models and simulate the behaviour of the systems of systems that are cities. To understand energy requirements, for example, it is insufficient to model the increased use of electric cars without understanding its implications for electricity supply.
The new innovation model shows that no single organization on its own – even one the size of IBM – has the depth and breadth of capability to tackle the problems facing urban systems. It recognizes how solutions to the problems of population growth, health, energy, and transportation are interconnected and global in reach.
How successful the Smart Planet strategy will be in helping solve these problems will depend on the resolution of significant technical, organizational, social and political questions. Simple, robust, fail-safe technological systems are needed. New skills are required to analyse the mountains of data, interpret patterns, and produce insights to assist in making near real-time decisions. New forms of participation have to be developed to engage people in developing and implementing innovations.
IBM's Smart Planet is the evolving strategy of one organization, whose success or failure has yet to be enacted. IBM, like all organizations, has made mistakes in the past and will do so again in the future, and we have yet to see whether and how the strategy will work. The concept of a smarter planet, however, is an enticing one when considering the broader role of innovation in the future, and IBM gives us some clues as to what directions we might move in. The company shows us how the interconnections encouraging innovation are based on talented and knowledgeable people, deeply engaged within innovation ecologies, guided by imaginative strategies, and facilitated by technology. A smarter planet would use resources more efficiently and effectively. It would use new organizational approaches and technologies in highly connected institutions, innovation processes, and ways of working, and be better positioned to address current and emerging challenges.
### Smarter institutions
#### Governments
As well as pursuing the innovation policies discussed in Chapter 4, encouraging the stock and flow of innovation, national governments need high levels of inter-governmental coordination: internationally, regionally, and locally.
The use of innovation to deal with many contemporary problems requires more resources and skills than can be mustered by individual nations. Some challenges, such as controlling greenhouse emissions, simply cannot have autonomous solutions and have to be addressed in international forums. Balancing national self-interest with the need for international approaches will pose an increasing innovation policy challenge. Furthermore, as social well-being and economic prosperity become driven ever more by the productive use of creativity and knowledge, there are profound implications for relationships and disparities between nations. Existing inequalities may become accentuated as the technologically, institutionally and organizationally rich nations pull further away from those not so advantaged. Inter-governmental institutions have to monitor and develop policies to address any such problems.
Many important decisions about innovation are made not at the national level but by increasingly powerful municipal authorities and regional governments that vigorously compete with each other domestically and internationally to attract investment and talent. Expertise in domestic intergovernmental coordination and collaboration is also essential to effective innovation policy.
The privatization in many countries of previously publicly held assets in energy, transportation, and telecommunications has removed a direct lever governments once possessed to improve innovation. Instead, new regulatory authorities have been established and their roles in supporting innovation in the private sector have to be explored and extended. Government's role is complicated by the ways boundaries of the public and private realms have become blurred with the creation of public/private partnerships. There are mutual benefits to be achieved from this form of organization, including access to resources for investment in innovation that may otherwise be unavailable. But the ownership and control of innovation assets and knowledge may be shaped by different incentives that may result in tensions between private gain and public good. Government innovation policy has to be formulated on the basis of deep engagement with business, and understanding of the strengths and shortcomings of the contributions it can make.
There are huge future opportunities for innovation in government services. These include, for example, telemedicine, using computers and the internet to assist medical diagnoses at home. Telemedicine is used in Australia to deliver health services to remote communities. In the UK, it is used to provide monitoring of elderly patients preventing them having to be in hospital. In India, mobile equipment is driven to impoverished villages where diagnoses are made via electronic links to urban hospitals, providing a level of health care to which the rural poor previously had no access.
As part of their contributions to innovation, governments have opportunities to use the new technologies that offer the means for more inclusiveness and participation in decision-making by citizens in designing and delivering the services we demand. By creating proposed new health centres in virtual worlds prior to their actual construction, for example, inputs can be elicited from health professionals and patients to produce better designs.
One of the most crucial areas of policy-making lies in the processes by which choices are made by governments about where to focus innovation investment to sustain future prosperity. No nation has the resources to innovate in all areas and trade-offs are needed between competing demands for scarce resources. Governments have to establish sophisticated approaches for making choices, whilst ensuring that enough is invested in a broad spread of areas to keep options open and allow nations to absorb useful ideas developed elsewhere. Decisions about what and what not to prioritize have to involve extensive discussions with business, social and environmental groups, and informed public debate in efforts to build consensus about the future.
The importance of innovation for government, and the difficulties in building the necessary connections and making good choices, requires broad and deep innovation policy-making skills. These extend comprehension of the importance and nature of innovation throughout the apparatus of government, and help develop a 'whole of government' approach. Greater appreciation of the contributions and difficulties of innovation will help address the very high aversion to risk in the public service. In recognition of its more widespread, distributed, and inclusive nature, public policy requires better forms of measurement of innovation -moving on from the partial and often misleading indicators of R&D expenditure and patenting performance – and new approaches and skills are needed in this area. Tools, such as social network analysis can be used, for example, to measure changing patterns of connectivity. Innovation policy-making has to recognize that innovation is a continuing challenge with no simple 'solutions'. As it evolves new issues arise and policies need to change in response.
### Universities
To contribute more effectively to innovation, universities have to be better at encouraging knowledge exchange and the internal and external flow of ideas. They should move beyond a restricted model of technology transfer in the form of formal intellectual property protection, licensing and start-up companies, and welcome the many opportunities collaboration provides for the creation and transfer of new educational and research services. Their strategies will find multiple ways of engaging with stakeholders in business, government, and the community, and yet still continue to be driven by scholarly values. They will educate and employ people able to work in multiple ways in research, business, and government, and build connections between different parts of innovation systems, encouraged by the mobility of broadly skilled graduates and enhanced through the use of eScience.
Few universities have the resources to provide universal service across all the academic disciplines, and most benefit from making choices and broadly specializing. Some may strengthen their positions by having a local flavour, while others see their role as nodes in global research and education efforts. Concentration in particular areas provides the depth of expertise attractive to the best collaborators in research and business, and partnerships can fill gaps where institutions have decided not to participate.
Universities have a continuing role to play in producing the large-scale research tools and instruments for science and engineering to encourage discovery, allow people to scout in unknown areas and see and measure the things that others cannot: perhaps increasingly in the design and delivery of services. They provide leadership in formulating common standards needed by innovators to help launch new products and services in dynamic industries.
The provision of 'rehearsal space' and collaborative laboratories for deep and sustained conversations and engagement in ideas generation and testing with business, government, and the community is one of the most important innovation supporting roles for universities. Researchers will continue to work with all the academic rigour and independence of their discipline, but through these conversations many will become comfortable as members of distributed teams exploring inter-disciplinary interfaces and the social and economic consequences of their work. Highly accomplished at providing the physical and organizational structures and incentives for scholarly recognition and career progression, universities will need to explore better spaces and methods for encouraging and rewarding such engagement.
### Business
When economies and technologies are rapidly changing and volatile, the value of firms' capacity to accept and implement radical and disruptive ideas increases. In such circumstances, the best strategies are those that are experimental and dynamic and achieve a judicious balance between exploiting existing ideas and exploring new ones. These strategies rely on continual investments in human capital and in research and technology.
**11. Some challenges of innovation may always be with us**
Innovation, in the words of Lou Gerstner, previous CEO of IBM, needs to be engrained in the DNA of the organization. As seen in the case of IBM, the performance of exceptional innovators and teams should be rewarded, but the responsibilities and opportunities for innovation are everyone's.
Continuing investments in R&D and the absorptive capacity it produces remain crucial, as does the ability to trade and broker knowledge within innovation ecologies. Corporate connections to sources of new ideas require long-term partnerships with universities around the world, deep embedding within innovative cities and regions, and effective management of supportive innovation technologies.
The breadth of these ecologies is extended as traditional distinctions between industries become blurred as knowledge, insights, and skills across sectors are transferred and combined to produce novel offerings. Much value creation in the manufacturing industry, for example, lies in design services. The services sectors and universities are collaborating in innovative ways, such as in IBM's SSME initiative. Innovation is a condition for success in the creative industries – such as new digital media, entertainment, and publishing – whose content is critical, for example, for innovative product and services companies involved in mobile telephony. The resource industries, such as agriculture and mining, depend on innovation to improve efficiencies and assist product improvements, and whose innovations in water management, for example, have broader applications.
The ideas for innovation in business will come from diverse and often unexpected sources in new and unforeseen combinations.
The effect of the global financial crisis of 2008/2009 on connections between innovators and sources of capital will take many years to be fully understood. In the short term, innovation investments will undoubtedly be adversely affected, and in the long-term trust between the financial and productive sectors will need to be rebuilt. New forms of governance of risk will be needed to oversee greater ethical and responsible decision-making and improve risk management of complex innovations.
Small and medium-sized organizations may increasingly be the progenitors of breakthrough technologies, using their advantages of speed, flexibility, and focus over larger ones. Compared to large, publicly traded companies, small firms can take and bear unusual risks. And not being so constrained by the organizational rigidities of large firms, they can more easily develop and trial novel business models and processes. Small and medium-sized organizations will combine their behavioural advantages with the greater resources found in large firms in new forms of innovation network and collaborative partnership. Larger organizations will continually experiment with attempts to emulate the entrepreneurial environments of smaller units, as seen in IBM's EBO process.
As Edison knew, innovation has to be organized in ways appropriate to its objectives. The benefits from the unbounded search for ideas, where chance and serendipity can produce so many rewards, have to be balanced with organizational focus and direction. There are far more opportunities than can be afforded, and choices need to be made that shape and direct the skills organizations use and resources they invest. Skills in the strategic management of innovation which help them make those choices will become amongst the most prized by business.
### Smarter innovation
As in the time of Wedgwood, innovation will result from the combinations of ideas, but these ideas are ever more widespread and distributed around the globe, and their integration can increasingly be assisted by the use of technology. As Wedgwood understood so well, innovation combines 'supply side' considerations – that is, sources of innovation such as research and technological developments – and deep appreciation of market demand. Smart innovators are immersed in understanding the changing patterns and meaning of consumption, and the values and norms that underlie decisions to purchase innovative products and services. These patterns are affected by globalization and are fluid in nature. A generation raised on conspicuous consumption, whatever its real costs, may be despised by another concerned for sustainability. Recognizing the capacities of new technologies to include increasing numbers of contributors to innovation, including communities of users, greater appreciation is needed of their motivations and how their energy and insights can be most effectively used.
Innovation strategy in companies of all sizes and sectors has to move beyond the planned, sequential models of the industrial age, and the forms of corporate R&D laboratories that produced Stephanie Kwolek's discovery. It has to account for opportunities that arise in unexpected places, high levels of uncertainty and great complexity, where organizational learning through collaboration is the key to survival and growth. The limited financial measurements and accounts used by business in the past – such as return on capital and quarterly reports to shareholders – have to be supplemented with indicators more meaningful to innovation and organizational resilience. What, for example, is the value of the options for the future that organizations possess through doing research? What innovations being explored and developed have the potential to account for major parts of the organization in 10 to 20 years? How has an organization's capacity to learn been improved through research investments? What is the value of being a trusted collaborator, and ethical employer and sustainable producer?
Economic thinking benefits from evolutionary approaches that see risk, uncertainty, and failure in innovation as normal and move us away from linear and planned to open, emergent, and highly connected systems. The value of ideas and learning become recognized as the most important drivers of economic growth and productivity. The importance of the exploration of new interdisciplinary combinations between science, arts, engineering, social sciences and humanities, and business is appreciated and the need for mechanisms and skills for building connections across organizational, professional, and disciplinary boundaries is emphasized. Attention is placed on improving the connections and performance of innovation systems and ecologies. These ecologies may form unimagined new combinations: anthropology may inform local energy production and distribution; philosophy may influence semiconductor circuit design; the study of music may affect the provision of financial services.
Innovation technologies intensify innovation. The instrumentation of trillions of devices and sensors embedded in the physical world contribute to the unimaginable amounts of data available to be used by the new technologies of design in the virtual world to create and improve the products and services we want and enhance the experiences we desire.
Innovation has to deliver non-damaging or environment-enhancing products and processes. Innovation and sustainable development will need to become two sides of the same coin. Many sustainability challenges – climate change, water resources management, genetically modified agriculture, waste disposal, marine ecosystem protection, and biodiversity loss – are persistent and have no complete solution, lacking a clear set of alternatives and little room for trial and error. They are characterized by contradictory certitudes among protagonists and strategies for dealing with them involve coping rather than solving and searching for what is feasible not optimal. Lessons from the study of innovation can be applied to deal with these persistent problems, including the facilitation, structure and management of cooperation and connectedness, the management of risk and assessment of options, and the use of collaboration tools such as social networking technologies. Furthermore, the use of innovation technologies can help model and simulate the implications of decisions, and their visualization capacities help communications and the informed involvement of diverse parties to assist participative decision-making.
### Smarter individuals
How will we deal personally with the way innovation is changing? Whether we work in the private or public sector, community groups, or as a member of the public, how can we be smarter in the way we develop and use innovation? Increased technological literacy will certainly improve our effectiveness in the massively connected world. But we shall also have to become more adept at encouraging creativity, dealing with change, communicating across boundaries, and putting ideas into practice. Intuition and judgement, tolerance and responsibility, diversity of interests and cross-cultural sensitivities are needed. Our capacities to think about new ideas, play with them by tinkering, testing, and prototyping, and rehearsing, implementing or doing them, have to be balanced. Our scepticism and critical faculties should be attuned to questioning: 'this is the way it is', and capacities accentuated for articulating: 'this is the way we want it'. We shall demand the rewards that Edison's laboratory workers experienced – although perhaps without the exhausting hours or fear of the corpse reviver. Indeed, with the wealth created through our knowledge, we expect job satisfaction in enriching workplaces, conducive to diversity, which fit in with our lifestyles and family circumstances and choices.
We should ensure the inventors and innovators that contribute so much – the Stephanie Kwolek's of the world – are recognized in a way that sport stars and entertainers are appreciated today.
Innovation is a restless process that brings with it continual uncertainty about its success and failure. It can be threatening as well as rewarding. How well we respond to it depends upon how open-minded and cooperative we are, how prepared we are to accept risk and give space for the unusual and work with others that are differently minded. This will be influenced by the culture of organizations and the quality of leaders that recognize that job security and toleration of failure are crucial to innovation, that no one holds all the answers, progress is collaborative, and reputation lies in modesty in claims and professionalism in delivery.
The outcomes of innovation are not always beneficial, and their consequences often cannot be foretold. Adding lead to petrol solved the problem of engine knocking, but left a disastrous environmental legacy. Thalidomide reduced morning sickness in pregnant mothers, but induced disabilities in their babies. The dangerous divorce between action and consequence was clearly seen in the global financial crisis of 2008/9 where financial innovations were introduced without any checks and balances or consideration for their implications. Concern for the implications of innovations have to be paramount amongst those who seek to introduce them.
The massive amounts of data available on individuals, to other people, corporations, and the state, also increase the responsibilities of those designing and managing innovation. Innovation – in information use and other areas such as genetics -requires deep ethical considerations and highly visible and accountable practices, and alert and responsive regulations. Simulation, modelling, and virtualization technologies provide huge opportunities for improving innovation processes, but their responsible use depends on the skills and judgement of people immersed in the theory and craft of their professions and trades. Innovation requires people to be informed, vigilant and responsible employees, customers, suppliers, collaborators, team members, and citizens. Andrew Grove, the founder of Intel, said that in our uncertain world only the paranoid survive, but it will be the discerning and informed, not the mistrusting and fearful, that will see us through. Immanuel Kant said that science is organized knowledge; wisdom is organized life. The future of innovation – where its benefits flow and costs are curtailed – lies in the wise organization of knowledge.
## **References**
W. Abernathy and J. Utterback, 'Patterns of Industrial Innovation', _Technology Review,_ 80(7) (1978): 40–7.
N. Baldwin, _Edison: Inventing the Century_ (New York: Hyperion Books, 1995).
W. Baumol, _The Free-Market Innovation Machine: Analyzing the Growth Miracle of Capitalism_ (Princeton, NJ: Princeton University Press, 2002).
C. Bilton, _Management and Creativity: From Creative Industries to Creative Management_ (Oxford: Blackwell, 2007).
M. Boden, _The Creative Mind: Myths and Mechanisms,_ 2nd edn. (London: Routledge, 2004).
T. Burns and G. Stalker, _The Management of Innovation_ (London: Tavistock Publications, 1961).
H. Chesbrough, _Open Innovation: The New Imperative for Creating and Profiting from Technology_ (Cambridge, MA.: Harvard Business School Press, 2003).
C. M. Christensen, _The Innovator's Dilemma: When New Technologies Cause Great Firms to Fail_ (Boston, MA.: Harvard Business School Press, 1997).
A. Davies, D. Gann, and T. Douglas, 'Innovation in Megaprojects: Systems Integration in Heathrow Terminal 5', _California Management Review,_ 51(2) (2009): 101–25.
M. Dodgson, D. Gann, and A. Salter, 'The Role of Technology in the Shift Towards Open Innovation: The Case of Procter & Gamble', _R &D Management,_ 36(3) (2006): 333–46.
M. Dodgson, D. Gann, and A. Salter,' '"In Case of Fire, Please Use the Elevator'': Simulation Technology and Organization in Fire Engineering', _Organization Science,_ 18(5) (2007): 849–64.
M. Dodgson and L. Xue, 'Innovation in China', _Innovation: Management, Policy and Practice,_ 11(1) (2009): 2–6.
G. Fairtlough, _Creative Compartments: A Design for Future Organisation_ (London: Adamantine Press, 1994).
C. Freeman, _The Economics of Industrial Innovation,_ 1st edn. (London: Pinter, 1974).
C. Freeman, _Technology Policy and Economic Performance: Lessons from Japan_ (London: Pinter, 1987).
C. Freeman and C. Perez, 'Structural Crises of Adjustment: Business Cycles and Investment Behaviour', in G. Dosi, C. Freeman,
R. Nelson, G. Silverberg, and L. Soete (eds.), _Technical Change and Economic Theory_ (London: Pinter, 1988).
D. Gann and M. Dodgson, _Innovation Technology: How New Technologies Are Changing the Way We Innovate_ (London: National Endowment for Science, Technology and the Arts, 2007).
L. Gerstner, _Who Says Elephants Can't Dance: Inside IBM's Historic Turnaround_ (New York: Harper Business, 2002).
S. Gu and B.-A. Lundvall, 'China's Innovation System and the Move Toward Harmonious Growth and Endogneous Innovation', _Innovation: Management, Policy and Practice,_ 8(1–2) (2006): 1–26.
A. B. Hargadon, _How Breakthroughs Happen: The Surprising Truth about How Companies Innovate_ (Cambridge, MA.: Harvard Business School Press, 2003).
C. Helfat, S. Finkelstein, W. Mitchell, M. Peteraf, H. Singh, D. Teece, and S. Winter, _Dynamic Capabilities: Understanding Strategic Change in Organizations_ (Malden, MA: Blackwell, 2007).
R. Henderson and K. B. Clark, 'Architectual Innovation: The Reconfiguration of Existing Product Technologies and the Failure of Established Firms', _Administrative Science Quarterly,_ 35(1) (1990): 9–30.
M. Josephson, _Edison_ (London: Eyre and Spottiswoode, 1961).
C. Kerr, _The Uses of the University_ (Cambridge, MA.: Harvard University Press, 1963).
R. K. Lester, _The Productive Edge_ (New York: W. W. Norton & Co., 1998).
B. A. Lundvall (ed.), _National Innovation Systems: Towards a Theory of Innovation and Interactive Learning_ (London: Pinter, 1992).
F. Malerba, _Sectoral Systems of Innovation: Concepts, Issues and Analyses of Six Major Sectors in Europe_ (Cambridge: Cambridge University Press, 2004).
K Marx, _Capital_ , Vol. 1 (Harmondsworth: Pelican, 1981).
A. Millard, _Edison and the Business of Innovation_ (Baltimore, MD: Johns Hopkins University Press, 1990).
F. Miller, _Thomas A. Edison: The Authentic Life Story of the World's Greatest Inventor_ (London: Stanley Paul, 1932).
R. Nelson and S. Winter, _An Evolutionary Theory of Economic Change_ (Cambridge, MA.: Belknap Press, 1982).
R. Nelson (ed.), _National Innovation Systems: A Comparative Analysis_ (New York: Oxford University Press, 1993).
D. F. Noble, _Forces of Production: A Social History of Industrial Automation_ (New York: Oxford University Press, 1986).
C. Paine, _Who Killed the Electric Car?,_ documentary film, Papercut Films (2006).
S. Palmisano, 'The Globally Integrated Enterprise', _Foreign Affairs,_ 85(3) (2006): 127–36.
J. Quinn, 'Interview with Stephanie Kwolek', American Heritage. com, 18(3) (2003).
E. M. Rogers, _Diffusion of Innovations,_ 4th edn. (New York: The Free Press, 1995).
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K. Sabbagh, _Twenty-First-Century Jet: The Making and Marketing of the Boeing 777_ (New York: Scribner, 1996).
J. A. Schumpeter, _The Theory of Economic Development: An Inquiry into Profits, Capital, Credit, Interest and the Business Cycle_ (Cambridge, MA.: Harvard University Press, 1934).
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## **Further reading**
### On Josiah Wedgwood:
B. Dolan, _Wedgwood: The First Tycoon_ (New York: Penguin, 2004).
### On Joseph Schumpeter:
T. McGraw, _Prophet of Innovation: Joseph Schumpeter and Creative Destruction_ (Cambridge, MA.: Harvard University Press, 2007).
### On the innovation process, and the ways it is organized, managed, and changing:
M. Dodgson, D. Gann, and A. Salter, _Think, Play, Do: Technology, Innovation and Organization_ (Oxford: Oxford University Press, 2005).
M. Dodgson, D. Gann, and A. Salter, _The Management of Technological Innovation: Strategy and Practice_ (Oxford: Oxford University Press, 2008).
### On the economics of innovation:
J. Fagerberg, D. Mowery, and R. Nelson (eds.), _The Oxford Handbook of Innovation_ (Oxford: Oxford University Press, 2006).
### On the history of innovation:
N. Rosenberg, _Inside the Black Box: Technology and Economics_ (Cambridge: Cambridge University Press, 1982).
D. Edgerton, _Shock of the Old: Technology and Global History since 1900_ (London: Profile Books, 2006).
### On innovation strategies:
M. Schilling, _Strategic Management of Technological Innovation_ (New York: McGraw-Hill/Irwin, 2005).
### On entrepreneurship:
G. George and A. Bock, _Inventing Entrepreneurs: Technology Innovators and their Entrepreneurial Journey_ (London: Prentice Hall, 2009).
M. Wright, B. Clarysse, P. Mustar, and A. Lockett, _Academic Entrepreneurship in Europe_ (Cheltenham: Edward Elgar, 2007).
### Data on international R&D and innovation performance:
National Science Foundation; Science and Engineering Statistics: <http://www.nsf.gov/statistics>
Organization for Economic Cooperation and Development; Science, Technology and Patents, Statistics Portal: <http://www.oecd.org>
## **Index**
adverse consequences , –, –, –
air transport industry , , , , –, –
analyses of innovation –
Apple –, ,
applied research , –,
automation , , –, –,
basic or pure research , –, ,
Baumol, William
Bayh–Dole Act
Bell Labs
Boeing ,
Bohr, Niels
bootlegging
Boulton, Matthew ,
boundary spanners –
British Airport Authority (BAA) –
broker connections ,
bullet-proof vests –
Bush, Vannevar
business –, , –, , –, –
Business Analytics Research Centre (IBM) –
business schools
capitalism , ,
car industry , , , –, –,
challenges of innovation –
China , , –, ,
cities –,
Clark, Graeme –
climate change ,
collaboration –, –
business –
Edison, Thomas
IBM
Japan
networks and communities , , –,
open innovation , , , –, ,
universities , –
communications , , –,
competition , , , , ,
complexity , –,
computers , –, , , – _see also_ Internet; software
congestion charges
coupling model –
craftsmanship –,
creativity and teamwork , , , –, –, –
customers , –, –, , –,
data processing , –
definition of innovation –
demand-pull model
design , , , –, –, , , , –, , , –, ,
destructive, innovation as –, ,
digital technology – _see also_ Internet
disruptive innovation –
division of labour , –, –, , –
DNA , ,
DuPont , –
dynamic capabilities theory ,
easyJet –
ecologies ,
economic development , –, –, , , –, ,
Edison, Thomas , , , –, –, –, –, , –
efficiency , ,
electricity , –, , ,
Emerging Business Opportunities (EBO) (IBM)
employment and work –, –
adverse effects , , ,
bootlegging and skunkworks
challenges –
creativity and teamwork , , –, –
division of labour , –, –, , –
Edison, Thomas , –, –
IBM –
immigration
organization
quality control
reward and recognition programmes
Wedgwood, Josiah ,
work environment –, –
energy –, , , , , –, ,
engineering –,
entrepreneurs , –, ,
environment –, , , , , –
Ericsson
evolutionary economics ,
experimental development , –, ,
failures –, –, –,
fast-followers –
fibre –
financial crisis 2008 , , ,
financial systems –
first-mover advantage –
flow of ideas , –
Ford, Henry , –, , –,
Frascati Manual –,
Freeman, Christopher
Genentech ,
General Electric (GE) , ,
globalization , ,
Google , ,
government , , , –, –, –, –
health –, , , –
Heathrow Airport –
Hollerith, Herman ,
housing , –
human capital , _see also_ employees
IBM –, –, , –,
IDEO –,
incremental innovation , –,
India ,
individuals, effect on – _see also_ employment and work
industrial clusters , ,
industrialization , , , –, –,
information, access to and use of , ,
infrastructure , –, , , , –
'Innovation Jam' (IBM)
innovation technology (IvT) –, –
'innovator's dilemma'
Intel , , ,
intellectual property , , _see also_ patents interests, threats to established –
Internet , , , –, –, –,
Japan –, , –, –
Kerr, Clark
Kevlar –
knowledge , , , –, , –
Kwolek, Stephanie –, ,
laboratories , , –, , , –
leadership , –, , –,
lean production
learning –,
legal systems –
Lunar Men
machinery and equipment , , –, –
managers
market failure –
market research
Marshall, Alfred
Marx, Karl , –
mass production , , –,
Massachusetts Institute of Technology (MIT) ,
mechanistic organizing –
medical instruments –
Menlo Park laboratory , ,
Microsoft , , ,
Millennium Bridge, London –
models of innovation –, ,
multi-factor productivity (MFP)
national systems –, –
networks and communities , , –,
new developments , , –, , –
open innovation , , , –, ,
operations and production , –
organic organizing –
organizational issues –, –, , , –, , , –,
partnerships and joint ventures –, –,
Pasteur, Louis , –,
patents –, , , , , , –, , ,
Perez, Carlota
polio vaccine
politics , –, –
polymers –
Post-It Notes
pottery –
practical and functional innovation
predict and provide strategies
privatization
processes –, ,
Procter and Gamble –
products –, , , , , –, ,
quality , , , , –
radical innovations , , –, , ,
radio frequency identification (RFID) –,
rates of innovation and diffusion
regions , –,
research and development (R&D) –, –, , –
China –
IBM –, , –
internal innovation –
Japan , –
profitability
structures –
reward and recognition programmes
risk , –, , , ,
Rolls Royce
Rothwell, Roy ,
Salk, Jonas
Schumpeter, Joseph , –, , , , , –
science and technology institutes , –
science push model
services , –, –, , –, –, ,
Silicon Valley , , –, , ,
simulation –, ,
Singapore
skills issues –
skunkworks
sleep apnoea
Sloan, Alfred
small businesses ,
Smart Planet strategy (IBM) –
Smith, Adam , –, –,
social and cultural factors –, ,
social networking ,
software , , –,
Sony
sources of innovation , , –, –, –,
South Korea ,
specialization –, , –
Staffordshire Pottery –, , , ,
stage-gate systems
steam power –, , ,
Stokes, Donald –
structures –
suppliers –, –
sustainable development , –
Tabulating Machine Company
tax credits , –
teaching ,
technical colleges and institutes –, –
technical standards , , , , ,
technological determinism
technology and organization, relationship between –
telemedicine
Tesco
Tesla, Nikola , , –
theories of innovation –
thinking –
time dimension –
Toyota , –, –, ,
transformational innovations
transport –, , , –, –,
trust –
typologies of innovation –
United States , –, , –, –, –, –, –, –
universities –, –, , , –
unpredictability , –, –, ,
venture capitalists , , –
virtual reality , ,
visualization technologies –, ,
Watson, Thomas –
Watson Jr, Thomas –
Wedgwood, Josiah –, , , , , –, , ,
West Orange laboratory ,
Woodward, Joan –
work _see_ employment and work
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Q: Correct way to concatenate string within GROUP_CONCAT I know this must be pretty basic, but I'm trying to make a query where I can show a continent and then show all countries with airport from that continent, so far, this query works for me:
$sql = "SELECT *, GROUP_CONCAT(DISTINCT ,country, SEPARATOR '')
as country
FROM airports
GROUP BY continent
ORDER BY continent, country ASC";
if($result = mysqli_query($conn, $sql)){
if(mysqli_num_rows($result) > 0){
while($row = mysqli_fetch_array($result)){
echo "<hr>" . $row['continent'] . "" . $row['country'] . "";
}
// Free result set
mysqli_free_result($result);
} else{
echo '<div class="autocomplete__item alert--warning"> No records matching your query were found.</div>';
}
} else{
echo "ERROR: Could not able to execute $sql. " . mysqli_error($conn);
}
It shows the continent name, and every country in it with one or more airports. So, I need that Country output to be surrounded by a lot of HTML. I guess the best way is to nest queries, but, I've done it in the same GROUP_CONCAT declaration and it works! Except for the fact that the HTML breaks.
I change GROUP_CONCAT to:
GROUP_CONCAT(DISTINCT '<div class=\"push-button\"><input type=\"checkbox\" id=\"',alpha_2,'\"><label for=\"',alpha_2,'\">',country,'</label></div>' SEPARATOR '')
That technically works, but it breaks the markup at some point:
<div class="push-button">
<input type="checkbox" id="bm">
<label <hr>Asia<div class="push-button"><input type="checkbox" id="cn"><label for="cn">China</label></div>
Or another the last row:
<div class="p </section>
That happens some 5 or 6 times in 56 countries.
Probably I should nest queries, not sure and not sure how to do it, but... Isn't this weird or am I missing something?
A: TL;DR:
SET SESSION group_concat_max_len = 10000;
Ok, Solved. The problem wasn't GROUP_CONCAT braking the markup, it happens that it has a max value, by default 1024 (I found out making test with MySQL Workbench).
So I found a solution here: In MySQL, is it possible to get more than 1024 characters back from GROUP_CONCAT
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{"url":"https:\/\/chemistry.stackexchange.com\/questions\/5665\/what-is-the-heat-energy-required-to-perform-electrolysis-equivalent","text":"# What is the heat energy required to perform electrolysis equivalent?\n\nThe question may be off-topic here. If my concept is flawed, please poke me in the rib (+:\n\nSources on the internet indicate a minimum 1.2Volts is adequate for electrolysis of water; nominally 6V. Applying a DC source to water provides adequate energy to break the hydrogen bond. Apparently some heat is also released in the process..\n\nSay, one were to collect 100ml (an arbitrary figure) oxygen by electrolysis applying 6V DC to 1 Litre of water.\n\nCan the energy to break the hydrogen bond be provided by applying heat to water? All other things being identical, how many calories would be required to collect the same amount of oxygen from the same volume of water?\n\nNomenclature: First, when you pass a current through water, you are not breaking the hydrogen bonds, which are intermolecular forces, but (a) breaking the bond between hydrogen and oxygen and (b) forming molecular oxygen and molecular hydrogen, $\\ce{2H2O -> 2H2 + O2}$, and this reaction can be broken down into two half reactions: $$\\ce{4H2O + 4 e- -> 2H2 + 4OH-}$$ $$\\ce{2H2O -> O2 + 4H+ + 4e-}$$","date":"2020-04-05 05:03:33","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\": 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\": 0.48669883608818054, \"perplexity\": 1030.044983330958}, \"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-2020-16\/segments\/1585370528224.61\/warc\/CC-MAIN-20200405022138-20200405052138-00163.warc.gz\"}"} | null | null |
{"url":"https:\/\/scicomp.stackexchange.com\/questions\/35593\/traction-boundary-conditions-in-elasticity","text":"# traction boundary conditions in elasticity\n\nI have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My question is: How is the node order\/connectivity for the traction elements defined? In 2D the boundary term $$({\\bf{w}},{\\bf{h}})_{\\Gamma_h}$$ is a line integral. Since the connectivity for elements 3 and 4 is (5,6,9,8) and (4,5,8,7) should the nodes for the traction elements be chosen in the same order? That is, should the four traction elements have the connectivity (6,9), (9,8), (8,7) and (7,4)? That means the boundary integral should be $$\\int_{node 6}^{node 9} {\\bf{w}}\\cdot{\\bf{h}} \\,d\\Gamma$$ and similarly for the others?\n\nWhat is the situation in 3D? In 3D the traction elements are bilinear quads (for trilinear hex elements). (Note: the faces of trilinear hexes in 3D may not be plane. See this.)Is the connectivity of traction elements in 3D defined so that the normal points outwards according to the right hand rule? So for the right most face element in the above figure, should the connectivity be (3,7,6,2) and for the topmost face element the connectivity should be (7,8,5,6)?\n\n\u2022 I'm pretty sure that the direction of integration is immaterial. For 2D elasticity, I think that the boundary term $(w,h)_\\Gamma_h$ is a line integral. The definition of a line integral renders the direction of integration immaterial. In 3D, the boundary term is a surface integral and I think it must be defined in a manner analogous to the line integral. \u2013\u00a0Nachiket Sep 8 '20 at 5:57","date":"2021-07-23 19:38:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8823764324188232, \"perplexity\": 405.6539617714456}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046150000.59\/warc\/CC-MAIN-20210723175111-20210723205111-00289.warc.gz\"}"} | null | null |
Кировский областной краеведческий музей имени П. В. Алабина — один из старейших региональных музеев России, расположенный в городе Кирове (Вятке).
История
Создание Вятского публичного музеума
В 1837 году в городе Вятке открылась выставка естественных и искусственных произведений Вятской губернии, подготовленная А. И. Герценом и приуроченная к приезду великого князя Александра Николаевича (будущего императора Александра II). Наследнику престола выставка понравилась, он приобрел некоторые предметы, в частности, деревянные карманные часы мастера Семена Бронникова, несколько каповых шкатулок, паровую кастрюлю, «для карандашей станок», «две машинки для чинения перьев» с Воткинского завода. Данная выставка послужила толчком к открытию музея в Вятской губернии, став одной из основ будущего Вятского публичного музеума.
