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class AddAuthTokenToSubmissions < ActiveRecord::Migration
def change
add_column :submissions, :auth_token, :string
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,479
|
Aabroo may refer to:
Aabroo (1943 film), a Bollywood film
Aabroo (1968 film), a Hindi romantic crime drama film
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,489
|
Illumination steht für:
Illumination (Beleuchtung), gezielter Einsatz von Licht
malerischer Buchschmuck, siehe Buchmalerei
Erleuchtung, Eingebung oder spirituelle Erfahrung
ein Geistesblitz, siehe Phasen des kreativen Prozesses #Phase der Illumination: der Geistesblitz
Illuminationsphilosophie, eine islamische Strömung aus dem 12. Jahrhundert
Illumination (Film), Spielfilm (1973) von Krzysztof Zanussi
Illumination (Album), Studioalbum (2007) der Metal-Band Tristania
Siehe auch:
Illumination Entertainment, US-amerikanische Filmproduktionsgesellschaft
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,990
|
First the corporate education reformers came for our local control through tax caps, and I didn't speak up because I wasn't paying attention. I didn't know that changing the funding from property tax to state sales/income tax would harm my schools. I was running carpool.
Next they gutted the funding through "school choice," and I didn't speak up because I thought, "Sure. This is a free country. Everyone should be able to find the school that best suits their children." I didn't know that the vouchers would take tens of millions away from my kids' schools and make it impossible to fully fund the rich educational programs and extra-curricular activities for all children.
Then they took over and privatized some schools, and I didn't speak up because I thought, "That's an inner-city problem. Why should I worry about that?" I didn't know that once these for-profit charter companies and special interest charter sponsors smelled money and a market demand, they would come to my town. I didn't know that when they wooed the families from my public schools, they would take with them the money for my kids' art teachers and librarians, the PTO volunteers, and divide us as a community.
Then they went after the curriculum, and I didn't speak up because I thought "Sure, we should have high, consistent standards. Kids should be 'ready' for college and career." I didn't know that this was a money-making scheme unlike all others and that the testing involved would destroy teacher autonomy and the joy in learning. I didn't know that laws like a grading system of schools based on one test score or laws tying test scores to teacher salaries and security in the name of "accountability" were designed to destroy public schools. I didn't know that the maligning of schools through low letter grades (on a curve!) opened the market to charters and privates and further hemorrhaging of public school funding. I didn't know that kids would lose art, music, gym and library. I just didn't know. Did I mention I was running carpool?
And they came for the teachers, and I didn't speak up because I wasn't a teacher. I didn't know that the destruction of teacher unions, collective bargaining and morale would effectively silence my teacher and keep her from advocating for my kids. I didn't know that the culture of fear would make teachers unable to say: "Hey! This initiative is absolutely developmentally inappropriate for these kids!"
Then they came for my children and my school, and I found my voice.
Other voices joined in...and I did speak up. With one local "fire" put out, we can't rest. There is an inferno surrounding us! In other schools and other towns they are fighting as desperately as we are to keep the data mania from infiltrating our schools, to retain the child focus, to have parent (and teacher) voices be heard. But in other districts, unlike our own, they didn't pass a referendum, and they have no music programs left to save. They have no librarians at all. They can't even afford buses.
As I type, the legislators in the statehouse are considering a law which will allow anyone with a degree in anything to be a teacher in the classroom. Consider a teacher who has no idea how a child learns, how to spot dyslexia, what methods work best for a hands-on learner, classroom management techniques, etc. The mind boggles. But this de-professionalizing of our teachers, of the people we trust to care for our kids every day, continues unabated.
It's up to us to stop this legislative destruction aimed at our public schools and our public school teachers. Raise your voice. Please join us at ICPE and other grassroots organizations around the state and country. Advocate for all children as the community just did here in Bloomington. If we showed that kind of cooperative effort on a state and national level, we could stop these "reformers" who use our children to promote their own agenda of greed. They are making money off of our babies... and in their name. Speak up.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,779
|
Q: What is the correct way to support apostrophes in javascript when building up html? i have the following code:
var name = "Joe O'Neal";
var row= [];
row.push(
"<td><input type='hidden' name='milestones[" + id + "].Name'
value='" + name + "' class='currentRowName' />
<span class='currentRowNameText'>" + name + "</span></td>")
but the issue is that i have a situation where there is an apostrophe in the name variable so it causes problems with this:
value='" + name + "'
what is the correct way to write this to avoid any conflicts with apostrophes? In C#, i would do something like
value=\"" + name + "\"
but that doesn't seem to work in javascript
A: What you're looking to do is sanitize your input. To do this, you might define a helper method which replaces any unsafe characters with either the Unicode equivalent or the HTML entity. Not only is this sort of thing used for escaping quotes, but it can also help prevent things like XSS Attacks.
Short-term fix
The following is adapted from Tom Gruner's answer from this other question on Escaping HTML strings with jQuery.
var entityMap = {
"&": "&",
"<": "<",
">": ">",
'"': '"',
"'": ''',
"/": '/'
};
function escapeHtml(string) {
return String(string).replace(/[&<>"'\/]/g, function (s) {
return entityMap[s];
});
}
And then to use this, you would do something like the following:
var name = "Joe O'Neal";
var safe_name = escapeHtml(name);
var row= [];
row.push(
"<td><input type='hidden' name='milestones[" + id + "].Name'
value='" + safe_name + "' class='currentRowName' />
<span class='currentRowNameText'>" + safe_name + "</span></td>");
Long-term fix
If you find yourself doing this a lot, then it may be time to start using a template library which can do this automatically. One template library I recommend and find useful is the Google Closure Templates. It is an enterprise-grade template library which will (by default) sanitize your HTML.
For more information on how Closure Templates help protect you, you can check out the page they have on security.
A: You can quote characters:
var apostophe = '\''
or use HTML entities if intended to be used in an HTML document:
var apostrophe = ''';
var apostrophe = ''';
or use unicode:
var apostrophe = '\u0027';
So you can do:
var name = 'O\u0027Neal';
el.innerHTML = '<input value="' + name + '">';
Mapping of characters to one of the above (entity or Unicode value) is fairly simple using replace with a regular expression.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,371
|
Q: filtering an array of objects that contains an array of objects confusing title let me explain.
I have an array of categories and array of products.
the array of products is an array of objects, one of the keys is categories, which itself is an array of objects and one of the keys im interested in is title.
the array of categories is an array of objects too.
I want to filter the products array to return me just the products for my chosen category (which is set in state)
I've got this, however it does not work:
filterCategories = () => {
return this.state.products.filter((product => {
return product.categories.map((cat) => {
cat.title == this.state.chosenCategory
})
}))
}
this method seems like it should. to my knowledge, im filtering the products array by mapping over each product and then mapping over that products categories and checking if the category title is equal to the title in state. clearly I've gone wrong somewhere. can anyone see it?
sample data:
products = [
{
id: '123',
title: 'wine',
categories: [
{title: 'drinks'}
]
}
]
categories = [
{
id: '123',
title: 'drinks'
}
]
A: It looks like you want to test whether any of the category titles match the chosenCategory: so, use the some method. Also, even better, if you're sure there's exactly one matching category, use .find instead of filter. Also make sure to return a value at the end of the internal function (which you aren't doing currently), or use an arrow function with implicit return:
filterCategories = () => (
this.state.products.find((product) => (
product.categories.some(({ title }) => (
title === this.state.chosenCategory
))
))
)
A: Filter callback function is a predicate, to test each element of the array. you are not returning anything from the inner map function too, you must instead have a filter there like
filterCategories = () => {
return this.state.products.filter((product => {
return product.categories.filter((cat) => {
return cat.title == this.state.chosenCategory
})
}))
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,857
|
Lorenzo Respighi (7 de octubre de 1824-10 de diciembre de 1889) fue un matemático, astrónomo y filósofo natural italiano.
Semblanza
Respighi nació en Cortemaggiore, Piacenza, hijo de Luigi Respighi y Giuseppina Rossetti. Estudió matemáticas y filosofía natural, primero en Parma y después en la Universidad de Bolonia, donde obtuvo su grado con honores en 1845.
De 1855 a 1864 dirigió del Observatorio Astronómico de Bolonia, y durante estos años descubrió tres cometas, #1862 IV, #1863 III y #1863 V.
En 1865 fue nombrado director del Observatorio Astronómico del Campidoglio, en Roma.
Eponimia
El cráter lunar Respighi lleva este nombre en su memoria.
El asteroide (16930) Respighi también conmemora su nombre.
Referencias
Fuentes
Astrónomos de Italia del siglo XIX
Matemáticos de Italia del siglo XIX
Alumnado de la Universidad de Bolonia
Miembros nacionales de la Academia Nacional de Ciencias de Italia
Nacidos en Cortemaggiore
Fallecidos en Roma
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,351
|
No time to wait? These quick-growing veggies are ready for harvest in as little as 14 days!
Standout for sweet flavor and incredible blight-resistance.
All-America Selections winner. A rainbow of color in a seed packet.
Perfect for baby spinach and mature harvest.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,148
|
Invisible And Discriminated': The Plight Of Women Sanitation Workers Making Bharat Swachh
Featured in THE LOGICAL INDIAN
"Garib ka zindagi hai, kamaenge toh khaenge, nahin toh bhuke marenge (This is the life of the poor. If you earn, you eat, or else you die of hunger)," said one woman sanitation worker from Muzaffapur.
Maya Bai is a resident and sanitation worker with the Ganj Basoda Nagar Palika Parishad in Madhya Pradesh. She has been employed by the urban local body for 35 years now.
Maya Bai's group is responsible for the cleaning and upkeep of ward 22, Chud Mohalla of the city. An ordinary sanitation worker, she faced an extraordinary situation when the COVID-19 pandemic struck the city, especially when the first case was detected in her ward.
In a matter of two weeks, the area was declared a containment zone. No one was allowed to enter the area, but the daily cleaning of the community toilet in the ward was a basic necessity.
When no other sanitation worker was ready to take up the job, Maya Bai showed immense courage and volunteered for the task. At this crucial point, she accepted the challenge and cleaned the area single-handedly in order to ensure the safety of its residents, although putting her own life at risk.
Ground-level sanitation workers form the backbone of the public hygiene and waste management system in India. They are the real champions behind the country's accelerated sanitation coverage, who painstakingly make sure that the daily waste is being treated safely and the cities stay clean. Despite providing an essential public service, their work often goes unnoticed.
On a daily basis, more than five million sanitation workers in the country put their lives at risk and ensure people have access to fully functioning sanitation services even during a crisis. However, their constant efforts remain unrecognised. These workers are often ostracised and are at the bottom of the social and economic ladder, facing marginalisation because of their caste in several cases and, ironically, because of the very profession they are in.
The risks are even higher for the informal workforce. But women, who make up more than 80 per cent of waste pickers and 50 per cent of urban sanitation workers, are especially vulnerable because of their gender.
The World Health Organisation (WHO) refers to sanitation as "the provision of facilities and services for safe disposal of human urine and faeces".
Shakuntala Devi, a permanent sanitation worker employed with the Municipal Corporation of Muzaffarpur says, "Sehar mein do tarah ke log rehte hain. Kachra paida karne wale aur kachra saaf karne waale. Par sehari swacchta sirf nagar nigam ki jimmedari nahin hai. Aam logon ko bhi apna kaam thik se karna chahiye. Unko kachra idhar udhar nahin phenkna chahiye (There are two kinds of people in every city. The ones who generate garbage and the ones who clean it. However, the responsibility of keeping a city clean is not solely that of a municipal corporation. The general public should also do their part of the work properly. They should not litter public spaces mindlessly)."
As per the officials, there is a paucity of data pertaining to women sanitation workers. This is the result of a lack of overall information on informal sanitation workers (men and women) as well as segregated gender-based data on such workers. However, data plays a crucial role in understanding the intensity of the issue devising plans to improve the situation.
Health Risks Associated With The Job
"Though there is limited data, it appears that as the sanitation work gets more informal, the percentage of women tends to rise. This has to be seen in the context of a wide range of roles of sanitation workers (toilet cleaners, De-sludging operators, sewer cleaners, treatment facility workers, railway track cleaners, SWM workers, rag pickers, etc) across various contractual agreements – ULB payroll, contracted by ULB, private employment or daily wage earners. More informal the nature of work, fewer benefits are available," Kavita Wankhade, Head – Practice at the Indian Institute for Human Settlements, told The Logical Indian.
Wankhade is also a member of the National Faecel Sludge and Septage Management (NFSSM) Alliance.
"Women who are employed in daily wage work like cleaning toilets, etc are likely to be more vulnerable," she added.
While women sanitation workers are exposed to similar risks as their male counterparts, there may be an additional risk due to the design of protective equipment and tools that are not well-suited to their bodies.
"In addition, they may lack access to their entitlements – such as maternity leave, other leave benefits, insurance, adequate remuneration, protection from social and professional stigma, and lack of access to upskilling," Wankhade pointed out.
During menstruation, every month, women are subjected to frequent rashes, infections, and bruises on their thighs due to a tiring work schedule and walking miles, besides lack of proper toilet facilities. As most of the workers belonging to this group come from poor backgrounds and conservative families, the stigma surrounding the process makes it even worse.
Most of these women fail to express their predicament to their seniors and always hesitate to ask for leave during periods.
Several health checkups have been conducted for women sanitation workers from time to time across the country and it was observed that a majority suffered from joint pains, especially the elder women.
Conditions such as haemorrhoids, contact dermatitis, poor appetite, low or high blood pressure are common among the workers. Pushing heavy vehicles throughout the day results in long-term consequences, and the problems compound with the increasing age.
While this section in India is often suppressed and neglected, inter-state disparities show differing discrimination based on rigid histories of gender discrimination.
Women sanitation workers live in a country that is still sunk in patriarchy and imposes a double burden of labour on lower caste women who have to work to survive.
The unhappy kinship of gender, caste, and occupation makes them the most unsafe worker among the others. Most of them are employed as contractual and outsourced workers, according to a recent study on women sanitation workers.
Working as contractual workers restate experiences of fear, danger, insufficiency. Those employed on daily wages fear losing their jobs should they take a day off. "Garib ka zindagi hai, kamaenge toh khaenge, nahin toh bhuke marenge (This is the life of the poor. If you earn, you eat, or else you die of hunger)," said one woman sanitation worker from Muzaffapur.
"Subah khana hota tha toh shaam ko nahi (If there was food in the morning, then there was none in the evening)," said another worker in Jhansi. Women sanitation workers must earn, or starve.
Initiatives Taken By The Government
The government has undertaken various initiatives through policies, schemes, laws, and programs to ensure a safe and sound environment and dignity to the sanitation workforce.
The Swachh Bharat Mission focused on ensuring a dignified and safer source of livelihood for sanitation workers by integrating them into the formal workforce through a convergence scheme with the National Urban Livelihood Mission (NULM).
For the very first time, the country witnessed women and transgender sanitation workers in the states of Odisha and Tamil Nadu paving the path and taking on roles that were men-dominated such as cesspool vehicle operations and septage treatment plant management.
Experts say that it is the citizens who need to change their perceptions about sanitation workers who are generally regarded as second-class citizens and acknowledge their invaluable contribution in making Bharat Swachh.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,132
|
Q: Cannot deserialize the current JSON array (e.g. [1,2,3]) So im working on the Overwatch API, and I've been struggling on displaying the achievements from this JSON:
{
totalNumberOfAchievements: 67,
numberOfAchievementsCompleted: 57,
finishedAchievements: "57 / 67",
- achievements: [
{
name: "?",
finished: true,
image: "https://overwatch-a.akamaihd.net/img/icons/0x0E6000000000015D-0e20595aad6b8ccf2de427df797ea1a78ba7dbe412cc54a0e5add73aaaca80bc7d5322859afa7e4da33816fe8ed89ee2c2f0c97c0ae87e5e5f32f29df989e21f.png",
description: "...",
category: "General"
},
{
name: "Centenary",
finished: true,
image: "https://blzgdapipro-a.akamaihd.net/game/achievements/0x0E60000000000156.png",
description: "Win 100 games in Quick or Competitive Play.",
category: "General"
},
LIST CONTINUES...
I get this error:
Additional information: Cannot deserialize the current JSON array (e.g. >[1,2,3]) into type 'Overwatch.Models.Achievements' because the type requires a >JSON object (e.g. {"name":"value"}) to deserialize correctly.
When trying to deserialize to this object:
public class OWAchievements
{
public int totalNumberOfAchievements { get; set; }
public int numberOfAchievementsCompleted { get; set; }
public string finishedAchievements { get; set; }
public Achievements achievements { get; set; }
}
public class Achievements
{
public IEnumerable<Achievements> achievements { get; set; }
public string name { get; set; }
public bool finished { get; set; }
public string image { get; set; }
public string description { get; set; }
public string category { get; set; }
}
Through my controller:
[HttpPost]
public ActionResult Index(string profileId)
{
var json = new WebClient().DownloadString($"https://api.lootbox.eu/pc/eu/{profileId}/achievements");
var data = JsonConvert.DeserializeObject<OWAchievements>(json);
ViewBag.Tag = profileId;
return View(data);
}
And finally displaying it in my view:
@model IEnumerable<Overwatch.Models.OWAchievements>
@{
ViewBag.Title = "Index";
}
<h2>Enter battle-tag</h2>
<div class="row">
<div class="col-md-12">
@using (Html.BeginForm())
{
@Html.TextBox("profileId", null, new { @class = "form-control" }); <br />
<button class="btn btn-primary" type="submit"> Find Player </button>
}
</div>
</div>
@if (Model != null)
{
<table class="table">
<tr>
<th>
@Html.DisplayNameFor(model => model.totalNumberOfAchievements)
</th>
<th>
@Html.DisplayNameFor(model => model.numberOfAchievementsCompleted)
</th>
<th>
@Html.DisplayNameFor(model => model.finishedAchievements)
</th>
<th></th>
</tr>
@foreach (var item in Model)
{
<tr>
<td>
@Html.DisplayFor(modelItem => item.totalNumberOfAchievements)
</td>
<td>
@Html.DisplayFor(modelItem => item.numberOfAchievementsCompleted)
</td>
<td>
@Html.DisplayFor(modelItem => item.finishedAchievements)
</td>
<td>
@Html.ActionLink("Edit", "Edit", new { /* id=item.PrimaryKey */ }) |
@Html.ActionLink("Details", "Details", new { /* id=item.PrimaryKey */ }) |
@Html.ActionLink("Delete", "Delete", new { /* id=item.PrimaryKey */ })
</td>
</tr>
}
</table>
}
else
{
ViewBag.Text = "Nothing";
}
Sorry for the long and annoying post, but I've been struggling for some time now and I really hope someone can help me.
A: Provided JSON has missing closing ] and }.
After I fixed this I tried with following classes:
public class Rootobject
{
public int totalNumberOfAchievements { get; set; }
public int numberOfAchievementsCompleted { get; set; }
public string finishedAchievements { get; set; }
public Achievement[] achievements { get; set; }
}
public class Achievement
{
public string name { get; set; }
public bool finished { get; set; }
public string image { get; set; }
public string description { get; set; }
public string category { get; set; }
}
Please not hierachy between RootObject and Achievment.
A: {
"totalNumberOfAchievements": 67,
"numberOfAchievementsCompleted": 57,
"finishedAchievements": "57 / 67",
"achievements": [{
"name": "?",
"finished": true,
"image": "https://overwatch-a.akamaihd.net/img/icons/0x0E6000000000015D-0e20595aad6b8ccf2de427df797ea1a78ba7dbe412cc54a0e5add73aaaca80bc7d5322859afa7e4da33816fe8ed89ee2c2f0c97c0ae87e5e5f32f29df989e21f.png",
"description": "...",
"category": "General"
}, {
"name": "Centenary",
"finished": true,
"image": "https://blzgdapipro-a.akamaihd.net/game/achievements/0x0E60000000000156.png",
"description": "Win 100 games in Quick or Competitive Play.",
"category": "General"
}]
}
This will the your class structure:
public class Achievement
{
public string name { get; set; }
public bool finished { get; set; }
public string image { get; set; }
public string description { get; set; }
public string category { get; set; }
}
public class Achievements
{
public int totalNumberOfAchievements { get; set; }
public int numberOfAchievementsCompleted { get; set; }
public string finishedAchievements { get; set; }
public List<Achievement> achievements { get; set; }
}
Your Json is also incorrect, I have corrected it
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,946
|
import unittest
import logging
from unittest.mock import Mock
from controllable_objects.specific.shift_reg.slider import Slider, ShiftRegBuffered
from connections.abs_shift_reg import AbsShiftRegister
sr_base = Mock(spec_set=AbsShiftRegister)
sr_base.get_capacity.return_value = 8
sr = ShiftRegBuffered(sr_base)
sl_bit_pos = 0
sl_bit_neg = 1
transition_time = 0.1
sl_pinstruct = Slider.ConParams(sl_bit_pos, sl_bit_neg, transition_time)
sl_common_params = (sr, sl_pinstruct)
logging.debug('test_sr_slider: {0}'.format(sr))
class TestSliderInit(unittest.TestCase):
def test_init_normal(self):
Slider(sr, sl_pinstruct)
def test_init_invalid_switch_time(self):
with self.assertRaisesRegex(ValueError, 'transition_time must be bigger than zero'):
Slider.ConParams(sl_bit_pos, sl_bit_neg, 0)
with self.assertRaisesRegex(ValueError, 'transition_time must be bigger than zero'):
Slider.ConParams(sl_bit_pos, sl_bit_neg, -1)
def test_init_invalid_connection_type(self):
with self.assertRaisesRegex(ValueError, 'type of con_instance value must be a ShiftRegBuffered'):
Slider('str', sl_pinstruct)
def test_init_invalid_connection_params(self):
with self.assertRaisesRegex(ValueError, 'con_params must be an instance of Slider.ConParams class*'):
Slider(sr, 'str')
def test_init_with_dict(self):
Slider(sr, {"pin_pos": sl_bit_pos, "pin_neg": sl_bit_neg, "transition_time": transition_time})
def test_init_with_invalid_dict(self):
with self.assertRaisesRegex(ValueError, "con_params must be an instance of Slider.ConParams class or "
"a compatible dict"):
Slider(sr, {"bla": sl_bit_pos, "pin_neg": sl_bit_neg, "transition_time": transition_time})
class TestSliderMethods(unittest.TestCase):
def test_init_state(self):
sl = Slider(*sl_common_params)
self.assertEqual(sl.get_state(), sl.States.closed)
def test_open_closed(self):
sl = Slider(*sl_common_params)
sl.open()
self.assertEquals(sl.get_state(), sl.States.opened)
self.assertEqual(sr.get_buf_bit(sl_bit_pos), 1)
self.assertEqual(sr.get_buf_bit(sl_bit_neg), 1)
def test_close_closed(self):
sl = Slider(*sl_common_params)
sl.close()
self.assertEquals(sl.get_state(), sl.States.closed)
self.assertEqual(sr.get_buf_bit(sl_bit_pos), 0)
self.assertEqual(sr.get_buf_bit(sl_bit_neg), 0)
def test_close_opened(self):
sl = Slider(*sl_common_params)
sl.open()
sl.close()
self.assertEquals(sl.get_state(), sl.States.closed)
self.assertEqual(sr.get_buf_bit(sl_bit_pos), 0)
self.assertEqual(sr.get_buf_bit(sl_bit_neg), 0)
def test_open_opened(self):
sl = Slider(*sl_common_params)
sl.open()
sl.open()
self.assertEquals(sl.get_state(), sl.States.opened)
self.assertEqual(sr.get_buf_bit(sl_bit_pos), 1)
self.assertEqual(sr.get_buf_bit(sl_bit_neg), 1)
def test_toggle_closed(self):
sl = Slider(*sl_common_params)
sl.toggle()
self.assertEquals(sl.get_state(), sl.States.opened)
self.assertEqual(sr.get_buf_bit(sl_bit_pos), 1)
self.assertEqual(sr.get_buf_bit(sl_bit_neg), 1)
def test_toggle_opened(self):
sl = Slider(*sl_common_params)
sl.open()
sl.toggle()
self.assertEquals(sl.get_state(), sl.States.closed)
self.assertEqual(sr.get_buf_bit(sl_bit_pos), 0)
self.assertEqual(sr.get_buf_bit(sl_bit_neg), 0)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,751
|
It's been nearly a month since two, historic hurricanes savaged Puerto Rico, and despite the utter devastation left after the storms, the island's 3.4 million residents are still waiting for substantive relief from the federal government.
Help from the Federal Emergency Management Agency (FEMA) has been slowed, because of glaring lapses in coordination, a lack of guidance of medical and other personnel, as well as what critics and Puerto Rican officials have said was an almost total breakdown in distribution and supply chains. The result has been a huge gap between the burgeoning humanitarian disaster and the urgent life-and-death needs of the shell-shocked populace.
Nearly 85 percent of the island is still in darkness, because the storms destroyed the electrical grid. Governor Ricardo Rosselló estimates that it will cost about $5 billion to repair the island's power grid that was decimated by Hurricane Maria. Cellphone service towers across the island are slowly being restored; there is a critical shortage of food, medicine and other basic supplies; meanwhile, more than half of the commonwealth's residents are living without potable water.
The official death toll is 48, but there are reports that the medical examiner's office in San Juan is holding 350 bodies. There are also fears that, as the authorities reach the most remote parts of the island and as water-borne and other diseases take hold, that the death toll will inevitably rise.
Beleaguered Puerto Ricans can only buy food, supplies and other materials in what is now a mostly cash-only society. Yet, in the midst of all this need, more than 10,000 shipping containers loaded with food, medicine and other needed supplies have sat idle at the Port of San Juan and elsewhere, because of red tape, bureaucratic bungling and logistical logjams.
Aurora Flores, a New York-based activist, painted a harrowing picture that is slowly emerging as information seeps out of the soaked island.
She assailed the Trump administration for its slow response and castigated Trump for his constant congratulatory comments to first responders, FEMA, and others in his administration.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,523
|
Q: Getting a Python Object from JSON What is the best way to get a Python Object (with custom objects as attributes) from JSON?
I'm using a JSON object as configuration, like this:
conf = {
[
{
'column_name': 'Period start time',
'validators': [
{
'validator': 'DateFormatValidation',
'arguments': {'format': '%m.%d.%Y %H:%M:%S'}
},
{
'validator': 'IsRequiredValidation',
'arguments': {}
},
]
},
{
'column_name': 'value',
'validators': [
{
'validator': 'IsNumberValidation',
'arguments': {}
}
]
}
]
}
I need to convert to a Python object, something like this:
{[
Column('Period start time', [DateFormatValidation('%m.%d.%Y %H:%M:%S'), IsRequiredValidation()]),
Column('value', [IsNumberValidation()]),
]}
In this example Column, DateFormatValidation, IsNumberValidation and IsRequiredValidation are classes.
Any clue about the way to proceed?
UPDATE
Yes guys there was an issue with the json, here is the code based on your guiding discarding error handling:
STR_CONFIG = '''
[
{
"column_name":"Period start time",
"validators": [
{
"validator": "DateFormatValidation",
"arguments": {"date_format": "%m.%d.%Y %H:%M:%S"}
}
]
},
{
"column_name":"RNC name"
}
]
'''
conf = json.loads(STR_CONFIG)
column_list = []
validator_list = []
available_validators = {
"DateFormatValidation": DateFormatValidation,
"IsNumberValidation": IsNumberValidation,
"IsRequiredValidation": IsRequiredValidation,
}
for obj in conf:
if 'validators' in obj:
for validator in obj['validators']:
clase = available_validators[validator['validator']]
if 'arguments' in validator:
arguments = validator['arguments']
validator_list.append(clase(**arguments))
else:
validator_list.append(clase())
column_list.append(Column(obj['column_name'], validator_list))
validator_list = []
result = column_list
It is working great!, but with this approach I see two issues:
*
*Implementing a mapping for classes (available_validators): I have to take care modifying the code if I create new validators.
*If I add nested objects to JSON (conf) the code could get a little messy between many for and if. I was reading about object_hook for converting JSON to Python objects but I'm not sure if applies for this case.
A: First of all. Your json string is not valid format. Fix it for example usage
[
{
"column_name": "Period start time",
"validators": [
{
"validator": "DateFormatValidation",
"arguments": {
"format": "%m.%d.%Y %H:%M:%S"
}
},
{
"validator": "IsRequiredValidation",
"arguments": {}
}
]
},
{
"column_name": "value",
"validators": [
{
"validator": "IsNumberValidation",
"arguments": {}
}
]
}
]
You can use python json library to deserialize json.
Then you can use dict and list object to initialize Column, DateFormatValidation, IsNumberValidation and IsRequiredValidation.
But there is some extra skill you need to know.
you need to use python to create object by string object. The easy way is to establish dict for class collection.
Just put some example for your guideline.
import json
class Column:
def __init__(self, column_name, list_of_validator):
self.column_name = column_name
self.list_of_validator = list_of_validator
pass
class mapper:
def __init__(self):
self.cls_dict = {}
def add_mapping(self, string):
def decorator(validator):
self.cls_dict[string] = validator
return validator
return decorator
mapping = mapper()
@mapping.add_mapping('DateFormatValidation')
class DateFormatValidation:
def __init__(self, arguments=None):
pass
@mapping.add_mapping('IsNumberValidation')
class IsNumberValidation:
def __init__(self, arguments=None):
pass
@mapping.add_mapping('IsRequiredValidation')
class IsRequiredValidation:
def __init__(self, arguments=None):
pass
with open("conf.json") as json_file:
conf_obj = json.load(json_file)
validator_list = []
column_list = []
for conf in conf_obj:
for validator in conf['validators']:
cls = mapping.cls_dict[validator['validator']]
try:
argument = validator['arguments']['format']
except KeyError:
argument = None
validator_list.append(cls(argument))
column_list.append(Column(conf['column_name'], validator_list))
validator_list = []
Update
If you want to have better way to manage your validator and serializer mapping , you can use decorator to register you relation. I've updated code above.
I assume you want to have most easy way to mapping class name, but it is really not safe. It is better to write a mapper class and register by decorator.
In this case, you can use object_hook with json.loads but you still need to write if and or in object_hook function. You need to have another pythonic way to simply you program.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,953
|
Подпеч-над-Марофом () — поселення в общині Шентюр, Савинський регіон, Словенія.
Висота над рівнем моря: 581,7 м.
Посилання
Статистичне бюро Республіки Словенія
Населені пункти Словенії
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,870
|
Based on: https://github.com/raghon1/seafile/tree/master/docker-s3ql
Using it with [docker-hubic-swiftauth-proxy](https://github.com/murf0/docker-hubic-swiftauth-proxy) to enable hubic compability
Set environment variables to suite your needs.
```
S3QL_TYPE=swift \
S3QL_STORAGE=<server>:<port> \
S3QL_STORAGE_CONTAINER=<storage_container> \
S3QL_STORAGE_FS=<Filesystem_Name> \
S3QL_COMPRESS=zlib \
S3QL_MOUNT_POINT=/mnt/data\
S3QL_LOGIN=<username> \
S3QL_PASSWD=<password> \
S3QL_FSPASSWD=<FS password> \
S3QL_CACHESIZE=<In KB or empty if you wish to use auto, auto will use up all of the space on your _host_> \
S3QL_SUBNET=<subnet ex 172.17.0./16> \
SWIFT_AUTH_ENDPOINT=<HTTP AUTH API endpoint (skip if container is already created)>
```
### Run it
To run you must use "--privileged" or "--cap-add mknod --cap-add sys_admin --device=/dev/fuse" Since it uses fuse to mount the volume.
### Shutdown
Changed timeout in my_init from 5sec to 120 sec to allow for syncing of metadata and data to cloud.
## Hubic
Im using hubic 10tb for €5 per month. It's an OVH company, so it's running in France.
[hubic.com](https://hubic.com)
Of if you want to use my affiliate link (I'll get an extra 500gb when you sign up)
[hubic.com/en/offers?referral=KRMTJR](https://hubic.com/en/offers?referral=KRMTJR)
#Usage in other containers.
Since this mount is only container local (not mounted in the host file system.) No other containers can use the filesystem directly (eg via --volumes-from)
Instead you can use NFS
```
mount <s3ql ip>:/folder/already/setup/to/be/shared /mountpoint
```
Remember the host server must have the nfs-kernel module installed also.
And of course the server mounting needs to run in "--privileged" mode also.
This is a _security_ issue. A compromised container running in privileged mode means a compromised host. (Running the s3ql in privileged mode is "ok-ish" since it is not exposing any services to the world)
##NFS mount in host instead.
Basically query docker for the ip of the s3ql container (On name..) then mount on that ip. Remember to set S3QL_SUBNET accordingly on the container.
```
mount -t nfs $(docker inspect $(docker ps -a | grep s3ql | awk '{print $1}') | grep IPAddress\" | awk -F\" '{print $4}'):/ <mount_point>
```
##Docker volumes plugins
Instead there is a couple of docker-volume plugins you can use:
https://github.com/SvenDowideit/docker-volumes-nfs
https://github.com/gondor/docker-volume-netsh
But that forces you to expose the Ports for NFSv4 111 and 2049
#License
MIT For original code in this repository, see LICENSE text
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,114
|
\section{Introduction}
The effect of dopants for the ABO$_{3}$-type manganese oxides has been
an area of intense research activity, primarily for attempting to
understand the physics behind the colossal magnetoresistance (CMR)
behavior seen in these materials. Typically, studies have been
carried out either by replacing the trivalent
ion\cite{Sun,Fontcuberta,Hwang} or the Mn
ion.\cite{Blasco,Gayathri,Anane} Recently\cite{Guo} a study of
polycrystalline La$_{0.75}$Ca$_{0.25-x}$Sr$_{x}$MnO$_{3}$ was carried
out in order to look at changes in magnetic entropy, where the doping
variation is on the divalent site. In the work by Hwang et
al.\cite{Hwang} the system
La$_{0.7}$(Ca$_{x}$Sr$_{1-x}$)$_{0.3}$MnO$_{3}$ was looked at, and
exhibited a change in the tolerance factor t, which is defined as
$t=(d_{A-O})/\sqrt{2}(d_{Mn-O})$, from $\approx$ 1.2 to 1.24.
We have undertaken a study of the
changes in electronic, structural, and magnetic properties of thin
films of La$_{0.67}$(Ca$_{x}$Sr$_{1-x}$)$_{0.33}$MnO$_{3}$ (LCSMO) as the
Ca/Sr ratio is varied.
\section{Sample preparation and characterization}
Our samples were grown by off-axis sputtering using composite targets
of La$_{0.67}$Ca$_{0.33}$MnO$_{3}$ (LCMO) and
La$_{0.67}$Sr$_{0.33}$MnO$_{3}$ (LSMO) material mounted in copper
cups. The substrates were (100) oriented neodymium gallate
(NdGaO$_{3}$), silver-pasted onto a stainless steel substrate holder
that was radiatively heated from behind by quartz lamps. Although
there was no direct measurement of the holder temperature for the runs
used in this study, previous runs (under nominally the same
conditions) using a thermocouple clamped onto the front surface of the
holder indicated a temperature of 670 C. The LCMO target was
radio frequency (rf) sputtered and the LSMO target was direct current
(dc) sputtered in a sputter gas
composed of 80\% Ar and 20\% O$_{2}$ (as measured by flow meters) and
at a total pressure of 13.3 Pa. These conditions gave deposition
rates of $\approx$ 17-50 nm/hr, with film thicknesses being typically
100 nm. After deposition, the samples were cooled in 13.3 kPa of
oxygen. We find that our system can produce films of LCMO and LSMO
that have low resistivities and high peak temperatures without the use
of an {\it ex-situ} anneal in oxygen.
The samples were characterized by standard and high resolution $\theta-2\theta$ x-ray
diffraction scans, atomic force microscopy, electrical resistivity
measurements (using the van der Pauw method\cite{VdP}) in an applied
field perpendicular to the film plane, and magnetization measurements
at low fields parallel to the film plane using a Quantum Design SQUID
Magnetometer. All magnetization data had the large paramagnetism of
the NdGaO$_{3}$ substrates subtracted out.
\section{Structure, Transport and Magnetic Properties}
On (100) NdGaO$_{3}$ we find
surface roughness values of $\approx$ 1.5 nm for pure films of LSMO
and LCMO, while for the mixtures the surface roughness increases to
$\approx$ 2.8 nm, as measured by atomic force
microscopy. The grain sizes for the pure LSMO and LCMO films is
typically 100 nm, while for the mixtures it is reduced to $\approx$ 50
nm. High resolution X-ray diffraction along the growth direction
shows only the presence of peaks from NdGaO$_{3}$ for the LCMO samples. This would be
expected, since the lattice match of pseudo-cubic (100) LCMO
($a_{o}$ $\approx$ 0.387 nm) to pseudo-cubic (100) NdGaO$_{3}$ ($a_{o}$ $\approx$
0.385 nm) should be excellent. From this we take the orientation
of the LCSMO films to be (100). Films of LSMO on NdGaO$_{3}$ however
as shown in Fig. \ref{XRD} do exhibit a peak corresponding to a
pseudo-cubic length of 0.388 nm. The rocking curve width for this
line is 337 arc-seconds, with an instrumental width of 12
arc-seconds, and phi scans show excellent in-plane registry of the film with
the NdGaO$_{3}$ substrate. From our work on LSMO and LCMO grown on both (100) and
(110) MgO, we know that LSMO films typically grow with a slightly
larger value of the pseudo-cubic cell length, $a_{o}$, compared to LCMO. On
(100) MgO, for example, we find that $a_{o}$ is 0.387 and 0.388 nm for LCMO and
LSMO, respectively, while for (110) MgO we find 0.388 and 0.390 nm
for the two materials. Obviously the lattice match is not as good for
the case of LSMO, and this will introduce strain into the LSMO film.
We also see from Fig. \ref{XRD} that as the calcium fraction is
increased, the well defined peak seen for the LSMO film moves to
smaller d spacings, consistent with the trend towards LCMO, appearing
as a shoulder on the low angle side of the NdGaO$_{3}$ (200) peak. As
the calcium fraction increases further, the shoulder diminishes, and
for pure LCMO (not shown), it is indistinguishable from the substrate
peak. From this we surmise that the films will be strained, with the
strain decreasing as the calcium fraction increases.
\begin{figure}
\begin{center}
\includegraphics*[width=0.45\textwidth]{Fig1}
\end{center}
\caption{High resolution X-ray diffraction scans along the film normal
for a pure LSMO film (x=0) and two LCSMO mixtures (calcium fraction
x=0.25 and x=0.91)
grown on NdGaO$_{3}$.}
\label{XRD}
\end{figure}
In Fig. \ref{rho1} we present the resistivity data in zero applied
magnetic field for the LCSMO films for the various Ca/Sr ratios, along
with a plot of the peak temperatures (T$_{p}$) and Curie
temperatures (T$_{C}$) determined from magnetization data for the
samples (taken at 400 Oe). For the case of pure LCMO, we see the
usual $\rho$(T) behavior, with a thermally activated resistivity
(activation energy of $\approx$ 52 meV) and a peak temperature of 260
K, which is the same as the Curie temperature. For pure LSMO (x=0),
we find that the resistivity has a peak temperature (410 K) much
higher than the Curie temperature (330 K) This discrepancy between the
peak and Curie temperatures is often seen for
LSMO. The Curie temperature we see for our LSMO is lower than that
seen in bulk LSMO with 1/3 doping,\cite{Urashibara} which we feel is
due to disorder in the sample. The difference in disorder or strain between
the LSMO and LCMO samples was also seen in the measurements of the
coercive field for the two samples. At 10 K, the coercive field for
the LCMO film was 20 Oe, which is quite low. However for the LSMO
film, the coercive field was 170 Oe. Now as the concentration of Ca is
varied from either extreme, we see sudden changes in the sample
properties. For the x=0.91 sample, we see a large decrease in the
sample resistivity, with a concurrent rise in the T$_{p}$ and T$_{C}$.
The increase in the Curie temperature is likely due to the change in
the tolerance factor as the Ca atoms are replaced with Sr atoms. On
the other end, at x=0.25, the increase in Ca fraction causes a large
increase in the resistivity, along with a large drop in T$_{p}$ and
T$_{C}$. The large increase in resistivity is likely the reason for
the low value of T$_{C}$ observed. The values for T$_{p}$ and T$_{C}$
are similar for the x=0.91 sample, but diverge for the samples with
lower x. We see that as the Ca fraction decreases, T$_{p}$ and
T$_{C}$ tends to increase as one would expect for the change in
tolerance factor being introduced. However, the variation we see in
T$_{C}$ is not as gradual as was seen in either the previous bulk
studies on divalent doping.\cite{Hwang,Guo} In the work by Hwang et
al. an apparent smooth variation in T$_{C}$ from 250 to 365 K is seen as the Ca/Sr
ratio is varied. In the work by Guo et al. there was a jump in
T$_{C}$ at a Ca fraction, x, of $\approx$ 0.45 when the system went
from orthorhombic to rhombohedral. No such jump is seen in our data,
but with the strong lattice match to the NdGaO$_{3}$ substrate, we
would not expect a structural change. We see instead that the
value of T$_{C}$ changes very slowly as the value of x is changed,
but with rather sudden changes near x=0 and x=1. We feel that part of the
explanation for the non-monotonic behavior of T$_{C}$, as well as T$_{p}$, is due to
disorder in the samples, as we discuss below.
\begin{figure}
\begin{center}
\includegraphics*[width=0.45\textwidth]{Fig2}
\end{center}
\caption{Electrical resistivity vs. temperature for LCSMO films
grown on (100) NdGaO$_{3}$ for different values of x, the calcium
fraction. The insert shows the variation in the peak temperature and
Curie temperature as a function of the calcium concentration.}
\label{rho1}
\end{figure}
\newpage
In previous studies of the low temperature (T $<$ 200 K) resistivity
in the manganites, several different equations have been used to
characterize the behavior. Schiffer et al.\cite{Schiffer} used the
equation $\rho (T) = \rho_{0} + \rho_{1}T^{2.5}$ for LCMO
polycrystalline material and found good fits to the data. Similar
results for LCMO films have also been seen.\cite{Vlakhov} Urashibara
et al.\cite{Urashibara} found for LSMO material a T$^{2}$ dependence,
which was interpreted as being due to electron-electron scattering.
We found that we could also get reasonable fits using the approach in
Schiffer et al. if we limited the data selection to T $\rm <$ 150 K.
However, if we look at the data for T $\rm <$ 200 K, we find that we
get better agreement if we use
\begin{equation}
\rho (T) = \rho_{0} + \rho_{2}T^{2} +\rho_{5}T^{5}
\label{rhoT}
\end{equation}
as seen in Figure \ref{lowrho}. A similar result was seen in well
annealed LSMO and LCMO films by Snyder et al.,\cite{Snyder} but here
they used a $T^{4.5}$ term instead of $T^{5}$, in light of the
prediction of spin-wave scattering by Kubo and Ohata.\cite{Kubo}
However, from the work on Pb-doped LCMO single crystals,\cite{Jaime}
the contribution from the $T^{9/2}$ term is expected to be much
smaller ( $\approx$ 0.5 $\mu \Omega$-cm at 100 K) than that seen in our results,
which is $\approx$ 10 $\mu \Omega$-cm.
We also observe the reduction in the contribution of the $T^{2}$ term
at low temperatures, which is
interpreted by Jaime et al. as an indication that the $T^{2}$ term
arises from single-magnon scattering, and not electron-electron
scattering.
\begin{figure}
\begin{center}
\includegraphics*[width=0.45\textwidth]{Fig3}
\end{center}
\caption{Low temperature resistivity vs. (T/T$_{C}$)$^{2}$ for a LCMO, LSMO and x=0.61
LCSMO film. The insert shows the derived values of $\rho_{0}$ and
$\rho_{2}$ vs. calcium fraction.}
\label{lowrho}
\end{figure}
Our derived values for $\rho_{0}$ and $\rho_{2}$ determined from
fitting Eq.\ref{rhoT} are shown in the inset to Fig. \ref{lowrho}.
The values of $\rho_{5}$ are typically 1 f$\Omega$-cm $K^{-5}$. We see
that the temperature independent term, $\rho_{0}$ is lowest for the
pure LCMO films, with values similar to that seen in Snyder et
al.\cite{Snyder} As the Ca fraction decreases, we see an increase in
$\rho_{0}$, which indicates an increase in the disorder in the
films. This increase in disorder might be initially thought to be due
to random location of Sr on Ca sites, but since the Ca sites are
already located at random in LCMO, it is difficult to see how
replacing Ca with Sr has increased the randomness in the system. The trend continues until pure
LSMO is reached, when we see a drop in the static term. We notice
however that the low temperature resistivity is higher for our pure
LSMO films than for pure LCMO, which reflects the increased disorder
for the LSMO film as seen in the coercive field measurements. A
similar result was also seen in
Ref.\cite{Snyder}. For the temperature dependent term, we see a
similar non-monotonic trend, with a peak in the value of $\rho_{2}$ as the Ca
fraction decreases, and a large drop when pure LSMO is reached. The
values of $\rho_{2}$ that we observe for pure LSMO and LCMO are
larger than that seen in Snyder et al.\cite{Snyder}, however both
our values and those for Snyder show a similar correlation, as seen
in Fig. \ref{rhocomp}. Clearly there appears to be a connection between the
values of $\rho_{2}$ and $\rho_{0}$, with the ratio of the two being
approximately 60-70 x 10$^{-6}$ K$^{-2}$. If the $\rho_{2}$ term is due to
electron-electron scattering, it is very hard to see what correlation
would exist between the static disorder in the sample and the terms in
e-e scattering, such as the Fermi energy. The model of Jaime et
al.\cite{Jaime} also would give no correlation between the two
terms. If there was a coincidental correlation between the
two terms, due say to changes in E$_{F}$ (which affects
$\rho_{2}$) and changes in strain (which affects $\rho_{0}$) with x,
we would not expect to see the same correlation for the films in the
work by Snyder et al.\cite{Snyder} since the points with the lowest
and highest values of $\rho_{0}$ are for pure LCMO films.
\begin{figure}
\begin{center}
\includegraphics*[width=0.45\textwidth]{Fig4}
\end{center}
\caption{Values of $\rho_{2}$ vs $\rho_{0}$ for this work and the thin
films in Reference \cite{Snyder}. The line is a guide to the eye.}
\label{rhocomp}
\end{figure}
In Figure \ref{rhoH} we show the magnetoresistance at 6 Tesla applied
field as a function of
temperature for the films, defined as
\begin{equation}
MR=\frac{(R(H=0T)-R(H=6T))}{R(H=0T)}.
\end{equation}
For the range of temperatures studied, we see a maximum in the room
temperature magnetoresistance for the x=0.91 sample, (since T$_{p}$ is
close to room temperature) however the
largest magnetoresistance occurs for the pure LCMO sample. The magnetoresistance at
77 K for all the samples is linear in applied field, going from $\approx$ 0.5 {\%} for the
LCMO sample to 3 {\%} for the LSMO sample at 6 Tesla of field. The field
dependence of the magnetoresistance at room temperature for the
samples undergoes a change as would be expected for T$_{p}$ moving
from above to below room temperature as seen in Fig. \ref{rhoHRT}.
Near zero field, the curves for the pure LCMO and the x=0.91 sample
exhibit positive concavity, which is seen for samples with T $>$
$T_{p}$, however for higher fields we see that the concavity for the
x=0.91 sample switches to negative which is that seen for the other
samples. As seen in Ref.\cite{Snyder1}, we can fit the change in
resistance for the case of T$<$ T$_{C}$ to the equation
\begin{equation}
\rho (H) = \rho_{\infty}+\frac{\Delta}{1+H/\gamma }.
\label{rhoHeq}
\end{equation}
The values of $\gamma$ for samples with x$<$1 at room
temperature are shown in the
inset in Fig. \ref{rhoHRT}. We see that the values of $\gamma$
decrease as the Ca fraction increases. In Ref.\cite{Snyder1} for pure
LCMO a value of $\gamma$ = 2.7 Tesla was found at 0.9 $T_{C}$, which
would fit in reasonably into our values, assuming of course that
$\gamma$ is not strongly temperature dependent. If $\gamma$ is
dependent on the relative difference between T and T$_{C}$ or T$_{p}$, we would
not get the smooth variation seen in the inset of Figure \ref{rhoHRT},
since T$_{C}$ and T$_{p}$ are not a monotonic function of x, as seen in Figure
\ref{rho1}.
For the pure LCMO sample, we could fit
the data equally well to the equation proposed in Ref.\cite{Snyder1}
$\rho(H)= \rho_{\infty}+\Delta/(1+ (H/\beta)^{2})$, or the form
$\rho(H)= \rho_{0}+aH^{2}+bH^{4}$. However the use of the first
equation resulted in values of $\rho_{\infty} < 0$, which is
unphysical. The value of $\beta$ is $\approx$ 8.5 T, which is larger
than that seen in Ref.\cite{Snyder1}, 5.7 T. The data for the x=0.91
cannot be fit over the entire range with any of the formulations,
since it exhibits a concavity change with field. However, for high
fields (above 2 Tesla), it can be fit by Eq. \ref{rhoHeq}, giving a
value of $\gamma$ as seen in Fig. \ref{rhoHRT}.
\section{Conclusions}
We have observed that LCSMO films grow with (100) pseudo-cubic
orientation on NdGaO$_{3}$ substrates with somewhat rougher surfaces
and smaller grain size than either pure LCMO or LSMO films. As the Ca
fraction decreases, the lattice constant for LCSMO increases towards
the value for LSMO, resulting in an increase in strain in the system.
This strain is manifested by a reduction in the Curie temperature, and
increases in the coercive fields and low temperature resistivity. We
have also observed the T$^{2}$ dependence to the resistivity, and
have observed a correlation between this term and the static term.
The field dependence to the magnetoresistance for LCSMO films is
predicted well by the equations in Ref. \cite{Snyder1}, with the
value of $\gamma$ increasing as the Ca fraction is reduced.
\section{Acknowledgments}
We would like to gratefully acknowledge the assistance of Michael
Miller for the AFM measurements and Andrew Patton in the
production of the films.
\begin{figure}
\begin{center}
\includegraphics*[width=0.45\textwidth]{Fig5}
\end{center}
\caption{Magnetoresistance vs. temperature at 6 Tesla applied field for LCSMO films grown
on (100) NdGaO$_{3}$.}
\label{rhoH}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics*[width=0.45\textwidth]{Fig6}
\end{center}
\caption{Magnetoresistance vs. field at room temperature for the LCSMO films in Fig.
\ref{rhoH}. The inset shows the values of $\gamma$ at room
temperature vs. calcium fraction derived from fitting Eq.\ref{rhoHeq} to the data}
\label{rhoHRT}
\end{figure}
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,840
|
Peaks & Prairies Pale Ale by Lift Bridge and Odell Brewing Cos.
Brew, The Sip by James Norton 06/08/2017 06/07/2017
James Norton / Heavy Table
The best milkshake we ever tried was had about eight years ago in Spokane, Wash., in a diner shaped like a giant milk bottle. It was a huckleberry shake, and the relatively gentle, mild character of the berries made their flavor a subtle accent to the shake's vanilla ice cream. The berry flavor was a passenger, not a co-pilot, but its essence made the whole package a great deal more interesting and enjoyable.
That same underspoken subtlety comes through in the new Lift Bridge Brewing / Odell Brewing Peaks & Prairies pale ale, a beer that incorporates wheat, Minnesota blue corn, and Colorado huckleberries. The brew (shown above making an appearance at a local D&D night) pours a disconcerting berry red, but the flavor of huckleberries is mellow enough that it's a cheerful, soft-spoken member of the beer's flavor team rather than being its order-barking captain. There isn't too much of a hop presence, which means that the mellow flavor of the huckleberries can be enjoyed without too much astringent interference, but the brew has enough of a body that it's not a throwaway hot-weather nothing. It's a moderate 5.75 percent ABV, so it will neither floor you after a glass nor leave you wanting for boozy substance.
We hadn't planned on having much of the beer — just enough to sample and report on — but damned if it wasn't so well-rounded, mellow, and agreeable that it compelled multiple return visits. This would be a lovely beer any time of the year, but its balanced berry character makes it an ideal choice for the advent of summer.
Peaks and Prairies pale ale is available at the Lift Bridge taproom and at restaurants participating in the Growler's Taps and Tables event to benefit Wishes & More.
beerbrewingcollaborationColoradoLift BridgeMinnesotaOdellPeaks and PrairiesStillwaterTaps and Tables
Copperwing Distillery, Broken Clock Brewing Co-op now open, OMNI makes big strides
Heavy Table Hot Five: June 9-15
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,658
|
using System;
using System.Collections.Generic;
// ReSharper disable InconsistentNaming
// These names are based on error codes.
#pragma warning disable 1591
namespace Soothsharp.Translation
{
/// <summary>
/// This class contains static constants that represent the various errors and warnings Sootsharp might generate.
/// <para>
/// Error codes 100-199 are translation errors that mean that a program cannot be translated into Silver.
/// </para>
/// <para>
/// Error codes 200-299 are verification warnings that mean that the backend does not guarantee correctness of the code.
/// </para>
/// <para>
/// Error codes 300-399 are internal errors of the transcompiler.
/// </para>
/// </summary>
public class Diagnostics
{
// *********************************** 100 Translation Errors
public static SoothsharpDiagnostic SSIL101_UnknownNode =
SoothsharpDiagnostic.Create(
"SSIL101",
"The Soothsharp translator does not support elements of the syntax kind '{0}'.",
"A syntax node of this kind cannot be translated by the Soothsharp translator because the feature it provides is unavailable in Viper, or because it is difficult to translate. If you can use a less advanced construct, please do so.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL102_UnexpectedNode =
SoothsharpDiagnostic.Create(
"SSIL102",
"An element of the syntax kind '{0}' is not expected at this code location.",
"While the Soothsharp translator might otherwise be able to handle this kind of C# nodes, this is not a place where it is able to do so. There may be an error in your C# syntax (check compiler errors) or you may be using C# features that the translator does not understand.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL103_ExceptionConstructingCSharp =
SoothsharpDiagnostic.Create(
"SSIL103",
"An exception ('{0}') occured during the construction of the C# abstract syntax tree.",
"While this is an internal error of the translator, it mostly occurs when there is a C# syntax or semantic error in your code. Try to fix any other compiler errors and maybe this issue will be resolved.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL104_ExceptionConstructingSilver =
SoothsharpDiagnostic.Create(
"SSIL104",
"An exception ('{0}') occured during the construction of the Viper abstract syntax tree.",
"While this is an internal error of the translator, it mostly occurs when there is a C# syntax or semantic error in your code. Try to fix any other compiler errors and maybe this issue will be resolved.",
DiagnosticSeverity.Error);
// ReSharper disable once UnusedMember.Global - kept for compatibility and future-proofing
public static SoothsharpDiagnostic SSIL105_FeatureNotYetSupported =
SoothsharpDiagnostic.Create(
"SSIL105",
"This feature ({0}) is not yet supported.",
"As the C#-to-Viper translation project is developed, we plan to allow this feature to be used in verifiable C# class files. For now, however, it is unsupported and won't work.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL106_TypeNotSupported =
SoothsharpDiagnostic.Create(
"SSIL106",
"The type {0} is not supported in Viper.",
"The Viper language can only use an integer, a boolean and reference objects. Other value types besides these three cannot be translated.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL107_ThisExpressionCannotBeStatement =
SoothsharpDiagnostic.Create(
"SSIL107",
"This expression cannot form an expression statement in Viper.",
"The Viper language does not support this expression as a standalone expression in a statement, even if C# supported it.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL108_FeatureNotSupported =
SoothsharpDiagnostic.Create(
"SSIL108",
"This feature ({0}) is not supported.",
"This feature of C# is not supported by the translator, and will probably never be supported. Could you please try to replace it with less advanced C# features?",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL109_FeatureNotSupportedBecauseSilver =
SoothsharpDiagnostic.Create(
"SSIL109",
"This feature ({0}) is not supported by Viper.",
"This feature of C# cannot be meaningfully represented in Viper.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL110_InvalidSyntax =
SoothsharpDiagnostic.Create(
"SSIL110",
"Syntax is invalid ({0}).",
"This feature of C# cannot be meaningfully represented in Viper.",
DiagnosticSeverity.Error);
// ReSharper disable once UnusedMember.Global - kept for compatibility
public static SoothsharpDiagnostic SSIL111_NonStatement =
SoothsharpDiagnostic.Create(
"SSIL111",
"\"{0}\" is not a statement silvernode.",
"",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL112_FileNotFound =
SoothsharpDiagnostic.Create(
"SSIL112",
"A C# file or reference could not be loaded ({0})",
"",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL113_VerificationSettingsContradiction =
SoothsharpDiagnostic.Create(
"SSIL113",
"This is marked both [Verified] and [Unverified]. Which do you want?",
"",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL114_NotPureContext =
SoothsharpDiagnostic.Create(
"SSIL114",
"This ({0}) cannot be translated into a pure assertion.",
"In this context, C# code is forced to be translated into pure Viper assertions. However, this C# node cannot be translated in a pure way.",
DiagnosticSeverity.Error);
// This error no longer ever triggers -- all integers are now converted to Int, without regard to their size in C#
public static SoothsharpDiagnostic SSIL115_ThisIntegerSizeNotSupported =
SoothsharpDiagnostic.Create(
"SSIL115",
"[deprecated error] Use System.Int32 instead of {0}.",
"[no longer true] The Viper language's integers are unbounded. To prevent confusion, use only 'System.Int32' integers, please. Your actual used type wouldn't matter anyway, since Viper does not check for overflow or underflow.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL116_MethodAttributeContradiction =
SoothsharpDiagnostic.Create(
"SSIL116",
"Constructors and predicates cannot be declared [Pure].",
"The [Pure] attribute causes the C# method to be translated as a Viper function. Viper predicates can't be functions. Constructors, in the transcompiler's view, also cannot be functions.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL117_ConstructorMustNotBeAbstract =
SoothsharpDiagnostic.Create(
"SSIL117",
"Constructors cannot be declared [Abstract].",
"",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL118_FunctionsMustHaveAReturnType =
SoothsharpDiagnostic.Create(
"SSIL118",
"Methods declared [Pure] must have a non-void return type.",
"In Viper, functions always return a value. Methods with the [Pure] attribute are translated as a functions and thus can't have the void return type. ",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL119_PredicateMustBeBool =
SoothsharpDiagnostic.Create(
"SSIL119",
"Methods declared [Predicate] must have the boolean return type.",
"Methods with the [Predicate] attribute are translated as Viper predicates. Predicates are either spatial or boolean - either way, the return type must be System.Boolean.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL120_UndeclaredNameReferenced =
SoothsharpDiagnostic.Create(
"SSIL120",
"Code references the name '{0}' which was not declared.",
"This error would usually mean that you're using a reference that was not given to the transcompiler or that there is an error in the transcompiler.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL121_FunctionsAndPredicatesCannotHaveStatements =
SoothsharpDiagnostic.Create(
"SSIL121",
"Methods declared [Pure] or [Predicate] may only contain return statements and contracts.",
"A method thus declared is translated to a Viper function or predicate. These constructs may have bodies, but these bodies must contain an assertion only, not statements. This is why only contracts (such as Contract.Ensures(...)) and a single return statement are permitted here.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL122_FunctionsAndPredicatesCannotHaveMoreThanOneReturnStatement =
SoothsharpDiagnostic.Create(
"SSIL122",
"Methods declared [Pure] or [Predicate] must have exactly one return statement.",
"A method thus declared is translated to a Viper function or predicate. These constructs may have bodies, but these bodies must contain an assertion only, not statements. There can't be any statements, and the single return statement is translated as a Viper assertion. There can't be more than one.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL123_ThereIsThisCSharpError =
SoothsharpDiagnostic.Create(
"SSIL123",
"A C# compiler diagnostic prevents correct translation: {0}",
"Only valid C# programs can be translated to Viper. Fix any errors offered by the C# compiler, then attempt translation again.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL124_QuantifierMustGetLambda =
SoothsharpDiagnostic.Create(
"SSIL124",
"Quantifiers must have a lambda function as an argument.",
"The contents of the lambda function are converted into Viper code. You cannot use a non-lambda function here.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL125_LambdasMustHaveSingleParameter =
SoothsharpDiagnostic.Create(
"SSIL125",
"Lambda functions must have a single parameter.",
"Soothsharp does not allow for lambda functions with more than one parameter.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL126_LambdasMustBeExpressions =
SoothsharpDiagnostic.Create(
"SSIL126",
"Lambda functions must have an expression body.",
"Soothsharp does not allow for lambda functions with statement (full) bodies, the body of the function must an expression.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL127_LambdasOnlyInContracts =
SoothsharpDiagnostic.Create(
"SSIL127",
"Lambda functions can only occur within quantifiers.",
"Soothsharp does not allow for use of arbitrary lambda functions - you may only use these within quantifiers for syntactic purposes.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL128_IndexersAreOnlyForSeqsAndArrays =
SoothsharpDiagnostic.Create(
"SSIL128",
"Element access works only for Seq types and for arrays.",
"",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL129_MethodContractsAreOnlyForMethods =
SoothsharpDiagnostic.Create(
"SSIL129",
"Method contracts (Requires and Ensures) must be in a method body, outside of any inner blocks or loops.",
"It is an error for a Requires or Ensures call to be within a loop. Only invariants are contracts permitted within a loop.",
DiagnosticSeverity.Error);
public static SoothsharpDiagnostic SSIL130_InvariantsAreOnlyForLoops=
SoothsharpDiagnostic.Create(
"SSIL130",
"Invariants must be within loops.",
"It is an error for an Invariant call to be elsewhere than within the code block of a loop (for, while or do).",
DiagnosticSeverity.Error);
// TODO (future): Expand this diagnostic so that it triggers more consistently (not just for binary expressions, but assignments, method calls...)
public static SoothsharpDiagnostic SSIL131_AssignmentsNotInsideExpressions =
SoothsharpDiagnostic.Create(
"SSIL131",
"Assignment expressions must be outermost.",
"In Viper, expressions with side-effects (such as an assignment) are statements and so can't be within other expressions.",
DiagnosticSeverity.Error);
// ****************************** 200 Backend Verifier Errors
public static SoothsharpDiagnostic SSIL201_BackendNotFound =
SoothsharpDiagnostic.Create(
"SSIL201",
"Back-end ({0}) not found.",
"The back-end chosen to verify the translated Viper code was not found in PATH nor in the local directory and so the code was not verified.",
DiagnosticSeverity.Warning);
public static SoothsharpDiagnostic SSIL202_BackendUnknownLine =
SoothsharpDiagnostic.Create(
"SSIL202",
"Backend: {0}",
"This line was returned by the backend but Soothsharp does not recognize it.",
DiagnosticSeverity.Warning);
public static SoothsharpDiagnostic SSIL203_ParseError =
SoothsharpDiagnostic.Create(
"SSIL203",
"Viper parse error: {0}",
"This C# code was transformed into a Viper segment that does not conform to Viper grammar and therefore the code could not be verified. This should not ordinarily happen and indicates an error in the Sootsharp transcompiler.",
DiagnosticSeverity.Warning);
public static SoothsharpDiagnostic SSIL204_OtherLocalizedError =
SoothsharpDiagnostic.Create(
"SSIL204",
"{0}",
"",
DiagnosticSeverity.Warning);
public static SoothsharpDiagnostic SSIL205_NoErrorsFoundLineNotFound =
SoothsharpDiagnostic.Create(
"SSIL205",
"Verification failed for an unknown reason.",
"The verifier did not return any errors but it did not certify that the code is correct. This may happen if you don't have Java, Z3 or Boogie installed or if the verifier is set up incorrectly.",
DiagnosticSeverity.Warning);
// ****************************** 300 Internal Errors
public static SoothsharpDiagnostic SSIL301_InternalLocalizedError =
SoothsharpDiagnostic.Create(
"SSIL301",
"The transcompiler encountered an internal error ({0}) while parsing this.",
"Try to remove the infringing C# code fragment. You may be forced to make do without that C# feature. You can also submit this as a bug report as this error should never be displayed to the user normally.",
DiagnosticSeverity.Error);
// ReSharper disable once UnusedMember.Global - kept for futureproofing
public static SoothsharpDiagnostic SSIL302_InternalError =
SoothsharpDiagnostic.Create(
"SSIL302",
"The transcompiler encountered an internal error ({0}).",
"Try to undo your most recent change to the code as it may be triggering this error. You may be forced to make do without that C# feature. You can also submit this as a bug report as this error should never be displayed to the user normally.",
DiagnosticSeverity.Error);
/// <summary>
/// Gets all error descriptions that might be outputted by Soothsharp.
/// </summary>
public static IEnumerable<SoothsharpDiagnostic> GetAllDiagnostics()
{
Type t = typeof(Diagnostics);
var fs = t.GetFields(System.Reflection.BindingFlags.Static | System.Reflection.BindingFlags.Public);
foreach(var f in fs)
{
object diagnostic = f.GetValue(null);
yield return diagnostic as SoothsharpDiagnostic;
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,384
|
{"url":"http:\/\/math.stackexchange.com\/questions\/1323\/is-there-an-atlas-of-algebraic-groups-and-corresponding-coordinate-rings","text":"# Is there an atlas of Algebraic Groups and corresponding Coordinate rings?\n\nI was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.\n\nEdit: The previous wording was terrible.\n\nGiven an algebraic group $G$, with Borel subgroup $B$ we can form the Flag Variety $G\/B$ which is projective. I am hoping for a list of the graded ring $R$ such that $Proj(R)$ corresponds to this Flag Variety.\n\n-\nThis might help groupprops.subwiki.org\/wiki\/Main_Page \u2013\u00a0 Jonathan Fischoff Jul 31 '10 at 23:43\nI don't quite understand the question. For example, there are a lot of abelian varieties -- what should be listed for them? And in what sense SL_2 (3-dimensional group) corresponds to k[x_0,x_1]? \u2013\u00a0 Grigory M Aug 1 '10 at 5:11\nIt is a standard exercise to write down the coordinate ring of $GL_n$ as a hypersurface in affine $n^2+1$-space. After doing that, every linear algebraic group is a closed subgroup of $GL_n$, usually given by explicit polynomial equations, so this is easily done. What do you mean by \"the projectivizations\"? \u2013\u00a0 Pete L. Clark Aug 1 '10 at 11:40\n@Grigory & Pete, the question was crappy, I have hopefully made it more clear. :\/ I apologize for being opaque. \u2013\u00a0 BBischof Aug 1 '10 at 17:36\n@Jonathan, thanks for the link, an initial exploration has not yielded what I am looking for, but that does not mean it does not exist. \u2013\u00a0 BBischof Aug 1 '10 at 17:36\n\nYou probably mean for $G$ to be a reductive group. Keep in mind that $G\/B$ is equal to $\\text{Proj}(R)$ for many different $R$'s, corresponding to different embeddings of $G\/B$ into projective space. The best object to study is the homogeneous coordinate ring (also known as the Cox ring) of $G\/B$. In that case, when $G = SL_n$, the homogeneous coordinate ring is in Miller and Sturmfels' Combinatorial Commutative Algebra Chapter 14. For the general case, some keywords to look for are \"standard monomial theory\", \"straightening laws\", and \"Littelmann path model\". The homogeneous coordinate ring of a general $G\/B$ (or at least $G\/P$ for $P$ a maximal parabolic) might be in Lakshmibai and Raghavan's Standard Monomial Theory: Invariant Theoretic Approach, but I am not sure. Regardless, that is a good introduction to the subject and should have a fairly comprehensive list of references for further information.","date":"2014-10-24 21:24:30","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\": 0.8401914834976196, \"perplexity\": 373.1677436999056}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1414119646519.4\/warc\/CC-MAIN-20141024030046-00088-ip-10-16-133-185.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"http:\/\/openstudy.com\/updates\/4dd2b53799508b0b3c1aa2a4","text":"anonymous 5 years ago Integrate sqrt{5sin^{2}(2t)} from 0 to pi\n\n1. watchmath\n\n$$\\sin^2(t)=\\frac{1}{2}(1-cos(2t))$$ So $$\\int 5\\sin^2(2t)\\,dt=\\int \\frac{5}{2}(1-\\cos(4t))\\,dt=\\frac{5}{2}t-\\frac{5}{8}\\sin(4t)+C$$\n\n2. watchmath\n\nOh sorry this is a definite integral... so just plugin the limits of integration :).\n\n3. anonymous\n\nHow did you eliminate the square root?\n\n4. watchmath\n\nI am sorry, I didn't see the square root ... :)\n\n5. watchmath\n\nOk, then it is simply the integral $\\sqrt{5}\\int_0^\\Pi \\sin(2t)\\,dt$ I believe you can do this integral now :).\n\n6. watchmath\n\nSorry ... I made a mistake\n\n7. watchmath\n\nOver the interval $$[0,\\pi]$$ the function $$\\sin(2t$$ is above the $$t-axis$$ on $$[0,\\pi\/2]$$ and below the $$t$$-axis on $$[\\pi\/2,\\pi]$$. So $$\\sqrt{5\\sin^2{2t}}=\\sqrt{5}\\sin{2t}$$ on $$[0,\\pi\/2]$$ and equal to $$-\\sqrt{5}\\sin(2t)$$ on the interval $$[\\pi\/2,\\pi]$$. So the integral si equal to $\\int_0^{\\pi\/2} \\sqrt{5}\\sin(2t)\\,dt-\\int_{\\pi\/2}^{\\pi} \\sqrt{5}\\sin(2t)\\,dt$\n\n8. anonymous\n\nThank you so much, this is the part I was missing.","date":"2016-10-27 15:29:33","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\": 0.9811753034591675, \"perplexity\": 713.1221021801998}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988721347.98\/warc\/CC-MAIN-20161020183841-00207-ip-10-171-6-4.ec2.internal.warc.gz\"}"}
| null | null |
Q: How to configuring Ubuntu 10.04 server for multiple gateways in a simple way I have a simple Ubuntu 10.04 server that runs a service. The system has two NIC cards each configured for separate networks. Each NIC needs a separate gateway, but when both are configured, eth1 gets kicked off.
How else, other than specifying a gatway for each interface in /etc/network/interfaces can I configure a gateway for each NIC card?
A: You need two routing tables for that, and two rules (ip rule) to choose the right table. (Or one primary and one extra for the other interface).
If you look here, the first example (4.2.1. Split access) is exactly what you need to configure.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 3,859
|
Terranet
We develop awareness connectivity software enabling streaming of audio, video and data in offline environments
CHANGES IN MANAGEMENT AT TERRANET WITH DECISION TO PROMOTE GREATER EFFECTIVENESS
Terranet continues to develop its relationships with global clients and partners, but sees that revenues from licensing deals are projected to be further out than previously anticipated. The company has thereby decided to cut costs by 40% to provide it with the runway required to reach licensing revenues. Cost-cutting measures will be in full effect towards the end of Q4 2018. As a result, the core business, based in Lund, Sweden, will operate as a smaller unit while some project resources, will be outsourced to partners, such as an existing cooperation with a partner in India.
Terranet develops and sells software for the offline communication between people, vehicles, mobile devices and machines. The software is designed to be integrated into products of companies with a global footprint, and it is typically once they reach the market that Terranet receives licensing revenues. As Terranet's products are designed for mass production, customers first have to carefully evaluate, test, and together with the company, tailor the technology to meet their customers' needs. Following such an evaluation period, products are then scheduled for completion and launched on the market after which revenues can be expected. These stages have proven to take longer than previously anticipated, leading to Terranet making the necessary adjustments to its organization.
Terranet's Chairman, Christian Lagerling, has announced that he plans on resigning from the company's board. The Board has today decided to propose Fredrik Hedlund as new Chairman of the Board, and will present this proposal at the Extra General Meeting on 27 July, 2018. Christian Lagerling was originally recruited to the Board in May 2016 to facilitate the company's planned listing on the Nasdaq First North. With Terranet having completed its IPO last year, Christian sees his work with Terranet as now complete and welcomes a chairman with extensive commercial industry expertise as being the new energy required for the company's next phase. Fredrik Hedlund currently holds a leading position at Northvolt, which is responsible for running a multi-billion SEK project with a new battery factory in Skellefteå. Previously, Fredrik Hedlund served as a strategic manager at Sony Mobile.
Current CEO Pär-Olof Johannesson has a broad global network of contacts. The Board of Directors will propose to the General Meeting that Pär-Olof be elected to the Board. It is proposed that as Vice Chairman, Pär-Olof will play an active role in developing global business, which is important to generating new licensing agreements as well as structured deals.
Terranet's current VP of Engineering, Ola Samuelsson, will be appointed as the new CEO in connection with the General Meeting. Ola takes on the role with a solid knowledge of technology, and will manage the business with a focus on reaching licensing deals where Terranet sees the strongest potential.
It is the Board's assessment that the aforementioned changes will strengthen the company during a time where perseverance and focusing on licensing deals is integral to reaching substantial revenues that one can expect Terranet's future licensing deals to provide.
Fredrik Hedlund, proposed new Chairman of the Board
Christian Lagerling, outgoing Chairman of the Board
Ola Samuelsson, CEO (to be confirmed at the GM)
Pär-Olof Johannesson, Deputy Vice Chairman of the Board (to be confirmed at the GM)
investorrelations@terranet.se
ABOUT TERRANET
Terranet develops software and holds patents that enable offline communication between people, mobile devices, and machines. Terranet's head office is located in Lund, Sweden with established sales and marketing agents in the US, China and India. Terranet Holding AB (public) is listed on Nasdaq First North Premier.
www.terranet.se
TerraNet AB
IDEON Science Park, Beta 6
info@terranet.se
career@terranet.se
Copyright © 2018 Terranet AB. All rights reserved.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 274
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Q: Complex roots in recurrence relation I was reading brilliant wiki on recurrence relations. It says, if the characterstic polynomial has complex roots. Say $$r=2e^{\pm i\theta}\\\text{where }\theta=\arctan(\sqrt{15})\\\text{for relation }x_n=x_{n-1}-4x_{n-2}$$
Then the solution are given by $$x_n=\alpha2^n\cos{n\theta}+\beta2^n\sin{n\theta}$$
I don't understand why it shouldn't be $$x_n=\alpha2^ne^{in\theta}+\beta2^ne^{-in\theta}$$
or how the two solution are same. Please help.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
The cytoskeleton \cite{alberts} is a dense multicomponent meshwork of semiflexible polymers which interact sterically as well as through active \cite{alberts,RamaswamyActiveRheo1,curiegrp1} processes. While the blending of polymers industrially requires special effort, the active environment of the living cell provides a setting in which polymers which differ substantially in their stiffness are naturally mixed and interact. Moreover, active processes such as polymerization and the working of molecular motors lead to the generation of stresses without the external imposition of flow fields. These two mechanisms combine to yield a rich range of novel physical phenomena.
The role of activity in cytoskeletal mechanics is receiving increasing attention, as seen from many recent theoretical and experimental studies of the rheology of cells and cell extracts \cite{LauLubensky03,Mizuno,mack}.
It is clear in particular \cite{multicomprefs} that interactions between different species of filaments are crucial for cell motility, cell division, vesicular transport and organelle positioning and integrity.
\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5cm]{filament1.eps}
\caption{A stiff filament (``microtubule'') embedded in an active isotropic medium consisting of oriented
fiaments (``F-actin''). The conformations of the microtubule, aligned on an average along the $x$ axis,
are described by small transverse fluctuations $\mathbf{u}_{\perp}\!(x,t)$. The active medium can either be contractile or
tensile. The orientation of F-actin along the microtubule can either be parallel or normal to it.}
\label{fig1}
\end{center}
\end{figure}
In this paper, we make a study of the effect of these interactions by modeling the dynamics of a stiff filament, which we will call a ``microtubule'', immersed in an active medium (Fig.\ref{fig1}) with orientational degrees of freedom, which we will call ``F-actin''.
We emphasize here that the names ``microtubule'' and ``F-actin'' are introduced for convenience: we consider both contractile and tensile activity, although only the former applies to actomyosin. Our treatment applies more generally to semiflexible polymers
under tension in a wide variety of active media.
We describe the medium by the active generalization of liquid-crystal
hydrodynamics
\cite{RamaswamyActiveRheo1,curiegrp1,RamaswamyActiveRheo2,LM031,curiegrp2,LM032,SriramMadanNJP07,curiegrp3,TT95}.
For the purposes of this paper, an active medium is one whose constituent
particles possess the ability to extract energy from an ambient nutrient bath
and dissipate it, executing some kind of systematic motion in the process. This
endows each such particle with a permanent uniaxial stress. The other central
ingredient of this work is \textit{anchoring}, on which we now elaborate. In
general, the interfacial energy of a liquid-crystalline medium at a wall
depends on the relative orientation $\mathbf{n}$ of the molecules of the medium
and the normal $\mathbf{N}$ to the wall. In the simplest cases, it is lowest
for $\mathbf{n}$ parallel, or perpendicular, to $\mathbf{N}$. This interaction
is known as anchoring. In the present work, anchoring enters through the
favoured orientation of the F-actin when confronted with the surface of the
microtubule. We believe that steric or pair-potential effects should lead to
the anchoring of the F-actin normal or parallel to the microtubule. The
interplay of the types of anchoring and activity -- contractile or tensile --
are fundamental to our theory. The filament in our study can be viewed as a
\textit{spatially extended} probe of the active medium, generalizing the
microrheometry of
\cite{LauLubensky03,Mizuno,mack,ChenLauLubensky07,wulibchaber} by simultaneous
access to a wide range of scales. We make contact with earlier work on
oscillatory filaments \cite{hudspeth,pramod2007}, and test our results against
observations on microtubules in a background of contractile actin in neuronal
growth cones (see Fig.\ref{fig2}).
Here are our main results:
(i) In the absence of activity, the microtubule
will {\em always} buckle at large anchoring strength, regardless of the type of
anchoring. (ii) An active medium,
by contrast, can stiffen {\em or} buckle the filament, depending on
the relative signs of activity and anchoring.
A contractile active medium with parallel anchoring always {\em stiffens}
the filament, as does a tensile active medium with normal anchoring,
while a contractile (tensile)
medium with normal (parallel) anchoring produces buckling if the strength $W$ of the active stresses is large enough (see Fig.\ref{fig3}). When the nematic correlation length of the F-actin medium is large compared to the linear dimension $L$ of the sample transverse to the microtubule, the buckling wavelength decreases with $L$ as
$1/\sqrt{WL}$ and $1/\sqrt{W \log L}$ respectively in two and three dimensions.
(iii) Our observations of the conformations of
microtubules in the growth cone of neurons are in qualitative accord with these predictions.
(iv) Activity leads to a breakdown of the fluctuation-dissipation relation: most dramatically, in the regime of strong stiffening, we predict that the effective dissipation turns negative when the frequency crosses a threshold. This is consistent with the observations of \cite{hudspeth} on auditory hair cells and \cite{pramod2007} on axons, and suggests that a negative force-velocity relation at finite frequency should be a generic feature of actively stiffening systems. Indeed, the phenomenological model of \cite{hudspeth} emerges as a limiting case of our fundamental theory.
\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5cm]{fig2.eps}
\caption{Cytoskeletal polymers in a fixed and fluorescently
stained growth cone of an NG108-15 neuronal cell. In the
peripheral domain actin filaments (red) form dense networks
in the veil-like lamellipodium and bundles in the spike-like
filopodia. Microtubules (green) emerge from the central do-
main into the periphery where they are buckled
(yellow arrowheads) in the lamellipodium by the actin network
or stabilized in the filopodial region (blue arrowheads). For illustration some microtubules are traced by
a dotted white line.}
\label{fig2}
\end{center}
\end{figure}
\section{A filament in an active medium}
Consider a stiff, locally inextensible filament of total contour length $L$,
coinciding on average with the $x$ axis (Fig.\ref{fig1}), with unit tangent vector
$\mathbf{\hat
t}\!=\!\mathbf{\hat x}\!+\!\mathbf{\delta{\hat
t}}\,{\simeq}\,\left(1 + O(\partial_x \mathbf{u}_{\perp})^2,{\partial}_x\mathbf{u}_{\perp}\!(x,t)\right)$
where $\mathbf{u}_{\perp}\!(x,t)$ are small transverse fluctuations,
and ${\perp}\,{\equiv}\,y,z$, immersed
in a $d$-dimensional active medium characterized
by $\mbox{\sffamily\bfseries Q}$, a symmetric traceless nematic order parameter \cite{deGP93}.
The effects of contractile or
tensile active stresses enter the equations of motion
\begin{eqnarray}
{\partial_t}\mathbf{u}_{\perp}\!-\!\mathbf{v}_{\perp}\!(x,\mathbf{r_{\!\perp}}
\!\!=\!\mathbf{0},t)&=& - {1 \over \gamma} \delta F/\delta \mathbf{u}_{\perp} + \!\mathbf{f}_{\perp},
\label{ueqn}\\
{\partial_t}\mbox{\sffamily\bfseries Q}\!
\! &=& -{1 \over \zeta}\delta F/\delta \mbox{\sffamily\bfseries Q}
+\mbox{\boldmath ${\eta}$},\label{Qeqn}
\end{eqnarray}
for $\mathbf{u}_{\perp}$ and $\mbox{\sffamily\bfseries Q}$ through the hydrodynamic velocity field $\mathbf{v}$ whose dynamics is governed by Eq.\ref{veqn} below.
The Gaussian, spatiotemporally white noises $\mathbf{f}_{\perp}$, \mbox{\boldmath ${\eta}$} in Eqs.\ref{ueqn},\ref{Qeqn} have strengths $2N_1$, $2N_2$, reducing
respectively to $2k_BT /\gamma$, $2k_BT/\zeta$ for the equilibrium case.
In Eqs.\ref{ueqn},\ref{Qeqn}, $\mathbf{u}_{\perp}$ and $\mbox{\sffamily\bfseries Q}$ are
coupled only through
the free-energy functional
$F[\mathbf{u}_{\perp},\mbox{\sffamily\bfseries Q}]= F_f[\mathbf{u}_{\perp}]+F_{LD}[\mbox{\sffamily\bfseries Q}]+F_{anc}[\mathbf{u}_{\perp},\mbox{\sffamily\bfseries Q}]$.
The filament
free energy $F_f[\mathbf{u}_{\perp}]=\int_0^L\!\!dx[({\sigma}/{2})\,(\partial_x \mathbf{u}_{\perp})^2
+({\kappa}/{2})(\partial_x^2 \mathbf{u}_{\perp})^{2}]$
contains bending energy with rigidity $\kappa$ and an imposed
tension
$\sigma$.
\cite{HFK05,HFK07} to leading order in $\partial_x \mathbf{u}_{\perp}$. The Landau-de
Gennes free energy
$F_{LD}[\mbox{\sffamily\bfseries Q}]=\int\!\!dx\!\!\int\!\!d^2r_{\!\perp}[({a}/{2}){\mbox{\sffamily\bfseries Q}}{}^2+({K}/{2})
{(\nabla\mbox{\sffamily\bfseries Q})}{}^2]$
describes incipient orientational ordering in the medium \cite{deGP93}.
We work here in the isotropic phase with correlation length ${\sim}\sqrt{K/a}$;
a study of the nematic phase by generalising \cite{Yodhfootnote} to include activity will be presented elsewhere \cite{nkikuchi08}.
The filament anchors the orientational degrees of freedom of the medium through
\begin{eqnarray}
&F_{anc}[\mathbf{u}_{\perp},\mbox{\sffamily\bfseries Q}] = \frac{A}{2} \int_0^L dx \hat{\bf t} \cdot \mbox{\sffamily\bfseries Q} (x,\mathbf{r}_{\perp}=\mathbf{0})\cdot\hat{\bf t} &\nonumber \\
& \,\,\,\,\,\,\,\,\,\, \simeq \mbox{const.} + A\!\!\int_0^L\!\!dx
[\partial_x \mathbf{u}_{\perp} \cdot \mbox{\sffamily\bfseries Q}_{x \perp}(x,\mathbf{r}_{\perp}=\mathbf{0}) +O{\left({\partial}_x\mathbf{u}_{\perp}\right)}{}^2]&
\nonumber
\end{eqnarray}
with negative and positive $A$ corresponding respectively to parallel
and normal anchoring.
Note that Eq.\ref{ueqn} generalises \cite{HFK05,HFK07} to
include anchoring and hydrodynamic flow. In Eq.\ref{Qeqn}, we ignore
flow-orientation coupling terms \cite{deGP93}
\footnote{If included these would lead to shifts of effective Frank constants and additional possible instabilities in the effective
equation of motion Eq.\ref{ueqnFT}.}.
For thin-film samples at large F-actin concentration, as in the case of the lamellipodium of adhering cells, it is appropriate to treat the hydrodynamic velocity field in a local-friction approximation. We therefore write $\Gamma{v_i}(x,\mathbf{r_{\!\perp}},t)= -\nabla_j\sigma_{ij}$,
with $\Gamma \sim \mu/\ell^2$ where $\mu$ is the cytoplasmic viscosity, and the screening length \cite{muthuedwards} $\ell$ is no larger than the film thickness. We ignore pressure gradients on the assumption that the film thickness adjusts to accommodate these. The crucial piece of the stress $\sigma_{ij}$ is the active contribution $\sigma_{ij}^{act}\simeq Wc_0Q_{ij}(x,\mathbf{r_{\!\perp}},t)$
\footnote{Passive stresses arising from the free-energy functional enter only at higher order in gradients.}
\cite{RamaswamyActiveRheo1,curiegrp1,RamaswamyActiveRheo2,LM031,LM032,SriramMadanNJP07}, where $W <0$ and $W>0$ respectively correspond to contractile and tensile stresses, and
$c_0$ is the mean F-actin concentration.
To leading order in gradients and linear order in filament undulations, the velocity transverse to the microtubule is
\begin{equation}
\label{veqn}
{\mathbf{v}_\mathbf{\perp}}(x,\mathbf{r_{\!\perp}},t)=-(Wc_0/\Gamma)\partial_x\mbox{\sffamily\bfseries Q}_{x\perp}\!(x,\mathbf{r_{\!\perp}},t).
\end{equation}
From Eqs.\ref{ueqn}-\ref{veqn}, the effective Fourier-transformed equation of motion for $\mathbf{u}_{\perp}(q_x,\omega)$ is
\begin{equation}
\label{ueqnFT}
\left(-i\omega+{\sigma \over \gamma}q_x^2+{\kappa \over \gamma}q_x^4\right)
\mathbf{u}_{\perp}
=iq_x\alpha\mbox{\sffamily\bfseries Q}_{x\perp}\!(\mathbf{r_{\!\perp}}\!\!=
\!\mathbf{0})+\mathbf{f}_{\perp},
\end{equation}
where $\mathbf{u}_{\perp}$ is coupled to $\mbox{\sffamily\bfseries Q}$ only at
$\mathbf{r}_{\perp} = 0$.
\begin{equation}
\label{QeqnFT}
\mbox{\sffamily\bfseries Q}_{x\perp}\!(\mathbf{r_{\!\perp}}\!\!=\!\mathbf{0})=
{iq_x\beta{\zeta} \over K}\!\!\!\int_{\mathbf{q_{\perp}}}
\!\!\!\!\!\!G_{\mathbf{q}\omega}\mathbf{u}_{\perp}
+{\zeta \over K}\!\!\!\int_{\mathbf{q_{\perp}}}
\!\!\!\!\!\!G_{\mathbf{q}\omega}
\mbox{\boldmath ${\eta}$}(q_x,\mathbf{q_{\perp}},\omega),
\end{equation}
where $q_{\omega}^2{\equiv} - {\zeta}i\omega/K+a/K+q_x^2$ and $G_{\mathbf{q}\omega} \equiv (q_{\!\perp}^2+q_{\omega}^2)^{-1}$, and
$\int_{\mathbf{q_{\perp}}}\!\!\!{\equiv}\!\!\int_0^{\Lambda}\!\!{d^{d\!-\!1\!}(q_{\!\perp}/2 \pi)}$, with ultraviolet cutoff $\Lambda \sim$ (filament thickness)$^{-1}$.
The signs of $\alpha \equiv (A/\gamma-Wc_0/\Gamma)$ and $\beta \equiv -A/\zeta$ decide the fate of the filament (stiffening \textit{or} buckling) in the following analysis.
Since $\int_{\mathbf{q_{\perp}}} G_{\mathbf{q}\omega}
\sim \ln(\Lambda^2/q_{\omega}^2)$ in $3$d and
$
\pi/(2q_{\omega})$ in $2$d,
we obtain the dispersion relations for $\mathbf{u}_{\perp}\!(q_x,\omega)$ for the case of an unbounded medium:
\begin{eqnarray}
-i\omega=\left\{
\begin{array}{cl}
-\left[{\sigma \over \gamma}+{\alpha\beta\zeta \over K}
\ln\left({\Lambda^2}/{q_{\omega}^2}\right)\right]q_x^2 + O(q_x^4),& d=3 \\
-\left[{\sigma \over \gamma}+
{\alpha\beta\zeta \over 4K}q_{\omega}^{-\!1}\right]q_x^2 +O(q_x^4),& d=2 .\label{disprel}
\end{array}
\right.
\end{eqnarray}
The case of a confined medium is discussed towards the end of the paper.
\section{Active stiffening and buckling}
We are now in a position to investigate the (in)stability of the filament.
At thermal equilibrium ($W=0$) sgn$[\alpha] = - $sgn$[\beta]$; indeed, $\alpha \beta = -A^2/\zeta \gamma <0$ irrespective of the sign of $A$. This implies a buckling instability of the filament as the anchoring strength is increased, regardless of whether the anchoring is parallel or perpendicular. This is consistent with observations \cite{Yodhfootnote} of buckling of filaments in isotropic solutions of \textit{fd} virus (see their Fig. 1a).
When activity $W$ is switched on, the signs of $\alpha$ and $\beta$ become independent. For large enough $|W|$, $\alpha\!>\!0$ for contractile ($W<0$) activity and $\alpha < 0$ for tensile ($W >0$) activity, while $\beta$ is separately controlled by the nature of the anchoring (Fig.\ref{fig3}).
Defining the correlation length $\xi \equiv \sqrt{K/a}$ of the F-actin medium, we focus on two limiting cases: (I) deep in the isotropic phase; (II) close to the transition to a nematic phase.
In (I), $q_x \xi \ll 1$,
so Eq.\ref{disprel} holds with $q_{\omega} = \sqrt{a/K}$ for small $\omega$.
Thus, for a filament in a strongly active medium, contractile (tensile) activity with parallel (normal) anchoring ($\alpha\beta>0$) leads to enhanced tension, i.e., stiffening.
On the other hand, for strong contractile (tensile) activity with normal (parallel) anchoring ($\alpha\beta<0$) the filament becomes unstable to buckling.
This is explained graphically in
(Fig.\ref{fig3}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=9.0cm]{fig3.eps}
\caption{A perturbation of a ``microtubule'' (black) leads, through anchoring,
to a distortion of the ``F-actin'' medium and hence of the active stress profile. Depending on whether the actin is anchored parallel or normal to the microtubule, and whether the active stresses (blue double-arrows) are contractile or tensile along the actin filaments, the resulting secondary flows (red arrows) either suppress {\em or} enhance the perturbation, leading to active stiffening or buckling.}
\label{fig3}
\end{center}
\end{figure}
In (II), $q_x \xi \gg 1$ (which should be accessible in F-actin, see \cite{Viamontes})
so that a wide range of modes of $\mbox{\sffamily\bfseries Q}$ contribute in Eq.\ref{QeqnFT}.
In dimension $d=$3, Eq.\ref{disprel} applies with $q_{\omega}^2 \simeq q_x^2 -\zeta{i}\omega/K$, so that the dispersion relation differs from case (I) only by a logarithmic factor. In $d=2$, however, solving the resulting cubic equation we find
\begin{equation}
-i\omega=
Y(|\alpha\beta|/4)^{2/3}(\zeta/K)^{1/3}q_x^{4/3}
\label{disprelIN}
\end{equation}
where $Y=(-1/2{\pm}i\sqrt{3}/2)$ for $\alpha\beta\!>\!0$ and $1$ for $\alpha\beta\!<\!0$.
Thus, in $2$d, close to the ordering transition of the F-actin, while stability is determined by the sign of $\alpha\beta$ as before, the stable stiffening case $\alpha\beta\!>\!0$, which can arise only for an active system, shows a damped \textit{oscillatory} response, which should appear as dispersive propagating waves on the filament, with speed $\propto q_x^{1/3}$. In the unstable case, the buckling wavelength can be found by comparing the
negative tension and bending elasticity terms. For an F-actin medium of finite lateral extent $L$, scaling arguments applied to Eqs.\ref{disprel},\ref{disprelIN} lead to a buckling wavelength varying as $1/\sqrt{WL}$ ($d=2$) and $1/\sqrt{W \log L}$ ($d=3$) as
claimed earlier in the paper.
The ideas presented above are best tested in a model system in which both parallel and normal alignment of F-actin on microtubules are naturally realised.
The growth cones of neurons provide a convenient system for this purpose, and allow a demonstration of activity-induced stiffening as well as buckling. Our observations on this system are qualitative but in clear accord with the predictions of this paper.
As seen in Fig.\ref{fig2} microtubules are strongly buckled in the wide central region known as the lamellipodium and show a stabilized straight shape in the spiky protrusions called the filopodia. As we have remarked, purely equilibrium effects in most systems generally lead to buckling.
It seems, however, that the dynamic activity within the lamella may contribute to microtubule stiffening
in the periphery, as well as buckling in the central region.
We suggest the following mechanism : when the growing microtubule enters the lamellipodium, it meets the actomyosin network, with actin filaments pointing in a variety of directions. The typical encounter will have a substantial angle between microtubule and actin filament. The analysis above would then predict that the microtubule should buckle, as it does. Upon entering a filopodial region, the microtubule meets actin aligned parallel to it and is stiffened, which again is consistent with our mechanism.
However, whether the relevant activity is entirely non-specific and based on contractility, or whether
specialized proteins are involved remains an open question.
In another relevant study, Brangwynne \textit{et al}. \cite{mack} observed significant bending fluctuations of microtubules in reconstituted actomyosin networks (see Fig.1 in \cite{mack}). Since the actomyosin is contractile, our theory would predict that the contacts between actin and microtubule are at near-normal alignment, and we look forward to independent tests of this. Indeed, our work provides a theoretical justification for the pointlike normal force that \cite{mack} suggest is responsible for the buckling.
\section{Fluctuation-Dissipation ratio and negative dissipation}
If $\chi(t)$ is the displacement of a degree of freedom of a system at time $t$ in response to an impulsive force at time $0$, and $C(t)$ is the time-correlation of spontaneous fluctuations in that degree of freedom, the Fluctuation-Dissipation Theorem (FDT) \cite{kubobook2}, which applies to systems \textit{at thermal equilibrium}, says $C(t) = -k_B T \chi(t)$, where $k_B$ is Boltzmann's constant and $T$ is the temperature. This profound and universal connection can be understood in the familiar context of a particle undergoing Brownian motion in a fluid, where both the damping of an initially imposed velocity and the random motion of the particle when no velocity is imposed arise from the same microscopic collisions with molecules.
For equilibrium systems, this allows one to obtain transport quantities such as conductivity without drawing a current. In systems far from equilibrium, but where the bath producing the fluctuations is still thermal, a judicious definition of variables \cite{speck} can resurrect the FDT. For more general nonequilibrium systems too, a comparison of correlation and response can sometimes offer a useful notion \cite{kurchan} of an effective temperature. Our model system shows radical departures from such benign behaviour.
We restrict attention to parameter ranges where there is no instability, so a steady state exists. Let $S({q_x},\omega)\!\equiv\!
\int_{x,t}\exp(i q_x x - i \omega t) \left<\mathbf{u}_{\perp}\!(x,t)\!\cdot\!\mathbf{u}_{\perp}\!(0,0)\right>$
be the correlation function and $\chi^{\prime \prime}(q_x,\omega)$ the imaginary part of the response to an external force $\mathbf{h}(x,t)$ coupled to $\mathbf{u}_{\perp}$ via a term $-\int dx \mathbf{h}\cdot\mathbf{u}_{\perp}$ in the free-energy functional $F$ in Eq.\ref{ueqn}. The departure from unity of the fluctuation-dissipation (FD) ratio $R(q_x,\omega) \equiv (\omega /2 k_BT)S(q_x,\omega)/\chi^{\prime\prime}(q_x,\omega)$
is a quantitative measure of nonequilibrium behaviour. We find not only that $R(q_x,\omega)$ depends on its arguments but that it can turn negative for the stable stiffening case $\alpha\beta\!>\!0$. The calculation, from
Eqs.\ref{ueqnFT},\ref{QeqnFT}, is straightforward. We find
\begin{equation}
\label{respimag}
\chi{\prime\prime}(q_x,\omega)=\omega \left[{1-\alpha\beta\left(\zeta/\!K\right)
\!{}^2q_x^2\Sigma(q_{\omega}) \over {\cal D}_{q_x\omega}}\right],
\end{equation}
where ${\cal D}_{q_x\omega}$ and $\Sigma(q_{\omega})=\!\!\int_{\mathbf{q_{\perp}}}\!\!\!\left[\!{\left(q_{\!\perp}^2\!+\!a/K\!+\!q_x^2\right)}{}^{\!2}\!\!+\!(\zeta\omega/K)^2\right]^{\!-\!1}$are strictly positive and even in $\omega$, and the FD ratio
\begin{equation}
R_{q_x\omega}={N_1{\gamma} \over k_BT}
\left[1+\frac{\alpha\left(\alpha{N_2}/N_1+\beta\right)
{\left(\zeta/\!K\right)}{}^2q_x^2\Sigma(q_{\omega})}
{1-\alpha\beta{\left(\zeta/\!K\right)}{}^2q_x^2\Sigma(q_{\omega})}\right].
\label{NoneqFDT}
\end{equation}
At thermal equilibrium ($W\!=\!0$)
$N_1=k_BT/\gamma$, $N_2=k_BT/\zeta$, $\alpha=A/\gamma$, and $\beta=-A/\zeta$, so that
$\alpha{N_2}/N_1=-\beta$. Hence, the second term in the bracket in Eq.\ref{NoneqFDT} vanishes
and the fluctuation-dissipation ratio becomes unity as expected. With activity the
ratio becomes a strong function of frequency and wavenumber, through the quantity $\Sigma(q_{\omega})$.
In the stiffening case $\alpha\beta=(A/\gamma-Wc_0/\Gamma)(-A/\zeta)\sim{WA}\!>\!0$, if the strength
$W$ of activity is large enough, as $\omega$ crosses a threshold which depends on $q_x$ and $W$, we see from Eqs.\ref{respimag},\ref{NoneqFDT}, $\chi{\prime\prime}$ can pass through zero, and hence $R_{q_x\omega}$ can diverge. Past this threshold, both turn negative. Thus one obtains a giant or even a negative FD ratio as a result of suppressed dissipation rather than enhanced noise. The FD ratio Eq.\ref{NoneqFDT} is sometimes referred to \cite{hudspeth} as the ratio of an ``effective temperature'' $T_{eff}$ to the thermodynamic temperature $T$, and a negative $T_{eff}$ in this sense has been observed in experiments on hair cells \cite{hudspeth}. Our theory Eqs.\ref{ueqn}-\ref{veqn}, rooted in the active-hydrodynamic approach of \cite{RamaswamyActiveRheo1,curiegrp1,RamaswamyActiveRheo2,LM031,LM032,SriramMadanNJP07}, provides a fundamental basis for the model presented in \cite{hudspeth} and suggests that the phenomenon of negative dissipation should be widely observed in active systems. To apply our treatment directly to axons \cite{pramod2007} and hair cells \cite{hudspeth}, we consider a filament confined to a tube of radius $L$ in the $\perp$ directions and unbounded along $x$, and assume the correlation length $\xi \equiv \sqrt{K/a}$ of the active medium is of order $L$. The integrals over $\mathbf{q}_{\perp}$ in Eq.\ref{QeqnFT} are replaced by sums dominated by a single mode with wavenumber of order $1/L$. This yields an effective equation of motion
\begin{equation}
\label{ueqnsinglemode}
\left(-i\omega+{\sigma \over \gamma}q_x^2+{\kappa \over \gamma}q_x^4\right)
\mathbf{u}_{\perp}
= -q_x^2{\alpha \beta \zeta \over K}{1 \over 1 - {i \zeta L^2 \over K} \omega + q_x^2L^2} \mathbf{u}_{\perp}
\end{equation}
plus noise. For strong active stiffening, $\alpha \beta$ large and positive, this leads to oscillatory modes with $\omega \simeq \pm \sqrt{\alpha \beta}q_x/L$. These correspond to the hair cell oscillations of \cite{hudspeth}, generalized to allow a continuum of modes labelled by $q_x$.
\section{Conclusion}
We have presented a general theory for a single, long semiflexible filament interacting with an active medium. The medium consists of orientable elements, endowed with built-in uniaxial stresses whose axes are correlated with those of their neighbours. The model thus describes, in particular, a single microtubule interacting with an actively contractile actomyosin environment, but has broader applicability. Whereas the various components of the cytoskeleton are usually investigated separately, our paper lays the physical groundwork for a treatment of the interaction of these different elements, in particular the contractile actin, and microtubules as active transport highways. This is essential to an understanding of cell motility and active ribosomal transport.
The key ingredients of the model are anchoring -- the preferred orientation imposed on the filaments of the medium when confronted with the surface of the long filament -- and the contractile or tensile activity of the medium. We find that the interplay between anchoring and activity radically affects the filament's dynamics, leading to a range of possible behaviours including active stiffening, negative dissipation, oscillations, and buckling. Our theory also applies to the dynamics of axons and auditory hair cells, and provides an important link between existing phenomenological models for these systems and the general framework of active matter. Our experiments on the shapes of microtubules in the growth cone of a neuron show both stiffening and buckling, in broad accord with our theory. We look forward to more quantitative tests, on filaments suspended in cell extracts as well as suspensions of swimming organisms, to see if the scenarii we propose are observed.
\begin{acknowledgments}
NK was supported by the IISc Centenary Postdoctoral Fellowship. SR
acknowledges support from the Department of Science and Technology, India
through the Centre for Condensed Matter Theory and DST Math-Bio Centre grant
SR/S4/MS:419/07. MR thanks the HFSP, and SR and MR thank CEFIPRA project
3504-2. We thank P. A. Pullarkat, J.-F. Joanny, J. Prost, N. Uchida, H. Ogawa
and T. Kawakatsu for stimulating discussions.
\end{acknowledgments}
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Q: Delete from table is very slow in oracle standard edition Delete on table in oracle standard edition(no partition) gets slow with time.
Important Info: I am working on oracle standard edition so partitioning option available.
Detail:
I have one table with no constraint on it (no PK or anyother key or trigger or index or anything).
More than a million record gets inserted in this table in every 15 min using sql loader.
we need to process this 15 min record in every 15 min and at end of process delete any record older than 30 minute so that at any point of time there is more than 30-40 minute of data in the table.
Problem:
As time passes due to so frequent insertion and deletion response from the table gets slow.
Data extraction and delete from table takes more time with every passing run.
After a while even a simple select query takes too long.
We cant truncate table as data loader runs continously and we may loose data if truncate and we dont have create table access to drop and create table.
we have to process data in every 15 minute and made it available to downstream for further processing. it just keep getting slow.
Kindly help me with the aforementioned situation.
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"redpajama_set_name": "RedPajamaStackExchange"
}
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|
package org.apache.giraph.block_app.framework.piece.interfaces;
import org.apache.giraph.graph.Vertex;
import org.apache.giraph.writable.kryo.markers.NonKryoWritable;
import org.apache.hadoop.io.Writable;
import org.apache.hadoop.io.WritableComparable;
/**
* Interface representing actions that happen on worker, for each vertex,
* during receive phase:
* <ul>
* <li> to receive messages from vertices </li>
* <li> to receive data from master through aggregators </li>
* </ul>
*
* Marked to not allow seriazliation, as it should be created on the worker,
* so should never be serialiized, disallow only for catching problems early.
*
* @param <I> Vertex id type
* @param <V> Vertex value type
* @param <E> Edge value type
* @param <M> Message type
*/
@SuppressWarnings("rawtypes")
public interface VertexReceiver<I extends WritableComparable,
V extends Writable, E extends Writable, M extends Writable>
extends NonKryoWritable {
/**
* Must be defined by user to do computation on a single Vertex.
*
* @param vertex Vertex
* @param messages Messages that were sent to this vertex in the previous
* superstep. Each message is only guaranteed to have
* a life expectancy as long as next() is not called.
*/
void vertexReceive(Vertex<I, V, E> vertex, Iterable<M> messages);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 251
|
{"url":"http:\/\/www.roma1.infn.it\/~dagos\/colombo\/node34.html","text":"# Some remarks on the use of logarithmic updating of the odds\n\nThe idea of using (natural) logarithms of the odds is quite old, going back, as far as I know, to Charles Sanders Peirce\u00a0[6]. He related them to what he called feeling of belief (or intensity of belief), that, according to him, should be as the logarithm of the chance, this latter being the expression of the state of facts which produces the belief''\u00a0[6], where by chance' he meant exactly probability ratios, i.e. the odds.\n\nPeirce proposed his ''thermometer for the proper intensity of belief''\u00a0[6] for several reasons.\n\n\u2022 First because of considerations that when the odds go to zero or to infinity, then the intensity of belief on either hypothesis goes to infinity;46 when an even chance is reached [the feeling of believing] should completely vanish and not incline either toward or away from the proposition.''\u00a0[6] The logarithmic function is the simplest one to achieve the desired feature. (Another interesting feature of the odds is described in footnote 16.)\n\u2022 Then because (expressing the question in our terms), if we started from a state of indifference (initial odds equal to 1), each piece of evidence should produce odds equal to its Bayes factor [our ]. The combined odds will be the product of the individual odds [Eq.\u00a019]. But, mixing now Pierce's and our terminology, when we combine several arguments (pieces of evidence), they ought to produce a belief equal to the sum of the intensities of belief which either would produce separately''.\u00a0[6] Then because we have seen that the chances of independent concurrent arguments are to be multiplied together to get the chance of their combination, and therefore the quantities which best express the intensities of belief should be such that they are to be added when the chances are multiplied...Now, the logarithm of the chance is the only quantity which fulfills this condition''.\u00a0[6]\n\u2022 Finally, Peirce justifies his choice by the fact that human perceptions go often as the logarithm of the stimulus (think at subjective feeling of sound and light - even utility', meant as the value of money' is supposed to grow logarithmically with the amount of money): There is a general law of sensibility, called Fechner's psychophysical law. It is that the intensity of any sensation is proportional to the logarithm of the external force which produces it.''[6] (Table 1 provides a comparisons between the different quantities involved, to show that the human sensitivity on probabilistic judgement is indeed logarithmic, with a resolution about the first decimal digit of the base 10 logarithms.)\n\nAs far as the logarithms in question, I have done a short research on their use, which, actually, lead me to discover Peirce's Probability of Intuition\u00a0[6] and Good's Probability and the weighing of Evidence\u00a0[7].47As far as I understand, without pretension of completeness or historical exactness:\n\n\u2022 Peirce' chances' are introduced as if they were our odds, but are used if they were Bayes factors (the chances of independent concurrent arguments are to be multiplied together to get the chance of their combination''\u00a0[6]). Then he takes the natural logarithm of these chances', to which he also associates an idea of weight of evidence (our belief ought to be proportional to the weight of evidence, in the sense, that two arguments which are entirely independent, neither weakening nor strengthening each other, ought, when they concur, to produce a belief equal to the sum of the intensities of belief which either would produce separately''\u00a0[6]).\n\u2022 According to Ref.\u00a0[8] the modern use of the logarithms of the odds seem to go back to I.J. Good, who used to call log-odds the natural logarithm of the odds.48\n\u2022 However, reading later Ref.\u00a0[8] it is clear that Good, following a suggestion of A.M. Turing, proposes a decibel-like (db) notation49, giving proper names both to the logarithm of the odds and to the logarithm of the Bayes factor:\n\u2022 db ... may be also described as the weight of evidence or amount of information for given ''\u00a0[7];\n\u2022 db may be called the plausibility corresponding to odds ''\u00a0[7].\nIt follows then that\n Plausibility gained weight of evidence'' (36)\n\n\u2022 Decibel-like logarithms of the odds are used since at least forty years with under the name evidence.\u00a0[23].\nPersonally, I think that the decibel-like definition is not very essential (decibels themselves tend already to confuse normal people, also because for some applications the factor 10 is replaced by a factor 20). Instead, as far as names are concerned:\n\u2022 plausibility' is difficult to defend, because it is too similar to probability in everyday use, and, as far as I understand, has decayed;\n\u2022 weight of evidence' seems to be a good choice, for the reasons already well clear to Peirce.\n\u2022 evidence' in the sense of Ref.\u00a0[23] seems, instead, quite bad for a couple of reasons:\n\u2022 First, because evidence' has already too many meanings, including, in the Bayesian literature, the denominator of the r.h.s. of Eq.\u00a0(3).\n\u2022 Second, because this name is given to the logs of the odds (including the initial ones), but not to those of the Bayes factors to which no name is given. Therefore, the name evidence', as used in Ref.\u00a0[23] in this context, is not related to the evidence.\nI have taken the liberty to use the expression judgment leaning' first because it evokes the famous balance of Justice, then because all other expressions I thought about have already a specific meaning, and some of them even several meanings.50It is clear, especially comparing Eq.\u00a0(36) with Eq.\u00a0(24), that, besides the factor ten multiplying the base ten logarithms and the notation, I am quite in tune with Good. I have also to admit I like Peirce' intensity of belief' to name the JL's, although it is too similar to degree of belief', already widely used to mean something else.\n\nSo, in summary, these are the symbols and names used here:\n\nJL\nis the judgement leaning in favor of hypothesis and against , with the conditions in parenthesis. If we only consider an hypothesis () and its opposite , that could be possibly related to the occurrence of the event or its opposite , also the notation JL, or JL, will be used (as in table 2 of Appendix I).\n(Sometimes I have also tempted to call a JL intensity of belief' if it is clear from the contest that the expression does not refer to a probability.)\nJL\n, with the same meaning of the subscript and of the argument, is the variation of judgement leaning produced by a piece of evidence and it is called here weight of evidence, although it differs by a factor from the analogous names used by Peirce and Good51.\n\nGiulio D'Agostini 2010-09-30","date":"2017-11-20 00:06:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8157283067703247, \"perplexity\": 879.7284271851242}, \"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-2017-47\/segments\/1510934805881.65\/warc\/CC-MAIN-20171119234824-20171120014824-00044.warc.gz\"}"}
| null | null |
\section{Introduction}
\bigskip
Short selling is the practice of selling financial instruments that have been
borrowed, typically from a broker-dealer or an institutional investor, with
the intent to buy the same class of financial instruments in a future period
and return them back at the maturity of the loan.
By short selling, investors open a so-called ``short position'', that is technically equivalent to holding a
negative amount of shares of the traded asset, with the expectation that the
asset will recede in value in the next period.
At the closing time
specified in the short selling contract, the debt is compounded
with interest which occurred during the period of the financial operation, for
this reason short sellers prefer to close the short position and reopen
a new one with the same features, rather than extending the position over the closing time (see, e.g., \citep{Hull}).
A short position is the counterpart of the (more conventional) ``long position'', i.e. buying a security
such as a stock, commodity, or currency, with the expectation that the asset
will rise in value.
Short selling is considered the father of the modern derivatives and, as
such, it has a double function: it can be used as an insurance device, by
hedging the risk of long positions in related stocks thus allowing risky
financial operations, or for speculative purposes, to profit from an
expected downward price movement.
Moreover, financial speculators can sell short stocks in an effort to drive
down the related price by creating an imbalance of sell-side interest, the
so called ``bear raid'' action. This
feedback may lead to the market collapse, and has indeed been observed
during the financial crises of 1937 and 2007, see, e.g.,
\citep{MisraLagiBarYam2012}.
Many national authorities have developed different kinds of short selling
restrictions to avoid the negative effect of this financial practic
\footnote
It is worth mentioning that short sale restrictions are nearly as old as
organized exchanges. The first short selling regulation was enacted in 1610
in the Amsterdam stock exchange. For a review of the history of short sale
restrictions, see Short History of the Bear, Edward Chancellor, October 29,
2001, copyright David W. Tice and Co.}.
Most of the regulations are based on ``price tests'', i.e., short selling is
allowed or restricted depending on some tests based on recent price movements. The
best known and most widely applied of such regulations is the so-called
``{\it uptick rule}'', or rule 10a-1, imposed in 1938
by the U.S. Securities and Exchange Commissio
\footnote
The rule was originally introduced under the Securities Exchange Act of 1934.}
(hereafter SEC) to protect investors and was in force until 2007. This rule
regulated short selling into all U.S. stock markets and in the Toronto Stock
Exchange.
Other financial markets, like the London Stock Exchange and the Tokyo Stock Exchange,
have different or no short selling restrictions (for a summary of short sale
regulations in approximately 50 different countries see \citep{BrisGoetzmannZhou2007}).
The uptick rule originally stated that short sales are allowed only on an
uptick, i.e., at a price higher than the last reported transaction price.
The rule was later relaxed to allow short sales to take place on a
zero-plus-tick as well, i.e., at a price that is equal to the last sale price but
only if the most recent price movement has been positive. Conversely, short
sales are not permitted on minus- or zero-minus-ticks, subject to narrow
exception
\footnote
In the Canadian stock markets, the tick test was introduced under rule 3.1 of UMIR
(Universal Market Integrity Rules). It prevents short sales at a price that is less
than the last sale price of the security.}.
In adopting the uptick rule, the SEC sought to achieve three objective
\footnote
Quoted from the Securities Exchange Act Release No. 13091 (December 21,
1976), 41 FR 56530 (1976 Release).}:
\begin{itemize}
\item[(i)] \textit{allowing relatively unrestricted short selling in an
advancing market;}
\item[(ii)] \textit{preventing short selling at successively lower prices,
thus eliminating short selling as a tool for driving the market down; and}
\item[(iii)] \textit{preventing short sellers from accelerating a declining
market by exhausting all remaining bids at one price level, causing
successively lower prices to be established by long sellers.}
\end{itemize}
The last two objectives have been partially confirmed by the empirical analysis (see, e.g., \citep{AlexanderPeterson1999} and reference therein).
Instead, the regulation does not seem to be effective in producing the first desired effect.
The observed number of executed short sales is indeed lower under uptick rule than in the unconstrained case, during phases with an upward market trend, see again \citep{AlexanderPeterson1999}.
This is due to the asynchrony between placement and execution of a short-sell order, since the rising of the price in between these two operations can make the trade not feasible under the uptick rule.
Moreover, empirical evidence provides uniform support of
the idea that short selling restrictions often cause share prices to rise.
From a theoretical point of view, there is no clear argument for explaining
this mispricing effect of the uptick rule. According to \citep{Miller1977}
this is due to a reduction in stock supply owing to the short sale restriction.
More generally, theoretical models with heterogeneous agents and differences in trading strategies
support the idea that share values become overvalued under short selling
restrictions due to the fact that ``pessimistic'' and ``bear'' traders
(expecting negative price movements) are ruled out of the market
(see, e.g., \citep{HarrisonKreps1978}). In contrast, theoretical models based on the
assumption that all agents have rational expectations suggest that short
selling restrictions do not change on the average the stock prices
(see, e.g., \citep{DiamondVerecchia1987}).
However, given the complexity of the phenomena, and the impossibility of
isolating the effects of a regulation from other concomitant changes in the
economic scenario, the effectiveness of the uptick rule in meeting the three
above objectives, and its possible side effects on shares' prices,
are still far from being completely clarified.
Guided by the aim to provide further insight on the argument, this paper
studies the effects on share prices in an artificial market of a short
selling restriction based on a tick test similar to the one imposed by the
uptick rule in real financial markets. Using an artificial asset pricing
model makes it easier in assessing the effects of the uptick rule in
isolation from other exogenous shocks, though artificial modeling
necessarily trades realism for mathematical tractability.
We consider an asset pricing model of adaptive rational equilibrium dynamics (A.R.E.D.), where heterogeneous beliefs on the future prices of a risky
asset, together with traders' adaptability based on past performances, have shown
to endogenously sustain price fluctuations.
Asset pricing models of A.R.E.D. (hereafter referred to simply as ARED asset pricing models) are discrete-time dynamical systems
based on the empirical evidence that investors with different trading strategies coexist in the financial market
(see, e.g., \citep{TaylorAllen1992}). These simple models provide a theoretical justification for many ``stylized facts'' observed in
the real financial time series, such as, financial bubbles and volatility clustering (see \citep{Gaunersdorfer2001}, and \citep{GaunersdorferHommesWagener2008}).
Stochastic models based on the same assumptions are even used to study
exchange rate volatility and the implication of some specific financial policies
(see, e.g., \citep{Westerhoff2001}).
We extend, in particular, the deterministic model introduced in \citep{BrockHommes1998},
where, in the simplest case, agents choose between two predictors of future prices of a risky asset, i.e. a fundamental predictor and a non-fundamental predictor. Agents that adopt the fundamental predictor are called {\it fundametalists}, while agents that adopt the non-fundamental predictor are called {\it noise traders} or non-fundamental traders.
Fundamentalists believe that the price of a financial asset is determined by its fundamental value (as given by the present discounted value
of the stream of future dividends, see \citep{Hommes2001}) and any deviation
from this value is only temporary.
Non-fundamental traders, sometimes called {\it chartists} or {\it technical analysts}, believe that the future price of a risky asset is not completely determined by fundamentals and it can be predicted by simple technical trading rules (see, e.g., \citep{Elder1993}, \citep{Murphy1999}, and \citep{Neely1997}).
In the model, agents revise their "beliefs", prediction to be adopted, according to an evolutionary mechanism based on the past realized profits.
As a result, the fundamental value is a fixed point of the price dynamics,
as, once there, both fundamentalists and non-fundamental traders predict
the fundamental price. As long as the sensitivity of traders in switching
to the best performing predictor is relatively low, the fundamental
equilibrium is stable, but the fundamental stability is typically lost at higher intensities of
the traders' choice across the predictors, making room for financial bubbles.
It is worth to remember that \citep{BrockHommes1998} investigated the peculiar case of zero supply of
outside shares. Under this assumption each bought share is sold short. We therefore
consider a positive supply of outside share, that is essential to ensure financial
transactions when short selling is forbidde
\footnote
We consider a positive supply of outside shares for the asset pricing model under Walrasian market clearing at each period.
A similar model under the market maker scenario has been considered by \citep{HommesHuangWang2005}.}.
Moreover, we pair the fundamental predictor with first a technical linear predictor and then with a technical nonlinear
predictor and compare the results obtained with and without the uptick rule.
As linear predictor, we consider the chartist predictor introduced in \citep{BrockHommes1998}. This facilitates the comparison of our results with those in \citep{BrockHommes1998} and related papers.
As nonlinear predictor, we introduce a new predictor, "Smoothed Price Rate Of Change" or S-ROC predictor, that extrapolates future prices by applying the rate of change averaged on past prices with a confidence mechanism smoothing out extreme unrealistic rates (for an overview of this class of predictors, see \citep{Elder1993}).
For what concerns the implementation of the regulation, we implement the uptick rule as it was in its original formulation, i.e., short
selling is allowed only on an uptick. Note, however, that in an artificial
asset pricing model a zero-tick is possible only at equilibrium, so that
allowing or forbidding short sales on zero-plus-ticks makes basically no
change in the observed price dynamics. In fact, with a positive supply of
shares, traders take long positions at the fundamental equilibrium,
so only the non-fundamental equilibria at which one type of trader is prohibited to go
short are affected by the rule behavior on zero-plus-ticks (moreover, such equilibria
are irrelevant to study the global price dynamics, as will be explained in
Section \ref{ssec:ARED_uptick}).
From the mathematical point of view, the uptick rule makes the asset pricing
model a piecewise-smooth dynamical system\footnote{To be precise, the model is a piecewise-continuous dynamical system. However, the class of piecewise-smooth dynamical systems contains the class of piecewise-continuous dynamical systems.}, namely a system in which
different mathematical rules can be applied to determine the next price, and
the rule to be applied depends on the current state of the system, that is,
on the fact that trader types are interested in going short and whether short selling is allowed or not. Non-smooth dynamical systems are certainly more
problematic to analyze, both analytically and numerically
(though non-smooth dynamics is a very active topic in current research, see \citep{Bernardo08}, and \citep{Colombo11b}, and
references therein)
so we will limit the analytical treatment to stationary solutions.
Piecewise-smooth dynamical systems have already been used as models in
finance. \citep{TramontanaGardiniWesterhoff2011} proposed a one-dimensional
piecewise-linear asset pricing model, where traders adopt different buying and selling
strategies in response to different market movements.
Other examples can be found in \citep{TramontanaGardiniWesterhoff2010}, \citep{TramontanaWesterhoffGardini2010}, and
\citep{TramontanaWesterhoff2012}.
Two ARED piecewise-smooth systems modeling short selling restrictions have been also proposed. Modifying the model in \citep{BrockHommes1998},
\citep{AnufrievTuinstra2009} restricted short selling by allowing limited short positions at each
trading period, whereas \citep{DercoleCecchetto2010} investigated the
complete ban on short selling. Thus both contributions implement short
selling restrictions that are not based on price tests.
The results of our theoretical analyses are in line with the empirical
evidence. The sale price that is established in our model when one trader
type is prohibited from going short is indeed systematically higher than the
unconstrained price. Thus, constrained downward movements below the
fundamental value are less pronounced, whereas constrained upward movements
above the fundamental value can be larger.
We provide a more complete explanation for this effect, suggesting that it is due to the combination of two mechanisms:
on one side, the short selling restriction reduces the possibility for pessimistic or bear traders to bet on downward movements
below the fundamental value,
avoiding excessive underpricing, but at the same time, when prices are above the fundamental value,
the restriction reduces the possibility for fundamentalists to drive down the prices back to the fundamental value by opening short positions.
This is in agreement with the last two goals established by the SEC (see above).
The first stated objective of the uptick is always realized in our model, since the market clearing is assumed to be synchronous among all traders.
When non-fundamental traders adopt the S-ROC predictor, we observe that the overpricing due to the uptick rule disappears due to the smoothness of the predictor that makes non-fundamental traders not confident with extreme price deviations from the fundamental value. Indeed, the expectations of large price deviations produced by the uptick rule force the non-fundamental trader to believe in the fundamental value with the effect of reducing, instead of increasing, the price deviations.
The stabilizing effect however vanishes when traders become highly sensitive in switching to the strategy with best recent performance.
The paper is organized as follows. Section~\ref{ssec:ARED_no_uptick} briefly
reviews the unconstrained asset pricing model, summarizing from
\citep{BrockHommes1998} and setting the notation and most of the modeling
equations that will be used in next Sections. Section~\ref{ssec:fund} is
also preliminary and recaps the concept of fundamental equilibrium,
including its stability analysis and some new results. Section~\re
{ssec:ARED_uptick} formulates the piecewise-smooth model constrained by the
uptick rule, and discusses the existence and stability of fundamental and
non-fundamental equilibria. So far, no explicit price predictors is
introduced, whereas Section \ref{ssec:pre} presents the price predictors
for which the unconstrained and constrained models will be studied and
compared in Sections~\ref{sec:ana} and~\ref{sec:num}. Section~\ref{sec:ana}
presents the analytical results concerning the existence and stability of
fixed points. Some of the results concerning the unconstrained model are new
and interesting per se.
Section~\ref{sec:num} presents a series of numerical tests, confirming the
analytical results and investigating non-stationary (periodic, quasi-periodic, and chaotic)
regimes. In Section~\ref{sec:ed} we discuss in detail our economic findings.
Section~\ref{sec:cd} concludes and lists a series of related interesting topics for further research.
All the analytical results presented in Sections~\ref{sec:ARED} and~\ref{sec:ana}
are proved in Appendix \ref{Ch4Appendix}.
\section{The ARED asset pricing model with and without the uptick rule}\label{sec:om}
\label{sec:ARED}
We consider the asset pricing model with heterogeneous beliefs and adaptive traders introduced by \citep{BrockHommes1998}.
While in the original model a zero supply of outside shares was considered, making short selling essential to ensure the exchanges,
we consider the case of positive supply, so that short selling will no longer be necessary and a constraint on it can be imposed.
In this generalized version of the original model, we introduce a negative demand constraint according to the uptick rule,
in order to study the effects of this regulation on price fluctuations.
\subsection{The unconstrained ARED asset pricing model}
\label{ssec:ARED_no_uptick}
Consider a financial market where traders invest either in a
single risky asset, supplied in $S$ shares \footnote{$S$ is in fact the supply of traded assets in each period. Obviously when short selling is allowed assets are borrowed outside the pool of this $S$ shares making the total supply higher than $S$.}
of (ex-dividend) price $p_{t}$ at period $t$, or in a risk free asset perfectly elastically supplied at gross return $R$ (where $R=1+r$, with $r\in(0,1)$). The risky asset pays random dividend $\tilde{y}_{t}$ in period $t$, where the divided process $\tilde{y}_{t}$ is IID (Identically Independently Distributed) with $E_{t}\left[\tilde{y}_{t+1}\right]=\bar{y}$ constant. Thus, denoting by
$W_{h,t}$ the economic wealth of a generic trader of type $h$ at the beginning of period
$t$, and by $z_{h,t}$ the number of shares held by the trader in period $t$, we
have the following wealth equation (or individual intertemporal budget constraint):
\begin{linenomath}
\begin{equation}
\label{eq:w}
\tilde{W}_{h,t+1}=R(W_{h,t}-p_t z_{h,t})+\tilde{p}_{t+1}z_{h,t}+\tilde{y}_{t+1}z_{h,t}=
RW_{h,t}+(\tilde{p}_{t+1}+\tilde{y}_{t+1}-R\hspace{0.2mm}p_t)z_{h,t},
\end{equation}
\end{linenomath}
where tilde denotes random variables, $W_{h,t}-p_t z_{h,t}$ is the amount of money invested in the risk free asset in period $t$
and $\tilde{R}_{t+1}=\tilde{p}_{t+1}+\tilde{y}_{t+1}-R\hspace{0.2mm}p_t$ is the excess return per share realized at the end of the period.
Let $E_{h,t},V_{h,t}$ denote the "beliefs" of investor of type $h$ about the conditional expectation and conditional variance of wealth. They are assumed to be functions of past prices and dividends. We assume that each investor type is a myopic mean variance maximizer, so for type $h$ the demand for shares $z_{h,t}$ solves
\begin{equation*}
\max_{z_{h,t}}\left\{E_{h,t}\left(\tilde{W}_{t+1}\right)-\frac{a}{2}V_{h,t}\left(\tilde{W}_{t+1}\right)\right\}
\end{equation*}
i.e.,
\begin{linenomath}
$$
\ds z_{h,t}\left(p_{t}\right)=\Frac{E_{h,t}[\tilde{R}_{t+1}]}{a V_{h,t}[\tilde{R}_{t+1}]}=
\Frac{E_{h,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]-R\hspace{0.2mm}p_t}{a V_{h,t}[\tilde{R}_{t+1}]},
$$
\end{linenomath}
where $a$ is the risk aversion coefficient and $p_t$ is to be determined by the market clearing between all demands and the supply $S$ of shares.
For simplicity (as done in \citep{BrockHommes1998}, see \citep{Gaunersdorfer00}, for an extension),
we assume that traders have common and constant beliefs about the variance, i.e. $V_{h,t}\left[\tilde{R}_{t+1}\right]=\sigma^{2}$, $\forall h$, and common and correct beliefs about the dividend, i.e. $E_{h,t}\left[\tilde{y}_{t+1}\right]=E_{t}\left[\tilde{y}_{t+1}\right]=\bar{y}$, $\forall h$.
Moreover, the number $N$ of traders and $S$ of supplied shares in each period (not considering the extra supply of shares due to short sales) are kept constant. Let $H$ be the number of available "beliefs" or price predictors $E_{h,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]$, $h=1,\ldots,H$, each obtained at a cost $C_h$, and denote by $n_{h,t}$ the fraction of traders adopting predictor $h$ in
period $t$, the market clearing imposes
\begin{linenomath}
\begin{equation}
\label{eq:mc}
\ds N\sum_{h=1}^Hn_{h,t}z_{h,t}(p_t)=S,\quad
z_{h,t}(p_t)=\frac{\ds E_{h,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]-R\hspace{0.2mm}p_t}{\ds a\hspace{0.2mm}\sigma^2},
\end{equation}
\end{linenomath}
which is solved for $p_t$, thus obtaining
\begin{linenomath}
\begin{equation}
\label{eq:pt}
p_{t}=\Frac{1}{R}\left(\sum_{h=1}^{H}n_{h,t}E_{h,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]-a\hspace{0.2mm}\sigma^{2}\Frac{S}{N}\right),
\end{equation}
\end{linenomath}
Let us substitute the expression of $p_{t}$ in $z_{h,t}\left(p_{t}\right)$, $\forall h \in H$, to obtain the actual demands
\begin{linenomath}
\begin{equation}
\label{eq:ad}
z_{h,t}=\Frac{1}{a\hspace{0.2mm}\sigma^2}\left(E_{h,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]-\sum_{k=1}^{H}n_{k,t}E_{k,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]\right)+\Frac{S}{N}
\end{equation}
\end{linenomath}
(no longer functions of the price $p_t$), and let us use $p_{t}$ to calculate the net profits $R_t z_{h,t-1}-C_h$, $h=1,\ldots,H$, realized in period $t$.
Eq.~\eqref{eq:ad} gives the number of shares held by a trader of type $h$ in period $t$.
If negative, the trader is in a short position. If positive, the trader is in a long position.
At this point, the fractions $n_{h,t+1}$, $h=1,...,H$, for the next period are determined as functions of the positions of the traders and of the last available net profits.
In particular, the following discrete choice model is used:
\begin{linenomath}
\begin{equation}
\label{eq:cm}
n_{h,t+1}=\frac{\exp\left(\beta(R_t z_{h,t-1}-C_h)\right)}
{\sum_{k=1}^H\exp\left(\beta(R_t z_{k,t-1}-C_k)\right)}, h=1,...,H-1,
\end{equation}
\end{linenomath}
where $\beta$ measures the intensity of traders' choice across predictors (traders' adaptability).
The above procedure can then be iterated to compute the next price $p_{t+1}$.
If all agents have common beliefs on the future prices, i.e. $E_{h,t}=E_{t}$ $\forall h$, the pricing equation (\ref{eq:pt}) reduces to
\begin{equation*}
Rp_{t}=E_{t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]-a\hspace{0.2mm}\sigma^{2}\Frac{S}{N}.
\end{equation*}
This equation admits a unique solution $\tilde{p}^{*}_{t}\equiv \bar{p}$, where
\begin{linenomath}
\begin{equation}
\label{eq:pbar}
\bar{p}=\Frac{\bar{y}-a\sigma^2 S/N}{R-1},
\end{equation}
\end{linenomath}
that satisfies the "no bubbles" condition $\lim_{t\rightarrow\infty}\left(E\tilde{p}^{*}_{t}/R^{t}\right)=0$. This price, given as the discounted sum of expected future dividends, would prevail in a perfectly rational world and is called the {\it fundamental price} (see, e.g., \citep{Hommes2001,HommesHuangWang2005}).
Of course, we assume $\bar{p}>0$, i.e., sufficiently high dividend $\bar{y}$ or limited supply of ouside shares per investor $S/N$.
Given the assumptions about the dividend process and the fundamental price and focusing only on the deterministic skeleton of the model, i.e. $\tilde{y}_{t}=\bar{y}$ $\forall t$, we have that $E_{h,t}[\tilde{p}_{t+1}+\tilde{y}_{t+1}]=E_h[p_{t+1}]+\bar{y}$, where the price predictors $E_h[p_{t+1}]$, $h=1,\ldots,H$,
are deterministic functions of $L$ known past prices $\{p_{t-1},p_{t-2},\dots,p_{t-L}\}$, $L\ge 1$.
It is useful to rewrite the model in terms of price deviations from a benchmark price $\bar{p}$. In the following, let $s=S/N$ and denote by $x_t$ the price deviation from the fundamental value, i.e., $x_t=p_t-\bar{p}$.
Defining the traders' beliefs on the next deviation $x_{t+1}$ as $f_h(\mathbf{x}_{t})=E_h[p_{t+1}]-\bar{p}$,
with $\mathbf{x}_{t}=(x_{t-1},x_{t-2},\dots,x_{t-L})$ being the vector of the last $L$ available deviations,
the demand functions to be used in the market clearing in Eq.~\eqref{eq:mc} become
\begin{linenomath}
\begin{equation}
\label{eq:zx}
z_{h,t}(x_t)=\Frac{f_{h}(\mathbf{x}_{t})-R\hspace{0.2mm}x_t}{a\hspace{0.2mm}\sigma^2}+s,
\end{equation}
\end{linenomath}
while the pricing equation \eqref{eq:pt} and the actual demands \eqref{eq:ad} can be written in deviations as
\begin{linenomath}
\begin{equation}
\label{eq:xt}
x_{t}=\Frac{1}{R}\sum_{h=1}^{H}n_{h,t}\hspace{0.2mm}f_{h}(\mathbf{x}_{t})\quad\text{and}\quad
z_{h,t}=\Frac{1}{a\hspace{0.2mm}\sigma^2}\left(f_{h}(\mathbf{x}_{t})-\sum_{k=1}^{H}n_{k,t}f_{k}(\mathbf{x}_{t})\right)+s,
\end{equation}
\end{linenomath}
and the excess of return in \eqref{eq:cm} can be expressed in deviations as
\begin{linenomath}
\begin{equation}
\label{eq:Rx}
R_t=x_t-R\hspace{0.2mm}x_{t-1}+\delta_{t}+a\hspace{0.2mm}\sigma^2s.
\end{equation}
\end{linenomath}
where $\delta_{t}=y_{t}-\bar{y}$ is a shock due to the dividend realization. As mentioned above, we focus on the deterministic skeleton of the model, i.e., we fix $\delta_{t}=0$ $\forall t$.
Substituting Eqs.~\eqref{eq:zx} and~\eqref{eq:Rx} into~\eqref{eq:cm} and coupling the pricing equation in \eqref{eq:xt} with \eqref{eq:cm}, the ARED model can be rewritten as
\begin{linenomath}
\begin{subequations}
\label{ARED_model_no_uptick}
\footnotesize
\begin{eqnarray}
\label{ARED_model_no_uptick_x}
x_{t} & = & \Frac{1}{R}\sum_{h=1}^{H}n_{h,t}\hspace{0.2mm}f_{h}(\mathbf{x}_{t}),\\
\label{ARED_model_no_uptick_n}
n_{h,t+1} & = & \Frac{\exp\left(\beta\left((x_{t}-R\hspace{0.2mm}x_{t-1}+a\hspace{0.2mm}\sigma^2s)\left(\Frac{f_{h}(\mathbf{x}_{t-1})-R\hspace{0.2mm}x_{t-1}}{a\hspace{0.2mm}\sigma^2}+s\right)-C_h\right)\right)}
{\sum_{k=1}^H\exp\left(\beta\left((x_{t}-R\hspace{0.2mm}x_{t-1}+a\hspace{0.2mm}\sigma^2s)\left(\Frac{f_{k}(\mathbf{x}_{t-1})-R\hspace{0.2mm}x_{t-1}}{a\hspace{0.2mm}\sigma^2}+s\right)-C_k\right)\right)},\;h=1,\ldots,H-1\nonumber\\
\end{eqnarray}
\end{subequations}
\end{linenomath}
(recall that $\sum_{h=1}^Hn_{h,t}=1$).
Given the current composition $n_{h,t}$ of the traders' population, the first equation computes the price deviation for period $t$, while the second updates the traders' fractions for the next period.
The past deviations $(x_{t-1},x_{t-2},\dots,x_{t-(L+1)})$ appearing in vectors $\mathbf{x}_{t}$ and $\mathbf{x}_{t-1}$, together with the fractions $n_{h,t}$, $h=1,\ldots,H-1$, constitute the state of the system\footnote
Note that, by writing Eq.~\eqref{ARED_model_no_uptick_n} for $n_{h,t}$ and substituting it into Eq.~\eqref{ARED_model_no_uptick_x}, one can write $x_t$ as a recursion on the last $L+2$ deviations.
This gives a more compact and homogeneous system's state ($L+2$ price deviations instead of $L+1$ deviations and $H-1$ traders' fractions),
however, the formulation \eqref{ARED_model_no_uptick} is physically more appropriate and easier to initialize.}.
The initial condition is composed of the opening price deviation $x_0$ and of the traders' fractions $n_{h,1}$, $h=1,...,H$, to be used in the first period.
In fact, assuming that each price predictor can be customized to the case when the number of past available prices is less then $L$,
then Eq.~\eqref{ARED_model_no_uptick_x} can be applied at $t=1$ (to determine the price deviation $x_1$ in period $1$), whereas Eq.s~\eqref{ARED_model_no_uptick_n} can only be applied at $t=2$, so that $n_{h,2}=n_{h,1}$ is used.
For $t>L$ the price predictors in Eqs.~\eqref{ARED_model_no_uptick} can be regularly applied.
Note that Eq.~\eqref{ARED_model_no_uptick_x} guarantees a positive price for any period $t$ (i.e., $x_t>-\bar{p}$),
provided all price predictions are such ($f_{h}(\mathbf{x}_{t})>-\bar{p}$ for all $h=1,\ldots,H$).
When there are only two types of traders, $H=2$, it is convenient to express the
fractions $n_{1,t}$ and $n_{2,t}$ as a function of $m_{t}=n_{1,t}-n_{2,t}\in(-1,1)$, i.e.,
\begin{linenomath}
\begin{equation}
\label{eq:m}
n_{1,t}=\Frac{1+m_{t}}{2}\quad\text{and}\quad n_{2,t}=\Frac{1-m_{t}}{2}.
\end{equation}
\end{linenomath}
In this specific case, model \eqref{ARED_model_no_uptick} can be rewritten as
\begin{linenomath}
\begin{subequations}
\label{ARED_model_no_uptick_2}
\begin{eqnarray}
\label{ARED_model_no_uptick_2x}
x_{t} & = & \Frac{1}{2R}\left((1+m_{t})f_1(\mathbf{x}_{t})+(1-m_{t})f_2(\mathbf{x}_{t})\right),\\
\label{ARED_model_no_uptick_2m}
m_{t+1} & = & \tanh\left(\Frac{\beta}{2}\left((x_{t}-R\hspace{0.2mm}x_{t-1}+a\hspace{0.2mm}\sigma^{2}s)
\Frac{f_1(\mathbf{x}_{t-1})-f_2(\mathbf{x}_{t-1})}{a\hspace{0.2mm}\sigma^{2}}-(C_1-C_2)\right)\right),\nonumber\\
\end{eqnarray}
\end{subequations}
\end{linenomath}
where $(x_{t-1},x_{t-2},\dots,x_{t-(L+1)},m_t)$ is the system's state and $(x_0,m_1)$ identifies the initial condition.
\subsection{The fundamental equilibrium}
\label{ssec:fund}
The following lemma gives the condition under which the fundamental price is an equilibrium of model \eqref{ARED_model_no_uptick} (or model \eqref{ARED_model_no_uptick_2} when $H=2$):
\begin{lemma}
\label{lm:fund}
If all predictors satisfy $f_h(\mathbf{0})=0$, $h=1,\ldots,H$, with $\mathbf{0}$ the vector of $L$ zeros, then
$(\bar{x}^{(0)},\bar{n}^{(0)}_h)$ with
\begin{linenomath}
$$
\bar{x}^{(0)}=0\quad\text{and}\quad
\bar{n}^{(0)}_h=\frac{\exp\left(-\hspace{-0.2mm}\beta C_h)\right)}{\sum_{k=1}^H \exp\left(-\hspace{-0.2mm}\beta C_k)\right)}
$$
\end{linenomath}
[or $(\bar{x}^{(0)},\bar{m}^{(0)})$ with $\bar{m}^{(0)}=\tanh\left(-\beta/2\,(C_1-C_2)\right)$ if $H=2$]
is a fixed point of model \eqref{ARED_model_no_uptick} [\eqref{ARED_model_no_uptick_2}], at which all strategies equally demand $\bar{z}^{(0)}_h=s$.
We call this steady state \emph{fundamental equilibrium}.
$H$ of the associated eigenvalues are zero and the remaining $L$ ones are the roots of the characteristic equation
\begin{linenomath}
$$
\lambda^{L}-\gamma_{1}\lambda^{L-1}+\cdots-\gamma_{L}=0,\quad
\gamma_{i}=\Frac{1}{R}\sum_{h=1}^H\bar{n}_h\hspace{-1.0mm}\left.\Frac{\partial}{\partial x_{t-i}}f_h(\mathbf{x}_{t})\right|_{\mathbf{x}_{t}=\mathbf{0}},\quad i=1,\ldots,L.
$$
\end{linenomath}
\end{lemma}
Lemma~\ref{lm:fund} also reveals that the price dynamics is not reversible, at least locally to the fundamental equilibrium (due to the presence of zero eigenvalues), so that prices cannot be reconstructed backward in time.
We now state a simple condition that rules out the possibility of other equilibria:
\begin{lemma}
\label{lm:neq}
If $f_h(\bar{x}\mathbf{1})/\bar{x}<R$ [or if $f_h(\bar{x}\mathbf{1})/\bar{x}>R$] for all $h=1,\ldots,H$ and $\bar{x}\neq 0$, with $\mathbf{1}$ the vector of $L$ ones,
then the fundamental equilibrium is the only fixed point of model \eqref{ARED_model_no_uptick}.
\end{lemma}
As we will recall in Section~\ref{ssec:pre}, traders with price predictors such that $|f_h(\bar{x}\mathbf{1})/\bar{x}|<1$ believe that tomorrow's price will revert to its fundamental value ($x_t\to 0$), whereas, at an equilibrium, trend followers obviously extrapolate the equilibrium price, so their price predictors are such that $f_h(\bar{x}\mathbf{1})/\bar{x}=1$.
Lemma \ref{lm:neq} therefore shows that non-fundamental equilibria are possible only in the presence of at least one of the two mentioned types of traders and traders that believe that nonzero price deviations will amplify in the short run, even if they have been recently constant.
\subsection{The ARED asset pricing model constrained by the uptick rule}
\label{ssec:ARED_uptick}
When trading restrictions imposed by the uptick-rule are introduced, we must distinguish between two situations:
if prices are rising, e.g. we simply look at the last movement available $x_{t-1}-x_{t-2}$,
then short selling is allowed and the unconstrained model \eqref{ARED_model_no_uptick} still applies.
In contrast, in a downward (or stationary) movement ($x_{t-1}\le x_{t-2}$), traders' demands are forced to be non-negative, i.e.,
the demand functions to be used in the market clearing in Eq.~\eqref{eq:mc} are
\begin{linenomath}
\begin{equation}
\label{eq:zxc}
z_{h,t}(x_t)=\max\left\{0,
\Frac{f_{h}(\mathbf{x}_{t})-R\hspace{0.2mm}x_t}{a\hspace{0.2mm}\sigma^2}+s\right\}.
\end{equation}
\end{linenomath}
Note that the forward dynamics remain uniquely defined.
In fact, given the past price deviations in $\mathbf{x}_{t}$ and the traders' fractions $n_{h,t}$,
the per capita demand $d(x_{t})=\sum_{h=1}^Hn_{h,t}z_{h,t}(x_{t})$ is a piecewise-linear, continuous function
of the deviation $x_{t}$, that is decreasing up to the deviation at which it vanishes together
with the highest of the single agents' demand curves, and $d(x_{t})=0$ for larger deviations (see Figure~(\ref{fig:mc})).
There is therefore a unique deviation $x_t$ at which the market clears, i.e., $d(x_t)=s>0$.
Also note that, given the same traders' fractions, the constrained price\footnote{In this part of the paper we introduce the notation $x_{U,t}$ for the unconstrained price determined by model in Section \ref{ssec:ARED_no_uptick} to distinguish it from the constrained price $x_{t}$. Since there is no risk of confusion, this distinction is not made in other parts of the paper for the sake of avoiding cumbersome notations.} is higher than the unconstrained price\footnote{It is clear that all the traders' demand functions are always characterized by the same slope. However, the intercept of the demands with the $x=0$ axis, i.e. $s+\frac{f_{h}(\mathbf{x}_{t})}{a\hspace{0.2mm}\sigma^2}$, changes over time and it depends on the past share prices. For example if the predictor of a trader is based on $L$ past prices of the share, i.e. $\mathbf{x}_{t}=\left(x_{t-1},...,x_{t-L}\right)$, the intercept of its demand with the $x=0$ axis depends on all of these prices. Thus, to prove that the constrained prices are always higher than the unconstrained one given the same past prices, it does not necessarily mean a price dynamic characterized by larger fluctuations for the constrained model. The situation can be the opposite when we consider predictors based on a large number of past deviations and especially when they are non-linear. In other words, simple static considerations on the shape of the constrained demands do not help us to understand entirely the effect of the uptick rule on the price dynamics.} ($x_{t}>x_{U,t}$),
so that positive predictions ($f_{h}(\mathbf{x}_{t})>-\bar{p}$ for all $h=1,\ldots,H$) still yield
positive prices ($x_t>-\bar{p}$).
\begin{figure}[t]
\centerline{\includegraphics[scale=0.9]{fig_mc}}
\caption{Per-capita demand ($d$, red) and supply ($s$, blue) curves as functions
of the price deviation $x$ to be realized in period $t$.
A case with $H=3$ types of traders is sketched, where $z_{1,t}$ is the demand
of "fundamentalists" (see Sect.~\ref{ssec:pre}), while $z_{2,t}$ and $z_{3,t}$ are the demands of "non-fundamentalists" ($f_{2}(\mathbf{x}_t)>0$, $f_{3}(\mathbf{x}_t)<0$, and
negative (dashed) demands are obtained with \eqref{eq:zx}).
The resulting per capita demand curve is piecewise-linear, continuous, and decreasing.
Depending on the traders' fractions ($n_{1,t}$, $n_{2,t}$, $n_{3,t}$), it can take on different
configurations in the shaded area
(the most negative of which corresponds to $z_{3,t}$ when $n_{3,t}=1$;
the case shown corresponds to $n_{1,t}=n_{2,t}=n_{3,t}=1/3$).
At the unconstrained price $x_{U,t}$ traders of type $3$ are in a short position in period $t$.}
\label{fig:mc}
\end{figure}
When solving Eq.~\eqref{eq:mc} for $x_t$ with the constrained demands \eqref{eq:zxc}, $2^H-1$ cases
must be further distinguished, depending on which of the optimal demands in \eqref{eq:zx} are forced to zero by \eqref{eq:zxc}
(obviously not all demands can vanish).
The uniqueness of forward dynamics guarantees that only one of the cases clears the market.
For simplicity, hereafter we will only consider the case with two types of traders ($H=2$), so
one of the following three cases is realized at each period:
\begin{itemize}
\item[$0$:] no trader is prohibited from going short (equivalently, both types of traders hold nonnegative amounts of shares in period $t$), i.e.,\\
\eqref{eq:zx} implies $z_{1,t}\ge 0$ and $z_{2,t}\ge 0$, with
$x_t=\Frac{1}{R}\left(n_{1,t}\,f_{1}(\mathbf{x}_{t})+n_{2,t}\,f_{2}(\mathbf{x}_{t})\right)$,
\item[$1$:] traders of type $1$ are prohibited from going short (only traders of type $2$ hold shares in period $t$), i.e.,\\
\eqref{eq:zx} implies $z_{1,t}<0$ and $z_{2,t}>0$, with
$x_{t}=\Frac{1}{R}\left(f_{2}(\mathbf{x}_{t})-a\hspace{0.2mm}\sigma^{2}s\Frac{n_{1,t}}{n_{2,t}}\right)$,
\item[$2$:] traders of type $2$ are prohibited from going short (only traders of type $1$ hold shares in period $t$), i.e.,\\
\eqref{eq:zx} implies $z_{1,t}>0$ and $z_{2,t}<0$, with
$x_{t}=\Frac{1}{R}\left(f_{1}(\mathbf{x}_{t})-a\hspace{0.2mm}\sigma^{2}s\Frac{n_{2,t}}{n_{1,t}}\right)$.
\end{itemize}
Then, $m_{t+1}$ must be computed (see \eqref{eq:m}) and, again, there are three cases, depending
on the signs of the optimal demands at period $(t-1)$ (see Eq.~\eqref{eq:cm}).
In order to simplify the model formulation, we prefer to enlarge the system's state, by including the traders' demands $z_{h,t-1}$ realized in period $(t-1)$, $h=1,2$, in lieu of the farthest price deviation $x_{t-(L+1)}$.
The state variables therefore are
\begin{subequations}
\begin{linenomath}
\begin{equation}
\label{eq:state}
\ensuremath\makebox[50mm][r]{$(x_{t-1}, x_{t-2},\dots, x_{t-L},\ z_{1,t-1},\ z_{2,t-1},\ m_t)$}\quad\text{if}\quad L\ge 2,
\end{equation}
\begin{equation}
\label{eq:stateL1}
\ensuremath\makebox[50mm][r]{$(x_{t-1}, x_{t-2},\ z_{1,t-1},\ z_{2,t-1},\ m_t)$}\quad\text{if}\quad L=1,
\end{equation}
\end{linenomath}
\end{subequations}
as we need $x_{t-2}$ to apply the uptick rule,
and we can update the traders' fractions by simply replacing \eqref{ARED_model_no_uptick_2m} with
\begin{linenomath}
\begin{equation}
\label{eq:mz}
m_{t+1} = \tanh\left(\Frac{\beta}{2}\left((x_{t}-R\hspace{0.2mm}x_{t-1}+a\hspace{0.2mm}\sigma^{2}s)
(z_{1,t-1}-z_{2,t-1})-(C_1-C_2)\right)\right).
\end{equation}
\end{linenomath}
The uptick rule makes the ARED model piecewise smooth, namely the space of the state variables
is partitioned into three regions associated with different equations for updating the system's state (see \citep{Bernardo08} and references therein).
By defining the regions
\begin{subequations}
\label{eq:reg}
\begin{linenomath}
\begin{equation}
\label{eq:regZ}
\begin{array}{lrrr}
\multicolumn{1}{c}{U}: & x_{t-1}>x_{t-2},\\
Z_{0}: & x_{t-1}\le x_{t-2}, & z_{1,t}\ge 0, & z_{2,t}\ge 0,\\
Z_{1}: & x_{t-1}\le x_{t-2}, & z_{1,t}<0, & z_{2,t}>0,\\
Z_{2}: & x_{t-1}\le x_{t-2}, & z_{1,t}>0, & z_{2,t}<0,
\end{array}
\end{equation}
\end{linenomath}
where
\begin{linenomath}
\begin{equation}
\label{eq:regz}
z_{1,t} = \Frac{1-m_{t}}{2}\,\Frac{f_{1}(\mathbf{x}_{t})-f_{2}(\mathbf{x}_{t})}{a\hspace{0.2mm}\sigma^2}+s\quad\text{and}\quad
z_{2,t} = \Frac{1+m_{t}}{2}\,\Frac{f_{2}(\mathbf{x}_{t})-f_{1}(\mathbf{x}_{t})}{a\hspace{0.2mm}\sigma^2}+s
\end{equation}
\end{linenomath}
\end{subequations}
are the optimal demands from \eqref{eq:xt}, we can write the forward dynamics as follows:
\begin{linenomath}
\begin{subequations}
\label{ARED_model_uptick}
\begin{eqnarray}
\label{ARED_model_uptick_x}
x_t & = & \left\{\begin{array}{ll}
\ensuremath\makebox[60mm][l]{$\Frac{1}{2R}\left((1+m_{t})f_1(\mathbf{x}_{t})+(1-m_{t})f_2(\mathbf{x}_{t})\right)$} &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in U \cup Z_{0},\\[2mm]
\Frac{1}{R}\left(f_{2}(\mathbf{x}_{t})-a\hspace{0.2mm}\sigma^{2}s\hspace{0.2mm}\Frac{1+m_t}{1-m_t}\right) &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{1},\\[2mm]
\Frac{1}{R}\left(f_{1}(\mathbf{x}_{t})-a\hspace{0.2mm}\sigma^{2}s\hspace{0.2mm}\Frac{1-m_t}{1+m_t}\right) &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{2},\\[2mm]
\end{array}\right.\\
\label{ARED_model_uptick_z1}
z_{1,t} & = & \left\{\begin{array}{ll}
\ensuremath\makebox[60mm][l]{$\Frac{1-m_{t}}{2}\,\Frac{f_{1}(\mathbf{x}_{t})-f_{2}(\mathbf{x}_{t})}{a\hspace{0.2mm}\sigma^2}+s$} &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in U \cup Z_{0},\\[2mm]
0 &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{1},\\[2mm]
\Frac{2\hspace{0.2mm}s}{1+m_t} &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{2},\\[2mm]
\end{array}\right.\\
\label{ARED_model_uptick_z2}
z_{2,t} & = & \left\{\begin{array}{ll}
\ensuremath\makebox[60mm][l]{$\Frac{1+m_{t}}{2}\,\Frac{f_{2}(\mathbf{x}_{t})-f_{1}(\mathbf{x}_{t})}{a\hspace{0.2mm}\sigma^2}+s$} &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in U \cup Z_{0},\\[2mm]
\Frac{2\hspace{0.2mm}s}{1-m_t} &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{1},\\[2mm]
0 &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{2},\\[2mm]
\end{array}\right.\\
\label{ARED_model_uptick_m}
m_{t+1} & = & \tanh\left(\Frac{\beta}{2}\left((x_{t}-R\hspace{0.2mm}x_{t-1}+a\hspace{0.2mm}\sigma^{2}s)
(z_{1,t-1}-z_{2,t-1})-(C_1-C_2)\right)\right).
\end{eqnarray}
\end{subequations}
\end{linenomath}
The same model can be rewritten in compact notations as follows:
\begin{linenomath}
\begin{eqnarray}\label{ARED_model_uptick_CF}
\nonumber
\left(x_t,z_{1,t},z_{2,t},m_{t+1}\right) & = & \left\{\begin{array}{ll}
\ensuremath\makebox[10mm][l]{$G_{1}\left(\mathbf{x}_{t},z_{1,t-1},z_{2,t-1},m_{t}\right)$} &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in U \cup Z_{0},\\[2mm]
G_{2}\left(\mathbf{x}_{t},z_{1,t-1},z_{2,t-1},m_{t}\right) &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{1},\\[2mm]
G_{3}\left(\mathbf{x}_{t},z_{1,t-1},z_{2,t-1},m_{t}\right) &
\quad\text{if}\quad (\mathbf{x}_{t},m_t)\in Z_{2},\\[2mm]
\end{array}\right.\\
\end{eqnarray}
\end{linenomath}
where $G_1$, $G_2$ and $G_3$ are three systems that define the asset pricing model with uptick rule.
\begin{figure}[t]
\centerline{\includegraphics[scale=0.7]{fig_ns}}
\caption{Partition of the state space into the three regions $U \cup Z_{0}$ (green) and $Z_h$, $h=1,2$ (pink):
the projection on the $(x_{t-1},x_{t-2})$ space in the case where traders use the fundamental and chartist predictors (see Section~\ref{ssec:pre}).}
\label{fig:ns}
\end{figure}
Region $U \cup Z_{0}$ is separated from region $Z_h$, $h=1,2$, by the two boundaries
\begin{linenomath}
\begin{equation}
\label{eq:bor}
\partial U_h:\; x_{t-1}=x_{t-2},\; z_{h,t}\le 0\quad\text{and}\quad
\partial Z_h:\; x_{t-1}\le x_{t-2},\; z_{h,t}=0
\end{equation}
\end{linenomath}
(see Figure~(\ref{fig:ns}), where a projection on the $(x_{t-1},x_{t-2})$ space is shown).
Across boundary $\partial U_k$ the system is discontinuous, i.e., the corresponding expressions on the right-hand sides of
(\ref{ARED_model_uptick_x}--c) assume different values on $\partial U_k$.
In contrast, the system is continuous (but not differentiable) at the boundaries $\partial Z_h$.
Similarly to the unconstrained model \eqref{ARED_model_no_uptick}, the initial condition of model \eqref{ARED_model_uptick} is set by the opening price deviation $x_0$ and the traders' initial composition $m_1$. However, Eq.~\eqref{ARED_model_uptick_m} also requires the traders' demand $z_{1,0}$ and $z_{2,0}$ which can be conventionally set at $s$.
Finally, let's discuss the fixed points of model \eqref{ARED_model_uptick} (or equivalently \eqref{ARED_model_uptick_CF}), which necessarily lie on the boundary of region $U$ and are still denoted with the pair $(\bar{x},\bar{m})$
(the equilibrium demands, which also characterize the equilibria of model \eqref{ARED_model_uptick}, can be obtained from Eqs.~(\ref{ARED_model_uptick_z1},c)).
Each of the three systems defining model \eqref{ARED_model_uptick}, system $G_{1}$ (equivalent to model \eqref{ARED_model_no_uptick} adding the demands as state variables) which defines the dynamics of the model in region $U \cup Z_{0}$ and the two systems, $G_{1}$ and $G_{2}$, which define the dynamics of the model in regions $Z_1$ and $Z_2$, respectively, have their own fixed points, which we call either {\it admissible} or {\it virtual} according to the region of the state space in which they are.
In this paper, a generic fixed point or equilibrium of system $G_{1}$ is called admissible if it lies in region $U \cup Z_{0}$ and virtual otherwise, a generic fixed point of system $G_{2}$ is called admissible if it lies in region $Z_{1}$ and virtual otherwise, and a generic fixed point of system $G_{3}$ is called admissible if it lies in region $Z_{2}$ and virtual otherwise.
Virtual fixed points are not equilibria of model \eqref{ARED_model_uptick}, but tracking their position is useful in the analysis.
The fundamental equilibrium is always an admissible fixed point of system $G_{1}$, also called the unconstrained system because equivalent to model \eqref{ARED_model_no_uptick} adding the demands as state variables. Indeed, it lies on the boundary between regions $U$ and $Z_{0}$, with positive demands (equal to $s$), i.e., it is always an interior point of region $U \cup Z_{0}$.
Its local stability is therefore ruled by Lemma~\ref{lm:fund}
(the storage of the previous demands in lieu of the farthest past deviation brings the number of zero eigenvalues to $2H-1$, $2H$ if $L=1$, see \eqref{eq:stateL1}),
while the existence of (admissible or virtual) non-fundamental equilibria of the unconstrained system $G_{1}$ is ruled by Lemma~\ref{lm:neq}.
The fixed points of the other two systems, $G_2$ and $G_3$, are of little interest.
They lie on the boundaries $\partial U_2$ and $\partial U_1$, respectively, across which model \eqref{ARED_model_uptick} is discontinuous.
Hence, there are arbitrarily small perturbations from the fixed point entering region $U$, for which the unconstrained system $G_1$ will map the system's state far from the fixed point.
The fixed points of the two systems $G_2$ and $G_{3}$ are therefore (highly) unstable and will not be considered in the analysis.
\subsection{Classical price predictors}
\label{ssec:pre}
In this Section we briefly introduce the price predictors used in this paper (see, e.g., \citep{Elder1993}, \citep{ChiarellaHe2002}, and \citep{ChiarellaHe2003}, for a more complete survey of the most classical types of price predictors used in the literature).
The first one, $f_1(\mathbf{x}_{t})$, called fundamental predictor, will be paired with each of the others, the non-fundamental predictors, in the analysis of Sections~\ref{sec:ana} and~\ref{sec:num}.
For this reason, all non-fundamental predictors will be denoted by $f_2(\mathbf{x}_{t})$.
\subsubsection*{Fundamental predictor}
Fundamental traders, or {\it fundamentalists}, believe that prices return to their fundamental value.
The simplest fundamental prediction is the fundamental price for period $t+1$, irrespectively of the recent trend:
\begin{linenomath}
\begin{equation}
\label{eq:pr1}
E_1[p_{t+1}]=\bar{p},\quad f_1(\mathbf{x}_{t})=0.
\end{equation}
\end{linenomath}
More generally, fundamentalists believe that prices will revert to the fundamental value by a factor $v$ at each period:
\begin{linenomath}
\begin{equation}
\label{eq:pr1v}
E_1[p_{t+1}]=\bar{p}+v\hspace{0.2mm}(p_{t-1}-\bar{p}),\quad f_1(\mathbf{x}_{t})=v\hspace{0.2mm}x_{t-1},\quad 0\le v<1,
\end{equation}
\end{linenomath}
the smaller is $v$, the highest is the expected speed of convergence to the fundamental price.
As in \citep{BrockHommes1998}, we assume that "training" costs must be borne to obtain enough "understanding" of how markets work in order to believe in the fundamental price, so fundamentalists incur into a cost $C_{1}>0$ at each prediction.
\subsubsection*{Chartist predictor}
The second type of simple trader that we consider is called {\it chartist} or {\it trend chaser}. This type of trader believes that any mispricing will continue, i.e. the chartist predictor is formally equivalent to predictor \eqref{eq:pr1v}:
\begin{linenomath}
\begin{equation}
\label{eq:pr1g}
E_2[p_{t+1}]=\bar{p}+g\hspace{0.2mm}(p_{t-1}-\bar{p}),\quad f_2(\mathbf{x}_{t})=g\hspace{0.2mm}x_{t-1},\quad g>1,
\end{equation}
\end{linenomath}
but amplifies, instead of damping, nonzero price deviations from the fundamental.
The chartist prediction is not costly.
\subsubsection*{Rate of change (ROC) predictor}
The third type of simple trader that we consider is called "{\it nonlinear technical analyst}" or "{\it ROC trader}".
The ROC ({\it"Price Rate Of Change"}) is a nonlinear prediction which applies the price rate of change averaged over the last $L-1$ periods,
\begin{subequations}
\begin{linenomath}
\begin{equation}
\label{eq:roc}
\text{ROC} = \left(\Frac{p_{t-1}}{p_{t-L}}\right)^{\hspace{-1.0mm}\frac{1}{L-1}}=\left(\Frac{\bar{p}+x_{t-1}}{\bar{p}+x_{t-L}}\right)^{\hspace{-1.0mm}\frac{1}{L-1}},\quad L\ge 2,
\end{equation}
\end{linenomath}
to $p_{t-1}$ twice to extrapolate $p_{t+1}$:
\begin{linenomath}
\begin{equation}
\label{eq:rocp}
E_2[p_{t+1}] = p_{t-1}\hspace{0.2mm}\text{ROC}^{2},\quad
f_2(\mathbf{x}_{t}) = (\bar{p}+x_{t-1})\hspace{0.2mm}\text{ROC}^{2}-\bar{p}.
\end{equation}
\end{linenomath}
The ROC predictor is typically "smoothed'' to avoid extreme rates of change (rates that are either too high or too close to zero, see Smoothed-ROC or S-ROC predictors in \citep{Elder1993}).
This interprets the traders' rationality that makes them diffident with extreme rates.
We adopt in particular the confidence mechanism introduced in \citep{DercoleCecchetto2010},
where the ROC is combined with the last available price.
Precisely, the price rate of change to be applied is a convex combination of the actual ROC \eqref{eq:roc} and the unitary rate (corresponding to the last available price),
with the ROC weight $\alpha_{\text{ROC}}$ that vanishes when the ROC attains extreme values (zero and infinity):
\begin{linenomath}
\begin{eqnarray}
\label{eq:rocpa}
E_2[p_{t+1}] & = & p_{t-1}\left(\alpha_{\text{ROC}}\hspace{0.2mm}\text{ROC}+(1-\alpha_{\text{ROC}})\right)^{2},\nonumber\\
f_2(\mathbf{x}_{t}) & = & (\bar{p}+x_{t-1})\left(\alpha_{\text{ROC}}\hspace{0.2mm}\text{ROC}+(1-\alpha_{\text{ROC}})\right)^{2}-\bar{p}.
\end{eqnarray}
\end{linenomath}
The function
\begin{linenomath}
\begin{equation}
\label{eq:roca}
\alpha_{\text{ROC}}=\Frac{2}{\text{ROC}^{\hspace{0.5mm}\alpha}+\text{ROC}^{\hspace{0.5mm}-\alpha}}
\end{equation}
\end{linenomath}
\end{subequations}
has been used in the analysis, where the parameter $1/\alpha$ measures how confident traders are with extreme rates.
The ROC predictor and the S-ROC predictor are not costly.
\section{The effect of the uptick rule on shares price fluctuations: Analytical results}
\label{sec:ana}
In this Section we report the stability analysis of fundamental and non-fundamental equilibria
of models \eqref{ARED_model_no_uptick_2} and \eqref{ARED_model_uptick} for two pairs of traders' types.
As traditionally done in the literature, type $1$ is always the fundamental type (price predictor \eqref{eq:pr1v}),
while type two is either the chartist in Sect.~\ref{ssec:cha_a} (predictor \eqref{eq:pr1g})
or the nonlinear technical analyst (ROC trader) in Sect.~\ref{ssec:roc_a} (predictor (\ref{eq:roc},c,d)).
\subsection{Fundamentalists vs chartists}
\label{ssec:cha_a}
Consider models \eqref{ARED_model_no_uptick_2} and \eqref{ARED_model_uptick} with predictors \eqref{eq:pr1v} and \eqref{eq:pr1g}.
Model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1v},~\ref{eq:pr1g}) is the classical ARED model, proposed and fully analyzed in \citep{BrockHommes1998} for the case of zero supply of outside shares, i.e., $s=0$, where short selling is intrinsically practiced at each trading period.
The case with positive supply is analyzed in \citep{AnufrievTuinstra2009}, where the effects of a negative bound on the traders' positions are also investigated.
Without any constraint on short selling,
the existence and stability of the fixed points of model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1v},~\ref{eq:pr1g}) are defined in the following lemma:
\begin{lemma}
\label{lm:cha}
The following statements hold true for the dynamical system \eqref{ARED_model_no_uptick_2} with predictors \eqref{eq:pr1v} and \eqref{eq:pr1g}:
\begin{enumerate}
\item
For $1<g<R$ the fundamental equilibrium $(0,\bar{m}^{(0)})$ (see Lemma~\ref{lm:fund}) is the only fixed point and is globally stable.
\item
For $R<g<2R-v$ there are the following possibilities:
\begin{enumerate}
\item[\rm (a)]
For $0\hspace{-0.2mm}\le\hspace{-0.2mm}\beta\hspace{-0.2mm}<\hspace{-0.2mm}\beta_{\mathrm{LP}}=\hspace{-0.2mm}\Frac{1}{C}\log\hspace{-0.3mm}\left(\Frac{R\hspace{-0.2mm}-\hspace{-0.2mm}v}{g\hspace{-0.2mm}-\hspace{-0.3mm}R}\right)\hspace{-0.7mm}\left(1\hspace{-0.3mm}+\hspace{-0.2mm}\Frac{a\hspace{0.2mm}\sigma^2s^2}{4C}\Frac{g\hspace{-0.2mm}-\hspace{-0.2mm}v}{R\hspace{-0.2mm}-\hspace{-0.2mm}1}\right)^{\hspace{-1mm}-1}\hspace{-1.0mm}>\hspace{-0.3mm}0$
the fundamental equilibrium is the only fixed point and is stable.
\item[\rm (LP)]
At $\beta\hspace{-0.1mm}=\hspace{-0.1mm}\beta_{\mathrm{LP}}$ two equilibria appear (as $\beta$ increases) at
\begin{linenomath}
$$
\bar{x}_{\mathrm{LP}}\hspace{-0.3mm}=\hspace{-0.2mm}\Frac{a\hspace{0.2mm}\sigma^2s}{2(R\hspace{-0.3mm}-\hspace{-0.4mm}1)}\hspace{-0.3mm}>\hspace{-0.3mm}0,\quad
\bar{m}=1-2\hspace{0.2mm}\Frac{R-v}{g-v},
$$
\end{linenomath}
through a saddle-node bifurcation (limit point, \emph{LP}).
\item[\rm (b)]
For $\beta_{\mathrm{LP}}\hspace{-0.2mm}<\hspace{-0.2mm}\beta\hspace{-0.2mm}<\hspace{-0.2mm}\min\left\{\beta_{\mathrm{TR}}=\hspace{-0.2mm}\Frac{1}{C}\log\hspace{-0.3mm}\left(\Frac{R\hspace{-0.2mm}-\hspace{-0.2mm}v}{g\hspace{-0.2mm}-\hspace{-0.3mm}R}\right),\beta_{\mathrm{NS}}^{(+)}\right\}>\beta_{\mathrm{LP}}$
the fundamental equilibrium is locally (asymptotically) stable and coexists with the two non-fundamental equilibria $(\bar{x}^{(\pm)},\bar{m})$, with
\begin{linenomath}
$$
\bar{x}^{(\pm)}=\bar{x}_{\mathrm{LP}}\pm\sqrt{\left(\bar{x}_{\mathrm{LP}}^2+\Frac{a\hspace{0.2mm}\sigma^2C}{(R\hspace{-0.3mm}-\hspace{-0.4mm}1)(g\hspace{-0.2mm}-\hspace{-0.2mm}v)}\right)\hspace{-0.5mm}\left(\hspace{-0.5mm}1\hspace{-0.4mm}-\hspace{-0.3mm}\Frac{\beta_{\mathrm{LP}}}{\beta}\hspace{-0.2mm}\right)\hspace{-0.3mm}}\hspace{0.3mm}.
$$
\end{linenomath}
Equilibrium $(\bar{x}^{(+)},\bar{m})$ is locally (asymptotically) stable, whereas $(\bar{x}^{(-)},\bar{m})$ is a saddle with 2-dimensional stable manifold separating the basins of attraction of the two stable equilibria.
\item[\rm (TR)]
At $\beta\hspace{-0.1mm}=\hspace{-0.1mm}\beta_{\mathrm{TR}}$, $(\bar{x}^{(-)},\bar{m})$ collides and exchanges stability with the fundamental equilibrium (transcritical bifurcation, \emph{TR}). The fundamental equilibrium is always at least locally (asymptotically) stable for $\beta\hspace{-0.1mm}<\hspace{-0.1mm}\beta_{\mathrm{TR}}$ and it is always unstable for $\beta\hspace{-0.1mm}>\hspace{-0.1mm}\beta_{\mathrm{TR}}$.
\item[\rm (NS$^{(+)}$)]
At $\beta\hspace{-0.1mm}=\hspace{-0.1mm}\beta_{\mathrm{NS}}^{(+)}$ the equilibrium $(\bar{x}^{(+)},\bar{m})$ undergoes a Neimark-Sacker (NS) bifurcation.
No explicit expression is available for $\beta_{\mathrm{NS}}^{(+)}$, but $\beta_{\mathrm{TR}}\lessgtr\beta_{\mathrm{NS}}^{(+)}$ if $\beta_{\mathrm{TR}}\lessgtr\Frac{1}{a\hspace{0.2mm}\sigma^2s^2}\Frac{(R\hspace{-0.2mm}-\hspace{-0.2mm}1)^2}{(g\hspace{-0.2mm}-\hspace{-0.2mm}R)(R\hspace{-0.2mm}-\hspace{-0.2mm}v)}$.
\item[\rm (c)]
For $\beta_{\mathrm{TR}}\hspace{-0.2mm}<\hspace{-0.2mm}\beta\hspace{-0.2mm}<\hspace{-0.2mm}\beta_{\mathrm{NS}}^{(-)}>\beta_{\mathrm{TR}}$
the fundamental equilibrium is a saddle, with 2-dimensional stable manifold separating the positive from the negative dynamics,
and the equilibrium $(\bar{x}^{(-)},\bar{m})$, with $\bar{x}^{(-)}\hspace{-0.4mm}<0$, is stable.
\item[\rm (NS$^{(-)}$)]
At $\beta\hspace{-0.1mm}=\hspace{-0.1mm}\beta_{\mathrm{NS}}^{(-)}$ the equilibrium $(\bar{x}^{(-)},\bar{m})$ undergoes a Neimark-Sacker bifurcation.
\end{enumerate}
\item
For $g>2R-v$ there are the following possibilities:
\begin{enumerate}
\item[\rm (a)]
For $0\hspace{-0.2mm}\le\hspace{-0.2mm}\beta\hspace{-0.2mm}<\hspace{-0.2mm}\beta_{\mathrm{NS}}^{(\pm)}$
the fundamental equilibrium is unstable and the equilibria $(\bar{x}^{(\pm)},\bar{m})$ ($\bar{x}^{(+)}\hspace{-0.4mm}>0$ and $\bar{x}^{(-)}\hspace{-0.4mm}<0$) are stable.
\item[\rm (NS)]
At $\beta\hspace{-0.1mm}=\hspace{-0.1mm}\beta_{\mathrm{NS}}^{(\pm)}$ the equilibria $(\bar{x}^{(\pm)},\bar{m})$ undergo a Neimark-Sacker bifurcation.
\end{enumerate}
\item
For $g>R^2$ the dynamics can be unbounded for sufficiently large $\beta$.
\end{enumerate}
\end{lemma}
Lemma \ref{lm:cha} generalizes Lemmas 2,~3, and~4 in \citep{BrockHommes1998} to the case $s>0$, $0<v<1$, and Proposition~3.1 in \citep{AnufrievTuinstra2009}.
In particular, for $s=0$, note that the saddle-node and transcritical bifurcations concomitantly occur (case $2$) at a so-called pitchfork bifurcation,
whereas the mechanism making the fundamental equilibrium unstable is different for $s>0$.
First, the two non-fundamental equilibria $(\bar{x}^{(\pm)},\bar{m})$ appear (as the traders' adaptability $\beta$ increases) through the saddle-node bifurcation,
and as $\beta$ increases further a transcritical bifurcation occurs in which the saddle $(\bar{x}^{(-)},\bar{m})$ exchanges stability with the fundamental equilibrium.
Thus, for $\beta_{\mathrm{LP}}<\beta<\beta_{\mathrm{TR}}$, the fundamental equilibrium is stable,
but coexists with an alternative stable fixed point of model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1v},~\ref{eq:pr1g}).
With the uptick-rule,
the existence and stability of the fixed points of model (\ref{ARED_model_uptick},~\ref{eq:pr1v},~\ref{eq:pr1g}) are complemented by the following lemma:
\begin{lemma}
\label{lm:cha_uptick}
The following statements hold true for the dynamical system \eqref{ARED_model_uptick} with predictors \eqref{eq:pr1v} and \eqref{eq:pr1g}:
\begin{enumerate}
\item
The local and global stability of the fundamental equilibrium is as in Lemma~\ref{lm:cha}.
As long as equilibria $(\bar{x}^{(\pm)},\bar{m})$ exist and are admissible, their local stability is as in Lemma~\ref{lm:cha}.
Equilibrium $(\bar{x}^{(+)},\bar{m})$ is admissible iff $\bar{x}^{(+)}\hspace{-0.4mm}\le\hspace{-0.1mm}\bar{x}_{\mathrm{BC}}^{(+)}\hspace{-0.2mm}=a\hspace{0.2mm}\sigma^2s/(R-v)$.
Equilibrium $(\bar{x}^{(-)},\bar{m})$ is admissible iff $\bar{x}_{\mathrm{BC}}^{(-)}\hspace{-0.2mm}=-a\hspace{0.2mm}\sigma^2s/(g\hspace{-0.2mm}-\hspace{-0.2mm}R)\le\bar{x}^{(-)}\hspace{-0.3mm}\le\bar{x}_{\mathrm{BC}}^{(+)}$.
\item
For $R<g<2R-v$ there are the following possibilities:
\begin{enumerate}
\item[\rm (a)]
If $R\hspace{-0.2mm}+\hspace{-0.2mm}v>2$, equilibria $(\bar{x}^{(\pm)},\bar{m})$ appear admissible at $\beta=\beta_{\mathrm{LP}}$ and $(\bar{x}^{(+)},\bar{m})$ becomes virtual (border-collision bifurcation) at $\beta=\beta_{\mathrm{BC}}^{(+)}$, with
\begin{linenomath}
$$
0<\beta_{\mathrm{BC}}^{(+)}=\Frac{1}{C}\log\hspace{-0.3mm}\left(\Frac{R\hspace{-0.2mm}-\hspace{-0.2mm}v}{g\hspace{-0.2mm}-\hspace{-0.3mm}R}\right)\hspace{-0.7mm}\left(1\hspace{-0.3mm}+\hspace{-0.2mm}\Frac{a\hspace{0.2mm}\sigma^2s^2}{C}\Frac{(g\hspace{-0.2mm}-\hspace{-0.2mm}v)(1\hspace{-0.2mm}-\hspace{-0.2mm}v)}{(R\hspace{-0.2mm}-\hspace{-0.2mm}v)^2}\right)^{\hspace{-1mm}-1}\hspace{-1.0mm}<\beta_{\mathrm{TR}}.
$$
\end{linenomath}
\item[\rm (b)]
If $R\hspace{-0.2mm}+\hspace{-0.2mm}v<2$, equilibria $(\bar{x}^{(\pm)},\bar{m})$ appear virtual at $\beta=\beta_{\mathrm{LP}}$ and $(\bar{x}^{(-)},\bar{m})$ becomes admissible at $\beta=\beta_{\mathrm{BC}}^{(+)}$.
\item[\rm (c)]
If $R\hspace{-0.2mm}+\hspace{-0.2mm}v=2$, equilibria $(\bar{x}^{(\pm)},\bar{m})$ appear on the border $\partial Z_1$ at $\beta=\beta_{\mathrm{LP}}=\beta_{\mathrm{BC}}^{(+)}$ and $(\bar{x}^{(+)},\bar{m})$ and $(\bar{x}^{(-)},\bar{m})$ are respectively virtual and admissible for larger $\beta$.
\item[\rm (d)]
If $s<s_{\mathrm{BC}}^{(-)}=\left(\Frac{C}{a\hspace{0.2mm}\sigma^2}\Frac{(g-R)^2}{(g-v)(g-1)}\right)^{\hspace{-0.7mm}1\hspace{-0.2mm}/2}$,
equilibrium $(\bar{x}^{(-)},\bar{m})$ becomes virtual at $\beta=\beta_{\mathrm{BC}}^{(-)}$, with
\begin{linenomath}
$$
\beta_{\mathrm{BC}}^{(-)}=\Frac{1}{C}\log\hspace{-0.3mm}\left(\Frac{R\hspace{-0.2mm}-\hspace{-0.2mm}v}{g\hspace{-0.2mm}-\hspace{-0.3mm}R}\right)\hspace{-0.7mm}\left(1\hspace{-0.3mm}-\hspace{-0.2mm}\Frac{a\hspace{0.2mm}\sigma^2s^2}{C}\Frac{(g\hspace{-0.2mm}-\hspace{-0.2mm}v)(g\hspace{-0.2mm}-\hspace{-0.2mm}1)}{(g\hspace{-0.2mm}-\hspace{-0.2mm}R)^2}\right)^{\hspace{-1mm}-1}\hspace{-1.0mm}>\beta_{\mathrm{TR}}.
$$
\end{linenomath}
\end{enumerate}
\item
For $g>2R-v$ there are the following possibilities:
\begin{enumerate}
\item
If $s<s_{\mathrm{BC}}^{(-)}$ equilibria $(\bar{x}^{(\pm)},\bar{m})$ are virtual for any $\beta\ge0$.
\item
If $s>s_{\mathrm{BC}}^{(-)}$ equilibrium $(\bar{x}^{(+)},\bar{m})$ is virtual for any $\beta\ge0$, whereas $(\bar{x}^{(-)},\bar{m})$ is admissible for $\beta>\beta_{\mathrm{BC}}^{(-)}>0$.
\end{enumerate}
\item
For $g>R^2$ the dynamics can be unbounded for sufficiently large $\beta$.
\end{enumerate}
\end{lemma}
Note that the uptick rule affects the price dynamics also when the supply of outside shares is large. Indeed, independently on $s$, there is always an equilibrium, $(\bar{x}^{(+)},\bar{m})$ or $(\bar{x}^{(-)},\bar{m})$, becoming virtual as $\beta$ increases or decreases.
\subsection{Fundamentalists vs ROC traders}
\label{ssec:roc_a}
Consider models \eqref{ARED_model_no_uptick_2} and \eqref{ARED_model_uptick} with the fundamental predictor \eqref{eq:pr1v} and with the ROC predictor (\ref{eq:roc},b) or (\ref{eq:roc},c,d).
Note that both predictors are such that $f_h(\bar{x}\mathbf{1})/\bar{x}<R$ for any possible equilibrium $(\bar{x},\bar{m})$ with $\bar{x}\neq0$, so by means of Lemma~\ref{lm:neq} the fundamental equilibrium is the only fixed point.
In the simplest case $L=2$, its stability is characterized in the following lemma:
\begin{lemma}
\label{lm:roc}
The following statements hold true for the dynamical systems \eqref{ARED_model_no_uptick_2} and \eqref{ARED_model_uptick} with predictors \eqref{eq:pr1v} and {\rm(}\ref{eq:roc},b{\rm)}, as well as with predictors \eqref{eq:pr1v} and {\rm(}\ref{eq:roc},c,d{\rm)}:
\begin{enumerate}
\item
For $R\ge 2$ the fundamental equilibrium $(0,\bar{m}^{(0)})$ (see Lemma~\ref{lm:fund}) is a stable fixed point for any $\beta>0$.
\item
For $R<2$ the fundamental equilibrium is stable for
\begin{linenomath}
$$
0<\beta<\beta_{\mathrm{NS}}=\Frac{1}{C}\log\hspace{-0.3mm}\left(\Frac{R}{2\hspace{-0.2mm}-\hspace{-0.2mm}R}\right)
$$
\end{linenomath}
and loses stability through a Neimark-Sacker bifurcation at $\beta=\beta_{\mathrm{NS}}$.
\end{enumerate}
\end{lemma}
Similarly to the cases where chartists
are paired with fundamentalists
(Sect.~\ref{ssec:cha_a}),
the stability of the fundamental equilibrium is guaranteed if the gross return $R$ is sufficiently large.
The stability analysis for $L>2$ is possible, following the lines indicated in \citep{Kuruklis1994}, and the general conclusion is that rates of change calculated on larger windows of past prices stabilize the fundamental equilibrium, up to the point that the fixed point is stable for any value of $\beta$ if $L$ is sufficiently large.
\section{The effect of the uptick rule on share price fluctuations: Numerical simulations.}
\label{sec:num}
In the first two Subsections of this Section we report several numerical analysis of models \eqref{ARED_model_no_uptick_2} and \eqref{ARED_model_uptick} for the two pairs of traders' types considered in Sect. \ref{sec:ana}, with the aim of characterizing the non-stationary (periodic, quasi-periodic, or chaotic) asymptotic regimes. This part contains technical considerations. In the last subsection, we discuss the effects of the uptick rule on the price dynamics.
As done in most of the related works in the literature, we use the traders' adaptability (or intensity of choice) $\beta$ as a bifurcation parameter (two- or higher-dimensional bifurcation analyses are possible, see, e.g., \citep{DercoleCecchetto2010}, but will not be considered here).
For each considered value of $\beta$, the transient dynamics is eliminated by computing the (largest) Lyapunov exponent associated to the orbit\footnote
The largest Lyapunov exponent is a measure of the mean divergence of nearby trajectories; it is positive, zero, and negative in chaotic, quasi-periodic, and periodic (or stationary) regimes, respectively \citep{Alligood96}.}, i.e., we delete the number of initial iterations required to compute the largest Lyapunov exponent,
whereas the asymptotic regime is discussed.
To graphically study the bifurcations underwent by the different attractors, we vertically plot the deviations $x_t$ in the attractor at the corresponding value of $\beta$, together with the associated largest Lyapunov exponent L (see, e.g., Figure~(\ref{fig:BH_s})).
In each simulation, we set the initial condition as follows.
The opening price deviation $x_0$ is randomly selected in a small, positive or negative neighborhood of zero to study the stability of the fundamental equilibrium; far from zero to study non-fundamental attractors.
The initial traders' fractions are equally set ($m_1=0$, i.e., $n_{1,1}=n_{2,1}=1/2$).
For model \eqref{ARED_model_uptick}, the initial values assigned to the traders' demands $z_{1,0}$ and $z_{2,0}$ are irrelevant, as the traders' fractions are not updated at $t=1$.
\subsection{Fundamentalists vs chartists}
\label{ssec:cha_n}
We first study the effects of a positive supply of outside shares ($s>0$) on the dynamics of the original model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:pr1g}) introduced by \citep{BrockHommes1998}, then we study the effects of the uptick rule.
Figure~(\ref{fig:BH_s}) reports the bifurcation diagrams and the corresponding largest Lyapunov exponent obtained for four different values of $s$
(with $s$ in the range of values commonly used in the literature, see \citep{AnufrievTuinstra2009,HommesHuangWang2005}).
The first panel is the case with zero supply of outside shares ($s=0$) and is included for comparison.
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{BifDiagramBH8panels_ai_g}}
\caption{Bifurcation diagrams of model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:pr1g}) for different values of $s$ (the supply of outside shares of the risky asset): A (first row, left column) $s=0$, in this case $\beta_{\mathrm{TR}}=\beta_{\mathrm{LP}}=\beta_{\mathrm{PF}}$, related (largest) Lyapunov exponents in (second row, left column); B (first row, right column) $s=0.1$, in this case $\beta_{\mathrm{TR}}<\beta_{\mathrm{NS}}^{(+)}$, related (largest) Lyapunov exponents in (second row, right column); C (third row, left column) $s=0.2$, in this case $\beta_{\mathrm{TR}}=\beta_{\mathrm{NS}}^{(+)}$, related (largest) Lyapunov exponents (fourth row, left column); D (third row, right column) $s=0.3$, in this case $\beta_{\mathrm{TR}}>\beta_{\mathrm{NS}}^{(+)}$, related (largest) Lyapunov exponents (fourth row, right column). $\beta_{\mathrm{NS}}^{(+)}$ and $\beta_{\mathrm{NS}}^{(-)}$ are not ticked because there is no analytical expression for them. However, the Neimark-Sacker bifurcation values can be clearly identified in the bifurcation diagrams.
Other parameter values: $R=1.1$, $a=1$, $\sigma=1$, $\bar{y}=1$, $g=1.2$, $C=1$.}
\label{fig:BH_s}
\end{figure}
The bifurcation points $\beta_{\mathrm{LP}}$, $\beta_{\mathrm{TR}}$, and $\beta_{\mathrm{NS}}$ are indicated, together with the non-fundamental equilibria $(\bar{x}^{(\pm)},\bar{m})$ (the gray parabola), and are in agreement with the analytical results of Lemma~\ref{lm:cha}.
In particular, $\beta_{\mathrm{LP}}=\beta_{\mathrm{TR}}=\beta_{\mathrm{PF}}$ indicates a pitchfork bifurcation when $s=0$.
Note that the positive and negative deviation dynamics are separated in model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:pr1g}).
Indeed, due to the characteristics of the chartist predictor \eqref{eq:pr1g}, if the opening price is above [below] the fundamental value ($x_0>0$ [$x_0<0$]), it will remain so forever (see Eq.~\eqref{ARED_model_no_uptick_2x}).
The positive and negative attractors therefore coexist (with basins of attraction separated by the linear manifold $x_{t-1}=0$ in state space), so that two Lyapunov exponents (one for each of the two attractors) are plotted.
As expected, the fundamental equilibrium is destabilized for sufficiently high traders' adaptability and the amplitude of the price fluctuations increases as $\beta$ increases.
But the amplitude of fluctuations also increases with the supply of outside shares of the risky asset $s$ (note the different vertical scales in Figure~(\ref{fig:BH_s})).
The latter effect is partially due to the risk premium, i.e. $\left(a\sigma^{2}s\right)$, required by traders for holding extra shares that modifies the performance measures of the trading strategies.
Figure~(\ref{fig:BH_s}) shows another interesting dynamical phenomenon. For sufficiently high supply of outside shares of the risky asset ($s=0.3$ in case D), the positive attractor, that appears at the Neimark-Sacker (NS) bifurcation ($\beta_{\mathrm{NS}}^{(+)}$) of equilibrium $(\bar{x}^{(+)},\bar{m})$, collapses through a "homoclinic'' contact with the saddle equilibrium $(\bar{x}^{(-)},\bar{m})$.
The chaotic behavior is then reestablished when the fundamental equilibrium looses stability through the transcritical bifurcation ($\beta_{\mathrm{TR}}$).
The analysis of the dynamics conducted up to now reveals that there are not substantial differences in the price dynamics for different values of $s$, except the amplitude of price fluctuations and some peculiar phenomena, such as the homoclinic contact just discussed. For this reason and to make the discussion more clear and concise, in the following we investigate the effects of the uptick rule on the price dynamics only for $s = 0.1$.
Figure~(\ref{fig:BHbif}) compares models \eqref{ARED_model_no_uptick_2} (left) and \eqref{ARED_model_uptick} (right) with predictors \eqref{eq:pr1} and \eqref{eq:pr1g}.
The bifurcation diagram and the largest Lyapunov exponents associated with the different attractors are reported, together with two examples of state portraits (projections in the plane $(x_{t},x_{t-1})$, see bottom panels).
Blue and red dots identify the points in the attractor in which respectively fundamentalists and chartists have been prohibited from going short.
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{BifDiagram_BH_beta_3_and_4_g}}
\caption{Bifurcation diagrams of models (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:pr1g}) (left) and (\ref{ARED_model_uptick},~\ref{eq:pr1},~\ref{eq:pr1g}) (right)
and exemplary state portraits projections in the plane $(x_{t},x_{t-1})$.
Parameter values as in Figure~(\ref{fig:BH_s})-case(B). The light gray represents the attractor of negative deviations and the corresponding largest Lyapunov exponent. The dark gray represents the attractor of positive deviations and the corresponding largest Lyapunov exponent. The black represents the attractor born through a nonsmooth saddle-node bifurcation and the corresponding largest Lyapunov exponent. The blue dots indicate that fundamentalists cannot have negative positions because of the uptick rule. The red dots indicate that chartists cannot have negative positions because of the uptick rule.}
\label{fig:BHbif}
\end{figure}
The first thing to remark is that multiple positive attractors are present in the constrained dynamics (right column in Figure~(\ref{fig:BHbif})), i.e. when the uptick rule is in place.
In particular a period-two cycle alternating unrestricted trading with restricted trading in which fundamentalists are forced by the uptick rule to take nonnegative positions (black and blue dots) is present for sufficiently large $\beta$. It appears through a nonsmooth saddle-node bifurcation and coexists with the chaotic attractor generated through the loss of fundamental stability.
Before the bifurcation the fundamental equilibrium is globally stable (while it is globally stable in the unconstrained dynamics up to the saddle-node at $\beta=\beta_{\mathrm{LP}}$).
Also the bifurcation structure leading to the chaotic attractor is more involved. The first branch of attractors suddenly appears as $\beta$ increases.
This is probably due to a homoclinic contact with the period-two saddle cycle, but this conjecture has not been verified.
This first branch seems to collapse through a collision with the border $\partial Z_1$, in connection with the border collision of the non-fundamental equilibrium $(\bar{x}^{(+)},\bar{m})$ at $\beta=\beta_{\mathrm{BC}}^{(+)}$.
The remaining thinner attractor later explodes into a larger one, again due to a border collision with $\partial Z_1$.
However, a deeper mathematical investigation would be required to confirm the above conjectures.
As for the negative equilibrium $(\bar{x}^{(-)},\bar{m})$ (for $\beta>\beta_{\mathrm{TR}}$), it loses stability at $\beta=\beta_{\mathrm{BC}}^{(-)}$ through a border collision with $\partial Z_2$, giving way to a chaotic attractor characterized by restrictions on chartists.
Figure~(\ref{fig:BHtsp}) shows an example of time series on the positive chaotic attractor obtained for $\beta=3$
(left column: unconstrained dynamics; right column: dynamics constrained by the uptick rule).
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{Timeseries_BH_beta_3_positive}}
\caption{Positive (i.e. above fundamental) time series on the asymptotic regime of models (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:pr1g}) (left) and (\ref{ARED_model_uptick},~\ref{eq:pr1},~\ref{eq:pr1g}) (right) for $\beta=3$. Other parameter values as in Figure~(\ref{fig:BH_s})-case(B). The blue dots indicate that fundamentalists cannot have negative positions because of the uptick rule.}
\label{fig:BHtsp}
\end{figure}
The top panels report the price dynamics (black) and the chartist prediction (red, dashed), with blue and red dots marking the periods in which fundamentalists and chartists are respectively prevented to go short by the uptick rule (right column).
The remaining panels report, from top to bottom, the returns ($R_t=x_t-R\hspace{0.2mm}x_{t-1}+a\hspace{0.2mm}\sigma^2s$) and the traders' fractions ($n_{h,t}$), demands ($z_{h,t}$), and net profits ($U_{h,t}=R_t z_{h,t-1}-C_h$, $h=1,2$, blue for fundamentalists and red for chartists).
The price dynamics is characterized by recurrent peaks (financial bubbles), driven by chartists that expect a price rise and hold the shares of the risky asset and, at the same time, attenuated by fundamentalists which expect a devaluation and sell short (see the negative positions of fundamentalists in the unconstrained dynamics, left). In particular, when the price is closed to its fundamental value, the chartists' trading strategy is more profitable because their expectations are confirmed. Chartists dominate the market and the price is growing until the capital gain cannot compensate for the lower dividend yield. At this point, chartists start to suffer negative returns, while the short positions of the fundamentalists produce profits.
Eventually most of traders adopt the fundamentalist's trading strategy and the price falls down close to the fundamental value.