13 июля 1863 г. в «Вятских губернских ведомостях» была опубликована записка управляющего делами Вятской публичной библиотеки П. В. Алабина «О необходимости устроить при Вятской публичной библиотеке публичный музеум». Материалы статьи были озвучены им и на заседании попечительного комитета библиотеки.
В разные концы России — по официальным инстанциям, коллекционерам, специалистам — отправлялись письма с просьбой о помощи в организации Вятского публичного музеума. В итоге в пользу музеума поступило большое количество уникальных предметов, положивших начало его богатейших коллекций, а также денежные средства в сумме более 2,7 тыс. руб. Устроителями музея закупалась необходимая мебель, уникальный микроскоп, отдельные экспонаты.
8 мая 1864 г. Александр II пожаловал богатую минералогическую коллекцию в количестве 645 образцов, стоимостью 423 руб. 32 коп., из фондов музея Горного института «с отнесением издержек на отправление коллекции в Вятку на счет сумм горного ведомства».
В целом на момент открытия музея в его залах насчитывалось около 7 тысяч экспонатов.
Музей во второй половине XIX — начале XX в.
Открытие Вятского публичного музеума состоялось 22 января (3 февраля) 1866 г. На торжестве присутствовало более 200 человек, в том числе вятский губернатор генерал-лейтенант В. Н. Струков, епископ Вятский и Слободской Агафангел (Соловьев), флигель-адъютант императора, гвардии ротмистр граф Н. Ф. Орлов-Денисов.
Первое время музей существовал при Вятской публичной библиотеке, занимая несколько её залов на ул. Копанской, и открывался для посетителей три раза в неделю, с 10 до 13 часов дня. В бесплатные воскресные дни число посетителей доходило до трехсот. Среди первых посетителей музея был юный семинарист, будущий известный художник Виктор Васнецов, в апреле 1866 г. купивший годовой абонемент.
В начале 1874 г. музей был передан земскому училищу Вятского губернского земства. В 1880 году земское училище было преобразовано в реальное училище, а музей — в кабинет наглядных пособий. В 1889 году земское собрание выделило на нужды музея по 500 рублей в год. Заведующим был назначен преподаватель реального училища С. Н. Косарев — селекционер, который явился основателем Вятской опытной станции и работал над выведением нового сорта ржи. Он же являлся учителем естественной истории реалиста Саши Гриневского — будущего великого русского писателя-романтика А. С. Грина. Много лет спустя, намечая основные темы в набросках «Книги воспоминаний», Александр Степанович напишет: «Музей при реальном училище».
Селекционную работу С. Н. Косарева завершил его ученик Н. В. Рудницкий, являвшийся заведующим музеем в 1904—1908 гг. При нём фонды пополнялись новыми коллекциями по природе, археологии, этнографии и сельскому хозяйству. В музее был создан небольшой актив из местных исследователей: археолога А. А. Спицына, Н. Г. Первухина, П. М. Сорокина.
Развитие музея в первой половине XX в.
В послереволюционный период музей, получивший название Вятский губернский музей местного края, значительно укрепил свой научный потенциал. В 1918 году при музее были созданы научная библиотека и естественно-научная лаборатория, открыт кружок юннатов. Свой вклад в развитие музея внес целый ряд крупных советских ученых, в частности, ихтиолог профессор Б. С. Лукаш (был директором музея в 1924—1929 годах), минеролог и почвовед профессор П. А. Земятченский. В 1919 году началась музейная карьера А. Д. Фокина — подвижника вятского музея, пришедшего на работу на должность младшего лаборанта ботанического кабинета и проработавшего с перерывами на должностях заведующего кабинетом, начальника отдела, заместителя директора по научной части, научного сотрудника и главного хранителя фондов вплоть до 1972 г.
Летом 1920 г. при музее работали одни из первых в стране специализированных курсов подготовки музейных работников. 1920—1930-е гг. были ознаменованы дальнейшей активизацией научной, экспедиционной, фондовой и выставочной работы. В середине 1930-х гг. произошло объединение с Музеем революции, что значительно обогатило фондовые коллекции музея.
Во время Великой Отечественной войны музей был законсервирован. На его площадях разместились общежития рабочих Коломенского паровозостроительного завода (затем здесь формировались новые воинские части).
Послевоенный период
Вторая половина XX века — время постоянного поступательного развития Кировского краеведческого музея.
В 1945 году научным сотрудником музея стал Л. В. Дьяконов, впоследствии — известный кировский писатель. Леонид Васильевич работал над созданием музейного «фольклария» — собрания записей вятского фольклора. К 1950 г. результатом этого труда стал сбор более 50 тысяч текстов — пословиц, поговорок, песен, загадок, частушек.
В музее постоянно работали новые выставки, периодически расширялась и постоянная экспозиция. Так, в 1955 году для посетителей была открыта экспозиция отдела досоветской истории. При музее действовала фотолаборатория, ученый совет, долгие годы возглавляемый профессором Кировского государственного педагогического института им. В. И. Ленина А. В. Эммаусским. Музей выступал в качестве организатора экспедиций, вел активную научную работу, собирались воспоминания участников революции 1917 г., Гражданской и Великой Отечественной войн.
В конце 1960-х гг. по всей области началось масштабное строительство новых музеев. Сотрудники музея приняли участие в восстановлении музея Ф. Э. Дзержинского в селе Кай Верхнекамского района, создании дома-музея маршала Советского Союза И. С. Конева в селе Лодейно Подосиновского района.
В 1968 году в г. Кирове в доме, где с 1848 по 1855 г. жил в ссылке сатирик М. Е. Салтыков-Щедрин, состоялось торжественное открытие литературного музея — филиала областного краеведческого музея. Его создание велось по инициативе известного библиофила и краеведа Е. Д. Петряева. Впоследствии учреждение было преобразовано в дом-музей М. Е. Салтыкова-Щедрина.
К ноябрю 1977 г. была смонтирована постоянная музейная экспозиция, посвященная истории развития народного прикладного искусства Вятки, в одном из старейших гражданских зданий города — Приказной избе (ныне музей «Вятские народные художественные промыслы» — отдел Кировского областного краеведческого музея). В декабре того же года начала работу кировская Диорама, полотно которой посвящено установлению советской власти в городе Вятке. Авторами диорамы выступили заслуженный деятель искусств РСФСР А. И. Интезаров и Н. Н. Соломин — ныне народный художник России. Их совместное творчество было отмечено золотой медалью имени М. Б. Грекова. В 1981 году архитектору Диорамы А. П. Павлову, скульптору В. А. Сонину, живописцу Н. Н. Соломину была присуждена Государственная премия РСФСР.
В августе 1980 г. состоялось открытие музея известно русского писателя-романтика А. С. Грина, родившегося на территории Вятской губернии. Работа по сбору материала для него была проведена сотрудником музея, краеведом А. В. Ревой. Значительное содействие оказали родственники А. С. Грина, его сестры и племянницы.
В 1988 году по инициативе летчика-космонавта СССР В. П. Савиных в г. Кирове состоялось открытие музея К. Э. Циолковского, авиации космонавтики (ныне это самостоятельный областной музей).
В 1992 году открыл свои двери для посетителей ещё один отдел музея — «Вятская кунсткамера», расположившийся в типичном купеческом особняке конца XIX — начала ХХ вв.
Современность
В 1989 году была впервые проведена конференция по итогам научно-исследовательской работы музея. В том же году состоялись естественно-научные краеведческие чтения, посвященные проблемам изучения и охраны природы Кировской области. В 1995 году были проведены первые Гриновские чтения, ставшие ныне традиционными. Со временем хорошей традицией стала организация конференций, приуроченных к различным юбилейным датам.
Активно развивается международное сотрудничество кировского музея. В 1996 году на открытии выставки шведского фотографа Л. Тунбьерка присутствовала делегация Швеции, а в 1998—1999 г. в г. Кассель (Германия) демонстрировалась выставка «Марионетки и матрёшки», на которую кировский музей предоставил 114 экспонатов. В 2013 году сотрудники музея участвовали в работе XIV Российско-Финляндского культурного форума.
18 мая 2006 г., к 140-летнему юбилею музея для посетителей было открыто современное главное здание музея, расположенное по адресу ул. Спасская, д. 6. В 2014 году заработал самый молодой на сегодня отдел краеведческого музея — Музей воинской славы. Открытие было приурочено к 25-летию вывода советских войск из Афганистана.
21-22 апреля 2016 г. состоялась Всероссийская научно-практическая конференция «Краеведческий музей: история, коллекции, люди», приуроченная к 150-летию со времени основания Кировского областного краеведческого музея. На конференцию поступило 150 заявок из 30 регионов России и из Республики Беларусь.
Кировский областной краеведческий музей — член Союза музеев России. Среди наград учреждения — почетный знак «За активную работу по патриотическому воспитанию граждан Российской Федерации», памятные медали — к 65-летию Победы в Великой Отечественной войне (диплом подписан Президентом РФ Д. А. Медведевым) и «Киров — город трудовой славы», памятный знак «80 лет Кировской области».
В начале XXI в. музей продолжает активно развиваться, принимает деятельное участие в проведении всероссийских акций «Ночь в музее» и «Ночь искусств». Неизменным успехов у кировчан пользуются организованные сотрудниками музея выставки.
С декабря 2022 года носит имя П. В. Алабина.
Музейные коллекции
Фонды Кировского областного краеведческого музея содержат свыше 250 тысяч музейных предметов, в том числе 159 тысяч единиц основного фонда.
Коллекция археологии хранит материалы раскопок на территории Кировской области, которые отражают все археологические периоды истории бассейна реки Вятки. Древнейшие из них датируются VII тыс. до н. э.
Одно из самых крупных музейных собраний — нумизматическое — насчитывает более 45 тыс. подлинников. Основу собрания составляют монеты, начиная с медных греческих оболов IV в. до н. э. и серебряных гривен Древней Руси XII в., заканчивая монетами и бумажными денежными знаками современной России, в том числе монетные клады VIII—XX вв., обнаруженные на территории Кировской области. Среди наград — Медаль «Золотая Звезда» (СССР) Героя Советского Союза Ф. А. Костина, ордена и медали генерал-лейтенанта Н. Т. Петрухина, спортивные награды многократной чемпионки мира по парашютному спорту Т. Н. Воиновой и других выдающихся представителей Вятской земли. Интересна коллекция медалей, жетонов, значков в память исторических событий XVIII—XX вв.
Большим разнообразием отличается музейное собрание металла: самовары М. Нечаева (г. Вятка), братьев Поповых (г. Слободской), колокольцы братьев Бакулевых, И. Каркина, В. Ситникова (г. Слободской), замечательные изделия из чугуна мастеров Холуницких заводов, орудия труда и инструменты, осветительные приборы, домашняя утварь, а также многое-многое другое.
В коллекции оружия наряду с предметами XIV—XV вв. (одни из первых экспонатов музея — пищали XVI в., палаш 1715 г.), в музее представлено стрелковое и холодное оружие периода Великой Отечественной войны. Особую ценность представляют снайперская винтовка Героя России Н. И. Галушкина, кортик генерал-лейтенанта Н. Т. Петрухина.
Собранные музеем авторские и массовые изделия из дерева, бересты, соснового корня, ивовой лозы, капа позволяют судить о высоком мастерстве вятских умельцев, а коллекция деревянной игрушки производства местных артелей 1920—1930-х гг. считается одной из самых интересных в стране.
Собрание текстиля составляют коллекции национального костюма народов, населяющих Вятскую землю: русского, удмуртского, марийского, татарского, коми XVIII—XX вв., образцов лицевого и орнаментального шитья XIX в., ткачества, вышивки, кружева XIX—XX вв. Бережно сохраняются в музее мемориальные вещи выдающихся земляков: гимнастерка С. М. Кирова и полотенце, вышитое им в г. Уржуме в подарок воспитательнице приюта; парадные кители Маршала Советского Союза, дважды Героя Советского Союза И. С. Конева, Главного маршала авиации, Героя Советского Союза К. А. Вершинина, генерал-полковника, Героя Советского Союза Н. Д. Захватаева. В коллекции кожи широко представлена продукция кожевенно-обувных и меховых предприятий области.
Фототека музея огромна: более 46 тысяч фотоснимков и негативов, датируемых периодом с 1860-х гг., времени появления фотографии на Вятки, до наших дней. На них запечатлены события, люди, места, значимые для истории родного края, работы выдающихся фотомастеров С. А. Лобовикова, А. В. Шишкина, А. В. Скурихина, А. М. Перевощикова, фотолетопись боевого пути 311-й Двинской стрелковой дивизии Д. Ф. Онохина периода Великой Отечественной войны. В коллекции также хранится дагеротипный портрет 1851—1854 гг. ссыльного князя А. П. Енгалычева, вятского сослуживца М. Е. Салтыкова-Щедрина, возможно прототипа Горехвастова из салтыковских «Губернских очерков».
Фонд письменных источников насчитывает 35 тыс. единиц хранения: редкие рукописные и печатные книги XVI—XIX вв., среди которых особо ценны Библия XVI в. первопечатника Ивана Федорова, «История вятчан» и «Вятская флора» 1807—1810 гг. первого вятского историка А. И. Вештомова, регистровая книга коммуны Ла Трините 1791—1795 гг., подаренная музею в 1927 году потомком французских революционеров Марком Бернардоном; а также автографы и документы людей, оставивших след в истории Вятского края и России.
Современное собрание изобразительных материалов включает в себя иконы XVII — начала XX в., живописные произведения, авторскую и полиграфическую графику, скульптуру. Особого внимания заслуживают работы вятских иконописцев, среди которых главная ценность — икона XVII в. «Преподобный Трифон Вятский перед Богоматерью». Среди живописных произведений — портрет писательницы М. Е. Селенкиной и этюды В. М. Васнецова, работы Е. Д. Чарушина, М. А. Демидова, А. В. Исупова, С. М. Мезенцева, П. С. Вершигорова, И. А. Широковой и др. Интересна серия этюдов, выполненных живописцами студии военных художников имени М. Б. Грекова А. И. Интезаровым и Н. Н. Соломиным при работе в 1976—1977 гг. над полотном диорамы «Установление Советской власти в г. Вятке» (с 1977 г. полотно экспонируется в МВЦ «Диорама» — отделе Кировского областного краеведческого музея).
В коллекции графики особой ценностью обладают чертежи и рисунки В. М. Васнецова — проекты мебели, которые были воплощены в жизнь его братом Аркадием, талантливым вятским мастером. Мебель, выполненная по этим проектам, ныне является украшением Дома-музея В. М. Васнецова в г. Москве. Интересен самодельный альбом с рисунками учащихся Вятской семинарии, преподнесенный в подарок епископу Лаврентию (Барановичу) в 1787 г. В данной коллекции также хранятся этнографические зарисовки художника-академиста А. В. Фищева, участника музейной экспедиции 1925 г. в Кумёнскую волость Вятской губернии, рисунки и эстампы художника-анималиста Е. И. Чарушина, выполненные им во время эвакуации в г. Киров в годы Великой Отечественной войны, рисунки Кукрыниксов, И. И. Бродского, рисунки и акварели А. И. Деньшина, Н. Н. Румянцева, С. А. Вшивцева, А. Е. Люстрицкого, С. Н. Мезенцева, Т. П. Дедовой, графика А. М. Колчанова. В 1988 году вместе с комплексом других материалов наш знаменитый земляк, солист Большого театра, народный артист СССР А. Ф. Ведерников подарил музею несколько автопортретов в костюмах своих оперных персонажей.
Печатная графика включает более 60 листов плакатов периода Великой Отечественной войны, большую подборку (около 800) открыток с видами г. Вятки и уездных городов губернии конца XIX — начала XX в., коллекцию экслибрисов, спичечных этикеток, марок.
Коллекция скульптуры содержит работы М. М. Кошкина. Ф. А. Шпак, Н. А. Захваткина, С. И. Крестьянинова, С. Лаптева и, конечно, знаменитую дымковскую игрушку, выполненную известными мастерицами дымковского промысла А. А. Мезриной, О. И. Коноваловой, Е. А. Кошкиной, Е. И. Косс-Деньшиной и др. Фонды музея хранят около 1000 дымковских барынь и кавалеров, зверей и птиц.
Ботаническая коллекция насчитывает более 80 тыс. листов гербарных сборов с 1848 г. по настоящее время — результат трудов И. П. Сележинского, А. И. Кардакова, А. Г. Зубарева, А. Д. Фокина и многих других исследователей. В 1939 и 1941 годах из ботанического музея университета г. Лунде (Швеция) поступили гербарные образцы, собранные в девяти странах Европы и Азии.
Немало находок хранит и палеонтологическая коллекция, наибольшее число которых относится к четвертичному периоду кайнозойской эры. Это фрагменты скелетов вымерших животных: пещерного бизона, мамонта, шерсистого носорога.
Жемчужиной палеонтологической коллекции является скелет короткорогого бизона, найденный в июле 1966 г. напротив с. Атары Лебяжского района Кировской области на левом берегу р. Вятки художником областной типографии О. К. Кобельковым. В августе того же года аспирантом Кировского государственного педагогического института им. В. И. Ленина Ю. М. Устюговым были произведены успешные раскопки. Было обнаружено 106 костей, в том числе полный позвоночник (без одного позвонка), нижняя челюсть со всеми зубами, правая сторона верхней челюсти, две конечности с копытами, тазовые кости, чешуя основной кости, части ребер, обломки лицевого скелета. Впоследствии скелет был выслан в г. Ленинград, где кости были реставрированы, а недостающие — смуляжированы в Зоологическом институте АН СССР.
Скелет короткорогого бизона был установлен летом 1969 г. в Кировском областном краеведческом музее выезжавшими для этой цели скульпторами из Ленинградского художественно-оформительского объединения И. Н. Хитровым и А. А. Колокольчиковым. Скелет являлся лучшим образцом из найденных к тому времени в СССР, в настоящее время он — один из наиболее полно представленных скелетов данного вида на территории России (экспонируется в Главном здании Кировского областного краеведческого музея).
Большая часть нашей энтомологической коллекции — результат сборов членов юннатского кружка, действовавшего в музее в 1920—1960-х гг. под руководством А. Д. Фокина.
Отделы
Главное здание музея (ул. Спасская, 6)
18 мая 2006 г. — в год 140-летия Кировского областного краеведческого музея — открылась экспозиция в здании по ул. Дрелевского (ныне ул. Спасская), 6. Здание бывших торговых рядов (начало XIX в., архитектор Ф. М. Росляков) — объект культурного наследия регионального значения.
Экспозиция «Россия-Вятка: особенности национальной истории» представляет развитие Вятского края с древнейших времен до начала XX в. Особенности расселения древнего человека, начиная с середины каменного века, представлены комплексами различных археологических культур. Здесь показано развитие отдельных компонентов материальной культуры: эволюция техники обработки камня и типов орудий; развитие орудий труда от кремнёвых до железных; украшения, предметы вооружения, глиняная посуда.
Формирование народностей, традиционно населяющих бассейн реки Вятки: марийцев, удмуртов, коми; заселение Вятской земли татарами, русскими; основание городов — все это представлено в экспозиции экспонатами, показывающими динамику и специфику развития Вятского края.
Жизнь крестьян Вятской губернии показана через их традиционные занятия, обряды, праздники. Развитие промыслов, что всегда выделяло Вятскую землю, годичный цикл тяжелого крестьянского труда показаны в фотографиях, домашней утвари, сельскохозяйственных орудиях, интерьерных комплексах. Языческие верования марийцев и удмуртов, сохранение старообрядчества наложило своеобразный отпечаток на традиционных уклад жизни вятских крестьян. Одна из самых ярких черт — торговая жилка населения Вятского края — проявилась в деятельности многочисленного и активного купеческого сословия в городах, в занятиях вятского населения промыслами.
В экспозиции посетители знакомятся с экспонатами, фотографиями XIX в., интерьерными комплексами.
Экспозиция «Природа Вятского края» включает два раздела. Первый показывает геологическое и палеонтологическое прошлое Кировской области в историческом развитии территории и органического мира в палеозойскую, мезозойскую и кайнозойскую эры. В разделе представлены уникальные экспонаты древней жизни на территории Кировской области — реконструированный скелет коротконогого бизона, многочисленные раковины, позвонки водных ящеров, кости мамонтов и многих других животных. Во втором разделе в оформленных диорамах и биогруппах показаны современные животные, растения, грибы, лишайники.
Музейно-выставочный центр «Диорама» (ул. Горького, 32)
Музей-диорама «Вятка. 1917» (с 1998 г. — МВЦ «Диорама») был открыт в декабре 1977 г. к 60-летию Октябрьской революции 1917 г.
Основным композиционным центром полотна диорамы стало изображение тревожных событий декабря 1917 г., происходивших в г. Вятке на перекрестке улиц Николаевской (ныне ул. Ленина) и Спасской.
Экспозиция «История Вятки в лицах» рассказывает о дореволюционной Вятке, о жизни купцов и традициях провинциального губернского города.
В МВЦ «Диорама» работает Детский центр. Детский центр является своеобразной лабораторией музейной педагогики, где экскурсии и занятия ориентированы на все возрастные группы — от самых маленьких до учащихся выпускных классов и студентов вузов. Они построены с учётом содержания школьных программ и прекрасно дополняют курсы по природоведению, истории, географии, литературе и краеведению, а рекреационные программы позволяют сделать досуг детей более интересным и познавательным.
Музей «Вятские народные художественные промыслы» (ул. Горького, 32)
Музей был открыт 11 ноября 1977 г. Здание, в котором он располагался до 01.09.2018, — одно из первых каменных строений гражданской архитектуры города первой половины XVIII в., "Питейная изба" (ул.Спасская 4 Б), объект культурного наследия федерального значения. С 1 сентября 2018 года экспозиция размещается в МВЦ «Диорама» (ул. Горького, 32).
В музее представлены изделия вятских мастеров, составляющие «золотой фонд» Кировского областного краеведческого музея. В Вятской губернии в XIX в. насчитывалось около 70 видов различных промыслов, а по количеству мастеров-ремесленников губерния занимала одно из первых мест в России. На сегодняшний день продолжают существовать традиционные художественные производства: изготовление глиняной игрушки, художественная обработка капа, резьба и роспись по дереву, кружевоплетение. Лучшие образцы этих и других промыслов представлены в экспозиции.
В музее проводятся мастер-классы по народным промыслам.
Дом-музей М. Е. Салтыкова-Щедрина (ул. Ленина, 93)
Одна из страниц культурного прошлого Вятки тесно связана с именем писателя-сатирика М. Е. Салтыкова-Щедрина. Семилетняя вятская ссылка (с 1848 по 1855 г.) стала важнейшей главой биографии писателя. Дом, где поселился М. Салтыков, принадлежал мастеру Медянской бумажной фабрики И. Х. Рашу. Он был построен в 1848 году по типовому проекту, с минимальным количеством комнат: столовая, гостиная, кабинет, спальня, людская. В настоящее время это объект культурного наследия федерального значения.
25 октября 1968 г. в г. Кирове в доме, где в годы ссылки проживал М. Е. Салтыков-Щедрин, открылся литературный музей на общественных началах (с 1971 г. — отдел Кировского областного краеведческого музея).
Основу современной экспозиции составляют подлинные предметы XIX в. из Вятского губернского правления, из архивов вятских знакомых великого писателя. В интерьере первого зала представлены виды Санкт-Петербурга и Вятки, книги, журналы, мебель, предметы быта середины XIX в., копии архивных документов. Во втором зале воссоздана обстановка гостиной: мебель, предметы интерьера, типичные для городской среды того времени. Интерьер третьего зала передает деловой стиль кабинета в квартире вятского чиновника середины XIX в.
Дом-музей А. С. Грина (ул. Володарского, 44)
Музей открыт 23 августа 1980 г., к 100-летию со дня рождения писателя А. С. Грина (Гриневского, 1880—1932 гг.). Одноэтажный каменный дом с мезонином принадлежал крестьянину А. Г. Морозову. Сейчас — объект культурного наследия федерального значения.
Главным замыслом авторов было стремление показать истоки творчества писателя, духовное становление личности. Большое место в экспозиции отведено детским годам будущего писателя, прошедшим в Вятке.
В «детской» и «гостиной» воссоздана типичная для семьи рядового служащего обстановка, где представлены подлинные и мемориальные предметы из семьи Гриневских: мебель, посуда, готовальня, книга, переплетенная Александром Грином, камень яшма, привезенный в подарок отцу. Эти и другие экспонаты раскрывают семейную атмосферу, мир увлечений школьника Саши Гриневского, широту его интересов, стремление к творчеству.
Экспозиция «Жизнь и творчество А. С. Грина» повествует о мире увлечений, интересах, трудных годах скитаний и творчестве писателя. Здесь показаны редкие прижизненные издания книг писателя, журналы с его рассказами, рукописи, документы, фотографии. Центральное место в экспозиции отведено феерии «Алые паруса» (1923) — произведению, ставшему классикой романтизма в отечественной литературе.
Музей воинской славы (ул. Красноармейская, 1 а)
Музей открыт 14 февраля 2014 г. к 25-летию вывода советских войск с территории Афганистана.
Экспозицию музея «Страницы ратной доблести» открывает раздел, рассказывающий об участии кировчан Великой Отечественной войне. Большой раздел экспозиции посвящен кировчанам-участникам войны в Афганистане в 1979—1989 гг. Эта часть экспозиции строилась при активном участии ветеранов Афганистана. Экспонатами стали подлинные вещи (документы, фотографии, письма, снаряжение, оружие, личные предметы), которые передали участники боевых действий.
Издания
Краеведческий музей: история, коллекции, люди (к 150-летию Кировского областного краеведческого музея): сб. статей и материалов: в 2 т. / ред.-сост. М. С. Судовиков, П. Н. Шарабаров. Киров: О-Краткое, 2016.
Кировский областной краеведческий музей: путеводитель / науч. ред. Т. А. Лебедева. Киров, 2006. — 20 с.
Дворецкая Т. А. По следам музея: очерки истории Кировского областного краеведческого музея. Киров, 2011. — 168 с.
Глиняная игрушка России: Буклет к выставке русской глиняной игрушки из коллекций К. А. Миронова, А. В. Быкова, В. А. Быкова и Кировского областного краеведческого музея. Киров, 2009. — 16 с.
Сокровища вятских музеев (к 145-летию музея) / сост. Л. А. Сенникова. Киров, 2011. — 7 с.
Кировский областной краеведческий музей: Путеводитель из собраний Кировского областного краеведческого музея / сост. Л. Н. Власова, Г. Г. Гузлаева, Н. Н. Злобина, О. В. Кротова, О. Л. Маклыгина, Е. Н. Махнёва, Н. Ю. Осколкова, Ю. Б. Порфирьев, Л. А. Сенникова. Киров, 2011. — 24 с.
Музей и дети: методический сборник музейных занятий и мероприятий / под ред. Т. А. Лебедевой (науч. ред.), Л. Н. Власовой, Д. В. Гришина, Н. Ю. Осколковой, Н. Н. Ходырева. Киров: О-Краткое, 2013. — 352 с.
Улицы старой Вятки: Фото из собрания Кировского областного краеведческого музея, Вятского епархиального архива, церковно-исторического центра Вятской епархии / сост. Л. А. Сенникова. Киров, 2013. — 23 с.
Александр Грин в экслибрисе: Альбом-каталог коллекции из собрания Кировского областного краеведческого музея / сост. М. А. Махнева. Киров, 2010. — 46 с.
Веселая потеха…: Альбом-каталог коллекции лубочных картинок в собрании Кировского областного краеведческого музея / сост. Т. А. Лебедева; Кировский областной краеведческий музей. Киров: О-Краткое, 2012. — 240 с.
В царстве минералов: Альбом-каталог минералов и горных пород их коллекции П. В. Алабина в собрании Кировского областного краеведческого музея / сост. Н. М. Меланина. Киров, 2013. — 160 с.
Вятка детства и юности А. С. Грина: Каталог в иллюстрациях художника Т. П. Дедовой (по «Автобиографической повести» и «Автобиографии» А. С. Грина) в собрании Кировского областного краеведческого музея / сост. Л. М. Ситникова. Киров, 2012. — 20 с.
Вятская деревянная игрушка (1920—1930 гг.): Каталог коллекции из собрания Кировского областного краеведческого музея / сост. Е. В. Налесникова. Киров, 2003. — 48 с.
Вятское кружево в трех веках: Альбом-каталог коллекции кружевных изделий в собрании Кировского областного краеведческого музея / сост. Л. И. Остапенко. Киров: О-Краткое, 2016. — 208 с.
«Детские игрушки, лошадки, безделушки…»: Альбом-каталог коллекции игрушек из папье-маше в собрании Кировского областного краеведческого музея / сост. Г. Н. Зельфиева. Киров, 2014. — 176 с.
И львы, и птицы, и цветы… Народная домовая роспись: Альбом-каталог из коллекции Кировского областного краеведческого музея / сост. В. А. Любимов. Киров, 2011. — 160 с.
Музей на рубеже веков: история, состояние, перспективы: мат-лы международной науч. конф., посв. 140-летию Кировского областного краеведческого музея / науч. ред. Т. А. Лебедева. Киров, 2006.
Политический плакат периода Великой Отечественной войны 1941—1945 гг.: Каталог коллекции из собрания Кировского областного краеведческого музея / сост. Т. Г. Семакова. Киров, 2005. — 35 с.
По Салтыковским местам города Вятки: Каталог в иллюстрациях худ. Т. П. Дедовой в собрании Кировского областного краеведческого музея/ сост. Л. Ю. Гулина, Е. А. Петрова, Л. Н. Самохвалова. Киров, 2013. — 20 с.
Сергей Александрович Лобовиков. Жанры, виды, портрет: Каталог из коллекции собрания Кировского областного краеведческого музея / сост. Л. В. Рыжакова. Киров, 2006. — 52 с.
Примечания
Ссылки
Появились в 1866 году в России
Краеведческие музеи России
Вятское краеведение
Музеи Кирова
История Кирова | {
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Narrowly dodging a life as a restaurateur, Julia Carey Arendell received her M.F.A. in Creative Writing with a minor in American Literature from Louisiana State University after completing a fellowship at Harvard. Pushcart Prize nominated for her fiction, she was also a winner of the William Faulkner-William Wisdom Prize in Poetry. Publications can be found at Heavy Feather Review, VIDA, Mason's Road, and Psychopomp Magazine, among others, as well as the anthologies Louisiana in Words and New Orleans: What Can't Be Lost. She has joined the editorial boards of the Early College Folio, The Dudley Review, The New Orleans Review, and New Delta Review and currently serves as the faculty advisor for the student run publication Global Commons, an international project sponsored by OSUN. Julia joined BECNO in 2014 and teaches creative writing coursework, First and Second Year Seminar, and manages the peer tutoring program on campus.
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Hierarchical clustering (or hierarchical merging) is the process by which larger structures are formed through the continuous merging of smaller structures. The structures we see in the Universe today (galaxies, clusters, filaments, sheets and voids) are predicted to have formed in this way according to Cold Dark Matter cosmology (the current concordance model).
The first structures in the Universe were small. Larger structures were later formed through the merging of these small, early objects. Further mergers over the lifetime of the Universe have resulted in the structures we see around us today.
For example, the formation of galaxies is thought to begin when small structures (perhaps no more massive than globular clusters) merge to form larger objects. These larger objects then merge to from even larger objects, which continue to merge until we arrive at the massive galaxies we see today in the local Universe.
Hickson Compact Group HCG92. Stephan's Quintet.
Small regions from the Hubble Ultra-Deep Field are full of examples of disturbed, interacting and merging galaxies.
The images above show a small number of the many examples of merger activity in the local Universe. In fact, the Milky Way itself is currently subject to at least two minor mergers, with both the Sagittarius and Canis Major dwarf galaxies in the process of being shredded and absorbed. However, according to Cold Dark Matter cosmology, the rate of mergers is expected to decline as the Universe ages. This means that although we see many mergers in the local Universe, mergers are expected to have been much more common in the past. Evidence to support this comes from observations of distant galaxies (e.g. those contained within the Hubble Deep and Ultra-Deep Fields), which show a much higher percentage of disturbed and interacting galaxies.
Since the merger process continues to the present day, galaxies should possess stars with a range of ages.
This is in direct contrast with the uniformly old ages for stars predicted by primordial collapse models of galaxy formation, and provides a means of determining the relative importance of the each of these processes. While young populations of stars have been observed in the centres of many galaxies, it is not yet clear whether these are a light 'frosting', perhaps the result of a secular evolution process, or whether they constitute evidence for major merger events. Astronomers are still investigating the relative importance of hierarchical clustering, primordial collapse and secular evolution in the formation of galaxies.
Although we have talked about hierarchical clustering almost exclusively in the context of the formation of galaxies, it is also an important mechanism on much larger scales – those of galaxy clusters and superclusters. These structures are again formed by the merging of smaller components. Galaxy groups form through the merger of individual galaxies. Groups then come together to form larger groups and clusters, which in turn merge to form large-scale filaments, walls and superclusters.
An example of the ongoing process of hierarchical merging can be initiated with the Milky Way. Right now the Milky Way is undergoing minor mergers with a couple of dwarf galaxies in the Local Group, and also appears to be interacting with both the Large and Small Magellanic Clouds. However, the ultimate fate of our Galaxy is to merge with Andromeda (the other large galaxy in the Local Group) to form a single, massive elliptical galaxy. The Local Group itself will eventually merge with the Virgo cluster, which it is believed will one day merge into the Shapley supercluster (the Great Attractor). This example, which traces the formation of a supercluster from a galactic scale, illustrates why hierarchical clustering is considered to be one of the most important processes in the formation of structure in the Universe. | {
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Home | Business in Action | Education | Career College Group
Career College Group
The Career College Group, an Ontario-based educational conglomerate, aims at supplying leading Canadian industries' with skilled, qualified workers. The group's two divisions – Medix College and North American Trade Schools – offer over 28 unique programs across seven campuses. Each program targets growing fields with high current employee demand.
For more than 40 years, Medix College has trained students for jobs within the healthcare service industry. Medix operates within six of the Career College Group Campuses and offers several online classes. North American Trade Schools, with two campuses within Ontario, provides students with the skills necessary to excel in hands-on trade areas.
Education with Purpose
The Career College Group puts the needs of the community first. They engage a group of employers, industry professionals, and post-secondary educators from each program discipline to assess community needs. These advisory boards help the Career College Group select which departments to expand, and decide what kind of future programs they could potentially offer. Through listening to community needs, the Career College Group can train students for in-demand positions and better equip Ontario businesses with qualified employees.