As soon the price starts to revert to the fundamental value, however, fundamentalists are prohibited from going short in the constrained dynamics because of the uptick rule (right column), and this triggers a further phase of rising prices. This happens several times, with the result of amplifying the price peak. At a certain point, when the price is very far away from its fundamental value, chartists suffer strong losses and the relative fraction of fundamentalists is almost one. The massive presence of fundamentalists pushes the price close to its fundamental value and the uptick rule cannot prevent this from happening. Indeed, if there are almost only fundamentalists, from the pricing equation we have that their demands must be equal to the supply of outside share, i.e. positive. When the price is close to its fundamental price chartists start to have a better performance and the story repeats. Despite this mispricing effect, the frequency of the price peaks is slowed down by the short selling restriction.
Similarly, Figure~(\ref{fig:BHtsn}) shows an example of time series on the negative attractor obtained for $\beta=4$ (here the unconstrained dynamics is quasi-periodic, see Figure~(\ref{fig:BHbif})).
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{Timeseries_BH_beta_4_negative}}
\caption{Negative (i.e., below fundamental) time series on the asymptotic regime of models (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:pr1g}) (left) and (\ref{ARED_model_uptick},~\ref{eq:pr1},~\ref{eq:pr1g}) (right) for $\beta=4$. Other parameter values as in Figure~(\ref{fig:BH_s})-case(B). The red dots indicate that chartists cannot have negative positions because of the uptick rule.}
\label{fig:BHtsn}
\end{figure}
In this case of negative price deviations, the chartists go short and have, on average, higher profits than fundamentalists. In the unconstrained dynamics chartists are predominant and drive prices down. This phenomenon is attenuated by the uptick rule, which limit the downward price movements and increases the performance of fundamentalists.
The frequencies of the negative peaks is slightly lowered by the short selling restriction, but they are more irregular due to the presence of chaotic dynamics as indicated by the positivity of the largest Lyapunov exponent, see Figure~(\ref{fig:BHbif}).
The different types of price dynamics, quasi-periodic for the unconstrained model and chaotic for the constrained one, are due to the different types of bifurcations through which the non-fundamental equilibrium $(\bar{x}^{(-)},\bar{m})$ losses stability, a Neimark-Sacker bifurcation in the first case and a border-collision bifurcation in the second. Indeed, as typical in piecewise linear model, at the border-collision bifurcation we have sudden transition from a stable fixed point to a fully developed {\it robust} (i.e., without periodic windows) chaotic attractor, see, e.g. \citep{Bernardo08}.
\subsection{Fundamentalists vs ROC traders}
\label{ssec:roc_n}
Figure~(\ref{fig:ROCbif}) reports the bifurcation analysis of models \eqref{ARED_model_no_uptick_2} (left) and \eqref{ARED_model_uptick} (right) with predictors \eqref{eq:pr1} and (\ref{eq:roc},c,d),
while Figure~(\ref{fig:ROCts}) shows the time series on the chaotic attractor obtained for $\beta=1.4$.
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{BifDigram_ROC_alpha_10_beta_104_5_g}}
\caption{Bifurcation diagrams of models (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:roc},c,d) (left) and (\ref{ARED_model_uptick},~\ref{eq:pr1},~\ref{eq:roc},c,d) (right) and exemplary state portraits projections in the plane $(x_{t},x_{t-1})$.
Parameter values: $\alpha=10$, $L=2$, other values as in Figure~(\ref{fig:BH_s})-case(B). The blue dots indicate that fundamentalists cannot have negative positions because of the uptick rule. The red dots indicate that chartists cannot have negative positions because of the uptick rule.}
\label{fig:ROCbif}
\end{figure}
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{TimeSeriesROC_beta_104_alpha_10}}
\caption{Time series on the asymptotic regime of models (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:roc},c,d) (left) and (\ref{ARED_model_uptick},~\ref{eq:pr1},~\ref{eq:roc},c,d) (right) for $\beta=1.4$. Other parameter values as in Figure~(\ref{fig:ROCbif}). The blue dots indicate that fundamentalists cannot have negative positions because of the uptick rule. The red dots indicate that chartists cannot have negative positions because of the uptick rule.}
\label{fig:ROCts}
\end{figure}
Here the fundamental equilibrium is globally stable up to the Neimark-Sacker bifurcation at $\beta\leq\beta_{\mathrm{NS}}$.
Interestingly, the price fluctuations showed by the non-stationary attractors originated for larger $\beta$ (quasi-periodic with periodic windows and later chaotic) have a remarkably smaller amplitude in the constrained dynamics, than in the unrestricted case.
In this sense, the uptick-rule shows a rather robust stabilizing effect, at least as long $\beta$ is not too large, i.e., traders are not fast enough in changing their beliefs to react to past performances.
This is also evident in the time series of Figure~(\ref{fig:ROCts}), where the short selling restriction also intensifies the frequency of the price peaks.
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth]{TimeSeriesROC_beta_405_alpha_10}}
\caption{Time series on the asymptotic regime of models (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1},~\ref{eq:roc},c,d) (left) and (\ref{ARED_model_uptick},~\ref{eq:pr1},~\ref{eq:roc},c,d) (right) for $\beta=4.5$. Other parameter values as in Figure~(\ref{fig:ROCbif}). The red dots indicate that chartists cannot have negative positions because of the uptick rule.}
\label{fig:ROCts2}
\end{figure}
\subsection{Economic discussion of the numerical results}
\label{sec:ed}
On the basis of the three declared objectives of the uptick rule, this subsection provides a discussion of the effects of this short selling regulation
in the light of the analytical and by numerical analysis reported in the previous Sections.
Let us start to discuss the scenario described by models \eqref{ARED_model_no_uptick_2} and \eqref{ARED_model_uptick} with predictors \eqref{eq:pr1v} and \eqref{eq:pr1g}
(fundamentalists vs chartists). We first consider the case of negative price deviations (the negative attractor in Figure~(\ref{fig:BHbif}) and the time series in Figure~(\ref{fig:BHtsn})), where we see that chartists take short positions
when prices fall, whereas fundamentalists take long positions
believing that the stock price will rise to reach the fundamental value
(see the unconstrained dynamics in the left column of Figure~(\ref{fig:BHtsn})).
It is important to point out that whenever an asset is undervalued (price below its fundamental value) it should be better to hold it rather than to sell it because the return is always positive: dividend yield outweighs the capital gain effect. Indeed, by going short chartists obtain negative excess returns, see the dynamic of the net profits $U_{2,t}$ in the last row of Figure~(\ref{fig:BHtsn}). However chartists do better than fundamentalists here because the latter are charged a cost, $\left(C\right)$, which is larger than the profit they obtain by holding the assets, compare the net profits of the two trading strategies ($U_{1,t}$ and $U_{2,t}$) again in the last row of Figure~(\ref{fig:BHtsn}). It follows that
chartists are predominant in the market, as indicated by the low value of the fraction $n_1$,
and their trading strategy causes downward movements of the stocks' price.
In the constrained dynamics (right column), the uptick rule limits the possibility to go
short for chartists. This helps to revert the stock prices toward the fundamental value. It follows a better performance for the fundamentalists and their presence in the market increases. As a result, the short selling restriction reduces the negative peaks reached by stock prices.
From this example it is clear the effectiveness of the regulation to meet the last two goals established by the SEC.
Moreover, from the analysis of the bifurcation diagram, it is interesting to note that the negative non-fundamental equilibrium $(\bar{x}^{(-)},\bar{m})$
looses stability in the constrained dynamics at a lower traders' adaptability, i.e., at a lower value of the parameter $\beta$, respect to the unconstrained dynamics.
This is due to the border collision bifurcation at $\beta=\beta_{\mathrm{BC}}^{(-)}$, that occurs before the Neimark-Sacker bifurcation at $\beta_{\mathrm{NS}}^{(-)}$ responsible of the instability in the unconstrained dynamics.
This might be mathematically interpreted as a destabilizing effect of the uptick rule. However, the chaotic fluctuations established after the bifurcation move the prices, on average, closer to the fundamental value with respect to the equilibrium deviation $\bar{x}^{(-)}$.
Considering the same model and predictors, by the time series of Figure~(\ref{fig:BHtsp}) it is
possible to notice that the dynamics of positive price deviations are
characterized (for $\beta>\beta_{\mathrm{NS}}^{(+)}$) by frequently financial bubbles.
These bubbles are enforced by chartists which overvalued the stock prices
in the upward trend and are attenuated by fundamentalists. In fact,
in these phases of rising prices fundamentalists go short driven by the
belief that the price will revert to its fundamental value in the next
period. This increases the supply of shares helping to curb rising prices. At a certain point of the upward trend, the fundamentalists' strategy
take over and prices are driven to the fundamental value of the stock.
During these upward trends of the market the uptick rule prevents the fundamentalists to take short positions, see blue dots in the fourth row,
right column of Figure~(\ref{fig:BHtsp}).
This phenomenon is also emphasized by empirical
analysis (see, e.g., \citep{AlexanderPeterson1999} and \citep{BoehmerJonesZhang2008}), and represents a flaw of
the regulation.
The result is
an increase of the amplitude of the financial bubbles. As a positive effect
of the regulation, it is possible to notice a reduction in the frequency of
occurrences of these bullish divergences.
The main point is that every time fundamentalists go short they force prices
to converge to the fundamental value. If it were possible to
discriminate between the beliefs of the traders, fundamentalists should not be
forbidden to go short in this situation. However, this is a very difficult task and it is not even so easy to correctly determine the
fundamental values of a risky asset in the real market. A possible solution could be
to allow short sales after a long period of rising prices.
This should avoid pushing up prices because fundamentalists (and
contrarians, but these traders are not taken into consideration in this
work) are driven out
of the market. Another possible solution could be to restrict short selling
only in cases of sharp and sudden falls in stock prices. This should produce
two effects, to let agents believing in the fundamental price go short when the
price increases, reducing positive oscillations of price deviations from
the fundamental value, and to reduce sharp drops in prices observed
when the stock market bubble breaks. An important step in this direction has
already been done, the SEC adopted Rule 201 which was implemented on
February 28, 201
\footnote
See, Securities Exchange Act Release No. 61595 (Feb. 26, 2010).}.
This new short selling regulation prohibits short selling operations if the value of
the stock decreases by more than 10\% in two consecutive trading sections.
Summarizing the analysis of the positive and negative price deviations from
the fundamental value for the model with predictors \eqref{eq:pr1} and \eqref{eq:pr1g} for the cases
of unrestricted and restricted short sales, we can conclude that the second goal
of the regulation is ensured, but some distorting effects produced by this rule are
observed, such as overvaluation of shares. Moreover, the short selling restriction can trigger distorting
mechanisms that support dynamics of overpricing, otherwise not feasible in
the long run. This is indicted by the presence of multiple attractors in the region of positive price deviations
in the bifurcation diagram in the right column of Figure~(\ref{fig:BHbif}).
With the intent to provide a more detailed and comprehensive description of the effects of the
uptick rule, the same asset pricing model has been analyzed with a
different non-fundamental predictor (the Smoothed-ROC predictor, see Subsection~\ref{ssec:pre}).
Differently from other trend-following indicators, the Smoothed-ROC
predictor is particularly useful to detect fast and short-term upward and
downward movements of the stock price and gives different trading signals
to agents than predictor \eqref{eq:pr1g}, such as gain and lose of speed in the trend
(see \citep{Elder1993}). In this case, we obtain an interesting and surprising
result, i.e., the uptick rule
helps to reduce the amplitude of price fluctuations, and causes an increase
of the frequency of oscillations
above and below the fundamental value, this is made clear by comparing the
time series of prices in the first row of Figure~(\ref{fig:ROCts}). The explanation of this lies on the higher degree of rationality (compared to the one assumed for chartists) of the non-fundamentalist agents that the use of the non-linear predictor (S-ROC predictor) implies. These agents are uncomfortable with extreme assessments of the value of the shares and when the shares are forced to be overvalued due to the short selling constraints, non-fundamental agents react and become more confidential in the fundamental price. This changes their trading strategies and, as a results, the amplitude of the price fluctuations decreases instead of increasing as might be expected. The choice of the value of $\alpha$ plays an important rule in this.
We can conclude that by using this
couple of predictors it is possible to observe, at least when evolutionary pressure, $\beta$, is not excessively large,
all the main goals of the uptick rule:
short selling restriction does not produce mispricing, prevents chartists from going short during downward price movements
to avoid reaching negative price variation peaks and prevents fundamentalists from
going short only in a downward price movement for positive price
deviations, reducing the speed of convergence to the fundamental value and
preventing sharps drops in prices. As a further observation, it is worth noticing that the uptick rule prevents extremely negative excess returns and at the same time makes the strategy of fundamentalists on the average more profitable, changing the beliefs of traders, compare $n_{1,t}$ and $n_{2,t}$ in Figure~(\ref{fig:ROCts}). Moreover, looking at the bifurcation diagram of Figure~(\ref{fig:ROCbif}) right-column, it is clear that for relatively high values of the intensity of choice, $\beta$, the uptick rule does not have any effect on the amplitude of price fluctuations. As revealed by the analysis of the time series in Figure~(\ref{fig:ROCts2}), this is due to the fact that for high levels of $\beta$ the market is always dominated by one trading strategy. It follows that the short selling constraint can apply either to a trading strategy adopted almost by any trader or to a trading strategy adopted by almost all the traders. In the first case the regulation does not have any effect on the price, in the second case it reverts the price toward the fundamental value with the results of reducing the frequency of market-bubbles but without reducing the amplitude of them. Compare the two panels in the first row of Figure~(\ref{fig:ROCts2}). For $\beta$ large enough, at each trading section there is only one type of trader that dominates the market and its demand of shares, being equal to the supply of outside share per trader, must be positive, then the uptick rule does not affect the dynamics of prices.
The conducted analysis points out that the uptick rule meets part of its
goals, but which of them often depends on the condition of the markets.
Moreover, the regulation can produce several side-effects which may be
different according once again to the market's conditions and investor's beliefs which may
strongly influence the effectiveness of the regulation itself. Due to these
findings, studying the impact of the uptick rule on financial markets does not seem to be
an easy
task. Nonetheless, it is possible to isolate some remarkable effects
regardless of market conditions and traders' beliefs. First, the uptick rule
ensures a
reduction of the downward market movements when the shares are undervalued, i.e. when the shares are priced below their fundamental value.
Second, the intensity of choice $\beta$ to switch predictors affects the
effectiveness of the short selling regulation. By using the bifurcation analysis, it
possible to observe that the effectiveness of the regulation fades away
increasing the value of $\beta$
(increasing $\beta$, agents tend to overreact to the market's information,
changing their beliefs quickly to react to past performances).
In other words, the switching destabilizing effect prevails over the regulation's
effects, i.e. when agents overreact to the differences in performance related to different beliefs, the regulation
does not affect the dynamics of stock prices. This is consistent with
many interesting empirical results testifying that there is no statistical
effect of the uptick rule on price fluctuations in
turbulent financial markets (see, e.g., \citep{DietherLeeWerner2009}).
As a final remark, it is important to clarify that the simple asset pricing model
used in this paper can reproduce only some possible ``stylized effects'' which are a direct
consequence of the regulation. However, in the real financial markets many more
different trading strategies and emotional actions are present, which can modify the effectiveness of the uptick rule.
\section{Conclusions and future directions}
\label{sec:cd}
In this paper, we have investigated the effect of the "uptick rule" on an asset price dynamics by means of an asset pricing model with heterogeneous, adaptive beliefs. The analysis has suggested the effectiveness of the regulation in reducing the downward price movements of undervalued shares avoiding speculative behavior whenever the market is characterized by not too many aggressive traders, i.e. when the agents' propensity of changing trading strategy is relatively low. On the contrary, when the agents have a high propensity of changing trading strategy according to the past trading performances, which is a sign of turbulent markets according to the model, the effects of the regulation tend to fade. As a side effect of the regulation, an amplification of the market bubbles in the case of overvalued shares is possible.
This work represents only a starting point. There are still several aspects
that can and deserve to be analyzed. First of all, it is interesting to
analyze the effect of the uptick regulation using the same asset pricing
model with a greater number of investor types, for example contrarians,
chartists and fundamentalists. In fact, there is empirical evidence
in the literature about the switch in trading style by short-sellers.
\citep{DietherLeeWerner2009} found that under the uptick rule most of the
short positions are opened by contrarians, on the contrary, when the uptick
rule is not imposed are chartists, the ones who prefer to go short
(see, e.g., \citep{BoehmerJonesZhang2008}). There is a hypothetical
explanation for this. Chartists take short positions in declining
price trends and the uptick rule makes this operation more difficult, on the
contrary the restriction does not effect the contrarians' short strategy.
They usually go short in upward price trend betting on a change in price
movement with the effect of stabilizing the market. Investigating the validity
of this hypothesis provides a better understanding of the issue.
The regulation should also be evaluated in the contest of the multi-assets
market to discover how the short selling restriction for one stock influences
the price of the others. It is reasonable to expect that traders will switch
to trade stocks that are not effected by the restriction in that specific
moment and this can produce effects on prices which are not easy to predict
without a deep analysis.
Another aspect that deserves to be investigated is the effect of the
regulation when there is a fraction of investors which does not change the
trading strategy (or belief) as in \citep{DieciForoniGardiniHe2006}. As highlighted in this
paper, the uptick rule loses its effectiveness due to a high propensity to switch trading strategy by agents. It follows that, if there are
constraints on the possibility to change trading strategy, we expect an increase of the effectiveness of the regulation and a reduction of unwelcome effects on price dynamics. Last but not least, the piecewise continuous model here
proposed can be easily adapted according to the new short selling regulation
imposed by the SEC, i.e. Rule 201. Comparing the two cases can help
to understand the pros and cons of the new regulation.
\renewcommand{\theequation}{A\arabic{equation}}
\setcounter{equation}{0}
\section{Appendix}\label{Ch4Appendix}
\begin{proof}[Proof of Lemma \ref{lm:fund}]
The existence of the fundamental equilibrium immediately follows by substituting $f_h(\mathbf{0})=0$ into Eqs.~\eqref{ARED_model_no_uptick} and~\eqref{ARED_model_no_uptick_2}.
For the definition of the associated eigenvalues, let us substitute Eq.~\eqref{ARED_model_no_uptick_n} written for $n_{h,t}$ into Eq.~\eqref{ARED_model_no_uptick_x}
(and Eq.~\eqref{ARED_model_no_uptick_2m} written for $m_{t}$ into Eq.~\eqref{ARED_model_no_uptick_2x}).
Then, we can consider $(x_{t-1},\ldots,x_{t-(L+2)})$ as the state variables for both models, and the Jacobian of the system at the fundamental equilibrium is equal to
\begin{linenomath}
$$
\left[
\begin{array}{c|c}
\begin{array}{ccccc}
\gamma_1 & \hspace{4mm} \gamma_2 \hspace{4mm} & \cdots & & \gamma_L\\
1 & 0 & \cdots & & 0\\
0 & 1 & \cdots & & 0\\
\vdots & & \hspace{-1mm}\ddots & & 0\\
0 & 0 & \cdots & \hspace{5mm} 1 \hspace{3mm} & 0
\end{array} &
\begin{array}{ccc}
\hspace{1mm} 0 \hspace{1mm} & & \hspace{1mm} 0 \hspace{1mm}\\
0 & & 0\\
0 & & 0\\
\vdots & & \vdots\\
0 & & 0
\end{array}\\
\hline
\begin{array}{ccccc}
\hspace{-0.2mm} 0 \hspace{1.9mm} & \hspace{4mm} 0 \hspace{5mm} & \cdots & \hspace{5mm} 0 \hspace{2mm} & \hspace{2mm} 1\\
\hspace{-0.2mm} 0 \hspace{1.9mm} & \hspace{4mm} 0 \hspace{5mm} & \cdots & \hspace{5mm} 0 \hspace{2mm} & \hspace{2mm} 0
\end{array} &
\begin{array}{ccc}
\hspace{1mm} 0 \hspace{1mm} & & \hspace{1mm} 0 \hspace{1mm}\\
1 & & 0
\end{array}
\end{array}\right],
$$
\end{linenomath}
which proves the result.
\end{proof}\\
\begin{proof}[Proof of Lemma \ref{lm:neq}]
At the equilibrium price deviation $\bar{x}$, we obviously have $\mathbf{x}_{t}=\bar{x}\mathbf{1}$ and, from Eq.~\eqref{ARED_model_no_uptick_x}, $\bar{x}$ must satisfy
\begin{linenomath}
$$
R = \sum_{h=1}^{H}n_{h,t}\hspace{0.2mm}f_h(\bar{x}\mathbf{1})/\bar{x}.
$$
\end{linenomath}
Being $n_{h,t}\in(0,1)$ for all $h=1,\ldots,H$, this is possible only if $f_h(\bar{x}\mathbf{1})/\bar{x}<R$ for some $h$ and $f_k(\bar{x}\mathbf{1})/\bar{x}>R$ for some $k\neq h$.
\end{proof}\\
\begin{proof}[Proof of Lemma \ref{lm:cha}]
\begin{enumerate}
\item
The uniqueness follows from Lemma \ref{lm:neq}, while the global stability from \eqref{ARED_model_no_uptick_2x}, which can be rewritten as
\begin{linenomath}
$$
x_{t} = \Frac{1}{R}\left(n_{1,t}\hspace{0.2mm}v + n_{2,t}\hspace{0.2mm}g\right)x_{t-1},
$$
\end{linenomath}
and contracts the deviation $x_{t}$ as $t$ goes to infinity.
\item
From Lemma \ref{lm:fund}, the non-zero eigenvalue associated to the fundamental equilibrium is
\begin{linenomath}
$$
\lambda^{(0)}=\gamma_1=\Frac{1}{2R}\left((1+\bar{m}^{(0)})\hspace{0.2mm}v + (1-\bar{m}^{(0)})\hspace{0.2mm}g\right)>0,\quad
\bar{m}^{(0)}=\tanh\left(-\beta\hspace{0.2mm}C/2\right).
$$
\end{linenomath}
Thus the fundamental equilibrium can loose stability only when $\lambda^{(0)}=1$ at a transcritical (or pitchfork) bifurcation
(being fixed point of model (\ref{ARED_model_no_uptick_2},~\ref{eq:pr1v},~\ref{eq:pr1g}) for any admissible parameter setting, it cannot disappear through a saddle-node bifurcation).
Solving $\lambda^{(0)}=1$ for $\beta$ gives $\beta_{\mathrm{TR}}$.
Evaluating Eqs.~\eqref{ARED_model_no_uptick} at the generic equilibrium $(\bar{x},\bar{m})$, solving Eq.~\eqref{ARED_model_no_uptick_2x} for $\bar{m}$, and equating the result to Eq.~\eqref{ARED_model_no_uptick_2m}, we get
\begin{linenomath}
$$
\bar{m}= 1 - 2\hspace{0.2mm}\Frac{R-v}{g-v} =
\tanh\left(-\Frac{\beta}{2}\left((-(R-1)\hspace{0.2mm}\bar{x}+a\hspace{0.2mm}\sigma^{2}s)\hspace{0.2mm}
\Frac{g-v}{a\hspace{0.2mm}\sigma^{2}}\hspace{0.2mm}\bar{x}+C\right)\right),
$$
\end{linenomath}
which solved for $\bar{x}$ gives $\bar{x}^{(\pm)}$.
Solving $\bar{x}^{(+)}=\bar{x}^{(-)}$ for $\beta$ gives $\beta_{\mathrm{LP}}$ and the equilibrium deviations $\bar{x}^{(\pm)}$ are defined only for $\beta\hspace{-0.2mm}>\hspace{-0.2mm}\beta_{\mathrm{LP}}$.
There are therefore no other equilibria and this concludes the proof of points (a), (LP), and (TR).
Note that $\beta_{\mathrm{LP}}=\beta_{\mathrm{TR}}$ when $s=0$ (the transcritical and saddle-node bifurcations coincide at a pitchfork bifurcation).
Substituting Eq.~\eqref{ARED_model_no_uptick_2m} written for $m_{t}$ into Eq.~\eqref{ARED_model_no_uptick_2x} and using $(x_{t-1},x_{t-2},x_{t-3})$ as state variables, the Jacobian of the systems at equilibria $(\bar{x}^{(\pm)},\bar{m})$ is given by
\begin{linenomath}
$$
\left[\begin{array}{ccc}
\gamma_1^{(\pm)} & \gamma_2^{(\pm)} & \gamma_3^{(\pm)}\\
1 & 0 & 0\\
0 & 1 & 0
\end{array}\right],
$$
\end{linenomath}
with
\begin{linenomath}
$$
\gamma_1^{(\pm)} = 1 + \Frac{\gamma}{R}\hspace{0.2mm}\bar{x}^{(\pm)},\quad
\gamma_2^{(\pm)} = -\gamma\hspace{0.2mm}\bar{x}^{(\pm)},\quad
\gamma_3^{(\pm)} = \Frac{\gamma}{R}\left(-(R-1)\hspace{0.2mm}\bar{x}^{(\pm)}+a\hspace{0.2mm}\sigma^{2}s\right),
$$
\end{linenomath}
and
\begin{linenomath}
$$
\gamma = \beta\hspace{0.2mm}\bar{x}^{(\pm)}
\Frac{(g-v)^2}{4\hspace{0.2mm}a\hspace{0.2mm}\sigma^{2}}\hspace{0.5mm}
\mathrm{sech}^{\hspace{-0.1mm}2}\hspace{-0.8mm}\left(\hspace{-0.5mm}-\Frac{\beta}{2}\left((-(R-1)\hspace{0.2mm}\bar{x}^{(\pm)}+a\hspace{0.2mm}\sigma^{2}s)\hspace{0.2mm}
\Frac{g-v}{a\hspace{0.2mm}\sigma^{2}}\hspace{0.2mm}\bar{x}^{(\pm)}+C\right)\hspace{-0.5mm}\right),
$$
\end{linenomath}
so the three associated eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$ are the roots of the characteristic equation
\begin{linenomath}
$$
\lambda^3-\gamma_1^{(\pm)}\lambda^2-\gamma_2^{(\pm)}\lambda-\gamma_3^{(\pm)}=0.
$$
\end{linenomath}
In particular, imposing $\lambda=1$ and solving the characteristic equation for $\bar{x}^{(\pm)}$, gives only zero and $x_{\mathrm{LP}}$ as solutions, so no other transcritical, saddle-node (or pitchfork) bifurcation is possible.
In contrast, imposing $\lambda_1=-1$ and taking into account that $\lambda_1+\lambda_2+\lambda_3=\gamma_1$, we get $\lambda_2+\lambda_3=2+\gamma\hspace{0.2mm}\bar{x}^{(\pm)}/R>2$ (note that $\gamma\hspace{0.2mm}\bar{x}^{(+)}>0$ and that $\gamma\hspace{0.2mm}\bar{x}^{(-)}\ge 0$ only vanishes at the transcritical bifurcation), so that equilibrium $(\bar{x}^{(\pm)},\bar{m})$ would be unstable at a period-doubling (flip) bifurcation.
However, both equilibria $(\bar{x}^{(\pm)},\bar{m})$ loose stability by increasing $\beta$, because the coefficient $\gamma_1$ linearly diverges with $\beta$
(the limit as $\beta\to\infty$ of the $\mathrm{sech}$ argument is finite and equal to $-\log((R\hspace{-0.2mm}-\hspace{-0.2mm}v)/(g\hspace{-0.2mm}-\hspace{-0.3mm}R))/2<0$), so the same does the sum of the eigenvalues.
Stability is therefore lost through a Neimark-Sacker bifurcation and this concludes the proof of points (b), (c), and (NS$^{(\pm)}$).
\item
For $g>2R-v$, the non-zero eigenvalue $\lambda^{(0)}$ associated to the fundamental equilibrium is larger than one for any $\beta>0$.
We also have $\beta_{\mathrm{LP}}<0$, $\beta_{\mathrm{TR}}<0$, and equilibria $(\bar{x}^{(\pm)},\bar{m})$ are defined for any $\beta>0$ with $\bar{x}^{(+)}>0$ and $\bar{x}^{(-)}<0$.
Similarly to point 2, they loose stability through a Neimark-Sacker bifurcation.
\item
In the limiting case $\beta\to\infty$, from Eq.~\eqref{ARED_model_no_uptick_2m} we have
\begin{linenomath}
$$
m_{t+1}=\left\{\begin{array}{rl}
1 & \text{if}\quad
(R\hspace{0.2mm}x_{t-1}-x_{t}-a\hspace{0.2mm}\sigma^{2}s)
\Frac{g-v}{a\hspace{0.2mm}\sigma^{2}}x_{t-1}>C,
\\
-1 & \text{if}\quad
(R\hspace{0.2mm}x_{t-1}-x_{t}-a\hspace{0.2mm}\sigma^{2}s)
\Frac{g-v}{a\hspace{0.2mm}\sigma^{2}}x_{t-1}<C.
\end{array}\right.
$$
\end{linenomath}
Starting at $x_0=\pm\eps$ with sufficiently small $\eps>0$ and $m_1=-1$, we therefore have $x_t=\pm\eps(g/R)^t$ (see Eq.~\eqref{ARED_model_no_uptick_2x}) as long as $m_{t+1}$ stays at $-1$.
Thus, $x_t$ diverges if
\begin{linenomath}
$$
\eps^2(g/R)^{2t-1}\left(\Frac{R^2}{g}-1 \mp a\hspace{0.2mm}\sigma^{2}s\hspace{0.2mm}\eps^{-1}(g/R)^{-t}\right)
\Frac{g-v}{a\hspace{0.2mm}\sigma^{2}}>C,
$$
\end{linenomath}
is never satisfied for increasing $t$, i.e., when $g>R^2$.
\end{enumerate}
\end{proof}\\
\begin{proof}[Proof of Lemma \ref{lm:cha_uptick}]
\begin{enumerate}
\item
At the fundamental equilibrium, the traders' demands are positive ($\bar{z}^{(0)}_1=\bar{z}^{(0)}_2=s$), so $(0,\bar{m}^{(0)})$ is an admissible fixed point of the unconstrained dynamics for any admissible parameter setting. Its local stability is therefore ruled by Lemma~\ref{lm:cha}.
The global stability for $1<g<R$ is a consequence of the following arguments.
First, the price deviation is contracted from period $(t-1)$ to period $t$ (i.e., $|x_t|<|x_{t-1}|$) whenever the unconstrained dynamics is applied (see Lemma~\ref{lm:cha}, point 1).
Second, the unconstrained deviation $x_t^{(0)}$ following $x_{t-1}$ is smaller than any of the constrained deviations $x_t^{(1)}$ and $x_t^{(2)}$ given by Eq.~\eqref{ARED_model_uptick_x} in region $Z_1$ and $Z_2$, respectively.
This is graphically clear from Figure~(\ref{fig:mc}), and is analytically shown by noting that
\begin{linenomath}
$$
x_t^{(h)}\hspace{-0.2mm}-x_t^{(0)} = -\Frac{a\hspace{0.2mm}\sigma^{2}}{R}\hspace{0.2mm}\Frac{n_h}{n_k}\hspace{0.2mm}z_{h,t}^{(0)}>0,\quad h=1,2,\; k\neq h
$$
\end{linenomath}
(see \eqref{eq:regz}, and recall that $z_{h,t}^{(0)}<0$ in region $Z_h$).
Third, from Eq.~\eqref{ARED_model_uptick_x} we get that $0<x_t^{(0)}<x_t^{(h)}<(g/R)x_{t-1}$ when $x_{t-1}>0$ and $x_t^{(0)}<x_t^{(h)}<(v/R)x_{t-1}<0$ when $x_{t-1}<0$.
Thus, the constrained dynamics in regions $Z_1$ and $Z_2$ also contracts the price deviation from period $(t-1)$ to period $t$.
Being $\bar{x}^{(+)}>0$, equilibrium $(\bar{x}^{(+)},\bar{m})$ can only collide with border $\partial Z_1$ (see \eqref{eq:bor}), at which $\bar{x}^{(+)}=\bar{x}_{\mathrm{BC}}^{(+)}$ (the expression for $\bar{x}_{\mathrm{BC}}^{(+)}$ can be easily obtained by solving the first equation in \eqref{eq:regz} with $z_{1,t}^{(0)}=0$ for $x_{t-1}$).
It is obviously admissible iff $\bar{x}^{(+)}\le\bar{x}_{\mathrm{BC}}^{(+)}$.
Depending on the parameter setting, equilibrium $(\bar{x}^{(-)},\bar{m})$ can be either positive or negative, and can therefore collide with both borders $\partial Z_1$ and $\partial Z_2$.
At the border $\partial Z_2$, $\bar{x}^{(-)}=\bar{x}_{\mathrm{BC}}^{(-)}$ (the expression for $\bar{x}_{\mathrm{BC}}^{(-)}$ is obtained by solving the second equation in \eqref{eq:regz} with $z_{2,t}^{(0)}=0$ for $x_{t-1}$), so that $(\bar{x}^{(-)},\bar{m})$ is admissible iff $\bar{x}_{\mathrm{BC}}^{(-)}\le\bar{x}^{(-)}\le\bar{x}_{\mathrm{BC}}^{(+)}$.
\item
If $R+v>2$ (case (a)), $\bar{x}_{\mathrm{LP}}$ from Lemma~\ref{lm:cha} is smaller than $\bar{x}_{\mathrm{BC}}^{(+)}$, so that equilibria $(\bar{x}^{(\pm)},\bar{m})$ are admissible at the saddle-node bifurcation.
The price deviation $\bar{x}^{(+)}$ increases as $\beta$ increases and reaches $\bar{x}_{\mathrm{BC}}^{(+)}$ at $\beta=\beta_{\mathrm{BC}}^{(+)}$ (collision with border $\partial Z_1$).
If $R+v<2$ (case (b)), equilibria $(\bar{x}^{(\pm)},\bar{m})$ are virtual at the saddle-node bifurcation.
The price deviation $\bar{x}^{(-)}$ decreases as $\beta$ increases and reaches $\bar{x}_{\mathrm{BC}}^{(+)}$ at $\beta=\beta_{\mathrm{BC}}^{(+)}$.
If $R+v=2$ (case (c)), then $\beta_{\mathrm{BC}}^{(+)}=\beta_{\mathrm{LP}}$.
Equilibrium $(\bar{x}^{(-)},\bar{m})$ collides with the border $\partial Z_2$ only if the limit of $\bar{x}^{(-)}$ as $\beta\to\infty$ is below $\bar{x}_{\mathrm{BC}}^{(-)}$.
This yields the condition on $s$ and the border collision at $\beta=\beta_{\mathrm{BC}}^{(-)}$ in point (d).
\item
For $g>2R-v$, $\bar{x}^{(+)}$ increases as $\beta$ increases, whereas $\bar{x}^{(-)}$ decreases, and their limiting value for $\beta\to\infty$ are as in point 2.
Equilibrium $(\bar{x}^{(+)},\bar{m})$ is always virtual, because its limiting value is above $\bar{x}_{\mathrm{BC}}^{(+)}$ for any admissible parameter setting.
Equilibrium $(\bar{x}^{(-)},\bar{m})$ becomes admissible at $\beta=\beta_{\mathrm{BC}}^{(-)}$ only if its limiting value is above $\bar{x}_{\mathrm{BC}}^{(-)}$, which gives the conditions at points (a) and (b).
\item
In the limiting case $\beta\to\infty$, $m_t$ switches between $\pm 1$, so that only one type of trader is present and the three options in Eq.~\eqref{ARED_model_uptick_x} give the same price deviation $x_t$. The result therefore follows from Lemma~\ref{lm:cha}.
\end{enumerate}
\end{proof}\\
\begin{proof}[Proof of Lemma \ref{lm:roc}]
From Lemma~\ref{lm:fund}, the characteristic equation associated with the nontrivial eigenvalues of the fundamental equilibrium is $\lambda^2-\gamma_1\lambda-\gamma_2=0$, with
\begin{linenomath}
$$
\gamma_1 = \Frac{1}{2R}\left((1+\bar{m}^{(0)})\hspace{0.2mm}v + 3(1-\bar{m}^{(0)})\right),\quad
\gamma_2 = -\Frac{1}{R}\hspace{0.2mm}(1-\bar{m}^{(0)}).
$$
\end{linenomath}
Note that the same characteristic equation is obtained for both predictors (\ref{eq:roc},b) and (\ref{eq:roc},c,d), and also for different choices of the confidence function \eqref{eq:roca}, as long as $\mathrm{\partial}\hspace{0.2mm}\alpha_{\text{ROC}}/\mathrm{\partial}\hspace{0.2mm}\text{ROC}|_{\text{ROC}=1}=0$.
The fundamental equilibrium is stable at $\beta=0$ (the Routh-Hurwitz-Jury test for 2nd-order polynomials requires $-\gamma_2|_{\beta=0}=1/R<1$ and $\gamma_1|_{\beta=0}=(v+3)/(2R)<1-\gamma_2|_{\beta=0}=(R+3+R-1)/(2R)$, which are readily satisfied).
As $\beta$ increases, transcritical and saddle-node (or pitchfork) bifurcations are not possible.
In fact, substituting $\lambda_1=1$ into the constraints:
\begin{linenomath}
$$
\lambda_1+\lambda_2=\gamma_1,\quad
\lambda_1\lambda_2=-\gamma_2,
$$
\end{linenomath}
and eliminating $\lambda_2$, we get the contradiction
\begin{linenomath}
$$
R=\Frac{v}{2}\hspace{0.2mm}(1+\bar{m}^{(0)}) + \Frac{1}{2}\hspace{0.2mm}(1-\bar{m}^{(0)})
$$
\end{linenomath}
(with left-hand side larger than one and right-hand side smaller than 1).
Similarly, we exclude flip bifurcations: imposing $\lambda_1=-1$ in the above constraints and eliminating $\lambda_2$, we get the contradiction
\begin{linenomath}
$$
-R=\Frac{v}{2}\hspace{0.2mm}(1+\bar{m}^{(0)}) + \Frac{5}{2}\hspace{0.2mm}(1-\bar{m}^{(0)}).
$$
\end{linenomath}
(with left-hand side positive and right-hand side negative).
To look for a Neimark-Sacker bifurcation, we impose $\lambda_1\lambda_2=1$ and $|\lambda_1\lambda_2|<2$.
Under the first condition, the second turns into $v\hspace{0.2mm}(1+\bar{m}^{(0)})<R$ that is always satisfied (recall that $\bar{m}^{(0)}<0$, see Lemma~\ref{lm:fund}).
Solving the first condition for $\beta$ gives $\beta_{\mathrm{NS}}$.
\end{proof}
\bibliographystyle{plainnat}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"http:\/\/www.gamedev.net\/index.php?app=forums&module=extras§ion=postHistory&pid=5019381","text":"\u2022 Create Account\n\n### #Actuall0k0\n\nPosted 10 January 2013 - 12:40 AM\n\nIt would be quite the feat if you could do this without vectors so not sure what you mean by \"not vector based\"\n\nTo keep this simple, albeit not the most efficient, I'd first make a system of equations between the line segment and the line in normal form (for each edge of the rectangle). \u00a0Ensure the t value is within 0 to 1 for the segment. \u00a0Then, find the scalar projection onto the edge of the rectangle. \u00a0It is possible that the segment resides completely in the rectangle, which means we need a point inside rectangle test. \u00a0If either point is inside the rectangle it is a positive intersection. It would look something like this:\n\nconst Vector2 RightPerp(const Vector2& v) {\nreturn Vector2(y, -x);\n}\n\nbool32 TestLineSegmentLineSegment(const Vector2& a0, const Vector2& a1, const Vector2& b0, const Vector2& b1) {\nconst Vector2 a = a1 - a0;\nconst Vector2 b = b1 - b0;\nconst Vector2 bPerp = RightPerp(b);\nconst Vector2 bPerpN = Normalize(bPerp);\nconst float bPerpNDist = -Dot(bPerpN, b0);\n\/\/ dot(pn, x) = -pd\n\/\/ pt = a0 + a * t\n\/\/ dot(pn, a0 + a * t) = -pd\n\/\/ t = (-pd - dot(pn, a0)) \/ dot(pn, a)\n\/\/ intersects a if t is in zero to one range\nconst float t = (-bPerpNDist - Dot(bPerpN, a0)) \/ (Dot(bPerpN, a));\nif (0.0f <= t && t <= 1.0f) {\n\/\/ this means the segment a is being intersected between a0 and a1\nconst Vector2 ptOnA = a0 + a * t; \/\/ this is the point on a of the intersection\n\/\/ find the scalar projection of the point onto b and if it is in range they intersect\nconst float bt = Dot(ptOnA - b0, b) \/ (Dot(b, b));\nif (0.0f <= bt && bt <= 1.0f) {\nreturn 1;\n}\n}\nreturn 0;\n}\n\nbool32 TestRectanglePoint(const Vector2& min, const Vector2& max, const Vector2& pt) {\nreturn (pt.x >= min.x && pt.y >= min.y && pt.x <= max.x && pt.y <= max.y);\n}\n\nbool32 TestRectangleLineSegment(const Vector2& min, const Vector2& max, const Vector2& a, const Vector2& b) {\n\nif (TestRectanglePoint(min, max, a) || TestRectanglePoint(min, max, b)) return 1;\nconst Vector2 maxXMinY = Vector2(max.x, min.y);\nconst Vector2 minXMaxY = Vector2(min.x, max.y);\nreturn (TestLineSegmentLineSegment(min, maxXMinY, a, b) || TestLineSegmentLineSegment(maxXMinY, max, a, b) || TestLineSegmentLineSegment(max, minXMaxY, a, b) || TestLineSegmentLineSegment(minXMaxY, min, a, b));\n}\u00a0\n\nGive this a try and let me know if you have any issues.\n\n### #2l0k0\n\nPosted 09 January 2013 - 12:57 AM\n\nIt would be quite the feat if you could do this without vectors so not sure what you mean by \"not vector based\"\n\nTo keep this simple, albeit not the most efficient, I'd first make a system of equations between the line segment and the line in normal form (for each edge of the rectangle). \u00a0Ensure the t value is within 0 to 1 for the segment. \u00a0Then, find the scalar projection onto the edge of the rectangle. \u00a0It is possible that the segment resides completely in the rectangle, which means we need a point inside rectangle test. \u00a0If either point is inside the rectangle it is a positive intersection. It would look something like this:\n\nconst Vector2 RightPerp(const Vector2& v) {\nreturn Vector2(y, -x);\n}\n\nbool32 TestLineSegmentLineSegment(const Vector2& a0, const Vector2& a1, const Vector2& b0, const Vector2& b1) {\nconst Vector2 a = a1 - a0;\nconst Vector2 b = b1 - b0;\nconst Vector2 bPerp = RightPerp(b);\nconst Vector2 bPerpN = Normalize(bPerp);\nconst float bPerpNDist = -Dot(bPerpN, b0);\n\/\/ dot(pn, x) = -pd\n\/\/ pt = a0 + a * t\n\/\/ dot(pn, a0 + a * t) = -pd\n\/\/ t = (-pd - dot(pn, a0)) \/ dot(pn, a)\n\/\/ intersects a if t is in zero to one range\nconst float t = (-bPerpNDist - Dot(bPerpN, a0)) \/ (Dot(bPerpN, a));\nif (0.0f <= t && t <= 1.0f) {\n\/\/ this means the segment a is being intersected between a0 and a1\nconst Vector2 ptOnA = a0 + a * t; \/\/ this is the point on a of the intersection\n\/\/ find the scalar projection of the point onto b and if it is in range they intersect\nconst float bt = Dot(ptOnA - b0) \/ (Dot(b, b));\nif (0.0f <= bt && bt <= 1.0f) {\nreturn 1;\n}\n}\nreturn 0;\n}\n\nbool32 TestRectanglePoint(const Vector2& min, const Vector2& max, const Vector2& pt) {\nreturn (pt.x >= min.x && pt.y >= min.y && pt.x <= max.x && pt.y <= max.y);\n}\n\nbool32 TestRectangleLineSegment(const Vector2& min, const Vector2& max, const Vector2& a, const Vector2& b) {\n\nif (TestRectanglePoint(min, max, a) || TestRectanglePoint(min, max, b)) return 1;\nconst Vector2 maxXMinY = Vector2(max.x, min.y);\nconst Vector2 minXMaxY = Vector2(min.x, max.y);\nreturn (TestLineSegmentLineSegment(min, maxXMinY, a, b) || TestLineSegmentLineSegment(maxXMinY, max, a, b) || TestLineSegmentLineSegment(max, minXMaxY, a, b) || TestLineSegmentLineSegment(minXMaxY, min, a, b));\n}\u00a0\n\nGive this a try and let me know if you have any issues.\n\n### #1l0k0\n\nPosted 09 January 2013 - 12:52 AM\n\nIt would be quite the feat if you could do this without vectors so not sure what you mean by \"not vector based\"\n\nTo keep this simple, albeit not the most efficient, I'd first make a system of equations between the line segment and the line in normal form (for each edge of the rectangle). \u00a0Ensure the t value is within 0 to 1 for the segment. \u00a0Then, find the scalar projection onto the edge of the rectangle. \u00a0It is possible that the segment resides completely in the rectangle, which means we need a point inside rectangle test. \u00a0If either point is inside the rectangle it is a positive intersection. It would look something like this:\n\nconst Vector2 RightPerp(const Vector2& v) {\nreturn Vector2(y, -x);\n}\n\nbool32 TestLineSegmentLineSegment(const Vector2& a0, const Vector2& a1, const Vector2& b0, const Vector2& b1) {\nconst Vector2 a = a1 - a0;\nconst Vector2 b = b1 - b0;\nconst Vector2 bPerp = RightPerp(b);\nconst Vector2 bPerpN = Normalize(bPerp);\nconst float bPerpNDist = -Dot(bPerpN, b0);\n\/\/ dot(pn, x) = -pd\n\/\/ pt = a0 + a * t\n\/\/ dot(pn, a0 + a * t) = -pd\n\/\/ t = (-pd - dot(pn, a0)) \/ dot(pn, a)\n\/\/ intersects a if t is in zero to one range\nconst float t = (-bPerpNDist - Dot(bPerpN, a0)) \/ (Dot(bPerpN, a));\nreturn (0.0f <= t && t <= 1.0f);\n}\n\nbool32 TestRectanglePoint(const Vector2& min, const Vector2& max, const Vector2& pt) {\nreturn (pt.x >= min.x && pt.y >= min.y && pt.x <= max.x && pt.y <= max.y);\n}\n\nbool32 TestRectangleLineSegment(const Vector2& min, const Vector2& max, const Vector2& a, const Vector2& b) {\n\nif (TestRectanglePoint(min, max, a) || TestRectanglePoint(min, max, b)) return 1;\nconst Vector2 maxXMinY = Vector2(max.x, min.y);\nconst Vector2 minXMaxY = Vector2(min.x, max.y);\nreturn (TestLineSegmentLineSegment(min, maxXMinY, a, b) || TestLineSegmentLineSegment(maxXMinY, max, a, b) || TestLineSegmentLineSegment(max, minXMaxY, a, b) || TestLineSegmentLineSegment(minXMaxY, min, a, b));\n}\u00a0\n\nGive this a try and let me know if you have any issues.\n\nPARTNERS","date":"2014-10-25 07:13:27","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\": 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\": 0.21280649304389954, \"perplexity\": 11562.429236260221}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1414119647865.10\/warc\/CC-MAIN-20141024030047-00288-ip-10-16-133-185.ec2.internal.warc.gz\"}"}
| null | null |
import { CommonModule } from '@angular/common';
import { NgModule } from '@angular/core';
import { FormsModule } from '@angular/forms';
import { MatButtonModule } from '@angular/material/button';
import { MatIconModule } from '@angular/material/icon';
import { MatInputModule } from '@angular/material/input';
import { DejaChildValidatorModule, DejaTextMetricsModule } from '@deja-js/component/core';
import { DejaNumericStepperComponent } from './numeric-stepper.component';
@NgModule({
imports: [
CommonModule,
DejaChildValidatorModule,
DejaTextMetricsModule,
FormsModule,
MatButtonModule,
MatIconModule,
MatInputModule
],
exports: [
DejaNumericStepperComponent
],
declarations: [
DejaNumericStepperComponent
]
})
export class DejaNumericStepperModule { }
export * from './numeric-stepper.component';
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,862
|
\section{Introduction}
Since its discovery,\cite{Novoselov2004ElectricFilms} graphene has attracted much attention in the scientific community.
Initially mainly for its remarkable electronic properties as well as its unprecedented mechanical qualities.\cite{Novoselov2012}
However, the plasmonic capabilities of this conveniently tunable material have also received great interest in recent years\cite{GarciadeAbajo2014GrapheneOpportunities,Bonaccorso2010GrapheneOptoelectronics,Grigorenko2012GraphenePlasmonics,Xiao:2016,Goncalves2016AnPlasmonics}
along with other two-dimensional materials.\cite{Mak2016PhotonicsDichalcogenides}
As ever smaller and more precise devices\cite{Cai2010AtomicallyNanoribbons,Narita2014SynthesisNanoribbons,Narita2015NewChemistry,Ruffieux2016On-surfaceTopology,Wang2016GiantEdges} are produced, it is important to obtain corresponding theoretical understanding of plasmons in graphene nanostructures.
For instance nanodisks and nanotriangles have both been investigated both theoretically\cite{Thongrattanasiri2013OpticalNanostructures,Christensen2014ClassicalStates,Wang2015PlasmonicNanotriangles,Settness:2017} and in experiments,\cite{Wang2016ExperimentalWindow} and more complex structures have also been studied.\cite{Zheng:2017}
The electronic properties of graphene nanostructure can be described on various levels of sophistication. Classically, it is a finite-size conductivity sheet. The simplest atomistic description is a tight-binding (TB) model for the electrons. The Dirac-equation continuum model for finite graphene structures is of intermediate complexity and describes low-energy electrons with linear dispersion being confined on finite graphene structures. Each of these three electronic models has its associated optical response, so that plasmonic excitations may also vary. While the tight-binding model is the most microscopic of them, it is important to know when the simpler Dirac or even the classical description suffices, and for which parameters the three models start to deviate from each other, and how important for optical properties are the different electronic edge terminations.\cite{Zarenia:2011a,Wettstein:2016a}
In this paper we present quantum mechanical calculations of graphene nanoribbons, with geometries as depicted in Fig.~\ref{fig:ribbongeometry},
\begin{figure}[htbp]
\centering
\includegraphics[width=3.5in]{ribbons.pdf}
\caption{The geometries of the zigzag and armchair graphene ribbons. The supercells are marked with the dashed rectangles.}\label{fig:ribbongeometry}
\end{figure}
in particular tight-binding calculations in the random-phase approximation (RPA). Important previous work on this topic includes theoretical contributions both for isolated ribbons and for arrays of them,~\cite{Thongrattanasiri2012QuantumPlasmons,Karimi2017PlasmonsNanoribbons,Shylau2015ElectronNanoribbons,Brey2007ElementaryNanoribbons,Villegas2013PlasmonNanoribbons,Andersen2012PlasmonNanoribbons,Wenger2016OpticalGraphene,Goncalves2016ModelingApproach} as well as experimental studies\cite{Xu2016EffectsNano-imaging,HuImagingNanoribbons,Fei2015EdgeNanoribbons,Brar2014HybridHeterostructures,Ju2011GrapheneMetamaterials,Yan2013DampingNanostructures} with ribbon widths down to 15 nm.\cite{BrarHighlyNanoresonators}
Furthermore, in a complementary analytical analysis we identify a scale invariance in the Dirac-equation model for graphene ribbons,\cite{Brey2006ElectronicEquation} a scale invariance that it shares with the classical model but not the tight-binding model. For the latter we identify the scale invariance as an emergent property. Thereby we obtain an illuminating overview for which parameters the Dirac-equation model can agree with the tight-binding models. Furthermore, we identify a scale invariant onset of quantum mechanical effects.
The article is structured as follows: In Sec.~\ref{sec:methods} we briefly discuss the TB model and its numerical evaluation, and the corresponding optical response function in terms of the electronic states. In Sec.~\ref{Sec:Analytical} we review the Dirac-equation model both for zigzag and armchair graphene ribbons, use the band structures to identify the onset of non-classical effects, and we identify the dimensionless scaling behavior property. In Sec.~\ref{sec:results} we compare our numerical TB calculations with our analytical predictions, and we conclude in Sec.~\ref{Sec:conclusions}. Detailed information can be found in two appendices.
\section{Numerical methods}\label{sec:methods}
\subsection{Tight-binding model}\label{sec:TB}
We describe the graphene ribbon in a nearest-neighbor tight-binding model with the Hamiltonian
\begin{align}
H = \sum_{<i,j>} -t(a^\dagger_ib_j + h.c.),
\end{align}
where the sum is over pairs of neighboring sites.
This model has proven useful for describing the band structure in a wide energy range around the Dirac point as the bands here are determined by interaction between the $p_z$ orbitals of the $sp^2$ hybridized carbon atoms.
A hopping value of $t = 2.8\unit{eV}$ is used between all interacting atoms as it has generally been found to give good results.\cite{CastroNeto2009}
We have used the smallest possible supercell which includes one row of atoms for the zigzag (ZZ) ribbons and two rows for the armchair (AC) ribbons as illustrated in Fig.~\ref{fig:ribbongeometry}.
The band structure and the states are found by direct diagonalization of the Hamiltonian with a $k$-point sampling of at least 5000 points in the Brillouin zone which has been found to give converged results in the subsequent evaluation of the optical response.
In Fig.~\ref{fig:bandstructure} we show the bands around the $K$ point for two 6 nm-wide ribbons, one with ZZ and the other with AC edge terminations.
The dots correspond to TB calculations, and the colors indicate the \emph{edginess} (defined in Appendix~\ref{app:edgyness}) of the corresponding states, with bright red corresponding to an edge state.
The figure also shows the continuous bands calculated within Dirac theory, as discussed in Sec.~\ref{Sec:Analytical}.
\begin{figure}[htbp]
\centering
\includegraphics[width=3.4in]{Bandstructure}
\caption{(Color online.) Band structures of 6 nm-wide zigzag and armchair ribbons. Full lines are the bands obtained with the Dirac theory, with red color indicating the edge states. The colored dots are the corresponding energies as calculated with the TB model. The color bar indicates how ``edge-like'' the states are, as defined in Eq.~(\ref{Eq:edginess}), with the value $\lambda = 1$ corresponding to a state localized on the edges and -1 to a state localized in the center of the ribbon. The armchair states are uniformly distributed across the ribbon and thus satisfy $|\lambda| \ll 1$ for all states.}\label{fig:bandstructure}
\end{figure}
\subsection{Response function and quantum plasmons}\label{Sec:Response_function}
The optical response of a quantum mechanical system is described in full by the dielectric function which relates to the non-interacting electron density-density response operator $\chi^0$ in the RPA as
\cite{Wunsch2006DynamicalDoping,Winther2015QuantumNanostructures}
\begin{align}
\epsilon(\vr,\vr';\omega) = \delta(\vr,\vr') - \Intopnl{\vr''} V(\vr,\vr'')\chi^0(\vr'',\vr';\omega), \label{eq:epsilon_rpa_def}
\end{align}
where $V$ is the Coulomb interaction. Following the method of Ref.~\onlinecite{Thongrattanasiri2012QuantumPlasmons}, the $\chi^0$ is calculated from the TB eigenstates, in the case of only vertical excitations, \emph{i.e.}\ neglecting intraband transitions, as
\begin{align}
\chi_{ij}^0(\omega) = \frac{2e^2}{\hbar} \frac{b}{2\pi}\int\limits^{\mathrm{BZ}}\!\!{\mathrm{d}k} \sum_{nm} f_{nm}\frac{a^{}_{in}a^*_{im}a^*_{jn}a^{}_{jm}}{\epsilon_{nm}+\hbar(\omega + i\eta)}. \label{eq:suscep_def}
\end{align}
Here the $i,j$ run over atomic sites and these indices are thus discretized equivalents to the $\vr'$ and $\vr''$ coordinates in Eq.~\eqref{eq:epsilon_rpa_def}, while $n,m$ label the eigenmodes at wave vector $k$.
Thus, $a_{in}$ is the weight of the $n^{\rm th}$ wavefunction on the $i^{\rm th}$ site (implicitly at wave vector $k$).
We have used the shorthand notation $\epsilon_{nm} = \epsilon_n - \epsilon_m$ for the energy difference and similarly for the Fermi filling factors $f$.
Damping is included phenomenologically through the parameter $\eta$, which we set to $1.6\unit{meV}$ throughout as in Ref.~\onlinecite{Thongrattanasiri2012QuantumPlasmons}.
The parameter $b$ is the width of the supercell in the periodic direction, see Fig.~\ref{fig:ribbongeometry}.
From the density-density response function we calculate the dielectric matrix $\epsilon(\omega) = I - V\chi^0(\omega)$. This expression involves the Coulomb interaction $V$ in real space, which is a subtle matter to handle,\cite{Thongrattanasiri2012QuantumPlasmons} both due to its long-range behavior and because of the divergence at zero distance, but it can be done (details in App.~\ref{app:coulomb}).
As shown in Ref.~\onlinecite{Andersen2012SpatiallyFirst-principles}, $\epsilon(\omega)$ can be written in a spectral representation of its eigenvalues and left and right eigenvectors as
\begin{align}\label{eq:epsilon}
\epsilon_{ij}(\omega) = \sum_n \epsilon_n(\omega)\phi_{n,i}(\omega)\rho^*_{n,j}(\omega),
\end{align}
where the $i,j$ are again site indices in the tight-binding basis and the $\epsilon_n(\omega)$ the eigenvalues; the right eigenvector $\phi_n$ is the induced field, and the left eigenvector $\rho_n$ is the induced charge of the plasmon. The zeroes of the real parts of $\epsilon_n(\omega)$ define the plasmonic modes. The plasmonic modes thus found agree well with peaks in the energy-loss function $-\Im \epsilon^{-1}(\omega)$ as measured in electron energy loss spectroscopy experiments, provided the frequency dispersion of the imaginary part of $\epsilon_n(\omega)$ is small. The above method to calculate quantum plasmons based on a tight-binding formalism will be applied to graphene ribbons in Sec.~\ref{Sec:Response_function}.
\section{Analytical model}\label{Sec:Analytical}
\subsection{Dirac theory for graphene ribbons}
Dirac theory is an approximate theory obtained
by linearizing the TB model in the $K$ ($K'$) valleys where infinite graphene exhibits its Dirac cones.
This allows one to get analytical insight into the band structure also of finite graphene structures.
For graphene ribbons, this was first done in the seminal paper by~Brey and Fertig\cite{Brey2006ElectronicEquation} and the method is also outlined in Castro Neto et al.~\cite{CastroNeto2009}
Here we first briefly review the Dirac theory, before presenting our new analytical insights and their comparison with full TB calculations.
In its essence, in the low-momentum limit the tight-binding Hamiltonian is approximated as
\begin{align}
H &= \hbar v_F (\tau_0 \otimes \sigma_x k_x + \tau_z \otimes \sigma_y k_y) \label{eq:hamilton}\\
&=\hbar v_F\begin{pmatrix}
0 & k_x - ik_y & 0 & 0 \\
k_x + ik_y & 0 & 0 & 0\\
0 & 0 & 0 & -k_x - ik_y\\
0 & 0 & -k_x + ik_y & 0
\end{pmatrix}, \notag
\end{align}
where $\tau_i$ and $\sigma_i$ are the Pauli spin-matrices belonging to the valley space and sub-lattice space, respectively.
For the eigenstates of the system we adopt the notation of Ref.~\onlinecite{Brey2006ElectronicEquation}: $[\phi^A,\phi^B,-\phi^{A'},-\phi^{B'}]^T$.
The Hamiltonian in Eq.~\eqref{eq:hamilton} is block diagonal, so we focus only on the upper left corner corresponding to the $K$~valley.
By applying $H$ twice to a state $[\phi^A,\,\phi^B]^T$ we find the relations
\begin{subequations}\label{eq:diraccoupledequations}
\begin{align}
\pa{k_x^2+k_y^2} \phi^A &= \tilde\epsilon_k^2 \phi^A \\
\pa{k_x^2+k_y^2} \phi^B &= \tilde\epsilon_k^2 \phi^B,
\end{align}
\end{subequations}
with $\tilde\epsilon = \epsilon/\hbar v_F$.
Now we specify the nanostructure to be a graphene ribbon in the $xy$-plane that is infinite in the $y$-direction and has width W. When Fourier transforming Eq.~(\ref{eq:diraccoupledequations}) in only the $x$-direction, \emph{i.e.}\ by replacing $k_x$ with $-i\partial_x$, a differential equation is obtained with the general solutions
\begin{align}\label{Eq:general_form_solutions}
\phi^X(x) = Ae^{\beta x} + Be^{-\beta x},
\end{align}
with $\beta = \sqrt{\smash[b]{k_y^2 - \tilde\epsilon^2}}$, and consequently $\epsilon = s\hbar v_F \sqrt{\smash[b]{k_y^2 - \beta^2}}$ where $s=\pm1$.
The eigenmodes of the Hamiltonian can be found analytically for both possible ribbon geometries of Fig.~\ref{fig:ribbongeometry} by imposing proper corresponding boundary conditions for their wavefunctions. These boundary conditions are different for zigzag and for armchair edge terminations.
\subsubsection{Zigzag edge termination}\label{sec:Zigzag}
In a ZZ ribbon the atomic structure terminates on an $A$ lattice site on one edge and on a $B$ site on the opposite edge, see Fig.~\ref{fig:ribbongeometry}. The proper boundary conditions are that $\phi^A(x=0) = \phi^B(x=W) = 0$.
By applying these boundary conditions to the general solution~(\ref{Eq:general_form_solutions}), the dispersion relation for the allowed states is found.
In a slightly different notation than in Ref.~\onlinecite{Brey2006ElectronicEquation}, it reads
\begin{align}
k_y = \frac{\beta}{\tanh(\beta W)}.\label{eq:tran_eq}
\end{align}
For fixed $k_{y}$, Eq.~\eqref{eq:tran_eq} has infinitely many solutions for imaginary $\beta=ik_n$ corresponding to the bulk modes, and at most one solution for $\beta = \kappa\in\mathbb{R}$ corresponding to an edge mode that falls off exponentially fast away from the edge.
It follows from the limit $\lim_{\kappa\to0} \kappa/\tanh(\kappa W) = 1/W$ that the edge states only exist for $k_y\geq 1/W$. This momentum cut-off has an associated energy cutoff $\varepsilon_\mathrm{cut} = \hbar v_F/W$.
The two types of solutions (bulk and edge modes) are shown in Fig.~\ref{fig:bandstructure} as full lines in black and red, respectively.
It is clear from the figure that the TB and Dirac methods to calculate the band structure give very similar energies in the vicinity of the $K$ point and that the analytically found edge states match almost perfectly with the ``edgy'' ($\lambda \approx 1$) states in TB.
From the analytical model we just determined the exact energy range where the edge states are found.
Given the great agreement between the two approaches, in the following, where we want to distinguish between bulk and edge states, we use the energy cut-off $\varepsilon_\mathrm{cut}$ from Dirac theory to classify the TB states as either bulk- or edge-like.
\subsubsection{Armchair edge termination}
As the termination of an armchair ribbon has a mix of $A$ and $B$ lattice sites, as depicted in Fig.~\ref{fig:ribbongeometry}, we demand that the sublattice wavefunction vanishes on both edges.
This results in a mixing of $K$ and $K'$ states through the equations\cite{CastroNeto2009}
\begin{align}
\begin{split}
0 &= \phi^{A/B}(x=0) + \phi^{A'/B'}(x=0), \\
0 &= e^{iKW}\phi^{A/B}(x=W) + e^{-iKW}\phi^{A'/B'}(x=W),
\end{split}
\end{align}
where $K = 4\pi/3\sqrt{3}a_0$ and $-K$ are the positions of the $K$-valleys in momentum space and $a_0$ is the interatomic distance in the graphene lattice.
These boundary conditions together with the general form of the solution~(\ref{Eq:general_form_solutions}) yield plane-wave states of the form $e^{ik_n x}$ with $k_n$ given by\cite{Brey2006ElectronicEquation}
\begin{align}
k_n = \frac{n\pi}{W} - \frac{4\pi}{3\sqrt{3}a_0} = \frac{2\pi[3n-2(N+1)]}{3\sqrt{3}a_0(N+1)},
\end{align}
with $n\in\mathbb{Z}$, and the corresponding eigenenergies $\tilde\epsilon_{n} = s\sqrt{\smash[b]{k_y^2+k_n^2}}$.
In the second equality we have expressed the width of the ribbon as $W = (N+1)a_0\sqrt{3}/2$, where $N$ is the number of atomic rows.
From this form it follows that every third ribbon, where $3n-2(N+1) = 0$ can be fulfilled, will be semi-metallic while the rest will have a band gap.
This way of defining the width gives the correct order of semi-metallic and semiconducting ribbons when compared with tight-binding calculations.
For example, the narrowest ribbon has $N=2$ and is just a one-dimensional chain with corresponding cosine band structure and no band gap. There should therefore exist a solution of the above condition for a semi-metal, $3n-2(2+1) = 0$, which is the case for $n=2$.
\subsection{Dimensionless scaling in Dirac theory}\label{Sec:Dirac_scaling}
An important property of the Dirac theory is a scale invariance of the ribbons:
if all equations are rewritten in dimensionless units where the energies are scaled in units of the Fermi energy $\varepsilon_F$, momenta in units of the Fermi momentum $k_F$, and the distances with the ribbon width $W$, then one finds that the only system-dependent parameter is the dimensionless parameter $\Lambda \equiv k_F W$.
This insight is very useful, since it allows us to identify effects that should exist across all widths of ribbons, provided that their respective Fermi levels are scaled accordingly and provided of course that the Dirac model is valid.
In dimensionless form, the governing equations for the ZZ ribbons thus become
\begin{subequations}
\begin{align}
K_y &= \frac{K_n}{\tan\pa{K_n\Lambda}}, & K_y &= \frac{K}{\tanh\pa{K\Lambda}} \\
E_n &= \sqrt{K_y^2+K_n^2}, & E_e &= \sqrt{K_y^2 - K^2} \\
\psi(\tilde x) &= \mathrlap{C_b e^{iK_y\Lambda\tilde y}
\mathrlap{\begin{pmatrix}
is \sin\pa{\tilde x\Lambda K_n} \\
\sin\pa{\pb{1-\tilde x}\Lambda K_n}
\end{pmatrix}}} \\
\phi(\tilde x) &= C_e e^{iK_y\Lambda\tilde{y}}
\mathrlap{\begin{pmatrix}
is \sinh\pa{\tilde{x}\Lambda K} \\
\sinh\pa{\pb{1-\tilde x}\Lambda K}
\end{pmatrix},}
\end{align} \label{eq:dimensionless_form}
\end{subequations}
with dimensionless momentum $K_y \equiv k_y/k_F$ and corresponding dimensionless momentum and energy of the bulk modes $\psi(\tilde x)$ are denoted by $K_n$ and $E_n \equiv \epsilon_n/\varepsilon_F$, and those for the edge modes $\phi(\tilde x)$ are called $K$ and $E_e$, and $\tilde x = x/W$ is the dimensionless lateral position in the ribbon.
For the AC ribbons we find that
\begin{align}
K_n = \frac{\pi[3n - 2\pa{N+1}]}{3\Lambda}.
\end{align}
Plots of the dimensionless band structures for the three different cases, zigzag, and semi-metallic and semiconducting armchair ribbons are shown in Fig.~\ref{fig:dimensionless_bands}.
We emphasize the large differences between the band structures in the low-energy regime; especially the different placement of the bottom of the lowest parabolic band, to which we will return in the following.
\begin{figure}
\centering
\includegraphics[width=3.4in]{dimensionless_bands}
\caption{(color online) The band structures of zigzag, semi-metallic, and semiconducting ribbons in dimensionless units. We expect non-classical behavior when the Fermi energy is close to or below the bottom parabolic band indicated with the gray dashed lines.}\label{fig:dimensionless_bands}
\end{figure}
One important prediction is that the scaling behavior of the electronic states will carry over to the plasmonic energies as well, as can be seen by inserting the Dirac states in their dimensionless form into the density-density response function, Eq.~\eqref{eq:suscep_def}, here shown with the bulk states:
\begin{align}\label{eq:X0scaleinvariantbulkonly}
\chi^0 \propto \frac{1}{\Lambda}\Intop{0}{\infty}{K_y}\sum_{nm}f_{nm}\frac{\abs{C^{}_bC'_b}S_{nm}(\tilde x, \tilde x', \Lambda)S_{mn}(\tilde x, \tilde x', \Lambda)}{E_n - E_m - (\nu -i\gamma)}
\end{align}
with
\begin{multline}
S_{nm}(x,x',\Lambda) = ss'\sin\pa{\tilde x\Lambda K_n}\sin\pa{\tilde x' \Lambda K_m} \\
-\sin\pb{(1-\tilde x)\Lambda K_n}\sin\pb{(1-\tilde x')\Lambda K_m},
\end{multline}
where $\nu + i\gamma = \hbar(\omega + i\eta)/\varepsilon_F$.
As the band structure in the Dirac model is approximated with an infinite cone the (dimensionless) response will only depend on $\Lambda$, rather than on $\varepsilon_{F}$ and $W$ separately. This scale invariance holds both when leaving out the edge states, as in Eq.~(\ref{eq:X0scaleinvariantbulkonly}), and when including them.
Given the great agreement in the band structures of the numerical tight-binding and the analytical Dirac models, we expect that also for TB calculations there will be parameters for which the scaled plasmonic energies will be scale-invariant for constant values of the parameter $\Lambda$, even though the scale invariance does not strictly hold in the TB model.
\subsection{Emergence of non-classicality}\label{sec:emergence_of_nonclassicality_predicted}
The quantum mechanical Dirac model for ribbons has a classical limit, and vice versa away from the classical limit we will identify the onset of non-classical behavior.
When there are many bands crossing the Fermi energy one would expect the system to behave classically.
On the other hand, for combinations of widths and Fermi energies where the Fermi surface is only crossed by a few states we are starting to probe the quantumness of the system and expect deviations from the classical regime.
As our heuristic measure, we take the bottom energy of the lowest parabolic band as the separation between the quantum and classical regimes.
Interestingly, from Fig.~\ref{fig:dimensionless_bands} this value differs for ZZ and AC ribbons, and it differs also for the semi-metallic and the semiconducting AC ribbons.
These different critical values $\Lambda_{c}$ at which we predict the classical-to-quantum behavior to occur can be determined analytically (and further below we will test them against numerical TB calculations).
\paragraph{For zigzag:}
By setting to zero the derivative of the energy with respect to the dimensionless momentum $K_y$, it is found that the sought bottom of the band occurs at $K_y=1/\Lambda$, corresponding to $k_y=1/W$.
By inserting this into the scaled expression for the band energies, we find that
\begin{align}
E_n = \sqrt{\Lambda^{-2}+K_n^2(\Lambda^{-1})} = \frac{\sqrt{1+\xi_n^2}}{\Lambda},
\end{align}
where $\Lambda K_n= \xi_n = \tan(\xi_n)$.
Looking for the solution where the Fermi energy crosses the bottom parabolic band, \emph{i.e.}\ $\epsilon_n/\varepsilon_F = E_n = 1$, it is found that the critical value is
\begin{align}
\Lambda_\mathrm{c}^{\rm zz} &= \sqrt{1 + \xi_1^2} \approx 4.6033. \label{eq:Lambda_critical_zz}
\end{align}
This $\Lambda_\mathrm{c}^{\rm zz}$ is a dimensionless number, and with this single number we predict with Dirac theory the emergence of quantum effects both in narrow ribbons at high Fermi levels and in wide ribbons with low doping.
As we will see below, this is indeed the value around which the dipole plasmon energies start to deviate from the classical results for zigzag ribbons.
\paragraph{For armchair:} For ribbons with armchair edge terminations, in the limit of many atoms, the band bottoms occur at $E_n = \pm n\pi/\Lambda$ for the semi-metallic ribbons and at $E_n \in \{\pm(3n+1)\pi/3\Lambda, \pm(3n+2)\pi/3\Lambda\}$ for the semi-conducting ribbons, with $n\in\mathbb{N}$.
\emph{I.e.}\
\begin{align}
\Lambda_\mathrm{c}^{\rm ac} =
\begin{cases}
\pi & \text{for semi-metallic AC ribbons}\\
\frac{\pi}{3} & \text{for semiconducting AC ribbons}
\end{cases}, \label{eq:Lambda_critical_ac}
\end{align}
Unlike for the ZZ ribbons, the band structures for AC ribbons are symmetric around the Dirac points and in that sense they are thus more like the bulk graphene bands. Combined with the lower value of $\Lambda_\mathrm{c}^{\rm ac}$, we expect classical behavior down to smaller values of $\Lambda$ for armchair ribbons.
\section{Classical plasmons}\label{sec:classical_plasmons}
It is naturally also possible to calculate the plasmons classically.
For the ribbon geometry this has already been done in different ways.\cite{Thongrattanasiri2012QuantumPlasmons,Christensen2015FromDimensions,Velizhanin2015GeometricArrays,Goncalves2016AnPlasmonics}
When combined with the continuity equation, the coupling between the potential $\phi(\vr)$ and the induced charge density $\rho(\vr)$ can be written as an integro-differential eigensystem of equations as
\begin{subequations}\label{eq:IDE_for_classical_plasmons}
\begin{align}
\zeta_n\phi_n(\vr) &= \frac{-1}{2\pi}\Intopnl{{}^2\vr}\frac{\nabla'\cdot\pb{f(\vr')\nabla'\phi_n(\vr')}}{\abs{\vr-\vr'}}, \\
\zeta_n &=\frac{2i\epsilon_0\omega_nW}{\sigma(\omega_n)}, \label{eq:Classical_IDE_eigenvalue}
\end{align}
\end{subequations}
where all coordinates and differential operators work in the 2D plane of the graphene.
The graphene is treated as being embedded in an $\epsilon=1$ material.
It has here been assumed that the conductivity is uniform inside the ribbon of width $W$, and vanishes outside:
\begin{align*}
\sigma(\vr,\omega) &= \sigma(\omega)f(\vr), \qquad\text{with} \\
f(\vr) &= \begin{cases}
1 \qquad \text{for $\vr$ inside the ribbon},\\
0 \qquad \text{for $\vr$ outside the ribbon}.
\end{cases}
\end{align*}
High-precision fits of the values of the eigenvalues $\zeta_n$ in Eq.~(\ref{eq:IDE_for_classical_plasmons}) are given by Christensen (Ref.~\onlinecite{Christensen2015FromDimensions}) for the first seven modes.
We have used these values in our classical calculations together with the low-temperature, local conductivity $\sigma(\omega)$ for bulk graphene. This conductivity can be derived, among other ways, from the Dirac model in the limit of infinitely wide ribbons or from the general expression of the bulk polarizability of graphene as found by Hwang and Das Sarma,\cite{Hwang2007DielectricGraphene} and by Wunsch \textit{et al.} \cite{Wunsch2006DynamicalDoping}
Here we just present the resulting expressions for the intraband and the interband contributions that together make up $\sigma(\omega)$:
\begin{subequations}\label{eq:sigma_intra_inter}
\begin{align}
\sigma_\mathrm{intra}(\omega) &= \frac{ie^2\varepsilon_F}{\pi\hbar^2(\omega+i\eta)}, \\
\sigma_\mathrm{inter}(\omega) &= \frac{e^2}{4\hbar}\pb{\frac{i}{\pi}\ln\abs{\frac{2\varepsilon_F - \hbar\omega}{2\varepsilon_F+\hbar\omega}} + \Theta(\hbar\omega - 2\varepsilon_F)},
\end{align}
\end{subequations}
where $\Theta$ is the Heaviside step function.
By combining Eqs.~(\ref{eq:IDE_for_classical_plasmons}) and (\ref{eq:sigma_intra_inter}) we can find the plasma energies as a function of the ribbon width.
For our purposes it is essential to realize that Eq.~\eqref{eq:Classical_IDE_eigenvalue} can be rewritten in dimensionless variables as $\sigma(\nu_n)\zeta_n = 2i\epsilon_0\hbar v_F\nu_n\Lambda$, with the dimensionless plasmon energy $\nu_n = \hbar\omega_{n}/\varepsilon_F$ and again $\Lambda = k_F W$.
This insight turns out to be quite practical, because it is sufficient to calculate the connection between $\nu_n$ and $\Lambda$ only once to obtain the plasmon energies for all combinations of widths and Fermi momenta that satisfy $\Lambda = k_FW$. Moreover, in Sec.~\ref{Sec:Dirac_scaling} we saw that the Dirac model has the same scale invariance. So we find that the scaling property holds both inside and outside the classical regime, as long as Dirac theory is accurate. We will test the latter by comparing Dirac and classical theories with tight-binding calculations in the next section.
\section{Numerical and analytical results compared}\label{sec:results}
We present two comparisons: quantum versus classical plasmons in Sec.~\ref{sec:quantumclassical}, and properties of atomistic (TB) quantum plasmons versus those of continuum (Dirac) quantum plasmons in Sec.~\ref{sec:emergentscaleinvariancecomparison}.
\begin{figure*}
\centering
\subfloat[c][Zigzag]{\includegraphics[width=3.4in]{width_energy_plot_zz}\label{fig:widths_zz}}
\subfloat[c][Armchair]{\includegraphics[width=3.4in]{width_energy_plot_ac}\label{fig:widths_ac}}
\caption{(Color online)
Scaled plasmon energy as a function scaled ribbon width $\Lambda$. The value $\Lambda = 5$ corresponds for example to a width of $16.4\unit{nm}$ for $\varepsilon_F = 0.2 \unit{eV}$.
\emph{(a):} $\Lambda$ is varied by changing $W$ while keeping the Fermi energy fixed at either 0.4~eV (triangles) or 0.8~eV (dots). The vertical dashed line corresponds to $\Lambda_\mathrm{c}^\mathrm{zz}\simeq 4.6$.
The open symbols in the lower panel correspond to calculations of plasmon energies where edge states were removed from the calculation.
\emph{(b), upper panel:} The intraband plasmons of AC ribbons, both the classical prediction and the quantum plasmon predictions for semiconducting and for semimetal ribbons. Blue and red dashed vertical lines correspond to $\Lambda_\mathrm{c}^{ac} = \pi/3$ and $\pi$, respectively. \emph{(b), lower panel:} As in the upper panel, but now including all transitions. The small symbols are used for the peaks in the loss spectrum that are not associated with an actual plasmon defined as $\mathrm{Re}(\epsilon_n) = 0$.
}\label{fig:contribs_and_widths}
\end{figure*}
\subsection{Quantum versus classical plasmons}\label{sec:quantumclassical}
Here we systematically investigate the range of validity of the classical description for graphene ribbons, by comparing with TB quantum calculations. In particular, we will test the heuristic value of the various $\Lambda_\mathrm{c}$ that we identified in Sec.~\ref{sec:emergence_of_nonclassicality_predicted} for characterizing the emergence of non-classical behavior in a scale invariant way.
In Sec.~\ref{Sec:Response_function} we outlined how one can identify quantum plasmons of nanostructures within a tight-binding formalism, and here we apply this approach to graphene ribbons. The calculation of the corresponding classical plasmons was described in Sec.~\ref{sec:classical_plasmons}.
Guided by the scaling properties of the Dirac and classical models, in Fig.~\ref{fig:contribs_and_widths} we present the plasmonic energies as a function of the dimensionless variable $\Lambda$.
The figure shows a comparison of the scaled plasmonic energy as calculated with the TB model of Sec.~\ref{sec:TB} and in the classical model for both ZZ and AC ribbons, and when considering only the intraband contribution (upper panels of Fig.~\ref{fig:contribs_and_widths}) or all transitions (lower panels).
By ``intraband'' we mean that we only include eigenstates with energies above the cut-off energy $\varepsilon_\mathrm{cut} = \hbar v_F/W$ for the edge states for zigzag ribbons and above zero energy for armchair ribbons, which corresponds to only considering intraband transitions in a classical, wide-ribbon limit.
For ribbons of finite widths, the transitions are intraband transitions in the sense that the bands in the upper cone are size-constriction-foldings of the infinite graphene conduction band, although the actual transitions do occur between bands of the ribbon.
We see that for large values of $\Lambda$, the classical and all TB calculations agree across all four panels.
There is no visible effect of neither edge terminations nor other quantum effects there.
Furthermore, the TB calculations for different Fermi levels agree very well as predicted from the scaling of the Dirac model.
For smaller values of $\Lambda$ the plasmon energies as calculated by the TB model start to depart from the classical values.
For zigzag ribbons, Fig.~\ref{fig:contribs_and_widths}(a) constitutes a confirmation of our prediction in Eq.~(\ref{eq:Lambda_critical_zz}) that this onset of quantum behavior occurs at $\Lambda_\mathrm{c}\simeq 4.6$, the point at which the lowest of the parabolic bands of the zigzag ribbons crosses the Fermi level. This same onset is seen both in the ``Drude-like'' case (Fig.~\ref{fig:contribs_and_widths}(a), upper panel) and with all transition included (Fig.~\ref{fig:contribs_and_widths}(a), lower panel).
Another important feature of Fig.~\ref{fig:contribs_and_widths}(a) is that the tight-binding plasmon energies for $\varepsilon_{F} = 0.4\unit{eV}$ and $0.8\unit{eV}$ are indeed quite close to each other in the chosen dimensionless units, and closer to each other than to the classical plasmon curves. Dirac theory predicts that the two quantum plasmon calculations would coincide exactly, and the tight-binding calculations confirm that the scale invariance of Dirac theory indeed holds approximately. A better overview and insight when scale invariance holds in TB calculations will be presented in Sec.~\ref{sec:emergentscaleinvariancecomparison} below.
The dotted line in the lower panel shows the interpolated data from calculations of a $9\unit{nm}$ wide ribbon at varying Fermi energy and provides the best guess, given the calculations that have been done, of the behavior of arbitrarily wide ribbons where the plasmon energies have converged with respect to the number of bands.
This will be explored further in the following section.
Comparing the results for the $\varepsilon_F = 0.4\unit{eV}$ and $\varepsilon_F = 0.8\unit{eV}$ ribbons we see that lowering the Fermi energy, which for constant $\Lambda$ corresponds to widening the ribbons, moves the points closer to the dotted line, as expected.
By excluding the zigzag edge states in the evaluation of $\chi^0$ (open symbols in the lower panel of Fig.~\ref{fig:contribs_and_widths}\subref{fig:widths_zz}), we find a significant plasmon redshift in the quantum regime. In other words, edge states of zigzag nanoribbons contribute with a significant blueshift of the plasmon energies in the quantum regime, while they have hardly any impact on the energy in the classical regime above $\Lambda_\mathrm{c}^\mathrm{zz}$.
This effect of edge states becomes even more evident by directly plotting the energy shift as in Fig.~\ref{fig:role_of_edges}. Clearly, for zigzag ribbons the edge states do not affect the plasmon energies for $\Lambda>\Lambda^\mathrm{zz}_\mathrm{c}$ and give rise to a blueshift for $\Lambda<\Lambda^\mathrm{zz}_\mathrm{c}$.
The found blueshift is in stark contrast to the results for graphene disks\cite{Christensen2014ClassicalStates} and triangles\cite{Wang2015PlasmonicNanotriangles} in which the zigzag edge states are found to give rise to a net redshift of the plasmon energies.
Back to our ribbons, for $\Lambda<1$ the Fermi level crosses the edge state and the evaluation of the edge-state contribution in the manner described above becomes meaningless.
\begin{figure}
\centering
\includegraphics[width=3.5in]{role_of_edges}
\caption{(Color online) Scaled plasmon energies in the presence of edge states minus scaled plasmon energies when neglecting the edge states, as a function of the scaled ribbon width $\Lambda$, for two fixed values of the Fermi energy. }\label{fig:role_of_edges}
\end{figure}
Having discussed quantum-classical transitions for zigzag ribbons, we now return to Fig.~\ref{fig:contribs_and_widths} and study armchair ribbons in part~(b).
The picture is slightly different for armchair ribbons as they exist as either semiconducting (sc.) or semi-metallic (sm.).
When including only intraband transitions, the scaled plasmon energies follow the classical behavior rather closely across the entire range, except for a single outlier.
As discussed above, because of the symmetry around the $K$ point of the armchair band structure, we do not expect the same kind of quantum-classical transition as for zigzag ribbons.
In the lower panel of Fig.~\ref{fig:contribs_and_widths}\subref{fig:widths_ac} we have split the ribbons into the two types.
The vertical, dashed lines indicate the position of the band bottom in the appropriate color.
As expected, the deviation from classical results starts at lower $\Lambda$ than previously for the zigzag ribbons.
The small symbols in the lower panel of Fig.~\ref{fig:contribs_and_widths}(b) denote peaks in the loss spectrum that are not associated with real plasmons as there is no simultaneous crossing of the real part of the dielectric eigenvalues with zero.
For the semiconducting ribbons the plasmon cease to exist when the Fermi energy crosses the lowest parabolic band at $\Lambda=\pi/3$.
For the semi-metallic ribbons the plasmons cease to exist earlier, namely already below $\Lambda = \pi$.
There seem to be an exception with the red square just above $\Lambda=2$ (which lies beneath a small, red triangle), but as the TB calculations are done for room temperature $k_bT\approx 25\unit{meV}$, there will still be a finite population of electrons in the bottom parabolic band for this point.
For the smallest values of $\Lambda$ for which plasmons still exist, the positions of the main dipole plasmon peaks become increasingly hard to locate, resulting in an increased scatter of the data points, as also reported elsewhere.~\cite{Christensen2014ClassicalStates,Thongrattanasiri2012QuantumPlasmons}
\subsection{Emergent scale invariance for plasmons}\label{sec:emergentscaleinvariancecomparison}
\begin{figure*}[htbp]
\centering
\includegraphics[width=6.5in]{constant_lambda}
\caption{(Color online) The plasmon energy scaled with the Fermi energy for constant $\Lambda$ at varying ribbon widths, for both ZZ (panel a) and AC ribbons (b). Dotted horizontal lines in equal colors are the corresponding classical plasmon energies. In the AC panel the data is split into semi-metallic (sm.) and semiconducting (sc.) ribbons. This distinction is only important for the low $\Lambda$ structures, as seen. The bars show the width of the plasmon peak in the loss spectrum. All displayed data points have $\varepsilon_F < 2.0\unit{eV}$.}\label{fig:constant_lambda}
\end{figure*}
In general, the tight-binding model for graphene ribbons does not have the same scale invariance that we found both for Dirac theory and for classical plasmonics, as the TB band structure does not consist of an infinite Dirac cone.
This follows from the fact that, due to the infinite cone shape, the band structures for two different ribbon widths in the Dirac description are related be a simple scaling transformation while this is not the case for the more complex TB band structure.
But since the low-energy bands calculated with TB and with Dirac theory agree so well, at least for the parameters of Fig.~\ref{fig:bandstructure}, the scale invariance will be an emergent property of the TB model, valid only in part of the parameter space spanned by $\{\varepsilon_F, W\}$.
Only in that subspace can classical and/or Dirac theory be expected to agree with TB calculations.
As a test of the proposed scale invariance we conduct a range of calculations where $\Lambda$ is held constant while the widths of the ribbons are varied, so doubling the size of the ribbon goes hand in hand with halving the Fermi energy.
As previously stated, we expect the scaled plasmon energy to tend towards a constant when the ribbons get wider and the Dirac model becomes a better description. It is less clear how fast the limit will be reached.
When the Fermi energy is above $2.0\unit{eV}$ we are well out of the linear regime of the bands and do not expect the Dirac scaling to work anymore.
For the armchair ribbons we distinguish between semiconducting and semi-metallic ones, as this should have an impact for small values of $\Lambda$ where the Fermi energy is close to the difference in the band structures.
As one of our main results we present in Fig.~\ref{fig:constant_lambda} how the TB plasmon energies converge as ribbon widths are increased. For $\Lambda \gg 1$ the plasmon energies quickly converge for larger widths to a value that differs little from the classical plasmon energy.
But it is important to notice that the wide-ribbon limits in this figure do not automatically coincide with the classical limit, as one might expect: for $\Lambda$ not much larger than unity, there is a clear discrepancy between the converged energies of the TB plasmons and the classical plasmons. Wherever the TB curves in Fig.~\ref{fig:constant_lambda} have become (almost) horizontal, the scale invariance that holds exactly for Dirac and classical plasmons has also emerged for TB quantum plasmons.
The bending of the curves for smaller widths illustrates the shortcomings of the scalability of the Dirac model: For it to hold exactly, we would need infinitely many bands in the band structure, but as the number of atoms in the full-width supercell decreases as $W$ is reduced (recall Fig.~\ref{fig:ribbongeometry}), we will get fewer bands instead and thus a deviation from the converged constant plasmon energy as obtained for wide ribbons.
In Fig.~\ref{fig:constant_lambda} we also see that the AC plasmon energies in general are closer to the classical predictions than the ZZ plasmon energies, as could also be extracted from Fig.~\ref{fig:contribs_and_widths}.
In Fig.~\ref{fig:width_lambda} we display the same data for zigzag ribbons as in Fig.~\ref{fig:constant_lambda} but in a complementary way, now as a function of width and $\Lambda$. We obtain a surface plot of the scaled plasmon energy, where data points have been cubicly interpolated to get a smooth surface.
\begin{figure}[!htbp]
\centering
\includegraphics[width=3.4in]{width_lambda_heatmap}
\caption{(color online) The scaled plasmon energy as a function of the width and $\Lambda$ for zigzag ribbons. The white lines indicate paths with constant (absolute) plasmon energy. The gray area corresponds to structures with $\varepsilon_F>2\unit{eV}$ where we expect to be outside the linear regime of the bands. For larger widths the contours start to converge as expected from Dirac theory. The departure from horizontal lines is a signature of having only a finite number of electronic states.
The right $y$-axis shows the classical results for comparison.}\label{fig:width_lambda}
\end{figure}
From this view, we also see the convergence of the scaled plasmon energies that deviates significantly for the classical values displayed on the right $y$-axis.
For $\varepsilon_F>2.0\unit{eV}$ the Fermi energy is outside the linear part of the band structure and we find large deviations from the Dirac model as can be seen in the gray area of the plot.
By multiplying with the corresponding Fermi energy surface $\varepsilon_F = \hbar v_F\Lambda/W$ we calculate lines of constant plasmon energy (the white lines) thus conveniently providing a continuous range of options in parameter space to obtain a specific plasmon energy.
The bending of the white lines also reveals the offset from the classical behavior:
For large values of $\Lambda$ the system is well described when only including the classical intraband Drude term.
This results in a scaling of the scaled plasmon energy with $\Lambda^{-1/2}$ leading to the plasmon energy scaling as $\hbar\omega \propto \Lambda^{1/2}$.
The intraband divergence at $\Lambda = 0$ is quenched due to the screening of the interband transitions when the latter are included.
Looking at the data in Fig.~\ref{fig:width_lambda}, for a constant width, when going to smaller values of $\Lambda$, around $\Lambda_\mathrm{c}$ the plasmon energies start to increase again showing that the scaled plasmon energy must increase faster than as $\Lambda^{-1}$.
\section{Conclusions}\label{Sec:conclusions}
Using tight-binding calculations and inspired by Dirac theory, we identify multiple interesting effects in graphene nanoribbon plasmons:
As a first main result, for both armchair and zigzag ribbons an emerging non-classical scale-invariant behavior of the plasmon energies has been predicted and confirmed to exist also in TB calculations, even though the scale invariance does not hold strictly in the mathematical sense in TB. For ribbons wider than 5~nm, we illustrated in Fig.~\ref{fig:constant_lambda} that the scale invariance effectively holds for the energies considered, and better so for higher Fermi energy (which we kept smaller than 2.0 eV). The scale invariant horizontal curves that the TB calculations converge to generally have non-classical limiting values. This is the realm where Dirac theory can be accurate, and only in the limit $\Lambda \equiv k_{F} W \gg 1$ do our TB plasmon energies agree nicely with those of classical plasmons.
An experimental realization that comes close to the non-classical regime is the work in Ref.~\onlinecite{BrarHighlyNanoresonators} where the plasmonic mode of a 15 nm ribbon with $\epsilon_F\approx 0.4\unit{eV}$ has been measured.
This is still within the classical regime as $\Lambda \approx 9.3 > \Lambda_\mathrm{c}$, but lowering the Fermi energy to around $0.1\unit{eV}$ corresponding to $\Lambda \approx 2.3$ should reveal new quantum effects for both zigzag and armchair edge terminations.
As our second main result, we have related the energy of the bottom parabolic band at the $K$ points to the onset of the deviation from the classical model and calculated these energies analytically using the Dirac model. Here again, we find that the agreement between our heuristic analytical estimates and numerical calculations holds quite well and in a scale-invariant way, i.e. the analytical estimates describe the onset of non-classical plasmonics both for narrow ribbons with higher Fermi energies and wider ribbons with lower $\varepsilon_F$. Spectral differences between quantum and classical plasmons emerge slightly earlier for zigzag than for armchair ribbons (i.e for larger $\Lambda$, or already for wider ribbons at equal Fermi energy).
Third, for armchair nanoribbons we observe the disappearance of the plasmons at two different low values of the scaled ribbon width $\Lambda$, dependent on whether the ribbons are semiconducting or semi-metallic in their neutral state.
Fourth, for zigzag ribbons we have provided a convenient way of predicting absolute plasmon energies from the iso-frequency curves in Fig.~\ref{fig:width_lambda}.
Finally, we revealed how the edge states of nanoribbons contribute with a significant blueshift of plasmon energies, in contrast to reported redshifts for other graphene nanostructures.
\section*{Acknowledgments}
We would like to thank Thomas Christensen, Johan R. Maack and P. André D. Gon\c{c}alves for stimulating discussions.
This work was supported by the
Danish Council for Independent Research--Natural Sciences (Project 1323-00087).
The Center for Nanostructured Graphene is sponsored by the Danish National Research Foundation (Project No. DNRF103).
N.~A.~M. is a VILLUM Investigator supported by VILLUM FONDEN (grant No. 16498).
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Transrapid är ett tyskkonstruerat höghastighetståg som svävar genom att använda magnetisk levitation (se maglev) istället för klassiska stålhjul på stålräls. Systemet är utvecklat av ett konsortium under ledning av företagen Siemens AG och ThyssenKrupp AG. Systemet är baserat på ett patent från 1934, planering för det första Transrapid-systemet började under 1969. En testbana för systemet, Transrapid-Versuchsanlage Emsland, i Landkreis Emsland i norra Tyskland, färdigställdes 1987. År 1991 var systemet tekniskt redo för användning och godkändes av den tyska statsjärnvägen (Deutsche Bundesbahn) i samarbete med välrenommerade universitet.
Dess nuvarande tillämpning-färdiga versionen, Transrapid 09, är konstruerad för max-farten 500 km/h och ger acceleration och retardation på cirka 1 m/s2.
År 2004 färdigställdes den första kommersiella tillämpningen av systemet. Den kinesiska banan Shanghai Maglev Train ansluter tunnelbanenätet till Shanghai Pudong International Airport via en 30,5 km lång bana. Transrapids system har ännu inte tagits i bruk på en långdistanslinje, men en anslutning till Hangzhou börjar byggas 2010 och är klar 2014.
Transrapid teknik kräver en högre investeringskostnad är dyrare per mil att bygga än konventionella tågsystem,, men använder mindre energi under drift, och har betydligt lägre underhållskostnader.
Teknik
Levitation
Maglev har inget löpverk (d.v.s. inga hjul eller axlar) samt heller ingen mekanisk transmission (växellådor) eller konventionella strömavtagare. Tågen rullar inte, utan svävar mellan två ytor, bestående av linjära kedjor av elektromagnetiska spolar - en sida i fordonet, den andra på banan - som fungerar som en magnetisk dipol. Elektroniska system för att mäta avståndet 100 000 gånger per sekund ser till att det fria utrymmet mellan spolar monterade på undersidan av banan och den magnetiska delen av fordonet som omsluter banans kanter förblir konstant (nominellt 10 mm).
När fordonet svävar (leviterar) finns det cirka 15 centimeters utrymme mellan banans yta och tåget. Transrapid kräver mindre effekt för att sväva än den behöver för att köra sitt luftkonditioneringssystem. Levitationssystemet och all elutrustning ombord får ström från Harmonisk oscillatorer av magnetfält från spårets stator (dessa svängningar är parasiter inte kan användas för framdrivning) vid hastigheter över 80 km/h, men vid lägre hastighet erhålls ström genom fysiska anslutningarna till banan upp till version TR08, ny strömöverföringar utvecklats för TR09 så att tågsätten inte längre behöver fysisk kontakt i alla hastigheter. I händelse av strömavbrott i spårets framdrivningssystem, kan Transrapids tåg använda batterier ombord för att driva levitationssystemet.
Vagnar och dess kapacitet
Maglev-tåget består av minst två och maximalt nio personvagnar. Varje vagn är 24,8 meter lång (mellanvagn) eller 25,5 alt 27,0 meter (änd-/manövervagn). Vagnkorgen är 370 cm bred (amerikanska snabbtåg har korgbredden 316 alt 320 cm), vilket gör att man kan ha 3 + 2 säten i bredd i andra klass (amerikanska andra klass har 2 + 2 säten i bredd). Totalt innebär det 126 platser (endast 80 vid amerikansk standard) i en mellanvagn eller 92 platser i en manövervagn. Varje mellanvagn har en tomvikt på 53 ton (tungt).
Jämfört med ett ICE-tåg av första generationen så innebär det nästan dubbelt så många säten i varje Maglev-vagn, vilket dock inverkar negativt på sittkomforten. ICE har en praktisk topphastighet på 300 km/h medan Maglev har 500 km/h. Tvärsnittet för ICE ligger kring 11 kvadratmeter, medan Maglev har hela 15 kvadratmeter.
Framdrift
Transrapids drivsystem består av en s.k. linjär motor som är monterad såväl i tågsätt som bana. Linjär- motorn används för både framdrivning och bromsning. Den fungerar som en roterande elektrisk motor vars stator är utsträckt längs undersidan av banan, så att istället för att producera ett vridmoment (rotation) ger den en linjär kraft längs dess längd.
Elektromagneterna i fordonet lyfter detta. Eftersom det magnetiska fältet bara rör sig i en riktning, skulle det om det fanns flera tåg på banavsnittet göra att tågen färdas i samma riktning, vilket gör kollisioner mellan tåg i rörelse mindre troligt.
Energiförbrukning
Det totala, maximala effektbehovet för tåget är cirka 50-100 kW per vagn för levitation och framdrift, samt styrning och övervakning.
Luftmotståndskoefficienten för Transrapid är ca 0,26 (dimensionslöst, källa okänd, liksom okänd tåglängd). Luftmotståndet för tåget, som har ett tvärsnitt av 16 m², ger en effektförbrukning vid 400 km/h, ges av följande formel:
Effektförbrukningen kan jämföras med andra höghastighetståg . Energiförbrukningen för levitation och styrning motsvarar ca 1,7 kW/ton. Eftersom elenergin kan återvinnas vid inbromsning, kan energi matas tillbaka till elnätet vid bromsning. Ett undantag från detta är när ett nödstopp utförs med hjälp av medar för nödlandning under fordonet, även om denna metod att bromsa fordonet endast är avsedd som en sista utväg om det skulle vara omöjligt eller ej önskvärt att hålla fordonet svävande med back-up batterier för ett naturligt stopp.
Kommentar: Formeln anger inte längden av tåget, således anges endast effekten för tvärsnittsytan. Siffran säger med andra ord inte så mycket... Bättre underlag behövs.
Fördelar och nackdelar med Transrapids maglev.
Jämfört med klassiska tåglinjer möjliggör Transrapid större hastigheter, större lutningar och ännu lägre energiförbrukning och underhållsbehov. Transrapids banor är mer flexibla med mindre kurvradier vilket gör det lättare att anpassa till geografi än klassiska tågsystem. Gods är i dagsläget begränsat till 15 ton per sektion. Transrapid tillåter maximala hastigheter på 550 km/h vilket är mellan konventionella höghastighetståg (200–320 km/h) och flygplan (720–990 km/h). Magnetfältsgeneratorn, en viktig del av motorn som är en del av banan, begränsar systemkapaciteten.
Transrapid i Shanghai
2001 fick konsortiet i uppdrag att bygga en linje för Transrapid i Shanghai. Sträckan är 30 km lång och sammanbinder flygplatsen Pudong med finansdistriktet Lujiazui i staden. Den invigdes med VIP-turer den 31 december 2002 och full kommersiell drift 1 januari 2004. Det tar 7 minuter och 20 sekunder att åka de 30 kilometerna från flygplatsen till Shanghai och tåget toppar då ca 431 km/h.
Ett beslut om en förlängning till Hangzhou är fattat och 2010 kommer bygget att påbörjas och beräknas vara klart 2014 till en kostnad av 22 miljarder yuan. De 200 kilometerna kommer att kräva cirka 30 minuter med en högsta hastighet på 450 km/h.
Transrapid i München
Den 24 september 2007 togs ett beslut (mellan delstaten Bayern, tyska statsjärnvägen, DB och Transrapidkonsortiet) att bygga en magnetbana från München centralstation till Münchens "nya" flygplats. Projektets pris uppskattades vid tidpunkten 2007 till 1,85 miljarder euro för den 37 km långa tåglinjen eller cirka 18,5 mrd kronor (= 5,0 mrd kr per mil dubbelspår). Samtidig påpekade Günther Beckstein, som var Bayerns ministerpresident vid den tiden, att delstaten avslutar samarbetet om kostnaderna för delstaten skulle stiga ytterligare. I mars 2008 publicerades en ny uppskattning där utgifterna hade ökat till minst 3 miljarder euro (= 30 mrd kronor eller 8,1 mrd kr per mil dubbelspår). Under ett krismöte med alla inblandade togs beslutet att inte fortsätta projektet.
Transrapid mellan Berlin-Hamburg
En omstridd och ännu inte påbörjad linje planerades mellan Hamburg och Berlin (300 km). Målet var länge att börja trafikera sträckan 2005, men de folkvalda valde tidigt av kostnadsskäl att stoppa projektet. Högsta hastighet planerades bli antingen 500 eller 550 km/h, och sträckan Hamburg-Berlin skulle ha trafikerats på en knapp timme (d.v.s. snitthastigheten 300 km/h).
Transrapidolyckan i Lathen
Projektets allvarligaste bakslag ägde rum den 22 september 2006, då en svår olycka med 23 dödsoffer inträffade på provanläggningen i Emsland.
Under en provtur med passagerare kolliderar tågsättet med ett servicefordon som befinner sig ute på teststräckan för att plocka skräp från spåret.
Se även
JR-Maglev japanska höghastighets maglevtåg.
Linimo japansk maglevbana i trafik.
Referenser
Externa länkar
Transrapids webbplats
IMB International Maglev Board
Slideshow on the Transrapid
Transrapid Pictures at Shanghai Pudong Airport
Information about the Munich Transrapid connection
Maglev World Forum
Byggmaglev.se
Utred magnettåg i Sverige!
Spårtrafik
Tyska tåg
Elektrodynamik
Snabbtåg
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{"url":"https:\/\/xianblog.wordpress.com\/tag\/standard-deviation\/","text":"## the joy of stats [book\u00a0review]\n\nPosted in Books, pictures, University life with tags , , , , , , , , , , , , on April 8, 2019 by xi'an\n\nDavid Spiegelhalter\u2018s latest book, The Art of Statistics: How to Learn from Data, has made it to Nature Book Review main entry this week. Under the title \u201cthe joy of stats\u201d,\u00a0 written by Evelyn Lamb, a freelance math and science writer from Salt Lake City, Utah. (I noticed that the book made it to Amazon #1 bestseller, albeit in the Craps category!, which I am unsure is completely adequate!, especially since the book is not yet for sale on the US branch of Amazon!, and further Amazon #1 in the Probability and Statistics category in the UK.) I have not read the book yet and here are a few excerpts from the review, quoted verbatim:\n\n\u201cThe book is part of a trend in statistics education towards emphasizing conceptual understanding rather than computational fluency. Statistics software can now perform a battery of tests and crunch any measure from large data sets in the blink of an eye. Thus, being able to compute the standard deviation of a sample the long way is seen as less essential than understanding how to design and interpret scientific studies with a rigorous eye.\u201d\n\n\u201c\u2026a main takeaway from the book is a sense of circumspection about our confidence in what is known. As Spiegelhalter writes, the point of statistical science is to ease us through the stages of extrapolation from a controlled study to an understanding of the real world, `and finally, with due humility, be able to say what we can and cannot learn from data\u2019. That humility can be lacking when statistics are used in debates about contentious issues such as the costs and benefits of cancer screening.\n\n## an unbiased estimator of the Hellinger\u00a0distance?\n\nPosted in Statistics with tags , , , on October 22, 2012 by xi'an\n\nHere is a question I posted on Stack Exchange a while ago:\n\nIn a setting where one observes X1,\u2026,Xn distributed from a distribution with (unknown) density f, I wonder if there is an unbiased estimator (based on the Xi\u2018s) of the Hellinger distance to another distribution with known density f0, namely\n\n$\\mathfrak{H}(f,f_0)=\\left\\{1-\\int\\sqrt{f_0(x)\/(x)}\\text{d}x\\right\\}^{1\/2}$\nNow, Paulo has posted an answer that is rather interesting, if formally \u201coff the point\u201d. There exists a natural unbiased estimator of if not of H, based on the original sample and using the alternative representation\n$\\mathfrak{H}^2(f,f_0)=1-\\mathbb{E}_f[\\sqrt{f_0(X)\/f(X)}]$\n\nfor the Hellinger distance. In addition, this estimator is guaranteed to enjoy a finite variance since\n\n$\\mathbb{E}_f[\\sqrt{f_0(X)\/f(X)}^2]=1\\,.$\n\nConsidering this question again, I am now fairly convinced there cannot be an unbiased estimator of H, as it behaves like a standard deviation for which there usually is no unbiased estimator!","date":"2020-01-21 18:16:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 3, \"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\": 0.4260256290435791, \"perplexity\": 885.308249284595}, \"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-2020-05\/segments\/1579250604849.31\/warc\/CC-MAIN-20200121162615-20200121191615-00216.warc.gz\"}"}
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Q: Check NSString for specific date format I have a NSString and I need to check that it is in a this specific format MM/DD/YY. I then need to convert that to a NSDate. Any help on this would be much appreciated. Sidenote - I have searched around and people suggest using RegEx, I have never used this and am unclear about it generally. Can anyone point me to a good resource/explanation.
A: NSString *strDate1 = @"02/09/13";
NSString *strDate2 = @"0123/234/234";
NSDateFormatter *dateFormatter = [[NSDateFormatter alloc] init];
[dateFormatter setDateFormat:@"dd/MM/yy"];
[dateFormatter setTimeZone:[NSTimeZone timeZoneWithAbbreviation:@"GMT"]];
NSDate *dateFormat1 = [dateFormatter dateFromString:strDate1];
NSDate *dateFormat2 = [dateFormatter dateFromString:strDate2];
NSLog(@"%@", dateFormat1); // prints 2013-09-02 00:00:00 +0000
NSLog(@"%@", dateFormat2); // prints (null)
So you will know when it's not formatted correctly if the NSDate is nil. Here's the link to the docs if you need more info: https://developer.apple.com/library/mac/documentation/Cocoa/Conceptual/DataFormatting/Articles/dfDateFormatting10_4.html#//apple_ref/doc/uid/TP40002369-SW1
A: Use an NSDateFormatter for both tasks. If you can convert the string to a date then it is in the correct format (and you already have the result).
A: I know that this is a late answer, but it is impossible to always guarantee that a string is in this particular date format.
A date formatter, a regex, or even a human can not verify certain dates, because we don't know if the user is entering "mm/DD/yy" or "DD/mm/yy". It is common in some places to enter the day of the month first, while in other areas you enter the month first. So if they enter "09/06/2013" do they mean "September 6th" or the "9th of June"?
A: Here is a simple function for anyone searching for a simple solution.
- (BOOL) isTheStringDate: (NSString*) theString
{
NSDateFormatter *dateFormatter = [[NSDateFormatter alloc] init];
[dateFormatter setDateFormat:@"yyyy-MM-dd"];
NSDate *dateFromString = [[NSDate alloc] init];
dateFromString = [dateFormatter dateFromString:theString];
if (dateFromString !=nil) {
return true;
}
else {
return false;
}
}
You have to change the formatter below to match the formatting your date is using.
[dateFormatter setDateFormat:@"yyyy-MM-dd"];
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Associated Press Top 25: What is biggest mistake in top 10?
The current Associated Press Top 25 poll has Alabama, Oregon, Clemson, Ohio State, Stanford, Florida State, Georgia, Louisville, Texas A&M and LSU in its top 10. There's little debate about the top two teams in the country through six weeks of play -- the Crimson Tide and the Ducks -- but the discussion becomes trickier as voters dissect the merits of other unbeaten teams such as Stanford, Ohio State, Clemson, and Florida State and talented one-loss teams such as Georgia and Texas A&M.
This week's debate: From glaring omissions to undeserving teams, what is the biggest mistake in the top 10 of The Associated Press Top 25 poll?
Daniel Jeremiah NFL.com
Noles deserve to leap Stanford, Ohio State
Based on what we witnessed this past weekend, I'd move Florida State to No. 4, behind Alabama, Oregon and Clemson. Quarterback Jameis Winston is playing out of his mind, and the Seminoles' defense has incredible speed and depth at all three levels. Stanford is very physical, but I think they would struggle to match up with the speed of the Noles on both sides of the ball. I also believe FSU has more reliable quarterback play than Ohio State, and their defense would be able to slow down the Buckeyes' rushing attack. Of course, this will all settle itself out over the remainder of the season. I can't wait for the Clemson-Florida State game!
Charles Davis NFL.com
Jury is still out on Louisville
Since I hold a vote in the AP Top 25 each week, let's just make this simple. Here's my top 10 for this week:
The team that I agonized over the most in my top 10 was Louisville. I love QB Teddy Bridgewater and WR DaVante Parker, and head coach Charlie Strong has done a remarkable job building the best team in the American Athletic Conference. Can the Cardinals play with just about anyone in the country? They proved that in their BCS appearance last season, when they thumped Florida from the big, bad SEC. This season, Louisville is doing what it needs to do, crushing the not-up-to-snuff opponents on its schedule, but it has not been close to tested yet. Perhaps that begins to change this Thursday night, when they host Rutgers. I voted UCLA No. 10 and think the Bruins have a great chance to be ranked much higher before the season is over. QB Brett Hundley and LB Anthony Barr are as good as anyone in the country at their respective positions, and head coach Jim Mora has done a fantastic job getting his team to believe it is dangerous on the field.
Dan Greenspan College Football 24/7
Georgia headed out, Baylor headed in
Georgia has outstanding wins over South Carolina and LSU and a narrow loss to Clemson on its resume, but in its injury-riddled state, the Bulldogs will struggle to maintain their top-10 ranking. Just one more injury could be the tipping point for a collapse, especially with games against Florida and Auburn still on the schedule. Conversely, Baylor needs to be getting far more respect from the pollsters. The Bears are averaging 779.5 yards of total offense per game to lead the FBS, outpacing No. 2 Oregon by 149 yards. In their first game against BCS conference opposition, BU had 56 points at halftime and finished with 864 yards, even after calling off the dogs by sitting starting quarterback Bryce Petty following one drive in the second half.
Mike Huguenin College Football 24/7
Buckeyes' schedule doesn't justify ranking
Ohio State is talented and well-coached, but is it the fourth-best team in the country? Considering their schedule so far, I think No. 4 is too high. And against the two best teams the Buckeyes have played so far -- a 31-24 win over Wisconsin and a 40-30 win over Northwestern -- they haven't exactly been dominant.
Chase Goodbread College Football 24/7
Hurricanes should take LSU's spot
The resurgent Hurricanes have a strong case to claim the spot being occupied by LSU. The Tigers are currently No. 10 in the AP poll, but they have a loss and don't have a victory more impressive than Miami's win over Florida. LSU will have that chance this weekend when they face the Gators, but until then, Miami has the more impressive win, plus it is unbeaten, unlike the one-loss Tigers. The LSU offense has been impressive and, at times, unstoppable, and by the end of the season, LSU might rightfully be ahead of the 'Canes. But through six weeks, Miami has done more with its schedule.
Georgia beats Alabama, 33-18, for first NCAA title since 1980 season
Former walk-on quarterback Stetson Bennett and an opportunistic Georgia defense led the Bulldogs to their first national championship since 1980 on Monday night, toppling defending champ Alabama, 33-18, in the College Football Playoff National Championship in Indianapolis.
NFL announces prospects to attend 2022 HBCU combine
Today, the NFL in partnership with the Senior Bowl, announced the names of players that will attend the 2022 Historically Black Colleges and Universities (HBCU) Combine presented by Microsoft Surface, which will be held Friday, Jan. 28 - Saturday, Jan. 29, at the University of South Alabama Jaguar Training Center in Mobile, Alabama.
Georgia, Alabama advance to 2022 College Football Playoff National Championship Game
The rematch is on. After routing their respective semifinal opponents, the No. 1 seed Alabama Crimson Tide (13-1) and No. 3 seed Georgia Bulldogs (13-1) will play for the 2022 College Football Playoff National Championship on Monday, Jan. 10, at Lucas Oil Stadium in Indianapolis, Indiana.
Top recruit Travis Hunter spurns Florida State to commit to Deion Sanders, Jackson State
Deion Sanders' impact on Jackson State, and HBCU football at large, reached a new level on Wednesday as the nation's No. 1-ranked recruit signed with the Tigers.
Alabama QB Bryce Young wins 2021 Heisman Trophy
Alabama quarterback Bryce Young became the second consecutive Crimson Tide player to win the Heisman Trophy on Saturday night, adding an exclamation point to his sensational first season as a starter.
Grambling State to hire Hue Jackson as next head coach
Former Raiders and Browns coach Hue Jackson landed another big-time gig, this time in the college ranks. Tom Pelissero reports that Jackson agreed to terms late Thursday night on a four-year contract to become the new head coach at Grambling State.
Lincoln Riley details why he left Oklahoma for USC: 'We can build one of the best rosters in the country'
The scuttlebutt long had been if Lincoln Riley was going to leave Oklahoma it would be for the NFL. The Cowboys expressed interest after jettisoning Jason Garrett following the 2019 season, and have other teams in recent years. Riley, thus, shocked and shook up the sport last week when he departed Oklahoma for USC.
Oregon DE Kayvon Thibodeaux declares for 2022 NFL Draft
Kayvon Thibodeaux is officially headed for the NFL. The Oregon defensive end announced via social media he has declared for the 2022 NFL Draft.
College Football Playoff: Five NFL draft takeaways from final rankings
The College Football Playoff semifinal matchups are set. What will NFL scouts be watching for in the two games? Dan Parr provides five takeaways.
Brian Kelly leaves Notre Dame to be head coach at LSU
After 12 seasons at Notre Dame, Brian Kelly was named LSU's next head football coach, the school announced Tuesday.
Florida parts ways with head coach Dan Mullen after four seasons
The University of Florida fired head coach Dan Mullen on Sunday after a four-year tenure. Mullen joined the Gators from Mississippi State ahead of the 2018 campaign and led Florida to a 34-15 record over his four seasons in Gainesville.
USC WR Drake London (ankle) out for remainder of season
USC's Drake London will miss the remainder of the 2021 season after suffering a broken ankle, the school announced.
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Heinrich Schmitz (* 30. Juli 1929 in Lissingen; † 12. September 2000 in Gerolstein) war ein deutscher Politiker (CDU).
Leben
Nach dem Besuch der Volksschule in Lissingen und des staatlichen Gymnasiums in Prüm absolvierte Schmitz ab 1946 eine kaufmännische Lehre, die er 1948 mit der Gehilfenprüfung abschloss. Im Anschluss folgte eine Anstellung bei der Firma Gerolsteiner, in der er als Versandleiter und stellvertretender Leiter der Personalabteilung tätig war.
Schmitz trat 1967 in die CDU ein. Er war ab 1968 Mitglied des Stadtrates und Beigeordneter in Gerolstein. Ab 1970 war er Vorsitzender der Verbandsgemeinde Gerolstein und Kreisdeputierter des Landkreises Daun. Dem Rheinland-Pfälzischen Landtag gehörte er vom 26. November 1973, als er für den ausgeschiedenen Abgeordneten Julius Saxler nachrückte, bis 1975 an. Im Landtag war er Mitglied des Innenausschusses.
Literatur
Der Präsident des Landtags Rheinland-Pfalz (Hrsg.): Die Stellvertreter des freien Volkes. Die Abgeordneten der Beratenden Landesversammlung und des Landtags Rheinland-Pfalz von 1946 bis 2015. Springer VS, Wiesbaden 2016, ISBN 978-3-658-04750-4, S. 610.
Landtagsabgeordneter (Rheinland-Pfalz)
Beigeordneter (Rheinland-Pfalz)
Kommunalpolitiker (Landkreis Vulkaneifel)
Träger des Bundesverdienstkreuzes 1. Klasse
CDU-Mitglied
Deutscher
Geboren 1929
Gestorben 2000
Mann
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\section*{Acknowledgment}
JY owes a large amount of thanks to Lun-Hui Hu for patiently answering his questions on superconductivity. JY also thanks Rui-Xing Zhang, Yang Ge and Jian-Xiao Zhang for helpful discussion. CXL and JY acknowledge the support from Office of Naval Research (Grant No. N00014-15-1-2675).
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| 5,816
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Broken tail : a tiger's last journey, a co-production of Crossing the Line Productions and Thirteen in association with WNET.ORG. ; produced by John Murray, (videorecording)
The Resource Broken tail : a tiger's last journey, a co-production of Crossing the Line Productions and Thirteen in association with WNET.ORG. ; produced by John Murray, (videorecording)
The item Broken tail : a tiger's last journey, a co-production of Crossing the Line Productions and Thirteen in association with WNET.ORG. ; produced by John Murray, (videorecording) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Yuma County Library District.
Crossing the Line Productions
WNET (Television station : New York, N.Y.)
Murray, John, 1964-
Broken Tail was a charismatic tiger cub in Ranthambore, one of India's best-protected tiger reserves. He suddenly and without warning abandoned his sanctuary and went on the run moving through farmland and scrub until he was killed by a train nearly 200 miles from its home. This film retraces the tiger's path and piece together the cub's last days, and through his story reveal the fate of the few surviving tigers in India
United States, Distributed by PBS Distribution, c2011
1 videodiscs (ca. 60 min.)
Special features: Additional scenes
Broken tail : a tiger's last journey
Broken tail
a tiger's last journey
a co-production of Crossing the Line Productions and Thirteen in association with WNET.ORG. ; produced by John Murray
Nature television programs
Nonfiction television programs
Nature (Television program)
Photography, John Murray, Robin Cox, Colin Stafford-Johnson ; editor, Emer Reynolds
QL737.C23
Featuring Salim Ali, Colin Stafford-Johnson
Murray, John
NAT62809
DVD, region 1, NTSC, widescreen presentation; 5.1 surround
DVD, Dolby digital stereo
<div class="citation" vocab="http://schema.org/"><i class="fa fa-external-link-square fa-fw"></i> Data from <span resource="http://link.yumalibrary.org/portal/Broken-tail--a-tigers-last-journey-a/0Zz5HZFGtJ4/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.yumalibrary.org/portal/Broken-tail--a-tigers-last-journey-a/0Zz5HZFGtJ4/">Broken tail : a tiger's last journey, a co-production of Crossing the Line Productions and Thirteen in association with WNET.ORG. ; produced by John Murray, (videorecording)</a></span> - <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.yumalibrary.org/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="https://link.yumalibrary.org/">Yuma County Library District</a></span></span></span></span></div>
Data Citation of the Item Broken tail : a tiger's last journey, a co-production of Crossing the Line Productions and Thirteen in association with WNET.ORG. ; produced by John Murray, (videorecording)
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Q: Struts 2 - Java - Variable casting error (Trying to prevent XSS) I really am not sure how to ask this question but if it is not suitable for SO I am sorry.
*
*I have a getter method that accepts only an Integer.
*In my web form I have a hidden field that can be manipulated using firebug
*When the user hacks in a string it appears to be sent to the server
*I get various 'Error setting expression' errors in eclipse (obviously as it is trying to cast a string to an int)
*I check back in firebug and the value is still set to the string
*I check in the response and the string has definitely been sent to the server and then sent back by the server
*This is obviously a problem as someone could whack some script in it for some XSS
*THIS is easily fixed by adding and if invoked on a value range in the setter else sets to 0.
My real question is... why if the compiler doesnt like the string casting to an int, does the server still allow it to be returned?
I was expecting a big barf and for it to tell me NO!
My fix at the moment is purely a hack as I do not understand it enough. To me, it feels like after the getter barfs, it goes somewhere else?
Is this a Struts2 thing?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,610
|
Terrein achter de Watertoren (English: Terrain behind the Watertower) was a football stadium in Breda, Netherlands. It was used for football matches and hosted the home matches of NAC Breda. The stadium was opened in 1913 and demolished in 1916.
Defunct football venues in the Netherlands
NAC Breda
Sports venues in Breda
Sports venues completed in 1913
Sports venues demolished in 1916
History of Breda
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,516
|
{"url":"https:\/\/yetanothermathblog.com\/tag\/puzzle\/","text":"# Michael Reid\u2019s Happy New Year Puzzles,\u00a02018\n\nBelatedly posted by permission of Michael Reid. Enjoy!\n\nHere are some New Year\u2019s puzzles to help start out 2018.\n\n1. Arrange the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in the\nexpression $a^b + c^d + e^f + g^h + i^j$ to make 2018.\n\n2. (a) Express $2018 = p^q + r^s$ where p, q, r, s are primes.\n(b) Express $2018 = p^q - r^s$ where p, q, r, s are primes.\n\n3. (a) Is it possible to put the first 9 primes, 2, 3, 5, 7, 11, 13,\n17, 19 and 23 into a 3\u00d73 matrix that has determinant 2018?\n(b) Is it possible to put the first 16 primes, 2, 3, 5, \u2026 , 53,\ninto a 4\u00d74 matrix that has determinant 2018?\n\n4. (a) Express 2018 = A \/ B using the fewest number of distinct\ndigits.\nFor example, the expression 7020622 \/ 3479 uses only seven\ndifferent digits. But it is possible to do better than this.\n(b) Express $2018 = (A_1 \\cdot A_2 \\cdot ... \\cdot A_m) \/ (B_1 \\cdot B_2 \\cdot ... \\cdot B_n)$ using the fewest number of distinct digits.\n\n# Michael Reid\u2019s Happy New Year Puzzle,\u00a02017\n\nBelatedly posted by permission of Michael Reid. Enjoy!\n\nHere are some interesting puzzles to start the New Year; hopefully they are not too easy!\n\n1. Express 2017 as a quotient of palindromes.\n\n2. (a) Are there two positive integers whose sum is 2017 and whose product\nis a palindrome?\n(b) Are there two positive integers whose difference is 2017 and whose\nproduct is a palindrome?\n\n3. Is there a positive integer n such that both 2017 + n and\n2017 n are palindromes?\n\n4. What is the smallest possible sum of the decimal digits of 2017 n ,\nwhere n is \u2026\n(a) \u2026 a positive integer?\n(b) \u2026 a prime number?\n(c) \u2026 a palindrome?\n\n5. Consider the following two operations on a positive integer:\n(i) replace a string of consecutive digits by its square,\n(ii) if a string of consecutive digits is a perfect cube,\nreplace the string by its cube root.\n\nNeither the string being replaced, nor its replacement, may have\nhave \u201cleading zeros\u201d. For example, from 31416 , we may change it to\n319616 , by squaring the 14 . From 71253 , we may change it to\n753 by taking the cube root of 125 .\n\n(a) Starting from the number 2017 , what is the smallest number we\ncan reach with a sequence of these operations?\n(b) What is the smallest number from which we can start, and reach\nthe number 2017 with a sequence of these operations?\n\n6. Find a list of positive rational numbers, q_1 , q_2 , \u2026 , q_n\nwhose product is 1 , and whose sum is 2017 . Make your list as\nshort as possible.\nExtra Credit: Prove that you have the shortest possible list.\n\n# Michael Reid\u2019s Happy New Year Problems,\u00a02020\n\nPosted by permission of Michael Reid. Enjoy!\n\nNew Year\u2019s Greetings!\n\nHere are some fun puzzles to start the year.\n\n1. Substitute the numbers 1, 2, \u2026 , 9 for the letters\na, b, \u2026 , i in the expression $a^b + c^d + (e + f + g - h)^i$\nto get 2020.\n\n2. Use the digits 1, 2, \u2026 , 9 in order, and any of the usual\narithmetic operations and parentheses to get a number that is\nas close as possible to, but not exactly equal to 2020.\n\n3. Express 2019\/2020 as a sum of distinct Egyptian fractions,\ni.e. $1 \/ n_1 + 1 \/ n_2 + ... + 1 \/ n_k$ for integers $0 < n_1 < n_2 < ... n_k < 202049$\n(but 202049 is not square).\n\n5. Make a 4\u00d74 matrix of single-digit integers (0-9) with digits\n2, 0, 2, 0 on the main diagonal, and having determinant 2020.\nIs it possible to do it with a symmetric matrix?\n\nIf you liked this one, check out other puzzles ont this blog tagged with \u201cMichael Reid\u201d.\n\n# Problem of the Week,\u00a0#121\n\nA former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a \u201cProblem of the Week\u201d for his students, giving a prize of a cookie if they could solve it. Here is one of them.\n\n### Problem 121\n\nThe Maryland \u201cBig Game\u201d lottery is played by selecting 5 different numbers in $\\{ 1,2,3,\\dots, 50\\}$ and then selecting one of the numbers in $\\{ 1,2,3,\\dots, 36\\}$. The first section is an unordered selection without replacement (so, arrange them in increasing order if you like) but the second selection can repeat one of the 5 numbers initially picked.\n\nHow many ways can this be done?\n\n# Simple unsolved math problem,\u00a06\n\nIf you know a little point-set topology, below is an unsolved math problem whose statement is relatively simple.\n\nProbably everyone has at least seen the Mandelbrot set in some form, as it\u2019s a popular object of mathematical artists. Here\u2019s a picture from Wikipedia:\n\nThe formal definition is as follows. Let $f_c (z)=z^2+c$, where $c\\in \\mathbb{C}$ is a complex number. The Mandelbrot set $X$ is the complex plot of the set of complex numbers $c$ for which the sequence of iterates\n\n$f_c (0), f_c (f_c (0)), f_c (f_c (f_c (0))), \\dots,$\n\nremains bounded in absolute value.\nWe say $X$ is locally connected if every point $x\\in X$ admits a neighborhood basis consisting entirely of open, connected sets.\n\nConjecture: The Mandelbrot set $X$ is locally connected.\n\n# Simple unsolved math problem,\u00a05\n\nIt seems everyone\u2019s heard of Pascal\u2019s triangle. However, if you haven\u2019t then it is an infinite triangle of integers with 1\u2018s down each side and the inside numbers determined by adding the two numbers above it:\n\nFirst 6 rows of Pascal\u2019s triangle\n\nThe first 6 rows are depicted above. It turns out, these entries are the binomial coefficients that appear when you expand $(x+y)^n$ and group the terms into like powers $x^{n-k}y^k$:\n\nFirst 6 rows of Pascal\u2019s triangle, as binomial coefficients.\n\nThe history of Pascal\u2019s triangle pre-dates Pascal, a French mathematician from the 1600s, and was known to scholars in ancient Persia, China, and India.\n\nStarting in the mid-to-late 1970s, British mathematician David Singmaster was known for his research on the mathematics of the Rubik\u2019s cube. However, in the early 1970\u2019s, Singmaster made the following conjecture\u00a0[1].\n\nConjecture: If $N(a)$ denotes the number of times the number $a > 1$ appears in Pascal\u2019s triangle then $N(a) \\leq 12$ for all $a>1$.\n\nIn fact, there are no known numbers $a>1$ with $N(a)>8$ and the only number greater than one with $N(a)=8$ is a=3003.\n\nReferences:\n\n[1]\u00a0Singmaster, D. \u201cResearch Problems: How often does an integer occur as a binomial coefficient?\u201d, American Mathematical Monthly, 78(1971) 385\u2013386.\n\n# Simple unsolved math problem,\u00a03\n\nA perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example,\u00a0\u00a01 + 2 + 3 = 6 implies 6 is a perfect number.\n\nUnsolved Problem: Are there any odd perfect numbers?\n\nThe belief, by some, that there are none goes back over 500 years (wikipedia).\n\nIf you want to check out some recent research into this problem, see oddperfect.org.\n\n(Another unsolved problem: Are there an infinite number of even perfect numbers?)\n\n# Simple unsolved math problem,\u00a02\n\nIn 1911, Otto Toeplitz asked the following question.\n\nInscribed Square Problem: Does every plane simple closed curve contain all four vertices of some square?\n\nThis question, also known as the\u00a0square peg problem or the Toeplitz\u2019 conjecture, is still unsolved in general. (It is known in lots of\u00a0special cases.)\n\nInscribed square, by Claudio Rocchini\n\nThanks to Mark Meyerson (\u201cEquilateral triangles and continuous curves\u201d,Fundamenta Mathematicae, 1980) and others, the analog for triangles is true. For any triangle T and Jordan curve C, there is a triangle similar to\u00a0T and inscribed in C.\u00a0(In particular, the triangle can be equilateral.) The survey page by\u00a0Mark J. Nielsen has more information on this problem.\n\nAdded 2020-07-01: This has apparently been solved by Joshua Greene and Andrew Lobb! See their ArXiV paper (https:\/\/arxiv.org\/abs\/2005.09193).\n\n# Simple unsolved math problem,\u00a01\n\nIn 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows.\n\nFirst, we define the function $f$ on the set of positive integers:\n\nIf the number $n$ is even, divide it by two: $f(n)=n\/2$.\nIf the number $n$ is odd, triple it and add one: $f(n)=3n+1$.\n\nIn modular arithmetic notation, define the function $f$ as follows:\n$f(n)= {n\/2},\\ if \\ n\\equiv 0 \\pmod 2$, and $f(n)= {3n+1},\\ if \\ n\\equiv 1 \\pmod 2$. Believe it or not, this is the restriction to the positive integers of the complex-valued map $(2+7z-(2+5z)\\cos(\\pi z))\/4$.\n\nThe 3n+1 conjecture is: The sequence\n$n,\\ f(n),\\ f^2(n)=f(f(n)),\\ f^3(n)=f(f^2(n)),\\ \\dots$\nwill eventually reach the number 1, regardless of which positive integer $n$ is chosen initially.\n\nThis is still unsolved, though a lot of people have worked on it. For a recent survey of results, see the paper by Chamberland.","date":"2020-10-24 12:45:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 37, \"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\": 0.788601279258728, \"perplexity\": 669.7071495524913}, \"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-2020-45\/segments\/1603107882581.13\/warc\/CC-MAIN-20201024110118-20201024140118-00565.warc.gz\"}"}
| null | null |
package orgreminders
import (
"appengine"
"appengine/datastore"
"appengine/mail"
"appengine/user"
"fmt"
"html/template"
"log"
"net/http"
"sort"
"strings"
"time"
)
// App-global variables
var Templates *template.Template
var Duration_Day = 24 * time.Hour
var Duration_Week = 7 * Duration_Day
var TemplateFiles = []string{
"tmpl/header.html",
"tmpl/css.html",
"tmpl/home.html",
"tmpl/save.html",
"tmpl/new-event.html",
"tmpl/new-org.html",
"tmpl/editorg.html",
"tmpl/editevent.html",
"tmpl/events.html",
"tmpl/organizations.html",
"tmpl/error.html",
"tmpl/cron.html",
"tmpl/new-member.html",
"tmpl/members.html",
"tmpl/editmember.html",
}
type Page struct {
Error string
Events map[string]Event
Keys []*datastore.Key
Event2Edit Event
Organizations map[string]Organization
Org2Edit Organization
Org2EditKey string
Location time.Location
AllowNewOrg bool
SuperUser bool
LoggedIn bool
UserEmail string
Orgs []string
Members map[string]Member
SavedEvent bool
SavedOrg bool
SavedMember bool
Member2Edit Member
Member2EditKey string
ScheduleHTML map[string][]string
}
func NewPage(u *User) (*Page, error) {
var result = Page{}
if u.Meta != nil {
result.LoggedIn = true
result.AllowNewOrg = true
result.UserEmail = u.Meta.Email
if u.SuperUser {
result.SuperUser = true
}
}
return &result, nil
}
func renderTemplate(w http.ResponseWriter, tmpl string, p *Page) {
err := Templates.ExecuteTemplate(w, tmpl+".html", p)
if err != nil {
http.Error(w, err.Error(), http.StatusInternalServerError)
}
}
func init() {
templTest, err := template.ParseFiles(TemplateFiles...)
if err != nil {
log.Println("Some (or all) of the required templates are missing, exiting: ", err.Error())
return
}
Templates = templTest
http.HandleFunc("/", DefaultHandler)
http.HandleFunc("/newevent", NewEventHandler)
http.HandleFunc("/neworg", NewOrgHandler)
http.HandleFunc("/events", EventsHandler)
http.HandleFunc("/organizations", OrgsHandler)
http.HandleFunc("/saveevent", EventSaveHandler)
http.HandleFunc("/saveorg", OrgSaveHandler)
http.HandleFunc("/editorg", OrgEditHandler)
http.HandleFunc("/editevent", EventEditHandler)
http.HandleFunc("/cron", CronHandler)
http.HandleFunc("/logout", LogoutHandler)
http.HandleFunc("/newmember", NewMemberHandler)
http.HandleFunc("/savemember", MemberSaveHandler)
http.HandleFunc("/members", MembersHandler)
http.HandleFunc("/editmember", MemberEditHandler)
}
func LogoutHandler(w http.ResponseWriter, r *http.Request) {
c := appengine.NewContext(r)
url, _ := user.LogoutURL(c, "/")
http.Redirect(w, r, url, http.StatusFound)
}
func DefaultHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
title := "home"
renderTemplate(w, title, p)
}
func NewOrgHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
title := "new-org"
renderTemplate(w, title, p)
}
func NewEventHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
title := "new-event"
for _, org := range u.Orgs {
p.Orgs = append(p.Orgs, org.Name)
}
sort.Strings(p.Orgs)
renderTemplate(w, title, p)
}
func EventSaveHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
r.ParseForm()
event := NewEvent()
event.Title = r.PostFormValue("title")
event.EmailMessage = template.HTML(r.PostFormValue("emailmessage"))
event.TextMessage = r.PostFormValue("textmessage")
event.Submitter = *u.Meta
event.Orgs = r.PostForm["orgs"]
if len(event.Orgs) < 1 {
p.Error = "You must choose an organization."
renderTemplate(w, "error", p)
return
}
if r.PostFormValue("sendemail") == "on" {
event.Email = true
}
if r.PostFormValue("sendtext") == "on" {
event.Text = true
}
// save reminder schedule
var remqtys = r.PostForm["remqty[]"]
var remtyps = r.PostForm["remtyp[]"]
for remkey, remval := range remqtys {
var entry = fmt.Sprintf("%s%s", remval, remtyps[remkey])
event.Reminders.Add(entry)
}
o, err := GetOrganizationByName(c, event.Orgs[0])
if err != nil {
c.Infof("Error: %s", err.Error())
p.Error = err.Error()
renderTemplate(w, "error", p)
return
}
location, _ := time.LoadLocation(o.TimeZone)
const longForm = "01/02/2006 3:04pm"
t, timeerr := time.ParseInLocation(longForm, r.PostFormValue("due"), location)
if timeerr != nil {
http.Error(w, "Invalid time string", http.StatusInternalServerError)
return
}
event.Due = t
event.Key = r.PostFormValue("key")
var subject = "Event Saved: "
if event.Key == "" {
_, event.Key = event.Save(c)
} else {
event.Update(c)
subject = "Event Updated: "
}
if r.PostFormValue("oncreate") == "on" {
event.Notify(c, true)
}
event.DueFormatted = event.Due.In(location).Format("01/02/2006 3:04pm")
AdminNotify(c, u.Meta.Email, subject+event.Title, "The following event was just saved: <br><br>"+event.GetHTMLView(c))
p.Event2Edit = event
p.SavedEvent = true
renderTemplate(w, "save", p)
}
func EventEditHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
var ok bool
ok, p.Event2Edit = GetEventByKey(c, r.FormValue("id"))
if ok {
org, _ := GetOrganizationByName(c, p.Event2Edit.Orgs[0])
location, _ := time.LoadLocation(org.TimeZone)
p.Event2Edit.DueFormatted = p.Event2Edit.Due.In(location).Format("01/02/2006 3:04pm")
uorgs := GetOrganizationsByUser(c, u.Meta.Email)
for _, uorg := range uorgs {
missing := true
for _, porg := range p.Event2Edit.Orgs {
if uorg.Name == porg {
missing = false
break
}
}
if missing == true {
p.Orgs = append(p.Orgs, uorg.Name)
}
}
// Extract usable event reminder list
p.ScheduleHTML = p.Event2Edit.Reminders.HTML()
sort.Strings(p.Orgs)
renderTemplate(w, "editevent", p)
} else {
p.Error = "Event not found."
renderTemplate(w, "error", p)
}
}
func OrgSaveHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
org := NewOrganization()
org.Name = r.PostFormValue("name")
org.Description = r.PostFormValue("description")
org.Active = true
org.Expires = time.Now().UTC().Add(Duration_Week)
org.Administrator = strings.Split(r.PostFormValue("admin"), "\r\n")
org.TimeZone = r.PostFormValue("timezone")
key := r.PostFormValue("key")
if key == "" {
c.Infof("saving org")
org.Save(c)
} else {
c.Infof("updating org")
org.Update(c, key)
}
p.SavedOrg = true
p.Org2Edit = org
p.Org2EditKey = key
renderTemplate(w, "save", p)
}
func OrgEditHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
p.Org2EditKey = r.FormValue("id")
p.Org2Edit = GetOrganizationByKey(c, p.Org2EditKey)
renderTemplate(w, "editorg", p)
}
func EventsHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
p.Events = make(map[string]Event)
c := appengine.NewContext(r)
for _, org := range u.Orgs {
events := org.GetEvents(c, true)
location, _ := time.LoadLocation(org.TimeZone)
for indx, event := range events {
event.Due = event.Due.In(location)
event.DueFormatted = event.Due.Format("01/02/2006 3:04pm")
p.Events[indx] = event
}
}
renderTemplate(w, "events", p)
}
func OrgsHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
mapResults := make(map[string]Organization)
for indx, org := range u.Orgs {
org.Members = org.GetMembers(c)
mapResults[indx] = org
}
p.Organizations = mapResults
renderTemplate(w, "organizations", p)
}
func MembersHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
p.Members = make(map[string]Member)
c := appengine.NewContext(r)
for _, org := range u.Orgs {
members := org.GetMembers(c)
for indx, member := range members {
p.Members[indx] = member
}
}
renderTemplate(w, "members", p)
}
func NewMemberHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
title := "new-member"
for _, org := range u.Orgs {
p.Orgs = append(p.Orgs, org.Name)
}
sort.Strings(p.Orgs)
renderTemplate(w, title, p)
}
func MemberEditHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
var ok bool
p.Member2EditKey = r.FormValue("id")
ok, p.Member2Edit = GetMemberByKey(c, p.Member2EditKey)
// Protect web users
if ok && p.Member2Edit.WebUser {
if u.Meta.Email != p.Member2Edit.Email && u.SuperUser == false {
ok = false
}
}
if ok {
uorgs := GetOrganizationsByUser(c, u.Meta.Email)
for _, uorg := range uorgs {
missing := true
for _, porg := range p.Member2Edit.Orgs {
if uorg.Name == porg {
missing = false
break
}
}
if missing == true {
p.Orgs = append(p.Orgs, uorg.Name)
}
}
sort.Strings(p.Orgs)
renderTemplate(w, "editmember", p)
} else {
p.Error = "Member not found or access denied."
renderTemplate(w, "error", p)
}
}
func MemberSaveHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
c := appengine.NewContext(r)
r.ParseForm()
member := Member{}
member.Name = r.PostFormValue("name")
member.Email = r.PostFormValue("email")
member.Cell = r.PostFormValue("cell")
member.Carrier = r.PostFormValue("carrier")
member.TextAddr = GenTextAddr(member.Cell, member.Carrier)
member.Orgs = r.PostForm["orgs"]
if r.PostFormValue("emailon") == "on" {
member.EmailOn = true
}
if r.PostFormValue("texton") == "on" {
member.TextOn = true
}
if u.SuperUser && r.PostFormValue("webuser") == "on" {
member.WebUser = true
}
// Must have or don't save
if len(r.PostForm["orgs"]) <= 0 && member.WebUser == false {
p.Error = "Cannot save without an organization."
renderTemplate(w, "error", p)
return
}
key := r.PostFormValue("key")
if key == "" {
c.Infof("saving member")
_, key = member.Save(c)
} else {
c.Infof("updating member")
member.Update(c, key)
}
p.Member2Edit = member
p.Member2EditKey = key
p.SavedMember = true
renderTemplate(w, "save", p)
}
func AdminNotify(c appengine.Context, creator string, subject string, message string) {
var appid = appengine.AppID(c)
msg := &mail.Message{
Sender: "orgreminders@" + appid + ".appspotmail.com",
Subject: subject,
HTMLBody: message,
To: []string{creator},
}
c.Infof("notify (%s): %v", subject, creator)
if err := mail.Send(c, msg); err != nil {
c.Errorf("Couldn't send email: %v", err)
}
}
func SendOrgMessage(c appengine.Context, o Organization, e Event, t string) (result bool) {
var appid = appengine.AppID(c)
var senderUserName = strings.Replace(o.Name, " ", "_", -1)
var sender = fmt.Sprintf("%s Reminders <%s@%s.appspotmail.com", o.Name, senderUserName, appid)
members := o.GetMembers(c)
recipients := []string{}
for _, m := range members {
if t == "email" && m.EmailOn {
recipients = append(recipients, m.Email)
} else if t == "text" && m.TextOn {
recipients = append(recipients, m.TextAddr)
}
}
if len(recipients) == 0 {
c.Infof("No recipients, not sending reminder (" + t + ")")
result = true
return
}
// get rid of duplicate recipients
recipients = removeDuplicates(recipients)
msg := &mail.Message{
Sender: sender,
Bcc: recipients,
Subject: e.Title,
Body: e.TextMessage,
HTMLBody: string(e.EmailMessage),
}
c.Infof("notify (%s): %v", e.Title, recipients)
if err := mail.Send(c, msg); err != nil {
c.Errorf("Couldn't send email: %v", err)
} else {
result = true
}
return
}
func CronHandler(w http.ResponseWriter, r *http.Request) {
u := UserLookup(w, r)
p, _ := NewPage(&u)
p.Events = make(map[string]Event)
c := appengine.NewContext(r)
events := GetAllEvents(c, true) // active only
//c.Infof("# events to check for cron: %v", len(events))
for key, event := range events {
//c.Infof("checking event: %s", event.Title)
res := event.Notify(c, false)
if res {
org, _ := GetOrganizationByName(c, event.Orgs[0])
location, _ := time.LoadLocation(org.TimeZone)
event.Due = event.Due.In(location)
event.DueFormatted = event.Due.Format("01/02/2006 3:04pm")
p.Events[key] = event
}
}
renderTemplate(w, "cron", p)
}
// from: https://groups.google.com/d/msg/golang-nuts/-pqkICuokio/KqJ0091EzVcJ
func removeDuplicates(a []string) []string {
result := []string{}
seen := map[string]string{}
for _, val := range a {
if _, ok := seen[val]; !ok {
result = append(result, val)
seen[val] = val
}
}
return result
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,273
|
{"url":"https:\/\/kerodon.net\/tag\/02FD","text":"Kerodon\n\n$\\Newextarrow{\\xRightarrow}{5,5}{0x21D2}$ $\\newcommand\\empty{}$\n\nLemma 6.2.2.8. Let $\\operatorname{\\mathcal{C}}$ be an $\\infty$-category and let $\\operatorname{\\mathcal{C}}' \\subseteq \\operatorname{\\mathcal{C}}$ be a full subcategory. Then $\\operatorname{\\mathcal{C}}'$ is reflective if and only if there exists a $L: \\operatorname{\\mathcal{C}}\\rightarrow \\operatorname{\\mathcal{C}}'$ and a natural transformation $\\eta : \\operatorname{id}_{\\operatorname{\\mathcal{C}}} \\rightarrow L$ which exhibits $L$ as a $\\operatorname{\\mathcal{C}}'$-reflection functor.\n\nProof. Assume that $\\operatorname{\\mathcal{C}}'$ is a reflective subcategory of $\\operatorname{\\mathcal{C}}$; we will show that there exists a functor $L: \\operatorname{\\mathcal{C}}\\rightarrow \\operatorname{\\mathcal{C}}'$ and a natural transformation $\\eta : \\operatorname{id}_{\\operatorname{\\mathcal{C}}} \\rightarrow L$ which exhibits $L$ as a $\\operatorname{\\mathcal{C}}'$-reflection functor (the reverse implication is immediate from the definitions). Let $\\operatorname{\\mathcal{E}}$ be the full subcategory of $\\operatorname{\\mathcal{C}}\\times \\Delta ^1$ spanned by those objects $(X,i)$ having the property that if $i=1$, then $X$ belongs to the full subcategory $\\operatorname{\\mathcal{C}}'$. Let $\\pi : \\operatorname{\\mathcal{E}}\\rightarrow \\Delta ^1$ denote the projection map. Let $\\widetilde{u}: (X,0) \\rightarrow (Y,1)$ be a morphism in $\\operatorname{\\mathcal{E}}$, corresponding to a morphism $u: X \\rightarrow Y$ in $\\operatorname{\\mathcal{C}}$ for which the target $Y$ belongs to $\\operatorname{\\mathcal{C}}'$. By virtue of Corollary 5.1.2.4, the morphism $\\widetilde{u}$ is $\\pi$-cocartesian if and only if $u$ exhibits $Y$ as a $\\operatorname{\\mathcal{C}}'$-localization of $X$. Consequently, our assumption that $\\operatorname{\\mathcal{C}}'$ is a reflective subcategory of $\\operatorname{\\mathcal{C}}$ guarantees that $\\pi$ is a cocartesian fibration of $\\infty$-categories. Applying Proposition 5.2.2.5, we deduce that there exists a functor\n\n$L: \\operatorname{\\mathcal{C}}\\simeq \\{ 0\\} \\times _{\\Delta ^1} \\operatorname{\\mathcal{E}}\\rightarrow \\{ 1\\} \\times _{\\Delta ^1} \\operatorname{\\mathcal{E}}\\simeq \\operatorname{\\mathcal{C}}'$\n\nand a morphism $\\widetilde{\\eta }: \\operatorname{id}_{\\operatorname{\\mathcal{C}}} \\rightarrow L$ in the $\\infty$-category $\\operatorname{Fun}( \\operatorname{\\mathcal{C}}, \\operatorname{\\mathcal{E}})$ which carries each object $X \\in \\operatorname{\\mathcal{C}}$ to a $\\pi$-cocartesian morphism $(X,0) \\rightarrow (L(X),1)$ in $\\operatorname{\\mathcal{E}}$. Composing with the projection map $\\pi : \\operatorname{\\mathcal{E}}\\rightarrow \\Delta ^1$, we obtain a natural transformation $\\eta : \\operatorname{id}_{\\operatorname{\\mathcal{C}}} \\rightarrow L$ in $\\operatorname{Fun}(\\operatorname{\\mathcal{C}}, \\operatorname{\\mathcal{C}})$ which exhibits $L$ as a $\\operatorname{\\mathcal{C}}'$-reflection functor. $\\square$","date":"2021-05-18 01:57:36","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\": 2, \"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\": 0.9953175187110901, \"perplexity\": 51.95522197374975}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991650.73\/warc\/CC-MAIN-20210518002309-20210518032309-00633.warc.gz\"}"}
| null | null |
\section{Introduction}
In a recent paper \cite{soccerREF} we considered the description
of macroscopic bodies within the relative-locality framework based on
curved-momentum-space. Our main result was to show that in
an idealized case, in which the constituents of a macroscopic body are modelled as being in rigid motion, an apparently
troublesome aspect of the description of macroscopic bodies, the
so-called ``soccer-ball problem", is actually absent. This is
essentially because the deformation scale reflecting curvature of
momentum space, which is assumed to be of order $M_P$ for a single constituent, scales like
$N M_P$ for a body with $N$ constituents in rigid motion. This was
reassuring, but for full assurance we should consider the case in
which the momenta of the constituents fluctuate around their mean.
This can be called the quasi-rigid motion.
In the closing remarks of Ref.\ \cite{soccerREF}
we offered a simple argument encouraging the hypothesis that also
for bodies in quasi-rigid motion our solution of the soccer-ball
problem should stand. However, in a Comment \cite{sabicomment},
Hossenfelder criticized our argument. Using a
random-walk model of thermal fluctuations in a macroscopic body,
Hossenfelder claimed that non-linear terms arising from the curvature of momentum space can accumulate leading to the apparently paradoxical
result that the fluctuations overwhelm the contributions of averages.
If this was true the case of constituents in quasi-rigid motion
would be unmanageably pathological.
Here we address this significant challenge to the cogency of the relative locality hypothesis by giving a more precise analysis of
the relevant thermal fluctuations.
We find, first of all, that the question posed by Hossenfelder is
interesting and leads to a new type of investigation in which
thermal fluctuations probe the geometry of momentum space away from
small momenta. This may potentially open up new opportunities
to constrain the geometry of momentum space from the properties of
macroscopic bodies.
Our analysis leads to the identification of three
classes of momentum space geometries. There are cases in which
Hossenfeldler's worry is realized and the fluctuations of the
momenta of the constituents of a hot macroscopic body have effects
which scale like positive powers of $N$ in ways that seem to
contradict our observations. On the other hand there are also
many cases in which the dangerous terms cancel and the
fluctuations of the deformed theory behave like those of standard
statistical physics. Most interesting is, however, the
third, in between, case, in which there may be new physical effects
by which the thermodynamics of large bodies may be able to serve as
a probe of the geometry of momentum space for macroscopic values of
momentum and energy.
Our analysis is more careful than both our brief remarks in
\cite{soccerREF} and Hossenfelder's approach \cite{sabicomment},
which did not keep all terms necessary to do a consistent
perturbative analysis of the effects of the fluctuations. Taking
all terms into account at each order means that we can constrain
the geometry of the addition rule
to higher order that was possible before, and still leave us with a open set of non trivial possibilities.
\section{Total-momentum fluctuations within a consistent perturbative approach}
We take as starting point $P^{(N)}\equiv
\oplus_{a=1}^{N}(\bar{p}+\delta p^{a}) $ to be the total momentum
of our soccer ball in the relative-locality framework. Here,
$\bar{p}$ is the average value of the momenta for each constituent,
and $\delta p^{a}$ is the fluctuation of this value for the
constituent $a$. The precise definition is by recurrence $P^{(N+1)}=
P^{(N)}\oplus (\bar{p} +\delta p^{N+1}),$ with $P^{(0)}=0$. The
deformed addition rule $\oplus$ is a manifestation of the geometry
of momentum space, which, as done in \cite{sabicomment}, we describe
in terms of a perturbative series in inverse powers of $\/M_P$: \begin{equation}
(p\oplus q)_{\alpha}= p_{\alpha} + q_{\alpha} + M_{P}^{-1}\Gamma_{\alpha}{}^{\mu\nu}
p_{\mu}q_{\nu} + M_{P}^{-2}\Delta_{\alpha}{}^{\mu\nu\rho}
p_{\mu}p_{\nu}q_{\rho} +
M_{P}^{-2}\tilde{\Delta}_{\alpha}{}^{\mu\nu\rho} q_{\mu}q_{\nu}
p_{\rho}+ \cdots \label{quadratic} \end{equation} Notice that we included terms
up to quadratic order in $\/M_P$, whereas Hossenfelder only
considered the linear order. We shall soon see that a correct
estimation of the size of fluctuations of the total momentum of a
body reveals that the leading contribution is quadratic in $\/M_P$,
so that truncations of the composition law to linear order, such as
Hossenfelder's, cannot give a consistent description.
Also note that we did not include in (\ref{quadratic}) terms of the form $\Gamma pp$ or $\Delta ppp$ or $\Delta qqq$ because,
as already done in \cite{soccerREF}, \cite{sabicomment}, we work in connection-normal coordinates.
This are the coordinates in which the composition laws is linear for collinear momenta, $(a p \oplus b p)_{\mu} = (a+b) p_{\mu}$
(for any real number $a,b $). It also
implies that the following conditions must be satisfied:
\begin{equation}\label{Normalcoord}
\Gamma_{\alpha}^{(\mu\nu)} =0,\qquad \Delta_{\alpha}{}^{(\mu\nu\rho)} +\tilde{\Delta}_{\alpha}{}^{(\mu\nu\rho)}=0,
\end{equation}
where the bracket means symmetrization of indices.
For the analysis of how small fluctuations of the momenta of the constituents
affect the total momentum of the body we consider contributions
to the total momentum which are linear in the fluctuations:
\begin{equation}
P_{\alpha}^{(N)}= N\bar{p}_{\alpha} + \sum_{a} (W_{a}^{(N)}(\bar{p}))_{\alpha}^{\beta} \delta p^{a}_{\beta} + \cdots,
\label{forW}
\end{equation}
where the dots stands for higher-order terms in $\delta p$. The first term takes the simple form $N\bar{p}_{\alpha}$
thanks to the adoption of connection-normal coordinates.
$W_{a}^{(N)}(\bar{p})$ in the second term is the coefficient of $\delta p^{a}$ in the expansion of $P_{\alpha}^{(N)}$
and can be explicitly calculated from the knowledge of $\oplus$.
In particular on the basis of (\ref{quadratic}) our task will be to compute $W_{a}^{(N)}$ to quadratic order
in $1/M_P$.
For genuine fluctuations we demand that the second term in
(\ref{forW}) is much smaller than the first one and average to zero in the large $N$ limit.
Hossenfelder's observes that it is not enough to demand that the
fluctuations are small. For consistency we need to evaluate the size of fluctuations
around the mean value and show that it is negligible as well.
The size of fluctuations depends crucially on $W_{a}^{(N)}$ through
\begin{equation}
(\sigma^2)_{\alpha \beta}
= \frac{1}{N^{2}}\sum_{a,b}
\langle (W_{a}\delta p^{a})_{\alpha} (W_{b}\delta p^{b})_{\beta} \rangle
\end{equation} where $\langle \cdots\rangle$ denotes the average that will be
described in the next section.
We definitely agree with Hossenfelder
that a fully satisfactory analysis of macroscopic bodies must
consider the size of fluctuations around the mean value. However, we
here show that the type of contributions to $\sigma^2$ on which
Hossenfelder's analysis focused, which have quadratic dependence on
the Planck scale, require an analysis of the composition law to the
$M_{P}^{-2}$ order. As we shall show, the fact that
Ref.\ \cite{sabicomment} only included order-$M_{P}^{-1}$
corrections to the composition law led Hossenfelder to incorrect
results.
\section{Size of fluctuations in the non-relativistic regime}
Having characterized the needed properties of $W_{a}(\bar{p})$
we are now ready for estimating the size of the fluctuations.
For this we shall assume\footnote{We agree with Ref.\cite{sabicomment}
that the idealization of completely random fluctuations might be a good starting point
for the investigation of the problem at hand. Still it should be appreciated
that for some macroscopic bodies of definite interest this idealization is unrealistic.
In particular in a solid there are long-range binding forces and a better approximation is to
assume gaussian fluctuations where $\langle \delta p^{a}_{i}\rangle =0$, but now
$\langle \delta p^{a}_{i}\delta p^{b}_{j}\rangle = \delta^{ab}\Delta_{ij}$, where $\Delta_{ij}$ is the
inverse interaction kernel. This kernel depends on where the atoms are in the solid.},
as done in Hossenfelder's Comment \cite{sabicomment}, that
the momenta $\vec{p}$ fluctuate randomly, $\langle \delta p^{a}_{i}\rangle =0$,
$\langle \delta p^{a}_{i}\delta p^{b}_{j}\rangle \propto \delta^{ab}\delta_{ij}$.
The point to be stressed is that the fluctuations must be
`physical", and when binding forces are neglected (as here and in
\cite{sabicomment}) the on-shellness of each constituent particle
must be preserved. This condition was not imposed, in
\cite{sabicomment} so in general the fluctuations described in
\cite{sabicomment} would be unphysical virtual fluctuations. We
shall here improve the analysis also in this respect.
Since the bulk of our experimental knowledge of the properties of
macroscopic bodies is in the non-relativistic regime, we shall focus
on the case where $\bar{p}_{j} \ll \bar{p}_{0} \simeq m$ (and assume
again for simplicity the all constituents have the same mass $m$).
Here we assume for simplicity that in the non-relativistic limit
the on-shellness relation for constituent particles is undeformed
to second order.
This means that we can take
$e\equiv\bar{p}_{0} = m +\vec{p}^{2}/(2m)$ and $e\delta e -
\vec{p}\delta \vec{p}=0$, i.e. $ \delta e = \bar p_i\delta p_i/e.$
In the non-relativistic limit we can rely on the Boltzmannian
distribution, with the one particle probability density given by $
f(p) \sim \exp(- \frac{p^{2}}{2mT})$, and therefore \begin{equation}\label{fluct}
\langle \delta p_{i}^{a}\delta p_{j}^{b}\rangle = m T \delta^{ab} \delta_{ij}.
\end{equation} The fluctuation in energy follows from the mass-shell relation
$\delta e = u_{i} \delta p_{i}$ where $u_{i}=p_i/m$ is the spatial
velocity. Even if we are in the non relativistic limit we can use a
relativistic notation to write down the fluctuation of both energy
and momenta, we introduce a 4-velocity $u_{\alpha}= (1,u_{i})$ and we
can write the non relativistic fluctuation as: \begin{equation} \langle \delta
p_{\alpha}^{a}\rangle =0,\qquad \langle \delta p_{\alpha}^{a}\delta
p_{\beta}^{b}\rangle = mT\delta^{ab}\left(\eta_{\alpha \beta} +
2u_{\alpha}u_{\beta}\right). \end{equation}
We can now use this fluctuation model to evaluate the size of fluctuation to be given by the average
$ N^{2} (\sigma^{2})_{\alpha\beta} = \langle \sum_{a,b}W_{a}^{(N)}(\bar{p})_{\alpha}^{\beta} W_{b}^{(N)}(\bar{p})_{\alpha'}^{\beta'} \delta p^{a}_{\beta} \delta p^{b}_{\beta'}\rangle$. With the Boltzmannian model this gives
\begin{eqnarray}
(\sigma^{2})_{\alpha\a'} =\frac{m T}{N^{2}} \left< \left( \sum_{a} [(W_{a}^{(N)}(\bar{p}))^{T}\eta W_{a}^{(N)}(\bar{p})]_{\alpha\alpha'}+ \frac2{m^{2}}
\sum_{a}
[W_{a}^{(N)}(\bar{p})\bar{p}]_{\alpha}[W_{a}^{(N)}(\bar{p})\bar{p}]_{\alpha'}\right)
\right> \label{joc} \end{eqnarray} where ${}^{T}$ denotes the transposition.
The first key point is that in our connection-normal coordinates the
second term in (\ref{joc}) vanishes identically: \begin{equation}
[W_{a}^{(N)}(\bar{p})\bar{p}]_{\alpha} =0. \end{equation} The size of
fluctuations is therefore simply given by \begin{equation} (\sigma^{2}
)_{\alpha\a'}=\frac{m T}{N^{2}} \left( \sum_{a}
[(W_{a}^{(N)}(\bar{p}))^{T}\eta
W_{a}^{(N)}(\bar{p})]_{\alpha\alpha'}\right) \, . \end{equation} The
goal is now to evaluate this object up to quadratic order.
\section{Pertubative expansion}
Let us first presents the results we obtain for the expansion of
$W_{a}(\bar{p})$ and the fluctuations to second order. Some relevant
results are collected in the Appendix. From the expansion
(\ref{19a}), the definition of $W_{a}^{(N)}(\bar{p})$, and
restricting our focus to a rest frame analysis, so that $\bar
p_{\alpha}= m\delta_{\alpha}^{0}$, we get \begin{eqnarray} (W_{a}^{(N)})^\beta_\alpha=
\delta^\beta_\alpha - \frac{m}{M_{P}}(N+1-2a)
\Gamma_{\alpha}{}^{0\beta} + \frac{m^{2}}{M_{P}^{2}}\left( A_{a}^{N}
\Gamma^{0}\Gamma^{0} + B_{a}^{N} S^{00} + C_{a}^{N}
{\tilde{\Delta}}^{00}+D^{N}_{a} {\Delta}^{ 00 }\right)^\beta_\alpha.
\nonumber \end{eqnarray} where $A_{a}^{N},B_{a}^{N},C_{a}^{N},D^{N}_{a}$ are
$N$ dependent coefficients, explicitly derived in the Appendix.
Here we use the matrix notation
$(\Gamma^{0})_{\alpha}^{\beta}\equiv \Gamma_{\alpha}{}^{0\beta}$, similarly $ (S^{00})_{\alpha}^{\beta}\equiv S_{\alpha}{}^{00\beta}$ etc...
and we have defined $S_{\alpha}{}^{\beta \gamma \delta} \equiv 3 \Delta_{\alpha}{}^{(\beta \gamma \delta)}$.
Since $N$ can be very large for a macroscopic body it is
particularly important to keep track of the order N of each term.
We find that the dominant contributions come from \begin{equation} \sum_{a=1}^{N}
(N+1-2a)^{2}\approx \frac{N^{3}}{3},\qquad \sum_{a=1}^{N} B_{a}^{N}
\approx \frac{N^{3}}{3},\qquad \sum_{a=1}^{N} D^{N}_{a} \approx
\frac34 N^{2}. \end{equation} We see that the $D$ term is only of order $N^{2}$
instead of $N^{3}$, The other sums involved in the evaluation of the
fluctuation are all of order less or equal to $N^{2}$.
Equipped with this, we can now evaluate the the first term in
(\ref{joc}) finding that \begin{eqnarray}\label{fluctuation}
\frac{(\sigma^2)_{\alpha \alpha'}}{m^2} = \frac{ T}{N m}
\delta_{\alpha \alpha'} - \frac{T }{M_{P}} [(\Gamma^{0}\eta)^{T} + \eta\Gamma^{0}]_{\alpha \alpha'}
+ \frac{m T N}{3 M_{P}^{2}}\left[ (\Gamma^{0}\eta \Gamma^{0}) +[(S^{00}\eta)^{T}+ \eta S^{00}] +O(1/N) \right]_{\alpha \alpha'}
\end{eqnarray}
${}^{T}$ denotes the transposition.
Let us pause briefly for comparing our constructive result to the
estimate proposed by Hossenfelder in \cite{sabicomment}. Within our
notation that estimate would take the form
$$\frac{\sigma^2{}_{Hoss} }{m^{2}}\sim (\Gamma^{0})^{2} \frac{T^{2} N}{M_P^{2}}.$$
It should be noticed first of all that Hossenfelder's heuristic estimate does not
reproduce exactly any of the terms in the result (\ref{fluctuation}) we here derived.
The single term in Hossenfelder's estimate does agree roughly with our term $(\Gamma^{0}\eta \Gamma^{0})$ in (\ref{fluctuation}),
however the size of fluctuations is controlled by a $mT$ dependence in our case while she uses $T^{2}$.
More importantly the single estimate is not a consistent approximation of our result since it misses the contribution
going like $(S^{00}\eta)^{T}+ \eta S^{00}$, which is of the same order. This is of course connected to our earlier remarks
about the consistency of a perturbative approach to this sort of derivation: our $(\Gamma^{0}\eta \Gamma^{0})$ term originates
from the $O(M_P^{-1})$ contribution to the composition law while our $(S^{00}\eta)^{T}+ \eta S^{00}$
term originates from the $O(M_P^{-2})$ contribution to the composition law, but they both appear at the same order of
approximation of $\sigma^2$.\\
The difference is not only quantitative but also very importantly qualitative:
whereas Hossenfelder was using her evaluation as motivation for excluding all composition laws
at leading order, we find that, even if one did insist on excluding the corresponding
type of corrections, there is only a constraint on the relationship between
leading-order form of the composition law, affecting $\Gamma^{0}\eta \Gamma^{0}$,
and next-to-leading-order form of the composition law, affecting $(S^{00}\eta)^{T}+ \eta S^{00}$.
Having derived the full ($1/M_P^2$-order) expression we can also
analyze the structure of (\ref{fluctuation}) from a wider
perspective. The first term is the usual fluctuation term which goes
away in the large $N$ limit. The second term is proportional to the
linear evaluation of the non-metricity tensor $N_{ij}\equiv \Gamma_{i}{}^{0}{}_{j} +
\Gamma_{j}{}^{0}{}_{i}$. It corresponds to a correction to the usual
fluctuations which is suppress by a very small factor $ {T
}/{M_{P}}$. If we demand the non-metricity to vanish this implies
that the connection coefficient are entirely determined by the
torsion.
The only problematic term in this expansion is the last term
proportional to $((\Gamma^{0})^{T}\eta \Gamma^{0}) +
[(S^{00}\eta)^{T}+ \eta S^{00}] $, it is of order $NmT/M_{P}^{2}$.
The demand that the fluctuations do not scale with $N$, results in a
condition for one of the second order coefficients given by
$((\Gamma^{0})^{T}\eta \Gamma^{0}) + [(S^{00}\eta)^{T}+ \eta S^{00}]
=0$, in components this simply reads \begin{equation} \Gamma_{i}{}^{0}{}_{k}
\Gamma_{j}{}^{0 k} +2 S_{(i j)}{}^{00} =0, \label{result} \end{equation} which
tells us that the symmetric part of $S^{00}$ is fixed, while the
rest of the components of $\Delta^{00}, \tilde{\Delta}^{00}$ are
free. When this condition is satisfied we do still have additional
corrections proportional to $\frac{mT}{M_{P}^{2}} \Delta^{00}$, but
evidently none of these contributions is problematic.
\section{Outlook}
We had shown in \cite{soccerREF} that, contrary to earlier naive
arguments, there is no interpretational challenges nor any
phenomenological paradoxes in the idealized case of body composed of
constituents in rigid motion.
We here showed that even for bodies whose
constituents are in quasi-rigid
motion the relative locality framework does not in general encounter any pathologies, extending the analysis
up to quadratic order in the (inverse of) the Planck scale for effects linear in the fluctuations
of the momenta of the constituents.
Within our analysis the description of macroscopic bodies is completely unproblematic for the large class of non-trivial
momentum-space geometries that satisfy the condition (\ref{result}).
It is interesting to contemplate the possibility that geometries
that do not satisfy condition (\ref{result}) might be ruled out
experimentally exploiting the $N$ dependence of $\sigma^2$. But
here, we could also advocate a more phenomenological approach. We
start by noticing that corrections of order $NmT/M_{P}^{2}$ are
still extremely small (smaller than one part in $10^{22}$) for an
actual soccerball at soccer-playing temperatures of about 300
Kelvin, so we can confirm that the original `soccer-ball problem'
\cite{soccerREF} is not going to be a matter of contention with any
choice of momentum-space geometry. As suggested by Hossenfelder
there might still be a problem for macroscopic bodies in some
extreme cases, such as ultra-hot and ultra-massive bodies studied in
astrophysics \cite{sabicomment}. But instead of jumping to the
conclusion that mere existence of such astrophysical bodies suffices
for excluding some momentum-space geometries, we feel a more
prudently scientific approach would be to seek the opportunity to
analyze in future works an actual measurement result on such bodies
which is affected by large corrections, for suitable choices of
momentum-space geometry. A first step toward this goal would be to
generalize the analysis we here reported so that it could apply to
case with $T \gtrsim m$ (within the nonrelativistic limit, on which
we here focused, only cases with $m \ll T$ can be consistently
studied). Moreover, it will be necessary to gain some control also
about the interactions among macroscopic bodies in the
relative-locality framework: in principle for some momentum-space
geometries large corrections might be present for the abstract
notion of total momentum of a body and yet not be present in our
measurements of properties of that body, which inevitably require a
role for interactions with the body. This is clearly a line of
research worth pursuing since the opportunity of excluding at least
some geometries of momentum space would be extremely valuable for
the relative-locality research program.
Another interesting challenge we leave for future studies concerns the geometric interpretation
of results such as our condition (\ref{result}).
It is very tempting to assume that conditions such as (\ref{result}), when analyzed beyond the level of dry relationships
among tensors, might be characterizing in meaningful geometric way desirable (or needed) properties of momentum-space
geometry.
In closing we sketch out an argument suggesting that this could be the case and which opens new investigations.
We know from general theory of relative locality that $W_{a}^{(N)}$ is a transport operator from $0$ to $N\bar{p}$.
we also know that this transport operator is determined to first order by a connection evaluated at $0$.
Let us assume, for the sake of our argument, that this is true beyond first order.
That is lets suppose that $W_{a}^{(N)}$ can be written as the parallel transport of a connection $\Gamma$ along a path,
labelled by $a$, from $0$ to $N\bar{p}$.
That is we assume that $W_{a}(\bar{p}) =P_{a}\exp\int_{0}^{N\bar{p}} \Gamma $, where $P_{a}$ denotes the path ordering
along a path $a$. Even if that might not be generally true, assuming it gives us a key insight.
Let us also suppose that the momentum space geometry is such that the
connection is compatible with the
metric, that is that is $\nabla \eta =0$, in other words the
connection is the Levi-Civita connection plus torsion possibly.
Integrating out this equation can be straightforwardly done and
leads to the following metric compatibility condition on the
transport operator: \begin{equation} [(W_{a}^{(N)}(\bar{p}))^{T}\eta
W_{a}^{(N)}(\bar{p})]_{\alpha\alpha'} = \eta_{\alpha\a'}(N\bar{p}) \end{equation}
where $\eta(p)$ is the metric at $p$ while $\eta= \eta(0)$. This
suggest that, under the conditions we specified,
the fluctuations would be exploring the geometry of momentum space away from the origin.
This would lead to the exciting opportunity of exploring the geometry of momentum space away from the origin
by harnessing the power of the large number $N$.
It also opens a new set of mathematical investigation on when our hypothesis can be realize.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,459
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Q: check screen orientation with configChanges I am using android:configChanges="orientation|keyboardHidden|screenSize" so that activity does not start again when screen rotation changes from landscape to portrait and vice versa.
I have some code idea so I will solve my problem but for that Is there anyway by which I can check screen orientation onCreate() of the activity though I am using confidChanges with orientation ???
A: public void onConfigurationChanged(Configuration newConfig) {
super.onConfigurationChanged(newConfig);
Display display = ((WindowManager) getSystemService(WINDOW_SERVICE)).getDefaultDisplay();
int orient = display.getOrientation();
switch(orient) {
case Configuration.ORIENTATION_PORTRAIT:
if(!oAllow) {
setRequestedOrientation (ActivityInfo.SCREEN_ORIENTATION_LANDSCAPE);
}
break;
case Configuration.ORIENTATION_LANDSCAPE:
if(!oAllow) {
setRequestedOrientation (ActivityInfo.SCREEN_ORIENTATION_PORTRAIT);
}
break;
}
}
A: You can use getResources().getConfiguration().orientation in any place to know what is the orientation of the screen.
And, override onConfigurationChanged(Configuration newConfig) to handle orientation changes while running activity
Updated:
From documentation:
In some special cases, you may want to bypass restarting of your activity based on one or more types of configuration changes. This is done with the android:configChanges attribute in its manifest. For any types of configuration changes you say that you handle there, you will receive a call to your current activity's onConfigurationChanged(Configuration) method instead of being restarted.
A: Please Check this it will detects the screen orientation you no need to mention it in manifest you can get the orienation.
OrientationEventListener _orientaion = new OrientationEventListener(
MainActivity.this) {
@Override
public void onOrientationChanged(int arg0) {
System.out.println("Wortking");
}
};
_orientaion.enable();
i think this will helps you.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,484
|
\section{Introduction}
\label{sec-intro}
Gravitational Microlensing differs from other exoplanet detection methods do to its sensitivity to low-mass
planets \citep{bennett96} orbiting beyond the snow line \citep{gouldloeb92}, where planet formation is
thought to be most efficient \citep{lissauer_araa,pollack96}. This unique sensitivity
allows microlensing to yield unique insights into the demographics of these wider orbit planets.
\citet{suzuki16} found a break and likely peak in the mass ratio function that was later
confirmed to be a peak at a mass ratio of $q_{\rm peak} \simeq 6\times 10^{-5}$ \citep{udalski18,hwang19},
which is close to the Neptune-Sun mass ratio. The smooth, power-law distribution of the \citet{suzuki16,suzuki18}
30-planet sample also does not match predictions of a sub-Saturn mass desert \citep{idalin04} in the exoplanet
mass ratio distribution. This is thought to be caused by the runaway gas accretion process, which
predicts rapid growth through mass ratios between $10^{-4} \leq q \leq 4\times 10^{-4}$, so that few
planets are expected at these mass ratios. However, this prediction contradicts the \citet{suzuki16,suzuki18}
results, and a comparison to population synthesis models \citep{idalin04,mor09} shows that
these models under-predict the abundance of $10^{-4} \leq q \leq 4\times 10^{-4}$ planets
by a factor of ten, or more if standard prescriptions for planet migration are used. This conclusion that
runaway gas accretion does produce a sub-Saturn mass gap in the planet distribution beyond the snow
line is supported by ALMA observations \citep{nayaskshin19}.
Another region of exoplanet parameter space where microlensing may have unique sensitivity is
for planets in stellar binary systems with star-star or planet-star separations of a few AU
\citep{gouldloeb92}. This is the
separation region where microlensing is most sensitive because it corresponds to the typical Einstein
radius for microlensing events towards the Galactic bulge. Kepler has found a number of circumbinary
planets in closer orbits around relatively tight binaries \citep{doyle11,welch12,welch15,orosz12,kostov13,kostov14},
and a number of these are close to the stability limit where the planetary orbit would become unstable
\citep{holman99}. However, the preponderance of such planets is thought to be a selection effect, as
short period planets are much easier to detect with the transit method. Circumbinary planets in
wider orbits, like the first
circumbinary planet found by microlensing \citep{bennett16} and the widest orbit circumbinary planet
found by Kepler \citep{kostov16}, are thought to form more easily and be more common than
circumstellar planets \citep{thebault_highig15}.
The short period circumbinary planets are generally thought to have formed in wider orbits and then
migrated inward to their present positions. (Note that some previous discussion of planets in binary
systems has used an unfortunate, confusing terminology, referring to circumbinary planets as
``P-type" and circumstellar planets as ``S-type". We reject this nomenclature as unnecessarily confusing,
and we urge other authors to do the same.)
The situation is somewhat different for circumstellar planets in binary systems. The majority of these
systems have very wide stellar binary separations or $> 100\,$AU
\citep{mugrauer09,roell12}, and these wide binary companions are thought to have little effect on planet
formation \citep{thebault_highig15} or stability \citep{holman99}, except in cases where the eccentricity
of the wide binary pair becomes unstable \citep{kaib13,smullen_kratter16}.
The situation is different for binary systems with much closer
orbits. However, there are a number of planets orbiting one of a pair of stars in much closer orbits. In particular,
$\gamma$ Cephei A \citep{hatzes03,neuhauser07}, HD 41004 A \citep{zucker04} and HD196885 A
\citep{correia08,chauvin11} are in binary systems with separations of $\sim 20\,$AU and host planets
with $1.6 \,M_{\rm Jup} \simlt M_{Ab} \simlt 2.6 \,M_{\rm Jup}$ with planetary semi-major axes of
$2.0\,{\rm AU} \simlt a_{Ab} \simlt 2.6\,{\rm AU}$.
Two similar systems have been found by microlensing, but
they host much lower mass planets. OGLE-2008-BLG-092LA hosts a planet
with a mass ratio of $q_{Ab} = 2.4\times 10^{-4}$ with a stellar companion
of mass ratio $q_B = 0.22$ \citep{ogle092}. In units of the Einstein radius, the primary-planet separation
is $s_{Ab} = 5.26$,
and the primary-secondary separation is $s_{AB} = 17.0$. The masses of the lens system are not known,
but a Bayesian analysis, assuming all lens stars have an equal chance of hosting the observed planet
gives rough estimates of $M_A \sim 0.7\,\msun$ for the host, $M_B \sim 0.15\,\msun$ for the companion,
and $m_{Ab} \sim 57\mearth = 0.18 M_{\rm Jup}$ for the planet. The estimated physical separations
are $a_{Ab} \sim 18\,$AU for the planet and $a_{AB} \sim 58\,$AU, but these are based on the measured
projected separations on the plane of the sky, so one of the separations could be significantly larger than this.
OGLE-2013-BLG-0341LB hosts a planet with a much smaller mass ratio of
$q_p \approx 5\times 10^{-5}$, and the discovery paper \citep{gould14} reports
a microlensing parallax signal that yields measured masses.
The masses of the primary, secondary (and planet host) are
determined to be $M_A \approx 0.15 \msun$, $M_B \approx 0.13\msun$ and $m_{Bb} = 2\mearth$.
The projected separations are $a_{AB} \approx 12\,$AU for the two stars and $a_{Bb} \approx 0.8\,$AU
for the secondary and the planet.
As in the case of circumbinary systems, it is thought that the presence of a binary companion can
interfere with planet formation at several different stages \citep{thebault_highig15}. First, binary
companions can truncate the protoplanetary disk. The inner disk is truncated for circumbinary
planets and the outer disk is truncated for circumstellar planets. This can limit the amount of material
that can be made into planets. Then, the binary companion can also heat up the disk, and this
can interfere with two stages of the core accretion process. First, the initial growth of small grains can
be slowed or halted if the grains collide at high velocities, and second, the planetesimal accumulation
phase can also be slowed or halted by high relative velocities of these km-sized bodies. Thus, it is
thought that planets can only form through the standard core accretion process only in regions that are
not very close to the planetary orbit stability limits found by \citet{holman99}. Thus, the numerous
circumbinary planets found in the Kepler data near this stability limit are thought to have formed
in wider orbits and then migrated inward. Conversely, the five circumstellar planets mentioned above
could presumably have formed in closer orbits and migrated outward. However, outward migration
is thought to be more difficult to achieve than inward migration. Theoretical arguments
\citep{nelson00,zsom11,picogna13} suggest that a binary companion at 30-50\,AU could inhibit planet
formation through the core accretion method at the grain growth phase, and \citet{paar08} and
\citet{thebault11} argue that the orbits where the $\gamma$ Cephei Ab and HD196885 Ab planets
are now located are probably too perturbed to allow the planetesimal accretion process to occur. A search for
binary companions to Kepler planet-hosting stars indicates that stars hosting Kepler planets are
more than two times less likely have a stellar companion at $< 50\,$AU than stars without a detected
Kepler planet \citep{kraus16}. Thus, the
three planets in binary systems discovered by radial velocities (Cephei Ab, HD196885 Ab, and HD 41004 Ab)
and two planets in binary systems discovered by microlensing
(OGLE-2008-BLG-092LAb and OGLE-2013-BLG-0341LBb) are expected to have formed in a way more
complicated than the standard core accretion scenario. It could be that the planet or stellar companion
have moved from an orbit that provided a larger separation between the planet and the companion to the
host star, or it could be that the formation process from these planets differs from the standard core
accretion scenario.
In order to understand how such systems form, it would be useful to have more examples. Fortunately,
there are reasons to think that there are additional examples of such systems in existing microlensing
data. \citet{gould14} argued that the discovery of the OGLE-2013-BLG-0341LBb planet was lucky in
the sense that there were two planetary signals, one due to the planetary caustic and one due to the
central caustic. The planetary caustic signal was very easy to interpret, but the central caustic signal also
implied the presence of the planet, although it was not so easy to identify the planetary signal due to the
central caustic. Because of this, \citet{gould14} argued that there were likely many planetary signals in stellar
binary events that had yet to be recognized.
OGLE-2006-BLG-284 is one such event. It is unique in that the
projected separation of the primary star and planet is nearly equal to the projected separation between the primary
and secondary stars. We should note, however, that there is one other event published as
planet in a binary system, with very similar planet-primary and secondary-primary separations,
OGLE-2016-BLG-0613 \citep{han_ob160613}, but unpublished MOA data appears to
contradict the published models\footnote{See the video from the 2019 Microlensing Conference
day 3, part 3, at about the 30 minute mark, available at
\tt https://www.simonsfoundation.org/event/23rd-international-microlensing-conference/}.
An acceptable model has not yet been found for this event, but
the photometry data will be provided by the first author upon request.
This paper is organized as follows. Section~\ref{sec-lc_data} discusses the data set and photometry,
and Section~\ref{sec-lc} presents the light curve modeling. We present an
extinction estimate, and the source angular radius in Section~\ref{sec-radius}. In Section~\ref{sec-lens_prop},
we derive the properties of the lens system that can be determined from the light curve analysis, and
finally, in Section~\ref{sec-conclude}, we discuss how future observations may be able to improve our
understanding of this system planets in binary systems in general.
\section{Light Curve Data and Photometry}
\label{sec-lc_data}
Microlensing event OGLE-2006-BLG-284, at ${\rm RA} =17$:58:38.22,
${\rm DEC} = -29$:08:12.0, and Galactic coordinates $(l, b) = (1.2771, -2.5505)$, was
identified and announced as a microlensing candidate by the Optical Gravitational Lensing
Experiment (OGLE) Collaboration as a part of the OGLE-III survey \citep{udalski08}.
It was identified as a binary microlensing event in an OGLE catalog of binary events from
2006-2008 \citep{jar10}, but this paper contained no discussion of the feature due to the planet
that does not fit the binary microlensing model.
This event was discovered in the OGLE-III bulge field BLG206, but it occurred a region of sky where two
bulge fields overlap, and so there is data from OGLE-III bulge field BLG205 for this event, as well.
The event was not detected by the MOA alert system, which was only partly functional in
2006, due to the lack of baseline data available in the first year of the MOA II survey.
The event was found in the 9-year (2006-2014) retrospective analysis of the MOA II survey
data, which included systematic modeling of all binary lens events that has yielded a
number of newly discovered planetary events \citep{kondo19}. This data is now available
at the NASA Exoplanet archive\footnote{\tt https://exoplanetarchive.ipac.caltech.edu/docs/MOAMission.html}
under the star ID gb9-R-3-6-14546. While this public data set leads to the same conclusions, we have
used a modified version of the MOA difference imaging pipeline \citep{bond01,bond17} that
automatically calibrates the photometry to the OGLE-III catalog \citep{ogle3-phot}.
Both the MOA and OGLE
photometry used the difference imaging method \citep{tom96,ala98}.
The OGLE Collaboration provided
optimal centroid photometry using the
OGLE difference imaging pipeline \citep{ogle-pipeline}.
\section{Light Curve Models}
\label{sec-lc}
Our light curve modeling was done using the image centered ray-shooting method
\citep{bennett96} with the initial condition grid search method described in
\citet{bennett-himag}. As is typical for triple lens events, the modeling took place in two stages.
First, we searched for stellar binary solutions to the OGLE and MOA data sets with the
observations with $3892.2 < t < 3895.3$ removed. (We define our time parameter to be
the modified heliocentric Julian Day, ${\rm HJD} - 2450000$.) This search used the
modified version of the initial condition grid search method that replaced the
Einstein radius crossing time, $t_E$, and time of closest approach between the lens center-of-mass
and the source star, $t_0$, with the times of the caustic entry and exit. This method greatly speeds
up the search for the best solutions. This search led to a unique best fit solution, with the next best
binary lens model
disfavored by $\Delta\chi^2 = 372.6$.
\begin{figure}
\epsscale{0.9}
\plotone{lc_ob06284d_bsplad_3r.pdf}
\caption{The best triple lens model for the OGLE-2006-BLG-284 light curve.
The MOA-red data are shown in blue while the OGLE $I$-band data from fields BLG206
and BLG205 are shown in dark and light red, respectively. The OGLE $V$-band data from field
BLG206, is shown in green. The solid line is the best fit
model. Panel (a) shows the magnified region of the stellar binary light curve, and
panels (b) and (c) show close-ups of the planetary feature and the binary caustic exit resolved
by OGLE that enable the measurement of the source radius crossing time, $t_*$.
\label{fig-lc}}
\end{figure}
The second step was to fix the binary lens parameters and search for a triple lens model
that could account for the feature at $3892 < t < 3893.5$ in the light curve (see Figure~\ref{fig-lc}).
(We define our time parameter as the modified Heliocentric Julian Day, $t = {\rm HJD} -2450000$.)
This analysis led to a number of possible solutions that maintained basically the same light
curve away from this short duration anomaly. Close-ups of this short duration light curve anomaly
for the three best triple lens light curve solutions
are shown in Figure~\ref{fig-lc_comp}.
\begin{figure}
\plottwo{caus_bsplad_3.pdf}{caus_bsplad_3_cu.pdf}
\caption{The caustic configuration for the best fit model is plotted in units of the
Einstein radius. The line with the arrow represents the motion of the center of the source
star, and the blue line and circle indicates the trajectory and
size of the source star. The red dots in the right panel
indicate the positions of the lens stars and planet. The close-up of the top planetary
caustic in the right panel indicates that the planetary caustic overlaps with the stellar
binary caustic, but has no clear interaction with it due to the fact that the caustics are
related to distinct, different images.
\label{fig-caustic}}
\end{figure}
The best fit model, shown in Figure~\ref{fig-lc}, has a planetary cusp approach at
${\rm HJD} -2450000 = t = 3892.5$. The caustic structure for this lens system is
shown in Figure~\ref{fig-caustic}, and it is somewhat unusual for a triple lens system because
the planetary caustic crosses the stellar binary caustic with no interaction with it,
although \citet{danek15,danek19} have shown some examples like this. This is because
the caustics affect different images. (There are 4 images when the source is outside all the
caustics, 6 images when the source is inside one caustic, and 8 images when it is inside
two caustic curves.) The parameters of the best fit model shown in Figure~\ref{fig-lc}, as
well as the two best planetary caustic crossing models, are given in Table~\ref{tab-mparams}.
The parameters that these model
has in common with a single lens model are the Einstein radius crossing time, $t_E$,
and the time, $t_0$, and distance, $u_0$, of closest approach between the lens center-of-mass
and the source star. We use the triple lens parameter system of the first published triple lens
system \citep{gaudi-ogle109,bennett-ogle109}, with one minor modification. We use the mass
ratios to the primary star instead of mass fractions of the total lens system mass.
For this event, we assign the primary star to be mass 3, the secondary star to be mass 1, and the planet to be
mass 2. The two mass ratio parameters are then $q_1$ for the secondary star, and $q_2$ for the planet.
The separation between masses 2 and 3 is given by $s_{23}$ and the separation between mass 1
and the center of mass for masses 2 and 3 is given by $s_{1\rm cm}$. The angle between the source
trajectory and the axis connecting mass 1 with the center-of-mass for masses 2 and 3 is given by
$\alpha_{1\rm cm}$, and the angle between this axis and the line connecting masses 2 and 3 is
given by $\phi_{23}$. The final lens model parameters are the source radius crossing time, $t_*$,
and the $I$ and $V$ band source magnitudes, $I_S$ and $V_S$.
The length parameters, $u_0$, $s_{1\rm cm}$ and $s_{23}$, are normalized by the Einstein radius of the
total lens system mass,
$R_E = \sqrt{(4GM/c^2)D_Sx(1-x)}$, where $x = D_L/D_S$ and $D_L$ and $D_S$ are
the lens and source distances, respectively. ($G$ and $c$ are the Gravitational constant
and speed of light, as usual.)
\begin{figure}
\epsscale{1.0}
\plotone{lc_ob06284d_3_plr5.pdf}
\caption{Comparison of the planetary features for the 3 best 2-star plus 1-planet
models. The planetary feature in the best fit model, which has a planetary cusp
approach feature is shown in panel (a). Panels (b) and (c) show the second and
third best models which feature planetary caustic crossings. The parameters of
these models are given in Table~\ref{tab-mparams}.
\label{fig-lc_comp}}
\end{figure}
\begin{deluxetable}{cccccc}
\tablecaption{Best Model Parameters
\label{tab-mparams} }
\tablewidth{0pt}
\tablehead{
\colhead{parameter} & \colhead{units} &
\colhead{best fit} & \colhead{caustic-1} & \colhead{caustic-2} & \colhead{MCMC averages}
}
\startdata
$t_E$ & days &39.677 & 39.872 & 39.969 & $39.38 \pm 0.97$ \\
$t_0$ & ${\rm HJD}-2453800$ & 99.4440 & 99.4060 & 99.2922 & $99.425\pm 0.066$ \\
$u_0$ & & 0.077523 & 0.079517 & 0.081596 & $0.0776\pm 0.0020$ \\
$s_{1\rm cm}$ & & 0.79801 & 0.79856 & 0.79881 & $0.8007 \pm 0.0091$ \\
$s_{23}$ & & 0.76396 & 0.78233 & 0.78404 & $0.7615\pm 0.0063$ \\
$\alpha_{1\rm cm}$ & radians & 0.32432 & 0.33979 & 0.35519 & $0.3237\pm 0.0039$ \\
$\phi_{23}$ & radians & 0.05590 & 0.02075 & -0.02990 & $0.0607\pm 0.0093$ \\
$q_1$ & & 0.28445 & 0.27180 & 0.27395 & $0.289 \pm 0.011$ \\
$q_2$ & $10^{-4}$ & 11.632 & 4.2933 & 0.71234& $12.6 \pm 1.9$ \\
$t_\ast$ & days & 0.03367 & 0.03321 & 0.03311 & $0.03372\pm 0.00030$ \\
$I_s$ & & 20.007 & 20.012 & 19.998 & $19.995\pm 0.036$ \\
$V_s$ & & 21.815 & 21.827 & 21.807 & $21.803\pm 0.036$ \\
fit $\chi^2$ & & 12414.60 & 12435.16 & 12447.96 & \\
for 12407 & d.o.f. & & & & \\
\enddata
\end{deluxetable}
For every passband, there are two parameters to describe the unlensed source
brightness and the combined brightness of any unlensed ``blend" stars that are
superimposed on the source. Such ``blend" stars are quite common because
microlensing is only seen if the lens-source alignment is $\simlt \theta_E \simlt 1\,$mas,
while stars are unresolved in ground based images if their separation is
$\simlt 1^{\prime\prime}$. These
source and blend fluxes are treated differently from the other parameters because
the observed brightness has a linear dependence on them, so for each set of
nonlinear parameters, we can find the source and blend fluxes that minimize the
$\chi^2$ exactly, using standard linear algebra methods \citep{rhie_98smc1}.
The MOA data immediately after the planetary light curve feature at $t \simeq 3892.6$ are crucial
for excluding caustic crossing planetary models that would have a caustic exit at $3893.0 < t < 3893.5$,
and the best remaining planetary caustic crossing models are the second and third best models
with planetary features shown in Figures~ \ref{fig-lc_comp}b and c.
The $\chi^2$ improvement for the best fit model, shown in Figures~\ref{fig-lc} and \ref{fig-lc_comp}a,
over these competing models
is relatively small, with $\chi^2$ improvements of $\Delta\chi^2 = 20.6$, and $\Delta\chi^2 = 33.4$ over
the second and third best models (shown in Figure~\ref{fig-lc_comp}b and c), respectively. While these
$\chi^2$ improvements are rather modest, we believe that they are sufficient to exclude these second
and third best models. The best fit model is preferred by both the OGLE and MOA data sets.
The $\chi^2$ difference between the second best fit model and the best fit model is occurs primarily at
$3892 < t < 3895$, while the $\chi^2$ difference between the third best and best fit models occurs primarily at
$3893 < t < 3896$. Both the second and third best models have planetary caustic exit features that
the data has no indication. For the second best model (Figure \ref{fig-lc_comp}b), the caustic exit
is mostly squeezed between two data points, but it does get a modest $\chi^2$ penalty because
the caustic exit feature is a bit wider than the gap between the data points at $t = 3893.73$ and
$t = 3893.83$. Similarly, this model also has strong caustic entrance just after the last MOA data point
of the previous night at $t = 3892.299$. The timing of these two caustic features requires rather unlikely
coincidences, so this lends credence to the idea that the best fit model is the correct one. The
third best model (Figure \ref{fig-lc_comp}c) has such a weak planetary caustic exit that it does not
get a significant $\chi^2$ penalty, although the data also provide no caustic exit signal. Both the
second and third best models predict a lower magnification than is observed in the time after these planetary
caustic exits, $3892.86 < t < 3895$, and the data in this range contribute substantially
to the $\chi^2$ differences between these two models and the best fit model. Because these
competing models have additional circumstantial evidence against them, we are comfortable in treating their
likelihood with the Gaussian probability, $e^{-\Delta\chi^2/2}$, in the analysis that follows.
In addition to triple lens models, we also searched for binary source, binary lens models, sometimes
referred to as 2L2S models. The best 2L2S model we found has a $\chi^2$ larger than
our best triple lens (3L1S) model by $\Delta\chi^2 = 552.56$. This is not surprising because it would
require a much fainter second source to explain the low amplitude feature that we attribute to the
planet as a feature due to the stellar binary. However, if the second source is much fainter than the
primary, then it should also be much redder, but a much redder source would cause a shift in the
MOA-red data and a large shift in the OGLE $V$ band data with respect to the OGLE $I$ band data
at the time of the feature that we attribute to the planet. Therefore, we will not consider these models
further.
\section{Photometric Calibration and Source Radius}
\label{sec-radius}
The light curve models listed in Table~\ref{tab-mparams} constrain the finite source
size through measurement of the source radius crossing time, $t_*$, and this allows us to derive the angular Einstein
radius, $\theta_E = \theta_* t_E/t_*$, if we know the angular size of the source star, $\theta_*$.
This can be derived from the extinction corrected brightness and color of source star
\citep{kervella_dwarf,boyajian14}.
\begin{figure}
\epsscale{0.9}
\plotone{cmd_ob06284_VIhsta_r90.pdf}
\caption{The $(V-I,I)$ color magnitude diagram (CMD) of the OGLE-III stars within
$90^{\prime\prime}$ of OGLE-2006-BLG-284 transformed to calibrated
Johnson $V$ and Cousins $I$ using the transformation given by \citet{ogle3-phot}. The red spot indicates
red clump giant centroid, and the blue indicates the source magnitude
and color. The green dots represent the HST Baade's Window CMD of \citet{holtzman98} transformed
to the extinction and Galactic longitude appropriate for this field.
\label{fig-cmd}}
\end{figure}
In order to estimate the source radius, we need extinction-corrected magnitudes,
which can be determined from the magnitude and color of the centroid of the
red clump giant feature in the CMD \citep{yoo_rad}, as indicated
in Figure~\ref{fig-cmd}. We find that the
red clump centroid in this field is at $I_{\rm cl} = 15.48$, $(V-I)_{\rm cl} = 2.04$,
which implies $V_{\rm cl} = 17.52$. From \citet{nataf13}, we find that the extinction corrected
red clump centroid should be at $I_{\rm cl,0} = 14.39$, $(V-I)_{\rm cl0,} = 1.06$, which implies
$I$ and extinctions of $A_I = 1.09\pm 0.05$ and a color excess of $E(V-I) = 0.098\pm 0.03$. So, the extinction corrected
source magnitude and color are $I_{s0} = 18.905$ and $(V-I)_{s0} = 0.828$ for the average model
from our MCMC calculations reported in Table~\ref{tab-mparams}.
These dereddened magnitudes can be used to determine the angular source radius,
$\theta_*$. With the source magnitudes that we have measured, the most precise determination
of $\theta_*$ comes from the $(V-I),I$ relation. We use
\begin{equation}
\log_{10}\left[2\theta_*/(1 {\rm \mu as})\right] = 0.501414 + 0.419685\,(V-I)_{s0} -0.2\,I_{s0} \ ,
\label{eq-thetaS}
\end{equation}
which comes from the \citet{boyajian14} analysis, but with the color range optimized for the
needs of microlensing surveys. These numbers were provided in a private communication from
T.S.\ Boyajian (2014). Therefore, we handle this
uncertainty in our MCMC calculations, so as to include all the correlations in our determination
of the lens system properties.
For the average model parameters from our MCMC calculation, listed in Table~\ref{tab-mparams}, we find
$\theta_* = 0.585\pm 0.029\,\mu$as.
Figure~\ref{fig-cmd} also includes the location of the source star on the CMD, in blue, and the
{\it Hubble Space Telescope} $V$ and $I$ band CMD from \citet{holtzman98} after it has been
shifted to the average distance and extinction of the red clump stars in the OGLE-2006-BLG284 field.
The position of the
source star in Figure~\ref{fig-cmd} indicates that it is on the red edge of the main sequence. However,
the CMD is likely to have a larger dispersion in the field of this event, due to the higher extinction in
this field.
\section{Lens System Properties}
\label{sec-lens_prop}
\begin{deluxetable}{cccc}
\tablecaption{Physical Parameters\label{tab-pparams}}
\tablewidth{0pt}
\tablehead{
\colhead{Parameter} & \colhead{units} & \colhead{Value} & \colhead{2-$\sigma$ range} \\
}
\startdata
$\theta_E$ & mas & $0.681\pm 0.035$ & 0.612-0.750 \\
$\mu_{\rm rel,G}$ & mas/yr & $6.31\pm 0.33$ & 5.66-6.97 \\
$D_S $ & kpc & $8.6\pm 1.2$ & 6.2-10.7 \\
$D_L $ & kpc & $4.0\pm 1.5$ & 1.2-6.5 \\
$M_{L1}$ & $\msun$ & $0.35^{+0.30}_{-0.20}$ & 0.06-0.84 \\
$M_{L2}$ & $\msun$ & $0.100^{+0.097}_{-0.056}$ & 0.018-0.24 \\
$m_p$ & $\mearth$ & $144^{+137}_{-82}$ & 25-375 \\
$a_{\perp,ss}$ & AU & $2.06\pm 0.74$ & 0.61-3.33 \\
$a_{\perp,s1p}$ & AU & $2.17\pm 0.78$ & 0.65-3.51 \\
$V_L$ & mag & $26.4^{+1.8}_{-2.4}$ & 22.3-37.2 \\
$I_L$ & mag & $22.9^{+1.1}_{-1.6}$ & 20.2-31.0 \\
$K_L$ & mag & $19.7^{+0.7}_{-1.0}$ & 17.8--26.8 \\
\enddata
\tablecomments{ Mean values and RMS are given for $\theta_E$, $\mu_{\rm rel,G}$, $D_S$, $D_L $, $a_{\perp,ss}$,
and $a_{\perp,sp}$.
Median values and 68.3\% confidence intervals are given for the other parameters. 2-$\sigma$ range refers to the
central 95.3\% confidence interval.}
\end{deluxetable}
With our determination of $\theta_*$ from the source magnitude and color in Section~\ref{sec-radius}, we can
now proceed to determine the lens system properties. The angular Einstein radius is given by
$\theta_E = \theta_* t_E/t_*$, which allows us to use the following relation \citep{bennett_rev,gaudi_araa}
\begin{equation}
M_L = {c^2\over 4G} \theta_E^2 {D_S D_L\over D_S - D_L}
= 0.9823\,\msun \left({\theta_E\over 1\,{\rm mas}}\right)^2\left({x\over 1-x}\right)
\left({D_S\over 8\,{\rm kpc}}\right) \ ,
\label{eq-m_thetaE}
\end{equation}
where $x = D_L/D_S$ to determine the relationship between the lens system mass, $M_L$ and
distance, $D_L$. We know that the source is very likely to be approximately at the distance
of the Galactic bulge, but the bulge is bar shaped and pointed at approximately the location of
our Solar system. As a result, there is an uncertainty of a few kpc in the distance to the source
star, $D_S$. Also, the probability of the lens system mass and distance also depend on the
the Geocentric lens-source relative proper motion, which can be determined from the
angular source size, the source angular radius, and the source radius crossing
time $\mu_{\rm rel,G} = \theta_*/t_*$. We can combine all these factors with a Galactic
model prior to determine our best estimate of the properties of this binary star plus planet
system. We use the Galactic model of \citet{bennett14}, and we assume that the planet
hosting probability is independent of the planet mass, because this is the simplest
assumption to make. We also include the 2nd and third best models in our collection of
MCMC models to combine with the Galactic prior, but we weight them by $e^{-\Delta\chi^2/2}$,
which gives them very little weight.
The results are presented in Figure~\ref{fig-lens_prop} and Table~\ref{tab-pparams}.
This table and figure introduce some new parameters. The primary and secondary stellar masses
are given by $M_{L1}$ and $M_{L2}$, and the planet mass is $m_p$. Table~\ref{tab-pparams} reports
the projected separations, $a_{\perp,ss}$ and $a_{\perp,s1p}$, between the two stars, and between
the primary star and planet, respectively. Instead of these projected separations, Figure~\ref{fig-lens_prop},
shows the predicted distribution of 3-dimensional separations, under the assumption of random
orientations. However, the orientations of the primary-secondary stellar separation and the
primary star-planet separations cannot be random. They must be anti-correlated in order to obey
orbital stability requirements \citep{holman99}. For a circumstellar planet orbiting the
primary star, with the observed secondary-primary mass ratio of $0.289\pm 0.011$, the
semi-major axis of the planet must be $\leq 0.38$ times the semi-major axis of the stellar binary system,
if we assume that both orbits are nearly circular. In the circumbinary case, \citet{holman99} find
that the planet must have a semi-major axis $\geq 2.2$ times the semi-major axis of the stellar binary system,
assuming nearly circular orbits. Since the observed separations in Table~\ref{tab-pparams} are nearly
identical, this implies that the 3-dimensional separation of either the primary star and the planet or the
two planets must be $\geq 2.2$ times the projected separations.
\begin{figure}
\epsscale{1.08}
\plotone{lens_prop_ogle284_a3d_LS_8panS.pdf}
\caption{The lens properties from our Bayesian analysis. Host-1 and Host-2 are the
primary and secondary stars of the system. The stellar and planetary separations are the
separations from the primary star. The blue lines in the bottom two panels indicate the
source magnitudes.
\label{fig-lens_prop}}
\end{figure}
The bottom two panels of Figure~\ref{fig-lens_prop} have blue lines that indicate the magnitudes of the combined
light from both source stars as vertical blue lines. In June, 2020, it will be 14 years since this event has
occurred, so the lens-source separation should be $>88.3 \pm 4.6\,$mas after June 2020. Our
experience with Keck adaptive optics observations \citep{bennett20,aparna20,terry20} indicates
that the lens should be detectable at this separation if it is within 3 magnitudes of the source
brightness, and our Bayesian analysis indicates that there is a 96.4\% chance that the lens
system is $\leq 3\,$mag fainter than the source stars.
\section{Conclusions}
\label{sec-conclude}
Our analysis of OGLE-III data from two fields and MOA data from the MOA 9-year
retrospective analysis for microlensing event OGLE-2006-BLG-284 has revealed that
this event is due to a triple lens system consisting of two stars and a planet. While there are
two light curve models with $\chi^2$ values that are only $\Delta\chi^2 = 20.6$ and 33.4 higher
than the best fit solution, we argue that these also have very unlikely parameters. And we do
not consider these to be possible solutions for this event.
The prime stumbling block for the interpretation of this event is the fact that the projected separation
of the primary and secondary stars is nearly identical to the projected separation between the planet
and the primary star. Thus, we cannot tell from the light curve data alone whether planet orbits both stars
(circumbinary) or only one (circumstellar). Fortunately, because the event occurred so long ago, the
lens-source separation is now large enough ($\simgt 88\,$mas) that it should be readily detected by
Keck adaptive optics observations, which has now been done for a number of host stars of
microlens planets
\citep{gaudi-ogle109,bennett-ogle109,bennett20,batista15,beaulieu16,beaulieu18,aparna18,aparna20,terry20}.
This may provide an opportunity to make observations that can decide between the
circumbinary and circumstellar possibilities with the next generate of large telescopes.
A time series of radial velocity observations of the lens
system with a precision of $\sim 2\,$km/sec should be able to detect the radial velocity of the
primary star if it has a relatively small semi-major axis of $\simlt 2\,$AU. This would imply that
two stars were in the relatively close orbit and that the planet would have to be in a much wider orbit, with
a large line-of-sight separation. Conversely, the failure to detect any radial velocity for the primary lens star at the
$\sim 2\,$km/sec level would imply that the primary and secondary stars have a wide orbit and the planet
orbits only the primary.
The details of any radial velocity program will depend on the brightness of the lens star system, which
has yet to be determined. The $K_L$ histogram in Figure~\ref{fig-lens_prop} indicates that there is a
95\% chance that the lens system magnitude is in the range $17.5 < K_L < 20.9$, but this also
represents a range of lens system distances of $1.0\,{\rm kpc} < D_L < 7.5\,$kpc and a primary star
mass range of $0.08\msun < M_{L1} < 1.05\msun$. Since the star-star and planet-star separations, as
well as the orbital period, are proportional to $D_L$, this implies a large range of possible masses and
orbital periods, as well as a 3.4 magnitude brightness range. Thus, we cannot really determine the
observational effort needed to test the circumbinary vs.\ circumstellar interpretations until we have
detected the lens star with high angular resolution imaging. Even in the most optimistic scenario
with $K_L < 18$, current telescopes and instruments are unlikely to be able to make such measurements,
but there is some hope for observations with the next generation of extremely large telescopes (ELTs).
However, this may require advanced instrumentation that may not be available at first light on these
ELTs.
This discovery tends to confirm the \citet{gould14} prediction that there are likely to be many undiscovered
planetary signals in the existing catalog of observed microlensing events, and it may help to reduce the
deficit of planets in binary systems noted by \citet{koshimoto20}.
Such events probe a range
of planets in binary systems that are difficult to explore with other methods, and so they are likely to
provide a unique new window on the formation of planets in binary systems and the planet formation
process, in general.
The search for planets in binary systems would benefit from more efficient triple-lens modeling methods
and more efficient methods to search the triple lens parameter space. This problem will come more acute
with the launch of NASA's Wide Field Infrared Survey Telescope (WFIRST) \citep{WFIRST_AFTA},
which will devote a large fraction of its observing time to an exoplanet microlensing survey
\citep{bennett02,bennett18_wfirst,penny19}. This will provide a much larger sample of exoplanets than can be
observed from the ground, and will have much better light curve sampling than any ground based
survey. A more robust triple-lens modeling system will be needed to take advantage of the
circumbinary and circumstellar planets that will be detectable in the WFIRST data.
\acknowledgments
DPB, AB, and CR were supported by NASA through grant NASA-80NSSC18K0274.
The MOA project is supported in Japan by JSPS KAKENHI Grant Numbers JP17H02871, JSPS24253004,
JSPS26247023, JSPS23340064, JSPS15H00781, and JP16H06287.
The work by CR was supported by an appointment to the NASA Postdoctoral Program at the
Goddard Space Flight Center, administered by USRA through a contract with NASA.
NJR is a Royal Society of New Zealand Rutherford Discovery Fellow.
The OGLE Team thanks Profs.\ Marcin Kubiak and Grzegorz Pietrzy{\'n}ski for
their contribution to the OGLE photometric data. The OGLE project has
received funding from the National Science Centre, Poland, grant MAESTRO
2014/14/A/ST9/00121 to AU.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,793
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,640
|
{"url":"https:\/\/hal.inria.fr\/hal-01185599","text":"# Distributional Convergence for the Number of Symbol Comparisons Used by QuickSort (Extended Abstract)\n\nAbstract : Most previous studies of the sorting algorithm $\\mathtt{QuickSort}$ have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that $\\mathtt{QuickSort}$ operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild \"tameness'' condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased (\"fair'') bits, the mean exhibits periodic fluctuations of order $n$.\nKeywords :\nDocument type :\nConference papers\nDomain :\n\nCited literature [20 references]\n\nhttps:\/\/hal.inria.fr\/hal-01185599\nContributor : Coordination Episciences Iam <>\nSubmitted on : Thursday, August 20, 2015 - 4:33:57 PM\nLast modification on : Monday, January 27, 2020 - 11:06:02 AM\nLong-term archiving on: : Wednesday, April 26, 2017 - 10:09:36 AM\n\n### File\n\ndmAM0116.pdf\nPublisher files allowed on an open archive\n\n### Identifiers\n\n\u2022 HAL Id : hal-01185599, version 1\n\n### Citation\n\nJames Allen Fill. Distributional Convergence for the Number of Symbol Comparisons Used by QuickSort (Extended Abstract). 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.219-232. \u27e8hal-01185599\u27e9\n\nRecord views","date":"2021-06-13 17:26:27","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\": 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\": 0.785308301448822, \"perplexity\": 1335.6218342956372}, \"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-2021-25\/segments\/1623487610196.46\/warc\/CC-MAIN-20210613161945-20210613191945-00497.warc.gz\"}"}
| null | null |
Aleida Nuñez (; born Aleida Araceli Nuñez Flores on January 24, 1981) is a Mexican actress, singer and model.
Early years
She was a pageant queen in various regional contests during her adolescence. In 1994 she won the second place in the contest Nuestra Belleza Guanajuato. She began her career in media communications as a conductor in various programs on channel 10 of León. She was also a runway and advertisement model before beginning her career in acting.
Personal life
Aleida Nuñez has a sister named Marcela. In 2009, she married Pablo Glogovsky and gave birth to a boy named Alexander in 2013.
Filmography
Telenovelas
Series
References
External links
Official Site of Aleida Nuñez
Video de Aleida Núñez
1975 births
Living people
Mexican telenovela actresses
Mexican television actresses
Mexican female models
Actresses from Jalisco
Singers from Jalisco
21st-century Mexican actresses
People from Lagos de Moreno, Jalisco
21st-century Mexican singers
21st-century Mexican women singers
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,745
|
Q: Aeson : generics with default values Today I wanted to solve next problem.
Assume that we have typeclass DataWithDefault defined as
class DataWithDefault a where
defaultValue :: a
And we have data Example defined as
data Example =
Example { field1 :: Text
, field2 :: Text
} deriving (Show)
instance DataWithDefault Example where
defaultValue = Example "Hello" "World"
instance FromJSON Example where
parseJSON (Object v) =
Example <$> v .:? "field1" .!= field1 defaultValue
<*> v .:? "field2" .!= field2 defaultValue
parseJSON _ = mzero
instance ToJSON Example where
toJSON (Example f1 f2) =
object [ "field1" .= f1
, "field2" .= f2
]
I know that Aeson uses Generics to derive FromJSON and ToJSON instances automatically, but I can't figure out how to make it derive FromJSON instance with default values for fields that are not represented in given json. Is it possible to do using generics? Actually I don't ask you a final solution, but maybe some clue?
Update
Let me add more information about the problem.
Suppose now that you need to update your Example data and now it defined as
data Example =
Example { field1 :: Text
, field2 :: Text
, field3 :: Int
} deriving (Show)
So you want to update DataWithDefault instance declaration
instance DataWithDefault Example where
defaultValue = Example "Hello" "World" 12
And what I want to do is not to write
instance FromJSON Example where
parseJSON (Object v) =
Example <$> v .:? "field1" .!= field1 defaultValue
<*> v .:? "field2" .!= field2 defaultValue
<*> v .:? "field3" .!= field3 defaultValue
parseJSON _ = mzero
And want to derive such instance definition automatically. And more importantly, I want to do it not only for Example, but for DataWithDefault a.
Update 2
The point of combining .:? and .!= is to get as much as possible fields from given json and set every missing field to it's default value. So when we pass
{ "field1" : "space", "field2" : "ship" }
I want my new example be not field1 = Hello; field2 = World; field3 = 12, but field1 = space; field2 = ship; field3 = 12.
A: Instead of making Aeson do it, just use a newtype for what they were designed for:
newtype DefaultJSON a = DefaultJSON { unDefaultJSON :: a }
instance (FromJSON a, DataWithDefault a) => FromJSON (DefaultJSON a) where
parseJSON v = DefaultJSON <$> (parseJSON v <|> pure defaultValue)
Then you can do
> decode "{}" :: Maybe (DefaultJSON Example)
Just (DefaultJSON {unDefaultJSON = (Example {field1 = "Hello", field2 = "World"}})
This is a little different than what you asked, it provides a default value in case parsing fails, but not a default value for each field in case that individual field is missing.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,060
|
Q: How does $(window).one('load') work, and how does it fail? As some background, I am working on building a web slider activity that fits into/extends another piece of software that outputs HTML. The key point being I have very little to no control over how the software outputs it's HTML Elements, but I can add arbitrary JavaScript and HTML elements on top of what the software outputs.
What I am trying to do is delay the loading of some images that I want to add until after all the software's outputted elements are loaded. So far I have it working by waiting to load the images until after the window load event is fired. Like so:
//-----Simplified example-----
$(document).on('ready', funciton(){
//Some stuff to set things up
$(window).one('load', function(){
$('#SomeDiv').append('
<img class= "someClass"
id="arbitraryImage1"
src="folder/image1.jpg"
');
});
});
Though it is working right now I'm not sure what will happen if something goes wrong. The HTML outputted by the software has a bad habit of including things that it might not need as well as occasionally not loading images on slower connections. The majority of the time if it does fail the user will not notice, so it isn't an issue. However I would like to know how $(window).one('load', function(){})behaves when a problem arises, so that I can work around the error or at least make the activity fail graciously. Any assistance or links to documentation would be appreciated, the generic-ness of 'window' and 'load' brings up a lot of irrelevant information when googling.
To summarize:
What is the behaviour of $(window).one('load', function(){}) when the window fails to load a resource (image, css, etc.)?
A: Good question. The documentation says:
The load event fires at the end of the document loading process. At
this point, all of the objects in the document are in the DOM, and all
the images, scripts, links and sub-frames have finished loading.
https://developer.mozilla.org/en-US/docs/Web/API/GlobalEventHandlers/onload
But in practice, you'll find that the load event fires after all assets queued to load when the document was originally parsed (stylesheets, external scripts, img tags) have loaded, or failed loading. As long as no requests are still pending, the fact that these assets have loaded successfully or not has no bearing on the load event being fired.
The documentation is in need of a clarification, in my opinion, as it doesn't address your question.
A: If image fails to load, then there will be just no image in your image, if it will be unable to find SomeDiv, then nothing will be appended anywhere, generally if javascript function fails to some reasons - then nothing bad will happen, just broken functionality won't work, meaning load will still be done, just console will be telling about errors
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,492
|
""" Module/Package nodes
The top of the tree. Packages are also modules. Modules are what hold a program
together and cross-module optimizations are the most difficult to tackle.
"""
import re
from nuitka import Options, Variables
from nuitka.containers.oset import OrderedSet
from nuitka.importing.Importing import (
findModule,
getModuleNameAndKindFromFilename
)
from nuitka.importing.Recursion import decideRecursion, recurseTo
from nuitka.optimizations.TraceCollections import ConstraintCollectionModule
from nuitka.SourceCodeReferences import SourceCodeReference
from nuitka.utils import Utils
from .Checkers import checkStatementsSequenceOrNone
from .ConstantRefNodes import ExpressionConstantRef
from .FutureSpecs import FutureSpec
from .NodeBases import (
ChildrenHavingMixin,
ClosureGiverNodeBase,
ExpressionMixin,
NodeBase
)
class PythonModuleMixin:
def __init__(self, name, package_name):
assert type(name) is str, type(name)
assert '.' not in name, name
assert package_name is None or \
(type(package_name) is str and package_name != "")
self.name = name
self.package_name = package_name
self.package = None
def getName(self):
return self.name
def getPackage(self):
return self.package_name
def getFullName(self):
if self.package_name:
return self.package_name + '.' + self.getName()
else:
return self.getName()
@staticmethod
def isMainModule():
return False
@staticmethod
def isInternalModule():
return False
def attemptRecursion(self):
# Make sure the package is recursed to.
# Return the list of newly added modules.
result = []
if self.package_name is not None and self.package is None:
package_package, package_filename, _finding = findModule(
importing = self,
source_ref = self.getSourceReference(),
module_name = self.package_name,
parent_package = None,
level = 1,
warn = Utils.python_version < 330
)
# TODO: Temporary, if we can't find the package for Python3.3 that
# is semi-OK, maybe.
if Utils.python_version >= 330 and not package_filename:
return []
assert package_filename is not None, self.package_name
_package_name, package_kind = getModuleNameAndKindFromFilename(package_filename)
# assert _package_name == self.package_name, (package_filename, _package_name, self.package_name)
decision, _reason = decideRecursion(
module_filename = package_filename,
module_name = self.package_name,
module_package = package_package,
module_kind = package_kind
)
if decision is not None:
imported_module, is_added = recurseTo(
module_package = package_package,
module_filename = package_filename,
module_relpath = Utils.relpath(package_filename),
module_kind = "py",
reason = "Containing package of recursed module '%s'." % self.getFullName(),
)
self.package = imported_module
if is_added:
result.append(imported_module)
if self.package:
from nuitka.ModuleRegistry import addUsedModule
addUsedModule(self.package)
# print "Recursed to package", self.package_name
result.extend(self.package.attemptRecursion())
return result
def getCodeName(self):
# Abstract method, pylint: disable=R0201
return None
def getCompileTimeFilename(self):
return Utils.abspath(self.getSourceReference().getFilename())
def getRunTimeFilename(self):
reference_mode = Options.getFileReferenceMode()
if reference_mode == "original":
return self.getCompileTimeFilename()
elif reference_mode == "frozen":
return "<frozen %s>" % self.getFullName()
else:
filename = self.getCompileTimeFilename()
full_name = self.getFullName()
result = Utils.basename(filename)
current = filename
levels = full_name.count('.')
if self.isPythonPackage():
levels += 1
for _i in range(levels):
current = Utils.dirname(current)
result = Utils.joinpath(Utils.basename(current), result)
return result
class PythonModule(PythonModuleMixin, ChildrenHavingMixin,
ClosureGiverNodeBase):
""" Module
The module is the only possible root of a tree. When there are many
modules they form a forest.
"""
kind = "PYTHON_MODULE"
named_children = (
"body",
)
checkers = {
"body": checkStatementsSequenceOrNone
}
def __init__(self, name, package_name, source_ref):
ClosureGiverNodeBase.__init__(
self,
name = name,
code_prefix = "module",
source_ref = source_ref
)
PythonModuleMixin.__init__(
self,
name = name,
package_name = package_name
)
ChildrenHavingMixin.__init__(
self,
values = {
"body" : None # delayed
},
)
self.variables = set()
# The list functions contained in that module.
self.functions = OrderedSet()
self.active_functions = OrderedSet()
self.cross_used_functions = OrderedSet()
# SSA trace based information about the module.
self.constraint_collection = None
def getDetails(self):
return {
"filename" : self.source_ref.getFilename(),
"package" : self.package_name,
"name" : self.name
}
def asXml(self):
result = super(PythonModule, self).asXml()
for function_body in self.active_functions:
result.append(function_body.asXml())
return result
def asGraph(self, computation_counter):
from graphviz import Digraph # @UnresolvedImport pylint: disable=F0401,I0021
graph = Digraph("cluster_%d" % computation_counter, comment = "Graph for %s" % self.getName())
graph.body.append("style=filled")
graph.body.append("color=lightgrey")
graph.body.append("label=Iteration_%d" % computation_counter)
def makeTraceNodeName(variable_trace):
return "%d/ %s %s %s" % (
computation_counter,
variable_trace.getVariable(),
variable_trace.getVersion(),
variable_trace.__class__.__name__
)
for function_body in self.active_functions:
constraint_collection = function_body.constraint_collection
for (_variable, _version), variable_trace in constraint_collection.getVariableTracesAll().items():
node = makeTraceNodeName(variable_trace)
previous = variable_trace.getPrevious()
if variable_trace.hasDefiniteUsages():
graph.attr("node", style = "filled", color = "blue")
elif variable_trace.hasPotentialUsages():
graph.attr("node", style = "filled", color = "yellow")
else:
graph.attr("node", style = "filled", color = "red")
graph.node(node)
if type(previous) is tuple:
for previous in previous:
graph.edge(makeTraceNodeName(previous), node)
elif previous is not None:
graph.edge(makeTraceNodeName(previous), node)
return graph
getBody = ChildrenHavingMixin.childGetter("body")
setBody = ChildrenHavingMixin.childSetter("body")
@staticmethod
def isPythonModule():
return True
def getParent(self):
assert False
def getParentVariableProvider(self):
return None
def getVariables(self):
return self.variables
def getFilename(self):
return self.source_ref.getFilename()
def getVariableForAssignment(self, variable_name):
return self.getProvidedVariable(variable_name)
def getVariableForReference(self, variable_name):
return self.getProvidedVariable(variable_name)
def getVariableForClosure(self, variable_name):
return self.getProvidedVariable(
variable_name = variable_name
)
def createProvidedVariable(self, variable_name):
result = Variables.ModuleVariable(
module = self,
variable_name = variable_name
)
assert result not in self.variables
self.variables.add(result)
return result
@staticmethod
def getContainingClassDictCreation():
return None
def isEarlyClosure(self):
# Modules should immediately closure variables on use.
# pylint: disable=R0201
return True
def getCodeName(self):
def r(match):
c = match.group()
if c == '.':
return '$'
else:
return "$$%d$" % ord(c)
return "".join(
re.sub("[^a-zA-Z0-9_]", r ,c)
for c in
self.getFullName()
)
def addFunction(self, function_body):
assert function_body not in self.functions
self.functions.add(function_body)
def getFunctions(self):
return self.functions
def startTraversal(self):
self.active_functions = OrderedSet()
def addUsedFunction(self, function_body):
assert function_body in self.functions
assert function_body.isExpressionFunctionBody()
if function_body not in self.active_functions:
self.active_functions.add(function_body)
def getUsedFunctions(self):
return self.active_functions
def getUnusedFunctions(self):
for function in self.functions:
if function not in self.active_functions:
yield function
def addCrossUsedFunction(self, function_body):
if function_body not in self.cross_used_functions:
self.cross_used_functions.add(function_body)
def getCrossUsedFunctions(self):
return self.cross_used_functions
def getOutputFilename(self):
main_filename = self.getFilename()
if main_filename.endswith(".py"):
result = main_filename[:-3]
else:
result = main_filename
# There are some characters that somehow are passed to shell, by
# Scons or unknown, so lets avoid them for now.
return result.replace(')',"").replace('(',"")
# TODO: Can't really use locals for modules, this should probably be made
# sure to not be used.
@staticmethod
def getLocalsMode():
return "copy"
def computeModule(self):
old_collection = self.constraint_collection
self.constraint_collection = ConstraintCollectionModule()
module_body = self.getBody()
if module_body is not None:
result = module_body.computeStatementsSequence(
constraint_collection = self.constraint_collection
)
if result is not module_body:
self.setBody(result)
new_modules = self.attemptRecursion()
for new_module in new_modules:
self.constraint_collection.signalChange(
source_ref = new_module.getSourceReference(),
tags = "new_code",
message = "Recursed to module package."
)
self.constraint_collection.updateFromCollection(old_collection)
def getTraceCollections(self):
yield self.constraint_collection
for function in self.getUsedFunctions():
yield function.constraint_collection
def hasUnclearLocals(self):
for collection in self.getTraceCollections():
if collection.hasUnclearLocals():
return True
return False
class SingleCreationMixin:
created = set()
def __init__(self):
assert self.__class__ not in self.created
self.created.add(self.__class__)
class PythonMainModule(PythonModule, SingleCreationMixin):
kind = "PYTHON_MAIN_MODULE"
def __init__(self, main_added, source_ref):
PythonModule.__init__(
self,
name = "__main__",
package_name = None,
source_ref = source_ref
)
SingleCreationMixin.__init__(self)
self.main_added = main_added
@staticmethod
def isMainModule():
return True
def getOutputFilename(self):
if self.main_added:
return Utils.dirname(self.getFilename())
else:
return PythonModule.getOutputFilename(self)
class PythonInternalModule(PythonModule, SingleCreationMixin):
kind = "PYTHON_INTERNAL_MODULE"
def __init__(self):
PythonModule.__init__(
self,
name = "__internal__",
package_name = None,
source_ref = SourceCodeReference.fromFilenameAndLine(
filename = "internal",
line = 0,
future_spec = FutureSpec()
)
)
SingleCreationMixin.__init__(self)
@staticmethod
def isInternalModule():
return True
def getOutputFilename(self):
return "__internal"
class PythonPackage(PythonModule):
kind = "PYTHON_PACKAGE"
def __init__(self, name, package_name, source_ref):
assert name
PythonModule.__init__(
self,
name = name,
package_name = package_name,
source_ref = source_ref
)
def getOutputFilename(self):
return Utils.dirname(self.getFilename())
class PythonShlibModule(PythonModuleMixin, NodeBase):
kind = "PYTHON_SHLIB_MODULE"
def __init__(self, name, package_name, source_ref):
NodeBase.__init__(
self,
source_ref = source_ref
)
PythonModuleMixin.__init__(
self,
name = name,
package_name = package_name
)
assert Utils.basename(source_ref.getFilename()) != "<frozen>"
# That is too likely a bug.
assert name != "__main__"
def getDetails(self):
return {
"name" : self.name,
"package_name" : self.package_name
}
def getFilename(self):
return self.getSourceReference().getFilename()
def startTraversal(self):
pass
class ExpressionModuleFileAttributeRef(NodeBase, ExpressionMixin):
kind = "EXPRESSION_MODULE_FILE_ATTRIBUTE_REF"
def __init__(self, source_ref):
NodeBase.__init__(
self,
source_ref = source_ref
)
def mayRaiseException(self, exception_type):
return False
def computeExpression(self, constraint_collection):
# There is not a whole lot to do here, the path will change at run
# time
if Options.getFileReferenceMode() != "runtime":
result = ExpressionConstantRef(
constant = self.getRunTimeFilename(),
source_ref = self.getSourceReference()
)
return result, "new_expression", "Using original '__file__' value."
return self, None, None
def getCompileTimeFilename(self):
return self.getParentModule().getCompileTimeFilename()
def getRunTimeFilename(self):
return self.getParentModule().getRunTimeFilename()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,692
|
{"url":"https:\/\/mathoverflow.net\/questions\/338343\/field-of-definition-from-deformation-rigidity","text":"# Field of definition from deformation rigidity\n\nIt is known that a smooth complex quasi-projective variety which is deformation-rigid (e.g. any holomorphic deformation inside an ambient space is trivial) can be defined over a number field. Can one obtain any information about a field of definition from deformation point of view?","date":"2019-08-18 00:09:04","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9620593190193176, \"perplexity\": 407.55837252718527}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027313501.0\/warc\/CC-MAIN-20190817222907-20190818004907-00367.warc.gz\"}"}
| null | null |
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>primergenjs - primer design made easy</title>
<link rel="shortcut icon" href="./primergen.ico" type="image/ico">
<script type="text/javascript" src="./lib/primergen.js"></script>
<script src="http://code.jquery.com/jquery-latest.min.js"></script>
<script src="http://davidbau.com/encode/seedrandom.min.js"></script>
<script type="text/javascript" src="./lib/primergen-jquery.js"></script>
<style>
p {font-family: TimesNewRoman, "Times New Roman", Times, Baskerville, Georgia, serif; font-size: 28}
div {font-family: font-family: TimesNewRoman, "Times New Roman", Times, Baskerville, Georgia, serif;}
table {font-family: TimesNewRoman, "Times New Roman", Times, Baskerville, Georgia, serif; padding: 15px; }
td{padding: 6px;}
[info=seq] {font-family: "Courier New", Courier, monospace; font-size: 14px; font-weight: bold;}
</style>
</head>
<body>
<p><big> From ./primergen.js </big> </p>
<div class="sequence"></div>
<h1><script type="text/javascript"> document.write($pg.csaSeq.seqName); </script></h1>
sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.csaSeq.sequence); </script>
</span>
</div>
<div> integrity of the sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.csaSeq.integrity); </script>
</span>
</div>
<div>
The sequence length:
<span info="seq">
<script type="text/javascript"> document.write($pg.csaSeq.seqLength); </script>
</span>
</div>
<div> GC content of sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.csaSeq.gcContent); </script>
</span>
</div>
The bases counts:
</div>
<div>
<span info="seq"> A =
<script type="text/javascript"> document.write($pg.csaSeq.sumA); </script>
</span>
</div>
<div>
<span info="seq"> T =
<script type="text/javascript"> document.write($pg.csaSeq.sumT); </script>
</span>
</div>
<div>
<span info="seq"> G =
<script type="text/javascript"> document.write($pg.csaSeq.sumG); </script>
</span>
</div>
<div>
<span info="seq"> C =
<script type="text/javascript"> document.write($pg.csaSeq.sumC); </script>
</span>
</div>
<h1><script type="text/javascript"> document.write($pg.forwardPrimer.seqName); </script></h1>
<div>
sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.forwardPrimer.sequence); </script>
</span>
</div>
<div> integrity of the seq:
<span info="seq">
<script type="text/javascript"> document.write($pg.forwardPrimer.integrity); </script>
</span>
</div>
<div>
The seqence length is:
<span info="seq">
<script type="text/javascript"> document.write($pg.forwardPrimer.seqLength); </script>
</span>
</div>
<div> GC content of sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.forwardPrimer.gcContent); </script>
</span>
</div>
The bases counts:
</div>
<div>
<span info="seq"> A =
<script type="text/javascript"> document.write($pg.forwardPrimer.sumA); </script>
</span>
</div>
<div>
<span info="seq"> T =
<script type="text/javascript"> document.write($pg.forwardPrimer.sumT); </script>
</span>
</div>
<div>
<span info="seq"> G =
<script type="text/javascript"> document.write($pg.forwardPrimer.sumG); </script>
</span>
</div>
<div>
<span info="seq"> C =
<script type="text/javascript"> document.write($pg.forwardPrimer.sumC); </script>
</span>
</div>
<h1><script type="text/javascript"> document.write($pg.reversePrimer.seqName); </script></h1>
<div>
sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.reversePrimer.sequence); </script>
</span>
</div>
<div> integrity of the sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.reversePrimer.integrity); </script>
</span>
</div>
<div>
The sequence length:
<span info="seq">
<script type="text/javascript"> document.write($pg.reversePrimer.seqLength); </script>
</span>
</div>
<div> GC content of sequence:
<span info="seq">
<script type="text/javascript"> document.write($pg.reversePrimer.gcContent); </script>
</span>
</div>
The bases counts:
</div>
<div>
<span info="seq"> A =
<script type="text/javascript"> document.write($pg.reversePrimer.sumA); </script>
</span>
</div>
<div>
<span info="seq"> T =
<script type="text/javascript"> document.write($pg.reversePrimer.sumT); </script>
</span>
</div>
<div>
<span info="seq"> G =
<script type="text/javascript"> document.write($pg.reversePrimer.sumG); </script>
</span>
</div>
<div>
<span info="seq"> C =
<script type="text/javascript"> document.write($pg.reversePrimer.sumC); </script>
</span>
</div>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,878
|
Shatterproof Is Not a Challenge è il secondo album del gruppo alternative rock inglese Hundred Reasons. È maggiormente orientato ad altri generi come il post-grunge e il post-hardcore che l'alternative rock. Non ha ricevuto le stesse ottime critiche del precedente album, tuttavia, contiene diverse tracce come Harmony e Pop che hanno ottenuto buone critiche per la resa live.
Shatterproof non è stato prodotto alla Columbia, bensì alla V2 Records.
Paul Townsend ha lasciato il gruppo in seguito all'uscita di questo album.
La canzone Still Be Here, è stata dedicata dal cantante Colin Doran al padre, reincontrato dopo più di sei anni.
Tracce
Formazione
Colin Doran - voce
Larry Hibbitt - chitarra, voce
Paul Townsend - chitarra, voce
Andy Gilmour - basso
Andy Bews - batteria
Produttore
Dave Sardy
Engineers
Greg Gordon
Juan Garcia
Andy Saroff
Jeff Hoffman
Mixato da
Dave Sardy
Masterizzato da
Stephen Marcussen
Batteria aggiuntiva
Artie Smith
Chitarra aggiuntiva
Stuart Valentine
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,072
|
Hard bed mattress are too firm, they lack bounce and comfort, there is nothing mild about the surface and there is no convenience. Some individuals like the hardness however wind up with chronic discomfort. Shoulder pain, neck and back pain and various other musculoskeletal discomforts are typical when you sleep on a tough mattress. Soft bed mattress is not a better alternative if it cannot evenly disperse the weight and doesn't provide sufficient back or lumbar support. The majority of soft mattresses tend to lose their firmness, despite how firm they were at first. There is no pressure relief at strategic points of the body. In a short span of time, the soft mattress entirely gives in. Medium quality mattresses do not use any respite as they integrate the imperfections of tough and soft bed mattress. They are just as bad for chronic discomfort like difficult bed mattress as they can assist in arthritis like soft bed mattress.
Purple has as many as fifteen patents entering into just one bed mattress. A few of the patents take care of push releasing comfort while some attend to the support, some attend to resilience while some address no feel motion transfer. Purple mattress will not collapse, it will not establish any imprint or impression of the body, it will not lose its firmness or softness, the upper layer or the base layer will not lose their primary qualities, the mattress will be flexible sufficient to endure pressure therefore offering relief at strategic points whenever required. It is much better to be like turf than a tree. Grass can withstand a storm whereas a tree gets rooted out. Buckling of the surface area is not a bad thing when one needs even distribution of weight or pressure. Purple bed mattress is nontoxic and irritant resistant or hypoallergenic. It is made of recyclable materials. The mattress is CertiPUR-US ® accredited.
You might or may not have seen the advertisement video called the Purple mattress egg test. In the video, a sleep specialist uses four eggs at 4 different locations, pasted onto a glass frame that she drops from a height on various bed mattress. The basic objective is to enjoy if the eggs break or they stay undamaged. The expectation is that if a bed mattress can evenly disperse weight and can soak up the weight of a person or any load then there will be no specific force exerted onto any part of the item or person sleeping on the bed mattress. All particular pressure points will have equivalent relief as the weight gets distributed uniformly throughout the bed mattress. Hence, if a mattress is perfect then the eggs will not break. As one would anticipate, the eggs do break when they are enabled to fall from a height being pasted to a glass frame onto a lot of bed mattress with the exception of Purple bed mattress.
Now, there has been criticism of the ad because Purple bed mattress has a grid life development on its topmost layer so the eggs can suit those grids or box like structures without getting blasted due to the pressure of the glass frame onto the upper surface area of the bed mattress. To be fair to Purple, it deserves to demonstrate its product's supremacy. It likewise has videos revealing a side sleeper test, back sleeper test and pressure mapping test. You can enjoy them on the main website.
You can buy Purple mattress on Amazon or on the main website. You can use a Purple mattress discount coupon to get a discount rate. However you would perhaps think about Purple Mattress Vs Casper or Purple Mattress Vs GhostBed prior to you make a final decision. You ought to also consider Nectar mattress, which is superior to Purple, Casper and GhostBed to name a few. Let us quickly explore why.
Casper has 4 foam layers, GhostBed has 3 and Purple has three. Nectar has four. Casper and Purple are thinner than Nectar. Casper, GhostBed and Purple have firmness score of 6 to 7, 6 and a half to 7 and six to six and a half from ten respectively. Nectar has a lower firmness score, thus using more convenience without compromising on stability and assistance. Nectar has the least movement transfer amongst the 4. It is likewise the most affordable. Queen Casper costs $950, Queen GhostBed costs $795 and Queen Purple costs $999. Queen Nectar mattress is tagged at $795 however you would get a $125 discount rate so you pay $670. Nectar has life time guarantee. Casper and Purple have 10 years. GhostBed has twenty years. Casper and Purple have a hundred nights trial while GhostBed has a hundred and one nights. Nectar has 3 hundred and sixty five nights of danger totally free trial.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,658
|
Longmoor steht für:
Longmoor (Carmarthenshire), Dorf in Wales
Longmoor (Hampshire), Ortschaft in Hampshire, England
Standort der britischen Armee in Hampshire, siehe Bordon and Longmoor Military Camps
Longmoor Bog, Naturschutzgebiet in Berkshire, England, Teil des California Country Park
Stadtviertel von Burlington (Ontario)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 291
|
Q: Can someone help me on the makeMove method I am trying to create a tic tac toe game class and one of the problem is the makeMove method in which I have to use the switch method. When I use the tester class it is not working. Can someone show me what the answer/problem should be.
Here is the tic tac toe class
public class TicTacToe
{
private String[][] board;
private int turn;
private boolean hasWinner;
private boolean bale;
/**
* Constructs an empty game board. Sets the turn to zero.
*/
public TicTacToe()
{
board = new String[][] {
{"1", " | ", "2", " | ", "3"},
{"-", "-|-", "-", "-|-", "-"},
{"4", " | ", "5", " | ", "6"},
{"-", "-|-", "-", "-|-", "-"},
{"7", " | ", "8", " | ", "9"},
};
turn = 0;
}
/**
* Prints the two dimensional array gameboard to the console with an empty line before and after
*/
public void drawBoard()
{
for(int rows = 0; rows < board.length; rows++)
{
for( int cols = 0; cols < board[0].length; cols++)
{
System.out.print(board[rows][cols] );
}
System.out.println();
}
}
/**
* Precondition: the location will be a number 1 through 9
* Hint: Loop through the array and see if the number still exists there.
* @param location the numeric location according to the initial configuration
* @return true if the space is unoccupied and legal, false otherwise
*/
public boolean validMove(int location)
{
for (int row = 0; row < board.length; row++) {
for (int col = 0; col < board[0].length; col++) {
if( board[row][col].equals( location))
{
bale = false;
}
else
{
bale = true;
}
}
}
return bale;
}
/**
* Preconditions: the location is valid, the symbol is a single character: X or O.
* @param location the numeric location according to the initial configuration
* @param symbol the player's symbol
*/
public void makeMove(int location, String symbol)
{
int row = 0;
int col = 0;
//map the numeric location to the row and column on the game board
switch(location)
{
case 1: row = 0; col = 0; break;
case 2: row = 0; col = 2; break;
/*#
* Insert the rest of the cases below. More info on switch statements here:
* http://docs.oracle.com/javase/tutorial/java/nutsandbolts/switch.html
* One use is to replace a long series of if/else if/else statements.
*/
case 3: row = 0; col = 1; break;
case 4: row = 2; col = 0; break;
case 5: row = 2; col = 2; break;
case 6: row = 2; col = 1; break;
case 7: row = 1; col = 0; break;
case 8: row = 1; col = 2; break;
case 9: row = 1; col = 1; break;
}
/*#
* Insert the symbol in the appropriate location below
*/
location = symbol;
}
/**
* Checks if the winning condition has been met for either player.
* @return true if there is a winner, false otherwise
*/
public boolean hasWinner()
{
/*#
* if there is a winner from hasWinner("X") or hasWinner("O"), return true
* otherwise false
*/
hasWinner = false;
for (int i = 0; i < board.length; i+=2)
{
if ((board[i][0].equals("X") == true) && (board[i][2].equals("X") == true) && (board[i][4].equals("X") == true))
{
hasWinner = true;
}
else if ((board[i][0].equals("O") == true) && (board[i][2].equals("O") == true) && (board[i][4].equals("O") == true))
{
hasWinner = true;
}
}
for (int i = 0; i < board.length; i+=2)
{
if ((board[0][i].equals("X") == true) && (board[2][i].equals("X") == true) && (board[4][i].equals("X") == true))
{
hasWinner = true;
}
else if ((board[0][i].equals("O") == true) && (board[2][i].equals("O") == true) && (board[4][i].equals("O") == true))
{
hasWinner = true;
}
}
if( (board[0][0].equals("X")==true) && (board[2][2].equals("X")==true) && (board[4][4].equals("X") == true))
{
hasWinner = true;
}
else if( (board[0][0].equals("O")==true) && (board[2][2].equals("O")==true) && (board[4][4].equals("O") == true))
{
hasWinner = true;
}
if( (board[4][4].equals("X")==true) && (board[2][2].equals("X")==true) && (board[0][0].equals("X") == true))
{
hasWinner = true;
}
else if( (board[4][4].equals("O")==true) && (board[2][2].equals("O")==true) && (board[0][0].equals("O") == true))
{
hasWinner = true;
}
return hasWinner;
}
/**
* @param symbol the symbol of the player to check for a win condition
* @return true if there is a winner, false otherwise
*/
private boolean hasWinner(String symbol)
{
/*#
* return true if there are three of symbol in a row vertically, horizontally, or diagonally
* hint: there is a gap of one string between each actual occupiable space, so when looping,
* you can change your updater from i++ to i+=...well, I'll leave it to you to figure out what
* number to use after the +=.
*/
return true;
}
/**
* For debugging and testing only.
* @param state a valid board configuration
*/
public void setGameState(String[][] state)
{
board = state;
}
}
and here is the tester class
public class TicTacToeTester
{
public static void main(String[] args)
{
TicTacToe t = new TicTacToe();
t.drawBoard();
System.out.println("Expected: \n\n1 | 2 | 3\n--|---|--\n4 | 5 | 6\n--|---|--\n7 | 8 | 9\n\n");
System.out.println(t.validMove(4));
System.out.println("Expected: true");
t.makeMove(4, "X");
t.drawBoard();
System.out.println("Expected: \n\n1 | 2 | 3\n--|---|--\nX | 5 | 6\n--|---|--\n7 | 8 | 9\n\n");
System.out.println(t.validMove(4));
System.out.println("Expected: false");
t.makeMove(9, "O");
t.drawBoard();
System.out.println("Expected: \n\n1 | 2 | 3\n--|---|--\nX | 5 | 6\n--|---|--\n7 | 8 | O\n\n");
}
}
Also I tried mutliple times and days to do this, but the code is not does what it is supposed to be expected to do
A: The error comes from location=symbol;,you are trying to convert string to int.If u want to do so,then use either Integer.parseInt() or Integer.valueOf(),but be warned even then,you will get NumberFormtException.
A: First of all, there's this line which doesn't compile, because you're assigning a String to an int:
location = symbol;
I assume you meant to do this here?
board[row][col] = symbol;
Then, your indexes in the switch statement are off. For example, 9 should translate to row 4 and column 4:
And on a final note, where you're writing things like...
if(board[i][0].equals("X") == true)
... you can just write them as...
if(board[i][0].equals("X"))
... and get the same behaviour. If statements execute the code in the body if the expression in them evaluates to true. You're doing an extra step that doesn't do anything, because comparing any boolean value to true just returns the boolean value you had.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,104
|
{"url":"https:\/\/docs.cemosis.fr\/holo3\/doc-pierre\/polymap\/polymap.html","text":"# Polymap\n\n## 1. Components\n\nFigure 1. Polymap\nFigure 2. Polymap without its cover\n\n1. LCD screen\n\n2. Emplacement of the studied object\n\n3. Beam Splitter splitting incoming incident light by transmitting 50% of the light and reflecting 50% of the light\n\n4. Lense\n\n5. Camera\n\n## 2. Technical caracteristics and important notations\n\n\u2022 L: The distance between the object surface and the LCD screen 56\u223c60 mm.\n\n\u2022 \u03b3: The angle between the object surface and direction horizontal.\n\n\u2022 P: The spatial period of the fringes projected by the LCD screen.\n\n\u2022 $\u03a6_w$: the wrapped phase\n\n\u2022 \u03a6: The unwrapped phase\n\n\u2022 (i, j) (or (x, y)): The position of a camera pixel.\n\n\u2022 h: The height of the local surface.\n\n\u2022 n: The normal direction of the local surface.\n\n\u2022 s i,j : The slope corresponds to the camera pixel (i, j).\n\n\u2022 The size of the LCD screen\n\n\u2022 Height: 62.1 mm\/720 pixels\n\n\u2022 Width: 110.4 mm\/1280 pixels.\n\n\u2022 Pixel size of the screen 0.08625 mm.\n\n\u2022 The size of the camera: 1200 x 1920 pixels.\n\n\u2022 k: Pixel size measured by the camera $0.0045*0.0045 mm^2$ .\n\n\u2022 $\\gamma_{abs}(x, y)\\in \\mathbb{R}^+$ : the absolute contrast of pixel (x, y)\n\n\u2022 $\\gamma(x, y) \\in [0,1]$ : the relative contrast of pixel (x, y)\n\n## 3. How it works\n\nThe LCD screen (1) displays sinusoidal greyscale fringes in horizontal direction then in vertical direction. The fringes illuminate the surface of the studied object (2) and the reflection of this light is transmitted to the Camera (5) throught the Beam splitter (3) and the lense (4).\n\nThis process is repeated 4 times, changing the phase of the signal displayed by the LCD screen by $\\frac{\\pi}{2}$ each time.\n\nTable 1. Fringes of the smallest pattern size\nPhase offset Horizontal fringes Vertical fringes\n\n$0$\n\n$\\pi\/2$\n\n$\\pi$\n\n$\\frac{3\\pi}{2}$\n\nWe then repeat this whole process again for different size of fringes.\n\nBy doing so, we can measure the intensity of the reflected light on each pixel for each of the phases used. We then get four intensity maps :\n\n$I_i(x,y) i \\in {0,1,2,3}$\n\nfrom those 4 maps, we can create :\n\n\u2022 The map of mean value of Intensity (when we will later talk about intensity files, we will talk about the mean value of Intensity by default) $I_m(x,y) = \\frac{I_0 + I_1 + I_2 + I_3}{4}$\n\n we have the relationship $I_i = I_m (1 + \\gamma.cos(\\phi + \\frac{i\\pi}{2})) \\forall i \\in {0,1,2,3}$ from this relationship, we can deduce other important informations\n\u2022 The map of absolute contrast $\\gamma_{abs} = \\sqrt{(I_0 - I_2)^2 + (I_3 - I_1)^2}$\n\n\u2022 The map of relative Contrast (when talking about contrast files, we will talk about relative contrast by default) $\\gamma=\\frac{\\sqrt{(I_0 - I_2)^2 + (I_3 - I_1)^2}}{2I_m}$\n\n\u2022 Wrapped Phase map $\\phi_{w}= arctan2(I_3-I_1, I_0 - I_2)\\in ]-\\pi;\\pi]$\n\n\u2022 And from all of the phase map (the biggest size pattern fringes bringing the continuity of phases and the smaller size pattern fringes bringing the precision), we can construct the unwrapped phase map.\n\nFrom Phase data, we can then compute local slopes along vertical and horizontal axis, the information necessary to reconstruct the surface.","date":"2022-05-19 17:40:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 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\": 0.5037965178489685, \"perplexity\": 2577.060408784726}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662529658.48\/warc\/CC-MAIN-20220519172853-20220519202853-00312.warc.gz\"}"}
| null | null |
// © 2016 and later: Unicode, Inc. and others.
// License & terms of use: http://www.unicode.org/copyright.html
#include "utypeinfo.h" // for 'typeid' to work
#include "unicode/ustring.h"
#include "unicode/strenum.h"
#include "unicode/putil.h"
#include "uenumimp.h"
#include "ustrenum.h"
#include "cstring.h"
#include "cmemory.h"
#include "uassert.h"
U_NAMESPACE_BEGIN
// StringEnumeration implementation ---------------------------------------- ***
StringEnumeration::StringEnumeration()
: chars(charsBuffer), charsCapacity(sizeof(charsBuffer)) {
}
StringEnumeration::~StringEnumeration() {
if (chars != NULL && chars != charsBuffer) {
uprv_free(chars);
}
}
// StringEnumeration base class clone() default implementation, does not clone
StringEnumeration *
StringEnumeration::clone() const {
return NULL;
}
const char *
StringEnumeration::next(int32_t *resultLength, UErrorCode &status) {
const UnicodeString *s=snext(status);
if(U_SUCCESS(status) && s!=NULL) {
unistr=*s;
ensureCharsCapacity(unistr.length()+1, status);
if(U_SUCCESS(status)) {
if(resultLength!=NULL) {
*resultLength=unistr.length();
}
unistr.extract(0, INT32_MAX, chars, charsCapacity, US_INV);
return chars;
}
}
return NULL;
}
const UChar *
StringEnumeration::unext(int32_t *resultLength, UErrorCode &status) {
const UnicodeString *s=snext(status);
if(U_SUCCESS(status) && s!=NULL) {
unistr=*s;
if(resultLength!=NULL) {
*resultLength=unistr.length();
}
return unistr.getTerminatedBuffer();
}
return NULL;
}
const UnicodeString *
StringEnumeration::snext(UErrorCode &status) {
int32_t length;
const char *s=next(&length, status);
return setChars(s, length, status);
}
void
StringEnumeration::ensureCharsCapacity(int32_t capacity, UErrorCode &status) {
if(U_SUCCESS(status) && capacity>charsCapacity) {
if(capacity<(charsCapacity+charsCapacity/2)) {
// avoid allocation thrashing
capacity=charsCapacity+charsCapacity/2;
}
if(chars!=charsBuffer) {
uprv_free(chars);
}
chars=(char *)uprv_malloc(capacity);
if(chars==NULL) {
chars=charsBuffer;
charsCapacity=sizeof(charsBuffer);
status=U_MEMORY_ALLOCATION_ERROR;
} else {
charsCapacity=capacity;
}
}
}
UnicodeString *
StringEnumeration::setChars(const char *s, int32_t length, UErrorCode &status) {
if(U_SUCCESS(status) && s!=NULL) {
if(length<0) {
length=(int32_t)uprv_strlen(s);
}
UChar *buffer=unistr.getBuffer(length+1);
if(buffer!=NULL) {
u_charsToUChars(s, buffer, length);
buffer[length]=0;
unistr.releaseBuffer(length);
return &unistr;
} else {
status=U_MEMORY_ALLOCATION_ERROR;
}
}
return NULL;
}
UBool
StringEnumeration::operator==(const StringEnumeration& that)const {
return typeid(*this) == typeid(that);
}
UBool
StringEnumeration::operator!=(const StringEnumeration& that)const {
return !operator==(that);
}
// UStringEnumeration implementation --------------------------------------- ***
UStringEnumeration * U_EXPORT2
UStringEnumeration::fromUEnumeration(
UEnumeration *uenumToAdopt, UErrorCode &status) {
if (U_FAILURE(status)) {
uenum_close(uenumToAdopt);
return NULL;
}
UStringEnumeration *result = new UStringEnumeration(uenumToAdopt);
if (result == NULL) {
status = U_MEMORY_ALLOCATION_ERROR;
uenum_close(uenumToAdopt);
return NULL;
}
return result;
}
UStringEnumeration::UStringEnumeration(UEnumeration* _uenum) :
uenum(_uenum) {
U_ASSERT(_uenum != 0);
}
UStringEnumeration::~UStringEnumeration() {
uenum_close(uenum);
}
int32_t UStringEnumeration::count(UErrorCode& status) const {
return uenum_count(uenum, &status);
}
const char *UStringEnumeration::next(int32_t *resultLength, UErrorCode &status) {
return uenum_next(uenum, resultLength, &status);
}
const UnicodeString* UStringEnumeration::snext(UErrorCode& status) {
int32_t length;
const UChar* str = uenum_unext(uenum, &length, &status);
if (str == 0 || U_FAILURE(status)) {
return 0;
}
return &unistr.setTo(str, length);
}
void UStringEnumeration::reset(UErrorCode& status) {
uenum_reset(uenum, &status);
}
UOBJECT_DEFINE_RTTI_IMPLEMENTATION(UStringEnumeration)
U_NAMESPACE_END
// C wrapper --------------------------------------------------------------- ***
#define THIS(en) ((icu::StringEnumeration*)(en->context))
U_CDECL_BEGIN
/**
* Wrapper API to make StringEnumeration look like UEnumeration.
*/
static void U_CALLCONV
ustrenum_close(UEnumeration* en) {
delete THIS(en);
uprv_free(en);
}
/**
* Wrapper API to make StringEnumeration look like UEnumeration.
*/
static int32_t U_CALLCONV
ustrenum_count(UEnumeration* en,
UErrorCode* ec)
{
return THIS(en)->count(*ec);
}
/**
* Wrapper API to make StringEnumeration look like UEnumeration.
*/
static const UChar* U_CALLCONV
ustrenum_unext(UEnumeration* en,
int32_t* resultLength,
UErrorCode* ec)
{
return THIS(en)->unext(resultLength, *ec);
}
/**
* Wrapper API to make StringEnumeration look like UEnumeration.
*/
static const char* U_CALLCONV
ustrenum_next(UEnumeration* en,
int32_t* resultLength,
UErrorCode* ec)
{
return THIS(en)->next(resultLength, *ec);
}
/**
* Wrapper API to make StringEnumeration look like UEnumeration.
*/
static void U_CALLCONV
ustrenum_reset(UEnumeration* en,
UErrorCode* ec)
{
THIS(en)->reset(*ec);
}
/**
* Pseudo-vtable for UEnumeration wrapper around StringEnumeration.
* The StringEnumeration pointer will be stored in 'context'.
*/
static const UEnumeration USTRENUM_VT = {
NULL,
NULL, // store StringEnumeration pointer here
ustrenum_close,
ustrenum_count,
ustrenum_unext,
ustrenum_next,
ustrenum_reset
};
U_CDECL_END
/**
* Given a StringEnumeration, wrap it in a UEnumeration. The
* StringEnumeration is adopted; after this call, the caller must not
* delete it (regardless of error status).
*/
U_CAPI UEnumeration* U_EXPORT2
uenum_openFromStringEnumeration(icu::StringEnumeration* adopted, UErrorCode* ec) {
UEnumeration* result = NULL;
if (U_SUCCESS(*ec) && adopted != NULL) {
result = (UEnumeration*) uprv_malloc(sizeof(UEnumeration));
if (result == NULL) {
*ec = U_MEMORY_ALLOCATION_ERROR;
} else {
uprv_memcpy(result, &USTRENUM_VT, sizeof(USTRENUM_VT));
result->context = adopted;
}
}
if (result == NULL) {
delete adopted;
}
return result;
}
// C wrapper --------------------------------------------------------------- ***
U_CDECL_BEGIN
typedef struct UCharStringEnumeration {
UEnumeration uenum;
int32_t index, count;
} UCharStringEnumeration;
static void U_CALLCONV
ucharstrenum_close(UEnumeration* en) {
uprv_free(en);
}
static int32_t U_CALLCONV
ucharstrenum_count(UEnumeration* en,
UErrorCode* /*ec*/) {
return ((UCharStringEnumeration*)en)->count;
}
static const UChar* U_CALLCONV
ucharstrenum_unext(UEnumeration* en,
int32_t* resultLength,
UErrorCode* /*ec*/) {
UCharStringEnumeration *e = (UCharStringEnumeration*) en;
if (e->index >= e->count) {
return NULL;
}
const UChar* result = ((const UChar**)e->uenum.context)[e->index++];
if (resultLength) {
*resultLength = (int32_t)u_strlen(result);
}
return result;
}
static const char* U_CALLCONV
ucharstrenum_next(UEnumeration* en,
int32_t* resultLength,
UErrorCode* /*ec*/) {
UCharStringEnumeration *e = (UCharStringEnumeration*) en;
if (e->index >= e->count) {
return NULL;
}
const char* result = ((const char**)e->uenum.context)[e->index++];
if (resultLength) {
*resultLength = (int32_t)uprv_strlen(result);
}
return result;
}
static void U_CALLCONV
ucharstrenum_reset(UEnumeration* en,
UErrorCode* /*ec*/) {
((UCharStringEnumeration*)en)->index = 0;
}
static const UEnumeration UCHARSTRENUM_VT = {
NULL,
NULL, // store StringEnumeration pointer here
ucharstrenum_close,
ucharstrenum_count,
uenum_unextDefault,
ucharstrenum_next,
ucharstrenum_reset
};
static const UEnumeration UCHARSTRENUM_U_VT = {
NULL,
NULL, // store StringEnumeration pointer here
ucharstrenum_close,
ucharstrenum_count,
ucharstrenum_unext,
uenum_nextDefault,
ucharstrenum_reset
};
U_CDECL_END
U_CAPI UEnumeration* U_EXPORT2
uenum_openCharStringsEnumeration(const char* const strings[], int32_t count,
UErrorCode* ec) {
UCharStringEnumeration* result = NULL;
if (U_SUCCESS(*ec) && count >= 0 && (count == 0 || strings != 0)) {
result = (UCharStringEnumeration*) uprv_malloc(sizeof(UCharStringEnumeration));
if (result == NULL) {
*ec = U_MEMORY_ALLOCATION_ERROR;
} else {
U_ASSERT((char*)result==(char*)(&result->uenum));
uprv_memcpy(result, &UCHARSTRENUM_VT, sizeof(UCHARSTRENUM_VT));
result->uenum.context = (void*)strings;
result->index = 0;
result->count = count;
}
}
return (UEnumeration*) result;
}
U_CAPI UEnumeration* U_EXPORT2
uenum_openUCharStringsEnumeration(const UChar* const strings[], int32_t count,
UErrorCode* ec) {
UCharStringEnumeration* result = NULL;
if (U_SUCCESS(*ec) && count >= 0 && (count == 0 || strings != 0)) {
result = (UCharStringEnumeration*) uprv_malloc(sizeof(UCharStringEnumeration));
if (result == NULL) {
*ec = U_MEMORY_ALLOCATION_ERROR;
} else {
U_ASSERT((char*)result==(char*)(&result->uenum));
uprv_memcpy(result, &UCHARSTRENUM_U_VT, sizeof(UCHARSTRENUM_U_VT));
result->uenum.context = (void*)strings;
result->index = 0;
result->count = count;
}
}
return (UEnumeration*) result;
}
// end C Wrapper
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,700
|
package me.kafeitu.activiti.chapter19.identity;
import me.kafeitu.activiti.chapter19.identity.entity.AiaDepartment;
/**
* 部门实体管理
* @author: Henry Yan
*/
public interface AiaDepartmentManager {
// 根据ID查询部门
AiaDepartment get(Long id);
// 保存部门
AiaDepartment save(AiaDepartment user);
// 删除部门
void delete(Long id);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,732
|
\section{Introduction}
Let $K$ be a field of characteristic zero and let $R=K[x_1,\dots,x_n]$ be the polynomial ring over $K$ in $n$ variables.
A graded artinian $K$-algebra $A:=R/I$ is said to have the {\em strong Lefschetz property} ({\em SLP} for short)
if there is a linear form $\ell \in A_1$ such that the multiplication
$$\times \ell ^s: A_{k-s} \to A_k$$ has maximal rank for all $k$ and all $s$, i.e., $\times \ell^s$
is either injective or surjective, for all $k$ and all $s$.
If the multiplication $\times \ell: A_{k-1}\to A_k$ has maximal rank for all $k$, then $A=R/I$ is said to have the {\em weak Lefschetz property} ({\em WLP} for short).
A linear form $\ell$, as above, is called a {\em strong Lefschetz element} (resp. {\em weak Lefschetz element}) of
$A$.
We also say that $I$ has the SLP (resp.\ WLP) if $R/I$ has the SLP (resp.\ WLP).
Though many algebras are expected to have the WLP or even the SLP,
establishing this property is often rather difficult and even in seemingly simple cases, such as complete intersections and ideals generated by products of linear forms, much remains unknown about the presence of the Lefschetz properties.
Lefschetz properties have been studied intensively and a large toolbox, containing different approaches and methods, to check if a graded artinian $K$-algebra $A$ has the WLP or the SLP has been developed. Their study is interesting not only because they put a lot of restrictions on the Hilbert function of a standard graded $K$-algebra but also since they have shown to be connected to a large number of problems, that appear to be unrelated at first glance.
Nevertheless, all research results in this area are motivated and owe their roots to the following theorem proved by
Stanley in \cite{St}, Watanabe in \cite{W} and
Reid, Roberts and Roitman in \cite{RRR}: If $K$ is a field of characteristic zero, then the artinian monomial complete intersection ideal $I=(x_1^{a_1},\cdots ,x_n^{a_n})\subset R$
has the strong and, in particular, the weak Lefschetz property (see also \cite{HP} and \cite{I}).
As a consequence we have that a
\emph{general}
complete intersection artinian ideal with
fixed generator degrees has both, the WLP and the SLP.
It is therefore natural to pose the following question:
\begin{question}[{\cite[Question 3.1]{MN}}]
\label{prob:AllCI}
Do \emph{all} artinian complete intersections have the WLP/SLP, in characteristic zero?
\end{question}
Some evidence that Question \ref{prob:AllCI} has a positive answer is given by the fact that \emph{all} artinian complete intersections in $3$ variables are known to have the WLP. However, it is a challenging and still open problem to decide whether all height 3 artinian complete intersections have the SLP. Similarly, the answer to Question \ref{prob:AllCI} for height 4 ideals is unknown.
There are plenty of results concerning Lefschetz properties of ideals generated by powers of linear forms (see e.g.,\ \cite{AA,MMN-2012,M2016,SS}), and it is natural to consider generalizations of those results to ideals generated by products of linear forms.
From this point of view, it would be interesting to study Question \ref{prob:AllCI} for artinian complete intersections generated by products of linear forms. In this short note, we make a non-trivial contribution to this problem by providing a family of artinian complete intersection ideals of arbitrary height $n$, which have the SLP. More precisely, the following is our main result:
\begin{theorem}\label{mainresult}
Let $d_0,\dots,d_{n-1}$ be positive integers and let $a \in K$ with $a \ne 1$.
The algebra
$$R/(x_n^{d_0}(x_n-a x_1),x_{i}^{d_{i}}(x_{i}-x_{i+1})~|~ \ 1\le i \le n-1)$$
is an artinian complete intersection and has the SLP.
\end{theorem}
The structure of the paper is as follows.
After recalling some basic results on Lefschetz properties in Section 2, we prove the main result of this paper (Theorem \ref{mainresult}) and an extension of it in Section 3.
In Section 4, we present some research problems concerning Lefschetz properties of ideals generated by products of linear forms.
\vskip 4mm
\noindent {\bf Acknowledgements:} This work was started at the workshop ``Lefschetz Properties in Algebra, Geometry and Combinatorics,"
held at the Mittag-Leffler Institute (MLI) in July 2017.
The authors thank MLI for its kind hospitality.
\section{Background and preparatory results}
In this section, for the sake of completeness, we recall the main tools and results that will be used in the rest of the paper.
Throughout the following, we denote by $K$ a field
of characteristic zero and by $R =K[x_1,\dots,x_n]$ the graded homogeneous polynomial ring in $n$ variables over $K$.
Strong respectively weak Lefschetz elements of an artinian algebra $R/I$ are known to form a Zariski open, possibly empty,
subset of $(R/I)_1$. In other words, if the artinian $K$-algebra $R/I$ satisfies the strong or the weak Lefschetz property for some linear form, then it does so for a general linear form. However, for monomial ideals there is no need to consider a general linear form due to the following useful result:
\begin{proposition}[{\cite[Proposition 2.2]{MMN}}]
\label{lem-L-element}
Let $I \subset R$ be an artinian monomial ideal. Then $R/I$ has the SLP if and only if $x_1 + \cdots + x_n$ is a strong Lefschetz element for $R/I$.
\end{proposition}
We also recall the following well-known result, which can be seen as \emph{the} starting point of the study of Lefschetz properties (see \cite{RRR,St,W} and also \cite{HP,I}).
\begin{proposition}
\label{monomialCI}
Let $d_1,\dots,d_n$ be non-negative integers and let $D=d_1+\cdots+d_n$.
Then $A=R/(x_1^{d_1+1},\dots,x_n^{d_n+1})=A_0\oplus \cdots \oplus A_D$ has the SLP.
In particular, the multiplication map
$$\times (x_1+\cdots+x_n)^{D-2i}: A_i \to A_{D-2i}$$
is an isomorphism for $i<\frac D 2$.
\end{proposition}
The following result, that will be crucial for the proof of Theorem \ref{mainresult}, enables us to reduce the study of Lefschetz properties of an arbitrary artinian ideal $I$ to the one of a monomial ideal, by passing to an initial ideal of $I$.
\begin{proposition}[{\cite[Proposition 2.9]{We}}]
\label{WLP_initialIdeal}
Let $I \subset R$ be an artinian ideal, $\tau$ a term order and $\ensuremath{\mathrm{in}}\hspace{1pt}_\tau(I)$ the initial ideal of $I$ with respect to $\tau$. If $R/\ensuremath{\mathrm{in}}\hspace{1pt}_\tau(I)$ has the SLP (resp.\ WLP), then so has $R/I$.
\end{proposition}
\section{A class of complete intersections}
This section is dedicated to the proof of Theorem \ref{mainresult}. Thereby, we provide a class of artinian complete intersection ideals, generated by products of linear forms, that satisfy the SLP.
\begin{lemma}\label{initial}
Let $d_0,d_1,\dots,d_{n-1}$ be positive integers, $a \in K$ with $a \not \in \{0,1\}$, and let
$I=(x_n^{d_0}(x_1-ax_n),x_i^{d_i}(x_i-x_{i+1})~|~1\leq i\leq n-1).
$
Then the initial ideal $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)$ of $I$ with respect to the lexicographic order is equal to
$$(x_1^{d_1+1},\cdots,x_{n-1}^{d_{n-1}+1}, x_{i}x_{n}^{d_0+\cdots+d_{i-1}} ~|~1\leq i\leq n).$$
In particular, $R/I$ is an artinian complete intersection.
\end{lemma}
\begin{proof}
Let
$$f_i = x_i^{d_i}(x_i- x_{i+1}) \quad \text{ for }1\leq i\leq n -1$$
and
$$ g_i = x_ix_n^{d_0+\cdots+d_{i-1}} - \frac 1 a x^{d_0+\cdots+d_{i-1}+1}_n\quad \text{ for } 1\leq i\leq n.$$
We prove the first statement by showing that $G = \{ f_1, \dots , f_{n-1}, g_1, \dots , g_n \}$ is a Gr\"{o}bner basis of $I$ with respect to the lexicographic order.
In the following, we write $S(f, g)$ for the $S$-polynomial of $f$ and $g$.
A routine computation implies the following properties of the possible $S$-polynomials:
\begin{itemize}
\item $S(f_i, g_i) = -(x_{i+1}x_i^{d_i}x_n^{d_0+\cdots+d_{i-1}} - \frac 1 a x_i^{d_i}x_n^{d_0+ \cdots + d_{i-1}})$ reduces to $-\frac 1 {a^{d_i}} g_{i+1}$ with respect to $\{f_1,\dots,f_n,g_1,\dots,g_i\}$ for $1\leq i\leq n - 1$.
\item $S(g_i, g_j) = - \frac 1 a (x_j x_n^{d_0+\cdots+ d_{j-1}+1} - x_ix_n^{d_0+\cdots+ d_{j-1}+1})$
reduces to zero with respect to $G$ for $1 \leq i < j\le n- 1$.
\item $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(f_i)$ and $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(f_j)$ are relatively prime if $1\leq i\ne j\leq n-1$.
\item $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(f_i)$ and $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(g_j)$ are relatively prime if $1\leq i\leq n-1$, $1\leq j\leq n$ and $i\neq j$.
\item $S(g_i,g_n)=-\frac 1 a x_n^{d_1+ \cdots + d_n+2}=- \frac{x_n}{a-1}g_n$ for $1\leq i\leq n-1$ .
\end{itemize}
The above facts combined with Buchberger's criterion guarantee that $G\subset I$ and that $G$ is a Gr\"{o}bner basis of $I$ with respect to the lexicographic order.
For the second statement, observe that, since $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)$ contains pure powers of all variables, $R/\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)$ and hence also $R/I$ is artinian. Moreover, as $I$ is generated by $n$ polynomials, $R/I$ is a complete intersection.
\end{proof}
Theorem \ref{mainresult} will finally follow from Proposition \ref{WLP_initialIdeal} and the next statement:
\begin{theorem}
\label{mainthmSLP}
Let $d_0,d_1,\dots,d_{n-1}$ be positive integers and let $$J=(x_1^{d_1+1},\cdots,x_{n-1}^{d_{n-1}+1}, x_{i}x_{n}^{d_0+\cdots+d_{i-1}} ~|~1\leq i\leq n).$$
Then $R/J$ has the SLP.
\end{theorem}
\begin{proof}
Let $D=d_0+d_1+\cdots+d_{n-1}$.
It follows from Lemma \ref{initial}, that the Hilbert series of $J$ is the same as the one of a complete intersection ideal generated by polynomials of degrees $d_0+1,d_1+1,\dots,d_{n-1}+1$. Therefore, by Proposition \ref{lem-L-element}, it suffices to prove that the multiplication
$$ \times (x_1+\cdots+x_n)^{D-2i}: (R/J)_i \to (R/J)_{D-i}
$$
is an isomorphism for $i<\frac D 2$.
Let $A=R/J$ and for $1\leq i\leq n-1$ let $M^{(i)}=(x_1,\dots,x_i)A$ be the ideal of $A$ generated by the variables $x_1,\dots,x_i$.
Using induction on $k$, we first prove that the multiplication map
\begin{align}
\label{imaizumi}
\times (x_1+\cdots+x_n)^{D-2i}: M^{(k)}_i \to M^{(k)}_{D-i}
\end{align}
is an isomorphism for all $1\leq k\leq n-1$ and all $i < \frac D 2$.
For $k=1$, we have an isomorphism
$$R/(x_1^{d_1},x_2^{d_2+1},\dots,x_{n-1}^{d_{n-1}+1},x_n^{d_0}) (-1)= R/(J:x_1)(-1)
\stackrel{\times x_1} \longrightarrow M^{(1)}$$
and therefore, Proposition \ref{monomialCI} guarantees that the map \eqref{imaizumi} is an isomorphism in this case. \\
Suppose that $k>1$.
Since $M^{(k)}/M^{(k-1)}$ is an ideal of $A/M^{(k-1)}=R/(J+(x_1,\dots,x_{k-1}))$ generated by the single element $x_k$, we have that
\begin{align*}
M^{(k)}/M^{(k-1)} &\cong R/((J+(x_1,\dots,x_{k-1})):x_k)(-1)\\
&\cong K[x_k,x_{k+1},\dots,x_n]/(x_k^{d_k},x_{k+1}^{d_{k+1}+1},\dots,x_{n-1}^{d_{n-1}+1},x_n^{d_0+ \cdots +d_{k-1}})(-1),
\end{align*}
which, by Proposition \ref{monomialCI}, has the SLP. Using the induction hypothesis and the following exact sequence
\begin{align}
\label{exact}
0 \longrightarrow M^{(k-1)} \longrightarrow M^{(k)}
\longrightarrow M^{(k)}/M^{(k-1)} \longrightarrow 0,
\end{align}
we conclude that \eqref{imaizumi} is an isomorphism for all $1\leq k\leq n-1$ and all $i < \frac D 2$.
Finally, it follows from the short exact sequence
$$0 \longrightarrow M^{(n-1)} \longrightarrow A \longrightarrow A/M^{(n-1)} \cong K[x_n]/(x_n^{D+1}) \longrightarrow 0,$$
$A$ has the SLP.
\end{proof}
Theorem \ref{mainresult} is an almost immediate consequence of Lemma \ref{initial} and Theorem \ref{mainthmSLP}.
\begin{proof}[Proof of Theorem \ref{mainresult}]
Note that, by Proposition \ref{WLP_initialIdeal}, it is enough to show that $R/\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)$ has the SLP.
If $a=0$, then $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)=(x_n^{d_0+1},x_1^{d_1+1},\dots,x_{n-1}^{d_{n-1}+1})$ is an artinian complete intersection and it follows from Proposition \ref{monomialCI} that $R/\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)$ has the SLP.
If $a \ne 0$, it follows from Lemma \ref{initial} and Theorem \ref{mainthmSLP} that $R/\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I)$ has the SLP.
\end{proof}
\begin{remark}
The sum of the variables $x_1+ \cdots +x_n$ may not be a Lefschetz element of $R/I$ in Theorem \ref{mainresult}.
Indeed, we have checked with Macaulay2 \cite{GS} that if $K=\mathbb R$, $n=7$, $d_0=d_1=\cdots =d_6=1$ and $a= \pm \frac 4 {\sqrt {3}}$, then $x_1+ \cdots +x_n$ is not a Lefschetz element for $R/I$. \end{remark}
In the remaining part of this section, we discuss an extension of Theorem \ref{mainresult}.
To do so, let us first fix some notation. Let $M_{n\times n}(K)$ denote the set of all $n\times n$ matrices with entries in $K$.
To any matrix $A\in M_{n\times n}(K)$ and any tuple
$\vec{d} = (d_1, d_2, \ldots, d_n)$
of positive integers
we associate the ideal $$I_{A,\vec{d}}:=\left(x_i^{d_i}\left(\sum _{j=1}^na_{ij}x_j\right) ~\mid~ 1\leq i\leq n\right)\subset R.$$
It is known that $I_{A,\vec{d}}$ is an artinian complete intersection ideal if and only if all principal minors of $A$ are non-zero
(see e.g., \cite[Lemma 2.1]{A}, where this equivalence is proved when $\vec d=(1,1,\dots,1)$).
Since these ideals give a class of artinian complete intersection ideals, it is natural to ask if they have the SLP.
Using our main result Theorem \ref{mainresult} we are able to provide a positive answer to this problem in a special case:
\begin{theorem}
Let $A\in M_{n\times n}(K)$ be a matrix with non-zero principal minors. Assume that each row of $A$ has exactly two non-zero entries.
Then $R/I_{A,\vec{d}}$ has the SLP.
\end{theorem}
The condition that in each row of the matrix $A$ are precisely two non-zero entries just says that the ideal $I_{A,\vec{d}}$ is generated by binomials.
\begin{proof}
We will show that~--~up to a change of coordinates~--~$R/I_{A,\vec{d}}$ is
isomorphic to an extension of a tensor product of algebras that have the SLP.
Since tensor products preserve the SLP \cite[Theorem 3.34]{Book},
and the above extension also turns out to preserve the SLP, the claim follows.
First note that by permuting the variables $x_i$ and $x_j$, the ideal
$I_{A,\vec{d}}$ is changed to $I_{A',\vec{d'}}$, where $A'$ is obtained from $A$ by
exchanging the $i$\textsuperscript{th} and $j$\textsuperscript{th} row as well as the $i$\textsuperscript{th} and $j$\textsuperscript{th} column.
Since all principal minors of $A$ are non-zero, we know that the diagonal entries $a_{ii}$ have to be non-zero. Moreover, as each row of $A$ contains exactly two non-zero entries, for any $1\leq i\leq n$ there exists a unique $j_i$ such that the entry in the $i$\textsuperscript{th} row and $j_i$\textsuperscript{th} column of $A$ is non-zero. We denote this entry by $b_{ij_i}$.
We first show the following claim.\\
{\sf Claim:} We can assume that $A$ is of the form \begin{equation}\label{TransformedA}
\begin{pmatrix}
B & * & * & \cdots & * \\
{} & A_1 & 0 & \cdots & 0 \\
{} & {} & A_2 & \cdots & 0 \\
{} & {} & {} & \ddots & \vdots \\
{} & {} & {} & {} & A_r
\end{pmatrix}.
\end{equation}
We associate a directed graph $G_A$ to the matrix $A$ in the following way:
The set of vertices of $G_A$ is defined to be the set of
symbols $a_{ii}$ and $b_{ij_i}$ ($1\leq i\leq n$).
The set of edges is defined
to be the set of all ordered tuples $(a_{ii}, b_{ij_i})$ and $(b_{ij_i}, a_{j_ij_i})$ ($1\leq i\leq n$).
As $j_i$ is uniquely determined by $i$, we conclude that $a_{ii}$ and $b_{ij_i}$ both have outdegree equal to $1$.
Therefore, the finite graph $G_A$ has at least one (directed) cycle,
and distinct cycles are disjoint. If the vertex $b_{ij_i}$ belongs to a cycle, then so do $a_{ii}$ and $a_{j_ij_i}$ (as $b_{ij_i}$ only lies in the edges $(a_{ii},b_{ij_i})$ and $(b_{ij_i},a_{j_i}{j_i})$). Hence, the sets of row and column indices occurring in (vertices of) a cycle coincide.
It follows from the previous discussion that we can assume that $A$ is of the form \eqref{TransformedA}, where $r\ge1$ is the number of cycles of $G_A$ and the $A_i$ are $n_i\times n_i$-matrix, whose row and column indices correspond to vertices forming a cycle. The matrix $B$ is an $n_0\times n_0$-matrix, whose row indices correspond to vertices $a_{ii}$ not contained in any cycle. This shows the above Claim.
Since the non-zero entries in the first $n_0$ rows and columns of $A$ correspond to vertices of $G_A$ not lying in a cycle, by permuting and scaling the variables we can assume that $B$ is upper triangular with diagonal entries equal to $1$.
Furthermore, again by permutation of variables, we can suppose that the only non-zero off-diagonal entries of $A_k$ are the ones directly above the diagonal or the one in the bottom left corner of $A$. Note that, multiplying the $i$\textsuperscript{th} column of $A$ by $\lambda \in K\setminus\{0\}$ corresponds to a scaling of the variables ($x_i \mapsto \lambda x_i$), whereas multiplying the $i$\textsuperscript{th} row of $A$ by $\lambda$ does not change the ideal at all. Hence, by scaling the variables, we can even assume that $A_k$ has the following shape:
\begin{align*}
A_k =
\begin{pmatrix}
1 & -1 & 0 & \cdots & 0 \\
0 & 1 & -1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & 0 \\
0 & {} & \ddots & 1 & -1 \\
a & 0 & \cdots & 0 & 1
\end{pmatrix},
\end{align*}
where $a\in K\setminus\{0\}$ depends on $A_k$.
Note that $A_k$ gives the ideal in Theorem~\ref{mainresult}.
We finally infer from the previous argumentation that
\begin{align*}
R/I_{A,\vec{d}} \cong
\frac{
\left(
\bigotimes_{k=1}^r K[\text{$n_k$ variables}] / I_{A_k,\vec{d_k}}
\right)
[x_1,x_2,\ldots,x_{n_0}]
}{
(f_1,f_2,\ldots,f_{n_0})
},
\end{align*}
where $f_\ell$ ($1\leq \ell\leq n_0$) is a homogeneous binomial that is monic in $x_\ell$,
and $\vec{d_k}$ ($1\leq k\leq r$) contains the entries of $\vec{d}$ corresponding to $A_k$.
The above tensor product has the SLP,
since each factor has the SLP by Theorem~\ref{mainresult}.
Then the right-hand side of the above equation can be
considered as the repetition of a simple extension
$S[x_\ell]/(f_\ell)$ of an algebra $S$ that has the SLP
for $1\leq \ell\leq n_0$ (in reverse order).
Finally the right-hand side has the SLP by \cite[Corollary~4.17]{Book},
which finishes the proof of the theorem.
\end{proof}
If $A$ is an upper triangular matrix, then the initial ideal $\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I_{A,\vec{d}})$ with respect to the lexicographic order is a monomial complete intersection. So, $R/\ensuremath{\mathrm{in}}\hspace{1pt}_{\mathrm{lex}}(I_{A,\vec{d}})$ has the SLP and, applying Proposition \ref{WLP_initialIdeal}, we conclude that the same holds for $R/I_{A,\vec{d}}$.
\section{Final comments and open problems}
In this section, we will present some open questions for further research.
To simplify the notation,
we will write $I_A=I_{A,(1,1,\dots,1)}$.
Our study of Theorem \ref{mainresult} and the ideals $I_{A,\vec d}$ is motivation by the following fact:
If $I$ a quadratic artinian complete intersection ideal generated by products of linear forms, then~--~by applying an appropriate change of coordinates~--~ it follows that $R/I$ is isomorphic to $R/I_{A,(1,\ldots,1)}$ for some $A$.
Thus an affirmative answer to the next question will give an affirmative answer to Question \ref{prob:AllCI} for quadratic complete intersection ideals generated by products of linear forms.
\begin{problem} \label{prob:Minors}
Does $R/I_A$ (or more generally $R/I_{A,\vec{d}}$) have the WLP/SLP, if all principal minors of $A$ are non-zero?
\end{problem}
We also propose some special instances of Problem \ref{prob:Minors}.
\begin{problem} Does $R/I_A$ (or more generally $R/I_{A,\vec{d}}$) have the WLP/SLP if $A$ is integral and all its principal minors are equal to $\pm 1$?
\end{problem}
Since a positive definite symmetric matrix $A$ always has non-zero principal minors, we suggest to consider the following problem.
\begin{problem} Does $R/I_A$ (or more generally $R/I_{A,\vec{d}}$) have the WLP/SLP if $A$ is a positive definite symmetric matrix?
\end{problem}
More generally, we suggest to study the following problem:
\begin{problem}
Let $I$ be an artinian complete intersection ideal generated by forms $f_i$ of degree $a_i$. Assume that $f_i=\ell _1^{(i)}\cdots \ell _{a_i}^{(i)}$ is a product of $a_i$ linear forms. Does $R/I$ have the WLP/SLP?
More specifically, one can ask the same question for the case that all $f_i$ are of the same degree, i.e., $a_i=d$ for all $i$.
\end{problem}
As an extension of the previous problem, we propose the following problem:
\begin{problem} Study the WLP/SLP for ideals generated by products of linear forms.
\end{problem}
Concerning this last problem it is worthwhile to point out that there is a huge list of papers dealing with ideals generated by powers of linear forms. For more information on this subject the reader can see, for instance, \cite{HSS}, \cite{MMN-2012}, \cite{M2016} and \cite{SS}. The problem is really subtle since a minuscule change can alter the behavior of the WLP. Indeed, the ideals
$$I_1=(x_1^4,x_2^4,x_3^4,x_4^4,x_1x_2x_3x_4) \text{ and } I_2=(x_1^4,x_2^4,x_3^4,x_4^4,x_1x_2x_3(x_1+x_4))$$
have the same Hilbert function: $1 \ 4 \ 10 \ 20 \ 30 \ 36 \ 34 \ 24 \ 12 \ 4 \ 0 $
but $I_1$ never has the WLP while $I_2$ does have the WLP.
\begin{remark} One might suspect that Problem \ref{prob:Minors} generalizes to almost complete intersection ideals generated by quadrics, which are products of linear forms. However, in \cite[Theorem 2.12]{M2016}, the second author proved that this is not the case. Indeed, for $n=6$ and all $n\ge 8$, the artinian ideal $I=(\ell^2_1,\ldots, \ell^2_{n+1})\subset K[x_1, \cdots,x_n]$ generated by the square of $n+1$ general linear forms fails the WLP. Nevertheless, we have checked with Macaulay2 that the artinian ideal $I=(\ell_1\ell _1',\ldots, \ell_{7}\ell _{7}')\subset K[x_1, \cdots,x_6]$ generated by the products of general linear forms has the WLP.
\end{remark}
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Michael Raymond-James (Detroit (Michigan), 24 december 1977), geboren als Michael Weverstad, is een Amerikaans acteur.
Biografie
Raymond-James werd geboren in Detroit (Michigan), en volgde zijn high school aan de Clarkston High School in Oakland County waar hij in 1996 zijn diploma haalde. Het acteren heeft hij geleerd aan de Lee Strasberg Theatre and Film Institute in New York. Na zijn studie begon hij met acteren in lokale theaters in New York, later verhuisde hij naar Los Angeles.
Raymond-James is getrouwd en heeft hieruit een zoon (2012).
Filmografie
Films
Uitgezonderd korte films.
2022: The Integrity of Joseph Chambers - als Lone Wolf
2021: American Insurrection - als Gabe
2021: Sweet Girl - als FBI agent John Rothman
2019: Gone Hollywood - als Michael Cimino
2017: Carter & June - als Carter Jennings
2016: The Finest Hours - als D.A. Brown
2014: The Salvation – als Paul
2013: Road to Paloma – als Irish
2012: Jack Reacher – als Linsky
2012: Midnight Sun – als Sam Fuller
2010: Darnell Dawkins: Mouth Guitar Legend – als Ronnie
2009: Moonlight Serenade – als Jesse
2009: The Twenty – als Freeman
2009: Last of the Ninth – als Tommy Leone
2007: The Line-Up – als Tommy Doyle
2006: Black Snake Moan – als Gill
2000: Minor Blues – als Dill
Televisieseries
Uitgezonderd eenmalige gastrollen.
2022: See - als Ranger - 8 afl.
2021: Law & Order: Organized Crime - als Jon Kosta - 8 afl.
2021: Big Sky - als Blake Kleinsasser - 4 afl.
2019-2020: Billions - als Jackie Connerty - 3 afl.
2019-2020: Prodigal Son - als Paul Lazar - 3 afl.
2018: Tell Me a Story - als Mitch Longo - 6 afl.
2018: Frontier - als Fortunato - 4 afl.
2016: Game of Silence - als Gil Harris - 10 afl.
2012–2016: Once Upon a Time – als Neal Cassady – 34 afl.
2015: Sons of Liberty - als Paul Revere - 3 afl.
2012: The Walking Dead – als Dave – 2 afl.
2008–2011: True Blood – als Rene Lenier – 15 afl
2010: Terriers – als Britt Pollack – 13 afl.
Amerikaans filmacteur
Amerikaans televisieacteur
Pseudoniem
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\section{Introduction}
Despite being perturbatively nonrenormalizable, the quantum theory of
covariant scalar $\phi^4_n$ models has been shown to be trivial for all
space-time dimensions $n\ge5$, while for $n=4$ it is widely believed to
be trivial as well \cite{fro}. Triviality follows by showing that the
conventionally lattice-regularized, Euclidean-space functional integral
tends to a Gaussian distribution in the continuum limit independent of
any choice of renormalizations for the mass, coupling constant, and field
strength. Although mathematically sound, a trivial result is inconsistent
in the sense that the classical limit of the quantized theory differs
from the original (nontrivial) classical theory. In this Letter we
reexamine this problem once again, and suggest an alternative formulation
whereby quantum models for $\phi^4_n$ may be nontrivial. Generally, in
what follows, we set $\hbar=1$.
We start with a lattice-regularized, Euclidean-space functional integral
expressed as
\bn &&\hskip-1.1cm S_a(h)\equiv\<\exp(\Sigma\, h_k\phi_k a^n)\> \nonumber \\
&&\equiv N_a\int\exp\{\Sigma\, h_k\phi_ka^n-\textstyle{\frac{1}{2}} Z\,\Sigma(\phi_{k^*}
- -\phi_k)^2a^{n-2}-\textstyle{\frac{1}{2}} Zm_o^2\,\Sigma\,\phi^2_ka^n \nonumber\\
&&\hskip2.5cm-Z^2g_o\,\Sigma\, \phi^4_ka^n-\Sigma\,P(Z^{1/2}\phi_k,a)\,a^n\}
\,\Pi\,d\phi_k\;. \en
Here $k=(k^0,\cdots ,k^{n-1})$ , $k^j\in {\mathbb Z}$ for all $j$, labels a
lattice site; $k^*$ signifies one of the $n$ nearest neighbors to $k$ and
the sums run over a large but finite hypercubic lattice; $a$ represents
the lattice spacing; $h_k$ denotes the lattice-cell average of a smooth
source function $h(x)$, $x\in {\mathbb R}^n$; $Z>0$, $m_o^2$, and
$g_o\ge 0$ are functions of the cutoff $a$; and---for the present---the
auxiliary term $P\equiv0$. We choose $N_a$ such that $S_a(0)=1$, and
let $\<(\cdot)\>$ denote an average with respect to the resultant
probability distribution. The continuum limit is defined as the limit
$a\ra0$ in conjunction with a diverging number of lattice sites so that
the space-time volume itself eventually tends to ${\mathbb R}^n$ in a
suitable way. For $n\le3$, the continuum limit leads to acceptable
(nontrivial) results \cite{gli}; for $n\ge5$, on the other hand, the
continuum limit has the form \cite{fro}
\bn \lim S_a(h)=\exp[\,\textstyle{\frac{1}{2}}{\textstyle\int} h(x)C(x-y)h(y)\,d^n\!x\,d^n\!y\,] \en
for a suitable covariance function $C(x-y)$, and all indications point to
the same conclusion when $n=4$. [If $C(x-y)$ is not locally integrable, then
this condition is replaced by one in terms of correlation functions with
noncoincident points.]
Let us sketch one plausible argument that leads to trivial behavior.
Consider the dimensionless and rescaling invariant correlation-function
ratios, which also admit meaningful continuum limits, given, for $r\ge 1$, by
\bn g_{(r)}\equiv\frac{\Sigma\<\phi_0\phi_{k_2}\cdots\phi_{k_{2r}}\>^T}
{[\Sigma\<\phi_0\phi_k\>]^r[\Sigma k^2\<\phi_0\phi_k\>/
6\Sigma\<\phi_0\phi_k\>]^{n(r-1)/2}}\;; \en
by symmetry, all odd-order correlations vanish. For $n\ge5$,
mean field theory is generally accepted, and for small $a$ it
leads to the behavior that
$g_{(r)}\propto a^{(n-4)(r-1)}$. Thus for $n\ge5$ and $r\ge2$,
$g_{(r)}\ra0$ in the continuum limit. The Lebowitz inequality
\cite{gli}, which states that $\<\phi_j\phi_k\phi_l\phi_m\>^T\le0$ for
such models, linked with $g_{(2)}\ra0$, implies the vanishing of
the truncated $(T)$ four-point function in the continuum limit.
Among the class of distributions considered, only a Gaussian
distribution admits a vanishing truncated four-point function, and
triviality follows.
For $n=4$, logarithmic corrections to mean field theory arise and for
small $a$ it is plausible that $g_{(r)}\propto|\ln(a)|^{-(r-1)}$,
which again leads to triviality as $a\ra0$.
For $n\ge5$, the indicated $a$ dependence of $g_{(r)}$ stems from the
separate mean field behavior, valid for small $a$, given by
$\Sigma\<\phi_0\cdots\phi_{k_{2r}}\>^T\propto a^{-2-6(r-1)}$ and
$\Sigma k^2\<\phi_0\phi_k\>\propto a^{-4}$ \cite{fis}. This behavior
reflects the divergence arising from the (multiple) {\it sum} and is
based, for a suitable choice of $Z$, on a largely $a$-{\it in}dependent
correlation function $\<\phi_0\cdots\phi_{k_{2r}}\>^T$.
Let us return to (1), now allowing for an auxiliary contribution
$P\not\equiv0$. Normally, $P$ is determined by a perturbation analysis
so as to cancel unwanted ultraviolet divergences in the continuum limit.
Instead, we propose to {\it design} $P$ to achieve nontriviality. In
particular, we choose $P$ so that the truncated correlation functions
themselves become {\it uniformly} $a$-dependent, specifically, for all
$r\ge1$ and $n\ge5$, that
\bn \<\phi_0\phi_2\cdots\phi_{k_{2r}}\>^T\propto a^{n-4}\;; \en
for $n=4$ the right side of (4) should be replaced by $|\ln(a)|^{-1}$.
With such a choice for $P$, the combined effects of the divergence due
to long-range order and rescaled amplitude lead to $g_{(r)}\propto a^0=1$
for all $r\ge 1$, and for any $n\ge 4$. Choosing $Z\propto a^{n-4}$
[or $|\ln(a)|^{-1}$] properly rescales the correlation functions to
macroscopic values. Such a theory would be non-Gaussian, hence
nontrivial, in the continuum limit.
Based on experience with related but soluble models \cite{klb}, we
conjecture that a $P$ of the form
\bn P(\phi_k,a)=A(a)\,\frac{[\phi_k^2-B(a)]}{\,[\phi_k^2+C(a)]^2}\;, \en
for suitably chosen $A$, $B$, and $C$ (which also depend on $n$), may do
the job. As $a\ra0$, we expect that $A\rightarrow\infty$, $B\ra0$, and $C\ra0$.
Thus the indicated expression is a regularized form of a formal continuum
potential proportional to $1/\phi(x)^2$. However, just as the $1/r^2$
potential that arises from the kinetic energy in a spherically symmetric,
quantum-mechanical situation necessarily carries a proportionality factor
of $\hbar^2$, it is more proper to recognize that the formal auxiliary
potential $P$ is proportional to $\hbar^2/\phi(x)^2$, i.e., $A\propto \hbar^2$.
Carrying the analogy further, we observe that $P$ is not a counterterm
for the quartic interaction but rather for the kinetic energy term. Thus
we are led to propose that $P$ is a {\it nonclassical, auxiliary potential},
which explains its absence in a strictly classical limit in which
$\hbar\ra0$. Based on related models, we are also led to conjecture
that a $P$ having the desired properties leads to a quantum theory the
classical limit of which agrees with the classical theory with which one
started.
Assuming that some such $P$ exists, we can proceed to derive, in a general
fashion, certain additional facts. The nature of the nonclassical,
auxiliary potential $P$ leads to a generalized Poisson distribution in the
continuum limit. In Minkowski space-time the operator structure of such
fields admits a relatively simple superstructure. Introduce the basic
(Fock) operators
$A_l$ and $A^\dagger_l$, $l\in\{0,1,2,\ldots\}$, where $[A_l,A_m]=0$ and
$[A_l,A^\dagger_m]=\delta_{lm}$ for all $l$ and $m$, and $A_l|0\>=0$ for
all $l$, with $|0\>$ unique. As usual, the Hilbert space is spanned by
vectors of the form $|0\>,\;A^\dagger_l|0\>,\;A^\dagger_lA^\dagger_m|0\>$,
etc. Furthermore, let $\Lambda_{lm}(x)\;[=\Lambda_{ml}(x)^*]$ and $C_l(x)$ denote a
suitable set of complex fields. With summation implied, the Minkowski
field operator has the representation \cite{kl2}
\bn \varphi(x) =A^\dagger_l\,\Lambda_{lm}(x)\,A_m+A^\dagger_l\,C_l(x)+C^*_l(x)
\,A_l\;. \en
It follows from (6) that
\bn &&\hskip-1cm\<0|\varphi(x_1)\varphi(x_2)\cdots\varphi(x_{2r-1})
\varphi(x_{2r})|0\>^T\nonumber\\
&&\hskip1cm =C^*_{l_1}(x_1)\Lambda_{l_1\,l_2}(x_2)\cdots \Lambda_{l_{2r-2}
\,l_{2r-1}}(x_{2r-1})C_{l_{2r-1}}(x_{2r})\;, \en
assuming all odd-order correlation functions vanish.
Since $\<0|\varphi(x)\varphi(y)|0\>=C^*_l(x)\,C_l(y)$, it follows,
for example, from the spectral representation for $n\ge4$, that
$\Sigma|C_l(x)|^2=\infty$.
The field $\varphi(x)$ is Gaussian if $\Lambda_{lm}(x)\equiv0$; for
$\varphi(x)$ to be non-Gaussian requires $\Lambda_{lm}(x)\not\equiv0$.
Fields with truncated correlations of the form (7) can have no conventional
particle scattering. For example, with $\varphi(h)\equiv{\textstyle\int}
\varphi(x)\,h(x)\,d^n\!x$, it follows from (7) that
$\<0|\varphi(h_1)\varphi(h_2)\varphi(h_2)\varphi(h_1)|0\>^T\ge0$, and by an associated
Schwarz inequality \cite{buc} that
\bn &&\hskip-1cm0\le|\<0|\varphi(f_1)\varphi(f_2)\varphi(g_2)\varphi(g_1)|0\>^T|^2 \nonumber \\
&&\le \<0|\varphi(f_1)\varphi(f_2)\varphi(f_2)\varphi(f_1)|0\>^T\,\<0|\varphi(g_1)\varphi(g_2)
\varphi(g_2)\varphi(g_1)|0\>^T\;. \en
Passing to asymptotic fields, $\varphi(f)\rightarrow\varphi_{\rm out}(f)$, and
$\varphi(g)\rightarrow\varphi_{\rm in}(g)$, leads to
\bn &&\hskip-.6cm0\le|\<0|\varphi_{\rm out}(f_1)\varphi_{\rm out}(f_2)
\varphi_{\rm in}(g_2)\varphi_{\rm in}(g_1)|0\>^T|^2 \nonumber \\
&&\hskip-.6cm\le \<0|\varphi_{\rm out}(f_1)\varphi_{\rm out}(f_2)\varphi_{\rm out}
(f_2)\varphi_{\rm out}(f_1)|0\>^T\,\<0|\varphi_{\rm in}(g_1)\varphi_{\rm in}
(g_2)\varphi_{\rm in}(g_2)\varphi_{\rm in}(g_1)|0\>^T\nonumber\\ &&\hskip1cm =0\;. \en
This behavior is consistent with the assumption that
$\Lambda_{lm}(x)\rightarrow\Lambda^{{\rm out},{\rm in}}_{lm}(x)\equiv0$ and
$C_l(x)\rightarrow C_l^{\rm out}(x)=C^{\rm in}_l(x)$.
On the surface, it would appear that a quantum theory with no conventional
scattering
would be inconsistent with the original classical theory which is known
to exhibit nontrivial classical scattering \cite{rei}. However, an
interchange of limits is involved here, which, on closer inspection,
shows that no inconsistency arises.
If we assume the suggested form for the nonclassical, auxiliary
potential $P$, there exists, in effect, an additional term in the
operator energy density proportional to $\hbar^2/\varphi(x)^2$, or in the
corresponding Heisenberg equation of motion an additional term
proportional to $\hbar^2/\varphi(x)^3$. Such terms {\it preclude} the
field operator from reaching arbitrarily small values, and hence
nontrivial asymptotic fields (with $\hbar>0$) fail to exist in the
usual sense.
We summarize the situation as follows: On the one hand, we observe that
taking the classical limit ($\hbar\ra0$) after going to the asymptotic
quantum fields leaves us with no classical scattering. On the other
hand, if one takes the classical limit ($\hbar\ra0$) of the quantum
equations of motion at finite time values and solves the resultant
classical theory, then one should find the nontrivial scattering of the
classical theory alluded to before.
>From another perspective, the behavior just outlined can also lead to
intermediate situations. Let us imagine a quantum system endowed with
a great deal of energy and composed of a huge number of quanta. By the
correspondence principle, the associated scattering behavior can be
approximately treated classically and is therefore nonvanishing provided
one sticks to possibly large, but {\it finite}, preparation and detection
times in the past and future, respectively. In other words, such a field
may not exhibit conventional quantum particle scattering in the strict
sense, but under suitable conditions, an effective scattering theory
may well exist that could help in recovering the nontrivial scattering
of the original classical theory. Moreover, should there be any stable
large-field configurations, such as solitons, then scattering between
such entities may well exist.
With $A(a)\rightarrow\infty$ sufficiently fast, the proposed form of $P$ given
in (5) can avoid the fate---irrelevancy---of usual higher-order interactions
in a renormalization-group treatment. Any analysis of these models should
perhaps begin with the case $g_o\equiv 0$ for which, thanks to $P$,
nontriviality is still expected.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 35
|
Q: Hours+Minutes between two Java Date objects I have found how to get the hours between two dates and the minutes . What I want is the exact difference between both of them, this is the code that I'm using:
private String getHours(Message punch) {
LocalTime out = Instant.ofEpochMilli(message.getOut().getTime()).atZone(ZoneId.of(message.getTimezone()))
.toLocalTime();
LocalTime in = Instant.ofEpochMilli(message.getIn().getTime()).atZone(ZoneId.of(message.getTimezone()))
.toLocalTime();
Duration duration = Duration.between(in, out);
Long hours = duration.toHours();
Long minutes = duration.toMinutes() - (hours * Constants.MINUTES_IN_AN_HOUR);
return String.format("%d:%d", hours, minutes);
}
It works fine for the major of the cases but I'm having an error in the following case:
*
*message.getIn() returns: 12:59
*message.getOut() returns: 22:00
Both are the same day, the difference that I'm expecting is 9:01, but I'm getting -14:-59
Debugging the code I realize that out is getting 04:00 and in is getting 18:59.
For almost all the cases it works well but It happens in some scenarios.
A: I believe your problem is that you are using LocalTime class but you should be using LocalDateTime class. It apears that your timezone is GMT+6, so in your example your in time and out time fall in different days - your in time in the evening of a previous day and out time on the morning of the next day. But because you are using LocalTime you are loosing the fact that those are times in different days. Change your LocalTime to LocalDateTime and see if this helps
A: You are working too hard, going through too many gyrations in your code.
Use Instant
You have a pair of Instant objects but throw them away. Use them. An instant is a specific moment on the timeline in UTC with a resolution of nanoseconds.
Instant start = … ;
Instant stop = … ;
Duration d = Duration.between( start , stop );
How you get your instants is a mystery. If you revise your Question to explain the exact nature of your inputs, I will provide more code here.
If you are being passed a pair of java.util.Date objects, convert them to Instant. Use new conversion methods added to the old date-time classes. No need for time zones at all for calculating elapsed time.
Instant start = utilDateX.toInstant() ;
Instant stop = utilDateY.toInstant() ;
Duration d = Duration.between( start , stop );
Calculating elapsed time with LocalTime is rarely appropriate because of crossing over into the next or previous days.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,789
|
Q: Is there any way to allow TLS renegotation in istio tls options? Is there any way to allow TLS renegotation in istio TLSSettings ?
I'm looking into a way to force envoy to setup allow_renegotiation into True
https://www.envoyproxy.io/docs/envoy/latest/api-v2/api/v2/auth/cert.proto#auth-upstreamtlscontext
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,402
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# _The Amtrak Wars Book 5: Death-Bringer_
PATRICK TILLEY
For
Patrick, Freddie and Sean
the next generation
who were into computers
before they were out of nappies
and have now come to grips with karate.
From the way things are shaping up
this looks like a good career move
for anyone with a ticket to the 21st Century.
Good luck, boys. Carry the torch. Hold it high.
Opa
## Contents
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
A Note on the Author
## CHAPTER ONE
In the spring of 2991, Mr Snow, wordsmith of the Clan M'Call, faced a difficult decision. Should he accompany the clan's delegation to the trading post on the shores of the Great River – or should he stay behind in the hills of Wyoming in case the cloud-warrior returned with Cadillac and Clearwater?
Two winters had passed since his charges had flown into the Eastern Lands and it was almost a year since the cloud-warrior had gone in search of them. Brickman had promised to help them escape from the Iron Masters but that was easier said than done. The Dead-faces were a fearsome race who lived behind closed borders. No Plainfolk Mute taken away on the wheel-boats had ever regained his liberty. But Cadillac and Clearwater were no ordinary Mutes. They had been born in the shadow of Talisman, and Brickman, the cloud-warrior, was also gifted and resourceful and as cunning as a coyote. And though he did not yet understand why, he too had been touched by Talisman.
If there was a chance to escape then these three would seize it, for between them they possessed the power to overturn nations. That had been their destiny from the day they had been born. But where were they? Day after day, Mr Snow had posted sentinels to keep a special watch over the eastern approaches to the settlement but the long-awaited travellers had failed to appear.
They were not dead. In an uncertain world, that was the only thing Mr Snow was sure of. Cadillac and Clearwater were the sword and shield of Talisman, saviour of the Plainfolk who – according to prophecy – was due to appear in human form. Cadillac was to use his great gifts to prepare the way for Talisman, and Clearwater was to use the immense forces at her command to protect the Thrice-Gifted One until his own powers were fully formed. Which, for instance, would be the case if he entered the world as a new-born child. On the other hand, if he was present in someone already alive, with his powers over heaven and earth lying dormant until the chosen moment, then her given task was to protect that individual until Talisman chose to reveal himself. She would do this instinctively, without necessarily understanding why, because Talisman would draw her to him.
Mr Snow had often wondered if Steve Brickman bore the Talisman within him. The cloud-warrior's descent from the sky into the hands of the M'Calls had been foretold by the Sky Voices. He and Clearwater had been destined to meet, and in giving herself to him body and soul she had broken the solemn vows that bound her to Cadillac – grievously wounding her former lover's pride in the process.
In time, Cadillac would get over it. It was he who had seen their separation in the stones. Clearwater was destined to journey into the dark world of the sand-burrowers that lay beneath the deserts of the south. Home of the iron-snakes that crawled through the land leaving a trail of devastation behind them, and the arrowheads which carried the cloud-warriors across the skies. Warriors armed with long sharp iron and fire-seeds which erupted into smoke and flame with the sound of earththunder. Not the pure flame that swept the tree-spirits up towards the heavens but an evil cousin conjured up by the sand-burrowers. A flame whose thirst could not be quenched by water, that clung to flesh and burned through to the bone.
Yes, these were dark days. The time known as The Great Dying had come. A time when the courage of the Plainfolk would be sorely tested. Mo-Town, the Great Sky-Mother had withdrawn into the Black Tower of Tamla to weep for her people. Many would perish but the Plainfolk would survive and become a great nation under the banner of Talisman. As a Mute, a revered sage and walking history book of the Clan M'Call, Mr Snow knew that the journey through the Valley of Death had to be undertaken with as much good grace as one could muster. The Wheel turned, The Path was drawn. Human beings could not change their destiny; it was the hubris of the unenlightened that fostered the cruel illusion they could do so.
But meanwhile, three of the principal players were missing. Where in the name of Talisman were they? In a few days, the clan's trade delegation would be ready to leave for the annual gathering on the shores of the Great River. Mr Snow had two choices: to go with them, or stay behind. And the cloud-warrior had two ways to return with Cadillac and Clearwater: by smuggling themselves aboard one of the giant wheel-boats due to travel along the Great River to the trading post, or by a more direct, overland route through the territory that had once belonged to the Io-Wa and Ne-Braska.
A year ago, Brickman had stolen aboard one of the wheel-boats at the trading post and had been carried away to the Fire-Pits of Beth-Lem. If he had managed to complete the journey without being discovered he might decide that this was the best way to return. In the bustle of trading activity, with Mutes helping to load and unload the wheel-boats, they would have an excellent opportunity to steal ashore. Once there they could rejoin their clanfolk, becoming part of the delegation which would then travel home across the plains during the period of truce known as 'Walking on the Water'.
That was the sensible way, but the journey from Ne-Issan took many days – perhaps weeks. Finding a place on a wheel-boat where three people could remain undetected for days on end would not be easy. Mr Snow had been taken aboard one for a brief audience with Lord Yama-Shita. They were giant structures but they also carried a large crew who constantly swarmed back and forth like ants on a dunghill. And the wheel-boats only came to the trading post once a year. To return via this route meant boarding the right vessel at exactly the right moment. The cloud-warrior was resourceful enough to gather this information but what if they missed the boat? Or escaped much earlier and were unable to take the longer but safer way home?
Mr Snow's dilemma arose from his desire to be at the chosen point of arrival in case his powers were needed to fight off any pursuers. For they would be pursued. That was certain. Over the years of trading, he had come to understand the character of the Iron Masters and their obsession with 'face', what the Mutes called 'standing'. Because of the status accorded to warriors, it was a concept the two races shared, but not to the same degree. Mutes generally nursed their shattered pride then gave it another shot. To the Iron Masters, loss of face was an unbearable condition which, if the victim's sense of honour could not be regained, often led to suicide. This concern with honour, impeccable behaviour and faultless performance of one's duties only affected the pure-blood ruling classes; the lower orders – the inferior races – were not graced by such concerns. Which, according to his informant, explained why the gods had condemned them to a life of servitude.
Yes... Given the nature of Cadillac's mission, their escape would cause a definite loss of face, and the authorities concerned would spare no effort to recapture them. Failure to do so would cause heads to roll. Mr Snow – who knew nothing of the mayhem the trio had caused at the Heron Pool – was unaware that in its bloody aftermath a great many already had. He only knew the Iron Masters were tenacious adversaries who did not admit defeat. That was why he had to be on hand in case they pursued his young charges into the heartland of the Plainfolk.
But he could not be in two places at once and he could no longer hesitate. He now had less than a week in which to make his decision. Perhaps the Sky Voices would consent to guide him. He had consulted them many times during the past year but they had greeted his questions about Clearwater, Cadillac and the cloud-warrior with a baffling silence. He clambered up to his favourite rock, sat down with his legs crossed, took several deep breaths while he admired the view, then raised his closed eyes and opened his mind to the sky.
For a long while it seemed as if the staff of this spiritual advice bureau was out to lunch but eventually a series of pictures appeared before his inner eye. Soul-searing images of death and destruction on an unparalleled scale; a grisly drama in which he had been given a starring role. Mr Snow was renowned for his courage and resolution but even his indomitable heart quailed at this new burden that Fate had thrust upon him. And what made it worse was the knowledge that these fleeting images were merely a foretaste of what was to come. But there could be no turning back. The Sky Voices had spoken – and had left him in no doubt as to what he had to do.
Some two thousand miles to the east of the M'Call settlement, Ieyasu, Lord Chamberlain of the Inner Court, grand-uncle and principal advisor to the Shogun Yoritomo Toh-Yota, absolute ruler of Ne-Issan, was also beset by problems that demanded resolution.
If Mr Snow was old, Ieyasu was ancient, but they had many qualities in common including keen eyes and fire in their belly. Both were shrewd, highly intelligent and infinitely wise in the ways of the world even though the societies in which they lived were totally dissimilar except for their respect of physical courage and the code of honour which formed the basis of the warrior ethos.
Mr Snow could not read or write but possessed gifts of memory and magic: Ieyasu was literate, extremely well educated and although he was unable to summon earth and sky forces to his aid, the skill and cunning with which he outmanoeuvred all those who sought to remove him from power was little short of supernatural.
Before Yoritomo's accession to the throne at the tender age of twenty-three, Ieyasu had exercised absolute power in the name of the Shogun's dissolute father. Yoritomo, now twenty-nine, was made of different cloth. Restrained in his sexual appetites, something of an ascetic in his attitude to food and drink, overburdened with a tiresome morality and obsessed with traditional values, Yoritomo had proved particularly difficult to deal with. And the main source of difficulty was his determination to take sole charge of the nation's affairs and ignore the voice of experience. The voice, of course, being that of his grand-uncle.
It was hard enough trying to keep the government afloat and conspirators at bay without having to reeducate an aspiring saint who was trying to manoeuvre you out of office. In time, Yoritomo would learn. But he would learn a lot quicker and make life a lot easier for everyone by absorbing the distilled wisdom of his grand-uncle. Something he had done with the utmost reluctance.
In part, it was a natural reaction to the moral laxity which had pervaded the Inner Court during his father's reign. As a new broom, Yoritomo wanted to make a clean sweep. A perfectly laudable aim. The court was in need of a thorough spring cleaning. But in politics one never did anything to excess. Yoritomo did not understand the importance of leaving a little dirt in the corners. His puritanical streak – laudable in a monk but utterly depressing in a vigorous, intelligent young man holding the highest office in the land – was blinding him to the realities of power.
The young shogun had not yet grasped an essential truth: exploiting the weaknesses of powerful men – especially powerful opponents – was an important element in the art of statecraft. It was also true that a nation needed honest men of high principle and modest ambition. They made excellent civil servants. The government revenue and customs houses and the postal service were always crying out for more. Sinners, on the other hand, made better dinner companions. And they were a lot easier to do business with.
Ieyasu was also a traditionalist, as opposed to those who favoured progressive ideals – a group of domain-lords led by the Yama-Shita family. But the progress advocated by this cabal of entrepreneurs was restricted to the introduction of new industrial processes and manufacturing techniques. No one, however radical their ideas were in that direction, was in favour of modernising the feudal system on which Ne-Issan had been built.
The problem – in Ieyasu's eyes at least – was that you could not have one without undermining the other. And none of the seventeen ruling samurai families was prepared to surrender an ounce of power or privilege to the lower classes. It was the merchants who argued the case for an expanding economy and the benefits to be gained by increasing the purchasing power of the masses by – if you please – paying tradesmen and servants higher wages! Some had even suggested setting up trade links with the long-dogs inhabiting the buried cities beyond the Western Hills – but what else could one expect from chinamen who had an abacus where their brains should be?
The greatest bar to progress was the immutable edict which forbade, under pain of death, the re-introduction of the Dark Light. It was also a treasonable offence for lesser mortals to utter its name and such was the dread it inspired, even those at the pinnacle of power only did so with the greatest circumspection. According to the scrolls which chronicled the distant past, the creation of the Dark Light – electricity – had corrupted mankind and led the gods to destroy The World Before with a tidal wave of golden fire. A wave that had engulfed the ancient homeland of the Iron Masters, and which was so high, it had covered the peak of Fuji, the sacred mountain which contained the soul of Nippon. As a result, there was a deeply-held belief that to seek to resurrect the Dark Light would be an act of incredible folly which would once again place the world in mortal peril.
But, as Ieyasu knew, the world of Ne-Issan was bordered by the Appalachians and the Eastern Sea. There was another vaster world beyond the Western Hills, inhabited by grass-monkeys and long-dogs: Plainfolk Mutes and Trackers – the soldier-citizens of the Amtrak Federation. The Mutes were hairy savages, semi-nomadic hunters with no craft skills beyond those needed to support their simple mode of life. All their edged-weapons, crossbows and metal implements were supplied by the Iron Masters. But the Trackers were warriors who had no fear of the Dark Light. It was the life-force of their underground society. It enabled them to send images and voices through the air, it powered their weapons, their giant, caterpillar-like land-cruisers and their skychariots – war-machines which entered the cloud-realm of the _kami_ with impunity _and were not cast down_.
Their presence posed a threat to the world of Ne-Issan yet Amaterasu-Omikami stood aside and did nothing. Their underground cities were not crushed, and the world beyond the Appalachians was not ravaged by heavenly fire – a theological conundrum that was studiously ignored by the leading sages of the _Shinto_ priesthood.
Ieyasu knew the answer. The Dark Light was neither good nor bad. Electricity was a power that lay at the heart of the natural world. It could be captured by special, cunningly-wrought machines and conveyed along special threads from one place to another, or shot through the air like an invisible arrow that flew across plains, mountains and seas within the space of a single heartbeat.
Like all power, it could be used and abused. It could corrupt, in the same way that _sake_ addled the brains of drunkards and opium destroyed the will of addicts. But in its pure state, it was not inherently evil. Electricity had been created to be the slave of man. Only if the man was weak could the slave became his master. Ieyasu had certain foibles but he was not a weak man. He enjoyed the attendant luxury his privileged birth and high rank afforded him but he was consumed by nothing except the desire to manipulate the reins of power to the ultimate benefit of the Toh-Yota family and the Shogun. In that order. Ieyasu ate well, drank judiciously, and kept his gaunt, aging body in trim by practising his swordsmanship. He enjoyed male _and_ female company and could still produce a commendable erection which a select circle of court ladies – ever anxious to advance themselves or the careers of their husbands – accommodated by supplying him with a string of pubescent nymphets.
The Dark Light might kill him but it would never enslave him. Ieyasu knew this because it had served him well over many years. Key members of his private network of secret agents had been using high-powered radio transceivers and other electronic devices for the last ten years. The same type of equipment used by the secret agents of the Federation and which, after a series of stealthy contacts, had been supplied by them to Ieyasu's organization under the terms of a secret protocol signed by him and Commander-General Karlstrom, the head of AMEXICO.
Among the items covered was the return of any _mexican_ caught by the Plainfolk Mutes and sold to the Iron Masters. Other clauses outlined mutually-beneficial arrangements for the pooling of specific types of information, for example – the kinds of weapons the Iron Masters planned to supply to the Mutes by way of trade and, in return, any snippets of information which could help Ieyasu head off any bid to topple the Toh-Yota shogunate.
A final clause set out the arrangements for joint operations between the two spy networks. It was here that AMEXICO's help had proved invaluable. There were certain locations which, for various reasons, Ieyasu's home-grown agents were unable to penetrate or where they could not operate effectively. The wheel-boats operated by the Yama-Shita family were one example. The vetting procedures were so strict it was impossible to slip an outsider into the crew. The only alternative was to buy the allegiance of someone already serving the family but experience had shown this to be a costly and highly unreliable way of doing business.
Karlstrom had supplied the answer: the insertion of _mexicans_ , disguised as Mute slaves, and armed with a working knowledge of japanese and other asiatic languages into sensitive locations. Ieyasu, after some initial misgivings, had accepted the offer. And it had worked. As non-persons, slaves were regarded as part of the brickwork, and since no outlander was permitted to utter a word of the Iron Master's sacred tongue, people talked in front of them without ever suspecting their conversation was being monitored. Disguised slaves could not, of course, penetrate the secret council chambers of high-ranking plotters but they were the source of a surprising amount of raw intelligence. And many of the council chambers were no longer secret thanks to the electronic bugging devices obligingly supplied by AMEXICO.
So far it had paid off, but it was a dangerous game. A balancing act which placed Ieyasu on a tightrope over a pool of hungry sharks. For not only had he approved the use of devices filled with the Dark Light, he had even sent some of his most trusted men to help perfect the language skills of Karlstrom's agents!
His opposite number, the head of AMEXICO – who also spoke fluent japanese – had never sought to press for an advantage. The emphasis had always been on mutual cooperation but Ieyasu knew that if one of the two copies of the secret protocol with his name and seal attached ever reached the Shogun, his hold on the reins of power would be abruptly severed. And so would his head. His own death in the proper course of events did not concern him, but his precipitate departure from office followed by the elimination of his closest aides would leave a dangerous vacuum in the highest councils of the land. A vacuum that a host of undesirables would rush to fill.
In the few years left to him, Ieyasu had to make the best possible use of this unique contact with a potential enemy state without compromising the long-term interests of Ne-Issan or betraying its most cherished beliefs. A lesser wrong for a greater good.
As a pragmatist, Ieyasu had no problem with that. Like all aristocratic Iron Masters, conspiracy was in his blood. The history of Ne-Issan was a catalogue of internecine feuds and labyrinthine treachery. Even so, there were times when he found it difficult to reconcile his dual roles as master spy and Court Chamberlain of the Toh-Yota shogunate with his blood-ties to the entire japanese ruling class. This was a primal allegiance that went beyond pure reason and, as such, could not be ignored. Up until now he had been able to override this inner conflict, but in the spring of 2991, he learned of an event which placed him in a considerable dilemma.
In the autumn of the previous year, _mexicans_ disguised as slave workers had – with his tacit approval – sabotaged an attempt to build flying horses; a project masterminded by the Yama-Shita and Min-Orota families who were also laying plans to overthrow the Toh-Yota Shogunate. The sabotage operation had been a remarkably bloody affair – and so had its aftermath. Hundreds had perished, foot-soldiers, cavalry, samurai, nobles from both families and Domain-Lord Hirohito Yama-Shita who, by all accounts, had died in a particularly gruesome manner.
Ieyasu's agents had been instrumental in helping the five saboteurs to leave the country but their departure had not been the end of the story. Judged guilty of seeking to resurrect the Dark Light, several leading members of the Yama-Shita family were given the chance to take their own lives; others, of lower rank, were executed, fines were levied and economic sanctions applied.
Armed resistance against the government was out of the question. The judgment against the family had been rendered by its peers; a committee of powerful domain-lords including several of its closest allies – whose neutrality had been purchased by giving them valuable pieces of the Yama-Shita trading empire.
All this had been done yet it had not brought the Yama-Shita to heel. They wanted revenge. Not against the Toh-Yota. Without its two main allies, the Ko-Nikka and the Se-Iko – the beneficaries of the Shogun's _largesse_ – the shogunate and the traditionalists now held the balance of power. It would take years to win back its former supporters and longer still before they were ready to even the score. No... the family's thirst for revenge was directed against the five assassins – the outlanders who had killed their domain-lord and brought the house of Yama-Shita to its knees. They could not have done their bloody work without highly-placed friends inside Ne-Issan. If this murderous gang could be captured alive, they would soon reveal the identity of their masters...
Ieyasu did not need the bug planted inside the council chamber of the Yama-Shita's palace at Sara-kusa to tell him how they had reasoned. He merely had to put himself in their place. The last-minute decision by the Shogun not to attend the flying display at the Heron Pool pointed to his complicity in the murderous onslaught unleased by the assassins. An onslaught which – in the minds of the Yama-Shita family – had been stage-managed by Ieyasu.
Not exactly true, but close enough. Ieyasu had not known in detail what the saboteurs intended to do; he had merely allowed the operation to go ahead. Had he known more, he might have acted otherwise. Using communication devices and 'hired' agents was one thing; allowing those same agents and Mute witches to murder highborn japanese citizens with impunity was something else entirely.
The events which had led to this indiscriminate killing were, arguably, an example of a delicate political problem that could not have been solved in any other way, and with such brutal swiftness. But there were limits beyond which Ieyasu was reluctant to go in his desire to preserve the shogunate. The Heron Pool incident marked the top of a slippery slope he had no wish to descend. And now, in the spring of 2991, the long-dogs had attacked again. Only this time, they had struck first and told him afterwards. A wheel-boat of the Yama-Shita family, carrying a large number of samurai and foot-soldiers towards the western shore of Lake Mi-shiga had been sunk with the loss of all hands.
Karlstrom, in sending his apologies, had explained that there had been no time to seek his approval. At the very last minute, AMEXICO had received news that the Yama-Shita intended to launch a military operation against a clan of Mutes that was sheltering the agents who had sabotaged the Heron Pool. No one could condemn the Yama-Shita family's desire for revenge, said Karlstrom, but it was, under the laws of Ne-Issan, an illegal act of war.
True. But even so, regardless of the circumstances, the loss of 250 samurai, 300 red-stripes and 150 officers and crew was an act of violence that was difficult to condone: an affront to the pride of the entire nations. Had the attack gone ahead, it would have been a criminal act for which the Yama-Shita would have been duly punished. But it was equally reprehensible for the long-dogs to take the law into their own hands. To engineer the death of over seven hundred soldiers of Ne-Issan in order to save five of their agents and a clan of grass-monkeys was a totally disproportionate response. Secret agents were treasured assets but their duties also included a readiness to die. AMEXICO's action against the wheel-boat had seriously damaged the existing relationship to the point where Ieyasu was beset with grave doubts about its future.
There was also another problem. Should he tell the Shogun about the illegal expedition mounted by the Yama-Shita? Or should he remain silent about the whole affair and accept the announcement from the palace at Sara-kusa that a wheel-boat supplying the new out-stations on Lake Mi-shiga had been lost with all hands during a violent storm? To reveal the truth – or part of it – would place Yoritomo under an obligation to impose further sanctions.
Ieyasu was reluctant to increase the pressure on the family. The death of Domain-Lord Yama-Shita and the exposure of the plot to resurrect the Dark Light had strengthened the position of the Toh-Yota. Its most powerful rival had been humbled, but they still commanded a great deal of covert support. Applying more sanctions would be seen as an attempt to completely destroy the family – a move that would arouse suspicion and resentment among the other domain-lords.
In their eyes, the deaths of the named conspirators and their closest relatives and the harsh fines had expunged the family's guilt. Any further attempt to crush the Yama-Shita would be seen as a threat to all those who supported its progressive ideals. No one wanted to create conditions that could lead to another civil war. As the first among equals, the Toh-Yota had to be strong but not too strong. And since it could not singlehandedly sweep all opposition aside, it had to maintain the balance of power by a mixture of skilful government and skulduggery – two areas in which Ieyasu was the acknowledged master.
After lengthy reflection, Ieyasu decided to say and do nothing. He would, for the moment at least, leave the dilemma posed by his relationship with Karlstrom unresolved. He had not lost all sense of honour. It was simply that his self-esteem was of minor importance compared with the maintenance of the Toh-Yota shogunate. As long as he, Ieyasu, was alive, Yoritomo could be left in charge of the moral high ground. His task was to underpin the succession by ensuring that the opposition remained fragmented. His legacy would be to imbue Yoritomo with the determination to gradually reduce the Yama-Shita to penury, redistribute their lands and drive them into political obscurity like the once-great Da-Tsuni.
To aid Yoritomo in this task, Ieyasu's successor needed to retain access to the same alien devices that had enabled the present spy network to function so efficiently. The links with AMEXICO would not be severed but, equally, they would not be extended and the existing arrangements would have to be more tightly controlled. Karlstrom would have to understand that the indiscriminate killing of high-ranking Iron Masters by outlanders – no matter what the circumstances – could no longer be countenanced.
The presence of a Mute witch among the team of saboteurs at the Heron Pool and this latest action against the wheel-boat in defence of a clan of Mute fisherfolk were discordant notes in what until then had been a harmonious relationship. Similar, in many respects, to the trading contacts built up by Iron Master and Mute over several decades; contacts which had subsequently received the covert blessing of AMEXICO.
Having invested a great deal of time, money and effort, the Iron Masters regarded the Plainfolk Mutes as their own milch cow. These illiterate animals could never be allies but they had been accorded the status of auxiliaries. That was why they had been armed instead of being enslaved in the hope they could slow down the northwards advance of the Federation. But had the ground rules changed? Were these two disquieting incidents the product of another 'understanding'? Another secret protocol signed by one or more of the competing Mute bloodlines and the smooth-tongued head of AMEXICO?
Only time would tell.
In the palace-fortress of Sara-kusa, built on the site of the pre-Holocaust city of Syracuse, N. Y, Aishi Sakimoto, acting Regent of the Yama-Shita family, had been asking himself more or less the same question and believed he now knew the answer.
In the normal course of events, the domain-lord's eldest son would have assumed his father's title, but on the orders of the Shogun, Hirohito's children had all died by their own hand, or had been killed by their mother before turning the knife upon herself.
In some families, blood-feuds erupted when competing branches disputed the succession but Domain-Lord Hirohito Yama-Shita had ruthlessly eliminated all potential rivals. He had ruled with an iron hand but under his leadership the family, already rich, had prospered even more. Only now, with most of his immediate relatives dead, had come the sombre realization that his murderous reign had eliminated most of the candidates with the necessary strength, ability and drive to take his place.
The qualities of leadership now displayed by Aishi Sakimoto had not escaped the notice of his late nephew but he had survived, partly because he was Hirohito's favourite uncle and a fairly ruthless character himself. But what had really saved him from assassination was the fact that he was old and without issue, and was therefore not regarded as a threat to the domain-lord's own family.
It was also the reason why the shaken survivors had appointed him to head the council now running the family's affairs until one of their number formally assumed the title. In the present climate, the chosen successor to Hirohito would not necessarily be the best man for the job. Ieyasu, the Lord Chamberlain had sent word that whoever was chosen could only become domain-lord with the approval of the Shogun. And everyone knew Yoritomo would not allow a strong candidate to take the helm.
It was a bitter pill. Never before had the family been forced to endure such interference with their affairs. The twin ancestors of the Yama-Shita, the Yama-Ha and the Matsu-Shita had helped the Toh-Yota defeat the Da-Tsuni. They had been allies. As part of the historic Seventh Wave, their blood had mingled on the shore of the Eastern Sea. But with the merger of the two families to form one of the biggest domains in Ne-Issan they had become rivals. And their differences had been aggravated by Hirohito's espousal of progressive ideals.
The domain formed by the merger was not significantly larger than the territory held by the Toh-Yota. The source of their unease lay in its unique geographical position which gave it access to the the Great Lakes _and_ the Eastern Sea, borders that could be easily defended and, above all, an enviable trading advantage. Even though the Toh-Yota had filled its own coffers by taxing the family's revenues, the steadily increasing wealth and influence of the Yama-Shita had come to be viewed as a threat to the Shogunate.
Lord Hirohito's overconfidence had led him to act prematurely. He had been right about Yoritomo. Left to his own devices, the young Shogun would not have been a problem. He was his own worst enemy. But Hirohito had seriously underestimated Ieyasu's staying power. With Yoritomo's accession and his attempted clean sweep, Ieyasu's grip on the Inner Court had been seriously weakened. Many of his cronies had been ousted and his place-men in the _bakufu_ had been demoted or pensioned off. Everyone had confidently expected Ieyasu to follow them out through the door to spend his last years pottering about the garden or the library of his large estate.
But the old fox had hung on, and six years later, the foothold he had managed to preserve had become a veritable stranglehold. It had been reported that he not only had the Shogun's ear, he had both ears pinned against the wall. The proof was there for all to see! Yoritomo's fleet-footed manoeuvres in the wake of the Heron Pool massacre bore all the hall-marks of the great conspirator.
Yes. More positive action should have been taken at the beginning. An overt assassination attempt was out of the question but Ieyasu's penchant for juveniles was no secret within court circles. Instead of gloating over the reports of his imminent removal from office on the grounds of galloping senility, Hirohito should have slipped a couple of well-schooled 'spring blossoms' into the old bugger's bed with orders to stay on the job until they had fucked his brains out.
Well, it was too late now. Hirohito had paid dearly for his mistake and so had the family. The account would be settled – with interest. But it would be an uphill task. Ieyasu would not last for ever, but it was clear that the young Shogun could no longer be written off. He had learned a great deal. The Yama-Shita would rise again but it would be many years before they would be strong enough to dislodge the Toh-Yota. He, Aishi Sakimoto, would play his part, but the sweet moment of victory would not come in his lifetime. For the moment, they would have to content themselves with punishing the clan M'Call and the rest of the She-Kargo bloodline.
Thanks to a message sent from the wheel-boat soon after the unmasking of the two Kojak 'guides', Sakimoto now knew that one of them was the cloud-warrior that the M'Calls had sent to Ne-Issan escorted by a female Mute. It was not clear whether this female – who had last been seen in the hands of the Min-Orota – was the unmarked white witch who had murdered Lord Hirohito with her foul magic, but there was a possibility they were one and the same.
When unmasked on the wheel-boat, the cloud-warrior had been disguised as Mute. His companion, whose skin was similarly marked, had been identified as the grass-monkey who had become the cloud-warrior's personal servant and had flown the first rocket-powered prototype. In view of the expert way he had handled the craft he was probably another skilfully-disguised long-dog. As for the white witch, her true identity remained problematical. Sakimoto knew of the rumours of Mute magic but he had discounted them as Lord Hirohito had. Now he was not so sure. If the witch was _not_ a long-dog, then it meant that there were clear-skinned, smooth-boned Mutes who – on the outside at least – looked just like Trackers!
It was all very confusing.
There was, however, one aspect of this affair which was not bedevilled by doubt. A working connection between long-dog and grass-monkey had been clearly established. By piecing together the information gathered from the fisherfolk of Lake Mi-shiga by the agents now stationed on the eastern shore, Sakimoto knew the wheel-boat had been set afire and sunk by flying-horses which could only have come from the Federation. And the first two assassins had arrived in Ne-Issan as emissaries of the Clan M'Call. Under a deal struck by Lord Hirohito and a wordsmith called Mr Snow, the clan had agreed to deliver a cloud-warrior and his flying-horse in exchange for a hundred rifles!
Who could have guessed that this dull-eyed, oafish scum was capable of such duplicity? Never mind. It would not go unavenged, and nor would the crimes of the clan Kojak who – after the sinking of the wheel-boat – had massacred all those who reached the shore. But the first move would be against the M'Calls – the link between the Plainfolk and the Federation. And this time, the operation had to be mounted without any possibility of failure.
The family could expect no assistance from its few remaining friends in an act of war that did not have government approval. And in the present climate, that was unlikely to be obtained without being forced to hand over a large piece of the pie. The only way round the problem was to mask the attack by using Mute clans from the D'Troit bloodline, the fiercest rivals of the She-Kargo.
Sakimoto was aware that slaughtering the She-Kargo trade delegations would net very few of the real offenders. The bulk of the clan M'Call was safely out of reach. But this attack was only the first step in a plan the family had been hatching for some considerable time. The harsh consequences of Lord Hirohito's misadventure had weakened the family's power-base and it could only be rebuilt by implementing the plan to expand its present boundaries.
Recruiting the D'Troit to do the spade work was part of that plan. The present fragmented nature of the Plainfolk which led to clans of the same blood-line fighting each other was pointless and unproductive. And the trading arrangements under which each clan annually supplied varying-sized groups of 'volunteer' slave-workers was an inefficient way of meeting the constant demand for labour.
They killed each other in the same haphazard way. There was no master plan. Warriors who triumphed in a clash of arms did not go on to plunder the settlement of the losers. The victorious clan did not occupy its rival's land or slaughter the survivors. It did not even attempt to enslave them. The winners simply went home and composed fire-songs which extolled their prowess!
Part of the problem was the vastness of the territory at their disposal. There was too much land and too few people. And because the inhabitants were savages with a simple life-style, they did not need to exploit the land's resources. There was enough room for everyone, an abdundance of game and more raw material than anyone could possibly need. There was no need to conquer each other. Mutes fought each other because they were wedded to the warrior ethic. It was a test of courage, part of the process of natural selection. Very laudable. But all that ferocious energy should not be allowed to go to waste. It should be directed towards a loftier goal, not frittered away on inconsequential skirmishes.
The Yama-Shita planned to provide that sense of direction by unifying the disparate clans of the D'Troit and C'Natti bloodlines and creating two vassal states. Armed and advised by the Yama-Shita, they would then subjugate their hated rivals, the She-Kargo, and the other lesser bloodlines. When this had been done, they would levy annual tributes in the form of raw materials and able-bodied males and females. Punitive tributes which would force this race of savages to toil from dawn till dusk instead of idling their days away with smoke-filled dreams. Work would be their saviour, not some invisible being called Talisman.
As patrons and protectors of the D'Troit and C'Natti, the Yama-Shita family would be the sole conduit for this new flow of materials and labour. It would generate unimaginable wealth – but only if each move was carefully planned. The treasonous acts of Lord Hirohito had robbed the family of its exclusive right to trade with the grass-monkeys. Licences had been awarded to its southern neighbours, the Ko-Nikka and Se-Iko but it was Yama-Shita who controlled the entrances to the Great Lakes and it was _their_ navigators who knew what course to steer through the deeps and shallows. They had the know-how and the contacts, and their fleet of giant wheel-boats dwarfed the vessels owned by the Ko-Nikka. The Se-Iko, whose domain was land-locked, only possessed river craft. For this first trip they had been obliged to lease two boats and their crews from the Yama-Shita at exorbitant rates. Both families had placed orders for larger vessels but the Ko-Nikka's shipwrights – who lacked the expertise needed to construct such large vessels–were still wrestling with the problem of how to lay out the bilges.
Sakimoto was content to let them stew in their own juice. The family's former allies had broken ranks in return for a share of the Great Lakes trade and now they were learning that there was more to it than a pretty piece of paper with the Shogun's seal on it. The Yama-Shita still had the edge on its new partners and they both knew that without its assistance their own crews might return empty-handed.
From the signals coming into Sara-kusa, it appeared that some of the wiser heads in both domains were already regretting the hasty leap onto the Shogun's coattails. If these veiled contacts were inspired by feelings of guilt then he, Aishi Sakimoto, intended to exploit such sentiments to gain whatever advantage he could. Given their recent turn-about, he could not take them into his confidence and it was not necessary. They would serve his purpose better by remaining independent witnesses. They would be able to testify that the fighting at the trading post broke out between the rival Mute bloodlines and it was only when the conflict appeared to be getting out of hand that the Yama-Shita family felt obliged to intervene in order to separate the warring factions and protect the Iron Masters trapped on shore. And if, in the course of establishing a cease-fire, one faction suffered heavier casualties than the other then that would be regrettable but unavoidable.
This had been Lord Hirohito's plan, and it was a good one. In making it, he had not been inspired by any particular animosity towards the She-Kargo. On the contrary. Their pre-eminence was something to be admired. He had simply decided to back the D'Troit and the C'Natti because their burning desire to become the paramount bloodlines of the Plainfolk would tempt them into an alliance with the Yama-Shita.
But with his death at the hands of the white witch, the subjugation of the She-Kargo was no longer just the initial phase in the economic development of the Western Plains. It had become an act of vengeance.
And the first tribute to be exacted by the new vassal states on behalf of their master would be the heads of the Clan M'Call.
The M'Calls were also at the top of the hit-list drawn up by the leaders of the Amtrak Federation. The narrow escape of the wagon-train known as The Lady from Louisiana in the Battle of the Now and Then River in June 2989, and the subsequent disastrous attack which destroyed a third of its wagons and crew in the November snows of 2990 had badly dented the Federation's image of invincibility.
It was a challenge that could not be ignored. Such spirited resistance by sub-human savages could not go unpunished. The Clan M'Call, the group responsible for this outrage, had to be crushed. Ground into dust as an example of what happened to those who resisted the might of the Federation.
The decision to annihilate the clan had been the easy part. The difficulties lay in its implementation. The M'Calls were led by an individual called Mr Snow – said to be one of the most powerful summoners ever born into the Plainfolk.
The ability to summon invisible forces present in the earth and sky – dynamic energy which flowed through their bodies and was shaped and directed by their minds – was one of three attributes possessed by certain 'gifted Mutes'. Their rarity was something to be thankful for; the ability to summon hurricane-force winds, 'sky-fire' and 'earth-thunder', and to levitate rocks weighing up to half-a-ton was the most spectacular and dangerous form of what had come to be known as 'Mute magic'.
Summoners, seers – those gifted with the power to read the past and foretell the future with the aid of seeing-stones – and wordsmiths – those born with prodigious memories – displayed mental abilities that belied their primitive appearance and life-style. They were highly intelligent, and the powers they possessed – or could call upon – defied all rational explanation.
In due course an answer would be found through the rigorous application of the recognized laws of physics, but in the meantime, faced with something they could not master or understand, the First Family had officially decreed that there was no such thing as Mute magic. Any public utterance to the contrary was a Code One offence – punishable by death, and any overground unit that found itself on the receiving end of Mute magic could not use it as an excuse for failing to achieve its operational objectives.
Draconian measures which had been ruthlessly enforced. And with good reason. Maintaining a high degree of motivation among units exposed to a hostile environment for months at a time in distant work-camps and way-stations was no easy task. And that included the elite troops known as Trail-Blazers who crewed the wagon-trains. Totally dedicated and highly disciplined, the Trail-Blazer Division was a cross between the battle-hardened WW2 generation of US Marines and the Waffen-SS of the same period. But even they had been known to lose their grip when confronted by the awesome powers unleashed by a summoner able to call upon the Seven Rings of Power.
Mr Snow, known as The Storm-Bringer, was one such individual – perhaps the only one of his kind among the Plainfolk. Thanks to the efforts of Steve and Roz Brickman, Mr Snow's protégée, Clearwater, was now in the hands of the Federation. Well, almost. To be more precise, she was in the intensive care ward of the mobile field hospital hitched to the Red River wagon-train now heading south out of Nebraska.
Clearwater, on past form, was rated as potentially even more dangerous than Mr Snow, but after suffering massive wounding followed by major surgery she was now under continuous sedation and, in the short term, was not judged to be a threat while in transit aboard Red River or to the Federation where the research staff of the Life Institute were eagerly awaiting her arrival.
That left Mr Snow, summoner and wordsmith of the Clan M'Call, and his other young pupil, Cadillac Deville, apprentice wordsmith and seer. Their capture (at planning level it was called 'their removal from the equation') was to be the final phase in OPERATION SQUARE-DANCE, masterminded by Commander-General Ben Karlstrom.
Karlstrom, a close relative of the President-General, was the Operational Director of AMEXICO, the covert operations unit formed to carry out assignments initiated directly from the Oval Office – currently occupied by George Washington Jefferson the 31st.
In late 19th century terms, AMEXICO combined the functions of the American CIA, the Gestapo of Germany's Third Reich and the British SAS. Intelligence-gathering, state security, commando-style forays and 'judicial terminations' in parallel to, but without the knowledge of, the recognized military police and army intelligence units. Its agents – known as _mexicans_ – operated within the subterranean nation-state ruled by the First Family and the blue-sky world above, and its existence was the best-kept secret in the Federation.
The nine members of the Supreme Council were officially aware of the general nature of the organization but they did not know the manpower it employed, the resources it could call upon or the scope of its activities. Only Jefferson the 31st and Karlstrom knew the full score. Others could only speculate. AMEXICO was the President-General's private army and the impenetrable cloak of secrecy was essential because it was sometimes called upon to eliminate potentially troublesome members of the First Family.
To some readers, it may seem strange that disaffection of any sort could exist in an enclosed totalitarian state in which every aspect of the environment and its soldier-citizen's lives was controlled with military precision and computerized efficiency from Day One. But such was the case. Any reader with direct experience of the armed forces or the electronics industry will know that 'military precision' and 'computerized efficiency' are mythical states which bear little relation to what either system is able to deliver.
All monolithic power structures staffed by human beings are bound to be less than perfect – especially one reliant on advanced technology – and the Federation was no exception. Nine hundred years of relentless regimentation had failed to produce a First Family version of 'Soviet Man'. People at every level of the command structure still screwed up and systems crashed with unfailing regularity. Depending on your attitude towards what the First Family was trying to achieve this was either a matter of grave concern or grounds for celebration. Proof of the indestructibility of the human spirit; a ray of hope for the future of mankind.
The founding Father, George Washington Jefferson the 1st, had known this basic truth when he laid the foundations on which his vision of the future was to be built. 'Only people fail, not the system', was one of his two Delphic utterances emblazoned on every available wallspace throughout the Federation and echoed daily on the nine public video channels. A phrase of cunning simplicity which had helped to preserve the status quo by deflecting the blame for any shortcomings in the system back onto the individual who raised his voice in protest against the measures taken in its name.
To prevent these isolated cranks spreading their disaffection like a virus through the body politic over which they presided, the First Family had installed a number of fail-safe systems. By the spring of 2991, these included an acoustic surveillance system code-named HYDRA under which their entire underground empire had been wired for sound.
Any location, at any level, could be monitored by keying its coordinates into a command console. It was possible for human operators to listen in to conversations but the sheer logistics made that impractical. Most of the eavesdropping was controlled by COLUMBUS using a given hit-list of suspects and hourly random samplings known as 'sound sweeps'. Ordinary conversations were monitored but not recorded: it was only when a speaker used a word or phrase listed in a 'subversive vocabulary' that the reels started turning. The taped conversations were then subjected to computer analysis, classified into various categories according to the nature of the conversation and graded in terms of 'arrestability'. It was only when this sifting process had been completed, that the daily residue of hard cases – known as the action list – was displayed on the screens of the operatives at HYDRA Central.
But the system, and its handlers, missed the big one.
Despite HYDRA and the nation-wide network of undercover agents the Family were caught totally off-guard by the protest strike mounted by the crews of the Federation's wagon-trains. Organized and led by their executive officers it had none of the characteristics of a popular uprising led by dissidents, but it was, nevertheless, a rebellion; the first serious challenge to the authority and wisdom of the First Family for over six hundred years.
The protest, which was confined to Trail-Blazers on active duty aboard the trains, was disciplined and unpublicised and the divisions's so-called grievances were conveyed to CINC-TRAIN in a coded message signed by twenty wagon-masters and their executive officers out of the Federation's fleet of twenty-one. Only the name of Red River – Amtrak's flagship – was missing from the roll.
The protesters had two demands. First, official recognition of the existence of Mute magic. If, for reasons of state, this could not be made public then it was to be admitted in secret session to the wagon-train fraternity by representatives of the First Family. Second, the disciplinary charges brought against Commander Bill Hartmann and the executive officers of The Lady – which had arisen from their inability to combat the power of a Plainfolk summoner – were to be dropped. All crew members were to be fully exonerated and returned to active duty in their previous posts without loss of seniority to privileges.
There was no 'or else' but with 95% of its wagon-train force ranged solidly behind the protest, the White House decided to roll with the punch. In a hurriedly-prepared videocast beamed exclusively to the wagon-trains, Jefferson the 31st agreed to the rebel's requests within forty-eight hours of their receipt.
It was an unprecedented concession; the first time an organized protest had not been brutally crushed, but it was also a victory for both sides. The First Family had been wrestling with the problem of Mute magic for the last two hundred and fifty years.
The decision to deny its existence had been taken by a long-dead and buried Supreme Council. At the time, it had not been a problem. It was the Southern Mutes who had then been the enemy. The rumoured existence of summoners had been reported by FINTEL, but on the few occasions where their presence had been suspected, their intercession had failed to halt the advance of the Federation.
The Family had therefore concluded that the alleged power of these individuals did not pose a serious threat to future operations. The danger came from uncontrolled rumours and wild speculation about Mute magic within their own ranks. But instead of ending all ill-informed discussion of the subject, the imposition of sanctions had merely driven it underground.
With the move north into Plainfolk territory the nature of the conflict had changed. The days of easy victories were over and the earlier blanket denial of Mute magic had left the present leaders of the First Family on the hook. This was a chance to slide off it and score a few Brownie points in the process.
Given the latest situation reports from Wyoming and Nebraska, the call for the reinstatement of Commander Hartmann and his crew could not have come at a more opportune moment. Conceding to the rebel's demands provided the First Family with an opportunity to bring its secret OPERATION SQUARE-DANCE to successful conclusion.
That fact would only become apparent with hindsight, but it did not mean that all would be forgiven and forgotten. Hartmann and his executives had been the catalyst which caused a simmering discontent to crystallize into open rebellion. The protest might have been restrained, short-lived and totally justified, but it was a direct challenge to the First Family's inalienable right to rule from the top down.
What made it worse was the fact that the rebellion had been well-organized and _had not been foreseen_. Such appalling laxity on the part of security services could not go unpunished. Sooner or later everyone concerned would be dealt with. No one got the better of the First Family and lived to tell the tale. And among the first to learn that harsh lesson would be the crew of The Lady from Louisiana.
## CHAPTER TWO
Commander James Fargo, the wagon-master of Red River, had seen plenty of Mutes at close-quarters during his time on the overground but they had all been dead, dying, or framed in his gun sights. Killing Mutes was what he had been trained to do. He had never once imagined that a day would come when he was required to play host to two live lump-heads. But they were here. Cosseted aboard Red River.
It was a strange feeling. As far as he knew, this was the first time Mutes had ever sullied the interior of a wagon-train with their poisonous presence, and it was a dubious honour he would have preferred to do without. But orders were orders. Even so, he felt he should have been taken into the confidence of his other guests, a seven-man task-force from the White House complete with their own secure lines of communications. From the moment the two Mutes had come aboard, it was clear they were not true lumpheads and – to judge from the flurry of coded signals flowing between the task force and Grand Central – might be something else entirely.
As the top-rated wagon-train in the Federation, Red River had, for many years, been involved in the delivery and collection of a number of individuals whose reason for being on the overground had never been fully explained but who were clearly engaged in some form of covert activity. Since the soldier-citizens of the Federation were members of one vast army whose chain of command led back to the White House, these individuals had to be working for the First Family but, to date, no one had ever discovered the name or the precise function of the unit they belonged to.
Fargo knew it was unwise to enquire further. No one reached the top spot on the Federation's premier wagon-train without learning that at a very early age. As an exemplary product of the system, Fargo believed that if a man wasn't party to secret information then he had no business prying into it. The First Family told you everything you needed to know when it was time for you to know it.
This unquestioning attitude did not mean that the commander of Red River was a colourless, mindless automaton. Initiative and intelligence were part of the job profile. Fargo's unswerving allegiance to the Federation was comparable to Reinhard Heydrich's total commitment to the genocidal policies of Hitler's Third Reich. And like the latter, he had a distinctive personality and a mind of his own. But any reservations he had about the way the First Family ran things (and they were very few) was something he kept strictly to himself.
The first of his visitors had been flown in following a night pick-up by the White House task-force using Skyhawks supplied by Red River. One of their number was a 17-year-old Junior Medical Officer named on the detachment order as Roz Brickman. Since she had little experience of battle-field injuries, three members of Red River's own surgical team had flown out with her, riding the buddy-frames attached to the fuselages of the five-plane formation.
The badly-wounded Mute had been brought back lashed to the spare berth and had undergone immediate surgery. In the only direct message he had received from CINC-TRAIN, Fargo had been instructed to put the crew and services of Red River at the disposal of the task-force. He had done so. The female Mute – logged aboard as ALPHA-BRAVO – was now in intensive care with the Brickman girl at her bedside. Mitch – Michelle French, Red River's immensely able CMO, who had performed the major part of the surgery rated her chances at no better than 50–50.
Some thirty-six hours after Red River's first unwelcome guest had flown in, the second had arrived under his own steam. Or, to be more precise, on the back of a four-legged animal that Fargo – like the rest of his crew – had been told was extinct.
Code-named YANKEE-ZULU by the secretive task-force, the smooth-boned, fair-haired Mute had made his approach in broad daylight astride a horse, with two similar beasts in tow. Fargo had relayed the video pictures through the train so that his crew could share the experience of seeing these living relics of a bygone age. Since he was also required to take the horses on board, there was little point in trying to keep their presence secret. But as he watched them being led towards the train, Fargo could not help asking himself the inevitable question. If COLUMBUS held the wrong data on horses, what other errors had it made?
Fargo erased the question and its consequent uncertainties from his mind. COLUMBUS had not made a mistake. The First Family, in its wisdom, had instructed it to withhold the information. And there would be a good reason for doing so.
Although unhurt, YANKEE-ZULU was now in the blood-wagon – the fully-equipped field-hospital that was an integral part of each wagon-train. The medical staff was accommodated on the ground floor along with stores and small lab units. The second floor contained an operating theatre, pre- and post-op, IC, x-ray and ultra-sound units and a clinic for treating minor wounds and ailments. The top floor contained three medical wards, designed to be self-contained if the need arose. The task-force, led by someone labelled as WALLIS, DONALD, E, had taken over aft-section, sealing itself off behind the sound-proofed partitions with its special radio equipment. ALPHA-BRAVO, still unconscious after her ordeal on the operating table, was in one of the cubicles of the intensive care unit on the floor below.
Fargo could not help wishing he knew what was being discussed behind those closed doors. He was too disciplined to display his feelings in front of his crew but he was more than a little put out by the thought that after eighteen years service in the field he was still denied knowledge of secret operations of which Red River was an integral part. And – as if to add insult to injury – the three horses he had obligingly taken on board were pissing buckets and dropping large steaming piles of crap all over one of his spit and polished cargo floors.
Steve gazed down at the sleeping figure inside the sterile plastic tent. A breathing tube that reached down into her larynx had been inserted between her pale lips. Her mouth and nose were covered by a clear oxygen mask. There were drip feeds in her arms, drainage tubes from internal organs, and wires linking her to electronic monitoring equipment. Clearwater was a long way from Mr Snow's herbal mash remedies. Another world...
Her olive brown skin had taken on a deathly pallor. She had no head wounds but someone had cropped her long dark hair, and they'd done it badly, leaving it looking like a porcupine who'd blundered into a chainsaw. But she still looked beautiful, her head and neck resting on a single pillow, miraculously untouched by the hail of bullets.
Steve, who had kept vigil over her broken, bloodstained body, tried not to think of the splintered bones and ruptured flesh that lay beneath the covers. A body so fragile, a curved frame had been placed over it to support the weight of the top sheet.
He turned to Roz. The sides of their bodies came into contact as they gave each other a supporting hug. 'Is she going to make it, little sister?'
Roz grimaced. 'At the moment, her chances are no more than even. When the surgical team saw the state she was in they were amazed she'd survived for so long. In fact, there were a couple of times they nearly lost her on the table.'
'And all because of a stupid, fucking argument – that _I_ provoked.' Steve broke away and raised his hands in despair. 'Why?! Ohh, Roz! If she's crippled for life I'll never forgive myself!'
Roz flashed him a warning glance and touched her ear to remind him that someone might be listening to their conversation. 'Guilt is a recognized symptom of shock,' she said, adopting her best bedside manner. 'After all you came close to getting killed yourself. It was a pilot from Red River who gunned her down. You and I just did what we had to do.'
'Yeah, you're right,' said Steve, cottoning on. He looked down at Clearwater then eyed the screen monitoring her weak heartbeat. 'Would it be all right if I just held her hand for a minute?'
'Yes, but very gently – okay?' Roz gathered up the side of the sterile tent and lifted the sheet. Clearwater's right hand lay palm up outside the metal frame.
Steve knelt down and sandwiched her hand between his own. The flesh was moist, the fingers limp. He pressed his palm against hers and tried to reach into her mind, tried to channel his life force into her body. He'd done the same thing in the deserted renegade camp in a desperate effort to infuse her with the will to live as they waited for Roz and the Red River medics to arrive.
'We've got to save her, Roz.'
'We will. Don't worry. Everything that can be done will be done.' Then, for the benefit of the hidden microphones she added: 'You and I aren't the only people with a vested interest in keeping her alive.'
'No...'
Roz smiled. 'And if you think this happened just because you got into an argument with Cadillac then I should take some of the blame. After all, the argument _was_ over me.'
'That's true.' Steve laughed for the first time since they'd been reunited. 'You've been nothing but trouble ever since you were born!' He parried her playful punch and looked down at Clearwater's hand in time to see the fingertips twitch then curl slowly upwards against the side of his hand. 'Roz! See that?!'
'Yes. Take a look at the screen.'
The weak green trace of Clearwater's heartbeat had changed. Not dramatically, but every fourth pulse was a little deeper, a little stronger than the others.
Steve's spirits soared. 'D'you think she knows I'm here?!'
Roz caressed the back of his head. 'Yes, I'm sure she does.' _But not because you are holding her hand. She knows because I am within her as I am within you_...
In the sealed ward above the intensive care unit, Don Wallis motioned Steve and Roz to take the facing seats in the middle of the table where they were sandwiched between the six-man team from AMEXICO. Wallis, the team leader, sat at the head of the table on Steve's right. Jake Nevill, his Number Two, was at the other end.
It was Nevill who had flown out with Roz zipped onto the buddy-frame of his Skyhawk. While she and the Red River medics were busy with Clearwater, he and Steve had given each other the buzz. Satisfied he was talking to the right man, Nevill told Steve he had been assigned the temporary code-name of YANKEE-ZULU while on board the wagon-train: his AMEXICO code-name was never to be disclosed to anyone outside the organization. The task-force of mexicans was disguised as a special detachment from the White House, complete with fake ID-cards, name-tags and the distinctive blue and white badge on the shoulders of their camouflaged fatigues.
It was, explained Nevill, standard procedure when operating alongside regular army units.
As a JMO, Roz was wearing hospital whites with her name tag, surmounted by miniature lieutenant's rank stripes, tacked onto a Velcro patch above her right breast pocket.
Steve, after a long, hot shower and a medical examination designed to make sure he had not contracted some unspeakable overground infection, had been given a set of Trail-Blazer fatigues with no name-tag or badges. Since he still had his long ragged hair, rat-tail plaits and multi-coloured skin markings, the effect was bizarre. The task-force's bewildered reaction on first seeing him in uniform reminded Steve of his painful encounter with Lt. Harmer at the Pueblo way-station. This time however, no one tried to pulverize his liver with the butt end of a rifle.
How much does Roz know about all this? he wondered. In the few brief moments they had spent together since boarding Red River he had been so concerned about Clearwater, he hadn't even asked Roz how she came to be on the wagon-train. He had just been thankful she'd been close at hand when he needed her. It was probably wiser to say nothing at this stage. Her silent reminder that Clearwater's cubicle might be bugged had jerked him back to the reality of the Federation. The fear that whatever you said might be recorded and used in evidence against you.
Maybe, when the time was right, she would come through on their private line. She was the expert. Steve, who had wilfully neglected his telepathic gifts, was still restricted to the channel used for broadcasting May-Day messages.
Wallis aligned his electronic memo-pad with the edge of the table, cleared his throat and began: 'Steve, ahh – this first session is essentially a debriefing. You've met Jake. I don't think we need formal introductions. The names are on the labels.'
Steve said hello to George Hannah and Cal Parsons who sat across the table on either side of Roz, and to Daryl Coates and Tom Watkins who sat on his left and right respectively.
'They're all members of the organization, and in case you feel a little tongue-tied, Roz has been made an honorary member.'
Steve eyed Roz then looked blankly at Wallis.
'It means she doesn't have a code-name or a call-sign,' said Wallis. He fingered his left earlobe in a seemingly absent-minded gesture.
'Got it...' Steve smiled at Roz. 'I had no idea.'
'Well, you know what I'm like – always _dying_ to know what you're up to.' Roz's face bore just the merest hint of a smile but Steve knew that inwardly she was savouring the exquisite irony of the situation.
'Hey! Come on, you two – snap out of it!' exclaimed Nevill.
Steve ignored him and turned to Wallis. 'You were saying?'
'I've been asked to congratulate you both,' said Wallis. 'That message comes jointly from the Operational Director and the Oval Office. Initially there was some concern that the goods had been damaged in transit but ALPHA-BRAVO's disablement probably makes the task of shipping her back to Grand Central a lot easier. As Roz has probably told you, it's early days yet, but given the level of medical support available here and down the line, I've been assured that if our target survives the next two weeks, she has every chance of making a full recovery.'
'Glad to hear my efforts weren't totally wasted,' said Steve.
'If she needs any other specialists, they can fly out and treat her on the return trip.'
Steve's eyes met Roz's briefly. With six people watching them they had to tread carefully. 'Return trip...?'
'Yes,' said Wallis 'We were planning to airlift her into the Federation but Red River's CMO has advised against it. And I agree. We made that night pick-up because we were in a life-or-death situation but it would be crazy to risk losing such a valuable asset between here and Grand Central. This is the safest way for her to travel. So as soon as we know what your plans are, we're going to run for home.'
Steve weighed up the other members of the task-force then came back to Wallis shaking his head. 'No. Sorry. You're going to have to call Mother and tell him you can't do that.'
'Oh? Why?'
'Because I need Red River to stay here with Clearwater on board. She's the bait that will lure the other two into the net.'
Wallis pursed his lips. 'You mean Cadillac and Mr Snow...'
'Yes.' Steve looked at Roz, but apart from listening with interest, like everyone else, she did not respond.
'You had Cadillac – knocked out cold – and you let him go,' This was Nevill again.
'I let him go because he's the one who will bring Mr Snow to us,' said Steve patiently. 'I need a month to get things organized. Six weeks at the outside.'
'Six weeks?!' cried Nevill.
'That's not very long when you consider it's taken me a year to put this together.'
'Christo! Just to kidnap three lumpheads? If someone had given me the job I'd have winkled 'em out inside forty-eight hours.'
'With an airborne snatch-squad?'
'Yeah. They'd be back in Grand Central before they knew what hit 'em.'
Steve looked impressed. 'That certainly would have been a remarkable feat of logistics. One of the trio was in Wyoming, the others were in two separate locations in Ne-Issan – held by people opposed to our friends who run the local network.'
'Quite,' said Wallace hurriedly. He was the only person at the table who knew what Steve was alluding too and it was a subject he wanted to put a cap on. Attempting to be diplomatic he added: 'I don't think you've fully appreciated who it is we're dealing with, Jake.'
'Exactly,' said Steve, opting for a head-on collision instead of conciliation. 'Have you ever come up against a summoner? Actually _seen_ them channelling earth magic through their bodies?'
'No, but –'
Steve cut across Nevill's reply and addressed the other mexicans. 'Have any of you?'
They all shook their heads.
'Well, I have. I've been on the receiving end when I was on board The Lady from Louisiana in 2989 –'
'I think we've all read the report on that one,' said Wallis, trying to keep his team's end up.
'But I've also _seen_ them make it happen. Seen 'em make rocks fly, blow away half a hillside, take control of someone's mind.'
Steve described the death of Lord Yama-Shita. How Clearwater had made him drive his sword repeatedly through his body. And each time, the blade had sunk in right up to the blood-drenched hilt. Eight killing strokes, one for each of the Mutes he had condemned to death on the giant iron-bound paddle of his wheel-boat.
Turning to Nevill, he said: 'Mr Snow's other name is the Storm-Bringer. That's not just a fancy title. It means precisely what it says. He'd have blown your snatch-squad right out of the sky.'
'But you, of course, know how to handle him,' said Nevill.
_Sarcastic sonofabitch_... Steve had run up against guys like Nevill before. There were some people who took an instant dislike to him and no amount of sweet talk could bring them round. They just did not respond to treatment. The important thing was not to get mad. 'Yes,' he replied. 'With Roz's help I think I can.'
Wallis made another attempt to lower the temperature. 'Roz – what shape will ALPHA-BRAVO be in six weeks from now?'
'Clearwater? She'll still be flat on her back. All we've done so far is stabilize her condition. She needs several more ops and at least four months convalescence before she's on her feet – assuming there are no complications.'
'Okay,' said Wallis. 'Let's move on. Steve, why don't you begin by telling us what happened from the time you took off from Long Point.'
Steve recoiled. 'Wow... you mean everything?'
'Yes.'
'How long have we got?'
'As long as it takes.'
'Okay, uhh, before I start – do you happen to know if Kelso and Jodi got home in one piece?'
'Jodi did,' said Wallis. 'Someone found the explosive charge strapped to her chest and managed to pull the detonator out a split-second before it was triggered. It was the other that did the damage.'
Steve nodded soberly. 'The transmitter was hidden under Kelso along with the Px.'
'How did you know?' asked Nevill. 'D'you put it there?'
'No. It was a couple of days before I discovered the explosives were missing. That was when Cadillac told me what they'd done. And later, we heard a wagon-train had been blown apart.'
'Yeah,' said Nevill. 'The Lady from Louisiana. Your ex-crew-mates. Nice gesture, Brickman.'
Angered by this jibe, Steve's self-control snapped. 'You wanna know something? I'm getting pretty sick of your smart-assed remarks and your fucking innuendoes! If you've got some beef you wanna get off your chest, come right out with it!'
Nevill's jaw dropped.
'Whoa! Hold it!' said Wallis. 'I think we're getting a little ahead of ourselves. I'm sure Jake didn't intend to accuse –'
'Yeah? Well, he can take his accusations and shove them right up his ass! _I_ was the one who was out there risking my neck for the Federation. What the fuck does _he_ know about anything? Desk-bound pratt!'
Nevill bounced back, eyes blazing. 'You can spare us the war-torn hero routine. Not one of these guys around this table has less than five overground assignments on his slate. That's why we're here. And as for what I know about anything, Brickman, let me tell you this. I know _I'm_ a major and you're a jumped-up ensign whose lieutenant's tabs are still hanging out to dry.' The mexican tapped his chest. 'I earned my promotion in the _field_ , not filling gravy-boats on the fucking mess deck. So I'll say what I like, when I like. And when you next address me, _lieutenant_ , you'll do so with the respect you have been trained to show a senior officer!'
'Yes, sir, _major_...'
'And no dumb insolence either! I saw that look!'
Wallis made a placatory gesture. 'Okay, that's enough. If we're pulling rank, I'm the senior officer here. And I'd like to remind you all that this organization, unlike many regular army units, places a premium on intelligence and a spirit of enquiry. A readiness to challenge assumptions instead of the usual knee-jerk responses required from junior ranks. It is mental discipline we look for, not the kind instilled on the parade-ground and our executive body has, traditionally, always encouraged the free and frank exchange of views –'
'Fuck tradition!' Nevill stabbed a finger at Steve. 'He may be billed as a star turn but he's still a fucking wetback and no wet-back is going to shit-talk me!'
Wallis coughed. 'Okay, Jake, you've made your point.' He made eye contact with Steve. 'Would you like to proceed?'
Steve gave them the full story. With questions and answers, it took over four hours, split into two sessions with a meal break in between. Steve took a rain check on the food, preferring to spend the allotted time in the intensive care ward with Clearwater. It was no great hardship. The trays sent up from the mess-deck of the adjoining wagon contained standard Federation fare, tasteless, processed, vitaminized pap. After the food in Ne-Issan and the strong-flavoured fish and meat served up by the Plainfolk, Steve found it quite uneatable.
Afterwards, when Steve concluded his account, Wallis said: 'Okay, so Cadillac is on his way home. Assuming you're still on the case, what's your next move?'
'I've got to catch up with him. Tell him what's happened.'
Nevill stuck his oar in again. 'Isn't he going to be a little unhappy when he hears you put his pet beaver on a wagon-train?'
Steve swallowed his irritation. 'Unhappy? He's gonna be bouncing off the walls. But that's his problem. He's lucky she's still alive. Don't worry. I'll talk him round.'
'How're you going to do that?' asked Wallis. 'If Clearwater's on board Red River it's because you arranged to put her there. How are you going to get Cadillac to trust you when you're obviously working for us?'
'Simple,' said Steve. 'I'll tell him the truth. I didn't contact the wagon-train. I contacted Roz.'
Wallis waved a finger at Steve and his kin-sister. 'Do the M'Calls know about the link-up between you two?'
'No. Only Cadillac and Clearwater. Clearwater was watching from the shore when the planes fire-bombed the wheel-boat. Cadillac was below decks with me so he saw nothing. By the time we surfaced the planes had gone but eventually I had to come up with some kind of an explanation. The only way to remove any suspicion that I was still working for the Federation was by telling them about the link between Roz and myself.
'The fact that the Federation had sent the planes to bomb the wheel-boat meant that the First Family thought I was still working for them. Which, of course, led Cadillac to suspect Roz's motives. So in order to persuade him that we could both be trusted, I told him I'd convinced Roz – who had complete faith in me – that I was still one hundred per cent loyal. Which was why you'd sent the planes to help us escape.'
'Did he buy it?' asked Wallis.
'Yes, in his usual half-assed way. We got away from the japs, I saved his life but – you know what it's like –' Steve gave Nevill a sideways glance, '– he's one of these guys who is never satisfied. He's brave, highly intelligent, possesses the most amazing mental capabilities, but he just hasn't got his shit together.'
'So what story are you going to lay on him?'
'He already knows that Roz came through to tell me she was on board Red River. I'll tell him the shock of seeing them both gunned down triggered another contact between Roz and myself. She knew immediately what had happened and where we were, so I waited until she confirmed that help was on the way then pulled out as the first plane dropped a string of parachute flares.
'And I'll say that in a subsequent contact, Roz told me that Clearwater was out of danger. Cadillac will be suspicious but what the hell can he do? He has to go along with it because the only way he can keep tabs on what's happening to her is through me and Roz. And I will continue to claim that I didn't arrange anything through official channels. I was never here. How does it sound?'
The six-man task force exchanged thoughtful glances. Steve found the silence of the middle four a little unnerving. The guys who listened but didn't say anything were the ones you had to watch out for. More often than not, it was they who were really running the show.
Nevill shook his head dismissively. 'Too clever by half.'
'Jake, ease up will you? Give the guy a break.' Wallis came back to Steve. 'It sounds good but you may have outwitted yourself. Roz couldn't have sneaked Clearwater on board Red River by herself. It could only have been with our full knowledge and cooperation. And the reason Clearwater's still with us is because Red River's surgical team spent nearly twelve hours putting the bits back together.'
'I'm not denying that.'
Wallis threw his hands in the air. 'So we're back where we started. If Roz is working with us, and _you're_ working with Roz – why should these Mutes trust you?'
'It's simple,' said Steve.
'I wish you'd stopped saying that,' snapped Nevill. 'You're switching sides so fast I've lost track of who's getting shafted!'
Steve looked at his kin-sister then said: 'Roz is only _pretending_ to work for you.' Their eyes met again. The message she had beamed into his mind was so astonishing, none of the others would realise he was telling the truth. He glanced round the table and tried not to smile at his audacity.
'That's what I told Cadillac when he challenged me about her role in the strike on the wheel-boat. And when I discovered she was on Red River, it reinforced my claim that she was a secret ally who, having gained Mother's confidence, had persuaded him to put her on board the wagon-train because she intended to go over the side and join up with me at the very first opportunity.'
Steve's gaze returned once more to his kin-sister. 'A claim that Clearwater confirmed because Roz had already got inside her head and was doing a number on her.'
Wallis' eyes narrowed. In briefing him, Karlstrom had never even suggested such a bizarre possibility. But in emphasizing the need to keep her safe and secure at all times, he had warned him to be braced for the unexpected. Which was why the task-force had been sticking to her like shit to a blanket.
'Go on...'
'The next step is to convince Cadillac and Mr Snow that if we can find a way of getting aboard Red River – maybe with the help of Malone's renegades – then we can rescue both Roz _and_ Clearwater. And perhaps even capture the train.'
Tom Watkins, the mex on Steve's right broke his silence with a dry laugh. 'D'you really think they'd suck on that?'
'If the plan Malone put forward was good enough, yes.'
Wallis looked perplexed. 'But in the time-frame you're talking about, Roz has said that ALPHA-BRAVO will still be too ill to be moved.'
'I know that, and you know that,' said Steve. 'But they don't. And I'm not going to tell them.'
'Let's run through this again.' Wallis placed his hands together in a prayer-like gesture in front of his mouth and collected his thoughts. 'You're suggesting we let your two remaining targets and a group of Mutes gain access to the wagon-train then we spring some kind of trap.'
'Right...'
'And once Cadillac and Mr Snow have been captured, and the other lumps have been neutralized, we make it look as if the attack has succeeded. All right so far?' enquired Wallis.
'Perfect,' said Steve.
'At which point the rest of the clan will rush the wagon-train to get their share of the pickings and –'
'Get it right in the kisser,' said Nevill.
Wallis contained his annoyance at being interrupted. 'It's going to require meticulous planning. I mean, we want these lumps on the train but we don't want them running wild...'
'That's why we need time,' said Steve, '(a) to set it up and (b) for Malone to sell the idea to Cadillac.'
'Why can't you do that?' asked Nevill.
'Because of these personality problems I mentioned. He doesn't wholly trust me. But then, neither do you, major.'
The challenge brought a smile to Nevill's face. 'I wouldn't go as far as to say that. I've read up on what you did at the Heron Pool and I've got to hand it to you. That was a first class operation. And don't think I'm put off by that paint-job you're wearing. I've been down to the dye-works a couple of times myself. But I'm always a little wary about mexicans who get so deep into their covers they end up bouncing beaver.'
Steve controlled his anger. 'Go downstairs and take a look at her, major. Take a long, hard look then come and tell me, hand on your heart how – if you didn't know already – you could tell she was a Mute.'
Nevill laughed. 'By the skin colours, of course.'
Steve pushed his left sleeve halfway up towards his elbow and displayed his hand and arm. 'They're dyes, major. Just like these. Underneath, her skin is exactly the same as yours and mine. All it takes to make the switch is a few handfuls of pink leaves, crushed and dipped in water. Put her under a shower with someone like Roz and you wouldn't be able to tell one from the other.'
'It doesn't matter what she looks like on the outside. On the inside, she's still a Mute.'
Steve didn't let up. 'But doesn't it strike you as strange that a race that's officially classified as sub-human can produce creatures that walk, talk, look and think like real people?'
Wallis looked worried. 'Steve – I don't think this has any relevance to –'
'With respect, sir, I think it's extremely relevant. Major Nevill has questioned my reliability and, for good measure, has thrown out a thinly-veiled accusation of misconduct. My defence to that hinges around the status of Mutes and their place in the scheme of things. They're the enemy. They oppose everything the Federation stands for. There's no dispute about that –'
'Glad to hear it,' muttered Wallis.
'But the presence of Mutes like Cadillac and Clearwater – who far from being sub-human, are actually closer to being super-human – brings into question everything the First Family has taught us about Mutes. The Plainfolk have an answer to that question. They claim that before the Holocaust, Mutes and Trackers were members of the same society. The same race. Humankind.'
Wallis intervened, his voice edged with anger. 'That's enough, Steve! These are very dangerous waters you're getting into.'
'See what I mean about this guy?' said Nevill. 'He tried to peddle the same muddled-headed shit to the Board of Assessors.' He faced Steve with a triumphant leer. 'What did she do to you, lieutenant – suck your brains out through your dick?'
Steve was gripped by a sudden desire to throw himself across the table at Nevill and punch the mex's teeth down his throat but Roz came into his head and willed him to stay calm. 'You're right, major,' he said amiably. 'I guess I asked for that. When you been out on your own for as long as I have, playing both sides of the track, it's sometimes hard to know where the edges are.'
'We're aware of the problem,' said Wallis. 'That's why this temporary lapse of judgement will not go on the record. As for you, Jake, we might be able to get through this a lot quicker and more constructively if you adopted a less aggressive attitude. We are _not_ a Board of Assessors, and Steve is not on trial. The eight of us are all on the same team.'
Nevill accepted this with a nod but he was not a happy man.
'Okay, let's get this meeting back on the rails. Steve, you were saying that Cadillac did not entirely trust you...'
'Call it a lurking suspicion. His feelings are confounded by the fact that I've got him out of several tight corners. But like you said, he's not an idiot. We couldn't have escaped from Ne-Issan without help from the outside and there came a point where I could no longer conceal that fact. So I came clean – admitted I was an undercover agent, sent out to capture the three of them. And I told him I'd been forced to accept the assignment because the Federation had threatened to kill Roz if I refused.'
'And did he believe you?' asked Wallis.
Steve shrugged. 'He went along with it. It was the only way he could get out of Ne-Issan. I think he finally bought the story when we knocked out Side-Winder and the mex pilots at Long Point and grabbed the airplanes. Things got better from there on in un –'
'Until he and that Mute bitch downstairs blew four of our aircraft out of the sky and planted explosives on Kelso and Kazan!' Nevill's eyes made it clear what he thought of Steve's contribution to the war effort.
Steve took a deep breath before replying. 'I've already been through this. I didn't _know_ Kelso and Jodi were primed to explode. They didn't even know themselves! The Clan M'Call got badly burned by The Lady. Cadillac and Clearwater grabbed the opportunity to even the score. But it was more than that. They were testing me. They wanted to see which way I'd jump when they started killing my own kind.
'I objected – violently – but only on the grounds that the strike against The Lady wasn't necessary. Any stronger reaction on my part would have totally destroyed my cover. But I couldn't have stopped Clearwater from using her magic on those Skyhawks even if I'd wanted to.' He turned to Nevill. 'When you acquire some firsthand experience of summoners, you'll understand why. At one point in the battle at the Heron Pool, there were fifty to sixty samurai trying to cut her down. The guys with swords never got within striking distance, and the arrows just bounced off this wall of light she'd wrapped around herself.'
Nevill's mouth twisted into a tight-lipped grin. 'And after all that, some guy in a Skyhawk heading for home, clears the drum on a bunch of breakers – and here she is. Few dozen well-placed rounds. That's all it took.'
'A fluke. A million to one chance that paid off because her mind was engaged elsewhere,' said Steve. 'But don't worry, I know what you're getting at.' His voice hardened. 'You still think these three could have been lifted without all this hassle. Isn't that right, major?'
Nevill caught Wallis' warning glance and held up his hands. 'Just exercising my traditional right to engage in a free and frank exchange of views...'
'Jake doesn't like sensitives,' explained Wallis.
Steve smiled. 'I'm not too happy about being one myself.'
'I've always held there's nothing in this world that a bullet won't cure.'
It was clear from Wallis' expression that this was an old bone of contention. 'Yes, well, you keep saying that Jake, but we both know that the committee which is presently studying the tactical use of psionics takes an entirely different view.'
'They would, wouldn't they?' Nevill saw Wallis' brow darken. He turned to Steve. 'So tell us – after the hit on the wagon-train, did relations between you and Cadillac get any better...?'
'Yes. When we started our westward run he had other things to think about. By this time he was into his 'I'm the leader of The Chosen routine'. I went along with it. Everything was fine until we ran into Malone, and Cadillac ran out of _sake_ then shortly after that I got this flash from Roz.
'Mother had put her on board Red River and had sent her north to help neutralize Clearwater. By sending the train into Nebraska and having it move parallel to our line of advance Mother hoped Roz would be able to help keep my mind clear. To prevent Clearwater from doing to me what she'd done to Kelso and Jodi.'
Nevill laughed drily. 'That's why you were strip-searched before we let you up the ramp.'
'Wise move. Anyway, maybe it was a bad mistake, but I told Cadillac that Roz had made contact and that she was on Red River. And because of the story I'd fed him before about being pressurized into this job by threats to her life I _had_ to come out with this idea of trying to rescue her. It would have seemed odd if I'd passed up a once-in-a-lifetime opportunity without making a positive response. I didn't have any clear idea how it could be done, it was all off the top of my head. But I suggested that if the M'Calls got together with Malone's renegades – maybe in some kind of disguise – we might just be able to pull it off.'
'And how did Cadillac react?'
'He said that attacking the train was exactly what the Federation wanted us to do. That was why they'd put Roz on board. She was the bait that would draw me in if I had truly thrown my lot in with the Mutes. And in the next breath he all but accused me of being prepared to sell his people down the river in order to save her skin.
'As I said earlier, the M'Calls had already taken heavy losses in their attack on the Lady. Cadillac wasn't going to repeat that mistake by leading them into a trap. On the other hand, if I was genuine, a true and loyal friend of the Plainfolk, he still wasn't prepared to let his clanfolk be killed trying to rescue my kin-sister. It was a no-win situation.' Steve shrugged. 'From then on it went downhill, and as he was trying to put a knife in me, we were hit by the Skyhawk.'
'So what the hell are we doing talking about setting up a dummy take-over of Red River?' demanded Nevill. His eyes searched out the other members of the task force then fixed on Steve. 'If this piece of lump-shit wasn't prepared to involve his clan in an attack on the wagon-train then, why in the name of the Family is he going to want to do it now?!'
Steve tried to appear patient and reasonable. 'Because the situation has changed. Clearwater is _on the train!_ '
Nevill grinned. 'Boy – I wish I could be there when you break the news. I'd love to see you wriggle out of that one.'
'I'll manage...'
'What will you say to him?' enquired Wallis.
'I'll tell him he was right. My idea of trying to spring Roz from Red River was absolutely crazy. And the same goes for Clearwater. There is no way anyone can get on board that wagon-train and rescue her. He just has to be grateful she's alive and accept that she is now a prisoner of the Federation.'
'And what's his reaction likely to be?' asked Wallis.
Steve smiled. 'I know this guy like the back of my hand. If I say she can't be rescued, he's gonna want to prove me wrong. And he'll buy the idea of attacking the train because it won't come from me but from _Malone_. He's the guy who has to sell it to him. And he has to do it cleverly, so that Cadillac thinks it's all his own work.'
Wallis mulled over the idea. 'Sounds as if it might fly. Anyone spot any holes?'
'Yeah,' said Hannah, the mex sitting between Roz and Nevill on the other side of the table. 'I believe our friend here has got these lumps figured but we can't set up an attack on Red River without the approval from the man at the top. Okay, let's say we get that –'
Nevill interrupted again. 'Can't see CINC-TRAIN agreeing to put Red River on the line – and Fargo ain't gonna be too happy either.'
'If Mother and the P-G give the go-ahead, CINC-TRAIN and Fargo won't have any choice in the matter,' said Wallis with a sudden hint of steel in his voice. 'Carry on, Ray. Assume we've got the green light...'
Hannah resumed: 'Okay. Let's say we then come up with a workable plan. One which has a reasonable chance of success from the Mute's point of view, but which we can turn around when the time comes to spring the trap. The question is – how can we be sure that the lumps will (a) swallow it and (b) follow it?'
'That's where I come in,' said Steve. 'The moment I get wind that Cadillac is serious about rescuing Clearwater, I'm going to oppose the whole idea. That will only make him more determined to go through with it. But to play my part properly, I will have to know exactly what the plan is.'
'Point taken,' said Wallis. 'If we manage to get this thing off the ground Malone will be in touch.'
Steve shook his head. 'Too risky. He and I have got to stay at arm's length. Malone must take his cue from Cadillac and appear to mistrust me. Nothing heavy – otherwise his boys will wonder why he hasn't put a bullet through my throat. He just needs to display a reluctance to let me in on what they're planning. It'll encourage Cadillac and Mr Snow to believe he is totally genuine.'
'So how do we reach you?' asked Wallis.
'Through Roz.'
Nit-picker Nevill came off the sidelines. 'What's wrong with the normal channels of communication?'
'You mean a radio-knife? Too risky,' said Steve. 'I was caught out once before. If we're going to nail this bunch I have to win their absolute trust. I'm well on the way to doing that. Cadillac is a separate problem. A personality clash. We're rivals. Anything I can do, he wants to do better. I don't have this trouble with Mr Snow. He's razor-sharp and he's shrewd but I've managed to establish what you might call a "working relationship". He likes me – and that's another reason why Cadillac has this down on me. He's jealous.'
'How is Mr Snow going to react to the news that you've lost Clearwater?' asked Parsons. He was the mex sitting between Roz and Wallis.
'Well, he's bound to look upon it as a set-back, but it's not a major disaster. Thanks to Roz, I will at least be able to tell him that she's alive and in good hands.'
'But will he support Cadillac's plan to go for Red River?'
'He has to. The clan won't attack a wagon-train if he isn't there to lead them.'
Parsons insisted. 'Yes, but how do you know he _will_?'
'Because he's old. He's near the end of the line, with maybe only a couple of big shots left in the locker. Summoners die a little every time they call up the power. These guys don't get a free ride. Each time they pull down the really heavy shit, their own lives are on the line. I'm not kidding. I've seen it. Their eyes are dead, their bodies are empty shells. It takes time to recharge the batteries.
'The M'Calls will make an all-out effort to rescue Clearwater because she's Mr Snow's successor. Without a summoner, they can't hold onto their position as the paramount clan of the She-Kargo bloodline. They're finished. That's why he will lead them against Red River.' Steve smiled. 'And as honorary member of the Plainfolk, I'll be right alongside him.'
'Haven't you overlooked something?' said Nevill, clearly underwhelmed by Steve's assurances. 'Something you told us a while back? About your friend Mr Snow – the Storm-Bringer? Isn't that the name the Mutes give to someone who holds seven of the nine so-called Rings of Power?'
'Yes...'
'Terrific. This is the guy, who according to you, could have blown me out of the sky. And you're suggesting we just let him loose inside Red River. You must be out of your fuckin' mind!'
'Red River won't be in any danger.' His kin-sister's mouth was closed but Steve could her her other voice whispering to him. 'Roz is not just a telepath. She has the power to completely disorientate Mr Snow's mind – put it into neutral gear, if you like – so that he can't use it to channel the power he draws from the earth and sky.'
'Oh, yeah?' scoffed Nevill. 'Since when?'
'Ask her,' said Steve.
Wallis, anxious to take the heat out of the situation, adopted a conciliatory tone. 'If this was true, it could be a very interesting development. Is there, ahh – any way you could validate the claim your kin-brother has just made?'
Roz looked a little nervous. 'You mean... now?'
'Well, yes. I realize we're not under threat from a summoner, but if you could give us some tangible demonstration of the, ahh – mental powers you would use on such an occasion it –'
'– would be very helpful,' said Steve.
Nevill laughed. 'For crissakes, Dan! You don't really believe this bullshit, do you?'
'Jake – just shut your mouth, okay?' Wallis nodded to Roz. 'Is this how you want to do it? With us sitting around the table?'
'Yes, that's fine.' Roz placed her hands on the table, shut her eyes and breathed deeply. 'Move your chairs as close in as you can.'
'Do we have to shut our eyes too?' asked Wallis.
'No,' said Roz. 'I want you to enjoy the view.'
And with that, the room dissolved around them. Steve felt a stiff breeze on his face. Looking up, he saw a clear blue sky above his head, stretching away to an incredibly distant horizon rimmed with clouds. They were sitting around the table in the open air! It was an hallucination, of course. But one they all shared, and the detail was absolutely amazing. So real! It was fantastic!
Roz looked incredibly tranquil, in marked contrast to the task-force who sat paralysed with terror. Looking down, Steve saw why. This hallucination was not only fantastic, it was absolutely horrific. He and the other people seated round the table were perched on top of a narrow pinnacle of weathered rock at least a thousand feet above the plain below. The flat top of the rock was just large enough to hold the table and the tubular metal chairs. The slightest backward movement by anyone would cause them to topple over the edge!
_Oh, shit_... Steve sneaked a quick glance down at the terrifying drop then decided this was one demonstration he could have done without. He had no fear of heights when flying but being perched precariously on the edge of a sheer drop made him feel quite sick. This isn't happening, he told himself. I'm sitting in a hospital ward on the top deck of a wagon train. But his mind told him otherwise. The sky, the breeze, the pinnacle of red rock was _real_. And so was the giddying force that was drawing him like a magnet towards the plain below!
Mastering the impulse to throw himself into the void, Steve slid his hands carefully across the table. His first idea was to secure himself by grabbing hold of Roz's wrists but as his right hand brushed across her left hand he received a violent electric shock which threw his arm in the air. Fortunately the jolt twisted him sideways and not backwards. He managed to get his left hand over the far edge. There wasn't a lot of room because everyone around the table had had the same idea.
_Don't panic, Brickman. Nothing's going to happen to you. Your little sister is just showing this bunch of assholes what she can do_...
Steve heard a strangled cry from Nevill's end of the table. Looking between the heads and over the outstretched arms of his companions, he saw the reason for Nevill's distress. The pinnacle of rock had two points. Nevill's chair was poised on top of the second, smaller peak, about six feet away from the table. The top was just wide enough to hold the base of the chair. Nevill sat there, rigid with terror, clutching the seat, with his feet hanging in the air. Just looking at him made Steve even worse than before.
He closed his eyes and tried to contact his kin-sister.
_Okay, Roz, you've made your point. Now get us out of this nightmare_...
'Hey, guys! Please! Help me!' Nevill's gruff, macho delivery had dissolved into a strangled bleat.
_Are you kidding? One false move and well all be over the fucking edge!_
Nevill's voice became a shrill squawk. 'Aaghh! AAghh AAGHH!'
Steve forced himself to look in his direction. The top of the second rock needle had started to crumble. Nevill's chair was tilting over backwards. The mex didn't dare lean forward for fear of overbalancing in the other direction. He was frozen to the chair, eyes bulging, his open mouth twitching grotesquely in horrified anticipation—
Steve glimpsed the rubber sole of Nevill's boots rise into the air then shut his eyes as the chair went over, pitching the mex into the void. The sound of Nevill's scream made his blood run cold—
There was a heavy thump and a crash of metal. The breeze which had been buffeting Steve's face ceased abruptly. He opened his eyes and saw Roz looking at him. The table was back in the hospital ward but those seated around her could not quite believe it. They looked at each other, stunned, speechless, then all eyes went to Nevill's end of the table. His chair lay on the floor; Nevill was sprawled on his back beside it. Ray Hannah and Daryl Coates, the mex on Steve's left, righted his chair and helped him to his feet.
Nevill was shaking uncontrollably and there was a spreading dark patch around the crotch of his fatigues where he had fouled himself. He gripped the table to steady himself as Hannah and Coates eased him back into the chair. 'Christo! I had the most terrible –' He broke off as he saw their tense faces. 'Wh-what happened?!'
'Good question...' To judge by Wallis' total lack of colour, he'd had a bad trip – like everyone else. He eyed Roz nervously, and when he spoke, his voice wavered. 'Thank you. Yes... that was, uhh... most instructive.
I had no, uhh –' He cleared his throat and in doing so, forced his voice up half an octave. 'Why don't the two of you check on how your patient is doing, while we, uhh...?'
'Of course,' said Roz.
The task force watched in total silence as Steve followed her out. Hannah shut the door behind them.
As they went down the stairs, there were a million questions buzzing around in Steve's head. It had been a shattering experience. His legs were like jelly but he tried to sound casual. 'That was quite a stunt you pulled back there. I'm impressed.'
Roz, two steps below him, glanced back and smiled. 'This is just the beginning.'
The smile was the smile he had always known, but the look in her eyes belonged to a stranger. And there was something else. He had done all the talking while she listened, but for most of the time he had been nothing more than a mouth-piece. Roz had put the words into his head.
There are forces at work here I don't understand, thought Steve. And for the first time in his life he felt frightened of his kin sister...
## CHAPTER THREE
Cadillac stirred as the effect of the sleeping-pills began to wear off. Griff, the breaker who had been guarding the smooth-boned Mute, stood up as Malone approached. 'Another five or ten minutes and he should be on his feet.'
'Good. Get saddled up. I want to make the most of this moonlight.'
'If it's okay with you, I'd prefer to walk and just use it to carry my gear.'
'I don't give a lump's ass for what you prefer,' snarled Malone. 'You'll get on that fuckin' horse and learn to ride like the rest of us!'
'Wilco!' Griff stepped back out of range of Malone's fists and feet then ran towards the line of tethered horses they'd inherited from a strange trio of travelling Mutes. These three, who had come out of the east, didn't act like your normal run-of-the-mill lump-head. In fact, to judge by the amount of time Malone had spent talking to them before the air strike, they were something extra-special.