"Our ultimate goal is that students come here, graduate, and get a job," explains Peter Dykstra, the Career College Group's Vice President. "Whatever the community needs is our focus. We want to get our students employed."
The most recent addition to the Career College Group's multi-disciplinary education opportunities is the Solar Energy Technician program. This six-month technical training program provides students with the necessary knowledge and experience to work with both private energy service companies and government institutions.
Learning Philosophy
The Career College Group educates students through maintaining strong values. They employ the "Five A's of Employment" within each program in order to prepare students for professional working environments. These five factors – Attitude, Attendance, Appearance, Academics, and Accountability – focus on developing the professional attitude needed to reach career success.
"These are the soft skills that really help our students secure employment and be excellent in their industry," explains Ryan Alary, Director of North American Trade Schools London.
Why Choose Career College Group?
Through working closely with the Ontario Ministry of Training Colleges and Universities, the Career College Group offers exceptional student support and financial aid services. The group aims at offering equal opportunities to all students regardless of background or monetary limitations.
The Career College Group looks to offer assistance on a level easily accessible to all potential students. Dykstra explains, "Our focus is that students who choose to attend a career college like ours have equal opportunities and access to support programs, credit transfer, grants and financial aid. We believe we have one of the best education systems in the world… We want to make sure our students have the same opportunity as any other college or university."
Both Medix College and North American Trade Schools offer students the option of making monthly payments or running a student line of credit. They are also registered with OSAP and operate in conjunction with the provincial government's Employment Ontario initiative.
The Career College Group also provides accelerated program opportunities in order to quickly bring students into the workforce. Their in-class time averages at about 20 to 25 hours per week, ten hours more than the alternative's average. Classes run daily, with no semester breaks. They also don't offer any elective classes, allowing students to focus solely on their chosen field.
Each Medix and North American Trade Schools program includes an Internship/Externship program, providing students with on-the-job training before entering their field. These opportunities can range anywhere from 240 to 600 hours per college program. There are also labs and practice facilities located within each campus.
"In terms of both our programs, we spend about 60 per cent of the time doing hands-on training and 40 per cent within the classroom," says Alary.
Many programs, such as HVAC Technician and Solar Energy Technician, include licencing and certification as part of the learning experience. This allows students to enter the workforce fully equipped to find meaningful employment within their field.
The Career College Group is currently collecting data with plans to publish each program's hiring percentages later this year. The findings will be available through the Ontario government.
"What I can guarantee you is that our focus is on getting every student a job. We don't stop till that happens."
Starting Family Legacies
Through providing financial assistance and job training, Career College Group hopes to assist students in achieving successful, long-term careers. They specifically work with people within social support systems in finding necessary job training and career counselling. These assistance programs help bring students out of welfare situations and allow them to make more meaningful contributions in their communities.
"I probably have the best job in the world," says Dykstra, "We help people reach their goals. A lot to the time our students are the first ones in their family to attend post-secondary. This can start a tradition of achieving higher education within families. There's nothing more rewarding than a student coming and letting us know their on a better track."
Why Hire a College Career Group Graduate?
The Career College Group aims at providing Canadian businesses with skilled, professional workers. Through extensive hands-on training and career-specific classes, Career College Group graduates can provide top growing companies to meet industry demands.
www.careercollegegroup.com
CBJ July 2019 | Volume 12 | Issue 7
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\section{Introduction}
In this paper we extend the results of
Robert Seeley concerning the complex powers of elliptic pseudodifferential
operators and the residues of their zeta function to operators acting on
vector orbibundles over orbifolds.
We present the theory of pseudodifferential operators acting on a vector
orbibundle over an orbifold, construct the zeta function of an elliptic
pseudodifferential operator and show the existence of a meromorphic extension
to $\mathbb{C}$ with at most simple poles. We give formulas for generalized
densities on the orbifold whose integrals compute the residues of the
zeta function.
\par
Recall that a pseudodifferential operator $A$ of order $d$ acting on a
hermitian vector bundle $E \overset{p}\to M$ over a Riemannian manifold $M$
of dimension $m$
becomes an unbounded operator on the space of $L^2$ sections $L^2(M;E)$.
If $A$ is an elliptic pseudodifferential
operator with spectrum in $(\epsilon , \infty)$ for a
sufficiently small $\epsilon > 0$, by using the functional calculus one can
define the complex powers $A^s$, $s\in \mathbb C$, as
\begin{equation}
A^s=\frac{1}{2\pi i}\int_{\gamma}\lambda^s(\lambda - A)^{-1} \,
d\lambda \qquad \text{when }Re(s) < 0
\end{equation}
(where $\gamma$ is a contour in the complex plane obtained by joining
two parallel half-lines to the negative real axis by a circle around
the origin) and
\begin{equation}
A^s=A^{s-k}A^k \qquad \text{for } Re(s)\ge 0
\end{equation}
for a large enough $k\in \mathbb Z$ that makes $s-k < 0$.
\par
The condition on the spectrum of $A$ can be relaxed; in order to apply
the above construction, it is sufficient for
the spectrum of $A$ to be inside an angle with the vertex the origin and
away from the negative real axis.
\par
In particular one can define the complex powers of a selfadjoint positive
pseudodifferential operator $A$. Seeley showed in his paper \cite{seeley}
that $A^s$ are pseudodifferential operators of complex order $sd$
and gave a local description of their symbols.
For $s$ in the half-plane $Re(s)<\frac{-m}{d}$, the operators $A^s$ are
of trace class. Seeley also showed that the zeta function of $A$ defined as
\begin{equation}
\zeta_A(s)=Tr(A^s)\qquad\text{ for } Re(s) < -\frac{m}{d}
\end{equation}
has a meromorphic extension to the complex plane with at most simple poles
at $\frac{-m+k}{d}$, with $k\in \{0,1,\dots\}$ and residues computable as
integrals
on $M$ of quantities that depend only on the total symbol of the operator $A$.
These results will be referred to as Seeley Theory.
\par
An $m$-dimensional orbifold $M$ is locally the quotient of $\mathbb{R}^m$ by a finite
group of diffeomorphisms $\Gamma$ and globally it is the quotient of a
manifold $\Tilde{M}^{m+q}$ by a compact Lie group $G$ of dimension $q$ with
finite isotropy groups.
The dimension of the Lie group can be a priori quite large, like for
example $\frac{m(m+1)}{2}$.
Similarly, a rank $k$ vector orbibundle $E\overset{p}\to M$ can be represented
locally as the quotient of a rank $k$ vector bundle by a finite group of
bundle diffeomorphisms $\Gamma$, and globally as the quotient of
a $G$ vector bundle of rank $k$,
$\Tilde{E}\overset{\tilde{p}}\to \Tilde{M}$, by a compact Lie group $G$
(cf. Proposition
\ref{gsection} and Theorem \ref{represent} ). We will refer to the local
description of the orbifolds and orbibundles
using the quotient by a finite group $\Gamma$ as {\it Perspective 1} and
to the global description using the quotient by a Lie group $G$ as
{\it Perspective 2}.
\par
From these perspectives the space of smooth sections in
a vector orbibundle can be thought of as the space of invariant smooth
sections in
a genuine vector bundle over a manifold. One defines a pseudodifferential
operator $A$
in a vector orbibundle as an operator acting on the space of invariant
sections in $\Tilde{E}\overset{\tilde{p}}\to \Tilde{M}$ which is the
restriction
of a $\Gamma$ (respectively $G$) equivariant pseudodifferential operator
$\Tilde{A}$
acting on $\Tilde{E}\overset{\tilde{p}}\to \Tilde{M}$. In the case of a finite
group $\Gamma$, the operator $\Tilde{A}$ is unique up to a smoothing
operator, cf. Proposition \ref{unique}. If $dim(G)\ge 1$ then the operator
$\Tilde{A}$ which induces $A$ is far from being unique.
\par
For the definition and the discussion of the basic properties of the
pseudodifferential operators we will use
perspective $1$, while for some global properties and spectral theory of the
elliptic pseudodifferential operators we will use perspective $2$.
Seeley theory is about elliptic operators which is a global theory. However,
to formulate and prove the results concerning the pseudodifferentiability
of the complex powers and the
residues of the zeta function we use the local theory.
Of particular importance are the explicit formulae in coordinate charts
for the residues of the zeta function. This makes clear the need of
perspective $1$.
Nevertheless, the use only of perspective $1$ to extend Seeley theory in the
context of orbibundles and orbifolds will require the repetition of the entire
theory of elliptic pseudodifferential -- a rather tedious and very lengthy
work.
We will considerably short-cut this work by the use of perspective $2$.
However, getting explicit formulae using perspective $2$
would be an almost impossible task
because of the large number of parameters introduced when constructing the
operator $\Tilde{A}$ and because of the
non-unique nature of $\Tilde{A}$.
In fact even the extension of Seeley theory to incorporate the presence of a
compact Lie group is rather complicated and very far from providing explicit
formulas (see \cite{bh}).
\par
We give a brief description of the results in our paper.
\par
In the first section we consider a pseudodifferential operator of classical type
$A$ on the space of smooth sections of a vector bundle $E\overset{p}\to M$
endowed with the a smooth action of a finite group $\Gamma$
such that $A$ commutes with this action and extend Seeley's results
to this case.
This extension, which is of independent interest, is also crucial for the study
of the pseudodifferential operators acting on vector orbibundles.
The results are probably implicit in the existing literature, but
not explicit enough for our needs in connection with the operators acting
in orbibundles.
\par
The $\Gamma$ action on $E\overset{p}\to M$ induces the decomposition:
\begin{equation}
\label{decomp}
C^{\infty}(M;E)=\bigoplus_{i=1}^l C^{\infty}(M;E)_i
\end{equation}
with
$C^{\infty}(M;E)_i\overset{\sim}=V_i \otimes \Hom_{\Gamma}(V_i;
C^{\infty}(M;E))$, where $(V_i, \rho_i)_{i=1}^l$ is a complete set of
irreducible representations of $\Gamma$, with $\rho_1$ being the trivial
$1$-dimensional representation. The $\Gamma$ equivariance of $A$ induces
the decomposition
\begin{equation}
A=\bigoplus_{i=1}^l A_i
\end{equation}
with $A_i:C^{\infty}(M;E)_i \to C^{\infty}(M;E)_i$. This decompositions
will be referred
to as the decomposition along the irreducible representations of $\Gamma$.
We restrict our attention
to an elliptic positive pseudodifferential $A$ of order $d>0$. We are
interested in the traces of the complex powers $(A^s)_i=(A_i)^s$ which
can be recovered from
$Tr(A^s\cdot \mathcal{T}_{\gamma})$ (as prescribed by formula
\eqref{equation:i}), where $\mathcal{T}_{\gamma}$ is the operator
by which $\gamma\in \Gamma$ acts on $C^{\infty}(M;E)$. Although
$\mathcal{T}_{\gamma}$
are not pseudodifferential operators, we showed that the trace functional
$\zeta_{A,\gamma}(s)=Tr(A^s\cdot \mathcal{T}_{\gamma})$, and therefore
$\zeta_{a,i}(s)=Tr(A^s_i)$, defined for $Re(s)<-\frac{m}{d}$ can be extended
to a meromorphic function on $\mathbb{C}$ with at most simple poles.
We construct a sequence of smooth densities $\eta^{\gamma}_k$
on the fixed point sets $M^{\gamma}=\{x\in M\,|\,\gamma x=x\}$ such that
the residue of the meromorphic extension of $\zeta_{A,\gamma}$ at
$s=\frac{-m+k}{d}$, $k\in\{0,1,\dots\}$, is computed as an integral on
$M^{\gamma}$ of the density $\eta^{\gamma}_k$ (see Theorem
\ref{residues:g} and
\ref{residues:i}). Here $m=dim(M)$ and $d=ord(A)$. These densities can
be interpreted as Dirac-type generalized densities on $M$(as in
\cite{gs}, Chap. VI). For $k<(dim(M)-dim(M^{\gamma}))$ we have
$\eta^{\gamma}_k=0$.
As a direct consequence we will show that the trace functionals
$\zeta_{A,i}(s)=Tr(A_i^s)$
can be extended to meromorphic functions with at most simple
poles at $s=\frac{-m+k}{d}$, $k\in\{0,1,\dots\}$ and compute the residues in
Theorem \ref{residues:g}.
Of particular interest is $Tr(A_1^s)$ -- the component
corresponding to the trivial one dimensional representation $\rho_1$, because
the $\Gamma$ equivariant operator $A$ induces an elliptic pseudodifferential
operator acting on sections
of the orbibundle $E/\Gamma\overset{\overline{p}}\to M/\Gamma$ which can
be identified with $A_1$. The trace functional $Tr(A_1^s)$ is the zeta
function of this operator.
\par
In the second section we present the basic definitions and properties
of the orbifolds and orbibundles.
It is a known fact that any
orbifold $M^m$ is the quotient of a smooth manifold $\Tilde{M}^{m+q}$ by a
compact Lie group $G$ of dimension $q$ acting on $\Tilde{M}$ with finite
isotropy groups.
We complete this result by proving that a vector orbibundle
$E\overset{p}\to M$
is the quotient by $G$ of a $G$ vector bundle $\Tilde{E}\overset{\Tilde{p}}
\to \Tilde{M}$ in Theorem \ref{represent}. This will allow us to give a global
characterization for the spaces of smooth sections in orbibundles in
Proposition \ref{gsection}. We also describe
the space of generalized Dirac-type densities on orbifolds and the
canonical stratification associated with an orbifold structure.
\par
In the third section we show how one can extend the theory of
pseudodifferential operators acting on smooth sections in a vector bundle
over a manifold to pseudodifferential operators
acting on smooth sections in a vector orbibundle over an orbifold.
The basic definitions and elementary properties were stated, though with
some gaps, in \cite{GN1} and \cite{GN2}. Next we discuss the complex powers
and the zeta function of such an operator which is selfajoint and elliptic
and extend Seeley theory to this context.
We show that the zeta function is well defined and has a meromorphic
extension to the complex plane with at most simple poles located at
$s=\frac{-m+k}{d}$, $k\in\{0,1,\dots\}$. In Theorem \ref{zeta:Dirac:density},
using the tools presented in the previous sections, we construct a sequence of
generalized Dirac-type densities $\eta_k$, $k\in\{0,1,\dots\}$, on the
base orbifold $M$
and compute the residues of the zeta function as integrals of these densities.
In Theorem \ref{zeta:strata:density} we reinterpret the computations we make
as integrals of genuine smooth densities on the strata of the canonical
stratification of $M$.
\par
The results of this paper provide the analytic foundation for the study of
indices, signatures and torsion for orbifolds equipped with a Riemannian
metric (Riemannian orbifolds), which will provide the topic of a subsequent
paper. We recall the reader that many of the moduli spaces have a canonical
structure of an orbifold (rather that manifold)
and their topological and geometric invariants are of legitimate interests.
\par
Finally, I want to thank D. Burghelea for the help and advice in preparing
this work. In many respects he is the coauthor of this paper.
\section{$\Gamma$ Equivariant Pseudodifferential Operators}
\subsection{$\Gamma$ Vector Bundles}
\par
\begin{defn}
Let $(\Gamma,\cdot)$ be a finite group of order $|\Gamma|$. A real or
complex smooth vector bundle $E\overset{p}\to M$ endowed with a left
smooth $\Gamma$ action by bundle isomorphisms is
called a $\Gamma$ vector bundle.
\end{defn}
\par
For each $\gamma \in \Gamma$ we have a diffeomorphism $t_\gamma : M\to M$
of the base space and a linear isomorphism $(T_\gamma )_{x}: E_x \to
E_{t_\gamma (x)}$ between the fiber above $x$ and the fiber above $t_\gamma
(x)$. As usual, we will denote $t_\gamma (x)$ by $\gamma x$. $\Gamma$
acts on the space of sections $C^{\infty}(M;E)$ by $(\gamma \cdot f)
(x)=( T_\gamma )_{\gamma ^{-1} x}f(\gamma ^{-1} x)$.
We will denote the action of $\gamma $ on the space of sections by
$\mathcal T_\gamma $.
\par
\begin{example}
\label{gammalinear}
A particular case of a $\Gamma_0$ vector bundle is the trivial vector bundle
$U\times V \overset{pr_1}\to U$ endowed with a product $\Gamma_0$ action
$\mu\times \rho$ on $U\times V$ and with the $\Gamma_0$ action $\mu$ on $U$
where $\mu:\Gamma_0\times U \to U$ is the restriction of a linear action $\mu:
\Gamma_0\times\mathbb{R}^m\to \mathbb{R}^m$ to an open invariant neighborhood
of the origin $U\subset \mathbb{R}^m$ and $\rho:\Gamma_0\times V\to V$ is a
linear representation on the vector space $V$.
\end{example}
\par
\begin{example}
\label{gammalinear_ext}
If $\Gamma_0\subset \Gamma$ is an inclusions of groups, then the bundle
$\Gamma\times_{\Gamma_0}(U\times V)\overset{pr_1}\to \Gamma\times_{\Gamma_0}U$
is a $\Gamma$ vector bundle, with $\Gamma$ acting on the total space and
base space by left translations.
\end{example}
\par
The following proposition states that any $\Gamma$ vector bundle is
locally diffeomorphic to a $\Gamma$ vector bundle as above.
\begin{prop}
\label{local}
Let $E\overset{p}\to M$ be a $\Gamma$ vector bundle and $x\in M$. Let
$\Gamma_x$ be the isotropy group of $x$. Then there exists $U$ a $\Gamma_x$
invariant neighborhood of $x$ in $M$ and $O$ a neighborhood of the origin in
$\mathbb{R}^m$ such that the restriction bundle $E_{|\Gamma U}\overset{p}\to
\Gamma U$ is isomorphic to the $\Gamma$ vector bundle $\Gamma\times_{\Gamma_x}
(O\times V)\overset {p_1}\to \Gamma\times_{\Gamma_x}O$ by a $\Gamma$
equivariant isomorphism.
\end{prop}
The proof of this proposition is elementary and rather standard. For the sake
of completeness we included it in the Appendix A.
Let $E\overset{p}\to M$ be a $\Gamma$ vector bundle. Let $(V_1,\rho_1),
(V_2,\rho_2),...,(V_l,\rho_l)$ be a complete set of irreducible
non-isomorphic complex representations of $\Gamma$, with $\rho_1$ being the
trivial one dimensional representation. Then $C^{\infty}(M;E)$
decomposes as the direct sum of multiples of $V_i$
\begin{equation}
C^{\infty}(M;E)=\bigoplus_{i=1}^l C^{\infty}(M;E)_i
\end{equation}
$C^{\infty}(M;E)_i$ is spanned by submodules of the
form $h(V_i)$ where $h: V_i\to C^ {\infty}(M;E)$ are $\Gamma$ equivariant
maps. We have the isomorphism
\begin{equation}
V_i \otimes \Hom_{\Gamma}(V_i; C^{\infty}(M;E))
\overset{e}\to C^{\infty}(M;E)_i
\end{equation}
where $e$ is the evaluation map, $e(v,\phi)=\phi(v)$.
If $M$ is a closed Riemannian manifold endowed with a $\Gamma$ invariant
metric, and we have a $\Gamma$ invariant Hermitian structure on the bundle
$E\overset{p}\to M$, then $\Gamma$ acts on the space of $L^2$ integrable
sections $L^2(M;E)$, and we get an analogous decomposition
\begin{gather}
L^2(M;E)=\bigoplus_{i=1}^{l}L^2(M;E)_i\\
\intertext{with}
L^2(M;E)_i=V_i\otimes \Hom_{\Gamma}(V_i; L^2(M;E))
\end{gather}
We will denote by $pr_i$ the projection on the $i^{th}$ factor $L^2(M;E)_i$
with respect to the above decomposition.
We will need the following statement that describes the projection $pr_i$ in
terms of the action and characters of the group $\Gamma$.
\begin{prop}
\label{pri}
If $\chi_i$, $i=1,\dots,l$ is the complete set of irreducible characters of the
group $\Gamma$ corresponding to the representations $(V_i,\rho_i)$ and $k_i=
dim_{\mathbb{C}}V_i$ are the corresponding dimensions, then the projection on
the $i^{th}$ factor is given by
\begin{equation}
pr_i=\frac {k_i}{|\Gamma|} \sum_{\gamma \in \Gamma} \chi_i(\gamma ^{-1})\cdot
\mathcal{T}_\gamma
\end{equation}
\end{prop}
The proof can be found in \cite{serre}, Part 1, Theorem 8.
\subsection{$\Gamma$ Equivariant Pseudodifferential Operators}
\begin{defn}
A pseudodifferential operator $A$ acting
on the space of sections $C^{\infty}(M;E)$ of a $\Gamma$ vector bundle is
called $\Gamma$ equivariant if $A$ commutes with the action of $\Gamma$ on
$C^{\infty}(M;E)$ i.e.
$\mathcal{T}_\gamma \cdot A= A\cdot \mathcal{T}_\gamma $ for any $\gamma \in
\Gamma$.
\end{defn}
Throughout this chapter we suppose that $A$ is elliptic
$\Gamma$ equivariant pseudodifferential operator of classical type
(as described in \cite{shubin}, Section 3.7), of positive order $d$.
We suppose that $\pi$ is an Agmon angle for $A$, i.e. there exists
$\varepsilon>0$ such that the spectrum of $A$ is disjoint from the region
in the complex plane $\{z\,|\, arg(z)\in(\pi-\varepsilon, \pi+\varepsilon)\}
\cup\{z\,|\,|z|<\varepsilon\}$.
Then $A$ and all its complex powers $A^s$, $s\in\mathbb C$, will
preserve the decompositions $C^{\infty}(M;E)=\bigoplus_{i=1,l}
C^{\infty}(M;E)_i$ and $L^2(M;E)=\bigoplus_{i=1}^{l}L^2(M;E)_i$.
Consequently, we can consider the restrictions $A^s_i:C^{\infty}(M;E)_i\to
C^{\infty}(M;E)_i$ and $A^s_i:L^2(M;E)_i\to L^2(M;E)_i$ for $s\in\mathbb C$.
The goal of this section is the study of the trace of these operators
$\zeta_{A,i}(s)=Tr (A^s_i)$. First observe that $A^s_i=A^s\circ
pr_i$. Using Proposition \ref{pri}, we get
\begin{equation}
\label{equation:i}
A^s_i=\frac {k_i}{|\Gamma|}\sum_{\gamma \in \Gamma}\chi_i(\gamma ^{-1})
A^s\circ\mathcal{T}_\gamma
\end{equation}
so, in order to study the trace of $A^s_i$, it is
convenient to study the trace of $A^s\circ\mathcal{T}_\gamma$. We will
show that these operators are of trace class for all complex numbers $s$ in a
half-plane $Re(s)<-K$, and that the associated trace function,
$Tr(A^s\circ\mathcal{T}_\gamma)$,
as a function of $s$, has a meromorphic extension to $\mathbb{C}$ with at most
simple poles.
\par
Let us fix an element
$\gamma\in \Gamma$ and denote by $ T = T_\gamma $, $t=t_\gamma
$ and $\mathcal{T}=\mathcal{T}_\gamma $
respectively the action of $\gamma $ on the total space, on the base
space and on the space of sections of the bundle. Let $M^{\gamma}$ be
the fixed point set of the
diffeomorphism $t$ and $M^{\gamma}=\bigcup_{i\in I}M^{\gamma}_i$ be the
decomposition in connected components of dimensions respectively $n_i$.
Let $m=dim(M)$.
\par
\begin{thm}
\label{residues:g}
The operators $A^s\circ\mathcal{T}$ are of trace class for $s$ in
the half-plane $Re(s)<\frac {-m}{d}$. The associated zeta function
$\zeta_{A,\gamma}(s)=Tr (A^s\circ\mathcal{T})$, defined for
$Re(s)<\frac {-m}{d}$, has a meromorphic continuation to the whole
complex plane $\mathbb{C}$ with at most simple poles at $\frac{-m+k}{d}$ for
$k\in\{0,1,2,...\}$. One can construct positive numbers $d^{\gamma}_i$ and
densities $\{\eta^{\gamma}_{i,k}\}$ for $k\in\{0,1,2,\dots\}$ on the submanifolds
$M^{\gamma}_i$ such that $\eta^{\gamma}_{i,k}=0$ for $k<m-n^{\gamma}_i$
and the residue of the function $\zeta_{A,\gamma}$ at $s=\frac{-m+k}{d}$ is
equal to
\begin{equation}
\label{res:g}
res_{|s=\frac{-m+k}{d}}\zeta_{A,\gamma}=\sum_{i}d^{\gamma}_i\int_{M^{\gamma}_i}
\eta^{\gamma}_{i,k}
\end{equation}
\end{thm}
\par
The details of the proof of this Theorem are contained in Appendix B.
We will give the description of the coefficients $d^{\gamma}_i$ and the
densities $\eta^{\gamma}_{i,k}$.
\par
For $\gamma\in \Gamma$ and $M^{\gamma}_i$ a connected component of the fixed
point set $M^{\gamma}$, let $x\in M^{\gamma}_i$. Using a $\Gamma$ invariant
metric in the tangent space $T(M)$ one decomposes $T_x(M)=T_x(M^{\gamma}_i)
\oplus T_x(M^{\gamma}_i)^{\perp}$ and the action of $\gamma$ on $T_x(M)$
as $Id\oplus \overline{t}$. Then we define:
\begin{equation}
\label{defn:d.gamma}
d^{\gamma}_i=|det(\overline{t}-Id)|^{-1}.
\end{equation}
The quantity above does not depend on the particular choice of
$x\in M^{\gamma}_i$
and decomposition $T_x(M)=T_x(M^{\gamma}_i)\oplus T_x(M^{\gamma}_i)^{\perp}$.
\par
In order to define the densities $\eta_{i,k}^{\gamma}$ near $x\in M^{\gamma}_i$
we choose a coordinate chart $\phi:(O,0)\to (\phi(O),x)$ in an open
neighborhood $\phi(O)$ of $x\in M$ such that the induced action
of $\Gamma_x$ on $O\subset \mathbb{R}^m$ is given by a linear orthogonal maps.
This can be realized with the help of
a $\Gamma$ invariant metric on $M$ and the associated exponential map at $x$.
Denote the action of $\gamma$ on $O$ by $t$.
Let $(x_1,x_2)$ be coordinates given by this
chart, $x_1\in O^t$--the fixed point set of $t$, and $x_2\in O^{\perp}$.
Let $(\xi_1,\xi_2)$ be the
corresponding coordinates in the cotangent bundle space. Then $t=Id\oplus
\overline{t}$. Observe that $d^{\gamma}_i=|det(\overline{t}-Id)|^{-1}$.
Let $a_s(x_1,x_2,\xi_1,\xi_2)$ be the total symbol of $A^s$ in $O$ and
\begin{equation}
a_s(x_1,x_2,\xi_2,\xi_2)\sim\sum_{k\ge 0}a_{s,k}(x_1,x_2,\xi_1,\xi_2)
\end{equation}
be its asymptotic expansion (as defined in
\cite{shubin}, Section 3.7). The component $a_{s,k}$ is homogeneous in
$(\xi_1,\xi_2)$ of degree of homogeneity $sd-k$.
\par
Let $\overline{a}_{s,k}(x_1,w,\xi_1\xi_2)=a_{s,k}(x_1,(\overline{t}-Id)^{-1}w,
\xi_1,\xi_2)$.
\par
Consider the homogeneous symbol $\Tilde{b}_{s,j}(x_1,\xi_1)$ of degree of
homogeneity $sd-j$ given by
\begin{equation}
\Tilde{b}_{s,j}(x_1,\xi_1)=\sum_{|\alpha|+k=j}\frac{1}{\alpha !}(D^{\alpha}_w
\partial^{\alpha}_{\xi_2}\overline{a}_{s,k})(x_1,0,\xi_1,0)
\end{equation}
Then the density on $O^t$ whose integral computes the residue of
$\zeta_{A,\gamma}$ at $s=\frac{-m+k}{d}$ is given by
\begin{align}
\label{defn:eta.def1}
\eta^{\gamma}_{i,k}(x_1)&=-\frac{1}{d}Tr(\int_{S^{n-1}}\Tilde{b}_{s,n_i-m+k}
(x_1,\overline{\xi})\,\dbar\overline{\xi}\circ T)\,dx_1&&\text{ if }
k\ge m-n_i\\
\label{defn:eta.def2}
\eta^{\gamma}_{i,k}&=0 &&\text{ if } k< m-n_i
\end{align}
Here $n_i=dim(O^t)=dim(M^{\gamma}_i)$ and $T=T_{\gamma,x}$ is the map by
which $\gamma$ acts in the fiber $E_x$ above $x\in M$. The $(n-1)$ form
$\dbar\overline{\xi}$ is the canonical volume form on $S^{n-1}$ induced from
$\mathbb{R}^n$, rescaled by a factor of $(2\pi)^{-n}$. $dx_1$ is the canonical volume
form on $O^t$.
Using Theorem \ref{residues:g} and the formula \eqref{equation:i}
linking the operators $A^s_i:C^{\infty}(M;E)_i\to C^{\infty}(M;E)_i$ and
$A^s\circ\mathcal{T}_{\gamma}$, for $\gamma \in \Gamma$, we can formulate
the following theorem
\begin{thm}
\label{residues:i}
The operator $A^s_i$ is of trace class and the trace functional
\begin{equation}
\zeta_{A,i}(s)=Tr(A^s_i)
\end{equation}
is a holomorphic function in $s$ on the half-plane $Re(s)<-\frac{m}{d}$.
The function $\zeta_{A,i}$ has a meromorphic continuation to the whole
complex plane with at most simple poles situated at $s=\frac{-m+k}{d}$
for $k\in\{0,1,2,\dots\}$. For each $k\in\{0,1,2,\dots\}$ there exist smooth
densities $\eta^{\gamma}_k$ on the fixed point set set $M^{\gamma}$
such that the residue of $\zeta_{A,i}$ at $s=\frac{-m+k}{d}$ is equal to
\begin{equation}
\label{res:i:g}
res_{|s=\frac{-m+k}{d}}\zeta_{A,i}=\frac {k_i}{|\Gamma|}
\sum_{\gamma \in \Gamma}
\chi_i(\gamma ^{-1})\int_{M^{\gamma}}\eta^{\gamma}_k
\end{equation}
The smooth density $\eta^{\gamma}_k$ depends only on a finite number
of terms in the asymptotic expansion of the total symbol of
the operator $A^s$ and the action of $\Gamma$ on a neighborhood
of $M^{\gamma}$.
\end{thm}
\begin{proof}
Formula \eqref{equation:i} implies:
\begin{equation}
\zeta_{A,i}(s)=\frac {k_i}{|\Gamma|}\sum_{\gamma \in \Gamma}\chi_i
(\gamma ^{-1}) \zeta_{A,\gamma}(s)
\end{equation}
The first part of the statement in the theorem follows directly from Theorem
\ref{residues:g}. If we define $\eta^{\gamma}_k=\sum_i d_i^{\gamma}
\eta^{\gamma}_{i,k}$ then \eqref{res:i:g} is a direct consequence of
the equality \eqref{res:g}
\end{proof}
\subsection{Dirac-Type Densities on $\Gamma$ Manifolds}
Let us consider $C^{\infty}(M)$ the space of smooth functions on a compact
manifold $M$ and endow it with the topology of uniform convergence together
with a finite number of derivatives on $M$.
\begin{defn} Let $N$ be a smooth submanifold which is a closed subset of $M$.
A smooth density $\eta$ on $N$ defines a continuous functional on the
space $C^{\infty}(M)$ by
\begin{equation}
\label{pairing}
<\eta, f>=\int_N f_{|N}\eta \qquad \text{ for } f \in C^{\infty}(M)
\end{equation}
We call such a functional a Dirac-type distribution on $M$.
\end{defn}
The singular support of this distribution is equal to $N$
if $dim(N)< dim(M)$ and it is empty if $dim(N)=dim(M)$.
If $N=M$ then a Dirac-type distribution on $M$ is given by smooth density
on $M$. If $N$ is a proper subset of $M$ then a Dirac-type distribution
on $N$ can be described locally by a density that is zero on open sets
disjoint from $N$ and by a smooth density on $N$. This will not be a
smooth density on $M$ anymore, the singular set being exactly $N$. We
call this density a Dirac-type density and denote it with the same letter
as the smooth density on $N$.
If $\phi:M\to M$ is a diffeomorphism and $\eta$ is a Dirac-type density
with singular support $N$, then the push-forward $\phi_*(\eta)$ is a
Dirac-type density associated with the smooth density on $\phi(N)$ which
is the push-forward of the smooth density $\eta$ on $N$ by the
diffeomorphism $\phi_{|N}:N \to\phi(N)$.
We have the following identity
\begin{equation}
<\phi_*(\eta), \phi_*(f)>=<\eta ,f>.
\end{equation}
\begin{defn}
\label{gamma:density}
If $\Gamma\times M \to M$ is a $\Gamma$ smooth manifold, a $\Gamma$
Dirac-type density on $M$ is a collection of Dirac-type densities $\eta=\{
\eta^{\gamma}\}_{\gamma\in\Gamma}$
indexed by the elements of the group $\Gamma$ with singular support
respectively the fixed point sets $M^{\gamma}$ and such that
$\gamma'_*(\eta^{\gamma})=
\eta^{\gamma'\gamma \gamma'{}^{-1}}$ for any $\gamma, \gamma'\in \Gamma$
(observe that, in general, $\gamma'(M^{\gamma})=(M^{\gamma' \gamma
\gamma'{}^{-1}})$).
\end{defn}
\begin{defn} If $\eta=\left\{\eta^{\gamma}\right\}_{\gamma\in\Gamma}$ is a
$\Gamma$ Dirac-type density on a $\Gamma$ manifold $M$ and $\chi=\chi_i$
is the character of an irreducible representation of $\Gamma$ we define
the associated distribution $\eta^{\chi}$ by
\begin{equation}
<\eta^{\chi}, f>=\frac{k_i}{|\Gamma|}\sum_{\gamma \in \Gamma}\chi_i
(\gamma ^{-1})\cdot \int_{M^{\gamma}}f_{|M^{\gamma}} \eta^{\gamma}
\end{equation}
\end{defn}
Using the above definitions, Theorem \ref{residues:i} can be reformulated as
follows:
\begin{thm}
\label{residues:d}
Let $A$ be a $\Gamma$ equivariant
elliptic pseudodifferential operator acting on the smooth sections of a
$\Gamma$ bundles manifold and $A_i$ be the restriction of $A$ to the
component of the space of sections $C^{\infty}(M;E)$ corresponding to
the irreducible representation $(V_i,\rho_i)$ with the character $\chi_i$.