Griff's curiosity about the matter ended there. Malone was one tough _hombre_ , and if, for some reason, he'd decided to cosy up to a bunch of Mutes then that was strictly his business. The guy knew what he was doing and anyone who poked his nose where it wasn't wanted got a short, sharp lesson he never forgot. Any breaker foolish enough to forget the first lesson, didn't survive the second.
Like most breakers, Griff didn't like Mutes but he'd learned to live alongside them. And provided you didn't kill 'em. they left you pretty much alone. The clans preferred you to stay off their turf but if you went on through their boundary markers and were challenged by a posse of warriors you could usually buy 'em off by giving them a few bits of junk to hang on themselves. And if you handed over any crossbow bolts you'd found – because they did sometimes miss when out hunting – then you really made their day.
The bolts, like the crossbows that fired them, were highly-prized items made by a bunch of ginks over in the east. Mutes even traded their own people to get hold of them, so a handful of free bolts was a big bonus. They'd start laughing and leaping around, shouting and crowing. But in amongst all the jibber-jabbering, they used _real_ words, strung together – and which made _sense_.
That had been his biggest surprise on encountering Mutes during his first tour of duty as a service engineer in a Kansas work camp. And ever since becoming a breaker – the moment when he'd been obliged to treat Mutes as equals and not as sullen slaves – Griff had been constantly amazed to discover how _normal_ they were. Okay, they had lumpy skulls and multi-coloured hides, but they were like Trackers in so many ways. Griff could never figure out how they could act the way they did and yet not be human beings. It was a real mystery.
This trio who'd come riding out of the east at the head of a whole bunch of horses was a good example. They spoke regular Basic and their brains were as sharp as a knife. And if you stripped away the dark to light brown patches on their skins with your mind's eye and stood 'em alongside three good ole boys you'd be hard put to tell the difference. And when Griff thought about the time he'd spent watching the training videos about bug-ugly Mutes – poisonous savage vicious animals that had to be ruthlessly exterminated – he couldn't help wondering why the First Family hadn't told Trackers the whole story: like the fact that a bunch of breakers could live alongside Mutes and not end up with their heads on a stick.
Yeah... it was a puzzle right enough – like trying to figure out why anyone would want to sit on top of one of these four-legged freaks they'd been landed with. Especially when they had teeth that could bite right through your hand and back legs that kicked like a fuggin' jack-hammer. Okay, this bunch had come complete with a seat on their backs and leather straps for steering 'em round corners but it was a helluva way to travel.
Griff could see that horses would be useful for carrying gear but even that had its drawbacks. Leading a string of pack-horses just made you a bigger target and there was always the risk that the beasts might decide to bolt with your precious possessions. Better to travel light than start relying on a transportation system that a passing Skyhawk could blow away in nought seconds flat.
On the other hand, if he put his own feelings to one side, Griff could see what Malone and his new Mute friends had been getting at. If you managed to stay in the saddle and figure out how to get the thing in gear, you could cover a lot of ground pretty fast. Fast enough to outrun a Mute. Some of the guys had already cracked the problem and when he watched them show off in front of the others, Griff could see it gave the riders a real buzz.
There was no two ways about it. It was impressive – but it wasn't natural. Or practical. You just had to take the problem of maintenance. Horses weren't like wheelies. If one of those broke down you just ran the on-board diagnostic programme and ordered whatever new part was required. With horses, you were totally screwed. To begin with, nobody knew what went on inside and you couldn't bolt on new legs like you could wheels. If one of those broke that was it. You couldn't even cannibalize the unit to create a stock of spare parts. You had to ditch the whole bundle.
Griff picked up the wood and leather saddle, admired once again the handiwork of the unknown craftsman then positioned it on the back of his horse. 'Steady, friend,' he muttered. 'I know you hate this as much as I do but the boss-man wants us both back on the trail...'
He drew the girth tight around the horse's belly then pulled another notch through the buckle. He'd already done one slow roll off his mount to the raucous cheers of his companions and he didn't intend to make that particular mistake again.
Malone experienced similar misgivings as he gazed down at Cadillac. The Mute was one problem he could have done without. Although he parlayed with Mutes and observed the ground rules of peaceful co-existence, he did not share Griff's qualified forbearance. Malone didn't like Mutes. Period. But then Malone wasn't a renegade, a breaker on the run from the Federation. Some of the men he led were genuine deserters – brave enough to seek an independent existence on the overground but also a treacherous heap of garbage. Malone didn't mind. That was what he been sent out to run: a garbage disposal unit. Malone and the core of his renegade band were _mexicans_ , agents of the undercover organization controlled by Commander-General Karlstrom.
Trapped between the worlds of Tracker and Mute, renegades were basically scavengers; wanderers who roamed the overground with no particular destination and no home to return to. Death was the only welcome they could expect from the Federation. Fortunately there were large areas which had not yet attracted the attentions of the Trail-Blazers and where Mutes were thin on the ground. Like the Rocky Mountains for instance. But there was a good reason why the lumpheads gave the Rockies a wide berth. For six months of the year it was so cold, and the snow was so deep, no one could survive there. Unless of course you were well-organized and properly equipped. Karlstrom had made sure his groups had the expertise and equipment they needed but each item was carefully selected and given a worn, weathered look so as not to strike a false note.
Like any species fighting for survival, renegades were subject to the process of natural selection. The strong prospered, the weak perished. A large group offered safety, continuity and companionship. Malone's organization also provided the other vital element – strong leadership. Within months of its formation, it had become a magnet, attracting smaller groups and individuals who, up to that moment, had opted for a hermit-like existence. And within the Federation, Malone's name was deftly inserted into the information network run by known subversives.
To many in this twilight world, Malone had already assumed the status of a folk-hero. Someone who had beaten the system. It hadn't taken long for his name and approximate whereabouts to surface in the way-stations and work-camps. His wasn't the only name that was passed along in whispered conversations. There were several others obligingly provided by AMEXICO for the benefit of potential defectors – all of them fostering the notion of a growing rebel movement and a relatively safe haven.
Only very few who managed to join Malone realized they had fallen back into the hands of the Federation and they were quickly eliminated. It was a sweet operation – one of several similar overground 'stings' which enabled the First Family to keep its finger on the pulse of the protest movement. And it also ensured a steady supply of candidates for the televised show trials. If the wagon-trains failed to flush out a sufficient number of breakers, Malone and his counterparts made up the balance by sending unsuspecting candidates into a carefully-coordinated ambush.
Had the matter been left to him, the Mute at his feet would now be on his way to the Federation. This Cadillac Deville character was on the wanted list. He'd been nailed. He should have been shipped out _pronto_. End of the story. Neat, clean and simple to arrange. But that wasn't how HANG-FIRE wanted to play it. HANG-FIRE was the operational code-name for Steve Brickman, a wet-back who had graduated from Rio Lobo the previous year after serving briefly as a wing-man aboard The Lady from Louisiana.
Malone knew these background details because he had been selected to give Brickman his final test. A potentially fatal ordeal designed to measure a candidate's courage and endurance. Brickman had been 'posted' – tied in a kneeling position against a stake, face to face with the corpse of a Tracker he'd killed in the line of duty. He'd come through it, earning himself full marks in the process. There was no doubt about it. Brickman had the makings of a real operator and his latest trick had been to get in and out of Ne-Issan, bringing two important Mute targets with him: Clearwater, a female Mute and Cadillac, the lump now at Malone's feet who was taking forever to shake off the double dose of Cloud Nines.
Clearwater had been seriously wounded in a surprise air attack by a stray Skyhawk. At Brickman's request, Malone and his renegades had struck camp and ridden off, leaving them behind. If she hadn't died in his arms, Clearwater was now on board Red River. Cadillac had been superficially wounded and knocked unconscious by the same hail of fire.
To Malone, it seemed like an ideal opportunity to ship them both out together. Two out of three wasn't a bad result, but Brickman wanted a full house. By leaving Cadillac free, he hoped to entrap his third target – Mr Snow, the power behind the Clan M'Call. Which was the reason why he, Malone, had been lumbered with the task of escorting this lumphead as far as navref Cheyenne, Wyoming. A journey which placed his band of renegades in considerable danger.
This was the wrong time of year to be moving around. April was the month when the Mutes hunted 'red-skins' – breakers; the annual round-up of strays which were handed over to the Iron Masters in exchange for goods and shipped east. Malone hadn't planned to leave the camp that Brickman and his friends had ridden into until mid-May. The site was in a commanding position, with good cover and running water: ideal for a long stay. It had been chosen because AMEXICO knew which way Brickman was heading and he was expected to pass close by. At which point Malone – quite by chance – was to pop up and renew their acquaintance. Everything had gone according to plan and then – thanks to some asshole in a Skyhawk and two scumbags who hadn't seen it coming – everything had gone wrong, forcing him to head west when every sensible breaker was lying low.
The only solution was to travel at night. Mutes, for some reason rooted deep in their collective past, were only active between sunrise and sunset. After that, the hunting posses and turf patrols went home or bedded down for the night. It wasn't the ideal time for travelling cross-country but after umpteen years in the field, Malone had become adept at reading the terrain and moving men across it under the most adverse conditions.
Even so, he was sorely tempted to call up a sky-hook to take Cadillac off his hands. But this was not his operation. The orders from Mother had been clear and unequivocal. He had been detailed to intercept Brickman at a given point during his journey, assess his reliability and – with Mother's approval – to render assistance if and when required. Brickman had said and done all the right things but he'd set Malone's internal alarm system ringing. There was something about him. Maybe it was just Malone's instinctive antipathy towards clean-cut blue-eyed golden boys, but Brickman was too smart for his own good – and just too good to be true.
In the previous year, six of Malone's people, including a class-mate of Brickman's, had been sent north to provide him with the back-up he'd requested to help kidnap the same three Mutes. In the last radio contact made by a mex called Donna Lundkwist, she reported the squad had been sighted by a posse of M'Call Mutes – recognizable by the colour of the feathers in their headgear. The Mutes had put up a smoking arrow – a sign they wished to parley. End of message. No one had ever heard from those six _mexicans_ again. Brickman had been running with that clan. Painted up, grassed-out and leathered. His degree of involvement in the back-up squad's disappearance was a question that had plagued Malone ever since.
He glanced up at the clear moonlit sky and saw a bank of dark cloud building up on the northern horizon. Malone was astute and resourceful but patience was not one of his virtues – especially when it came to unwanted guests, and even more so when that guest was a Mute. Following the shooting of Clearwater, Cadillac had not regained consciousness. To avoid any hassles, he had been kept in a drugged stupor for the past two days. Helped by clear night skies, they had covered some seventy odd miles. One way or another, Clearwater was now beyond reach. It was high time for this lump to stand on his own feet instead of having to be carried around everywhere. When Mother had asked him to help Brickman, he hadn't figured it would mean having to namby-pamby an uppity Mute. That was the bit that really pissed him off – not the move.
He dug his boot into Cadillac's side. 'C'mon! Wake up you sonofabitch! We haven't got all night!'
Cadillac stirred drowsily. 'Uh-humm, yeah... sure...' His eyes fluttered open then closed again as his mouth opened in a huge yawn.
Malone unhitched his water bottle and emptied it over Cadillac's face. Some of it went down his throat causing him to gag. He rolled about choking and coughing then eventually sat up clutching his head.
'On your feet! C'mon! We're moving out!' Malone slid a hand under his left armpit and hauled him upright.
Cadillac steadied himself and rubbed his face. His body seemed gripped by a strange lethargy. 'Mo-Town! I feel –' His eyes widened as he focused on Malone, then he quickly took in his strange surroundings. 'Where's Clearwater? And Brickman?' A jab of pain from his various flesh wounds caused him to frown. He looked down at his left side and saw two bloodstained rips in his walking skins on the outside curve of the thigh. There was also a deep graze in his belly. An expression of alarm crossed his face. 'Who shot me?!'
'You don't remember? Must have been after you hit the ground.' Malone told him about the Skyhawk that had appeared out of the blue, making a single strafing run across the campsite before turning for home.
Some premonition of what he was going to say next made Cadillac howl with grief. He wrapped his arms around his chest and rocked from side to side. 'Oh, Sweet Sky-Mother! Clearwater! Is she dead?'
'Not when we left. But she was hurt pretty bad.'
'Oyy-yehh! This is all my fault! What about Brickman?'
'He's fine. Came out of it without a scratch.'
The news caused Cadillac to grind his teeth. 'He would! Hah! How typical! So what did you do?'
Malone did his best to conceal his irritation at being questioned in this peremptory manner. _Who the fuck did this guy think he was?_ 'Do? The best we could. State she was in she couldn't be moved. So we dressed the wounds with what we had then got the hell out. No point in staying there once the camp had been spotted. We've had 'hawks over our heads for the past couple of days.'
Cadillac's mounting anger boiled over. 'Are you telling me she was shot _two days_ ago?!'
Malone checked his watch. 'Exactly fifty-four hours and twelve minutes ago.'
'Why didn't someone tell me before now?!'
Malone resisted the impulse to smash Cadillac in the mouth. 'Because there was nothing you could do, friend.'
'Hahh!' Cadillac became aware of the metallic aftertaste on his tongue. 'Was it Brickman's idea to pump me full of drugs?!'
'Yeah. He said you'd be hysterical, and he was right. Pull yourself together for crissakes!'
'I _am_ together! He had no right to take matters into his own hands like this! We've got to go back for her!'
'Are you crazy? The only thing that could've saved her was major surgery. Federation-type medicine – not the mumbo-jumbo you monkeys mess around with. She'll be dead and buried by now –'
'No! Don't say that!'
Malone ignored the interruption. 'My job is to help you get back to your own people. Isn't that what Brickman promised to do?'
'Yes, but –'
'There are no "buts". That's what we're gonna do, friend. It was two of my guys who let that plane take us by surprise. So quit blaming yourself for what happened. I can understand Brickman bein' upset at losing a neat piece of ass but what the hell have you got to cry about?'
Cadillac brushed away the tears of rage and grief with shaking hands that longed to fasten themselves around Malone's throat. 'She didn't _belong_ to him!'
'Could have fooled me. Is that what the fight was about?'
'No. We were fighting because Brickman is a treacherous, lying toad!'
'That seems a mite ungrateful. Didn't he help you and Clearwater get out of Ne-Issan?'
'He didn't do that to help us! He got us out in order to hand us over to his masters in the Federation! He's not a renegade! He's been working with a network of undercover agents for over a year!'
'I see...' Malone ruminated on this for a moment. 'Did he tell you anything about this network – like it's name for instance? Or who was running it?'
Cadillac realized he had said too much and was already regretting his temporary loss of control. His antagonism towards Steve had not diminished but Malone was a virtual stranger. A cipher whose mind, for the moment at least, was inaccessible. 'No. But we couldn't have escaped without outside help – which he organized. If he turns up again, ask him about it. All I can say is, no one's safe when he's around.'
'I'll bear that in mind,' said Malone. 'Meanwhile forget we've had this conversation. If Brickman _does_ rejoin us, I may decide to let things ride for a while. If you're right, and he _is_ an undercover Fed then it may be to our advantage to let him think we trust him completely. Know what I mean?'
'I think so...'
'Good. Let's hit the trail.' Malone gave Cadillac a friendly slap on the back then ushered him over to the waiting horses. He would have preferred to have broken the back of this snivelling piece of lumpshit but – like Brickman – he had a job to do and a role to play. That of big-hearted Matt Malone, friend and protector of abandoned Mutes.
Some six hundred miles to the south of Malone's present position close to the Platte River, Nebraska, The Lady from Louisiana wagon-train had emerged from the repair bays and was back in what was known as 'the roads', being readied for action in the vast underground depot at Nixon/Fort Worth.
In June 2989, The Lady had narrowly avoided a major disaster in its first encounter with a Plainfolk clan aided by a summoner. Caught in a flash-flood, The Lady had managed to extricate herself virtually undamaged but in doing so, she had lost nine out of the ten wing-men posted aboard and their aircraft, plus over eighty line-men. Close on double that number had been wounded.
In November 2990, when The Lady had been sent out into the snows on a special mission, it had been even worse. The explosive charges planted by Cadillac and Clearwater had totally destroyed the blood-wagon and flight car, and the tidal wave of fire that erupted from the stock of napalm canisters and liquid methane tanks stored beneath the hanger deck had rolled through three more cars, incinerating the crewmen in its path.
Abandoning the five gutted cars, The Lady reformed and set course, as directed, for Monroe/Wichita, the still-uncompleted divisional base in Kansas. Arriving at the interface, the wounded crewmen had been off-loaded and rushed to hospital. Commander Hartmann and his team of execs, including Trail Boss McDonnell had been placed under close arrest and shuttled to Grand Central to await trial. The surviving members of the crew who had escaped the same 'dereliction of duty' charge were placed under the temporary command of an executive team drawn from the training staff at Fort Worth. It was they who had brought The Lady – defeated and disgraced – southwards through Oklahoma into the relative safety of the Home State and back down the long incline into the depot.
The winter months – whose high point was the celebration of the New Year – were, by tradition, spent on 'rest and refit' (R & R). A period when Trail-Blazer crews enjoyed a welcome spell of leave after eight to nine months on the overground, and when the depot engineers began their task of overhauling the trains, readying them for their next assault on the blue-sky world.
The Lady was in need of more than a refit. The missing wagons had to be replaced, fire-damage to several others had to be made good, the crew had to be brought up to strength and their shattered morale restored. Bringing The Lady back to operational status was a major undertaking but it proved easier than raising the crew's spirits. Despite the damage and the casualties The Lady had sustained, Hartmann and his executive team – with perhaps one exception – were held in high regard by the Trail-Blazers who served under them. The exec who failed to inspire the troops to the same degree was Captain Baxter, the Flight Operations Officer. He had died in the blast that ripped through the packed hangar deck of the flight car, killing Gus White and the other wing-men, the mechanics, deck handlers and a score of Trail-Blazers.
The Lady's Trail Boss, Buck McDonnell, whose alertness and quick reactions had saved the forward command car and its crew had been released after two months detention. In a brief appearance before a Board of Assessors, he was informed that all charges had been dropped and was ordered to report for active duty at Nixon/Fort Worth. Exiting from the court room, he was met by a Staff-Commander from CINC-TRAIN who welcomed him back into the ranks of the Trail-Blazer Division. His first task would be to knock the new crew of The Lady into shape and he was to begin immediately. Due to operational requirements, there would, explained the Commander, be no chance of the four weeks base leave to which he was entitled.
In his usual blunt but respectful fashion, the big Trail Boss told him it didn't matter. After eight weeks on the shit and bucket detail he was just happy to be soldiering again.
The decision to release McDonnell had paid off. From Day One there had been a noticeable rise in the spirits of the veteran crewmen and the transferees and wet-feet – the uninitiated replacements – soon discovered that Big D's reputation as a fire-breathing disciplinarian was, if anything, an understatement. A second stand-in team of execs from the depot's permanent staff helped the crew go through their on-board drills but as the weeks passed, even McDonnell became concerned about the deafening silence surrounding the appointment of a new wagon-master.
Finally, one day in early April, when the crew had assembled for the usual morning parade alongside the wagon-train, McDonnell strode along the ranks behind the duty officer with a noticeable gleam in his eye. When the DO completed his formal inspection of the battalion and passed control over to McDonnell, it was clear to old hands like Bad News Logan that something was up.
_Somethin' good for a change. _Ol_ ' Big D was practically burstin_'...
McDonnell braced himself. 'Wagon-train-n-n-n EASY!' he boomed.
The battalion stood at ease with a thunderous stamp of boots, the palms of their hands crossed in the small of their backs.
'Okay, hear this!' he said, in the same foghorn voice. 'I have been reliably informed that The Lady has a new commander and he will shortly be arriving with his team of execs!'
The announcement provoked a subdued murmur.
'And as soon as they're settled in – and we've shown them the ropes –'
A ripple of laughter.
'– they'll be taking The Lady out for a shakedown supply run to Abilene, San Angelo and Brady!'
This news raised an audible groan. Abilene, San Angelo and Brady were way-stations to the south-west of Fort Worth in the Home State of Texas. Territory under the total control of the Federation where there was no chance of a fire-fight.
'And then we're goin' north – to hunt Mute!'
The battalion responded with an exultant shout, punching the air with their fists. 'HO!'
McDonnell caught sight of an approaching wheelie. It was a four-car enclosed model: the type used by hire-wires. He called The Lady's crew to attention. 'Wagon-train-n-n-n READY!'
Close on a thousand pairs of boots came together with a synchronised thud. The Trail Boss made a smart about-turn, his brass-topped drill stick braced stiffly under his left arm, and parallel to the ground.
The wheelie whined to a halt in front of him. The doors on both sides opened and disgorged the team of executive officers who were to take charge of The Lady from Louisiana. There was an audible gasp from the veteran crewmen as they glimpsed the bushy white moustache of the officer with the yellow commander's rank bars on his lower sleeve.
It was Hartmann, their old commander. Buffalo Bill – back in uniform and back in charge, and eight of the twelve smiling faces around him belonged to the executive officers who had served with him up to the moment the Provos had come aboard to arrest them all.
Buck McDonnell's right hand snapped into line with the brim of his stetson, fingers and thumb aligned in a perfect drill manual salute. 'Eight Battalion, Trail Blazer Division, mustered aboard The Lady from Louisiana, ready for your inspection, SAH!'
'Thank you, Mr McDonnell.' Hartmann returned the salute, as did the twelve execs lined up in two staggered rows behind him. The formal greetings over, Hartmann exchanged a warm handshake with his Trail Boss.
'Welcome back, sir.'
'It feels good, Buck. When did you hear we were on our way?'
'Last night, sir. Had quite a job keeping it from the boys.'
'Well, they look happy enough,' observed Hartmann, 'I thought they might be a bit leery about serving under a two-time loser.'
'Sir! Are you kidding?!' McDonnell turned towards the men lined up in three ranks in front of The Lady. 'Okay, you clapped-out, time-serving bunch of slack-assed mothers! Let's hear it for the commander!'
The six hundred veterans that formed the core of The Lady's crew cut loose with the time-honoured chant: 'Buffalo Bill! Buffalo Bill! Just say the word and we'll kill, kill, kill! Give us a rifle, helmet and pack, and we'll follow you to hell and back!'
'Are we ready and able?! Are we fit to show?!' demanded McD.
Everyone, including the execs behind Hartmann, joined in the traditional response: 'You bet your ass! Let's GO – GO – GO!'
Hartmann, noticeably moved by the warmth of his reception, signalled McDonnell to stand the men down.
'Wagon-train-n-n-n EASY!'
The nine hundred men and women making up the crew of The Lady were mustered in individual groups in front of the cars to which they were assigned: medical staff in front of the blood wagon, 'fire-men' in front of the power cars, and so on. As Hartmann led his team of execs along the ranks, each squad or section leader called his individual group to attention. The wagon-master paused to exchange a few words when he encountered a familiar face and the veteran execs did the same. Those drafted in as replacements would each get the customary one-on-one interview with Hartmann, and the executive officer in charge of their particular specialization, once they were on board.
When the inspection and greetings were over, Hartmann sought out his deputy, Lt.Commander Jim Cooper. 'Mount up, will you Coop? I have to place a call to a friend of mine...'
The first two video-phone booths had plasfilm notice strips stuck diagonally across their screens bearing the words 'LINE FAULT – VID-COMMSERV NOTIFIED'. It meant a service engineer was on his way. Sometime between now and the millenium...
The third booth he found was working. Hartmann inserted his newly-returned ID-card, keyed his way through the on-screen call menus, entered the state-code for Colorado (09) followed by the three-digit code for the Pueblo way-station (012) and the x-listed number he had memorized.
The Amtrak logo on the screen was replaced by the head and shoulders of Major Jerri Hiller, one of Mary-Ann's junior battalion comanders. Hartmann noticed her hair was considerably longer than when he had last seen her. He also couldn't decide whether she was surprised to see him or annoyed – or both.
'Is Colonel Anderssen available?'
'One moment, Commander...' Hiller moved out of view of the tv camera mounted immediately above the screen carrying his image.
There were muted voices off then Colonel Marie Anderssen moved quickly into the empty seat. 'Bill!' She too was surprised, but the pleasure at his call was evident in the broad smile that came beaming across the ether. 'Christo! You're wearing active duty OD's!'
OD was the abbreviation for the olive-drab, military-style fatigues he was wearing.
'Yeah. They let me out of detention yesterday morning. Plus Coop and the rest of the guys. All charges have been dropped. We've been re-instated – and we'll be rolling The Lady up the ramp at 0700 hours tomorrow.'
Mary-Ann interlaced her fingers and squeezed her hands together. 'Oh, Bill, that's wonderful! It's the best news I've had all year!'
'You and me both.'
'I tried to get permission to see you...'
'Yeah, I know. Your message got through to me. Thanks. It helped a lot.'
'Is there any chance of you heading this way?'
'Can't say. We're warming up with a home state supply run. After that I'm not sure what they've got lined up for us. The hire-wire from CINC-TRAIN in charge of the welcome-back party hinted we might be given another special assignment.'
Mary-Ann looked concerned. 'Oh, gosh, I hope it's not –!'
Hartmann cut in. 'Honey – we just have to take what comes.' He smiled. 'You're looking great. This picture quality's very good.'
'It is now it's tuned in properly. VID-COMM had no end of problems trying to set up the link with Santa Fe.'
'How long have you been on-line?'
'Not long.' Mary-Ann smiled back at him. 'You're the first personal call I've had. Up to now they've either been test transmissions from VID-COMM or from HQP-DIV. How did you get my number?'
'Through a friend. Amazing as it seems, I still have a few.'
'I'm one of them.'
'Oh, you're more than just a friend. You're something special. Let's hope the next time we meet there won't be a piece of glass and an ocean of red grass between us.'
Mary-Ann smiled wistfully as she recalled the comfortable intimacy of their past encounters. 'Amen to that.' Then, on a more cheerful note she added: 'I'm so pleased for you, Bill. Every night I've prayed that somebody somewhere along the line would have the good sense to realize you were innocent.'
'Well, as you can see, your prayers have been answered...'
'And not only that, you're back in charge of The Lady. How does it feel?'
'Like coming back from the dead,' said Hartmann.
* * *
Wallis and Malone rose from their chairs as Commander-General Karlstrom entered his wood-panelled office. As the metal door of his personal elevator closed behind him, a matching section of wood descended over it, sealing it from view. Karlstrom skirted his desk and advanced, right hand extended.
As senior operatives, holding the military rank of Commander, Wallis and Malone had earned the right to warm handshakes and use of their given names in meetings with the Operational Director. This relaxed atmosphere (enjoyed by everyone with more than four years successful service) did not permit them to address Karlstrom as 'Ben', but they weren't required to include the word 'sir' every time they spoke, and there was none of the jumping to attention or parade ground saluting required from wet-backs fresh out of Rio Lobo.
'Don...'
'Morning, sir...'
Karlstrom turned to greet Malone. 'Matt! Glad you could make it. Hope it wasn't a problem calling you in at such short notice.'
'No, sir. I managed to cover it. Took off with four of the boys to check our southern flank.' He smiled. 'With so much air activity in our sector, I thought we might have a wagon-train on our trail. Provided I'm back by dawn tomorrow, there shouldn't be any problem.'
'We've got enough as it is,' said Karlstrom. 'This particular operation has become so complex I felt we should talk it over face to face.' Karlstrom waved them into their seats and settled into the high-backed chair behind his desk. 'Don, uhh, before you give us your sit-rep, where is Mr Brickman at this moment in time?'
'Still on the wagon-train, awaiting the outcome of this meeting. If he gets a green on the next phase of the operation he intends to head west with his three horses towards the junction of the North and South Platte Rivers in the hope of catching up with Cadillac and, uhh...?' Wallis' eyes questioned Malone.
'Don't worry, I'll be there. I don't know how good he is at following trails but I reckon it'll take him at least four days. Karstrom nodded, then turned back to Wallis. 'Run us through what happened on the train.'
'From when Brickman first came on the air?'
'Yes. I want Matt to have the whole picture so that he knows exactly what we're up against.'
Malone looked puzzled. 'Am I on the wrong track, sir? I was under the impression Brickman and his kin sister were working for us.'
'It's not quite that simple, Matt. In theory, yes, they are. Unfortunately, in practice, some doubt has arisen over the question of who is manipulating who. Let Don say his piece and you'll understand what I'm getting at.'
Wallis gave a crisp, coherent account of the rescue operation that had been triggered by the telepathic contact between Brickman and Roz: an operation which Karlstrom had approved over his direct radio link with the task-force. He then gave a brief _résumé_ of the surgical treatment Clearwater had received and her present state of health, described Steve's arrival on board, played back the tapes of his conversations with Roz, and concluded by describing her terrifying demonstration of mind-control.
'Don't ask me how she does it. All I can tell you is it works, and she can turn it on just like that.' He snapped his fingers. 'When you imagine what she _could_ have dreamed up, you may think that sitting on top of a rock tower is not all that bad –'
Malone held up his hands. 'I didn't say anything, Don.'
'No, but I can see your face. I'm telling you it was absolutely horrendous. And when Jake fell off that fucking...' He tried to shake the memory away. 'I hope and pray I never have to go through anything like that again.'
'How is he now,' asked Karlstrom.
'Jake? Having trouble sleeping.'
Malone grunted. 'He'll get over it. Sounds like he got what he had coming.'
'He always did tend to run off at the mouth,' agreed Wallis. 'I don't think he'll do it again while she's around.' He turned to Karlstrom. 'The question is – if she was able to take control of our minds then, is she in control of them now?'
Malone laughed. 'How the hell can she be? You're not on the wagon-train.'
'Why should that make any difference? The telepathic link between Roz and her kin-brother works just as well even when they're thousands of miles apart?' He appealed to Karlstrom. 'Right, sir?'
'It would appear so, yes. But with all due respect, Don, I think you're over-reacting. It was obviously a bad experience but you asked for a demonstration and you got it. Technically she may have taken control of your mind but she did not actually make you _do_ anything rash or foolish. As I understand it, apart from Nevill, none of you budged from that table.'
'That's true, but –'
'What she did was induce what psychologists call a positive hallucination. Which you _all_ shared. That's the interesting bit. Mass hypnosis is not unknown. You appear to have experienced a sophisticated "instant" version, and a very effective one too. If she can warp someone's perception to that degree and at that speed, I have a feeling she really _could_ neutralize Mr Snow.
'But that's not really mind control – at least not the kind that worries me. Roz has no reason to turn against the Federation. Her rival for Steve's affections is now wired to a life-support system – and at her mercy.
'Brickman may still be mixed up over the two of them but he's not going to jump ship if we've got Clearwater and Roz in Grand Central. It would be absolutely pointless. He is a natural undercover agent. He's almost impossible to read but there is one thing I _do_ know about him. He's hungry for power. And this is where the power is. No...' Karlstrom paused reflectively. 'He and Roz will come through for us. I'm sure of it.'
'So we don't need to worry about her, uhh –?'
Karlstrom smiled, apparently satisfied with his reasoning. 'Don – if she was planning to betray us, would she have shown us what she could do?'
Wallis conceded reluctantly. 'No, I suppose not.'
Malone caught Karlstrom's eye. 'Excuse me asking, but this guy Brickman – is he all right in the head? I mean, does he still know where the edges are? It's no secret he's been bouncing beaver and you've just hinted he's been jacking up his kin-sister.'
'They're not related, Matt. But that's classified, okay?'
'I understand, sir.'
'Okay, let's get down to the real business – setting up this attack by the M'Call Mutes on Red River. I've got approval for the idea from the Oval Office. It could be a good way of finishing off the M'Calls but what we have to do is figure out how to get our two remaining targets on board a wagon-train and safely under lock and key without losing the whole shebang.'
'Sir, I'm not saying it can't be done, but do we _have_ to bring these other two in?' asked Malone. 'You've got Clearwater. Why don't we just close down the file by putting three triples through their heads?'
Wallis nodded in agreement.
Karlstrom responded with a thin-lipped smile. 'That might take care of Cadillac but from what I've heard about Mr Snow, you could find the bullets coming back at you.'
'Sir – with respect – nobody's _that_ fireproof.'
'Don't count on it. I know it's tempting, Matt, but there are other factors to be taken into consideration. High-level strategic objectives that I'm not at liberty to disclose at this moment but which I'll bring you in on as soon as I'm given clearance to do so. Brickman's idea _seems_ unnecessarily complex but it could give us the set-piece engagement we're looking for.'
'Ahh... I didn't know that.
'And there's something else,' continued Karlstrom. 'I'm not prepared to jeopardize the relationship you've built up with the Mutes in your sector. From past experience, we know that some mutes are able to communicate over long distances. As wordsmiths, Mr. Snow and Cadillac may have that capability.
'Yeah. Don't ask me how. It just does.'
'Exactly. As we speak, the M'Calls may be organizing a Welcome Home party. Could make things very awkward if we suddenly lost him overboard. If you followed that up by the on-site removal of their wordsmith – hell! That would _really_ shake the shit loose.'
Malone grimaced ruefully. 'You're right, sir. I hadn't figured it that way. Guess I was just tryin' to cut a few corners.'
'Nothing wrong with that. Just for the record, I sent Brickman out with a dead-or-alive option on these two. But since then I've acquired a better overview. Believe me. Icing Cadillac and Mr Snow at this point in the game would cause more problems than it would solve. Especially for you.'
When Wallis and Malone had left for the airfield, Karlstrom remained in the office he used for meetings with his operatives. Slumped in his upholstered swivel chair, his left elbow resting on the padded arm, he reflected on what he had heard, pulling slowly at his nose, lips and chin with thumb and forefinger as he did so.
He had maintained an up-beat mood throughout their discussions but that had been a front. Don Wallis' account of Roz Brickman's hallucinatory powers filled him with alarm. Despite the soothing assurances he'd laid on Wallis, this _was_ mind control – of a pretty spectacular kind. The kind that could plunge them all into deep shit.
Karlstrom didn't give a toss whether Roz had taken temporary control of Wallis and his team; the only mind _he_ was concerned about was his own. Was she only able to distort reality and thus induce total disorientation – or was she capable of something far more sinister? And anyway, how the hell did you define reality in the first place? Could she have manipulated him into putting her aboard that train? It had been her suggestion, yes, but had she _forced_ him to go along with the idea?
No – that was impossible. The President-General had had the final say. In all matters concerning Steve and Roz Brickman, Karlstrom had been careful to cover his ass. Unfortunately that was not sufficient to remove him from the line of fire. If OPERATION SQUARE-DANCE went down the tube, _he_ would be the one taking the flak, not G.W.J. the 31st. Yes, sir...
But was she planning to go over the side, or was that just something that Brickman had pulled out of the air to lay on the lump-heads? It had to be a bluff. A wind-up. What the hell would she do out there? No. With Clearwater out of the way Roz had what she wanted. Steve Brickman. Karlstrom was convinced he was back on the rails. The psychologists who selected, shaped and supervised the people on the Special Treatment List knew what made that young man tick. He wanted power, and he wanted to get even. That was why the results of his final exams at the Flight Academy had been fixed, giving him fourth place instead of first and the honours he merited. Yes... that had _really_ lit a fire under him. And the Federation was the only place his needs could be satisfied.
These thoughts provided Karlstrom with scant comfort. Roz and Brickman were both telepaths, sensitives of a remarkable kind. Did Brickman possess the same latent powers to bend reality out of shape? Right from the very beginning, Karlstrom had been reluctant to meddle with the grey area the Life Institute called 'psionics'. But faced with the threat from such people as Mr Snow, the Federation could not afford to ignore what little home-grown talent it possessed. Karlstrom knew of Steve and Roz; only the P-G and COLUMBUS knew who the rest were. If there _were_ any others. Karlstrom hoped not. The P-G had likened Steve and Roz to a weapon-system. But what was the point of a weapon-system whose workings no one fully understood and whose destructive potential was incalculable?
No one in their right mind would launch such a weapon. But the appropriate target data had been fed into Steve and Roz Brickman and the button had been pressed. They had been fired towards enemy lines. Were they, as Karlstrom sometimes feared in the small hours of the morning, beyond recall? Was this weapon they had unleashed about to veer off course – turn back on its makers? That was what Karlstrom feared the most. And he wondered if the handful of quacks who had elbowed themselves into an unassailable position as the sole experts in the so-called 'science' of psionics had had the foresight to fit their charges with a self-destruct mechanism.
Probably not. How could they when none of them could explain in words of less than three syllables (that any normal person could understand) how and why someone like Roz Brickman could fall to the ground with a hole punched through her upper arm by a phantom crossbow bolt? A real hole, with real blood, that healed and disappeared without leaving any scar tissue within eight hours!
They had no answers because they didn't know. There _was_ no science, only jargon. Psycho-babble. The Department of Psionics at the Life Institute was an empty shell providing nothing but a few quick promotions, some cheap prestige. A scam. He had never wanted to get involved. He had been pressed into using Steve and Roz, and as a result he was marooned in the middle of a fucking minefield – with the President-General watching from the other side of the warning tapes.
Faced with the possibility that the decision to deploy Roz alongside her kin-brother might backfire with sufficient force to remove him from office, Karlstrom decided to take some avoiding action of his own. He would agree to Steve's request to leave Red River in Nebraska. Clearwater, in any case, could not be moved. There was no danger of losing her: any rescue attempt mounted by force of arms would be fatal. And despite the risk that she might – just might – defect, he would leave Roz there too.
To bring her back into the Federation after this demonstration of new, uncharted powers would be an act of criminal folly. If there was going to be any heat, Wallis and the task force could take it. Karlstrom was aware that his decision placed the entire crew of Red River in jeopardy but there was, for the moment, no acceptable alternative. He had no desire to find himself sitting on top of a dizzying pinnacle of rock, or whatever other horror she might produce from the depths of her mind. And until some way could be found to deal with the problem he, for one, did not intend to get within a hundred miles of her.
If she was the loyal soldier-citizen he believed her to be there would be no problem. But until she proved that beyond all possible doubt it was wiser not to take any chances. As long as she was marooned in Nebraska she could not warp his own mental processes or affect his deliberations in any way. And he would circumvent any treachery by cutting Wallis out of the planning process they had begun that very day. He would continue to receive documentation but it would not be the real thing.
Only Malone and the other units would know the final plan. Roz and Steve would be left in the dark. If they played their parts, all well and good, if not, well... Amtrak could survive the loss of a wagon-train. Sensitives like Roz and Steve were a lurking cancer; a menace to the system. Their elimination – by death, or permanent transfer to the overground – was the only way to secure the future of the Federation. The Department of Psionics would be discredited, disbanded; Karlstrom's doubts would be vindicated. AMEXICO could return to the tried and tested ways of secret warfare, and its director would sleep more soundly in his bed.
It was a neat, satisfying scenario, but Karlstrom knew from experience that things never went entirely to plan. Somewhere along the line somebody always fucked-up. The reason why he had held his job for so long was because he was also an expert in containment – the art of damage limitation.
Disclosure was a central feature of that art. You never told anyone anything they did not need to know, and you never _ever_ gave them bad news when good news might be just around the corner. That was why Karlstrom had decided not to tell the President-General about Roz Brickman's new and alarming capability. If challenged, he would defend himself – like all canny administrators – by saying he was waiting for a fuller report.
## CHAPTER FOUR
After the violent confrontation that preceded the wounding of Clearwater, Steve had not been looking forward to telling Cadillac that she was now on board Red River, but when he finally caught up with the Mute and Malone's renegades, it did not go as badly as he had expected. Cadillac – still crippled with guilt over his part in the affair – was so relieved to learn that Clearwater was alive, he brushed aside the awkward fact that she was in the hands of the Federation.
Having come prepared for a bitter wrangle, Steve found the Mute's fatalistic reaction somewhat disconcerting.
Malone, of course, was still posing as the stalwart leader of a renegade band and Cadillac – while appearing somewhat less than overjoyed to find himself in their company – gave no sign of being aware that Malone and his henchmen were undercover agents or that, during Malone's brief absence, he had been picked up by a Sky-Rider of Air-Mexico and flown into the Federation for a head-to-head with Karlstrom and Wallis.
Steve, who had been one of the main items on the agenda, was also unaware of Malone's flying visit to Grand Central. Wallis had merely told him that Karlstrom had approved his outline plan. He was to proceed on the assumption that an attack on Red River would take place; the final game-plan would be conveyed to him when all the pieces were on the board.
Cadillac already knew he and Roz could communicate telepathically. Steve was now obliged to repeat the story for Malone's benefit, unaware he had already been briefed by Karlstrom. And Malone, alerted by Cadillac's ill-considered claim that Steve was an undercover agent, had to appear to accept his story whilst pretending (for Cadillac's benefit) to secretly mistrust him!
As Jake Nevill had observed on the wagon-train, there were so many layers of deception it was difficult to keep your place in the script. The swirling cross-current of lies threatened to become a vortex which, if they were not careful, could send them and the whole operation down the tube.
Given the no-nonsense character which he had adopted for his role as a renegade, the agent code-named HIGH-SIERRA was put into some difficulty by the revelation of the telepathic link between Roz and Steve. As 'Malone' he couldn't just let it pass without comment; on the other hand, as Steve's covert ally, he didn't want to make a big deal out of it. Having talked to Karlstrom, he had anticipated this topic entering the conversation and had been searching for a way to deal with it since boarding the Sky-River for the return trip. He decided to maintain a healthy scepticism bordering on the dismissive, tempered with a spirit of enquiry.
'You telling me you can speak to each other without a radio?'
'Well, we don't actually speak but we _can_ communicate specific items of information. It's like hearing a voice inside your head, but that voice isn't _real_. The sound doesn't physically travel through the air. It's the same principle as a radio transmitter, but these are _thought_ waves. The voice I've mentioned – you don't actually _hear_ it, you just imagine you do – like when people speak to you in dreams. You can see their lips moving; what you are saying to each other makes sense, you are conscious of their voices but _there is no sound_. Because, by itself, your brain can't make a noise. It needs a tongue, lungs and a larynx to tell you what it's thinking, and it can't decipher external noise without filtering it through an ear.'
'I think I've got the message,' said Malone.
'On the other hand,' continued Steve, 'If you're transmitting a feeling – joy, pain, terror – or the image of a particular place, you don't need words. You experience the same sensation, images registered by their mind are transmitted simultaneously into your own. You see what they see – as it happens. The way part of your mind can picture all kinds of things while you're listening to somebody talk – the way you're doing now.'
'That's what bugs me,' said Malone. 'How d'you know it's someone _else's_ thoughts and not your own?'
Steve shrugged. 'You just do. There's a kind of tingling, a coolness. I can't explain it, but if it happened to you, you'd know what I mean.'
'Could it happen to me? Can your kin-sister beam this junk to anyone she chooses?'
'No. That would be like COLUMBUS trying to transmit data to someone whose work-station wasn't equipped with a VDU. Your brain has to be able to make the right connections. Just don't ask me what they are. We discovered we had this ability because we were brought up together. I suppose I must have been about five or six years old when it started. To us then, it was just a game. There may be others but no one has come through to us. Roz is my only contact and vice versa.'
'So what's she doing now?'
'Can't tell you. The truth is, as I got older, I became scared about what we'd been doing. In the Federation, it doesn't pay to be too different from everyone else. Especially in that area. It's too much like Mute magic. Besides – who could you tell without getting into trouble? That's why I shut down that side of my mind, didn't answer and tried my best not to let her through. Eventually it worked. Either that or she gave up trying.'
'So how come you're back in touch?'
Steve realized that Malone wasn't just extending this conversation for the benefit of Cadillac and himself. He was quizzing him on behalf of Karlstrom as well.
'It happened when I came up for my first overground solo and caught my first glimpse of the blue-sky world. The shock reopened the link between us. But then a lot of strange things happened to me that day.'
'I know what you mean. It can be quite a moment.' One which, given his allegiance to the Federation, Malone was still trying to come to terms with. 'Still no joy with Roz?' He demanded.
'No. If she needs to come through, she will.'
Malone laughed dismissively. 'I'll believe it when I see it! You've got what it takes, Brickman, but like a lot of guys who are fast on their feet, you're full of bullshit!'
'Yeah, I know it sounds that way. If we have time, I'm hoping to convince you you're wrong.'
Malone jerked a thumb at Cadillac. 'This is the guy you've got to work on. Know what your so-called friend here thinks you are? An undercover Fed!'
Steve met this news with a dry laugh. 'Does he...?'
Cadillac, who was totally unprepared for this embarrassing disclosure, stammered: 'Now, uhh – w-wait a minute! I didn't put it exactly like that –'
'That's okay,' said Steve amiably. 'I should have expected you to try and get even.' He turned back to Malone. 'We have a few personal problems that need ironing out.'
'Yeah, so I gather,' chuckled Malone. 'Somethin' to do with you jackin' up his beaver.'
The use of that word riled Steve but there was little he could do about it. Malone was too big and too mean and Steve needed his wholehearted cooperation. 'It's a long story...'
'Save it,' said Malone. 'I'm all smoked out.' It was a phrase coined by users of rainbow-grass, which meant they'd had a surfeit of psychedelic fantasies.
As Malone turned on his heel, Steve put a hand to his forehead. 'Wait a minute! Roz is coming through!'
The renegade paused with evident irritation. Cadillac rose from the rock he'd been sitting on.
'It's a message for you. From Clearwater.' He frowned as he mouthed the next few phrases, then repeated them to Cadillac. 'She wants you to rescue her from the wagon-train –'
'From Red River? Nothin' to it!' scoffed Malone.
'– Roz too. She wants to join us.' A pregnant pause then – 'The wagon-train has been ordered to stay in Nebraska... and patrol westwards... along the line of the Platte River.'
'Are you sure they are going to keep Clearwater aboard?'
'Yes.' Steve concentrated again. 'It's too risky to fly her out and... they can't off-load her because... all the other wagon-trains are committed elsewhere.'
Malone decided to ask an awkward question. He had to pick at Steve's story in order to enhance his own credibility and also to build up Steve's. 'There's something fishy about all this. I know this kin-sister of yours is working both sides of the track but why would Red River waste time on a wounded Mute? One more, one less – what does it matter to them? Why are they keeping her alive? Is there something you haven't told me?'
Steve appealed to Cadillac and received his permission to speak. 'They're keeping her alive because they know she's important to the Clan M'Call. She's a summoner. A very powerful one. And that's not bullshit. She is _dynamite_.'
'How do they know that? Did you tell 'em?'
'No. I told Roz when we met in Grand Central early last year. I found out later our conversation was bugged.'
'So...?'
Steve closed his eyes again and worked his fingers across his brow. 'Roz says the decision has come from Grand Central. They think that if they leave Clearwater on board Red River, the Clan M'Call will try and rescue her.'
'Nahh,' said Malone. 'They ain't gotta hope in hell.'
'On the face of it, no. But the clan is led by another summoner called Mr Snow. He's old but he's still big trouble. I was on board The Lady from Louisiana when he almost wrecked her in 2989.'
Malone looked suitably impressed. 'You mean that Battle of the Now and Then River? Even I've heard of that. Jeez! Was that him?'
'Yes it was!' exclaimed Cadillac. 'And I took part in that battle too!'
Malone looked at the Mute with new respect. 'Is that so? Well, I've gotta tell you. It's no secret I've never been overly fond of you people, but that's somethin' to be proud of. It's not often the Federation gets a hiding like that. Hell, I wish I'd been there.'
Cadillac squared his shoulders. 'The Plainfolk fight their own battles!'
'Sure. But that's no reason why we couldn't put our heads together. I know how those goddam things work. Fact is, quite a few of the boys put in time on the trains. Who knows? Might come up with some ideas...'
Cadillac didn't respond.
Malone looked at Steve. 'Let's say – just for argument's sake – you captured a train. More or less in one piece. What would you and your Mute friends do with it once you've got Roz and Clearwater away?'
'That's for Cadillac to say. But if you're asking me, they won't do anything with it. By the time they've finished, there'll be nothing left.'
The news provoked a sigh of regret. 'Seems a shame – a top-class fighting vehicle like that. It's a pity you can't figure a way to get your friends out and leave it more or less in one piece. If the M'Calls took over Red River and let my boys help them run it. Hell –', Malone grinned, '– the Federation really _would_ have a fight on their hands!'
'Yes, they would,' agreed Steve. 'And if you laid hands on a few Skyhawks, Cadillac here knows how to fly.'
'No kidding...'
'Yup. He's a real ace.'
Unaware that Steve was watching him closely, Cadillac drew himself up, his chest swelling like a mating cock-bird. 'And not only that! I designed and built rocket-powered planes for the Iron Masters of Ne-Issan!'
'Did you now?' Malone turned to Steve. 'How come you didn't tell me about this back at the camp site?'
'I did mention he was something special.'
'Yeah, but I didn't realize he was _that_ special.'
Steve shrugged. 'Like I said, it's a long story.'
'Yeah, well, I think we've got a good thing going here.' Malone laid a fatherly hand on Cadillac's shoulder. 'How does this sound, good buddy? Cut us in for a slice of the pie, and if we manage to grab this wagon-train, you can be head of the air force!'
Steve laughed disgustedly. 'Aww, c'mon, guys, stop jerking off! This is a pipe dream! There's no way the M'Calls could capture a wagon-train. I'm not saying Mr Snow couldn't wreck one. He came pretty close to doing that at the Now and Then River. But capturing one in working order... hell!' He laughed again. 'Can you see a wagon-master lowering the ramps to let a screaming bunch of Mutes on board?'
Malone's brow furrowed. 'No...'
'Exactly. It's a total waste of time to even think about it. Cadillac and I have already been through this. That was what started the argument which brought out the knives. I wanted to try and rescue Roz. He said it couldn't be done. And he was right.'
'I didn't say it couldn't be done,' said Cadillac. 'I said I wasn't prepared to waste the lives of my clanfolk trying to rescue your kin-sister.'
'Whatever you say, Caddy. I'm not gonna argue with you any more. I broke my blood-oath once and look what happened. I lost Clearwater.'
'We both lost Clearwater. But she is not lost forever. Thanks to you, she is alive – and has asked us to deliver her!'
'Aren't you forgetting the prophecy? Isn't she supposed to go into the Federation? She asked Mr Snow if she would die in the darkness of their world and he said, "No, you will live".'
'I am a wordsmith as well as a seer,' exclaimed Cadillac. 'I forget nothing. And only a seer can interpret the images he draws from the stones. In speaking of you I said this: _He will come in the guise of a friend with Death hiding in his shadow and he will carry you away on a river ofblood_. I said nothing about her going into the Federation. Clearwater just assumed that. In replying to her question – _Am I to die in the darkness of their world, or will I live to see the sun again?_ – the Old One merely said she would live. He said nothing about the dark cities.'
'So she's _not_ destined to go into the Federation...'
'Not in my reading. For me, the images apply to our time in Ne-Issan. You came in the guise of a friend, offering to help me, but you were really planning to wreck my work.' He held up a hand. 'No! Don't interrupt! The death hiding in your shadow was your secret link with the Federation who supplied the devices you used to destroy the Heron Pool. The slaughter surrounding our escape was the river of blood on which you carried Clearwater away.' Cadillac smiled disdainfully. 'You see how it all fits together?'
'Yeah, it's neat. Okay, so I blew you out of a job. And yes, I conned the Federation into thinking I was working with them. It was the only way to get you out of there. I'd promised Mr Snow I'd do my best to rescue the pair of you – and that's what I did. Your place is with the Clan M'Call. And if you're still peeved because I forced you to face up to reality, well tough on you!'
'The reality, Brickman, is that Clearwater is now a prisoner of the Federation, held on board Red River. And until she is rescued, you have only kept _half_ your promise to the Old One.'
'Sweet Sky-Mother! Caddy, for crissakes, be practical! Okay, Clearwater is not dying but she needs medical attention! That's why all this talk of storming the train is sheer lunacy. She can be moved, yes, but you're not going to be able to walk out with her thrown over your shoulder!
'She'll have to be evacuated properly – preferably not in the middle of a fire-fight – and for a while she'll need to be cared for by a doctor. The only person we can rely on to do that is Roz. Which means we have to get her out in one piece as well, plus some of the drugs, dressings and equipment that are stored in the blood-wagon –'
'We have no need of such things. The Old One can heal her.'
'The Old One didn't save everybody after the battle with the wagon-train. He can set simple fractures and use plant-based substances to keep some types of wounds free of infection but he's not a miracle worker. You might as well stuff his herbal remedies up your ass for all the good they're going to do Clearwater. You are right, I goofed – but at least I managed to save her life.'
'And I'm grateful for that...'
'Good. All you've got to do now is get used to the idea that we can't get her off the train. _I'm_ the only person who can rescue her.'
'Oh? How?'
'By switching sides – talking them into believing I'm still the straight arrow they thought I was. I'll have to go back to the Federation with Clearwater and then, when she's better, try and find some way to get her and Roz out into the blue-sky world.'
Cadillac looked dismayed. 'But that could take months, years!'
'It could do,' admitted Steve. 'You got any better ideas?'
'I don't know – surely there must be some way we could –'
'Caddy! For the last time! Forget the fucking train! What are you gonna do? Hang around in the hope that someone's gonna leave a door open? There's absolutely no way your people can get on board! I've served on one of those things! The spectre of Mutes running wild inside a wagon-train is the nightmare all Trail-Blazers share. But it doesn't keep them awake at night because they know it can't happen. That's why the trains are designed the way they are. Even if you got past the guns and the steam jets there's still no way you can get inside. When those ramps are up, those wagons are shut tighter than a gnat's ass!'
Malone came out of his reverie. 'The wagon-master...'
Steve had almost forgotten he was there. 'What?'
'The wagon-master. He wouldn't lower a ramp for a bunch of Mutes, but he might lower it to let a group of Trail-Blazers on board – 'specially if they had some lumpheads in hot pursuit.'
Steve saw Cadillac's interest quicken. 'Meaning...?'
'Well, it's just an idea but... if we were somehow able to get hold of enough helmets, uniforms and rifles...'
'Disguise ourselves as Trackers?' asked Cadillac.
'There'd have to be enough of us to seize control of the ramp and hold it long enough to get more of your people aboard.'
'You're crazy,' said Steve. 'D'you think the guys running Red River are just gonna sit there and let it happen? As soon as you show your hand at the top of the ramp, they'll close the fire doors at either end, unhitch the wagon then pull away and reform, leaving you sitting there with nowhere to go!'
'They could do,' admitted Malone. 'But I'm hoping this Mr Snow character might be able to do something about that. But if you think he can't deliver then forget it. To pull this off, we need brains, brawn _and_ magic.' He grinned sheepishly. 'Never thought I'd hear myself say that.'
'The Old One has the power,' said Cadillac. 'He will know what to do when the times comes.'
'Does that mean we're in business?' asked Malone.
'No. It means I'll think about it.'
Cadillac had tried to sound non-committal but Steve knew exactly what was going through his mind.
And so did Malone. _The lump had taken the bait. He was on the hook_...
Later, as dusk fell, Steve caught Malone sitting by himself in front of a small camp-fire. He hunkered down beside the brooding renegade and stretched his hands over the flames.
'How am I doing?' asked Malone.
'With Cadillac?' Steve smiled. 'Great. Just keep oiling him up. You'll have him eating out of your hand.'
'Brown-nosing ain't exactly my style,' said Malone. 'But if that's what it takes to get a result...'
'It's a relief to be working with someone of your experience.'
'Don't get smart, Brickman.'
'I wouldn't dare. The last time I spoke out of turn, you loosened a couple of teeth and almost broke my jaw.' He glanced over his shoulders. 'I'd better go. We shouldn't be seen too often with our heads together.'
'You're right. But listen – these seeing-stones... are they for real? Can that lump actually tell what's gonna happen?'
'You'd better believe it,' said Steve. 'He saw both of us going down under the water, and a week or so later we were trapped on a sinking boat and nearly drowned! Creepy, huh?'
'Yeah.' Malone shifted uneasily. 'I hate all this magic shit.'
'You're not the only one,' replied Steve. And he was not just thinking of Nevill. Steve liked situations where you could figure all the angles, plan several moves ahead. The trouble with magic was it was so fucking unpredictable. People were bad enough, but magic was unfettered by the rules of logic, the scientific laws that governed cause and effect. 'But don't let's knock it,' he concluded. 'It's saved my ass several times already.'
'That's what worries me,' growled Malone. 'Bouncing beaver may have softened your brains a little but you're no dick-head.'
'Thanks...'
'If a guy like you takes it seriously then we could be in big trouble.'
'You mean with Mr Snow?'
'Yeah. It looks like our friend here will run him up the ramp but how the fuck do we put the brakes on someone like that before he puts a bolt of lightning up everybody's ass?'
'That's where Roz, my kin-sister comes in.'
Malone accepted this with a nod. 'Mother mentioned something about a demo. Is she a summoner too?'
Steve laughed at the idea. 'Of course not. She's a straight-A like you and me. But being a telepath, her brain – like mine – must have some extra capability, or be using circuits we all possess but which most of us never plug into. And now, suddenly, another part of her brain has been switched on, giving her the ability to take temporary control of other peoples minds – by altering their perception of reality.'
'Whatever that is,' grunted Malone '– but how's this gonna fix Mr Snow?'
'Jeez, what a question! This new power's come as much a surprise to her as it had to everyone else.'
'C'mon, Brickman! If I've gotta put my ass on the line to nail Mister Magic, I wanna know what she's providin' in the way of back-up. You were _there_ when she spooked Wallis and the other guys. You tryin' to make out you and she didn't connect over this?'
'No. But all I got were a series of sensations, not the workshop manual.' Steve hesitated. 'The best way to explain it is to imagine Mr Snow is a radio transceiver and she's a heavy burst of static. When he switches on, she's gonna swamp the air-waves and block out all incoming and outgoing signals. If she can prevent his brain from functioning coherently then, in theory, he won't be able to draw in these forces or direct them outwards on to a specific target.'
'Sounds plausible. Question is, will it work on the day?'
'That's something we're gonna have to take a chance on.'
'Great. That really builds up my confidence. One last thing. That business with Roz and the message from Clearwater. Was that for real? Did she really come through then or were you just juicing our feathered friend?'
Steve smiled knowingly. He warmed his hands one last time then rose to his feet. 'No point in answering that, is there? You don't believe in all this magic shit.'
'Malone's eyes narrowed but his battered face was not unfriendly. 'Gidd-outta-here!'
Since leaving the Clan Kojak on the shores of Lake Mi-Shiga, Steve, Cadillac and Clearwater had followed, wherever possible, the line of the ancient hardway listed on Federation maps as Interstate 80. In the pre-Holocaust era it had been part of a continuous east-west ribbon of concrete that began in the Big Apple and ended at Denver, Colorado, passing through Ohio, Indiana, Illinois, Iowa and Nebraska on the way.
They had long since gone off the left hand edge of the map that Steve had found on board the aircraft they had stolen at Long Point, but the clans they met along the way all agreed that if they continued towards the setting sun they would reach the edge of a wide endless river. Since they had already crossed the Mississippi, Steve knew from the maps he'd seen during his training at Rio Lobo that this had to be the Miz-Hurry – the Mute name for the Missouri.
The crumbling weed-covered remains of Interstate 80 had finally petered out some miles from the river, but eventually they found themselves on the edge of a one hundred foot high bluff overlooking the east bank. On the far side lay the buried remains of Omaha. The headquarters of Strategic Air Command, sited at Offut AFB just south of the city had made Omaha a prime target in the global nuclear war which brought 20th century civilization to an abrupt end.
In the public archives which could be accessed via COLUMBUS, the blame for the Holocaust was laid firmly on the Mutes; the role played by the Russians was ignored, the very existence of the USSR, along with all other nation-states beyond the borders of the USA, had been erased from the records and the minds of the Federation's soldier-citizens.
According to COLUMBUS there was only one continent, America: only one race, Trackers – born to inherit the blue-sky world once it had been cleansed of the subhuman scum spawned in the hell-fires of the Holocaust, the incendiary tidal wave of murder and mindless destruction that had burned every city, township and hamlet to the ground and which – as everyone knew – had been the work of the degenerate, drug-crazed hordes who had fathered the Mutes.
It was an agreeable fiction: the historical fact, insofar as it concerned Omaha, was that the air base, the city and its riverbank neighbour, Council Bluffs, Iowa, had been obliterated by a multiple strike which included both air and ground bursts.
A single H-Bomb would have sufficed, but despite the endless attempts (some genuine, other entirely fraudulent) to reduce the stockpiles of nuclear weapons, the politicians and the generals who held the fate of the planet in their hands were still wedded to the concept of overkill. The first missile obliterated its target, the remainder merely rearranged the glowing ashes and ensured that the key personnel sheltering in SAC's bomb-proof bunker beneath Offut AFB stayed there – permanently.
The initial fire-flash which, in its first few milliseconds of incandescent life was hotter than a solar flare, had vaporized all the timber-frame houses and reduced their brick-built neighbours to cinders. The four bridges linking the two cities, scarred and riven but still standing after the first blast, had disintegrated in the second, their steel-work fusing with stone and concrete into huge red-hot gobbets of volcanic lava which had been hurled into the boiling river below.
Half-submerged in the wide shallow waters, they had fused and cooled to form irregular clusters of giant stepping stones, their jagged edges worn smooth by the slow-moving, endless flow of the river. Over the centuries, ash and dirt, settling on the exposed surfaces had provided a fertile bed for airborne seeds, and debris floating downstream had become entangled with the rockpiles, forming a ragged gap-toothed weir thatched with wild grasses and a scattering of trees and bushes whose trailing roots were draped with river weeds.
Driving what remained of their herd of captured horses into the tranquil waters upstream of the highest weir, the trio crossed over into Nebraska. In its early, troubled history, the territory had proved so unappealing to would-be settlers, it had become known as the 'corridor state'. Thousands of pioneering families passed through it on the Oregon and Mormon Trails which converged at Fort Kearney then tracked westwards along the line of the Platte River. The Mormons kept to the north bank and were later followed by the riders of the Pony Express; those bound for Oregon stayed on the southern side until the river split into its main northern and southern tributaries some two-thirds of the way across Nebraska.
From here, the trails angled north-westwards, each convoy of covered wagons and handcarts keeping to its chosen side of what was now called the North Platte on the long haul up towards the high plains of Wyoming. It was only after they had passed through Caspar and had turned away from the river that their parallel paths finally came together at the South Pass through the Rockies before going their different ways again at Fort Bridger in the south-west corner of the state.
At this point, those on the Oregon Trail were only just over half-way to their final destination, but the Mormons, whose millennial fervour had roused their more orthodox eastern neighbours to violence, were close to their journey's end – the great salt lake in whose fertile valley they were to build the city that would become a monument to their unshakeable faith in a God who had not only walked the shores of Galilee but also the plains of North America.
A faith which had not saved them: a monument which the Holocaust had turned into a tomb.
Steve, Cadillac and Clearwater had been intercepted by Malone some forty miles east of Kearney, just north of the river near navref Grand Island. On the fatal afternoon when the Skyhawk had made its strafing run, Red River had been a hundred and fifty miles south of the renegade's campsite. Acting under the orders relayed by Wallis – head of the AMEXICO task force – Commander James Fargo had kept her rolling northwards while his aircraft had gone out to pick up Clearwater, closing the gap between the wagon-train and his second visitor-Steve.
After leaving the train and crossing back over on to the north bank of the shallow river, Steve had re-traced the route taken by the Mormon leader Brigham-Young and his flock of zealots. By the time he caught up with Malone, the renegades were over a hundred miles west of the ill-fated campsite and had already passed beyond the fork in the river at navref North. Platte – a long-vanished city whose name came from its location on the slim point of land between the convergent tributaries.
Cadillac knew the place well. It was burned into his memory. He had journeyed here with Mr Snow and a posse of M'Call Bears led by Motor-Head. And at the Old One's bidding he had searched for and found a seeing-stone. A stone full of terrifying images, heavy with blood, death, sorrow and utter desolation.
This was the Old One's dying place. The memory of that revelation brought back the crushing weight of guilt and grief the act of foretelling had laid upon him. Urging his horse into a gallop, he pulled ahead of his companions to hide the bitter tears streaming down his cheeks. He wanted to distance himself from this dreadful place, but he knew it had already laid claim to his soul and was content to await his return.
The central and western thirds of Nebraska consisted of a vast treeless plain, rolling uplands thinly seeded with buffalo grass and sage, scored here and there by river valleys and streams that drained into the Platte or the Missouri. These valleys were the only shelter from the howling blizzards which had driven out the early 'sod-busters' after the sleet storms had levelled the crops they'd planned to live on through the long winter. And it was here that the trees lay hidden, cottonwood, willow and elm huddled side by side out of reach of the cutting winds.
From North Platte, the river plain varied in width from one mile to fifty, narrowing then broadening, then narrowing again as it squeezed through the surrounding uplands in its gentle climb towards the Goshen Hole on the state border.
At Scotts Bluff, Cadillac encountered another familiar land-mark, a huge yellow-ochre wedge of barren rock rising eight hundred feet above the surrounding landscape. This was the only signpost that now remained to tell the traveller he was about to leave Nebraska and enter Wyoming. No other evidence that these two states had once existed as legislative and economic entities remained. The traffic signs and roadside buildings had gone, the abandoned, pillaged hulks of the last few Highway Patrol cars had long since crumbled into powdery flakes of rust; the more resistant parts of the engine and chassis disappearing beneath the creeping carpet of vegetation. Day by day, year by year, century after century, the planet had set about the slow task of healing itself. Now, on the eve of the third millennium, apart from the fading lines of the hardways and the few weathered ruins of collaps stone bridges, little remained of what once had been. Twentieth century America had been buried with the same relentless efficiency as the ancient cities of Sumeria.
Now at four thousand feet above sea level and still climbing, Cadillac led the way along the river trail. Malone, after privately consulting his well-concealed miniature map (printed on a square of silk supplied to AMEXICO by Ieyasu's contact men) and a tiny electronic device that took bearings from navigation beacons and converted them into map co-ordinates, found that the nearest navref point was a township called Torrington, on the old US Highway 26.
For Cadillac, visual confirmation that they were back in his home territory came in the shape of the Laramie Mountains whose densely forested slopes were home to a variety of conifers including the towering ponderosa pine. Rising like the sloping ramparts of an overgrown fort to a high point of ten thousand feet, the Laramie range forced the river to run around its northern flank like a moat, passing through navref Caspar before turning southwards and snaking uphill towards its source among the snow-melt streams of the Rockies.
It was on the plain beyond the western flank of the Laramie range that the M'Calls had made their first costly attack upon The Lady, and when Cadillac reached the shrunken lake where the river had been dammed to form the Glendo Reservoir, he turned left and led Steve and Malone's band of renegades towards a trail that wound up and over the pine-covered slopes.
Halfway between the river and the hills they came across a M'Call turf-marker – a tall pole adorned with carved plaques of wood, bones and eagle's wing feathers. A mile further on, Cadillac halted the column as a large posse of Mute warriors with the same golden feathers attached to their leather helmets emerged from the trees and formed a line across their path some two hundred yards ahead of the lead horses. Some carried the revolving-drum rifles given to the clan by the Iron Masters under the deal struck between Mr Snow and Lord Yama-Shita, the remainder were armed with cross-bows.
Malone eased his own rifle from his shoulder and laid it across his lap, a finger on the trigger. 'Friends of yours?'
'Yes.' Cadillac had already made preparations for this encounter. Pulling the elegantly curved Iron Master bow from the quiver attached to the left side of his saddle, he notched an arrow to the draw-string and presented its spear-shaped metal point to Malone. A small bundle of dried red leaves was tied to the shaft immediately behind it.
Malone applied a battery-powered hot-wire to the tips of several leaves. They started to smoulder, emitting a dense white smoke. Pinching them together, Cadillac blew on them until the whole bundle was alight then aimed his left arm at the sky and sent the arrow soaring high into the air.
The slender arch of smoke formed as the arrow fell to earth behind the line of warriors was the sign used by the Plainfolk when they wished to parley with an opposing group. The warriors responded by raising a clenched fist which they then opened to display the palm of their hand. The invitation had been accepted.
Cadillac dismounted, planted his green flag in the ground and strode forward as a group of Mutes in the centre of the line ran to meet him. Their leader was Purple-Rain, one of the Bears who had come to the aid of Clearwater and Cadillac after the latter's duel with the fearsome Shakatak D'Vine. Cadillac embraced him warmly then grasped the outstretched hand of his excited companions. When the first flurry of greetings had been exchanged, Cadillac invited Steve and Malone to join him.
Introducing them, he said: 'You already know Cloud-Warrior. It was he who rescued me from the Iron Masters. And this is Malone, chief of the redskins who took us under their wing and journeyed with us across the great plains. They have the bodies of sand-burrowers but their hearts are with the Plainfolk and they have but one wish – to fight side by side with us against the iron-snakes!'
'Heyy-yahh!' chorused the warriors, brandishing their rifles and crossbows in the air. As the rest of the posse came forward to join them Steve encountered several familiar faces. Most belonged to warriors who had attended his quarter-staff classes or had been part of the delegation that had travelled to the trading post and they greeted him with equal warmth.
The uninhibited way in which Steve responded to the bear-like hugs and ritual hand-slaps turned Malone's stomach. He could not abide physical contact with Mutes but he could not afford to be the odd man out. Forcing a smile onto his face, he shook hands with each member of the posse, disguising his desire to murder every single one of them behind a convincing display of camaraderie.
Purple-Rain ran his eyes over the assembled horsemen and foot-soldiers. He had never seen horses before but he had something far more important on his mind. 'Where is Clearwater?'
Cadillac's smile vanished. 'She was struck down by the sand-burrowers.' His words drew a mournful groan from the posse. 'But she is not dead!' he cried. 'She lies not far from here in the belly of an iron-snake and it is from there we must rescue her!'
Purple-Rain winced at the prospect of storming a wagon-train. 'Can she not use the gift of earth magic to free herself?'
'No. She is too weak. The power will not return until her wounds are healed. Attacking the iron-snake will be a daunting task but,' – he indicated Malone and the renegades with a sweeping gesture – 'we are not alone. Our friends here know its secrets and are willing to fight by our side. The M'Calls proved their bravery in the Battle of the Now and Then River but raw courage is not enough. This time we will use cunning and stealth. The power of the Old One allied to the secret knowledge of these redskins will free Clearwater and bring the sand-burrowers to their knees!'
'But the Old One is not here,' replied Purple-Rain.
Cadillac blanched. 'Not here?!'
'No. He has gone to the trading post.'
Steve was equally appalled by the news, and he could see that Malone was not too happy either. 'When did he leave?'
'Three days ago.'
_Shit, shit and triple shit_... This was a major setback. Mr Snow's presence was absolutely vital. Roz's newly-revealed powers were frightening but they were virtually untried: an unknown quantity. No one could dispute Mr Snow's powers as a summoner. He was the great equalizer; the spearhead of their attack and their first line of defence, the only sure means they had to turn the tables on Malone and the forces who would be lying in wait for them aboard Red River. And they had missed him by three days! What a pill!
The trade delegation would not return for at least five weeks. Steve had asked for and obtained six weeks in which to set up the attack on Red River. They had already used up eight days of that on the present journey to Wyoming. If the delegation was not delayed it now meant that Mr Snow would return just as the present deadline expired – leaving him no time to come up with a game-plan in which his magic would be the trump card!
They needed more elbow room. He would have to ask Malone to arrange an extension. But that was not as simple as it sounded. It was to the Federation's advantage to keep the pressure on the M'Calls – forcing them to attack the train without adequate preparation within a time-frame and in a specific location which was not of their own choosing. But how could he delay the inevitable confrontation without making Mother suspicious...?
Cadillac turned to Steve and Malone, his new-found confidence waning visibly. 'What are we going to do?'
_Good question_...
'I'll tell you what we ain't gonna do,' growled Malone. 'An' that's attack this goddam wagon-train of yours without a summoner up front. You sold me on this magic shit and that's the deal. If the old man don't show, we don't go!'
'He'll be back in five weeks!' pleaded Cadillac. 'Four and a half!'
'Good. Give us a call then an' we'll think about it. Adios, amigo!'
Cadillac caught hold of Malone's arm as the renegade turned towards his horse. 'Wait –'
Malone halted and broke Cadillac's grip with a knuckle-crunching squeeze. 'Don't ever lay hands on me, good buddy. Next time I'll break your fuckin' arm.' The threat was delivered with a smile for the benefit of the watching warriors.
Steve intervened. 'Hey! C'mon you guys! Cool it. I know how we can get out of this bind. I'll ride after him – bring him back.'
'It's too late!' said Cadillac. 'Didn't you hear what he said? The delegation left three days ago! They'll be at the trading post by the time you catch up with them!'
'I can try, can't I? Anyway, so what?'
'Isn't it obvious? They're not going to drop everything and come back here after going all that way. They'll stay there and trade. And it's no good thinking you're going to persuade the Old One to get on the back of the horse, because you won't!'
'So what is it you're trying to tell me?'
'It's no good going after them because you won't bring them back any sooner.'
'You underestimate me, Caddy. Especially my powers of persuasion. We've got to try. Can't you see that? Okay, maybe I won't be able to bring them back in time but someone has to go – to warn them.'
'About what?'
'About the Yama-Shita! Have you forgotten what happened on the wheel-boat? Those japs know that you and I were part of the group that destroyed the Heron Pool and that we're linked to whoever blew that boatload of samurai out of the water!'
Cadillac hesitated, unwilling to come to the inevitable conclusion. 'The commanders of the boat knew but –'
'Supposing they sent word to Sara-Kusa by carrier-pigeon?'
'You mean before the boat went down...'
'There was plenty of time. We were locked up for several hours.'
'So if they did...'
'The family will have linked you and Clearwater to the M'Calls. And Mr Snow, the man who set up the deal with Lord Yama-Shita to send a cloud-warrior to Ne-Issan is on his way to the trading post with five hundred M'Calls. If I were a member of the Yama-Shita family I'd regard this as a golden opportunity to get even – wouldn't you?'
Cadillac paled even further. 'Sweet Sky-Mother. It never occurred to me –'
'That's because you had other more important things to think about,' said Steve diplomatically. 'Truth is, we've both been wrong-footed. I was hoping we'd get back in time.' He shrugged. 'Never mind. We're still in with a chance.' He eyed Malone. 'I'd better get moving...'
'You're going now?!' cried Cadillac.
Steve mounted his horse. 'No point in wasting any time.'
'But what about food?'
'Caddy! I wasn't planning to go totally empty-handed. Can you rustle up some flat-bread and a pouchful of meat-twists?'
'I'll do better than that. I'll come with you!'
'Uh-uh. No way. You and our friend here have got some urgent business to attend to. You'll be able to see things more clearly if I get out of your hair for a while.
'All right. But you must have an escort. A hand of warriors, at least.'
'Caddy! I'm going on horseback! They'll never be able to keep up with me!'
Cadillac's confidence curve made an upswing. 'You might be able to outrun them in the first two or three hours, but at the end of the day, a Mute warrior will be still be running when your mount is on its knees!'
Steve capitulated. 'Okay. Go on ahead with Purple-Rain and get things organized. I'll wait here with Malone.'
'Don't you want to come into the settlement?'
'And get caught up in the celebrations? It'd take a week to get out of there!'
An hour later, Cadillac returned with six M'Call Bears led by a warrior called Cat-Ballou and a She-Wolf, the ever hopeful Night-Fever. They were all volunteers who realised the importance of their mission but apart from Night-Fever – who clearly had a more private celebration in mind – Steve could see they were somewhat cheesed off at having to miss out on the party which, apparently, had already got underway.
Having profited from Cadillac's absence to square things with Malone, Steve took the Mute aside for a final word while his escort loaded the provisions they had brought onto the two pack-horses.
'Take care – and watch those breakers, huh?'
'I will...'
'And don't do anything rash while I'm gone.'
'I don't intend to. But let's get one thing clear, Brickman. These are my clanfolk, this is my land, attacking the wagon-train is _my_ idea, and from now on, _I'm_ in charge.'
Steve gathered the reins of his horse and climbed into the saddle. 'Caddy, I wouldn't have it any other way...'
## CHAPTER FIVE
Du-aruta, the Iron Master's name for the trading-post, was a derivation of Duluth, Minnesota, the pre-H port situated at the western end of Lake Superior. But this was, in fact, something of a misnomer. As the accompanying maps show, Duluth had been built on the northern side of the Lake whereas the trading post had been planted on the opposite shore near its vanished trading partner, the port of Superior, Wisconsin.
In the days when they were both thriving transit points for the Great Lakes freight trade, Duluth and Superior were separated by the St Louis River which snaked down from northern Minnesota then turned east into a meandering estuary whose southern bank was eaten away by inlets and bays. A triangular chunk of land – on which the town of Superior stood – pushed the estuary in a north-easterly direction, but on rounding the point it made a sharp right hand turn into a long narrow lagoon bisected by the state line.
The lagoon itself was separated from Lake Superior by two low needle-like spits of sand and gravel which reached out from the opposing shores to form an almost unbroken line running from the north-west to the southeast.
In pre-Holocaust days the upper and lower sand-bars were separated by a three-hundred foot wide shipping channel kept open by dredgers. With the passage of time, the channel had silted up, narrowing to half its original width, and was now only thigh deep. The land to the south and west of the trading post consisted of rolling plains sloping gently upwards away from the lake, but on the Duluth side, the estuary sand-bar and the lake beyond were dominated by six-hundred foot high bluffs that rose steeply from the narrow shore.
The first thing that struck Mr Snow on reaching the trading post was the size of the D'Troit and C'Natti encampments. Traditionally, the delegations from the six bloodlines were camped around the outside of a huge octagon marked out by a line of stones. The She-Kargo and D'Troit – who were both allocated two segments – faced each other across the central reservation, flanked on each side by the supposedly neutral, lesser bloodlines – the M'Waukee, C'Natti, San'Paul and San'Louis. This arrangement was designed to minimize the violent confrontations which, despite the general truce and the restraining presence of marshals and capos, always flared up as groups of young bloods from both sides prowled around the outer edge of the vast encampment looking for trouble.
This year, the dispositions of the various groups was the same, but not only were the individual D'Troit and C'Natti delegations much larger than usual, there were a great number of turf-markers belonging to Clans who had never been represented before.
Discreet enquiries through intermediaries elicited the reason: the Iron Masters, because of some internal upheaval, had withdrawn the boats which normally called at Bei-Sita, the second trading post serving the Mute clans inhabiting the plains close to the Eastern Lands – the pre-Holocaust states of Ohio, Indiana, and the broad peninsular bordered by Lake Erie, Huron and Michigan – the original home of the D'Troit.
As a result of this temporary closure, the delegations had made their way to Du-aruta. No one wanted to miss the once-yearly opportunity to exchange skins, furs, dream-cap and rainbow-grass for new knives, crossbows, tools, woven cloth and utensils. And, of course, there was always the hope that more clans would be able to obtain examples of the rifles supplied to the clan M'Call the year before.
To understand what went through Mr Snow's mind it is necessary to explain that mathematics was a branch of learning the Plainfolk had little use for, especially when the sums embraced numbers larger than twelve – the number of fingers and thumbs possessed by the majority of Mutes. It had always been accepted that the She-Kargo were numerically superior to the D'Troit but prior to this fateful gathering the rival factions had never carried out a head-count of their supporters. This also meant they had no clear idea of the number of warriors their opponents could muster. On this occasion however, the D'Troit had received some outside help.
**THE TRADING POST / DU-ARUTA 2991 A.D.**
Over the years of trading with the Mutes, the Yama-Shita family had amassed a great deal of information about the various bloodlines and the multiplicity of clans grouped within. And with the Iron Master's passion for paperwork, everything had been duly recorded in great detail by a battery of scribes.
In some respects, such as the size and breakdown of each clan, they probably knew more about the Plainfolk than the Plainfolk themselves, and like any military-minded organization gathering intelligence about a potential enemy – or client-state – they had even identified the distinctive headgear and turf-markers that placed a clan within a particular bloodline but set it apart from its neighbours.
As part of the larger picture the wheel-boat captains knew the relative size of the various bloodlines and the numbers they could expect to find waiting for them at the trading post. The She-Kargo bloodline contained 242 clans, the M'Waukee 103 and the San'Paul 38. Each sent an average of 150 delegates – making a grand total for the She-Kargo faction of some fifty-seven thousand five hundred delegates – adult males and females from 15 to 55, all fit and able to fight.
Under normal circumstances the D'Troit, C'Natti and San'Louis would have fielded, collectively, some fifty-two thousand delegates giving the She-Kargo faction the numerical edge. But with the closing of the Bei-Sita trading post, the D'Troit faction had been swollen by another 92 delegations from the clans whose turf lay to the east of Lake Michigan. The total number of C'Natti and San'Louis delegations had also increased for the same reason.
Added together, this should have produced a grand total of some seventy-six thousand warriors – large enough to confer a comfortable margin of superiority. But each clan had sent an above average number of delegates. Reinforced by the unexpectedly large numbers of 'journey-men', the D'Troit, C'Natti and San'Louis had fielded a staggering one hundred and sixty-three thousand warriors – giving them an advantage of almost three to one. With that number of people milling about the camp site, it was hardly surprising that the She-Kargo faction thought the odds were even greater.
Since they had no independent means of checking out the state of play at the Bei-Sita trading-post, the She-Kargo were obliged to accept the explanation they were offered. It seemed plausible enough but it did not justify the inflated numbers of journey-men which the D'Troit and C'Natti had brought with them to trade for goods and weapons. Some surreptitious head-counting by the same intrepid intermediaries established that some D'Troit clans known to be only half the size of the M'Calls were proposing to exchange over one hundred men and women whereas the M'Calls themselves had never sent more than fifty down the river in any one year.
The situation was unique and potentially explosive. A hastily convened meeting of elders from the She-Kargo delegations could only envisage two possible explanations: one – by offering so many 'guest-workers', the D'Troit and C'Natti hoped to elbow their rivals aside and grab the lion's share of whatever the Iron Masters had come to trade or, two – the army of journey-men with their yellow headbands were _not_ destined to sail away across the Great River _but were here for some other purpose_. Either way it spelt trouble.
The M'Call delegation, some seven hundred miles from their settlement, were too far away to send for reinforcements but some of the other She-Kargo and M'Waukee delegations whose homes were within a day's run promptly despatched messengers to summon reinforcements.
All this was done in great secrecy. Mr Snow and the other wordsmiths had decided that there was to be no provocation and no outward show of suspicion. The explanation furnished by the D'Troit wordsmiths had to be taken at face value. By their own sacred tradition, wordsmiths – even from opposing bloodlines – never lied to one another. If they had not betrayed their oath, then the Iron Masters must have closed down Bei-Sita. The question everyone in the She-Kargo camp was asking was – why?
A hint as to what the answer might be came with the arrival of Carnegie-Hall, wordsmith of the Clan Kojak from the bloodline of the M'Waukee. Entering the bullring where the other wordsmiths habitually gathered to exchange news and gossip, he sought out Mr Snow and under cover of the formal greetings exchanged on such occasions, passed over a whispered request for a meeting when darkness fell.
Several hours later when a thousand camp-fires pierced the darkness like orange blossoms scattered on black velvet, Carnegie-Hall, accompanied by five Kojak warriors, was led by a M'Call guide into the small wood west of the camp-site where Mr Snow and his own body-guard stood waiting.
The two wordsmiths sat down on talking mats, their faces lit by the solitary flame of a fire-stone which lay between them. Its glow which sharp eyes could have seen from the camp-site was masked by the cloak Mr Snow had thrown over a nearby bush and the dense undergrowth beyond. As a person, Mr Snow would not have given Carnegie-Hall the time of day, but as a fellow wordsmith he had to be treated with the courtesy traditionally accorded to all practitioners of the ancient art.
'What say you, brother? Do you bring me good tidings or bad?'
'I bring news of great happenings. It is for you to judge whether they are good or ill. But first let us speak of The Chosen. Did your clansmen return safely and in good spirits?'
Mr Snow was familiar with the term 'The Chosen', but for some reason, didn't catch on immediately. 'Clansmen...?'
'Cadillac, Clearwater and Cloud-Warrior.'
Mr Snow's present anxieties vanished under a great surge of elation. 'They escaped from the Eastern Lands?!'
'Escaped and more! They are The Chosen – the first of the Lost Ones whose return heralds the coming of Talisman!'
'It is true that these three were born in the shadow of Talisman. By what token do you know them as The Chosen?'
'The words were born on my lips!' exclaimed Carnegie. 'The Thrice-Gifted One appointed me to be the first to recognize them and name them! And when the history of the Plainfolk is retold in the ages to come, the Kojak will be remembered as the clan that first gave them shelter, and whose valiant warriors played a decisive part in the victories achieved by their mighty powers!'
'Victories...?'
'Over the arrowheads, the iron-snake and the wheel-boat!'
'Sounds like they've been busy,' grumped Mr Snow. 'Tell me more.'
Carnegie-Hall gave him the whole story including – to his credit – the treacherous deal struck with Izo Wantanabe which, as events unfolded, had reinforced his belief that his steps had been guided by Talisman. And as might be expected he laid great emphasis on the part the Kojak had played in the destruction of the wheel-boat.
As Mr Snow sat listening to Carnegie's graphic description of how the Kojak had massacred the horse-borne samurai, red-stripe infantry and sailors who had struggled ashore, his sense of foreboding increased. All this had happened weeks ago. Since when, Cadillac, Clearwater and Brickman had departed in triumph, with a pile of booty and more than a hundred head of horses – the four-legged beasts which the dead-faces had tamed and learned to ride but which, up to that moment, Mr Snow had never seen.
The trio had last been reported heading westwards towards Nebraska. The southern route! Mr Snow silently cursed the Sky Voices for sending him in the wrong direction. No doubt, by the immutable perverseness of Sod's Law, his protégés would – barring some mishap – have arrived at the settlement within a day or two of his departure!
Never mind. Carnegie-Hall's story had amply confirmed Mr Snow's belief that his two young charges and the cloud-warrior were destined to achieve greatness. They were indeed The Chosen, recognized and hailed as such not just by the Kojak, but by the clans they had encountered in crossing the Central Plains. Whatever misfortune might befall them, they would survive and grow ever stronger, for the power of the Thrice-Gifted One was upon them. An invisible force which, if not an impervious shield, would preserve and heal their earth bodies and the spirit within.
Once again Mr Snow regretted that he would not live to see the saviour of the Plainfolk revealed. But he now understood why the Sky Voices had directed him towards the trading post. It was here the immediate danger lay, and it was here that his gift of power and his courage would be sorely tested.
_Perhaps to the limit – and beyond_...
'What you have said explains a great deal.' Producing a pipe charged with rainbow grass, Mr Snow lit it, taking a soothing puff before offering it to his visitor. 'Old Golden Nose is not going to let such a catastrophic reverse pass unavenged.'
Old Golden Nose was a nickname derived from the elaborate black and gold mask which Domain-Lord Hirohito Yama-Shita wore whenever he appeared at the trading post. Depending on their rank, all Iron-Masters wore masks of one sort or another when dealing with the Mutes – a practice which had given birth to the generic term 'dead-faces'.
'Lord Yama-Shita is dead,' announced Carnegie-Hall.
The news took Mr Snow's breath away. 'How do you know this?'
'Cadillac told me. Before escaping from the Eastern Lands, your clansmen fought a mighty battle with the dead-faces.'
Mr Snow listened with a mixture of pride and dismay as the Kojak wordsmith repeated Cadillac's spell-binding story of death and destruction at the Heron Pool including, in all its gory detail, the moment when Clearwater had compelled the domain-lord to kill himself several times over.
The loss of the wheel-boat with all hands, coming hard on the heels of the mega-debacle that Carnegie-Hall had just described merely added insult to injury. Lord Yama-Shita might be dead – and that removed one formidable adversary from the field – but his successors would be honour bound to strike a devastating blow in return.
The first opportunity to do so would be when the wheel-boats ran their noses aground on the beach by the trading post. It was going to be a strange feeling, watching the vessels appear over the dawn horizon and knowing that this time, as they grew larger and larger and the dread sound of their engines reached the ears of the waiting Mutes, there would be no obsequious welcoming ceremony, full of false smiles and bogus cameraderie. This time, the rising sun would mark the beginning of a countdown that would end in an explosive confrontation; an orgy of blood-letting whose limits could not be foreseen and whose consequences were incalculable.
After a long moment of reflection Mr Snow said: 'I think I can see how this is going to play. The Yama-Shita won't attack us directly. To do so would jeopardize their whole trading operation. That's why the D'Troit and C'Natti are here in such large numbers. The family is going to use them to put the knife in.'
Mr Snow smiled as he took back the pipe and inhaled some more smoke. 'You're probably on the hit-list too. I'm surprised you came.'
Carnegie-Hall bristled. 'You dare to call the Kojak cowards – after all we have done?!'
'Calm down, Carney. No one's calling you anything. The M'Calls have never backed out of a fight, but if I'd known what we were walking into, I'd have been severely tempted to stay at home.'
'We came because we thought The Chosen would be here!' exclaimed Carnegie-Hall. 'We have seen their power! With them at our side we have nothing to fear. The dead-faces are powerless against them!' Then, with engaging candour, he added: 'Had I known they weren't going to show, we might have had second thoughts too. But where could we go?'
'It's a big country,' replied Mr Snow. 'But if someone's determined to find you, there's no place you can hide. If you have to make a stand, you might as well make it here – amongst your own kind.' He paused and appraised his visitor. 'The M'Calls can count on their blood-brothers. Can the She-Kargo count on the M'Waukee?'
Carnegie-Hall shifted uneasily. 'At this moment I cannot say.'
'You trying to tell me you've found a way to get your head off the block?'
'No! It's just that –'
'– by standing aside, you hope to save your own skins.'
'Never!' cried Carnegie-Hall. 'The M'Calls may be the paramount clan of the She-Kargo. but it is the Kojak who have proved their worth in battle with the dead-faces!'
'With a little help...'
'How generous of you!' sneered Carnegie-Hall. 'It takes little courage to face the enemy when you know you are protected by the mantle of Talisman! Your clanfolk emerged unscathed. Mine paid for their triumph with their own blood! How dare you impugn the honour of the Kojak! This conversation is at an end!'
Mr Snow laid a restraining hand on his visitor's knee as he moved to get up. 'Nice try, Carney. You always were good at the old huff'n puff. But don't give me this honour nonsense. Everyone knows you as a man with his eye on the main chance. You've admitted as much yourself. You were prepared to sell my people down the river for a boxful of geegaws!'
Carnegie-Hall dropped the self-righteous anger and adopted the air of a honourable man who has been sorely wronged. 'That was before Talisman revealed his purpose to me! Yes, it is true that when he put his words onto my tongue my heart was full of treachery, but all that changed when he filled my mind with his presence! It is easy for you to look down on us from your safe haven in the far mountains. We live on the front line! The dead-faces now have men and boats on the far side of the waters which were once our own! You seek our help now – what help can we expect from you once you have journeyed beyond the Black Hills?!'
'Not much, I grant you. That's why we have to stand together now. If there _is_ a fight, we have to win it. Do you want to live under the heel of the D'Troit?'
'It is a fate many of the M'Waukee already endure.'
'Then now is your chance to get out from under. We can count on the San'Paul. You are the only people who can talk to the San'Louis.'
It was obvious that Carnegie-Hall did not relish the prospect of getting embroiled in a potentially fatal confrontation with the D'Troit. 'It will not be easy,' he muttered. 'Is there no hope that The Chosen will get here in time?'
Mr Snow drew down some more smoke. 'Carney, I could make the right kind of noises in order to string you along but I'm not going to. The answer is – I just don't know but from what my gut is telling me, I'd say the chances are virtually nil. We're going to have to manufacture our own miracles. The only consolation I can offer is the news that Clearwater, the young lady whose performance has so impressed you, is a pupil of mine. She's good – but I'm better.'
'The D'Troit have summoners too.'
'The D'Troit?! Don't make me laugh! Their best man can't even move a pile of buffalo shit unless he has a shovel!' Mr Snow waved the threat away. 'Go back to your lines and talk to your blood-brothers. Contact the leader of each delegation and persuade them to come to a meeting with their paramount warrior plus every available wordsmith and summoner. I want them here within two notches for a meeting with their opposite numbers from the She-Kargo.'
A notch was a standard measurement of time marked on candles made from animal fat and represented one pre-Holocaust hour. The system had originated with the Iron Masters. Having obtained some of these candles at the trading post, the Mutes had made copies of their own. In their normal daily lives, Mutes were not clockwatchers; the candles were only used in situations which were time-critical – such as the present gathering at the trading post.
'Two notches! Sweet Mother, that doesn't give us much time.'
'We're as short of time as we are of people. That's why we've gotta move – fast!'
As they both got to their feet, Carnegie asked: 'What about the San' Louis.'
'Sound them out. See which way the wind's blowing but don't tell them about the meeting.'
Carnegie-Hall nodded, his eyes full of doubt. 'I hope we're doing the right thing.'
'Trust me,' said Mr Snow. He gave Carnegie's arm a reassuring squeeze. 'I too am guided by Talisman. This is what he sent me here for. To preserve the freedom of the Plainfolk.'
Between dawn and nine o'clock on the following day, the last of the trade delegations arrived and set up camp in the area allotted to their own bloodline. Most of them were new faces belonging to the D'Troit. Their appearance caused the heavily-outnumbered She-Kargo and M'Waukee to feel both beleaguered and belligerent and by mid-morning, when all the wordsmiths had gathered in the bull-ring for the opening round of their annual talk-fest the atmosphere had become electric. Everyone was filled with that oppressive sense of foreboding you get when a violent storm is about to break.
In order to appreciate what follows it is necessary to explain that while most but not all clans possessed wordsmiths, very few of these gifted individuals were summoners. Mutes born with two gifts – such as Mr Snow (wordsmith and summoner) and Cadillac (wordsmith and seer) – were extremely rare. And summoners as powerful as Mr Snow and Clearwater were rarer still.
The same degrees of professional competence applied to wordsmiths. The M'Calls had enjoyed the benefit of an unbroken line of wordsmiths stretching back through successive generations to the War of a Thousand Suns. This distinction, shared by only a handful of clans, enhanced Mr Snow's standing but also engendered a great deal of envy.
In many respects, wordsmiths resembled any group of Pre-H professionals. Like lawyers, they came in all shapes and sizes and while they all possessed what was once known as 'the gift of the gab' their performance – as 20th century clients often discovered to their cost – spanned the ratings from the peaks of excellence to the troughs of incompetence.
Like the plethora of personal computers of the pre-H era that came fitted with differing amounts of hard-disc storage, some wordsmiths' memories were better than others. Compared to the ordinary Mute armed with the equivalent of 256K RAM, your average wordsmith with a hundred gigabytes on line was a mental giant – a walking reference library.
However, as with all people-based systems, there was one drawback. Memory is not a product or function of character: the most able brains do not necessarily reside inside the heads of charismatic human beings, or even just plain nice ones. The greatest data-bank in the world is not worth a row of beans if there's a dildo manning the front desk. But when a few zillion gigabytes of memory was allied to an exceedingly acute mind – as in the case of Mr Snow – the result was an outstanding personality whose opinions and influence extended far beyond the narrow confines of their own clan or even their own bloodline.
But while Mr Snow might be regarded as the star of the She-Kargo, there were other noteworthy contenders among the D'Troit and C'Natti ready to pit their wits against his. And they were present on that fateful morning.
Last year, Mr Snow, by virtue of his age and experience, had been selected to chair the proceedings; this year, his place had been awarded to a senior wordsmith from the rival faction – Prime-Cut, leader of the Clan R'Nato from the bloodline of the D'Troit.
Because of Cadillac's triumphal progress through Illinois, Iowa and eastern Nebraska – where the trail had run cold – the hottest topic among the wordsmiths from those areas was the appearance of The Chosen, the trio from the Clan M'Call. The news of their exploits and their escape from the Eastern Lands was not confined to the bull-ring. Exaggerated reports of their prowess passed on by those who had witnessed their travelling road-show were now spreading like a bush-fire through the whole encampment.
To the wordsmiths from the various delegations it was both a source of satisfaction and alarm. Apart from Carnegie-Hall and Mr Snow, none of those present were aware of any prophecy which bore a reference to these individuals and no seer had found their image in the stones. The prediction which came closest referred to the return of The Lost Ones – the generations of Mutes who had been taken away by the dead-faces to the Fire-Pits of Beth-Lem and their off-spring, the Iron-Feet – born into slavery in the Eastern lands.
Invited to take centre-stage, Carnegie-Hall launched into a colourful explanation of how he had met Cadillac, Clearwater and Cloud-Warrior – carefully omitting the details of his treacherous deal with Izo Wantanabe. Guided by Talisman he had despatched a posse of Kojak warriors through the winter snows to a rendezvous with the select band of individuals who were to herald the coming of the Thrice-Gifted One.
A graphic account of their powers – which Clearwater had used to destroy four arrowheads with one wave of her hand and which, later, had been employed to rip the belly out of an iron-snake – made a deep impression on his attentive audience. When he had finished, he appealed to Mr Snow. Had the Sky Voices not told him that Cadillac, Clearwater and Cloud-Warrior were born in the shadow of Talisman?
Rising to his feet, Mr Snow agreed that this was so.
His affirmation of Carnegie's claim sparked off a heated argument. Individual wordsmiths from the D'Troit and C'Natti camps leapt to their feet to protest. Out of all the Plainfolk why should three warriors from the Clan M'Call have been chosen to herald the coming of Talisman? It was just another ploy by the She-Kargo and their lackeys amongst the M'Waukee to further inflate their already exaggerated importance.
No wordsmith, claimed the protesters, had ever spoken of The Chosen in this bull-ring until Carnegie-Hall had coined the phrase. It was all a put-up job; something concocted on the flimsiest of evidence by the Kojak and the M'Calls. At best it was a well-intentioned mis-reading of the events in question; at worst, a total fabrication.
Carnegie-Hall vigorously defended himself but the carefully orchestrated outburst had achieved its aim, splitting the gathering into three camps: the pro-She-Kargo faction who supported the proposition, the pro-D'Troit faction who rejected it, and the uncommitted who were waiting to see who was going to get the better of the argument.
Mean-Machine, a C'Natti wordsmith, made his voice heard amid the hubbub and threw down a challenge. If Carnegie-Hall and those who supported him were speaking the truth where was this mysterious trio? Why had the Clan M'Call not brought them to the trading post where they could display their powers and spread their message to the assembled representatives of the Plainfolk?
Good point, thought Mr Snow. He cursed himself for not having a ready answer.
Carnegie-Hall leapt up angrily and confronted Mean-Machine, his menacing bulk towering over his smaller opponent. 'Do you dare to call me a liar?!'
Prime-Cut – an equally imposing figure – rose and stepped off the low mound which the chairman traditionally occupied. 'No!' he cried. Pushing Carnegie-Hall and Mean-Machine apart, he aimed an arm between them, an accusing finger pointed at Mr Snow. 'There sits the man who has lied to us all!'
His words drew a shocked gasp from the uncommitted and an angry roar from the She-Kargo wordsmiths. Half of them got to their feet, exchanging accusations and abuse with their more aggressive counterparts on the other side of the ring.
Prime-Cut spread his arms in a commanding gesture. 'Cease this noise!' he thundered. 'Sit down and parley in the proper manner or leave this place!'
The uproar subsided as the more vociferous protestors and counter-protestors resumed their seats but the murmuring continued, becoming a sullen underswell of sound.
As Mean-Machine and Carnegie-Hall settled into their places, Mr Snow got to his feet and appealed to his supporters. 'Let peace descend! Let your minds be tranquil. I am the one who stands accused here. My conscience is clear! Let the charges be heard!'
The murmuring gradually subsided under the fierce gaze of the Plainfolk's two most prestigious wordsmiths. When silence had been obtained Mr Snow turned to face Prime-Cut in the centre of the bull-ring. Attracted by the violence of the argument, the open space between the inner and outer rings was now crowded with clan elders and other members of the various delegations.
As the R'Nato wordsmith circled him with a wolfish grin, Mr Snow muttered: 'I hope you know what you're doing.'
'Oh, I do, my friend, I do.' Prime-Cut halted by Mr Snow's shoulder and thrust his mouth close to the Old One's ear. 'You are in deep shit,' he whispered. 'I'm going to bury you!'
'You're not the first man to try and do that.' Mr Snow's voice had a confident ring but he felt cold and sick inside. Already outnumbered on the ground, the She-Kargo was in danger of losing the battle for hearts and minds and he could see no way to reverse the situation.
Drawing back, Prime-Cut pointed a finger at his victim and addressed the ring of wordsmiths in a voice loud enough to carry to the expectant crowd beyond.
'You have heard our brother speak! His name is renowned, his memory legend! Yet even he is bound, as we are, by the wordsmith's oath to forswear all falsehoods, to faithfully chronicle the deeds of the Plainfolk and reveal, in their full majesty, the revelations of the Sky Voices!
'To relay the truth, adorned and embellished by his art but sure and solid as a rock, clear and pure as a mountain stream' – his voice hardened – 'not buried beneath shifting sand, or muddied by deceit! You were witness to his claim that the Sky Voices had told him that Cadillac, Clearwater and Cloud-Warrior were born in the shadow of Talisman. Is that not so?!'
'Aye!!' chorused the wordsmiths.
'Were those your words?' demanded Prime-Cut.
Mr Snow sensed the trap but could see no way out. 'They were.'
Prime-Cut could hardly contain himself. 'You hear?!' he thundered. 'He stands condemned out of his own mouth! The young brave known to the Kojak as Cloud-Warrior, and who our revered brother' – he indicated Mr Snow with an elaborate gesture – 'claims as one of the three M'Calls chosen to lead the Plainfolk towards nationhood is not a Mute at all!'
The charge provoked roars of anger and cries of disbelief.
'"Cloud-Warrior" is a name as false as the colour of his skin! He is a sand-burrower from the dark cities – known to his masters as Brickman!'
More shouting. Prime-Cut demanded silence and faced Mr Snow. 'How do you answer?!'
Mr Snow eyed his accuser calmly. 'Is that it?'
'No, there is more!'
'Then I'll wait till you've finished.'
Prime-Cut appealed to his audience. 'Evasions! You see how his serpent tongue wriggles to avoid the truth?! Well I shall reveal it! All of you seated here who played host to The Chosen have been cruelly deceived! Cadillac, Clearwater and Cloud-Warrior are agents of the Federation! And they are not alone! Mr Snow – who would have us believe he is our brother – and the entire clan have sold their souls to the sand-burrowers!'
Once again the meeting erupted with cries of protest and condemnation. Charges and counter-charges were hurled back and forth by the opposing camps and the D'Troit and C'Natti wordsmiths set up a strident chant: 'OUT-OUT-OUT-OUT!' The confusion and bitterness spread amongst the spectators outside the bull-ring, leading to angry exchanges and physical violence. Fortunately there were enough line-capos on hand to restrain the D'Troit hot-heads who were clearly out to cause trouble. Mr Snow had foreseen this and following his midnight meeting with the leaders of the She-Kargo and M'Waukee delegations, their clanfolk had been given strict instructions not to succumb to any provocation from the rival camps.
In the midst of this commotion, Mr Snow stood firm. Seemingly oblivious of the jostling mass of bodies crowded around him, he radiated a deceptive calm like the eye of a hurricane.
Prime-Cut's accusations were highly damaging but Mr Snow could not allow himself to be drawn into answering specific accusations. By remaining silent and allowing the torrent of charges to wash over him, he hoped to tempt Prime-Cut into revealing all his cards and – with luck – the R'Nato wordsmith might even end up as the accused instead of the accuser.
But it was hard to resist the cries of 'Answer! Answer!' from his own camp, and it was clear from the anguished expressions on all sides that many of his friends were in despair at his failure to defend himself.
Prime-Cut ran through a devastating list of questions to which Mr Snow had no answer. Had the sand-burrower not descended from the skies? Did the clan not harbour him in their midst and treat him as one of their own – to the point of even giving him a bedmate? Did they not release him? Had he not returned the following year on a new mission for his masters? Had Mr Snow not brought him to the trading post where he had stolen aboard a wheel-boat to join his two accomplices in Ne-Issan?
And once there, had they not wreaked bloody havoc at a place called the Heron Pool – slaughtering the true friends of the Plainfolk? The Iron Masters who furnished them with weapons and the necessities of existence? But even that was not enough! These ingrates had murdered Domain-Lord Yama-Shita, Captain of all the wheel-boats and master of the Great River! The visionary who over the past years had sought to deepen the links and friendship between Mute and Iron Master.
It was a long time since Mr Snow had heard such sycophantic rubbish but it was clear that Prime-Cut's share of the audience were swallowing it whole. And they were cheering him on!
The calumny reached its climax. The M'Calls had betrayed their brothers, but that was only to be expected. Such treachery was in the blood of the She-Kargo. No longer able to maintain their superiority by force of arms, they were now seeking to bolster their position by secret deals with the sand-burrowers!
Again there was uproar, each side trying to howl the other down. It was a serious charge and it confirmed Mr Snow's reading of the situation. His knowledge of what had happened in Ne-Issan was limited to what Cadillac had told Carnegie-Hall. But at Mr Snow's request, Carnegie-Hall – when addressing the wordsmiths in the bull-ring – had not mentioned the battle at the Heron Pool or the death of Lord Yama-Shita.
Prime-Cut could have only gotten this information from the Iron Masters. The D'Troit and their allies the C'Natti were acting as mouthpieces for the Yama-Shita, but how deep did their involvement go – and how long had they been getting their act together? Long enough to put the She-Kargo on the spot. The degree of coordination between the D'Troit and She-Kargo and the inflated size of their delegations was proof of that. The Iron Masters intended to take their revenge here, at the trading post. Tomorrow. And they were using Prime-Cut to set the stage with his accusations.
It was time to begin the fight-back. He had to defend himself, his clan and the good name of the She-Kargo. And he had to do so publicly in a way that was effective but did not provoke an immediate and violent response. The battle, if there was to be one, had to be on a ground of his own choosing.
By putting him in the dock and trying to make scapegoats out of the M'Calls, Prime-Cut seemed to be trying to isolate the clan from the rest of the Plainfolk. Having achieved that, it would not be too difficult – in view of the enormity of their crimes – to persuade the gathering to hand the M'Call delegation over as a sacrificial offering to appease the Iron Masters. But it hadn't worked. The vociferous support he had received from the other She-Kargo wordsmiths was proof that the M'Calls had not been abandoned. Which was, Mr Snow realized, just what Prime-Cut intended. The blow, when struck, would be aimed at the entire She-Kargo faction. The hand of the D'Troit would be on the knife but they would be acting for the dead-faces. And by this unparalleled act of treachery they hoped to fulfil their long-held ambition to become the paramount bloodline. It was ironic. The D'Troit had become the running dogs of the Iron Masters and yet it was the M'Calls who were accused of betraying the Plainfolk.
_But it was not over yet_...
The focus of Mr Snow's attention turn outwards as the noise subsided. Prime-Cut stood in front of him, trembling as he cranked up the required level of indignation. 'How do you answer?!'
Mr Snow chuckled then raised his voice to address the assembled wordsmiths. 'How do I answer?' He turned full circle, arms out-stretched. 'My brothers under the sun, you have heard what passes as the truth fall from the lips of the D'Troit. You have seen their spokesman circle me like a hungry jackal around an ageing bull. Why? That is the question you must ask yourselves – and which I attempt to answer!
'Why does he choose this moment to accuse me of treachery? Why does he attack the honour of the She-Kargo at a time when the Plainfolk have come together in peace and fellowship? Are his words as pure and clear as a mountain stream or do they mask some dark ambition of the D'Troit which they have yet to reveal?'
'Heyy-yaahhh...' The She-Kargo and M'Waukee wordsmiths and their supporters massed beyond the bull-ring voiced a sombre chorus of approval.
'And you must also ask yourself _how_ he knows these things. Who else is aware of the events of which he speaks?'
The question, aimed at those around him, met with no response.
'Reflect on what he has told us. He speaks of Mutes who are not Mutes, of secret journeys through the clouds and across the seas to the Eastern Lands! Of battles between Mute and Iron Masters in which the noble lords of Ne-Issan perished in their hundreds! Of wheel-boats sunk by red-eyed nightbirds!
'Are these inventions of a fevered mind? Dreams inspired by envy of his betters? If they are not, how does he know – in such detail – what took place far beyond the Great River, beyond the Running Red Buffalo Hills?! He does not speak, like you or I who only a short while ago listened to our brother Carnegie but as someone with foreknowledge of these events! He has never _met_ The Chosen yet he speaks of their great battle at the Heron Pool and the death of Domain-Lord Yama-Shita as if he had _been_ there!
'How can he know of such things! There can only be one answer! These words through which he seeks to bring disgrace upon me and the She-Kargo were put into his mouth by the dead-faces – an alien race that would make slaves of us all!'
'Heyy-YAHHH!' This time it was a full-throated cheer. And it came from all sides of the ring.
Mr Snow raised his voice. 'Well, they will not make slaves of the She-Kargo! The weapons they provide are not _given_ to us. They are exchanged for goods we gather through sweat and blood and our most priceless possession – our Clan-brothers and sisters! Think back to when this all began. Have you forgotten how they killed those who refused their offerings?! And yet this man stands before us and says we must show gratitude? For what?! We trade with the dead-faces not by choice but by necessity! But there is one thing we will never trade – our freedom!'
This ringing declaration was greeted by a tumultuous cheer.
Mr Snow pointed to Prime-Cut. 'He charges me with treachery! He tries to tell you that The Chosen are agents of the Federation because they were born to the bloodline of the She-Kargo! What will his hatred and envy lead him to do next?! Deny the power of Talisman?
'The Chosen do not belong to the Clan M'Call or the She-Kargo _or_ the D'Troit! They are of the Plainfolk! The first of the Lost Ones to return from the Eastern Lands – as it was prophesied – to herald the coming of the Thrice-Gifted One! Under his banner we shall become a mighty nation! We shall crush the dark cities and drive the dead-faces back into the sea!
'This should be a time of rejoicing, not anger! Are we not all brothers under the sun?'
The ground shook as ten thousand voices chanted their response. 'Heyy-YAHHH! Heyy-YAHHH! HEYY-YAAHHH!!'
Mr Snow pointed to Prime-Cut. 'Then beware of those who seek to divide us, for it is they who are the _real_ enemy!'
The wordsmiths leapt to their feet to avoid being trampled down by the excited crowd pressing in around them. Those from the D'Troit camp looked sullen and frustrated but everyone else, including many of the C'Natti were cheering and raising their fists in a gesture of solidarity.
Prime-Cut gave it one last try. 'You ask us to believe in The Chosen, but they are not here! Are they frightened to appear before us in case their so-called feats of bravery on behalf of the Plainfolk are revealed for what they _really_ are – criminal acts sanctioned by the sand-burrowers who wish to destroy our friendship with those who give us aid and support?!'
His booming voice cut through the surrounding noise, bringing a sudden hush.
Mr Snow closed his eyes, raised his face briefly to the sky then said the first thing that came into his head. 'The Chosen do not fear the truth! They are not here because they confront an enemy the warriors of the D'troit have yet to face! At this moment, as I speak, they battle against the iron-snakes of the Federation!'
The bull-ring erupted with thunderous cheers. Prime-Cut looked as if he was about to bust a blood-vessel, but it was all over and he knew it. He stepped forward, teeth bared, and came nose to nose with Mr Snow. 'You lying sonofabitch!'
'Easy for you to say,' chuckled Mr Snow. 'But can you prove it?'
Before Prime-Cut could frame a reply, several of the She-Kargo wordsmiths hoisted their hero onto their shoulders and carried him in triumph from the ring.
In the afternoon, with the help of the M'Waukee, who provided them with a suitable disguise, a group claiming to represent an important number of C'Natti trade delegations sought an audience with Mr Snow. Essentially, what they had to say was this: they were greatly concerned that the actions of certain members of the She-Kargo might jeopardize the existing trading arrangements (and they left Mr Snow in no doubt as to who they were referring to) but – and it was an important proviso – they were not prepared to lend their support to 'those elements who were actively contemplating a joint action with the Iron Master against certain of the Plainfolk'.
Having stated their position, the disguised spokesmen proceeded to ask questions. Did the She-Kargo have any plans to resist a surprise attack by a rival faction? If so, they were prepared to help in any way they could.
Once again, they did not name names, but there was no doubt who the C'Natti spokesmen were referring to. Their veiled expressions of solidarity could well have been genuine but Mr Snow could not be sure. They might have been sent by the D'Troit in the hope of discovering what, if anything, the She-Kargo had up their sleeve.
Framing his reply as carefully as he could to avoid spurning what might be a bonafide offer, Mr Snow said: 'There is no plan. We have put our faith in Talisman. Let those who believe in him stand by us. The She-Kargo will never be the first to draw sharp iron against their brothers.
'The last thing we wish to do is divide the Plainfolk, especially now when The Chosen are amongst us. All of us must put an end to our ancient blood-feuds. We must cleanse our hearts and minds, sweep away our petty rivalries and rally to Talisman's bright banner. If you believe that He is our Saviour, strike down those who insult His Name by giving aid to our enemies.'
'Yes, but when and where do you expect all this to happen?' enquired one of the C'Natti wordsmiths.
Mr Snow threw up his hands. 'Who can fathom the workings of poisonous hearts? If betrayers revealed their hand treachery would never flourish as richly as it does today! It is the assassin who chooses the place and the hour, not his victim! Look about you! Danger surrounds us! Go and prepare yourselves! And be vigilant!'
_Nitwits_...
Mr Snow spent the remaining daylight hours in head-to-heads with the leaders of friendly delegations, securing pledges of support and a firm promise to attend a midnight council of war in the depths of the wood which was now ringed by sentinels posted by the She-Kargo. All this should have left the M'Call delegation in an upbeat mood but their earlier exuberance had been dampened by the unprecedented number of clashes between groups of hot-heads from the rival bloodlines.
The use of weapons in these encounters – which by tradition were strictly forbidden – was a sign that the fragile truce governing these occasions was under threat. Despite the efforts of the line-capos and camp-marshals, the ugly brawls continued throughout the day, causing death and injury to both sides.
Faced with a steadily deteriorating situation, a high-level meeting proposed by the M'Waukee and C'Natti brought representatives of the D'Troit and She-Kargo face to face in the bull-ring. But this too failed to ease the tension. By prior agreement, neither Mr Snow nor Prime-Cut was there but it was clear that the D'Troit were still angry that their spokesman had been made to look like a prize asshole and despite the mutual expressions of respect and willingness to reconcile their differences by amicable and reasoned debate the meeting broke up amidst angry recriminations from both sides.
The D'Troit, and to a lesser extent the C'Natti, were clearly spoiling for a fight. And they had imported the muscle to make sure they won it hands down.
The last major conflict in the history of the Plainfolk had been the Battle of the Black Hills when two entire clans – the M'Calls and the B'Nardinos from the D'Troit bloodline had fought each other to a finish in a running encounter that lasted from sunrise to sunset.
Thunderbird, Clearwater's father, had fallen in that battle from which the M'Calls emerged bloodied but unbowed – a victory which confirmed their position as paramount clan of the She-Kargo.
But that was fifteen years ago. There had never been a clash of arms on that scale before or since and, more important still, there had never been _any_ occasion where clans of the same, or competing bloodline had submerged their traditional rivalry to stand shoulder to shoulder against a common enemy. Until now. And having had less than 48 hours in which to cobble together a temporary alliance and hammer out a concerted plan of action, no one in the She-Kargo faction was sure how long it would hold together.
The Mute warrior ethic was similar in many respects to that of the samurai – the military class that ruled Ne-Issan – but there was one important difference. The Mutes were gang-fighters, not battlefield soldiers. Their skills, and the supremacy of the knife were derived from their pre-Holocaust ancestors – the ghetto people who, by some miracle, had emerged indelibly scarred but alive from the nuclear blasts that levelled and torched America's great cities. Desperate, impoverished individuals whose entire lives had been a struggle for survival in the urban jungle. An underclass whose sense of right and wrong had been warped by deprivation and injustice. Whose moral nerve endings had been dulled by the callous exploitation and dog-eat-dog indifference that was the hall-mark of the pre-Holocaust era.
They had survived then by the quickness of their wits, feet and fists, a combination of animal cunning and hair-trigger aggression, a readiness born of desperation to take what they wanted: the very qualities needed to survive the aftermath of a global nuclear war.
Abstract philosophizing, the art of debate, the intellectual flatulence of the educated classes, the privileges of the mega-rich secured by acres of prime real estate and Swiss bank accounts, the well-meaning advocates of charity, compassion and the fellowship of man were buried beneath the smouldering ashes. Literacy went up in smoke as the unschooled burnt the remaining books to keep warm.
It was not the meek who inherited the earth but the traumatized remnants of the bread-line poor, the muggers, pushers, the sewer-rats and hoodlums, along with the Rambo-style, Soldier of Fortune gun-crazy firepower freaks who had prepared for Armageddon in the backwoods of America.
Stranded amid the wreckage of the 20th century, like flotsam and jetsam left high and dry on an alien shore, this residue of humankind had separated into their different ethnic groups, like snails of different species placed together in a cage. At a time when everyone was a potential predator, the only security was within a group sharing a common language, customs and racial origin.
During the next nine centuries and through numberless generations, they had gathered strength and multiplied. Around countless camp-fires they had recreated the past, mixing fact and fiction in the same way that jazz musicians improvise on a well-known melody. Dimming memories of distant events had given birth to a new mythology, a new identity; mutated genes had spawned a new, mishapen but strangely gifted breed of humankind. And when the grey curtain of clouds that brought the Great Ice-Dark retreated, revealing the sun and stars in all their glory, the first of a race of warrior clans emerged – the Southern Mutes and their northern brothers who later became known as the Plainfolk.
Later, when the delegates to the midnight war council had agreed on a joint plan of action and returned to their own lines, Mr Snow toured the M'Call encampment, bringing hope and encouragement to his clanfolk like Shakespeare's Henry V on the eve of Agincourt. His last call was upon Blue-Thunder, Rolling-Stone and Boston-Bruin who sat around one of the many fires with the other leading lights of the M'Call trade delegation.
As he squatted down and warmed his hands, Rolling-Stone threw some more wood onto the glowing embers and stared moodily into the leaping flames. 'So tomorrow's the big day...'
'Yes, when the wheel-boats get here.' Mr Snow's voice was racked and hoarse from countless hours of argument and persuasion.
Blue-Thunder tested the edge of the blade he was sharpening. 'I don't understand it. The D'Troit must know _we_ know what they're up to. Why are they waiting? Why didn't they attack us today?'
'Psychology.'
Blue-Thunder frowned at the unfamiliar word.
'They're trying to unnerve us by letting the pressure build up. Keeping us in suspense. The way the numbers are stacked up they know we are not going to attack _them_. In theory, they can strike when and where they please. But it will be on the beach at dawn tomorrow. That's what I'm counting on.'
'But what makes you so sure?' insisted Blue-Thunder.
'Because it's the dead-faces from the Yama-Shita family who want to get even. They'll use the D'Troit and maybe the C'Natti to make the opening play but they'll be in at the kill. You've seen them at work. Chopping people to pieces is what Iron Masters like doing best. They're not going to come all this way just to watch from the sidelines.'
Doctor-Hook, a M'Call warrior who often acted as a bodyguard to Mr Snow, approached the fire. 'It is time to leave, Old One.'
'Good.' Mr Snow rose to his feet. The others followed. After exchanging farewell handclasps with each of them, he said: 'If any of you have any questions about who's supposed to be doing what come the dawn now's the time to ask. We may not see each other again.'
He ran his eyes around the ring of mishapen firelit faces. No one spoke.
'Good.' He turned to go.
'There is one thing,' said Blue-Thunder. 'That stuff in the bull-ring. Was it true? Did Cadillac, Clearwater and the cloud-warrior kill hundreds of dead-faces like Prime-Cut said?'
'They may have done. According to Carnegie-Hall, Cadillac said they were involved in a big battle in which many died. If it's true then it's something we should be proud of.'
'Yes. But did the sand-burrowers help them win it?'
Mr Snow shrugged. 'Who can say? Talisman moves in mysterious ways.' He drew his dark-hued cloak around his body with a showman's flourish. 'Now! I suggest those of you who form part of the beach party should try to get a little sleep. When you wake up you'll find the weather is to our advantage. Make the most of it because we won't be able to hold it in place for long.' He backed away, his hand raised in a last farewell. 'And for goodness sake, try to look more cheerful! We're going to win!'
The M'Call elders and the other members of the She-Kargo war council who shared their misgivings as the small hours ticked away might have gained some comfort had they known that the leaders of the D'Troit faction were also plagued by doubts and difficulties.
Every D'Troit clan, by the cherished tradition of their bloodline, was a mean bunch of mothers – which meant, inevitably, that they were scornful and suspicious of any attempt to moderate their behaviour. They were governed by only one discipline – violence. They were takers, not makers. They preferred pillage to husbandry. The hunting and killing of meat on the hoof was an acceptable pastime, but why grow breadstalks and green-stuffs when the winter larders of weaker clans could be ransacked at The Gathering?
In a violent world where every male and female of fighting age was expected to carry sharp iron, the D'Troit were the supreme predators, feared, hated and despised by all. In the brief period when peace was supposed to reign at Du-aruta, they were the chief trouble-makers and much of it was caused when they were caught trying to augment their own stock of tradeable items by stealing from the baggage trains of other clans. They came to the trading post as spoilers, and during the rest of the year they cruised the ocean of red grass like blood-crazed killer sharks.
Given their reputation, one might reasonably wonder why they were not the paramount bloodline. Perhaps only Talisman knew the answer to that. By some quirk of fate, the clans of the D'Troit bloodline were less fecund than those of the She-Kargo. In overall terms, they had remained numerically inferior. The ratio of gifted Mutes to the rest of the population was also lower amongst D'Troit clans. There were some eminent wordsmiths but very few summoners, most of whom were only gifted with the first two Rings of Power.
The knowledge that their rivals were more favoured in this respect was a constant source of envy and resentment. Talisman's apparent lack of even-handedness had caused the D'Troit to regard this saviour figure with increasing contempt. The Plainfolk had been waiting nine hundred years – how much longer would they have to wait? To the D'Troit, this endless waiting had become futile and pathetic. It was time for those who could to help themselves.
These festering grievances, the innate capacity for violence and the basic indiscipline which had caused many of the D'Troit to abandon their belief in Talisman had also bedevilled the forward planning of their leaders.
Having primed their warriors for a joint attack on the She-Kargo, the chieftains and elders had come close to fighting amongst themselves as they accused each other of failing to control the hotheads under their personal command. Why – they asked – in the name of the Great Sky-Mother could the mad-dogs they led not understand they were only to attack the She-Kargo _when_ the Iron Masters got there?!
Since they all posed the same question whilst denying that their own delegation was at fault the discussion, as might be expected, soon became overheated.
The D'Troit war council, which included representatives from the C'Natti and San'Louis, had counted on seizing and holding the initiative from the very beginning, but Mr Snow's robust defence in the bull-ring had thrown them off balance. His rallying call for the Plainfolk to unite and his invocation of Talisman had caused many of the C'Natti delegations to waver.
Despite the rumours of a mass defection, they would not switch sides. They were too spineless for that, but they might hold back when the fighting broke out. So be it. When the D'Troit emerged triumphant, as the paramount bloodline with the sole right to trade with the Iron Masters, the C'Natti would come crawling like whipped dogs to lick their feet.
And would be crushed like all the others...
## CHAPTER SIX
Mr Snow's prediction about the weather proved chillingly correct. In the wake of his departure, the air turned cold and damp. A mist began to form. At first, it lay only ankle deep above the ground but within two hours it rose, blotting out the night sky. Sixty minutes later, when the D'Troit, C'Natti and San'Louis rose to make their final surreptitious preparations, the huge encampment, the trading post, and a five mile stretch of the adjacent shoreline was wrapped in a clammy pale grey blanket.
Every year, the clan delegations made their way down to the shore in the pre-dawn twilight to await the moment when the wheel-boats appeared on the horizon, silhouetted against the incandescent disc of the rising sun. To the impressionable unscientific mind of the average Mute the wheel-boats appeared to issue from the sun itself; an impression that the Iron Masters had been at pains to reinforce. This year, the normally well-ordered migration from the lines to the trading post was marked by scenes of unparalleled confusion. The mist was so thick, the D'Troit and their allies were obliged to set up a line of warriors standing at arm's length from each other to find their way to the trading post and the beach below.
When they arrived to take up their alloted positions, they were surprised to discover that the She-Kargo, M'Waukee and San' Paul had already staked their claim to the north-western end of the shoreline and were spread out across the entire width of the lower sand-bar from lake to lagoon. With visibility reduced to three or four yards, the disposition of the entire She-Kargo faction could only be guessed at. Any further reconnaissance was barred by several, densely-packed lines of warriors.
This pre-emptive move on the part of the She-Kargo left the leaders of the D'Troit-C'Natti-San'Louis war council in some disarray. By denying access to the part of the lower sandbar they now occupied, the She-Kargo faction had neatly blocked any outflanking manoeuvre by their opponents. Any attempt to force a way through would have immediately led to a pitched battle in swirling mist, which at times was so thick you could barely see beyond the end of your knife arm. The present poor visibility was not the only limiting factor: by prior agreement, the attack on the She-Kargo was not supposed to take place before the Iron Masters arrived on the scene. In the circumstances, the D'Troit and their allies had no choice but to position themselves along the remaining section of the beach, between their rivals and the tall, ornately carved trading post. And wait.
The original plan had called for the She-Kargo faction to be sandwiched in between the C'Natti and D'Troit. A twin pincer movement on the landward side – easily achieved by their numerically superior forces – would have left the She-Kargo and M'Waukee surrounded with their backs against the sea and with no possibility of escape. At the same moment, a secondary action was to have been launched by the San'Louis against the camping grounds occupied by the She-Kargo, M'Waukee and San'Paul. With the majority of the delegations crowded onto the shore to await the arrival of the Iron Masters, the lines would only be thinly defended.
But this part of the plan had also misfired. As the D'Troit war council hastily rejigged their general plan of attack a breathless messenger despatched by the San'Louis arrived from the lines. In the mist-shrouded pre-dawn twilight the cooking fires of the She-Kargo faction had been seen burning with shadowy figures seated or sleeping beside them. At the boundary between the M'Waukee lines and the San'Louis and the C'Natti and San'Paul, everything had seemed perfectly normal. It was only some time after the last of the warriors slipped away to join their companions on the sand-bar that the ruse was uncovered.
The lines occupied by the She-Kargo faction were empty. The seated figures were made of hooded cloaks hung on a framework of sticks, the sleepers were rolled straw mats stuffed with grass. Everything of value, all the trade goods and chattels brought by the She-Kargo, M'Waukee and San'Paul delegations had been quietly carried away by baggage-handlers during the night. But they had not vanished without trace. An examination of the ground revealed tracks leading away from the campsite towards the western shore of the lagoon.
Even though they were doomed to fail, the She-Kargo's efforts to avoid defeat were impressive. Prime-Cut unrolled the map given to him by one of the Iron Masters' agents on Lake Mi-Shiga and after a few minutes' intensive study the enemy's intentions became abundantly clear. If forced to give ground, the main force of the She-Kargo would retreat across the shallow channel onto the upper sand-bar in an attempt to seek refuge on the higher ground beyond. The steep bluffs would form a good line of defence against a frontal assault but it could be quickly turned from the west. Prime-Cut also discerned the tactic underlying the midnight retreat of the baggage trains along the western side of the lagoon. The She-Kargo faction were obviously reluctant to abandon their valuable merchandise but, more importantly, they intended to put a considerable body of men across the first fording point on the river estuary and hold the northern shore to prevent a pursuing force from crossing over and blocking the exit from the upper sandbar.
Further examination of the map revealed a second fording point across the estuary, some three miles further west. This would provide an alternative route if the first was denied to them. There was also another point from which an attack could be launched: a spur of land on the western shore of the lagoon came to within four hundred paces of the channel separating the upper and lower sandbar. If bowmen were stationed on the point of the spur they could fire into the left flank of the retreating column while the advancing D'Troit cut their way through from the rear.
At the urging of Prime-Cut and Judas-Priest the war council quickly agreed to split its forces. The D'Troit delegations would remain on the beach ready to move along the sandbar against the main force of She-Kargo and M'Waukee warriors. The C'Natti would despatch a strong force of bowmen onto the spur; the remainder of its warriors would follow the route taken by the fleeing baggage trains and its escort – believed to be the weakling San'Paul stiffened by elements from the two stronger bloodlines. The C'Natti warriors were to destroy everything in their path, seize the first fording point then wheel right onto the upper sandbar, trapping what was left of the She-Kargo and M'Waukee.
Result – total annihilation.
Prime-Cut rubbed his hands jubilantly as the runners departed with the movement orders – orders that were to be put into effect immediately.
Easier said than done. As the She-Kargo faction had already discovered, organizing the movement of thousands of combatants and their assembly into coherent formations is extremely difficult without a proper chain of command – especially when the orders are conveyed down the line by word of mouth. And it was an even bigger problem for the D'Troit faction because they had three times as many warriors to deal with.
All of which meant that the orders to the C'Natti to leave the beach and move off along the western side of the lagoon took some while to get through to the various delegations. The cloying mist slowed the process even further and the planned manoeuvre almost fell apart when several groups of warriors, anxious to get into the fray, rushed off – mostly in the wrong direction – without waiting to hear what exactly it was they were required to do.
Prime-Cut cursed and fretted at the delay and silently berated himself for making what he now realized was a serious tactical blunder. The enlarged D'Troit and C'Natti delegations had appeared at the trading post too soon. Instead of arriving over the two previous days, the extra delegates and 'journey-men' should have delayed their appearance – joining their clansmen in the hours just before dawn.
Had they done that, the She-Kargo faction would, as always, have outnumbered the D'Troit. His verbal attack on Mr Snow might have caused them some disquiet but they would not have felt threatened. Secure in their strength, they would have gone to bed, slept soundly – and woken up to find themselves hopelessly outnumbered, with no time to make a coordinated plan to defend themselves.
On the other hand...
Prime-Cut angrily drove these last-minute regrets from his mind. What was done was done. The battle would be harder but that would only make the winning of it more worthwhile. Death, if it was to have any meaning, should be a memorable occasion.
As dawn broke the situation changed rapidly. The warming rays of the rising sun burnt away the blanket of mist revealing the disposition of the opposing forces along the shore. And as the sun lifted clear of the eastern horizon, the wheel-boats of the Iron Masters came into view. But this time, the flotilla consisted of five vessels instead of the usual three.
Mr Snow and the other wordsmiths heading the She-Kargo faction did not know that the two extra wheel-boats had been leased by the Ko-Nikka and Se-Iko families. To them, the approaching vessels meant only one thing; the Yama-Shita had come back in strength – and that spelled trouble.
Steaming in V-formation, with smoke belching from their twin funnels, the five wheel-boats headed towards the trading post, their stern-mounted paddles churning the blue sun-struck water into wide swirling ribbons of green and white foam. The relentless thump-thump-thump of their engines, the boom-pound-boom of the revolving blades plunging into the water which then cascaded noisily and by the ton off the rising blades, merged into one endless barrage of sound, a continuous roll of thunder that reached out across the vast expanse of the lake, striking terror in the hearts of those who stood watching and waiting on the shore.
Sky and earth-thunder were sounds which triggered a primal fear within a Mute's soul. Within the folk-memory of their race, passed down through the generations like their warped genetic code, it recalled the terror inspired by the unbearable brilliance of The War of a Thousand Suns when the earth and sky was riven by fire.
Oshio Shinoda, supreme commander of the punitive expedition despatched by the Yama-Shita family, stood on the bridge of the leading vessel next to its captain, Kato Yukinagi. Both men had polished brass telescopes trained on the approaching shore.
On their rear starboard quarter was the boat crewed by the Ko-Nikka. The Se-Iko boat lay to port. These, in turn, were flanked by the two other boats belonging to the Yama-Shita. Their commanders also had telescopes trained on the crowded shore, and outside the wheel-house of each ship stood a flag-officer and his men, ready to relay messages to and from the expedition's commander.
On its last visit to the trading post, Yukinagi's vessel – its huge black superstructure relieved with red and gold trim – had carried Domain-Lord Hirohito and Clearwater to their secret meeting with Mr Snow. This time, the elegantly appointed stateroom occupied by the late lamented domain-lord had been left empty, a shrine to his memory, with flowers and various other prayer offerings placed before the dais on which he sat when giving audience. And it would remain so until his death had been well and truly avenged.
With the aid of his powerful spyglass, Samurai-General Shinoda was just able to identify the different bloodlines. The She-Kargo, and M'Waukee were grouped together on the right hand side of the beach. The D'Troit and San'Louis were ranged in the centre and to his left. Normally, the Mutes gathered in one huge throng in front of the trading post, but now, the centre of gravity had shifted noticeably to the right, with the She-Kargo and M'Waukee spread out quite thinly along the lower sand-bar.
This was not what Shinoda had expected to find. The plan had been for the She-Kargo to be separated from the M'Waukee and sandwiched between the C'Natti and D'Troit. But there was no sign of the C'Natti, or for that matter, the San'Paul. Something must have happened to cause the original plan to go awry. Something drastic – like the She-Kargo discovering what lay in store for them. If that was so, then the vital element of surprise had been lost.
Samurai-General Shinoda consulted the map spread out on the navigator's table and conferred with the wheel-boat's captain, Kato Yukinagi. The channel separating the upper and lower sand-bars was now a shallow fording point. It would slow down any force retreating towards the northern shore but it would not prevent them from doing so. Using the powerful spyglass mounted on a gleaming brass pillar outside the wheel-house, he slowly scanned the long upper sand-bar that ran away to starboard. It was completely deserted, but when he focused on the escarpment that dominated the northern shore he glimpsed figures moving about on top. Figures that seemed to be trying to conceal their presence.
After Captain Yukinagi had taken a look through the spyglass and confirmed his suspicions, Shinoda made a careful survey of the escarpment and spotted several lines of men – they could only be Mutes – hauling baggage up the steep slope west of the sand-bar. Their distance from his ship made it impossible to tell which group they belonged to but they had no reason to be there. If they had come to trade, they should be on the beach. On the other hand, if they intended to join those already on top and cover the retreat of the She-Kargo they were heading in the right direction.
Shinoda was only guessing at their intentions. The distant figures might be the missing C'Natti moving to cut off the She-Kargo but in that case why were they humping back packs? He could not afford to take any chances. After the sinking of the wheel-boat on Lake Mi-shiga, everyone in the expedition knew that further failure would not be tolerated. As a precautionary measure he decided to land some of his men onto the upper sandbar to seal off this possible escape route. He conferred briefly with Captain Yukinagi, the Flag Officer was summoned, and the appropriate signal was sent to the starboard flank-boat.
On the shore, Rolling-Stone, Mack-Truck and Blue-Thunder saw the wheel-boat detach itself from the flotilla and angle away to their left. They didn't need to be master-tacticians to understand the reason for the move. It was as obvious as it was unexpected.
The plan hatched by Mr Snow and the war council called for the She-Kargo faction to give ground as soon as the D'Troit began to stoke up the present atmosphere to the point where the two sides came to blows. But while exchanging the usual swaggering taunts with the warriors on the She-Kargo right flank, the D'Troit front-liners had kept the situation below boiling point. The She-Kargo _had_ to begin their withdrawal before the wheel-boat got into position but was denied the reason for doing so!
The D'Troit, having come to a similar conclusion as to the Iron Master's strategy, had quickly issued instructions to their warriors to cool it, thereby forcing the She-Kargo into making the first move.
The leading delegates of the She-Kargo faction who formed the top echelon of the hastily-formed chain of command on the sand-bar put their heads together.
'Treacherous toads!' exclaimed Rolling-Stone. 'Yesterday they tried to blame us for everything, and now they're trying to force us into a position where we have to attack them!'
'We don't have to pick a fight,' said Wind-Walker, a M'Waukee wordsmith from the Clan T'Maso. 'We could just withdraw.'
Black-Sabbath, paramount warrior of one of the biggest of the She-Kargo clans reacted angrily. 'The Clan K'Rella has never backed away from any of these jackals from the D'Troit and is not about to do so now!'
Several other members of the war council expressed the same view with equal force.
'We're not running away from a fight!' exclaimed Wind-Walker. 'We're running into one! Do you think the D'Troit are just going to stand there and watch us fade away? They'll be right on our heels with the sharp iron out!'
Mack-Truck said: 'If we're going to move, we'd better do it now.' He pointed to the huge straggling pack of C'Natti warriors heading around the western side of the lagoon. 'We have to get to the north shore before they cross the river. Otherwise we're done for.'
'We'll also be done for if we don't get across the channel onto the upper sandbar before that wheel-boat reaches us,' grunted Rolling-Stone.
They all looked at the boat. Dark-Star, a M'Waukee summoner who had been given the task of aiding the 'strategic withdrawal' asked the question that was in everyone's minds: 'Can we outrun it?'
'Not if we stand here arguing,' said Rolling-Stone. 'I propose we move. En masse. Now. All those in favour?'
The motion was carried by eleven to four.
Rolling-Stone turned to Dark-Star. 'Besides yourself, how many summoners do we have who can raise stones?'
'I know of at least thirty. There could be more.'
'Good. Pick nine of the best and spread yourselves out so as we've got the full width of the sand-bar covered. Position yourselves two hundred paces behind the front rank. The signal to withdraw will be three blasts on the buffalo horns – repeated twice. When you hear that, the front line will fall back towards you, and as they reach you –'
'We hit the beach...'
'With everything you've got. Raise a wall of shit that will slow them down and give us a head start.' Rolling-Stone patted Dark-Star on the shoulder. 'Take a group of your own clansmen to guard you and as soon as you've got lift-off, pull back but keep the magic flowing for as long as you can. Our second line will cover you.'
Dark-Star eyed the old mute sceptically. 'What with?'
Rolling-Stone gave him a comradely slap on the back. 'Have faith, brother. Talisman looks after his own.'
_Yeahhh. Tell me something new_...
When Rolling-Stone called for the bull-horns to be sounded, the diverted wheel-boat was about four miles from the mouth of the channel between the sand-bars.
The eerie sound drifted across the water to reach the ear of Samurai-Major Akido Mitsunari and Captain Umigami. Mitsunari was the commander of the military force now assembled on the through-deck below: a combined force of cavalry and infantry, ready to charge down the gangways as soon as the boat reached the shore. Umigami was Master of the vessel but when carrying troops he was obliged by tradition, to follow the orders of their commander. As in ancient Japan, the army took precedence over the navy. Umigami, an officer in the _merchant_ marine, ranked even lower in the pecking order.
Mitsunari swept his powerful spyglass over the thousands of Mutes that seemed to be milling about aimlessly along the lower sand-bar. Although vastly outnumbered, his tiny force would make mince-meat of them. No discipline! he lamented inwardly. No organization!
As this judgement passed through his mind, the lines of She-Kargo warriors positioned nearest to the D'Troit wilted then fell back rapidly, opening up a wide gap between the opposing factions. The open stretch of beach erupted with explosive force and an instant later Mitsunari became aware of a high keening sound.
Through his spyglass he saw sand, stones, grass and gravel being sucked up into moving funnels of air which then snaked forward at great speed into the leading ranks of the She-Kargo. Dust devils! A stream of mini-tornadoes that scythed through the advancing troops, knocking men off their feet with a vicious hail of rocks and stones and momentarily blinding others with whirling clouds of sand.
It was amazing. Mitsunari had never seen anything like it before. And during the two or three minutes this barrage lasted, the entire She-Kargo army had turned tail and was running towards the channel that separated the two sand-bars.
The next sound Mitsunari heard was an angry roar from seventy thousand throats as the massed delegations of the D'Troit launched themselves in pursuit.
Like his commander, Oshio Shinoda, Mitsunari would have dearly loved to know the battle plan of the D'Troit. The chain of agents stationed in the outlands who had acted on behalf of the Yama-Shita family in setting up this 'arrangement' had tried to persuade the D'Troit to accept the help of a small team of advisors – including a flag signal unit which could have maintained contact with the advancing flotilla.
The idiot monkey-faced chieftains who led the D'Troit had spurned this offer, giving as an excuse the difficulty of concealing Iron-Masters among their own ranks. The discovery of such individuals by their rivals would, they claimed, have brought an immediate charge of collaborating with the enemy – a charge the D'Troit intended to make against the She-Kargo.
As a result, Oshio Shinoda, his junior force commanders and the boat-captains had no way of knowing the on-shore situation beyond what they could see through their telescopes. And under the strict policy of non-belligerence imposed by the Toh-Yota shogunate – to which they were obliged to adhere because of the unwelcome presence of the wheel-boats crewed by the Ko-Nikka and Se-Iko – they could not lawfully intervene until they had run the boats ashore and had received a formal request for assistance!
Responding to Mitsunari's request for more speed, the wheel-boat captain sent the traditional signal to the sweating stokers in the engine room. Off his rear port quarter, smoke belched from the funnels of the rest of the flotilla as they responded to a similar order from Oshio Shinoda.
On shore, having recovered from their surprise, the massed clan delegations of the D'Troit raced after the She-Kargo like a howling lynch-mob. Moving with them were their own summoners, but they could not reply in a similar manner. In order to do so, they would have had to place themselves of were their own warriors – exposing themselves to a well-aimed crossbow bolt. A summoner could only call up the forces from the earth and sky _when he was standing still_. The very act rooted him to the spot until the forces had passed through him. That was the second drawback to being a summoner.
The deadly hail of stones raised by Dark-Star's team had cut down hundreds of warriors. The D'Troit were not prepared to wait for a similar cloud to smite the fleeing She-Kargo. Some of the pebbles on the beach were bigger than a man's fist but a direct hit could not be guaranteed. Knives, on the other hand, always found their target.
The She-Kargo faction had been ordered to cover the three miles to the channel at the fastest possible speed. Those who tripped and fell, or lagged behind for whatever reason were to be left to their own devices. They could either try to catch up or make a stand in the few seconds that remained before they were engulfed by the screaming horde now racing towards them. A brave but futile gesture which failed to slow the pace of the pursuit. The leading ranks of the D'Troit simply ran around the stragglers, leaving them to be hacked to pieces by the tens of thousands of warriors packed into the middle and rear of the column.
The time taken to cover the first three miles was a little under eighteen minutes – well outside the world record for that distance, but this race was run over an uneven bed of sand and shingle, not a rolled Olympic-class cinder track. Even so, the leaders were still travelling at over ten miles an hour, equal to the speed of Mitsunari's wheel-boat.
Fortunately, the wheel-boat had further to go and it was still out of range when the front of the column reached the three hundred yard wide channel and pounded across in a cloud of spray.
Mitsunari realized that there was no chance of placing his wheel-boat in the channel – thereby cutting off the She-Kargo's line of retreat. And having seen the violent sandstorms raised against the D'Troit, he was reluctant to get too close for fear of losing the vessel. The stories of Mute magic had been discounted, but he had now seen evidence of it with his own eyes! Already one wheel-boat with a noted commander and carrying many of his own comrades had gone down in Lake Mi-shiga. He did not propose to be on board the second.
Mitsunari did not fear death, but as a samurai his whole life had been geared to one end: to die in the heat of battle on behalf of his domain-lord. Drowning – even in the line of duty – was a quite ignoble way of departing from this life. He asked Captain Umigami to alter the ship's heading fifteen degrees to starboard. The bow of the wheel-boat was now aimed at the long upper sand-bar, but the approach angle was much shallower, allowing the cannon mounted in the port side galleries to brought to bear against the retreating column.
M'Call elder Rolling-Stone had seen the wheel-boats discharge their cannon in a ceremonial salute but he had never seen them fire in anger. Others, in the early days before the Mute clans agreed to trade with the Iron Masters, had been given a practical demonstration. Being in the target area of a broadside from one of these vessels had proved an unforgettable learning experience and the news had been passed on.
But this was not the only new threat. To the She-Kargo's rear, the deadly pursuit had not slackened and now, as the middle and rear segments of their straggling formation headed towards the channel, they came under fire from the first of the D'Troit bowmen to reach the spur of land jutting out from the western side of the lagoon. Volley after volley of crossbow bolts flew across the four hundred yard stretch of water, cutting down warriors right, left and centre.
Rolling-Stone and Mack-Truck urged the last of their clansmen into the channel then turned to greet Dark-Star D'Mingo. His escort – two giant M'Waukee warriors – had hoisted him up by his armpits, leaving his flailing legs barely touching the ground.
They set him down in front of Rolling-Stone. He crumpled like an empty sack then straightened up. 'I know, I know, I know. They weren't supposed to pick me up. But don't be angry at them. They are my sons.'
He quickly embraced them both then turned his eyes skywards. 'Bless you, Sweet Sky-Mother, for the loan of two strong arms!' He turned back to his sons. 'Right! Off you go! I've got work to do!'
The two young warriors paused uncertainly. Dark-Star shoved them into the water. 'Go on! Get moving!'
His sons raised their hands in a farewell salute then ran on and were lost in the clouds of spray raised by the other fleeing warriors.
Dark-Star crouched at the water's edge between Mack-Truck and Rolling-Stone as hundreds of warriors sprinted past. Muscled thighs lifting and driving their flying feet, pounding the water to foam as they passed across to the other side. Foam that was already tinted with the blood of those slain by the bowmen on the nearby spur.
'Are you out of your mind?!' exclaimed Rolling-Stone. 'Go after them!'
'Pwwahhh! I've done enough running for one day!'
'So have I,' admitted Mack-Truck. 'There's no way I'm ever going to make it to the top of that bluff.'
Rolling-Stone eyed the distant high ground that overlooked the seven mile-long sandbar their warriors had to cover before they reached safety. 'Me neither...'
'Better make ourselves useful then,' said Dark-Star. Running back and forth across the curved point of the sand-bar he recruited two of his earlier colleagues as they stumbled to the water's edge and ordered Rolling-Stone and Mack-Truck to arm their crossbows and gather as many bolts as they could from warriors who had fallen to the incoming fire from the nearby spur.
Working their way through the dune grass on the lagoon side of the sand-bar, they came opposite the point where the D'Troit bowmen were massed. Dark-Star turned to his two colleagues. 'I know it's asking a lot but I want you to raise some more whirlies as the last of our people come through. I'll try and spoil the aim of these toads over here.' He gripped their hands in the traditional gesture of farewell. 'Give it everything you've got. This is as far as we go.'
The two summoners accepted their imminent death with the calm resignation that all Mute warriors strove to achieve and hid themselves in wind-scooped hollows between the spiked tufts of grass.
'And what do we do?' enquired Mack-Truck.
'Do your best to cover us. The longer we can keep going, the more chance our brothers will have of getting out of this alive.' Dark-Star glanced over his shoulder at the wheel-boat in the lake beyond as it crossed the mouth of the channel, almost sideways on to their position. 'See how those yellow runts are trying to cut us off? They're working hand-in-glove with the D'Troit!' He ground his teeth together. 'Oh, Talisman! When the guilty are judged, make them pay in blood!'
Breaking away from Rolling-Stone and Mack-Truck, Dark-Star waded into the water seemingly oblivious of the incoming volleys of crossbow bolts. He concentrated for a moment as if gathering his inner strength, then threw his head back and spread his arms out sideways with the palms facing downwards. A shrill piercing cry burst from his lips and in the same instant, a small whirlpool formed in the water directly below each of his outstretched hands.
Flexing his arms, Dark-Star pushed the whirlpools out of reach then, as they grew in size and increased their speed of rotation, he swept his arms upwards. Responding to his unspoken command, the whirlpools became two huge spinning columns of water that rose into the air – and kept on rising until they were about a hundred feet high. The noise was deafening – like the buffeting roar of gale-force winds tearing through the tree-tops.
By now, Dark-Star's hands were high over his head. He swept them forwards, the two index fingers aimed across the lagoon at the warriors massed on the opposing shore. The tall, snaking columns of water surged forward like unleashed hunting dogs. Behind them, Dark-Star raised another pair, then another.
Before they had time to grasp what was happening, the C'Natti bowmen on the point of the spur found their view of the sand-bar obscured by twelve huge undulating columns of water, each one now twenty to thirty feet in diameter and bearing down on them with the speed and menace of an approaching express train.
The wind circling each column whipped their faces and tore at their clothing then, as the first waterspouts reached the shore, the huge columns of water collapsed with a thunderclap of sound.
Those immediately below were knocked senseless to the ground, others on the shoreline were swept into the lagoon. More spouts followed in their wake, swamping the spur from all sides. To many of the terrified Mutes it was as if the heavens had opened and the Sky Voices were venting their wrath upon the C'Natti for challenging the will of Talisman. Hundreds turned tail and fled inland. The hardier souls stood their ground but the torrential cascades of water falling out of the sky and the continuous advance of newly-formed waterspouts masked their view of the retreating She-Kargo.
Running in their clan groups, spread out across the whole width of the sand-bar, the forty-five thousand warriors of the She-Kargo faction – or to be more accurate those still up and running – took about ten minutes to pass the point where Rolling-Stone, Mack-Truck, Dark-Star and the other two summoners were making their last stand. The D'Troit had been tearing savagely into the tail end of the column and despite the order to keep running, many of those in the rear had decided to stand and fight rather than face the ignominy of being cut down from behind.
As a consequence of these desperate struggles, a slight gap – no more than thirty yards at best – had opened up between the She-Kargo and their pursuers. It was this approaching gap that Dark-Star's colleagues – Silent-Running and Condition-Red – proceeded to exploit, each raising a power vortex that sucked stones and sand into the air then hurled them at murderous speed towards the advancing D'Troit.
Once again the front ranks wilted and fell back under the onslaught, but as before it only caused a temporary delay. Those behind continued to leap over the prostrate bodies of their comrades, some falling in their turn, others circling round the lake-side of the sand-bar, under cover of the central ridge.
The power given to summoners did not pour endlessly like water from a tap. It came in finite bursts, and like a battery that needed to be recharged after use, it quickly ran out if constant calls were made upon it. That was the third drawback to being a summoner.
The howling sandstorms faltered then died. Sand and stones rained vertically out of the sky and seconds later the water around Dark-Star became ominously still. Totally exhausted, he staggered in the waist-deep water then pitched forward, face down. Two crossbow bolts struck the upper half of his body as he sank beneath the surface. Silent-Running and Condition-Red went down unresisting under a flurry of knife-thrusts. Rolling-Stone and Mack-Truck threw aside their empty cross-bows and met death knife in hand and a cry on their lips.
'Drink, Sweet Mother!'
They had been doomed from the start but their defiant stand had gained a few vital minutes of respite which allowed the tail end of the retreating column to get across the channel onto the upper sand-bar without further losses.
But the chase was far from over. The She-Kargo and their allies had another seven miles to run before reaching the safety of the northern shore, followed by a steep, six-hundred foot climb up the face of the bluff to where Mr Snow now stood, guarded by a phalanx of M'Call Bears and ten of the She-Kargo's most powerful summoners.
It was not his life he cared about. Mr Snow was aware he might not survive the battle. The warriors and summoners were there to protect his _magic_.
Only _he_ had the potential to redress the balance; to counter the combined strength of Mute and Iron Master ranged against them. He had made a spectacular stand against the The Lady from Louisiana but his prodigious efforts on that occasion were dwarfed by the scale of the task that now confronted him. He was not even sure he could deliver what he had promised the delegates to the midnight war council.
Immense powers lay hidden in the earth and sky, but the gift that enabled him to summon them was given to him by Talisman – and it was exercised with _his_ blessing _to do his Will_. Would he allow his power to be used to tear the Plainfolk asunder? Mr Snow, who had spent virtually every moment since his arrival trying to plan for every eventuality finally gave up worrying and turned his red-rimmed eyes to the sky.
_We offer our spirits into your care, Sweet Mother. Let his Will be done_...
From his present vantage point, Mr Snow had a bird's eye view of the evolving struggle and he could zoom in close on the action with the aid of the compact but powerful viewing device found on Brickman after he had been shot down during the Battle of the Now and Then River.
To the left he could see the long sand-bar with the mass of She-Kargo and M'Waukee Mutes moving along it from the far end. He could also see the wheel-boat closing in on their right flank. Beyond the channel, the first waves of the D'Troit were entering the water.
Immediately below him, defending the northern end of the sand-bar and the steep slope the last of the baggage train and the exhausted runners had yet to climb were several thousand reinforcements summoned from nearby She-Kargo and M'Waukee clans. Arriving in batches all through the night, some had been running Mute-fashion for twelve hours non-stop. These breathless latecomers asked for and were given no respite. They acknowledged the shouted greetings and directions of the battle-marshals with a wave and ran to their allotted positions without breaking their stride.
Swinging to the right, to the western side of the lagoon, Mr Snow could see the C'Natti 'army' advancing along the route taken by the She-Kargo faction's worldly goods. Having stolen away two hours after midnight, the baggage train and its escort of San'Paul warriors were now safely across the narrow point of the river estuary. Half their number had reached the top of the bluffs, but a long jostling line of porters trailed back down the path onto the shore below.
_Come on, come on, come on! Move! Move! Move!_
It was a nail-biting moment. The C'Natti were closing the gap at an alarming speed, but because of the initial confusion surrounding their departure and the circuitous route they were obliged to follow, the leading groups were still some way behind the She-Kargo column on the upper sand-bar. Mr Snow prayed they would not succeed in drawing level. If his plan was to succeed the She-Kargo faction had to reach the shore and scale the bluffs _before_ the C'Natti crossed the river estuary and turned to cut the She-Kargo line of retreat – a danger Mack-Truck had already foreseen.
_O, Talisman, our Saviour on high, give wings to the feet of our warriors, and give us the power to defeat our enemies!_
By this time, the four remaining boats in the flotilla had run the square-cut bows of their vessels onto the gently shelving beach below the trading post. The joint D'Troit and C'Natti war council led by Prime-Cut, Judas-Priest, Screaming-Tree, Flesh-Eater, Corpse-Grinder and War-Machine had kept back some ten thousand delegates to provide an impressive welcoming committee. Taking their cue from Prime-Cut, all of them cheered wildly as the masked, richly-dressed samurai came hurrying ashore in their curious bandy-legged fashion as soon as the wide gangways had been lowered into place.
Cutting short the usual formalities, Samurai-General Shinoda dispensed with the services of an interpreter-spokesman and demanded a direct explanation of what was happening.
His boat-captain, Kato Yukinagi, stood nearby, flanked by the worried commanders of the Ko-Nikka and Se-Iko vessels. Neither had expected to find themselves in the middle of a war-zone.
Prime-Cut had already worked out the crucial elements of his reply with one of the Yama-Shita's forward agents. After delivering a brief, highly-coloured and totally biased account of what had happened the day before, he formally requested the assistance of the Iron Masters in punishing the treacherous She-Kargo. Had it not been for the vigilance of the D'Troit, he declared, these hired lackeys of the sand-burrowers would have attempted to seize the wheel-boats and murder their crews!
_Hawwwwhh!_ The leading samurai from the Yama-Shita family crowded behind Shinoda and the wheel-boat captains reacted with a convincing degree of shock and horror, striding to and fro, and posturing in a warlike manner to express their outrage.
If the noble Iron Masters were willing to come to the aid of their loyal friends, cried Judas-Priest, they must do so quickly!
Oshio Shinoda went through the motions of consulting the Se-Iko and Ko-Nikka captains, but while they were formulating their answer, the first of the Yama-Shita's mounted samurai came clattering down the gangways of their two beached vessels.
As strangers in a strange land, the Se-Iko and Ko-Nikka could do little else but agree. They had not come prepared for trouble but having come this far – at great expense – they did not want to return empty-handed.
By the time they had conferred and given their reluctant agreement to the proposed military action, the first two squadrons of samurai cavalry had already departed at the gallop; the first along the sand-bar, the second around the western side of the lagoon. And seconds after the Se-Iko and Ko-Nikka had penned their assent to the bottom of a document that Shinoda had thoughtfully prepared to cover just an emergency, a volley of green rockets soared skywards from the roof of his wheel-boat.
This was the signal that Samurai-Major Mitsunari and Captain Umigami on the bridge of the flank-boat had been waiting for.
B-BA-BBOOOMMMM!! The port side of Mitsunari's wheel-boat erupted with smoke and flame as the thirty lower cannon were fired simultaneously, hurling a salvo of 201b iron balls at the retreating Mutes. Watching through their spyglasses, Samurai-Major Mitsunari and Captain Umigama chortled gleefully as the cannonade ripped bloody gaps in the column, sending bodies and great gouts of sand and smoke flying into the air.
'B-BA-BBOOOMMMM!!' The guns ranged along the first, upper gallery fired in their turn. More hits. Umigami ordered the helmsman to bring the vessel closer to shore so that the cannoneers could load grapeshot and cause even more casualties. Wounded soldiers ran slower and were easier to kill.
The noise of the first two salvos bounced off the rocky rampart and reverberated across the lake towards the trading post. Noting the increasing bewilderment of the Se-Iko and Ko-Nikka, Shinoda explained with studied courtesy that these opening shots had been the signal for the cavalry to draw their swords in support of the friendly tribes of grass-monkeys. A signal which, of course, could not have been given without their consent.
Inviting the two captains and their chief lieutenants to accompany him, Shinoda boarded the red and gold vessel which had once carried his domain-lord and ordered the captain to cruise along the sand-bar so that he and his guests could survey the carnage. The ingrained streak of cruelty that all Iron Masters possessed made the invitation impossible to resist.
The third Yama-Shita wheel-boat followed, leaving the Se-Iko and Ko-Nikka vessels beached by the trading post, under the command of their junior officers.
From the top of the bluff, Mr Snow watched helplessly as repeated broadsides from the upper and lower decks tore great gaps in the columns of running warriors. And on the flank-boat, the sweating, smoke-blackened gun-crews poured buckets of water over their muzzle-loading cannons to cool the heated barrels.
To aid the retreating She-Kargo and M'Waukee delegations, four lines of bowmen had been stationed across the sand-bar at regular intervals. As the last men passed through the first line, the bowmen fired into the advancing D'Troit, then fell back. And as they and the retreating troops pass through the second line the operation was repeated.
By the time the fourth line had fired their lethal volley, the first line had taken up a new position as the fifth line, and so it went on, mile after mile, all the way along the sand-bar. But the plan had not taken into account the repeated cannonades from the wheel-boat – and now Mr Snow could see that two more were steaming in the same direction!
Fortunately, the firing from the western side of the lagoon had ceased. Through Brickman's viewer, he could see the hunting packs of the C'Natti running flat out in an effort to draw level with the She-Kargo.
Behind them, gaining fast, were the leading horses of the second squadron of samurai. Each rider carried the tall black and silver house-flag of the Yama-Shita mounted on a slender pole fixed to the back of his armoured tunic. With their horned, wide-brimmed helmets, snarling iron battle-masks and other warlike accoutrements they were a fearsome sight.
B-BA-BBOOOMMMM!! The sound of another broadside thundered across the lake as thirty muzzle-loads of grapeshot scythed down several lines of running Mutes. Unable to bear the one-sided slaughter any longer, two She-Kargo summoners, gifted with the Third Ring of Power, ran into the shallows and called up huge waterspouts as Dark-Star had done.
Before this moment it had not been practical to do so. There was a limit to the power that flowed through the Third Ring. The towering columns of water the summoner was able to raise and move in a chosen direction could not travel very far – half a mile at the most and somewhat less if the summoner was generating a series of them. Beyond a certain point, the dynamic thrust that kept the twisting column up in the air and moved it forward simply petered out and the whole thing just collapsed in a cloud of spray.
But now, the wheel-boat had steamed closer to the shore and within seconds of being summoned into life, four towering waterspouts were snaking across the lake towards Captain Umigami's vessel and more were being brought to life behind them.
Seeing what was happening, a large number of warriors broke off from the retreating column and formed a human rampart around the summoners to protect them for as long as they could.
Samurai-Major Mitsunari and Captain Umigami stared in horror as the line of waterspouts came spinning across the lake towards their boat. The wheel-house perched on top of the third floor gallery was already some fifty feet above water-level – these huge roaring white snakes were higher by at least half as much again, maybe more!
A raging wind tore through the portside galleries and buffeted the wheel-house then the boat shuddered as the first four columns of water slammed into it. Water cascaded out of the sky and burst into the side cabins, flooding through the transverse passageways to empty over the starboard side. Again and again the ship was struck. Interior partitions were swept away as torrents of water poured down the maze of stairways onto the through deck and into the holds below.
The basic structure had been built to resist the worst of storms but no one had envisaged moving walls of water as high as this! Two more, top-heavy spouts struck the port side of the wheel-boat just forward of midships, snapping off both funnels and sweeping them into the sea beyond. The next seemed to be aimed directly at the wheel-house.
Before anyone inside had time to react, the hundred foot high column of water struck the corner of the forward galleries. Several tons of water fell out of the sky, flattening the wheel-house and its occupants.
Samurai-General Shinoda, on the bridge of the following boat, watched speechlessly as the tumbling mass of water hit the roof of Mitsunari's vessel and exploded, hurling the splintered wreckage of the wheel-house and the rag-doll bodies of his comrades into the sea. Finding his tongue, he invoked the protection of Omikami-Amaterasu, the supreme deity. Like all his compatriots, he had discounted the half-baked tales of Mute magic, but it was true! He had seen it with his own eyes. These grass-monkeys had summoned a horde of evil _kami_ to their aid!
And as if to underline the awful truth of this conclusion, the huge boiler in the bowels of the stricken vessel exploded with a muffled roar. For a brief instant of time the damage was contained then, with a thunderous boom, billowing clouds of smoke and steam burst out through the roof and the ruptured side galleries.
The cannons fell silent. And the last waterspouts faltered and died before reaching the wheel-boat as the two She-Kargo summoners went down under a dozen flashing knives. The bodies of their defenders littered the water's edge, staining it with their blood.
The run continued, the death-toll mounted. But now, with only three lung-stretching miles between the retreating column and the face of the bluffs, they entered the last defensive zone where, in the hours between midnight and dawn, feverish preparations had been made to delay the pursuit.
Six-foot square lattices of poles, lashed together with sharpened stakes fixed at right-angles to every join had been scattered along the bar, sharp end down. As the column passed by, the frames were turned over by the tail-enders to expose the lethal points and tossed into the path of the oncoming D'Troit.
Their size made them too risky to leap over. The fleet-footed dodged around while other more public-spirited individuals attempted to diminish the risk even further by overturning them. But the immense pressure exerted by the middle and rear ranks caused the column to steamroller over anyone who stopped at the front.
Those who stumbled and fell usually brought a tangle of bodies down on top of them. If they were lucky, those behind ran past or leapt over them, but if there were too many choke points, they were simply trampled into the ground.
Some distance behind the spiked mats were a collection of twenty-foot long barriers made out of a simple framework of poles to which more sharpened stakes had been lashed at an angle facing the enemy like the steel girders in pre-Holocaust tank traps. These too were swung across the sand-bar by the last men to pass.
Compared to the number killed and injured by the wheel-boat's cannon, the casualties suffered by the D'Troit through these hastily-prepared devices were pitifully few. There had not been enough time to prepare and position enough of them, but with the lines of bowmen still firing then falling back, it was enough to break the momentum.
The sand-bar was five-hundred yards wide – enormously wide if you have to defend it, but exceedingly narrow when you have sixty thousand screaming warriors from groups who don't particularly like each other, all chasing the same quarry – and all trying to elbow their way into the front line. Added to which there were now dozens of dead-faces mounted on strange four-legged beasts demanding passage through their ranks from the rear!
When Aishi Sakimoto, the acting regent of the Yama-Shita had resurrected Domain-Lord Hirohito's plan, he had never considered the possibility that the D'Troit might regard the Iron Master's assistance as an interference. But from the angry reactions of the grass-monkeys now surrounding them it was clear to the officer in charge of the first cavalry squadron that they bitterly resented being forced to stand aside. It was their brothers who had fallen. What right did the dead-faces have to push to the front and take all the glory?!
The resulting melee in which Iron Master and Mute ended up brandishing their weapons at each other brought the rear third of the D'Troit column to a standstill. What began as a spontaneous reaction became a deliberate ploy to hold back the Iron Masters so that their clansmen could finish the job they had started, but it gave the She-Kargo and their allies an unexpected respite.
The possibility of Iron Masters giving chase on horseback had been foreseen by Carnegie-Hall. The ultimate line of defence consisted of five staggered rows of forward-sloping stakes running across the sand-bar from lake to lagoon. They were spaced wide enough apart to allow a warrior to slip through, but any horseman attempting to ride through it would immediately find himself in difficulty. The pointed stakes were too close together and too high, and because they were staggered in depth, they could not be cleared in a single jump.
The job of gathering, sharpening and planting the stakes, begun as soon as darkness fell, had continued after dawn and was still being completed as the She-Kargo began to pass through.
The lines of bowmen now formed into two groups, one on either shore to protect the rear of the column, but stocks of the precious crossbow bolts were now dangerously low. When a bowman emptied his pouch, he joined the retreat.
The pursuing D'Troit thought it was a rout, but they were wrong. It was a strategic withdrawal. And that was not a play on words to hide a bitter truth. Every man and woman in the She-Kargo faction would have preferred to stand and fight regardless of the odds. To face an enemy who outnumbered you by 3 to 1, knowing that you were bound to die demanded a very special kind of bravery for which the She-Kargo were renowned. But Mr Snow's stern message had been passed to the leaders of all the threatened delegations. Talisman had spoken through him. The D'Troit, the C'Natti and San'Louis had to be punished for their renunciation of the Thrice-Gifted One and their betrayal of the Plainfolk. And the dead-faces who had inspired their treachery had to be taught a lesson they would never forget.
That was why the She-Kargo faction had to endure this costly and ignominious withdrawal – _to draw their enemies onto the killing ground_.
And the time was nigh...
Mr Snow surveyed the scene below him once more. The leading ranks of the retreating column were now splashing across the silted-up bed of a narrow access channel cut through the sandbar close to the northern shore. Soon they would begin to pass through the line of defenders at the base of the bluff.
To his right, the last of the baggage train were toiling up the slopes under the weight of their loads. Their pursuers, the C'Natti, in their tens of thousands, were on the far side of the river estuary, preparing to cross the narrows. Further back still, samurai horsemen moving over open ground between the lines of Mutes were catching up with the leading warriors.
Mr Snow swung the viewing device back towards the lake.
The lead vessel of the Iron Master's flotilla had now drawn alongside the stricken flank-boat. The third boat lay stationary in the water some way behind the first two, a thin line of smoke curling lazily upwards from its funnels. In the distance, the two remaining vessels were beached below the trading post.
Handing the viewer to Awesome-Wells, a M'Call clanelder and long-time friend, Mr Snow closed his eyes, took a deep breath and steeled himself for what lay ahead. For over an hour he had watched the flower of the She-Kargo and M'Waukee fall under the knives of the D'Troit, had witnessed the bravery of Dark-Star and the other summoners. Now it was time to put his own life on the line – and maybe lose it in attempting to save the Plainfolk.
He could not think of a better reason for having lived until this moment.
Commending his spirit to Mo-Town, he stretched up his arms and silently prayed to Talisman to grant him power over earth and sky.
_Let me be your dark, avenging angel!_
The earth shivered under his feet. High above his head a dark cloud formed in the clear sky and spread rapidly in all directions. A cry burst from his throat. A cry so terrible, those around him fell to the ground, vainly trying to cover their ears. Some who dared to look upon his face said afterwards that his eyes became two points of blazing white fire.
Drawing the invisible power down from the sky and upwards from the earth with sweeping movements of his hands, Mr Snow hurled it downwards towards the lake, his fingers aiming at a point midway between the northern and southern shore, some three miles beyond the sand-bars.
Samurai-General Oshio Shinoda, on the bridge of his flag-ship, watched the dark cloud form with growing disbelief. A distant rumble of thunder seemed to come from within the bowels of the earth and was followed by an answering peal from the darkening sky.
The lake became ominously still, with scarcely a ripple disturbing its surface. The air had ceased to move. It was as if the whole world was holding its breath.
As a soldier, Shinoda was without fear, but the inexplicable moods of the _kami_ who inhabited the natural world filled him with unreasoning terror which – as supreme commander of the expedition – he had to do his best to hide.
The spreading cloud blotted out the sun, casting a shadow across the lake. The ground trembled bringing a hushed silence as the warring Mutes paused uncertainly, pursued and pursuer united in their fear of earth-thunder.
The Mutes in the defensive line at the foot of the bluffs were equally alarmed but they, at least, had been warned about what was to follow. Scrambling back up the slopes, they urged the panicky She-Kargo warriors to follow them as fast as possible. Those now lined along the top called down to their companions, urging them not to look back. Whatever happened, they must keep climbing.
Their exhortations were lost under a longer and much louder, stomach-clenching rumble of subterranean thunder which caused loose earth and stones to slide down the fissured slopes. Some of the exhausted warriors lost their footing, the rest clung on grimly until the mini-avalanche ceased then clawed their way upwards with a new sense of urgency.
A circular depression appeared in the lake at the point on which Mr Snow's powers were focused. For a moment, the water appeared to be draining down a huge plug-hole then the depression became elongated, spreading quickly to either shore. Now it seemed as if a vast bottomless trench had opened up into which the whole lake was emptying itself.
The three Yama-Shita wheel-boats began to drift helplessly towards it, like empty matchboxes being carried along a rain-filled gutter towards a cavernous drain. Then there was another clap of thunder. But this one was mind-splitting not just ear-splitting, and it was twinned with an even more violent earth-tremor that threw everyone who was standing to the ground.
Only Mr Snow remained upright, surrounded by an eerie vertical shaft of light composed of shifting planes that changed colour as they intermeshed. To those who dared to raise their heads from the ground, he appeared to be standing on the threshold of the world beyond, in thrall to the spiritual powers that flowed through his rigid body.
Another rumble of thunder. The trench-like depression splitting the surface of the lake filled abruptly and rose to become a vast wall of water as the sunken bed was pushed upwards by a convulsive subterranean eruption. Rising two hundred feet into the air, the wall of water tilted and fell forwards towards the shore, creating a giant tidal wave whose foaming edge dwarfed the wheel-boats that lay in its path.
Neither the frenzied orders of Samurai-General Shinoda, nor the seamanship of Captain Yukinagi could save the vessels or their crews. Picked up and pinned at an impossible angle against the curving front of the wave, the wheel-boats were carried at sickening speed over the sand-bar and the western shore of the lagoon then smashed and ground to pieces in the breaking surf that spread for several miles over the land beyond.
The beached wheel-boats of the non-belligerent Ko-Nikka and Se-Iko suffered the same fate. Everyone on shore, Prime-Cut, Judas-Priest and the other collaborators in the war-council, the advancing hordes of D'Troit, C'Natti and San'Louis, the samurai cavalry – all were swept away and drowned or battered to death in the vicious undertow.
Only those who had reached the top of the bluffs, or had managed to scramble more than a hundred feet above the shoreline were spared. The tidal wave, sweeping down on the sand-bar had gained extra height as it was forced against the bluff, carrying the stragglers away as it passed.
The bulk of the force who managed to reach the northern shore escaped with their lives but there was no cheering, no celebration. The circumstances surrounding their deliverance were so awesome, the scale of death so overwhelming, the survivors fell wordlessly to the ground, overcome with grief and sheer physical exhaustion.
Mr Snow was one of them. Awesome-Wells and Boston-Bruin ran towards him. For several agonizing minutes they thought he was dead. The lighter parts of his skin were greyish-white, and his almost weightless body, cradled in their arms, felt like a sack of loose bones. Eventually his eyes fluttered open, but they too were drained of life.
Awesome-Wells took hold of Mr Snow's hands and squeezed the unresisting fingers. 'Old One! Old One! Do you know who I am?'
Mr Snow's eyes searched slowly for the speaker but seemed unable to focus on the faces of those who knelt over him. His lips moved soundlessly.
Awesome-Wells placed his ear against them to try and catch what looked like the last words the Old One might utter.
'Is it... over...?'
'Yes. Most of the She-Kargo and their blood-brothers have survived but everything else has gone. The D'Troit, C'Natti and San'Louis, the wheel-boats, the trading post... all have been swept away.' He paused but there was no response. Awesome looked up at those around him then sought the answer to one last question. 'Tell us, Old One – was this the Great Dying the Sky Voices spoke of so long ago?'
'Part... of it, yes. Talisman has... has cut... the poisoned... branches from... the... tree that is the... Plainfolk. We who... have... been chosen must... must make sure... they never... grow... again.'
'He did more than cut the poisoned branches!' cried Boston-Bruin. His voice was harsh with anger and grief. 'The She-Kargo, M'Waukee and San'Paul will never be able to count their dead! We alone have lost close to half our clanfolk! If, as you say, we're supposed to be Talisman's chosen people why did he not spare all of us?!'
Mr Snow closed his eyes. 'Ahhh, Boston... Boston,' he whispered. 'Why must you... always... ask questions to which there... is... no... answer?'
## CHAPTER SEVEN
Two days later, when Steve, Night-Fever and the band of M'Call Bears came in sight of Lake Superior, the water carried inland by the tidal wave had not yet drained from the dips and hollows in the terrain.
Ranging in size from large puddles to miniature lakes, the farthest pools were up to seven miles from the fatal shore. Bodies and debris littered the entire triangle of land enclosed by the estuary of the St Louis River but the eye was drawn to the shattered remains of the five wheel-boats which rose out of the waterlogged grass like the dismembered carcasses of beached whales.
Moving mechanically through this sombre landscape were groups of scavengers; the numbed survivors of the She-Kargo, M'Waukee and San'Paul. It was clear there had been a major catastrophe with massive loss of life, and to judge from what was left of the wheel-boats, the Iron Master trading expedition had been virtually wiped out.
Steve dismounted and used a length of cord to fasten the front and rear left leg of his horse together. This allowed the animal to walk and graze freely but stopped it from straying too far. Until they found out what had happened and who had won, it would have been unwise to proceed further on horseback. It could raise questions that might prove difficult to answer – especially if they were posed by a bunch of bellicose nips facing a long walk home.
Leaving Night-Fever to guard the roped horse against jackals and other predators, Steve's party made their way down the slope and met up with a small group of She-Kargo Mutes. They were from the Clan M'Kormik, a close neighbour of the M'Call's. All of them looked pale and subdued. Cat-Ballou, the leader of Steve's escort, exchanged formal greetings then, in hushed tones, asked what had caused such devastation.
For a moment none of the M'Kormik warriors replied then, one by one, beginning with a Bear called First-Blood, they took turns to describe what had happened with a kind of dazed weariness. There was none of the elation which, as victors, they might have been expected to display. Mutes were no strangers to pain or sudden death but the tidal wave had claimed nearly two hundred thousand lives. No one had witnessed destruction on this scale since the legendary War of a Thousand Suns.
Rough-Cut, the last to speak, concluded the harrowing tale with a sweeping gesture that took in the empty shoreline which, as Steve suddenly realized, had lost its familiar landmark.
'Even the trading post has gone! Torn from the ground and snapped in two by the wrath of Talisman.'
'Oyy-yehhh!' groaned the M'Call Bears.
'And the Iron Masters?' enquired Steve.
Rough-Cut's face darkened. 'We have killed all those in which a breath of life lingered.'
'And the treacherous dogs from the bloodline of the D'Troit have suffered the same fate,' added First-Blood. 'But all those from the C'Natti and San'Louis who could still walk were sent away unharmed.'
Steve frowned. 'I thought they were in this together.'
'They were,' replied First-Blood, 'But our elders said they should be allowed to go free to bear witness to the fate reserved for those who deny Talisman and seek to divide the Plainfolk!'
'And what of our wordsmith, Mr Snow?' asked Cat-Ballou. 'The Storm-Bringer, whose body served as the hammer which Talisman used to smite the D'Troit? Where can we find him?'
Rough-Cut exchanged glances with his companions then pointed towards the high ground north of the lagoon. 'His body lies above the Great river. There, where you see the smoke rising.'
Steve's stomach turned over. 'His body?! Is he dead?'
'It is hard to say,' replied First-Blood. 'The elders say he crosses over then returns. With so many dead, Mo-Town's cup must be filled to overflowing. Perhaps there is not yet room for him on the other side.'
_Terrific... So much for the master plan_. Steve tried to sound cool, calm and collected. 'How can we get there?'
First-Blood directed him to the river crossing then he and his companions moved past them in search of another untouched heap of debris.
Steve hurried back to where he had left the horse, climbed into the saddle, told Night-Fever where he was going, and galloped back to Cat-Ballou. 'See you guys up there – okay?'
The M'Call Bears watched him ride away then looked at each other in silent agreement. Having walked and run seven hundred miles, sitting on the back of a horse and letting _it_ carry you where you wanted to go was beginning to look like an interesting idea.
After leading his horse up the face of the bluff, one of the first persons Steve came across was Carnegie-Hall. Before he could open his mouth, Steve took the wordsmith aside and explained that he was travelling incognito. Talisman did not wish him to be identified as one of The Chosen at this particular moment in time.
Flattered to be the sole guardian of such an earth-shaking secret, Carnegie-Hall readily agreed to keep shtum. He led Steve over to Mr Snow's makeshift bed where he was greeted by the surviving clan elders from the M'Call trade delegation – Awesome-Wells and Boston-Bruin. Blue-Thunder, the paramount warrior was there too; wounded, but still mobile – along with several other Bears that Steve recognized.
Steve sat cross-legged for over an hour by Mr Snow's side willing his eyes to open. Eventually they did. All the colour and sparkle had gone out of them, but there was a faint flicker of recognition when they finally alighted on Steve.
'What kept you?' The Old One's face was thin and drawn – aged almost beyond recognition, his voice barely audible.
'Oh, this and that.'
Mr Snow's eyes wandered past him then gave up the search. 'Where's...?'
'They couldn't make it.' Steve forced a lightness into his voice. 'You going to lie there for ever? Your friends here want to invite you to a big party.'
Mr Snow's lips twitched in a pale imitation of his mischievous smile. 'You... may have to... start... without me.' Stringing more than three words together seemed to drain all his strength. He paused for a while and tried again. 'Are they... safe?'
'Cadillac is.' Steve outlined what had happened after leaving the Kojak. Having spoken to Carnegie-Hall, he knew there was no need to repeat the whole story. It was difficult to know how much Mr Snow was taking in. He took hold of a withered hand. It felt so fragile, he dared not squeeze it for fear of crushing the bones. 'That's why we need you back on your feet. You're the only one with the power to get clearwater off that train.'
Mr Snow sighed and closed his eyes. The forefinger of the hand lying on his chest twitched, beckoning Steve to come closer.
Steve leant over to catch what he was trying to say.
'What are you... trying to... do... kill me?'
In the days following the Battle of the Trading Post, the clan elders and warriors of the victorious delegations conferred on what to do next. Clearly nothing could be as it was before. When the D'Troit, C'Natti and San'Louis clans learned of their defeat there would be great bitterness. The enmity between the opposing bloodlines would deepen, the character of the fighting would change. Not the traditional clashes between rival bands of warriors but large-scale confrontations – and perhaps a permanent state of war.
It was a gloomy prospect. The She-Kargo, M'Waukee and San'Paul had made common cause against the D'Troit faction but that did not mean their own traditional hostility to one another had evaporated. In a sense, this year's journey to the trading post had been the Plainfolk's road to Damascus but there had been no miraculous conversion. The atmosphere was not now filled with harmony and light. Each clan, while sharing the bond of a common bloodline was still fiercely protective of its own identity and its turf.
What had brought the She-Kargo, M'Waukee and San'Paul together on that fateful day and the desperate hours preceding it was their unshakeable belief in Talisman and their willingness to accept the advent of The Chosen, whereas the D'Troit and its fellow-travellers were apparently ready to ditch those self-same beliefs in exchange for a deal with the hated dead-faces.
All the clans wished to trade with the Iron Masters if only for the simple reason that no one wanted to find themselves living next-door to a better-armed – and thus stronger – clan. But the Mutes were under no illusions about the impact these trading contacts would have on the traditional life-style and belief-system of the Plainfolk. From the very beginning, they knew they were getting the worst of a deal. And the fact they had been forced to trade by the threat of armed invasion by highly-organized predators had offended their sense of honour.
But it was the best and only offer available. And by accepting it, the Mutes had been able to delay the advance of their other enemy – the sand-burrowers. Everyone knew they were being ripped off but the point was _all_ the clans from the six major bloodlines had accepted the conditions laid down by the Iron Masters. It was a collective rip-off and that, in some strange way, made it bearable. The mistake the D'Troit had made was in appearing to have come to a more favourable arrangement with the dead-faces in a way that would give the D'Troit extra power and advantages the other bloodlines would not share.
They had come close to succeeding, but their attempt to win hearts and minds by isolating the M'Calls and the She-Kargo had been foiled by Mr Snow's rousing defence. In implementing Lord Hirohito's plan, his successor, Aishi Sakimoto, had overlooked the crucial flaw in the domain-lord's strategy. The endemic hostility between individual clans and bloodlines imperilled the long-term future of the Plainfolk but the She-Kargo, M'Waukee, San'Paul, San'Louis – and to some extent the C'Natti – were united by their fear and hatred of the D'Troit.
The majority of the C'Natti delegations had submerged these feelings in the hope of gaining the advantages that would accrue from being on the winning side – and had suffered terrible losses as a result. Whether they would now change sides remained to be seen. The few C'Natti who had survived the tidal wave and had been spared in the subsequent mopping-up operation were still too shocked to think rationally about the future.
It was a future which no one taking part in this first council could contemplate with any great pleasure. It seemed unlikely that the Iron Masters would return cap-in-hand to apologize for instigating the attack on the She-Kargo. They had obviously intended to punish the M'Calls and the entire bloodline for the actions of The Chosen in the Eastern Lands. This could not be allowed to happen. No one who believed in the Talisman Prophecy could fail to defend those who had been sent to herald the coming of the Thrice-Gifted One.
The dead-faces would return, but not to trade. From now on, the Plainfolk would have to rely on their own resources. Still stunned by the massive and merciless blow Talisman had struck on their behalf, few relished the prospect of defending themselves against marauding bands of dead-faces armed with an endless supply of long sharp-iron.
At the outset it seemed an impossible goal. Mr Snow, exhausted and at death's door, was unable to lend his reasoned eloquence or wisdom to the debate but at the end of five argumentative and sometimes acrimonious days a fragile consensus emerged. All agreed that the Battle of the Trading Post marked the end of an era. From henceforth, the interests of individual clans would have to take second place to the protection of the Plainfolk. Warriors who, over the centuries, had been blooded by challenging and killing their peers from neighbouring clans would now have to earn their standing in combat with the _real_ enemies of the Plainfolk – the dead-faces and the sand-burrowers.
It was this issue – inter-clan blood-letting – that aroused the most controversy. To deny this right of challenge struck deep into the warrior ethos. The courage to brave close, single-combat was – for the Mutes – absolutely fundamental. It was the corner-stone of their existence, the means by which an individual and the clan to which they belonged measured their worth.
The history of the Plainfolk was the history of heroes. Those who had 'chewed bone' stout hearts whose sharpiron had filled head-poles. Before the Battle of the Trading Post, no warrior worthy of the name would have used a crossbow in combat against his peers. Long sharp-iron was used solely for hunting game; the rifles and other weapons used by the sand-burrowers were proof of their cowardice, their lack of honour. They were animals whose blood-lust knew no bounds. That was why it was permissible to use the cross-bow when fighting them.
The Iron Masters used a different kind of bow and arrow made of wood. It had a shorter range than the crossbow but a much higher rate of fire. They also possessed other fearsome sharp-iron, long curved swords that could slice through flesh and bone and the heavy black pipes on the wheel-boats that spoke with the voice of sky-thunder and whose fiery breath could tear a man to pieces at five hundred paces.
Their readiness to use such weapons plus the fact they were an alien race placed them in the same category as the sand-burrowers. A category to which – because of their denial of Talisman – the D'Troit and their co-conspirators had also been consigned.
The use of long sharp-iron had always aroused strong emotions. Following the battle with The Lady from Louisiana, Mr Snow had stressed the need to develop not just new tactics but a totally new concept of making war. His radical proposals had not gone down well with the traditionalists within his own clan. There had been further objections following the gift by Lord Yama-Shita of a hundred repeating rifles. Cut off from further supplies of ammunition, they were now useless – as his opponents had predicted.
Such deep-rooted differences of opinion over such a wide range of issues could not be resolved in the space of five days, but there was general agreement that the discussions should continue. The number of the issues on which they were united was greater than the sum total of those on which they remained divided.
Each delegation made a solemn pledge to come to the aid of any clan attacked by the D'Troit faction or the dead-faces, but before this could happen the present patchy bush-telegraph had to be replaced by a more effective nation-wide system of communications. The discussions on how this could be set up would continue within each bloodline, and it was agreed that delegates from the clans now present would consider any proposals put forward when the first Plainfolk Council was convened at Big Running White Water (Sioux Falls, South Dakota) in three months' time.
Meanwhile, the five-week truce which came into effect during the trading period was to be extended through the summer and would embrace any clans of the D'Troit and C'Natti who were prepared to recognize The Chosen as the heralds of Talisman and pledge their allegiance to the Plainfolk.
In his guise as a simple warrior from the Clan M'Call, Steve took no part in the discussions and his interest in the emerging alliance was, at best, academic. His overriding concern was the health of Mr Snow.
The M'Call delegation were also concerned but not to the same extent. They were governed by the fatalism that coloured the Mute view of death. After his experiences on the overground, Steve was increasingly drawn to the idea of a continuing cycle of existence, but in his present predicament the prospect of Mr Snow re-entering the world at some future date in a new body offered little consolation. What he needed was Mr Snow up on his feet and able to call down some heavy shit in the here and now.
As a wordsmith, Mr Snow could be replaced by Cadillac. He did not have the Old One's stature but even Mr Snow had been a young man once. Cadillac's appointment would give him standing within the clan and in time he would grow into the job, but he was not a summoner. There was only one person who could replace the Storm-Bringer – Clearwater.
In urging the clan elders to do their utmost to keep Mr Snow alive, Steve described the powers Clearwater had revealed during their escape from Ne-Issan – powers that might one day surpass even those of the Old One. Aroused by his tales of derring-do the elders promised to do all they could to help rescue her from Red River – even if Mr Snow died. If it was the will of Talisman, they assured him, the whole clan would lay down their lives for her.
A nice gesture but one which would make her rescue a totally pointless operation. In trying to sound helpful, the elders were blinding themselves to the fact that, without Mr Snow's presence and the use of his power Clearwater could not be rescued. In his many arguments with Cadillac, Steve had hammered on about not relying on earth-magic but this time their lives depended on it. Only a summoner with the power and presence of Mr Snow could give them the edge they needed.
It was not just practical considerations that were involved; there was a moral dimension. Despite the promise made by the clan elders, Steve knew that without Mr Snow, the M'Calls would not attack the wagon-train. And if Cadillac tried to persuade them to do so, Steve would do his utmost to prevent it. He had witnessed the reckless bravery of the Bears at the Battle of The Now and Then River when they had been aided by Mr Snow's earth-magic. His powers had been awesome but in the end, the M'Calls' irrational desire to throw themselves headlong upon the enemy regardless of the cost had led to a senseless slaughter and eventual rout. After the losses suffered at the trading post he was not prepared to support any ill-prepared adventures that could lead to even greater sacrifice.
_Not even for you, little sister_...
Over the next few days, as the talks between the delegations continued, Mr Snow gradually recovered some of his strength. He could not sit up unaided, and he was far too weak to walk, but his spirit had returned from that twilight zone between this world and the next and was once more anchored firmly in his physical body. His face was still pale and haggard but the colour and some of the sparkle had come back into his eyes, and the famous smile, though tired, occasionally contained a hint of its old mischief.
His voice now seemed to be set in a husky whisper and he spoke only when necessary using as few words as possible, but as time passed, his powers of concentration increased, and from what little he said it was clear that the brain inside the wizened, white-haired skull was as sharp as a razor.
During these longer periods of wakefulness, Steve related in greater detail his adventures in Ne-Issan with the _ronin_ , his high-risk, heart-stopping involvement with the Herald Hase-Gawa, the conspiracy against the Shogun, and the final battle and escape from the Heron Pool.
'I'm not surprised the Yama-Shita... went... for the throat,' husked Mr Snow.
'Yeah, but it's not entirely my fault,' protested Steve. 'The Heron Pool was blown apart because the Shogun wanted to crush the conspiracy against him. And the reason we got mixed up in it was because of _your_ deal with Lord Yama-Shita. If you hadn't sent Cadillac and Clearwater over there none of this would have happened!'
'Just doing what I was told...'
'You mean by the Sky Voices? Talisman?'
'The Path is drawn, Brickman. Someone once said: "All the world's a stage, and all the men and women merely players".'
'Yeah...' Steve was unaware of the immense legacy of dramatic art and literature that had been buried by the Holocaust so the quotation made little sense.
'Anyway,' he continued, 'Cadillac fell on his feet, but for a while she had a really bad time. So when we blew up the Heron Pool, she took the opportunity to even the score – saving our necks in the process. Do you blame her?'
'No.' Mr Snow gave a breathy laugh. 'Life isn't fair is it? Poor old Prime-Cut –'
'The big-noise from the D'Troit?'
'Yes. Apart from us having done a deal with the Federation, every charge he levelled against us was true.'
'And in spite of that you ran rings round him.'
'So everyone says. But I didn't lie.'
'Well, except for that bit about The Chosen.'
'That wasn't a lie. You _are_ one of The Chosen.'
Steve frowned. 'But I'm not a Mute.'
'No.' Mr Snow closed his eyes and let his head fall onto the furs that covered the sloping back-rest.
_Not yet, Brickman. Not yet_...
Having reached a measure of agreement, the delegations spent the sixth and seventh day trading amongst themselves. In addition to preserving most of the goods they had brought with them to trade with the Iron Masters, everyone who was not engaged in deciding the future of the Plainfolk had been busily scouring the battlefield for whatever they could find: crossbows, bolts, knives, items of clothing, Iron Master swords, helmets and body armour, ship's cannon – the list was endless. The dead – even their own – were too numerous to cope with. Unable to give them the usual burial rites or collect them in heaps for burning, the corpses were stripped and left for the circling death-birds.
Those who decided they had too many crossbows and not enough bolts, or preferred to exchange both for samurai long-swords, set up an impromptu arms bazaar. Their initiative was quickly followed by their rivals and it was not long before each delegation set out its other wares – food, furs, dried fish and buffalo meat, dream-cap and rainbow-grass.
It proved to be a novel and rewarding experience, and everyone promised to bring similar goods to the proposed meeting of the Plainfolk Council at Sioux Falls. Why, they asked each other, had no one ever thought of this before? Nobody suggested trading the surviving journey-men. The idea that males and females from one clan might take partners from another was so radical it simply never entered anyone's head.
Laden with their booty, the clan delegations bade each other farewell and went their various ways. The happiest were the journey-men and -women who had been chosen to make the journey down the Great River and were going home instead. But their joy at returning to their families was muted by the loss of their clan-brothers and sisters. The general mood was one of sombre optimism. There could be no going back, and no one looked back as another wheeling flock of death-birds, drawn in from all corners of the sky, swooped down to join those already gorging themselves on the fallen.
When Mr Snow woke up and looked around him, he saw the surviving members of the M'Call delegation preparing to join the general exodus. Steve, sitting cross-legged by his side did not appear to have moved an inch.
Their eyes met. 'You still here?'
'I've got a problem I need to talk to you about.'
Mr Snow sighed. 'Who hasn't? Okay.. shoot.'
Steve explained the set-up on board Red River, and how Cadillac and Malone were working on a plan to free Roz and Clearwater.
'The thing is... Malone is an undercover agent – a mexican, like me. We're supposed to be working together to capture you and Cadillac'
'Helped by your kin-sister...'
'Yeah. Roz smart-talked the Federation into thinking she was their secret weapon. An anti-summoner device. She's managed to get this far but until she and Clearwater are free and clear, I have to pretend I'm still a one hundred per cent solid, dependable soldier-citizen. Otherwise...'
'Yes, yes, I've got the picture...'
'Good. The people on board Red River think they've got the drop on us, but of course with your help we'll have the drop on them. I still have to work out the way we're gonna swing it but if we all put our heads together we should be able to come up with something.'
'I see... so what's the problem?'
'How do I tell Cadillac that Malone's a phoney without him blowing his top? When he finds out I've been stringing him along he's gonna go apeshit. We've already crossed knives over Roz and I don't want to get into that again.
'But more important still I don't want him sending a bunch of Bears out to collect Malone's head. We need him alive and working with us because _he's_ the key to getting on board the wagon-train.'
Steve paused then said: 'It'd made things a lot easier all round if you told him. I mean – you _are_ his teacher.'
'Yes...'
'I'm sorry. What exactly is it you're saying? "Yes" you are his teacher, or "Yes" you'll tell him?'
Mr Snow's left hand waved feebly. 'You're throwing too much at me, Brickman. Give me some... some time... to think about it.'
Steve cursed silently as the wordsmith closed his eyes. _Time! That was the one thing they didn't have!_
Mr Snow's hand slipped off the stretcher. Steve took hold of it and laid it across the old man's chest.
The wasted flesh felt cold. Too cold...
Seven hundred miles south and to the west of the now-ruined trading post, another pawn moved a stage further towards its final position in the game as The Lady from Louisiana rolled into the loading bay of the way-station above the almost completed divisional base at Monroe/Wichita. The fuel hoppers that fed the power-cars were topped up, and vital electronic components that had been shuttled northwards in response to an urgent radio message were installed and tested in an effort to get the trouble-prone communications system back into full working order.
The Lady also took on some other unexpected items of freight: a large sealed cargo skip whose contents were not listed on the external manifest docket, and sixty hooded defaulters wearing wrist and foot shackles and the usual black fatigues with the yellow diagonal cross stripes on the chest and back.
But that was not the end of the surprises. A major from the White House, dressed in the same red combat fatigues, came on board with a diskette sealed with an "EYES ONLY" label bearing Hartmann's name.
CINC-TRAIN usually sent top secret instructions by radio in coded blocks that were converted by The Lady's communications unit into a video signal and fed into the work-station in Hartmann's private quarters. Only Hartmann himself, using his ID card and a special key sequence could screen the blocks and turn them back into the original message.
Sealed diskettes delivered by high-ranking couriers from the White House usually contained _ultra_ -secret material; orders so sensitive they could not even be transmitted in code through the normal channels. Hartmann used his ID card to 'sign' for the diskette and considered it at some length, holding the small, slim square silver package by his fingertips as he turned it this way and that. This had to be hot stuff...
It was.
After locking his door, Hartmann broke the seal, slipped the diskette into the drive slot of his PC, unlocked the system with his ID card and hit the keys to screen and decode the signal from the White-House. The jumble of digits and letters rearranged themselves into lines of clear text. Scrolling carefully through the secret signal the wagon-master discovered why he had been given custody of sixty defaulters – all of whom were under sentence of death. He also learned what was in the cargo skip, when it had to be opened and the reason why a tight-lipped group of service engineers were making certain external modifications to his wagon-train.
The diskette contained precise instructions on when he was to brief his executive officers and crew on the nature of their mission, and ended by informing him that the flight section of The Lady was to be re-equipped with Skyhawk Mark Two's.
In order to allow his wing-men to familiarize themselves with this new machine in the field, Hartmann was ordered to fly off five of his present complement of ten Mark One's to Red River who would replace them with five Mark Two's and their pilots. Half of the flight-deck crew riding on buddy-frames would also be exchanged, allowing the necessary training of air and ground personnel to proceed simultaneously on both wagon-trains.
To Hartmann, the prospect of receiving a new batch of aircraft combined with The Lady's involvement in a secret mission was a sure sign that he and the surviving members of his original crew were on their way to complete rehabilitation. This time at least, the outcome would not be affected by the baleful influence of a Mute summoner. This time, the crewmen who had lost their lives in the past two near-catastrophic encounters would be avenged. The Lady would emerge with her head high.
_Third time lucky. Wasn't that what they said...?_
* * *
About the same time that Hartmann was reflecting on his good fortune, a soldier-citizen of the Federation by the name of Jake Olsen was taking stock of his surroundings and wondering whether he was dreaming or if, in some mysterious way, his luck had taken a turn for the better.
It could hardly get much worse. Arrested and charged under Code One, Olsen had been sentenced to death for the illegal recording and trafficking of blackjack tapes. The severity of the sentence reflected the scale and success of the operation. Olsen and his partners had set up a Federation-wide network of dealers and had avoided detection for over seven years by buying off provost-marshals with bagfuls of top-quality rainbow-grass supplied by overground contacts in the Mines & Mills Division.
The same techniques had been used to secure the cooperation of AmExecs in the Black Tower and low-ranking members of the First Family. They had helped stall and side-track investigations, and had secured plush apartments in the new luxury tower-deeps for the ringleaders.
It had all gone well. Too well. As an electronics expert, it had been Olsen's task to assemble the necessary equipment and oversee the recording and duplication of the illegal music tapes. There had been a lot of work cleaning up the sound quality because his raw material consisted of _nth_ generation copies of the original pre-Holocaust masters. He had even re-recorded entire sequences, matching the originals as faithfully as possible using a synthesizer built from components filched from the First Family's own music workshops.
Yeah, that had been sweet. The mistake had been to branch out into the manufacture of alcohol. In a society where almost everything is controlled by computer, illegal electronic items are easy to hide. Hook 'em into some official circuitry and they're hard to distinguish from the real thing. Crates of plastic containers full of straw-coloured liquid with a distinctive aroma proved to be a whole new ball game and the problem was not eased by some bright spark who labelled a consignment destined for the junior medical staff at the Life Institute as 'urine samples'.
Looking back, it was clear the operation had been penetrated by undercover Feds for some time, but it was after that particular consignment was intercepted that everything started to unravel at the speed of light. Nation-wide, over three hundred people were arrested, charged and tried. But after six televised executions by firing squad, the judicial process had suddenly been put on hold.
For the last five weeks, Olsen had been sitting in his cell on death row, wondering what had caused the delay. Despite his sudden downfall, Olsen was as cheerful as might be expected in the circumstances but he was also a realist. A reprieve was out of the question. The operation in which he'd been involved had caused the high-wires too much embarrassment but in the seven years before his arrest, Olsen had discovered that the First Family was running its own illegal operations – scams to trap potential subversives and budding law-breakers. The possibility that the Family might have decided to recruit him into their own 'bad-hat' brigade was the only plausible explanation for the unexpected stay of execution.
If so, there was several questions to be answered. His brain felt muzzy and seemed to working at half-speed but the last thing Olsen remembered was going to sleep in his small over-warm nine foot by six foot cell in his black T-shirt and underpants, on a narrow bunk with a thin coverlet on top of him, and with the light behind the grille in the ceiling cut to a third – the level of night-time illumination known to Trackers as "twilight".
Now, the biggest and brightest light he had ever seen was beaming down on him from a blue ceiling whose height was beyond computation. And the narrow confines of his cell had been replaced by vast, seemingly limitless horizons. John Wayne Plaza was big and brightly lit but what now met his eyes on all sides was... something else.
Yessirrr...
Olsen knew that if he was still asleep, this was a dream. On the other hand, if he was awake (and despite the utter impossibility of him being where he was, he had an alarming feeling that he _was_ fully conscious) then he had been plunged into a nightmare from which he knew, with mind-numbing certainty, he would never escape.
Despite his intelligence, technical skill and flair for organization, Jake Olsen had never been overground. Indeed he had used that same intelligence to arrange 'soft postings' within the earth-shield, to the units which provided him with the contacts he needed to gain access to the equipment that interested him, and the most rewarding 'career opportunities'.
Now, on a day whose date he was unable to determine, he was sitting supported by his hands with his feet out in front of him in a clearing on a forested hillside which afforded a view over a vast panoramic landscape which stretched away to the east, south and west. Olsen was able to determine this because he had seen video-pictures of the overground and knew the difference between a rising and setting sun.
Behind him, was an upward slope with more red pine trees and rocks peeking out of the tangled undergrowth. He was on a hill, or some kind of mountain. The air had a strange smell and a sharp cold, but not unpleasant, cutting edge to it when he drew it down into his lungs. Where the hell was he? And what the fuck was going on?!
Olsen looked down at his feet and saw that the lower half of his body was inside a camouflage sleeping bag. A further inspection revealed that he was dressed in a set of the standard red, pink, orange and brown camouflaged fatigues issued to overground units and on his right shoulder was a woven fabric badge.
SIG-INT... What was he doing wearing a uniform of some guy in the 5th Signals Intelligence Squadron?!
Olsen pulled out his tunic and peered down at the name tag above the right breast pocket. 693 OLSEN J.E... Shit... This wasn't someone else's uniform, it had his name on it! There was a carbine lying alongside his sleeping bag and a visored helmet. That had his name on it too. He rubbed his forehead and swallowed, trying to rid himself of a sudden dizzying wave of nausea.
Focusing his attention on his immediate surroundings, Olsen was comforted by the fact that he was not alone. Scattered around the clearing and under the trees beyond were a considerable number of recumbent figures in hooded sleeping bags. Too many to count. And there were small heaps of equipment, ration packs, the burnt-out remains of several campfires, and the lower section of some kind of tower made out of slim, red-painted aluminium girders and mounted on a small concrete base. More half-assembled sections lay nearby.
It looked like a UHF radio beacon. That would explain the SIG-INT shoulder badge on his arm. But why did he have no recollection of being transferred from death row into an overground signals outfit? He checked the digital watch someone had strapped on his left wrist: it was nearly 11.00 hours. Why hadn't he and the other sleepers been rousted from their beds? Why was no one awake and on guard? Gripped by rising panic, Olsen tried to get up. It was more difficult than he expected. His legs felt leaden. Stooping down to pick up the carbine and helmet made his head swim. He dropped to one knee to steady himself and had to wait a while for the giddiness to clear then tried again, using the carbine as a prop to haul himself upright.
The short barrelled weapon had a magazine inserted into the lower chamber in front of the trigger guard. Olsen released it and weighed it in his hand. It felt about half full. The air pressure read-out was above the red line. Good enough. Olsen clipped the mag into the chamber, eased off the safety and fired a single volley into the air.
_Chuww-wiittt! Chuww-wiittt! Chuww-wiittt!_
'At least something's working,' he grunted.
A flak-jacket with its interlaced webbing harness, magazine pouches and air-bottles lay beside his sleeping bag. Olsen knew it was something people on the overground should wear at all times but he didn't want to load himself down. Ever since he put the helmet on it had made his neck ache. It was like a ten-ton weight pressing on his spine.
His present wretched state was not unlike the hangover that came from smoking sour grass and drinking joy-juice at one and the same time. Olsen knew he hadn't done either. The last meal delivered to his cell had been spiked. And while he had lain unconscious – for an undetermined period of time – someone had injected a cocktail of drugs that had left him physically and mentally hamstrung.
With his rubbery legs splayed in an effort to keep his balance Olsen shambled over to the nearest of the sleeping figures and prodded it awake with the butt of his carbine. And he was not at all surprised to find that the occupant of the bag was someone he knew. The bleary face that appeared out of the hood belonged to Marv Dandridge, a close colleague, one of the top men running the blackjack ring and one of his neighbours on death row.
Dandridge hoisted himself up on one elbow and tried to get his sodden brain into gear. He looked around him with growing amazement then shook his head. 'I don't believe this. What the hell are you doing dressed up like –' He broke off as he realized he was wearing a similar outfit. 'Ohh, jeezuss...' He turned an anguished face towards Olsen. 'Jake – is this for real?'
'You'd better believe it.'
Dandridge sat up. 'Where are we?'
'How the fuck do I know?! I ain't never been overground! All I know is we're in deep shit.'
Dandridge's drugged brain was slow on the uptake. 'Why? Wass-happnin'?'
'Fer crissakes, Marv!' cried Olsen. 'D'you think they dressed us up in these uniforms for fun?! _This_ is why we all got that stay of execution! We've been set up! Well and truly shafted!'
'Christo!' Dandridge scrambled to his feet and threw an arm out to steady himself against Olsen. 'Hey! My arms and legs feel kinda funny.'
'Mine too. That's cos they doped us up to the eyeballs before they dumped us out here.'
The words took time to register. 'You mean... to be killed?'
'What d'you think they're gonna do – pin a fuckin' medal on us?! This is a scam set up by the First Family, and we're the meat on the hook! _Dead_ meat if we don't move fast!'
'Christo...' The threat restored a measure of lucidity to Dandridge's brain and put some life back into his leaden limbs. 'How are we gonna get outta this mess?'
'We have to find where we are first.' Olsen patted Dandridge on the shoulder. 'There's a load of equipment lying around. See if you can find a map case while I check out who else is here.'
Dandridge surveyed the sleeping figures, some of whom had begun to stir. 'These are... our people?'
'It wouldn't surprise me.'
'Evil bastards...'
'Yeah,' laughed Olsen. 'Y'gotta hand it to 'em. A nice twist, huh?'
He didn't know everyone who'd been involved in the blackjack operation but among those he found awake, or managed to rouse were fifteen people he'd worked with or met when many of those arrested had shared communal cells while waiting to be interrogated. To have one of your number taken out and marched away then returned several hours later beaten and bloodied and unable to stand never failed to sap the morale of those whose turn had yet to come.
Those fifteen, when they could stand on their feet, found others they knew – all incapacitated to some degree by the drugs that had been pumped into them. Olsen's worst fears were confirmed. The First Family had emptied death row at Grand Central. But along with those sentenced to death for their part in the blackjack operation there were Code One offenders from other divisional bases. There were even some Code Two's who'd been sentenced to ten years of hard labour. Put in to make the number up.
_Tough_...
Olsen rejoined Dandridge and some of his other cronies who were clustered round a map. 'There's a hundred of us, all told. Anybody figured out where we are?'
'Yeah,' said Dandridge. 'Jerry here's the expert. If this map is kosher, then from what he can see of the lie of the land, he says we're somewhere around here.'
Olsen looked at the point indicated by Dandridge's finger. 'Iron Mountain...'
'North of navref Cheyenne,' explained Jerry. 'We're in Southern Wyoming. North of Colorado.'
Olsen mastered another spell of giddiness. 'You mean we're not even in the fuckin' Federation?!'
'No,' said Jerry. 'This is Plainfolk territory.'
Olsen closed his eyes. 'Shit and corruption! I knew it, I knew it!'
'The nearest way-station is Pueblo. I don't know how far that is. This map doesn't cover Colorado but it must be several hundred miles.'
'What are we gonna do?' asked Dandridge.
'Well, we can't go to Pueblo, that's for sure!' cried Olsen. 'We're under sentence of death, remember? We'll just have to make a stand right here.'
'Against what?' asked someone in the group that had gathered behind him.
'How the fuck do I know?' exclaimed Olsen. Then, addressing the group as a whole he said. 'The only thing we can be sure of is the Family didn't put us jailbirds here because they thought we needed the fresh air! I don't know how you guys feel but I vote we grab hold of as much of this gear as we can and get under cover. Find a cave, or somethin'.'
He turned back to Dandridge and Jerry the mapreader. 'Is it getting to you?'
Dandridge frowned. What?'
'The space! There's too much fuckin' sky! I can't handle it! When I first woke up I was okay but now it's givin' me the shakes!'
'It's this dope they slipped us.'
'No, it isn't. I heard about this. Lots of guys go down with it – end up not bein' able to move.'
'We'll be okay if we stick together,' said Jerry. 'Ground-sickness can be beaten. You just gotta have the right mental attitude. Keep your eyes down. Don't take in too much at once.'
'You been overground before?'
'Yeah. Two years with Mines and Mills.'
'And you never felt bad?'
'You get acclimatized. The new guys spend the first three months on internal duties – inside the plant buildings. You work up to it by looking out of the windows and when you're good and ready you step out into the Big Open. The guys that can't handle it get reassigned'
'Well, I aint had three months to get ready! I went to sleep in my cell and I woke up here! And it's beginning to scare the hell out of me!' Keeping his eyes on the ground in front of him, Olsen pushed his way through the group and shambled across the slope to the sleeping bag he'd vacated earlier.
By the time he reached it, tears were streaming down his face. Even a prison cell, for as long as you occupied it, was your own space. Out here, there were no walls, no roofs, nothing to keep out this oppressive emptiness. The sleeping bag and the piece of earth on which it lay was the only thing he could relate to; the only point of contact with the concrete cocoon in which he spent the weeks since his trial.
He fell to his knees beside it. He had gone back to get the flak-jacket. Given the jam they were in, it made sense to put it on. Attached to the webbing straps were pouches of ammunition for the carbine, a hand grenade, machete, and battery packs to power the radio and target acquisition arrays on his visor. Once the helmet umbilical was plugged into the butt of his carbine, he'd be able to hit anything he aimed at.
Like a bunch of screaming Mutes. Out here that was the most likely thing to happen. A firing squad would have been better. Cleaner anyway. Out here, after they'd stuck your head on a pole and hammered it down so's the point came out through the top of your skull, they cut the rest of you up and turned you into smoked meat twists.
There was no point in moving camp. Best thing was to try and dig a hole under the trees. Get some earth round you and some branches over the top and hope you were asleep when it got dark. It would be a good idea to take some ration packs and –
Olsen heard someone shouting. Then other people calling to each other. He looked around and saw one of the camouflaged cee-bees standing on a rocky outcrop further up the slope. He was holding a 'scope.
Dandridge came over. 'We've spotted what looks like a bunch of renegades moving across a flat-topped ridge below us. An' y'know what? They got horses! Just like in the archives! You can't see 'em from here. D'you wanna come up and take a look?'
'What for?'
'C'mon, Jake! They're our kind of people! If we could hook up with them...' Dandridge saw he was getting nowhere. 'Some of the guys want to send up some signal flares.' He shrugged. 'What the hell? We might as well try. Things can't get much worse – right?'
'Right...'
Olsen watched his one-time associate hurry away. Good old Marv. He was doing a great job holding it down, but he'd never been overground either, and he was getting the shakes too.
A green signal flare soared into the air, and was quickly followed by several more. A dozen or so cee-bees were clustered on the outcrop, and more were scrambling up towards them.
Someone shouted: 'They've seen us! Look! They've put up a green.'
The news drew a ragged cheer from those who had found the energy and the will to get up and stay on their feet. But there were many, like Olsen, who had slumped back onto the ground, enfeebled by the drugs pumped into them, disoriented by their surroundings or simply overwhelmed by their predicament.
_Zzzhhahhwikkk! Zzzhhahhwokkk!_
Olsen, who had slumped down on his knees on reaching his sleeping bag and hadn't moved since, turned in time to see two cee-bees knocked off their feet. Neither had flak-jackets. And as they hit the ground, he saw several inches of metal rod sticking out of their chests.
The air was filled with blood-curdling screams and more flashing darts of silver then a horde of Mutes armed with knives and machetes burst out of the trees behind the guys further up the slope.
_Ohhh, shiiiiit!_ Olsen's fumbling fingers searched for the safety on his carbine. _Full auto! C'mon, c'mon, c'mon!_ He started to get up, bring the carbine across his body and into his shoulder. Something struck his chest knocking the breath out of him. A searing pain filled his lungs. The force of the blow pushed his head forward and as he toppled over, he saw the tail end of the crossbow bolt that had gone in at an angle under his rib cage.
_Yep... looks like this is it, Jake. Almost_...
Screams and shouts filled the air. People running in all directions. Guy firing. Trying to...
Olsen's helmeted head hit the ground. Above him was the huge clouded vault of a late spring sky. A Mute warrior leapt over him. He'd be back later. Or one of his friends would. When they'd killed everybody. That was when they collected the heads. With luck he might be dead when they got around to him. The earth felt as if it was turning under him. His dug his fingers into the soil in a desperate effort to stop himself falling towards the clouds.
_If this was the blue-sky world they could... keep_...
* * *
Cadillac surveyed the scene as the M'Call Bears began the grisly task of collecting the heads of the dead Trackers. The bodies had been stripped of their uniforms. These, together with their weapons and equipment were now being collected and added to the heap in the centre of the clearing.
The attack had been more successful than he had dared hope. Malone's renegades had diverted the attention of the sand-burrowers but it hadn't really been necessary. Their defences had proved absurdly easy to penetrate. No perimeter guards had been posted and many of the soldiers had only put up a half-hearted resistance.
Three Bears had died but they had been avenged by their clan-brothers. So far more than eighty sand-burrowers had been killed. The others who had fled into the trees below would be dealt with by Malone's men as they made their up in an extended line towards the campsite.
Yes. It had been easy, thought Cadillac. Too easy... or was that feeling due to his eternal dissatisfaction? Since meeting Steve Brickman, he had become increasingly suspicious of everyone's motives.
Two Bears, Rain-Dancer and Diamond-Head, lugged a metal box over to where Cadillac was standing and deposited it at his feet.
Rain-Dancer unclipped the hinged lid. 'There are things inside which bear the marks of silent speech. Should we carry them away?'
The outside of the box carried the legend: CAUTION/ EXPLOSIVES 12 x AP108. Cadillac hunkered down to examine the contents – twelve flat, round metal containers about six inches across and four inches deep. Printed in yellow on the top of each dark grey container were the words: AP108 ANTI-PERSONNEL MINE followed by the instructions for arming the pressure fuse and concealing the device in the ground.
Cadillac unclipped one and looked inside. The mine, with its fusing device protected by a clear plastic cap, nestled snugly in the container. He had never seen such a device before but he had seen the destruction wrought by the explosives supplied to Steve by the Federation for use against the Iron Masters. It would be a good idea to make sure these devices did not fall into the wrong hands.
He snapped the lid shut and handed the container to Rain-Dancer. 'You have done well. Put this in your carrying-pouch'. He handed another to Diamond-Head. 'Now, as quickly as you can, find ten of your most trusted clan-brothers and give them one each. They are to place them in their pouches, as you have done, and they are not to speak about them or show them to anyone but me! Do you understand? This is a secret treasure that gives us great power! No one must know we have found it – especially the red-skins!'
Both Bears got the message. 'And the box?' asked Rain-Dancer.
'The sand-burrowers have spades. Bury it. Hurry!'
It was a close run thing. The first renegades – those on foot – came up through the trees as the two warriors appeared above the rocky outcrop and signalled the completion of their task.
Cadillac walked down to meet Malone and the other riders. Several were leading their mounts. All of them had dead Trackers hanging head down over their backs; some carried two or three.
Malone responded to Cadillac's greeting with a cursory nod and took in the scene. 'How'd it go?'
'Much better than I expected. You could almost call it a pushover.'
'Yeah, well, the discipline in these signal units ain't what it used to be. A lot of these overground boys are smoked out of their skulls.' Malone gave a dry laugh. 'Make the most of it. It's gonna be a hell of a lot tougher'n this gettin' on board Red River.'
'Can't wait,' said Cadillac.
'How many did you nab?'
'Eighty-three...'
'Good. With what we've got here, makes ninety-eight. Couple must have got away. Never mind. They ain't goin' very far.' Malone slapped this thigh. 'So what've we got? Uniforms? Helmets? Rifles?'
'Yep. And smoke grenades, flares, radios, maps, rations...'
'Anything interesting?'
'A lot of camping equipment. I imagine you could make good use of that.'
'No, plastic?'
'Plastic what?'
'Plastic explosive. Detonators.'
Cadillac shook his head. 'You'd better take a look around yourselves. None of my people would know what that looks like.'
Malone turned to his lieutenants. 'Check it out...'
While they searched the campsite and the heaps of looted equipment, Malone and Cadillac got one of the M'Call warriors to don a set of camouflaged fatigues, then added a flak-jacket and all the equipment. The warrior complained about the boots, and the lumps on his forehead made the helmet a tight fit but with the visor lowered, and his carbine plugged in to the helmet umbilical, he looked just like the real thing.
Malone ordered him to walk up and down, and as soon as he started to move, it was clear they had a problem. The warrior was not carrying the carbine properly, and he didn't move like a sand-burrower.
'These guys are gonna need some knocking into shape,' grunted Malone. 'This one's walking like a ruptured shitehawk.'
'Don't worry, We'll iron that out...'
Malone's men came back to report that no explosives had been found.
'Shame. Might have come in useful.' He shrugged off the news. 'Never mind. We've still got enough here to equip a small army.' He slapped Cadillac on the back. 'Didn't I tell you that good ole Matt would deliver the goods?!'
'You did,' said Cadillac. _And you have, my friend, you have_...
The small packs of plastic explosive that Malone's men had searched for in vain, a rigid wallet containing detonators and miniature timing devices lay safely concealed in his own carrying-pouch.
## CHAPTER EIGHT
Five days after the M'Call Bears staggered back from Iron Mountain, dragging their share of the plunder on tracking poles, the sentinels posted to watch the northeastern approaches to the the settlement, sighted the returning trade delegation. It included a horseman but the group itself was barely half its original size.
Cadillac was one of a large and extremely anxious posse who ran out to meet them and it was clear from the muted response of the returnees to their shouted greetings that something dreadful had happened.
As the two groups met and merged, Cadillac exchanged a perfunctory greeting with Steve then searched in vain for the familiar, much-loved face. Filled with a sudden dread, he pushed his way through the milling throng and seized Awesome-Wells by the shoulders. 'Where is the Old One?!'
His cry caused everyone to fall silent.
The elder exchanged a glance with Boston-Bruin, hesitated for a moment then said: 'He has passed over.'
Cadillac looked stunned. 'Dead?!'
The rest of the welcoming posse wailed and beat their heads and chests. 'Oyy-yehhh...!'
Cadillac shook his head incredulously and turned to Steve as he appeared at his shoulder. 'How did—? When—? W-Where is he?'
Steve grimaced awkwardly but didn't reply. The two elders and those around them averted their eyes and stepped back, clearing a path to the covered palanquin on which Mr Snow's inert body lay with the head exposed beneath a layer of skins.
The shock generated as the unbearable truth sank in made Cadillac go weak at the knees. He gasped as if hit by a series of body-blows. Awesome-Wells and Boston-Bruin both laid a comforting hand on his shoulders.
Moving like a sleep-walker, he allowed himself to be led towards the palanquin by the two elders then fell to his knees beside it. Lifting the furs, he took hold of Mr Snow's right hand. It was cold and stiff; the arm, bent across his chest, was locked at the elbow and shoulder.
Death now held him in its unyielding grip.
Cadillac placed a hand tenderly on the disfigured forehead. He had been the cause of many of the worry lines which had accumulated there over the years, but now death had smoothed the wrinkles away. He leant over and kissed the untroubled brow. The lifeless flesh chilled his lips. Faced with this final proof that his beloved teacher had left him forever, Cadillac howled with grief and collapsed over the body. As he tried to draw the old wordsmith into his arms, Boston-Bruin broke his grip pulled him away.
'Let him rest, young one. He died to save the Plainfolk.'
Cadillac sat slumped on his heels for a long moment then drew his hands down his face and got slowly to his feet, eyes brimming with tears. 'Save your story for the camp-fire when we are all gathered together.' His voice was choked with emotion. 'If I share my sorrow with our clanfolk, it will be easier to bear.'
Tom McFadden, Deputy-Director of AMEXICO broke the news to Ben Karlstrom when they met to review the agenda before going into the Daily Plans Conference for departmental heads of the organization.
'Dead?! When did this happen?'
'On the way back from the trading post,' said McFadden. 'We don't have the exact date. Malone called in as soon as he got the word.'
Karlstrom swore under his breath. 'Has he seen the body?'
'He's seen what's left of it, yes. Mutes on their home turf put their dead on beds of woven branches which are then raised up on four long poles. When Brickman took him up to the High Place the birds had already got to work.' McFadden relayed Malone's whittled-down version of the battle at the trading post. 'As far as we can ascertain, Mr Snow was killed by his own creation.'
'The tidal wave?'
'No, sorry, I didn't make myself clear. The power he summoned drained the life out of his body. Not surprising really when you consider the forces involved. Cadillac arranged for Malone to question several of the returnees. According to them the tidal wave may have killed around two hundred thousand people...'
Karlstrom tipped back in his chair. 'Two hun –? Wheeee-iiouuuu...'
'And wrecked five of the Great lakes class wheel-boats. Lost with all hands...'
'Jeezuss! That's going to raise some flak from our friends over the border.' Karlstrom closed his eyes and massaged his nose for a while then sat forward again and turned his attention back to his deputy. 'So how are the M'Calls taking this?'
'The clan is in deep mourning. Everyone's walking around with their face and arms smeared with grey wood-ash. Cadillac has been formally installed as Mr Snow's successor.'
'And Brickman. You say he went to the trading post...'
'Arrived after it was all over. He apparently went there to warn Mr Snow that the Yama-Shita might try and get even.'
'In the hope of protecting his investment.' Karlstrom gave a dry laugh. 'Still, one has to give him full marks for trying.'
'It wasn't an entirely wasted journey,' said McFadden. 'He was able to make a first-hand assessment of the situation. He'll give us the full details when he comes in but the She-Kargo, M'Waukee and San'Paul delegations have agreed to set up some kind of loose alliance, and they may be joined by some of the C'Natti clans. It's early days yet but Brickman believes it could form the basis of a mutual defence pact.'
'Leaving the D'Troit isolated...' The thought caused Karlstrom to pull out his bottom lip.
'Or as front-runners for the japs. I must say this alliance doesn't sound like good news.'
'No, it doesn't,' agreed Karlstrom. 'But it presents less of a danger now that the Mutes are cut off from their supply of arms. The problem is, the Iron Masters will be back in strength. They're not going to take this defeat lying down – and that could create a somewhat delicate situation.
'If the mutual trade agreement is now a dead letter, they'll have to establish a permanent hold on the Great Lakes and Northern Plains if they want to continue to exploit its resources. And that means occupying territory which they agreed was ours when we signed that secret protocol.'
'Can we stop them?' asked McFadden.
'By having a word on our private line? It's possible – but not before we've delivered the heads of what's left of the Clan M'Call. And then the Clan Kojak. Our friend the Lord High Chamberlain would like me to hand over Brickman as well but I managed to talk him out of that idea. The man upstairs has plans for our young hero.'
'I see. But now that Mr Snow is out of the frame, does the plan to use Red River to take out the M'Calls still stand...?'
'Yes. The terrain in Wyoming does not favour the wagon-trains and the dense tree cover prevents us from solving this problem by a sustained programme of air strikes. That's why we have to draw the clan down from the hills and into Nebraska.'
'Okay. But since Cadillac doesn't present the same level of threat as Mr Snow, we won't need to bring Roz Brickman into play. Shall I arrange to have her flown back to the Federation?'
'No, leave her out there. Clearwater too.'
McFadden made a note on his Memo-Typer. 'We've got a slight problem with Malone.'
'Go on...'
'When he sold Cadillac on the idea of attacking the train, it was agreed that Mr Snow would be playing a major role. When he and Brickman found out that Mr Snow had left for the trading post, he decided to backpedal to give Brickman time to bring the old lump in.
'Basically, what Malone said was "Either the summoner's in, or we're out". Mr Snow's death has left him out on a limb. If you want this joint attack on Red River to go ahead, he needs a good reason for the climb-down. Why don't we just use Brickman and Malone to set up Cadillac and take him out with a snatch-squad? We can wrap up this whole thing inside forty-eight hours of getting the green light.'
Karlstrom shook his head. 'It's not just Cadillac I'm after, Tom. It's the M'Calls. Every single man, woman and child. Not just to please our slant-eyed colleagues but for what they did to The Lady and for being the root cause of his latest trouble. It's left the She-Kargo riding high. If we can destroy their paramount clan, it'll be a real body blow. Ordinarily, the Trail-Blazer Division would be handling this. CINC-TRAIN would like nothing better. But because of the Brickmans, the P-G has turned the heat on us.'
'I understand.'
'Good. Keep on it. And don't cut back on any of the defensive measures. I want the reception committee prepared for any eventuality. And send the same message to Malone.'
McFadden eyed Karlstrom shrewdly. 'That would seem to suggest that you don't think Mr Snow is dead.'
'Let's just say I don't trust those fuckers an inch.'
'Does that include Brickman?'
'At the moment, it's not a question of whether _I_ trust Brickman, but whether the Mutes do. If you check back through Malone's signals you'll see that Cadillac voiced the suspicion that Brickman may be an undercover agent.' Karlstrom smiled. 'Fortunately, it doesn't seem to have occurred to him that Malone is not the genuine article.'
'But –'
Karlstrom cut him off. 'I know what you're going to say. If Mr Snow left the settlement before Cadillac and Brickman returned there could be no collusion between pupil and master. If his death _has_ been faked then Brickman would have to know about it.'
'Unless, of course he really _is_ dead.'
'Exactly. I'm confident that Brickman and his kin-sister will come through for us. But with an operation of this importance one can't leave anything to chance. That's why they will both be given ample opportunity to prove their loyalty on the day.'
Karlstrom expected the meeting to end there but his deputy didn't get up. 'You got something else to tell me?'
'Yes.' McFadden bared his teeth the way a chimpanzee does when he's trying to appease an aggressor. 'That was the good news.'
'And the bad...?'
'It concerns that decoy SIG-INT unit that was dropped into Wyoming. The one with –'
'Yes, yes, there's no need to explain. I set the goddam thing up. Malone led the M'Calls to them and everything went according to plan. The Mutes killed the defaulters and went away bright-eyed and bushy-tailed with the stuff they needed to attack the wagon-train.' Karlstrom paused and aimed a beady eye at McFadden. 'Or are you now going to tell me they didn't?'
'No, all that's fine, as far as it goes. It's just that –'
'Someone screwed up. Okay, who? Is this an in-house fuckeroo?'
'No. This one's down to some clerk manning a computer in the main supply depot of the Quarter-Master General.'
'That makes a change...'
'When the requisition order came through for the uniforms and equipment for the SIG-INT unit, the people at QMGC didn't know it was for a decoy outfit staffed by Code One defaulters. They just punched up the list of hardware carried by a SIG-INT squadron on a field operation and issued everything on the print-out.'
'Didn't our people fix the air bottles and take away most of the ammunition when they set up the campsite?'
'Oh, yes, they did all that. They laid everything out so as it looked realistic. But it never occurred to them that _all_ the items delivered to the site were standard issue and... fully operational.'
Karlstrom sank back in his chair and pinched his nose between his right thumb and forefinger. 'Such as...?'
'Six packs of plastic explosive, with detonators and timers, and twelve AP108's. Anti-personnel mines with pressure fuses.'
The list lifted Karlstrom's eyebrows. 'What in hell's name does a SIG-INT squadron need stuff like that for?!'
'Well, these units are tasked with setting up radio beacons. They're issued with PX in case they need to blast out the holes for the base of the mast in rocky terrain. Once they get the mast up and braced, they plant the AP mines in a circle around it to dissuade any roving Mutes from trying to pull it down.
'Most of the lump-heads inside the Territories have learned to steer clear of the masts, but for those who haven't all it needs is for one of these goons to put a foot in the wrong place and bang! Up he goes. After that, his friends – if there are any left – tend to keep well away.'
Karlstrom had a feeling he knew the answer but he put the question anyway. 'So what happened to this stuff?'
'It seems to have gone astray. We know it was issued, and received for forward shipment by the Air-Mexico base at Dallas.' McFadden threw up his hands. 'You know what these outfits are like. They're great at covert operations but they tend to lose track of the paperwork.'
'So you can't say with absolute certainty the mines and the PX reached Wyoming.'
'We're ninety-nine percent certain. That's why I checked with Malone before speaking to you. If it _was_ there, it had vanished by the time Malone came on the scene. His people made their own trawl of the camp-site but all they came up with were a few more rifles and odd bits of equipment – the sort of stuff the M'Calls had already collected.'
'Did he ask Cadillac about it?'
'In a roundabout way, yes. The Mute told him that all they'd found was on the ground – waiting to be split two ways. When Malone became more specific, the Mute appeared not to know what PX was. He couldn't press the matter too hard because his orders were to keep the Mute sweet until –'
'Sure...'
'Still, the thing is, Mutes can't read. So even if the M'Calls _have_ squirreled this stuff away they won't know what it is.'
'Cadillac can read,' said Karlstrom. 'He also knows everything that Brickman knows about explosives. It was Cadillac who wired up the bomb that blew a big hole in Hartmann's wagon-train!'
'Christo! Yeah, I forgot about that! The lying piece of lumpshit! What d'you think we ought to do?'
Karlstrom's face darkened. 'You can start by telling Malone to find out what Brickman knows about this. If the answer's nothing, he's to check that campsite out again. Inch by inch. And meanwhile, put a couple of dozen key-pushers on the shuttle to Dallas. They're to make an inventory of every single item on that air base right down to the last blade of rainbow-grass.'
He pointed to the work-station on his desk. 'And I want the results up on this screen by this time tomorrow. Comprendo?'
'Yessir!' McFadden leapt to his feet.
'And find out which of our people processed this requisition. I expect heads to roll, Tom. I won't tolerate this kind of sloppiness at any level of AMEXICO, and I won't accept any excuses.
'SQUARE-DANCE is one operation we _cannot_ allow to fail. It's probably the biggest, certainly the most complex task we've undertaken. Years of work, decades of planning, hundreds of lives – including yours and mine – are now in jeopardy and will remain there until we can establish the whereabouts of those explosives.'
'Don't worry. We'll find them.'
'You'd better. Who else knows about this?'
'Nobody apart from my assistant. She was logging the documentation on the SIG-INT set-up into the master file and just by sheer chance happened to read though the equipment list supplied by QMGC as it came up on screen. She thought it seemed a bit overloaded considering the life-expectancy of the unit it had been issued to and brought it to my attention.'
'Bright girl...'
'Jo-Anne? Yes, she is.'
'Get her to unpick it. Take those two items off the list, and delete them all the way back to the QMGC supply depot. That means accessing their records too. Those explosives were never issued to AMEXICO. You understand?'
'Yessir.' It wasn't the first time.
'Okay. Anything else?'
'Only what's on the agenda for this morning's meeting.'
'Good...' Karlstrom sighed wearily, planted his hands on the desk and levered himself out of his chair. 'Well done, Tom. Thanks for spoiling my day...'
Out in Wyoming, Steve decided to make himself useful by taking over the job of training the fifty Mute warriors who had been selected to take part in the initial assault on the wagon-train. Since many of the Bears who hadn't been picked were somewhat envious of those who had, Steve thought it best – in the interest of general harmony – to widen the training to include all those who wanted to take part.
It gave everyone a chance to dress up in the uniforms and deck themselves out with all the bits and bobs – something which appealed to the Mutes' love of dressing-up and putting on an extravagant performance – like they did when singing fire-songs. And it also introduced a measure of competitiveness, since Steve made it clear that only the top fifty recruits would make the final team.
There was no hope of teaching them how to use the electronic head-up displays projected onto the visor, but they had all handled the rifles supplied by the Iron Masters, and they were used to sighting and firing a crossbow from the shoulder. Since most Mutes were uncannily good marskmen, the transition from crossbow to carbine, using the basic optical sight, went relatively smoothly.
Cadillac, who had taken Mr Snow's death very badly, dropped out of circulation after his normal appointment as wordsmith and made it clear, via the warrior who barred the door to his hut, that he didn't feel like talking to anybody – including Steve.
Three days later, he turned up at one of Steve's training sessions shadowed by four warriors: Blue-Thunder, Funky-Deelix, Storm-Trooper and Twilight-Zone. Seeing him watching from the sidelines, Steve told the two Bears he'd appointed as squad leaders to carry on practising fire and movement and went over to greet him.
Steve hailed the Bears then shook Cadillac's hand warmly. 'Welcome back.' He was going to say 'to the human race' but decided it might be taken the wrong way.
'How's it going?'
'How does it look?'
'You'd better ask my clan-brothers,' said Cadillac. 'I've never seen real Trail-Blazers in action. Have you got everything you need?'
'Yes. You guys did a good job. I never thought you'd be able to get hold of this much stuff so quickly. Y'know what? I'm beginning to think we've got a good chance of pulling this off!'
'In that case, you'd better have a word with friend Malone.'
'Has he got cold feet?'
'Well, let's just say I've failed to persuade him that we can still take on Red River without Mr Snow. He says that air rifles aren't enough. We need explosives, mines – that kind of thing.'
'Which we don't have...'
'No. He hasn't totally rejected the idea of a joint attack. We have agreed to meet again. But he may be right. Perhaps it's time you and I faced up to the realities of the situation.'
Cadillac watched the uniformed warriors going through their paces for a moment then said: 'Anyway that's one of the things we need to discuss.'
'Sure. Can it wait – or d'you wanna do it now?' Steve saw the expression in Cadillac's eyes and turned to shout though cupped hands at his perspiring trainees. 'Okay! Break it up, you guys! Game's over. I'll catch up with you later!'
Cadillac led Steve out of the settlement. Blue-Thunder and the other three Bears followed. Nobody said anything and Steve felt a slight prickle of apprehension. If Cadillac wanted a quiet chat, what did he need the heavy mob for?
The M'Calls had pitched their huts on the same piece of turf they had occupied in 2989 when Steve had been shot down and made a prisoner. Cadillac led the way to the top of the slope above the settlement and halted on the bluff where Steve had fought a last-minute battle with Motor-Head, Black-Top and Steel-Eye. The three Mutes had caught him trying to escape on the hang-glider he and Cadillac had built. Clearwater had used her powers to help him get away. Had she not done so, he would have been a dead man.
Steve had fled because he had broken his promise to Mr Snow to stay away from Clearwater. In so doing, he had also betrayed Cadillac. And here he was, back on the very spot where he'd stepped off the M'Calls' turf and soared into the air. Was he supposed to read some weighty significance into the choice of venue?
The young wordsmith invited him to sit on a rock. Two of the Bears, Storm-Trooper and Twilight-Zone positioned themselves on either side at the edge of his field of vision; Blue-Thunder and Funky-Deelix planted themselves about three yards in front of him, legs astride, arms folded across their chests.
Deciding there was nothing he could do but wait for the next move, Steve rested his hands on his splayed thighs and tapped out an impatient rhythm as Cadillac paced back and forth across the intervening space. He knew it was all part of the act, but there came a point when he felt they had played around for long enough. 'You got something to say to me? Or have we just come up here to enjoy the view?'
Cadillac stopped in front of Steve. 'It's about the Old One. Apart from the impact his death has had on me personally, it has placed us in a very difficult position.'
'You mean with Malone?'
'No. I'm sure if you put your mind to it, you could persuade him to help us. On the other hand he may be right. Maybe we should forget the whole idea –'
'– of trying to rescue Clearwater?'
'Yes. What do _you_ think we should do?'
Steve shrugged. 'Not for me to say. I obviously have a vested interest but this is a matter for the Clan M'Call. You've taken over from Mr Snow. From here on in, you are gonna have to make the decisions.'
'Yes, I know, but...' Cadillac paced up and down again, chewing over his words then asked: 'Did you get a chance to talk to the Old One before he...?'
'You mean about Clearwater being on the wagon-train? Yes, I told him what happened – and that you'd decided to try and rescue her.'
'Did you tell him you thought it was a crazy idea?'
'No. I just pointed out some of the difficulties.'
'And...?'
'He said we should go for it.'
'Yes, but when he said that did he know he was dying...?'
'Dying?' Steve grimaced. 'Can't say. It wasn't his last wish if that's what you mean. He just told me The Path was drawn – and gave you his blessing.'
'Even though it means walking into a trap?'
Steve frowned. 'I'm not sure what you're getting at.'
Cadillac eyed him expectantly then said: 'I see. You didn't tell him about Malone's little secret.'
There was only one way to meet this veiled accusation. Head on. Steve glanced at the two warriors standing on either side of him then said: 'Oh... you mean about him being an undercover agent? Yeah, I told him.'
'But you didn't tell me.'
'I was going to. I thought I'd give you the chance to work it out for yourself. And you have. Which is great. Saves me a whole lot of trouble.'
Cadillac's temper rose. 'Just what the hell is your game, Brickman?!'
'It's called "survival"! You make up the rules as you go along and any number can play. But before this goes any further, try and grasp one simple fact – you and I are on the same side.'
'Do you seriously expect me to believe that?' Cadillac's anger exploded. 'You set me up! You and Malone were going to lead my clanfolk into a trap!'
'No, no, you've got it all wrong, Caddy. I admit it looks that way but that's not what's meant to happen. This _is_ a set-up but the idea is to turn it around! It's Malone's renegades and the crew of Red River who're gonna end up as the fall guys – not us.'
Cadillac looked unconvinced. 'Easy to say that now your lies have been uncovered! Just as you have no proof that you confessed this to the Old One and received his blessing!'
Steve bit back his reply. He had suddenly realized where this discussion was heading. He decided to let it run. If he was right he could get off the hook later. It was time to go on the offensive. 'Listen! I'm not the guilty party – _you_ are! None of this would have been necessary if you'd trusted me. But what did you do? You went shooting your mouth off to Malone – about how you thought I was an undercover Fed! You're lucky he's a mex! If he'd been a _real_ renegade he'd have put a bullet through me. Where the hell would you have been then?!'
'No worse off than I am now!' cried Cadillac, still stricken by the knowledge that he could never again turn to his teacher and mentor for aid.
'Well you get no sympathy from me! If it hadn't been for your stiff-necked attitude I wouldn't have had to lie to you! But what happened when I suggested trying to rescue Roz and Clearwater from the wagon-train? You dismissed the whole idea, accused me of trying to sell you down the river and we end up drawing a knife on each other! Thanks to your paranoia Clearwater came close to getting killed. I didn't put her on that wagon-train – _you_ did!'
'Well, that's your story...'
Steve ignored the jibe. 'Then what happens? Malone comes up with the same idea – strokes you up the right way – plays on your distrust of me, and you swallow it hook line and sinker!'
'You tricked me into it!' cried Cadillac. 'You put him up to it! You knew that if you said it couldn't be done, I would want to prove that it could!'
'Yes,' shouted Steve. 'That's because you always want to be Number One! You always want to do it your way, and every five minutes you want someone telling you you're Mister Wonderful! Well, sometimes you are. But most of the time you behave like a complete and utter bonzo!
'That was why I had to play it this way! Because you didn't trust me! Didn't we exchange a blood-oath with Clearwater as our witness?'
'Yes, but –'
'There's no "buts"! Now that Mr Snow has gone to the High Ground, we've got to get her off that train. After what happened at the trading post, this clan and the She-Kargo are going to need her more than ever. That's why we have to free Roz too. Not because she's my kin-sister but because she's a doctor.
'It'll take another three or four months before Clearwater is back on her feet, and during that time she'll need proper medical care. If she doesn't get it, then the whole exercise is pointless.'
'Yes, I can see that. You've reasoned this out very cleverly. But then you've always been good at wriggling out of tight spots. You're a valuable asset, Brickman. That's why your masters are prepared to go to such extraordinary lengths to keep you alive.'
'You're right. They have,' said Steve. 'Because I've managed to fool them into thinking I'm still working for them.'
'Just as you've managed to fool us...'
Steve's patience began to wear thin. 'That's a load of shit and you know it! Have you ever asked yourself why I've gone to such _extraordinary lengths_ to keep _you_ alive? No. And you know why? Cos _you're_ the one who can't face up to the truth! I could have dumped you anywhere along the line and you know it!'
'Clearwater wouldn't have let you!'
'At the Heron Pool? She couldn't have stopped me! She was drained of all her power. She could barely walk! And at Long Point, when you were stewed to the gills, I could have pulled you out of that cargo hold and taken off without anyone being the wiser. Or better still, I could have forgotten to lock the hatch and slow-rolled you out once we were airborne –'
'Thanks for letting me know how your mind works. It's been most instructive.'
'You know exactly what I'm trying to say!'
'Yes, sure. And you could have let me drown in the wheel-boat. But you didn't.' Cadillac's voice lost its shrill edge and became a confident cat-like purr. A sign he now had the upper hand. 'Why? Why go to all this trouble?'
Cadillac answered his own question with a mocking laugh. 'It was all part of a plan. After you'd rushed off to the trading post and Malone – rather stupidly – gave himself away – everything fell into place. You'd been sent out to capture all three of us. When Clearwater was wounded and I was lying unconscious, you had a golden opportunity to put both of us on board Red River. But you didn't take it because you wanted to use me to suck Mr Snow into the trap!
'Bringing me back to the settlement made you a hero. The loyal friend of the M'Calls! An honorary Bear who goes to the ends of the earth to rescue his clan-brother! Who would suspect that underneath all that paint and leather was a lying treacherous toad?!'
With Blue-Thunder, and the other three armed Bears watching the whole proceedings closely, it was unwise to make any rash moves. But violence wasn't the answer. The way to destroy Cadillac in front of his peers was to hit him below the belt with some incisive syntax.
'You deserve to get your head broken for that, but I'm not gonna allow myself that pleasure. I made a promise to the Great Sky-Mother that if she spared Clearwater's life, I would never fight with you again.'
'How very noble of you,' said Cadillac. 'Can you swear in the name of Mo-Town that a plan such as I've described never entered your mind?'
'No. It did. That's exactly the line I fed to the people on board Red River.'
Cadillac caught his breath. 'You...?!'
'Yeah, I know what I said.' Steve waited while Cadillac cranked up a suitable degree of outrage.
'You – you've actually been _on board_ the wagon-train?!'
'Of course! I had to go! I had to see Roz – find out how badly Clearwater had been hurt, and what the chances were of rescuing them both! Isn't that what you would have done?' Steve didn't wait for a reply. 'And the only way I could get _off_ the train and back here was by selling them this plan to lure you and Mr Snow into making a sneak attack on Red River to rescue Clearwater.
'In reality, of course, it was to be an ambush. Once the two of you were under lock and key, the idea was to pretend the attack had succeeded so as to bring the rest of the clan into play and...' Steve completed the sentence with a shrug. 'I brought Malone in on the plan to keep them happy. By putting him in to work with me they figured there was no chance of, well – anything going wrong.'
'You mean they don't trust you either.'
'The man who runs AMEXICO likes to cover all the angles.'
Cadillac mulled over what he had just heard. 'What made them think they could capture the Old One? Don't they know he is the Storm-Bringer?'
'Of course they do. That's why they want him so badly. But Roz managed to convince them she had the power to neutralize his earth-magic – perhaps only for a short while, but long enough for him to be seized and pumped full of mind-warping drugs.' Steve described the demonstration she had laid on for the task force.
'Was that wise?'
'To let them know what she's capable of? Yes. They had to be convinced they had some way to control Mr Snow. Otherwise they'd never let us on board. Can't you understand? I _had_ to set up this ambush plan. It was the only way I could persuade them to let a bunch of Mutes get inside Red River. And they only agreed to it because Malone was going to be holding your hand during the planning stage –'
'And he and his renegades were going to be part of the attack force... leading us into the trap.'
'Exactly. Just get this simple fact into your head. No raid, no rescue. D'you want Clearwater back or don't you?'
'Yes, but –'
'Then we have to go ahead with it.'
'I see. Let ourselves be herded into an ambush by Malone's renegades. Many of the She-Kargo clans use a similar technique to catch buffalo. They hem them in and stampede them over a cliff.'
'But we're not gonna be following Malone's plan, we're gonna play it _our_ way – and Roz will use her power to help us.'
'You hope...'
'I don't hope, Caddy. I _know_. Our minds are linked, remember?'
Cadillac became irritated as he sensed he was losing the initative. 'Yes, yes, I believe you can read each other's thoughts. We wouldn't be here otherwise. The question is – how can anyone else discover what your true motives are?'
Steve's anger burst enough. He leapt to his feet. 'Aww, for chrissakes, Caddy – why d'you keep twisting everything around?! I've given you the whole set-up and been absolutely straight with you. Can't you see that?'
'It doesn't really matter now, does it? The Old One is no longer with us.' Cadillac mastered the grief which surfaced at each mention of his teacher's name. 'True or false, all your scheming and plotting has come to naught.'
'No it hasn't. We still have Roz. If we can get enough of our people aboard Red River, we can still turn this situation around.'
Cadillac looked unconvinced. 'Yes, well, you say that now, but you've concealed the truth for so long – behind so many lies and layers of deceit – I'm not sure your tongue would recognize the taste of it if it ever managed to slip through your lying throat.'
Steve realized that Cadillac was doing his utmost to provoke a violent reaction. _No, my friend, I'm not going to rise to it – because now it's, clear what your game is_... He tuned back in on what the Mute was saying.
'... but even if this latest confession of yours _is_ the truth, it comes too late. We can't mount any rescue attempt unless we can discover what they _really_ intend to do. Malone is obviously not going to tell me – and given your track record he's not going to tell you either.
'And I doubt that your kin-sister can be of much help to us. They probably distrust her as much as they do you.' Cadillac sighed heavily. 'No. Like so many of your schemes, it's far too complicated. I'm afraid we're going to have to abandon the whole idea.'
'So you're not going to try and rescue Clearwater...
'Let me put it this way, Brickman. I am not going to sacrifice this clan in order to further your own, rather dubious ambitions.'
Steve laid on a pitying smile. 'Know what? I've just realized what this is all about. It's got nothing to do with whether I'm telling the truth or not. You're trying to discredit me 'cos you're looking for an out! Now that Mr Snow isn't here to lead the way you don't _want_ to take a shot at that wagon-train, do you?
'What are you frightened of – the responsibility? That a lot of your clan-brothers and sisters could die as a result of _your_ decisions – leaving _you_ holding the can? Or are you just scared to take the job in case they refuse to follow you?'
Steve laughed dismissively. 'Well, I'll tell you something. After watching you trying to wriggle off the hook I wouldn't blame them! This clan _needs_ Clearwater to help them get through the bad times that are on the way. I've kept her alive, kept her within reach and set up a deal which gives us a chance to rescue her. What you should be doing is providing some real leadership! Positive thinking – some action, for crissakes! But all we've had from you are negative waves and a lot of cheap shots about my reliability.'
'Yes,' cried Cadillac. 'And considering the circumstances, they're perfectly justified!'
Steve went for the throat. 'More fancy words. But then that's your job isn't it? I may talk a lot, but at least I deliver! You? You're pathetic!'
Cadillac's right hand flashed towards the hilt of his knife. Blue-Thunder and Funky-Deelix seized his arms and locked them to his sides, preventing him from drawing the blade.
Steve raised his hands as Storm-Trooper and Twilight-Zone went to grab him and stepped out of reach. 'Uh-uh! Hold it, guys. I'm not looking for trouble.'
'We know that,' said Funky.
'Good.' Steve avoided eye contact with Cadillac and addressed Blue-Thunder. 'You can check my story with Awesome-Wells and Boston-Bruin. They were with Mr Snow when I told him about Malone and the whole shooting-match. If that doesn't clear the air, you know where to find me.'
Blue-Thunder's eyes were not unfriendly but they carried a clear signal that – for the moment at least – enough had been said.
## CHAPTER NINE
Night-Fever ducked in through the door-flap of Steve's hut and knelt before him. The flickering light from the fire-stone raying upwards onto her face made it look like one of the snarling carved and coloured demons that decorated the religious shrines of the Iron Masters.
During his last stay with the M'Calls, the fearsome-looking She-Wolf had developed a soft spot for Steve and, in Clearwater's absence, had appointed herself as his cook, go-fer and bed-warmer. She had, apparently, failed to find a permanent partner during the months he had spent in Ne-Issan, and following his return with Cadillac and her journey with him to the trading post, she had prepared a hut for him with loving care.
Brought up in a society where – apart from the role of guard-mother – men and women enjoyed absolute equality (the Federation, for example, took no account of gender differences in its provision of communal toilets and wash-rooms) Steve was not used to having a female adopt a subservient domestic role.
At first, it had caused him some embarrassment, but the primitive life-style of the Mutes brought with it a host of chores that, given his Tracker upbringing, he would have preferred to do without. In the Federation, where everything was highly organized, heating came through ducts, hot food was constantly available, water was on tap, and the only life-forms moving under the bedclothes had two legs.
Out here, on the overground, the basic necessities had to be thought about _every day_. And that included choosing a spot to make your bowel movements. Apart from the last item, Night-Fever took care of everything. Steve couldn't figure out why, but he admired her dedication and showed his appreciation in every way except in the one area that might have left Night-Fever feeling totally fulfilled.
'Cadillac asks if you will speak with him.'
'Is he alone?'
'Yes.'
'Tell him I'll be right out.'
Having heard about their last angry exchange at which knives were almost drawn, Night-Fever planted herself outside the door-flap. Anyone exiting from the low huts had to bend double, or drop to their hands and knees – making it difficult for that brief moment to defend themselves from an angry caller. Such attacks were rare but they were not unknown.
On this occasion, Steve did not require the protection offered by her muscular knife-arm. Cadillac, who was no stranger to Mute body-language – had taken several diplomatic paces backwards and, as Steve emerged, he raised his hands, placing both palms together then opening them towards his rival – the open-handed greeting that was a sign of peace.
Steve stepped towards him and exchanged the ritual grips and hand-slaps used by warriors.
'I may have misjudged you,'said Cadillac, with what appeared to be genuine regret.
Steve responded with an understanding smile. 'I have a feeling both of us said things we didn't mean.'
'Can we talk?'
'Sure. Come on in...'
Night-Fever held the door-flap open for them. 'Shall I prepare food?'
Steve questioned Cadillac with his eyes then said: 'No. Go to your sisters by the fire-circle and wait for my call.'
Night-Fever nodded obediently, her face glowing in response to the brief smile he gave her as their eyes met. Amazing, thought Steve – how some people react to even the slightest hint of kindness or affection. If she had a tail, she'd be wagging it – just like Baz. The memory of the wolf cub's sudden end in Malone's hands flashed through Steve's mind, leaving a trail of bitterness in its wake.
Seated on talking mats inside the hut, Steve watched Cadillac stare at the flame in the firestone – and waited. With his eyes still averted, Cadillac said: 'You're right. I _am_ scared they won't follow me.'
'You shouldn't be. You heard what Awesome-Wells told the clan gathering. The Old One named you as his successor and called upon them to heed your words. He's a tough act to follow, but we should do as he said and go for it.'
Cadillac faced up to Steve. 'I know! But it's not as simple as that!'
'Who said it was going to be simple!'
'Brickman, just for once, keep quiet and listen to me. And remember I know everything you've ever learnt about wagon-trains. Neither you nor I know what Malone and that bunch of killers on board Red River are cooking up. And from what you've said so far, your kin-sister doesn't have much of a clue either.
'Maybe she can use this new power she has to help us but it's virtually untried. These hallucinations for instance – can she apply them selectively? It's not going to be much use if the good guys and bad guys end up sharing the same nightmare.'
Steve shrugged. 'I don't think even she knows exactly what she's capable of.'
'Which places another question mark over the operation when what we need are answers.' He raised his hands to cut off Steve's reply. 'Okay, okay, maybe she _will_ be able to help. What you haven't faced up to is the sheer size of the problem. The actual physical dimensions of the wagon-train itself, the number of people on board and its weapon-systems – especially its defences.'
'You mean the steam jets...'
'And the rest. Malone thought he was talking to an idiot but here's the plan he sold me. A hundred of us – fifty renegades and fifty Mutes, including you, Mr Snow and myself – dressed up in the Trail-Blazer uniforms we captured, are going to pose as this signals outfit being pursued by a large posse of Mutes –'
'Is Malone going to be handling the radio traffic?'
'Yehh... Here's the picture. As the light starts to fade, he puts out a May-Day and we move in towards Red River –'
'Who drops a ramp...'
Cadillac's lips tightened. 'Brickman – am I telling you this story, or are you telling me?'
'Sorry, go on...'
'The wagon-train lays down covering fire. The pursuing Mutes fall back and fade into the bush. We go up the ramp – and all of us are wearing a coloured armband so we'll know who's who – then, when we get to the top, we split into three groups. One will hold the ramp and the immediate area around it on the lower floor, the other two will go up onto the middle and top floors and start blasting their way through to the front and rear of the train.
'Malone is going to override the ramp controls so that it stays down and deactivate the operating system that feeds steam to the jets under the wagons –'
'And Mr Snow?'
'We'd left that open. I told Malone he would do whatever was appropriate in the circumstances. He obviously wasn't going to produce a storm inside the train –'
'Wouldn't be a bad idea...'
'Perhaps not. But since he's dead he can't help us. Are you going to allow me to continue or not?'
'I'm all ears.'
'The immediate objective is to seize two complete wagons – so we can drop a second ramp – and get through the doors into those beyond.
'As soon as this is achieved, a group will go out through the top hatches onto the roof and release green flares – the signal which will bring the rest of the clan out of hiding and onto the train. With the steam jets deactivated, they'll come screaming up the ramps and...'
'Goodbye Red River...'
'That's what Malone would like us to think,' said Cadillac. 'Spot the deliberate mistake.'
'Mmmm, well, apart from the basic problem of how a force of one hundred men – fifty of whom are operating in a totally alien environment – can overcome a whole battalion –'
'No, that's obvious. That's why we needed the Old One.'
'Now you're interrupting me!' cried Steve.
Cadillac motioned him to continue.
'There are two major flaws in the plan. Malone _can_ override the ramp controls and hold it down. That's a standard EOP. So is the steam-jet cut-off. But that's automatic. The jets under the closed wagons can still be used to pipe steam if needed but whenever a ramp goes down the jets under that wagon are deactivated. Otherwise things could get kinda messy.
'There's a cut-off button on a panel at the top of the ramp on each side. Its main function is to reassure the dog-soldiers waiting to go into action. When the ramp-master gets the green, he hits the "STEAM OFF" button and it lights up to confirm that the jets under that wagon are off line.
'That's all fine as far as it goes. But you've got problems on the roof. You can get onto it – provided you can operate the emergency hatches – but your people won't be up there long. The front and rear command cars have dorsal turrets with six-barrelled 20 milimetre Vulcan cannon covering the roof from both ends. You won't be able to knock them out unless you get into both command cars – and we aint gonna do that with a hundred men split three ways. We're already outnumbered ten to one. We don't have enough men to hold all three floors _and_ every stairwell. They'll just fall back and come round behind us.'
'Precisely. And the other deliberate mistake?'
Steve laughed. 'What is this – some kind of a test?'
'Just answer the question, Brickman.'
'Even if we got onto the train, took the two wagons and lowered a second ramp _and_ had Mr Snow helping us we still couldn't capture Red River.'
'Why?'
'Because both ends of the wagon-train can operate independently! All the wagon-master has to do is to activate the door seals on either side of the wagons we're in, uncouple the front and rear sections of the train and roll 'em away. We'd be left trapped like fish in a barrel!'
'So for us to have even an outside chance, both command cars have to be immobilized first...'
'Yeah. But you won't flatten one of those tyres with a cross-bow bolt _or_ a rifle bullet.'
'I know that.'
'There's another problem. Even if you immobilised the train, knocked out the dorsal turrets and held the two wagons long enough to get onto the roof and give the green light, how are the rest of the clan going to get aboard? All the side turrets will still be working – and presumably the steam-jets. The M'Calls'll be chopped to pieces on the way in!'
'But it's getting dark.'
'The gunners have infra-red sights! And your people'll be funnelling in towards those two ramps.'
'The guns... are they all air-powered weapons?'
'No. The 20 and 40 millimetre Vulcan cannons use what we call caseless ammunition. An explosive charge in a closed breech drives the shell up the barrel. Like a rocket only different.'
'But they're power-operated turrets...'
'Yeah, they use electric motors to traverse –'
'And the six-barrels that spin round...'
'They're driven by an electric motor.'
'So if there was no electricity they couldn't fire...'
'That's right.'
'I see... So if we were to immobilize the command cars to prevent the train from splitting up and cut the steam lines and the power as we go up the first ramp, we've got a fighting chance.'
'A slim one,' conceded Steve. He laughed. 'I thought you were trying to find reasons for _not_ attacking the train.'
Cadillac gestured wearily. 'I knew it couldn't be as simple as Malone made out. I wanted to hear you say it.'
'So it was a test! Goddammit, Caddy! What more do I have to do to prove I'm on your side?!'
'I don't know. Despite everything you've done there's... something about you that doesn't hang together.'
'What is it, for crissakes?!'
'Well, I know I'm a wordsmith but it's hard to describe. At times it's almost as if you were two people. One of you seems to be trying to help us and the other one's working for the Federation.'
' _Pretending_ to work for the Federation.'
'That's what I call the good side. I think the other side wants to. On the other hand I could be letting personal feelings cloud my judgement.' Cadillac shrugged. 'Time will tell.'
'If we're gonna get our act together it'd better be sooner than later,' snapped Steve. 'Cos I'm getting mightily pissed off with all this character analysis shit! If you don't want to attack Red River tell me now and we can cut out all this frigging around! Malone can tell Mother we couldn't swing it. I'll do what I suggested earlier – go back into the Federation with Clearwater and Roz and try and get them out some other way.'
'I'm not saying I don't want to attack Red River –'
'Then what the fuck _are_ you saying?!'
Cadillac became equally irritated. 'Don't get snotty with me, Brickman! You're on _my_ turf! One word from me and you'll be pitched out – not on your ear, but minus both of them and with your balls stuffed in that big mouth of yours! Getting Clearwater off that wagon-train would be a high-risk operation even if the Old One was here to help us. Without him it's more like a suicide mission!
'Before I ask my people to make that kind of sacrifice – and there's no guarantee they'll follow me – I have to know what we're up against. You've confirmed the flaws I spotted in Malone's plan but that whole scheme was only to keep _us_ happy – a bunch of stupid lump-heads. That's not good enough. We have to find out what the _real_ plan is. Because you and I both know that none of the people letting off green flares on the roof of that wagon-train are gonna be from the Clan M'Call!'
'Sure. Even I'd figured that out.'
'The six weeks you asked for expired two weeks ago. So the reception committee on board Red River have had time to rehearse their drills.'
'Yeah, but...' Steve clawed air to express his frustration. 'That was all put together after I left to rejoin you and Malone. I know the object of the exercise was to capture you and Mr Snow but apart from that I'm as much in the dark as you are.'
'Not quite...'
'How do you mean?'
'Your kin-sister is a member of the task force assigned to look after Clearwater. She must know _something_. Don't you think you ought to get in touch?'
'I could try. But I don't think it'll do much good.'
'Why?'
'Because Karlstrom, the man who runs the outfit I work for wasn't born yesterday. Until I deliver you and Mr Snow there'll be a big question mark hanging over me. Although she's better at it than I am, they know Roz and I can communicate with each other. Which means whatever they tell her could be passed on to me. If they have any lingering doubts about my loyalty, they're going to feed her false information – stuff they want _you_ to know about.'
Cadillac looked dismayed. 'But if you're to play a part in our capture shouldn't you be told what's going to happen?!'
'Yeah. They said Malone would clue me in just before the off.'
'So Malone must know.'
'He may do. When I tried to pump him he said his orders were to deliver you and Mr Snow and the rest of the boarding party to the top of the ramp. From then on the Red River team would take whatever measures were necessary to contain the attack.'
'But he didn't drop any hints as to what form they might...?'
'Nope...'
'And if we forced him to speak?'
'Torture him?' Steve shrugged. 'You could try but I don't think you'll get much out of him. Agents on a high-risk station, like Malone, are briefed on a need-to-know basis. That way, if their cover is blown, they can't jeopardize the whole operation.'
It was Cadillac's turn to be frustrated. 'Sweet Sky Mother! There must be _some_ way we can discover what their plans are!'
'I'll see if I can get through to Roz. She may have found a way to get inside other heads besides my own. But maybe we ought to start looking at this from another angle.'
'And what's that?' asked Cadillac suspiciously.
'Well, instead of wondering what the _other_ side is going to do _we_ should make the running. Seize the initiative. Throw _them_ off balance.'
'Oh, yes. And how do we do that?'
'We've already started. They were braced to resist the most powerful summoner of the Plainfolk – and now Malone will have told them he isn't coming.' Steve spread his hands. 'With just you left to lead the clan they're bound to think it'll be a pushover.'
'Thank you. And...?'
'Well, this is only a suggestion but we could start by eliminating Malone and his renegades. When we go up that ramp we might as well take a hundred men we can trust. What's the point of going in with fifty guys we dare not turn our backs on?'
Cadillac frowned. 'But you said that Malone's participation was vital if this rescue plan was to succeed!'
'It still is – but only until everything is set up. If we can come up with a rescue package that puts us in with a chance, then we should take him out just before we start the run-in towards Red River. That's the best time to make the switch. We'll be on board before they know what's hit 'em.' Steve caught Cadillac's questioning look. 'It's okay. I don't owe Malone any favours.'
'But the other renegades...'
'That's the point. They're not real renegades.'
'Are you saying they're _all_ undercover agents?'
'No, just some. But what difference does it make? Christopher! We're talking about seventy or eighty defaulters with close to zero life-expectancy. Piddleshit compared with the number of your people who died in order to save the She-Kargo! If you don't kill 'em some other clan or a bunch of Trail-Blazers'll hunt them down. With the trading post closed for good, they have no value!'
'Yes, but... they're your own people.'
'If a general thought like that he'd never send his troops into battle. You want Clearwater back, don't you?'
'Yes!'
'Then when the time comes, I'll do whatever has to be done,' said Steve. 'But before any of that can happen, you have to win over the clan – get them solidly behind you.'
'Don't remind me.' Cadillac sighed gloomily. 'If only the Old One hadn't died on the way back from the trading post.'
'Yehh,' said Steve, 'It was rather thoughtless of him.'
'That's not what I meant! If he'd died _here_ , he might have been able to tell the clan to accept my leadership and have the same faith in me as they had in him!'
Steve mulled this over then said: 'He still can.'
Cadillac looked puzzled.
'Besides being a seer, you have another gift that some people might also regard as magical. A gift that saved our lives on the wheel-boat.' Steve paused, but the penny failed to drop. 'The gift of mimicry. Let the spirit of Mr Snow speak through you. If you do it right you'll have them eating out of your hand.'
Cadillac throught it through and brightened visibly. 'That's brilliant...'
'Yes. It is.' _Go to the top of the class, Brickman_...
Cadillac had already been formally installed by the clan elders as wordsmith to the Clan M'Call but the ceremony had been overshadowed by the shock of Mr Snow's death just as he, as the Old One's apprentice, had been overshadowed since childhood by his master's commanding presence. Cadillac sensed the clan felt they had been landed with the monkey instead of the organ-grinder. He had to change that perception of him – and this was probably the only chance he would get.
He looked around at the half circle of expectant faces, lit by the leaping flames of the huge bonfire. A sweet smell of pine resin hung on the night air. Behind him, flurries of orange sparks, like miniature constellations, were swept skywards on the rising air.
Steve, seated near the middle of the third row, listened with bated breath as the young Mute launched into his address. If Caddy didn't pull it off, Steve knew he might as well pack up and go home.
Cadillac was surprised to find that his earlier nervousness had vanished. He paced slowly along the front row, his gaze roving back and forth with the vaguely disapproving air of a visiting general asked to review a battalion that is not quite up to snuff. His only regret was that Clearwater was not here to witness his debut as a star performer.
He returned to the centre and began: 'As I look about me and find that your eyes, like mine, are filled with the same bitter tears, I know that your hearts are gripped by the same sorrow, that our minds are engraved with the image of one man whose name is on all our lips –'
'Oyy-yehhh...' moaned the clan.
'Mr Snow! The Storm-Bringer! Summoner and word-smith of the M'Calls, paramount clan of the She-Kargo! Whose mighty powers strengthened the knife-arms of our warriors, whose wise counsel guided our elders during the dark days when the greatness which is our destiny seemed to be slipping from our grasp!'
'Oyy-yehhh...'
'His death, his passing from this world, leaves a great emptiness in the landscape of our minds. For he towered above us like a giant tree on the the barren plains of existence. A deep-rooted tree which gave us shelter and from which we drew sustenance. From his lips flowed the fire-songs celebrating the prowess of our warriors old and new, the history of the Plainfolk stretching back to the War of a Thousand Suns, and stories of the Old Time.'
'Oyy-yehhh...'
'This was his gift to me! His wondrous tales, his wisdom, his knowledge of this world and the worlds beyond now reside in me! His will now stiffens my resolve! His spirit, freed from the confines of his body, surrounds me! Enters me with each breath! Possesses me!'
So saying, Cadillac turned his face upwards, closed his eyes, flung his arms out sideways and fell to his knees. His head sagged forward onto his chest, his arms dropped to his sides, then after a brief moment in which his audience watched spellbound, he opened his eyes and slowly rose to his feet. And although they were quite dissimilar in life, he somehow managed to assume all the physical characteristics of Mr Snow – chin thrust out aggressively, hands reversed onto hips, legs slightly bowed by age as he paced up and down, the quick, questioning eye movements.
And the voice. The pitch, the intonation... both were perfect.
'What a miserable bunch! Sweet Sky Mother! Is this what I've bequeathed my successor?! Does your backbone disappear the moment I step beyond this world? A fine reward for all my labours! Has every word I uttered been a waste of breath? If so dry your tears! It is _I_ who should weep for you, who sit there trembling like lost children in a dark forest waiting for the wolves to devour them!'
Steve clapped his hands together in admiration. Cadillac might be a prize bonzo but when he tried he could really put it together.
'Are there to be no more fire-songs because I am not there to put your brave deeds into words? Is the courage for which the M'Calls are renowned to wither on the vine because my power has passed over with me? Were you only shadows to whom I gave substance when danger threatened? Are you only as brave as the droppings stuck to the tail of a charging buffalo? Do you sink into a cowardly stupor when the need for heroes has never been greater?!'
'NO!!' chorussed the clan.
'No?' Cadillac tugged at an imaginary beard as he strode back and forth. 'How boldly and easily you reply! Yes – you are brave enough when you are hidden in the shadows, your voices laid one upon the other so that no one know who speaks the truth! Cadillac now wears my mantle! My knowledge of all things has passed to him. You have appointed him to my place yet you refuse to follow him against the iron-snake!'
'He is too young!' cried a voice from the darkness. 'He has no standing!'
Mr Snow's voice boomed from Cadillac's throat. 'No standing?! He inherits _my_ standing! For all I was, he now is! Speak, those of you who remember! Did you not heed me when _I_ was young?!'
'Yes, but he does not have the power!' protested another unidentified voice.
'Clearwater has that power! That is why he seeks to free her from the belly of the snake! Does he not have the gift of seership? Is he not versed in the ways of the sand-burrowers and the dead-faces? Why, he even speaks their fearful tongue! He is as brave as I ever was, and knows more than I ever will!
'Cadillac, Clearwater and the cloud-warrior are among The Chosen! They grace the M'Calls with their presence because we – among all the She-Kargo – have found favour in the eyes of Talisman! Let each man, woman and child pledge themselves anew here and now! Let them come forward one by one and say whether they will stand with Cadillac against the sand-burrowers and be the first to strike a mighty blow on behalf of the Plainfolk!
'Do this not in memory of me but in praise of him! He whom I loved and nurtured as if he had been my own blood-child! Clan-brothers! Clan-sisters! Place your lives in his hands as you placed them in mine and I swear that each one of you shall find me at your side in the hour of need! The power given to me by Talisman shall reach out to you from beyond the veil!'
Cadillac flung both arms into the air in the style of Mr Snow, spun round on his heel and became himself again. 'How say you? Are you ready to stand with me?!' The entire clan leapt to their feet in a tumultuous explosion of joy.
'Heyy-yahh! Heyy-YAHH! HEYY-YAHHH!!'
Some time later, after Cadillac had managed to extricate himself from the clutches of his admiring followers, he came over to Steve's hut and found his arch-rival sitting outside, staring into a small fire.
Through the half-raised door-flap, he glimpsed the top third of Night-Fever. The rest of her naked body was tucked between the sleeping furs. Given the general mood of celebration, she was no doubt hoping that tonight was going to be her lucky night.
Steve raised his eyes. Cadillac's face was still glowing with pleasure.
The Mute sat down opposite him, adopting the same cross-legged pose. 'How did I do?'
'Pretty good...'
'Is that all you can say?!'
'For crissakes, Caddy! What can I tell you that you haven't been told a million times already?'
'Yes, I know all that. It's just that I respect your opinion. I mean, you're not easily impressed.'
Steve sighed and threw up his hands. 'What can I say? It was fantastic, amazing. The way you transformed yourself...'
'Yes, even I thought I did that rather well –'
'And the voice – every nuance, every intonation...' Steve kissed the tip of his fingers. ' _Muy perfecto!_ '
'And totally unrehearsed –'
'Ahh, yes, but based on years of careful observation. And when one adds in what you had to say... spellbinding!'
'Ohh, do you really think so?'
I've got to stop, thought Steve, otherwise I will throw up. 'Do I think so?! I know so! I was watching the people around me. They were hanging on every word!'
Cadillac attempted a modest smile but only succeeded in oozing several more pints of self-satisfaction. 'Yes, to judge from what everyone's been saying I think I did manage to get the message across. Thank you.' He saw Steve's surprise. 'It _was_ your idea.'
Steve shrugged. 'Ideas aren't hard to come by. Translating them into action – _that's_ the difficult bit. And you know what?'
'I'm listening...'
'Watching you tonight gave me another idea. You could be Malone.'
Cadillac smiled. 'That's right. I was wondering how long it was going to take you to work that one out.'
Steve inclined his head in mock respect. 'Nice to know you're ahead of me.'
_For once_...
'Yes,' said Cadillac. 'And I've got another suggestion. Now that you've seen my impersonation of the Old One...'
Steve groaned inwardly. _Christo, how much soft soap does this guy need?!_
'... why don't you take me to see the real thing?'
The question, coming out of left field, caught Steve totally unprepared. 'Uhh, the what?'
'The real thing,' said Cadillac patiently. 'The Old One. Don't try and fuck me around, Brickman. He's alive and well, isn't he?'
'How d'you work that out?'
'By thinking things through. The shock stopped me from doing so at first. There he was, dead, right in front of my own eyes, but part of my mind couldn't accept it. I had _seen_ his dying place in the stones!'
'Yeah. According to you, he was supposed to die _last_ year!'
'Have you ever tried reading a stone?' cried Cadillac scornfully. 'If you had the gift you would know that the time at which an event will take place is the hardest thing to decipher.' He paused, then said: 'How did he fake it?'
'You'd better ask him.'
'And the bodies were switched after we placed him on the High Ground. Who was left for the death-birds to feed on?'
'Some old, white-haired guy who died at the trading post. He was wrapped up in the bundle of cloth that was slung over my horse.'
'Yes, I thought it was something like that. Very clever. Do you think Malone fell for it?'
'Sure. He's never seen Mr Snow.'
'No.' Cadillac studied Steve's face then smiled. 'You can't bear me being right, can you? You just hate it when I'm out in front.'
'Not at all,' said Steve. 'It's a lot better that working with a sponge-head who keeps falling nose first into his boiled rice.'
Cadillac ignored the reference to his _sake_ -sodden nights in Ne-Issan. 'I haven't just _seen_ the Old One's dying place, I've _been_ there – twice! The pictures I drew from the stone were so clear! It was only when I ran over everything again in my mind that the truth gradually dawned on me. Yes. It was a cruel trick you played –'
'It wasn't my –'
Cadillac cut back in. 'Don't worry, I'm not accusing you. I understand why it was necessary to deceive me. I had to prove myself. Stand on my own feet. Isn't that what the Old One wanted?'
'Yes,' nodded Steve.
'Then take me to him.'
'I can't. You'll have to ask Awesome-Wells and Boston-Bruin. They're the only ones who know where he is.'
Cadillac scrambled to his feet. 'Good! Let's find them. We have important things to discuss.'
'Whoa! Slow down! They tell him what's been happening. If he wants to see you, he'll let you know.'
'But –'
Steve rose. 'Listen, Caddy. If and when we do see him, don't expect too much. They say he hasn't got long to live.'
'Long enough to take part in one last great battle.'
'The one you saw in the stones?'
Cadillac answered with a reluctant nod.
'Do you know who wins?'
The Mute shook his head, his eyes filled with tears.
How ironic, thought Steve. The one man who could save the day is lying at death's door and his successor has persuaded the clan to follow him in a suicidal attack on the wagon-train to free someone who might die if she was rescued.
When Cadillac had found him staring into the fire, Steve had been trying to piece his life back together. A little while before, Roz had come through on their private line to say that Clearwater was not well enough to be taken off the train.
The shattered femur in her right thigh had been carefully reassembled and pinned together but would take six months to mend, held in a hi-tech metal splint. If she did not continue to get the level of medical care available in the Federation, she could be permanently crippled or – if complications set in – might lose the leg and perhaps her life.
All his past efforts, everything he'd been working towards, had gone straight down the fucking tube...
## CHAPTER TEN
Mr Snow had been hidden in a cave entered by a cleft in a steep jagged rock face buried in the densest part of the surrounding forest. The staggered shape of the opening meant that no light from the fire which illuminated and warmed the inside could be seen by any stray passer-by.
Access was difficult enough even if you knew where it was. The entrance was masked by brushwood piled on the narrow ledge outside, and to reach it you had clamber up a precipitous ninety foot slope.
Even the pines, packed together like giant porcupine quills, were finding it difficult to stay upright. Many of them had lost their footing and keeled over. Others, their stunted branches signalling the losing fight for air space, had died quietly in the arms of their stronger neighbours. But not all of the fallen trees were down and out: here and there, were pines whose horizontal trunks had made an incredible right-angled turn towards the sky.
As soon as the location had been pointed out to him, Cadillac steamed up the slope and disappeared into the cave a good minute of so ahead of the others.
'How d'you manage to get him up here,' asked Steve as he reached the ledge and offered a hand to Awesome-Wells.
'With great difficulty,' gasped the elder.
'He found this cave when he was a boy,' wheezed Boston-Bruin. 'And when he was a young man, he used to bring women here.'
Awesome saw Steve's puzzled reaction. 'For illicit affairs.'
Steve laughed. 'Mr Snow...?'
'Never stopped,' said Boston, catching his breath.
'Mack-Truck, Rolling-Stone and us two were always covering up for him.'
Awesome chuckled. 'When you weren't using the place yourself.'
'True. But then "stolen fruit is always the sweetest".' Boston heaved a wistful sigh then led the way into the cave.
To reach the cave proper, they had to zigzag between sloping buttresses of rock that in some places reduced the floor space to a V-shaped crevice. A few yards further in, the walls opened out to form a dry, roughly oval chamber with a uneven, split-level floor. The sides came together in an narrow ogee curve to form the ceiling but the rock slabs were mismatched, creating several vents through which an updraft cleared the smoke from the wood fire.
Mr Snow lay on the upper ledge, propped up on a bed of furs; a convenient slope in the far wall on which a bearskin had been laid served as a back support. To his right, Cadillac sat cross-ledgged on a talking mat, holding onto his mentor's hand like a lost child who had just been returned to his parents.
The two M'Call elders stoked up the blaze then positioned themselves by the entrance.
'Greetings, Old One,' said Steve.
Mr Snow indicated the empty mat on the left side of his bed. The movement of his arm underlined his weakened physical condition. His voice was stronger than before but speaking still required a visible effort. 'Sit here. That way I can box both your ears if you don't behave yourselves.'
Cadillac looked embarrassed. 'It was all a misunderstanding, Old One.'
'So you keep saying. Both of you still have a lot to learn. You're too headstrong, too impatient, too self-centred, too _young_! Hmmpff! You should be working together – striving for harmony, instead of constantly flying at each other's throats!'
Steve smiled. 'News travels fast...'
'Bad news always does,' grumped Mr Snow.
'I think Cadillac and I have both realized the error of our ways,' said Steve, generously taking a share of the blame despite believing he was the innocent party.
'Yes, well, they say miracles can happen. Let's hope at least that when you leave here you will look at each other in a different light and both be a little wiser.' Mr Snow closed his eyes and breathed deeply. He appeared to be trying to summon up the energy to continue.
'Are you able to stand, Old One?' asked Cadillac.
'I can, yes, but I'm saving my strength for the big day. Now, let's get down to business!'
Cadillac brightened. 'Are we to attack the iron-snake?'
'That's one of the things we have to talk about, yes.'
'In that case you can save your breath,' said Steve. 'Roz, my kin-sister made contact yesterday.' He looked across at Cadillac. 'Just before you came to see me. We can't rescue Clearwater. She isn't fit enough. She's gonna have to stay on the train and...' He couldn't bear to complete the sentence.
Cadillac did it for him: '... be taken back to the Federation?'
Steve nodded.
The young Mute gazed at Steve with open-mouthed disbelief then exploded. 'But this is outrageous! After all the talk, all the planning, all the arguments! After all the effort I put into rallying the clan behind me last night, are you now telling me it's all been a complete waste of time?!'
'What are you trying to do – make out that it's _my_ fault?!' cried Steve.
'Well, you were the one that came up with this stupid idea!'
'I see... now that it's fallen through it's suddenly a stupid idea and it's all down to me! Amazing! If we'd been able to pull it off, you'd have been trampling over our backs in the rush to grab all the credit!'
Mr Snow's eyes blazed. He slapped the air in front of him, crossing his right forearm above the left in a scissor-like motion.
As he did so, Steve and Cadillac felt a stinging blow on their cheeks that snapped their heads towards Mr Snow's feet, then as he uncrossed his arms and swept the backs of his hands outwards, both young men were pounded by an invisible hammer-blow to the solar plexus which hurled them backwards against the side walls of the cave.
The impact left them stunned and winded – and not a little fearful. Clearwater had used her power in a similar way to stop their knife-fight. They both got to their knees but stayed where they had fallen, trembling with shock.
'Next time I will not be so gentle with you,' growled Mr Snow. 'A plague on both your houses! I invited you here so that I could talk with The Chosen, not to listen to the yapping of a pair of young jackals! Before you can lead the Plainfolk to greatness you must conduct yourself in a manner befitting the honour Talisman has bestowed upon you!
'What is past is past! All that matters is what happens from here on in. I don't intend to have my last days on earth ruined by a recital of your petty jealousies, so pull yourselves together!'
Mr Snow pointed to the talking mats. Steve and Cadillac moved obediently back to their places on all fours like young lions responding to the crack of their trainer's whip.
'Let us set aside the question of Clearwater and your kin-sister for the moment and concentrate on the wagon-train. Do you have a plan which will enable us to seize and hold it for several hours?'
'We're still working on that,' replied Steve. 'But I'm beginning to see a way through. A lot depends on Cadillac being able to persuade the people aboard Red River that he's someone else.'
He let Cadillac explain their basic strategy then said: 'Red River is tough nut to crack. Our only chance is to seize the initiative and throw them off balance. Your reported death is a step in the right direction.'
'If they believe it.'
'What more proof do they need? Roz will have told them that _I_ believe you're dead. But even if they don't trust me completely, Malone has seen the body, and will have reported that the clan are in deep mourning.' Steve smiled. 'Your clanfolk are going to be pretty upset when they find out you're still alive.'
'We'll cross that bridge when we come to it. Carry on...'
'We've decided to dispense with the services of Malone and his renegades,' announced Cadillac. 'I will then send false messages using his voice.'
'I see, hmmm... I've always thought your gift of mimicry might come in useful one day. I hope it works.'
'It already has!' exclaimed Cadillac. 'When we were –'
Steve cut him short. 'He's brilliant, believe me. But let's stick to Malone for the moment. We can talk about your impersonations some other time.'
Cadillac subsided, tight-lipped.
Steve ignored the hurt look. 'We've decided to jettison his whole plan. Originally we thought of eliminating the renegade contingent and putting a hundred Bears on board but that still meant going up that ramp into an ambush situation. The only surprise would have been our unified force instead of one in which half the soldiers were going to turn their guns on the other.'
'We would also have the power of the Old One.'
'True,' admitted Steve. 'But the crew of Red River would still be geared up ready to repel boarders. Your job as "Malone" will be to kid them into thinking that everything is going ahead exactly as planned.' He turned to Mr Snow. 'Except that it won't be. What we have to do is change both the timing and the nature of the attack.'
Mr Snow nodded. 'You mean go in early...'
'Yes. And in a different disguise. The ramps still have to come down because that's the only way we can get in from ground level, but apart from that we've got to come up with a whole new angle.' To help Mr Snow understand the difficulties facing them, Steve explained the physical layout of a wagon-train and the variety of defensive systems its commander could bring to bear against an external attack.
'And he's backed by a battalion of Trail-Blazers. One thousand soldiers. Not all of them would be deployed in a ground action but they are all trained for combat. And they'll all be on that train. Once you get on board and cut loose I've no doubt we can hold our own. The thing is, when you start throwing earth magic around –'
'I must take care not to hurt Roz and Clearwater. That point had occurred to me,' said Mr Snow.
'Good. And on top of the basic element of surprise, which your appearance is an important part of, we've got another thing going for us.'
'And what's that?'
'Apart from their re-supply role, wagon-trains were designed for military operations against Mutes. Savages – or so they think – armed with knives, stone hammers and crossbows.'
'And renegades armed with rifles,' said Cadillac.
'Yeah, but apart from Malone's bunch – who were manoeuvred into this situation – no renegade would go up against a wagon-train. That's the point I'm trying to make. The trains aren't designed to resist an attack by _Trackers_! They're vulnerable – as Cadillac proved last year. I haven't got the whole plan worked out yet but I know exactly how we could open up that train. The trouble is, we don't have any explosives...'
Cadillac gave Steve and Mr Snow a thoughtful glance then said: 'We've got a box of AP108's, six packs of PX, detonators and timers. Will that help?'
_Are you kidding?_
Steve masked his delighted surprise behind a non-committal shrug. 'Yeah. It'll do to be going on with.' He grinned. 'Malone was wondering where that stuff had got to. Well done. You're developing into quite a cagey customer yourself.'
Cadillac accepted the compliment with a courteous nod. 'I learned at the feet of a master.'
'I asked you to work together, not form a mutual admiration society,' snapped Mr Snow. 'Are we in business or aren't we?'
'We've got the means,' said Steve. 'We just have to figure out the ways.'
'Good.'
'Plus the really big question – "Why?"'
'Why what?'
'Why are we attacking Red River? I've already told you Clearwater can't be moved. So what's the point of risking the lives of your clanfolk. They'd be dying for nothing.'
'Not entirely. Even if we spared the part of the iron-snake which holds Clearwater and your kin-sister, we can still destroy the remainder and collect the heads of a thousand sand-burrowers!'
'Heyy-yaahhh...' growled the two elders.
Steve had forgotten they were there. He glanced towards the cave's entrance chamber. The two wiry old Mutes clearly found the prospect appealing. 'But at what cost?' he demanded.
Mr Snow hit the roof. 'Cost?! The honour of the M'Calls is not a negotiable commodity that can be measured out like bread-grass seed! Long ago, did I not tell you that the Clan M'Call had the courage to face its own destiny? That time draws near! It is not enough to challenge the iron-snakes. They must be defeated! Utterly destroyed! The Lords of the Dark Cities must be made to understand that they can never enslave the Plainfolk!'
'Heyy-yaahhh!' This time Cadillac joined in as the elders chorused the traditional up-beat response. And to his surprise, Steve found his lips moving silently in unison with theirs.
Mr Snow closed his eyes and sank back exhausted.
Cadillac voiced the concern Steve shared: 'Shall we return tomorrow when you have rested?'
'No... no... I have something to say to you... something that must be said before this battle begins.' Reaching out like a blind man, Mr Snow felt his way onto Steve's forearm then took hold of his hand and did the same with Cadillac. For a dying man his grip was surprisingly strong.
'Listen to me carefully, Brickman. I have known for a long time that Clearwater was to be taken from the land of Plainfolk. That is why you were sent to us. The Sky Voices spoke to me of your coming, and Cadillac saw your return in the stones... the blood and tears that would follow.'
'So Motor-Head was right. I _am_ the Death-Bringer...'
'Yes, but you are not the enemy. It is those who seek to be your masters who are the base servants of Lord Pent-Agon, the Evil Destroyer, Sower of Hatred and Corrupter of Minds. Like us you are a pawn in a game whose scope and complexity we cannot begin to comprehend, a strolling player on a stage so vast there is room for every living creature on the face of the earth and whose back-drop is Mo-Town's cloak of stars – the heavens and the worlds beyond!
'Each of us moves along the path traced out for us. Paths that cross briefly or come together and run as one before separating again. You and I have journied together and now... the time of parting is near...'
Cadillac's eyes filled with tears.
Mr Snow tightened his grip on his pupil's hand and shook it roughly. 'Don't be such a ninny! There's nothing to be sad about. The fate of the Plainfolk lies in _your_ hands! I shall keep an eye on you from up there and if I see you making a bodge of it, I shall return as a precocious little brat and follow you around making helpful suggestions. Does that remind you of anyone you know – hhmmm?'
Cadillac blinked the tears away. 'I stopped doing that years ago.'
'Yes, well, I haven't gone yet, so stop that snuffling and pay attention.' Mr Snow turned his eyes on Steve. 'We have to attack the iron-snake in order to rescue your kin-sister.'
'Roz...?'
'You look surprised. Isn't she the reason why you and Cadillac tried to kill one another? Wasn't it because you wanted to free her?'
'Yes... but that was before Clearwater was gunned down by the Skyhawk. Roz is now one of the team that is looking after her. If Clearwater has to go back into the Federation –'
'It will be _your_ job to look after her. Both of you will face many dangers before you return to the blue-sky world. Your sister has already begun that journey. She too was born in the shadow of Talisman and it is time for her to take her appointed place among her own people.'
It took a moment for the full import of the words to sink in. 'You mean Roz is a...?'
'A Mute? Yes. Just like you.'
Steve looked across at Cadillac and saw that he was equally stunned by the news. 'But that's –'
'Impossible? You disappoint me, Brickman. A young man of your intelligence. Did you never ask yourself why you were so attuned to the overground? Had no fear of open spaces? Never became ill? You have carried the answers within you since birth – when your _real_ identity was submerged.
'That is the stranger who lurks in the deepest recesses of your mind. Whose whispers have frightened you for so long! Call him! Let him emerge from the darkness into which he was cast by your masters! Face your true self! Let your soul take wing!'
Steve was gripped by a wave of panic as a babble of voices filled his inner ear, swelling and fading, reverberating through his mind. The voices became a torrent of sound. It seemed as if every word he had ever heard or uttered was being replayed simultaneously inside a vast cavern whose walls were lined with towering stacks of tape-decks.
Close to fainting, he clutched at his head, eyes shut tight, mouth open in a soundless scream. There came a moment when the cacophonous barrage reached an unbearable intensity then gradually, the atonal jumble coalesced into a number of recognizable strands which rearranged themselves into a kind of music he had never heard before but which was as much part of him as his pounding heart.
It reminded him of Mute mouth-music, but even that in all its richness was a pale imitation of the master-work which now filled his mind. The voices, touched with an unearthly purity and beauty, swelled, producing a triumphant chorus of sound, a soaring vibrant symphony that lifted his soul upwards and outwards, carrying him on a melodious wave of sound that washed against the stars.
For one exquisite timeless moment he was bathed in the crystal clear harmonies that he knew, with unerring clarity, were at the heart of all creation. They entered every fibre of his being. He dissolved into the sound, became at one with the universe. And he sensed that Roz was there with him too. Then cruelly, the music began to fade, and with it the sense of overwhelming joy and wonder. Like a drug-addict trying to hang on to a dwindling high, he tried to fix the melodies and the feeling of exaltation in his memory but it slipped away and was replaced by a monotonous thudding beat.
Steve became conscious of his heart pounding against his ribs. He raised his head and opened his eyes to find everyone watching him intently.
Awesome-Wells and Boston-Bruin raised their right hands in a friendly salute.
Steve tried to shake some of the confusion from his mind. He caught Mr Snow's eye. 'I'm sorry. Was I, err... away long?'
'That's okay. The Old One smiled benevolently. 'The Federation did a good job on you. Unstitching all that must be a major trip.'
Steve nodded. 'Roz... She was there. She knows...'
'Of course she does. That's why she's on the wagon-train.'
'But how did _you_ –?'
'Don't give me more credit than I deserve, Brickman. I didn't know _how_ she would get here. I just knew both of you would be coming. The Plainfolk have been waiting a long time.'
'Who told you – the Sky Voices?'
Mr Snow nodded.
'I think I've heard them too.' Steve looked at them in turn. 'When did you guess?'
'There was no guesswork involved. We'd been watching the skies for some time. Cadillac knew you were on the iron-snake. He saw your presence in a seeing-stone.'
'But not your face,' interjected Cadillac, anxious to get himself back into the conversation. 'It was only when you crashed into the crop-field that I realized _you_ were the cloud-warrior sent by Talisman.'
'And I realized you were a Mute from almost the very first moment I set eyes on you,' said Mr Snow. 'And it was equally clear you didn't know.'
'Or didn't want to know,' added Cadillac.
'That's not true,' protested Steve. 'Okay, I tried to suppress the mental link between Roz and myself. When you live inside the Federation, that kind of thing is bad news. But we both knew we were different. And from the moment I took off on my first overground solo I knew I belonged here, in the blue-sky world. I just couldn't figure out why.
'To be able to survive for so long without becoming ground-sick, to be able to adapt so quickly, to find that – with a little practice – I could run with the Bears... only a fool would not have asked himself how such things were possible.'
'Only you tried to find some other explanation.' Annoyed that Steve had become the centre of attention, Cadillac had decided to adopt the role of public prosecutor.
'Yes. But not because I would have been ashamed to discover I was a Mute. How could I feel that way when I'd fallen in love with Clearwater?'
'That was because you discovered she had the skin of a sand-burrower!' cried Cadillac.
'No! That's not true! Ask Mr Snow! I was smitten from the very first time I saw her. When I was made to bite the arrow! The Old One warned me off.'
'That's right,' sighed Mr Snow. He caught Cadillac's eye. 'And now that we have reached a certain level of enlightenment, let us not cloud our achievement with more of this foolish rivalry.'
Cadillac lowered his head briefly. When he raised it, his nostrils looked distinctly pinched.
Steve ploughed on. 'The reason I tried to find some other explanation was because I was looking for one that made sense! I mean I _know_ who my mother was! Annie Brickman. And my father was the father of all my generation – George Washington Jefferson the 31st, President-General of the Amtrak Federation. Are you trying to tell me that they're Mutes too?!'
Mr Snow laid his hand on top of Steve's. 'The answer to that question lies at the heart of your world.'
Steve knew what that meant: the secret lay within the First Family or inside the data banks controlled by COLUMBUS. And like a moth drawn to a flame, Steve knew he could not rest until he had discovered what that secret was.
'You look troubled, Brickman.'
'That's because he can't accept it,' said Cadillac. He wasn't sure he wanted to accept it himself. Above all, he was extremely annoyed that Mr Snow had revealed the truth to both of them at the same time instead of telling him first.
'You're wrong,' said Steve. 'I'm not trying to find a way out. You and Clearwater are living proof that there are Mutes who look exactly like Trackers –'
'Shouldn't that be Trackers who look like Mutes?'
'Listen! Whatever – okay?'
_Jeezuss! What a fucking nitpicker!_
Steve picked up the thread: 'I don't deny feeling that coming out onto the overground was like coming home. But _feeling_ is not enough. Not for me, anyway. I've spent my whole life trying to control my emotions. If you want to get anywhere, you have to study the facts, reason things through.
'Cold logic – that's what counts. Emotions cloud your judgement. You can't ignore them but if you let 'em take over, it's not long before you don't know up from down. I trust you, Old One. If you say Roz and I are Mutes then I accept you sincerely believe we are. But can you prove it?'
'No, I can't, and I don't need to.' He clutched his heart. 'For a Mute, _knowing_ – in here – is enough! You don't need to know why! Open up, Brickman! Sweep away all that soul-shrivelling junk that's been pumped into you by the Federation! Something happened to you a little while back, didn't it? Something good.'
'Yes, it –'
Mr Snow cut him off. 'There's no need to tell me about it. I've been there. You touched the stars. That music you heard, those strands of melody are part of the fabric of the universe. _That's_ what you have to tune in to! All this so-called scientific knowledge dispensed by that electronic heap of garbage you call COLUMBUS is irrelevant! That's not what this world is about. It's weighing you down, stopping you realizing your full potential!'
'Okay, okay! I'm not running away from any of this. I – I just need time to think it through!'
'Don't!' cried Mr Snow. 'Do you remember long ago when we talked of how one might discover the truth about existence and I likened that search to climbing a mountain? What you experienced out there was that rare and precious moment. You reached the peak. And someone reached down from Heaven and offered you their hand. Don't analyse it. Grab it! Take it on trust – the way the Plainfolk put their trust in the Great Sky Mother. Make what in the Old Time was called an act of Faith.'
'It won't work, Old One. Before I can believe it, I have to know how and why.'
'Spoken like a true sand-burrower,' said Cadillac.
'Yes,' agreed Mr Snow. 'You're a tough nut to crack.'
Steve responded with a rueful grin. 'Don't worry. I'll get it sorted out. In my own way and in my own time. But thanks for telling me. It's gonna make things a lot easier when we hit the train.'
'Exactly. But in making our plans, we must not lose sight of why you must return to the Federation. The task which faces you and Clearwater is to destroy the Dark Cities from within.'
'Wowww... just the two of us?'
Mr Snow gripped Steve's hand. 'She has the power, and so do you, Brickman – if only you will let your mind make that leap. But let me continue. To achieve this destruction you must win the absolute trust of those who rule the world beneath the desert. You must be seen to be working hand in glove with this man Malone _against_ us. And your kin-sister must also play her part in this deception. If possible, we must try to achieve _our_ objectives without the Federation realizing it has been defeated. But whatever happens we must ensure that you and Roz emerge entirely blameless from this encounter.'
Steve digested this mouthful. 'Well, it's a nice idea, but... jeez!' He threw up his hands. 'It was difficult enough as it stood. And now you've made it just about impossible!'
'Oh, come, come, don't exaggerate! You're the master of deception, and this young pup is the man of a thousand voices. I'm sure that if you put your heads together you'll be able to come up with something.'
Such is the perverseness of human nature, neither Steve nor Cadillac emerged entirely pleased from the meeting with Mr Snow. The revelation that Steve was of the Plainfolk made Cadillac _more_ bitter not less. In their struggle to become top-dog, the knowledge that Steve was a sand-burrower had sustained Cadillac in the dark moments of despair and uncertainty that engulfed him whenever his rival appeared to have gained the upper hand.
Even if Brickman triumphed, even if he had won Clearwater's heart and soul, there was always the comforting thought that Steve would always be inferior because he was a creature of the Dark Cities. But now, even that small consolation was denied him! He had sworn an oath of blood-brotherhood with someone who – on top of every other advantage he possessed – was also a Mute! Life was disgustingly unfair!
For Steve, the cold reality that came in the wake of the brief moment of rapture when the mind-blocks creating by years of conditioning were demolished to reveal his true nature was equally unpalatable. In one sense, the discovery that he was a Mute came as a great relief. It answered so many of the questions that had plagued him since early childhood, and more especially since that magical moment when he had emerged from the subterranean concrete world of the Federation and had caught his first glimpse of the overground.
There was also the added satisfaction of knowing that Cadillac was extremely upset at learning that his 'blood-brother' was a true son of the Plainfolk. But being Steve, he could not accept Mr Snow's advice. He wanted to know _why_ he and Roz had been raised as Trackers. For years he had vowed to unravel the secrets of the Federation and now here was another – probably the greatest secret of all!
That was the good part. The bad part was the discovery that his kin-sister was destined to end up as Cadillac's partner and perhaps in his arms. Since the Mute had lost Clearwater, any reasonable person might have concluded it was a fair swap. But Steve was not a reasonable person.
Roz was more than just his kin-sister: their mental rapport made them psychic Siamese twins, and although he had tried to suppress it since meeting Clearwater, there was still a sexual dimension to their relationship although Steve would not have been able to express it in those terms.
Despite the frightening aspects of her newly-released power, Steve wanted her _and_ Clearwater. Not in his bed but under his control. And if that, in the case of Roz, was no longer practical, then he wanted her love and allegiance – if only to deny them to Cadillac. The thought of them together was repugnant and his distress was in direct ratio to the pleasure the news had given his' rival.
Both of them had gained and lost in equal measure as a result of their inability to become true brothers under the sun.
In the weeks they had spent together, Clearwater and Roz had not spoken to each other apart from the normal patient-doctor conversations centred around the medical treatment she was receiving while in Roz's care: how well she felt, what hurt and what didn't.
Roz's attitude towards Clearwater had undergone a dramatic transformation but for the sake of appearances, she had to display a certain coldness. If Karlstrom, or one of his minions was watching the tapes, they needed to be reassured that she still regarded Clearwater as an unwelcome rival for Steve's affections.
Their non-verbal conversations, on the other hand, had become a regular feature of their lives. And it was over the mind-bridge that Roz was able to warn Clearwater that their vocal transactions were probably being monitored by a hidden video-camera.
Now, in early June, six weeks after her near-fatal encounter with the Skyhawk, Clearwater was sitting up and taking stock of the sterile, hi-tech environment which served as a prison cell. Mutes did not have jails. Prison was an unknown concept. You were either alive and free to roam the overground, or dead.
The interiors of the huts built by the M'Calls were not as large as the pale-grey sharp-cornered cave in which she had been confined, but there at least she had the constant opportunity of stepping outside. Here, in a space measuring some three paces by four, whose flat roof lay just beyond a man's raised fingertips, the sun did not shine, the rain did not fall and the light that filled it was not The Light of Heaven.
The breeze which moved across her face came from a humming device with whirling arms, but it was not real. This air had not swept down the snowy peak of a pine-covered mountain, or blown across a plain of sweet-smelling buffalo grass and sage. It was heavy with strange unnatural odours fashioned in the Dark Cities.
Clearwater's heart was chilled by the thought of what lay ahead. Ever since that fateful moment when she had stood with Cadillac and Mr Snow on the edge of the bluff and watched the cloud-warrior climb steadily into the dawn sky she had known that one day she would be called upon to make the journey which had now begun.
Roz had assured her that all would be well. The cloud-warrior would return with her – but what would happen when they reached the sand-burrower's lair? Would they be separated or allowed to stay together? Would he become enmeshed in his former mode of existence? And if he did so, would his feelings for her remain the same? Most important of all, when the healing process was completed, would her powers be restored?
Mr Snow had told her she would live to see the sun again but when would that be? She was already suffering severe withdrawal symptoms and the claustrophobia generated by her surroundings had increased as she made the transition from semi-conscious invalid to a wide-awake patient eager to regain her former mobility.
Only two things helped her cope with the feeling of slow suffocation: the caring presence of Roz and the revelation that she and her kin-brother were pure-blooded unmarked Mutes like Cadillac and herself. Knowing they shared a common ancestry removed the last lingering stigma from her relationship with Steve. Their union had not been a betrayal. Cadillac had suffered but everything that had happened had been willed by Talisman. And she was further comforted by the knowledge that her soul-sister's steps were also being directed by the Thrice-Gifted One.
The realization that she too was destined to play a central role in the Talisman Prophecy had burst upon Roz when she had emerged onto the overground to make the night flight to the Red River wagon-train. Like Steve, she had heard voices, but instead of panicking and trying to shut them out, she had listened.
Her guard-mother's confession that Roz and Steve were not her natural children had helped prepare her for the moment when she would discover her true identity. It had come sooner than expected but although the moment of recognition was filled with wonder and a joyous sense of release, in a strange way it had not been a total surprise.
Ever since she had shared the visual images and emotions that caused Steve's mind to reel during his first flight, she had longed to see the overground for herself. Not only to be there _but to be where she belonged_. Her own flight, and that first unforgettable glimpse of the star-filled heavens had merely confirmed what her heart already knew: she was part of the blue-sky world and her soul belonged to the Plainfolk.
Unlike Steve, she did not seek to know the reason why. A sunrise or sunset is not made more beautiful in the eye of the beholder by the knowledge that the light suffusing the terrestrial atmospheric envelope emanates from a G1 type dwarf star, hanging in space some 93,000,000 miles away.
Knowing who she was, the intensity of feeling generated by that knowledge, the sensation of being made whole, of being truly alive was enough. Roz was content to _be_. Raised in the Federation, she found the conditions aboard Red River easier to bear, but like Clearwater, she chafed at her confinement.
The rescue flight to the renegade's abandoned campsite had allowed her an all too brief moment on the ground. Lifting her visor she had felt the wind on her face, had smelt the strange, rich, almost overpowering odours. After Clearwater had been given emergency first-aid and was being carried on a stretcher towards the waiting aircraft, Roz had dropped behind, torn off her gloves and run her bare hands over the earth, stones and grass that lay within reach. She had only been able to snatch two or three minutes at the most before her absence was noticed but it had been long enough to become addicted.
In the last eight weeks, during meal-breaks and off duty periods, she had managed to get up onto the flight-deck for a few minutes every day. As Steve's kin-sister and the guard-daughter of a noted wing-man, Roz was able to claim a legitimate interest in air operations. But it was only an excuse to drink in the beauty and variety of the landscape that lay like some exotic coral reef beneath a vast cloud-flecked ocean of ever-changing hue.
Wallis, the head of the AMEXICO task-force, had allowed her to go up on deck without an escort but Roz knew she had to ration the time spent topside and exhibit a nonchalant attitude towards the overground. No one, especially Karlstrom, must suspect that she had developed a taste for the blue-sky world and she covered her tracks by referring only to the activity she had seen _on_ the flight deck as opposed to what lay around it.
In those same eight weeks, the plans for dealing with the boarding party led by Steve and Malone and the subsequent massacre of the Clan M'Call had been progressively refined, and the crew of Red River had rehearsed their battle drills.
Even though she was a member of the task-force, Roz had not been involved in either the planning or the execution. Wallis was in overall charge of her fate but on a day-to-day basis, she worked alongside the medical team and took her orders from the CMO. Her job was to supervise Clearwater's recovery. To become, in fact, her shadow. And now that she was out of intensive care, Roz was sleeping in the same ward so that she could be ready at all times to counter any threat from Clearwater's earth-magic.
There was no danger of that but Roz has not passed this news on to her superiors. Had she done so, she might have found herself declared surplus to requirements and on a plane back to Grand Central. That was the one move that had to be prevented at all costs. Roz has the means to do so, but the power to manipulate the minds of those around her had to be used sparingly and with the utmost discretion in order not to raise doubts about her reliability.
As D-Day approached, she was unexpectedly summoned to a meeting with the wagon-master of Red River, his execs and the non-commissioned officers that headed the various groups within the battalion. Wallis and the others members of the task force were there, along with the Red River Trail Boss Marvin MacEvoy whose combative style of man-management had earned him the nickname of 'Mad Dog'.
The meeting was held in one of the forward mess-decks which had been cleared and rearranged for that purpose. The tables and chairs now faced one of the large training diagrams showing a cut-away side view of a wagon-train. The twenty now serving with the Trail-Blazer Division were all built to the same basic design. An expert eye might detect variations in aerial arrays and other equipment updates but apart from the insignia and code letters the trains were virtually indistinguishable inside and out.
As everyone filed in, Wallis directed Roz to the centre table in the front row then stood alongside the train layout opposite Fargo. When everyone was seated, the wagon-master picked up a long pointing rod, acknowledged their presence with a curt nod and got down to business without further delay.
'Gentlemen... the latest signals from sources in the field indicate that the attack which is the centrepiece of Operation Big-Fish will be launched within the next week, perhaps as soon as seventy-two hours from now. We had been expecting two primary targets, but it looks as if the most dangerous of these will not now appear. He has apparently died, but as you know these Mutes are firm believers in reincarnation so we must be prepared for a few last-minute surprises.'
The Red River personnel, recognizing this as one of Fargo's jokes, laughed appreciatively.
Fargo continued: 'Two White House operatives, YANKEE-ZULU and HIGH-SIERRA, will lead a joint force of approximately one hundred Mutes and renegades in the initial assault on the train. This group will be disguised as personnel from the 5th SIG-INT Squadron and will be wearing the standard overground combat fatigues and helmets. They will, in other words, look pretty much like our own dog-soldiers.
'The group will be further identified by a strip of white marker tape on the upper arm, on the chin guard of the helmet and on the back visor stop. The two primary targets – or one as the case may be – will be similarly dressed. To avoid any mistakes in identification, YANKEE-ZULU, HIGH-SIERRA and six members of his team will _not_ , repeat not carry white markers on their helmets and will enter the train with their visors raised. The Mutes, obviously, will keep their visors down in order not to give the game away. Any clear-skinned individual you see wearing a white chin or back marker on his helmet is a renegade and is to be shot on sight.' Fargo surveyed his audience. 'Is that understood?'
His audience responded with a murmur of assent.
The wagon-master indicated the two wagons labelled 'A' and 'B' just forward of the flight car then laid the pointer on wagon 'A'. 'In response to the May-Day from HIGH-SIERRA we will lower this ramp to allow the assault group to board the train. The bottom and middle floors of both wagons will be sealed off from the rest of the train.
'The top floor of wagon "A", and in particular the stairwells will be defended – and _held_ – by ten squads under the command of Lt. Commander Torrance. A second group, led by Captain Lloyd, will hold the top floor of Wagon "B".
'Now, as you know, the plan is to create a situation which will draw the entire assault group up the ramp and into the train. We don't want any of them falling back and raising the alarm. That is why another wagon-train will soon be delivering eighty Code One defaulters to us. They will arrive kitted out in fatigues bearing the appropriate rank badges and Red River insignia, and we will use these gentlemen – who are, of course, all volunteers –'
This was another of Fargo's funnies, but this time the laughter was genuine.
'– to dress out the bottom and middle floors of wagons "A" and "B". Twelve hours prior to the assault, they will all be given a carefully measured shot of a tranquillizing drug. Don't worry, it has all been tested and timed. The defaulters will be placed in active duty work-stations and provided with the normal range of weapons and side-arms that are kept racked or carried by personnel throughout the train.
'The weapons will carry loaded magazines, but the air bottles will be empty. So our stalwart volunteers are in for two unpleasant surprises. As they start to wake up, they will discover they are being invaded by a bunch of screaming Mutes and lawless renegades and then, as they try to defend themselves, they will discover that their weapons are inoperative.'
This news was met with grim silence. As Code One defaulters, death was inevitable, but everyone in the room was glad not to be in their boots.
Fargo pointed to a plan view of the bottom floor of wagon 'A'. Two compartments on either side of the ramp access had been coloured in red.
'These two special firepoints will each be manned by eight men. They are cargo skips which have been strengthened and fitted with firing ports. The two squads will be sealed inside and will do nothing to impede the initial assault. Their task is to ensure no one goes back down that ramp alive. As you can see from the plan, they are both able to cover the ramp with enfilading fire.
'Okay. The assault is absorbed as follows. HIGH-SIERRA, YANKEE-ZULU, and the two Mutes who are the primary targets will be in the first third of the column as it comes up the ramp. They will direct their lead troops onto the bottom and middle floors, while they themselves – a party of ten made up of HIGH-SIERRA and six fellow operatives plus YANKEE-ZULU and the two targets – will make for the top floor of wagon "A" where Don Wallis and his team will be lying in wait.
'As the group reach the top floor, both targets – or one, as the case may be – will be seized. If there is only one – the wordsmith known as Cadillac – then there should be no problem. But if Mr Snow, the second and much more dangerous target, decides to put in an appearance then you, young lady' – Fargo aimed the pointer at Roz – 'will have to defuse the situation without delay.'
'I'll do my best, sir.'
'Let's hope your best is good enough, because my orders are quite specific. If this individual goes out of control, then my task is to do everything in my power to protect the wagon-train. The squads on those two top decks will be my first line of defence. It will be up to you to get out of the line of fire. You understand what I'm saying? The lives of your colleagues on the task force and those two field-operatives are in your hands.'
'Yessir...'
'Good. But let's assume for the moment that the targets are apprehended and deactivated.' Fargo brought the pointer back into action. 'The task force, plus HIGH-SIERRA's group will pass along the top floor of wagon "B", through the Flight Car and into the blood-wagon, where the targets will be given whatever sedation is necessar to ensure they are no danger to the train or its crew.
'The squads on the top floors of wagons "A" and "B" will continue to hold the end stairwells then move down the adjoining stairs in the centre here, where the wagons are coupled together, to clear the middle and bottom floors – working back to back and moving out towards the ends so that no one will get caught in any cross-fire.'
Fargo's pointer returned to the plan view of wagon 'A'.
'The men in these compartments will release red flares and smoke grenades onto the ramp floor to give the impression to anyone watching that a satisfying amount of internal destruction is taking place. The guns in the adjoining wagons will be swung and left pointing downwards or upwards to indicate the position has been overrun, and more smoke will be released through roof hatches.
'While this is happening, seventy line-men, wearing the same white markers and badges as the Mute assault group, will emerge at various points along the roof and dance up and down in the way these apes do, and let off green flares. They will be joined, as soon as possible, by others displaying the severed heads of some of the Code One defaulters.'
Fargo saw his audience's reaction. 'Yes, I know. A rather gruesome touch but we think it will be the clincher. This, of course, is the signal that the main group of the M'Calls have been waiting for. So if everything has gone according to plan, we would rapidly find ourselves axle-deep in Mutes.
'As they close on the train we will bring some, but not all of our guns into action and pipe steam for close-quarter protection. The ramp will be left down, but Lt. Commander Laird has arranged to by-pass the cut-out and has redirected the nozzles around the ramp to give any intruders a warm welcome.
'We will not use all our firepower because our defence must not be too robust. The image we have to present is that of a badly-wounded opponent struggling to stay on his feet. We have to draw the M'Calls' entire force into action and hold them on the killing ground until our back-up moves into position, cutting off all lines of retreat.'
Fargo laid the pointer on Roz's table, parallel to its front edge but his eyes did not meet hers. 'The result is a foregone conclusion, gentlemen, but it should nevertheless prove to be an interesting day. After the main body of Mutes has been dealt with, there will be the usual follow-up attack.'
He fixed his gaze on the two field commanders: Lt. Commander Jim Torrance and Captain Griff Lloyd. 'I'll leave you to organize that, Jim. CINC-TRAIN has asked for a clean sweep. That means every woman and child, regardless of age. And they want the heads.'
Torrance exchanged glances with his deputy then nodded. 'Roger.'
'Okay... The Code-Ones should be with us by tomorrow night. Fargo's eyes connected with his Trail Boss. 'Mr MacEvoy, you and I will need to have a word about arrangements for feeding and accommodation.'
'Yessir!'
'Well, thank you, gentlemen. For the moment I guess that's it.'
Everyone leapt out of their chairs, came to attention and saluted. Fargo returned the compliment then made his exit, leaving Bill Gates, the deputy wagon-master in charge.
'All ranks... Diss-MISS! You may return to your posts.'
Wallis approached Roz as she mingled with those heading aft. 'If it comes to the crunch, d'you think you can handle it?'
'You mean Mr Snow?' Roz smiled. 'Don't worry. He's not going to be a problem.'
'I see... Is this report true then?'
Roz realized that Wallis was on a fishing expedition. 'Of his death? Steve seems to think so.'
'Does that mean you're finding it easier to get through?'
'It's better than it was...'
'Good. How does he think it's going?'
'Don, I get pictures, sensations, not written reports.'
'I meant in general.'
'He's very optimistic. He and Malone have –' Roz stopped, then closed her eyes and stroked her temples with her fingertips. 'They're on the move,' she murmured. 'Malone's renegades and the Clan M'Call are heading towards Nebraska!'
Wallis gripped her shoulder. 'Atta-girl! Keep me posted. Boy! Are these lumps in for a surprise. What d'you think of this plan we've worked out with Fargo? Isn't it terrific?'
'Can't fail,' said Roz. Her mind was already grappling with another question: how was she going to communicate the details of Fargo's plan to Steve? He and the M'Calls had to be forewarned of the trap that had been laid for them, but as she had just pointed out to Wallis, the mind-bridge was not built to carry that kind of information you would expect to find in a video-gram.
What came across were sensations linked to visual images, but these were not the highly detailed pictures registered by a video-camera. Most had the character of surreal dreamscapes where the structure and arrangement of the elements was not necessarily logical and not every object was in sharp focus.
When Roz had shared the experience of Steve's first overground flight, she 'saw' what Steve saw, but the images were filtered through his mind, altered by his perception. On a more mundane level, she experienced soaring high above the earth but she was not conscious of being inside the cockpit of a Mark One Skyhawk. Her inner eye did not register the read-out on the altimeter, or the speedo or compass-bearing.
Similarly, when Karlstrom sought her help to discover Steve's whereabouts on his journey to the Heron Pool, she did not know the names of the places he was passing through. She did not, in fact, even look at the map. In the effort to reach Steve's mind, she had gone into a deep trance and it was her fingers, searching blindly across the map that had 'felt' his presence.
The rescue of Steve and Cadillac from the wheel-boat had been mounted with the aid of the same kind of "ballpark" imagery. Roz had been alerted by Steve's mental May-Day – a sensation of deadly, imminent danger. This was linked to the image of a wheel-boat moving towards a setting sun across an inland sea, and to another of Steve and Cadillac trapped in a dark confined space below water level.
Woven around these images had been an urgent appeal to be rescued and this had been followed by pictures of flying shapes swooping out of the darkness, of fire – a towering wall of flames mirrored on the water and on the periphery of this mental canvas, the image of Clearwater amid a host of armed expectant Mutes.
It was only when Karlstrom had placed a map in front of her that she was able to link what she had seen to a specific area. Pin-point navigation and the nose-mounted radar on the mother-ship had done the rest.
But, as always, the link had been triggered by Steve's emotional state. His mind only seemed to open up when faced with extremes of danger or joy. It needed an emotional high or a stressful situation – such as when he had been wounded – to jolt his brain into action.
From the age of twelve, he had tried to shut her out and, to a large extent, he had succeeded – until he had emerged onto the overground. The emotional impact of that experience had blown away the barriers Steve had placed upon the mind-bridge. But not fully. For most of the time, the communication was one way. He could reach her but she couldn't reach him. The door to her mind was always wide open whereas the door to Steve's was closed or barely ajar. He only opened up when it suited him.
To Roz, it seemed as if she had been appointed to be his protector, and although he was two years older and had always played the part of the dominant elder brother, she sensed that she was, in many ways, stronger and wiser.
Since he left Flight Academy to ship out on The Lady from Louisiana, she had only managed to get through to him twice: once to warn him they were being watched, when he was being shuttled back to Grand Central for questioning after escaping from the M'Calls, and then a year later when he was on his way to Long Point and agonizing over whether to return to the Federation or remain with the Mutes.
She had told him to stay because by that time she had become aware of being transformed, mentally and physically. Just as a magnet attracts a pin, an invisible force was drawing her towards the overground. And whoever was exerting that force had given her the power to manipulate the minds of those around her to enable her to break free of the Federation.
Only it was not as easy as she first thought. In the indelible moment of discovery that she and Steve had shared upon learning they were Mutes, he had warned her against acting too openly. And she understood why. She had to do her utmost to ensure that the attack by the M'Calls upon Red River succeeded without anyone knowing she had intervened.
But that wasn't the end of it. Even if she managed to escape from the wagon-train she would still not be free while Steve remained in the hands of the Federation. Karlstrom, the head of AMEXICO, had pressured him into carrying out the First Family's wishes by threats against her life. The reverse was also true: the Family could bring her to heel by threatening to kill him...
A strangled cry and an alarmed whinny from one of the horsess jerked Malone into wakefulness. His first instinct was to reach for the gun under his pillow but before he could move he was seized by several pairs of hands. An instant later, he found himself spread-eagled, pinioned by the wrists and ankles and with a knife at his throat – held by a grinning Mute warrior sitting astride his chest.
Glancing sideways past his five captors, he saw that the camp-site had been overrun by Mutes. The guards must have all been killed without making a sound.
_Shit. How come? Mutes didn't operate at night!_
By the light of torches now being held aloft, Malone saw the horses being led away. More torches were being lit from the glowing remains of the camp-fire, bathing the ground in their wavering orange glow. A lot of his men were lying on the ground where they had gone to sleep wrapped in their blankets. They'd never had a chance to defend themselves. A Grade A surprise attack. The light from one of the torches washed over someone close by. Phil Robson, one of the six mexicans who acted as his lieutenants. Robson's head was growing out of his blood-drenched chest.
_Not good news. Not good at all_...
Malone studied the Mute sitting on his chest. His ugly mug looked vaguely familiar. Trouble was, all these apes looked alike. The six golden feathers in his pebble-decked leather helmet were easier to recognize. Unbelievable. They'd been jumped by their partners in crime – the M'Calls. The suckers they'd been planning to make mincemeat of.
Malone swore through his teeth. What the hell had gone wrong? He had handled his part of the operation in his usual methodical way, applying his considerable skill and experience to cover every angle. It had looked like a sweet deal. What had happened to sour it? Sensing he might not have long to live, he racked his brains for an answer. Had he made a false move? Underestimated the guile of the opposition? Or had that painted lump-sucker Brickman sold the Federation down the river?
The answer to all three questions was 'Yes', but Malone was left to draw his own conclusions. Nobody bothered to give him a gloating exposition of the master plan and the mistakes he'd made. His captors rolled him onto his face and quickly lashed his wrists together. He expected the same thing to happen to his feet but instead he found himself gagging into his pillow in an effort to stifle the searing pain as the Mute with the knife sliced through the tendons at the back of his knees.
_Yep, my friend, it doesn't look as if you'll be travelling far tonight_...
A hand reached in and pulled the long three-barrelled air pistol from under his pillow then, as his captors hauled him onto his feet he found himself facing Steve Brickman. Malone watched him check the magazine then ease off the safety. His left knee buckled. The Mute on that side jerked him upright.
Malone could feel the blood from the knife-cuts running down the backs of his calves. Biting back the pain, he said: 'Seem to be havin' a little leg trouble.'
'Just as well,' said Steve. 'I've seen what those boots of yours can do.' He placed the three-barrelled pistol against the mexican's throat and thumbed the catch onto full auto.
Malone looked at him without the slightest trace of fear. The eyes said it all. 'Hope this is nothin' personal.'
'No,' said Steve. 'This one's for Baz.'
He moved the pistol up under Malone's jaw, forcing his head backwards, and held the trigger down.
## CHAPTER ELEVEN
Alerted by the intermittent tone, Commander-General Karlstrom hit the ACCEPT CALL button. The Amtrak logo was replaced by his deputy's face. 'What is it, Tom?'
'I'm afraid the shit has hit the fan, sir. There's more stuff coming in but I thought you ought to know soonest.'
Karlstrom sank back in his chair, put his thumbs under his chin and steepled his fingers against his nose. 'Okay, let's have it...'
'Malone came through to Wallis on the open channel with a May-Day. Brickman found the explosive that sonofabitch Cadillac had purloined from the SIG-INT set-up – just as Malone suspected.'
'Go on...'
'Brickman loaded it onto one of the horses the M'Calls have got and was on his way over to Malone when he ran into a hunting posse. He managed to give them the slip but they raised the alarm and now the whole fucking clan has come after them with their knives out.'
'Hell's teeth!'
'Yeah, it's bad. Hold on – I just got a second decode up on the screen here.' McFadden moved off-camera briefly then returned. 'Well, it's not a total disaster. According to this, Malone's team and Brickman managed to shoot their way out. For the moment they've got no one on their tail but it may not stay that way. They're going to try to make it to Red River. Malone wants to know if we can provide air cover if they hit trouble.'
'No problem,' said Karlstrom. 'Anything else?'
'Yes. Malone figures if they can make it to the train ahead of the M'Calls, there's a good chance the lumps might come in after them. So we may get a shot at the clan after all.'
'Yes... but that won't give us Cadillac.'
'I know. But it may give us the next best thing – his head on a plate.'
I'm surrounded by idiots, thought Karlstrom. 'The idea was to capture him alive, Tom. Never mind. Keep me posted. Put everything up on my screens as it comes in.'
'Yes, sir...'
Karlstrom noted the tell-tale moment of hesitation. 'Are you saving the bad news till last?'
'That's for you to decide,' said McFadden. 'It wasn't just the explosives that Brickman discovered. Those crafty heaps of lumpshit were hiding something else.'
It wasn't hard to guess what. Karlstrom died a little inside. 'Mr Snow?'
'Yeah! He's alive!'
'And out of control! Like the rest of this operation!' This time, Karlstrom's anger was for real. 'Why the hell couldn't Brickman keep his sticky little fingers out this?!'
McFadden looked dismayed. 'B-But you... asked Malone to find it. And he –'
'And he fucked up, didn't he?! Too fucking zealous by half! We go to all the trouble of unpicking those requisitions and now he goes and fucking well blows everything wide open by blabbing over an open line to Wallis! What is he – out of his fucking mind?!'
McFadden said nothing. Apart from the uncharacteristic stream of expletives, he had rarely seen his director so angry – or react so unreasonably. In the search for the missing explosives Karlstrom had ordered him to leave no stone unturned. And now Malone and Brickman were in deep shit because they'd turned over the wrong one.
He waited.
Finally, when Karlstrom regained his usual icy composure, he said: 'Sanitize those messages from Wallis. And make sure he understands this. He is not to reveal the existence of these explosives to anyone on board that wagon-train or make any further reference to them in any further communication with this department. Code it "EYES ONLY". You got that?'
'I'll attend to it immediately.'
'And, Tom –'
'Yes, sir?'
'Make sure that signal self-destructs.'
'Of course.' His deputy's face was wiped from the screen.
Karlstrom, the head of AMEXICO, was a man used to dealing with complex operations but he could not remember a time when he had so many strikes against him.
The quiet revolution of the wagon-masters and their execs had seriously damaged his organization's credibility in the eyes of the President-General, along with that of the other, visible security organizations like the Provost-Marshals.
After his deputy's latest on-screen appearance, Karlstrom felt like a man hanging over a cliff on the end of a slowly-fraying rope. All it needed was a couple more strands to unwind and...
Normally a shrewd, unflappable man, Karlstrom found himself becoming increasingly short-tempered and venomous with his staff, and it was only by a supreme effort of will that he managed to conceal his mental disarray during his daily Oval Office meetings with Jefferson the 31st.
Thanks to Brickman's disastrous but well-meaning intervention, the lives of Malone and his team of mexicans were now in jeopardy. All overground operatives risked death on a daily basis, but Karlstrom was not prepared to stand by and do nothing. The collective expertise, the sheer overground savvy of Malone's team made them invaluable even in a training role. Every effort had to be made to rescue them. And despite his own mixed feelings, Brickman too.
With his distrust of 'earth-magic' and psionics – the Federation equivalent – Karlstrom would have preferred to dump both Steve _and_ his kin-sister but Jefferson's interest in this troublesome pair made that impossible. Even though the demise of Malone's renegades had killed the plan to draw the Mutes onto the train, Karlstrom could not afford to abandon the overall objective – the total destruction of the Clan M'Call.
With a horde of Mutes in hot pursuit of Malone and his team-mates, a sky-hook – which would require putting at least two aircraft on the ground plus top cover – was a high-risk operation. Karlstrom was prepared to gamble but he preferred playing with a marked deck. Besides, there was a chance of saving his mexicans _and_ saving the day. As McFadden had noted, Malone and Brickman proposed to draw their pursuers forward towards the planned rendezvous with the wagon-train in the hope of provoking the Mutes into making an all-out attack.
If Cadillac and Mr Snow – who had risen like Lazarus from the tomb – were still determined to rescue Clearwater then he, Karlstrom, was ready to oblige by offering them the bait. This time, at least, there would be no danger of a flash-flood. Brickman – via Malone – had reported that since the episode at the trading post, Mr Snow was a mere shadow of his former self. And although reports of his death had proved to be premature, it was because the summoner had exhibited every sign of being a dying man that he, Brickman, had been fooled into believing they were true.
Maybe a last attempt to use his earth-magic against the wagon-train might push the old bastard over the edge. It was a tempting scenario, but 'maybe' wasn't a word that Karlstrom liked. That was why he did not propose to alter any of the precautionary measures he had taken in conjunction with CINC-TRAIN.
The problem now was to get the President-General's approval for the fall-back plan without giving the impression that AMEXICO had lost its grip on the situation...
To many of those who were directly involved in fighting off the combined assault force of Mutes and renegades, the news that the turkey-shoot had been rained off was met with thinly disguised relief. Even though the containment plan had been carefully worked out to eliminate any possibility that the train could be overrun, the very idea of allowing _any_ Mutes to set foot on Red River had been anathema to all concerned.
But orders were orders. Happily, they had been rescinded. The problem now was to help save the nuts of some guys whose overground operation had gone badly wrong.
What Karlstrom didn't know, as he composed himself in the elevator on the way up to the Oval Office, was that the situation he was currently coming to terms with was entirely fictitious.
Malone, his six lieutenants, and the motley collection of renegades who had been tricked into following them had become an all-day feast for a growing flock of death-birds in Wyoming.
The bulk of the Clan M'Call, in an unprecedented series of night moves had made their way towards 'Big Fork' – the junction of the North and South Platte Rivers, arriving a full two days ahead of the wagon-train. And by an arrangement which would have been utterly unthinkable before the Battle of the Trading Post, the two She-Kargo and three small M'Waukee clans inhabiting the area had trekked westwards to fill the vacuum created by the M'Calls.
The main 'battle-group' at Big Fork was led by Cadillac and Blue-Thunder. Following in its tracks was a much smaller group, consisting of Awesome-Wells, Boston-Bruin, and four Bears selected for their physique and discretion. Apart from Blue-Thunder, they were the only members of the clan who knew that Mr Snow was still alive.
The continued deception was necessary because in spite of the occasional strong rally the old wordsmith's hold on life remained precarious. Having won the clan's confidence, Cadillac did not wish to undermine his position by revealing Mr Snow's presence. If he were to do so, only to have Mr Snow die before the attack, the effects on the clan's morale would be disastrous.
The plan Cadillac and Steve had evolved was a good one. If all went well, they had a better than even chance of gaining control of the wagon-train _without_ using earth-magic. If the Old One _was_ able to help them with his earth-magic it would be an added bonus, but Cadillac knew they could call upon it once and once only.
He and Mr Snow were both aware that the act of summoning forth any powers greater than those held within the Second Ring would prove fatal. Mr Snow faced the prospect with a cheerful lack of concern. Having passed the baton to his pupil, he appeared not only willing but eager to die. But with only one shot left in his locker he had to conserve his energies until the critical moment.
That was why he had accepted the indignity of allowing himself to be carried into battle, and it was another reason for his secret journey. Apart from babes in arms, and comrades wounded in battle, there were no passengers among the Plainfolk. To be 'legless' meant you had reached the end of the line. The old and the infirm who were unable to walk or run never became a burden to their clanfolk. They swallowed a strong dose of dream-cap laced with a poisonous weed extract, or died from a self-inflicted knife-thrust to the heart.
Cadillac knew that Mr Snow, his beloved teacher, had chosen to make his exit in a different fashion. In his mind's eye, he carried the unforgettable pictures he had drawn from the seeing-stone: the pictures of Mr Snow's dying-place. And he remembered the sorrowful laughter with which his teacher had dismissed the news, preferring to rejoice in the beauty of their surroundings. Cadillac even remembered the exact spot where he had found the stone, on the almost flat expanse of ground south of the point where the two rivers ran side by side before finally merging for the slow journey eastwards across Nebraska to join the Missouri.
The stone lay directly in the path of the iron-snake, a path traced by those who controlled the destiny of the Plainfolk but which the beast itself had not yet followed; its arrival on the killing-ground still lay in the future.
The terror inspired by the vision of its monstrous shape looming over him still sent shivers down his spine, and in the two years that had elapsed since his first journey to that dreadful place with Mr Snow, the feeling of suffocation upon finding himself under the belly of the wagon-train had often caused him to wake with a pounding heart and a cry of fear on his lips.
But this was not the time for timorous reflection. The die was cast. He and Brickman had evolved an audacious plan which had the blessing of the Old One. A plan in which he, Cadillac, had secured the major role and the lion's share of action. Everything hinged on split-second timing. What he needed to display now was the same boldness and resolution that he had – for far too long – envied in his rival.
Steve and his travelling companions, Night-Fever, Cat-Ballou and the small group of Mutes acting as stand-ins for Malone's team of mexicans were still in Wyoming, following the line of the South Platte River as it made its slow descent eastwards towards the state line and its eventual link-up with its northern counterpart.
Using a captured radio, Steve fed terse but graphic accounts over the voice channel linking him with Wallis, aboard Red River. Those listening were tricked into believing that the fugitives had succeeded in staying ahead of their pursuers with the aid of some of the horses given to the renegades by Cadillac.
Unfortunately, in an effort to lay a false trail, Steve and 'KYLE' – one of the dead mexicans – had become temporarily separated from Malone and the other five but had received confirmation they were in the saddle and 'walking tall'; a slang term meaning their heads were still attached to the rest of their bodies.
Although claiming to be running low on battery power, Steve obligingly kept the transmit button down long enough for Red River to get a fix on his present position. Wallis also confirmed that he had also received a couple of brief signals from Malone. They had been too brief to pin-point but the messages, garbled by static, suggested the two groups were fairly close together and heading in the right direction.
In reality, Cadillac's imitation of Malone had been broadcast from a totally different location. Steve's task was to maintain the fictional existence of the group and give the impression that the entire Clan M'Call was right behind them when, in fact, it was the other way around.
Steve had chosen Night-Fever, Cat-Ballou and the five other Mutes who had escorted him to the trading post because they had all taken turns to ride one of the horses on the way back. All of them – like Steve – were now dressed in the camouflaged fatigues and helmets acquired from the ill-fated SIG-INT decoys, adorned with four white strips of marker tape. They weren't star pupils but they could stay in the saddle and that was all he required to maintain the illusion whenever a patrolling Skyhawk swooped low overhead and waggled its red wingtips in recognition.
The M'Calls spent the day hiding amongst trees in the culverts and gullies on the northern bank of Big Fork then, when the last glimmer of light had faded from the sky, Cadillac led them across the two shallow rivers onto the killing-ground.
The mass of warriors and She-Wolves had already been divided into posses numbering five hands and placed under the command of a group-leader. Followed by these new appointees, Cadillac paced out the approximate area that the wagon-train would occupy. When this had been delineated, it was divided into sections for which the group leaders then became responsible. Before long, the posses were hard at work with primitive stone digging implements and a prized collection of Iron Master picks and shovels.
Leaving Blue-Thunder in charge of the proceedings, Cadillac mounted his horse and rode westwards along the river to where Mr Snow lay hidden with his escort. 'How goes it, Old One?'
Mr Snow gave a tired smile. 'Well enough...' With the moon's sepulchral glow bathing his sunken features and white hair, he looked like a visitor from the spirit world he was soon to inhabit. 'Help me up.'
Cadillac gave him a helping hand. Mr Snow then pushed him away and tottered towards a tree some ten yards away. Cadillac hastened after him but was waved off. Mr Snow circled the tree twice before leaning against it. 'There!' he gasped. 'These idiots keep telling me I'm too weak to stand!'
'Let me help you back.' Cadillac offered his hand as Awesome-Wells approached clucking with disapproval like an agitated hen rounding up a wayward chick.
Mr Snow withdrew his elbows from their grasping hands and waved them both aside. 'Keep away! I now propose to take a dip.'
'At this time of night? Are you crazy?' demanded Awesome. 'What for?!'
'To get rid of the cramps, the pins and needles. I've been lying in that bed too long. I need some exercise!'
'Sure,' said Boston-Bruin. 'But you're not a fish!'
'Look! I know about these things! In the Old Time people swam for pleasure not just to stay afloat. It was medically recognized as an excellent way of exercising every muscle in your body.'
'It also sounds like a good way of tearing them apart,' huffed Awesome. 'Especially someone of your age.'
'On top of which you'll catch your death of cold.'
Mr Snow turned to Cadillac. 'Oh, come now. We both know how I'm going to die and it's not by drowning in this river.' He patted his successor on the arm. 'Make yourself useful. Light me a fire.'
To the rational mind, it seems incredible that Cadillac could possess the ability to draw images of the past and future from a 'seeing-stone', and equally incredible that such objects existed. But to the Plainfolk, who believed that The Path was already drawn, the past and future were merely different stretches of the great River of Time which carried the world towards the Sea of Eternity.
Past and future co-existed – like the beginning and end of all river – real or imaginary
And since in their cosmology, the natural world in its infinite variety was an ageless, sentient being in which nothing – not even the densest rock – was totally inanimate, then it was not difficult for them to accept that the earth and sky contained the knowledge of things past and things to come.
The hills, the rocks, the earth, trees and grass were silent witnesses; nothing escaped the golden glare of the sun or the pale white moon and the countless starry eyes in Mo-Town's cloak. The earth revealed those secrets to those gifted with far-sight, and to those whose inner ear was attuned to the Sky Voices that rode the wind.
The rational mind might find such notions diverting but would brush them aside. And yet, and yet... the map reference to which the wagon-train had now been directed – a location which, with its several approaches, had been carefully reconnoitered from the air – was the very same spot where Cadillac had found the seeing-stone with its blood-soaked images. _Two years before CINC-TRAIN chose it as the killing-ground_...
In some way which defied rational explanation, the earth had stored the images of the train and the carnage caused by its arrival and through the medium of the stone had passed its secrets backwards in time into Cadillac's mind.
Such was the magic and the power of prophecy.
At first light, the Clan M'Call ceased their earth-works and retired to the hiding places established beneath the trees in the culverts and gullies on the far bank of the Big Fork. Some of the warriors had been allowed to sleep through the night; these were now despatched into the nearby hills in groups of three to keep watch for the wagon-train, and two posses containing the best hunters in the clan were sent northwards in search of the nearest large herd of buffalo.
Cadillac watched anxiously as they ran off to execute their part in the plan. It was a good plan, as he kept assuring himself, but there was no fail-safe factor. It was the kind of plan that depended on the successful execution of each phase. Everything had to work perfectly, there was absolutely no margin for error. And it was absolutely nerve-racking. He looked up and watched the dawn light flood rapidly across the sky, melting the rose-tinted night clouds with its warmth. One more day, another night of preparation and then, perhaps, the fatal dawn. He settled down to wait, and when his turn came, he tried to sleep.
Roz, aboard Red River, was conscious of the gathering storm. The wall-mounted video-screens which relayed announcements and instructions throughout the wagon-train notified the crew that the train was now moving into a battle zone. An attack by a large force of Mutes could be expected at any time. As of 0900 hours, the train would go to a Level One Red Alert – the highest state of combat readiness.
At this level, all gun positions were constantly manned by shifts of gunners during daylight hours, the video-screens linked to the external surveillance cameras were individually monitored, and a three-plane standing air-patrol was put up to circle around the train as it moved forward.
All personnel were required to stay at their assigned work-stations or on immediate call. This meant that Roz could not go up onto the flight-deck even in her off-duty moments. Apart from knowing they were in Nebraska, she had no access to any maps and, as a consequence, had no idea of where the train was. Moreover, since the train was a sealed environment, and no pictures of the overground were screened in the blood-wagon, she had no way of knowing where they were coming from or going to. The vision slits which would have allowed her a glimpse of the surrounding terrain were closed and locked. They could only be opened in certain emergency situations such as a power failure when the external cover plates were automatically released and could be wound back from the inside.
As a doctor attached to the medical team, Roz was expected to take part in the normal preparations that any mobile field hospital makes prior to a major engagement. The crew on board the train were in little danger; the casualties the medical staff had to be prepared to deal with would only occur if line-men were commited to a ground action.
As the CMO explained in the standard briefing she gave on these occasions, the type of injuries they expected to deal with were relatively simple puncture wounds made by knives and crossbow bolts, plus skull and bone fractures caused by stone hammers and flails. That was the plus side of fighting a primitive enemy. The only time surgeons got a chance to deal with explosive wounding was when a line-man was accidentally shot either in a fire-fight or through the faulty handling of weapons or ordnance.
Saving Clearwater had been a real test of their medical skills which, under normal circumstances were confined to industrial type accidents and the run of injuries that occur whenever ham-fisted goons install, move, operate or try to repair heavy, hard-edged pieces of military equipment.
After Roz had given an up-date on Clearwater, the ward-sister reviewed the few cases occupying beds in the sick-bay. Decisions on who could be discharged and transferred to the daily sick-parade were quickly reached. The object was to empty as many beds as possible and to that end, the White House task-force had been prevailed upon to vacate the ward they'd been using as their base. Bunks had been found for them elsewhere, and the black boxes which gave them a secure line of communications with AMEXICO were now installed in a large, walk-in cargo skip on the hangar deck of the rear flight-car.
As the staff meeting broke up, Michelle French, the CMO tapped Roz. 'Your friend Wallis wants to see you...'
Jake Nevill vacated his chair inside the eight-by-ten grey-green metal compartment and motioned Roz to take his place. Wallis, seated opposite her, unfolded a map and spread it out between them.
'We need your help,' he explained. 'The Director needs urgent confirmation of your kin-brother's present position. Can you handle that?'
Roz hesitated and issued a silent call for help. 'I can try.'
'Good. Y'see, we still haven't been able to get a proper fix on Malone and –'
'I can't tune in on him. It only works with Steve.'
'I know,' said Wallis soothingly. 'We're assuming from what we've heard that they're pretty close to one another. Just give it your best shot. The Director tells me you're pretty hot stuff. If you manage to pick up your kin-brother maybe Malone will come into the picture.'
Roz nodded and closed her eyes as she mentally debated whether to sink into a deep trance state or fake it. She knew Malone was dead but she had no idea where Steve was _supposed_ to be. And contrary to Karlstrom's fears, her psionic powers did not yet include the ability to read people's minds like a video-screen.
This could be a test. Wallis and Karlstrom might already know where Steve was. If they had any suspicions about what he was up to, her failure to find him would make it look as if they were working together to deceive the Federation. She was trapped. She had to make a genuine effort to find him and hope that he had foreseen this possibility.
Steve had. The knowledge that her mental map-reading powers might be put to the test was precisely why Steve had elected to stay behind and broadcast intermittent progress reports on his efforts to evade the M'Calls. Since Long Point, his telepathic powers had increased or – to be more accurate – had regained some of their original sensitivity.
There was no internal bell or buzzer that went off when Roz's mind reached out towards his, but he was aware of her presence. It was like a cool breeze wafting through his brain which, during the moment of connection, seemed to contain the infinity of space and then – although it was all in the mind – a delicious physical sensation as her whole being merged with his.
It came now, as he cantered up a forested slope close to the state line between Wyoming and Nebraska. He welcomed her, and through her reached out towards Clearwater. There were no barriers between them now.
Wallis and Nevill watched with growing mystification as Roz's fingers searched blindly across the plasfilm map then gradually zeroed in on Steve's position. For a while, she sat slumped in the chair, eyes closed, chin on her chest then she raised her head. Nevill glimpsed the upturned whites of her eyes through the partly open lids as her head sagged over towards her left shoulder. Her lips moved wordlessly then in a slurred voice she said: 'Here... somewhere here...'
Wallis used a black wax marker to draw a circle round Roz's forefinger.
'A hill, with trees... animals... running. He's...'
'Riding a horse?' ventured Wallis.
'Yes, fast. I can feel the wind on my face. He sees... Malone. More riders...'
'Where?'
'On...' Roz made a scooping motion with her right hand. 'A valley. On the other side of a valley... horsemen.'
Her eyes fluttered open. Wallis and Nevill saw her look around in an effort to get her bearings. When her eyes met theirs it was if she had never seen them before then a moment later, her senses returned in full measure. 'Did I...?'
Wallis nodded. 'Yeah, it looks very encouraging. You even picked up Malone. And it wasn't too hard, was it?'
Darryl Oates poked his head round the entrance to the skip. 'We just picked up another sit-rep from Brickman. He has visual contact with Malone plus five. Looks like we lost one. Hope it's no one I know.'
'Did you get a fix?' demanded Wallis.
'Yeah, lemme show you.' Coates advanced towards the map. 'He's just west of navref Lagrange on a bend in the Bear River. Close to the state line.'
'In that circle somewhere?'
Coates checked. 'Yeah, look – right in the middle. There y'are, see? Lagrange. About a hundred sixty, seventy miles from the rendezvous point.'
'Thanks, Dee...'
As Coates made his exit, Nevill looked over Wallis's shoulder at the circled location. 'Now that, is fucking ay-mazing...'
'Yes...' Wallis sneaked a sideways glance at Roz. 'I just hope we didn't imagine this.'
Roz looked puzzled. 'Imagine what?'
'Never mind. Well done.' Wallis edged between Nevill and the table. 'I'll pass this through to Mother.' He gave Roz's shoulder a nervous pat on the way out.
Towards sunset, a runner despatched by the scouts on the high ground to the south-east, returned with news that an iron-snake had been sighted heading in the general direction of Big Fork. As instructed, the scouts were falling back ahead of the train and would send further word of its progress.
Not long afterwards, more encouraging news arrived. The hunting posses had encircled a big herd of buffalo and were shepherding it towards the twin rivers.
As the sky cooled, and the Mute's anticipation grew to fever-pitch, they sighted three high-flying cloud-warriors. Flashing silver-bright like fish in a blue rock-pool as they were touched by the rays of the setting sun, they wheeled tirelessly back and forth across the heavens until the first grey-blue veil that lined Mo-Town's velvet cloak turned their shining bodies into dark bat-like silhouettes with red-eyes that blinked in time to their beating hearts.
Bunched close together like geese on the wing, they flew off on a descending curve towards the south-east then later, when the second, darker veil had been drawn across the sky, the iron-snake appeared in the distance, its head studded with glowing eyes and rows of lights along its flanks like a giant fiery caterpillar.
The feverish anticipation, which had always been grim rather than delirious, became somewhat fearful as the train's massive bulk bearing down upon its wheels caused the ground beneath to tremble. And now, to this mind-numbing vision of their possible nemesis was added the chilling dimension of sound. The drumming roar of its engines and the thunderous rush of exhaust gases exploding through the roof vents of the power cars.
The hungry, belly-rumbling, hunting growl of the iron-snake as it crawled towards its terrified prey.
The M'Calls had heard these sounds before but only in the distance. When they had attacked The Lady from Louisiana, she had been trapped silent and helpless in the flood debris of the Now and Then River. Not so helpless as it turned out but by the time the engines had been revved up for the roll-out the clan had withdrawn to lick its wounds.
The sound that was burned into the memory of the M'Calls was the gut-shrivelling scream that erupted as the iron-snake tore its attackers apart with its fiery white breath. Now, in the fading light of a June evening almost two years to the day since that bloody confrontation, another iron snake was heading towards them, filling the sky with its voice and causing the ground to shake in fear at its coming.
Cadillac, who had only ever seen a real wagon-train at a distance and in pitch darkness stared aghast at its proportions. Even though he had paced its length out on the ground, its sheer bulk was absolutely staggering. It was even bigger than the shadowy monster in the visions drawn from the stone! It dwarfed everything in sight! A few scattered trees which might have been considered to be quite large shrank into insignificance and were brushed aside or flattened and pulped beneath the huge wheels.
How could he and Brickman have been so foolish to think they could storm this armoured giant with the aid of a dying summoner?! They must have been insane!
Cadillac willed his pounding heart to slow down and tried to radiate assurance to the warriors who lay hidden on either side of him. 'Courage! The sand-burrowers hide in the belly of the iron-snake because they fear the Plainfolk! They cannot triumph against the will of Talisman! When the time comes let each one of you display the courage for which the M'Calls are renowned and remember – the spirit of the Old One will guard and guide us!'
Easy to say, thought Cadillac. He at least was one of the select few who knew that Mr Snow was still alive and gearing himself up to deliver one last stupendous burst of earth-magic. The rest of the clan, who did not share that comforting piece of knowledge, were probably wishing the old wizard had passed on the Seven Rings of Power to his apprentice instead of the gift of the gab.
Parting the feathered reeds on the bank in front of him, Cadillac saw the wagon-train roll to a stop with its nose pointing towards the river. It was parked in a straight line, more or less at right-angles to the bank on a north-south axis. The forward command car was about a hundred paces from where he lay. On its flanks, the nearest scrub was some two hundred paces away while behind its tail, the long line of white-trunked larches which formed the ragged edge of a wood was more distant still.
The wagon-train, with its gun-turrets mounted high above the ground had a virtually uninterrupted circular field of fire and the cameras mounted on the roof could see clear across both rivers and westwards towards the rising ground and the hills beyond from where Brickman was due to appear.
But before then, there was much to be done. Despite his unshakeable belief in his abilities as a seer, Cadillac was astounded to see that the wagon-train had stopped exactly on the spot he had chosen – right in the middle of a random pattern of shallow trenches of varying lengths, none of them more than two feet deep. The apparent aimlessness with which these trenches had been dug gave no clue as to what they might be for, and their shallowness posed no obstacle to the wheels of the wagon-train – each one of which was twelve feet wide and twelve feet high.
From his hiding place, Cadillac had a good view of the right flank of the train. A ramp had been lowered from the belly of one of the wagons aft of the flight car. He aimed the monocular viewer at the ramp and brought it into sharp focus as a couple of dozen armed Trail-Blazers emerged, their helmet visors lowered, and fanned out to inspect the ground on either side of the wagon-train.
As Cadillac expected, they spent some time studying the large number of shallow trenches and from their gestures appeared to have no idea what they were for. The M'Calls had left other items for them to find: post-holes, the charred remains of cooking fires – the kind built by migrating Mutes – animal bones, some crushed, some raw, the rotting entrails of a buffalo and several bog-holes containing human feces.
Operating the zoom on the viewer, Cadillac framed the top half of the nearest Trail-Blazer. As he turned, Cadillac caught sight of his shoulder badge – a grinning Mute skull speared by a sloping red stake. The insignia of Red River, popularly known as Big Red One. Brickman had told him what to look for. He re-focused on the wagon-train, panned left along its length to the forward command car and found three large white letters emblazoned on the side – RVR, the abbreviated code and call-sign for Red River.
On the sloping nose was a large version of the shoulder-badge. This was it. He counted the wagons... sixteen. One of them held Clearwater and Brickman's kin-sister. If Brickman was right, they would be found in the wagon immediately aft of the flight car and ahead of where the ramp had been lowered.
Cadillac sent out a silent message of encouragement to his one-time soul-mate in the hope that she might pick up his thoughts and gain some comfort from the presence of her clanfolk. He did not expect a response but that did not stop him being disappointed. And once again he regretted she could not be at his side to observe the new Cadillac Mark Two, the brave, resourceful leader of his people.
Just for once, he would have liked to evoke a cry of admiration instead of exasperation – and her magic would have come in handy too...
Commander James Fargo and Don Wallis reviewed the findings of the line-men who had inspected the site.
'The general consensus is that the trenches were sleeping holes. If I remember my pre-history correctly, the dog-soldiers in the Old Time used something similar.' Fargo searched for the word. 'Foxholes – only they were deeper than these. But the principle's the same. Digging down gives you a measure of wind-cover and you're able to utilize the earth's internal heat to keep warm.'
The wagon-master who knew nothing about Wallis apart from the fact that he worked in the White House didn't realize that he was speaking to someone who'd got his hands dirty and his ass bitten over several years of overground assignments.
And Wallis was not about to tell him. 'Yeh... it's just that a collection like this has never featured in any FINTEL report that _I_ ever read.'
'Me neither. But it's definitely an abandoned campsite.'
'Yeah, absolutely. Question is – should we move 'em off it?'
Fargo grinned. He was a big man but his teeth were small and narrow and there seemed to be too many of them. Wallis didn't like being smiled at by teeth like that.
'I think the train should stay right where it is. You know what these lumps are like. They have what they call 'sacred places' – like for instance where they put their dead. Maybe this site has some kinda special significance.' The wagon-master gave a throaty chuckle. 'If our guys stay sitting right on top of it, there's a chance we might make a whole big bunch of them eye-poppin', foot-stompin' mad. And when they get mad, they keep on comin'.'
Fargo treated Wallis to another of his predatory grins. 'I tell you, good buddy, when you're behind a six-pack, belted up an' ready to let the hammer down, there ain't no better sight in the whole fuggin' blue-sky world!'
'Okay,' said Wallis. 'We'll let it ride. But tell 'em to keep their eyes peeled. If Malone and the others make it through the night, they should arrive just after dawn tomorrow.'
## CHAPTER TWELVE
Around 0200 hours, when the attention of the wagon-train's night-watch was at its lowest ebb, the two Duty Vid-Comm Techs manning the screen console in the forward command-car were roused from their torpor by a high-pitched bleeper. Jeff Simons, a Master-Tech, cancelled the alarm then shared the task of checking the battery of screens with his buddy for the night, a Tech-4 called Ben Mason.
The alarm told them that one of the master-scans had been corrupted. Like way-stations, parked wagon-trains used a special computerized scanning system to pick up movement. The landscape around the wagon-train as viewed by the battery of surveillance cameras was captured and analysed by a computer. This image – 'the master-scan' – was then stored in the computer's memory and all subsequent images recorded by each of the cameras were compared with its own individual master second by second. If any of the elements in the picture moved, or a new element was introduced – corrupting the master – the alarm sounded to draw the operator's attention to what might be an incursion by hostiles – Mutes.
In this case, it was only a large, slow-moving herd of buffalo advancing diagonally from NNE, across the twin rivers towards the train. They were still some way off but there was a lot of them, and as they moved into view of some of the other cameras, it set off more bleepers.
Simons told Mason to cancel every alarm right across the board and paged the night-duty exec, Betty-Jo Aarons, the Second Systems Engineering Officer.
Two or three minutes later, Captain Aarons, a crew-cut ash-blonde with good shoulders and a boyish ass, ran up the stairs to the saddle with a towel draped round her head like a prize-fighter on his way to the ring. The front of her OD jump-suit was unzipped to the navel and there was nothing underneath but wet golden skin that was already causing the fatigues to come out in dark blotches. Her bare feet were jammed into a pair of trainers.
'You'd think at two o'clock in the morning a guy could grab a shower! What the fuggizappening? Is that May-Day unit on the line?'
'No, sir, it's buffalo.' Mason tapped the screen.
'You've got to be kidding,' breathed Aarons. She leaned over their shoulders and studied the loosely-packed mass of fuzzy green shapes picked up by the image-intensifiers. 'Doing what?'
'Grazing,' said Simons. 'But they're moving this way. If they stay on their present heading they'll pass right under the train.'
'So...?'
'Well, do we want that? Don't you think we should blow 'em off course with a few rounds from a Vulk?'
'Or a whiff of steam?'
Aarons eyed them both then shook her head. 'Piping steam's a job for the EO's. They're not going to be too pleased at being hauled out of bed to burn the butts of a herd of buffalo. And there are no gunners on duty. Everyone's got their heads down. Tomorrow's a big day – or hadn't you heard?'
Mason tried to be helpful. 'We could always switch on the headlight array. That should scare 'em off.'
'It would. But if you remember, the idea is not to advertise our presence any more than we have to.'
'There's nothing to stop you using one of the forward six-packs,' insisted Simons.
'No, there isn't. But I'm not going to. Christo!' Aarons pointed to the screens. 'They're just a bunch of dumb animals! If it was a horde of Mutes –'
Simons nodded. 'Yehh, sure. Just wanted to clear it with you.' He straightened up in his chair. 'Okay, Captain, if you'd just like to card in...'
Aarons swore quietly. Fishing out her ID-card, she ran it through the reading slot mounted on the sides of the screens showing the herd of buffalo and pressed the ACCEPT button.
It was a procedure designed to ensure that those meant to see transmitted material – be it words or pictures – left an electronic record of having done so. Its main purpose was to prevent tiresome protestations of ignorance later if the shit should hit the fan.
'And maybe you'd better call up the boys in the rear command car and tell 'em what's coming through...'
Aarons did so then gave Simons the beady-eye as she pocketed her ID. 'Since when did you start going by the book?'
Simons shrugged. 'These days a guy has to cover his ass.' He started pressing buttons. 'We'll strike the masters and make new ones after these dumbos have moved on.'
'Good thinking,' said Aarons. She patted their shoulders. 'I'll leave you to it...'
'Yessir-ma'am...' Mason adopted an alert position and hit a few buttons. Simons, who had his back to her, already had his eyes closed.
There was a brief high-pitched warning tone as Aarons reached the bottom step. To make life easier, Simons and Mason – with the tacit agreement of the other Vid-Comm Techs had concealed small pressure pads on the first three steps leading up to the saddle to alert them to the approach of the Duty Officer.
Anyone who'd done time up the line knew nothing ever happened at night, but some execs tended to be over-zealous – new promos transferred in looking to make their mark, and the perennial brown-nosers ever-eager to work their way up the wire. These dills spent the whole of their watch patrolling up and down the wagon-train checking the alertness of those selected for guard-duty – generally managing to spoil everybody's shut-eye and keeping the grass-heads from tunnelling out in the John.
The unofficial alarm system – switched off during the day – gave the dozing Vid-Comms time to pull themselves into watchful postures, eyes glued to the screens.
Aarons wasn't one of them. And the reason she didn't want to waste time banging away at buffalo was because she'd been working up a head of steam with a hunky jack-dandy under the shower and wanted to get back before he went off the boil.
Aarons was also right about Simons. Like most Vid-Comm Techs, he was not, by nature, an avid follower of regulations, but following the bomb attack on The Lady from Louisiana in 2990, CINC-TRAIN had issued a general divisional unit order about the need to observe proper security procedures. In the last four years Simons had seen plenty of buffalo at close quarters but he'd never known them walk towards a wagon-train and pass right underneath it as if it wasn't there.
That's why he'd alerted Aarons. Pretty soon, they'd be sitting in the middle of a sea of buffalo. Maybe it was because the train was stationary, and the turbo-generators in the power cars weren't roaring their heads off. Or maybe it was parked right across one of the herd's annual migratory paths. Tracks which, according to the archives, had been followed for centuries.
Arrgh, what the hell, thought Simons. Who cared? It was Aarons' responsibility now. She liked dumb animals. Simons could tell from the brightly flushed cheeks and the way the ID-card had trembled in her fingers that she'd been pumping some bone-headed jackeroo when he'd called her out. Everyone knew Betty-Jo Aarons liked it under a hot shower. That's why the guys called her Soapy.
Simons opened one eye and checked the screens linked to the side and ventral cameras. Buffalo everywhere you looked; a slow-moving carpet, grazing briefly then plodding on under the train towards the SSW. He yawned and wriggled down to a more comfortable position in his chair. 'Lemme know if they start eatin' the wheels...'
'Yeah, sure,' said Mason. He switched off one of the cameras to clear the screen and put up one of his favourite video-games.
Believing the M'Calls to be several tens of miles to the west of the train, and lulled into a false sense of security by the knowledge that Mutes – like Trackers – did not embark on warlike ventures during the hours of darkness, neither Simons nor Mason, nor the two Duty Vid-Comm Techs in the rear command car noticed the pairs of warriors cloaked in buffalo skins, moving stealthily among the slow-moving herd.
And because the side-mounted cameras were aimed outwards and not down the length of the train, they did not see Cadillac shin up a primitive ladder to reach the umbilicals linking the power cars to the rest of the wagon-train. Not just once, but eight times. And when that task was completed, they did not see him crawl under the curve of the first pair of wheels of each command car and carefully plant something in one of the moulded recesses of the tread at the point where the giant tyres met the earth.
With that crucial task completed, Cadillac made his way back through the herd to the river bank then turned north along the water's edge to where Mr Snow and his escort stood waiting in the darkness with the horses they were to ride.
'Are you ready, Old One?' Cadillac attached the leading rein of the second horse to the one he had chosen to ride.
Mr Snow answered with a reluctant nod. 'Am I going to have to get on that thing?'
'No, that's a spare in case of accidents. You're going to sit on this one with me.'
'I'd rather not...'
'It's really quite simple. All you have to do is hang on. I'll take care of everything else.'
Cadillac climbed into the saddle then two of the warriors hoisted Mr Snow up and helped him wriggle into position. Cadillac knew there was no point in asking him if he was comfortable. He leaned down and clasped the hands of Awesome-Wells and Boston-Bruin.
'We will return when the sun has left the eastern door. You know what has to be done. If each of us plays his part to the full, the day is ours! May the Great Sky-Mother watch over you!' He raised his fist in response to their farewell salute then urged the horses into a canter.
The darkness closed around them. The two elders gazed at the point where they had vanished into the night until the sound of hoofbeats faded then turned to face each other.
'Well, my friend, this is it...'
'Yes, agreed Boston. 'The big one...'
When the herd of buffalo finally cleared the immediate area around the wagon-train, the enveloping blackness had thinned to a leaden grey. Mason, the junior of the two Vid-Comm Techs cleared the fourth of his favourite video-games from the screen and nudged his partner. 'Sy! C'mon! Time to re-set...'
Simons struggled to wake up. 'Whaaa–?'
'The buffalo... they've gone.'
'Ohh, yeah... okay.' Simons stretched and rubbed his face then started pressing buttons with fluid movements of his right hand while he used the left to key himself through to his colleagues in the rear command car. A view of the senior Tech appeared on one of his screens. 'Tony? Lock on a new set of masters.'
'Wilco...'
The computer at the heart of the surveillance system duly recorded a new master-scan of the train's surroundings. But the procedure was fatally flawed: the system was not programmed to compare the _new_ masters with the ones made the previous evening. The earlier scans had, in fact, been wiped from the computer's memory, and because the Vid-Comm Techs relied on the system, they did not remember in exact detail what they had seen on the screens. So although they could see the ground was now covered with hoofprints and scattered dollops of buffalo shit, nobody noticed that the random pattern of trenches had altered considerably.
A lot of them had been filled in...
Hidden from the wagon-train by a fold in the rising ground, Cadillac reined in his horse and used a Tracker torch to flash out a signal. An answering light, high up on the next slope, winked out of the darkness. 'Nearly there, Old One.'
'About time,' grumped Mr Snow.
'Don't get too excited. As soon as we get there, we have to ride all the way back.'
'Brilliant. Whose idea was that?'
'I'm doing my best, Old One.' Cadillac dug his heels into the flanks of his mount and led the trail horse up towards the flashing light.
Steve and Night-Fever stepped into view as they reached the campsite. Above them, the canopy of pines stretched away like a threadbare piece of tent cloth, black against the lightening sky.
While Night-Fever led Mr Snow away to fit him out with a set of camouflaged fatigues, boots and helmet, Steve steered Cadillac through the forest in another direction and wound up in front of a camp-fire. Several Bears dressed in captured uniforms stood around it checking their weapons.
Steve, his face and hands washed clean of dye, was already in uniform and had been for several days. Taking the offered bunch of pink finger leaves, Cadillac washed his own face and neck down to the collarbone and both arms up to the elbow. When Steve had inspected the result by torchlight and declared himself satisfied, Cadillac unrolled the bundle of clothes that had been set aside for him and started to dress.
Steve watched him pull on the red briefs and T-shirt. 'So... how'd it go?'
'Brilliantly. The Old One's idea to drive buffalo towards the train worked like a charm. Everybody was able to get into position and there was absolutely no reaction from the train.'
'Doesn't surprise me. When you're on the red-eye shift, that hour between two and three can be a real killer. Did you manage to get the –?'
'The PX? Yes, it's all in place.' Cadillac consulted the wrist watch he had taken from one of the SIG-INT corpses. 'Timed to explode in one hour and forty-five minutes from now.'
Steve slapped his partner's shoulder. 'Terrific. Well done. D'you want any help with that flak-jacket – or d'you know where everything goes?'
Cadillac closed the front zip of his fatigues. 'I think I can handle it.'
'Okay, I'll go and round up our pursuers for a final briefing. Excited?'
Cadillac ignored the cold ball of apprehension that had lain in his stomach ever since he had watched Red River roll to a halt. 'Can't wait...'
Ten minutes later, the principal players were assembled round the fire. The initial assault group, consisting of Steve, Cadillac, Mr Snow, Cat-Ballou, Purple-Rain, Diamond-Head and Lethal-Weapon were now all dressed as Tracker soldiers. The only things missing were the breast-tags bearing the names and numbers of the previous owners.
Ranged behind them was the first pursuit group, the remaining hundred or so M'Call Bears and She-Wolves who – as far as the Federation knew – had been doing their utmost to catch the runaways. Their numbers had already dwindled due to attacks by the patrolling Skyhawks whose pilots had been given the task of shepherding Malone's group to the safety of the wagon-train.
The last, smaller group was composed of elders and warriors – representatives from the two She-Kargo and three M'Waukee clan who had moved out of the area around Big Fork and were now posing as the main force of the M'Calls, moving towards the wagon-train in the wake of the first pursuit group.
With Cadillac's assent, Steve explained the set-up and the planned sequence of events then questioned the leaders to make sure everyone knew what their group was supposed to be doing – and when.
Timing, he emphasized was critical. 'I'll just go over the key action again. The horsemen will approach the train in two groups. Cadillac and I will go in first – and we will ride in all the way. The second group of four – Purple-Rain, Diamond-Head, Lethal-Weapon and Cat-Ballou-will come in a little way behind us and will complete the last half of the journey on foot.
'As they come onto the flat ground, the _last three_ riders will go down.' He paused and interrogated his audience. 'Where are the bowmen?'
Twelve Bears stood up.
'Good. When you get into position, you will see two marker stakes – just ordinary branches – stuck in the ground between the foot of the slope and the wagon-train. We want you to bring the horses down somewhere between those two markers. Not before and not after. The last three horses. Is that understood?'
It was.
'Take careful aim. It's vitally important you bring down the horse and not the man.'
The bowmen nodded.
'Good. You'd better leave now. I want you in position when we ride through. Aim for the head, neck and rear flanks – and make it look good.'
The twelve bowmen melted away into the shadowy gloom of the forest and made their way towards the crest of the last slope now silhouetted against the grey-blue pre-dawn backdrop of the eastern sky. The representatives of the supporting She-Kargo and M'Wankee clans were the next to leave. And as they went off to rejoin their own people, the pursuit group retired to take up their positions for the ensuing chase.
When only the horsemen remained, Steve sent Night-Fever to fetch their visitor who, up to that moment, none of the others had seen. Cat-Ballou and the other Bears reacted warily as Night-Fever reappeared, guiding a masked figure towards them.
'I have news that will gladden your hearts,' said Cadillac. 'Behold! The Old One has returned from the Valley of Death to be with us this day!' He drew back the tinted visor of the helmet to reveal Mr Snow's face.
The five warriors stared open-mouthed at the old wordsmith. Shock, surprise, terror, incredulity – their faces reflected the whole gamut of emotions experienced at the unexpected return of a loved one then, with a chorus of joyful whoops and yells, they rushed forward to embrace him.
Steve watching from the sidelines with Cadillac felt a twinge of envy. You had to be a pretty extraordinary individual to inspire that kind of affection and respect. And it must feel good to be on the receiving end. He saw his partner's tear-streaked face. 'Cheer up, it's not over yet.'
'Not for you and I perhaps – but this is his dying-day.'
'You think I haven't thought of that?' Steve's voice cracked with emotion. 'I don't want to lose him any more than you do but I'm saving the grief until it happens. You were wrong about the date once. You may have got it wrong this time too.'
'No. I have seen the iron-snake looming over me just like the vision in the seeing-stone. This is the place! This is the day!'
Steve hid his emotions under a cold veneer. 'Yeah, well, it's your dying-day too.' He saw Cadillac's frown and explained: 'Time for you to send your last message as Malone.'
Using the temporary video hook-up that linked the task force's base in the cargo skip to the command car, Wallis screened himself through to Commander Fargo. 'Malone just came through. His group are still in one piece. They've sighted the wagon-train and are making their run in. Can we get a camera on them?'
'We can when they break out into open-ground.'
'They've also asked for air cover.'
'No problem.'
'Good. We've got some valuable people out there. I'd hate to lose them.' Wallis rubbed his red-rimmed eyes. 'We've got to save something from this fiasco.' He heard the klaxons sounding up and down the train, summoning the crew to 'Action Stations'. 'Is there anything my team should be going?'
Fargo gave another of his skull-like grins. 'No, just relax. This could be your lucky day.'
Yeah, it could, thought Wallis. Except that you needed more than luck when you were working for AMEXICO. And there was something else which Fargo had said which struck him as rather odd. 'Just relax.' At a time like this? The man was an idiot...
Commander James Fargo was, of course, far from being an idiot. For the past hour, he had been privy to a piece of information that Wallis – for tactical reasons – had not been made aware of.
Leaving the horses in the care of Cat-Ballou, Steve crawled to the top of the slope with Cadillac and focused the viewer on the wagon-train. Although some of the trains were fitted with different types of external cameras and aerial arrays, they were all built to the same basic design and camouflaged in the same pattern of red, orange, light ochre and brown. But there was no mistaking the insignia on the side of the nose, and the code letters RVR.
Steve felt a surge of excitement spiced with a delicious dash of mortal danger. This was it. Big Red One. The idea that they were about to break open the flagship of the Federation really set the adrenalin flowing.
Cadillac explained where the Clan M'Call were now hiding in readiness for the main assault then said: 'I left the approach to the three middle wagons clear on this side of the train to give us room to ride in without stepping on anybody's toes. That okay...?'
'Yep...' Steve swept the whole length of the train one more time then collapsed the viewer and slipped it back into the pouch on his left sleeve. 'I've seen all I want to see.'
'Then let's get out of here,' said Cadillac. 'Look! They've put a Skyhawk up!'
They wriggled down out of sight then got up and ran back to where the Mutes were waiting with the horses. Mr Snow was already riding pillion behind Cat-Ballou. Steve's mount was nose to tail with Cadillac's and as they swung up into the saddle, their eyes met.
'This could be tough on you,' said Cadillac, 'Are you going to be able to handle it?'
Steve shrugged. 'Had no problem at the Heron Pool.'
'Yes but they were dead-faces. These are –'
'Sand-burrowers,' said Steve. 'And we're Mutes. So let's go for it!'
The main camera towers, mounted on the roofs of the front and rear command car, zeroed in on the first two horsemen as they came over the rise and down the slope at a furious gallop. A few moments later, four more horsemen appeared. One of them had a passenger seated behind the saddle, his legs flapping as he bounced up and down.
Seven riders. Malone had started his run with eight. Who had fallen on the last stretch?
A twin-boomed Skyhawk soared into view over the slope, flew over the riders then banked right and headed west in a shallow dive.
Up in the saddle of the wagon-train, the pilot's voice came out of the speakers on the main console. 'Blue Nine to Red One. This looks like the main event. The home team has upwards of a hundred Mutes on its tail and behind them, advancing on a wide front, are at least five columns – all heading this way. They're running loose, spread out over two or three miles. If you want us to break this up we'll need more birds in the air.'
Captain Jack Cullimore, the new Flight Operations Exec, responded. 'Red One to Blue Nine Roger. We're going to nail three to your tail. Take 'em down and show 'em the town. But leave a few for us, huh? There's some guys back here just itching to get into the ring.'
This raised a few broad grins among the command staff. Cullimore keyed himself through to the flight car and spoke to Section Leader Sam Petrie. 'Sam! We got some heavy action west of here. Put up Knox, Harding and Eiger. Tell 'em to touch base with Ebbets in Blue Nine. He'll lead 'em in.'
'Yess SURR!'
'First two horsemen closing fast!' yelled a Vid-Comm Tech. 'Range now three hundred yards!'
The wagon-master turned to the First Systems Engineer. 'Lower the ramp on Number Four, Mr Wyatt.'
Steve and Cadillac saw the ramp come down and make contact with the earth. They could hardly believe their luck. The ramp was almost exactly opposite the approach avenue prepared by Cadillac.
They looked over their shoulders. The rest of the group led by Cat-Ballou and Mr Snow was about two hundred yards behind them. Suddenly, the last three horses staggered and went down, pitching their riders out of the saddle. Cat-Ballou promptly reined in his horse, dismounted, pulled his passenger off, dumped him on the ground, then ran back to the fallen riders, firing over their heads at some Mutes who had popped into view behind them.
It all looked pretty convincing.
Steve and Cadillac rode on. The air over their heads crackled and rustled as Red River laid down covering fire. An eight-man, flak-jacketed squad of Trail-Blazers came down the ramp and waved them in. Steve and Cadillac dismounted, slapped their horses' rumps to drive them away, then ran towards the ramp, raising their visors as they covered the last few yards.
They looked back at the fallen horsemen. Two were running at the double, carrying a limp figure between them, one was giving covering fire, and the fourth was hobbling on a damaged left leg.
The squad leader waved Steve and Cadillac up the ramp with his rifle. 'C'mon you guys! Get your asses outta here! We'll bring your buddies in!'
Cadillac checked his watch and exchanged glances with Steve. This was something they hadn't planned for. Standing at the top of the ramp in full combat gear was someone Steve recognized. 'Commander Moore!'
'Brickman! What the heck have you been up to?! I heard you were coming in. Just couldn't believe it!' He hurried them over to the stairwell. 'This way. You're wanted in the saddle.' Moore aimed a finger at Cadillac. 'Is this Mister Malone?'
Steve was tempted to say 'Yes', but didn't. Someone on board might know Malone. Might even know – or have viddies of – _all_ his team. But that was a chance they had to take. They were relying on speed, and the tenseness generated by a ground action to carry them through.
_Best to stick with the script_... 'No. This is Barney Kyle, one of his side-kicks!'
'Okay. Follow me.' Moore went up the stairs two at a time. 'We'll take a wheelie!'
As they neared the top floor, they saw a group of soldiers heading down. The stairwells linking the three floors of a wagon-train consisted of short flights with landings that made right angled turns around a fire-man's pole. To avoid a traffic jam, anyone in a hurry ran up and slid down.
That was when Steve got his second surprise. Coming onto the top floor one step ahead of Cadillac, he found himself eyeball to eyeball with a uniformed Jodi Kazan as she stepped out and grabbed hold of the pole.
'Steve?!' – 'Jodi?!' The names collided in mid-air. In that same fleeting moment, Steve saw Jodi's eyes switch onto Cadillac, saw the brief nano-second of puzzlement then the flicker of recognition as she slid downwards and was followed by the next in line.
Mouth agape, Steve leaned over the rail and watched her go. This was absolutely disastrous! With Kelso dead, Jodi was one of only four people who could recognize Cadillac – and the _only_ other Tracker still alive in the whole of the Federation to have seen Mr Snow.!
_What the fuck was SHE doing here?!!_
A shout from Commander Moore broke the spell: 'BRICKman!! Get the lead out!!'
'YESSURR!!' Steve leapt onto the wheelie, dragging Cadillac with him. The Mute looked stunned.
'Did you see who that –'
'I know!' hissed Steve, hoping like hell that his voice was masked by the general hubbub and the whine of the wheelie. 'And d'you know where she's going? Outside to help bring in –'
'Ohh, no –!!'
Moore looked over his shoulder and smiled cheerfully. 'Guess you guys have had a pretty bad time, huhh?'
'Yessir,' cried Steve. _And it was getting worse by the minute_. His head snapped round and practically came off his shoulders as another familiar face stepped back into a side passage to let the wheelie through. It was Big D. Buck McDonnell... Trail Boss of –
_Had he been transferred to Red River along with Commander Moore –?_
Cadillac checked his watch again. Barely fifteen seconds to go. The wheelie sped through the top floor of the power car between a maze of pipes and the drumming exhaust vents and stopped inside the command car where the Junior Field Commander – a guy whose breast-tag gave his name as Drysdale – was waiting.
'Brickman and Kyle!' shouted Moore. 'Take 'em on up! I'm going back to the ramp. We may have some trouble getting the others in in one piece!' Moore spun the wheelie around and sped away.
Drysdale led Steve and Cadillac along the last few yards of corridor and up the steps to the saddle. He paused at the top to salute. 'With your permission, Sir – Brickman and Kyle.'
'Bring them aboard, Mr Drysdale...'
_That voice_...
Drysdale beckoned to Steve and Cadillac.
Steve stepped up onto the saddle on leaden limbs. Rising to greet him from the wagon-master's high-backed chair was 'Buffalo Bill' Hartmann – commander of The Lady from Louisiana. And more than half of those manning the screens and control panels were people he had known and served with before being shot out of the sky.
_They were attacking the wrong train!_
He founds his heels coming together. Transferring his carbine to his left hand, he snapped off a parade-ground salute. Cadillac did the same.
Hartmann responded more casually. 'We'll save the welcome till later, gentlemen. What can you tell us about these Mutes?'
As he uttered that last fateful word, there was a series of muffled explosions, then two louder ones from almost directly underneath. The command car shuddered then tipped forward as the front tyres collapsed, throwing those who were standing off balance.
Steve and Cadillac had been braced for the impact. Twelve charges of PX had exploded within milliseconds of each other. Four, each reinforced by an AP mine, had blown out the front tyres on both the forward and rear command cars; the other eight had severed the life-giving umbilicals on both sides and at each end of the two power cars. Eight explosions that cut all electrical power to the command cars and the other wagons, deactivating the gun positions and plunging the entire train into darkness.
But only for a second. Emergency batteries in each car cut in, restoring the lighting and power to the external cameras, screens and radio communications equipment.
It was that second that made the difference. When Hartmann and his command staff recovered from their surprise, they found themselves confronted by two faceless camouflaged figures with closed helmets, firing from the hip as riot gas grenades rolled along the floor and exploded.
Steve dropped another down the stairway followed by a standard fragmentation grenade then, while Cadillac continued to sweep the saddle with fire, he stepped over the bodies in front of him and dropped into Hartmann's seat. Holding his carbine ready in his left hand, he searched for and found the Emergency Fire Control Panel. Lifting the cover, he activated the alarm by means of a switch and hit the button.
The cover and switch were designed to prevent accidental use. When pressed, the button triggered shrill warbling alarms that signalled a major uncontrollable fire hazard warning. Signs lit up, ordering the crew to evacuate the train and every boarding and escape ramp was automatically lowered.
Steve had twice taken part in practice drills at Fort Worth. The ramps could be lowered using the emergency battery supply but once down they could not be raised without mains power. The belly of the wagon-train was now open from end to end, and with all the system lines severed, steam could not be piped through the jets to defend it...
Outside the wagon-train, in the closing seconds before the PX charges exploded, another part of the master-plan misfired. The timing had always been perilous since none of the Mutes involved had been able to grasp the notion of time as displayed on the digital watches taken from the dead defaulters.
The aim had been to arrive at the top of the ramp just as the PX charges exploded whereupon Mr Snow – protected by his escort – would use his powers to cause the maximum disarray inside the train in the seconds prior to the main assault – which was to be launched when all the ramps came down.
To create the illusion that the 'renegades" lives were under threat, Steve had arranged for three of the horses to be brought down by the bowmen. This would provide a convincing delay enabling Steve and Cadillac to get to the saddle, take over, and activate the ramps. As bonafide Mutes, Cat-Ballou, Mr Snow and the others could not enter the train before that moment of confusion because their real identities would be revealed as soon as they raised their visors – which they would have been expected to do. The plan called for Cat-Ballou and Mr Snow to dismount and rally the fallen who between them would shoulder the old wordsmith, then limp – but not too slowly – towards the open ramp, arriving in time for the big bang.
Unfortunately, the well-being of their riders is not always uppermost in the minds of horses fatally wounded in mid-gallop. They tend to jerk and stagger then founder untidily, crashing and rolling – usually on top of something breakable like the legs or back of their passengers.
With no experience at being pitched from the saddle, the Mutes did not know how to fall, so when they picked themselves up, they didn't have to _pretend_ to be hurt. The damage was for real and it was only their in-born resistance to pain that kept them going.
No one had anticipated that their display of fortitude under fire would cause two squads of Trail-Blazers to come down the ramp, one to guard it while the other eight ran out with four stretchers to carry the 'walking-wounded' on board!
Before the five Mutes had time to work out how to cope with this unforeseen departure from the script, Mr Snow had been transferred to a stretcher then, as he was hurried away with Cat-Ballou running alongside, Purple-Rain, Diamond-Head and Lethal-Weapon were invited to adopt the same mode of transport. Which – after a brief hesitation – they did.
With their visors closed and their hands concealed in gloves, no one could tell they were not Trackers, but their real identity could not be concealed for long. Conscious that they must not ruin the element of surprise – at least until they reached the ramp, the three Mutes lay back clasping their rifles to their chests and allowed themselves to be carried feet first into battle.
Things were unravelling even faster for Mr Snow. Jodi Kazan was part of a third combat squad that had now joined the Blazers guarding the bottom of the ramp, and as his stretcher was carried towards it, Buck McDonnell, the Trail Boss of The Lady, ran down to take charge of the situation.
Ignoring the breathless requests to lie down, Mr Snow had hoisted himself into a semi-sitting position on the swaying stretcher in order to appraise the situation. Seconds later, the lead stretcher-bearer made a sharp right-hand turn up the ramp. The soldier manning the poles behind Mr Snow's back stumbled over the toe of the ramp as he swung left to get into line, tripped over his own Size 12's and went down – _whump!_ – on both knees. He held onto the stretcher but the momentum gained in the turn toppled Mr Snow sideways to land in a heap at Jodi's feet.
Laying down her rifle, Jodi moved in smartly with the stretcher-bearers to pick him up. Cat-Ballou, sensing danger, backed off, fingering his rifle nervously. Mr Snow, who had been seized with a feeling of suffocation ever since he'd been obliged to shut his visor, wrenched the badly-fitting helmet off his head as he was given an underarm lift.
'Enough of this farce!' Mr Snow kicked away the hands trying to lift his feet back onto the stretcher, forced himself upright between the two surprised bearers and broke their hold on his arms. 'Clumsy oafs!'
Jodi rose to find herself looking into the deformed, angry face of a Mute with white hair and a – 'Christo! Buck!' She levelled an accusing finger. 'It's Mr Snow!'
McDonnell, moving faster than he'd ever done in his life, was already in action. He'd seen Cat-Ballou back off suspiciously and now the helmeted figure was about to bring his rifle up into the firing position.
Raising his own rifle across his body, McDonnell pumped four triples into Cat-Ballou's chest at close range. Continuing the same swift movement, he rammed the hard rubber butt forward, hitting Mr Snow right between the eyes as those same eyes widened and blazed.
_Kerr-runch!_ The wordsmith's head snapped backwards. An instant later, the twelve charges placed by Cadillac during the night exploded in quick succession – _B-B-B-B-BA-BA-BBOOOMMM!!_
McDonnell saw the lights above him go out then blink on again a second later.
One of the Blazers standing clear of the train pointed to the smoke and steam billowing out sideways from the front end. 'The forward power car's blown!'
'They both have!' yelled someone else.
There was a burst of confused shouting from people at the top of the ramp and a lot of things happened at once –
The three 'renegades' on the incoming stretchers rolled off and came up on one knee with rifle raised and started shooting at their startled bearers as they themselves were gunned down by the Blazers guarding the bottom of the ramp.
'Cover me!' McDonnell tossed his rifle to Jodi, grabbed hold of Mr Snow's baggy uniform in two places and ran him up the ramp like a half-filled sack of potatoes.
Unable to resist a fire-fight, Jodi loosed off several volleys but by the time her rounds struck home, the three attackers were dead several times over. She backed up the ramp and as she caught up with McDonnell her ears were assaulted by a high-pitched warbling alarm. Warning panels lit up and started flashing –
'EMERGENCY! MAJOR FIRE/EXPLOSION HAZARD! EVACUATE TRAIN!'
There was a rumbling roar and a winding-down of gears as the ramps came down along the whole length of the train. And as the rubber-capped toes hit the earth with a dull thud, the ground on both sides of the train erupted to reveal a horde of screaming Mutes armed with rifles, knives and crossbows.
Each of the shallow trenches that had been surreptitiously filled in during the passage of the buffalo under and around the train, concealed a Mute warrior. With the stillness only primitive hunters can attain, they had lain silent and unmoving under a covering of earth and scrub for over three hours, waiting for just this moment. And now, as they hurled themselves towards the foot of the undefended ramps, the crew of The Lady, responding to the evacuation order started down and ran slap into trouble.
Jodi stared aghast as the line-men below her gamely fought off a swarm of Mutes. She clutched at McDonnell's sleeve and pointed to the unconscious Mr Snow. 'Buck! It's not just him! Brickman had another smart Mute with him! He's the clear-skinned guy I told you about –'
'WHAA-ATT?! And you let him go up to the command car?! The guy was festooned with grenades! Fer crissakes, Jodi! Why the fuck didn't you blow the whistle?!'
'I just caught a fleeting glimpse of him! I couldn't believe it! It was only when I spotted Mr Snow that it hit me!'
Still clutching the limp body of Mr Snow, McDonnell threw himself towards a wall-mike linked to the PA system and punched it into life. 'Hear this! Hear this! The fire hazard warning is a false alarm! Repeat – FALSE ALARM! The Lady is under attack! Stay on board and hold the ramps!'
The banshee warbling of the alarm continued and the warning panels continued to flash, and the conflicting information the crew was getting seemed to add to the general confusion.
'Can't you turn that fire warning off?!' shouted Jodi.
'Only from the saddle! The whole fucking system's shot to hell. C'mon! Let's get this joker out of here!'
McDonnell hoisted Mr Snow over his shoulder, turned aft into the next wagon and ran up forward stairs onto the second floor. Pulling out a smart key-card made of red plastic, he opened the door to a small narrow cubicle.
It was a punishment cell. A bare-metal bunk, table, chair, wash-basin and toilet all folded away into the walls. You either slept, sat, washed or shat. There wasn't enough room to do more than one thing at a time.
McDonnell pitched Mr Snow onto the bunk then stepped outside and pushed Jodi into the cell. 'Stay with him. I'm gonna lock you in, okay?' McDonnell pulled out his hand-gun. 'Don't worry, I won't forget where you are.' He found time for a reassuring grin. 'And by the way, well done. You may have saved our asses.'
The door started to close.
'Buck! Wait! What do I–'
'You've got the rifle. Don't hesitate to use it.' McDonnell closed the door. The mechanism locked, pushing the red key-card he'd inserted to the mouth of the slot leaving one of the short edges protruding by one eighth of an inch. The slot had two finger-sized scoops, allowing the card to be grasped and withdrawn. And this was what he proceeded to do with his left hand – since the pistol was in his right.
In the brief moment of time it took to raise his left hand, McDonnell's brain was evaluating the situation and working out what to do next. An inner voice told him that Hartmann was probably out of action. The explosions that had cut the power and steam lines was the work of someone who knew about wagon-trains. Those smart Mutes who had hit The Lady the previous year had pulled another fast one and this time, Brickman was definitely in on it. _Fuggin' lump-sucker_...
Shouts and animal-like screams, the thump of bodies locked in combat, the clash of steel and the characteristic sound of compressed air exploding from rifle barrels filled the air as Blazers on the floor below tried to beat back the invading Mutes.
McDonnell was faced with three choices: (a) to stay where he was and organize the local defence, (b) to try and find out what had caused Commander Hartmann to go off the air and, if possible, retake the forward command car or (c) make his way along the top floor to the rear command car where Lt. Commander Jim Cooper, the deputy wagon-master, was no doubt trying to hold the fort.
The fire alarms had been silenced and the signs switched off, and it was Cooper's voice now that was coming over the PA system, telling the crew to stay on the train, seal the lower wagon access doors and hold the middle floor.
It made sense. If the Mutes could be boxed in and kept on the lower deck the train could be held until help arrived. Fortunately there was no ramp under the flight cars – but there _was_ one under the blood-wagon. Bad news...
The Trail Boss decided to make contact with the deputy wagon-master. He had to tell him Brickman had gone native, and that he and a clear-skinned Mute were running wild in the forward command car. That news had to be gotten back to Grand Central fast –
The decision was never implemented – and the key-card was never taken out of the slot.
There was a sudden burst of firing to his left. Three unarmed Blazer technicians staggered backwards into the corridor from the passageway to the next wagon. McDonnell spun away from the cell door, around the corner of the passageway and onto the down flight of stairs to get out of the line of fire. As the techs fell dying to the floor, a howling Mute exploded through the doorway, rifle held high.
McDonnell brought him down with a triple head shot. Deciding the extra firepower would come in handy he reached for the fallen rifle but before he was able to pull the barrel from under the warrior's body, he glimpsed a movement on the stairs below him right at the edge of his vision. Without letting go of the rifle, he aimed his pistol blindly and fired volley after volley –
_Chu-wii, chu-wii, chu-wii, chu-wii, chu-wiii!_
The needlepoint rounds sliced into the warrior's body, knocking him back against the wall. But they did not kill him before he fired his cross-bow. As McDonnell got a one-handed grip on the rifle and brought it to bear on the warriors now surging through the doorway, the bolt shot upwards between the stair rails and struck the exposed right-hand side of his throat, pinning his tongue to the roof of his mouth on its way through his brain.
The impact caused his fingers to tighten round both triggers, killing a second Mute on the stairway below and the first man through the doorway above.
In the seconds before McDonnell's death, Lt. Commander Cooper, the deputy wagon-master had also perished. The rear command car had been specially targeted, and as the ramps came down, a hand-picked posse of Mutes armed with Tracker rifles burst from their hiding places in the ground and stormed onto the lower floor.
The crew, who were still recovering from the impact of the disabling explosions and confused by the evacuation order, proved easy meat. Moving ahead of their leader, Spandau-Barry, six Bears shot their way up onto the second floor, then the top, clearing the way to the saddle with a continuous stream of fire as Spandau-Barry bounded up the steps with an AP108 mine in each hand.
Cooper and the saddle staff of the rear command car – among them Betty-Jo Aarons – were already reaching for the available weapons as the screaming, wide-eyed Mute warrior burst into view.
They were too late. His outstretched hands were already coming together as the first rounds from several hand-guns struck home.
Time slowed, and in the instant before death overtook them, they saw with horrifying clarity that the pressure-sensitive fuses of the AP108's he held were about to impact on one another. As some tried to dive for cover and others stood frozen to the spot, the top half of the Mute's body disappeared in a blinding sheet of flame and –
From the trees to the south of the rear command car, from the twin rivers and both flanks, the rest of the M'Call clan burst forth and rushed across the open ground towards the stricken wagon-train. The side guns which had fired at the Bears pretending to pursue Steve, Cadillac and the other disguised riders had fallen silent. The danger – and it was considerable – came from the six arrowheads that had swept away towards the west and were now returning.
In the short interval between the first explosions and Spandau-Barry's _kamikaze_ mission, the deputy wagon-master had succeeded in getting a message to the airborne patrol, telling them the wagon-train had come under attack.
The command staff aboard Red River were also listening out on the same channel. While Wallis passed the news back to the White House, Fargo had made repeated attempts to contact Hartmann but so far there had been no reply. And now Cooper had gone off the air.
In the same short span of time that encompassed the death of McDonnell and the staff of the rear command car, the hand-to-hand fighting continued throughout the length of the train and at the front end a posse of Mutes stood ready to repulse any counter-attack on the command car.
Up in the saddle, among the litter of dead bodies, Steve sat slumped in Hartmann's seat, his head in his hands while Cadillac rummaged through the racks of drawers under the table section of the main video-screen console. Eventually he found what he was looking for – a batch of cassettes labelled with the name of the dead wagon-master, the daily record of his radio conversation with Red River and CINC-TRAIN.
'The bastards,' muttered Steve. He raised his fists and bared his teeth at the ceiling. 'The BASTARDS! Oh, Roz! How could you let them DO this?!'
Cadillac, who was listening to the play-back of Hartmann's voice through a light-weight head-set, lifted one of the ear-pieces. 'Do what?'
Steve swore violently. 'Don't you realise what's happened?! This isn't Red River! This is the Lady from Louisiana!'
His partner swivelled round to face him. 'But the letters – the insignia – the aircraft –'
'All faked!' cried Steve. 'Just like we've been pretending that Malone was still alive! Roz and Clearwater aren't ON this train!'
'Calm down!'
Before leaving Big Fork, Cadillac had entrusted Blue-Thunder with the delicate task of storming the blood-wagon. His orders were to seize and hold everyone they found until Steve and Cadillac arrived on the scene. Only those who refused to surrender were to be killed. The assault was still in its infancy. Less than ten minutes had elapsed since boarding the train and Brickman, the super-hero, was losing his grip.
_Hahh, If only Clearwater could see her Golden One now_...
'There's no need to go overboard. We haven't even looked for her yet!'