The trace functional
$\zeta_{A,i}(s)=Tr(A_i^s)$ defines a holomorphic function on the half-plane
$Re(s)<\frac{-m}{d}$ which has a meromorphic extension to the whole
complex plane with at most simple poles situated at $s=\frac{-m+k}{d}$, for
$k\in\{0,1,2,\dots\}$.
There exists a family $\{\eta_k\}_{k=0}^{\infty}$ of $\Gamma$ Dirac-type
densities on $M$ with $\eta_k=\{\eta^{\gamma}_k\}_{\gamma\in\Gamma}$
so that the residue of $\zeta_{A,i}$ at $s=\frac{-m+k}{d}$ is equal to
$<\eta^{\chi_i}_k,1>$.
\end{thm}
\begin{proof}
We only have to prove the existence of the representation of the residues
of $\zeta_{A,i}$ as stated in the theorem.
For $k\in\{0,1,2,\dots\}$ and $\gamma\in\Gamma$ let $\eta_k^{\gamma}$ be the
Dirac-type distribution given by the sum of the smooth densities
$\sum_{i}d_i^{\gamma} \eta_{i,k}^{\gamma}$ each defined on the connected
component $M^{\gamma}_i$ of the fixed point set $M^{\gamma}$ as described
in Theorem \ref{residues:g} and by the formulas \eqref{defn:d.gamma},
\eqref{defn:eta.def1} and \eqref{defn:eta.def2}.
We have $\gamma^{-1}(M^{\gamma\gamma'\gamma^{-1}})=M^{\gamma'}$ for any
$\gamma, \gamma'\in\Gamma$, and after a
convenient reindexing of the connected components of each fixed point set, we
can suppose that $\gamma'{}^{-1}(M^{\gamma'\gamma\gamma'{}^{-1}}_i)=
M^{\gamma}_i$ as well. Then $d_i^{\gamma'\gamma\gamma}=d_i^{\gamma}$
because, as defined by formula \eqref{defn:d.gamma}, they are determinants of
two linear maps conjugated by a diffeomorphism defined at the tangent
space level by $\gamma'$. Because the operator $A$ is $\Gamma$ equivariant,
a straightforward computation
shows that the smooth densities $\eta_{i,k}^{\gamma}$ and $\eta_{i,k}^{\gamma'
\gamma\gamma'{}^{-1}}$ are conjugated by the the map induced at the cotangent
space level by $\gamma'$. Then the collection $\{\eta_k^{\gamma}\}_{\gamma
\in\Gamma}$ is a $\Gamma$ Dirac-type distribution, as described in the
definition \ref{gamma:density}.
Theorem \ref{residues:i} states that the residue of $\zeta_{A,i}$ at
$s=\frac{-m+k}{d}$ is equal to
\begin{equation}
\frac{k_i}{|\Gamma|}\sum_{\gamma\in\Gamma}\chi_i(\gamma^{-1})
(\sum_{i}d_i^{\gamma} \int_{M_i^{\gamma}}\eta_{i,k}^{\gamma})
\end{equation}
which can be rewritten as $<\eta^{\chi_i}_k, 1>$.
\end{proof}
\vfill\eject
\section{Orbifolds and Orbibundles}
\subsection{Orbifolds}
Let $M$ be a Hausdorff topological space.
\begin{defn}
An orbifold chart on $M$ is
given by $\mathcal{R}=(\Tilde U, \Gamma, \mu, U, \pi)$
where $U$ is an open set in $M$, $\Tilde U$ is an open set in the
$n$-dimensional
Euclidean space $\mathbb{R}^n$, $\Gamma$ is a finite group, $\mu:\Gamma\times
\Tilde{U}\to\Tilde{U}$ is a faithful smooth action of $\Gamma$ on $\Tilde{U}$
and $\pi :\Tilde U \to U$
is a continuous map that factors through a homeomorphism
$\overline\pi$ from the orbit space $\Tilde U /\Gamma$ to $U$ and
makes the following diagram commutative:
$$
\xymatrix{
\Tilde{U} \ar[rr]^{\pi} \ar @{->>}[dr]& &U\\
&\Tilde{U}/\Gamma\ar @{-->}[ur]^{\overline\pi}_{\sim}&}
$$
The open set $U$ is called a coordinate neighborhood.
\end{defn}
If the group action $\mu$ is the restriction of a linear representation of
$\Gamma$ on $\mathbb{R}^n$ to a neighborhood of the origin $\Tilde{U}$ and
$x=\pi(0)$, we call $\mathcal{R}$ a linear chart at $x$.
\begin{defn} Two orbifold charts $\mathcal{R}_i=(\Tilde U_i, \Gamma_i, \mu_i,
U_i, \pi_i)$, $i=1,2$ are called compatible if for any two points
$\tilde x_i \in \Tilde U_i$ such that $\pi_1(\tilde x_1)=\pi_2(\tilde x_2)=x
\in U_1\cap
U_2$ there exists a diffeomorphisms $h$ from a neighborhood of $x_1$ in
$\Tilde{U}_1$ to a neighborhood of $x_2$ in $\Tilde{U}_2$ such that
$\pi_2\circ h=\pi_1$.
\end{defn}
\begin{remark}
\label{linchart}
If $\mathcal{R}=(\Tilde U, \Gamma,\mu,U, \pi)$ is an orbifold
chart and $x\in U$, there exist a linear orbifold chart
$\mathcal{R}_x$ at $x$ which is compatible with $\mathcal{R}$. The proof follows
the same idea as the one in the proof of Proposition \ref{local},
contained in Appendix A. Let $\tilde x \in \Tilde{U}$
such that $\pi(\tilde x)=x$ and $\Gamma_{\tilde x} \subset\Gamma$ be
the isotropy group of $\tilde x$.
Let $\Tilde{V}$ be a $\Gamma_{\tilde x}$ invariant neighborhood of
$\tilde x$ such that $\Tilde{V}\cap \gamma\cdot\Tilde{V} = \emptyset$
for any $\gamma \in \Gamma \backslash \Gamma_{\tilde x}$. Then
$(\Tilde{V}, \Gamma_{\tilde x}, \mu, V=\pi(\Tilde{V}), \pi_{|\Tilde{V}})$
is an orbifold chart which is compatible with $\mathcal{R}$.
The linearization of the action of $\Gamma_{\tilde x}$ on $\Tilde{V}$
gives us an action $\mu'$ of $\Gamma_{\tilde x}$ on $T_{\tilde x}(\Tilde{V})$,
the tangent space of
$\Tilde{V}$ at $\tilde x$ . The exponential map $exp:T_{\tilde x}(\Tilde{V})
\to \Tilde{V}$ associated with a $\Gamma_{\tilde x}$
invariant metric on $\Tilde{V}$ is $\Gamma_{\tilde x}$ equivariant.
One can choose a smaller $\Tilde{V}$ so that
$exp$ is an equivariant diffeomorphism between a neighborhood
$\Tilde{W}$ of $0$ in $T_{\tilde x}(\Tilde{V})$ and $\Tilde{V}$. Then
$(\Tilde{W}, \Gamma_x, \mu', V, \pi_{|\Tilde{V}}\circ exp)$
is an linear orbifold chart at $x$ which is compatible with $\mathcal{R}$.
\end{remark}
\begin{defn} An orbifold atlas $\mathcal{A}$ on $M$ is a collection of
compatible orbifold charts on $M$ such that the corresponding coordinate
neighborhoods form an open cover of $M$.
Two orbifold atlases $\mathcal{A}_i,\;i=1,2$ on $M$ are compatible if their
reunion is an orbifold atlas of $M$.
\end{defn}
\begin{defn} An orbifold structure on $M$ is given by a maximal orbifold
atlas $\mathcal{A}$ on $M$.
\end{defn}
Though an orbifold is given by a topological space and a maximal atlas,
in the future we will drop the atlas from the notation of an orbifold and use
only the letter designated for the underlying topological space. In order to
keep our notations simple we will also drop the representation $\mu$
from the notation of an orbifold chart.
\begin{example}
If $W$ is an open subset of $M$, then $W$ inherits an orbifold
structure whose charts are all orbifold charts $(\Tilde{U}, \Gamma, U,
\pi)$ of $M$ for which $U\subset W$.
\end{example}
\begin{example}
\label{cano}
Let $M$ be a differentiable manifold and $\Gamma$ a
finite group of diffeomorphisms of $M$. Let $\pi$ be the projection map
onto the orbit space $M/\Gamma$. Then
$M/\Gamma$ has a canonical orbifold structure (cf. \cite{haef}). An atlas of
$M/\Gamma$ can be obtained as follows:
let $x\in M$ and $\overline{x}\in M/\Gamma$ its orbit. Let $\Gamma_x
\subset\Gamma$ be the isotropy group of $x$. Because $\Gamma$ is
finite, there exists an open neighborhood $\Tilde{U}$ of $x$ in $M$
which is $\Gamma_x$ invariant and $\Tilde{U}\cap \gamma\cdot\Tilde{U}
= \emptyset$ for $\gamma \in \Gamma\backslash \Gamma_x$. The set
$U=\Tilde{U}/\Gamma_x \subset M/\Gamma$ is an open neighborhood of
$\overline{x}$. We call a neighborhood like $\Tilde{U}$ a slice at $x$. Choose $\Tilde{U}$
small enough such that $\Tilde{U}\overset{\phi}
\to O\subset \mathbb{R}^n$ is a smooth chart for $M$. The action of the group
$\Gamma$ can be transported to a smooth action on $O$ via $\phi$
and the collection $(O, \Gamma_x, U, \pi\circ\phi^{-1})$ is an
orbifold chart of $M/\Gamma$. All the charts obtained as above are compatible
(cf. \cite{haef}) and the maximal atlas containing them defines the
canonical orbifold structure on $M/\Gamma$.
\end{example}
We have a generalization of the previous example:
\begin{prop}
\label{liecano}
Let $M$ be a differentiable manifold and $\mu:G\times M \to M$ a smooth action
of a compact Lie group $G$ with finite isotropy groups $G_x\subset G$ for any
$x\in M$. Then the quotient topological space $M/G$ has a canonical structure
of an orbifold.
\end{prop}
\begin{proof}
Let us fix a $G$ invariant metric on $M$. This can be done by averaging any
metric over the compact group $G$.
Let $\overline{x}\in M/G$ be a point in the quotient space and $x\in M$ such
$\overline{x}=Gx$. Let $G_x$ be the isotropy group at $x$. Then $G_x$ acts on the
tangent space $T_x(M)$ and keeps invariant the tangent space $T_x(Gx)$ to the
$G$-orbit of $x$. Let $V\subset T_x(M)$ be the orthogonal complement of $T_x(Gx)$
in $T_x(M)$, $V\oplus T_x(Gx)=T_x(M)$. Because the metric on $M$ is $G$
invariant, $G_x$ will act on $V$ by restriction. Also, all its translations
$gV\subset T_{gx}(M)$ will be $G_{gx}$ invariant and $gV\oplus T_{gx}(Gx)=
T_{gx}(M)$.
Let $T(M)_{|Gx}\to Gx$ be the restriction of the tangent vector bundle to the orbit
$Gx$, and $\mathcal{V}\to Gx$ be the subbundle whose fiber above $gx$ is
$gV$. This subbundle has a natural $G$ vector bundle structure coming from
the action of $G$ on $M$.
Let us consider the principal bundle $G_x\hookrightarrow G\to G/G_x$, where
$G_x$ acts by right translations on $G$, and the associated vector bundle
$V\hookrightarrow G\times_{G_x}V \to G/G_x$. This vector bundle has a natural
$G$ vector bundle structure, $G$ acting by left translations on $G$ and
$G/G_x$.
The maps $\Phi: G\times_{G_x} V \to \mathcal{V}$, $\Phi(g,v)=gv$ and
$\phi:G/G_x \to Gx$, $\phi(gG_x)=gx$ give a $G$-equivariant
isomorphism $(\Phi,\phi)$ between the two $G$ vector bundles considered above.
If $exp:T(M) \to M$ is the exponential map associated with the $G$
invariant metric
on $M$, then $exp$ realizes a $G$ equivariant diffeomorphism between a
neighborhood $\mathcal{U}$ of the zero section in $\mathcal{V}$ and an
open tubular neighborhood
$N$ of the orbit $Gx$ in $M$. Because $G$ is compact one can find an open
$G_x$ invariant neighborhood $U$ of the origin in $V$ such that
$exp\circ \Phi: G\times_{G_x} U\to N$ is a $G$ equivariant diffeomorphism.
We will then take $\mathcal{U}=\Phi(G\times_{G_x} U)$.
Passing to the $G$-orbit spaces, we get a homeomorphism $\overline
{exp\circ \Phi}:(G\times_{G_x}U)/G \to N/G$, where $N/G$ is an open
neighborhood of $\overline{x}=Gx$ in $M/G$. We will construct an orbifold chart
over $N/G$ around $\overline{x}$.
The map $\iota: U\to G\times_{G_x}\!
U$, $\iota(u)=(e,u)$ gives a homeomorphism when passing to the orbit spaces
$\overline{\iota}: U/G_x \overset{\sim}\to (G\times_{G_x}\!U)/G$.
Denote by $\pi$ the composition $U\overset{\iota}\to G\times_{G_x}U \overset{
exp\circ\Phi}\longrightarrow N \overset{proj}\longrightarrow N/G$. Then
$(U, G_x, N/G, \pi)$ is a linear orbifold chart at $\overline{x}$, where the
action of $G_x$ on $U\subset V$ is the restriction of the linear
representation of $G_x$ on $V$.
As shown above, the induced map to the orbit spaces $\overline{\pi}:
U/G_x \to N/G$ is a homeomorphism.
We have to show that any two different charts defined as above are compatible.
Let $\mathcal{R}_i=(U_i, G_{x_i}, N_i/G, \pi_i)$ be two orbifold
charts around $\overline{x}_i$, $i=1,2$. Let $u_i\in U_i$ such that
$\pi_1(u_1)=\pi_2(u_2)=\overline{x}
\in M/G$. Then, using the definition of $\pi$ in the construction of the
charts $\mathcal{R}_i$, one can find $g_i\in G$, $i=1,2$ and $x\in M$,
$Gx=\overline{x}$, so that we have $exp_{g_1 x_1}(g_1 u_1)=exp_{g_2 x_2}
(g_2 u_2)=x\in M$. By replacing $x$ with $g_1^{-1}x$ we can assume that $g_1=e\in G$.
Moreover, the map $G\times_{G_{x_i}}\!U_i \ni (g,u)\mapsto
exp_{gx_i}(gu)\in M$ is a $G$ equivariant local diffeomorphism, so one gets
a local $G$ equivariant diffeomorphism $\Psi$ between a neighborhood
of $(e,u_1)$ in $G\times_{G_{x_1}}\!U_1$ and a neighborhood of
$(g_2,u_2)$ in $G\times_{G_{x_2}}\!U_2$. The tangent space to $G\times_{G_
{x_i}}\!U_i$ at $(g_i,u_i)$ is equal to $T_{g_i}(G)\oplus T_{u_i}(U_i)$
and the derivative of $\Psi$ will have a block decomposition corresponding
to this direct sum as $T_{(g_1,u_1)}(\Psi)=\left[\begin{smallmatrix}
A&B\\C&D \end{smallmatrix}\right]$. Because $G$ acts by left translations
on the first component of $G\times_{G_{x_i}}\!U_i$ and $\Psi$ is $G$
equivariant, a straightforward computation shows that $A=Id$ and $C=0$,
so $D$ is a diffeomorphism between $T_{u_1}(U_1)$ and $T_{u_2}(U_2)$.
Then there exist neighborhoods $U'_i\subset U_i$ of $u_i$ such that
the map $U'_1\ni u \mapsto pr_2\circ\Psi(e,u) \in U'_2$ is a
diffeomorphism, whose derivative at $u_1$ is $D$. Denote this map by $h$.
The horizontal dotted lines in the following diagram represent locally defined
maps that are local diffeomorphisms:
\begin{equation}
\xymatrix{
&G\times_{G_1}\!U_1 \ar@{-->}[rr]^{\Psi}_{\sim}\ar[rd]_{exp\circ\Phi_1}
&&G\times_{G_2}\!U_2 \ar[ld]^{exp\circ\Phi_2}\ar@{-->}[rddd]^{pr_2}&\\
&&M\ar@{->>}[d]^{proj}&&\\
&&M/G&&\\
U_1\ar[uuur]^{\iota_1} \ar[urr]_{\pi_1} \ar@{-->}[rrrr]^{h}_{\sim}&&&&
U_2\ar[llu]_{\pi_2}
}
\end{equation}
The map $pr_2$ is locally defined in a neighborhood of $(g_2,u_2)\in G\times_{G_2}\!U_2$
as $pr_2(g,u)=u$.
Using the fact that the upper triangle, the left and right quadrilaterals and
the large trapezoid are commutative, we get that
$\pi_1=\pi_2\circ h$ on a neighborhood near $u_1$. Because $u_1$ was chosen
arbitrarily, we conclude that the orbifold charts $\mathcal{R}_1$ and $\mathcal
{R}_2$ are compatible.
\end{proof}
Later we will show that any connected orbifold can be obtained as the orbit space
of a smooth manifold endowed with a smooth action of a compact Lie group.
\begin{remark}
If $M$ is an orbifold and $x\in M$ consider an orbifold chart
$(\Tilde{U}, \Gamma, U, \pi)$ in a neighborhood of $x$
and $\Tilde{x}\in \Tilde{U}$ such that $\pi(\Tilde{x})=x$. The isomorphism class
of the isotropy group $\Gamma_{\Tilde{x}}$ depends only on $x$ and not
on a particular choice of $\Tilde{x}$ or chart around $x$.
We will denote this isomorphism class by $\mathcal{G}_x$.
\end{remark}
\begin{defn} A point $x\in M$ is called smooth if $\mathcal{G}_x$ is the isomorphism
class of the trivial group $(1)$ and it is called singular otherwise.
\end{defn}
We will denote by $M_{reg}$ the set of regular points of $M$ and by $M_{sing}$ the set of
singular points. $M_{reg}$ is an open and dense subset of $M$ whose induced orbifold
structure is a genuine manifold structure.
\begin{defn} Let $M$ and $N$ be two orbifolds. A map $\phi:M\to N$ is
an orbifold diffeomorphism if $\phi$ is a homeomorphism and for any
orbifold chart $\mathcal{R}=(\Tilde{U}, \Gamma, U, \pi)$ of $M$, the
collection $\phi(\mathcal{R})=(\Tilde{U}, \Gamma,\phi(U),
\phi\circ\pi)$ is an
orbifold chart of $N$ and for any orbifold chart $\mathcal{R}'=(\Tilde{U}',
\Gamma', U', \pi')$ of $N$, $\phi^{-1}(\mathcal{R}')=(\Tilde{U'},
\Gamma', \phi^{-1}(U'), \phi^{-1}\pi')$ is an orbifold chart of $M$.
\end{defn}
To see whether a homeomorphism $\phi$ is a diffeomorphisms one must
check that $\phi$ and $\phi^{-1}$ take orbifold charts of an atlas (not
necessarily the {\it maximal} atlas) into orbifold charts.
\begin{example} If $M$ is an orbifold and $(\Tilde{U}, \Gamma, U, \pi)$ is
an orbifold chart, then $\Tilde{U}/\Gamma$ has a canonical orbifold
structure as described in Example \ref{cano} and $U$ has an induced orbifold
structure from the orbifold structure of $M$. Then the induced map
$\overline{\pi}:\Tilde{U}/\Gamma \to U$ is an orbifold diffeomorphism.
\end{example}
\begin{defn} Let $M$ and $N$ be two orbifolds. A continuous map $f:M\to N$ is
smooth if for any $x \in M$ one can find orbifold charts $\mathcal{R}=
(\Tilde{U}, \Gamma, U, \pi)$ around $x$ and $\mathcal{R}'=(\Tilde{U}',
\Gamma', U',\pi')$ around $f(x)$ and a smooth map $\Tilde{f}:\Tilde{U}\to
\Tilde{U}'$ such that the following diagram is commutative:
\begin{equation}
\xymatrix{
\Tilde{U}\ar[r]^{\Tilde{f}}\ar[d]_{\pi}&\Tilde{U}'\ar[d]^{\pi'}\\
U\ar[r]^{f} &U'
}
\end{equation}
\end{defn}
\begin{remark} The spaces $\mathbb{F}^k$ with $\mathbb{F}=\mathbb{R}$ or
$\mathbb{C}$ have a canonical structure of an orbifold, given by the atlas
with a unique chart $\mathcal{R}=(\mathbb{F}^k, \Gamma=(e), \mathbb
{F}^k, \pi=Id)$. Then a continuous map $f:M\to \mathbb{F}^k$ is called
smooth if it is smooth as a map between orbifolds.
\end{remark}
\begin{remark}
\label{glue}
If $M_i$, $i=1,2$ are two orbifolds, $U_i\subset M_i$ are open subsets
and $\phi:U_1\overset{\sim}\to U_2$ is an orbifold diffeomorphism, then
the topological space $M_1\amalg M_2/\sim$ where we identify $x\in U_1$
with $\phi(x)\in U_2$ has a canonical structure of an orbifold, and
the inclusion maps $\iota_i:M_i\to M_1\amalg M_2/\sim$ are smooth.
An atlas for $M_1\amalg M_2/\sim$ is given by the reunion of two atlases
for respectively $M_1$ and $M_2$.
This procedure allows us to create an orbifold by gluing orbifold charts
along orbifold diffeomorphisms.
\end{remark}
We will need the following statement, whose proof can be found in \cite{GN1},
Theorem 2.1.
\begin{prop}
Let $M$ be a smooth orbifold and $\{U_{\alpha}\}$ an open cover
of $M$. Then there exists a countable partition of unity $\{h_i,\;i\in
\mathbb{N}\}$ subordinated to $\{U_{\alpha}\}$ such that each $h_i$ is a
smooth function with compact support.
\end{prop}
\subsection{Vector Orbibundles}
Let $p:E\to M$ be a continuous map between two orbifolds. Let $V$ be a fixed
vector space over $\mathbb{R}$ or $\mathbb{C}$.
\begin{defn} A vector orbibundle chart is a collection
$\mathcal{R}=(\Tilde{U}, V, \Gamma,U, \Pi, \pi)$
such that $(\Tilde U, \Gamma, U, \pi)$ is an orbifold chart for $M$,
$(\Tilde U\times V, \Gamma, p^{-1}(U), \Pi)$
is an orbifold chart for $E$ such that the induced action of $\Gamma$ on
the trivial vector bundle $(\Tilde U\times V\overset{pr_1}\to \Tilde{U})$
is by vector bundle isomorphisms, and the following diagram is commutative:
$$
\xymatrix{
\Tilde{U}\times V \ar[r]^{\Pi} \ar[d]_{pr_1}& p^{-1}(U)
\ar[d]^{p}\\
\Tilde{U} \ar[r]^{\pi}& U}
$$
\end{defn}
Though every vector orbibundle chart comes with the actions of a group $\Gamma$ on
$\Tilde{U}$ and $\Tilde{U}\times V$, for the sake of simplicity we will not
show them explicitly in the notation of the chart.
\begin{defn}
A vector orbibundle chart $\mathcal{R}=(\Tilde{U}, V, \Gamma, U, \Pi, \pi)$
is called a linear chart at $x\in M$ if $(\Tilde U, \Gamma, U, \pi)$ is a
linear orbifold chart at $x\in M$ with $\mu:\Gamma\times \Tilde{U}\to \Tilde{U}$
the restriction of a linear representation of $\Gamma$ and there exists a linear
representation $\rho:\Gamma\times V\to V$ such that the action of $\Gamma$ on the trivial
bundle $\Tilde{U}\times V \overset{pr_1}\to \Tilde{U}$ is equal to the
diagonal action $\mu\otimes\rho$.
\end{defn}
\begin{defn} Two vector orbibundle charts $\mathcal{R}_i=
(\Tilde U_i, V, \Gamma_i, U_i, \Pi_i, \pi_i)$,
$i=1,2$, are compatible if for any two points $\tilde x_i \in \Tilde U_i$
such that $\pi(\tilde x_1)=\pi(\tilde x_2)=x \in U_1\cap U_2$ there exists a
neighborhood $\Tilde W_1$ of $\tilde x_1$ in $\Tilde U_1$, a neighborhood
$\Tilde W_2$ of $\tilde x_2$ in $\Tilde U_2$ and a vector bundle
diffeomorphism $(H,h)$ between $(\Tilde W_1\times V \overset{pr_1}\to
\Tilde W_1)$ and
$(\Tilde W_2\times V \overset{pr_1}\to \Tilde W_2)$ such that
$\pi_2\circ h=\pi_1$ and $\Pi_2 \circ H = \Pi_1$.
\end{defn}
\begin{defn} A vector orbibundle atlas on $E\overset{p}\to M$ is a collection
of compatible vector orbibundle charts such that the corresponding coordinate
neighborhoods of $M$ form an open cover of $M$.
A structure of vector orbibundle on $E\overset{p}\to M$ is given by a maximal
vector orbibundle atlas.
\end{defn}
Using the ideas in the proof of Proposition \ref{local} and Remark
\ref{linchart}, one can prove that for any vector orbibundle chart
$\mathcal{R}= (\Tilde{U}, V, \Gamma, U, \Pi, \pi)$ and $x\in U$ there exist a
linear vector orbibundle chart at $x$, $\mathcal{R}_x$, which is compatible
with $\mathcal{R}$. As a consequence, any vector orbibundle has an atlas
consisting of linear vector orbibundle charts.
\begin{remark} A vector orbibundle $E\overset{p}\to M$ is usually not a
vector bundle.
If $x\in M_{\text{sing}}$ is a singular point, then $p^{-1}(x)$
might not have a vector space structure. If $(\Tilde U, V, \Gamma,
U, \Pi, \pi)$ is a chart around $x$ then $p^{-1}(x)$ is the
quotient of $V$ by the action of the isotropy group $\Gamma_{\Tilde{x}}$
with $\pi(\Tilde{x})=x$. The isomorphism
class of the representation of $\Gamma_{\Tilde{x}} \in \mathcal{G}_x$ on $V$ depends
only on $x$ and not on a particular choice of vector orbibundle chart and $\Tilde{x}$.
Denote this isomorphism class by $\mathcal{V}_x$. Then the restriction of
$E\overset{p}\to M$ to $\{x\in M\,|\, \mathcal{V}_x \text{ is trivial }\}$ is a
genuine vector bundle.
\end{remark}
\begin{example}
\label{vcano}
Let $E\overset{p}\to M$ be a smooth vector bundle
and suppose the finite group $\Gamma$ acts on the vector bundle by bundle
diffeomorphisms. Denote by $\Pi$, resp. $\pi$, the canonical projections
onto the orbit spaces $E\overset{\Pi}\to E/\Gamma$ and $M\overset{\pi} \to
M/\Gamma$. For $x\in M$, let $\overline{x}=\Gamma x\in M/\Gamma$ be its orbit and
$\Gamma_x$ the isotropy group of $x$. Choose $\Tilde{U}$ a $\Gamma_x$ invariant
neighborhood of $x$ in $M$, as we did in the Example \ref{cano}, such
that $\Tilde{U}\cap\gamma\Tilde{U}=\emptyset$ for $\gamma\in\Gamma\backslash
\Gamma_x$. The group $\Gamma_x$ acts by diffeomorphisms on the restriction
of the initial bundle $E_{|\Tilde{U}}=p^{-1}(\Tilde{U})\overset{p}\to\Tilde{U}$
and Proposition \ref{local} provides us with a linear vector orbibundle chart at
$x$, $\mathcal{R}_x=(O, E_x, \Gamma_x, \Tilde{U}/\Gamma_x, \Pi,\pi)$.
\end{example}
We also have a generalization of the above example, analogous to Proposition
\ref{liecano}
\begin{prop}
\label{lievcano}
Let $E\overset{p}\to M$ be a smooth vector bundle endowed with a
smooth action of a compact Lie group $G$ such that any $x\in M$ has a finite
isotropy group $G_x\subset G$. Then
$E/G \overset{\overline{p}}\to M/G$ has a canonical structure of a vector
orbibundle.
\end{prop}
\begin{proof}
Let us fix a $G$ invariant Riemannian metric on the base space and a
$G$ invariant linear connection $\nabla$ in the vector bundle.
For a fixed $\overline{x}=Gx\in M/G$ with $x\in M$, consider, as described in
the proof of Proposition \ref{liecano}, the isotropy group $G_x$ and the direct sum
decomposition of the tangent space at $x$ as $G_x$ modules $V\oplus T_x(Gx)=
T_x(M)$. Denote by $\mu_x:G_x\times V\to V$ the action of $G_x$ on the vector
space $V$. Let $\mathcal{V}\to Gx$ be the restriction to $Gx$ of the subbundle
of the tangent bundle to $M$
whose fiber above $gx$ is $gV$. The map $\Phi:G\times_{G_x}\!V\to \mathcal{V}$
, $\Phi(g,v)=gv$ is $G$ equivariant and realizes an isomorphism between the
vector bundles $G\times_{G_x}\!V \to G/G_x$ and $\mathcal{V}\to Gx$. Let
$U\subset V$ be an open $G_x$ invariant neighborhood of the
origin such that the map $G\times_{G_x}\! U\ni(g,u) \mapsto exp\circ\Phi(g,u)=
exp_{gx}(gu)\in M$ is a $G$ equivariant map and realizes a diffeomorphism onto
its image $N$. We
showed in the proof of Proposition \ref{liecano} that $(U, G_x, N/G, \pi)$ is a
linear orbifold chart at $\overline{x}$
for $M/G$. We will construct a vector orbibundle chart for $E/G \overset
{\overline{p}}\to M/G$ at $\overline{x}$.
Let $\Tilde{\mathcal{V}}\overset{\Tilde{p}}\to \mathcal{V}$ be the
pull-back of the vector bundle $E_{|Gx}\overset{p}\to Gx$ via the natural
projection map $T(M)_{|Gx}\overset{proj}\longrightarrow Gx$ restricted to
$\mathcal{V}$.
The vector bundle
$\Tilde{\mathcal{V}}\overset{\Tilde{p}}\to\mathcal{V}$ has a
$G$ vector bundle structure coming from the action of $G$ on $E$ and $M$;
indeed \linebreak
$\Tilde{\mathcal{V}}=\{(v,Y)| v\in \mathcal{V},Y\in E_{|Gx}\text{ with }
proj(v)=p(Y)\in Gx\}$ and $g\in G$ acts by the diagonal action
$g\cdot(v,Y)=(gv,gY)$.
The group $G_x$ fixes the point $x$, so it acts on the fiber above $x$ by
a linear representation $\rho_x:G_x\times E_x\to E_x$. Let $G_x$ act on
$V\times E_x$ by the product action $\mu_x\otimes\rho_x$.
Consider the
vector bundle $G\times_{G_x}\!(V\times E_x)\overset{pr_1}\to G\times_{G_x}\!
V$ with fiber $E_x$. The group $G$ acts on this bundle by left translations.
We already showed that $\Phi:G\times_{G_x}\!V\to\mathcal{V}$ given by
$\Phi(g,v)=gv \in \mathcal{V}_{gx}$ is a $G$ equivariant diffeomorphism.
Let $\Tilde{\Phi}:G\times_{G_x}\!(V\times E_x)\to\Tilde{\mathcal{V}}$
defined as $\Tilde{\Phi}(g,v,Y)=(gv,gY)$. Then we have the following
commutative diagram in which the horizontal maps are $G$ equivariant
diffeomorphisms
\begin{equation}
\xymatrix{
G\times_{G_x}\!(V\times E_x)\ar[rr]^{\Tilde{\Phi}}_{\sim}\ar[d]_{pr_1} &&
\Tilde{\mathcal{V}}\ar[d]^{\Tilde{p}}\\
G\times_{G_x}\!V\ar[rr]^{\Phi}_{\sim} && \mathcal{V}}
\end{equation}
Indeed $\Tilde{p}\circ\Tilde{\Phi}(g,v,Y)=\Tilde{p}(gv,gY)=gv=
\Phi\circ pr_1(g,v,Y)=\Phi(g,v)$. $\Tilde{\Phi}$ is surjective because
any $(v',Y')\in \Tilde{\mathcal{V}}$ with $proj(v')=p(Y)=gx\in Gx$
is of the form $\Tilde{\Phi}(g, g^{-1}v', g^{-1}Y')$. Also, if
$\Tilde{\Phi}(g,v,Y)=\Tilde{\Phi}(g',v',Y')$ then $proj(gv)=proj(g'v')
\in Gx$, so $g^{-1}g'=g''\in G_x$ and $g'=gg''$. But $gv=g'v'=gg''v'$ and
$gY=g'Y'=gg''Y'$ so $v=g''v'$, $Y=g''Y'$ with $g''\in G_x$, $gg''=g'$.
We conclude that $(g,v,Y)=(g',v',Y')\in G\times_{G_x}(V\times E_x)$, so
$\Tilde{\Phi}$ is injective as well. The pair $(\Tilde{\Phi}, \Phi)$
defines a $G$ equivariant isomorphism of $G$ vector bundles.
In the proof of Proposition \ref{liecano} we considered the exponential map
$exp:\mathcal{V}\to M$ with respect to the $G$ invariant metric on $M$,
which realizes a $G$ equivariant diffeomorphism
between a neighborhood $\mathcal{U}$ of the zero section in $\mathcal{V}\to
Gx$ and $N$--a tubular neighborhood of $Gx$ in $M$.
For $v\in gV\subset T_{gx}(M)$ and $X\in E_{gx}$, let $s(t)$, $t\in [0,1]$
be the path that realizes the parallel transport in $E\overset{p}\to M$
with respect to the $G$ invariant connection $\nabla$ above the path
$exp_{gx}(tv)\in M$, with $s(0)=X$. Then we define $\widetilde{exp}(v,X)=s(1)$,
$\widetilde{exp}:\Tilde{\mathcal{V}}\to E$. The map $\widetilde{exp}$ is
$G$ equivariant and $\widetilde{exp}(v,\cdot)$ is a linear isomorphism.
Then the restriction of $\widetilde{exp}$ to ${\Tilde{p}}^{-1}(\mathcal{U})
\subset \Tilde{\mathcal{V}}$ together with the restriction of $exp$ to
$\mathcal{U}$ give us a $G$ equivariant isomorphism between the restriction
of the bundle $\Tilde{\mathcal{V}}\overset{\Tilde{p}}\to\mathcal{V}$ to
$\mathcal{U}$ and the $G$ vector bundle $E_{|N}\overset{p}\to N$.