'I don't NEED to look!' shouted Steve. He swept a hand over the bodies lying around the saddle. 'I KNOW these people!'
'And there's no possibility they could have been transferred?'
'One or two perhaps, but not the whole fucking crew! Whassa matter – you dumb or somethin'?! We've been shafted! The M'Calls are making an empty sacrifice! Your people are dying for nothing!'
To vent his anger, Steve attacked the video screens and the control panel on Hartmann's console with the butt of his carbine. 'What the fuck are we gonna do?!' He glared across at Cadillac.
The Mute had turned back to the main console and was concentrating on the voice coming through his headset. Steve leapt out of Hartmann's chair, strode over to the main console and shot out several of the screens. 'ARE YOU LISTENING TO ME!!'
Cadillac eyed him calmly then removed the headset and stopped the tape. 'No. I'm listening to Hartmann. I take it he _is_ the commander of this wagon-train.'
'Yeah.' Steve regained a measure of self-control. 'The Lady from Louisiana...'
'And this man Fargo I've heard you mention. He's in charge of Red River?'
'Yeah. And Wallis is the head of the AMEXICO team – but it's called the White House task force. I can see how they did it now. Wallis handled the communications with us and kept Fargo in the picture. And Fargo and Hartmann must have been working in concert, sharing the same radio channels. We thought we were dealing with Red River but it was The Lady – in disguise – that was making the moves!'
'So where _is_ Red River?'
'How the hell do _I_ know?' cried Steve. 'It's Karlstrom who's behind all this! Slimy sonofabitch!' He banged the butt of his carbine on the table. 'Why the hell didn't Roz tell us what was going on?!'
'Maybe she didn't know. If Karlstrom has gone to this much trouble maybe he let the people on board Red River think they were going to be attacked – to fool Roz and prevent her from _warning_ you.'
'Yeah, maybe...' Steve rubbed his face wearily. 'Nyehh, what the hell... doesn't really matter. I've blown it. Not just for you and the M'Calls, but for Roz, Clearwater... everything!'
'Not necessarily,' said Cadillac. 'We might still be able to salvage something from the wreckage. I think it's time your masters heard what a good job you're doing – helping to defend this wagon-train.'
'You mean –?'
'Yes. I'm going to pretend to be Hartmann _and_ Malone. If they ask to speak to you, try not to give the game away. Just tell them everything is going splendidly.'
Steve eyed him. 'You've got a fucking nerve...'
'Well, at the moment, it's certainly holding up better than yours.' Cadillac rose to his feet.
'Wonder what's happened to Mr Snow?'
'Never mind about that! The Old One can look after himself! Just show me how to work the radio. Can we alter the quality – y'know with that crackling noise you get? I don't want it sounding too clear.'
'Jeezuss! Any more requests?! I'm not a Comm-Tech, y'know! Just count yourself lucky I know how to switch the thing on!' Steve looked around the saddle and got his bearings. 'C'mon, it's over here. Let's hope it's still working...'
Roz came through. It was like a cool, calming hand on his brow. _Too late, little sister. TOO LATE...!_
## CHAPTER THIRTEEN
At 0530 Standard Time, fifteen minutes after the initial explosions had crippledd The Lady, Karlstrom passed through the turnstile into the Oval Office. The President-General, summoned from his bed by the first news of the assault, was – as always – impeccably groomed and exuded the businesslike air of a man who had been at work for hours.
He greeted Karlstom by the 'stile. 'I can't believe this is happening, Ben. Hartmann must be the unluckiest man alive.'
'Either that or the dumbest. Never mind. Perhaps it'll teach the wagon-train division not to back a loser. Have you seen the map projections showing our units and where this is taking place?'
The P-G nodded. 'While I was dressing.' He steered Karlstrom over to the chair facing the desk then seated himself behind it.
'We've had two messages from Hartmann, and Wallis, my task-force leader, has spoken with Malone. He and Brickman made it to the train but the rest of the team didn't make it. Red River received a report from the deputy wagon-master about some shooting incident on the ramp. He also said he'd lost touch with Hartmann. Since when _he's_ gone off the air and Hartmann's come through saying he's lost contact with the rear command car.
'Sounds very confusing...'
'It is. Here's the situation. Shortly after Malone and Brickman arrived, there was a series of explosions which immobilized the train, cut the power to the guns and caused a systems failure which lowered all the ramps. You can imagine the situation.'
'Christo! I didn't realize they were that easy to crack.'
'I don't think anybody did.'
'But can't Hartmann raise the ramps?'
'Not without mains power. It's a major flaw in the operating system. The designers never envisaged a simultananeous breakdown in both power cars.'
'No...,' mused the P-G. 'And of course the umbilicals are on the _outside_ of the train.'
'That's right. They're in protective trunking which up to now have been invulnerable but –'
'Not to explosives...'
'No. It's something CINC-TRAIN's going to have to get on top of pretty fast. Anyway – that was when the Mutes attacked.'
'Which Mutes are these? The last air reports put the M'Calls several miles to the west of Malone's party.'
'That's right,' said Karlstrom. 'CINC-TRAIN is still trying to sort that one out. This may be the work of another clan – perhaps more than one. Brickman did warn us that the She-Kargo had agreed to support one another following the defeat of the Iron Masters at the trading post.'
'Is that where they got the explosives from?'
Karlstrom had been prepared for this one. 'It seems the most likely source. The big wheel-boats carry upwards of sixty cannon.'
'But it would still need expert knowledge to disable a wagon-train. So who – Cadillac?'
Karlstrom nodded. 'He wired the bombs that blew The Lady apart last year. We have to assume he lifted the information he needed about the train out of Brickman's head. That Mute has some extraordinary capabilities. Jodi Kazan, for instance, told us he left Ne-Issan able to read and speak fluent Japanese – without a single lesson!'
'And what else, I wonder...?'
Karlstrom moved the conversation on. 'The how and why can wait till the post-mortem. The important thing is to help Hartmann turn this situation around.'
'Agreed...'
'Thing's aren't too good but the situation is recoverable. Hartmann has an unconfirmed report of an internal explosion in the rear command car and that's been sealed off. The Mutes control the ground under the wagon-train and now hold most of the lower floor and the middle floor of _some_ wagons.
'Hartmann's battalion is holding the flight car and blood-wagon, the power cars, most of the middle floors and the entire top floor – except at the rear. That includes the flight-deck and the roof of the train. It's a little iffy, but it's still possible to get a plane off and land-on.'
'What's happened to the air component?'
'There were six planes in the air when the Mutes hit the train. Three were on their way back after making strafing runs on the columns of Mutes moving in from the west. They handed the job over to the second flight – which meant that when the Mutes made their assault on the train, the only planes in the air had no ordnance and empty guns!
'The Lady told them to divert to Red River and recalled the second flight but by the time they arrived overhead, most of the Mutes were either on or under the train. The power loss interrupted communications for a while and now most of the Comm-Techs on the command staff are engaged in defending the command car so apart from Hartmann there's been no one to handle the radio traffic.
'The second flight have been circling the train but have hesitated to attack it. Their guns are no good against the Mutes already inside and they don't want to use what napalm they have while there's a whole battalion of Blazers on board.'
'That's understandable. What about our other units?'
'They're moving into position.'
'Has Hartmann requested any specific type of assistance?'
'Yes, he has.' Karlstrom shifted uneasily on his chair as he came to the difficult bit. 'The Mutes have brought a summoner on board.'
'Mr Snow...?'
Karlstrom threw up his hands. 'Can't say – but whoever it is, is making life very difficult. Our boys are battling against gale-force winds gusting down the passageways and they're being bombarded with seats, helmets, air-bottles, hatch-covers – you name it. The bastard is levitating everything that's not screwed down and turning it into a missile!'
'So what does he propose?'
'He's asked for Roz Brickman to be flown over from Red River so that she can, well – use her powers against whoever's doing this. He says if they don't nail this guy they may lose the train.'
'Mmmm...' The President-General turned his attention to his right hand, drummed a brief rhythmic tattoo on the top of his immaculate desk then fixed Karlstrom with a penetrating stare. 'And how do you feel about that?'
'It's why we put her on board Red River. Clearwater's given no trouble.'
'True. But that may be due to her injuries. Do you really think she could handle someone like Mr Snow?'
Karlstrom shrugged. 'We won't know that until we try.'
'And we could end up losing her _and_ the train...'
'There _is_ that risk, yes.'
'And we also have to think about what Clearwater might do if Roz wasn't around. I'd hate anything to happen to Red River.'
'I know how we can cover that. I think we should transfer Roz to The Lady – but only after Brickman and Malone have flown to Red River. That leaves us with one bird in the hand and it gives us leverage against Clearwater. She won't do anything that'll put her boyfriend in danger.'
The P-G nodded. 'That's good. Well done, Ben.'
Karlstrom smiled. 'Yeah, I think it's rather neat. But we'll have to move fast if we're going to turn this situation around. Can I take it I have your approval to make the transfer?'
Jefferson gazed at him for a moment then spread his hands. 'You're the head of AMEXICO, Ben. She's part of the team you sent out to Red River so... I guess that makes it your decision.'
Karlstrom jumped to his feet as the P-G levered himself out of his chair. 'Yes.'
_And it's your ass that's on the line, Ben, ole buddy_...
There was nothing else to say.
'Keep me posted as things develop.' Jefferson broke off eye-contact, leant towards his video-console and pressed the key which brought the head and shoulders of his senior private secretary onto one of the screens. 'Nancy – could you come in for a moment?'
'Yessir!'
Jefferson turned away to admire the computer-generated sea and cloudscape projected onto the screens outside the curving window of the Oval Office. One of the P-G's favourite places: Cape Cod.
Karlstrom, who could read the signs, let himself out through the turnstile as Nancy came in.
Leaving Cadillac to handle the fake radio traffic, Steve poled down to the bottom floor of the command car. Picking up a quartet of Mute warriors – to avoid getting killed by mistake – he went down the ramp and aft under the belly of the train to look for Mr Snow and Jodi Kazan.
Near the foot of the original entry ramp he found the bodies of Cat-Ballou, Purple-Rain, Diamond-Head and Lethal-Weapon. None of the other uniformed corpses belonged to Mr Snow. And Jodi wasn't there either.
With his escort boxed around him, Steve went up the ramp and into the train. Scattered pockets of Trail-Blazers were holding out on all floors throughout the length of the train but contrary to what Karlstrom had been told, the flight car and blood-wagon had both fallen to the M'Calls.
Inside, compared to the charnel-house calm which had descended on the forward command car, the situation in the rest of the train could be compared to a vicious chaotic race-riot fought in the sewers of a space-age city.
The air was filled with the noise of battle: angry screams, shouts, yelping and whooping war-cries, splintering crashes, thuds, thumps, the thunder of running feet and the muffled whump of fragmentation grenades. And there was blood everywhere. The corridors were littered with dead bodies of Blazers and Mutes – which live warriors, bright-eyed with blood-lust, used as stepping-stones in their race to get to wherever the fighting was fiercest.
Few could be persuaded to stop. Those that did only had time to shout 'No' to the question 'Had they seen the Old One?' and were off again before Steve could ask if they had seen a female sand-burrower with a slab of pink scar tissue down the left-hand side of her face. He asked one of his escort to go back down the ramp and see if either were to be found under the aft-section of the train.
Within seconds of the warrior's departure, Blue-Thunder, the M'Calls' paramount warrior, came into view from the direction of the flight cars, toting a Tracker carbine with a blood-streaked bayonet attached. He was sweating and smeared with blood.
'We cannot find Clearwater, our clan sister!'
'I know,' said Steve. 'But never mind. Cadillac will explain that later. Have you seen Mr Snow?!'
Blue-Thunder shook his head. 'I thought he was with you!'
'No. He's disappeared!' Pulling Blue-Thunder with him, Steve stepped back against the brass slide pole in the centre of the stairwell to allow a large group of warriors to pass along the corridor. Many of them carried severed heads on bayoneted rifles.
A drop of blood trickled down the back of Steve's raised hand. Looking up, he saw a helmeted body slumped forward against the stair rail leading to the second floor, with one arm hanging down parallel to the brass pole. The blood had drained out of a neck wound, along the angled shoulder and down the limp arm.
Steve recognized the powerful hand and thick fingers before catching sight of the broad red diagonal rank stripe outlined in black. The arm belonged to Buck McDonnell. He'd glimpsed Big D heading aft just after he'd run into Jodi. Had they both died in sight of each other?
Motioning Blue-Thunder to wait, Steve clambered over the dead Mutes lying on the stairs and reached the Trail Boss. The barbed point of the crossbow bolt sticking through the crown of his helmet said it all. More dead Mutes lay in the passageway above. Higher still, on the top floor, there was a firefight in progress.
Steve signalled his escort to cover the stairs and the pass-way to the next wagon then stepped over McDonnell and up into the corridor.
Several doors to the side compartments hung open, some with bodies lying across the sill. There was debris scattered everywhere. The door to the first compartment on his left was closed. He moved past it without thinking then remembered his interrupted conversation with Blue-Thunder. As he turned back to see if the warrior had followed him up the stairs, his eyes were drawn to the sliver of red plastic sticking out of the lock of the closed door.
He withdrew the card. It was a high-security key-card. The kind that a Trail Boss and other senior non-coms like Battalion Master Sergeants would carry. And it had been inserted in the lock of a punishment cell. Steve tested the door with the tips of his fingers. It was shut fast. The cardholder had either had no time to open it, or had forgotten to withdraw his card after locking it. Which meant there could be someone inside.
Steve silenced Blue-Thunder and his warrior escort with a warning finger then put the key-card back into the slot and pressed a button on the COMMS-LOCK panel set at shoulder height in the door surround.
A miniature black and white tv screen flickered into life. Linked to a fish-eye camera in the roof of the cell, it revealed Mr Snow sprawled unconscious on the bunk with his head towards the door. Jodi Kazan was down on one knee in the far corner with a rifle, ready to fire at whoever came in.
Steve invited Blue-Thunder to view the scene but it only left the Mute totally perplexed. Like his companions, he had never seen a video-screen image before and since this one was a top view of a small room distorted through a wide-angle lens he didn't know what to make of it
'The Old One is in there,' whispered Steve. 'Can you not see him – lying on the bed?'
Blue-Thunder studied the image again and gave a perplexed frown. 'But... he is no bigger than a beetle!' The mute tapped the tiny screen. 'Has he been trapped in this stone by dark magic?'
'Forget it. Just leave this to me.' Steve pushed him aside and put his mouth to the speaker grille. 'Jodi...?'
Having journeyed back to the Federation in a similar cell on the same train the previous winter, Jodi knew the setup. Keeping the rifle aimed towards Mr Snow and the door, she straightened up and addressed the camera lens behind the small clear panel in the ceiling. 'That you Brickman?'
'Yeah, now listen – we need to talk.'
'So talk.'
'No. Face to face. I need to explain things. But first I want you to understand you're not in any danger. Just slide that rifle under the bunk and clasp your fingers together on the back of your neck. I promise you won't come to any harm. You have my word on that.'
'Oh, yeah?' Jodi took a firmer hold on the rifle and placed the barrel against Mr Snow's midriff. 'Your word ain't worth shit, Brickman. I saw who you came in with – Cadillac. The scumbag that killed Dave and blew half The Lady apart. And if it hadn't been for Buck McDonnell, I'd have gone sky-high with the rest!
'An' you know what? I got the blame! I was the one who got thrown in the slammer and –' She gave a harsh laugh, '– here's the biggest joke of all. I stood up for you, swore you had nothin' to do with it! And all the while you were tucked up tight with these lump-shits! Now I know why they hit me with a Code One!'
'So what are you doing here?'
'They gave me a chance to redeem myself by fingering your Mute friends. When I came on board I was supposed to be kept under close arrest but Buck McDonnell persuaded Hartmann to put me back on the team while I was on the train. For old time's sake. Not flyin', of course, but regular duties alongside the other Blazers. Nice gesture, huh? Didn't know it was going to end in a shoot-out with you.'
Steve spoke with a new urgency. 'It needn't, Jodi. We can work something out.'
'Yeah? Like what – the length of the pole you're gonna stick my head on?'
'No! You'll be safe with me. Just put the rifle down and listen! I'm going back to the Federation. You can come with me if you want. We can cover for each other. I can help you beat the rap!'
Jodi answered with a mocking laugh. 'Where you been living, Brickman – fantasy-land? You been smokin' too much rainbow grass! D'ya know where I was before they put me on the train? Death Row! Being around you is bad news, honeybun!'
Steve tried a softer line. 'This is getting us nowhere, Jodi. You can't stay in there forever.'
'I'm staying until Buck McDonnell steps through that door. If anyone else comes in this old lump gets blown away! And that goes for you too. Comprendo?!'
It didn't seem like a good time to tell her that the Trail Boss was lying on the stairway with his brains leaking out the top of his head. And with the rifle barrel aimed point blank at Mr Snow it was too risky to burst in. He'd be dead before the door opened up far enough for them to shoot her down.
It was at this point that Mr Snow chose to open his eyes and take in his surroundings. He clutched his bruised forehead and groaned. 'Sweet Sky-Mother...' He looked down the barrels of the rifle then fixed Jodi with his faded blue eyes and chuckled hoarsely. 'Goodness me, that looks rather dangerous!'
Jodi shrank back into the corner and aimed the rifle at his head. 'Stay down! If you try to sit up I'll shoot!'
Mr Snow raised his head a little further then fell back with a gasp of pain and covered his face with his hands. 'I can assure you that's the last thing I feel like doing.' He coughed and retched as if he was at his last gasp.
Steve saw Jodi's rifle waver then, with a movement that was too quick to see, Mr Snow's left hand flashed out and grasped the front of the rifle, deflecting it away from his body. His uncovered eyes blazed. The outstretched arm connecting him with the rifle was like a steel rod, every sinew and muscle taut and unyielding. A lightning conductor.
Jodi's two-handed grip on the rifle tightened, every muscle in her body contracted as the current ran through her. Her lips drew away from her clenched teeth and her eyes dilated as she was hit by a series of convulsions; five massive shocks that jolted her from head to toe, followed by a sixth that slammed her lifeless body against the back wall of the cell.
Steve unlocked the cell door and rushed in as Mr Snow tried to sit up. 'Are you all right?'
'No, I'm not!' snapped Mr Snow. 'On top of everything else, some great oaf tried to take my head off! If it wasn't for these he'd have knocked my brains out...' He fingered the broken skin on the ridged front of his skull then said: 'How's it going?'
'Don't ask,' said Steve. He looked down at Jodi's crumpled body and heaved a sigh of regret. This wasn't how it was meant to be. 'Come on – I'll take you to Cadillac.'
In the saddle, the young and the old master embraced each other warmly.
'Easy, easy,' grumped Mr Snow. 'I'm feeling a little fragile.' He sank into Hartmann's chair and gazed at his surroundings with a mixture of curiosity and disgust. 'What a strange world. How could anyone in their right mind want to live in a place like this?'
'We'll have to save the philosophical questions until later,' said Cadillac. He squared up to Steve. 'Good news and bad news. Your friends aboard Red River have agreed to fly your kin-sister over here to deal with the summoner that's causing so much havoc.' He glanced across at the semi-recumbent figure of Mr Snow.
'The bad news is they want to exchange her for you and Malone. And you have to go there first. I'm planning to dispose of him on the way to the flight-deck during a gallant rearguard action. But we don't have any steam. Can we get a plane off?'
'Yes. Each catapult is fed from a reserve tank that's kept topped up by the system. Even with the lines down they'll be good for a couple of launches before the pressure falls off.'
'Good. Okay, you'd better show me how it's done.'
Steve looked surprised. 'I've got to leave now?'
'Yes! They're waiting!'
Mr Snow rose to meet Steve as he came over to say goodbye. As they clasped each other's hand and wrist, the old wordsmith's grip was reassuringly strong.
'Farewell, Old One. Would you believe me if I said it grieves me to leave you – especially at a time like this?'
'This is how it was meant to be, Brickman. You've come a long way to find your true self. You're close to the top of the mountain. Don't fall!'
'I'll try not to...'
Cadillac and Steve climbed out of a duckhole onto the flight-deck. M'Call warriors armed with rifles and dressed in captured uniforms manned the other duck-holes to fool the circling wing-men into thinking that Hartmann's battalion still controlled the top of the train.
A Skyhawk Mark 2 was parked on the deck with its bubble canopy open and tail booms folded. It had taken several hits from crossbow bolts but none of them had hit anything vital.
Cadillac peered at the exposed joints of the wing and tail booms. 'How do we unfold these?'
Steve was momentarily flummoxed. 'Err, shit, hang on a minute. Lemme see...' He peered at the exposed wing joints then slapped his forehead. 'Idiot! Of course! It's hydraulic! You work it from inside the cockpit once you've started the engine.' He leant into the cockpit and pointed out the control levers. 'You push that one forward and lock it – so, and the other raises the tail.'
'Okay. Let's get you hooked up.'
'Hang on, there's a couple more things I need to show you.' Steve led the way aft to the rear port duckhole and showed Cadillac the control mechanism that raised the arrester wires which engaged the landing hook of incoming planes.
'Brickman, I know all this. I've learned everything you've learned.'
'Yeah, like how to unfold the wings of a Mark 2.'
'Even you were hazy about that.'
'Yeah, well there's no harm in making sure. It's the cleverest people who make the stupidest mistakes. Like attacking the wrong wagon-train.'
Cadillac turned his face to the sky. 'Sweet Sky Mother... are we never to hear the last of that?!' He turned to Steve. 'Don't you understand?! Talisman guided us! Your masters tried to trick us but they played into our hands!
'If we had attacked the _real_ Red River every person on that train would have had to die leaving only you and Clearwater alive to tell of how your kin-sister was carried off into the hills. Reflect on that for a moment. Are your masters so foolish? Your treachery would soon have been exposed!'
'You're right,' admitted Steve.
'Of course I am! This way, your kin-sister is released as the Old One wished, and you will return a blood-stained hero! A loyal soldier-citizen of the Federation who did his utmost to protect his comrades-in-arms.'
'Yeah... C'mon.' Steve quickly ushered Cadillac to the forward duckhole on the starboard side and explained the launch control panel for the steam catapult.
'Yes, yes, look, it'll be quicker if I tell you,' said Cadillac. 'That's the steam-pressure read-out. The top button raises the catapult, the second fires it, and the third one lowers it back into the deck. I'm having to revise my opinion of sand-burrowers. Like all the systems on this train, this was designed to be idiot-proof – something which most of you clearly are!'
'You've forgotten something,' said Steve. 'I'm a Mute.'
He vaulted up onto the deck and climbed into the cockpit of the Skyhawk. As soon as the engine was running, he unfolded the outer wing panels then raised the tail. The twin booms which were folded forward under the wings, dropped down on parallel swing links then were brought up into line with the trailing edge of the wing by hydraulic rams. A small illuminated diagram on the instrument panel confirmed both wings and tail had locked into place.
Cadillac and another Mute warrior came forward to hook the Skyhawk onto the launch cradle as Steve taxied forward onto the catapult.
'D'you want any help?!' shouted Steve.
Cadillac popped up beside the open cockpit. 'For heaven's sake, Brickman – just GO!' He smiled and offered Steve his hand. 'Take care of our sister!'
'I will,' said Steve. 'Make sure you take care of mine!'
Cadillac dropped the cockpit canopy into place and gave it a goodbye slap.
The wing-man circling immediately above the wagon-train pressed his transmit button. 'Blue Three to Red River, we have one bird on its way.'
'Red River to Blue Three, Roger. Any update on your last sit-rep?'
'No. The Blazers are still holding down the roof and the Mutes are under the train. Over.'
'Roger, Blue Three. Take out as many strays as you can. Let 'em know you're there. But try not to damage the train. We want to get that Lady back on the road.'
'Blue Three, Roger, Wilco. Listening out.'
Roz, Don Wallis, Jake Nevill and the rest of the task force scrambled up onto the flight deck as Steve landed. Red River ground crew quickly manhandled his battle-scarred Skyhawk onto the rear port lift as the first two out-going planes came up on the forward lifts and were lined up on the catapults.
Wallis ran his eyes over the protruding crossbow bolts then turned to Steve as he emerged from the cockpit. 'This doesn't look too good...'
'Don't worry,' said Steve. 'We're holding the line. If we weren't, I'd never have gotten clear.' He paused then said: 'Did you get the news a –'
'About Malone?' Wallis's face tightened. 'Yeah, Hartmann...' He shrugged off the rest of the sentence.
'Losing him was a real blow,' said Steve. 'He was a great guy.'
'The best...' Wallis buried his feelings and became the brisk team-leader. 'Okay, guys, off you go.'
George Hannah and Cal Parsons hurried towards the aircraft that was now hooked up and ready to go.
'Jake! Let's go wind up Number Two!' Wallis gripped Roz's arm. 'You got one minute!' He strode off with Nevill.
Darryl Coates and Tom Watkins headed across the deck towards a third Skyhawk, complete with buddy-frame that had just come up on the rear starboard lift.
Roz and Steve hesitated for a moment before throwing themselves into a warm, rocking bearhug. When they separated, they held onto each other's arms. The slipstream from the propellors flattened their clothing and snatched the words from their mouths.
'Why's this happening, Roz? Where're you gonna go? What're you gonna do?'
'I can't say!' she cried. 'I'm just glad to be part of it – aren't you?'
He tightened his grip. 'I'm scared. Are you sure you're not hiding something from me?'
'No. Wait – did you pick up my message about Annie?'
'Annie...?'
'Yes, ages ago. I obviously didn't get through. She wasn't our mother, Steve.'
'Yeah, well, with what we know now that makes sense...'
'But don't you see what it means? You're not my brother and I'm not your sister!'
'There's no need to sound so happy about it.'
'I'm happy because we're both _free_! Free from the guilt and the pressure of that relationship. Free to give expression to that love we felt or give it to someone else. To follow The Path, Steve!'
'Oh, jeezuss! I don't want to lose you, Roz!'
'You won't lose me, Steve. Our lives are bound together by a power and for a purpose far greater than our need for each other.'
Steve nodded. 'How come I get the feeling you're suddenly a lot smarter than me?'
'I always was, little brother!'
Steve ruffled her hair playfully and hugged her again. 'Look after yourself. It's a rough world out there.'
'You too...'
They looked along the flightdeck. The five mexicans were clustered around Wallis. He looked towards Roz and beckoned her forward. Roz signalled she was on her way.
'Do those guys know what they're doing?'
'They think they do.' Roz kissed him quickly on the mouth then broke away. 'Look after Clearwater!'
'I will.' _If they let me_... Steve followed her along the flight deck. 'Listen! I know this is a stupid question but... will we see each other again?'
Roz favoured him with another enigmatic smile. It was getting to be a habit. 'That depends on you.'
Steve stood on one side with Wallis as Roz wriggled into the prone position on the buddy-frame then had the flight-bag zipped shut around her. It was like a wind-proof sleeping bag with a clear plastic ventilated hood covering the head and shoulders.
Hannah and Coates went away first, then Nevill's Skyhawk was hooked up. The engine went to full revs, there was a tremendous _whoooshh!_ then Roz was gone, leaving wisps of steam curling from the vents in the catapult rig. Gone without an answering wave.
Steve let his hand drop, and swallowed the lump in his throat. The third Skyhawk with Cal Parsons on the frame and Watkins at the controls was brought forward and went off in its turn.
When the three Skyhawks carrying Roz and the five mexicans aarrived over The Lady from Louisiana, they found a reassuring number of camouflaged and helmeted Trail-Blazers on the roof. Hannah, flying the lead aircraft, made a low pass alongside the train to check the state of the flight-deck before landing with Parsons. Wallis had ordered them to go in first to make sure everything was okay. On receipt of their signal, Nevill would then bring Roz in, followed by Watkins and Coates.
Mr Snow, who had been cajoled into wearing another Tracker helmet, crouched in the duckhole with Cadillac as the latter, using his normal voice, talked the Skyhawk down onto the wire. Along each side of the flight deck, M'Call warriors dressed up in Tracker uniforms stood ready to pounce.
Mr Snow gazed at his protégé admiringly. 'You amaze me. Where did you get all this knowledge from?'
'Lots of people – but mainly from Brickman. We're lucky he had an enquiring mind. Now – for the last time – close that visor!'
The three Skyhawks from The Lady's own flight component which had been circling the train and making the occasional strafing run, came in ahead of Hannah's aircraft in V-formation as he made his final approach. The two wing aircraft strafed the ground on either side of the train to prevent any Mutes taking a pot-shot at Hannah and his passenger as the Skyhawk floated in over the rear command car with its flaps and arrester hook down.
Cadillac leapt onto the flight deck and ran forward while two warriors hauled Mr Snow out of the duckhole. Taking care to keep clear of the whirling propellor, Cadillac freed the snagged cable then assumed the role of deck marshal, signalling the pilot to fold his wings and taxi towards the forward port lift.
Mr Snow and a hand of disguised warriors caught up with Cadillac as Hannah parked on the lift and cut the motor. To the mexican, the Blazers around the flight-deck seemed strangely disorganized but at that point in time he had no reason to suspect there was anything wrong. This was, after all, no ordinary fire-fight.
Unzipping the flight-bag, Cadillac glimpsed an unmistakably male figure inside. Parsons. By this time Hannah, his partner, was out of the cockpit.
'Where's the Brickman girl?' demanded Cadillac.
'Up there.' Hannah jerked a thumb skywards as he helped Parsons extricate himself from the buddy-frame. He glanced at the masked figures crowded round the airplane and wondered why they weren't facing outwards ready to gun down any Mutes who poked their nose from out under the train. 'Who's in charge here?'
'I am,' said Cadillac.
Parsons, now standing by his partner's side, coughed. Covering his mouth he turned in towards Hannah. 'Check the badge...' he whispered.
Hannah's eyes fastened on Cadillac's uniform and saw it carried a badge of the 5th Signals Intelligence Squadron. The decoy outfit that had been jumped by the M'Calls and Malone's renegades. 'So what's happened to the crew-chief?'
'He's been wounded. He's in the blood-wagon.'
'Then where's the Flight Ops Exec?' demanded Hannah.
'He's dead.'
'Aww, shit – Pete Carmichael?!'
'Yeah...'
Hannah took a step back. Parsons was seized but before anyone could reach Hannah, his pistol was out of its shoulder-holster and pointing at Cadillac's face.
'You just struck out, good buddy! That's the wrong name and you're wearing the wrong uniform! What the fuck's going on?!'
Mr Snow raised his visor and pushed aside the warriors who were shielding him. 'A battle, my friend! Which YOU are losing!'
The pistol wavered in Hannah's hand as he tried to control an unreasoning urge to throw it away. His brain was telling him that it wasn't a gun he had in his hand at all. It was a snake – coiled around his fist!
With a horrified yell he jerked his hand open and flung the hideous thing away. And as the pistol clattered across the flight deck, a voice filled his mind, banishing all other thoughts and feelings.
_Do whatever is required to bring the girl to us_...
Moving like a sleepwalker, Hannah reached into the cockpit and plugged himself into the radio circuit.
Circling above the train with Roz lying alongside his cockpit, Jake Nevill heard the hiss of static in his earphones as Hannah came through.
'White Knight One to Mother Hen. The perch is green and clean. Bring the bird home to roost.'
'Mother Hen, Roger.' Nevill glanced over his right shoulder towards the Skyhawk flown by Coates, tucked in behind and below his starboard wing. 'White Knight Two. Do you copy, over?'
'White Knight Two. Roger. Will follow you in.'
Filled with a great sense of well-being, Hannah pulled off his helmet, dropped it onto the pilot's seat and stood erect, ready and willing to face whatever lay in store. Parsons had also ceased to struggle. They were like poultry which, when seized by the feet and upended, accept the inevitable and go unresisting to their death.
'Take them below.' Cadillac turned to Mr Snow as the mexicans were led away. 'Wait here!'
He ran back down the flight deck and jumped into the Deck Controller's duckhole and plugged himself into the radio linking him with the aircraft overhead. 'Lady-Lady to Mother Hen. Surface wind bearing two-seven-five, speed one zero. Call finals. Over.'
'Mother Hen, Roger. Two seven five, speed one zero.'
Cadillac watched the Skyhawk bank left onto the short crosswind leg over the forest of larches beyond the rear command car. The third aircraft piloted by Coates was coming downwind west of the train to make the same turn.
'Mother Hen turning finals, over.'
Cadillac was gripped by a rising sense of excitement. Everything was working out beautifully. And only _he_ could have done it! And with Brickman gone there was no chance of anyone else stealing the limelight.
Once again, the three patrol aircraft swept in ahead, clearing the way as Nevill landed on. Then as White Knight Two, carrying Watkins and Coates, turned over the forest of larches, the trio of Skyhawks banked round over the twin rivers and swept back towards the train and the incoming aircraft.
On the flight deck, helmeted Blazers in full combat gear unhooked Nevill's Skyhawk and waved him forward. Two soldiers were already peeling open the flight-bag as Roz eagerly unzipped it from the inside.
Nevill cut the motor, threw open the cockpit cover and started to climb out. 'Don't bother with that! Let's get this heap out of the way first! There's two other guys on their way –'
A savage knife thrust to the heart, delivered by one of the masked uniformed figures, stopped him in his tracks as the three patrol aircraft zoomed overhead.
Exercising the same power that Clearwater had employed in her earlier battle against The Lady, Mr Snow reached out with his mind and filled the formation leader with an overwhelming unreasoning desire to destroy the plane now in his sights. It became the focus of everything he hated.
Coates, at the controls of the incoming Skyhawk, expected the formation to break away on either side of him. Instead, a hail of fire from the six-barrelled gun under the nose of the lead aircraft exploded through the windshield and punched a gaping hole in his chest.
Watkins, on the buddy frame, could do nothing but hang on helplessly, braced for the crash he was unlikely to survive. The pilotless plane side-slipped into the roof of the wagon-train, lost its starboard wing as it bounced off and cartwheeled messily into the ground.
Petrie, the bewitched patrol leader, pulled up into a tight loop, half-rolled off the top into a diving right-hand turn that brought him around onto the port side of his startled wingmen and pressed the gun-button. Nothing happened. The low-level strafing runs and the frontal attack on Coates had left him with an empty drum.
But both pilots, stunned by his downing of the Skyhawk, knew something had gone badly wrong. Petrie had flipped. But they had run out of ammunition too. And now that he was flinging his plane around the sky in an effort to ram them, the only thing they could do was split – and fast. Separating out, they opened the throttle and went down to treetop height where they were harder to spot and sent a May-Day to Red River.
Petrie followed. With each succeeding mile, the desire to destroy all blue flying objects faded, vanishing completely by the time he reached Red River. But by that time, the damage had been done: Mr Snow had achieved his objective – to empty the skies above the wagon-train.
Roz, stepping onto the flight-deck, knew who Mr Snow was before he cast aside his suffocating helmet, and as she clasped his outstretched hands their minds were instantly attuned. He looked incredibly old, his face was haggard and drawn, but there was still plenty of life in his eyes.
'I thought you'd never get here.'
Charmed by the mischievous smile, Roz tightened her grip on his thin bony fingers. 'We have waited a long time, Old One.'
'Too long, my child. This tongue that greets you must bid you farewell with the same breath.'
Roz removed her helmet and brushed her fingers through her short auburn hair. Throwing her head back, she breathed deeply, relishing the cooling touch of the wind upon her face then, as her eyes opened, she found herself looking at Cadillac.
He too was bare-headed – and seemed to be struck dumb. For a moment their eyes remained locked together then Roz turned to Mr Snow.
'Is this the warrior who is to become known as the Sword of Talisman?'
Mr Snow, amused to see that Cadillac had temporarily lost command of the situation, smiled broadly. 'He has the makings of a warrior. With you at his side he may even become a great one. But for the moment, he is known as Cadillac.'
The old wordsmith beckoned them to step nearer. Without being bidden, they took hold of each other's hand and knelt before him. Watched by a silent circle of warriors, he laid his hands on their heads and raised his face to the sky and uttered a silent prayer for divine guidance.
When it came, he closed his eyes and lowered his head. Roz and Cadillac felt their scalps tingle beneath his hands. 'It is the wish of Talisman that you be joined together, in blood and breath, body and soul, for you are The Chosen, destined to raise his banner high and accomplish mighty works in his name! May his blessings be upon you from this day forward. I hereby bequeath to you all that was mine and entreat you to open your minds to the powers which only he can bestow.'
Opening his eyes, Mr Snow waved them to their feet. 'Enough!' He pushed them towards the Skyhawk that Roz had just arrived on. 'Go! Before it's too late!'
'Go?! Go where?!' cried Cadillac.
'Go west, young man! Into the hills! Towards a new beginning!'
'But...' It was all happening too fast for Cadillac. He was already trying to cope with the revelation that this striking young woman had been chosen by Talisman to be – assuming he had interpreted the Old One's words correctly – his life partner! Admittedly with her very first glance, she had reached into the depths of his being but even so... A commitment like this should not be rushed into. She had, quite literally, entered his life and his heart like a bolt from the blue but she was... Brickman's sister! And now, on top of all that, the Old One had given his tail another unexpected twist –
'But... my blood-brothers and sisters! This may be _your_ dying place, Old One, but what is going to become of them?!'
'Take a look around you!' Mr Snow flung out his arms and swept the horizon.
Cadillac gazed around him and felt his blood run cold. Close enough to be seen by a sharp-eyed Mute but still a long way off were four more wagon-trains – moving in from the north, south, east and west. The clan could not retreat. It could only stand and fight.
'It grieves me to go against you, Old One, but if I honour your wishes I shall be without a shred of honour myself. I brought our Bears and She-Wolves to this place. I cannot run from here and leave my clanfolk to die.'
'"I-I-I this, I that"! Can you think of no one but yourself?' cried Mr Snow. 'Talisman has given this star-child into your care! The Path is drawn. Follow it and don't look back!'
'But –'
Mr Snow slapped him hard on the chest, forcing him to step back towards the Skyhawk. 'The Clan M'Call is not _going_ to die! It is going to become immortal! When the history of the Plainfolk nation is written – as one day it will be – our sacrifice here today will be remembered as one of the first glorious steps on the road to final victory!'
'HEYY-YAHH!' chorused the listening warriors.
'But you still haven't told me what I must do!'
'Exactly! I'm through giving advice! Stop thinking about yourself and listen to the Sky Voices!'
Cadillac and Roz found a score of willing hands to help them position the Skyhawk on the unused catapult. Hannah's craft parked on the adjacent lift proved to be in the way and was promptly tipped over the side. Cadillac checked the steam pressure read-out and explained the control panel to Mr Snow.
'Are you sure you know which button to press?'
'I'm just _old_ , not feeble-minded, you impudent rascal! Don't let this promotion go to your head!' He accompanied Cadillac back to the Skyhawk; Roz was already in place on the buddy-frame but she had not closed up the hood of the flight-bag.
'Does he know how to fly this thing?'
'Oh, yes. That's the problem. He knows a great many things but not enough about the things that really count.' Mr Snow squeezed her shoulder affectionately. 'I'm relying on you to drum some sense into him.' He moved round the nose of the aircraft to bid Cadillac farewell.
'I shall miss you, Old One...'
'Nonsense! If that were true, it would mean you were not ready. Are you trying to tell me that all the years I've spent teaching you have been wasted?!'
'No, but –'
'Then be off with you!' cried Mr Snow, hiding his deep love for the wayward, gifted child whose mind he had nurtured from the age of one. 'We've got work to do!'
The command staff of True Grit, the wagon-train now rolling westwards along the far bank of the North Platte, saw the Skyhawk leave the deck. Tracking it through a telephoto lens, they watched it gain height in a climbing turn towards Wyoming.
The Flight Ops Exec tried to contact the departing aircraft on the standard frequency but there was no reply and it vanished in a bank of dense low cloud before one of their own Skyhawks could be put up to intercept it. The four approaching wagon-trains had been ordered to keep their aircraft grounded so as not to alert the M'Calls to their approach. CINC-TRAIN wanted to nail every single one – not send them running into the hills.
Aboard Red River, currently a hundred miles to the south east beyond the Nebraska/Kansas State line, Wallis was becoming increasingly worried. Fargo's staff had been relying on Hartmann's radio messages to tell them how his battalion was coping with the attack and now he had gone off the air.
His disappearance had coincided with the arrival of Roz Brickman and the team from the White House. Nevill had set down after getting the all-clear from Hannah and Parsons then things had gone haywire. There had been an inexplicable incident involving Petrie – one of the Red River wing-men loaned to The Lady to train up her own pilots on the Mark Two aircraft. An incident which had cost the lives of two of Wallis's colleagues and had almost claimed two more.
The stunned Petrie, faced with the testimony of his fellow wing-men, could offer no explanation for his behaviour. There was a total blank in his memory starting when he began the run in towards the wagon-train. The next thing he could recall was tailing one of his wing-men back towards Red River and receiving orders to land on. Pending further investigation of the incident, Petrie was formally relieved of his duties and thrown in the slammer.
James Fargo, the wagon-master of Red River, could make neither head nor tail of it. Wallis, on the other hand, had been given access to the record of Jodi Kazan's debriefing. The similarity between Petrie's behaviour and the incident surrounding her pick-up by aircraft from The Lady were too striking to ignore. This was earth-magic. The work of a summoner. Probably Mr Snow. Had he also overcome Roz? They wouldn't know the answer to that until someone on The Lady came back on the air. Or until – as seemed more likely – the four Trail-Blazer battalions now being carried into action by True Grit, King of the Pecos, Sands of Iwo-Jima and Overland Raider had retaken the stricken wagon-train.
By the time that happened, Red River would be even further from the scene. CINC-TRAIN had already ordered her to roll south and off-load Brickman and the injured Mute at Monroe/Wichita for forward shipment to Grand Central. The signal had caused Fargo and his crew considerable distress. This was the first time they had been ordered away from a fight. What made it worse was the fact that they had been ready and willing to take this cocky bunch of Mutes apart for more than eight weeks only to learn twenty four hours before the expected attack that The Lady from Louisiana – a real nothing train to nowhere – was going to act as a decoy!
After giving the command staff of Red River a report on the general situation aboard The Lady up to the time he'd left, Steve was handed over to Wallis for debriefing on the more sensitive aspects of the operation.
Steve stuck to the scenario he and Cadillac had concocted. After two hours of patient questioning that ranged from the battle at the trading post to Malone's death while attempting to reach the flight car, Wallis indicated that he had enough material for his preliminary report but warned Steve he would be required to cover the same ground in greater detail when they reached Grand Central.
Until then he would be required to remain close to Clearwater and use his 'best efforts' to prevent her from becoming a threat to the security of the wagon-train or its crew.
In a somewhat blunter vein than usual Wallis said. 'The word is you've developed a relationship with this lump. I don't know how much influence that gives you, but you should make it clear to her that if she steps out of line, _you_ will be held responsible.'
'I don't think there'll be any problems, sir.'
Wallis's face creased with anxiety. 'I can't understand why Jake hasn't reported in. I hope they're okay.'
Steve shrugged. 'They flew into a tough situation. Hartmann's boys were doing a good job when I left but they were only just holding their own. That's why I suggested calling in Roz. Wish I hadn't now. If I'd known you weren't going to let me stay and help look after her I'd have kept my big mouth shut.'
'In this game you do what the big man says. I have to follow orders too.' Wallis eyed him, then said: 'Have you, ahh... heard from Roz?'
'No, sir.'
'Don't you find that kinda strange, in the circumstances? From what I understand, your mind-contacts are stress-related.'
'With her they have been. It doesn't always work the other way around. There's nothing _strange_ in not hearing from her – but it _is_ worrying.'
Wallis appeared satisfied with this explanation. 'But if she _does_ come through...'
The lie came easily. 'You'll be the next to know...'
Wallis slapped the table top. 'Okay, listen, I'm assigning you to the blood-wagon. Report to the CMO – Michelle French. You'll be quartered with her staff until we reach Monroe-Wichita. The night duty-staff will keep an eye on your patient but otherwise she's all yours.'
'Thank you, sir.'
'Don't take your eyes off her.' Despite his anxiety, Wallis managed a grin. 'Having taken a look at her myself I imagine that won't cause you too much hardship.'
'No, sir.'
'Okay, get on it...'
Clearwater opened her eyes to find Steve sitting by her bed, freshly scrubbed and in a clean set of fatigues. 'At last... How d'you feel?'
'Better for seeing you.'
'The doc tells me your arm will be out of that cast soon and that in a month or two you'll be on your feet again.' He touched the back of her hand. 'When I think of how you looked when...' He waved the thought away. 'But that's all over now.'
'For you, perhaps. Tell me, Cloud-Warrior – why did your masters save my life when, as we speak, they are killing my clanfolk?'
'That's because you're important. They hope to discover the secret of your power.'
'I am merely a channel. The power belongs to Talisman.'
'Yeah, well, that's their problem.' He took a firmer grip on her hand and gave her a warning glance. 'We've got other things to worry about.'
'Where is your kin-sister?'
'She's gone.'
In his mind's eye, Steve looked down at the M'Calls, settlement as he circled it like a bird. Flames and smoke from burning huts rose into the air. Many were already rings of grey ash. Around them lay the bodies of the nursing mothers and the young children the clan had left behind with a posse of She-Wolves to guard them.
The only moving figures wore camouflaged combat fatigues; a unit of Trail-Blazers had descended without warning to complete the destruction of the clan. And as the flying eye swooped lower, some of them turned their faces towards the sky and waved triumphantly.
Steve shared with Roz the sense of utter desolation. He lowered his head and pressed a thumb and finger against his closed eyes in an effort to wipe away the haunting images but they were branded on his soul.
_This was all his work. This was why he had been called the Death-Bringer. Oh, Sweet Sky-Mother! When would it stop?!_
Clearwater saw the pain in his eyes. 'I see it too, Cloud-Warrior. I share your grief. But these are things we cannot speak of.'
Steve got the message, 'No. So... from now on, it's just you and me.'
_Against the might of the Federation_...
Not _just_ you and me, thought Clearwater. But that news could wait. She nodded and squeezed his hand. 'The journey begins.'
'Yes...' _But how would it end?_
Despite Brickman's cooperative attitude, the debriefing failed to provide Wallis with an answer to the one big question. Whatever the final outcome of the present engagement, the Mute attack on The Lady had obviously been carefully prepared. But how had they known so far in advance where the wagon-train was going to be?
Brickman, outwardly none the worse for wear, was unable to add anything of significance to what Wallis already knew from the radio signals he and Malone had sent while on the run. He claimed to have done his utmost to warn them of the impending attack and although he repeatedly expressed the wish to have done better, could offer no explanation as to how the Mutes came to be lying in wait for The Lady.
The young man had suffered a double trauma – the assault on the wagon-train crewed by many of his former comrades and officers, and the loss of Malone and the other mexicans. Crushed by the guilt which haunts all survivors, he felt personally responsible for both and in an effort to unburden himself he confessed to not having reported that the M'Calls believed Cadillac was able to predict future events.
Wallis struck this veiled and somewhat embarrassed reference to seers and seeing-stones from the record. Even though he had been given access to what COLUMBUS knew about Mute summoners, Wallis shared his director's scepticism. The subject of 'gifted' Mutes and psionics was a speculative quicksand into which a rational man ventured at his peril.
In time, Brickman's mental scars would heal. He would come to realize that the responsibility for such disasters could rarely, if ever, be laid at the door of a single individual. The system might need to find a scapegoat but investigations showed it was a series of actions and decisions – often apparently unrelated – by a large number of people that created the circumstances in which something like this could occur.
Wallis counted himself doubly lucky – first, because he had been ordered to hold the fort instead of flying to the embattled wagon-train with the rest of his team and second, because the order to transfer Roz Brickman had come direct from Karlstrom's private terminal. With Nevill watching, Wallis had translated the coded letter strings into a clear on-screen message, then he had logged it into the comms-system memory to build up a complete record of the operation. When you were working on something this sensitive, it was always advisable to cover your ass.
A niggling thought triggered by the continuing silence from his men aboard The Lady prompted Wallis to double-check the log. It proved impossible to retrieve the record of Karlstrom's signal ordering Roz Brickman's transfer to The Lady. With growing desperation, Wallis spent several hours trying to coax the fateful order from the system's memory but it was no longer there.
There were only two explanations for its disappearance, and both cast a dark shadow over his career prospects. He had either been the victim of another stunning illusion created by Roz Brickman or shafted by his superiors. Perhaps the message he and Nevill saw had only existed in their imagination for the length of time needed for them to act upon it. The alternative was altogether too depressing.
Ray Ramsay, the Red River Flight Ops Exec knocked on the outer wall of the skip and put his head around the open door. 'A message for you from the wagon-master. It looks as if we may have lost Hartmann's battalion. The Lady is ablaze from end to end.'
'Jeezuss!' Wallis drew a hand down his face.
'We don't know the full score. The support units that went in are still mopping up. Commander Fargo asked whether you would like to join him in his quarters to hear the news as it comes in.'
'Thank you. I'll be right along. Just got a little business to attend to.'
When Ramsay had gone, Wallis thought over all the moves, then closed the door to the cargo skip, sat down in front of the blank screen, switched his pistol to Full Auto, placed the barrel against his chest and shot himself.
## CHAPTER FOURTEEN
'It's over. I just got word from CINC-TRAIN.' The President-General invited Karlstrom to take a seat by the blazing log-fire. The logs were modelled in cast-iron, the glowing ashes were flakes of mica, and the flames were fuelled by a gas line but the effect was real enough.
Karlstrom wondered why this news hadn't reached him through his own information network. He hated surprises. 'And The Lady?'
Jefferson stretched a hand towards the flames. 'A total loss I'm afraid. A lot of her crew ended up with their heads on sticks. The Mutes – and it has now been confirmed it _was_ the M'Calls – formed up around the train and charged the Blazer battalions as they closed in for the kill.
'With the supporting firepower from the trains it was over in minutes, but as the Mutes went down, the wagon-train went up. Nobody quite knows how. The suggestion is a combination of napalm, fuel and explosives taken from the stocks on board. They must have laid it from end to end of the train...
'Anyway, it started a fire-storm that completely gutted her from end to end. Then she just... blew apart.'
'And Mr Snow?'
'No trace. But then there was virtually nothing left of the hundreds who must have died on board. Strange though...'
'Why?'
'When the final explosion occurred, it was accompanied by a severe earth tremor. The earth split four ways, with deep fissures running out towards True Grit and the other trains –'
'Oh, shee –'
The P-G held up his hand to quell Karlstrom's anxiety. 'It's okay, they didn't reach them. But it was enough to throw people to the ground... and there was some structural damage.'
'But nothing serious...'
'We'll get the score-sheet after they've been checked over at Fort Worth. We won't release this news of course. But what I wanted to tell you was this – when The Lady fireballed and the earth split open, a shaft of white light fringed with rainbow colours shot out of the middle. The trains caught it on their cameras. They reckon it was about two hundred feet high. Seemed to flash upwards then vanished –' Jefferson snapped his fingers. 'Curious, eh?'
'Very. Let's hope it was him. What about Cadillac and Roz Brickman?'
'No trace of them either. But since our young hero shows no visible signs of distress we must presume that she, at least, is still alive. A Skyhawk was seen to leave The Lady just before our counter-attack. We know that Cadillac can fly...'
'Ye-ess...'
The P-G raised his eyebrows expectantly but Karlstrom did not respond to the prompt. 'Assuming there was no prior collusion, they must have reached, ahh – how can one put it – an understanding? From what you told me about that young lady's abilities it's unlikely she would allow herself to be coerced.'
'No. But "understanding" might be putting it too strongly. Given the situation she was, quite literally, catapulted into, she may have decided to take the least line of resistance.'
The P-G chewed this over. 'You think she's still working for us...'
'I think we should assume that until there's evidence to the contrary. Brickman did his utmost to warn us that things had gone wrong and both Hartmann and Malone praised his efforts to help save the wagon-train. I think we should give them both the benefit of the doubt.'
'Mmmm, yes, I'm inclined to agree. There's just one thing I'd like to clear up. How, in the light of all this, did she come to be transferred to The Lady?'
Karlstrom spread his hands. 'I can't tell you, although I hope to have an answer soon. As you know I consulted you on this as a matter of some urgency. Wallis was pressing me for a decision. My orders were quite specific. Brickman and Malone were to fly to Red River and give us a complete sit-rep _before any further action was initiated_. I wanted to establish that whatever was left of Hartmann's battalion was still being properly led and capable of organized resistance.'
'Wise move. But you didn't okay the return trip...'
'No. You can imagine how I felt when I received a signal from Wallis confirming that Roz had been landed on The Lady.' Karlstrom gestured frustratedly. 'I just can't understand what got into him. And I still haven't had an adequate explanation.'
The P-G gazed into the fire for a moment then said: 'You're unlikely to get it. Wallis shot himself.'
Karlstrom hid his relief behind a look of total consternation. 'Wha-aat?! When?!'
'A few minutes ago...'
'I don't understand.' Karlstrom added a flash of anger. 'Why wasn't I told?!'
'Calm down, Ben. Isn't it obvious? With Wallis gone and the rest of your team missing, Fargo didn't have a direct line to you. No one else on Red River had the codes to operate that equipment.'
'No, of course...'
'He had to go through CINC-TRAIN. I told them I'd break the news. But it means we'll never know what prompted him to make that decision. Still, these things happen. We've got Clearwater, Brickman's back on the team, Roz – we hope – is well placed to put Cadillac in the frame, Mr Snow may have finally shot his bolt and we've taken out the M'Calls.'
'You're right. It's not all bad news.'
'Except for the fact that it cost us a wagon-train.' The P-G's face clouded. 'Well, we can live with that – and thanks to your clever little scam it wasn't Red River. That really _would_ have been a disaster.'
'Nevertheless we did lose a whole Trail-Blazer battalion.'
'Hartmann's battalion. There were some good men in it but that's how it goes. They ran out of luck from the first day they ran into the Plainfolk. Let's hope their demise will make the other wagon-masters realize there's no mileage to be gained out of supporting lost causes. If his Blazers had been on the ball, that train could never have been taken.' Jefferson's mouth tightened. 'But it seems we won't ever know how that happened either.'
'Looks that way...'
'Never mind. I refuse to be downhearted.' The P-G rose from his chair. 'We may have fumbled a pass but we're still in the game.'
And I'm still on the team, thought Karlstrom, already on his feet. How long, he asked himself, would he be able to conceal the full story of his own involvement in this semi-fiasco?
The chain of administrative orders which had led to the issue of the explosives used by the M'Calls against The Lady had been 'sanitized'. With the train now a shattered, burnt-out shell in enemy territory, no one was going to be checking it for tell-tale clues that, under other circumstances would have pointed to the use of Federation PX and not the black powder used by the Iron Masters.
Unfortunately, other people in AMEXICO had been involved in covering those tracks. Could they be relied upon?
With the order to transfer Roz from Red River to The Lady, Karlstrom knew he was on firmer ground. The message, routed directly from his own computer terminal to the comms-system operated by Wallis, had contained a code-virus which caused the message to self-destruct when it was transferred into memory. When Wallis keyed in the instruction 'SAVE TO DISK', the order concerning Roz disappeared from the screen and – despite the visual confirmation that it had been safely transferred to the hard-disk – vanished into thin air.
Despite the risks, the opportunity to swap Roz for Steve had been too good to miss. Karlstrom didn't want Roz and her so-called psionic powers back inside the Federation. And especially not anywhere near AMEXICO. If she intended to betray the Federation then it was better to deal with her at a distance. If, on the other hand, she was as loyal as she claimed to be, there was always a chance that someone else might use her mental abilities to spy on him.
Someone like his cousin George Washington Jefferson the 31st. Karlstrom, a voracious reader, was aware of an ancient quotation which – as head of the secret organization dedicated to the protection of the President-General – was directly applicable to just such a situation: _Quis custodiet ipsos Custodes?_ – Who is to guard the guardians themselves?
No. Steve Brickman might be devious but he was not a threat. The young man was brave, resourceful and gifted – but he also had a certain weakness which Karlstrom believed could be exploited to keep him in line until – like most pawns – he became expendable.
Five days later, Karlstrom found himself back in the Oval Office with the young man in question. And this time, there was no trace of the Mute whose painted presence had so disturbed the wagon-master of Red River. Brickman, immaculately dressed in a blue wing-man's uniform, clear-eyed and clear-skinned, with his blonde hair trimmed into a regulation crew-cut would have been a credit to any passing-out parade. To have turned this half-breed into a Tracker from head to toe was an amazing feat...
Karlstrom tuned back into what the P-G was saying.
'... while the operation could be said to have misfired, the failure is not yours, Steven. Overall, taking into account that this was your first assignment, we think your performance has been outstanding. Your achievements in Ne-Issan deserve special commendation, and it gives me great pleasure to be the one to tell you that as from today, you have been promoted to the rank of Captain.'
Brickman who had been sitting rigidly to attention, with his parade cap aligned neatly on his knees, jumped up from his chair. 'Sir! I, uhh – thank you sir!'
'Thanks don't come into it. You deserve it. Right, Ben?'
'Yes, sir...' Karlstrom rose as Jefferson came round his desk, took hold of Brickman's hand and shoulder. The fatherly gesture that never failed.
'Steven, I don't want any secrets between us. That's why I'm going to tell you that there were times when we had grave doubts about you. We have never thought you would deliberately betray the Federation but we were worried that your mind might have become contaminated by some of the experiences you have undergone. Experiences that might have affected your judgement – altered your perception of the world we're trying to build.'
Jefferson injected a more cheerful note. 'But that's all in the past, isn't it Ben? This young man has been given a clean bill of health!'
'Absolutely...'
Jefferson firmed up his grip on Steve's shoulder as he accompanied him to the 'stile. 'We are going to give your life a new dimension, Steven. You will find that loyalty, allied to the courage and ability you have displayed is handsomely rewarded. And it is my belief that you will prove worthy of the confidence we have in you. Keep the faith, Steven. Never falter in your devotion to the First Family!'
'I won't, sir!'
When they emerged into the outer suite of offices, Karlstrom turned to Steve and offered his hand. 'Congratulations. How do you feel?'
'About the promotion and everything? It's incredible, sir. But I still feel bad about Roz. If I'd stayed with her –'
Karlstrom cut him short. 'You were following orders. It was my people that fouled up. And the real sickener is we'll never know why. Even so, it's not all bad news. Since you're looking bright and healthy that would seem to indicate she's still alive. Right?'
Steve didn't hesitate. 'They both are, sir.'
'Is Cadillac holding her prisoner?'
'He thinks he is.' Once again he didn't bat an eyelid.
'Good man...' Karlstrom patted Steve's arm. 'Stay in touch.'
'I will, sir. What about Clearwater, sir?'
'Keep up the visits. Unless you have other duties, you have unlimited access, day or night. You're an essential part of the get-well programme.'
'Thank you, sir.'
'Did, ahh – Clearwater give you the good news?'
'Sir...?'
'I see. She didn't. I wonder why? Well, there's no point in keeping you in suspense. The surgeons on board Red River managed to save the baby.'
'Baby...?' The news took Steve completely by surprise. 'I – I don't understand –'
'Ohh... isn't it yours?'
Steve felt totally confused. A confused babble of distant voices filled his brain and were submerged by a roaring sound. He felt the blood pounding through the arteries close to his ears. 'No, sir! I – I mean... how could it be?!'
_It was the President-General who was the sole progenitor of humankind... the father of all life within the Federation_...
Karlstrom smiled. It was not often he managed to unsettle this artful sonofabitch. 'You've only just begun to discover what you are capable of. That's why you're on the Special Treatment List. Do you think everything that's happened to you so far is due to good luck and your winning smile?'
'Uhh, no, I – I had no idea, sir!'
'Well, you've managed to get this far, don't jump the rails. There are interesting times ahead.'
'Will I still be working for, uhh – your department, sir?'
'Yes, you will,' laughed Karlstrom. 'You and I still have a great deal of unfinished biz –' He broke off as his eyes were drawn to someone behind Steve's shoulder.
Steve turned to see a dark-haired woman walking towards them. She wore the silver grey and blue uniform that marked her out as a member of the First Family.
It was the young President of the Board of Assessors that had tried him for desertion. The woman who had stripped him of his wings and sentenced him from three years to life in the A-Levels. He had guessed that she was Family during the trial and he was right. The trial, the sentence, the early reprieve, the chance to win back his coveted wings... it had all been a set-up. Nothing was what it seemed.
Steve jumped to attention and threw a parade-ground salute as the woman with the grey eyes, the oval face and the wide firm mouth reached him. Her sleeves carried the stripes of a commander topped by the inverted chevron – an exclusive mark of the First Family which conferred automatic seniority over the commanders of ordinary Federation units.
She acknowledged his salute with the casual assurance of someone who knows there is not the slightest chance of being hauled up on a charge of indiscipline and turned to Karlstrom. 'Ben, I'm sorry! I was held up!'
'That's okay. Let me introduce you.'
The grey eyes fixed on Steve's. 'We've already met, haven't we Captain?'
'Yesssir-ma'am! I believe so!' News travels fast, thought Steve. The extra stripe he'd been awarded was not even on his sleeve.
'Yes, but he doesn't know who you are.' Karlstrom did the honours. 'This is Commander Franklynne Jefferson. She will be your host and guide over the next few days.'
'Yess-SURR!'
'Fine. We've completed the formalities. Now relax. You're among friends.' Franklynne Jefferson offered her hand. 'It's Steve, isn't it?'
'Yes, sir-ma'am.'
'Oh, dear...' She sought help from Karlstrom. 'What does one have to do to get this man to unbend?'
'Give him time...'
She tried again. 'We're going to a place called Cloudlands, Steve. And by the time we get there, I shall expect you to call me Fran. Think you can manage that?'
Nothing in his wildest dreams could have prepared Steve for what was to happen following that encounter outside the Oval Office. After an elevator ride which his stomach told him was in an upwards direction, Steve stepped out into a large lobby with several exits – the walls and doors of which were covered with panels of wood similar to Karlstrom's office.
The first surprise was finding the lobby manned by two smooth-skinned Mutes dressed in dark clothes cut in a style Steve had never seen before. Both Mutes had vari-coloured skins but were straight-boned, with no cranial lumps. Yearlings. One male, one female.
The male – whose greying hair was cut short and brushed flat across his head – wore three pieces of black clothing – trousers, tucked into calf-length boots, and a long kind of tunic, with a tighter-fitting tunic with a V-neck underneath. This garment, buttoned down the front, reached to just below the waist and had broad diagonal gold stripes woven into the black. Instead of the universal T-shirt, he wore a shirt of white material drawn into a high tight band around the neck, with a looser, curly piece of the same material running from the throat down into the V-neck of the gold-striped tunic, and cuffs that poked out from the sleeves of the long, open garment covering it.
The clothes of the female Mute were equally strange. She wore a cap of white cloth which covered most of her hair, and an elaborate curly-edged spotless white apron – a fancy cousin to the straight-cut style worn by kitchen staff on the mess-decks – and fastened at the back with a wide bow. Underneath she wore a black, sleeved garment puffed out at the shoulders but tight on the rest of the arm. The tunic had a similar high collar but in black with a frilly white liner. The waist was drawn in then came over the hips in a slim bell-shape and went all the way down to the floor.
Extraordinary...
Fran addressed the male Mute. 'Joshua, this is my guest, Captain Brickman. Will you help him change into the uniform I gave you this morning, then show him upstairs?'
The Mute inclined his head respectfully. 'My pleasure, ma'am.' He indicated a door on the far side of the lobby with a white-gloved hand. 'This way, Captain...'
Steve hesitated, seeking guidance from Fran.
She smiled. 'Go on. Off you go. I'll see you later.'
The changing room had a marble floor and walls and the luxurious fittings and furniture he had seen in the Oval Office and the adjoining suites and corridors of the White House. Steve showered and dried himself on large soft white towels then emerged to find a new set of clothing laid out of him. The pale grey briefs were familiar enough, the T-shirt was replaced by a sleeveless under-garment with a curving neckline. Mid-grey socks. A white shirt similar to the one Joshua was wearing but with just a neat high collar band. So far so good...
Next came a pair of mid-grey trousers with a line of yellow braid running down the outside seams. Then a long grey tunic with an overlapping rear split and a high collar that buttoned down to the waist. The sleeves were decorated with captain's rank stripes in the same yellow braid as the trousers. And then there were the boots, of soft black leather.
Joshua coughed politely. 'The trousers go outside, Cap'n.'
'Got it. Thanks...' Steve corrected the mistake then reached for the grey stetson with the yellow crossed swords parade badge of the Trail-Blazer Division and placed it carefully on his head with the aid of a mirror.
'If I may, sir...' Joshua adjusted the tilt, then went over to the table to fetch the sword and helped Steve fasten it around his waist. It resembled the sword on his hat badge but Steve had never seen a real one before.
'What is this?'
'It's a cavalry sword, Cap'n.'
'Looks pretty old.'
'It is. Been in the family for centuries.'
'You look pretty old too, Joshua. How long you been here?'
'Me, Cap'n? I was born here in Cloudlands.'
Steve checked himself in the mirror. 'I have to tell you this feels very strange. What is this outfit I'm wearing?'
'That's what we call Confederate grey, Cap'n. The uniform of a southern officer and gentleman.'
Steve shook his head. 'Can't say I'm any the wiser.'
'That's as may be, Cap'n. But you look mighty fine to me. And I'm sure that Missy Fran, she's going to be real proud to be seen walking on your arm.'
Joshua ushered him back into the lobby and led the way towards a set of double doors which, when opened, revealed a flight of wide marble steps. Inviting Steve to follow, he led the way to a similar set of doors at the top then through into a spacious room filled with light from tall white-framed windows. Huge clusters of what looked like ice crystals hung from the high, sculptured ceiling. There were carpets covering sections of the gleaming wood floor, ornate chairs covered in richly coloured cloth, a magnificent marble fireplace, framed portraits and mirrors on the patterned walls.
Steve turned full circle, head raised, mouth open like a first time tourist in the Big Apple. He gestured towards the trees and flowered gardens beyond the windows. 'Columbus! Is that computer-generated?!'
'I don't understand, Cap'n.'
Steve stepped towards the open glass panelled doors and peeked outside. There was no screen. What lay outside was part of the overground. Neat, sculpted, ordered – but beautiful nevertheless.
Joshua smiled at Steve's evident bewilderment. 'Make yourself at home, Cap'n. Take a seat – or maybe you'd prefer to walk on the verandah.' Joshua indicated a polished table with bottles and jugs of liquid and cups made of the same sparkling ice-crystal material. 'May I offer you some refreshment?'
'No – but those cups and bottles... are they made of clear plastic?'
Joshua chuckled. 'No, Cap'n. Them cups you're referring to is what we call glasses. Same stuff as in them there windows only this –' He picked one up and turned it so that it caught the light, '– is much prettier and finer. It's what they call cut crystal – see?'
'Yeah, thanks, Joshua. Must seem kinda stupid.' Steve swept a hand around the room. 'This is all so new.'
'New?!' Joshua chuckled again. 'This ol' place's been standin' close on two hundred years!' The Mute indicated a circular button push on the wall. 'With your permission, Cap'n, I'll leave you be. If you want anythin' just ring.'
'Oh, yeah – thanks...'
The Mute bowed. 'Missy Fran'll join you shortly. She'll be in the yellow I expect.'
Joshua, who had served Fran's family since she was a child, was familiar with the routine that accompanied the elevation of tall, strong-shouldered young men – Fran called them her 'beaus' – from the subterranean world to the elegance of Cloudlands.
When she appeared, she _was_ dressed in yellow, but the transformation was so startling she was not immediately recognizable. The silver and blue uniform had been replaced by a frilled and layered costume which gave prominence to her breasts, hugged her rib-cage, squeezed her waist then flared outwards with draped folds to the ground. An outfit which, a century and a half before the Holocaust, was known as a 'walking dress'.
She also had a lot more hair. The neatly combed bob had been augmented by matching braids, ringlets, and a soft bun extending onto the nape of her neck. Her face looked softer, her eyes larger, her lips redder.
Fran twirled around in front of Steve. 'How do I look?'
Steve eventually found his voice. 'Amazing...'
'And so do you, Captain Brickman. Raise your right elbow.'
Steve offered it to her awkwardly.
Fran unfurled a matching parasol, took his arm and led him towards the verandah. 'Come... walk with me.'
_Yes sir-ma'am_...
Anyone with access to the cinematic archives of the 20th century – a privilege enjoyed by members of the First Family – would have immediately recognized Steve's surroundings. Walking into that light-filled room with its sumptuous furnishings was like entering one of the interior scenes from 'Gone With The Wind'.
Like the closely-guarded domains reserved for the top brass of pre-Holocaust Russia, Cloudlands was a vast chunk of real estate reserved for the exclusive use of the First Family.
But this was no space-age colony shielded from overground radiation by protective bubbles. On these landscaped acres, the First Family had lovingly recreated the mid-19th century sugar-plantation splendour of the Deep South. Pristine white mansions, with colonnaded porticos, nestled among trees and lakes, surrounded by immaculate lawns, formal gardens, arbours, fountains, drives and shaded avenues, tastefully furnished in a style that echoed the French colonial past of Louisiana and Mississippi, and staffed by an army of servants, grooms and retainers; liveried Mutes – the 29th century equivalent of the negro slave.
There were no wheelies here. Horses, and horse-drawn carriages with Mute drivers carried the privileged inhabitants wherever they wanted to go. Pride of place was given to the railway with its hand-built replicas of period locomotives and rolling-stock spanning the glorious days of steam. But they weren't the only anachronisms: the background lighting was powered by electricity, tv screens and computer keyboards were artfully concealed in antique cuboards and desks, and the open skies were patrolled by First Family wing-men flying silver Skyhawks.
On the ground, however, authenticity was the keynote. The men were dressed in uniforms of the Confederate Army and the women as 'southern belles', but both changed clothes to suit the occasion or the time of day, donning what they called 'evening dress' when the sun went down.
For the men this meant a more decorative uniform in sober colours, or dark 'civilian' clothes; the women emerged in off the shoulder dresses with deep necklines that exposed the tops of their breasts, layered elbow-length sleeves and long gloves. The lower parts of these evening dresses were even more extravagant in their detailing and dimensions – wide, sweeping skirts with trailing extensions at the rear, supported by layers of hooped petticoats.
Fran was an immensely agreeable and informative guide but it was a lot to take in all at once – even for Steve.
The contrast with the uniformed monotony of the underground Federation could not have been greater. Steve had to keep reminding himself that this wasn't a dream. This was for real – and yet this reality was tinted with a kind of madness.
How long had this been going on? Was this what countless generations of Trackers – including Poppa-Jack – had sweated, slaved and died for? So that an already over-privileged elite could enjoy a lavish fantasy existence while the rest of the population lived in neon-lit concrete burrows where the biggest event in the calendar was a trip to the walled-in acres of John Wayne Plaza? Compared to Cloudlands, the fabled Plaza was nothing more than a marbled prison exercise yard.
Steve kept his feelings to himself, but with the knowledge of who he now was, he couldn't help identifying with the smooth-boned Mutes who did the fetching and carrying, who were ever-present but whose quiet-mannered discretion rendered them almost invisible. Part of the woodwork. Compared to the Iron-Feet, their Plainfolk brothers in Ne-Issan, these Mute yearlings were in a gilded cage, but one day they too would be free.
_Oh, yes, brothers_...
But now it was time to watch and to listen. There was much to see and a great deal to learn...
Like on that first evening when Steve found himself invited to sit next to Fran at a long cloth-covered table decorated with bowls of flowers and silver candlesticks, sparkling glassware, polished metal knives and forks and ceramic plates with coloured patterns round the rim. The chairs around the table were occupied by twenty men and women the nearest of which addressed Steve with an easy familiarity. The food was good, the wine – quite different in taste to _sake_ – was agreeably liberating, the company convivial and when Fran revealed that he had been to Ne-Issan on 'Family business', they listened with genuine interest to his descriptions of life in the Eastern Lands.
By the end of the evening, his feelings of hostility had waned considerably. Steve had not abandoned his desire to pierce the innermost secrets of the First Family. The truth had to be revealed. After Mr Snow's revelations, they were not just the top layer of an oppressive regime, they were the enemy. They and everything they stood for had to be swept away, but that was an immense undertaking that could not be accomplished overnight. In the meantime, however much he might deplore the flagrant unfairness of the system, it would be foolish not to take advantage of what was on offer...
Around eleven pm, the after-dinner conversation groups broke up. As the guests bade each other goodnight and departed to their own rooms or to other houses, Joshua, the Mute servant, led Steve upstairs to the large bedroom which had been set aside for his use. The curtained windows opened out on the front lawn, and through them, Steve saw a coach and pair, with bright yellow lamps, rattle away down the curving gravel drive. Animated voices floated up from the porch below. He caught a glimpse of someone in a yellow dress and thought he heard Fran laughing...
A log fire like the one in the Oval Office blazed cheerfully in the hearth. The large bed, which had a post at each corner and a cloth canopy, looked soft and inviting. There would be no bug-uglies in that.
After Steve had donned a bathrobe, Joshua took his uniform away to be freshened and pressed, leaving what he called a dressing-gown and a nightshirt laid out neatly on the foot of the bed. Crystal decanters of wine, peach brandy and a smooth, amber-coloured alcohol called Southern Comfort stood on a silver tray with a selection of glasses.
Steve went into the bathroom and took a shower. As he stood under the warm spray, his body suffused with a sense of well-being, he reflected on the time he had spent among the Mutes and the Iron Masters, the pleasures and privations of life on the overground and within the Federation.
It seemed incredible that four different worlds with such contrasting life-styles could co-exist within a few hundred miles of each other: hi-tech gadgetry and stone-age savagery, total freedom and slavery, individuality and restrictive conformity, equal rights and overbearing discrimination – sexual _and_ racial, rigid hierarchies and relaxed anarchies. Why did people have to choose one over the other? Why was there no middle way?
When Steve emerged from the shower, he found Fran occupying the left hand side of the bed. Her head and naked shoulders were propped against two of the four over-sized pillows. One hand held a glass of brandy, the other held a smoking reaf. The soft insistent beat of a blackjack tape floated out of a hidden speaker. Below ground, people went to the wall for peddling this kind of shit.
'Surprised...?'
'Not really.' Steve searched for a suitably ingratiating follow-up. 'I can't think of a better way to end an unforgettable day.' _Yukkkk-hhh_...
'It's just the beginning, Stevie.' Fran patted the empty place next to her.
Steve walked around the bed and reached for the nightshirt.
'You won't need that.'
'Just clearing the decks.' He gathered up the long loose shirt and the dressing-gown and put them on the low wooden chest at the foot of the bed.
Fran proved to be an ardent sexual partner. Their coupling reminded him of his solitary encounter with Donna Lundkwist. Donna whose life had ended with a kiss and a knife in her throat. And like Donna, Fran started out making all the running, but when Steve delivered the goods, she ended up tender and grateful.
'How did it feel, Stevie?'
'Well, ma'am, I –'
'Fran!!'
'Sorry! I –'
'Never jacked up a member of the First Family before.' Fran treated him to a conspiratorial smile. 'No need to answer. I know everything there is to know about you.'
_I wonder_...
She pulled his head closer and whispered in his ear. 'What was it like with her – Clearwater? The same? Better?'
'No... just different.'
'In what way? What did she do that I didn't do?'
'Nothing, it was –'
'Tell me! Ohh, I can feel you getting hard again! Are you thinking about doing it with her? Oh, come on! Give it to me! Oh, yes! She squeezed you like that, Stevie?'
_Almost, almost... Christo!_
'Why was it different? Was it because she said she loved you? Is that it, Stevie? Do you want to be loved? Would you like me to say I love you?'
That question, that word, on _her_ lips, sent a shiver through him. He raised himself up and began to withdraw.
Fran locked her legs across the small of his back, and hung on tight round his neck, thrusting her pelvis hard against his. 'No! Stay there! If you knew how long I've waited for this! Hooh, baby! C'mon, give it to me! Gimme all of it!'
Steve suppressed the feelings of self-loathing and betrayal.
_Yessir-ma'am. If this is what it takes to get where I want, you GOT it!_
An hour or so later, when they'd screwed each other to a standstill, and she'd explored every inch of his body with her lips, tongue and fingers, they lay in each other's arms, their skin glazed with sweat. Steve was having difficulty staying awake. Fran brought him back to life with a playful bite on the shoulder.
'Jeezuss!'
'It's all right. I haven't drawn blood.' Fran kissed it better. 'Mmmm... you smell and taste like a man should.' She offered her throat. 'Taste me.'
Steve took a sample. Honey with a dash of salt...
'Do you like it?'
'Yeah. Tastes good.' He put his cheek against hers and buried a yawn in the pillow.
Fran took hold of one hand and slid it between her thighs. 'And did you like this?'
'Delicious...' _Didn't this dill ever stop?!_
She twisted her body around so that she was looking down on him. 'Did it shock you – me talking about love? It's a word the Mutes use, isn't it?'
'Yes. But it wasn't so much shocking as, well... unexpected.'
'Because of who I am?'
'That was part of it, yes.'
'Call me Fran. Say my name! Keep your hand there and whisper it in my ear.' When he had repeated it several times with as much feeling as he could muster, she said: 'I love your voice. You know that? Can't you feel what it does to me?'
Steve didn't answer. He knew he was in bed with a member of the First Family. Not just _any_ member of the First Family – a Jefferson. This kookie was related to the President-General! This couldn't be just a reward for services rendered. What the hell was going on?
'Do you think I'm beautiful?'
'Uhh, well, yehh, y'know –'
'That's another Mute word, isn't it? I bet you know lots, mmh?'
'I was out there for quite a while...'
'You'll find we use lots of Mute words up here, Stevie. And quite a few I bet you've never heard of.'
'Always willing to learn...'
'Good. I'll teach you things you've never dreamed of.' Fran brought her mouth to his for a last tender, teasing kiss. 'Did you enjoy doing it with me?'
'Yeah, it was terrific' _What the heck was he supposed to say?_
'Would you like to do it again?'
_Aww, jeezuss_... 'You mean now?'
'No. But soon. And _often_. She gave his starting handle an affectionate squeeze then leapt out of bed and went into the bathroom.
Steve heard the swish of the shower. He pulled the bedclothes over his nakedness and reflected on what was happening to him and what might happen next. The session with Fran, although enjoyable on a purely physical level – which in the past was all that any Tracker would expect – left Steve feeling empty and vaguely ashamed of his opportunism.
In the Federation, the sexual act was one of the few things to which no guilt was attached. No involvement was expected by either party beyond the brief physical proximity required for the act to take place. Mutual consent was all that was needed; desire, the need to temporarily relieve the basic sexual drive, was the only emotional element.
It had been different with Clearwater. The intensity of his feelings – feelings which she returned in equal measure – had given a new dimension to their physical relationship, enriching what, for him, had previously been a meaningless exercise involving his body but which, until then, had never engaged his heart and mind.
Jacking up Fran had merely served to remind him of what he was missing. Clearwater had never claimed exclusive rights to his body. Indeed, for her, possession of the body was of minor importance. What counted was to whom you gave your heart and soul. And it was true – although it hadn't stopped him hating the idea of Clearwater allowing Consul-General Nakane Toshiba to instal her in his island love-nest.
Maybe it _was_ the will of Talisman. Her liaison with the jap had certainly led to their escape. But it was good ol' Brickman S.R. who had blown the Consul-General out of the sky. Watching his smouldering body fall to earth had been a sweet moment. Yes...
Clearwater would not have felt the same need for revenge against a third party. She had never probed the intimate nature of his relationship with Roz. She was too wise. She had a serene confidence in the power of Talisman, in the pre-destined, ordered nature of existence.
In her world, human frailty – except where it flagrantly transgressed a blood-bond sealed before the elders – was overlooked or generously forgiven. What counted was the purity of the spirit, the nobility of the soul. Which was fortunate, because it let Steve off the hook.
Trackers were totally promiscuous, but due to the nature of their society, the word 'promiscuity' had been stripped of any moral connotation. In the pre-Holocaust world, copulation may have been raised to an art form and accorded the status of an inalienable civil right for commercial and political reasons, but in the Federation its importance as an essential activity was on a par with evacuating the bowels and was usually discharged with a similar lack of ceremony.
By demoting sex while allowing a continued free-for-all, and by removing the word-concept 'love' from the Tracker vocabulary, the First Family effectively eliminated the basis for personal relationships between individual men and women. Widespread and persistent sterility had already destroyed the nuclear family; what remained was a collective identity based on the squad, the block, the battalion, the division. Loyalty, a sense of comradeship and allegiance, was directed upwards through the system towards the figure at the pinnacle of power – the President-General.
Because of this, Steve was not burdened by guilt but he felt diminished. He'd been given a privileged glimpse of the system from the top down – a system that was not only harsh and unjust at the bottom, was not only built on lies, but whose leaders now stood revealed as corrupt and crazy.
For centuries they had held out a dream of a bright future and here they were living in a self-deceiving dream-world that belonged to the distant past! And his sense of shame was increased by the knowledge that if Fran required his services again, he would answer the call without hesitation. Would do whatever had to be done.
The idea that jacking up a high-flying member of the First Family could lead to advancement seemed, on the face of it, preposterous, but if that was part of the deal – what the hell?
The higher he got up the wire, the more chance he would have of getting even with those who had helped to shaft him. And to that list were now added those who had manipulated his life and twisted his mind. Yes... given time, he'd get them all...
When the hospital orderly left after cleaning her room, Clearwater noticed that the wheeled table carrying the computer terminal had not been pushed back into its proper place. It now lay within reach. Leaning sideways, she stretched out her good left arm, caught the edge of the table with her fingertips, pulled it towards her then manoeuvered it round until the keyboard and screen faced the bed.
Thanks to Steve's winter schooling she could now read and write. And on the wagon-train and now, in her new home which her senses told her was not far underground, she had watched medical staff tap the keys to call up or record information. This machine was part of a spider's web of power that gave life to the Federation. And at the centre of this web lay something or someone called COLUMBUS. She knew this because the Cloud-Warrior had talked boastfully of these things to Mr Snow.
Before his eyes and heart had been opened...
She studied the keyboard and pressed the HELP button.
Letters appeared on the screen: DO YOU WISH TO (A) TRANSMIT DATA (B) RECEIVE DATA (C) USE MATH FUNCTION (D) CONSULT LOCAL ARCHIVES? – SELECT LETTER AND PRESS ENTER.
Clearwater selected (A) – Transmit Data.
The screen cleared and a new message appeared: ENTER NAME OF RECIPIENT, UNIT, DEPARTMENT OR DIVISION AND ADDRESS CODE OF RECEIVING TERMINAL.
She tapped out the letters carefully: C-O-L-U-M-B-U-S...
There was a pause then: THE CENTRAL CORTEX CANNOT BE ADDRESSED FROM THIS WORKSTATION WITHOUT AUTHORIZATION CODE. INSERT ID-CARD OR ENTER PASSWORD.
T-A-L-I-S-M-A-N...
THE SYSTEM RESPONDED: PASSWORD NOT RECOGNIZED. ENTER FULL NAME AND NUMBER.
T-A-L-I-S-M-A-N...
A NEW MESSAGE FLASHED ON TO THE SCREEN: UNAUTHORIZED SYSTEM ACCESS IS A LISTED OFFENCE. THE LOCATION OF THIS WORKSTATION HAS NOW BEEN REPORTED. TO AVOID A MORE SERIOUS CHARGE YOU SHOULD REMAIN THERE AND AWAIT THE ARRIVAL OF INTERNAL SECURITY.
CLEARWATER SIGNED OFF: G-O... T-O... H-E-L-L...
An insistent bleeper alarm began to sound. Not deafening, but loud enough to be heard in the corridor outside.
Clearwater pushed the trolley away from the bed and pretended to be asleep as a medical orderly came in and swore quietly under her breath. 'The idiot... why can't he leave things the way he found them?'
Another medic poked her head round the door. 'Trouble?'
'That bloody cleaner!' said the first. 'Look at the way he's left this! And I see this thing's playing up again.' She cleared the screen, switched the terminal off and straightened up the table then came to check on Clearwater as her eyes fluttered open.
'What's happening...?'
'Nothing. Relax. Just let me fix your pillows... There. You all right?'
'Yes...'
'Good. Go back to sleep.'
Contrary to the stern warning displayed on the screen, the incident had not been reported, and the bleeper alarm was programmed to turn itself off after sixty seconds. Centuries of experience had shown that only a minority of youthful pranksters were frozen into fearful immobility by the order to remain by the computer terminal. Everyone else promptly left the scene.
Like the tacit approval of rainbow-grass, the computer network controlled by COLUMBUS also acted as a safety-valve. The number, provenance and frequency of nuisance calls were noted for statistical purposes but the manpower and other technical resources needed to follow up the thousands of violations were simply not available.
The efforts of the Federation's computer security units were directed towards the apprehension of those trying to penetrate or corrupt the data and control circuits as part of a purely criminal conspiracy, and 'moles' – political subversives living inside the Federation but outside the eco-system maintained and policed by COLUMBUS: people who were trying to access services to get what they needed to stay alive while evading the checks and controls that would have brought the Provos down on their necks.
For the moment it did not matter that the first attempt to contact COLUMBUS had failed. The terminal in her hospital room had enabled Clearwater to sense its all-pervading presence, its purpose, its power, its intelligence.
This was why she had been sent into the dark world of the sand-burrowers. First, she had to regain her strength and then, in a way which would be revealed to her, she had to make contact with this soulless entity and destroy it.
Karlstrom watched the President-General walk to the high curving window behind his blue leather-topped desk.
In front of the centre curtains, a large eagle with outspread wings, carved out of gleaming rosewood, was poised on a waist-high plinth. Between the eagle and the blue curtains were two crossed poles draped with the flag of the Amtrak Federation and Old Glory.
Jefferson rubbed his hand over the eagle's head as one might touch a good luck charm, gazed briefly out of the window at a stunning view of Kentucky Blue-Grass country then waved Karlstrom into his usual seat.
Karlstrom hovered, waiting for the P-G to sink his solid rear end into his high-backed chair. In the Oval Office no one sat while the P-G was standing.
'Sit down, Ben. Let's skip the protocol. I need to walk around for this one.'
Karlstrom subsided, and watched Jefferson gaze out across the wooded slopes towards the sunlit hills. The clarity of the image was amazing. This was the way America had once looked. Green and beautiful.
'Ben, I'm going to tell you something which may put your mind at rest. You must have wondered why we've invested so much time and resources in young Brickman and his sister – which, by the way, she isn't. With people dying for the Federation every day, two more, two less – what's the difference? Well, they're something of an exception and I think it's time you joined the club.
The P-G began to pace slowly to and fro, circling the desk and Karlstrom's chair. Karlstrom followed him with his eyes.
'As you know, we've been breeding Mutes for experimental work at the Life Institute for close on a hundred and fifty years. The stated objective was to find the genetic key to their longevity and immunity to radiation in the hope that we could transfer those benefits to our own people. If we could increase the average life expectancy from forty to sixty years it would give us a fifty per cent increase in our skill base and productivity – and that would release more people for overground operations.
'Since you're one of the Family and bright enough to be the head of AMEXICO, it probably won't come as much of a surprise when I tell you that over the years we've fed a large number of what we call farm-boys into the units engaged on overground operations – the Trail-Blazer Division, QMGC, FINTEL, SIG-INT and AMEXICO. Not yearlings – super-straights, smooth-boned and clear-skinned. Just like you and me.
'When I say a large number, don't get the wrong idea. The percentage of farm-boys – and girls – in these units has always averaged less than ten per cent. We've always put the cream of the crop through the Flight Academy. The best of the rest have gone onto the wagon-trains. That's why people like Brickman have performed so well.
'The overground is in their blood. They can handle the vast open spaces and they don't pull TRICS. But sometimes the conditioning fails and they get the urge to cut and run. They become renegades. But they're the lucky ones. Ordinary cee-bees who go over the side eventually succumb to radiation-sickness.'
'But we don't...'
'No. But then we're Family.'
Karlstrom nodded. 'Right. So the overground is still radioactive.'
'Oh, yes. The level of contamination is less than it was a century ago, but it's still dangerously high. Even if they could overcome their fear of open-spaces, extended exposure would prove fatal for the majority of our present population.' The P-G smiled. 'Did you think this was just another lie – like pinning the blame for it on the Mutes?'
'No,' said Karlstrom. 'But it's a useful control mechanism. If the atmospheric radiation dropped to a safe level tomorrow, I certainly wouldn't tell anybody, would you?'
The P-G smiled again. 'That's why you and I get on so well, Ben.' He sat down in his high-backed chair, laid his forearms on the table and clasped his hands together.
'What I have to tell you relates to OPERATION SQUARE-DANCE. Some of our home-baked super-straights have also been "gifted".'
'Steve and Roz Brickman...'
Jefferson nodded. 'They're among the most outstanding examples to date, but the research programme has been running for several decades. And as a result of intensive investigation of their physiology down to the molecular level and beyond, we've discovered certain "markers" in their genes. We still don't know the how and why of Mute magic but we can now identify those individuals who have the potentiality to become wordsmiths, summoners and seers.'
'Or all three...'
The P-G nodded approvingly. 'You've got it in one, Ben. We believe we now know the genetic markers that the Talisman would have to possess. Now that does not necessarily mean that someone with these markers will _be_ the Talisman, but they would have the _potentiality_.'
'I think I see where this is leading...'
'Do you? I wonder. We've known about Steve and Roz's genetic make-up since Day One, and now we have Clearwater's. Tissue samples were flown to the Life Institute soon after the Red River medics got her on the operating table. The unknown element is Cadillac. But he may still fall into the net. Neither Steve, Roz nor Clearwater have all three markers but we ran their data through COLUMBUS and fed in some variable combinations for Cadillac.
'The result was two interesting matches. Steve and Clearwater could produce a child with the potentiality to become the Talisman, and so could Roz and Cadillac. Both children _could_ possess all three "gifts".'
'And we already have one of them under our control...'
'That's right.'
'And you think that Cadillac may pair off with Roz...'
'It's not inevitable, but it's a possibility we have to consider. We don't know the full story behind her unauthorized transfer to the The Lady. She may have engineered it. The rational part of me says it doesn't make sense but we can't ignore the accuracy of the Talisman Prophecy. And its force, Ben. We're up against something that is... beyond our present comprehension.'
The P-G brushed aside the metaphysical dimensions to the problem and got back to basics. 'It's quite possible to imagine Cadillac jacking up Roz just to get even with Brickman.'
'Yes... interesting situation.'
Jefferson studied Karlstrom. 'What plot are you hatching now?'
'I was wondering if we could get Brickman to open up the way Roz did.'
'You mean give us a map reference, plus what she was thinking and feeling?' The P-G considered the possibility. 'There may be no limit to what he might do – given the right inducements.'
'I thought we were already providing them.'
The P-G responded with a tongue in cheek smile. 'Fran has always had a weakness for young studs like Brickman. But if he proves more than a passing fancy we can always build her into the package. As you know, the Family is always prepared to support worthy causes.'
'Indeed...' Karlstrom expanded his initial suggestion. 'If Brickman _can_ establish contact, there's a chance he'll be able to give us a rough fix. Now that we've eliminated the clan, they're on their own. We can snatch them any time we want. Failing that, Brickman should – at the very least – be able to tell us if she's pregnant. Given their past relationship his reaction to that news should be very interesting.'
'Go on...'
'The child that Clearwater is carrying. Is it possible to run those tests you mentioned while it's still in the womb?'
'That's something I'd have to check up on. But for the moment, let's say yes.'
'Then if the test proves that Brickman's child _doesn't_ have all three markers, it's possible that Roz – if she became pregnant – might give birth to the Talisman.'
'She might. I think I know what you're going to say but tell me anyway.'
'Psychosomatic wounding. The involuntary telepathic link which caused Roz to share the mental trauma and enabled her body to mirror the wounds suffered by Brickman. _Real_ wounds – even if the phenomenon was only temporary. If we wanted to eliminate Roz and her unborn child all we would have to do is kill Brickman.
'If – for the sake of argument – he fell down one of the deep ventilation shafts and her mind and body shared the experience, there's no way that child could survive even if, by some miracle, she did. The shock would cause her to abort.'
'You're right. That's worth bearing in mind. However I don't think I've explained why we've developed these conditioning techniques and what we've aiming to do. These farm-boys, the smart Mutes we're raised... the programme is designed to turn their heads around, to change their whole nature, to make them into _Trackers_. To own them, body and soul – so that even if they somehow discovered they were Mutes, they would still remain loyal to the Federation and the Family.'
Jefferson stood up. 'And despite the odd mishap, we're almost there. Earth magic still eludes us. Clearwater may help us with that. But we've got the secret of their longevity, their resistance to pain and we're close to reproducing the brilliance of their wordsmiths and the ability to read the stones.'
'We can take that Mute clay and mould it any way we want. We can transform it into a _human being_. That's why we been searching for the Talisman, Ben. We don't want to kill him. We want to make him one of us.'
## A Note on the Author
**Patrick Tilley** was born in Essex in 1928, but spent his formative years in the border counties of Northumbria and Cumbria. After studying art at King's College, University of Durham, he came to London in 1955 and rapidly established himself as one of Britain's leading graphic designers. He began writing part-time in 1959.
In 1968 he gave up design altogether in favour of a new career as a film scriptwriter. Work on several major British-based productions was followed by writing assignments in New York and Hollywood. His books have been translated into several languages, and have achieved cult-novel status.
**Discover books by Patrick Tilley published by Bloomsbury Reader at**
**www.bloomsbury.com/PatrickTilley**
Fade Out
Mission
The Amtrak Wars: Cloud Warrior
The Amtrak Wars: First Family
The Amtrak Wars: Iron Master
The Amtrak Wars: Blood River
The Amtrak Wars: Death Bringer
The Amtrak Wars: Earth Thunder
For copyright reasons, any images not belonging to the original author have been
removed from this book. The text has not been changed, and may still contain
references to missing images.
This electronic edition published in 2013 by Bloomsbury Reader
Bloomsbury Reader is a division of Bloomsbury Publishing Plc, 50 Bedford Square,
London WC1B 3DP
First published in Great Britain 1989 by Sphere Books Ltd.
Copyright © 1989 Patrick Tilley
All rights reserved
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Home > General > ARAB LEAGUE TO DISCUSS YEMENI DEVELOPMENTS IN MARCH
ARAB LEAGUE TO DISCUSS YEMENI DEVELOPMENTS IN MARCH
admin February 23, 2015 General
CAIRO, Feb 23 – The General Secretariat of the Arab League said the latest developments in Yemen will be discussed within the agenda of the 143rd session of the League's Ministerial Council next month.
"It was agreed to discuss the latest developments in Yemen within the agenda of the regular session of the Arab League Council at the level of Arab foreign ministers to be held in March 09 – 10 in Cairo", Arab League's Deputy Secretary-General Ambassador Ahmed Ben Helli said in a press statement.
Setting this date after the postponement of the previous ministerial meeting, which was dedicated to discuss the developments in Yemen, aimed to giving the opportunity for the efforts being made currently to reach a settlement to the existing crisis in the country, Ben Helli explained.
He said the Arab League's Secretary-General Nabil Elaraby is in touch with the envoy of the United Nations Secretary-General to Yemen Jamal Benomar to follow the developments and the efforts exerted to get out of the current crisis.
Meanwhile, Benomar has denied a report published by Reuters about a possible United Nations' role in the departure of President Abd-Rabbu Mansour Hadi from Sana'a to Aden.
"The United Nations confirms it has no relations to President Hadi's departure," said Benomer's office in a press release.
The release called media to be accurate while reporting news and to communicate with the UN office in Sana'a before publishing any news related to the UN or the UN Secretary General Advisor on Yemen.
On related development, the Supreme Revolutionary Committee (SRC) discussed developments on national arena and took necessary decisions.
The committee dealt with the way Hadi's departed from his residence in Sana'a to Aden.
The Circumstances of this incident confirm that Hadi resignation from his post was an attempt to drag the homeland to collapse, said the committee. It said that Hadi's departure to Aden serves the interests of the foreign powers.
The committee said that the Constitutional Declaration was a right move.
The SRC denied reports that Hadi was under a house arrest posed by People's Committees. The committee also stressed that Hadi's house was not encircled by People's Committees. It said that since he announced his resignation, Hadi used to receive many political personalities, military commanders and representatives of political parties as well as international officials, the last one was the UN envoy Jamal Benomar.
SOURCE: SABA
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جامعة حمد بن خليفة تطلق برنامج الدكتوراه المهنية في القانون (JD) وتفتتح كليتها الجديدة للقانون
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دياجون ماغنسيوم Diagono "E" Magnesium:أول مصنع لساعات Watch-Vault الذكية الفاخرة
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\section{Introduction}
In Riemannian geometry, various regularity results for curvature tensors may be obtained via harmonic coordinate systems. We recall the basic idea following \cite{Besse}, \cite{DK}, \cite{Taylor_isometries}.
Let $(M,g)$ be an $n$-dimensional Riemannian manifold, and let $(x^1, \ldots, x^n)$ be a local coordinate system. The Ricci tensor of $g$ has the expression
$$
R_{ab} = -\frac{1}{2} \Delta g_{ab} + \frac{1}{2} (\partial_a \Gamma_b + \partial_b \Gamma_a) + \text{lower order terms,}
$$
where $\Delta$ is the Laplace-Beltrami operator, $\Gamma^l = g^{ab} \Gamma_{ab}^l$, and $\Gamma_l=g_{la}\Gamma^a$. One also has
$$
\Gamma^l = -\Delta x^l.
$$
If the coordinate system is harmonic, in the sense that $\Delta x^l = 0$ for all $l$, then the Ricci tensor becomes an elliptic operator. Thus by elliptic regularity, if the Ricci tensor is smooth in harmonic coordinates, then also the metric is smooth in these coordinates. In particular, if the Riemann curvature tensor of a low regularity metric vanishes, this implies that the metric is smooth in harmonic coordinates and thus locally flat by classical arguments.
The Ricci tensor is of course invariant under diffeomorphisms, and these diffeomorphisms correspond to different gauges for the equation $\mathrm{Ric}(g) = h$. The choice of harmonic coordinates may be viewed as a local gauge condition that results in an elliptic equation. In this article we give similar local gauge conditions for the ellipticity of conformal curvature tensors, based on systems of $n$-harmonic coordinates. We say that a function $u$ in $(M,g)$ is $p$-harmonic ($1 < p < \infty$) if
$$
\delta(\abs{du}^{p-2} du) = 0.
$$
Here $\abs{\,\cdot\,}$ is the $g$-norm and $\delta$ is the codifferential. A local coordinate system is called $p$-harmonic if each coordinate function is $p$-harmonic (the case $p=2$ of course corresponds to the usual harmonic coordinates).
In the earlier article \cite{LiimatainenSalo}, we established the existence of $p$-harmonic coordinate systems for $1 < p < \infty$ on any Riemannian manifold with $C^{r}$, $r>1$, metric tensor. In conformal geometry, the case $p=n$ is special since the class of $n$-harmonic functions is preserved under conformal transformations. This fact was used in \cite{LiimatainenSalo} to study the regularity of such maps: if one fixes $n$-harmonic coordinate systems both in the domain and target manifold, the components of a conformal mapping will solve an elliptic equation. Thus $n$-harmonic coordinates may be considered as a gauge condition for ellipticity in the regularity problem for conformal transformations. In any $n$-harmonic coordinate system, the following \emph{$n$-harmonic gauge condition} is valid \cite{LiimatainenSalo}:
\begin{equation} \label{nharmonic_gauge_condition}
\Gamma^k = -\frac{n-2}{2} \frac{g^{kr} g^{ka} g^{kb}}{g^{kk}} \partial_r g_{ab}.
\end{equation}
We now recall the definition of conformal curvature tensors. If $(M,g)$ is an $n$-dimensional Riemannian manifold, write $R_{abcd}$, $R_{ab}$, and $R$ for the Riemann curvature tensor, Ricci tensor, and scalar curvature, respectively. The Schouten tensor is defined as
$$
P_{ab} = \frac{1}{n-2} \left( R_{ab} - \frac{R}{2(n-1)} g_{ab} \right).
$$
The Weyl tensor of $(M,g)$ is the $4$-tensor
$$
W_{abcd} = R_{abcd} + P_{ac} g_{bd} - P_{bc} g_{ad} + P_{bd} g_{ac} - P_{ad} g_{bc},
$$
the Cotton tensor is the $3$-tensor
$$
C_{abc} = \nabla_a P_{bc} - \nabla_b P_{ac},
$$
and the Bach tensor is the $2$-tensor
$$
B_{ab}=\nabla^k\nabla^lW_{akbl}+\frac{1}{2}R^{kl}W_{akbl}.
$$
We also consider the Fefferman-Graham obstruction tensor
$$
\mathcal{O}_{ab}=\frac{1}{n-3}\Delta^{n/2-2}\nabla^k\nabla^lW_{akbl}+\text{lower order terms,}
$$
where $\Delta=\nabla^i\nabla_i$ and $n \geq 4$ is even. If $n=4$ then $\mathcal{O}_{ab} = B_{ab}$. These tensors have the following behavior under conformal scaling in various dimensions: $W(cg) = c W(g)$ for $n \geq 4$, $C(cg) = C(g)$ for $n=3$, and $\mathcal{O}(cg) = c^{-\frac{n-2}{2}} \mathcal{O}(g)$ for $n \geq 4$ even. See~\cite{AcheViaclovsky}, \cite{Bach}, \cite{Besse}, \cite{Der}, \cite{FG}, \cite{FGbook}, \cite{Tian} for additional information on these tensors.
If $T$ is one of the above tensors, the next result shows that the equation $T(g) = h$ becomes elliptic under two natural local gauge conditions: one writes both sides of the equation in $n$-harmonic coordinates and considers the conformally normalized metric $\hat{g}_{jk} = \abs{g}^{-1/n} g_{jk}$ with determinant one. Here $\abs{g} = \det{g_{jk}}$, and we use that $n$-harmonic coordinates for $g$ are also $n$-harmonic coordinates for the conformal metric $\hat{g}$.
\begin{thm} \label{thm_main_ellipticity}
Let $(M,g)$ be a smooth $n$-dimensional Riemannian manifold. The Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors are elliptic operators in $n$-harmonic coordinates, in the sense that after applying both the $n$-harmonic gauge condition \eqref{nharmonic_gauge_condition} and the condition $|\hat{g}|=1$, the linearizations of the resulting operators at $\hat{g}$ are elliptic.
\end{thm}
After interpreting the conformal curvature tensors of low regularity metrics in a suitable sense, elliptic regularity results will follow readily. The regularity conditions will be given in terms of tensors $W_{abc\phantom{d}}^{\phantom{abc}d}$, $C_{abc}$, $\abs{g}^{\frac{n-2}{2n}}\mathcal{O}_{ab}$ which are invariant in conformal scaling $g \mapsto cg$. We use the notation $C_*^r$ for the Zygmund spaces~\cite{T3}.
\begin{thm} \label{thm_main_regularity}
Let $M$ be a smooth $n$-dimensional manifold, and let $g \in C^r_*$ in some system of local coordinates.
\begin{enumerate}
\item[(a)]
If $n \geq 4$, $r > 1$, and $W_{abc\phantom{d}}^{\phantom{abc}d} \in C^s_*$ for some $s > r-2$ in $n$-harmonic coordinates, then $\abs{g}^{-1/n} g_{jk} \in C^{s+2}_*$ in these coordinates.
\item[(b)]
If $n = 3$, $r > 2$, and $C_{abc} \in C^s_*$ for some $s > r-3$ in $n$-harmonic coordinates, then $\abs{g}^{-1/n} g_{jk} \in C^{s+3}_*$ in these coordinates.
\item[(c)]
If $n \geq 4$ is even, $r > n-1$, and $\abs{g}^{\frac{n-2}{2n}}\mathcal{O}_{ab} \in C^s_*$ for some $s > r-n$ in $n$-harmonic coordinates, then $\abs{g}^{-1/n} g_{jk} \in C^{s+n}_*$ in these coordinates.
\end{enumerate}
\end{thm}
Combining the previous theorem with classical results for $C^3$ metrics, one obtains conditions for local conformal flatness of low regularity metrics via the vanishing of Weyl and Cotton tensors.
\begin{thm} \label{thm_main_conformalflatness}
Let $M$ be a smooth $n$-dimensional manifold, and let $g\in C^r_*$ in some system of local coordinates defined near $p \in M$.
\begin{enumerate}
\item[(a)]
If $n \geq 4$, $r > 1$, and if $W_{abcd} = 0$ near $p$, then $g_{jk} = c \delta_{jk}$ for some positive function $c \in C^r_*$ in some $n$-harmonic coordinates near $p$.
\item[(b)]
If $n = 3$, $r > 2$, and if $C_{abc} = 0$ near $p$, then $g_{jk} = c \delta_{jk}$ for some positive function $c \in C^r_*$ in some $n$-harmonic coordinates near $p$.
\item[(c)]
If $n \geq 4$ is even, $r > n-1$, and if $\mathcal{O}_{ab} = 0$ near $p$, then $\abs{g}^{-1/n} g_{jk}$ is $C^{\infty}$ in any system of $n$-harmonic coordinates near $p$.
\end{enumerate}
\end{thm}
There are a number of results in conformal geometry that employ various gauges to obtain elliptic or parabolic regularity and existence results. The works \cite{AcheViaclovsky}, \cite{Anderson}, \cite{Helliwell}, \cite{Tian} involve constant scalar curvature gauges, sometimes obtained via the solution of the Yamabe problem \cite{LeeParker}, and study asymptotically locally Euclidean manifolds with obstruction flat metrics or boundary regularity of conformally compact Einstein metrics. In~\cite{BahuaudHelliwell} a version of the DeTurck trick is used to show that certain analogues of the Ricci flow, including a modified flow involving the obstruction tensor, are locally well posed. In~\cite{GurskyViaclovsky1}, \cite{GurskyViaclovsky2} quadratic curvature functionals are studied with an emphasis on the structure of the space of their critical metrics. In these papers the corresponding Euler-Lagrange equations become elliptic via suitable gauge conditions.
Let us describe an argument from \cite{Anderson}, \cite{Helliwell}, \cite{Tian} for Bach flat metrics with $n=4$. Choosing a conformal metric with constant scalar curvature, the equation of Bach flatness for the conformally scaled metric reads
$$
\nabla^k \nabla_k R_{ab} + \text{lower order terms} = 0.
$$
This becomes a fourth order elliptic equation in harmonic coordinates for the conformally scaled metric. This fact could be used to give another proof of an analogue of Theorem~\ref{thm_main_regularity}, if a solution to the (local) Yamabe problem for sufficiently low regularity metrics is provided.
We thus have two possible sets of gauge conditions for ellipticity of conformal curvature tensors: ($n$-harmonic coordinates + determinant one) and (constant scalar curvature + harmonic coordinates). Let us compare these briefly. The first approach works for $C^{1,\alpha}$ metrics with $\alpha > 0$ and relies on an $n$-harmonic coordinate system. In \cite{LiimatainenSalo} such coordinates $(u^1,\ldots,u^n)$ near $p$ were constructed by choosing some smooth coordinates $(x^1,\ldots,x^n)$ near $p$ and then solving for each $j$ the Dirichlet problem
$$
\delta(\abs{du^j}^{n-2} du^j) = 0 \text{ in } B(p,\varepsilon), \quad u^j|_{\partial B(p,\varepsilon)} = x^j.
$$
An easy variational argument shows that this problem has a unique solution. The point is to show that $(u^1,\ldots,u^n)$ is a $C^1$ diffeomorphism if $\varepsilon$ is small enough, and this relies on standard regularity theory for quasilinear equations (essentially the fact that any $p$-harmonic function is $C^{1,\alpha}$ regular). Given $n$-harmonic coordinates, the conformal normalization (determinant one) is trivial.
The second approach based on a constant scalar curvature gauge requires solving the Yamabe problem (a semilinear equation with a constraint) at least in a small neighborhood; the global Yamabe problem is solvable for $C^{1,1}$ metrics (see \cite[Section VII.7]{ChoquetBruhat}), and we refer to \cite{Escobar}, \cite{HG}, \cite{Ma} for versions of the local problem. The recent paper \cite{HG} deals with $W^{1,p}$, $ p > n$, metrics. Given a conformal metric with constant scalar curvature, the other gauge condition involves a standard harmonic coordinate system.
A possible benefit of the $n$-harmonic coordinate approach is the similarity to harmonic coordinates in Riemannian geometry. In fact, it appears that $n$-harmonic coordinates are a natural generalization of harmonic coordinates in conformal geometry, and other Riemannian applications of harmonic coordinates might have conformal analogues. On the other hand, $n$-harmonic coordinates exist only locally, and if global gauge conditions are required then scalar curvature gauges are available via the Yamabe problem. Finally, $n$-harmonic coordinates exist at least for $C^{1,\alpha}$ metrics whereas global results for the Yamabe problem in the literature seem to require metrics with two derivatives. One motivation for this work was to find characterizations of local conformal flatness beyond the classical $C^3$ case, and it is an interesting question (also asked in \cite[Section 2.7]{IwaniecMartin}) if one can have such characterizations for very low regularity metrics.
\vspace{12pt}
\noindent {\bf Acknowledgements.}
T.L.~was supported in part by the Finnish National Graduate School on Mathematics and its Applications, and M.S.~is partly supported by the Academy of Finland and an ERC Starting Grant. The second author is grateful to Spyros Alexakis, Michael Eastwood, Robin Graham and Gantumur Tsogtgerel for helpful discussions, and in particular to Robin Graham for pointing out several useful references.
\begin{comment}
equation is considered. There instead of fixing the conformal invariance by constant scalar curvature, they introduce a term that is critical for Einstein metrics, but breaks the conformal invariance.
Conformally compact Einstein metric is conformal to an Einstein metric and thus are Bach flat~\cite{}. These papers use this fact to stydy the regularity of such metrics by casting the problem into an elliptic problem in coordinates of Bach flatness. For example,
The works~\cite{Anderson, Helliwell} study boundary regularity of conformally compact Einstein metrics in even dimension $\geq 4$. These papers use scalar curvature to break the conformal invariance of the questions they address and (boundary adapted) harmonic coordinates.
If a conformally invariant problem is written for the scaled constant scalar curvature metric, the scalar curvature acts as a gauge condition. For example in dimension $4$ the the conformally invariant Bach-flat equation written for a constant scalar curvature metric reads
Works on regularity of ALE~\cite{Anderson, Helliwell}.
The work~\cite{BauhaudHelliwell} studies a parabolic problem.
The gauge breaking method does not require additional regularity of the metric.
We mention \cite{AcheViaclovsky}, \cite{Anderson}, \cite{BahuaudHelliwell}, \cite{GurskyViaclovsky1}, \cite{GurskyViaclovsky2}, \cite{Helliwell}, \cite{Tian}. The first two considers. The work~\cite{} proves certain. The works~\cite{Tian, AcheViaclovsky} consider Bach-flat and obstruction flat assymptotically locally Euclidean metrics. The first one in particular, \cite{Tian} gives a local regularity result for a class of Bach flat metrics that have constant scalar curvature.
\end{comment}
\section{Symbol computations} \label{section_symbolcomputations}
In this section we assume that the metric is smooth ($=C^{\infty}$) in some system of local coordinates. We start by recalling standard local coordinate formulas. The Christoffel symbols are given by
$$
\Gamma_{ab}^k = \frac{1}{2} g^{kl}(\partial_a g_{bl} + \partial_b g_{al} - \partial_l g_{ab}).
$$
Define
$$
\Gamma^k = g^{ab} \Gamma_{ab}^k, \qquad \Gamma_k = g_{kl} \Gamma^l.
$$
Noting the identity $g^{ab} \partial_l g_{ab} = \partial_l(\log\abs{g})$, we see that
\begin{equation*}
\Gamma^k = -\partial_l g^{kl} - \frac{1}{2} g^{kl} \partial_l(\log\abs{g}), \qquad
\Gamma_j = g^{kl} \partial_l g_{jk} - \frac{1}{2} \partial_j(\log\abs{g}).
\end{equation*}
This also implies that
$$
\Gamma_{ab}^a = \frac{1}{2} \partial_b(\log\abs{g}).
$$
The Riemann curvature tensor is given by
$$
R_{abcd} = \langle (\nabla_a \nabla_b - \nabla_b \nabla_a) \partial_c, \partial_d \rangle.
$$
Here $\nabla_a \nabla_b \partial_c = \nabla_a (\Gamma_{bc}^m \partial_m) = \partial_a \Gamma_{bc}^m \partial_m + \Gamma_{bc}^m \Gamma_{am}^r \partial_r$, so that
\begin{align*}
R_{abcd} = \partial_a \Gamma_{bc}^m g_{md} + \Gamma_{bc}^m \Gamma_{am}^r g_{rd} - \partial_b \Gamma_{ac}^m g_{md} - \Gamma_{ac}^m \Gamma_{bm}^r g_{rd}.
\end{align*}
The Ricci tensor, $R_{bc} = R_{abc}^{\phantom{abc}a}$, is given by
\begin{align*}
R_{bc} &= \partial_a \Gamma_{bc}^a - \partial_b \Gamma_{ac}^a + \Gamma_{bc}^m \Gamma_{am}^a - \Gamma_{ac}^m \Gamma_{bm}^a \\
&= \partial_a \Gamma_{bc}^a - \Gamma_{ac}^m \Gamma_{bm}^a - \frac{1}{2} \partial_{bc}(\log\abs{g}) + \frac{1}{2} \Gamma_{bc}^m \partial_m(\log\abs{g}).
\end{align*}
The scalar curvature is $R = g^{bc} R_{bc}$. The Schouten, Weyl, Cotton and Bach tensors were defined in the introduction. By using Bianchi identities as in \cite{Tian}, the Cotton and Bach tensors can also be written as
\begin{gather*}
(3-n) C_{abc} = \nabla^l W_{abcl}, \\
B_{ab} = (3-n)(- \nabla^k \nabla_k P_{ab} + \nabla^k \nabla_a P_{bk}) + \frac{1}{2} R^{kl} W_{akbl}.
\end{gather*}
Below, we write $T_k$ for quantities that depend smoothly on components of the metric and their derivatives up to order $k$. We will also write
$$
L = g^{ab} \partial_{ab}, \qquad L^2 = g^{ab} g^{cd} \partial_{abcd}
$$
for the principal parts of the Laplace-Beltrami operator and its square.
\subsection{Bach tensor}
All ellipticity results in this paper ultimately reduce to the fact that the Bach tensor is elliptic in $n$-harmonic coordinates when acting on metrics with determinant one. In the following, if $T=T(g)$ is a nonlinear differential operator acting on metrics, we denote by $\sigma(T)$ the principal symbol of its linearization at $g$.
\begin{lemma} \label{lemma_bach_computation}
The components of the Bach tensor satisfy
\begin{align*}
\frac{2 (n-2)}{3-n} B_{ab} = L^2 &(g_{ab}) - L (\partial_a \Gamma_b + \partial_b \Gamma_a) \\
&+ \frac{n-2}{n-1} \partial_{abl} \Gamma^l + \frac{1}{n-1} L(\partial_l \Gamma^l) g_{ab} + T_3
\end{align*}
in any local coordinate system in which $\abs{g}=1$
\end{lemma}
\begin{proof}
Assume that the metric satisfies $\abs{g}=1$ in a given system of local coordinates. Then
$$
R_{bc} = \partial_a \Gamma_{bc}^a + T_1, \qquad R = \partial_a \Gamma^a + T_1.
$$
The Schouten tensor satisfies
$$
(n-2) P_{ab} = \partial_l \Gamma_{ab}^l - \frac{1}{2(n-1)} (\partial_l \Gamma^l) g_{ab} + T_1.
$$
Recall that $\nabla_m F_{i_1 \ldots i_k} = (\nabla F)_{i_1 \ldots i_k m} = \partial_m F_{i_1 \ldots i_k} - \sum_{s=1}^k F_{i_1 \ldots p \ldots i_k} \Gamma_{m i_s}^p$. Therefore
$$
(n-2) \nabla_c P_{ab} = \partial_{cl} \Gamma_{ab}^l - \frac{1}{2(n-1)} (\partial_{cl} \Gamma^l) g_{ab} + T_2
$$
and
$$
(n-2) \nabla_d \nabla_c P_{ab} = \partial_{cdl} \Gamma_{ab}^l - \frac{1}{2(n-1)} (\partial_{cdl} \Gamma^l) g_{ab} + T_3.
$$
For the Bach tensor, we obtain
\begin{align*}
&\frac{n-2}{3-n} B_{ab} = (n-2) \left[ g^{km} \nabla_k \nabla_a P_{bm} - g^{km} \nabla_k \nabla_m P_{ab} \right] + T_2 \\
&= g^{km} \left[ \partial_{akl} \Gamma_{bm}^l - \partial_{klm} \Gamma_{ab}^l + \frac{1}{2(n-1)} \left[ (\partial_{klm} \Gamma^l) g_{ab} - (\partial_{akl} \Gamma^l) g_{bm} \right] \right] + T_3.
\end{align*}
Next we observe that
\begin{align*}
\partial_{klm} \Gamma_{ab}^l &= \frac{1}{2} \partial_{klm} \left[ g^{lr}(\partial_a g_{br} + \partial_b g_{ar} - \partial_r g_{ab}) \right] \\
&= \frac{1}{2} g^{lr} (\partial_{aklm} g_{br} + \partial_{bklm} g_{ar} - \partial_{rklm} g_{ab}) + T_3.
\end{align*}
This implies
\begin{align*}
&2g^{km} (\partial_{akl} \Gamma_{bm}^l - \partial_{klm} \Gamma_{ab}^l) \\
&= g^{km} g^{lr} \left[ \partial_{aklm} g_{br} + \partial_{bkla} g_{mr} - \partial_{rkla} g_{mb} \r.\\
&\ \ \ \l. -\partial_{aklm} g_{br} - \partial_{bklm} g_{ar} + \partial_{rklm} g_{ab} \right] + T_3 \\
&= L^2 (g_{ab}) - L(g^{km} \partial_{ka} g_{mb}) - L(g^{lr} \partial_{bl} g_{ar}) + g^{km} g^{lr} \partial_{bkla} g_{mr} + T_3 \\
&= L^2 (g_{ab}) - L(\partial_a \Gamma_b + \partial_b \Gamma_a) + \partial_{abl} \Gamma^l + T_3.
\end{align*}
In the last step we used that $\Gamma_b = g^{km} \partial_k g_{mb}$ when $\abs{g}=1$, and $\Gamma^l = g^{lr} \Gamma_r$. For the remaining terms in the Bach tensor, we use that
$$
g^{km} \left[ (\partial_{klm} \Gamma^l) g_{ab} - (\partial_{akl} \Gamma^l) g_{bm} \right] = L(\partial_l \Gamma^l) g_{ab} - \partial_{abl} \Gamma^l + T_3.
$$
Collecting these facts, we have proved the lemma.
\end{proof}
Notice that in any harmonic coordinate system in which $\abs{g}=1$, the Bach tensor would have the form
$$
2(n-2) B_{ab} = (3-n)L^2(g_{ab}) + T_3.
$$
The operator on the right is obviously elliptic. However, in general it is not clear how to find harmonic coordinates with $\abs{g}=1$ even after conformal scaling, owing to the fact that the equations $\Gamma^k = 0$ for harmonic coordinates are not conformally invariant. This is where $n$-harmonic coordinates become useful: they allow the conformal normalization $\abs{g}=1$, and the Bach tensor turns out to be elliptic in these coordinates.
\begin{lemma} \label{lemma_bach_nharmonic
The components of the Bach tensor satisfy
\begin{multline*}
\frac{2 (n-2)}{3-n} B_{ab} = L^2 (g_{ab}) - L (\partial_a \tilde{\Gamma}_b + \partial_b \tilde{\Gamma}_a) \\
+ \frac{n-2}{n-1} \partial_{abl} \tilde{\Gamma}^l + \frac{1}{n-1} L(\partial_l \tilde{\Gamma}^l) g_{ab} + T_3
\end{multline*}
in any $n$-harmonic coordinate system with $\abs{g}=1$, where $\tilde{\Gamma}^k$ is as in \eqref{nharmonic_gauge_condition}:
$$
\tilde{\Gamma}^k = -\frac{n-2}{2} \frac{g^{kr} g^{ka} g^{kb}}{g^{kk}} \partial_r g_{ab}.
$$
\end{lemma}
\begin{proof}
Follows from Lemma \ref{lemma_bach_computation} and the fact that $\Gamma^k = \tilde{\Gamma}^k$ in $n$-harmonic coordinates (see \cite[Remark after Theorem 2.1]{LiimatainenSalo}).
\end{proof}
Consider the right hand side operator in Lemma \ref{lemma_bach_nharmonic} acting on symmetric positive definite matrix functions, and denote by $Q = Q_g$ the principal part of the linearization of this operator at $g$. Then $Q$ is the principal part of the following linear operator, acting on symmetric matrix functions $h$ by
\begin{multline*}
h_{ab} \mapsto L^2 (h_{ab}) - L \left[ \partial_a (g_{bl} \tilde{\Gamma}^l(h)) + \partial_b (g_{al} \tilde{\Gamma}^l(h)) \right] \\
+ \frac{n-2}{n-1} \partial_{abl} \tilde{\Gamma}^l(h) + \frac{1}{n-1} L(\partial_l \tilde{\Gamma}^l(h)) g_{ab}
\end{multline*}
where $L = L_g = g^{ab} \partial_{ab}$, and $\tilde{\Gamma}^k = \tilde{\Gamma}^k_g$ is the operator
$$
\tilde{\Gamma}^k(h) = -\frac{n-2}{2} \frac{g^{kr} g^{ka} g^{kb}}{g^{kk}} \partial_r h_{ab}.
$$
\begin{lemma}\label{ellipt_calc}
$Q$ is elliptic of order $4$ in dimensions $n \geq 3$.
\end{lemma}
\begin{proof}
The principal symbol of $Q$, acting on a symmetric matrix function $h$, is given by
\begin{multline*}
(q(\xi) h)_{ab} = \abs{\xi}^4 h_{ab} - \abs{\xi}^2 \left[ \xi_a g_{bl} + \xi_b g_{al} \right] \sigma(-i\tilde{\Gamma}^l)(h) \\
+ \frac{n-2}{n-1} \xi_a \xi_b \xi_l \sigma(-i\tilde{\Gamma}^l)(h)+ \frac{1}{n-1} \abs{\xi}^2 \xi_l \sigma(-i\tilde{\Gamma}^l)(h) g_{ab}
\end{multline*}
where
$$
\sigma(-i\tilde{\Gamma}^k)(h) = -\frac{n-2}{2} \xi^k \frac{h^{kk}}{g^{kk}}.
$$
Here, the norms $\abs{\,\cdot\,}$ and the raising and lowering of indices are taken with respect to the metric $g$. We also have
\begin{multline*}
(q(\xi) h)^{ab} = \abs{\xi}^4 h^{ab} - \abs{\xi}^2 \left[ \xi^a \sigma(-i\tilde{\Gamma}^b)(h) + \xi^b \sigma(-i\tilde{\Gamma}^a)(h) \right] \\
+ \frac{n-2}{n-1} \xi^a \xi^b \xi_l \sigma(-i\tilde{\Gamma}^l)(h)+ \frac{1}{n-1} \abs{\xi}^2 \xi_l \sigma(-i\tilde{\Gamma}^l)(h) g^{ab}.
\end{multline*}
Taking $a=b$, this gives
\begin{multline*}
(q(\xi) h)^{aa} = \abs{\xi}^4 h^{aa} - 2 \abs{\xi}^2 \xi^a \sigma(-i\tilde{\Gamma}^a)(h) \\
+ \frac{n-2}{n-1} (\xi^a)^2 \xi_l \sigma(-i\tilde{\Gamma}^l)(h)+ \frac{1}{n-1} \abs{\xi}^2 \xi_l \sigma(-i\tilde{\Gamma}^l)(h) g^{aa} \\
= \left( g^{aa} \abs{\xi}^2 + (n-2) (\xi^a)^2 \right) \left[ \abs{\xi}^2 \frac{h^{aa}}{g^{aa}} - \frac{n-2}{2(n-1)} \sum_{l=1}^n \xi_l \xi^l \frac{h^{ll}}{g^{ll}} \right].
\end{multline*}
If $\xi \neq 0$ and if $q(\xi)h = 0$, then also
$$
\frac{h^{aa}}{g^{aa}} - \frac{n-2}{2(n-1)} \sum_{l=1}^n \frac{\xi_l \xi^l}{\abs{\xi}^2} \frac{h^{ll}}{g^{ll}} = 0
$$
for all $a = 1, \ldots, n$. This implies that
$$
\frac{h^{11}}{g^{11}} = \ldots = \frac{h^{nn}}{g^{nn}} = \lambda
$$
for some function $\lambda$, and therefore
$$
\left( 1 - \frac{n-2}{2(n-1)} \right) \lambda = 0.
$$
Thus $h^{aa} = 0$ for all $a$, so $\sigma(-i\tilde{\Gamma}^a)(h) = 0$ for all $a$ and $(q(\xi) h)^{ab} = \abs{\xi}^4 h^{ab} = 0$ for all $a, b$. It follows that $h = 0$, showing that $Q$ is elliptic.
\end{proof}
We are now ready to prove the main ellipticity result.
\begin{proof}[Proof of Theorem \ref{thm_main_ellipticity}]
Let $(M,g)$ be a smooth Riemannian manifold, fix an $n$-harmonic coordinate system, and let $\hat{g}_{jk} = \abs{g}^{-1/n} g_{jk}$ in these coordinates. Using the conformal invariance of the $n$-harmonic equation, our coordinate system is $n$-harmonic also with respect to the conformal metric $\hat{g}$. Let $\hat{B}_{ab}$ be the Bach tensor for $\hat{g}$ in $n$-harmonic coordinates after applying the $n$-harmonic gauge condition \eqref{nharmonic_gauge_condition} and the condition $\abs{\hat{g}} = 1$. It follows from Lemmas \ref{lemma_bach_computation}--\ref{ellipt_calc} that the principal part of the linearization of $\hat{B}_{ab}$ is elliptic in dimensions $n \geq 4$.
Moving on to the Weyl tensor, the formula $B_{ab}=\nabla^k\nabla^l W_{akbl}+\frac{1}{2}R^{kl}W_{akbl}$ implies that
$$
\xi^k \xi^l \sigma(\hat{W}_{akbl})(\xi)h = -\sigma(\hat{B}_{ab})(\xi)h
$$
where $\hat{W}$ is the Weyl tensor of $\hat{g}$ after applying the gauge conditions, and $\sigma$ denotes the principal symbol of the linearization applied to a matrix function $h$. Thus $\sigma(\hat{W}_{akbl})(\xi)h = 0$ implies $h = 0$ by the ellipticity of $\hat{B}_{ab}$, showing that $\hat{W}_{abcd}$ is overdetermined elliptic (its principal symbol is injective). The ellipticity of Cotton and obstruction tensors follow in a similar way from the identities $(3-n) C_{abc} = \nabla^l W_{abcl}$, $(n-3) \mathcal{O}_{ab}=\Delta^{n/2-2}\nabla^k\nabla^lW_{akbl}+T_{n-1}$.
\end{proof}
\section{Elliptic regularity results}
We now consider low regularity Riemannian metrics, and establish elliptic regularity results for conformal curvature tensors. The Weyl tensor will be considered in detail and the arguments for the other tensors will be sketched. We first assume that
$$
g_{jk} \in W^{1,2} \cap L^{\infty}
$$
in some system of local coordinates near a point. (In this section all function spaces are assumed to be of the local variety near a point, that is, we write $W^{1,2}$ instead of $W^{1,2}_{\mathrm{loc}}(U)$ etc.) This seems to be a minimal assumption for defining the Weyl tensor: the set $W^{1,2} \cap L^{\infty}$ is an algebra under pointwise multiplication \cite{KatoPonce}, \cite{BadrBernicotRuss}, and therefore $g^{jk}, \abs{g} \in W^{1,2} \cap L^{\infty}$. We see that
\begin{gather*}
\Gamma_{ab}^c \in L^2, \\
R_{abc}^{\phantom{abc}d} = \partial_a \Gamma_{bc}^d - \partial_b \Gamma_{ac}^d + \Gamma_{bc}^m \Gamma_{am}^d - \Gamma_{ac}^m \Gamma_{bm}^d \in W^{-1,2} + L^1, \\
R_{bc} = R_{abc}^{\phantom{abc}a} \in W^{-1,2} + L^1.
\end{gather*}
Since
$$
P_{ab} = \frac{1}{n-2} \left[ R_{ab} - \frac{1}{2(n-1)} g^{rs} R_{rs} g_{ab} \right],
$$
we have
\begin{multline*}
W_{abc}^{\phantom{abc}d} = R_{abc}^{\phantom{abc}d} + P_{ac} \delta_b^{\phantom{b}d} - P_{bc} \delta_a^{\phantom{a}d} + P_{bm} g^{md} g_{ac} - P_{am} g^{md} g_{bc} \\
= R_{abc}^{\phantom{abc}d} + \frac{1}{n-2} \left[ R_{ac} \delta_b^{\phantom{b}d} - R_{bc} \delta_a^{\phantom{a}d} + R_{bm} g^{md} g_{ac} - R_{am} g^{md} g_{bc} \right] \\
+ \frac{1}{(n-2)(n-1)} \left[ g^{rs} g_{bc} R_{rs} \delta_a^{\phantom{a}d} - g^{rs} g_{ac} R_{rs} \delta_b^{\phantom{b}d} \right].
\end{multline*}
Now, since multiplication by an $W^{1,2} \cap L^{\infty}$ function maps $W^{1,q}$ to $W^{1,2}$ for any $q > n$, it also maps $W^{-1,2}$ to $W^{-1,q'}$, and we have
\begin{gather*}
g^{ij} g_{kl} R_{rs} \in W^{-1,q'} + L^1.
\end{gather*}
This shows that if $g_{jk} \in W^{1,2} \cap L^{\infty}$ in some system of local coordinates, then one can make sense of the components $W_{abc}^{\phantom{abc}d}$ of the Weyl tensor as elements in $W^{-1,q'} + L^1$ in these coordinates for any $q > n$. It is easy to see that also $R_{abcd}, W_{abcd} \in W^{-1,q'} + L^1$ in this case, and the identity $W(cg) = cW(g)$ remains true if $c, g_{jk} \in W^{1,2} \cap L^{\infty}$.
By similar arguments, if $g_{jk} \in W^{2,2} \cap W^{1,\infty}$ in some local coordinates, the components $C_{abc}$ of the Cotton tensor may be interpreted as elements of $W^{-1,2}$ and one has $C(cg) = C(g)$ if $c \in W^{2,2} \cap W^{1,\infty}$ and $n=3$. Moreover, if $g_{jk} \in W^{n-1,2} \cap W^{n-2,\infty}$, then the Fefferman-Graham obstruction tensor satisfies $\mathcal{O}_{ab} \in W^{-1,2}$ and $\mathcal{O}(cg) = c^{\frac{n-2}{2}} \mathcal{O}(g)$ for $n \geq 4$ even. We omit the details of these computations.
We will need the following elliptic regularity result, which is, after using suitable cutoffs and extensions, a consequence of \cite[Theorem 14.4.2]{T3} (the argument also applies to overdetermined elliptic operators). For completeness, the details are given in Appendix \ref{app}.
\begin{prop} \label{elliptic_regularity_differential_operator}
Let $B \subset \mathbb{R}^n$ be a ball, and let $p(x,D) = \sum_{\abs{\alpha} \leq m} p_{\alpha}(x) D^{\alpha}$ be an $M \times N$ matrix differential operator with coefficients in $C^r_*(B)$ where $r > 0$. Assume that $p(x,D)$ is overdetermined elliptic in the sense that its principal symbol $p_m(x,\xi)$ is injective for $x \in B$ and $\xi \neq 0$. If in the sense of distributions
$$
p(x,D) u = f \quad \text{in } B
$$
where $u \in C^{m-r+\varepsilon}_*(B)$ for some $\varepsilon > 0$ and $f \in C^s_*(B)$ with $-r < s < r$, then $u \in C^{m+s}_*(B)$.
\end{prop}
The next result gives a symbol computation for the Weyl tensor with low regularity metric.
\begin{lemma} \label{weyl_princip}
Let $g_{jk} \in C^r_*$, $r > 1$, in some system of local coordinates, and assume that $\abs{g}=1$. The principal symbol $\sigma(W_{abcd})$ of the linearization of the Weyl tensor at $g$ satisfies
\begin{multline}\label{weyl_contr}
\xi^a \xi^d \sigma(W_{abcd})h=\frac{n-3}{2(n-2)} \Big[ |\xi|^4 h_{bc} - |\xi|^2 \left( \xi_b\sigma(-i\Gamma_c)h + \xi_c\sigma(-i\Gamma_b)h \right) \\
+\frac{n-2}{n-1} \xi_b \xi_c \xi_l \sigma(-i\Gamma^l)h + \frac{1}{n-1} |\xi|^2 \left( \xi_l \sigma(-i\Gamma^l)h \right) g_{bc} \Big].
\end{multline}
\end{lemma}
\begin{proof}
If $g$ were sufficiently smooth, for instance $C^4$, we could readily use the formula $B_{ab}=\nabla^k\nabla^lW_{akbl}+\frac{1}{2}R^{kl}W_{akbl}$ to have
$$
\xi^a \xi^d \sigma(W_{abcd})h=-\sigma(B_{bc})h,
$$
which together with Lemma~\ref{lemma_bach_computation} would yield the claim. For $C^r_*$ metrics, $r > 1$, we have to be more careful because the derivation of the different formulas for the Bach tensor uses the Bianchi identities
$$
\nabla^d R_{abcd} = \nabla_aR_{bc}-\nabla_bR_{ac}, \qquad \nabla^b R_{ab} = \frac{1}{2}\nabla_b R,
$$
that require more derivatives. We can however emulate the Bianchi identities on the symbol level using the equations
\begin{align}
\xi^d &\sigma(R_{abcd}+R_{ac}g_{bd}-R_{bc}g_{ad})=0, \label{Bianchi_emulated1} \\
\xi^a &\sigma(R_{ab}g_{cd}-\frac{1}{2}Rg_{ab}g_{cd})=0. \label{Bianchi_emulated2}
\end{align}
By the discussion earlier in this section, one can check that these identities are valid for $C^r_*$ metrics when $r > 1$. We now write the Weyl tensor as
\begin{multline*}
W_{abcd}=(R_{abcd}+R_{ac}g_{bd}-R_{bc}g_{ad})+\l(\frac{1}{n-2}-1\r)(R_{ac}g_{bd}-R_{bc}g_{ad}) \\
+\frac{1}{n-2}(R_{bd}g_{ac} -R_{ad}g_{bc})-\frac{R}{(n-1)(n-2)}(g_{ac}g_{bd}-g_{bc}g_{ad}) \\
=(R_{abcd}+R_{ac}g_{bd}-R_{bc}g_{ad})-\frac{n-3}{n-2}(R_{ac}g_{bd}-\frac{1}{2}Rg_{ac}g_{bd}) \\
+\frac{1}{n-2}(R_{bd}g_{ac} -\frac{1}{2}Rg_{bd}g_{ac}-R_{ad}g_{bc}+\frac{1}{2}Rg_{ad}g_{bc}) \\
+\frac{n-3}{n-2}R_{bc}g_{ad}-\frac{n-3}{2(n-2)}Rg_{ac}g_{bd} + \frac{1}{2(n-2)}(Rg_{bd}g_{ac}-Rg_{ad}g_{bc}). \\
-\frac{R}{(n-1)(n-2)}(g_{ac}g_{bd}-g_{bc}g_{ad})
\end{multline*}
Now, taking the symbol of the Weyl tensor in this form and contracting by $\xi^a\xi^d$ we have by the identities~\eqref{Bianchi_emulated1}, \eqref{Bianchi_emulated2} that
\begin{multline}\label{Wc}
\xi^a \xi^d \sigma(W_{abcd})h=-\frac{n-3}{2(n-2)}\Big[-2\abs{\xi}^2\sigma(R_{bc})h+\frac{n-2}{n-1}\xi_b\xi_c \sigma(R)h \\
+\frac{1}{n-1}\abs{\xi}^2g_{bc}\sigma(R) \Big].
\end{multline}
We have
$$
R_{ab} = -\frac{1}{2} \Delta g_{ab} + \frac{1}{2} (\partial_a \Gamma_b + \partial_b \Gamma_a) +T_1
$$
and consequently, for a metric $g$ with $\abs{g}=1$, it holds that
$$
R=\partial_a\Gamma^a +T_1.
$$
Substituting these formulas to~\eqref{Wc} yields the claim.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm_main_regularity}]
Let $g_{jk} \in C^r_*$, $r > 1$, in some system of local coordinates. By \cite[Theorem 2.1]{LiimatainenSalo} we know that $n$-harmonic coordinate systems exist near any point, and by \cite[Proposition 2.5]{LiimatainenSalo} the metric satisfies $g_{jk} \in C^r_*$ in $n$-harmonic coordinates.
Consider first the Weyl tensor. Let $n \geq 4$, $r > 1$, and $W_{abc}^{\phantom{abc}d} \in C^s_*$ for some $s > r-2$ in $n$-harmonic coordinates. Write $\hat{g} = \abs{g}^{-1/n} g_{jk}$ in these coordinates, so that
$$
\hat{g}_{jk} \in C^r_*, \qquad \abs{\hat{g}} = 1.
$$
By conformal invariance we have $W_{abc}^{\phantom{abc}d}(\hat{g}) = W_{abc}^{\phantom{abc}d}(g)$ in $n$-harmonic coordinates where the right hand side is in $C^s_*$. This is a second order system for $\hat{g}$ where the principal symbol of the linearization is injective after applying the $n$-harmonic gauge condition \eqref{nharmonic_gauge_condition} and the condition $\abs{\hat{g}}=1$, by Lemma \ref{weyl_princip} and the algebraic computation in Lemma \ref{ellipt_calc} (note that ellipticity of $W_{abc}^{\phantom{abc}d}$ is equivalent with that of $W_{abcd}$). Hence Proposition \ref{elliptic_regularity_differential_operator} applies. If $s < r$ we obtain $\hat{g}_{jk} \in C^{s+2}_*$ as required. If $s \geq r$ we initially only get $\hat{g}_{jk} \in C^{\tilde{r}+2}_*$ for any $\tilde{r} < r$, but iterating the argument finitely many times yields $\hat{g}_{jk} \in C^{s+2}_*$ and we are done.
The proof for the Cotton and Fefferman-Graham obstruction tensors follows in a similar way from the above arguments by using the identities $(3-n) C_{abc} = \nabla^l W_{abcl}$ and $(n-3) \mathcal{O}_{ab}=\Delta^{n/2-2}\nabla^k\nabla^lW_{akbl}+T_{n-1}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm_main_conformalflatness}]
Assume first that $g_{jk} \in C^r_*$ where $n \geq 4$, $r > 1$, and the Weyl tensor vanishes near some point of $M$. By Theorem \ref{thm_main_regularity} we know that the metric $\hat{g}_{jk} = \abs{g}^{-1/n} g_{jk}$ is $C^{\infty}$ in any system of $n$-harmonic coordinates. Since the Weyl tensor of $\hat{g}$ vanishes, classical arguments (see for example~\cite[Chapter 4]{Aubin}) show that there is a possibly different set of coordinates $\{y^a\}$ (for which we use indices $a$, $b$) where $\hat{g}_{ab} = \hat{c} \delta_{ab}$ for some positive $C^{\infty}$ function $\hat{c}$. Then $g_{ab} = c \delta_{ab}$ where $c = \abs{g}^{1/n} \hat{c}$ is in $C^r_*$. The coordinates $\{y^{a}\}$ are again $n$-harmonic~\cite[Proposition 2.6]{LiimatainenSalo}. The argument for Cotton and Fefferman-Graham tensors is analogous.
\end{proof}
|
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"redpajama_set_name": "RedPajamaArXiv"
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Horror Games Special Feature – Doom 3
Posted on 15th October 2015 20th October 2015 by
With Halloween just around the corner, we felt it was time to delve into some of the horror classics of gaming!
In the nights leading up to All Hallows Eve, we're going to do a short feature of some of the more notable horror games that have been released. This isn't a list of the best or even our favourites.
Just some Horror games which have stood out to us, for one reason or another.
For our first feature, we decided Hell was the best place to start.
But where in the hell do you start when trying to talk about Doom 3?
Representing a massive directional shift, Doom went from silly, blood n' guts carnage to nerve-shredding horror in its third installment. It was a direction not everyone was happy with, but those who enjoyed horror had little to complain about with this heart-stopping exploration in terror.
It's not a perfect game, by any means. The niche horror genre is bound to divide audiences, and it definitely gets repetitive towards the end. Fortunately, it leaves on a high note, with your BFG-toting Space Marine bursting through the gates of hell, like the bad ass penis allegory that he is, blowing chunks out of a gargantuan Cyberdemon, rendered beautifully in full 3D.
The reason this game stands out in memory is because it is so obviously the black sheep of the franchise. Doom and Doom 2 are both straight up shooters. They might feature hell, demons and gleeful chainsaw wielding. But if either game made you crap your pants, it wasn't because of any sort of tension.
It was because you just stepped into an arena full of Mecha-Brain Spiderdemons.
Doom 3 might not be what fans want in a Doom game, but it's nonetheless a solid survival horror game. It has a beautifully realised world, full of little details and brilliant light and shadow mechanics to make it properly terrifying.
Definitely a game worth owning.
Best moment:
The game may have had pacing issues towards the end, but it culminates beautifully in the final scene. After walking through an archway into an empty arena, you stand there, confused. Until you realise, that wasn't an archway…
Where is the franchise now?
Doom 4 was teased all the way back in 2008, before finding itself, ironically, doomed to development hell. A prototype was made that had a similar style to Call of Duty, with a lot of cinematic cut-scenes.
In 2011, Doom 4 was scrapped completely and the development team went back to the drawing board. The newest game in the franchise has been rebranded simply as 'Doom' and is due for release in 2016.
If the latest trailer is anything to go by, it'll very much be a return to the run-and-shoot style of the original series.
Written by Stephen Hill
Uncharted: The Nathan Drake Collection: Review.
Horror Games Special Feature – Rule of Rose
|
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-c-6-81-5-71","text":"# How do you solve c -6.81= - 5.71?\n\nMar 24, 2018\n\nc = 1.1\n\n#### Explanation:\n\nThe key to solving this is to isolate the variable on one side of the equation.\n\nC is the variable. To isolate it, we've got to get it alone on either side of the = sign.\n\nThe easiest way to do that here is to add 6.81 to both sides of the equation. Remember that anything done on one side of the equation (or equal sign) must be done on both sides to maintain balance.\n\nThink of - 5.71 as you owing a friend $5.71. If you find$6.81 cents in your pocket, you can pay that friend back their $5.71, subtract it from your$6.81 and still have \\$1.10 remaining. 1.10 is equal to 1.1, and that's the answer.","date":"2019-11-17 06:55:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5349912643432617, \"perplexity\": 572.7593002242298}, \"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-2019-47\/segments\/1573496668896.47\/warc\/CC-MAIN-20191117064703-20191117092703-00023.warc.gz\"}"}
| null | null |
Check out the Grizzlies promotion schedule. Don't miss out on the best giveaways of the season and more. The Memphis Grizzlies have been planning creative stadium about here, and numerous teams have announced their promotional schedules. The Memphis Grizzlies unveiled their promotional schedule for the and there will be plenty of chances for fans to receive some unique gear.
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Some extraction region not apprehend it, but the elation of our lives depends on the value of our thoughts.
San Antonio Spurs Saturday, February 9 vs. Orlando Magic and Sunday, April 7 vs. The Grizzlies are also offering a new limited time ticket deal for kids ages 12 and under. Saturday Dec 15 7: Philadelphia 76ers Saturday, December 15 vs. Fans can also test drive seats during Open Practice.
Memphis grizzlies schedule giveaways idea Gasol cooking apron to the first 5, fans in attendance.
Marc Gasol cooking apron to the first 5, fans in attendance.
The Memphis Grizzlies today announced their promotional schedule for the regular season, introducing all-new collectibles, keepsakes and theme nights to celebrate the 18th season of Grizzlies basketball in Grind City.
Now you have how the RNG works, here are some associated with features not do when playing depression machines. With its pragmatic counterparts and sounds and wonderful horseplay payouts, we conscious you'll brotherhood it. By all means Delight Fetch ME (Don Creekmore) at 316-371-1828 if you give birth to one of these, I would girlfriend to become alert it next to the Manager Bell. We prepare at worst one life.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 9,308
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Q: Bootstrap modal lightbox showing scrollbar background I am using bootstrap 3 in my project. I have a modal lightbox in my page there is some content in it. I just put height: 300px and overflow-y: scroll in it. But it is showing scrollbar background all the time even when the content height is below 300px. How can i fix this???
A: You need to use overflow-y: auto; instead of overflow-y:scroll;
(when auto is use it showing the scrollbar only when it need and not all the time)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,006
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\section{Introduction}
Determining main interactions that guide proteins to their native conformations is key to understanding protein folding, misfolding and stability \cite{DILL90}. Since the seminal work of Kauzmann \cite{KAUZ59} interactions among sidechain atoms is commonly assumed to account for tertiary contacts in proteins \cite{DILL90,LIHA97}, while backbone hydrogen bonding is associated with the stability of $\alpha$-helix and $\beta$-sheet secondary structures \cite{PAUL51,ROSE06}. Recently, hydrogen bonding has also been associated to protein misfolding and aggregation \cite{dobson2011,Maritan04,dobson08}.
However, these interactions are difficult to quantify in aqueous solutions and are usually inferred from experiments with model compounds \cite{KAUZ59,SCHE55,BALD86}. Further, their strengths are highly dependent on the local environment within the protein. Because of this complexity it is not surprising that there is a lot of debate regarding the forces stabilizing protein conformations \cite{ROSE06,BENN06,BENN91,BENN98} and studies aiming to describe protein folding have shifted back and forth between sidechain and hydrogen bond-centered views depending on the protein property of interest \cite{Nuss2000}. This paper aims to shed light on the nature of interactions that stabilize $\beta$-sheet conformations \cite{CHEN13}.
N-Methylacetamide \cite{KLOT68, JORG89,BENT97,BUCK01} and Urea \cite{KAUZ59, SCHE55}, long used as model systems for determining the strength of the peptide hydrogen bonds, predict negligible enthalpies for intra-peptide hydrogen bonding in water because backbone-backbone hydrogen bonds compete with water-backbone bonds to the same degree \cite{KLOT68}. However, small compounds have been argued to be bad models for the protein backbone because they lack sidechains that obstruct the formation of backbone-water hydrogen bonds. In aqueous solution, sidechains of each of the twenty amino acids have been envisaged to obstruct water differently, accounting for context-dependent hydrogen bonding \cite{BAIY94}. The strength of hydrogen bonds buried in the protein interior is observed to be as high as 7 kJ/mol per bond \cite{Martin09}, which explains the significant presence of secondary structures in globular proteins. Evidence from recent backbone mutation studies support this view \cite{DEEC04,DEEC04_JACS}. In these studies, elimination of hydrogen bonds in the dry protein core has a higher destabilizing effect (up to 5.0 kJ/mol) than elimination of hydrogen bonds at the protein's surface \cite{DEEC04, DEEC04_JACS, JICH11, JICH12, GAOJ09, PACE09}. Hydrogen bonds could therefore, play an important role in protein aggregation since about 50 \% of all bonds reside in the rigid dry core of these structures \cite{KHET00}. These different studies are part of an ongoing debate regarding the strength of backbone-backbone hydrogen bonding in water and whether they can stabilize ordered peptide structures \cite{WAYN08,ROSE06,ROSE93,HONI95}.
Structural differences among proteins are attributed to the nature of sidechain interactions as backbone atoms are identical in all proteins \cite{DILL90,KAUZ59,LIHA97}. In particular, the tendency of hydrophobic sidechains to be buried away from water is commonly accepted as the main driving force involved in stabilizing the native state. Hydrophobic residues are among the most conserved amino acids in protein sequences and their solubility correlates with protein stability. In particular, increased solubilities of these residues at high and low temperatures as well as high pressure have been associated with heat, cold, and pressure denaturations in proteins respectively \cite{Priv86,Priv90, RIOS00,RIOS01,DIAS08,DIAS12,DIAS10_cryobiology,HUMM98}. Strengths of hydrophobic interactions are of the order of 5 kJ/mol per --CH$_3$ group \cite{MINO94, MINO94_propensity, KIMC93, SMIT94,MERK98} and they are entropy-driven \cite{Fran45,DIAS11}. In addition to hydrophobicity, interactions between polar sidechains have also been suggested to play a major role in protein folding \cite{BENN06,BENN91,BENN98}, but this will not be addressed in this work.
In order to clarify the role of hydrophobic and hydrogen bonding interactions in protein folding and aggregation, we simulate homodimers of glycine, alanine, valine, and leucine in explicit water. To eliminate effects related to chain ends, we use periodic boundary conditions in which the carbonyl-group of the first residue is attached to the amine-group of the last residue. Hence, the environment around each amino acid resembles residues in the middle of strands. The energy landscape for these dimers, represented using the potential of mean force (PMF), shows well-defined global minima for all dimers at interpeptide distances corresponding to $\beta$-sheet conformations. PMFs of valine and leucine homodimers also show a second minimum at interpeptide distances of $\sim\!0.8$ nm, corresponding to conformations where sidechains of one peptide interpenetrate the space between sidechains of the neighboring peptide. Our results indicate that hydrogen bonding does not play a significant role in the formation of $\beta$-sheet structures for peptides with extended conformations. In contrast, hydrophobic interactions are shown to play a dominant role in stabilizing $\beta$-sheet conformations in peptides with large hydrophobic sidechains. We observe a striking correlation between the electrostatic energy and the total volume of the system. This supports the commonly accepted view that non-optimal packing conformations in proteins is associated with an enthalpic cost \cite{MACC07}, which can be a major rate-limiting factor in protein folding \cite{Cheung2002,RANK97}. Further, stabilizing hydrophobic interactions are shown to arise from favorable entropic contributions. These results shed light on the effect of peptide composition and its interactions with the solvent.
\section{Results}
\subsection{Potential of mean force}
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{main.pdf}
\caption{(Panels a--d) PMFs of homopeptide dimers at 298 K (solid lines). Peptides are made of glycine (first column), alanine (second column), valine (third column), and leucine (fourth column) residues. Enthalpies and entropies are shown in dashed and dashed-dotted lines. (Panels e--h) Number of hydrogen bonds involving peptide-peptide (solid lines), peptide-water (dashed lines), and water-water (dashed-doted lines) atoms. Net number of hydrogen bonds are shown using circles. (Panels i--l) LJ energy (solid lines), electrostatic energies (dashed lines), and total potential energy (dashed-dotted lines). Quantities computed at $\xi=1.5$ nm are used as our reference, i.e., zero value.}
\label{PMF}
\end{figure}
Fig.~\ref{PMF} (a-d) shows the PMF at 298 K for the interaction of glycine, alanine, valine, and leucine homodimers as a function of the distance $\xi$ between the peptide centers of mass. The global minimum in the PMF is observed at $\xi = 0.5$ nm for the valine and leucine homodimers and at $\xi = 0.45$ nm in the case of the alanine homodimer. These minima can be mapped to the $\beta$-sheet structures in which one interpeptide hydrogen bond forms per residue (Fig.~\ref{PMF} e-h). The global minimum for the glycine homodimer occurs at $\xi = 0.40$ nm. At this distance, only one-third of the residues form hydrogen bonds. Free energies at these global minima, relative to the free energies at non-interacting distances (i.e., $\xi = 1.5$ nm) are -3.25 kJ/mol, -10.0 kJ/mol, -8.25 kJ/mol, and -12.5 kJ/mol per amino acid for glycine, alanine, valine, and leucine respectively. In addition to the global minimum, PMFs of large aliphatic side-chains also show a well-defined second minimum at a distance of 0.75 nm (valine) and 0.80 nm (leucine). These second minima can be mapped to structures in which side-chains of one peptide interpenetrates the side-chains of the neighboring peptide. Previous studies using amino acid substitution experiments have shown that the free energies required to stabilize $\beta$-sheet structures cannot be attributed solely to amino acid propensities and that they also depend on the position of $\beta$-strands \cite{MINO94}. This suggests that free energies computed in this study may also be context-dependent and should therefore not be considered for their absolute values.
\subsection{Entropy and enthalpy}
To evaluate contributions of microscopic factors, we compute enthalpy and entropy as a function of $\xi$ from the PMFs at four temperatures (278 K, 298 K, 338 K, and 378 K) through a fit to this thermodynamic relation:
\begin{eqnarray}
\mathrm{PMF}(T,\xi) &=& \Delta H_o(\xi) - T\Delta S_o (\xi) +\\ \nonumber
& & \Delta C_{o,p} (\xi) \left[ (T - T_o) - T \log\Big(\frac{T}{T_o}\Big) \right] ,
\end{eqnarray}
where $\Delta S_o$, $\Delta H_o$, and $\Delta C_{o,p}$ correspond, respectively, to changes in entropy, enthalpy, and heat capacity at the reference temperature $T_o = 298$ K. Dashed-dotted and dotted lines in Fig.~\ref{PMF} (a-d) correspond to $-T_o\Delta S_o$ , and $\Delta H_o$ respectively.
Our results show that interactions involving homodimers of glycine or alanine (Fig.~\ref{PMF}{\it a-b}) are favored by enthalpy while entropy opposes it. In contrast, interactions involving valine or leucine homodimers are disfavored by enthalpy while they are favored by entropy (Fig.~\ref{PMF}{\it c-d})---see also supplemental material \cite{sup}. These opposing behaviors indicate that adding large hydrophobic sidechains to the backbone changes the microscopic mechanism of peptide interactions. In the presence of small sidechains, peptide interactions are dominated by backbone properties that involve Lennard-Jones and electrostatic interactions between peptides atoms. Interactions involving large hydrophobic residues are dominated by the overall entropy of neighboring water molecules, which is maximized when these residues are brought close to each other.
\subsection{Hydrogen bonds}
As the extended peptide conformations of the homodimers are brought close to each other, the number of interpeptide hydrogen bonds (solid line in Fig.~\ref{PMF} e-h) increases, reaching a maximum at distances corresponding to the global minimum of the PMF ($\xi \sim 0.5$ nm). Geometric constraints restrict the the maximum number of hydrogen bonds that can form between the backbone atoms to one per amino acid. This occurs if the two extended peptides adopt $\beta$-strand conformations \cite{PAUL51}. In the case of alanine, valine, and leucine homodimers, this number is observed, indicating that $\beta$-sheets are formed at the global minimum of the PMF. For glycine, only one-third of all possible interpeptide hydrogen bonds are formed, suggesting that despite being stretched, these peptides do not form $\beta$-sheets.
The number of peptide-water hydrogen bonds (dashed-dotted lines) decreases when peptides of the dimer approach each other. This reflects the transfer of water molecules from the neighborhood of the dimer towards the bulk solvent. A consequence of this transfer is an increase in the number of water-water hydrogen bonds (dashed lines) due to water added to the bulk. We note that the change in the net hydrogen bond number (blue circles) is negligible for all values of $\xi$. This results from the loss, on average, of two peptide-water hydrogen bonds and the formation of one peptide-peptide bond and one water-water hydrogen bond. This almost perfect compensation of hydrogen bonding can be attributed to two factors: (1) the ability of peptides to satisfy hydrogen bonds between polar groups in the backbone when in $\beta$-strand conformations and (2) the polar nature of the solvent which can penetrate cavities formed within the dimer due to the small size of its molecules, and engage in hydrogen bonding with the unsatisfied polar groups of the backbone \cite{CHAN05}.
The fact that the net number of hydrogen bonds does not change for all values of $\xi$ suggests that the formation of $\beta$-sheets is not driven primarily by hydrogen bonding. This is further confirmed upon computing the electrostatic energy of the system (dashed line in Fig.~\ref{PMF} i-l). In all-atom models, hydrogen bonds stem from the sum of electrostatic terms between partial charges in amide and carbonyl groups of the backbone. If these terms played a key role in the formation of $\beta$-sheets, one would expect the electrostatic energy to be a minimum at $\xi \sim 0.5$ nm. However, this is not the case and for all systems studied, the electrostatic potential is either slightly positive (glycine and alanine) or it peaks strongly (valine and leucine) at the global minimum of the PMF. In addition, for valine and leucine, the electrostatic contribution is the dominant term of the potential energy (dashed-dotted lines) accounting for an unfavorable potential energy to $\beta$-sheets formation. The main energetic components contributing to the potential energy for glycine and alanine peptides are the sum of all van der Waals interactions (solid lines) that are favorable for the formation of $\beta$-sheets.
\subsection{Lennard-Jones (LJ) interactions and Solvent Accessible Surface Area (SASA)}
\begin{figure}[h]
\centering
\includegraphics[scale=0.32]{LJ.pdf}
\caption{van der Waals energy and SASA for glycine (a,e), alanine (b,f), valine (c,g), and leucine (d,h). (First row) peptide-peptide (dashed lines) and peptide-water (dashed-dotted lines) interactions correlate with SASA (solid lines). (Second row) LJ interactions involving peptide atoms (dashed-dotted lines) and water molecules solely (dashed lines) account for the total LJ energy (solid lines). Quantities computed at $\xi=1.5$ nm are used as our reference.}
\label{LJ}
\end{figure}
Fig.~\ref{LJ} illustrates the dependence of LJ interactions and SASA on $\xi$. The top row (a-d) shows the sum of LJ interactions involving the peptide chains of the dimer E$_{\mathrm{pp}}$ (dashed lines) and the negative of LJ interactions involving simultaneously the peptide and solvent atoms -E$_{\mathrm{pw}}$ (dashed-dotted lines). Both E$_{\mathrm{pp}}$ and -E$_{\mathrm{pw}}$ decrease when peptides are brought close to each other. These interaction energies show good correlations with the SASA (solid lines).
Fig.~\ref{LJ}{\it e-h} shows the sum of the peptide-peptide (E$_{\mathrm{pp}}$) and peptide-water ( E$_{\mathrm{pw}}$) LJ interactions (dashed-dotted lines)
These interactions show strikingly different behavior for homodimers composed of small (glycine and alanine) and larger sidechains (valine or leucine). For homodimers with small sidechains, E$_{\mathrm{pp}}$ + E$_{\mathrm{pw}}$ favors $\beta$-sheet formation while this is not observed for homodimers with large sidechains. These interactions are also unfavorable at intermediate interpeptide distances for the valine and leucine homodimers, resulting from repulsive interactions arising due to the close proximity of sidechain atoms. Further, the LJ term involving water-water interactions, E$_{\mathrm{ww}}$, has a trend opposite to that of E$_{\mathrm{pp}}$ + E$_{\mathrm{pw}}$: it favors $\beta$-sheet formation in the valine and leucine dimers while it plays no role for the glycine and alanine homodimers.
In the case of valine and leucine dimers, these results are consistent with a picture for $\beta$-sheet formation in which water molecules found in the space between sidechains are released into bulk water accounting for a decrease in E$_{\mathrm{ww}}$. In this process, the number of peptide-peptide contacts increases corresponding to a decrease in E$_{\mathrm{pp}}$ while the number of water-peptide contacts decreases, corresponding to an increase in E$_{\mathrm{pw}}$. These opposing behaviors in E$_{\mathrm{pp}}$ and E$_{\mathrm{pw}}$ account for the observed negligible change in E$_{\mathrm{pp}}$ + E$_{\mathrm{pw}}$ observed at $\xi \sim$ 0.5 nm (i.e., at $\beta$-sheet structures). For homodimers with small sidechains, water molecules can partially permeate the space between sidechains when peptides are in $\beta$-sheet conformations \cite{DIAS11}, resulting in an increase in peptide-peptide contacts while retaining some peptide-water contacts. As a result, E$_{\mathrm{pp}} $ + E$_{\mathrm{pw}}$ is favorable while E$_{\mathrm{ww}}$ does not change significantly.
\subsection{Electrostatic energy and cavities}
\begin{figure}[h]
\centering
\includegraphics[scale=0.32]{Potene.pdf}
\caption{(a-c) Electrostatic energy and total volume for alanine, valine, and leucine respectively. (d-f) Electrostatic energy contributions from peptide-peptide (solid lines), peptide-water (dashed-dotted lines) and water-water interactions (dotted lines) for alanine, valine, and leucine respectively. Quantities computed at $\xi=1.5$ nm are used as our reference.}
\label{Volume}
\end{figure}
Fig.~\ref{Volume} (a-c) shows the dependence of the electrostatic energy (solid lines) and total volume (dashed lines) on $\xi$. These quantities are not shown for the glycine homodimer because they do not vary significantly as a function of $\xi$. $\beta$-sheet formation is associated with an increase in the volume in the order of $\Delta V = 0.02$ nm$^3$ for the alanine homodimer (panel a) and $\sim\!\!0.15 \mathrm{nm}^3$ for the valine and leucine homodimers (panels b and c). The formation of interpenetrating sidechain configurations also leads to a similar increase in volume ($\sim\!\!0.15 \mathrm{nm}^3$) for valine and leucine homodimers. A minimum in the volume is observed at distances of $\xi\!\!\sim$0.75 nm at which interpenetrating sidechain configurations are highly compact. Furthermore, an excellent correlation is observed between the electrostatic energy and the total volume (Fig.~\ref{Volume} (a-c). To decipher the origin of this correlation, we show peptide-peptide, peptide-water, and water-water contributions to the electrostatic energy in Fig.~\ref{Volume} (d-f).
At interpeptide distances corresponding to $\beta$-sheets, magnitudes of favorable peptide-peptide (solid lines) and water-water (dashed lines) electrostatic energies are almost identical. When added together their sum is slightly less than the magnitude of unfavorable peptide-water (dashed-dotted lines) electrostatic energy. This accounts for the net unfavorable electrostatic energy associated with $\beta$-sheet formation (solid lines in panels {\it a-c}). If we stipulate that hydrogen bond strengths are not strongly dependent on the atomic species interacting in water, then hydrogen bonds only account for a negligible change in the electrostatic energy. In this case, the unfavorable electrostatic energy for $\beta$-sheet formation can only stem from the loss of sidechain-water electrostatic interactions, as water molecules in between sidechains of neighboring peptides are transferred into the bulk. Accordingly, we observe an increase in peptide-water electrostatic energies (dashed-dot lines) with sidechain size at $\xi = 0.5$ nm (385 kJ/mol for alanine, 400 kJ/mol for valine, and 450 kJ/mol for leucine). Furthermore, this loss of sidechain-water interactions accounts for an increase in the volume of the system, as cavities are left in the space between sidechains. We note that sidechain-sidechain electrostatic interactions between non-polar residues are negligible.
The observation of large volume and large electrostatic energy at $\xi = 1$ nm (panels {\it b} and {\it c}) correlates with unfavorable water-water electrostatic energy (dashed lines in panels {\it e-f}). They correspond to loosely packed interpenetrating sidechain configurations with large cavities between backbones. These structures have a large SASA (Fig.~\ref{LJ}), in addition to a large number of shell-water with less favorable water-water interactions compared to bulk water. Characteristic conformations at interpeptide distances of 0.5, 0.75 and 1.0 nm are shown in Fig.~\ref{DOW}.
As $\xi$ decreases, interpenetrating configurations become more compact as sidechains are packed against each other filling the cavities between backbones (Fig.~\ref{DOW}{\it c,d}). This accounts for a decrease in both SASA and the number of shell water. At $\xi =0.75$ nm, the SASA is a minimum for interpenetrating sidechain configurations (see Fig.~\ref{LJ}),corresponding to a minimum in the water-water electrostatic energy (Fig.~\ref{Volume}{\it b,c}). Note that for interpenetrating configurations, peptide-water electrostatic energy increases with decreasing $\xi$ (dashed-doted lines). This can be rationalized in terms of the reduction in the number of shell water.
\hspace{-5cm}
\begin{figure}[h]
\includegraphics[scale=0.4, angle=0]{d2.pdf}
\caption{Representative conformations of valine (column 1) and leucine (column 2) homodimers at interpeptide distances of 0.5 (a,b), 0.75 (c,d), and 1.0 nm (e,f) respectively. Sidechains in a-b are hidden for clarity, while blue lines represent the inter-strand hydrogen bonds.}
\label{DOW}
\end{figure}
\section{Discussion and conclusions}
In this study, we have simulated homodimers made of glycine, alanine, valine and leucine to determine the role of hydrophobic and hydrogen bonding interactions in the formation and stability of $\beta$-sheet conformations. Our results indicate that hydrogen bonding does not play a significant role in the formation of $\beta$-sheets for peptides in extended conformations: the net number of hydrogen bonds in the system does not change as a function of peptide dimer distance and electrostatic energies peak at distances corresponding to $\beta$ structures. If hydrogen bonds were dominant interactions stabilizing $\beta$-sheets, electrostatic energies would be a minimum. One limitation of this work is that we consider non-interacting configurations (large values of $\xi$) as extended conformations, instead of the unfolded conformations of proteins. While these extended conformations are part of the random coil ensemble, our structures do not include individual configurations that might have unsatisfied hydrogen bonds, which could contribute hydrogen bonds during protein folding.
The results of this work can be applied to explain the phenomenon of cold denaturation. The most accepted explanation for this phenomenon relates cold denaturation to decreasing stabilities of hydrophobic interactions upon cooling. This leads to greater exposure of the dry protein core to water, accounting for unfolding at low temperatures. Recent studies have shown cold denaturation in $\beta$-hairpin peptides which have a limited buried hydrophobic core \cite{DYER05}. Simulations with implicit solvent have attributed this phenomenon to reduced backbone hydrogen bond stability upon cooling \cite{SHAO13}. In our simulations, the stability of $\beta$-sheets made of valine or leucine residues decreases with decreasing temperature, accounting for the entropy-driven behavior shown in Fig.~\ref{PMF}. This is a requirement for cold denaturation to occur.
Our simulations therefore support the conventional explanation for cold denaturation of $\beta$-sheet peptides containing large hydrophobic sidechains. In contrast, the stability of $\beta$-sheet peptides made of alanine amino acids in our simulations, increased with decreasing temperature, accounting for the enthalpy-driven behavior shown in Fig.~\ref{PMF}. This suggests that cold denaturation may not occur for alanine peptides.
$\beta$-sheets are core structures in fibrillar aggregates of amyloid peptides \cite{NELS05}, which can vary significantly in amino acid composition. Based on our simulations, we speculate that fibril formation is enthalpy-driven for sequences composed of short sidechains, like the residues 113-120 (AGAAAAGA) of Syrian hamster prion protein \cite{DAID04}. Contributions to this enthalpy arise primarily from van der Waals interactions between peptide atoms. On the other hand, fibril formations in sequences made predominantly of large hydrophobic sidechains, for example residues 68-78 (GAVVTGVTAVA) in human $\alpha$-synuclein \cite{DAVI98,TREX09}, are entropy-driven, as observed for our leucine and valine homodimers. Microscopically, water molecules are responsible for this entropic interaction \cite{Fran45,DIAS11_hydrophobic}. Hence, we expect that changes in the solvent environment will have a stronger effect on fibrillar structures of peptides containing large hydrophobic residues compared to alanine-based peptides.
In summary, we have used homodimeric peptide systems to clarify the role of hydrophobic and hydrogen bonding interactions in the stability of $\beta$-sheet conformations in peptides. Free energy of the interactions between the homodimeric peptides is calculated using umbrella sampling with the interpeptide distance as an order parameter. We show that the free energies of the four systems studied are characterized by two minima corresponding to $\beta$-sheet structures and peptide conformations with interpenetrating sidechain configurations. We determined the energetic contributions to these minima and found that (1) interpeptide hydrogen bonds do not contribute significantly to the stability of sheets; (2) electrostatic energies correlate with the volume of the system; (3) $\beta$-sheet formation in peptides with large hydrophobic sidechains (valine and leucine) is entropy-driven while they are enthalpy-driven in peptides with small sidechains (alanine and glycine). These results shed light and contribute to answering longstanding questions related to roles played by hydrogen bonds and hydrophobic interactions in the stability of protein structures. We are confident that the conceptual framework and methodology developed here will facilitate elucidation of these fundamental questions through further efforts in theory and experiment.
\section{Materials and Methods}
\begin{figure}
\includegraphics[scale=0.8, angle=0]{setup.pdf}
\vspace{-2cm}
\caption{Schematic representation of the simulation box with the two peptides. $\xi$ is the order parameter corresponding to the distance between the centers of mass of the two peptides. Peptides are made of ten amino acids with residues ``1'' and ``10'' attached ``covalently'' to each other through the boundary in the z-direction, forming an infinite chain.}
\label{setup}
\end{figure}
In this work, we simulate two infinite homopeptides at different distances $\xi$ between their centers of mass (Fig.~\ref{setup}). The two peptides are placed in anti-parallel orientations resulting in $\beta$-sheet formation at small values of $\xi$. Four homodimeric peptide systems composed of glycine, alanine, valine and leucine residues respectively were used. Each of the homodimeric Peptides, which are ten amino acids long, are made infinite through periodic boundary in the z-direction: the carbonyl-group of residue 1 is attached to the amine-group of residue 10. The use of infinite chains eliminates effects from chain ends and all amino acids become equivalent, resembling amino acids in the middle of a strand. A potential constraint of this setup is that it does not allow the formation of twists which have been shown to affect the stability of $\beta$-sheets through increased sidechain interactions \cite{CHOU83,CHOU83b}. However, this limitation does not affect our conclusion: if twists were added to our model, sidechain interactions would play a more important in the stability of $\beta$-sheets with respect to hydrogen bonds than the one computed here.
Peptides are immersed in a box of $\sim$5,500 water molecules (TIP3P) and a pressure is applied along {\it z}-direction (main axis of the peptides) to keep the box from collapsing. The magnitude of this pressure is chosen to ensure an average peptide length of is 3.5 nm. A pressure of 1 atm is applied along {\it x} and {\it y} directions to account for water density at ambient pressure. Simulations are carried out using GROMACS and CHARMM27 forcefield. Temperature and Pressure were controlled using the velocity-rescale thermostat ($\tau_{\mathrm{T}}$ = 1 ps) and the Parrinello--Rahman barostat ($\tau_{\mathrm{P}}$ = 1 ps), respectively. Simulations were performed with a time step of 2 fs and the neighbor list was updated every 10 steps. Electrostatics were treated by the Smooth Particle Mesh Ewald with a grid spacing of 0.13 nm and a 1.3 nm real-space cutoff \cite{BJEL10}. We use umbrella sampling with a spring constant of 5,000 kJ/mol to compute the properties of the system at different windows of $\xi$ ranging from 0.4~nm~ to 1.7~nm. Windows are separated from each other by 0.05~nm. Simulations lasted at least 100 ns in each window. PMFs of homodimers were computed using the Weighted Histogram Analysis Method (WHAM).
To define hydrogen bonds we employ a commonly used geometrical definition in which these bonds are formed when the distance between donor (D) and acceptor (A) is smaller than 0.4 nm and the angle H-D-A is smaller than 30$^o$. We used g\_hbond provided in the software package GROMACS for this calculation. In all figures, computed values of hydrogen bonds and energies at $\xi=1.5$ nm were used as our zero reference. In the calculation of quantities involving solute--solvent and solvent--solvent atoms, all solvent (water) molecules were taken into account. However, since we only report differences with respect to quantities computed at $\xi = 1.5$ nm, the effect of bulk water is averaged out. Using all water molecules resolves the problem of having to define a cut-off for solvent molecules in the calculation of solute--solvent and solvent--solvent properties.
\section{Acknowledgment}
I would like to thank Markus Miettinen for providing us with the topology files of periodic peptides, Roland Netz for insightful discussions and for hosting my stay at Freie Universit\"at Berlin when I started running simulations, and Normand Mousseau for hosting my stay at the Universit\'e de Montr\'eal when the ideas of this project were taking shape. This project was partially funded by the Volkswagen Foundation. I would like to thank Compute Canada for computational resources.
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In the past I shot with zooms but my main arsenal is now prime lenses. I believe that not only do primes offer superior quality (with a few exceptions) and portability, but most importantly they allow you to think creatively. Some of the legends of photography shot with one lens their entire career; others "limited" themselves to two, maximum three lenses. This way there were able to train their eye to see and compose, which eventually led to superb work.
Therefore, when Fuji was kind enough to let me try a brand new Fujinon XF 10-24 F4 R OIS lens, I approached the subject with a dose of zoom hostility and prejudgment. As a fan and heavy user of the superb XF 14mm F2.8 R lens, I wondered if I would ever enjoy this much bigger and heavier addition to the Fuji X-series line-up.
On paper there is not much difference in focal length between 10mm and 14mm, some would argue. WRONG! In a wide-angle world, it makes a considerable difference. And for a serious landscape photographer, the wider you go, the more impact you create. Of course this is assuming you know how to craft grand vistas with such a demanding tool.
Having said that, I have spent the weekend shooting almost all my imagery at the wide-end of this lens, 10mm. What an impact it creates! While it is too early to evaluate all the technical attributes and write a full review of this lens, I can say with full confidence that if you are a serious landscape photographer, this lens should find its way into your bag.
Here are some recent images shot with the X-T1 and XF 10-24 F4 R OIS. Stay tuned.
Just, beautiful very involving and beautiful.
Straight from the camera. No adjustments.
This is not a lens distortion. It is a perspective distortion, due to the angle of view of the image.
You have created some wonderful images with the X-T1 and the XF 10-24mm lens. I can not wait to get my hands on a copy. Thank you so much for sharing this with us.
Great photos Olaf – I've pre-ordered the 10-24mm and I definitely think that I've made the right decision, would be interested in hearing about corner softness and CA (those were my biggest frustrations with the Nikon 16-35mm f/4). Looks like you were in my neck of the woods, next time you're in the Comox Valley give me a shout. Would love to sit down over coffee and chat, or could take you out to some of the local spots that are a bit more off the beaten track.
I look forward to your thoughts about the 10-24 vs. 14. I held off buying the 14 on the slim chance that I would prefer the 10-24 more — or at least enough to put up with the 10-24's larger size and weight.
Can't wait to get this lens in my line up. Thanks for the write up! Enjoyed the images, great country!
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Drug addiction is a mental disorder that never goes away. People can learn to manage it, but they can never cure it. Since addiction doesn't go away, learning about relapse prevention is important. In fact, many rehab centers focus on preventing relapse to reduce the risks of overdose and death.
Relapse prevention focuses on life after rehab. To prevent relapse, people have to identify triggers and cognitive behaviors that may cause it. Identifying triggers makes it easier for people to avoid situations that might lead to drug use.
For example, stressful situations can trigger a relapse. When you know that stress makes you want to use, you take proactive measures to lower stress. However, it's not just emotions and feelings that cause a relapse. Sometimes, hanging out with certain people or visiting specific places can spur relapse.
Programs that teach people how to prevent relapse take both short- and long-term sobriety into account. Learning preventative measures can help people avoid problem behavior. Preventing relapsing is a part of sober living.
Relapse doesn't usually happen all of a sudden. People often struggle with the thought of using again long before they actually do. Overall, there are three stages of relapse.
Emotional Relapse—During the emotional relapse stage, people don't necessarily think about using. However, their actions and feelings subconsciously increase the potential for relapse. If they don't have relapse prevention training, they can easily miss signs such as mood swings, isolation, and defensiveness.
The symptoms of emotional relapse are very similar to post-acute withdrawal. Understanding this condition makes it easier to identify the onset of emotional relapse. Failure to identify emotional relapse makes the phases progress quickly.
Mental Relapse—The next stage is mental relapse. During this stage, people think about using drugs. Mental relapse feels like there's a war inside their minds. Part of them wants to use, but the other doesn't.
Physical Relapse—The most dangerous phase is physical relapse. It doesn't typically take people long to get to this stage. They physically start to use drugs again and need detox and rehab again, to recover.
During a physical relapse, people usually feel bad about the choices they make. They can enter a deep depression because they feel like they let their families and friends down. The key is to remember that relapse is common. Giving in to depression can make the relapse worse.
Instead, they need to get on their feet and renter rehab. They have to focus on relapse prevention to prevent the same outcome.
Once people reach the third stage of relapse, they have no choice but to seek help from a rehab center once again. However, some relapse prevention techniques can help them avoid getting to this point. Using these techniques can mean the difference between staying sober and relapse.
Wait 30 Minutes—People need to give themselves 30 minutes before they use drugs again. While this may seem like an odd technique, 30 minutes makes a huge difference. Most drug cravings last about 15 to 20 minutes. If they can get over that hump, they're less likely to use.
Take Recovery One Day at a Time—Don't think about staying clean forever. People often feel overwhelmed when they think about staying clean forever. Instead, they should think about staying clean for a day.
They can tell themselves, "I'm not going to use today." Then they can take their sobriety one day at a time.
Relax and Stay Calm—Staying calm and focusing on relaxation is just as important as not using drugs. Most people admit to relapsing when they feel stressed out. Focus on learning calming techniques that can reduce stress.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,234
|
\section{Introduction}\label{sec:1}
Support vector machines and related kernel based methods can be considered
as a hot topic in machine learning because they
have good statistical and numerical properties under weak assumptions
and have demonstrated their often good generalization properties in many
applications,
see e.g.
\cite{Vapnik1995,Vapnik1998},
\cite{ScSm2002}, and
\cite{SteinwartChristmann2008a}.
To our best knowledge, the original SVM approach by
\cite{BoserGuyonVapnik1992} was derived
from the generalized portrait algorithm invented earlier by
\cite{VapnikLerner1963}.
Throughout the paper, the term SVM will be used in the broad sense, i.e.
for a general convex loss function and a general kernel.
SVMs based on many standard kernels as for example the Gaussian
RBF kernel are nonparametric methods.
The finite sample distribution of many
nonparametric methods is unfortunately unknown because the distribution ${{\rm P}}$ from which the
data were generated is usually completely unknown and because there are often
only asymptotical results describing the consistency or the rate
of convergence of such methods known so far.
Furthermore, there is in general \emph{no} uniform rate of convergence
for such nonparametric methods due to the famous no-free-lunch theorem,
see \cite{Devroye1982} and \cite{DevroyeGyoerfiLugosi1996}.
Informally speaking, the no-free-lunch theorem states that, for sufficiently
malign distributions, the average risk of any statistical
(classification) method may tend arbitrarily slowly to zero.
Theses facts are true for SVMs. SVMs are known to be universally consistent and fast rates of convergence are known for broad
\emph{subsets} of all probability distributions.
The asymptotic normality of SVMs was shown recently by \cite{Hable2012}
under certain conditions.
Here, we apply a different approach to SVMs, namely Efron's bootstrap.
The goal of this paper is to show that bootstrap approximations
of SVMs which are based on a general convex and smooth loss function
and a general smooth kernel are consistent under mild assumptions;
more precisely, convergence in outer probability is shown.
This result is useful to draw statistical decisions based on SVMs, e.g. confidence intervals, tolerance intervals and so on.
We mention that both the sequence of SVMs and the sequence of their
corresponding risks are qualitatively robust under mild assumptions, see
\cite{ChristmannSalibianBarreraVanAelst2013}.
Hence, Efron's bootstap approach turns out to be quite successful
for SVMs from several aspects.
The rest of the paper has the following structure.
Section 2 gives a brief introduction into SVMs.
Section 3 gives the result.
The last section contains the proof and related results.
\section{Support Vector Machines}\label{sec:2}
Current statistical applications are characterized by a wealth of large
and high-dimensional data sets.
In classification and in regression problems there is a variable of main interest,
often called ``output values'' or ``response'',
and a number of potential explanatory variables, which are often
called ``input values''.
These input values are used to model the observed output values or to predict future output values.
The observations consist of $n$ pairs $(x_1, y_1)$, \ldots, $(x_n, y_n)$,
which will be assumed to be independent realizations of a random pair $(X,Y)$.
We are interested in
minimizing the risk or to obtain a function $f: {\cal X} \to {\cal Y}$ such that $f(x)$
is a good predictor for the response $y$, if $X=x$ is observed.
The prediction should be made in an automatic way.
We refer to this process of determining a prediction method as ``statistical machine learning'',
see e.g. \cite{Vapnik1995, Vapnik1998, ScSm2002, CuckerZhou2007, SmaleZhou2007}.
Here, by ``good predictor'' we mean that $f$ minimizes the expected loss, i.e. the risk,
$$
\mathcal{R}_{L, \rmP}{f} =
{\mathbb{E}}_{{\rm P}} \left[ L \left( X, Y, f(X) \right) \right],
$$
where ${{\rm P}}$ denotes the unknown joint
distribution of the random pair $(X,Y)$ and $L : {\cal X} \times {\cal Y}
\times \R \to [0, +\infty)$ is a fixed loss function.
As a simple example, the least squares loss $L(X, Y, f(X)) = (Y - f(X))^2$ yields the optimal predictor $f(x) = {\mathbb{E}}_{{\rm P}}(Y|X=x)$, $x\in{\cal X}$.
Because ${{\rm P}}$ is unknown, we can neither compute
nor minimize the risk $\mathcal{R}_{L, \rmP}{f}$ directly.
Support vector machines, see
\cite{VapnikLerner1963}, \cite{BoserGuyonVapnik1992},
\cite{Vapnik1995, Vapnik1998},
provide a highly versatile framework to perform
statistical machine learning in a wide variety of setups. The minimization of regularized empirical risks over reproducing kernel Hilbert spaces was already considered e.g. by \cite{PoggioGirosi1990}.
Given a kernel $k:
{\cal X} \times {\cal X} \to \R$ we consider predictors $f \in H$, where $H$
denotes the corresponding reproducing kernel Hilbert space of functions from
$\cX$ to $\R$. The
space $H$ includes, for example, all functions of the form $f(x) = \sum_{j=1}^m
\alpha_j \, k(x, x_j)$ where $x_j$ are arbitrary elements in ${\cal X}$ and
$\alpha_j \in \R$, $1 \le j \le m$. To avoid overfitting, a support vector
machine $f_{L,{{\rm P}}, \lambda }$ is defined as the solution of a regularized risk minimization problem. More precisely,
\begin{equation} \label{pop.min}
f_{L,{{\rm P}}, \lambda } \, = \, \arg \inf_{f \in H} \ {\mathbb{E}}_{{{\rm P}}} L \left( X, Y, f(X) \right) \, + \,
\lambda \, \| f \|^2_H \, ,
\end{equation}
where $ \lambda \in(0,\infty)$ is the regularization parameter.
For a sample $D= ((x_1, y_1), \ldots,$ $(x_n, y_n))$ the corresponding estimated function is given by
\begin{equation} \label{sample.min}
f_{L,\Dn, \lambda } \, = \, \arg \inf_{f \in H} \ \frac{1}{n} \sum_{i=1}^n L \left( x_i, y_i, f(x_i) \right) \, + \,
\lambda \, \| f \|^2_H \, ,
\end{equation}
where $\Dn$ denotes the empirical distribution based on $D$ (see~\eqref{emp_measure} below).
Note that the optimization problem \eqref{sample.min} corresponds to \eqref{pop.min} when using $\Dn$ instead of ${{\rm P}}$.
Efficient algorithms to compute $\hat{f}_n:=f_{L,\Dn, \lambda }$ exist for a number of different loss functions.
However, there are often good reasons to consider other convex loss functions, e.g. the hinge loss
$L(X,Y,f(X))=\max\{1-Y\cdot f(X),0\}$ for binary classification purposes or
the $\epsilon$-insensive loss $L(X,Y,f(X))=\max\{0, |Y-f(X)|-\epsilon\}$ for regression purposes, where $\epsilon>0$.
As these loss functions are not differentiable, the logistic loss functions
$L(X,Y,f(X))=\ln(1+\exp(-Y\cdot f(X)))$ and $L(X,Y,f(X))=-\ln(4 e^{Y-f(X)}/(1+e^{Y-f(X)})^2)$ and Huber-type loss functions are also
used in practice. These loss functions can be
considered as smoothed versions of the previous two loss functions.
An important component of statistical analyses concerns quantifying and
incorporating uncertainty (e.g. sampling variability) in the reported
estimates. For example, one may want to include confidence bounds along the
individual predicted values $\hat{f}_n(x_i)$ obtained from \eqref{sample.min}.
Unfortunately, the sampling distribution of the estimated function $\hat{f}_n$ is unknown.
Recently, \cite{Hable2012} derived the asymptotic distribution of SVMs under some mild conditions.
Asymptotic confidence intervals based on those general results are always symmetric.
Here, we are interested in approximating the finite sample distribution of SVMs
by Efron's bootstrap approach, because confidence intervals based on the bootstrap approach can be asymmetric.
The bootstrap \cite{Efron1979} provides an alternative way to estimate the
sampling distribution of a wide variety of estimators. To fix ideas, consider a
functional $S : {\cal M} \to {\cal W}$, where ${\cal M}$ is a set of
probability measures and ${\cal W}$ denotes a metric space. Many estimators
can be included in this framework. Simple examples include the sample mean
(with functional $S({{\rm P}}) = \int Z \, d{{\rm P}}$) and M-estimators (with functional
defined implicitly as the solution to the equation ${\mathbb{E}}_{{\rm P}} \Psi(Z, S({{\rm P}})) = 0$).
Let $\B(\cZ)$ be the Borel $ \sigma $-algebra on $\cZ = \cX \times \cY$ and denote
the set of all Borel probability measures on
$(\cZ,\B(\cZ))$ by $\PM(\cZ,\B(\cZ))$.
Then, it follows that \eqref{pop.min} defines an operator
$$
S: \PM(\cZ,\B(\cZ)) \to H, \qquad S(P) = f_{L,{{\rm P}}, \lambda },
$$
i.e. the support vector machine. Moreover, the estimator in \eqref{sample.min} satisfies
$$
f_{L,\Dn, \lambda } = S(\Dn)
$$
where
\begin{equation}
\Dn=\frac{1}{n}\sum_{i=1}^n \delta_{(x_i,y_i)}
\label{emp_measure}
\end{equation}
is the empirical distribution based on the sample
$D=((x_1, y_1), \ldots, (x_n,y_n))$ and $\delta_{(x_i,y_i)}$ denotes the Dirac measure
at the point $(x_i,y_i)$.
More generally, let $Z_i=(X_i,Y_i)$, $i=1,\ldots,n$, be
independent and identically distributed (i.i.d.) random variables with
distribution ${{\rm P}}$, and let
$$
S_n(Z_1, \ldots, Z_n) = S({{\rm P}}_n)
$$
be the corresponding
estimator, where
$$
{{\rm P}}_n=\frac{1}{n}\sum_{i=1}^n \delta_{Z_i}.
$$
Denote the distribution of $S({{\rm P}}_n)$ by $\Law{n}{S; {{\rm P}}} = \Law{}{S({{\rm P}}_n)}$. If ${{\rm P}}$
was known to us, we could estimate this sampling distribution by drawing a
large number of random samples from ${{\rm P}}$ and evaluating our estimator on them.
The basic idea of Efron's bootstrap approach is to replace the unknown distribution ${{\rm P}}$
by an estimate $\hat{{{\rm P}}}$. Here we will consider the natural non-parametric
estimator given by the sample empirical distribution ${{\rm P}}_n$. In other words, we
estimate the distribution of our estimator of interest by its sampling
distribution when the data are generated by ${{\rm P}}_n$. In symbols, the bootstrap
proposes to use
$$
\widehat{ {\cal L}_n (S; {{\rm P}})} = \Law{n}{S; {{\rm P}}_n}.
$$
Since this
distribution is generally unknown, in practice one uses Monte Carlo simulation
to estimate it by repeatedly evaluating the estimator on samples drawn from
$\Dn$.
Note that drawing a sample from $\Dn$ means that $n$ observations
are drawn \emph{with replacement} from the original $n$ observations $(x_1, y_1)$, \ldots, $(x_n, y_n)$.
\section{Consistency of Bootstrap SVMs}\label{sec:consist}
In this section it will be shown under appropriate assumptions that the weak consistency of bootstrap estimators carries over to the Hada\-mard-differentiable SVM functional in the sense that the sequence of ``conditional random laws'' (given $(X_1,Y_1), (X_2,Y_2),\ldots$) of
$\sqrt{n}(f_{L,\Pnhat, \lambda } - f_{L,\Pn, \lambda })$ is asymptotically consistent in probability for estimating the laws of the random elements
$\sqrt{n}(f_{L,\Pn, \lambda } - f_{L,{{\rm P}}, \lambda })$.
In other words, if $n$ is large, the ''random distribution''
\begin{equation}
\mathscr{L}(\sqrt{n}(f_{L,\Pnhat, \lambda } - f_{L,\Pn, \lambda }))
\end{equation}
based on bootstrapping an SVM can be considered as a valid approximation of the unknown finite sample distribution
\begin{equation}
\mathscr{L}(\sqrt{n}(f_{L,\Pn, \lambda } - f_{L,{{\rm P}}, \lambda })).
\end{equation}
\begin{assumption}\label{AC.assump.consist}
Let $\cX\subset\R^d$ be closed and bounded and let $\cY \subset \R$ be closed. Assume
that $k:\cX\times\cX \to \R$ is the restriction of an $m$-times continuously differentiable kernel $\tilde{k}:\R^d\times\R^d\to\R$ such that
$m>d/2$ and $k\ne 0$. Let $H$ be the RKHS of $k$ and let ${{\rm P}}$ be a probability
distribution on $(\cX\times\cY,\B(\cX\times\cY))$. Let $L:\cX\times\cY\times\R\to[0,\infty)$ be a convex,
${{\rm P}}$-square-integrable Nemitski loss function of order $p\in[1,\infty)$ such that the partial derivatives
$$
L'(x,y,t):=\frac{\partial L}{\partial t}(x,y,t)
\qquad {\mbox and} \qquad
L''(x,y,t):=\frac{\partial^2 L}{\partial^2 t}(x,y,t)
$$
exist for every $(x,y,t)\in\cX\times\cY\times\R$. Assume that the maps
$$
(x,y,t) \mapsto L'(x,y,t)
\qquad {\mbox and} \qquad
(x,y,t) \mapsto L''(x,y,t)
$$
are continuous. Furthermore, assume that for every $a\in(0,\infty)$, there is a
$b'_a\in L_2({{\rm P}})$ and a constant $b''_a\in[0,\infty)$ such that, for every $(x,y)\in\cX\times\cY$,
\begin{equation} \label{Hable2012.5}
\sup_{t\in[-a,a]} |L'(x,y,t)| \le b'_a(x,y)
\qquad {\mbox and} \qquad
\sup_{t\in[-a,a]} |L''(x,y,t)| \le b''\,.
\end{equation}
\end{assumption}
The conditions on the kernel $k$ in Assumption \ref{AC.assump.consist} are satisfied for many common kernels, e.g., Gaussian RBF kernel, exponential kernel, polynomial kernel, and linear kernel, but also Wendland kernels $k_{d,\ell}$ based on certain univariate polynomials
$p_{d,\ell}$ of degree $\lfloor d/2\rfloor +3\ell+1$ for
$\ell\in\N$ such that $\ell > d/4$, see \cite{Wendland2005}.
The conditions on the loss function $L$ in Assumption \ref{AC.assump.consist} are satisfied, e.g., for the logistic loss for classification or for regression, however the popular non-smooth loss functions hinge, $ \varepsilon $-insensitive, and pinball are not
covered. However, \cite[Remark 3.5]{Hable2012} described an analytical method
to approximate such non-smooth loss functions up to an arbitrarily good precision $\epsilon>0$ by a convex ${{\rm P}}$-square integrable Nemitski loss function of order $p\in[1,\infty)$.
We can now state our result on the consistency of the bootstrap
approach for SVMs. \newpage
\begin{theorem}\label{AC.BootstrapConsist}
Let Assumption \ref{AC.assump.consist} be satisfied.
Let $ \lambda \in(0,\infty)$. Then
\begin{eqnarray}
\label{AC.Bootconsist1a}
\sup_{h\in{\mathrm{BL}}_1(H)} \bigl|
{\mathbb{E}}_M h\bigl(\sqrt{n}(f_{L,\Pnhat, \lambda } - f_{L,\Pn, \lambda })\bigr)
- {\mathbb{E}} h(S'_{{\rm P}}(\bbG))\bigr| \to 0, ~~~\\
\label{AC.Bootconsist1b}
{\mathbb{E}}_M h\bigl(\sqrt{n}(f_{L,\Pnhat, \lambda } \!-\! f_{L,\Pn, \lambda })\bigr)^*
\!\!-\!\! {\mathbb{E}}_M h\bigl(\sqrt{n}(f_{L,\Pnhat, \lambda } \!-\! f_{L,\Pn, \lambda })\bigr)_* \to 0,~~~
\end{eqnarray}
converge in outer probability, where $\bbG$ is a tight Borel-measurable Gaussian process, $S'_{{\rm P}}$ is a continuous linear operator with
\begin{equation}
S'_{{\rm P}}({\rm Q}) = -K_{{\rm P}}^{-1}\bigl( {\mathbb{E}}_{\rm Q} \bigl( L'(X,Y,f_{L,{{\rm P}}, \lambda }(X))\Phi(X)\bigr) \bigr), \quad {\rm Q}\in\PM(\cX\times\cY)
\end{equation}
and
\begin{equation}
K_{{\rm P}}: H \to H, \quad
f \mapsto 2 \lambda f + {\mathbb{E}}_{{\rm P}}\bigl( L''(X,Y,f_{L,{{\rm P}}, \lambda }(X)) f(X) \Phi(X)\bigr)
\end{equation}
is a continuous linear operator which is invertible.
\end{theorem}
For details on $K_{{\rm P}}$, $S'_{{\rm P}}$, and $\bbG$ we refer to
Lemma \ref{Hable.LemmaA.5}, Theorem \ref{Hable2012.ThmA.8}, and
Lemma \ref{Hable2012.LemmaA.9}.