If we choose a small enough $G_x$ equivariant neighborhood $U$ of the
origin in $V$ as in Proposition \ref{liecano}, the pair of maps $(\widetilde{exp},
exp)\circ (\Tilde{\Phi}, \Phi)$ gives us a $G$ equivariant isomorphism
between the $G$ vector bundles $G\times_{G_x}\!(U\times E_x)\overset{pr_1}
\to G\times_{G_x}\!U$ and $E_{|N}\overset{p} \to N$.
Passing to the $G$ orbits we get a homeomorphism between $G\times_{G_x}\!(
U\times E_x)/G=(U\times E_x)/G_x\overset{\overline{pr}_1}\to
G\times_{G_x}\!U/G=U/G_x$ and $E_{|N}/G=\overline{p}^{-1}(N/G)
\overset{\overline{p}}\to N/G$.
We will describe a linear vector orbibundle chart around $\overline{x}=
Gx\in M/G$. Let $\iota:U\to G\times_{G_x}U$, $\iota(u)=(e,u)$ and $\pi$ be
the composition $U\overset{\iota}\to G\times_{G_x}\!U\overset{exp\circ \Phi}
\longrightarrow N \to N/G$. Also let $I:U\times E_x\to G\times_{G_x}
(U\times E_x)$, $I(u,X)=(e,u,X)$ and $\Pi$ be the composition
$U\times E_x\overset{I}\to G\times_{G_x}\!(U\times E_x) \overset{\widetilde{exp}
\circ \Tilde{\Phi}}\longrightarrow E_{|N}\overset{proj}\longrightarrow E_{|N}/G$.
Then $(U, E_x, G_x, N/G, \Pi, \pi)$ is a linear vector orbibundle chart around
$Gx$.
As in Proposition \ref{liecano}, we have to prove that any two vector
orbibundle
charts defined above are compatible.
Let $\mathcal{R}_i=(U_i, E_{x_i}, G_{x_i}, N_i/G, \Pi_i, \pi_i)$, $i=1,2$
be two
vector orbibundle charts around $\overline{x}_i$. Let $u_i\in U_i$ such that
$\pi_1(u_1)=\pi_2(u_2)=\overline{x}\in M/G$. As described in Proposition
\ref{liecano}, we can find $g_2\in G$ and $x\in M$ so that
$exp_{x_1}(u_1)=exp_{g_2 x_2}(g_2 u_2)=x$ and $Gx=\overline{x}$.
The maps $G\times_{G_{x_i}}\!(U_i\times E_{x_i})\ni(g,u,X)\mapsto
\widetilde{exp}\circ
\Tilde{\Phi}(g,u,X)\in E$ for $i=1,2$ define $G$ equivariant diffeomorphisms.
Then
one can choose $G$ invariant neighborhoods $W_i$ of $(g_i,u_i)$ in $G\times_
{G_{x_i}}\!U_i$ so that the composition of the previous diffeomorphisms
defines a $G$ equivariant diffeomorphism $\Tilde{\Psi}$ between
${\overline{pr}}^{-1}_1(W_1)\subset G\times_{G_{x_1}}\!(U_1\times E_{x_1})$
and
${\overline{pr}}^{-1}_1(W_2)\subset G\times_{G_{x_2}}\!(U_2\times E_{x_2})$.
$\Tilde{\Psi}$ induces a $G$ equivariant diffeomorphism $\Psi$ between
$W_1$ and $W_2$. Using the fact that $\Tilde{\Psi}$ is $G$ equivariant,
one can show, as in the proof of Proposition \ref{liecano}, that the
composition
\begin{equation}
H=(U_1\times E_{x_1}\overset{I}\to G\times_{G_{x_1}}\!(U_1\times E_{x_1})
\overset{\Tilde{\Psi}}\to G\times_{G_{x_2}}\!(U_2\times E_{x_2})
\overset{pr_2}\to U_2\times E_{x_2})
\end{equation}
is a local diffeomorphism which together with the induced local diffeomorphism
$h:U_1\to U_2$ between neighborhoods of $u_1$ in $U_1$ and $u_2$ in $U_2$
define a vector bundle diffeomorphism $(H,h)$ which realizes the compatibility
of the charts $\mathcal{R}_1$ and $\mathcal{R}_2$ in a neighborhood of $u_1$.
\end{proof}
\begin{defn}
Let $E_i\overset{p_i}\to M_i$, $i=1,2$, be two vector orbibundles.
The pair $(\Phi,\phi)$
defines a vector bundle diffeomorphism if $\Phi:E_1\to E_2$ and
$\phi:M_1\to M_2$ \linebreak
are smooth diffeomorphisms, $p_2 \Phi= \phi p_1$, for any vector orbibundle
chart \linebreak
$\mathcal{R}=(\Tilde{U},V,\Gamma, \Pi, \pi)$ of
$E_1\overset{p_1}\to M_1$
the collection
$(\Phi,\phi) (\mathcal{R})=(\Tilde{U}, V, \Gamma, \phi(U),
\Phi\Pi, \phi\pi)$ \linebreak
is a vector orbibundle chart of $E_2\overset{p_2}\to M_2$
and for any vector orbibundle chart \linebreak
$\mathcal{R'}=(\Tilde{U'},V,\Gamma', \Pi', \pi')
$ of $E_2\overset{p_2}\to M_2$ the collection \newline
\centerline{$(\Phi^{-1},\phi^{-1}) (\mathcal{R'})=(\Tilde{U'}, V', \Gamma',
\phi^{-1}(U), \Phi^{-1}\Pi, \phi^{-1}\pi)$}\newline
is a vector orbibundle chart of $E_1\overset{p_1}\to M_1$.
\end{defn}
\begin{example}
If $E\overset{p}\to M$ is a vector orbibundle and
$(\Tilde{U},V,\Gamma,U,\Pi,\pi)$ is a chart, then
$((\Tilde{U}\times V)/\Gamma \overset{pr_1}\to \Tilde{U}/\Gamma)$ is a
vector orbibundle which is isomorphic to the restriction to $U$ of the initial
orbibundle $E_{|U}\overset{p}\to U$ by an isomorphism induced by
$(\Pi,\pi)$ to the $\Gamma$ orbit spaces.
\end{example}
\begin{defn}
Let $E_i\overset{p_i}\to M_i$, $i=1,2$ be two vector orbibundles. The pair
$(\Phi, \phi)$ defines a vector orbibundle morphism if $\Phi:E_1\to E_2$
and $\phi:M_1\to M_2$ are smooth, $p_2 \Phi= \phi p_1$, and for any
$x\in M_1$ one can find vector orbibundle charts $\mathcal{R}=(\Tilde{U},
V,\Gamma,U,\Pi,\pi)$ around $x$ and $\mathcal{R}'=(\Tilde{U}',V',
\Gamma',U',\Pi',\pi')$ around $\phi(x)$ and a smooth vector bundle map
$(\Tilde{\Phi},\Tilde{\phi})$ from $\Tilde{U}\times V \overset{pr_1}\to
\Tilde{U}$ to $\Tilde{U}'\times V' \overset{pr_1}\to \Tilde{U}'$ such
that $(\Pi',\pi')\circ(\Tilde{\Phi},\Tilde{\phi})=(\Phi, \phi)\circ
(\Pi, \pi)$.
\end{defn}
\begin{remark} If $E_i\overset{p_i}\to M_i$, $i=1,2$ are two vector
orbibundles and $U_i\subset M_i$ are two open sets such that there
exists a vector orbibundle diffeomorphism $(\Phi, \phi)$ between
$E_1{}_{|U_1}\overset{p_1}\to U_1$
and $E_2{}_{|U_1}\overset{p_2}\to U_2$ then $E_1\amalg E_2/{}_{\sim}\,
\overset{p}\to M_1\amalg M_2/{}_{\sim}$ has a canonical structure
of a vector orbibundle, an atlas being given by the reunion of two
atlases $\mathcal{A}_i$ for $E_i\overset{p_1}\to M_i$, $i=1,2$.
In view of the above observation, one can apply to vector orbibundles
the usual
operations that were done on ordinary vector bundles: duality, direct
sum, tensor
product, symmetric and exterior powers. If $E_i \overset{p_i}\to M_i$,
$i\in I$,
is a collection of vector orbibundles and if $\mathcal{R}_{i,\alpha}$
are vector
orbibundle charts, then we can apply the operations on the vector bundles
$\Tilde{U}_{i,\alpha}\times V_i\overset{pr_1}\to\Tilde{U}_{i,\alpha}$
and extend the action of the group $\Gamma_{i,\alpha}$ to these
vector bundles. We will get new vector orbibundle charts and
the vector orbibundles of orbit spaces associated to these new
charts can be glued together by diffeomorphisms prescribed by the
way the initial vector orbibundle charts were glued to yield the
initial vector orbibundles. Because the initial spaces were Hausdorff, the
underlying topological spaces for the resulting orbibundle will be
Hausdorff as well.
\end{remark}
\begin{example}
Let $M$ be an orbifold. The tangent vector orbibundle $T(M)$ can be
constructed
starting from an atlas $\mathcal{A}=(\mathcal{R}_i)_i$. If $\mathcal{R}_i=
(\Tilde{U}_i,\Gamma_i, U_i, \pi_i)$ is an orbifold chart, with $\Tilde{U}
\subset\mathbb{R}^m$, then $(\Tilde{U}_i, \mathbb{R}^m, \Gamma_i, U_i, \pi'_i,\pi_i)$
will be a vector orbibundle chart for $T(M)$, where the action of $\Gamma_i$
on $\Tilde{U}\times\mathbb{R}^m\overset{\sim}=T(\Tilde{U}_i)$ is the induced
action to the tangent space level. The vector orbibundles $\Tilde{U}_i
\times\mathbb{R}^m/\Gamma_i \overset{pr_1}\to\Tilde{U}_i/\Gamma_i$ are glued together
along diffeomorphisms prescribed by the identifications of $\Tilde{U}_i/\Gamma
\overset{\sim}=U_i$ inside $M$ to form the tangent vector orbibundle $T(M)$.
\end{example}
\begin{defn}
A metric on the orbifold $M$ is a smooth map $\mu:T(M)\otimes T(M)\to
\mathbb{R}$ such that for each orbifold chart $(\Tilde{U},\Gamma,U,\pi)$
the map $\mu$ is induced by a $\Gamma$ invariant metric $\Tilde{\mu}:
T(\Tilde{U})\otimes T(\Tilde{U})\to \mathbb{R}$.
An orbifold endowed with a metric is called a Riemannian orbifold.
\end{defn}
On any paracompact orbifold one can construct a metric by first defining
metrics on orbifold charts and then gluing them together using a
smooth partition of unity.
\begin{defn}
A hermitian structure on a complex vector orbibundle $E\overset{p}\to M$
is given by a smooth map $<,>:E\otimes E \to \mathbb{C}$ such that
for each vector orbibundle chart $(\Tilde{U}, V,\Gamma, \Pi, \pi)$ there
exists a $\Gamma$ invariant hermitian structure on $(\Tilde{U}\times
V \overset{pr_1}\to \Tilde{U})$ which induces $<,>$ above
$\Tilde{U}/\Gamma\overset{\sim}=U$.
\end{defn}
\begin{remark} We can think of a vector orbibundle as of a smooth family
of linear representations of groups $\Gamma_x\in \mathcal{G}_x$ on
$\Tilde{E}_x$ parameterized by $x\in M$. Then $E_x=\Tilde{E}_x/\Gamma_x$.
A hermitian structure is a smooth family of hermitian products in
$\Tilde{E}_x$ which are $\Gamma_x$ invariant and induce $<,>_x$ on
$E_x=\Tilde{E}_x/\Gamma_x$.
\end{remark}
\begin{defn}
Let $E\overset{p}\to M$ be a vector orbibundle. A map $f:M\to E$ is a smooth
section if $p\circ f= Id_M$ and if $f$ is smooth as a map between the orbifolds
$M$ and $E$. We will denote the space of smooth sections by $C^{\infty}(M;E)$.
\end{defn}
\begin{remark}
If $\mathcal{A}=({\mathcal{R}_i})$ is a vector orbibundle atlas with $\mathcal{R}_i
=(\Tilde{U}_i, V, \Gamma_i, U_i, \Pi_i, \pi_i)$ then the vector orbibundles
$\Tilde{U}_i\times V/\Gamma_i\overset{pr_1}\to \Tilde{U}/\Gamma_i$ and
$E_{|U_i}\overset{p}\to U_i$ are isomorphic via the map induced by $(\Pi_i,\pi_i)$ to
the orbit spaces. A map $f:M\to E$ is a smooth section if the restriction to each
$U_i$ can be identified via the above vector bundle diffeomorphisms to sections
induced to the $\Gamma_i$ orbit spaces by $\Gamma_i$ invariant smooth sections $\Tilde{f}_i$
of the $\Gamma_i$ vector bundles $\Tilde{U}_i\times V\overset{pr_1}\to \Tilde{U}_i$.
\end{remark}
In Proposition \ref{lievcano} we showed that starting with a $G$ vector bundle
$E\overset{p}\to M$ where the Lie group $G$ acts on $M$ with finite isotropy groups
and passing to the $G$ orbit spaces we obtain a vector orbibundle $E/G \overset
{\overline{p}}\to M/G$. In this case we have a global characterization of the
smooth sections of the vector orbibundle.
\begin{prop}
\label{gsection}
The canonical map
\begin{equation}
Sect(M;E)^G\to Sect(M/G;E/G)
\end{equation}
identifies the $G$ invariant
smooth sections of the $G$ vector bundle $E\overset{p}\to M$ with the smooth sections
of the vector orbibundle of $G$ orbits $E/G \overset{\overline{p}}\to M/G$.
\end{prop}
\begin{proof}
It is obvious that any $G$ invariant section $f\in Sect(M;E)^G$ defines a section
$\overline{f}\in Sect(M/G;E/G)$ by $\overline{f}(Gx)=Gf(x)$.
Let $f_1$ and $f_2$ be two $G$ invariant sections in $Sect(M;E)^G$ which induce
equal sections $\overline{f}_1=\overline{f}_2\in Sect(M/G;E/G)$. If $f_1\ne f_2$
then there exists $x\in M$ such that $f_1(x)\ne f_2(x)$. Nevertheless, $\overline
{f}_1(\overline{x})=\overline{f}_2(\overline{x})\in E/G$. Then there exists $g\in G$
such that $f_1(x)=gf_2(x)\in E_x$. But $gf_2(x)=f_2(gx)\in E_x$ so $gx=x$. So
$f_1(x)=f_1(g^{-1}x)=g^{-1}f_1(x)=g^{-1}gf_2(x)=f_2(x)$. We reached a contradiction so
the map $Sect(M:E)^G\to Sect(M/G;E/G)$ is injective.
Let $h\in C^{\infty}(M/G;E/G)$ be a smooth section. We need to construct a $G$ invariant
smooth section $f\in C^{\infty}(M;E)^G$ such that $\overline{f}=h$.
We will construct $f$ from local data. As we showed in Proposition \ref{lievcano},
the local model of passing from a $G$ vector bundle
to a vector orbibundle is given by passing from the $G$ vector bundle $G\times_{G'}
\!(U\times V) \overset{p}\to G\times_{G'}\! U$ to the vector orbibundle
$(U\times V)/G' \overset{\overline{p}}\to U/G'$. $G'\subset G$ is a finite group
which acts on the left on the open set $U\subset \mathbb{R}^n$ and on a vector
space $V$ and by right translations on $G$. The action of $G'$ on $G\times U$ and
$G\times(U\times V)$ is given by $g' \cdot (g,u)=(g g'{}^{-1}, g'u)$ and
$g' \cdot (g,u,v)= (gg'{}^{-1}, gu, gv)$. In fact the vector bundle $G\times(U\times V)
\overset{\Tilde{p}}\to G\times U$ is a $G'$ vector bundle whose associated vector
orbibundle of $G'$ orbits is the local model $G\times_{G'}\!(U\times V)\overset{p}
\to G\times_{G'}\!U$, which happens to be a genuine vector bundle because the action
of $G'$ is free. A section $h\in C^{\infty}(U/G';
(U\times V)/G')$ is induced by a smooth section $\Tilde{h}\in C^{\infty}(U; U\times V)$.
Let $\Tilde{f}\in C^{\infty}(G\times U; G\times (U\times V))$ be defined by
$\Tilde{f}(g,u)=(g,h(u))$. This is a $G$ invariant smooth section which is also $G'$
invariant and it induces a smooth section $f\in C^{\infty}(G\times_{G'}\! U;
G\times_{G'}\!(U\times V))$. It is obvious that the map $f$ induces $h$ when passing
to the $G$ orbit spaces. Because of the first part of the proof such an $f$ is
unique, so we can glue different sections $f$ to get a global $G$ invariant smooth
section in $C^{\infty}(M;E)$ which will induce $h\in C^{\infty}(M/G; E/G)$.
The problem of proving that if $f\in Sect(M;E)^G$ is smooth then its image
$\overline{f}\in Sect(M/G;E/G)$ is smooth is a local one. A smooth
section $f\in C^{\infty}(G\times_{G'}\! U;G\times_{G'}\!(U\times V))$ is induced by a $G'$
invariant smooth section $\Tilde{f}\in C^{\infty}(G\times U; G\times(U\times V))^{G'}$.
If $f$ is $G$ invariant then $\Tilde{f}$ is $G$ invariant so
there exists a $G'$ smooth section $h$ of the $G'$ vector bundle $U\times
V\overset{pr_1}\to U$ such that $\Tilde{f}(g,u)=(g,h(u))$.
The induced section $\overline{f}$ of the vector orbibundle $(U\times V)/G' \overset
{\overline{p}}\to U/G'$ is the same as the one induced by $h$.
So $\overline{f}$ is smooth.
\end{proof}
\begin{remark}
\label{globalize}
Let $\Tilde{E}\overset{\Tilde{p}}\to \Tilde{M}$ be a $\Gamma$ vector
bundle and $E\overset{p}\to M$ be the associated vector orbibundle of
$\Gamma$ orbits. If $\Tilde{U}\subset\Tilde{M}$ is an open subset and
$\Gamma'\subset\Gamma$ a subgroup such that $\Tilde{U}$ is $\Gamma'$ invariant
and $\gamma\Tilde{U}\cap\Tilde{U}=\emptyset$ for $\gamma\in\Gamma
\setminus\Gamma'$, then $\Tilde{E}_{|\Tilde{U}}\overset{p}\to \Tilde{U}$ is
a $\Gamma'$ vector bundle and the associated vector orbibundle of
$\Gamma'$ orbits is a subbundle of $E\overset{p}\to M$. Let $U=\Tilde{U}/
\Gamma'$. Denote by
$\iota$ the inclusion map between the two vector orbibundles.
$\iota$ induces a map between the spaces of compactly supported sections
$\iota_*: C^{\infty}_0(U;E_{|U})\to C^{\infty}_0(M;E)$ given by extension
by zero on $M\setminus U$ and a restriction map $\iota^* :C^{\infty}
(M;E)\to C^{\infty}(U;E_{|U})$. We have $C^{\infty}(M;E)\overset{\sim}=
C^{\infty}(\Tilde{M};\Tilde{E})^{\Gamma}$ and $C^{\infty}(U;E_{|U})\overset
{\sim}=C^{\infty}(\Tilde{U};\Tilde{E}_{|\Tilde{U}})^{\Gamma'}$.
Then the map $\iota^*$ is induced by the restriction map $C^{\infty}
(\Tilde{M};\Tilde{E})\to C^{\infty}(\Tilde{U};\Tilde{E}_{|\Tilde{U}})$.
The map $\iota_*$ between $C^{\infty}(\Tilde{U}; \Tilde{E}_{|\Tilde{U}})$ and
$C^{\infty}(\Tilde{M};\Tilde{E})$ is given by
\begin{align}
\iota_*(\Tilde{f})&=\sum_{i=1}^l \gamma_i \Tilde{f}\\
\intertext{where $\{\gamma_1=e,\gamma_2,\dots,\gamma_l\}$ with $l=[\Gamma:\Gamma']$
is a complete system of left coset representatives for $\Gamma/\Gamma'$, or}
\iota_*(\Tilde{f})&=\frac{1}{|\Gamma'|}\sum_{\gamma\in\Gamma}\gamma \Tilde{f}
\end{align}
for $\Tilde{f}\in C^{\infty}(\Tilde{U};\Tilde{E}_{|\Tilde{U}})^{\Gamma'}$.
\end{remark}
\subsection{The Representation Theorem}
In this section we will state and prove the reciprocal of
Proposition \ref{lievcano}.
\begin{thm}
\label{represent}
Any vector orbibundle $E\overset{p}\to M$ is isomorphic to the vector orbibundle
of $G$ orbits associated with a $G$ vector bundle $\Tilde{E}\overset{\Tilde{p}}\to
\Tilde{M}$, with $G$ a compact Lie group acting on $M$ with finite isotropy groups.
In particular any orbifold $M$ is diffeomorphic to the orbifold of $G$ orbits
associated with a $G$ manifold $\Tilde{M}$ where the action of the compact Lie group
$G$ has finite isotropy groups.
\end{thm}
\begin{proof}
Let us fix a metric $\mu$ on the base space $M$. Let $G=O(m)$ be the
orthogonal group, $m=dim(M)$. We will describe the orthonormal frame
bundle $\mathcal{F}(M)\overset{proj}\longrightarrow M$.
We will show that $\mathcal{F}(M)$ is a manifold and that the group
$G$ acts on it with finite isotropy groups. The $G$ vector bundle we need
to construct will be essentially the pull-back of $E\overset{p}\to M$
to $\mathcal{F}(M)$ via the canonical projection.
The description of the space of orthonormal frames uses an atlas
$\mathcal{A}=(\mathcal{R}_i)$ of the vector orbibundle $E\overset{p}\to M$.
Let $\Tilde{\mu}_i$ be the $\Gamma_i$ invariant metric on
$\Tilde{U}_i\subset \mathbb{R}^m$ which induces the metric $\mu$ via the map
$\Tilde{U}_i\overset {\pi_i}\to U_i$. Let $\mathcal{F}(\Tilde{U}_i)$ be the
space of orthonormal frames of $\Tilde{U}_i$. Then the orthogonal group
$O(m)$ acts by right translations on $\mathcal{F}(\Tilde{U}_i)$. Because
the metric $\Tilde{\mu}_i$
is $\Gamma_i$ invariant, the action of $\Gamma_i$ on $\Tilde
{U}_i$ extends to $\mathcal{F}(\Tilde{U}_i)$. Observe that this action
is free. Indeed, if $\gamma\in \Gamma_i$ and $(x,\mathcal{X})\in \mathcal{F}
(\Tilde{U}_i)$ such that $\gamma(x,\mathcal{X})=(x,\mathcal{X})$ then
$\gamma\in \Gamma_x$ and $\gamma$ acts on the tangent space $T_x(\Tilde{U}_i)$
as the identity map. But the order of $\gamma$ is finite so $\gamma=
e\in \Gamma_i$. Then the space of orbits $\mathcal{F}(\Tilde{U}_i)/
\Gamma_i$ is a smooth manifold. The local diffeomorphisms between open
sets $\pi_i^{-1}(U_i\cap U_j)\subset \Tilde{U}_i$ and $\pi_j^{-1}
(U_i\cap U_j)\subset \Tilde{U}_j$ preserve the metrics so they will
induce local diffeomorphisms between $\mathcal{F}\pi_i^{-1}(U_i\cap
U_j)$ and $\mathcal{F}\pi_i^{-1}(U_i\cap U_j)$ and a diffeomorphism
between their images in $(\mathcal{F}(\Tilde{U}_i))/\Gamma_i$ respectively
$\mathcal{F}(\Tilde{U}_j)/\Gamma_j$. The space of orthonormal
frames $\mathcal{F}(M)$ is obtained by gluing the manifolds $(\mathcal{F}
(\Tilde{U}_i))/\Gamma_i$ along these diffeomorphisms and it has a natural
structure of smooth manifold. The right action of $O(m)$ on $\mathcal{F}
(\Tilde{U}_i)$ commutes with the action of $\Gamma_i$ and with the local
diffeomorphisms induced between different $\mathcal{F}(\Tilde{U}_i)$
and $\mathcal{F}(\Tilde{U}_j)$ so we will get an induced right action of
$O(m)$ on $\mathcal{F}(M)$.
Let us consider the $\Gamma_i$ vector bundle $\Tilde{U}_i\times V
\overset{pr_1}\to\Tilde{U}_i$ corresponding to the chart $\mathcal{R}_i$
and the pull-back to the space of orthonormal frames
$\mathcal{F}(\Tilde{U}_i)\times V \overset{pr_1}\to\mathcal{F}
(\Tilde{U}_i)$ together with the induced left action of the group
$\Gamma_i$ on it and the right action of group $O(m)$. The action of
$\Gamma_i$ is free and commutes with the action of $O(m)$ so if we pass
to the spaces of $\Gamma_i$ orbits we
get a genuine vector bundle $\mathcal{F}(\Tilde{U}_i)\times V/\Gamma_i
\overset{\overline{pr}_1} \to\mathcal{F}(\Tilde{U}_i))/\Gamma_i$ endowed
with the action of $O(m)$. By gluing these vector bundles together using
diffeomorphisms induced by the identifications made on the initial vector orbibundle
charts we get an $O(m)$ vector bundle $\Tilde{E}\overset{\Tilde{P}}\to \mathcal{F}(M)$.
We will show that the action of $O(m)$ on $\mathcal{F}(M)$ has finite isotropy
groups and if we pass to the space of $O(m)$ orbits we get a vector orbibundle
isomorphic with the initial vector orbibundle $E\overset{p}\to M$.
Any point in $\mathcal{F}(M)$ is the $\Gamma_i$ orbit of a point $(x,\mathcal{X})
\in \mathcal{F}(\Tilde{U}_i)$ for some index $i$. The group $O(m)$ acts freely
on $\mathcal{F}(\Tilde{U}_i)$.
If $g\in O(m)$ fixes the $\Gamma_i$ orbit $\overline{(x,\mathcal{X})}$ then there exists
$\gamma\in \Gamma_i$ such that $(x,\mathcal{X})g=\gamma(x,\mathcal{X})$. This is
equivalent to $x=\gamma x$ and $\mathcal{X}g=\gamma\mathcal{X}$. The
group $O(m)$ acts freely and transitively on the frames at $x\in\Tilde{U}_i$.
The group $(\Gamma_i)_x$ acts freely on the frames at $x\in\Tilde{U}_i$ as well.
Then the above equalities imply that the cardinality of the isotropy group
of $\overline{(x,\mathcal{X})}\in\mathcal{F}(M)$ with respect to the action of
$O(m)$ is equal to the cardinality of the group $(\Gamma_i)_x$, which is finite.
Passing to the vector orbibundle of $O(m)$ orbit spaces $\Tilde{E}/O(m)\overset
{\overline{\Tilde{p}}}\to \mathcal{F}(M)/O(m)$ can be realized by first passing to the
$\Gamma_i$ vector bundles $\mathcal{F}(\Tilde{U})\times V/O(m)\overset{\overline{pr}_1}
\to \mathcal{F}(\Tilde{U})/O(m)$ and then gluing together the resulting vector
orbibundles of $\Gamma_i$ orbits. But the $\Gamma_i$ vector bundles
$\Tilde{U}\times V\overset{pr_1}\to\Tilde{U}$ and $\mathcal{F}(\Tilde{U})\times V/O(m)
\overset{\overline{pr}_1}\to \mathcal{F}(\Tilde{U})/O(m)$ are canonically isomorphic
and the resulting vector orbibundles obtained by gluing the vector orbibundles of
$\Gamma_i$ orbits will be isomorphic. So $E \overset{p}\to M$ and $\Tilde{E}/O(m)
\overset{\Tilde{p}}\to \mathcal{F}(M)/O(m)$ are isomorphic.
\end{proof}
\subsection{Sobolev Spaces}
Consider the Euclidean space $\mathbb{R}^m$ with the canonical scalar product
$\left<x,y\right>=\sum_{i=1}^m x_iy_i$ and let the finite group $\Gamma$ act on $\mathbb{R}^m$ by
linear isometries. We have an induced action of $\Gamma$ on the space of function
on $\mathbb{R}^m$ defined by $\gamma f(x)=f(\gamma^{-1}x)$.
For compactly supported smooth functions $f\in C^{\infty}_0(\mathbb{R}^m)$ the Fourier
transform $\Hat{f}:\mathbb{R}^m\to \mathbb{R}$ is defined by
$\Hat{f}(\xi)=\int e^{-i<x,\xi>}f(x) \, dx$.
Then $\Hat{(\gamma f)}(\xi)=\int e^{-i<x,\xi>}f(\gamma^{-1}x)\,dx$ and if we replace
$x$ by $\gamma y$ we get
\begin{equation}
\begin{split}
\Hat{(\gamma f)}(\xi)&=\int e^{-i<\gamma y,\xi>}f(y)det(\gamma)\,dy=\\
&=\int e^{-i<y, \gamma^{-1} \xi>}f(y)\, dy=
\Hat{f}(\gamma^{-1}\xi)=\gamma \Hat{f}(\xi)
\end{split}
\end{equation}
(we used the fact that $\gamma$ was an isometry and $det(\gamma)=1$).
This shows that the Fourier transform commutes with the action of $\Gamma$ on the space
of functions.
For $s\in\mathbb{R}$ and $f\in C^{\infty}(\mathbb{R}^m)$ the Sobolev norm $||\cdot||_s$
is defined by $||f||^2_s=\int (1+|\xi|^2)^s |\Hat{f}(\xi)|^2\,d\xi$. The Sobolev space $H_s
(\mathbb{R}^m)$ is the completion of $C^{\infty}_0(\mathbb{R}^m)$ with respect to the norm
$||\cdot||_s$. Because the group $\Gamma$ acts by isometries on $\mathbb{R}^m$ and
the Fourier transform is $\Gamma$ equivariant we conclude that the action of $\Gamma$ on
the space of smooth functions preserves the Sobolev norms and extends to an action by
isometries on the Sobolev spaces $H_s(\mathbb{R}^m)$. The closure of the space of smooth
$\Gamma$ invariant functions $C^{\infty}_0(\mathbb{R}^m)^{\Gamma}$ coincides with the
subspace of $\Gamma$ invariant elements $H_s(\mathbb{R}^m)^{\Gamma}=\{f\in H_s(\mathbb{R}^m)\,
|\, \gamma f=f \text{ for all } \gamma \in \Gamma \}$.
In order to define the Sobolev norm of the smooth functions on the orbifold $\mathbb{R}/
\Gamma$ it is convenient to replace the Sobolev norm of the $\Gamma$ invariant
smooth functions with an equivalent norm.
\begin{defn}
For $f \in C^{\infty}_0(\mathbb{R}^m)^{\Gamma}\overset{\sim}=
C^{\infty}_0(\mathbb{R}^m/\Gamma)$ let $||f||^{\Gamma}_s=\frac{1}{|\Gamma|}||f||_s$.
The Sobolev space $H_s(\mathbb{R}^m/\Gamma)$ is the completion of $C^{\infty}
(\mathbb{R}^m/\Gamma)$ with respect to the norm $||\cdot||^{\Gamma}_s$.
If $\Tilde{U}\subset \mathbb{R}^m$ is a $\Gamma$ invariant neighborhood of the origin
with compact closure
let $H_s(\Tilde{U}/\Gamma)$ be the closure of $C^{\infty}_0(\Tilde{U})^{\Gamma}$
in $H_s(\mathbb{R}^m/\Gamma)$.
\end{defn}
We observe that $H_s(\mathbb{R}^m/\Gamma)\overset{\sim}=H_s(\mathbb{R}^m)^{\Gamma}$.
Our goal is to define the Sobolev spaces $H_s(M)$ for $M$ a compact orbifold. The above
definitions will describe the elements of the Sobolev space $H_s(M)$ whose support is
included in the image of a linear chart $(\Tilde{U}, \Gamma, \Tilde{U}/\Gamma\overset
{\sim}=\pi(\Tilde{U}), \pi)$. In order for this description to be independent
of the chosen orbifold chart we need to show that the definition of the spaces
$H_s(\Tilde{U}/\Gamma)$ is compatible with the following two operations occurring
in the compatibility of the linear orbifold charts:
\begin{enumerate}
\item If $\phi:\Tilde{U}_1\to \Tilde{U}_2$ is a diffeomorphism intertwining the
two linear actions of the group $\Gamma$ on two open neighborhoods of the origin
$\Tilde{U}_i$, $i=1,2$ in $\mathbb{R}^m$ then the two orbifold charts
$(\Tilde{U}_i, \Gamma, \Tilde{U}_i
/\Gamma, \pi_i)$, $i=1,2$ are compatible with the map $\phi$ inducing a diffeomorphism
between the orbifolds $\Tilde{U}_1/\Gamma$ and $\Tilde{U}_2/\Gamma$.
\item If $\Gamma_x\subset \Gamma$ is the isotropy group of $x\in \mathbb{R}^m$
and $\Tilde{V}\subset \Tilde{U}$ is a $\Gamma_x$ invariant neighborhood of $x$ such that
$\gamma \Tilde{V}\cap \Tilde{V}=\emptyset$ for $\gamma \in \Gamma \backslash \Gamma_x$ then
the restricted chart $(\Tilde{V}, \Gamma_x, \pi(\Tilde{V}), \pi_{|V})$ and the initial chart
$(\Tilde{U}, \Gamma, \Tilde{U}/\Gamma, \pi)$ are compatible via the canonical
inclusion map $\iota:\Tilde{V}\hookrightarrow \Tilde{U}$ inducing a smooth map between
the associated orbifolds $\Tilde{V}/\Gamma_x$ and $\Tilde{U}/\Gamma$.
\end{enumerate}
In the first case the diffeomorphism $\phi$ induces a continuous bijection with a
continuous inverse between the spaces of smooth functions $C^{\infty}_0(\Tilde{U}_i)$,
for $i=1,2$, endowed with the Sobolev norms, (cf. \cite{gilkey} Lemma [1.3.3]) and permutes
with the action
of the group $\Gamma$. As a consequence $\phi$ will induce a continuous bijection
with a continuous inverse between the Sobolev spaces $H_s(\Tilde{U}_i/\Gamma)\overset
{\sim}=H_s(\Tilde{U}_i)^{\Gamma}$, $i=1,2$.