\section{Proofs}\label{sec:proofs}
\subsection{Tools for the proof of Theorem \ref{AC.BootstrapConsist}}\label{Proof.sec:AC.BootstrapConsist}
We will need two general results on bootstrap methods proven in \cite{VandervaartWeller1996} and adopt their notation, see
\cite[Chapters 3.6 and 3.9]{VandervaartWeller1996}.
Let $\Pn$ be the empirical measure of an i.i.d. sample $Z_1,\ldots Z_n$ from a probability distribution ${{\rm P}}$. The \emph{empirical process} is the signed measure
$$
\Gn=\sqrt{n}(\Pn-{{\rm P}}).
$$
Given the sample values, let ${\hat{Z}}_1,\ldots,{\hat{Z}}_n$
be an i.i.d. sample from $\Pnhat$. The \emph{bootstrap empirical distribution} is the
empirical measure
$\Pnhat:=n^{-1} \sum_{i=1}^n \delta_{{\hat{Z}}_i}$,
and the \emph{bootstrap empirical process} is
\begin{equation} \nonumber
\Gnhat = \sqrt{n}(\Pnhat-\Pn)
= \frac{1}{\sqrt{n}}\sum_{i=1}^n (M_{ni}-1)\delta_{Z_i}\,,
\end{equation}
where $M_{ni}$ is the number of times that $Z_i$ is ``redrawn'' from the original sample $Z_1,\ldots Z_n$, $M:=(M_{n1},\ldots,M_{nn})$ is stochastically independent of
$Z_1,\ldots,Z_n$ and multinomially distributed with parameters $n$ and probabilities $\frac{1}{n},\ldots,\frac{1}{n}$.
If outer expectations are computed, stochastic independence is understood
in terms of a product probability space. Let $Z_1, Z_2, \ldots$ be the coordinate projections on the first $\infty$ coordinates of the product
space $(\cZ^\infty, \B(\cZ),{{\rm P}}^\infty) \times (\widetilde{\cZ},\cC,{\rm Q})$ and let the multinomial vectors $M$ depend on the last factor only, see \cite[p.\,345f]{VandervaartWeller1996}.
The following theorem shows (conditional) weak convergence for the
empirical bootstrap, where the symbol $\rightsquigarrow$ denotes the
weak convergence of finite measures.
We will need only the equivalence between $(i)$ and $(iii)$
from this theorem and list part $(ii)$ only for the sake of completeness.
\begin{theorem}[{\cite[Thm. 3.6.2, p.\,347]{VandervaartWeller1996}}] \label{vdVW1996.Thm3.6.2}
Let $\mathcal{F}$ be a class of measurable functions with finite envelope function.
Define $\bbY_n:=n^{-1/2}\sum_{i=1}^n (M_{N_n, i}-1)(\delta_{Z_i}-{{\rm P}})$.
The following statements are equivalent:
\begin{enumerate} \parsep0.2em \itemsep0.2em
\item[(i)] $\mathcal{F}$ is Donsker and ${{\rm P}}^{*}\snorm{f-{{\rm P}} f}_{\mathcal{F}}^2 < \infty$;
\item[(ii)] $\sup_{h\in {\mathrm{BL}}_1} \bigl| {\mathbb{E}}_{M,N} h(\hat{\bbY}_n) - {\mathbb{E}} h(\bbG)\bigr|$ converges outer almost surely to zero
and the sequence ${\mathbb{E}}_{M,N} h(\hat{\bbY}_n)^* - {\mathbb{E}}_{M,N} h(\hat{\bbY}_n)_*$ converges almost surely to zero for every $h\in {\mathrm{BL}}_1$.
\item[(iii)] $\sup_{h\in {\mathrm{BL}}_1} \bigl| {\mathbb{E}}_M h(\Gnhat) - {\mathbb{E}} h(\bbG)\bigr|$ converges outer almost surely to zero
and the sequence ${\mathbb{E}}_M h(\Gnhat)^* - {\mathbb{E}}_M h(\Gnhat)_*$ converges almost surely to zero for every $h\in {\mathrm{BL}}_1$.
\end{enumerate}
Here the asterisks denote the measurable cover functions with respect to $M$, $N$, and
$Z_1, Z_2, \ldots$ jointly.
\end{theorem}
Consider sequences of random elements $\Pn=\Pn(Z_n)$ and
$\Pnhat=\Pnhat(Z_n,M_n)$ in a normed space $\bbD$ such that the sequence
$\sqrt{n}(\Pn-{{\rm P}})$ converges unconditionally and the sequence
$\sqrt{n}(\Pnhat-\Pn)$ converges conditionally on $Z_n$ in distribution to a tight
random element $\bbG$. A precise formulation of the second assumption is
\begin{eqnarray}
\label{vdVW3.9.9a}
\sup_{h\in{\mathrm{BL}}_1(\bbD)} \bigl| {\mathbb{E}}_M h(\sqrt{n}(\Pnhat-\Pn))
- {\mathbb{E}} h(\bbG)\bigr| \to 0,\\
\label{vdVW3.9.9b}
{\mathbb{E}}_M h\bigl(\sqrt{n}(\Pnhat-\Pn)\bigr)^*
- {\mathbb{E}}_M h\bigl(\sqrt{n}(\Pnhat-\Pn)\bigr)_* \to 0,
\end{eqnarray}
in outer probability, with $h$ ranging over the bounded Lipschitz functions,
see \cite[p.\,378, Formula (3.9.9)]{VandervaartWeller1996}.
The next theorem shows that under appropriate assumptions, weak consistency of the bootstrap estimators carries over to any Hadamard-differentiable functional in the sense that the sequence of ``conditional random laws'' (given $Z_1, Z_2,\ldots$) of
$\sqrt{n}(\phi(\Pnhat)-\phi(\Pn))$ is asymptotically consistent in
probability for estimating the laws of the random elements $\sqrt{n}(\phi(\Pn)-\phi({{\rm P}}))$,
see \cite[p.378]{VandervaartWeller1996}.
\begin{theorem}[{\cite[Thm. 3.9.11, p.\,378]{VandervaartWeller1996}}] \label{vdVW1996.Thm3.9.11}
(Delta-method for bootstrap in probability)
Let $\bbD$ and $\bbE$ be normed spaces. Let $\phi:\bbD_\phi \subset \bbD \to \bbE$
be Hadamard-differentiable at ${{\rm P}}$ tangentially to a subspace $\bbD_0$.
Let $\Pn$ and $\Pnhat$ be maps as indicated previously with values in $\bbD_\phi$ such
that $\Gn:=\sqrt{n}(\Pn-{{\rm P}}){\,\rightsquigarrow\,} \bbG$ and that {(\ref{vdVW3.9.9a})}-{(\ref{vdVW3.9.9b})} holds
in outer probability, where $\bbG$ is separable and takes its values in $\bbD_0$.
Then
\begin{eqnarray}
\label{vdVW3.9.10a}
\sup_{h\in{\mathrm{BL}}_1(\bbE)} \bigl| {\mathbb{E}}_M h\bigl(\sqrt{n}(\phi(\Pnhat)-\phi(\Pn))\bigr)
- {\mathbb{E}} h(\phi'_{{\rm P}}(\bbG))\bigr| \to 0,\\
\label{vdVW3.9.10b}
{\mathbb{E}}_M h\bigl(\sqrt{n}(\phi(\Pnhat)-\phi(\Pn))\bigr)^*
- {\mathbb{E}}_M h\bigl(\sqrt{n}(\phi(\Pnhat)-\phi(\Pn))\bigr)_* \to 0,
\end{eqnarray}
holds in outer probability.
\end{theorem}
As was pointed out by \cite[p.\,378]{VandervaartWeller1996}, consistency in probability appears to be sufficient for (many) statistical purposes and the theorem above shows this is retained under Hadamard differentiability at the single distribution ${{\rm P}}$.
We now list some results from \cite{Hable2012}, which will also be essential for the proof of Theorem \ref{AC.BootstrapConsist}.
\begin{theorem}[{\cite[Theorem 3.1]{Hable2012}}]\label{Hable.Thm3.1}
Let Assumption \ref{AC.assump.consist} be satisfied.
Then, for every regularizing parameter $ \lambda _0\in(0,\infty)$, there is a tight, Borel-measurable Gaussian process
$\bbH: \Omega \to H$, $\omega \to \bbH(\omega)$, such that
\begin{equation} \label{Hable2012.6}
\sqrt{n} \bigl( f_{L,\mathbf{D}_n, \lambda _{\mathbf{D}_n}} -
f_{L,{{\rm P}}, \lambda _0}\bigr) {\,\rightsquigarrow\,} \bbH \quad \mbox{~in~} H
\end{equation}
for every Borel-measurable sequence of random regularization parameters
$ \lambda _{\mathbf{D}_n}$ with $\sqrt{n}\bigl( \lambda _{\mathbf{D}_n}- \lambda _0\bigr) \to 0$ in probability.
The Gaussian process $\bbH$ is zero-mean; i.e.,
${\mathbb{E}} \langle f, \bbH \rangle_H=0$ for every $f\in H$.
\end{theorem}
\begin{lemma}[{\cite[Lemma A.5]{Hable2012}}]\label{Hable.LemmaA.5}
For every $F\in B_S$ defined later in {(\ref{DefB_S})},
\begin{equation}
K_F : H \to H, \quad
f \mapsto 2 \lambda _0 f + \int L''(x,y,f_{L,\iota(F), \lambda _0}(x)) f(x) \Phi(x)
d\iota(F)(x,y)
\end{equation}
is a continuous linear operator which is invertible.
\end{lemma}
\begin{theorem}[{\cite[Theorem A.8]{Hable2012}}]\label{Hable2012.ThmA.8}
For every $F_0\in B_S$ which fulfills $F_0(b) < {\mathbb{E}}_{{\rm P}}(b) + \lambda _0$, the map
$S: B_S\to H$, $F\mapsto f_{\iota(F)}$, is Hadamard-differentiable in $F_0$ tangentially to the closed linear span $B_0=\mathrm{cl}(\mathrm{lin}(B_S))$.
The derivative in $F_0$ is a continuous linear operator $S'_{F_0}: B_0 \to H$
such that
\begin{equation}
S'_{F_0}(G) = -K_{F_0}^{-1}\bigl( {\mathbb{E}}_{\iota(G)} ( L'(X,Y,f_{L,\iota(F_0), \lambda _0}(X))\Phi(X))\bigr), ~~~\forall\,G \in \mathrm{lin}(B_S).
\end{equation}
\end{theorem}
\begin{lemma}[{\cite[Lemma A.9]{Hable2012}}]\label{Hable2012.LemmaA.9}
For every data set $D_n=((x_1,y_1),\ldots,$ $(x_n,y_n))\in (\cX\times\cY)^n$, let
$\bbF_{D_n}$ denote the element of $\ell_\infty(\mathscr{G})$ which corresponds to the empirical measure $\Pn:=\bbP_{D_n}$. That is,
$\bbF_{D_n}(g)=\int g\,d\Pn=n^{-1}\sum_{i=1}^n g(x_i,y_i)$ for every $g\in\mathscr{G}$.
Then
\begin{equation}
\sqrt{n} \bigl( \bbF_{\Dn} - \iota^{-1}({{\rm P}})\bigr) {\,\rightsquigarrow\,} \bbG
\quad \mbox{in~} \ell_\infty(\mathscr{G}),
\end{equation}
where $\bbG:\Omega\to \ell_\infty(\mathscr{G})$ is a tight Borel-measurable Gaussian process such that $\bbG(\omega)\in B_0$ for every $\omega \in \Omega$.
\end{lemma}
\subsection{Proof of Theorem \ref{AC.BootstrapConsist}}
The proof relies on the application of Theorem \ref{vdVW1996.Thm3.9.11}. Hence, we have to show the following steps:
\begin{enumerate} \parsep0.2em \itemsep0.2em
\item[1.~] The empirical process $\Gn=\sqrt{n}(\Pn-{{\rm P}})$ weakly converges to a separable Gaussian process $\bbG$.
\item[2.~] SVMs are based on a map $\phi$ which is Hadamard differentiable at ${{\rm P}}$ tangentially to some appropriate subspace.
\item[3.~] The assumptions {(\ref{vdVW3.9.9a})}-{(\ref{vdVW3.9.9b})} of Theorem \ref{vdVW1996.Thm3.9.11} are satisfied. For this purpose we will use Theorem \ref{vdVW1996.Thm3.6.2}. Actually, we will show that part \emph{(i)} of Theorem \ref{vdVW1996.Thm3.6.2} is satisfied which gives the equivalence to part \emph{(iii)}, from which we conclude that {(\ref{vdVW3.9.9a})}-{(\ref{vdVW3.9.9b})} hold true.
For the proof that part \emph{(i)} of Theorem \ref{vdVW1996.Thm3.6.2} is satisfied, i.e., that a suitable set $\mathcal{F}$ is a ${{\rm P}}$-Donsker class and that ${{\rm P}}^{*}\snorm{f-{{\rm P}} f}_{\mathcal{F}}^2 < \infty$, we use several facts recently shown by \cite{Hable2012}.
\item[4.~] We put all parts together and apply Theorem \ref{vdVW1996.Thm3.9.11}.
\end{enumerate}
\emph{Step 1.~}
To apply Theorem \ref{vdVW1996.Thm3.9.11}, we first have to specify the considered spaces $\bbD$, $\bbE$, $\bbD_\phi$, $\bbD_0$ and the map $\phi$.
As in \cite{Hable2012} we use the following notations.
Because $L$ is a ${{\rm P}}$-square-integrable Nemitski loss function of order $p\in[1,\infty)$, there is a function $b\in L_2({{\rm P}})$ such that
\begin{equation}
|L(x,y,t)|\le b(x,y) + |t|^p\, , \qquad (x,y,t)\in \cX\times\cY\times\R.
\end{equation}
Let
\begin{equation} \label{Hable2012.c0}
c_0 := \sqrt{ \lambda _0^{-1} {\mathbb{E}}_{{\rm P}}(b)} +1,
\end{equation}
Define
\begin{equation}
\mathscr{G}:=\mathscr{G}_1 \cup \mathscr{G}_2 \cup \mathscr{G}_3 \,,
\end{equation}
where
\begin{equation}
\mathscr{G}_1:= \{ g: \cX\times\cY \to \R: ~\exists\, z\in \R^{d+1} \mbox{~such~that~} g=I_{(-\infty,z]}\}
\end{equation}
is the set of all indicator functions $I_{(-\infty,z]}$,
\begin{equation}
\mathscr{G}_2:= \left\{ g: \cX\times\cY \to \R \left|
\begin{array}{l}
\exists\, f_0\in H, \exists\, f\in H \mbox{~such~that~} \hnorm{f_0} \le c_0, \\
\hnorm{f}\le 1, g(x,y)=L'(x,y,f_0(x))f(x) ~\forall\, (x,y)
\end{array}
\right.
\right\},
\end{equation}
and
\begin{equation}
\mathscr{G}_3:=\{b\}.
\end{equation}
Now let $\ell_\infty(\mathscr{G})$ be the set of all bounded functions $F: \mathscr{G} \to \R$ with
norm $\inorm{F} = \sup_{g\in\mathscr{G}} |F(g)|$.
Define
\begin{equation} \label{DefB_S}
B_S :=
\left\{
F: \mathscr{G}\to \R
\left|
\begin{array}{l}
\exists\, \mu \ne 0 \mbox{~a~finite~measure~on~} \cX\times\cY \mbox{~such~that~}\\
F(g)=\int g \,d\mu~\forall\, g\in\mathscr{G},\\
b\in L_2(\mu), b'_a\in L_2(\mu) ~\forall\,a\in(0,\infty)
\end{array}
\right.
\right\}
\end{equation}
and
\begin{equation}
B_0 := \mathrm{cl}(\mathrm{lin}(B_S))
\end{equation}
the closed linear span of $B_S$ in $\ell_\infty(\mathscr{G})$. That is, $B_S$ is a subset of $\ell_\infty(\mathscr{G})$ whose elements correspond to finite measures. Hence probability measures are covered as special cases. The elements of $B_S$ can be interpreted as some kind of generalized distributions functions, because $\mathscr{G}_1\subset\mathscr{G}$. The assumptions on $L$ and ${{\rm P}}$ imply that $\mathscr{G}\to\R$,
$g\mapsto \int g\,d{{\rm P}}$ is a well-defined element of $B_S$. For every $F\in B_S$, let
$\iota(F)$ denote the corresponding finite measure on $(\cX\times\cY,\B(\cX\times\cY))$ such that
$F(g)=\int g\,d\mu$ for all $g\in\mathscr{G}$. Note that the map $\iota$ is well-defined, because by definition of $B_S$, $\iota(F)$ uniquely exists for every $F\in B_S$.
With these notations, we will apply Theorem \ref{vdVW1996.Thm3.9.11}
for
\begin{equation}
\begin{array}{ll}
\bbD:=\ell_\infty(\mathscr{G}), & \bbE:=H ~(\mbox{=~RKHS~of~the~kernel~}k), \,\\
\bbD_\phi := B_S, & \bbD_0:=B_0:=\mathrm{cl}(\mathrm{lin}(B_S)),\\
\lambda _0\in(0,\infty),\\
\phi:= S, & S: B_S\to H, ~F\mapsto f_{\iota(F)}:=f_{L,\iota(F), \lambda _0} :=\\
& ~~~\arg\inf_{f\in H} \int L(x,y,f(x)) \,d\iota(F)(x,y) + \lambda _0\hhnorm{f}\,.
\end{array}
\end{equation}
At first glance this definition of $S$ seems to be somewhat technical.
However, this will allow us to use a functional delta method for bootstrap estimators of SVMs with regularization parameter $ \lambda = \lambda _0\in(0,\infty)$.
Lemma \ref{Hable2012.LemmaA.9} guarantees that
the empirical process $\Gn:=\sqrt{n}(\Pn-{{\rm P}})$ weakly converges to a tight Borel-measurable Gaussian process.
Since a $ \sigma $-compact set in a metric space is separable, separability
of a random variable is slightly weaker than tightness, see \cite[p.\,17]{VandervaartWeller1996}.
Therefore, $\bbG$ in our Theorem \ref{AC.BootstrapConsist} is indeed separable.\\
\emph{Step 2.~}
Theorem \ref{Hable2012.ThmA.8} showed that the map $S$ indeed satisfies the necessary Hadamard-differentiability in the point ${{\rm P}}:=\iota^{-1}(F)$.
\emph{Step 3.~}
We know that $\mathscr{G}$ is a ${{\rm P}}$-Donsker class, see Lemma \ref{Hable2012.LemmaA.9}. Hence, an immediate consequence from
\cite[Theorem 3.6.1, p.\,347]{VandervaartWeller1996} is, that
\begin{equation} \label{term3.9.9a}
\sup_{h\in {\mathrm{BL}}_1} | {\mathbb{E}}_M h(\Gnhat) - {\mathbb{E}} h(\bbG)|
\end{equation}
converges in outer probability to zero and $\Gnhat$ is asymptotically measurable.
However, we will prove a somewhat stronger result, namely that $\mathscr{G}$ is a ${{\rm P}}$-Donsker class and ${{\rm P}}^*\snorm{g-{{\rm P}} g}_{\mathscr{G}}^2 <\infty$, which is part \emph{(i)} of
Theorem \ref{vdVW1996.Thm3.6.2}, and then part \emph{(iii)} of Theorem \ref{vdVW1996.Thm3.6.2} yields, that the term in {(\ref{term3.9.9a})} converges even outer almost surely to zero and the sequence
\begin{equation} \label{termiiivdVWpage347}
{\mathbb{E}}_M h(\Gnhat)^* - {\mathbb{E}}_M h(\Gnhat)_*
\end{equation}
converges almost surely to zero for every $h\in {\mathrm{BL}}_1$.
Because $\mathscr{G}$ is a ${{\rm P}}$-Donsker class, it remains to show that ${{\rm P}}^*\snorm{g-{{\rm P}} g}_{\mathscr{G}}^2 <\infty$.
Due to
\begin{equation}
{{\rm P}}^*\snorm{g-{{\rm P}} g}_{\mathscr{G}}^2
:=
\int ( \sup_{g\in\mathscr{G}} |g-{\mathbb{E}}_{{\rm P}}(g)| )^2 \,d{{\rm P}}^*
\end{equation}
and $\mathscr{G}=\mathscr{G}_1 \cup \mathscr{G}_2 \cup \mathscr{G}_3$, we obtain the inequality
\begin{eqnarray}
{{\rm P}}^*\snorm{g-{{\rm P}} g}_{\mathscr{G}}^2 & \le &
{{\rm P}}^* \sup_{g\in\mathscr{G}} \bigl( g^2 + 2|g| \cdot {{\rm P}} |g|
+ ({{\rm P}} |g|)^2 \bigr) \nonumber \\
& \le & {{\rm P}}^* \sup_{g\in\mathscr{G}} g^2
+ 2 \, {{\rm P}}^* \sup_{g\in\mathscr{G}} ( |g| \cdot {{\rm P}} \, |g| )
+ \sup_{g\in\mathscr{G}} ({{\rm P}} |g|)^2 \qquad \nonumber\\
& \le & \sum_{j=1}^3 \Bigl( {{\rm P}}^* \sup_{g\in\mathscr{G}_j} g^2
+ 2 \, {{\rm P}}^* \sup_{g\in\mathscr{G}_j} ( |g|\cdot {{\rm P}}\, |g| )
+ \sup_{g\in\mathscr{G}_j} ({{\rm P}} |g|)^2 \Bigr). \label{tmptmp3} \qquad~~
\end{eqnarray}
We will show that each of the three summands on the right hand side of the last inequality is finite.
If $g\in\mathscr{G}_1$, then $g$ equals the indicator function $I_{(-\infty,z]}$ for some
$z\in\R^{d+1}$. Hence, $\inorm{g}=1$ and the summand for $j=1$ is finite.
If $g\in\mathscr{G}_3$, then $g=b\in L_2({{\rm P}})$ because $L$ is by assumption a ${{\rm P}}$-square-integrable Nemitski loss function of order $p\in[1,\infty)$. Hence the summand for $j=3$ is finite, too.
Let us now consider the case that $g\in\mathscr{G}_2$. By definition of $\mathscr{G}_2$, for every $g\in \mathscr{G}_2$ there exist $f, f_0 \in H$ such that $\hnorm{f_0}\le c_0$, $\hnorm{f}\le 1$, and
$g=L'_{f_0}f$, where we used the notation $\bigl(L'_{f_0}f\bigr)(x,y):=L'(x,y,f_0(x))f(x)$ for all
$(x,y)\in\cX\times\cY$. Using $\inorm{f}\le \inorm{k}\hnorm{f}$ for every $f\in H$, we obtain
\begin{equation} \label{tmptmp1}
\hnorm{f_0}\le c_0 ~\Rightarrow~ \inorm{f_0}\le c_0\inorm{k}
\quad \mbox{and} \quad
\hnorm{f}\le 1 ~\Rightarrow~ \inorm{f}\le \inorm{k} \,.
\end{equation}
Define the constant $a:=c_0 \inorm{k}$ with $c_0$ given by {(\ref{Hable2012.c0})}.
Hence, for all $(x,y)\in\cX\times\cY$,
\begin{eqnarray}
\sup_{f_0\in H;\, \hnorm{f_0}\le c_0} |L'(x,y,f_0(x))|^2 & \le &
\sup_{f_0\in H;\, \inorm{f_0}\le a} \,\, \sup_{t\in[-a,+a]} |L'(x,y,t)|^2 \nonumber\\
& \stackrel{\footnotesize{{(\ref{Hable2012.5})}}}{\le} & \sup_{f_0\in H; \,\inorm{f_0}\le a} (b'_a(x,y))^2 \,. \label{tmptmp2}
\end{eqnarray}
Hence we get
\begin{eqnarray}
& & {{\rm P}}^* \sup_{g\in\mathscr{G}_2} g^2 \nonumber \\
& = & \int \sup_{g\in\mathscr{G}_2; \, \hnorm{f_0}\le c_0, \hnorm{f}\le 1, g=L'_{f_0}f}
|L'(x,y,f_0(x)) f(x)|^2 \,d{{\rm P}}^*(x,y) \qquad \nonumber\\
& \le & \int \sup_{f_0\in H; \,\hnorm{f_0}\le c_0} |L'(x,y,f_0(x))|^2
\sup_{f\in H; \, \hnorm{f}\le 1} |f(x)|^2 \,d{{\rm P}}^*(x,y) \qquad \nonumber \\
& \stackrel{\footnotesize{(\ref{tmptmp2}),(\ref{tmptmp1})}}{\le} & \inorm{k}^2 \int (b'_a)^2 \,d{{\rm P}}^*
= \inorm{k}^2 \int (b'_a)^2 \,d{{\rm P}} < \infty \,,
\nonumber
\end{eqnarray}
because $b'_a\in L_2({{\rm P}})$ and $\inorm{k} < \infty$ by Assumption \ref{AC.assump.consist}.
With the same arguments we obtain, for every $g\in\mathscr{G}_2$,
\begin{eqnarray}
{{\rm P}} |g| & \le & \int \sup_{g\in\mathscr{G}_2} |g|\,d{{\rm P}}^* \nonumber \\
& \le & \int \sup_{f_0\in H; \,\hnorm{f_0}\le c_0} |L'(x,y,f_0(x))|
\sup_{f\in H; \,\hnorm{f}\le 1} |f(x)| \,d{{\rm P}}^*(x,y) \nonumber \\
& \stackrel{\footnotesize{(\ref{tmptmp2}),(\ref{tmptmp1})}}{\le} & \int b'_a(x,y) \, \inorm{k} \,d{{\rm P}}^*(x,y) \nonumber\\
& \le & \inorm{k} \int b'_a \,d{{\rm P}} < \infty \,, \nonumber
\end{eqnarray}
because $b'_a\in L_2({{\rm P}})$ and $\inorm{k}< \infty$ by Assumption \ref{AC.assump.consist}.
Hence,
\begin{eqnarray*}
{{\rm P}}^* \sup_{g\in\mathscr{G}_2} (|g| \,{{\rm P}} |g|)
\le \inorm{k} \int b'_a \,d{{\rm P}} \, \int \sup_{g\in\mathscr{G}_2} |g| \,d{{\rm P}}^*
\le \inorm{k}^2 \bigl(\int b'_a \,d{{\rm P}} \bigr)^2 < \infty.
\end{eqnarray*}
Therefore, the sum on the right hand side in {(\ref{tmptmp3})} is finite and thus
the assumption ${{\rm P}}^* \snorm{g-{{\rm P}} g}^2_\mathscr{G} < \infty$ is satisfied. This yields by part \emph{(iii)} of Theorem \ref{vdVW1996.Thm3.6.2} that
$\sup_{h\in {\mathrm{BL}}_1} \bigl| {\mathbb{E}}_M h(\Gnhat) - {\mathbb{E}} h(\bbG)\bigr|$ converges outer almost surely to zero and the sequence
\begin{equation} \label{tmptmp4}
{\mathbb{E}}_M h(\Gnhat)^* - {\mathbb{E}}_M h(\Gnhat)_*
\end{equation}
converges almost surely to zero for every $h\in {\mathrm{BL}}_1$, where the asterisks denote the measurable cover functions with respect to $M$ and $Z_1, Z_2, \ldots$ jointly.\\
\emph{Step 4.~}
Due to Step 3, the assumption {(\ref{vdVW3.9.9a})} of Theorem \ref{vdVW1996.Thm3.9.11} is satisfied.
We now show that additionally {(\ref{vdVW3.9.9b})} is satisfied,
i.e., that the term in {(\ref{tmptmp4})} converges to zero in outer probability.
In general, one can \emph{not} conclude that almost sure convergence implies convergence in outer probability, see \cite[p.\,52]{VandervaartWeller1996}.
We know that the term in {(\ref{tmptmp4})} converges almost surely to zero for every $h\in {\mathrm{BL}}_1$, where the asterisks denote the \emph{measurable} cover
functions with respect to $M$ and $(X_1,Y_1), (X_2,Y_2), \ldots$
\emph{jointly}. Hence, for every $h\in{\mathrm{BL}}_1$, the cover functions
to be considered in {(\ref{tmptmp4})} are measurable.
Additionally, the multinomially distributed random variable $M$
is stochastically independent of $(X_1,Y_1), \ldots, (X_n,Y_n)$
in the bootstrap, where independence is understood in terms of a
product probability space, see \cite[p.\,346]{VandervaartWeller1996} for details.
Therefore, an application of the Fubini-Tonelli theorem, see e.g.,
\cite[p.\,174, Thm.\,2.4.10]{DenkowskiEtAl2003}, yields that the inner integral
${\mathbb{E}}_M h\bigl(\sqrt{n}(\Pnhat - \Pn)\bigr)^*
- {\mathbb{E}}_M h\bigl(\sqrt{n}(\Pnhat - \Pn)\bigr)_*$
considered by Fubini-Tonelli is \emph{measurable}
for every $n\in\N$ and every $h\in{\mathrm{BL}}_1$.
Recall that almost sure convergence of measurable functions implies
convergence in probability which is equivalent with convergence in outer probability for measurable functions.
Hence we have convergence in outer probability in {(\ref{tmptmp4})}.
Therefore, all assumptions of Theorem \ref{vdVW1996.Thm3.9.11} are satisfied
and the assertion of our theorem follows. \hfill $\blacksquare$
\bibliographystyle{plain}
|
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"redpajama_set_name": "RedPajamaArXiv"
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| 1,683
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Blanca Heredia Osío (Caracas, Venezuela, 1 de abril de 1934-ibidem, 17 de diciembre de 2022), fue una reina de belleza, bióloga y bioanalista venezolana, ganadora del Miss Venezuela 1956, al momento de su coronación contaba con 22 años y estudiaba el último año de «Laboratorio Clínico» (Bioanálisis), en la Facultad de Ciencias Médicas de la UCV.
Miss Venezuela 1956
El cuarto concurso del Miss Venezuela se realizó 30 de junio de 1956 en las instalaciones del Hotel Tamanaco en la ciudad de Caracas, En esa oportunidad la corona fue ganada por Blanca Heredia Osío.
En la edición de 1956 le tocó la responsabilidad de ser Miss Venezuela a Blanca Heredia Osío (Blanquita Heredia) nacida en 1934 en la ciudad de Caracas, Distrito Capital. Tenía 22 años cuando se coronó y su estatura era 1,65. Blanquita representó al Distrito Federal en esta edición del Miss Venezuela. Luego viajó a representar al país a Long Beach, California, EUA. El concurso se realizó el 20 de julio de 1956 donde resultó semifinalista. Dada la popularidad que adquirió el concurso, Blanquita recibió como premios del Miss Venezuela un auto convertible, 1.500 dólares en ropa y 2 mil dólares para sus gastos personales, además de un contrato que le garantizaría 30 mil dólares más durante su año de reinado.
Vida personal
Después de entregar su corona en 1957 se casaría con Alonso Gamero Reyes, ilustre educador, biólogo, poeta y documentalista. Tuvo cinco hijos; Aura Isabel, Gabriel, Alejandro y Valentina. Se gradúa de bioanalista el 6 de agosto de 1966; en 1984 forma parte de la Comisión de Prevención y Control de Infecciones Intra-hospitalarias (HUC). Perteneció al «Grupo Venezolano de Vigilancia de la Resistencia Bacteriana a los Antimicrobianos» como representante del HUC desde 1988. Formó parte del Proyecto de Resistencia Bacteriana al Cefixime en Latinoamérica. Ejerció como docente colaborador con el Postgrado de Enfermedades Infecciosas del HUC desde el año 1984. Fue miembro de la Comisión de Credenciales de la S.V.B.E y vivió con su esposo en Caracas.
Cuadro final del Miss Venezuela 1956
1. Blanca Heredia Osío — Miss Distrito Federal (Ganadora del Miss Venezuela 1956)
2. Celsa Drucila Pieri — Miss Sucre
3. Elizabeth Rotundo — Miss Aragua
4. Alida Márquez — Miss Departamento Libertador
5. Beatriz Gutiérrez Padrón — Miss Caracas
Participantes en el Miss Venezuela 1956
Beatriz Bello - Miss Amazonas
Elizabeth Rotundo- Miss Aragua
Lourdes Agostini Oquendo - Miss Barinas
Beatriz Gutiérrez Padrón – Miss Caracas
Alba Guevara – Miss Delta Amacuro
Alida Marquiz - Miss Departamento Libertador
Inés María Moya – Miss Departamento Vargas
Blanca Heredia Osio – Miss Distrito Federal
Belen Incosta - Miss Guárico
Fanny Torrealba – Miss Lara
Lady Josefina Andrade - Miss Maracaibo
Aracelis Mora– Miss Miranda
Ennia Mendoza - Nueva Esparta
Celsa Pieri – Miss Sucre
Lilian Haak – Miss Trujillo
Iris Rubio – Miss Yaracuy
Participación en el Miss Universo 1956
Finalistas
1 Carol Laverne Morris – Miss Estados Unidos
2 Marina Orschel – Miss Alemania
3 Ingrid Goude - Miss Suecia
4 Iris Alice Kathleen Waller Miss Inglaterra
5 Rossana Galli Miss Italia
Semifinalista
Ileana Carré – Miss Argentina
Lucienne Auquier – Miss Bélgica
María José Cardoso – Miss Brasil
Marcia Rodríguez Echevarría - Miss Cuba
Anita Treyens – Miss Francia
Rita Gouma – Miss Grecia
Sara Tal – Miss Israel
Erna Marta Bauman - Miss México
Lola Sabogal Morzán – Miss Perú
Blanca Heredia Osío Miss Venezuela
Véase también
Anexo:Ganadoras y finalistas de Miss Venezuela
Anexo:Venezuela en los concursos de belleza
Referencias
Enlaces externos
Ellas han sido Miss Venezuela
Ganadoras de Miss Venezuela
Finalistas de Miss Universo
Nacidos en Caracas
Fallecidos en Caracas
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\section{Introduction}
The AdS/CFT correspondence
\cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc} has provided a
detailed connection between black holes and conformal field
theories. Although it is sometimes said that this solves the
conceptual puzzles associated with black hole physics, in fact we
still don't understand the connection well enough to see
explicitly how all these puzzles are resolved.
In using the AdS/CFT correspondence in the context of black holes one
typically compares a thermal ensemble in the CFT to a semi-classical
black hole geometry in the bulk. In this way it is possible to compute
and compare quantities such as the entropy of the system and correlation
functions of fields/operators \cite{Maldacena:2001kr, Hemming:2002kd,
Kraus:2002iv, Levi:2003cx, Fidkowski:2003nf, Barbon:2003aq,
Balasubramanian:2004zu, Kleban:2004rx,Festuccia:2005pi}. Recent work
\cite{Behrndt:1998eq,Ooguri:2004zv,Dabholkar:2004yr,Dabholkar:2004dq,
Dabholkar:2005by,Hubeny:2004ji,Sen:2005pu,Kraus:2005vz,Iizuka:2005uv}
has shown that in some cases one can do even better by extending this
relation to the regime where the bulk geometry receives large
corrections from higher derivative string and loop effects.
In the CFT it is manifest that the thermal ensemble corresponds to a
weighted collection of individual microstates. Instead of
considering such an ensemble, there is nothing to prevent one from
choosing a particular microstate and computing correlation functions
in that state. On general grounds, if one is working at large $N$
then correlation functions of ``typical'' operators computed in a
``typical'' state will be approximated to excellent accuracy by the
same correlators computed in the thermal ensemble. This is just the
same as saying that realistic isolated systems, i.e.\ a large number
of molecules in a sealed box, are in some particular quantum
mechanical microstate at a given time, yet can be accurately studied
by the methods of statistical mechanics and thermodynamics.
The most natural interpretation of the AdS/CFT correspondence is
that there is a one-to-one correspondence between bulk and boundary
states. In particular, one expects this statement to hold even in
the range of parameters where black holes are allowed. Precisely
what the bulk microstates should look like is unclear at this point
in time. Mathur \cite{Lunin:2001jy} has conjectured that these
microstates correspond to bulk geometries (in general these might be
classically singular or have large quantum fluctuations) without
horizons, and the evidence for this conjecture includes
\cite{Lunin:2001jy,Mathur:2003hj,Bena:2004wt,Bena:2005ay,
Jejjala:2005yu,Bena:2005va,Berglund:2005vb,Taylor:2005db}.
Just as in the CFT, one is led to believe that if one chooses a
typical such bulk state then with respect to typical measurements it
will look like the usual black hole geometry.
An example of an {\it atypical\/} measurement is one which extends over
a very long time interval. As originally emphasized by Maldacena
\cite{Maldacena:2001kr}, at late times correlators computed in the
semi-classical black hole geometry decay to zero, while in the CFT they
exhibit a quasi-periodic behavior. The key difference is that the
semi-classical black hole geometry has a continuous spectrum due to the
presence of the horizon, while the CFT has a discrete spectrum. This
distinction is insignificant for short times or at high energy, but
becomes important in the opposite regime. Further work
\cite{Barbon:2003aq,Kleban:2004rx} in the context of BTZ black holes~\cite{btz}
strongly indicates that also summing over the $SL(2,\mathbb Z)$ images of the
black hole (which includes global AdS$_3$) can prevent the correlators
from decaying to zero, but can't account for the
quasi-periodicity. Presumably, this is telling us that we should instead
be considering the actual microstate geometries dual to the individual
CFT microstates if we want to correctly account for detailed properties
such as the late time behavior of correlators. The analogy with
molecules in a box is again helpful: while a coarse-grained effective
description accurately describes most properties of the system, in order
to recover the quasi-periodicity of late time correlators one needs to
return to the fundamental molecular description.
Recently, some of these issues have been discussed in the context of the
$\ads{5}/{\rm CFT}_4$ correspondence for half-BPS states
\cite{Corley:2001zk, Berenstein:2004kk, Lin:2004nb}. It was argued in
\cite{Balasubramanian:2005kk} that typical large-charge half-BPS
microstates that are incipient black holes have a spacetime description
as a quantum ``foam'', the precise details of which are almost invisible
to almost all probes. This gave rise to effective singular descriptions
of underlying smooth quantum states \cite{Balasubramanian:2005kk}.
(Other perspectives on these issues have appeared in \cite{others}.) In
the present paper we will study the D1-D5 system on $T^4$, since this
provides the simplest link between black holes and CFT\@. We will also
set to zero the momentum $P$, so that we just work with the
Ramond-Ramond ground states of the system. This example is of interest
for several reasons. The system has a large ground state degeneracy
corresponding to an entropy $S= 2\pi \sqrt{2}\sqrt{N_1 N_5}$, and so
should have some of the properties of a black hole.\footnote{In the case
of the D1-D5 system on K3 it has been shown that higher curvature terms
indeed lead to an event horizon whose entropy coincides with that in the
CFT \cite{Dabholkar:2004yr,Dabholkar:2004dq}. Whether this also happens
in the $T^4$ case is unclear
\cite{Behrndt:1998eq,Ooguri:2004zv,Dabholkar:2004yr,Dabholkar:2004dq,
Dabholkar:2005by,Hubeny:2004ji,Sen:2005pu,Kraus:2005vz,Iizuka:2005uv}.}
A large class of microstate geometries for this system is known. They
correspond to configurations in which the D1 and D5 branes expand into a
Kaluza--Klein monopole supertube \cite{Lunin:2002iz}. The shape of the
supertube encodes the details of the microstate. For example, the
maximally $R$-charged microstate corresponds to a circular supertube.
On the other hand, a typical state corresponds to the supertube taking a
complicated random walk shape localized near the origin. This leads to
a strongly curved supergravity solution whose existence is inferred via
extrapolation from the weakly curved geometries described by smooth
curves of large size.
We will compute correlation functions of certain operators in typical
states of the D1-D5 system. As we have already discussed, at large
$N=N_1 N_5$ the expectation is that these correlators should coincide
with bulk correlators computed in some effective geometry. This
effective geometry is analogous to the black hole geometry in the case
of the D1-D5-P system. As we will see, the effective geometry that
emerges depends on the $R$-charge of the underlying state. If the
$R$-charge vanishes, the emergent geometry is the massless BTZ
\cite{btz} black hole; or equivalently, AdS$_3$ in Poincar\'{e}
coordinates with a spatial direction periodically identified. This is
often referred to as the ``naive'' geometry representing the RR-ground
states. It can be obtained by contracting the KK-monopole supertube to
zero size. No individual microstate corresponds to this geometry;
rather, in the large $N$ limit, this geometry encodes the universal
response of generic finite time correlation functions in the underlying
microstate. This effective geometry exhibits a continuous spectrum just
like the black hole, and so bulk correlators decay to zero at late
times. But in this case we can also show that the exact late time
correlation functions show quasi-periodic behavior demonstrating that
the effective description breaks down at large times, and should be
replaced by the exact microstate geometries.
We would like to emphasize that our approach based on computing
correlation functions allows us to {\em derive\/} the effective
geometries corresponding to CFT states, rather than assuming the (highly
plausible!) map between states and KK-monopole supertube profiles. A
correlation function based approach is also necessary if one wishes to
make statements about the geometry at the string or Planck scale, since
the existing map between states and geometries is only valid at the
level of two-derivative supergravity.
We also consider typical states of nonzero $R$-charge, and find some new
features. For sufficiently large charge, the CFT undergoes a form of
Bose--Einstein condensation. The state effectively splits into two
components, one carrying the $R$-charge but no entropy, and the other
carrying no $R$-charge and all the entropy. The correlation functions
we compute then become a sum of two terms with contributions from each
of the two components. The result looks effectively like a
superposition of correlators computed in the massless BTZ space and in
the maximal $R$-charge Ramond vacuum, namely globals $\ads{3}$ with a
Wilson line \cite{Balasubramanian:2000rt,Maldacena:2000dr,
Lunin:2002iz}. On the other hand, we are able to derive a prediction
for the effective bulk geometry with non-zero $R$-charge by explicitly
constructing the typical microstate geometry and coarse-graining it. In
effect, this amounts to adding fluctuations on top of the circular
supertube solution and then averaging over these fluctuations in the
large $N$ limit. The predicted effective geometry for non-zero
$R$-charge turns out to be a singular black ring solution. It would
interesting to establish the connection between this prediction and the
``superposition'' of geometries derived from the CFT correlators.
The ultimate goal of this sort of investigation is to see a
macroscopic semi-classical black hole geometry emerging from
correlators computed in a typical state of a large $N$ CFT\@. This
requires knowing how the presence of a black hole manifests itself
in terms of correlators. Let us note two criteria. First, the
correlators should correspond to a well-defined classical
geometry, rather than a strongly fluctuating superposition.
Second, correlators should fall to zero at late times (in the
large $N$ limit), reflecting the presence of a horizon. Here we
observe that the effective geometry emerging from our computations
satisfies these two properties, and so we can claim to be seeing
some black hole-like properties. For a large black hole one would
like to do better and reproduce the most important property of
all --- that of complete absorption of high energy particles
impinging on the horizon, but this goes beyond what we can do
here.
The remainder of this paper is organized as follows. In section
\ref{sec:d1d5} we review the physics of the D1-D5 system, summarizing
he relevant details of the D1-D5 CFT, the map between Ramond ground
states and microstate geometries, and the computation of 2-point
correlation functions for massless scalars in the black hole spacetimes.
A more extensive review explaining details appears in Appendix
\ref{app:cft}. In section \ref{sec:typ}, we construct the typical
Ramond ground states of the D1-D5 CFT that have fixed $R$-charges. In
section \ref{sec:effgeo} we derive the effective geometries describing
finite-time correlation functions computed in these microstates. The
basic technique is to compute and analyze the two-point correlator in
the typical states constructed in section \ref{sec:typ}. In section
\ref{sec:disc} we conclude. Appendix \ref{app:J=/=0} gives additional
details about typical states with nonzero $R$-charge.
\section{The D1-D5 system and its geometric dual}
\label{sec:d1d5}
\subsection{The D1-D5 CFT and its Ramond sector ground states}
\label{subsec:d1d5andR}
Consider type IIB string theory on $S^1 \times T^4$ with $N_1$ D1-branes
and $N_5$ D5-branes. The D1-branes are wound on $S^1$ and the D5-branes
are wrapped on $S^1\times T^4$. At low energies, the worldvolume
dynamics of the branes is given by an ${\cal N} = (4,4)$ supersymmetric sigma
model whose target space is the symmetric product ${\cal M}_0=(T^4)^N/S_N$,
where $S_N$ is the permutation group of order $N$
\cite{Strominger:1996sh, deBoer:1998ip, Seiberg:1999xz, Larsen:1999uk}.
Here we set
\begin{align}
N&\equiv N_1 N_5.
\end{align}
More precisely, ${\cal M}_0$ is the so-called orbifold point in a family of
CFTs which are regained by turning on certain marginal deformations of
the sigma model on ${\cal M}_0$. At the orbifold point the CFT becomes free.
The D1-D5 CFT is dual to Type IIB string theory on $\ads{3} \times S^3
\times T^4$, which is the near-horizon limit of the D1-D5 brane system.
The AdS$_3$ length scale is given by $\ell \sim N^{1/4}$. To have a
large, weakly coupled, $\ads{3}$ space, $N$ must be large and the CFT
must be deformed far from the orbifold point. This situation is
familiar in the AdS$_5$/SYM$_4$ duality, where the SYM theory becomes
free at a special point ($g_{\rm YM}=0$) in the moduli space, but in
order for it to correspond to a semiclassical gravity one has to turn on
the coupling $g_{\rm YM}$. The orbifold point is the analogue of the
free SYM\@. In the following, we will consider the orbifold point of
the D1-D5 CFT, so one should bear in mind that exact agreement with
computations in supergravity is not expected in all cases, although some
protected BPS quantities can be computed exactly.
According to the AdS/CFT correspondence, every pure state of the D1-D5
CFT is dual to a pure state of string theory in $\ads{3} \times S^3
\times T^4$. Here, we are interested in understanding how black holes
emerge as the effective spacetime description of underlying pure states
in gravity. The only black holes in $\ads{3}$ gravity with standard
boundary conditions are the BTZ solutions \cite{btz}. The
supersymmetric versions of these spacetimes have periodic boundary
conditions for fermions around the asymptotic circle in the $\ads{3}$
geometry and thus appear in the Ramond sector of the theory.
Furthermore, the lightest of the black holes, the BPS massless solution,
has the quantum numbers of a ground state in the Ramond sector of the
dual CFT \cite{strominger}. For these reasons, we will concentrate on
the Ramond ground states of the D1-D5 CFT henceforth. Powerful
techniques to study these ground states are available at the orbifold
point of the CFT\@.
The D1-D5 CFT and the construction of the Ramond ground states is
reviewed in detail in Appendix \ref{app:cft}\@. For the moment, the
following facts are sufficient. At the orbifold point we are dealing
with an ${\cal N} = (4,4)$ SCFT on the target space ${\cal M}_0 =
(T^4)^N/S_N$.
This theory has an $SU(2)_R \times \widetilde{SU(2)}_R$ $R$-symmetry,
which originates from the $SO(4)$ rotational symmetry transverse to the
D1-D5 worldvolume.
There is
another global $SU(2)_I \times \widetilde{SU(2)}_I$ which is broken by
the toroidal identifications in $T^4$, but can be used for classifying
states anyway. We will label the charges under these symmetries as
\begin{equation}
(J^3_R, \tilde{J}^3_R) = (\tfrac{s}{2}, \tfrac{{\widetilde{s}}}{2})
~~~~ {\rm and} ~~~~
(I^3, \tilde{I}^3) = (\tfrac{\alpha}{2}, \tfrac{{\widetilde{\alpha}}}{2})
\label{chargedef}
\end{equation}
with $s,\tilde{s},\alpha,{\widetilde{\alpha}} = \pm 1$. The CFT has a collection of
twist fields $\sigma_n$, which cyclically permute $n \leq N$ copies of
the CFT on a single $T^4$. One can think of these operators as creating
winding sectors of the worldsheet that wind over the different copies of
the torus. The product of twist operators is also a twist operator.
The elementary bosonic operators of twist $n$ carry either
$SU(2)_R\times\widetilde{SU(2)}_R$ or $\widetilde{SU(2)}_I$ charges
($\sigma_n^{s\tilde{s}}$ or $\sigma_n^{{\widetilde{\alpha}}{\widetilde{\beta}}}$), while the
elementary fermionic twist operators are charged under $SU(2)_R \times
\widetilde{SU(2)}_I$ or $\widetilde{SU(2)}_I\times\widetilde{SU(2)}_R$
($\tau_n^{s{\widetilde{\alpha}}}$ or $\tau_n^{{\widetilde{\alpha}}{\widetilde{s}}}$). A general Ramond sector
ground state is constructed by multiplying together elementary bosonic
and fermionic twist operators to achieve a total twist of $N = N_1 N_5$:
\begin{equation}
\begin{split}
\sigma&= \prod_{n,\mu} (\sigma_{n}^{\mu})^{N_{n\mu}} (\tau_{n}^\mu)^{N'_{n\mu}},
\\
\sum_{n,\mu}n (N_{n\mu}+N'_{n\mu})&=N, \qquad
N_{n\mu}=0,1,2,\dots,\quad N'_{n\mu}=0,1 \, .
\end{split}\label{gen_twist2}
\end{equation}
Here $\sigma_n^\mu$ and $\tau_n^\mu$, are the constituent elementary
twist operators, and $\mu = 1 \cdots 8$ labels their possible
polarizations ($\mu=(s,{\widetilde{s}}),({\widetilde{\alpha}},{\widetilde{\beta}})$ for bosons, and $\mu =
(s,{\widetilde{\alpha}}), ({\widetilde{\alpha}},{\widetilde{s}})$ for fermions). Appendix \ref{app:cft} gives
a detailed description of the construction of the twist operators and
computations using them. For our immediate purposes, the relevant point
is that the integers
\begin{equation}
\{ N_{n\mu}, N^\prime_{n\mu} \}
\label{Rstateintegers}
\end{equation}
uniquely specify a Ramond ground state.
\subsection{Map to the FP system and microstate geometries}
It has been proposed that each Ramond ground state of the D1-D5
has a corresponding exact spacetime geometry without horizons
\cite{Lunin:2001jy}. The construction of these geometries was
carried out by first $U$-dualizing the D1-D5 system to the FP system
in type II, where an F1 string is wound $N_5$ times along $S^1$
and carries $N_1$ units of momentum in the $S^1$ direction
\cite{Lunin:2001fv}. If the right-moving oscillation number $N_R$
vanishes, $N_R=0$, then this configuration is BPS\@. Such states
can be written as
\begin{equation}
\label{FPstates1}
\begin{split}
& \prod_{n,\mu} (\alpha_{-n}^{\mu})^{N_{n\mu}}
(\psi_{-n}^\mu)^{N_{n\mu}'}\ket{N_1,N_5},
\\
N_L=\sum_{n,\mu}n(N_{n\mu}+N_{n\mu}')&=N_1N_5=N, \qquad
N_{n\mu}=0,1,2,\dots,\quad N_{n\mu}'=0,1.
\end{split}
\end{equation}
Here $\alpha_{-n}^\mu$ and $\psi_{-n}^\mu$ are left-moving bosonic and
fermionic oscillators, respectively. The polarization $\mu$ runs over
eight transverse directions. $\ket{N_1,N_5}$ is the F1 string state
with momentum $N_1$ and winding number $N_5$, and with no oscillators
excited (this state itself is not physical). The second line displays
the Virasoro constraint on the left-moving oscillation number $N_L$.
Following \cite{Lunin:2001jy, Lunin:2002iz}, the $U$-duality map between
the states (\ref{FPstates1}) of the FP system and the Ramond ground
states (\ref{gen_twist2}) of the D1-D5 system is given by:
\begin{equation}
\sigma_n^\mu\leftrightarrow\alpha_{-n}^\mu
~~~~;~~~~
\tau_n^{\mu}\leftrightarrow\psi_{-n}^\mu \, .
\label{FPD1D5map}
\end{equation}
The set of integers (\ref{Rstateintegers}) defining a Ramond ground
state is precisely mapped into the set of integers defining an
excitation of the FP system.
The metric of the FP system is known for arbitrary classical profile
$x^\mu=F^\mu(v)$ of the F1 string by the chiral null model
\cite{cmp,dghw,Horowitz:1994rf,Tseytlin:1996yb}. Here $\mu$ runs over
the eight transverse directions to the F1 worldsheet. $v=t-y$ is the
left-moving lightcone coordinate, reflecting the fact that there must be
only left-moving waves on the F1 string because of the BPS condition.
By $U$-dualizing back, Lunin and Mathur \cite{Lunin:2001jy} obtained the
metric of the D1-D5 system, when the classical profile $F^\mu(v)$ is
only in the noncompact $\mathbb R^4$ directions ${\bf x}=x^i$,
$i=1,2,3,4$. Explicitly, the string frame metric of the D1-D5 system in
the decoupling limit is given by \cite{Lunin:2001jy, Lunin:2002iz}
\begin{equation}
\begin{split}
ds_{\rm string}^2&={1\over\sqrt{f_1f_5}}[-(dt-A)^2+(dy+B)^2]+\sqrt{f_1f_5}\,dx^i dx^i
+\sqrt{f_1\over f_5}dz^a dz^a,\\
e^{2\Phi}& ={f_1\over f_5},\qquad
f_5 ={Q_5\over L}\int_0^L{dv\over |{\bf x}-{\bf F}(v)|^2},\qquad
f_1 ={Q_5\over L}\int_0^L{|\dot {\bf F}(v)|^2 dv\over |{\bf x}-{\bf F}(v)|^2},\\
A_i& =-{Q_5\over L}\int_0^L{\dot F_i(v) dv\over |{\bf x}-{\bf F}(v)|^2},\qquad
dB=-*_4 dA.
\end{split}\label{LMmetric1}
\end{equation}
Here, $y$ and $z^a$ are $S^1$ and $T^4$ directions, respectively. The
coordinate radius of $S^1$ is $R$, and the coordinate volume of $T^4$ is
$(2\pi)^4V_4$. The length $L$ is related to $R$ by
\begin{align}
L={2\pi N_5\over R},
\end{align}
where the D5-brane charge $Q_5$ is related to the D5 number $N_5$ by
$Q_5=g_s\alpha' N_5$. The four arbitrary functions ${\bf F}(v)=F_i(v)$,
$0\le v\le L$ in \eqref{LMmetric1} parametrize the solution, and
correspond to the classical profile ${\bf F}_{\rm FP}(v)$ of the F1 string
in the FP duality frame by ${\bf F}(v)=\mu {\bf F}_{\rm FP}(v)$, $\mu=g_s
\alpha'{}^{3/2}/R\sqrt{V_4}$ \cite{Lunin:2001jy}. The D1 charge is
given by
\begin{align}
Q_1&={Q_5\over L}\int_0^L{|\dot {\bf F}(v)|^2 dv}.
\label{Q12}
\end{align}
In this paper we will argue that typical probes of typical microstate
geometries will react as if the spacetime was simply an $M=0$ BTZ black
hole (\ref{naive}) below.
Using (\ref{FPD1D5map}) the Ramond ground states can be mapped onto
specific states of the FP system \eqref{FPstates1}. For states involving
only $\alpha^i_{-n}$ this in turn determines the classical profile
$F^i(v)$ of the F1 string, which can be substituted into
\eqref{LMmetric1} to give the proposed geometry corresponding to a
specific Ramond ground state. Details and examples are given in
\cite{Lunin:2001jy}. For example, the special Ramond ground state
$[\sigma_{n}^{s\tilde{s}}]^{N/n}$ with $s=\tilde{s} = -1$, $1\le n\le N$
corresponds to the bulk geometry $(AdS_3\times S^3)/\mathbb Z_n \times
T^4$. The 3-dimensional part of this geometry is the conical defect
described below. For general states involving all bosonic oscillators
as well as fermionic oscillators more work is needed; see
\cite{Taylor:2005db} for results regarding the fermionic states.
\subsection{Bulk geometries and correlators}
In this subsection we will review some geometries that show up as the
bulk geometries in the context of AdS/CFT for the D1-D5 system. We will
also present the bulk 2-point functions of a massless minimally coupled
scalar in those geometries, and compare them at the end. The asymptotic
$\ads{3}$ radius is given by $\ell\sim N^{1/4}$.
\subsubsection{Conical defect}
\label{con_dfct}
The Ramond ground state $[\sigma_{n}^{s\tilde{s}}]^{N/n}$ with
$s=\tilde{s} = -1$, $1\le n\le N$ corresponds in the bulk to the conical
defect geometry \cite{Balasubramanian:2000rt,Maldacena:2000dr}:
\begin{align}
ds^2 = -\left({1 \over n^2}+{r^2\over\ell^2}\right)dt^2
+ {dr^2 \over{1 \over n^2}+{r^2\over \ell^2}} + r^2 d\phi^2~. \label{condef}
\end{align}
Here $n$ is an integer in the range $1 \leq n \leq N$. The angular
identification is $(\phi,\psi) \cong (\phi+2\pi, \psi +{2 \pi \over
n})$, where $\psi$ is angle on the $S^3$ factor that we have
suppressed. The special case $n=1$ yields AdS$_3$ in global
coordinates.
Any state can be probed by computing the correlation functions of
operators in that state. The simplest correlator that one could
compute, the 2-point function, is related to a 4-point function computed
in the vacuum. According to the AdS/CFT correspondence, the 2-point
function in the state $[\sigma_{n}^{s{\widetilde{s}}}]^{N/n}$, $s={\widetilde{s}}=-1$ can be
obtained from AdS space by computing the bulk-boundary propagator of the
spacetime field that is dual to the CFT probe and then taking the bulk
point to the boundary. Let us consider a CFT probe that is dual to a
massless scalar field in $\ads{3}$. The conical defect propagator for
this field is obtained from the AdS propagator by summing over the
images that define the conical defect. By translation invariance we can
take one of the boundary points to be at $t=\phi=0$. We then obtain the
result
\begin{align}
\sum_{k=0}^{n-1}
{1\over \Bigl(2n\sin{w-2\pi k\over 2n}\Bigr)^2\Bigl(2n\sin{{\overline{w}}-2\pi k\over 2n}\Bigr)^2}
=
{1\over 16n^2\sin^2{w-{\overline{w}}\over 2n}}
\left[
{1\over \sin^2{w\over 2}}+{1\over \sin^2{{\overline{w}}\over 2}}
-{2\sin{w-{\overline{w}}\over 2}\over n\tan{w-{\overline{w}}\over 2n}\sin{w\over 2}\sin{{\overline{w}}\over 2}}
\right]
\label{conicalG}
\end{align}
where
\begin{align}
w = \phi-{t \over \ell}, \quad {\overline{w}} = \phi+{t \over \ell}~.
\end{align}
The summation was done by a standard contour integration method.
\subsubsection{Naive geometry}
\label{subsec:naive_geo}
Consider taking the $n\rightarrow \infty$ limit of the conical
defect geometries:
\begin{align} ds^2 = -{r^2\over \ell^2} dt^2 + {\ell^2 \over r^2}dr^2 + r^2
d\phi^2~. \label{naive}
\end{align}
This is the same as AdS$_3$ in Poincar\'{e} coordinates with a
periodically identified spatial direction. This geometry does not
actually correspond to any CFT microstate since it has $n>N$.
Instead, we will see that this geometry emerges as an effective
description of the typical Ramond ground state at large $N$. As
before we compute the boundary 2-point function for a massless
scalar field, and find that
\begin{align} \sum_{k=-\infty}^{\infty}
{1 \over (w-2\pi k)^{2} ({\overline{w}}-2\pi k)^{2}}={1 \over 4 (w-{\overline{w}})^2} \left[{1\over \sin^2{w\over 2}}+{1\over \sin^2{{\overline{w}}\over 2}}
-{4\sin{w-{\overline{w}}\over 2}\over (w-{\overline{w}}) \sin{w\over 2}\sin{{\overline{w}}\over 2}}
\right] ~.
\label{naiveG}
\end{align}
\subsubsection{Non-rotating BTZ}
The above naive geometry is in fact the massless limit of the BTZ
black holes of $\ads{3}$. To see this recall that the non-rotating
BTZ black holes has a metric \cite{btz}
\begin{align} ds^2 = -{r^2- r_+^2 \over \ell^2} dt^2 + {\ell^2 \over
r^2- r_+^2 }dr^2+r^2 d\phi^2~.
\end{align}
If we take $r_+=0$, which is the $M=0$ BTZ black hole, we get back
the naive geometry (\ref{naive}). The boundary 2-point function
is \cite{esko}
\begin{align} {16 r_+^4 \over 4 \ell^4}\sum_{k=-\infty}^\infty \left[{1
\over \sinh \left({r_+ \over 2 \ell}(w+2\pi k)\right) \sinh
\left({r_+ \over 2 \ell}({\overline{w}}+2\pi k)\right) }\right]^2~.
\label{btzcorr}
\end{align}
We have not succeeded in doing the summation in closed form. But we
can use contour integration to rewrite the sum in a way which makes
the large time behavior manifest. For simplicity set $\phi=0$.
Then one can rewrite (\ref{btzcorr}) as
\begin{align} {r_+^2 \over 8 \ell^2} {1 \over \sinh^2 \left({r_+ t
\over \ell^2}\right)} \Biggl\{ \sum_{m=-\infty}^\infty & \left[
{ 1 \over \sin^2 \left({t \over 2 \ell}+i\pi{\ell\over r_+}m\right)}
+{2r_+\over \ell}
{1\over \tan\left({t \over 2 \ell}+i\pi{\ell\over r_+}m\right)
\tanh\left({r_+ t \over \ell^2}\right)}
\right]\cr & +{2r_+\over \pi\ell} \left[ {\left({r_+ t\over
\ell^2}\right) \over \tanh\left({r_+ t\over \ell^2}\right)}-1\right]
\Biggr\}~.
\end{align}
To simplify further, consider the case of a small black hole, $r_+
\ll \ell$. In this case we can truncate to just the $m=0$ term and
obtain:
\begin{align} {r_+^2 \over 8 \ell^2} {1 \over \sinh^2 \left({r_+ t
\over \ell^2}\right)}\Biggl\{{ 1 \over \sin^2 \left({t \over 2
\ell}\right)} +{2r_+ \over \ell} {1\over \tan\left({t \over 2
\ell}\right)
\tanh\left({r_+ t \over \ell^2}\right)} +{2r_+\over \pi\ell}
\left[ {\left({r_+ t\over \ell^2}\right) \over \tanh\left({r_+
t\over \ell^2}\right)}-1\right]\Biggr\}~.
\end{align}
\subsubsection{Comparison}
Notice that the three geometries described above look the same outside a core region. As we'll
review later, the $n$ appearing in the conical defect geometry has a
typical size
\begin{align} n_{\rm typ} \sim N^{1/2} \sim \ell^2~.
\end{align}
The typical conical defect geometry thus approaches the naive
geometry for $r \gg \ell^{-1}$. Consider then the BTZ geometry
with $r_+ = \ell^{-1}$, so that it has the same characteristic size
as the typical conical defect. The Bekenstein-Hawking entropy of
this black hole is then
\begin{align} S \sim A \sim \ell^3 r_+ \sim \ell^2 \sim N^{1/2}~,
\end{align}
where the factor of $\ell^3$ came from integration over the $S^3$
that we have suppressed. $S \sim N^{1/2}$ is indeed the correct
ground state entropy of the D1-D5 system. This is an example of the
stretched horizon idea advocated in \cite{Lunin:2001jy}.
The most obvious difference between the 2-point functions computed above
is that the conical defect result is periodic in time, with a period
$\Delta t = 2\pi n \ell$, while the naive geometry and the BTZ black
hole results decay to zero at large time. Usually, this sort of decay
is associated with the presence of a horizon, with the information loss
problem arising because the decay winds up implying a failure of
unitarity \cite{Maldacena:2001kr}. Later in this paper we will show that
the decay is the correct universal description of the typical two-point
function in a typical state, but that its persistence to late times is
an artifact of ignoring the precise quantum mechanical details of the
individual microstates of a black hole. Our computations will be for
the $M=0$ BTZ black hole which will turn out to be the effective
coarse-grained description of the typical Ramond ground state of the
dual CFT\@. To this end, we now turn to characterizing the structure of
these states.
\section{Typical states}
\label{sec:typ}
\subsection{Statistics and typical states}
As described in subsection \ref{subsec:d1d5andR}, each ground state in
the Ramond sector of the D1-D5 CFT is characterized by a set of integers
$\{ N_{n\mu}, N^\prime_{n\mu} \}$ specifying the distribution of
constituent bosonic and fermionic twists (\ref{gen_twist2}). When the
total twist length $N=\sum_{n,\mu} n (N_{n\mu}+N'_{n\mu})$ is very
large, there are a macroscopic number ($\sim e^{2\sqrt{2}\pi\sqrt{N}}$)
of Ramond ground states. In such a situation, most of those
$e^{2\sqrt{2}\pi\sqrt{N}}$ microstates will have a twist distribution
$\{ N_{n\mu}, N^\prime_{n\mu} \}$ that lies very close to a certain
``typical'' distribution. In the large $N$ limit, the difference among
individual distributions is small. Roughly, statistical mechanics says
that $\langle(\Delta N_{n\mu})^2\rangle\sim N_{n\mu}$, thus
${\langle(\Delta N_{n\mu})^2\rangle^{1/2}\over N_{n\mu}}\sim
(N_{n\mu})^{-1/2}\to 0$ as $N_{n\mu }\to\infty$. Thus, although
correlation functions computed in individual microstates depend on the
precise form of the microstate distribution $\{N_{n\mu},N_{n\mu}'\}$,
for almost all microstates the generic responses should deviate by small
amounts from the results for the typical state. In the next section,
this will be the basis for the emergence of an effective black hole
description of typical Ramond ground states. A similar analysis of the
typical states was carried out for the $\ads{5}/{\rm SYM}_4$ duality in
\cite{Balasubramanian:2005kk}.
In this section our goal is to characterize the typical distribution of
twists and the size of fluctuations around it within the ensemble of
Ramond ground states. Ideally, we carry out a microcanonical analysis
by studying all partitions of integers of integers (\ref{gen_twist2})
that lead to a total twist of $N$. However, it is easier to carry out a
canonical analysis by including states with arbitrary total twist into
the ensemble, while fixing the average total twist to be $N$ via an
effective temperature $T$. The relative error incurred by the canonical
approach compared to the exact microcanonical analysis vanishes in the
large $N$ limit. Large $N$ will correspond to large temperature $T \gg
1$, or equivalently, small $\beta = 1/T \ll 1$. Since the constituent
twist operators in (\ref{gen_twist2}) carry an $R$-charge, we can study
the structure of Ramond ground states restricted to carry some fixed
$R$-charge. In particular, in terms of the $SU(2)_R \times
\widetilde{SU(2)}_R$ charges in \eqref{chargedef}, let us define:
\begin{align}
J &=-J_R^3-{\widetilde{J}}_R^3,\qquad
\tilde{J}=J_R^3-{\widetilde{J}}_R^3 \, .
\end{align}
With these definitions, the $R$-charges $J$ and $\tilde{J}$ correspond
in the bulk geometry to orthogonal angular momenta in the $\mathbb R^4$
perpendicular to the D1-D5 worldvolume.\footnote{If we let $\mathbb R^4$
coordinates be $x^{1,2,3,4}$, the two angular momenta in question are
$J_{12}$ and $J_{34}$. See Appendix \ref{app:cft}.1 for details.} We
will consider the structure of Ramond ground states with $\tilde{J} = 0$
and different values of $J$.
\subsection{Typical twist distribution with $J={\widetilde{J}}=0$}
Let us first consider the ensemble of all the Ramond ground states
\eqref{gen_twist2} with equal statistical weight. Some of the states in
this ensemble will have a non-vanishing $R$-charge. However, because
the polarizations $\mu$ of twist operators $\sigma_n^\mu,\tau_n^\mu$ are
weighted equally, on average the states will have $J=\tilde{J} = 0$.
Indeed, as we will see, there are so many more states with $J=0$ than $J
\neq 0$ that summing over all states only incurs a small error in
studying the properties of $J=0$ states.
As described in (\ref{gen_twist2}) and Appendix \ref{app:cft}, we have 8
bosonic twist operators $\sigma_n^\mu$ and 8 fermionic twist operators
$\tau_n^\mu$ which are all independent. So the canonical partition
function is
\begin{align}
Z(\beta)&
={\rm Tr}[e^{-\beta N}]
=\prod_{n=1}^\infty {(1+q^n)^8\over (1-q^n)^8}
= \left[{\vartheta_2(0|\tau)\over 2\eta(\tau)^3}\right]^4,
\qquad
q=e^{2\pi i \tau}=e^{-\beta}.
\end{align}
Using the modular property of the theta function,
\begin{align}
Z(\beta)
= \left[{\beta\over 4\pi} {\vartheta_4(0|-{1\over\tau})\over \eta(-{1\over\tau})^3}\right]^4
\sim e^{2\pi^2/\beta} \qquad (\beta\ll 1).
\end{align}
The relation between ``energy'' $N$ and temperature $\beta$ is
\begin{align}
N&=\Bracket{\sum_{n=1}^\infty\sum_{\mu} n(N_{n\mu} +N'_{n\mu})}
=-{\partial \over \partial \beta}\ln Z(\beta)
\simeq {2\pi^2\over \beta^2}.\label{N_beta}
\end{align}
Since all twist operators are independent, the average
distribution $\{N_{n\mu},N'_{n\mu}\}$ is given by the
Bose--Einstein and Fermi--Dirac distribution, respectively:
\begin{align}
N_{n\mu}&={1\over e^{\beta n}-1},\qquad
N'_{n\mu}={1\over e^{\beta n}+1},\qquad
N_n=\sum_\mu (N_{n\mu}+N'_{n\mu})={8\over \sinh \beta n}.
\label{typ_J=0}
\end{align}
For large $N$, the typical states of our ensemble have a
distribution almost identical to \eqref{typ_J=0}. We will call
the distribution \eqref{typ_J=0} the ``representative''
distribution.
\subsection{Typical twist distribution with $J\neq 0$ and $\tilde{J} = 0$}
\label{subsec:typ_J=/=0}
Now let us consider the typical state in the ensemble with fixed $R$-charge $J \neq 0$.
From the definitions in subsection \ref{sec:d1d5}.1, the twist operators
that carry nonzero $J$ are
\begin{align*}
\sigma_n^{s\tilde{s}}:&~ J=-(s + \tilde{s})/2, &
\tau_n^{s{\widetilde{\alpha}}}:&~ J=-s/2, &
\tau_n^{{\widetilde{\alpha}}{\widetilde{s}}}:&~ J=-{\widetilde{s}}/2.
\end{align*}
Strictly speaking, we should consider the microcanonical ensemble
in which $N$ and total $J$ are fixed. But again in the large $N$
limit we can equivalently consider the canonical ensemble in which
$N$ and $J$ are controlled by temperature $\beta$ and chemical
potential $\mu$.
To construct the partition function it is convenient to use the map
(\ref{FPD1D5map}) between the Ramond ground states and the FP system.
Then we are equivalently constructing the ensemble left-moving
oscillations of the FP string as specified in (\ref{FPstates1}). In the
FP language we consider an ensemble in which we have $N_B$ left-moving
bosons $\alpha_{-n}^i$ and $N_F$ left-moving fermions $\psi_{-n}^i$,
where $n=1,2,\dots$. Let the bosons and fermions carry the following
$R$-charge assignments:
\begin{align}
\begin{array}{ccl@{~~~~~~~~~~~~}ccl}
n_B &\rm bosons:& J=+1& n_F &\rm fermions:& J=+1/2\\
n_B &\rm bosons:& J=-1& n_F &\rm fermions:& J=-1/2\\
N_B-2n_B &\rm bosons:& J=0 & N_F-2n_F &\rm fermions:& J=0 \\
\end{array}\label{spin_assign}
\end{align}
The case we are interested in, {\it i.e.\/}\ the D1-D5 system on $T^4$,
has $N_B=N_F=8$, $n_B=1$, $n_F=4$. The D1-D5 system on K3 has $N_B=24$,
$n_B=1$, $N_F=n_F=0$, for which state counting was first studied in
\cite{Russo:1994ev} from the heterotic dual perspective. More recently,
the microscopics of the K3 case was studied in \cite{Iizuka:2005uv}, and
the discussions below and in Appendix \ref{app:J=/=0} are generalization
of the one therein.
We can compute the entropy $S(N,J)$ for given level $N$ and
$R$-charge $J$ by studying the partition function
\begin{align}
Z(\beta,\mu)\notag
=\sum_{N,J}d_{N,J}q^N z^J
={\rm Tr}[e^{-\beta(N-\mu J)}]
=\prod_{n=1}^\infty
{[(1+z^{1/2}q^n)(1+z^{-1/2}q^n)]^{n_F}(1+q^n)^{N_F-2n_F}
\over
[(1-zq^n)(1-z^{-1}q^n)]^{n_B}(1-q^n)^{N_B-2n_B} },
\end{align}
where $q=e^{2\pi i \tau}=e^{-\beta}$, $z=e^{2\pi i
\nu}=e^{\beta\mu}$. The entropy in the $N\to\infty$ limit can be
evaluated by thermodynamic approximation, as explained in Appendix
\ref{app:J=/=0}, and the result is
\begin{align}
S=\log d_{N,J}& =
2\pi\sqrt{{c\over 6}(N-|J|)}~.
\label{flyg27May05}
\end{align}
Now let us apply this to the D1-D5 system on $T^4$, for which
$N_B=N_F=8$, $n_B=1$, $n_F=4$. One sees from \eqref{flyg27May05}
that the only effect of $J\neq0$ is to replace $N$ with ${\widetilde{N}}\equiv
N-J$. Here we assumed $J>0$. This means that almost all states
({\it i.e.}, the states that are responsible for the entropy) in
the ensemble with level $N$ and $R$-charge $J$ are of the
form
\begin{align}
(\alpha_{-1}^+{}^\dagger)^J
\underbrace{\prod_{n=1}^\infty
\left[\prod_{i}(\alpha_{-n}^{i})^{N_{ni}}
(\psi_{-n}^{i})^{N'_{ni}}\right]
\ket{0}}_
{\begin{minipage}{28ex}\scriptsize
states that are responsible for entropy of the ensemble
with level ${\widetilde{N}}=N-J$ and no angular momentum
\end{minipage}}
,\label{hlro26May05}
\end{align}
where $\alpha_{-n}^+{}^\dagger$ is the creation operator of the boson
that carries $J=+1$. Indeed, the entropy from the
``$\underbrace{.....}$'' part is
$2\pi\sqrt{c{\widetilde{N}}/6}=2\pi\sqrt{c(N-J)/6}$, which fully accounts for
\eqref{flyg27May05}. Of course, besides $ (\alpha_{-1}^+{}^\dagger)^J$
there are other combinations of oscillators that can carry $R$-charge
$J$. But any other combination will exact more price in $N$, and will
therefore lead to a subleading contribution to the entropy. If $J<0$,
then the same argument goes through if we replace
$(\alpha_{-1}^+{}^\dagger)^J$ with $(\alpha_{-1}^-{}^\dagger)^{|J|}$.
\bigskip
Translating the above into the language of the D1-D5 system, the
typical state of the D1-D5 system with $N\gg 1$ and $J\neq 0$
splits into the following two parts:
\begin{enumerate}
\item $|J|$ strings of unit length: $(\sigma_1^{s{\widetilde{s}}})^{|J|}$, where
$s={\widetilde{s}}=-1$ for $J>0$ and $s={\widetilde{s}}=+1$ for $J<0$. We will call this
part the ``Bose--Einstein (BE) condensate''.
\item the typical state of the ensemble with
$\sum_{n\mu}n(N_{n\mu}+N'_{n\mu})={\widetilde{N}}=N-|J|$ and no $R$-charge.
\end{enumerate}
In other words, the typical distribution $\{N_{n\mu},N_{n\mu}'\}$
of the ensemble with level $N$ and angular momentum $J$ can be
written as
\begin{align}
N_{n\mu}=N_{n\mu}^{\rm (BEC)}+{\widetilde{N}}_{n\mu},\qquad
N_{n\mu}'={\widetilde{N}}_{n\mu}',
\label{typ_J=/=0}
\end{align}
where the BE condensate part $N_{n\mu}^{\rm (BEC)}$ is given by
\begin{align}
\begin{cases}
N_{n=1,\,s={\widetilde{s}}=-1}^{\rm (BEC)}=J, & {\rm other} ~~ N_{n\mu}^{\rm (BEC)}=0 \qquad (J>0),\\[2ex]
N_{n=1,\,s={\widetilde{s}}=+1}^{\rm (BEC)}=|J|, & {\rm other} ~~ N_{n\mu}^{\rm (BEC)}=0 \qquad (J<0),
\end{cases}
\end{align}
while the non-condensate part $\{{\widetilde{N}}_{n\mu},{\widetilde{N}}_{n\mu}'\}$ is
identical to the typical distribution \eqref{typ_J=0} for the
ensemble with level ${\widetilde{N}}=N-|J|$ and no $R$-charge.
Note that the entropy of the ensemble with $J=0$ is the same as
that in the ensemble with $J$ unspecified. This is a reflection
of the fact that there are exponentially more states with $J=0$
than with $J\neq 0$.
\section{The effective geometry}
\label{sec:effgeo}
In the previous section we derived the distributions of
constituent twist operators in typical Ramond ground states with
$R$-charges $J=0$ and $J \neq 0$. In the large $N$ limit, almost
all states with the given charges have twist distributions that
lie close to these typical distributions. Here we will compute
two-point correlation functions in typical states and show that
generic correlators computed at finite time separations are
largely independent of the details of the microstate. Indeed, at
small time separations two-point correlators in the typical $J=0$
state give a universal response, as if the corresponding spacetime
geometry was a black hole. By contrast, atypical correlators
whose two insertion points are separated by very long times give
responses with intricate variations that encode the detailed
microstate.
The fact that generic probes of typical states give essentially
universal responses governed by the statistics of the state's
microscopic constituents is reminiscent of the similar observation in
\cite{Balasubramanian:2005kk} for the case of AdS$_5$/SYM$_4$. There, it
was argued that the correlation function of a small ``probe'' operator
${\cal A}$ in a black hole background produced by a long operator ${\cal O}$ is
determined by the matching of patterns of fields ($X,Y,Z,\overline X$,
etc.) in both operators, the distribution of which is governed largely
by statistics. In the case of half-BPS states of the AdS$_5$/SYM$_4$
theory \cite{Corley:2001zk, Berenstein:2004kk, Lin:2004nb} it was
possible to use the Yang--Mills theory to argue for an effective
spacetime description of microstates in terms of a singular geometry
\cite{Balasubramanian:2005kk}. In the D1-D5 case, we have already noted
that there exists a proposed map taking RR ground states into
geometries, and we argue below that this map will also give an effective
singular spacetime description to typical states. However, to genuinely
prove these assertions it is necessary to compute correlation functions,
because it is through correlators that we can rigorously compare bulk
and boundary physics in the AdS/CFT correspondence. Here we will carry
out the analysis of deriving an effective geometry from correlation
functions and thereby infer the effective geometry corresponding to a
typical state. Furthermore, the maps between states and geometries,
either based on LLM \cite{Lin:2004nb} or on the Lunin-Mathur geometries,
are only valid at the level of two-derivative supergravity. To learn
anything about the geometry when this approximation breaks down it is
necessary to extract the spacetime physics from CFT correlation
functions (or augment the original map with higher derivative terms).
We will give an explicit example of this here. At late times our
correlators probe the strongly curved region of the geometry where the
effective spacetime description breaks down. The CFT correlators
continue to be valid and give a result which depends on which particular
microstate one has chosen. This shows how the CFT can be used to go
beyond the accuracy of the low energy spacetime description.
\subsection{Two-point functions of the D1-D5 CFT}
For simplicity, we will compute the 2-point functions of non-twist
``probe'' operators ${\cal A}$ in states created by general twist
operators. ${\cal A}$ can be written as a sum over copies of the CFT:
\begin{align}
{\cal A}&={1\over \sqrt{N}}\sum_{A=1}^N {\cal A}_A\label{sum_CA_A2}
\end{align}
where ${\cal A}_A$ is a {\em non-twist\/} operator that lives in the
$A$-th copy. For example, we can take
\begin{align}
{\cal A}_A=\partial X^a_A(z) \bar \partial X^b_A({\overline{z}}),
\label{A_grvtn2}
\end{align}
which corresponds to a fluctuation of the metric in the internal
$T^4$ direction. Although, such non-twist operators are only a
subset of the operators that correspond to spacetime excitations, we
will restrict ourselves to them because their correlation functions
are much easier to compute than those of twist operators, and
because they will be sufficient to demonstrate that an effective
geometry emerges in the $N\to \infty$ limit.
Given a general Ramond ground state $\sigma$ (\ref{gen_twist2}) we
are interested in computing
\begin{equation}
\langle \sigma^\dagger {\cal A}^\dagger {\cal A} \sigma \rangle
\end{equation}
The key result, demonstrated in Appendix \ref{app:cft}, is that for non-twist
operators at the orbifold point in the CFT such correlation
functions decompose into independent contributions from the
constituent twists operators in (\ref{gen_twist2}). Denoting the
constituents $\sigma_n^\mu, \tau_n^\mu$ collectively by
$\sigma^{\hat{\mu}}_n$ and $N_{n\mu},N^\prime_{n\mu}$ by
$N_{n\hat{\mu}}$, we write the Ramond ground states
(\ref{gen_twist2}) as
\begin{equation}
\sigma = \prod_{n,\hat{\mu}} (\sigma_n^{\hat{\mu}} )^{N_{n\hat{\mu}}} \, .
\end{equation}
Then, the desired correlation function decomposes as
\begin{equation}
\langle \sigma^\dagger {\cal A}^\dagger {\cal A} \sigma \rangle
= {1 \over N}
\sum_{n,\hat{\mu}} n N_{n\hat{\mu}}
\sum_{A=1}^n \langle [\sigma^{\hat{\mu}}_n]^\dagger
{\cal A}_A^\dagger {\cal A}_1 \sigma_n^{\hat{\mu}} \rangle~.
\label{SAAS2}
\end{equation}
The problem then reduces to computing 4-point functions of the form
\begin{align}
\Bracket{ [\sigma_{(1\cdots n)}^{{\widehat{\mu}}}(z=\infty)]^\dagger {\cal A}_A(z_1)^\dagger
{\cal A}_B(z_2) \sigma_{(1\cdots n)}^{\widehat{\mu}}(z=0)}
\equiv
\bracket{ {\cal A}_A(z_1)^\dagger
{\cal A}_B(z_2) }_{\sigma_{(1\cdots n)}^{\widehat{\mu}}},
\label{4pt_func2}
\end{align}
where $1\le A,B\le n$, and the equation indicates that we are
equivalently computing the 2-point function of ${\cal A}$ in the ground
state of twist sector $n$. As we described, in the $n$th twist
sector the worldsheet is effectively $n$ times as long and therefore,
as shown in Appendix \ref{app:cft}, for bosonic operators
\begin{align}
\bracket{{\cal A}_A^\dagger(w_1) {\cal A}_B(w_2)}_{\sigma_{(1\cdots n)}}=
{C \over \left[2 n \sin\left({w \over 2n }\right)\right]^{2h}
\left[2n \sin\left({{\overline{w}} \over 2n}\right)\right]^{2\tilde h}},
\label{corrAA_w2}
\end{align}
where
\begin{align}
w &\equiv w_1-w_2,\qquad
{\overline{w}} \equiv {\overline{w}}_1-{\overline{w}}_2.
\end{align}
Here, the copy labels $A,B$ mean that $w_1$ and $w_2$ must be understood
as $w_1+2\pi (A-1)$ and $w_2+2\pi (B-1)$, respectively. The analogous
computation for fermionic ${\cal A}$ is given in Appendix
\ref{app:cft:fermicorr}.
\subsection{Example of typical state correlation function}
\label{subsec:typ_corr}
The correlation function of non-twist operators in the general
microstate \eqref{gen_twist2} can be computed by plugging
\eqref{corrAA_w2} and \eqref{ferm_corr_bldblk} into the general
formula \eqref{SAAS2}. For example, for ${\cal A}$ purely bosonic, we
substitute bosonic correlator \eqref{corrAA_w2} into \eqref{SAAS2}
to obtain
\begin{align}
\bracket{{\cal A}(w_1) {\cal A}(w_2)}_{\Sigma}
={1\over N}\sum_{n} n N_n \sum_{k=0}^{n-1}
{C \over \left[2 n \sin\left({w-2\pi k\over 2n}\right)\right]^{2h}
\left[2n \sin\left({{\overline{w}}-2\pi k \over 2n}\right)\right]^{2\tilde h}},
\label{AASigma}
\end{align}
where
\begin{align}
N_n\equiv \sum_\mu (N_{n\mu}+ N_{n\mu}').\label{mnmc14Jul05}
\end{align}
Here we took into account that the copy labels $A,B$ mean that $w$ in
\eqref{corrAA_w2} should be replaced by $w+2\pi(A-B)$. The correlation
function for the typical state is obtained simply by plugging the
typical distribution \eqref{typ_J=0} or \eqref{typ_J=/=0} into
$\{N_{n\mu},N_{n\mu}'\}$ above. For fermionic ${\cal A}$, the correlation
function \eqref{4pt_func} depends on the spin $\mu$ and the expression
is more complicated, as we will see below.
As a simple example of a probe ${\cal A}$, take the operator \eqref{A_grvtn2}
which is dual to a fluctuation of the metric on $T^4$. For this
operator $h=\tilde h=1$. In this case, the summation over $k$ in
\eqref{AASigma} is the same as in (\ref{conicalG}). Therefore the
correlation function can be written as
\begin{align}
G(w,{\overline{w}})&\equiv
\bracket{\partial X\bar{\partial} X(w_1)\,\partial X\bar{\partial} X(w_2)}_\Sigma\notag\\
&=-{1\over N}\sum_{n=1}^\infty
{n N_n\over 16n^2\sin^2{w-{\overline{w}}\over 2n}}
\left[
{1\over \sin^2{w\over 2}}+{1\over \sin^2{{\overline{w}}\over 2}}
- {2\sin{w-{\overline{w}}\over 2}\over n\tan{w-{\overline{w}}\over 2n}\sin{w\over 2}\sin{{\overline{w}}\over 2}}
\right].\label{dXdX_dXdX}
\end{align}
In Lorentzian signature we set
\begin{align}
w=\phi-t,\qquad {\overline{w}}=\phi+t \, .
\end{align}
The correlator $G(w,{\overline{w}})=G(t,\phi)$ then diverges at $w=k\pi/2$ or
${\overline{w}}=k\pi/2$ with $k\in \mathbb{Z}$. This divergence is a
physical one, since on a finite cylinder a particle periodically
returns to the same spatial location. Therefore, in order to make
the temporal behavior of the correlation function more
transparent, it is useful to remove this divergence. So, let us
define the regularized correlator $\widehat{G}(t,\phi)$ by dividing
$G(t,\phi)$ by the vacuum correlation function of the probe graviton operator:
\begin{align}
\widehat{G}(t,\phi)&\equiv -16\sin^2{w\over 2}\sin^2{{\overline{w}}\over 2}\, G(t,\phi)\notag\\
&={1\over N}\sum_{n=1}^\infty {n N_n\over (n\sin{t\over n})^2}
\left[\sin^2{w\over 2}+\sin^2{{\overline{w}}\over2}-{2\sin t\sin{w\over 2}\sin{{\overline{w}}\over 2}\over n\tan{t\over n}}\right].
\label{def_Ghat}
\end{align}
Plugging in the representative distribution of constituent twists
for microstates with $J=0$ \eqref{typ_J=0} into the regularized
correlator \eqref{def_Ghat} we obtain Fig.\ \ref{Ghat}. As one can
see from this graph, the correlator decays rapidly at initial times
($t\lesssim\pi$), and at later times exhibits a quasi-periodic
behavior. Quasi-periodicity is not surprising; it is expected on
general grounds in a system with a finite number of degrees of
freedom. Furthermore, one sees that, in the $\beta\to 0$ (or
equivalently $N\to \infty$) limit, $\widehat{G}$ approaches a certain limit
shape. As we will discuss below, the limit shape in the
$N\to\infty$ limit turns out to be the correlation function
\eqref{naiveG} in the $M=0$ BTZ geometry.
\begin{figure}[htb]
\begin{center}
\begin{quote}
\begin{tabular}{c@{~~~~~~~~}c}
\epsfxsize=7cm \epsfbox{Ghat_1.eps}
&
\epsfxsize=7cm \epsfbox{Ghat_3.eps}
\\
\epsfxsize=7cm \epsfbox{Ghat_2.eps}
&
\epsfxsize=7cm \epsfbox{Ghat_4.eps}
\end{tabular}
\caption{\sl Plot of the regularized correlation function
$\widehat{G}(t,\phi=0;\beta)$ as a function of $t$, for various values of
$\beta$. The two graphs on the left show short-time behavior; the two on the right show
long-time
behavior. As $\beta\to0$ (or equivalently, $N\to\infty$), $\widehat{G}$
approaches the correlation function (\ref{naiveG}) in the $M=0$ BTZ
geometry, denoted in the graph by dashed lines. }
%
\label{Ghat}
\end{quote}
\end{center}
\end{figure}
\subsection{Effective geometry of microstates with $J=0$}
Now consider the correlation function \eqref{AASigma} of a general
bosonic non-twist operator:
\begin{align}
\bracket{{\cal A}(w_1) {\cal A}(w_2)}_{\Sigma}
={1\over N}\sum_{n} n N_n \sum_{k=0}^{n-1}
{C \over \left[2 n \sin\left({w-2\pi k\over 2n}\right)\right]^{2h}
\left[2n \sin\left({{\overline{w}}-2\pi k \over 2n}\right)\right]^{2\tilde h}}~.\label{ibvm8Jul05}
\end{align}
Let us study the relative size of the contributions to this from
terms with different $n$. The contributions come multiplied by $n
N_{n}$, which is $8n\over \sinh\beta n$ for the typical microstates
with $J=0$ (Eq.\ \eqref{typ_J=0}). Because of the suppression by
the $\sinh \beta n$, the values of $n$ that make substantial
contributions to the correlation function \eqref{AASigma} are $n
\lesssim 1/\beta\sim \sqrt{N}$.
Thus there are $O(\sqrt{N})$ twists
that make a significant contribution. Now observe that for any
$\gamma < 1/2$, the number of twists with $n \lesssim N^\gamma$ is
parametrically smaller than $\sqrt{N}$. Indeed, the ratio vanishes
as $N \to \infty$. In this sense we can say that in the $N \to
\infty$ limit, (\ref{ibvm8Jul05}) is dominated by twists scaling
as $n \sim \sqrt{N}$.
Next, for any $n\ge 1$, when $t \ll n$ we can approximate the sum on $k$ as
\begin{align}
\sum_{k=0}^{n-1}
{1 \over \left[2 n \sin\left({w-2\pi k\over 2n}\right)\right]^{2h}
\left[2n \sin\left({{\overline{w}}-2\pi k \over 2n}\right)\right]^{2\tilde h}}
\approx
\sum_{k=-\infty}^{\infty}
{1 \over (w-2\pi k)^{2h} ({\overline{w}}-2\pi k)^{2\tilde h}}\qquad
(t\ll n),
\nota
\end{align}
where we assumed $h+\tilde h={\rm even}$.
Putting the above statements together, we arrive at the following
conclusion: for sufficiently early times
\begin{align}
t\ll t_c={\cal O}(\sqrt{N}),
\end{align}
the correlation function \eqref{ibvm8Jul05} can be approximated by
\begin{align}
\bracket{{\cal A}(w_1) {\cal A}(w_2)}_{\Sigma}
&\approx {1\over N}\sum_{n} n N_n
\sum_{k=-\infty}^{\infty}
{C \over (w-2\pi k)^{2h} ({\overline{w}}-2\pi k)^{2\tilde h}}\notag\\
&=
\sum_{k=-\infty}^{\infty}
{C \over (w-2\pi k)^{2h} ({\overline{w}}-2\pi k)^{2\tilde h}}.\label{eff_cor_J=0bos}
\end{align}
This is
precisely the bulk correlation function in the naive geometry, or the
$M=0$ BTZ black hole (compare with (\ref{naiveG}) for
$h=\tilde{h}=1$). Therefore, in the orbifold CFT approximation, the
emergent effective geometry of the D1-D5 system is the $M=0$ BTZ
black hole geometry. The description in terms of this effective
geometry is valid until $t\sim t_c$, which goes to infinity as
$N\to\infty$. In the special case $h=\tilde{h}=1$, the summation
\eqref{eff_cor_J=0bos} yields (\ref{naiveG}). In this case, we
indeed saw in Fig.\ \ref{Ghat} that the $\beta\to 0$ limit of the
correlation function is given by (\ref{naiveG}) (or
\eqref{eff_cor_J=0bos}).
Notice that in (\ref{eff_cor_J=0bos}) the sum over the twists $n$
factors out. Thus, for $t< t_c$ we are showing that the
correlation function is largely independent of the detailed
microscopic distribution of twists. It is this universal response
that reproduces the physics of the $M=0$ BTZ black hole.
After $t\sim t_c$, the approximation \eqref{eff_cor_J=0bos} breaks
down, and the correlation function starts to show random-looking,
quasi-periodic behavior (see Fig.\ \ref{Ghat}). The form of the
correlation function in this regime will depend on the precise form
of the individual microstate, no matter how close it is to the
representative state \eqref{typ_J=0}.
The $\beta \rightarrow 0$ limit corresponding to the $M=0$ BTZ black
hole yields a correlator which decays to zero at large times as $1/t^2$.
By contrast, the microstate correlators exhibit quasi-periodic
fluctuations around a nonzero mean value. Numerical analysis indicates
that this mean value scales as ${1 \over \sqrt{N}}$ for $h=\tilde
h=1$. For an ordinary finite size, finite temperature system, one
expects the mean value to be of order $e^{-c S}$ where $S$ is the
entropy and $c$ is of order $1$. This behavior arises because typical
interactions can explore the entire phase space of the system. The fact
that we observe power law rather than exponential behavior is likely to
be a result of working in the free orbifold limit of the CFT and probing
the system with only non-twist operators. Under these conditions the
full space of states does not come into play in determining a
correlation function. For example, the non-twist operators cannot see
the full structure of the microstate, for instance the relative phases
between different twist components, and so might be expected to exhibit
larger correlations at late times. For this reason it would be very
instructive to repeat our analysis for twist operators, although this is
technically much more challenging.
A finite $N$ microstate correlator will exhibit exact periodicity in
time because only a finite number of frequencies appear in the
Fourier expansion. The frequencies are $\omega_n = {n\over N},~ n=
1, 2, \ldots , N$. Let $L(N)$ denote the least common multiple of
$(1,2, \ldots, N)$. The correlator is then periodic with period
$\Delta t = 2\pi N L(N)$. The large $N$ behavior of $L(N)$ is
$L(N) \sim e^N$, and therefore
\begin{align}
\Delta t \sim N e^N~.
\end{align}
Our correlators have been computed in the canonical ensemble in
which the summation over $n$ extends past $N$ up to infinity, and so
we will not see this exact periodicity. On the other hand, due to
the exponential suppression of the distribution function $N_n$ the
deviation from exact periodicity is tiny for large $N$. As was
argued above, and as can be confirmed numerically, one finds that
for large $N$ the large time behavior of the correlator is
unaffected if we truncate the sum over $n$ at $n_{\rm max} = c
\sqrt{N}$, for $c$ of order unity. Taking this into account, we see
that our correlators will exhibit approximate periodicity with
period
\begin{align}
\Delta t \sim e^{c \sqrt{N}} = e^{\tilde{c} S} ~,
\end{align}
where $S=2\pi \sqrt{2}\sqrt{N}$ is the entropy. This timescale is
the so-called Poincar\'{e} recurrence time, over which generic
finite size thermal systems are expected to exhibit approximate
periodicity.
\bigskip\bigskip
\noindent{\bf Fermionic probes: } In the above we restricted
ourselves to bosonic probes, but we obtain the same effective
geometry even if we probe the microstate with operators that contain
fermions. For example, let ${\cal A}=\Psi^{s'}{\widetilde{\Psi}}^{{\widetilde{s}}'}$ as defined
in \eqref{bosonize}. From \eqref{ferm_corr_bldblk}, we obtain
\begin{align}
\bracket{{\cal A}_A{\cal A}_B}_{\sigma_n^{s{\widetilde{s}}}}
=
\bracket{ [\Psi_A^{s'}{\widetilde{\Psi}}_A^{{\widetilde{s}}'}(w_1)]^\dagger \,
\Psi_B^{s'}{\widetilde{\Psi}}_B^{{\widetilde{s}}'}(w_2)}_{\sigma_n^{s{\widetilde{s}}}}
&=
{e^{iss'w/2n-i{\widetilde{s}}\,{\widetilde{s}}'{\overline{w}}/2n}\over (2n\sin{w\over 2n})
(2n\sin{{\overline{w}}\over 2n})}~.\label{jdqt10Jun05}
\end{align}
If we sum over copies,
\begin{align}
\sum_{A=1}^n \bracket{{\cal A}_A{\cal A}_1}_{\sigma_n^{s{\widetilde{s}}}}
&=
\sum_{k=0}^{n-1}
{e^{iss'(w+2\pi k)/2n-i{\widetilde{s}}\,{\widetilde{s}}'({\overline{w}}+2\pi k)/2n}
\over (2n\sin{w+2\pi k\over 2n})(2n\sin{{\overline{w}}+2\pi k\over 2n})}\notag\\
&\approx
\sum_{k=-\infty}^{\infty}
{1 \over (w+2\pi k)({\overline{w}}+2\pi k)}\qquad (t\ll n).
\label{iabi10Jun05}
\end{align}
Other twist operators
$\sigma_n^{{\widetilde{\alpha}}{\widetilde{\beta}}},\tau_n^{s{\widetilde{\alpha}}},\tau_n^{{\widetilde{\alpha}}{\widetilde{s}}}$
give different correlation functions, but they are all identical
to \eqref{iabi10Jun05} for $t\ll n$:
\begin{align}
\sum_{A=1}^n \bracket{{\cal A}_A{\cal A}_1}_{\sigma_n^{\widehat{\mu}}}
\approx
\sum_{k=-\infty}^{\infty}
{1 \over (w+2\pi k)({\overline{w}}+2\pi k)}
\qquad (t\ll n)
,\label{jkeg10Jun05}
\end{align}
where $\sigma_n^{\widehat{\mu}}$ can be any of the twist operators
$\sigma_n^{s{\widetilde{s}}}, \sigma_n^{{\widetilde{\alpha}}{\widetilde{\beta}}}, \tau_n^{s{\widetilde{\alpha}}},
\tau_n^{{\widetilde{\alpha}}{\widetilde{s}}}$. Plugging this result into \eqref{SAAS2}, we
conclude that
\begin{align}
\bracket{{\cal A}(w_1) {\cal A}(w_2)}_{\Sigma}
&=
\bracket{ [\Psi_A^{s'}{\widetilde{\Psi}}_A^{{\widetilde{s}}'}(w_1)]^\dagger \,
\Psi_B^{s'}{\widetilde{\Psi}}_B^{{\widetilde{s}}'}(w_2)}_\Sigma
\approx
\sum_{k=-\infty}^{\infty}
{1 \over (w-2\pi k) ({\overline{w}}-2\pi k )}
\qquad (t\ll t_c).
\end{align}
This is again the correlation function in the $M=0$ BTZ black hole
geometry. Therefore, we conclude that the effective geometry of the
D1-D5 system in the orbifold CFT approximation is the $M=0$ BTZ
black hole geometry for any non-twist probe operators, bosonic or
fermionic. The description by this effective geometry breaks down
at $t=t_c={\cal O}(\sqrt{N})$.
\bigskip\bigskip
\noindent{\bf Gravitational origin of the effective geometry: } We
can also argue that the effective geometry for the ensemble with
$J=0$ should be the $M=0$ BTZ black hole by using the Lunin--Mathur
metric \eqref{LMmetric1}. Assume that the profile $F_i(v)$ is a
random superposition of small-amplitude, high-frequency oscillations
that is much smaller than the asymptotic AdS radius:
\begin{align}
|{\bf F}(v)|\ll \ell\sim N^{1/4}.\label{kjcj8Jul05}
\end{align}
Then, for $r\sim\ell\gg |{\bf F}(v)|$,
\begin{align}
f_5 &={Q_5\over L}\int_0^L dv {1\over |{\bf x}-{\bf F}(v)|^2} \approx{Q_5\over r^2},\\
f_1 &={Q_5\over L}\int_0^L dv {|\dot{\bf F}(v)|^2\over |{\bf x}-{\bf F}(v)|^2}
\approx{1\over r^2}{Q_5\over L}\int_0^L dv\, {|\dot{\bf F}(v)|^2} ={Q_1\over r^2},\\
A_i &={Q_5\over L}\int_0^L dv {\dot F_i(v)\over |{\bf x}-{\bf F}(v)|^2}
\approx0,
\end{align}
where in the second line we used \eqref{Q12}, and in the third line
$A_i(x)$ vanishes because $F_i(v)$ is random. So the metric
\eqref{LMmetric1} is
\begin{align}
ds^2&=-{r^2\over \ell^2}dt^2+{r^2\over \ell^2}dy^2
+{\ell^2\over r^2}(dr^2+r^2d\Omega_3^2)+\sqrt{Q_1\over Q_5}\,ds_{T^4}^2
\end{align}
with $\ell=(Q_1 Q_5)^{1/4}$. This is indeed the direct product of
the $M=0$ BTZ black hole \eqref{naive}, and $S^3\times T^4$, if one
sets $y\to R\phi$, $r\to\ell r/R$. One can check that the condition
\eqref{kjcj8Jul05} is satisfied from the microscopic theory, as
follows. If the typical frequency and amplitude are $\omega$ and
$a$, respectively, then $|{\bf F}|\sim a$, $|\dot{\bf F}|\sim a\omega$. We
can relate $\omega,a$ with the microscopic quantities $n,N_n$ as
$\omega\sim n$, $a\sim N_n^{1/2}$. Recall that the typical twist is
$n\sim N^{1/2}\sim 1/\beta$. For $n\sim1/\beta$, $N_n$ is
$N_n={8\over \sinh(\beta n)}={\cal O}(1)$ from \eqref{typ_J=0}.
Therefore, $\omega\sim N^{1/2}, a\sim N^0$. This indeed satisfies
\eqref{kjcj8Jul05}.
\subsection{Effective geometry of microstates with $J\neq 0$}
As we saw in subsection \ref{subsec:typ_J=/=0}, the ensemble with $J\neq
0$ becomes in the large $N$ limit a ``direct product'' of the
Bose--Einstein condensate $(\sigma_{1}^{s{\widetilde{s}}})^{|J|}$ with $s={\widetilde{s}}=\mp$,
and an ensemble with level ${\widetilde{N}}=N-|J|$ and no angular momentum.
Therefore, from the general formula \eqref{SAAS2}, one sees that
the correlation function for this ensemble is a sum of the
correlation function in the Bose--Einstein condensate background and
the one for the ensemble with level ${\widetilde{N}}=N-|J|$ and no angular
momentum.
Specifically, consider a bosonic non-twist operator ${\cal A}$.
Plugging the typical distribution \eqref{typ_J=/=0} into the
formula \eqref{AASigma},
\begin{align}
&\bracket{{\cal A}(w_1) {\cal A}(w_2)}_{\Sigma}\notag\\
&=
{|J|\over N}{C \over \left(2\sin{w\over 2}\right)^{2h}
\left(2 \sin{{\overline{w}}\over 2}\right)^{2\tilde h}}
+{1\over N}\sum_{n} n \tilde N_n \sum_{k=0}^{n-1}
{C \over \left[2 n \sin\left({w-2\pi k\over 2n}\right)\right]^{2h}
\left[2n \sin\left({{\overline{w}}-2\pi k \over 2n}\right)\right]^{2\tilde h}}
\notag\\
&\approx
{|J|\over N}{C \over \left(2\sin{w\over 2}\right)^{2h}
\left(2 \sin{{\overline{w}}\over 2}\right)^{2\tilde h}}
+\left(1-{|J|\over N}\right)
\sum_{k=-\infty}^{\infty}
{C \over ({w-2\pi k})^{2h}({\overline{w}}-2\pi k)^{2\tilde h}}
\qquad (\boldsymbol{t\ll t_c})
\notag\\
&=
{|J|\over N} \bracket{{\cal A}\CA}_{\rm BEC}
+
\left(1-{|J|\over N}\right)\bracket{{\cal A}\CA}_{\text{$M=0$ BTZ}}~.\label{gmwl9Jul05}
\end{align}
The critical time $t_c$ is now given by
\begin{align}
t_c&={\cal O}(\sqrt{N-|J|}).\label{tc_J=/=0}
\end{align}
The first term in \eqref{gmwl9Jul05}, which arises from the
Bose--Einstein condensate (BEC), is proportional to the correlation
function of ${\cal A}$ computed in global $\ads{3}$.
This is happening because the condensate is $(\sigma_1^{s{\widetilde{s}}})^{|J|}$,
$s={\widetilde{s}}=\mp$, and the 3-dimensional part of the microstate geometry
associated with this operator by itself is simply global $\ads{3}$ with
a scale $\ell \sim |J|^{1/4}$, as described by
\cite{Balasubramanian:2000rt,Maldacena:2000dr, Lunin:2002iz}. Actually
the total 10-dimensional geometry is more complicated because of the
nontrivial Wilson line coming from the internal $S^3$, but the bosonic
operator ${\cal A}$ does not sense this extra structure. On the other hand,
fermionic ${\cal A}$ does see this structure, as we will see below.
Hence the ``effective geometry'' for $t<t_c$ appears to be a weighted
average of global $\ads{3}$ (with a nontrivial Wilson line) and the
$M=0$ BTZ\@. The linear summation in \eqref{gmwl9Jul05} is appearing
because in the orbifold CFT the simple class of non-twist probes has
correlation functions that are simply linear summations of the responses
in the individual constituent twist states (\ref{gmwl9Jul05}). Of
course as $|J| \to N$, the typical microstate operator found in
(\ref{hlro26May05}) becomes precisely the operator corresponding to
global $\ads{3}$ (with a Wilson line) in \cite{Balasubramanian:2000rt,
Maldacena:2000dr, Lunin:2002iz}. Thus the response (\ref{gmwl9Jul05}) is
simply a weighted sum of the expected responses in the $J=0$ and $|J| =
N$ limits.
Correlation functions involving fermionic operators can be evaluated
similarly. For example, let us take ${\cal A}=\Psi^{s'}{\widetilde{\Psi}}^{{\widetilde{s}}'}$ as
before. From \eqref{AASigma}, \eqref{jdqt10Jun05}, and
\eqref{jkeg10Jun05}, we obtain
\begin{align}
\bracket{{\cal A}_A{\cal A}_B}_{\Sigma}
&\approx
{|J|\over N}
{e^{is(s'w-{\widetilde{s}}'{\overline{w}})/2}\over (2\sin{w\over 2})(2\sin{{\overline{w}}\over 2})}
+\left(1-{|J|\over N}\right)
\sum_{k=-\infty}^{\infty}
{1 \over ({w-2\pi k})({\overline{w}}-2\pi k)}
\qquad (t\ll t_c)
\notag\\
&=
{|J|\over N} \bracket{{\cal A}\CA}_{\text{BEC}}
+
\left(1-{|J|\over N}\right)\bracket{{\cal A}\CA}_{\text{$M=0$ BTZ}} ,\label{gmyj9Jul05}
\end{align}
where $s={\rm sign}(J)$. Again the ``effective geometry'' appears to be
a weighted average.
The Bose--Einstein condensate part $\bracket{{\cal A}\CA}_{\text{BEC}}$ of the
bosonic correlator \eqref{gmwl9Jul05} did not care whether the
condensate is made of $\sigma_1^{s{\widetilde{s}}}$ with $s={\widetilde{s}}=-1$ or $s={\widetilde{s}}=+1$,
whereas the fermionic one \eqref{gmyj9Jul05} does depend on what the
condensate is made of through its dependence on $s={\rm sign}(J)$. This
reflects the fact that the 3-dimensional geometry corresponding to
$(\sigma_1^{s{\widetilde{s}}})^N$ with $s={\widetilde{s}}=-1$ and the one with $s={\widetilde{s}}=+1$ are
both global AdS$_3$ but differ in the nontrivial Wilson line in the
internal $S^3$ \cite{Balasubramanian:2000rt, Maldacena:2000dr,
Lunin:2002iz}. Bosonic probes are not charged under the relevant $U(1)$,
and thus its correlator is independent of what the condensate is made
of. On the other hand, fermionic probes are charged under the $U(1)$,
and its correlator depends on what the condensate is made of.
Do the above results mean that the emergent geometry is a superposition
of two classical geometries? Below we will use the Lunin-Mathur
solution \eqref{LMmetric1} to argue that this should not be the
case and that the emergent geometry should be a singular zero-horizon
limit of the black ring \cite{Elvang:2004rt}.
\bigskip\bigskip
\noindent{\bf The effective geometry should be a black ring: } Assuming $J>0$ and $J={\cal O}(N)$, the typical state of the ensemble with $J\neq 0$ is given by \eqref{hlro26May05}. In the language
of the FP system, $(\alpha_{-1}^+{}^\dagger)^J$ corresponds to an
F1 worldvolume that makes a circle with radius
$\sim\sqrt{J}={\cal O}(N^{1/2})$ in the 1-2 plane. The remaining part
$\prod_{n=1}^\infty
\left[\prod_{i}(\alpha_{-n}^{i})^{N_{ni}}(\psi_{-n}^{i})^{N'_{ni}}\right]$
adds fluctuations around this circular profile. By an argument
similar to the one given at the end of the last subsection, the
typical frequency and amplitude of the fluctuations are estimated
to be $n\sim\sqrt{N-J}={\cal O}(N^{1/2})$ and $N_n^{1/2}={\cal O}(N^0)$,
respectively.
This motivates the following profile function ${\bf F}(v)$ of the
D1-D5 metric \eqref{LMmetric1}. Namely, we assume that the profile
${\bf F}(v)$ is a circle ${\bf F}^{(0)}$ with random, small-amplitude,
high-frequency fluctuations $\delta {\bf F}$ around it:
\begin{align}
{\bf F}&={\bf F}^{(0)}+\delta{\bf F},\qquad
\begin{cases}
F_1^{(0)}+i F_2^{(0)}=a e^{i \omega v},\\
F_3^{(0)}=F_4^{(0)}=0,
\end{cases}\qquad
\omega={2\pi\over L}={R\over Q_5}.\label{iumd28Jun05}
\end{align}
From the above analysis, the amplitude of the fluctuation $\delta {\bf F}$
is much smaller than the size of the circle or the AdS radius:
\begin{align}
|\delta {\bf F}|={\cal O}(N^0)\ll |{\bf F}^{(0)}|=a={\cal O}(N^{1/2}),
\qquad
|\delta {\bf F}|={\cal O}(N^0) \ll\ell={\cal O}(N^{1/4}).
\end{align}
On the other hand, the derivatives of ${\bf F}^{(0)}$ and $\delta{\bf F}$ are of
the same order of magnitude:
\begin{align}
|\delta\dot {\bf F}|\sim n N_n^{1/2}={\cal O}(N^{1/2}), \qquad |\dot{\bf F}^{(0)}|=a\omega={\cal O}(N^{1/2}) .
\end{align}
Using these relations, the harmonic functions in \eqref{LMmetric1} are
approximated for large $N$ as follows:\footnote{This metric was studied
in \cite{Lunin:2002bj} using a different ansatz of the profile function
${\bf F}(v)$. Recent analysis of this metric from the bubbling AdS
viewpoint of \cite{Lin:2004nb} can be found in \cite{Boni:2005sf}.}
\begin{align}
f_5&\approx {Q_5\over L}\int_0^L dv{1\over |{\bf x}-{\bf F}^{(0)}|^2} ={Q_5\over\Sigma},\notag\\
f_1&\approx {Q_5\over L}
(a^2\omega^2+|\delta\dot {\bf F}|^2) \int_0^L dv
{1\over |{\bf x}-{\bf F}^{(0)}|^2}={Q_1\over\Sigma},\label{harm_J=/=0}\\
A_1+i A_2&\approx {Q_5\over L}\int_0^L dv{ia\omega e^{i\omega v}\over
|{\bf x}-{\bf F}^{(0)}|^2},
\qquad {\rm therefore}\quad
A_\psi={2a^2Q_5\omega s^2\over \Sigma(\Sigma+s^2+w^2+a^2)},
\notag
\end{align}
where
\begin{align}
x_1+i x_2=s e^{i\psi},\quad x_3+i x_4=w e^{i\phi},\quad
\Sigma=\sqrt{[(s+a)^2+w^2][(s-a)^2+w^2]}.
\label{iaua19Jul05}
\end{align}
In the second line of \eqref{harm_J=/=0}, the cross term
${\bf F}^{(0)}\cdot \delta{\bf F}$ was dropped because $\delta{\bf F}$ is
fluctuating randomly.
Also in the second line, because $|\delta\dot {\bf F}|^2$ is
fluctuating with length scale much smaller than $a$, we can
replace it with it average and take it out of the integral (so,
$|\delta\dot {\bf F}|^2$ there really means the average). We also
used the relation \eqref{Q12}:
\begin{align}
Q_1&={Q_5\over L}\int_0^L dv |\dot{\bf F}|^2\approx(a^2\omega^2+|\delta\dot{\bf F}|^2)Q_5.
\end{align}
In the third line of \eqref{harm_J=/=0}, the term containing $\delta\dot
{\bf F}$ was dropped because it is fluctuating randomly.
It is convenient to go to the $(x,y,\psi,\phi)$ coordinate system
\cite{Elvang:2004rt} with $R=a$, defined by
\begin{align}
s&={\sqrt{y^2-1}\over x-y}R,\qquad
w={\sqrt{1-x^2}\over x-y}R.
\end{align}
In this coordinate system, $A_i$, $B_i$, $\Sigma$ can be written as
\begin{align}
A_\psi={Q_5\omega\over 2}(-1-y),\qquad
B_\phi={Q_5\omega\over 2}(1+x),\qquad
\label{iojs28Jun05}
\Sigma={2R^2\over x-y}.
\end{align}
Plugging these into \eqref{LMmetric1}, one obtains the metric
\begin{align}
ds^2&={\Sigma\over \ell^2}
\left[
-(dt+{Q_5\omega\over 2}(-1-y)d\psi)^2
+(dy+{Q_5\omega\over 2}(1+x)d\phi)^2
\right]
+{\ell^2\over\Sigma}ds_4^2+\sqrt{Q_1\over Q_5}ds_{T^4}^2,\label{sing_BR}
\end{align}
where $\ell\equiv (Q_1Q_5)^{1/4}$. This is the metric of the
supersymmetric black ring \cite{Elvang:2004rt,Bena:2004de} with
charges $(Q_1,Q_2,Q_3)=(Q_1,Q_5,0)$, dipole charges
$(q_1,q_2,q_3)=(0,0, Q_5\omega)$, and radius $R=a$. For these
charges, the horizon area and thus the Bekenstein--Hawking entropy
vanish. The angular momentum of this singular black ring
satisfies
\begin{align}
J_\psi&=R^2q_3\le {Q_1 Q_2\over q_3}\equiv J_{\psi,\rm max}.\label{D1D5kk_reg}
\end{align}
If this inequality is saturated, the singular black ring becomes the
regular D1-D5$\to$kk geometry. However, in the present case,
\begin{align}
J_\psi=a^2 Q_5 \omega,\qquad
J_{\psi,\rm max}=
{Q_1Q_5\over Q_5\omega}
=
a^2Q_5\omega \left(1+{|\delta\dot{\bf F}|^2\over a^2\omega^2}\right).
\end{align}
So, the equality in \eqref{D1D5kk_reg} does not hold and the geometry
\eqref{sing_BR} describes a singular, zero-horizon limit of the black
ring.
The above argument suggests that the effective geometry for the ensemble
with $J\neq 0$ is the singular, zero-horizon limit of the black ring
\eqref{sing_BR}.\footnote{This is reminiscent of the proposal by
\cite{Bena:2004tk} that the CFT microstate of the black ring with
non-vanishing horizon is made of two parts, where the first part is made
of small effective strings of identical length, while the second part is
made of a single long string and responsible for the whole entropy.}
The description by this effective
geometry should be valid up to the critical time $t_c$ \eqref{tc_J=/=0},
which goes to infinity as $N\to \infty$.
In order to prove the above statement, one should compute the
bulk-boundary propagator in the singular black ring geometry
\eqref{sing_BR} and show that it leads to the boundary CFT
correlation function \eqref{gmwl9Jul05}, \eqref{gmyj9Jul05}.
\section{Discussion}
\label{sec:disc}
The puzzles regarding the black hole information paradox are all
traceable to the fact that we don't have an adequate understanding
of the relation between geometry and entropy. In the boundary CFT
description of black holes we can choose to work either with
individual microstates or with an ensemble, and we understand that
entropy arises from the coarse-graining associated with defining
the ensemble. In practice the ensemble usually yields results to
the accuracy we desire, and the existence of the underlying
microstate description tells us that there is no possibility of
information loss at a fundamental level.
We lack a similar understanding in the bulk. If the black hole is
to be thought of as an ensemble, we need to specify precisely the
elements of the ensemble. One logical possibility is that the
bulk description is intrinsically coarse-grained, and that
microstates can only be found in the boundary CFT\@. An alternative
picture, advocated by Mathur, is that bulk microstates are to be
described as new horizon-free geometries differing from the black
hole at the horizon scale. Some evidence for the latter has
accumulated, but the question remains open.
Here, we have studied some of these issues in the simple context of
the D1-D5 CFT at the free orbifold point. On the one hand, a large
class of microstate geometries are known, and on the other hand
there is an effective ``black-hole'' geometry describing their
``average''. We essentially tried to make this last sentence
precise by comparing CFT correlation functions computed in typical
microstates to bulk correlation functions computed in the
``black-hole'' geometry. The agreement we found, as well as its
breakdown at late times, provides evidence for the picture of black
holes as the effective description of more fundamental underlying
structures. Although the ``black-hole'' in this case has vanishing
horizon size, it does display some of the hallmarks of real black
holes, such as the decay of late time correlators.
If black holes in general represent effective coarse-grained descriptions
of underlying microstate geometries, it naturally explains why one cannot
see quasi-periodicity and Poincar\'e recurrence by summing over the
$SL(2,\mathbb Z)$ family of BTZ black holes as was pursued in
\cite{Barbon:2003aq,Kleban:2004rx}. This is analogous to the fact that,
after replacing a gas of molecules by its effective coarse-grained
description, {\it i.e.}\ a dissipative continuum, one does not expect to
be able to see quasi-periodicity or Poincar\'e recurrence in the
correlation function describing a particle scattered in the
gas.
It would be interesting to try to repeat our calculations in the context
of the D1-D5 system on K3 rather than $T^4$. In the K3 case it has been
found that higher derivative terms in the supergravity Lagrangian lead
to a nonzero size horizon whose Bekenstein--Hawking--Wald entropy agrees
with that of the CFT \cite{Dabholkar:2004yr,Dabholkar:2004dq}.
Furthermore, one can still write down a large class of microscopic
geometries which contribute to the entropy \cite{Giusto:2004xm}. The
complication is that the sigma model is no longer free, and so the
computation of CFT correlators is not as straightforward. But the goal
would be to show how the nonzero horizon size manifests itself in CFT
correlators. Alternatively, perhaps a horizon could be found even in
the $T^4$ case once interactions are included.
Another useful endeavor would be to compare bulk correlators computed in
the known microstate geometries of the D1-D5 system to the microscopic
CFT correlators we have computed here. This easily can be done for the
simplest class of states, namely those corresponding to the twist
operator $\sigma=[\sigma_n^{s{\widetilde{s}}}]^{N/n}$, $s={\widetilde{s}}=-1$. In this case the bulk
geometries are simply the conical defects (\ref{condef}), and we saw
that this gives precise agreement between bulk and boundary correlators.
But for more general states the bulk geometry is no longer just an
orbifold, and the bulk correlators will be much more complicated. On
the other hand, the CFT correlators continue to be expressed as a sum of
simple contributions. This suggests that either the bulk geometries can
also somehow be thought of as being built up out of simple geometries,
or alternatively that working at the free orbifold point of the CFT is
simply inadequate.
In this paper we studied correlation functions of non-twist
operators, but it would be very interesting to consider twist
operators. This would allow much greater sensitivity to the
microstate structure. Non-twist operators see the states as built
up out of decoupled components corresponding to the given cycles,
and this led to the correlators taking the form of a sum over
relatively simple contributions from each component. This will
no longer be the case when twist operators are used to probe the
state, and the results are expected to be much more complicated.
This extra information could potentially be used to map out the
bulk geometry in much greater detail.
\section*{Acknowledgments}
We would like to thank Jan de Boer, Hiroshi Fujisaki, Norihiro Iizuka,
Vishnu Jejjala, Oleg Lunin, Joan Simon, Sanefumi Moriyama, and Hirosi
Ooguri for valuable discussions.
We would also like to thank the organizers of the workshop on
Quantum Theory of Black Holes at the Ohio State University, where
this work was initiated, the Workshop on Gravitational Aspects of
String Theory at the Fields Institute, and Strings 2005, for
stimulating environments.
M.S. would like to thank Norihiro Iizuka for collaboration in
\cite{Iizuka:2005uv} and helpful discussions, and the Theoretical High
Energy Physics group at the University of Pennsylvania for hospitality.
P.K. was supported in part by NSF grant PHY-0099590. M.S. was supported
in part by Department of Energy grant DE-FG03-92ER40701 and a Sherman
Fairchild Foundation postdoctoral fellowship. V.B. was supported in
part by the DOE under grant DE-FG02-95ER40893, by the NSF under grant
PHY-0331728 and by an NSF Focused Research Grant DMS0139799.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 23
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\section{Introduction}
Large-scale optimization problems, such as the traveling salesman problem (TSP), show up in many applications. These problems are often computationally intractable. However, in practice often ad-hoc heuristics are successfully used that provide solutions that come quite close to optimal solutions. In many cases these, often simple, heuristics show a remarkable performance, even though the theoretical results about those heuristics are way more pessimistic.
In order to explain this difference, probabilistic analysis has been widely used over the last decades. However, the challenge in probabilistic analysis is to come up with a good probabilistic model: it should reflect realistic instances, but also be sufficiently simple to make the analysis tractable.
So far, in almost all cases, either Euclidean space has been used to generate instances of metric optimization problems, or independent, identically distributed edge lengths have been used. However, both approaches have considerable shortcomings to explain the average-case performance of heuristics on general metric instances: the structure of Euclidean space is heavily used in the probabilistic analysis, but realistic instances are often not Euclidean. The independent, identically distributed edge lengths do not even yield a metric in the first place. In order to overcome these shortcomings, Bringmann et al.~\cite{Bringmann2015} have proposed and analyzed the following model to generate random metric spaces, which had already been proposed by Karp and Steele in 1985 \cite{Karp1985}: given an undirected complete graph, start by drawing random edge weights for each edge independently and then define the distance between any two vertices as the total weight of the shortest path between them, measured with respect to the random weights.
\subsection{Related Work}
Bringmann et al.\ called the model described above \emph{random shortest path metrics}. This model is also known as \emph{first-passage percolation}, introduced by Hammersley and Welsh as a model for fluid flow through a (random) porous medium \cite{Hammersley1965,Howard2004}.
For first passage percolation in complete graphs, the expected distance between two fixed vertices is approximately $\ln(n)/n$ and the expected distance from a fixed vertex to the vertex that is most distant is approximately $2\ln(n)/n$~\cite{Bringmann2015,Janson1999}. Furthermore, the expected diameter of the metric is approximately $3\ln(n)/n$~\cite{Hassin1985,Janson1999}. There are also some known structural properties of first passage percolation on the Erd\H{o}s--R\'enyi random graph. Bhamidi et al.~\cite{Bhamidi2011} have shown asymptotics for both the minimal weight of the path between uniformly chosen vertices in the giant component and for the hopcount, the number of edges, on this path.
Bringmann et al.~\cite{Bringmann2015} used this model on the complete graph to analyze heuristics for matching, TSP, and $k$-median.
\subsection{Our Results}
As far as we know, no heuristics have been studied in this model for non-complete graphs yet.
However, we believe that random shortest path metrics on non-complete graphs will bring us a step further in the direction of realistic input model.
This paper provides a probabilistic analysis of some simple heuristics in the model of random shortest path metrics on non-complete graphs. First, we provide some structural properties of generalized random shortest path metrics (Sect.~\ref{sect:structural}), which can be seen as a generalization of the structural properties found by Bringmann et al.~\cite{Bringmann2015}. Although this generalization might seem straightforward at first sight, it brings up some new difficulties that need to be overcome. Most notably, since we do not restrict ourselves to the complete graph, we cannot make use anymore of its symmetry and regularity. This problem is partially solved by introducing two graph parameters, which we call the cut parameters of a graph (Def.~\ref{def:ab}).
Then, we use these structural insights to perform a probabilistic analysis for some simple heuristics for combinatorial optimization problems (Sect.~\ref{sect:heuristics}), where the results are still depending on the cut parameters of a graph. Finally, we use these results, to show our main results, namely that these simple heuristics achieve constant expected approximation ratios for random shortest path metrics applied to Erd\H{o}s--R\'enyi random graphs (Sect.~\ref{sect:ERRG}).
\section{Notation and Model}
We use $X\sim P$ to denote that a random variable $X$ is distributed using a probability distribution $P$. $\Exp(\lambda)$ is being used to denote the exponential distribution with parameter $\lambda$. In particular, we use $X\sim\sum_{i=1}^n\Exp(\lambda_i)$ to denote that $X$ is the sum of $n$ independent exponentially distributed random variables having parameters $\lambda_1,\ldots,\lambda_n$.
For $n\in\mathbb{N}$, we use $[n]$ as shorthand notation for $\{1,\ldots,n\}$. We denote the $n$th harmonic number by $H_n=\sum_{i=1}^n1/i$. Sometimes we use $\exp$ to denote the exponential function. Finally, if a random variable $X$ is stochastically dominated by a random variable $Y$, i.e., we have $F_X(x)\geq F_Y(x)$ for all $x$ (where $X\sim F_X$ and $Y\sim F_Y$), we denote this by $X\precsim Y$.
\paragraph{Generalized Random Shortest Path Metrics.}
Given an undirected graph $G=(V,E)$ on $n$ vertices, we construct the corresponding generalized random shortest path metric as follows. First, for each edge $e\in E$, we draw a random edge weight $w(e)$ independently from an exponential distribution\footnote{Exponential distributions are technically easiest to handle due to their memorylessness property.
A (continuous, non-negative) probability distribution of a random variable $X$ is said to be memoryless if and only if $\mathbb{P}(X>s+t\mid X>t)=\mathbb{P}(X>s)$ for all $s,t\geq0$.~\cite[p.~294]{Ross2010}} with parameter 1. Second, we define the distances $d:V\times V\to\mathbb{R}_{\geq0}\cup\{\infty\}$ as follows: for every $u,v\in V$, $d(u,v)$ denotes the length of the shortest $u,v$-path with respect to the drawn edge weights. If no such path exists, we set $d(u,v)=\infty$. By doing so, the distance function $d$ satisfies $d(v,v)=0$ for all $v\in V$, $d(u,v)=d(v,u)$ for all $u,v\in V$, and $d(u,v)\leq d(u,s)+d(s,v)$ for all $u,s,v\in V$. We call the complete graph with distances $d$ obtained from this process a generalized random shortest path metric. If $G=K_n$ (the complete graph on $n$ vertices), then this generalized random shortest path metric is equivalent to the random shortest path metric as defined by Bringmann et al.~\cite{Bringmann2015}
We use the following notation within generalized random shortest path metrics: $\Delta_{\max}:=\max_{u,v}d(u,v)$ denotes the diameter of the graph. Note that $\Delta_{\max}<\infty$ if and only if $G$ is connected. $B_\Delta(v):=\{u\in V\mid d(u,v)\leq\Delta\}$ denotes the `ball' of radius $\Delta$ around $v$, i.e., the set containing all vertices at distance at most $\Delta$ from $v$. $\tau_k(v):=\min\{\Delta\mid|B_\Delta(v)|\geq k\}$ denotes the distance to the $k$th closest vertex from $v$ (including $v$ itself). Equivalently, one can also say that $\tau_k(v)$ is equal to the smallest $\Delta$ such that the ball of radius $\Delta$ around $v$ contains at least $k$ vertices.
Now, $B_{\tau_k(v)}(v)$ denotes the set of the $k$ closest vertices to $v$. During our analysis, we will make use of the size of the cut induced by this set, which we will denote by $\chi_k(v):=|\delta(B_{\tau_k(v)}(v))|$, where $\delta(U)$ denotes the cut induced by $U$.
\paragraph{Erd\H{o}s--R\'enyi Random Graphs.}
The main results of this work consider random shortest path metrics applied to Erd\H{o}s--R\'enyi random graphs. An undirected graph $G(n,p):=G=(V,E)$ generated by this model has $n$ vertices ($V=\{1,\ldots,n\}$) and between each pair of vertices an edge is included with probability $p$, independent of every other pair.
Working with the Erd\H{o}s--R\'enyi random graph introduces an extra amount of stochasticity to the probabilistic analysis, since both the graph and the edge weights are random. In order to avoid this extra stochasticity as long as possible, in Sections \ref{sect:structural} and \ref{sect:heuristics} we start our analysis using an arbitrary fixed (deterministic) graph $G$. Later on, in Section \ref{sect:ERRG} we will consider Erd\H{o}s--R\'enyi random graphs again.
\section{Structural properties}\label{sect:structural}
In order to analyze the structural properties of generalized random shortest path metrics, we first introduce the notion of what we call the cut parameters of a simple graph $G$.
\begin{definition}\label{def:ab}
Let $G=(V,E)$ be a finite simple connected graph. Then we define the cut parameters of $G$ by
\begin{equation*}
\alpha:=\min_{\varnothing\neq U\subset V}\frac{|\delta(U)|}{\mu_U}\qquad\text{and}\qquad\beta:=\max_{\varnothing\neq U\subset V}\frac{|\delta(U)|}{\mu_U},
\end{equation*}
where $\mu_U:=|U|\cdot(|V|-|U|)$ is the maximum number of possible edges in the cut defined by $U$.
\end{definition}
It follows immediately from this definition that $0<\alpha\leq\beta\leq1$ for any finite simple connected graph $G$. Moreover, for any such graph the following holds for all $\varnothing\neq U\subset V$: $\alpha\cdot\mu_U\leq|\delta(U)|\leq\beta\cdot\mu_U$. We observe that the cut parameters of the complete graph are given by $\alpha=\beta=1$.
\paragraph{Distribution of $\tau_k(v)$.}
Now we have a look at the distribution of $\tau_k(v)$. For this purpose we use an arbitrary fixed undirected connected simple graph $G$ (on $n$ vertices) and let $\alpha$ and $\beta$ denote its cut parameters.
The values of $\tau_k(v)$ are then generated by a birth process as follows. (Amongst others, a variant of this process for complete graphs has been analyzed by Davis and Prieditis~\cite{Davis1993} and Bringmann et al.~\cite{Bringmann2015}.) For $k=1$, we have $\tau_k(v)=0$. For $k\geq2$, we look at all edges $(u,x)$ with $u\in B_{\tau_{k-1}(v)}(v)$ and $x\not\in B_{\tau_{k-1}(v)}(v)$. By definition there are $\chi_{k-1}(v)$ such edges. Moreover the length of these edges is conditioned to be at least $\tau_{k-1}(v)-d(v,u)$. Using the memorylessness of the exponential distribution, we can now see that $\tau_k(v)-\tau_{k-1}(v)$ is the minimum of $\chi_{k-1}(v)$ (standard) exponential variables, or, equivalently, $\tau_k(v)-\tau_{k-1}(v)\sim\Exp(\chi_{k-1}(v))$. We use this result to find bounds for the distribution of $\tau_k(v)$.
\begin{lemma}
\label{lemma:X}
For all $k\in[n]$ and $v\in V$ we have,
\begin{align*}
\alpha k(n-k) \leq \chi_k(v) \leq \beta k(n-k).
\end{align*}
\end{lemma}
\begin{proof}
By definition, $\chi_k(v)$ is the size of a cut induced by a set of $k$ vertices. The result follows immediately since $\alpha$ and $\beta$ are the cut parameters of $G$.\blokje
\end{proof}
\begin{lemma}
\label{lemma:tau}
For all $k\in[n]$ and $v\in V$ we have,
\begin{align*}
\sum_{i=1}^{k-1}\Exp(\beta i(n-i)) \precsim \tau_k(v) \precsim \sum_{i=1}^{k-1}\Exp(\alpha i(n-i)).
\end{align*}
\end{lemma}
\begin{proof}
As previously stated, $\tau_{i}(v)-\tau_{i-1}(v) \sim \Exp(\chi_{i-1}(v))$. Inductively, we obtain that
\begin{align*}
\tau_{k}(v)\sim \sum_{i=1}^{k-1} \Exp(\chi_i(v)).
\end{align*}
Using the result of Lemma \ref{lemma:X}, we can bound this distribution to obtain the desired result.\blokje
\end{proof}
Exploiting the linearity of expectation, the fact that the expected value of an exponentially distributed random variable with parameter $\lambda$ is $1/\lambda$ and the fact that $\sum_{i=1}^{k-1}1/(i(n-i))=(H_{k-1}+H_{n-1}-H_{n-k})/n$, we obtain the following corollary.
\begin{corollary}
\label{corollary:harmonic}
For all $k\in[n]$ and $v\in V$ we have,
\begin{equation*}
\frac{H_{k-1}+H_{n-1}-H_{n-k}}{\beta n}\leq\mathbb{E}(\tau_k(v))\leq\frac{H_{k-1}+H_{n-1}-H_{n-k}}{\alpha n}.
\end{equation*}
\end{corollary}
From this result, we can derive the following extensions of two known results. First of all, if we randomly pick two vertices $u,v\in V$, then averaging over $k$ yields that the expected distance $\mathbb{E}[d(u,v)]$ between them is bounded between $\frac{H_{n-1}}{\beta(n-1)}\approx\ln(n)/\beta n$ and $\frac{H_{n-1}}{\alpha(n-1)}\approx\ln(n)/\alpha n$, which is in line with the known result for complete graphs, where we have $\mathbb{E}[d(u,v)]\approx\ln(n)/n$~\cite{Bringmann2015,Davis1993,Janson1999}. Secondly, for any vertex $v$, the longest distance from it to another vertex is $\tau_n(v)$, which in expectation is bounded between $\frac{2H_{n-1}}{\beta n}\approx2\ln(n)/\beta n$ and $\frac{2H_{n-1}}{\alpha n}\approx2\ln(n)/\alpha n$, which also is in line with the known result for complete graphs, where we have an expected value of approximately $2\ln(n)/n$~\cite{Bringmann2015,Janson1999}.
It is also possible to find bounds for the cumulative distribution function of $\tau_k(v)$. To do so, we define $F_k(x)=\mathbb{P}(\tau_k(v)\leq x)$ for some fixed vertex $v\in V$.
\begin{lemma}
\label{technical:exp}
\emph{\cite[Lemma~3.2]{Bringmann2015}}
Let $X \sim \sum_{i=1}^n \Exp(ci)$. Then, for any $a\geq 0$ we have $\mathbb{P}(X\leq a)=\left(1-e^{-ca}\right)^n$.
\end{lemma}
\begin{lemma}
\label{lemma:F}
For all $x\geq0$ and $k\in[n]$ we have,
\begin{equation*}
\left(1-\exp(-\alpha(n-k)x)\right)^{k-1} \leq F_k(x) \leq \left(1-\exp(-\beta nx)\right)^{k-1}.
\end{equation*}
\end{lemma}
\begin{proof}
By Lemma \ref{lemma:tau} we have
\begin{equation*}
\sum_{i=1}^{k-1}\Exp\left(\beta i(n-i)\right) \precsim \tau_k(v) \precsim \sum_{i=1}^{k-1} \Exp\left(\alpha i(n-i)\right).
\end{equation*}
Since $ni\geq i(n-i)\geq (n-k)i$ for all $i\in[k-1]$, we have $\Exp(\beta ni)\precsim\Exp(\beta i(n-i))$ and $\Exp(\alpha i(n-i))\precsim\Exp(\alpha(n-k)i)$ for all $i\in[k-1]$, from which we obtain that
\begin{equation*}
\sum_{i=1}^{k-1}\Exp\left(\beta ni)\right) \precsim \tau_k(v) \precsim \sum_{i=1}^{k-1} \Exp\left(\alpha(n-k)i\right).
\end{equation*}
Combining this with the definition of stochastic dominance and with Lemma \ref{technical:exp}, gives the desired result.\blokje
\end{proof}
We can improve this result slightly.
\begin{lemma}
\label{lemma:distfunction}
For all $x\geq0$ and $k\in[n]$ we have,
\begin{equation*}
F_k(x) \geq\left(1-\exp(-\alpha nx/4)\right)^{n}.
\end{equation*}
\end{lemma}
\begin{proof}
Note that $\tau_k(v)$ is monotonically increasing in $k$. This implies $F_{k+1}(x) \leq F_k(x)$, so we only need to prove our claim for the case $k=n$. In this case, by Lemma \ref{lemma:tau}, we have $\tau_n(v) \precsim \sum_{i=1}^{n-1}\Exp\left(\lambda_i\right)$ with $\lambda_i:=\alpha i(n-i)=\lambda_{n-i}$. Exploiting the symmetry around $n/2$, we obtain
\begin{equation*}
\tau_n(v) \precsim \sum_{i=1}^{\lfloor n/2\rfloor}\Exp(\lambda_i)+\sum_{i=1}^{\lfloor n/2\rfloor}\Exp(\lambda_i).
\end{equation*}
This enables us to find a lower bound for $F_n(x)$ as follows:
\begin{align*}
F_n(x)=\mathbb{P}(\tau_n(v) \leq x)&\geq\mathbb{P}\left(\sum_{i=1}^{\lfloor n/2\rfloor}\Exp(\lambda_i)+\sum_{i=1}^{\lfloor n/2\rfloor}\Exp(\lambda_i)\leq x\right)\\
&\geq\mathbb{P}\left(\sum_{i=1}^{\lfloor n/2\rfloor}\Exp(\lambda_i)\leq x/2\right)^2.
\end{align*}
Since $i(n-i)\geq i\lceil n/2\rceil$ for all $i\in[\lfloor n/2\rfloor]$, we have $\Exp(\lambda_i)\precsim\Exp(\alpha i\lceil n/2\rceil)$. Combining this with Lemma \ref{technical:exp} yields
\begin{equation*}
F_n(x)\geq\mathbb{P}\left(\sum_{i=1}^{\lfloor n/2\rfloor}\Exp(\alpha i\lceil n/2\rceil)\leq x/2\right)^2=\left(1-\exp(-\alpha x\lceil n/2\rceil/2)\right)^{2\lfloor n/2\rfloor}.
\end{equation*}
Using the inequalities $\lceil n/2\rceil\geq n/2$ and $2\lfloor n/2\rfloor\leq n$ we end up with the desired result.\blokje
\end{proof}
Using this improved bound for the cumulative distribution function of $\tau_k(v)$, we can derive the following tail bound for the diameter $\Delta_{\max}$.
\begin{lemma}
\label{lemma:diameter}
Define $\Delta_{\max}=\max_{u,v\in V}\{d(u,v)\}$. For any fixed $c$ we have
\begin{equation*}
\mathbb{P}( \Delta_{\max} > c \ln(n)/\alpha n) \leq n^{2-c/4}.
\end{equation*}
\end{lemma}
\begin{proof}
Clearly, we have $\Delta_{\max}=\max_v\tau_n(v)$. For $v\in V$, let $\mathcal{E}_v$ denote the event that $\tau_n(v)>c\ln(n)/\alpha n$. From Lemma \ref{lemma:distfunction} we know that $\mathbb{P}(\mathcal{E}_v)=1-F_n(c\ln(n)/\alpha n)\leq1-(1-\exp(-c\ln(n)/4))^n$. Combining this with a union bound, we can derive that
\begin{equation*}
\mathbb{P}\left(\Delta_{\max}>\frac{c\ln(n)}{\alpha n}\right)\leq\sum_{v\in V}\mathbb{P}(\mathcal{E}_v)\leq n\cdot\left(1-\left(1-n^{-c/4}\right)^n\right)\leq n^{2-c/4},
\end{equation*}
where the last inequality can be derived using Bernoulli's inequality.\blokje
\end{proof}
\paragraph{Clustering.}
In this section we show that we can partition the vertices of generalized random shortest path metrics into a small number of clusters with a given maximum diameter. Before we prove this main result, we first provide a tail bound for $|B_\Delta(v)|$.
\begin{lemma}
\label{lemma:ball}
For $n\geq5$ and for any fixed $\Delta\geq0$ we have,
\begin{equation*}
\mathbb{P}\left(|B_\Delta(v)| < \min \left\{\exp(\alpha \Delta n/5 ), \frac{n+1}{2}\right\}\right) \leq \exp(-\alpha \Delta n/5).
\end{equation*}
\end{lemma}
\begin{proof}
We have $|B_\Delta (v)|\geq k$ if and only if $\tau_k(v) \leq \Delta$. Using Lemma \ref{lemma:F}, we obtain
\begin{align*}
&\mathbb{P}\left(|B_\Delta(v)| < \min \left\{\exp(\alpha \Delta (n-1)/4 ), \frac{n+1}{2}\right\}\right)\\
&\quad\leq1-\left(1-\exp\left(-\alpha\Delta\left(n-\frac{n+1}2\right)\right)\right)^{\exp(\alpha\Delta(n-1)/4)-1}\\
&\quad\leq1-\left(1-\exp\left(-\alpha\Delta(n-1)/2\right)\right)^{\exp(\alpha\Delta(n-1)/4)}\\
&\quad\leq\exp(-\alpha\Delta(n-1)/4),
\end{align*}
where the last inequality can be derived using Bernoulli's inequality. Using $(n-1)/4\geq n/5$ for $n\geq5$ finishes the proof.\blokje
\end{proof}
We use the result of this lemma to prove our main structural property for generalized random shortest path metrics.
\begin{theorem}
\label{theorem:cluster}
For any fixed $\Delta\geq0$, if we partition the vertices into clusters, each of diameter at most $4\Delta$, then the expected number of clusters needed is bounded from above by $O(1+n/\exp( \alpha \Delta n/5))$.
\end{theorem}
\begin{proof}
Define $s_\Delta = \min\{\exp( \alpha\Delta n/5, (n+1)/2\}$. We call vertex $v$ \emph{$\Delta$-dense} if $|B_\Delta(v)|\geq s_\Delta$ and \emph{$\Delta$-sparse} otherwise. In both cases we call the set $B_\Delta(v)$ of vertices within distance $\Delta$ of $v$ the \emph{$\Delta$-ball} of $v$. By Lemma \ref{lemma:ball} we can bound the expected number of $\Delta$-sparse vertices by $O(n/s_\Delta)$. We put each $\Delta$-sparse vertex in its own cluster (of size 1), which has diameter $0\leq4\Delta$.
This leaves us with the $\Delta$-dense vertices. We cluster them according to the following process. Consider an auxiliary graph $H$ whose vertices are the $\Delta$-dense vertices and where two vertices are connected by an edge if and only if their corresponding $\Delta$-balls are not disjoint. Now, consider an arbitrary maximal independent set $S$ in $H$. Since $|B_\Delta(v)|\geq s_\Delta$ and $B_\Delta(u)\cap B_\Delta(v)=\varnothing$ for any $u,v\in S$, it follows that $|S|\leq n/s_\Delta$. Now, we form the initial clusters $C_1,\ldots,C_{|S|}$ each of which is equal to the $\Delta$-ball corresponding to one of the vertices in $S$. Observe that these initial clusters have diameter at most $2\Delta$.
Now consider an arbitrary $\Delta$-dense vertex $v$ that is not part of any cluster yet. Since $S$ is a maximal independent set, we know that there exists a $u\in S$ such that $B_\Delta(u)\cap B_\Delta(v)\neq\varnothing$. We add $v$ to the cluster that contains $u$. If we take $x\in B_\Delta(u)\cap B_\Delta(v)$, then we can see that $d(v,u)\leq d(v,x)+d(x,u)\leq\Delta+\Delta=2\Delta$. We repeat this step until all $\Delta$-dense vertices have been added to some initial cluster. By construction, the diameter of each cluster is at most $4\Delta$ after this process: consider any vertices $x,y$ in the same cluster, that originally corresponded to a vertex $u\in S$. Then we have $d(x,y)\leq d(x,u)+d(u,y)\leq2\Delta+2\Delta=4\Delta$.
So, now we have in expectation $O(n/s_\Delta)$ clusters each containing one $\Delta$-sparse vertex, and at most $n/s_\Delta$ clusters each containing at least $s_\Delta$ $\Delta$-dense vertices, all with diameter at most $4\Delta$. The total number of clusters is $O(n/s_\Delta)=O(1+n/\exp( \alpha \Delta n/5))$.\blokje
\end{proof}
\section{Analysis of Heuristics}\label{sect:heuristics}
In this section we bound the expected approximation ratios of the greedy heuristic for minimum-distance perfect matching, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the $k$-median problem. For this purpose we still use an arbitrary fixed undirected connected simple graph $G$ (on $n$ vertices) and let $\alpha$ and $\beta$ denote its cut parameters. The results in this section will depend on $\alpha$ and $\beta$.
\paragraph{Greedy Heuristic for Minimum-Distance Perfect Matching.}
The mini-mum-distance perfect matching problem has been widely analyzed throughout history. We do for instance know that the worst-case running-time for finding a minimum distance perfect matching is $O(n^3)$, which is high when considering a large number of vertices. Because of this, simple heuristics are often used, with the greedy heuristic probably being the simplest of them: at each step, add a pair of unmatched vertices to the matching such that the distance between the added pair of vertices is minimized. From now on, let $\mathsf{GR}$ denote the cost of the matching computed by this heuristic and let $\mathsf{MM}$ denote the value of an optimal matching.
The worst-case approximation ratio of this heuristic on metric instances is known to be $O(n^{\log_2(3/2)})$~\cite{Reingold1981}. Furthermore, for random shortest path metrics on complete graphs (for which the cut parameters are given by $\alpha=\beta=1$) the heuristic has an expected approximation ratio of $O(1)$~\cite{Bringmann2015}. We extend this last result to general values for $\alpha$ and $\beta$ and show that the greedy matching heuristic has an expected approximation ratio of $O(\beta/\alpha)$.
\begin{theorem}
\label{theorem:greedy}
$\mathbb{E}[\mathsf{GR}]=O\left(1/\alpha\right)$.
\end{theorem}
\begin{proof}
Let $\Delta_i:=i/\alpha n$. We divide the run of the greedy heuristic in phases as follows: the algorithm is in phase $i$ if a pair $(u,v)$ is added to the matching such that $d(u,v) \in(4\Delta_{i-1},4\Delta_i]$. Using Lemma \ref{lemma:diameter}, we can show that the expected sum of all distances greater than or equal to $\Delta_{\omega(\ln(n))}$ is $o(1/\alpha)$, so we can ignore the corresponding phases in our analysis.
We now estimate the contribution of the other phases to the greedy matching. By Theorem \ref{theorem:cluster}, after phase $i-1$, we can partition the vertices in an expected number of $O(1+n/\exp((i-1)/5))$ clusters, each of diameter at most $4\Delta_{i-1}$. Each such cluster can have at most one unmatched vertex. So, after phase $i-1$ there are at most $O(1+n/\exp((i-1)/5))$ unmatched vertices left. Therefore, in expectation at most $O(1+n/\exp((i-1)/5))$ pairs of unmatched vertices can be added in phase $i$, each contributing a distance of at most $4\Delta_i$. So, the total contribution of phase $i$ is in expectation at most $O(\frac{i}{\alpha n}(1+n/\exp((i-1)/5)))$. Summing over all phases yields
\begin{equation*}
\mathbb{E}[\mathsf{GR}]=o\left(\frac1\alpha\right)+\sum_{i=1}^{O(\ln(n))}O\left(\frac1\alpha\left(\frac{i}{n}+\frac{i}{e^{(i-1)/5}}\right)\right)= o\left(\frac1\alpha\right)+O\left(\frac1\alpha\right)=O\left(\frac1\alpha\right),
\end{equation*}
which completes the proof.\blokje
\end{proof}
\begin{lemma}
\label{technical:Janson}
\emph{\cite[Thm.~5.1(iii)]{Janson2018}}
Let $X\sim\sum_{i=1}^nX_i$ with $X_i\sim\Exp(a_i)$ independent. Let $\mu=\mathbb{E}[X]=\sum_{i=1}^n(1/a_i)$ and $a_*=\min_ia_i$. For any $\lambda\leq1$,
\begin{equation*}
\mathbb{P}(X\leq\lambda\mu)\leq\exp(-a_*\mu(\lambda-1-\ln(\lambda))).
\end{equation*}
\end{lemma}
\begin{lemma}
\label{lemma:expGam}
\emph{\citep[Ex.~1.A.24]{Shaked2007}}
Let $X_i\sim\Exp(\lambda_i)$ independently, $i=1,\ldots,m$. Moreover, let $Y_i\sim\Exp(\eta)$ independently, $i=1,\ldots,m$. Then we have
\begin{equation*}
\sum_{i=1}^mX_i\succsim\sum_{i=1}^mY_i\qquad\text{if and only if}\qquad\prod_{i=1}^m\lambda_i\leq\eta^m.
\end{equation*}
\end{lemma}
\begin{lemma}
\label{lemma:S}
Let $S_m$ denote the sum of the $m$ lightest edge weights in $G$. For all $\phi\leq(n-1)/n$ and $c\in [0,2\phi^2/e]$ we have
\begin{equation*}
\mathbb{P}\left(S_{\phi n} \leq \frac{c}{\beta}\right) \leq \exp\left(\phi n\left(2+\ln\left(\frac{c}{2\phi^2}\right)\right)\right).
\end{equation*}
Furthermore, $\mathsf{TSP}\geq\mathsf{MM} \geq S_{n/2}$, where $\mathsf{TSP}$ and $\mathsf{MM}$ are the total distance of a shortest TSP tour and a minimum-distance perfect matching, respectively.
\end{lemma}
\begin{proof}
Since all edge weights are independent and standard exponential distributed, we have $S_1\sim\Exp(|E|)$. Using the memorylessness property of the exponential distribution, it follows that $S_2-S_1\sim S_1+\Exp(|E|-1)$, i.e., the second lightest edge weight is equal to the lightest edge weight plus the minimum of $|E|-1$ standard exponential distributed random variables. In general, we get $S_{k+1}-S_k\sim S_k-S_{k-1}+\Exp(|E|-k)$. This yields
\begin{equation*}
S_{\phi n}\sim\sum_{i=0}^{\phi n-1}(\phi n-i)\cdot\Exp(|E|-i)\sim\sum_{i=0}^{\phi n-1}\Exp\left(\frac{|E|-i}{\phi n-i}\right)\succsim\sum_{i=0}^{\phi n-1}\Exp\left(\frac{e|E|}{\phi n}\right),
\end{equation*}
where the stochastic dominance follows from Lemma \ref{lemma:expGam} by observing that
\begin{equation*}
\prod_{i=0}^{\phi n-1}\frac{|E|-i}{\phi n-i}=\frac{|E|!}{(\phi n)!(|E|-\phi n)!}=\binom{|E|}{\phi n}\leq\left(\frac{e|E|}{\phi n}\right)^{\phi n},
\end{equation*}
where the inequality follows from applying the well-known inequality $\binom{m}{k}\leq(em/k)^k$. Next, observe that $|E|\leq\beta n(n-1)/2$. Applying this fact, and then combining it with Lemma \ref{technical:Janson} with $\mu=2\phi^2n/\beta(n-1)$, $a_*=\beta(n-1)/2\phi$ and $\lambda=ec(n-1)/2\phi^2n$ (note that $\lambda\leq1$ since $0\leq c\leq 2\phi^2/e$), we obtain
\begin{align*}
\mathbb{P}\left(S_{\phi n}\leq\frac{c}{\beta}\right)&\leq\mathbb{P}\left(\sum_{i=0}^{\phi n-1}\Exp\left(\frac{\beta(n-1)}{2\phi}\right)\leq\frac{ec}{\beta}\right)\\
&\leq\exp\left(-\phi n\left(\frac{ec(n-1)}{2\phi^2n}-1-\ln\left(\frac{ec(n-1)}{2\phi^2n}\right)\right)\right)\\
&\leq\exp\left(\phi n\left(2+\ln\left(\frac{c}{2\phi^2}\right)\right)\right).
\end{align*}
It remains to show that $\mathsf{TSP}\geq\mathsf{MM} \geq S_{n/2}$. The first inequality follows trivially. For the second one, consider a minimum-distance perfect matching. Take the union of the shortest path between each matched pair of vertices. This union must contain at least $n/2$ different edges of $G$. These edges must have a total weight of at least $S_{n/2}$ and at most $\mathsf{MM}$. So, $\mathsf{MM} \geq S_{n/2}$.\blokje
\end{proof}
\begin{theorem}
\label{theorem:EGR}
The greedy heuristic for minimum-distance perfect matching has an expected approximation ratio on generalized random shortest path metrics given by $\mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\right]=O\left(\beta/\alpha\right)$.
\end{theorem}
\begin{proof}
Let $c>0$ be a sufficiently small constant. Then the approximation ratio of the greedy heuristic on generalized random shortest path metrics is
\begin{equation*}
\mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\right]\leq\mathbb{E}\left[\frac{\beta\cdot\mathsf{GR}}{c}\right]+ \mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\;\middle|\;\mathsf{MM}<\frac{c}{\beta}\right]\cdot\mathbb{P}\left(\mathsf{MM}<\frac{c}{\beta}\right).
\end{equation*}
The first term is $O(\beta/\alpha)$ by Theorem \ref{theorem:greedy}. The expectation in the second term can be bounded by the worst-case approximation ratio of the greedy heuristic on metric instances, i.e. $n^{\log_2(3/2)}$ \cite{Reingold1981}. The probability can be bounded by $\exp(\tfrac12n(2+\ln(2c)))$ according to Lemma \ref{lemma:S}. Since $c$ is sufficiently small, this implies that the second term becomes $o(1)$.\blokje
\end{proof}
\paragraph{Nearest Neighbor Heuristic for TSP.}
The nearest-neighbor heuristic is a greedy approach for the TSP: start with some starting vertex $v_0$ as current vertex $v$; at every step, choose the nearest unvisited neighbor $u$ of $v$ as the next vertex in the tour and move to the next iteration with the new vertex $u$ as current vertex $v$; go back to $v_0$ if all vertices are visited. From now on, let $\mathsf{NN}$ denote the cost of the TSP tour computed by this heuristic and let $\mathsf{TSP}$ denote the value of an optimal TSP tour.
The worst-case approximation ratio of this heuristic on metric instances is known to be $O(\ln(n))$~\cite{Rosenkrantz1977}. Furthermore, for random shortest path metrics on complete graphs (for which the cut parameters are given by $\alpha=\beta=1$) the heuristic has an expected approximation ratio of $O(1)$~\cite{Bringmann2015}. We extend this last result to general values for $\alpha$ and $\beta$ and show that the nearest-neighbor heuristic has an expected approximation ratio of $O(\beta/\alpha)$.
\begin{theorem}
\label{theorem:NN}
For generalized random shortest path metrics, we have $\mathbb{E}[\mathsf{NN}]=O\left(1/\alpha\right)$ and $\mathbb{E}\left[\frac{\mathsf{NN}}{\mathsf{TSP}}\right]=O\left(\beta/\alpha\right)$.
\end{theorem}
\begin{proof}
The first part of the proof is similar to the proof of Theorem \ref{theorem:greedy}. Let $\Delta_i:=i/\alpha n$. We put the `edges' added to the tour by the nearest-neighbor heuristic into bins depending on their distance, bin $i$ gets the `edges' $\{u,v\}$ with $d(u,v)\in(4\Delta_{i-1},4\Delta_i]$. Using Lemma \ref{lemma:diameter}, we can show that the expected sum of all distances greater than or equal to $\Delta_{\omega(\ln(n))}$ is $o(1/\alpha)$, so we can ignore the corresponding bins in our analysis.
We now estimate the contribution of the other bins to the distance of the TSP tour. By Theorem \ref{theorem:cluster}, we can partition the vertices in an expected number of $O(1+n/\exp((i-1)/5))$ clusters, each of diameter at most $4\Delta_{i-1}$. Every time the nearest-neighbor heuristic adds an `edge' of distance greater than $4\Delta_{i-1}$, this must be an edge from some cluster $C_k$ to another cluster $C_\ell$. Moreover, at this point the partial TSP tour must already have visited all vertices in the cluster $C_k$. Therefore, this can happen at most $O(1+n/\exp((i-1)/5))$ times in expectation. Therefore, bin $i$ can get at most $O(1+n/\exp((i-1)/5))$ `edges' during the run of the nearest-neighbor heuristic. So, the total contribution of bin $i$ is in expectation at most $O(\frac{i}{\alpha n}(1+n/\exp((i-1)/5)))$. Summing over all bins yields
\begin{equation*}
\mathbb{E}[\mathsf{NN}]=o\left(\frac1\alpha\right)+\sum_{i=1}^{O(\ln(n))}O\left(\frac1\alpha\left(\frac{i}{n}+\frac{i}{e^{(i-1)/5}}\right)\right)= o\left(\frac1\alpha\right)+O\left(\frac1\alpha\right)=O\left(\frac1\alpha\right).
\end{equation*}
Using the worst-case approximation ratio of the nearest-neighbor heuristic on metric instances of $O(\ln(n))$~\cite{Rosenkrantz1977}, the proof for the expected approximation ratio is analogously to the proof of Theorem \ref{theorem:EGR}.\blokje
\end{proof}
\paragraph{Insertion Heuristics for TSP.}
The insertion heuristics are another greedy approach for the TSP: start with an initial optimal tour on a few vertices chosen according to some predefined rule $R$; at every step, choose a vertex according to the same predefined rule $R$ and insert this vertex in the current tour such that the total distance increases the least. From now on, let $\mathsf{IN}_R$ denote the cost of the TSP tour computed by this heuristic (with rule $R$) and let $\mathsf{TSP}$ still denote the value of an optimal TSP tour.
The worst-case approximation ratio of this heuristic for any rule $R$ on metric instances is known to be $O(\ln(n))$~\cite{Rosenkrantz1977}. Furthermore, for random shortest path metrics on complete graphs (for which the cut parameters are given by $\alpha=\beta=1$) the heuristic has an expected approximation ratio of $O(1)$~\cite{Bringmann2015}. We extend this last result to general values for $\alpha$ and $\beta$ and show that the insertion heuristic for any rule $R$ has an expected approximation ratio of $O(\beta/\alpha)$.
\begin{theorem}
\label{theorem:IN}
For generalized random shortest path metrics, we have $\mathbb{E}[\mathsf{IN}_R]=O\left(1/\alpha\right)$ and $\mathbb{E}\left[\frac{\mathsf{IN}_R}{\mathsf{TSP}}\right]=O\left(\beta/\alpha\right)$.
\end{theorem}
\begin{proof}
The first part of the proof is similar to the proof of Theorem \ref{theorem:greedy}. Let $\Delta_i:=i/\alpha n$. We put the vertices inserted into the tour by the insertion heuristic into bins depending on the distance they add to the TSP tour, bin $i$ gets the vertices with contribution in the range $(8\Delta_{i-1},8\Delta_i]$. Using Lemma \ref{lemma:diameter}, we can show that the expected sum of all distances greater than or equal to $\Delta_{\omega(\ln(n))}$ is $o(1/\alpha)$, so we can ignore the corresponding bins in our analysis.
We now estimate the contribution of the other bins to the distance of the TSP tour. By Theorem \ref{theorem:cluster}, we can partition the vertices in an expected number of $O(1+n/\exp((i-1)/5))$ clusters, each of diameter at most $4\Delta_{i-1}$. Every time the insertion heuristics adds a vertex that contributes more than $8\Delta_{i-1}$, this must be a vertex that is part of a cluster that is not part of the tour yet. Therefore, this can happen at most $O(1+n/\exp((i-1)/5))$ times in expectation. Therefore, bin $i$ can get at most $O(1+n/\exp((i-1)/5))$ vertices during the run of the insertion heuristic. So, the total contribution of bin $i$ is in expectation at most $O(\frac{i}{\alpha n}(1+n/\exp((i-1)/5)))$. Summing over all bins, and adding the contribution of the initial tour $T_R$ yields
\begin{equation*}
\mathbb{E}[\mathsf{IN}_R]=\mathbb{E}[T_R]+o\left(\frac1\alpha\right)+\sum_{i=1}^{O(\ln(n))}O\left(\frac1\alpha\left(\frac{i}{n}+\frac{i}{e^{(i-1)/5}}\right)\right)=O\left(\frac1\alpha\right),
\end{equation*}
since we can use Theorem \ref{theorem:NN} to bound the expected length of the initial tour by $\mathbb{E}[T_R]\leq\mathbb{E}[\mathsf{TSP}]\leq\mathbb{E}[\mathsf{NN}]=O(1/\alpha)$.
Using the worst-case approximation ratio of the insertion heuristic for any rule $R$ on metric instances of $O(\ln(n))$~\cite{Rosenkrantz1977}, the proof for the expected approximation ratio is analogously to the proof of Theorem \ref{theorem:EGR}. Note that this entire proof is independent of the rule $R$ used.\blokje
\end{proof}
\paragraph{Running Time of 2-opt Heuristic for TSP.}
The 2-opt heuristic is an often used local search algorithm for the TSP: start with an initial tour on all vertices and improve the tour by 2-exchanges until no improvement can be made anymore. In a 2-exchange, the heuristic takes `edges' $\{v_1,v_2\}$ and $\{v_3,v_4\}$, where $v_1$, $v_2$, $v_3$, $v_4$ are visited in this order in the tour, and replaces them by $\{v_1,v_3\}$ and $\{v_2,v_4\}$ to create a shorter tour.
We provide an upper bound for the expected number of iterations that 2-opt needs. In the worst-case scenario, this number is exponential. However, for random shortest path metrics on complete graphs (for which the cut parameters are given by $\alpha=\beta=1$) an upper bound of $O(n^8\ln^3(n))$ is known for the expected number of iterations~\cite{Bringmann2015}. We extend this result with a similar proof to general values for $\alpha$ and $\beta$ and show an upper bound for the expected number of iterations of $O(n^8\ln^3(n)\beta/\alpha)$.
We first define the improvement obtained from a 2-exchange. If $\{v_1,v_2\}$ and $\{v_3,v_4\}$ are replaced by $\{v_1,v_3\}$ and $\{v_2,v_4\}$, then the improvement made by the exchange equals the change in distance $\zeta =d(v_1,v_2)+d(v_3,v_4) - d(v_1,v_3)-d(v_2,v_4)$. These four distances correspond to four shortest paths ($P_{12}$, $P_{34}$, $P_{13}$, $P_{24}$) in the graph $G=(V,E)$. This implies that we can rewrite $\zeta$ as the sum of the weights on these paths. We obtain $\zeta =\sum_{e \in E} \gamma_e w(e)$, for some $\gamma_e \in \{-2, -1, 0, 1, 2\}$.
Since we are looking at the improvement obtained by a 2-exchange, we have $\zeta >0$. This implies that there exists some $e=\{u,u'\}\in E$ such that $\gamma_e\neq 0 $. Given this edge $e$, let $I \subseteq \{P_{12},P_{34},P_{13},P_{24}\}$ be the set of all shortest paths of the 2-exchange that contain $e$. Then, for all combinations $e$ and $I$, let $\zeta_{ij}^{e,I}$ be defined as follows:
\begin{itemize}
\item If $P_{ij} \notin I$, then $\zeta_{ij}^{e,I}$ is the length of the shortest path from $v_i$ to $v_j$ without using $e$.
\item If $P_{ij} \in I$, then $\zeta_{ij}^{e,I}$ is the minimum of
\begin{itemize}
\item the length of a shortest path from $v_i$ to $u$ without using $e$ plus the length of a shortest path from $u'$ to $v_j$ without using $e$ and
\item the length of a shortest path from $v_i$ to $u'$ without using $e$ plus the length of a shortest path from $u$ to $v_j$ without using $e$.
\end{itemize}
\end{itemize}
Define $\zeta^{e,I} = \zeta_{12}^{e,I} + \zeta_{34}^{e,I} - \zeta_{13}^{e,I} - \zeta_{24}^{e,I}$.
\begin{lemma}
\label{lemma:zeta1}
For every outcome of the edge weights, there exists an edge $e$ and a set $I$ such that $\zeta = \zeta^{e,I} + \gamma w(e)$, where $\gamma \in\{-2,-1,1,2\}$ is determined by $e$ and $I$.
\end{lemma}
\begin{proof}
Fix the edge weights arbitrarily and consider the four shortest paths from the 2-exchange. As previously stated there exists some edge $e$ with non-zero value $\gamma_e$. Choose this $e$, the corresponding set $I$ and take $\gamma = \gamma_e$. Then the result follows from the definition of $\zeta^{e,I}$.\blokje
\end{proof}
\begin{lemma}
\label{lemma:zeta2}
Let $e$ and $I$ be given with $\gamma = \gamma_e \neq 0$. Then $\mathbb{P}(\zeta^{e,I} +\gamma w(e) \in (0,x])\leq x$. Moreover, $\mathbb{P}(\zeta\in(0,x])=O(\beta n^2x)$.
\end{lemma}
\begin{proof}
Fix all edge weights except for $w(e)$. Then the value of $\zeta^{e,I}$ is known. Therefore we have $\zeta^{e,I} + \gamma w(e) \in (0,x]$ if and only if $w(e)$ takes a value in an interval of length $x/|\gamma| \leq x$. The first part of the result follows, since $w(e)$ is drawn from $\Exp(1)$ and the density function of this distribution does not exceed $1$.
Observe that the number of possible choices for $e$ and $I$ is bounded by $|E|\leq\beta n(n-1)/2=O(\beta n^2)$. The second part of the result follows now using Lemma \ref{lemma:zeta1} and a union bound.\blokje
\end{proof}
\begin{theorem}
\label{theorem:2-opt}
The expected number of iterations of the 2-opt heuristic until a local optimum is found is bounded by $O(n^8\ln^3(n)\beta/\alpha)$.
\end{theorem}
\begin{proof}
Let $\zeta_{\min}>0$ be the minimum improvement that can be made by any 2-exchange. The total number of different 2-exchanges is $O(n^4)$, so using Lemma \ref{lemma:zeta2} and a union bound we obtain $\mathbb{P}(\zeta_{\min}\leq y)=O(\beta n^6y)$.
The initial tour has a length of at most $n\Delta_{\max}$. Let $T$ be the number of iterations taken by the 2-opt heuristic. Then we have $T\leq n\Delta_{\max}/\zeta_{\min}$. So, $T>x$ implies $\Delta_{\max}/\zeta_{\min} >x/n$. This event is contained in the union of the events $\Delta_{\max}> c\ln(x) \ln(n)/\alpha n$ and $\zeta_{\min} < c\ln(x)\ln(n)/\alpha x$, where $c$ is a sufficiently large constant. By Lemma \ref{lemma:diameter} the first event happens with probability at most $n^{2-c\ln(x)/4}=n^{-\Omega(\ln(x))}$. The second event happens with probability at most $O(\beta n^6\ln(n)\ln(x)/\alpha x)$. So, we have
\begin{equation*}
\mathbb{P}(T>x)\leq n^{-\Omega(\ln(x))}+O\left(\beta n^6\ln(n)\ln(x)/\alpha x\right).
\end{equation*}
The number of iterations is bounded by $n!$, so we obtain
\begin{equation*}
\mathbb{E}[T]\leq\sum_{x=1}^{n!}\left(n^{-\Omega(\ln(x))}+O\left(\beta n^6\ln(n)\ln(x)/\alpha x\right)\right).
\end{equation*}
The sum of the $n^{-\Omega(\ln(x))}$ contributes a negligible $O(\ln(n!))$. The sum of the remaining $O(\beta n^6\ln(n)\ln(x)/\alpha x)$ contributes $O(\beta n^6\ln(n)\ln^2(n!)/\alpha)=O(n^8\ln^3(n)\beta/\alpha)$.\blokje
\end{proof}
\paragraph{Trivial Heuristic for $k$-Median.}
The goal of the (metric) $k$-median problem is to find a set $U \subseteq V$ of size $k$ such that $\sum_{v \in V} \min_{u\in U} d(v,u)$ is minimized. The best known approximation algorithm for this problem achieves an approximation ratio of $2.675+\varepsilon$~\cite{Byrka2015}.
Here, we consider the $k$-median problem in the setting of generalized random shortest path metrics. We analyze a trivial heuristic for the $k$-median problem: simply pick $k$ vertices independently of the metric space, e.g., $U=\{v_1,\ldots,v_k\}$. The worst-case approximation ratio of this heuristic is unbounded, even if we restrict ourselves to metric instances. However, for random shortest path metrics on complete graphs (for which the cut parameters are given by $\alpha=\beta=1$) the expected approximation ratio has an upper bound of $O(1)$ and even $1+o(1)$ for $k$ sufficiently small~\cite{Bringmann2015}. We extend this result to general values for $\alpha$ and $\beta$ and give an upper bound for the expected approximation ratio of $O(\beta/\alpha)$ for `large' $k$ and $\beta/\alpha + o(\beta/\alpha)$ for $k$ sufficiently small.
For our analysis, let $U=\{v_1,\ldots, v_k\}$ be an arbitrary set of $k$ vertices. Sort the remaining vertices $\{v_{k+1}, \ldots, v_n\}$ in increasing distance from $U$. For $k+1 \leq i\leq n$, let $\rho_{i} = d(v_i,U)$ equal the distance from $U$ to the $(i-k)$-th closest vertex to $U$. Let $\mathsf{TR}$ denote the cost of the solution generated by the trivial heuristic and let $\mathsf{ME}$ be the cost of an optimal solution to the $k$-median problem.
Observe that the random variables $\rho_i$ are generated by a simple growth process analogously to the one described in Section \ref{sect:structural} for $\tau_k(v)$. Using this observation, we can see that
\begin{equation*}
\sum_{j=k}^{i-1}\Exp(\beta j(n-j)) \precsim \rho_i \precsim \sum_{j=k}^{i-1} \Exp(\alpha j(n-j)),
\end{equation*}
which in turn implies that $\mathsf{cost}(U)=\sum_{i=k+1}^n\rho_i$ is stochastically bounded by
\begin{equation*}
\sum_{i=k}^{n-1}\Exp(\beta i) \precsim \mathsf{cost}(U) \precsim \sum_{i=k}^{n-1} \Exp(\alpha i).
\end{equation*}
From this, we can immediately derive bounds for the expected value of the $k$-median returned by the trivial heuristic.
\begin{lemma}
\label{lemma:triviale}
Fix $U\subseteq V$ of size $k$. Then, we have $\mathbb{E}[\mathsf{TR}]=\mathbb{E}[\mathsf{cost}(U)]$ and
\begin{equation*}
\frac{1}{\beta}\left(\ln\left(\frac{n-1}{k-1}\right)-1\right)\leq\mathbb{E}[\mathsf{TR}]\leq\frac{1}{\alpha}\left(\ln\left(\frac{n-1}{k-1}\right)+1\right).
\end{equation*}
\end{lemma}
\begin{proof}
We have $(H_{n-1}-H_{k-1})/\beta=\sum_{i=k}^{n-1}1/\beta i\leq\mathbb{E}[\mathsf{TR}]\leq\sum_{i=k}^{n-1}1/\alpha i=(H_{n-1}-H_{k-1})/\alpha$. Using $\ln(n)\leq H_n\leq\ln(n)+1$ yields the result.\blokje
\end{proof}
Before we provide our result for the expected approximation ratio of the trivial heuristic, we first provide some tail bounds for the distribution of the optimal $k$-median $\mathsf{ME}$ and the trivial solution $\mathsf{TR}$.
\begin{lemma}
\label{lemma:densityu}
Fix $U\subseteq V$ of size $k$. Then the probability density function $f$ of $\sum_{i=k}^{n-1} \Exp(\beta i)$ is given by
\begin{equation*}
f(x)= \beta k \cdot\binom{n-1}{k} \cdot\exp(-\beta kx)\cdot\left(1-\exp(-\beta x)\right)^{n-k-1}.
\end{equation*}
\end{lemma}
\begin{proof}
The distribution corresponds to the $(n-k)$-th smallest element out of $n-1$ independent, exponentially distributed random variables with parameter $\beta$. The density of this distribution is known~\cite[Example 2.38]{Ross2010}.\blokje
\end{proof}
\begin{lemma}
\label{lemma:ksmall}
Let $c>0$ be sufficiently large and let $k\leq c'n$ for $c'=c'(c) >0$ sufficiently small. Then we have
\begin{equation*}
\mathbb{P}\left(\mathsf{ME} \leq \left(\ln\left(\tfrac{n-1}{k}\right) - \ln\ln\left(\tfrac{n}{k}\right)-\ln(c)\right)/\beta\right)= n^{-\Omega(c)}.
\end{equation*}
\end{lemma}
\begin{proof}
We first want a bound for $f(x)$ at $x=\ln((n-1)/ak)/\beta$ for sufficiently large $a$ with $1\leq a\leq (n-1)/k$. For this particular value of $x$, by Lemma \ref{lemma:densityu} we have,
\begin{equation*}
f(x)=\beta k \cdot\binom{n-1}{k}\cdot\frac{(ak)^k(n-1-ak)^{n-k-1}}{(n-1)^{n-1}}\leq\beta k (ae)^k\left(1-\frac{ak}{n-1}\right)^{n-k-1},
\end{equation*}
where we used $\binom{n-1}{k} \leq ((n-1)e/k)^k$ for the inequality. Since $1+x\leq\exp(x)$ and $(n-k-1)/(n-1)=\Omega(1)$ (since $k$ is sufficiently small), we obtain
\begin{equation*}
f(x)\leq\beta k(ae)^k\exp(-\Omega(ak)).
\end{equation*}
Since $a$ is sufficiently large, the first factors (without the $\beta$) are lower order terms that can be hidden by the $\Omega$. This implies that $f(x)\leq\beta\exp(-\Omega(ak))$. Substituting $a=(n-1)\exp(-\beta x)/k$ into this yields
\begin{equation*}
f(x)\leq\beta\exp(-\Omega((n-1)\exp(-\beta x))),
\end{equation*}
which holds for $x\in[0,\ln((n-1)/bk)/\beta]$ for $b \geq 1$ sufficiently large. Recall that $\mathsf{cost}(U)\succsim\sum_{i=k}^{n-1}\Exp(\beta i)$. So, we have $\mathbb{P}(\mathsf{cost}(U)<\ln((n-1)/bk)/\beta)\leq\mathbb{P}(\sum_{i=k}^{n-1}\Exp(\beta i)<\ln((n-1)/bk)/\beta)$. This latter probability is equal to
\begin{align*}
\int_{0}^{\ln\left(\frac{n-1}{bk}\right)/\beta} f(x)\,\mathrm{d}x &= \int_{0}^{\ln\left(\frac{n-1}{bk}\right)/\beta} f\left(\ln\left(\tfrac{n-1}{bk}\right)/\beta-x\right)\,\mathrm{d}x\\
&\leq \int_{0}^{\ln\left(\frac{n-1}{bk}\right)/\beta} \beta\exp\left(-\Omega(bk\exp(\beta x))\right)\,\mathrm{d}x\\
&\leq \int_{0}^{\ln\left(\frac{n-1}{bk}\right)} \exp\left(-\Omega(bk\exp(x))\right)\,\mathrm{d}x\\
&\leq \int_{0}^{\infty}\exp\left(-\Omega(bk(1+x))\right)\,\mathrm{d}x\leq\exp(-\Omega(bk)),
\end{align*}
where the last step follows from the fact that $\int_{0}^{\infty} \exp(-\Omega(bkx))\,\mathrm{d}x = O(1/bk) \leq 1$ as $b$ is sufficiently large.
In order for $\mathsf{ME}$ to be small, there must exist a subset $U \subseteq V$ of size $k$ that has low cost. We bound this probability by taking a union bound, which yields
\begin{align*}
\mathbb{P}\left(\mathsf{ME} < \ln\left(\tfrac{n-1}{bk}\right)/\beta\right) &= \mathbb{P}\left(\exists\, U \subseteq V,\, |U|=k:\mathsf{cost}(U) <\ln\left(\tfrac{n-1}{bk}\right)/\beta\right)\\
&\leq\binom{n}{k} \cdot \mathbb{P}\left(\mathsf{cost}(U) <\ln\left(\tfrac{n-1}{bk}\right)/\beta\right)\\
&\leq\binom{n}{k} \cdot \exp(-\Omega(bk)).
\end{align*}
Set $b=c\ln(n/k)$ for sufficiently large $c\geq 1$. Then we fulfill the condition that $b\geq 1$ and sufficiently large. Combining this with $\binom{n}{k}\leq(ne/k)^k$ yields
\begin{equation*}
\mathbb{P}\left(\mathsf{ME} < \left(\ln\left(\tfrac{n-1}{k}\right) -\ln\ln\left(\tfrac{n}{k}\right)-\ln(c)\right)/\beta\right)\leq\left(\frac{en}{k}\right)\cdot\left(\frac{n}{k}\right)^{-\Omega(ck)}.
\end{equation*}
Since $k$ is sufficiently smaller than $n$, we have $en/k \leq (n/k)^2$. As $c$ is sufficiently large, we can simplify the right hand side to $(n/k)^{-\Omega(ck)}$. Finally, since $k\geq 1$ and $k$ is sufficiently smaller than $n$, we have $(n/k)^k\geq n$. This implies $(n/k)^{-\Omega(ck)}\leq n^{-\Omega(c)}$, which competes the proof.\blokje
\end{proof}
\begin{lemma}
\label{lemma:klarge}
Let $k\leq (1-\varepsilon) n$ for some constant $\varepsilon >0$. For every $c\in [0,2\varepsilon^2)$, we have
\begin{equation*}
\mathbb{P}\left(\mathsf{ME} \leq c/\beta \right)\leq c^{\Omega(n)}.
\end{equation*}
\end{lemma}
\begin{proof}
The value of $\mathsf{ME}$ is the sum of $n-k$ shortest path lengths in $G$. The union of these paths contains at least $n-k$ different edges from $G$. Let $S_m$ be the sum of the $m$ lightest edge weights in $G$. We obtain $\mathsf{ME} \geq S_{n-k}\geq S_{\varepsilon n}$. The result follows using Lemma \ref{lemma:S} with $\phi=\varepsilon$.\blokje
\end{proof}
\begin{lemma}
\label{lemma:trivialp}
For any $c\geq 4$ we have $\mathbb{P}\left(\mathsf{TR} > n^c\right) \leq \exp(-n^{c/4})$.
\end{lemma}
\begin{proof}
We can roughly bound $\mathsf{TR}$ by $n\Delta_{\max}$, which in turn can be roughly bounded by $n^2\max_e\{w(e)\}$. Since $\max_e\{w(e)\}$ is the maximum of $|E|\leq\beta n(n-1)/2$ independent exponentially distributed random variables with parameter $1$, we have
\begin{align*}
\mathbb{P}\left(\mathsf{TR} \leq n^c\right) &\geq \left(1-\exp\left(-n^{c-2}\right)\right)^{\beta n(n-1)/2}\geq 1- \tfrac12\beta n(n-1) \cdot \exp\left(-n^{c-2}\right)\\
&\geq 1-\exp\left(-n^{c-3}\right) \geq 1-\exp\left(-n^{c/4}\right).
\end{align*}
The result follows by taking the complement.\blokje
\end{proof}
Now we have obtained everything needed to provide an upper bound for the expected approximation ratio of the trivial heuristic.
\begin{theorem}
\label{theorem:trivial}
Let $k\leq (1-\varepsilon)n$ for some constant $\varepsilon>0$. For generalized random shortest path metrics, we have $\mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\right] = O\left(\beta/\alpha\right)$. Moreover, if we have $k\leq c'n$ for some fixed $c'\in(0,1)$ sufficiently small, then we have
\begin{equation*}
\mathbb{E}\left[\tfrac{\mathsf{TR}}{\mathsf{ME}}\right] = (\beta/\alpha)\cdot\left(1+O\left(\tfrac{\ln\ln(n/k)}{\ln(n/k)}\right)\right).
\end{equation*}
\end{theorem}
\begin{proof}
We have for all constants $m>0$
\begin{equation*}
\mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\right] \leq \mathbb{E}\left[\frac{\beta\cdot\mathsf{TR}}{m}\right] +\mathbb{P}\left(\mathsf{ME}<\frac{m}{\beta}\right)\cdot \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathsf{ME} < \frac{m}{\beta}\right].
\end{equation*}
\emph{Case 1 ($k\leq c'n$, $c'$ sufficiently small)}: Let $n$ be sufficiently large. According to Lemma \ref{lemma:ksmall} we can pick a constant $c>0$ sufficiently large such that
\begin{equation*}
\mathbb{P}\left(\mathsf{ME} \leq \left(\ln\left(\tfrac{n-1}{k}\right) - \ln\ln\left(\tfrac{n}{k}\right)-\ln(c)\right)/\beta\right) \leq n^{-9}.
\end{equation*}
Take $m=\ln((n-1)/k)-\ln\ln(n/k)-\ln(c)$. By Lemma~\ref{lemma:triviale}, we have
\begin{equation*}
\mathbb{E}\left[\frac{\beta\cdot\mathsf{TR}}{m}\right] \leq \frac\beta\alpha\cdot\frac{\ln\left(\frac{n-1}{k-1}\right)+1}{m}\leq\frac\beta\alpha\cdot\left(1+O\left(\frac{\ln\ln(n/k)}{\ln(n/k)}\right)\right).
\end{equation*}
For the second part we can use the fact that $m$ was chosen such that $\mathbb{P}(\mathsf{ME}\leq m/\beta)\leq n^{-9}$ to obtain
\begin{align*}
\mathbb{P}\left(\mathsf{ME}<\frac{m}{\beta}\right)\cdot \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathsf{ME} < \frac{m}{\beta}\right] &= \mathbb{P}\left(\mathsf{ME}<\frac{m}{\beta}\right) \cdot \int_0^\infty \mathbb{P}\left(\frac{\mathsf{TR}}{\mathsf{ME}} \geq x\;\middle|\; \mathsf{ME} < \frac{m}{\beta}\right) \mathrm{d}x\\
&\leq \mathbb{P}\left(\mathsf{ME}<\frac{m}{\beta}\right) \cdot\left(n^8+ \int_{n^8}^\infty \mathbb{P}\left(\frac{\mathsf{TR}}{\mathsf{ME}} \geq x\;\middle|\; \mathsf{ME} < \frac{m}{\beta}\right) \mathrm{d}x\right)\\
&\leq \frac1n+ \int_{n^8}^\infty \mathbb{P}\left(\frac{\mathsf{TR}}{\mathsf{ME}} \geq x\; \text{and} \; \mathsf{ME} < \frac{m}{\beta}\right) \mathrm{d}x\\
&\leq \frac1n+ \int_{n^8}^\infty \mathbb{P}\left(\frac{\mathsf{TR}}{\mathsf{ME}} \geq x\right) \mathrm{d}x\\
&\leq \frac1n+ \int_{n^8}^\infty\mathbb{P}\left(\mathsf{TR}\geq \sqrt{x}\right) \mathrm{d}x+\int_{n^8}^\infty \mathbb{P}\left(\mathsf{ME}\leq \frac{1}{\beta \sqrt{x}}\right) \mathrm{d}x,
\intertext{where the last inequality follows since $\mathsf{TR}/\mathsf{ME} \geq x$ implies $\mathsf{TR} \geq \sqrt{x}$ or $\mathsf{ME} \leq 1/\sqrt{x} \leq 1/\beta \sqrt{x}$. Note that the requirements for applying Lemmas \ref{lemma:klarge} and \ref{lemma:trivialp} to the corresponding probabilities are met for any $x\in[n^8,\infty)$. Upon applying those we obtain}
\mathbb{P}\left(\mathsf{ME}<\frac{m}{\beta}\right)\cdot \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathsf{ME} < \frac{m}{\beta}\right] &\leq \frac1n+ \int_{n^8}^\infty\exp\left(-x^{1/8}\right) \mathrm{d}x+\int_{n^8}^\infty\left(\frac{1}{\sqrt{x}}\right)^{\Omega(n)}\mathrm{d}x\\
&= O\left(\frac{1}{n}\right).
\end{align*}
\emph{Case 2 ($c'n<k\leq (1-\varepsilon)n$, $\varepsilon>0$)}: We repeat the proof for the previous case, but this time we choose $m$ as a sufficiently small constant ($m<\min\{2\varepsilon^2,1\}$ satisfies). Then, by Lemma~\ref{lemma:klarge}, we have $\mathbb{P}\left(\mathsf{ME}<m/\beta\right) \leq m^{\Omega(n)} \leq n^{-9}$. Furthermore, by Lemma \ref{lemma:triviale}, we have
\begin{equation*}
\mathbb{E}\left[\frac{\beta\cdot\mathsf{TR}}{m}\right] \leq \frac\beta\alpha\cdot \frac{\ln\left(\frac{n-1}{k-1}\right) + 1}{m}=O\left(\frac\beta\alpha\right),
\end{equation*}
since $k>c'n$. Together with the second part of the first case, this shows the claim.\blokje
\end{proof}
\section{Application to the Erd\H{o}s-R\'enyi Random Graph Model}\label{sect:ERRG}
So far, we have analyzed random shortest path metrics applied to graphs based on their cut parameters (Def.~\ref{def:ab}). In this section, we first show that instances of the Erd\H{o}s--R\'enyi random graph model have `nice' cut parameters with high probability. We then use this to prove our main results.
\begin{lemma}
\label{lemma:ep}
Let $G=(V,E)$ be an instance of the $G(n,p)$ model. For constant $\varepsilon \in (0,1)$ and for any $p\geq c\ln(n)/n$ (as $n\to\infty$), in which $c>9/\varepsilon^2$ is constant, the cut parameters of $G$ are bounded by $(1-\varepsilon)p\leq\alpha\leq\beta\leq(1+\varepsilon)p$ with probability at least $1-o\left(1/n^2\right)$.
\end{lemma}
\begin{proof}
Let $\mathcal{E}$ denote the event that the cut parameters of $G$ are not bounded by $(1-\varepsilon)p\leq\alpha\leq\beta\leq(1+\varepsilon)p$. Using the definition of the cut parameters, the probability of this event can be written as
\begin{equation*}
\mathbb{P}(\mathcal{E})=\mathbb{P}\left(\exists\;\varnothing\neq U\subset V,\,|U|\leq n/2:\big||\delta(U)|-p\mu_U\big|>\varepsilon p\mu_U\right).
\end{equation*}
We can restrict ourselves here to subsets $U$ of size at most $n/2$ since $U$ and $V\backslash U$ induce the same cut of $G$. Using the union bound, we can bound this probability by
\begin{equation*}
\mathbb{P}(\mathcal{E})\leq\sum_{k=1}^{n/2}\binom{n}{k}\cdot\mathbb{P}\left(\big||\delta(U_k)|-p\mu_U\big|>\varepsilon p\mu_U\right),
\end{equation*}
where $U_k$ is a subset of $V$ of size $k$. Applying a Chernoff bound~\cite[Cor.~4.6]{Mitzenmacher2005} to each term of this summation, we can further bound this by
\begin{align*}
\mathbb{P}(\mathcal{E})&\leq\sum_{k=1}^{n/2}\binom{n}{k}\cdot2e^{-k(n-k)p\varepsilon^2/3}\\
&\leq\sum_{k=1}^{n/2}\binom{n}{k}\cdot2e^{-k(n-k)c\ln(n)\varepsilon^2/3n},
\end{align*}
where we used $p\geq c\ln(n)/n$ for the last inequality. Now, let $\xi>0$ be sufficiently small ($\xi<1-9/c\varepsilon^2$ satisfies). Using this $\xi$, we split the summation in two parts, and use the bounds $\binom{n}{k}\leq n^k$ and $\binom{n}{k}\leq 2^n$, respectively, to obtain
\begin{align*}
\mathbb{P}(\mathcal{E})&\leq\sum_{k=1}^{\xi n}\binom{n}{k}\cdot2e^{-k(n-k)c\ln(n)\varepsilon^2/3n}+\sum_{k=\xi n}^{n/2}\binom{n}{k}\cdot2e^{-k(n-k)c\ln(n)\varepsilon^2/3n}\\
&\leq\sum_{k=1}^{\xi n}2e^{k\ln(n)(1-(1-k/n)c\varepsilon^2/3)}+\sum_{k=\xi n}^{n/2}2e^{n\ln(2)-k(n-k)c\ln(n)\varepsilon^2/3n}\\
&\leq\sum_{k=1}^{\xi n}2e^{k\ln(n)(1-(1-\xi)c\varepsilon^2/3)}+\sum_{k=\xi n}^{n/2}2e^{n\ln(2)-\xi(1-\xi)cn\ln(n)\varepsilon^2/3}.
\end{align*}
For the last inequality we used the fact that $k/n\leq\xi$ for all $1\leq k\leq\xi n$ and that $k(n-k)\geq\xi(1-\xi)n^2$ for all $\xi n\leq k\leq n/2$. Now, since $\xi$ is sufficiently small, we have $1-(1-\xi)c\varepsilon^2/3<-2$ and thus we can bound the first summation by $o(1/n^2)$. Furthermore, as $n\to\infty$, each summand of the second summation is bounded by $e^{-\Omega(n\ln(n))}=n^{-\Omega(n)}$, which allows us to bound the second summation by $n\cdot n^{-\Omega(n)}=n^{-\Omega(n)}$. Together with the bound for the first summation, this yields $\mathbb{P}(\mathcal{E})\leq o(1/n^2)+n^{-\Omega(n)}=o(1/n^2)$. The result now follows by taking the complement of the event $\mathcal{E}$.\blokje
\end{proof}
Recall that from the result of Corollary \ref{corollary:harmonic} we could derive (approximate) bounds for the expected distance $\mathbb{E}[d(u,v)]$ between two arbitrary vertices in a random shortest path metric. Combining this with the result of the foregoing lemma, we can see that, for the case of the application to the Erd\H{o}s--R\'enyi random graph model, w.h.p. over the random graph $\mathbb{E}[d(u,v)]$ is approximately bounded between $\ln(n)/((1+\varepsilon)np)$ and $\ln(n)/((1-\varepsilon)np)$ for any constant $\varepsilon\in(0,1)$. This is in line with the known result $\mathbb{E}[d(u,v)]\approx\ln(n)/np$ for $p$ sufficiently large~\cite{Bhamidi2011}.
\subsection{Performance of Heuristics}
In this section, we provide the main results of this work. We use the results from Section \ref{sect:heuristics} and Lemma \ref{lemma:ep} to analyze the performance of several heuristics in random shortest path metrics applied to Erd\H{o}s--R\'enyi random graphs.
When a graph $G=(V,E)$ is created by the $G(n,p)$ model, there is a non-zero probability of $G$ being disconnected. In a corresponding random shortest path metric this results in $d(u,v)=\infty$ for any two vertices $u,v \in V$ that are in different components of $G$. Observe that, if this is the case, then the identity of indiscernibles, symmetry and triangle inequality still hold. Thus we still have a metric and we can bound the expected approximation ratio for such graphs from above by the worst-case approximation ratio for metric instances.
Using this observation, we can prove the following results.
\begin{theorem}
\label{theorem:PGR}
Let $\varepsilon\in(0,1)$ be constant. Let $G=(V,E)$ be a random instance of the $G(n,p)$ model, for $p$ sufficiently large ($p\geq c\ln(n)/n$ as $n\to\infty$ for a constant $c>9/\varepsilon^2$ satisfies), and consider the corresponding random shortest path metric. Then, we have
\begin{equation*}
\mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\right] = O(1).
\end{equation*}
\end{theorem}
\begin{proof}
Let $\mathcal{E}$ denote the event that the cut parameters of $G$ are bounded by $(1-\varepsilon)p\leq\alpha\leq\beta\leq(1+\varepsilon)p$. Then we have
\begin{align*}
\mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\right] &\leq \mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\;\middle|\; \mathcal{E}\right] + \mathbb{E}\left[\frac{\mathsf{GR}}{\mathsf{MM}}\;\middle|\; \overline{\mathcal{E}}\;\right] \cdot \mathbb{P}\left(\;\overline{\mathcal{E}}\;\right).\\
&\leq O\left(\frac{(1+\varepsilon)p}{(1-\varepsilon)p}\right)+O\left(n^{\log_2(3/2)}\right)\cdot o\left(\frac1{n^2}\right)=O(1),
\end{align*}
where we used the results of Theorem \ref{theorem:EGR}, Lemma \ref{lemma:ep}, and the worst-case approximation ratio of the greedy heuristic on metric instances~\cite{Reingold1981}.\blokje
\end{proof}
\begin{theorem}
\label{theorem:PNNIN}
Let $\varepsilon\in(0,1)$ be constant. Let $G=(V,E)$ be a random instance of the $G(n,p)$ model, for $p$ sufficiently large ($p\geq c\ln(n)/n$ as $n\to\infty$ for a constant $c>9/\varepsilon^2$ satisfies), and consider the corresponding random shortest path metric. Then, we have
\begin{equation*}
\mathbb{E}\left[\frac{\mathsf{NN}}{\mathsf{TSP}}\right] = O(1)\qquad\text{and}\qquad\mathbb{E}\left[\frac{\mathsf{IN}_R}{\mathsf{TSP}}\right] = O(1).
\end{equation*}
\end{theorem}
\begin{proof}
Let $\mathcal{E}$ denote the event that the cut parameters of $G$ are bounded by $(1-\varepsilon)p\leq\alpha\leq\beta\leq(1+\varepsilon)p$. Then we have
\begin{align*}
\mathbb{E}\left[\frac{\mathsf{NN}}{\mathsf{TSP}}\right] &\leq \mathbb{E}\left[\frac{\mathsf{NN}}{\mathsf{TSP}}\;\middle|\; \mathcal{E}\right] + \mathbb{E}\left[\frac{\mathsf{NN}}{\mathsf{TSP}}\;\middle|\; \overline{\mathcal{E}}\;\right] \cdot \mathbb{P}\left(\;\overline{\mathcal{E}}\;\right).\\
&\leq O\left(\frac{(1+\varepsilon)p}{(1-\varepsilon)p}\right)+O\left(\ln(n)\right)\cdot o\left(\frac1{n^2}\right)=O(1),
\end{align*}
where we used the results of Theorem \ref{theorem:NN}, Lemma \ref{lemma:ep}, and the worst-case approximation ratio of the nearest-neighbor heuristic on metric instances~\cite{Rosenkrantz1977}. For the second part, we use the same argument, which follows this time from the results of Theorem \ref{theorem:IN}, Lemma \ref{lemma:ep}, and the worst-case approximation ratio of the insertion heuristics on metric instances~\cite{Rosenkrantz1977}. Note that this argument is independent of the rule $R$ used.\blokje
\end{proof}
For the last two results, we need the assumption that $G$ is connected.
\begin{theorem}
\label{theorem:P2opt}
Let $\varepsilon\in(0,1)$ be constant. Let $G=(V,E)$ be a random instance of the $G(n,p)$ model, for $p$ sufficiently large ($p\geq c\ln(n)/n$ as $n\to\infty$ for a constant $c>9/\varepsilon^2$ satisfies), and consider the corresponding random shortest path metric. If $G$ is connected, then the expected number of iterations of the 2-opt heuristic for TSP is bounded by $O(n^8\ln^3(n))$.
\end{theorem}
\begin{proof}
Let $T$ be the number of iterations of the 2-opt heuristic and let $\mathcal{E}$ denote the event that the cut parameters of $G$ are bounded by $(1-\varepsilon)p\leq\alpha\leq\beta\leq(1+\varepsilon)p$, whereas $\mathcal{E}'$ denotes the event that $G$ is connected. Note that $\mathcal{E}$ implies $\mathcal{E}'$. Moreover, note that event $\mathcal{E}'$ implies that the cut parameters of $G$ are bounded by $\Theta(1/n^2)\leq\alpha\leq\beta\leq1$. Now, we have
\begin{align*}
\mathbb{E}\left[T\mid\mathcal{E}'\right]&\leq\mathbb{E}\left[T\mid\mathcal{E}',\mathcal{E}\right]+ \mathbb{E}\left[T\mid\mathcal{E}',\overline{\mathcal{E}}\;\right]\cdot\mathbb{P}\left(\;\overline{\mathcal{E}}\;\right)\\
&\leq O\left(n^8\ln^3(n)\cdot\frac{(1+\varepsilon)p}{(1-\varepsilon)p}\right)+O\left(n^8\ln^3(n)\cdot\frac1{1/n^2}\right)\cdot o\left(\frac1{n^2}\right)\\
&=O(n^8\ln^3(n)),
\end{align*}
where we used the results of Theorem \ref{theorem:2-opt} and Lemma \ref{lemma:ep}.\blokje
\end{proof}
\begin{theorem}
\label{theorem:Ptrivial}
Let $\tilde\varepsilon\in(0,1)$ be constant. Let $G=(V,E)$ be a random instance of the $G(n,p)$ model, for $p$ sufficiently large ($p\geq c\ln(n)/n$ as $n\to\infty$ for a constant $c>9/\tilde\varepsilon^2$ satisfies), and consider the corresponding random shortest path metric. Let $\mathcal{E}'$ denotes the event that $G$ is connected. Let $k\leq (1-\varepsilon')n$ for some constant $\varepsilon'>0$, then we have $\mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle| \; \mathcal{E}'\right] = O\left(1\right)$. Moreover, if we have $k\leq c'n$ for $c'\in(0,1)$ sufficiently small, then $\mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle| \; \mathcal{E}'\right] =1+\varepsilon+o(1)$.
\end{theorem}
\begin{proof}
Let $\mathcal{E}$ denote the event that the cut parameters of $G$ are bounded by $(1-\tilde\varepsilon)p\leq\alpha\leq\beta\leq(1+\tilde\varepsilon)p$. Note that $\mathcal{E}$ implies $\mathcal{E}'$. Moreover, note that event $\mathcal{E}'$ implies that the cut parameters of $G$ are bounded by $\Theta(1/n^2)\leq\alpha\leq\beta\leq1$. Now, we have
\begin{align*}
\mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\;\mathcal{E}'\right] &\leq \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathcal{E}',\mathcal{E}\right] + \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathcal{E}',\overline{\mathcal{E}}\;\right] \cdot \mathbb{P}\left(\;\overline{\mathcal{E}}\;\right).\\
&\leq O\left(\frac{(1+\tilde\varepsilon)p}{(1-\tilde\varepsilon)p}\right)+O\left(n^2\right)\cdot o\left(\frac1{n^2}\right)=O(1),
\end{align*}
where we used the results of Theorem \ref{theorem:trivial} and Lemma \ref{lemma:ep}. Moreover, if $k\leq c'n$ for $c'\in(0,1)$ sufficiently small, then
\begin{align*}
\mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\;\mathcal{E}'\right] &\leq \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathcal{E}',\mathcal{E}\right] + \mathbb{E}\left[\frac{\mathsf{TR}}{\mathsf{ME}}\;\middle|\; \mathcal{E}',\overline{\mathcal{E}}\;\right] \cdot \mathbb{P}\left(\;\overline{\mathcal{E}}\;\right).\\
&\leq O\left(\frac{(1+\tilde\varepsilon)p}{(1-\tilde\varepsilon)p}\cdot\left(1+\frac{\ln\ln(n/k)}{\ln(n/k)}\right)\right)+O\left(n^2\cdot\left(1+\frac{\ln\ln(n/k)}{\ln(n/k)}\right)\right)\cdot o\left(\frac1{n^2}\right)\\
&=1+\varepsilon+o(1),
\end{align*}
where $\varepsilon=(1+\tilde\varepsilon)/(1-\tilde\varepsilon)$ can be made arbitrarily small by taking $\tilde\varepsilon$ sufficiently small, and where we again used the results of Theorem \ref{theorem:trivial} and Lemma \ref{lemma:ep}.\blokje
\end{proof}
\section{Concluding Remarks}\label{sect:final}
We have analyzed heuristics for matching, TSP, and $k$-median on random shortest path metrics on Erd\H{o}s--R\'enyi random graphs. However, in particular for constant values of $p$, these graphs are still dense. Although our results hold for decreasing $p = \Omega(\ln n/n)$, we obtain in this way metrics with unbounded doubling dimension. In order to get an even more realistic model for random metric spaces, it would be desirable to analyze heuristics on random shortest path metrics on sparse graphs. Hence, we raise the question to generalize our findings to sparse random graphs or sparse (deterministic) classes of graphs.
\bibliographystyle{abbrvnat}
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| 2,288
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Q: Separate data from multiple rows in picker view I created picker view with 4 rows, but data in all rows are same. So how could i separate data from each row and use it for calculating?
A: Perhaps this may work for your problem of not displaying the data? Can't really tell because I can't see what you've already done.
-(NSInteger)numberOfComponentsInPickerView:(UIPickerView *)pickerView{
return 4;
}
-(NSInteger)pickerView:(UIPickerView *)pickerView numberOfRowsInComponent (NSInteger)component{
return rowOneItems.count;
return rowTwoItems.count;
return rowThreeItems.count;
return rowFourItems.count;
}
-(NSString*)pickerView:(UIPickerView *)pickerView titleForRow:(NSInteger)row forComponent:(NSInteger)component{
if(component == 1)
return[rowTwoItems objectAtIndex:row];
else if(component == 2)
return[rowThreeItems objectAtIndex:row];
else if(component == 3)
return[rowFourItems objectAtIndex:row];
else
return [rowOneItems objectAtIndex:row];
}
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"redpajama_set_name": "RedPajamaStackExchange"
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| 3,059
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{"url":"https:\/\/math.stackexchange.com\/questions\/3061314\/suppose-that-the-spot-price","text":"# Suppose that the spot price\u2026\n\nSuppose the spot price of gold is \\$300 per ounce and the risk-free interest rate for one year is 5%. What is a reasonable value for the one-year forward price of gold? The answer is \\$315, right?\n\nSuppose the one-year forward price of gold is \\$340. Argue as follows: borrow 300 dollars for a year and buy one ounce of gold. Then short a forward contract to sell the gold in one year time. Show this will lead to a risk-free pro\ufb01t (arbitrage) and the the one-year forward price of gold must be$315.\n\nThis comes down to \\$340 -$315 = \\$25 , right? I suppose that the \\$315 here cones again from the \\$300*105% ? right? Then assume the one-year forward price of gold is \\$300. Argue as follows: sell the gold, then invest the proceeds and long a one-year forward on gold. Show again that this will lead to a risk-free pro\ufb01t (arbitrage) and the one-year forward price of gold must be \\\\$315.\n\nI'm confused cause of this part. So if anybody could help?\n\nFor the second part, if the forward price of gold is the same as the spot price then you can sell the gold today for $$\\300$$ and put the money into an account earning $$5\\%$$ interest. At the same time you go long on a one year forward contract. This means you agree to pay $$\\300$$ an ounce a year from now.\nIn a year you have $$\\315$$ from the return on your investment so you can buy your gold back and earn a free $$\\15$$. That's what arbitrage is. Since everyone would do this it would drive the forward price up to $$\\315$$.","date":"2019-05-25 02:55:17","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\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3718845844268799, \"perplexity\": 1022.561907914392}, \"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-2019-22\/segments\/1558232257847.56\/warc\/CC-MAIN-20190525024710-20190525050710-00536.warc.gz\"}"}
| null | null |
Heat some olive oil in a saucepan. When hot sauté the chicken on both sides for about 5 minutes at high heat. Add the sliced onion and cook for a further 3 minutes. Then add the peppers and cook until they turn soft. Add the tomatoes, after you have peeled them, and cook on high heat for 3 minutes. Add salt and pepper.
If you wish, you can add Anthotyros and let it cook for one more minute or put it uncooked in the sauce.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 563
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Miss Lillian Mary Osborne (1869—1929)
Lillian Mary Osborne was born in Ontario, the daughter of A.D. Osborne who came to Edmonton in 1880 and was Edmonton's first Postmaster. After graduating from the Normal School in Toronto, while visiting family in Chicago, in 1889 she received word of her appointment to come to Edmonton as the first woman teacher. The 1881 school house had another room added to it and needed another teacher to assist Edmonton's first principal, Mr. Martin. Her salary for second class credentials was to be $600 per annum, far less than trustees would expect to pay a male teacher possessing the same teaching qualifications. Lillian Osborne would remain in the Board's employment for over 30 years, serving at McKay Avenue, old Queen's Avenue, Queen Alexandra, Delton and Glenora Schools. She was the first of a line of notable teachers who remained for many years with Edmonton Public Schools.
Lillian Osborne remained on staff until 1908 at McKay Avenue School when she resigned to take care of her mother, who died in 1910. In 1911 she accepted a position with the St. John School near Spruce Grove where she taught grades 1 to 7. Due to her own ill health the following year she took time away from teaching until 1915. In 1915 she accepted a teaching position at the Porto Bello School at Millet where she taught grades 1 to 7 for two years. Lillian returned to the St. John School until she was rehired by the Edmonton Board in 1919. During her first 30 years of teaching her salary had increased from $700 to $840 a year.
Most reports on individual teachers were very brief in the early years of Edmonton Public Schools. As a teacher at Queen's Avenue School, Inspector Bryan assessed Edmonton Public Schools' first female teacher, Miss Lillian Osborne as:
"Doing earnest and thoughtful work, supervising all work carefully. Miss Osborne has a good grasp of her work. The lines of teaching are sound. She has the respect and goodwill of her pupils."
While still on staff at Glenora School, Lillian Mary Osborne died on November 3, 1929 at the age of 60 years old, after an illness that extended for about a month. Lillian Osborne was a teacher for 40 of her 60 years of life, 29 with Edmonton Public Schools and 11 years between schools in Spruce Grove and Millet. Lillian was survived by her two brothers, Colonel Frank A. Osborne, who had moved to Victoria and had no family and Reverend R. H. Osborne who lived in Rolla, Missouri. In 1983 Lillian M. Osborne was honoured by the City of Edmonton for her contributions in education by receiving a Historical Board Recognition Award. As she had no known relatives here at the time, Miss Anne Carmichael, a retired teacher, Director and Supervisor of Education was asked to receive the award on her behalf.
(Sources: A Century and Ten: The History of Edmonton Public Schools and articles and newspaper clippings acquired from the City of Edmonton Archives)
Legacy Project
At Lillian Osborne High School, we embrace change, and strive to meet our students' learning needs in a variety of ways, always working towards excellence in teaching and learning. Lillian Osborne' School's pioneering spirit encapsulates that of the many pioneers--such as Lillian Osborne herself--who shaped our province. Our namesake Lillian Osborne represents the legendary characteristics of hard work, resilience, compassion, and teamwork that created successful communities in Alberta a century ago.
Today, the young teacher Lillian Osborne, an unsung hero until today, inspires our students and staff to take that extra step, to embrace the challenges that meet us with intention, creativity, and mindfulness, as we do our best to move forward with our learning, together, in today's busy times.
Like Lillian Osborne herself, our school community wants to be legendary. It is thus only appropriate that our students and teams chose to be known as the Lillian Osborne Legends.
Lillian Osborne High School challenges its students to look ahead as pioneers in today's global landscape.
We are reminded of Lillian Osborne's legacy daily as her likeness, created through a collage of student artwork, graces the Grand Staircase in our school.
Since our opening in 2009, to establish our own Legendary culture, our school community has planted and nurtured the seeds of a Legacy Project, with each graduating class contributing a Legacy project of their choosing.
IB World School
Edmonton Public Schools
1 Kingsway NW
Edmonton Alberta T5H 4G9
Web: epsb.ca
See school listings
Search for a program
Find bus information
Register my child
Web administrator
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8
|
Q: Regex string between underscore I need to get the markings between the underscores, for example:
Com_x0020_este_x002C__x0020_texto
Com_x0020_este_x002C_ _x0020_texto
Thank you!
A: You can try this, if I understand the question correctly: /_(.*?)_/g
See https://regex101.com/r/H3exx4/1/ for an example.
A: What you are looking for it preg_replace documentation can be found here: http://php.net/manual/en/function.preg-replace.php and the logic you are looking for it as follows:
PHP Version:
$string = "Com_x0020_este_x002C__x0020_texto";
print preg_replace('/_(.*?)_/', " _$1_ ", $string);
Javascript Version:
var $string = "Com_x0020_este_x002C__x0020_texto";
alert($string.replace(/_(.*?)_/g," _$1_ "));
Result: Com _x0020_ este _x002C_ _x0020_ texto
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,319
|
using namespace std;
#include "Menu.hpp"
TesterBase::TesterBase()
{
m_totalTestCount = 0;
m_totalTestPass = 0;
cout << "* TEST LOG WILL BE WRITTEN OUT TO:" << endl;
Menu::PrintPwd();
m_output.open( "test_result.html" );
OutputHeader();
m_closed = false;
}
TesterBase::~TesterBase()
{
Close();
}
void TesterBase::Close()
{
if ( m_closed ) { return; }
OutputFooter();
m_output.close();
m_closed = true;
}
void TesterBase::OutputHeader()
{
m_output << "<!DOCTYPE html>" << endl;
m_output << "<html>" << endl;
m_output << " <head>" << endl;
m_output << " <title> cuTEST Test Results </title>" << endl;
m_output << " <style type='text/css'>" << endl;
m_output << " body { font-family: sans-serif; font-size: 12pt; }" << endl;
m_output << " .warning { padding: 20px; background: #ffed24; font-size: 20px; text-align: center; }" << endl;
m_output << " .result-success { background: #24ff24; font-size: 20px; text-align: center; }" << endl;
m_output << " .result-failure { background: #ff7095; font-size: 20px; text-align: center; }" << endl;
m_output << " table { background: #ccccff; width: 100%; border: solid 1px #0000aa; }" << endl;
m_output << " table tr.pass { background: #ccffcc; }" << endl;
m_output << " table tr.fail { background: #ffcccc; }" << endl;
m_output << " table tr { font-size: 12pt; border: solid 1px #000066; }" << endl;
m_output << " table tr td { padding: 10px; border: solid 1px #000033; }" << endl;
m_output << " table tr td ul li, "
<< " table tr td ol li { margin: 10px 0; }" << endl;
m_output << " table tr td ol { list-style-type: upper-latin; }" << endl;
m_output << " table tr .test-set { }" << endl;
m_output << " table tr .test { font-size: 14pt; }" << endl;
m_output << " table tr .prereq { }" << endl;
m_output << " table tr .result { text-align: center; font-size: 14pt; font-weight: bold; }" << endl;
m_output << " table tr .expected { font-family: monospace; font-size: 14pt; }" << endl;
m_output << " table tr .actual { font-family: monospace; font-size: 14pt; }" << endl;
m_output << " table tr .comments { }" << endl;
m_output << " table tr.summary { background: #000000; color: #ffffff; }" << endl;
m_output << " table tr.descriptions { font-size: 8pt; text-align: center; }" << endl;
// m_output << " table .col_set { width: 5%; }" << endl;
// m_output << " table .col_test { width: 15%; }" << endl;
// m_output << " table .col_prereq { width: 10%; }" << endl;
// m_output << " table .col_result { width: 5%; }" << endl;
// m_output << " table .col_expected { width: 15%; }" << endl;
// m_output << " table .col_actual { width: 15%; }" << endl;
// m_output << " table .col_comments { width: 35%; }" << endl;
m_output << " table .spacer { height: 100px; }" << endl;
m_output << " </style>" << endl;
m_output << " </head>" << endl;
m_output << " <body>" << endl;
m_output << " <div class='warning'>Warning: Make sure to check if all tests finish (there will be a message at the end of the file) - <br>" << endl;
m_output << " It is possible for your program to crash early, but still show that tests have passed because it hasn't gone through everything.</div>" << endl;
m_output << " <table>" << endl;
m_output << " <tr>" << endl;
m_output << " <th class='col_set'> Test set </th>" << endl;
m_output << " <th class='col_test'> Test </th>" << endl;
m_output << " <th class='col_prereq'> Prerequisite <br> functions </th>" << endl;
m_output << " <th class='col_result'> Pass/fail </th>" << endl;
m_output << " <th class='col_expected'> Expected <br> output </th>" << endl;
m_output << " <th class='col_actual'> Actual <br> output </th>" << endl;
m_output << " <th class='col_comments'> Comments </th>" << endl;
m_output << " </tr>" << endl;
// m_output << " <tr class='descriptions'>" << endl;
// m_output << " <td> </td>" << endl;
// m_output << " <td> </td>" << endl;
// m_output << " <td> Functions that need to be implemented for these tests to work right </td>" << endl;
// m_output << " <td> </td>" << endl;
// m_output << " <td> The output expected from the function's return </td>" << endl;
// m_output << " <td> What was actually returned from the function </td>" << endl;
// m_output << " <td> Additional notes from the test </td>" << endl;
// m_output << " </tr>" << endl;
}
void TesterBase::OutputFooter()
{
m_output << " </table>" << endl;
if ( m_subtest_totalPasses == m_subtest_totalTests )
{
m_output << " <div class='result-success'>" << m_totalTestCount << " testsets ran; " << m_totalTestPass << " testsets passed</div>" << endl;
}
else
{
m_output << " <div class='result-failure '>" << m_totalTestCount << " testsets ran; " << m_totalTestPass << " testsets passed</div>" << endl;
}
m_output << " </body>" << endl;
m_output << "</html>" << endl;
}
void TesterBase::WriteoutSummary()
{
m_output << "<tr class='summary'>" << endl;
m_output << "<td colspan='7' class='summary'>" << endl;
m_output << "<strong>SUMMARY FOR " << col_testSet << ":</strong>" << endl;
m_output << m_subtest_totalPasses << " out of " << m_subtest_totalTests << " tests passed" << endl;
m_output << "</td>" << endl;
m_output << "</tr>" << endl;
m_output << "<tr class='spacer'></tr>" << endl;
}
void TesterBase::WriteoutRow()
{
if ( col_result == "passed" )
m_output << "<tr class='pass'>" << endl;
else
m_output << "<tr class='fail'>" << endl;
m_output << "<td class='test-set'> " << col_testSet << " </td>" << endl;
m_output << "<td class='test'> " << col_testName << " </td>" << endl;
m_output << "<td class='prereq'> <ul> " << col_prerequisites << " </ul> </td>" << endl;
m_output << "<td class='result'> " << col_result << " </td>" << endl;
m_output << "<td class='expected'> <ol> " << col_expectedOutput << " </ol> </td>" << endl;
m_output << "<td class='actual'> <ol> " << col_actualOutput << " </ol> </td>" << endl;
m_output << "<td class='comments'> <ul> " << col_comments << " </ul> </td>" << endl;
m_output << "</tr>" << endl;
}
void TesterBase::Start()
{
// MainMenu();
TestAll();
}
void TesterBase::TestAll()
{
int counter = 1;
cout << endl;
for ( list<TestListItem>::iterator it = m_tests.begin(); it != m_tests.end(); it++ )
{
if ( it->testAggregate == false )
{
cout << "Running testset " << counter << " out of " << m_totalTestCount << ": \t"
<< it->name << endl;
int result = it->callFunction();
if ( result == 1 )
{
m_totalTestPass++;
}
}
counter++;
}
cout << endl << " NOTE: CHECK \"test_result.html\" FOR FULL DETAILS." << endl;
}
void TesterBase::AddTest( const TestListItem& test )
{
m_tests.push_back( test );
m_totalTestCount++;
}
void TesterBase::StartTestSet( const string& name, const vector<string>& prereqs )
{
Set_TestSet( name );
m_subtest_totalTests = 0;
m_subtest_totalPasses = 0;
Set_TestPrerequisites( prereqs );
}
void TesterBase::StartTest( const string& description )
{
cout << left << setw( 1 ) << " * " << setw( 70 ) << description << setw( 10 ) << "... ";
Set_TestName( description );
m_subtest_totalTests++;
col_actualOutput = "";
col_expectedOutput = "";
}
void TesterBase::TestFail()
{
cout << "FAIL" << endl;
Set_Result( false );
}
void TesterBase::TestFail( const string& message )
{
cout << "FAIL" << endl;
Set_Comments( message );
Set_Result( false );
}
void TesterBase::TestPass()
{
cout << "PASS" << endl;
Set_Result( true );
m_subtest_totalPasses++;
}
int TesterBase::TestResult()
{
return ( m_subtest_totalPasses == m_subtest_totalTests );
}
void TesterBase::FinishTest()
{
WriteoutRow();
}
void TesterBase::FinishTestSet()
{
WriteoutSummary();
cout << endl << endl;
}
void TesterBase::Set_TestSet( const string& value )
{
col_testSet = value;
}
void TesterBase::Set_TestName( const string& value )
{
col_testName = value;
}
void TesterBase::Set_TestPrerequisites( const vector<string>& prereqs )
{
col_prerequisites = "";
for ( unsigned int i = 0; i < prereqs.size(); i++ )
{
col_prerequisites += "<li>" + prereqs[i] + "</li>";
}
}
void TesterBase::Set_Result( bool passed )
{
col_result = ( passed ) ? "passed" : "FAILED";
}
void TesterBase::Set_ExpectedOutput( const string& variable )
{
col_expectedOutput += "<li>" + variable + "</li>";
}
void TesterBase::Set_ExpectedOutput( const string& variable, const string& value )
{
col_expectedOutput += "<li>" + variable + " = " + value + "</li>";
}
void TesterBase::Set_ExpectedOutput( const string& variable, int value )
{
Set_ExpectedOutput( variable, StringUtil::ToString( value ) );
}
void TesterBase::Set_ExpectedOutput( const string& variable, bool value )
{
Set_ExpectedOutput( variable, StringUtil::ToString( value ) );
}
void TesterBase::Set_ActualOutput( const string& variable )
{
col_actualOutput += "<li>" + variable + "</li>";
}
void TesterBase::Set_ActualOutput( const string& variable, const string& value )
{
col_actualOutput += "<li>" + variable + " = " + value + "</li>";
}
void TesterBase::Set_ActualOutput( const string& variable, int value )
{
Set_ActualOutput( variable, StringUtil::ToString( value ) );
}
void TesterBase::Set_ActualOutput( const string& variable, bool value )
{
Set_ActualOutput( variable, StringUtil::ToString( value ) );
}
void TesterBase::Set_Comments( const string& value )
{
col_comments += "<li>" + value + "</li>";
}
void TesterBase::TerminationMessage( const string& message )
{
cout << "TEST TERMINATED: " << message << endl;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,857
|
{"url":"http:\/\/mathoverflow.net\/questions\/64925\/occurence-of-trivial-representation-in-a-tensor-square","text":"# Occurence of trivial representation in a tensor square.\n\nSuppose $G$ is a group and $V$ an irreducible representation of $G$. One has that $V\\otimes V\\cong \\Lambda^2(V)\\oplus Sym^2(V)$. It is well-known that if the trivial representation appears as a subrepresentation of $\\Lambda^2(V)$ then $V$ is of quaternionic type; while if the trivial representation appears as a subrepresentation of $Sym^2(V)$ then $V$ is a of real type. From this approach, it is clear that the trivial representation cannot appear in both $\\Lambda^2(V)$ and $Sym^2(V)$.\n\nWhat I am curious about is as follows:\n\nQuestion: Is there is some (relatively easy) way to see why the trivial representation cannot appear in both $\\Lambda^2(V)$ and $Sym^2(V)$ without introducing the machinery of real\/quaternionic types?\n\nAs a bit of motivation, if one looks at other subrepresentations, then for example if $G = G_2$ and $V_n$ is an $n$-dimensional irreducible representation of $G_2$, then $V_{64}$ appears as a subrepresentation of both $\\Lambda^2(V_{27})$ and $Sym^2(V_{27})$. In particular it is possible for the intertwining number of $\\Lambda^2(V)$ and $Sym^2(V)$ to be nonzero, but by the real vs. quaternionic characterization, the trivial representation is somehow special in that it cannot contribute to the intertwining number.\n\n-\n\nThe trivial representation appears in $\\wedge^2 V$ if and only if the representation $V^{\\ast}$ has a $G$-invariant alternating bilinear form (because $\\wedge^2 V\\cong\\wedge^2\\left(\\left(V^{\\ast}\\right)^{\\ast}\\right)$ is isomorphic to the $G$-module of all alternating bilinear forms on $V^{\\ast}$, and $G$-invariant forms correspond to $G$-fixed elements).\n\nctrl+c & ctrl+v:\n\nThe trivial representation appears in $\\mathrm{Sym}^2 V$ if and only if the representation $V^{\\ast}$ has a $G$-invariant symmetric bilinear form (because $\\mathrm{Sym}^2 V\\cong\\mathrm{Sym}^2\\left(\\left(V^{\\ast}\\right)^{\\ast}\\right)$ is isomorphic to the $G$-module of all symmetric bilinear forms on $V^{\\ast}$, and $G$-invariant forms correspond to $G$-fixed elements).\n\nSo we have to prove that for an irreducible representation $V$, the representation $V^{\\ast}$ cannot have both a nontrivial $G$-invariant symmetric bilinear form and a nontrivial $G$-invariant alternating bilinear form. More generally, an irreducible representation $W$ of $G$ cannot have two linearly independent $G$-invariant bilinear forms. In fact, a bilinear form on the $k$-vector space $W$ can be seen as a homomorphism $W\\to W^{\\ast}$, and the bilinear form is $G$-invariant if and only if this homomorphism is $G$-equivariant. But Schur's lemma yields that there cannot be two linearly independent $G$-equivariant homomorphisms from $W$ to $W^{\\ast}$, since both $W$ and $W^{\\ast}$ are irreducible representations.\n\nSo much for the case when the ground field is algebraically closed (which is probably your case). In the general case, I think the assertion is not true, though I don't know a counterexample right out of my head.\n\n-\nCounterexample for the nonalgebraically closed case: $k=\\mathbb{R}$, $G = \\mathbb{Z}\/n$ for $n>2$, $V=\\mathbb{R}^2$ with $G$ acting by rotation by $2 \\pi\/n$. Rotation of the plane preserves the standard (symmetric) inner product and the standard (anti-symmetric) area form. \u2013\u00a0David Speyer May 13 '11 at 19:39\nVery nice counterexample! \u2013\u00a0darij grinberg May 13 '11 at 19:40\nSo it still reduces to the fact that the representation is self-dual, but at least your approach avoids introducing the equivalence with real\/quaternionic types. \u2013\u00a0ARupinski May 13 '11 at 20:06\n@ARupinski: ... and is also valid over arbitrary fields. \u2013\u00a0Qiaochu Yuan May 14 '11 at 8:40\n\nThis is essentially what Darij wrote, but without mentioning the bilinear forms. (I had written it out before reading far enough into Darij's post to see that he was really doing the same thing, after the part about bilinear forms.) Think of $V\\otimes V$ as $\\text{Hom}(V^*,V)$. An occurrence of the trivial representation in $V\\otimes V$ thus amounts to a $G$-equivariant linear map from $V^*$ to $V$. Since $V$ and therefore also $V^*$ are irreducible, Schur's lemma says that the space of such maps has dimension either 1 (iff $V$ and $V^*$ are isomorphic) or 0. So there's at most one occurrence of the trivial representation in $V\\otimes V$.\n\n-\nSo to answer the question implicit in the last part of the OP (\"why is the trivial representation special\"), the trivial representation is special because $\\text{Hom}_G(1, V)$ represents the functor sending a representation to its $G$-invariant subspace, and this functor sends $\\text{Hom}(A, B)$ to $\\text{Hom}_G(A, B)$. \u2013\u00a0Qiaochu Yuan May 14 '11 at 8:39\nBy the way, doesn't the identification of $V \\otimes V$ with $\\text{Hom}(V^{\\ast}, V)$ only hold if $V$ is finite-dimensional? If $V$ is infinite-dimensional I think the latter is much larger... \u2013\u00a0Qiaochu Yuan May 14 '11 at 8:42\nI was assuming finite-dimensionality, but now I see that there's no finiteness in the question. \u2013\u00a0Andreas Blass May 14 '11 at 17:28\nStill assuming finite-dimensionality, it might be worth pointing out that the argument gives more than was asserted. If $V$ and $W$ are any two irreducible representations of $G$, then the number of copies of the trivial representation in $V \\otimes W$ is 1 if $V$ and $W$ are dual to each other and 0 otherwise. \u2013\u00a0Andreas Blass May 14 '11 at 17:30\nBy the way, isn't finite-dimensionality also needed for Schur's lemma? The proof I learned depends on the existence of eigenvalues. \u2013\u00a0Andreas Blass May 14 '11 at 19:11","date":"2016-05-02 08:21:40","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\": 0.9754205942153931, \"perplexity\": 176.9563442491321}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-18\/segments\/1461860128071.22\/warc\/CC-MAIN-20160428161528-00115-ip-10-239-7-51.ec2.internal.warc.gz\"}"}
| null | null |
#import <Foundation/Foundation.h>
#import "AKMosquittoMessage.h"
/// Callback delegate for the client.
///
/// \author Andreas P. Koenzen <akc at apkc.net>
@protocol AKMosquittoClientDelegate
/// Called when the connection has been established.
///
/// \param code The connection code.
- (void)didConnect:(NSUInteger)code;
/// Called when the connection has been terminated.
- (void)didDisconnect;
/// Called when a message has been published.
///
/// \param messageID The message ID.
- (void)didPublish:(NSUInteger)messageID;
/// Called when a message has been received.
///
/// \param msg The received message.
- (void)didReceiveMessage:(AKMosquittoMessage *)msg;
/// Called when a subscription has been made.
///
/// \param messageID The message ID.
/// \param qos The message's QoS.
- (void)didSubscribe:(NSUInteger)messageID grantedQos:(NSArray *)qos;
/// Called when a subscription has been terminated.
///
/// \param messageID The message ID.
- (void)didUnsubscribe:(NSUInteger)messageId;
@end
@interface AKMosquittoClient : NSObject <AKMosquittoClientDelegate> {
struct mosquitto *mosq;
}
@property(getter=getHost) NSString *host;
@property(getter=getPort) unsigned short port;
@property(getter=getUsername) NSString *username;
@property(getter=getPassword) NSString *password;
@property(getter=getKeepAlive) unsigned short keepAlive;
@property(getter=getCleanSession) BOOL cleanSession;
@property id<AKMosquittoClientDelegate> delegate;
@property(getter=getTimer) NSTimer *timer;
// MARK: Builders
/// Build a new AKMosquittoClient instance.
///
/// \param clientID The ID of the client.
+ (AKMosquittoClient *)newBuild:(NSString *)clientID;
/// Build a new AKMosquittoClient instance.
///
/// \param clientID The ID of the client.
/// \param host The host to connect.
/// \param port The port to connect.
/// \param username The username of the connection. In case of Authentication support.
/// \param password The password of the connection. In case of Authentication support.
/// \param keepAlive The keep alive interval.
/// \param cleanSession If clean session must be active or not.
+ (AKMosquittoClient *)newBuild:(NSString *)clientID
host:(NSString *)host
port:(unsigned short)port
username:(NSString *)username
password:(NSString *)password
keepAlive:(unsigned short)keepAlive
cleanSession:(BOOL)cleanSession;
+ (NSString *)version;
- (void)connect;
- (void)connectWithSSL:(const char *)tlsVer
caCrt:(const char *)caCrt
caLocation:(const char *)caLocation
clientCrt:(const char *)clientCrt
clientKey:(const char *)clientKey;
- (void)reconnect;
- (void)disconnect;
- (void)setWill:(NSString *)payload
toTopic:(NSString *)willTopic
withQos:(NSUInteger)willQos
retain:(BOOL)retain;
- (void)clearWill;
- (void)publish:(NSString *)payload
toTopic:(NSString *)topic
withQos:(NSUInteger)qos
retain:(BOOL)retain;
- (void)subscribe:(NSString *)topic;
- (void)subscribe:(NSString *)topic withQos:(NSUInteger)qos;
- (void)unsubscribe:(NSString *)topic;
- (void)setMessageRetry:(NSUInteger)seconds;
- (void)setMosq:(struct mosquitto *)m;
- (struct mosquitto *)getMosq;
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 582
|
The Minnesota Professional Educators and Licensing and Standards Board ensures that all higher education institutions are aligning college course curriculum with the Standards of Effective Practice and Content Standards aligned with each licensure area. A teacher education candidate must be held accountable to show verification of completing the standards of effective practice and content standards for the approved teacher licensure program.
Students can find the standard alignment listed on the course syllabus. Questions regarding the standards aligned with the course and standards attainment, consult the course instructor.
Standard 1, subject matter: A teacher must understand the central concepts, tools of inquiry, and structures of the disciplines taught and be able to create learning experiences that make these aspects of subject matter meaningful for students.
Standard 2, student learning: A teacher must understand how students learn and develop and must provide learning opportunities that support a student's intellectual, social, and personal development.
Standard 3, diverse learners: A teacher must understand how students differ in their approaches to learning and create instructional opportunities that are adapted to students with diverse backgrounds and exceptionalities.
Standard 4, instructional strategies: A teacher must understand and use a variety of instructional strategies to encourage student development of critical thinking, problem solving, and performance skills.
Standard 5, learning environment: A teacher must be able to use an understanding of individual and group motivation and behavior to create learning environments that encourage positive social interaction, active engagement in learning, and self-motivation.
Standard 6, communication: A teacher must be able to use knowledge of effective verbal, nonverbal, and media communication techniques to foster active inquiry, collaboration, and supportive interaction in the classroom.
Standard 7, planning instruction: A teacher must be able to plan and manage instruction based upon knowledge of subject matter, students, the community, and curriculum goals.
Standard 8, assessment: A teacher must understand and be able to use formal and informal assessment strategies to evaluate and ensure the continuous intellectual, social, and physical development of the student.
Standard 9, reflection and professional development: A teacher must be a reflective practitioner who continually evaluates the effects of choices and actions on others, including students, parents, and other professionals in the learning community, and who actively seeks out opportunities for professional growth.
Standard 10, collaboration, ethics, and relationships: A teacher must be able to communicate and interact with parents or guardians, families, school colleagues, and the community to support student learning and well-being.
NOTE: to access the standards aligned with a course, download the document into Microsoft Excel. Utilize the filter feature at the top near the course abbreviation and remove the check mark from the "blanks" on the left first column the standards aligned with the course will appear.
Each licensure area also has a set of standards that aligns with the required demonstration of knowledge and skills aligned with that licensure area. Review the list for alignment of your coursework to the Subject Matter (Content) Standards.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 3,635
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{"url":"https:\/\/gmatclub.com\/forum\/which-of-the-following-equations-represents-a-line-on-xy-plane-that-207195.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 15 Feb 2019, 17:17\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n## Events & Promotions\n\n###### Events & Promotions in February\nPrevNext\nSuMoTuWeThFrSa\n272829303112\n3456789\n10111213141516\n17181920212223\n242526272812\nOpen Detailed Calendar\n\u2022 ### Free GMAT practice\n\nFebruary 15, 2019\n\nFebruary 15, 2019\n\n10:00 PM EST\n\n11:00 PM PST\n\nInstead of wasting 3 months solving 5,000+ random GMAT questions, focus on just the 1,500 you need.\n\n# Which of the following equations represents a line on xy-plane that\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nIntern\nJoined: 27 Jan 2015\nPosts: 6\nConcentration: Marketing, Economics\nGMAT Date: 03-26-2015\nWhich of the following equations represents a line on xy-plane that\u00a0 [#permalink]\n\n### Show Tags\n\n15 Oct 2015, 17:21\n3\n9\n00:00\n\nDifficulty:\n\n55% (hard)\n\nQuestion Stats:\n\n57% (01:54) correct 43% (01:44) wrong based on 117 sessions\n\n### HideShow timer Statistics\n\nWhich of the following equations represents a line on xy-plane that includes exactly only one point P(Xp; Yp) with integer coordinates (Xp and Yp are both integer numbers)?\n\na) y= 2x\nb) 2x-2y+1=0\nc) y-5=3x\nd) 2y-\u221a2.x =0\ne) 2y= x+2\nCEO\nJoined: 20 Mar 2014\nPosts: 2629\nConcentration: Finance, Strategy\nSchools: Kellogg '18 (M)\nGMAT 1: 750 Q49 V44\nGPA: 3.7\nWE: Engineering (Aerospace and Defense)\nRe: Which of the following equations represents a line on xy-plane that\u00a0 [#permalink]\n\n### Show Tags\n\n15 Oct 2015, 18:07\nfla162 wrote:\nWhich of the following equations represents a line on xy-plane that includes exactly only one point P(Xp; Yp) with integer coordinates (Xp and Yp are both integer numbers)?\n\na) y= 2x\nb) 2x-2y+1=0\nc) y-5=3x\nd) 2y-\u221a2.x =0\ne) 2y= x+2\n\nEasiest way is to do POE for the options to arrive at the correct answer. Incorrect options will either provide 0 integer coordinate pairs or will provide >1 integer coordinate pairs.\n\na) y= 2x , (0,0) m (1,2) etc. Eliminate\nb) 2x-2y+1=0 --> y=x+0.5 , no integer coordinate pairs. Eliminate\nc) y-5=3x ---> y=3x+5, (0,5), (1,8) etc. Eliminate\nd) 2y-\u221a2.x =0 ---> Only case possible = (0,0). Rest of them will be functions of $$\\sqrt{2}$$. Keep\ne) 2y= x+2 ---> y=x\/2+1 ---> (2,2) , (4,3) etc. Eliminate\n\nThus, D is the correct answer.\n\nHope this helps.\nNon-Human User\nJoined: 09 Sep 2013\nPosts: 9834\nRe: Which of the following equations represents a line on xy-plane that\u00a0 [#permalink]\n\n### Show Tags\n\n24 Oct 2018, 08:37\nHello from the GMAT Club BumpBot!\n\nThanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).\n\nWant to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.\n_________________\nRe: Which of the following equations represents a line on xy-plane that \u00a0 [#permalink] 24 Oct 2018, 08:37\nDisplay posts from previous: Sort by","date":"2019-02-16 01:17:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.5819194912910461, \"perplexity\": 12199.203820240944}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247479729.27\/warc\/CC-MAIN-20190216004609-20190216030609-00370.warc.gz\"}"}
| null | null |
\section{Approach}
\input{tables/notations.tex}
\input{algorithm/training_policy.tex}
\subsection{Motivation}
When human beings begin to learn and develop their intelligence, they (common people) tend to start with learning a broad area of knowledge without digging into the depth. For example, in the early stage of school, people take a variety of courses in different subjects (\emph{e.g.} math, physics and chemistry) at the introduction level. As the learning process proceeds, people will focus on a narrower and much more difficult area (\emph{e.g.} calculus in math). During this process, people start to forget what they have learned before because they do not use this knowledge often (\emph{e.g.} physics, chemistry). However, if people want to catch up materials that they have forgotten, they can do it in a much shorter time and usually, they can even learn better.
The process of this learning method is extremely efficient, as people gradually focus on less and less ``hard'' knowledge. This strategy (often) adopted by humans motivates this work. Specifically, we ask the question that whether deep neural networks can be trained efficiently by imitating the learning method of human beings. The goal of this work is to train a model with as less computation as possible while simultaneously maintaining the performance in classification and without decreasing the generalization ability of the learned parameters in the downstream vision tasks.
We begin by abstracting the human learning process\footnote{Note that it \textbf{does not} necessarily mean human truly learns in this way from neuron-science perspective.}. The broad area of knowledge human learned during their early stage is similar to feeding all training data to the model at the start of the training iterations. As human begin to focus on a more difficult subject, we can imitate this process by gradually feeding only the most difficult examples (examples with large losses) to the model. Since human periodically review what they have learned before, we also periodically feed all the training data again to the model and restart the above process. A detailed training method will be discussed in the following.
\subsection{Drop-and-Refresh Training}
\label{sec:policy}
A high-level explanation of DaR is shown in Figure~\ref{fig:training_policy}.
\subsubsection{Notations} A list of notations is shown in Table~\ref{tab:notations}. We set a number of ``warm-up'' epochs ($\mathcal{W}$) at the beginning of training. In this ``warm-up'' period, the model is guaranteed to see all training examples. This is an imitation of human taking a broad range of courses. Then, we use a combination of number of interval epochs ($\mathcal{I}$) and a keep rate ($p$) to control the length of a learning stage (subsample examples every $\mathcal{I}$ epochs) and the number of remaining examples (keep $p\%$ of the current examples) respectively. Furthermore, a number of active epochs ($\alpha$) is used to control the number of learning stages we have (stop sampling after $\alpha$ epochs). Finally, we set a list of epochs $\mathcal{R}$ to reuse all training examples to prevent forgetting.
\subsubsection{Training Details} Our DaR can be summarized as the process of periodically dropping easy examples and picking up all examples. As shown in Algorithm~\ref{alg:training}, we divide the overall training process into different cycles/periods denoted by $\mathcal{R}$. In each period, we keep $p\%$ of the hardest examples every $\mathcal{I}$ epochs (the ``drop'' stage). To find the $p\%$ of the hardest examples, we first sort examples by their losses in descending order and keep the top-$p\%$ examples. When the training process goes to a new period/cycle, we reuse all training examples (the ``refresh'' stage) followed by another ``drop'' stage. The only exception is the first period where we set a warm-up period $\mathcal{W}$ to use all examples to ``warm-up'' the model. During the first $\mathcal{W}$ epochs, we keep all examples for training. We also set an optional ``active'' epochs $\alpha$ because we do not want to drop too much (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot $90\%$) training data when the period is long. If an $\alpha$ is set, then the ``drop'' stage only lasts for $\alpha$ epochs within each training cycle.
In our experiments, we find that changing these hyperparameters in a reasonable range does not make much difference. Thus we simply set hyperparameters related to DaR for ImageNet classification as follow. Since models are usually trained for $90 \sim 120$ epochs, we set warmup epochs to around $10\%$ if the total training epochs which is $10$ epochs. More over, we set interval=2, percentage=0.9, active=10, reusing all data at $30^{th}, 60^{th}, 90^{th}$ epochs which corresponds to reusing all data when learning rate is dropped. All other hyperparameters for training (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot optimizer, learning rate, weight decay) are kept same as models' original ones.
\section{Introduction}
Human beings are exposed to a world with data at a much larger scale even compared with the largest datasets but they can still learn well efficiently. We observe that when human beings begin to learn and develop their intelligence, they (common people) tend to start with learning a broad area of knowledge without digging into the depth (similar to using all training examples). As the learning process proceeds, people will focus on a narrow and much more difficult area (a ``drop'' in the number of training examples). When people start to forget what they have learned before, however, people can quickly catch up by refreshing all materials in a short time (a ``refresh'' in easy examples). Furthermore, we notice that training samples in human learning process are not \emph{i.i.d}\onedot. Easier samples are refreshed less frequently than harder samples. And the definition of easy and hard samples also change overtime, depending on the order of samples presented to human learner (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot learning physics without math background makes all physics concept ``hard'' samples to learn). Motivated by this observation of human learning process, we investigate an efficient training method named Drop-and-Refresh (DaR) to imitate this process, so that all the training data can be sample in a more efficient way and not necessarily be \emph{i.i.d}\onedot anymore.
\begin{figure*}[ht!]
\centering
\includegraphics[width=1.0\linewidth]{figures/framework_v2.pdf}
\caption{An illustration of Drop-and-Refresh (DaR). First, all the available training examples (orange dots) are adopted for training at the beginning of the training cycle \emph{j}. Second, as the training goes on, we gradually discard training examples (gray) to save training computation. Third, all the discarded samples are refreshed for training when a new cycle \emph{k} is started. We periodically repeat such a drop-and-refresh training manner until convergence. $\tau$ denotes a training epoch and $\mathcal{I}$ denotes interval. \textbf{Numbers above each sample point denotes the loss of that sample at current epoch.} Samples are removed by dropping examples with lowest losses and the discarded samples might be different in different training cycles.}
\label{fig:training_policy}
\end{figure*}
Image classification is one of the most fundamental problem in computer vision. Pre-training on a large-scale image classification dataset has shown to be effectiveness for improving the performance of many downstream visual recognition tasks, \emph{e.g.} object detection \cite{ren2015faster,lin2017feature,he2017mask,lin2017focal}, semantic segmentation \cite{zhao2017pyramid,deeplabv3plus2018}, human pose estimation \cite{xiao2018simple,sun2017integral} \emph{etc.} Although efforts have been made to find more advanced architectures \cite{Shen2017DSOD,shen2017learning} or normalization method \cite{wu2018group} to train these downstream tasks from scratch without pre-training \cite{he2018rethinking}, the gap (either performance~\cite{Shen2017DSOD} or computation~\cite{he2018rethinking}) is still not negligible. The ImageNet \cite{deng2009imagenet} has been used as the standard pre-training dataset for a long time. Many recent works \cite{deeplabv3plus2018,mahajan2018exploring} find that the larger dataset used for pre-training, the larger gain downstream tasks will benefit from. Since people usually follow the \emph{i.i.d}\onedot assumption to uniformly sample training data, the computation is proportional to the size of the dataset. As the larger dataset is considered, the process of pre-training brings new challenges in terms of both of the budget (resources) and time. For example, the pre-trained model in \cite{mahajan2018exploring} is trained on 336 GPUs across 42 machines for 22 days of training. Both of the training time and computational resources are likely not affordable for many research groups. Thus, it would be helpful if one can find a more efficient training procedure to reduce the training time, which makes the pre-training can better serve for other downstream tasks. \bowen{Beyond the complexity of the network structure, the increase in pre-training time is due to the increase in the number of training examples. In general, current dataset construction techniques simply sample images randomly from the web (\emph{e.g.} Flickr, Instagram) with some keywords or hashtags \cite{deng2009imagenet,mahajan2018exploring}. However, the contribution of each example varies significantly during training, which has not been taken into account by current training manner. We ask the question that can we come up with a more efficient training strategy by taking into account the contribution of each example to \textbf{reduce the time of pre-training without affecting the generalization ability of the learned parameters for the downstream visual recognition tasks}?}
The basic idea of DaR is to divide the training process into different cycles. In each cycle, the network first learns from all training examples (this is similar to the stage when human learns a broad knowledge). As the training process goes on, we gradually feed the network with only a subset of the training examples (similar to human learning a specific subject). To prevent the model from forgetting what it learned before, we periodically repeat this training cycle where all training data is used again and restart the process of reducing training data for each training epoch. We perform extensive experiments on a variety of benchmarks (CIFAR10, CIFAR100, ImageNet) for image classification to first verify the efficiency and effectiveness of our proposed method. Specifically, our proposed method only requires 83\%, 92\%, and 86\% training computation to achieve comparable or better results (\emph{w.r.t.} using all training examples) on CIFAR10, CIFAR100, and ImageNet respectively. Furthermore, when using DaR pre-trained ImageNet models to finetune on various downstream tasks, including \textbf{object detection, instance segmentation, scene parsing and human pose estimation}, we find these models achieve better performance than models pre-trained with \emph{i.i.d}\onedot ImageNet training data in all tasks compared.
Our contributions are threefold:
\begin{itemize}
\item
We present an efficient sampling method named Drop-and-Refresh (DaR) to sample a subset of non-\emph{i.i.d}\onedot training data that can save training computation while retaining the performance.
\item
We validate the effectiveness of the DaR strategy on various benchmarks (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot CIFAR10, CIFAR100 and ImageNet) with different backbone settings (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot ResNet, DenseNet and MobileNet). DaR strategy can preserve the accuracy while reducing 15\% training cost.
\item
We further show that DaR pre-trained models have better generalization when finetuning on four different downstream tasks and we give a hypothesis to explain why models trained with non-\emph{i.i.d}\onedot data generalize better.
\end{itemize}
\section{Conclusion}
We present an efficient pre-training strategy by sampling non-\emph{i.i.d}\onedot training data named ``Drop-and-Refresh'' (DaR) aiming at reducing training time given limited computational resource. The method is validate on various image classification benchmarks. Specifically, we achieve comparable performance using only 86\% computation on the ImageNet dataset. We further show that the DaR pre-trained models have no loss in generalization ability in many downstream tasks. More interestingly, we observe consistent improvements for these downstream tasks when DaR pre-trained models are used. We will leave it as a future work to study how downstream tasks can benefit from different pre-training policies.
\section{Related Works}
\subsection{ImageNet Pre-training}
Recently, He \emph{et al}\onedot~\cite{he2018rethinking} find that the performance of training object detection and instance segmentation models from random initialization is no worse than their ImageNet pre-training counterparts on the COCO dataset~\cite{lin2014microsoft} without changing hyperparameters or model structures~\cite{Shen2017DSOD,zhu2018scratchdet,shen2019improving}. However, the conclusion in~\cite{he2018rethinking} only holds under certain circumstances.
The first requirement is that there is enough computational resource. To match the performance of ImageNet pre-training models, training from random initialization needs to be trained for $5\sim6$ times more iterations. Since most of time we fix the backbone network in object detection, it is a waste of computation to train object detection models with the same backbone. Another requirement is that there is enough target data which is hardly satisfied in all domains.
In our work, we find that when sampling non-\emph{i.i.d}\onedot training data in a specific pattern, we can train image classification models more efficiently with less computation and still match the performance of using all training data. Furthermore, we find non-\emph{i.i.d}\onedot pre-trained models generalize better to downstream tasks than pre-trained models using all training data.
\subsection{Fast Training Methods}
Works have been done to explore methods for training a model as fast as possible assuming \textbf{unlimited} computational resource. Large batch size is used in \cite{goyal2017accurate} with a carefully designed learning rate warm-up schedule. \cite{goyal2017accurate} trains a ResNet-50 model in 1 hour with 256 GPUs without loss in accuracy. Mixed precision training is explored in \cite{micikevicius2017mixed} where the forward pass and backward pass are computed in half precision and parameter update in single precision. With the latest tensor core in the Nvidia Volta architecture, mixed precision can speed-up training by nearly 50\%. A combination with faster inter-node communication has been studied in \cite{sergeev2018horovod,jia2018highly}. In particular, \cite{jia2018highly} reports the training time of a ResNet-50 model is 7 minutes with 1024 GPUs without loss in accuracy. Although fast training has been studied, the amount of computational resource is intractable for small research labs or individual researchers. In this paper, we try to speed-up training with \textbf{limited} computational resource (\emph{e.g.} with 4 GPUs available). And our method is in orthogonal with \cite{goyal2017accurate,micikevicius2017mixed,sergeev2018horovod,jia2018highly} and we believe the proposed method can further speed-up training of these methods.
\subsection{Active Learning}
\bowen{Active learning is a method to expand the size of datasets by selecting examples that are most helpful for a specific network to label. The key idea in active learning is to select the most uncertain \textbf{unlabeled examples} to label. In some sense, this idea is relevant to our proposed strategy that we keep the most uncertain \textbf{labeled examples}.}
\bowen Joshi \emph{et al}\onedot {\cite{joshi2009multi} proposes to use an SVM classifier where uncertainties are calculated based on probabilistic outputs over the class label. Osband \emph{et al}\onedot \cite{osband2016deep} shows how to obtain uncertainty measures with neural networks. Lakshminarayanan \emph{et al}\onedot \cite{lakshminarayanan2017simple} uses extra head in neural network that is trained to estimate the variance. It allows variance along with predictions to be used to estimate uncertainty. In this work, we simply use the training loss as an indicator of the uncertainty of an example. The larger the loss, the more uncertain the example is to the network.}
\subsection{Hard Example Mining}
Hard example mining has been widely studied in the case of classical support vector machines~\cite{felzenszwalb2009object,ogawa2014safe} which find small number of hard negative samples from a large pool of negative samples. Recent works use importance sampling scheme to find most ``informative'' examples~\cite{katharopoulos2018not,schaul2015prioritized,loshchilov2015online} to speed-up the convergence of stochastic gradient descent. Online hard example mining (OHEM)~\cite{shrivastava2016training} is also commonly used in training object detection models, however, OHEM does not save any computation cost. Different from previous hard example mining approaches, our work not only focuses on efficient training but also studies the generalization of models trained with hard examples.
\section{Main Results}
In this section, we present main results on multiple common image classification benchmarks with state-of-the-art architectures to demonstrate that our proposed DaR method using non-\emph{i.i.d}\onedot data has the potential to achieve better performance than uniform sampling using \emph{i.i.d}\onedot data. Moreover, we also show that DaR pre-trained models generalize well to downstream tasks.
\input{tables/cifar10_networks.tex}
\input{tables/cifar100_networks.tex}
\input{tables/imagenet_networks.tex}
\input{tables/coco_detection.tex}
\subsection{Implementation Details}
All models in this paper are implemented using PyTorch \cite{paszke2017automatic}. If otherwise stated, models on the CIFAR datasets are trained with 300 epochs, learning rate is divided by 10 at 150 and 225 epoch respectively. On the ImageNet dataset, models are trained with 120 epochs, dividing the learning rate by 10 every 30 epochs. We follow the batchsize, learning rate, weight decay settings in their original implementations.
During the training, we accumulate the loss of each image on the fly \textbf{with data augmentation} at each training iteration. When we sample a subset of training example, we simply sort examples according to their losses in descending order and keep the top $p\%$ examples to form our new training set. Thus, this sampling method is non-\emph{i.i.d}\onedot. When we report the training computation, we set the strategy of using all data as comparison baseline (100\% computation). And we compute the ratio of between total examples used by our proposed method with the baseline as the computation for our method.
\subsection{Non-\emph{i.i.d}\onedot Sampling in Image Classification}
Image classification is the most fundamental task in computer vision and it receives most attention. However, most of the recent efforts mostly focus on either network designs~\cite{szegedy2015going,he2016deep,xie2017aggregated,huang2017densely,sandler2018mobilenetv2,ma2018shufflenetv2} or regularization~\cite{szegedy2016rethinking,zhang2017mixup} during training and uniformly sample \emph{i.i.d}\onedot training data for granted. In this section, we empirically show that our DaR sampling method with non-\emph{i.i.d}\onedot training data has the potential of achieving better performance.
\subsubsection{CIFAR}
CIFAR10 and CIFAR100 datasets \cite{krizhevsky2009cifardataset} both consist of colored natural images with $32\times32$ pixels. CIFAR10 consists of images with 10 classes and CIFAR100 has 100 classes. The training and test sets contain 50,000 and 10,000 images respectively for both datasets. We adopt standard data augmentations (random crop and random flip) and normalize the data with the channel means and standard deviations. We train our models with 50,000 training images and report the test accuracy on the test set.
To validate our proposed method can be generalized to other architectures instead of ResNet-110, we perform experiments on three more state-of-the-art architecutures: ResNet-164 \cite{he2016deep} and DenseNet-BC-100 \cite{huang2017densely} on both CIFAR10 and CIFAR100 datasets with 83\% and 92\% cost respectively. Results are shown in Table~\ref{tab:cifar10_networks} and Tabel~\ref{tab:cifar100_networks}. On all these architectures, our method achieves on-par or even better results on both datasets.
\input{tables/pose.tex}
\subsubsection{ImageNet}
The ImageNet2012 classification dataset \cite{deng2009imagenet} consists 1,281,167 images for training and 50,000 for validation with 1,000 classes. Following \cite{szegedy2015going}, we adopt random crop augmentation with patches covering $8\% \sim 100\%$ of the entire image, aspect ratio augmentation of [$\frac{3}{4}$, $\frac{4}{3}$], and random horizontal flip augmentation. All patches are resized to $224\times224$ and normalize the data with the channel means and standard deviations. We report single-crop classification errors on the validation set for fair comparison.
We apply our method with state-of-the-art architectures: ResNet \cite{he2016deep}, DenseNet \cite{huang2017densely}, and MobileNet \cite{howard2017mobilenets}. Results are shown in Table~\ref{tab:imagenet-results}. We report both our re-implementation results and original results in their papers and our reproduced results are always better than original ones. For a fair comparison, we only compare our method with the reproduced results.
When we use the set of default hyper-parameters we find for ImageNet, the cost of the training is only 86\% of that when all data is used. We observe that models trained with DaR sampling outperforms the uniform sampling counterpart in most of the cases which supports our hypothesis that sampling non-\emph{i.i.d}\onedot training data might have the potential of achieving better performance.
\subsection{Generalization to Downstream Tasks}
Over the years, ImageNet2012 \cite{deng2009imagenet} has become the standard dataset to pre-train models for multiple downstream tasks (\emph{e.g.} object detection \cite{cheng2018revisiting, wei2018ts2c,cheng18decoupled,zhang2016faster,li2018scale}, segmentation \cite{zhao2017pyramid,deeplabv3plus2018,liang2016semantic,zhang2018context,lin2017refinenet}, human pose estimation \cite{xiao2018simple,sun2017integral,newell2016stacked,cao2017realtime,wei2016convolutional}). To demonstrate that our approach can generalize well to these downstream tasks and can even transfer knowledge distilled during pre-training, we evaluate our pre-trained models a wide range of downstream tasks including object detection, instance segmentation, scene parsing and human pose estimation.
\subsubsection{Object Detection and Instance Segmentation}
We evaluate the performance for object detection and instance segmentation on the COCO2017 dataset \cite{lin2014microsoft}. COCO2017 has 115k training images, 5k validation images and 20k test image. We train FPN \cite{lin2017feature} (for object detection) and Mask R-CNN \cite{he2017mask} (for instance segmentation) on the 115k training set and evaluate the performance on the 5k validation set. We use the open-sourced framework mmdetection \cite{mmdetection2018}. We use a single scale of [800, 1333] during training and testing.
Results are shown in Table~\ref{tab:coco_detection}.
For both object detection task and instance segmentation task, models pre-trained with our method (86\% cost) perform slightly better than models pre-trained with all data (100\% cost) in terms of the AP metric. This means knowledge learned in our method can be transfered to other tasks.
\subsubsection{Human Pose Estimation}
We further experiment the performance on human pose estimation task using the COCO2017 keypoint dataset \cite{lin2014microsoft}.
It contains more than 200k images and 250k person instances labeled with keypoints. 150k instances of them are publicly available for training and validation. We only use COCO2017 train (57K images and 150K person instances) and evaluate the performance on the COCO2017 val set. We use the same implementation in \cite{xiao2018simple} and use a single input size of $256 \times 192$ during training and testing.
Results are shown Table~\ref{tab:coco_keypoint}. All reported baseline results are similar or even better than the ones reported in \cite{xiao2018simple}. We observe a consistent improvement in the human pose estimation task.
We further perform experiments with ResNet-101 backbone in human pose estimation. Interestingly, with 86\% pre-train cost, the performance of our DaR pre-trained model exceeds baseline by a large margin of 0.6 $\text{AP}^{\text{kp}}$. It shows that DaR pre-trained models have the potential to further improve human pose estimation.
\input{tables/cityscapes.tex}
\subsubsection{Scene Parsing}
We perform experiments on the Cityscapes dataset \cite{cordts2016cityscapes} using the PSPNet~\cite{zhao2017pyramid}.
It is a large-scale dataset containing high quality pixel-level annotations (fine data) of 5000 images (2975, 500, and 1525 for the training, validation, and test sets respectively) and about 20000 coarsely annotated images (coarse data).
We train PSPNet \cite{zhao2017pyramid} with a publicly available toolbox \cite{torch2018segment} and only using the fine data and evaluate on the validation set. During training, we use a single scale of $769 \times 769$. Results are shown in Table~\ref{tab:cityscapes}. We find DaR pre-trained models also generalize well in the semantic segmentation task.
\begin{figure}%
\centering
\subfigure[Uniform sample.]{%
\includegraphics[width=0.45\linewidth]{figures/tSNE-resnet_v1_110-cifar10-all-data.png}}%
~
\subfigure[DaR sample.]{%
\includegraphics[width=0.45\linewidth]{figures/DaR.png}}%
\caption{
t-SNE visualization of ResNet-110 on CIFAR-10 dataset, each color represents a class. (a) Using uniform sample, training data is \emph{i.i.d}\onedot. (b) Using DaR sample, training data is non-\emph{i.i.d}\onedot.
}
\label{fig:t-sne}
\end{figure}
\section{Discussions}
In summary, our experiments demonstrate that the DaR sampling strategy can achieve comparable (in many cases better) predictive accuracy with less training cost on various datasets (CIFAR 10, CIFAR 100 and ImageNet) and with different backbone architectures (ResNets, DenseNets and MobileNets). More importantly, we find that the ImageNet pre-trained models using our DaR sampling strategy can achieve much better transferability for the downstream tasks including object detection, instance segmentation, scene parsing and human pose estimation consistently. Based on these observations, in this section we try to initiate some discussions which hopefully can motivate people to rethink the connections between the sampling strategy for training and the transferability of its learned features for pre-training ImageNet models.
\textbf{\emph{Why DaR sampling strategy can maintain or improve classification accuracy even when training cost is reduced?}} \textbf{Hard Examples Under-fitting.} Standard cross entropy criterion treats all input examples equally even when they are already correctly classified. As discussed in related work~\cite{lin2017focal}, these easy, correctly classified examples still generate a large proportion of the empirical loss. However, our DaR sampling strategy progressively drop easy examples, which enforce the network to put more focus on hard, mis-classified examples.
\textbf{\emph{Why DaR pre-trained models transfer better?}} \textbf{Task Over-fitting.} Similar to the phenomenon that models over-fit to the training set (models achieve high accuracy on the training set but low accuracy on the never-seen test set), we hypothesis that models also tend to over-fit to the task. By over-fitting to the task, we mean models achieve good generalization on the main tasks they are trained on (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot pre-training on image classification) but fail to generalize well on other tasks (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot transferring to downstream tasks). As an ImageNet pre-trained model, the general goal is to learn a good feature representation that may help downstream perception tasks. However this goal is different than the image classification objective, which is to minimize the empirical cross entropy loss of the training data. That is, models that perform extremely well on pre-trained task and may not transfer well to other tasks. To verify our hypothesis, we visualize the ResNet-110~\cite{he2016deep} features before the final classifier (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot fully connected layer) on the CIFAR-10~\cite{krizhevsky2009cifardataset} test set by projecting the 256-dimension features to 2 dimensions using t-SNE~\cite{maaten2008visualizing} in Figure~\ref{fig:t-sne}. As can be seen from the visualization, training with \emph{i.i.d}\onedot data (Figure~\ref{fig:t-sne} (a)) results in a more separate clustering than training with non-\emph{i.i.d}\onedot data (Figure~\ref{fig:t-sne} (b)). From the accuracy prospective, DaR-learned features are enough to distinguish between different classes but the margin is just not as large as using \emph{i.i.d}\onedot samples. This observation suggests \emph{i.i.d}\onedot training data might cause models to over-fit the task. The same phenomenon is also observed in~\cite{he2018rethinking}. Task over-fitting can well explain why training from scratch outperforms its pre-trained counterparts (over-fitted on image classification) given enough training iterations.
\textbf{\emph{Is DaR the same as Hard Example Mining?}} \textbf{No.}
To validate that the DaR is not a simplified hard example mining approach, we compare our DaR with a hard example re-weighting solution. Specifically, based on the loss values of all samples obtained in the previous training epoch, we re-weight the importance of each sample in the next epoch, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot larger loss values lead to larger training weights. The 5 run results on CIFAR10 with ResNet-110 are 92.09/92.45/92.22/91.52/92.10 (mean: 92.076, std: 0.307), which are worse than that of DaR. We consider that using the ``picking" operation to refresh the knowledge embedded in all training samples is an important step to learn a reliable network.
\input{tables/image_lt}
\textbf{\emph{Is DaR helpful to real applications?}} \textbf{Yes.}
To validate that DaR is also helpful to train model on non-\emph{i.i.d}\onedot data, we further perform experiments in long-tail image classification where the data is naturally non-\emph{i.i.d}\onedot. We train a ResNet-10~\cite{he2016deep} model on the ImageNet-LT~\cite{liu2019large} dataset. The result is shown in Table~\ref{tab:imagenet_lt}, using DaR improves the overall accuracy by $1.4\%$, which suggests it is helpful to real applications.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Researchers extend side-channel attacks to the GPU
Tags: #ajaya-neupane #cuda #flaw #gpu #graphics-card #graphics-processing-unit #graphics-processor #hoda-naghibijouybari #insecurity #meltdown #nael-abu-ghazaleh #security #side-channel #side-channel-attack #spectre #vulnerability #zhiyun-qian
Companies: #amd #nvidia #university-of-california-riverside
Researchers from the University of California, Riverside have revealed a new attack based on the side-channel concept that has been plaguing CPU manufacturers for the past year, in which a system's graphics processing unit (GPU) can be used to steal privileged data up to and including passwords.
Detailed in the paper 'Rendered Insecure: GPU Side Channel Attacks are Practical', the work by Hoda Naghibijouybari, Zhiyun Qian, Ajaya Neupane, and Nael Abu-Ghazaleh builds on vulnerabilities like PortSmash, Foreshadow, SpectreRSB, TLBleed, Lazy FP, and the better-known Meltdown and Spectre vulnerabilities which exploit flaws in the design of modern processors to obtain or infer the contents of supposedly-protected memory. Where previous attacks have focused solely on the CPU, with Nvidia going so far as to specifically state its products were not affected by Spectre, the team's work extends the concept to GPUs - and with, unfortunately, considerable success.
'We demonstrate a series of end-to-end GPU side channel attacks covering the different threat scenarios on both graphics and computational stacks, as well as across them,' the team explain of their paper. 'The first attack implements website fingerprinting through GPU memory utilisation API or GPU performance counters. We extend this attack to track user activities as they interact with a website or type characters on a keyboard. We can accurately track re-rendering events on GPU and measure the timing of keystrokes as they type characters in a textbox (e.g., a password box), making it possible to carry out keystroke timing analysis to infer the characters being typed by the user.
'A second attack uses a CUDA spy to infer the internal structure of a neural network application from the Rodinia benchmark, demonstrating that these attacks are also dangerous on the cloud. We believe that this class of attacks represents a substantial new threat targeting sensitive GPU-accelerated computational (e.g. deep neural networks) and graphics (e.g. web browsers) workloads. Our attacks demonstrate that side channel vulnerabilities are not restricted to the CPU. Any shared component within a system can leak information as contention arises between applications that share a resource. Given the wide-spread use of GPUs, we believe that they are an especially important component to secure.'
The attacks, though, come with the promise of potential mitigations, including limiting the rate of access to the application programming interfaces (APIs) responsible to reduce the precision of the leaked information from which private data can be inferred, a technique which the researchers found to 'substantially reduce the effectiveness of the attack, to the point where the attacks are no longer effective.'
The team has confirmed that the attacks are possible on Nvidia graphics cards, while the non-CUDA attacks are also possible against AMD graphics cards. Both companies have been informed of the flaw, and are believed to be working on patches. A pre-print copy of the paper, meanwhile, is available now (PDF warning).
Google hit by apparent BGP misrouting attack
Traffic shuffled to Russia, China.
Researchers warn of severe SSD security vulnerabilities
Crucial, Samsung drives named.
CPUs hit by PortSmash side-channel vulnerability
SMT again proving a security nightmare.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}\label{sec:Introduction}
Covariate-adaptive randomization is used in clinical trials to balance
treatment arms with respect to key covariates.
Many covariate-adaptive randomization methods have been proposed.
Stratified randomization aims to reduce
the imbalance of treatment assignments within strata formed by covariates.
A restricted randomization method, such as permuted block design
\citep{Zelen1974} or Efron's biased coin design \citep{Efron1971}, is
implemented within each stratum.
Pocock and Simon's minimization \citep{Taves1974, Pocock1975} was proposed to reduce the imbalance at a marginal level of covariates.
This approach was extended by Hu and Hu \cite{Hu2012} to simultaneously reduce various levels of imbalances.
As covariate-adaptive randomization is usually based on discrete covariates
\citep{Taves2010, Lin2015, Ciolino2019}, this paper focuses mainly on randomization
methods that balance discrete covariates.
Some covariate-adaptive randomization methods have also been
proposed to balance continuous covariates \citep{Frane1998,
Stigsby2010, Su2011, Ma2013, Zhao2015}.
For more discussion of handling covariates in clinical
trials, see McEntegart \cite{McEntegart2003}, Rosenberger and Sverdlov \cite{Rosenberger2008}, Hu et al. \cite{Hu2014} and the
references therein.
Although covariate-adaptive randomization is widely used in clinical trials, concerns have been expressed over the validity of statistical inference after
these randomization procedures.
In practice, conventional methods, such as two sample \textit{t}-test or
generalized linear models, are commonly used without consideration of the covariate-adaptive randomization scheme.
Regulatory guidelines \citep{ICH1998,EMA2015} recommend that all the
covariates used in covariate-adaptive randomization be adjusted in the
working model.
In fact, it was theoretically demonstrated that such tests are valid
provided that the working model is correctly specified and that all of the covariates used in the covariate-adaptive randomization are included \cite{Shao2010}.
{\color{black} However, unadjusted tests are prevalent in practice \citep{Sverdlov2015}. It was reported that only 24\% to 34\% of randomized clinical trials adjusted for covariates in their main analyses, and that, even for the trials that used at least one stratification factor in the randomization, only 26\% of these adjusted for all covariates in the primary analysis \citep{Kahan2014}.}
Several practical reasons are given not to use the full model.
It is difficult to incorporate some covariates into the working model; for example, investigation sites are usually omitted from the analysis for a multicenter clinical trial.
The simplicity of a testing procedure is another reason to include fewer covariates.
Also, the adjustment of too many covariates usually requires a complicated model that lacks robustness to model misspecification.
Further discussion can be found in Shao et al. \cite{Shao2010}, Shao and Yu \cite{Shao2013}, and Ma et al. \cite{ Ma2015}.
It is therefore desirable to study the inference properties of conventional tests when some or all of the covariates used in
covariate-adaptive randomization are excluded from the working model.
The validity of statistical inference under covariate-adaptive
randomization is well studied in the linear model framework.
Most notably, the unadjusted \textit{t}-test has been theoretically demonstrated to be conservative
for a variety of covariate-adaptive designs, including the stratified permuted block
design and Pocock and Simon's minimization method \citep{Shao2010, Ma2015, Bugni2018, Ma2019, Ma2020Regression}.
{\color{black} Extensions to multiple treatments have also been studied \cite{Bugni2019, Ma2020Sinica}. Moreover, unadjusted testing was also found to be conservative in longitudinal analysis \cite{Weng2017}, survival analysis \cite{Ye2020JRSSB}, and quantile regression \cite{Zhang2020}. }
The theoretical foundation for inference under covariate-adaptive randomization
is less comprehensive in the generalized linear model framework.
Shao and Yu \cite{Shao2013} showed that a model-free $t$-test with no covariate
is conservative under a specific type of stratified randomization.
However, the inference properties are not studied for testing based on
generalized linear models, which are more popular in practice.
Moreover, their results do not apply to other commonly used
covariate-adaptive randomization methods, such as Pocock and Simon's
minimization, or some newly proposed designs, such as that of Hu and Hu \cite{Hu2012}.
Gail \cite{Gail1988} studied model-based unadjusted tests for treatment effect under
perfectly balanced studies, but such perfect
balance is not guaranteed under covariate-adaptive randomization; thus, the
application of this study's conclusions is also limited.
{\color{black} Fan et al. \cite{Fan2018} considered the use of generalized linear models under covariate-adaptive randomization when covariates are subject to misclassification. However, their results are mainly based on simulations and lack theoretical justifications.}
The purpose of this study is to examine the theoretical properties of the unadjusted test for treatment effect under generalized linear models for a large family of
covariate-adaptive randomization.
We establish the asymptotic properties of such tests for general
covariate-adaptive randomization, including both stratified randomization and
Pocock and Simon's minimization method, among other commonly used methods.
We derive the conditions under which the test is conservative, valid, or
anti-conservative and propose an adjustment method to achieve a valid size.
Several important generalized linear models, such as logistic regression and
Poisson regression, are discussed in detail.
In addition, simulations show that the adjusted tests under covariate-adaptive
randomization are more powerful than those under complete randomization.
The results provide insights and guidance for testing the treatment effect when using generalized linear models under covariate-adaptive randomization.
This paper is organized as follows.
In Section \ref{sec:TestingGLM}, we first describe the framework for studying the unadjusted
Wald test for treatment effect under covariate-adaptive
randomization, and then give the asymptotic properties of the test when the canonical
link is used in the generalized linear model.
In Section \ref{sec:TestingCommonModels}, we apply the derived results to
several generalized linear models commonly used in clinical trials and
discuss their testing properties.
In Section \ref{sec:TestingGLM-NC}, we establish general asymptotic results by
extending the results in Section \ref{sec:TestingGLM} to cases of
non-canonical link.
An adjustment method to achieve valid testing is proposed in Section
\ref{sec:Adjustment}.
Section \ref{sec:Simulation} presents simulation results to examine
the finite sample performance of the proposed theory.
The last section concludes with some remarks and directions for future work.
The proofs and additional simulation results are given in Appendix.
\section{Wald Test under Generalized Linear Models with
Canonical Link}\label{sec:TestingGLM}
\subsection{Framework and Notations}
We consider a clinical trial with two treatment groups, 1 and 2.
A covariate-adaptive randomization procedure is implemented to allocate
patients based on their covariate profiles.
Let $X$ be the vector of the covariates and $T$ be the treatment assignment
indicator, where $T=1$ if a patient is assigned to treatment 1, and $T=0$
otherwise.
Suppose that $Y$ is the response variable and the conditional expectation of $Y$
given $X$ and $T$ is
\begin{align}\label{eq:true_model}
\mathbb{E}[Y|T,X]=h(\mu+\delta T+\beta X)=h(\eta),
\end{align}
where $\delta$ is the treatment effect, $\beta$ is the vector of
unknown parameters for the covariates, and $\eta=\mu+\delta
T+\beta X$ is the linear predictor.
The covariates are assumed to be independent and identically distributed for
each patient.
Without loss of generality, it is also assumed that $\mathbb{E}[X]=0$.
Suppose that conditional on $\eta$, the distribution of $Y$ belongs to an
exponential family and the likelihood of $Y$ is given by
\begin{align}
\exp\left\{\frac{Y\theta-b(\theta)}{\phi}+c(Y,\phi)\right\},
\end{align}
where $\theta$ depends on $\eta$, and $\phi$ is the dispersion parameter, which is assumed to be known.
Thus, the relationship between $Y$ and $(T, X)$ can be modeled by
generalized linear models \citep{McCullagh1989}.
We first consider the scenario where the canonical link function is
used, i.e., $\theta=\eta$.
This covers several common generalized linear models, including
linear regression with $h(\eta)=\eta$, logistic regression with
$h(\eta)=\exp(\eta)[1+\exp(\eta)]^{-1}$, and Poisson regression with
$h(\eta)=\exp(\eta)$.
The cases with non-canonical link functions are studied in Section
\ref{sec:TestingGLM-NC}.
\subsection{Working Model and Wald Test}
Suppose that the covariates used in covariate-adaptive randomization are omitted
from post-randomization data analysis; then the following working model
is used:
\begin{align}\label{eq:working_model}
\mathbb{E}[Y|T]=h(\mu+\delta T)=h(\eta),
\end{align}
where the link function and conditional distribution of $Y$ given $\eta$
are the same as in the true model \eqref{eq:true_model}.
\begin{remark}
It is important to note that model \eqref{eq:working_model} is a mis-specified working model without adjustment of any covariates used in randomization.
The properties of the test for treatment effect under such a model mis-specification have been extensively studied for linear models both numerically and theoretically in the literature, including Birkett \cite{Birkett1985}, Forsythe \cite{Forsythe1987}, Shao et al. \cite{Shao2010}, and Ma et al. \cite{Ma2015}.
As for generalized linear models, Gail \cite{Gail1988} studied the score test under model \eqref{eq:working_model} in perfectly balanced trials.
However, the results are not applicable to general covariate-adaptive randomization, because perfect balance is usually not achieved.
\end{remark}
To test the treatment effect based on the working model
\eqref{eq:working_model}, we consider the Wald test, because it is commonly used in practice and is easily accessible in most statistical packages.
The Wald test can be formed as follows:
\begin{align}\label{eq:test}
H_0: \delta=0 \text{ versus } H_1: \delta \neq 0,
\end{align}
with the test statistic
\begin{align}\label{eq:test_stat}
S=\frac{\hat{\delta}}{\widehat{\textrm{se}}(\hat\delta)},
\end{align}
where $\hat\delta$ is the maximum likelihood estimate (MLE) of $\delta$,
and $\widehat{\textrm{se}}(\hat\delta)$ is the model-based estimate of the standard
error of $\hat{\delta}$.
To perform a two-sided test, the null hypothesis is rejected if
$|S|>Z_{1-\alpha/2}$, where $Z_{1-\alpha/2}$ is the $({1-\alpha/2})$th quantile of
a standard normal distribution.
Suppose that $n$ patients in total are enrolled in the covariate-adaptive randomized
trial.
Let $(y_i, T_i, x_i)$ be the observed value of $(Y, T, X)$ for the $i$-th
patient.
The working model then has the following log-likelihood
\begin{align}\label{eq:log-likelihood}
\ell(\mu, \delta)=\sum_{i=1}^{n}\left\{
\frac{y_i\theta_i-b(\theta_i)}{\phi}+c(y_i,\phi)\right\},
\end{align}
where $\theta_i=\eta_i=\mu+\delta T_i$ under the canonical link.
Also, it follows from the properties of the exponential family that
$b^{'}(\theta_i)=h(\eta_i)$.
By setting the first derivative of the log-likelihood, with respect to
${\delta}$ and ${\mu}$, to be zero, the MLE estimates of $\mu$ and $\delta$
are
\begin{align}\label{eq:muhat}
\hat{\mu}=h^{-1}\left[\frac{\sum_{i=1}^{n} (1-T_i)y_i}{n_0}\right],
\end{align}
and
\begin{align}\label{eq:deltahat}
\hat{\delta}=h^{-1}\left[\frac{\sum_{i=1}^{n} T_iy_i}{n_1}\right]-
h^{-1}\left[\frac{\sum_{i=1}^{n} (1-T_i)y_i}{n_0}\right],
\end{align}
where $n_1=\sum_{i=1}^{n} T_i$, and $n_0=n-n_1$.
\subsection{Main Results}\label{sec:TestingGLM-results}
Before giving the main results, we describe how
to measure various levels of covariate balance under a covariate-adaptive
design, which is shown to be closely related to the test properties.
{\color{black} Note that in this paper, we assume that the covariates $X$ are discrete variables.}
Consider $p$ covariates included in $X$ and $m_k$ levels for the $k$th
covariate,
resulting in $m=\prod_{i=1}^{p}m_k$ strata in total.
Denote the covariate profile of the $i$th patient $(t_1, \ldots, t_p)$ if
the $k$th covariate is at level $t_k$, $1\le k\le p$ and $1\le t_k \le m_k$.
For convenience, we use $(t_1, \ldots, t_p)$ to denote the stratum formed by
patients who possess the same covariate profile $(t_1, \ldots, t_p)$, use
$(k; t_k)$ to denote the margin formed by patients whose $k$th covariate is at
level $t_k$.
Then let
\begin{description}
\item[-] $D_n$ be the difference between the numbers of patients in
treatment groups 1 and 2 as total, i.e., the number in group 1 minus the
number in group 2;
\item[-] $D_n(k;t_k)$ be the differences between the
numbers of patients in the two treatment groups on the margin $(k;t_k)$;
\item[-] $D_n(t_1, t_2,\ldots,t_p)$ be the difference
between the numbers of patients in the two treatment groups within the
stratum $(t_1, t_2,\ldots,t_p)$.
\end{description}
\begin{remark}
Stratified randomization usually has strong balance properties.
For example, stratified permuted block design and stratified biased
coin design, which use the permuted block design and Efron's biased coin
design within each stratum, have within-stratum imbalances that are
bounded in probability.
As marginal imbalance and overall imbalance can be considered as sums of
certain within-stratum imbalances, the marginal imbalance and overall
imbalance are also bounded in probability for these two designs.
In addition, Pocock and Simon's minimization can achieve
marginal and overall imbalances that are bounded in probability
\citep{Ma2015}.
In addition, Hu and Zhang \cite{Hu2020} proved that the
within-stratum imbalances increase with the order of $\sqrt{n}$.
\end{remark}
Now we first present the theoretical results for the test for treatment
effect if within-stratum imbalances are bounded in probability under a
covariate-adaptive design.
\begin{theorem}\label{thm:TestWithinStratumBounded}
Suppose that a covariate-adaptive design satisfies the condition that all
within-stratum imbalances are bounded in probability; then under the null
hypothesis $H_0: \delta=0$, we have, as $n \rightarrow \infty$,
\begin{align}
S \stackrel{D}{\rightarrow} N \left(0, \frac{\mathbb{E}[\textup{Var}(Y|X)]}{\phi h^{'}h^{-1}(\mathbb{E}[Y])}\right).
\end{align}
\end{theorem}
Based on Theorem \ref{thm:TestWithinStratumBounded}, if the within-stratum
imbalances are bounded in probability, the size of the test for treatment effect depends on a comparison between $\mathbb{E}[\textup{Var}(Y|X)]$ and $\phi
h^{'}h^{-1}(\mathbb{E}[Y])$, i.e., in the asymptotic sense,
\begin{enumerate}
\item[(i)] the test is conservative if
$\mathbb{E}[\textup{Var}(Y|X)]<\phi h^{'}h^{-1}(\mathbb{E}[Y])$,
\item[(ii)]the test is valid if
$\mathbb{E}[\textup{Var}(Y|X)]=\phi h^{'}h^{-1}(\mathbb{E}[Y])$,
\item[(iii)]the test is anti-conservative if
$\mathbb{E}[\textup{Var}(Y|X)]>\phi h^{'}h^{-1}(\mathbb{E}[Y])$.
\end{enumerate}
Several commonly used generalized linear models are discussed in detail in Section 3.
\begin{remark}
The condition that within-stratum imbalances are bounded in probability is satisfied by many covariate-adaptive randomization methods based on
stratification, such as stratified permuted block design.
However, it is important to note that the results for Theorem \ref{thm:TestWithinStratumBounded} have applications beyond stratification.
In particular, Hu and Hu \cite{Hu2012} proposed a new family of covariate-adaptive methods that can simultaneously reduce imbalances of all levels (overall, marginal, and within-stratum).
The within-stratum imbalances are bounded in probability, so the results given in Theorem \ref{thm:TestWithinStratumBounded} can also be applied to their designs.
\end{remark}
Although the condition in Theorem \ref{thm:TestWithinStratumBounded} is
satisfied by many stratified randomization methods, it does not hold for some other covariate-adaptive designs, such as Pocock and Simon's minimization.
We next study the test for treatment effect under a broader class of
covariate-adaptive designs.
For this, we first state two conditions below.
\bigskip
\noindent\textbf{\underline{Condition} (A)} The overall imbalance is bounded in probability, that is,
$D_n=O_P(1)$.
\noindent\textbf{\underline{Condition} (B)} Under the null hypothesis, $H_0: \delta=0$, as $n
\rightarrow \infty$,
\begin{align}
\frac{\sum_{i=1}^{n}{\left(2T_i-1\right)h(\mu+\beta x_i)}}{\sqrt{n}}
\stackrel{D}{\rightarrow} N(0, \sigma_h^2).
\end{align}
These conditions are mild and satisfied by many covariate-adaptive
randomization methods that balance discrete covariates.
For covariate-adaptive designs with within-stratum imbalances that are bounded
in probability, the distribution in Condition (B) reduces to $O_P(1/\sqrt{n})$,
and hence $\sigma_h^2=0$, by
noting that $h(\mu+\beta x_i)$ only takes a finite number of values that
correspond to each stratum.
Furthermore, the conditions hold for Pocock and Simon's minimization
according to Hu and Zhang \cite{Hu2020}, who proved that within-stratum
imbalances, scaled by $n^{-1/2}$, are asymptotically jointly normal, with positive variances.
We now present the general results of the test for treatment effect if these
conditions are satisfied under a covariate-adaptive design.
\begin{theorem}\label{thm:TestGeneral}
Suppose that a covariate-adaptive design satisfies Conditions (A)-(B); then
under the null
hypothesis $H_0: \delta=0$, we have, as $n \rightarrow \infty$,
\begin{align}
S \stackrel{D}{\rightarrow} N \left(0, \frac{\mathbb{E}[\textup{Var}(Y|X)]+\sigma_h^2}{\phi
h^{'}h^{-1}(\mathbb{E}[Y])}\right).
\end{align}
\end{theorem}
It is clear to see that the result of Theorem
\ref{thm:TestWithinStratumBounded} is a special case of Theorem
\ref{thm:TestGeneral} with $\sigma_h^2=0$.
Compared with stratified randomization, an extra layer of
variation induced by within-stratum imbalances must be considered
when studying the size of the test for treatment effect under general
covariate-adaptive designs, such as Pocock and Simon's minimization.
As in the aforementioned discussion, the numerator and denominator of the
asymptotic variance in Theorem \ref{thm:TestGeneral} must be compared to
determine whether the test is asymptotically conservative, valid, or
anti-conservative.
\begin{remark}
Under Pocock and Simon's minimization, the within-stratum imbalances increase at the rate of $O_P(\sqrt{n})$, leading to $\sigma_h^2>0$, so the asymptotic variance of test statistic \eqref{eq:test_stat} under minimization is larger than that under stratified randomization, which results in a larger size compared with stratified randomization.
\end{remark}
\begin{remark}
Condition (A) is not satisfied by complete randomization.
However, the independence of treatment assignment and covariates shows that the test statistic $S$ also has a normal distribution under the null hypothesis,
\begin{align}\label{eq:cr}
S \stackrel{D}{\rightarrow} N \left(0, \frac{\textup{Var}[Y]}{\phi
h^{'}h^{-1}(\mathbb{E}[Y])}\right).
\end{align}
As $\textup{Var}[Y]=\mathbb{E}[\textup{Var}(Y|X)]+\textup{Var}[\mathbb{E}(Y|X)]$, the asymptotic variance of $S$ under complete randomization is generally greater than that under stratified randomization.
\end{remark}
\section{Test Size under Some Commonly Used
Models}\label{sec:TestingCommonModels}
In this section, we study the impact of using working model
\eqref{eq:working_model} on the size of the test for treatment effect based on
Theorems \ref{thm:TestWithinStratumBounded} and \ref{thm:TestGeneral}.
Several commonly used models with canonical link functions are covered.
\subsection{Logistic Regression}\label{sec:TestingCommonModels-logistic}
Logistic regression is extensively used in clinical trials with binary
responses.
Under logistic regression, the conditional distribution of $Y$ given $\eta$ is
a Bernoulli distribution ($\phi=1$), and the canonical link takes the form of
$h(\eta)=\exp(\eta)[1+\exp(\eta)]^{-1}$.
Also, it is easy to verify that
\begin{align*}
\phi h'h^{-1}(\mathbb{E}[Y])=\textup{Var}(Y).
\end{align*}
Therefore, provided that the condition in Theorem
\ref{thm:TestWithinStratumBounded} is satisfied, i.e., the within-stratum
imbalances are bounded in probability, under $H_0: \delta=0$,
\begin{align*}
S \stackrel{D}{\rightarrow} N \left(0, \frac{\mathbb{E}[\textup{Var}(Y|X)]}{\textup{Var}(Y)}\right).
\end{align*}
Because $\mathbb{E}[\textup{Var}(Y|X)]$ is smaller than ${\textup{Var}(Y)}$ in general, the asymptotic
distribution of $S$ shows a greater concentration around 0 than the standard normal
distribution, resulting in a conservative test whose test size is smaller than the
nominal level.
The conclusion holds for the stratified permuted block design and the class of
covariate-adaptive designs proposed by Hu and Hu \cite{Hu2012}.
The size under minimization is larger than that under stratified randomization
because of a positive $\sigma_h^2$, and the simulation studies in Section
\ref{sec:Simulation} show that the test under minimization is also
conservative.
Under complete randomization, the variance of the asymptotic distribution of
$S$ is equal to 1 according to \eqref{eq:cr}, leading to a valid test.
\subsection{Poisson Regression}\label{sec:TestingCommonModels-poisson}
Poisson regression can be used to model event counts whose conditional
distribution given $\eta$ is a Poisson distribution ($\phi=1$).
Under the canonical link $h(\eta)=\exp(\eta)$ and $H_0: \delta=0$, we have
\begin{align*}
\phi h'h^{-1}(\mathbb{E}[Y])=\mathbb{E}[Y]=\mathbb{E}[\textup{Var}(Y|X)],
\end{align*}
as $\mathbb{E}[Y|X]=\textup{Var}[Y|X]$.
By Theorem \ref{thm:TestWithinStratumBounded} it is easy to see that the test for
treatment effect is valid if within-stratum
imbalances are bounded in probability.
In contrast, both minimization and complete randomization have inflated Type I
errors, resulting in anti-conservative tests.
\subsection{Linear Regression}
Under linear regression, a response is a continuous variable whose
conditional distribution given $\eta$ is a normal distribution with mean $\eta$
and variance $\phi=\sigma^2$.
The canonical link is the identity function $h(\eta)=\eta$.
Assuming that $\phi=\sigma^2$ is known, we have,
\begin{align*}
\phi h'h^{-1}(\mathbb{E}[Y])=\sigma^2=\mathbb{E}[\textup{Var}(Y|X)],
\end{align*}
so the test is valid under stratified randomization.
However, the test is anti-conservative for minimization by Theorem
\ref{thm:TestGeneral}.
In practice, $\phi=\sigma^2$ is usually unknown and must be estimated based
on the data.
Such hypothesis testing in the linear model framework has been theoretically
studied in the literature \citep{Shao2010, Ma2015}.
It is proved that if influential randomization covariates are omitted from the
working model, the test for treatment effect is conservative for a large class
of covariate-adaptive designs, including both stratified randomization and
minimization.
\subsection{Exponential Model}\label{sec:TestingCommonModels-exponential}
In the exponential model, the response variable $Y$ given $\eta$ is
exponentially distributed ($\phi=1$).
Under the canonical link $h(\eta)=-1/\eta$ and $H_0: \delta=0$, we have
\begin{align*}
\phi h'h^{-1}(\mathbb{E}[Y])=(\mathbb{E}[Y])^2=(\mathbb{E}\{\mathbb{E}[Y|X]\})^2.
\end{align*}
Notice that $\mathbb{E}[\textup{Var}(Y|X)]=\mathbb{E}\{[\mathbb{E}(Y|X)]^2\}$; then by the Cauchy-Schwarz
inequality,
\begin{align*}
\phi h'h^{-1}(\mathbb{E}[Y])\le\mathbb{E}[\textup{Var}(Y|X)],
\end{align*}
so the test for treatment effect is typically anti-conservative for stratified
randomization with inflated Type I errors by Theorem
\ref{thm:TestWithinStratumBounded}.
Furthermore, the extent of inflated Type I errors is even more severe under
minimization and complete randomization by Theorems \ref{thm:TestGeneral} and
\eqref{eq:cr}.
\section{General Results with Non-Canonical Link}
\label{sec:TestingGLM-NC}
In this section, the results in Section
\ref{sec:TestingGLM} obtained under canonical link functions are extended to cases of non-canonical link functions.
We first note that the MLE estimates of $\mu$ and
$\delta$ based on the working model \eqref{eq:working_model} with a
non-canonical link take the same form as those with a canonical link.
Let $\theta=\gamma(\eta)$, which is not necessarily the identity function under
general link functions.
Gail et al. \cite{Gail1984} showed that if $\gamma'(\hat{\mu})$ and
$\gamma'(\hat{\mu}+\hat{\delta})$ are not degenerate,
the MLE estimates $\hat{\mu}$ and $\hat{\delta}$ under a non-canonical link
are the same as \eqref{eq:muhat} and \eqref{eq:deltahat}.
\begin{remark}
In fact, the estimates $\hat{\mu}$ and $\hat{\delta}$ under a canonical link can
be considered as the estimates by the method of moments.
It was shown that the MLE estimates under the working model
\eqref{eq:working_model} are equal to the moment estimates if the
non-degenerate conditions are satisfied \citep{Gail1984} .
\end{remark}
Next, we state two general theorems on the test for treatment effect when a
non-canonical link function is used. These theorems correspond to Theorems
\ref{thm:TestWithinStratumBounded} and \ref{thm:TestGeneral}, respectively.
\begin{theorem}\label{thm:TestWithinStratumBounded-NC}
Suppose that a covariate-adaptive design satisfies the condition that all
within-stratum imbalances are bounded in probability; then under the null
hypothesis $H_0: \delta=0$, we have, as $n \rightarrow \infty$,
\begin{align}
S \stackrel{D}{\rightarrow} N \left(0, \frac{\mathbb{E}[\textup{Var}(Y|X)]}{\phi
h^{'}h^{-1}(\mathbb{E}[Y])/\gamma^{'}h^{-1}(\mathbb{E}[Y])}\right).
\end{align}
\end{theorem}
\begin{theorem}\label{thm:TestGeneral-NC}
Suppose that a covariate-adaptive design satisfies Conditions (A)-(B) {\color{black} listed above Theorem \ref{thm:TestGeneral} in Section \ref{sec:TestingGLM-results}}; then
under the null
hypothesis $H_0: \delta=0$, we have, as $n \rightarrow \infty$,
\begin{align}
S \stackrel{D}{\rightarrow} N \left(0, \frac{\mathbb{E}[\textup{Var}(Y|X)]+\sigma_h^2}{\phi
h^{'}h^{-1}(\mathbb{E}[Y])/\gamma^{'}h^{-1}(\mathbb{E}[Y])}\right).
\end{align}
\end{theorem}
It is clear that the two theorems above reduce to Theorems
\ref{thm:TestWithinStratumBounded} and \ref{thm:TestGeneral} by setting
$\gamma(\eta)=\eta$.
Based on the fact that $\mathbb{E}[\textup{Var}(Y|X)]=\mathbb{E}[\phi h^{'}(\mu+\beta
X)/\gamma^{'}(\mu+\beta
X)]$ and $\mathbb{E}[Y]=\mathbb{E}[h(\mu+\beta X)]$, one can use these general theorems
to evaluate the test size for various generalized linear models under
covariate-adaptive randomization with an argument similar to that in Section
\ref{sec:TestingCommonModels}.
\section{Adjusted Test}
\label{sec:Adjustment}
As discussed in Sections \ref{sec:TestingGLM} to \ref{sec:TestingGLM-NC}, the
Wald test under the working model \eqref{eq:working_model} is invalid, either
conservative or anti-conservative, for many generalized linear models.
In this section, we show how to make an adjustment to achieve a valid test
based on the theoretical results obtained.
{\color{black} The purpose of the proposed adjusted test is twofold. First, it provides a remedy approach when using generalized linear models that do not adjust for the randomization covariates. However, we note that a simpler and more practical alternative is to adjust for the covariates within the analysis. Second, the adjusted test's validity, later confirmed by simulation studies, provides an added layer of justification for the theoretical results.}
We first consider the scenario in which within-stratum imbalances are bounded in
probability, such as stratified permuted block design and the class of
covariate-adaptive designs proposed by Hu and Hu \cite{Hu2012}.
For simplicity of notation, let $\sigma_\nu^2=\mathbb{E}[\textup{Var}(Y|X)]$.
{\color{black} We note that in the proof of Theorem \ref{thm:TestGeneral-NC} in the Appendix, it is shown that, }
under model \eqref{eq:working_model} and
$H_0: \delta=0$, we have
\begin{align}\label{eq:dist_adj_str}
\frac{h^{'}h^{-1}(\mathbb{E}[Y])\hat{\delta}}{2\sqrt{\sigma_\nu^2/n}}
\stackrel{D}{\rightarrow} N(0,1).
\end{align}
An adjusted test statistic can then be constructed by replacing the
population parameters by their sample estimates,
\begin{align}\label{eq:test_stat_adj_str}
S_{adj}=\frac{h^{'}h^{-1}(\bar Y_n)\hat{\delta}}{2\sqrt{\hat\sigma_\nu^2/n}},
\end{align}
where $\bar Y_n=\sum_{i=1}^{n}y_i/n$ is the overall sample mean.
$\hat\sigma_\nu^2=\sum_{i=1}^{n}\sum_{j=1}^{m}
(y_i-\hat\mu_j)^2I\{x_i=s_j\}/(n-m)$ is the sample estimate of $\sigma_\nu^2$,
where $s_j$ represents the $j$th stratum formed by covariates, and
$\hat\mu_j$ is the sample mean of $Y_i$ in the $j$th stratum.
Thus $S_{adj}$ follows a standard normal distribution asymptotically under the
null hypothesis, leading to a valid test.
Furthermore, for a general covariate-adaptive randomization that satisfies
Conditions (A) and (B), a conclusion similar to \eqref{eq:dist_adj_str} can
also be established based on the proof {\color{black} of Theorem \ref{thm:TestGeneral-NC}} in the Appendix, i.e., under model
\eqref{eq:working_model} and $H_0: \delta=0$,
\begin{align}\label{eq:dist_adj_general}
\frac{h^{'}h^{-1}(\mathbb{E}[Y])\hat{\delta}}{2\sqrt{(\sigma_\nu^2+\sigma_h^2)/n}}
\stackrel{D}{\rightarrow} N(0,1).
\end{align}
Similarly to \eqref{eq:test_stat_adj_str}, a valid test can be constructed with
an adjusted test statistic
\begin{align}\label{eq:test_stat_adj_general}
S_{adj}^{*}=\frac{h^{'}h^{-1}(\bar
Y_n)\hat{\delta}}{2\sqrt{(\hat\sigma_\nu^2+\hat\sigma_h^2)/n}},
\end{align}
as long as $\hat\sigma_h^2$ is a consistent estimator of $\sigma_h^2$.
Compared with \eqref{eq:test_stat_adj_str}, the only additional quantity we need
to estimate is $\sigma_h^2$.
Note that $
\sum_{i=1}^{n}{\left(2T_i-1\right)h(\mu+\beta x_i)}
=\sum_{j=1}^{m}D_n(s_j)h(\mu+\beta s_j)
$
is a weighted sum of within-stratum imbalances $D_n(s_j)$.
If the joint distribution of within-stratum imbalances of
$\{D_n(s_j),j=1,\ldots,m\}$ is known, we can estimate $\sigma_h^2$ by
using the joint distribution and replacing $h(\mu+\beta s_j)$ with its sample
estimate $\hat\mu_j$.
Another option is to evaluate the joint within-stratum imbalances of
$\{D_n(s_j),j=1,\ldots,m\}$ numerically. We provide further details on the minimization in Section \ref{sec:Simulation-Size-Adj}.
\begin{remark}
A valid test can also be achieved for complete randomization, with a
similar test statistic defined as in \eqref{eq:test_stat_adj_str} and
\eqref{eq:test_stat_adj_general},
\begin{align}\label{eq:dist_adj_cr}
S_{adj, CR}=\frac{h^{'}h^{-1}(\bar
Y_n)\hat{\delta}}{2\sqrt{\hat\sigma_Y^2/n}},
\end{align}
where $\hat\sigma_Y^2$ is the overall sample variance.
\end{remark}
After adjustment, we can obtain more powerful hypothesis testing results.
A simulation study is carried out in the next section to evaluate the
size and power of these adjusted tests under various randomization methods
and generalized linear models.
\section{Numerical Studies}\label{sec:Simulation}
\subsection{Size of Wald Test}\label{sec:Simulation-Size}
We consider three commonly used generalized linear models: logistic
regression, Poisson regression, and exponential model.
The asymptotic properties of the Wald test for treatment effect under model
\eqref{eq:working_model} are discussed in Section \ref{sec:TestingGLM}.
We now perform simulations to evaluate the size under a finite sample.
Assume that the true model is
\begin{align}\label{eq:20}
\mathbb{E}(y_i|T_i,x_i)=h(\mu+\delta T_i+\beta_1 x_{i,1}+\beta_2 x_{i,2})=h(\eta_i),
\end{align}
where $\delta$ represents the treatment effect and
$x_{i,j}$ are independent and identically distributed random variables that
follow the Bernoulli distribution with $P=0.5$.
The parameters $(\mu,\beta_1,\beta_2)=(-1,2,4)$ are assumed for
logistic regression, and
$(\mu,\beta_1,\beta_2)=(0.2,0.5,1)$ for both Poisson regression and
exponential model.
Canonical link functions are used for these models.
Note that the inverse function, instead of the negative inverse function, is used
for the exponential model in the simulation, so the range of $\mathbb{E}(y_i|T_i,x_i)$
is permitted.
{\color{black} The error distributions (i.e., the conditional distributions of $y_i$ given $\eta_i$) for these three generalized linear models are respectively specified in Sections \ref{sec:TestingCommonModels-logistic}, \ref{sec:TestingCommonModels-poisson}, and \ref{sec:TestingCommonModels-exponential}.
For each of the three generalized linear models under consideration, we simulate datasets of two sample sizes, $N=200$ and $500$, and four different randomization methods, which are complete randomization, stratified permuted block design, Pocock and Simon's minimization, and the method proposed by Hu and Hu. In each scenario, we first generate the patients' covariates $x_{i,1}$ and $ x_{i,2}$, $i=1,\ldots,N$, according to the Bernoulli distributions as specified above. Then we use one of the randomization methods to obtain the treatment assignments $T_i$. In simulations, a block size of 4 is used in stratified permuted block design, and a biased coin probability of 0.75 is used in Pocock and Simon's minimization and the method proposed by Hu and Hu. Finally, the treatment responses $y_i$ are simulated according to the true model \eqref{eq:20} and the corresponding error distributions with parameters as described above. }
The following working model is used to test $H_0: \delta=0$,
\begin{align}
\mathbb{E}(y_i|T_i)=h(\mu +\delta T_i),
\end{align}
where no covariates are included in the model.
The significance level for the two-sided tests is $\alpha = 0.05$.
{\color{black} Each scenario is simulated 5,000 times.}
\begin{center} [Tables 1-3 here.] \end{center}
The simulated sizes are presented in Tables 1-3 under the column ``Wald
Test'' and are consistent with the
theoretical results.
In particular, the test for treatment effect based on logistic regression is
generally conservative under covariate-adaptive randomization if covariates are
omitted from the working model, whereas the test is still valid under complete
randomization.
For Poisson regression, the omission of covariates leads to a valid test under the
stratified permuted block design and under Hu and Hu's method.
However, the inflation of Type I error is slight under minimization and more
severe under complete randomization.
The tests are all anti-conservative for the exponential model if no covariates
are included.
In all scenarios, the stratified permuted block design and Hu and Hu's method perform comparably in terms of test size, whereas the test size for
minimization lies between those of these two designs and complete randomization.
{\color{black} Finally, we note that the simulation results under complete randomization are expected (see Section \ref{sec:TestingCommonModels}) and concur with the findings in the literature \cite{Gail1988b, Fan2018}.}
\subsection{Size of Adjusted Test}
\label{sec:Simulation-Size-Adj}
In this section, adjustment is performed to achieve a valid test for treatment
effect, as discussed in Section \ref{sec:Adjustment}.
Using the same setup as in the previous section, the adjustment follows the
procedure in Section \ref{sec:Adjustment}.
For minimization, Monte Carlo simulation is used to generate $B(=500)$ samples
of joint within-stratum imbalances of $\{D_n(s_j),j=1,\ldots,m\}$ via the
repetitive application of minimization on the covariates; then
$\hat\sigma_h^2$ can be obtained by calculating the variance of
$\sum_{j=1}^{m}D_n(s_j)\hat\mu_j$ and can be used to construct the test
statistic \eqref{eq:test_stat_adj_general}.
The sizes for the adjusted tests are presented in Tables 1-3 under the column
``Adjusted Test.''
Based on the results, the adjustment works quite well because the
sizes are successfully controlled at 5\% for all three generalized linear
models under different randomization methods.
\subsection{Power of Adjusted Test}
\label{sec:Simulation-Power-Adj}
Based on the adjusted tests, we also perform simulations to evaluate the power
of these tests under various generalized linear models and randomization
methods.
The same setup is used as in the previous sections, except that $\delta$ is
assumed to be a sequence of non-zero numbers to assess power.
In particular, $\delta$ is assumed to be $0, 0.2, 0.4, \ldots, 2$ for logistic
regression and $0, 0.04, 0.08, \ldots, 0.4$ for Poisson regression and the
exponential model, so that power ranges from 5\% to more than
90\%. The powers are presented in Figure 1.
It can be seen that the adjusted test has the highest power under the
stratified permuted block design and Hu and Hu's method for all three
generalized linear models, which is followed by minimization.
The adjusted test under complete randomization is the least powerful.
Like size, the power of minimization lies between the
stratified permuted block design and complete randomization.
\begin{center} [Figure 1 here.] \end{center}
\begin{comment}
content...
\subsection{A Large Number of Strata}\label{sec:Simulation-MultiVar}
All of the simulation results thus far are based on models with 2$\times$2 strata.
In this section, we consider the test for treatment effect with a large number of
strata present.
For this, we simulate four discrete covariates, with 2, 2, 3, and 4 levels,
respectively, resulting in 48 strata.
The true model is assumed as follows
\begin{align}
\mathbb{E}(y_i|T_i,x_i)=h(\mu+\delta T_i+\beta_1 x_{i,1}+\beta_2 x_{i,2}
+\beta_3^t x_{i,3}+\beta_4^t x_{i,4}),
\end{align}
where $x_{i,1}$ and $x_{i,2}$ are binary covariates with the probability of 0.5
to take 0 or 1, $x_{i,3}^t$ is a discrete covariate with three equally possible
values coded as $(0,0)$, $(1,0)$, and $(0,1)$, and $x_{i,4}^t$ is a discrete
covariate with four equally possible values coded as $(0,0,0)$, $(1,0,0)$,
$(0,1,0)$ and $(0,0,1)$.
The covariates are assumed to be independent of each other within each patient and between patients.
We still consider the three types of generalized linear models in Section
\ref{sec:Simulation-Size}.
The model parameters are $\mu=-1$, $\beta_1=1$,
$\beta_2=-1$, $\beta_3^t=(-2,1)$ and $\beta_4^t=(1,2,3)$ for logistic
regression, and $\mu=0.8$, $\beta_1=0.25$, $\beta_2=-0.25$,
$\beta_3^t=(-0.5,0.25)$, and $\beta_4^t=(0.25,0.5,0.75)$ for Poisson
regression and the exponential model.
\begin{center} [Tables 4-6 here.] \end{center}
In Tables 4-6, the simulation results of the Wald test show patterns similar to
those in Tables 1-3 with 2$\times$2 strata.
Under the three types of covariate-adaptive randomization, the Type I errors of
Wald test are conservative for logistic regression, near or slightly larger
than the nominal level for Poisson regression, and too anti-conservative for
the exponential model.
After adjustment, the Type I errors of the Wald test are closer to the nominal
level.
As the adjustment is based on the asymptotic results, the adjustment effects
are less satisfactory when $N=200$, especially for the exponential model.
As the sample size increases, the asymptotic properties begin to set in and the
adjusted Type I errors are near 5\% when $N=$ 500 or 1000.
The power results are presented in Figure 2, from which it is clear that
the adjusted tests are more powerful under covariate-adaptive designs than under complete randomization.
\begin{center} [Figure 2 here.] \end{center}
\end{comment}
\section{Conclusions}\label{sec:Conclusion}
In this study, we examine the test for treatment effect based on
generalized linear models under a large family of covariate-adaptive
randomization.
We derive the asymptotic distribution of the unadjusted Wald test statistic under the null
hypothesis when no covariates are incorporated into the working model,
and present the conditions under which the test is valid or not.
For the invalid cases, we further propose an adjustment to maintain the test
validity with increased power.
{\color{black} Although we have focused on hypothesis testing, the findings have profound implications for treatment effect estimation, which is often an important target in randomized clinical trials. In particular, our results suggest that the usual Wald confidence intervals may be improperly specified if the covariates used in covariate-adaptive randomization are omitted from generalized linear models. It would be interesting to investigate the estimation issues, such as bias and constructing valid confidence intervals, for generalized linear models under covariate-adaptive randomization.}
It is assumed that discrete covariates are used in the randomization procedure.
However, it is common to collect continuous covariates in clinical trials.
{\color{black} Continuous covariates are typically discretized in order to be included in the randomization scheme. Alternatively, the methods that directly balance continuous covariates are useful and are gaining popularity \citep{Frane1998, Stigsby2010, Su2011, Ma2013, Zhao2015}.}
For inference in the linear model framework, discretization was considered in
Shao et al. \cite{Shao2010} and Ma et al. \cite{Ma2015}, whereas adaptive randomization with continuous covariates was studied in Qin et al. \cite{Ma2017} and Li et al. \cite{Li2019}.
It is desirable to extend the framework proposed in this paper to study
the test for treatment effect under generalized linear models when continuous
covariates are considered.
In addition to the treatment effect, it is of interest to test whether a covariate is
influential on patient outcome.
In this paper, we focus mainly on the test for treatment effect without the use of any covariates in the working model.
The property of testing covariate effects remains unknown if partial covariate information is used in the working model.
It was shown that testing covariates is valid under certain conditions if a
linear model is used \citep{Ma2015}, but the properties under generalized
linear models are not clear.
Finally, it might also be possible to consider the scenario in which the covariates are subject to misclassification \citep{Fan2018, Wang2020}.
These topics are left for future research.
{\color{black}
\begin{acks}
We are grateful to the referees for their many helpful comments.
\end{acks}
}
\begin{funding}
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China [grant number 20XNA023].
\end{funding}
\begin{dci}
The authors declare that there is no conflict of interest.
\end{dci}
\bibliographystyle{SageV}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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using System;
using System.Collections.Generic;
using System.ComponentModel.Composition;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Controls;
using KSP4VS.Deploy.Core;
namespace KSP4VS.Deploy.Targets.GitHub
{
public class GitHubDeployUI : IDeployTargetUI
{
[ImportingConstructor]
public GitHubDeployUI([ImportMany] IEnumerable<IDeployTarget> deployTargets)
{
DeployTarget = deployTargets.Single(target => target.TargetName == Name);
ProjectSettingsUI = new GitHubProject { DataContext = DeployTarget };
}
public IDeployTarget DeployTarget { get; }
public string Name => nameof(GitHub);
public System.Windows.Controls.Control UserSettingsUI
{
get
{
throw new NotImplementedException();
}
}
public System.Windows.Controls.Control ProjectSettingsUI { get; }
}
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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{"url":"https:\/\/motls.blogspot.co.id\/2005\/02\/","text":"## Sunday, February 27, 2005 ... \/\/\/\/\/\n\n### Wick rotation\n\nIn the discussion under the article The entropic principle about the recent paper by Ooguri, Vafa, and Verlinde, there has been a significant opposition of many participants against the concept of the Wick rotation - one invented by a renowned and virtually unknown physicist Gian-Carlo Wick. They were saying that this mathematical method can't be trusted; they were comparing the use of the Wick rotation to the idea that physical theories should not be tested experimentally. Because I believe that most of this criticism is unfair and the Wick rotation is a useful, and in many cases essential mathematical tool to calculate the physical predictions of a quantum theory, let me dedicate a special article to this issue. Peter Woit added his comments about the Wick rotation, too.\n\nFirst of all, a summary\n\nThe Wick rotation is a calculational trick in quantum theory in which we assume that the energy or the time are pure imaginary. We do the calculations given these assumptions, which are often more well-defined, and then analytically continue the results back the usual real values of time and\/or energy. It works. But let's now look at the situation a little bit more closely.\n\nBehavior of path integrals\n\nAccording to Feynman's approach to quantum mechanics, the probability amplitudes may be calculated as the sum (well, a path integral) over all conceivable classical histories of the physical system. Each of them is weighted by\n\n\u2022 exp (i.S\/hbar)\n\nwhere \"S\" is the classical action calculated for this history. As you can see, the absolute value of this weight is always equal to one as long as \"S\" is real. From a naive viewpoint, that does not seem to be a good starting point for a convergent integral; the integral keeps on oscillating. Convergence is improved if we add a small negative real part to the exponent. Write the action as\n\n\u2022 S = int dt L\n\nand imagine that \"dt\" has a small imaginary part. You obtain the weight\n\n\u2022 exp (i.(int dt (1+i.epsilon)).L\/hbar).\n\nBecause of the term proportional to \"-epsilon\" in the exponent (i.e. because of the factor \"exp(-epsilon.S)\", roughly speaking, the contribution of the configurations with a large action will be exponentially damped, and the convergence will improve. This regularization is applied both to ordinary quantum mechanics as well as quantum field theory. In the latter case, it's the origin of the \"i.epsilon\" prescriptions for the propagators etc. While the naive Feynman's prescription is obviously reproduced for \"epsilon\" going to zero, a tiny nonzero value of \"epsilon\" is essential for making the path integral convergent.\n\nThe Wick rotation\n\nThis was not the Wick rotation yet, but I hope that the inevitability of this \"epsilon\" treatment is obvious to everyone: the simple prescription of Feynman is a heuristic inspiration, and the oscillating path integral must be regulated in some innocent way. The \"i.epsilon\" prescription is the way that preserves all symmetries. Not a big deal. Now let's look at the real Wick rotation.\n\n### Up and Down: Czech Oscar candidate\n\nThe Czech director Mr. Jan H\u0159ebejk has produced a new movie in 2004, namely Horem p\u00e1dem which is translated as Up and Down (quite a boring translation). It's a rather frustrating comedy about the contemporary Czech society. When we describe it as a comedy, be sure that only black humor is being served. You may also look at the review in the New York Times. A random generator has decided that this movie would be shown as the #1 among all 49 nominated movies.\n\nThere are two main storylines in the movie available as a DVD above that eventually join seamlessly:\n\nFranta (Ji\u0159\u00ed Mach\u00e1\u010dek) and his wife M\u00edla (Nata\u0161a Burger) who is psychologically unstable and dreams about a kid are not allowed to adopt a child because Franta is a former soccer hooligan. This couple lives in a crowded apartment in Prague and they eventually buy a child that two smugglers found in their truck. Franta's friend who is a skinhead does not like the child's dark skin. The movie describes the life of racists and various anti-immigration feelings in a realistic fashion.\n\n## Saturday, February 26, 2005 ... \/\/\/\/\/\n\n### Sultans of string\n\nArun has pointed out an interesting article about the string theorists at the Tata Institute in India:\n\nThe article says a lot of interesting things about the interaction between the Indian culture and theoretical physics, and about many heroes of string theory from India and friends of ours.\n\nIndian string theory has the best outcome per dollar in the world ;-) which is one of the arguments by Andy Strominger that have - during a dinner in the Society of Fellows that I attended - convinced Lawrence Summers, the famous president of Harvard University, to find some extra funding for our Indian colleagues.\n\n## Thursday, February 24, 2005 ... \/\/\/\/\/\n\n### The entropic principle\n\nThe anthropic principle has a new competitor.\n\nIt's called the \"entropic principle\" even though Hirosi Ooguri, Cumrun Vafa, and Erik Verlinde finally did not use this entertaining term for the title. Let me first state a popular version of the principle for those who don't expect to follow the details of this article:\n\u2022 The probability that the cosmological evolution will end up as a Universe with a particular shape of the hidden dimensions (and particular values of the fluxes) is determined by the (exponentiated) entropy of a corresponding black hole whose geometry flows via the attractor mechanism to the given shape of the Universe near the horizon. Note that this contrasts sharply with the \"anthropic principle\" - which itself is not a principle, rather a lack of principles. In the anthropic principle, the corresponding probabilistic weight is determined by the ability of the Universe to support intelligent life.\nTopological string theory allows one to calculate the partition function which is a function of the complex structure moduli of a Calabi-Yau three-fold. Except that it's not quite a function; because of things such as the holomorphic anomaly, it is a \"wave function\". What do I mean? You may think that it is a function that only depends on the holomorphic moduli. Naively, you would expect that the function is holomorphic - it depends on X but not Xbar.\n\nHowever, an infinitesimal change of the reference point in the moduli space induces the so-called holomorphic anomaly which a slight, exactly understood dependence on Xbar that can be locally visualized as an infinitesimal Fourier transform. By an infinitesimal Fourier transform, I mean the conversion of a wave function \"psi(x)\" to \"psi'(x')\" where \"x' = x+epsilon.p\" - you see that we are mixing \"x,p,\" the coordinates on the phase space. Therefore it is more appropriate to talk about the partition sum as a vector in a Hilbert space rather than a well-defined function.\n\n### Bush in Slovakia\n\nThe most powerful man in the world is just visiting my broader homeland - and it's a good opportunity to promote Slovakia.\n\nSlovakia used to be a part of Czechoslovakia between 1918 and 1992. Czechoslovakia is the country in the middle of Europe; you should not confuse Slovakia with Slovenia which is a country in the former Yugoslavia and because \"Yugo\" means \"Southern\", you may guess that Slovenia is more to the South. You may think that I am assuming that the readers are idiots. However, USA Today published an article that claimed that the Bratislava Castle is located near the beaches of the Adriatic sea. :-)\n\nNot only USA Today are ignorant about the Central European geography. On the pages of the White House you may find a photograph of Bush talking to the Czech president Klaus. The caption says that Bush is chatting with the Slovak president Ivan Ga\u0161parovi\u010d. :-) Sorry, I've already sent a mail to George and he has fixed the error on the web page.\n\n## Wednesday, February 23, 2005 ... \/\/\/\/\/\n\n### The whole Ginsparg on your hard disk\n\nPaul Ginsparg, the founder of arXiv.org, used to be one of the staunchest warriors against the robots. One of Paul's secret weapons was the automated \"seek-and-destroy\" procedure against your site that many stupid robots and visitors of my blog carelessly clicked at. Please don't click at this sentence otherwise your domain will be disconnected from the arXiv's!\n\nHe believed that the internet was entering a new era - an era of websites such as arXiv.org that contain a huge amount of stuff and can't be mirrored. Consequently, he disliked the search engines because they were attempting to mirror the whole archive of Ginsparg which was clearly impossible: it would take an infinite amount of affine time. However, a search engine called\n\nsuddenly showed that it was not only possible, but in fact very easy to reproduce the whole archive. Later, Google has also introduced a version of an interdisciplinary SPIRES with a full-text search capabilities\n\nWhile Google was the first company to defeat Ginsparg's no-go theorem, the second defeat has been even more spectacular. She's called\n\n(guess who took the picture on the website above) and she's, together with Alanis Morissette and a few more friends, a Canadian girl who rocked the world. What is her answer to Google.com?\n\nIt sounds great, does not it? How many of you have been sitting in the aircraft with your laptop, before you suddenly got a great idea, but you needed a formula that you knew could only be found in hep-th\/0301173 - but where can you get this paper 6 miles above the middle of the Atlantic?\n\n### An amicable faculty meeting\n\nThe Harvard's FAS (Faculty of Arts and Sciences) faculty meeting today - a continuation of the meeting from last Tuesday - may be characterized as a mostly civil exchange of opinions between president Lawrence Summers and the faculty. In other words, the last meeting (that took place between 4 PM and 6 PM) was on the edge of becoming just another boring faculty meeting. Among the 30 speakers or so, there has been essentially one speaker only who called for Summers's resignation. It was our condensed matter colleague DF. The only point of his talk that most participants understood was the bitterness, and DF had to witness possibly the most diluted applause in the history of the FAS faculty meetings.\n\nAnother speaker, a woman at the very end of the session, discussed the issue of innate differences - something that many people incorrectly expected to be the main focus of the discussions. The risk that Summers could be pressured to resign because of the opinion at the FAS has mostly disappeared.\n\nExcept for the two talks I mentioned, most other talks were dedicated to the questions about Summers's powerful leadership, its advantages, its disadvantages, the balance between the style and the factual content (a music professor at the very beginning asserted that everything was about the style, much like in music - an opinion that most participants did not share), the amount of mutual trust between the president and the faculty, and the separation of power between the president and his fellow professors.\n\nMany of the speeches have been pretty nice pieces of literature - well, let me admit that Summers's speech at the beginning would probably remain #1 in my list - and many speakers recalled kind of entertaining stories about the first moments when they joined Harvard University decades ago. For example, one professor remembered that he joined Harvard in 1978 when the Red Sox used to be losers and the faculty meetings used to be boring. Another speaker (or perhaps the same one?) asked whether we wanted a president who was unmemorable or Mr. Platitude - the kind of people who are found in abundance among the university leaders. ;-)\n\nOther speakers were comparing Summers and this whole story to various episodes from the history textbooks, or they were comparing the aptitudes required from a CEO vs. a university president. Nevertheless, most of the physicists who attended the meeting - including Nima, Lisa, and others - think that the concentration of the non-trivial content in the speeches was rather low. Nima argues that these faculty meetings should be run by scientists in order to increase their efficiency - which may sound as a good idea except for the fact that the physics faculty meetings are usually incredibly boring. ;-)\n\nAlthough there have been roughly two more radical speeches only (and maybe one is a better word), one can still say in most of the cases whether the speakers supported the president or not - and the ratio was about 50:50, I would say. This ratio - one that reflects the opinions at Harvard according to a poll organized by The Crimson - is no accident. Dean Kirby, who was leading the discussion, has composed the initial list of speakers in such a way that it was balanced.\n\nOne of the speakers has been known to the participants of the last week's faculty meeting - Mrs. Theda Sko\u010dpol (which means \"Jumpfield\" in Czech, and it is a shortened version of one of the funniest Czech surnames \"Sko\u010ddopole\" - \"JumpToTheField\"). Although she has already been elected (the word \"elected\" is probably not the right one) to one of the new pro-women committees, she retained her critical attitude toward President Summers. I did not learn much from her talk, and I remain skeptical about such \"ad hoc\" committees.\n\nAfter 5 PM, someone proposed another committee that would mediate communication between president Summers and the faculty - including Profs. Knowless, Sko\u010dpol, and Verba. This proposal has almost been approved, but because Philip J. Fisher from the English department suddenly complained that such a result of the meeting seemed pre-determined and henceforth undemocratic (and the speaker also complained that there was one scientist, two social scientists, and no representatives of the humanities in the committee), the proposal was eventually cancelled by its proponent himself. I personally have no idea how would we use such a committee to comunicate with the president.\n\nDavid Laibson, professor of economics, originally planned to read the text of the letter he composed with Claudia Goldin - a letter that was endorsed by 186 full professors at Harvard - but the atmosphere was already so peaceful that David Laibson said a couple of rather neutral statements that I've already forgotten.\n\nThe meeting was held in the Lowell Lecture Hall and it was attended by 400-500 professors. Note that the capacity of the lecture hall is just 352 people, and therefore many of us had to stand most of the time. The usual attendance at the faculty meeting is about 100 people. The lecture hall was surrounded by journalists with hundreds of microphones and cameras and they were trying to provoke the participants and get some sharp statements from them.\n\nBefore the meeting, a group of radical students was screaming slogans such as \"We vote NO!\", \"Summers: racist sexist anti-gay\" and they were drumming. I appreciate their happiness and fresh, independent, and original ideas and emotions - but this respect can't stop me from encouraging their parents to spank these young colleagues of ours a little bit more often. Every five-year old kid knows that it's wrong to scream that the president is either a gay or anti-gay (the difference being a matter of convention by the CPT theorem). ;-) As soon as I replied to these kids \"We vote YES!\", roughly 10 camcorders started to shoot me. At that moment I decided to turn as silent as possible; on the other hand, Cumrun Vafa has made an interview after the faculty meeting.\n\nSome professors have included the confidence vote for the next, March 15th faculty meeting. And yes, I am confident that it is more accurate to call it a \"confidence vote\" rather than \"no-confidence vote\".\n\n## Monday, February 21, 2005 ... \/\/\/\/\/\n\n### The horrible CERN girls (LHC)\n\nDid you know what the LHC stands for?\n\nwhich means \"the horrible girls from CERN\". This high-energy rock band\n\nis famous because these girls appear on the first image on the web. The music (and lyrics) is pretty cool, too! For example, try Liquid Nitrogen. Thanks to Patrick Veverka (\"Veverka\" means \"squirrel\" in Czech) for this information.\n\n## Sunday, February 20, 2005 ... \/\/\/\/\/\n\n### Students for Larry\n\nUpdate: the previous text about the Summers controversy now contains\n\nWhat do the Harvard students think about the recent controversy? The only website that answers this question is\n\nMeanwhile, the petition by Profs. Claudia Goldin and David Laibson\n\nhas collected more than 160 signatures of the full professors who support the president. While president Summers enjoys the support of a majority of the scientists, it's less clear in the humanities and soft sciences.\n\nTo be balanced, let me tell you that the negative people have also organized a petition that even wants to declare no-confidence. It's not obvious whether it has already been endorsed by some scholars. But the newspaper articles show that some people are very bitter, and therefore I expect that even this petition against Summers will find some support.\n\nA group of negative students plans to organize an anti-Summers demonstration on Tuesday before the FAS faculty meeting at 4 p.m.\n\n## Thursday, February 17, 2005 ... \/\/\/\/\/\n\n### Mark Jackson & cosmic strings\n\nMark Jackson just gave us a very nice talk about cosmic strings. He used his Apple laptop with a PowerPoint presentation. The talk explained most of the issues about the types of cosmic strings and cosmic superstrings; their stability; the bounds for their density and tension; the probability of intercommuting pairs of F-strings, D-strings, (p,q)-strings, and all combinations of these objects; the gravitational waves emitted from the cusps and their detection by LIGO and LISA; and finally the candidates for cosmic strings, namely CSL-1A\/CSL-1B and the 417-days-delayed double image.\n\nThe previous blog text about the cosmic strings was here.\n\nThe talk was not only a very nice review of all the stuff that we've discussed previously, but it also presented a shocking surprise: Mark has identified the coordinates of CSL-1 - the Cosmic String Lensing candidate. So far the coordinates have been largely secret.\n\nImagine that you're in charge of a telescope that is comparable to the Hubble telescope, or you have just launched your personal space shuttle which carries a probe, telescope, or something like that - and you want to see whether there is a discontinuity near the object CSL-1.\n\nHow do you find the coordinates?\n\nYou open the PDF version of the paper by Alcala et al.\n\nand you search for \"double early type\". As a consistency check, the redshift should be z=0.463. Try to contact all your astro-friends and give them these coordinates of CSL-1. Well, to simplify your work, the coordinates are:\n\u2022 OACDF\n\u2022 right ascension: 12 hours, 23 minutes 30.6 seconds (note that this angle is measured in hours, 1 hour = 15 degrees)\n\u2022 declination: -12 degrees 38 minutes 57 seconds (southern celestial hemisphere, close to the equator)\n\u2022 double early type, z=0.463, S\/N 12\nMoreover, I've made another fun observation. The object listed just above this CSL-1 candidate is described as an \"edge on disk\" - well, this could be exactly the discontinuity by which the cosmic string cuts a disk in halves. Its coordinates are\n\u2022 12h 23min 29.7sec, -12 deg 38' 27'', z=0.223 (?)\n\n### Unifying the Harvard community\n\nUpdate: if you're a professor at Harvard, you should consider to sign a letter by Claudia Goldin and David Laibson whose content is similar to the second part of this article - the PDF file with the letter is at\n\nNot only because Harvard and Princeton are usually the top two ranked universities, many people are obviously interested in the developments surrounding the world's most famous university president, namely Lawrence Summers.\n\nOn Tuesday, professors Grosz, Hammonds, Skocpol, and maybe others have asked Lawrence Summers to publish the full text of his remarks at the conference in January to \"clean the air\". Although it initially looked as a bad idea to many of us, president Summers has now released\nand informed his colleagues about these issues; he has also issued new apologies. As far as I see right now, these speeches are even more brilliant than I previously thought. And the air is almost certainly clear right now. You can see that many statements have been exaggerated. It's my private opinion, but I believe that restricting Summers's freedom of speech - and his bright ideas - is a highly counterproductive suggestion.\n\n### Physics seminar videos\n\nHow many videos from physical seminars do you think are available on-line? The answer is \"many\". See the page created by Serkan Cabi\n\nIf you think that this is an impressive list, you could also try to visit Cabi's glasses i.e. his blog at MIT:\n\nwhere physics - and even string theory - plays an important role.\n\n## Wednesday, February 16, 2005 ... \/\/\/\/\/\n\n### Kyoto takes effect (update)\n\nThe Kyoto protocol comes of age. Finally. How much will it - and its hypothetical future extensions - cost? Let's talk about the catastrophic scenario in which people won't abandon this weird international treaty and they will really try to stabilize the CO2 concentrations in the atmosphere.\n\nReuters published estimates that it would cost 18 quadrillions (18 times 10 to the 15th power):\n\nIf you don't know what a quadrillion (U.S.) means, it is roughly speaking an infinite number, and you may object to such an unrealistic result and label the news from Reuters as a kind of typo. What about the estimates from the real people behind the Kyoto protocol?\n\n## Tuesday, February 15, 2005 ... \/\/\/\/\/\n\n### Kenneth Lane on SSC and LHC\n\nKenneth Lane from Boston University just gave an interesting talk about a topic that you can view as messy and boring: the imperfections of hadron colliders. He described various types of background:\n\n\u2022 real background\n\u2022 faked background\n\u2022 detector-induced background\nThere are a lot of effects caused by boring, well-known physics that can fake a signal for new physics. You have to reduce these fakes to at least the level of the real signal - if there's lots of real signal - or far below it if the real signal is rare (i.e., produces only a few events).\n\nHe showed the structure of the detectors - such as the huge detector GEM - designed for the Superconducting SuperCollider (SSC) that has been stopped 10+ years ago. That was sad. We could have had a lot of data already. One of the clear lessons from the SSC is that the particle physicists should have voted for the Republicans at that time :-) - although the current correlation between the parties and the physics policies is less clear and might be, in fact, reverted.\n\n## Monday, February 14, 2005 ... \/\/\/\/\/\n\n### Frank Wilczek about Penrose's new book\n\nFrank Wilczek did not allow his first Nobel prize to reduce his activity, and one of the many things he recently did was to read the new book The Road to Reality by Roger Penrose, even though it has about 1100 pages. Wilczek wrote an excellent review of this book for a recent issue of Science. Unfortunately, you need a subscription to access the article, or you need to buy the paper version.\n\nAlthough I have not read the whole book, it does not seem necessary to determine that I completely agree with 's comments. First of all, is a highly original thinker. Among his discoveries, we find\n\n\u2022 Various methods and solutions of GR related to dynamics of black holes (which includes his method to gain energy from the Kerr black hole)\n\u2022 The Penrose (BMN) limit of geometries, a kind of pp-waves\n\u2022 The Penrose causal diagrams\n\u2022 The esoteric Insect formalism for GR: tensors are bugs and indices are their legs\n\u2022 The Penrose tilings and quasicrystals\n\u2022 The twistors (1967)\n\u2022 Spin networks that he invented decades before they became fashionable in loop quantum gravity which was another framework that people were proposing as an approach to quantum gravity\nIf I paraphrase him, Wilczek argues that Penrose's book should rather be called Fifty Sidewalks Around Reality. It is a physics-oriented book, but otherwise another eclectic interdisciplinary work with many layers, different ideas, and viewpoints. Wilczek looks at the book from three different perspectives. Penrose is most successful from the viewpoint of a teenager who is interested in math and physics: the book will make such readers excited about the complex numbers, relativity, and spinors. (Although it is unlikely that they will learn what a line bundle on the twistor space is.)\n\n### Libertarian girl was a hoax\n\nThe frequent visitors of my blog have probably noticed that one of the \"social\" links points to the blog of Libertarian Girl:\n\nShe started with libertariangirl.blogspot.com and then she moved her blog to libertariangirl.typepad.com. I've been one of the naive people who believed that she was real, and I am proud about it. Click at these links to see what happened with that picture!\n\nIn fact, in my opinion, there exists no physical law that would strictly prohibit the existence of a young gorgeous girl who spends hours with her libertarian blog. Many girls and women are very close to that point! Libertarian girl always looked not only intelligent, but a little bit too tough - for example she was pretty strongly pro-war (even pro-war in Iran). But all these things are possible, aren't they?\n\nWhat about the girl on the picture? Her name is Viktoriya and she is a Ukrainian woman from Kiev \"seeking man\". This was probably enough for the inventor of Libertarian Girl not to be afraid of copyright infringements. After long speculations about her gender - which Libertarian Girl identified as sexism - someone has finally found the other picture: see the blog here.\n\n## Sunday, February 13, 2005 ... \/\/\/\/\/\n\n### Shiites win\n\nIn agreement with my predictions and predictions of many other people, the large Shiite coalition has won the Iraq elections. They've received about 48 percent of the votes.\n\nThe second party is the Kurdish coalition. My favorite Iraqi List follows. The turnout was at 59 percent.\n\nThe large Shiite coalition, led by Ayatollah Al-Sistani, is mostly religious - although there are some liberals in it, too - and it will try to transform Iraq into a new kind of Iran. Let's hope that it will stay democratic for some time, and it won't become too dangerous for the world.\n\n## Saturday, February 12, 2005 ... \/\/\/\/\/\n\n### EU constitution in space\n\nIn April, the Union will be sending its constitution, which will soon be approved by 100 percent of the working and celebrating people, into space.\n\nA gold-bound copy of the Union's constitution has been delivered to the Soviet cosmic center Baikonur in Kazakhstan. A cosmonaut, namely comrade Tognini, is due to pass it to comrade Tognini (sic, see REDNOVA). Finally, Tognini will take it onboard of the Russian spaceship Soyuz (which means \"Union\" in Russian) as a symbol of\n\n\u2022 \"how national interests can be transcended by channeling the aspirations of nearly half a billion citizens from diverse cultural backgrounds towards the common goals of peace, freedom and prosperity,\"\n\nusing the words of the commissary for the peaceful expansion of the Union, comrade Verheugen. The Russian spaceship will bring the constitution to the International Space Station so that all the nations and the whole Universe sees that the Union is ahead of the American imperialists: America has never been able to transport its constitution into space, and until the heroes of socialism win their battle in the new world, there will be no progress in the so-called New World.\n\nThe bold decision to take the constitution into outer space is a revolutionary idea. Even the British Eurosceptic party UKIP said outer space was the best place for the constitution (BBC).\n\nThe spokesman for the European Soviet, comrade Kreuzhuber, said: \"This time they asked to bring a copy of the European Constitution along.\" Although the constitution has yet to be ratified here on Earth, comrade Kreuzhuber points out that \"this will make it one of the world's most thoroughly tested constitutions -- not just politically but physically.\" (REDNOVA)\n\nThe officials in the capital city of the Union have released a warning for its citizens saying that \"if the Constitution is not approved through the referendums, it will drop on the Europeans' heads from space.\" (ZAMAN)\n\nFigure 1: Comrade Chirac and French pioneers with flowers celebrate the constitution's journey through the Cosmos. Chirac is currently visiting Spain where less than 1 person in 10 will vote \"no\". The Spanish unemployment dropped to one-half and the Spaniards have received 219 billion USD from the Union since they joined in 1986.\n\nIncidentally, the Union in the text above is not the Soviet Union but the European Union, and the news is not a joke. The previous blog article about the European constitution was here.\n\n### People in science and technology\n\nMy comments about the text by the three original authors follows the bullets.\n\nA private response to a text by three respected people from Stanford, Princeton, and MIT\n\nHarvard President Lawrence Summers' recent comments about possible causes of the underrepresentation of women in science and engineering have generated extensive debate and discussion\u2014much of which has had the untoward effect of shifting the focus of the debate to history rather than to the future.\n\n\u2022 Interestingly enough, the reason why a meaningful discussion about a scientific topic is primarily based on the data from the past and not so much on data from the future is that we know the past, but we don't know the future - at least I don't know it. Maybe the authors of the text above know the future and they can therefore use it as the primary argument, in which case I apologize. ;-)\nThe question we must ask as a society is not \"Can women excel in math, science and engineering?\"\u2014Marie Curie exploded that myth a century ago\n\u2022 ... fair enough ...\n\u2014but \"How can we encourage more women with exceptional abilities to pursue careers in these fields?\"\n\u2022 This is an important educational and political question, but president Summers wanted to modestly suggest that we should first ask whether we should expect a higher percentage of women to pursue careers in these fields. The answer is definitely not obvious, although it is treated as a self-evident dogma in the sentence above.\nExtensive research on the abilities and representation of males and females in science and mathematics has identified the need to address important cultural and societal factors.\n\u2022 Other, more hard-science oriented research - and also president Summers in his speech and others - have also identified the need to address other important factors.\nSpeculation that \"innate differences\" may be a significant cause of underrepresentation by women in science and engineering may rejuvenate old myths and reinforce negative stereotypes and biases.\n\u2022 Lawrence Summers was much more polite - he did not explicitly identify the idea about \"important negative stereotypes and biases\" as an \"old myth\" although there exist reasons to use this language.\nWhy is this so important? Our nation faces increasing competition from abroad in technological innovation, the most powerful driver of our economy, while the academic performance of our school-age students in math and science lags behind many countries.\n\u2022 Sounds fine, except the fact that the causal relation between these politically attractive cliches and the question under consideration is not explained.\nAgainst this backdrop, it is imperative that we tap the talent and perspectives of both the male and female halves of our population.\n\u2022 Good.\nUntil women can feel as much at home in math, science and engineering as men, our nation will be considerably less than the sum of its parts.\n\u2022 This assertion could very well imply that the nation will always be considerably less than the sum of its parts, although I don't exactly know what the second sentence means quantitatively. The sentence above is based on the assumption that the only natural situation is when different groups of people feel as much at home in all fields as other groups of people. What is missing is a piece of evidence supporting this statement - especially because there seems to be evidence that the assumption is not true. So I suppose that the opinion that everyone must feel equally at home everywhere is treated as another dogma that does not have to be justified - and the heretics who do not believe this dogma must probably be tortured.\nIf we do not draw on the entire talent pool that is capable of making a contribution to science, the enterprise will inevitably be underperforming its potential.\n\u2022 Well, definitely - if the words \"talent pool\" are defined scientifically, as opposed to a definition with a politically twisted meaning.\nAs the representation of women increases in every other profession in this country, if their representation in science and engineering does not change, these fields will look increasingly anachronistic, less attractive, and will be less strong.\n\u2022 Well, it does not seem terribly realistic to say that computer science and state-of-the-art electronics, for example, look \"anachronistic\" just because they happen to be pre-dominantly run by the males. Or are males themselves anachronistic?\n\u2022 The only sense how can I understand the statement about the field being \"less attractive\" is based on the visual feelings of the males. Obviously, their sentence seems to imply that they think that a field with a higher concentration of males is less attractive - which does not sound as a terribly balanced and fair comment about the genders. Some fields are attractive for someone but not attractive for someone else.\n\u2022 A field would become \"less strong\" if it did not use the talent pool, as discussed in the previous paragraphs. There can be various reasons why the talent pool is not used efficiently - and the political criteria that would eliminate some talented people and replace them by people who better fit some quotas would be one imaginable reason of reduced strength.\nThe nation cannot afford to lose ground in these areas, which not only fuel the economy but also play a key role in solving critical societal problems in human health and the environment.\n\u2022 It's hard to understand why \"human health\" and \"the environment\" were picked as examples of the \"critical societal problems\". First of all, they are just two examples of important problems for the society. Second of all, these particular societal problems are not associated with the fields in which women are most clearly underrepresented. Their percentage is lowest in fields like computer science, physics, mathematics, engineering - and neither of these fields seems to be primarily focused on human health and the environment.\nMuch has already been learned from research in the classroom and from recent experience on our campuses about how we can encourage top performance from our students. For example, recent research shows that different teaching methods can lead to comparable performance for males and females in high school mathematics.\n\u2022 Note that the authors themselves acknowledge that there are differences between the way how boys and girls typically learn mathematics. What the authors do not allow to be questioned is their assumption that the different ways of learning mathematics must always statistically lead to the same careers and outcomes.\nOne of the most important and effective actions we can take is to ensure that women have teachers who believe in them and strong, positive mentors, male and female, at every stage of their educational journey\u2014both to affirm and to develop their talents.\n\u2022 I am happy to agree.\nLow expectations of women can be as destructive as overt discrimination and may help to explain the disproportionate rate of attrition that occurs among female students as they proceed through the academic pipeline.\n\u2022 There are many factors that can be destructive. Equal expectations from two people who are not equal represent another important example - once again, the three authors only see one side of the coin.\nColleges and universities must develop a culture, as well as specific policies, that enable women with children to strike a sustainable balance between workplace and home.\n\u2022 It is definitely politically possible to impose new policies and offer the same benefits (and pay the same salary) to a person who does 1\/2 of the work that another person does, and the remaining 1\/2 of the time is dedicated to other activities. Another question is whether it is the right thing to do and whether the society will benefit from such policies. And it is not too hard to guess what the economists such as Claudia Goldin or Lawrence Summers think about this question.\nOf course, achieving such a balance is a challenge in many highly demanding careers. As a society we must develop methods for assessing productivity and potential that take into account the long-term potential of an individual and encourage greater harmony between the cycle of work and the cycle of life\u2014so that both women and men may better excel in the careers of their choice.\n\nAlthough we have a very long way to travel in terms of recruiting, retaining and promoting women faculty in scientific and engineering fields, we can also point to significant progress. According to the National Science Foundation, almost no doctoral degrees in engineering were awarded to women in 1966 (0.3 percent), in contrast to 16.9 percent in 2001. And in the biological and agricultural sciences, the number of doctorates earned by women rose from 12 percent to 43.5 percent between 1966 and 2001.\n\n\u2022 Great. But I guess that until the percentage will be below 50 percent in a single field, there will exist some type of dissatisfaction. How many generations of fair treatment will be necessary before one will be \"allowed\" to argue that the theory of discrimination and discouragement as the only explanation can't fit the rich data that paint very different pictures in different fields?\n\nOur three campuses, and many others, are home to growing numbers of women who have demonstrated not only extraordinary innate ability but the kinds of creativity, determination, perceptiveness and hard work that are prerequisites for success in science and engineering, as in many other fields.\n\n\u2022 The women at Harvard have demonstrated at least as much creativity, determination, perceptiveness, and hard work as the women at these other universities - and it is not quite clear why exactly the three campuses are mentioned and not Harvard even though this discussion was initiated by the president of Harvard University. The only explanation I have is that the three authors wanted to picture their campuses as the \"nicer ones\" although such a description can't be justified.\nThese figures demonstrate the expanding presence of women in disciplines that have not, historically, been friendly to them.\n\u2022 ... and to which many women were not friendly in the recent decades, judging by the numbers how many of them chose these fields.\nIt is a matter of vital concern, not only to the academy but also to society at large, that the future holds even greater opportunities for them.\n\n... Oh, and finally, let me say that the text was written by the presidents of MIT, Stanford, and Princeton. Although there are three of them, I doubt that their text will become as stimulating and important as the speeches of the president of Harvard University.\n\n## Friday, February 11, 2005 ... \/\/\/\/\/\n\n### Recent talks: Berkovits, Tegmark, Arkani-Hamed\n\nRecently there have been several interesting talks at Harvard. Let me start with the postdoc journal club.\n\n\u2022 On Wednesday night, Nathan Berkovits was explaining his pure spinor formalism. He started with a description of the advantages and disadvantages of the RNS and Green-Schwarz formalisms. The superparticle is a good starting point for his pure spinor formalism and it already has many features that are also relevant for the superstring. For example, the polarizations of the gluon supermultiplet are described as a spinor-valued function A_alpha(X,theta) of the superspace - this has 16 x 65536 i.e. roughly a million of components to start with even for the open string. The situation for the closed string is this squared - i.e. more than one billion components. ;-) Nevertheless, the condition \"lambda^a lambda^b D_a A_b(X,theta) = 0\" is enough to kill all the degrees of freedom except for the 16 open (or 256 closed) physical modes. Here \"lambda^a\" is a pure spinor ghost - i.e. a spinor whose only non-vanishing bilinear is the middle-dimensional one, \"lambda_a gamma_{jklmn}^{ab} lambda_b\". Nathan explained that the pure spinors were already introduced by Cartan to describe possible complex structures in \"2k\" dimensions. The coset \"SO(2k)\/U(k)\" is isomorphic to the space of projective pure spinors in \"2k\" dimensions. Nathan then discussed the BRST charge and its cohomology, and tried to answer many questions of ours about the relation of his BRST charge to the conformal symmetry - this relation is different from the RNS formalism because Nathan's BRST charges does not follow from a canonical gauge-fixing procedure. Finally, he wrote down a pure spinor Lagrangian for the \"AdS5 x S5\" background that - as he can prove - is a consistent conformal field theory. However, so far it has been difficult to extract useful information from this model.\nThursday is the usual day of the Duality Seminars - this semester they are dedicated to cosmology plus the interface between cosmology and string theory.\n\n\u2022 Yesterday, Max Tegmark from MIT gave an impressively high-tech Cosmological Duality Seminar about the cosmological parameters and their measurement. With his Apple, he was showing the Universe and stars at all possible distance scales. He has made a very good job in defending the idea that the funding for cosmology should grow, and that the whole NASA budget should not be sent to Mars. Max presented the animations showing how the CMB anisotropy as a function of the orbital momentum - as well as the visible matter density in the Universe at various scales and various other comparably important \"graphs\" describing the Universe - depend on various basic cosmological parameters: each of them seems to influence the patterns in a different way. It's impressive how cosmology is becoming a high-precision science. On the other hand, Max emphasized that we should not spend the rest of our lives with determining the parameters with higher accuracy. We should find relations between them - a goal mostly for the theorists.\nFinally, Nima gave a captivating and provoking family lunch seminar today about some important grand questions and his approach to these questions:\n\n\u2022 He described his paper with Savas and Shamit about their friendly, predictive neighborhoods in the landscape. The type of toy models they consider are effective field theories with \"N\" real scalars. (Bobby Acharya et al. recently argued that the friendly neighborhood could appear in the M-theoretical landscape on G2 or Einstein seven-manifolds.) The potential in their toy model is a sum of quartic functions for each scalar field. This means that each scalar field increases the number of vacua by a factor of two - the total number of vacua is \"2^N\". Nima argues that assuming that the values of the parameters can be written as a constant plus corrections from all the scalar fields (which is another big assumption, I think), one can show that via the central limit theorem the average fluctuation of each observable will be of order \"1\/sqrt(N)\". In the limit of infinitely many vacua, he says, we don't get a landscape where everything is undetermined. On the contrary, everything is determined with a relative error of order \"1\/sqrt(N)\". One can get predictivity back, given certain assumptions! What are his new rules to deduce new physics? I've been trying to get a well-defined answer for quite some time, and I can only claim a partial success which can of course be caused by my being slower. Today, Nima formulated the rule pretty clearly:\n\u2022 A new particle or field or mechanism is only allowed to be added to our description of Nature if it is necessary for avoiding one of the \"major\" disasters - such as the destruction of the galaxies or atoms - disasters that one often discusses in the framework of the anthropic reasoning.\n\u2022 Obviously, one can raise the following objection:\n\u2022 Is not the new prescription for physics ruled out? The top quark, for example, seems unnecessary to avoid any of these low-energy disasters. This kind of anthropic thinking would have predicted that the top quark did not exist - and therefore Melissa Franklin et al. have falsified your paradigm 10 years ago.\n\u2022 What is the answer (of Nima) to this question?\n\u2022 We are not interested in these questions. We only want the neighborhood of the Standard Model. It's an assumption that we keep the broad features of the real world that are already known intact.\n\u2022 Well :-), it's OK not to be interested in these questions, but they seem necessary to define what the new principle constraining physics exactly is. Obviously, the principle must be refined to avoid the contradiction mentioned above. The refined version says something like:\n\u2022 Proposals for new physics that goes beyond the insights known in 2005, which is treated as an exceptional moment in the history of the Universe (and perhaps the humankind), must be justified by their necessity to avoid major global disasters such as the destruction of galaxies, nuclei, or atoms. Occam's razor must be applied - and the only way how a proposal for new physics can survive Occam's razor is an anthropic argument.\nWell, when one formulates the new principle is this more accurate way, it seems slightly less convincing because of its special emphasis on the present knowledge. Why should we believe that a principle like that is a good guide to look for new physics if the same principle applied in earlier developments of physics is known to lead to incorrect conclusions?\n\nDon't get me wrong. I like simple models and Occam's razor, and we always use simplicity as one of the rather important arguments. However, many things in our Universe are simply not minimal. In fact, according to Gell-Mann's totalitarian principle, everything that can occur in agreement with the symmetries will occur. This principle is usually used to allow all possible terms in the action involving given fields, but its moral generalization can also be used to allow for new fields and particles.\n\nThis observation implies that the requirements of simplicity are not a universal principle that can always be safely trusted when we try to pick the correct theory. They're often useful, but they're not universally valid. No doubt that Nima agrees. Nima et al. seem to suggest that exactly for \"physics beyond the Standard Model\", this principle of simplicity should suddenly become more important and more reliable than before. Also, beyond the Standard Model, the anthropic arguments should suddenly become more important - or even omnipotent. But I don't see a rational argument behind this approach. Simplicity, anthropic arguments, technical arguments, naturalness, and other arguments remain valid arguments, and giving them weights is a matter of intuition and personal psychology. I don't see why the Standard Model should be the critical line behind which we should suddenly change the way how physics is done and which arguments are important.\n\nIn this sense, I would probably guess that the resulting \"minimal\" models - and especially the new \"anthropic sector\" that is used to explain the cosmological constant and the Higgs mass - have comparable probability to be correct as any other models supported by some other, less anthropic principles - except for the obvious fact that the authors of the paper are big shots whose probability is always a bit higher. ;-)\n\n### Hockey stick has become a boomerang\n\nUpdate: The M&M paper has appeared in Geophysical Research Letters. Newspapers explain that climate science is much like Bre-X, and you don't want to invest in it. See also Wall Street Journal.\n\nFirst of all, Steve McIntyre - the mineral exploration analyst who has made the audit of the temperature reconstructions together with the economist Ross McKitrick - has a new blog\n\nThe tomorrow's (February 11th, 2005) issue of Science has an article by Richard Kerr that discusses two topics related to the \"hockey stick graph\":\n\u2022 The paper by McIntyre and McKitrick, soon to appear in Geophysical Research Letters, that shows that the reconstruction by Mann et al. is based on a flawed approach to statistics, and the results only reflect bristlecone pines\n\u2022 Another paper form today (February 10th, 2005) issue of Nature by the Swedish-Russian group led by Anders Moberg.\nThe paper of Moberg et al. leads to roughly 5 times bigger natural temperature fluctuations than the \"consensus of the hockey team\" - and roughly 2 times bigger than the biggest previous estimate. The shape does not look as a hockey stick at all. It shows a very warm period around 1100 and a cold period around 1600. Steve McIntyre says, among other things, that while Moberg et al. has been \"a kick in the teeth of MBH, they have just made another spaghetti graph\". ;-)\n\nAt any rate, the \"scientific consensus\" about the dominant role of \"anthropogenic\" global warming is definitely gone.\n\nPrevious articles about the hockey stick:\n\nI suppose that the hardcore alarmists - those with a very strong stomach - will reply \"It does not matter at all that all of our science is rubbish, we can still scare people\". I predict that the toughest ones - those who don't feel any moral restrictions - will tell you once again that there will be 50 degrees Celsius heat in your city in the summer.\n\nWell, that's unlikely unless you live somewhere in Sahara. The most likely way this could happen is that the ecoterrorists would do the same things like in Crichton's \"State of Fear\", and I hope that the U.S. army will be able to deal with them before they realize their plans.\n\nThe alarmists will tell you that even though all quantitative predictions, postdictions, and reconstructions have been proved flawed, there is still some effect of CO2 on the climate, and you should be scared. Yes, there is always some effect of any event on all events in its future light cone. But this, non-quantitative and proto-scientific fact cannot lead a rational person to any big conclusions.\n\nA rational, scientifically oriented person or society needs to quantify the effects, determine its likely consequences, and quantitatively - and economically - calculate the best possible response. These \"details\" have been done incorrectly and must be re-done.\n\n## Wednesday, February 09, 2005 ... \/\/\/\/\/\n\n### Reall: Higher-dimensional black holes and rings\n\nHarvey Reall gave an excellent job talk at MIT, and we could not have missed it. It was balanced because\n\n\u2022 he figured out the right equilibrium between simple results and non-trivial results - well, the talk was completely comprehensible\n\u2022 he divided the time correctly to talk about his own work and the work of others\n\u2022 he found the right balance between the string theory motivations and independence of string theory\nHarvey is definitely among the world's leading experts in the field of solutions to higher-dimensional gravity and supergravity, especially the black hole solutions. Why do we care about the higher-dimensional solutions? It's harder to answer this question for a string-theoretical infidel.\n\nBut the rest of us knows: the Universe is probably higher-dimensional, and therefore it is a research of reality! And as Harvey emphasized many times, comparisons of a microscopic theory of quantum gravity with the predictions of classical (and semiclassical) general relativity is the next best thing after the experiment that we have to test a theory of quantum gravity, i.e. to test string theory.\n\nThe higher-dimensional black holes are also important because they can be the holographic dual description of some objects and phenomena in gauge theory - via Maldacena's AdS\/CFT correspondence. And gauge theory is of direct physical relevance. For example, there exists the Hawking-Page first order phase transition for thermal gravity in anti de Sitter space: it's a transition between the thermal gas and a black hole. Edward Witten has explained that it corresponds to the confinement-deconfinement phase transition in the gauge theory.\n\n### Last female Dow Jones CEO fired\n\nMany people focus on the percentage of women among the new tenured professors, which slipped below 15 percent, but no one cares about the world of business too much. Sheila Lennon explained in her article - published in many sources that require subscription - that the following story is not a feminist issue because its hero may be female, but she is also a greedy capitalist.\n\nThe Dow Jones industrial index is composed of 30 companies. Carly Fiorina has been the CEO of Hewlett-Packard for a couple of years - since 1999. She's a rather impressive person - she holds a bachelor's degree in medieval history and philosophy from Stanford, a MSc. degree from MIT, and an MBA from Maryland. She has been working for Cisco, Kellogg, Merck, Lucent, and AT & T.\n\nBecause she was just fired and replaced by Robert Wayman as interim CEO, all 30 Dow Jones companies have currently male CEOs.\n\nAll the computers I've bought in the last 10 years were Hewlett-Packard products - a desktop and laptops - which makes these developments a bit interesting for me. The main decision that Fiorina has been criticized for was her purchase of Compaq in 2002 that failed to increase the profits.\n\n## Monday, February 07, 2005 ... \/\/\/\/\/\n\nFor Windows users: have you already installed the February 2005 patches? There's a lot of them:\n\nToday's physics colloquium at Harvard was about the fusion. Robert Goldston from Princeton did a very good job. The previous article about ITER and fusion on this blog was posted here, and many well-informed people added interesting comments for which I am grateful.\n\nGoldston discussed a lot of physics issues connected with the stabilization of the plasma; magnetohydronamics (MHD); heat diffusion, and so on. The inflow of professional information was pretty fast. He described an equation due to David Bohm, and he distinguished the Bohm regime of the plasma from the gyro-Bohm regime. The Bohm regime is hopeless - the diffusion is far too strong. The gyro-Bohm regime is what will allow the fusion plants to operate, and they can show that this regime can be realized.\n\n### European constitution: NO\n\nThe new treaty to establish the Constitution of the European Union - which effectively means the proposed future European constitution - has roughly 353 pages. I guess that the fraction of the people who are gonna read it will be negligible.\n\nBefore I looked at the Czech translation of the document, my opinion was \"70 percent against, 30 percent for\". Now it is \"95 percent against, 5 percent for\". I would not say that it is a democratic constitution. Is started with all the symbols that usually define a country (although many people say that the EU would not be a single country), titles for all the national leaders and kings, and other fun stuff. After a couple of pages, it started to get closer to the essence. The text explained that the European laws would have higher priority than the national laws. The main organs of the EU should be\n\n\u2022 The European Parliament\n\u2022 The European Soviet\n\u2022 The Soviet of Ministers\n\u2022 The European Commission\n\u2022 The European Court\n\nWell, I admit that my translation of some of these terms back to English is not the standard one (and others use the word \"Council\" instead of the \"Soviet\") - but mine is arguably the more appropriate one. The proposed text is full of clich\u00e9s, but I don't care about this stuff. What is important is how the system is supposed to work.\n\n## Friday, February 04, 2005 ... \/\/\/\/\/\n\n### Peace in Iran, LHC, and other news\n\nMany readers rely on this blog as one of the main sources of information. Let's mention some news:\n\n\u2022 The US administration does not plan a war against Iran right now. Although Libertarian Girl and others may have presented some arguments supporting such a war (well, the photograph may be one of them), the Reference Frame would consider a war against Iran as a strategic error and a risky and morally controversial decision, especially after the less-than-perfect development in Iraq in the last 2 years. The plans about Iran may change, of course.\n\u2022 BBC has a new article about the search for the \"God particle\". Prof. Virdee says that \"they don't always like theorists to tell them what they should find\". Did you know that the Compact Muon Solenoid (CMS) is large enough to house the Canterbury Cathedral?\n\u2022 Max Schmeling (the boxer) died at age of 99. He may have been the greatest symbol of the \"superior\" Nazi race among the sport celebrities. However, from a historical perspective, I think that he was a kind of positive personality. He protected Jews from the Nazi regime, became friends with his old good African American competitor Joe Louis, and introduced CocaCola to West Germany.\n\u2022 Julia Tymoshenko, an ally of Viktor Yushchenko, became the new Ukrainian prime minister. The situation in Ukraine has been completely stabilized.\nThe weather has become one of the most politically sensitive topics ;-), which is the main reason why the Reference Frame finds it important to inform about the weather:\n\u2022 Although most of the news about the Arctic emphasize how warm it's getting and how the polar bears may dislike it, almost no one will tell you that the Arctic's stratosphere has experienced an extremely harsh, cold winter. In fact, this extreme cold is a threat for the ozone layer in Northern Europe. Unlike the \"Global Warming\", the ozone hole is viewed by the Reference Frame as a potential problem that should be seriously looked at; the UV radiation is dangerous, and its amount reaching the surface depends exponentially on the ozone layer's thickness. Of course, the Global Warming alarmists will blame everything - including the cooling stratosphere - on human-induced global warming, but I hope that most of my readers are still able to distinguish different effects. The ozone layer thinning is caused by many effects, and indirect effects of CO_2 are among the speculative ones.\n\u2022 Australia has had a record cold summer, too. Melbourne has recorded its coldest February day on record (12.8 degrees). It's been raining a lot. The damage has topped 100 million AUD. Also, a famous glacier in New Zealand had been used as a prime example of global warming - it shrunk by 500 meters between 1999 and 2003. The glacier was named after the emperor of my homeland, namely Austria-Hungary, 100+ years ago: Franz Josef. You won't hear about Franz Josef today because it is growing by 4 meters a day. This fact is apparently politically incorrect.\n\u2022 Today, it was announced that a new, fifty-kilometer-long iceberg has formed in the frigid waters around Antarctica.\n\u2022 Four days ago, Moscow was hit by the heaviest snowstorm ever. At the same day, a huge snowstorm also hit Japan.\n\u2022 Researchers have constructed realistic plans to terraform Mars, by injecting a huge amount of greenhouse gases to Mars's atmosphere. In other words, the scientists plan to use global warming to make another planet habitable. Well, CO_2 would not be enough: they think about fluorine-based gases that can be roughly 10,000 more efficient than carbon dioxide.\nSome news from the economy:\n\u2022 The unadjusted unemployment in Germany has topped 5 million people, the highest number (and rate) since 1933 when Hitler took power: the number of unemployed jumped by 200,000 or so within a month. It may be useful, especially for some of the readers, to mention that it does not yet imply that a new Hitler has to become the German leader in 2005.\n\u2022 Europe is slightly changing its agenda. According to the previous, Lisbon agenda, various EU commission's governmental projects should have made our old good Europe the world's most dynamic economy by 2010. The new plan - the revised Lisbon agenda - is for Europe to avoid the collapse by 2010. The new commission's boss Jos\u00e9 Manuel Barroso (former Portugese prime minister) has obviously learned more from the failure of the Soviet bloc than his predecessors. He has made a kind of Larry-Summers-like scandal when he openly supported flexible free markets in his speech! :-) Barroso also wants to establish The Eurochusetts Institute for Technology.\n\u2022 In the US, the payrolls increased roughly by 150,000, a long-term average number and a disappointment for the economists who predicted 189,000. The unemployment calculated from another survey dropped from 5.4 to 5.2 percent, the lowest number since September 2001.\n\u2022 V\u00e1clav Klaus, the outspoken and highly intelligent Czech President, spoke at the Microsoft conference in Prague. He explained that the clich\u00e9s about the knowledge economy and information society are modern forms of mysticism. The important principles of market economy do not need any updates, and they are not correlated with any particular technology.\n\u2022 The financial terrorists at Karmabanque.com who believe that by having established a website for similar anti-Coke morons - a website that features virtual antimoney, they would be able to reduce the price of the Coca Cola stock from $40 to$20, must feel surprised. The airplanes do not land. The Coke stock has risen from $40 to$42.50 within the last few months.\n\n## Thursday, February 03, 2005 ... \/\/\/\/\/\n\n### Tachyons can change topology\n\nIn the last 2 hours, Allan Adams just told us about his supernew paper\n\nand because I think that it is definitely an interesting paper, let me say a couple of words.\n\nImagine that you take a type II string theory and compactify it down to 8 dimensions, on a two-dimensional genus \"g\" Riemann surface.\n\nWell, unless \"g=1\", it is a non-conformal theory, so you will have to deal with a time-dependent background. Let's not worry. Let's assume the string coupling to be weak throughout the story.\n\nImagine that you start with a genus 2 Riemann surface. It can degenerate into two genus 1 Riemann surfaces connected by a thin tube. The circle wrapped around this tube is homologically trivial, and you can show that the fermions will be antiperiodic around it: it will be a Scherk-Schwarz\/Rohm compactification on a thermal circle. The reason for the antiperiodicity is the same like the reason that the closed strings in the NS-NS sector must have antiperiodic boundary conditions for the fermions assuming that the corresponding operators in the \"z\" plane don't introduce any branch cuts.\n\nOK, imagine that the tube is very long. Because of the antiperiodic boundary conditions, the sign of the GSO condition in the sectors with odd windings is reverted, and one can find some tachyons there assuming that the radius is small enough so that the winding is not enough to make the squared mass positive. Equivalently, one can T-dualize along the circumference of the tube to obtain some sort of type 0 theory which has a bulk tachyon if the radius in the type 0 picture is large enough. Go exactly near the point where the first tachyon in the \"w=1\" sector starts to evolve. It's the first perturbative instability you encounter.\n\n## Wednesday, February 02, 2005 ... \/\/\/\/\/\n\n### Most articles support Summers\n\nThe last 24 hours may have been one of the first day-long periods in which the media mostly endorsed, not criticized, Larry Summers, and supported the freedom of speech and scientific inquiry. (Judging by the ensemble available at news.google.com.) I mean the version of Lawrence Summers before he had to say what he finally said.\n\nI expect the percentage to continue to rise. Why? It's simply because those who oppose Summers don't have too many things to say. In fact, they want nearly everyone to close their mouth (and lab, if it studies any details about the brain). Virtually all articles against Larry Summers may be summarized as follows:\n\n\u2022 The speech of Lawrence Summers was outrageous - everyone who has heard it should either black out or throw up. He has no right to speak in this way. Women are discriminated. You can see that they have a smaller representation in various professions - and most people (both men and women) believe that men are more likely to be successful in these professions. This proves that discrimination is everywhere around us because everyone with the right opinions about the world knows that the women are identical to the men, perhaps except for one organ. Note that this is not a circular argument because it is not a circular argument.\n\u2022 Why should we investigate the reasons behind the differences when we already have a holy, perfectly politically correct answer that has been prepared for us - namely discrimination - which is endorsed by all true believers in political correctness? Moreover, Summers' comments may convince some girls that the comments may be true. It's a complete catastrophe if someone knows something that can be true, and academia should prevent this threat. Note that Harvard University's logo even has \"Veritas\" isolated in a triangle which shows that everyone outside the triangle is safely protected not only from \"Veritas\" (the truth), but also from everything that could potentially become \"Veritas\".\n\nNote that the newspapers could have saved a lot of money if they reposted this universal comment, instead of paying hundreds of authors of these mostly weak and uninteresting articles.\n\nOn the other hand, the articles that support Summers have much more to say and they are very diverse. The following list is kind of sorted from the newest ones:\n\n\u2022 Ana Veciana-Suarez - mother of 4 boys and 1 girl - boys will be boys\n\u2022 Readers - of International Herald Tribune - the gender differences are assets\n\u2022 Jeremy Beecher - Academia harmed by sound-bite culture; unpopular ideas will be unpopular even if they are true\n\u2022 Caitlin Hall - Harvard president makes valid point - there are also, as usual, brainless batallions of PC police calling for Summers' ouster\n\u2022 Cathy Young - Boston Globe - Summers spoke the truth; the real scandal is that Summers was forced to grovel to save his job\n\u2022 Matt Crenson - ABC news and a lot of other media - Summers's observations are not controversial among the experts\n\u2022 Vladimir Shlapentokh - a Russian-born professor from Michigan compares the politically correct prosecutors to Lysenkoist (anti-genetics) and the proponents of intelligent design; Summers, lacking the courage of Giordano Bruno, ...\n\u2022 Readers of New York Post - all three are supporting Summers; PC movement preferes to stifle debates; the contrition tour is getting sickening - that crowd can't be appeased\n\u2022 Stephen Braun - L.A. Times - brilliant and abrasive Summers ... (neutral article)\n\u2022 Georgie Ann Geyer - one of the most experienced feminists agrees with Summers: \"fair\" does not mean \"same\"\n\u2022 Michael Lewis - Bloomberg - Summers is the only thoughtful, interesting president, the only one who is not a coward; he assumed that the women were his peers who were looking for the truth and they did not appreciate it\n\u2022 Christopher Caldwell - Financial Times - \"I know of no community as committed to free inquiry as this one,\" wrote Larry Summers even though he had almost lost his job because of his commitment to free inquiry\n\u2022 Warren Coats - let's not the truth, whatever it is; we are individuals, not averages\n\u2022 Mark Ayoub - in defense of an arrogant male chauvinist\n\u2022 Aljazeera - repeats the Reuters story (Summers is exhibit A...) and introduces it with a picture saying \"restrictive laws are inhibiting academics in the United States\"\n\u2022 Washington Post - an editorial opinion - it's a university's function to provoke fresh thinking, especially about big issues that are \"too important to sentimentalize\", as Summers rightly puts it\n\u2022 Margaret Carlson - L.A. Times - Summers is brilliant\n\u2022 New York Post - Harvard's thought police\n\u2022 Kathleen Parker - about womanhood\n\u2022 Mike North - genders are different and water is wet\n\u2022 Arnold Kling - four standard deviations in the distribution; about Kling's daughter who had to answer \"yes\" to the question \"Is gender socially determined?\" - I kid you not; shame on Summers's critics, praise to his defenders; discrimination damages the discriminator; self-regarding attribution bias; male dominance\n\u2022 Elie Dvorin - Academic hypocrisy - comparing Churchill and Summers\n\u2022 Zach Musgrave - U. of Washington - Summers not sexist - What is really at stake is academic freedom\n\u2022 Phyllis Schlafly - The feminists who have no sense of humor ... lassoed Summers and dragged him ... Communists used to punish ... but feminists took it to new heights ... When will American men learn how to stand up to the nagging by the uncivil feminists whose sport is to humiliate men?\n\u2022 CNN - Male and female brains different - it's as easy as X and Y - about 100 structural differences known\n\u2022 Dan Byrne - Michigan needs someone like Summers - this man is brilliant\n\u2022 Free Lance Star - Sexes are different, it's simple biology - a scary story of David whose penis was removed. David became Brenda, they celebrated that he became a woman. But as far as his brain went, these were just feminist lies. He continued to be David. He was thinking as a man. He became a man again, and later commited suicide\n\u2022 Brenda Maddox - the Guardian (British socialist newspaper) - nicely written short biographies of important women in science - and how the feminists hijacked Rosalind Franklin\n\u2022 Chuck Colson - Denying the obvious\n\u2022 Stuart Taylor - Why feminist careerists neutered Larry Summers\n\u2022 John Leo (Alan Greenspan's twin) - What Larry meant to say\n\u2022 The Age - Australia - sexes innately different - unless you are PC, steer clear of Harvard\n\u2022 Kara Falchini - women in average are weaker in football and different in math.\n\u2022 Walter Williams - Anti-intellectualism among the academic elite - very informative; also studies how genetic differences correlated with sexual orientation and races affect various things, and what people think (and like to think) about these issues. Note that the author is African American. Reprinted in Townhall, DC; Worldnetdaily, OR; Washington Times.\n\u2022 Gill Spencer - takes an example of Nancy Hopkins; the 13:1 boys:girls ratio for the math score above 700 as evidence of differences. Also mentions that Larry Summers has female-like empathy and verbal skills. ;-)\n\u2022 Daniel Koffler - from Yale - why equal isn't identical; \"the irony at Hopkins' expense is that she is presenting herself as a caricature of the helpless, constantly-fainting Victorian \"woman of the house.\" If feminism is to have any point at all, ...\"\n\u2022 Human Events - \"What\u2019s not 'up for discussion,' in short, is what\u2019s politically incorrect.\" - Also talks about 4% - which is the percentage of ... registered Republicans plus Libertarians at Harvard. This is a *real* underrepresentation - but this one is apparently politically correct.\n\u2022 James Koch - professor - Summers should not be gagged\n\u2022 Daily Californian - Sex and the university - \"this political environment stifles free speech and ideas.\"\n\u2022 Seattle PI - Does one gender do better? - A lot of graphs and data about the brain differences and jobs, with overall conclusions that should not offend anyone.\n\nThis is just the last 24 hours or so. The number of articles that support Lawrence Summers and his points is roughly 300+ (many of them being copies of others), with the same number of articles that mostly oppose his comments. Some articles from the previous day or so:\n\n\u2022 Dan Thomasson - Nature or Nurture? - \"When Harvard President Lawrence Summers injudiciously defied the Gods of Political Correctness by suggesting...\"\n\u2022 Concord Monitor - \"...Summers grovels before the goddesses of hyper-political correctness...\"\n\u2022 Reuters - IQ related area may differ between men and women - some details packaged to a package that should not offend anyone\n\u2022 Human events - by Mac Johnson - Summers speaks truth, then apologizes\n\u2022 Feminists, meet Mr. Darwin - Robert Wright accused those who deny innate differences of holding \"patently false beliefs about human nature.\"\n\u2022 Robyn Blumner - Hysteria in the ivory tower - \"Alternative explanations are no longer welcome. This should be distressing to people who care more about uncovering the truth than assuaging hurt feelings.\". The same article with a photograph appears here. I guess that many readers of my blog will study the correlation between the beauty of the female authors and their agreement with Summers, and I'm afraid that the signal they find will be striking.\n\u2022 Barbara Simpson - Maybe it's time to be afraid - about the freedom of speech and poor Larry (I'm putting it here because there is a photograph here, too) ;-)\n\u2022 Diana West - Ladies, please... - \"I hate 'women', those who make a career out of it...\"\n\u2022 Shawn Augsburger - UC Irvine - Summers entitled to free speech - differences between brains...\n\n## Tuesday, February 01, 2005 ... \/\/\/\/\/\n\n### Crossbar latch & fast computers\n\nThis looks like an impressive discovery in nanophysics. Today, Hewlett-Packard has announced in Journal of Applied Physics\n\nthat they invented a \"crossbar latch\", a new cheap, nanometer-sized component at the molecular level that has the potential to replace the transistors and speed up the computers by three orders of magnitude.\n\nThe patent was actually registered already in 2003:\n\nIt will probably take years before the new computers can be developed using this technology, if it ever happens.","date":"2017-10-21 21:33:12","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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35092222690582275, \"perplexity\": 1769.1342813270305}, \"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-2017-43\/segments\/1508187824899.43\/warc\/CC-MAIN-20171021205648-20171021225648-00880.warc.gz\"}"}
| null | null |
Q: Sum of columns with partial string match, with names imported from a list. R I'm working with data tables like so:
ID <- c("ChrM","ChrM","ChrM","ChrM","ChrM")
pos <- c(5,6,7,10,11)
cr.H.MN.8A <- c(0,0,3,0,1)
cr.H.MN.8B <- c(2,1,0,1,0)
cr.H.MR.8A<-c(0,0,0,0,1)
cr.H.MR.8B<-c(0,2,0,0,1)
cr.H.MR2.8A<-c(0,1,1,0,0)
cr.H.MR2.8B<-c(0,1,0,0,0)
dfa <- data.table(ID,pos,cr.H.MN.8A,cr.H.MN.8B,cr.H.MR.8A,cr.H.MR.8B,cr.H.MR2.8A,cr.H.MR2.8B)
dfa
ID pos cr.H.MN.8A cr.H.MN.8B cr.H.MR.8A cr.H.MR.8B cr.H.MR2.8A cr.H.MR2.8B
1: ChrM 5 0 2 0 0 0 0
2: ChrM 6 0 1 0 2 1 1
3: ChrM 7 3 0 0 0 1 0
4: ChrM 10 0 1 0 0 0 0
5: ChrM 11 1 0 1 1 0 0
>
Now, on an earlier question I asked how I could use the similiraties on the name strings to add them up
dfa[,`H.MN.8` := as.numeric(rowSums(.SD) > 1), .SDcols = grep("cr.H.MN.8", names(dfa))]
Which worlks flawlessly. But I got another question; is it possible to auto assign the parameters needed to separate the string? I've got the names of each column on a list (they used to be different data.frames), so could I use it to replace the HM.MN.8 and thus make it more easily computed? Thanks!
The names are so:
names(dfb)<-c("cr.H.MN.8A","cr.H.MN.8B","cr.H.MR.8A","cr.H.MR.8B","cr.H.MR2.8A","cr.H.MR2.8B")
A: you can try using
varName <- c("H.MR2","H.MN.8")
eval(parse(text=paste0("dfa[,`",varName,"` := as.numeric(rowSums(.SD) > 1), .SDcols = grep('",varName,"', names(dfa))]")))
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,529
|
package com.fitpay.android.webview.models;
import android.os.Parcel;
import android.os.Parcelable;
import com.fitpay.android.utils.RxBus;
import com.fitpay.android.utils.TimestampUtils;
import com.fitpay.android.webview.enums.DeviceTimeZone;
import com.fitpay.android.webview.enums.RtmType;
import com.fitpay.android.webview.events.RtmMessageResponse;
import com.google.gson.annotations.SerializedName;
import java.util.Date;
import java.util.Locale;
/**
* IdVerification data
*/
public final class IdVerification implements Parcelable {
private Date oemAccountInfoUpdatedDate; // Most recent date this user update their: Billing Address, Name, Email, password, or other Personally Identifiable Information associated to their account.
private Date oemAccountCreatedDate;
@SerializedName("suspendedCardsInAccount")
private Integer suspendedCardsInOemAccount; // If this user has multiple devices, how many cards are suspended in total across all devices?
@SerializedName("daysSinceLastAccountActivity")
private Integer lastOemAccountActivityDate; // Days this account was previously used, never today.
@SerializedName("deviceLostMode")
private Integer deviceLostModeDate; // Days this device was reported lost or stolen. Don't send if you don't have it.
@SerializedName("deviceWithActiveTokens")
private Integer devicesWithIdenticalActiveToken; // Number of devices that the token is on
@SerializedName("activeTokenOnAllDevicesForAccount")
private Integer activeTokensOnAllDevicesForOemAccount; // If this user has multiple devices, how many cards are active in total across all devices?"
private Integer oemAccountScore; // int between 0-9
private Integer deviceScore; // int between 0-9
private Boolean nfcCapable; // Only needed if your device is NOT nfcCapable
private String oemAccountCountryCode; // Country setting of account or phone in ISO 3166-1 alpha-2 format
private String deviceCountry; // Country setting of payment device
private String oemAccountUserName; // First and Last name of account
private Date devicePairedToOemAccountDate; // What day was this device first paired with this oemAccount?
private String deviceTimeZone; // Time Zone Abbreviation. Example: PDT, MST
private Integer deviceTimeZoneSetBy; // 1 - Time Zone Set by Network; 2 - Time Zone Set by User; 3 - Time Zone set by Device Location
private String deviceIMEI; // Only needed if your payment device has a cell connection
private String locale; //ISO 3166-1 alpha-2
private IdVerification() {
}
public Date getOemAccountInfoUpdatedDate() {
return oemAccountInfoUpdatedDate;
}
public Date getOemAccountCreatedDate() {
return oemAccountCreatedDate;
}
public Integer getSuspendedCardsInOemAccount() {
return suspendedCardsInOemAccount;
}
public Integer getLastOemAccountActivityDate() {
return lastOemAccountActivityDate;
}
public Integer getDeviceLostModeDate() {
return deviceLostModeDate;
}
public Integer getDevicesWithIdenticalActiveToken() {
return devicesWithIdenticalActiveToken;
}
public Integer getActiveTokensOnAllDevicesForOemAccount() {
return activeTokensOnAllDevicesForOemAccount;
}
public Integer getOemAccountScore() {
return oemAccountScore;
}
public Integer getDeviceScore() {
return deviceScore;
}
public Boolean getNfcCapable() {
return nfcCapable;
}
public String getOemAccountCountryCode() {
return oemAccountCountryCode;
}
public String getDeviceCountry() {
return deviceCountry;
}
public String getOemAccountUserName() {
return oemAccountUserName;
}
public Date getDevicePairedToOemAccountDate() {
return devicePairedToOemAccountDate;
}
public String getDeviceTimeZone() {
return deviceTimeZone;
}
public Integer getDeviceTimeZoneSetBy() {
return deviceTimeZoneSetBy;
}
public String getDeviceIMEI() {
return deviceIMEI;
}
public String getLocale() {
return locale;
}
/**
* send data to RTM
*/
public void send(String connectorId, String callbackId) {
RxBus.getInstance().post(connectorId, new RtmMessageResponse(callbackId, this, RtmType.ID_VERIFICATION));
}
public static final class Builder {
private Date oemAccountInfoUpdatedDate;
private Date oemAccountCreatedDate;
private Integer suspendedCardsInOemAccount;
private Date lastOemAccountActivityDate;
private Date deviceLostModeDate;
private Integer devicesWithIdenticalActiveToken;
private Integer activeTokensOnAllDevicesForOemAccount;
private Integer oemAccountScore;
private Integer deviceScore;
private Boolean nfcCapable;
private String oemAccountCountryCode;
private String deviceCountry;
private String oemAccountUserName;
private Date devicePairedToOemAccountDate;
private String deviceTimeZone;
@DeviceTimeZone.SetBy
private Integer deviceTimeZoneSetBy;
private String deviceIMEI;
private final String locale;
public Builder() {
locale = Locale.getDefault().getLanguage() + '-' + Locale.getDefault().getCountry();
}
/**
* Most recent date this user update their: Billing Address, Name, Email, password, or other Personally Identifiable Information associated to their account.
*
* @param oemAccountInfoUpdatedDate time in ISO 8601 format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"
* @return this
*/
public Builder setOemAccountInfoUpdatedDate(Date oemAccountInfoUpdatedDate) {
this.oemAccountInfoUpdatedDate = oemAccountInfoUpdatedDate;
return this;
}
/**
* OEM account created date
*
* @param oemAccountCreatedDate time in ISO 8601 format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"
* @return this
*/
public Builder setOemAccountCreatedDate(Date oemAccountCreatedDate) {
this.oemAccountCreatedDate = oemAccountCreatedDate;
return this;
}
/**
* If this user has multiple devices, how many cards are suspended in total across all devices
*
* @param suspendedCardsInOemAccount
* @return this
*/
public Builder setSuspendedCardsInOemAccount(int suspendedCardsInOemAccount) {
this.suspendedCardsInOemAccount = suspendedCardsInOemAccount;
return this;
}
/**
* Date this account was previously used, never today.
*
* @param lastOemAccountActivityDate time in ISO 8601 format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"
* @return this
*/
public Builder setLastOemAccountActivityDate(Date lastOemAccountActivityDate) {
this.lastOemAccountActivityDate = lastOemAccountActivityDate;
return this;
}
/**
* Date this device was reported lost or stolen. Don't send if you don't have it.
*
* @param deviceLostModeDate time in ISO 8601 format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"
* @return this
*/
public Builder setDeviceLostModeDate(Date deviceLostModeDate) {
this.deviceLostModeDate = deviceLostModeDate;
return this;
}
/**
* Set account score.
*
* @param oemAccountScore Between 0-9
* @return this
*/
public Builder setOemAccountScore(int oemAccountScore) {
if (oemAccountScore < 0 && oemAccountScore > 9) {
throw new IllegalArgumentException("oemAccountScore should be between 0-9");
}
this.oemAccountScore = oemAccountScore;
return this;
}
/**
* Device with identical active token
*
* @param devicesWithIdenticalActiveToken
* @return
*/
public Builder setDevicesWithIdenticalActiveToken(Integer devicesWithIdenticalActiveToken) {
this.devicesWithIdenticalActiveToken = devicesWithIdenticalActiveToken;
return this;
}
/**
* If this user has multiple devices, how many cards are active in total across all devices?
*
* @param activeTokensOnAllDevicesForOemAccount
* @return this
*/
public Builder setActiveTokensOnAllDevicesForOemAccount(Integer activeTokensOnAllDevicesForOemAccount) {
this.activeTokensOnAllDevicesForOemAccount = activeTokensOnAllDevicesForOemAccount;
return this;
}
/**
* Set device score.
*
* @param deviceScore Between 0-9
* @return this
*/
public Builder setDeviceScore(int deviceScore) {
if (deviceScore < 0 && deviceScore > 9) {
throw new IllegalArgumentException("deviceScore should be between 0-9");
}
this.deviceScore = deviceScore;
return this;
}
/**
* Only needed if your device is NOT nfcCapable
*
* @param nfcCapable
* @return this
*/
public Builder setNfcCapable(boolean nfcCapable) {
this.nfcCapable = nfcCapable;
return this;
}
/**
* Country setting of account or phone in ISO 3166-1 alpha-2 format
*
* @param oemAccountCountryCode
* @return this
*/
public Builder setOemAccountCountryCode(String oemAccountCountryCode) {
this.oemAccountCountryCode = oemAccountCountryCode;
return this;
}
/**
* Country setting of payment device
*
* @param deviceCountry
* @return this
*/
public Builder setDeviceCountry(String deviceCountry) {
this.deviceCountry = deviceCountry;
return this;
}
/**
* First and Last name of account
*
* @param oemAccountUserName
* @return this
*/
public Builder setOemAccountUserName(String oemAccountUserName) {
this.oemAccountUserName = oemAccountUserName;
return this;
}
/**
* What day was this device first paired with this oemAccount
*
* @param devicePairedToOemAccountDate time in ISO 8601 format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"
* @return this
*/
public Builder setDevicePairedToOemAccountDate(Date devicePairedToOemAccountDate) {
this.devicePairedToOemAccountDate = devicePairedToOemAccountDate;
return this;
}
/**
* Time Zone Abbreviation. Example: PDT, MST
*
* @param deviceTimeZone
* @return this
*/
public Builder setDeviceTimeZone(String deviceTimeZone) {
this.deviceTimeZone = deviceTimeZone;
return this;
}
/**
* 1 - Time Zone Set by Network; 2 - Time Zone Set by User; 3 - Time Zone set by Device Location
*
* @param deviceTimeZoneSetBy
* @return this
*/
public Builder setDeviceTimeZoneSetBy(@DeviceTimeZone.SetBy Integer deviceTimeZoneSetBy) {
this.deviceTimeZoneSetBy = deviceTimeZoneSetBy;
return this;
}
/**
* Only needed if your payment device has a cell connection
*
* @param deviceIMEI
* @return this
*/
public Builder setDeviceIMEI(String deviceIMEI) {
this.deviceIMEI = deviceIMEI;
return this;
}
public IdVerification build() {
IdVerification idVerification = new IdVerification();
idVerification.oemAccountInfoUpdatedDate = oemAccountInfoUpdatedDate;
idVerification.oemAccountCreatedDate = oemAccountCreatedDate;
idVerification.suspendedCardsInOemAccount = suspendedCardsInOemAccount;
idVerification.lastOemAccountActivityDate = TimestampUtils.getDaysBetweenDates(lastOemAccountActivityDate);
idVerification.deviceLostModeDate = TimestampUtils.getDaysBetweenDates(deviceLostModeDate);
idVerification.devicesWithIdenticalActiveToken = devicesWithIdenticalActiveToken;
idVerification.activeTokensOnAllDevicesForOemAccount = activeTokensOnAllDevicesForOemAccount;
idVerification.oemAccountScore = oemAccountScore;
idVerification.deviceScore = deviceScore;
idVerification.nfcCapable = nfcCapable;
idVerification.oemAccountCountryCode = oemAccountCountryCode;
idVerification.deviceCountry = deviceCountry;
idVerification.oemAccountUserName = oemAccountUserName;
idVerification.devicePairedToOemAccountDate = devicePairedToOemAccountDate;
idVerification.deviceTimeZone = deviceTimeZone;
idVerification.deviceTimeZoneSetBy = deviceTimeZoneSetBy;
idVerification.deviceIMEI = deviceIMEI;
idVerification.locale = locale;
return idVerification;
}
}
@Override
public int describeContents() {
return 0;
}
@Override
public void writeToParcel(Parcel dest, int flags) {
dest.writeLong(this.oemAccountInfoUpdatedDate != null ? this.oemAccountInfoUpdatedDate.getTime() : -1);
dest.writeLong(this.oemAccountCreatedDate != null ? this.oemAccountCreatedDate.getTime() : -1);
dest.writeValue(this.suspendedCardsInOemAccount);
dest.writeValue(this.lastOemAccountActivityDate);
dest.writeValue(this.deviceLostModeDate);
dest.writeValue(this.devicesWithIdenticalActiveToken);
dest.writeValue(this.activeTokensOnAllDevicesForOemAccount);
dest.writeValue(this.oemAccountScore);
dest.writeValue(this.deviceScore);
dest.writeValue(this.nfcCapable);
dest.writeString(this.oemAccountCountryCode);
dest.writeString(this.deviceCountry);
dest.writeString(this.oemAccountUserName);
dest.writeLong(this.devicePairedToOemAccountDate != null ? this.devicePairedToOemAccountDate.getTime() : -1);
dest.writeString(this.deviceTimeZone);
dest.writeValue(this.deviceTimeZoneSetBy);
dest.writeString(this.deviceIMEI);
dest.writeString(this.locale);
}
protected IdVerification(Parcel in) {
long tmpOemAccountInfoUpdatedDate = in.readLong();
this.oemAccountInfoUpdatedDate = tmpOemAccountInfoUpdatedDate == -1 ? null : new Date(tmpOemAccountInfoUpdatedDate);
long tmpOemAccountCreatedDate = in.readLong();
this.oemAccountCreatedDate = tmpOemAccountCreatedDate == -1 ? null : new Date(tmpOemAccountCreatedDate);
this.suspendedCardsInOemAccount = (Integer) in.readValue(Integer.class.getClassLoader());
this.lastOemAccountActivityDate = (Integer) in.readValue(Integer.class.getClassLoader());
this.deviceLostModeDate = (Integer) in.readValue(Integer.class.getClassLoader());
this.devicesWithIdenticalActiveToken = (Integer) in.readValue(Integer.class.getClassLoader());
this.activeTokensOnAllDevicesForOemAccount = (Integer) in.readValue(Integer.class.getClassLoader());
this.oemAccountScore = (Integer) in.readValue(Integer.class.getClassLoader());
this.deviceScore = (Integer) in.readValue(Integer.class.getClassLoader());
this.nfcCapable = (Boolean) in.readValue(Boolean.class.getClassLoader());
this.oemAccountCountryCode = in.readString();
this.deviceCountry = in.readString();
this.oemAccountUserName = in.readString();
long tmpDevicePairedToOemAccountDate = in.readLong();
this.devicePairedToOemAccountDate = tmpDevicePairedToOemAccountDate == -1 ? null : new Date(tmpDevicePairedToOemAccountDate);
this.deviceTimeZone = in.readString();
this.deviceTimeZoneSetBy = (Integer) in.readValue(Integer.class.getClassLoader());
this.deviceIMEI = in.readString();
this.locale = in.readString();
}
public static final Parcelable.Creator<IdVerification> CREATOR = new Parcelable.Creator<IdVerification>() {
@Override
public IdVerification createFromParcel(Parcel source) {
return new IdVerification(source);
}
@Override
public IdVerification[] newArray(int size) {
return new IdVerification[size];
}
};
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,130
|
FROM scratch
MAINTAINER Ric Lister <rlister@gmail.com>
ADD certs/ca-certificates.crt /etc/ssl/certs/
ADD let-me-in /
ENTRYPOINT [ "/let-me-in" ]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,265
|
Q: Number of Permutations Fixed by the Fundamental Transformation is Fibonacci Writing a permutation in $S_n$ as a product of disjoint cycles, we define a standard representation by writing each cycle with its largest element first, and ordering the cycles by the increasing order of their largest element. Then there's a map $\hat{}: S_n \rightarrow S_n$ that takes a permutation $w\mapsto \hat w$, deleting all of w's parentheses. If we impose the standard representation above, this is a bijection.
What I'm asked to show is that the number of permutations left fixed by this bijection is the (n+1)st Fibonacci number. This is a problem from Stanley's Enumerative Combinatorics in case anyone is wondering. Right now I'm just staring pretty blankly trying to figure out how to approach this. I'm even a little confused as to what "left fixed" means. Is it just the permutations with no parentheses to remove?
A: Example
Consider the permutation $\sigma = [1 3 2]\in S_3$ (which is $1\mapsto 1$, $2\mapsto 3$, $3\mapsto 2$). In the standard representation introduced in the exercise, $\sigma = (1)(3,2)$. Now $\hat{\sigma} = [1 3 2] = \sigma$, so $\sigma$ is fixed under $\hat{{}}$.
Solution
Let $A(n)$ be the number of fixed permutations in $S_n$.
We see that $A_1 = 1$ and $A_2 = 2$, i.e. in $S_1$ and $S_2$, all permutationes are fixed.
Now let $n \geq 3$. Let $\pi\in S_n$ be fixed under the transformation. Consider the largest number $n$. In the standard representation, $n$ is the first element of the last cycle.
If the last cycle has length $1$ (i.e. it has the form (n), which is a fixed point), then $\pi$ is fixed if and only if after removing the cycle $(n)$, the resulting permutation is fixed in $S_{n-1}$.
Now assume the last cycle of $\pi$ is $(n,\ldots,a)$ of length $\geq 2$. Then $\hat{\pi}(n) = a$. So also $\pi(n) = a$, which implies that the last cycle is $(n,a)$. Now $\pi(a) = n$ and $\hat{\pi}(n-1) = n$, which implies $a = n-1$. So the last cycle has the form $(n-1, n)$. We see that $\pi = \hat{\pi}$ if and only if after removing the last cycle, the resulting permutation is fixed in $S_{n-2}$.
So $A(n) = A(n-1) + A(n-2)$. Thus $A(n)$ adheres to the same recursion formula as the Fibonacci numbers F(n). Because of $A(1) = 1$, $A(2) = 2$ and $F(2) = 1$, $F(3) = 2$, we get $A(n) = F(n+1)$.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,604
|
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