In the second case the inclusion $\iota:\Tilde{V}\to \Tilde{U}$ induces the map
$\iota_*:C^{\infty}_0(\Tilde{V})^{\Gamma_x}\to C^{\infty}_0(\Tilde{U})^{\Gamma}$ with
$\iota_*(f)=\sum_{i=1}^k\gamma_i f$ where $\{\gamma_i\}_{i=1}^k$ is a complete system
of left coset representatives for $\Gamma/\Gamma_x$. We extend the compactly
supported function $f$ by zero outside $\Tilde{V}$. Then $||\iota_*(f)||_s^{\Gamma}
=\frac{1}{|\Gamma|}||\iota_*(f)||_s=\frac{1}{[\Gamma:\Gamma_x]|\Gamma_x|}
\sum_{i=1}^k ||\gamma_i f||_s=\frac{1}{|\Gamma_x|}||f||_s=||f||_s^{\Gamma_x}$.
So $\iota_*$ induces an isometric embedding of $H_s(\Tilde{V}/\Gamma_x)$ into
$H_s(\Tilde{U}/\Gamma)$.
We will need the following statement as well:
\begin{lemma}
If $f\in C^{\infty}_0(\Tilde{U})^{\Gamma}\overset{\sim}=C^{\infty}_0(\Tilde{U}/\Gamma)$
is a compactly supported smooth function then the multiplication operator by $f$ is
continuous operator on $C^{\infty}_0(\Tilde{U}/\Gamma)$ endowed with the Sobolev norm
$||\cdot||_s$.
\end{lemma}
\begin{proof}
The multiplication operator by $f$ is a continuous map on $C^{\infty}_0(\Tilde{U})$
endowed with the Sobolev norm $||\cdot||_s$, being a differential operator of order
zero. The the restriction to the $\Gamma$
invariant smooth functions remains a continuous operator.
\end{proof}
We will define the Sobolev spaces $H_s(M)$ for a compact orbifold $M$.
Let $\mathcal{A}=(\Tilde{U}_i, \Gamma_i, U_i, \pi_i)_i$ be a finite atlas of the orbifold $M$
consisting of linear charts and $(\phi_i)_i$ be a smooth partition of unity associated with
the open cover $(U_i)_i$. For $f\in C^{\infty}(M)$ define the Sobolev norm
\begin{equation}
||f||^{\mathcal{A},(\phi_i)}_s=\sum_i||\pi_i^*(\phi_i f)||^{\Gamma_i}_s
\end{equation}
where $\pi_i^*(\phi_i f)\in C^{\infty}_0(\Tilde{U}_i)^{\Gamma_i}$ is
the unique $\Gamma_i$
invariant map inducing $\phi_i f\in C^{\infty}(U_i)$. As in the case of
manifolds, this
norm depends on the choice of the atlas and the partition of unity, but
two different choices
lead to two equivalent norms. This is a direct consequence of the
equivalence of Sobolev norms for equivalent orbifold charts as described above
and the continuity of the multiplication operators by smooth functions.
The Sobolev space $H_s(M)$ is the completion of the space of smooth functions
with respect to any of the norms $||\cdot||^{\mathcal{A},(\phi_i)}_s$.
In the case of vector orbibundles, we define the Sobolev norms
$||\cdot||^{\Gamma}_s$
for the local model $(\Tilde{U}\times \mathbb{R}^k)/\Gamma \overset{pr_1}
\to \Tilde{U}/\Gamma$, with$\Tilde{U}\subset \mathbb{R}^m$,
by $||f||^{\Gamma}_s=\frac{1}{|\Gamma|}||f||_s$ where $f\in C^{\infty}_0
(\Tilde{U}/\Gamma; (\Tilde{U}\times \mathbb{R}^k)/\Gamma))\overset{\sim}=
C^{\infty}_0(\Tilde{U};\Tilde{U}\times
\mathbb{R}^k)^{\Gamma}$ and $||\cdot||_s$ is the usual Sobolev norm restricted
to the $\Gamma$ invariant sections. If $E\overset{p}\to M$ is a
vector orbibundle, we use a vector orbifold atlas and a partition of
unity to define the Sobolev norm of smooth sections as in the case of
smooth functions. All Sobolev norms defined using different atlases
and partitions of unity
will be equivalent and the Sobolev space $H_s(M;E)$ is the
completion of $C^{\infty}(M;E)$
with respect to any of these equivalent norms.
\begin{remark}
We showed in Proposition \ref{represent} that any vector orbibundle
$E\overset{p}\to M$ is isomorphic to the associated vector orbibundle
of a $G$ vector bundle $\Tilde{E} \overset{\Tilde{p}}\to \Tilde{M}$
with the compact Lie group $G$ acting on $\Tilde{M}$
with finite isotropy groups. Then $C^{\infty}(M;E)\overset{\sim}=
C^{\infty}(\Tilde{M};\Tilde{E})^G$. Also
$H_s(M;E)=H_s(\Tilde{M};\Tilde{E})^G$. To prove this, we use
a $G$ invariant Sobolev norm $||\cdot||_s$ on $C^{\infty}(\Tilde{M};\Tilde{E})$.
The fact that the restriction of the Sobolev norm $||\cdot||_s$ to
the $G$ invariant smooth sections is equivalent to the Sobolev norm on
$C^{\infty}(M;E)$ can be seen directly when
looking at the local model $G\times_{G'}\!(U\times V) \overset{p}
\to G\times_{G'}\! U$
and passing to the vector orbibundle $(U\times V)/G' \overset{\overline{p}}
\to U/G'$. Then the completion of $C^{\infty}(\Tilde{M};\Tilde{E})^G
\overset{\sim}=C^{\infty}(M;E)$ with respect to the induced Sobolev norm
$||\cdot||_s$ is equal to $H_s(M;E)$.
\end{remark}
\subsection{Dirac-Type Densities on Orbifolds}
In this chapter we will construct a certain family of Dirac-type densities and
the associated Dirac-type distributions.
\begin{defn}
Let $\Gamma\times \Tilde{M}\to \Tilde{M}$ be a faithful action of a finite
group on a smooth manifold and $M=\Tilde{M}/\Gamma$ the associated orbifold.
A Dirac-type density $\eta$ on the orbifold $M$ is given by a $\Gamma$
Dirac-type density $\{\eta^{\gamma}\}_{\gamma\in\Gamma}$ on the $\Gamma$
manifold $\Tilde{M}$.
The Dirac-type density $\eta$ defines a continuous functional on $C^{\infty}_0
(M)$ by
\begin{equation}
<\eta,f>=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}
\int_{\Tilde{M}^{\gamma}}\Tilde{f}_{|\Tilde{M}^{\gamma}} \eta^{\gamma}
\end{equation}
where $\Tilde{f}\in C^{\infty}_0(\Tilde{M})^{\Gamma}$ induces $f\in
C^{\infty}_0(M)$. We call the above functional the Dirac-type distribution
associated with $\eta$.
\end{defn}
We want to extend the above notions to a general orbifold $M$, which is not
the quotient of a smooth manifold by a finite group.
\begin{remark}
Let $\Gamma_i$, $i=1,2$ be two isomorphic groups acting respectively on two
smooth manifolds $\Tilde{M}_1$ and $\Tilde{M}_2$ as above. Suppose that
the actions are conjugated by a diffeomorphism $\Tilde{\phi}:\Tilde{M}_1
\to\Tilde{M}_2$. Then the orbifolds $M_1=\Tilde{M}_1/\Gamma_1$ and
$M_2=\Tilde{M}_2/\Gamma_2$ are diffeomorphic by the diffeomorphism
$\phi$ and for any Dirac-type density $\eta_1=\{\eta^{\gamma}\}
_{\gamma\in\Gamma_1}$ on $M_1$ the collection $\phi_*(\eta_1)=
\{\Tilde{\phi}_*(\eta^{\gamma})\}_{\gamma\in\Gamma_1}$ is a Dirac-type
density on $M_2$. The corresponding Dirac-type distributions on
$C^{\infty}_0(M_i)$, $i=1,2$ are conjugated by the isomorphism
induced by $\phi$ between the spaces of smooth functions.
\end{remark}
\begin{remark}
\label{restrict:density}
Let $\eta$ be a Dirac-type density on the orbifold $M=\Tilde{M}/\Gamma$.
Let $\Tilde{U}\subset \Tilde{M}$ be an open subset and $\Gamma'\subset\Gamma$
a subgroup such that $\Tilde{U}$ is $\Gamma'$ invariant and $\gamma\Tilde{U}
\cap \Tilde{U}=\emptyset$ for $\gamma\in\Gamma\setminus\Gamma'$. Then
we have a $\Gamma$ equivariant diffeomorphism $\Gamma\times_{\Gamma'}\Tilde{U}
\overset{\sim}=\Gamma\Tilde{U}$. The orbifold $U=\Tilde{U}/\Gamma'$ is
an open suborbifold of $M$. Denote by $\iota:U\to M$ the inclusion map.
The restriction of the Dirac-type densities
$\eta^{\gamma'}$ on $U$ for $\gamma'\in\Gamma'$ define a $\Gamma'$
Dirac-type density on $\Tilde{U}$ and so a Dirac-type density on the orbifold
$U$ which we denote by $\iota^*(\eta)$ or simply $\eta_{|U}$.
Let $\iota_* :C^{\infty}_0(U) \to C^{\infty}_0(M)$ be the map given by the
extension by zero on $M\setminus U$. Then
\begin{equation}
\label{restrict:dens:f}
<\eta_{|U}, f>=<\eta,\iota_*(f)> \qquad\text{ for any } f\in C^{\infty}_0(U)
\end{equation}
Indeed, if $\Tilde{f}\in C^{\infty}_0(\Tilde{U})$ induces $f$, then
$\iota_*(\Tilde{f})=\sum_{i=1}^{l}\gamma_i \Tilde{f}$, with $\{\gamma_1,
\gamma_2,\dots,\gamma_l\}$ a complete system of left coset representatives
for $\Gamma/\Gamma'$, as proved in Remark \ref{globalize}.
The left-hand side of \eqref{restrict:dens:f} is equal to
\begin{equation}
<\eta_{|U}, f>=\frac{1}{|\Gamma'|}\sum_{\gamma'\in\Gamma'}
\int_{\Tilde{U}^{\gamma'}}\Tilde{f}_{|\Tilde{U}^{\gamma'}} \eta^{\gamma'}
\end{equation}
and the right-hand side is equal to
\begin{align}
<\eta,\iota_*(\Tilde{f})>&=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}
\int_{\Tilde{M}^{\gamma}}
\sum_{i=1}^{l}\gamma_i\Tilde{f}_{|\Tilde{M}^{\gamma}} \eta^{\gamma}=\\
&=\frac{1}{|\Gamma|}\sum_{i=1}^{l}(\sum_{\gamma\in\Gamma}\int_{\gamma_i^{-1}
\Tilde{M}^{\gamma}}f(\gamma_i^{-1}\eta^{\gamma}))=\\
&=\frac{1}{|\Gamma|}\sum_{i=1}^{l}(\sum_{\gamma\in\Gamma}\int_{(\Tilde{M}
^{\gamma_i^{-1}\gamma\gamma_i}\cap \Tilde{U})}f\eta^{\gamma_i^{-1}\gamma
\gamma_i})=\\
&=\frac{1}{|\Gamma|}\sum_{i=1}^{l}(\sum_{\gamma_i^{-1}\gamma\gamma_i\in\Gamma'}
\int_{\Tilde{U}^{\gamma_i^{-1}\gamma\gamma_i}}f
\eta^{\gamma_i^{-1}\gamma\gamma_i})=\\
&=\frac{1}{|\Gamma|}\sum_{i=1}^{l}(|\Gamma'|<\eta_{|U},f>)=\\
&=<\eta_{|U},f>
\end{align}
\end{remark}
Let $\phi:\Tilde{M}_1/\Gamma_1\to\Tilde{M}_2/\Gamma_2$ be a diffeomorphism
between two
orbifolds. We will define the action of $\phi$ on Dirac-type densities.
Because $\phi$ is a diffeomorphism, there exist open covers
$\{U_{i,\alpha}\}_{\alpha}$ of the two orbifolds, finite groups
$\Gamma_{\alpha}$ acting on open subsets $\Tilde{U}_{i,\alpha}$ of the
Euclidean space $\mathbb{R}^m$ such that $\Tilde{U}_{i,\alpha}/\Gamma_{\alpha}
\overset{\sim}=U_{i,\alpha}$. The diffeomorphism $\phi$ between
$U_{1,\alpha}$ and $U_{2,\alpha}$ is induced by a diffeomorphism
$\Tilde{\phi}_{\alpha}:\Tilde{U}_{1,\alpha}\to\Tilde{U}_{2,\alpha}$
which is $\Gamma_{\alpha}$ equivariant.
If $\eta$ is a Dirac-type density on $M_1=\Tilde{M}_1/\Gamma_1$
then the Dirac-type density $\phi_*(\eta)$ on $M_2=\tilde{M}_2/\Gamma_2$
is given locally on each $U_{2,\alpha}$ by the Dirac-type density
$\phi_{\alpha}{}_*(\eta_{|U_{1,\alpha}})$. Because of the two remarks we made
above, the Dirac-type distributions associated with $\eta$ and $\phi_*(\eta)$
are conjugated by the isomorphism between the spaces of functions
induced by $\phi$.
\begin{defn}
Let $M$ a smooth orbifold. A Dirac-type density on $M$ is given by a
collection of Dirac-type densities $\eta_{\alpha}$ on the orbifolds
$U_\alpha=\Tilde{U}_\alpha/\Gamma_{\alpha}$ such that
$\phi_{\alpha\beta}{}_*(\eta_{\alpha}{}_{|U_{\alpha}\cap U_{\beta}})=
\eta_{\beta}{}_{|U_{\alpha}\cap U_{\beta}}$, where
$\phi_{\alpha\beta}$ are the transition maps associated with
an orbifold atlas $\mathcal{A}=(\Tilde{U}_{\alpha},\Gamma_{\alpha},
U_{\alpha}, \pi_{\alpha})_{\alpha}$.
\end{defn}
The distribution associated with the Dirac-type density $\eta$ on
$M$ is defined using a partition of unity associated with the orbifold atlas;
if $f\in C^{\infty}_0(M)$ then
\begin{equation}
<\eta, f>=\sum_{\alpha}<\eta_{\alpha},
\psi_{\alpha}f>
\end{equation}
where $\{\psi_{\alpha}\}_{\alpha}$ is a partition of
unity associated with the above atlas. We will denote $<\eta, f>$ by
$\int_M f\eta$.
\begin{defn}
The integral of the Dirac-type density $\eta$ on $M$ is equal to \linebreak
$<\eta, 1>$.
\end{defn}
\subsection{The Canonical Stratification of an Orbifold}
The goal of this section to describe the canonical stratification $\mathcal{S}(M)$
of an orbifold $M$.
Let $\Gamma$ be a finite group.
Let $\mathcal{O}(\Gamma)$ be the set of equivalence classes of subgroups
of $\Gamma$ with respect to the conjugation relation: $\Gamma_1\sim\Gamma_2$ if
there exists $\gamma\in\Gamma$ such that $\gamma\Gamma_1\gamma^{-1}=\Gamma_2$.
Denote by $[\Gamma']$ the class of $\Gamma'\subseteq \Gamma$.
On $\mathcal{O}(\Gamma)$ we consider the order relation $[\Gamma_1]\preceq[\Gamma_2]$
if there exists $\gamma\in\Gamma$ such that $\gamma\Gamma_1\gamma^{-1}\subseteq
\Gamma_2$. Then $(\mathcal{O}(\Gamma), \preceq)$ is a partially order set.
Let $\Gamma\times \Tilde{M}\to \Tilde{M}$ be a faithful action on a manifold
$\Tilde{M}$. Let $M=\Tilde{M}/\Gamma$ be the associated orbifold of $\Gamma$ orbits.
Let $\mathcal{O}(M)\subseteq \mathcal{O}(\Gamma)$ given by
\begin{equation}
\mathcal{O}(M)=\{[\Gamma']\,|\,\text{ there exists }\Tilde{x}\in\Tilde{M}
\text{ such that }\Gamma_{\Tilde{x}}\sim \Gamma'\}
\end{equation}
\begin{defn}
$\mathcal{O}(M)$ is called the set of orbit types of $M$.
\end{defn}
We have the map $\mathcal{O}:M\to\mathcal{O}(M)$ given by $\mathcal{O}(x)=
[\Gamma_{\Tilde{x}}]$
where $\Tilde{x}\in\Tilde{M}$ is a representative for the class $x\in\Tilde{M}/\Gamma$.
The image of $x$ in $\mathcal{O}(M)$ is called the orbit type of $x$.
We will define the set of orbit types of a general orbifold $M$. Let $\mathcal{A}
=(\mathcal{R}_{\alpha})$ an atlas with $\mathcal{R}_{\alpha}=(\Tilde{U}_{\alpha},
\Gamma_{\alpha},U_{\alpha}, \pi_{\alpha})$. On the disjoint reunion $\coprod_{\alpha}
\mathcal{O}({U}_{\alpha})$ we consider the equivalence relation generated by
$[\Gamma'_{\alpha}]\sim[\Gamma'_{\beta}]$ if $\Gamma'_{\alpha}=(\Gamma_{\alpha})_
{\Tilde{x}_{\alpha}}\subseteq\Gamma_{\alpha}$ and $\Gamma'_{\beta}=(\Gamma_{\beta})
_{\Tilde{x}_{\beta}}\subseteq\Gamma_{\beta}$ where $\Tilde{x}_{\alpha}\in
\Tilde{U}_{\alpha}$ and $\Tilde{x}_{\beta}\in\Tilde{U}_{\beta}$ are such that
$\pi_{\alpha}(\Tilde{x}_{\alpha})=\pi_{\beta}(\Tilde{x}_{\beta})=x\in U_{\alpha}\cap
U_{\beta}$. Let $\mathcal{O}(M)=\coprod_{\alpha}\mathcal{O}({U}_{\alpha})/\!{\sim}$
be the set of orbit types of $M$. This definition does not depend on the choice of
the atlas because any addition of an extra chart to $\mathcal{A}$ will not
change $\mathcal{O}(M)$. There exists a map $\mathcal{O}:M\to\mathcal{O}(M)$
which associates to each $x\in M$ its orbit type, given by $\mathcal{O}(x)=
[(\Gamma_{\alpha})_{\Tilde{x}}]$
where $\Tilde{x}\in U_{\alpha}$ with $\pi_{\alpha}(\Tilde{x})=x$. We have a partial
order $\preceq$ on $\mathcal{O}(M)$ induced by the partial order relations on
$\mathcal{O}(U_{\alpha})\subseteq\mathcal{O}(\Gamma_{\alpha})$.
For each $\upsilon\in\mathcal{O}(M)$ let
\begin{equation}
M_{\upsilon}=\{x\in M\,|\, \mathcal{O}(x)=\upsilon\}
\end{equation}
Observe that $M_{[(e)]}$ is equal to the set of regular points $M_{reg}$.
\begin{lemma}
For any $\upsilon\in \mathcal{O}(M)$ the set $M_{\upsilon}$ is a
smooth submanifold of $M$.
\end{lemma}
\begin{proof}
It is sufficient to prove that any point $x\in M_{\upsilon}$ has a neighborhood
$U$ such that $M_{\upsilon}\cap U$ is a submanifold in $U$. Let $(\Tilde{U},\Gamma,
U,\pi)$ be an linear orbifold chart at $x$, $\pi(0)=x$. Observe that $[\Gamma]=\upsilon$.
Then for any $\Tilde{y}\in\Tilde{U}$ we have $\Gamma_{\Tilde{y}}
\subset \Gamma=\Gamma_0$. So $M_{[\Gamma]}\cap U=\pi(\Tilde{U}^{\Gamma})$.
The fixed point set $\Tilde{U}^{\Gamma}\subset\Tilde{U}$ is a submanifold given by
linear equations on which $\Gamma$ acts trivially. Then $M_{[\Gamma]}\cap
U\overset{\sim}=\Tilde{U}^{\Gamma}$ is a smooth submanifold of $U$.
\end{proof}
\begin{defn}
The decomposition
\begin{equation}
M=\bigcup_{\upsilon\in\mathcal{O}(M)} M_{\upsilon}
\end{equation}
is called the canonical stratification on $M$ and will be denoted by $\mathcal{S}(M)$.
\end{defn}
A strata $M_{\upsilon}$ is not necessarily connected. Let $M_{\upsilon}=\cup_{\alpha}
M_{\upsilon}^{\alpha}$ be the decomposition in connected components of $M_{\upsilon}$.
\begin{prop}
\label{eta:strata}
Let $\eta$ be a Dirac-type density on $M$. There exist densities
$\eta_{\upsilon}$ on the strata $M_{\upsilon}$ such that for any $f\in C^{\infty}_0(M)$
we have:
\begin{equation}
<\eta, f>=\sum_{\upsilon\in\mathcal{O}(M)}\int_{M_{\upsilon}} f_{|M_{\upsilon}}
\eta_{\upsilon}
\end{equation}
\end{prop}
\begin{proof}
We will give the construction of the density $\eta_{[\Gamma']}$ on a linear orbifold
chart $\mathcal{R}=(\Tilde{U},\Gamma, U, \pi)$. Let $\{\eta^{\gamma}\}_{\gamma
\in\Gamma}$ be the $\Gamma$ Dirac-type density on $\Tilde{U}$ that represents
$\eta_{|U}$.
For $\Gamma'\subseteq\Gamma$ let $\Tilde{U}_{\Gamma'}=\{\Tilde{x}\,|\, \Tilde{x}
\in\Tilde{U}\text{ and }\Gamma_x=\Gamma'\}$. Then $\Tilde{U}_{\Gamma'}$ is a
submanifold of $\Tilde{U}$. We have $\pi(\Tilde{U}_{\Gamma'})=U_{[\Gamma']}$
and $\pi^{-1}(U_{[\Gamma']})=\Gamma\Tilde{U}_{\Gamma'}=\cup_{\gamma\in\Gamma}
\Tilde{U}_{\gamma\Gamma'\gamma^{-1}}$.
We will need the following lemma:
\begin{lemma}
The restriction of the projection map
\begin{equation}
\pi_{|\Gamma\Tilde{U}_{\Gamma'}}:\Gamma\Tilde{U}_{\Gamma'}\to U_{[\Gamma']}
\end{equation}
is a covering map with $[\Gamma:\Gamma']$ sheets.
\end{lemma}
\begin{proof}
$\Gamma_{\Tilde{x}}$ is locally constant on $\Gamma\Tilde{U}_{\Gamma'}$,
being either $\Gamma'$ or a conjugate of it by an element in $\Gamma$.
For each $\Tilde{x}\in \Tilde{U}_{\Gamma'}$ one can
choose a small neighborhood $\Tilde{V}_{\Tilde{x}}$ of $\Tilde{x}$ in
$\Tilde{U}_{\Gamma'}$ such that $\Gamma'$ acts trivially on $\Tilde{V}_
{\Tilde{x}}$ and $\gamma\Tilde{V}_{\Tilde{x}}\cap \Tilde{V}_{\Tilde{x}}=
\emptyset$ for $\gamma\in\Gamma\setminus\Gamma'$. Let $x=\pi(\Tilde{x})$
and $V_x=\pi(\Tilde{V}_{\Tilde{x}})$.
The map $\pi_{|\Tilde{V}_{\Tilde{x}}}:\Tilde{V}_{\Tilde{x}}\to V_x$ is
a homeomorphism. For any other $\Tilde{y}=\gamma\Tilde{x}\in
\pi^{-1}(x)$ the map $\pi_{|\gamma\Tilde{V}_{\Tilde{x}}}:\gamma
\Tilde{V}_{\Tilde{x}}\to V_x$ is a homeomorphism between a neighborhood of
$\Tilde{y}$ and $V_x$. Observe also that $\pi^{-1}=\Gamma\Tilde{x}$ has exactly
$[\Gamma:\Gamma']$ points.
\end{proof}
Let us fix $\gamma\in\Gamma$. Then $\Tilde{U}^{\gamma}=\{\Tilde{x}\,|\,
\gamma\Tilde{x}=\Tilde{x}\}$ is a smooth submanifold of $\Tilde{U}$ and $
\Tilde{U}^{\gamma}=\bigcup\limits_{\gamma\in\Gamma'}\Tilde{U}_{\Gamma'}$
is a stratification of $\Tilde{U}^{\gamma}$. Then
\begin{equation}
\bigcup\limits_{\substack{\gamma\in\Gamma' \\ dim(\Tilde{U}^{\gamma})=
dim(\Tilde{U}_{\Gamma'}^{\alpha})}}\Tilde{U}_{\Gamma'}^{\alpha} \subset
\Tilde{U}^{\gamma}
\end{equation}
is an open and dense set in $\Tilde{U}^{\gamma}$ ($\{\Tilde{U}_{\Gamma'}
^{\alpha}\}_{\alpha}$ is the decomposition in connected components
of $\Tilde{U}_{\Gamma'}$).
Let $\eta^{\gamma}_{\Gamma', \alpha}$ be the restriction of the density $\eta^
{\gamma}$ to $\Tilde{U}_{\Gamma'}^{\alpha}$ and
\begin{equation}\label{eta_Gamma'}
\eta_{\Gamma',\alpha}=\frac{1}{|\Gamma'|}\sum_{\substack{\gamma\in\Gamma' \\
dim(\Tilde{U}^{\gamma})=dim(\Tilde{U}_{\Gamma'}^{\alpha})}}\eta^{\gamma}
_{\Gamma',\alpha}
\end{equation}
Let $U_{[\Gamma']}^{\alpha}=\pi(\Tilde{U}_{\Gamma'}^{\alpha})$. We showed in the
previous lemma that $\pi$ realizes a local diffeomorphism between
$\Tilde{U}_{\Gamma'}^{\alpha}$ and $U_{[\Gamma']}^{\alpha}$.
Let $\eta_{[\Gamma']}^{\alpha}(x)=\pi_*(\eta_{\Gamma',\alpha})(\Tilde{x})$
for some $\Tilde{x}\in \Tilde{U}_{\Gamma'}^{\alpha}$ such that
$\pi(\Tilde{x})=x$.
This definition is independent of the choice of $\Tilde{x}$ and
of $\Gamma'\subset\Gamma$. Indeed, let $\gamma\in\Gamma$ be such that
$\gamma\Gamma'\gamma^{-1}$ is another
representative for $[\Gamma']$ and $\pi(\gamma\Tilde{x})=x$.
After a reordering of the indices $\alpha$
we can suppose that $\Tilde{U}_{\gamma\Gamma'\gamma^{-1}}^{\alpha}=
\gamma\Tilde{U}_{\Gamma'}^{\alpha}$. Because $\{\eta^{\gamma'}\}_
{\gamma'}$ is a $\Gamma$ Dirac-type density on $\Tilde{U}$ we have
$\eta_{\gamma\Gamma\gamma^{-1},\alpha}(\gamma\Tilde{x})=\gamma_*(\eta_
{\Gamma',\alpha} (\Tilde{x}))$ and because $\pi$ commutes with
the action of $\Gamma$ on $\Tilde{U}$ we get the independence
of $\eta_{[\Gamma']}(x)$ of the choices we made.
If $\eta_{[\Gamma']}^{\alpha}$ was not defined on $U_{[\Gamma']}^{\alpha}$
because there were no $\gamma\in\Gamma$ such that $\Tilde{U}_{\Gamma'}^{\alpha}$
is an open submanifold in $\Tilde{U}^{\gamma}$, we take
$\eta_{[\Gamma']}^{\alpha}=0$.
We define $\eta_{[\Gamma']}$ on $M_{[\Gamma']}$ to be the density whose restriction
to $U_{[\Gamma']}^{\alpha}$ is equal to $\eta_{[\Gamma']}^{\alpha}$.
We need to show that
\begin{equation}
\label{eta:strata:eq}
<\eta, f>=\sum_{[\Gamma']\in\mathcal{O}(M)}
\int_{M_{[\Gamma']}} f_{|M_{[\Gamma']}} \eta_{[\Gamma']}\quad
\text{ for any } f\in C^{\infty}_0(M).
\end{equation}
It is sufficient to prove this for $f\in C^{\infty}_0(U)$.
If $\Tilde{f}\in C^{\infty}_0(\Tilde{U})^{\Gamma}$ is such that
$\Tilde{f}=\pi f$ then
\begin{align}
<\eta, f>&=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\int_{\Tilde{U}^
{\gamma}} \Tilde{f} \eta^{\gamma}=\\
&=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}(\sum_{\substack{\Gamma'
\ni \gamma \\ dim(\Tilde{U}^{\gamma})= dim(\Tilde{U}^{\alpha}_{\Gamma'})}}
\int_{\Tilde{U}^{\alpha}_{\Gamma'}}\Tilde{f}\eta^{\gamma}_
{\Gamma',\alpha})=\\
&=\frac{1}{|\Gamma|}\sum_{\alpha,\Gamma'<\Gamma}(\sum_{\substack{
\gamma\in\Gamma' \\ dim(\Tilde{U}^{\gamma})= dim(\Tilde{U}^{\alpha}_{\Gamma'})}}
\int_{\Tilde{U}^{\alpha}_{\Gamma'}}\Tilde{f}\eta^{\gamma}_{\Gamma',\alpha})=\\
&=\frac{1}{|\Gamma|}\sum_{\upsilon\in\mathcal{O}(\Gamma)}
(\sum_{\alpha, \Gamma'\in\upsilon} \int_{\Tilde{U}^{\alpha}_{\Gamma'}}
\Tilde{f}(|\Gamma'| \eta_{\Gamma',\alpha}))=
\qquad\text{(by \eqref{eta_Gamma'})}\\
&=\frac{1}{[\Gamma:\Gamma']}\sum_{[\Gamma']\in\mathcal{O}(U)}
\int_{\pi^{-1}(U^{\alpha}_{[\Gamma']})} \Tilde{f} \eta_{\Gamma',\alpha}=
\sum_{[\Gamma']\in\mathcal{O}(U)}\int_{U_{[\Gamma']}} f \eta_{[\Gamma']}
\end{align}
which is equal to the right side of \eqref{eta:strata:eq}. In the last step
we used the fact that $\pi:\pi^{-1}(U^{\alpha}_{[\Gamma']})\to
U_{[\Gamma']}^{\alpha}$ is a covering map with $[\Gamma:\Gamma']$ leaves.
\end{proof}
\vfill\eject
\section{Pseudodifferential Operators in Orbibundles}
\subsection{Pseudodifferential Operators -- The Local Model}
We remind the reader that locally any vector orbibundle over an orbifold is
obtained as $\Gamma$ orbits of a $\Gamma$ vector bundle, with
$\Gamma$ finite. We begin by analyzing the case where $\Gamma$ acts freely on
a vector bundle $E\overset{p}\to M$, in which case the associated vector
orbibundle of $\Gamma$ orbits $E/\Gamma \overset{\overline{p}}\to B/\Gamma$
is a genuine vector bundle over a manifold.
Let $R:C^{\infty}(M;E)\to C^{\infty}(M/\Gamma;E/\Gamma)$ be defined as
\begin{equation}
R(f)(\overline{x})=\frac1{|\Gamma|}\sum_{x\in \overline{x}}\overline{f(x)}
\qquad\text{ for } f\in C^{\infty}(M;E)\text{ and }\overline{x}\in M/\Gamma
\end{equation}
and $I:C^{\infty}(M/\Gamma;E/\Gamma)\to C^{\infty}(M;E)$ by
\begin{equation}
I(f)(x)=v \in E_x \quad \text{ for }f\in C^{\infty}(M/\Gamma;E/\Gamma)
\text{ and }x\in M
\end{equation}
where $v\in E_x$ is such that $f(\Gamma x)=\Gamma v$.
$v$ is unique in $p^{-1}(x)=E_x$ because the action of $\Gamma$ is free.
$I$ identifies $C^{\infty}(M/\Gamma;E/\Gamma)$ with the space of
$\Gamma$ invariant sections $C^{\infty}(M;E)^{\Gamma}$ and then $R$ is
the averaging operator over the group $\Gamma$. Obviously, $R\cdot I=Id$ and
$I\cdot R$ is the averaging operator over the group $\Gamma$. If we endow
the quotient vector bundle with the trivial $\Gamma$ action then
both $R$ and $I$ are $\Gamma$ equivariant.
Let $\Psi(M;E)$ and $\Psi(M/\Gamma;E/\Gamma)$ be the spaces of
pseudodifferential operators on $C^{\infty}(M;E)$ respectively
$C^{\infty}(M/\Gamma;E/\Gamma)$. Let
$\mathbi{R}:\Psi(M;E)\to \Psi(M/\Gamma;E/\Gamma)$ given by
\begin{equation}
\mathbi{R}(A)=R A I\quad \text{ for }A\in \Psi(M;E)
\end{equation}
$\mathbi{R}(A)$ is a pseudodifferential operator acting on $C^{\infty}
(M/\Gamma;E/\Gamma)$. To see this, observe that if the distributional kernel
of the operator $A$ is equal to $K_{A}(x,y)$ then the distributional kernel of
$\mathbi{R}(A)$ is given by the formula
\begin{equation}
K_{\mathbi{R}(A)}(\overline{x},\overline{y})=\frac1{|\Gamma|}
\sum_{\substack{x\in
\overline{x}\\ y\in \overline{y}}}\overline{K_{A}(x,y)}
\end{equation}
In a coordinate chart $(U,\phi)$ the total symbol of the operator
$\mathbi{R}(A)$ is given by
\begin{equation}
\overline{a}(\overline{x},\overline{\xi})=\dfrac{1}{|\Gamma|}\sum_
{\substack{x\in\overline{x} \\ \xi\in\overline{\xi}}}a(x,\xi)
\end{equation}
where $a(x,\xi)$ is the total symbol of $A$ in the chart $(p^{-1}U, \phi p)$.
If $B$ is the average
of $A$ with respect to the action of $\Gamma$ by conjugation on $\Psi(M;E)$,
$B=\dfrac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\gamma A \gamma^{-1}$, then
$\mathbi{R}(A)=\mathbi{R}(B)$. Moreover, $\mathbi{R}(B)$ can be seen as the
restriction of the $\Gamma$ invariant operator $B$ to $C^{\infty}(M;E)^{\Gamma}
\overset{\sim}=C^{\infty}(M/\Gamma;E/\Gamma)$.
\begin{lemma}\label{bold:I}
There exists a $\Gamma$ equivariant operator $\mathbi{I}:
\Psi(M/\Gamma;E/\Gamma)\to
\Psi(M;E)$ so that $\mathbi{R}\mathbi{I}=Id$ on $\Psi(M/\Gamma;E/\Gamma)$.
\end{lemma}
\begin{proof}
If we define $\mathbi{I}$ by $\mathbi{I}(A)=IAR$ then
$\mathbi{R}\mathbi{I}=Id$, but $IAR$ will not be, in general,
a pseudodifferential operator, unless $A$ is smoothing and then
the smooth kernel $K_{\mathbi{I}(A)}$ of $\mathbi{I}(A)$ is
defined uniquely by $\overline{K_{\mathbi{I}(A)}(x,y)}=
K_A(\overline{x},\overline{y})$.
We will define $\mathbi{I}$ using a partition
of unity of $M/\Gamma$ with smooth functions $\{\phi_{\lambda}\}_
{\lambda\in \Lambda}$ subordinated to a cover of $M/\Gamma$ with
coordinate chart neighborhoods $(U_i)_{i\in I}$ which
are slices with respect to the $\Gamma$ action on $M$. We will
choose $\phi_{\lambda}$ so that for $\lambda, \lambda' \in \Lambda$
at least one of the
following conditions holds:
\begin{enumerate}
\item[(i)]$supp(\phi_{\lambda})\cap supp(\phi_{\lambda'})=\emptyset$
\item[(ii)]$supp(\phi_{\lambda})\cup supp(\phi_{\lambda'})$ is included in
the same coordinate neighborhood.
\end{enumerate}
If $A\in \Phi(M/\Gamma;E/\Gamma)$
then $A=\sum_{\lambda,\lambda'}\phi_{\lambda}A\phi_{\lambda'}$. Denote
$A_{\lambda\lambda'}=\phi_{\lambda}A\phi_{\lambda'}$. We will define
$\mathbi{I}_{\lambda\lambda'}(A)=\mathbi{I}(A_{\lambda\lambda'})$ and
then $\mathbi{I}=\sum_{\lambda,\lambda'}\mathbi{I}_{\lambda\lambda'}$.
If $supp(\phi_{\lambda})\cap supp(\phi_{\lambda'})=\emptyset$, then
$A_{\lambda\lambda'}$ is smoothing and we define
$\mathbi{I}_{\lambda\lambda'}(A)=IA_{\lambda\lambda'}R$.
Otherwise, choose $i\in I$ such that $supp(\phi_{\lambda})\cup
supp(\phi_{\lambda'}) \subset U_i$. Because $U_i$ is a slice,
its preimage $\Tilde{U}_i\subset M$ can be
identified with $\Gamma\times U_i$ endowed with the $\Gamma$ action by
left translations. We have the induced $\Gamma$ equivariant isomorphism
\begin{equation}\label{block}
C^{\infty}(\Tilde{U}_i ;E_{|\Tilde{U}_i})\overset{\sim}=\Gamma\times
C^{\infty}(U_i;E/\Gamma_{|U_i})
\end{equation}
The operator $A_{\lambda\lambda'}$ is localized above $U_i$ and we define
$\mathbi{I}(A_{\lambda\lambda'})\in \Psi(\Tilde{U}_i ;E_{|\Tilde{U}_i})$
to be the block-diagonal operator with diagonal cells equal to
$A_{\lambda\lambda'}$, where the block representation is with
respect to the decomposition given by the isomorphism \eqref{block}.
Then $\mathbi{I}(A_{\lambda\lambda'})$ is a $\Gamma$ invariant
pseudodifferential operator on $C^{\infty}(M;E)$ localized above
$\Tilde{U}_i$ and
$\mathbi{R}\mathbi{I}(A_{\lambda\lambda'})=A_{\lambda\lambda'}$.
Define $\mathbi{I}$ to be the sum $\sum_{\lambda,\lambda'}\mathbi{I}_
{\lambda\lambda'}$. Then $\mathbi{R}\mathbi{I}=Id$ and $Im(\mathbi{I})$ is a
subset of the $\Gamma$ equivariant pseudodifferential operators in
$C^{\infty}(M;E)$.
So $\mathbi{I}$ is $\Gamma$ equivariant.
\end{proof}
Thus every pseudodifferential operator $A$ in $C^{\infty}(M/\Gamma;E/\Gamma)$
can be represented by the restriction to the $\Gamma$ invariant sections
of a $\Gamma$ equivariant pseudodifferential operator in $C^{\infty}(M;E)$
given by $\mathbi{I}(A)$.
\begin{remark}
\label{gamma:prime}
In the presence of the action of another finite group $\Gamma'$ on the vector
bundle $E\overset{p}\to M$ which is free and commutes with the action of
$\Gamma$, we have an induced natural action of $\Gamma'$ on the
quotient vector bundle
$E/\Gamma \overset{p}\to M/\Gamma$; $\gamma'\in\Gamma'$ acts on $\Gamma x$
by $\gamma'\Gamma x=\Gamma \gamma' x$. This induced action might not be free.
The maps $R$ and $I$ between the spaces of smooth
sections in the two vector bundles are $\Gamma'$ equivariant.
Indeed, the $\Gamma'$ action on the $\Gamma$ invariant sections is
the restriction of the $\Gamma'$ action on all sections, and because
$R$ and $I$ are essentially the projection onto the $\Gamma$
invariant sections
and the inclusion of the invariant sections into the space of all sections,
the two maps are $\Gamma'$ equivariant.
As a consequence, $\mathbi{R}$ is $\Gamma'$ invariant. The components
$\mathbi{I}_{\lambda,\lambda'}$ of $\mathbi{I}$ for $supp(\phi_{\lambda})\cap
supp(\phi_{\lambda'})=\emptyset$ are $\Gamma'$ equivariant as well.
To show that $\mathbi{I}_{\lambda,\lambda'}$ are $\Gamma'$ equivariant
for $supp(\phi_{\lambda})\cup supp(\phi_{\lambda'})
\subset U_i$ for a slice $U_i\subset M/\Gamma$, observe that the
identification of the preimage $\Tilde{U}_i\subset M$ of $U_i$
with $\Gamma\times U_i$ is
$\Gamma'$ equivariant as well. The map $\mathbi{I}_{\lambda,\lambda'}$
from $\Psi(U_i,E_{|U_i})$ to $\Psi(\Tilde{U}_i ;E_{|\Tilde{U}_i})$ given by
the block-diagonal construction as above is $\Gamma'$ equivariant.
So $\mathbi{I}$ is $\Gamma'$ equivariant.
\end{remark}
We will use the above facts to define the pseudodifferential operators
acting on the space of smooth sections of a vector orbibundle. We will start
with the local construction.
Let $\Tilde{U}\times V\overset{pr_1}\to \Tilde{U}$ be a $\Gamma$ vector
bundle, with $\Gamma$ a finite group. Suppose that $\Tilde{U}\subset
\mathbb{R}^m$ is an open subset with compact closure. The group $\Gamma$
acts on the space of sections $C^{\infty} (\Tilde{U}, \Tilde{U}\times V)$.
Let $\Gamma$ act on the space of operators on $C^{\infty}(\Tilde{U},
\Tilde{U}\times V)$ by conjugation: if $A$ is an operator then
$(\gamma A)(f)=\gamma A(\gamma^{-1} f)$ for any $f\in C^{\infty}(\Tilde{U},
\Tilde{U}\times V)$ and $\gamma \in \Gamma$.
Let $(\Tilde{U}\times V)/\Gamma\overset{\overline{pr}_1}\to \Tilde{U}/\Gamma$
be the associated vector orbibundle.
\begin{defn} An operator $A$ acting on $C^{\infty}_0(\Tilde{U}/\Gamma;
(\Tilde{U}\times V)/\Gamma)\overset{\sim}=C^{\infty}_0(\Tilde{U};
\Tilde{U}\times V)^{\Gamma}$ is called a pseudodifferential operator
if $A$ is the restriction to the $\Gamma$ invariant sections
of a $\Gamma$ invariant pseudodifferential operator $\Tilde{A}$ acting
on $C^{\infty}_0
(\Tilde{U}; \Tilde{U}\times V)$.
\end{defn}
We will show later in Proposition \ref{unique}, in a greater generality, that
the operator $\Tilde{A}$ whose restriction to the invariant
sections is equal to $A$ is unique up to smoothing operators.
We define the order of $A$ to be equal to the order of $\Tilde{A}$.
We denote the space of pseudodifferential operators of order $d$ by
$\Psi^d(\Tilde{U}/\Gamma; (\Tilde{U}\times V)/\Gamma)$ or simply by $\Psi^d$
and the space of all pseudodifferential operators by $\Psi$. The space $\Psi$
is a filtered algebra with respect to the composition of operators, where the
filtration is given by the degree of the operators.
An operator $A$ is smoothing if the operator $\Tilde{A}$ is smoothing.
The space of smoothing operators will be denoted by $\Psi^{-\infty}$.
\begin{defn}
$A$ is called a classical pseudodifferential operator if $\Tilde{A}$
is classical. (as described in \cite{shubin}, Section 3.7).
\end{defn}
Let $\Tilde{a}(x,\xi)$ be the total symbol of $\Tilde{A}$. $\Tilde{a}$
is a section of the
pull-back of the endomorphism vector bundle to the tangent space, $\Tilde{a}\in
C^{\infty}(T^*(\Tilde{U}), T^*(\Tilde{U})\times End(V))$. The principal symbol
$\Tilde{a}_{pr}$ is a section of the same vector bundle.
The group $\Gamma$ acts naturally on the vector bundle $T^*(\Tilde{U})
\times End(V)\overset{pr_1}\to T^*(\Tilde{U})$. Then $\Gamma$ acts
on the sections of this vector bundle and $\gamma \Tilde{a}_{pr}$
is the principal symbol of the operator $\gamma \Tilde{A}$ (cf. Lemma 1.1.3
\cite{gilkey}). This is the direct consequence of the invariance of
the principal symbol with respect to changes of coordinates. Because
$\Tilde{A}$ is $\Gamma$ invariant we conclude that $\Tilde{a}_{pr}$ is
a $\Gamma$ invariant section so it defines a smooth section in the
vector orbibundle
$(\Tilde{U}\times End{V})/\Gamma\overset{\overline{pr}_1}\to\Tilde{U}/\Gamma$.
Because $\Tilde{A}$ is unique up to smoothing operators,
$\Tilde{a}_{pr}$ depends only on $A$ and will be called the principal symbol
of the operator $A$.
In the case when the $\Gamma$ action on the vector bundle $\Tilde{U}\times V
\overset{pr_1}\to \Tilde{U}$ is given by the restriction to $\Tilde{U}\subset
\mathbb{R}^m$ of a linear action of $\Gamma$ on $\mathbb{R}^m$ and
by a $\Gamma$ representation $\rho:\Gamma\times V\to V$, we will be
able to define the total symbol of the pseudodifferential operator $A$.
This is the consequence of the invariance of the total symbol of
$\Tilde{A}$ with respect to linear changes of coordinates. The total
symbol of $A$ is the given by $\Tilde{a}\in C^{\infty}_0((T^*(\Tilde{U})
\times V)/\Gamma; T^*(\Tilde{U}))$ and is unique up to a smoothing symbol.
If $\Tilde{A}$ is classical and
\begin{equation}
\label{asymptotic:exp}
\Tilde{a}(x,\xi)\sim\sum_{i\ge 0} \Tilde{a}_{d-i}(x,\xi)
\end{equation}
is an asymptotic expansion of the total symbol with $\Tilde{a}(x, \xi)_{d-i}$
being homogeneous symbols of degree of homogeneity $d-i$, then each
homogeneous component is $\Gamma$ invariant and will define the
homogeneous component $a_{d-i}$ of the
asymptotic expansion of the total symbol of the operator $A$. The formal
sum in the right-hand side of the equality \eqref{asymptotic:exp} is
called the asymptotic symbol of the operator $A$.
If a pseudodifferential operator $A$ of order $d$ acts on the sections of the
vector orbibundle $(\Tilde{U}\times V)/\Gamma \overset{\overline{pr}_1}
\to \Tilde{U}/\Gamma$ then, as in the case of genuine vector bundles,
it extends to a continuous operator between Sobolev spaces
$A:H_s(\Tilde{U}/\Gamma; (\Tilde{U}\times V)/\Gamma)
\to H_{s-d}(\Tilde{U}/\Gamma; (\Tilde{U}\times V)/\Gamma)$ (cf. Lemma 1.2.1
\cite{gilkey})
\begin{remark}
\label{smoothing}
If $A\in \Psi^{-\infty}$ then there exists a $\Gamma$ invariant
smoothing operator $\Tilde{A}$ whose restriction to the $\Gamma$ invariant
sections is
equal to $A$. Because $\gamma \Tilde{A}=\Tilde{A} \gamma$ for all
$\gamma\in \Gamma$,
the kernel $\Tilde{K}(\Tilde{x}, \Tilde{y})$ of $\Tilde{A}$ satisfies
the equality
$\gamma\Tilde{K}(\gamma^{-1}\Tilde{x}, \gamma^{-1}\Tilde{y})=\Tilde{K}
(\Tilde{x},\Tilde{y})$ so
it defines a $\Gamma$ invariant section in the vector bundle $(\Tilde{U}
\times\Tilde{U})\times\End(V)\overset{pr_1}\to (\Tilde{U}\times\Tilde{U})$.
Here
$\Gamma$ acts by the diagonal action on $\Tilde{U}\times\Tilde{U}$.
If we define
\begin{equation}
\Tilde{K}'(\Tilde{x}, \Tilde{y})=\frac {1}{|\Gamma|}\sum_{\gamma\in \Gamma}
\gamma \Tilde{K}(\Tilde{x}, \gamma^{-1}\Tilde{y})
\end{equation}
then $\Tilde{K}'$ is the kernel of the $\Gamma$ invariant smoothing operator
$\Tilde{A}'=\frac{1}{\Gamma}\sum_{\gamma\in \Gamma}\Tilde{A}\gamma$ whose
restriction to the $\Gamma$ invariant sections induces $A$. $\Tilde{K}'$
defines
a $\Gamma\times\Gamma$ invariant smooth section of the vector bundle
$(\Tilde{U}
\times\Tilde{U})\times\End(V)\overset{pr_1}\to (\Tilde{U}\times\Tilde{U})$
where
$\Gamma\times\Gamma$ acts by the product action on $\Tilde{U}\times\Tilde{U}$.
Then $\Tilde{U}\times\Tilde{U}/\Gamma\times\Gamma\overset{\sim}=U\times U$ and
$\Tilde{K}'$ defines a smooth section of the endomorphism vector orbibundle
$(\Tilde{U}\times\Tilde{U})\times \End(V)/\Gamma\times\Gamma\overset{pr_1}\to
U\times U$ which will be the smooth kernel of the operator $A$.
\end{remark}
\begin{defn}
\label{elliptic:local}
The pseudodifferential operator $A$ of order $d$ is called elliptic above
an open set
$U_1\subset \Tilde{U}/\Gamma$ if the operator $\Tilde{A}$ is elliptic above
the preimage of $U_1$ in $\Tilde{U}$.
\end{defn}
Let $\Tilde{U}_1\subset \Tilde{U}$ be the preimage of $U_1$.
The above definition implies the existence of a parametrix $\Tilde{B}'$
for $\Tilde{A}$
above $\Tilde{U}_1$. Then $\Tilde{B}=\frac{1}{|\Gamma|} \sum_{\gamma\in
\Gamma} \gamma
\Tilde{B}'\gamma^{-1}$ is a $\Gamma$ invariant parametrix for $\Tilde{A}$
and defines
a parametrix $B\in \Psi^{-d}$ for $A$, $BA-Id\in \Psi^{-\infty}$, $AB-Id\in
\Psi^{-\infty}$.
\subsection{Pseudodifferential Operators in Orbibundles}
Let $E\overset{p}\to M$ be a vector orbibundle and $A$ be a linear operator
acting on the space of smooth sections $C^{\infty}(M;E)$.
If $\mathcal{R}=(\Tilde{U},\,V,\,\Gamma,U,\Pi,\pi)$ is
an orbibundle chart for $E\overset{p}\to M$ above $U\subset M$ then
$\Tilde{U}\times V\overset{pr_1}\to \Tilde{U}$ is a $\Gamma$ vector bundle and
passing to the orbit space yields the orbibundle
$\Tilde{U}\times V/\Gamma \overset{\overline{pr}_1}\to \Tilde{U}/\Gamma$.
This orbibundle is isomorphic to the
restriction of $E\overset{p}\to M$ to $U$ via the orbibundle map
$(\overline{\Pi},\overline{\pi})$:
$$
\xymatrix{
\Tilde{U}\times V/\Gamma\ar@{-->}[r]^{\overline{\Pi}} \ar[d]_{\overline{pr}_1}
& E_{|U}\ar[d]^{p}
\\
\Tilde{U}/\Gamma\ar@{-->}[r]_{\overline{\pi}}&U}
$$
The orbibundle $E\overset{p}\to M$ can be covered by such orbibundle maps.
Let $\phi$ and $\psi$ two smooth functions on $M$ such that their support
is included in $U$. We will denote by the same letter the multiplication
operator by the respective functions. Then the operator $\phi\circ A\circ\psi$
has support in $U$ and takes sections with support in $U$ to sections
with support in $U$. Using the orbibundle isomorphism $\overline{\pi}$,
one can define the operator $A_{\mathcal{R}}$ acting on the sections
of $\Tilde{U}\times V/\Gamma \to \Tilde{U}/\Gamma$ as
\begin{equation}
\label{localize:op}
A_{\mathcal{R}}\,f=\overline{\pi} (\phi\circ A\circ\psi){\overline
{\pi}}^{-1}\,f.
\end{equation}
\begin{defn} $A$ is a pseudodifferential operator if $A_{\mathcal
{R}}$ is a pseudodifferential operator for any orbibundle chart $\mathcal{R}$.
\end{defn}
In order for the previous definition to be consistent, we need to show that
this property is independent of the choice of a chart. More precisely:
\begin{prop}
\label{pdounique}
Let $\Tilde{E}_i\overset{p_i}\to\Tilde{M}_i$ be two
$\Gamma_i$ vector bundles, $i=1,2$, such that the corresponding orbibundles
are isomorphic via an isomorphism $( T, t)$. Let $A_1$ be an operator
acting on sections of $\Tilde{E}_1/\Gamma_1\overset{p_1}\to
\Tilde{M}_1/\Gamma_1$ and $A_2= (T,t)_*\circ A_1\circ (T,t)^*$ the
corresponding
operator acting on $\Tilde{E}_2/\Gamma_2\overset{p_2}\to
\Tilde{M}_2/\Gamma_2$.
Then there exists a $\Gamma_1$ equivariant pseudodifferential operator
$\Tilde{A}_1$ acting on sections of $\Tilde{E}_1\overset{p_1}\to
\Tilde{M}_1$ whose restriction to $\Gamma_1$
invariant sections is equal to $A_1$ if and only if there exists a
$\Gamma_2$ equivariant
pseudodifferential operator $\Tilde{A}_2$ acting on sections of
$\Tilde{E}_2\overset{p_2}\to\Tilde{M}_2$
whose restriction to $\Gamma_2$ invariant sections is equal to $A_2$.
\end{prop}
\begin{proof} For each $x_i \in \Tilde{M}_i/\Gamma_i$ such that $ t(x_1)=
x_2$ choose $\tilde{x}_i \in \Tilde{M}_i$ with $\pi_i(\tilde{x}_i)=x_i$.
Because the orbibundles are isomorphic, there exists a group isomorphism
$\alpha$ between $(\Gamma_i)_{\tilde{x}_i}$--the corresponding isotropy
groups of $\tilde{x}_i$ in $\Gamma_i$ and one can find neighborhoods
$\Tilde{W}_i$ of $\tilde{x}_i$ in $\Tilde{M}_i$ which are $(\Gamma_i)_
{\tilde{x}_i}$ invariant and an $\alpha$-equivariant bundle diffeomorphism
$( L, l)$ that makes the following diagram commutative:
\begin{equation}\label{local:diffs}
\xymatrix{
\Tilde{E}_1 \ar[dd]_{p_1}& & & \Tilde{E}_2\ar[dd]^
{p_2}\\
&\Tilde{W}_1\times V \ar@{_{(}->}[ul] \ar[r]^{ L}_{\sim} \ar[dd]_{p_1} &
\Tilde{W}_2 \times V \ar@{^{(}->}[ur] \ar[dd]^{p_2}& \\
\Tilde{M}_1 \ar[dd]_{\pi_1}& & & \Tilde{M}_2\ar[dd]^{\pi_2}\\
&\Tilde{W}_1 \ar@{_{(}->}[ul] \ar[r]^{ l}_{\sim} \ar@{-}'[d]_{\pi_1}
\ar[dl]_{\pi_1} &
\Tilde{W}_2 \ar@{^{(}->}[ur] \ar[dr]^{\pi_2} \ar@{-}'[d]^{\pi_2}& \\
\Tilde{M}_1/\Gamma_1 \ar[rrr]^{ t}_{\sim} & \ar[d]&\ar[d] & \Tilde{M}_2/
\Gamma_2\\
&\Tilde{W}_1/(\Gamma_1)_{\tilde{x}_1}\ar@{_{(}->}[ul]
\ar[r]^{\overline{l}}_{\sim} & \Tilde{W}_2/(\Gamma_2)_{\tilde{x}_2}
\ar@{^{(}->}[ur]&
}
\end{equation}
Let $\{\phi_{\lambda}\}_{\lambda\in\Lambda}$ be a partition of unity of
$\Tilde{M}_i/\Gamma_i$ subordinated
to the open cover $\Tilde{W}_i/(\Gamma_i)_{\tilde{x}_i}$
(because $\Tilde{M}_i/\Gamma_i$ are diffeomorphic and the diffeomorphism $t$
permutes the two open covers, we will denote the two partition of
unity with the
same letters, thought we will refer to functions on two different,
but diffeomorphic
orbifolds). We can choose this partition so that for any $\lambda,\,
\lambda' \in \Lambda$ at least one of the following conditions holds:
\begin{enumerate}
\item[(i)]$supp(\phi_{\lambda})\subset \Tilde{W}_i/(\Gamma_i)_{\tilde{x}_i}$,
$\quad supp(\phi_{\lambda'})\subset \Tilde{W'}_i/(\Gamma_i)_{\tilde{x}_i'}$ and
\newline
$\Tilde{W}_i/(\Gamma_i)_{\tilde{x}_i}\cap \Tilde{W'}_i/(\Gamma_i)_
{\tilde{x}_i'}
=\emptyset\qquad$
\item[(ii)]$supp(\phi_{\lambda})\cup supp(\phi_{\lambda'})\subset \Tilde{W}_i/
(\Gamma_i)_{\tilde{x}_i}$.
\end{enumerate}
We will show only one implication of the proposition, the other implication
can be proved similarly.
Let $\Tilde{A}_1$ be a $\Gamma_1$ equivariant pseudodifferential
operator acting on the smooth sections of $\Tilde{E}_1
\overset{p_1}\to \Tilde{M}_1$ whose restriction to the
invariant sections is equal to $A_1$. We will construct a lift $\Tilde{A}_2$
as required in the proposition.
Because
\begin{equation}\label{lambda:decomp}
A_1=\sum_{\lambda,\lambda'\in\Lambda}\phi_{\lambda}A_1 \phi_{\lambda'}
\end{equation}
and because a lift for the operator $\phi_{\lambda}A_1
\phi_{\lambda'}$ is provided by $\tilde{\phi_{\lambda}}
\Tilde{A}_1\tilde{\phi_{\lambda'}}$, we reduced our problem to the
operators of the form $\phi_{\lambda}A_1 \phi_{\lambda'}$ which we
will denote by $(A_1)_{\lambda,\lambda'}$ for simplicity (we denoted by
$\tilde{\phi}$
the lift of the smooth function $\phi$).
Denote the lifts $\tilde{\phi_{\lambda}}\Tilde{A}_1\tilde{\phi_{\lambda'}}$
by $(\Tilde{A}_1)_{\lambda,\lambda'}$.
In the first case the operator $(\Tilde{A}_1)_{\lambda,\lambda'}$
is smoothing.
In the second case the operator $(\Tilde{A}_1)_{\lambda,\lambda'}$
is localized to the
open set $\Gamma_1\cdot\Tilde{W}_1\overset{\sim}=\Gamma_1\times_
{(\Gamma_1)_{\tilde{x}_1}}\Tilde{W}_1$. We will treat the two cases separately.
In the first case define:
\begin{equation}\label{bi:average}
\overline{A}_1=\sum_{\gamma\in\Gamma_1}(\Tilde{A}_1)_{\lambda\lambda '}\,\gamma
\end{equation}
Because $(\Tilde{A}_1)_{\lambda\lambda '}$ was smoothing and $\Gamma_1$
equivariant, $\overline{A}_1$ is smoothing and $\Gamma_1$ bi-invariant:
\begin{equation}\label{bi:invariant}
\mu \overline{A}_1=\overline{A}_1\nu=\overline{A}_1\qquad \text{ for any }
\mu,\nu\in\Gamma_1
\end{equation}
Moreover $\overline{A}_1$ induces the same operator $(A_1)_{\lambda\lambda'}$
at the orbibundle level because its action on the $\Gamma_1$ invariant
sections is not changed.
If we chooose a $\Gamma_1$ invariant metric on $M_1$ then the
kernel $\overline{K}_1(\tilde{x},\tilde{y})$ of the operator $\overline{A}_1$
is a smooth section in $C^{\infty}(\Tilde{M}_1\times \Tilde{M}_1;
\End(\Tilde{E}_1))$
and because of the equalities \eqref{bi:invariant} it satisfies the condition
$\mu K_1(\mu^{-1}\tilde{x},\tilde{y})=\nu K(\tilde{x},\nu^{-1}\tilde{y})$
for all $\mu,\nu\in\Gamma_1$.
This implies that $K_1$ is a $\Gamma_1\times\Gamma_1$ invariant smooth
section in the
$\Gamma_1\times\Gamma_1$ vector bundle $\End(\Tilde{E}_1)\to\Tilde{M}_1
\times\Tilde{M}_1$ and so it defines a smooth section in the vector orbibundle
$\End(\Tilde{E}_1/\Gamma_1)\to \Tilde{M}_1/\Gamma_1\times \Tilde{M}_1/
\Gamma_1$.
Because the vector orbibundles $\Tilde{E}_i/\Gamma_i\overset{p_i}\to
\Tilde{M}_i/\Gamma_i$,
$i=1,2$, are isomorphic, there exists a unique smooth $\Gamma_2\times
\Gamma_2$ invariant
section $\overline{K}_2$ in $\End(\Tilde{E}_2)\to\Tilde{M}_2\times
\Tilde{M}_2$ which induces
the same section via the isomorphism $(T,t)$ at the orbibundle level as
$\overline{K}_1$.
The smoothing operator $\overline{A}_2$ with kernel $\overline{K}_2$ is
$\Gamma_2$ bi-invariant,
and so it is $\Gamma_2$ equivariant. The operator $\overline{A}_2$ induces
at the orbibundle level
the operator $(T,t)_* \circ (A_1)_{\lambda,\lambda '} \circ (T,t)^*$.
The statement of the proposition is proved in the case (i).
In the case (ii) $(\Tilde{A}_1)_{\lambda \lambda'}$ is a $\Gamma_1$
equivariant operator in
$\Psi( \Gamma_1\Tilde{W}_1;\Tilde{E}_1{}_{|\Gamma_1\Tilde{W}_1})$.
Let us denote
$\Gamma_1'= (\Gamma_1)_{\tilde{x}_1}$. The $\Gamma_1$
vector bundle $\Tilde{E}_1{}_{|\Gamma_1\Tilde{W}_1}\to\Gamma_1\Tilde{W}_1$
is isomorphic with
$\Gamma_1\times_{\Gamma_1'}(\Tilde{W}_1\times V)\to\Gamma_1\times_{\Gamma_1'}
\Tilde{W}_1$.
Observe that $\Gamma_1\times_{\Gamma_1'}\Tilde{W}_1=(\Gamma_1\times
\Tilde{W}_1)/\Gamma_1'$.
We will use the observations regarding the operators $\mathbi{R}$
and $\mathbi{I}$ made at the beginning of
section 3.1.
Cf. Lemma \ref{bold:I}, there exists a $\Gamma_1'$ equivariant
pseudodifferential operator
$\mathbi{I}((\Tilde{A}_1)_{\lambda \lambda'}) \in \Psi(\Gamma_1\times
\Tilde{W}_1;
\Gamma_1\times(\Tilde{W}_1\times V))$ which is $\Gamma_1$ equivariant as well.
An element $\gamma'\in\Gamma_1'$ acts on $\Gamma_1\times\Tilde{W}_1$
by right multiplication on $\Gamma_1$ by $\gamma'{}^{-1}$ and left
multiplication on $\Tilde{W}_1$ by $\gamma'$.
Let $\mathbi{R}\mathbi{I}((\Tilde{A}_1)_{\lambda \lambda'})$ be the
induced operator to the vector
bundle of $\Gamma$ orbits $\Gamma\times(\Tilde{W}_1\times V)/\Gamma
\to \Gamma\times
\Tilde{W}_1/\Gamma$ which is isomorphic to $\Tilde{W}_1\times V\to
\Tilde{W}_1$. Because
$\mathbi{I}((\Tilde{A}_1)_{\lambda \lambda'})$ was $\Gamma_1'$ equivariant,
cf. remark \ref{gamma:prime},
the operator $\mathbi{R}\mathbi{I}((\Tilde{A}_1)_{\lambda \lambda'})$
is $\Gamma'$ equivariant.
We have the natural isomorphisms between the spaces of invariant sections
\begin{multline}
C^{\infty}(\Tilde{W}_1;\Tilde{W}_1\times V)^{\Gamma_1'}\overset{\sim}=
C^{\infty}(\Gamma_1\times\Tilde{W}_1;\Gamma_1\times(\Tilde{W}_1\times V))^
{\Gamma_1\times\Gamma_1'}\overset{\sim}=\\
\overset{\sim}=C^{\infty}(\Gamma_1\times_{\Gamma_1'}\Tilde{W}_1;\Gamma_1
\times_{\Gamma_1'}(\Tilde{W}_1\times V))^{\Gamma_1}
\end{multline}
which can be identified with the space of smooth sections in the vector
orbibundle
$\Tilde{E}_1/\Gamma_1 \overset{p_1}\to \Tilde{M}_1/\Gamma_1$ above the
open set $\Tilde{W}_1/\Gamma_1'$.
The operators $\mathbi{R}\mathbi{I}((\Tilde{A}_1)_{\lambda \lambda ' })$,
$\mathbi{I}((\Tilde{A}_1)_{\lambda \lambda ' })$
and $(\Tilde{A}_1)_{\lambda \lambda ' }$ restricted to the above spaces
of invariant sections will be equal via the natural isomorphisms.
In particular they induce $(A_1)_{\lambda \lambda '}$ at the level of sections
in the associated vector orbibundle
$\Tilde{E}_1/\Gamma_1 \overset{p_1}\to \Tilde{M}_1/\Gamma_1$.
We define the $\Gamma_2'$ equivariant operator
$\overline{A}_2=(L,l)_*\circ\mathbi{R}\mathbi{I}((\Tilde{A}_1)_{\lambda
\lambda'})\circ
(L,l)^*$ in $\Psi(\Tilde{W}_2;\Tilde{W}_2\times V)$ using the isomorphisms
$(L,l)$ provided in the diagram \eqref{local:diffs}.
(here, as before, $\Gamma_2'=(\Gamma_2)_{\tilde{x}_2}$). Let
$\mathbi{I}(\overline{A}_2)\in\Psi(\Gamma_2\times\Tilde{W}_2;\Gamma_2\times
(\Tilde{W}_2\times V))$
be the $\Gamma_2$ equivariant operator which induces $\overline{A}_2$ at the
$\Gamma_2$ orbit level, and let $\mathbi{R}\mathbi{I}(\overline{A}_2)
\in\Psi(\Gamma_2\times_{\Gamma_2'}\Tilde{W}_2;\Gamma_2
\times_{\Gamma_2'}(\Tilde{W}_2\times V))$ be the
operator induced at the $\Gamma_2'$ orbit level. Cf. remark \ref{gamma:prime},
this operator is $\Gamma_2$ equivariant. Using the sequence of isomorphisms:
\begin{multline}
C^{\infty}(\Tilde{W}_2;\Tilde{W}_2\times V)^{\Gamma_2'}\overset{\sim}=
C^{\infty}(\Gamma_2\times\Tilde{W}_2;\Gamma_1\times(\Tilde{W}_2\times V))^
{\Gamma_2\times\Gamma_2'}\overset{\sim}=\\
\overset{\sim}=C^{\infty}(\Gamma_2\times_{\Gamma_2'}\Tilde{W}_2;\Gamma_2
\times_{\Gamma_2'}(\Tilde{W}_2\times V))^{\Gamma_2}
\end{multline}
observe that the operators $\mathbi{R}\mathbi{I}(\overline{A}_2)$ and
$\overline{A}_2$ act the same way on the
corresponding spaces of invariant sections and induce $(A_2)_{\lambda
\lambda '}=(T,t)_*\circ ({A}_1)_
{\lambda \lambda '}\circ (T,t)^*$ at the vector orbibundle level. If we extend
$\mathbi{R}\mathbi{I}(\overline{A}_2)$
by $0$ outside the neighborhood $\Gamma_2\Tilde{W}_2$ we obtain a lift of
$(A_2)_{\lambda \lambda '}$.
We proved the proposition in the case (ii) as well.
\end{proof}
We also have the uniqueness up to smoothing operators of the lift of a
pseudodifferential operator.
\begin{prop}
\label{unique}
Let $\Tilde{E}\overset{p}\to\Tilde{M}$ a $\Gamma$ vector bundle,
with $\Gamma$ a finite group and $E\overset{\overline{p}}\to M$ the
associated orbibundle. If $\Tilde{A}_i$, $i=1,2$ are two classical
pseudodifferential operators on $C^{\infty}(\Tilde{M};\Tilde{E})$ which are
$\Gamma$ equivariant and induce the same pseudodifferential operator on
$C^{\infty}(M;E)$, then $\Tilde{A}_1-\Tilde{A}_2$ is a smoothing operator.
\end{prop}
\begin{proof}
It is sufficient to prove that if $\Tilde{A}$ is a classical
pseudodifferential operator that induces the zero operator on $C^{\infty}
(M;E)$, then $\Tilde{A}$ is smoothing.
Let $\tilde{x}\in\Tilde{M}$ be a point such that its projection $\pi
(\tilde{x})$ onto
$M=\Tilde{M}/\Gamma$ is a smooth point. We will show that the total symbol
of the operator $\Tilde{A}$ on a neighborhood of $\tilde{x}$ is smoothing
(though the total symbol of the operator is not well-defined, the property
of being a smoothing symbol is independent of the chart around $\tilde{x}$).
Let $\Tilde{W}$ be a slice at $\tilde{x}$ with respect to the action of the
group $\Gamma$. Because the point $x=\pi(\tilde{x})$ is smooth, the isotropy
group $(\Gamma)_{\tilde{x}}$ is trivial. Let $\phi$ be a smooth function
on $\Tilde{M}$ with support inside $\Tilde{W}$ which is equal to 1
on a neighborhood of a smaller slice $\Tilde{V}$ at $\tilde{x}$.
Let $\psi$ be a smooth function on $\Tilde{M}$ which is equal to 1 on
a neighborhood of $supp(\phi)$ and zero outside $\Tilde{W}$.
$\phi\Tilde{A}\phi$ is a pseudodifferential operator which is
localized on $\Tilde{W}$. We will prove that this operator is smoothing.
Then $\Tilde{A}$ will be smoothing above $\Tilde{V}$ where $\phi=1$.
Let $\tilde{\phi}=\sum_{\gamma\in\Gamma}\gamma\phi$ be the $\Gamma$ invariant
extension of $\phi$ to $\Tilde{M}$. Then $\tilde{\phi}\Tilde{A}\tilde{\phi}$
is a $\Gamma$ equivariant extension to $C^{\infty}(\Tilde{M};\Tilde{E})$
of the zero operator on $C^{\infty}(M;E)$. Let $f\in C^{\infty}(\Tilde{M};
\Tilde{E})$ be an arbitrary section with support in a
neighborhood of $supp(\psi)$ that vanishes outside the set $\{\tilde{x}|
\psi(\tilde{x})=1\}$.
Then $\tilde{f}=\sum_{\gamma\in\Gamma}\gamma f$ is a $\Gamma$ invariant
section that extends $f$.
Because $\tilde{\phi}\Tilde{A}\tilde{\phi}$ restricted to the $\Gamma$
invariant section is equal to zero, we have:
\begin{align}
0&=\tilde{\phi}\Tilde{A}\tilde{\phi}(\tilde{f})=
\psi\tilde{\phi}\Tilde{A}\tilde{\phi}(\sum_{\gamma\in\Gamma}\gamma f)=\\
&=\psi\tilde{\phi}\Tilde{A}\tilde{\phi}\psi (\sum_{\gamma\in\Gamma}\gamma f)+
\psi\tilde{\phi}\Tilde{A}\tilde{\phi}(1-\psi)(\sum_{\gamma\in\Gamma}
\gamma f)=\\
&=\phi\Tilde{A}\phi(f)+(\sum_{\gamma\in\Gamma}\phi\Tilde{A}\tilde{\phi}(1-\psi)
\gamma)(f)=0
\end{align}
But $\tilde{\phi}(1-\psi)$ has support inside $\displaystyle\cup_{\gamma\ne e}
\gamma\Tilde{W}$ which is disjoint from $\Tilde{W}$ so $\phi\Tilde{A}
\tilde{\phi}
(1-\psi)$ and so $\sum_{\gamma\in\Gamma}\phi\Tilde{A}\tilde{\phi}(1-\psi)
\gamma$ are smoothing. Because $f$ was chosen arbitrary, we conclude that
$\phi\Tilde{A}\phi$ is smoothing so $\Tilde{A}$ is smoothing above $\Tilde{V}$.
Then $\Tilde{A}$ is smoothing on the regular set $\Tilde{M}_{\text{reg}}=
\{\tilde{x}|(\Gamma)_{\tilde{x}}=(e)\}$.
If $\tilde{x}_1\in\Tilde{M}_{\text{sing}}$ is a singular point then, on an
Euclidean chart around $\tilde{x}_1$ the total symbol of $\Tilde{A}$ will
have the asymptotic expansion
\begin{equation}
a(\tilde{x},\xi)\sim\sum_{k\ge 0}a_{d-k}(\tilde{x},\xi)
\end{equation}
But $\Tilde{M}_{\text{reg}}$ is dense in $\Tilde{M}$ so $a_{d-k}(\tilde{x},
\xi)=0$ on a dense set because $a(\tilde{x},\xi)$ is smoothing on $\Tilde{M}_
{\text{reg}}$. We conclude that $a_{d-k}(\tilde{x},\xi)=0$ on a
neighborhood of $\tilde{x}_1$ so $\Tilde{A}$ is smoothing on that neighborhood
and thus on the whole manifold $\Tilde{M}$.
\end{proof}
We denote by $\Psi(M;E)$ or simply by $\Psi$ the space of pseudodifferential
operators acting on the smooth sections of the vector orbibundle
$E\overset{p}\to M$. Most of the definitions and constructions for the
pseudodifferential operators acting on genuine vector bundles can be extended
to the operators in $\Psi(M;E)$.
The symbol of a pseudodifferential operator $A$ in an linear vector
orbibundle chart $\mathcal{R}$ is a section in the vector orbibundle
$\End(E)\overset{p}\to T^*(U)$ and is unique up to a smoothing symbol.
The order of a pseudodifferential operator
is the maximum of the orders of the restrictions to vector orbibundle charts
and denote by $\Psi^d$ the subspace of operators of order less or equal to $d$.
$\Psi$ becomes a filtered algebra with respect to the compositions of
operators.
\begin{defn} A pseudodifferential operator $A$ of order $d$ is called elliptic
if it is elliptic when restricted to all vector orbifold charts.
\end{defn}
An elliptic pseudodifferential operator $A\in \Psi^d$ has a parametrix
$B\in\Psi^{-d}$ such that $AB-Id$ and $BA-Id$ are both smoothing. The
construction of $B$ can be done locally, in each vector orbibundle chart,
as in the case of pseudodifferential operators on manifolds.
In Proposition \ref{represent} we showed that any vector orbibundle
$E\overset{p}\to M$ can be realized as a map between the orbit spaces
of a $G$ vector bundle
$\Tilde{E}\overset{\Tilde{p}}\to \Tilde{M}$, with $G$ a compact Lie group.
If $G$ was finite, Proposition \ref{pdounique} shows that any
pseudodifferential operator acting on $C^{\infty}(M;E)$ can be realized
as a $G$ equivariant pseudodifferential operator acting on
$C^{\infty}(\Tilde{M};\Tilde{E})$. The following proposition extend
this result to the case of Lie groups.
\begin{prop}
\label{pdolift}
Let $\Tilde{E}\overset{\Tilde{p}}\to \Tilde{M}$ be a $G$ vector bundle
over a compact smooth manifold $M$ with $G$ a compact
Lie group such that any $x\in M$ has a finite isotropy group $G_x\subset G$
and denote by $E\overset{p}\to M$ the associated vector orbibundle of
$G$ orbits.
For any pseudodifferential operator $A$ acting on $C^{\infty}(M;E)$ there
exists a
$G$ equivariant pseudodifferential operator $\Tilde{A}$ acting on
$C^{\infty}(\Tilde{M};
\Tilde{E})$ of the same order as $A$ such that the restriction to the
$G$ invariant sections
$C^{\infty}(\Tilde{M};\Tilde{E})^G \overset{\sim}=C^{\infty}(M;E)$ is
equal to $A$.
If $A$ is classical then $\Tilde{A}$ can be chosen classical.
\end{prop}
\begin{proof}
As described in Proposition \ref{lievcano} one can find an open cover
$(U_{\alpha})
_{\alpha}$ of the orbifold $M$, a family of finite groups
$(G_{\alpha})_{\alpha}$
, $G_{\alpha}\subset G$, and a vector space $V$ such that $E_{|U_{\alpha}}
\overset{p}\to U_{\alpha}$ is isomorphic to the associated vector orbibundle
of $G_{\alpha}$ orbits of the $G_{\alpha}$ vector bundle $\Tilde{U}_{\alpha}
\times V \overset{pr_1}\to\Tilde{U}_{\alpha}$ and $\Tilde{M}$ can be
covered with a family of open sets $(N_{\alpha})_{\alpha}$
which are $G$ equivariant diffeomorphic to $G\times_{G_{\alpha}}
\!U_{\alpha}$ and such that the restriction of the $G$ vector bundle $\Tilde{E}
\overset{\Tilde{p}}\to \Tilde{M}$ to $N_{\alpha}$ is diffeomorphic with the
$G$ vector bundle $G\times_{G_{\alpha}}\!(U\times V)\overset{pr_1}\to G
\times_{G_{\alpha}}\!U$ by a $G$ equivariant vector bundle diffeomorphism.
The cover $(U_{\alpha})_{\alpha}$ can be chosen to be finite because
$\Tilde{M}$ is compact.
\par
Using a partition of unity, any pseudodifferential operator $A$ acting on
sections
of the vector orbibundle $E\overset{p}\to M$ can be written as
$A=\sum_{\alpha}A_{\alpha}+K$ where $K$ is a smoothing operator with
smooth kernel
which is zero on a neighborhood of the diagonal and $A_{\alpha}$ are
pseudodifferential operators which vanish on sections that are
supported outside the
open set $U_{\alpha}$ and take sections with support inside $U_{\alpha}$
into sections with support inside $U_{\alpha}$.
For each $\alpha$ we will construct a $G$ equivariant pseudodifferential
operator $\Tilde{A}_{\alpha}$ acting on sections of $\Tilde{E}
\overset{\Tilde{p}}
\to \Tilde{M}$ whose restriction to the $G$ invariant sections induces
$A_{\alpha}$.
\par
First let us fix a $G$ bi-invariant pseudodifferential operator $Q$ acting
on $C^{\infty}(G)$ of order $d=ord(A)$ such that $Qf=0$ on constant
functions $f$.
In all the situations $Q$ can be taken to be the $d/2$ power of the
Laplacian $\Delta$ on $G$ with respect to a bi-invariant metric on $G$.
Using Proposition \ref{pdounique} we can find a $G_{\alpha}$ equivariant
pseudodifferential operator $B_{\alpha}$ acting on the space of sections
$C^{\infty}(\Tilde{U}_{\alpha};V)$ of
the $G_{\alpha}$ bundle $\Tilde{U}_{\alpha}\times V\overset{pr_1}\to
\Tilde{U}_{\alpha}$ which induces $A_{\alpha}$ when restricted to
$G_{\alpha}$ invariant sections.
The space of smooth sections of the $G$ vector bundle $G\times
(\Tilde{U}_{\alpha}\times V)\overset{pr_1}\to G\times \Tilde{U}_{\alpha}$
is equal to $C^{\infty}(G\times \Tilde{U}_{\alpha}; V)\overset{\sim}=
C^{\infty}(G)\hat{\otimes}C^{\infty}(\Tilde{U}_{\alpha};V)$ so the
pseudodifferential operator $Q\otimes 1+1\otimes
B_{\alpha}$ acts on this space and it has order $d$. It is $G$ equivariant.
The finite group $G_{\alpha}$ acts by a diagonal action on $G\times
(\Tilde{U}\times V)$ and $G\times\Tilde{U}$ so we can replace the above
pseudodifferential operator with the $G_{\alpha}$ equivariant
pseudodifferential operator
$\frac{1}{|G_{\alpha}|}\sum_{g\in G_{\alpha}} gQ\otimes 1 + 1
\otimes B_{\alpha}$.
This will induce a pseudodifferential operator on the sections
of the vector bundle of $G_{\alpha}$ orbits $G\times_{G_{\alpha}}\!(\Tilde{U}
_{\alpha}\times V)\overset{pr_1}\to G\times_{G_{\alpha}}\!\Tilde{U}_{\alpha}$.
We will transport it using the isomorphism between the previous $G$
vector bundle and
$\Tilde{E}_{|N_{\alpha}}\overset{\Tilde{p}}\to N_{\alpha}$ to a
pseudodifferential
operator $\Tilde{A}_{\alpha}$ acting on $\Tilde{E}\overset{\Tilde{p}}\to M$,
extending it by $0$ outside $N_{\alpha}$.
\par
We have to compare the actions of $A_{\alpha}$ and $\Tilde{A}_{\alpha}$.
Let $f$ be a $G$ invariant smooth section in $C^{\infty}(\Tilde{M};\Tilde{E})$
which induces $\overline{f}\in C^{\infty}(M;E)$. It is enough to take $f$
with support in $N_{\alpha}$. In this case $f$ is induced by a $G_{\alpha}$
invariant
section $\Tilde{f}\in C^{\infty}(G\times \Tilde{U}_{\alpha}; G\times
(\Tilde{U}_{\alpha}\times V))\overset{\sim}=C^{\infty}(G)\hat{\otimes}
C^{\infty}
(\Tilde{U};V)$ which is $G$ invariant as well. Then $\Tilde{f}(g,u)=gh(u)$
with $h\in C^{\infty}(\Tilde{U};V)$, $h$ induces $\overline{f}\in C^{\infty}
(M;E)$ and $\Tilde{A}_{\alpha}f$ corresponds via the $G$ equivariant
vector bundle diffeomorphism to $(Q\otimes 1 + 1\times B_{\alpha})
\Tilde{f}=B_{\alpha}(h)$ and this section induces $A_{\alpha}\overline{f}
\in C^{\infty}(M;E)$.
\par
We will construct a smoothing operator $\Tilde{K}$ in $C^{\infty}
(\Tilde{M};\Tilde{E})$ whose action on $G$ invariant sections induces the
smoothing operator $K$. The kernel of $K$ is given by a smooth section
$S(x,y)$ of the endomorphism bundle $\End(E)\to M\times M$ which
corresponds to a smooth $G$ invariant section $\Tilde{S}(\Tilde{x},\Tilde{y})$
of the endomorphism bundle $\End(\Tilde{E})\to\Tilde{M}\times\Tilde{M}$.
$\Tilde{S}$ is the kernel of a smoothing operator $\Tilde{K}$, which
induces $K$ when restricted to the $G$ invariant sections of $\Tilde{E}
\overset{p}\to \Tilde{M}$.
\par
The operator $\Tilde{A}$ will be the sum of the operators
$\Tilde{A}_{\alpha}$ and $\Tilde{K}$.
\par
If the operator $A$ is classical, we can choose $Q$ to be classical, then
$B_{\alpha}$, $Q\otimes 1+1\otimes
B_{\alpha}$ and $\Tilde{A}_{\alpha}$ are classical so $\Tilde{A}$
is classical.
\end{proof}
\subsection{Zeta Function of an Elliptic Pseudodifferential Operator}
Let $A$ be a pseudodifferential operator acting on the space of smooth sections
in a vector orbibundle $E\overset{p}\to M$. Suppose that $M$ is compact and it
is endowed with a Riemannian metric $g$ and that we have a hermitian
structure $<,>$ in the vector
orbibundle. As in the case of hermitian vector bundles over a closed Riemannian
manifold, we define the scalar product on $C^{\infty}(M;E)$ by
the formula
\begin{equation}
<f,g>=\int_M<f_1(x),f_2(x)>_x \,d\,vol \qquad\text{ for }f_1,f_2\in
C^{\infty}(M;E).
\end{equation}
The integration with respect to the volume form $d\,vol$ can be defined
locally on vector orbibundle charts. If $(\Tilde{U}, \Gamma , U, \pi)$
is such a chart then the metric $g$ and the hermitian structure $<,>$
above $U$ are
induced by $\Gamma$ invariant metric $\Tilde{g}$ and respectively a $\Gamma$
invariant hermitian structure $\tilde{<,>}$ in the $\Gamma$ vector
bundle $\Tilde{U}\times V\overset{pr_1}\to\Tilde{U}$. Let $d\,\Tilde{vol}$ be
the volume form on $\Tilde{U}$ associated with $\Tilde{g}$. Then define
\begin{equation}
\int_U \alpha(x)\,d\,vol=\frac{1}{|\Gamma|}\int_{\Tilde{U}} \Tilde{\alpha}
(\Tilde{x})\,d\,\Tilde{vol}
\end{equation}
for $\Tilde{\alpha}\in C^{\infty}(\Tilde{U})^{\Gamma}$ that induces
$\alpha\in C^{\infty}(U)$ (we chose $\alpha=<f_1,f_2>$). It is easy to see
that the above formula is
consistent with respect to the equivalence of charts.
The $L^2$ completion of $C^{\infty}(M;E)$ with respect to the above scalar
product will be denoted by $L^2(M;E)$. A pseudodifferential operator $A$
defines an unbounded operator on $L^2(M;E)$.
\par
\begin{defn}
The pseudodifferential operator $A$ is called symmetric if
\begin{equation}
<Af_1,f_2>=<f_1,Af_2> \qquad\text{ for all } f_1,f_2\in C^{\infty}(M;E).
\end{equation}
A is positive if
\begin{equation}
<Af, f>\, \ge 0 \qquad\text{ for any } f\in C^{\infty}(M;E).
\end{equation}
\end{defn}
\begin{remark} If $A$ is symmetric, then the restriction $A_{\mathcal{R}}$
to any vector orbibundle chart $\mathcal{R}$ as defined in
\ref{localize:op} is symmetric. This implies that $A_{\mathcal{R}}$ is
induced by a $\Gamma$ equivariant pseudodifferential operator $\Tilde{A}_
{\mathcal{R}}$ which can be chosen to be symmetric with respect to the
$\Gamma$ invariant metric $\Tilde{g}$ and $\Gamma$ invariant hermitian
structure inducing $g$ and $<,>$ above $U$. Indeed, the formal adjoint
$\Tilde{A}^*_{\mathcal{R}}$ of the operator $\Tilde{A}_{\mathcal{R}}$
induces $A_{\mathcal{R}}$ as well, so the symmetric operator $\frac{1}{2}
(\Tilde{A}_{\mathcal{R}}+\Tilde{A}^*_{\mathcal{R}})$ induces
$A_{\mathcal{R}}$.
Then the principal symbol of $A$, which is induced by the $\Gamma$ invariant
principal symbol of $\Tilde{A}$, is selfadjoint.
\end{remark}
\begin{thm}
\label{selfadjoint}
Let $A$ be a positive symmetric elliptic pseudodifferential operator acting
on the space of sections of a vector orbibundle $E\overset{p}\to M$
with compact base space. Then the operator $A$ acting on $L^2(M;E)$
is essentially selfadjoint and its spectrum is discrete.
\end{thm}
\begin{proof}
Let $m=dim(M)$ and denote by $G$ the orthogonal group $O(m)$.
Let $\Tilde{E}\overset{\Tilde{p}}\to \Tilde{M}$ be the $G$ vector bundle,
with $\Tilde{M}=\mathcal{F}(M)$,
such that the vector orbibundle of orbits is canonically isomorphic
to $E\overset{p}\to M$, as in Proposition \ref{represent}. Let us
fix a $G$ bi-invariant metric $g_m$ on $G$ so that $\int_G\,dg_m=1$ and
a $G$ bi-invariant positive selfadjoint pseudodifferential operator $Q$
of order $d$ acting on $C^{\infty}(G)$. As a metric $g_m$ we can choose
the left translations of the opposite of the Killing form on the
Lie algebra $o(m)$ and $Q=\Delta^{d/2}$ where $\Delta$ is the Laplace
operator associated with the metric $g_m$.
We will construct a $G$ invariant metric on $\Tilde{M}$ which induces
the given metric on $M$, a $G$ invariant hermitian
structure on $\Tilde{E}\overset{\Tilde{p}}\to \Tilde{M}$ which
induce the given hermitian metric on $E\overset{p}\to M$
and a $G$ equivariant elliptic positive selfadjoint pseudodifferential
operator $\Tilde{A}$ acting on $C^{\infty}(\Tilde{M};\Tilde{E})$ whose
restriction to the invariant sections is equal to $A$.
\par
Let us fix a finite atlas $\mathcal{A}=(\mathcal{R}_{\alpha})_{\alpha}$
of the vector orbibundle $E\overset{p}\to M$, with
$\mathcal{R}_{\alpha}=(\Tilde{U}_{\alpha}, V,
\Gamma_{\alpha}, U, \Pi_{\alpha}, \pi_{\alpha})$.
Then $\Tilde{\mathcal{R}}_{\alpha}=(G\times\Tilde{U}_{\alpha}, V, \Gamma
_{\alpha}, G\times_{\Gamma_{\alpha}}\Tilde{U}_{\alpha}, Id\times\Pi_{\alpha},
Id\times\pi_{\alpha})$ forms an atlas for $\Tilde{E}\overset{\Tilde{p}}\to
\Tilde{M}$.
Let $\Tilde{g}_{\alpha}$ be the $\Gamma_{\alpha}$
invariant metric on $\Tilde{U}_{\alpha}$ which induces $g$ on $U$. Then the
collection of $\Gamma_{\alpha}$ invariant metrics $g_m\otimes \Tilde{g}
_{\alpha}$ on $G\times\Tilde{U}_{\alpha}=\mathcal{F}(\Tilde{U})$ induces
the $G$ invariant
metric $\Tilde{g}$ on $\Tilde{M}$. Because $\Tilde{E}\overset{\Tilde{p}}
\to \Tilde{M}$ is the pull-back of $E\overset{p}\to M$ with respect
to the canonical map $\Tilde{M}=\mathcal{F}(M)\overset{pr}\to M$,
we define the hermitian structure on $\Tilde{E}\overset{\Tilde{p}}\to
\Tilde{M}$ to be the pull-back hermitian structure. It will be
$G$ invariant by construction and induces the initial hermitian structure
when passing to the $G$ orbit spaces. Consider the $G$ invariant
scalar product on
$C^{\infty} (\Tilde{M};\Tilde{E})$ associated with hermitian structure and the
metric $\Tilde{g}$ constructed above. When restricted to the $G$ invariant
sections, this scalar product is equal with the scalar product on
$C^{\infty}(M;E)$ constructed with the help of the metric $g$ and the
hermitian structure on $E\overset{p}\to M$. Also $L^2(\Tilde{M};\Tilde{E})^G
\overset{\sim}=L^2(M;E)$.
The pseudodifferential operator $\Tilde{A}$ acting on $C^{\infty}(\Tilde{M};
\Tilde{E})$ constructed as in Proposition \ref{pdolift} will be
symmetric with respect to the scalar product on $C^{\infty}(\Tilde{M};
\Tilde{E})$. By construction, the principal symbol of $\Tilde{A}$ is equal to
$q_{pr}\otimes Id+a_{pr}$ where $q_{pr}$ is the principal symbol of $Q$ and
$a_{pr}$ is the principal symbol of $A$. Because $Q$ and $A$ are positive
symmetric elliptic pseudodifferential operators, the operator $\Tilde{A}$
is elliptic as well.
Because $\Tilde{M}$ is compact, the operator $\Tilde{A}$
acting on $L^2(\Tilde{M};\Tilde{E})$ is essentially selfadjoint.
The restriction of $\Tilde{A}$ to the closed subspace of $G$ invariant
$L^2$ sections $L^2(\Tilde{M};\Tilde{E})^G\overset{\sim}=L^2(M;E)$ which
is equal to $A$ is essentially selfadjoint. The spectrum of $\Tilde{A}$ is
real and discrete so the spectrum of $A$ is a discrete subset of $\mathbb{R}$.
\end{proof}
To show that the complex powers of the elliptic positive selfadjoint
operator $A$ are
pseudodifferential operators, one can apply the same local approach as in
\cite{seeley}, and use approximations of the resolvent of $(A-\lambda)$
in order to show that $A^s$ are, up to a smoothing operator, pseudodifferential
operators of complex order $s$. Because all the proofs follow in the same
way with minimal complications, we will not repeat them in our paper.
For $Re(s)<\frac{-m}{d}$ the distributional
kernel of $A^s$ is continuous and $A^s$ is of trace class.
We can relax the conditions imposed on the operator $A$, and prove an
analogous
statement to Theorem \ref{selfadjoint}. Suppose that $\pi$ is an Agmon
angle for $A$. Then one can define the complex powers of $A$ as
\begin{equation}
\label{pow:neg}
A^s_{\pi}=\frac{1}{2\pi i}\int_{\wp}\lambda^s(\lambda - A)^{-1} \,
d\lambda \qquad \text{when }Re(s) < 0
\end{equation}
(where $\wp$ is a contour in the complex plane obtained by joining
two parallel half-lines to the negative real axis by a circle around
the origin) and
\begin{equation}
\label{pow:pos}
A^s_{\pi}=A^{s-k}_{\pi}A^k_{\pi} \qquad \text{for } Re(s)\ge 0
\end{equation}
for large enough $k\in \mathbb Z$ that makes $s-k < 0$.
Obviously, if $A$ is positive selfadjoint then $\pi$ is an Agmon angle for $A$.
\begin{prop}
\label{Agmon}
Let $A$ be an elliptic pseudodifferential operator of positive order $d$
acting on the space of sections of a vector orbibundle
$E\overset{p}\to M$ with compact
base space. Suppose that $\pi$ is an Agmon angle for $A$.
Then the operator $A$ acting on $L^2(M;E)$ has a discrete spectrum
and its complex powers $A^s_{\pi}$ are pseudodifferential operators of
complex order $sd$.
\end{prop}
\begin{proof}
The proof is analogous to the proof of Theorem \ref{selfadjoint}
We construct the operator $\Tilde{A}$ acting on sections of the $G$
vector bundle
$\Tilde{E}\overset {\Tilde{p}}\to \Tilde{M}$ with the help of a $G$
bi-invariant elliptic selfadjoint pseudodifferential operator $Q$ acting
on $C^{\infty}(G)$.
The principal symbol of $\Tilde{A}$ is equal to $\Tilde{a}_{pr}=q_{pr}\otimes
Id+ a_{pr}$ with $q_{pr}$ the principal symbol of $Q$ and $a_{pr}$
the principal symbol of $A$.
Because $\pi$ is an Agmon angle for $A$, the spectrum of $a_{pr}(x,\xi)$ is
disjoint from the region in the complex plane
$\mathcal{C}=\{z\,|\, arg(z)\in(\pi-\varepsilon, \pi+\varepsilon)\}
\cup\{z\,|\,|z|<\varepsilon\}$ for a small enough $\varepsilon>0$. Because
$q_{pr}(x,\xi)$ is selfadjoint the spectrum
of $\Tilde{a}(x,\xi)$ is disjoint from $\mathcal{C}$. We conclude that $\pi$ is
an Agmon angle for $\Tilde{A}$. Then $\Tilde{A}$ is elliptic and has a discrete
spectrum. $A$ being the restriction to the $G$ invariant sections will have a
discrete spectrum as well.
The complex powers $\Tilde{A}^s_{\pi}$ are well
defined and are pseudodifferential operators of order $sd$ so the restriction
to the $G$ invariant sections, which are equal to $A^s_{\pi}$, are
pseudodifferential operators of order $sd$.
\end{proof}
Throughout the rest of the chapter we suppose that $\pi$ is an Agmon
angle for the elliptic pseudodifferential operator $A$.
The complex powers $A^s=A^s_{\pi}$ will be defined as in
\eqref{pow:neg} and \eqref{pow:pos}.
\begin{defn} The zeta function of the operator $A$ is equal to
\begin{equation}
\zeta_A(s)=\Tr (A^s) \qquad\text{ for }Re(s)<\frac{-m}{d}.
\end{equation}
\end{defn}
We will prove the following theorem:
\begin{thm}
\label{zeta:Dirac:density}
The zeta function of an elliptic pseudodifferential operator $A$
acting on the sections of a vector orbibundle $E\overset{p}\to M$
can be extended to a meromorphic function on $\mathbb{C}$ with at most
simple poles at $s=\frac{-m+k}{d}$, $k=0,1,2,\dots$. The residues are
integrals on $M$ of Dirac-type densities which can be explicitly computed
in terms of the asymptotic expansion of the total symbol of the operator $A$.
\end{thm}
\begin{proof}
Let $\mathcal{A}=(\Tilde{U}_{\alpha}, V,\Gamma_{\alpha},U_{\alpha},
\Pi_{\alpha},\pi_{\alpha})_{\alpha}$ be a fixed finite atlas of linear
orbibundle charts . Using a partition of unity we can decompose the
complex powers of the operator $A$ as
\begin{equation}
A^s=\sum_{\alpha}A^s_{\alpha} + K_s
\end{equation}
where $A^s_{\alpha}$ are pseudodifferential operators of order $sd$
with support inside
$U_{\alpha}$ and $K_s$ is a smoothing operator. We can arrange for
$A^s_{\alpha}$ to take smooth sections with support inside $U_{\alpha}$
into smooth sections with support inside $U_{\alpha}$.
The trace $Tr(K_s)$ is defined for $s\in \mathbb{C}$ and it is a holomorphic
function on the complex plane. In order to prove the theorem, we will
show that the statement in the theorem holds for the trace of the
operators $A^s_{\alpha}$.
We will show that $Tr(A^s_{\alpha})$ is a meromorphic function on $\mathbb{C}$ with
at most simple poles at $s=\frac{-m+k}{d}$, $k=1,2,\dots$, and that
the residues are computed as integrals on $M$ of Dirac-type densities.
\par
$A^s_{\alpha}$ can be seen as a holomorphic family of pseudodifferential
operators of order $sd$ acting on the smooth sections of the vector orbibundle
$E_{|U_{\alpha}}\overset{p}\to U_{\alpha}$. For the sake of simplicity,
we will drop the index $\alpha$ and denote $A^s_{\alpha}$ by $A_s$. There
exists a holomorphic family of $\Gamma$ equivariant pseudodifferential
operators $\Tilde{A}_s$ of order $sd$ acting on the sections of the
$\Gamma$ vector bundle $\Tilde{U}\times V
\overset{pr_1}\to \Tilde{U}$ which induce $A_s$ on the $\Gamma$
invariant sections.
This family is unique up to smoothing operators, cf. Proposition \ref{unique}.
Then the component corresponding to the trivial representation of the trace
of $\Tilde{A}_s$ and of the residues will be equal to respectively the
trace functional of $A_s$ and its residues.
Theorem \ref{residues:i} shows that the component of the trivial
representation
of trace functional is a meromorphic function with at most simple
poles at $s=\frac{-m+k}{d}$ for $k=0,1,2,\dots$. From Theorem
\ref{residues:d}, the component of the residue of the trace
functional $Tr(\Tilde{A}_s)$ corresponding to the trivial irreducible
representation is:
\begin{equation}
\label{residues:o}
res_{|s=\frac{-m+k}{d}}Tr(\Tilde{A}_{s,\chi_0})=\frac{1}{|\Gamma|}
\sum_{\gamma\in\Gamma}\int_{\Tilde{U}^{\gamma}}\eta_k^{\gamma}
\end{equation}
where $\{\eta_k^{\gamma}\}_{\gamma\in\Gamma}$ are $\Gamma$ Dirac-type
densities on $\Tilde{U}$ constructed from the total symbol of $\Tilde{A}_s$
which on the chosen linear vector orbibundle chart coincides with the
total symbol of the pseudodifferential operator $A_s$ on $U$.
Though the statements of Theorems \ref{residues:i} and \ref{residues:d}
refer to the complex powers of a pseudodifferential operator, their proofs can
be slightly changed and adapted to include the case of holomorphic families
of pseudodifferential operators and so the conclusions listed above are
still true.
The $\Gamma$ Dirac-type density $\{\eta_k^\gamma\}_{\gamma\in\Gamma}$
defines a Dirac-type density on the orbifold $U$ which we will denote
by $\eta_k$. Observe that the formula \eqref{residues:o} is equivalent to
\begin{equation}
\res_{|s=\frac{-m+k}{d}}Tr(A_s)=\int_U \eta_k
\end{equation}
The Dirac-type density whose integral on $M$ is equal to the residues of
the zeta function $\zeta_A$ at $s=\frac{-m+k}{d}$ will be equal to
the sum of the Dirac-type densities $\eta_k$ constructed as above on the
charts of the vector orbibundle.
\end{proof}
Using Proposition \ref{eta:strata}, we can reformulate
Theorem \ref{zeta:Dirac:density} and express the residues of the zeta
function of $A$ in terms of integrals of smooth densities on submanifolds of
$\mathcal{S}(M)$ associated with the canonical stratification.
\begin{thm}
\label{zeta:strata:density}
The zeta function of an elliptic pseudodifferential operator $A$
acting on the sections of a vector orbibundle $E\overset{p}\to M$
can be extended to a meromorphic function on $\mathbb{C}$ with at most
simple poles at $s=\frac{-m+k}{d}$, $k=0,1,2,\dots$.
For each $k\ge 0$ there exist densities $\eta_{\upsilon,k}$ on the strata
$M_{\upsilon}$ of the canonical
stratification of $M$ such that the residue of the zeta function
at $s=\frac{-m+k}{d}$ is the sum of
integrals of these densities.
On each strata $M_{\upsilon}$ the density $\eta_{\upsilon,k}$ can be
explicitly computed in terms of the asymptotic expansion of total symbol
of the operator $A$ in a neighborhood of $M_{\upsilon}$.
\end{thm}
\begin{proof}
For each $\upsilon\in\mathcal{O}(M)$ and $k\ge 0$, the density
$\eta_{\upsilon,k}$
can be obtained as follows: let $x\in M_{\upsilon}$ and $(\Tilde{U}, V,
\Gamma, U, \Pi, \pi)$ be a linear vector orbibundle chart centered at $x$.
Observe that $\Gamma\in\upsilon$ and the map $\pi: \Tilde{U}^{\Gamma}
\to U\cap M_{\upsilon}$ is a local diffeomorphism.
Theorem \ref{zeta:Dirac:density} gives us Dirac-type densities $\eta_k$ on $M$
whose integrals are equal to the residues of the zeta function and can
be computed explicitly in terms of the total symbol of $A$. These
densities can be described above $U$ as families of densities
$\{\eta^{\gamma}_k\}$ such that $\eta^{\gamma}_k$ is a smooth density on
$\Tilde{U}^{\gamma}$.
Consider the density on $\Tilde{U}^{\Gamma}$ given by
\begin{equation}
\eta_{\Gamma,k}=\frac{1}{|\Gamma|} \sum_{\substack{\gamma\in\Gamma \\
dim(\Tilde{U}^{\Gamma})=dim(\Tilde{U}^{\gamma})}} \eta^{\gamma}_k
{}_{|\Tilde{U}^{\Gamma}}
\end{equation}
Define $\eta_{\upsilon,k}$ on $M_{\upsilon}$ to be the density whose
restriction to $U$ is equal to $\pi_*(\eta_{\Gamma,k})$.
As proved in Proposition \ref{eta:strata}, the integral on $M$ of the
Dirac-type density $\eta_k$ and the sum of the integrals of the
densities $\eta_{\upsilon,k}$ on the strata
$M_{\upsilon}$ for $\upsilon\in\mathcal{O}(M)$ are equal, so
they are equal to the residue of $\zeta_A$ at $s=\frac{-m+k}{d}$.
\end{proof}
\vfill\eject
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,213 |
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